Continuous Advances In QCD 2004: Proceedings Of The Conference, William I. Fine Theoretical Physics Institute, Minneapolis, USA, 13 A» 16 May 2004

CONTINUOUS ADVANCES IN QCD 2004

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Proceedings of the Conference on

CONTINUOUS ADVANCES IN QCD 2004 William I. Fine Theoretical Physics Institute 13 - 16 May 2004 Minneapolis, USA

Editor

T. Gherghetta University of Minnesota, USA

NEWJERSEY * LONDON

-

K0World Scientific SINGAPORE * BElJlNG

*

SHANGHAI

-

HONG KONG

TAIPEI

-

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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CONTINUOUS ADVANCES IN QCD 2004 Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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FOREWORD The William I. Fine Theoretical Physics lnstitute hosted a workshop on the “Continuous Advances in QCD 2004”, from May 13-16, 2004. This biennial workshop was the sixth meeting of the series held at the University of Minnesota, Minneapolis, and marked the loth anniversary since the first such workshop was organised by the Institute in 1994. The workshop gathered together over sixty leading experts in the field to discuss the latest results and exchange ideas in Quantum Chromodynamics and non-Abelian gauge theories in general. In fact this year there were a record number of participants. The talks were organized into plenary sessions in the morning, which included reviews of the newest and most interesting research topics, while for the first time parallel sessions were scheduled in the afternoon devoted to shorter, original presentations. This lead to a broad scope of topics discussed. Amongst the hot topics this year included the discussion of the experimental evidence for pentaquarks, strong interactions under extreme conditions related to the ongoing experiments at RHIC, and new string theory inspired approaches to gauge theories. There were also very interesting presentations associated with perturbative and nonperturbative dynamics, heavy hadron decays, topological field configurations, as well as supersymmetry and novel theoretical methods. This Proceedings volume represents the current state of the QCD research frontier. It is hoped that the articles presented here convey some of the excitement of these topics as well as provide a useful review and reference for the reader in the years to come. Finally, on behalf of the organizers, I would like to take this opportunity to express my gratitude to Sally Menefee and Catharine Grahm for their dedicated efforts on the organization of numerous aspects of the workshop. T . Gherghetta

V

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CONTENTS 1. Perturbative and Nonperturbative QCD

1

Advances in Generalized Parton Distribution Study A . V. Radyushkin

3

QCD Dirac Spectra and the Toda Lattice K. Splittorff und J. J. M. Verbaarschot

15

Conformal Symmetry as a Template for QCD S.J. Brodsky

30

A Variational Look a t QCD J. Guilherme Milhano

44

Baryons, I N c . R.F. Lebed

54

Quark Correlations and Single-Spin Asymmetries M . Burkardt

64

2. Heavy Quark Physics

75

Soft-Collinear Factorization and the Calculation of the B M. Neubert

-+

X,y Rate 77

Status of Perturbative Description of Semileptonic Quark Decays A . Czarnecki

91

Heavy Quark Expansion in Beauty: Recent Successes and Problems N. Uraltsev

100

Polarization, Right-Handed Currents, and CP Violation in B A.L. Kugan

115

vi i

VV

viii

Lifetimes of Heavy Hadrons A.A. Petrov

129

Inclusive B-Decay Spectra and IR Renormalons E. Gardi

139

Summing Logs of the Velocity in NRQCD and Top Threshold Physics 150 A.H.-Hoang

B

+ 7r,K , r] Decay Formfactors from Light-Cone Sum Rules

160

P. Ball and R. Zwicky Understanding D$(2317), D,~(2460) F. De Fazio Search for Dark Matter in B M. Pospelov

+ S Transitions with Missing Energy

170

180

3. Exotic Hadrons

189

Exotica R.L. Jaffe

191

Quark Structure of Chiral Solitons D. Diakonov

215

Do Chiral Soliton Models Predict Pentaquarks? I.R. Klebanov and P. Ouyang

227

Baryon Exotics in the l/Nc Expansion E. Jenkins

239

Large N, QCD and Models of Exotic Baryons T.D. Cohen

251

4. QCD Matter at High Temperature and Density

259

Baryon and Anti-Baryon Production at RHIC J.I. Kapusta

261

ix

QCD at the Boiling Point: What Does Hadron Production at RHIC Tell Us? R. J. Fries

273

Nuclear Physics a t Small z D.E. Kharzeeu

283

Saturation Physics Meets RHIC Data Y. V. Kovchegou

293

Confinement and Chiral Symmetry A. ~ ~ ' c s y

303

Gapless Superconductivity in Dense QCD I . A . Shovkouy

313

Inhomogeneous Color Superconductivity G. Nardulli

323

Spontaneous Rotational Symmetry Breaking and Other Surprises in c-Model at Finite Density V.A. Miransky

335

Analytical Approach to Yang-Mills Thermodynamics R. Hofmann

346

Exotic Superfluids: Breached Pairing, Mixed Phases and Stability E. Gubankoua

357

5. Topological Field Configurations

367

Quantum Weights of Monopoles and Calorons with Non-trivial Holonomy D. Diakonou

369

Noncommutative Solitons and Instantons F. A. Schaposnik

381

X

Breather Solutions in Field Theories V. Khemani

393

Quantum Nonabelian Monopoles K. Konishi

403

Dilute Monopole Gas, Magnetic Screening and K-Tensions in Hot Gluodynamics C.P. Korthals Altes

416

Supersizing Worldvolume Supersymmetry: BPS Domain Walls and Junctions in SQCD A. Ritz

428

New Topological Structures in QCD F. Bruckmann, D. Ndgra'di and P. van Baal

440

6. Supersymmetry and Theoretical Methods

453

Viscosity of Strongly Coupled Gauge Theories P. Kovtun, D. T. Son and A . 0. Starinets

455

AdS/CFT Duality for States with Large Quantum Numbers A.A. Tseytlin

464

Planar Equivalence: From Type 0 Strings to QCD A. Armoni, M. Shifman and G. Veneziano

478

Gauge Theory Amplitudes, Scalar Graphs and Twistor Space V. V. Khoze

492

Weak Supersymmetry and its Quantum-Mechanical Realization A. V. Smilga

504

Nonperturbative Solution of Yukawa Theory and Gauge Theories J. R. Hiller

515

Fermionic Theories in Two-Dimensional Noncommutative Space E.F. Moreno

527

xi

Nonabelian Superconformal Vacua in Theoreis

= 2 Supersymmetric

537

R. Auzzi Predictions for QCD from Supersymmetry F. Sannino

547

Self-Duality, Helicity and Background Field Loopology G. V. Dunne

557

The Optical Approach to Casimir Effects A. Scardicchio

569

At The Crossroad: Perpetual Questions of the Fundamental Physics Here and Now A. Loseu

579

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SECTION 1. PERTURBATIVE AND NONPERTURBATIVE QCD

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ADVANCES IN GENERALIZED PARTON DISTRIBUTION STUDY

A. V. RADYUSHKIN172>* Physics Department, Old Dominion University, Norfolk, VA 23529, USA Theory Group, Jefferson Lab, Newport News, VA 23606, USA E-mail: [email protected]

The basic properties of generalized parton distributions (GPDs) and some recent applications of GPDs are discussed.

1. Introduction

The concept of Generalized Parton distribution^^^^^^ is a modern tool to provide a more detailed description of hadronic structure. The need for GPDs is dictated by the present-day situation in hadron physics, namely: i) The fundamental particles from which the hadrons are built are known: quarks and gluons. ii) Quark-gluon interactions are described by QCD whose Lagrangian is also known. iii) The knowledge of these first principles is not sufficient a t the moment, and we still need hints from experiment to understand how QCD works, and we must translate information obtained on the hadron level into the language of quark and gluonic fields. One can consider projections of combinations of quark and gluonic fields onto hadronic states IP) : ( 0 I qa(zl)qp(z2) I P ) , etc., and interpret them as hadronic wave functions. In principle, solving the bound-state equation H I P ) = E I P ) one should get complete information about hadronic structure. In practice, the equation involving infinite number of Fock components has never been solved. Moreover, the wave functions are not directly accessible experimentally. The way out is to use phenomenological functions. Well known examples are form factors, usual parton densities, and distribution amplitudes. The new functions, Generalized Parton D i s t r i b u t i o n s l ~ ~(for > ~ recent reviews, see4i5), are hybrids of these “old” *Also at Laboratory of Theoretical Physics, JINR, Dubna, Russia

3

4

functions which, in their turn, are the limiting cases of the ‘hew” ones. 2. Form factors, usual and nonforward parton densities

The nucleon electromagnetic form factors measurable through elastic eN scattering (Fig. 1, left) are defined through the matrix element

where r = p - p‘, t = r 2 . The current is given by the sum of its flavor eaF1,2,(t). components J,”(z) = e,&(z)rp$,(z), hence, F1,2(t)=

c,

P

P’

Figure 1. Left: Elastic e N scattering in one-photon approximation. Right: Lowest order pQCD factorization for DIS.

The parton densities are defined through forward matrix elements of quark/gluon fields separated by lightlike distances. In the unpolarized case,

and fa(a) (x)is the probability to find a (a)-quark with momentum xp in a nucleon with momentum p . One can access fa(a) (x)through deep inelastic scattering (DIS) y*N -+ X . Its cross section is given by imaginary part of the forward virtual Compton scattering amplitude. For large Q 2 E - q 2 , the perturbative QCD (pQCD) factorization works, and the leading order handbag diagram (Fig. 1, right) measures parton densities at the point x = X B ~E Q 2 / 2 ( p q ) . Note, that form factor deals with a point vertex instead of a light-like separation for the parton densities, and that p # p‘. Let us now “hybridize” form factors with parton densities by writing form factor components FI,(t) as integrals over the momentum fraction x

(3)

5

k

P = ( p + p ' ) /2 Figure 2.

t

J

Form factor and WACS amplitude in terms of nonforward parton densities.

(see Fig. 2, left). The nonforward parton densities (NPDs) Fa(a)(z,t), coincide in the forward t = 0 limit with the usual densities: t = 0) = fa(ii) (x). A nontrivial question is the interplay between z and t dependence. The simplest factorized ansatz F,(z, t ) = fa(x)F1(t) satisfies both the forward constraint and the local constraint (3). However, using the Gausk~1/ziX2]suggests7i6 sian light-cone wave functions !Q(zi,k i l ) exp[- C i Fa(%, t ) = f a ( z ) e Z t / 2 z XTaking Z. fa(z)from existing parametrizations and X2 generating the standard value (k:) M (300MeV)2 for quarks gives a reasonable description6 of FP(t) for -t 1 - 10GeV2. For small z, the usual parton densities have a Regge behavior f(z) zPa(O).For t # 0, this suggests F ( z , t ) z-a(t)or, for a linear Regge trajectory .F,(x, t ) = fa(z) z-~'~ With . the Regge slope a' 1GeV2, this model (Fig. 3, dotted lines) allows to obtain correct charge radii for the proton and neutron'. At large t , the form factor behavior is determined by This correthe z 1 behavior of f,(z),giving t-(n+')if fa(z) (1 - z)~. lation is different from the Drell-Yan-West relation, which gives t - ( n + 1 ) / 2 . One can conform with DYW without changing small-z behavior by taking = f,(~)z-"'('-")~. To apply this model to Fz(t), modified ansatz Fa(z,t) one needs unknown magnetic parton densities K , ( X ) . To produce a faster large-t fall-off of Fz(t) compared to Fl(t),one can take functions R , ( x ) having extra powers of (1 - z). With ~ ~ ( z (1 ) - z)qafa(z)one gets F2,(t)/Fla(t) l/t"'. The choice' qZl= 1.52, q d = 0.31 allows to fit the JLab polarization transfer datag on the ratio FZ(t)/Fl(t) for the proton, and also provides rather good fits for all four nucleon electromagnetic form factors, see solid line curves on Fig. 3.

-

-

N

-

-

-

-

-

3. Wide-angle Compton scattering

NPDs also appear in the wide-angle real Compton scattering (WACS). The handbag term (Fig. 2, right) is now given by the 1/z moment of F ( z ,t )

6 "0

.

L B

[3

1.2 1

0.8

0.6 0.4

0.08 0.07 0.06 0.05 0.04

0.03 0.02 0.01 0

10

-'

1

10

-t (GeV')

Figure 3.

Nucleon form factors in Regge-type models for nonforward parton densities.

and the amplitude of the Compton scattering off an elementary fermion. The cross section then can be expressed in terms of F a ( x ,t) and the KleinNishina (KN) cross section for the Compton scattering off an electron:

The approach6?l0based on handbag dominance gives (with the Gaussian NPDs fixed from the F l ( t ) form factor fitting) the results close both to old Cornell data" and the new preliminary d ata l2 ~ l3of JLab E-99-114 experiment. The predictions based on pQCD two-gluon hard exchange mechanism depend on the proton wave function and the value of a,. For the standard choice as = 0.3, the pQCD curves (see Ref.14 for the latest calculation) are well below the data even if one uses extremely asymmetric distribution amplitudes (DAs). Increasing a, to 0.5 gives a better agreement, but then pQCD predictions for Fl (t) form factor overshoot the data. To remove the overall normalization uncertainty, one can consider the ratio [ ~ ~ d u / d t ] / [ t ~ F 1sensitive ( t ) ] ~ only to the shape of the proton DA. The pQCD results for this ratio presented in Ref.14 are a n order of magnitude below the data for all DAs considered: unlike the GPD approach, pQCD cannot simultaneously describe form factor and WACS cross section data.

7

I Polarizationtransfer coefficient K~~ I

I Cross section scaling parameter 9.5

rn

E99-114(preliminsr/) Cornell

6.5

two duon exchange

5.5 60

Figure 4.

m

80

w

Ldeg

100

110

I 0

Comparison of preliminary JLab data with theoretical predictions.

Furthermore, hard pQCD and soft handbag mechanism give drastically different predictions1°~14for the polarization transfer coefficient KLL. The preliminary results (Fig. 4, left) of E-99-114 experiment13 strongly favor handbag mechanism that predicts a value close to the asymmetry for the Compton scattering on a single free quark. Another ratio-type prediction of pQCD is based on the dimensional quark counting rules, which give for WACS d a / d t s-"f(Oc,) with n = 6 for all center-of-mass angles OCM. The handbag mechanism corresponds to a power n depending on O C M , in agreement with the preliminary E-99-114 data13 (see Fig. 4, right). N

4.

Distribution amplitudes and pion form factors

Distribution amplitudes describe the hadron structure in situations when pQCD factorization is applicable for exclusive processes. They are defined through matrix elements (01.. . lp) of light cone operators. For the pion,

1 1

( 0 I d,i(-Z/2)rsy"$,zl(z/2)

I r + ( p ) ) = i#fir

-1

e-i"(pz)/2p,(a)

da 7 (5)

with 21 = (1+ a ) / 2 , 2 2 = (1 - a ) / 2 being the fractions of the pion momentum carried by the quarks. The simplest case is y*y -+ ro transition. Its large-Q2 behavior is light-cone dominated: there is no competing Feynmantype soft mechanism. The handbag contribution for y*y-+ ro (Fig. 5 , left) is proportional to the 1/(1- a 2 )moment of p x ( a )which allows for an experimental discrimination between the two popular models: asymptotic cpv(a)= i ( 1 - a 2 )and Chernyak-Zhitnitsky DA cpZz(a)= y a 2 ( 1 - a 2 ) . Comparison with data favors DA close to pF;s(cr). An important point is

8

Figure 5. Lowest-order pQCD factorization for y'y the pion EM form factor.

-+ K O

transition amplitude and for

-

that pQCD works here from rather small values Q2 2 GeV2, just like in DIS, which is also a purely light-cone dominated process. Another classic application of pQCD to exclusive processes is the pion electromagnetic form factor. With the asymptotic pion DA, the hard pQCD contribution (Fig. 5, right) to Q2Fn(Q2)is 2a,/n x 0.7GeV2, less than 1/3 of experimental value which is close to VMD expectation l / ( l / Q 2 l/mz). The suppression factor 2a,/n reflects the usual a,/n per loop penalty for higher-order corrections. The competing soft mechanism is zero order in a, and dominates over the pQCD hard term at accessible Q2. Just like in the case of F f ( t ) , the soft contribution for Fn(Q2) can be modeled by nonforward parton densities and easily fits the data (see Ref.15).

+

5.

Hard electroproduction processes and generalized parton distributions

A more recent attempt to use pQCD to extract information about hadronic structure is the study of deep exclusive photon2i3 or meson3J6 electroproduction. When both Q2 and s ( p q)2 are large while t ( p - P ' ) ~is small, one can use pQCD factorization of the amplitudes into a convolution of a perturbatively calculable short-distance part and nonperturbative parton functions describing the hadron structure. The hard subprocesses in these two cases have different structure (Fig. 6). For deeply virtual Compton scattering (DVCS), hard amplitude has structure similar to that of the y*yno form factor: the pQCD hard term is of zero order in a,, and there is no competing soft contribution. Thus, we can expect that pQCD works from Q2 2GeV2. On the other hand, the deeply virtual meson production process is similar to the pion EM form factor: the hard term has 0(as/7r) 0.1 suppression factor. As a result, the dominance of the hard pQCD term may be postponed to Q2 5 - 10GeV2. Just like in case of pion and nucleon EM form factors, the competing soft mechanism

+

-

=

-

-

9

Figure 6.

Hard subprocesses for deeply virtual photon and meson production.

can mimic the power-law Q2-behavior of the hard term. Hence, a mere observation of a “correct” power behavior of the cross section is not a proof that pQCD is already working. One should look a t several characteristics of the reaction to make conclusions about the reaction mechanism. To visualize DVCS’s specifics, take the y*N center-of-mass frame, with the initial hadron and the virtual photon moving in opposite directions along the z-axis. Since t is small, the hadron and the real photon in the final state also move close to the z-axis. This means that the virtual photon momentum q = q/ - z ~ j (where p X B ~= Q2/2(pq) is the same Bjorken variable as in DIS) has the component - z ~ j pcanceled by the momentum transfer T . In other words, T has the longitudinal component T + = z ~ j p + , and DVCS has skewed kinematics: the final hadron’s “plus” momentum (1 - [)p+ is smaller than that of the initial hadron (for DVCS, [ = X B ~ ) . The plus-momenta Xp+ and ( X - [)p+ of the initial and final quarks in DVCS are also not equal. Furthermore, the invariant momentum transfer t in DVCS is nonzero. Thus, the nonforward parton distributions (NFPDs) F c ( X ; t ) describing the hadronic structure in DVCS depend on X , the fraction of p+ carried by the initial quark, on [, the skewness parameter characterizing the difference between initial and final hadron momenta, and on t , the invariant momentum transfer. In the forward T = 0 limit, we have a reduction formula Ff=o(X,t = 0) = f a ( X ) relating NFPDs with the usual parton densities. The nontriviality of this relation is that &(X;t ) appear in the amplitude of the exclusive DVCS process, while the usual parton densities are extracted from the cross section of the inclusive DIS reaction. In the limit of zero skewness, NFPDs correspond to nonforward parton densities F f z 0 ( X ,t ) = P ( X ,t). The local limit results in a formula similar to Eq.(3) : X integral of F f ( X ,t ) - FF(X,t ) gives Fl,(t). The NFPD convention uses the variables most close to those of the usual parton densities. To treat initial and final hadron momenta symmetrically, Ji proposed2 the variables in which the plus-momenta of the hadrons are (1+ [)P+ and (1 - [ ) P + , and those of the active partons are (z [)P+ and (x - [)P+, with P = ( p p‘)/2 (Fig. 7). Since [p+ = T+ = 2[P+,

+

+

10

Figure 7. Comparison of N F P D s and OFPDs.

<

we have = C / ( 2 - C). To take into account spin properties of hadrons and quarks, one needs 4 off-forward parton distributions H , E , H , g , all being functions of x , <,t. Each OFPD has 3 distinct regions. When < z < 1, it is analogous to usual quark distributions; when -1 < z < -[, it is similar to antiquark distributions. In the region -5 < x < <, the “returning” quark has negative plus-momentum, and should be treated as an outgoing antiquark with momentum (< - x)P+. The total qtj pair momentum T+ = 2
<

-<

<

The E function, like F 2 , comes with the rP factor, hence, it is invisible in DIS described by exactly forward T = 0 Compton amplitude. However, the limit Ea*’(x,<= 0 ; t = 0 ) ~ ~ , ‘ ( exists. x) These functions give the proton anomalous magnetic moment K ~ and, , through Ji’s sum rule2, the total quark contribution Jq into the proton spin

Only valence quarks contribute to K ~ while , Jq involves also sea quarks. The determination of the tc-contribution to Ji’s sum rule is one of the original motivations2 to study the GPDs.

11

6. Double distributions

To model GPDs, two approaches are used: a direct calculation in specific dynamical models (bag model, chiral soliton model, light-cone formalism, etc.) and phenomenological construction based on the relation of SPDs to usual parton densities f a ( x ) , A f a ( x )and form factors Fl(t),F2(t),G A ( ~G) p, ( t ) . The key question is the interplay between x,( and t dependencies of GPDs. There are not so many cases in which the pattern of the interplay is evident. One example is the function E ( z ,(; t ) that is related to G p ( t ) form factor and is dominated for small t by the pion pole term l / ( t - m:). It is also proportional to the pion distribution amplitude (p(a)M $fT(l - a 2 )taken a t (Y = x/(. The construction of self-consistent models for other GPDs is performed using the formalism of double distributions’>l7. The main idea behind the double distributions is a %uperposition” of P+ and T+ momentum fluxes, i.e., the representation of the parton momentum Ic+ = PP+ + (1 a)r+/2 as the sum of a component PP+ due to the average hadron momentum P (flowing in the s-channel) and a component (1 a)r+/2 due to the t-channel momentum T . Thus, the double distribution f(P,a ) (we consider here for simplicity the t = 0 limit) looks like a usual parton density with respect to ,B and like a distribution amplitude with respect to a (Fig. 8). Using T+ = 2(P+ gives the connection x = P ( a between DD variables P, a and OFPD variables x,(.

+

+

+

Comparison of GPD and DD descriptions.

Figure 8.

The forward limit ( = 0 , t = 0 corresponds to x = relation between DDs and the usual parton densities

P, and gives the

1-181

1

f a ( P , a;t = 0) da: = f a ( P ) * (9) -1+101 The DDs live on the rhombus lal+lPI 5 1 and they are symmetric functions of the “DA” variable a: fa(@, a;t ) = fa(P,--a;t ) (“Munich” symmetry18).

12

These restrictions suggest a factorized representation for a DD in the form of a product of a usual parton density in the P-direction and a distribution amplitude in the a-direction. In particular, a toy model for a double distribution

f(P,a)= 3[(1- l P V - .”w4 + IPI I 1) and the corresponds to the toy “forward” distribution f(P) = 4(1 a-profile like that of the asymptotic pion distribution amplitude.

linefor

-

1

f ( B, a ) - f ( B )

:*on

line

~

producing H(x& )

-I

-I

Figure 9.

Scanning pattern for DD

+ SPD conversion.

To get usual parton densities from DDs, one should integrate (scan) them over vertical lines P = z = const. To get OFPDs H(z,J) with nonzero J from DDs f ( P , a ) ,one should integrate (scan) DDs along the parallel lines a = (x - ,B)/,$ with a t-dependent slope. One can call this process the DD-tomography. The basic feature of OFPDs H ( z ,,$) resulting from DDs is that for 5 = 0 they reduce to usual parton densities, and for = 1 they have a shape like a meson distribution amplitude. In fact, such a DD modeling misses terms proportional to the momentum transfer, and thus invisible in the forward limit. These include meson exchange contributions and so-called D-term, which can be interpreted as cT-exchange. The inclusion of the D-term induces nontrivial behavior in the central 1x1 < ,$ region (for details, see Ref.4).

7. Conclusions Hadronic structure is a complicated subject, it requires a study from many sides, in many different types of experiments. The description of specific aspects of hadronic structure is provided by several different functions:

13

form factors, usual parton densities, distribution amplitudes. Generalized parton distributions provide a unified description: all these functions can be treated as particular or limiting cases of GPDs H ( x ,<,t). Usual Parton Densities f (x)correspond to the case = 0, t = 0. They describe a hadron in terms of probabilities 1!DI2. But QCD is a quantum theory: GPDs with # 0 describe correlations !P:!P2. Taking only one point t = 0 corresponds to integration over impact parameters b l information about the transverse structure is lost. Form Factors F ( t ) contain information about the distribution of partons in the transverse plane, but F ( t ) involve integration over momentum fraction x - information about longitudinal structure is lost. Nonforward parton densities. A simple "hybridization" of usual densities and form factors in terms of NPDs F ( x ,t ) (GPDs with = 0) shows that behavior of F ( t ) is governed both by transverse and longitudinal distributions. NPDs provide adequate description of nonperturbative soft mechanism, they also allow to study transition from soft to hard mechanism. Distribution Amplitudes p(x) provide quantum level information about longitudinal structure of hadrons. Information about DAs is accessible in hard exclusive processes, when asymptotic pQCD mechanism dominates. GPDs have DA-type structure in the central region 1x1 < <. Generalized Parton Distributions H ( x ,<; t ) provide a 3-dimensional picture of hadrons. GPDs also provide some novel possibilities, such as "magnetic distributions" related to the spin-flip GPDs &(%, <,t ) . In particular, the structure of the nonforward density &(x, t ) E,(z, = 0, t ) determines the t-dependence of F2(t). Recent JLab data on the ratio F ! ( t ) / F I ( t ) can be explained within a GPD-based model' by assuming an extra (1- x)" suppression of &(x,t ) . The forward reductions tcE"(x) of E,(x, <,t ) look as fundamental as fa(%) and Afa(x): Ji's sum rule involves na(x) on equal footing with f"(z). Magnetic properties of hadrons are strongly sensitive to dynamics, thus providing a testing ground for models. A new direction is the study of flavor-nondiagonal distributions: protonto-neutron GPDs accessible through exclusive charged pion electroproduction process, proton-to-A GPDs (they appear in kaon electroproduction); proton-to-Delta - this one can be related to form factors of PA+ transition (another puzzle for hard pQCD approachable by the NPD model8). The GPDs for N + N + soft T processes4 can be used for testing the soft pion theorems and physics of chiral symmetry breaking. A challenging problem is the separation and flavor decomposition of GPDs. The DVCS amplitude involves all 4 types: H , E , H , E of GPDs, SO

<

<

N

N

<

=

<

--

14

we need to study other processes involving different combinations of GPDs. An important observation is that, in hard electroproduction of mesons, the spin nature of the produced meson dictates the type of GPDs involved, e.g., for pion electroproduction, only H , E appear, with?! , dominated by the pion pole at small t. This gives access to (generalization of) polarized parton densities without polarizing the target.

--

8. Acknowledgements

I thank the organizers for invitation, support and hospitality in Minneapolis. This work is supported by the US Department of Energy contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson Accelerator Facility. References 1. D. Muller, D. Robaschik, B. Geyer, F. M. Dittes and J. Horejsi, Fortsch. Phys. 42, 101 (1994). 2. X. D. Ji, Phys. Rev. Lett. 78, 610 (1997), Phys. Rev. D 55, 7114 (1997). 3. A. V. Radyushkin, Phys. Lett. B 380, 417 (1996), Phys. Lett. B 385, 333 (1996), Phys. Rev. D 56, 5524 (1997).

4. K. Goeke, M. V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47, 401 (2001). 5. M. Diehl, Phys. Rept. 388, 41 (2003) 6. A. V. Radyushkin, Phys. Rev. D 58, 114008 (1998). 7. V. Barone, M. Genovese, N. N. Nikolaev, E. Predazzi and B. G. Zakharov, Z. Phys. C 58, 541 (1993). 8. M. Guidal, M. V. Polyakov, A. V. Radyushkin and M. Vanderhaeghen, in preparation (2004). 9. 0. Gayou et al. [Jefferson Lab Hall A Collaboration], Phys. Rev. Lett. 88, 092301 (2002) [arXiv:nucl-ex/0111010]. 10. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur. Phys. J. C 8, 409 (1999). 11. A. Shupe et al., Phys. Rev. D 19,1921 (1979). 12. A. Nathan, “Real Compton scattering f r o m the proton”, in LLExclusive processes at high momentum transfer”, World Scientific, Singapore, (2002) pp. 225-232. 13. B. Wojtsekhowski, private communication (2004). 14. T. C. Brooks and L. J. Dixon, Phys. Rev. D 62, 114021 (2000) [arXiv:hepph/0004143]. 15. A. Mukherjee, I. V. Musatov, H. C. Pauli and A. V. Radyushkin, Phys. Rev. D 67, 073014 (2003) [arXiv:hep-ph/0205315]. 16. J. C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. D 56, 2982 (1997). 17. A. V. Radyushkin, Phys. Rev. D 59, 014030 (1999). 18. L. Mankiewicz, G. Piller and T. Weigl, Eur. Phys. J. C 5, 119 (1998).

QCD DIRAC SPECTRA AND THE TODA LATTICE*

K. SPLITTORFF Nordita, Blegdamsvej 17, DK-2100, Copenhagen, Denmark [email protected]. dk

J. J. M. VERBAARSCHOT Department of Physics and Astronomy, S U N Y at Stony Brook, Stony Brook, N Y l l 7 9 4 , US jacobus. verbaarschot Ostonybrook. edu

We discuss the spectrum of the QCD Dirac operator both at zero and at nonzero baryon chemical potential. We show that, in the ergodic domain of QCD, the Dirac spectrum can be obtained from the replica limit of a Toda lattice equation. At zero chemical potential this method explains the factorization of known results into compact and noncompact integrals, and at nonzero chemical potential it allows us to derive the previously unknown microscopic spectral density.

1. Introduction

Because of the spontaneous breaking of chiral symmetry and confinement, QCD at low energy is a theory of weakly interacting Goldstone bosons. In the spontaneously broken phase, the QCD partition function is a nonanalytic function of the quark mass with a chiral condensate that is discontinuous as the quark mass crosses the eigenvalue axis of the QCD Dirac operator. The strength of this discontinuity is proportional to the eigenvalue density, a relation known as the Banks-Casher formula l. A theory with spontaneously broken chiral symmetry that is much simpler than QCD is chiral Random Matrix Theory This is a theory with the global flavor symmetries of QCD in which the matrix elements of the Dirac operator are replaced by random numbers. Although, this theory is zero dimensional, chiral symmetry is broken spontaneously in the limit

'.

*This work is supported in part by US DOE grant No. DE-FG-88ER40388.

15

16

of infinitely large matrices, and the mass of the non-Goldstone modes diverges in the limit N -+ m, where N is the matrix size. Therefore, in the thermodynamic limit, N + m, chiral Random Matrix theory reduces to a theory of Goldstone bosons for which, in the limit of small quark masses, the Lagrangian is just the mass term of the chiral Lagrangian. This is the main reason why Random Matrix Theories have been so successful in this context. One of the questions we have been asking is whether we can identify a parameter domain where QCD and chiral Random Matrix Theory reduce to the same theory of Goldstone bosons. The affirmative answer to this question is that this is the case if the Compton wavelength of the Goldstone bosons is much larger than the linear size L of the box. This requires that , is an unphysical domain of QCD, the quark masses m f << F 2 / ( C L 2 )which so that the kinetic term of the chiral Lagrangian can be ignored (C is the chiral condesate and F is the pion decay constant.) However, even for realistic quark masses we can identify a parameter domain where QCD and chiral Random Matrix Theory behave the same, namely the domain where eigenvalues of the Dirac operator X << F 2 / ( C L 2 )which is known as the ergodic domain '. The reason is that the generating function of the Dirac spectrum is a QCD-like partition function with additional ghost quarks with mass A. The condition for the validity of the Random Matrix Theory description of the Dirac spectrum is then that the Compton wavelength of Goldstone bosons composed out of ghost quarks is much larger than the size of the box. Indeed, such behavior has been observed in numerous lattice QCD simulations (as discussed in detail elsewhere 6,7). At nonzero chemical potential the eigenvalues of the Dirac operator are scattered in the complex plane. It has been shown that the Dirac spectrum remains in the ergodic domain if the inverse chemical potential is much larger than the size of the box *. In this domain the Dirac spectrum can be described by a chiral Random Matrix Model that has been extended with a chemical potential However, until recently, this random matrix model had only been solved at the mean field level '. The standard methods to derive nonperturbative results such as the supersymmetric method l1 and complex orthogonal polynomial methods l2,l3>l4 were not successful in this case. So far the supersymmetric method failed because of technical problems in calculating the graded integrals, while the method of complex orthogonal polynomials failed because of the absence of an eigenvalue representation at non-zero chemical potential. The mean field analysis of the random matrix model was performed using the replica trick l5 which was 394.

9110.

17

widely believed to only work for the derivation of perturbative results 16. However, it was shown recently that if a family of partition functions has certain integrability properties it is possible to obtain exact nonperturbative results by means of the replica trick 17718. We will show that the ergodic limit of the phase quenched QCD partition function at nonzero chemical potential has the required integrability properties l9 so that the hierarchy of phase quenched partition functions with a different number of flavors are related by a recursion relation l 9 which is known as the Toda lattice equation. The spectral density is then obtained from the replica limit of this Toda lattice equation. In the first part of this lecture we discuss the Dirac spectrum at zero chemical potential. We show that in the ergodic domain the spectral density can be obtained from the replica limit of the Toda lattice equation. This result explains a factorization property of the resolvent. In the second half of this lecture we study the quenched Dirac spectrum at nonzero chemical potential. Using the replica limit of the Toda lattice equation we derive the analytical result for the microscopic spectral density in the ergodic domain. 2. The Dirac Spectrum in QCD

The eigenvalues { x k } of the anti-hermitian Dirac operator are determined by the eigenvalue equation D$k

= i&$k.

(1)

Because of the axial symmetry the nonzero eigenvalues occur in pairs f&. The number of zero eigenvalues is almost always equal to the topological charge of the gauge field configuration. The average spectral density, defined by

can be obtained from the discontinuity of the average resolvent

where, in both cases, the average is over gauge field configurations distributed according to the QCD action. This can be seen by considering the rectangular contour in Fig. 1. Assuming that the average spectral density does not vary substantially along this contour, the average total number of

18 1.2

1

'

9

I

'

"'

I

I

1 ' 0 '

"

.-

1 C(z)/Z:v

0.8

-

0.8

-

.

Columbia Data -

z=O 0.4

-

0

0.2

0

I

I

,

,

I

,

,

I

I

I

,

8=5.245

-

8=5.265

-

8=5.270

-

,

I

> , ,

Figure 1. A typical Dirac spectrum (left) and the average resolvent G(z) in units of CV compared with lattice QCD data.

eigenvalues inside this contour is p(X)Z, where X is a point on the imaginary axis inside this contour. Therefore, if we integrate the resolvent along this contour, we obtain

fG(z) = iZ(G(iX +

E)

- G(iX - E ) ) = 2nip(X)Z.

(4)

Using the symmetry of the spectrum we find Re[G(iX

+ E ) ] = n.p(X).

(5)

In QCD, the chiral condensate defined by C = G(e)/V where V is the volume of space time, is nonzero because of the spontaneous breaking of chiral symmetry. The average level spacing near zero is therefore given by

which is also the scale of the smallest nonzero eigenvalue. Asymptotically, for large A, the Dirac eigenvalues approach the spectrum of a free Dirac operator with eigenvalue density given by VX3. N

19

Because the eigenvalues near zero are spaced as 1/CV it makes sense to introduce the microscopic spectral density 1

(7)

In the ergodic domain of QCD this is a universal function that can also be derived from chiral Random Matrix Theory. 2.1. Ergodic Domain of QCD

The low-energy limit of QCD for Nf fermionic flavors is described by a theory of weakly interacting Goldstone bosons parametrized by the unitary If their mass matrix U(Nf).

(with F the pion decay constant) the kinetic term of the chiral Lagrangian factorizes from the partition function and the mass dependence of the partition function in the sector of topological charge v is given by 2o

where the quark mass matrix is defined by M = diag(m1,. . . m N , ) . This is the ergodic domain of QCD. As we will explain below partition functions with bosonic quarks are essential for obtaining the resolvent and average spectral density of the Dirac operator. For bosonic quarks, the Goldstone bosons cannot be parameterized by a unitary matrix. The reason is that symmetry transformations have to be consistent with the convergence of the bosonic integrals. Let us consider the case of one bosonic flavor. Then

so that the exponent is purely imaginary for m = 0 and convergent for Re(m) > 0. The most general flavor transformation of the action in (10) is a GZ(2) transformation that can be parameterized as

U =eHV

with Ht = H

and VVt = l .

(11)

20

For U to be a symmetry transformation for m = 0 we require that

0 id idt 0 so that H has to be a multiple of ug and V has to be a multiple of the identity. The V part of U is not broken by the mass term and is thus a vector symmetry. Only the symmetry transformation exp(sa3) is broken by the mass term so that the axial transformations are parameterized by

with

0 e-'

s E

(--oo,w).

For Nf bosonic flavors the axial transformations are parameterized by with

Ht = H.

The Goldstone manifold thus is GZ(Nf)/U(Nf). In the domain were the kinetic term of the chiral Lagrangian can be ignored, the effective bosonic (indicated by the subscript -Nf)partition function in the sector of topological charge v is given by 21922323

ZENf

(mf) =

1

detvUeiCVTr[MU-'+Mt

W N f) I U ( N f1

U]

(15)

The resolvent can be obtained from a supersymmetric generating function that contains one additional fermionic ghost quark and one additional bosonic ghost quark, det(D z ) G(z) = & . det(D z') Therefore the low energy limit of this generating function contains additional ghost Goldstone bosons and fermions with mass given by 2 z C / F 2 . For z <( F 2 / C L 2 the z-dependence of this generating function is given by

(

+ +

>I

21,22

+

with M = diag(m1,. . . ,m N f ,z , z') and G l ( N f 111) are super-matrices with a unitary U ( N f 1) upper left block, a GZ(l)/U(l) lower right block and Grassmann valued matrix elements everywhere else. The number of QCD Dirac eigenvalues that is described by this partition function is of the order F2 - F ~ L ~ . -CL2A

+

21

This number increases linearly in N, for Nc + 0;) which has been studied recently by lattice simulations 24. An alternative to the supersymmetric method is to use the replica trick to calculate the resolvent. It comes in two different versions: the fermionic replica trick defined by 1 G ( z ) = lim -logZf;,(z), Nf+O

Nf

and the bosonic replica trick defined by

If we take the replica limit of the fermionic (19) or bosonic (20) partition functions directly, we will obtain a result that differs from the supersymmetric calculation 16. For almost two decades there were no methods to do reliable nonperturbative calculations with the replica trick. In the next section we will show that these problems with the replica trick can be avoided if the take the replica limit of the Toda lattice equation. 2 . 2 . Toda Lattice Equation

We now consider bosonic and fermionic partition functions with all masses equal to z which only depend on the combination, cf. (9) and (15), 2

= ZEV.

(21)

The unitary integral in the fermionic partition function (9) can be evaluated by decomposing U = Vdiag(eiek)Vt and choosing the {Ok} and V as new integration variables. By expanding the Jacobian of this transformaI exp(iOk)- exp(iOl)I2,the different terms factorize into tion, given by products of modified Bessel functions which can be combined again into a single determinant. The final result is given by

n,,,

25726

Gf (2)= det [ L + k - l ( 2 ) 1 k , l = 1 , . . .

,Nf

.

(22)

By using recursion relations for the Bessel functions, this result can be rewritten as ,Nf-l

'

Next we use the Sylvester identity 27,28 which is valid for determinant of an arbitrary matrix A. It is given by CijC,, - CipCpj= det(A)Cij;,,,

(24)

22

where the Cij are cofactors of the matrix A and the CijiPqare double cofactors. By applying this identity to the determinant in (23) for i = j = N f - 1 and p = q = N f , we easily derive the Toda lattice equation 29130131

This equation has also been derived as a consistency condition for QCD partition functions 3 2 . It also occurs in other application such as for example in self-dual Chern-Simons theory 33. In the case of bosonic quarks (15) the positive definite matrix is dias new inagonalized as U = Vdiag(esk)Vt. Choosing the { s k } and tegration variables, we obtain 23 after expanding the Jacobian given by rIk- exP(sl))(exP(-sk) - exp(-sd), ZENf (x)= det [ K u + k - l ( ~ ) l k , l = l , . . . ,Nf*

(26)

As in the fermionic case, this result can be written in the form of a r function

where we have used that I,, and (-1)”KV satisfy the same recursion relations. Therefore, the bosonic partition function also satisfies the Toda lattice equation ( 2 5 ) . The two semi-infinite hierarchies are connected by 1 lim -(xCa,)2 log z;, (z). N/+O N f By extending the Toda lattice hierarchy to include an additional spectator boson, it can be shown that 34 1 lim - ( z & ) ~ l o g ~ k(x) , = lim xCaz(xCa, yay)log ZF,-,(xly) N/+O N f Y+X = zd, lim d,log Z;,-,(Z\Y)

+

Y+X

= zd,zG(x).

(29)

Taking the replica limit of the Toda lattice equation (25) we thus obtain the identity

xd,xG(x) = 2 x 2 2 ; ( ~ ) 2 ! 1 ( ~ ) ,

(30)

which explains this factorization property of the resolvent. In the same way we can show the factorization of the susceptibility in a bosonic and a fermionic partition function 1 9 .

23

Inserting the expressions for

21

and 2-1 we find U

G(z) = z(Kv(z)Iv(z)+ Kv-~(z)Iv+l(z)) + ;-

(31)

For u = 0 this result is shown as G(z)/CV by the solid curve in the right figure of Fig. 1. Agreement with lattice data 35 is found in the ergodic domain of QCD. 3. Dirac Spectrum at Nonzero Chemical Potential

At nonzero baryon chemical potential the Dirac operator is modified according to D

+ D +p ~ o .

(32)

This Dirac operator does not have any hermiticity properties and its eigenvalues are scattered in the complex plane For small p we p2. The average expect that the width of the cloud of eigenvalues 36 spectral density is given by

-

36937338939740.

and the average resolvent is defined as usual by (3). They are related by

8,. G(z)Iz=x = T P ( A).

(34)

The quenched spectral density is therefore given by the replica limit

4194299

with generating function given by (note that n counts pairs of quarks)

+

Zn(z,z*)= (detn(D puyo

+ z)detn(-D + p70 + z * ) ).

(36)

The low-energy limit of this generating function is a chiral Lagrangian which is determined by its global symmetries and transformation properties. By writing the product of the two determinants as Iid+p

det(D

+ py0 + z)det(-D + p~~ + z * ) =

0

0 id-p

z 0

0 z*

z 0

0

z* idt+p 0 0 idt - p

1

24

we observe that the U ( 2 )x U ( 2 ) flavor symmetry is broken by the chemical potential term and the mass term. Invariance is recovered by transforming the mass term as in the case of zero chemical potential and the chemical potential term by a local gauge transformation. In the domain of p and z where we can neglect the kinetic terms, the partition function is given by 8,19

where

B = ( ’ , 0 -1,

)’

M = ( ’ l n0 z * l n

).

(39)

3.1. Integration Formula

We have proved the following integration formula

l9

where

and c, is an n-dependent constant. For example, consider the case n = 1 and all a p = 0. Then the integral is given by Zl(x,y) which is a known integral given by

Z,V(x,Y ) l a p = O =

I’

XdXIv(XX)L(-XY)

which is a known result for the QCD partition function with two different masses 44. 3 . 2 . Toda Lattice Equation

Using the integration formula of previous section we find that the lowenergy limit of the phase quenched QCD partition function at nonzero chemical potential is given by det [(.Wk(.*&*)lZ,Y(z, z * ) ] 0 5 k , 1 < n,- 1(43) z ; ( z , z * ) = (zz*)n(n--l) cn

25

where

z,v(z,z * ) =

/

1

XdXe-2"F2P2(X2-1) IIu(XzVC)12.

(44)

Using the Sylvester identity as at zero chemical potential we obtain the Toda lattice equation

The spectral density (35) follows from the replica limit of this equation. Using Z i ( z ,z*) = 1 we find the simple expression 1 zz* log Z i ( z , z * ) = -Zz,Y(z, z*)Z!,(z, z * ) . (46) xn 2 What remains to be done is to calculate the partition function with one bosonic and one conjugate bosonic quark which will be completed in the next subsection. p(z, z * ) = lim -d,& n+O

3.3. The Bosonic Partition Function

In this subsection we evaluate the low-energy limit of the QCD partition function at nonzero chemical potential for one bosonic quark and one conjugate bosonic quark. Because of convergence requirements it is more complicated to derive the chiral Lagrangian in this case. By a careful analysis we find l9 zyl(z,z*) =

dUdetUUe- w T r [ U , B ] [ U - ', B ] + q T r C T (U-1U-l L 2 ) , u ( 2 )

1) 7

(47) where B is the baryon number matrix defined in (39) and the mass matrix and the anti-symmetric matrix I are defined as

<

<=(") z* E

and I = ( "-1 ) .0

Although this integral is convergent for Imc > 0, it diverges logarithmically for c + 0. We have also l9 derived the partition function (47) starting from a chiral Random Matrix Theory at nonzero chemical potential and using the Ingham-Siege1 integral 45. The integral (47) can be evaluated analytically by using an explicit parameterization of positive definite 2 x 2 matrices. We find

26

The final result for the quenched spectral density is given by

l9

The constant has been chosen such that the p + 0 limit of p (z, y) for large y is given by C V / r (see below). The solid curve in Fig. 2 shows a graph of this result for y = 0 and p 2 F 2 V = 16. We plot the ratio

versus xCV at y = 0. The dotted curve shows the result which is obtained when the Bessel function K , is replaced by its asymptotic expansion. This result that was obtained from a nonhermitian eigenvalue model l 4 is not in the universality class of QCD. An important difference between the two results is that the spectral density (50) for y = 0 is quadratic in z for z + 0, whereas the result given by the dotted curve is linear in z for z -+ 0. Taking the thermodynamic limit at fixed z and p the Bessel functions can be approximated by their asymptotic limit. This results in VC2 P(GY) =

for

2 F2 p 2 C .

1x1 < -

and p(z,y) = 0 outside this strip in agreement with a mean field analysis For the integrated eigenvalue density we then find '7*.

in agreement with the eigenvalue density at p = 0. 4. Conclusions

In this work we have analyzed the QCD partition function in the ergodic domain where the pion Compton wavelength is much larger than the size of the box. In this domain the QCD partition function reduces to a theory of weakly interacting Goldstone bosons for which the kinetic term in the chiral Lagrangian can be ignored. Independent of the quark masses, the generating function for the Dirac spectrum is in this domain for sufficiently small eigenvalues. We have shown that fermionic partition functions, bosonic partition functions and the supersymmetric partition function are connected by a

27 0.008 P.(X,O)

5

'

I

"

'

I

'

I

'

"

'

s

'

___... .......... ....__________,,

0.006 -

0

.,..." ._..

2

4

6

xcv

8

10

0

10

20

30

40

xXV

Figure 2. The quenched spectral density at nonzero chemical potential in the ergodic domain of QCD (full curve). Also shown is a result derived from an eigenvalue model (dotted curve). The left hand plot is a zoom in of the right hand one.

Toda lattice equation. This recursion relation makes it possible to derive nonperturbative results using the replica trick. In particular, this reveals the factorization of the resolvent and the susceptibility into products of simple bosonic and fermionic partition functions. In this article, we have considered the resolvent a t zero chemical potential and the spectral density a t nonzero chemical potential. The resolvent for quenched QCD a t nonzero chemical potential approaches zero linearly as a function of Re(z). In the unquenched theory, where the argument of the resolvent is also the mass in the fermion determinant, we expect a discontinuity in the resolvent as Re(z) crosses the imaginary axis. Can we understand the differences between these two theories in terms of the spectrum of the Dirac operator? As is suggested by earlier Random Matrix Theory simulations 46 there are significant differences between the two. For example, in the unquenched theory, because of the phase of the fermion determinant, there is no reason that the spectral density is positive definite or even real. The first analytical results for the unquenched spectral density where recently obtained by James Osborn 47 for a nonhermitian Random Matrix Model that is in the universality class of QCD at nonzero chemical potential. In the thermodynamic limit his results shows strong oscillations but the connection with broken chiral symmetry is still a mystery. We hope to address this issue in a future publication 4 8 .

28

The replica limit of the Toda lattice equation is a powerful method that is also applicable to other partition functions with an integrable structure. For example, we mention the Ginibre ensemble 49 , parametric correlations, the two-point function of the Gaussian Unitary Ensemble. Our experience tells us that all universal results that can be derived from a complex random matrix theory (B = 2) can also be obtained from the replica limit of a Toda lattice equation. Acknowledgments

We thank Gernot Akemann, Gerald Dunne and James Osborn for useful discussions. References 1. T. Banks and A. Casher, Nucl. Phys. B169 (1980) 103. 2. E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. A 560, 306 (1993); J.J.M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994). 3. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984); Nucl. Phys. B 250, 465 (1985); H. Leutwyler, Ann. Phys. 235, 165 (1994). 4. H. Leutwyler and A. Smilga, Phys. Rev. D 46, 5607 (1992). 5. J.J.M. Verbaarschot, Phys. Lett. B 368 (1996) 137. 6. J . J. M. Verbaarschot and T. Wettig, Ann. Rev. Nucl. Part. Sci. 50, 343 (2000). 7. P. H. Damgaard, Nucl. Phys. Proc. Suppl. 106, 29 (2002). 8. D. Toublan and J.J.M. Verbaarschot, Int. J. Mod. Phys. B 15, 1404 (2001). 9. M.A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996). 10. M. A. Halasz, J. C. Osborn and J. J. M. Verbaarschot, Phys. Rev. D 56, 7059 (1997). 11. K.B. Efetov, Phys. Rev. Lett. 79, 491 (1997); Adv. Phys. 32, 53 (1983), Supersymmetry in disorder and chaos, (Cambridge University Press, Cambridge, 1997). 12. F. Di Francesco, M. Gaudin, C. Itzykson and F. Lesage, Int. J. Mod. Phys. A 9 4257 (1994). 13. Y.V. Fyodorov and H.-J. Sommers, J. Phys. A 36, 3303 (2003). 14. G. Akemann, Phys. Rev. Lett. 89, 072002 (2002); J.Phys. A 36, 3363 (2003). 15. S.F. Edwards and P.W. Anderson, J. Phys. F5, 965 (1975). 16. J.J.M. Verbaarschot and M.R. Zirnbauer, J. Phys. A 18, 1093 (1985). 17. E. Kanzieper, Phys. Rev. Lett. 89, 250201 (2002). 18. K. Splittorff and J.J.M. Verbaarschot, Phys. Rev. Lett. 90, 041601 (2003). 19. K. Splittorff and J.J.M. Verbaarschot, NucLPhys. B 683, 467 (2004). 20. J. Gasser and H. Leutwyler, Phys. lett. B 188, 477 (1987). 21. J.C. Osborn, D. Toublan and J.J.M. Verbaarschot, Nucl. Phys. B 540, 317 (1999). 22. P.H. Damgaard, J.C. Osborn D. Toublan, and J.J.M. Verbaarschot, Nucl. Phys. B 547, 305 (1999).

29 23. D. Dalmazi and J.J.M. Verbaarschot, Nucl. Phys. B 592, 419 (2001). 24. R. Narayanan and H. Neuberger, arXiv:heplat/0405025. 25. R.C. Brower, P. Rossi and C-I. Tan, Nucl. Phys. B 190, 699 (1981); R.C. Brower and M. Nauenberg, Nucl. Phys. B 180, 221 (1981). 26. J. B. Kogut, M. Snow and M. Stone, Nucl. Phys. B 200, 211 (1982). 27. J.J. Sylvester, Compt. Rend. Acad. Sc. 54, 129 (1862). 28. P. Forrester, Log-gases and Random matrices, Web Book - available at http://uwu.ms.unimelb.edu.au/~matpjf/matpjf.html.

29. R. Dijkgraaf, H. Verlinde and E. Verlinde, Nucl. Phys. B 348, 435 (1991). 30. S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Nucl. Phys. B 380, 181 (1992). 31. A. Mironov, A. Morozov and G. Semenoff, Int. J. Mod. Phys. A 11, 5031 (1996). 32. G. Akemann and P. H. Damgaard, Phys. Lett. B 432, 390 (1998). 33. G. V. Dunne, R. Jackiw, S. Y . Pi and C. A. Trugenberger, Phys. Rev. D 43, 1332 (1991) [Erratum-ibid. D 45, 3012 (1992)l; G. V. Dunne, Selfdual Chern-Simons theories, Lect. Notes Phys. M36, 1 (1995). 34. K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 695 (2004) 84. arXiv:hep-th/0402177. 35. S. Chandrasekharan and N. Christ, Nucl. Phys. Proc. Suppl. 47 (1996) 527. 36. I.M. Barbour, N.E. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone, and H.W. Wyld, Nucl. Phys. B 275 (1986) 296; M.-P. Lombardo, J.B. Kogut, and D.K. Sinclair, Phys. Rev. D 54 (1996) 2303. 37. H. Markum, R. Pullirsch and T. Wettig, Phys. Rev. Lett. 83, 484 (1999). 38. E. Bittner, M. P. Lombardo, H. Markum and R. Pullirsch, Nucl. Phys. Proc. Suppl. 106, 468 (2002). 39. E. Bittner, S. Hands, H. Markum and R. Pullirsch, Prog. Theor. Phys. Suppl. 153, 295 (2004). 40. G. Akemann and T. Wettig, Phys. Rev. Lett. 92 (2004) 102002. 41. V.L. Girko, Theory of random determinants (Kluwer Academic Publishers, Dordrecht, 1990). 42. A. Goksch, Phys. Rev. D 37 (1988) 1014. 43. J. Feinberg and A. Zee, Nucl. Phys. B 504 (1997) 579; Nucl. Phys. B 501 (1997) 643. 44. A.D. Jackson, M.K. Sener and J.J.M. Verbaarschot, Phys. Lett. B 387, 355 (1996). 45. Y. V. Fyodorov, Nucl. Phys. B 621, 643 (2002). 46. M. A. Halasz, A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 56, 5140 (1997). 47. J. C. Osborn, arXiv:hep-th/0403131. 48. G. Akemann, J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, in preparation. 49. E. Kanzieper, cond-mat/0312006.

CONFORMAL SYMMETRY AS A TEMPLATE FOR QCD*

S. J. BRODSKY Stanford Linear Accelerator Center 2575 Sand Hill Road Menlo Park, CA 94025, USA E-mail: [email protected]

Conformal symmetry is broken in physical QCD; nevertheless, one can use conformal symmetry as a template, systematically correcting for its nonzero function as well as higher-twist effects. For example, commensurate scale relations which relate QCD observables to each other, such as the generalized Crewther relation, have no renormalization scale or scheme ambiguity and retain a convergent perturbative structure which reflects the underlying conformal symmetry of the classical theory. The “conformal correspondence principle” also dictates the form of the expansion basis for hadronic distribution amplitudes. The AdS/CFT correspondence connecting superstring theory to superconformal gauge theory has important implications for hadron phenomenology in the conformal limit, including an all-orders demonstration of counting rules for hard exclusive processes as well as determining essential aspects of hadronic light-front wavefunctions. Theoretical and phenomenological evidence is now accumulating that QCD couplings based on physical observables such as T decay become constant at small virtuality; i.e., effective charges develop an infrared fixed point in contradiction to the usual assumption of singular growth in the infrared. The near-constant behavior of effective couplings also suggests that QCD can be approximated as a conformal theory even at relatively small momentum transfer. The importance of using an analytic effective charge such as the pinch scheme for unifying the electroweak and strong couplings and forces is also emphasized.

1. Introduction: The Conformal Correspondence Principle The classical Lagrangian of QCD for massless quarks is conformally symmetric. Since it has no intrinsic mass scale, the classical theory is invariant under the S0(4,2) translations, boosts, and rotations of the Poincare group, plus the dilatations and other transformations of the conformal group. Scale invariance and therefore conformal symmetry is destroyed *This work is supported by the Department of Energy under contract number D E AC03-76SF00515.

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in the quantum theory by the renormalization procedure which introduces a renormalization scale as well as by quark masses. Conversely, Parisi has shown that perturbative QCD becomes a conformal theory for /3 -+ 0 and zero quark mass. Conformal symmetry is thus broken in physical QCD; nevertheless, we can still recover the underlying features of the conformally invariant theory by evaluating any expression in QCD in the analytic limit of zero quark mass and zero p function:

This conformal correspondence limit is analogous to Bohr’s correspondence principle where one recovers predictions of classical theory from quantum theory in the limit of zero Planck constant. The contributions to an expression in QCD from its nonzero P-function can be systematically identified order-by-order in perturbation theory using the Banks-Zaks procedure 4 . There are a number of useful phenomenological consequences of near conformal behavior of QCD: the conformal approximation with zero P function can be used as template for QCD analyses 5,6 such as the form of the expansion polynomials for distribution amplitudes The near-conformal behavior of QCD is the basis for commensurate scale relations which relate observables t o each other without renormalization scale or scheme ambiguities By definition, all contributions from the nonzero p function can be incorporated into the QCD running coupling a,(Q) where Q represents the set of physical invariants. Conformal symmetry thus provides a template for physical QCD expressions. For example, perturbative expansions in QCD for massless quarks must have the form 233

778.

’.

n=O

where the Cn are identical to the expansion coefficients in the conformal theory, and Q: is the scale chosen to resum all of the contributions from the nonzero p function at that order in perturbation theory. Since the conformal theory does not contain renormalons, the Cn do not have the divergent n! growth characteristic of conventional PQCD expansions evaluated at a fixed scale. 2. Effective C h a r g e s

One can define the fundamental coupling of QCD from virtually any physical observable lo. Such couplings, called “effective charges”, are all-order resummations of perturbation theory, so they correspond to the complete

32 theory of QCD. Unlike the MS coupling, a physical coupling is analytic across quark flavor thresholds In particular, heavy particles will contribute to physical predictions even at energies below their threshold. This is in contrast to mathematical renormalization schemes such as M S , where mass thresholds are treated as step functions. In addition, since the QCD running couplings defined from observables are bounded, integrations over effective charges are well defined and the arguments requiring renormalon resummations do apply. The physical couplings satisfy the standard renormalization group equation for its logarithmic derivative, dc\l,hys/dlnlc2 = ~ p h y s [ 3 , h y s ( k 2 ) ] , where the first two terms in the perturbative expansion of ,&ys are scheme-independent at leading twist; the higher order terms have to be calculated for each observable separately using perturbation theory. Commensurate scale relations are QCD predictions which relate observables to each other at their respective scales. An important example is the generalized Crewther relation 13: 11i12.

where the underlying form at zero p function is dictated by conformal symmetry 14. Here ~ R ( s ) / Tand - a g 1 ( Q 2 ) ) / nrepresent the entire radiative corrections to Re+,- (s) and the Bjorken sum rule for the gI(x, Q 2 ) structure function measured in spin-dependent deep inelastic scattering, respectively. The relation between s* and Q2 can be computed order by order in perturbation theory, as in the BLM method 15. The ratio of physical scales guarantees that the effect of new quark thresholds is commensurate. Commensurate scale relations are renormalization-scheme independent and satisfy the group properties of the renormalization group. Each observable can be computed in any convenient renormalization scheme such as dimensional regularization. The MS coupling can then be eliminated; it becomes only an intermediary In such a procedure there are no further renormalization scale ( p ) or scheme ambiguities. In the case of QED, the heavy lepton potential (in the limit of vanishing external charge) is conventionally used to define the effective charge a,,d(q2). This definition, the Dyson Goldberger-Low effective charge, resums all lepton pair vacuum polarization contributions in the photon propagator, and it is analytic in the lepton masses. The scale of the QCD coupling is thus the virtuality of the exchanged photon. The extension of this concept to non-abelian gauge theories is non-trivial due to the self

'.

33 interactions of the gauge bosons which make the usual self-energy gauge dependent. However, by systematically implementing the Ward identities of the theory, one can project out the unique self-energy of each physical particle. This results in a gluonic self-energy which is gauge independent and which can be resummed t o define an effective charge that is related through the optical theorem t o differential cross sections. The algorithm for performing the calculation a t the diagrammatic level is called the “pinch technique” The generalization of the pinch technique to higher loops has recently been investigated Binosi and Papavassiliou have shown the consistency of the pinch technique to all orders in perturbation theory, thus allowing a systematic application t o the QCD and electroweak effective charges a t higher orders. The pinch scheme is in fact used to define the evolution of the couplings in the electroweak theory. The pinch scheme thus provides an ideal scheme for QCD couplings as well. 16117118919.

20921,22723924,25,

23124v25

3. Effective Charges and Unification

Recently Michael Binger and I have analyzed a supersymmetric grand unification model in the context of physical renormalization schemes 2 6 . Our essential assumption is that the underlying forces of the theory become a t the unification scale. We have found a number of qualitative differences and improvements in precision over conventional approaches. There is no need to assume that the particle spectrum has any specific structure; the effect of heavy particles is included both below and below the physical threshold. Unlike mathematical schemes such as dimensional reduction, the evolution of the coupling is analytic and unification is approached continuously rather than at a fixed scale. The effective charge formalism thus provides a template for calculating all mass threshold effects for any given grand unified theory. These new threshold corrections are important in making the measured values of the gauge couplings consistent with unification. A comparison with the conventional scheme based on dimensional regularization scheme is summarized in Fig. 1.

m,

4. The Infrared Behavior of Effective QCD Couplings

It is often assumed that color confinement in QCD can be traced t o the singular behavior of the running coupling in the infrared, ie. “infrared slavery.” For example if as(q2)4 l / q 2 a t q2 -+ 0, then one-gluon exchange leads to a linear potential a t large distances. However, theoretical and phenomenological evidence is now accumulating 27,28t29,30131

32933134

34

27

I

Asymptotic Unification I ' "'I I

A 5

Q(GeV) Figure 1. A s y m p t o t i c Unification. An illustration of strong and electroweak coupling unification in an SU(5) supersymmetric model based on the pinch scheme effective charge. The solid lines are the analytic pinch scheme PT effective couplings, while the dashed lines are the couplings. For illustrative purposes, a 3 ( M z ) has been chosen so that unification occurs at a finite scale for and asymptotically for the PT couplings. Here M s u s y = 200GeV is the mass of all light superpartners except the wino and gluino which have values + m g x = Msusy = amurs.

that the QCD coupling becomes constant at small virtuality; ie., as(Q2) develops an infrared fixed point in contradiction to the usual assumption of singular growth in the infrared. Since all observables are related by commensurate scale relations, they all should have an IR fixed point 30. A recent study of the QCD coupling using lattice gauge theory in Landau gauge in fact shows an infrared fixed point 35. This result is also consistent with Dyson-Schwinger equation studies of the physical gluon propagator The relationship of these results to the infrared-finite coupling for the vector interaction defined in the quarkonium potential has recently been discussed by Badalian and Veselov 36. Menke, Merino, and Rathsman 33 and I have considered a physical coupling for QCD which is defined from the high precision measurements of the hadronic decay channels of the r- + u,h-. Let R, be the ratio of the hadronic decay rate to the leptonic rate. Then R, = R! [1+ %], where R! is the zeroth order QCD prediction, defines the effective charge a,. The 27928.

35 data for r decays is well-understood channel by channel, thus allowing the calculation of the hadronic decay rate and the effective charge as a function of the r mass below the physical mass. The vector and axial-vector decay modes which can be studied separately. Using an analysis of the r data from the OPAL collaboration 37, we have found that the experimental value of the coupling Q,(S) = 0.621 f 0.008 a t s = m: corresponds to a value of a z ( M 2 ) = (0.117-0.122) f 0.002, where the range corresponds to three different perturbative methods used in analyzing the data. This result is in good agreement with the world average a z ( M z ) = 0.117f0.002. However, from the figure we also see that the effective charge only reaches a,(s) 0.9f0.1 a t s = 1GeV2, and it even stays within the same range down t o s 0.5 GeV2. This result is in good agreement with the estimate of Mattingly and Stevenson 32 for the effective coupling Q R ( S ) 0.85 for ,b < 0.3 GeV determined from e+e- annihilation, especially if one takes into account the perturbative commensurate scale relation, a,(m$) = a ~ ( s *where ), s* N 0.10 m:, . This behavior is not consistent with the coupling having a Landau pole, but rather shows that the physical coupling is close to constant a t low scales, suggesting that physical QCD couplings are effectively constant or “frozen” at low scales. Figure 2 shows a comparison of the experimentally determined effective charge a,(s) with solutions t o the evolution equation for a, at two-, three-, and four-loop order normalized at m,. At three loops the behavior of the perturbative solution drastically changes, and instead of diverging, it freezes to a value a, 21 2 in the infrared. The infrared behavior is not perturbatively stable since the evolution of the coupling is governed by the highest order term. This is illustrated by the widely different results obtained for three different values of the unknown four loop term P7,3which are also shown. The values of P,,3 used are obtained from the estimate of the four loop term in the perturbative series of R,, K p = 25 f50 38. It is interesting to note that the central four-loop solution is in good agreement with the data all the way down to s 21 1GeV2. The results for a, resemble the behavior of the one-loop “time-like” effective coupling 39,40,41 N

N

-

a,E(s) =

1 { Po

47T

1

- - - - arctan 2

7r

[$

In

$1 }

(4)

which is finite in the infrared and freezes to the value a , ~ ( s )= 41r/Pa as s + 0. It is instructive t o expand the “time-like” effective coupling for

36

d 1.75 1.5 1.25 1 0.75 0.5 0.25

0 -0.25 -0.5

0

0.5

1

1.5

2

2.5

3 s (GeV')

Figure 2. The effective charge oT for non-strange hadronic decays of a hypothetical T lepton with m:, = s compared to solutions of the fixed order evolution equation for oT at two-, three-, and four-loop order. The error bands include statistical and systematic errors.

large s,

This shows that the "time-like77effective coupling is a resummation of (7r2ppo"a:)n-corrections to the usual running couplings. The finite coupling aeffgiven in Eq. (4)obeys standard PQCD evolution a t LO. Thus one can have a solution for the perturbative running of the QCD coupling which

37 obeys asymptotic freedom but does not have a Landau singularity. The near constancy of the effective QCD coupling at small scales illustrates the near-conformal behavior of QCD. It helps explain the empirical success of dimensional counting rules for the power law fall-off of form factors and fixed angle scaling. As shown in the references 42i43, one can calculate the hard scattering amplitude TH for such processes 44 without scale ambiguity in terms of the effective charge aT or CURusing commensurate scale relations. The effective coupling is evaluated in the regime where the coupling is approximately constant, in contrast t o the rapidly varying behavior from powers of a, predicted by perturbation theory (the universal two-loop coupling). For example, the nucleon form factors are proportional at leading order t o two powers of a, evaluated at low scales in addition to two powers of l / q 2 ; The pion photoproduction amplitude at fixed angles is proportional at leading order t o three powers of the QCD coupling. The essential variation from leading-twist counting-rule behavior then only arises from the anomalous dimensions of the hadron distribution amplitudes.

5. Light-Front Quantization The concept of a wave function of a hadron as a composite of relativistic quarks and gluons is naturally formulated in terms of the light-front Fock expansion at fixed light-front time, T = X.W. The four-vector w , with w 2 = 0, determines the orientation of the light-front plane; the freedom to choose w provides an explicitly covariant formulation of light-front quantization 45. Although LFWFs depend on the choice of the light-front quantization direction, all observables such as matrix elements of local current operators, form factors, and cross sections are light-front invariants - they must be independent of wp. The light-front wave functions (LFWFs) &(xi, k l z ,Xi), with xi = Cr=,xi = 1, Cr=,k l l = 0 1 , are the coefficient functions for n partons in the Fock expansion, providing a general frame-independent representation of the hadron state. Matrix elements of local operators such as spacelike proton form factors can be computed simply from the overlap integrals of light front wave functions in analogy to nonrelativistic Schrodinger theory. In principle, one can solve for the LFWFs directly from fundamental theory using nonperturbative methods such as discretized light-front quantization (DLCQ), the transverse lattice, lattice gauge theory moments, or BetheSalpeter techniques. The determination of the hadron LFWFs from phenomenological constraints and from QCD itself is a central goal of hadron

z,

38

and nuclear physics. Reviews of nonperturbative light-front methods may be found in the references A potentially important method is to construct the qq Green’s function using light-front Hamiltonian theory, with DLCQ boundary conditions and Lippmann-Schwinger resummation. The zeros of the resulting resolvent projected on states of specific angular momentum J , can then generate the meson spectrum and their light-front Fock wavefunctions. The DLCQ properties and boundary conditions allow a truncation of the Fock space while retaining the kinematic boost and Lorentz invariance of light-front quantization. One of the central issues in the analysis of fundamental hadron structure is the presence of non-zero orbital angular momentum in the bound-state wave functions. The evidence for a “spin crisis” in the Ellis-Jaffe sum rule signals a significant orbital contribution in the proton wave function The Pauli form factor of nucleons is computed from the overlap of LFWFs differing by one unit of orbital angular momentum AL, = 51. Thus the fact that the anomalous moment of the proton is non-zero requires nonzero orbital angular momentum in the proton wavefunction 51. In the lightfront method, orbital angular momentum is treated explicitly; it includes the orbital contributions induced by relativistic effects, such as the spinorbit effects normally associated with the conventional Dirac spinors. In recent work, Dae Sung Hwang, John Hiller, Volodya Karmonov 52, and I have studied the analytic structure of LFWFs using the explicitly Lorentz-invariant formulation of the front form. Eigensolutions of the Bethe-Salpeter equation have specific angular momentum as specified by the Pauli-Lubanski vector. The corresponding LFWF for an n-particle Fock state evaluated a t equal light-front time T = w . x can be obtained by integrating the Bethe-Salpeter solutions over the corresponding relative light-front energies. The resulting LFWFs $:(xi, k l i ) are functions of the light-cone momentum fractions xi = ki . w / p ’ w and the invariant mass squared of the constituents M: = (CZ,k f ) 2 = Cy=l and the light-cone momentum fractions xi = k . w / p w each multiplying spinvector and polarization tensor invariants which can involve wp. The resulting LFWFs for bound states are eigenstates of the Karmanov-Smirnov kinematic angular momentum operator 53 and satisfy all of the Lorentz symmetries of the front form, including boost invariance. 46145947948.

49t50.

[k:Lma]i

39 6. A F S / C F T Correspondence and Hadronic Light-Front

Wavefunct ions

As shown by Maldacena 54, there is a remarkable correspondence between large N c supergravity theory in a higher dimensional anti-de Sitter space and supersymmetric QCD in 4-dimensional space-time. String/gauge duality provides a framework for predicting QCD phenomena based on the conformal properties of the AdS/CFT correspondence. The AdS/CFT correspondence is based on the fact that the generators of conformal and Poincare transformations have representations on the five-dimensional antideSitter space Ads5 as well as Minkowski spacetime. For example, Polchinski and Strassler 55 have shown that the power-law fall-off of hard exclusive hadron-hadron scattering amplitudes at large momentum transfer can be derived without the use of perturbation theory by using the scaling properties of the hadronic interpolating fields in the large-r region of AdS space. Thus one can use the Maldacena correspondence to compute the leading power-law falloff of exclusive processes such as high-energy fixed-angle scattering of gluonium-gluonium scattering in supersymmetric QCD. The resulting predictions for hadron physics effectively coincide with QCD dimensional counting r ~ l e s : ~ ~ ~ ~ ~ ~ ~ ~ 55956157

where n is the sum of the minimal number of interpolating fields in the initial and final state. (For a recent review of hard fixed Q C M angle exclusive processes in QCD see reference 6 1 . ) As shown by Brower and Tan 56, the non-conformal dimensional scale which appears in the QCD analysis is set by the string constant, the slope of the primary Regge trajectory A’ = aL(0) of the supergravity theory. Polchinski and Strassler 55 have also derived counting rules for deep inelastic structure functions at z 4 1 in agreement with perturbative QCD predictions 62 as well as Bloom-Gilman exclusive-inclusive duality. The supergravity analysis is based on an extension of classical gravity theory in higher dimensions and is nonperturbative. Thus analyses of exclusive processes 44 which were based on perturbation theory can be extended by the Maldacena correspondence to all orders. An interesting point is that the hard scattering amplitudes which are normally or order a: in PQCD appear as order a:’2 in the supergravity predictions. This can be understood as an all-orders resummation of the effective potential 54~63. The superstring theory results are derived in the limit of a large NC 64.

40

For gluon-gluon scattering, the amplitude scales as 1/ N s . For color-singlet bound states of quarks, the amplitude scales as 1/Nc. This large Nccounting in fact corresponds to the quark interchange mechanism 6 5 . For example, for K+p 4 K+p scattering, the u-quark exchange amplitude scales approximately as $, which agrees remarkably well with the measured large BCM dependence of the K+p differential cross section 66. This implies that the nonsinglet Reggeon trajectory asymptotes to a negative integer 67, in this case, lim-+,m a ~ ( t ) -1. De Teramond and I 68 have shown how to compute the form and scaling of light-front hadronic wavefunctions using the AdS/CFT correspondence in quantum field theories which have an underlying conformal structure, such as N = 4 super-conformal QCD. For example, baryons are included in the theory by adding an open string sector in A d s 5 x S5 corresponding t o quarks in the fundamental representation of the SU(4) symmetry defined on S5 and the fundamental and higher representations of S U ( N c ) . The hadron mass scale is introduced by imposing boundary conditions a t the A d s 5 coordinate r = ro = A Q C D R ~ The . quantum numbers of the lowest Fock state of each hadron including its internal orbital angular momentum and spin-flavor symmetry, are identified by matching the fall-off of the string wavefunction 8 ( x ,r ) at the asymptotic 3 1 boundary. Higher Fock states are identified with conformally invariant quantum fluctuations of the bulk geometry about the Ads background. The scaling and conformal properties of the AdS/CFT correspondence leads t o a hard component of the LFWFs of the form:

+

where gs is the string scale and A, represents the basic QCD mass scale. The scaling predictions agree with perturbative QCD analyses 69,44, but the AdS/CFT analysis is performed at strong coupling without the use of perturbation theory. The near-conformal scaling properties of lightfront wavefunctions lead to a number of other predictions for QCD which are normally discussed in the context of perturbation theory, such as constituent counting scaling laws for the leading power fall-off of form factors and hard exclusive scattering amplitudes for QCD processes. The ratio

41

-

of Pauli t o Dirac baryon form factor have the nominal asymptotic form F2(Q2)/F1(Q2)1/Q2, modulo logarithmic corrections, in agreement with the perturbative results ' O . Our analysis can also be extended t o study the spin structure of scattering amplitudes at large transverse momentum and other processes which are dependent on the scaling and orbital angular momentum structure of light-front wavefunctions.

Acknowledgements This talk is based on collaborations with Michael Binger, Gregory Gabadadze, Einan Gardi, Georges Grunberg, John Hiller, Dae Sung Hwang, Volodya Karmanov, Andre Kataev, Hung Jung Lu, Sven Menke, Carlos Merino, Johan Rathsman, and Guy de TBramond. I am also thankful t o the William I. Fine Theoretical Physics Institute at the University of Minnesota which sponsored this meeting. This work was supported by the U.S. Department of Energy, contract DE-AC03-76SF00515.

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A VARIATIONAL LOOK AT QCD*

J. GUILHERME MILHANO CENTRA, Instituto Superior Te‘cnico (IST), Av. Rovisco Pais, P-1049-001 Lisboa, Portugal E-mail: [email protected]

I discuss the applicability of variational methods to the study of non-perturbative aspects of QCD. An illustration of the capabilities of the method pioneered by Kogan and Kovner is given through the analysis of the deconfinement phase transition in gluodynamics in 3+1 dimensions.

1. Introduction

There is very little doubt that Quantum Chromodynamics (QCD) is the correct theory of the strong interactions. Understanding the properties of the vacuum sector of QCD - phenomena such as confinement and chiral symmetry breaking - remains, in spite of years of attempts, one of the main problems in modern quantum field theory. Amongst the variety of approaches used to tackle these problems, there have been several attempts to apply versions of the variational RayleighR t z method [l].The crux of my argument is that such methods, in particular Kogan and Kovner’s version [l,21, offer a possibility to analytically obtain non-perturbative dynamical information directly from QCD. Although this information is partial and incomplete, the variational approach provides significant qualitative understanding, and, to a limited extend, quantitative estimates. When attempting to apply the variational principle to quantum field theory (QFT) one is faced with the discussed by Feynman [3]. In what follows I will describe the construction of a variational trial state which, to a certain extent, overcomes these difficulties, and then illustrate the variational approach’s capabilities by discussing its application to the study *Thiswork is supported by the Fundqiio para a C i h c i a e a Tecnologia (Portugal) under contract SFRH/BPD/12112/2003.

44

45

of the deconfining phase transition in gluodynamics in 3+1 dimensions [4,5].

2. The variational Ansatz The first criterion to be considered when constructing a trial variational state is that the trial state ought to be general enough to allow, through variation of its parameters, the relevant physics to be spanned. Then there is the problem of calculability. In QFT, our ability to evaluate path integrals is, to say the least, limited. The requirement of calculability almost unavoidably restricts the possible form of the trial state to a Gaussian. Another serious problem is that of “ultraviolet modes”. The main motivation of a variational calculation in a strongly interacting theory is to learn about the distribution of the low momentum modes of the field in the vacuum wave functional. However, the VEV of the energy (and all other intensive quantities) is dominated by contributions of high momentum fluctuations, for the simple reason that there are infinitely more ultraviolet modes than modes with low momentum. Therefore, even if one has a clear idea of how the WF at low momenta should look, if the ultraviolet part of the trial state is even slightly incorrect the minimization of energy may lead to absurd results. Due to the interaction between the high and low momentum modes, there is a good chance that the infrared (IR) variational parameters will be driven to values which minimize the interaction energy, and have nothing to do with the dynamics of the low momentum modes themselves. Finally, in gauge theories there is the additional complication of gauge invariance. Allowable wave functionals must be invariant under the time independent gauge transformations. If one does not impose the Gauss’ law on the states exactly, one is not solving the right problem. The QCD Hamiltonian is only defined on the gauge invariant states, and its action on non gauge invariant states can be modified at will. Thus, by minimizing a particularly chosen Hamiltonian without properly restricting the set of allowed states, one is taking the risk of finding a “vacuum” which has nothing to do with the physical one, but is only picked due to a specific form of the action of the Hamiltonian outside the physical subspace. The trial variational state proposed by Kogan and Kovner [2] explicitly satisfies Gauss’ law, has the correct UV behaviour built-in, and allows for analytical calculations - yielding non-perturbative results - to be carried out.

46

In order to illustrate the capabilities of this variational method, I now turn t o the analysis of the deconfinent transition in gluodynamics.

3. Deconfinement phase transition In the almost 25 years since the pioneering work of Polyakov [6] and Susskind [7] much effort has been devoted t o attempts to understand both the basic physics and quantitave features of the deconfining phase transition of QCD. The high temperature phase is becoming well understood, and is widely believed to resemble a plasma of almost free quarks and gluons. However, the transition region, Tc < T < 2Tc, is very poorly understood. This region is the most interesting since it is there that the transition between 'hadronid and 'partonic' degrees of freedom occurs. The study of the transition region is a complicated and inherently nonperturbative problem which has mostly evaded treatment by analytical methods. Recently, the method introduced by Kogan and Kovner [2] was applied to the study of the phase transition in S U ( N ) gluodynamics [4,5]. Following [4,5],we minimize the relevant thermodynamic potential a t finite temperature, i.e. the Helmholtz free energy, on a set of suitably chosen trial density matrix functionals. We consider density matrices which, in the field basis, have Gaussian matrix elements and where gauge invariance is explicitly imposed by projection onto the gauge-invariant sector of the Hilbert space

p[A, A'] =

/

DU exp

{ / -

[A:(z)G,i'"b(~,y)A~(y)

X,Y

+ A ~ U a ( ~ ) G i j l a ~)Ai.'~(y) b(~, - 2 A ; ( ~ ) H $ ~ ( z , y ) A y ~ ( y ) ] } , (1)

s,,

s

where = d3x d3y, DU is the S U ( N ) group-invariant measure, and under an S U ( N ) gauge transformation U

A:(z)

.+

Aya(z) = Sab(z)A!(z)

+ X~(X),

(2)

with 1 sub= -tr(TaUtTbU) , 2

i A: = - t r ( r U U t & U ) ,

(3)

9

and f form an N x N Hermitian representation of S U ( N ) : [$, i f a b c $ with normalization tr(TaTb)= 2hab.

$1

=

47

We take the variational functions diagonal in both colour and Lorentz indices, and translationally invariant

GTlab(z,y) Y = 6ab6i.jG-1(z- y) , HGb(z,y)= Pb&.jH(z- y) .

(4)

Further, we split the momenta into high and low modes with k >( M and restrict the kernels G-l and H to the one parameter momentum space forms

The logic behind this choice of ansatz is the following. At finite temperature we expect N ( k ) to be roughly proportional to the Bolzmann factor exp{-E(k)P}. In our ansatz, the role of one particle energy is played by the variational function G-'(k) and its form is motivated by the propagator of a massive scalar field, i.e. ( k 2 M 2 ) l / ' . We will be interested only in temperatures close to the phase transition, and those we anticipate to be small, T, 5 M . For those temperatures one particle modes with momenta k 2 M are not populated, and we thus put H ( k ) = 0. For k 5 M the Bolzmann factor is non-vanishing, but small. Further, it depends only very weakly on the value of the momentum. With the above restrictions on the kernels, only two variational parameters, M and H , remain. Importantly, the density matrix functional in eq. (1) describes, for H = 0, a pure state p = (@[A] >< @ [ A ]where ) @[A]are Gaussian wave functionals. For H # 0, eq. (1) describes a mixed state with IHI proportional to the entropy of the trial density matrix. The expectation value of a gauge invariant operator in the variational state eq. (1)is then given by

+

( # ) A , U = z-lTr(p#)

= 2-l

J

V U V A # ( A , A').

where Z is the normalization of the trial density matrix p, i.e. 2 = T r p = J V U D A exp

{ f [AG-lA + AUG-lAU -

- ,,HAu]}

(7)

To evaluate the above expressions we first perform, for fixed U ( z ) ,the Gaussian integration over the vector potential A. For 2 we get, in leading

48

order in H ,

z=JOUexp{

- ~ A ( ~2 + ~ ( S + S T ) ) A + ~ H G t r ( S + S T )

We now integrate out the high momentum, k 2 > M 2 , modes of U perturbatively t o one-loop order. This effects a renormalization group transformation on the low modes, replacing the bare coupling g2 by the running coupling g 2 ( M ) [2,9]. To one-loop accuracy, the coupling g 2 ( M ) runs identically to the Yang-Mills coupling [9]. The normalization 2 can be then interpreted as the generating functional

z= T r p =

J

vue-S(U)

(9)

for an effective non-linear a-model for the low momentum modes ( k 2 < M 2 ) defined by the action

M S ( U ) = -tr(auaut) 2g2

-

" [

-tr 8g2

( U ~ B U- X J ~ U ) ( ~ U-UU~X J ~ ) ] -

1 4 9

-HM2trUttrU,

(10)

where U independent pieces have been dropped. The matrix U plays the same role as Polyakov's loop P at finite temperature - the functional integration over U projects out the physical subspace of the large Hilbert space on which the Hamiltonian of gluodynamics is defined. = This a-model has a phase transition [2]at the critical point (for A Q ~ D 150Mev, N = 3 and with the one-loop Yang-Mills ,B function) 24

M C= AQcDerr

= 8 . 8 6 A ~ c o= 1.33Gev.

(11)

R For M < Mc, the a-model is in a disordered, S U ( N ) L @ S U ( N ) symmetric, phase with massive excitations and where ( U ) = 0. Since U is the Polyakov loop, this corresponds to a confined state. When M > Mc, the a-model is in an ordered, SU(N)" symmetric, phase with massless Goldstone bosons for which ( U ) # 0, corresponding to a deconfined state. With this analysis we have established a correspondence between the a-model phase transition and the deconfinement transition in SU(N ) gluodynamics. In fact, this correspondence can be argued in rather general terms. Let us ask ourselves what would happen if we did not restrict H to be small, and more generally did not restrict the functional forms of G ( k ) and H ( k )

49

in our variational ansatz. We could still carry on our calculation for a while. Namely we would be able to integrate over the vector potentials in all averages, and would reduce the calculation to a consideration of some non-linear a-model of the U-field. This a-model quite generally will have a symmetry breaking phase transition as the variational functions G ( k ) and H ( k ) are varied. Since at this transition the Polyakov loop U changes its behavior, the disordered phase of the a-model corresponds to the confining phase of the Yang-Mills theory, while the ordered phase of the o-model represents the deconfined phase. Thus, in order to study deconfinement in the S U ( N ) Yang-Mills theory, we should analyze the physics of each a-model phase as accurately as possible and calculate the transition scale M , (or rather G c ( k ) ) . We then calculate the free energy of the a-model in each phase at temperature T and extract the minimal free energy. The deconfinement transition occurs at the temperature for which the free energies calculated in the ordered and disordered phases of the sigma model coincide. The Helmholtz free energy F of the density matrix p is given by F = (H) - T(S) ,

where H is the standard Yang-Mills Hamiltonian

with

1

+

B ~ ( x=) ~ ~ j k { d j A $ &A?(X) (~) g f a b c A ~ ( X ) A ~ ( x ) }(14) ,

S is the entropy, and T is the temperature. Thus

In the disordered phase, no progress seems possible without restricting the arbitrary kernels. Following [4],we adopt the forms eq. ( 5 ) . For small H , we consider only the first non-trivial order in H , that is a term of o(H1nH) in the entropy. This term can be written as a product of left S U ( N ) and right S U ( N ) currents and does, therefore, vanish in the disordered, S U ( N ) L@ S U ( N ) R symmetric, phase [4].The remaining contribution to the free energy, the average of the Hamiltonian, is evaluated

50

in the mean field approximation 121. The free energy is minimized for M = Mc 2~ 1.33GeV

N ~ M , ~ Fdis = --. 30n2 The simplest option to evaluate the free energy in the disordered phase is to use perturbation theory. Perturbation theory is certainly appropriate for large enough values of M , where the expectation value of the U field is close to unity. From numerical studies [lo] it is known that the transition occurs when the expectation value of U is greater than 0.5. We can thus expect perturbation theory to be qualitatively reliable down to the transition point. In the leading order perturbation theory approximation to the ordered phase of the a-model, however, minimisation with respect to arbitrary kernels G-' and H for both high and low modes is possible. Further, the analysis can be carried out to all orders in the thermal disorder kernel H. In this approximation, the U matrices can be parameterized in the standard exponential form and expanded in the coupling g

Hence at leading order one can take

Thus, the gauge transformations ( 2 ) reduce to

and the Hamiltonian (13) reduces to

These last two equations describe the theory U(1)N2-1: in the leading order of a-model perturbation theory, the S U ( N ) Yang-Mills theory reduces to the U(1)N2--'free theory. Moreover, the density matrix eq. (1) becomes Gaussian again, because the gauge transformations are linear:

p[A,A'] =

/

Dp exp

{

-

[AG-lA

+ (A' - dp)G-'(A' -

- dp)

2AH(A' - dp)]

}.

(21)

51

+

The theory of N 2 - 1 U(1) free fields in 3 1 dimensions is completely tractable; the variational analysis for the U(1) theory (with Gaussian ansatz (21)) was discussed in [12]. The free energy in momentum space in terms of the arbitrary kernels G-l and H is

(

In G ][:- -In [;HI -

-4T

$1

’ (22)

<

where Q = 1 - (1 - (GW)2)1/2and = (1 - (GH)2)1/2- (1 - G H ) . The kernels which minimize the free energy are

and the minimal value of the free energy a t temperature T is

F=

,

7r2 N 2 T 4

45

(24)

So we see that the free energy of S U ( N ) is minimised with M = M , in the disordered phase of the a-model for temperatures from zero up to a temperature T, where

which in turn implies

Tc --

(i)1’4T

- II 470MeV.

Although the actual value of the transition temperature is considerably larger than the lattice estimate it makes more sense t o look a t dimensionless quantities. In particular, if we identify 2Mc with the mass of the lightest glueball [ll](see also the discussion in [l]),we find

This is in excellent agreement with the lattice estimate for S U ( 3 ) pure gauge theory [8].

52 4. Conclusions

In 3+1 dimensional gluodynamics, this variational method gives results which on the qualitative level at least, conform with our intuition about the structure of the ground state. We find a first order phase transition which corresponds to the Polyakov loop acquiring a non-zero average. Although we have not calculated the string tension directly, the behavior of the Polyakov loop is very much indicative that this is indeed the deconfining phase transition. The value of the critical temperature (in units of glueball mass) we find is in good agreement with lattice results. We also found that in the low temperature phase the entropy remains zero all the way up to the transition temperature. This is a rather striking result, which has not been built into our variational ansatz, but rather emerged as the result of the dynamical calculation. An important lesson we learned is that the projection of the Gaussian trial state onto the gauge invariant Hilbert subspace dictates most, if not all, of the important aspects of the non-perturbative physics. It was absolutely essential to perform the projection non-perturbatively, fully taking into account the contribution of the overlap between gauge rotated Gaussians into the variational energy prior to minimization. We have seen that from the point of view of the effective a-model the energy is minimized in the disordered phase. In other words, the low momentum fluctuations of the field U are large, unlike in the perturbative regime, where U is close to a unit matrix. From the point of view of the trial wave functional, this means that the off-diagonal contributions, coming from the Gaussian WF gauge rotated by a slowly varying gauge transformation, are large. It is these “off diagonal” contributions to the energy that lowered the energy of the best trial state below the perturbative value. In the low temperature phase the vanishing of the entropy was also a direct consequence of the effective a-model being in the disordered phase, and thus of the non-perturbative nature of the gauge projection. This variational method appears to be a good candidate for a useful calculational scheme for strongly interacting gauge theories. However , many outstanding questions remain. Is the best variational state confining? How do we calculate the interaction potential between external sources? How do we understand better the relation between the variational parameter and the glueball masses? Can we extend the Ansatz to include (massless) fermions? Personally, I believe that these results are genuine and that there is

53 enough scope for further development of t h e approach which warrants continuing active investigations.

Acknowledgments This talk is based upon work done in collaboration with Ben Gripaios Alex Kovner and Ian Kogan.

,

References 1. A. Kovner and J. G. Milhano, in M. Shifman, J. Wheater and A. Vainstein (eds.): Circumnaviagting Theoretical Physics (World Scientific, Singapore, 2004), [hep-ph/0406165]. 2. I. Kogan and A. Kovner, Phys. Rev. D52, 3719 (1995); e-Print Archive: hepth/9408081; 3. R. Feynman, in Wangerooge 1987, Proceedings, Variational calculations in quantum field theory, L. Polley and D. Pottinger, eds (World Scientific, Singapore, 1988). 4. I. I. Kogan, A. Kovner and J. G. Milhano, JHEP 0212, 017 (2002) [arXiv:hepph/0208053]. 5. B. M. Gripaios and J. G. Milhano, Phys. Lett. B 564 (2003) 104 [arXiv:hepph/0302172]. 6. A. Polyakov, Phys. Lett. B 72 (1978) 477 7. L. Susskind, Phys. Rev. 20 (1979) 2610 8. M. J. Teper, arXiv:hep-th/9812187. 9. W. E. Brown and I. I. Kogan, Int.J.Mod.Phys.A14:799,1999; e-Print Archive: hep-th/9705136; 10. J. B. Kogut, M. Snow and M. Stone Nucl.Phys.B200:211,1982 11. B. M. Gripaios, Int. J. Mod. Phys. A 18 (2003) 85 [arXiv:hep-ph/0204310]. 12. B. M. Gripaios, Phys. Rev. D 67,025023 (2003) [arXiv:hep-th/0211104].

BARYONS, INc.

RICHARD F. LEBED Department of Physics & Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA E-mail: [email protected] Excited baryons may be analyzed in the l/Nc expansion as true resonances in scattering amplitudes. The key idea making this program possible is a generalization of methods originally applied to chiral soliton models in the 1980's. One finds model-independent relations among amplitudes that impose mass and width degeneracies among resonances of various quantum numbers. Phenomenological evidence demonstrates that patterns of resonant decay predicted by l/Nc agree with data. The analysis can be extended to subleading orders in l/Nc, where again agreement with data is evident.

1. Introduction Although 100 of the 1000 pages in the Review of Particle Properties' consist of compiled measured properties of baryon resonances, these states have resisted a model-independent analysis for decades. No one really understands ab initio from QCD why baryon resonances exist at all, much less their peculiar observed multiplicities, spacings, and branching fractions. Even the unambiguous existence of numerous resonances remains open to debate, as evidenced by the infamous 1- to 4-star classification system. Baryon resonances are exceptionally difficult to study precisely because they are resonances, i. e., unstable under strong decay. As a particular example, quark potential models are strictly speaking applicable only when qq pair production from the vacuum is suppressed, but this mechanism is the means by which baryon resonances are produced in meson-baryon scattering. The most natural description of excited baryons in large N , employs the N, valence quark baryon picture: Since the ground-state baryon multiplets ( J p = $' and for N, = 3) neatly fill a single multiplet completely symmetric under combined spin-flavor symmetry [the SU(6) 561, one may suppose that the ground state of O(N,) quarks is also completely spin-

'4

54

55 flavor symmetric.a Then, in analogy to the nuclear shell model, excited states may be formed by promoting a small number [O(N:)]of quarks into orbitally or radially excited orbitals. For example, the generalization of the SU(6)xO(3) multiplet (70,1-) consists of N , - 1 quarks in the ground state and one in a relative l = 1 state. One may then analyze observables such as masses and axial-vector couplings by constructing a Hamiltonian whose terms possess definite transformation properties under the spin-flavor symmetry and are accompanied by known powers of N,. By means of the Wigner-Eckart theorem, one then relates observabIes for different states in each multiplet. This approach has been extensively s t ~ d i e (see d Ref. 9 for a short review), but it falls short in two important respects: First, a Hamiltonian formalism is not entirely appropriate to unstable particles, since it refers t o matrix elements between asymptotic external states. Indeed, a resonance is properly represented by a complex-valued pole in a scattering amplitude, for which the real and imaginary parts indicate the mass and width, respectively. Moreover, the Hamiltonian approach makes no reference t o resonances as excitations of ground-state baryons. Second, even if one were to construct a Hamiltonian respecting the instability of the resonances, it is not clear that the simple quark-shell baryon multiplets would be eigenstates of this Hamiltonian. Just as in the nuclear shell model, the possibility of configuration mixing means that the true eigenstates might consist of mixtures with 1, 2, or more excited quarks. In contrast t o quark potential models, chiral soliton models naturally accommodate baryon resonances as excitations resulting from scattering of mesons off ground-state baryons. Such models are consistent with the large N, limit because the solitons are heavy, semiclassical objects compared to the mesons. As has been known for many years,1° a number of predictions following from the Skyrme and other chiral soliton models are independent of the details of the soliton structure, and may be interpreted as grouptheoretical, model-independent large N, results. Indeed, the equivalence of group-theoretical results for ground-state baryons in the Skyrme and quark models in the large N, limit was demonstrated" long ago. In the remainder of this paper I discuss how the chiral soliton picture may be used to study baryon resonances as well as the full scattering amplitudes in which they appear. It is based upon a series of papers written in collaboration with Tom Cohen (and more recently his s t ~ d e n t s ) . ~ aThis is reasonable because SU(6) spin-flavor symmetry for ground-state baryons becomes exact in the large Nc limit.2

56

2. Amplitude Relations In the mid-1980’s a series of papers17~18~19~20~2’ uncovered a number of linear relations between meson-baryon scattering amplitudes in chiral soliton models. It became apparent that these results are consistent with the large N, limit because of their fundamentally group-theoretical nature. Standard N , counting22 shows that ground-state baryons have masses of O(N:), but meson-baryon scattering amplitudes are O(N;). Therefore, the characteristic resonant energy of excitation above the ground state and resonance widths are both generically expected to be O ( N 2 ) . To say that two baryon resonances are degenerate to leading order in l/Nc thus actually means that their masses are equal at both the O(N:) and O(N;) level. An archetype of these linear relations was first derived in Ref. 19. For a ground-state (N or A) baryon of spin = isospin R scattering with a meson (indicated by the superscript) of relative orbital angular momentum L (and primes for analogous final-state quantum numbers) through a combined channel of isospin I and spin J , the full scattering amplitudes S may be expanded in terms of a smaller set of “reduced” scattering amplitudes s:

~ K 6L( L R J );3

SzRJ =

,

(2)

K

where [XI = 2X+1, and d ( j l j 2 j 3 ) indicates the angular momentum triangle rule.b The fundamental feature inherited from chiral soliton models is the quantum number K (grand spin), with K - I + J , conserved by the underlying hedgehog configuration, which breaks I and J separately. The physical baryon state consists of a linear combination of K eigenstates that is an eigenstate of both I and J but no longer K . K is thus a good (albeit hidden) quantum number that labels the reduced amplitudes s. The dynamical content of relations such as Eqs. (1)-(2) lies in the s amplitudes, which are independent for each value of K allowed by b ( I J K ) . In fact, K conservation turns out to be equivalent to the l/Nc limit. The proof12 begins with the observation that the leading-order (in l/Nc) t-channel exchanges have It = Jt,24 which in turn is proved using large N, consistency ~onditions~~-essentially,unitarity order-by-order in 1IN, in meson-baryon scattering processes. However, (s-channel) K conservation bBoth are consequences of a more general formulaz3 involving 9 j symbols that holds for mesons of arbitrary spin and isospin, which for brevity we decline t o include.

57 was found-years earlier-to be equivalent to the (t-channel) It = Jt rule.21 The proof of this last statement relies on the famous Biedenharn-Elliott sum rule,26 an SU(2) identity. The significance of Eqs. (1)-(2) lies in the fact that more full observable scattering amplitudes S than reduced amplitudes s exist. Therefore, one finds a number of linear relations among the measured amplitudes holding at leading [O(N;)] order. In particular, a resonant pole appearing in some physical amplitude must appear in at least one reduced amplitude; but this same amplitude contributes to a number of other physical amplitudes, implying a degeneracy between the masses and widths of resonances in several channels." For example, we apply Eqs. (1)-(2) to negative-parityc I = J = and states (called N1/2, N3/2) in Table 1. Noting that neither the orbital angular momenta L , L' nor the mesons n , that ~ comprise the asymptotic states can affect the compound state except by limiting available total quantum numbers ( I , J , K ) , one concludes that a resonance in the S r p N channel ( K = 1) implies a degenerate pole in DTTN, because the latter contains a K = 1 amplitude. One thus obtains towers of degenerate

4,

Table 1. Application of Eqs. (1)-(2) to sample negativeparity channels. State Nl/2

N3/2

Quark Model Mass n o , ml

m l , m2

Partial Wave, K-Amplitudes SfiNN

=

SYOO

DTfA

-

sY22

s:,""

= =

so"

=

a2

sy,p

=

STOO

DyeA

= =

DTNN 13 DTNA 13

DVNN 13

(ST22 1 (ST22

a

+ SZ22)

-G22)

(ST22 + s ; 2 2 )

5:

negative-parity resonance multiplets labeled by K : N1/27 A3/2,

" '

N1/2, A1/2? N3/2, A3/2, A5/27

* '

A1/27 N3/2, A3/2, N5/2, A5/21 A7/2,

" '

(3;)

7

(S';OOr

(sk!2,s;)

s722)

.

7

(3)

It is now fruitful to consider the quark-shell large Nc analogue of the first excited negative-parity multiplet [the (70,1-)]. Just as for N, = 3, Varity enters by restricting allowed values of L ,

58 there are 2 Nl/2 and 2 N3/2 states. If one computes the masses to O(N;) for the entire multiplet in which these states appear, one finds only three distinct e i g e n v a l u e ~ ,which ~ > ~ ~are ~ ~labeled ~ mo, m l , and m2 and listed in Table 1. Upon examining an analogous table containing all the states in this multiplet,12 one quickly concludes that exactly the required resonant poles are obtained if each K amplitude, K = 0,1,2, contains precisely one pole, which is located at the value mK. The lowest quark-shell multiplet of negative-parity excited baryons is found to be compatible with, i.e., consist of a complete set of, multiplets classified by K . One can prove13 this compatibility for all nonstrange multiplets of baryons in the S U ( 6 ) ~ 0 ( 3shell ) picture.d It is important to point out that compatibility does not imply SU(6) is an exact symmetry at large N, for resonances as it is for ground states.2 Instead, it says that S U ( 6 ) ~ 0 ( 3 ) multiplets are reducible multiplets at large N,. In the example given above, m0,1,2 each lie only O(N;) above the ground state, but are separated by amounts of O(N,O). We point out that large N, by itself does not mandate the existence of any resonances at all; rather, it merely tells us that if even one exists, it must be a member of a well-defined multiplet. Although the soliton and quark pictures both have well-defined large N, limits, compatibility is a remarkable feature that combines them in a particularly elegant fashion.

3. Phenomenology Confronting these formal large N, results with experiment poses two significant challenges, both of which originate from neglecting O(l/Nc) corrections. First, the lowest multiplet of nonstrange negative-parity states covers quite a small mass range (only 1535-1700 MeV), while O(l/N,) mass splittings can generically be as large as O(100 MeV). Any claims that two such states are degenerate while two others are not must be carefully scrutinized. Second, the number of states in each multiplet increases with N,, meaning that a number of large N, states are spurious in N, = 3 phenomenology. For example, for N, 2 7 the analogue of the 70 contains 3 A3/2 states, but only 1 [A(1700)] for N, = 3. As N, is tuned down from large values toward 3, the spurious states must decouple through the appearance of factors such as (1 - 3/N,), which in turn requires one t o understand simultaneously leading and subleading terms in the l/Nc expansion. dStudies to extend these results t o flavor SU(3) are underway; while the group theory is more complicated, it remains tractable.

59 Nevertheless, it is possible t o obtain testable predictions for the decay channels, even with just the leading-order results. For example, note from Table 1 that the K = O(1) N I p resonance couples only to ~(n). Indeed, despite lying barely above the qN threshold, the N(1535)resonance decays through this channel 30-55% of the time, while the N(1650), which has much more comparable phase space for n N and v N , decays t o qN only 310% of the time. This pattern clearly suggests that the n-phobic N(1535) should be identified with K=O and the q-phobic N(1650) with K = l , the first fully field theory-based explanation for these phenomenological facts. 4. Configuration Mixing

As mentioned above, quark-shell baryon states with a fixed number of excited quarks are not expected to be eigenstates of the full QCD Hamiltonian. Rather, configuration mixing is expected t o cloud the situation. Consider, for example, the statement that baryon resonances are expected to have generically broad [O(N:)] widths. One may ask whether some states might escape this restriction and turn out to be narrow in the large N, limit. Indeed, some of the first work5 on excited baryons combined large N , consistency conditions and a quark description of excited baryon states to predict that baryons in the 70-analogue have widths of O(l/N,), while states in an excited negative-parity spin-flavor symmetric multiplet ( 5 6 ’ ) have O(N;) widths. In fact there exist, even in the quark-shell picture, operators responsible for configuration mixing between these multiplets. l4 The spin-orbit and spin-flavor tensor operators (respectively l s and l(’)g G, in the notation of Refs. 6,7,27),which appear a t O ( N 2 ) and are responsible for splitting the eigenvalues mo, ml, and m2,give nonvanishing transition matrix elements between the 70 and 56’. Since states in the latter multiplet are broad, configuration mixing forces at least some states in the former multiplet to be broad as well. One concludes that the possible existence of any excited baryon state narrow in the large N , limit requires a fortuitous absence of significant configuration mixing. 5 . Pentaquarks

The possible existence of a narrow isosinglet, strangeness +1 (and therefore exotic) baryon state 0+(1540),claimed to be observed by numerous experimental groups (but not seen by several others), was much discussed at this meeting. Although the jury remains out on this important question,

60 one may nevertheless use the large Nc methods described above to determine what degenerate partners a state with these quantum numbers would possess.15 For example, if one imposes the theoretical prejudice JQ = then there must also be pentaquark states with I = 1 , J = $ and I = 2, J = $, with masses and widths equal that of the O+, up to O(l/N,) corrections. The large N, analogue of the “pentaquark” actually carries the quantum numbers of N,+2 quarks, consisting of (N,+1)/2 spin-singlet, isosinglet ud pairs and an S quark. The operator picture, for example, shows the partner states we predict to belong to SU(3) multiplets 27 (I=1) and 35 (I=2).28 However, the existence of partners does not depend upon any particular picture for the resonance or any assumptions regarding configuration mixing. Since the generic width for such baryon resona.nces remains O(NZ), the surprisingly small reported width (
3,

5,

i,

6. l/Nc Corrections

All the results exhibited thus far hold at the leading nontrivial order (N:) in the l / N c expansion. We saw in Sec. 3 that l / N c corrections are essential not only to explain the sizes of effects apparent in the data, but in the very enumeration of physical states. Clearly, if this analysis is to carry real phenomenological weight, one must demonstrate a clear path to characterize l/Nc corrections to the scattering amplitudes. Fortunately, such a generalization is possible: As discussed in Sec. 2, the constraints on scattering amplitudes obtained from the large N, limit are equivalent to the t-channel requirement It = Jt. In fact, Refs. 24 showed not only that this result holds in the large N, limit, but also that exchanges with IIt - Jt I =n are suppressed by a relative factor l / N F . This result permits one to obtain relations for the scattering amplitudes including all effects up to and including O(l/N,):

-k

1 R’ I , [ R 1 It=J

] [L’R R’

J, L Jt=J+l

]

t(+) sgLL‘

61

One obtains this expression by first rewriting s-channel expressions such as Eqs. (1)-(2) in terms of t-channel amplitudes. The S j symbols in this case contain It and Jt as arguments (which for the leading term are equal). One then introduced6 new O(l/N,)-suppressed amplitudes st(*) for which Jt - It = f l . The square-bracketed S j symbols in Eq. (4) differ from the usual ones only through normalization factors, and in particular obey the same triangle rules. Relations between observable amplitudes that incorporate the larger set s t , st(+), and st(--) are expected to hold a factor of 3 better than those merely including the leading O(N;) results. Indeed, this is dramatically evident in cases where sufficient data is available, n N t nA scattering (Fig. 1). For example, (c) and (d) in the first 4 insets give the imaginary and real parts, respectively, of partial wave data for the channels sD31 (0) and (l/&)DS13 (0),which are equal up to O(l/N,) corrections; in (c) and (d) of the second 4 insets, the 0 points again are 5 0 3 1 data, while 0 represent -fiDS33, and by Eq. (4) these are equal up to O(l/N:) corrections.

7. Conclusions The purpose of this talk has been to convince you that there now exist reliable calculational techniques to handle not only long-lived ground-state baryons, but also unstable baryon resonances and the scattering amplitudes in which they appear. This approach, originally noted in chiral soliton models but eventually shown to be a true consequence of large N , QCD, is found to have phenomenological consequences [such as the large 77 branching fraction of the N(1535)] that compare favorably with real data. The first few steps into studying and classifying l/Nc corrections to the leading-order results, absolutely essential to obtaining comparison with the full data set, have begun. The measured scattering amplitudes appear t o obey the constraints placed by these corrections, and more work along these lines will be forthcoming. For example, the means by which the spurious extra resonances of large N, decouple as one tunes the value of N, down to 3 is crucial and not yet understood. All the results presented here, as mentioned in Sec. 2, have used only relations among states of fixed strangeness. Moving beyond this limitation means using flavor SU(3) group theory, which is rather more complicated than isospin SU(2) group theory. Nevertheless, this is merely a technical complication, and existing work shows that it can be o v e r ~ o m e . ~

62 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 1.3 1.2 1 0.8 0.6 0.4 0.2

0.1 0

-0.1 -0.2

1.4

1.5

1.6

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1.8

1.9

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1.4

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1.9

2

0.3 0.2 0.1

0 -0.1 -0.2

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I._

1.3

-0.3 -n-._ A 1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

-0.4 1.3

(d) 1.4

1.5

W (GeV)

W (GeV) 0.1

0 0.6 0.4 o.2 0

-0.1

-0.2

t

-0.3 -n

A

I._

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"._

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W (GeV)

1.3 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 1.3

-

1.4

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1.4

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W (GeV)

Figure 1. Real and imaginary parts of 7rN 7rA scattering amplitudes. The first 4 insets give two particular partial waves equal to leading order [hence indicating the size of O ( l / N c )corrections]. The second 4 insets give two particular linear combinations of the same data good to O ( l / N : ) .

At the time of this writing, all of the necessary tools appear to be in place to commence a full-scale analysis of baryon scattering and resonance parameters. One may envision a sort of resonance calculation factory, Baryons I N c . With sufficient time and researchers, the whole baryon resonance spectrum can be disentangled using a solid, field-theoretical approach based upon a well-defined limit of QCD. Acknowledgments

I would like to thank the organizers for their hospitality and inviting me to this most lively conference. The work described here was supported in

63 part by t h e National Science Foundation under Grant No. PHY-0140362,

References 1. Review of Particle Properties (K. Hagiwara et al.), Phys. Rev. D66,010001 (2002). 2. R.F. Dashen and A.V. Manohar, Phys. Lett. B315,425 (1993). 3. C.D. Carone, H. Georgi, L. Kaplan, and D. Morin, Phys. Rev. D50, 5793 (1994). 4. J.L. Goity, Phys. Lett. B414,140 (1997). 5. D. Pirjol and T.-M. Yan, Phys. Rev. D57, 1449 (1998); D57,5434 (1998). 6. C.E. Carlson, C.D. Carone, J.L. Goity, and R.F. Lebed, Phys. Lett. B438, 327 (1998); Phys. Rev. D59,114008 (1999). 7. J.L. Goity, C. Schat, and N. Scoccola, Phys. Rev. Lett. 88, 102002 (2002); Phys. Rev. D66,114014 (2002); Phys. Lett. B564,83 (2003). 8. C.E. Carlson and C.D. Carone, Phys. Rev. D58,053005 (1998); Phys. Lett. B441,363 (1998); B484,260 (2000). 9. R.F. Lebed, in NStar 2002: Proceedings of the Workshop on the Physics of Excited Nucleons, ed. by S.A. Dytman and E.S. Swanson, World Scientific, Singapore, 2003, p. 73 [hep-ph/0301279]. 10. E. Witten, Nucl. Phys. B223, 433 (1983); G.S. Adkins, C.R. Nappi, and E. Witten, Nucl. Phys. B228, 552 (1983); G.S. Adkins and C.R. Nappi, Nucl. Phys. B249,507 (1985). 11. A.V. Manohar, Nucl. Phys. B248,19 (1984). 12. T.D. Cohen and R.F. Lebed, Phys. Rev. Lett. 91,012001 (2003); Phys. Rev. D67,012001 (2003). 13. T.D. Cohen and R.F. Lebed, Phys. Rev. D68,056003 (2003). 14. T.D. Cohen, D.C. Dakin, A. Nellore, and R.F. Lebed, Phys. Rev. D69, 056001 (2004). 15. T.D. Cohen and R.F. Lebed, Phys. Lett. B578,150 (2004). 16. T.D. Cohen, D.C. Dakin, A. Nellore, and R.F. Lebed, hep-ph/0403125. 17. A. Hayashi, G. Eckart, G. Holzwarth, H. Walliser, Phys. Lett. B147,5 (1984). 18. M.P. Mattis and M. Karliner, Phys. Rev. D31,2833 (1985). 19. M.P. Mattis and M.E. Peskin, Phys. Rev. D32,58 (1985). 20. M.P. Mattis, Phys. Rev. Lett. 56,1103 (1986); Phys. Rev. D39,994 (1989); Phys. Rev. Lett. 63,1455 (1989). 21. M.P. Mattis and M. Mukerjee, Phys. Rev. Lett. 61,1344 (1988). 22. E. Witten, Nucl. Phys. B160,57 (1979). 23. M.P. Mattis, Phys. Rev. Lett. 56,1103 (1986). 24. D.B. Kaplan and M.J. Savage, Phys. Lett. B365, 244 (1996); D.B. Kaplan and A.V. Manohar, Phys. Rev. C56,76 (1997). 25. R.F. Dashen, E. Jenkins, and A.V. Manohar, Phys. Rev. D49,4713 (1994). 26. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, NJ, 1996) [Eq. (6.2.12)]. 27. D. Pirjol and C. Schat, Phys. Rev. D67,096009 (2003). 28. E. Jenkins and A.V. Manohar, hep-ph/0401190 and hep-ph/0402024.

QUARK CORRELATIONS AND SINGLE-SPIN ASYMMETRIES*

M. BURKARDT Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA E-mail: [email protected]

We illustrate, in a semi-classical picture, how the Wilson line phase factor in gauge invariantly defined unintegrated parton density can lead to a nonzero single-spin asymmetry (Sivers effect).

1. Introduction

Many hadron production reactions give rise to significant I single-spin asymmetries (SSA) which persist at high energies. For example, hyperons produced in hadronic collisions typically show a large polarization perpendicular to the production plane. More recently, the HERMES collaboration found a significant SSA (left-right asymmetry in the transverse momentum distribution of the produced mesons in the directions perpendicular to the nucleon spin) in semi-inclusive production of 7r and K mesons l. Theoretically, two mechanisms have been proposed to explain the asymmetry: the Sivers mechanism, where the final state interactions (FSI) give rise to a 1momentum asymmetry of the active quark already before it fragments into hadrons ', and the Collins mechanism, where the asymmetry arises when a Ipolarized quark fragments into mesons (see Ref. and references therein). Both mechanisms have recently been observed by the HERMES collaboration (they can be disentangled by also measuring the scattering plane of the electron) Simple model calculations have revealed that, even at high energies, the FSI can indeed give rise to a non-vanishing transverse single-spin asymmetry for the active quark (Sivers mechanism). Those FSI can formally be incorporated into a definition of unintegrated parton densities by

'.

495

*this work was supported by the department of energy (de-fg03-96er40965)

64

65 introducing appropriate Wilson line phase factors The Sivers mechanism is interesting for a variety of reasons 617.

it vanishes under naive time-reversal and a non-trivial phase from the FSI is needed to give a nonzero effect V the Sivers mechanism provides information about the space-time structure of the target 0 A nonzero Sivers mechanism implies a nonzero Compton amplitude involving nucleon helicity flip without quark helicity flip L) requires (nonperturbative) helicity nonconservation in the nucleon state (xSB!) 0 in model calculation the mechanism requires interference between phases of wave function components that differ by one unit of orbital angular momentum L I the effect may provide novel insights about orbital angular momentum in the proton. 0

The paper is organized as follows: in Section 2 we will introduce gauge invariant unintegrated parton densities. in Section 3, we will discuss the physics associated with the Wilson line gauge links in gauge invariantly defined parton densities, and in Section 4 we will investigate the asymmetry in light-cone gauge and explain how the Sivers mechanism can be related to quark correlations in the Iplane. We will then conclude with a simple semi-classical picture that relates I deformations of the quark densities with the average sign of the Sivers mechanism. 2. Unintegrated Parton Densities

The naive definition of a unintegrated (i.e. kl-dependent) parton density reads

This density is not only gauge dependent but also leads to a vanishing asymmetry J d2klqnaive(x,k l ) k l = 0. due to time-reversal invariance. At first one might be tempted to render Eq. (1) gauge invariant by connecting the quark field operators by a straight line gauge string (Wilson line), but this does not change the time-reversal argument and the resulting asymmetry is still zero. More importantly, the choice of path for the gauge string is not arbitrary but should be determined by the physical quantity that is observed in the experiment. If one is interested in the transverse

66

momentum of the knocked out quarks (i.e. mesons after fragmentation) then the choice of path should be such that it reflects the FSI of the outgoing quark. Since the active quark in DIS is ultra-relativistic, the correct way to render Eq. (1) gauge invariant is by including Wilson lines from the position of the active quark to infinity along the light-cone. In addition, the two Wilson lines need to be connected at light-cone infinity. The choice of path for the segment at x- = 00 is arbitrary as long as the gauge field at x- = 03 is pure gauge. These considerations suggests the following definition for unintegrated parton densities relevant for semi-inclusive DIS

with

= ~[m-,0,;0-,0,]4(0)

4u@)

qu (Y 1 3 q(Y P [ Y -

(3)

J, ioo- , Y l l *

The U's are Wilson line gauge links, for example

connecting the points 0 and

E. With the gauge links to x-

= 00 included,

Figure 1. Illustration of the gauge links in gauge invariant Sivers distributions (2).

time reversal invariance no longer implies a vanishing asymmetry. Indeed, under time reversal the direction of the gauge link changes and the only consequence of time reversal invariance for Eq. (2) is opposite signs for the asymmetries in semi-inclusive pion production and Drell-Yan experiments 3

While in principle all three gauge links in Fig. 1 contribute, the gauge link segment at x- = 03 is important only in light-cone gauge '. However, while the introduction of Wilson line phase factors in gauge invariant parton densities has helped to understand why there can be a

67 nonzero Sivers mechanism a t high energies, it has at the same time made the underlying physics of the mechanism rather obscure: the asymmetry arises from interference between phase factors from different partial waves and in addition require a additional nontrivial phase contribution from the Wilson line. In the rest of these notes we will illustrate how the Wilson line, together with the nucleon ground state wave function, conspires t o provide a transverse asymmetry.

3. Physics of the Wilson Line Phase Factor In order t o illustrate the physics of the Wilson line factors, we focus now on the average transverse momentum for quarks of flavor q

(kl,)

=

/ / dz

d2klq(z,k l ) k l .

(5)

We evaluate ( k l q )from Eq. (2) and integrate by parts, yielding

In this expression, the derivative can act either on the quark field operator or the gauge links. The term where the derivative acts on the quark field operator vanishes again due to time reversal invariance [it corresponds to the asymmetry that one would obtain starting from Eq. (l)].The interesting term is obtained by acting on the gauge links, which yields (modulo light-like gauge links) after some algebra 819*10

where terms that vanish because of time-reversal invariance have been dropped. Here Gb” is the gluon field strength tensor. This result has a simple semi-classical interpretation: Gf’ ( q ) is the I component of the force from the spectators on the active quark. Integrating this force along the trajectory (for a ultrarelativistic particle, time=distance) of the outgoing quark then yields the I impulse Somdq-G+’(v) which the active quarks acquires from the FSI as it escapes the hadron. The average k l is then obtained by correlating the quark density with the Iimpulse. However, although this result nicely illustrates the physics of the contribution from the Wilson lines, it still does not tell us the sign/magnitude of the asymmetry. Indeed, early estimates for Eq. (7) concluded that the resulting asymmetry should be very small

’.

68

4. Sivers Mechanism in A+ = 0 Gauge

There are several reasons why one is interested to proceed with light-cone gauge A+ = 0. First of all this is the most physical gauge for a lightcone description of hadrons, which is in turn the natural framework to describe DIS. Secondly, all light-like Wilson lines become trivial in this gauge. Furthermore, there exists already a rich phenomenology for lightcone wave functions of hadrons. However, if one neglects the gauge link at x- = 00 in Eq. (2) then the I asymmetry vanishes in A+ = 0 gauge: as we mentioned above the light-like gauge links are trivial in this gauge and without the gauge link at 2- = co, Eq. (2) reduces t o Eq. (1)which yields a vanishing asymmetry due to timereversal invariance. In fact, as was revealed by explicit model calculations in Ref. 4 , very careful regularization of the light-cone zero-modes (see Ref. l 1 and references therein) is required if one wants to calculate the asymmetry in A+ = 0 gauge. Starting again from Eq. ( 2 ) and setting A+ = 0 one finds

4.1. Finiteness conditions

In order t o make further progress, we need to express the gauge field at x- = 00 in terms of less singular degrees of freedom a t finite x-. We will do this in several steps. First me make use of the time reversal invariance and replace Al(co-,Ol) by [Al(co-,Ol) - A~(-oo-,Ol)] in Eq. (8). Then we use the fact that the gauge field at x- = f c o must be pure gauge, i.e. we impose as a condition on the states G+-( x - _- ~ C QXI) , = G 12 (x-= ~ C OXI) , = 0. (9) In light-cone gauge, G:- = 6'- A-a , and therefore Eq. (9) implies d-A-(z- = f c q x l ) = 0. Integrating the constraint equation for A-

-@A,

- d-diAS

- gfabcAiG:+ = j:,

we thus find the first finiteness condition

(10)

l2

@ d ( X l ) = -Pa(xL),

(11)

1 2

(12)

where .

clii(xl) = - [ A ~ ( c o - , x ~ ) - A ~ ( - w - , x ~ ) ]

69

Igauge field at the boundary and

is the anti-symmetric piece of the Pa(xl) =g

1

Xa

dx- [ q T + T q

+ .fadtG:+

1

(13)

is the total (quark+glue) color charge density integrated along x-. If one imagines the Lorentz contracted proton as a pizza then p a ( x ~is) the color charge density operator at position XI on the pizza. Finally, imposing G1‘(x- = fco, XI) = 0 implies i

Aj (00- ,XI) = - Ut (xi)@ U ( X ~ ) 9

(14)

and similarly at x- = -co with a different U . It is instructive to solve these constraints perturbatively. To lowest order one finds the “abelian” solution

corresponding to a Lorentz boosted Coulomb field integrated along the iaxis. Inserting this result into (8) yields

(ki) =

-+J *&

(P,S

1 ~ ( o > y f ~ t L Q ( O ) P a ( Y I )7I p , s )(16)

which has again a very physical interpretation: the average kl is obtained by summing over the I impulse caused by the color-Coulomb field (since we solved the constraint equations only to lowest order) of the spectators. One immediate consequence of this result is that the total Sivers effect (for the gluon Sivers effect see Refs. 13914) summed over all quarks and gluons with equal weight is zero

c

(kf) = 0

c=q,g

(by symmetry). One can show l5 that this result holds beyond lowest order in perturbation theory. It should emphasized that Eq. (17) is not a trivial consequence of momentum conservation since kl in the Sivers effect is not the momentum of the partons before the collision (which also enters the Nother momentum). Instead the I momentum in the Sivers effect is the sum of the momentum the partons have before being ejected plus the momentum they acquire due to the FSI. Since the momenta before the partons are ejected add up to zero, Eq. (17) is thus a statement about the net momentum due to the FSI: the net (summed over all partons) I momentum due to the FSI is zero, which is a nontrivial result since what one adds up here is not the Imomenta of all fragments in the target but

70 only the I momenta in the current fragmentation region. Eq. (17) is therefore a nontrivial and useful constraint on parameterizations of Sivers distributions Eq. (16) is also very useful for practical evaluation of the Sivers effect from light-cone wave functions. The original expression (8) involved the gauge field a t x- = 00, which is very sensitive to the regularization procedure, we have succeeded t o express the asymmetry in terms of degrees of freedom at finite x-, i.e. I color density-density correlations in the I plane. Eq. (16) can be directly applied t o light-cone wavefunctions, without further regularization. If we want to proceed further, we need a model for the light-cone wave function. Here we do not want to consider a specific model, but rather the whole class of valence quark models, which may be useful for intermediate and larger values of x. In a valence quark model, since the color part of the wave function factorizes, one can replace the color density-density correlations by neutral density-density correlations 14315y16.

and therefore

4.2. Connection with GPDs

From studies of generalized parton distributions (GPDs) it is known that the distribution of partons in the Iplane q(x,b l ) is significantly deformed for a transversely polarized target 17. The mean displacement of flavor q (Iflavor dipole moment) is given by

d;

's

IEP

- dxE,(x,O,O) = 2. (20) '-2M 2M

5Jdxpblq(x,bl)b

-

The I E ~= O(1- 2 ) are the anomalous magnetic moment contribution from each quark flavor to the anomalous magnetic moment of the nucleon (with charge factors taken out), i.e. Fz(0)= 3nU 2 - ?1 I E ~ Z1 K , .... This yields Id:[ = O ( 0 . 2 f m ) , where u and d quarks have opposite signs. This is a sizeable effect as is illustrated in Fig. ( 2 ) . The physical origin of this distaortion is that due t o the kinematics of DIS it is the j + = j o + j " density of the quarks which couples to the

71

Figure 2. Distribution of the j + density for u and d quarks in the Iplane ( x ~ =j 0.3 is fixed) for a nucleon that is polarized in the x direction in the model from Ref. 17. For other values of x the distortion looks similar.

electron: the electron in DIS couples more strongly to quarks which move towards the electron rather than away from it because if the quarks move towards (collision course) the electron the electric and magnetic forces add up, while if they move away the electric and magnetic forces act in opposite directions. For relativistic particles electric and magnetic forces are of the same magnitude. As a consequence, if the i axis is in the direction of the momentum of the virtual photon then the virtual photon couples only to the j + component of the quark current. Even though the j o component of the current density is the same on the +6 and -6 sides of the nucleon, the j z component has opposite signs on the +$ and -6 sides if the quarks have orbital angular momentum. Therefore the reason for the distortion is a combination of the fact that the electron ‘sees’ oncoming quarks better and the presence of orbital angular momentum. While Eq. (20) is a rigorous result regarding the average distortion of quarks with flavor q relative to the center of momentum, it still does not tell us exactly what the density density correlations are. However, qualitatively we expect that the sign and magnitude of the distortion is correlated with the sign of the density-density correlation. Using Eq. (19) we therefore expect for the resulting Sivers effect

for a proton polarized in +5 direction and we expect them to be roughly of the same magnitude.

72

Figure 3. The transverse distortion of the parton cloud for a proton that is polarized into the plane, in combination with attractive FSI, gives rise to a Sivers effect for u ( d ) quarks with a Imomentum that is on the average up (down).

The interpretation of these results is as follows: the FSI is attractive and thus it “translates” position space distortions (before the quark is knocked out) in the +fj-direction into momentum asymmetries that favor the -6 direction and vice versa (Fig. 3) 18. At least in a semi-classical description, this appears to be a very general observation, which is why we expect that the signs obtained above are not affected by higher order effects. 5 . Summary

Wilson line gauge links in gauge invariant Sivers distribution are a formal tool to include the final state interaction in semi-inclusive DIS experiments. The average transverse momentum due to these Wilson lines is obtained as the correlation between the quark density and the impulse from the spectators on the active quark is it escapes along its (almost) light-like trajectory. In light-cone gauge A+ = 0 only the gauge link at infinity contributes and careful regularization of the zero-modes is necessary. However, we succeeded in expressing the net asymmetry in terms of color density-density correlation in the Iplane. For a transversely polarized target the quark distribution in impact parameter space is transversely distorted due to the presence of quark orbital angular momentum: the j + current density is enhanced on the side where the quark orbital motion is head-on with the virtual photon. As the struck quark tries t o escape the target, one expects on average an attractive force from the spectators on the active quark, i.e. the FSI convert a left-right asymmetry for the quark distribution in impact parameter space into a right-left asymmetry for the I momentum of the active quark (Sivers effect). The sign of the distortion in impact parameter space, and hence the sign of the Sivers effect, for each quark flavor is determined by the sign of the anomalous magnetic moment contribution (of course with the electric

73 charge factored out) from t h a t quark flavor t o t h e anomalous magnetic moment of t h e nucleon.

References 1. N.C.R. Makins (HERMES collaboration), talk at eRHIC workshop, BNL, Jan. 2004; see also http://www-hermes.desy.de 2. D.W. Sivers, Phys. Rev. D 43,261 (1991). 3. J.C. Collins, Acta Phys. Polon. B 34,3103 (2003). 4. S.J. Brodksy, D.S. Hwang, and I. Schmidt, Phys. Lett. B 530,99 (2002). 5. M. Burkardt and D.S. Hwang, Phys. Rev. D 69, 074032 (2004); F. Yuan, Phys. Lett. B575,45 (2003); A. Bachetta, A. Schafer, and J.-J. Yang, Phys. Lett. B 578,109 (2004). 6. X. Ji and F. Yuan, Phys. Lett. B 543,66 (2002); A. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B 656,165 (2003). 7. J.C. Collins, Phys. Lett. B 536, 43 (2002); D. Boer, P.J. Mulders, and F. Pijlman, Nucl. Phys. B 667,201 (2003); see also R.D. Tangerman and P.J. Mulders, Phys. Rev. D 51,3357 (1995). 8. J.W. Qiu and G. Sterman, Phys. Rev. Lett. 67,2264 (1991). 9. A. Schafer et al., Phys. Rev. D 47,1 (1993). 10. D. Boer, P.J. Mulders, and F. Pijlman, Nucl. Phys. B 667,201 (2003). 11. M. Burkardt, Adv. Nucl. Phys. 23, l(1996). 12. M. Burkardt, Phys. Rev. D 69,057501 (2004). 13. P.J. Mulders and J. Rodrigues, Phys. Rev. D 63,094021 (2001). 14. D. Boer and W. Vogelsang, hep-ph/0312320. 15. M. Burkardt, Phys. Rev. D 69,091501 (2004). 16. M. Anselmino, M. Boglione, and F. Murgia, Phys. Rev. D 60,054027 (1999); M. Anselmino, U. D’Alesio, and F. Murgia, Phys. Rev. D 67,074010 (2003). 17. M. Burkardt, Int. J. Mod. Phys. A 18, 173 (2003). 18. M. Burkardt, Nucl. Phys. A 733,185 (2004).

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SECTION 2. HEAVY QUARK PHYSICS

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SOFT-COLLINEAR FACTORIZATION AND THE CALCULATION OF THE B -+ X,-y RATE

MATTHIAS NEUBERT* Institute for High-Energy Phenomenology Newman Laboratory for Elementary-Particle Physics, Cornell University Ithaca, IVY 14853, U . S . A .

Using results on soft-collinear factorization for inclusive B-meson decay distributions, a systematic study of the partial B --t X,y decay rate with a cut E, 2 Eo on photon energy is performed. For values Eo 5 1.9 GeV the rate can be calculated without reference to shape functions. The result depends on three large scales: mb, and A = m b - 2Eo. The sensitivity to the scale A M 1.1GeV (for Eo M 1.8 GeV) introduces significant uncertainties, which have been ignored in the past. Our new prediction for the B + X,y branching ratio with E, 2 1.8 GeV is where the errors refer to perturbative Br(B -+Xsy) = ( 3 . 4 4 f 0 . 5 3 f 0 . 3 5 ) x and parameter uncertainties, respectively. The implications of larger theory uncertainties for New Physics searches are explored with the example of the type-I1 two-Higgs-doublet model.

a,

1. Introduction

Given the prominent role of B + X,y decay in searching for physics beyond the Standard Model, it is of great importance to have a precise prediction for its inclusive rate and CP asymmetry in the Standard Model. The total inclusive B + X,y decay rate can be calculated using a conventional operator-product expansion (OPE) based on an expansion in logarithms and inverse powers of the b-quark mass. However, in practice experiments can only measure the high-energy part of the photon spectrum, E-, 2 Eo, where typically EO = 2GeV or slightly below (measured in the B-meson rest frame).li2 With E-, restricted to be close to the kinematic endpoint at M B / ~the , hadronic final state X , is constrained to have large energy EX M B but only moderate invariant mass Mx (MBAQCD)lI2. In this kinematic region, an infinite number of leading-twist terms in the OPE need

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*Work supported by the National Science Foundation under grant PHY-0355005

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to be resummed into a non-perturbative shape function, which describes the momentum distribution of the b-quark inside the B m e s ~ n . ~ ? ~ Conventional wisdom based on phenomenological studies of shapefunction effects says these effects are important near the endpoint of the photon spectrum, but they can be ignored as soon as the cutoff EO is lowered below about 1.9 GeV. In other words, there should be an instantaneous transition from the “shape-function region” of large non-perturbative corrections to the “OPE region”, in which hadronic corrections to the rate are suppressed by at least two powers of AqcDlmb. Below, we argue that this notion is based on a misconception. While it is correct that once the cutoff EO is chosen below 1.9GeV the decay rate can be calculated using a local short-distance expansion, we show that this expansion involves three “large” scales. In addition to the hard scale T n b , an intermediate scale corresponding to the typical invariant mass of the hadronic final state X,, and a low scale A = mb - 2Eo related to the width of the energy window over which the measurement is performed, become of crucial importance. The precision of the theoretical calculations is ultimately determined by the value of the lowest short-distance scale A, which in practice is of order 1GeV or only slightly larger. The theoretical accuracy that can be reached is therefore not as good as in the case of a conventional heavy-quark expansion applied to the B system. More likely, it is similar to (if not worse than) the accuracy reached, say, in the description of the inclusive hadronic decay rate of the T lepton. While we are aware that this conclusion may come as a surprise to many practitioners in the field of flavor physics, we believe that it is an unavoidable consequence of our analysis. Not surprisingly, then, we find that the error estimates for the B + X,y branching ratio that can be found in the literature are, without exception, too optimistic. Since there are unknown a : ( A ) corrections a t the low scale A 1GeV, we estimate the present perturbative uncertainty in the B -+X,y branching ratio with Eo in the range between 1.6 and 1.8GeV to be of order 10-15%. In addition, there are uncertainties due to other sources, such as the b- and c-quark masses. The combined theoretical uncertainty is of order 15-20%, about twice as large as what has been claimed in the past. While this is a rather pessimistic conclusion, we stress that the uncertainty is limited by unknown, higher-order perturbative terms, not by non-perturbative effects, which we find to be under good control. Therefore, there is room for a reduction of the error by means of well-controlled perturbative calculations.

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79 2. QCD factorization theorem

Using recent results on the factorization of inclusive B-meson decay distribution^,^?^ it is possible to derive a QCD factorization formula for the integrated B + X,y decay rate with a cut E7 2 EO on photon energy. In the region of large Eo, the leading contribution to the rate can be factorized in the form7

where AE = M B - 2Eo is twice the width of the window in photon energy over which the measurement of the decay rate is performed. The variable P+ = E x - IFXI is the “plus component” of the 4-momentum of the hadronic final state X,, which is related to the photon energy by P+ = M B - 2E7. The endpoint region of the photon spectrum is defined by the requirement that P+ 5 A E <( M B , in which case P p is called a hard-collinear momentum.8 mb is a hard scale, while pi In the factorization formula, ph ,/= is an intermediate hard-collinear scale of order the invariant mass of the hadronic final state. The precise values of these matching scales are irrelevant, since the rate is formally independent of ph and pi. The hard corrections captured by the function H y ( p h ) result from the matching of the effective weak Hamiltonian of the Standard Model (or any of its extensions) onto a leading-order current operator of soft-collinear effective theoryg (SCET). At tree level, H 7 ( p h ) = C;y(ph) is equal to the “effective” coefficient = C77 - C5 - CS. The expression valid a t next-to-leading order can be found in Ref. 7. The function H y ( p h ) is multiplied by the running b-quark mass m b ( p h ) defined in the MS scheme, which is part of the electromagnetic dipole operator Q77. The jet function J(mb(P+- G ) , p i ) in (1) describes the physics of the final-state hadronic jet. An expression for this function valid at next-toleading order in perturbation theory has been derived in Refs. 5, 6. The perturbative expansion of the jet function can be trusted as long as ps m,A with A = mb - 2Eo << MB. Note that the “natural” choices ph 0: mb and ps = mbjii with jii independent of mb remove all reference to the bquark mass (other than in the arguments of running coupling constants) from the factorization formula. The shape function S(h,pi) parameterizes our ignorance about the soft N

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5

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physics associated with bound-state effects inside the B m e s ~ n .Its ~ ?naive ~ interpretation is that of a parton distribution function, governing the distribution of the light-cone component k+ of the residual momentum of the b quark inside the heavy meson. Once radiative corrections are included, however, a probabilistic interpretation of the shape function breaks down.5 For convenience, the shape function is renormalized in (1) at the intermediate hard-collinear scale pi rather than at a hadronic scale phad. This removes any uncertainties related to the evolution from pi to phad. Since the shape function is universal, all that matters is that it is renormalized at the same scale when comparing different processes. The last ingredient in the factorization formula is the function U1 (ph, p i ) , which describes the renormalization-group (RG) evolution of the hard function IHYl2from the high matching scale ph down to the intermediate scale p i , at which the jet and shape functions are renormalized. The exact expression for this quantity and its perturbative expansion valid at next-to-next-to-leading logarithmic order can be found in Ref. 7. As written in ( l ) , the decay rate is sensitive to non-perturbative hadronic physics via its dependence on the shape function. This sensitivity is unavoidable as long as the scale A = mb - 2Eo is a hadronic scale, corresponding to the endpoint region of the photon spectrum above, say, 2 GeV. Here we are interested in a situation where EOis lowered out of the shape-function region, such that A can be considered large compared with AQCD. For orientation, we note that with mb = 4.7GeV and the cutoff EO = 1.8GeV employed in a recent analysis by the Belle Collaboration’ one gets A = 1.1GeV. The values of the three relevant physical scales as functions of the photon-energy cutoff Eo are shown in Figure 1. This plot illustrates the fact that the transition from the shape-function region to the region where a conventional OPE can be applied is not abrupt but proceeds via an intermediate region, in which a short-distance analysis based on a multi-stage OPE (MSOPE) can be performed. The transition from the shape-function region into the MSOPE region occurs when the scale A becomes numerically (but not parametrically) large compared with A Q ~ D . Then terms of order a:(A) and ( A Q ~ D / A )which ~ , are non-perturbative in the shape-function region, gradually become decent expansion parameters. Only for very low values of the cutoff (Eo< l G e V or so) it is justified to as scales of order mb. treat A and Separating the contributions associated with these scales requires a sophisticated multi-step procedure. The first step, the separation of the hard scale from the intermediate scale, has already been achieved in (1). To

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0.5

0

1

1.5

Eo [GeV]

2

2.5

a

Figure 1. Dependence of the three scales p h = mb (solid), pi = (dashed), and pa = A (dash-dotted) on the cutoff Eo, assuming mb = 4.7GeV. The gray area a t the bottom shows the domain of non-perturbative physics. The light gray band in the center indicates the region where the MSOPE must be applied.

proceed further, we use that integrals of smooth weight functions with the shape function S(b,p ) can be expanded in a series of forward B-meson matrix elements of local operators in heavy-quark effective theory” (HQET), provided that the integration domain is large compared with A Q C D . ~The ’~ perturbative expansions of the associated Wilson coefficient functions can be trusted as long as p A. In order to complete the scale separation, it is therefore necessary to evolve the shape function in (1) from the intermedidown to a scale po A. This can be achieved using ate scale pi the analytic solution to the integro-differential RG evolution equation for the shape function in momentum s p a ~ e . ~ J ~ As a final comment, we stress that the main purpose of performing the scale separation using the MSOPE is not that this allows us to resum Sudakov logarithms. Indeed, the “large logarithm” ln(mb/A) M 1.5 is only parametrically large, but not numerically. What is really important is to disentangle the physics at the low scale po A, which is “barely perturbative”, from the physics associated with higher scales, where a short-distance treatment is on much safer grounds. The MSOPE allows us to distinguish M 0.29, between the three coupling constants a,(mb) M 0.22, a,(-) and as(A) x 0.44 (for A = 1.1GeV), which are rather different despite the fact that there are no numerically large logarithms in the problem. Given the values of these couplings, we expect that scale separation between A and mb is as important as that between mb and the weak scale M w .

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82 3. Calculation of the shape-function integral

The scale dependence of the renormalized shape function is governed by an integro-differential RG evolution equation, whose exact solution in momentum space can be found using a technique developed in Ref. 11. The result takes the remarkably simple form

The exact expression for the evolution function Uz(pi,110) can be found in Ref. 7, and

(Pi 1

is given in terms of the cusp anomalous dimension.12 Relation ( 2 ) accomplishes the evolution of the shape function from the intermediate scale down to the low scale PO A. The remaining task is to expand the integral over the shape function in (1)in a series of forward B-meson matrix elements of local HQET operators of increasing dimension, multiplied by perturbative coefficient functions. This can be done whenever A = AE - A = mb - 2Eo is large compared with A Q C D .The ~ result is

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leading

rB+X,y(EO)=

G%Q

3Iv,bv,*,12

mi E 3 P h ) IH.y(Ph)12U l ( P h ,P i ) U2(Pi,Po)

where

A 5T2 + 4 [2h(q)- 11In - 4h2(q) + 4h(q) + 4h’(q) - , (5) PO 6 and h ( ~=) $(l + 77) + Y E is the harmonic function generalized to non2A

S ( A ) = -41n PO

integer argument. Even though it is parametrically larger than ordinary power corrections of order (AQcD/mb)2,the “enhanced” A1 /A2 correction in (4) remains small in the region of “perturbative” values of A, where the MSOPE can be trusted. The net effect amounts to a reduction of the decay rate by less than 5%.

a3

The rate in (4) is formally independent of the three matching scales, at which we switch from QCD to SCET ( p h ) , from SCET to HQET ( p i ) , and finally at which the shape-function integral is expanded in a series of local operators ( P O ) . In practice, a residual scale dependence arises because we have truncated the perturbative expansion. Varying the three matching scales about their default values provides some information about unknown higher-order terms. In the limit where the intermediate and low matching scales pi and po are set equal to the hard matching scale p h , our result reduces to the conventional formula used in previous analyses of the B + X,y decay rate. However, this choice cannot be justified on physical grounds. In (4) we have accomplished a complete resummation of (parametrically) large logarithms at next-to-next-to-leading logarithmic order in RGimproved perturbation theory, which is necessary in order to calculate the decay rate with O(a,)accuracy. Specifically, it means that terms of the form ayLk with k = ( n - I), . . . ,2n and L = ln(mb/A) are correctly resummed to all orders in perturbation theory. To the best of our knowledge, a complete resummation a t next-to-next-to-leading order has never been achicvcd bcforc. Finally, we stress that the various next-to-leading order terms in the expression for the decay rate should be consistently expanded to order a, before applying our results to phenomenology. Up to this point, the b-quark mass mb entering the formula (4) for the decay rate is defined in the on-shell scheme. While this is most convenient for performing calculations using heavy-quark expansions, it is well known that HQET parameters defined in the pole scheme suffer from infra-red renormalon ambiguities. It is necessary to replace them in favor of some physical, short-distance parameters. For our purposes, the “shape-function scheme” provides for a particularly suitable definition of the heavy-quark mass.5 The idea is that a good estimate of a shape-function integral can be obtained using the mean-value theorem, replacing L2 with

Here mb(A,po) is the running shape-function mass, which depends on a hard cutoff A in addition to the renormalization scale po. The quantity A in the shape-function scheme is defined by the implicit equation A = AE - A(A, po) = mb(A, po) - 2Eo. The shape-function scheme provides a physical definition of mb, which can be related to any other shortdistance definition using perturbation theory. Based on various sources

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of phenomenological information including T spectroscopy and moments of inclusive B-meson decay spectra, the value of the shape-function mass at a reference scale p* = 1.5GeV has been determined as mb(p*,p*)= (4.65 f 0.07) GeV.’ The results discussed so far provide a complete description of the B + X,y decay rate at leading order in the l/mb expansion, where the two-step matching QCD + SCET + HQET is well understood. For practical applications, however, it is necessary to include corrections arising a t higher orders in the heavy-quark expansion. Most important are “kinematic” power corrections of order (A/mb)n,which are not associated with new hadronic parameters. Unlike the non-perturbative corrections, these effects appear already at first order in A/mb, and they are numerically dominant in the region where A >> AQCD. Technically, the kinematic power corrections correspond to subleading jet functions arising in the matching of QCD onto higher-dimensional SCET operators, as well as subleading shape functions arising in the matching of SCET onto HQET operators. The corresponding terms are known in fixed-order perturbation theory, without scale separation and RG r e ~ u r n m a t i o n . ’ ~To > ~perform ~ a complete RG analysis of even the first-order terms in A/mb is beyond the scope of our discussion. Since for typical values of Eo the power corrections only account for about 15% of the B + X,y decay rate, an approximate treatment suffices at the present level of precision. Details of how these corrections are implemented can be found in Ref. 7. 4. Ratios of decay rates

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The contributions from the three different short-distance scales entering the central result (4) and the associated theoretical uncertainties can be disentangled by taking ratios of decay rates. Some ratios probe truly shortdistance physics (i.e., physics above the scale p h mb) and so remain unaffected by the new theoretical results presented above. For some other ratios, the short-distance physics associated with the hard scale cancels to a large extent, so that one probes physics a t the intermediate and low scales, irrespective of the short-distance structure of the theory.

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Ratios insensitive to low-scale physics: Physics beyond the Standard Model may affect the theoretical results for the B -+ X,y branching ratio and CP asymmetry only via the Wilson coefficients of the various operators in the effective weak Hamiltonian. As a result, the ratio of the B + X,y decay rate in a New-Physics model relative to that in the Standard Model

85 remains largely unaffected by the resummation effects studied in the present work. F'rom (4),we obtain rB--tX.TlNP rB--tX,-ylSM

- IWPh)I2,P IH'i'(Ph)l$M

+

power corrections.

(7)

The power corrections would introduce some mild dependence on the intermediate and low scales pi and PO, as well as on the cutoff Eo. Another important example is the direct CP asymmetry in B 3 X,y decays, for which we obtain

. .

where p7 is obtained from H , by CP conjugation, which in the Standard Model amounts to replacing the CKM matrix elements by their complex conjugates. It follows that predictions for the CP asymmetry in the Standard Model and various New Physics scenarios15 remain largely unaffected by our considerations.

Ratios sensitive to low-scale physics: The multi-scale effects studied in this work result from the fact that in practice the B + X,y decay rate is measured with a restrictive cut on the photon energy. These complications would be absent if it were possible to measure the fully inclusive rate. It is convenient to define a function F(E0) as the ratio of the B + X,y decay rate with a cut EO divided by the total rate,

Because of a logarithmic soft-photon divergence for very low energy, it is con~entional'~ to define the "total" inclusive rate as the rate with a very low cutoff E, = rnb/20. The denominator in the expression for F(E0) can be evaluated using the standard OPE, which corresponds to setting all three matching scales equal to ph. The numerator is given by our expression in (4),supplemented by power corrections. Another important example of a ratio that is largely insensitive to the hard matching contributions is the average photon energy (E,), which has been proposed as a good way to measure the b-quark mass.16 The impact of shape-function effects on the theoretical prediction for this quantity has been investigated and was found to be significant.14J7 Here we study the average photon energy in the MSOPE region, where a modelindependent prediction can be obtained. It is structurally different from

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the one found using the conventional OPE in the sense that contributions associated with different scales are disentangled from each other. We stress that the hard scale ph mb affects the average photon energy only via second-order power corrections. This shows that it is not appropriate to compute the quantity ( E y )using a simple heavy-quark expansion a t the scale mb, which is however done in the conventional approach.16 This observation is important, because information about moments of the B + X,y photon spectrum is sometimes used in global fits to determine the CKM matrix element IVcbl. Keeping only the leading power corrections, which is a very good approximation, we find that ( E 7 ) only depends on physics a t the intermediate and low scales pi and po. For EO = 1.8GeV, we obtain ( E 7 ) NN [2.222 0.2540,(=) 0.009as(A)] GeV M 2.30 GeV.

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5 . Numerical results

We are now ready to present the phenomenological implications of our findings. A complete list of the relevant input parameters and their uncertainties is given in Ref. 7, where we also explain our strategy for estimating the perturbative uncertainty as well as the uncertainty due to parameter variations. We begin by presenting predictions for the CP-averaged B -+ X,y branching fraction with a cutoff Ey 2 EO applied on the photon energy measured in the B-meson rest frame. Lowering Eo below 2GeV is challenging experimentally. The first measurement with EO = 1.8GeV has recently been reported by the Belle Collaboration.' It yieldsa E,>1.8 GeV

Eo=1.8 GeV

= (3.38 f 0.30 f 0.29) . l o v 4 , = (2.292 f 0.026 f 0.034) GeV

For Eo = 1.8GeV we have A M 1.1GeV, which is sufficiently large to apply the formalism developed in the present work. (For comparison, the value EO= 2.0 GeV adopted in the CLEO analysis' implies A x 0.7 GeV, which we believe is too low for a short-distance treatment.) We find E0=1.8GeV

= (3.44 f0.53 [pert.] f0.35 [pars.]) x

, (11)

where the first error refers to the perturbative uncertainty and the second one to parameter variations. The largest parameter uncertainties are due &To obtain the first result we had to undo a theoretical correction accounting for the effects of the cut E, > 1.8GeV, which had been applied to the experimental data.

87 to the b- and c-quark masses. Our result is in excellent agreement with the experimental value shown in (10). Comparing the two results, and naively assuming Gaussian errors, we conclude that Br(B

+ X s ~ ) e x-pBr(B + X,Y)SM< 1.4. lop4

(95% CL) .

(12)

Mainly as a result of the enlarged theoretical uncertainty, this bound is much weaker than the one derived in Ref. 18, where this difference was found to be less than 0.5. Hence, we obtain a much weaker constraint on New Physics parameters. For instance, for the case of the type-I1 twoHiggs-doublet model, we may use the analysis of Ref. 19 to deduce mH+

> (slightly below) 200GeV

(95% CL) ,

(13)

which is significantly weaker than the constraints mH+ > 500 GeV (at 95% CL) and mH+ > 350GeV (at 99% CL) found in Ref. 18. The function F(E0) provides us with an alternative way to discuss the effects of imposing the cutoff on the photon energy. In contrast to the branching ratio, it is independent of several input parameters (e.g., mb(%b), Iv,',&bl,T B , XI,^), and it shows a very weak sensitivity to variations of the remaining parameters. We obtain F(1.8GeV) = (92f,; [pert.] f 1[pars.])%.

(14)

This is the first time that this fraction has been computed in a model independent way. The result may be compared with the values (95.8?:':)% and (95 f 1)%obtained from two studies of shape-function model^,^^^^^ in which perturbative uncertainties have been ignored. We obtain a significantly smaller central value with a much larger uncertainty. The last quantity we wish to explore is the average photon energy. As mentioned above, this observable is very sensitive to the interplay of physics at the intermediate and low scales. The study of uncertainties due to parameter variations exhibits that the prime sensitivity is to the b-quark mass, which is expected, since (E-,) = mb/2 . . . to leading order. The nextimportant contribution to the error comes from the HQET parameter X1. To a very good approximation, we have

+

where the error accounts for the perturbative uncertainty. The quantities 6mb and 6x1 parameterize possible deviations of the relevant input parameters from their central values mb = 4.65 GeV and A 1 = -0.25 GeV2. Our

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prediction is in excellent agreement with the Belle result in (10). This finding provides support to the value of the b-quark mass in the shape-function scheme extracted in Ref. 5. We stress, however, that the large perturbative uncertainties in the formula for (E T ) impose significant limitations on the precision with which mb can be extracted from a measurement of the average photon energy. Our estimate above implies a perturbative uncertainty of bmb[pert.] = ':::MeV. This is in addition to twice the experimental error in the measurement of ( E y ) ,which at present yields bmb[exp.] = 86 MeV.

6. Conclusions and outlook

We have performed the first systematic analysis of the inclusive decay B -+ X,y in the presence of a photon-energy cut ET 2 Eo, where EOis such that A = mb - 2Eo can be considered large compared to AQCD, while still A <( mb. This is the region of interest to experiments at the B factories. The first ensures that a theoretical treatment without shape condition (A >> A,,,) functions can be applied. However, the second condition (A << mb) means that this treatment is not a conventional heavy-quark expansion in powers of ffs(mb) and hQcD/mb. Instead, we have shown that three distinct shortdistance scales are relevant in this region. They are the hard scale mb, the hard-collinear (or jet) scale and the low scale A. To separate the contributions associated with these scales requires a multi-scale operator product expansion (MSOPE). Our approach allows us to study analytically the transition from the AQCD, into the MSOPE region, where shape-function region, where A AQCD <( A <( mb, into the region A = U(mb), where a conventional heavyquark expansion applies. This is a significant improvement over previous work. For instance, it has sometimes been argued that exactly where the transition to a conventional heavy-quark expansion occurs is an empirical question, which cannot be answered theoretically. Our formalism provides a precise, quantitative answer to this question. In particular, for B + X,y with a realistic cut on the photon energy, one is not in a situation where a short-distance expansion at the scale mb can be justified. The analysis makes it evident that the precision that can be achieved in the prediction of the B + X , y branching ratio is, ultimately, determined by how well perturbative and non-perturbative corrections can be controlled at the lowest relevant scale A, which in practice is of order 1GeV. Consequently, we estimate much larger theoretical uncertainties than previous authors. These

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uncertainties are dominated by yet unknown higher-order perturbative effects. Non-perturbative, hadronic effects at the scale A appear to be small and under control. This is not the first time in the history of B + X,y calculations that issues of scale setting have changed the prediction and error estimate for the branching ratio (see, e.g., the discussion in Ref. 18). In our case, however, the change in perspective about the theory of B + X,y decay is more profound, as it imposes limitations on the very validity of a short-distance treatment. If the short-distance expansion at the scale A fails, then the rate cannot be calculated without resource to non-perturbative shape functions, which introduces an irreducible amount of model dependence. In practice, while A x 1.1GeV (for Eo M 1.8GeV) is probably sufficiently large to trust a short-distance analysis, it would be unreasonable to expect that yet unknown higher-order effects should be less important than in the case of other low-scale applications of QCD. Obtaining a precise prediction for the B + X,y decay rate in the Standard Model is an important target of heavy-flavor theory. The present work shows that the ongoing effort to calculate the dominant parts of the next-to-next-to-leading corrections in the conventional heavy-quark expansion is only part of what is needed to achieve this goal. Equally important will be to compute the dominant higher-order corrections proportional to as (A) and as and to perform a renormalization-group analysis of the leading kinematic power corrections of order A/m,. In fact, our error analysis suggests that these effects are potentially more important that the hard matching corrections at the scale mb.

(m),

Acknowledgments: I am grateful to the organizers for the invitation to deliver this talk. It is a pleasure to thank Alex Kagan and Bjorn Lange for useful discussions.

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References 1. S. Chen et al. [CLEO Collaboration], Phys. Rev. Lett. 87, 251807 (2001) [hep-ex/0108032]. 2. P. Koppenburg et al. [Belle Collaboration], Phys. Rev. Lett. 93, 061803 (2004) [hep-ex/0403004]. 3. M. Neubert, Phys. Rev. D 49, 3392 (1994) [hep-ph/9311325]; Phys. Rev. D 49, 4623 (1994) [hep-ph/9312311]. 4. I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein, Int. J. Mod. Phys. A 9, 2467 (1994) [hep-ph/9312359]. 5. S. W. Bosch, B. 0. Lange, M. Neubert and G. Paz, hep-ph/0402094, to appear in Nuclear Physics B. 6. C. W. Bauer and A. V. Manohar, hep-ph/0312109. 7. M. Neubert, hep-ph/0408179. 8. S. W. Bosch, R. J. Hill, B. 0. Lange and M. Neubert, Phys. Rev. D 67, 094014 (2003) [hep-ph/0301123]. 9. C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001) [hep-ph/0011336]; C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 65, 054022 (2002) [hep-ph/0109045]. 10. For a review, see: M. Neubert, Phys. Rept. 245, 259 (1994) [hepph/9306320]. 11. B. 0. Lange and M. Neubert, Phys. Rev. Lett. 91, 102001 (2003) [hepph/0303082]. 12. G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys. B 283, 342 (1987); I. A. Korchemskaya and G. P. Korchemsky, Phys. Lett. B 287, 169 (1992). 13. C. Greub, T. Hurth and D. Wyler, Phys. Rev. D 54, 3350 (1996) [hepph/9603404]. 14. A. L. Kagan and M. Neubert, Eur. Phys. J. C 7, 5 (1999) [hep-ph/9805303]. 15. A. L. Kagan and M. Neubert, Phys. Rev. D 58, 094012 (1998) [hepph/9803368]. 16. A. Kapustin and Z. Ligeti, Phys. Lett. B 355, 318 (1995) [hep-ph/9506201]. 17. I. Bigi and N. Uraltsev, Int. J. Mod. Phys. A 17, 4709 (2002) [hepph/0202175]. 18. P. Gambino and M. Misiak, Nucl. Phys. B 611, 338 (2001) [hep-ph/0104034]. 19. F. M. Borzumati and C. Greub, Phys. Rev. D 58, 074004 (1998) [hepph/9802391].

STATUS OF PERTURBATIVE DESCRIPTION OF SEMILEPTONIC QUARK DECAYS

ANDRZEJ CZARNECKI* Department of Physics, University of Alberta Edmonton, AB, Canada T6G 2J1

Perturbative QCD corrections to semileptonic decays of heavy quarks are reviewed. Beyond the next-to-leading order, exact analytical results are difficult to obtain and various approximations have been developed. Expansion methods have been proposed, starting primarily in various zero-recoil configurations and, more recently, also away from the zero-recoil. First three-loop QCD corrections have also recently been found in a simple kinematics.

1. Introduction

Radiation of photons and gluons, and exchange of virtual quanta, modify decay rates and distributions. We have t o understand such dynamics before we can accurately determine parameters of weak interactions or search for new flavor changing forces. Studies of radiative corrections, especially in higher orders, are difficult for quark decays, because of the radiation from the decaying massive particle. However, over the last decade, powerful technical tools have been developed to facilitate such calculations: asymptotic expansions in various kinematical configurations, algebraic relations for reducing multiloop Feynman diagrams t o small basic sets, and methods for computing those basic “master integrals.” In this talk, some of the resulting progress in the description of semileptonic decays is described. The focus of this talk will be on effects beyond the next-to-leading order (NLO): there is now a variety of next-to-next-to-leading order (NNLO) results and even one NNNLO calculation. In the NLO, complete description of all decay distributions is possible, including energy, angular, and polarization information of various final particles. Technical methods for such studies have been known since rnid-fiftie~,’>~?~ when they were developed in *Work partially supported by Science and Engineering Research Canada.

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92

the context of QED corrections t o the muon decay. They were successfully applied t o semileptonic quark decays (see for example for results and further references). 4159637

2. Expansions around the zero recoil Among various kinematic configurations which can occur in semileptonic decays, the easiest one from the point of view of radiative corrections is the zero recoil limit. In the rest frame of the decaying quark, the final state quark also remains at rest and the neutrino and the charged lepton are emitted in opposite directions, carrying away all released energy. There is no phase space available for radiation of real gluons and only virtual corrections have t o be calculated, see Fig. 1. In the NNLO, the latter were first found using an expansion around the case of equal masses of the initial and final state quarks and subsequently analytically for any mass of the final state q ~ a r k . ~ ~ ' ~ The knowledge of corrections in the zero recoil limit was a starting point of various expansions beyond that kinematics, which will be briefly reviewed in the next section. Similar program can, if need arises, be carried out in the NNNLO. The first calculation in the limit of equal masses of the decaying and the produced quarks is summarized in the subsequent section.

b

b-k

- k - k

c

Figure 1. Th e only NLO radiative correction in case of zero recoil. In this kinematics, b and c quarks remain at rest and t h e virtual W carries away energy equal t o their mass difference, and no momentum. No real radiation of gluons is possible.

2.1. Next-to-next-to-leadingorder: a: Fig. 2 shows the possible kinematics of semileptonic decays. Masses of the final-state quark and of the lepton pair, shown respectively on the horizontal and the vertical axes, cannot exceed the mass of the decaying quark. If they add up to it, we deal with the diagonal boundary line indicating the zero

93

recoil limit. Along that line, the NNLO corrections are known analytically and it is possible t o obtain corrections in the vicinity of zero recoil by expanding in some small parameter, describing the deviation of the recoil from zero. Two such expansions were motivated by a study of the total b quark lifetime: maximum and intermediate l2 recoil, shown in Fig. 2 as dotted arrows starting, respectively, in the lower right vertex and in the middle of the zero recoil line. In that figure we also see a vertical dashed line, corresponding t o the physical mass of the c quark and the possible range of the lepton invariant mass in the decay b -+ clu. The expansions, together with the exact correction in the upper (zero recoil) end of the dashed line, give three points, sufficient to interpolate and estimate corrections to the total rate of that decay.13

q-quark mass Figure 2. Kinematic bounds of semileptonic quark decays Q + q+leptons. On the horizontal axis: mass of the final state quark; on the vertical: invariant mass of the lepton pair. Also shown are locations of known expansions of NNLO QCD corrections, discussed further in the text. The vertical dashed line corresponds to the decay of a b quark into a c.

More recently,14 a third expansion starting from zero recoil was constructed, for decays into a massless quark, shown with the vertical dotted

94

arrow in Fig. 2. That study gave the differential rate of the decay b -+ ulv describing the distribution of the lepton invariant mass, with an 0 (a:) accuracy. This was the first NNLO result for a differential decay distribution and has recently been complemented with an expansion along the same vertical line, but starting from its bottom end, to be discussed in section 3.

2.2. Next-to-next-to-next-to-leading order: a: Very recently, the first three-loop (NNNLO) correction to semileptonic decays has been computed.15 In this order, no corrections had previously been studied to any charged particle decay, partially because they are not expected to be large, and also because of lacking technical tools for solving multi-loop massive Feynman diagrams. It is however possible to compute such corrections in the limit of equal masses of the decaying and the produced quarks. In this “extreme zero recoil” limit, there is no momentum transfer to the leptons, and no change in the quark four momentum. Thus, the amplitude of this process can be calculated using only propagator-like diagrams, which are known a t this order.16>17i1s An example of a contributing diagram is shown in Fig. 3.

Figure 3. Example of three-loop corrections to the semileptonic b -+ c decay. Other possible points of the W emission are denoted by crossed circles. In addition to such QED-like diagrams, there are also diagrams with three and four gluon vertices.

QCD corrections to the heavy quark transitions mediated by W boson can be expressed, at zero recoil, using two functions, ’Yp(1 - 75)

+

p ‘ ?

[ T V ( q 2 )- T A ( q 2 ) y 5 ].

(1)

In the extreme zero recoil limit we are considering, the momentum transfer q vanishes. In this case, the vector part of the interaction does not obtain any QCD corrections, qv(q2 = 0) = 1. The axial part of the interaction does receive finite corrections even in the q2 = 0 limit. They are expressed

95

as a power series in as,

The first two nontrivial terms of this expansion have been known for a long time,19)20,8

+TRNH

($

1

- ?n2)

7 + -TRNL. 36

(3)

In SU(3), we have CA= 3, CF = 4/3, TR = 1/2. NH,Ldenote the number of heavy and light fermions in closed loops. The result in the following order consists of ten gauge-invariant parts, depending in various ways on the SU(3) color factors and the numbers of light and heavy fermions. One of those parts, corresponding to abelian diagrams with two light fermion loops, can be obtained using a technique of a paper written a decade ago in Minnesota.21 In fact, knowing the coefficient of the square of the number of light fermions, N i , is enough t o estimate the leading part of the NNNLO correction, enhanced by p,", where PO= 11- $ N L is the leading coefficient of the QCD beta f ~ n ~ t i o n . ~This ~ t ~ alternative derivation of the leading correction is outlined below. Following 21 we write the renormalized gluon propagator, including only effects of a chain of fermion loops, as a subtracted dispersion integral. Evaluation of fermion loop corrections to any process involving a gluon exchange requires two steps, (1) computation of the gluon exchange diagrams with a mass A assigned to the gluon; (2) and an integration of the dispersion relation over A.

A similar procedure has been carried out for numerous observables, including for example semileptonic decay rate,26i27,28hadronic 7 decays,29 or threshold quark pair p r o d u ~ t i o n . ~ ~ We use the MS scheme to renormalize the strong coupling constant and take the momentum transfer equal to minus the heavy quark mass -m2 as the normalization point. The effects of any number of fermion loop

96

insertions into a gluon propagator can be summed up as a geometric series. The vacuum polarization becomes

and satisfies the following dispersion relation,

I t is this form that allows us t o evaluate the effect of fermion bubbles on an observable, using a formal analogy with calculations with a massive gluon. As an example, consider the fermion loop effects on the correction to the axial current. Here we are interested in ~ A ( N Lresulting ), from inserting in the gluon propagator one or more fermion bubbles,

In this equation w(k) is a function determined by the structure of the diagram apart from the gluon exchange, described by the last factor. Substituting in this equation the representation of eq. (5) we find

Here q i ) ( X ) is the one loop correction calculated with a massive gluon, which we will study below, and qA (1)(0) is its standard value at A = 0, q i ’ ( 0 ) = -1/2.

To compute the one loop correction with a gluon of mass A, we use (1) dimensional regularization, with D = 4 - 26. The correction, qA (A) = V(X) + Z,(X), is a sum of the vertex correction and the wave function

97 renormalization constant, for which we find

1 V(X)= 4E

-

1 + -12 - -A2 + 41 4

-(A2

X (A4 - 6 X 2

- 2)

2

log(X)

+ 12) arctan (9) 4dc-P

1

The complete correction t o the axial current is the sum of these two, which we write in a form suitable for subsequent integration,

&l)(X)

1 2

= -- - X2 - A 2 (1 - X2) log(X)

( q) m arctan ( q)

arctanh

+A3 (3 - X2)

< 2)

4EP

We now have all ingredients necessary for finding any correction of the form aYNFn-’. Here, we will be interested in the first two terms, resulting from an expansion of eq. (4) in a,,

IrnP,(X) = -

a ~ 3

2 a2 N -~- LT( N ~L T R ) ~ ~ ~ ( x ~ ~ .-.~. / ~ ) (10) 97T

+

Using in eq. (7) this expansion and the “gluon-mass dependent” correction found in eq. (9), we find

in agreement with the explicit three-loop evaluation. l5 This fermionnumber dependent correction is important because it determines the coefficient in front of a given power of Po = 11 - ~ N Lwhich , is a large number. In this way, the large corrections can be absorbed in the running of the coupling constant. The complete study l5 found that the remaining three-loop corrections enhance V A by only about 0.2%.

98 3. Beyond the zero recoil

It should be mentioned that in addition t o the NNLO results obtained using expansions starting from the zero recoil, also the total decay rates for the muon decay and for the decay b -+ulu were obtained a n a l y t i ~ a l l y . These were ground-breaking results derived by introducing a fictitious large mass for the decaying fermion in its virtual lines, expanding the diagrams in the ratio of external to internal fermion mass, and (the most challenging point) determining the form of a general term of such expansion. In the resulting exact formula one can set the expansion parameter equal t o its physical value of unity. This approach seems t o be very labor intensive and finding alternatives is desirable. Recently 33 NNLO results were obtained for the first time by constructing an expansion away from the zero recoil limit. In Fig. 2 it is indicated by a solid arrow starting at the lower left vertex of the triangle. That vertex itself corresponds t o a decay of a heavy quark into a massless quark and a massless pair of leptons (or a massless boson), as is nearly the case for the decay of a top quark. Sufficiently many terms were found to match smoothly with the expansion for b u decays around its zero recoil limit, described in section 2.1. --f

4. Summary

It is fair t o say that the NNLO description of semileptonic decays is mature in the sense that nowhere are the perturbative NNLO effects limiting accuracy of determination of any physical quantity of interest. This is a satisfactory progress compared with a few years ago, when the unknown 0 (a:) corrections were among the dominant errors in IV&l. The methods developed in this process can now be applied to studies of radiative decays, heavy neutral meson mixing, and to determinations of Wilson coefficients of higher dimensional operators in effective lagrangians. Such studies are most fruitful when motivated by experimental demands. The next qualitative step that should be made is an extension of the methods developed for semileptonic decays to hadronic decays. Those are complicated by the number of radiating particles in the tree level and the resulting complexity of diagram topologies in the NNLO. For such studies, it is particularly important to develop novel approaches to the evaluation of master integrals.

99 Acknowledgements

I t h a n k J o h n P a u l Archambault, Ian Blokland, Kirill Melnikov, Maciej Slusarczyk, and Fyodor Tkachov for collaboration on various aspects of research reported here.

References 1. R. E. Behrends, R. J. Finkelstein, and A. Sirlin, Phys. Rev. 101,866 (1956). 2. S. M. Berman, Phys. Rev. 112,267 (1958). 3. T. Kinoshita and A. Sirlin, Phys. Rev. 113,1652 (1959). 4. G. Altarelli et al., Nucl. Phys. B208,365 (1982). 5. M. Jezabek and J. H. Kuhn, Nucl. Phys. B314,1 (1989). 6. A. Czarnecki and M. Jezabek, Nucl. Phys. B427,3 (1994). 7. J. G. Korner and M. C. Mauser, (2003), hep-ph/0306082. 8. A. Czarnecki, Phys. Rev. Lett. 76,4124 (1996). 9. A. Czarnecki and K. Melnikov, Nucl. Phys. B505,65 (1997). 10. J . Franzkowski and J. B. Tausk, Eur. Phys. J. C5,517 (1998). 11. A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 78,3630 (1997). 12. A. Czarnecki and K. Melnikov, Phys. Rev. D59,014036 (1999). 13. A. Czarnecki and K. Melnikov, in Proceedings of the 13th Lake Louise Winter Institute, edited by A. Astbury et al. (World Scientific, Singapore, 1998), pp. 84-91, hepph/9806258. 14. A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 88,131801 (2002). 15. J. P. Archambault and A. Czarnecki, 2004, preprint Alberta Thy 17-04. 16. S. Laporta and E. Remiddi, Phys. Lett. B379,283 (1996). 17. K. Melnikov and T. van Ritbergen, Phys. Rev. Lett. 84,1673 (2000). 18. K. Melnikov and T. van Ritbergen, Nucl. Phys. B591,515 (2000). 19. M. Voloshin and M. Shifman, Sov. J. Nucl. Phys. 47,511 (1988). 20. J . E. Paschalis and G. J. Gounaris, Nucl. Phys. B222,473 (1983). 21. B. H. Smith and M. B. Voloshin, Phys. Lett. B340,176 (1994). 22. S. J. Brodsky, G. P. Lepage, and P. B. Mackenzie, Phys. Rev. D28, 228 (1983). 23. M. Beneke and V. M. Braun, Phys. Lett. B348,513 (1995). 24. M. Neubert, Phys. Rev. D51,5924 (1995). 25. N. Uraltsev, Nucl. Phys. B491,303 (1997). 26. M. E. Luke, M. J. Savage, and M. B. Wise, Phys. Lett. B343,329 (1995). 27. E. Bagan, P. Ball, V. M. Braun, and P. Gosdzinsky, Phys. Lett. B342,362 (1995). 28. N. G. Uraltsev, Int. J. Mod. Phys. All,515 (1996). 29. P. Ball, M. Beneke, and V. M. Braun, Nucl. Phys. B452,563 (1995). 30. 0. I. Yakovlev and A. Yelkhovsky, Phys. Lett. B438,341 (1998). 31. T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82,488 (1999). 32. T. van Ritbergen, Phys. Lett. B454,353 (1999). 33. I. Blokland, A. Czarnecki, M. Slusarczyk, and F. Tkachov, (2004), hepph/0403221, in press in Phys. Rev. Lett.

HEAVY QUARK EXPANSION IN BEAUTY: RECENT SUCCESSES AND PROBLEMS

N. URALTSEV INFN, Sezione d i Milano, Milano, Italy Department of Physics, University of Notre Dame, Notre Dame, I N 46556, USA Petersburg Nuclear Physics Institute, Gatchina, 188300 Russia The status of the QCD-based heavy quark expansion is briefly reviewed. A good agreement between properly applied theory and new precision data is observed. Critical remarks on certain recent claims from HQET are presented. Recent applications to the exclusive heavy flavor transitions are addressed. The '$ > $' problem for the transitions into the charm P-wave states is discussed.

Heavy quark physics, in particular electroweak decays of beauty particles, is now a well developed field of QCD. The most nontrivial dynamic predictions are made for sufficiently inclusive heavy flavor decays admitting the local operator product expansion (OPE). These predictions are phenomenologically important - they allow to reliably extract the underlying CKM mixing angles lV&,l and lvubl with record accuracy from the data, or the fundamental parameters like mb and m,. At the same time heavy quark theory yields informative dynamic results for a number of exclusive transitions as well. Recent years have finally witnessed a more united approach to inclusive and exclusive decays which previously have been largely isolated. In this talk I closely follow the nomenclature of the review' where the principal elements of the heavy quark theory can be found. For a number of years there has been a wide spread opinion that the predictions of the dynamic QCD-based theory were not in agreement with the data, a sentiment probably still felt today by many. The situation, in fact, has changed over the past few years. A better, more robust approach to the analysis has been put forward', made more systematic3 and applied in p r a ~ t i c e ~ !The ~ . perturbative corrections for all inclusive semileptonic characteristics have finally been ~ a l c u l a t e d Experiments ~~~. have accumulated data sets of qualitatively better statistics and precision. Critically reviewing the status of the theory when confronted with the

100

101

data, we find that the formerly alleged problems are replaced by impressive agreement. Theory often seems to work even better than can realistically be expected, when pushed to the hard extremes. Old problems are left in the past. 1. Inclusive semileptonic decays: theory vs. data

The central theoretical result’ for the inclusive decay rates of heavy quarks is that they are not affected by nonperturbative physics at the level of AQcD/rnQ (even though hadron masses, and, hence the phase space itself, are), and the corrections are given by the local heavy quark expectation values - p: and p& to order l/m$, etc. Today’s theory has advanced far beyond that and allows, for instance, to aim at an 1% accuracy in lvcbl extracted from &@). A similar approach to Ivubl is more involved since theory has to conform with the necessity for experiment to implement significant cuts which discriminate against the b + cCu decays. Yet the corresponding studies are underway and a 5% accuracy seems realistic. There are many aspects theory must address to target this level of precision. One facet is perturbative corrections, a subject of controversial statements for many years. The reason goes back to rather subtle aspects of the OPE. It may be partially elucidated by Figs. 1 which shows the relative weight of gluons with different momenta Q affecting the total decay rate and the average hadronic recoil mass squared ( M i ) ,respectively. The contributions in the conventional ‘pole’-type perturbative approach have long tails extending to very small gluon momenta below 500MeV, especially for ( M i ) ; the QCD coupling a,(&) grows uncontrollably there. These tails would be disastrous for precision calculations manifest, for instance, through a numerical havoc once higher-order corrections are incorporated. Yet applying literally the Wilsonian prescription for the OPE with an explicit separation of scales in all strong interaction effects, including the perturbative contributions, effectively cuts out the infrared pieces! Not only do the higher-order terms emerge suppressed, even the leading-order corrections become small and stable. This approach, applied to heavy quarks long agog implies that the precisely defined running heavy quark parameters mb(p),K ( p ) , p ; ( p ) , ... appear in the expansion, rather than ill-defined parameters like pole masses, K, --A1 employed by HQET. Then it makes full sense to extract these genuine QCD objects with high precision.

102

Figure 1. The role of the gluons with different momenta in rSland in

(M:),for b - t c t v .

The most notable of all the alleged problems for the OPE in the semileptonic decays was, apparently, the dependence of the fi0.4 m nal state invariant hadron mass on 0.2 the lower cut EfUtin the lepton Lepton Energy Cut (GeV) 0 energy: theory seemed to fall far 0.8 1 1.2 1.4 1.6 off lo of the experimental data, see Figure 2. Ref.lo predictions for (M;) Fig. 2. The robust approach, On (red triangles), with the authors' theory the contrary appears to describe error bars. Black squares are preliminary it well1', as illustrated by Figs. 3. (2002) BaBar data points. The second moment of the same distribution also seems to perfectly fit theoretical expectation^^.^ obtained using the heavy quark parameters extracted by BaBar from their data5. The second moment in the 'inapt' calculations by Bauer et aL1', on the contrary showed unphysical growth with the increase of E&, in clear contradiction with expectations and data.

,

I

Figure 3.

Hadron mass moments dependence on the lepton energy cut.

The comprehensive data analysis is now in the hands of professionals (experimentalists) armed with the whole set of the elaborated theoretical expressions. They are able to perform extensive fits of all the available data from different experiments, and arrive at rather accurate values of the heavy quark parameters, still observing a good consistency of data with

103 theory. A number of such analyses are underway12. Another possible discrepancy between data and theory used to be an inconsistency between the values of the heavy quark parameters extracted from the semileptonic decays and from the photon energy moments13 in B + X , fy. It has been pointed out, however,l' that with relatively high experimental cuts on E, the actual 'hardness' Q significantly degrades compared to mb, thus introducing the new energy scale with Q N 1.2 GeV at E&, = 2 GeV. Then the terms exponential in Q left out by the conventional OPE, while immaterial under normal circumstances, become too important. This is illustrated by Figs. 3 showing the related 'biases' in the extracted values of mb and p:. Accounting for these effects appeared to turn discrepancies into a surprisingly good agreement between all the measurements".

Figure 4. 'Exponential' in Q biases in mb and p : due to the lower cut on photon energy inB-tX,+y.

The problem of deteriorating hardness with high cuts and of the related exponential biases raised in'' was well taken by many experimental groups. BELLE have done a very good job14 in pushing the cut on E, down to 1.8 GeV, which softens the uncertainties in the biases:

(E,) = 2.292 f 0.026,tat f 0.034,,,

GeV

((E,-(E,))2) = 0.0305 5 0.0074,tat f 0.0063,,, GeV2.

(1)

The theoretical expectations based on the central BaBar values of the parameters with mb = 4.612GeV, p: = 0.40GeV2, for the moments with E&, = 1.8 GeV are

(E,) N 2.316GeV, ( ( E , - ( E , ) ) 2 )cz 0.0324GeV2 , (2) again in a good agreement. Although the heavy quark distribution functions governing the shape of the decay distribution in the b c and b 4 u or b + s transitions are different, the Wilsonian OPE ensures that the nonperturbative part of the moments in all these decays is given by the same heavy quark expectation values. This fact appears very important in practical studies aimed

104

at extracting IVubl from the inclusive B Xulv rates, since the accuracy in constraining the heavy quark parameters achieved in the b + c l u measurements is significantly higher than direct constraints from the radiative decays. According to experimental analyses, incorporating the former information brings the currently achievable accuracy for extracting 1VubI close to the 5% goal. As a brief summary, the data show good agreement with the properly applied heavy quark theory. In particular, it appears that 0 Many underlying heavy quark parameters have been accurately determined directly from experiment. 0 Extracting lVcbl from r,l(B) has high accuracy and rests on solid grounds. 0 We have precision checks of the OPE-based theory at the level where nonperturbative effects play the dominant role. In my opinion, the most nontrivial and critical test for theory is the consistency found between the hadronic mass and the lepton energy moments, in particular ( M i )vs. (El). This is a sensitive check of the nonperturbative sum rule for M ~ - m b , at the precision level of higher power corrections. It is interesting to note in this respect that a particular combination of the quark masses, mb-Q.74mc has been determined in the BaBar analysis with only a 17MeV error bar! This illustrates how lVCblcan be obtained with high precision: the semileptonic decay rate rs1(B)is driven by nearly the same combination15.

1.1. Comments on the literature a) Semileptonic decays The developed, thorough theoretical approach to the inclusive distributions has not escaped harsh criticism from Ligeti et al. which amounted to the strong recommendations not to use it for data analysis, with the only legitimate approach assumed to be that of Ref.l'. It was claimed, in particular that the observed correct E&,-dependence of ( M ; ) is lost once the complete cut-dependence of the perturbative corrections is included, being offset by the growth in the latter. We showed these claims were not true: the perturbative corrections remain small for the whole interval of Ecutup to 1.4GeV, and actually are practically flat, Fig. 1 of Ref.6. Moreover, the figure shows that these perturbative corrections with the full Ecutdependence in the traditional pole scheme decrease for larger Ecut,in agreement with intuition. The problems in the calculations of Ref.1° have not been traced in detail; its general approach has a number of vulnerable elements, and the

105

calculations themselves were not really presented. There are reasons to believe they actually contained plain algebraic mistakes. There is a deeper theoretical reason to doubt the validity of the approach adopted in Ref.” based on the so-called “1s”scheme which is pushed for the analysis of B decays somewhat beyond reasonable limits. In respect to the OPE implementation, it differs little, if any from the usual pole scheme. Only a t the final stage are the observables, like the total width which depends on the powers of mb, re-expressed in terms of the so-called ‘1,”’b mass. The latter is basically in perturbation theory. Since no ‘ T ( 1 S )c-quark mass’ exists, for b+c decays the scheme intrinsically relies on the pole mass relations, in particular to exclude m, from consideration. Use of the l / m , expansion certainly represents a weak point whenever precision predictions are required. In fact, there is a more serious concern about the legitimacy of the perturbative calculations in the ‘1s’scheme, whose working tool is the socalled ‘Upsilon expansion’16. Surprisingly, it is often not appreciated that this is not the conventional perturbative expansion based on the algebraic rules for the usual power series in an expansion parameter like as(mb). This framework rather involves more or less arbitrary manipulations with the conventional perturbative series, referred to as a ‘modified’ perturbative expansion. The rationale behind such manipulations is transparent: the Coulomb binding energy of two massive objects starts with a: terms, hence mis differs from the usual pole mass only to the second order in a , :

The last IR divergent terms 0; a: ln a, simply signify that the Coulomb bound state of a heavy quark Q has an intrinsically different, lower momentum scale a,mQ; in a sense, this scale is zero in the conventional perturbative expansion which assumes a series expansion around a, +0. Hence the relation is not infrared-finite, in the conventional terminology. On the other hand, since the leading, O(a:) term in Eq. (3) comes without PO,the ‘1s’ b quark mass in all available perturbative applications to B decays has to be equated with the pole mass m;’le, whether or not a few BLM corrections are included. To get around this obvious fact, the ‘Texpansion’ postulated considering a number of terms appearing to the lc-th order in perturbation theory, ckat(mb),to be actually of a lower order, c,ay(mb) with n < k . Since the power of the strong coupling is explicit, this is done by introducing an ad hoc factor 6 = 1 , making use of the property that unity remains unity raised

106

to arbitrary power. This ad hoc reshuffling constitutes the heart of the ‘T expansion’ and of using the ‘1s’b quark mass mis in B decays.a The scale a,mQ naturally appears in bound-state problems for heavy quarks since the perturbative expansion parameter for nonrelativistic particles is not necessarily a,, but rather runs in powers of a , / ~ where , v is their velocity. The analogue of the ‘bound-state’ mass then naturally appears there, since powers of velocity make up for the missing powers of a,. Yet nothing of this sort is present in B mesons or in their decays, the E parameters introduced by the ‘Yexpansion’ is unity and can be placed ad hoc at any arbitrary place. One clearly should not make up for the numerically larger than a,(mb) value of a,(&) at the smaller momentum scale Q = a,mb by equating at will terms of explicitly different orders in as in the usual perturbative expansion. The ‘Yexpansion’ would be meaningless already in the simplest toy analogue of the B decays, muon @-decay. It is then difficult to count on this approach to be sensible for more involved B decays where real OPE has to be used for high precision. More recently, when this contribution was in writing, the new paper by Bauer et al. appearedlg. Claiming now to describe the cut-dependence of (M:), this paper came up with new statements aimed at discrediting the Wilsonian approach and its implementation. The authors assert that the approach we follow suffers from a large ‘scale-dependence’ when varying the Wilsonian separation scale p. In addition, the authors state they cannot reproduce the hadronic moments calculated in ref^.^>^ used by experimental groups for the data analysis. Once again I have to refute the criticism the p-dependence turns out weak, actually far below the expected level, as illustrated, for instance by Figs. 5 for ( E l ) and ( M i ) . The change in the moments from varying p corresponds to the variation in, say mb of only 4 MeV and 1MeV, respectively! (Ref.3 allowed an uncertainty of 20 MeV due to uncalculated higher-order perturbative corrections.) It looks probable that the authors of Ref.lg simply were not able to perform correctly the calculations in the Wilsonian ‘kinetic’ scheme, at least for the hadronic mass moments. In fact, the suppressed dependence of the observables on the separation scale p is a routine check applied to the calculations. The two facts together are then rather suggestive. Varying p represents a useful - if limited - probe of the potential impact *The original paper16 presented some arguments calling upon the so-called “large n f expansion” supposed to justify reshuffling the orders. I believe that the reasoning was wrong ab initio missing the basics of the renormalon c a l c ~ l u s ’ ~ ~ ~ ~ .

107

-7Y 1.387

1.3861 1.386

\

1.384

0.6

0.8

1.2

1.4

Figure 5. Dependence of ( E l ) (left) and of ( M i ) (right) on the separation scale p . The green vertical bars show the change in the moments when mb is varied by f l MeV.

of the omitted higher-order corrections. Clearly, not varying p but fixing its value once and for all, one does not see any scale-dependence (the pole scheme simply amounts to setting p=O). In this respect hints a t an absent p-dependence in the pole-type schemes like ‘IS’smell suspiciously. And, certainly, the absence of an explicit separation scale is not an advantage. The analogous sensitivity to the actually used scale is of course present in the approach of ref^.^^,^^, and the related uncertainties can be easily revealed. The ‘1s’scheme ad hoc postulates using mtS, a half of the “(1s)mass. However, on the same grounds m;b, half of the mass of the ground-state bottomonium, qb(lS) can be used. Even accepting the arbitrary counting rules of the ‘T-expansion’ , all the theoretical expressions used in the analyses, are identical for mp and mis - the masses differ only to order a: (without DO). At the same time, the two b quark masses do differ numerically by at least 20 to 30MeV! This is significantly larger than the criticized p-dependence of the ‘kinetic’ Wilsonian scheme, and it, in any case, should be included as the minimal theory uncertainty of every calculation based on the ‘T’-mass of the b quark (it has not been, of course). The theory error estimates of ref^.^^!^^, upon inspection, look unrealistic, significantly underestimating many potential corrections. The numerical outcome of the fit for lV&l looks close to the value obtained by experimental groups in our approach, within the error estimates we believe are right. This impression would be superficial - the two calculations share many common starting assumptions; therefore, they must yield - if performed correctly - much closer results. One can state they do differ a t a level which is significant theoretically. In this respect I would urge experiments to refrain from averaging the results obtained in the two approaches. It is never a good idea to combine correct results with those based on a potentially flawed calculations. In my opinion, those relying on the ‘Upsilon expansion’ can be considered as such. For instance, the authors of Ref.”, according to their Eq. (26) and

108

Table I increase the value of lVcal due to electroweak corrections (the same appears t o apply t o the recent Ref.lg). The fact is the electroweak factor qaEDincreases the width and, therefore suppresses the extracted value of lVcbl by an estimated 0.7%. It is curious to note that, assuming this is just a mistake rather than yet another ad hoc postulate of the ‘Upsilon expansion’, correcting for it would make the lVcbl value of Ref.lg nearly identical to the result obtained by BaBar. Whether such a correspondence is inevitable, or is a matter of coincidence, is not obvious at the moment. The existing PDG reviews on the subject have been so far based exclusively on the questionable papers ignoring more thorough existed analyses, and may therefore represent a not too trustworthy source of information.

b) b + s+-y and b + u+Lv There is a subtlety in accounting for the perturbative effects in the heavy-to-light decays which we do not see in b c &. The radiated gluons can be emitted with sufficiently large energy yet at a very small angle, so that their transverse momentum is only of order Phadr or even lower. This is a nonperturbative regime, and it may generate a new sort of the nonperturbative corrections. These are physically distinct from the Fermi motion encoded in the distribution function of the heavy quark inside the B meson. A dedicated discussion can be found in the recent paper20. Such contributions may indicate that the so-called ‘soft-collinear effective theories’ (SCET), in all their variety, may not truly represent an effective theory of actual QCD, not having the identical nonperturbative content. It has been shown in Ref.20 that this physics, nevertheless do not affect the moments of the decay distributions, in particular the photon energy moments (at a low enough cut). The relation of the moments to the local heavy quark expectation values remains unaltered: the perturbative corrections have the usual structure and include only truly short-distance physics. In this respect, we do not agree with the recent claims found in the literature that the usual OPE relations for the moments in the light-like distributions do not hold where perturbative effects are included. Our analysis does not support large uncertainties in the b -+ s y moments reported by Neubert at the Workshop, see also Refs.21. (It is curious to note the increase in ( E 7 )when lowering Ecutobtained by the author). I would disagree already with the starting point of that approach. On the contrary, applying the Wilsonian approach we find 22 quite accurate, stable (and physical as well) predictions whenever the cut on the photon energy is sufficiently low to cover the major part of the distribution function domain.

+

109

2.

A ‘BPS’ expansion

The heavy quark parameters as they emerge from the fit of the data are close to the theoretically expected values, mb(1 GeV) N 4.60 GeV, p:(1 GeV) 21 0.45 GeV2, p& (1 GeV) N0.2 GeV3. The precise value, in particular of p:, is of considerable theoretical interest. It is essentially limited from below by the known chromomagnetic expectation value 23 : &(p) > &(p), & ( l GeV) 11 0.35::;; GeV2, (4) and experiment seem to suggest that this bound is not too far from saturation. This is a peculiar regime where the heavy quark sum rules’, the exact relations for the transition amplitudes between sufficiently heavy flavor hadrons, become highly constraining. One consequence of the heavy quark sum rules is the lower bound24 on the slope of the 1W function e2> They also provide upper bounds which turn out quite restrictive once is close to &, say

i.

e2-

i 5 0.3

if pE(l GeV)-&(l GeV) 6 0.1 GeV2. (5) This illustrates the power of the comprehensive heavy quark expansion in QCD: the moments of the inclusive semileptonic decay distributions can tell us, for instance, about the formfactor for B+ D or B - i D*decays. Another application is the B + D e u amplitude near zero recoil. Expanding in & - & an accurate estimate was obtained25

f+(O) = 1.04 f 0.01 f 0.01 MB MD In fact, p: 1~& is a remarkable physical point for B and D mesons, since the equality implies a functional relation ZbbjiblB) = 0. Some of the Heavy Flavor symmetry relations (but not those based on the spin symmetry) are then preserved to all orders in l/mQ. This realization led to a ‘BPS’ e x p a n ~ i o nwhere ~~>~ the ~ usual heavy quark expansion was combined with an expansion around the ‘BPS’ limit &bjib(B)=O. There are a number of miracles in the ‘BPS’ regime. They include e2= 2 and p i s = -p$; a complete discussion can be found in Ref.25. Some intriguing ones are2’: No power corrections to the relation M p = mQ + and, therefore to mb-m, = M B - M D . For the B + D amplitude the heavy quark limit relation between the two formfactors 2m

+

110

does not receive power corrections. 0 For the zero-recoil B -+ D amplitude all b l l m k terms vanish. 0 For the zero-recoil formfactor f+ controlling decays with massless leptons

holds t o all orders in l/mQ. 0 At arbitrary velocity, power corrections in B + D vanish,

so that the B-+ D decay rate directly yields the Isgur-Wise function [(w). Since the ‘BPS’ limit cannot be exact in actual QCD, we need to understand the accuracy of its predictions. The dimensionless parameter ,B describing the deviation from the ‘BPS’ limit is not tiny, similar in size to the generic l/m, expansion parameter, and relations violated to order ,B may in practice be more of a qualitative nature. However, the expansion parameters like p: -& 0: ,B2 can be good enough. One can actually count together powers of l/m, and ,B to judge the real quality of a particular heavy quark relation. In fact, the classification in powers of ,B to a l l o r d e r s in l / m is ~ possible.25 Relations (7) and (9) for the B -+ D amplitudes at arbitrary velocity can get first order corrections in ,B, and may be not very accurate. Yet the slope e2 of the IW function differs from only a t order ,B2. Some other important ‘BPS’ relations hold up to order ,B2: M B - M D = mb-m, and M D = m c + x Zero recoil matrix element (DlEyoblB) is unity up to O(,B2) The experimentally measured B -+ D formfactor f+ near zero recoil receives only second-order corrections in ,B to all orders in l/mQ:

2

The latter is an analogue of the Ademollo-Gatto theorem for the ‘BPS’ expansion, and is least obvious. The ‘BPS’ expansion turns out more robust than the conventional l / m Q one which does not protect the decay against the first-order corrections. As a practical application, Ref.25 derived an accurate estimate for the formfactor f+(O) in the B + D transitions, Eq. ( 6 ) , incorporating terms through l/mz,b. The largest correction, +3% comes from the short-distance

111

perturbative renormalization; power corrections are estimated to be only about 1%.

3. The

‘5 > %’problem

So far mostly the success story of the heavy quark expansion for semileptonic B decays has been discussed. I feel obliged to recall the so-called ‘ $ > $, puzzle related to the question of saturation of the heavy quark sum rules. It has not attracted due attention so far, although it had been raised independently by two teams28*1t29 including the Orsay heavy quark group, and it has been around for quite some time. A useful review was recently presented by A. Le Yaouanc30; here I briefly give a complementary view. There are two basic classes of the sum rules in the Small Velocity, or Shifrnan-Voloshin (SV) heavy quark limit. First are the spin-singlet sum rules which relate e2, p:, p g , ... to the excitation energies E and transition amplitudes squared 1rI2for the P-wave states. Both and $ P-wave states, i.e. those where the spin j of the light cloud is 3 or $, contribute to these sum rules. The second class are ‘spin’ sum rules, they express similar relations for e2- h-2C7 p:-p’$, etc. These sum rules include only states. The spin sum rules strongly suggest that the $ states dominate over states, having larger transition amplitudes ~ 3 / 2 In . fact, this automatically happens in all quark models respecting Lorentz covariance and the heavy quark limit of QCD; an example are the Bakamjian-Thomas-type quark models developed at 0rsayB1,or the covariant models on the light front32. The lowest $ P-wave excitations of D mesons, D1 and 05 are narrow and well identified in the data. They seem to contribute to the sum rules too little, with 1 ~ ~ / ~ 1 ~according ~ 0 . 1 5to Ref.33. Wide states denoted by D,*and D; are possibly produced more copiously; they can, in principle, saturate the singlet sum rules. However, the spin sum rules require them to be subdominant to the states. The most natural solution for all the SV sum rules would be if the lowest $ states with ~ 3 / 2N 450 MeV have 1 ~ ~ M/ 0.3, while for the states 1 ~ ~ M/ 0.07 ~ 1 to0.12 ~ with ~ 3 / 2% 300 to500 MeV. Strictly speaking, higher P-wave excitations can make up for the wrong share between the contributions of the lowest states. This possibility is disfavored, however. In most known cases the lowest states in a given channel tend to saturate the sum rules with a reasonable accuracy. It should be appreciated that the above sum rules are exact for heavy quarks. Likewise, the discussed consequences rely on the assumptions most

x,

3

i,

3

3

3

112

robust among those we usually employ in dealing with QCD. Therefore, the problem we examine is not how in practice 7 3 1 2 might turn out less than r1/2. Rather it is why, in spite of the actual hierarchy between 7312 and r112the existing extractions seem to indicate the opposite relation. In fact, the recent pilot lattice indicated the right scale for both ~ 3 / 2and r112and, taken at face value, showed a reasonable saturation of the spin sum rule by the lowest P-wave clan. Similar predictions had been obtained in the relativistic quark model from Orsay31 and in the light-cone quark models32. Experimentally 7312 and r112can be extracted from either nonleptonic decays B + D** T assuming factorization and the absence of the final state interactions, or directly from their yield in B -+ D**lv decays. The former way suffers from possible too significant corrections to factorization, in particular for the case of excited charm states. Such decays also depend on the amplitude at the maximal recoil, kinematically most distant from the small recoil we need the amplitude at; we know that the slopes of the formfactors are quite significant even with really heavy quarks. The safer approach is the direct yield in the semileptonic decays. The data interpretation is obscured, however by the significant corrections to the heavy quark limit for charm mesons. For instance, the classification itself over the light cloud angular momentum j relies on the heavy quark limit. However, one probably needs a good physical reason to have the hierarchy between the finite-m, heirs of the $ and states inverted, rather than only reasonably modified compared to the heavy quark limit. Yet, as has been shown, these corrections are generally significant and may noticeably affect the extracted 1r3p12. It has also been routinely assumed that the slope of the formfactors are similar for the and for the D**, something which is not expected to hold in QCD. The existing models likewise predict a large slope for the $ mesons and a moderate one for the states. This clearly enhances the actual extracted value of 1 ~ ~ / ~ ( ~ . The experimental situation in respect to the wide $ charm states still remains uncertain. It cannot be excluded that their actual yield is smaller, and a t the same time it can be essentially enhanced compared to the largem, limit. To summarize, we do not have a definite answer to how this apparent contradiction of theory with data is resolved. Considering all the evidence, the scenario seems most probable where all the above factors contribute coherently, suppressing the yield of the states more than expected and

+

3

113

i

enhancing the production of the states. First of all, this refers to the size of the power corrections in charm. Secondly, the effect of significant formfactor slopes for the $ states. Finally, it seems possible that the actual branching fraction of the P-wave states in the semileptonic decays would be eventually below 1%level. In my opinion it is important to clarify this problem.

i

Conclusions. The dynamic QCD-based theory of inclusive heavy flavor decays has finally undergone and passed critical experimental checks in the semileptonic B decays at the nonperturbative level. Experiment finds consistent values of the heavy quark parameters extracted from quite different measurements once theory is applied properly. The heavy quark parameters emerge close to the theoretically expected values. The perturbative corrections to the higher-dimension nonperturbative heavy quark operators in the OPE have become the main limitation on theory accuracy; this is likely to change in the foreseeable future. Inclusive decays can also provide important information for a number of individual heavy flavor transitions. The B + D lv decays may actually be accurately treated. The successes in the dynamic theory of B decays put a new range of problems in the focus; in particular, the issue of the saturation of the SV sum rules requires close scrutiny from both theory and experiment. Acknowledgments

I am grateful to D. Benson, I. Bigi, P. Gambino, M. Shifman, A. Vainshtein, 0. Buchmueller and P. Roudeau, for close collaboration and discussions. This work was supported in part by the NSF under grant number PHY0087419. References 1. N.Uraltsev, in Boris I o f e Festschrift “At the Frontier of Particle Physics -- Handbook of QCD”, Ed. M. Shifman (World Scientific, Singapore, 2001), Vol. 3, p. 1577; hep-ph/0010328. 2. N.Uraltsev, Proc. of the 31st International Conference on High Energy Physics, Amsterdam, The Netherlands, 25-31 July 2002 (North-Holland Elsevier, The Netherlands, 2003), S. Bentvelsen, P. de Jong, J. Koch and E. Laenen Eds., p. 554; hep-ph/0210044. 3. P. Gambino and M. Uraltsev, Europ.Phys. Journ. C 34 (2004) 181. 4. M. Battaglia e t al., Phys. Lett. B 556 (2003) 41. 5. B. Aubert et al., BaBar Collaboration, Phys.Rev. Lett. 93 (2004) 011803.

114

6. N. Uraltsev, hep-ph/0403166; to appear in IJMPA. 7. M. Trott, hep-ph/0402120. 8. I. Bigi, N. Uraltsev and A. Vainshtein, Phys. Lett. B293 (1992) 430; B. Blok and M. Shifman, Nucl. Phys. B399 (1993) 441 and 459. 9. N.G. Uraltsev, Int. J . Mod. Phys. A l l (1996) 515; Nucl. Phys. B491 (1997) 303; I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Phys. Rev. D56 (1997) 4017. 10. C.W. Bauer et al., Phys. Rev. D67 (2003) 054012. 11. 1.1.Bigi and N. Uraltsev, Phys. Lett. B579 (2004) 340. 12. 0. Buchmueller, private communication. 13. Z. Ligeti, M. E. Luke, A. V. Manohar and M. B. Wise, Phys. Rev. D60 (1999) 034019. 14. P. Koppenburg et al., Belle collaboration, Phys. Rev.Lett. 93 (1999) 061803. 15. D. Benson, I. Bigi, Th. Mannel and N. Uraltsev, Nucl. Phys. B665 (2003) 367. 16. A.H. Hoang, Z. Ligeti and A.V. Manohar, Phys. Rev. Lett. 82 (1999) 277. 17. A.I. Vainshtein and V.I. Zakharov, Phys. Rev.Lett. 73 (1994) 1207. 18. Yu.L. Dokshitzer and N.G. Uraltsev, Phys. Lett. B380 (1996) 141. 19. C.W. Bauer et al., hep-ph/0408002. 20. N. Uraltsev, hep-ph/0407359. 21. M. Neubert, hep-ph/0408179; hep-ph/0408208. 22. D. Benson, I. Bigi and N. Uraltsev, UND-HEP-04-BIGO3, LNF04/17(P), in preparation. 23. I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Int. Joum. Mod. Phys. A9 (1994) 2467; M. Voloshin, Surv. High En. Phys. 8 (1995) 27. 24. N. Uraltsev, Phys. Lett. B501 (2001) 86; J . Phys. G27 (2001) 1081. 25. N. Uraltsev, Phys. Lett. B585 (2004) 253. 26. N. Uraltsev, Phys. Lett. B545 (2002) 337. 27. N. Uraltsev, hep-ph/0309081, eConf C030603, JEU05 (2003). Talk at Int. Conference “Flavor Physics & CP Violation 2003” , June 3-6 2003, Paris. 28. I. Bigi, M. Shifman and N.G. Uraltsev, Ann. Rev.Nucl. Part. Sci. 47 (1997) 591. 29. A. Le Yaouanc et al., Phys. Lett. B480 (2000) 119. 30. A. Le Yaouanc, hep-ph/0407310. 31. A. Le Yaouanc, L. Oliver, 0.Pene and J.C. Raynal, Phys. Lett. B365 (1996) 319; V. Morenas et al., Phys. Rev. D56 (1997) 5668. 32. H.Y. Cheng, C.K. Chua and C.W. Hwang, Phys. Rev. D69 (2004) 074025. 33. A.K. Leibovich, Z. Ligeti, I.W. Stewart and M.B. Wise, Phys. Rev.Lett. 78 (1997) 3995; Phys. Rev. D57 (1998) 308. 34. D. Becirevic et al., hep-lat/0406031.

POLARIZATION, RIGHT-HANDED CURRENTS, AND CP VIOLATION I N B VV ---f

ALEXANDER L. KAGAN Department of Physics University of Cincinnati Cincinnati, Ohio 45221, U.S.A. A detailed analysis of B ---t VV polarization in QCD factorization reveals that the low longitudinal polarization fractions f ~ ( 4 K ' = ) 50% can be accounted for in the SM via a QCD penguin annihilation graph. The ratio of transverse rates rl/rll % provides a sensitive test for new right-handed currents in the four-quark operators. CP violation measurements in B -+ VV decays can discriminate between new contributions to the dipole and four quark operators

1. Introduction

Polarization measurements in B -+ VV decays should be sensitive to the V - A structure of the Standard Model. This issue has recently been studied in the QCD factorization framework in and the results are summarized in this contribution. We have found that certain non-factorizable annihilation graphs can easily account for the small B -+ 4K* longitudinal polarization fractions observed a t the B factories. A direct test for right-handed currents in the four-quark operators emerges from the ratio of the transverse perpendicular and parallel transversity rates. In the event that non-Standard Model CP-violation is confirmed, e.g., in the B -+ q5K3 time-dependent CP asymmetry, an important question will be whether it arises via New Physics contributions to the four-quark operators, the b -+ sg dipole operators, or both. We will see that this question can be addressed by comparing CP asymmetries in the different transversity final states in pure penguin B + VV decays, e.g., B -+$K* and Bh -+ K*'p+. The underlying reason is large suppression of the transverse dipole operator matrix elements. It is well known that it is difficult to obtain new O(1) CP violation effects a t the loop-level from the dimension-six four-quark operators. Thus, this information could help discriminate between scenarios in which New Physics effects are induced via loops versus a t tree-level.

115

116

Extensions of the Standard Model often include new b -+ S R righthanded currents. These are conventionally associated with opposite chirality effective operators Qi which are related to the Standard Model operators Qi by parity transformations, 0

QCD Penguin operators Q3,5 Q4,6

0

= ( S ~ ) V - A( ( ? ? ) v F A = (3ibj)V-A (4jQi)VrA

= (Sb)v+A (qq)VfA

Q4,6

-+

(%bj)V+A (qjqi)V&A

Chromo/Electromagnetic Dipole Operators

+~5)biFpu + y5)tabGEu

= &mb%a’l’(l &ag = ambSaPu(1 Q7y

0

Q3,5

+

Q7y

-+

-i

Qgg

= &mbSial’lv(l = &mbSafi’(l

-

ys)U‘p”

- y5)tabGEu

Electroweak Penguin Operators

Q7,9

&S,lO

= Z ( S ~ ) V - Aeq ( & ) V f A = ; ( S i b j ) V - A eq ( q j % ) V * A

+

+

Q7,9 Q8,lO

= ; ( S ~ ) V + A eq ( q Q ) v r A = ;(SZbj)V+A eq ( q j q i ) V r A

Examples of New Physics which could give rise to right-handed currents include supersymmetric loops which contribute to the QCD penguin or chromomagnetic dipole operators. Figure 1 illustrates the well known squarkgluino loops in the squark mass-insertion approximation. For example, the down-squark mass-insertion bm? - (bm!*- ) would contribute to Qsg bRSL

SRbL

whereas bmiLiL(SmiRLR) would contribute t o Q3,..6 (Q3,..,6). Righthanded currents could also arise via tree-level contributions to the QCD or electroweak penguin operators, e.g. , due t o flavor-changing 2’ couplings 2 , R-parity violating couplings 3, or color-octet exchange 4 . (Qgg),

Figure 1. Down squark-gluino loop contributions t o the Standard Model and opposite chirality dipole operators in the squark mass insertion approximation.

We recall that there are three helicity amplitudes Ah ( h = 0, -, +) in VV decays: A’, in which both vectors are longitudinaly polarized; A-, in which both vectors have negative helicity; and A+, in which both

B

-+

117

vectors have positive helicity. In the transversity basis are given by,

A l J = (A- F A + ) / & ,

5,

the amplitudes

A0 = A 0

(1)

In B decays, Al,ll = (A+ ~ A - ) / f i . The polarization fractions are defined as fi = ri/rtota1, i = 0, I,11, where rtotal is the total decay rate. Under parity, the effective operators transform as Qi c) Qi. The New Physics amplitudes, for final states f with parity Pf, therefore satisfy

(fIQiP)= -(-)Pf(fIQilB)

* ANp(B

-+

f) K CifTp(pb)- (-)pfCFp(pb),

(2)

where CY' and 6Yp are the new Wilson coefficient contributions to the i'th pair of Standard Model and opposite chirality operators 6 . In B -+ VV decays the Itransversity and 0, 11 transversity final states are P-odd and P-even, respectively, yielding AiN P ( B + VV)o,ll K CifTP(pb)-C'FP(pb), ANp(B -+ V V ) I K CyP(pb)+CyP(pb).

(3) The modes of particular interest in the search for new physics in rare B decays are those which are pure-penguin or penguin-dominated in the Standard Model. This is because they necessarily have 0 0

null decay rate CP-asymmetries, A c p ( f ) 1%,or null deviations of the time-dependent CP-asymmetry coefficient Sf,, from (sin 2P)J/,pK, in decays to CP-eigenstates, (->cpsfcpl 1% or l(sin2P)J/*Ks null B + VV triple-product CP-asymmetries, A$ll(f) 1%in the Standard Model. N

+

0

N

The CP-violating triple-products

2. Polarization in

B

.--)

(related t o

4'.

x

Z2)

are given by

VV decays

A discussion of polarization in B -+ VV decays has been presented in in the framework of QCD factorization. Here we summarize some of the results. Sensitivity to the V - A structure of the Standard Model is due to

118

the power suppression associated with the ‘helicity-flip’ of a collinear quark. For example, in the Standard Model the factorizable graphs for B 4 4K* are due to transition operators with chirality structures (Sb)V-A(Ss)V?A, see Figure 2 . In the helicity amplitude A- a collinear s or 3 quark with positive helicity ends up in the negatively polarized 4, whereas in A+ a second quark ‘helicity-flip’ is required in the form factor transition. Collinear quark helicity flips require transverse momentum, kl , implying a suppression of O(AQcD/mb)per flip. In the case of new right-handed currents, e.g., ( S ~ ) V + A ( S S ) V ~ the A , helicity amplitude hierarchy would be inverted, with A+ and A- requiring one and two helicity-flips, respectively.

Figure 2. Quark helicities (short arrows) for the B -+ q5K* matrix element of the operator ( B b ) v - A ( S s ) v - A in naive factorization. Upward lines form the q5 meson.

In naive factorization the B -+ $K* helicity amplitudes, supplemented by the large energy form factor relations 8, satisfy 2

A’

0: f$mB(11

(r

and

K’ 1

-f@mq5mB2cf*,

A-

c,” are the B

t

A+ 0: -fdm+mB2cLK’rlK’ .

V form factors in the large energy limit

8.

(5) Both

scale as mb3’2 in the heavy quark limit, implying A-/A’ = O ( m + / m g ) . r l parametrizes form factor helicity suppression. It is given by

where Al,2 and V are the axial-vector and vector current form factors, respectively. The large energy relations imply that r l vanishes at leading power, reflecting the fact that helicity suppression is O ( l / m b ) . Thus, A+/A- = O(f!QcD/mb). Light-cone QCD sum rules ’, and lattice form

119

factor determinations scaled to low q2 using the sum rule approach l o , give rf' M 1 - 3%; QCD sum rules give r f * M 5 % ll; and the BSW model gives rf' M 10%12. The polarization fractions in the transversity basis (1) therefore satisfy

fi/flJ = 1 + 0 (l/mb>

1 - f L = 0 (1/mi) I

(7)

7

in naive factorization, where the subscript L refers to longitudinal polarization, fi = J?i/rtotal, and f~ f i fll = 1. The measured longitudinal fractions for B -+ pp are close to 1 This is not the case for B -+ +K*O for which full angular analyses yield

+ + 16117.

f~ = 0 . 5 2 5 . 0 7 f .02, fr. = .52 f .05 rt .02, For B*

-+$K**,

f i= . 3 0 f fi = .22 rt

. 0 7 f .03

.05 & .02

'* 19.

(8) (9)

Belle measures l8

fL = .49 f

.13 f .05,

fi = .12::

f .03.

(10)

and BaBar measures f~ = .46 f 0.12 f 0.03 17. Naively averaging the Belle and BaBar $K*O measurements (without taking correlations into account) yields f i / f l l = 0.92 f .31. Finally, the longitudinal fraction for B* -+ K*'p* has also been measured at BaBar 2o and Belle21, yielding f~ = 0.79 f 0.08 f 0.04 f 0.02, and 0.5 f o.19tbqi7, respectively, which averages t o f~ = 0.74 f .08. We must go beyond naive factorization in order to determine if the small values of ~ L ( $ K *could ) simply be due to the dominance of QCD penguin operators in A S = 1 decays, rather than New Physics. In particular, it is necessary to determine if the power counting in (7) is preserved by non-factorizable graphs, i.e., penguin contractions, vertex corrections, spectator interactions, annihilation graphs, and graphs involving higher Fock-state gluons. This question can be addressed in QCD factorization '. In QCD factorization exclusive two-body decay amplitudes are given in terms of convolutions of hard scattering kernels with meson light-cone distribution amplitudes At leading power this leads to factorization of short and long-distance physics. This separation breaks down a t subleading powers with the appearance of logarithmic infrared divergences, 1 e.g., dxlx InrnBlAh, where x is the light-cone quark momentum fraction in a final state meson, and Ah AQ~D is a physical infrared cutoff. Nevertheless, the power-counting for all amplitudes can be obtained. The extent to which it holds numerically can be determined by assigning large 13314115.

so

N

N

120 uncertainties t o the logarithmic divergences. Fortunately, certain polarization observables are less sensitive t o this uncertainty, particularly after experimental constraints, e.g., total rate or total transverse rate, are imposed.

b

Figure 3. Quark helicities in B -+ q5K* matrix elements: the hard spectator interaction for the operator ( S b ) v - A ( S S ) V F A (left), and annihilation graphs for the operator ( ( t b ) S - p ( B d ) S + p with gluon emitted from the final state quarks (right).

Examples of logarithmically divergent hard spectator interaction and QCD penguin annihilation graphs are shown in Figure 3, with the quark helicities indicated. The power counting for the helicity amplitudes of the annihilation graph, including logarithmic divergences, is

The logarithmic divergences are associated with the limit in which both the s and 3 quarks originating from the gluon are soft. The annihilation topology implies an overall factor of l / r n b . Each remaining factor of l/mb is associated with a quark helicity flip. In fact, adding up all of the helicity amplitude contributions in QCD factorization formally preserves the naive factorization power counting in (7) Recently, the first relation in (7) has been confirmed in the soft collinear effective theory 23. However, as we will see below, it need not hold numerically because of QCD penguin annihilation. 22y1.

2.1. Numerical results f o r polarization

'.

The numerical inputs are given in The logaritmic divergences are modeled as in l4?l5. For example, in the annihilation amplitudes the quantities

121

X A are introduced as

This parametrization reflects the physical ~ ( R Q c Dcutoff, ) and allows for large strong phases E [0,2n] from soft rescattering. The quantities X A (and the corresponding hard spectator interaction quantities X H ) are varied independently for unrelated convolution integrals. The predicted longitudinal polarization fractions f~(p-po) and f ~ ( p - p + ) are close to unity, in agreement with observation and with naive power counting (7). The theoretical uncertainties are small, particularly after imposing the branching ratio constraints, due to the absence of (for p-po) or CKM suppression of (for p-p+) the QCD penguin amplitudes. Averaging the Belle and BaBar B -+ 4K*O measurements yields = 0.52 h 0.04 and 1O6BreXP= 9.7 f 0.9, or lo6 B r y P = 5.1 f 0.6 and lo6 B r y P = 4.7f0.6. BrL and BrT = B r l +Brll are the CP-averaged longitudinal and total transverse branching ratios, respectively. In the absence l61l7

18,19124

fFp

+6.79+.88

of annihilation, the predicted branching ratios are lo6 BrL = 5.15- 4.66-.81 and lo6 BrT = .61?:2:?:;,, where the second (first) set of error bars is due to variations of XH (all other inputs). However, the ( S P ) ( S - P ) QCD penguin annihilation graph in Figure 3 can play an important role in both 21' and 21- due to the appearance of a logarithmic divergence squared ( X z ) , the large Wilson coefficient c6, and a l/Nc rather than 1/NZ dependence. Although formally O(l/m2), see (ll),these contributions can be O(1) numerically. This is illustrated in Figure 4, where BrL and BrT are plotted versus the quantities p; and p i , respectively, for B ---f q5K*O. p i and p i enter the parametrizations (12) of the logarithmic divergences appearing in the longitudinal and negative helicity ( S P ) ( S - P ) annihilation amplitudes, respectively. As p 2 - increase from 0 to 1, the corresponding annihilation amplitudes increase by more than an order of magnitude. The theoretical uncertainties on the rates are very large. Furthermore, the largest input parameter uncertainties in BrL and BrT are a priori unrelated. Thus, it is clear from Figure 4 that the QCD penguin.annihilation amplitudes can account for the q5K*O measurements. Similarly, the measurements of ~ L ( $ K * *M) 50% can be accounted for. Do the QCD penguin annihilation amplitudes also imply large transverse polarizations in B + pK* decays? The answer depends on the pattern of s U ( 3 ) flavor ~ symmetry violation in these amplitudes. For light mesons containing a single strange quark, e.g., K * , non-asymptotic effects

+

+

122

10

10

5

5

0

0

0.2

0.6

0.4

0.8

1

0

0

0.2

0.6

0.4

0.8

1

PA

POA

Figure 4. BrL(dK*O) vs. p i (left), BrT(dK*O) vs. p a (right). Black lines: default inputs. Blue bands: input parameter variation uncertainties added in quadrature, keeping default annihilation and hard spectator interaction parameters. Yellow bands: additional uncertainties, added in quadrature, from variation of parameters entering logarithmically divergent annihilation and hard spectator interaction power corrections. Thick line: BrFax under simultaneous variation of all inputs.

shift the weighting of the meson distribution amplitudes towards larger strange quark momenta. As a result, the suppression of ss popping relative to light quark popping in annihilation amplitudes can be 0(1),which is consistent with the order of magnitude hierarchy between the B -+ Doro and B -+ D$K- rates 25. (See 26 for a discussion of other sources of S U ( 3 ) violation). In the present case, this implies that the longitudinal polarizations should satisfy f ~ ( p * K * ' ),< f ~ ( q 5 K *in) the Standard Model '. Consequently, f ~ ( p * K * ' )M 1 would suggest that U-spin violating New Physics entering mainly in the b + sss channel is responsible for the small f ~ ( q 5 K * ) . One possibility would be right-handed vector currents; they could interfere constructively (destructively) in A 1 (&) transversity amplitudes, see (3). Alternatively, a parity-symmetric scenario (CTp M a t the weak scale) would only significantly affect A*. A more exotic possibility would be tensor currents; they would contribute to the longitudinal and transverse amplitudes a t sub-leading and leading power, respectively, opposite to the vector currents. The current experimental average for f ~ ( K * ~ p is * )2 , . 5 ~ larger than f ~ ( q 5 K *but ) , it lies significantly below the naive factorization prediction. We should mention that our treatment of the charm (and up) quark loops in the penguin amplitudes follows the usual perturbative approach used in QCD factorization The authors of 23 have argued that the

eTp

13914,15.

123

region of phase space in which the charm quark pair has invariant mass q2 4mz, and is thus moving non-relativistically, should be separated out into a long-distance 'charming penguin' amplitude 27. NRQCD arguments are invoked to claim that such contributions are O ( v ) ,where u M .4-.5, so that they could effectively be of leading power. Furthermore, it is claimed that the transverse components may also be of leading power, thus potentially accounting for f ~ ( c $ K * )Also . see 28. However, a physical mechanism by which a collinear quark helicity-flip could arise in this case without power suppression remains t o be clarified. Arguments against a special treatment of this region of phase space l 3 ? l 4are based on parton-hadron duality. More recently, arguments supporting the power suppression of long-distance charming penguin effects have been presented in 29. N

3. A test for right-handed currents Does the naive factorization relation fi/fll= 14- O(hQcD/mb)(7) survive in QCD factorization? This ratio is very sensitive t o the quantity TI defined = .05 f .05 in (6). As TI increases, f i / f l l decreases. The range spanning existing model determinations is taken in '. In Figure 5 (left) the resulting predictions for fi/fll and BrT are studied simultaneaously for in the Standard Model. Note that the theoretical uncertainty B -+ for f i / f l l is much smaller than for f ~ Evidently, . the above relation still holds, particularly a t larger values of BrT where QCD penguin annihilation dominates both B r l and Brll. A ratio for f ~ / f l l in excess of the Standard Model range, e.g., f i / f l l > 1.5 if TI > 0, would signal the presence of new right-handed currents. This is due to the inverted hierarchy between A- and A+ for righthanded currents, and is reflected in the sign difference with which the Wilson coefficients (?i enter and All. For illustration, new contributions to the QCD penguin operators are considered in Figure 5 (right). At the New Physics matching scale M , these can be parametrized as

TI"'

(-)

(-1

(-)

(-)

CS = -3C5 = -3C3 =

(-)

. For simplicity, we take M

MW and consider two cases: K = -.007 or new left-handed currents (lower bands), and R = -.007 or new right-handed currents (upper bands), corresponding to c,",;;(mb) or (?Gc)(mb)M .18Cfg(mb),and c,",c)(mb)or (?,",$(mb) M .25 C,"t;',(mb). Clearly, moderately sized right-handed currents could increase f i / f l l well beyond the Standard Model range if TI 2 0. However, new left-handed currents would have little effect. The experimental average for f l / f j r in B -+ c$K*' lies close to unity, and thus gives no C4 =

K

M

124

indication for sizable right-handed currents.

1.5

1

3

4

5

6

l o 6 BrT

7

I

I

8

3

I

3.5

4

4.5

5

5.5

6

10 BrT

Figure 5. f l / f l l vs. BTT in the SM (left), and with new RH or LH currents (right). Black lines, blue bands, and yellow bands are as in Figure 4. Thick lines: (f~/fll)"'"" in the Standard Model for indicated ranges of rf' under simultaneous variation of all inputs. Plot for r y * > 0 corresponds to BrFaX in Figure 4.

4. Distinguishing four-quark and dipole operator effects

The O(a,) penguin contractions of the chromomagnetic dipole operator Qsg are illustrated in Figure 6. a4 and a6 are the QCD factorization coefficients of the transition operators ($)"-A 8 (.&)V-A and (Qb)s-p 8 (Dq)S+p1 respectively, where q is summed over u , d , s l49l5. Only the contribution on the left ( a q ) to the longitudinal helicity amplitude A' is non-vanishing In particular, the chromo- and electromagnetic dipole operators QS9 and Qyy do not contribute to the transverse penguin amplitudes a t O(a,) due to angular momentum conservation: the dipole tensor current couples to a transverse gluon, but a 'helicity-flip' for q or in Figure 6 would require a longitudinal gluon coupling. Formally] this result follows from Wandura-Wilczek type relations among the vector meson distribution amplitudes, and the large energy relations between the tensor-current and vector-current form factors. Transverse amplitudes in which a vector meson contains a collinear higher Fock state gluon also vanish at O(a,), as can be seen from the vanishing of the corresponding partonic dipole operator graphs in the same momentum configurations. Furthermore, the transverse O(a:) contributions involving spectator interactions are highly suppressed.

'.

125

Figure 6. Quark helicities for the O(cys)penguin contractions of &Q. The upward lines form the q4 meson in B -+ q4K* decays.

This has important implications for New Physics searches. For example, in pure penguin decays to CP-conjugate final states f , e.g., B .+ 4(K*' -+ KSn-O),if the transversity basis time-dependent C P asymmetry parameters (S,)l and ( S j ) , ,are consistent with ( s i n 2 P ) ~ , ~ K ,and , ( S ~ )isOnot, then this would signal new CP violating contributions to the chromomagnetic dipole operators. However, deviations in (Sf)lor ( S f )11 would signal new CP violating four-quark operator contributions. If the triple-products A$ and A,11 (4) do not vanish and vanish, respectively, in the pure-penguin decays B #K*'lf, K*Op*, then this would also signal new CP violating contributions to the chromomagnetic dipole operators. This assumes that a significant strong phase difference is measured between All and A l l for which there is some experimental indication However, non-vanishing A,,II or non-vanishing transverse direct CP asym--f

19118.

metries, e.g.,Acp(4K*'~*)ll,l,A C P ( K * ' ~ * ) I ( Iwould , ~ , signal the intervention of four-quark operators. The above would help to discriminate between different explanations for an anomalous S+K, or S q / ~ which ,, fall broadly into two categories: radiatively generated dipole operators, or tree-level induced four-quark operators. Finally, a large value for f i / f l l would be a signal for right-handed four-quark operators. 5. Conclusion

Polarization measurements in B decays to light vector meson pairs offer a unique opportunity to probe the chirality structure of rare hadronic B decays. A Standard Model analysis which includes all non-factorizable graphs in QCD factorization shows that the longitudinal polarization formally satisfies 1 - f~ = O ( l / m 2 ) ,as in naive factorization. However, the contributions of a particular QCD penguin annihilation graph which is formally O ( l / m 2 )can be O(1) numerically in longitudinal and negative helicity A S = 1 B decays. Consequently, the observation of f~(qW*'?-) M 50%

126 can be accounted for, albeit with large theoretical errors. The expected pattern of s U ( 3 ) ~ violation in the QCD penguin annihilation graphs, i.e., large suppression of sS relative t o uii or dd popping, implies that the longitudinal polarizations should satisfy f ~ ( p * K * ' ),< f ~ ( 4 K *in)the Standard Model. Consequently, f ~ ( p * K * ' )F=: 1 would suggest that U-spin violating New Physics entering mainly in the b SSS channel is responsible for the small values of f ~ ( 4 K * )There . is currently no experimental indication for such effects a t the B factories. The ratio of transverse rates in the transversity basis satisfies f ~ / f r = 1 O(l/m), in agreement with naive power counting. A ratio in excess of the predicted Standard Model range would signal the presence of new righthanded currents in dimension-6 four-quark operators. The maximum ratio attainable in the Standard Model is sensitive to the B -i V form factor combination r L , see (6), which controls helicity suppression in form factor transitions. All existing model determinations give a positive sign for r l , which would imply f i ( $ K * ) / f l l ( + K *<) 1.5 in the Standard Model. The magnitude and especially the sign of rf* is an important issue which should be clarified further with dedicated lattice studies. The experimental average fi(4K*)/fll($IT*) is consistent with unity, and thus gives no indication for the existence of such right-handed currents. Contributions of the dimension-5 b -+ sg dipole operators t o the transverse B -+ VV modes are highly suppressed, due to angular momentum conservation. Comparison of CP violation involving the longitudinal modes with CP violation only involving the transverse modes in pure penguin A S = 1 decays could therefore distinguish between new contributions to the dipole and four-quark operators. More broadly, this could distinguish between scenarios in which New Physics effects are loop induced and scenarios in which they are tree-level induced, as it is difficult to obtain 0(1) CP-violating effects from dimension-6 operators beyond tree-level. Again, a high luminosity B factory will be required in order to obtain the necessary level of precision in CP violation measurements. Acknowledgments: I would like to thank Martin Beneke, Alakhabba Datta, Keith Ellis, Guy Engelhard, Andrei Gritsan, Yuval Grossman, Matthias Neubert , Uli Nierste, Yossi Nir, Dan Pirjol, and Ian Stewart for useful discussions. This work was supported by the Department of Energy under Grant DE-FG02-84ER40153. ---f

+

127 References 1. A. L. Kagan, University of Cincinnati preprint UCTP-102-04 [arXiv:hepph/0405134]. 2. Y. Grossman, M. Neubert and A. L. Kagan, JHEP 9910, 029 (1999) [arXiv:hep-ph/9909297]; V. Barger, C. W. Chiang, P. Langacker and H. S. Lee, Phys. Lett. B 580, 186 (2004) [arXiv:hep-ph/0310073]. 3. A. Datta, Phys. Rev. D 66, 071702 (2002) [arXiv:hep-ph/0208016]; B. Dutta, C. S. Kim and S. Oh, Phys. Rev. Lett. 90 (2003) 011801 [arXiv:hepph/0208226]. 4. G. Burdman, Phys. Lett. B 590, 86 (2004) [arXiv:hep-ph/0310144]. 5. I. Dunietz, H. R. Quinn, A. Snyder, W. Toki and H. J. Lipkin, Phys. Rev. D 43, 2193 (1991). 6. A. L. Kagan, talks at SLAC Summer Institute, August 2002, and SLAC Workshops, May and Oct 2003. 7. G. Valencia, Phys. Rev. D 39, 3339 (1989); A. Datta and D. London, arXiv:hep-ph/0303159. 8. J. Charles, A. Le Yaouanc, L. Oliver, 0. Pene and J. C. Raynal, Phys. Rev. D 60, 014001 (1999) [arXiv:hep-ph/9812358]. 9. P. Ball and V. M. Braun, Phys. Rev. D 58, 094016 (1998) [hep-ph/9805422]. 10. L. Del Debbio, J. M. Flynn, L. Lellouch and J. Nieves, Phys. Lett. B 416, 392 (1998) [hep-lat/9708008]. 11. P. Colangelo, F. De Fazio, P. Santorelli and E. Scrimieri, Phys. Rev. D 53, 3672 (1996) [Erratum-ibid. D 57, 3186 (1998)] [arXiv:hep-ph/9510403]. 12. M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34, 103 (1987). 13. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999) [hep-ph/9905312]; Nucl. Phys. B 591, 313 (2000) [hepph/0006124] ; 14. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B 606, 245 (2001) [arXiv:hep-ph/0104110]. 15. M. Beneke and M. Neubert, Nucl. Phys. B 675, 333 (2003) [arXiv:hepphj03080391. 16. J. Zhang el al. [BELLE Collaboration], Phys. Rev. Lett. 91, 221801 (2003) [arXiv:hep-ex/0306007]. 17. B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 91, 171802 (2003) [arXiv:hep-ex/0307026]; B. Aubert el al. [BABAR Collaboration], arXiv:hepex/0404029. 18. K. Abe et al. [BELLE Collaboration], arXiv:hep-ex/0307014, contributed to ICHEP 04, Beijing, August 2004. 19. B. Aubert et al. [BABAR Collaboration], arXiv:hep-ex/04008017, contributed to ICHEP 04, Beijing, August 2004. 20. B. Aubert et al. [BABAR Collaboration], arXiv:hep-ex/04008093], contributed to ICHEP 04, Beijing, August 2004. 21. K. Abe et al. [BELLE Collaboration], arXiv:hep-ex/0408102, contributed to ICHEP 04, Beijing, August 2004. 22. A. L. Kagan, talks at Super B Factory Workshop, Honolulu, January 2004;

128

Les Rencontres de Physique de la Valle d’Aoste, February 2004. 23. C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, arXiv:hepph/0401188 v2. 24. K.-F. Chen et al. [BELLE Collaboration], Phys. Rev. Lett. 91, 201801 (2003) [arXiv:hep-ex/0307014]. 25. S. Mantry, D. Pirjol and I. W. Stewart, Phys. Rev. D 68, 114009 (2003) [arXiv:hep-ph/0306254]. 26. M. Beneke, eConf C0304052 (2003) FOOOl [arXiv:hep-ph/0308040]. 27. M. Ciuchini, E. Franco, G. Martinelli, and L. Silvestrini, Nucl. Phys. B 501, 271 (1997). 28. P. Colangelo, F. De Fazio, and T. N. Pham, arXiv:hep-ph/0406162. 29. T. Feldmann and T. Hurth, arXiv:hep-ph/0408188.

LIFETIMES OF HEAVY HADRONS*

ALEXEY A. PETROV Department of Physics and Astronomy Wayne State University Detroit, MI &20l, USA E-mail: [email protected]

We review current status of theoretical predictions of lifetimes of heavy hadrons in heavy-quark expansion. We present a calculation of subleading l/mb corrections to spectator effects in the ratios of beauty hadron lifetimes. We find that these effects are sizable and should be taken into account in systematic analyses of heavy hadron lifetimes. In particular, the inclusion of l/mb corrections brings into agreement the theoretical predictions and experimental observations of the ratio of lifetimes of Ab-baryon and B d meson. We obtain T ( B , ) / T ( B ~=) 1.09 f0.03, T ( B , ) / T ( B = ~) 1.00 f 0.01, T ( A ~ ) / T (=B0.87 ~ ) & 0.05.

1. Introduction The hierarchy of lifetimes of heavy hadrons can be understood in the heavyquark expansion (HQE), which makes use of the disparity of scales present in the decays of hadrons containing b-quarks. HQE predicts the ratios of lifetimes of beauty r n e s o n ~ which ~ ~ ~ agree ~ ~ , with the experimental observations well within experimental and theoretical uncertainties. Most recent experimental analyses give4i5 ?(Bu)/T(B,j)lez = 1.085 f 0.017,

T(B,)/T(B~)J,, = 0.951 f 0.038,

(1)

T ( & , ) / T ( B ~= ) I 0.797 ~ ~ f0.053. The most recent theoretical predictions show evidence of excellent agreement of theoretical and experimental result^^^^. This agreement also provides us with some confidence that quark-hadron duality, which states that *This work is supported by the U.S. National Science Foundation under Grant PHY0244853, and by the U.S. Department of Energy under Contract DE-FG02-96ER41005.

129

130

smeared partonic amplitudes can be replaced by the hadronic ones, is expected to hold in inclusive decays of heavy flavors. It should be pointed out that the low experimental value of the ratio 7 ( & ) / 7 ( B d ) has long been a puzzle for the theory. Only recent next-to-leading order (NLO) calculations of perturbative QCD‘ and l/mb corrections7 to spectator effects significantly reduced this discrepancy. Of course, the problem of 7 ( & , ) / 7 ( B d ) ratio could reappear again if future measurements a t Fermilab and CERN would find the mean value to stay the same with error bars shrinking. Upcoming Fermilab measurements of Ab lifetime could shed more light on the experimental side of this issue. This talk reports on the calculation of subleading contributions to spectator effects in the l/mb expansion7 to study their impact on the ratios of lifetimes of heavy mesons. We also discuss the convergence of the l/mb expansion in the analysis of spectator effects. 2. Formalism

The inclusive decay rate of a heavy hadron Hb is most conveniently computed by employing the optical theorem to relate the decay width to the imaginary part of the forward matrix element of the transition operator: 1 r(Hb)= -(HblTIHb), T = Im i d42T{He~(~)Heff(O)}.( 2 ) 2MHb Here Heffrepresents an effective A B = 1 Hamiltonian,

J

~ G F Heff = -Kb

Jz

C

v z d ‘ [c1(P)Qfd’(P)

iG(P)&;”’~’(P)] -k h.c.,

d’=d,s,u‘=u,c

(3) where the four-quark operators I

Q”1

,

-

Q1

= d i y p u i FLypbL,

and Q2 are given by Q f d ’ = Ehypu’, d”7’lbL.

(4)

In the heavy-quark limit, the energy release is large, so the correlator in Eq. (2) is dominated by short-distance physics. An Operator Product Expansion (OPE) can be constructed for Eq. (2), which results in a prediction of decay widths of Eq. (2) as a series of local operators of increasing dimension suppressed by powers of l/mb:

In other words, the calculation of I?(&) is equivalent to computing the matching coefficients of the effective A B = 0 Lagrangian with subsequent

131

computation of its matrix elements. Indeed, at the end, the scale dependence of the Wilson coefficients in Eq. ( 5 ) should match the scale dependence of the computed matrix elements. It is customary to make predictions for the ratios of lifetimes (widths), as many theoretical uncertainties cancel out in the ratio. In addition, since the differences of lifetimes should come from the differences in the "brown mucks" of heavy hadrons, at the leading order in HQE all beauty hadrons with light spectators have the same lifetime. Thus, all ratios converge to unity in the heavy quark limit. The difference between meson and baryon lifetimes first occurs at order l / m 2 and is essentially due to the different structure of mesons and baryons. In other words, no lifetime difference is induced among the members of the meson multiplet at this order. The ratio of heavy meson and baryon lifetimes receives a shift which amounts to at most 1 - 2010, not sufficient to explain the observed pattern of lifetimes2. The main effect appears at the l/m3 level and comes from the set of dimension-six four-quark operators, whose contribution is enhanced due to the phase-space factor 167r2. They are thus capable of inducing correc~ These operators, which are tions of order 1 6 ~ ~ ( A ~ ~ ~ =/ r0 (n5 b- )10%). commonly called Weak Annihilation (WA) and Pauli Interference (PI), introduce a difference in lifetimes for both heavy mesons and baryons. Their effects have been computed2?*at leading order in perturbative QCD, and, more recently, including NLO perturbative QCD corrections6 and l/mb corrections7. The contribution of these operators to the lifetime ratios are governed by the matrix elements of A B = 0 four-fermion operators Tspec

= T G e c + T sd'p e c + TsSdeci

where the Ti are TGec

=

d'

Tspec

=

Ggmflhc(2(1- z ) ~

{ (4 + 4)02"+

2lT - G",tll&12(1

{ c:

-z ) ~

47r

+ (Nee; +

2ClC2)

+ (N& + 2

[

"1

+

(1 z ) 5 f

c 4 [

2ClC252"

+ z)Of + -23( 1 + +

2 -(1+ 3 2z)5$]

I + sp;,} 240$

1 1

6;' + -2( 1 + 245;' + s;& 3

+ 61u/,},

.

, (7)

132

Here the terms Sf,, refer to limb corrections to spectator effects, which we discuss below. Note that we include the full z = m:/mi dependence, which is fully consistent only after the inclusion of higher limb corrections. The operators Oi and 6i in Eq. (7) are defined as

0; = b i T p ( l

- T s ) b i q j j y p ( l - ~ 5 ) q j ,0; = b i y P 7 5 b i C i j 7 p ( 1 - 7 5 ) ~ j ,

6; = b i Y ( 1 - % i ) b j @ ' Y p ( 1 -

-

Q

~ 5 ) ~ 0j 2,

--& .z~

P

75

b j. q- i y p ( 1 - 7 5 ) ~ . (8)

In order to assess the impact of these and other operators, parameterizations of their matrix elements must be introduced. The meson matrix elements are

Here the parameters Bi and ~i are usually referred to as "singlet" and "octet" bag parameters'. Expressed in terms of these parameters, the lifetime ratios of heavy mesons can be written as

where the coefficients G were computed at NL06. S 1 / , represents spectator corrections of order l/mb and higher, which we discuss below. Calculations of the matrix elements of four-fermion operators in baryon decays are not straightforward. Similar to the meson matrix elements above, we relate them to the value of &,-baryon wave function a t the origin, (Ablo~lAb)= - g ( A b l @ l A b )

-

B

= -6f i , m B , m A b T ,

( A b 10;IA b ) = - g ( Ab 10; I h b ) =

E mB, m A b S. -fi, 6

(11)

Here the parameter k accounts for the deviation of the Ab wave function from being totally color-asymmetric' (B = 1 in the valence approximation), 2

and the parameter T = (t,bt:(0)12 / lt,bz(O)/ is the ratio of the wave functions at the origin of the Ab and B, mesons. Note that 6 = O ( l / m b ) , which

133

follows from the heavy-quark spin symmetry. It needs to be included as we consider higher-order corrections in l/mb. While these parameters have not been computed model-independently, various quark-model arguments suggest that the meson and baryon matrix elements are quite different. Thus a meson-baryon lifetime difference can be produced. In general, one can parametrize the meson-baryon lifetime ratio as

+ d4c2) +d ~ B 2 ) 0.98 - mE(d', + $ E ) r - m? [(dhel + d : ~ + ) (dkB1 + d i B z ) ] , (12)

r(&,)/r(Bd) _N

N

0.98 - (dl

+ d2@r

- (d3c1

(dSB1

where in the last line we scaled out the coefficient m: emphasizing the fact that these corrections are suppressed by 1Jmi compared to the leading mi effect. The scale-dependent parameters are di(mb) = (0.023, 0.028, 0.16, -0.16, 0.08, -0.08) at N L 0 2 . It is interesting t o note that in the absence of limb corrections to spectator effects, it would be equally correct to substitute the b-quark mass in Eq. (12) with the corresponding meson and baryon masses, so

which reflects the fact that WS and PI effects occur for the heavy and light quarks initially bound in the Bd meson and I i b baryon, respectively. While correct up to the order l / m i , these simple substitutions reduce the ratio of lifetimes by approximately 3-4%! We take this as an indication of the importance of bound-state effects on the spectator corrections, represented by subleading l/mb corrections to spectator operators.

3. Subleading corrections to spectator effects

.,"

We computed the higher order corrections, including charm quark-mass effects, to Eq. (7) in the heavy-quark expansion, denoted below as The l/mb corrections to the spectator effects were computed7 by expanding the forward scattering amplitude of Eq. (2) in the light-quark momentum and matching the result onto the four-quark operators containing contributions can be derivative insertions (see Fig. 1). The resulting,,:S

134 d

U

X

U

Figure 1. Kinetic corrections to spectator effects. The operators of Eqs. (14) are obtained by expanding the diagrams in powers of spectator's momentum.

written in the following form:

Sy/m = -2 (c: d' 6,/,=c:

l + Z + c;) -R;" 1-2

= C?

1-2

-$)I,

[ X R f +21+Z+1022Rf+2(l+2z)(R~ 1-Z 3 1-2 3 8z2 ~f 2 1 + z + 10z2 2ClC2) 1-2 3 1-2 16z2 1 - 22 1 6 ~ ~2 1 -42 3 1-42 Rf 3(1+ 22) ( R i - R f ) ]

[

+ pee; + 'l/m S'

l+2-

-~c~c~-RY,

+

Ri 2

[

+

+ (Nee; + 2 C l C 2 )

16a2 -R i [1-4z

+

+

I)';.-

- 2 + -32 1-22+16z2 R< + ?(1+ 22) (@ 1-42

where the following operators contribute 1R,Q = -biy'~56~biIjjyp(l - 75)daqj, m; 1RY = -biyp(l - y5)6j"biqjyp(l - 75)zaqj, m; 1 R,Q = -biyp(l - y5)6"biqjj^/v(l- y5)fiPqj, m; m R l = -%(l - y5)biIjj(l - y5)qj.

(15)

mb

Here @' denote the color-rearranged operators that follow from the expres-

(14:

135 sions for R: by interchanging color indexes of bi and qj Dirac spinors. Note that the above result contains fulZ QCD b-fields, thus there is no immediate power counting available for these operators. The power counting becomes manifest at the level of the matrix elementsg. In order to include above corrections into the prediction of lifetime ratios a calculation of meson and baryon matrix elements of the operators in Eq. (15) must be performed. We use factorization to guide our parameterizations of Ab and meson matrix elements, but keep matrix elements which vanish in factorization. This is important, as the Wilson coefficients of these operators are larger than the ones multiplying the operators whose matrix elements survive in the N , + 03 limit. Our parameterizations for meson and baryon matrix elements can be found in7, where it was shown that a set of l/mb-corrections to spectators effects can be parametrized by eight new parameters pi and pi (i = 1,..., 4) for heavy mesons and eight new parameters ,@ and ,&' for heavy baryons. Although model-independent values of these parameters will not be known until dedicated lattice simulations are performed, we presented an estimate of these parameters based on quark model arguments. In our numerical results we assume the value of the b-quark pole mass to be mb = 4.8 f 0.1 GeV and f B = 200 f 25 MeV, as well as lattice-inspired values of Bi and Q parameterslo. Numerically, the set of l/mb corrections does not markedly affect the ratios of meson lifetimes, changing the T(B,)/T(Bd) and the T(B,)/T(Bd) ratios by less than half a percent. The effect is more pronounced in the ratio of Ab and B d lifetimes, where it constitutes a 40 - 45% of the leading spectator (WA plus PI) contribution, or an overall correction of about -3% to the T ( A b ) / T ( B d ) ratio. While such a sizable effect is surprising, the main source of such a large correction can be readily identified. While the individual l/mb corrections to WS and P I are of order 20%, as expected from the naive power counting, they contribute to the Ab lifetime with the same (negative) sign, instead of destructively interfering as do WS and 20% effects produces such a sizable shift P13>7.This conspiracy of two in the ratio of the Ab and B-meson lifetimes. Since all three heavy mesons belong to the same S U ( 3 ) triplet, their lifetimes are the same at order l / m t . The computation of the ratios of heavy meson lifetimes is equivalent to the computation of U-spin or isospin-violating corrections. Both l/mz-suppressed spectator effects and our corrections computed in the previous sections arise from the spectator interactions and thus provide a source of U-spin or isospin-symmetry

-

-

-

136

400

200 0

0.75 0.8 0.85 0.9 0.950.75 0.8 0.85 0.9 0.95

0.75 0.8 0.85 0.0 0.95

T(Ab)/T(Bd)

Histograms sh_owing the random distributions around the central values of the and p; parameters contributing to T ( R b ) / T ( B d ) . Three histograms are f ~ m b~, B,, 6, shown for the scales p = m b / 2 (a), p = m b (b), and p = 2 m b ( c ) . Figure 2.

-1.04 1.06 1.08

1.1 1.12 1.14

1.04 1.08 1.08 1.1 1.12

1.04 1.08 1.08 1.1 1.12

T(Bu)/T(Bd)

Figure 3.

Same as Fig. 2 for T(B,)/T(B~).

breaking. We shall, however, assume that the matrix elements of both l / m i and l / m t operators respect isospin. The ratio of lifetimes of B, and Bd mesons involves a breaking of U-spin symmetry, so the matrix elements of dimension-6 operators could differ by about 30%. We shall introduce different B- and €-parameters to describe B, and Bd lifetimes. In order to obtain numerical estimate of the effect of l/mb corrections to spectator effects, we adopt the statistical approach for presenting our results and generate 20000-point probability distributions of the ratios of lifetimes obtained by randomly varying the parameters describing matrix elements within a &30% interval around their “factorization” values7, for three different scales p. The decay constants . f ~ and~b-quark pole mass mb are taken to vary within la interval indicated above. The results are presented in Figs. 2, 3, and 4. These figures represent our main result. We also performed studies of convergence of l/mb expansion by computing a set of l/mt-corrections to spectators effects and estimating their size in factorization7. The expansion appears to be well-convergent for the bflavored hadrons. Due to the relative smallness of m, (and thus vicinity

137

400

200 0 0.940.980.98

11.021.04 0.940.980.98

1 1.021.04 0.940.980.98

1 1.021.04

T(B*)/T(Bd)

Figure 4.

Same as Fig. 2 for T(B~)/T(B~).

of the sector of QCD populated by the light quark resonances'') it is not clear that the application of these findings to charmed hadrons will produce quantitative, rather than qualitative results. 4. Conclusions

We computed subleading l/mb and l / m t corrections to the spectator effects driving the difference in the lifetimes of heavy mesons and baryons. Thanks to the same 16r2 phase-space enhancement as l/mz-suppressed spectator effects, these corrections constitute the most important set of l/m:-suppressed corrections. The main result of this talk are Figs. 2, 3, and 4, which represent the effects of subleading spectator effects on the ratios of lifetimes of heavy mesons and baryons. We see that subleading corrections to spectator effects affect the ratio of heavy meson lifetimes only modestly, at the level of a fraction of a percent. On the other hand, the effect on the Ab-Bd lifetime ratio is quite substantial, at the level of -3%. This can be explained by the partial cancellation of WS and P I effects in hb baryon and constructive interference of l/mb corrections to the spectator effects. There is no theoretically-consistent way to translate the histograms of Figs. 2, 3, and 4 into numerical predictions for the lifetime ratios. As a useful estimate it is possible to fit the histograms to gaussian distributions and extract theoretical predictions for the mean values and deviations of the ratios of lifetimes. Predictions obtained this way should be treated with care, as it is not expected that the theoretical predictions are distributed according to the gaussian distribution. This being said, we proceed by fitting the distributions to gaussians and, correcting for the small scale uncertainty, extract the ratios T(B~)/T(B~) = 1.09 & 0.03, 7(Bs)/7(Bd) = 1.00 f 0.01, and T(Ab)/T(Bd) = 0.87 f 0.05. This brings the experimental

138

and theoretical ratios of baryon and meson lifetimes into agreement. I would like to thank F. Gabbiani and A. I. Onishchenko for collaboration on this project, and N. Uraltsev and M. Voloshin for helpful discussions. It is my pleasure to thank the organizers for the invitation to this wonderfully organized workshop. References I. I. Bigi et al., arXiv:hep-ph/9401298; M. B. Voloshin, arXiv:hep-ph/0004257. M. Neubert and C. T. Sachrajda, Nucl. Phys. B 483,339 (1997). J. L. Rosner, Phys. Lett. B 379,267 (1996). S. Eidelman et al., Phys. Lett. B592, 1 (2004); E. Barberio, presented at the Workshop on the CKM Unitary Triangle, http://ckm-workshop.web.cern.ch; M. Battaglia et al., arXiv:hep-ph/0304132. 5. J. Rademacker [On behalf of the CDF Collaboration], arXiv:hep-ex/0406021. See also: LEP B Lifetime Working Group, http://lepbosc.web.cern.ch/LEPBOSC/lifetimes/lepblife.html. 6. M. Ciuchini, E. Franco, V. Lubicz, and F. Mescia, Nucl. Phys. B 625, 211 (2002); E. Franco, V. Lubicz, F. Mescia, and C. Tarantino, ibid. 633, 212 (2002); M. Beneke, G. Buchalla, C. Greub, A. Lenz, and U. Nierste, Nucl. Phys. B 639,389 (2002). 7. F. Gabbiani, A. I. Onishchenko and A. A. Petrov, arXiv:hep-ph/0407004; F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, Phys. Rev. D 68,114006 (2003). 8. M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. 41,120 (1985) [Yad. Fiz. 41, 187 (1985)]; I. Bigi, M. Shifman, N. Uraltsev, and A. Vainshtein, Phys. Rev. Lett. 71,496 (1993); B. Guberina, S. Nussinov, R. D. Peccei, and R. Ruckl, Phys. Lett. B 89,111 (1979); N. Bilic, B. Guberina, and J. Trampetic, Nucl. Phys. B 248, 261 (1984); B. Guberina, R. Ruckl, and J. Trampetic, Z. Phys. C 33, 297 (1986); B. Guberina, B. Melic, and H. Stefancic, Phys. Lett. B 484,43 (2000). 9. M. Beneke, G. Buchalla, and I. Dunietz, Phys. Rev. D 54, 4419 (1996). M. Beneke, G. Buchalla, A. Lenz, and U. Nierste, Phys. Lett. B 576, 173 (2003). 10. J. Chay, A. F. Falk, M. E. Luke, and A. A. Petrov, Phys. Rev. D 61,034020 (2000); see also4 and M. Di Pierro and C. T. Sachrajda [UKQCD Collaboration], Nucl. Phys. B 534, 373 (1998); M. Di Pierro, C. T. Sachrajda, and C. Michael [UKQCD collaboration], Phys. Lett. B 468, 143 (1999); P. Colangel0 and F. De Fazio, ibid. 387,371 (1996); M. S. Baek, J. Lee, C. Liu, and H. S. Song, Phys. Rev. D 57,4091 (1998); J. G. Korner, A. I. Onishchenko, A. A. Petrov, and A. A. Pivovarov, Phys. Rev. Lett. 91,192002 (2003). 11. A. F. Falk et. al, Phys. Rev. D 69, 114021 (2004); A. F. Falk et. al, Phys. Rev. D 65,054034 (2002); E. Golowich and A. A. Petrov, Phys. Lett. B 427, 172 (1998); A. A. Petrov, Phys. Rev. D 56,1685 (1997).

1. 2. 3. 4.

INCLUSIVE B-DECAY SPECTRA AND IR RENORMALONS

E. GARDI Cavendish Laboratory, University of Cambridge Madingley Road, Cambridge, CB3 OHE, UK I illustrate the role of infrared renormalons in computing inclusive B-decay spectra. I explain the relation between the leading ambiguity in the definition of Sudakov form factor exp(NAlA4) and that of the pole mass, and show how these ambi-

-

guities cancel out between the perturbative and non-perturbative components of the b-quark distribution in the meson.

1. Introduction

-

B-decay physics is gradually turning into a field of precision phenomenology. Inclusive decay measurements provide some of the most robust tests of the standard model. Classical examples are the rate of B X,y decays [l] and constraints on the unitarity triangle through the measurement of Vub from charmless semileptonic decays [2]. The advantage of inclusive measurements over exclusive ones is that the corresponding theoretical predictions are, to large extent, free of hadronic uncertainties. QCD corrections to total decay rates are dominated by short distance scales, of order of the heavy-quark mass m, and are therefore primarily perturbative. Confinement effects appear as power corrections in A / m . Moreover, the Operator Product Expansion (OPE) allows one to estimate these power corrections by relating them to specific matrix elements of local operators between B-meson states, which are defined in the infinite-mass limit in the framework of the heavy-quark effective theory (HQET) [3]. These matrix elements can either be computed on the lattice or extracted from experimental data. In reality, however, experiments cannot perform completely inclusive measurements. Precise measurements are restricted t o certain kinematic regions where the background is sufficiently low. The experimentally accessible region in B X,y is where the photon energy E7 in the B rest frame is close t o its maximal possible value, M / 2 , (A4is the B meson mass),

-

139

140

-

or, equivalently, x = 2E,/M is near 1,which is the endpoint. Similarly, the accessible region in the CKM-suppressed B X,lV decay is where the lepton energy fraction is near maximal, or where the invariant mass of the hadronic system is small. Out of this region this decay mode is completely overshadowed by the decay into charm. As a consequence, precision phenomenology must rely on detailed theoretical understanding of the spectrum [4].Of particular importance is the spectrum near the endpoint. It turn out, however, that the endpoint region is theoretically much harder to access as both the perturbative expansion and the OPE break down there. In the large-z region gluon emission is restricted to be soft or collinear to the light-quark jet. While the associated singularities cancel with virtual corrections (decay spectra being infrared and collinear safe) large Sudakov logarithms of (1- x) appear in the expansion, which must therefore be resummed. Moreover, the OPE breaks down since the hierarchy between operators scaling with different powers of the mass is lost when (1 - z ) M become as small as the QCD scale. Physically this reflects the fact that the spectrum in the endpoint region is driven by the dynamics of the light degrees-of-freedom in the meson. The lightcone-momentum distributiona of the b-quark in the B-meson has a particularly important role in the endpoint region [5-131. It has been shown that up to subleading corrections O(A/m) the physical spectrum can be obtained as the convolution between a perturbatively-calculable coefficient function and the QDF, where the latter essentially determines the shape for x 1. The key point is that the QDF is a property of the B meson, not of the particular decay mode considered, so it can be measured in one decay and used in another. Moreover, a systematic analysis of the QDF in the HQET highlights the significance of a few specific parameters which constitute the first few moments of this function: most importantly = M - m, the difference between the meson mass and the quark pole mass, and then XI corresponding to the kinetic energy of the b quark in the meson. Nevertheless, the dependence of theoretical predictions for the spectra on the QDF is still a major source of uncertainty. Apart from identifying its first few moments, very little is known about this function, so the phenomenology of decay spectra in the immediate vicinity of the endpoint

-

*We shall define this function in full QCD, and call it Quark Distribution Function (QDF). This should be distinguished from the common practice to define it directly in the HQET, where the name “Shape Function” is often used.

-

141

(z 1) remains, to large extent, model dependent. On the other hand, successful precision phenomenology can well be expected for more moderate (yet large) z values, corresponding to the region where the distribution peaks. Here the main obstacle has been in combining [10,14] perturbative Sudakov effects with the HQET-based non-perturbative treatment discussed above. It has recently been shown [15] that the resolution of this problem is firmly connected with infrared renormalons (for general review of renormalons see [16]). Since the formulation of the HQET as well as the perturbative calculation of decay spectra rely on the concept of an on-shell heavy quark, both ingredients suffer from renormalon ambiguities. These ambiguities cancel out, of course, in the physical spectra. It is therefore useful to traceb the precise cancellation of ambiguities: the use of the HQET brings about dependence on the quark pole mass, which has a linear renormalon ambiguity [17-201. This ambiguity cancels against the leading renormalon ambiguity in the Sudakov exponent [15]. In order to achieve power-like separation between perturbative and non-perturbative contributions to decay spectra, one must therefore compute the Sudakov exponent as an asymptotic expansion, thus replacing the standard Sudakov resummation with fixed logarithmic accuracy by Dressed Gluon Exponentiation (DGE) [15,21-261. In what follows we illustrate the role of renormalons in the QCD description of decay spectra. We begin by briefly reviewing the HQET analysis for the QDF where we identify dependence on the quark pole mass. We recall that the pole mass suffers from an infrared renormalon ambiguity and show how this affects the QDF [15].We then consider inclusive B-meson decays within perturbation theory, review the relevant results on large-z factorization and Sudakov resummation [13], and then show that renormalon ambiguities appear in the Sudakov exponent [15], which, we emphasize, is a general phenomenon rather than a peculiarity of B decays. Finally, we combine the perturbative and non-perturbative ingredients recovering an unambiguous answer for the QDF in the meson and consequently for decay spectra. We conclude by shortly discussing the implications for precision phenomenology in inclusive decays.

bThis can be understood in analogy with factorization scale dependence, the main difference being that here the interest is in power terms.

142

2. Heavy-quark effective theory and the QDF

We define the QDF f ( z ;p ) as the Fourier transform of the forward hadronic matrix element of two heavy-quarks fields on the lightcone (y2 = 0):

where a path-ordered exponential between the fields is understood, p~ is the B-meson momentum ( p i = M 2 ) , z is the fraction of the momentum component carried by the b-quark field and p is the renormalization scale of the operator. Decay spectra can be computed as a convolution between a perturbatively calculable coefficient function and f ( z ; p ) . Let us first analyze f ( z ;p ) non-perturbatively - we denote it fNp(z)- suppressing any perturbative corrections. These will be recovered later on. Since the b-quark mass is large, the heavy quark is not far from its mass shell. This observation is the basis of the HQET. The momentum of the heavy quark is p = mu Ic where u is the hadron four velocity, v = p~ J M , and Ic is a residual momentum, llcl << m. The effective field is defined by scaling out the dependence on the quark mass: h,(z) = eimv.z;(1+$) @(x). It then follows from the definition (1) that in the heavy-quark limit [5-151

“+”

+

-

1

f2 . + -(-ZU. 2!

y)2

f3 . + -(-ZU. 3!

y) 3

+. . = 3 ( - i u . y), *

(2)

where we inverted the Fourier transform and defined A E M - m; in the second line we expanded the lightcone operator in the HQET in terms of local operators, where e.g. -3f2

= XI

1

E

-(B(Mu) 2M

Ih,(0)(gpv - U , V , ) ~ D ~ ~h,(0)1 D ” B ( M u ) ) . (3)

-

The HQET matrix elements fn O(Rn) do not depend on the definition of the mass. On the other hand the vanishing of the linear term in (-iv.y), fl = 0, (and the absence of additional, mass dependent terms in front of higher powers of ( 4 u . y ) ) in the second line of Eq. ( 2 ) are due to the HQET equation of motion for the heavy quark. Thus Eq. ( 2 ) relies on using the pole mass to define the HQET‘. ‘The use of the pole mass in the field redefinition can be avoided if a residual mass term is introduced. This, however, does not change any of the conclusions [15].

143

Expanding the exponential on the 1.h.s of Eq. (2) we obtain:

so the moments are fixed by the local matrix elements in the HQET. Mellin moments are defined by

=lo P l

FF

dZ.ZN-'fNp(Z).

(5)

For the first few Mellin moments we have F,Np = 1 and

It is apparent that all the moments depend on the quark pole mass. They satisfy M d F F / d m = ( N - 1 ) FF-l. At large N they are given by [15]

FNN P - e -(N-l)K/MF( ( N - 1 ) / M ) + W / N ) , (7) where the exponential factor depends on the pole mass through fi while F ( ( N - l ) / M ) , defined in (2), is entirely quark-mass independent. Note that large N corresponds to asymptotically large lightcone separations. 3. IR renormalon ambiguity in the pole mass

The result of the previous section indicates inherent dependence of the non-perturbative component of the QDF in the heavy-quark limit on A = M - m, or, equivalently, on the pole mass m. We recall that the pole mass is defined in perturbation theory by requiring that the inverse quark propagator $ - m z - C ( p ,m m ) vanishes a t p 2 = m2. At any given order in a, one can solve the resulting equation obtaining a unique relation between the pole mass and the MS mass (or any other renormalized short-distance mass). However, when considered to power accuracy this definition remains ambiguous [17-201. The on-shell condition brings about linear sensitivity t o long-distance scales. In the perturbative expansion (in schemes such as MS) this sensitivity translates into non-alternating factorial divergence making the sum of the series ambiguous - the well known infrared renormalon. Specifically, with a single dressed gluon - thus to leading order in the large-po limit - the relation between the pole mass and r n s is given by the following Bore1 sum [18]:

m -I--mm

PO

3e3u(1 - u)r(u)r(i -2 ~ ) 3 -4u

+R d u ) ] (8)

1

144

where Rcl (u)is free of singularities and Po = ~ C - gAN f . The singularity of the integrand a t u = $ translates into an O ( R / m ) ambiguity, which directly affects the QDF moments F F . As one would expect, the pole-mass renormalon ambiguity cancels out whenever the pole mass is used to compute an observable quantity. A wellknown example [19] is the calculation of the total semileptonic decay rate, which explicitly depends on the fifth power of the mass. Another example is the total energy of quarkonia [27]. Here we review this cancellation for the QDF in the meson and consequently for decay spectra [15].

4. Large-a: factorization in B decay

Let us now consider B-decay spectra in perturbation theory. By taking the initial state t o be an on-shell b quark we neglect non-perturbative effects associated with the meson structure. The decay rate is infrared and collinear safe, so the partonic calculation yields finite perturbative expansion when expressed in terms of the renormalized coupling and mass. It should be noted, however, that this finiteness is owing to cancellation of logarithmic singularities between real and virtual corrections. As usual, the singularity leaves a trace in the form of Sudakov logarithms of (1 - z). Since such logarithms appear a t any order in the perturbative expansion, they must be resummed. The resummation of Sudakov logarithms takes the form of exponentiation in Mellin space. This is a consequence of the factorization property of QCD matrix elements in the soft and collinear limits together with the factorization of phase space [13,29,30]. Up to 0 ( 1 / N ) corrections the perturbative expansion of inclusive decay spectra can be written in Mellin space as a product of three functions [13]: a soft function depending on m / N , a jet function depending on m 2 / N and a hard function depending on m - see Fig. 1. Furthermore, the resummation can be formulated as DGE, resumming running-coupling effects in the Sudakov exponent. The result is most conveniently expressed as a Bore1 sum. Explicitly, ford B X,y we have [15]:

-

dThe corresponding formula for the semileptonic decay appears in [15].

145

Hard

Jet

Hard

2

Figure 1. Large-a: factorization of inclusive decays into soft ( m / N ) , jet ( m 2 / N ) and hard (m) functions.

where

Here S ~ ( r np;) and J ~ ( r np;) are the soft and jet functions, respectivelye. These functions were both normalized to unity - the exponents vanish at N = 1 - so they acquire dependence on the hard scale. Bs(u),B g ( u )and B d ( U ) are Bore1 representations of anomalous dimensions of the soft, jet and cusp functions, respectively. In the large-Po limit

+

Bs(u) = ecu (1 - u) 0 ( 1 / P 0 ) ,

m.

where c = 5/3 in Beyond this limit the anomalous dimensions are known only as an expansion in u (through "LO). Terms that are subleading in 1//30,which appear first at O ( u l ) , are not small in QCD. The advantage of the Iarge-Po limit, where an analytic function is known, is that it allows one to verify the exact cancellation of renormalon ambiguities. e T ( u )depends [15]on the approximation used for the coupling T ( u )= 1.

function; for one-loop running

146 Wilson linc

4'-

On shell b quark Figure 2. Process-independent calculation of the QDF in an on-shell heavy quark in the large-po limit: the gluon is dressed by any number of fermion-loop insertions and -6po. In the A+ = 0 axial gauge only this diagram contributes. then N f

-

The soft function is the large-N limit of the lightcone momentum distribution of a b-quark field in an on-shell b quark. In can be computed in a process-independent manner from the QDF definition (l),replacing the external meson states IB(PB))by on-shell quark states Ib(p)). For example, the large-po limit result of Eqs. (10) and (12) can be obtained from the diagram of Fig. 2. It is related by crossing to the perturbative heavy-quark fragmentation function analyzed in [25]. The jet function describes the radiation associated with an unresolved jet of invariant mass m2/N. It is a universal object appearing in many observables including deep inelastic structure functions [22-241 , single-particle inclusive cross sections [22,25] and event-shape distribution [21,26,28]. In both the soft and jet functions there are renormalon ambiguities owing to the singularities of I'(-2u) and r'(-u), respectively. They appear as a result of integrating over the longitudinal momentum fraction near the endpoint and reflect the sensitivity of the exponent to large-distance scales through the running of the coupling. The ambiguity indicates the presence of non-perturbative power corrections at the exponent for each of the functions: powers of NA/m in the soft function and powers of NR2/m2 in the jet function. 5 . Cancellation of renormalon ambiguities in the exponent

When considered to power accuracy, the perturbative soft function of Eq. (10) becomes ambiguous. This is not surprising since its definition involves the on-shell quark state ( b ( p ) ) . Its non-perturbative analog, the QDF in the meson defined in Eq. (l),should be well defined. Yet, at large N

147

these two functions differ just by (an infinite set of) non-perturbative power corrections on the scale M / N :

F N ( M ; p )= SN(m;p)FY,

(14)

wheref FF is given by (7). The corresponding non-perturbative large-x factorization in B-meson decay is:

where the soIe difference from the perturbative formula (9) is the replacement of the QDF in the quark, S ~ ( r np;) , by that in the meson, F N ( M ;p ) . Since the QDF F N ( M ;p ) directly enters the measurable moments M N , it must be well defined. This will be the case only if renormalon ambiguities cancel in (14) between the perturbative and non-perturbative components. Indeed, such cancellation is expected because both SN(rn;p) and F F involve the concept of an on-shell heavy quark, while F N ( M ; ~does ) not. Putting together (7) and (10) we obtain [15]:

+ ($) B ~ ( u ) l n N ] }+ 0 ( 1 / N ) , 21

[

x Bs(u)r(-2u) ( N 2 u- 1 )

(16)

ambiguities in the which can be explicitly verified t o be free of u = large-po limit by substituting A = M - rn and using Eqs. (8) and ( 1 2 ) .

6. Prospects for precision phenomenology The results of Secs. 4 and 5 have direct implications for the calculation of inclusive decay spectra: they open up the way for consistent power-like separation between perturbative and non-perturbative contributions depending on N R / M . QCD predictions for decay spectra in the peak region require Sudakov resummation as well as non-perturbative corrections depending on the meson structure. However, conventional Sudakov resummation with fixed logarithmic accuracy does not deal with the problem of separation between perturbative and non-perturbative contributions (the immediate price is Landau singularities). The perturbative coefficients get significant contributions from small momentum scales. contributions that increase with ‘We systematically neglect 0(1/N), or, equivalently, O ( A / M )effects. Eq. (14) does not hold for small N.

148

increasing logarithmic accuracy [22]. DGE addressed the source of this problem. Using Borel summation - with a principal-value (PV) prescription for example - it systematically separates non-perturbative power-like terms in the Sudakov exponent from perturbative contributions. This procedure uniquely defines the non-perturbative power terms - this is precisely the meaning of Eq. (16): consider for simplicity the hypothetical case where 3 2~ 1 so fNp(z) N 6(z - rn/M). The shape of the QDF is then determined by the perturbative Sudakov form factor; it is just shifted non-perturbatively by A / M toward smaller z values (in the general case F leads to some smearing). Precise control of this shift is, of course, crucial; an ambiguity of order A in A would be a catastrophe. However, based on Eq. (16) A is uniquely fixed: if the principal value of the Borel sum is used t o define the perturbative Sudakov exponent, the same prescription must be used t o relate the pole mass to any (well measured) short-distance mass when computing A, so A = M - mpv. It should be emphasized that quantitative control of power-like contributions by means of Borel summation requires more information than available either from fixed-order calculations of the anomalous dimensions or from the large-po limit alone. For example, the value of Bs(u)near u = becomes relevant. While challenging, this question can still be addressed within perturbation theory.

Acknowledgments

I would like t o thank Gregory Korchemksy for illuminating discussions. This work is supported by a European Community Marie Curie Fellowship, HPMF-CT-2002-02112.

References 1. R. Barate et al. [ALEPH Collaboration], Phys. Lett. B429 (1998) 169; S. Chen et al. [CLEO Collaboration], Phys. Rev. Lett. 87 (2001) 251807 [hep-ex/0108032]; B. Aubert et al. [BaBar Collaboration], [hep-ex/0207076]; P. Koppenburg et al. [Belle Collaboration], [hep-ex/0403004]. 2. A. Bornheim et al. [CLEO Collaboration], Phys. Rev. Lett. 88 (2002) 231803 [hep-ex/0202019]; B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 92 (2004) 071802 [hep-ex/0307062]; B. Aubert et al. [BaBar Collaboration], Phys. Rev. D69, 111104 (2004) [hep-ex/0403030];H. Kakuno et al. [Belle Collaboration], Phys. Rev. Lett. 92 (2004) 101801 [hep-ex/0311048]. 3. M. Neubert, “Heavy-quark effective theory,” hep-ph/9610266. 4. M. Luke, “Applications of the heavy quark expansion: IV(ub)l and spectral moments,” eConf C0304052 (2003) WG107 [hep-ph/0307378].

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5. I. I. Y . Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein, Phys. Rev. Lett. 71 (1993) 496 [hep-ph/9304225]. 6. A. V. Manohar and M. B. Wise, Phys. Rev. D 49 (1994) 1310 [hepph/9308246]. 7. M. Neubert, Phys. Rev. D49 (1994) 3392 [hep-ph/9311325]. 8. A. F. Falk, E. Jenkins, A. V. Manohar and M. B. Wise, Phys. Rev. D 49 (1994) 4553 [hep-ph/9312306]. 9. M. Neubert, Phys. Rev. D49 (1994) 4623 [hep-ph/9312311]. 10. I. I. Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein, Int. J. Mod. Phys. A9 (1994) 2467 [hep-ph/9312359]. 11. T. Mannel and M. Neubert, Phys. Rev. D50 (1994) 2037 [hep-ph/9402288]. 12. A. L. Kagan and M. Neubert, Eur. Phys. J . C7 (1999) 5 [hep-ph/9805303]. 13. G. P. Korchemsky and G. Sterman, Phys. Lett. B340 (1994) 96 [hepph/9407344]. 14. S. W. Bosch, B. 0. Lange, M. Neubert and G. Paz, “Factorization and shapefunction effects in inclusive B-meson decays,’’ [hep-ph/0402094]. 15. E. Gardi, JHEP 0404,049 (2004) [hep-ph/0403249]. 16. M. Beneke, Phys. Rept. 317 (1999) 1; M. Beneke and V. M. Braun, “Renormalons and power corrections,”, in the Boris Ioffe Festschrift, At the Frontier of Particle Physics / Handbook of QCD, ed. M. Shifman (World Scientific, Singapore, 2001), vol. 3, p. 1719 [hep-ph/0010208]. 17. I. I. Y . Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein, Phys. Rev. D50 (1994) 2234 [hep-ph/9402360]. 18. M. Beneke and V. M. Braun, Nucl. Phys. B426 (1994) 301 [hep-ph/9402364]. 19. M. Beneke, V. M. Braun and V. I. Zakharov, Phys. Rev. Lett. 73 (1994) 3058 [hep-ph/9405304]. 20. M. Neubert and C. T. Sachrajda, Nucl. Phys. B438 (1995) 235 [hepph/9407394]. 21. E. Gardi and J. Rathsman, Nucl. Phys. B609 (2001) 123 [hep-ph/0103217]; Nucl. Phys. B638 (2002) 243 [hep-ph/0201019]. 22. E. Gardi, Nucl. Phys. B622 (2002) 365 [hep-ph/0108222]. 23. E. Gardi, G. P. Korchemsky, D. A. Ross and S. Tafat, Nucl. Phys. B636 (2002) 385 [hep-ph/0203161]. 24. E. Gardi and R. G. Roberts, Nucl. Phys. B653 (2003) 227 [hep-ph/0210429]. 25. M. Cacciari and E. Gardi, Nucl. Phys. B664 (2003) 299 [hep-ph/0301047]. 26. E. Gardi and L. Magnea, JHEP 0308 (2003) 030 [hep-ph/0306094]. 27. A. H. Hoang, M. C. Smith, T. Stelzer and S. Willenbrock, Phys. Rev. D59 (1999) 114014 [hep-ph/9804227]. 28. C. F. Berger and L. Magnea, “Scaling of power corrections for angularities from dressed gluon exponentiation,” [hep-ph/0407024]. 29. G. Sterman, Nucl. Phys. B281 (1987) 310; J . C. Collins, D. E. Soper and G. Sterman, Adv. Ser. Direct. High Energy Phys. 5 (1988) 1, published in ‘Perturbative QCD’, A.H. Mueller, ed. (World Scientific Publ., 1989). 30. S . Catani and L. Trentadue, Nucl. Phys. B327 (1989) 323; S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Nucl. Phys. B407 (1993) 3.

SUMMING LOGS OF THE VELOCITY IN NRQCD AND T O P THRESHOLD PHYSICS

ANDREH. HOANG Max-Planck-Institut fur Physik (Werner-Heisenberg-Institut), Fohringer Ring 6, 80805 Munchen, Germany E-mail: [email protected]

To achieve reliable predictions of the topantitop threshold cross section at a future e+e- Linear Collider logarithms of the top velocity need to be resummed. I review the issues that make this problem particularly complicated and show how the task can be achieved by renormalization in an effective theory using the so called velocity renormalization group. The most recent NNLL order results are discussed.

1. Introduction

The so-called “threshold scan” of the total cross section line-shape of top pair production constitutes a major part of the top quark physics program at a future e+e- collider. From the location of the rise of the cross section a precise measurement of the top quark mass with uncertainties at the level of 100 MeV will be possible, while from the shape and the normalization of the cross section one can extract the top quark Yukawa coupling yt (for a light Higgs), the top width or the strong coupling. In the threshold region, ,b N 2mt f 10 GeV, the top quarks move with nonrelativistic velocity in the c.m. frame. Due to the large top width, rt M 1.5 GeV, toponium resonances cannot form and the cross section is a smooth function of the c.m. energy. This also means that non-perturbative effects can be neglected for predictions of the total cross section. It is convenient to define the top velocity by mtu2 3 ,b - 2mt. Because in the loop expansion one encounters terms proportional to ( a , / ~from ) ~ the instantaneous exchange of n gluons, one has to count v as,and one has to carry out an expansion in as as well as in u.Schematically the perturbative expansion of the R-ratio for the total top-antitop threshold cross section,

-

150

151

R = atr/op+p-,has the form 03

R =u k=O

k

(5) x 21

{

1 (LO) ; a,, u (NLO) ; a:, a,v,u2 (NNLO)

}.

(1)

The expansion scheme in Eq. (1) is called fixed-order expansion, although it involves summations of the terms proportional to (a,/v)" t o all orders. The scheme can be implemented systematically using the factorization properties of Non-relativistic QCD (NRQCD) based on the zero-distance Green function of the non-relativistic Schrodinger equation. The NNLO QCD corrections to the total cross section were calculated already some time a g o . 4 ~ 5 ~ 6 ~ 7 However, ~ s ~ g ~ 1 0the corrections were found to be as large as the NLO corrections, and the normalization of the cross section was estimated to have a t least 20% theoretical uncertainty.12 It was shown l1>l2that this normalization uncertainty does not affect significantly the top mass measurements (in a threshold mass scheme) which is mainly sensitive to the energy were the cross section rises. However, it jeopardizes competitive measurements of the top width, strong top coupling, and the top Yukawa coupling. Figure l a shows the dominating vector-current-induced cross section o(e+ey* 4 tEj at LO, NLO and NNLO in the fixed-order expansion for typical choices of parameters and renormalization scales. One way to understand the large NNLO corrections is to recall that the tf pair a t threshold is non-relativistic and that its dynamics is governed by vastly different energy scales, the top mass (mt 175 GeV), the top threemomentum (p N mu 2: 25 GeV, soft) and the top kinetic or potential energy ( E N mu2 P 3 - 4GeV, ultrasoft). The top width, which protects against non-perturbative effects (and which is the main reason rendering the top threshold a precision observable) is of order E , but can be neglected for most of the following conceptual considerations. This hierarchy of scales is the basis of NRQCD factorization, which separates short-distance physics at the scale m from long-distance physics a t the non-relativistic scales p and E. NRQCD has one quantum field for each of the non-relativistic quarks and antiquark and the low-energy gluons. NRQCD matrix elements and Wilson coefficients therefore involve three types of logarithmic terms, ---f

N

but they cannot be rendered small (or summed) for any single choice of the renormalization scale p. This can be severe: for example, a,(mt) ln(mz/E2) 2i 0.8 for p = mt. One also cannot distinguish uniquely

152

the scale a t which to evaluate a, terms. These problems cannot be addressed systematically within NRQCD theory and a more sophisticated effective field theory (EFT) framework has to be devised which allows to resum all logarithmic terms. In this talk I discuss some of the conceptual aspects of an effective theory that is capable of summing consistently all logarithms of the velocity that can appear in the description of the non-relativistic quark-antiquark dynamics. The tool needed is called the velocity renormalization group (VRG) and it is the basis of a renormalization group improved perturbative expansion. To be specific I will concentrate mostly on an E F T known as vNRQCD. At the end I will come back to the top threshold cross section in order t o review the present status. 13114,15116.

2. The Proper Effective Theory

To begin with, it is useful t o remind us of the properties one desires from an effective theory of non-relativistic quark-antiquark pairs: (1) All IR (on-shell) fluctuations, including IR divergences, in the full theory are reproduced by EFT fields. This avoids large logarithms in matching conditions. (2) There is a well-defined and systematic power counting scheme (in w). This allows to uniquely identify what to compute a t a given order and renders all E F T loop being governed by one single scale. (3) There is a consistent renormalization prescription t o treat UV divergences and to formulate anomalous dimension which, eventually, will allow t o sum all "large" logarithmic terms. (4) All symmetries (spin, gauge, etc.) are implemented. This gives the most predictive power. (5) The Lagrangian is formulated in the regulator-independent way.

The original NRQCD mentioned above does actually not have property (2), because there is one single gluon field describing soft and ultrasoft gluon effects, while the power counting for soft and ultrasoft gluon effects differs. For the task of summing logarithms this is a severe obstacle. For example, soft gluons are participating in the binding of the quark pair while ultrasoft gluons are responsible for retardation effects such as the Lamb shift, and their interactions have t o be multipole expanded. Within dimensional regularization this also leads to problems with point (3). The situation is, however, even more complicated because, although w << 1 en-

153

sures that m >> mu >> mu2,the soft and ultrasoft scales are correlated by the dispersion relation of the quarks, E = p2/m. One approach to resolve these issues is to ignore the correlation of soft and ultrasoft scales at the beginning and to account for the scale hierarchy m >> m u >> mu2 in the traditional Wilsonian (step-wise) way. One starts with NRQCD having soft gluons and then “integrates out” effects at the scale mu. This results in non-local quark interactions, the potentials, and interactions with the remaining ultrasoft gluons. The approach is called pNRQCD l7 (p for “potential”) and naturally avoids any double counting of gluon effects. A v power counting only exists in pNRQCD, while there is no unique power counting in NRQCD. A particular property of this approach is that the correlation of the soft and ultrasoft scales in pNRQCD needs to be imposed by hand. In an alternative approach one accounts for the correlation of the soft and the ultrasoft scales from the very beginning which forbids using a stepwise approach for the hierarchy of the momentum and the energy scale. (Anyone disagreeing with this statement should step forward!) Rather, there is only one EFT below the scale m which simultaneously contains (v soft and ultrasoft gluons. This approach is called vNRQCD for “velocity”) and allows for a consistent power counting at all scales below m. There is an ultrasoft renormalization scale, p u , for loops dominated by the ultasoft energy scale, and a soft renormalization scale, p s , for loops dominated by the soft momentum scale. Both renormalization scales are correlated, p u = p i / m = mu2. The running of coefficients and operators in the EFT is then expressed in terms of the dimension-less scaling parameter u. The resulting renormalization group scaling, describing the correlated running coming from soft and ultrasoft effects is called the VRG l 3 and is now sometimes also referred to as the non-relativistic renormalization group in the literature. The effective vNRQCD Lagrangian (defined in the c.m. frame) is build xp),soft gluons, ghosts, from heavy potential quarks and antiquarks ($J~, and massless quarks ( A t , c, y,) and ultrasoft gluons, ghosts, and massless quarks (AP, c, y z l s ) .Double counting for the gluon effects is avoided since ultrasoft gluons reproduce only the physical gluon poles where ko k mu2, while soft gluons only have poles with ko k mu. All soft loops are made infrared-finite and at the same time all ultrasoft divergences in ultrasoft loops are made correspond to the hard scale m by the pull-up mechanism. l8 It is essential that both soft and ultrasoft gluons are included at all scales below m because the heavy quark equation of motion corre13,14315916

N

-

N

N

154

lates the soft and ultrasoft scales. The dependences on soft energies and momenta of the heavy quark and soft gluon fields appear as labels on the fields, while only the lowest-energy ultrasoft fluctuations are associated by an explicit coordinate dependence. Formally this is achieved by a phase redefinition for the potential and soft fields 13, $(z) + X I ,e-ik.z&(z), where k denotes momenta N mv and apq5k(z) mv2q5k(x). N

3. The Effective Theory Action

The effective vNRQCD Lagrangian for a tf angular momentum S-wave and color singlet state has terms 13914115

(P-~D)~ 2mt

+ . . .]Qp + (Ic, -+ X I }

P

where color and spin indices have been suppressed and g, = g,(mtv), gu = gs(mtv2).All coefficients are functions of the renormalization parameter v, and all explicit soft momentum labels are summed over. Massless quarks and ghost terms are not displayed and there are other terms needed for renormalization purposes 19,20 not shown here. In d = 4 - 26 dimensions powers of & and & are uniquely determined by the mass dimension and the v counting of each operator. The covariant derivative contains only the ultrasoft gluon field, DP = d@ ip;guA,. The first line contains the heavy quark kinetic terms and their interaction with ultrasoft gluons. There are 4-quark potential-like interactions of the form (k = (p - p’))

+

vc

V(p,p’) = -

+v k 2

VT.(p2+p’2)

v, + - + -s2, v 2

(4) k2 mlk( 2rn!k2 m: m: where S is the total tf spin operator. At NNLL order for the total cross section the coefficient V , of the l / k 2 +

potential has to be matched at two loops l8 because it contributes at the LL level, whereas the coefficients of the order l/m: potentials have t o matched

155

a t the Born level14. The l/(mlkl)-type potentials are of order a: and have to be matched a t two loops. l9 There are also 4-quark interactions with the radiation of an ultrasoft gluon (last line) and interactions between quarks and soft gluons (second line). Due t o momentum conservation a t least two soft gluons are required. The sum of the potential terms shown in Eq. (4) and time-ordered products of soft interactions contribute to the instantaneous interactions between the top quarks (that are traditionally called “potentials”). Particularly important for the phenomenological application are the effective operators involving the top quark width and the residual mass term Smt (first line). The width term arises by including the absorptive electroweak contribution from the W-b final state of the top 2-point function into the vNRQCD matching conditions a t v = 1. The width is of order E mu2 and has to be included in the heavy quark propagator, explicitly serving as an infrared cutoff. Practically it corresponds t o a shift of the c.m. energy into the positive complex plane by iI’t.21The residual mass term is non-zero in the threshold mass schemes, which avoids the order A Q ~ Drenormalon problem of the pole mass that would otherwise destabilize thc pcrturbative behavior of the location where the threshold cross section rises. l2 Besides the interactions contained in the effective Lagrangian that describe the dynamics of the tz pair we also need external currents that describe the production of the top quarks. For efe- annihilation these currents are induced by the exchange of a virtual photon or a Z boson. At NNLL order we need the vector S-wave currents JE = c1(v)OP,1 c2(v)OP,2, where Op,i = 1Clptn(im)xYplOp,2 = $$ptp2a(io2)x?, and the axial-vector P-wave current JE = cg(v)Op,gl where Op,3 = [a,n . p](il~2)xT_~. The currents O,J and op,glead to contributions in the total cross section that are v2-suppressed with respect to those of the current O,,J. Thus, a t NNLL order, two-loop matching is needed for c1 and Born level matching for c2 and cg.

-

+

G$,+

4. The Computations

The computations in the EFT are performed in the three steps, i) matching to QCD (including in principle electroweak interactions) a t p s = p u = m (v = 1); ii) determination of the anomalous dimensions and running with the VRG from v = 1 to v = V O , were vo is the typical heavy quark

156 velocity (WO M 0.15 for ti? threshold production); iii) computation of the EFT matrix elements a t Y = WO.

Conceptually, the most novel aspect of the computations is the summation of the logarithms of the correlated soft and ultrasoft scales using the VRG. Although technically these calculations are standard once renormalization is expressed in terms of the renormalization parameter v , they are non-trivial conceptually. This is because the correlation of soft and ultrasoft renormalization scales in the evolution is crucial to achieve the proper summation of logarithms. This was demonstrated in QED, 22 where higher order logarithmic terms could only be reproduced correctly using the VRG. A nice demonstration at the technical level about the difference between correlated and uncorrelated running was presented in. 23 It was also shown 16,20 that subdivergences in the NNLL (%loop) renormalization of the current Op,l could be treated consistently in the VRG, while a uncorrelated treatment of soft and ultrasoft renormalization scales fails. In fact, the necessity of the VRG is apparent whenever UV-divergences from potential and ultrasoft or soft loops contribute simultaneously to an anomalous dimension. Through steps i) and ii) all logarithmic terms involving w are summed into the operator coefficients and the matrix elements in iii) involve ln(v/v) terms that are not parametrically large. For the renormalization group improved ti? threshold cross section all necessary components in the steps i), ii) and iii) in vNRQCD are known a t NNLL order with the exception of step ii) for the current coefficient c1. The running of the potential coefficients Vi was determined in Refs. and of the current coefficients ci in Refs.24i19. (In Ref.19 a missing set of spin-independent operators was accounted for which affected the running of the spin-independent potential coefficients and, through mixing, the running of c1. The numerical change is insignificant for phenomenological applications.) Up to notational differences there is agreement with the potential coefficients obtained in pNRQCD. 25 The running of c1 is fully known a t NLL order, 24,19 while at NNLL order only the non-mixing contributions coming from genuine three-loop diagrams have been determined. 2o A non-trivial cross check for this computation, which checked the In u term in c1(v) was also carried out. 26 14918924,19

02

157 5 . The Top Pair Total Cross Section at Threshold The total cross section for eSe- -+ y*, Z* + tz at threshold at NNLL order in renormalization group improved perturbation theory has the form

4m2 Q&) = [ F"(s) R"(s)+ F a ( s ) R a ( s ) ], 3s

(5)

where F"va are trivial functions of the electric charges and the isospin of the electron and the top quark and of the weak mixing angle. The vector and axial1.6

,

I

,

1.4

1.2 1.0

0.4

0.2 346

347

348

349

350

351

352

353

354

352

353

354

fi(GeV) 1.6 1.4

1.2 1.0

0.4 0.2

346

347

348

349

350

351

&(Gel9 Figure 1. Panel a) shows the results for Q:Rv with M I s = 175 GeV and rt = 1.43 GeV in fixed-order perturbation theory at LO (dotted lines), NLO (dashed lines) and NNLO (solid lines). Panel b) shows the results for Q:Rv with the same parameters in renormalization group improved perturbation theory at LL (dotted lines), NLL (dashed lines) and NNLL (solid lines) order. For each order curves are plotted for v = 0.15, 0.20, and 0.3. The effects of initial state radiation, beamstrahlung and the beam energy spread at a e+e- collider are not included.

158 vector R-ratios have the form

R"(s) =

fIm [ c?(v) dl (w,m, v) f 2 ci (v)cz(v) .Az(v,mt,v) ] ,

(6)

I

~3(v)A3(w,mt,v) , (7) 47r S [ z where the di's are T-products of the effective theory currents described before. As mentioned before, c1 is only partly know at NNLL order. These T-products are related t o the zero-distance Green function of the equation of motion of the 4-quark 4-point functions obtained in the EFT. (At NNLL order this is a common 2-body Schrodinger equation, since higher Fock quark-antiquark-gluon states only contribute t o the renormalization of operators.) I use the 1s massgyz7since it is the threshold mass that is most closely related to the peak position visible in the theoretical prediction. Other threshold masses are also viable12 but not discussed here further. In Fig. l b I have displayed the photon induced cross section Q;R" at LL (dotted blue lines), NLL (dashed green lines) and NNLL (solid red lines) order for v = 0.15,0.2 and 0.3. Figure l a shows the corresponding results in fixed-order perturbation theory already discussed earlier. Compared to the fixed-order results with the same scales the improvement is substantial, particularly around the peak position and for smaller energies, but the somewhat inconvenient behavior that the NLL and the NNLL order predictions do not overlap remains. At present the uncertainty of the normalization of the total cross section is butE/utf N f 6 % due to this visibly large shift between the NLL and the NNLL order predictions. (A comparison of the NNLL prediction excluding the three-loop non-mixing contributions t o c1, which also demonstrated that the shift originates from the three-loop non-mixing contributions t o c1, can be found in Ref. 28.) In fact, an uncertainty not larger than 3% would be required t o have the theoretical errors of measurements of the top Yukawa coupling, rt and cy3 match with the ones presently expected from experiments. It is, however, premature to draw definite conclusion as long as the NNLL mixing contributions to c1 have not yet been determined.

Ra(s)= -Im

References 1. R. D. Heuer, D. J. Miller, F. Richard and P. M. Zerwas, hep-ph/0106315.

T. Abe et al. [American Linear Collider Working Group Collaboration], in Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001) , ed. N. Graf, arXiv:hep-ex/0106057. 2. M. Martinez and R. Miquel, Eur. Phys. J. C 27, 49 (2003) [arXiv:hepph/0207315]. 3. W.E. Caswell and G.P. Lepage, Phys. Lett. 167B,437 (1986). G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D51, 1125 (1995), ibid. D55,5853 (1997).

159 4. A. H. Hoang and T. Teubner, Phys. Rev. D58, 114023 (1998) [hepph/9801397]. 5. K. Melnikov and A. Yelkhovsky, Nucl. Phys. B528, 59 (1998) [hepph/9802379]. 6. 0. Yakovlev, Phys. Lett. B457, 170 (1999) [hep-ph/9808463]. 7. M. Beneke, A. Signer and V. A. Smirnov, Phys. Lett. B454, 137 (1999) [hep-ph/9903260]. 8. T. Nagano, A. Ota and Y. Sumino, Phys. Rev. D60, 114014 (1999) [hepph/9903498]. 9. A. H. Hoang and T. Teubner, Phys. Rev. D60, 114027 (1999) [hepph/9904468]. 10. A. A. Penin and A. A. Pivovarov, Nucl. Phys. B550, 375 (1999) [hepph/9810496]; Phys. Atom. Nucl. 64, 275 (2001) [Yad. Fiz. 64, 323 (2001)] [hep-ph/9904278]. 11. D. Peralta, M. Martinez and R. Miquel, talk presented at the 4th Znternational Workshop on Linear Colliders, Sitges, Barcelona, Spain, April 28 May 5 1999. 12. A. H. Hoang et al., in Eur. Phys. J. direct C3, 1 (2000) [hep-ph/0001286]. 13. M. Luke, A. Manohar and I. Rothstein, Phys. Rev. D61, 074025 (2000) [arXiv:hep-ph/9910209]. 14. A.V. Manohar and I.W. Stewart, Phys. Rev. D 62, 014033 (2000) [arXiv:hepph/9912226]. 15. A.V. Manohar and I.W. Stewart, Phys. Rev. D62,074015 (2000) [arXiv:hepph/0003032]. 16. A. H. Hoang, A. V. Manohar, I. W. Stewart and T. Teubner, Phys. Rev. D 65, 014014 (2002) [arXiv:hep-ph/0107144]. 17. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B 566, 275 (2000) [arXiv:hep-ph/9907240]. 18. A.H. Hoang, A.V. Manohar and I.W. Stewart, Phys. Rev. D 64, 014033 (2001) [arXiv:hep-ph/0102257]. 19. A. H. Hoang and I. W. Stewart, Phys. Rev. D 67, 114020 (2003) [arXiv:hepph/0209340]. 20. A. H. Hoang, arXiv:hep-ph/0307376. 21. V. S. Fadin and V. A. Khoze, JETP Lett. 46, 525 (1987). 22. A. V. Manohar and I. W. Stewart, Phys. Rev. Lett. 85, 2248 (2000) arXiv:hep-ph/00004018; 23. A. V. Manohar, J. Soto and I. W. Stewart, Phys. Lett. B 486, 400 (2000) arXiv:hep-ph/0006096; 24. A.V. Manohar and I.W. Stewart, Phys. Rev. D63,054004 (2001). [arXiv:hepph/0003107]. 25. A. Pineda, Phys. Rev. D 65, 074007 (2002) [arXiv:hep-ph/0109117]. 26. B. A. Kniehl, A. A. Penin, M. Steinhauser and V. A. Smirnov, Phys. Rev. Lett. 90, 212001 (2003) [arXiv:hep-ph/0210161]. 27. A. H. Hoang, Z. Ligeti and A. V. Manohar, Phys. Rev. Lett. 82, 277 (1999) [hep-ph/9809423]; Phys. Rev. D59, 074017 (1999) [hep-ph/9811239]. 28. A. H. Hoang, Acta Phys. Polon. 34, 4491 (2003) [arXiv:he~-ph/0310301].

B + T ,K ,

DECAY FORMFACTORS FROM LIGHT-CONE SUM RULES

P. BALL Department of Physics University of Durham Durham DH1 3LE, UK E-mail: Patricia. [email protected]. uk

R.ZWICKY William I. Fine Theoretical Physics Institute University of Minnesota Minneapolis, M N 55455, USA E-mail: [email protected] We present an improved calculation of all B -+ light pseudoscalar formfactors from light-cone sum rules, including one-loop radiative corrections t o twist-2 and twist-3 contributions, and leading order twist-4 corrections. The total theoretical uncertainty of our results at zero momentum transfer is 10 to 13%. The dependence of the formfactors on the momentum transfer q2 is parametrized in a simple way that is consistent with their analytical properties and is valid for all physical q 2 . The uncertainty of the extrapolation in q2 on the semileptonic decay rate I'(B -+ r e v ) is estimated to be 5%.

1. Introduction and Definitions In a recent paper' we have reported a new calculation of B + 7r, K , Q decay formfactors from QCD sum rules on the light-cone (LCSRS). The paper improves upon our previous publication^^^^ by: 0

0 0 0

including radiative corrections to twist-3 contributions to one-loop accuracy, for all formfactors; a precisely defined method for fixing sum rule specific parameters; using updated values for input parameters; a careful analysis of the uncertainties of the formfactors a t zero momentum transfer; a new parametrization of the dependence of the formfactors on

160

161

momentum transfer, which is consistent with the constraints from analyticity and heavy-quark expansion; a detailed breakdown of the dependence of formfactors on nonperturbative hadronic parameters describing the T , K , r] mesons, which facilitates the incorporation of future updates of their numerical values and also allows a consistent treatment of their effect on nonleptonic decays. The key idea of LCSRs is to consider a correlation function of the weak current and a current with the quantum-numbers of the B meson, sandwiched between the vacuum and, in the present context, the pseudoscalar meson P , i.e. T , K and r]. For large (negative) virtualities of these currents, the correlation function is, in coordinate-space, dominated by distances close to the light-cone and can be discussed in the framework of light-cone expansion. In contrast to the short-distance expansion employed by conventional QCD sum rules A la SVZ4, where nonperturbative effects are encoded in vacuum expectation values of local operators with vacuum quantum numbers, the condensates, LCSRs rely on the factorisation of the underlying correlation function into genuinely nonperturbative and universal hadron distribution amplitudes (DAs) 4 that are convoluted with process-dependent amplitudes T , which are the analogues to the Wilson-coefficients in the short-distance expansion and can be calculated in perturbation theory. Schematically, one has correlation function

C T ( ~ )4(n).

(1)

n

The sum runs over contributions with increasing twist, labelled by n, which are suppressed by increasing powers of, roughly speaking, the virtualities of the involved currents. The same correlation function can, on the other hand, be written as a dispersion-relation, in the virtuality of the current coupling to the B meson. Equating dispersion-representation and the lightcone expansion, and separating the B meson contribution from that of higher one- and multi-particle states, one obtains a relation (QCD sum rule) for the formfactor describing the B + P transition. The particular strength of LCSRs lies in the fact that they allow the inclusion not only of hard-gluon exchange contributions, which have been identified, in the seminal papers that opened the study of hard exclusive processes in the framework of perturbative QCD (PQCD)~,as being dominant in light-meson form factors, but that they also capture the so-called Feynman-mechanism, where the quark created at the weak vertex carries

162

nearly all momentum of the meson in the final state, while all other quarks are soft. This mechanism is suppressed by two powers of momentumtransfer in processes with light mesons, but there is no suppression in heavy-to-light transitions6, and hence any reasonable application of pQCD to B meson decays should include this mechanism. It is precisely LCSRS that accomplish this task and have been applied to a variety of problems in heavy-meson p h y s i c ~ . ~ A l ~more > ~ > detailed ~ discussion of the rationale of LCSRS and of the more technical aspects of the method can be found e.g. in Ref.8. The formfactors in question can be defined as ( q = p~ - p )

The starting point for the calculation of e.g. f; is the correlation function

i

/

d4YeiqY(~(p)lT[~,Y~b](y)[mbbzysq](0))0) =

+ ...,

(4)

where the dots stand for other Lorentz structures. For a certain configuration of virtualities, namely mg - p i 2 O ( A Q c D m b ) and mi - q2 L O ( h Q c D m b ) , the integral is dominated by light-like distances and can be expanded around the light-cone:

As in Eq. (l),n labels the twist of operators and p~ denotes the factorisation scale. The restriction on q 2 , mz - q2 2 O ( L i Q c D m b ) , implies that is not accessible at all momentum-transfers; to be specific, we restrict ourselves to 0 5 q2 5 14GeV2. As 11+ is independent of p ~ the , above formula implies that the scale-dependence of T(n)must be canceled by that of the DAs # ( n ) . In Eq. (5) it is assumed that 11+ can be described by collinear factorisation, i.e. that the only relevant degrees of freedom are the longitudinal momentum fractions u carried by the partons in the T , and that transverse momenta can be integrated over. Hard infrared (collinear) divergences occurring in T(")should be absorbable into the DAs. Collinear factorisation is trivial at tree-level, where the b quark mass acts effectively as regulator, but

f?

163 can, in principle, be violated by radiative corrections, by so-called "soft" divergent terms, which yield divergences upon integration over u.Actually, however, it turns out that for all formfactors calculated in Ref.' the T are nonsingular a t the endpoints u = 0,1, so there are n o soft divergences, independent of the end-point behavior of the distribution amplitudes. In Ref.' Eq. ( 5 ) has been demonstrated to be valid t o O(a,) accuracy for twist-2 and twist-3 contributions for all correlation functions II+,o,Tfrom which to determine the formfactors f+,O,T. As for the distribution amplitudes (DAs), they have been discussed intensively in the literatureg. For pseudoscalar mesons, there is only one DA of leading-twist, i.e. twist-2, which is defined by the following light-cone matrix element ( x 2 = 0):

-XI

where C = 2u - 1 and we have suppressed the Wilson-line [ x , to ensure gauge-invariance. The higher-twist DAs are of type

needed

where u is a number between 0 and 1 and r a combination of Dirac matrices. The sum rule calculations performed in Refs.ll2y3include all contributions from DAs up t o twist-4. The DAs are parametrized by their partial wave expansion in conformal spin, which t o NLO provides a controlled and economic expansion in terms of only a few hadronic parametersg. The LCSR for f; is derived in the following way: the correlation function II+,calculated for unphysical p i , can be written as dispersion relation over its physical cut. Singling out the contribution of the B meson, one has

H+ = f ; ( q 2 )

m;fB 2

mg

-Pi

+ higher poles and cuts,

(7)

where f B is the leptonic decay constant of the B meson, fBm$ = mb(Bl6iysdlO). In the framework of LCSRs one does not use (7) as it stands, but performs a Borel transformation, l / ( t - p i ) + 2l / ( t - p $ ) = 1/M2 exp(-t/M2), with the Borel parameter M 2 ; this transformation enhances the ground-state B meson contribution to the dispersion representation of II+ and suppresses contributions of higher twist to the light-cone expansion of II+. The next step is to invoke quark-hadron duality to approximate the contributions of hadrons other than the ground-state B me-

164

son by the imaginary part of the light-cone expansion of II+, so that

Subtracting the 2nd term on the right-hand side from both sides, one obtains

Eq. (9) is the LCSR for f;. SO is the so-called continuum threshold, which separates the ground-state from the continuum contribution. At tree-level, the continuum-subtraction in (9) introduces a lower limit of integration in u,the momentum fraction of the quark in the 7r: u 2 (m: - q 2 ) / ( s o - q 2 ) , in (5), which behaves as 1 - h q c D / m b for large m b and thus corresponds t o the dynamical configuration of the Feynman-mechanism, as it cuts off low momenta of the u quark created a t the weak vertex. At O ( a s ) ,there are also contributions with no cut in the integration over u,which correspond to hard-gluon exchange contributions. The task now is to find sets of parameters M 2 (the Bore1 parameter) and SO (the continuum threshold) such that the resulting formfactor does not depend too much on the precise values of these parameters. 2. Results For a detailed discussion of the procedure used to determine the hadronic and sum rule specific input parameters we refer to Ref.l. One main feature is that f ~ the , decay constant of the B meson entering Eq. (9) is calculated from a sum rule itselflo, which reduces the dependence of the resulting formfactors on the input parameters, in particular mb, which is the one-loop pole mass and taken to be (4.80 f 0.05) GeV. This procedure does not, however, reduce the formfactors’ dependence on the parameters describing the twist-2 DAs, which turns out to be rather crucial. Despite much effort spent on both their calculation from first principles and their extraction from experimental data, these so-called Gegenbauer moments, a1 (only for K ) , a2 and a4 (for all P ) are not known very precisely. Figure 1 shows the dependence of f+(O) on a2 and a4; the dots represent different determinations of these parameters and illustrate the resulting spread in values of the formfactor. The situation is even more disadvantageous for

165

Figure 1. Dependence of fT(0) on a2 and a 4 , for central values of input parameters. The lines are lines of constant fT(0). The dot labeled BZ denotes our preferred values of a2,4, BMS the values from the nonlocal condensate modelll and BF from sum rule caIculationsQ.

Figure 2. (a) Dependence of f f ( 0 ) on the Gegenbauer moment a l . (b) f 2 ( q 2 ) as function of q2 for different values of a l : solid line: a? = 0.17, short dashes: uf = 0, long dashes: a? = -0.18.

the K , whose formfactors depend on the SU(3) breaking parameter u f , whose size and even sign are under discussion12: at present, values as different as -0.18 and +0.17 (at p = 1GeV) are being quoted. Figure 2 shows the dependencies of (a) f$(O) and (b) f F ( q 2 ) on this parameter; evidently it is very important to determine its value more precisely. Summarizing the detailed analysis of the uncertainties induced by both external input and LCSR parameters, the final results for the formfactors at zero momentum transfer obtained in Ref.l are:

fT(0) = 0.258 f0.031, f+K(O) = 0.331 f 0.041 0.256a1, fT(0) = 0.275 f0.036,

+

f;(O)

= 0.253 f 0.028,

f$(O)

= 0.358 f 0.037

f;(O)

= 0.285 f 0.029.

+ 0.31&, ,

daI is defined as u f ( 1 GeV) - 0.17, i.e. the deviation of uf from the central value used in Ref.l. For f T > q the total theoretical uncertainty ranges between 10% to 13%, for f K it is 12%, plus the uncertainty in u l , which hopefully will be clarified through an independent calculation in the not

166

Figure 3. f+ (solid lines), fo (short dashes) and f~ (long dashes) as functions of q2 for n, K and q. The renormalisation scale of f T is chosen t o be m b .

too far future. The intrinsic, irreducible uncertainty of the sum rule calculation is related to the dependence of the result on the sum rule specific parameters M 2 and SO and estimated to be 7%. Turning t o the q2-dependence of formfactors, it has to be recalled that LCSRS are only valid if the energy Ep of the final state meson, measured in the rest frame of the decaying B , is large, i.e. if q2 = m i - 2 r n ~ E pis not too large; specifically, we choose Ep > 1.3 GeV, i.e. q2 5 14 GeV2. The resulting formfactors are plotted in Fig. 3, using central values for the input parameters. In order to allow a simple implementation of these results in actual applications, and also in order to provide predictions for the full physical regime 0 5 q2 5 (mB - m p ) 2 M 25GeV2, it is necessary to find parametrizations of f ( q 2 ) that N

0

reproduce the data below 14 GeV2 with good accuracy; provide a n extrapolation to q2 > 14GeV2 that is consistent with the expected analytical properties of the formfactors and reproduces the lowest-lying resonance (pole) with J p = 1- for f+ and fT .a

aFor fo, the lowest pole with quantum numbers O+ lies above the two-particle threshold

167 Table 1. Fit parameters for f ( q 2 ) . ml is the vector m e o n mass in the corresponding channel: m:9q = mgr = 5.32 GeV and mf = mB; = 5.41 GeV. The scale of fT is p = 4.8 GeV.

f;

I Tl 1 0.744 0 1.387 0.162 0 0.161 0.122 0 0.111

rz -0.486 0.258 -1.134 0.173 0.330 0.198 0.155 0.273 0.175

(m~)'

(my)' (m?)'

(rn7)'

m?, 40.73 33.81 32.22

37.46 (mf)2 (rn?)' 31.03 (m:)2

As shown in Ref.', the following parametrizations are appropriate: 0

for f-;,T:

0

where my is the mass of B*(l-), my = 5.32 GeV; the fit parameters are r-1, 1-2 and mfit; for f+,;.: Kv

0

where ml is the mass of the 1- meson in the corresponding channel, i.e. 5.32 GeV for 77 and 5.41 GeV for K; the fit parameters are r1 and 7-2; for fo:

the fit parameters are 7-2 and mfit. The central results for the fit parameters are collected in Tab. 1. The quality of all fits is very good and the maximum deviation between LCSR and fitted result is 2% or better. The impact of the extrapolation of the fit formulas to q2 > 14GeV2 is of phenomenological relevance mainly for B --+ r e v , relevant for the determination of lVubl from experiment. We have starting at (mB

+ mp)'

and hence is not expected to feature prominently.

168

estimated the effect of the extrapolation on the decay rate by implementing different parametrisations for f;, which all fit the LCSR result very well for q2 < 14 GeV2, but differ for larger q 2 , the main distinguishing feature being the positions of the poles. We find that for reasonable parametrisations, that is such that do not exhibit too strong a singularity a t q2 = mp, the total rates differ by not more than 5%, the difference becoming smaller if an cut-off on the maximum invariant mass of the lepton pair is implemented, which implies that the extrapolation is well under control.

3. Summary & Conclusions LCSRs provide accurate results for weak decay formfactors of the B meson into light mesons, in particular 7r, K and q. The results depend on sum rule specific input parameters which generate an irreducible “systematic” uncertainty of the approach estimated to be 7%. Additional uncertainties are induced by imprecisely known hadronic input parameters, in particular the Gegenbauer moments a1,2,4 describing the leading-twist light-meson distribution amplitudes. An improved determination of these parameters would be very welcome. The present total uncertainty of the formfactors a t zero momentum transfer varies between 10 and 13%, but becomes smaller a t larger q 2 . LCSR calculations require the energy of the final state meson to be large in the rest-frame of the decaying B and hence are valid only for not too large momentum transfer q2; the maximum eligible q2 is chosen to be 14 GeV2. The q2-dependence of the formfactors can be cast into simple parametrizations in terms of two or three parameters, which also capture the main features of the analytical structure and are expected to be valid in the full kinematical regime 0 5 q2 5 ( m-~ m ~ )The ~ total . uncertainty introduced by the extrapolation of the formfactors to q2 larger than the sum rule cut-off 14 GeV2 is estimated to be 5% for the semileptonic rate N

N

r ( B t 7rev). Ref.l also contains a detailed breakdown of the dependence of the formfactors on the Gegenbauer moments, which not only allows one to recalculate the formfactors once these parameters are determined more precisely, but also makes it possible t o consistently assess their impact on nonleptonic decay amplitudes ( e g . B + m )treated in QCD factorisation. The LCSR approach is complementary to standard lattice calculations, in the sense that it works best for large energies of the final state meson (i.e. small q 2 ) , whereas lattice calculations work best for small energies - a situation that may change in the future with the implementation of moving

169

NRQCD13. Previously, the LCSR results for f-;,o at small and moderate q2 were found to nicely match14 t h e lattice results obtained for large q2. T h e situation will have to be reassessed in view of our new results and i t will be very interesting to see if a n d how i t will develop with further progress in both lattice a n d LCSR calculations. References 1. 2. 3. 4.

5.

6. 7.

8. 9. 10.

11. 12.

13. 14.

P. Ball and R. Zwicky, arXiv:hep-ph/0406232. P. Ball, JHEP 9809 (1998) 005 [hep-ph/9802394]. P. Ball and R. Zwicky, JHEP 0110 (2001) 019 [arXiv:hep-ph/0110115]. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385; ibd. 147 (1979) 448. V.L. Chernyak and A.R. Zhitnitsky, J E T P Lett. 25 (1977) 510 [Pisma Zh. Eksp. Teor. Fiz. 25 (1977) 5441; Sov. J. Nucl. Phys. 31 (1980) 544 [Yad. Fiz. 31 (1980) 10531; A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94 (1980) 245; Theor. Math. Phys. 42 (1980) 97 [Teor. Mat. Fiz. 42 (1980) 1471; G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87 (1979) 359; Phys. Rev. D 22 (1980) 2157; V.L. Chernyak, A.R. Zhitnitsky and V.G. Serbo, J E T P Lett. 26 (1977) 594 [Pisma Zh. Eksp. Teor. Fiz. 26 (1977) 7601; Sov. J. Nucl. Phys. 31 (1980) 552 [Yad. Fiz. 31 (1980) 10691. V.L. Chernyak and I.R. Zhitnitsky, Nucl. Phys. B 345 (1990) 137. P. Ball and V.M. Braun, Phys. Rev. D 55 (1997) 5561 [arXiv:hepph/9701238]; A. Khodjamirian et al., Phys. Lett. B 402 (1997) 167 [arXiv:hep-ph/9702318]; E. Bagan, P. Ball and V.M. Braun, Phys. Lett. B 417 (1998) 154 [hep-ph/9709243]; A. Khodjamirian, R. Ruck1 and C.W. Winhart, Phys. Rev. D 58 (1998) 054013 [arXiv:hep-ph/9802412]; P. Ball and V. M. Braun, Phys. Rev. D 58 (1998) 094016 [arXiv:hep-ph/9805422]; A. Khodjamirian et al., Phys. Rev. D 62 (2000) 114002 [arXiv:hepph/0001297]; P. Ball and E. Kou, JHEP 0304 (2003) 029 [arXiv:hepph/0301135]; P. Ball, arXiv:hep-ph/0308249. P. Colangelo and A. Khodjamirian, hep-ph/0010175; A. Khodjamirian, hepph/0108205. V. M. Braun and I. E. Filyanov, Z. Phys. C 44, 157 (1989) [Sov. J. Nucl. Phys. 501; P. Ball, JHEP 9901 (1999) 010 [hep-ph/9812375]. For instance, T. M. Aliev and V. L. Eletsky, Sov. J. Nucl. Phys. 38 (1983) 936 [Yad. Fiz. 38 (1983) 15371; E. Bagan et al., Phys. Lett. B 278 (1992) 457. A. P. Bakulev et al., arXiv:hep-ph/0405062. P. Ball and M. Boglione, Phys. Rev. D 68 (2003) 094006 [arXiv:hepph/0307337]. C.H. Davies, talk at UK BaBar meeting, Durham April 2004. D. Becirevic, arXiv:hep-ph/0211340.

UNDERSTANDING 0; (2317), O,j(2460)

F. DE FAZIO Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Orabona, 4 I-70186 Bari, ITALY E-mail: fuluia. defazioQba.infn.it

I briefly review the experimental observations concerning the charmed mesons D:J(2317), 0,~(2460) and survey on some of the interpretations proposed in order to understand their nature. I present an analysis of their decay modes in the hypothesis that they can be identified with the scalar and axial vector sf = states of cS spectrum (D:o, Oil). The method is based on heavy quark symmetries and Vector Meson Dominance ansatz. Comparison with present data supports the interpretation.

4'

1. Introduction

In April 2003, the BaBar Collaboration reported the observation of a narrow peak in the D$r0 invariant mass distribution, corresponding to a state of mass 2317 MeV, denoted as DtJ(2317) The state is produced from charm continuum and the observed width is consistent with the resolution of the detector, I? 5 10 MeV. A possible quantum number assignment to D,*,(2317) is J p = Of, as suggested by the angular distribution of the meson decay with respect to its direction in the e+ - e- center of mass frame. This assignment can identify the meson with the scalar D:o state in the spectrum of the cS system. Considering the masses of the other observed states belonging to the same system, D,1(2536) and D,~ (2 5 7 3 ),the mass of the scalar DZ0 meson was expected in the range 2.45 - 2.5 GeV, therefore 150 MeV higher than the observed 2.317 GeV. A D;, meson with such a large mass would be above the threshold M D K = 2.359 GeV to strongly decay by S-wave K m n emission to D K , with a consequent broad width. For a mass below the D K threshold the meson has to decay by different modes, namely the isospinviolating D,7ro mode observed by BaBar, or radiatively. The J p = O+ assignment excludes the final state D,y, due to angular momentum and

'.

-

170

171

parity conservation; indeed such a final state has not been observed. On the other hand, for a scalar cs meson the decay DQ0 4 D,*y is allowed. However, no evidence is reported yet of the D,yy final state resulting from the decay chain D:o t DGy + D,yy. Later on, in May 2003, CLEO Collaboration confirmed the BaBar observation of D,*,(2317) with the same features outlined above; furthermore, CLEO Collaboration observed a second narrow peak, corresponding to a state with mass 2460 MeV decaying t o DGxo '. Again, the width is compatible with the detector resolution. Evidence of this second state was present in the first analysis by BaBar Collaboration, which gave subsequent confirmation of the CLEO observation3. BELLE Collaboration has confirmed both states4, observing their production both from charm continuum, both in B decays; more recently also FOCUS Collaboration5 has detected of a narrow peak at 2323 f 2 MeV, slightly above the values obtained by the other three experiments for Di,(2317). The observation of the decay D,~ (2 4 6 0 ) -+ D,*xo suggests that D,~(2460)has J p = I f . This assignment is supported also by the observation of the mode D,~ (2 4 6 0 )-+ D,y, forbidden to a O+ state, and by the angular analysis performed by BELLE6. Such an analysis was carried out for D,~(2460)produced in B decays and favours the identification of D,~ (2 460)with an axial-vector particle. Production of D0,5(2460)in B decays was observed also by BaBar'. However, as in the case of D,*,(2317), the measured mass is below theoretical expectations for the 1+ cg state DLl and the narrow width contrasts with the expected broadness of the latter. These peculiar features of D,*,(2317) and D,~ (2 4 6 0 )have prompted a number of analyses, aimed either at refining previous results in order to , at explaining support the cg interpretation of D,*,(2317) and D,~ ( 2 4 6 0 )or their nature in a different context. The various interpretations are reviewed in Ref. 8. Among the non standard scenarios, it has been often considered the possibility of a sizeable four-quark component in D,*,(2317) and D,J( 2460). Four-quark states could be baryonum-like or molecular-like, if they result from bound states of quarks or of hadrons, respectively, and examples of the ) insecond kind of states are the often discussed fo(980) and ~ ( 9 8 0 when terpreted as K r molecules. In the molecular interpretation, the D,*,(2317) could be viewed as a D K moleculeg, an interpretation supported by the fact that the mass 2.317 GeV is close to the DK threshold, or as a D,x atomlo. Analogously, D , ~( 246 0 )would be a D * K molecule. Mixing between ordinary cS state and a composite state has also been considered". No definite

172

answer comes from lattice QCD, since, according t o Ref. 12, lattice predictions are inconsistent with the simple qtj interpretation for DiJ(2317), while in Ref. 13 no exotic scenario is invoked to interpret this state. QCD sum rules are compatible with the cB interpretati~n'~. To understand the structure of a particle one needs to analyse its decay modes under definite assumptions and compare the result with the experimental measurements. In the following we present an analysis based on such a strategy t o discuss whether the identification of D,',(2317) and D , ~ ( 2 4 6 0 )with the two states (D,*,,DL,) is supported by data. To this end, we compute the decay modes of a scalar and an axial-vector particle with masses of 2317 MeV and 2460 MeV respectively, and check whether they can be predicted in agreement with the experimental findings presently available. In particular, the isospin violating decays t o D$*).rroshould proceed a t a rate larger than the radiative modes, though not exceeding the experimental upper bounds on the total widths. 2. Hadronic Modes In order to analyze the isospin violating transitions DHo -+ D,ro and DLl t D,'ro, one can use a formalism that accounts for the heavy quark spin-flavour symmetries in hadrons containing a single heavy quark, and the chiral symmetry in the interaction with the octet of light pseudoscalar states. In the heavy quark limit, the heavy quark spin ZQ and the light degrees of freedom total angular momentum Z l are separately conserved. This allows t o classify hadrons with a single heavy quark Q in terms of se by collecting them in doublets the members of which only differ for the relative orientation of ZQ and Ze. The doublets with J p = (0-, 1-) and J p = (0+, 1+) (corresponding to = 1- and = ;+, respectively) can be described by the effective fields

SF

SF

where v is the four-velocity of the meson and a is a light quark flavour index. In particular in the charm sector the components of the field Ha are P,'*' = D(*)O,D(*)+and D?' (for a = 1 , 2 , 3 ) ;analogously, the components of Sa are P,*, = D:', D;', D,*, and Pi, = Die, D';t, DLl. In terms of these fields it is possible to build up an effective Lagrange

173

density describing the low energy interactions of heavy mesons with the pseudo Goldstone 7r, K and q b o ~ ~ n ~ ~ ~ * ~ ~ ~

-

+

L = i Tr{HbZfPDpbaHa} cTr{6’PCapct} f Tr{Sb (i

V’Dpba

-

8

dba A)sa}

+ + [i h T ~ ( S b 7 p ~ 5 A i ~ +Z ah.c.1 )

9 Tr{HbypY5Atapa} i 9’ Tr{SbYpY5Atasa}

sa

(3)

In (3) Ba and are defined as p a = yoHAyo and 3, = yoSfLyo;all the heavy field operators contain a factor and have dimension 312. The parameter A represents the mass splitting between positive and negative parity states. The T , K and q pseudo Goldstone bosons are included in the effective lagrangian (3) through the field t = e y that represents a unitary matrix describing the pseudoscalar octet, with

The strong interactions between the heavy Ha and Sa mesons with the light pseudoscalar mesons are thus governed, in the heavy quark limit, by three dimensionless couplings: g, h and 9’. In particular, h describes the coupling between a member of the Ha doublet and one of the Sa doublet to a light pseudoscalar meson, and is the one relevant for our discussion. Isospin violation enters in the low energy Lagrangian of 7r, K and q mesons through the mass term

with m, the light quark mass matrix:

mu 0 0 m q = ( 0 m0d m, 0 )

~

174

In addition to the light meson mass terms, L,,, contains an interaction term between no ( I = 1) and r] ( I = 0) mesons: LmiIing = $ m d ~ m ~ n O r ] which vanishes in the limit mu = m d . Let us focus on the mode D,*o -+ D,no. As in the case of D,* ---f D,n' studied in Ref. 19, the isospin mixing term can drive such a transition". The amplitude A(DZ0 -+ D,no) is simply written in terms of A(D,*,4 D,r]) obtained from (3), A(v + no)from (6) and the r] propagator that puts the strange quark mass in the game. The resulting expression for the decay amplitude involves the coupling h and the suppression factor (md - mu)/(m,- ""$"Y) accounting for isospin violation, so that the width I'(Dz0 + D,no) reads:

As for h, the result of QCD sum rule analyses of various heavy-light quark current correlators is Ih( = 0.6 f 0.2 17. Using the central value, together with the factor (md -mu)/(ms- m d $ m = ) fi 1 43.7 2o and f = ffl= 132 MeV we obtain21:

I'(Dzo 3 D,no) = 7 f 1 K e V .

(8)

The analogous calculation for Oil -+ D,*noprovides the result':

rp;, + D,*T") = 7 f 1 K e V .

(9)

3. Radiative Modes Let us now turn to the calculation of radiative decay rates. We describe the procedure considering the mode DQ0 + D,*y, the amplitude of which has the form:

A(DBo -D,*y) = e d [ ( ~ * . r ] * ) ( p . I c ) - ( r ] * . p ) ( ~ * . k ) ] , (10) where p is the Dz0 momentum, E the D,* polarization vector, and Ic and r] the photon momentum and polarization. The corresponding decay rate is:

I'(D,*, -+ D z y ) = ald121L13 .

(11)

The parameter d gets contributions from the photon couplings to the light quark part e,%y,s and to the heavy quark part e,Cy,c of the electromagnetic aElectromagnetic contributions to Dfo -+ D s x o are expected to be suppressed with respect t o the strong interaction mechanism considered here.

175

current, e, and e, being strange and charm quark charges in units of e. Its general structure is:

where A, (a = c, s) have dimension of a mass. Such a structure is already known from the constituent quark model. In the c a e of M1 heavy meson transitions, an analogous structure predicts a relative suppression of the radiative rate of the charged D* mesons with respect t o the neutral onez2123~z4y25, suppression that has been experimentally confirmedz6. From (11,12) one could expect a small width for the transition D& --+ DZr,t o be compared t o the hadronic width D,*o-+ D,r0 which is suppressed as well. In order to determine the amplitude of DE0 -+ DQ*y we follow a method based again on the use of. heavy quark symmetries, together with the vector meson dominance (VMD) a n ~ a t z We ~ ~first ~ ~consider ~ . the coupling of the photon t o the heavy quark part of the e.m. current. The matrix element (DH(v‘, ~)l~?~~clD (v,~v’~meson ( v ) ) four-velocities) can be computed in the heavy quark limit, matching the QCD Ey,c current onto the corresponding HQET e x p r e ~ s i o n ~ ~ :

JfLHQE= T&,[v,

i + -( i +a, - ca,) + B , , , ( ~+ ~ %,,) + . . . ] h ,

(13) 2mQ 2mQ where h, is the effective field of the heavy quark. For transitions involving DHo and DQ*, and for v = v’ (v . v’ = l), the matrix element of vanishes. The consequence is that dh)is proportional to the inverse heavy quark mass mQ and presents a suppression factor since in the physical case v . v’ = (m2 m & ) / 2 m p o m ~ D=: 1.004. Therefore, we neglect d h )in

JFQET

D:O

+

(12).

To evaluate the coupling of the photon to the light quark part of the electromagnetic current we invoke the VMD ansatz and consider the contribution of the intermediate 4(1020): (Dg*(v’,f)JSrPslD;o(v)) =

(14)

with k 2 = 0 and (OJSy,sl+(k, €1)) = M4 f461,. The experimental value of f4 is f+ = 234 MeV. The matrix element (D;(v’,~ ) $ ( kE, ~ ) I D Q * ~describes (V)) the strong interaction of a light vector meson ( 4 ) with two heavy mesons (0: and DQo). It can also be obtained through a low energy lagrangian in which the heavy fields Ha and S, are coupled, this time, t o the octet

176 of light vector mesonsb. The Lagrange density is set up using the hidden gauge symmetry method16, with the light vector mesons collected in a 3 x 3 matrix ipanalogous to M in (4). The lagrangian' reads as28:

L'

=if

+

i ~ r { S , ~ b a ~ ~ ~ ~ ,h.c., (p)~,}

(15)

+

with V x u ( p ) = &,pu - dypx [px,py] and px = i%bx, gv being fixed to gv = 5.8 by the KSRF rule2'. The coupling fi in (15) is constrained to fi = -0.1GeV-1 by the analysis of the D -+ K' semileptonic transitions induced by the axial weak current28t1s. The resulting expression for is:

&

The numerical result for the radiative width2,:

l?(Dt04 Dty) = 0.85 f 0.05 KeV

(17)

shows that the hadronic D& -+ D,*7ro transition is more probable than the radiative mode D,*o + Dfy, a t odds with the case of the D,* meson, where the radiative mode dominates the decay rate. In particular, if we assume that the two modes essentially saturate the DZO width, we have l?(DfO)= 8 5 1 KeV. As for the two radiative modes allowed for D,~(2460), one findss:

r ( D ; , -+ D,y) = 3.3 f 0.6 KeV

I'(D;,

-+

D t y ) = 1.5 KeV (18)

which in turn give a total width r(DA,) = 12 f 1 KeV. 4. Comparison with other approaches

The results of the previous two sections show that, within the described approach, the observed hierarchy of hadronic versus radiative modes is realized, supporting the identification of D,*,(2317) and D,.~(2460)with ( D & ,D t l ) . Other analyses have followed the same strategy of computing decay rates of the two narrow states in order t o understand their structure. In Table 1 we compare our results with the outcome of other approaches based on the c s picture as well. Analyses in which the states are assumed to have an exotic structure provide larger values for the widths (O(10') KeV)30. bThe standard W 8 - wo mixing is assumed, resulting in a pure 3s structure for 4. 'The role of other possible structures in the effective lagrangian contributing to radiative decays is discussed in Ref. 25.

177 Table 1. Estimated width (KeV) of DS'o and DL1, using the cB picture. The results in column [35]are obtained using experimental inputs from Belle (Focus). Decay mode

[31]

D:, -+D,T'

21.5

D,*o4 D:?

1.74

[32]

[21,81

10

751

21

1.9

[33]

l6 0.851k0.05 0.2

D ~ , - + D , ' T ~ 21.5

2110

7fl

D:,

+ D,y

5.08

6.2

3.3f0.6

Dil

-+

D:y

4.66

5.5

1.5

32

1341 129 f 43 (109616) < 1.4 187 f 73 (7.4312.3) 5 5

In particular, we observe that conclusions analogous to those presented above have been reached in Ref. 31, which is based on the observation that heavy-light systems should appear as parity-doubled, i.e. in pairs differing for parity and transforming according to a linear representation of chiral symmetry. In particular, the doublet composed by the states having J p = (Of, 1+) can be considered as the chiral partner of that with J p = (0-, 1-) '. Since our calculation is based on a different method, the sep = 1- and s c ='f doublets being treated as uncorrelated multiplets, we find the agreement noticeable. 5 . Conclusions and perspectives

We presented the calculation of hadronic and radiative decay rates of D,*,(2317) and D,~(2460)in a framework based on heavy quark symmetries and on the Vector Meson Dominance ansatz. This analysis shows that the observed narrow widths and the enhancement of the DL*)7rodecay modes are compatible with the identification of Dz,(2317) and D,~(2460) with the states belonging to the J E = (O+, l+); doublet of the cs spectrum. Nevertheless, unanswered questions remain, such as the near equality of the masses of DzJ(2317) and D,~(2460)with their non-strange partners. The missing evidence of the radiative mode D:,(2317) -+ D:-y is another puzzling aspect deserving further experiment a1 investigations. The quantum number assignment has a rather straightforward consequence concerning the doublet of scalar and axial vector mesons in the b3 spectrum. Since the mass splitting between B and D states is similar to the corresponding mass splitting between B, and D, states, such mesons dThis idea was first suggested in Ref. 35 in order to obtain a consistent implementation of chiral symmetry and reconsidered also in Ref. 36.

178

should be below t h e B K and B*K thresholds, thus producing narrow peaks in B,xo and B,*nomass distributions.

Acknowledgments I warmly t h a n k the Organizers for inviting m e to present this talk and for their very kind hospitality. I also thank P. Colangelo and R. Ferrandes for collaboration o n t h e topics discussed above. Partial support from t h e EC Contract No. HPRN-CT-2002-00311 (EURIDICE) is acknowledged. References B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 90,242001 (2003). D. Besson et al. [CLEO Collaboration], Phys. Rev. D68, 032002 (2003). B. Aubert et al. [BABAR Collaboration], Phys. Rev. D69,031101 (2004). K. Abe et al., Phys. Rev. Lett. 92,012002 (2004). E. Vaandering, arXiv:hepex/0406044. P. Krokovny et al. [Belle Collaboration], Phys. Rev. Lett. 91,262002 (2003). P. Krokovny, arXiv:hep-ex/0310053. 7. G. Calderini [BaBar Collaboration], arXiv:hep-ex/0405081. 8. P. Colangelo, F. De Fazio and R. Ferrandes, arXiv:hepph/0407137. 9. T. Barnes, F. E. Close and H. J. Lipkin, Phys. Rev. D68, 054006 (2003); H. J. Lipkin, Phys. Lett. B580,50 (2004); P.Bicudo, arXiv:hep-ph/0401106. 10. A. P. Szczepaniak, Phys. Lett. B567,23 (2003). 11. T. E. Browder, S. Pakvasa and A. A. Petrov, Phys. Lett. B578,365 (2004); S. Nussinov, arXiv:hepph/0306187. 12. G. S. Bali, Phys. Rev. D68,07150 (2003). 13. A. Dougall, R. D. Kenway, C. M. Maynard and C. McNeile [UKQCD Collaboration], Phys. Lett. B569,41 (2003). 14. P. Colangelo, F. De Fazio and N. Paver, Phys. Rev. D58, 116005 (1998); Y. B. Dai, C. S. Huang, C. Liu and S. L. Zhu, Phys. Rev. D68, 114011 (2003). 15. M. B. Wise, Phys. Rev. D45,R2188 (1992); G.Burdman and J. F. Donoghue, Phys. Lett. B280, 287 (1992); P. Cho, Phys. Lett. B285, 145 (1992); H.Y.Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin and H.-L. Yu, Phys. Rev. D46, 1148 (1992). 16. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B292,371 (1992). 17. P. Colangelo, F. De Fazio, G. Nardulli, N. Di Bartolomeo and R. Gatto, Phys. Rev. D52, 6422 (1995); P.Colangela and F. De Fazio, EZLT. Phys. J. C4, 503 (1998). 18. For a review see: R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281,145 (1997). 19. P. L. Cho and M. B. Wise, Phys. Rev. D49,6228 (1994). 20. J. Gasser and H. Leutwyler, Nucl. Phys. B250,465 (1985).

1. 2. 3. 4. 5. 6.

179 21. P. Colangelo and F. De Fazio, Phys. Lett. B570, 180 (2003). 22. E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D21, 203 (1980). 23. J. F. Amundson et al., Phys. Lett. B296,415 (1992); P. L. Cho and H. Georgi, Phys. Lett. B296,408 (1992) [Erratum-ibid. B300, 410 (1993)]. 24. P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B334, 175 (1994). 25. P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B316, 555 (1993). 26. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D66, 010001 (2002). 27. A. F. Falk, B. Grinstein and M. E. Luke, Nucl. Phys. B357, 185 (1991). 28. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Lett. B299, 139 (1993). 29. K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16,255 (1966); Riazuddin and Fayazuddin, Phys. Rev. 147, 1071 (1966). 30. H. Y. Cheng and W. S. Hou, Phys. Lett. B566, 193 (2003); S. Ishida, M. Ishida, T. Komada, T. Maeda, M. Oda, K. Yamada and I. Yamauchi, arXiv:hep-ph/0310061. 31. W. A. Bardeen, E. J. Eichten and C. T. Hill, Phys. Rev. D68,054024 2003). 32. S . Godfrey, Phys. Lett. B568, 254 (2003). 33. Fayyazuddin and Riazuddin, arXiv:hep-ph/0309283. 34. Y. I. Azimov and K. Goeke, arXiv:hep-ph/0403082. 35. M. A. Nowak, M. Rho and I. Zahed, Phys. Rev. D48, 4370 1993); W. A. Bardeen and C. T. Hill, Phys. Rev. D49, 409 (1994). 36. M. A. Nowak, M. Rho and I. Zahed, arXiv:hep-ph/0307102.

SEARCH FOR DARK MATTER IN B + S TRANSITIONS WITH MISSING ENERGY

M. POSPELOV Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada E-mail: pospelov@uvic. ca Dedicated underground experiments searching for dark matter have little sensitivity to GeV and sub-GeV masses of dark matter particles. We show that the decay of B mesons to K ( K * ) and missing energy in the final state can be an efficient probe of dark matter models in this mass range. We analyze the minimal scalar dark matter model to show that the width of the decay mode with two dark matter scalars B + K S S may exceed the decay width in the Standard Model channel, B ---t KuD, by up to two orders of magnitude. Existing data from B physics experiments almost entirely exclude dark matter scalars with masses less than 1 GeV. Expected data from B factories will probe the range of dark matter masses up to 2 GeV.

1. Introduction

Although the existence of dark matter is firmly established through its gravitational interaction, the identity of dark matter remains a big mystery. Of special interest for particle physics are models of weakly interacting massive particles (WIMPS), which have a number of attractive features: well-understood mechanisms of ensuring the correct abundance through the annihilation a t the freeze-out, milli-weak t o weak strength of couplings t o the “visible” sector of the Standard Model (SM), and as a consequence, distinct possibilities for WIMP detection. The main parameter governing the abundance today is WIMP annihilation cross section directly related to the dark matter abundance. In order to keep WIMP abundance equal or smaller than the observed dark matter energy density, the annihilation cross section has t o satisfy the lower bound, oannvre1 >, 1 pb, (see e.g. I ) . In all WIMP models studied to date the annihilation cross section is suppressed in the limit of very large or very small mass of a WIMP particle S . This confines the mass of a stable WIMP within a certain mass range, mmin 5 ms 5 mmax,which we refer to as the Lee-Weinberg window ’.

180

181

This window is model-dependent and typcally extends from a few GeV to a few TeV. If the neutralino is the lightest stable supersymmetric particle, mmin_N 5 GeV but in other models of dark matter mmin can be lowered 495

Recently, WIMPs with masses in the GeV and sub-GeV range have been proposed as a solution to certain problems in astrophysics and cosmology. For example, sub-GeV WIMPs can produce a high yield of positrons in the products of WIMP annihilation near the centres of galaxies 6 , which may account for 511 KeV photons observed recently in the emission from the Galactic bulge GeV-scale WIMPs are also preferred in models of self-interacting dark matter that can rectify the problem with over-dense galactic centers predicted in numerical simulations with non-interacting cold dark matter. Dedicated underground experiments have little sensitivity to dark matter in the GeV and sub-GeV range. Direct detection of the nuclear recoil from the scattering of such relatively light particles is very difficult because of the rather low energy transfer to nuclei, AE v2mi/mNuc15 0.1 KeV, which significantly weakens experimental bounds on scattering rates below r n of ~ few GeV, especially for heavy nuclei. Indirect detection via energetic neutrinos from the annihilation in the centre of the Sun/Earth is simply not possible in this mass range because of the absence of directionality. Therefore, the direct production of dark matter particles in particle physics experiments stands out as the most reliable way of detecting WIMPs in the GeV and sub-GeV mass range. The purpose of this work is to prove that B decays can be an effective probe of dark matter near the lower edge of the Lee-Weinberg window. K decays can also be used for this purpose, but B decays have far greater reach, up to m s 2.6 GeV. In particular, we show that pair production of WIMPs in the decays B -+ K ( K * ) S Scan compete with the Standard Model mode B -+ K(K*)vF. In what follows, we analyze in detail the “missing energy” processes in the model of the singlet scalar WIMPS and use the existing data from B physics experiments to put important limits on the allowed mass range of scalar WIMPs. The main advantage of the singlet scalar model of dark matter is its simplicity,

’.

-

-

4,9110

xs 4

= -S

4

1

-t- -(m: 2

x + Xv‘&,)S2 + XvEWS’h + -S2h2, 2

182

where H is the SM Higgs field doublet, WEW = 246 GeV is the Higgs vacuum expectation value (vev) and h is the field corresponding t o the physical Higgs, H = (0, (WEW h ) / a ) . It is important to recognize that the physical mass of the scalar S receives contributions from two terms, m i = m; and can be small, even if each term is on the order ~ O ( W ’ &Although ~). admittedly fine-tuned, the possibility of low ms is not a priori excluded and deserves further studies as it also leads t o Higgs decays saturated by the invisible channel, h 4 SS and suppression of all observable modes of Higgs decay a t hadronic colliders ‘. The minimal scalar model is not a unique possibility for light dark matter, which can be introduced more naturally in other models. If for example, the dark matter scalar S couples to the Hd Higgs doublet in the two-Higgs modification of (l), XS2HlHd, the fine-tuning can be relaxed if the ratio of the two electroweak vevs, t a n p = (Hu)/(Hd) is a large parameter. A well-motivated case of tanp 50 corresponds t o ( H d ) 5 GeV, and only a modest degree of cancellation between mg and X(Hd)’ would be required to bring m S in the GeV range. More model-building possibilities open up if new particles, other than electroweak gauge bosons or Higgses, mediate the interaction between WIMPs and SM particles. If the mass scale of these new particles is smaller than the electroweak scale 5, sub-GeV WIMPs are possible without fine-tuning.

+

+

N

-

Figure 1. Feynman diagrams which contribute t o B meson decays with missing energy.

2. Pair-production of WIMPs in B decays The Higgs mass mh is heavy compared to ms of interest, which means that in all processes such as annihilation, pair production, and elastic scattering of S particles, X and mh will enter in the same combination, X2mh4. In what follows, we calculate the pair-production of S particles in B decays in terms of two parameters, A2/mi and ms, and relate them using the dark matter abundance calculation, thus obtaining the definitive prediction for

183 the signal as a function of ms alone. At the quark level the decays of the B meson with missing energy correspond to the processes shown in Figure 1. The SM neutrino decay channel is shown in Figure l a and l b . The b -+ s Higgs penguin transition, Figure l c , produces a pair of scalar WIMPS S in the final state, which likewise leave a missing energy signal. In this section, we show that this additional amplitude generates b --f sSS decays that can successfully compete with the SM neutrino channel. A loop-generated b - s-Higgs vertex at low momentum transfer can be easily calculated by differentiating the two-point b s amplitude over D E W . We find that t o leading order the b + sh transition is given by an effective interaction --f

Using this vertex, Eq. (1) and safely assuming mh >> mb, we integrate out the massive Higgs boson to obtain the effective Lagrangian for the b --+ s transition with missing energy in the final state: Lb,,&

1 2

= -CDMmbsLbRS2

-

C,gLy,bLfi~,v 4-(h.c.).

(3)

Leading order Wilson coefficients for the transitions with dark matter scalars or neutrinos in the final state are given by

where xt = m:/M&. We would like to remark at this point that the numerical value of CDM is a factor of few larger than C,,

-

if AmL2 O(g&MG2). This happens despite the fact that the effective bsh vertex is suppressed relative to bsZ vertex by a small Yukawa coupling mb/vEw. The 1 / v in~ (2) ~ is compensated by a large coupling of h to S2, proportional to AVEW, and mb is absorbed into the definition of the dimension 6 operator mbBLbRS2. N

184

We concentrate on exclusive decay modes with missing energy, as these are experimentally more promising than inclusive decays and give sensitivity to a large range of ms. A limit on the branching ratio has recently been at 90% c.1. 'I, reported by BABARcollaboration, BrB+--rK+vu< 7.0 x which improves on a previous CLEO limit 12, but is still far from the SM prediction Br(B + KvP) N (3 - 5) x (See, e.g. 1 3 ) . We use the result for Lb,,@ along with the hadronic form factors determined via light-cone sum rule analysis in l4 to produce the amplitude of B 4 K S S decay,

) ~ the form factor for B -+ K transition is where q2 = d = ( p - ~ p ~ and approximated as fo N 0.3exp{0.63dMB2 - O.095i2ME4 O.591d3Mi6}. The differential decay width to a K meson and a pair of WIMPS is given by

+

where I ( ; ) reflects the available phase space,

+

I ( ; , m s ) = [i2 - 2 2 ~ ~M2; ) +

+ [I- 4mS/d] +.

( M B - hfK)2]

From Eq. (7) and the prediction for the SM neutrino channel, we obtain the total branching ratio for the B+ to K+ decay with missing energy in the final state,

+

Brg+-+K++F= B ~ B + - + K +B~B++K+ss ~~

4

+ 2.8 x 1 0 - 4 K 2 ~ ( m s ) .

Eq. (8) uses the parametrization of X2mi4, K 2 = A 2 ( 100mGeV h

(8)

), 4

(9)

and the available phase space as a function of the unknown m s , F ( m s ) = / ' m ~ ( d ) 2 1 ( d , m s > dd Smin

[

iz!(i)21(d, 0) dd

1

-l

Notice that F ( 0 ) = 1 and F ( m s ) = 0 for m s > $(mB - mK) by construction. Similar calculations can be used for the decay B K*SS,

185

with an analogous form factor. For light scalars, ms few 100 MeV, and K O(1) the decay rates with emission of dark matter particles are 50 times larger than the decay with neutrinos in the final state! This is partly due to a larger amplitude, Eq. (5), and partly due to phase space integral that is a factor of a few larger for scalars than for neutrinos if m S is small. N

N

N

3. Abundance calculation and Comparison with

Experiment The scalar coupling constant X and the scalar mass ms are constrained by the observed abundance of dark matter. For low ms, as shown in 4 , the acceptable value of K is K O(1). Here we refine the abundance calculation for the range 0 < m S < 2.4 GeV in order to obtain a more accurate quantitative prediction for K . The main parameter that governs the energy density of WIMP particles today, which we take to be equal to the observed value of RoMh2 0.13 15, is the average of their annihilation cross section a t the time of freeze-out. This cross section multiplied by the relative velocity of the annihilating WIMPS is fixed by S ~ D Mand can be conveniently expressed as N

N

Here rGx denotes the partial rate for the decay, 4 X , for a virtual Higgs, h, with the mass of mh = 233s N 2ms. Notice that Eq. (11) contains the same combination X2rnh4as (8). The zero-temperature width rGx was extensively studied two decades ago in conjunction with searches for light Higgs 16,17,18 For ms larger than mT the annihilation to hadrons dominates the cross section, which is therefore prone t o considerable uncertainties. At a given value of ms, we can predict rhx within a certain range that reflects these uncertainties. With the use of (ll),this prediction translates into the upper (A) and the lower (B) bounds on K(ms),which we insert into Eq. (8) and plot the resulting Brg++K++e in Figure 2 . In the interval 150 MeV 5 ms <, 350 MeV the annihilation cross section is dominated by continuum pions in the final state, and can be calculated with the use of low-energy theorems l6 t o good accuracy. Requiring < 4 7 ~allows to determine the lower end of the Lee-Weinberg window in our model, mmin 350 MeV. In the interval 350 MeV - 650 MeV N

186

the strangeness threshold opens up, and annihilation into pions via the schannel fo resonance become important. The strength of this resonance and its width and position a t freeze-out temperatures, T (0.05 - O.l)ms, are uncertain. Curve B in this domain of Figure 2 assumes the fo resonance is completely insensitive to thermal effects and has the minimum width quoted by PDG l9 which maximizes Firx, whereas curve A corresponds to a complete smearing of the fo resonance by thermal effects and much lower value of r L x . Above ms =1 GeV, curve B takes into account the annihilation into hadrons mediated by a, (GCLV)’ with the two-fold enhancement suggested by charmonium decays l6 whereas curve A uses the perturbative formula. The charm threshold is treated simply by the inclusion of open charm quark production a t a low threshold (m, 21 1.2 GeV) in B curve and at a high threshold (ms > mo) in curve A. Both curves include the T threshold. There are no tractable ways of calculating the cross section in the intermediate region 650 MeV 5 ms 5 1 GeV. However, there are no particular reasons to believe that the annihilation into hadrons will be significantly enhanced or suppressed relative to the levels in adjacent domains. In this region, we interpolate between high- and low-energy sections of curves A and B. Thus, the parameter space consistent with the required cosmological abundance of S scalars calculated with generous assumptions about strong interaction uncertainties is given by the area between the two curves, A and B. Figure 2 presents the predicted range of BrB++K++F as a function of ms and is the main result of our paper. The SM “background” from B --t KuF decay is subdominant everywhere except for the highest kinematically allowed domain of ms. To compare with experimental results ”~”, we must convert the limit on BrB+-+K+VD t o a more appropriate bound on Brg+,K++F according to the following procedure. We multiply the experimental limit of 7.0 x by a ratio of two phase space integrals, F ( m s ,imin)/F(ms,iexp), where seXpis determined by the minimum Kaon momentum considered in the experimental search, namely 1.5 GeV. This produces an exclusion curve, nearly parallel to the ms axis at low ms, and almost vertical near the experimental kinematic cutoff. The current BABARresults (curve I) exclude ms < 430 MeV, as well as the region 510 MeV< ms < 1.1 GeV, and probe the allowed parameter space for dark matter up t o ms 1.5 GeV. Generalized model with N component dark matter scalar gives N2-fold increase in the branching ratio 4 , and thus greater sensitivity to ms. The B factories will soon have larger data samples and can extend the

-

N

187

w

+ 1: 1 0 - ~

M

a

6i

10-~ 0

0.5

1

1.5

2 MdGeV)

Figure 2. Predicted branching ratios for the decay B + K+ missing energy, with current limits from Babar (I) CLEO (111) and expected results from BABAR(11). Parameter space above curves I and I11 is excluded. The horizontal line shows the SM B -+ KvD signal. Parameter space to the left of the vertical dashed line is also excluded by K i + T+$.

search to lower Kaon momenta. The level of sensitivity expected from an integrated luminosity L of 250 fb-I and momentum cutoff of 1 GeV is shown by curve 11, which assumes that the sensitivity scales as L-l/’, as suggested by the analysis in In reality, the experimental limit will extend to Kaon momenta below 1 GeV where the sensitivity will gradually degrade due t o increasing backgrounds; however, we expect the implication of curve I1 t o remain valid, namely that the B factories will probe scalar dark matter up to 2 GeV. If ms 5 150 MeV, the decay K+ -+ T+SS becomes possible. The width for this decay can be easily calculated in a similar fashion to b 4 s transition. The concordance of the observed number of events with the SM prediction 2o rules out scalars in our model with ms < 150 MeV. This exclusion limit is shown by a vertical line in Figure 3. It is below mminof 350 MeV. 4. Conclusions

To conclude, we have demonstrated that the b 4 s transitions with missing energy in the final state can be an efficient probe of dark matter when pair production of WIMPS in B meson decays is kinematically allowed. In particular, we have shown that the minimal scalar model of dark matter

188

with the interaction mediated by the Higgs particle predicts observable rates for B f -+K+ and missing energy. A large portion of the parameter space 1 GeV is already excluded by current BABARlimits. New with m S experimental data should probe a wider range of masses, up t o ms 2 GeV. The limits obtained in this paper have important implications for Higgs searches, as the existence of relatively light scalar WIMPs leads t o the Higgs decays saturated by invisible channel. Given the astrophysical motivations for GeV and sub-GeV WIMPs combined with insensitivity of dedicated dark matter searches in this mass range, it is important t o extend the analysis of b -+ s transition with missing energy onto other models of light dark matter. N

Acknowledgments We thank Misha Voloshin for valuable discussion. This research is supported in part by NSERC of Canada and PPARC UK.

References 1. E.W. Kolb and M.S. Turner, The Early Universe, Addison-Wesley (1990). 2. B. W. Lee and S. Weinberg, Phys. Rev. Lett. 39, 165 (1977). 3. G. Belanger, F. Boudjema, A. Cottrant, A. Pukhov and S. Rosier-Lees, JHEP 0403, 012 (2004). 4. C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B 619, 709 (2001). 5 . P. Fayet, Phys. Rev. D 70,023514 (2004). 6. C. Boehm, D. Hooper, J. Silk and M. Case, Phys. Rev. Lett. 92, 101301 (2004). 7. P. Jean et al., Astron. Astrophys. 407, L55 (2003). 8. D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760 (2000). 9. V. Silveira and A. Zee, Phys. Lett. B 161, 136 (1985). 10. J. McDonald, Phys. Rev. D 50, 3637 (1994). 11. B. Aubert et al. [BABAR Collaboration], arXiv:hep-ex/0304020. 12. T. E. Browder et al. [CLEO Collaboration], Phys. Rev. Lett. 86, 2950 (2001). 13. G. Buchalla, G. Hiller and G. Isidori, Phys. Rev. D 63, 014015 (2001). 14. A. Ali, P. Ball, L. T. Handoko and G. Hiller, Phys. Rev. D 61,074024 (2000). 15. D. N. Spergel et al., Astrophys. J . Suppl. 148, 175 (2003). 16. M. B. Voloshin, Sou. J . Nucl. Phys. 44, 478 (1986). 17. S. Raby and G. B. West, Phys. Rev. D 38, 3488 (1988). 18. T. N. Truong and R. S. Willey, Phys. Rev. D 40, 3635 (1989). 19. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66, 010001 (2002). 20. S. Adler et al. [E787 Collaboration], Phys. Rev. Lett. 88, 041803 (2002).

SECTION 3.

EXOTIC HADRONS

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EXOTICA

R. L. JAFFE Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology Cambridge, M A 02139, USA E-mail: jaffeamit. edu The sudden appearance of narrow, prominent exotic baryons has re-invigorated light hadron spectroscopy. At present the experimental situation is confused; so is theory. The recent discoveries are striking. So too is the complete absence of exotic mesons, and, except for the recent discoveries, of exotic baryons as well. Whether or not the new states are confirmed, the way we look at complicated states of confined quarks and gluons has changed. Perhaps the most lasting result, and the one emphasized in these notes, is a new appreciation for the role of diquark correlations in QCD.

1. Introduction The absence of exotics is one of the most obvious features of QCDa. In the early years experimenters searched hard for baryons that cannot be made of three quarks or mesons that cannot be made of qq. Exotic mesons seemed entirely absent. Controversial signals for exotic baryons known as 2's came, and usually went, never rising to a level of certainty sufficient for the Particle Data Group's tables2. In the 1990's the subject of exotic baryons did not receive much attention except from a small band of theorists motivated by the predictions of chiral soliton model^^^^^^. Then, in January of 2003 evidence was reported of a very narrow baryon with strangeness one and charge one, of mass M 1540 MeV, now dubbed the O+, with minimum quark content u ~ d d S ~The > ~ first . experiment was followed by evidence for other exotics: a strangeness minus two, charge minus two particle now officially named the @-- by the PDG, with minimum quark content ddsscg at 1860 MeV , and an as-yet nameless charm exotic (uuddc)" at 3099 MeV. Theorists, myself included, descended upon these aThis paper is a condensation of material that can be found in Ref.

191

'.

192

reports and tried to extract dynamical insight into QCD. Other experimental groups began searches for the O+ and its friends. As time has passed the situation has become more, rather than less, confusing": several experiments have now reported negative results in searches for the O+13; no one has confirmed either the @--(1860) or the uuddc(3099); and theorists have yet to find a compelling (to me at least) explanation for the low mass or narrow width of the O f . The existence of the O+ is a question for experimenters. Theorists simply do not know enough about QCD t o predict without doubt whether a light, narrow exotic baryon exists. Whether or not the Of survives, it is clear that exotics are very rare in QCD. Perhaps they are entirely absent. This remarkable feature of QCD is often forgotten when exotic candidates are discussed. The existence of a handful of exotics has to be understood in a framework that also explains their overall rarity. Along the same line, the aufbau principle of QCD differs dramatically from that of atoms and nuclei: to make more atoms add electrons, to make more nuclei, add neutrons and protons. However in QCD the spectrum - with the possible exception of a few states like the O+ - seems to stop at qqq and qij. Thinking about this problem in light of early work on multiquark correbegan t o re-examine the role of lations in QCD 14, Frank Wilczek and 115b diquark correlations in QCD. Diquarks are not new; they have been chamWe l~~. pioned by a small group of QCD theorists for several d e c a d e ~ ~ ~ already knew14 that diquark correlations can naturally explain the general absence of exotics and predict a supernumerary nonet of scalar mesons which seems to exist. We quickly learned that they can rather naturally accommodate exotics like the Of. They also seem to be important in dense quark matter20, to influence quark distribution21 and fragmentation functions, and to explain the systematics of non-leptonic weak decays of light quark baryons and mesons22. Whether or not the O+ survives, diquarks are here to stay. In the first part of this talk, after looking briefly at the history of exotics, I assume that the O+ exists, and see how well it fits with other features of light quark spectroscopy. I will take a look at the O+ from several perspectives: general scattering theory, large N,, chiral soliton models, quark models, and lattice calculations. In general these exercises raise more questions than they answer. In brief A light, narrow exotic is inconvenient but not impossible for QCD spectroscopy. bClosely related ideas have been explored by NussinovlG and by Karliner and LipkinL7.

193 The later half of the talk is devoted to diquarks. I define them carefully and then describe how diquark correlations in hadrons can explain qualitatively most of the puzzles of exotic hadron spectroscopy: first, why exotics are so rare in QCD; next, why there is an extra nonet of scalar mesons; third, why an exotic baryon antidecuplet containing the O+ would be the only prominent baryon exotic; and finally, why non-strange systems of 6, 9, 12, . . .quarks form nuclei not single hadrons. “Qualitatively” is an important modifier, however: like all quark model ideas, this one lacks a quantitative foundation: the need for a systematic and predictive phenomenological framework for QCD spectroscopy has never been greater. Spectroscopy was at the cutting edge of high energy physics in the ‘60’s and ‘70’s. A great deal of effort and sophisticated analysis was brought to bear on the study of the hadron spectrum, and the conclusions remain important. In the decade that followed the first conjectures about quarks experimental groups studied meson-baryon and meson-meson scattering, and extracted the masses and widths of meson and baryon resonances. Resonances were discovered in nearly all non-exotic meson and baryon channels, but no prominent exotics were found. The zeroth order summary prior to January 2003 was simple: no exotic mesons or baryons. In fact the only striking anomaly in low energy scattering was the existence of a supernumerary (ie not expected in the quark model) nonet of scalar, (J” = Of) mesons with masses below 1 GeV: the f0(600),n(800), f0(980), and ao(980) that is now widely considered to contain important ijGqq components23 When the Of was first reported, several groups re-examined the old K N scattering data and interpreted the absence of any structure near 1540 MeV as an upper limit on the width of the O+’6924>25>26. The limits range from 0.8 MeV through “a few” MeV. It is important to remember that these are not sightings of a narrow O+, rather they are reports of negative results expressed as an upper limit on the width of the O+. Space and time do not permit me to present and review all the reports of exotics since January of 2003. Summaries of the experimental sightings of the O+ can be found in many recent reviews27. Table 1 is a summary of the properties of the reported states. The baryons in Table 1 can be classified in the i-6or 8 and representations of SU(3)f Although they could be in higher representations, the is the simplest that can accomodate both the O+ and the @--. I have not attempted to summarize the searches that fail to see the Of or the other new exotics. These require a careful discussion, which can be found, for example in Ref. 13.

s

194

Isospin

Decays

K+n, KSP

I

a--

I

I

I

~0/~0141

a- /=-I41

1860 1860 1855

I

< 18 < 18 < 18

From analysis of K N scattering; 1’ dence for J

I

I I

?

I

?

I ? E*O( 1 5 3 0 ) ~ -

from direct detection of 8 ; 131

Weak evi-

= 112, no information on parity; 141 QifI = 3/2, 9 i f I = l/2;[’]I

=

112favored if decay to 8’0(1530) is correct.

There are several puzzling aspects of the data: the variation of the O+ mass, the claims of HERMES and ZEUS to have measured a non-vanishing width, and the apparent inconsistency of several positive experiments with the limits obtained in other experiments, for example. The interested reader should consult the talks by Nakano” and Dzierba13 and other presentations at QNP2004. There are many predominantly experimental issues that I have not covered here: proposals to measure the parity of the O+, limits on the production of other exotics like a @++, and reports of other bumps with the quantum numbers of the O+ at higher mass, to name only a few. 2. Theoretical perspectives 2.1. Insight

from scattering theory

The small width of the O+ poses a challenge for any theoretical interpretation. The O+ is unique among hadrons in that its valence quark configuration, uudds, already contains all the quarks needed for it to decay into K N . Non-exotic hadrons like the ~ ( 7 7 0 or ) A(1520) in their valence quark configuration can only couple to their decay channels (TT for the former, KN for the latter) by creating quark pairs. The suppression of quark pair creation, known as the Okubo, Zweig, Iizuki (OZI) Rulez8, is often invoked as an explanation for the relative narrowness of hadronic resonances. Some other, as yet unknown mechanism would have to be responsible for the narrowness of the O+. General principles of scattering theory allow one to get at least a qualitative answer to the question: ‘ L Hunusual ~ ~ is the width of the 0+?’’29.

195

There are two ways t o make a resonance in low energy scattering, either (a) the resonance is generated by the forces between the scattering particles, or (b) it exists in another channel, which is closed (or confined), and couples to the scattering channel by some interaction. Potential scattering resonances (case (a)) are generated by the interplay between attraction due to interparticle forces and repulsion, usually due t o the angular momentum barrier. These resonances subside into the continuum as the interaction is turned off. An example of case (b) is a bound state in a closed or confined channel that couples to a scattering channel by an interaction. These states decouple, i e their widths vanish, as the interaction is turned off. In the case of a qqq baryon coupling to the meson(@q)-baryons(qqq)continuum, quark pair creation is the interaction. The O+ is unusual: Because its valence quark content, uudds, is the same as the valence quark content of K N , the possibility that it arises from the K N potential cannot be excluded a priori and has to be analyzed. The lesson of this exercise is qualitative (and here I quote from Ref. ”): If the O+ appears in a low partial wave (l = 0 or l), schemes which hope to produce it from K N forces seem doomed from the start; schemes which introduce confined channels (quark models are an example, where reconfiguration of the quark substructure of the Of could be required for it to decay) are challenged to find a natural physical mechanism that suppresses the Of decay more effectively than the OZI rule suppresses the decay of the A(1520).

2.2. Large Nc a n d chiml soliton models

The number of colors is the only conceivable parameter in light quark QCD, so it is natural t o consider exotic baryon dynamics as an expansion in l,”,. It is known from the work of ‘t Hooft, Witten, and many others, that as N, + 00 QCD reduces t o a theory of zero width @q mesons with AQCLI~’and heavy baryons with masses N,AQCO in which masses quarks move in a mean (Bartree) field31. This is far from a complete description even at the heuristic level: Almost nothing is known about the spectrum of 4q mesons from large N, except for the pseudoscalar Goldstone bosons required by chiral symmetry. The dynamics of baryons is accessible only through a loose association with the chiral soliton model (CSM). It is important to remember that the CSM has not be derived from large N, QCD. Its appeal is based on its proper implementation of chiral symmetry and anomalies, and on the resemblance of collectively quantized solitons to N

N

196

the lightest positive parity baryons. Dashen, Jenkins, and M a n ~ h a made r ~ ~ the application of large N, ideas to baryons more precise and extended their work t o exotic baryons33. Their ideas were presented a t this conference by E. Jenkins. They cannot determine whether the @+ is light enough to be narrow and prominent, or even whether it is the lightest qqqqtj state. However, if the existence of the @+ is fed into their machinery, one can predict its properties and the masses and properties of other states in the qqqqtj spectrum. One result of their analysis that is particularly troubling is also a general feature of all quark model/QCD treatments of the exotic baryons: the most natural candidate for the @+ (all quarks in the lowest Hartree level) has negative parity, corresponding to the K N s-wave, where it is very hard to explain its narrow width. The positive parity @+, which Jenkins and Manohar study in detail, require one quark to be in an excited orbital. This state has internal structure which might sequester it from the K N channel enough to help explain its long lifetime. So all QCD approaches consistent with large N , should share a fundamental difficulty a t the outset: If the quark configuration is “natural”, it’s hard to explain why the O+ should be narrow. To have any hope t o obtain a narrow @+, strong quark forces must make a state of mixed spatial symmetry the lightest. Understanding these strong quark correlations and their consequences then becomes a central issue. Much of the discussion of exotic baryons over the past decade4i6 and, in particular, a remarkable prediction of the mass and width of the @+ by Diakonov, Petrov, and Polyakov 5 , has been carried out in the context of chiral soliton models. This is not surprising since CSM’s are teeming with exotics. Collective quantization of a classical soliton solution t o a chiral field theory of pseudoscalar bosons in SU(2)f or SU(3)f, consistent with anomaly constraints, yields towers of baryons only the lightest of which are not e x o t i ~ ~ The ~ i ~simplest ~. case is SU(2)f, where the spectrum of baryons begins with a rotational band of positive parity states with I = J = 1/2,3/2,5/2,. . .. Other excitations, radial for example, are heavier, separated from the ground state band by order ( N : ) . The lack of evidence for a I = 5/2 baryon resonance led most workers to dismiss the heavier states as artifacts of large N,. The generalization of the CSM to three flavors with broken SU(3)f has always been controversial. Guadagnini’s original approach (the “rigid rotor” (RR) approach) was to quantize in the SU(3)f limit and introduce SU(3)f violation p e r t ~ r b a t i v e l y ~ Alternatively, ~. Callan and Klebanov

197

quantized the S U ( 2 ) f soliton and constructed strange baryons as kaon bound states (the "bound state (BS) a p p r ~ a c h ) ~Although ~. different in principle, the two approaches give roughly the same spectrum for the octet and decuplet. When generalized to three flavors the rotational band of the 3+

1 '

I+

3+

103 , 10' , 275 ' 5 , . . .. Diakonov, Petrov, and Polyakov5 estimated its mass and width of the --1/2+ first exotic multiplet in this tower, 10 . They argued that it should be light and narrow37. Their work stimulated the experimenters who found the first evidence for the Of7. Soon afterwards Weigel showed that it is inconsistent in the RR approach to ignore the mixing between the and radial excitations excited by order (Nf) above the ground state6. After the first reports of the Of, other groups found the mass splitting between the ground state and the 10 to be order (Nf)40739.Cohen pointed out that the width of the Of does not vanish as N, -+ ca in contrast to non-exotic states like the A39, making it hard to understand the very small value of I'(O)/r(A) obtained in Ref. 5 . Furthermore, the O+ does not exist in the BS approach unless the mass of the kaon is of the order of 1 GeV6y4O. In fact, in the BS approach the force between the collectively quantized two-flavor soliton and the kaon is repulsive for physical kaon masses. Chiral soliton models describe at best a piece of QCD: Their picture of the nucleon and A (or 84' and 10;' in S U ( 3 ) f )is internally consistent and predictive. Some progress has been made in the description of baryon resonances41. However the incorporation of strangeness is not satisfactory and still c o n t r o ~ e r s i a land ~ ~ , the CSM gives no insight at all into the meson spectrum. As for exotics, the candidate for the O+ is controversial and there is no insight into the striking absence of exotic mesons and baryons in general. The prediction of a narrow width for the O+ is very controversial.

RR approach becomes B;',

2.3. Quark models

The quark model in its many variations has been by far the most successful tool for the classification and interpretation of light hadrons. It predicts the principal features and many of the subtleties of the spectrum of both mesons and baryons, and it matches naturally onto the partonic description of deep inelastic phenomena. The limitations of the quark model are, however, as obvious as its successes. It has never been formulated in a way that is fully consistent with confinement and relativity. Of course quarks can move relativistically, governed by the Dirac equation, in first quantized models like the MIT bag, but

198

there is no fully relativistic, second quantized version of the quark model. Furthermore, quark models are not the first term in a systematic expansion. No one knows how to improve on them. Nevertheless all hadrons can be classified as relatively simple configurations of a few confined quarks, and there is no reason to be expect the Of to be an exception. So looking for a natural quark description of the Of is a high priority, and if there were none, it would be most surprising.

2.3.1. Generic features of a n uncorrelated quark model Although quark models differ in their details, the qualitative aspects of their spectra are determined by features that they share in common. They need not be correct but they form the context in which other proposals have to be considered. Certainly they do a good job for mesons, baryons, and even tetraquark (ie. tjtjqq) spectroscopy. Here is a summary of the basic i n g r e d i e n t ~ that ~~t~ can ~ be applied to qqqqq states. (1) The spectrum can be decomposed into sectors in which the numbers of quarks and antiquarks, ng and nq, are good quantum numbers - the OZI rule28. (2) Hadrons are made by filling quark and antiquark orbitals in a hypothetical mean field. The total angular momentum of the (relativistic) quark in the lowest orbital is 1/2. Its parity is even (relative to the proton). Typically the first excited orbitals have negative parity and total angular momentum 1/2 and 3/2 because orbital excitations are invariably less costly than radial. (3) The ground state multiplet is constructed by putting all the quarks and antiquarks in the lowest orbital - the “single mode configuration”. This is a natural assumption, but one which must fail for qqqqq if the O+ has positive parity (see below). (4) As the number of quarks and antiquarks grows, the number of p c q n q eigenstates proliferates wildly. Although the qqqq and qqqqq states are stationary states in a potential or bag, they do not in general correspond to stable hadrons or even resonances. In a fairly precise way the @?jqq state can be considered a piece of the mesonmeson continuum that has been artificially confined by an inappropriate confining potential or boundary condition or potential43. Unless the multiquark state is unusually light or sequestered (by the spin, color and/or flavor structure of the wavefunction) from the scattering channel, it is just an artifact of a silly way of enu-

199

merating the states in the continuum.

2.3.2. Pentaquarks an the uncorrelated quark model The parity of the ground state of nq quarks and nq antiquarks in the lowest orbital is (-l)ng. When a quark or antiquark is excited, the parity flips. Light meson and baryon multiplets do (roughly) alternate in parity - one of the remarkably simple and successful predictions of the quark model. Accordingly the pentaquark ground state has negative parity. This makes the existence of the @+ embarrassing for this model for several reasons: first, there is no evidence for a negative parity, non-strange @-analogue state, uudd(ti, 2) which should be lighter than the @+. Second, the ground state multiplet of qqqqq contains 1260 states. Although most will be heavy enough to disappear into the continuum, it is hard to imagine that only the antidecuplet should be seen. Finally, a negative parity @+ would have to appear in the s-wave of K N scattering, where its narrow width would be very hard t o explain. All in all, the uncorrelated quark model gives little reason to expect a light, narrow, exotic baryon with the quantum numbers of the @+. Later I will describe the somewhat more positive perspective of correlated quark models.

2.4. Early lattice results

The first lattice studies of the @+ appeared soon after the first experimental r e p o r t ~ ~ ~Both v ~ ~groups . agreed that there is evidence for the K N threshold and for a state in the uuddg J" = 1/2- channel. They also reported evidence for a state in the 1/2+ channel, but a t a higher mass. Subsequent work by the Kentucky does not find a state in either parity channel. These results are troubling: lattice calculations have, in the past, got the quantum numbers of the ground state correct in each sector of QCD. In this case the calculations support negative parity (Ref. 46 excepted), which, as we have seen, is hard to reconcile with the narrow width of the @+. Lattice studies of qqqqq are only beginning. They are still far from the chiral limit and confined to small lattices. Advocates of a positive parity @+ can hope that better sources and better approximations to the chiral limit will reverse the order of states.

200

3. Diquarks An uncorrelated quark model leads to a negative parity ground state multiplet, which contains 1/2- and 3/2- candidates for the However the very narrow width of the Of seems to be an insuperable difficulty. So a quark description of the O+ must look to some correlation to invert the naive ordering of parity supermultiplets. This is where diquarks enterl5iI6. QCD phenomena are dominated by two well known quark correlations: confinement and chiral symmetry breaking. Confinement hardly need be mentioned: color forces only allow quarks and antiquarks correlated into color singlets. Chiral symmetry breaking can be viewed as the consequence of a very strong quark-antiquark correlation in the color, spin, and flavor singlet channel: [qq]lclfo.The attractive forces in this channel are so strong that [qq]lClf0condenses in the vacuum, breaking SU(Nf)r,x SU(Nf)R chiral symmetry. The “next most attractive channel” in QCD seems to be the color antitriplet, flavor antisymmetric (which is the 3f for three light flavors), spin singlet with even parity: [qq]zc3fo+. This channel is favored by one gluon exchange and by instanton interactions. It will play the central role in the exotic drama to follow. The classification of diquarks is not entirely trivial. Operators that will create a diquark of any (integer) spin and parity can be constructed from two quark fields and insertions of the covariant derivative. We are interested in potentially low energy configurations, so we omit the derivatives. There are eight distinct diquark multiplets (in colorxflavor xspin) that can be created from the vacuum by operators bilinear in the quark field’. However, the interesting candidates can be pared down quickly: Color 6, diquarks have much larger color electrostatic field energy. All models agree that this is not a favored configuration. Odd parity diquarks require quarks to be excited relative to one another. This leaves only two diquarks consistent with fermi statistics,

e+.

(hl3)c c 4 S f ( 4Of@)) ) { 4 d %(A) 6f(S>l+(S))

(1) where A or S denotes the exchange symmetry of the preceding representation. Both of these configurations are important in spectroscopy. In what follows I will refer to them sometimes as the “scalar” and “vector” diquarks, or more suggestively, as the “good” and “bad” diquarks. Remember, though, that there are many “worse” diquarks that we are ignoring entirely.

201 Models universally suggest that the scalar diquark is lighter than the vector. For example, one gluon exchange evaluated in a quark model gives rise t o a color and spin dependent interaction that favors the scalar diquark. The matrix elements of this interaction in the “good” and “bad” diquark states are -8MoO and +8/3Moo respectively, where Moo is model dependent. To set the scale, the A-nucleon mass difference is 16MO0, so the energy difference between good and bad diquarks is $ ( M A- M N ) 200 MeV. Not a huge effect, but large enough to make a significant difference in spectroscopy. After all, the nucleon is stable and the A is 300 MeV heavier and has a width of 120 MeV!

-

N

3.1. Characterizing diquarks

The good scalar and bad vector diquarks are our principal subjects. Since the good diquarks are antisymmetric in flavor they lie in the 3 representation of S U ( 3 ) f . We will denote them by [ q l ,421 : { [u, d] [d, s] [s, a ] } when flavor is important and by when it is not. Under flavor S U ( 3 ) transformations they behave exactly like antiquarks, [u, d] c-) 3, [d,s] t--f ‘L1, [s, u] H 2. The bad diquarks are symmetric in flavor, forming the 6 representation of SU(3)f. The notation {q1,q2} : {{u,u}{u,d} (64( 4 s ) {s,s} { s , u } } will do. Although diquarks are colored states, their properties can be studied in a formally correct, color gauge invariant way on the lattice. To define the non-strange diquarks, introduce an infinitely heavy quark, Q, ie a Polyakov line. Then study the qqQ correlator with the qq quarks either antisymmetric ([u, d ] Q ) or symmetric ( { u ,d } Q ) in flavor. The results, M [ u ,d] and M { u , d } - labelled unambiguously - are meaningful in comparison, for example, with the mass of the lightest qQ meson, M ( u ) = M ( d ) . M { u , d } - M [ u ,d] is the good-bad diquark mass difference for massless quarks. It is a measure of the strength of the diquark correlation. The diquark-quark mass difference, M [ u , d ]- M ( u ) , is another. The same analysis can be applied to diquarks made from one light and one strange quark giving M [ u ,s] and M { u , s}. These mass differences are fundamental characteristics of QCD, which should be measured carefully on the lattice. In practice we can estimate these masses by replacing the infinitely heavy quark by the physical charm or bottom, or even the strange quark. The analysis is complicated by the fact that the spin interactions between the light quarks and the s, c or b quark are not negligible. Of course the scalar diquark has no spin interaction with the spectator heavy quark (Q),

202 but the vector diquark does. We parameterize it by K(Q, {ql, q z } ) , for Q = s, c, b. The light antiquark and heavy quark in a 4Q meson has a similar interaction, K(Q, q ) . In order to obtain estimates of diquark mass differences, it is necessary to take linear combinations of baryon and meson masses that eliminate these spin interactions'. Among the non-strange quarks, we obtain 1 M { u , d)lQ - M [ u ,dllQ = 3 (2M(C;)

+ M ( C Q ) )- M(AQ)

1 = M(AQ)- 4 ( M ( D Q )+3M(DG)) 1 K(Q,{ u , d ) ) = j ( M ( C ; ) - M ( ~ Q ) )

M[u,dllQ - M(u)IQ

(2)

To obtain useful information from the BQ and RQ (0 = ( Q s s ) ~ = ' / ~ ) states, it is necessary to assume that both the bad diquark mass and the spin interaction are linear functions of the strange quark mass, M { s , S } ~ Q M { u ,~ ) I Q = 2M{u, S ) ! Q and K(Q,{s, S}) + K(Q,(21, d ) ) = 2K(Q, {u, s)), amounting to first order perturbation theory in m,. With this we can deduce,

+

hf{'LL, S } / Q

- hf[u, S ] ~ Q=

2

j ( h f ( E 6 ) + h f ( C Q ) + hf(RQ))

M [ u , s ] l Q - M(s)lQ = M ( E Q )

- h f ( E Q )-

hf(z:(Q)

1

+ M ( Z b ) - Z ( M ( C Q )+ M(RQ))

1 - -4( M ( D ~ Q )+ 3 M ( D , * Q ) ) 1 K(Q,('4 s)) = - (2M(EG) - M ( ~ Q-)M ( ~ Q .) ) 6

(3)

When we substitute numbers into eqs. (2)-(3), quite a consistent picture of diquark mass differences and diquark-spectator interactions emerges: First,

M[u,d]l,- M ( u ) l s = 321 MeV M { u , d ) l , - M[u , d] J= , 205 MeV M[u,dlIc- M(u )l c = 312 MeV M{u,d)Ic - M[u,d]Ic= 212 MeV M[u,dllb - M(u)lb = 310 MeV

(4) shows that the properties of hypothetical non-strange diquarks are the pretty much the same when extracted from the charm and bottom, and

203 even strange, baryon sectors. Second,

M { u , s}lc - M [ u ,s]Ic = 152 MeV M [ u , s ] lc- M ( s ) l , = 498 MeV

(5)

shows that the diquark correlation decreases when one of the light quarks is strange. This is certainly to be expected, since it originates in spin dependent forces. As the correlation decreases, the mass difference between the scalar and vector diquarks decreases (-210+-150 MeV) and the mass difference between the scalar diquark and the antiquark increases (-310+-500 MeV). Finally,

K ( s ,{ u , d } ) = 64 MeV K ( c ,{u,d } ) = 21 MeV K ( c ,{u,s}) = 24 MeV shows that the non-strange vector diquark interaction with the spectator charm quark is significantly weaker than with a spectator strange quark, as expected from heavy quark theory. The only mildly surprising result is that the { u , s} and { u , d } vector diquarks have roughly the same interaction with the charm spectator. It will be very interesting to compare these results with further measurements in the b-quark sector and, of course, with the results of lattice calculations. 4. Diquarks and Exotics

4.1. A n overview

I want to look at exotics assuming little more than that two quarks prefer to form the good, scalar diquark when possible. States dominated by that configuration should be systematically lighter, more stable, and therefore more prominent, than states formed from other types of diquarks. This qualitative rule leads to qualitative predictions - all of which seem to be supported by the present state of experiment. This is clearly an idealization - a starting place for describing exotic spectroscopy. To learn the real dominance will require more models and more information from experiment. The qualitative ideas explored here are not powerful enough to fix the overall mass scale of any given sector in QCD. So we cannot predict the existence of (nearly) stable exotic pentaquarks. As was the case of the large N,-dynamics of Jenkins and Manohar, once a particle like the O+ is found, it sets the scale, and leads to many interesting predictions.

204

dominance are simple, and striking. The predictions that follow fro of exotic spectroscopy and provide They capture all the important fe the conceptual basis of a unified description of this sector of QCD. ould be no (light, prominent) exotic mesons: The good flavor 3, just like the antiquark. Tetraquarks, 4444, poexotics in 27,10, and 10 representations of flavor S U ( 3 ) . contains only non-exotic representations, 1 and 8, just like q @ q: q3 @ q3 = ( t j q ) l @ compared with Other diquark-antidiquark mesons are heavie in the meson-meson continuum. As described in Section 1I.C probably they are not just “broad”, but in fact absent43. [b] The only prominent tetraquark mesons should be an SU(3) nonet with Jn = Of. This prediction - a simple corollary of the one just above - dates back to the late 1 9 7 0 ’ ~Since ~ ~ . the good diquarks, bosons, the spinparity of the lightest nonet is J” = O+. Over the years evidence has accumulated that the nine O+-mesons with masses below 1 GeV (the f0(600), ~ ( 8 0 0 )f0(980), , and ~ ( 9 8 0 )have ) important tetraquark componentsz3. Space does not permit me to present the evidence here. The interested reader can find more in Ref. [c] If there are any exotic pentaquark baryons, they lie in a positive parity of SU(3)f. This is also a simple consequence of combining good diquarks. To make pentaquarks it is necessary to combine two diquarks and an antiquark. The result is 3€338 3 = 1 @ 8 @ 8 @ 10. The only exotic is the 10. Other exotic flavor multiplets, like the 27 and 35, which occur in the uncorrelated quark picture and/or the chiral soliton models, should be heavier and most likely lost in the meson-baryon continuum. [d] Nuclei will be made of nucleons. To a good approximation, nuclei are made of nucleons - a fact which QCD should explain. If diquark correlations dominate, systems of 3A quarks should prefer to form individual nucleons, not a single hadron. The argument is based on statistics: Good diquarks are spinless color anti-triplet bosons. Only one, [u,d], is non-strange. A six-quark system made of three of these, antisymmetrized in color to make a color singlet, would have to have fully antisymmetric space-wavefunction to satisfy Bose statistics. The simplest would be a triple-scalar product, $1 . $2 x $3, which should be much more energetic than two separate, color-singlet nucleons in an s-wave (eg. the deuteron). The argument generalizes to heavy nuclei. Of course it does not explain nuclear binding or the rich phenomena of nuclear physics.

’.

205 4.2. Pentaquarks from diquarks I: The general idea

Again, we assume that the scalar diquark dominates the spectrum. The rest follows from rather simple considerations of the symmetry of the function in color, flavor, and space (the spi function is trivial)15J6. The good diquark is a spinless boson so the wavefunction must be symmetric under interc The two diquarks must couple to a color 3,, wavefunction is antisymmetric in color. Two choices remain: It can be (a) antisymmetric in flavor and symmetric in space; or (b) symmetric in flavor and antisymmetric in space. Symmetric in flavor means 6 and antisymmetric means 3: p C3 = p 8 A = 3. Symmetry in space means even parity and a tower of states presumably beginning with e = 0. Antisymmetry in space means odd parity and a tower beginning with l= 1. So the candidates for light pentaquarks in the diquark scheme fall into two categories: (a) A negative parity nonet with J" = 1/2-. Overall the four quarks carry no spin and form a flavor triplet. [The three states are [[u, d], [d, s]], [[d,s ] , [s,u],and [[s, u], [u, 611.3 They combine with the antiquark to form nine states with J" = 1/2+; and (b) A positive parity 18-plet (an octet and antidecuplet) with J" = 1/2+ & 3/2+. With e = 1 the four quarks couple to the antiquark to make J" = 1/2-, or 3/2-. The six four quark states, [u, d2,{[u, d], [u, s ] } , etc, combine with the antiquark to make eighteen states that include both a flavor octet and antidecuplet. The quark content of the eighteen states is summarized in Fig. 1, where ideal mixing has been assumed. In Fig. 1 the SU(3)f weight diagrams of the unmixed octet and antidecuplet are shown on the left. The results after ideal mixing are shown on the right. The exotics in the antidecuplet do not mix with the octet. Isospin symmetry precludes mixing between the A and the Cos or between the Eoy- and the a0>-. The other states, the N's and the C's, mix to diagonalize the number of 5s pairs. One set has hidden strangeness, the other does not. Which multiplet, the odd parity nonet or the even parity 18-plet, is lighter depends on the quark model dynamics. This is exactly the same question we encountered in the large N, classification of Jenkins and Manohar. Color-spin interactions modeled after one gluon exchange favor the s-wave, ze the odd parity nonet, but there may be Pauli blocking in this state as in the H . This effect would elevate the mass of the negative parity nonet. Flavor-spin interactions, modeled after pseudoscalar meson exchange48,

q

s,

q

206

I

*+

1

(‘1)

Octe\ and Anticlccupirt

(b) Ideally hlivcd Quark Contcnt

Figure 1. Even parity pentaquark 18-plet: diquark pairs in the Gf combine with an The S U ( 3 ) weight diagram for the 8fand antiquark in the & to make a 8fand i&. is shown at left, where the unmixed states are named (the decuplet in black, the octet in grey). The ideally mixed states, some with their valence quark content, are shown a t right. The exotics (O+, @--, and @+)and certain octet states (A, So, Z-) do not mix if isospin is a good symmetry.

mf

apparently favor the pwave (in contrast to the qq, qqq, and qQq sectors where the ground state is always the s-wave), making the even-parity 18plet the lightest. Whichever way, the diquark picture leads to clear predictions for the light pentaquarks: The only potentially light, prominent exotic multiplet is the antidecuplet, which contains candidates for the O+, the @--, and an as yet unreported @+. The exotics are accompanied by an non-exotic octet, which mixes with the antidecuplet to give several non-exotic (or “cryptoexotic”) analogue states, for example a [u,dl[u,d&and [u,dl[u,d]dpair, which should be lighter than the O+. There are no other light, prominent exotics, like the 27 that figures prominently in the chiral soliton model. The O+ should have positive parity. The exotics should come in spin-orbit pairs with J“ = and %.

207 More predictions include SU(3)f mass splittings and the existence of charm and bottom analogue states discussed in detail in Ref. I will have little further to say about the negative parity nonet. These states couple strongly to the meson-nucleon s-wave. The non-strange members of the multiplet contain an Ss pair and should therefore couple to N q and A K , not to N n . Unless these states were below fall apart decay threshold they would be lost in the meson-nucleon continuum. The absence of candidates in the PDG tables should not be surprising. A word about complications that I have ignored in this presentation: First are the states constructed from the other diquarks: Residual interactions will certainly mix them into the Qq states, but a t zeroth order the goodxbad and badxbad states are -200 and -400 MeV heavier than the goodxgood states. If the lightest states in each family are the s-waves - as QCD based interactions prefer -then these states are all well above threshold to fall apart into meson and baryon, and disappear into the continuum. Among them are many exotics, but only one candidate for a negative parity antidecuplet. Goodxbad states lie in the 3@6@3 of SU(3)f which includes the exotic 27. Badxbad states lie in the 6 @ 6 @ 5 and include a negative parity antidecuplet (as well as the 35). So the first candidate for a negative parity 0’ lies in the “bad-bad” sector and furthermore is created by the same operator that creates K N in an s-wave1. So the diquark picture is quite firm that a negative parity O+ is much heavier and strongly coupled to the K N s-wave continuum. Second is mixing between qqqqq states and ordinary qqq baryons. Mixing is possible when the qqqqq states are not exotic, especially if there are qqq states with the same quantum numbers nearby. Mixing will alter both the spectrum and the decay widths that would otherwise be determined by S U ( 3 ) flavor symmetry.

’.

4.3. Pentaquarks from diquarks 11: A mom detailed look at the positive parity octet and antidecuplet

If the O+ and its brethren are confirmed, and if they have positive parity, then the diquark based pentaquark picture seems like a strong candidate for a quark description of the structure. The diquark model predictions for masses, mixings, and flavor selection rules can be found in Refs. and are reviewed in Ref. l. The spectrum is summarized in Table 2. The SU(3) violating effects of 15747,49

208

the s-quark mass have been included to lowest order in perturbation theory, which has been perfectly adequate for all qqq baryons and Qq mesons in the past. For reasons discussed in Ref. it is reasonable, to lowest order, to assume that the symmetry breaking Hamiltonian acts independently in the four quark 6 and the antiquark 3. This leaves a mass formula that depends mass, p the matrix on only three parameters, Mo, the unperturbed 8 and element of m,ss, and a , the mass difference M [ u , s ]- M[u,d]. In Ref. l5 (labelled Mass I in the table) p was taken from the @-Ropermass difference. a was taken from a full quark model analysis of the C - A mass difference, which goes beyond the diquark hypothesis, and led to a prediction of 1750 MeV for the Q,--. At the time the prediction of a relatively light Q, was rather daring. Now that the Q,-- has reported a t 1860 MeV, it seemed appropriate to re-examine these predictions and assignments. In retrospect taking a from the A-C system may not have been particularly appropriate. The value extracted in Ref. l5 assumed the full color-spin Hamiltonian of the quark model. Instead in Ref. 49 Wilczek and I propose to identify C(l660), a 1/2+ resonance given three stars by the PDG12, with the C states in the 18-plet. This choice is motivated by a global fit to baryon resonances5’ and is labelled “Mass II” in Table 2.

There is an important, qualitative difference between the diquark picture with its 18-plet and other models of the @+ with an antidecuplet alone, which will help sort out the correct physical picture of the exotics. In an antidecuplet-only picture the @+, NE, CE, and Q, must be spaced a t equal mass intervals. In their original paper5 Diakonov, Petrov, and Polyakov identify the Nm with the N(1710), which puts the Q, at 2070 MeV, much higher than the quark content (uudds + ddssti) would suggest15. If the

209 @ lies at 1860, the N m and Cm must lie near 1650 MeV and 1750 MeV respectively. In their revised discussion of the spectrum, Diakonov and Petrov, fit the @(1860)and predict 1/2+ N and C resonances in the intervals 1650-1690 and 1760-1810 respectivelys1. In contrast with the 1 - o n l y , the 18-plet picture suggested by diquark arguments allows the O+ and to be interior to the multiplet, with the C,(uussS) and N(uuddG) at the top and bottom respectively. As noted in the Table, there are possible candidates for all the 18-plet states.' Time will tell which of these qualitatively different spectra are closest to Nature - provided, of course, the exotics survive the next round of experiments. 4.4. Pentaquark from diquarks 111: Charm and bottom

analogues Charm and bottom analogues of the O+ can be obtained by substituting the heavy or 6 quark for the 3 in the O+, @: =

I[u,dl"L1,dlc)

@- = I[% dlb, 46)

(7)

=

The existence of the 0: O+(1540) fixes the mass scale for exotics and leads to rather robust predictions of the masses of the O: and O t . The simplest, though not necessarily the least accurate approach, is to find an Q system. The obvious analogy among qqQ baryons and apply it to the choice is the AQ, which has the quark content The heavy anti-quark inglet, color 3,spin singlet pair in the OQ sits in the background of an sits in a background identical of diquarks. The heavy quark in the in isospin, color, and spin. The only difference is that the spin of the 0 in the OQ can interact with the orbital angular momentum (! = 1) in the OQ and this interaction is not would expect the relations M ( is M ( @ ) - M ( O $ ) = M ( A , ) - M ( A ) to be nearly exact. QCD spin-orbit interactions are not strong, so these rules should not be badly violated. The differences among the predictions of various QCD based quark models reflect the different ways that the residual interactions are treated. From the relation quoted Wilczek and I estimated, Ad(@:) = 2710 MeV and M ( O z ) = 6050 MeV. CAlthough the width of the Roper presents a problem5', suggesting that non-exotic qqqqg states may mix significantly with qqq states.

210

If these estimates are correct, the 0: and 0: will be stable against strong decay. The lightest strong decay channel for the 0: is N D with a threshold at 2805 MeV, and for the @ ,: it is N B with a threshold at 6220 sec. MeV. They would have to decay weakly with lifetimes of order How did this happen? The 0: is light, but it is not stable. The reason lies not in the linear scaling of the masses of the heavy pentaquarks with the heavy quark mass, but rather in the non-linear scaling of the pseudoscalar meson masses, which determine the strong decay thresholds. Consider the four analogue states, [u,d][u,d]Q, with Q = u , s , c , ~and , identify the 00, with the Roper as I advocated earlier. Then

+ N7r 0: + N K 0: + N D 0: + N B 0;

has decay Q-value

Q

M

350 MeV

has decay Q-value

Q

M

100 MeV

has decay Q-value

Q

M

-100 MeV

has decay Q-value

Q x -150 MeV

(8)

The 0;, ie the Roper, is unstable because the pion is anomalously light, a consequence of approximate chiral symmetry. The effect is still significant enough for the kaon to make the @$unstable. The D and B-meson masses are not significantly lowered by chiral symmetry, the thresholds are proportionately higher, and the @: and @: are stable. The details are model dependent. Other model estimates are generally higher than the simple scaling law described here53, some predict stable c and b-exotics, others predict light and narrow, but not stable states. The exotic charm baryon reported by H1 is not bound. With a mass of 3099 MeV, it is much too heavy to be the 0: as I have described it. The width is reported to be less than 12 MeV. It has been observed through its strong decay into D*-p, into which it has a Q value of 150 MeV. If it were the @!, and if it has J" = 1/2+, it would have a significant decay into D-p (which would not have been seen at H l ) , with a partial width that can be related to the width of the 0: by scaling pwave phase space. The result is l?(3099)/r(0$) M 15, barely consistent with the H1 limit if the width of the 0: is 1 MeV. An interesting possibility - if the 3099 state should be confirmed - is that it is an L = 2 Regge excitation of the 0: with J" = 312-. This object can decay into D*-p in the s-wave, but D-p in the d-wave, accounting perhaps for its surprisingly narrow width. Should this assignment prove correct, there must be many other excited charm exotic baryons awaiting discovery. Clearly, if the initial reports are confirmed, there is a fascinating spec-

21 1

troscopy of heavy exotic baryons awaiting us. But it is a big “if”!

5 . Conclusions

There are two distinct, but related issues at the core of this discussion: first, a question: are there light, prominent exotic baryons, and if so, what is the best dynamical framework in which to study them? and second, a proposal: diquark correlations are important in QCD spectroscopy, especially in multiquark systems, where they account naturally for the principal features. I believe the case for diquarks is already quite compelling. There are many projects ahead: re-evaluating the 444 spectrum5’; systematically exploring the role of diquarks in deep inelastic distribution and fragmentation functions, and in scaling violation; seeing if diquarks can help in other areas of hadron phenomenology like form-factors, low p~ particle production, and polarization phenomena; developing a more sophisticated treatment of quark correlations, recognizing that diquarks are far from pointlike inside hadrons; establishing diquark parameters and looking for diquark structure in hadrons using lattice QCD; and - the holy grail of this subject -seeking a more fundamental and quantitative phenomenological paradigm for light quark dynamics at the confinement scale. Diquark advocates have considered many of these issues in the pastlg. No doubt many other important contributions, like the diquark analysis of the AI = 1 / 2 - r ~ l e have ~ ~ , already been accomplished. We can hope eventually to have as sophisticated an understanding of diquark correlations as we have of iiq correlations, as expressed in chiral dynamics. The situation with the O+ is less clear. Of course it will eventually be clarified by experiment - a virtue of working on QCD as opposed to string theory! However, theorists’ attempts to understand the O+ have raised more questions than they have answered. To wit: (a) A negative parity ( K N s-wave) O+ is intolerable to theorists, but that is what lattice studies find, if they find anything at all. (b) No one has come up with a simple, qualitative explanation for the exceptionally narrow width of the O+. (c) The original prediction of a narrow, light O+ in the chiral soliton model does not appear to be robust. (d) Quark models can accomodate the O+, but only by reversing the naive, and heretofore universal, parity of the qnqQllq- ground state. It is necessary to excite the quarks in order to capture the correlation energy of the good diquarks. This does not sound like a way to make an exceptionally light and stable pentaquark. (e) When

212

models are adjusted to accomodate the O+, they predict the existence of other states that should have been observed by now: The diquark picture wants both a of+and a O$+; the CSM and large N, want a relatively light 27, which includes an I = 1 triplet: O*O, O*+, O*++. None of these problems seems insuperable. Indeed, there are papers appearing every day that propose an interesting solution to one or another. Taken together, however, they are an impressive set. They leave us in limbo: Either the O+ will go away, or it will force us to rewrite several chapters of the book on QCD. 6. Acknowledgements

Many of these ideas were developed in collaboration with Frank Wilczek. This work is supported in part by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.

References 1. R. L. Jaffe, hep-ph/0409065. 2. M. Roos et al, “Review Of Particle Properties. Particle Data Group,” Phys. Lett. B 111,1 (1982). 3. G. P. Yost et al, “Review Of Particle Properties: Particle Data Group,” Phys. Lett. B 204, 1 (1988). 4. A. V. Manohar, Nucl. Phys. B 248, 19 (1984); M.Chemtob; L. C. Biedenharn and Y . Dothan, Print-84-1039 (DUKE) in E. Gotsman, G. Tauber, From SU(3) To Gravity p. 15; M. zalowicz, in Skyrmions and Anomalies, M. Jezabek and M. Praszalowicz, eds., World Scientific (1987), p. 112;H. Walliser, Nucl. Phys. A 548 649 (1992). 5. D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359, 305 (1997) [arXiv:hep-ph/9703373]. 6. H. Weigel, Eur. Phys. J. A 2, 391 (1998) [arXiv:hep-ph/9804260]. 7. T. Nakano et a1 [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003) [arXiv:hep-ex/0301020]. 8. For references, see for example, Ref. 9. C. Alt e t a1 “A49 Collaboration], Phys. Rev. Lett. 92, 042003 (2004) [arXiv:hep-ex/0310014]; but see also H. G. Fischer and S. Wenig, arXiv:hepex/0401014 and K. T. Knopfle, M. Zavertyaev and T. Zivko [HERA-B Collaboration], J. Phys. G 30,S1363 (2004) [arXiv:hep-e~/0403020]. 10. A. Aktas e t a1 [Hl Collaboration], arXiv:hep-ex/0403017, but see also K. Lipka [Hl Collaboration], arXiv:hep-ex/0405051. 11. For a recent overview of the experimental situation, see G. Trilling in Ref. l 2 12. S. Eidelman et al [Particle Data Group Collaboration], “Review of particle physics,” Phys. Lett. B 592, 1 (2004).

’.

213 13. For a recent summary of negative results see A. Dzierba, talk presented at at the International Conference on Quarks and Nuclear physics, QNP2004, http://k9.physics.indiana.edu/.ueric/QNP/QNP /QNP-2004-talks/plenary-Friday /dzierba5q-qnp2004.pdf. 14. R. L. Jaffe and K. Johnson, Phys. Lett. B 60,201 (1976). R. L. Jaffe, Phys. Rev. D14 267, 281 (1977). 15. R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003) [arXiv:hepph/0307341]. 16. S. Nussinov, arXiv:hep-ph/0307357. 17. M. Karliner and H. J. Lipkin, arXiv:hep-ph/0307243. 18. M. Ida and R. Kobayashi, Prog. Theor. Phys. 36 (1966) 846; D.B. Lichtenberg and L.J. Tassie, Phys. Rev. 155 (1967) 1601. 19. For a review an further references, M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson and D. B. Lichtenberg, Rev. Mod. Phys. 65, 1199 (1993), or M. Anselmino, E. Predazzi, eds. International Workshop on Diquarks and Other Models of Compositeness: Diquarks III, Turin, Italy, 28-30 Oct 1996 (World Scientific, 1998). 20. M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B 422,247 (1998) [arXiv:hep-ph/9711395]; for a review with further references, see K. Rajagopal, “Color superconductivity,” Prepared f o r Cargese S u m m e r School o n Q C D Perspectives o n Hot and Dense Matter, Cargese, France, 6-18 Aug 2001. 21. See, for example, F. E. Close and A. W. Thomas, Phys. Lett. B 212, 227 (1988). 22. M. Neubert and B. Stech, Phys. Lett. B 231,477 (1989), Phys. Rev. D 44, 775 (1991). 23. C. Amsler and N. A. Tornqvist, Phys. Rept. 389, 61 (2004); F. E. Close and N. A. Tornqvist, J. Phys. G 28, R249 (2002) [arXiv:hep-ph/0204205]; For a recent reconsideration, see L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, arXiv:hep-ph/0407017. 24. R. N. Cahn and G. H. Trilling, Phys. Rev. D 69,011501 (2004) [arXiv:hepph/0311245]. 25. R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 68,042201 (2003) [Erratum-ibid. C 69,019901 (2004)] [arXiv:nucl-th/0308012]. 26. A. Sibirtsev, J. Haidenbauer, S. Krewald and U. G. Meissner, arXiv:hepph/0405099. 27. See, for example, T. Nakano, talk presented at the International Conference on Quarks and Nuclear physics, QNP2004, http: //www.qnp2004.org/5q~talks/T_Nakano.ppt. 28. S. Okubo, Phys. Lett. 5, 165 (1963); G.Zweig, CERN Report No. 8419 T H 412, 1964 (unpublished); reprinted in Devel- opments in the Quark Theory of Hadrons, edited by D. B. Lichtenberg and S. P. Rosen (Hadronic Press, Massachusetts, 1980); J. Iizuka, Prog. Theor. Phys. Suppl. 37,21 (1966). 29. This section is based on R. L. Jaffe and A. Jain, arXiv:hep-ph/0408046, where more details can be found . 30. G. ’t Hooft, Nucl. Phys. B 72,461 (1974).

214 31. E. Witten, Nucl. Phys. B 160,57 (1979). 32. R. F. Dashen, E. Jenkins and A. V. Manohar, Phys. Rev. D 49,4713 (1994) [Erratum-ibid. D 51,2489 (1995)l [arXiv:hep-ph/9310379]. 33. E. Jenkins and A. V. Manohar, Phys. Rev. Lett. 93, 022001 (2004) [arXiv:hep-ph/0401190]; JHEP 0406,039 (2004) [arXiv:hep-ph/0402024]. 34. G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228,552 (1983). 35. E. Guadagnini, Nucl. Phys. B 236,35 (1984). 36. C. G. Callan and I. R. Klebanov, Nucl. Phys. B 262,365 (1985). 37. Ref. quotes a width of order 15 MeV, however the calculation on which that claim is based contains an error, which corrected yields an estimate of -30 MeV 38. 38. R. L. Jaffe, Eur. Phys. J. C 35, 221 (2004) [arXiv:hep-ph/0401187]; D. Diakonov, V. Petrov and M. Polyakov, arXiv:hep-ph/0404212; R. L. Jaffe, arXiv:hep-ph/0405268. 39. T. D. Cohen, Phys. Lett. B 581, 175 (2004) [arXiv:hep-ph/0309111], Phys. Rev. D 70, 014011 (2004), arXiv:math-ph/0407031. For further discussion, see A. Cherman, T. D. Cohen and A. Nellore, arXiv:hep-ph/0408209. 40. N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B 684,264 (2004) [arXiv:hep-ph/0309305]; I. R. Klebanov and P. Ouyang, arXiv:hepph/0408251. 41. M. P. Mattis and M. Karliner, Phys. Rev. D 31, 2833 (1985), Phys. Rev. Lett. 56,428 (1986), Phys. Rev. D 34,1991 (1986). 42. R. L. Jaffe, K. Johnson and Z. Ryzak, Annals Phys. 168,344 (1986). 43. R. L. Jaffe and F. E. Low, Phys. Rev. D 19,2105 (1979). For a pedagogical introduction, see R. L. Jaffe, “HOWto analyse low energy scattering”, in H. Guth, K. Huang, and R. L. Jaffe, eds. Asymptotic Realms of Physics, Essays in Honor of Rancis E. Low, (MIT Press, Cambridge, 1983). 44. S. Sasaki, arXiv:hep-lat/0310014. 45. F. Csikor, Z. Fodor, S. D. Katz and T. G. Kovacs, JHEP 0311,070 (2003) [arXiv:hep-lat/0309090]. 46. N. Mathur e t al, arXiv:hep-ph/0406196. 47. R. Jaffe and F. Wilczek, arXiv:hep-ph/0401034. 48. F. Stancu and D. 0. Riska, Phys. Lett. B 575, 242 (2003) [arXiv:hepph/0307010]. 49. R. Jaffe and F. Wilczek, Phys. Rev. D 69, 114017 (2004) [arXiv:hepph/0312369]. 50. A. Selem and F. Wilcxek, to be published. 51. D. Diakonov and V. Petrov, Phys. Rev. D 69, 094011 (2004) [arXiv:hepp h/ 03 102 121. 52. T. D. Cohen, arXiv:hep-ph/0402056. 53. For references and a review of heavy exotic baryons and their excitations, see K. Maltman, arXiv:hep-ph/0408145.

QUARK STRUCTURE OF CHIRAL SOLITONS

DMITRI DIAKONOV Thomas Jefferson National Accelerator Facility, Newport News, V A 23606, USA NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark St. Petersburg Nuclear Physics Institute, Gatchina, 188 900, St. Petersburg, Russia There is a prejudice that the chiral soliton model of baryons is something orthogonal to the good old constituent quark models. In fact, it is the opposite: the spontaneous chiral symmetry breaking in strong interactions explains the appearance of massive constituent quarks of small size thus justifying the constituent quark models, in the first place. Chiral symmetry ensures that constituent quarks interact very strongly with the pseudoscalar fields. The “chiral soliton” is another word for the chiral field binding constituent quarks. We show how the old S U ( 6 ) quark wave functions follow from the %oliton”, however, with computable relativistic corrections and additional quark-antiquark pairs. We also find the 5-quark wave function of the exotic baryon O+.

1. The necessity of quantum field theory

It has been known since the work of Landau and Peierls (1931) that the quantum-mechanical wave function description, be it non-relativistic or relativistic, fails a t the distances of the order of the Compton wave length of the particle. Measuring the electron position with an accuracy better than cm produces a new electron-positron pair, by the uncertainty principle. One observes it in the Lamb shift and other radiative corrections. Fortunately, the atom size is cm, therefore there is a gap of three orders of magnitude where we can successfully apply the Dirac or even the Schrodinger equation. In baryons, we do not have this luxury. Measuring the quark position with an accuracy higher than the p i o n Compton wave length of 1.4fm produces a pion, i.e. a new quark-antiquark (QQ) pair, whereas the baryon size is 0.8 fm. Therefore, there seems t o be no room for the quantum-mechanical wave function description of baryons a t all. To describe baryons, one needs a quantum field theory from the start, with a varying number of QQ pairs, because of the spontaneous chiral symmetry breaking which makes pions light.

215

216

Ignoring quantum field theory where it cannot be ignored, causes multiple problems. Let me mention just two paradoxes of the standard constituent quark models, out of many. The first is the value of the so-called nucleon sigma term It is experimentally measured in low-energy nN scattering, and its definition is the scalar quark density in the nucleon, multiplied by the current (or bare) quark masses,

’.

mu

+m d

+

< Nlau &IN >= 67 f 6 MeV. 2 The standard values of the current quark masses are mu N 4MeV, md N 7MeV (and m, 2: 150 MeV). In the non-relativistic limit, the scalar density is the same as the vector density; therefore, in this limit the matrix element above is just the number of u, d quarks in the nucleon, equal to 3. If u,d quarks are relativistic, the matrix element is strictly less than three. Hence, in the naive constituent quark model o=

cquarks

MeV * 3 = 17.5 MeV, 5 MeV -I2

that is four times less than experimentally! Three quarters of the (T term is actually residing not in the three constituent quarks but in the additional quark-antiquark pairs in the nucleon. The second paradox which is probably less known, arises when one attempts t o extract quark distributions as function of Bjorken z from a constituent quark model, be it any variant of the bag model or any variant of the potential models with any kind of correlations between quarks. If the three quarks are loosely bound, their distribution function is just 6 (z each quark carrying 1/3 of the nucleon momentum in the infinite momentum frame. As quarks become more bound, this &function is smeared around 1/3. However, higher quark velocities imply that the “lower” component of the Dirac bispinor wave function increases (it is zero in the extreme non-relativistic case), a t the expense of the decrease of the “upper” component. It means that if quarks are moving inside a nucleon, there are less than three quarks in the nucleon. Since the number of quarks minus the number of antiquarks is the conserved baryon number, it automatically means that the number of antiquarks is negative ’. It is a n inevitable mathematical consequence of the Dirac equation. The paradox is cured by adding the Dirac sea t o valence quarks; only then the antiquark distribution becomes positive-definite, and satisfies the general sum rules Thus, a field-theoretic description of baryons is a must if one does not wish to violate general theorems, and also for practical reasons.

i),

’.

217

I present below a relativistic field-theoretic model of baryons where the above paradoxes are resolved, together with the well-known “spin crisis” paradox. Actually, one has to be surprised not by why the constituent quark approach is a failure but rather why does it work a t all in a variety of cases. The model will answer this question, too. 2. The chiral quark - soliton model The most important happening in QCD from the point of view of the light hadron structure is the Spontaneous Chiral Symmetry Breaking (SCSB): as its result, almost massless u , d, s quarks get the dynamical momentumdependent masses MU,d,+(p),and the pseudoscalar mesons 7r,K , q become light (pseudo) Goldstone bosons. At the same time, pseudoscalar mesons are themselves bound QQ states. How to present this queer situation mathematically? There is actually not much freedom here: the interaction of pseudoscalar mesons with constituent quarks is dictated by chiral symmetry. It can be written in the following compact form 3:

L,ff = 4 [i@-

M exp(i y5 7rAXA/F,)] q,

7rA

= 7 r , K , 77.

(1)

Since Eq.(l) is an effective low-energy theory, one expects formfactors in the constituent quark - pion interaction; in particular, M ( p ) is momentumdependent and provides an UV cutoff. In fact, Eq.( 1)is written in the limit of zero momenta. A possible wave-function renormalization factor Z ( p ) can be also admitted but it can be absorbed into the definition of the quark field. Notice, that there is no kinetic-energy term for pseudoscalar fields in Eq.(l). It is in accordance with the fact that pions are not “elementary” but a composite field, made of constituent quarks. The kinetic energy term (and all higher derivatives) for pions appears from integrating out quarks, or, in other words, from quark loops, see Fig. 1.

Figure 1. The effective chiral lagrangian is the quark loop in the external chiral field, or the determinant of the Dirac operator (1). Its real part is the kinetic-energy term for pions, the Skyrme term and, generally, an infinite series in derivatives of the chiral field. Its imaginary part is the Wess-Zumino-Witten term (with the correct coefficient), plus also an infinite series in derivatives *.

218

An interesting question is, how does the effective lagrangian (1) “know” about the confinement of color? One writes Eq.(l) from the general chiral symmetry considerations, and only the formfactors e.9. the dynamical mass M ( p ) are subject to dynamical details. The difference between a confining and a non-confining theory is hidden in the subtleties of the analytical behavior of M ( p ) and possible other formfactor functions in the Minkowski domain of momenta. Specifically, the instanton model of the spontaneous chiral symmetry breaking leads to such M ( p ) that there is no real solution of the mass-shell equation p 2 = M 2 ( - p 2 ) , meaning that quarks cannot be observable, only their bound states! However, this is not the only confinement requirement. Unfortunately, the instanton model’s M ( p ) has a cut at p 2 = 0 corresponding to massless gluons left in the model. In the true confining theory there should be no such cuts. In the bound states problems, however, quarks’ momenta are space-like. Therefore, one can use any reasonable falling function M ( p ) reproducing the phenomenological value of F, constant and of the chiral condensate ‘. As a matter of fact, instantons do it phenomenologically very satisfactory. Constituent u,d, s quarks necessarily have to interact with the 7r, K , 77 fields according to Eq.(l), and the dimensionless coupling constant is actually very large: gTqq(0)= M(O) 21 4, where the constituent quark mass F, M ( 0 ) N_ 350MeV and F, N_ 93MeV are used. The chiral interactions of constituent quarks in baryons, following from Eq.(l), are schematically shown in Fig. 2. Antiquarks are necessarily present in the nucleon as pions propagate through quark loops. The nonlinear effects in the pion field are essential since the coupling is strong. I would like to stress that this picture is a model-independent consequence of the spontaneous chiral symmetry breaking. One cannot say that quarks get a constituent mass but throw away their strong interaction with the pion field. In principle, one has to add perturbative gluon exchange on top of Fig. 2. However, a , is never really strong, such that gluon exchange can be disregarded in the first approximation. The large value of the pion-quark coupling suggests that Fig. 2 may well represent the most essential forces

Figure 2. Quarks in the nucleon (solid lines), interacting via pion fields (dash lines).

219

inside baryons. No ‘‘confiningstrings” are expected in the real world where it is energetically favorable to break an expanded string by creating light pions. Although the low-momenta effective theory (1) is a great simplification as compared to the microscopic QCD, as it uses the right degrees of freedom appropriate a t low energies, it is still a strong-coupling relativistic quantum field theory. Summing up all interactions inside the nucleon of the kind shown in Fig. 2 is a difficult task. Maybe some day it will be solved numerically, e.g. by methods presented by John Hiller in these Proceedings ‘. In the meanwhile, it can be solved exactly in the limit of large number of colors N,. With N, colors, the number of constituent quarks in a baryon is N,, and all quark loop contributions in Fig. 2 are also proportional t o N,. Therefore a t large N,, quarks inside the nucleon create a large, nearly classical pion field: quantum fluctuations about the mean field are suppressed as l / N c . The same field binds the quarks; therefore it is called the self-consistent field. [A similar idea is exploited in the shell model for nuclei and in the Thomas-Fermi approximation to atoms.] The problem of summing up all diagrams of the type shown in Fig. 2 is reduced to finding a classical self-consistent pion field. As long as l / N c corrections to the mean field results are under control, one can use the large-N, logic and put N , to its real-world value 3 a t the end of the calculations. The model of baryons based on these approximations has been named the Chiral Quark Soliton Model (CQSM) ‘. The “soliton” is another word

-

- - _ - - - - - ---_, a_ _ _ - - _ - - _ _ _ -

-- - --

-

i* BI

E-sea

r

i

m

--

E = -lM

Dirac sea

Figure 3. If the trial pion field is large enough (shown schematically by the solid curve), there is a discrete bound-state level for three ‘valence’ quarks, Eval.One has also to fill in the negative-energy Dirac sea of quarks (in the absence of the trial pion field it corresponds to the vacuum). The continuous spectrum of the negative-energy levels is shifted in the trial pion field, its aggregate energy, as compared to the free case, being Es,,,. The nucleon mass is the sum of the ‘valence’ and ‘sea’ energies, multiplied by three The self-consistent pion field binding quarks colors, M N = 3 (E,,l[n(z)] Esea[n(z)]). is the one minimizing the nucleon mass.

+

220 for the self-consistent pion field in the nucleon. However, the model operates with explicit quark degrees of freedom, which enables one to compute any type of observables, e.g. relativistic quark (and antiquark!) distributions inside nucleons 2 , and the quark light-cone wave functions ‘. In contrast to the naive quark models, the CQSM is relativistic-invariant. Being such, it necessarily incorporates quark-antiquark admixtures to the nucleon. Quark-antiquark pairs appear in the nucleon on top of the three valence quarks either as particle-hole excitations of the Dirac sea (read: mesons) or as collective excitations of the mean chiral field. There are two instructive limiting cases in the CQSM: 1. Weak T ( Z ) field. In this case the Dirac sea is weakly distorted as compared to the no-field and thus carries small energy, E,,, 21 0. Few antiquarks. The valence-quark level is shallow and hence the three valence quarks are non-relativistic. In this limit the CQSM becomes very similar to the constituent quark model remaining, however, relativistic-invariant and well defined. 2. Large T ( X ) field. In this case the bound-state level with valence quarks is so deep that it joins the Dirac sea. The whole nucleon mass is given by E,,, which in its turn can be expanded in the derivatives of the mean field, the first terms being close to the Skyrme lagrangian. Therefore, in the limit of large and broad pion field, the model formally reduces to the Skyrme model. The truth is in between these two limiting cases. The self-consistent pion field in the nucleon turns out to be strong enough to produce a deep relativistic bound state for valence quarks and a sufficient number of antiquarks, so that the departure from the non-relativistic constituent quark model is considerable. At the same time the self-consistent pion field is spatially not broad enough to justify the use of the Skyrme model which is just a crude approximation to the reality, although shares with reality some qualitative features. The CQSM demystifies the main paradox of the Skyrme model: how can one make a fermion out of a boson-field soliton. Since the %oliton” is nothing but the self-consistent pion field that binds quarks, the baryon and fermion number of the whole construction is equal to the number of quarks one puts on the valence level created by that field: it is three in the real world with three colors.

3. Baryon excitations There are excitations related to the fluctuations of the chiral field about its mean value in the baryons. In the context of the Skyrme model many

221 resonances were found and identified with the existing ones in Ref. 7,8 and quite recently in Ref. '. As I said before, the Skyrme model is too crude, and one expects only qualitative agreement with the Particle Data. The same work has to be repeated in the CQSM but it has not been done so far. There are also low-lying collective excitations related to slow rotation of the self-consistent chiral field as a whole both in ordinary and flavor spaces. The result of the quantization of such rotations was first given by Witten lo. The following SU(3) multiplets arise as rotational states of a chiral soliton: , (10, , , (27, , (27,;') ... They

$+) (m, a+)

(8,i')

:+)

are ordered by increasing mass, see Fig. 4. The first two (the octet and the decuplet) are indeed the lowest baryons states in nature. They are also the only two that can be composed of three quarks. However, the fact that one can manage to obtain the correct quantum numbers of the octet and the decuplet combining only three quarks, does not mean that they are made of three quarks only. The difficulties of such an interpretation have been mentioned in the beginning.

__

4

_ _ _ _ .z

..__ .__ &-. _ ._ 4_ ..._. _ _ ._ _ 4_ _

f

-=

,

,

I:

z+ Q

_-=i - , , - =o

i

-

=+

n(8,1/2)

(10:3/2)

(iO,l/2,

Figure 4. The lowest baryon multiplets which can be interpreted as rotational states in ordinary and 3-flavor spaces, shown in the Y - T3 axes.

Therefore, one should not be a priori confused by the fact that higherlying multiplets cannot be made of three quarks: even the lowest ones are not. A more important question is where to stop in this list of multiplets. Apparently for sufficiently high rotational states the rotations become too fast: the centrifugal forces will rip the baryon apart. Also the radiation of pions and kaons by a fast-rotating body is so strong that the widths of the corresponding resonances blow up Which precisely rotational excitation is the last to be observed in nature, is a quantitative question: one needs to compute their widths in order to make a judgement. If the width turns out to be in the hundreds of MeV, one can say that this is where the rotational sequence ceases to exist. An estimate of the width of the lightest member of the antidecuplet,

222

shown at the top of the right diagram in Fig. 4, the Of, gave a surprisingly small result: I?@ < 15MeV 12. This result obtained in the CQSM, immediately gave credibility to the existence of the antidecuplet. It should be stressed that there is no way to obtain a small width in the oversimplified Skyrme model. In pentaquarks forming the antidecuplet shown on the right of Fig. 4, the additional QQ pair is added in the form of the excitation of the (nearly massless) chiral field. Energy penalty would be zero, had not the chiral field been restricted to the baryon volume. Important, the antidecupletoctet splitting is not twice the constituent mass 2M but less. In the case of a large-size baryon it costs a vanishing energy to excite the antidecuplet 13. 4. Quark wave functions

The wave function of baryons in the CQSM has been derived recently by Petrov and Polyakov in the infinite momentum frame. Here I translate it to the baryon rest frame. We shall see how easily one can get the nonrelativistic SU(6) wave functions for ordinary octet and decuplet baryons “from the soliton”. Next, I derive the new result for the antidecuplet 5quark wave functions. Let a , at(p) and b, b+(p)be the annihilation-creation operators of quarks and antiquarks (respectively) satisfying the usual anticommutator algebra. The vacuum (00> is such that a,b(Oo >= 0. According to the CQSM, a baryon is N, “valence” quarks on a discrete level created by the selfconsistent pion field, plus the negative-energy Dirac sea of quarks, distorted as compared to the free case by the same self-consistent pion field, see Fig. 2. At large N,, the Dirac sea is given by the coherent exponent coherent exponent = exp (/(dp)(dp’) at (p)W(p, p’)bt (p’)) 100>, (2) where (dp) = ~ i ~ p / ( 2 and 7 ~ )W(p1,pz) ~ is the finite-time quark Green function at equal times in the static external field of the chiral “soliton”, to be specified below. The valence quark part of the wave function is given by a product of N , quark creation operators that fill in the discrete level: NC

valence =

IT /@PI

F(P) J(P),

(3)

color= 1

~ ( p=/(dp’) ) [ . * ( P > f i e v ( P ) ( 2 ~ ) 3 ~ ( p - p ’ ) - ~ (P’)V*(P‘)fiev(-P)] pi ,(4)

223 where fiev(p) is the Fourier transform of the wave function of the level. The second term in Eq.(4) is the contribution of the distorted Dirac sea to the one-quark wave function; I shall neglect it for simplicity in what follows. With the same accuracy, the discrete level’s wave function can be approximated by the upper component (as if it was non-relativistic):

where h(r) is the L = 0 solution of the bound-state Dirac equation with energy E E [-M, M ] for the given profile function of the soliton P ( T )4:

+ ( E + Mcos P )j , j ’ + -2 = ( M cos P - E ) h + M s i n P j . r Ph

h’ = -&‘sin 2.

In the non-relativistic limit the L= 1 function j ( r ) is neglected in Eq.(5). In Eq.(5) i = 1 , 2 are spin and j = 1 , 2 are isospin indices; e i j is the antisymmetric tensor. The QQ pair wave function W(p1, p2) determines the structure of the Dirac continuum; it is also a matrix in both spin and isospin indices. I denote by ( i , j ) those of the quark and by (z’,j’) those of the antiquark. We shall need the Fourier transforms of all odd (IT) and all even (C) powers of the self-consistent pion field:

J

I!, (9)= dr e--i(q.r)(n. TI;,sin ~ ( r, ) 3

c:, (9) = /dr

e--i(q.r)d;,

(cos P ( r ) - 1).

+

Correspondingly, W = W(”) W(’) can be divided into two pieces,

W ; ; p ) ( p , p ’ ) = ul:!“’”(p,p’)~(C);,(p

+ p’),

where iW,’) =

Wi’

+ E’)

2(€ l

MM‘ d €E’(M + € ) ( M I+ € 1 )

[(P.P’) - ( M

+

+ € ) ( M I+ €91bit + iEpqrTpP;(OT):’,

with E = dM2(p) p2, the primed variables being related to the antiquark. In the coordinate space the pair wave function is given by a convo-

224 lution of the self-consistent chiral field and the Fourier transforms of

~ ( ~ 1 ’ ) :

These functions can be computed numerically once the profile function of the self-consistent chiral field is known. Eqs.(9,10) give the amplitudes of various spin, isospin and orbital QQ states inside a baryon. The partial waves depend on the QQ coordinates (r,r‘) with respect to the baryon center of mass. To get quark wave functions inside a particular baryon, one has to rotate all the isospin indices j’s, both in the discrete level and in the QQ pairs, by an SU(3) matrix R f , f = 1,2,3, j = 1 , 2 , and to project it to the

4’)

(m, 4’).

, (10,;’) or “Project” means specific baryon from the (8, integrating over the SU(3) rotation matrices R with a Haar measure normalized to unity. In full glory, the quark wave function inside a particular baryon B with spin projection k is given by Q.~B =

s

~ R D ,*(R)E--aNc B

fj/(dpn)

Rf;Fi”jqpn)a; ,, ,i,(pn)

n= 1

.exp ( / ( d p ) ( d p ’ ) a t u , i ( p ) f~w j jj‘~ i , ( p , p ’ ) R ~ ~ ’ ~ t ~ f lao> ’ z ’ ~ p. ’ (11) )) Here Q stands for color, f for flavor and i for spin indices. Let me give a few examples of the baryons’ (conjugate) rotational wave functions DB* (R): neutron, spin projection IC : 3 A++ , spin projection - : 2

+

A’, spin projection

+ -21 :

o+,spin projection k :

D;* = d ‘ i i E k l ~ I 1 ~ , 3 , DA++ TT *

- mRf,2Rf,2Ri2,

D p o* = O R ; 2(2Rf,2Ri

D? * = d36R;R;R;.

(12)

(13)

+Ri’Rf, ’),( 14) (15)

If the coherent exponent with Q o pairs is ignored, one gets from the general Eq.(ll) the 3-quark Fock component of the octet and decuplet baryons. It depends on the quark “coordinates”: the position in space (r), the color ( a ) ,the flavor ( f ) and the spin (i), and also on the baryon spin

225

k. For example, the neutron 3-quark wave function turns out to be ( in >k ) f 1 f 2 f 3 ,i1izi3 (rl, r2, r3) = ~ f l f zeiliz bzf3 :6 h ( ~ l ) h ( ~ 2 ) h ( ~ 3 ) +permutations of 1,2,3,

(16)

antisymmetrized in color. It is better known in the form InT> = 2 d f (rl)df ( T 2 ) u l (T3)-dT permutations of T I , ~ 2 , ~

+

(T1)uf ( T 2 ) d i

(T3)-uT ( r l ) d i (r2)dT ( T 3 )

3 ,

(17)

which is the well-known non-relativistic SU(6) wave function of the nucleon! Petrov and Polyakov have obtained the corresponding SU(6) function in the infinite-momentum frame. Performing the group integration with the decuplet rotational functions (13,14) one also gets the well-known S U ( 6 ) wave functions in the nonrelativistic limit. Relativistic corrections to those wave functions are easily computable from Eq.(4), as are the 5-quark Fock components of the usual octet and decuplet baryons. To find those, one needs to expand the coherent exponent in Eq.(ll) to the linear order in the additional QQ pair, and perform the S U ( 3 ) projecting. The result will be given in a subsequent publication. Here I shall go straight to the 0+. Projecting the three quarks from the discreet level on the 0 rotational function (15) gives an identical zero, in accordance with the fact that the 0 cannot be made of 3 quarks. The non-zero projection is achieved when one expands the coherent exponent to the linear order. One gets then the 5-quark component of the 0 wave function: f fz f3 f4,ii iz i3 i4 (rl . . . r 5 ) = E f fz ef3 f4 63 + >f5,i5 1

lQk

I

€ i i2~

f5

. h ( ~ l ) h ( ~ 2 ) hWi:2(rq, ( ~ 3 ) r5) + permutations of 1,2,3.

(18)

The color structure of the antidecuplet wave function is ~ ~ ~ ~ Indices ~ ~ ~ 1 to 4 refer to quarks and index 5 refers to the antiquark, in this case S thanks to 6j5. The quark flavor indices are f1-4 = 1 , 2 = u, d. Naturally, we have obtained 0+ = uudd3. We see that the first two valence u , d quarks from the discrete level form a spin- and isospin-singlet diquark (although not correlated in space), like in the nucleon, see Eq.(16). However, the other pair of quarks do not form a similar spin-zero diquark. For example, in the “C” part of the wave function the 0 spin k is determined by the spin of the third quark from the discrete level. Since in the CQSM the functions h(r1,2,3)and W(r4, r5) are known, Eq.(18) gives the complete color, flavor, spin and space 5-quark

b

226

wave function of the O+ in its rest frame. The 5-quark wave functions of other members of the antidecuplet can be obtained in a similar manner. For the computation of the 0 width, this wave function is, however, inadequate as a matter of principle. As explained in Refs. the only consistent way to compute the width is using the infinite momentum frame where there is no pair creation or annihilation, and the Fock decomposition is well defined. In that frame, the decay of the Of goes into the fiwe-quark component of the nucleon only. It is first of all suppressed to the extent the 5-quark component of the nucleon is less than its 3-quark component. An additional suppression comes from the spin-flavor overlap. A preliminary crude estimate shows that the O+ width can be extremely small. 14915,

I thank the organizers of the Continuous Advances for hospitality, and V. Petrov and M. Polyakov for numerous discussions. This work has been supported in part by the DOE under contract DE-AC05-84ER40150. References 1. M.M. Pavan, R.A. Arndt, 1.1. Strakovsky and R.L. Workman, TN Newslett. 16, 110 (2002), hep-ph/0111066. 2. D. Diakonov, V. Petrov, P. Pobylitsa, M. Polyakov and C. Weiss, Nucl. Phys. B 4 8 0 , 341 (1996), hep-ph/9606314; Phys. Rev. D 5 6 , 4069 (1997), hep-ph/9703420. 3. D. Diakonov and V. Petrov, Nucl. Phys. B272, 457 (1986). 4. D. Diakonov and V. Petrov, JETP Lett. 43, 75 (1986) [PismaZh. Elcsp. Teor. Fiz.43, 57 (1986)l; D. Diakonov, V. Petrov and P.V. Pobylitsa, Nucl. Phys. B306, 809 (1988); D.Diakonov and V. Petrov, in Handbook of QCD, M. Shifman, ed., World Scientific, Singapore (2001), vol. 1, p. 359, hep-ph/0009006. 5. J . Hiller, these Proceedings, hep-ph/0408131. 6. V. Petrov and M. Polyakov, hep-ph/0307077. 7. A. Hayashi, G. Eckart, G. Holzwarth and H. Walliser, Phys. Lett. B147, 5 (1984). 8. M. Karliner and M.P. Mattis, Phys. Rev. D31, 2833 (1985); ibid. 34, 1991 (1986). 9. N. Itzhaki, I.R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B684, 264 (2004), hep-ph/0309305. 10. E. Witten, Nucl. Phys. B160, 433 (1983). 11. D. Diakonov and V. Petrov, hep-ph/0312144. 12. D. Diakonov, V. Petrov and M. Polyakov, 2. Phys. A359, 305 (1997), hep-ph/9703373; hep-ph/0404212. 13. D. Diakonov and V. Petrov, Phys. Rev. D69, 056002 (2004), hep-ph/0309203. 14. D. Diakonov and V. Petrov, Phys. Rev. D69, 094011 (2004), hep-ph/0310212. 15. D. Diakonov, hep-ph/0406043.

DO CHIRAL SOLITON MODELS PREDICT PENTAQUARKS ?

IGOR R. KLEBANOV AND PETER OUYANG Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544, USA We reconsider the relationship between the bound state and the S U ( 3 ) rigid rotator approaches to strangeness in chiral soliton models. For non-exotic S = -1 baryons the bound state approach matches for small m K onto the rigid rotator approach, and the bound state mode turns into the rotator zero-mode. However, for small m K , there are no S = -tl kaon bound states or resonances in the spectrum. This shows that for large N and small m K the exotic state is an artifact of the rigid rotator approach. An S = +1 near-threshold state with the quantum numbers of the O+ pentaquark comes into existence only when sufficiently strong S U ( 3 ) breaking is introduced into the chiral lagrangian. Therefore, pentaquarks are not generic predictions of the chiral soliton models.

1. Introduction These lecture notes are largely based on our paper with N. Itzhaki and L. Rastelli'. Recently there has been a flurry of research activity on exotic pentaquark baryons, prompted by reports2v3y4of the observation of the S = tl baryon O+ (1540). The original photoproduction experiment2 was largely motivated by theoretical work5 in which chiral soliton models were used to predict a rather narrow I = 0, J P = $+ exotic S = +1 baryon whose minimal quark content is uudds. The method used in t o predict the baryon spectrum is the S U ( 3 ) collective coordinate quantization of chiral soliton^^?^. This approach predicts the well-known 8 and 10 S U ( 3 ) multiplets of baryons, followed by an exotic 10multiplets~g~'O whose S = +1 member is the Of. The fact that the exotic 10multiplet is found simply by exciting the soliton to the next rotational energy level after the well-known decuplet has led to a widespread belief that the pentaquarks are a robust prediction of chiral soliton models, independent of assumptions about the dynamics. In this talk we argue that this belief is not well-founded. In-

227

228 stead, we will conclude that in soliton models the existence or non-existence of the pentaquarks very much depends on the details of the dynamics, i.e. the structure of the chiral lagrangian. Thus, the soliton models do not produce any miracles that are not obvious from general priciples of QCD. Neither in QCD nor in chiral soliton models is there anything that a priori guarantees the existence of narrow pentaquarks. Indeed, we will show that with the standard set of the Skyrme model parameters, a resonance with the quantum numbers of the O+ does not form. This fact should be kept in mind as some of the more recent searches" have failed to confirm the existence of the O+ or other pentaquarks. Our theoretical discussion follows the basic premise6 that the semiclassical quantization of chiral solitons corresponds to the 1/N expansion for baryons in QCD generalized to a large number of colors N . It is therefore important to generalize the discussion of exotic collective coordinate states carried out for N = 3 in to large N . The allowed multiplets must contain states of hypercharge N / 3 , i.e. of strangeness S = 0. In the notation where S U ( 3 ) multiplets are labeled by ( p , q ) , the lowest multiplets one finds8~14~15~16~17~18 are (1,n) with J = ?jand ( 3 , n - 1) with J = $. These are the large N analogues of the octet and the decuplet. Exactly the same multiplets appear when we construct baryon states out of N quarks. The splittings among them are of order 1/N, as is usual for soliton rotation excitations. The large N analogue of the exotic antidecuplet is the representation (0,n 2) with J = and one finds that its splitting from the lowest multiplets is O ( N o )in the rotator approximation. The fact that the mass splitting is of order one, comparable to the energy of mesonic fluctuations, raises question^^^^^^^^^ about the validity of the rigid rotator approach to these states. Instead, a better treatment of these states is provided by the bound state approach20 where one departs from the rigid rotator ansatz and adopts more general kaon fluctuation profiles; in this approach one describes the O+ as a kaon-skyrmion resonance or bound state of S = +1, rather than by a rotator state (a similar suggestion was made independently by Cohen17.) In the non-exotic sector, as we take the limit m K 4 0 the bound state description of low-lying baryons smoothly approaches the rigid rotator description, and the bound state wavefunction approaches a zeromode. However, in contrast with the situation for S = -1, for S = +1 there is no fluctuation mode that in the m~ + 0 limit approaches the

+

i,

229

rigid rotator mode. Thus, for large N and small S U ( 3 ) breaking, the rigid rotator state with S = +1 is an artifact of the rigid rotator approximation (we believe this to be a general statement that does not depend on the details of the chiral lagrangian.) Next we ask what happens as we increase the SU(3) breaking by varying parameters in the effective lagrangian (such as the kaon mass and the weight of the Wess-Zumino term) and find that a substantial departure from the SU(3)-symmetric limit is necessary to stabilize the kaon-skyrmion system. We reach a conclusion that, at least for large N, the exotic S = +1 state exists only due to the S U ( 3 ) breaking and disappears when the breaking is too weak. 2. The rigid rotator vs. the bound state approach

Our discussion of chiral solitons is mainly carried out in the context of the Skyrme model, but our conclusions will not be tied to a specific model. The Skyrme approach to baryons begins with the Lagrangian12

+ T r ( M ( U + Ut - 2 ) ) ,

(2.1)

where U ( x p ) is a matrix in SU(3) and M is proportional to the matrix of quark masses. There is an additional term in the action, called the Wess-Zumino term, whose normalization is proportional to N. Skyrme showed that there are topologically stabilized static solutions in which the radial profile function F ( r ) of hedgehog form, Uo = ei.r..FF(T), satisfies the boundary conditions F ( 0 ) = r,F(cm) = 0. The non-strange low-lying excitations of this soliton are given by rigid rotations of the pion field A ( t ) E S U ( 2 ) :

U ( Z ,t ) = A(t)UoA-'(t).

(2.2)

For this ansatz the Wess-Zumino term does not contribute. By expanding the Lagrangian about Uo and canonically quantizing the rotations, one finds that the Hamiltonian is

where J is the spin and the c-numbers Mcl and R are functionals of the

230

soliton profile. For vanishing pion mass, one finds numerically that 107 R2Mcl 21 3 6 . 5 5 , e3fn For N = 2n 1, the low-lying quantum numbers are independent of the integer n. The lowest states, with I = J = and I = J = %,are identified and e with the nucleon and A particles respectively. Since fn 1 / a , the soliton mass is N, while the rotational splittings are 1/N. Adkins, Nappi and Witten13 found that they could fit the N and A masses with the parameter values e = 5.45, fT = 129 MeV. In comparison, the physical value of f n = 186 MeV. A generalization of this rigid rotator treatment that produces SU(3) multiplets of baryons is obtained by making the collective coordinate A ( t ) an element of SU(3). Then the WZ term makes a crucial constraint on allowed m u l t i p l e t ~ ~The ~ ~ large-N ~ ~ ~ ~ .treatment of this 3-flavor Skyrme model is more subtle than in the 2-flavor case. When N = 2n 1 is large, even the lowest lying (1,n) S U ( 3 ) multiplet contains (n 1)(n 3) states with strangeness ranging from S = 0 to S = -n - 1 16. When the strange quark mass is turned on, it introduces a splitting of order N between the lowest and highest strangeness baryons in the same multiplet. Thus, SU(3) is badly broken in the large N limit, no matter how small m, is 16. It is helpful to think in terms of SU(2)x U(1) flavor quantum numbers, which do have a smooth large N limit.. In other words, we focus on low strangeness members of these multiplets, whose I , J quantum numbers have a smooth large N limit, and identify them with observable baryons. N strange quarks, Since the multiplets contain baryons with up to the wave functions of baryon with fixed strangeness deviate only an amount 1 / N into the strange directions of the collective coordinate space. Thus, to describe them, one may expand the SU(3) rigid rotator treatment around the SU(2) collective coordinate. The small deviations from SU(2) may be assembled into a complex SU(2) doublet K ( t ) . This method of 1/N expansion was implemented in 19, and reviewed in 16. From the point of view of the Skyrme model the ability to expand in small fluctuations is due to the Wess-Zumino term which acts as a large magnetic field of order N. The method works for arbitrary kaon mass, and has the correct limit as m K -+ 0. To order O(No) the Lagrangian has the formlg

+

- a, -

N

+

N

+ +

N

-

N L = 4akW + i - ( ~ +-kk 2

+~ - r) KtK.

(2.5)

231 The Hamiltonian may be diagonalized:

where

N2 M,2 = 1 6 . ~ (2.7) The strangeness operator is S = btb - at,. All the non-exotic multiplets contain at excitations only. In the SU(3) limit, w- -+ 0, but w+ t $ N o . Thus, the “exoticness” quantum number mentioned in l8 is simply E = btb here, and the splitting between multiplets of different “exoticness” is &, in agreement with results found from the rigid rotator14?15f16p17>18. The O ( N o )splittings predicted by the rigid rotator are, however, not exact: this approach does not take into account deformations of the soliton as it rotates in the strange direction^^^^'^^^^. Another approach to strange baryons, which allows for these deformations, and which proves to be quite successful in describing the light hyperons, is the so-called bound state method“. The basic strategy is to expand the action to second order in kaon fluctuations about the classical hedgehog soliton. Then one can obtain a linear differential equation for the kaon field, incorporating the effect of the kaon mass, which one can solve exactly. The eigenenergies of the kaon field are then the O ( N o ) differences between the masses of the strange baryons and the classical Skyrmion mass. It is convenient to write U in the form U = f l U ~ a where , U, = exp[2iXj7rj/fT] and UK = exp[2iXaKa/f,] with j running from 1 to 3 and a running from 4 to 7. The A, are the standard SU(3) Gell-Mann matrices. We will collect the K“ into a complex isodoublet K : Wf

=

N (J1+ 8@

( r n K / M 0 ) 2 f 1)

,

-

K=’(

K 4 - iK5 K6 - iK7

)=(::).

Expanding the Wess-Zumino term to second order in K , we obtain

ZN

L~~ = -w

f:

( K ~ D , K- ( D , K ) ~ K )

where

and B, is the baryon number current. Now we decompose the kaon field into a set of partial waves. Because the background soliton field is invariant

232

+

under combined spatial and isospin rotations T = I L, a good set of quantum numbers is T ,L and T,, and so we write the kaon eigenmodes as K = k(r, t)YTLT,. Substituting this expression into L S k y r m e LWZ we obtain an effective Lagrangian for the radial kaon field k ( r ,t ) of the form21

+

d

d

-h(r)-kt-k dr dr

- ktk(m;

+I&(?-))

The formula for the effective potential V,ff(r)appears in equation of motion for k is

-f(r)k

The resulting

21y1.

+ 2iA(r)k + Ok = 0 ,

(2.12)

1

C? E --arh(r)r2ar - m& - ~ ( r ) . r2

The eigenvalue equations are

+ 2X(T)W, + O)kn = 0 ( f ( r ) G z - 2 ~ ( r )+~ o$, , =o (f(r)w:

( S = -1)

,

( S = +I),

(2.13)

with w,,G, positive. Crucially, the sign in front of A, which is the contribution of the WZ term, depends on whether the relevant eigenmodes have positive or negative strangeness. It is possible t o examine these equations analytically for m~ = 0. Then one finds that the S = -1 equation has an exact solution with w = 0 and k ( r ) sin(F(r)/2), which corresponds to the rigid rotator zero-mode21. As m K is turned on, this solution turns into an actual bound state 20,21. On the other hand, the S = +1 equation does not have a solution with G= and k(r) sin(F(r)/2). This is why the exotic rigid rotator state is not reproduced by the more precise bound state approach to strangeness. In section 3 we further check that, for small m ~there , is no resonance corresponding to the rotator state of energy in the S U ( 3 ) limit. The lightest S = -1 bound state is in the channel L=l, T = $, and its mass is Mcl 0.218 efr = 1019 MeV. This state gives rise to the A(1115), C(1190), and C(1385) states, where the additional splitting arises from S U ( 2 ) rotator corrections20y21.There is also a L = 0, T = bound state corresponding to the negative parity hyperon A(1405). The natural appearance of the A(1405) is a major success of the bound state a p p r ~ a c h ~

-

&

N

2

+

233 6 4

1 ~

0

0.2

0.4

0.6

'

0.8

1

W

s,

Figure 1. Phase shift as a function of energy in the L = 2, T = S = -1 channel. The energy w is measured in units of e f n (with the kann mass subtracted, so that w = 0 at threshold), and the phase shift 6 is measured in radians. Here e = 5.45 and fT = 129 MeV.

The same method can be applied also to states above threshold. Such states will appear as resonances in kaon-nucleon scattering, which we may identify by the standard procedure of solving the appropriate kaon wave equation and studying the phase shifts of the corresponding solutions as a funcbion of the kaon energy. In the L = 2, T = channel there is a resonance a t Mcl 0.7484 ef,=1392 MeV (see Figure 1). Upon the SU(2) collective coordinate quantization, it gives rise t o three states22with ( I ,J ) given by (0, :), (1, (1, with masses 1462 MeV, 1613 MeV, and 1723 MeV respectively (see Table 2). We see that these correspond nicely to the known negative parity resonances h(1520) (which is Do3 in standard notation), C(1670) (which is 0 1 3 ) and C(1775) (which is 0 1 5 ) . As with the bound states, we find that the resonances are somewhat overbound (the overbinding of all states is presumably related to the necessity of adding an overall zero-point energy of kaon fluctuations), but that the mass splittings within this multiplet are accurate to within a few percent. In fact, we find that the ratio

+

4

g), g),

(2.14) while its empirical value is 1.70.

234 3. Baryons with S=+1?

For states with positive strangeness, the eigenvalue equation for the kaon field is the same except for a change of sign in the contribution of the WZ term. This sign change makes the WZ term repulsive for states with 3 quarks and introduces a splitting between ordinary and exotic baryons20. In fact, with standard values of the parameters (such as those in the previous section) the repulsion is strong enough to remove all bound states and resonances with S = 1, including the newly-observed O+. It is natural to ask how much we must modify the Skyrme model to accommodate the pentaquark. The simplest modification we can make is to introduce a coefficient a multiplying the WZ term. Qualitatively, we expect that reducing the WZ term will make the S = +1 baryons more bound, while the opposite should happen to the ordinary baryons. The most likely channel in which we might find an exotic has the quantum numbers L = 1, T = f , as in this case the effective potential is least repulsive near the origin. For fn= 129, 186, and 225 MeV, with e 3 f n fixed, we have studied the effect of lowering the WZ term by hand. Interestingly, in all three cases we have to set a 21 0.39 to have a bound state at threshold. If we raise a slightly, this bound state moves above the threshold, but does not survive far above threshold; it ceases to be a sharp state for a N 0.46. We have plotted phase shifts for various values of a in Figure 2.a Assuming that the parameters of the chiral lagrangian take values such that the Of exists, we can then consider the SU(2) collective coordinate quantization of the state, in a manner analogous to the treatment of the S = -1 bound states2'. Here we record our results, assuming that a = 0.39 and fn = 129 MeV, and refer the reader to our original paper' for details. and positive The lightest S = +1 state we find has I = 0, J = parity, i e . it is an S = +1 counterpart of the A. This is the candidate O+ state. Its first S U ( 2 ) rotator excitations have I = 1, J p = and I = 1, Jp = (a relation of these states to O+ also follows from general large N relations among baryon^^^^'^). The counterparts of these J p = $+, states in the rigid rotator quantization lie in the 27-plets of SU(3) We find that the I = 1, J p = $' state is 148 MeV heavier than the W,

i

:+

4'

:

26y27.

N

aWhen the state is above the threshold, we do not find a full 7r variation of the phase. Furthermore, the variation and slope of the phase shift decrease rapidly as the state moves higher, so it gets too broad to be identifiable. So, the state can only exist as a bound state or a near-threshold state.

235

:+

while the I = 1 , J p = state is 289 MeV heavier than the O+. We may further consider I = 2 rotator excitations which have J p = ;+,:+. Such states are allowed for N = 3 (in the quark language the N

charge +3 state, for example, is given by uuuus). The counterparts of these J p = states in the rigid rotator quantization lie in the 35plets of SU(3)26t27.We find

z+, 5’

M ( 2 , $) - M ( 0 ,

i)

M(2,g)- M(0,;)

N

N

494 MeV, 729 MeV

.

(3.15)

Although the specific mass splittings which we have computed depend on the choice of parameters in the chiral lagrangian, it turns out that we may form certain combinations of masses of the exotics which rely only the existence of the S U ( 2 ) collective coordinate:

+ M ( 1 ,!j)- 3M(O,3) = 2(MA - M N ) = 586 MeV , z M ( 2 , z) + M ( 2 , ; ) - g M ( 0 , i)= MA - M N ) = 1465 MeV , M ( 2 , S) - M ( 2 , E) = 4 ( M ( 1 ,3) - M ( l , z)) , (3.16) 2M(l,z)

3

where we used M a - M N = &. These “model-independent” relations have also been derived using a different method 2 5 . 6

a=0.4 1.5

/----

---______-

---

a=O.6

0.5

a

Figure 2. Phase shifts 6 as a function of energy in the S = +1, L = 1, T = channel, for various choices of the parameter a (strength of the WZ term). The energy w is measured in units of efir ( e = 5.45, fir = f~ = 129 MeV) and the phase shift 6 is measured in radians. w = 0 corresponds to the K - N threshold.

As another probe of the parameter space of our Skyrme model, we may vary the mass of the kaon and see how this affects the pentaquark. As

236 observed in Section 3, in the limit of infinitesimal kaon mass, there is no resonance in the S = +1, L = l , T = channel. We find that to obtain a bound state in this channel, we must raise m K to about 1100 MeV. Plots of the phase shift vs. energy for different values of m K may be found in l. 4. Discussion

The main implication of our analysis is that in chiral soliton models there is no “theorem” that exotic pentaquark baryons exist, nor is there a theorem that they do not exist. The situation really depends on the details of the dynamics inherited from the underlying QCD. The statements above apply to general chiral soliton models containing various lagrangian terms consistent with the symmetries of low-energy hadronic physics. In the bound state approach to the Skyrme model we saw that an S = +1 near-threshold state is absent when we use the standard parameters, but comes into existence only at the expense of a large reduction in the Wess-Zumino term. It is doubtful that such a reduction is consistent with QCD. However, one can and should explore other variants of chiral soliton models. For example, in 28 the exotic S = +1 resonances were studied in a model containing explicit K* fields. This model contains a coupling constant which is, roughly speaking, the analogue of the coefficient of the WZ term, a , in our approach. The findings of 28 are largely parallel to ours. For a wide range of values of this coupling, the repulsion is too strong, and no S = +1 resonances can form. When this coupling is very small, then there exists an S = +1 bound state. There is also a narrow intermediate range where this bound state turns into a near-threshold resonance. An important question is whether chosing parameters to lie in this narrow range is consistent with the empirical constrains on the effective lagrangian. If not, then one may have to conclude that chiral soliton models actually predict the absence of pentaquarks.

Acknowledgments We are grateful to N. Itzhaki and L. Rastelli for collaboration on a paper reviewed here, and to E. Witten for discussions. This material is based upon work supported by the National Science Foundation Grants No. PHY0243680 and PHY-0140311. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

237 References 1. N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, “Is Theta(1540)+ a kaon Skyrmion resonance?,” Nucl. Phys. B 684, 264 (2004) [arXiv:hep-

ph/0309305]. 2. T. Nakano et al. [LEPS Collaboration], “Evidence for a narrow S = +1 baryon resonance in photoproduction from the neutron,” Phys. Rev. Lett. 91,012002 (2003) [arXiv:hep-ex/0301020]. 3. S. Stepanyan et al. [CLAS Collaboration], “Observation of an exotic S = +1 baryon in exclusive photoproduction from the deuteron,” arXiv:hepex/03070 18. 4. J. Barth et al. [SAPHIR Collaboration], “Observation of the positivestrangeness pentaquark Theta+ in photoproduction with the SAPHIR detector at ELSA,” arXiv:hep-ex/0307083. 5 . D. Diakonov, V. Petrov and M. V. Polyakov, “Exotic anti-decuplet of baryons: Prediction from chiral solitons,” Z. Phys. A 359, 305 (1997) [arxiv:hep-ph/9703373]. 6. E. Witten, “Global Aspects Of Current Algebra,” Nucl. Phys. B 223, 422 (1983); “Current Algebra, Baryons, And Quark Confinement,” Nucl. Phys. B 223,433 (1983). 7. E. Guadagnini, “Baryons As Solitons And Mass Formulae,” Nucl. Phys. B 236,35 (1984). 8. A. V. Manohar, “Equivalence Of The Chiral Soliton And Quark Models In Large N,” Nucl. Phys. B 248, 19 (1984). 9. M. Chemtob, “Skyrme Model Of Baryon Octet And Decuplet,” Nucl. Phys. B 256,600 (1985). 10. M. Praszalowicz, “SU(3) Skyrmion,” TPJU-5-87 Talk presented at the Cracow Workshop on Skyrmions and Anomalies, Mogilany, Poland, Feb 20-24, 1987

11. K. T. Knopfle, M. Zavertyaev and T. Zivko [HERA-B Collaboration], “Search for Theta+ and Xi(3/2)- pentaquarks in HERA-B,” J. Phys. G 30, S1363 (2004) [arXiv:hep-ex/0403020]; R. Mizuk [Belle collab.], talk at PENTA04; V. Halyo [Babar collab.], talk at PENTA04. 12. T. H. Skyrme, “A Nonlinear Field Theory,” Proc. Roy. SOC.Lond. A 260, 127 (1961); “A Unified Field Theory Of Mesons And Baryons,” Nucl. Phys. 31,556 (1962). 13. G. S. Adkins, C. R. Nappi and E. Witten, “Static Properties Of Nucleons In The Skyrme Model,” Nucl. Phys. B 228,552 (1983). 14. V. Kaplunovsky, unpublished. 15. Z. Dulinski and M. Praszalowicz, “Large N(C) Limit Of The Skyrme Model,” Acta Phys. Polon. B 18, 1157 (1988). 16. I. R. Klebanov, ‘[Strangeness In The Skyrme Model,” PUPT-1158 Lectures given at N A T O A S 1 on Hadron and Hadronic Matter, Cargese, fiance, Aug 8-18, 1989.

17. T. D. Cohen, “Chiral soliton models, large N(c) consistency and the Theta+

238

18. 19. 20.

21.

22. 23. 24.

25. 26. 27. 28.

exotic baryon,” arXiv:hep-ph/0309111; T. D. Cohen and R. F. Lebed, “Partners of the Theta+ in large N(c) QCD,” arXiv:hep-ph/0309150. D. Diakonov and V. Petrov, “Exotic baryon multiplets at large number of colours,” arXiv:hep-ph/0309203. D. B. Kaplan and I. R. Klebanov, “The Role Of A Massive Strange Quark In The Large N Skyrme Model,” Nucl. Phys. B 335,45 (1990). C. G. Callan and I. R. Klebanov, “Bound State Approach To Strangeness In The Skyrme Model,” Nucl. Phys. B 262,365 (1985); C. G. Callan, K. Hornbostel and I. R. Klebanov, “Baryon Masses In The Bound State Approach To Strangeness In The Skyrme Model,” Phys. Lett. B 202, 269 (1988). N. N. Scoccola, “Hyperon Resonances In SU(3) Soliton Models,” Phys. Lett. B 236, 245 (1990). D. 0. Riska and N. N. Scoccola, “Anti-charm and anti-bottom hyperons,” Phys. Lett. B 299, 338 (1993). C. L. Schat, N. N. Scoccola and C. Gobbi, “Lambda (1405) in the bound state soliton model,’’ Nucl. Phys. A 585, 627 (1995) [arXiv:hep-ph/9408360]. E. Jenkins and A. V. Manohar, “l/N(c) expansion for exotic baryons,” JHEP 0406,039 (2004) [arXiv:hep-ph/0402024]. H. Walliser and V. B. Kopeliovich, “Exotic baryon states in topological soliton models,” arXiv:hep-ph/0304058. D. Borisyuk, M. Faber and A. Kobushkin, “New family of exotic Theta baryons,” arXiv:hep-ph/0307370. B. Y . Park, M. Rho and D. P. Min, “Bound state approach to pentaquark states,” arXiv:hep-ph/0405246.

BARYON EXOTICS IN THE 1/Nc EXPANSION

ELIZABETH JENKINS Department of Physics, 9500 Gilman Drive, University of California San Diego, La Jolla, CA 92093-0319, USA E-mail: [email protected] The implications of the l / N c expansion for the spin-flavor properties of qqqqq pentaquark exotics are presented. The masses, axial couplings and decay widths of pentaquark baryons containing only light quarks are discussed. In addition, qqqqQ pentaquark baryons containing a single heavy antiquark are studied.

1. Introduction Normal QCD baryons are three-quark qqq states which are completely antisymmetric in the color indices of the quarks. In the past year and a half, experimental evidence has been reported for exotic baryon^^^^^^ - baryons whose flavor quantum numbers forbid their classification as qqq states. All presently observed exotic baryons can be classified as qqqqq E q44 pentaquarks. Experimental evidence has been reported for three different pentaquark and 0,-(3099). The Of baryons: 0+(1540), @(1860)(also designated +) is thought to be a u u d d s bound state with strangeness S = +1 (since it contains a strange antiquark), isospin I = 0 and spin J = :. The @ has strangeness S = -2, isospin I = and spin J = so the manifestly exotic @-- state is an ssddii bound state. The 0, is the anticharmed analogue of the Of:it is an isosinglet u u d d c bound state. The @ has been observed by a single experiment2. This observation has not been confirmed, and indeed a number of experiments have failed to observe the states even though they have greater statistical sensitivity. Thus, it does not appear that the evidence for the Q, will survive. The 0,-reported by the H1 experiment3 also is unconfirmed. Thus, the predominant evidence for pentaquarks is for the state Of. The 0+ is seen by eight different experiments'. The experimental situa-

4

3,

239

240

tion is confused, however, since a number of experiments (ALEPH, BaBar, CDF) do not see it. The experiments which do report evidence for the O+ see a narrow state with a width I? < 15 MeV. A lower bound on the width has not been set; however, studies of K N scattering data and other data indicates that the width must be of order 1 MeV to have escaped d e t e ~ t i o n ~The > ~ O+ . is seen primarily in photoproduction, but it has also been seen in deep inelastic scattering. The parity of the state is unmeasured. The O+ is observed to decay to nucleon and kaon, either n K + or pKg. The experimental data in support of the O+ is summarized in Table 1. Table 1. Experiments reporting evidence of the Q+. The mass and width measurements, observed decay channel, production mechanism and statistical significance of the reported signal axe given. Expt . LEPS DIANA CLAS SAPHIR SVD HERMES COSY-TOF ZEUS

Mass (MeV)

Width (MeV)

1540 f 10 1539 f 2 1542 f 5 1540 f 4 f 2 1526 f3 f3 1528 f 2.6 f 2.1 1530 f 5 1521.5 f 1.5:;:;

< 25 <9

< 21 < 25 < 24 <19f5f2 <18f4 8 f 4

Decay

Production

U

y n -+ Q+K4.6 K+Xe 4.4 yd -+ K+K-pn 5.3 f 0.5 yp -+ nK+Kg 4.8 5.6 P N P Ki ~X 4-6 7d 4-6 p p --t C+Kop 4.6 efp -+

Whether pentaquark baryons indeed exist is an experimental issue which needs to be sorted out. If the eight experiments in Table 1 have not seen a pentaquark, then it remains to be understood what exactly they are seeing. In this proceedings, I explore the implications of the l/Nc expansion of QCD for the spin-flavor properties of pentaquarks, should they e ~ i s t ~ The l/Nc expansion of QCD is a systematic expansion which yields modelindependent results for the properties of baryonsg. Large-N, baryons have a contracted spin-flavor symmetry S U (2F), for F flavors of light quarkdo. The consequences of contracted spin-flavor symmetry and its breaking account for all successful group theoretic relations for normal baryons, as spinflavor symmetry remains a good approximate symmetry for QCD baryons with N, = 3. The l/Nc expansion of QCD gives a power counting in the expansion parameter l / N c = 1/3, which is comparable to flavor S U ( 3 ) breaking in QCD. By considering the l/Nc and flavor-symmetry breaking power countings of all spin @ flavor structures, one obtains a hierarchy of baryon symmetry relations which are evident in experimental data”,

241

Exotic baryons are studied for N, colors and F flavors, where N, = 3 and F = 3 in QCD. It is possible to define a quantum number called exoticness E for baryon exotics. Baryon exotics with exoticness E have flavor and spin quantum numbers which require that they be q N c f E q E= q N c ( q q ) E states, where E is the minimum number of additional 44 pairs required to build a baryon with given flavor and spin quantum numbers. The pentaquark baryons are E = 1 baryons. The l / N c expansion is used t o study the masses, axial couplings and widths of pentaquark baryons. Much has been said about exotic baryons in the chiral solition model and the quark model. Both of these models satisfy the same contracted spin-flavor symmetry as QCD in the N, -+ 00 limit, and satisfy the l/Nc power counting of the spin-flavor analysis presented here. In order to make contact with these models, a brief review of a few salient points is in order. It has been known for some time that rigid rotator quantization of the chiral soliton model yields exotic baryon multiplets with positive parity in the same spin-flavor tower as the normal 81 and 10; baryon^'^^^^. The rule is that one obtains all SU(3) flavor representations ( p , q ) which satisfy p

+ 2q = Nc + 3r

(1)

for F = 3 flavors. Thus, the integer T is a natural variable in the Skyrme model. The normal baryons 84 and 10; satisfy Eq. ( 1 ) with T = 0. In addition, there are exotic baryons which satisfy Eq. (1) with non-vanishing T . The lowest-lying states have T = 1 and consist of the 274, 27;,

...

mi,

Figure 1. Flavor S U ( 3 ) representations ( p ,q ) obtained from rigid rotator quantization in the Skyrme model.

In the quark model, exotic baryons appear as q44 pentaquark bound states. There are two choices for the qqqq wavefunction: the mixed symmetry of the 4-quark color wavefunction can be compensated by mixed symmetry of the spin-flavor wavefunction (so that the orbital wavefunction is completely symmetric) OR by mixed symmetry of the orbital wavefunction (so that the spin-flavor wavefunction is completely symmetric). The q44 states obtained from these two choices have opposite parity, see Table 2. For the completely symmetric quark orbital wavefunction, the pentaquarks have negative parity, whereas for the completely symmetric

242

spin-flavor wavefunction, the pentaquarks have positive parity. Whether the positive or negative parity pentaquarks are the low-lying exotics is a dynamical issue. Table 2.

Symmetry of qqqq wavefunction.

Color

Orbital

SDin-Flavor

Paritv

Note: The qqqqq pentaquark wavefunctions have a parity flip relative to the qqqq wavefunctions, so the positive parity pentaquarks have wavefunctions which are completely symmetric in spin-flavor.

There are reasons to believe that the positive parity pentaquark states are lower in energy than the negative parity ~ t a t ewhich ~ ~means ~ that ~ complete symmetry of the qqqq spin-flavor wavefunction is more important energetically than having the quarks in the same Is orbital wavefunction. Under the breakdown of spin-flavor S U ( 4 ) to its SU(2) 8 SU(2) spin and isospin subgroups, the completely symmetric qqqq spin-flavor representation decomposes into the spin-isospin tower Jq = Iq = 0,1,2. The generalization to flavor S U ( 3 ) yields the spin and flavor representations given in Fig. 2. Tensoring in the antiquark yields the pentaquark representations lo;, 27;, 274, 35; and 359.

Figure 2.

S U ( 2 ) €3 S U ( 3 ) spin and flavor representations of qqqq, Q and qqqqq.

243 It is non-trivial that the positive parity baryon states of the chiral soliton and quark models are the same6. Below, exotic baryons are studied in the l / N c expansion without recourse to models.

2. Exotic Baryons in Large N , Large-N , exotic baryons obey a U ( 2 F ) , x U ( 2 F ) , spin-flavor symmetry6y7i8. The symmetry is exact in the N, + 00 limit which has no qQ pair creation and annihilation. For exotic baryons at finite N,, U ( 2 F ) , x U ( 2 F ) , is a good approximate symmetry which is broken down to the spin and flavor subgroup S U ( 2 ) x S U ( F ) by l/Nc suppressed effects. A baryon observable transforming under a given S U ( 2 ) x S U ( F ) representation can be written in terms of a complete basis of operators arising at various orders in the l/Nc expansion. The ( N , E ) quarks of the exotic baryon are in the completely symmetric S U ( 2 F ) , representation ( N , E , 0 , . . . , O ) , and the E antiquarks are in the completely symmetric S U ( 2 F ) , representation (0,. . . ,O, E ) . Nonexotic baryons, which correspond to the special case E = 0, have N , light quarks in the completely symmetric spin-flavor representation. This representation decomposes under spin and flavor into the tower of baryon states (J,(p,q))= (;,(3,-)), * ' * , (+,(NC,O)). E x o t i c E = 1 baryons contain N, 1 quarks and a single antiquark. The N, 1 quarks decompose into the spin and flavor tower (0, (0, , (1,( 2 , . .., , ( N , 1 , O ) ) . The exotic baryon states are obtained by tensoring in the antiquark (0,l)). It is easiest to study exotic baryons for F 2 5 flavors. The N , E quarks are in the S U ( 2 ) x S U ( F ) representations (121, n2, 0 , . . . , 0,O)) with quark spin j , = n 1 / 2 and quark number N, = nl 2122 = N, E. The E antiquarks are in the S U ( 2 ) x S U ( F ) representations ( O , O , . . ,0 , 72-2, n-1)) with antiquark spin j , = n-1/2 and antiquark number Nq = n-1 2n-2 = E . The qNc+EqE exotic baryons are in the tensor product S U ( 2 ) x S U ( F ) representations ( j , 8 j q , (n1,n2,O1. . . ,O , n - 2 , n - I ) ) ,with no contractions on qq flavor indices. Contraction on qq flavor indices is not allowed because flavor singlet qq pairs correspond to higher Fock components of baryons with lesser E . For example, the proton is normally regarded as a uud bound state although it is actually a superposition of states uud(qtj)Ewith extra flavor-singlet qtj pairs. Below, exoticness E is defined as the minimum number of qq pairs

+

+

(;,(LW)), + F + (3,

+

v)) v)) (u,

+

(y,

+

+

+

244

required to create a baryon with given quantum numbers. Thus, the leading Fock component of an exotic baryon is qNc+EQE. Normal non-exotic baryons have E = 0, and the pentaquarks have E = 1. The S U ( F ) representation (nl,n2, 0, * * , 0, n-2, n-1) satisfies the identities

-

n1+

2n2

+ ( F - 2)n-2 + ( F - 1)n-1 = N , + rF, n1+

2n2 - 2n-2 - n-1 = N,,

(2) where r = n-1 n-2 is the natural variable defined in the chiral soliton or Skyrme model. The Dynkin indices of the S U ( F ) flavor representation also satisfy

+

n1+

n-1

2n2 = N g = N ,

+ 2n-2

+ El

= N, = E ,

(3) where E is the natural variable of the quark model. The connection between E and r is given by E = r n-2, with r 5 E 5 2r. It is an important point6 that E # r. This distinction arises because 72-2 columns of F antisymmetrized boxes are dropped in making the SU(F ) representation (nlrn2,0 , . . . ,o, n - 2 , n - l ) . Three flavors is a special case. The S U ( 3 ) representation ( p , q ) is given by (n1 n-2,n-l n ~ ) The . four integers n1, 122, n-2 and n-1 are no longer known separately. The integers p and q satisfy

+

+

+

+ 2q = N, + 3r, (4) which again defines an integer r = n-1 + The integers p , q and r can p

72-2.

be expressed in terms of N,, El j , and j , by

E =

.

2 +3G.

These three integers are invariant under the redefinitions E --f E - 2X, j , + j, X and j , 4 j , X, so E is not uniquely defined. To break this ambiguity, exoticness E is defined as the minimum value allowed. For F = 3 flavors, it can be shown6 that the exotic baryon states consist of the flavor representations ( p , q ) satisfying p 2q = N, 3r with spins

+

+

+

+

245

For E = 1, the pentaquark representations are given by 1 0 3 , 271, 27;, 35; and 353. The flavor representations of the light pentaquark multiplets are exhibited in Figs. 3, 4 and 5.

m

e

e r n e

e

e

e

e

O+

S=+1

N

s=o

c

s=-1

@

S = -2

Figure 3. Antidecuplet weight diagram.

Figure 4. 27 weight diagram.

Figure 5.

35 weight diagram.

uudds

ssddii

246

2.1. Pentaquark Masses The pentaquark masses are described by the Hamiltonian

----) Ji Jl

E N:'N,2' N z 7 Nc ' J2

for arbitrary function gives

Ho =c0Nc

(7)

f. Expanding f in a polynomial in its arguments

1 + c1E + ( ~ J2i + Nc

~3 J;

+

cq J 2

+ csE 2 ) + . . .

(8)

where the ellipsis represents terms suppressed by 1/N: and higher powers of l / N c . The above Hamiltonian can be used to relate the masses of pentaquarks in the same spin-isospin tower. It yields the mass relations7

20'

+

(02)

+

= 3 (01) 0 ( l / N ; ) ,

(ol)-o+=-23 ( A - N ) + o ( ~ / N , ~ ) , (02)

- 0'

= 2 (A - N )

(9)

+ 0 (1/N:) ,

to given orders in the l / N c expansion. These same mass relations also were obtained in the context of the bound state approach t o the Skyrme model17. Using 0' = 1540 MeV and the experimental value of the (A - N ) mass difference yields the predictions (01)= 1735 MeV and ( 0 2 ) = 2126 MeV for the spin-averaged masses of the isovector and isotensor S = +I pentaquark states. The error on these predictions is order l / N ; , which is 30 MeV. N

2.2. Pentaquark Axial Couplings and Decay Widths The axial couplings of pentaquark baryons also can be studied. The possible transitions are depicted in Fig. 6. The lightest isosinglet pentaquark 0' cannot decay within the pentaquark tower, and so it naturally has a narrower decay width than the isovector and isotensor states. I consider the analysis using only isospin flavor symmetry. The l / N c expansion implies that all pion couplings for baryons with a given E are related up to corrections of relative order 1/N;. In addition, the pion couplings of the E = 1 tower are related to the pion coupling g A of the normal baryon tower up t o corrections of relative order l / N c . Thus, a t leading order, all the vertical transitions can be related. The diagonal transitions between the pentaquark and normal baryon towers are

247

A

N Figure 6. The pion (vertical) and kaon (diagonal) decay modes of the S = +1 pentaquarks. The lightest pentaquark @+ can only decay via kaon emission to the ground state baryons.

all related by spin-isospin symmetry in terms of a single coupling constant

go, which can be obtained from the @+ decay width

when measured. The following predictions are made for the decay widths of the isovector and isotensor pentaquarks:

r(@1,j=1/2) 2 30 MeV -t- 1.2r(@), 2 30 MeV + 4.9r(O), r(02,j=3/2) 2 560 MeV + 5 . i r ( @ ) , r(02,jz5/2) 2 560 MeV + 13.6r(O). r(Ol,j=3/2)

(11)

In the above equations, the unknown hyperfine mass splittings of the different isovector states and isotensor states have been ignored. The first number is an estimate of the pion decay width of the pentaquark to decay within the pentaquark tower, whereas the second number gives an estimate of the decay width of the pentaquark t o normal baryons. Clearly, for a O+ width of order an MeV, the isovector and isotensor states decay predominately within the pentaquark tower rather than to normal baryons. Note that the isotensor decay widths are very broad, so these states are not expected t o be observable.

248

2.3. Heavy qqqqQ Pentaquarks

The properties of heavy qqqqQ pentaquarks also can be studied. The positive parity heavy pentaquarks consist of the representations G + , 156,15+, 15'; and 15's. These flavor representations are depicted in Figs. 7, 8 and 9.

OQ e

e

s= -1 s= -2

e

e

e

Figure 7.

S=O

6 weight diagram.

%,Q

Figure 8.

15 weight diagram.

Figure 9.

15' weight diagram.

uuddQ qqqsQ

qqssQ

249

T h e mass relations derived in the l/Nc expansion7 are given by

+

2 (00)

(02,Q)

(0 - ) - @ 1,Q ( 0 2 , ~ )

= 3 (@,,Q)

-

Q -

2

-(A 3

+ 0 (1/N,3)

-N,

+

(l/N?)

0 9 = 2(A - N ) + 0 (1/N?)

where

1

(@l,Q)

3 ( @ l , Q , j = 1 / 2 + 201,Q,j=3/2) ,

(@2,Q)

5 (2@2,Q,j=3/2

1

-/- 3@2,Q,j=5/2)

I

denote spin-averaged masses. In addition, results have been derived for t h e decay widths of the heavy pentaquarks7. It is possible t h a t the lightest isosinglet state O E is stable, with only weak decays. The exotic observed by H1 is unlikely t o be this lightest state because of its large mass.

Acknowledgments I thank the organizers for the invitation t o give a plenary talk. This work was supported in part by the Department of Energy under Grant DE-FGOS97ER40546. References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91, 012002 (2003); V. V. Barmin et al. [DIANA Collaboration], Phys. Atom. Nucl. 66, 1715 (2003); S. Stepanyan et al. [CLAS Collaboration], Phys. Rev. Lett. 91, 252001 (2003); J. Barth et al. [SAPHIRCollaboration], Phys. Lett. B572, 127 (2003); A. Aleev et al. [SVD Collaboration], arXiv:hep-ex/0401024; A. Airapetian et al. [HERMES Collaboration], Phys. Lett. B585, 213 (2004); M. Abdel-Bary et al. [COSY-TOF Collaboration], arXiv:hep-ex/0403011; S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B591, 7 (2004) [arXiv:hep-ex/0403051]. 2. C. Alt et al. "A49 Collaboration], Phys. Rev. Lett. 92, 042003 (2004)

[arXiv:hep-e~/0310014]. 3. A. Aktas et al. [Hl Collaboration], arXiv:hep-ex/0403017. 4. R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C68, 042201 (2003) [Erratum-ibid. C69, 019901 (2004)] [arXiv:nucl-th/0308012]. 5 . R. N. Cahn and G. H. Trilling, Phys. Rev. D69, 011501 (2004). 6. E. Jenkins and A. V. Manohar, Phys. Rev. Lett. 93,022001 (2004) [arXiv:hepph/0401190].

250 7. E. Jenkins and A. V. Manohar, JHEP 0406, 039 (2004) [arXiv:hepph/0402024]. 8. E. Jenkins and A. V. Manohar, arXiv:hep-ph/0402150. 9. R. F. Dashen, E. Jenkins and A. V. Manohar, Phys. Rev. D49,4713 (1994) [Erratum-ibid. D51, 2489 (1995)] [arXiv:hep-ph/9310379]. 10. R. F. Dashen and A. V. Manohar, Phys. Lett. B315,425 (1993) [arXiv:hepph/9307241]; R. F. Dashen and A. V. Manohar, Phys. Lett. B315,438 (1993) [arXiv:hep-ph/9307242]; E. Jenkins, Phys. Lett. B315,441 (1993) [arXiv:hepph/9307244]. 11. For a review, see E. Jenkins, Ann. Rev. Nucl. Part. Sci. 48, 81 (1998). [arXiv:hep-ph/9803349]. 12. A. V. Manohar, Nucl. Phys. B248,19 (1984). 13. M. Chemtob, Nucl. Phys. B256,600 (1985). 14. R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003) [arXiv:hepph/0307341]. 15. L. Y . Glozman, Phys. Lett. B575,18 (2003) [arXiv:hep-ph/0308232]. 16. C. E. Carlson, C. D. Carone, H. J. Kwee and V. Nazaryan, Phys. Lett. B579, 52 (2004) [arXiv:hepph/0310038]. 17. N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B684, 264 (2004) [arXiv:hep-ph/0309305].

LARGE N c QCD AND MODELS OF EXOTIC BARYONS

THOMAS D.COHEN* Department of Physics University of Maryland College Park, MD 20742 USA E-mail: [email protected]

Exotic baryons have been predicted in the context of the chiral soliton model. These states have been identified with strangeness +1 resonances reported in a number of experiments. In this talk it is pointed out that the technique used to quantize these solitons in most conventional treatments depends on dynamical assumptions beyond those in standard large Nc physics. These additional assumptions have never been justified.

2003 saw the first report of a narrow resonant baryon state with strangeness f l and a mass of approximately 1.54 GeVl; this state has been denoted the B+. This claim stimulated an extraordinarily high level of both experimental and theoretical research. The reason for this intense interest stems largely from the fact that such a state is manifestly exotic: it cannot be made as a three quark state. The minimum number of quarks to construct a state with these quantum numbers is 5 (four quarks and one anti-quark) and accordingly such states are often referred to as pentaquarks. I will not review the experimental situation here except to note that it is quite confused: numerous experiments appear to confirm the existence of a narrow exotic state at a similar mass while many others see no indication of it. While it is extremely important that this experimental situation be sorted out-as will presumably happen in the next couple of years-the issues of relevance here do not strongly depend on how things are ultimately resolved experimentally. Instead, this talk focuses on the theory side and, in particular, on how large N , QCD can help understand and the situation. The work discussed here summarizes work in refs. 213

*Work supported in part by the 40762.

U.S.

department of energy under grant de-fg02-93er-

251

252

for a more complete discussion the reader is referred to the original work. Most theories concerning pentaquarks is based on modeling QCD rather than QCD itself. Of course, the number of models is quite large and I will not attempt to review the full spectrum of models. Instead, I will focus on one class of models-the chiral soliton models-since they have a rather special status. In the first place, a chiral soliton model4 was used to predict a narrow pentaquark state with virtually the same mass as was ultimately reported experimentally. In contrast, most other models were postdictions with parameters fit with a knowledge of the mass. Clearly, true predictions carry much greater weight: as has been said,“Predictions are difficultespecially about the future”. (It is interesting t o note that an internet search to find the origin of this quote led to a site attributing it to Yogi Berra-the former New York Yankee catcher-and another site attributing it to Neils Bohr-the Danish physicist.) Perhaps more importantly, the prediction was quite insensitive to the details of the model; the functional form of the soliton profile function played no role. All that mattered dynamically was the structure of the model, the values of SU(3) breaking parameters (fixed in the non-exotic sector) and the identification of the nucleon state of the multiplet. jF’rom a theory perspective this model insensitivity is crucial. It has long been known that there are predictions of soliton models for which the details of the soliton model are totally irrelevant (such as the ratio g n N N / g n N h 5 ) . All known cases where predictions of soliton models do not depend on the dynamical details are also known to be true in large N , QCD. This can be demonstrated by the use of large N , consistency rules6. This suggests that the predicted O+ properties might also be model-independent predictions of large N , QCD. If true, this would be really important: we would then essentially understand the structure of the 8+ modulo SU(3) symmetry breaking effects and higher-order l/Nc corrections. Unfortunately, however, this simply is not the case. The apparent insensitivity to the model details arises from an inconsistent treatment of the quantization of the soliton models in the sense of large N , counting. To understand why this is so, we need to review the standard methods for quantizing solitons. To begin, consider a topological soliton model which has SU(2) flavor symmetry and is based on a nonlinear sigma model. The chiral fields are given in terms of a matrix U = exp 2where i? are the pions. Large N , QCD justifies solving the theory classically and the lowest energy solution having winding number unity (corresponding to

(3

253 baryon number unity) is a "hedgehog" of the form UO = exp (i7. .^f(r)) with f(0) - f ( w ) = 7r. Note that such a configuration correlates isospace and ordinary space and thereby breaks both rotational and isospin symmetries. Accordingly such a classical configuration cannot correspond directly to a single physical state. Rather, the hedgehog corresponds to a band on nearly degenerate states-in much the same way as a deformed HartreeFock solution in nuclear physics corresponds to a band of low-lying states. To obtain physical states one needs a method to project from the hedgehog onto states with physical quantum numbers. Adkins, Nappi and Witten (ANW) introduced a semiclassical method (justified at large N,) for doing this '. One introduces an ansatz for a time-dependent U given by U(F,t ) = At(t)Uo(r')A(t),where A ( t ) is a time-dependent, spaceindependent, SU(2) matrix. Inserting this into the lagrangian density of the model and integrating over space gives a collective lagrangian which only depends on A and &A. This can be Legendre transformed into a collective hamiltonian:

-+

+

-

= 1. The key to what follows with the constraint laoI2 (all2 la2I2-t is the N, scaling: MO N , and I N,. This corresponds to a slowly rotating hedgehog a t large N, and its motion can be separately quantized:

MI=J = Mo +

+

J ( J 1) 21

Note all states have I = J and one can consistently quantize the system to have I=J being half integers. The lowest two states have I=J=1/2 (nucleon) and I = J = 312 (A) and thus M A - M N = 3/I l/Nc. The collective wave functions are just the Wigner D matrices (appropriately normalized). It should be stressed that this approach is based on an ansatz and as such one must check the self-consistency of the approach. Ultimately it is justified a t large N, since the collective motion is adiabatic: angular velocities go like J / I NT1 and are truly collective, covering the full angular space (of order N:) yielding frequencies (and hence excitation energies of order N;'. This, in turn, implies a Born-Oppenheimer separation of the slow collective motion from the faster modes associated with vibrations of the meson fields (with time scales of order N," and hence energies of order N t ) . Because of this scale separation the collective modes can be quantized separately from the intrinsic vibrations. N

N

254

Now let us turn to the problem of relevance to 8+ physics: SU(3) solitons. Shortly after the ANW quantization was introduced it was extended by a number of workers to solitons in models with SU(3) flavor*. Extending the models from SU(2) to SU(3) flavor is intellectually straightforward. The only essential new feature is the inclusion of a topological Wess-ZuminoWitten term which builds in the anomaliesg. For simplicity, first consider the case of exact SU(3) symmetry. The standard semiclassical approach is then to first solve the problem classically which yields a hedgehog configuration in a two-flavor subspace (which we will take to an intrinsic u-d subspace by convention). Following the ANW ansatz one introduces a time-dependent, space-independent, global SU(3) rotation A(t). At this stage the parallel of the two flavor case is virtually exact. However, the Wess-Zumino-Witten term introduces a constraint: In the body-fixed (corotating) frame the hypercharge must be N,/3. This constraint plays a critical role in what follows. Going through this procedure yields a collective rotation hamiltonian of the form:

where the prime indicates the generator as measured in a co-rotating frame, and I1 (I2) is the moment of inertia for motion within (out of) the original SU(2) subspace. Note that there is kinetic energy in the intrinsic 8 direction as it leaves the original hedgehog unchanged. The physics associated with this direction is encoded in the constraint. As noted by Wittenlo this is quite analogous to the problem of a charge particle moving in the field of a magnetic monopole. This procedure yields masses given by

with

CZ= (p2 + q2 + p q -t- 3(p + q ) ) /3 .

(4)

C2 is the quadratic Casimir, and is labeled by p , q which specify the SU(3) representation. The constraint due to the Wess-Zumino-Witten term imposes the restriction that the representation must have a state with Y = N,/3 and implies that angular momentum is determined by the condition that 2 5 + 1 equals the number of states with S=O. Plugging in N , = 3, one sees the lowest representations are (p,q)=(l,l) (spin 1/2 octet),

255 (p,q)=(3,0) (spin 3/2 decuplet) and (p,q)=(0,3) (spin 1/2 anti-decuplet). This last representation is clearly exotic. Diakonov, Petrov and Polyakov proposed to take these anti-decuplet states seriously4. The predicted value depends on 12 which was fit by identifying the N(1710) as a member of the multiplet and by using SU(3) breaking effects included perturbatively with all parameters determined in the non-exotic sector. The fact the width of the state came out as relatively small was taken as a self-consistent justification for not treating the B+ as a large N , artifact. The question which needs to be addressed is whether the rigid-rotor type semiclassical projection used here is kosher for the exotic states. Since the question ultimately comes down to whether the Born-Oppenheimer separation is justified a t large N, for these states, care should be taken to keep N, arbitrary and large throughout the analysis. In particular one ought not set N, = 3 when imposing the constraint of the Witten-WessZumino term, at least when doing formal studies of the N , dependence. In doing this one sees that the flavor SU(3) representations at large N, differ from their N , = 3 counterparts. The lowest-lying representation consistent with the Witten-Wess-Zumino term constraint has ( p , q ) = (1,v and) has J = 1/2. Thus, it is a clear analog of the octet representation and will be denoted as the “8” representation. (The quotes are to remind us that it is not in fact an octet representation.) Similarly the next lowest and has J = 3/2. It is the large N , representation, has ( p , q ) = (3, analog of the decuplet and will be denoted “10”. The salient feature of the anti-decuplet is that its lowest representation contains an exotic S=+l state. Thus its large N, analog is the lowest representation containing manifestly exotic states. This representation is (p,q ) = (0, and has J = 1/2; it will be denoted Now let us look a t the excitation energy of the exotic states which can be computed from eq. (4):

y)

“m’.

-

9)

Noting that I2 N,, one sees that this implies that the excitation energy of the exotic state is of order N,“. This may be contrasted to the excitation of the non-exotic “10” representation which is of order l / N c . The fact the standard semiclassical quantization gave excitation energies of order N,“ for exotic states means that the approach is not justified for such states. Recall that the analysis was justified self-consistently via a Born-Oppenheimer scale separation. For the non-exotic states this is

256 justified. However, for the exotic states the characteristic time associated with the excitations (one over the energy difference) is order N,“. This is the same characteristic time as the “fast” vibrational excitation of the meson fields. One cannot therefore justify treating the “collective” degrees of freedom separately. Thus the prediction of 8+ properties via this collective quantization procedure cannot be justified from large N,. In fact, there are many other ways to see that approach is not justified by large N,. There is an extensive discussion of these in ref. 3 . Here we will briefly mention a couple of these. As discussed above, ref. stressed the small numerical value of the width to justify the treatment self-consistently. However, from the perspective of formal large N, consistency, this numerical value is essentially irrelevant. The key question is how does the width depend parametrically on N,? If the standard collective quantization procedure outlined above had been justified by large N , physics, then it should become exact in the large N,limit in the sense of giving an exact value for the mass. If this is not the case then some kind of ad hoc correction must be added on and there is no a priori reason for it to be small unless it vanishes a t large N,. This in turn means the width must go to zero a t large N,. One way to see this is simply that if the width is non-zero, then an asymptotic state doesn’t exist and the concept of an exact mass becomes ill defined. Alternatively one can view the width as originating from an imaginary contribution to the mass. However, the standard collective quantization gives a real value, and had it become exact a t large N , one would have zero widths in this limit. The upshot of all this is that if the approach is justified, then at a formal level the width must approach zero at large N,. Of course, for non-exotic states such as the decuplet, this is true. The reason is simply phase space. As N, grows, the excitation energy for these states drops thereby killing the phase space for decay. In contrast, as shown recently by Praszalowiczll the width of the 8+ as calculated via the standard collective approach is order N,“. This demonstrates that the procedure is not self consistent. Another perspective is given by the spin-flavor contracted SU(6) symmetry which can be derived on general grounds from QCD using large N, consistency rules It is precisely the existence of such a symmetry that requires predictions independent of details in soliton models to arise as true large N , model-independent results of QCD. However, for three flavors the results of such an analysis are quite well known. One predicts exactly the states in a large N, naive quark modelLexotic collective states are not obtained in this model-independent approach. Thus, a priori there is no

‘.

257

fundamental reason to believe that the quantization procedure discussed above which gives rise to collective exotic states is consistent with QCD. In contrast, the nonexotic states clearly are. For another perspective on problems with the standard method used to quantize these solitons see Igor Klebanov’s talk in this volume and his work with collaborators a t Princeton in ref. 1 2 . While there appears to be compelling evidence that the approach fails there is a n obvious question as to why. Of course, at certain level there is no mystery. The approach is based on an ansatz and the ansatz needs to be shown to be self consistent. While the scales are such that the properties of non-exotic states can be self-consistently described, they are also such that the self-consistency fails for the exotic states. At a deeper level the failure is due to the mixing of intrinsic vibrational modes with the collective rotational modes. For systems without velocity dependent forces such modes are orthogonal and hence cannot mix at leading order. However, as discussed in ref. 3 , velocity-dependent forces spoil this orthogonality and allow mixing. The Witten-Wess-Zumino term gives rise to precisely such velocity-dependent interactions and spoils the orthogonality. In conclusion, the standard collective quantization method as applied to exotic states such as the 8+ cannot be justified from large N , QCD. It is logically possible, of course, that such a procedure can be justified for some other reason, but at present no such justification is known. Thus one should view any predictions based on this procedure with real caution. References T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). T. D. Cohen, Phys. Lett B581 175 (2004) T. D. Cohen, hep-ph/0312191, to be published in Phys. Rev. D. D. Diakonov, V. Petrov and M. Polyakov, 2. Phys. A 359, 305 (1997). While this paper was not the first to touch on the possibility of the 0’ in a soliton model, it is probably the first to take the possibility seriously and was of considerable importance in prompting experimental studies. The earliest published work on the subject appears to be M. Praszalowicz, talk at “Workshop on Skyrmions and Anomalies”, M. Jezabek and M. Praszalowicz, editors (World Scientific, 1987), page 112. However, Praszalowicz’s work was not published in a refereed journal. An updated analysis based on this work may be found in Phys. Lett. B575 234 (2003). 5. G. S. Adkins and C. R. Nappi, Nucl. Phys. B249, 507 (1985). 6. J.-L. Gervais and B. Sakita Phys. Rev. Lett. 52, 87 (1984); Phys. Rev. D 30, 1795 (1984);R.F. Dashen, and A.V. Manohar, Phys. Lett. 315B, 425 (1993); Phys. Lett. 315B, 438 (1993);E. Jenkins, Phys. Lett. 315B, 441

1. 2. 3. 4.

258

7. 8.

9.

10. 11. 12.

(1993);R.F. Dashen, E. Jenkins, and A.V. Manohar, Phys. Rev. 49, 4713 (1994): Phys. Rev. D 51,3697 (1995). G . Adkins, C. Nappi and E. Witten, Nucl. Phys. B228, 522 (1983). E. Guadagnini, Nucl. Phys. B236, 35 (1984); P. 0. Mazur, M. A. Nowak and M. Praszalowicz, Phys. Lett. B147,137 (1984); A. V. Manohar, Nucl. Phys. B248, 19 (1984); M. Chemtob, Nucl. Phys. B256, 600 (1985); S. Jain and S. R. Wadia, Nucl. Phys. B258, 713 (1985). E. Witten, Nucl. Phys. B223, 433 (1983). E. Witten, Nucl. Phys. B223, 422 (1983). M. Praszalowicz, Phys. Lett. B583 96 (2004). N. Itzhaki, I. R. Klebanov, P. Ouyang and L. Rastelli, Nucl. Phys. B684, 264 (2004).

SECTION 4.

QCD MATTER AT HIGH TEMPERATURE

AND DENSITY

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BARYON AND ANTI-BARYON PRODUCTION AT RHIC

J. I. KAPUSTA School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA E-mail: [email protected] We use a form of the fluctuation-dissipation theorem to derive formulas giving the rate of production of the spin-1/2 baryon octet in terms of the fluctuations of vector and axial-vector currents at finite temperature. An estimate is made of the corresponding rates for producing the spin-3/2 baryon decuplet. Rate equations are solved for hydrodynamically expanding matter to obtain the final observed baryon ratios. The results are compared to RHIC data.

1. Introduction It has been a challenge to understand the relatively high abundances of baryons and anti-baryons produced in high energy gold-gold collisions at the Brookhaven National Laboratory RHIC (Relativistic Heavy Ion Collider). The center of mass collision energies have ranged between 56 and 200 GeV per nucleon pair. This energy is so high that baryonlanti-baryon pairs can readily be created. To very good accuracy the relative abundances, including multiply strange hyperons, are consistent with them being in chemical equilibrium at a temperature of 170 f 10 MeV and a baryon chemical potential on the order of tens of MeV l ~ ~Furthermore, . the ratio of baryons to mesons grows with increasing transverse momentum, reaching one at p r x 2 GeV/c ’. A frequently given explanation of the first result is that the hadrons are created in chemical equilibrium at some temperature and chemical potential, possibly at the end of a quark-gluon to hadron phase transition. In the absence of detailed information on the dynamics it is quite reasonable to postulate that phase space is filled randomly. For a big system, this is equivalent mathematically to momentary or instantaneous thermal and chemical equilibrium. A possible explanation of the second result is that since the transverse momentum is proportional to mass, collective fluid-like flow velocity of the expanding matter would boost heavier

261

262 particles to higher transverse momentum. A nice overview of these results can be obtained from the Proceedings of Quark Matter 2002 '. The results on baryon production at RHIC are exciting and stimulating. A parametrization of them in terms of temperature T, baryon chemical potential p arising from the initial baryon number of the gold nuclei, and transverse flow velocity UT is a first step. The next step is to understand them in terms of microscopic, dynamical processes. I will present a network of rate equations to describe the production of baryon/anti-baryon pairs (such as ph and ZA) including all members of the octet and decuplet. Also included are transformations among them (such as A+ H p r o ) . The rate of production of octet-octet pairs was derived with Shovkovy while the rates for octet-decuplet and decuplet-decuplet was estimated with Huovinen G . The network of rate equations in a 2+1 dimensional hydrodynamical model of the expanding high energy density matter in heavy ion collisions, which has been tuned to reproduce the measured pion multiplicity 3 , was solved with Huovinen Since this is a set of coupled differential equations initial conditions are required. We begin to solve the rate equations in the mixed phase of a first order quark-gluon to hadron phase transition with three values of T,, namely, 165, 180, and 200 MeV. In one scenario we assume that all baryons are in chemical equilibrium initially, and in the other we assume no baryons initially. Other scenarios are certainly possible. We plot the resulting abundances as a function of the local (freeze-out) temperature T f and compare to the measured numbers of p , A, E-, and s2 in central gold-gold collisions at mid-rapidity at a beam energy of 65 GeV per nucleon at FtHIC.

'.

2. Network of rate equations

We need to specify the dynamical variables, measure or calculate the microscopic rates, and then solve the resulting network of rate equations. We shall do each of these in turn. Various members of the baryon (or anti-baryon) octet and decuplet have been or will be measured in central heavy ion collisions at the CERN Super Proton Synchrotron (SPS) and at RHIC. Therefore the variables will be all members of those multiplets. There are many mesons that are much lighter than the baryons, such as the pion and the p and w vector mesons to name just a few. The baryons are also much heavier than the three lightest species of quarks when the latter are given their current quark masses. Therefore it ought to be a good approximation to consider the mesons and/or the quarks

263 and gluons as providing a thermal bath in which baryonlanti-baryon pairs can be created or destroyed. Since kinetic equilibrium is usually reached much quicker than chemical equilibrium it is reasonable to take the kinetic energy distribution of the baryons and anti-baryons as approximately thermal. We assume that only the absolute number of each species deviates from local chemical equilibrium. These approximations can be relaxed but, as they are, the calculations become increasingly complex. Ultimately one would reach the point where microscopic cascade calculations would be necessary, and then one encounters the problem of multi-particle initial states in the quasi-localized interactions. That problem is largely avoided in the present approximation because of the thermal averaging of the microscopic rates (see below). Under the above conditions, the rate equation for the spatial density of a baryon species b is

b'

The first term on the right involves the rate for producing the specified baryonlanti-baryon pair by strong interaction currents, R(b6') = the number of such pairs produced per unit volume per unit time, and is the driving term in the rate equation. Of course, the same pair can annihilate each other, and this is related to the production rate by detailed balance. The factor in square brackets enforces detailed balance. The n;Pui1 is the equilibrium density of the species b' as represented by the temperature and chemical potentials at the time t. The second term is a dilution term; the density will decrease in inverse proportion to the volume for an expanding system. The third term arises if there exists a baryon species that can decay via the strong interactions into the baryon species of interest. The quantity in square brackets following it allows for the inverse reaction and satisfies detailed balance. The last term arises if the baryon species of interest can decay into another baryon via the strong interactions. There is one first order in time, nonlinear, rate equation for each species of baryon. This makes a coupled network of differential equations.

264

The decay rate r(b‘ + b + X ) is just the inverse lifetime for the specified decay, namely, 1/T(b’ --f b X ) . These are taken from the Particle Data Tables ’. We include the following decays: A .+ N T , C* + h T ,

+

+

+

C*-tC+.rr,Z*--tZ+T.

There will always be more baryons than anti-baryons present in the final state of a heavy ion collision because the colliding nuclei have a net baryon number. However, this asymmetry is reduced as the beam energy increases because produced baryons always come with an anti-baryon due to baryon number conservation. The pair production is an increasing function of beam energy and so the initial number of baryons from the nuclei becomes a smaller and smaller fraction of the total. At the highest energies at RHIC the baryon chemical potential is usually estimated to be on the order of 10 MeV when the temperature is 150 MeV. For simplicity of calculation and presentation we shall take the baryon chemical potential to be exactly zero in this paper. To remove the difference between the experimental data and our calculation due to this approximation, we compare our results to the observed average yield of a baryon and antibaryon, for example ( p p ) / 2 , not t o the observed baryon yield as such.

+

3. The Rates The rate R is the crucial ingredient in the network of rate equations. It was derived in quite some detail by Shovkovy and me for the case of baryons and anti-baryons in the lowest octet. The basis for that calculation involved an effective current-current interaction between the baryons and strong interaction currents such as the vector and axial-vector. As is well known there are three types of SU(3) invariant couplings: the F and D types, so-called because they involve the correspondingly labeled group structure constants, and the singlet coupling, which involves the trace of the meson matrix. r

+ a v n (BY’ { V p ,B } ) + ~

’

A ~(BY A YD{ & B ) )

+ g ~ ( 1 -c ~ A ) T(BY’Y~ ~. [ A p a]) , + SAPATI. (BY’Y~B)Tr (A,)] (2) The overall normalization of this interaction Lagrangian is determined by the coupling of the nucleons. Four parameters are introduced: av and

265 Q A , which determine the relative contributions of the D and F type COUplings in the vector and axial-vector channels, respectively, and ,& and PA, which determine the corresponding singlet contributions. These constants are fixed by data to be approximately: QV = 0, Pv = 1, Q A = 2/3, and PA = -113. A combination of hadron phenomenology and measured cross sections allow the rates to be expressed in terms of the spectral densities of the strong interaction currents. Since the f i is so large, with a minimum value determined by threshold production of the baryonlanti-baryon pair, these spectral densities can be taken from perturbative QCD. For the production of baryons in the octet, the rate is expressed as

Here

where zi = mi/T, T is the temperature, and FiN,(s) is an annihilation form factor evaluated at the average value 3 = (ml m2)2 3(m1-k m2)T. A simple parametrization of this form factor which is consistent with nucleon/anti-nucleon annihilation data was found to be

+

+

with A = 1.63 GeV. According to the latest data analysis the strong interaction coupling is a,(m:) = 0 . 3 5 f 0 . 0 3 . The z t correspond to vector/axialvector contributions to the rate. The numerical coefficients C*(b1&) are given in the paper with Shovkovy 5; they are all of order one. There is unlikely ever to be sufficient experimental information from hadron-hadron scattering to pin down the production rates for baryons in the decuplet since they are all unstable with lifetimes less than 10-l' seconds. Therefore, Huovinen and I estimated the rates and parametrize them as follows '.

We take the coefficients of the vector and axial-vector contributions to be

266

equal which means that

+ 4Z2Kl(ZZ)KZ(zl) + 221z2Kl(zl)Kl(&?)

x {4zlKl(zl)K2(z2)

+ [16 + .$+ &]K2(z1)K2(~2>} J’&N(S).

(7)

The coefficients C8,10(blb2) and Clo,lo(b1b2) are tabulated in the paper with Huovinen ‘. Exact isospin symmetry within SU(2) multiplets was used. Flavor SU(3) symmetry was also used, but it cannot predict all of the coefficients uniquely. We made our best estimates of the coefficients when symmetry and hadron phenomenology was not sufficient. The uncertainty in the coefficients is unlikely to be more than a factor of 2, which is probably good enough for our purposes. 4. Comparison to data

Baryon and anti-baryon production cannot begin until the local energy density is low enough, that is, when the matter is in the hadronic phase as opposed to the quark-gluon phase. This will occur during the expansion stage of a high energy nucleus-nucleus collision. This stage is frequently modeled with hydrodynamics ’. We will use a 2+1 dimensional description of the final stage expansion that takes into account transverse expansion. The details of these calculations have been given many times before and so we just refer the reader to those papers ’. Briefly, the expansion begins in the quark-gluon plasma phase with an energy density adjusted to reproduce the measured pion rapidity density at mid-rapidity. The phase transition is first order; we have chosen T, to be 165, 180, and 200 MeV and test the sensitivity of baryon production to it. The network of rate equations is solved within each co-moving cell in coordinate space. Different cells evolve somewhat differently in space and time, and this makes for a very computationally intense task. At each point in time, as measured by an observer at rest in the center-of-momentum frame of the central gold-gold collision, each cell has its own temperature. This local temperature enters in the production rates and in the local chemical equilibrium densities. In order to concisely display the results of our calculations we plot the solutions to the network of rate equations as a function of the local temperature and not as a function of the local time. Because each fluid cell has its own temperature at any given time, we must integrate over a constant temperature hypersurface to get the baryon rapidity density at this temperature. Thus the value at a fixed temperature

267 does not correspond to any particular time. This choice is rather natural since baryon production begins at a system-wide value of the temperature equal to T, and ends at a supposed freeze-out temperature Tf where the hadrons lose local thermal equilibrium and begin their free-streaming stage. In figure 1 we display the results of numerical solution of the rate equation network for p , A, 2- and R. Plotted is the ratio of the calculated

lo2

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\

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.

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T, (MeV) The ratio of the calculated abundance of the indicated species to the chemical equilibrium value as a function of the local temperature. The upper set of curves start with the baryons in equilibrium at T,,the lower set start with no baryons.

density to the equilibrium density a t that particular temperature. No weak decays of unstable baryons have been allowed for a t this point. The rate equations require an initial condition and we have chosen two: all baryons in chemical equilibrium at T,, and no baryons present a t all a t T,. The former are represented in the figure by the upper set of curves while the latter is represented by the lower set of curves. Obviously this calculation has used T, = 180 MeV, but it is representative of other choices. When baryons are present initially with their equilibrium abundances, they evolve

268

in such a way that they always stay above the equilibrium abundance at each temperature below T,. The reason is that the system expands more rapidly than chemical reactions can keep up with. In other words, the typical annihilation rates are smaller than the expansion rate. When no baryons are present initially, their abundance at first builds up rapidly. The reason that there is a finite abundance at T, already is that the cells generally remain within the mixed phase at T, for a finite time span, thus allowing a buildup before the temperature drops below T,. In the range from 120 to 130 MeV, these baryon species have caught up to the equilibrium abundance at that temperature. Thereafter they are above the equilibrium values for the same reason as stated above, namely, that annihilation rates are generally smaller than the expansion rate. Naturally, the abundances with no baryons present initially never catch up with the abundances where baryons were initially created in chemical equilibrium. In figures 2-5 we show the rapidity density at mid-rapidity as a function of freeze-out temperature for p , A, E- and R, respectively. The upper set of curves result from having baryons in chemical equilibrium initially, while the lower set of curves result from having no baryons present initially. The three sets of curves correspond to the three different choices of T,. Also shown in these figures are the experimental data from RHIC experiments. Because we use the approximation of zero baryon chemical potential, we compare our result to the measured average yield of baryon and antibaryon. The ( p p)/2 and (A i ) / 2 data come from PHENIX lo and the (Z%+)/2 and (R 0 ) / 2 data come from STAR ll. The calculation and the PHENIX data are for 5% most central collisions whereas the STAR data are for 10% most central collisions. By comparing the hydrodynamically calculated pion yields at different centralities we have estimated that this leads to about 10% larger yield in our calculation compared to the data. The darker central band in each figure represents statistical errors only while the lighter outer band includes systematic errors as well. (For R the systematic error is smaller than the statistical error.) We have performed decays of unstable baryons appropriate to the particular measurement. For example, protons coming from the weak decay A + p + r - are not included in figure 2, but protons from the decay C+ 4 p no are.

+

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T, (MeV) Figure 2. The number of protons at central rapidity in 5% most central gold-gold collisions as a function of local temperature. The solid curves start with baryons in equilibrium, the lower set with no baryons. With the exception of A, baryons unstable to strong or weak decays have been decayed. The data are from the PHENIX collaboration lo for the same centrality. The dark band represents statistical and the light band systematic errors.

20 15 h -d

\ 10

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T, (MeV) Figure 3.

Same as Fig. 2 except for lambdas.

200

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T, (MeV) Figure 4. The number of cascades at central rapidity in 5% most central gold-gold collisions a s a function of local temperature. The solid curves start with baryons in equilibrium, the lower set with no baryons. Baryons unstable to strong decays have been decayed. The data are from the STAR collaboration l1 for 10% most central collisions. The dark band represents statistical and the light band systematic errors.

1.0

0.8 h

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\

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Figure 5. Same as Fig. 4 except for omegas. The dark band represents statistical errors only; the systematic errors are smaller than the statistical ones.

271

What we learn from this set of figures is that the baryon production rates are too small in comparison to the expansion rate to reproduce any of the experimental data when there are no baryons present initially. There must be a significant abundance of baryons present already at the beginning of the hadronic phase. Assuming the baryons to be produced in chemical equilibrium and then evolving them during the expansion of the matter provides agreement with the data on p , A, and Z- within systematic error bars for 120 5 T f 5 T,. However, the theoretical calculations produce about 50% more R than observed. It is interesting that this magnitude of discrepancy does not occur for the Z-, even though it has two valence s-quarks. Oftentimes in purely statistical models an s-quark fugacity factor is used to fit the data. A typical value might be 0.9, so that the A yield would be multiplied by 0.9, the E- yield by (0.9)2,and the R yield by (0.9)3. Any such suppression factor should be an outcome of the results of a rate equation calculation like the present one, and cannot be done arbitrarily at the end of the calculation to fit data. On the other hand, if one had confidence that there was a first-order phase transition with a known value of T, and that the expansion was described adequately by hydrodynamics, then one could adjust the initial conditions to match the experimental data. Presently, however, there is no value of T, which provides agreement between the calculated abundance and the observed one for all four species of baryons.

5. Conclusions and Tasks

We have calculated the production of spin-112 octet and spin-312 decuplet baryon/anti-baryon pairs through fluctuations in the strong interaction currents. We evaluated them in thermal equilibrium and made quantitative predictions for the rates. A network of rate equations were solved in a 2+1 dimensional relativistic hydrodynamic expansion model of central Au+Au collisions at RHIC. Comparison to PHENIX and STAR data suggest that baryons and anti-baryons must have appeared near chemical equilibrium at a temperature of about 170 MeV. There is insufficient time to create them from zero initial abundances. Two extensions of this work immediately suggest themselves. First, including the exchange reactions m b --+ m’ b’ will redistribute the baryon number and may improve agreement with data. Second, one should calculate the transverse momentum distributions of baryons and anti-baryons as this will provide a more refined comparison between theory and experiment.

+

+

272 Acknowledgements This work was done in collaboration with I. Shovkovy a n d P. Huovinen '. This work is supported by t h e US Department of Energy under grant DE-

FG02-87ER40328.

References 1. P. Braun-Munzinger, D. Magestro, K. Redlich, and J. Stachel Phys. Lett. B518, 41 (2001). 2. J. Rafelski, J. Letessier, and G. Torrieri, Phys. Rev. C64, 054907 (2001); Erratum-ibid. C65, 069902 (2002). 3. K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 8 8 , 242301 (2002). 4. Proceedings of Quark Matter 2002, Nantes, France, July 2002, Nucl. Phys. A715, l(2003). 5. J. Kapusta and I. Shovkovy, Phys. Rev. C68, 014901 (2003). 6. P. Huovinen and J. Kapusta, Phys. Rev. C69, 014902 (2004). 7. Particle Data Group, D. E. Groom et al., Eur. Phys. J . C15, 1 (2000). 8. P. Huovinen, arXiv:nucl-th/0305064; P. F. Kolb and U. Heinz, arXiv:nuclt h/0305084. 9. P. F. Kolb, P. Huovinen, U. W. Heinz and H. Heiselberg, Phys. Lett. B500, 232 (2001); P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Rev. C62,054909 (2000). 10. K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 89, 092302 (2002). 11. J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92, 182301 (2004).

QCD AT T H E BOILING POINT: WHAT DOES HADRON PRODUCTION AT RHIC TELL US?

R. J. FRIES School of Physics and Astronomy, University of Minnesota, 11 6 Church Street SE, Minneapolis, MN 55455, USA E-mail: [email protected]. edu The Relativistic Heavy Ion Collider (RHIC) has provided us with large amounts of data and some unexpected discoveries. I discuss some of the recent results on hadron production, how they can be interpreted and how this can help us to establish the existence of a phase transition in QCD.

1. Introduction The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab1 was built to create matter at densities and temperatures that have not existed in our universe since the first instance after the Big Bang. By smashing completely stripped gold ions with a center of mass energy of & = 200 GeV per nucleon pair, a total energy of 39.4 TeV is concentrated into a very small region in space (several fm3 in the lab frame). Most of the energy will remain kinetic energy of the beam fragments, but a considerable fraction is used to heat the collision zone, create momentum transverse to the beam axis and produce new particles in copious amounts - several thousand new hadrons emerge from a central AufAu collision together with a flash of y-rays and neutrinos. Behind these impressive numbers is an ambitious physics goal to find new states of matter at ultra high temperatures. We believe that the early universe had to undergo several phase transitions while expanding and cooling, to arrive at the state we know it today. One of these phase transitions is related to the Strong Interaction. Massive numerical calculations using a discretized version of quantum chromodynamics (QCD) have shown that at a critical temperature T, around 170 MeV (- 2,000 billion K) and not too high baryon density a deconfinement phase transition together with

273

274

a restoration of chiral symmetry takes place'. While at low temperatures quarks and gluons are forced to appear only as part of colorneutral states in hadrons - above T, quarks and gluons can be viewed as free particles. If an ensemble of hadrons is subject to heating above T,,the individual hadrons will eventually melt and the freed partons will form a state of matter termed the quark gluon plasma (QGP). In the other direction, the hot soup of partons that filled the early universe had to condense into hadrons upon reaching the critical temperature.

Quark gluon plasma

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

PB

Figure 1. A sketch of the QCD phase diagram as a function of temperature T and baryon chemical potential pg . The confinement-deconfinement phase transition and a speculative high density phase as it could appear in the interior of neutron stars are shown. The freeze-out conditions of several heavy ion experiments are indicated.

RHIC was built to confirm the existence of a quark gluon plasma, to create it routinely in Au+Au collisions and to study its properties. This is a task more difficult than it appears since we do not have a direct probe of the short-lived (- lo-'' s) plasma phase. All that we can study are the hadronic and electromagnetic endproducts of the collision. In these notes I will review some of the results on hadron production delivered by the four RHIC experiments and discuss how they might help us learn about the existence of a QGP.

275 2. Thermalization and Flow

Our first piece of evidence is the finding that the abundances of hadrons produced at RHIC are perfectly described by statistical ensembles. The yields of hadrons are given by Fermi-Dirac or Bose-Einstein distributions

with a Temperature T and chemical potentials for baryon number (and possibly strangeness and isospin) The temperature is found to be around 175 MeV. Hence hadrons seem to be produced in a thermal equilibrium with the temperature very close to the alleged phase transition temperature. Interestingly, heavy ion experiments at the CERN SPS, conducted at much lower energies & = 17.6 GeV result in hadron spectra with almost the same temperature. In other words, increasing the initial energy by an order of magnitude by going from SPS to RHIC did not result in higher temperatures for our hadron cocktail. This is a strong indication that 175 MeV is something like a maximal temperature for hadronic matter, but it is not a proof for a deconfinement phase transition. If one does a more differential measurement, e.g. in transverse momentum PT of the hadrons, the PT spectra are again well described by the momentum distribution of the statistical ensemble up to PT N 1.5 GeV/c. But there is one more interesting observation. The source of hadrons is not a thermal ensemble sitting at rest in the lab frame. Instead, parts of the source are accelerated and moving out of the collision zone with large velocity u p . The radial velocity can be as large as 0 . 6 in ~ the lab frame. This phenomenon, called radial flow, is not present in collisions of elementary particles. Radial flow is an effect of bulk matter, not of single particles and its existence can be explained by pressure gradients in the hot bulk matter created in the collision. Indeed, it turns out that the dynamics of the hot fireball created in central AufAu collisions at RHIC obeys the laws of hydrodynamics6. The initial heating and build-up of pressure' is followed by a phase of expansion and cooling of the system that is nicely described by numerical simulations of relativistic hydrodynamics. We should note that in an ensemble of particles the hydrodynamic limit is the limit of vanishing mean free path of the particles. Hence the system we are looking at is dense and strongly interacting. Hydrodynamics is also successful to describe how the initial spatial anisotropy of a collision at finite impact parameter is translated into an 3,475.

276 anisotropy in the observed momentum spectra of hadrons. The originally ellipsoidal collision zone of the two nuclei exhibits higher pressure gradients where the overlap zone is thinner, leading to a more rapid expansion and higher flow velocities in this direction, see Fig. 2. This is usually analyzed by using a Fourier decomposition7

The coefficient v2 is also called elliptic flow and gives the approximate shape of the anisotropy. We restrict our discussion to midrapidity where the odd-order coefficients vanish. We conclude that signs of thermalization and collectivity have been seen at RHIC. Therefore we definitely created bulk matter which is necessary to have a phase transition. But we still lack a hint whether the bulk matter had partonic degrees of freedom at some point.

Figure 2. Azimuthal anisotropy originating from a finite impact parameter of the colliding nuclei. The pressure gradient is larger where the overlap zone is narrower.

3. J e t Quenching and RHIC Puzzles We have seen that PT spectra of hadrons are well explained by hydrodynamics or simple blastwave models of an expanding thermal source up to 1-2 GeV/c. However, there are also hard QCD processes taking place that produce partons with 10 GeV/c or more and that could develop into a jet. We even expect hard processes to scale more than linearly with the mass number A of the nucleus, more precisely with the number of binary nucleon-nucleon collisions Ncoll A4/3. It had been argued for a long time that these jets, embedded in the large fireball, should not behave like jets

-

277

+

+

developing in the vacuum (like the ones created in e+ e- or p f j collisions). It was suggested that hard partons lose energy by radiation induced by the hot medium and that jets should therefore be quenched’. The RHIC experiments have confirmed that the yield of hadrons with PT above 2 GeV/c is indeed suppressed by up to a factor of 49. The interesting quantity is the nuclear modification factor

which gives the yield of hadrons relative to a simple superposition of the corresponding number of nucleon-nucleon collisions. The suppression of high-& partons is another indication that a hot and dense medium is created at RHIC. Arguments have been presented why the observed jet quenching can only be partonic in naturelo. (Jets could also be quenched in hadronic matter.) This is providing a first but indirect indication that the initial degrees of freedom in the bulk matter are quarks and gluons. At the intersection of soft and hard QCD physics, i.e. for hadrons produced at several GeV/c transverse momentum, several puzzling observations were made by the RHIC experiments. First, the ratio of protons to pions is about one for PT > 1.5 GeV (Fig. 3)11. This is at odds with ex0.2 as it has pectations from pQCD, that the ratio should be roughly e- collisions at LEP. It turns been observed at similar energies in e+ out that this is a consistent feature of baryon enhancement at intermediate PT (from roughly 1.5 to 5 GeV/c) and is also confirmed for jj, A and Ll2. The anomalous baryon enhancement can also be seen in the nuclear suppression factor RAA. While pions and kaons show values as low as 0.2 at intermediate PT, protons and baryons have values between 0.8 and 1. One immediately asks the question whether energy loss, which is believed to be responsible for the suppression, can be partonic in nature, if baryons and mesons are affected differently. The standard theory is that a fast parton is created in a hard QCD process, propagates through the medium and loses energy while interacting with it, and finally fragments into hadrons. If this picture applies the energy loss should be equal for all kinds of hadrons.

+

-

4. Hadronization by Quark Recombination

As a solution, several groups suggested, that hadronization at intermediate PT in Au+Au collisions is not governed by fragmentation of hard partons, as it is in e+ + e- and p f j collisions, but by recombination of quarks from the surface of the firebal113~14~15~16. The basic idea is that if partons

+

278

1.8 1.6 1.4 1.2 +k 1.0 \ Q 0.8 0.6 0.4 0.2 0.0

'

'

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 PT (GeV)

Figure 3. The ratio of p and 7 ~ f in central (filled circles) and very peripheral (open circles) Au+Au collisions. The ratio in central collisions, where we expect a quark gluon plasma, follows predictions from a purely thermal model (dash-dotted line). In peripheral collisions, where thermal and chemical equilibrium is not reached, the data is closer to the pQCD prediction (solid line). Recombination can explain why protons 4 GeV/c. Data from the PHENIX are so effectively produced thermally at high PT collaboration. N

are available in phase space, in particular in a situation close to kinetic equilibrium, they might simply coalesce or recombine into hadrons. This idea was first advocated outside of heavy ion physics to describe hadron production, in particular the so called leading particle effect17. In a heavy ion collision, the partonic degrees of freedom, when approaching T, from above, are essentially constituent quarks, since the average virtualities are below any perturbative scale. So gluons and sea quarks are taken care of by dressing the constituent quarks and the data indeed seems to indicate that hadrons are only recombining using the lowest Fock state, i.e. valence quarks. How can recombination help to explain the anomalous baryon enhancement seen at RHIC? Let us first see how the hadron yield is usually calculated in the case of a meson. The Wigner function W of the valence quarks in the parton phase is convoluted (using relative momentum and relative distance) with the Wigner function of the meson. The spectrum is then

279

obtained by integrating over the hadronization hypersurface C

E dN d3P

d a u ( a )" ( 2 ~ ) ~

It is easy to prove that recombination is more effective than fragmentation on a thermal parton distribution, while a power-law shaped parton spectrum favors fragmentation at large enough PT. It is also easy to see that recombination from a thermal distribution treats mesons and baryons on equal footing, leading to a thermal distribution of hadrons in the final state, while it is known that fragmentation disfavors baryons. When doing the numerics one finds that a thermal parton distribution, recombining at a temperature of T = 175 MeV and with a radial flow of 0 . 5 5 ~together with fragmentation from hard processes gives an excellent description of all hadron production data above a PT of 1.5 GeV/c. Recombination from the fireball is dominant up to 4 GeV/c for mesons and up to about 6 GeV/c for baryons, while fragmentation from hard processes takes over at higher PT. The calculations show good agreement with the measured hadron spectra, hadron ratios and RAAratios. If already dominant at intermediate PT, recombination should be even more important at low PT where the bulk of hadrons is produced. This is probably true, but it is obscured by several things. Firstly, it can be shown that at intermediate PT and for thermal parton distributions the hadron Wigner functions (or equivalently the hadron wave function) can be integrated trivially regardless of their shape. This means that the nonperturbative dynamics of the bound state is not important in this case. This is very different at low PT. Even worse, the kinematics must be different to accommodate energy conservation, involving at least 2 -+ 2 and 3 -+ 2 processes instead of 2 -+ 1 and 3 -+ 1 as in the case of the simple minded coalescence. This would also cure the problem of entropy non-conservation. Finally, interactions also occur in the hadronic phase after the phase transition. They are very important at low PT and have important consequences on the observed hadron spectra.

5. A New Quark Counting Rule for Elliptic Flow In the last section we have seen that we have a theory involving partonic degrees of freedom, thermalization and collectivity, and which can nicely describe the data. Nevertheless it would be good to have more direct evidence

280 for subhadronic degrees of freedom showing thermalization or collectivity. This was provided by a stunning result coming from measurements of the elliptic flow coefficient u2 as a function of PT. Measurements carried out at intermediate PT were a t first disappointing, since different hadrons led to very different results, making this another RHIC puzzle. However, it soon became clear that there is again a systematic pattern, distinguishing between mesons and baryons. It was proposed that this can again be understood in terms of parton recombinationls. Starting from the expansion in Eq. (2) one finds that the elliptic flow of a hadron is connected to the elliptic flow of the partons before hadronization by a simple scaling law13918

where n is the number of valence quarks. One can rephrase this t o make it look like one of the classical quark counting rules

As Fig. 4 shows, the experimental data follows this quark counting rule with impressive p r e c i s i ~ n Deviations ~ ~ ~ ~ ~ . for the pion are understood since most of them, even at intermediate PT, are coming from resonance decays15. Besides being a strong direct indication of quark degrees of freedom, the scaling law permits, for the first time, direct access to a n observable in the QGP phase, namely the elliptic flow of quarks.

6. Summary

I have discussed evidence from measurements of hadrons, that matter created in central Au+Au collisions a t RHIC exhibits partonic degrees of freedom which show signs of thermalization and collectivity. This is an indication for a deconfinement phase transition a t temperatures around 170-180 MeV. Further signatures including jet quenching, the absence of back-to-back jet correlations, saturation of the hydrodynamic limit, d+Au control experiments and others confirm this picture. However, they could only be touched shortly or had to be completely omitted in these notes. An overview of recent developments can be found in the proceedings of the last Quark Matter conference21.

281

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Figure 4. The scaled elliptic flow coefficient v z / n as a function of scaled transverse momentum P T / n for different hadron species where n is the number of valence quarks. The v2 data have been measured by the PHENIX and STAR collaborations at RHIC. The data points for all hadron species lie on top of each other after scaling. This is interpreted as the universal curve for the elliptic flow of u , d and s quarks (and their antiquarks) before hadronization. Pions are pushed to lower PT because they come mainly from decays of p mesons.

Acknowledgments I want to t h a n k t h e organizers for a n inspiring workshop. T h e author is supported by DOE grant DE-FG02-87ER40328. References 1. Official web site http://www. bnl.gov/rhic. 2. F. Karsch, Nucl. Phys. A698, 199 (2002); Lect. Notes Phys. 583,209 (2002); C.R. Allton et al., Phys. Rev. D66, 074507 (2002). 3. W. Broniowski and W. Florkowski, Phys. Rev. Lett. 87, 272302 (2001); W.Broniowski, A. Baran and W. Florkowski, Acta Phys. Polon. B33, 4235 (2002). 4. P. Braun-Munzinger, D. Magestro, K. Redlich and J. Stachei, Phys. Lett. B518, 41 (2001). 5. N. Xu and M. Kaneta, Nucl. Phys. A698, 306 (2002). 6. P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanen and S. A. Voloshin, Phys. Lett. B503, 58 (2001); P.F. Kolb, U. W. Heinz, P. Huovinen, K. J. Eskola and K. Tuominen, Nucl. Phys. A696, 197 (2001); D. Teaney, J. Lauret and E. V. Shuryak, preprint nucl-th/0110037; T. Hirano, J. Phys. G30,(2004). 7. S. Voloshin and Y. Zhang, Z. Phys. C70,665 (1996).

282 8. J. D. Bjorken, FERMILAB-PUB-82-059-THY (1982); M. H. Thoma and M. Gyulassy, Nucl. Phys. B 351,491 (1991); X. N. Wang and M. Gyulassy, Phys. Rev. Lett. 68,1480 (1992); M. Gyulassy and X. Wang, Nucl. Phys. B 420,583 (1994); R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. B 483,291 (1997); B.G. Zakharov, J E T P Lett. 65,615 (1997) M. Gyulassy, P. Levai and I. Vitev, Phys. Rev. Lett. 85,5535 (2000); U.A. Wiedemann, Nucl. Phys. B 588,303 (2000); R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, JHEP 0109,033 (2001). 9. K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88, 022301 (2002); C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 90,082302 (2003). 10. X. N. . Wang, Phys. Lett. B579,299 (2004). 11. S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 172301 (2003). 12. H. Long [STAR Collaboration], J. Phys. G30,S193 (2004). 13. R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys. Rev. Lett. 90,202303 (2003); Phys. Rev. C68,044902 (2003). 14. R. J. Fries, J . Phys. G30,S853 (2004). 15. V. Greco, C. M. KO and P. Levai, Phys. Rev. Lett. 90,202302 (2003); Phys. Rev. C68,034904 (2003). 16. R. C. Hwa and C. B. Yang, Phys. Rev. C67,034902 (2003); preprint hepph/0312271; preprint nucl-th/0401001. 17. J. D. Bjorken and G. R. Farrar, Phys. Rev. D9, 1449 (1974). K. P. Das and R. C. Hwa, Phys. Lett. B68,459 (1977); Erratum-ibid. B73,504 (1978); E.Braaten, Y. Jia and T. Mehen, Phys. Rev. Lett. 89,122002 (2002). 18. S. A. Voloshin, Nucl. Phys. A715, 379 (2003); Z. W. Lin and C. M. KO, Phys. Rev. (265,034904 (2002); D.Molnar and S. A. Voloshin, Phys. Rev. Lett. 91,092301 (2003). 19. S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 182301 (2003). 20. J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92,052302 (2004). 21. Proceedings of Quark Matter 2004, Oakland, J. Phys. G30.

NUCLEAR PHYSICS AT SMALL X*

D. E. KHARZEEV Physics Department Brookhaven National Laboratory, Upton, NY 11777, USA E-mail: [email protected]

Some of the recent work on the theory of nuclear processes at high energies and small Bjorken 2 is reviewed, with an emphasis on the physics of nucleus-nucleus and hadron-nucleus collisions. In particular, we discuss the influence of parton saturation in the Color Glass Condensate on the nuclear dependence of hard processes, approach to thermalization and the recent insights on the properties of the Quark-Gluon Plasma.

1. Introduction

Recent years have seen an increase of interest in high energy nuclear physics. To large extent, this is due to an intense stream of new data from Relativistic Heavy Ion Collider and elsewhere; moreover, for a theorist nuclear physics a t small Bjorken z is appealing because it is placed at the intersection of three different, and equally interesting, directions in contemporary theoretical research: i) high parton density QCD; ii) non-equilibrium field theory; and iii) phase transitions in strongly interacting matter. Indeed, understanding the evolution of a heavy ion collision requires a working theory of initial conditions, of the subsequent evolution of the produced partonic system, and of the phase transition(s) to the deconfined phase. This talk is an attempt to capture some of the recent changes and developments in the theoretical picture of these phenomena.

*Work supported by the US Department of Energy under under Contract No. DE-ACOZ98CH10886.

283

284

2. Initial conditions and global observables

2.1. The r81e of coherence

Not so long ago, before the advent of RHIC, it was widely believed that at collider energies the total multiplicities will become dominated by hard incoherent processes. The very first data from RHIC (see and references

1" 0

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200

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Figure 1. Centrality dependence of the charged particle multiplicity near mid-rapidity Au collisions at fi = 20 and 200 GeV; fromlO. in Au

+

therein in this volume) provided a lot of food for new thought: the measured charged hadron multiplicities in Au - Au collisions appeared much smaller than expected on the basis of incoherent superposition of hard processes. Given that any inelastic rescatterings in the final state can only increase the multiplicity", we have an experimental proof of a high degree of coherence in multi-particle production in nuclear collisions at RHIC energies. 2.2. Semi-classical QCD and hadron multiplicities

Combining the idea of coherence with the parton model, we have to consider the initial parton wave functions of the colliding nuclei as coherent superpositions of the wave functions of the constituent nucleons. Since at small Bjorken x all of the partons in the nucleus a t a fixed transverse coordinate participate in a hard scattering process, this treatment naturally leads to the notion of parton density in the transverse plane Q: - a new dimensionful scale of the problem. Once this scale becomes comparable to the resolution scale determined by the kinematics of the hard scattering, the amplitude of the process is severely affected by the coherence. The aFor statistical systems, this is due to the second law of thermodynamics

285 limit on the parton density is reached when the occupation numbers of the gluon field modes with transverse momenta p~ < Q, reach the value nk l/cys(Qs), characteristic for classical gauge fields - this is the phenomenon known as ”parton saturation”2, leading to a coherent state of gluons - Color Glass Condensate (for reviews, see Since the integrated multiplicities are dominated by momenta p~ 5 Q, and parton density in the transverse plane scales as Q: Npart 113 (where N

314,5t6,7).

N

NpaTtis the number of nucleons which participate in the process), Color Glass Condensate leads to a simple predictions for the centrality dependence of hadron multiplicity in heavy ion collisions:

Combined with the dependence of the gluon structure function on Bjorken x known from HERA, which implies Q:(x) l / d ,one can generalize this formula to predict the energy, centrality, rapidity, and atomic number dependencies of hadron multiplicities8. Additional information on the dynamics of the collision can be inferred from the numerical lattice simulationsg. So far this approach has been quite successful in predicting the multiplicities measured a t RHIC; a recent important example is given at Fig.1 which shows the evolution of centrality dependence with energy in the entire RHIC range between fi = 20 and 200 GeV. One can see that the shape of the centrality dependence changes very little over a large energy range, in which the perturbative minijet cross section grows by over an order of magnitude. The prediction of the saturation model is seen to agree with the data reasonably well; this indicates the possibility that parton saturation sets in in heavy ion collisions already at moderate energies. We do not expect the method to apply below fi = 20 GeV however, since at lower energies the coherence length becomes shorter than the nuclear radius. N

2.3. High p~ hadron suppression at forward rapidities, and quantum evolution i n the Color Glass Condensate

Parton saturation at transverse momenta kT 5 &, at sufficiently small x appears to have non-trivial consequences also for the nuclear dependence of the semi-hard processes. At very small 5, when a , In 1/x 1, a semiclassical description has to be modified due to the quantum evolution. Small x evolution introduces anomalous dimension y 21 1/2 in the gluon densities, so that the dependence on the momentum scale Q is modified, to Q 2 -+ Q2’. Since in the vicinity of the saturation boundary the only dimensionful scale N

286

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Figure 2. Nuclear modification factor in dAu collisions as a function of transverse momentum for different rapidities; the data from BRAHMS Collaboration”, theoretical calculations from16.

characterizing the system is the saturation momentum Q:, the cross section of semi-hard scattering should scale as a function of Q:/Q2 - it was found is consistent with HERA data that this ”geometrical scaling” on deep-inelastic scattering. Combining these two observations with the A dependence of the saturation momentum Q: All3 we come to the c o n c l ~ s i o nthat ~ ~ ~at~sufficiently ~ small z and moderate kT the nuclear dependence of hard processes in A A collisions should change from S A Q ~ (where SA N,”$ is the overlap area) to SAQ;? Npart. In p A (or dA) collisions the nuclear dependence is then SAQ?? A5I6,so there has to be a suppression as well. This suppression has also been found17 in the numerical solution of the Balitsky-Kovchegov equation, as well as in18; for recent work, see also The experimental test of these ideas has been performed shortly af11112913314

-

-

--

191’20721.

-

287 terwards - it has been established (for a review, see 43 in this volume) that at mid-rapidity y = 0 there is no high kT suppression in dAu data; this means that the suppression observed in AuAu collisions has to come from the final-state effects, which will be discussed below. The data thus rule out the possibility1' that 2 is small enough for quantum evolution to develop already at mid-rapidity at RHIC. Nevertheless, the presented arguments should apply at sufficiently small z. This is why the data on high kT hadron production at forward rapidities giving access to much smaller values of IC were eagerly awaited. The results from the BRAHMS experiment22 demonstrated a strong suppression of high kT hadrons; moreover, the centrality dependence appeared consistent with the predicted &A N~::{~scaling, in a dramatic contrast to the increasing Cronin enhancement observed at mid-rapidity. Complementary results have been reported in23127124.STAR Colaboration has also reported2' on an observation of a predicted26 nuclear-dependent weakening of the back-to-back correlations for hadrons separated by several units of rapidity. It will be interesting to check if the suppression extends to charm hadrons at forward rapidities28; at mid-rapidity, the first results have been reported inz9. Alternative explanations based on "conventional" shadowing and multiple scattering (for a review, see30) are also being explored.

-

3. Approach to thermalization, and the r6le of classical fields

There is by now an ample evidence of the importance of final state interactions in heavy ion collisions. Among the bulk observables, the azimuthal anisotropy of hadron production is a most spectacular evidence of this - indeed, if all of the elementary nucleon-nucleon collisions were independent, the produced hadrons would not be correlated with the nucleus-nucleus reaction plane. The observed azimuthal anisotropy (or the "elliptic flow", for a review) indicates the existence in the parlance of the field; see of a correlation between the geometry of the nucleus-nucleus collision and the momenta of the emitted hadrons. An economical way of describing the evolution of a large number of particles in space and momentum is provided by relativistic hydrodynamics, which transforms the gradients of the initial parton density into the momentum flow of the produced hadrons. Hydrodynamical description is valid when the mean free path of partons is much smaller than the size of the system, i.e. when the system is sufficiently thermalized. The free expansion (or "inflation") of the produced system 31932

288

with time reduces the density gradients, so the magnitude of the elliptic flow crucially depends on the thermalization time when a hydrodynamical calculation is initiated. It appears that to describe the elliptic flow of the

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Figure 3. Elliptic flow near mid-rapidity in Au + Au collisions as a function of centrality (left) and transverse momentum (right); from Ref.31.

observed magnitude 31, one has to assume that the thermalization time is Such a short very short, about Ttherm N_ 0.5 fm (for a review, see thermalization time presents a problem both for the traditional perturbative and non-perturbative treatments. Indeed, in perturbative QCD the rescattering amplitudes are suppressed by powers of the coupling a,, so the thermalization time appears long, on the order of 10 fm, which makes the description of the elliptic flow problematic 35. In non-perturbative approaches, the interactions can be assumed strong, but the typical time scale of an interaction is l / h ~ c 1~fm, so it is difficult to expect that several interactions needed for thermalization will occur during qherm 21 0.5 fm. The coherent classical fields present in the Color Glass Condensate (CGC) scenario may eventually provide a solution to this puzzle, since in this case the multi-gluon scattering amplitudes A ( n + rn) from n l / a , to m l / a , gluons are not suppressed. An approach to thermalization in this scenario was explored in Ref. 36; recently, an attention was brought also to the role of instabilities in the equilibration process 3 7 . The use of CGC initial conditions of Ref. * in a hydrodynamical approach 38 has led to a successful description of the RHIC data. Nevertheless, much work will have to be done to understand the thermalization process. 33134).

N

-

-

-

4. Hydrodynamical evolution: more fluid than water

Since hydrodynamical description relies on the direct use of the equation of state, the data can be used to extract an information on the properties

289

of the medium. It appears that the quark-gluon plasma equation of state as measured on the lattice (for a review, see 39) is successful in describing the data. However the data can tell even more about the properties of the medium, if one considers the influence of viscous corrections on various observables 40. Viscosity of the medium appears to affect the observables in a very significant way; in fact, one can deduce an upper limit 40,34 on the ratio of shear viscosity r] to the entropy density s, r]/s 5 0.1 - much smaller than the same ratio for the water! Such a small value of viscosity, which reflects the dissipation of energy in a hydrodynamical evolution, contradicts the picture of weakly coupled quark-gluon plasma, and is more indicative of a strongly coupled quark-gluon liquid. A calculation of shear viscosity in the strong coupling regime of QCD is still beyond the reach; however it has been made in N = 4 supersymmetric Yang-Mills theory 41 - the result is a small ratio of r]/s = 1/4n, comparable to the one inferred from RHIC data. A small value of viscosity in the strongly coupled quark-gluon plasma a posteriori justifies the use of the approach to hadron multiplicities assuming the proportionality of the number of measured hadrons to the number of the initially produced partons. This assumption would be unnatural if the evolution of the plasma were accompanied by parton multiplication, but is justified if the viscosity is small and evolution of the system is close to isenthropic. 5. High p~ hadron suppression, jet quenching, and heavy quarks The suppression of high p~ hadrons in Au - Au collisions is certainly one of the most spectacular new results at RHIC (for a comprehensive review of the data, see 43 in this volume). Such an effect has not been seen at lower energiesb; moreover, the results from the dAu run at RHIC indicate that at pseudo-rapidity r] = 0 the observed suppression is entireIy due to the final state effects, very likely a jet quenching in the quark-gluon plasma (for an overview, see 4 4 ~ 4 5 ) . Alternative scenarios, e.g. the absorption in a dense hadron gas, seem unlikely in view of the high energy density 6 20 GeV/fm3 (see e.g. 8, achieved in the collisions. Nevertheless, additional experimental checks have to be performed; an important additional test of the jet quenching scenario involves the measurement of the suppression

-

bA moderate amount of suppression in the SPS results however cannot be excluded due to uncertainties in the reference pp data 42

290 for heavy hadrons containing c or b quarks. If the suppression of high pt particles is indeed due to the induced radiation of gluons by fast partons, heavy quarks should lose significantly less energy than the light ones due to the "dead cone" effect 46. This prediction seems to be in accord with the first RHIC data, which within the error bars indicate no quenching effect on the spectra of open charm, as inferred from the decay electrons 47; however more precise data are desirable. Several other calculations of the energy loss of heavy partons have been performed (see Refs.48~49~50 and papers in this volume); while they differ in the formalisms used, they all find a reduced energy loss for the heavy quarks. On the other B , ...) have a typical large size determined hand, since heavy mesons (D, by the presence of the light quark in their wave functions, in the hadronic absorption mechanism one would expect that heavy mesons interact with about the same probability as the light ones. 51352

Figure 4. Nuclear modification factor for charged hadrons and neutral pions (left and right panels, respectively) in Au Au collisions a t b , = 200 GeV; from Ref.43

+

6. Summary The first years of the experiments at RHIC have changed in a dramatic way the theoretical picture of dense and hot parton systems. The evidence for the existence of new states of QCD matter is mounting53, and a consistent description of the observed phenomena has started to emerge. Nevertheless, hot and dense QCD is still in its infancy - and we have every reason to expect new surprises ! I am grateful to the Organizers for a most stimulating and inspiring meeting.

291

References 1. 2. 3. 4. 5. 6. 7. 8.

9.

10. 11. 12. 13. 14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

D. J . Hofman et al., arXiv:nucl-ex/0406026. L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rept. 100,1 (1983). L. McLerran, Acta Phys. Polon. B 34,5783 (2003). E. Iancu and R. Venugopalan, arXiv:hep-ph/0303204. A. H. Mueller, arXiv:hep-ph/0111244. E. Levin, arXiv:hep-ph/0105205. D. Kharzeev, arXiv:hep-ph/0204014. D. Kharzeev and M. Nardi, Phys. Lett. B 507 (2001) 121 D. Kharzeev and E. Levin, Phys. Lett. B 523,79 (2001) D.Kharzeev, E. Levin and M. Nardi, arXiv:hep-ph/Oll1315; Nucl. Phys. A 730,448 (2004) A. Krasnitz and R. Venugopalan, Phys. Rev. Lett. 84,4309 (2000); A. Krasnitz, Y. Nara and R. Venugopalan, Phys. Rev. Lett. 87,192302 (2001); Phys. Lett. B 554, 21 (2003); T. Lappi, Phys. Rev. C 67,054903 (2003). B. B. Back et al. [PHOBOS Collaboration], arXiv:nucl-ex/0405027. A. M. Stasto, K. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett. 86,596 (2001). E. Iancu, K. Itakura and L. McLerran, Nucl. Phys. A 708,327 (2002). A. H. Mueller and D. N. Triantafyllopoulos, Nucl. Phys. B 640, 331 (2002). N. Armesto, C. A. Salgado and U. A. Wiedemann, arXiv:hep-ph/0407018. D. Kharzeev, E. Levin and L. McLerran, Phys. Lett. B 561,93 (2003). D. Kharzeev, Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 68, 094013 (2003); arXiv:hep-ph/0405045. J. L. Albacete, N. Armesto, A. Kovner, C. A. Salgado and U. A. Wiedemann, Phys. Rev. Lett. 92,082001 (2004). R. Baier, A. Kovner and U. A. Wiedemann, Phys. Rev. D 68,054009 (2003). J. P. Blaizot, F. Gelis and R. Venugopalan, arXiv:hep-ph/0402256. 3. Jalilian-Marian, arXiv:nucl-th/0402080. E. Iancu, K. Itakura and D. N. Triantafyllopoulos, arXiv:hep-ph/0403103. I. Arsene et al. [BRAHMS Collaboration], arXiv:nucl-ex/0403005. M. X. Liu et a1 [PHENIX Collaboration], arXiv:nucl-ex/0405034. L. S. Barnby [STAR Collaboration], arXiv:nucl-ex/0404027. A. Ogawa [STAR Collaboration], Talk at the XI1 Int. Workshop on Deep Inelastic Scattering, Strbske Pleso, Slovakia, 2004. D. Kharzeev, E. Levin and L. McLerran, arXiv:hep-ph/0403271. B. B. Back et al. [PHOBOS Collaboration], arXiv:nucl-ex/0406017. D. Kharzeev and K. Tuchin, Nucl. Phys. A 735,248 (2004). A. Tai [STAR Collaboration], arXiv:nucl-ex/0404029. M. Gyulassy, I. Vitev, X. N. Wang and B. W. Zhang, arXiv:nucl-th/0302077. R. Snellings, arXiv:nucl-ex/0407010. S. A. Voloshin, Nucl. Phys. A 715,379 (2003). U. W. Heinz, arXiv:nucl-th/0407067. E. V. Shuryak, arXiv:hep-ph/0405066. D. Molnar and M. Gyulassy, Nucl. Phys. A 697,495 (2002). R. Baier, A. H. Mueller, D. Schiff and D. T. Son, Phys. Lett. B 502, 51

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46. 47. 48. 49. 50. 51. 52. 53.

(2001). P. Arnold, J. Lenaghan and G. D. Moore, JHEP 0308,002 (2003). T. Hirano and Y. Nara, arXiv:nucl-th/0403029. F. Karsch and E. Laermann, arXiv:hep-lat/0305025. D. Teaney, Phys. Rev. C 68,034913 (2003). G. Policastro, D. T. Son and A. 0. Starinets, Phys. Rev. Lett. 87, 081601 (2001). D. d’Enterria, arXiv:nucl-ex/0403055. D. d’Enterria, arXiv:nucl-ex/0406012. M. Gyulassy, arXiv:nucl-th/0403032. P. Jacobs and X. N. Wang, arXiv:hep-ph/0405125. Y. L. Dokshitzer and D. E. Kharzeev, Phys. Lett. B 519, 199 (2001). K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88,192303 (2002). M. Djordjevic and M. Gyulassy, Phys. Lett. B 560,37 (2003). N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D 69,114003 (2004). R. Thomas, B. Kampfer and G. Soff, arXiv:hep-ph/0405189. N. Armesto, C. A. Salgado and U. A. Wiedemann, arXiv:hep-ph/0405184. A. Dainese [ALICE Collaboration], arXiv:nucl-ex/0405008. M. Gyulassy and L. McLerran, arXiv:nucl-th/0405013.

SATURATION PHYSICS MEETS RHIC DATA

YURI V. KOVCHEGOV Department of Physics, University of Washington Seattle, W A 98195-1 560, USA E-mail: [email protected]. edu We review the derivation of the expression for the cross section of single inclusive gluon production in proton-nucleus collisions ( P A )and in deep inelastic scattering (DIS) at very high energy, when the parton saturation/Color Glass Condensate effects become important in the target’s wave function, both at the classical and quantum levels. We study the resulting particle spectra concentrating on the nuclear modification factor R P A for gluon production. We show that a t moderately high energy/rapidity the nuclear modification factor R P A exhibits Cronin enhancement at transverse momenta p~ of the order of the saturation scale Q s . As the energy/rapidity increases, RPA decreases. At sufficiently high energy/rapidity R P A becomes less than 1 for all values of p~ indicating the onset of suppression of gluon production due to small-s evolution effects. Our prediction of suppression is confirmed by the recently reported BRAHMS collaboration data on particle production at forward rapidity in dAu collisions a t RHIC.

1. Introduction

The results presented here are based on the work done in collaboration with Dmitri Kharzeev and Kirill Tuchin in Refs. 1, 2, 3. Recently there has been a surge of interest in particle production in proton-nucleus (PA)and deuteron-nucleus (dA) collisions at high energies. The interest was inspired by the new data produced by the d A program at RHIC, which should enable one to separate the contributions of the initial state effects, such as parton saturation, from the final state effects, such as jet quenching and energy loss in the quark-gluon plasma (QGP), to the suppression of high transverse momentum particles observed in Au - Au collisions at RHIC. However, particle production in p ( d )A collisions is very interesting by itself. The transverse and longitudinal momentum spectra of the produced particles are related to the small-a: wave function of the nucleus. One can, therefore, gain important insights in the dynamics of small-x partons in the

293

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nuclear wave function by studying particle production in p ( d ) A collisions. The hadronic and nuclear wave functions at small-a: are characterized by extremely high parton density. The density of gluons in it can become very large, corresponding to very strong gluonic fields, A, l/g, which is the strongest possible gluon field in QCD. At the same time the strong coupling constant in these systems is small, because of the presence of the saturation scale Q,, which is a large momentum scale arising due to a high density of color charges (partons) in the transverse plane. Therefore, the partonic systems in the hadronic and nuclear small-a: wave functions have very high density with weak interactions at the partonic level. This means that high parton density leads to complicated nonlinear effects, which can nevertheless be studied analytically due to the weakness of individual partonic interactions characterized by the small parameter a,(&,) << 1. This high parton density system is called Color Glass Condensate. The situation is similar to other systems where strong interactions can be studied under extreme conditions, which, in turn, make the strong coupling small, such as QCD at high temperature and/or baryon density. Below we will first describe gluon production in p A and in DIS in the saturation/Color Glass Condensate framework. As we have already mentioned, the gluons in the small-a: region of hadronic and nuclear wave functions are described by a large momentum scale Q , making the strong coupling constant small a s ( Q s )<< 1. Therefore, the leading gluonic field in a proton-nucleus collision is classical and quantum effects come in as corrections to it. Below we will first describe gluon production in the quasiclassical limit, and then we will show how to include leading logarithmic (asIn 1/a:) quantum corrections in the obtained classical expression. For both the quasi-classical limit and for the case of included quantum evolution we will study the nuclear modification factor RPA given by the derived gluon production cross sections. We will present our prediction for the dependence of RPA on energy and transverse momentum. The prediction has been recently confirmed by the new data reported by the BRAHMS c ~l l a bor atio n .~

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2. Gluon Production in the Quasi-Classical Limit in pA and DIS 2.1. Gluon PFooduction Cross Section

The problem of gluon production in proton-nucleus collisions in the quasiclassical approximation (McLerran-Venugopalan model5) was solved in

295 Ref. 6. For a quark-nucleus scattering the production cross section reads6

- e-Y

2

2

Q.0

ln(l/YTA)/4

+ e-(Z-y)2&?o

ln(1/(E-2)TA)/4

(1)

which then has to be convoluted with the light cone wave function of a quark in a proton to obtain the gluon production cross section for protonnucleus scattering. The quasi-classical saturation scale in Eq. (1) is given by

QZo

Q:,(b) = 47r

P T(b),

(2)

with p the atomic number density in the nucleus with atomic number A, T(h)the nuclear profile function at impact parameter b (such that T(b)= for a spherical nucleus of radius R) and A some infrared cutoff. It is possible to rewrite Eq. (1) in a kT-factorized form.7 Repeating the steps outlined in Sect. IV of Ref. 2 we first perform one of the transverse coordinate integrations in Eq. (1) rewriting it as

2 d m

where NG(z,b, 0) is the forward scattering amplitude of a dipole with transverse size at impact parameter b on a nucleus, which is given by

N G ( z? -7b y = 0) = 1- e - t 2

2

(4)

Using the fact that NG(Z= O,b,O) = 0 we write Eq. (3) as

Let us denote the forward scattering amplitude of a gluon dipole of transverse size r on a single nucleon (proton) integrated over the impact parameter b' of the dipole measured with respect to the proton by

s

d2b'nG(?-,$,y

= 0) = 7rasc 2 2 In- 1

(6) rTA' Equation (6) is obtained by expanding Eq. (4) to the leading order and taking A = 1. It corresponds to the two gluon exchange interaction between the dipole and the proton. In the quasi-classical Glauber-Mueller approximation in which Eq. (4) is derived each nucleon exchanges only two gluons with the dipole. Therefore Eq. (6) is a natural reduction of Eq. (4) to a single nucleon case.

296 With the help of Eq. ( 6 ) we rewrite Eq. ( 5 ) as2

Now is the impact parameter of the proton with respect to the center of the nucleus and b is the impact parameter of the gluon with respect to the center of the nucleus. Equation (7) is the expression for gluon production one would write in the LT-factorization approach. To see this explicitly let us rewrite Eq. (7) in terms of the unintegrated gluon distribution functions of the nucleus

and the proton

One easily derives

which is the same formula as obtained in kT-factorization approach '. Equation (10) demonstrates that the gluon production cross section in pA with all the multiple rescatterings included can still be expressed in kT-factorized form.2 2.2. Cronin Eflect

Now let us study the properties of the quasi-classical gluon production cross section (1). Our goal here is to construct the ratio of the number of gluons produced in a pA collision over the number of gluons produced in a pp collision scaled by the number of collisions dcPA

The gluon production cross section in p p scaled up by A is given by

297

which can be obtained for instance by taking the kT/QSo >> 1 limit of A l l 3 . For a cylindrical nucleus the Eq. (1) and using the fact that Q:o impact parameter b integration would just give a factor of SA. As was shown in Ref. 6 in the approximation when the logarithmic dependence of the powers of the exponents in Eq. (1) on the transverse size is neglected, x21n(l/xTA) x g2, the x~ and y~ integrations in Eq. (1) can be done exactly. (Note that Eq. (12) was obtained in the same approximation.) Using the result of the integrations along with Eq. (12) in Eq. (11) we then obtain' N

+ Ei ( g)]} , (13) Q:o

where Ei(z) is the exponential integral. The ratio R P A ( k ~is ) plotted in Fig. 1for A = 0.2 QSo.It clearly exhibits an enhancement at high-kT typical of Cronin effect (see also Ref. 8). Analyzing Eq. (13) and remembering

RPA t

2-

1.5.

1. 0.5. -.

~ the o quasi-classical Figure 1. The ratio RPA for gluons plotted as a function of k ~ / Q in McLerran-Venugopalan model as found in Ref. 6. The cutoff is A = 0.2Qs.

--

that Q20 one can easily see that the height of the Cronin maximum, scaling like In Q s 0 , and its position, scaling like Qs0, are both increasing functions of A, or, equivalently, of the centrality of the collision.' N

298 3. Gluon Production Including Small-z Evolution

3.1. Gluon Production Cross Section

As the energy of the collisions increases quantum evolution corrections become important. The evolution equation in rapidity Y for the forward dipole-nucleus scattering amplitude N ( r ,b, Y) closes only in the large-N, limit of QCD and readsg

with the initial condition given by Eq. (4)with the saturation scale Q:o replaced by Q;suark2 = ( C F / N ~02,. ) Equation (14) resums leading powers of aSY and all multiple rescatterings on the target.g Incorporating the effects of evolution (14) in the expression for the single inclusive gluon production cross section is rather intricate and tedious. It has been done for the case of DIS in Ref. 2 and we refer the interested reader to Ref. 2 for more details. The result, however, is very straightforward and can be easily generalized to the case of p A to give'

-duPA -

d21c d y

-

1

cF / d 2 B d 2 b d 2 z V ~ n G ( z l b - - D , Y -y)ePik'" as7r ( 2 ~ b2 ) ~

v: NG (2,b, 9 )

7

(15)

where Y is the total rapidity interval between the proton and the nucleus. In Eq. (15) n ~ ( z , Y - y ) is evolved by the linear part of Eq. (14) (the BFKL equation") with Eq. ( 6 ) as the initial condition and NG = 2N - N 2 with N taken from Eq. (14). The effect of quantum evolution (14) is to leave Eq. (7) essentially unchanged evolving only the amplitudes NG and in it. The IcT-factorization expression (10) is still valid after inclusion of evolution (14) (see also Ref. 11).

b-a,

3.2. High-pT S u p p r e s s i o n in p A Collisions We will analyze RPA given by Eq. (15) in three separate regions of the transverse momentum of the produced gluon. We will first note that, as

299

was shown in Ref. 12, apart from the saturation scale Qs(y) there exists another scale in the problem - the geometric scaling scale, which was It signifies the boundary approximately found to be kgeom x Q;(y)/Q;,,. of the region where geometric scaling is valid: for k ~ s thek gluon ~ distribution is the function of a single variable k ~ / Q $ ( y ) .We will, therefore, consider three cases: (i) double logarithmic region k~ > kgeom,(ii) extended geometric scaling region Q,(y) < k~ s k g e o mand (iii) the saturation region kT

s

Qs(Y).

3.2.1. Double Logarithmic Region In the double logarithmic region k~ >> Qs(y) and the logarithm of the transverse momentum becomes important as well. At very high k~ only the linear part of the evolution equation (14) contributes. The resulting evolution is equivalent to the double logarithmic approximation (DLA) to the BFKL evolution, the solution of which is known exactly. In the end we write for the nuclear modification factor1

A1/3A2, for a large enough nucleus the power of the expoSince QZo nent in Eq. (16) becomes negative and large, giving R P A ( k ~ , y < ) 1 and, thus, leading to suppression of the gluon production in the double logarithmic region. Note that here saturation effects come in only through the logarithmic cutoff Qso, which is still sufficient to cause suppression. N

3.2.2. Extended Geometric Scaling Region In the extended geometric scaling region Qs(y) < kT 5 kgeom12 the effects of gluon anomalous dimension become important. One can show that in this region we obtain the following asymptotic expression for the nuclear modification factor calculated for simplicity at mid-rapidity ('y = Y/2)l

In Eq. (17) we assume that the collision energy is high enough for the gluon distribution function in the proton to also be in the extended geometric scaling region. From Eq. (17) we conclude that in the extended geometric scaling region at asymptotically high energies, RPA saturates to

~

~

300 a parametrically small lower bound, RPA A-l16 << 1, which is independent of energy and is a decreasing function of A , or, equivalently, centrality. This bound was originally predicted in Ref. 13. N

3.2.3. Saturation Region Deep inside the saturation region, for ICT << Q,(y), one always has a suppression of RpA, as can be seen even in the quasi-classical case of Fig. 1. The only remaining question is what happens to Cronin maximum as the evolution is turned on. As we have seen the Cronin peak occurs in Fig. 1 around ICT QSo. Therefore, to study the evolution of the Cronin peak with energy we need to find RPA at ~ C T= Q s ( y ) . Using the scaling property of the distribution function” one can derive’ N

RPA(Qs(y),y)

o(

exp

{

4 g y (1

-

/=)}

2%Y

< 1.

(18)

We observe from Eq. (18) that in the course of quantum evolution the Cronin maximum of the ratio RPA decreases with energy until, at very high energy, it saturates at the lowest value RPA A-l16 << 1. The height of the Cronin peak is also a decreasing function of collision centrality/atomic number A , as can be seen from Eq. (18). (A more careful estimate yields a power -0.124 instead of -1/6.’) Our prediction for the evolution of RPA with energy/rapidity is summarized in Fig. 2: at moderate energy/rapidity R P A is given by the top solid curve and exhibits Cronin enhancement at high-pT. As the energy increases RPA gradually decreases (dash-dotted and dashed lines) finally leveling off at RPA A-l16 (lower solid curve). Similar conclusions have been reached by the authors of Ref. 14. Figure 3 contains the data recently reported by BRAHMS collaboration on charged particle RdAas a function of pseudorapidity 7) and transverse momentum p~ for the dAu collisions studied at RHIC. The data in Fig. 3 confirms our qualitative predictions from Fig. 2! N

-

4. Comparison with Data

To make quantitative predictions for RdA we model the dipole scattering amplitude from Eq. (14) by (for more details see Ref. 3)

301

1.75. 1.5. 1.25. 1.

0.75. 0.50.25. I

1

2

3

4

5

Figure 2. The ratio RPA plotted as a function of / C T / Qfor ~ (i) McLerran-Venugopalan model, which is valid for moderate energies (upper solid line); (ii) our toy model for very high energies/rapidities (lower solid line); (iii) an interpolation to intermediate energies (dash-dotted and dashed lines).

where the anomalous dimension y(y, z$) interpolates between the DLA and LLA region^.^ Using Eq. (19) in Eq. (15) and adding the valence quark production contribution15 we obtain the fit shown in Fig. 33. It demonstrates that the data4 is compatible with the saturation approach described above.

Acknowledgments This work was supported in part by the U S . Department of Energy under Grant No. DE-FG02-97ER41014, preprint number NT@UW-04-016.

References 1. D. Kharzeev, Y . V. Kovchegov and K. Tuchin, Phys. Rev. D 6 8 , 094013 (2003)

[arXiv:hep-ph/0307037]. 2. Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 65, 074026 (2002) [arXiv:hepph/0111362]. 3. D. Kharzeev, Y . V. Kovchegov and K. Tuchin, arXiv:he~-ph/0405045. 4. I. Arsene et a1 [BRAHMS Collaboration], arXiv:nucl-ex/0403005; R. Debbe [BRAHMS Collaboration], arXiv:nucl-ex/0403052. 5. L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994) [arXiv:hepph/9309289]; Phys. Rev. D 49, 3352 (1994) [arXiv:hep-ph/9311205]; Phys. Rev. D 50, 2225 (1994) [arXiv:hep-ph/9402335]. 6. Yu. V. Kovchegov and A. H. Mueller, Nucl. Phys. B 529, 451 (1998) [arXiv:hep-ph/9802440]. 7. L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rept. 100, 1 (1983).

302

1.4

1A

1.2

4

u&

u&

1

1

0.8

0.1

0.6

0.6

:::

0.4 0.2

0

0

1

2

3

4

5

0

6

0

1

2

5

4

Pr f&V)

5

6

P, f-V)

2

'1 = 3.2

ld

1.8 1.4

1.4

1.2

9

3

1

U" 0.8

1.2 1

0.8

0.6

0.8

0.4

0 0.4 2

0.2 0

0

1

2

3

4

5

6

Pr f-Vly)

Figure 3. Nuclear modification factor RdAu of charged particles for different rapidities. Data is from Ref. 4 and the lines are given by our model from Ref. 3.

8. B. Z. Kopeliovich, J. Nemchik, A. Schafer and A. V. Tarasov, Phys. Rev.

Lett. 88, 232303 (2002) [arXiv:hep-ph/0201010]; R. Baier, A. Kovner and U. A. Wiedemann, Phys. Rev. D 68,054009 (2003) [arXiv:hep-ph/0305265]. 9. I. Balitsky, Nucl. Phys. B 463,99 (1996) [arXiv:hep-ph/9509348]; arXiv:hepph/9706411; Phys. Rev. D 60, 014020 (1999) [arXiv:hep-ph/9812311]; Y . V. Kovchegov, Phys. Rev. D 60, 034008 (1999) [arXiv:hep-ph/9901281]. Y . V. Kovchegov, Phys. Rev. D 61,074018 (2000) [arXiv:hep-ph/9905214]. 10. E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys. J E T P 45, 199 (1977) [Zh. Eksp. Teor. Fiz. 72,377 (1977)]; I. I. Balitsky and L. N. Lipatov, Sov. J . Nucl. Phys. 28,822 (1978) [Yad. Fiz. 28, 1597 (1978)]. 11. M. A. Braun, Phys. Lett. B 483, 105 (2000) [arXiv:hep-ph/0003003]. 12. E. Iancu, K. Itakura and L. McLerran, Nucl. Phys. A 708, 327 (2002) [arXiv:hep-ph/0203137]. E. Levin and K. Tuchin, Nucl. Phys. B 573, 833 (2000) [arXiv:hep-ph/9908317]. 13. D. Kharzeev, E. Levin and L. McLerran, Phys. Lett. B 561, 93 (2003) [arXiv:hep-ph/0210332]. 14. J. L. Albacete, N. Armesto, A. Kovner, C. A. Salgado and U. A. Wiedemann, Phys. Rev. Lett. 92,082001 (2004) [arXiv:hep-ph/0307179]. 15. A. Dumitru and J. Jalilian-Marian, Phys. Rev. Lett. 89, 022301 (2002) [arXiv:hep-ph/0204028].

CONFINEMENT AND CHIRAL SYMMETRY

A. MOCSY Institut fur Theoretische Physik, J. W. Goethe-Uniuersitat Postfach 11 19 32, 60054 Frankfurt, Germany First we discuss how hadronic states feel deconfinement. Then analyze the relation between deconfinement and chiral symmetry restoration. Our framework is a generalized Ginzburg-Landau effective field theory based on the underlying symmetries.

1. Introduction The theory of strong interactions, Quantum Chromodynamics (QCD), has been thoroughly investigated in its high temperature and high baryon density, as well as low temperature and low baryon density limit. In these two limits, respectively, perturbative QCD and chiral perturbation theory are successfully applicable. These limits of QCD are connected with a highly non-perturbative region characterized by the phenomena of confinement and chiral symmetry breaking. In order to theoretically investigate this region effective field theories need t o be constructed. These effective, QCDlike theories allow us to study generic properties of transitions between the low and high temperature/density phases. The predictions that result from such theories can be tested via the direct analysis of QCD performed on the lattice, and also against experimental results from the collision of relativistic heavy ions. Symmetries are used to constrain effective Lagrangian theories. It is customary to attach order parameter fields t o exact symmetries, such that their expectation value is zero in the symmetric phase and nonzero when the symmetry is spontaneously broken. Order parameters have been conveniently and successfully used to investigate phase transitions. The choice of these fields however, is a phenomenological matter, which is not always obvious. Even when one can identify an order parameter associated with the underlying symmetry of the theory, often this is not a physical observable, and as such, it is hard to access it experimentally. Furthermore,

303

304 most of the fields present in nature are not order parameter fields. Led by these thoughts, we have developed a new way to study phase transitions, by analyzing the behavior of, what we call non-order parameter fields in the transition region. The main ideas and results of our approach are presented in Section 3. Their application for pure gluonic Yang-Mills theory is discussed in Section 4. Application for theories with quarks and solution to the question raised in Section 2 is presented in Section 5. Extension of the results for density-driven transitions is discussed in Section 6. Concluding thoughts are given in Section 7.

2. Confinement and Chiral Symmetry During the last few decades there has been lots of interest for understanding deconfinement and chiral symmetry restoration as function of temperature, quark chemical potential, number of flavors, and number of colors. Accordingly, there have been lots of models developed, capturing different aspects of deconfinement and chiral symmetry restoration at finite temperature and/or density. Our goal is t o provide a unifying point of view, while remaining consistent with already acquired results from these models. In the meantime, we have developed the grounds for a new way t o study not only deconfinement and chiral symmetry restoration, but also phase transitions in general. In pure Yang-Mills gauge theory a t non-zero temperature the ZN center of S U ( N ) is a global symmetry of the theory. This symmetry is intact a t low temperatures and is broken a t high temperatures. The associated order parameter is the Polyakov loop I , which is the trace of the thermal Wilson line. Under the action of ZN the Polyakov loop transforms as I -i ZI with z E Z N . Accordingly, the order parameter ( I ) = 0 a t low temperatures T < Te , whereas ( I ) # 0 for T > Te . Since the Polyakov loop is related to the free energy of static (infinitely heavy) fundamental quarks, its condensation is associated with deconfinement. Another interesting sector of the pure Yang-Mills gauge theory is the hadronic sector, constrained by trace anomaly. At a certain temperature there is a drop in the gluon condensate, which is customarily associated with deconfinement. But the gluon condensate is not linked to an exact symmetry, so strictly speaking is not an order parameter. Therefore, there is no a priori reason why its drop should happen a t the same temperature as deconfinement does. Furthermore, the glueball theory is not aware of the number of colors, whereas the order of the phase transition in pure

305

gluonic theory is known to be sensitive to the number of colors. Based on we show in Section 4 that there exists a non-trivial information transfer from the ZN symmetric Polyakov loop to the non-order parameter hadronic fields, the glueballs. Accordingly, the drop in the gluon condensate must happen a t the deconfining transition. Moreover, also the order of the transition embedded in the Polyakov loop is transferred to the gluon condensate. When dynamical quarks in the fundamental representation of the gauge group are added into the theory, that is for any finite, including zero, quark mass the Z N symmetry is explicitly broken. However, when quarks are added in the adjoint representation, the obtained QCD-like theory still maintains Z N symmetry. QCD with massless quarks exhibits chiral symmetry. The associated order parameter is the chiral condensate u (-J $$J), which is nonzero for T < To,and vanishes at high temperatures T > T, , where chiral symmetry is restored. It is wellknown, that any finite quark mass breaks chiral symmetry explicitly. For realistic quark masses thus there is no exact symmetry and so there is no true order parameter either. One can still construct the Polyakov loop and the chiral condensate, and study their behavior with changing temperature or chemical potential. This has been done numerically on the lattice. The raise of the Polyakov loop with increasing temperature is generally thought of as signal of deconfinement. Similarly, the gradually vanishing chiral condensate is identified with chiral symmetry restoration. The analysis of the theory with quarks in the fundamental representation of the gauge group performed on the lattice found that the critical temperatures of the deconfinement and of the chiral symmetry restoration coincide, Te = T, . This is evidence for a single QCD phase transition. The same was found also in '. Also from lattice results we learned that when quarks are in the adjoint representation the situation is different. In adjoint QCD chiral symmetry is restored at a much greater temperature than that of deconfinement, T, N 8Te . As further noticed on the lattice, there is a jump in the behavior of the chiral condensate at Te . So even if the two transitions happen separately the chiral condensate knows about deconfinement. For QCD with quarks then a question naturally arises: Why, for matter in the fundamental representation deconfinement and chiral symmetry restoration appear t o be linked with a single phase transition, while for matter in the adjoint representation there are two phase transitions, well separated in temperature? There have been some recent attempts to anly2l3

306 306

swer this question, a1 least partially 7. Based on we show in Sections 5.1 and 5.2 how our effective Lagrangian description can offer a simple and unifying way of addressing this question. 3. The General Theory

In we constructed an effective generalized Ginzburg-Landau theory. The main idea is not to integrate out all heavy fields, as is customary in a traditional Ginzburg-Landau theory. When keeping besides the lightest field, which is the order parameter, also a heavy field, and investigating its behavior in the transition region, allows for studying generic properties of phase transitions using the non-order parameter field. The ingredients of our most general Lagrangian are two scalar fields constrained only by symmetries and renormalizability. The light field, that knows about the symmetry is the order parameter. As such, its correlation length is divergent at the transition point, and its mass is assumed to display lT-Tcl” . the characteristic critical behavior near the transition, rn,.,(T) The heavy field is singlet under the symmetries of the order parameter. We have shown that there exist a transfer of the critical properties from the order parameter to non-critical fields. A detailed approach is presented in 3 , and the initial investigation in l . For a complete review see g. In what follows we just state our results. The transfer of information is possible due to the presence of a trilinear interaction between the light order parameter and the heavy non-order parameter field. Due to this interaction, the expectation value of the order parameter field in the symmetry broken phase induces a variation in the expectation value of the singlet field. The information about the order of the transition is also encoded in a nontrivial way in the profile of the expectation value of the singlet field. Furthermore, also due to this interaction the spatial correlators of the non-critical fields are infrared dominated. This manifests in a finite drop at the transition in the screening mass of the non-critical field. These conclusions are generic, and as such are valid not only for the QCD phase transition that we focus on currently, but also to transitions that appear in other, for example different condensed matter systems. N

4. Pure Glue

First, we consider an effective theory that links two different sectors of pure gluonic Yang-Mills theory at finite temperature the hadronic sector and ‘1’:

307 the Polyakov loop. The hadronic states of pure glue are the glueball fields, and their effective theory at tree level is constrained by the Yang-Mills trace anomaly. The potential of the lightest glueball state H is lo:

A is the confining scale of the theory. This potential correctly saturates the trace anomaly when H 0: Tr [ G & G ~ ” ~ a , ]where Gpy,ais the gluon field strength and a = 1,..., N 2 - 1 the gauge indices. This potential encodes properties of the Yang-Mills vacuum at zero temperature and it has been used to deduce phenomenological results at finite temperature l1 and density 12. In pure glue the ZN symmetry is exact. At non-zero temperature the Yang-Mills pressure can be written in terms of the field l according to the Polyakov Loop Model 13. The free energy has the general form:

&[el = T ~ F [. ~ I

(2)

e . Its coefficients are temperature dependent, allowing for a mean field description of the Yang-Mills phase transitions 14. When linking these two sectors of the gluonic theory, in the proposed effective potential a general interaction term can be constructed without spoiling the zero temperature trace anomaly and respecting ZN symmetry F [ l ]is a ZN invariant polynomial in

l&[H,C] = H P [ l ]2 Hb1l2 for 2 2 . (3) This trilinear interaction term turns out to be crucial. In the low temperature region, where ZN symmetry is exact, ( l )= 0 and the two fields are decoupled. When crossing the critical temperature, symmetry is spontaneously broken and we found that temperature dependence of the gluon condensate is: A4 ( H ) = - exp [ - 2 b 1 ( l ) 2 ] . (4) e The gluon-condensates drops exponentially. This drop is triggered by the rise of the Polyakov loop and it happens at the deconfining critical temperature. Although the drop might be sharp it is continuous in temperature. This is related to the fact that the phase transition is second order. Our findings strongly support the above mentioned common picture according to which the drop of the gluon condensate signals, in absence of quarks, deconfinement.

308 For three colors the theory gets somewhat modified since the 2 3 symmetry allows for cubic terms too, which render the transition first order. In this case we have found that also the information about the order of the transition is transmitted to the profile of the condensate drop. Thus the gluon condensate, formally unaware of the number of colors, indirectly carries information about these. So this simple theory is able to communicate the information about the center group symmetry to the hadronic states. It also provides the link between deconfinement and conformal anomaly. A more involved treatment that includes also the kinetic terms was considered in We showed that there is a finite drop in the screening mass of the hadronic degrees of freedom near the transition, when approaching the transition point both from above and below. Such behavior for the glueballs has been already seen on the lattice for both the two color l5ll6 and the three color l7 theories. 293.

5. Including Massless Quarks As another application, we added massless quarks into the theory and investigated the relation of chiral symmetry restoration and deconfinement in temperature driven transitions. When the massless quarks are in the fundamental representation the ZN symmetry is broken, only chiral symmetry is exact. When quarks are in the adjoint representation, both symmetries are exact, providing an opportunity to study these transitions separately. In the following we discuss the two color and two flavor situation. This is our choice since with only minor modifications of the Lagrangian we can describe both of the above cases. Also, the extension to finite chemical potential becomes straightforward and can be compared to recent lattice results. For this see Section 6.

5.1. Fundamental Representation The global symmetry group for N f massless flavors is S U ( 2 N f ) , with the symmetry breaking pattern S U ( 2 N f ) -+ S p ( 2 N f ) . For N f = N, = 2 the chiral potential is 18: Vch[o,7ra] =

+

+ X l T r ["+MI2 + +Tr 4

g T r ["t"] 2

[ M f M M t M ] (5)

with 2 A4 = u i 2ara X",a = 1,., . , 5 and the generators X" given in equation (A.5) and (A.6) of The Polyakov loop l is now a heavy mass dimension one field, singlet under chiral symmetry. Its contribution to the

'*.

309 potential is

me2 v,"]=goe+-e 2

2

3 Q4 4 +-eQ3 3 + - e4 .

The allowed interaction terms are

Knt",

u,

7.4= (g1e + g#) n [ M +M3 = (g1e + g2P) (2+ na7ra). (7)

The g1 term plays a fundamental role. This term has been dropped in previous investigations 1 4 . In the low temperature phase, where chiral symmetry is spontaneously broken, 0 acquires a non-zero expectation value, which in turn induces a modification also for (e). The extremum of the linearized potential near the transition, with the usual choice of (n)= 0 , is at

(e)

"v

91 e, - -(0)2, m:

eo

N

90

--

m:

'

This equation shows that for g1 > 0 and go < 0 the expectation value of e behaves oppositely to that of u : As the chiral condensate starts to decrease towards chiral symmetry restoration, the expectation value of the Polyakov loop starts to increase, signaling the onset of deconfinement. Note here that the couplings go and g1 can in principle be determined from lattice data 19. When applying the analysis presented in the general behavior of the spatial two-point correlator of the Polyakov loop can be determined. Accordingly, there is a drop in the screening mass of the Polyakov loop, when approaching the transition from either sides. The size and slope of the drop depend on the resummation that is used ', and its future investigation on the lattice could clarify this. 'i3,

5 . 2 . Adjoint Representation

The two color adjoint QCD with N f massless Dirac quark flavors has global S U ( 2 N j ) symmetry, which breaks via a bilinear quark condensate to O ( 2 N j ) . The chiral part of the potential for N f = N, = 2 is given by (5) with 2 M = u i 2fi.rra X", a = 1,.. . ,9 and the generators X" given by equation (A.3) and (A.5) of I*. The potential for the Polyakov loop is

+

yo[!]

>e2 + -eQ44 2

= m2

4

,

(9)

31 0 and the only interaction term allowed by symmetries is vnt[l, a,..] = g 2 P

Tr [M'tM ] = g&a2

+ .%a) .

(10)

Due to the 2 2 symmetry the relevant interaction term g11a2 is now forbidden. So one might expect no communication at this level between the fields. As we discuss below, this turns out not to be the case '. The effective Lagrangian does not know which transition happens first but this is irrelevant for the validity of our general results. Consider the physical case in which the deconfinement phase transition happens first 6. For Te < T < T, both symmetries are broken, and the expectation values of the two order parameter fields are linked to each other: 1 ( a ) 2= (m2 2 g 2 ( l ) 2 ) ,

-1

(C)2

+

=

Furthermore, on both sides of Te a relevant interaction term g 2 ( a ) a 1 2 emerges. Similarly, on both sides of T, the interaction term g 2 ( l ) ! a 2 is generated. These interactions become the relevant trilinear interactions that govern the interdependence of the two fields. Thus when Te << T,, the two order parameter fields, a priori unrelated, do feel each other near the respective phase transitions. We have also predicted the existence of substructures near these transitions '. Such possible structures must be determined via first principle lattice calculations. 6. Finite Density

Our analysis can be extended for phase transitions driven by a chemical potential. This can be done in a straightforward manner for two color QCD. When quarks are in the pseudoreal representation there is a phase transition from a quark-antiquark condensate to a diquark condensate 2 0 , Based on our predictions, in two color QCD when diquarks form at a chemical potential of p = m,, also the Polyakov loop will know about presence of the phase transition in the same manner as we described for a temperature driven phase transition. Recent lattice simulations 21 support this prediction.

7. Concluding Thoughts Within an effective Lagrangian approach we have shown how deconfinement, i.e. the rise in the Polyakov loop, is a consequence of chiral sym-

31 1 metry restoration in the presence of massless quarks in the fundamental representation. We expect this t o hold for small quark masses. If quark masses were very large then chiral symmetry would be badly broken, and could not be used to characterize the phase transition. But then the 2, symmetry becomes more exact, and its breaking would drive the (approximate) restoration of chiral symmetry. In the non-perturbative regime the amount of ZN breaking is unknown, so we cannot a priori establish which symmetry is more broken for a given quark mass. Which of the underlying symmetries leads the transition could be determined directly from the critical behavior of the spatial correlators of hadrons and/or of the Polyakov loop 2 a 3 , and by extending our analysis to the systematic study of effects that have not been regarded here. Currently ongoing investigations on the effect of quark masses and flavors will allow to extract the crucial couplings from lattice data 19. This in turn allows us t o make some more quantitative predictions. Such analysis can be relevant not only to further our understanding of the nature of the different symmetry breakings and the QCD phase transition, but also possibly phenomenologically, when analyzing heavy ion collisions.

Acknowledgments My thanks go kindly to Francesco Sannino and Kimmo Tuominen for collaboration on this work and careful reading of this manuscript. I thank the Organizers of the "Advances in QCD" meeting for the kind hospitality. Support from the Alexander von Humboldt Foundation is gratefully acknowledged.

References 1. F. Sannino, Phys. Rev. D 66, 034013 (2002) [arXiv:hep-ph/0204174]. 2. A. Mocsy, F. Sannino and K. Tuominen, Phys. Rev. Lett. 91, 092004 (2003)

[arXiv:hep-ph/0301229]. 3. A . Mocsy, F. Sannino and K. Tuominen, JHEP 0403,044 (2004) [arXiv:hepph/0306069]. 4. F. Karsch, Lect. Notes Phys. 583,209 (2002) [arXiv:hep-lat/0106019]. 5 . S. Digal, E. Laermann and H. Satz, Eur. Phys. J. C 18,583 (2001) [arXiv:hepph/0007175]. 6. F. Karsch and M. Lutgemeier, Nucl. Phys. B 550, 449 (1999) [arXiv:heplat /98 120231. 7. Y. Hatta and K. Fukushima, Phys. Rev. D 69, 097502 (2004) [arXiv:hepph/0307068].

312

8. A. MOCSY, F. Sannino and K. Tuominen, Phys. Rev. Lett. 92, 182302 (2004) [arXiv:hep-ph/0308135]. 9. A. Mocsy, F. Sannino and K. Tuominen, arXiv:hep-ph/0401149. 10. J. Schechter, Phys. Rev. D 21,3393 (1980); A.A. Migdal and M. A. Shifman, Phys. Lett. B 114,445 (1982). 11. F. Sannino and J. Schechter, Phys. Rev. D 57, 170 (1998) [arXiv:hepth/9708113]; Phys. Rev. D 60, 056004 (1999) [arXiv:hep-ph/9903359]; G. W. Carter, 0. Scavenius, 1. N. Mishustin and P. J. Ellis, Phys. Rev. C 61,045206 (2000) [arXiv:nucl-th/9812014]. 12. G. E. Brown and M. Rho, Phys. Rept. 363, 85 (2002) [arXiv:hepph/0103102]. 13. R. D. Pisarski, hepph/0112037; R.D. Pisarski, Phys. Rev. D62, 111501 (2000).

14. A. Dumitru and R. D. Pisarski, Phys. Lett. B 504,282 (2001) [arXiv:hepph/0010083]. 15. P. Bacilieri et al., Phys. Lett. B 220, 607 (1989). 16. G. Boyd, S. Gupta, F. Karsch and E. Laermann, Z. Phys. C 64,331 (1994) [arXiv:hep-lat/9405006]. 17. S. Datta and S. Gupta, arXiv:hep-ph/9809382. 18. T. Appelquist, P. S. Rodrigues d a Silva and F. Sannino, Phys. Rev. D 60, 116007 (1999). 19. E. S. Fraga and A. Mbcsy, in preparation. 20. For a review on 2 color QCD see S. Hands, Nucl. Phys. Proc. Suppl. 106, 142 (2002). 21. B. Alles, M. D’Elia, M. P. Lombard0 and M. Pepe, arXiv:hep-lat/0210039.

GAPLESS SUPERCONDUCTIVITY IN DENSE QCD*

IGOR A. SHOVKOVY~ Institut fur Theoretische Physik, J. W. Goethe- Universitat, 0-60054 Frankurt a m Main, Germany E-mail: [email protected]

The construction and properties of the recently proposed gapless color superconducting phases of dense quark matter are briefly reviewed. Such phases of matter may naturally exist in cores of compact stars.

1. Introduction

Sufficiently cold and dense quark matter is a color superconductor.' The corresponding ground state is characterized by a condensate of Cooper pairs made of quarks. Because of color charges carried by quarks, the SU(3), color gauge group of QCD breaks down through the Anderson-Higgs mechanism. At asymptotic densities, this phenomenon was studied from first principle^.^>^>^ The studies suggest that dense QCD has a very rich phase ~tructure.~ It is natural to expect that some color superconducting phases may exist in the interior of compact star^.^>^ The estimated central densities of such stars might be as large as lop0 (where po M 0.15 fm-3 is the saturation density), while their temperatures are in the range of tens of keV. This could provide ideal conditions for the diquark Cooper pairing that leads to color superconductivity. I start my discussion here by pointing that matter in the bulk of a compact star should be neutral with respect to electrical as well as color charges. Also, such matter should remain in (chemical) /%equilibrium. Satisfying these requirements imposes nontrivial relations between the chemical 'This presentation is based on work done in collaboration with Mei Huang. +On leave of absence from Bogolyubov Institute for Theoretical Physics, 03143, Kiev, Ukraine. The work is supported by Gesellschaft fur Schwerionenforschung (GSI) and by Bundesministerium fur Bildung und Forschung (BMBF).

31 3

314

potentials of different quarks.' In turn, such relations influence the pairing dynamics between quarks, for instance, by suppressing conventional twoflavor color superconducting (2SC) phase and favoring the so-called gapless 2SC (g2SC) phase.g The description of such pairing dynamics is the main topic of this presentation. 2. Model

At present, the knowledge of dense QCD at baryon densities that are relevant for central regions of compact stars is not available. Because of the famous sign problem, lattice calculations do not help either. Therefore, one has to rely on models. In the case of dense quark matter, one may use a model that contains only the most important features of dense QCD. For the purposes of this presentation, it suffices to use a simple SU(2) NambuJona-Lasinio model,1° defined by the following lagrangian density:

+

L = q(iy"a, - m o ) ~ Gs -k G D

C

[(k' E E

b

[ ( 4 d 2+ ( @ i ~ 5 ? q ) ~ ]

'Y5Q)(@&cb'Y5qC)]

7

(1)

where qc = CQT is the charge-conjugate spinor and C = iy2yo is the charge conjugation matrix. The quark field is a 4-component Dirac spinor that carries flavor (i = 1 , 2 ) and color (a = 1 , 2 ,3 ) indices. The Pauli matrices in the flavor space are 7' = (T' ,T ~T ,~ ) The . antisymmetric tensors in the flavor and color spaces are ( E ) ~ ~cikand (eb)"fl capb , respectively. In this model, there are two independent coupling constants, Gs and G o , that describe pairing in the quark-antiquark and diquark channels. Without much limitation, only the chiral limit (mo = 0) is considered below. The values of the parameters in the NJL model are as follows:'1 Gs = 5.016 G e V 2 and the cut-off is given by A = 653 MeV. The strength of the diquark coupling GD is taken to be proportional to the quark-antiquark coupling constant, i.e., GD = qGs with q being around 0.75. In ,&equilibrium, the diagonal matrix of quark chemical potentials is given in terms of baryon ( p ~ 3p), electrical ( p e ) and color (,us) chemical potentials,

where Q and T8 are the generators of U(l)em of electromagnetism and the U(1)s subgroup of SU(3), group.

31 5

In the mean field approximation, the free energy of quark matter in /?-equilibrium (with electrons") takes the form:

R=Ro--

(p;

+ 27r2T2p2+ "T4)

+ 4Gs m2

15

12n2

A2

2T In (1

+ e - E a / T ) ],

(3)

where 00is a constant added to make the pressure of the vacuum zero. The sum in the second line of Eq. (3) runs over all (6 quark and 6 antiquark) quasiparticles. The dispersion relations and the degeneracy factors of the quasiparticles read

Here the following shorthand notation was introduced:

= JiG.2, J g ( P ) = J [ E ( P )f PI2 + A2, E(p)

The thermodynamic potential that determines the pressure of quark matter, 02phys = - P , is obtained from R in Eq. (3) after substituting pg, pe, m and A that solve the color and electrical charge neutrality conditions, i.e.,

as well as the gap equations, i.e.,

dR =0, dm

~

and

dR dA = O .

-

~~

assume that there are no neutrinos in the system. This properly describes the situation inside compact stars after deleptonization when the neutrino mean free path becomes larger than a typical star size.

316

3. Charge neutrality condition

Here I discuss the charge neutrality condition that plays a paramount role in producing a stable gapless phase of quark matter.b To start with, let me note that creating a macroscopic chunk of matter with a nonzero electrical charge density is very costly energetically. This is because of a large Coulomb potential energy associated with homogeneously charged matter. Indeed, the corresponding energy d e n s i t y grows with increasing the volume of the system (it is proportional to n;V2/3 where n~ is the charge density and V is the volume of the system). As a result, it is impossible for matter inside stars to remain charged over macroscopic distances. The neutrality in a macroscopic system could be imposed in two different ways: (i) globally (i.e., on average), or (ii) locally. For example, matter is globally neutral inside a mixed phase that is made of alternating layers of different coexisting components, each of which is charged, but their charges cancel on average. In such a mixed phase, the Coulomb energy density is nonzero but it does not grow with the volume of the system. A finite contribution t o the total energy density should still be taken into account. There is also a contribution to the total energy density that comes from creating the surfaces separating different components of the mixed p h a ~ e . ~If J ~ the corresponding surface tension is large, the mixed phase cannot appear. With this assumption in mind, below I do not consider such phases of dense quark matter in detail. (In fact, it was suggested recently that the surface tension between the 2SC phase and the normal phase of dense quark matter may be rather ~ m a 1 l . l ~ ) In a homogeneous (one component) phase of matter, the neutrality condition is enforced locally. In this case, there is no extra energy cost to compensate the Coulomb energy or the surface tension. By imposing the local neutrality in the model at hand, one finds that the gapless 2SC phase is the ground state of dense quark matter provided the diquark coupling constant is neither too strong nor too weak.g By omitting the details of the derivation, I note that the approximate solutions to the electrical and color charge neutrality conditions in Eq. (11)

bNote, however, that there may exist a chrornornagnetic instability in the g2SC phase.12

31 7

are given by p~ x 0 and

It is remarkable that the solutions in the regions of parameters A > dp and A 5 dp are very different. One could check that this reflects a qualitatively different nature of the quasiparticle spectra in the two regions. While all the modes described by the dispersion relations in Eq. (6) are gapped when A > 6p, there appear two gapless modes when A _< 6 p . Now, the effective potential in Eq. (3) in the case of neutral matter should be considered as a function of the gap parameter A only (here it is taken into account that, in the chiral limit, the dynamical Dirac mass m is vanishing a t sufficiently large densities). Indeed, after the neutrality is imposed, the chemical potentials pe and p8 are not free parameters, but functions of A. To avoid a confusion, note that the corresponding functions pe = p , ( A ) and p8 = ps(A) may not necessarily be given by explicit analytical expressions [see, for example, the relation in Eq. (14)]. The effective potential as a function of A for neutral matter is shown graphically in Fig. 1 (solid line). Q (MeVAm3 )

-81 - 82 - 83 - 84 - 85 - 86 - 87

A Figure 1. The effective potential as a function of the diquark gap A calculated at a fixed value of the electrical chemical potential pe M 148 MeV (dashed line), and the effective potential defined along the neutrality line (solid line). The results are plotted for p = 400 MeV and G D = 7Gs with q = 0.75.

To emphasize the role of the neutrality condition, in Fig. 1, the result for the effective potential of non-neutral quark matter at a fixed value of

318

the electrical chemical potential pe x 148 MeV is also shown (dashed line in Fig. 1). The chemical potential pe is chosen so that the electrical charge density of matter vanishes when the value of the gap corresponds to the maximum of the potential. This effective potential (labelled “pe = 148 MeV”) describes negatively charged matter to the left from the maximum, and positively charged matter to the right from the maximum. On both sides, it is irrelevant because of a large (infinite in an infinite volume) contribution of the Coulomb energy that was not included yet. Of course] the two charged minima of the dashed line would become physically important if one allows for mixed p h a ~ e s . ~ J ~ J ~ From the location of the minimum of the effective potential for neutral quark matter (solid line in Fig. l), one determines the value of the gap parameter in the ground state, A x 68 MeV. It appears that A < 6p pe/2 x 74 MeV in such a ground state. This is a very interesting solution because it corresponds to the so-called gapless supercondu~tivity.~ I discuss some unusual properties of this phase of matter in the next section. 4. Gapless superconductivity

Here I review some properties of the gapless 2SC phase of quark matter. By definition, this is a ground state that corresponds to a solution with A < 6p. The low energy quasiparticle spectrum in this phase contains four gapless and only two gapped modes, see Eqs. (4)-(6) and Fig. 2. This is in contrast to the ordinary 2SC phase (that appears in the strong coupling regime) whose spectrum contains only two gapless and four gapped modes. The most remarkable property of the quasiparticle spectrum in the g2SC phase is that the low energy excitations ( E << 6p - A) are very similar to those in the normal phase. The only difference is that the values of the chemical potentials of the free up and down quarks are replaced by fi f This simple observation may suggest that the p* low energy (large distance scale) properties of the g2SC phase look similar to those in the normal phase.g For example, the Debye screening mass of the gluons of the SU(2), subgroup should be nonzero. The latter, in fact, should be proportional to the density of the gapless modes, i.e.,

d

m

.

Note that the corresponding value for the Debye screening mass in the ordinary 2SC phase is vanishing.15 The overall coefficient here can be fixed

319

Figure 2. The schematic modification of the red and green quasi-quark dispersion relations in the g2SC phase a t low energies (dashed line), and the corresponding relations in the normal phase (solid line).

by matching the value of the Debye screening mass with the known result in the normal phase. Also, this can be obtained from a direct calculation.12 It appears that the magnetic screening properties of the g2SC phase are nothing like those in the normal phase, or like those in the gapped 2SC phase. In fact, it was shown recently that several values of the Meissner screening masses squared are negative in the gapless phase, indicating a chromomagnetic instability in dense quark matter.12 This instability may lead to a gluon condensation in the g2SC phase. Specific details of such a condensation still remain to be clarified. The fact that the low energy spectrum of quasiparticles in the g2SC phase looks nearly the same as in the normal phase has yet another consequence. Namely, at sufficiently low temperatures, there should appear two additional spin-1 condensates on top of the g2SC ground state. This conclusion is a result of a simple observation. Around the effective Fermi momenta pgff = p*, the gapless quasiparticles are double degenerate. This is the reflection of the degeneracy with respect to the red and green colors of the unbroken SU(2), subgroup. By taking into account the high degeneracy of states a t pgff = p* and the attraction in the color antisymmetric channel, one concludes that the Cooper instability should develop in the g2SC ground state. This instability is removed spontaneously by the appearance of two secondary condensates of Cooper pairs. In view of the Pauli principle, these are spin-1 condensates of the so-called polar phase.

320

The magnitude of spin-1 gaps is expected to be several orders of magnitude smaller than the spin-0 gap parameter A in the g2SC phase. The finite temperature properties of the g2SC phase are very unusual too. The most striking are the results for the temperature dependence of the gap parameter for various values of the diquark coupling strengths. This is demonstrated by the plot in Fig. 3. The most amazing are the results for weak coupling. It appears that the gap function could take finite values at nonzero temperature even if it is exactly zero at zero temperature. This possibility comes about only because of the strong influence of the neutrality condition on the choice of the ground state in quark matter.

60

a 20

Figure 3. The temperature dependence of the diquark gap in neutral quark matter calculated for several values of the diquark coupling strength 11 = G D / G ~ .

By looking a t the results in Fig. 3, one should immediately realize that the value of the ratio of the critical temperature T, to the value of the gap at zero temperature A, is not a universal number in the g2SC phase. In contrast to the Bardeen-Copper-Schrieffer (BCS) type theories, the ratio T,/Ao in the g2SC phase depends on the diquark coupling constant. Moreover, it can be arbitrarily large, and even remain strictly infinite for a range of coupling strengths. 5 . Summary

In this contribution, I reviewed some unusual zero and finite temperature properties of the g2SC phase of neutral two-flavor quark matter in P-eq~ilibrium.~ This new state of dense quark matter combines the properties of color superconducting and normal phases in a subtle way. As in

321 a 2SC superconductor, the original color symmetry SU(3), is broken down to SU(2), through the Higgs mechanism. At the same time, the low energy spectrum of quasiparticles looks very similar to the spectrum in the normal phase. As a reflection of this, the Debye screening masses for the gluons from the SU(2), subgroup are nonzero. In addition, instead of the standard Meissner effect, the magnetic (Meissner) screening masses are imaginary suggesting a chromomagnetic instability in the g2SC phase.12 The g2SC phase has rather unusual finite temperature properties. For example, the ratio of the critical temperature to the gap at zero temperature is not a universal number. It is determined by the value of the coupling constant and, in general, it is larger than the ratio rBcs x 0.567 in the BCS theory. Note that the corresponding ratio in high-T, superconductors is also larger than rBcs. It remains to be clarified whether this is a coincidence. At sufficiently low temperature, the situation gets even more complicated when the Cooper instability develops around the effective Fermi surfaces, and two additional spin-1 condensates develop on top of the gapless ground state. These spin-1 condensates would affect the transport properties of quark matter. It should be noted that a gapless phase, the so-called gapless color-flavor locked phase (gCFL), can also exist in strange quark matter when one takes into account that the strange quark mass is not very ~ m a l l . ' ~ItJ ~is the color neutrality condition, controlled by a color chemical potential, which is responsible for the appearance of the gCFL phase.16 This is in contrast to g2SC superconductivity which results from the electrical n e ~ t r a l i t y . ~ In nature, gapless phases of quark matter may exist in cores of compact stars. This possibility was the main motivation for the original study.g It should be mentioned, however, that some gapless phases may also be prepared in nonrelativistic systems, such as cold gases of fermionic atoms.l* In view of the recent results on the magnetic screening properties,12 the fate of the gapless phases is not obvious. Unless some nonperturbative effects stabilize the system, the g2SC ground state is expected to develop gluon condensates. Assuming that such condensates have only a weak effect on the underlying structure of the gapless phase, one may get additional interesting features in the g2SC phase. Of course, it may also happen that the corresponding condensation has strong effect on the g2SC phase. In either case, the search for the true ground state is of prime importance.

322 References 1. D. Bailin and A. Love, Phys. Rep. 107,325 (1984); M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422,247 (1998); R. Rapp, T. Schafer, E. V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81,53 (1998). 2. D.T. Son, Phys. Rev. D59, 094019 (1999); T.Schafer and F. Wilczek, Phys. Rev. D60, 114033 (1999); D.K. Hong, V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, Phys. Rev. D61, 056001 (2000); R.D. Pisarski and D.H. Rischke, Phys. Rev. D61, 051501 (2000); S.D.H. Hsu and M. Schwetz, Nucl. Phys. B572,211 (2000); W.E. Brown, J.T. Liu, andH.-C. Ren, Phys. Rev. D62,054016 (2000); V. A. Miransky, I. A. Shovkovy and L. C. Wijewardhana, Phys. Rev. D62,085025 (2000). 3. LA. Shovkovy and L.C.R. Wijewardhana, Phys. Lett. B470, 189 (1999); T. Schafer, Nucl. Phys. B575, 269 (2000); V. A. Miransky, I. A. Shovkovy and L. C. Wijewardhana, Phys. Rev. D63, 056005 (2001). 4. D. K. Hong, Nucl. Phys. B582,451 (2000); T.Schaefer, Nucl. Phys. A728, 251 (2003). 5. K. Rajagopal and F. Wilczek, hep-ph/0011333; M. G. Alford, Ann. Rev. Nucl. Part. Sci. 51,131 (2001); T. Schafer, hep-ph/0304281; D. H. Rischke, Prog. Part. Nucl. Phys. 52,197 (2004). 6. M. Alford and S. Reddy, Phys. Rev. D67, 074024 (2003); S. Banik and D. Bandyopadhyay, Phys. Rev. D67, 123003 (2003); G. Lugones and J. E. Horvath, Astron. Astrophys. 403, 173 (2003); M. Baldo, M. Buballa, F. Burgio, F. Neumann, M. Oertel and H. J. Schulze, Phys. Lett. B562,153 (2003); D. Blaschke, S. Fredriksson, H. Grigorian and A. M. Oztas, Nucl. Phys. A736, 203 (2004); S. B. Ruster and D. H. Rischke, Phys. Rev. D69, 045011 (2004). 7. I. Shovkovy, M. Hanauske and M. Huang, Phys. Rev. D67, 103004 (2003); eConf (2030614,039 (2003). 8. M. Alford and K. Rajagopal, JHEP 0206,031 (2002). 9. I. Shovkovy and M. Huang, Phys. Lett. B564,205 (2003); Nucl. Phys. A729, 835 (2004). 10. M. Huang, P. F. Zhuang and W. Q. Chao, Phys. Rev. D67, 065015 (2003). 11. T. M. Schwarz, S. P. Klevansky and G. Papp, Phys. Rev. C60,055205 (1999). 12. M. Huang and I. A. Shovkovy, hep-ph/0407049. 13. F. Neumann, M. Buballa and M. Oertel, Nucl. Phys. A714,481 (2003). 14. S. Reddy and G. Rupak, nucl-th/0405054. 15. D. H. Rischke, Phys. Rev. D62,034007 (2000). 16. M. Alford, C. Kouvaris and K. Rajagopal, Phys. Rev. Lett. 92,222001 (2004); hep-ph/0406137. 17. S. B. Ruster, I. A. Shovkovy and D. H. Rischke, hep-ph/0405170, Nucl. Phys. A,to appear. 18. W. V. Liu, F. Wilczek and P. Zoller, cond-mat/0404478; M. M. Forbes, E. Gubankova, W. V. Liu and F. Wilczek, hep-phJ0405059.

INHOMOGENEOUS COLOR SUPERCONDUCTIVITY

G. NARDULLI Physics Department, University of Bari, Italy, INFN, Sezione di Bari, Italy, E-mail: [email protected] I review some of the properties of the Larkin-Ovchinnikov-Flde-Ferrell phase of QCD.

1. Introduction Inhomogeneous crystalline superconductivity was predicted forty years ago by Larkin, Ovchinnikov, Fulde and Ferrell'i2 (LOFF). It arises when the Fermi surfaces of the two species participating in the Cooper pairing are different. For small separation 6 p the usual homogeneous BCS pairing occurs, while for large separation pairing is not possible. However in a window of values of 6 p the energetically favored ground state might be characterized by a non zero value of the total momentum of the Cooper pair Ptot = 2q. Translational invariance is lost and the resulting gap parameter has a space dependence that in the simplest case is a plane wave with momentum 2q, as first discussed in 2. Realistic conditions for experimental investigations of the LOFF phase became available only a few years ago. In condensed matter the separation of the Fermi surfaces is obtained by a Zeeman splitting due to an exchange interaction produced by a magnetic field. However the needed field strength is such to destroy superconductivity due to diamagnetic effects. The way to avoid the problem is t o use layered superconductors (e.g. organic compounds). If the magnetic field is parallel to the layers, the orbital effects can be controlled. There has been a remarkable experimental activity in this field, but in condensed matter the existence of the LOFF phase, with its characteristic space modulation of the energy gap, still awaits complete confirmation. New opportunities have recently arisen t o detect the LOFF phase in atomic physics (ultracold atomic gases), nuclear physics and especially quark matter To this last aspect the present review is devoted 3 ~ 4 ~ 5 ~ 6 9 8 ~ 7 ~ 9 .

323

324 (a recent more general survey of the LOFF phase is in lo). The study of the LOFF phase of QCD has been generated by the study of QCD at high density and small temperature. Due to the existence of an attractive channel in the color interaction, diquark condensates might be formed. These Cooper pairs break the color gauge symmetry and therefore the phenomenon gets the name of color superconductivity. In the last few years it has become an active field of research (for reviews see Exactly as in condensed matter, in a window of values of h p color superconductivity might realized by the LOFF phase. In this context it would be generated by the difference in quark chemical potentials induced by weak interactions and/or by the mass difference between the strange and the up/down quarks. I will review the LOFF phase of QCD in Section 2. An important point to be settled is the space dependence of the condensate. As already noticed, it might have the form of a single plane wave or a more complicated structure, as given, for example, by a sum of plane waves. The condensate would have a crystalline structure whose geometric properties have to be determined by the gap equation. I review this subject in Section 3, where I discuss both the Ginzburg-Landau approach and the effective gap equation proposed in ’. Color superconductivity has implications in astrophysics because the baryon densities necessary for color superconductivity can probably be reached in the inner core of pulsars. Therefore inhomogeneous color superconductivity might provide a mechanism for the explanation of glitches in pulsars. If pulsars are neutron stars with a core made by color superconducting matter, this mechanism would be complementary to the standard models of glitches. If pulsars are strange stars, then the crystalline structure of the condensate would provide the possibility for pinning the superfluid vortices and eventually creating the glitches. I will briefly review this subject in Section 4. 11712>13).

2. LOFF phase

To simplify the problem I will consider the case of two massless quarks with chemical potentials pu and pU given by

where up and down refer to flavor. The condensate has the form

325 where a , p = 1 , 2 , 3 and i, j = 1 , 2 are respectively color and flavor indices. It must be so because the attractive color channel is antisymmetric and since the spin 0 channel is favored, the condensate must be antisymmetric in flavor as well. Let A0 be the value of the homogeneous BCS condensate. For Sp < Ao/fi, the so called Clogston-Chandrasekhar limit, the energetically favored state is the homogeneous one. To show that for Sp > A o / f i the Cooper pair might prefer to have non zero total momentum, we consider a four-fermion interaction modelled on one-gluon exchange, that is 3 -

L = --g$'Y'lXa$ $yw%+b, (3) 8 where A" are Gell-Mann matrices. In the mean field approximation it reduces to 1

L = - -2- ~ ~ a 3 ~ z j ( $ z a $ ~ A e+ ~ ~c.c.) q"

+ ( L+ R ) ,

(4)

where we have defined

rse2iq.r = _ _21 EaP3€ij ($:$;)

,

(5)

$ are left-handed fields and A = gFs. The gap equation has the form

where p = 4p2/.rr2 is the density of states, S the ultraviolet cut-off and t the component of the quark momentum parallel t o the Fermi velocity v, measured from the Fermi surface. Moreover ji = dp - v . q.

(8)

Performing the integration over the energy we get

where

e(€ -

= 1 - e(-€ - p ) - e ( - E

+ ji) .

(10)

It can be shown that there is a first order transition in dp, between the homogenous state and the normal state and a second order transition between the LOFF state and the normal one. The first order transition occurs near

326

-

Bpl A,/& while the second order phase transition is at the critical point bp2 = 0.754Ao (see below). Near 6p2 one has

ALOFF= J1.7576p2(6p2 - b p ) = 1.15 A,

/F,

(11)

which is non vanishing, and the grand potential is given by Q L O F F - Rnormal =

- 0 . 4 3 9 ~ ( 6-~6

~ 27 ) ~

(12)

showing that the LOFF phase is indeed favored. 3. Gap equation for more complicated structures

Let us now consider the more general case P m=l

(the previous case corresponds to P = 1). Only approximate solutions can be given for the gap (13) either near the second order phase transition, by the Ginzburg-Landau method, or by an effective gap equation that captures the main aspects of the structure (13).

3.1. Gap equation in the Ginaburg-Landau approximation In the Ginzburg-Landau approximation the gap equation for condensate (13) is as follows

-aa =o aA

with

327 Here S(qk - qn) means the Kronecker delta: definitions have been adopted:

We have put w

Sn,k

and the following

= V F Gand 1

f i ( E ,"7

{q')

=E

+ if signE - fip + (-1)i\t- 2 ~ : = ~ ( - 1 ) +. qk] ;

(20) M moreover the condition Ek=l(-l)kqk = 0 holds, with M = 2 , 4 , 6 respectively for II,J and K . The critical value 6p2 is obtained by the condition that, at 6p = 6112, a vanishes. This approximation is adequate to deal with most of the structures that can be obtained summing several plane waves. This analysis is contained in8 and some of their results are reported in Table 1. Table 1. Crystalline structures with P plane waves. $ = 6p2 p, 7 = 6p47 , is the (dimensionless) minimum free energy computed at b p = 0.754.40. The phase transition (first order for $ < 0 and 7 > 0, second order for $ > 0 and 7 > 0) occurs at Sp*.

6

7

One-planewave ( P = 1)

0.569

1.637

0

Antipodal plane waves ( P = 2)

0.138

1.952

0

0.754

bcc ( P = 6 )

-31.466

19.711

-13.365

3.625

fcc ( P = 8)

-110.757

-459.242

Structure

Qmin

bp*/Ao

~~~

0.754

The most interesting case is offered by the cubic structures. They are formed either by six plane waves pointing to the six faces of a cube (the so-called body-centered-cube, bcc) or by eight plane waves pointing to the vertices of a cube (face-centered-cube, fcc). In the former case the grand potential turns out to be bounded from below and smaller than all the other

328 crystalline structures. In the latter case the grand potential is unbounded from below, because the coefficient y is negative. This represents a problem, because it shows that the analysis is incomplete and higher order terms, e.g. A8, should be included. Had these terms be taken into account, it is reasonable to expect that, given the large and negative value of y,the minimum of the grand potential would be obtained by the face-centeredcube. N

3.2. Eflective gap equation

The idea of the effective gap equation is t o perform an average over the original lagrangian so that the new lagrangian has some similarity with the one-plane wave case, which, as we have shown above, can be solved in the whole (Sp1,bp2) range and not only near the second order phase transition. The smoothing procedure is described in detail in and we can recall here only the main points. The method employed is the so called High Density Effective Theory (for a review see 1 3 ) . This method uses effective quark fields where the large part of the quark momentum pv has been extracted by a factor exp(ipv). Therefore in the condensate term of the lagrangian one has a factor 41719

exp i(puVu -tpdvd

+ 2%)

*

r

.

(21)

m

Next we multiply the lagrangian by some appropriate function g(r) and make an average over a crystal cell. In the gap equation the relevant integration momenta are small. Therefore one can assume that the fields are almost constant in the averaging procedure. Averaging the factor (21) produces as a result that the two velocities are antiparallel up to terms of the order of Sp/p; moreover the term (21) is a substituted by its average P

AE(v, t o ) =

C Aeff

(V . nm, l o )

,

(22)

m= 1

!,

where is the quasiparticle energy and nm = q,/q. It can be shown that this procedure is valid for A not too small, i.e. far away from the second order phase transition. The exact form of the function A,,, depends on the function g(r) chosen for the average. In particular g(r) can be chosen in such a way that

A for

a,,,

(E,v)

E PR

=AO(E,)O(E~) = 0

elsewhere

.

(23)

329 where for each plane wave of wave number nm one has &,d

= f6p

q n m v 4-

d

m

(24)

and the pairing region ( P R ) is defined by the condition Eu > 0, E d > 0. This procedure simplifies the gap equation that assumes the form

The energy integration is performed by the residue theorem and the phase space is divided into different regions according t o the pole positions, defined by

Therefore we get

where the regions Pk are defined as follows

and we have made use of the equation

relating the BCS gap A0 to the four fermion coupling g and the density of states. The first term in the sum, corresponding to the region P I , has P equal contributions with a dispersion rule equal to the F‘ulde-Ferrel P = 1 case. This can be interpreted as a contribution from P non interacting plane waves. In the other regions the different plane waves have an overlap. The free energy R is obtained by integrating in A the gap equation. At fixed b p , R is a function of A and q , therefore the energetically favored state satisfies the conditions

an = 0 , aA

aR -= 0, a9

and must be the absolute minimum. Assuming the values p = 400 MeV and 5 = p / 2 one gets the results in Table 2 for 6p = Spl = Ao/&.

330 One can repeat the analysis for Sp # 6111.The results of is that the body-centered-cube (bcc) is the favored structure up to 6p M .95A0. For larger values of 6p < 1.32A0 the favored structure is the face-centeredcube (fcc). In Table 3 numerical results for Spz are reported together with the computed order of the phase transition between the LOFF and the normal phases. Also the values of SpIq and of the discontinuity in A/A, at Sp = Sp2 - O+ are reported. Table 2. The gap, Sp/q and the free energy at 6,u = 6pi = A o / f i for different crystalline structures.

One-planewave ( P = 1)

0.78

0.24

Antipodal plane waves ( P = 2)

1.0

0.75

-0.0018 -0.08

bcc ( P = 6)

0.9

0.28

-0.11

fcc ( P = 8)

0.9

0.21

-0.09

Table 3. The values of 6 ~ 2 the , ratio 6,u/q, the discontinuity of AlAo and the order of the phase transition between the LOFF and the normal phases for different crystalline structures.

2

order

One-plane-wave ( P = 1)

0.754

I1

0.83

0

Antipodal plane waves ( P = 2)

0.83

I

1.0

0.81

bcc ( P = 6)

1.22

I

0.95

0.43

fcc ( P = 8)

1.32

I

0.9'

0.35

Structure

% &

One can incidentally notice that the regions Pk do not represent a partition of the phase space since it is possible to have at the same point quasi-particles with different gaps. The conclusions of this analysis, showing the dominance of the bcc structure up to a maximum value of Sp and the dominance of the fcc condensate for larger values of Sp, are in qualitative agreement with the results of the Ginzburg-Landau study of the gap equation. 4. Astrophysical implications

Inhomogeneous color superconductivity is obviously interesting for the study of the phase structure of QCD. It may also result of interest for

331 astrophysical dense systems, in particular in the explanation of the glitches in the pulsars. The pulsars are rapidly rotating stellar objects, characterized by the presence of strong magnetic fields and by an almost continuous conversion of rotational energy into electromagnetic radiation. Due to this loss of energy the periods increase slowly and never decrease except for occasional glitches, when the pulsar spins up with a variation in frequency that can be SR/R M or smaller. In the standard explanation pulsar are identified with neutron stars; these compact stars are characterized by a rather complex structure comprising a core, an intermediate region with superfluid neutrons and a metallic crust. The ordinary explanation of the glitches is based on the idea that these sudden jumps of the rotational frequency are due to the movements outwards of rotational vortices in the neutron superfluid and their interaction with the crust. This is one of the main reasons that allow the identification of pulsars with neutron stars, as only neutron stars are supposed to have a metallic crust. A possible model for the glitches is based on the simultaneous presence of a normal and a superfluid component in the compact star. The superfluid velocity is

ti v, = -V@ m

(31)

where @ is the phase of the condensate. Due t o the presence of points with a normal (not superfluid) component, an integration around these points correspond t o integration over a domain which is not simply connected. Therefore one has

f.,

.d l

= 2 r n ,~

(32)

where the integer n is a winding number counting the number of times the curve y goes around the singular point, and IF, is a constant with dimensions of vorticity, i.e. [LI2 . Clearly K = h/m ( K is called quantum of vorticity). These singular points form lines called vortex lines. Due t o (32), and the vortex motion, the superfluid component participates in the rotation of the vessel with an angular velocity R and the number N of vortex lines for unit of area is related t o R by

It can be shown the angular velocity 52 can change only by a radial motion, of the vortex lines. However they are in general pinned at the crust, which

332 forbids their motion. The maximum pinning force is obtained when the vortex passes through one layer of the lattice; therefore the maximum force per unit length of vortex line is

with

SE, = F,

- F,

C(

pA:

,

(35)

where F, and F, are the free energies of the superfluid neutrons and the nucleons in the crust; A, is the gap for superfluid neutrons and one can neglect A,, the gap of superfluid neutrons in the crust since A, << A,. These results imply that neutrons tend to remain in the volume V of the vortex core because they experience a force repelling them from the superconductive phase (if neutron rich nuclei are present, the repulsion will be less important). Typical values for the pinning energy per nucleus 6E, a t densities 3 x lOI3 - 1.2 x 1014 g/cm3 are SE, = 1 - 3MeV while b = 25 - 50fm and R,:

6v=(f2-f2,)r\r. (36) The interaction between the normal matter in the core of the vortex line and the rest of normal matter (nuclei in the lattice, electrons, etc.) produces a Magnus force per unit length given by

f =pkASv,

(37)

where the direction of k coincides with the rotation axis and its modulus is equal to the quantum of vorticity. f is the force exerted on the vortex line; as it cannot be larger than f, there is a maximum difference of angular velocity that the system can maintain:

If w < w,, the vortices remain pinned a t the lattice sites instead of flowing with the superfluid as they generally do in superfluid (see discussion above).

333

On the contrary, if w > w,,, the hydrodynamical forces arising from the mismatch between the two angular velocities ultimately break the crust and produce the conditions for the glitch. Let us discuss the possible role of the LOFF phase in this context. The QCD LOFF phase provides a lattice structure which is independent of the crust. Therefore it meets one of the two requirements of the model for glitches in pulsars we have outlined above, the other being the presence of a superfluid. The only existing calculations for the LOFF phase in color superconductivity have been performed for two flavors. In this case, however, in the homogeneous case, there is no superfluid, since there are no broken global symmetries. Superfluidity is on the other hand manifested by color superconductive QCD with three flavors. Therefore for a realistic application to QCD one should implement a calculation of the QCD LOFF phase with three flavors. Let us give however some order of magnitude estimates ’. Let us take AZSC= 40 MeV, ALOFFx 8 MeV, corresponding to the Fulde-Ferrel state; since q x 1 . 2 6 ~x 0.7Azsc, one would get for the average distance between nodal planes b = ~ / ( 2 l q l )x 9 fm and for the superconducting coherence length SO = 6 fm. The free energy per volume unit is of the order of

~ F L O F= F 8~ x ( 1 0 ~ V e V ) ~

(39)

and the pinning energy of the vortex line is

6Ep = ~ F L O FxFb3 ~ = 6 MeV

.

(40)

An order of magnitude estimate of the pinning force is therefore fp M

3 x lo3 MeV3 .

(41)

A comparison between these numerical values and the typical values mentioned above shows that they are similar. Therefore it is possible that some of the glitches in neutron stars may be generated well inside the star by vortices related t o the LOFF phase of QCD. Future studies will hopefully clarify this issue. Acknowledgments

I am deeply grateful to M. Shifman and A. Vainshtein for the kind hospitality extended t o me in Minneapolis. It is a pleasure to thank R. Casalbuoni, M. Ciminale, R. Gatto, M. Mannarelli and M. Ruggieri for a very fruitful collaboration.

334

References 1. A. J. Larkin and Yu. N. Ovchinnikov, Zh. Exsp. Teor. Fit. 47, 1136 (1964). 2. P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). 3. M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D63, 074016 (2001) hep-ph/0008208. 4. R. Casalbuoni, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Lett. B 511, 218 (2001) hep-ph/0101326. 5. A. K. Leibovich, K. Rajagopal and E. Shuster, Phys. Rev. D 64,094005 (2001) hep-ph/0104073. 6. J. Kundu and K. Rajagopal, Phys. Rev. D 65,094022 (2002) hep-ph/0112206. 7. R. Casalbuoni, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Rev. D 66, 014006 (2001) hep-ph/0201059. 8. J. A. Bowers and K. Rajagopal, Phys. Rev. D66, 065002 (2002) h e p ph/0204079. 9. R. Casalbuoni, M. Ciminale, R. Gatto, M. Mannarelli, G. Nardulli and M. Ruggieri Phys. Rev. D, to appear, hep-ph/0404090. 10. R.Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004) h e p ph/0305069. 11. K. Rajagopal and F. Wilczek, in At the frontier of particle physics (M. Shifman and B. Ioffe), 2061 (2001) hep-ph/0011333. 12. M. G. Alford, Ann. Rev. Nucl. Part. Sci. 51, 131 (2001) hepph/0102047. 13. G. Nardulli, Riv. Nuovo Cim. 25N3, 1 (2002) hep-ph/0202037.

SPONTANEOUS ROTATIONAL SYMMETRY BREAKING AND OTHER SURPRISES IN u-MODEL AT FINITE DENSITY

V.A. MIRANSKY* Department of Applied Mathematics, University of Western Ontario , London, Ontario N6A 5B7, Canada The linear SU(2)r. x S u ( 2 ) a-model ~ occupies a unique place in elementary particle physics and quantum field theory. It has been recently realized that when a chemical potential for hypercharge is added, it becomes a toy model for the description of the dynamics of the kaon condensate in high density QCD. I review recent results in nonperturbative dynamics obtained in the ungauged and gauged versions of this model.

1. Introduction

The linear s u ( 2 ) x~s u ( 2 ) R a-model occupies a unique place in elementary particle physics and quantum field theory. Introduced back in 1960 by GellMann and Levy,' the model is a t the heart of particle and nuclear dynamics a t very different scales. At the scale of order 10 MeV, the model with sigma, pion and nucleon fields is relevant for nuclear structure calculations. Then, at the scale of order 100 MeV, the model with nucleon fields replaced by quark ones, is an effective theory of chiral dynamics of hadrons (for a review, see Ref. [ 21). And at the scale of order 100 GeV, the model is nothing else but the Higgs sector of the electroweak theory. In this talk, I will describe new and quite surprising phenomena taking place in the h e a r su(2)~ x s u ( 2 ) R a-model a t finite density or, more precisely, at the finite chemical potential p for the hypercharge Y. The interest to this model is connected with that it is a toy model for the description of the dynamics of the kaon condensate3 of high density QCD that may exist in cores of compact stars. We will consider dynamics a t finite p both in the linear a-model itselFy5 and in its gauged version, when the subgroup s U ( 2 ) x~ U ( 1 ) y is being gauged.6 The central results are as follows: (1) In the ungauged a-model, the spontaneous breakdown of S U ( 2 ) x 335

336 U ( 1 ) y symmetry, caused by the chemical potential, leads to a lesser number of Nambu-Goldstone bosons than that required by the Goldstone theorem. One of the consequences of this phenomenon is that the system is not a superfluid despite the presence of a condensate. ( 2 ) In the gauged version of the model, the spontaneous breakdown of S U ( 2 ) x U ( 1 ) y symmetry, caused by the chemical potential, is always accompanied by spontaneous breakdown of both rotational symmetry and electromagnetic U (l)em. (3) The spectrum of excitations in the gauged model is very rich. In particular, there exist excitation branches that behave as phononlike quasiparticles for small momenta and as roton-like ones for large momenta. These roton-like excitations are present because of gauge vector fields. This can shed light on microscopic nature of roton-like excitations, which is an old problem in the theory of s u p e r f l ~ i d i t y . ~ ? ~ I t also suggests that this model can be relevant for superfluid and superconducting systems. 2. Dynamics with abnormal number of Nambu-Goldstone bosons

Recently a class of relativistic models with a finite density of matter has been revealed in which spontaneous breakdown of continuous symmetries leads to a lesser number of Nambu-Goldstone (NG) bosons than that required by the Goldstone t h e ~ r e m . The ~ > ~simplest representative of this class is the linear s U ( 2 ) x~ s U ( 2 ) o-model ~ with the chemical potential for the hypercharge Y ,

L = (80

+ i p ) @ t ( a o- i p ) -~ ai++ai@- m2@t@ - ~(@t@)’,

(1)

where @ is a complex doublet field. The chemical potential p is provided by external conditions (to be specific, we take /I > 0). For example, in the case of dense QCD with the kaon condensate, p is p = m 2 / 2 p ~ where , m, is the current mass of the strange quark and p~ is the quark Fermi r n ~ r n e n t u m Note .~ that the terms with the chemical potential reduce the initial S u ( 2 ) x~ s U ( 2 ) symmetry ~ to the s U ( 2 ) x~ U ( 1 ) y one. Henceforth we will omit the subscripts L and R, allowing various interpretations of the S U ( 2 ) [for example, in the dynamics of the kaon condensate, it is just the conventional isospin symmetry S U ( 2 ) 1 and aT = ( K + ,KO)]. The terms containing the chemical potential in Eq. (1) are

i p d a o a - ipaoata

+p W @ .

(2)

337 The last term in this expression makes the mass term in Lagrangian density (1) to be (p2 - m 2 ) @ t @Therefore . for supercritical values of the chemical potential, p 2 > m 2 , there is an instability resulting in the spontaneous breakdown of S U ( 2 ) x U ( l ) y down to U(l)emconnected with the electrical charge Q,, = I 3 i Y . One may expect that this implies the existence of three NG bosons. The explicit calculation shows that this is not the case. To see this, we represent the field in the following form:

+

By analyzing the quadratic forms for the two pairs of the fields ( 9 1 , ~ 2 and ($1, $ 2 ) , we arrive at the explicit dispersion relations for the charged and neutral degrees of f r e e d ~ m , ~ ~ ~ w1,2

=

d

m f p,

respectively. As is easy to check, here there are only two NG bosons with the following dispersion relations,

which carry the quantum numbers of K+ and KO mesons. The third wouldbe NG boson, with the quantum numbers of K - , is massive in this model. This happens despite the fact that the potential part of Lagrangian density (1) has three flat directions in the broken phase, as it should. The splitting between K+ and K - occurs because of the seesaw mechanism in the kinetic part of the Lagrangian density (kinetic seesaw m e ~ h a n i s m This mechanism is provided by the first two terms in expression (2) which, because of the imaginary unit in front, mix the real and imaginary parts of the field @. Of course this effect is possible only because C , CP, and CPT symmetries are explicitly broken in this system at a nonzero p. The latter point is reflected in the spectrum of K mesons even for subcritical values of the chemical potential p < m. In that case, there is a splitting of the energy gaps (“masses”) of ( K - , ko)and ( K + ,KO) doublets. While the gap of the first doublet is equal to m + p, the gap of the second one is m - p. Another noticeable point is that while the dispersion relation for KO is conventional, with the energy w k as the momentum k goes to zero,

-

)

338

-

the dispersion relation for K+ is w k2 for small k [see Eqs. ( 6 ) and (7)]. This fact is in accordance with the Nielsen-Chadha counting rule, N G / H = 121 2n2.' Here nl is the number of NG bosons with the linear dispersion law, w Ic, 122 is the number of NG bosons with the quadratic dispersion law w k 2 , and N G I H is the number of the generators in the coset space G I H (here G is the symmetry group of the action and H is the symmetry group of the ground state). The dispersion relation w k2 also implies that, despite the presence of the condensate, the Landau criterion' fails in this model and the system is not a superfluid. Indeed, recall that, according to the Landau criterion, superfluidity takes place for velocities Y < Y,, where the critical velocity u, is the minimum of the ratios w i ( k ) / k taken over all excitation branches and over all values of momentum k. Therefore even the presence of a single branch with w k2 implies that u, = 0. Does the conventional Anderson-Higgs mechanism survive in the gauged version of this model despite the absence of one out of three NG bosons? This question has motivated the work where the gauged g-model with the chemical potential for hypercharge was considered. The answer to this question is positive and we will consider it in the following sections.

+

--

-

N

3. Gauged a-model We will consider the dynamics in the gauged version of model (l),i.e., the model described by the Lagrangian density

+ [ ( D , - ipS,o)@]t(Dp - ipS@)@

- V(@t@),

(8) where V(@t@)= m2@t@+X(@t@)2 and the covariant derivative D , = 8, igA, - (ig'/2)B,. The field @ could be taken in the same form as in Eq. (3) with cpo being the expectation value that is determined by minimizing the effective potential. The SU(2) gauge fields are given by A, = A ; T ~ / where T~ are three Pauli matrices, and the field strength FLZ) = 8,AP) &At' g c a b c A f ) A f )B, . is the U ( l ) y gauge field with the strength B,, = a,B, - &B,. The hypercharge of the doublet @ equals +l. This model has the same structure as the electroweak theory without fermions and with the chemical potential for hypercharge Y . We consider two different cases: the case with g' = 0, when the hypercharge Y is connected with the global U ( l ) y symmetry, and the case with a nonzero g ' , when the U ( l ) y symmetry is gauged. The main characteristics of these dynamics are the following.6 For m2 > 0, the spontaneous

+

339 breakdown of the S U ( 2 ) x U(1)y symmetry is caused solely by a supercritical chemical potential p2 > m2. In this case spontaneous breakdown of the S U ( 2 ) x U(1)y is always accompanied by spontaneous breakdown of both the rotational symmetry SO(3) [down to SO(2)] and the electromagnetic U ( l ) e m connected with the electrical charge. Therefore, in this case the S U ( 2 ) x U(1)y x SO(3) group is broken spontaneously down to SO(2). This pattern of spontaneous symmetry breakdown takes place for both g‘ = 0 and g’ # 0, although the spectra of excitations in these two cases are different. Also, the phase transition at the critical point p2 = m2 is a second order one. The realization of both the NG mechanism and the Anderson-Higgs mechanism is conventional, despite the unconventional realization of the NG mechanism in the original ungauged model (1). For g’ = 0, there are three NG bosons with the dispersion relation w k, as should be in the conventional realization of the breakdown S U ( 2 )x U ( l ) y x SO(3) + SO(2) when U(1)y x SO(3) is a global symmetry. The other excitations are massive (the Anderson-Higgs mechanism). For g’ # 0, there are two NG bosons with w k, as should be when only SO(3) is a global symmetry (the third NG boson is now “eaten” by a photon-like combination of fields A; and B, that becomes massive). In accordance with the Anderson-Higgs mechanism, the rest of excitations are massive. Since the residual SO(2) symmetry is low, the spectrum of excitations is very rich. In particular, the dependence of their energies on the longitudinal momentum k3, directed along the SO(2) symmetry axis, and and on the transverse ones kl = (kl,kz), is quite different. A noticeable point is that there are two excitation branches, connected with two NG bosons, that behave as phonon-like quasiparticles for small momenta (i.e., their energy w k) and as roton-like ones for large momenta 163, i.e., there is a local minimum in w(k3) for a value of ks of order m (see the plots of the dispersion relations in Secs. 4 and 5). On the other hand, w is a monotonically increasing function of the transverse momenta. The existence of the roton-like excitations is caused by the presence of gauge fields [there are no such excitations in ungauged model (l)]. The connection of roton excitations with gauge fields in the gauged a-model is intriguing and could shed light on their microscopic nature. The presence of these excitations also suggests that the present model could be relevant for anisotropic superfluid systems.

-

-

-

340 4. Model with global U ( l ) y symmetry

We begin by making the following general observation. Let us consider a theory with a chemical potential p connected with a conserved charge Q. Let us introduce the quantity

Rmin E min(m2/Q2),

(9)

where on the right hand side we consider the minimum value amongst the ratios m2/Q2 for all bosonic particles with Q # 0 in this same theory but without the chemical potential. Then if p2 > Rmin,the theory exists only if the spontaneous breakdown of the U(1)Q symmetry takes place there. Indeed, if the U ( ~ ) Qwere exact in such a theory, the partition function, 2 = Tr[exp(pQ - H)/T], would diverge. Examples of the restriction p2 < Rmin in relativistic theories were considered Ref. [ 101. We begin by considering the case with g' = 0 and m2 > 0. When p2 < m2, the SU(2) x U(1)y x SO(3) symmetry is exact. Of course in this case a confinement dynamics for three SU(2) vector bosons takes place and it is not under our control. However, taking p2 m2 and choosing m to be much larger than the confinement scale Asu(2),we get controllable dynamics at large momenta k of order m. It includes three massless vector bosons A; and two doublets, ( K + , K o )and ( K - , K o ) . The spectrum of the doublets is qualitatively the same as that in model (1): the chemical potential leads to splitting the masses (energy gaps) of these doublets and, in tree approximation, their masses are m - p and m p, respectively (see Sec. 2). In order to make the tree approximation to be reliable, one should take X to be small but much larger than the value of the running coupling g4(m) related to the scale m [smallness of g2(m) is guaranteed by the condition m >> A s u ( ~ ) assumed above]. The condition g4(m) << X << 1 implies that the contributions both of vector boson and scalar loops are small, i.e., there is no Coleman-Weinberg (CW) mechanism (recall that one should have X g4 for the CW mechanism)." Let us now consider the case with p2 > m2 > 0 in detail. Since m2 is equal to Rmin (9), there should be spontaneous U ( l ) y symmetry breaking in this case. For g' = 0, the equations of motion derived from Lagrangian density Eq.(8) read N

+

-

= 0,

(10)

=0

(11)

341

(since now the field B, is free and decouples, we ignore it). Henceforth we will use the unitary gauge with a* = (0, cpo @~/fi). It is important that the existence of this gauge is based solely on the presence of SU(2) gauge symmetry, independently of whether the number of NG bosons in ungauged model (1) is conventional or not. We will be first looking for a homogeneous ground state solution (with cpo being constant) that does not break the = 0 where A?) = A ( A C ) f iAF)). rotational invariance, i.e., with Then we find that, besides the symmetric solution with PO = 0, the system of equations (10) and (11) allows the following solution:

+

We recall that in the unitary gauge all auxiliary, gauge dependent, degrees of freedom are removed. Therefore in this gauge the ground state expectation values of vector fields are well defined physical quantities. Solution (12), describing spontaneous SU(2) x U (1)y symmetry breaking, exists only for negative m2. On the other hand, the symmetric solution with cpo = 0 cannot be stable in the case of p 2 > Rmin = m2 > 0 we are now interested in. This forces us to look for a ground state solution that breaks the rotational invariance. Let us now consider the effective potential V . It is obtained from Lagrangian density Eq.(8), V = -L, by setting all field derivatives to zero. Then we use the ansatz with nonzero values for aT = (0,cpO) along with

A$+) = (A$-))*= C, Af) = D,

(13)

that breaks spontaneously both rotational symmetry [down to SO(2)] and SU(2) x U ( l ) y (completely). Substituting this ansatz into the potential, we arrive at the expression

One can always take both g and the ground state expectation value cpo to be positive (recall that we also take p > 0). Then we find the following nontrivial solution:

It is not difficult to show that for p 2 > m2 > 0 both expression (15) for cpo and expression (16) for ICI2 are positive and, for g2 5 8 X , this solution

342

corresponds to the minimum of the potential. The phase transition at the critical value p = m is a second order one. The situation in the region g2 > 8X is somewhat more complicated: in that region, the potential V becomes unbounded from below. Henceforth we will consider only the case with g2 5 8X when the potential is bounded g2(m)the inequality g2 5 8X is from below. Notice that for small g2 consistent with the condition g4 << X necessary for the suppression of the contribution of boson loops, as was discussed above. In order to derive the spectrum of excitations, one has to introduce small fluctuations a t ) and $1 about the ground state solution in Eq. (13) [i.e., A t ) = ( A p ) ) + a f )and @T = (0, (po+$l/fi)] and then make the expansion in Lagrangian density (8) keeping only quadratic fluctuation terms. This analysis of the quadratic form was done by using MATHEMATICA.6 Since in the subcritical phase, with p2 < m2, there are 10 physical states (6 states connected with three massless vector bosons and 4 states connected with the doublet a), there should be 10 physical states (modes) also in the supercritical phase. Out of the total 10 modes there exist 3 massless (gapless) NG modes, as should be in the conventional realization of the spontaneous breakdown of S U ( 2 ) x U(1)y x SO(3) + S0(2), when U ( l ) y x SO(3) is a global symmetry. The dispersion relations for them in the infrared region are

=

w

N-

”-

2P + 34

k2 + O(Ic;),

(17)

where w E ko. The infrared dispersion relations for the other seven excitations are rather complicated and can be found in the original paper.6 While the analytical dispersion relations in the infrared region are quite useful, we performed also numerical calculations to extract the corresponding dispersion relations outside the infrared region. The results are as follows. In the near-critical region, p + m 0 , the ground state expectation 4 becomes small. In this case, one gets 8 light modes. The results for their dispersion relations are shown in Fig. 1(the two heavy modes with the gaps of order 2p are not shown there). The solid and dashed lines represent the energies of the quasiparticle modes as functions of the transverse momentum k l = ( k 1 , O ) (with kg = 0) and the longitudinal momentum Icg (with k l = 0), respectively. Bold and thin lines correspond to double degenerate

+

343 0.8

//

0.7

’

0.4

0.3

a \

*.”

//

/

0.5

./’,

//

a/ /

0.6

‘.

/

8

// /

/

,

/ ’ / /

/

//

+

/

0.2

0.4

0.6

0.8

1

1.2

1.4

012

014

016

/

9‘

018

1

/

112

li4

‘’

Figure 1. The energy w of the 8 light quasiparticle modes as a function of Icl (solid lines, left panel) and k3 (dashed lines, right panel). The dispersion relations of two heavy modes are outside the plot range. The energy and momenta are measured in units of m. The parameters are p / m = 1.1 and # / m = 0.1

and nondegenerate modes, respectively, There are the following characteristic features of the spectrum. a) The spectrum with kl = 0 (the right panel in Fig. 1) is much more degenerate than that with k3 = 0 (the left panel). This point reflects the fact that the axis of the residual SO(2) symmetry is directed along k3. Therefore the states with kl = 0 and k3 # 0 are more symmetric than those with kl # 0. b) The right panel in Fig. 1 contains two NG branches with local minima at k3 m, i.e., roton-like excitations. Because there are no such excitations in ungauged model (l),they occur because of the presence of gauge fields. Since roton-like excitations occur in superfluid systems, the present model could be relevant for them. c) The NG and Anderson-Higgs mechanisms are conventional in this system. In particular, the dispersion relations for three NG bosons have the form w k for low momenta.

-

-

5. Model with gauged U ( l ) y symmetry

Let us now describe the case with g‘ # 0. In this case the U ( l ) y symmetry is local and one should introduce a source term Bo JO in Lagrangian density (8) in order to make the system neutral with respect to hypercharge Y . This is necessary since otherwise in the system with a nonzero chemical potential p the thermodynamic equilibrium could not be established. The value of the background hypercharge density JO (representing very heavy particles) is determined from the condition (Bo)= 0.l’ After that, the analysis follows closely to that of the case with g’ = 0. Because of the additional vector boson B,, there are now 12 quasiparticles in the spectrum. The sample of spontaneous SU(2) x U ( 1 ) y x SO(3)

344

symmetry breaking is the same as for g‘ = 0. However, for supercritical values of the chemical potential, there are now only two gapless NG modes (the third one is “eaten” by a photon-like combination of fields A; and B, that becomes massive). The rest 10 quasiparticles are gapped. The dispersion relations for these 10 quasiparticles are quite complicated. Therefore in order to extract the corresponding dispersion relations, numerical calculations were performed.6 In this case, the two branches connected with gapless NG modes contain roton-like excitations at lc3 m. Other characteristic features of the spectrum are also similar to those of the spectrum for the case with g’ = 0 shown in Fig. 1.

-

6. Summary

The dynamics of the linear a-model with a finite chemical potential for hypercharge is new, rich and, that is very important, controllable. In the ungauged version of the model, a dynamics with an abnormal number of NG bosons realizes and, despite the presence of a condensate, there is no superfluidity in the system. The richness of the spectrum of excitations in the gauged version of the model is provided by the coexistence of spontaneous breakdown of rotational symmetry and that of the electromagnetic U(l)em. It is noticeable that the spectrum contains roton-like excitations. Their connection with gauge fields is intriguing and deserves further study. The coexistence of spontaneous breakdown of the rotational symmetry with that of the electromagnetic U(l)em is provided by a condensate of vector charged bosons. The possibility of a condensation of vector bosons was considered in the literature in models different from the present 0ne.12913,14 The advantage of the linear a-model without fermions is in its simplicity. The model admits a controllable dynamics that puts the creation of a vector condensate on a solid ground and allows to study the spectrum of excitations in detail. One can expect that the phenomena discussed here should exist in a wide class of relativistic models at finite density. In connection with that, it is noticeable that recently a general approach to the description of systems with an abnormal number of NG bosons has been developed by Nambu.15 The linear a-model continues to teach and surprise us. Acknowledgments

I am grateful to the organizers of this Workshop for their warm hospitality. I acknowledge support from the Natural Sciences and Engineering Research Council of Canada.

345 References

* On leave of absence from Bogolyubov Institute for Theoretical Physics, 252143, Kiev, Ukraine. 1. M. Gell-Mann and M. Levy, Nuovo Cim. 16, 705 (1960). 2. T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994). 3. P. F. Bedaque and T. Schafer, Nucl. Phys. A 697, 802 (2002); D. B. Kaplan and S. Reddy, Phys. Rev. D 65, 054042 (2002). 4. V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett. 88, 111601 (2002). 5. T. Schafer, D. T . Son, M. A. Stephanov, D. Toublan and J. J . Verbaarschot, Phys. Lett. B 522, 67 (2001). 6. V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Lett. B 581, 82 (2004). 7. L. D. Landau, J. Phys. USSR 11,91 (1947); R. P. Feynman, Phys. Rev. 94, 262 (1954). 8. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 (Pergamon, New York, 1980). 9. H. B. Nielsen and S. Chadha, Nucl. Phys. B 105, 445 (1976). 10. J. I. Kapusta, Phys. Rev. D 24, 426 (1981); H. E. Haber and H. A. Weldon, Phys. Rev. D 25, 502 (1982). 11. S. Coleman and E. Weinberg, Phys. Rev. D 7,1888 (1973). 12. A. D. Linde, Phys. Lett. 86B, 39 (1979); I. Krive, Sov. Phys. J E T P 56, 477 (1982) [Zh. Eksp. Teor. Fiz. 83, 849 (1982)l. 13. E. J. Ferrer, V. de la Incera, and A. E. Shabad, Phys. Lett. B 185, 407 (1987); Nucl. Phys. B 309, 120 (1988); J. I. Kapusta, Phys. Rev. D 42, 919 (1990). 14. J. T. Lenaghan, F. Sannino and K. Splittorff, Phys. Rev. D 65,054002 (2002); F. Sannino and W. Schafer, Phys. Lett. B 527, 142 (2002); F. Sannino, Phys. Rev. D 67, 054006 (2003). 15. Y. Nambu, J. Stat. Phys. 115, 7 (2004).

ANALYTICAL APPROACH TO YANG-MILLS THERMODYNAMICS

R. HOFMANN Institut f u r Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany, E-mail: [email protected]

We propose a macroscopic approach to SU(N) Yang-Mills thermodynamics. We start by assuming the existence of an adjoint Higgs field generated by dilute trivialholonomy calorons at large temperatures (electric phase). Domain boundaries and isolated magnetic monopoles generate a nontrivial holonomy on the macroscopic level. We derive a thermodynamical evolution equation for the gauge coupling in the effective theory which predicts a second-order like transition to a magnetic phase (assuming maximal gauge symmetry breaking in the electric phase) where monopoles are condensed and off-Cartan excitations decoupled. This phase is not confining. The temperature dependence of the magnetic gauge coupling predicts the transition to the confining phase at Tc. Center-vortex loops are condensed in this phase, and the thermodynamics of the confining phase cannot be smoothly connected to that in the deconfining phases. We compute the temperature dependence of thermodynamical quantities throughout the electric and the magnetic phase and compare with lattice results.

1. Set-Up In this talk I would like to propose a macroscopic, effective theory for SU(N) Yang-Mills thermodynamics. This approach is constructed in the same spirit as the macroscopic U(l) Higgs theory of superconductivity is It predicts the phase structure of the theory, the (quasiparticle) spectrum of the excitations, and the thermodynamical quantities. In particular, we conclude that an SU(2) Yang-Mills theory comes in three rather than two phases a. We predict negative pressure P throughout the magnetic phase and a vanishing entropy density and an equation of state p = -P at T, (transition to confining phase). A nonthermal phase boundary sepaThere is magnetic phase inbetween the confining and the high-temperature electric phase

346

347 arates the confining from the magnetic phase. On the confining side an over-exponentially growing density of states (intersecting and single centervortex loops) allows to exceed the limiting temperature h y only ~ at the expense of destroying the spatial homogeneity of the system 3, on the magnetic side the pressure reaches the minimal possible (negative) value. Interestingly, the situation of two disconnected thermodynamical regimes in a pure SU(N) gauge theory has a counterpart in the N = 1 supersymmetric case where the exactly known beta function suggests the existence of two disconnected dynamical regimes ‘. Let me start with the basic assumption that a dilute-gas ensemble of calorons with trivial-holonomy (THC) forms a macroscopic, nonfluctuating, composite and adjoint Higgs field 4 a t very high temperatures. By dilute gas we mean that for the minimal, local definition (lowest possible mass dimension) of 4, N

the average is performed over a single THC and its zero-mode deformations only. Fluctuations, that would lift a given THC configuration A T C above the BPS bound, are not taken into account in Eq. (1). These fluctuations mediate interactions between calorons and will we considered a t a macroscopic level a t a later stage. The above assumption would be superfluous if it could be shown that at a given temperature T nontrivial solutions to the gap-equation (1) with 141 = l4l(T) exist for a certain range of values of the fundamental gauge coupling tj b. The phase, where the ground state of the theory is characterized by 4 # 0, is called as electric phase. The (nonfluctuating, see below) field 4 generates a (temperature-dependent) mass spectrum for topologically trivial gauge-field modes (thermal quasiparticles) and therefore evades the infrared divergences of ordinary thermal perturbation theory ‘. Moreover, the value of 141 represents a compositeness scale which governs the maximal off-shellness of these quasiparticles and the maximal center-of-mass energy allowed in their interactions. A fixed color orientation of $ forms a finite domain induced by a ‘seed’ caloron. At the points where at least four domains meet isolated zeros of 4 exist. The associated magnetic monopoles are mappings from 5’2 onto the coset spaces {SU(2)/U(1)} with winding numbers E Z), and the broken groups (SU(2)) are subgroups in SU(N) 7. Microscopically, isolated monopoles are associated with nontrivial-holonomy calorons bThe integration over the instanton scale

p in Eq. (1) is cut off at 14l-I in the UV.

348

(Ao(lZ1-+ GO) # 0 and copies under regular gauge transformations) which may form due to domain collisions. The latter have constituent BPS monopoles As it was shown long ago lo, configurations with nontrivial holonomy have a one-loop effective action that scales with the three-volume V. For an infinite volume this means complete suppression and thus isolated calorons with nontrivial holonomy should not play a role at large temperatures. Since the volume, over which the A,, component of the caloron is constant, is finite (existence of domains generated by trivial-holonomy calorons) configurations with nontrivial holonomy do contribute to the partition function. It was shown in that SU(2) nontrivial-holonomy calorons are unstable w.r.t. one-loop quantum fluctuations: for small (large) holonomy there is an attractive (repulsive) potential between the constituent monopole and antimonopole. In the former case the monopole and the antimonopole annihilate, in the latter case the caloron dissociates into a monopole or an antimonopole which, in turn, form isolated zeros of 4, see l3 for a lattice measurement of the magnetic-charge screening of a monopole in SU(2) Yang-Mills theory and l2 for an explanation of the surface tension of the spatial Wilson loop in terms of screened and free magnetic monopoles. Thc proposed effective action for the electric phase is 899.

In Eq. (2) GEv z 8,aE - 8,a; - efabca:aE denotes the field strength of a topologically trivial fluctuation a p , e denotes the effective gauge coupling constant, Dp4 3 8 p ie[q5,ap]and tr, tatb= 1/2dab. Here a fluctuation Ap in the fundamental theory is decomposed into a minimal-action topologically nontrivial part A;,,, which builds the ground-state structure, and ap which is associated with excitations: Ap = AgHC u p . In Eq. (2) the potential VE ( 4 )is constructed as to admit spatially homogeneous, periodicin-euclidean-time solutions (0 T 1/T) to the BPS equation 8,d = W E The first rewith a time independent modulus and VE($) f trNuEuE. t quirement derives from the zero-energy property of a THC of which 4 is composed (in absence of any other gauge-field fluctuations), the other requirements follow from the assumed thermodynamical equilibrium. For the sake of brevity Llet me only discuss the case of even N although we will later also present results for N=3, see l 4 for a discussion of odd N. The potential is uniquely determined by the above requirements as

+

+

< <

VE = trNwEwE t = A: tr, (q52)-1.

(3)

349 We can always work in a gauge where $ is SU(2) block diagonal (winding gauge). In Eq. (3) AE denotes a dynamically generated mass scale which is determined by a boundary condition to the thermodynamical evolution. Up to a global gauge transformations of the SU(2) blocks, the 'square-root' YE is uniquely given as

where X i , (i = 1 , 2 , 3 ) , denote the Pauli matrices. Solutions t o the BPS equation &$ = YE are labelled by nonzero winding numbers K(1) E Z. We have

Assuming that the solution 4 breaks SU(N) maximally t o U(l>"-l and minimizes the potential VE to V, = $ AiTN(N+2), we must have { K ( l ) , . . . , K(N/2)} = {1,2,. . . ,N/2} or a permutation thereof. Since 1411 represent compositeness scales in the effective theory Eq. (2) quantum fluctuations of 41 may not be further away from the mass-shell than 1411. From

we then conclude that $1 does not fluctuate quantum mechanically, and since XE E ~ T T / I ! E is substantially larger than unity (see below) we also conclude that thermal fluctuations of are negligible. Interactions between THC a t the domain boundaries are taken into account macroscopically by using the rigid configuration $ as a background in the macroscopic equation of motion D,G,, = 2ie[$, D,$]. On the macroscopic level, there should be not net field strength associated with a thermal ground state, and thus we are interested in a solution with G,, = 0. Such a solution (with Dp$ = 0) exists:

41

0 0 0 2x1 0

A1

=

T

-TJoj34 e

0

0

..

..

*.

'.. (7)

350 For SU(2) and K(1) = 1 the ground-state configuration a; implies a Polyakov loop P = -1. To asign (T dependent and 7 independent) masses rn; = -2e2 tr [d,t k ] [ 4t,k ] t o the fluctuations a: a singular gauge transformation to unitary gauge must be performed 14. This implies the change P = -1 -+ P = +1 for SU(2). It is important t o note that this gauge transformation does not change the periodicity of the fluctuations u p , and thus it is irrelevant whether one integrates out up in winding or unitary gauge (loop expansion of thermodynamical quantities) c : the two ground states P = f l # 0 are physically equivalent and signal deconfinement. Another important observation is that by virtue of Do$ = 0 the vanishing energy density (pressure) of a hypothetical ground state, composed only of noninteracting THC, is shifted to (-)VE by THC interaction a t the domain boundaries and by the presence of magnetic monopoles. As a consequence, the covariant BPS equation DT+ = V E is not satisfied by the above configurations: a fact which expresses the incompleteness of the ground-state thermodynamics (gauge-field excitations are emitted and absorbed by the ground state). Knowing the quasiparticle spectrum and the ground-state pressure, an expression for the total pressure P in one-loop approximation can be derived. It turns out that the part induced by quantum fluctuations can be neglected 1 4 . To assure consistent thermodynamics (as in the underlying theory) the requirement of stationarity of the pressure under variations of thermal masses needs to be implemented: 8,P. Here the mass parameter a is defined as a = 2 ~ e X , ~ / The ~ . requirement &P enforces an evolution of the effective gauge coupling with T which, in implicit form, is governed by the following equation

where a k = ck a , the numbers ck are derived from the mass spectrum of the gauge modes 14, and the function aD(a) is the derivative of the thermal component of the one-loop pressure for a bosonic field of mass T a . It can be shown l5 that the two-loop corrections t o P self-consistently are <2% of the one-loop values despite the fact that the effective gauge coupling e is larged, for N=2,3 we have ePlateau = 17.15,9.9, respectively. The righthand side of Eq. (refeeq) has zeros a t a = 0, 00, implying the existence of a =The latter gauge choice is obviously much more handy. dThe above kinematical constraints (existence of compositeness scales) imply this result.

351 highest and lowest attainable temperature in the electric phase. 2. Results

In Fig. 1 the evolution of e with temperature is shown. The strong rise of e e(hE)

10000

logarithmic divergences

6000

0

200

400

600

800

1000 J-E

Figure 1. The evolution of the gauge coupling e in the electric phase for N=2 (thick grey line), N=3 (thick black line), N=4 (dashed line), and N=10 (thin solid line). The gauge coupling diverges logarithmically, e cc - bg(xE - &E), a t A E , ~= (9.92,6.9,8.6,7.87}.

in the vicinity of the temperature Tp, where THC are assumed to ‘condense’ and e is close t o the fundamental gauge coupling, makes this assumption self-consistent. It can be shown that the plateau value of e, which signals the existence of isolated and conserved magnetic charge e , is independent of Tp if Tp is sufficiently larger than A E 14. So UV physics decouples from IR physics in the effective thermal theory: a result which is also observed in T = 0 perturbation theory as a consequence of the renormalizability of the underlying theory. At a temperature T E ,e~diverges logarithmically, e c( - lOg(xE - & E ) . It follows that gauge modes, which are massive on tree level, decouple (rn elq5l), magnetic monopoles condense ( m screening masses vanish 14, and the theory settles into a new (magnetic) phase with reduced gauge symmetry. This transition is of second order since the modes that are massless on tree level in the electric phase turn

-

- ?),

eThis agrees with the magnetic quasiparticle model l 2 where the spatial string tension is shown to be saturated by a dilute gas of screened magnetic charges.

352

into dual modes whose thermal mass increases continuously as T is lowered 14. This is a consequence of the continuous behavior of the magnetic gauge coupling g which is allowed since local magnetic charge conservation is violated in a condensate of magnetic monopoles, for the construction of an effective theory for the magnetic phase see 14. At T, < T,,E a nonthermal phase transition t o the confining phase takes place that is driven by the condensation of center vortices. The classical action of the latter vanishes at this point, see 14. This phase transiton is characterized by a complete decoupling of all gauge modes. For SU(2) we thus obtain a ‘decoupling’ of Wf bosons at T,,E and a ‘decoupling’ of a 20boson at T, f . No fundamental Higgs field is needed to generate a triplet of heavy vector bosons which occurs if magnetic and electric phase coexist in separate spatial regions. The confining phase is characterized by a vanishing ground-state pressure, for a construction see 14, whose excitations are intersecting and nonintersecting center-vortex loops with an equidistant mass spectrum set by the YangMills scale AYM g. Due to the existence of an over-exponentially growing density of state (proven by counting the number of vacuum diagrams in a $4 theory a t a given number N of vertices) in the confining phase there is a Hagedorn temperature TH (expressing the fact that the theory generates a mass scale dynamically) for the thermodynamics in the confining phase. TH can only be exceeded if the spatial homogeneity of the system is sacrificed (nonthermal phase transition). As a consequence, thermodynamics in the nonconfining phases is disconnected from thermodynamics in the confining phase, and the relation dP = S d T , which predicts a monotonous behavior of the pressure in a homogenous system, is violated close to T,, see Fig. (3). Let me now present results for the temperature dependence of thermodynamical quantities. In Fig.2 a comparison is made between the SU(3) lattice results of Ref. l7 (differential method, universal part of perturbative beta function) for the normalized entropy density and the results of our one-loop calculation. Since the entropy density is an infra-red safe quantity (there is no ground-state contribution) the use of a perturbative beta function in the lattice simulation should be justified. The quantitative

&

-

‘In reality, where there is not only a single SU(2) gauge theory, the electric and magnetic Tc, are not infinite but possibly of the order e, g lo6 (electroweak couplings e, g at T c , ~ theory)). The large value of the coupling constant g would then parametrize the mass hierarchy m,, : me : mz, N 9-l : go : gl,where me is set by the Yang-Mills scale AYM see below. gAn electron neutrino would be a single loop, an electron two loops with an intersection which provides for the charge 16.

353

agreement between lattice and our result is quite striking. The jump in S , which in our calculation is due to the electric-magnetic phase transition, was interpreted in l7 as the transition to the confining phase. The latter transition, however, happens in our approach at the point where S vanishes. At this point the equation of state is p = -P. The relative ‘duration’ of is about 0.9 (0.34) for SU(2) (SU(3)). the magnetic phase A M T 3 In Figs. 3 and 4 the T dependence of the pressure P and the energy density

Tc$LTc

Figure 2. $ as a function of p obtained in SU(3) lattice gauge theory using the differential method and a perturbative beta function and analytical result. The simulations were performed on (a) 163 x 4, (b) 243 x 4, (c) 163 x 6 (open circles) and 203 x 6 (closed circles), and (d) 243 x 6 lattices. Using the 243 x 6 lattice, the critical value of lo is between 5.8875 and 5.90.

p are depicted. The Stefan-Boltzmann (SB) limit is reached rather quickly, albeit with an extra polarization for slightly massive gauge bosons. The fast approach to SB was also observed in lattice simulation using the differential

354 method l8 h . In contrast, the convergence to SB seems to much slower in lattice results which used the integral method 19. For temperatures not far above T, there is even a qualitative disagreement with the integral method. We obtain negative values of P for SU(3) while only positive values were obtained throughout the deconfining regime in 19. However, the using the differential method (perturbative beta function) negative values of P were obtained close to T, 18. Both energy density and pressure should zero a t T = 0. For a more detailed comparison of lattice and analytical results see 14

Pt -2.5 -5 -7.5 -10

-12.5 -15 -17.5

10

15

20

25

30

hE

&

as a function of temperature for N=2,3. The horizontal lines denote the Figure 3. respective asymptotic values. For N=2 we obtain X E , ~= 9.92 and for N=3 X E , ~= 6.9.

7

6 5

4 3 2

10

15

20

25

30

hE

&

as a function of temperature for N=2,3. The horizontal lines denote the Figure 4. respective asymptotic values.

hFor SU(2) P is within 2% of the SB value for T

N

18Tc.

355 3. Conclusion & Outlook In this talk I have outlined an analytical, niacroscopic approach to SU(N) Yang-Mills thermodynamics which is based on the assumption that trivialholonomy calorons effectively form an adjoint Higgs field a t asymptotically high temperatures. I hope that this assumption will soon prove to be superfluous (availability of solutions to the gap equation Eq. (1)). The main results of our approach are: the existence of three phases - two deconfining phases (electric, magnetic) and the confining phase and disconnected thermodynamics of the two former as compared to the latter phase, the generation of negative pressure for temperatures not far above T,, zero ground-state pressure in the confining phase (due t o a spontaneously broken, local and magnetic 2, symmetry 14), a converging loop expansion of thermodynamical quantities despite a large value of the effective gauge coupling in the electric phase, and the prediction of a large hierarchy in the mass of excitations. For SU(2) the pattern of gauge symmetry breaking is SU(2) -+ U(1) -+ Z y -+ nothing, for higher N intermediate phases with unbroken nonabelian symmetry are possible. The above results together with the higgsparticle-free breakdown of S U ( 2 ) suggest that the existence of the first lepton family of the Standard Model and the known weak interactions of these leptons are the result of nonperturbative gauge dynamics subject to a single SU(2) group.

Acknowledgments The author would like t o thank the organizers for providing the conditions for constructive, stimulating discussions and for their financial support.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

V. L. Ginzburg and L. D. Landau, JETP 20, 1064 (1950). A. A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957). R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965). V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Phys. Lett. B 166, 329 (1986). B. J. Harrington and H. K. Shepard, Phys. Rev. D17,105007 (1978). A. D. Linde, Phys. Lett. B 108, 389 (1982). T. W. B. Kibble, J. Phys. A 9, 1387 (1976). W. Nahm, Phys. Lett. B 90, 413 (1980). W. Nahm, Lect. Notes in Physics. 201, eds. G. Denaro, e.8. (1084) p. 189. T:. C. Kraan and P. van Baal, Nucl. Phys. B 533,627 (1998). T. C. Kraan and P. van Baal, Phys. Lett. B 435,389 (1998).

358 10. D. J. Gross, R. D. Pisarski, and L. G. Yaffe, Rev. Mod. Phys. 53,43 (1981). 11. D. Diakonov, N. Gromov, V. Petrov, and S. Slizovskiy, hep-th/0404042. 12. P. Giovannangeli and C. P. Korthals Altes, Nucl. Phys. B 608,203 (2001). 13. Ch. Hoelbing, C. Rebbi, and V. A. Rubakov, Phys. Rev. D63,034506 (2001). 14. R. Hofmann, hep-ph/0404265. 15. U. Herbst, R. Hofmann, J. Rohrer, work in progress. 16. R. Hofmann, hep-ph/0312051. 17. F. R. Brown et a]., Phys. Rev. Lett. 61,2058 (1988). 18. Y. Deng, in BATAVIA 1988, proc. LATTICE 88, 334. J. Engels et al., Phys. Lett. B 252, 625 (1990). 19. G. Boyd et al., Phys. Rev. Lett. 75, 4169 (1995). G. Boyd et al., Nucl. Phys. B469, 419 (1996).

EXOTIC SUPERFLUIDS: BREACHED PAIRING, MIXED PHASES AND STABILITY *

E. GUBANKOVA Center f o r Theoretical Physics, Department of Physics, MIT, Cambridge, Massachusetts 02139

We review properties of gapless states. We construct a model where a stable breached pair (gapless) state is realized.

1. Introduction

Motivated by recent experiments in cold atoms and by questions in QCD at high densities 2 , we consider here superfluid fermion systems, in particular exotic superfluids. As suggested by Liu and Wilczek, exotic phases of matter in superfluid fermion systems involve coexistence of normal Fermi liquid and superfluid components 3. Superfluid properties are described by a nonzero condensate, A = ( T ( ~ t ( x ) $ t ( z ‘ )#) 0, ) being the order parameter, while a single quasiparticle dispersion crosses the momentum axis (free Fermi surface) leading to a gapless mode; thus this phase is sometimes called a “gapless” superconductive phase. Due to nonzero Fermi condensate the ground state is a superfluid in a classical sense ( i e . it has zero viscosity). On a microscopic scale, one can envision a momentum separation in exotic superfluids. For species with noticeably different Fermi momenta, Cooper pairing takes place around the Fermi surfaces, but there is no pairing in the momentum region between surfaces (the breach). Thus in this work we use the term “breached pair” (BP) superfluidity. To obtain exotic superfluids we consider pairing between two different fermion species whose Fermi surfaces do not match. This possibility arises in several situations: (1) Spin-up spin-down electrons in an ordinary superconductor placed in a uniform magnetic field undergo Zeeman splitting, leading to a mismatch in Fermi momentum. As found by Sarma in the ‘This work was performed in collaboration with M. Forbes, W. V. Liu and F. Wilczek.

357

358 1960’s 4 , an exotic superconducting ground state should arise at large momentum mismatch when ~ B > H A. Before that, however, the first-order phase transition from the superconducting state to the normal state takes place at p g H = A/&. Placing a superconductor in a spacially varying magnetic field or adding paramagnetic impurities with a strong spin-flip electron-impurity scattering amplitude stabilizes the gapless superconductor ’. ( 2 ) Recent experiments in cold atomic fermion gases trapped in an optical lattice and operating near Feshbach resonance deal with a mixture of two hyperfine spin components of alkali atoms. By changing the scattering length one can go from the regime of Bose-Einstein condensation to BCS superfluidity, which is of interest in this work. Laser lattice involves counterpropagating laser beams, that together generate a standing light wave leading to different AC Stark shifts for the spin-up and spin-down components. Using methods of ’engineering’ various lattice systems and by tuning effective masses one can produce exotic phases 6 . (3) In strongly interacting quark matter at high baryon densities and low temperatures, different flavors of quarks pair and form color superconductors 2 . Here a mismatch in Fermi momenta arises due to a nonzero strange quark mass, m, # 0, and is triggered by imposing a charge neutrality condition. At intermediate density (2-3 nuclear densities), an exotic state may arise which links the CFL and nuclear matter phases. Plan: First, we give a general analysis of gapless states. Then, we show how to realize a stable BP superfluid state. 2. Gapless state and its stability

We consider the mean-flield analysis of a model with two species of fermions A, and B of differing masses mA < mB and in the absence of interaction with different Fermi momenta P A < p g . The Hamiltonian is H = d3p/(2~)3 H I with attractive interac-

( ~ $ $ f i +~ $ ~ ~

+

tion H I = - g S d 3 p / ( 2 n ) 3 d 3 q / ( 2 , ) 3 $ a p $ ~ - p $ B s $ ~ - q ( g > 0) and E;;? = p 2 / 2 m -~P A , E: = p 2 / 2 m B - P B , where P A = d G ,P B = JG At the mean-field level, the condensate A = s d 3 p / ( 2 n ) 3 ( $ ~ p $ ~ - pis) a c-number, which permits to diagonalize the Hamiltonian. As a result, the

&--+

EP~

= (E; Z ~ E ; ) / Z ; quasiparticle excitations are E; = E; f E + ~ A2 with they contain mixture of A-particle and B-hole excitations. We minimize the thermodynamic potential s2 = H - P A n A - PBrLB, i.e. %l/i3A = 0, to find the gap parameter A. There are two non-trivial solutions. First one with larger gap corresponds to a fully gapped BCS state, where E; have oppo-

359

n

n

Figure 1. Dispersion relations and occupation numbers for the BCS (left) and Sarma (right) states.

site signes for all momenta. Though nA # nB at g = 0, in the presence of interaction particles redestribute so that the occupation numbers become

(

7)

n with n = 1/2 1- E : / cPf2 + A 2 ; the BCS condensation energy is the largest for $ A = $B. Second solution, obtained by Sarma in ~ O ’ S has , smaller gap. Interaction is not strong enough to pull both Fermi surfaces together, leaving a breach region with single occupancy by Bparticles. Pairing and superfluidity takes place primarily around the smaller Fermi surface, while there is normal component for momenta in the breach region where E: have the same signs (separation in momentum). Points where E; = 0, i.e., p f , 2 = ( p i + p i ) / 2 f 1 / 2 d ( & - p i ) 2 - 1 6 r n ~ r n ~ A ~ , give the free Fermi surfaces leading to gapless modes and to free fermion liquid component. Occupation numbers are j i = ~ j i = ~ n for 0 I p I p l and p z I p , and jig = 1 j i = ~ 0 for p l I p 5 p 2 , Figure 1. We consider nontrivial superfluid solutions of the gap equation in grand canonical and canonical ensembles. I. Grand canonical ensemble. We fix the chemical potentials P A , P B and minimize the thermodynamic potential 0 s = H - PAnA - PBnB over all ground state superconducting wave functions, rnin(9slR)Qs). We obtain, apart from the trivial solution A = 0, two nontrivial solutions: the fully gapped BCS and the gapless equal

j i = ~ j i = ~

360

Figure 2. Solutions of the gap equation and corresponding pressures a s functions of the Fermi momenta mismatch at fixed chemical potentials (left), and gap equation solutions and energies at fixed particle numbers (right).

',

Sarma solution As a function of the Fermi momenta mismatch Sp = (p; - p i ) / 4 d m , they are A ~ s = c A0 and As,,.,,/Ao = J 2 x with x = S p / A o , Figure 2 . Since at a given bp and 9, ABCS> A S a r m a , BCS state wins, i.e., system prefers the BCS ground state over the Sarma state. At fixed Fermi momenta p ~ , pthe ~ ,thermodynamic potential as a function of the gap has two minima - normal (N) and the BCS (absolute min) states, and one maximum - the Sarma state. Hence the Sarma state is metastable. We consider the pressure versus the Fermi momentum mismatch for states which are solutions of the gap equation, where normalized pressure is defined through the condensation free energy as Ps = - ( ( a s ) (Ro))/l( ( 0 ~ ~ (Ro))~ s ) where ( 0 0 ) is the free energy of the normal state at Sp = 0 and the BCS condensation energy is (OBCS)- (no)= - N ( O ) A ; / 2 , A0 is the BCS gap and N ( 0 ) is the density of states at the Fermi surface. In the leading order A Sp << p ~ , (Le., p ~when all quantities are written as expansions near the Fermi surface), pressures are PBCS= 1 for 0 5 x 5 1, Ps,,,, = 2x2 - (1 - 2 ~ for) 1 /~2 5 x 5 1 and PN = 2x2 for x 2 0 with x = S p / A o , Figure 2. State with maximum pressure (or minimum condensate energy) wins. For P A = p~ there is the BCS state. As we add B-particles, we create stress, and pressure of the BCS state PBCS N

361 drops relatively to the pressure of the normal unpaired state PN 6p2. When PBCS - PN 5 0, there is a first order phase transition at A # 0 from the superconducting to the normal state. The BCS superconductor is destroyed when win from condensation energy besomes less than loss in energy needed to pull the Fermi surfaces together to create the BCS. Sarma state is tangent to normal state at A = 0, and has always lower pressure than normal state; hence Sarma state is unstable. 11. Canonical ensemble. We fm the particle numbers n A , n B , allowing the chemical potentials to change, and minimize the Helmholtz energy over all possible superconducting ground states with constraint of fixed ni i = A , B , min(QsIHIQS)ni-const At a single point when n A = n B there is the BCS state with ABCS = A,. At n A # n B , there is Sarma state with decreasing gap as bn = ng - n A increases, ASarmalAO = v’i-75 where x = Sp/& 6n. The energy of Sarma state is lower than the energy of the normal state, Esarma< E N ;thus Sarma state is stable. Imposing neutrality condition a stable gapless superconducting state was obtained in the QCD context by Shovkovy et. al. ’. We came to different conclusions about stability of Sarma state using grand canonical and canonical ensembles. This difference in stability analysis can be resolved by considering mixed phase, which is a mixture in space of two (or more) homogeneous states. Bedaque et. al. lo suggested to consider a mixture of the BCS and normal states, separated in x-space. They found that the energy of the mixed state is lower than the energy of Sarma state, Emixed < Esarma;thus Sarma state is unstable with respect to decay into a mixed state. We confirm their findings. We define the normalized condensation energy as Es = ( ( H s ) - ( H N ) ) / I ( ( H B C S )- ( H N ) )where ~ ( H B C S )- ( H N ) = -N(O)Ag/2. In the leading order A 6p << p ~ , p ~ , condensation energies are E B C = ~ -1 for x = 0, Esarma = -(1 - 2 ~ for) ~ 0 5 x 5 112, Emixed = -(1 - f i x ) 2 and EN = 0, x = 6 p / & and A0 is the BCS gap, Figure 2. Allowing mixed states in our ansatz of trial ground state wave functions Qs, metastable Sarma state decays (rolls down) into a mixture of the BCS and normal states, Figure 3. (In a mixed state, pressures and chemical potentials of composite states are equal, hence there are two equal minima RBC= ~ O N ) . Generally, it is difficult to include mixed states in variational ansatz in the grand canonical ensemble. We conclude, that in grand canonical and canonical ensembles Sarma branch of the gapless solutions is unstable. N

’.

N

N

362

I

H-bun

H-\mu n

-1Td BCS

Gap

Gap

Figure 3. Thermodynamic potential as a function of the gap, and positions of each state. Sarma state decays into a mixed BCS and Normal states depicted on the right panel.

In 11, we show that conclusion about stability of a state is the same in any ensemble used. In particular, there is one-to-one correspondence between a state at fixed particle number(s) and the state that minimizes the thermodynamic potential R in grand canonical ensemble. Thus, there is always a stable state in the grand canonical ensemble that satsfies the constraint. Imposing constraint (over particle numbers) cannot stabilize the system. Practical guide is to look for a stable (gapless) solution in the grand canonical ensemble.

3. Breached paired superfluid state for a finite-range interaction Our goal is to construct a stable breached paired state which has coexisting superfluid and gapless components. We use an idea that existance of the gapless modes depend on the momentum structure of the gap A@). There should be two distinct regions in momentum space: first one where A p is large enough to support the superfluid, and second one where Ap is small enough that pairing does not appreciably affect the normal free-fermion behavior. To garantee stability, the phase must also have higher pressure than the normal state. We realize such BP states in two examples ll. I. Cut-off interaction. We impose a cut-off interaction such that it supports the BCS-like pairing for p < p~ and it allows free dispersion relations for p > PA, accomodating the excess of B-particles and leading to gapless modes. We construct this state by minimizing the thermodynamic potential, rnin(8slH - p ~ -npgnglQs), ~ where H includes the cut-off interaction -9 d 3 p / ( 2 n ) 3 d 3 4 / ( 2 ~ ) 3 f ( P ) f ( q ) ~ ~ p ~ ~ with - p ~ Bfb) - q ~= ~ q1 for p > p~ and f(p) = 0 for p 5 PA. The gap parameter, defined

363

Gap

20 E

30 E

n

n

nb

nb

Figure 4. Gap, dispersions and occupation numbers for the cut-off (left) and the twobody potential (right) interactions.

as A

s

=

g s d 3 p / ( 2 n ) 3 f ( p ) ( $ ~ p $ ~ - psatisfies ), the gap equation A =

1/29 d 3 q / ( 2 n ) 3 A f ( q ) / d G where momentum integration is performed outside the breach region. The occupation numbers n A , TZBshow the evidance that it is a breached paired state, Figure 4. This state is an absolute minimum of the thermodynamic potential, hence we obtained a stable BP state. 11. Spherically symmetric static two-body potential. With attractive potential V ( x - d), interaction is HI = J d 3 p / ( 2 n ) 3 d 3 q / ( 2 n ) 3 V ( p- q ) $ l p $ L . - p $ B - q $ A q and the gap parameter acquires a momentum dependence, Ap = d 3 q / ( 2 n ) 3 V ( p- Q ) ( + B ~ $ A - ~ ) .

s

The gap equation is written Ap = 1 / 2 J d ’ q / ( 2 ~ ) ~ w (-p q ) A , / J G ,

+

and quasiparticle dispersion relations are Epf = E; f JE$ A;. We take a gaussian potential for numerical simulations. Due to the BCS instability,

364

A p picks at the effective Fermi surface given by the pole of the gap equation at A = 0, = 0. Therefore A p supports the BCS-like pairing around PO, and allows free dispersion relations, and hence free Fermi surfaces, outside the breached region, Figure 4. It is, however, difficult to varify that this state is an absolute minimum of the thermodynamic potential since instead of a number, A, we have a function Ap in the variational ansatz. We performed minimization of the thermodynamic potential numerically using different potentials. Generally, there is a central strip of fully gapped BCS phase about P A = p~ with normal unpaired phase outside. Depending on the parameters of interaction, these phases may be separated by a region of gapless BP superfluid phase, Figure 5. Conditions to have BP phase are as follows. At P A = p g there is standard BCS, which is a stable fully gapped solution. By adjusting the chemical potentials so as to increase the Fermi surface p g , we stress the system and low the pressurerelative to the normal phase. Eventually, either before or after a transition to a BP state, the pressure becomes negative and there is a first order phase transition t o the normal phase. At the point just before transition: if Ape is sufficiently large, the state is fully gapped (BCS) and no BP state will occur; if Ape is small, then it will not appreciably affect the dispersions and one finds a gapless Fermi surface coexisting with the superfluid phase. As long as A p falls off sufficiently quickly, one can choose large ratio mB/mA >> 1 so that the transition will occur with Ape small enough to support the BP phase. States shown at Figure 5 have mg/mA = 10. For a wider in q-space interaction, larger mass ratio is needed.

&to

I

Pb

I

I

pb N

I

/

20

Figure 5. Phase diagram of possible homogeneous phases in coordinates of the Fermi momenta (PA,p ~ for) the cut-off (left) and two-body potential (right) interactions.

365 4. Conclusion

We considered a Fermi system with weak attractive interaction between species A and B, where the BCS state forms at equal number densities, = n B . What is the ground state of this system when n A

# nB?

There is a range of parameters, where a breached pair phase exist. Breached pair state is a homogeneous phase where superfluid and normal components coexist. This state is stable and can be found provided there is a momentum structure of interaction and large enough mass ratio of two species. nA

Acknowledgments This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DF-FC0294ER40818.

References 1. For a review, see Nature 416, 205 (2002). 2. For a review, see K Rajagopal and F. Wilczek, hep-ph/0011333. 3. W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002 (2003), condmat/0208052. 4. G. Sarma, Phys. Chem. Solid 24, 1029 (1963); A. A. Abrikosov, “Foundations of the theory of metalls”. 5 . A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, “Methods of quantum field theory in statistical physics”, Dover Publications, Inc., New York. 6. W. V. Liu, F. Wilczek, and P. Zoller (2004), cond-mat/0404478. 7. M. Alford, C. Kouvaris, and K. Rajagopal (2003), hep-ph/0311286. 8. E. Gubankova, W. V. Liu, and F. Wilczek, Phys. Rev. Lett. 91,032001 (2003), hep-ph/0304016. 9. I. Shovkovy and M Huang, Phys. Lett. B564,205 (2003), hep-ph/0302142. 10. P. F. Bedaque, H. Caldas, and G Rupak, Phys. Rev. Lett. 91, 247002 (2003), cond-mat/0306694. 11. M. M. Forbes, E. Gubankova, W. V. Liu, and F. Wilczek, hep-ph/0405059.

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SECTION 5. TOPOLOGICAL FIELD CONFIGURATIONS

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QUANTUM WEIGHTS OF MONOPOLES AND CALORONS WITH NON-TRIVIAL HOLONOMY *

DMITRI DIAKONOV Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA NORDITA, Blegdamsuej 17, DK-2100 Copenhagen, Denmark St. Petersburg Nuclear Physics Institute, Gatchina, 188 300, St. Petersburg, Russia

Functional determinant is computed exactly for quantum oscillations about periodic instantons with non-trivial values of the Polyakov line at spatial infinity (or holonomy). Such instantons can be viewed as composed of the BogomolnyiPrasad-Sommerfeld (BPS) monopoles or dyons. We find the weight or the probability with which dyons occur in the pure Yang-Mills partition function. It turns out that dyons experience quantum interactions having the familiar “linear plus Coulomb” form but with the “string tension” depending on the holonomy. We present an argument that at temperatures below the critical one computed from A,,, , trivial holonomy becomes unstable, with instantons “ionizing” into separate dyons. It may serve as a microscopic mechanism of the confinement-deconfinement phase transition.

1. Introduction

Several years ago a new self-dual solution of the Yang-Mills equations at non-zero temperatures has been found independently by Kraan and van Baal [2] and Lee and Lu [3]. I shall call them for short the KvBLL calorons. In the case of the simplest SU(2) gauge group to which I restrict myself in this paper, the KvBLL caloron is characterized by 8 parameters or collective coordinates, as it should be according to the general classification of the self-dual solutions with a unity topological charge. The most interesting feature of the KvBLL calorons is that they can be viewed as composed of two “constituent” dyons; one is the standard BPS monopole [4,5] and the other is the so-called Kaluza-Klein monopole [6]. I denote them as M , L dyons; explicitly their fields can be found e.g. in the Appendix of Ref. [7]. *Based on the work in collaboration with Nikolay Gromov, Victor Petrov and Sergey Slizovskiy [ 11.

369

370

The collective coordinates or the moduli spa,ceof the KvBLL caloron can be chosen in various ways, however the physically most appealing choice is the six coordinates of the two dyons’ centers in space 2 1 , 2 , and two compact “time” variables. When the spatial separation of two constituent dyons 7-12 = 1 2 1 - 7 2 1 is larger than the compactification circumference 1/T the caloron action density becomes static and is reduced to the sum of the static action densities of the two dyons. At r l 2 T 5 1 the two dyons merge, and the action density becomes a time-dependent 4d lump, see Fig. 1.

Figure 1. The action density of the KvBLL caloron as function of z , t at fixed x = y = 0, with the asymptotic value of A4 at spatial infinity v = O.QnT,V = 1.lnT. It is periodic 2 in the t direction. At large dyon separation the density becomes static (left, ~ 1 = 1.5/T). As the separation decreases the action density becomes more like a 4d lump 2 0.6/T). In both plots the dyons are centered at 21 = -vr12/2nT, z2 = (right, ~ 1 = vr12/2nT, % i , z = y i , z = 0. The axes are in units of temperature T.

We use the gauge freedom to choose the gauge where the A4 component of the Yang-Mills field is static and diagonal at spatial infinity, A4 -+ ~ T ~ vvE, [0,27rT].The Polyakov line or the holonomy at spatial infinity is then -1T r L = - T1r P e x p 2 2

(

ii1’TdtA4)

4 C O SV 5

E [-1,1].

(1)

At the end points (v = 0, 2x7’) the holonomy belongs to the group center, iTrL = fl,and is said to be ‘trivial’. In this case the KvBLL caloron is reduced to the standard periodic instanton, also called the HarringtonShepard caloron [8]. It has been known since the work of Gross, Pisarski and Yaffe [9] that gauge configurations with non-trivial holonomy, iT r L # f l , are strongly suppressed in the Yang-Mills partition function. Indeed, the 1-loop effective action obtained from integrating out fast varying fields where one keeps all

371

powers of

A4

but expands in (covariant) derivatives of

A4

has the form [lo]

~~(2nT-v)' mod 27rT

where the perturbative potential energy term P(A4) has been known for a long time [9,11],see Fig. 2. The zeros of the potential energy correspond to 4Tr L = fl,ie. to the trivial holonomy. If a dyon has v # 27rTn a t spatial infinity the potential energy is positive-definite and proportional to the 3d volume. Therefore, dyons and KvBLL calorons with non-trivial holonomy seem to be strictly forbidden: quantum fluctuations about them have an unacceptably large action. PT '

Figure 2. Potential energy as function of v/T. Two minima correspond t o $TrL = fl,the maximum corresponds to TrL = 0. The range of the holonomy where dyons experience repulsion is shown in dashing.

Meanwhile, precisely these objects determine the physics of the supersymmetric YM theory where in addition to gluons there are gluinos, i.e. Majorana (or Weyl) fermions in the adjoint representation. Because of supersymmetry, the boson and fermion determinants about dyons cancel exactly, so that the perturbative potential energy (2) is identically zero for all temperatures, actually in loops. Therefore, in the supersymmetric theory dyons are openly allowed. [To be more precise, the cancellation occurs when periodic conditions for gluinos are imposed, so it is the compactification in one (time) direction that is implied, rather than physical temperature which requires antiperiodic fermions.] Moreover, it turns out [12] that dyons generate a non-perturbative potential having a minimum a t v = 7rT, i e . where the perturbative potential would have the

372 maximum. This value of A4 corresponds to the holonomy Tr L = 0 at spatial infinity, which is the “most non-trivial” ; as a matter of fact < Tr L >= 0 is one of the confinement’s requirements. In the supersymmetric YM theory there is a non-zero gluino condensate whose correct value is reproduced by saturating it by the L , M dyons’ zero fermion modes [12]. On the contrary, the saturation of the (square of) gluino condensate by instanton zero modes gives the wrong result, namely that of the correct value [13]. Recently it has been observed [7] that the square of the gluino condensate must be computed not in the instanton background but in the background of exact solutions “made of” L L , M M and LM dyons. The first two are the double-monopole solutions and the last one is the KvBLL caloron. As the temperature goes to zero, the L L and M M solutions have locally vanishing fields, whereas the KvBLL LM solution reduces to the instanton field up to a locally vanishing difference. Therefore, naively one would conclude that the dyon calculation of the gluino condensate, which is a local quantity, should be equivalent to the instanton one, but it is not. The fields vanishing as the inverse size of the system have a finite effect on such a local quantity as the gluino condensate! This is quite unusual. The crucial difference between the (wrong) instanton and the (correct) dyon calculations is in the value of the Polyakov loop, which remains finite. In the N = 1 SUSY theory, as one increases the compactification circumference 1/T, the average < A4 >= 7rT at infinity decreases, however the theory always remains in the Higgs phase, in a sense. Instantons do not satisfy this boundary conditions whereas dyons and calorons with the non-trivial holonomy do satisfy them. In the supersymmetric YM theory configurations having Tr L = 0 at infinity are not only allowed but dynamically preferred as compared to those with L = fl.In a non-supersymmetric theory it looks as if it is the opposite. Nevertheless, it has been argued in Ref. [14] that the perturbative potential energy (2) which forbids individual dyons in the pure YM theory might be overruled by non-perturbative contributions of an ensemble of dyons. For fixed dyon density, their number is proportional to the 3d volume and hence the non-perturbative dyon-induced potential as function of the holonomy (or of A4 at spatial infinity) is also proportional to the volume. It may be that at temperatures below some critical one the nonperturbative potential wins over the perturbative one so that the system prefers < Tr L >= 0. This scenario could then serve as a microscopic mechanism of the confinement-deconfinement phase transition [14]. It should be

373 noted that the KvBLL calorons and dyons seem to be observed in lattice simulations below the phase transition temperature [15,16,17]. To study this possible scenario quantitatively, one first needs to find out the quantum weight of dyons or the probability with which they appear in the Yang-Mills partition function. At-the - second stage, one has to consider the statistical mechanics of the L , M , L , M dyons for fixed Ad at infinity and find the free energy of the system as function of v = Finally, one has t o study this free energy as function of v a t different temperatures, to see what value of v (equivalent to the holonomy according to the formula L = cos(v/2T)) is preferred by the theory. Unfortunately, the single-dyon measure is not well defined: it is too badly divergent in the infrared region owing to the weak (Coulomb-like) decrease of the fields. What makes sense and is finite, is the quantum determinant for small oscillations about the KvBLL caloron made of two dyons with zero combined electric and magnetic charges. Knowing the weight of the electric- and magnetic-neutral KvBLL caloron one can read off the individual L , M dyons’ weights and their interaction as one moves the two dyons apart. The problem of computing the effect of quantum fluctuations about a caloron with non-trivial holonomy is of the same kind as that for ordinary instantons (solved by ’t Hooft [18]) and for the standard HarringtonShepard caloron (solved by Gross, Pisarski and Yaffe [9]) being, however, technically much more difficult. I remind the results of the above two calculations in the next two sections. In Section 4 I report on the very recent result for the KvBLL caloron [l].Remarkably, the quantum weight of the KvBLL caloron can be computed exactly. It becomes possible because we are able t o construct the exact propagator of spin-0, isospin-1 field in the KvBLL background, which is some achievement by itself. It turns out from the exact calculation of the KvBLL weight that dyons experience a familiar “linear plus Coulomb” interaction at large separations. That is why the individual dyon weight is ill-defined: their interaction grows with the separation. The sign of the interaction depends critically on the value of the holonomy. If the holonomy is not too far from the trivial such that 0.787597 < iITrLI < 1, corresponding t o the positive second derivative P”(v) (see Fig. 2) the L and M dyons experience a linear attractive potential. Integration over the separation 7-12of dyons inside a caloron converges. We perform this integration in Section 6 assuming calorons are in the “atomic” phase, estimate the free energy of the neutral caloron gas and conclude that the trivial holonomy (v = 0,27~T)is unstable a t temperatures

~~l,l,+m.

iTr

374 below T, = 1.125Rm, despite the perturbative potential energy P(v). In the complementary range $ITrLI < 0.787597, the second derivative P”(v) is negative, and dyons experience a strong linear-rising repulsion. It means that for these values of v, integration over the dyon separations diverges: calorons with holonomy far from trivial “ionize” into separate dyons. 2. Ordinary instantons at

T =0

The usual Belavin-Polyakov-Schwartz-Tyupkin instanton [19]has the field

A, = AEta =

-ip2U[a,(z - z)+ - (z - Z),]U+ , .( - z)2[p2 (z - z)2]

+

of = (l,*i?).

The moduli or parameter space is described by the center z, (4), size p (l), and orientation U (3). The action density TrF$ is O(4) symmetric, see Fig. 3. As it is well known [18],the calculation of the l-loop quantum weight of a Euclidean pseudoparticle consists of three steps: i) calculation of the metric of the moduli space or, in other words, calculation of the Jacobian composed of zero modes, needed to write down the pseudoparticle measure in terms of its collective coordinates, ii) calculation of the functional determinant for non-zero modes of small fluctuations about a pseudoparticle, iii) calculation of the ghost determinant resulting from background gauge fixing in the previous step. In fact, for self-dual fields problem ii) is reduced to iii) since for such fields Det(W,,) = Det(-D2)4, where W,, is the quadratic form for spin-1, isospin-1 quantum fluctuations and D2 is the covariant Laplace operator for spin-0, isospin-1 ghost fields [20]. Symbolically, one can write

J

pseudoparticle weight = d(col1ective coordinates). Jacobian.Det-l(-D2), The functional determinant is normalized to the free one (with zero background fields) and UV regularized by the standard Pauli-Villars method. The l-loop quantum weight of the BPST instanton has been computed by ’t Hooft [18]. If p is the Pauli-Villars mass, i.e. the UV cutoff, and g 2 ( p ) is the gauge coupling given at this cutoff, the instanton weight is

CO

= exp

16 log2 [-9 3 + -9- - - 3

210g(2n)

3

375 The last factor (in the curly brackets) is due to the regularized smalloscillation determinant; all the rest is actually arising from the 8 zero 22

-*

modes. The combination p T e 9 ( P ) = A? is the scale parameter which is renormalization-invariant at one loop. Since the action density is O ( 4 ) symmetric, the quantum weight depends only on the dimensionless quantity p A where p is the instanton size, and even this dependence follows trivially from the known renormalization properties of the theory. Therefore, only the overall numerical constant Co is the non-trivial result of the calculation. The prefactor g(p)-8 is not renormalized at one loop. At two loops the instanton weight can be obtained without further calculations [21] if one demands that it should be invariant under the simultaneous change of the cutoff p and g2(p) given at this cutoff, such that the 2-loop scale parameter

11 , b l = -N 3

b2

34 = -N

3

2

,

remains fixed. The 2-loop instanton weight computed from this requirement is [22]

3. Quantum weight of the periodic instanton with trivial holonomy

The Harrington-Shepard caloron [8] is an immediate generalization of the ordinary Belavin-Polyakov-Schwartz-Tyupkin instanton [19],continued by periodicity in the time direction. The l-loop quantum weight of the periodic instanton with the trivial holonomy has been computed by Gross, Pisarski and Yaffe [9]. The weight is a function of the instanton size p, temperature T and, after the renormalization, of the scale parameter A. In fact, the dependence on A follows from the renormalization properties of the theory, therefore the caloron weight is a non-trivial function of one dimensionless variable, p T . At p T < I it reduces to the 't Hooft's result for the ordinary BPST instanton [18]. At p T > l the caloron weight is [9]

376

0

Figure 3. Action densities of the ordinary BPST instanton (left) and of the periodic Harrington-Shepard instanton with trivial holonomy (right) as function of z, t at fixed 2 = y = 0. The former is O(4) and the latter is O ( 3 ) symmetric. The size of the latter is p = 0.8 2T. ’

The last factor suppresses large calorons: it is the consequence of the Debye screening mass which is nothing but the second derivative of the potential energy P(A4) a t zero. The vacuum made of these calorons was built using the variational principle in Ref. [23]. It turns out that the average of the Polyakov line grows rapidly from 0 to 1 near T M [24], however strictly speaking, there is no mass gap and no confinement-deconfinement phase transition. 4. Q u a n t u m weight of the caloron w i t h non-trivial

holonomy We define the quantum weight of the KvBLL caloron in the same way as it is done in the case of ordinary instantons and the periodic instantons, see eq.(3). The problem of writing down the Jacobian related to the caloron zero modes has been actually solved already by Kraan and van Baal [2]. Therefore, to find the quantum weight of the KvBLL caloron, only the ghost determinant needs to be computed. The KvBLL caloron has only the O(2) symmetry corresponding to the rotation about the line connecting the dyon centers, and the determinant is a non-trivial function of three variables: the holonomy at spatial infinity encoded in the asymptotic value

377 of ~ ~ ~ l = ~v, the l temperature + m T, and the separation between the two dyons 7-12. Computing this function of three variables looks like a formidable problem, however it has been solved exactly in Ref. [l]. We have followed Zarembo [26] and first found the derivative of the determinant Det(-D2) with respect to the holonomy or, more precisely, to v. [The holonomy is $TrL = cos(v/2T)]. The derivative dDet(-D2)/av is expressed through the Green function of the ghost field in the caloron background [26]. If a self-dual field is written in terms of the AtiyahDrinfeld-Hitchin-Manin-Nahm construction, and in the KvBLL case it basically is [2,3], the Green function is generally known [27-291 and we build it explicitly for the KvBLL case. Therefore, we are able to find the derivative dDet(-D2)/dv. Next, we reconstruct the full determinant by integrating over v using the determinant for the trivial holonomy (4) as a boundary condition. This determinant at v = 0 is still a non-trivial function of the caloron size p related t o the dyon separation according to 7-12 = Iz1-221 = np2T, and the fact that we match it from the v # 0 side is a serious check. Actually we need only one overall constant factor from Ref. [9] in order to restore the full determinant a t v # 0, and we make a minor improvement of the Gross-Pisarski-Yaffe calculation as we have computed the needed constant analytically. Depending on the holonomy, the M , L dyon cores are of the size $ and 1 c, respectively, where v = 2nT - v. At large dyon separations, 7-12 >> 1/T, the l-loop KvBLL caloron weight can be written in a compact form in terms of the coordinates of the dyon centers [l]:

= 1.031419972084,

P ( v ) = -v2ij-2, 12.rr2T

P”(v)

=

d2 -P(v). dv2

This expression is valid a t 7-12T>> 1 but arbitrary holonomy, v, V E [0,2.rrT1, meaning that it is valid also for overlapping dyon cores.

378

5. Dyon interaction At large separations, dyons (curiously) have the familiar “linear plus Coulomb” interaction:

V(r12)= r12T22n

(-

4

7-12

37r

+

log [ v ( l - v ) ( 2 ~ 1 2 T ) ~1.946 ]

+.. . ,

v

V =27rT E [0,11.

This interaction is a purely quantum effect: classically dyons do not interact a t all since the KvBLL caloron is a classical solution whose action is independent of the dyon separation. When the holonomy is not too far from trivial, 0.788 < LI < 1, such that P”(v) > 0, dyons inside the KvBLL caloron attract each other, and calorons can be stable. At v -+ 0 this attraction is in fact the wellknown effect of the suppression of large-size calorons owing to the non-zero Debye mass, cf. eq.(4):

iITr

More generally, the coefficient in the linear term is the second derivative of the potential energy P”(v). Therefore, in the complementary range, +ITr LI < 0.788, where the second derivative changes sign, dyons experience a strong linearly rising repulsion, see Fig. 2. For these values of v, integration over the dyon separation diverges: calorons with holonomy far from trivial “ionize” into separate dyons. 6. Caloron free energy and instability of the trivial

holonomy Let us make a crude estimate of the free energy of the non-interacting N+ calorons and N- anti-calorons a t small v < 7rT 1 - - where the integral over dyon separation inside the KvBLL caloron converges:

(

1

03

fugacity C N = exp

drl2

[-VT3F(v,T ) ]

(dyon weight) exp

2

379 where 4x2

V

v =27rT ’

F(u,T ) = - 3- - - v ~ ( ~ - u-)2~<(T,u), is the free energy of the non-interacting caloron gas. 0.025

,._..’.__--’.

-0.025

-0.05

.

I~

-------

Figure 4. Free energy of the caloron gas in units of T3V at T = 1.3A (dotted), T = 1.125A (solid) and T = 1.05A (dashed) as function of the asymptotic value of A4 = v, in units of 2nT.

At high temperatures the free energy is dominated by the perturbative energy P(A4). Calorons are lowering the free energy but their nonperturbative effect is small. Therefore, the minimum corresponds to the holonomy being close to the trivial one. However, below the critical temperature T, = 1.125 A trivial holonomy becomes unstable, and the system rolls to large v where dyons repulse each other. To find the free energy a t large values of v, one has t o consider the -statistical mechanics of the interacting system of L , M , L , M dyons carrying all four possible combinations of the electric and magnetic charges. This has not been done. One can expect, however, that the minimum of the full free energy below T , will occur at v = 7rT corresponding to T r L = 0. Probably it will mean confinement, with the correlator of two Polyakov lines decaying as an exponent of the separation times the second derivative of the free energy a t v = T T ,and with the area law for the spatial Wilson loop determined by the magnetic screening length.

7. Summary 1. The KvBLL caloron is “made of’ two Bogomolnyi-Prasad-Sommerfeld monopoles or dyons and characterized by non-trivial holonomy, f Tr L # 1 , -1 (in SU(2)).It is self-dual, and has one unit of topological charge.

380 2. T h e quantum weight, or t h e probability with which KvBLL calorons appear in the Yh4 partition function has been computed exactly at 1-loop. 3. At large separation of constituent dyons, they experience a linear rising attraction if I T ~ L ~ M 1 or repulsion if f I T ~ L I M 0. 4. At very high temperatures only calorons with trivial holonomy survive (f lTr Ll = 1). At temperatures below critical T, 2~ 1.125R trivial holonomy becomes unstable, and calorons “ionize” into separate dyons. It may be t h a t at this point t h e confinement-deconfinement transition takes place.

+

References 1. D. Diakonov, N. Gromov, V. Petrov and S. Slizovskiy, hep-th/0404042. 2. T.C. Kraan and P. van Baal, Nucl. Phys. B533, 627 (1998), hep-th/9805168. 3. K. Lee and C. Lu, Phys. Rev. D58, 025011 (1998), hep-th/9802108. 4. E.B. Bogomolnyi, Yad. Fit. 24,861(1976) [Sov.J. Nucl. Phys. 24,449 (1976)]. 5. M.K. Prasad and C.M. Sommerfeld, Phys. Rev. Lett. 35,760 (1975). 6. K. Lee and P. Yi, Phys. Rev. D56 (1997), hep-th/9702107. 7. D. Diakonov and V. Petrov, Phys. Rev. D67, 105007 (2003), hep-th/0212018. 8. B.J. Harrington and H.K. Shepard, Phys. Rev. D18, 2990 (1978). 9. D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53,43 (1981). 10. D. Diakonov and M. Oswald, Phys. Rev. D68, 025012 (2003), hepph/0303129; hep-phJ0403108. 11. N. Weiss, Phys. Rev. D24, 475 (1981); Phys. Rev. D25, 2667 (1982). 12. N.M. Davies, T.J. Hollowood, V.V. Khoze and M.P. Mattis, Nucl. Phys. B559, 123 (1999), hep-th/9905015; N.M. Davies, T.J. Hollowood and V.V. Khoze, hep-th/0006011. 13. V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B229, 394, 407 (1983). 14. D. Diakonov, Prog. Part. Nucl. Phys. 51,173 (2003), hep-ph/0212026. 15. R.C. Brower et al., Nucl. Phys. Proc. Suppl. 73,557 (1999), hep-lat/9810009. 16. E.M. Ilgenfritz et al., hep-lat/0402010. 17. C. Gattringer and R. Pullirsch, hep-lat/0402008. 18. G. ’t Hooft, Phys. Rev. D14, 3432 (1976), D18, 2199 (1978) (E). 19. A. Belavin et al. Phys. Lett. 59,85 (1975). 20. L.S. Brown and D.B. Creamer, Phys. Rev. D18, 3695 (1978). 21. A. Vainshtein, V. Zakharov, V. Novikov and M. Shifman, Sov. Phys. Uspekhi 136,553 (1982). 22. D. Diakonov and V. Petrov, Nucl. Phys. B245, 259 (1984). 23. D. Diakonov and A. Mirlin, Phys. Lett. B203, 299 (1988). 24. D. Diakonov and V. Petrov, unpublished. 25. K. Zarembo, Nucl. Phys. B463, 73 (1996), hep-th/9510031. 26. S. Adler, Phys. Rev. D18, 411 (1978); D19, 2997 (1979). 27. P. Rossi, Nucl. Phys. B149, 170 (1979). 28. W. Nahm, Phys. Lett. 90B,413 (1980).

NONCOMMUTATIVE SOLITONS AND INSTANTONS

F.A.SCHAPOSNIK* Departamento de Fisica Universidad Nacional de La Plata C.C.@67 - 1900 La Plata - Argentina E-mail:[email protected]. edu. ar

Noncommutative gauge field theories have a rich variety of soliton and instanton solutions. Appart from smooth deformations of the well-known solutions in ordinary space, there are those existing only when the parameter governing noncommutativity is different from zero. Using the Weyl connection between Moyal product in even dimensional configuration space and operator product in Fock space, I describe in this talk how equations of motion (and in certain cases BPS equations) can be easily solved leading t o new noncommutative vortex, monopole and instanton solutions.

1. Introduction Recent developments in noncommutative field theories1y2were triggered by the construction of noncommutative instantons4 and solitons5, solutions t o the equations of motion or self-duality equations of noncommutative gauge theories (for a complete list of references see for example the review [3] and references therein). The present talk covers this subject describing various vortex, monopole and instanton solutions6i10obtained by exploiting a connection between noncommutative configuration space and Fock space. 2. Connection between Moyal product in configuration

space and operator product in Fock space The use of the Moyal product approach was in part at the root of the revival of activity in noncommutative field t h e ~ r i e s ~particularly ?~, in the unravelling of the connection between string theories and noncommutative Yang-Mills theory. However, since the Moyal product introduces an infinite number of field derivatives, it could in principle make the search of classical *Associated with CICBA.

381

382

solutions of the equations of motion extremely difficult. It is through the use of the Weyl connection that one can avoid this problem, passing from complicated differential equations (of infinite order) t o very simple algebraic equations in Fock space. It was this connection which allowed t o find a plethora of noncommutative soliton and instanton solutions even in cases in which they do not exist in the commutative case. Let us then start by stating the above mentioned Weyl connection. Let us call x p ( p = 1 , 2 , ...d) the coordinates of d-dimensional Euclidean spacetime. Given two ordinary functions in Rd, 4(z) and ~ ( z )their , Moyal product is defined aszo

with 8,” a constant antisymmetric matrix of rank 2r 5 d and dimensions of (length)2. One can easily see that (1) defines a noncommutative but associative product. Moreover, under certain conditions, integration over Rd of Moyal products has all the properties of the the trace in matrix calculus. Denoting with

id, XI = 4(x) * X ( X ) - X ( X ) *

(2)

the Moyal bracket, one can see from (1) that the commutator for two coordinates, d , z2 reads

[d, z2] = ie12 = ie

(3)

The Weyl connection is an isomorphism between the algebra of functions with the noncommutative Moyal product and the algebra of operators on some Hilbert space. To see this, consider (22) as defining the coordinate commutation relation in two dimensional space and introduce complex coordinates

Changing the coordinate normalization,

one ends with noncommutative coordinates satisfying

383 Then h and 6t realize the algebra of annihilation and creation operators. One then considers a Fock space with a basis In) provided by the eigenfunctions of the number operator N ,

Icr = ~ + i i

filn)= nln)

(7)

+

Since N = 6th x (x2 y2)/20 = r2/28 when 0 -+ 0 one can associate configuration space at infinity to n -+ 00 in Fock space. In order to give the precise formula for the Weyl connection, consider a field $ ( z , 2 ) in configuration space and take its Fourier transform

~ ( kI ,) =

1

+

c~.z$(z,2 ) exp ( i ( ~ z k i ) )

(8)

If one defines the associated operator

one can then prove that 040,

v

Operator product

=oqh*x v product

(10)

I

Hence, the complicated star product of fields in configuration space becomes just a simple operator product in Fock space. One can either work using Moyal products or operator products and pass from one language to the other one just by Fourier (anti)transforming the results. 3. Noncommutative field theories

As an example of how a noncommutative field theory is defined through the use of the Moyal product, we write below the action for a massive self-interacting scalar field takes, in the noncommutative case, the form

We will need to couple scalars to gauge fields. Given a gauge connection A, and a gauge group element g E G, the gauge connection should transform, under a gauge rotation as

Ag,(z) = g(x) * A,(x)

2

* 9 - l ( 4 + ;s-W * $dz)

(12)

Note that even in the U(1) case, due to noncommutative multiplication, the second term in the r.h.s. has to be present in order to have a consistent

384

definition of the curvature. Also, the expression for g ( x ) as an exponential should be understood as 1 g(s)= exp,(ic(z)) = 1 k ( s )- -c(s) * E(Z) . . . (13) 2 Accordingly, even in the U(1) case the curvature F,, necessarily includes a gauge field commutator,

+

FPv= a,Av

+

- &A, - ie (A,

* A, - A , * A,)

(14)

and then, as it happens for non Abelian gauge theories in ordinary space, the field strength F,, is not gauge invariant but gauge covariant,

F,,

-+

* F,, * 9

9-1

(15)

However, due to the trace property of the integral, the Maxwell action is gauge invariant. Concerning the coupling of matter to gauge fields, one can write

DE$ = 8,

+ iA, * $

“fundamental”

but also

DE$ = 8, - i$

DEd$ = a,$

* A,

- ie(A,

“anti - fundamental”

* $ - $ * A,)

“adjoint” (16)

Noncommutative non-Abelian gauge theories pose a problem. Consider for example the case of G = S U ( N ) and call ta the Lie algebra generators. If one computes the gauge field commutator

[A,, A,] = A f * ALtatb- A; * AEtbta and uses 2

tatb= 2i fabctC+ --6,bI N

+ 2dabctc

(18)

one easily verifies that the commutator does not belong to the Lie algebra of S U ( N ) but has also a component along the identity,

F,, = (l&’;vta+(2)F,,I

(19)

and hence F,, @ S U ( N ) . Of course, the problem can be avoided if one includes the identitnty among the generators, by considering U ( N )as gauge group.

385 4. Noncommutative solitons

In order to understand the difficulties and richness of noncommutative soliton moduli space, let us disregard the kinetic energy term in action (11) and just consider the scalar potential, 1 x 2 4 ( m / f i ) $the equation for its extrema is given by

V[$*+]= -mZdJ*$- - $ * $ * f $ * $ After the shift $ 4

4 b ) * 4(z) * $(x) = 4 ( x )

(21)

A subset of solutions of this equation can be found from a simpler one, (22) dJo(z) * $o(z) = d J O ( 2 ) which evidently satisfies (21). Although simpler than eq.(21), eq.(22) implies, through Moyal star products, derivatives of all orders and then only a few solutions can be found straightforwardly or with some little work. To obtain more general solutions requires a new angle of attack. A very fruitful approach was developed in [5] by exploiting the Weyl connection. We shall describe this procedure below in a simple two-dimensional example (but any even dimensional space can be treated identically). First, observe that equation (22) can be seen, in Fock space, as that obeyed by a projector In)(nl.Then, a general solution for the minima is p4 = CXnln)(nl

in Fock space :

4=

in configuration space :

C h m z ,2) (23)

with A, = 0 , f l and $; easily constructed through the use of the Weyl connection,

Concerning derivatives entering in kinetic energy terms, they become, in operator language,

A last useful formula for the connection relates integration in configuration space with trace of operators in Fock space:

J dzdy$(z, 1 ~ )

4

2 d TI-o4

(26)

386 From here on we shall abandon the notation 0 4 for operators and just write 4 both in configuration and Fock space.

5. Noncommutative vortices Let us briefly review how vortex solutions were found in the Abelian Higgs model in ordinary space''^^^. The energy for static, z-independent configurations is, for the commutative version of the theory,

Here i = 1 , 2 and hence one can consider the model in two dimensional Euclidean space with

4 = 41 + i&

Di4 = & - ~ A i 4, ’

(28)

The Nielsen-Olesen strategy to construct topologically regular solutions to the equations of motion starts from a trivial (constant) solution and implies the following steps: -Trivial solution

141 = r] , Ai

=0

- Topologically non-trivial but singular solution (fluxon with

N units of

magnetic flux) which in polar coordinates reads

4 = r]exp(iNp) , Ai

= n&p

j ~TNS(~)(?) , ~ i j F i=

- Regular Nielsen-Olesen vortex solution

4 = f ( ~exp(iNp) ) , Ai = a ( r ) & p

(29)

with f(0) = a ( 0 ) = 0 , f ( w ) = r] ,a(oo) = N For the particular value X = X ~ p = s 2 one can establish the Bogomol’nyi bound E 2 ~ T Nattained , when first order “Bogomol’nyi” equations hold

Fzz = q2 - $4

-F

zz-

- q2 - $4

D &= J 0

DZ4= 0

Selfdual

Antiselfdual

(30)

One can copy this strategy, to attack the noncommutative problem. The energy for the noncommutative version of the model is obtained replacing ordinary productos by * products in eq.(27). Taking the scalars in the

387 fundamental representation (but the other possibilities go the same) one gets

+m*

1 x 4-~')~ * Fij Di$+ 2(4* (31) 2 The three steps in Nielsen-Olesen demarche become now(we take N = 1 for simplicity)

E = -Fij

-

ordinary space

-

+

noncommutative space

141 = rl trivial

141 = rlexp(icp) = rl-

0,rtl

z

IzI

singular

regular

Note that in the last two lines we have identified z with ( 1 / 4 ) & . The difference between the two expansions is that in the second one coefficients are f l while in the third one the fm coefficients should be adjusted using the equations of motion (or Bogomol'nyi equations) and boundary conditions. Of course (32) should be accompanied by a consistent ansatz for the gauge field. One easily finds

Differential equations become, in Fock space, algebraic recurrence relations which can be easily solved. The appropriate condition at infinity in configuration space (f(l.1) + 1 as IzI -+ m) translates to fn -+ 1 for n oa in Fock space. Then, using fo as a shooting parameter, one determines fl, fz, . . . so that fm 4 1 for large m and then one can compute the magnetic field, the flux, etc, either in Fock space or, after use of the Weyl connection, in configuration space, -+

For small 8 one re-obtains the Nielsen-Olesen regular vortex solution. Exploring the whole range of the dimensionless parameter 8q2, one finds that the vortex solution with +1 units of magnetic flux exists in all the 8 range. Anti-selfdual (negative flux) solutions can be trivially obtained in the commutative case from selfdual ones, just by making B -B, 4 4 --f

4.

388 Now, the presence of the noncommutative parameter 8, breaks parity and the moduli space for positive and negative magnetic flux vortices differs drastically. One has then to carefully study this issue in all regimes, not only for X = A s p s but also for X # A s p s , when Bogomol'nyi equations do not hold and the second order equations of motion should be analyzed. We give below a summary of the main results Positive flux There exist BPS and non-BPS solutions in the whole range of v28. Their energy and magnetic flux are: For BPS solutions, E ~ p = s 27rq2N , @ = 27rN , N = 1 , 2 , . . . For non-BPS solutions, Enon-sps > 27rq2N , @ = 27rN , N = 1 , 2 , . . . For q28 -., 0 solutions become, smoothly, the known ordinary space regular vortices. In the non-BPS case, the energy of an N = 2 vortex compared to that of two N = 1 vortices is a function of 8. As in the commutative case, if one compares the energy of an N = 2 vortex to that of two N = 1 vortices as a function of X one finds that for X > XBPS N > 1 vortices are unstable (vortices repel) while for X < Asps they attract. Negative flux BPS solutions only exist in a finite range: 0 5 q28 5 1. Their energy and magnetic flux are: E ~ p = s 27rq2N , @ = 27rN , N = 1 , 2 , . . . When q28 = 1 the BPS solution becomes a fluxon, a configuration which is regular only in the noncommutative case. The magnetic field of a typical fluxon solution is B exp(-r2/8). There exist non-BPS solutions in the whole range of 8 but Only for 8 < 1 solutions are smooth deformations of the commutative ones. For 8 4 1 they tend to the fluxon BPS solution. For 8 > 1 solutions coincide with the non-BPS fluxon. 6t7:

-

6. Instantons The well-honored instanton equation

Fpu = &Ppu

(34) was studied in the noncommutative case by Nekrasov and Schwarz4 who showed that even in the U(1) case one can find nontrivial solutions. The approach followed in that work was the extension of the ADHM construction, successfully applied to the systematic construction of instantons in ordinary space, to the noncommutative case. This and other approaches are discussed in [3] and references therein. Here we shall describe a method developed in [9] which has the advantage leading to explicit noncommutative U ( 2 ) multi-instanton solution from which BPS monopole solutions

389 can be also inferred. Our method extends Witten’s approachlg reducing the four dimensional problem to a two dimensional one through an axially symmetric N-instanton ansatz. That is, one passes from 4 dimensional Euclidan space to 2 dimensional space, (d, x 2 ,x3,x4 + r, t ) but this last with a nontrivial metric g i j = r2bij, i , j = 1,2. To proceed in this way, one needs a noncommutative setting for curved %dimensional space, where 8 can in principle depend on x,

[ x i , d ]= ei+) (35) Now, handling such a commutator is not trivial since not all functions &(x) will guarantee a noncommutative but associative product. One can see, however, that associativity can be achieved whenever

Vk@j = 0

(36)

In the present 2 dimensional case, these equations have as solution

with 80 a constant. Then, given the metric in which the instanton problem with axial symmetry reduces to a vortex problem, an associative noncommutative product should take the form

[r,t]= r2e0; all other [., .] = o

(38)

with r and t defining the variables in two dimensional curved space. A further simplification occurs after noting that r*t-t*r=r2eo + t * -r1- - *r1t = e o (39) Then, calling y1 = t and y2 = 1/r we have the usual flat space Moyal product instead of (38). The Bogomol’nyi equations for vortices take the form (1 - ; ( z P)2) D,$J = (1 -(% 1 i ) 2 ) Dz$ 2 1 iF,z = 1 - -[$,$I+ (41) 2 1 iF,z = --[$,$I (42) 2 with z = y1 i y 2 . We can at this point apply the Fock space method detailed above for constructing vortex solutions. In the present case, consistency of eqs.(40)-(42) imply

+

+

+

+

$$=

1

(43)

390 and hence the only kind of nontrivial ansatz should lead, in Fock space, to a scalar field of the form n=O

where q is some fixed positive integer. With this, it is easy now to construct a class of solutions analogous to those previously found for vortices in flat space. It takes the form n=O

.

0-1

One can trivially verify that configurations (45) satisfy eqs.(40)-(42) provided 00 = 2. In particular, both the 1.h.s. and r.h.s of eq.(40) vanish separately. The field strength associated to our solution reads, in Fock space, 1

ZF,, = --2 (lO)(Ol

+. . . + 14 - l ) ( q - 11) E B

(46)

This curved vortex configuration can be related with a 4-dimensional (axially symmetric in time) selfdual configuration of the form The axially symmetric ansatz for the gauge field components is

A', = &(r, t)Q6,cp) , At

= At(r,t ) W , cp)

+ (1+42(.,t))fi(6,cp) A a d ( 1 9 , ' p ) 'p) + (1+ 42(T, t ) )Q.19, 'p) A a,q19, cp)

A 9 = 41(r,t)dofi(.19,cp) A p

= 41(r,t)a,fi(s,

(47)

with 6(6,'p) = (sin 6 cos cp, sin 6 sin 'p, cos 6) One gets for the selfdual configuration +

gt, = B(r)fi ,

Fo, = B(r)sin 6fi

Ft", = B(r) ,

F&, = B(r) sin19

(48)

As before, starting from (46) for B in Fock space, we can obtain the explicit form of B(r) in configuration space in terms of Laguerre polynomials. Concerning the topological charge, it is then given by

1 Q = -tr 32r2

1

d 4 5 ~ p y Fpv a P Fap = IT

s"

-m

d u l l d t B 2 = 2TrB2 = Q 2

391 We thus see that Q can be in principle integer or semi-integer, and this for an ansatz which is formally the same as that proposed in ordinary spacelg and which yielded in that case to an integer. The origin of this difference between the commutative and the noncommutative cases can be traced back to the fact that in the former case, boundary conditions were imposed on the half-plane and forced the solution t o have an associated integer number. In order t o parallel this treatment in the noncommutative case one should impose the condition q = 2N. 7. Monopoles

In ordinary space Manton developed an approach leading from the axially symmetric Witten N-instanton t o a spherically symmetric BPS monopole solution2". This author observed that if one identifies A0 with a scalar 4 in the adjoint and then takes a certain N + co limit that wipes out the dependence on Euclidean time, selfdual equation (34) becomes the BPS monopole equation &21 . .k

Dk4= Fij

(49)

Manton procedure can be extended to the noncommutative case by starting from the instanton solution discussed above. The outcome is a Wu-Yang g = 1 monopole solution with electromagnetic field strength

Details of the derivation and applications can be found in [24]

Acknowledgments

I am grateful t o the organizers of the conference "Continuous Advances in QCD-2004" and especially M. Shifman and A. Vainshtein for their invitation to participate in the conference and present this report. F.A.S is partially supported by UNLP, CICBA, ANPCyT (PICT 03-05179). References 1. A. Connes, M.R. Douglas and A. Schwarz, JHEP 9802, 003 (1998). 2. N. Seiberg and E. Witten, JHEP 9909, 032 (1999). 3. M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73,977 (2001). 4. N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198, 689 (1998). 5 . R. Gopakumar, S. Minwalla and A. Strominger, JHEP 0005, 020 (2000). J. E. Moyal, Proc. Cambridge Phil. SOC.45, 99 (1949).

392 6. G. S. Lozano, E. F. Moreno and F. A. Schaposnik, Phys. Lett. B 5 0 4 , 117 (2001); G. S. Lozano, E. F. Moreno and F. A. Schaposnik, JHEP 0102, 036 (2001). 7. G. S. Lozano, E. F. Moreno, M. J. Rodriguez and F. A. Schaposnik, JHEP 0311, 049 (2004). 8. D. H. Correa, G. S. Lozano, E. F. Moreno and F.A.Schaposnik, Phys. Lett. B515, 206 (2001). 9. D. H. Correa, E. F. Moreno and F. A. Schaposnik, Phys. Lett. B543, 235 (2002). 10. D. H. Correa, P. Forgacs, E. F. Moreno, F. A. Schaposnik and G. A. Silva, arXiv: hep-t h/04040 15. 11. H. B. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973). 12. H. J. de Vega and F. A. Schaposnik, Phys. Rev. D14, 1100 (1976). 13. E. B. Bogomolny, Sou. J. Nucl. Phys. 24, 449 (1976). 14. D. P. Jatkar, G. Mandal and S. R. Wadia, JHEP, 018 0009 (2000). 15. A. P. Polychronakos, Phys. Lett. B B495, 407 (2000). 16. J. A. Harvey, P. Kraus and F. Larsen, JHEP 0012, 024 (2000). 17. D. Bak, Phys. Lett. B 4 9 5 , 2 5 1 (2000). A. Khare and M. B. Paranjape, JHEP 0104,002 (2001); K. Hashimoto and H. Ooguri, Phys. Rev. D64, 106005 (2001); 0. Lechtenfeld and A. D. Popov, JHEP 0111, 040 (2001). D. Tong, J. Math. Phys. 3509 44, (2003); A. Hanany and D. Tong, JHEPO307,037 (2003); K. Furuuchi, Prog. Theor. Phys., 103, 1043 (2000); A. Schwarz, Commun. Math. Phys. 221, 433 (2001); S. Parvizi, Mod. Phys. Lett. A17, 341 (2002); K. Y. Kim, B. H. Lee and H. S. Yang, Phys. Lett. B523,357 (200l);O. Lechtenfeld and A. D. Popov, JHEP0203,040 (2002);F. Franco-Sollova and T. A. Ivanova, J. Phys. A36, 4207 (2003); Z. Horvath, 0. Lechtenfeld and M. Wolf, JHEP 0212, 060 (2002). 18. A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Y. S. Tyupkin, Phys. Lett. B59, 85 (1975). 19. E. Witten, Phys. Rev. Lett. 38, 121 (1977). 20. N. S. Manton, Nucl. Phys. B135, 319 (1978).

BREATHER SOLUTIONS IN FIELD THEORIES

V. KHEMANI Center for Theoretical Physics Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge, MA 02139 E-mail: uishesh@mit. edu

I review breathers - classical solutions that are localized in space and periodic in time - in continuum field theories and discrete systems. I also report on our ongoing work on approximate breathers in several continuum field theories, including the electroweak theory. These approximate breathers challenge the notion of naturalness with their unexpectedly long lifetimes.

1. Introduction

Certain field theories admit solutions to the classical equations of motion that are spatially localized and temporally periodic. These solutions are called breathers (among other things). In the first half of this talk (Sec. 2), I briefly review breathers in one spatial dimension in the sine-Gordon model, the $4 theory and in discrete systems. In the second half (Sec. 3), I report on our1 ongoing work on understanding and demonstrating the existence of approximate breathers in a variety of non-linear field theories, including the physically relevant electroweak theory. An approximate breather is a configuration that locally oscillates and confines energy in the region of oscillation for a period of time many orders of magnitude larger than all scales in the theory. Their existence challenges the notion of naturalness in field theories. On a phenomenological front, if approximate breathers exist in the electroweak theory, then they would create out-of-equilibrium regions in space after the electroweak phase transition and could drive baryogenesis.

393

394 2. Breathers Review

2.1. The sine-Gordon Theory i n 1 + 1 Dimensions I introduce breathers by briefly reviewing them in the sine-Gordon model in 1+1 dimensions. (See Rajaraman’ and references therein for details.) This pedagogical illustration provides a firm foundation for the approximate breathers I present later in the talk. The sine-Gordon breathers manifest themselves as pionic breathers3 in the chiral Lagrangian of low-energy QCD, and may be physically interesting on their own, but I will not discuss this any further. The theory has a real scalar field 4 with the action

where the potential is

The classical equations of motion, obtained by extremizing the action, have several solutions, all of which are known in this “perfect theory”. I focus on only those that illuminate the path to breather solutions. The soliton (antisoliton) solutions 4 = (-)I-- m tan-’ emx (3)

A

interpolate between neighboring vacua (0, f 2 n ) as z goes from --oo to 00. Interactions between solitons and antisolitons are available in analytical form. For example, collisions between a soliton-antisoliton pair are described by the time-dependent solution sinh(mut/dm) (4) JT; uc o s h ( m z / d m ) ’ where u denotes the speed of the soliton and antisoliton in the distant past and future. The only residual effect of the collision is a negative time delay, which indicates that the interaction is attractive. Soliton-antisoliton bound states may be obtained by setting the speed to be imaginary:

4 = 4-

m

tan-’

(5) is real. This gives the family of breather solutions parametrized by

u= i / E ,

where

E

E:

4 = 4-

m

A

tan-’

sin(mt/Jm) cosh (Emz/ d m E

)

‘

395 These are spatially localized and temporally periodic with angular frequency m,

which ranges from 0 to m. The family of breather solutions can be quantized exactly (using the WKB method, for example) to obtain a finite tower of breather states with masses na M B ,= ~ 2Ms sin - , 16 where

x

-1

n = 1 , 2 , . . . < 87r/a,

(9)

and the quantized soliton/antisoliton mass 8m

Ms=-.

CY

The upper limit on n corresponds to W B = m after which we run out of classical breathers to quantize. Note that M B , < ~ 2Ms, which is consistent with the breathers being bound states of a soliton and antisoliton. To summarize, the sine-Gordon theory has a continuous family of known breather solutions, which may be thought of as bound configurations of a soliton and an antisoliton. The breathers are exponentially localized in space and are periodic in time with an angular frequency between 0 and m, where m is the mass of the lightest particle in the theory. They can be quantized to give a finite tower of states. 2 . 2 . The (p4 Theory i n l + l Dimensions

Next I review another well-studied theory in 1+1 dimensions on the route towards more physical theories. It is defined by the symmetric double-well potential

where A denotes the field self-coupling and v denotes its vacuum expectation value. In contrast to the sine-Gordon model, this theory is not integrable and that makes all the difference.

396 The kink (antikink) solutions

mx 2 interpolate between the vacua (h) as x goes from -00 to 00. These are not solitons in the strict sense of the word because they do not emerge unchanged from collisions. The question arises whether the theory admits bound state configurations of kink (antikink) in the form of breather solutions, in analogy with the sine-Gordon theory. Numerical investigations4 of kink-antikink collisions indicate that for small relative velocities, they indeed form a trapped state which is approximately periodic and localized (an approximate breather). For large relative velocities, they reflect inelastically. For intermediate velocities (0.193.. . < u < 0.2598.. . , where u is the speed of the kink and antikink in the distant past), there are windows of alternating trapping and reflection. The existence of the trapped state for soft collisions suggests that if the initial conditions are tweaked just right, then an exact breather may result. Furthermore, a multiple-scale expansion5 for small-amplitude configurations indicates that to all orders in the expansion there is an exact breather solution. However, it turns out that there are no ex& breathers (at least in the small amplitude limit) in the 44 theory. There are non-perturbative effects (beyond-all-orders in the multiple-scale expansion)6 that delocalize the configuration. In the sine-Gordon case all the nonlocal non-perturbative terms are absent. So exact breathers are special to the integrable sineGordon theory and persist only approximately in the less perfect 44 theory. (See Ref.7 for a numerical algorithm to determine whether a scalar field theory supports an exact breather or not.)

4 = (-)w tanh -

2.3. Discrete Breathers

Now I briefly leave the continuum realm and review breathers in discrete systems. This will provide crucial insights into what allows approximate breathers to exist in continuum theories like the 44. Theoretical developments in solid-state physics in the late 1980s led to the realization that localized modes do not have to originate exclusively from extrinsic disorder that spoils a perfect crystal lattice. Intrinsic localized modes (discrete breathers) are typical excitations in perfectly periodic but strongly non-linear systems. This has been confirmed by a flood of experimental observations since the late 1990s (phonon modes and spin waves

397 in solids, photonic crystals, etc.). See Ref.8 for a comprehensive review. In order to probe the mechanism that allows discrete breathers to exist, consider a chain of anharmonic oscillators (with a double-well potential) that are coupled to nearest neighbors with a coupling strength l/(Ax)' . The equation of motion for oscillator 'i' is

(This system has been chosen because it goes to the continuum q54 scalar field theory in the strong inter-oscillator coupling limit.) First consider small amplitude oscillations around q5i = w. Transforming to momentum space, we get the linear spectrum for each oscillator:

where

mzJzxw The above linear spectrum is bounded below by m which corresponds to the mass of phonon excitations. The upper bound

< Jm2 + AX)^

(16) is solely due to the discretization. Now consider large amplitude oscillations around w and choose the inter-oscillator coupling to be small (i.e. large Ax) so that we may treat the oscillator as decoupled. For large enough deviations of $ ~ i towards 0 (the peak of the double-well potential), the fundamental frequency of oscillation, W B , lies below the spectrum of linear oscillations. (As the choice of initial q5i is made closer to 0, the frequency approaches 0.) It is possible to choose an amplitude in such a way that WB lies below the linear spectrum and all harmonics lie above the linear spectrum. This makes the oscillation stable against linear decay (emission of phonons) leading to a discrete breather. The crucial ingredients in the system that allow the breathers to exist are: WL

0

WL

0

WB

0

> m i.e. there is a mass gap in the linear spectrum. < m i.e. the potential is non-linear in such a way

that it supports oscillations with fundamental frequency in the mass gap. 2wB > Jm2 AX)^ i.e. the discreteness of the system imposes an upper bound on the linear spectrum, which allows the harmonics of the non-linear oscillation to lie beyond the linear spectrum.

+

398 I refer to this stabilization scenario as the frequency-mismatch mechanism. 3. Approximate Breathers in Continuum Field Theories

We1 have fruitfully adopted the ideas of discrete breathers into continuum field theory and find that the frequency-mismatch mechanism persists in a variety of models. Consider the strong inter-oscillator coupling limit of the chain of anharmonic oscillator discussed in the previous section. This corresponds to the continuum #4 theory with the equation of motion

4- + A#(& #I/

- w2) = 0 .

(17)

The mass gap in the linear spectrum persists and we have a theory of elementary bosons with mass

rn=dSw.

(18)

However, the linear spectrum has no upper bound now. For small curvature configurations, we may treat #(O,t) as a decoupled oscillator in a double-well potential. Large amplitude oscillation of #(O, t ) has a fundamental frequency, W B , in the mass gap suggesting stability against decay by elementary boson radiation. However, the very non-linearity that stabilizes the oscillation has a destabilization mechanism built in - harmonics of the fundamental frequency must be present and these can couple to linear modes. So, we retain two out of the three crucial ingredients that drive breathers as we transition to the continuum and at best we can expect to find approximate breathers in the continuum. These will not be absolutely stable, but as I show later, they are unnaturally long-lived. This heuristic of frequency mismatch with the linear spectrum does not require solitons to exist in the theory. This allows us to explore a large class of theories with massive particles and appropriate non-linear scalar potentials, including the physically interesting case of gauge theories with the Higgs mechanism of spontaneous symmetry breaking in 3+1 dimensions. In the rest of the talk I demonstrate how these ideas do indeed work and argue that there could be approximate breathers in the electroweak theory. 3.1. 1+1 Dimensions First consider scalar field theories in one spatial dimension with the potential

x ($2 - w2) 2 + -.w2$($ 1 Va($)= 4 2

- w)2 .

(19)

399 This describes a self-interacting scalar with mass

m=J X ' v . We choose the dimensionless vev v to be 1. We also choose X = 1 and SO all dimensionful quantities are measured in units of 6. For Q = 0, the asymmetry vanishes and the potential reduces to the symmetric doublewell, in which case we already know of approximate breathers from soft kink-antikink collisions. In order to emphasize that the frequency-mismatch mechanism of stability does not rely on underlying solitons forming bound configurations, we consider the case of CY > X/4 for which we have an asymmetric single-well at q5 = u and there are no solitonic solutions. We choose CY = XI2 and the initial static configuration

+(x,0) = tanh2(x/2).

(21)

Then we numerically evolve the configuration according to the equation of motion. q5(0,t) starts with a large deviation from Y and so the nonlinear potential could drive a low frequency oscillation (wg < m) with the configuration relaxing to an approximate breather. Since the initial configuration is a n even €unctions of z, it remains even throughout the time evolution according to the equation of motion. So we consider only the positive half-line with vanishing spatial derivative boundary condition at the origin. (In higher dimensions this condition is required for regular configurations.) In Fig. 1 I display some characteristics of the evolution of the initial configuration. In the left panel I plot the value of the field a t the origin as a function of time, towards the end of the time of evolution considered. This oscillates about q5 = u (asymmetrically because the potential is asymmetric). The fundamental period of oscillation is approximately obtained by measuring the interval between peaks and the corresponding fundamental frequency turns out to be W B M 1.43. This is lower than the mass of the scalar particle, m M 1.58, as expected. In the right panel I plot the total energy, Et, on the half-line as a function of time. This is of course conserved through the evolution. I also plot the energy, El, localized between x = 0 and 2 = 10. This falls very slowly and after a time of 10,000, only about 4% of the total energy has dissipated out of the local region. We do not yet understand what sets the scale for this decay rate. But it is clear that naturalness arguments are violated in this example and the energy is localized for time periods much longer than all dimensionfull scales in

400

the problem. The existence of such approximate breathers seems to be a generic feature of the theory, and we have successfully constructed several such configurations with different frequencies and amplitudes.

1.05 1.1

1.5

-

1 -

-*

0

8

W

1

0.9 -

0.95

L

0.5 0.85

I 9970

9980

9990

-

2-------I,

0.8

10000

t

Figure 1. Time evolution of the initial configuration in Eq. 21 in a 1+1 dimensional scalar theory with the potential in Eq. 19. In the left panel is the oscillatory d(0, t ) . In the right panel is the total energy, Et, on the positive half-line and the local energy, El, between z = 0 and x = 10, as a function of time. All dimensionful quantities are in units of fi,the vev v is 1, and the asymmetry parameter a is 0.5X.

Next consider 0 < a < X/4, which corresponds to an asymmetric double-well potential with again no solitonic solutions. We find that the initial configuration in Eq. 21 again relaxes to an approximate breather with all the signatures of the frequency-mismatch mechanism. The same happens in the $4 theory ( a = 0). Thus, instead of colliding kink-antikink to find an approximate breather (as discussed in Sec. 2.2), we obtain them using large amplitude, low frequency oscillations of $(O, t ) as above. We extend the $4 theory by considering complex $ and gauging the global U(1) symmetry to obtain the Abelian Higgs model in one spatial dimension. It is described by the action

S[$,A,] = / d 2 z [ D p $ t D p $ -

- v")'

-

;FpuFpu l

l

,

(22)

where

Spontaneous symmetry breaking results in a Higgs scalar with mass mH = 6

v ,

(24)

401

and the gauge boson eats the Goldstone degree of freedom to acquire mass

So the linear spectrum has mass gaps, as required. We find that the breathers in the real q54 theory are stable against perturbations in the complex gauged theory, as long as the U(1) charge is large enough so that the fundamental breather frequencies are lower than the mass of the gauge boson. However, when we reduce the charge so that the fundamental breather frequency falls within the spectrum of small gauge field oscillations, the breathers dissipate their energies rapidly. Again this is completely consistent with our picture. 3 . 2 . Towards the Electroweak Theory

The existence of approximate breathers is not a low-dimensional, toy-model peculiarity. We have found them within a spherical ansatz in 3+1 dimensions in the 44 theory, with all the signatures of the frequency-mismatch mechanism. In fact Gleiserg has demonstrated their existence in scalar field theories in 3+1 dimensions, for both symmetric and asymmetric double-well potentials. Now that we understand the mechanism that allows long-lived, approximately spatially localized, temporally periodic configurations to exist in a classical field theory, we can ask if there is any possibility for such objects to exist in the Standard Model. The answer is yes. When we ignore the hypercharge gauge fields in the electroweak theory and consider the S U ( 2 ) Higgs theory, there is no unbroken subgroup, and all three W gauge bosons are massive. So, the theory fulfills the first requirement of having a mass gap in the linear spectrum. Also, the Higgs potential is the usual doublewell potential and so it seems likely that we could set up large-amplitude configurations with oscillatim frequencies in the mass gap. The theory appears to have all the ingredients required for the existence of approximate breathers. When we include the U(1) sector, the photons remain massless after symmetry breaking. However, it is conceivable that the breathers can be made electrically neutral so that they don’t dissipate by electromagnetic radiation. 4. Conclusions and Discussion

We are currently investigating whether approximate breathers exist in the electroweak theory. They could have many significant implications. Firstly,

402

since the lifetimes of the breathers is expected to be many orders of magnitude larger than all scales in the problem, they could shed light on the notion of naturalness in physics. Secondly, if these breathers exist just after the electroweak phase transition, they would set up out-of-equilibrium regions in space, with implications for electroweak baryogenesis. There are several other intriguing questions to be answered. What sets the lifetime of approximate breathers? Is there a quantum particle association with these approximately periodic configurations? We continue to explore these issues.

Acknowledgments This work has been done in collaboration with E. Farhi and N. Graham. I am supported in part by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.

References 1. E. Farhi, N. Graham, and V. Khemani. In preparation. 2004. 2. R. Rajaraman. Solitons and Instantons. Elsevier Science B. V, The Netherlands, paperback, fourth impression edition, 2003. 3. James N. Hormuzdiar and Stephen D. H. Hsu. Pion breather states in QCD. Phys. Rev., C59:889-893, 1999, hep-phJ9805382. 4. David K. Campbell, Jonathan F. Schonfeld, and Charles A. Wingate. Resonance structure in kink - antikink interactions in phi**4 theory. Physica, 9D:1, 1983. 5. Roger F. Dashen, Brosl Hasslacher, and Andre Neveu. The particle spectrum in model field theories from semiclassical functional integral techniques. Phys. Rev., D11:3424, 1975. 6. H. Segur and M. D. Kruskal. Nonexistence of small amplitude breather solutions in phi**4 theory. Phys. Rev. Lett., 58:747-750, 1987. 7. James N. Hormuzdiar and Stephen D. H. Hsu. On spherically symmetric breathers in scalar theories. 1999, hep-th/9906058. 8. D. K. Campbell, S. Flach, and Y. S. Kivshar. Localizing energy through nonlinearity and discreteness. Physics Today, page 43, January, 2004. 9. Marcel0 Gleiser. Pseudostable bubbles. Phys. Rev., D49:2978-2981, 1994, hepph/9308279.

QUANTUM NONABELIAN MONOPOLES

K. KONISHI Dipartimento d i Fisica, ‘%. Fermi” Universitci d i Pisa, Via Buonamoti, 2, Ed. C 56127 Pisa, Italy E-mail: [email protected]

We discuss quantum mechanical and topological aspects of nonabelian monopoles. Related recent results on nonabelian vortices are also mentioned.

1. Prologue There are several reasons to be interested in quantum, nonabelian monopoles. First, if confinement of QCD is a sort of dual superconductor, it is likely to be one of nonabelian variety. Then the effective degrees of freedom involve nonabelian, and not, better understood abelian monopoles. Second, the phenomenon of confinement has t o do with fully quantum mechanical, and not semi-classical, behavior of the monopoles. Thirdly, the very concept of nonabelian monopoles is, as we shall see, intrinsically quantum mechanical, in contrast t o that of the ’t Hooft-Polyakov monopole carrying an abelian charge only. A semi-classical consideration only might easily lead us astray. Finally, some recent developments on nonabelian BPS vortices provide further hints on the subtle nature of nonabelian monopoles and related dual gauge transformations. These considerations are sufficient motivations to give a renewed look on the topological as well as dynamical aspects of these soliton states, in particular in relation t o N = 2 gauge theories.

2.

Confinement in S U ( N ) YM Theory

The test charges in S U ( N ) YM theory take values in ( Z L ! ) ,ZLE))where ZN is the center of S U ( N ) and ZkM’,Z?) refer t o the magnetic and electric center charges. (ZLM’,Z?’) classification of phases follows Namely, 192.

403

404

(1) If a field with z

=

( a ,b ) condenses, particles X = ( A ,B ) with

( z , X )3 U B- b A # 0 ( m o d N ) are confined. (e.g. (q5(o,1)) # 0 -+ Higgs phase.) ( 2 ) Quarks are confined if some magnetically charged particle x condenses, (X(1,O)) # 0. (3) In the softly broken N = 4 (to N = 1) theory (often referred to as N = 1”)all different types of massive vacua, related by SL(2,Z ) , appear; the chiral condensates in each vacua are known. (4) Confinement index is equal to the smallest possible r E Zf’for which Wilson loop displays no area law. For instance, for S U ( N ) YM, r = N in the vacuum with complete confinement; r = 1 in the totally Higgs vacuum, etc. (5) In softly broken N = 2 gauge theories, dynamics turns out to be particularly transparent. We are particularly interested in questions such as: What is x in QCD? How do they interact? Is chiral symmetry breaking related to confinement? A familiar idea is that the ground state of QCD is a dual superconductor Although there exist no elementary nor soliton monopoles in QCD, monopoles can be detected as topological singularities (lines in 4 0 ) of Abelian gauge fixing, SU(3) + U(1)2. Alternatively, one can assume that certain configurations close to the Wu-Yang monopole ( S U ( 2 ) )

’.

A;

N

(a,n x n)a+ . . . ,

r n(r)= r

*

rj

AUa = caz3 r3

dominate 4 . Although there is some evidence in lattice QCD for “Abelian dominance”, there remain several questions to be answered. Do abelian monopoles carry flavor? What is L , f f ? What about the gauge dependence of such abelian gauge-fixed action? Most significantly, does dynamical S U ( N ) -+ U ( l ) N - ’ breaking occur? That would imply a richer spectrum of mesons (TI # T2, etc) not seen in Nature and not expected in QCD. Both in Nature and presumably in QCD there is only one “meson” state, EL1I qi qi), i e . , 1 state vs states. Note that it is not sufficient to assume the symmetry breaking S U ( N ) -+ U ( l ) N - ’ x Weyl symmetry, with an extra discrete symmetry, to solve the problem: the multiplicity would be wrong. If nonabelian degrees of freedom are important, after all, how do they manifest themselves?

[g]

405 3. “Semiclassical” Nonabelian Monopoles

Let us review briefly the standard results about nonabelian monopoles 9-18. One is interested in a system with gauge symmetry breaking

where H is non abelian. Asymptotic behavior of scalar and gauge fields (for a finite action) are:

4 A:

-

U . &Ut

N

4

U . ( 4 ) .U-’ Fij

-

N

rk

EijkT(D.

II2(G/H)= IIl(H); Hi E Cartan S.A. of G.

H),

Topological quantization then leads to 2 a . P c Z,

cfr. 2gegrn = n

pi = weight vectors of

(1)

(= dual of H ) ,

namely, the nonabelian monopoles are characterized by the weight vectors of the dual group H . A general formula for the semiclassical monopole solutions (set ( $ 0 ) = h . H) is given in terms of various broken S U ( 2 ) subgroups, 1

s1 = m(Ea

+ E-a);

S2 =

i --(Ea

- E-a);

S3

1CY*

.H;

the nonabelian monopoles are basically an embedding of the ’t HooftPolyakov monopoles in such S U ( 2 ) subgroups:

-

A i ( r ) = Aq(r,h * a ) Sa; $(r) = Xa(r,h a ) S,

+ [ h - (h . a )a*] H, (2) *

where (a*= a / ( a .a ) )

r-l A:(r) = ~ , i j7 A ( r ) ;

ra X”(r) = T x ( r ) l

x(00)

= h . a.

The mass and U(1) flux can be easily calculated:

M=

I

dS*Tr4B,

U ( 1 ) flux (for instance, for S U ( N

+ 1) + S U ( N ) x U ( 1 ) )is

Example of dual groups (defined by a H a*)are:

406

'

I

-

\

,

4. Some Examples The simplest system with nonabelian monopoles involves the gauge symmetry breaking,

The monopole solutions are

-;vo

0

$(r) and A ( r ) are BPS 't Hooft's functions with $(m) = 1, $(O) 0, A(m) = -l/r, where S is an S U ( 2 ) subgroup

=

or an analogous one in the (2 - 3) raws and columns. So in this case there are two degenerate S U ( 3 ) solutions. The generalization to the case with symmetry breaking SU(N /u

is straightforward. rows/ columns: then

0

0

+ 1)

...

+

S U ( N ) x U(1) ZN

,

0 \

Consider a broken S U ( 2 ) , Si living in (1,N

.. .

...

0

0

.. ...

+(N+I)u(S.F)+(r),

+ 1)

407

+

gives a monopole solution of SU ( N 1) equations of motion. By considering various SU(2) subgroups living in (ilN 1) rows/columns, i = 1 , 2 , . . . N , one is led t o N degenerate solutions. 5. Homotopy Groups in Sytems G

+

H

----t

Let us consider now the relevant homotopy groups. The short exact sequence

0 -+ .rrz(G/H)f .rrl(H)

-+

.rrl(G)

-+

0.

tells us that regular (BPS) monopoles represent .rr2(G/H) C n l ( H ) c 7r1 (G). Alternatively (Coleman) one can say that regular monopoles correspond to the kernel of mapping .rrl(H) -+ .rrl(G).In general, BPS monopoles belong t o a Ic t h tensor irrep of Ic E .rrl(H). The relation between 't Hooft-Polyakov (regular) monopoles and Dirac (singular) monopoles is illustrated in Figure l , which schematically represents the exact sequence above.

-

'T HooftlPolyakov

/

Dirac Figure 1

6 . Monopoles are multiplets of H

A crucial fact for us is that monopoles are multiplet of H and not of the original gauge group H . This is most clearly seen in the case of USp(2N 2 ) - 4 USp(2N) x U(1) where we find 2N 1 degenerate monopoles (of USp(2N)= SO(2N 1) !), or in the system SO(2n 3) -+ SO(2n 1) x U(l), where the multiplicity of degenerate monopoles is 2N (a right number for the fundamental representation of SO(2N 1 ) = USp(2N).) We have recently re-examined the possible irreducible representation (of the dual group H ) to which monopoles belong, in various cases. The results are shown in Table 1 taken from 19.

+

+

+

+

+

+

408

7. Why Nonabelian Monopoles are Intrinsically Quantum Mechanical Nonabelian monopoles turn out to be essentially quantum mechanical. In fact, finding semiclassical degenerate monopoles, as reviewed above, is not sufficient for us t o conclude that they form a multiplet of H , as H can break itself dynamically at lower energies and break the degeneracy among the monopoles. We must ensure that this does not take place. Nonabelian monopoles are in this sense never really semi-classical, even if (4) >> AH : (e.g., Pure N = 2, SU(3) ). In this connection, there is a famous “no go theorem” which states that there are no “colored dyons”16. For instance, in the background of the monopole arising from the breaking s usu ( 2 3) ) ~ u ( 1no ) , global T 1 ,T 2 ,T 3 isomorphic t o SU(2) can be shown to exist. These results have somewhat obscured the whole issue of nonabelian monopoles for some time. Do they not exist? Are they actually inconsistent? The way out of this impasse is actually very simple: nonabelian monopoles are multiplets of the dual H group, and the results of l6 does not exclude existence of sets of monopoles transforming as members of a dual multiplet (even if a.t present the explicit form of such nonlocal transformations are not known; see however below). Nevertheless, the no-go theorem implies that the true gauge group of the system is not

Ggauge #H8

409

as sometimes suggested, but H or H or something else, according t o which degrees of freedom are effectively present. (See also 12). 8. Phases of Softly Broken N = 2 Gauge Theories

Fully quantum mechanical results about the phases of SU(n,), U S p ( 2 n C ) and SO(n,) theories with nf hypermultiplets (quarks), perturbed by the superpotential

W ( 4 ,Q , Q ) = p”rTrQr2+ miQiQi, are known

20,21.

Deg.Freed. monopoles monopoles NA monopoles rel. nonloc.

mi -+

0

(See Table). Eff. Gauge Group U (1 y - l U (1 p - 1 S U ( T )x U ( l ) n c - r

Phase Confinement Confinement Confinement Confinement

Global Symmetry

Wnf) U ( n f - I) x U(1) U ( n f - T-) x U ( T ) U ( n f / 2 )x U ( n f / 2 )

NA monopoles

SU(&) x U ( l ) n c - n ~ Free Magnetic

Deg.Freed. rel. nonloc. dual quarks

Eff. Gauge Group

Phase

Global Symmetry

USp(25,) x U(l)nc-nc

Confinement Free Magnetic

SO(2 n f )

U(nf)

Wn,)

From these results we learn, in particular, that the spectrum of the “dual quarks” in the infrared theory (charges, multiplicity, flavor) is identical to what is expected from the semiclassical abelian or nonabelian monopoles. We note in particular that the r- vacua (i.e. vacua with a low-energy effective SU(T) gauge group) exist only for T < %, namely as long as the sign-flip of the beta function occurs:

bpa’)-2r + nf > 0 , 0:

bo

0:

-2n,

+ n f < 0.

Indeed, analogous r vacua exist semiclassically for all values up to min(nf,nc),but quantum mechanically, only those with T 5 n f / 2 give rise to vacua with nonabelian gauge symmetry. Also, when the sign flip is not possible (e.g. N = 2 YM or on a generic point of the quantum moduli space) dynamical Abelianization is expected and does take place! These observations led us to conclude that the “dual quarks” belonging to the fundamental representation of the infrared SU(T) gauge group,

410

actually are the Goddard-Nuyts-Olive-Weinberg monopoles, which have become massless by quantum effects 22. Most importantly, we are led to the general criterion for nonabelian monopoles to survive quantum effects: the system must produce, upon symmetry breaking, a sufficient number of massless flavors to protect H from becoming too strongly-coupled. Natural embedding in N = 2 systems for various cases in Table 1 has been discussed in Ref.19 A very subtle hint about the nature of the nonabelian monopoles come from the recent discovery of nonabelian vortices. 9. Vortices

Vortices occur in a system where a gauge group H is broken to some discrete group

such that I I l ( H / C ) is not trivial. Gauge field behaves far from the vortex axis as

Quantization condition reads a! . p dual of H Some known cases are: 0

c Z where pi

are weight vectors of H ,

H = U(1): in this case vortices correspond to the wellknown Abrikosov-Nielsen-Olesen vortices, representing elements of IIl(U(1)) = Z. According to the parameters appearing in the system they yield Type I, Type I1 or BPS superconductors; The case H = S U ( N ) / Z N yields ZN vortices. These are non BPS and are difficult to analyse (model dependence), although there are some interesting work on the tension ratios, the sine formula ( T k 0: sin $), etcZ3

10. Nonabelian Vortices

Truely nonabelian vortices (ie., with a nonabelian flux) have recently been constructed In the simplest case, we consider the system 24125.

41 1

The high-energy theory has monopoles; the low-energy theory (monopoles heavy) has vortices. We are here mainly interested in the low-energy theory ( s u ( 2ZZ) x u ( 1 )3 0,). We embed the system in a N = 2 model with number of flavor, 4 5 n f 5 5, so as to maintain the “unbroken” subgroup SU(2) non asymptotically free. We shall take the bare mass m and the adjoint scalar mass p a2,so that 212 = E = &Ti << 211 = m. The scalar VEVs are

where only nonvanishing color (vertical) and flavor (horizontal) components of squarks are shown. Set Q, = (@); q = $; and q -+ i q , then the action density is

11. Nonabelian Bogomolnyi Equations

The tension reads

c = vi q A fi€ijVj q A , leading to the nonabelian Bogomolnyi Equations, A, = B = C = 0. The vortex flux ( S U ( N ) )is

This matches exactly the monopole flux Eq. (3) 28. A crucial fact is that there is an unbroken global symmetry, SU(2)c+p (see Eq. (4)), broken only by the vortex configuration (to a U(1)). This implies the existence of exact zeromodes (moduli) labelling

SU(2)/U(l) = s2= CP1. The vortex of Generic Orientation (Zero Modes) can be explicitly constructed as

41 2

A i ( x )=

r3

U[--

2

x.

~ i2 j

r2

1 2. [l - f3(r)]]U-' = -- naraEij-2 [l- f 3 ( r ) ] , 2 r2

2.

A:(x) = -& ~ i 3 j [l - f s ( r ) ] r2

where

na = (sin a cos p, sin a sin p, cos a ), The tension T = 27rt is independent of

u = e-iP

T33/2

e-ia

T2/2

U.

Remarks: In more general S U ( N 1) 4 S u ( NZ)Nx u ( ' ) 3 8 systems with flavors 2N+ 2 > N f 2 2N, there appear vortices with 2 ( N - 1) - parameter family of zero modes, parametrizing

+

they nicely match the space of (quantum) states of a particle in the fundamental representation of an S U ( N ) . (Actually, for N f > N there are other vortex zero modes (semilocal strings), not related to the unbroken, exact S U ( N ) ~ + symmetry. F Those are related to the flat directions.) )x U(1) Furthermore, vortex dynamics ( S U ( 2zz --f 8) n -+ n(z,t)

can be shown to be equivalent to:

Spsl) = p

/

1

d2x 5

+ fermions :

an O ( 3 ) = CP' sigma model It has two vacua; no spontaneous breaking of S U ( ~ ) C +occurs; F Also, there is a close connection between the 2D vortex sigma model dynamics and the 4D gauge theory dynamics: they are dual to each other 29. In N = 2 theory, due to the presence of two independent scales which we take very different ( p << m), we can study monopoles (HE theory) and vortices (LE theory) separately in the effective theories valid at respective scales. Physically, of course, it is perfectly sensible to consider both together; the only problem is that it becomes very difficult to disentangle the two if the two scales are of the same order. Nevertheless, there remains the fact that monopoles (of the HE theory) and vortices (of the LE theory) are actually incompatible 26127!

413

as static configurations. In fact, the monopoles of HE theory represent II2(SU(N l ) / S U ( N )x U ( l ) / z ~ ) ; but in the full theory, l&(SU(N+ 1)) = 0, so no topologically stable monopoles exist. On the other hand, vortices represent Ill( S U ( NzN) x U ( l ) ) of LE theory; no vortices exist in the full theory since IIl(SU(N 1))= 0. What happens is that monopoles of G / H are confined by magnetic vortices of H --+ 0, leading to monopole-vortex-antimonopole bound states, which are not stable as static configurations. They could however give rise to rotating and dynamically stable states. After all, the mesons in QCD are systems of this kind! Note that the restriction on the number of flavors, 2 N + 2 > N f 2 2 N , is fundamental. If N j were less than 2 N , the subgroup S U ( N ) would become strongly coupled, and break itself dynamically. Nonabelian vortices do not exist quantum mechanically in such a system. We are then led to the following relation between the vortex zeromodes and fi transformation of monopoles. Consider a configuration consisting of a monopole (of G / H ) and an infinitely long vortex, which carries away the full monopole flux. At small distances r from the monopole center, T O(l/m), HE theory is a good approximation and the monopole flux looks isotropic; at a much larger distances of order of, T > one sees the vortex of LE theory. The energy of the configuration is unchanged if the whole system is rotated by the exact HC+F transformation. This is a nonlocal transformation. The end point monopole is apparently transformed by the HC part only, but, since in order to keep the energy of the whole system unchanged it is necessary to transform the whole system, it is not a simple gauge transformation H of the original theory. It is in this sense that the nonlocal, global HC+F transformations can be interpreted as the dual transformation H acting on the monopole, at the endpoint of the vortex. -

+

+

-

i,

414

12. To conclude: where do we stand ? 0 0

0 0

0

Nonabelian monopoles are intrinsically quantum mechanical; Massless flavors are important for (i) keeping H unbroken; and (ii) for providing enough global symmetry giving rise to exact vortex zeromodes: these can be interpreted as the dual gauge transformation acting on the monopoles at the ends of the vortex; One has a nice ”model” of monopole confinement by vortices. Light nonabelian monopoles appear as IR degrees of freedom (examples in N = 2 models). Are there light nonabelian monopoles in some other N = 1 theories? Do some vacua of N = 2 theories, especially those based on “almost superconformal vacua” 30 provide a good model of confinement in

QCD? Acknowledgment The results reported here are fruits of enjoyable collaboration I had with Roberto Auzzi, Stefan0 Bolognesi, Jarah Evslin, Alexei Yung and Hitoshi Murayama. I thank Arkady Vainshtein and the organizers of the Workshop for their kind hospitality and for providing us with an occasion for interesting discussions with many participants.

References 1. G. ’t Hooft, Nucl. Phys. B138 (1978) 1; ibid B153 (1979) 141, B190 (1981) 455. 2. G. ’t Hooft, Nucl. Phys. B190 (1981) 455; S. Mandelstam, Phys. Lett. 53B (1975) 476. 3. F. Cachazo, N. Seiberg and E. Witten, JHEP 0302 (2003) 042, hepth/0301006. 4. T.T. Wu and C.N. Yang, in “Properties of Matter Under Unusual Conditions”, Ed. H. Mark and S. Fernbach, Interscience, New York, 1969, Y.M Cho, Phys. Rev. D21 1080 (1980), L.D. Faddeev and A.J. Niemi, Phys. Rev. Lett. 82 (1999) 1624, hep-th 9807069. 5. L. Del Debbio, A. Di Giacomo and G. Paffuti, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 231; A. Di Giacomo, hep-lat/0206018. 6. R. Dijkgraaf and C.Vafa, hep-th/0208048; F. Cachazo, M. R. Douglas, N. Seiberg and E. Witten, JHEP 0212 (2002) 071, hep-th/0211170. 7. T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. 8. G. ’t Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, JETP Lett. 20 (1974) 194. 9. E. Lubkin, Ann. Phys. 23 (1963) 233. 10. C. Montonen and D. Olive, Phys. Lett. 72 B (1977) 117. 11. P. Goddard, J. Nuyts and D. Olive, Nucl. Phys. B125 (1977) 1.

41 5 12. F.A. Bais, Phys. Rev. D18 (1978) 1206; B.J. Schroers and F.A. Bais, Nucl. Phys. B512 (1998) 250, hep-th/9708004; Nucl. Phys. B535 (1998) 197, hep-th/9805163. 13. E. J. Weinberg, Nucl. Phys. B167 (1980) 500; Nucl. Phys. B203 (1982) 445. 14. S. Coleman, “The Magnetic Monopole Fifty Years Later,” Lectures given at Int. Sch. of Subnuclear Phys., Erice, Italy (1981). 15. Chan Hong-Mo and Tsou Sheung Tsun, Nucl. Phys. B189 (1981) 364. 16. A. Abouelsaood, Nucl. Phys. B226 (1983) 309; P. Nelson and A. Manohar, Phys. Rev. L e t t . 50 (1983) 943; A. Balachandran, G. Marmo, M. Mukunda, J. Nilsson, E. Sudarshan and F. Zaccaria, Phys. Rev. Lett. 50 (1983) 1553; P. Nelson and S. Coleman, Nucl. Phys. B227 (1984) 1. 17. K. Lee, E. J. Weinberg and P. Yi, Phys. Rev. D 54 (1996) 6351, hepth/9605229. . 18. C. J. Houghton, P. M. Sutcliffe, J.Math.Phys.38 (1997) 5576, hepth/9708006. 19. R. Auzzi, S. Bolognesi, Jarah Evslin, K. Konishi, H. Murayama, hepth/0405070. 20. A. Hanany and A. Oz, Nucl. Phys. B452 (1995) 283, hep-th/9505075, P. Argyres, M. Plesser and N. Seiberg, Nucl. Phys. B471 (1996) 159, hep-t h/9603042, 21. G. Carlino, K. Konishi and H. Murayarna, JHEP 0002 (2000) 004, hepth/0001036; Nucl. Phys. B590 (2000) 37, hep-th/0005076; G. Carlino, K. Konishi, Prem Kumar and H. Murayama, Nucl. Phys. B608 (2001) 51, hep- th/Ol04064. 22. S. Bolognesi and K. Konishi, Nucl. Phys. B645 (2002) 337, hepth/0207161. 23. M. Douglas and S. Shenker, Nucl. Phys. B447 (1995) 271-296, hepth/9503163, A. Hanany, M. Strassler and A. Zaffaroni, Nucl.Phys. B513 (1998) 87, hep-th/9707244. B. Lucini and M. Teper, Phys. Rev. D64 (2001) 105019, hep-lat/0107007, L. Del Debbio, H. Panagopoulos, P. Rossi and E. Vicari, Phys. Rev. D65 (2002) 021501, hep-th/0106185; JHEP 0201 (2002) 009, hep-th/0111090, C. P. Herzog and I. R. Klebanov, Phys. L e t t . B526 (2002) 388, hep-th/0111078, R. Auzzi and K. Konishi, New J. Phys. 4 (2002) 59 hep-th/0205172. 24. A, Hanany and D. Tong, JHEP 0307 (2003) 037, hep-th/0306150. 25. R. Auzzi, S. Bolognesi, Jarah Evslin, K. Konishi and A. Yung, Nucl. Phys. B to appear, hep-th/0307287. 26. V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rep C 116 (1984) 103. 27. K. Hori and C. Vafa, hep-th/0002222. 28. R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Nucl. Phys. B686 (2004) 119, hep- t h/03 12233. 29. A. Hanany and D. Tong, JHEP 0404 (2004) 066, hep-th/0403158; M. Shifman and A. Yung, hep-th/0403149. 30. R. Auzzi, R. Grena and K. Konishi, Nucl. Phys. B653 (2003) 204, hepth/0211282, R. Auzzi and R. Grena, hep-th/0402213.

DILUTE MONOPOLE GAS, MAGNETIC SCREENING AND K-TENSIONS IN HOT GLUODYNAMICS

C. P. KORTHALS ALTES

Centre Physique The‘orique au CNRS, Case 907, Luminy, F13288, Marseille, France

E-maihltes @cpt.univ-rnrs.fr A dilute monopole gas explains, in quarkless gluodynamics, the small ratio 6 between the square of magnetic screening mass mM and spatial Wilson loop tension. This ratio is 0.0895 for T = 0 to 0.0594 a t T = ca for any number of colours with order N-’ corrections and equals up to a numerical factor of 0(1)the diluteness. The monopoles have a size 1~ = m;’. The GNO classification tells us they are in a representation of the magnetic SU(N) group. Choosing the adjoint for the dilute gas predicts the k-tensions to scale as k ( N - k ) , within a percent for high T, and a few percent for low T for the seven ratios determined by lattice simulation. The transition is determined by the transition of the dilute Bose gas at Tc = 0 . 1 7 4 m ~ , and the transition is that of a non- or nearly-relativistic Bose gas.

1. Introduction

An ancient idea in QCD is that monopoles are responsible for flux tube formation through a dual superconductor mechanism. Another mechanism, that of the “Copenhagen vacuum” of the early eighties 2 , proposes macroscopic Z(N) Dirac strings or Z(N) vortices. In this talk I will discuss a specific model of the first type 4 , that works surprisingly well in its most direct applications. A straightforward way to see its workings is to start from the plasma phase. That sounds at first sight self-defeating as there is no confinement in this phase. But it will turn out that precisely the absence of confinement renders the detection of these monopoles, or perhaps more appropriately, magnetic quasi-particles, so straightforward. We know that electric quasi-particles, the gluons, are approximately free at very high temperature. This follows from the Stephan-Boltzmann law. We may guess that the same happens to the magnetic monopoles that were condensed in the cold phase. What kind of monopoles can we expect? In the absence of any spon-

41 6

417

taneous breaking the GNO classification applies. For a theory with only Z(N) neutral fields, like the adjoint, the magnetic group is S U ( N ) . SO we can have monopoles in the fundamental, adjoint or any representation we like. One of the consequences of the idea of a monopole condensate is the screening of the magnetic Coulomb force between two static magnetic sources. This is a straightforward generalization of the screening of the Coulomb force between two heavy quarks. There is now ample affirmation of screening l 3 21 from lattice simulations. It is reasonable to think of the monopole as having a size of about the screening length 111.1 = rn;. We will assume that its size is much smaller than the inter-monopole distance. That, together with the choice of multiplet, will fix the ratios of spatial string tensions. The reader who is only interested in how one computes these ratio’s should read sections 2,4 and 5. A more complete version appeared in hep-ph/0408301. 2. Electric flux loops

In this section we discuss the behaviour of electric flux loops in the plasma, as it will explain the basic features that are permeating the talk. 2.1. &ED

Consider a plasma of ions and electrons. We will take the ions to have the same but opposite charge of the electron. Suppose we want to compute the electric flux going through some large (with respect to the atomic size) closed loop L with area A(L). Normalize the flux @ = Js ds’ . l? by the electron charge e and define :

V ( L )= exp i27r@/e.

(1)

Of course, at T below the ionization temperature no flux would be detected by the loop, because there are only neutral atoms moving through the loop. Let us now raise the temperature above Tionisation. What will happen? Both electrons and ions are screened. For simplicity we will take the ions to have the opposite of one electron charge. We are going to make the following simplification. The charged particles are supposed to shine their flux through the loop if they are within distance ID from the minimal area of the loop. This defines a slab of thickness 210. Here we will keep the factor 2.

418

Then one electron (ion) on the down side of the loop will contribute +1/2(-1/2) to the flux, and with opposite sign if on the up side of the loop. That is: V(L)lonecharge = -1. This result is independent of the sign of the charge! The plasma is overall neutral, the loop is sensitive only to charge fluctuations. For 1 charges inside the slab the flux adds linearly and we find:

V(L)II = (-l)!

(2)

Assuming that all charges move independently, the average of the flux loop V ( L ) is determined by the probability P(1) that 1 electrons (ions) are present in the slab of thickness 21D around the area spanned by the loop. Taking for P(1) the Poisson distribution exp -1- dis the average number of electrons (ions) in the slab- we find for the thermal average of the loop due to both ions and electrons:

i(d)'

(v(L))T

=

C P ( ~ ) V ( L )=J ~C ~ ( l ) ( - ) ' I

= exp -41.

(3)

1

Now = A ( L ) ~ Z D ~ ( Tso) , the electric flux loop obeys an area law exp -p(T)A(L), with a tension

p(T) = 81~7%(T).

(4)

This area law distinguishes the behaviour of the loop in the plasma from that in the normal unionized state. The flux loop has a very useful alternative formulation as a Dirac flux along the border L. 2.2. Electric flux in gluodynarnics

Is there an S U ( N ) generalization of the QED case? In fact yes. The closed Dirac string in QED is replaced by a Z(N) vortex of strength k, the 't Hooft loop3. We introduce in the Lie algebra of traceless hermitean N x N matrices a basis for the Cartan subalgebra, the (N-1) x (N-1) dimensional subspace of diagonal matrices. This basis of N - 1 diagonal matrices Yk is chosen such that in exponentiated form exp ( - i27~Yk)it gives the N - 1 centergroup elements e x p i k 9 . A simple choice is 1 N

Yk = --diag(N

- k, . . , ...., N

- k, -k,. . ....., -k).

(5)

The entry N - k comes in k times, and -k comes in N - k times, so the trace is 0.

419

The flux operator becomes in this notation:

It does correspond to the vortex operator of 't Hooft with strength Ic. The next question is: does the gas of deconfined gluons induce an area law in this operator? The answer is yes, and the reasoning is as before. The charge of a gluon with respect to the charge Y k is found from the form Y k takes in the adjoint representation. This is easy: we have charge f l like in QED, but unlike in QED we have 2k(N - Ic) gluon species in the gluon multiplet with such a charge. All other gluons have vanishing charge so do not contribute to the flux. Since we take the gluons to be statistically independent, the charged ones all contribute a factor to ( v k ) , and this factor is the same by S U ( N ) symmetry: exp -21~n. Conclusion: the expectation value of the loop is

with the tension

So this is the k-scaling law for the electric flux loop. It does obey large N factorization:

This computation is corroborated by low order perturbation theory for the loop. The expectation value of the loop can be computed from a tunneling process between adjacent vacua of an effective potential with a Z ( N ) symmetry 1 9 , 2 0 . This potential has, because of the Z ( N ) symmetry, degenerate minima in the centergroup elements. The propagators feel the coloured background, so the double line representation does not apply. So the expectation of pure gluodynamics that only 1 / N 2 corrections appear is not valid. 3. Some basic facts in quarkless Yang-Mills

In this section we gather a few facts, partly empirical, from lattice simulations, and partly from simple physics arguments.

420

3.1. Magnetic screening

A feature that sets gluodynamics apart from QED is the appearance of magnetic screening. When we put two heavy monopoles far apart at a distance T we find a Yukawa type potential: C

V ( T )= - exp -mMr. T

(10)

A simple argument shows

that the magnetic mass is the mass of a scalar excitation of a Hamiltonian in a world with two large space dimensions and one periodic mod 1/T. The space symmetries in such a world are SO(2) x P x C x R. We have the rotation group in the two large dimensions, C is charge conjugation, P is parity in the large 2d space, and R is the parity in the periodic direction. As T + 0 the rotation group becomes S 0 ( 3 ) ,and R and P become related through a rotation, so at T = 0 the Of+ glueball mass results. 3.2. Spatial tension

The spatial tension is defined through a spatial loop L, on which an ordered product P exp i $ d$.AR lives. R labels the representation for the vector potential we have chosen. The spatial Wilson loop is then defined as the trace over colour: WR(L) = T r P e x p i

dZ.AR.

The path integral average gives an area law: (WdL)) = Cexp ( - d T ) A ( L ) )

(12)

for all temperatures. At zero temperature the spatial tension equals the string tension a from a time-space loop because of Euclidean rotation invariance. But as function of temperature they behave differently. The string tension suffers a correction at low temperature of -:T2 due to the excitations contributing the Luescher term. Above T, it vanishes. The spatial tension stays flat, see fig. (1). Like the magnetic mass it does not feel the Z ( N ) transition and starts to scale like (g2T)2well above the critical temperature. The dimensionless ratio of the spatial tension in the fundamental representation and the square of the magnetic mass is small, on the order of a few percent. In two extreme cases the ratio 6(T) is accurately known from simulations : at T = 00 in the 3d theory, and at aIn what follows we reserve the name string tension for the time-space loop.

42 1

T = 0 lo. In between there are simulations l 3 21 that are consistent with a slowly varying S(T). It is very important to have this verified with more precision.

TC

Tg

T

Magnetic mass mM and tension u as function of temperature, schematically. mo++and from ref. (13): Tc = 0.174 mo++.The temperature Tq 5 mo++ is where the de Broglie thermal wave length becomes equal to the magnetic screening length. For the calculation of the tension it is below Tq that quantum statistics applies, above classical statistics applies as in section 5. :

3.3. Dependence of the tension on the representation, and N-allity

There is widespread agreement on the dependence of the tension and the string tension on the representation R. It is only its N-allity k that counts. If we think of R being built up by f fundamental and f anti-fundamental representations, the N-allity is just the difference:

k = f -J.

(13)

To get a more precise idea, let us imagine the Wilson loop is formulated in a periodic box, with the 3 space dimensions being of macroscopic length L. The fourth dimension is of length l/T. The loop is now replaced by

422

two Polyakov lines of opposite orientation in the x-direction. They are a distance r apart- and carry the reprsentation R. The expectation value of the loop is then parametrized as:

( W R ( L )= ) Cexp -LVT(T).

(14)

If the distance T is very small, asymptotic freedom tells us:

where C2(R) is the quadratic Casimir operator. In three dimensions the Coulomb force is logarithmic. The Casimir operator can be related to our N x N matrices Y k if R corresponds to a Young tableau, with the first row having w1 more boxes than the second. The second row has w2 more boxes than the third, and so on. By definition, the numbers 'Wk are never negative. There is a one-to-one correspondence between Young diagrams and irreducible representations of SU(N ) . Define the highest weight HR of R as k

as we already alluded to at the end of section 2.2 , Then, if Y = Yk : 1 C2(R) = ~ ( T T H ; ~ T T Y H R ) .

ck

+

(17)

If the distance is long enough ( a somewhat ambiguous criterion!) the string regime sets in and we expect: v(T)

=ak(T)T.

(18)

The tension Ck only depends on the N-allity k. String formation between the two sources renders this dependence very plausible. The tension (TI of the string formed in between two fundamental sources can form a bound state in the string with tension u2. We can go on like this and the question is, what is the dependence on k? It should obey certain a priori criteria. For large number of colours we expect factorization ffk

= ka1.

From general arguments one also expects ffk

= kffl

(19) l4

in that same limit:

+ o(l/N2).

This is discussed in depth by Misha Shifman in this volume.

(20)

423

3.4. Flux representation of the Wilson loop

If we want to proceed with the Wilson loops, as we did with the 't Hooft loops, we need clearly a representation similar to eq.(6). Of course a natural guess is to take this electric flux formula, replace E' by 6, a = g 2 / 4 r by l / a to get:

This formula cannot be true 20. But it is true in a dynamical sense: consider the limit of a path integral average in the 3d theory with an adjoint Higgs, the electrostatic QCD Lagrangian. For a given representation R with highest weight H R with stability group S (i.e. S H R S - ~= HR)one finds: (wR)

=< J DRexp (ig

J~s.(z,

- -1f a b c f i n b A f i j n , ) Xa ~ r - - ~ ~ R> ~ .+ )

9

2

(22) with na = T T % R H R R ~the Higgs field's angular part that parametrizes the coset S U ( N ) / S ( H R ) .The second term in the exponent is the source term for the monopoles in the sense that it carries no long range effects in the original Higgs phase. The average of the r.h.s is the 3d Yang-Mills average. The integration is over all gauge transforms with an arbitrary number of hedge hog configurations. The r.h.s. is obtained l6 by taking the average of the magnetic charge operator in the 3d Higgs phase characterized by H R . Then the the VEV of the Higgs field is let to zero, which introduces the fluctuations over na, the angular components of the Higgs field. Finally one lets the mass of the Higgs become infinite, which suppresses the radial integrations. On the other hand l 5 the integrand of the r.h.s., without the brackets, was shown by Diakonov and Petrov to equal WR,by one dimensional quantum mechanics methods. It involves a limiting procedure which is procured in the path integral by starting out from the Higgs phase and moving to the symmetic phase as described above. With this specification we will use eq. (21) without quotation marks in what follows. 4. Predictions for the Wilson loops

Once we are given a dilute gas of screened monopoles, we only need to specify the representation of the magnetic group for our monopoles. Then the calculation of the tension is done with our flux representation for the

424

average of the loop, eq.(22). But in practice we can use the simple formula eq.(21). The reason is that we put the contribution of the monopole gas in by hand. That takes care of the singular gauge transforms in eq.(22). The remaining regular ones can be dropped because we compute something gauge invariant. So the average of a k-loop in the totally anti-symmetric representation is computed from (23) For the dilute gas in the adjoint representation the computation is not different from the one with gluons in section 2.2. The adjoint monopoles have a magnetic charge equal to f?, or 0. Only the former contribute and their individual contribution to the loop in eq. (23) is -1. The diluteness, and classical statistics (as the thermal wave length is T-l and the typical interparticle distance is ( g 2 T ) - l this is justified, see also fig. 1.) then give for every charged species the same contribution exp -2lMnM, nM being the common density of a given species. As there are 2k(N - k) charged species in the adjoint, the total results in the k-tension being: LTk

= 4k(N

- k)lMnM.

(24)

One can do a similar calculation for monopoles in the fundamental representation. The counting goes now as follows. Recall eq.(5), Y k = $diag(N - Ic, ....N - k,- k , .... - k) with trace zero. The Y k charge of the highest Ic components of the N components of the column spinor is The remaining N - k components have charge given by It then follows from the use of the Poisson distribution that the flux of a given component is contributing c o s ( n 9 ) or cos(r6). Taking into account the degeneracies the final result for the tension becomes:

9.

-6.

k N

ck = dMnM(N - k cos(np ( N - k)) - ( N - k) cos(n-)).

N

Both tensions give ratios

L T ~ / L Tthat ~

(25)

behave like k for large N and finite

k. This factorization is expected for a k-loop tension caused by screened particles , But for large N and 5 = N/2 the ratio for the fundamental multiplet is a factor 2 larger. bFactorization is not obvious if uncorrelated Z(N) vortices cause the area law: uk N (1 - c o s ( 2 k a / N ) ) . This is due to the macroscopic size of the vortex perimeter causing long range correlations between loops.

425

5. Comparison to lattice simulations

We have been discussing a model at very high temperature. Hence it is tested in 3d lattice simulations. The ratios found l 2 for the totally antisymmetric irreps are close - within a percent for the central value - as far as the adjoint multiplet of magnetic quasi-particles is concerned : SU(4) : 02/01

= 1.3548 f 0.0064

adjoint : 1.3333

fundamental : 1.8182

SU(6) : 02/01

= 1.6160 f 0.0086

adjoint : 1.6000

fundamental : 1.9686

adjoint : 1.8000

fundamental : 2.3635

uS/u1 = 1.808 f 0.025

The results are that precise, that you see a two standard deviation from the adjoint, except for the second ratio of SU(6). This deviation is natural, since the diluteness of the magnetic quasi-particles is small, on the order of a couple of percent, as we will explain at the end of this subsection. So we expect corrections on that order to our ratios. There is a less precise determination of the ratio 02/01 = 1.52 f 0.15 in SU(5) 17. But the central value is within 1 to 2% of the predicted value 3/2 from the adjoint. The fundamental gives a ratio 1.8231. The SU(8) ratios are known on a rather course lattice l7 and using a different algorithm: 02/01

= 1.692(29)

adjoint : 1.714

fundamental : 2.106

03/01

= 2.160(64)

adjoint : 2.143

fundamental : 2.958

04/01

= 2.26(12)

adjoint : 2.286

fundamental : 3.256

In conclusion: the seven measured ratios are consistent with the quasiparticles being independent, as in a dilute gas and in the adjoint representation. The number of quasi-particle species contributing to the k-tension is 2k(N - k). This number happens to coincide with the quadratic Casimir operator of the anti-symmetric representation. The fundamental monopoles are clearly disfavoured by the data. 6. Conclusions

There is remarkable agreement with an accuracy of about a percent between numerical simulation at high T and the dilute adjoint monopole gas. This diluteness appears as a small parameter for every SU(N) theory. Some parameters, like the mass of the magnetic quasi-particle, are still within a

426

large range, although its size is 1 ~ Is. it heavy, on the order of the lowest , the transition is that of a non-relativistic glueball mass (i.e. m ~ )then Bose gas. Is it light (with respect to its size) then the transition is that of a near-relativistic BE transition. Perhaps the best way to summarize our approach is to return to fig. 1. It is clear that the relation: CJ

-

(26)

ZMnM

implies that Z&CJ is the diluteness 6, and must be small for consistency. And lattice data have borne that out! Now , once we accept the idea, that on the high temperature side of the transition a dilute gas describes tensions so well, the constancy of that diluteness down till T = 0 suggests that all what changes is the thermal wavelength AB(T). At temperatures on the order of the glueball mass or higher it is clear that the interparticle distance O(l/g2T)is larger than the thermal wave length 1/T, because the coupling is so small. But at a temperature Tq the thermal wave length takes over with Tq mmonopo~e. It is very unlikely that Tq will be below T,,since that would imply the BE transition as a second one below T,. So the monopole mass should not be lower than T,, so the transition will then be a non- or near-relativistic BE transition. Below Tq the dilute gas gives a contribution to our ratios, but now determined by Bose statistics. For an analysis of the cold phase ratios see ref. 18. Of course for most observables in the low T phase the Bose statistics is all-important. Thus the string tension will become non-zero below T,, the character of the transition, i.e. a jump or continous behaviour in the occupation fraction of the p’ = 0 states will be crucial to know and and is calculable in this model. Realistic QCD involves quarks, and there is all reason t o believe they couple strongly to our monopoles. After all, the latter are bound states of magnetic gluons. Note that a real time picture of our quasi-particles is lacking. According to our Euclidean picture they become, at very high T , a 3d gas of particles with small size ZM and small interparticle distance. The real time picture is a challenge. Adi Armoni, Luigi del Debbio, Pierre Giovannangeli, Christian Hoelbling, Dima Kharzeev, Alex Kovner, Mikko Laine, Biagio Lucini, Harvey Meyer, Misha Shifman, Rob Pisarski, and Mike Teper provided me with

-

427 useful comments. I thank t h e organizers for their invitation, a n inspiring meeting, and for wonderful hospitality.

References 1. G. 't Hooft, in High Energy Physics, ed. A. Zichichi (Editrice Compositori Bologna, 1976); S. Mandelstam, Phys. Rep. 23C (1976), 245 2. H. B. Nielsen, P. Olesen, Nucl. Phys.B61, 45,(1973); Nucl. Phys.Bl60, 380 (1979). 3. G.'t Hooft, Nucl.Phys.Bl38, 1 (1978). 4. P. Giovannangeli, C.P. Korthals Altes, Nucl.Phys.B608:203-234,2001; hepph/0102022. 5. P. Goddard, J. Nuyts, D. Olive, Nucl.Phys B125,1 (1977). S.Coleman, Erice lectures 1981. 6. G. 't Hooft, Nucl. Phys.B 105, 538 (1976). 7. P. de Forcrand, C. P. Korthals Altes, 0. Philipsen, in preparation 8. C. P. Korthals Altes, 2003 Zakopane lectures,hep-ph/0406138. 9. M. Teper, Phys.Rev.D59, 014512, 1999; hep-lat/9804008. 10. B. Lucini, M. Teper, JHEP 0106,050 (2001), M. Teper, hep-th/9812187 . 11. B. Lucini, M. Teper, U. Wenger, JHEP 0401:061,2004, hep-Iat/0307017. 12. M. Teper, B. Lucini, Phys.Rev. D64 (2001) 105019. 13. C. Hoelbling, C. Rebbi, V.A. Rubakov, Phys. Rev.D63 :034506,2001; heplat/0003010. 14. A. Armoni, M. Shifman, Nucl.Phys.BB71, 67, 2003; hep-th/0304127, hepth/0307020. 15. D. Diakonov, V. Petrov, Phys. Lett.B242, 425 (1990); see also heplat/0008004. 16. C.P. Korthals Altes, A. Kovner, Phys.Rev.D62:096008, 2000; hepph/0004052 17. H. Meyer, private communication and hep-lat/0312034. 18. C. P. Korthals Altes, H. Meyer, to be published. 19. T. Bhattacharya, A. Gocksch, C. P. Korthals Altes and R. D. Pisarski, Phys.Rev.Lett.66, 998 (1991); Nuc1.Phys.B 383 (1992), 497. 20. C.P. Korthals Altes, A. Kovner and M. Stephanov, hep-ph/99, Phys.Lett. B469, 205 (1999), hep-ph/99095 16. 21. Ph. de Forcrand, M. D'Elia, M. Pepe, Phys.Rev.Lett.86:1438,2001;heplat /0007034. 22. For a related problem see: A. Kovner, B. Rosenstein, 1nt.J.Mod. Phys. A7(1992), 7419.

SUPERSIZING WORLDVOLUME SUPERSYMMETRY: BPS DOMAIN WALLS AND JUNCTIONS IN SQCD

A. RITZ Theory Division, Department of Physics, CERN, Geneva 23, CH-1211, Switzerland We study the worldvolume dynamics of BPS domain walls in N= 1 SQCD with gauge group SU(N) and Nf = N flavors, and exhibit an enhancement of supersymmetry for the reduced moduli space associated with broken flavor symmetries. The worldvolume superalgebra corresponds to an N= 2 Kahler sigma model in 2+1D deformed by a potential, given by the norm squared of a U(1) Killing vector, resulting from the flavor symmetries broken by unequal quark masses. This framework leads t o a description of 1/4-BPS two-wall junction configurations as l/P-BPS worldvolume kinks.

1. Introduction and Summary

Solitonic solutions of supersymmetric field theories generally exhibit a moduli space M which locally admits the decomposition

M

2~

Msusy x

G.

(1)

Here Msusy refers t o the sector associated with bosonic generators in the supersymmetry (SUSY) algebra which are broken by the soliton, by virtue of the introduction of central charges [l],and in flat space always includes a translational component Rd c M S U S Ywhere , d is the codimension. The realization of supersymmetry in this sector, associated with the unbroken generators, is then fixed by the kinematics of the bulk superalgebra. In contrast, G - the ‘reduced moduli space’ - is not directly associated with broken super-translation generators. This implies that the realization of worldvolume supersymmetry in this sector is less constrained and, as discussed below, can exhibit more supersymmetry (at least in the twoderivative sector) than the translation sector associated with Msusy. The origin of this supersymmetry enhancement is that not all the supercharges which are realized on the worldvolume of the soliton lift to supercharges in the full theory. The additional charges arise purely due to geometric

428

429

features, e.g. a Kahler structure, of the reduced moduli space. As a simple exmaple, we will consider BPS kinks in an N = 1 sigma model on S3, where the kink profile lies entirely within an S1 fibre of S 3 . The kinks then 21 S2 which is the base of this exhibit a nontrivial reduced moduli space fibration. The latter manifold is Kahler and thus the low energy dynamics exhibits enhanced N=2 supersymmetry, despite the soliton being 1/2-BPS. A primary motivation for studying this phenomenon is that it arises rather naturally in the context of BPS domain walls in N=1 SQCD with a sufficiently large number of flavors. With gauge group S U(N) accompanied by N f = N funadamental flavors, with masses small compared to the dynamical scale AN, the low energy description on the Higgs branch in terms of meson and baryon moduli corresponds to a massive Kahler sigma model on the manifold determined by the constraint [2]

c

detM - B B

= ACN

.

(2)

The theory possesses N quantum vacua, distinguished by the phase of the superpotential which is a multiple of 27r/N [3,4,5,6,7], and between which 1/2-BPS domain walls can interpolate [8,11,12,13,14,15]. In this theory, BPS k-walls (walls which interpolate between vacua differing in phase by 27rk/N) were shown [9] to exhibit a nontrivial classical reduced moduli space c k due to localized Goldstone modes associated with the flavor symmetries which are broken by the wall solution. The corresponding coset is a complex Grassmannian [9],

One can then formally deduce that the multiplicity of k-walls, vk [lo], is given by the worldvolume Witten index for this Grassmannian sigma model, which depends only on the topology of the space, and is given by the Euler characteristic (see also [16]),

The fact that the reduced moduli space is Kahler is significant here. Since the worldvolume theory lives in 2+1D, the dual constraints of (i) a Kahler target space, and (ii) Lorentz invariance, imply that the low energy dynamics must preserve N= 2 supersymmetry, namely four supercharges! Since only two of the bulk supercharges are preserved by a 1/2-BPS state, we see that this implies sueprsymmetry enhancement on M k . We will show

-

430 below that, for the simplest SU(2) example, this feature can be straightforwardly understood as an extension of the enhancement observed for kinks in the S3 sigma model. The relevant point is that the nontrivial part of the Higgs branch (2) is T * ( S 3 )and , BPS walls can be understood as kinks embedded in the S3 base of this manifold. In the next three sections we describe several aspects of this phenomenon (see [17] for further details). In section 2, supersymmetry enhancement on the reduced moduli space of BPS kinks in the perturbed S3sigma model is illustrated in detail. We then turn in Section 3 t o BPS walls in hl= 1 SQCD and describe how a similar structure arises for gauge group SU(2) with two flavors. In this case, detuning the two quark masses, as is required for the vacua to remain a t weak coupling, has the effect of inducing a potential on the moduli space given by the norm-squared of a U(l) Killing vector. This potential is nonetheless consistent with the enhanced supersymmetry. Finally, in Section 4 we consider the worldvolume interpretation of intersecting domain walls, as worldvolume BPS kinks, and verify that the tension deduced in this way is equal to the bulk central charge. Although we will focus here just on BPS kinks and walls, it seems likely that this mechanism for supersymmetry enhancement will arise in other contexts. For example, N= 1 perturbations of the vortices recently studied in N= 2 SQCD [18], should still preserve a Kahler reduced moduli space, and it would be intersting to verify whether the worldvolume theory realizes (0,2) or an enhanced (2,2) supersymmetry in this case. More generally, the “unwinding” of the target space fibration by the soliton described above as the underlying mechanism for the enhancement in the models considered here has some analogy with gauging the U(1) isometry of the fibre. This again leads to a Kahler sigma model a t low energies, albeit now in the same dimension. This unwinding is also known to occur under T-duality, and it would be interesting t o explore this connection in more detail. 2. Supersymmetry enhancement and worldvolume moduli We can exhibit the enhancement phenomenon for the low energy dynamics on M rather transparently in a simple lfl-dimensional model. Consider an N= 1 sigma model with target space S 3 , with coordinates qP = (13,E , 4 } , ds2 = T [do2

+ sin2 I3 (dJ2 + sin2 Jdq52)] .

(5)

We also turn on a (real) superpotential,

W ( $ )= mcosl9

,

(6)

43 1

which depends on only one of the angular coordinates parametrizing the S 3 . The theory then has two vacua a t 0 = 0 , ~ . Classical BPS kinks exist which interpolate between the two vacua, having mass Msol = 2 = 2rn and satisfying the Bogomol’nyi equation, = g a b a b w ( $ ) . Solutions have the sine-Gordon form

a,@

osoi ( z ) = 2 arctan

[exp

m (- r

(2

- .a))],

tsoi =

to,

= $0

, (7)

exhibiting three bosonic moduli { ZO,6 ,$o}. These bosonic moduli are Goldstone modes for the symmetries broken by the wall: zo is associated with the breaking of translation invariance; and $0 arise from the SO(3) global symmetry of the target space which is preserved in the vacua but broken to SO(2) by the kink solution. &, and $0 thus coordinatize the coset S 0 ( 3 ) / S 0 ( 2 ) 2~ S 2 , as may be verified by computing the induced metric for the bosonic zero modes [19,20],

d.sL = 2m d z i

+ hijdxidxi

= 2m d z i

2r2 +[dci + sin2 Jd&] m

,

(8)

where hij is the metric of the reduced moduli space %. The bosonic moduli space is thus

M

=R

x

xi = IR x S 2 ,

(9)

with the natural metric on each factor. are Let us now consider the fermionic sector. The S3 coordinates partnered under N=1 SUSY by a set of two-component Majorana spinors, $:, a = 1,2. For each bosonic zero mode 22, one finds a corresponding (one-component) fermionic partner vi in the lower component of $,:

$tol= r l i z (;) +- non-zero modes . Only one of these modes is guarunteed to exist by virtue of the fact that the solution is classically 1/2-BPS and thus breaks one of the two supercharges. The broken supercharge is realized as &I = 2 2 f in terms of this ‘goldstino’ mode. Here $ is the superpartner of zo. The novel feature of this system is that the reduced moduli space % is a Kahler manifold and, since the bosonic and fermionic zero-modes are paired, exhibits N= 2 supersymmetry! One of these supercharges is Q 2 , the unbroken charge present in the bulk theory, while the second which we will call &2 exists only due to the complex structure J associated with %.

432

We can represent the supercharges in the rest frame as

and, noting that {qi,$} = hij, one can verify that they satisfy the algebra of N = 2 SQM,

{Q',

eJ>=

WSQMGI~

,

(12)

where WSQM = ( M - Z ) is the worldline Hamiltonian. Introducing the complex coordinate CO w = eido tan -

(13)

2

on

c, and its fermionic partner (14)

we can rewrite the algebra in the form

{ Q , Q*> = WSQM ,

(Q)2

= ( Q * l 2=

o,

(15)

where

In this specific example, one can show that on quantization there are no supersymmetric vacua, and thus no quantum BPS kinks, since ( Q 2 ) 2 is bounded from below by the scalar curvature R of which is clearly positive [20]. Nonetheless, this argument for SUSY enhancement clearly generalizes straightforwardly to other sigma models with targets which are (real) fibrations over Kahler manifolds. One particular generalization will be relevant here, where we embed this model in a Kahler N = (2,2) sigma model with target space T * ( S 3 ) which , arises in context of N= 1 SQCD with gauge group SU(2). 3. Domain Wall Moduli in n/=1 SQCD

Worldvolume supersymmetry enhancement has an interesting application in the context of BPS domain walls in SQCD. Restricting ourselves to the simplest example, consider N= 1 SQCD with gauge group SU(2) and two

433 fundamental flavors. This matter content is sufficient to fully Higgs the gauge group in any vacuum in which the matter fields have a nonzero vacuum expectation value. One can then write down a low energy description in terms of meson moduli fields. Limiting attention to the 'hen-baryonid' branch of the moduli space, the low energy superpotential is given by [2]

W = Tr(rizM)+ X (detM - A:),

(17)

in terms of the meson matrix M , the dynamical scale A2, and a Lagrange multiplier X which enforces a reduced (non-baryonic) form of the general quantum moduli space constraint [2]. The mass matrix 7j7. = (ml,mz} should be hierarchical, m2 >> ml, to ensure that the ensuing vacua (M:) = f(mz/m1)1/2A; lie at weak coupling. However, for part of the analysis below, we will temporarily ignore this constraint in order to focus on the most symmetric regime. These two vacua allow for the presence of BPS domain walls interpolating between them. To proceed, it is useful to introduce a dimensionless meson field Z = MAZ2, and a convenient basis is then provided by the following decomposition,

z= u,,-,, (zon+ ~ Z ~ O ~ ) U , ~ - , ~ ,U, = exp (i a c 3 ) ,

(18)

where the (axial) rotation angle is the relative phase of the two quark masses; m k = lmkleiak for k = 1,2. In this basis,

W

+ iArn&] + X

= eiYh; [%ZO

+

where y = (a1 a 2 ) / 2 and and Am = lm2l - Irnll. Cz=,2," = 1, describing as the deformed conifold.

(1,

22

-

)

1

,

(19)

+

the (real) mass parameters are m = Iml I Im2 I The moduli space constraint takes the form, a smooth complex submanifold of C4, known This manifold is symplectically equivalent to

T*(S3). We now observe that this system contains the simple model of the previous section as a subsector. Setting Am = 0, the two vacua, 2 0 = f l , now lie at the poles of the S3 which forms the real section of the surface C;=,Z," = 1. As explained in [17], it is sufficient here to use the classical metric on the base S3,i.e. the metric induced by the classical Kahler potential, and thus in spherical polar coordinates (0, E , +}, ds&,,

= A; (do2

+ sin20 (dE2 + sin2 @id2)),

(20)

434

while

W = eiYpA;TrZ

-

2eaYpA; cos 0 ,

(21)

where p = d m = f i / 2 , and we have set Am = 0. This is equivelant, up to normalization, to the superpotential of the S3 model analyzed in Sec. 2. We conclude that the bosonic moduli space for BPS walls in this theory is the same as that obtained for kinks within the S3 model, namely MN=2= R x CP’. Matching the scales, we obtain

d s h = T I d.zi

+ h i j d x i d x j = T Id z i + R,-

( d J i 3- sin2 J0d&)

,

(22)

with T I = 4 4 ; the wall tension and R,- = A ; / p the scale of the reduced moduli space. The underlying Kahler structure of the bulk theory does however leave an imprint on the realization of supersymmetry in the reduced moduli space. The first point to note, following the comments at the end of Sec. 2, is that the present system has twice as many fermions as the S3 model considered earlier. The second set of fermions arise from the cotangent directions of T * ( S 3 ) .We can choose a basis where the complex fermions lying in the chiral multiplet Z decompose into two (real) sets, one $1, the N = 1 partner of the S3 coordinates of the base, and the other $2, the N= 1partner of the cotangent directions. One then finds that a second set of fermionic zero modes arise from $21. The fermionic mode decomposition takes the form

where {&}, for A = 1 , 2 , are two sets of fermionic operators satisfying

1 {&,rljB} = hZj6AB , (24) Tl where hij is the reduced moduli space metric. Thus we now find in full a one-to-two matching between the number of bosonic versus fermionic zero modes. It is important that since the worldvolume is now 2+1-dimensional, this matching condition is a requirement of Lorentz invariance - a constraint that was not present in our earlier discussion of 1+1D kinks. The complex structure on CP1 again leads to an enhancement of supersymmetry. Indeed, it is clear that essentially the same construction as before, now augmented with two-component spinors q’, will lead to the dynamics admitting N= 2 supersymmetry in 2+1D, or four supercharges, only two of which can be identified with the unbroken generators of the bulk superalgebra.

{772,&}

= - ~ A ,B

435

If we drop the translational zero modes, and restrict Qa to the reduced moduli space, with xi = {&,,$o}, then we discover that there is a second unbroken spinor supercharge, existing by virtue of the complex structure J associated with .6?= S2. We can then form a complex spinor charge

Qi

and these charges satisfy the algebra of N= 4 SQM or more importantly, when lifted back to 2+1D, the N=2 superalgebra. The worldvolume theory is then an N = 2 @PI sigma model and the Witten index for this theory is equal to two, implying that there are two supersymmetric vacua and thus two BPS domain walls. The doubling of fermions in this system relative to the worldline theory of the S3 kink is crucial in generating these supersymmetric vacua. In the preceding discussion, we abstracted slightly in ignoring the deformation imposed by considering a hierarchical mass matrix for the quarks. The effect of this deformation at linear order in

corresponds to "twisting" the worldvolume supercharges by a U( 1 ) Killing vector G = Gi& for rotations in $ 0 ,

1 Gi = -.lrAmdi40 . (27) 2 An important feature of this particular deformation on the worldvolume is that it preserves the enhanced N = 2 SUSY [21]. In particular, the worldvolume supercharges persist but pick up corrections due to this twist [17],and we can write them in the form

+

Q L = hut@[@~$JL TEma$R],

Q R = bur@

[W$JR - I T E ~ ~ ~ $ L ] ,

(28)

with the remaining supercharges given by Q L and QR. The Fubini-Study metric is

In the Lagrangian, this twist amounts to the introduction of a potential given by the norm-squared of the Killing vector, AV = hijGiGj/2, although strictly speaking this is now a second order effect and thus could be accompanied by further corrections.

436

4. 1/4-BPS Wall Intersections

As an application of the worldvolume theory described above, we can explore the structure of wall intersections - namely 1/4-BPS junctions of two walls which are possible by virtue of the nonzero multiplicity in each charge sector. These configurations thus constitute a novel class of junctions, distinct from those generically present in theories where the vacua spontaneously break a ZN symmetry. Junction sources are supported by two types of central charge, 2 w and 2 s , transforming respectively in the (0,l) and (1/2,1/2) representations of the Lorentz group. The first of these is supported by BPS walls, while the second is associated with string-like sources. The Bogomol'nyi bound in 2+1D, with one dimension compactified on a circle of circumference L , takes the form [22,23]

M > J2wlL+-% ,

(30)

and 1/4-BPS junction configurations are required t o saturate this bound.

4.1. Junction tension for N f = 1 We first consider the hierarchical regime for gauge group SU(2), and integrate out the second flavor. It is then useful to introduce another dimensionless field Y in the form Y = a ( R : m , 1 ) - 1 / 4 , such that after decoupling

W=

Jm,h: (Y' + Y-') ,

and

Kclass =

d=yY

.

(31)

Provided we take ml << hl, the vacua (Y') = f l lie a t weak coupling and one can construct two BPS wall configurations which we reproduce here in the form (with 8 E [ - 7 r , 7 r ] ) ,

The corresponding trajectories are illustrated in Fig. 1. Having two degenerate walls, we can contemplate the possibility of a 2-wall junction in the form illustrated in Fig. 1. Qualitatively, we see that at large lyl, remote from the junction, the field profiles are essentially those for the wall trajectories (32), i.e. W* for y positive or negative. However, the evolution in y must interpolate smoothly between W+ and W-. A (presumably rapid) transition necessarily occurs in the strong-coupling region near x = 0, y = 0 where the junction is located, where the low-energy description breaks down.

437

Figure 1. The wall junction geometry, indicating the field profiles in the W+ and Wcomponents.

+

The Bogomol'nyi equation for the intersection, &Y = -Y P-3 where l r n ~ l ( ~ Zy), is thus insufficient to determine the field trajectory near the core of the junction. Nonetheless, we can still deduce the junction tension associated with the (1/2,1/2) central charge [24],

2

=

Tj =

+

-id-'

c*

jakdza,

ak=iP&Y,

k=1,2,

(33)

where the integral runs over a large contour in the x y plane. The crucial point is that one may use a large rectangular contour in the plane of Fig. 1, for which only the horizontal sides will contribute, and along which only the phase 6 of Y changes, and so ak = It then follows that,

-a$.

f

akdxk = -A6,

with A6 = 27r,

(34)

where the numerical result holds for the field configuration depicted in Fig. 1. Thus,

This tension is determined [24] by the (real) central charge depends non-holomorphically on the mass parameter. 4.2. Wall intersections as worldvolume

ZS,and thus

@P1 kinks

We can also try to understand these configurations directly from the worldvolume point of view, by adding an additional light flavor so that the 2

438

component walls arise from the dynamics of a CP' sigma model, or more precisely a massive sigma model where the mass term is identified with IrAm1/2 as discussed in the previous section. This theory possesses 1/2BPS kink solitons, and it is natural to identify these kinks as the worldvolume description of 1/4-BPS 2-wall junctions. We will provide evidence for this identification by verifying that the kink tension reproduces the tension of the junction, given in (35), in the appropriate limit. Using complex coordinates for S 2 , as introduced in (13), the bosonic sector of the massive @PI sigma model becomes [25,26]

When lAml is large relative to any dynamically generated scale, the theory has classical vacua at w = 0,m. Using the coordinate relation w = tan $ei9O, one finds that classical BPS kink solutions exist which satisfy another sine-Gordon equation [as], ay[0

1 = f-nlAml sinto, 2

ayq50 = 0

.

(37)

The corresponding tension of the junction is given by 1

T . - -rlAm1(2R,-) 3 - 2

+ O(A,,)

,

(38)

where for the moment we assume Am is large and so provides the dominant mass scale, and Awv is any dynamically generated scale as would arise on compactification to 1+1D. In the present case R,- = A i / p , and we obtain

A simple check on this result follows on integrating out one of the flavors. On sending m2 -+ 00,we must keep A: = m2R; fixed, so that

which agrees precisely with the result obtained earlier in Eq. (35) from a direct analysis of the l-flavor model. The resulting junction tension again depends non-holomorphically on ml as one expects from the bulk point of view.

439

Acknowledgments

It is a pleasure to t h a n k M. Shifman and A. Vainshtein for a fruitful and enjoyable collaboration on these and related issues, and also for their kind invitation to present this work at such a lively a n d stimulating meeting. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

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NEW TOPOLOGICAL STRUCTURES IN QCD*

FALK BRUCKMANN, DANIEL NOGRADI AND PIERRE VAN BAAL Instituut-Lorentz for Theoretical Physics, University of Leiden, P. 0.Box 9506, NL-2900 R A Leiden, The Netherlands.

We review the recent progress made in understanding instantons at finite temperature (calorons) with non-trivial holonomy, and their monopole constituents as relevant degrees of freedom for the confined phase.

1. Introduction There has been a revived interest in studying instantons a t finite temperature T , so-called calorons.'Y2 The main reason is that new explicit solutions could be obtained in the case where the Polyakov loop a t spatial infinity (the so-called holonomy) is non-trivial, necessary t o reveal more clearly the monopole constituent nature of these c a l ~ r o n sTrivial . ~ ~ ~ holonomy, i.e. with values in the center of the gauge group, is typical for the deconfined phase. Non-trivial holonomy is therefore expected to play a role in the confined phase (i.e. for T < T,) where the trace of the Polyakov loop fluctuates around small values. The properties of instantons are therefore directly coupled t o the order parameter for the deconfining phase transition. At finite temperature A0 plays in some sense the role of a Higgs field in the adjoint representation, which explains why magnetic monopoles (in the BPS limit)5>6occur as constituents of calorons. Since A0 is not necessarily static it is better to consider the Polyakov loop as the analog of the Higgs field,

which, like an adjoint Higgs field, transforms under a periodic gauge transformation g(z) t o g ( z ) P ( z ) g - l ( x ) . Here ,O = l / k T is the period in the imaginary time direction, under which the gauge field is assumed to be 'Talk given by the last author

440

44 1

periodic. Finite action requires the Polyakov loop at spatial infinity to be constant. For SU(n) gauge theory this gives

Fa

= 18lim 1+00 P(O,?) = gt exp(27ridiag(pl1p2,

. . . ,p n ) ) g ,

(2)

where g is chosen t o bring Fa t o its diagonal form, with the n eigenvalues being ordered according to

C?='pi = 0,

p1

I p2 I . . . I p.n I pn+1 = 1 + p1.

(3)

In the so-called algebraic gauge, where Ao(z) is transformed to zero at spatial infinity, the gauge fields satisfy the boundary condition

A,(t +,B,?)

= PooA,(t,Z)F&,'.

(4)

Caloron solutions are such that the total magnetic charge vanishes. The "force" stability of these solutions in terms of its constituent monopoles is based, as for exact BPS multi-monopole solutions, on balancing the electromagnetic with the scalar (Higgs) f o r ~ eexcept , ~ ~ ~that for calorons repulsive and attractive forces are interchanged as compared t o multi-monopoles. A single caloron with topological charge one containsg n - 1 monopoles with a unit magnetic charge in the i-th U( 1) subgroup, which are compensated by the n-th monopole of so-called type (1,1,.. . , I), having a magnetic charge in each of these subgroups. This special monopole is also called a KaluzaKlein (KK) monopole.1° At topological charge k there are kn constituents, k monopoles of each of the n types. The monopole of type j has a mass n 87r2vj/,B,with vj = pj+l - p j . The sum rule Cj='vj = 1 guarantees the correct action, 87r2k, for calorons with topological charge k. Prior to their explicit construction, calorons with non-trivial holonomy were considered irrelevant12because the one-loop correction gives rise to an infinite action barrier. However, the infinity simply arises due to the integration over the finite energy density induced by the perturbative fluctuations in the background of a non-trivial Polyakov loop." The nonperturbative contribution of calorons (with a given asymptotic value of the Polyakov loop) to this energy density as the relevant quantity t o be considered, was first calculated in supersymmetric theories,12 where the perturbative contribution vanishes. I t has a minimum where the trace of the Polyakov loop vanishes, i.e. at maximal non-trivial holonomy. Recently the calculation of the non-perturbative contribution was performed in ordinary gauge theory at high temperature^.'^ When added to the perturbative contribution with its minima at center elements," these minima turn unstable for decreasing temperature right around the expected value of T,. This

442

lends some support to monopole constituents being the relevant degrees of freedom which drive the transition from a phase in which the center symmetry is broken at high temperatures to one in which the center symmetry is restored at low temperatures. Lattice studies, both using cooling14 and chiral fermion zero-modes15 as filters, have also conclusively confirmed that monopole constituents do dynamically occur in the confined phase. 2. Properties of caloron solutions

Well-separated constituents can be shown to act as point sources for the socalled far field, that is far removed from any of the cores, where the gauge field is abelian.16 When constituents of opposite charge ( n constituents of different type) come together, the action density no longer deviates significantly from that of a standard instanton. Its scale parameter p is related to the constituent separation d through .rrp2/,6 = d. A typical example for a charge one SU(2) caloron with far and nearby constituents is shown in Fig. 1 (left). When p << ,6 no difference would be seen with the HarringtonShepard solution,' the gauge field is nevertheless vastly different, as follows from the fact that within the confines of the peak there are n locations where two of the eigenvalues of the Polyakov loop ~ o i n c i d e . ~This ~ ~ ~difference ' is also seen to play an important role in the supersymmetric case.lg When, on the other hand, constituents of equal charge come together (which requires llcl > 1) an extended core structure appears.16 For two coinciding

On the left are shown two charge one SU(2) caloron profiles at t = 0 with = 0.125, for p = 1.6 (bottom) and 0.8 (top) on equal logarithmic scales, cutoff below an action density of 1/(2e2). On the right we show the action density (on a linear scale) of a typical SU(2) caloron with topological charge 2 and pz = -p1 = 0.25 for which two constituents of equal magnetic charge are closer than their individual sizes (but not exactly on top). When the other two constituents are far away, as for the case shown here, this becomes a charge two monopole solution. Figure 1.

p = 1 and

pz = - p i

443

constituents this gives rise to the typical doughnut structure also observed for monopoles,20 see Fig. 1 (right). 2.1. Fermion tero-modes

An essential property of calorons is that the chiral fermion zero-modes, whose existence is guaranteed by the index theorem,22 are localized to constituents of a certain charge only. The latter depends on the choice of boundary condition for the fermions in the imaginary time direction (allowing for an arbitray U(l) phase exp(2~Zz)).~' This result can be understood in terms of the Callias index theorem,23 and provides an important signature for the dynamical lattice studies, using chiral fermion zero-modes as a filter.15 To be precise, the zero-modes are localized to the monopoles of type m provided pm < z < p,+1 (most of the time we use the classical scale invariance to set p = 1). Denoting the zero-modes by G:(z), where a = 1,.. . , k for a caloron of topological charge lc, we can write

G:(z)+G;(z)

= -(2n)-2a33z7

2).

(5)

pab(*,

where z') is a Green's function that appears in the construction to be discussed below. The trace, i.e. the sum over the zero-mode densities, has a remarkably simple form in the far field limit (denoted by ff and defined by neglecting terms that decay exponentially with the distance to any of the constituent cores) n . f : ( z , z ) = 47r2vm(2), for

pm

< z < p,+1.

(6)

As is implicit in the notation, V , is static and independent of z within its interval of definition. It has been calculated explicitly for k = 1 and 2. In addition V , has to be harmonic (up to singularities), because the zeromodes decay exponentially as long as z # pj (for any j ) , and therefore do not survive in the far field limit. For k = 1 and g, the constituent location one simpIy has V m ( 2 )= 1/(4nl2- gml),whereas for k = 2 we found16

where y'(p) = ( d s c o s p, 0, sin p), up to an arbitrary coordinate shift and rotation. Here V is a scale and k a shape parameter to characterize arbitrary SU(2) charge 2 solutions. In this representation it is clear that V m ( 2 )is harmonic everywhere except on a disk bounded by an ellipse with minor axes 2 V d m and major axes 22). Although not directly obvious,

444

when k + 1 the support of the singularity structure is on two points only, separated by a distance 2D. Taking an arbitrary test function f (Z) one can prove that16 - lim k+ 1

J ~ ( z ) ~ : v ~ (f(o,z ) ~ ~+zf(o, =

0, D)

0 , -D).

(8)

Monopoles of different charges have to adjust to each other to form an exact caloron solution, such that k and D are in general not independent. So far we constructed two classes of solutions illustrated in Fig. 2 for both of which a large value of D implies that k approaches 1 exponentially. Hence we find point-like constituents, a necessary requirement to describe the field configurations at larger distances in terms of these objects. When all constituents of other types are sent to infinity, we recover the exact multimonopole solutions of a given type of magnetic charge, and our results therefore also provide explicit solutions for the monopole zero-modes, which were not known before for the multi-monopole configurations. Surprisingly, the charge distribution that gives rise to the abelian field far from any of the constituent cores (even when extended due to overlap) can be calculated exactly from V m ( Z ) . H u r t u b i ~ efound ~ ~ earlier for the asymptotic behavior of the abelian field exactly the same result through his study of the algebraic tail of the Higgs field, using twistor methods. Let us first consider SU(2), for which we can parametrize the holonomy by ’Pa= e x p ( 2 ~ G . 7(p2 ) = -p1 = Id/), where T, are the usual Pauli matrices. I Y

Figure 2. Illustration of the location of the disk singularities (light and dark shaded according to magnetic charge) for a so-called “rectangular” (left) and “crossed” (right) configuration, as used for the k = 2 solutions shown in resp. Fig. 1 and Fig. 3 . The curves for the “crossed” case represent a one parameter family of solutions interpolating between two axially symmetric solutions for which k = 1 independent of V , giving point-like constituents without the need to take 2) large. See Ref. 16 for more details.

445 The field asymptotically becomes abelian, and is necessarily proportional to Lj .7‘. One can therefore write AF(5) = 27riL3.7‘ - 4iLj . ?@(Z), where the constant term (absent in the algebraic gauge) shows again how A0 plays the role of a Higgs field. We found for k = 1 and 2 that @(Z) = 27r(V1(5)- U z ( Z ) ) . Therefore, the singularity structure in the zero-mode density agrees exactly with the abelian charge distribution, as given by a:@(?). Clearly our expectation is that this result will hold for arbitrary k and will generalize to SU(n) with each Um(Z)simply associated t o the U(1) generator with respect to which the magnetic charge of the monopole of type m is defined. Since the electric field is given in terms of the gradient of Ao, which due to the self-duality is equal to the magnetic field, a particularly simple formula results for the action density S. Using in addition that outside the cores there is no source for the abelian field, implying A0 to be harmonic in the far field (as is also clear from the relation to Urn), we find

Sff(5)= -2tr(aiA0(2))~= 2@trAi(Z) = -a@2(Z).

(9)

Therefore, the algebraic tail of the action density is known to all orders in 1/1.’1, which is of course equivalent to the fact that the charge distribution, a:@(?), giving rise to this asymptotic field is also exactly known, even though the formula for the action density can of course only be used outside any of the constituent cores. 3. The construction - in brief

Our construction of caloron solutions is in outline simple, and in its practical implementation relies heavily on the Atiyah-Drinfeld-HitchinManin constructionz5 of multi-instantons and on the closely related Nahm transformation.26 For the k = 1 Harrington-Shepard solution’ one simply takes a periodic array of R4 instanton solutions, which can be easily constructed through the ’t Hooft ansatz, provided the relative color orientation between subsequent periods is trivial. This necessarily gives trivial holonomy. Here the algebraic gauge discussed earlier is useful, as it shows that with non-trivial holonomy, shifting over each period the gauge field rotates in color space by an amount exactly given by the holonomy. It necessitates the use of the full ADHM formalism. The crucial observation has been that the “twisted” shift symmetry lends itself very well to Fourier transformation and makes contact with the Nahm transformation for calorons.26 The variable z we introduced in formulating the generalized boundary conditions for the chiral fermions is precisely the dual o f t under this transformation.

446

3.1. From ADHM to Nahm The ADHM construction for charge k instantonsZ5 starts with a 2k x 2k complex matrix B = 0, @IB, (each B, is a hermitian k x k matrix, whereas t~, = (12,ZT)) and a k dimensional vector X = ( X I , . . . , A,), where X i is a two-component spinor in the f i representation of SU(n) (i.e. X is a n x 2k complex matrix). These are combined to form a ( n + 2 k ) x 2k dimensional matrix A ( x ) , which has n normalized eigenvectors with vanishing eigenvalue, A t ( x ) u ( z )=O. The matrix u ( z ) is of size (n+2k) x n and one has

where the quaternion x = x p o p (a 2 x 2 matrix with spinor indices) defines the position. The hermitian positive definite n x n matrix $(x),

$(z)=n,+x

( ~ t ( z p ( X ) ) - l~

t ,

(11)

is required to normalize u(z). The gauge field A,(z) = V ~ ( ~ ) ~ ' ~ V U ( Z C ) is self-dual provided

(12)

= 12 @ fz,

where fz is a hermitian k x k matrix. This condition states that A t ( z ) A ( z ) is invertible and commutes with the quaternions, and is known as the quadratic ADHM constraint, turning a set of non-linear partial differential equations into a quadratic matrix equation. To construct a charge k caloron with non-trivial holonomy we place k instantons in the time interval [0,p[, performing a color rotation with Pa for each shift o f t over 0,cmp. Eq. (4). This is implemented by requiring (suppressing color and spinor indices and scaling p to 1) Xpk+k+a

=PcaXpkfa,

Bpk+a,qk+b

= Bpk-k+a,qk-k+b

Solutions to these equations are parametrized by &k+a

= PLCal

Bpk+a,qk+b

Ca

+ aO6pqdab.

(13)

and A;b,

= pgO6pq6ab

+ ebq,

(14)

with A to be determined so as to satisfy the quadratic ADHM constraint. To perform the Fourier transformation we introduce the n projectors P,, = EmeZaipmPm, such that &+a = Eme z a a P p m P m c a . This gives

P

P

m

447

<:

Here is again a two-component spinor in the fi representation of SU(n) and A"b(z) = all@(.), with All(,) an anti-hermitian k x k matrix. In terms of the latter

Ij$(z)

d = W ( z ) - 27rixsab = qp+ Aab(*) - 27rix6"b, dz

which is the positive chirality Dirac (i.e. Weyl) operator for the U(k) gauge field ~,(z)-27rzxplk defined on the circle, z E [0,1],i.e. with periodicity 1 (B-' in case p # 1). Introducing and Ggb,through

Sg

12Sz - 7" ckb,

27rciPm
(17)

the quadratic ADHM constraint gives precisely Nahm's equation,26

3 . 2 . From Green's function to solution

For topological charge 1 the dual gauge field is abelian, and the commutator terms in Eq. (18) vanish, such that A,(,) is simply piecewise constant, jumping a t the singularities. On the other hand the power of the ADHM formalism lies in the "magic" f ~ r m u l a e ~that ~ > ~express * the gauge field, fermion zero-mode, its density, and the action density in terms of fz as defined in Eq. (12). Fourier transformation turns the matrix fz into a Green's function fz(z,2 ) = it(z)fz(z,z')ij(z'), where fz(z, z') satisfies

1

{ -2

-I- V ( Z ; 2)

with

V ( z ;2) = 47r222(z; 2)

f&,

z') = 42nk6(z-z'),

(19)

+ 27r C S ( z - pm)Sm, sm= ij(/Lm)Sm$(pm), m

R j ( z ; Z )3 " j - (27ri)-lij(*)Aj(z)ij+(z).

(20)

Note that the Sm play the role of "impurities" and that

$ ( z ) 3 exp(27ri(
(21)

defined in terms of the dual holonomy exp(27rifo) = Pexp(JtAo(z)dz), can be used to transform A 0 - 27rixoIlk to zero, in order to simplify as much as possible the Green's function equation. This is a t the expense of

448

introducing periodicity up to a gauge transformation. Although fz(z, 2’) is periodic in z and z’ with period 1 (for /3 = l),fz(z,z’) no longer is. Given a solution for the Green’s function, there are straightforward expressions for the gauge fieldz9 only involving the Green’s function evaluated at the “impurity” locations and for the fermion zero-modes.16 For the zeromode density see Eq. (5). As an example we give the Green’s function a t z’ = z , which formally can be expressed as (x dependence of .Fz is implicit)

(22) where the (1,2) component on the right-hand side of the first identity is with respect to the 2 x 2 block matrix structure. This has allowed us to find a particularly compact expression for the action density,2g

S ( x ) = -+trF$(x)

= -+6’;8Zlogdet

(ie-nizo(12k - F z ) ) ,

(23)

which can be shown to be independent of z . The formal expression for .Fz can be made more explicit by a decomposition into the “impurity” contributions T, at z = p, and the “propagation” Hm wm(prn+1,p m ) between p m and pm+i, with for z,z’ E ( p m ,pm+i)

The kxk matrices (Z) are defined for

and behave as as

where be arbitrary non-singular matrices. STogether these form the 2k solustions of the homogeneous Green's function equation,

($-

) 6(2;2)

47r29(z; 2)

= 0.

We have been able to use this “mix” of the ADHM and Nahm formalism both in making powerful approximations, like in the far field limit (based on our ability t o identify the exponentially rising and falling terms), and for finding exact solutions through solving the homogeneous Green’s function equation for k = 2. What makes this case tractable is the fact that the Nahm equations on each interval can be solved in terms of elliptic function^,^' 4(z)Aj(Z)et(Z)E27ri

[ajn2+DRjbfb (47T2)(2-20))

htTbh] ,

(26)

449

where h, R is a (gauge) rotation, a’ is a coordinate shift and

Jz-

with snk, cnk and dnk(z) = the standard Jacobi elliptic functions. This does not yet address matching of & ( z ) on different intervals, where some difference between the monopole and caloron application appears. For the caloron, apart from the axially symmetric solutions constructed in Ref. 29, we found two sets of non-trivial solutions that interpolate between overlapping and well-separated constituents, see Fig. 2. To resolve the full structure of the cores we had to find the exact solutions to Eq. (25). For this task we could make convenient use of an existing analytic results for charge 2 monopoles,31 adapting it to the case calorons. Essential is that once the solutions f $ ( ~ )are known, everything else can be easily determined in terms of these, cmp. Eq. (24). We conveniently expressed the explicit solutions for Eq. (25) in terms of the elliptic integral of the third kind, but refer to Ref. 16 for further details. A sample of the results can be found in Fig. 1 (right) and Fig. 3.

Figure 3. In the middle is shown the action density in the plane of the constituents at t = 0 for an SU(2) charge 2 caloron with trP, = 0 in the “crossed” configuration of Fig. 2, where all constituents strongly overlap. On a scale enhanced by a factor 1 6 ~ ’are shown the densities for the two zero-modes, using either periodic (left) or anti-periodic (right) boundary conditions in the time direction. Note that these still are able to identify four individual, albeit deformed, constituents.

450

4. Conclusions We have seen that instantons at finite temperature are composed of constituent monopoles. Of course, the hope is t o develop a calculational scheme t o address questions like monopole condensation in the popular scenario of the dual ~ u p e r c o n d u t o r However, .~~ this is very much complicated by the fact t h a t at low temperature the coupling constant is large, and instantons form a dense ensemble. This does imply that the monopoles form a dense ensemble as well. One may then have some hope that the confining electric phase could be characterized by a dual deconfining magnetic phase, where the dual deconfinement is due t o the large monopole density, in a similar spirit t o high density induced quark deconfinement .

Acknowledgements

PvB is grateful t o the organisers for inviting him again and for continuously creating stimulating workshops with a broad spectrum of informative talks. The research of F B is supported by FOM. References 1. B.J. Harrington and H.K. Shepard, Phys. Rev. D17 (1978) 2122; Phys. Rev. D18 (1978) 2990. 2. D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43. 3. T.C. Kraan and P. van Baal, Phys. Lett. B428 (1998) 268 [hep-th/9802049]; Nucl. Phys. B533 (1998) 627 [hep-th/9805168]. 4. K. Lee, Phys. Lett B426 (1998) 323 [hep-th/9802012]; K. Lee and C. Lu, Phys. Rev. D58 (1998) 025011 [hep-th/9802108]. 5. G. ’t Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, JETP Lett. 20 (1974) 194. 6. E.B. Bogomol’ny, Sov. J. Nucl. Phys. 24 (1976) 449; M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 (1975) 760. 7. N.S. Manton, Nucl. Phys. B126 (1977) 525. 8. C. Montonen and D. Olive, Phys. Lett. 72B (1977) 117. 9. T.C. Kraan and P. van Baal, Phys. Lett. B435 (1998) 389 [hep-th/9806034]. 10. K.Lee and P. Yi, Phys. Rev. D56 (1997) 3711. 11. N. Weiss, Phys. Rev. D24 (1981) 475. 12. N.M. Davies, T.J. Hollowood, V.V. Khozeand M.P. Mattis, Nucl. Phys. B559 (1999) 123 [hep-th/9905015]. 13. D. Diakonov, N. Gromov, V. Petrov and S. Slizovskiy, “Quantum weights of dyons and of instantons with non-trivial holonomy,” hep-th/0404042. 14. E.-M. Ilgenfritz, B.V. Martemyanov, M. Miiller-Preussker,S. Shcheredin and A.I. Veselov, Phys. Rev. D66 (2002) 074503 [hep-lat/0206004].

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15. C. Gattringer and S. Schaefer, Nucl. Phys. B654 (2003) 30 [hep-lat/0212029]; C. Gattringer, R. Pullirsch, Phys.Rev. D69 (2004) 094510 [hep-lat/0402008]. 16. F. Bruckmann, D. N6grbdi and P. van Baal, Nucl. Phys. B666 (2003) 197 [hep-th/0305063]; hep-th/0404210. 17. M. Garcia PBrez, A. Gonzblez-Arroyo, A. Montero and P. van Baal, JHEP 06 (1999) 001 [hep-lat/9903022]. P. van Baal, in: “Lattice fermions and structure of the vacuum”, eds. V. Mitrjushkin and G. Schierholz (Kluwer, Dordrecht, 2000), p. 269 [hep-th/9912035]. 18. E.M. Ilgenfritz, B.V. Martemyanov, M. Muller-Preussker and A.I. Veselov, Phys. Rev. D69 (2004) 114505 [hep-lat/0402010]. 19. D. Diakonov and V. Petrov, Phys. Rev. D 67 (2003) 105007 [hep-th/0212018]. 20. P. ForgBcs, Z. HorvBth and L. Palla, Nucl. Phys. B192 (1981) 141; M.F. Atiyah, and N.J. Hitchin, “The Geometry and Dynamics of Magnetic Monopoles”, (Princeton Univ. Press, 1988). 21. M. Garcia PBrez, A. Gonzblez-Arroyo, C. Pena and P. van Baal, Phys. Rev. D60 (1999) 031901 [hep-th/9905016]; M.N. Chernodub, T.C. Kraan, P. van Baal, Nucl. Phys. B(ProcSupp1.)83-84 (2000) 556. 22. M.F. Atiyah and I.M. Singer, Ann. Math. 87 (1968) 484; Ann. Math. 93 (1971) 119. 23. C.J. Callias, Comm. Math. Phys. 62 (1978) 213. 24. J . Hurtubise, Comm. Math. Phys. 97 (1985) 381. 25. M.F. Atiyah, N.J. Hitchin, V. Drinfeld and Yu.1. Manin, Phys. Lett. 65A (1978) 185; M.F. Atiyah, “Geometry of Yang-Mills fields”, Fermi lectures, (Scuola Normale Superiore, Pisa, 1979). 26. W. Nahm, “Self-dual monopoles and calorons”, in: Lecture Notes in Physics, 201 (1984) 189. 27. E.F. Corrigan, D.B. Fairlie, S. Templeton and P. Goddard, Nucl. Phys. B140 (1978) 31. 28. H. Osborn, Ann. Phys. (N.Y.) 135 (1981) 373; Nucl. Phys. B159 (1979) 497. 29. F. Bruckmann and P. van Baal, Nucl. Phys. B645 (2002) 105 [hepth/0209010]. 30. W. Nahm, “Multi-monopoles in the ADHM construction”, in: Gauge theories and lepton hadron interactions, eds. Z. HorvBth, e.a., (Budapest, 1982); S.A. Brown, H. Panagopoulos and M.K. Prasad, Phys. Rev. D26 (1982) 854; A S . Dancer, Comm. Math. Phys. 158 (1993) 545. 31. H. Panagopoulos, Phys. Rev. D28 (1983) 380. 32. S. Mandelstam, Phys. Rept. 23 (1976) 245. G.’t Hooft, in: High Energy Physics, ed. A. Zichichi (Editrice Compositori, Bolognia, 1976); Nucl. Phys. B138 (1978) 1.

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SECTION 6. SUPERSYMMETRY AND THEORETICAL METHODS

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VISCOSITY O F S T R O N G L Y C O U P L E D G A U G E THEORIES

P. KOVTUN* Department of Physics, University of Washington, Seattle, WA 98195-1560, USA Email: [email protected]

D. T. SON+and A. 0. STARINETS$ Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA Email: [email protected], [email protected]

We extend the correspondence between black hole physics and thermodynamics to non-equilibrium processes. The shear viscosity is shown t o correspond to the absorption cross section of low-energy graviton by the black hole. We show that the ratio of shear viscosity to volume density of entropy in theories with gravity duals is equal to a universal value of h/(47r). We conjecture that this value is a lower limit on the ratio of shear viscosity to entropy density for a very wide class of systems, Possible applications of this visconsity bound conjecture are mentioned.

1. Introduction

The discovery of Hawking radiation1 confirmed that black holes are endowed with thermodynamic properties such as entropy and temperature, as first suggested by Bekenstein2 based on the analogy between black hole physics and equilibrium thermodynamics. For black branes, i.e. , black holes with translationally invariant horizons, thermodynamics can be extended to hydrodynamics-the theory that describes long-wavelength deviations from thermal equilibrium. Thus, black branes possess hydrodynamic properties of continuous fluids and can be characterized by kinetic coefficients *Supported by uppercasedoe grant DOE-FG03-96ER40956 t Supported by DOE grant DOE-FG02-00ER41132 and the Alfred P. Sloan Foundation. SSupported by DOE grant DOEFG02-00ER41132

455

456

such as viscosity, diffusion constants, etc. From the perspective of the holographic principle3v4, the hydrodynamic behavior of a black-brane horizon is identified with the hydrodynamic behavior of the dual theory. In this talk, we argue that in theories with gravity duals, the ratio of the shear viscosity t o the volume density of entropy is equal to a universal value of fi/4n. Moreover, we argue that this is the lowest possible value for the ratio of shear viscosity to entropy density in all isotropic systems described by a finite-temperature local quantum field theory. 2. Dimension of q / s

The standard textbook definition of the shear viscosity is as follows. Take two large parallel plates separated by a distance d. The space between the two plates is filled with a fluid. Let one plate moves relative to the other with a velocity u. Then the drag force acting on a unit area of the plate is

_F - v A

-'z

which defines the viscosity 7. It is measured in kg m-l s-'. In d spatial dimensions, shear viscosity q is measured in kg m2-d s-'. The volume density of entropy s is measured in m-d (in units where the Boltzmann constant ICE is set to one). The ratio q / s thus has the dimension of kg m2 s-l, i.e., the same as the Planck constant ti. This observation is more than merely a curiosity, as we shall see shortly. 3. Viscosity from dual gravity description

Consider a field theory dual to a black-brane metric. One can have in mind, as an example, the Af = 4 supersymmetric Yang-Mills theory dual to the the near-extremal D3 brane in type IIB supergravity,

but our discussion is not tied t o any specific form of the metric. All black branes have an event horizon ( r = rg for the metric (2)), which is extended along several spatial dimensions (2,y, z in the case of (2)). The dual field theory is at a finite temperature, equal to the Hawking temperature of the black brane. The entropy of the dual field theory is equal to the entropy of the black brane, which is proportional to the area of its event horizon, s = -A (3) 4G '

457

where G is the Newton constant (we set ti = c = 1). For black branes A contains a trivial infinite factor V equal to the spatial volume along directions parallel to the horizon. The entropy density s is equal to a/(4G), where a = A/V. The shear viscosity of the dual theory can be computed from gravity in a number of approaches5t6l7.Here we use Kubo's formula, which relates viscosity to equilibrium correlation functions. In a rotationally invariant field theory,

Here Txyis the zy component of the stress-energy tensor (one can replace Txyby any component of the traceless part of the stress tensor). We shall now relate the right hand side of (4) to the absorption cross section of low-energy gravitons. According t o the AdS/CFT correspondence8, the stress-energy tensor T,, couples to metric perturbations at the boundary. Following Klebanovg, let us consider a graviton of frequency w,polarized in the xy direction, and propagating perpendicularly to the brane. In the field theory picture, the absorption cross section of the graviton by the brane measures the imaginary part of the retarded Greens function of the operator coupled to hxy, i.e., Tzy, Oabs(w)

2 2

= --ImG w

R(W) = K,2 w /dtdxe"t

([Tx,(t,x),Txy(0,O N 1 (5)

where K , = appears due the normalization of the graviton's action. Comparing (4) and (5), one finds

The absorption cross section g a b s , on the other hand, is calculable classically by solving the linearized wave equation for h:. It can be shown (see Appendix) that under rather general assumptions the equation for h; is the same as that of a minimally coupled scalar. The absorption cross section for the scalar is constrained by a which states that in the low-frequency limit w 4 0 this cross section is equal to the area of the horizon, Cabs = a. Since s = a/4G, one immediately finds that

-q _- - ti s

4n'

458

where h is now restored. This shows that the ratio q / s does not depend on the concrete form of the metric within the assumptions ofloill. Indeed, explicit calculations of the viscosity using the AdS/CFT correspondence or the “membrane paradigm” technique show that the ratio q / s is the 1/(47r) for Dp5l7, M2 and M512 branes and N = 2* deformations of the D3 metric7>l3. 4. A viscosity bound

Dual gravity description of gauge theories is valid in the regime of infinitely strong coupling. As Eq. (7) shows, in this regime the ratio q / s appears to be universal (independent of the coupling constant and other microscopic details of the theory). Let us now argue that the ratio q / s approaches infinity in the limit of vanishing coupling. The entropy density s of a weakly coupled system is proportional to the number density of quasiparticles n,

s-n.

(8)

The shear viscosity is proportional to the product of the energy density and the mean free time (time between collisions) T

-

q

nu,

(9)

where E is the average energy per particle (which is of the order of the temperature T ) . Therefore rl - € T . S

Now, in order for the quasiparticle picture to be valid, the width of the quasiparticles must be small compared to their energies, i.e., one should have

h

- << E T

which means that rl

->h S

The observation that q / s is a constant in strongly coupled theories with gravity dual and is large in weakly coupled theories prompts us t o formulate the “viscosity bound” conjecture: in any finite-temperature field theory, the

459 ratio of shear viscosity to entropy density cannot be smaller than the value of this ratio in theories with gravity duals:

-r 2] -f .i s

47T

As we have seen, the bound (13) can be understood as a consequence of the uncertainty principle: the product of the energy and the mean free time of a quasiparticle cannot be smaller than ti. The precise numerical coefficient 1/(47~)cannot, however, be obtained from the uncertainty principle alone. 5 . Checks of the viscosity bound

The viscosity bound has a verifiable consequence for thermal N = 4 supersymmetric Yang-Mills theory. It implies any value of the ’t Hooft coupling g t M N the ratio r]/s is larger than 1/(47~).In particular, it implies that the Erst correction in the strong-coupling expansion of r]/s is positive. Corrections in inverse power of ’t Hooft coupling correspond to stringtheory corrections to supergravity. For the entropy density, it has been computed by Gubser, Klebanov and T~eytlin:’~

1 The correction to St arinet s: l5

r],

.

and hence r]/s, was found by Buchel, Liu and

The first correction to r]/s is positive, in agreement with the conjectured bound. Moreover, since the viscosity bound does not contain the speed of light c, one can try to check it on nonrelativistic laboratory liquids. First let us check the conjecture for the most ubiquitous fluid-water. Under normal conditions ( P = 0.1 MPa, T = 298.15 K) the viscosity of water is r] M 0.89 x Pa s and the entropy density is s M 2.8 x lo2’ m-3. The ratio r]/s is 380 times larger than ti/(47r). Using standard tables16 one can find r]/s for many liquids and gases at different temperatures and pressures. Figure 1 shows temperature dependence of r]/s, normalized by ti/(47r), for a few substances at different pressures. It is clear that the viscosity bound

460 I

I

I

I

, 1 1 1 ,

471 n

I

I'

'

,

4,

II1 1 ' 1

I t : " '

I

I

loot

01

1

I

10

, , ,,*I

100

I

1N

T,K Figure 1.

The viscosity-entropy ratio for some common substances.

is well satisfied for these substances. Liquid helium reaches the smallest value of q / s , but this value still exceeds the bound by a factor of about 10. It is important to avoid some common misconceptions which a t first sight seem to invalidate the viscosity bound. One might think that an ideal gas has an arbitrarily small viscosity, which violates the bound. However, the viscosity of a gas diverges when the interaction between molecules is turned off. This is because for gases viscosity is proportional to the mean free path of the molecules. The second common misconception involves superfluids, which seem to have zero viscosity. However, according to Landau's two-component theory, superfluids have finite and measurable shear viscosity associated with the normal component. 6. Conclusion

It is useful t o compare the viscosity bound with two other bounds widely discussed in the literature: the entropy bound (e.g., in its covariant form~l a t i on' ~) which states that the entropy of a region of space is limited by the area A of the region's boundary, c3 A S<---, hG 4

461

and the Bekenstein boundla, which states that the entropy of a system is limited by the product of its linear size R and mass M , n

S

< 427rRM. ii

(18)

The viscosity bound can also be written as an upper bound on the entropy,

where V is the total volume. It is similar t o the Bekenstein bound in the sense that it covers non-gravitating systems despite having the origin in the theory of gravitation. The viscosity bound stands apart as the only bound that does not involve the speed of light. Because of this feature, it is relevant for non-relativistic systems, in contrast to the other bounds. One may hope that it is possible to relate the viscosity bound to the two other bounds, and eventually t o the generalized second law of thermodynamics. It will be important to learn whether the viscosity bound suffers from the same problems as the other bounds such as the species problem. The bound (13) is most useful for strongly interacting systems where no theoretical calculation of viscosity is possible. One of such systems is the quark-gluon plasma (QGP) created in heavy ion collisions which behaves in many respects as a droplet of a liquid. As the viscosity of the QGP is not computable reliably except a t inaccessibly high temperatures, the information provided by the conjectured bound is extremely valuable. There are experimental hints that the viscosity of the QGP at temperatures achieved by the Relativistic Heavy Ion Collider is surprisingly small, possibly close t o saturating the viscosity boundlg. Further investigations may reveal whether the QGP conforms to the viscosity bound. Another possible application of the viscosity bound is trapped atomic gases. By using the Feshbach resonance, strongly interacting Fermi gases of atoms have been created recently. These gases have been observed t o behave hydrodynamically20. Currently the viscosity bound is the only source of information about the viscosity of these gases. It would be very interesting t o test the bound experimentally using these gases.

Appendix A. In this Appendix, we elaborate on the statement made after Eq. (6) that under rather general assumptions the equation for h; is identical t o the one obeyed by a minimally coupled massless scalar.

462

Consider a D-dimensional solution to Einstein’s equations of the form

ds2 = gg&dxMdxN = f ( z ) (dx2

+ dy2) + g,,(Z)

dPdzy.

(A.1)

We shall be interested in small perturbations around the metric, g,, = g,,,( 0 ) h,,. We assume that the only non-vanishing component of h,, is

+

h,,, and that it does not depend on x and y: h,, = hx,(z). This field has spin 2 under the O(2) rotational symmetry in the xy plane, which implies that all other components of h,, can be consistently set to zero6. We now show that h; = hxY/f obeys the equation for a minimally coupled massless scalar in the background (A.1). Einstein’s equations can be written in the form

T RMN= T M N- D -2 gMN’

(-4.2)

where the stress-energy tensor T M Nand its trace T depend on other fields such as the dilaton and various forms supporting the background (A.l) (for example, the fields appearing in the low energy type I1 string theory). Again, O(2) xy rotational symmetry implies that all perturbations of matter fields can be set to zero consistently. Thus when M and N are x or y, the right hand side of Einstein’s equations reads

where C is the Lagrange density of matter fields and T(O)is the trace of the unperturbed stress-energy tensor. Substituting the unperturbed metric (A.l) into Einstein’s equations, one finds

Expanding Einstein’s equations to linear order in h,,, one has

Combining Eqs. (A.4) and (A.5)’ we obtain an equation for h,,

Changing the variable from h,, to h; = hxy/f , one can see that h&indeed satisfies the equation for a minimally coupled massless scalar: Oh&= 0.

463 References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17. 18. 19. 20.

S. W. Hawking, Commun. Math. Phys. 43,199 (1975). J. D. Bekenstein, Phys. Rev. D7, 2333 (1973). G. 't Hooft, gr-qd9310026. L. Susskind, J . Math. Phys. 36,6377 (1995). G. Policastro, D. T. Son and A. 0. Starinets, Phys. Rev. Lett. 87,081601 (2001). G. Policastro, D. T. Son and A. 0. Starinets, J . High Energy Phys. 0209, 043 (2002). P. Kovtun, D. T. Son and A. 0. Starinets, J. High Energy Phys. 0310,064 (2003). For review, see: 0. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y . Oz, Phys. Rep. 323,183 (2000). I. R. Klebanov, Nucl. Phys. B496,231 (1997). S. R. Das, G. W. Gibbons and S. D. Mathur, Phys. Rev. Lett. 78,417 (1997). R. Emparan, Nucl. Phys. B516,297 (1998). C. P. Herzog, J . High Energy Phys. 0212,026 (2002). A. Buchel and J. T. Liu, hep-th/0311175. S. S. Gubser, I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B534, 202 (1998). A. Buchel, J. T. Liu and A. 0. Starinets, hep-th/0406264. E. W. Lemmon, M. 0. McLinden and D. G. Friend, NIST Chemistry WebBook (http://webbook.nist.gov); J. D. Cox, D. D. Wagman and V. A. Medvedev, CODATA Key Values for Thermodynamics (http://www.codata.org). R. Bousso, J. High Energy Phys., 9907,004 (1999). J. D. Bekenstein, Phys. Rev. D23, 287 (1981). E. Shuryak, Prog. Part. Nucl. Phys. 53,273 (2004). K. M. O'Hara et al., Science 298,2179 (2002).

ADS/CFT DUALITY FOR STATES WITH LARGE QUANTUM NUMBERS

A.A. TSEYTLIN* Smith Laboratory The Ohio State University Columbus, OH 43210-1106 USA

We review recent progress in quantitative checking of AdS/CFT duality in the sector of “semiclassical” string states dual to “long” scalar N = 4 super YangMills operators. In particular, we describe the effective action approach, in which the same sigma model type action describing coherent states is shown to emerge from the Ads5 x S 5 string action and from the integrable spin chain Hamiltonian representing the SYM dilatation operator.

1. Introduction

The N = 4 SYM theory is a remarkable example of 4-d conformal field theory. In the planar ( N -+ co) limit it is parametrized by the ’t Hooft coupling X = g:MN, and the major first step towards the solution of this theory would be t o determine the spectrum of anomalous dimensions A(A) of the primary operators built out of products of local gauge-covariant fields. That this may be possible in principle is suggested by the AdS/CFT duality implying the existence of hidden integrable 2-d structure corresponding to AdSs x S5 string sigma model. The AdS/CFT duality implies the equality between the A d s energies of quantum closed string states as functions of the effective string tension T = 5 and quantum numbers like S5 angular momenta Q = ( J , ...) and dimensions of the corresponding local SYM operators. To give a quantitative check of the duality one would like t o understand how strings cLemerge’l from the field theory, in particular, which (local, single-trace) gauge theory operators correspond to which “excited” string states and how one may verify the matching of their dimensions/energies beyond the well-understood ‘Also a t Imperial College London and Lebedev Physics Institute, Moscow.

464

465

BPS/supergravity sector. We would like to use the duality as a guide t o deeper understanding of the structure of quantum SYM theory. In particular, results motivated by comparison t o string theory may allow one t o “guess” the general structure of the SYM anomalous dimension matrix or dilatation operator. This may shed light on the form of interpolation from weak t o strong coupling and may also suggest new methods of computing anomalous dimensions in less supersymmetric gauge theories. Below we shall review recent progress in checking AdS/CFT correspondence in a subsector of string/SYM states with large quantum numbers. Let us start with brief remarks on SYM and string sides of the duality. The SYM theory contains a gauge field, 6 scalars 4, and 4 Weyl fermions, all in adjoint representation of S U ( N ) . It has global conformal and Rsymmetry, i.e. is invariant under S 0 ( 2 , 4 ) x SO(6). To determine (in planar limit) dimensions of local gauge-invariant operators one in general needs t o find the anomalous dimension matrix t o all orders in A and then to diagonalize it. The special case is that of chiral primary or BPS operators (and their descendants) tr(Q1{,, ...Q1mh}) whose dimensions are protected, i.e. do not depend on A. The problem of finding dimensions appears to simplify also in the case of “long” operators containing large number of fields under the trace. One example is provided by “near-BPS” operators like tr(@{@y...) ... where J >> n, and @ k = 4 k ?&+3, k = 1 , 2 , 3 . Below we will consider “far-from-BPS” operators like t r ( @ p@$ ...) ... where J1 JZ >> 1. The type IIB string action in Ad& x S5 space has the following structure

+

+

+

-

1 I = --T/dT 2

27r

+

du ( d P Y p d P Y V ~ , , dPXmdpXnG,,

+++

+ ...)

, (1)

where YpY”qpV= -1, XmX”S,, = 1 , qpV = (+-), T = and dots stand for the fermionic terms that ensure that this model defines a 2-d conformal field theory. The closed string states can be classified by the values of the Cartan charges of the obvious symmetry group S0(2,4) x S 0 ( 6 ) ,i.e. (El5’1,s~; J1, Jz, J3), i.e. by the Ad& energy, two spins in Ads5 and 3 spins in S 5 . The mass shell condition gives a relation E = E ( Q ,T ) . Here T is the string tension and Q = (S1,S2, 51, J2, J3; Tzk) where stand for higher conserved charges (analogs of oscillation numbers in flat space). According to AdS/CFT duality quantum closed string states in Ads5 x S5 should be dual to quantum SYM states at the boundary R x S 3 or, via radial quantization, to local single-trace operators at the origin of R4.

466

The energy of a string state should then be equal to the dimension of the corresponding SYM operator, E ( Q ,T ) = A(Q, A), where on the SYM side the charges Q characterise the operator. By analogy with flat space and ignoring a‘ corrections (i.e. assuming R -+ co or a‘ + 0) the excited string states are expected to have energies E 1 A l l 4 which represents a non-trivial prediction for strong-coupling asymptotics of SYM dimensions. In general, the natural (inverse-tension) perturbative expansion on the string side will be given by En &, while on the SYM side the usual planar perturbation theory will give the eigenvalues of the anomalous dimension matrix as A = C n u n A n . The AdS/CFT duality implies that the two expansions are to be the strong-coupling and weak-coupling asymptotics of the same function. To check the relation E = A is then a non-trivial problem. On the symmetry grounds, this can be shown in the case of 1 / 2 BPS (chiral primary) operators dual to supergravity states (“massless” or ground state string modes) since their energies/dimensions are protected from corrections. For generic non-BPS states the situation looked hopeless before a remarkable suggestion ‘v4 that progress is checking duality can be made by concentrating on a subsector of states with large ( LLsemiclassical’l) values of quantum numbers, Q T f i (here Q stands for generic quantum number like spin in Ads5 or S5 or a n oscillation number) and considering the new limit limit

-a-

N

-

A

A == fixed , Q2

where on the string side

5 -& plays the role of the semiclassical pa=

rameter (like rotation frequency) which can then be taken to be large. The energy of such states happens to be E = Q + f ( Q , A). The duality implies that such semiclassical string states as well as near-by fluctuations should be dual to “longrrSYM operators with large canonical dimension, i.e. containing large number of fields or derivatives under the trace. In this case the duality map becomes more explicit. The simplest possibility is to start with a BPS state that carries a large quantum number and consider small fluctuations near it, i.e. a set of nearBPS states all characterised by a large parameter ’. The only non-trivial example of such BPS state is described by a point-like string moving along geodesic in S5 with large angular momentum Q = J = J1. Then E = J and the dual operator is t r a J , = & + id’. The small (nearly pointlike) closed strings representing near-by fluctuations are ultrarelativistic,

467

i.e. their kinetic energy is much larger than their mass. They are dual t o SYM operators of the form tr(@J . . . ) where dots stand for a small number of other fields and/or covariant derivatives (and one needs to sum over different orders of the factors to find an eigenstate of the anomalous dimension matrix). The energies of the small fluctuations happen to be E =J N, O( -$). One can argue in general and check explicitly that higher-order quantum string sigma model corrections to the leading square root term in the fluctuation energy are indeed suppressed in the limit (2), i.e. in the large J , fixed 5 = -$= A’ limit. The remarkable feature of this expression is that E is analytic in i, suggesting direct comparison with perturbative SYM expansion in A. Indeed, it can be checked directly that the first two and terms in the expansion of the square root agree precisely with the one and two lo (and also three loop terms in the anomalous dimensions of the corresponding operators. there is also a general argument l3 (for a 2-impurity case) suggesting how the full expression can appear on the perturbative SYM side. However, the general proof of the consistency of the BMN limit on the SYM side (i.e. that the usual perturbative expansion can be rewritten as an expansion in iand remains to be given; also, to explain why the string and SYM expressions match one should show that the string limit (first J -+ 00, then i= -+ 0) and the SYM limit (first X + 0, then J + m) produce the same expressions for the dimensions (cf. l4,l5>l6). If one moves away from the near-BPS limit and considers, e.g., a nonsupersymmetric closed string state with a large angular momentum Q = S in AdS5 4 , a direct quantitative check of the duality is no longer possible: here the classical energy is not analytic in X and quantum corrections are no longer suppressed by powers of $. However, it is still possible t o demonstrate the remarkable qualitative agreement between the string energy and SYM anomalous dimension as far as the dependence on the spin S is concerned. The energy of a folded closed string rotating a t the center of Ads5 which is dual t o the twist 2 operators on the SYM side (tr(@:D’@h), D = D1 +iDz and similar operators with spinors and gauge bosons that mix at higher loops 19,20) has the form (when expanded at large S): E = S+f(X)lnS+ .... On the string side f(X)x,l = c o f i + c l + a + ..., Jj; In 2 is the l-loop coefficient. where co = is the classical and c1 = On the gauge theory side one finds the same S-dependence of the anomalous dimension with the perturbative expansion of the In S coefficient being f(X),,, = a1X azX2 a 3 X 3 ..., where a1 = 18, a2 = 19, and ‘9’

+ Jm +

697

899

x2

11i12)

J

m

3)

-2

2

+

+

+

&

-&

468

a3 = -20. Like in the case of the SYM entropy 21, here one expects the existence of a smooth interpolating function f ( X ) that connects the two perturbative expansions. One could still wonder if examples of quantitative agreement between string energies and SYM dimensions observed for near-BPS (BMN) states can be found also for more general non-BPS string states. Indeed, it was noticed already in that a string state that carries large spin in Ads5 as well as large spin J = 0 in S5 has, in contrast to the above J = 0 case, an analytic expansion of its energy in i= -$, just as in the BMN case with N,, S. It was observed in 22 that in general semiclassical string states carrying several large spins (with at least one of them being in S 5 ) have regular expansion of their energy in square of effective tension or in powers of iand it was suggested, by analogy with the near-BPS case, that the corresponding coefficients can be matched with the coefficients in the perturbative expansion for the SYM dimensions. For a classical rotating closed string solution in S5 one has E = f i & ( w i ) , J; = f i w ; so that E = E ( J ; , X ) and the key property is that there is no fifactors in the expression of the energy (as it was in the case of a single spin in AdS5)

-

X

A2

E = J+ci-+c:!-+...= J J3

J [1+c1i+czi2+...

1,

(3)

xi=l

A where J = J i , X- = 7 and c, = c n ( $ ) are functions of ratios of the spins which are finite in the limit Ji >> 1 , X =fixed. The simplest example of such a solution is provided by a circular string rotating in two orthogonal planes in S3 part of S5 with the two angular momenta being equal J1 = J2 22: XI XI i x 2 = cos(na) eiwT, ~2 G ~3 i x 4 = sin(na) eiwT, with the global Ads5 time being t = ICT (Y5 iY0 = eit). The conformal gauge constraint implies K' = w2 n2 and thus E = dor E = J(1+ ...), where J = J1+ J2 = 2J1. For fixed J the energy thus has a a regular expansion in tension (in contrast t o what happens in

+

+

i n 2 i in4x2+

+

F

+

flat space where E = 7J ) . Similar expressions (3) are found also for more general multispin circular strings In particular, for a folded string string rotating in one plane of S5 and with its center of mass orbiting along big circle in another plane 24 the coefficients c , are transcendental functions (expressed in terms of elliptic integrals). More generally, the 3-spin solutions are described by a n integrable Neumann model and the coefficients c, in the energy are expressed in terms of genus two hyperelliptic functions. 22f23124925926~27.

25726

469

To be able t o compare the classical energy to the SYM dimension one should be sure that string a’ corrections are suppressed in the limit J + 00, 5 =fixed. Formally, this should be the case since a’ 2 - but, fi J f i ’ what is more important, the $ corrections are again analytic in i23, i.e. the expansion in large J and small is well-defined on the string side,

-

N

di d2 l + X ( c i + j + ...)+ X 2 ( c 2 + -J+

1

...)+... ,

(4)

with the classical energy (3) being the J -+00 limit of the exact expression.a Similar expressions are found for the energies of small fluctuations near a given classical solution: as in the BMN case, the fluctuation energies are ...) i 2 ( k 2 7 suppressed by extra factor of J , i.e. 6E = X ( K 1

+ + +

...) + ....

+ +

Assuming that the same limit is well-defined also on the SYM side, one should then be able to compare the coefficients in (4) to the coefficients in the anomalous dimensions of the corresponding SYM operators 22 t r ( @ F @ 2 @ 2 ) ... (and also do similar matching for near-by fluctuation modes). In practice, it is known at least in principle how t o compute the dimensions in a different limit: first expanding in X and then expanding in -$. One may expect that this expansion of anomalous dimensions takes the form equivalent to (4), i.e.

+

A = J + X ( -a1 + J

bl

a2 -+...) + A 2 ( - + J2 J3

b2 +...) +... , J4

(5)

and, moreover, the respective coefficients in (4) and (5) agree with each other. The subsequent work did verify the structure of (5) and moreover established the general agreement between the two leading coefficients cl, c2 in (4) and the “one-loop” and “two-loop” coefficients a l l a2 in ( 5 ) . To compute (5) one is first to solve a technical problem of how to diagonalize anomalous dimension matrix defined on a set of long scalar operators. The crucial step was made in 28 where it was observed that the one-loop 28329130931732j33*15*34335

aThe reason for this particular form of the energy (4) can be explained as follows we are computing string sigma model loop corrections to the mass of a stationary solitonic solution on a 2-d cylinder (no IR divergences). This theory is conformal (due to the crucial presence of fermionic fluctuations) and thus does not depend on UV cutoff. The As a result, the inrelevant fluctuations are massive and their masses scale as w :’2p‘

- 3.

verse mass expansion is well-defined and the quantum corrections should be proportional to positive powers of X.

470

planar dilatation operator in the scalar sector can be interpreted as a Hamiltonian of an integrable SO(6) spin chain and thus can be diagonalized even for large length L = J by the Bethe ansatz method. In the simplest case of (closed) “SU(2)” sector of operators t r ( @ p @ $ ) ... built out of two chiral scalars the latter can be interpreted as spin up and spin down states of periodic XXX1/2 spin chain with length L = J = J1 J2, Then the 1-loop dilatation operator becomes equivalent t o the Hamiltonian of the ferromagnetic Heisenberg model

+

+

Using this relation and considering the thermodynamic limit ( J + m) of the Bethe ansatz the proposal of 22 was confirmed at the leading order of expansion in 5 in 29,30. Namely, it was found that for eigen-operators with both J1 and J2 being comparable and large A - J = A? .,. and a remarkable agreement was found between a1 and the coefficient c1 (which both are non-trivial functions of $) in the energies of various 2-spin string solutions. As in the BMN case, it was possible also to match the energies of fluctuations near the circular J1 = J2 with the corresponding eigenvalues of ( 6 ) 29. Similar leading-order agreement between string energies and SYM dimensions was observed also in other sectors of states with large quantum numbers: (i) for specific solutions in the SU(3) sector of states with 3 spins in S5 dual to tr(@?@$@$) ... operators (ii) for a folded string state belonging t o the SL(2) 37 sector of states with one spin in Ads5 and one spin in S5 (with E = J + S -i-$c1(5) ... 6 * 2 2 ) dual to t r ( D S a J ) ... 30; (iii) in a “subsector” of SO(6) states containing pulsating (and rotating) solutions which again have regular energy expansion in the limit of large oscillation number, e.g., E = L c1$ ... 38.

+

22925926

+

32936;

+

+

29932

+

+

2. Effective actions for coherent states

The observed agreement between energies of particular semiclassical string states and dimensions of the corresponding “long” SYM operators leaves many questions, in particular: (i) How to understand this agreement beyond specific examples, i.e. in a universal way? Can one derive a relevant limit of string sigma model action directly from SYM dilatation operator? (ii) Which is a precise relation between profiles of string solutions and the structure of the dual SYM operators? (iii) How to characterise the set

47 1

of semiclassical string states and dual SYM operators for which the correspondence should work? (iv) Why agreement works, i.e. why the two limits (first J + 00, and then i-+ 0, or vice versa) taken on the string and SYM sides give equivalent results? Should it work to all orders in expansion a n alin i(and $)? The questions (i),(ii) were addressed in ternative approach based on matching the general solution (and integrable structure) of the string sigma model with that of the thermodynamic limit of the Bethe ansatz was developed in 34. The question (iii) was addressed in 43,44,45,42 , and the question (iv) - in One key idea was that instead of comparing particular solutions one should try t o match effective sigma model actions which appear on the string side (in the limit J -+ 00, i-+ 0) and the SYM side (in the limit 5 -+ 0, J -+m). Another related idea was that since “semiclassical” string states carrying large quantum numbers are represented in the quantum theory by coherent states, one should be comparing coherent string states t o coherent SYM (spin chain) states. The crucial point is that because of the ferromagnetic nature of the dilatation operator (6) in the thermodynamic limit J = J1 J2 -+ m with fixed number of impurities it is favorable to form large clusters of spins and thus a “low-energy” approximation and continuum limit should apply, leading t o an effective “non-relativistic” sigma model for a coherent-state expectation value of the spin operator. Taking the “large energy” limit directly in the string action gives a reduced “non-relativistic” sigma model that describes in a universal way the leading-order O ( i ) corrections to the energies of all string solutions in the two-spin sector. The resulting action agrees exactly 33 with the semiclassical coherent state action describing the S U ( 2 ) sector of the spin chain in the J -+ 00, i=fixed limit. This demonstrates how a string action can directly emerge from a gauge theory in the large-N limit and provides a direct map between the “coherent” SYM states and all two-spin classical string states, bypassing the need to apply the Bethe ansatz to find anomalous dimension for each particular state. Furthermore, the correspondence established at the level of the action implies also the matching of fluctuations around particular solutions. Let us briefly review the definition of coherent states. Starting with the SU(2) algebra [S3,S*] = fS*, [S+,S-] = 2S3 and considering the s = 1/2 representation where S = f 3 one can define spin coherent state as a linear superposition of spin up and spin down states: Iu) = R(u)JO), where R = eus+-u’s- , 10) = 1 f , and u is a complex number that can be parametrized as u = i 8 e i @ . An equivalent way to label the coherent state 33135*40941142;

15116917.

33935942

2

+

-4

4)

472

is a by a unit 3-vector 5 defining a point of S 2 . Then 15) = R(5)lO)where 10) corresponds t o a 3-vector (0, 0 , l ) along the 3-rd axis ( 5 = UtSU, U = (211,212)) and R(5) is an SO(3) rotation from a north pole to a generic point of S2. The key property of the coherent state is that ii determines the expectation value of the spin operator: (filslfi)= f f i . In general, one can rewrite the usual phase space path integral as a path integral over the overcomplete set of coherent states (for the harmonic oscillator this is simply the change of variables u = ’ ( q + i p ) ) : 2 = J [ d u ]eiSI”].The action J;i is S = J d t ( ( u l i ~ l u ) - ( u I H I u ) ) where , the first (WZ or ‘‘Berry phase”) term is the analog of the usual pq term in the phase-space action. Applying this t o the case of the Heisenberg spin chain Hamiltonian (6) one ends up with with the following action for the coherent state variables & ( t )at sites 1 = 1,..., J (see also, e.g., 39): S = C{=,I d t [ ( ? ( n l ) . % - & (a) fi~+l - i i l ) ’ ] . Here d C = e i j k n i d n j A d n k , i.e. (? is a monopole potential on s2.In the limit J + 00 with fixed = -$ we are interested in one can take a continuum limit by introducing the 2-d field 5(t,u ) = {5(t,?I)}. Then

S=J/dt12a

2 [B .

1&fi - -8X ( & f i ) 2

1

+ ... ,

(7)

5.

where dots stand for higher derivative terms suppressed by Since in the limit we are interested in J -+ 00 all quantum corrections are also suppressed by $, and thus the above action can be treated classically. The corresponding equation of motion n; = i i e ; j k n j n i are the Landau-Lifshitz equation for a classical ferromagnet. The action (7) should be describing the coherent states of the Heisenberg spin chain in the above thermodynamic limit. One may wonder how a similar “non-relativistic” action may appear on the string side where one starts with a usual sigma model (1)which is quadratic in time derivatives. However, to obtain an effective action that can be compared t o the spin chain one is first t o perform the following procedure (i) isolate a “fast” coordinate (Y whose momentum p , is large in our limit; (ii) gauge fix t 7- and p , J or ii u where ii is “T-dual” to a; (iii) expand the action in derivatives of “slow” or ‘transverse” coordinates (to be identified with G ) . To see how this procedure can be implemented explicitly let us consider the SU(2) sector of string states carrying two large spins in S5, with string motions restricted t o S3 part of S5. The relevant part of the A d s 5 x S5 metric is then ds2 = -dt2 d X i d X f , with X ; X f = 1, and we can set X 1 = X1 iX2 = uleiLI, X2 = X3 iX4 = u3eia, u;uf = 1. Here (Y will be a collective coordinate associated to the total (large) 33135*42:

N

-

N

+

+

+

473 spin in the two planes (which in general will be the sum of orbital and internal spin). ui will be “slow” coordinates determining the “transverse” string profile (ui are defined modulo U(1) gauge transformation). Then dXidX,t = (da C)’ DuiDuf, where C = -iuTdui, Dui = dui - iCui and the second term represent the metric of C P 1 (this parametrisation corresponds t o Hopf fibration S3 S 1 x S 2 ) . Introducing 5 = UtZU, U = ( 2 ~ 1 , ~ ’ we ) get d X i d X r = (Da)’ ;(dii)’, Da = da! C ( n ) . Writing the resulting sigma model action in phase space form and imposing the (non-conformal) gauge t = T, p , =const= J one gets the action (7) with the WZ term 6 . 8,s originating from the p,Da term in the phase-space Lagrangian. This conforms to its origin on the spin chain side as an analog of the ‘pq’ in the phase space action. Equivalent approach is based on first applying a 2-d duality (or “T-duality”) a + 6 and then choosing the “static” gauge t = 7, 5 = ( f i ) - ’ o , = Applying T-duality ~~~) Thus we get L = - ~ J - 9 9 P ~ ( - a , t ~ , t + d , 5 ~ , 5 + D , ~ T D +@%‘,aq& the “T-dual” background has no off-diagonal metric component but has a non-trivial NS-NS 2-form coupling in the (6,ui) sector. Eliminating the 2-d metric g p q we then get the Nambu-type action L = P C p d , 6 where h = ldet h,,l and h,, = -aptaqt 8,6aq6 D~,uTD,)ui. If we now fix the static gauge we finish, to the leading order in A, with

+

+

-+

+

(A)-’ 5.

+

+

6,

which is the same as the C P 1 Landau-Lifshitz action ( 7 ) when written in terms of 5. We thus uncover 42 the origin of the string-theory counterpart of the WZ term in the spin-chain coherent state effective action (7): it comes from the 2-d NS-NS WZ term upon the static gauge fixing in the “T-dual” 6 action. The agreement between the low-energy actions on the spin chain and the string side explains not only the matching between energies and coherent states for configurations with two large spins (and near-by fluctuations) but also the matching of integrable structures observed on specific examples in 31,32. To summarize: (i) (t15)play the role of longitudinal coordinates that are gauge-fixed (with 5 playing the role of string direction or spin chain direction on the SYM side); (ii) U = (ul,u2) or 6 = U t X J are “transverse” coordinates that determine the semiclassical string profile and also the structure of the coherent operator on the SYM side, t r IIo(ui@i). This leading-order agreement in SU(2) sector has several generalizations. First, we may include higher-order terms on the string side. System-

474

atically expanding in iand eliminating higher powers of time derivatives by field redefinitions (note that leading-order equation of motion is 1st order in time) we end up with 35

The same i2 term is obtained 35 in the coherent state action on the spin chain side by starting with the sum of the 1-loop dilatation operator (6) and the 2-loop one l1

This explains the matching of energies and dimensions t o the first two orders, as first observed on specific examples using Bethe ansatz in 15. Equivalent general conclusion about 2-loop matching was obtained in the integrability-based approach in 34. The order-by-order agreement seems to break down at h3 (%loop) order, which has a natural explanation 15*16in that the string limit (first J -+ 00, then X -+ 0) and the SYM limit (first X -+ 0, then J + 00) need not be the same.b One can also generalize the above leading-order agreement to the SU(3) sector of states with three large S5 spins J i , i = 1,2,3, finding the CP2 analog of the CP1 “LandauLifshitz” Lagrangian in (7),(8) 41 C = --iufd~ui- iiIDlui12 on both string and spin chain sides. Similar conclusion is found 41 in the SL(2) sector of ( S ,J ) states. Finally, one can also consider pulsating string states 42. One may try also to go further and use the duality t o string theory as a tool t o determine the structure of planar SYM theory to all orders in X by imposing the exact agreement with particular string solutions. For example, demanding the consistency with the BMN scaling limit determines (along with the superconformal algebra) the structure of the full %loop SYM dilatation operator in the SU(2) sector l1sl2. One can also use the BMN limit t o fix a part of the dilatation operator but to all orders in X 47. Generalizing (6),(10) and the 3- and 4-loop expressions in l1>l2one can organize the dilatation operator as an expansion in powers of &k,l = I - a‘k . & which reflects interactions between spin chain sites, D = C Q C Q Q + C QQQ+ ..., where the products Q...Q are “irreducible”, i.e. each 40t41

+

bSuggestions how t o “complete” the gauge-theory answer t o restore the agreement with disagreement string theory appeared in l 6 , I 7 . This may also resolve the order i3 corrections t o the BMN spectrum. between string a n d gauge theory predictions for

5

475

site index appears only once. The QQ-terms first appear at 3 loops, QQQterms - a t 4 loops, etc. l 1 > l 2 Concentrating . on the order Q part D(’) of D and demanding the BMN-type scaling limit (and agreement with the BMN square root spectrum) one finds, in the limit of large L , i.e. when D acts on “long” operators,

goes rapidly to zero at large k , so we effectively have a spin chain with short range interactions. The function fk(X) smoothly interpolates between the usual perturbative expansion at small X and 6 at strong X (which is the expected behaviour of anomalous dimensions of LLlong’l operators dual to states). Similar interpolating functions are expected to appear in anomalous dimensions of other SYM operators. Acknowledgments We are grateful to M. Kruczenski, A. Ryzhov and B. Stefanski for collaborations on the work described above. This work was supported by the DOE grant DE-FG02-91ER40690, the INTAS contract 03-51-6346 and the RS Wolfson award. fk

N

References 1. A. M. Polyakov, “Gauge fields and space-time,” Int. J. Mod. Phys. A 17S1, 119 (2002) [hep-th/0110196]. 2. D. Berenstein, J. M. Maldacena and H. Nastase, “Strings in flat space and pp waves from N =4 super Yang Mills,” JHEP 0204,013 (2002) [hep-th/0202021]. 3. R. R. Metsaev and A. A. Tseytlin, “Type IIB superstring action in AdS(5) x S(5) background,” Nucl. Phys. B 533, 109 (1998) [hep-th/9805028]. 4. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A semi-classical limit of the gauge/string correspondence,” Nucl. Phys. B 636,99 (2002) [hep-th/0204051]. 5. R. R. Metsaev, “Type IIB Green-Schwarz superstring in plane wave RamondRamond background,” Nucl. Phys. B 625, 70 (2002) [hep-th/0112044]. R. R. Metsaev and A. A. Tseytlin, “Exactly solvable model of superstring in plane wave Ramond-Ramond background,” Phys. Rev. D 65, 126004 (2002) [hepth/0202109]. 6. S. Frolov and A. A. Tseytlin, “Semiclassical quantization of rotating superstring in Ads5 x S5 ,” JHEP 0206, 007 (2002) [hep-th/0204226]. 7. A. A. Tseytlin, “Semiclassical quantization of superstrings: Ads5 x S5 and beyond,” Int. J. Mod. Phys. A 18,981 (2003) [hepth/0209116].

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477 Mills,” JHEP 0303,0 13 (2003) [hep-t h/02 122081. 29. N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, “Stringing spins and spinning strings,” JHEP 0309,010 (2003) [hep-th/0306139]. 30. N. Beisert, S. Frolov, M. Staudacher and A. A. Tseytlin, “Precision spectroscopy of AdS/CFT,” JHEP 0310,037 (2003) [hep-th/0308117]. 31. G. Arutyunov and M. Staudacher, “Matching higher conserved charges for strings and spins,” JHEP 0403,004 (2004) [hep-th/0310182]. “Two-loop commuting charges and the string / gauge duality,” hep-th/0403077. 32. J. Engquist, J. A. Minahan and K. Zarembo, “Yang-Mills duals for semiclassical strings on Ads5 x S5 ,” JHEP 0311,063 (2003) [hep-th/0310188]. 33. M. Kruczenski, ‘‘Spin chains and string theory,” hepth/0311203. 34. V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, “Classical/quantum integrability in AdS/CFT,” hep-th/0402207. 35. M. Kruczenski, A. V. Ryzhov and A. A. Tseytlin, “Large spin limit of Ads5 x S5 string theory and low energy expansion of ferromagnetic spin chains,” h e p th/0403 120. 36. C. Kristjansen, “Three-spin strings on AdS(5) x S**5 from N = 4 SYM,” Phys. Lett. B 586, 106 (2004) [hep-th/0402033]. L. Freyhult, “Bethe ansatz and fluctuations in SU(3) Yang-Mills operators,’’ hep-th/0405167. C. Kristjansen and T. Mansson, “The Circular, Elliptic Three Spin String from the SU(3) Spin Chain,” hep-th/0406176. 37. N. Beisert and M. Staudacher, “The N=4 SYM integrable super spin chain”, Nucl. Phys. B 670, 439 (2003) [hep-th/0307042]. 38. J. A. Minahan, “Circular semiclassical string solutions on Ads5 x S5 ,” Nucl. Phys. B 648, 203 (2003) [hepth/0209047]. 39. E. H. Fradkin, “Field Theories Of Condensed Matter Systems,” Redwood City, USA: Addison-Wesley (1991) 350 p. (Frontiers in physics, 82). 40. R. Hernandez and E. Lopez, “The SU(3) spin chain sigma model and string theory,” JHEP 0404,052 (2004) [hep-th/0403139]. 41. B. J . Stefanski, Jr. and A. A. Tseytlin, “Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations,” JHEP 0405, 042 (2004) [ h e p th/0404133]. 42. M. Kruczenski and A. A. Tseytlin, “Semiclassical relativistic strings in S5 and long coherent operators in N = 4 SYM theory,” arXiv:hepth/0406189. 43. D. Mateos, T. Mateos and P. K. Townsend, “Supersymmetry of tensionless rotating strings in Ads5 x S5 , and nearly-BPS operators,” hepth/0309114. “More on supersymmetric tensionless rotating strings in Ads5 x S5 ,” hepth/0401058. 44. A. Mikhailov, “Speeding strings,” JHEP 0312,058 (2003) [hep-th/0311019]. 45. A. Mikhailov, “Slow evolution of nearly-degenerate extremal surfaces,” h e p th/0402067. “Supersymmetric null-surfaces,” hepth/0404173. 46. J . A. Minahan, “Higher loops beyond the SU(2) sector,” hep-th/0405243. 47. A. V. Ryzhov and A. A. Tseytlin, LLTowards the exact dilatation operator of N = 4 super Yang-Mills theory,” arXiv:hep-th/0404215.

PLANAR EQUIVALENCE: FROM TYPE 0 STRINGS TO QCD

A. ARMONI Department of Physics, Theory Division C E R N , CH-1211 Geneva 23, Switzerland M. SHIFMAN * William I. Fane Theoretical Physics Institute, University of Minnesota, Minneapolis, M N 55455, USA G. VENEZIANO Department of Physics, Theory Division C E R N , CH-1211 Geneva 23, Switzerland

This talk is about the planar equivalence between N = 1 gluodynamics (superYang-Mills theory) and a non-supersymmetric “orientifold field theory.” We outline an “orientifold” large N expansion, analyze its possible phenomenological consequences in one-flavor massless QCD, and make a first attempt a t extending the correspondence to three massless flavors. An analytic calculation of the quark condensate in one-flavor QCD starting from the gluino condensate in N = 1 gluodynamics is thoroughly discussed.

1. Genesis of the idea

Kachru and Silverstein studied various orbifolds of R6 within the framework of the AdS/CFT correspondence. Starting from N = 4,they obtained distinct four-dimensional (daughter) gauge field theories with matter, with varying degree of supersymmetry, N = 2,1,0, all with vanishing ,8 functions. Shortly after, Bershadsky and Johansen abandoned the string theory set-up altogether. They proved that non-supersymmetric large-N orbifold *conference speaker

478

479

field theories admit a zero beta function in the framework of field theory per se. The first attempt to apply the idea of orbifoldization to non-conformal field theories was carried out by Schmaltz who suggested a version of Seiberg duality between a pair of non-supersymmetric large-N orbifold field theories. After a few years of a relative oblivion, the interest in the issue of planar equivalence was revived by Strassler 4 . In the inspiring paper entitled “On methods for extracting exact non-perturbative results in nonsupersymmetric gauge theories” he shifted the emphasis away from the search of the conformal daughters, towards engineering QCD-like daughters. Unfortunately, it turned out that Strassler conjecture could not be valid. The orbifold daughter theories “remember” that they have fewer vacua than the parent one, which results in a mismatch in low-energy theorems. In string theory language, the killing factor is the presence of tachyons in the twisted sector. This is clearly seen in light of the calculation presented in Ref. ‘. 596

2. Orientifold field theory and

N = 1 gluodynamics

Having concluded that, regretfully, the planar equivalence of the orbifold daughters does not extend to the non-perturbative level, we move on to another class of theories, called orientifolds, which lately gave rise to great expectations. We will argue that in the N + 03 limit there is a sector in the orientifold theory exactly identical to N = 1 SYM theory, and, therefore, exact results on the IR behavior of this theory can be obtained. This sector is referred to as the common sector. The parent theory is N = 1 SUSY gluodynamics with gauge group SU(N). The daughter theory has the same gauge group and the same gauge coupling. The gluino field A; is replaced by two Weyl spinors 77[ijl and [rijl, with two antisymmetrized indices. We can combine the Weyl or !PIij]. Note that the number spinors into one Dirac spinor, either 9[ij] of fermion degrees of freedom in is N2 - N , as in the parent theory in the large-N limit. We call this daughter theory orientifold A . There is another version of the orientifold daughter - orientifold S. Instead of the antisymmetrization of the two-index spinors, we can perform symmetrization, so that A; -+ ( r ] { i j ) , [iij)). The number of degrees of freedom in P{ij) is N 2 N. The field contents of the orientifold theories

+

480

is shown in Table 1. We will mostly focus on the antisymmetric daughter since it is of more physical interest; see Sect. 4. Table 1. The field content of the orientifold theories. Here, 77 and E are two Weyl fermions, while A, stands for the gauge bosons. In the left (right) parts of the table the fermions are in the two-index symmetric (antisymmetric) representation of the gauge group SU(N).

Adj

0

0

The hadronic (color-singlet) sectors of the parent and daughter theories are different. I n the parent theory composite fermions with mass scaling as N o exist, and, moreover, they are degenerate with their bosonic SUSY counterparts. In the daughter theory any interpolating color-singlet current with the fermion quantum numbers (if it exists at all) contains a number of constituents growing with N . Hence, at N = co the spectrum contains only bosons. 2.1. Perturbative equivalence

Let us start from perturbative considerations. The Feynman rules of the planar theory are shown in Fig. 1. The difference between the orientifold theory and N = 1 gluodynamics is that the arrows on the fermionic lines point in the same direction, since the fermion is in the antisymmetric representation, in contrast to the supersymmetric theory where the gaugino is in the adjoint representation and the arrows point in opposite directions. This difference between the two theories does not affect planar graphs, provided that each gaugino line is replaced by the sum of q[,,] and <[.I. There is a one-to-one correspondence between the planar graphs of the two theories. Diagrammatically this works as follows (see, for example, Fig. 2). Consider any planar diagram of the daughter theory: by definition of planarity, it can be drawn on a sphere. The fermionic propagators form closed, non-intersecting loops that divide the sphere into regions. Each time we cross a fermionic line the orientation of color-index loops (each one producing a factor N ) changes from clock to counter-clockwise, and vice-

48 1

a

b

C

Figure 1. (a) The fermion propagator and the fermion-fermion-gluon vertex. (b) N = 1 SYM theory. (c) Orientifold daughter.

versa, as is graphically demonstrated in Fig. 2c. Thus, the fermionic loops allow one t o attribute to each of the above regions a binary label (say *1), according to whether the color loops go clock or counter-clockwise in the given region. Imagine now that one cuts out all the regions with a -1 label and glues them again on the sphere after having flipped them upside down. We will get a planar diagram of the SYM theory in which all color loops go, by convention, clockwise. The number associated with both diagrams will be the same since the diagrams inside each region always contain an even number of powers of g, so that the relative minus signs of Fig. 1 do not matter.

Figure 2. (a) A typical planar contribution to the vacuum energy. (b and c) The same in the 't Hooft notation for (b) N=1 SYM theory; (c) orientifold daughter.

Thus, all perturbative results that we are aware of in N = 1 SYM theory

482

apply in the orientifold model as well. For example, the orientifold field theory is

p function of the

p=-- 1

3Na2 27r 1 - (Na)/(27r)

’.

In the large-N limit it coincides with the N = 1 SYM theory result Note that the corrections are 1/N rather than 1/N2. For instance, the exact first coefficient of the ,B function is -3N - 4/3 versus -3N in the parent theory. 2.2. Non-perturbative equivalence proof

Now we will argue that the perturbative argument can be elevated to the non-perturbative level in the case at hand. It is essential that the fermion fields enter bilinearly in the action, and that for any given gauge-field configuration in the parent theory there is exactly the same configuration in the daughter one. Our idea is to integrate out fermion fields for any fixed gluon-field configuration, which yields respective determinants, and then compare them. Consider the partition function of N = 1 SYM theory, 20=

I

DA DX exp (iS[A, A,

4) ,

where J is any source coupled to color-singlet gluon operators. (Appropriate color-singlet fermion bilinears can be considered too.) For any given gluon field, upon integrating out the gaugino field, we obtain

Z~= J

J

DA exp (iS[A, J ] )det (i

g+ PT&j) ,

(2.3)

where Tidjis a generator in the adjoint representation. If one integrates out the fermion fields in the non-supersymmetric orientifold theory, at fixed A, one arrives at a similar expression, but with the generators of the antisymmetric (or symmetric) representation instead of the adjoint, T i d j + TSymm or TGmm. Since the theory, being vector-like, is anomaly free, the determinant in Eq. (2.3) is a gauge-invariant object and, thus, can be expanded in Wilsonloops operators

483

For other representations, the Wilson-loop operator is defined in a similar manner. Thus, one can write

D E det (i &t &Tidj - m) =

X

QC WC[AAdj].

(2.5)

n

Using

D = X a c T r P exp

(il

A; (T" €3 1+ 18 7 " ) d z p

C

Moreover, since the commutator is such that

[(T"€3 l), (1 €3 F")] = 0 , the determinant (2.7) can be rewritten as

D = X a c T r P exp ( i l A L T a d x p ) T r P exp ( i L A i 7 " d x p )

. (2.8)

C

As a result, the partition function takes the form 20

=

CQC(Wc[AglWE[&]) .

(2.9)

C

One of the two most crucial points of the proof is the applicability of factorization in the large-N limit, C

C

In the second equality in Eq. (2.10) we used the second most crucial point, the reality of the Wilson loop,

(Wc) = ( W E )*

(2.11)

The partition function (2.10) is exactly the same as that obtained in the (extended) orientifold theory upon exploiting factorization, Zorientifold

=

QC

( WC (b) ) ( WC (k) ).

(2.12)

C

The third ingredient is independence of the expansion coefficients QC of the fermion representation.

484

At this point we can take the square root of the determinants of the two theories. Owing to the non-vanishing mass, the determinants do not vanish and no sign ambiguities arise. In the parent theory we recover the (softly broken) super-Yang-Mills determinant, while in the daughter theory, given the planar equivalence of the symmetric and antisymmetric representations, we recover either one of them. We finally take the (supposedly) smooth massless limit and, thus, prove our central result. 3. Orientifold large-N expansion

3.1. General features (Nf

> 1 fixed, N

large)

We will now abandon for a while the topic of planar equivalence, and look at the SU(N) orientifold theories with N j flavors from a more general perspective. We will focus on the antisymmetric orientifold theories, assuming that N j does not scale with N at large N , say, N f = 1, 2 or 3. We will return to the issue of planar equivalence later. If N = 3 (i.e. if the gauge group is SU(3)) the two-index antisymmetric quark is identical to the standard quark in the fundamental representation. Therefore, it is quite obvious that extrapolation to large N , with the subsequent 1/N expansion, can have distinct starting points: (i) quarks in the fundamental representation; (ii) quarks in the two-index antisymmetric representation; (iii) a combination thereof. The first option gives rise to the standard ’t Hooft 1/N expansion ’,’, while the second and third lead to a new expansion, to which we will refer as the orientifold large-N expansion.a It is clear that the ’t Hooft expansion underestimates the role of quarks. This was noted long ago, and a remedy was suggested 11, a topological expansion. The topological expansion (TE) assumes that the number of flavors N f scales as N in the large-N limit, so that the ratio N j / N is kept fixed. The graphs that survive in the leading order of T E are all planar diagrams, including those with the quark loops. This is easily seen by slightly modifying l 1 the ’t Hooft double-line notation - adding a flavor line to the single color line for quarks. In the leading (planar) diagrams the quark loops are “empty” inside, since gluons do not attach to the flavor line. Needless to say, obtaining analytic results in TE is even harder than in the ’t Hooft case. a It

is curious that Corrigan and Ramond suggested lo to replace the ’t Hooft model by a model with one two-index antisymmetric quark \k[ijl and two fundamental ones q\,2, as early as 1979.

405

The orientifold large-N expansion opens the way for a novel and potentially rich large-N phenomenology in which the quark loops (i.e. dynamical quarks) do play a non-negligible role. An additional bonus is that in the orientifold large-N expansion, one-flavor QCD gets connected to supersymmetric gluodynamics, potentially paving the way to a wealth of predictions. In order to demonstrate the difference between the standard large-N expansion and the orientifold large N expansion we exhibit a planar contribution to the vacuum energy in two ways in Fig. 3. Moreover, we will illustrate the usefulness of the orientifold large-N expansion at the qualitative, semiquantitative and quantitative levels by a few examples given below.

-.('I-. a .......

.................

b ........... ';

..............

............. __..

f"\* +Q:+ 1 ",,..,

\,-<-(J<-; ;.>........................

c ,..,.'

................ ,..'

.......

.;

............ ..........

Figure 3. (a.) A typical contribution to the vacuum energy. (b.) The planar contribution in 't Hooft large-N expansion. (c.) The orientifold large-A' expansion. The dotted circle represents a sphere so that every Line hitting the dotted circle gets connected "on the other side."

Probably, the most notable distinctions from the 't Hooft expansion are as follows: (i) the decay widths of both glueballs and quarkonia scale with N in a similar manner, as l/N2; this can easily be deduced by analyzing appropriate diagrams with the quark loops of the type displayed in Fig. 3c; (ii) unquenching quarks in the vacuum produces an effect that is not suppressed by 1/N; in particular, the vacuum energy density does depend on the quark masses in the leading order in 1/N. 3.2. Qualitative results for one-flavor QCD from the orientifold expansion

In this subsection we list some predictions for one-flavor QCD, keeping in mind that they are expected to be valid up to corrections of the order of 1/N = 1/3 (barring large numerical coefficients): (i) Confinement with a mass gap is a common feature of one-flavor QCD and SUSY gluodynamics. (ii) Degeneracy in the color-singlet bosonic spectrum. Even/odd parity

486

mesons (typically mixtures of fermionic and gluonic color-singlet states) are expected to be degenerate. In particular, m2,

one-flavor QCD , L L = 1 + 0(1/N), m2, where r]‘ and (T stand for the lightest 0- and O+ mesons, respectively. This follows from the exact degeneracy in n/= 1 SYM theory. Note that the (T meson is stable in this theory, as there are no light pions for it to decay into. The prediction (3.1) should be taken with care: a rather large numerical coefficient in front of 1/N is not at all ruled out, since the r]’ mass is anomaly-driven (the Witten-Veneziano (WV) formula 12*13), whereas the (T mass is more “dynamical.” In more general terms the mass degeneracy is inherited by all those “daughter” mesons that fall into one and the same supermultiplet in the parent theory. The accuracy of the spectral degeneracy is expected to improve at higher levels of the Regge trajectories, as 1,” corrections that induce splittings are expected to fall off (see e.g. ll). (iii) Bifermion condensate. N = 1 SYM theory has a bifermion condensate. Similarly we predict a condensate in one-flavor QCD. A detailed calculation is discussed in Sect. 4. (iv) One can try to get an idea of the size of 1 / N corrections from perturbative arguments. In one-flavor QCD the first coefficient of the ,f3 function is b = 31/3, while in adjoint QCD with N f = 1 it becomes b = 27/3. One can go beyond one loop too. As was mentioned, in the very same approximation the ,B function of the one-flavor QCD coincides with the exact NSVZ ,B function, 1 9a2 2~ 1 - 3 ~ / 2 ~ ‘ Thus, for the (relative) value of the two-loop &function coefficient, we predict +3a/27r, to be compared with the exact value in one-flavor QCD, 134 a a! M 4.32 - . 31 2~ 27r We see that the orientifold large-N expansion somewhat overemphasizes the quark-loop contributions, and, thus, makes the theory less asymptotically free than in reality- the opposite of what happens in ’t Hooft’s expansion. Parametrically, the error is 1/N rather than 1/N2. This is because there are N 2 - 1 gluons and N 2 - N fermions in the orientifold field theory.

p=--

+--

487

4. Calculating the quark condensate in one-flavor QCD

from supersymmetric gluodynamics The gluino condensate in SU(N) SYM theory is (XtXal") = -6NA3.

(4.1)

It is not difficult to show that the correspondence between the bifermion operators is as follows: (X;X"q

c)

(9 Q ) .

(4.2)

The left-hand side is in the parent theory, the right-hand side is in the orientifold theory A, and they project onto each other with the unit COefficient. It is worth emphasizing that Eq. (4.2) assumes that (XEXat") is real and negative in the vacuum under consideration (which amounts to a particular choice of vacuum). It also assumes that the gluino kinetic term is normalized non-canonically, as in Eq. (4.4). Thus, the planar equivalence gives us a prediction for the quark condensate in the one-flavor orientifold theory at N = 00, for free. Our purpose here is to go further, and to estimate the quark condensate at N = 3, i.e. in one-flavor QCD. Table 2. Comparison of the anomalous dimensions and the first two coefficients of the /3 function. If-QCD

Orienti A

Coeff.

3(N-2)(N+1

The anomalous dimension y of the fermion bilinear operators $Q is normalized in such a way that

(4.3)

and p and Q denote the normalization points. For our present purposes we can limit ourselves to the two-loop P functions and the one-loop anomalous dimensions. We can easily check that the various coefficients of the orientifold theory A go smoothly from those of YM (N = 2) through those of lf-QCD ( N = 3) to those of SYM theory at N + 00. Note, however, that some corrections are as large as 2 / N . This is an alarming signal. The gluino condensate (XzXa @) is renormalization-group-invariant (RGI) at any N , and so is A. This is not the case for the quark condensate in the non-SUSY daughter (i.e. the orientifold theory) at finite N . Unlike SUSY theories, where it is customary to normalize the gluino kinetic term as

-

1 --xipx,

(4.4) g2 the standard normalization of the fermion kinetic term in non-SUSY theories is canonic,

GipQ.

(4.5)

Hence, in fact, the correspondence between operators is

(x:x"y

-3( g 2 G

Q) .

(44

In the canonic normalization (4.5), the RGI combination is

GQ = - (g2)1-6(N)

(g2)YIP0

-

(4.7)

!€jQ 7

where

(4.8) and

6(N)

Y = 1- = 0(1/N), PO

19 6(3) = -.

(4.9)

31

Combining Eqs. (4.1), (4.6) and (4.7) we conclude that [S2(PU>]6 ( N ) ( ( 9 2 )

1-6(N)

-

QQ) =

-6(N - 2) A$

wN)

7

(4.10)

where p is some fixed normalization point; the correction factor

K ( p , N = 00) = 1 ,

(4.11)

simultaneously with 6 ( N = a)= 0. At finite N the correction factor K ( p , N ) - 1 = 0(1/N) and K depends on p in the same way as [ g 2 ( p ) ]6 ( N ).

489

-

1-6(N)

The combination (g2) QQ on the left-hand side is singled out because of its RG invariance. Equation (4.10) is our master formula. The factor N - 2 on the right-hand side of Eq. (4.10), a descendant of N l--b(N) in Eq. (4.1), makes (g2) QQ vanish at N = 2. This requirement is obvious, given that at N = 2 the antisymmetric fermion loses color. In fact, we replaced N in Eq. (4.1) by N - 2 by hand, assembling all other 1/N corrections in K, in the hope that all other 1/N corrections collected in K are not so large. There is no obvious reason for them to be large. Moreover, we can try to further minimize them by a judicious choice of p . It is intuitively clear that 1/N corrections in K will be minimal, provided that p presents a scale “appropriate to the process”, which, in the case at hand, is the formation of the quark condensate. Thus, p must be chosen as low as possible, but still in the interval where the notion of g 2 ( p ) makes sense. Our educated guess is

(

)

[g2(p)]6(3)M 4.9.

(4.12)

As a result, we arrive at the conclusion that in one-flavor QCD (4.13) Empiric determinations of the quark condensate with which we will confront our theoretical prediction are usually quoted for the normalization point 2 GeV. To convert the RGI combination on the left-hand side of Eq. (4.13) to the quark condensate at 2 GeV, we must divide by [g2(2 GeV)]1-6(3) x 1.4. Moreover, as has already been mentioned, we expect non-planar corrections in K to be in the ballpark f l / N . If so, three values for K ,

K

(4.14)

= {2/3, 1, 4/31 7

give a representative set. Assembling all these factors together we end up with the following prediction for one-flavor QCD:

( $[ijlQ[ijl)

2 GeV

= - (0.6 to 1.1)

Ak.

(4.15)

Next, our task is to compare it with empiric determinations, which, unfortunately, are not very precise. The problem is that one-flavor QCD is different both from actual QCD, with three massless quarks, and from quenched QCD, in which lattice measurements have recently been carried out 1 4 . In quenched QCD there are no quark loops in the running of a,; thus, it runs steeper than in one-flavor QCD. On the other hand, in threeflavor QCD the running of a, is milder than in one-flavor QCD.

490

To estimate the input value of X m (the ’t Hooft coupling) we resort to the following procedure. First, starting from a,(M,) = 0.31 (which is (3) close to the world average) we determine Am Then, with this A used as an input, we evolve the coupling constant back to 2 GeV according to the one-flavor formula. In this way we obtain X(2 GeV) = 0.115.

(4.16)

A check exhibiting the scatter of the value of X(2 GeV) is provided by lattice measurements. Using the results of Ref. l5 referring to pure YangMills theory one can extract a,(2 GeV) = 0.189. Then, as previously, we find Am, ( 0 ) and evolve back to 2 GeV according to the one-flavor formula. The result is X(2 GeV) = 0.097.

(4.17)

The estimate (4.17) is smaller than (4.16) by approximately one standard deviation CJ. This is natural, since the lattice determinations of a , lie on the low side, within one o of the world average. In passing from Eq. (4.13) to Eq. (4.15) we used the average value X m ( 2 GeV) = 0.1. One can summarize the lattice (quenched) determinations of the quark condensate, and the chiral theory determinations extrapolated to one flavor, available in the literature, as follows: 2 GeV, “empiric”

= - (0.4 to 0.9) A&.

(4.18)

We put empiric in quotation marks, given all the uncertainties discussed above. Even keeping in mind all the uncertainties involved in our numerical estimates, both from the side of supersymmetry/planar equivalence 1/N corrections, and from the “empiric” side, a comparison of Eqs. (4.15) and (4.18) reveals an encouraging overlap. References 1. S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998) [hepth/9802183]. 2. M. Bershadsky and A. Johansen, Nucl. Phys. B 536, 141 (1998) [hepth/9803249]. 3. M. Schmaltz, Phys. Rev. D 59, 105018 (1999) [hep-th/9805218]. 4. M. J. Strassler, On methods for extracting exact non-perturbative results in non-supersymmetric gauge theories, hep-th/0104032, unpublished. 5. A. Gorsky and M. Shifman, Phys. Rev. D 67, 022003 (2003) [hepth/0208073].

491

6. D. Tong, JHEP 0303,022 (2003) Ihep-th/0212235]. 7. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229, 381 (1983); Phys. Lett. B 166,329 (1986). 8. 0. 't Hooft, Nucl. Phys. B 72, 461 (1974). 9. E. Witten, Nucl. Phys. B 160,57 (1979). 10. E. Corrigan and P. Ramond, Phys. Lett. B 87, 73 (1979). 11. G. Veneziano, Nucl. Phys. B 117,519 (1976). 12. E. Witten, Nucl. Phys. B 156,269 (1979). 13. G. Veneziano, Nucl. Phys. B 159,213 (1979). 14. L. Giusti, C. Hoelbling and C. Rebbi, Phys. Rev. D 64,114508 (2001), (E) D65, 079903 (2002) [hep-lat/0108007]. 15. M. Liischer, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 413, 481 (1994) [hep-lat/9309005]; M. Guagnelli, R. Sommer and H. Wittig [ALPHA collaboration], Nucl. Phys. B 535,389 (1998) [hep-lat/9806005].

GAUGE THEORY AMPLITUDES, SCALAR GRAPHS AND TWISTOR SPACE

VALENTIN V. KHOZE Department of Physics a n d IPPP University of Durham. Durham, DH1 3LE United Kingdom E-mail: [email protected]

We discuss a remarkable new approach initiated by Cachazo, Svrcek and Witten for calculating gauge theory amplitudes. The formalism amounts to an effective scalar perturbation theory which in many cases offers a much simpler alternative t o the usual Feynman diagrams for deriving n-point amplitudes in gauge theory. At tree level the formalism works in a generic gauge theory, with or without supersymmetry, and for a finite number of colours. There is also a growing evidence that the formalism works for loop amplitudes.

1. Introduction

In a recent paper1 Cachazo, Svrcek and Witten (CSW) proposed a new approach for calculating scattering amplitudes of n gluons. In this approach tree amplitudes in gauge theory are found by summing tree-level scalar diagrams. The CSW formalism' is constructed in terms of scalar propagators, l/q2, and tree-level maximal helicity violating (MHV) amplitudes, which are interpreted as new scalar vertices. This novel diagrammatic approach follows from an earlier construction2 of Witten which related perturbative amplitudes of conformal N = 4 supersymmetric gauge theory to Dinstanton contributions in a topological string theory in twistor space. The key observation2y1is that tree-level and also loop diagrams in SYM posess a tractable geometric structure when they are transformed from Minkowski to twistor space. The results2i1 have been tested and further developed in gauge theory3>4,5,6,897,9and in string theoryl0,11,12,13,14,15,16 The CSW diagrammatic approach' was extended to gauge theories with fermions3, and it was also shown that supersymmetry is not required for

492

493

the construction to work. At tree level the scalar graph formalism works in supersymmetric and non-supersymmetric theories, including QCD. Using this approach all tree-level antianalytic MHV (or googly) aplitudes were calcualted4 in complete agreement with known results. Recursive relations for constructing generic tree-level non-MHV amplitudes in the CSW formalism were derived5 and further progress was made in Refs.6f8where generic n-point amplitudes with three negative helicities were calculated at treelevel. In fact, all tree-level amplitudes in 0 N _< 4 gauge theories can be obtained directly from the scalar graph approach of CSW. By starting with amplitudes containing fermions', the reference spinor for the negative helicity gluons can be chosen to be that of the negative helicity fermion. As a consequence, the amplitudes are free of unphysical singularities for generic phase space points and no further helicity-spinor algebra is required to convert the results into an immediately usable form. The gluons only amplitudes can then be simply obtained as sums of fermionic amplitudes using the supersymmetric Ward identity. These amplitudes are therefore also immediately free of unphysical poles. Expressions for n-point amplitudes with three negative helicities carried by fermions and/or gluons were derived' in this way. The next logical step is to extend the formalism to the computation of loop graphs. When gauge theory amplitudes at 1-loop level are Fourier transfomed to twistor space, their analytic structure again acquires geometric meaning7. In Ref.g the CSW diagramatic approach' is used at loop level leading to a remarkable agreement with the known results17 for 1-loop MHV amplitudes in N = 4 SYM. It will be very interesting to extend the resultsg and to see how the CSW formalism will work in general settings, i.e. at 1-loop and beyond, for N 5 4 supersymmetry, and for non-MHV amplitudes.

<

1.1. Amplitudes i n the spinor helicity formalism We will consider tree-level amplitudes in a generic S U ( N ) gauge theory with an arbitrary finite number of colours. S U ( N ) is unbroken and all fields are taken to be massless, we refer to them generically as gluons, fermions and scalars. Using colour decomposition, an n-point amplitude M , can be represented as a sum of products of colour factors T, and purely kinematic partial amplitudes A,. The latter have the colour information stripped off and hence do not distinguish between fundamental quarks and adjoint

494

gluinos. The scalar graph method’ is used to evaluate only the purely kinematic amplitudes A,. Full amplitudes are then determined uniquely from the kinematic part A,, and the known expressions for T,. We will first consider theories with N 5 1 supersymmetry. Gauge theories with extended supersymmetry have a more intricate behaviour of their amplitudes in the helicity basis and their study will be postponed to sections 3 and 4. Theories with N = 4 (or N = 2 ) supersymmetry have N different species of gluinos and 6 (or 4) scalar fields. For now we concentrate on tree level partial amplitudes A, = A1+zrn with 1 gluons and 2 m fermions in the helicity basis, and all external lines are defined to be incoming. In N 5 1 theory a fermion of helicity is always connected by a fermion propagator to a helicity -$ fermion hence the number of fermions 2 m is always even. This statement is correct only in theories without scalar fields. In the N = 4 theory, a pair of positive helicity fermions, A’+, A2+, can be connected to another pair of positive helicity fermions, A3+, A4+, by a scalar propagator. In N 5 1 theory a tree amplitude A, with less than two opposite helicities vanishes” identi~ally’~.First nonvanishing amplitudes contain n - 2 particles with helicities of the same sign20>21 and are called maximal helicity violating (MHV) amplitudes. In the spinor helicity formalism22~20~21 an on-shell momentum of a massless particle, p,p, = 0, is represented as

++

pa& = p,afa = A a i a ,

(1)

where A, and i& are two commuting spinors of positive and negative chirality. Spinor inner products are defined by (X,X’) = EabXaAtb and - [X,A’] = ~,~;\ai’b.A scalar product of - -two null vectors, pa& = A a i & and qa& = A b i 6 , becomes p,q, = $(A, A’)[A, A’]. An MHV amplitude A, = A1+2rnwith 1 gluons and 2 m fermions in N 5 1 theories exists only for m = 0,1,2. This is because it must have precisely n - 2 particles with positive and 2 with negative helicities, and our fermions always come in pairs with helicities ff.Hence, there are three types of MHV tree amplitudes in N 5 1 theories:

An(g,,gs)

7

A n ( g t , K , A b )7

Suppressing the overall factor,

A n ( A t , A Q , K , A : ).

(2)

( 2 ~6(4)(Cy=1 ) ~ A i a i i & ) , the MHV

aIn the N = 1 theory this is also correct to all orders in the loop expansion and nonperturbatively.

495

purely gluonic amplitude reads20i21:

The MHV amplitude with two external fermions and

where the first expression corresponds to T < s and the second to s < T (and t is arbitrary). The MHV amplitude with four fermions and n - 4 gluons on external lines is

This corresponds to t < s < r < q , and there are other similar expressions, obtained by permutations of fermions, with the overall sign determined by the ordering. Expressions (4), (5) can be derived from supersymmetric Ward i d e n t i t i e ~ l ~ t ~and ~ > we ' ~ ,will have more to say about this in section 3. The MHV amplitude can be obtained, as always, by exchanging helicities H - and (i j ) ++[i j ] .

+

2. Gluonic NMHV amplitudes and the CSW method

The formalism of CSW was developed' for calculating purely gluonic amplitudes at tree level. In this approach all non-MHV n-gluon amplitudes (including MHV) are expressed as sums of tree diagrams in an effective scalar perturbation theory. The vertices in this theory are the MHV amplitudes (3), continued off-shell as described below, and connected by scalar propagators l / q 2 . It was s h ~ w nthat ~ ? the ~ same idea continues to work in theories with fermions and gluons. Scattering amplitudes are determined from scalar diagrams with three types of MHV vertices, (3),(4) and (5), which are connected to each other with scalar propagators l / q 2 . When one leg of an MHV vertex is connected by a propagator to a leg of another MHV vertex, both legs become internal to the diagram and have to be continued off-shell. Off-shell continuation is defined as follows': we pick an arbitrary spinor and define A, for any internal line carrying a momentum qaa by A, = qaa[R,f. External lines in a diagram remain onshell, and for them X is defined in the usual way. For the off-shell lines, the same tRef is used in all diagrams contributing to a given amplitude.

tief

496

&f

For practical applications CSWl have chosen to be equal to x a of one of the external legs of negative helicity, e.g. the second one, JRef = @. This corresponds to identifying the reference spinor with one of the kinematic variables of the theory. The explicit dependence on the reference spinor disappears and the resulting expressions for all scalar diagrams in the CSW approach are the functions only of the kinematic variables X i a and This means that the expressions for all individual diagrams automatically appear to be Lorentz-invariant (in the sense that they do not depend on an external spinor J&,) and also gauge-invariant (since the reference spinor corresponds to the axial gauge fixing n,A, = 0, where nab = J R e f a J R e f b ) . There is a price to pay for this invariance of the individual diagrams. Off-shell continuation described above leads to unphysical singularities which occur for the whole of phase space and which have to be cancelled between the individual diagrams. The result for the total amplitude is, of course, free of these unphysical singularities, but their cancellation and the retention of the finite part requires some work, see' and section 3.1 of Ref.3. However, these unphysical singularities are specific to the three-gluon MHV vertices and, importantly, they do not occur in any of the MHV vertices involving a fermion field3t8. Using supersymmetric Ward identitieslg one can obtain purely gluonic amplitudes from a linear combination of amplitudes with one fermionantifermion pair':


it.

(77 Wn(g;,g;,gJ

= (rl . 1 ) A n ( ~ L ~ ; , g & 7 ; )

+ (777-2)An(Akf,g;,A;,g;)

f (77 7-3)An(A:,S;,S&Q

.

(6)

After choosing an a priori arbitrary spinor 77 to be one of the three rj we find from (6) that the purely gluonic amplitude with three negative helicities is given by a sum of two fermion-antifermion-gluon-gluonamplitudes. These fermionic amplitudes were calculated in Ref.8 using the scalar graph method. The amplitudes are free of unphysical singularities for generic phase space points and the gluons only amplitudes are therefore also immediately free of unphysical poles.

3. The Analytic Supervertex

So far we have encountered three types of MHV amplitudes (3), (4) and ( 5 ) . The key feature which distinguishes these amplitudes is the fact that they depend only on ( X i X j ) spinor products, and not on [ x i xi]. We will call such amplitudes analytic.

497

All analytic amplitudes in generic 0 5 N 5 4 gauge theories can be combined into a single N = 4 supersymmetric expression of Nair18,

Here q t are anticommuting variables and A = 1 , 2 , 3 , 4 . The Grassmannvalued delta function is defined in the usual way,

Taylor expanding (7) in powers of qi, one can identify each term in the expansion with a particular tree-level analytic amplitude in the N = 4 theory. (qi)lCfor k = 0 , . . . , 4 is interpreted as the ithparticle with helicity hi = 1$. This implies that helicities take values, (1, ;,O, -+, -l}, which precisely correspond t o those of the N = 4 supermultiplet, { g - , X i , 4AB, AA+,g+}. It is straightforward t o write down a general rule3 for associating a power of q with all component fields in N = 4, -

gi

1 2 3 4

N

qiqiqiqi

A,

i

2 3 4

N

-qiqiqi

>

4fB

N

VAVB

7

A?+

N

1:

i

gi+

1 7

N

(9) with expressions for the remaining hi with A = 2 , 3 , 4 written in the same manner as the expression for A, in (9). The first MHV amplitude (3) is derived from (7) by using the dictionary (9) and by selecting the (qr)4(q8)4 term in (7). The second amplitude (4) follows from the (qt)4(qT)3(qs)1 term in (7); and the third amplitude (5) is an (qr)3(q8)1(qt)3(qq)1 term. There is a large number of such component amplitudes for an extended susy Yang-Mills, and what is remarkable, not all of these amplitudes are MHV. The analytic amplitudes of the N = 4 SYM obtained from ( 7 ) , (9) are8: An(g-,g-) , An(g-,A,,AA+) > An(A,,A,,AA+,AB+) 7 A3+, A4+) , A,(g-, A'+, A2+, A3+, A4+) , A n ( A i ,AA+,A1+,A2+, An(A 1-k 7 A2+ 7 AS+, Ad+,A1+,A2+, A4-k) An(?AB,A A+ > A B + A l + , A 2 f , A 3 + , A 4 + ) , An(g-,$AB,$AB)

7

An(g-r$AB,AA+iAB+)

An(&, 4 A B , $ B C , A C + )1 An@,

4, $, 4)

1

,

An($,

41 $7

7

An(Ai,Aii4AB)

I

A n ( ~ , l $ B C I A A + , ~ B + l ~ C7+ )

A+,A+)

7

An($,

$1

A+,A+,A+,A+)

7

(10)

498 where it is understood that $AB = ~ E A B C D $ In ~ ~Eqs. . (10) we do not distinguish between the different particle orderings in the amplitudes. The labels refer to supersymmetry multiplets, A , B = 1,. . . , 4. Analytic amplitudes in (10) include the familiar MHV amplitudes, (3), (4), ( 5 ) , as well as more complicated classes of amplitudes with external gluinos hA,RBZAl etc, and with external scalar fields $ A B . The second, third and fourth lines in (10) are not even MHV amplitudes, they have less than two negative helicities, and nevertheless, these amplitudes are non-vanishing in N = 4 SYM. The conclusion we drawg is that in the scalar graph formalism in N 5 4 SYM, the amplitudes are characterised not by a number of negative helicities, but rather by the total number of q’s associated to each amplitude via the rules (9). All the analytic amplitudes listed in (10) can be calculated directly from (7), (9). There is a simple algorithm for doing thiss. (1) For each amplitude in (10) substitute the fields by their 7expressions (9). There are precisely eight q’s for each analytic amplitude. (2) Keeping track of the overall sign, rearrange the anticommuting 7’s into a product of four pairs: (sign) x qfq; qZ$q$q: q:qt. (3) The amplitude is obtained by replacing each pair q$$ by the spinor product (i j ) and dividing by the usual denominator,

The vertices of the scalar graph method are the analytic vertices (10) which are all of degree-8 in q and are not necessarily MHV. These are component vertices of a single analytic supervertexb (7). The analytic amplitudes of degree-8 are the elementary blocks of the scalar graph approach. The next-to-minimal case are the amplitudes of degree-12 in q, and they are obtained by connecting two analytic vertices” with a scalar propagator l / q 2 . Each analytic vertex contributes 8 q’s and a propagator removes 4. Scalar diagrams with three degree-8 vertices give the degree-12 amplitude, etc. In general, all n-point amplitudes are characterised by a degree 8,12,16,. . . , (4n - 8) which are obtained from scalar diagrams with bThe list of component vertices (10) is obtained by writing down all partitions of 8 into groups of 4, 3, 2 and 1. For example, A,(g-,qAB,AA+,AB+)follows from 8 = 4 + 2 + 1 1.

+

499

1,2,3, . . . analytic vertices.c In Ref.8 we have derived a simple expression for the first iteration of the degree-8 vertex. This iterative process can be continued straightforwardly to higher orders. 4. Calculating Simple Antianalytic Amplitudes

To show the simplicity of the scalar graph method and to test its results, in this section we will calculate simple antianalytic amplitudes of r]-degree-12. We work in N = 1, N = 2 and N = 4 SYM theories, and study

using the scalar graph method with analytic vertices. The labels N = 1 , 2 , 4 on the three amplitudes above corresponds to the minimal number of supersymmetries for the given amplitude. In this section the N-supersymmetry labels A , B are shown as ( A ) and ( B ) . In all cases we will reproduce known results for these antianalytic amplitudes, which implies that at tree level the scalar graph method appears to work correctly not only in N = 0 , l theories, but also in full N = 2 and N = 4 SYM. In particular, the N = 4 result (34) for the amplitude (12c) will verify the fact that the building blocks of the scalar graph method are indeed the analytic vertices (lo), which can have less than 2 negative helicities, i.e. are not MHV. = as in the section We will be using the of-shell prescription tRef 3. Since in our amplitudes, the reference spinor always corresponds to a gluino A-, rather than a gluon g-, there will be no singularities in our formulae at any stage of the calculation. 4.1. Antianalytic N = 1 amplitude

There are three diagrams contributing to the first amplitude, Eq. (128). The first one is a gluon exchange between two 2-fermion MHV-vertices. This diagram has a schematic form,

=In practice, one needs to know only the first half of these amplitudes, since degree(4n- 8) amplitudes are anti-analytic (also known as googly) and they are simply given by degree-8* amplitudes, similarly degree-(4n - 12) are given by degree-12*, etc.

500

Here gf and g I I are off-shell (internal) gluons which are Wick-contracted -

+

via a scalar propagator, and I = (3,4), which means, X I = (p3 p 4 ) * i 2 . The second and the third diagrams involve a fermion exchange between a 2-fermion and a 4-fermion MHV vertices. They are given, respectively by

with I = (2,4), and

with I = (3,5). Both expressions, (14) and (15), are written in the form which is in agreement with the ordering prescription3 for internal fermions, ket+ ket-. All three contributions are straightforward to evaluate using the relevant expressions for the component analytic vertices. These expressions follow from the algorithm (11). 1. The first contribution, Eq. (13), is

-(1 2)2

1

((2 3 ) 31~+ (2 4 ~ 42 1 1 ~ ) 11~ (3 4 ~ 413 [2 412(1 2)2

-

(4 3)[2 412 (16)

-

1341((2 3 ) 31~+ (2 4 ) 4 ~1 ~ 15 ) 11~* 2. The second diagram, Eq. (14), gives

-(2 3 ) 2 1 ((2 3)[2 31 + (2 4 ) P 41)(3 4) (5 1)[5 11 -[2 512(2 3)2 [2 11[5 11((2 3 ) 31~+ (2 4 ~ 42 1 ~ 4) 3 3. The third contribution, Eq. (15), is

-(2. 1)

(5 ~2 5i2 [2 11

(17)

*

(3 1)2[2 11 (3 (18) [2 11 (1 2 ) 21~ (3 4 ~ 1)5 [2 11(3 4)(5 1) . Now, we need to add up the three contributions. We first combine the expressions in (16) and (17) into 1

using momentum conservation identitites, and the fact that (2 3)[2 31 + (2 4)[2 41 = -(3 4)[3 41 (5 1)[5 11. Then, adding the remaining contribution (18) we obtain the final result for the amplitude,

+

501

which is precisely the right answer for the antianalytic 5-point 'mostly minus' diagram. This can be easily verified by taking a complex conjugation (parity transform) of the corresponding analytic expression. 4.2. Antianalytic N = 2 amplitude

There are three contributions to the amplitude (12b) The first contribution is a scalar exchange between two analytic vertices,

4p4)are off-shell (internal) scalars which are Wick- =-

Here $-I(12) and 4 ( qI

-

contracted. The external index I = (1,2), which implies XI = (p1+pz).i2 = pl . i z . The second contribution to (12b) is a fermion exchange,

with external index I = (3,4), that is, XI = ( p 3 +p4) . x 2 . The final third contribution is again a fermion exchange,

+

with I = (2,3), and XI = (pz p3) . i 2 = p3 . XZ. As before, all three contributions are straightforward to evaluate using the rules (11). 1. The first contribution, Eq. (21), is

1

1 (1 2)[1 21

-(3 5)(3 1 ) P 11 - (3 5)(3 1) . (4 5)(5 1)[1 21 (4 5)(5 1) [2 11 2. The second contribution (22) gives

( l 2,

*

(3 4)'[2 41' 1 4 1 2) (4 3 ) 31~ (3 4 ~ 413 (5 1 ) 11~ 3. The third contribution (23), is

(1 2) P 4i2 . (5 1) [1 21[2 3113 41

(24)

(25)

Now, we add up the three contributions in Eqs. (24), (25), (26) and using the momentum conservation identities obtain

which is the correct result for the antianalytic amplitude.

502 4.3. Antianalytic N = 4 amplitude

The amplitude Af=4(Ac, 1, AG, 2 , A& 3 , A& 4 , g z ) receives contributions only from diagrams with a scalar exchange. There are three such diagrams. The first one is

d?;) and 4y) are off-shell (internal) scalars which are Wickcontracted and XI = (PI + p 2 + p 5 ) . x 2 = (PI + p 5 ) . i2. Here

The second contribution to (12c) is

with external index I = (1,2), that is, XI = (pl + p2) . The third diagram gives,

x2 = p l . x2.

+

with I = (2,3), and XI = (p2 p 3 ) 1 2 = p3 . x 2 . 1. The first contribution, Eq. (28), is

2. The second contribution (29) gives 1

2,

.

(12;p 21

.

3. The third contribution ( 1 4)2 ( 1 4)2 (33) (4 5)(5 1) (4 5)(5 1)[2 31 ’ We add up the three contributions (31), (32), (33) and using the momentum conservation identities obtain (2 3,

’

1 (2 3)[2 31

which is again the correct answer for this amplitude, as it can be easily seen from taking a complex conjugation of the corresponding analytic expression.

Acknowledgements. I am grateful to George Georgiou and Nigel Glover for an enjoyable collaboration and for greatly contributing to my

503 understanding of these topics. I thank t h e organizers for a n excellent conference and Zvi Bern, Lance Dixon, Misha Shifman, Andrei Smilga, Gabriele Travaglini and Arkady Vainshtein for useful discussions and comments. This work is supported by a PPARC Senior Fellowship.

References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17.

18. 19.

20. 21. 22.

23. 24.

F. Cachazo, P. Svrcek and E. Witten, hep-th/0403047. E. Witten, hep-th/0312171. G. Georgiou and V. V. Khoze, JHEP 0405 (2004) 070, hep-th/0404072. C. J. Zhu, JHEP 0404 (2004) 032 hep-th/0403115; J. B. Wu and C. J. Zhu, hep-th/0406085. I. Bena, Z. Bern and D. A. Kosower, hep-th/0406133. D. A. Kosower, hep-th/0406175. F. Cachazo, P. Svrcek and E. Witten, hep-th/0406177. G. Georgiou, E. W. N. Glover and V. V. Khoze, hep-th/0407027. A. Brandhuber, B. Spence and G. Travaglini, hep-th/0407214. N. Berkovits, hep-th/0402045; N. Berkovits and L. Motl, JHEP 0404 (2004) 056 hep-t h/0403 187. R. Roiban, M. Spradlin and A. Volovich, JHEP 0404 (2004) 012 hepth/0402016; hep-th/0403190. A. Neitzke and C. Vafa, hep-th/0402128; N. Nekrasov, H. Ooguri and C. Vafa, hepth/0403167. E. Witten, hep-th/0403199. S. Gukov, L. Motl and A. Neitzke, hep-th/0404085. W. Siegel, hep-th/0404255. N. Berkovits and E. Witten, hep-th/0406051. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425 (1994) 217 hep-ph/9403226; Nucl. Phys. B435:59 (1995) hep-ph/9409265; Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. Proc. Suppl. 51C:243 (1996) hep-ph/9606378. V. P. Nair, Phys. Lett. B 214 (1988) 215. M. T. Grisaru, H. N. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D 15 (1977) 996; M.T. Grisaru and H. N. Pendleton, Nucl. Phys. B 124 (1977) 81. S. J. Parke and T. R. Taylor, Phys. Rev. Lett. 56 (1986) 2459. F. A. Berends and W. T. Giele, Nucl. Phys. B 306 (1988) 759. F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T. T. Wu, Phys. Lett. B 103 (1981) 124; P. De Causmaecker, R. Gastmans, W. Troost and T. T. Wu, Nucl. Phys. B 206 (1982) 53; R. Kleiss and W. J. Stirling, Nucl. Phys. B 262 (1985) 235; J. F. Gunion and Z. Kunszt, Phys. Lett. B 161 (1985) 333. M. L. Mangano and S. J. Parke, Phys. Rept. 200 (1991) 301. L. J. Dixon, hep-ph/9601359.

WEAK SUPERSYMMETRY AND ITS QUANTUM-MECHANICAL REALIZATION

A. V. SMILGA SUBATECH, Universite‘ de Nantes 4 rue Alfred Kastler, BP 20722, Nantes 44307,France. * E-mail: smilgaOsubatech.in2p3.fr

We explore “weak” supersymmetric systems whose algebra involves, besides Poincare generators, extra bosonic generators not commuting with supercharges. This allows one to have inequal number of bosonic and fermionic 1-particle states in the spectrum. Coleman-Mandula and Haag-Lopuszanski-Sohnius theorems forbid the presence of such extra bosonic charges in interacting theory for d 2 3. However, these theorems do not apply in one or two dimensions. For d = 1, we construct a nontrivial interacting system characterized by weak supersymmetric algebra. It is related to “n-fold” supersymmetric systems and to quasi-exactly solvable systems studied earlier.

1. Introduction

The basic defining feature of any standard supersymmetric system is double degeneracy of all excited states. This follows from the minimal supersymmetry algebra Q2

= Q 2 = 0,

{Q,Q}+ = 2H ,

(1)

where Q is a complex conserved [this is a corollary of Eq.(l)] supercharge. If the superalgebra describing symmetries of the system includes (1) as a subalgebra, double degeneracy of all excited levels (if supersymmetry is spontaneously broken, also the ground state is doubly degenerate) follows. An interesting question is whether some other “weak” supersymmetric algebras involving Poincare generators and conserved supercharges, but not including (1) as subalgebra, are possible. At the algebraic level, the answer is trivially positive. It is easy also to construct Lagrangians enjoying weak * On leave of absence from ITEP, Moscow, Russia.

504

505 supersymmetry. Indeed, the Lagrangian 1 c = -(aw$)2+i$ffwaw$ (2) 2 (4 is a real scalar and $ is a Weyl spinor) is invariant with respect to supersymmetry transformations a

The corresponding supercharges are

Now, Q , and Q" are conserved, but their anticommutators involve besides Pa" also extra terms. In particular, { Q a , &a} # 0. The resulting superalgebra does not include the subalgebra (1) and the number of bosonic and fermionic 1-particle states might be different. And it is: the Lagrangian (2) describes a free real boson (one state IB) for each 3-momentum 3 and a free Weyl fermion (two states IF*)). It is interesting (and important !) that, in the sector with given 6, the state pairing is restored for two-particle excitations and higher. Thus, at the two-particle level, there are two boson states IBB) and IF+F-) and two fermion states IBF+) and IBF-). Actually, any Lagrangian involving some number of free bosonic and some number of free fermionic fields is supersymmetric. There are a lot of such supersymmetries: each bosonic field can be mixed with each fermionic field independently of the others. However, this is only true for free theory. As soon as the interaction is switched on, supersymmetry (strong or weak) is broken. Indeed, nonvanishing (Q,, &a} implies the presence of an extra conserved bosonic charge in the representation (1,O) of the Lorentz group. It is none other than the self-dual part of the fermion spin operator S,p. Spin is not conserved, however, in interacting theories. Actually, in use the standard Weyl notation where the dotted and undotted indices are raised and lowered with eao(&o) and eap(e&b) = - & ( e a o ) ; $6 = (&)t and $& = - ( Q a ) t ;

4' = $&$&; up = (1,a)and ot = (1,-m). of the consequencies of this is the presence of the so called quasigoldstino branch in the spectrum of collective excitations in quark-gluon plasma l. 11' = &$" and

506 any theory involving mass gap and a nontrivial S-matrix, the presense of extra nonscalar conserving charges is ruled out by the Coleman-Mandula theorem 2 . Interacting supersymmetric theories can only involve, besides the Poincare generators P,, M,, and supercharges Q i , central charges commuting with everything and some extra global symmetry generators, which can have nontrivial commutators with Q and between themselves, but they cannot appear in this case in the anticommutators of supercharges 3 . This is true if the dimension of space-time is 3 or more. In two dimensions, where scattering can be only forward or backward, the theorems do not apply. In particular, one can have an infinite number of conserved bosonic charges (like, e.g., it is the case in the Sine-Gordon model). Seemingly, nothing prevents one to have an interacting 2d theory enjoying a version of weak supersymmetry. We tried to construct one, but failed. Probably, one should try harder. What we were able to construct is a weakly supersymmetric quantum mechanical system. It has two complex conserved supercharges with nontrivial anticommutators involving besides H four other bosonic generators which are not central charges - their commutators with supercharges and between themselves do not vanish. The boson-fermion degeneracy is there starting from the second excited level. But not for the first excited level and not for vacuum. In Sect. 2, we describe the simplest such system - the weak supersymmetric oscillator. In Sect. 3, we present a nontrivial weak supersymmetric Hamiltonian. We find that previously studied quantum systems with socalled “2-fold supersymmetry” are in fact weak supersymmetric systems in disguise. We briefly discuss their relationship to quasi-exactly solvable models Sect. 4 is reserved for discussion, conclusions, and acknowledgements. 2t3

‘.

2. Weak supersymmetric oscillator

Consider the Lagrangian

a = 1,2. It can be obtained out of the massive version of the free field theory Lagrangian (2) by dimensional reduction. It is invariant with respect ~~

CFormassless theories, Poincare group can be extended to conformal and super-Poincare - to superconformal.

507

to supersymmetry transformations

The corresponding supercharges are

where p = j: is the bosonic canonical momentum. The canonical Hamiltonian is 1 2

H = -p2

l , , m - + -m x + y($a$a +$a$a) 2

.

(8)

The quantum Hamiltonian can be written by replacing p by -idIda: and by a/d$,. Supercharges commute with the Hamiltonian. On the other hand,

where

+ $'p$a) = y($a$a +$a$a)

zap = m($a$p

y

m - -

9

(10)

The operators Zap and Y commute with H and with each other, but the cammutator s

and also

508

are nontrivial. The algebra can be presented in a little bit more convenient form if introducing S, = Q , - Q,, Sa = Q" Qff. Then {S,

S p } = 4H6:

+ - 2Y6: + 22:

,

[Sa,2071 = m ( E m a s 7 + GYrSL3) , [Sa,~ p , ] = m (6Z.S; + 6,"Sp) ,

[So,YI = -m&, [Sa,Y]= mS"

,

(13)

to which the commutator (12) should be added. The subalgebra (12) is none other than sZ(2) , which can be readily seen by identification 211

2ima+,

Z22

= 2ima-,

212

-mn3 .

= 221

All other commutators and the anticommutator {S,, So} vanish. It is not difficult to find the spectrum of H . The eigenstates are ;

@: = +,In)

,

(14)

where In) are the bosonic oscillator eigenstates. Their energies are

E F = ( - - ; + n ) m , E: =

(: ) -+n

m , E;4 =

(i +

n ) m .(15)

The spectrum is drawn in Fig. 1. We see that there is one vacuum state (its energy can be brought to zero by adding the constant m/2 to the Hamiltonian, but for the weak supersymmetric systems with algebra (13), Eva,= 0 is an option rather than requirement). There are three first excited states: a bosonic and two fermionic. Starting from the second excited state, there are 2 bosonic and 2 fermionic states at each level. The eigenvalues of the operator Y is -m for the leftmost tower, 0 for two central and +m for the rightmost one. In other words, the operator Y / m (or rather Y / m 1 ) plays the role of the fermionic charge. The operators Zap annihilate the states @$while the states @: form doublet representations of the s l ( 2 ) algebra (12). To acquire further insights, it is instructive to write the action of the operators S,, Sa on the states (14):

+

S"@? = 0 ,

S,@? = 2Jmn+:-l,

so@; = -2&€,p@y-l, S,@T = 0

Sv; =2 , Sa@; = 2

J m 6 p n f l , J

m

~

~

~

.@

g( 1 6+)

~

509

Figure 1. Spectrum of weak supersymmetric oscillator.

We see that S, annihilates the states from the rightmost column and brings the states from the ieft columns to the right. The action of is opposite. Now, one can divide all eigenstates in two sets: (2) the states CP? and and (ii) the states CP; and CP?. The states from the subset (i) form the Hilbert space of the = 1 supersymmetric oscillator, with S1 playing the role of the supercharge. The same applies to the subset (iz) with the supercharge 5'2. For ordinary N = 2 supersymmetric quantum mechanics, the Hilbert space can also be divided into two N = 1 subspaces, but the specifics of a weak supersymmetric system is that two sets of states are shifted with respect to each other, i.e. the Harniltonian for the right subset differs from the Hamiltonian for the left subset by a constant. 3. A class of interactive weak supersymmetric systems

Consider the Lagrangian j.2

-

L = - + i$"$, 2

.

v2

V'

2

2

B' 2

- - - - ($2 + 42) - - $ 2 4 2

,

(17)

510

where V(x) is an arbitrary function and C is an arbitrary constant. One can observe that the corresponding action is invariant with respect to the supersymmetry transformations

s x = S$

S$, SG"

+

$€

,

+ S,(V + B $ J -~ 2B ) + ($E)$"] = iPX - E,(V + B q 2 )- 2B [ P ( $ $+ ) $a(S$)]

,

= -i~aX

.

(18)

When V(x) = mx and C = m, Eq.(17) is reduced to the oscillator Lagrangian (5) considered above. The first four terms in Eq.(17) represent a rather natural generalization of Eq.(5), like in Witten's supersymmetric quantum mechanics 6 . In our case, we are obliged to add also a 4-fermion term in the Lagrangian and extra nonlinear terms in the transformation law. The canonical classical supercharges and Hamiltonian are

Qa = p$, Q" = pq"

+ iV$, + iB$'&

, + iV$" + iBq2$," ,

{au,

(19)

The Poisson brackets {Q,, H } ~ . Band . H}P.B.vanish. A certain care is required when quantizing this theory. We want to fix the ordering ambiguities in Q and H such that classical supersymmetry were not spoiled at the quantum level. An experience acquired by fiddling with supersymmetric a-models and gauge theories teaches us that proper quantum supercharges should be obtained by Weyl ordering procedure from the classical expressions (19). In other words, (19) should be Weyl symbols of the quantum supercharges. This does not necessarily apply to the Hamiltonian. Indeed, if choosing the quantum Hamiltonian such that (20) represents the Weyl symbol of &qu, the Weyl symbol of the commutator [ Q a ,&] would be given by the Moyal bracket of Q$ and Hc', 877

[Qa,&] W =

{Q,,H}M.B.=

51 1

TOcompensate that, we should add to Hc' the term B2/2. The quantum supercharges and Hamiltonian thus obtained are Qa

= Inl,

+ iV& + iB

4"= p$" + iV$"

+ iB

+ &)

,

- $a)

(22)

and

v2 V' B' B2 C 2 (q2 q2) 2 2 ($'q' - 2$4 1) 2 2 (23) where we added for convenience the constant C/2 in the Hamiltonian. Direct calculation of the commutators (or, which is simpler, of the Moyal brackets of the classical expressions) leads to a remarlable conclusion: the algebra (13, 12) derived for the oscillator is valid also in the interactive case, with m C, H -+ H - C/2 and Z,p,Y having the same form (10) as before. As earlier, the quantum states can be divided into three classes: (i) the states I-) 0: 1 - $'/2, (ii) the states la) o( qff (they are present in two copies as the Hamiltonian (23) is not sensitive to the index a of the fermion state) and (iii) the states I+) oc 1+$2/2. These states are characterized by a definite value of the "fermion charge" Y : Y* = f m and Ya = 0. In each such sector, we have an ordinary Schrodinger equation with the potentials

a

=

+y

+

+

1 u" ---(W"WL) 2

u,

1

= z(w:

+ +

+

=

+

1 -cw:-w;>+c, 2

+ w;) + c ,

where W*=V&B.

(25)

It is clear now that we are dealing with two superimposed ordinary Witten's SQM systems. The states I-) and 11) are described by such system with superpotential W- and the states 12) and I+) - by the system with superpotential W+, with the constant C added to the Hamiltonian. Excited states are mostly 4-fold degenerate as for usual N = 2 SQM. The ground state is not necessarily degenerate. If exp{ - J W- (z)ds} is normalizable, this (being multiplied by 1 - +'/2) determines the wave function of the unique vacuum state. With the chosen normalization of the Hamiltonian [the term C/2 in Eq.(23) !] it has zero energy. Further,

512 if exp{ - 1W+(z)dx} is normalizable, there is also a unique zero-energy ground state for Witten’s Hamiltonian with superpotential W+. Thus, we obtain a state in the sector 12) with energy C. Due to 12) +) 11) and 11) +) I-) degeneracies, we have altogether three states with energy C at the first excited level, and the picture is the same as for the oscillator (see Fig.1). We have obtained free of charge a wide class of quasi-exactly solvable potentials U - ( x ) for which the energy of the ground state (Eo = 0) and of the first excited state El = C are exactly known. They depend on an arbitrary function V ( x )and an arbitrary constant C with a certain restriction: both exp{W - ( x ) d z } and exp{- W+(x)dz} should be normalizable. Probably, the simplest nontrivial choice is V ( z )= mx ax3 with C = m (m,a > 0). Note that the potential U-(x)is not polynomial in this case. The potentials U* in Eqs.(24, 25) were discussed before (see Eq.(4.18) in Ref.g and Eq.(50) in Ref.” ) in association with the so called 2-fold supersymmetry construction developped in 4 . N-fold supersymmetry is a supersymmetry where supercharges are not linear in momentum, but present polynomials of power N . In our case, one can define the quadratic in p operator

s

s

& =

+

s1s2

el+)

a

(26)

with the action = Q ( a )= 0 and &I-) = I+). acts in the opposite direction: GI+) = I-), Q1a) = GI-) = 0. The operators &, commute with H (as S1,2 and S1>’do). If disregarding the states la) annihilated by both Q and and considering only the sectors I+) and I-), one can deduce

a

{Q,Q} = 16H(H- C) .

(27)

The quadratic polynomial of H appearing on the right side is characteristic of 2-fold supersymmetry. The full algebra of the weak supersymmetry (12, 13) displays itself only if the “central” sector la) is brought into consideration. 4. Discussion

Our original motivation was the quest for nontrivial supersymmetric systems with mismatch between bosonic and fermionic degrees of freedom. Let us note here that, while it is difficult to find such field theory systems, their presence in quantum mechanics was known for a long time. Most popular SQM systems (Witten’s quantum mechanics, standard supersymmetric n models, etc) have an equal number of bosonic and fermionic phase

513

space coordinates. But the SQM system describing planar motion in transverse magnetic field involves two pairs of bosonic variables and only one pair of fermionic variables. A class of nonstandard “symplectic” N = 2 supersymmetric B models involving 3r bosonic variables and 2r fermionic variables (r is an integer) was constructed in Ref.”. The Diaconescu-Entin N = 4 symplectic B model l2 generalized in l 3 involves 5r bosonic and 4r fermionic variables. An industrial method to construct SQM models where the number of bosonic variables is less than the number of fermionic ones was suggested in 1 4 . In SQM, an equal number bosonic and fermionic degrees of freedom is not required by supersymmetry. What is required is the equal number of bosonic and fermionic quantum states. But in field theory, any bosonic or fermionic dynamical field correspond to an asymptotic state (a particle), and bosons and fermions should normally be matched. We notice that this matching can be absent if relaxing the requirement that the anticommutator of supercharges involves only the Hamiltonian, momentum, and central charges. A lot of free weak supersymmetric models can be written, but, for d > 3, interactive weak supersymmetric theories do not exist. This follows from the Haag-Lopuszanski-Sohnius theorem. This theorem does not apply to 2 dimensions, however, and the existence of interactive weak supersymmetric theories cannot be ruled out. Our quest for such theories was not successful, but it certainly pays to try harder. The main positive result of this paper is the system (17) which enjoys a weak supersymmetry algebra (12), (13). It describes quantum systems which were studied before, but from a different perspective. It would be interesting to construct and study other, more complicated weak supersymmetric models, especially the models involving several bosonic degrees of freedom. This would allow one to construct new examples of multidimensional quasi-exactly solvable models. I am indebted to N. Dorey, M. Henneaux, N. Nekrasov, M. Plyushchay, M. Shifman, and A. Vainshtein for illuminating discussions and correspondence.

References 1. V.V. Lebedev and A.V. Smilga, Ann. Phys. 202, 229 (1990). 2. S.R. Coleman and J. Mandula, Phys. Rev. 159,1251 (1967). 3. R.Haag, J.T. Lopuszanski, and M. Sohnius, Nucl. Phys. B88,257 (1975).

4. A.A. Andrianov, M.V. Ioffe, and V.P. Spiridonov, Phys. Lett. A174, 273 (1993). 5. A. Turbiner, Commun. Math. Phys. 118,467 (1988); A. Ushveridze, Sov. J .

514

Part. Nucl. 20, 504 (1989); for a review see M. Shifman, ITEP lectures on particle physics and field theory, World Scientific, 1999, p. 775. 6. E. Witten, Nucl. Phys. B188, 513 (1981). 7. A.V. Smilga, Nucl. Phys. B292, 363 (1987). 8. I.E. Moyal, Proc. Cambr. Phil. SOC.45,99 (1949). 9. H. Ayoama, M. Sato, and T. Tanaka, Nucl. Phys. B619, 105 (2001) [arXive: quant-ph/0106037]. 10. A.A. Andrianov and A.V. Sokolov, Nucl. Phys. B660, 25 (2003) [arXive: hep-th/0301062]. 11. A.V. Smilga, Nucl. Phys. B291, 241 (1987); E.A. Ivanov and A.V. Smilga, Phys. Lett. B257, 79 (1991). 12. D.-E. Diaconescu and R. Entin, Phys. Rev. D56, 8045 (1997) [arXive: hepth/9706059]. 13. A.V. Smilga, Nucl. Phys. B652, 93 (2003) [arXive:hep-th/0209187]. 14. E.A. Ivanov, S.O. Krivonos, and A.I. Pashnev, Class. Quant. Grav. 8 , 19 (1991); A. Losev and M. Shifman, Mod. Phys. Lett. A16, 2529 (2001) [arXive: hep-th/0108151].

NONPERTURBATIVE SOLUTION OF YUKAWA THEORY AND GAUGE THEORIES

JOHN R. HILLER Department of Physics University of Minnesota-Duluth Duluth, M N 55812 USA E-mail: [email protected]

Recent progress in the nonperturbative solution of (3+1)-dimensional Yukawa theory and quantum electrodynamics (QED) and (l+l)-dimensional super Yang-Mills (SYM) theory will be summarized. The work on Yukawa theory has been extended to include two-boson contributions to the dressed fermion state and has inspired similar work on QED, where Feynman gauge has been found surprisingly convenient. In both cases, the theories are regulated in the ultraviolet by the inclusion of Pauli-Villars particles. For SYM theory, new high-resolution calculations of spectra have been used to obtain thermodynamic functions and improved results for a stress-energy correlator.

1. Introduction Numerical techniques can be successfully applied to the nonperturbative solution of field theories quantized on the light c ~ n e Unlike . ~ ~lattice ~ ~ gauge ~ theory: wave functions are computed directly in a Hamiltonian formulation. The properties of an eigenstate can then be computed relatively easily. There have been a number of successes in two-dimensional t h e ~ r i e s ,but ~ in three or four dimensions the added difficulty of regulating and renormalizing the theory has until recently limited the success of the approach. Here we discuss recent progress with two different yet related approaches to regularization. One is the use of Pauli-Villars (PV) r e g u l a r i ~ a t i o n ~ ~ and the other supersymmetry.1° The particular applications to be discussed are to Yukawa theory and &ED in 3fl dimensions with P V fields and to super Yang-Mills (SYM) theory in 1+1 dimensions. In the latter case, extension to 2+1 dimensions has already been done;" however, the most recent developments have used two dimensions as a proving ground. There we consider in particular a stress-energy c ~ r r e l a t o r land ~ ~ analysis ~ ~ > ~ ~of finite-temperature effects.15

515

516

The light-cone coordinates1 that we use are defined by xf = xo f x3, 21 = (x1,x2),with the expression for xf divided by & in the case of supersymmetric theories. Light-cone three-vectors are denoted by a n underline: p = (p+ ,pi). The keyelements of the PV approach are the introduction of negative metric P V fields to the Lagrangian, with couplings only to null combinations of P V and physical fields; the use of transverse polar coordinates in the Hamiltonian eigenvalue problem; and the introduction of special discretization of this eigenvalue problem rather than the traditional momentum grid with equal spacings used in discrete light-cone q u a n t i ~ a t i o n . The ~ ? ~ choice of null combinations for the interactions eliminates instantaneous fermion terms from the Hamiltonian and, in the case of QED, permits the use of Feynman gauge without inversion of a covariant derivative. The transverse polar coordinates allow use of eigenstates of J, and explicit factorization from the wave function of the dependence on the polar angle; this reduces the effective space dimension and the size of the numerical calculation. The special discretization allows the capture of rapidly varying integrands in the product of the Hamiltonian and the wave function, which occur for large PV masses. For supersymmetric theories, the technique used is supersymmetric discrete light-cone quantization (SDLCQ) ,l6>lowhich is applicable to theories with enough supersymmetry to be finite. .This method uses the traditional DLCQ grid in a way that maintains the supersymmetry exactly within the numerical approximation. The symmetry is retained by discretizing the supercharge Q- and computing the discrete Hamiltonian P- from the superalgebra anticommutator {Q- , Q - } = Z A P - . To limit the size of the numerical calculation, we work in the large-l\r, approximation; however, this is not a fundamental limitation of the method. 2. Yukawa theory

The Yukawa action with a PV scalar and a PV fermion is

517

From this we obtain the light-cone Hamiltonian (2)

+

c

/d&{

[v--*2s(p,g)+ V z s ( p + q-, g ) ] bJ,s(p)a:(g)bi,-s(E+g)

i,j,k,s

+ [W_P,g)+

+ g 4 ] b;,,(E)ab(g)bi,s(p + g) + h.c.}

7

where antifermion terms have been dropped. No instantaneous fermion terms appear because they are individually independent of the fermion mass and cancel between instantaneous physical and P V fermions. The vertex functions are

with

-32,

- L ( 2 s , i ) . The nonzero (anti)commutators are fi

We construct a dressed fermion state, neglecting pair contributions; it takes the form

The wave functions f...s(xn,fin) satisfy the coupled system of equations that results from the Hamiltonian eigenvalue problem P+P-@+ = M2@+. Each wave function has a total L, eigenvalue of 0 (1)for s = +1/2 (-1/2). The coupled equations are P+

mSzi

+ C(-~)~’+jp+ J d q { f i j j - ( g ) [ V + ( L :-g,g> + v-*(~fl~)I (6) i’ , j

+ fitj+(Q)[Ui,(P - g,g) + K(P,Q)l)= M 2 G ,

518

and

We consider truncations of the system.

A truncation to one boson leads to an analytically solvable problem dThe one-boson wave functions are

Substitution into Eq. (6) yields

519

with

The presence of the P V regulators allows 10 and J to satisfy the identity & J ( M 2 ) = M210(M2). With M held fixed, the equations for zi can be viewed as a n eigenvalue problem for g2. The solution is

_ z1 -F mo = - ( M F mo)(M F m1) (14) (m1- mo)(Pol1 f M I o ) zo M F 7% * An analysis of this solution is given in Ref. 7. In a truncation to two bosons, we obtain the following reduced equations for the one-boson-one-fermion wave functions:8 92

= f i j + ( y , q ~ ) ,@fij-(g) = fij-(y,ql)ei', I is an where mfij+(g) analytically computable self-energy, and J(")is a kernel determined by nboson intermediate states. These reduced integral equations are converted to a matrix equation via quadrature in y' and q y . The matrix is diagonalized to obtain g2 as an eigenvalue and the discrete wave functions from the eigenvector. A useful set of quadrature schemes is based on Gauss-Legendre quadrature and particular variable transformations. The transformation for y' is motivated by the need for an accurate approximation to the integral J . This integral appears implicitly in the product of the Hamiltonian and the eigenfunction and is largely determined by contributions near the endpoints whenever the P V masses are large. The transformation for the transverse integral is chosen to reduce the range from infinite to finite, so that no momentum cutoff is needed.

520 From the wave functions we can extract a structure function fBs(y),

n=l

n=l

defined as the probability density for finding a boson with momentum fraction y while the constituent fermion has spin s. Typical results are plotted in Fig. 1 .

0.10

0.08 0.06 >\

W

l4

cc

0.04 0.02 0.00 0.0

0.2

0.4

0.6

0.8

1 .o

Y Figure 1. Bosonic structure functions in Yukawa theory, with a two-boson truncation (fs+:solid; fs-: long dash) and a one-boson truncation ( f ~ + :short dash; fs-: dotted). The constituent masses had the values mo = -1.7~0, ml = p1 = 1 5 ~ 0 .The resolutions used in the Gauss-Legendre method are K = 20 and N = 30.

521 3. Feynman-gauge QED We apply these same techniques to QED.g The Feynman-gauge Lagrangian is

which is independent of A and can therefore be solved without inverting a covariant derivative. We then obtain the Hamiltonian without antifermion terms as being

+ bf,s(P)b3.-s(Q)~~,-zs(p_,g)]

- p ) + h.c.} ,

where e p = (-1, 1 , 1 , 1 ) . The vertex functions U and V are given in Ref. 9. The dressed electron state, without pair contributions and truncated to one photon, is

I+) = C z i b i + ( P ) I O ) + i

c1

dkf? - %is (k)bf,(kb!”(P - IG)IO),

S,p,iJ

(21)

522 with one-photon-one-electron wave functions

Substitution into P+P-I$) = M21$) yields

This is the same form as in the one-boson Yukawa problem, with g2 -+ 2e2 and I1 + -2I1, and a n analytic solution is again obtained. From this solution we can compute various quantities, including the anomalous magnetic m ~ m e n t . ~ 4. A correlator in n / = ( 2 , 2 ) SYM theory Reduction of N=1 SYM theory from four to two dimensions provides the action we need. In light-cone gauge ( A - = 0) it is

+-(a-A+)2 1 + gA+ J+ 2

+ h g B T ~ ~ P r [ OR] x r ,+ g XI, 2 XJ]’

1

.

Here the trace is over color indices, the X I are the scalar fields and the remnants of the transverse components of the four-dimensional gauge field A,, the two-component spinors OR and BL are remnants of the right-moving and left-moving projections of the four-component spinor in the fourdimensional theory. We also define J+ = i [ X ~ , d - X r ] 2 @ 3 ~ , PI (TI, P2 ( ~ 3 and , €2 -ia2. The stress-energy correlation function for N=(S,S)SYM theory can be calculated on the string-theory side:12 (T++(x)T++(O))= (N2’2/g)x-5. We find numerically that this is almost true in N=(2,2)SYM theory.14 To compute the correlator,13 we fix the total momentum P+, compute the Fourier transform, and express the transform in a spectral decomposed

+

=

523 form

1 F(P+,x+) = -(T++(P+, x+)T++(-P+, 0)) 2L

The position-space form is recovered by Fourier transforming with respect to the discrete momentum P+ = K7r/L, where K is the integer resolution and L the length scale of DLCQ.2 This yields

We then continue to Euclidean space by taking r = d E to be real. The is independent of L. Its form can be matrix element (L/7r)(OIT++(K)li) substituted directly to give an explicit expression for the two-point function. The correlator behaves like l / r 4 at small r :

For arbitrary r , it can be obtained numerically by either computing the entire spectrum (for “small” matrices) or using Lanczos iterations (for large).13 In Fig. 2, we plot the log derivative of the scaled correlation function14 -

f

2

47r25-4 (3) N,”(2% + nf)

(T++(X)T++(O))

’

At small r , the results for f match the expected (1 - 1 / K ) behavior. At large r the behavior is different between odd and even K , but as K increases, the differing behavior is pushed to larger r . For even K , there is exactly one massless state that contributes to the correlator, while there is no massless state for odd K . The lowest massive state dominates for odd K a t large r ; however, this state becomes massless as K -+ co. In the intermediate-r region, the correlator behaves like r-4,75,or almost T - ~ . The size of this intermediate region increases as K gets larger. 5.

N=(l,l)SYM theory

at finite temperature

In this case, the Lagrangian is L: = Tr (-$F,,Fp” + i$’y,D”iP), with F,, = a,A, - &A, ig[A,,A,] and D, = 8, ig[A,]. The supercharge in light-cone gauge is &- = 23/49 dx- (i[+, &+] 2 $ 4 ~ a:)’$.

+

s

+

+

524

log,&)

Figure 2. The log derivative of the scaled correlation function f defined in Eq. (30) of the text. The resolution K ranges from 3 to 12. For even K , f becomes constant at large r and the derivative goes to zero.

From the discrete form we can compute the spectrum, which at large-Nc represents a collection of noninteracting modes. With a sum over these modes, we can construct the free energy at finite temperature from the partition function17J5 e - p o l T . The one-dimensional bosonic free energy is

and the fermionic free energy is

The contributions from the K - 1 massless states in each sector are

The total free energy, with the logs expanded as sums and the PO integral already performed, is

The sum over 1 is well approximated by the first few terms. We can represent the sum over n as an integral over a density of states: E n -+

525

s p( M ) d M and approximate p by a continuous function. The integral over

-

M can then be computed by standard numerical techniques. We obtain p by a fit to the computed spectrum of the theory and find p ( M ) exp(M/TH), with TH 0 . 8 4 5 d m , the Hagedorn temperature.18 From the free energy we can compute various other thermodynamic functions up to this temperature.15 N

6 . Future work

Given the success obtained to date, these techniques are well worth continued exploration. In Yukawa theory, we plan to consider the two-fermion sector, in order to study true bound states. For QED the next step will be inclusion of two-photon states in the calculation of the anomalous moment. For SYM theories, we are now able to reach much higher resolutions, by computing on clusters. This will permit continued reexamination of theories where previous calculations were hampered by low resolution, particularly in more dimensions. Earlier work on inclusion of fundamental matterlg can be extended to three dimensions and modified to include finite-N, effects, such as baryons with a finite number of partons and the mixing of mesons and glueballs. For all of this work, the ultimate goal is, of course, the development of techniques sufficient to solve quantum chromodynmics. Acknowledgments

The work reported here was done in collaboration with S.J. Brodsky, G. McCartor, V.A. Franke, S.A. Paston, and E.V. Prokhvatilov, and S. Pinsky, N. Salwen, M. Harada, and Y. Proestos, and was supported in part by the US Department of Energy and the Minnesota Supercomputing Institute. References P.A.M. Dirac, Rev. Mod. Phys. 21,392 (1949). H.-C. Pauli and S.J. Brodsky, Phys. Rev. D 32, 1993 (1985); 2001 (1985). S.J. Brodsky, H.-C. Pauli, and S.S. Pinsky, Phys. Rep. 301,299 (1997). I. Montvay and G. Miinster, Quantum Fields on a Lattice (Cambridge U. Press, New York, 1994); J. Smit, Introduction t o Quantum Fields on a Lattice (Cambridge U. Press, New York, 2002). 5. W. Pauli and F. Villars, Rev. Mod. Phys. 21,4334 (1949). 6. S.J. Brodsky, J.R. Hiller, and G. McCartor, Phys. Rev. D 5 8 , 025005 (1998) [arXiv:hep-th/9802120]; 60, 054506 (1999); 64, 114023 (2001): Ann. Phys. 296,406 (2002). 7. S.J. Brodsky, J.R. Hiller, and G. McCartor, Ann. Phys. 305,266 (2003).

1. 2. 3. 4.

526

8. S.J. Brodsky, J.R. Hiller, and G. McCartor, in preparation. 9. S.J. Brodsky, V.A. Franke, J.R. Hiller, G. McCartor, S.A. Paston, and E.V. Prokhvatilov, arXiv:hep-ph/0406325. 10. 0. Lunin and S. Pinsky, in New Directions in Quantum Chromodynamics, edited by C.-R. Ji and D.-P. Min, AIP Conf. Proc. No. 494 (AIP, Melville, NY, 1999), p. 140, [arXiv:hep-th/9910222]. 11. F. Antonuccio, 0. Lunin, and S. Pinsky, Phys. Rev. D 59, 085001 (1999); P. Haney, J.R. Hiller, 0. Lunin, S. Pinsky, and U. Trittmann, Phys. Rev. D 62,075002 (2000);J.R. Hiller, S. Pinsky, and U. Trittmann, Phys. Rev. D 64, 105027 (2001). 12. F. Antonuccio, A. Hashimoto, 0. Lunin, and S. Pinsky, JHEP 9907, 029 (1999). 13. J.R. Hiller, 0. Lunin, S. Pinsky, and U. Trittmann, Phys. Lett. B 482,409 (2000); J.R. Hiller, S. Pinsky, and U. Trittmann, Phys. Rev. D 63, 105017 (2001). 14. M. Harada, J.R. Hiller, S. Pinsky, and N. Salwen, to appear in Phys. Rev. D, arXiv:hep-ph/0404123. 15. J.R. Hiller, Y. Proestos, S. Pinsky, and N. Salwen, arXiv:hep-th/0407076. 16. Y. Matsumura, N. Sakai, and T. Sakai, Phys. Rev. D 52,2446 (1995). 17. S. Elser and A.C. Kalloniatis, Phys. Lett. B 375,285 (1996). 18. R. Hagedorn, Nuovo Cimento Suppl. 3, 147 (1965); Nuovo Cimento 56A, 1027 (1968). 19. J.R. Hiller, S.S. Pinsky, and U. Trittmann, Nucl. Phys. B 661, 99 (2003); Phys. Rev. D 67,115005 (2003).

FERMIONIC THEORIES IN TWO-DIMENSIONAL NONCOMMUTATIVE SPACE

E. F. MORENO* Departamento de Fisica Universidad Nacional de La Plata C.C. “ 6 7 - 1900 La Plata - Argentina E-mail:[email protected]

We analyze the connection between Wess-Zumino-Witten and free fermion models in two-dimensional noncommutative space. Starting from the computation of the determinant of the Dirac operator in a gauge field background, we derive the corresponding bosonization recipe. Concerning the properties of the noncommutative Wess-Zumino-Witten model, we construct an orbit-preserving transformation that maps the standard commutative WZW action into the noncommutative one.

1. Introduction

Noncommutative field theories have recently attracted much attention in connection with the low-energy dynamics of D-branes in the presence of a background B field Concerning two-dimensional noncommutative field theories, both bosonic and fermionic models have been recently investigated 4-9. In this paper we pursue the analysis of two-dimensional noncommutative models by carefully studying the fermion effective action. ll2i3.

2. Fermionic and gauge fields in d = 2 noncommutative

space We work in two-dimensional Euclidean space and define the tween functions $(x) and x ( x ) in the form

*Associated with CONICET.

527

* product

be-

528

where 6,, = OED, with 6 a real constant. Then, the Moyal bracket is defined as {+(Z)I

x ( x ) ) = +(XI

* X(”)

- x(x)

* +(x)

which implies a noncommutative relation for space-time coordinates

{x,, 5,)

= ie,,

(2) zp,

(3)

In the case of gauge theories, noncommutativity leads to the definition of the curvature F,, in the form

F,, = a,A, - &A, - ie{A,, A,}

(4)

Gauge transformations are defined in the form

* g-’(x) + ig-’(x)a,y(z)

A:(%)= g(x) * A,(x) where g(x) is represented by a

g ( x ) = exp,(iX(z))

(5)

* exponential, 1 = 1+ iX(x) - -X(x) * X(x) + . . . 2

(6)

with X = X a t a taking values in the Lie algebra of U ( N ) . The covariant derivative D, [A]implementing gauge transformations takes the form

D,X = a X ,

- ie{A,, A}

(7)

Given a fermion field $(x), three alternative infinitesimal gauge transformations can be considered lo

* $(x) z$(x) * 4x1

S,$ S,$

= i+)

(8)

=

(9)

S,$

= i{+),

-’

$(.))

(10)

In this respect, we should refer to fermions in the fundamental f (eq.(8)), ‘anti-fundamental’ f (eq.(9)) and ‘adjoint’ ad (eq.(lO)) representations. The associated covariant derivative are defined accordingly,

Dj$+”)

= a D $ ( 4 - ieA,(x)

D $ 4 1 $ ( 4 = a,$(.) D 3 A I $ ( x ) = D,$(x)

* $(x)

+ ie$(x)A,(x)

(11) (12)

(13)

Using each one of these three covariant derivatives we can construct three different gauge invariant Dirac actions for fermions

529

The corresponding gauge effective action is given by

3. Perturbative effective action

Let us start by computing the quadratic part of the effective action defined by eq.(15). The interaction term SI of the action S[$,$,A] (eq.(14)) takes the form, in momentum representation,

sr = 2ie

l,

q ( P ) y p $ ( q ) ~ , ( - p- 4 ) f ( q A P )

(16)

with f(p A q) =

{

fund ei P A q -e-i pAq antif 22 sin(p A q ) adj

,

P A q = 8’l”Ppq”

(17)

The quadratic part r(’) of the effective action is given by the vacuum polarization tensor, thus giving

Clearly, for the fundamental and anti-fundamental representation, If ( q A p)I2 = 1 and we obtain the standard (commutative) result. For the adjoint representation, using the identity sin(a)2 = 1/2 cos(2a)/2 we can extract form eq. (18) the so-called planar contribution (corresponding to the factor 1/2) and the non-planar contribution (corresponding to the factor - cos(2a)/2:

The planar contribution to the diagram is the standard (commutative) one and can be computed using for example dimensional regularization (in this case the infrared an ultraviolet divergences cancel each other). One has

It is worthwhile to mention that this result is twice the effective action in the fundamental and anti-fundamental representations (that is, taking the Dirac operator as defined either by (11) or by (12)), as it can be easily

530 seen by noticing that in the later the diagram has a vertex contribution of eipAq e-apAq = 1 (there is no non-planar contribution) while in the former we have a contribution -(2isin(p A q ) 2 = 4(1/2 - 1/2cos(2 p A 4)). Thus, for the planar part of the diagram we have (2) Adj = 2 r(2)F u n d (21) ‘planar Concerning the non-planar contribution to I?(’) (see for a detailed computation) , it can be shown using Schwinger parameters that it vanishes. That is, up to quadratic order in the fields, the effective action is given by the planar part:

Therefore, assuming that the higher point contributions to the effective action are the minimal necessary to recover gauge invariance (we will prove this statement in the next section), the effective action in the adjoint representation is twice the effective action in the fundamental representation. Hence, a relation like (21) should hold for the complete effective action I? rAdj = 2 rFund (23) This computation can be easily generalized to U ( N ) . The planar and non-planar contributions are the same as in the U(1) case except for the group theoretical prefactors. Taking into account that the non-planar contribution vanishes we finally have for U ( N )in the adjoint representation

This result is 2N times the quadratic effective action in the fundamental representation. 4. The fermion determinant in the U(1) case

Here we shall briefly describe the exact calculation of the effective action for noncommutative U(1) fermions in the fundamental and adjoint representations by integrating the chiral anomaly. In two dimensions a gauge field A, can always be written in the form 415

1 .

4 = - -e ( V w 4 ,771) * m 4 , 771

(25)

with

+

U [ 4 ,rll = exP*(Y54

277)

7

(26)

531 so one can relate the fermion determinant in a gauge field background A, with that corresponding to A, = 0 by making a decoupling change of variables in the fermion fields. The appropriate change of fermionic variables is

llt = U[+>rll * x

4 = x * u-l[+,rll

7

(27)

in the fundamental representation or

llt = ey5{4?'}* x = x + r5{+, XI

1 + Ti+, {+, X I ) +

* *

in the adjoint. One gets

where J f [t+,A] is the Fujikawa Jacobian associated with a transformation Ut such that (0 5 t 5 1)

uo = 1

7

Ul = U[+,rll

Let us briefly describe the calculation presented in 4,5 based in the the evaluation of the chiral anomaly. Considering an infinitesimal local chiral transformation of the fermion fields the chiral anomaly A = A?", 13,j,"" = A"[A] ,

(30)

where j: is the chiral current

j: =

{

llta * $p (r5 Y,)Ba *

fund (31)

(r5 r?pa

adj

can be calculated from the F'ujikawa Jacobian J [ EA,] , associated with infinitesimal chiral transformation, log J [ E A,] , = -2tr'

sd

'ZA[[A]E(Z)

(32)

Using the heat-kernel regularization, the anomaly can be written as

A[A]= lim Tr M2+Co

(33)

The covariant derivative in the regulator has to be chosen among those defined by eqs.(ll)-(13) according to the representation one has chosen for

532 the fermions. Concerning the fundamental representation, it can be shown but that the anomaly gives a result similar to the commutative case with the noncommutative version of FPu given in (4). e A f [ A ]= -&PUFP,, 4n (34) 111475

Analogously, one obtains for the anti-fundamental representation e A f [ A ]= --&P"FPU (35) 4n For the adjoint representation, however we have different result. In fact, expanding equation (33) in a planar wave basis, keeping the proper powers of M , taking the trace and integrating over d 2 k we arrive to the following expression for the anomaly

Notice in this expression that if we take the 8 -+ 0 limit before taking the M -+ 00 limit Aadjvanishes and the Jacobian is 1, so we recover the standard (commutative) result which corresponds to a trivial determinant for the trivial U(1) covariant derivative. However the limits 8 -+ 0 and M + 00 do not commute so that if one takes the M 00 limit at fixed 8 one has the result -+

"s

Aadj= _2n _ d2x&PuFLu* (I

(37)

which is twice the result of the fundamental representation. The integration in t can be done in a very similar way to the commutative case (see and for details). The result can be put in a more suggestive way in the light cone gauge where

Indeed, in this gauge one can see that the logarithm of the Jacobian becomes log

(w) det

iC

= log J [ $ ,A] =

+--&ijktrC 12n

d3yg-'

C

--trc 8n

/

d 2 x (dPg-')

* (aPg)

* ( a i g ) * 9-l * ( a j g ) * g - l *

(&g)

(39)

with c = 1 in the fundamental and anti-fundamental representations and c = 2 in the adjoint representation. Here we have written d 3 y = d 2 x d t so

533

that the integral in the second line runs over a 3-dimensional manifold B which in compactified Euclidean space can be identified with a ball with boundary S 2 . This is precisely the noncommutative extension of the WessZumino-Witten action. This result can be straightforwardly generalized to the group U ( N ) ,with the only difference that a trace over the color indices has to be performed and c = 2N in the adjoint representation4. 5. The bosonization rules Once one has gotten an exact result for the fermion determinant, one can derive the path-integral bosonization recipe. That is, a mapping from the two-dimensional noncommutative fermionic model onto an equivalent noncommutative bosonic model. This implies a precise relation between the fermionic and bosonic Lagrangian, currents, etc. The basic procedure to obtain this bosonization recipe parallels that already established in the ordinary commutative case. For the noncommutative case, it was developed in detail in (see also 5 ) . Summarizing we obtain, analogously to the commutative case, the following bosonization rules

We have then an equivalence between a two-dimensional noncommutative free fermion model and the non-commutative Wess-Zumino-Witten model. Now, on one hand we know that, being quadratic, the action for noncommutative free fermions coincides with that for ordinary (commutative) ones. On the other hand we know that ordinary free fermions are equivalent to a bosonic theory with ordinary WZW action. The situation can be represented in the following figure

Jd2x

4i$$

I

-

WZW[g]

534

So, we were able to pass from the noncommutative WZW theory with action WZW[ij]to the commutative one, WZW[g]by going counterclockwise through the fermionic equivalent models. One should then be able to find a mapping ij -+ g , analogous to the one introduced in for noncommutative gauge theories. In the present case, the mapping should connect exactly WZW[ij]with WZW[g]filling the cycle in the figure. Next section is devoted to the construction of such a mapping. 6. Mapping of the Wess-Zumino-Witten actions

The Seiberg-Witten map is a mapping between gauge theories defined in different noncommutative spaces, so is natural to try adapt this construction to noncommutative WZW fields. Consider a WZW action defined in a non-commutative space with deformation parameter 0. The action is invariant under chiral holomorphic and anti-holomorphic transformations ij

-+

iI(5)ij R(z)

(41)

so in analogy with the Seiberg-Witten mapping we will look for a trans-

formation that maps respectively holomorphic and anti-holomorphic orbits into orbits. Of course the analogy breaks down at some point as this holomorphic and anti-holomorphic “orbits” are not equivalence classes of physical configurations, but just symmetries of the action. However we will see that such a requirement is equivalent, in some sense, to the “gauge orbits preserving transformation condition” of Seiberg-Witten. Thus, we will find a transformation that maps a group-valued field g’ defined in noncommutative space with deformation parameter 0’ to a groupvalued field i j , with deformation parameter 0. We demand this transformation to satisfy the condition

iI’(z)*’ 4’

*’ R‘(z) + iI(z)* ij * R(z)

where the primed quantities are defined in a O‘-noncommutative space and the non primed quantities defined in a 0-noncommutative space. In particular this mapping will preserve the solutions of the equations of motion:

4’ = a’(z)*’ pyz)

ij = a(5)* p ( z )

(43) After some trial and error it can be shown that the following transformations -+

535 preserve condition (42) for functions R(z) and deed we have, for example

= (i* O ( z ) ) * a,

n(z)independent of 8. In-

(4 * R(z))-l * &(i* R(z))

(45)

and a similar equation for the anti-holomorphic transformation. The next step is to see how does the WZW action transforms under this mapping. Let us define w = 4-1

* dij

(46)

where 6 is any variation that does not acts on 8, and the holomorphic current j ,

j , = 4-1

* a,ij

(47)

It can be shown that

dw _ de - - a , ~* j ,

-j,

* aZw

4, = -a& -

* j , - j , * a& = -az(jZ) (49) d8 Similarly we can find the variations for a = dij * 4-l and the antiholomorphic current j,. Now, instead of studying how does the &map acts on the WZW action, it is easy to see how does the mapping acts on the variation of the WZW action with respect to the fields. In fact, we have k

dW[4 = 7T

/

d2x tr ( & j , w )

(50)

where j , and w are the quantities defines in eqs.(46) and (47) and there is no *-product between them in eq.(50) in virtue of the quadratic nature of the expression. Thus, a simple computation shows that

and we have a remarkable result: the transformation (44),integrated between 0 and 6 maps the standard commutative WZW action into the noncommutative WZW action. That is, we have found a transformation mapping orbits into orbits such that it keeps the form of the action unchanged. This should be contrasted with the 4 dimensional noncommutative YangMills case for which a mapping respecting gauge orbits can be found (the

536

Seiberg-Witten mapping) but the resulting commutative action is not the standard Yang-Mills one. However, one can see that the mapping (44) is in fact a kind of Seiberg-Witten change of variables. Indeed if we consider the WZW action as the effective action of a theory of Dirac fermions coupled t o gauge fields, as we did in previous sections, instead of an independent model we can relate the group valued field g to gauge potentials. As we showed in eq.(38), this relation acquires a very simple form in the light-cone gauge A+ = 0 where

A-

*

= ij(z) a-ij-’(z)

.

(52)

But notice that in this gauge, A- coincides with jzrso we have

6 ~ = - 68 ( a z ~*-A-

+ A- * a Z ~ - )

(53)

which is precisely the Seiberg-Witten mapping in the gauge A+ = 0.

Acknowledgments

I am very grateful to the organizers of the conference “Continuous Advances in QCD-2004” for their invitation to participate in the conference. References 1. A.Connes, M.R. Douglas and A.S. Schwarz, JHEP 02 (1998) 003. 2. M.R. Douglas and C. Hull, JHEP 02 (1998) 008. 3. N. Seiberg and E. Witten, JHEP 09 (1999) 032. 4. E. Moreno and F.A. Schaposnik, JHEP 0003 (2000) 032.

5. 6. 7. 8. 9.

E. Moreno and F.A. Schaposnik, Nucl. Phys. B596 (2001) 439. C.S. Chu, Nucl.Phys. B580 (2000) 352. L. Dabrowski, T. Krajewski and G. Landi, hep-th/00003099. K. F’uruta and T. Inami, hep-th/0004024. C. Nhiez, K. Olsen and R. Schiappa, JHEP 0007 (2000) 030. 10. J.M. Garcia-Bondia and C.P. Martin, Phys.Lett. B479 (2000) 321. Phys. Rev. D21 (1980) 2848. 11. F. Ardalan and N. Sadooghi, hep-th/0002143

NONABELIAN SUPERCONFORMAL VACUA IN n/ = 2 SUPERSYMMETRIC THEORIES

ROBERTO AUZZI Scuola Normale Superiore, Pisa Istituto Nazionale d i Fisica Nucleare - Sezione d i Pisa Piazza dei Cavalieri 8, Pisa, Italy E-mail: [email protected] The dynamics of some confining vacua which appear as deformed superconformal theories with a nonabelian gauge symmetry is studied by taking concrete examples in N = 2, S U ( 3 ) and USp(4) gauge theories with n f = 4 and equal quark masses. The low-energy degrees of freedom consist of some nonabelian monopoles and dyons with relatively non-local charges. The mechanism of cancellation of the beta function is studied, in analogy to an abelian superconformal vacuum studied by Argyres and Douglas. Study of our SCFT theories as a limit of colliding vacua suggests that confinement in the present theories occurs in an essentially different manner from those vacua with dynamical Abelianization, and involves strongly interacting nonabelian magnetic monopoles.

1. Introduction

The most interesting vacua found in the moduli space of N = 2 theories are the superconformal ones. The low-energy degrees of freedom are relatively nonlocal dyons. Electric particles couple to the vector potential A , magnetic particles couple to the magnetic potential AD and there is no simple local relation between A and A D . The first example of these vacua has been discovered by Argyres and Douglas in N = 2 S U ( 3 ) Super Yang-Mills (see 14); other examples have been discovered in S U ( 2 ) theories with flavor for particular values of quarks bare masses (see "). A classification of these vacua has been developed in 12. In the Argyres-Douglas vacuum the degrees of freedom are relatively non-local abelian dyons. We will study two examples in which relatively non-local nonabelian dyons are involved. Perturbing the theory with a small adjoint mass p, confinement and chiral symmetry breaking ensue, as can be demonstrated indirectly through various considerations, such as the study of large p effective action '. Confinement is described in this way in

537

538 many vacua of n/ = 2 theories with N j quarks hypermultiplets. The analysis of these nonabelian vacua has been developed in 5: these results are reviewed here.

and in

2. A nonabelian vacuum in S U ( 3 ) Nf = 4 theory

We study the r = 2 vacuum of S U ( 3 ) N = 2 gauge theory with 4 quarks hypermultiplets. The Seiberg-Witten curve 1,2 of this theory for equal bare quark masses is (by setting 2 h = 1) 3

y2 =

U(Z- 4i12 - (x + m14 = (x3 - uz - v ) -~ + m14. (Z

(1)

i=l

The T = 2 vacuum correspond to the point, diag 4 = (-m, -m, 2m),i.e.,

u = 3m2; v = 2m3,

(2)

corresponding to an unbroken S U ( 2 ) symmetry. This vacuum splits to six separate vacua when the quark masses are taken to be slightly unequal and generic. The conformally invariant vacua occur at the points where more than one singularity loci in the u - u plane, corresponding to relatively nonlocal massless states, coalesce l 1 > l 2 . The nature of the low-energy degrees of freedom at the SCFT vacua can be determined by studying the monodromy matrices around each of the singularity curves. In the case of the r = 2 vacua of S U ( 3 ) theory with n j = 4, it is necessary to study the behavior of the theory near the point Eq.(2). Set

u=3m2+u,

v=2m3+u,

The discriminant of the curve factorizes

l2

x+m-+x.

(3)

as

A = A, A+ A _ ,

(4)

where the squark singularity” corresponds to the factor

A,

= (mu - u ) ~ ;

(5)

where the fourth zero represents the flavor multiplicity n f = 4;

A+ = 4 m u

+ 36m2u + 108m3u+ 108m4u + u2 + 24mu2+ 36 m2u2+

4~~-4u-36mu-108m~u-108m~u-18uu-27u~,

(6)

aAt large m hence at large U and V it represents massless quarks and squarks; as is well known, at small m it becomes monopole singularity, due to the fact that the corresponding singularity goes under certain cuts produced by other singularities 15.

539

A- = - 4 m u

+ 36m2u - 108 m3u + 108m4u + u2 - 24mu2 + 36m2u2 + + 108m2w - 108m3w + 18uv - 27v2, (7)

4u3 + 4 w - 36mw

represent the loci where some other dyons become massless. The m = 0 case will be studied in some detail in the remaining part of this section. In order t o define uniquely the monodromies around the three curves near the SCFT point we consider the intersections of these curves with the S3 sphere 1612

+ ( G I 2 = 1.

(8)

It is convenient to make first a stereographic projection from S3 --t R3 after which the intersection curves take the form of the three linked rings (see figure 1).

Figure 1. Zero loci of the discriminant of the curve of N = 2, S U ( 3 ) , nf = 4 theory at m = 0.

We consider now various closed paths in the space (u,w), starting from a fixed reference point (for instance lying above the page), encircling various parts of the rings and coming back to the original point. As the branch points move, the integration contour cycles ai’s and pi’s over the SeibergWitten curve get entangled in a non-trivial way. The relation between the monodromy matrix and the corresponding massless charge (with magnetic charge m and electric charge q ) is the following (I6):

540

Indeed, various monodromy transformations are related by the conjugations, for example: = Mg1A5M6, A2 = kfT1Mikf2,

. ..

(10)

as can be easily verified by looking at Fig. 1. By knowing any three of them, for instance MI,M2, M6 above, these relations yield uniquely all the other monodromy matrices. The self-consistency of these relations is a non-trivial check for the numerical calculation of the first three matrices MI,Mz, M6. Using formula (9), it’s possible to calculate the the magnetic and electric charges {ml, m2;91,92): Table 1. Matrix

Charge

Note that only the members of the same doublet are relatively local, i.e., have a vanishing relative Dirac unit

c( 2

NO =

- qAi g B i

gAi 9Bi

).

(11)

i= 1

It’s interesting at this point to compute the low-energy abelian effective coupling matrix r i j (see for the details of the calculation). For m = 0 we have the following Riemann surface: y 2 = (x3 - ux - w)2 - x4 = (x3

+ x2 - ux - w)(x3 - x2 - ux - w).

(12)

At u = 0, w = 0 the branch points are at: 21

= x2 = 2 3 = 2 4 = 0,

25

= -1,

x6 = 1.

(13)

A similar degeneration of genus two Riemann surfaces was studied in the period matrix in a particular base r i j splits as:

17;

541 7-22 is the modulus of the “large” torus (given by the modulus of the genus one surface with two colliding branch points in 0 and other two branch points in 51; in our case 5-22 = 0 because this U(1) factor is infrared free) and 5-11 is the modulus of the “small” torus (given by the modulus of the genus one surface formed by the four colliding branch points; it is well defined only if in our limit the angles formed by the four colliding points are kept constant). It is convenient to introduce the variables ~ , (see p 14, 4 ) , given by:

2 v = E ;

u=Ep.

(15)

In these coordinates 5-11near the conformal point is an holomorphic function which depends only on p and can be written in terms of Jacobi Theta functions (see for the details). The function 5-11(p) has the following singularities:

p=

00

+ v = 0.

(17)

Thus around the points +2, -2,2i, -2i, 00 the function 7-11(p) has nontrivial monodromies, which correspond to the Ui(1) c S U ( 2 ) charges of the previous section.

Figure 2.

The p plane.

A crucial observation is that in the (u, v) space there are infinite number of copies of p plane, corresponding to different phases of E . Namely, by varying the phase of E the linked rings in Fig. 1 are cut in different sections (different copies of p plane). It is therefore natural to identify the set of singularities in each section with the monodromy matrices of Table 1. Physics near the conformal vacuum depends only on p; different sections of the moduli space at different phase of E give rise to the same dynamical

542

picture in different basis. In a given section at constant E the degrees of freedom are the following: a SU(2) x U(1) gauge theory, with 4 magnetic monopole doublets (61, @I) = ( f l , 0), a non-abelian electric doublet with charges (rh1,Ql) = ( 0 , f l ) and a non-abelian dyon doublet with charges (fi1,gl) = ( f 2 , hl) . The four magnetic doublets have U(1) abelian magnetic charge equal to f i z = 1; the other two doublets have abelian magnetic charge equal to f i 2 = 2. Changing the phase of E the p plane cuts the knot in Figure 1 at different points and the charges of our degrees of freedom can be different, but each od these description is equivalent to the one we have given modulo an S p ( 4 , Z)transformation. The SU(2) factor defines an interacting conformal theory. The p function cancellation is reproduced by the following guess, which is a generalization of an idea of Argyres and Douglas 14. The four magnetic monopole doublets cancel the contribution of the dual SU(2) gauge bosons and of their supersymmetric partners as in a local N = 2, SU(2) gauge theory with n f = 4. The non-trivial part of the cancellation occurs between the non-abelian electric and dyonic doublets. By considering the U(1) subgroup of the SU(2), their contribution to the first term of the beta function cancel if 14:

a

With the dyon charges works if

(fi1,Ql) =

(0, f l ) ,( f 2 , f l ) at our disposal, this

(19)

This calculation reproduces the value of ~ 1 1 ( p = 0) (see 4 ) . p = 0 seems to define the right superconformal limit; note that this value of p is the only one which does not break the anomaly free 2 4 symmetry (this discrete symmetry is contained in the anomalous U ( ~ ) R symmetry, which should be indeed somehow restored in all N = 2 scale invariant theories). 3. A nonabelian vacuum in U S p ( 4 ) N f = 4 theory

In a similar analysis is developed for a SU(2) superconformal vacuum in the U S p ( 4 ) theory with Nf = 4 with m = 0. With m # 0 the situation is very similar to the S U ( 3 ) vacuum analised in the previous section: the structure of the monodromies is the same as in Figure 1. In the m -+ 0 limit something new happens.

543

Figure 3. m + 0 limit for the monodromies structure near the superconformal vacuum of USp(4), Nf = 4 theory.

The Seiberg-Witten curve of the U S p ( 4 ) theory quarks, is (setting A = 1) y2

= x(x2 -

(16),

ux - vy - 4x3.

with nf massless (20)

where

u = $I + &,

v = -$&g*

(21)

$1 and $2 are the two components of the adjoint scalar field that breaks the gauge symmetry:

d i w d = ($1,

$2, -41, - 4 2 ) .

(22)

The behavior of the curve is highly singular at the two points U = f2, V = 0, where four of the five branchpoints coalesce. In these vacua we have $1 = 0, and so the gauge symmetry is broken to U S p ( 2 ) x U(1) 2: S U ( 2 ) x U(1). Giving equal masses to all of the quarks (ma = m ) the

544

U = -2 singularity splits into three singularities: two of them have an unbroken U(1)2, while the third has four colliding branchpoints. Considering the intersections of the zero-discriminant part of the moduli space with a little 3-sphere centered in the superconformal vacuum, the two rings shown in the third drawing of figure 3 are obtained. Monodromies around different elements of this knot configuration have been computed in 5 ; the results is that these elements are not in the form (9) and so charges cannot be computed directly. The approach used is to compute charges in the m 4 0 limit shown in figure 3: giving a mass to the quark hypermultiplets, one of the two rings splits in three disconnected pieces; monodromies for m # 0 are in the form (9) and so it is possible to compute the charges involved (see 5). The degrees of freedom are the following: a SU(2) x U(1) gauge theory, with 4 magnetic monopole doublets (rii1,Gl) = ( f l , 0), a non-abelian electric doublet with charges (riil,Q1) = (0, f l ) and two non-abelian dyon doublets with charges (rii1,Qll) = ( f 2 , f l ) . The four magnetic doublets have abelian magnetic charge equal to 1; the electric doublet and one of the dyonic doublet have abelian magnetic charge equal to 2; the other dyonic doublet has abelian magnetic charge equal to 0. The P-function cancellation condition (18) gives the following value of 7:

This result does not agree with the direct calculation done using the Seiberg-Witten curve (see 5 ) . With this approach the following value of 7 is found: 1 (24) 711 = -2' Neglecting the contribution of the electric doublet in (18) the condition becomes 1 7 - _ _ c (25) 2' and it would be satisfied by this vacuum; however, there is no a priori reason to neglect this particle in the calculation. 4. Superconformal Vacuum as Limit of Colliding Vacua

The superconformal limit may be approached breaking the flavor symmetry explicitly by unequal bare quark masses. For example, in the T = 2

545 vacuum of N = 2 S U ( 3 ) Nf = 4 theory with generic and small quark hypermultiplets masses, there are six nearby singularities. In the equal mass limit, these singularities coalesce and become the conformal vacuum. Each of the theories before the equal mass limit is taken is a local U(1)2 gauge theory, with precisely two massless hypermultiplets, each of which carrying only one of the U(1) charges. A partial support comes from the observation that the massless states at one singularity and those at another singularity are relatively non-local to each other (4). Nonetheless, there are reasons to believe that the mechanism of confinement in the superconformal theory cannot be understood this way. Adding an adjoint mass term p T r a 2 into each of these vacua the low energy degrees of freedom (monopoles, dyons) condense. However, in the superconformal limit all the condensates become zero (see the discussion of the SU(3) case in for more details). This is analogous to the phenomenon discussed in l8 at the Argyres-Douglas point of JV= 2, S U ( 2 ) gauge theory with nf = 1.

5 . Conclusions

The charges of the magnetic and dyonic degrees of freedom have been determined in two different superconformal vacua in SU(3) and USp(4) gauge theories; the direct calculation of some monodromies and the monodromomies algebra have been used and the result is self-consistent (this is a non-trivial check). These charges have been interpreted as non abelian degrees of freedom (in our vacua the gauge symmetry is broken to S U ( 2 ) x U(1)). The effective gauge coupling found with the Seiberg-Witten curve and the one found from the condition of the @-function cancellation at one loop have been compared. In the SU(3) case a perfect agreement is found, as in the abelian vacua studied by Argyres and Douglas. In the USp(4) case we have not found this agreement; the p function cancellation condition reproduces the value found with the Seiberg-Witten only if we neglect one of the dyons in the calculation. If an adjoint mass term in the superpotential AW = p T r Q 2 is added, confinement and chiral symmetry breaking occur. This can be seen in the limit p >> AN=^ ’. It’s difficult to study confinement and chiral symmetry breaking in the limit p << hN=2 because we have no Lorentz-invariant local effective lagrangian. In the limit of different quarks masses the superconformal vacuum splits in a finite number of local abelian vacua; the monopoles VEV in each of these vacua goes to zero in the superconformal limit in which they collide.

546

T h e microscopic mechanism of confinement and dynamical symmetry breaking in these vacua should be essentially different from t he Abelian confinement mechanism found in t he original Seiberg-Witten paper ' v 2 , and involves strongly interacting non-Abelian magnetic degrees of freedom.

Acknowledgments T h e author thanks Kenichi Konishi, Roberto Grena, Jarah Evslin, Stefan0 Bolognesi for many useful discussions.

References 1. N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19; Erratum ibid. Nucl.Phys. B430 (1994) 485, hep-th/9407087. 2. N. Seiberg and E. Witten, Nucl. Phys. B431 (1994) 484, hep-th/9408099. 3. G. Carlino, K. Konishi and H. Murayama, Nucl. Phys. B590 (2000) 37, hep-th/0005076; G. Carlino, K. Konishi, P. S. Kumar and H. Murayama, hepth/0104064, Nucl. Phys. B608 (2001) 51. 4. R. Auzzi, R. Grena and K. Konishi, Nucl. Phys. B653 (2003) 204, hepth/0211282. 5. R. Auzzi and R. Grena, arXiv:hep-th/0402213 6. P. Goddard, J. Nuyts and D. Olive, Nucl. Phys. B125 (1977) 1, E. Weinberg, Nucl. Phys. B167 (1980) 500; Nucl. Phys. B203 (1982) 445. 7. S. Bolognesi and K. Konishi, Nucl. Phys. B645 (2002) 337, hep-th/0207161. 8. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and H. Murayama, arXiv:hepth/0405070. 9. G. 't Hooft, Nucl. Phys. B190 (1981) 455. 10. S. Mandelstam, Phys. Lett. 53B (1975) 476; Phys. Rep. 23C (1976) 245. 11. P. C. Argyres, M. R. Plesser, N. Seiberg and E. Witten, Nucl. Phys. 461 (1996) 71, hep-th/9511154. 12. T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Nucl. Phys. B471 (1996) 430, hep-th/9603002. 13. T. Eguchi and Y. Sugawara, JHEP 0305 (2003) 063 [arXiv:hep-th/0305050]. 14. P. C. Argyres and M. R. Douglas, Nucl. Phys. B448 (1995) 93, hepth/9505062. 15. A. Bilal and F. Ferrari, Nucl. Phys. B516 (1998) 175, hep-th/9706145. 16. P. C. Argyres and A. F. Faraggi, Phys. Rev. Lett 74 (1995) 3931, hepth/9411047; P. C. Argyres, M. R. Plesser and A. D. Shapere, Phys. Rev. Lett. 75 (1995) 1699, hep-th/9505100; 17. A. Lebowitz, Israel J . Math. 12 (1972) 223. 18. A. Gorsky, A. Vainshtein and A. Yung, Nucl. Phys. B584 (2000) 197, hep- th/0004087.

PREDICTIONS FOR QCD FROM SUPERSYMMETRY

F. SANNINO N O R D I T A , Blegdamsvej 17, Copenhagen 0 DK-2100, Denmark We review the construction of the effective Lagrangians of the VenezianoYankielowicz (VY) type for two non-supersymmetric theories containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group (orientifold theories). Since these theories are planar equivalent, at N --f 00 to super Yang-Mill their effective Lagrangians coincides with the bosonic part of the VY Lagrangian. We depart from the supersymmetric limit in two ways. First, we consider finite but still large values of N . Then 1/N effects break supersymmetry. We suggest a minimal modification of the VY Lagrangian which incorporates these 1/N effects, leading to a non-vanishing vacuum energy density. We then analyze the spectrum at finite N . For N = 3 the two-index antisymmetric representation (one flavor) is equivalent to one-flavor QCD. We show that in this case the scalar quark-antiquark stat‘e is heavier than the corresponding pseudoscalar state, “ 7”’. Second, we add a small fermion mass term. The fermion mass term breaks supersymmetry explicitly. The vacuum degeneracy is lifted. The parity doublets split. We evaluate the splitting. Finally, we include the 0-angle and study its implications.

1. Introduction

Recently it has been argued that non-supersymmetric Yang-Mills theories with a fermion in the two index symmetric or antisymmetric representation are nonperturbatively equivalent to supersymmetric Yang-Mills (SYM) theory a t large N, so that exact results established in SYM theory (e.g. 3,4) should hold also in these “orientifold” theories. For example, the orientifold theories, at large N, must have an exactly calculable bifermion condensate and an infinite number of degeneracies in the spectrum of colorsinglet hadrons. The phenomenon goes under the name of planar equivalence; it does not mean, however, the full parent-daughter coincidence. For instance, at N -+ 00 the color-singlet spectrum of the orientifold theories does not include composite fermions. The planar equivalence relates corresponding bosonic sectors in the corresponding vacua of the two theories. Some predictions for one-flavor QCD (which is the antisymmetric orientifold daughter at N = 3) were made along these lines in 2 , 5 . Here we ‘i2

547

548

review the construction of the effective Lagrangians for the orientifold field theories which are able to capture relevant 1/N corrections '. Our starting point is the effective Lagrangian for supersymmetric Yang-Mills. The name of orientifold field theory is borrowed from string-theory terminology 7. 2. Reviewing SYM effective Lagrangian

The effective Lagrangian for supersymmetric gluodynamics was found by Veneziano and Yankielowicz (VY) '. In terms of the composite color-singlet chiral superfield S,

it can be written as follows:

Lvy = 94Na2 J d % d % ( S S t ) ' +

:/d2tJ

{Sln

($)N

- N S } +H.c.,

where A is a parameter related to the fundamental SYM scale parameter a. We singled out the factor N 2 in the Kahler term to make the parameter (Y scale as N o , see Eq. (3) below. With our definitions, the gluino condensate scales as N . Requiring the mass of the excitations to be N independent one deduces

a-NO.

(3)

Indeed, the common mass of the bosonic and fermionic components of S is M = 2 a A / 3 . The chiral superfield S at the component level has the standard decomposition S(y) = p(y) & 9 x ( y ) B2F(y), where y p is the chiral coordinate, y l = x p - itJap8, and

+

+

where f.t. stands for fermion terms. aThe Grassmann integration is defined in such a way that

tI2 d28 = 2.

549

The complex field 'p represents the scalar and pseudoscalar gluino-balls while x is their fermionic partner. It is important that the F field must be treated as auxiliary. This lagrangian is not complete as pointed out in '. Recently an extended VY Lagrangian which passes a number of non trivial consistency checks while naturally yielding the VY effective theory augmented by the missing terms pointed out in has been constructed l o . Such an extension requires the introduction of a glueball superfield, i.e. a chiral superfield with zero R charge. Earlier attempts of generalizing the VY Lagrangian containing glueball degrees of freedom are discussed in the literature l 2 J 3 J 4 > ? . These extensions were triggered, in part, by lattice simulations of SYM spectrum 16. Since in this paper we are interested in the mesonic degrees of freedom we will not consider the generalized version of the VY l o although it is now straightforward to generalized our results to the improved VY theory. For our purposes of most importance are the scale and chiral anomalies, "

(5) where J,, is the chiral current and W U is the standard (conserved and symmetric) energy-momentum tensor. In SYM theory these two anomalies belong to the same supermultiplet l7 and, hence, the coefficients are the same (up to a trivial 3 / 2 factor due to normalizations). In the orientifold theory the coefficients of the chiral and scale anomalies coincide only at N = 00; the subleading terms are different. Summarizing, the component form of the VY Lagrangian is LVY =

N2 7 (cpp)-2 1 3 8@dP'p-

4aN2

('p p ) 2 / 3In p In cp + fermions , (6) 9

where we set A = 1 to ease the notation. 3. Effective Lagrangians in orientifold theories

In the theory with the fermions in the two index-antisymmetric representation the trace and the chiral anomalies are

550 where

and F is given in Eq. (4).The gluino field of supersymmetric gluodynamics is replaced in this theory by two Weyl fields, @ ~ [ i ~ fand $a,[i,j~ , which can be combined into one Dirac spinor. The color-singlet field cp is now bilinear in and $ & , [ i , j ~ . Note the absence of the color-singlet fermion field x which was present in supersymmetric gluodynamics. In the infinite N limit we will deal with the same coefficient in both anomalies, much in the same way as in SUSY gluodynamics. In fact, in this limit the boson sector of the daughter theory is identical to that of the parent one l , and, hence, the effective Lagrangian must have exactly the same form as in Eq. ( 6 ) , with the omission of the fermion part and the obvious replacement of X"X" by in the definition of 'p. The dynamical degrees described by this Lagrangian are those related t o cp, i.e. scalar and pseudoscalar quark mesons. Hence we recover all supersymmetry-based bosonic properties such as degeneracy of the opposite-parity mesons. Moreover, in this approximation the vacuum energy vanishes. In the following we will concentrate on the two index antysimmetric representation. The analysis for the two index symmetric is presented in '. Recently theories with fermions in higher representations -in particular the two index symmetric representation- were shown to play a relevant role when used as the underlying strong dynamics triggering electroweak symmetry breaking 2@1[i3f$a,[i,jl

18t19.

3.1. Effective Lagrangians i n the orientifold theories at

finite N The effective Lagrangians approach turns to be very useful since it is hard t o compute 1/N corrections in the underlying theory. What changes must be introduced at finite N? First of all, the overall normalization factor N 2 in Eq. (6) is replaced by some function f ( N ) such that f ( N ) + N 2 at N + 00. Moreover, the anomalous dimension of the operator no more vanishes. In fact, the renormalization-group invariant combination is

-

$a"i7f$a,[i,j~

It is just this combination that should enter in the definition of the variable

551 cp replacing Eq. (9),

In passing from Eq. (9) to Eq. ( l l ) , in addition to taking account of the anomalous dimension, we replaced N in the denominator by N - 2. The distinction between these two factors is a subleading 1/N effect. The physical motivation for the above replacement is as follows. At N = 2 the antisymmetric quark field looses color, and, thus, ( @ i [ 2 i j ] , $ a , [ ~must , j ~ ) vanish. The definition (11) guarantees, that it does vanish. As constraints we require: (a) the finite-N effective Lagrangian to recover (6) once the 1/N corrections are dropped and (b) the scale and chiral anomalies (7), (8) to be satisfied. The first requirement means, in particular, that we continue to build L,tf on a single (complex) dynamical variable cp. Equations (7) and (8) tell us that we cannot maintain the “supersymmetric” structure of the potentail term. We have to “untie” the chiral and conformal dimensions of the fields in the logarithms, see Eq. (6). They cannot be just powers of cp since in this case the chiral and conformal dimensions would be in one-to-one correspondence, and the coefficients of the chiral and scale anomalies would be exactly the same, modulo the normalization factor 3/2. At this stage a non-holomorphicity must enter the game. Let us introduce the fields @

where

~ 1 , are 2

= ($+El

(pp,

6 = (pl+€l c2 7

(12)

parameters 0(1/N), €1

7 = -9”

€2=--.

11 9N

The scale and chiral dimensions of $ and @ are such that using 6 and @ in the logarithms, we will solve the problem of distinct 1/N corrections in the coefficients of the scale and chiral anomalies. The above replacement (12) is minimal (see 6 ) . An important point to recall here is that (*) our replacement does not spoil the fact that G2 and G 6 are real and imaginary parts of a certain field. (The operators G2 and GG will be identified through the non-invariance of the Lagrangian under the scale and chiral transformations, see below.) The vacuum expectation value (VEV) of GG vanishes in any gauge theory with massless fermion field while this is not the case for the vacuum expectation value of G2, which develops a VEV at the subleading order

552

in 1/N. Preserving the property (+) above leaves open a single route: the 0(1/N) term to be added to Len which will give rise to (G') must be scale invariant by itself. As for the kinetic term to begin with, we will make the simplest assumption and leave the kinetic term the same as in L v y . Other choices do not alter the overall picture in the qualitative aspect. The effective Lagrangian in the finite-N orientifold theory reads:

Len = f ( N ) where

{1 ff

(cp p)-2/3

4ff 9

app~ c - p- (cp p)2/3 (In i In a - p ) } ,

p is a numerical (real) parameter, p = 0(1/N), and ~ ( N ) + N ' at N + m .

(14) (15)

The variations of this effective action under the scale and chiral transformations (i.e. cp + (1+ 3y)cp and cp + (1 2iy)cp, respectively, with real 7 ) are

+

where the parameters €1,' are defined in Eq. (13). Comparing with Eqs. (7) and (8) we conclude that

G"P v

~~t

pv 0: -N

Ga G a , w llv

0: - N

(cp (p)2/3 (In i

+ In a) ,

i (cp pl2l3(In i - In a) .

(17)

Minimizing the potential term in the Lagrangian (14) we find that the minimum occurs at 2

+

lncp = - p O ( I / N ~ ) , 3 and the minimal value of the potential energy - i.e. the vacuum energy density - is

afp+~(~~). (19) 9 Here the value of Eva, is determined by the non-logarithmic term in the potential energy. The logarithmic term enters only at the level O ( N o ) . Vmin

4 = Eva, = --

553

There is a very important self-consistency check. One can alternatively define the vacuum energy density as a(19:), where the trace of the energy momentum tensor is in turn proportional to GE,,GaiP”, see Eq. (17). In this method Eva, will be determined exclusively by the logarithmic term. In fact, it is not difficult to see that 3 N 413 Em, = (G;,G”> p U ) 1 2 8 ~ ~

+

in complete agreement with Eq. (19). From the above consideration it is clear that the (infrared part of the) vacuum energy density is negative if p > 0. In we argued that this is indeed the case. 3.2. Lifting the spectrum degeneracy at finite N and gluino mass m At N 4 00 the orientifold theory inherits from its supersymmetric parent an infinite number of degeneracies in the bosonic spectrum. At the effective Lagrangian level this property manifests itself in the degeneracy of the scalarfpseudoscalar mesons. At finite N we expect this degeneracy to be lifted by 1 / N effects as well as the explicit presence of a gluino mass. The leading 1 / N and m corrections can be considered simultaneously using the Lagrangian found in 6: 4a f(N) a (cpp)-2/3dPcpdpcp- 9 ( ‘ ~ $ 5 )( I~n~6 ~In@ - p)

{

m N ( N - 2) + 4 3X (cp+P) (87r2 A)*

*

(21)

To explore the scalar/pseudoscalar splitting one must study excitations near the vacuum in the Lagrangian (14). Let us define 1 c ~ = ( c p ) ~ , ( l + a h, ) h = - ( ( a + i $ ) ,

Jz

and 7’are two real fields and a is a constant which is determined by requiring the standard normalization of the kinetic term for the complex field h, a a2 = - ~(cp)l-+ , (23) D

f

554 The vacuum expectation value reads:

+ 0 (m2,N - 2 , mN-’)

,

(24)

yielding the following vacuum energy density:

8N2 0 (m2,N o ,m N ) . Eva, = Vmin = - S P A 4 - -mA3 (25) 9 3x For the spectrum we predict the following ratio of the pseudoscalar to scalar mass:

+

The gluon condensate is:

8 (G’vGa’r””)- N m A3 + -a NpA4 + 0 ( m 2N , - l , m N o ) . (27) 64r2 3x 27 These results show that the contribution of the fermion mass reinforces the effect of the finite N contribution. Interestingly the scalar state becomes even more massive than the pseudoscalar state when considering finite both N and m. The &angle dependence of the vacuum energy for the fermions in the two-index antisymmetric representation of the gauge group is

8N2 Eva,= -mA3mink 3x

The N - 2-fold vacuum degeneracy is lifted due to the presence of a mass term in the theory, yielding a unique vacuum. As was mentioned, for N = 3 the two-index antisymmetric representation is equivalent to one-flavor QCD (with a Dirac fermion in the fundamental representation). We then predict that in one-flavor QCD the scalar meson made of one quark and one anti-quark is heavier than the pseudoscalar one (in QCD the latter is identified with the v’ meson). 4. Conclusions

We constructed the effective Lagrangians of the Veneziano-Yankielowicz type for orientifold field theories, starting from the underlying S U ( N )gauge theory with the Dirac fermion in the two-index antisymmetric (symmetric) representation of the gauge group. These Lagrangians incorporate “important” low-energy degrees of freedom (color singlets) and implement (anomalous) Ward identities. At N + 00 they coincide with the bosonic part of the

555

SYM

mdoo

YM

QCD+ 1 Massive Flavor

QCD+ 1 Massless Flavor Figure 1. Schematic diagram which summarizes the link between different gauge theories.

VY Lagrangian. The orientifold effective Lagrangians at N = 00 display the vanishing of the cosmological constant and the spectral degeneracy (i.e. the scalar-pseudoscalar degeneracy). The most interesting question we addressed is the finite-N/finite-m generalization. To the leading order in 1/N we demonstrated the occurrence of a negative vacuum energy density, and of the gluon condensate. We first derived these results at m = 0 and then extended them to include the case m # 0. At N = 3 the theory with one Dirac fermion in the two-index antisymmetric representation of the gauge group is in fact one-flavor QCD. Our analysis of the finite-N effective Lagrangian illustrates the emergence of the gluon condensate in this theory. The vacuum degeneracy typical of supersymmetric gluodynamics does not disappear a t finite N . However, introduction of mass m # 0 lifts the vacuum degeneracy, in full compliance with the previous expectations. Both effects, N # 00 and m # 0 conspire to get lifted the scalar-pseudoscalar degeneracy. We evaluated the ratio M v ~ l M , . Finally, we studied the effects of finite 0, as they are exhibited in the orientifold effective Lagrangian. In the diagram presented in figure we sketch how the different theories can be related as function of the mass of the gluino and 1/N corrections.

Acknowledgments

It is a pleasure for me to thank M. Shifman for careful reading of the manuscript and for a very enjoyable collaboration.

556 References 1. A. Armoni, M. Shifman and G. Veneziano, Nucl. Phys. B 667, 170 (2003) [hep-th/0302163]. 2. A. Armoni, M. Shifman and G. Veneziano, Phys. Rev. Lett. 91, 191601 (2003) [arXiv:hep-th/0307097]. 3. M. A. Shifman and A. I. Vainshtein, Instantons versus supersymmetry: Fifteen years later, in M. Shifman, I T E P Lectures on Particle Physics and Field Theory, (World Scientific, Singapore, 1999) Vol. 2, p. 485-647 [hep-th/9902018]. 4. M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 296, 445 (1988). 5. A. Armoni, M. Shifman and G. Veneziano, Phys. Lett. B 579, 384 (2004) [arXiv:hep-th/0309013]. 6. F. Sannino and M. Shifman, Phys. Rev. D 69, 125004 (2004) [arXiv:hepth/0309252]. 7. See P. Di Vecchia, A. Liccardo, R. Marotta and F. Pezzella, arXiv:hepth/0407038 for a recent work on the subject and a complete list of references. 8. G. Veneziano and S. Yankielowicz, Phys. Lett. B 113, 231 (1982). 5. A. Kovner and M. A. Shifman, Phys. Rev. D 56, 2396 (1997) [hep-th/9702174]. 10. P. Merlatti and F. Sannino, arXiv:hep-th/0404251. 11. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229, 381 (1983); Phys. Lett. B 166, 329 (1986). 12. G. M. Shore, Nucl. Phys. B 222, 446 (1983). 13. 0. Kaymakcalan and J. Schechter, Nucl. Phys. B 239, 519 (1984). 14. G. R. Farrar, G. Gabadadze and M. Schwetz, Phys. Rev. D 60,035002 (1999) [hep-th/9806204]. D. G. Cerdeno, A. Knauf and J. Louis, hep-th/0307198. 15. F. Sannino and J. Schechter, Phys. Rev. D 57, 170 (1998) [arXiv:hepth/9708113]. S. D. H. Hsu, F. Sannino and J. Schechter, Phys. Lett. B 427, 300 (1998) [arXiv:hep-th/9801097]. 16. I. Montvay, Nucl. Phys. Proc. Suppl. 63, 108 (1998) [hep-lat/9709080]; A. Donini, M. Guagnelli, P. Hernandez and A. Vladikas, Nucl. Phys. B 523, 529 (1998) [hep-lat/9710065]; A. Feo, hep-lat/0210015. 17. S. Ferrara and B. Zumino, Nucl. Phys. B87,207 (1975). 18. F. Sannino and K. Tuominen, arXiv:hep-ph/0405209. 19. D. K. Hong, S. D. H. Hsu and F. Sannino, arXiv:hep-ph/0406200. To appear in Physics Letters B.

SELF-DUALITY, HELICITY AND BACKGROUND FIELD LOOPOLOGY

GERALD V. DUNNE Department of Physics University of Connecticut Stows, CT 06269-3046, USA E-mail: [email protected] I show that helicity plays an important role in the development of rules for computing higher loop effective Lagrangians. Specifically, the two-loop Heisenberg-Euler effective Lagrangian in quantum electrodynamics is remarkably simple when the background field has definite helicity (i.e., is self-dual). Furthermore, the twoloop answer can be derived essentially algebraically, and is naturally expressed in terms of one-loop quantities. This represents a generalization of the familiar “integration-by-parts’’ rules for manipulating diagrams involving free propagators to the more complicated case where the propagators are those for scalars or spinors in the presence of a background field.

1. Basic Strategy The basic strategy of the approach described in this talk is as follows: 0 There has been dramatic progress in recent years in computing multiloop amplitudes in both gauge and gravitational theories, for small numbers of external legs1*’. A fundamental role has been played by helicity a m p l i t ~ d e s ~ 7 ~Other i ~ . key ideas’?’ include color decompositions, master diagrams, recurrence relations and differential equations, as well as the development of efficient algebraic manipulation programs such as FORM‘. 0 Multiloop effective actions are generating functionals for multiloop amplitudes. They therefore encapsulate information about multiloop amplitudes with any number of external legs. Unfortunately, very little is known about such effective actions beyond one loop7. 0 The main goal here is to report on the development of computational rules for higher-loop effective Lagrangians, along the lines of the rules, such as “integration-by-parts”8, developed for computing amplitude diagrams. This approach requires computing vacuum diagrams using propagators in background fields, rather than diagrams involving only free propagators.

557

558 0 Here we show that for two-loop effective actions helicity plays an important role, leading to dramatic simplifications. In particular, the rules for manipulating diagrams involving propagators in a self-dual background are surprisingly simpleg. 0 In light of recent advances by Witten and collaborators" concerning a twistor-space approach t o (tree and one-loop) helicity amplitudes, it would be interesting to see if these methods might be naturally extended to higher-loop amplitudes and possibly to higher-loop effective Lagrangians. As is familiar from instanton physics", twistor space is the natural language in which to describe propagators in self-dual backgrounds, so such an extension appears natural.

2. Brief review of one loop r e s u l t s

In classical field theory the Lagrangian encapsulates the relevant classical equations of motion and the symmetries of the system. In quantum field theory the effective Lagrangian encodes quantum corrections to the classical Lagrangian, corrections that are induced by quantum effects such as vacuum polarization. The seminal work of Heisenberg and Euler12, and Weisskopf13 produced the paradigm for the entire field of effective Lagrangians by computing the nonperturbative, renormalized, one-loop effective action for quantum electrodynamics (QED) in a classical electromagnetic background of constant field strength:

i m ) = -- Indet(@ + m 2 ) , 2 where the Dirac operator is $ = y ' (a, ieA,), A, is a fixed classical gauge potential with field strength tensor F,, = a,A, - &A,, and m is the electron mass. This one-loop effective action has a natural perturbative expansion in powers of the external photon field A,, as illustrated diagrammatically in Figure 1. By Furry's theorem (charge conjugation symmetry of QED), the expansion is in terms of even numbers of external photon lines.

s,';b,,= -ilndet(i$-

+

Figure 1. The diagrammatic perturbative expansion of the one loop effective action (1).

559 In the low energy limit for the external photon lines, in which case the background field strength Fpv could be taken to be constant, Heisenberg and Euler12 found a simple closed-form expression for the effective Lagrangian, which generates all the perturbative diagrams in Figure 1: e2a b t2 e2t2 - 1 - -(b2 3 tanh(ebt) tan (eat)

- a')} .(2)

Here a and b are related to the Lorentz invariants characterizing the background electromagnetic field strength: a2 - b2 = - g2,and a b = 2. B'. Weisskopf13 computed the analogous quantity for scalar QED

i

J $= -2 lndet(D2 ~ ~ +~ m 2 )~ , ~

(3)

which involves the Klein-Gordon operator rather than the Dirac operator:

'" 1) -

-

1

O3

dt

-m"t

Fe

{

e2ab t2 -1 sinh(ebt) sin(eat)

e2t2 + -(b' 6

- a 2 ) } . (4)

The Heisenberg and Euler result (2) leads immediately to a number of important physical insights and applications, such as the low energy limit of light-light scattering, and the existence of vacuum pair production in a background electric field.

3. Two-Loop Heisenberg-Euler effective Lagrangian In principle, the computation of the two-loop Heisenberg-Euler effective Lagrangian in QED is completely straightforward, as we only need to compute a single vacuum diagram (see Figure 2) with an internal photon line and a single fermion (or scalar) loop, where these spinor (or scalar) propagators are in the presence of the background field. These background field propagators are well-known14. However, at two-loop we need to perform mass

Figure 2. The two loop diagram for the two loop effective Lagrangian. The double line refers to a propagator in the presence of the constant background field, while the wavy line represents the internal virtual photon.

renormalization in addition to charge renormalization. Ritus" found exact

560 integral representations for the fully renormalized two loop HeisenbergEuler effective Lagrangian in both spinor and scalar QED. These are impressive computations, but unfortunately the answers are very complicated double-parameter integrals. 3.1. Self-dual magic at two-loop

Consider restricting the constant electromagnetic background to be selfdual: Fpv = Fpv, where Fpv= ~ E is the standard ~ dual ~ electromag~ netic field strength. Then the fully renormalized two-loop Heisenberg-Euler effective Lagrangian takes a remarkably simple closed-form (involving simple functions and no integrals!), for both spinor and scalar QED16:

Here f is the field strength parameter, iFpvFp” = f 2 , and K is the natural dimensionless parameter K = d. The ubiquitous function ( ( 6 ) is 2ef essentially the Euler digamma function +(K) = InI’(6):

2

The subtraction of the first two terms of the asymptotic expansion of +(K) correspond to renormalization subtractions, as shown below. It is also interesting to note that in such a self-dual background, the one-loop HeisenbergEuler effective Lagrangians (2) and (4)for spinor and scalar QED are also naturally expressed16 in terms of this same function ~ ( I E ) . The dramatic simplicity of the two loop results (5,6), compared to the complicated forms obtained by Ritus15, raises three obvious questions. (1) Why are these expressions so simple? (2) Why are the spinor and scalar expressions so similar? (3) Why is the particular function C(K) so special?

The answers lie in the three-way relationship between self-duality, helicity and (quantum mechanical) supersymmetry.

~

561 3.1.1. Simplicity of self-dual results Self-dual fields have definite helicity17. Indeed, the self-duality condition is just another way of writing the helicity projection:

For anti-self-dual fields the other helicity projection vanishes, so that the photon field has the opposite helicity. It is well-known that scattering amplitudes for external field lines with like helicities are particularly simple314. Since the effective action for a self-dual field is the generating function for like-helicity amplitudes, it is consistent that the effective action in a self-dual background should be simple. However, almost all of these helicity amplitude results are for massless particles on internal lines (but see for example''), while here we see a generalization to massive particles. A more prosaic reason for the simplicity of the two-loop expressions ( 5 ) and (6) is that for a self-dual field the square of the matrix F,, is proportional to the identity matrix: F,,F,, = -f25,,. This dramatically simplifies the propagators of spinors or scalars in the background field. For example, for a scalar particle O3

0

dt e--m2t-$ cosh2(eft)

tanh(eft)

(9)

Note that this propagator is a function of p 2 , rather than of individual components of the momentum, which greatly simplifies the background field computations. The propagator satisfies a simple differential equation: (p2

+ rn2)G(p)= 1+

3.1.2. Similarity of spinor and scalar results in self-dual background Another consequence of the self-duality of the background is that the corresponding Dirac operator has a quantum mechanical supersymmetry. That is, apart from zero modes, the Dirac operator has the same spectrum (but with a multiplicity of 4) as the corresponding scalar Klein-Gordon operator1g320.At one-loop this implies

562 2

(2)

where No = is the zero mode number density. The logarithmic term in (11) corresponds to the zero mode contribution. Renormalizing on-shell ( i e . , p 2 = m 2 ) ,we find that the spinor and scalar effective Lagrangians (2) and (4) are proportional to one another for a self-dual background, in such a way that the SUSY combination vanishes: C~~],,,, 2C!i!, = 0. Now consider the implications of self-duality of the background at the two loop level. The two-loop the effective action is not simply a log determinant, so the situation is more complicated. Nevertheless, the quantum mechanical SUSY of the Dirac operator relates the spinor propagator to the scalar propagator via simple helicity projections, which has the consequence that after just doing the Dirac traces in the spinor two loop effective Lagrangian, one finds that it can be written as the sum of two terms involving matrix elements of the scalar propagator. Moreover, these are the same two matrix elements of the scalar propagator that appear in the scalar QED effective Lagrangian, but with different numerical coefficients21. This structure explains why the two loop answers (5) and (6) for spinor and scalar QED have such a similar form, involving just two terms with different numerical coefficients.

+

3.1.3. Significance of

5

and

I'

An important question to address is why are the simple expressions (5) and (6) for the two loop Heisenberg-Euler effective Lagrangians in a selfdual background expressed in terms of the particular function [ ( K ) and its derivative [recall that 5 was defined in (7) as essentially the Euler digamma function]. The first hint comes from the following facts that for a selfdual background the following scalar propagator loops, evaluated using the background field propagator (9), are simply related to the [ ( K ) function:

Here the double lines refer to scalar propagators in the self-dual background and the single line is the free scalar propagator, while the dot on a propagator refers to the propagator being squared. Thus, 5 and [' are natural one loop traces for the propagator in a self-dual background. Given (12) and (13), we can write the closed-form expressions (5) and

563 (6) for the two loop effective Lagrangians in diagrammatic form: spinor QED :

(14)

scalar QED :

[@ @] -

=

: [O 01' $g[00 1 e2

-

-

-

*

(15) Here the notation is that the triple line loop on the LHS of (14) refers to a spinor propagator in a self-dual background, while the double-line loops [including those on the RHS of (14)] refer to a scalar propagator in the self-dual background. This shows the remarkable result that the two loop fully renormalized answers are expressed naturally in terms of one loop quantities. Qualitatively, we can write: two loop = (one loop)

2

+ (one loop)

(16)

Interestingly, such a relation with two loop quantities being expressed as squares of one loop quantities plus a one loop remainder has been found recentlyz2 in the amplitudes of 4 dimensional super Yang-Mills theory, which is a very different theory from QED. Also, the same function < ( K ) , and its derivatives appear naturally in recent studies of n/ = 2 SUSY QED and YM effective Lagrangians at t ~ o - l o o p This ~ ~ . suggests something deeper is at work here. 4. Background field loopology

It is natural to ask if the remarkable simplifications of the two loop results for a self-dual background might extend to even higher loops. To go beyond two loops one should take advantage of the great progress that has been made recently in understanding the structure of higher-loop quantum field theory (without background fields). The general strategy is to manipulate diagrams to reduce the number to a much smaller set of so-called "master diagrams'' which need to be computed. This has led, for example, to many new two-loop results for QCD scattering I conclude this talk with some comments and speculations about how these techniques can be extended to incorporate background fields"

564 Indeed, we can go further than the qualitative statement (16) and derive the results (14) and (15) by algebraic means. First, we identify the source of the coefficient factors -6e2 and $e2 which appear in front of the (one 1 0 0 ~ ) ~ terms in (14) and (15). Note that in free QED ( i e . , with no background field) it is a straightforward exercise to show that in 4 dimensions

spinor QED :

@ = -6 e2 [0 1,

(17)

(The loop on the RHS is a scalar loop in each case.) Thus, the two loop free vacuum diagram can be expressed in terms of a simpler one loop diagram. These results can either be derived by computing each side using dimensional regularization, or a quicker proof follows from an integration-by-parts argument (see below). Notice that the coefficients of the (one 1 0 0 ~ parts )~ in this free case are exactly the same as the corresponding coefficients in the background field expressions (14) and (15), for both spinor and scalar QED. This is no accident, as I now illustrate for the case of scalar QED (for spinor QED the argument is similar). Consider the derivation of (18) using dimensional regularization and integration-by-parts By purely algebraic manipulations

’.

where the dotted line denotes a massless scalar propagator. The first term has been written as the square of a one loop diagram but the second term is apparently still two loop. However, using simple integration-by-parts manipulations, this two loop diagram can also be written as a square of a

565 one loop diagram:

I)([])([

-( d - 3 ) [ 0 ]

-

Thus, the apparently 2-loop term on the RHS of (19) is a square of a one loop diagram, leading to

This reduces to (18) in d = 4, and I stress that this result has been derived without doing any integrations, only making simple algebraic manipulations on the integrands. Now consider the analogous manipulation in a self-dual backgroundg. First, we extend the scalar propagator in a self-dual background to arbitrary dimensions by taking multiple copies of the block diagonal structure of FDv.This is equivalent to dimensional regularization in the worldline formalism24. Then the scalar propagator (9) becomes =

Jm o

dt coshd'2(eft)

e-rn2t-$

tanh(eft)

Then we can repeat the algebraic steps in (19) to obtain

." ("-' d -) 3

= 2

[O]+

0(f2) ,

where we have chosen to isolate this particular coefficient of the square of the one loop propagator trace motivated by the free-field result (21).

566

The advantage of the manipulation in (23) is that it makes the mass renormalization (which was a very difficult part of previous two loop computations) almost trivial. To see this we subtract the free field two loop diagram from the background field diagram

[@

-

@]

=

f (E) { [ OI2[Ol’) W] -

+

,

and then simply complete the square in the first terms:

[O]’[O]”= [O-O]”i[O] [O-01. (25)

The cross-term in (25) is immediately identified with the mass renormalization because

which can also be derived algebraically. Furthermore,

The f 2 term in (27) contributes to the charge renormalization, and so (24) can be written as

Observe that the first term is now completely finite, so that we can set d = 4, and by (12) we obtain the first term of the final answer (15), the (one 1 0 0 ~ ) ~ piece, without doing any integrals a t all! The second term is manifestly the mass renormalization term, and so is absorbed by mass renormalization. The only remaining divergence can be proportional to the bare Maxwell Lagrangian f 2 , which is then subtracted by charge renormalization. It is simple to isolate and subtract this piece, leaving an 0(f4) term, whose kernel in the d -+ 4 limit reduces t o a momentum delta function:

567 where in the last step we have used (13). Thus, we have derived the diagrammatic form (15) of the fully renormalized two loop scalar QED effective Lagrangian in a self-dual background by essentially algebraic manipulations. It would be interesting to develop these background field “integration-by-parts” rules into a fully systematic set of rules that might be applied to even higher loop order. 5. Conclusions To conclude, I reiterate that self-duality appears to be playing a remarkable simplifying role in the computation of higher-loop effective Lagrangians. Analogously, self-duality is an important simplifying principle in the computation of higher-loop amplitudes. This talk is a first attempt to bring together these two aspects of higher-loop computations. The main goal is the development of computational techniques for computing higher-loop vacuum diagrams involving propagators in the presence of background fields, generalizing the integration-by-parts rules developed for higher-loop diagrams involving free propagators. Such background field vacuum diagrams are the building blocks of higher-loop effective Lagrangians. For a selfdual background this can be done explicitly to the two-loop level, and it is suggested that expansion about the self-dual background will provide a starting point for more general backgrounds. Furthermore, for a self-dual background, such algebraic rules for manipulating diagrams should facilitate the computation of even higher-loop effective Lagrangians. Finally, it would be interesting to investigate whether Kreimer’s Hopf algebra structure underlying Feynman diagramsz5 can be usefully applied to these background field computations. Acknowledgments

I thank Christian Schubert and Holger Gies for collaboration, and the US DOE for support through grant DE-FG02-92ER40716. References 1. E. W. Glover, Nucl. Phys. Proc. Suppl. 116, 3 (2003) [arXiv:hep-ph/0211412]. 2. Z. Bern, Nucl. Phys. Proc. Suppl. 117,260 (2003) IarXiv:hep-ph/0212406]. 3. R. Gastmans and T. T. Wu, The ubiquitous photon: helicit9 method f o r QED and QCD, (Oxford Univerity Press, New York, 1990). 4. M. L. Mangano and S. J. Parke, Phys. Rept. 200, 301 (1991); 5 . Z. Bern, G. Chalmers, L. J. Dixon and D. A. Kosower, Phys. Rev. Lett. 7 2 , 2134 (1994) [arXiv:he~-ph/9312333].

568

6. J. A. M. Vermaseren, “New features of FORM,” arXiv:math-ph/0010025. 7. G. V. Dunne, “Heisenberg-Euler Effective Lagrangians: Basics and Extensions”, arXiv:hep-th/0406216, to appear in Ian Kogan Memorial Collection, From Fields t o Strings: Circumnavigating Theoretical Physics ’, M. Shifman, A. Vainshtein and J. Wheater (Eds), (World Scientific, Singapore). 8. K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192,159 (1981). 9. G. V. Dunne, JHEP 0402,013 (2004) [arXiv:hep-th/0311167]. 10. E. Witten, “Perturbative gauge theory as a string theory in twistor space,” arXiv:hep-th/0312171; F. Cachazo, P. Svrcek and E. Witten, “MHV vertices and tree amplitudes in gauge theory,” arXiv:hep-th/0403047; “Twistor space structure of one-loop amplitudes in gauge theory,” arXiv:hep-th/0406177. 11. E. Corrigan, D. B. Fairlie, S. Templeton and P. Goddard, Nucl. Phys. B 140, 31 (1978); N.H. Christ, E. J. Weinberg and N. K. Stanton, Phys. Rev. D 18, 2013 (1978). 12. W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714. 13. V. Weisskopf, Kong. Dans. Vid. Selsk. Math-fys. Medd. XIV No. 6 (1936) 14. J. Schwinger, Phys. Rev. 82 (1951) 664. 15. V. I. Ritus, Sov. Phys. JETP 42 (1975) 774; Sov. Phys. JETP 46 (1977) 423. 16. G. V. Dunne and C. Schubert, Phys. Lett. B 526, 55 (2002) [arXiv:hepth/0111134]; JHEP 0208,053 (2002) (arXiv:hep-th/0205004]; JHEP 0206, 042 (2002) [arXiv:hep-th/0205005]. 17. M. J. Duff and C. J. Isham, Phys. Lett. B 86, 157 (1979); Nucl. Phys. B 162,271 (1980). 18. Z. Bern and A. G. Morgan, Nucl. Phys. B 467, 479 (1996) [arXiv:hepph/9511336]. 19. G. ’t Hooft, Phys. Rev. D 14,3432 (1976) (Erratum-ibid. D 18,2199 (1978)l. 20. R. Jackiw and C. Rebbi, Phys. Rev. D 16,1052 (1977). 21. G. V. Dunne, H. Gies and C. Schubert, JHEP 0211,032 (2002) [arXiv:hepth/0210240]. 22. C. Anastasiou, Z. Bern, L. Dixon and D. A. Kosower, Phys. Rev. Lett. 91, 251602 (2003) [arXiv:hep-th/0309040]. 23. S. M. Kuzenko and I. N. McArthur, Phys. Lett. B 591,304 (2004) [arXiv:hepth/0403082], “Relaxed super self-duality and N = 4 SYM at two loops,” arXiv:hep- t h/0403240. 24. C. Schubert, Phys. Rept. 355 (2001) 73, hep-th/0101036. 25. D. Kreimer, Adv. Theor. Math. Phys. 2, 303 (1998) [arXiv:q-aIg/9707029]; A. Connes and D. Kreimer, Commun. Math. Phys. 210,249 (2000) [arXiv:hepfh/9912092];

THE OPTICAL APPROACH TO CASIMIR EFFECTS

A. SCARDICCHIO* Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge, M A 02139, USA

We propose a new approach to the Casimir effect based on classical ray optics. We define and compute the contribution of classical optical paths to the Casimir force between rigid bodies. Our approach improves upon the proximity force approximation. It can be generalized easily to arbitrary geometries, different boundary conditions, to the computation of Casimir energy densities and to many other situations.

1. Introduction The last 10 years have witnessed quite a revolution in the experimental techniques used to prove Casimir effect.’ Casimir’s original prediction for the force between grounded conducting plates due to modifications of the zero point energy of the electromagnetic field has already been verified to an accuracy of a few percent. Progress has been slower on the theoretical side. Beyond Casimir’s original study of parallel plates,2 we are only aware of useful calculations for a corrugated plate3 and for a sphere and a plate.4 Simple and experimentally interesting geometries like two spheres, a finite inclined plane opposite an infinite plane, and a pencil point and a plane, remain elusive. The Proximity Force Approximation5 (PFA) was shown by Gies et d4to deviate significantly from their precise numerical result for the sphere and plane. Thus at present neither exact results nor reliable approximations are available for generic geometries. It was in this context that we recently proposed a new approach to Casimir effects based on classical optics.6 The basic idea is extremely simple: first the Casimir energy is recast as a trace of the Green’s function; then the Green’s function is approximated by the sum over contributions from optical paths labelled *Joint work with R. L. JaEe

569

570 by the number of (specular) reflections from the conducting surfaces. The integral over the wave numbers of zero point fluctuations can be performed analytically, leaving a formula which depends only on the properties of the paths between the surfaces. This approach will give an approximation (though a surprisingly accurate one) which is valid when the natural scales of diffraction are large compared to the scales that measure the strength of the Casimir force. In practice this will typically be measured by the ratio of the separation between the conductors, a , t o their curvature, R. It generalizes naturally to the study of Casirrlir thermodynamics, energy and pressure, t o various b.c., t o fermions, and t o compact and/or curved manifolds. In the optical approximation the cutoff dependent terms of the Casimir energy can easily be isolated and shown to be independent of the separation between conductors. They therefore do not contribute to forces and can be dropped.

2. Derivation

Most studies of Casimir energies do not consider approximations. Instead they focus on ways to regulate and compute the sum over modes, These methods have proved very difficult to apply to geometries other than parallel plates. The main reason for this impasse lies in the requirement of an analytic knowledge of the spectrum of the Laplace operator for the given geometry. However, the knowledge of this spectrum for a family of boundaries (like a sphere facing a plane) would have the most non-trivial implications for the same family of quantum billiards and hence of classical billiards. 30 years of work on the ergodicity of classical billiards and their quantum counterparts suggest this task is hopeless.8 A numerical knowledge of the spectrum does not represent a reliable solution to the problem either. The force, indeed, is given by the small oscillatory ripple in the density of state numerically shadowed by the ‘bulk’ contributions which give rise t o distance-independent divergencies. So we focused our attention on ways to get approximate solutions of the Laplace-Dirichlet problem which are apt to capture the oscillatory contributions in the density of states, providing physical insights and accurate numerical estimates.

ifiw.7

571 2.1. The Optical Approximation for the propagator There is a strong connection between the Casimir energy for a field 4 obeying Dirichlet boundary conditions (b.c.) on the boundary of the domain D and the propagator G(x’,x,k ) of the Helmholtz equation on the same domain with the same b.c. . The knowledge of the latter allows one to calculate the density of states p ( k ) and from this we can obtain the Casimir energy by quadratures. Indeed, from the well-known definition of the Casimir energy in terms of a space and wave-number dependent density of state^,^ P(X1 k ) ,

where w ( k ) = c d n , and the density of states p ( x , k ) is related to the propagator G(z;, x,k ) by P(X1

2k k ) = - Im G(x,2,k ) .

(2)

lr

where the usual density of states is p ( k ) = . [ d N z p ( zk, ) . We must choose G to be analytic in the upper-half k2-plane; in the time domain (see later) this means we are taking the retarded propagator. The Casimir energy depends on the b.c. obeyed by the field 4 and on the arrangement of the boundaries, S E d D (not necessarily finite), of the domain D. From the outset we recognize that E must be regulated, and will in general be cutoff dependent. We will not denote the cutoff dependence explicitly except when necessary. p and G are the familiar density of states and propagator associated with the problem

(A+k2)$(x) = O

for x E D ,

+(x) = O

for x E S.

(3)

We can regard this problem as the study of a quantum mechanical free particle with li = 1, mass m = 1 / 2 , and energy E = k 2 , living in the domain D with Dirichlet b.c. on dD. Dirichlet b.c. are an idealization for the interactions which prevent the quantum particle from penetrating beyond the surfaces 5’. This is adequate for low energies but fails for the divergent, i. e. cutoff dependent, contributions to the Casimir energy. lo However, the divergences can be simply disposed of in the optical approach, and the physically measurable contributions to Casimir effect are dominated by k l/a, where a , a typical plate separation, satisfies 1 / a << A whit A being the momentum cutoff characterizing the material. So the boundary conditions idealization is quite adequate for our purposes. Following this N

572 quantum mechanics analogy we introduce a fictitious time, t , and consider the functional integral representation of the propagator.” The space-time propagator G(x’,x , t ) obeys the free Schrodinger equation in D bounded by S. It can be written as a functional integral over paths from x’ to x with action S(x’, x , t ) = dtx2. The optical approximation is obtained by taking the stationary phase approximation of the propagator G in the

a6

fictitious time domain T

The classical action is

S,(X’, 5 , t ) =

lT(X’,

x)z

4t

(5)

and K , is the van Vleck determinant

KT(x‘,x , t ) o( det

axpxj

With some manipulations6 we can turn this determinant into

where N is the number of spatial dimensions. This approximation is exact to the extent one can assume the classical action of the path ST to be quadratic in x’,x. This is the case for flat and infinite plates. Thus the non-quadratic part of the classical action comes from the curvature or the finite extent of the boundaries, which we parameterize generically by R,

d3S/dx3 l / R t . N

In k-space the corrections hence will be 0 ( l / k R ) , and the important values of k for the Casimir energy are of order l / a , where a is a measure of the separation between the surfaces. Thus the figure of merit for the optical approximation is a/R. Certainly some of the curvature effects are captured by the van Vleck determinant but at the moment there is no good way to estimate the order in a / R of the corrections t o the optical approximation (possibly fractional, plus exponentially small terms). This is topic for further investigation. Putting all together we find the space-time form of the optical propagator to be

573 When dealing with infinite, parallel, flat plates this approximation becomes exact. For a single infinite plate, for example, the length-squared of the only two paths going from x to x’ are e2irect = 1 ) x ’ - ~ 1 ) ~ ,efreflection= ~ ~ X ’ - where 53 is the image of x. Both are quadratic functions of the points x, x’ and the optical approximation is indeed exact. Gopt(x’,x, k) is obtained by Fourier transformation and can be expressed in terms of Hankel functions, giving us the final form for our approximation

where A, is the enlargement factor (see6 for details)

and we have suppressed the arguments x and x’ on e, and A, in (9). This can be thought of as a particular case of a more general result.12 2.2. The Optical Casimir energy

The substitution of (9) into (2) and then in (1)gives rise to a series expansion of the Casimir energy associated with classical closed paths Eopt

=

c

paths

E,, T

where each term of this series will be in the form of

Here the integration has been restricted to the domain 0,c D where the given classical path T exists. If the length of the path is bounded from below (this is the case for more than one reflection) the z and k integral can be switched safely. The k-integral can be performed exactly for any N and p, but it is particularly simple for the massless case, w(k) = ck,

This is the central result of our work and associates a Casimir energy contribution to each optical path T > 1. The series (11) has a very fast convergence, usually 98% of the contribution is contained in the first 4 terms, as we will see in the examples below.

574 2.3. Divergencies

The paths r = 1B that bounce only once on a given body B must be treated with particular care because their contribution is divergent. The x and k integrals cannot be inverted without regulating the divergencies. To do so we insert a simple exponential cutoff in k. For a massless field the k-integration in E ~ can B be performeda giving

Notice that for l l ~ >> A 1 we reobtain the standard result, eq. (13). Hence it is convenient to rewrite this integral as

(15) where D ~ is B the domain where the path does not exist. The first integral does not depend on the position of the other bodies and in the second we can take the limit A + OCI safely because l 1 ~ ( x>) 2a > 0 , where a is the minimum distance between the bodies. Hence we will write E ~ B = EIB,div E 1 ~ , f i the ~ , finite part being the integral over D ~ (notice B the extra minus sign). The first term contains all the divergencies arising for l 1 -+~ 0. Notice that since when l l B A = & the sign of the integrand changes, the divergence is negative rather than positive, as one could have argued from ( 13).15 Of course the bulk contribution to the vacuum fluctuation energy comes from the zero-reflection term, which is positive. Using the expression of A113 for a general surface with principal radii of curvature Ra,b we find

+

-87r hcA3-h2-hc

/

-+-

+

dS O(1nA). (16) 32n2 (;a d b ) These terms do not contribute to the forces between rigid objects. The form of eq. (16) invites comparison with the work of Balian and B10ch.l~ One finds agreement in the surface terms but not in the curvature terms. ElB,div

3. Parallel Plates Parallel plates provide a simple, pedagogical example which has many features - fast convergence, trivial isolation of divergences, dominance of the "For simplicity we specialize to N = 3 although the analysis is completely general.

575 even reflections - that occur in all the geometries we analyzed. We assume for simplicity that the two plates have the same area S. The relevant paths are shown in Fig. 1 where the points x and x', which should be equal, are separated for ease of viewing. For the even paths &,(z) = 2na, n = 1 ,2 ,. . ., independent of z (here z is the distance from the lower surface). For the odd paths & , - ~ , ~ ( z )= 2(n - 1). 2c, where = z , a - z respectively if a = down, up and n = 1 ,2 ,. . . . For planar boundaries the enlargement factor is given by A, = I/!:.

+

I ," 2

4

6

...

... 2'

<

-1

3

5

(4 Figure 1. a) Even and b) odd optical paths for parallel plates. Initial and final points have been separated for visibility.

The sum over even reflections,

is trivial because it is independent of z. The result is the usual Dirichlet Casimir energy.' The sum over odd reflections, after being regulated by point splitting, before removing the infinite part, gives

The divergence as E 4 0 is precisely what is expected on the basis of the general analysis of the density of states in domains with b o u n d a r i e ~ . ~ ~ > ' ~ Moreover, since it is independent of a, it does not give rise to a force. F'rom Fig. 1 b) it is evident why this must happen. The total sum of the off reflection contributions is just the integral of the one reflection path extended up to KI and hence does not depend on a. The fact that the odd reflections sum up to a divergent constant is universal for geometries with planar boundaries, and to a good approximation is also valid for curved boundaries. Note also that the sum over n in eq. (17), converges rapidly: 92% of the effect comes from the first term (the two

576 reflection path) and > 98% comes from the two and four reflection paths. This rapid convergence persists for all the geometries we have analyzed due to the rapid increase in the length of the paths. 4. Sphere and Plane

We calculated the Casimir energy for the sphere and the plane up t o four reflections. El and E3 can be found analytically while E2 and E4 must be computed numerically. Henceforth a will be the distance between the sphere and the plate and R the radius of the sphere. Comparison with the parallel plate case as a -+ 0 suggests the error due to neglecting the fifth and higher reflections to be 2%. Hence we have plotted our results as a band 2% in width in Fig. 2. Since the fractional contribution of higher reflections decreases with a, we believe this is a conservative estimate for larger a. The proximity force approximation has been the standard tool for estimating the effects of departure from planar geometry for Casimir effects for many years7. In this approach one views the sphere and the plate as a superposition of infinitesimal parallel plates (i.e. the sphere is substituted by a stairlike surface). The resulting expression is N

It is custom to factor out the most divergent term of the Casimir force in the limit a / R 4 0 as predicted by the PFA, in this case -7r3hcR/720a3, so to write in general

which is the definition o f f . Modern experiments are approaching accuracies where the deviations of f (a/R) from unity are important. PFA predicts that

while the optical approximation data predict

Beyond the limit of small a/R, one must notice that the optical approximation to the Casimir energy and the data of Ref. both fall like

577 l/a2 a t large a/R. In fact both are roughly proportional to l/a2 for all a. In contrast the PFA estimates of the energy falls like l / a 3 a t large a and departs from the Gies et aL4 data at relatively small a/R. For pur-

poses of display we therefore scale the estimates of the energy by the factor -l440a2/.rr3Rh. The results are shown in Fig. 2. The dominant contri-

1.4 1.2 1.0

‘

0.6 0.4

0.2

1/256

I 1/64

1

I

1/16

1/4

I 1/2

I 1

I

1

2

4

a/R

Figure 2. Sphere facing a plane case. Comparison between different methods. Results for Ref.4 (stars with error bars), data from 6 , superseded by this work (triangles), optical approximation (thick grey line), PFA (broken line).

bution, always greater than 92%, comes from the second reflection. The fourth reflection contributes about 6% for a / R << 1 and less as a / R increases. The contributions of the first and third reflections are very small for all a / R . A relevant result, confirmed by the analytical analysis on the energy momentum tensor (within the optical approximation16) is that the asymptotic behavior of E as a / R >> 1 predicted by the optical approximation is oc l / a 2 . This is in contrast with the Casimir-Polder law which predicts E 0: l/a4 at large a , the discrepancy is to be attributed to settling in of diffraction effects. 5 . Conclusions

We have proposed a new method for calculating approximately Casimir energies between conductors in generic geometries. We use an approximation imported from studies of wave optics that we have therefore named the “optical approximation”. In this paper, we have outlined the derivation and applied it to two examples: the canonical example of parallel plates

578 and the experimentally relevant situation of a sphere facing a plane. Our results are in agreement with the Proximity Force Approximation only to leading order in the small distances expansion. The first order correction is found t,o be different. This is of particular importance in the example of the sphere and the plane because the first order correction in a / R ( a is the distance sphere-plate and R is the radius of the sphere) will soon be measured by new precision experiments. We would like to thank the organizers of the conference ‘Continuous Advances in QCD 2004’. This work is supported in part by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DFFC02-94ER40818 and in part by the INFN-MIT ‘Bruno Rossi’ fellowship. References 1. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys. Rept. 353, 1 (2001); G. Bressi,

G. Carugno, R. Onofrio and G. RUOSO,Phys. Rev. Lett. 88,041804 (2002);

R.S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. L. Lopez 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13.

14. 15. 16.

and V. M. Mostepanenko, Phys. Rev. D 68,116003 (2003). H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51,793 (1948). R. Golestanian and M. Kardar, Phys. Rev. A 58, 1713 (1998). H. Gies, K. Langfeld and L. Moyaerts, JHEP 0306,018 (2003). B. V. Derjagin, Kolloid Z. 69 155 (1934). R. L. Jaffe and A. Scardicchio, Phys. Rev. Lett. 92, 070402 (2004); A. Scardicchio and R. L. Jaffe, [arXiV:quant-ph/O406041]. V. M. Mostepanenko and N. N. Trunov, Casmir Effect and Its Applications, Oxford University Press, (1997). M. C. Gutswiller, J. Math. Phys.12, 343 (1971); Chaos in Classical and Quantum Mechanics, Springer, Berlin (1990). For a discussion, see Appendix A in Ref.15. N. Graham, R. L. Jaffe, V. Khemani, Ivl. Quandt, 0. Schroeder and H. Weigel, Nucl. Phys. B 677,379 (2004). L. S. Schulman, Techniques and Applications of Path Integration, John Wiley & Sons; (1981). M. V. Berry and K. E. Mount, Reps. Prog. Phys 35,315 (1972). R. Balian and C. Bloch, Annals Phys. 69,401 (1970); 63,592 (1971); 69,76 (1972). D. Deutsch and P. Candela, Phys. Rev. D 20, 3063 (1979). N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, M. Scandurra and H. Weigel, Nucl. Phys. B 645,49 (2002). A. Scardicchio, R. L. Jaffe, to be published.

AT THE CROSSROAD: PERPETUAL QUESTIONS OF THE FUNDAMENTAL PHYSICS HERE AND NOW

A. LOSEV ITEP, B. Cheremushkinskaya 25 Moscow, Russia E-mail: [email protected] I review the state of affairs in modern fundamental physics. I mention some missed opportunities and concentrate on String theory, proposing the modified Zwiebach approach as the only self-consistent one.

1. The quest for fundamental degrees of freedom

Let me begin by recalling that things are always different from what they look like. Stars are not rotating around the Earth, gases and liquids are not continuous, the weak interaction is not 4-fermionic and pions are not fundamental fields in the theory of strong interactions. The aim of fundamental physics is to discover these elusive fundamental degrees of freedom; our experience tells us that it was always possible in the past, therefore , we have to do it again and again. The main scientific discovery (in fundamental physics) of the last century is the Standard Model. Nevertheless, we can still pose the question whether this theory is fundamental. Some people may say that it is possible (if we ignore gravity for a while) quoting the renormalizability of the theory. 1.1. Renomalizable theories as effective theories

We have to admit that renormalizability allows computation of correlation functions, but it looks very unnatural for me. Let us look at previous renormalizable "theories" that were proposed during last 5000 years. First theory was an attempt to solve the problem "what holds the Earth" ( it was experimentally absolutely clear that everything falls down). One had to imagine the infinite set of animals standing on the top of each other. Here renormalizability is in the fact that each animal stands on something.

579

580 Another old example of renormalizability is the theory of the sky. It was clear that sky was holding stars and planets but it was absolutely unclear how far is it from the Earth. In some sense the size of the sky was like the size of cut-off in modern quantum field theory. Just imagine someone declaring that the sky is an auxiliary notion and the proper ideal notion of the star is the flux of light coming from infinity. But I know geople who seriously think about renormalizable quantum field theory as a fundamental in similar terms. All this makes me believe that renormalizability is a strong indication that the Standard model is an effective theory. Note, that effective does not mean wrong or bad, effective just means it is not fundamental. I admit that it may be hard or even useless for humanity to look for fundamental theory today - but one day we will have to. This has to be compared with the understanding that stars are interesting solar-like systems with planets ... and with understanding that in coming thousands years we are definitely not going to fly there (by the way, we must go somewhere in the future in order to save humanity from the unavoidable Sun collapse).

1.2. Possible fundamental theories

To the best of my knowledge ( as we use to write in the tax declaration) all candidates for fundamental theories could be decomposed into two classes. Theories from the first class have less degrees of freedom that one may expect from the effective theory and theories from the second class have additional degrees of freedom. Crystals for solid states and various attempts to put field theory on a lattice are examples of the fundamental theories of the first class. I am going to discuss them somewhere else (not because I do not consider them seriously, just because of the space limitations). The second class of fundamental theories includes theories with additional degrees of freedom, such as W-bosons for the effective theory of the 4-fermion interactions. They form a 3-level tower with levels distinguished by the cardinality of additional fields one adds to the effective field theory. The first level is occupied by theories containing the finite number of additional fields. It is well-known that all gauge theories with matter may be embedded in the finite gauge theory with matter fields ( N = 4 SYM is the most known example). It is possible to give mass to additional fields (in

581 supersymmetry this procedure is known as soft supersymmetry breaking). Thus, for the energies above this mass we will see finite theory while for the energies below it we will have the original theory. In this scheme the mass of additional fields plays the role of the cut-off and these additional fields serve as a modern version of the good old Pauli-Willars cut-off. The second and the third levels are occupied by theories with infinite number of additional fields. These fields in the theories of the second level are coming from adding to the our world a finite number of additional dimensions (Kaluza theory is the most prominent example of such theories). Among recent examples I would like to mention harmonic superspace theory,' theory of the superparticle in the superspace enlarged by the space of pure spinors2 and holomorphic Chern-Simons theory. All these theories may be considered as fundamental for the SYM (with various number of supersymmetries) . The third level is occupied by various string theories. Fields in the theories are functions on the infinite-dimensional space (on the loop space). The string program implies that theories of the second level do not provide finite unified theories - therefore we have to work on the third level. However, to tell the truth, we have not worked hard enough on the second level. In particular, twistor program somehow means that Lorentz invariant theory in 4 dimensions could come out from the non-Lorenz invariant theory in higher dimensions (like holomorphic CS in modern approach to twistor theory). Honestly speaking, nobody really studied renormalization of Lorentz-noninvariant theories! Therefore, there is a lot of work to be done. Another remarkable conjecture of the theory on the second level are Berkovits CS-like theories for d=10 SYM and even for d = l l supergravity (the latter theory was hidden under the misleading title of quantization of the supermembrane in M-theory). Finally, there are reasons for CS-like theories to have better renormalization properties. For example, the ordinary CS theory is finite while it should be logarithmically divergent by naive power counting arguments.

1.3. Gravity as a theoqt of the background

The discussion of various levels of theories looks rather abstract, but it is brought t o life by the question of quantum gravity and the lack of corresponding fundamental theory on the first level. The gravity theory plays a special role among all theories. To begin with, it is the theory of the background for all matter the-

582

ories. Therefore, some mathematician say that gravity theory should be considered as the theory of deformations of the matter theories. From this point of view we should first understand what are the matter theories, and then gravity laws should be unambiguously deduced out of it! Such exotic point of view is somehow supported by the string theory where matter is something like open strings while gravity is something like closed strings. Another example of this point of view is the B-world (sometimes called topological strings of type B),5 where matter theory is the theory of complexes of holomorphic bundles while gravity is the theory of deformations of complex structures. From this point of view the Einstein approach to gravity follows from the differential geometric understanding of field theory where fields themselves are something like maps of smooth manifolds ( "from the worldsheet into the target space") or connections in various bundles. Experience in differential geometry indicates the special role of the notion of Riemann manifold, i.e. manifold equipped with the metric, in construction of various functions (actions) on various field space. It is this understanding that made people to say that gravity is a beautiful theory and this beauty is a sign that this theory is fundamental. Nowadays, we should admit that being beautiful does not mean being fundamental. What we consider as beauty is just the feeling that in the world of differential geometry (the world of physics in the most of last century) the Riemann geometry is the universal background. So, it is just the feeling of harmony, nothing else! If we consider seriously other worlds, such as the world of complex manifolds (in some version of twistor program) or string theory, we will feel that in each of these worlds there is another candidate on the role of gravity, the theory of the background. In order to really feel its beauty we have to develop our understanding of these worlds to extend where they would seem to us as natural as the world of differential geometry. In particular, in order to feel the beauty of complex geometry one has to study rather seriously algebraic geometry. Therefore, each world contains its own theory of backgrounds, i.e. "gravity", and gravity of the effective world should be considered as an induced gravity (like effective world is induced, say, in the twistor approach or in the string theory).

583 1.4. Is the gravity findarnental?

The argument in the previous subsection opened the possibility that gravity can be considered not as a fundamental but as an induced theory. This idea could be pushed further even if someone does not believe in the aristocratic idea that ”gravity is very distinguished as a theory of deformations”. From the democratic point of view gravity is just a special gauge theory, one among the others, and, being a gauge theory, it could be induced! However, the aristocratic opponent would say that gravity has a very special gauge group, the group of diffeomorphisms. The democrat would say that it is not quite true, since in supergravity and in string theory this group is enlarged. Moreover, he would say that in Batalin-Vilkovisky (BV) approach to gauge theories one does not distinguish between the gauge invariant action and the gauge group. These two notions (sometimes!) are parts of the unifying object - the master action, and if the master action is of the general form one just cannot extract gauge group out of it. He would also say that it is what really happens in bosonic string theory (in the form of Zwiebach4) where vertex operators naturally correspond to fields in BV formalism. In the bosonic string theory it is completely clear that the group of diffeomorphisms (as a gauge group) is induced! Following the democratic pattern we may ask whether something like this may happen in the simpler cases. In particular, consider the 3 dimensional gravity as a CS theory. The fundamental theory is a theory of connections (in the bundle with the structure group being the group of 3dimensional motions) and gauge symmetry apriory had nothing to do with diffeomorphisms ’. In the process of construction of effective theory we may pass to the space of rotations. However, when we pass to the space of invariants with respect to the gauge group, we find the symmetric tensor! This phenomena is well-known in gauge theory - quark fields are fundamental and meson fields are effective. While quarks are in fundamental representations of both the color and flavor groups, mesons are colorless and in the adjoint representation of the flavor group. In our example local Lorentz is a n analogue of the color and group of the diffeomorphisms is like a flavor. The connection corresponding to translations (generalized weilbein) is charged with respect to both of the groups. The bilinear invariant is like a meson, i.e. it is charged with respect to diffeomorphism like a

=It is pleasure to thank D.Diakonov for discussions on this point

584

symmetric tensor. And we call this symmetric tensor a metric. It could be degenerate - we are leaving the Riemann geometry. Moreover, one can take it to be equal to zero - and this is an unbroken phase of the gravity theory. It looks interesting to study how ghosts related to the 3-dimensional translations turn into the diffeomorphism ghosts (in BV formalism). We know that it happens, and it opens up a whole world of possibilities to induce diffeomorphisms to please the democratic point of view. 2. What is the string theory?

Nowadays, in the period of string revolutions, people often use the word "string theory", but it has at least three different meanings. Here we will outline them in order to fix what we are talking about. 2.1. Theory with strings

People sometimes call by "string theory" a theory whose spectrum of extended objects contains strings. These people also call 11-dimensional phase of M-theory the "membrane theory". At first glance this looks stupid since the spectrum of extended objects generally contains a lot of different objects having different dimensions. For example, any 4-dimensional theory with several vacuums contains stable domain membranes (in previous life they were called domain walls) and any 3-dimensional theory with several vacuums contains domain strings. However, people argue that it is quite reasonable to look at the object with the smallest tension and claim that it is this object that is responsible for interactions. It is a version of the famous Yukava argument about pions and nuclear forces. However, in almost all examples of such theories there are massless fields - gauge fields or gravitons (that have particles with zero masses) therefore it is not clear why should people concentrate their attention on the particular extended object with the nonzero tension. Thus, I propose to call such theories "theories with strings". While the spectrum of extended objects is an important characteristic of a theory, such spectrum definitely does not characterize the dynamics of the theory. 2.2. Perturbative string theory The second understanding of what is "string theory" is in the prescription for calculation of amplitudes. We will present here the modern version of the definition, namely, the prescription with D-branes.

585

An n-point amplitude is given as a formal sum of integrals over moduli spaces of complex structures of surfaces with boundaries and n marked points. The measure of integration is the correlator of n vertex operators at marked points in the conformal QFT; the boundary conditions are determined by what is called "the states of the D-branes". Let us discuss this definition for a while. 2.2.1. Imminent perturbativity

To begin with, this definition is purely perturbative. There is a similar perturbative definition in quantum field theory, however, in QFT we expect that it is nothing but expansion of the functional integral. Here we just have no clue where could string prescription came from. Therefore, nonperturbative string theory in this approach is not defined at all. The expected existence of the NS-brane makes the situation rather confusing. The D-branes could be considered as nonperturbative phenomena in the open string sector but not in the closed one. It could be that some integral localizes on the space embedded Riemann surfaces - but such point of view would imply that string theory is effective and not fundamental. On the other hand it is definitely not clear why the two-dimensional generalization of a graph (i.e. of CW complex, not a smooth manifold) is a smooth Riemann surface and not a two dimensional CW complex. 2.2.2. Divergences in the perturbative definition

The great advantage of string prescription is the absence of the UV divergences. Really, we assume that the measure (that we integrate over the moduli space) is smooth on the moduli space, therefore the only divergences are coming near the boundary of the moduli space. In the case of ordinary QFT and graphs the analogue of the moduli space was the space of lengths of edges of the graphs. This space contained two types of the boundary - the UV boundary, where the length tends to zero, and IR boundary, where the length tends to infinity. The UV boundary causes problems when the shrinking edges form a loop - these are the famous loop divergences of QFT. It is remarkable that perturbative string theory is free from such divergences. However, nobody is perfect. The standard perturbative string theory has imminent divergences at the IR boundary - it comes from the zero energy string states that propagate along the long cylinders(See discussion,

586 in 6 ) . Note, that QFT generically (being put in a box) do not have such divergences. The reason for this is the following. In QFT states with zero energy correspond to zero modes, i.e. to flat directions. It is well known that the functional integral in the presence of flat direction could be considered perturbatively only in directions orthogonal to the flat ones. The integral along the flat directions should be taken nonperturbatively. But this is impossible in the perturbative formulation of string theory! People had found two ways out. One way is to consider scattering in the flat space and take momenta of scattering particles in such a way that states of zero energies do not propagate through long cylinders - but this can be done only at tree level (it is the honest way to calculate Veneziano amplitude in bosonic string). Then the general amplitude is reconstructed by analytical continuation in Mandelstam variables. If you ask me, I would say that such a prescription is disgusting in the theory that pretends to describe gravity, i.e. to work in the general curved space. Moreover, such prescription fails in loop amplitudes - here it is supersymmetry that (sometimes) saves the game. The only string theories that are free from this problem are topological strings on compact manifolds where the Hodge property of laplacians protects zero modes from propagating through handles (the so call QG-Lemma, known in physics as N = 2 supersymmetric quantum mechanics). I think that this is the fundamental difficulty of the perturbative approach and here we should say goodbye to this approach and either move to String Field Theory approach (in the sense of Zwiebach) or to invent something else.

2.2.3. Amplitudes or correlators Perturbative approach was often criticized for its ability to describe only amplitudes and not correlators ’. Here we will show that it is not so. Really, in ordinary perturbative QFT amplitudes correspond to insertion of vertex operators in the body of a Feynman graph, while correlators of local operators correspond to graphs with one-valent vertex. Note, that in the amplitude prescription there is no edge between the vertex and the rest of the graph, while in the off-shell correlator there is an edge, and we have bThis critics used the slogan :”string theory works only on-shell, while OFT works offshell”; some people promoted it to a version of holographic principle, saying that local correlators are intrinsically unobservable. I confess that I once was among these confused people

587 to integrate along the length of this edge.

On the string theory side the correlator of local operators corresponds to D-branes! Really, we place vertex operators at infinite distance - or just at the body of the Riemann surface. At the same time boundary could be considered as attachment to the Riemann surface of a cylinder of finite length, and in the process of integration along moduli of Riemann surfaces we do integrate over this length. It works - in the type B theory partition function in the presence of D(-1) branes placed at points on a Riemann surface equals to correlator of holomorphic fields at corresponding points (as it was computed in works of Polyshchuk, see7 . He also showed that this has a mirror description in terms of type A topological strings). It would be interesting to observe similar effects in other string theories. 2.3. Modified Zwiebach String Field Theory

Our idea will be to combine advantages of QFT and of String theory together. We will slightly modify the work of Zweibach who constructed special metric on a surface, such that some surfaces started to contain in itself standard flat cylinders of various length. Then he cut out these cylinders, and declared all the rest is coming from the vertex, while integral along the length of the cylinder is coming from the propagator. He cut the moduli space of Riemann surfaces with the boundaries into several pieces. Surfaces corresponding to the middle piece did not have long cylinders. Other surfaces could be constructed from several middle piece surfaces by gluing them with the help of standard cylinders of various length. He considered integral over the moduli space of the middle piece surfaces as new vertices in the string field theory, so he had to include a composite vertexes numbered by a pair consisting of the number of loops and number of boundaries. He used the BRST operator of the string theory as a kinetic term and claimed that such action on the space of string states reproduces perturbative definition of the amplitude. He made gauge fixing by imposing bo 60 conditions on the space of states, and obtained as a propagator

&.

+

Lo+Lo

I propose to exclude the states that are annihilated by LO+LO from the functional integration - they correspond to valleys and should be treated nonperturbatively. Moreover, we would like to eliminate the IR problem of a perturbative approach rather then to reproduce it! Let us call it Modified Zwiebach construction (MZ).

588 The MZ procedure has no divergences at all, it is a field theory with the action that satisfies BV master equations (as it was shown by Zwiebach). However, it depends on the details of determining what the middle piece in the Zweibach construction is. The point is that one may enlarge the middle piece in the Zwiebach construction by calling the new middle surfaces the old ones with glued cylinders with length less then T . So there is a family of constructions depending on T. Note, that when T goes to infinity all surfaces become of the new middle type and Zweibach vertices just compute perturbative amplitudes. Originally Zwiebach proved the renormgroup theorem, that says that the computation of the physical scattering amplitude is independent of T . However, this is not so in the modified Zwiebach theory - result depends on T - and this T serves as a regulator. Still we can console ourselves with the fact that the tree level amplitudes are T independent. But it seems that loop amplitudes do depend on T . I have to conclude with : love it or leave it - but it is the String theory. If you think that it is to complicated you may take one of the side paths and study some of the theories of the second level.

Acknowledgments This research was partially supported by Federal Program 40.052.1.1.1112, by Volkswagen Stiftung and by the Grants NSh-1999f2003.2 and RFFI-0402-17227

References 1. A.Galperin, E.Ivanov, S.Kalitzin, V.Ogievetsky and E.Sokatchev Class.Quantum Grav. 1,469 (1984). 2. N.Berkovits, Towards covariant quantization of the Supermembrane , hep-th 0201151. 3. E.WittenPerturbative gauge theory a string theory in twistor space, hep-th 0201151. 4. B.Zwiebach,Closed String Field Theory: Quantum Action and the BV Master Equation, Nucl.Phys B390,33 (1993). 5. M.Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematics, V01.1,2 (Zurich,l994),120-139, Birkhauser , Basel, 1995. 6. A.Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, 1987 7. A.Polishchuk,A, structures on an elliptic curve, math.AG 0001048.

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