Particle Physics Phenomenology, 5th Intl Workshop

Proceedings of the Fifth International Workshop

World Scientific

Particle Physics Phenomenology

Proceedings of the Fifth International Workshop

Particle Physics Phenomenology Chi-Pen, Taitung, Taiwain

8-11 November 2000

editors

Hsiang-nan Li National Cheng-Kung University, Taiwan, R.O.C.

Guey-Lin Lin National Chiao-Tung University, Taiwan, R.O.C.

Wei-Min Zhang National Cheng-Kung University, Taiwan, R.O.C.

V f e World Scientific wl

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PARTICLE PHYSICS PHENOMENOLOGY Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4645-5

Printed in Singapore.

PREFACE

The fifth International Workshop on Particle Physics Phenomenology was held at Chi-Pen, Taitung, Taiwan, from Nov. 8 to 11, 2000. About 90 participants attended the workshop. In this meeting there were three pedagogical lectures on QCD phenomenology, neutrino physics, and B physics, and eleven invited talks covering the above subjects. Recent theoretical and experimental progresses were reviewed. The workshop was supported by the National Center for Theoretical Sciences of R.O.C., the Ministry of Education of R.O.C., the National Science Council of R.O.C., and sponsered by National Cheng-Kung University and Academia Sinica.

Hsiang-nan Li Guey-Lin Lin Wei-Min Zhang

CONTENTS

Preface

v

Lectures Hadronic Light-Front Wavefunctions and QCD Phenomenology S. J. Brodsky

3

Lectures on the Theory of Non-Leptonic B Decays M. Neubert

43

Neutrino Physics P. Vogel

97

Invited Talks Recent Results from Lattice QCD on CP-PACS S. Aoki

131

QCD on a Transverse Lattice M. Burkardt and S. Seal

141

QCD at the Tevatron and LHC J. Huston

151

Rare B Physics Results from BELLE C. H. Wang

161

Recent BCP Progress in Taiwan H.-n. Li

169

QCD-Improved Factorization in Nonleptonic B Decays J. Chay

177

Rare Radiative B Decays in Perturbative QCD D. Pirjol

185

Neutrino Experiments: Highlights H. T.-K. Wong

196

Neutrinos and Cosmology S. Pakvasa

210

Embed Zee Neutrino Mass Model into SUSY K. Cheung

218

viii

Electroweak Sudakov Corrections at 2 Loop Level H. Kawamura

229

List of Participants

239

Lectures

H A D R O N I C LIGHT-FRONT W A V E F U N C T I O N S A N D Q C D PHENOMENONOLOGY S. J. BRODSKY Stanford Linear Accelerator Center, Stanford, CA 94309 E-mail: [email protected] A fundamental goal in QCD is to understand the non-perturbative structure of hadrons at the amplitude level—not just the single-particle flavor, momentum, and helicity distributions of the quark constituents, but also the multi-quark, gluonic, and hidden-color correlations intrinsic to hadronic and nuclear wavefunctions. A natural calculus for describing the bound-state structure of relativistic composite systems in quantum field theory is the light-front Fock expansion which encodes the properties of a hadrons in terms of a set of frame-independent n-particle wavefunctions. Light-front quantization in the doubly-transverse light-cone gauge has a number of remarkable advantages, including explicit unitarity, a physical Fock expansion, the absence of ghost degrees of freedom, and the decoupling properties needed to prove factorization theorems in high momentum transfer inclusive and exclusive reactions. A number of applications will be discussed in these lectures, including semileptonic B decays, two-photon exclusive reactions, and deeply virtual Compton scattering. The relation of the intrinsic sea to the light-front wavefunctions is discussed. A new type of jet production reaction, "self-resolving diffractive interactions" can provide direct information on the light-front wavefunctions of hadrons in terms of their quark and gluon degrees of freedom as well as the composition of nuclei in terms of their nucleon and mesonic degrees of freedom.

1

Introduction

In principle, quantum chromodynamics provides a fundamental description of hadron and nuclear physics in terms of quark and gluon degrees of freedom. QCD has been developed and successfully tested extensively, particularly in inclusive and exclusive processes involving collisions at large momentum transfer where factorization theorems and the smallness of the QCD effective coupling allow perturbative predictions. However, despite its empirical successes, many fundamental questions about QCD have not been resolved. These include a rigorous proof of color confinement, the behavior of the QCD coupling at small momenta, a rigorous description of the structure of hadrons in terms of their quark and gluon degrees of freedom, the problem of asymptotic n! growth of the perturbation theory (renormalon phenomena), the nature of the pomeron and Reggeons, the nature of shadowing and anti-shadowing in nuclear collisions, the apparent conflict between QCD vacuum structure and the small size of the cosmological constant, and the problems of scale and scheme ambiguities in perturbative QCD expansion. 3

4

In these lectures I will focus on one of the central questions in QCD the non-perturbative description of the proton and other hadrons as composites of confined, relativistic quark and gluon quanta. The goal is a frameindependent, quantum-mechanical representation of hadrons at the amplitude level capable of encoding multi-quark, hidden-color and gluon momentum, helicity, and flavor correlations in the form of universal process-independent hadron wavefunctions. For example, the measurement and interpretation of the basic parameters of the electroweak theory and CP violation depends on an understanding of the dynamics and phase structure of B decays at the amplitude level. As I will discuss in these lectures, light-front quantization allows a unifying wavefunction representation of non-perturbative hadron dynamics in QCD. Remarkably, it is now possible to measure the wavefunctions of a relativistic hadron by diffractively dissociating it into jets whose momentum distribution is correlated with the valence quarks' momenta.. 2 ' 3 ' 4,5 It is also particularly important to understand the shape of the gauge- and process-independent meson and baryon valence-quark distribution amplitudes 1 4>M(X,Q), and (f>B(xi,Q). These quantities specify how a hadron shares its longitudinal momentum among its valence quarks; they control virtually all exclusive processes involving a hard scale Q, including form factors, Compton scattering and photoproduction at large momentum transfer, as well as the decay of a heavy hadron into specific final states. 8,9 What do we mean by a hadronic wavefunction, and what are its degrees of freedom? In the chiral effective Lagrangian approximation, a baryon can be represented as a classical soliton solution as in the Skyrme model. On the other hand, bag models, the observed baryon spectroscopy, and magnetic moment phenomenology suggests that baryons are composites of three "constituent" quarks or perhaps a quark bound to an effective spin-0 or spin-l diquark. The cloudy bag model 11 takes into account the effect of mesonbaryon fluctuations such as nir+ and AK+ in the proton. This in turn implies the existence of sea quarks with an u(x) ^ d(x) and s(x) ^ s(x) asymmetries in the momentum and spin distributions of the quark sea. The n— particle Schrodinger momentum space wavefunction tpNiPi) of a composite system is the projection of the exact eigenstate of the equaltime Hamiltonian on the n-particle states of the non-interacting Hamiltonian, the Fock basis. It represents the amplitude for finding the constituents with three-momentum pi, orbital angular momentum, and spin, subject to threemomentum conservation and angular momentum sum rules. The constituents are on their mass shell, E{ = y/pf + m ? but do not conserve energy J2™=i &* > E = i/p 2 + M 2 . However, in a relativistic quantum theory, a bound-state cannot be represented as a state with a fixed number of constituents. For

5

example, the existence of gluons which propagate between the valence quarks necessarily implies that the hadron wavefunction must describe states with an arbitrary number of gluons. Thus a hadronic wavefunction must describe fluctuations in particle number n, as well as momenta and spin. One has to take into account fluctuations in the wavefunction which allow for any number of sea quarks, as long as the total quantum numbers of the constituents are compatible with the overall quantum numbers of the baryon. Gluon exchange between quarks of different nucleons in a nucleus will leave the nucleons in an exited octet state of SU(3)c, thus requiring the existence of "hidden color" configurations in a nuclear wavefunction. What is worse is that the description of the state at a given time io depends not only on the choice of gauge but also on the Lorentz frame of the observer since two observers will differ on their definition of simultaneity. A number of non-perturbative formalisms have been developed which in principle could provide the exact spectra and wavefunctions of bound states in relativistic quantum field theory, including QCD. These include Euclidean lattice gauge theory, the covariant Bethe-Salpeter bound state equation, and Hamiltonian methods based on equal-time or light-front quantization. Lattice gauge theory, particularly in the heavy quark approximation, has been an enormously helpful guide to the hadronic and glueball spectrum of QCD; however, progress in evaluating hadron wavefunctions has been so far limited to the evaluation of certain moments of hadronic matrix elements. 14 The Bethe-Salpeter formalism in principle provides a rigorous and systematic, renormalizable, and explicitly covariant treatment of bound states of two interacting fields. There has been recent progress applying model forms of Dyson-Schwinger equations to the Bethe-Salpeter formalism in order to incorporate the effects of dynamical symmetry breaking in QCD. 15 However, the actual interaction kernel in the bound state equation and the evaluation of current matrix elements requires a sum over an infinite number of irreducible graphs; the use of a single kernel (the ladder approximation) breaks gauge invariance, crossing symmetry, and has an improper heavy quark limit. It is also intrinsically difficult to apply the Bethe Salpeter to bound states of three or more constituents. There has been progress applying equal-time Hamiltonian methods to QCD in Coulomb gauge; 16 however, the renormalization procedure in this formalism is not well-defined and the results are typically frame-dependent. The evaluation of current matrix elements in the equal-time formalism is especially problematic since one must include contributions to the current from vacuum excitations. For similar reasons, the expansion of an eigenstate on the equal-time Fock basis of free quanta is not well-defined.

6

In the light-front quantization, one takes the light-cone time variable t + z/c as the evolution parameter instead of ordinary time t. (The z direction is an arbitrary reference direction.) The method is often called "light-front" quantization rather than "light-cone" quantization since the equation x+ = T = 0 defines a hyperplane corresponding to a light-front. The light-front fixes the initial boundary conditions of a composite system as its constituents are intercepted by a light-wave evaluated at a specific value of x+ = t + z/c. In contrast, determining an atomic wavefunction at a given instant t = to requires measuring the simultaneous scattering of Z photons on the Z electrons. An extensive review and guide to the light-front quantization literature can be found in ref.17. I will use here the notation A** = {A+, A—, A±), where A± = A°± Az and the metric A • B = \{A+B~ + A~B+) - A± • B±. The origins of the light-front quantization method can be traced to Dirac 18 who noted that 7 out of the 10 Poincare' generators, including a Lorentz boost K3, are kinematical (interaction-independent) when one quantizes a theory at fixed light-cone time. This in turn leads to the remarkable property that the light-front wavefunctions of a hadron are independent of the hadron's total momentum, whether it is at rest or moving. Thus once one has solved for the light-front wavefunctions, one can compute hadron matrix elements of currents between hadronic states of arbitrary momentum. In contrast, knowing the rest frame wavefunction at equal time, does not determine the moving hadron's wavefunction. Light-front wavefunctions are related to momentum-space Bethe-Salpeter wavefunctions by integrating over the relative momenta k~ = k° — kz since this projects out x+ = 0. The light-front Fock space is the eigenstates of the free light-front Hamiltonian; i.e., it is a Hilbert space of non-interacting quarks and gluons, each of which satisfy k2 = m2 and k~ = m £+ x > 0. The light-front wavefunctions are the projections of the hadronic eigenstate on the light-front Fock basis, the complete set of color singlet states of the free Hamiltonian. An essentially equivalent approach, pioneered by Weinberg, 19 ' 20 is to evaluate the equal-time theory from the perspective of an observer moving in the negative z direction with arbitrarily large momentum Pz —> —00. The light-cone fraction x = ^r of a constituent can be identified with the longitudinal momentum x = j£- in a hadron moving with large momentum Pz. It is convenient to define the invariant light-front Hamiltonian: Hfjc = p+p- _ p2 w h e r e p± = po ± pz T h e operator P~ = i£ generates lightcone time translations. The P+ and Pj_ momentum operators are independent of the interactions, and thus are conserved at all orders. The eigen-spectrum

7

of HLQD in principle gives the entire mass squared spectrum of color-singlet hadron states in QCD, together with their respective light-front wavefunctions. For example, the proton state satisfies: H£CD\^P) = M p | \ t p ) . The projection of the proton's eigensolution | $ p ) on the color-singlet B = 1, Q = 1 eigenstates {\n}} of the free Hamiltonian H2C°{9 = °) g i v e s t n e light-front 23 Fock expansion: tf, n>3,Ai

J

16***ll-S*il* (a) (E*-L<J i; Z j P + , XiP± + k±it A ^ ipn/p(xi, kxi, A,)It is especially convenient to develop the light-front formalism in the lightcone gauge A+ = A0 + Az = 0. In this gauge the A~ field becomes a dependent degree of freedom, and it can be eliminated from the gauge theory Hamiltonian, with the addition of a set of specific instantaneous light-cone time interactions. In fact in QCD(1 + 1) theory, this instantaneous interaction provides the confining linear x~ interaction between quarks. In 3 + 1 dimensions, the transverse field A1- propagates massless spin-one gluon quanta with polarization vectors 1 which satisfy both the gauge condition e^ = 0 and the Lorentz condition k • e = 0. Thus no extra condition on the Hilbert space is required. The light-front Fock wavefunctions tpn/H(xi^±i,Xi) thus interpolate between the hadron H and its quark and gluon degrees of freedom. The lightcone momentum fractions of the constituents, Xj = kf/P+ with J2Z=i x* = *> and the transverse momenta k_n with X ^ i ^J-» = "J- a P P e a j : as the momentum coordinates of the light-front Fock wavefunctions. A crucial feature is the frame-independence of the light-front wavefunctions. The X{ and kxi are relative coordinates independent of the hadron's momentum P1*. The actual physical transverse momenta are pxi = XiP\_ + kxi- The A* label the lightfront spin Sz projections of the quarks and gluons along the z direction. The physical gluon polarization vectors eM(fc, A = ±1) are specified in light-cone gauge by the conditions k • e = 0, r] • e = e+ = 0. Each light-front Fock wavefunction satisfies conservation of the z projection of angular momentum: J" — S"=i &i + S j = i lj • The sum over Sf represents the contribution of the intrinsic spins of the n Fock state constituents. The sum over orbital angular momenta Z| = -i(fcj g§j- - fcfafr) derives from the n - 1 relative momenta. This excludes the contribution to the orbital angular momentum due

to the motion of the center of mass, which is not an intrinsic property of the hadron. 21 The interaction Hamiltonian of QCD in light-cone gauge can be derived by systematically applying the Dirac bracket method to identify the independent fields.22 It contains the usual Dirac interactions between the quarks and gluons, the three-point and four-point gluon non-Abelian interactions plus instantaneous light-front-time gluon exchange and quark exchange contributions

nint =

-at-fA*v + | fabc {d„A\ -

d^^A^A™

2

+

where

a fabc jade !L

AbiiAdnAcvAev

o2 1 - — 7 + — — i+ a 2 (B I 2 a j+a = ^ V ( « a W

+ fabc(d-Abli)A°»

(D

(2)

In light-cone time-ordered perturbation theory, a Green's functions is expanded as a power series in the interactions with light-front energy denominators ^initial K - ^intermediate K +ie replacing the usual energy denominators. [For a review see ref.23] In general each Feynman diagram with n vertices corresponds to the sum of n! time-ordered contributions. However, in light-cone-time-ordered perturbation theory, only those few graphs where all kf > 0 survive. In addition the form of the light-front kinetic energies is rational: k~ = ±^71 , replacing the nonanalytic A;0 = Vk2 + m2 of equal-time theory. Thus light-cone-time-ordered perturbation theory provides a viable computational method where one can trace the physical evolution of a process. The integration measures are only three-dimensional cPk±dx; in effect, the k~ integral of the covariant perturbation theory is performed automatically. Alternatively, one derive Feynman rules for QCD in light-cone gauge, thus allowing the use of standard covariant computational tools and renormalization methods including dimensional regularization. Prem Srivastava and I 22 have recently presented a systematic study of light-front-quantized gauge theory in light-cone gauge using a Dyson-Wick S-matrix expansion based on

9

light-cone-time-ordered products. The gluon propagator has the form (0\T(A%(x)AU0))\0)

= ^fd4*e-ikx

§

^

(3)

where we have defined

^(«=)

= ^W

=- ^

+

" ^ * - ^

V

,

(4)

Here n^ is a null four-vector, gauge direction, whose components are chosen to be nM = <5M+, n^ = <5M_. Note also D„x(k)D\{k)

= Dti±(k)D±v(k) *"£>„„(*) = 0,

=

DXli(q) £>""(*) Dvp(q') = 2

-D„v{k), n"DM„(fc) = D-V{k) = 0, -£>A/J(?)£>'"V)-

(5)

The gauge field propagator iDliV(k)/(k + ie) is transverse not only to the gauge direction nM but also to k^, i.e., it is doubly-transverse. This leads to appreciable simplifications in the computations in QCD. For example, the coupling of gluons to propagators carrying high momenta is automatic. The absence of collinear divergences in irreducible diagrams in the light-cone gauge greatly simplifies the leading-twist factorization of soft and hard gluonic corrections in high momentum transfer inclusive and exclusive reactions 1 since the numerators associated with the gluon coupling only have transverse components. The renormalization factors in the light-cone gauge are independent of the reference direction nM. Since the gluon only has physical polarization, its renormalization factors satisfy Z\ = Z3. Because of its explicit unitarity in each graph, the doubly-transverse gauge is well suited for calculations identifying the "pinch" effective charge. 25,26 The running coupling constant and QCD 0 function have also been computed at one loop in the doubly-transverse light-cone gauge. 22 It is also possible to effectively quantize QCD using light-front methods in covariant Feynman gauge. A remarkable advantage of light-front quantization is that the vacuum state 10) of the full QCD Hamiltonian evidently coincides with the free vacuum. The light-front vacuum is effectively trivial if the interaction Hamiltonian applied to the perturbative vacuum is zero. Note that all particles in the Hilbert space have positive energy A;0 = |(fe + + k~), and thus positive light-front k±. Since the plus momenta £} kf is conserved by the interactions, the perturbative vacuum can only couple to states with particles in which all kf = 0; i.e., so called zero-mode states. In the case of QED, a massive electron cannot have k+ = 0 unless it also has infinite energy. In a remarkable

10

calculation, Bassetto and collaborators 27 have shown that the computation of the spectrum of QCD{1 +1) in equal time quantization requires constructing the full spectrum of non perturbative contributions (instantons). In contrast, in the light-front quantization of gauge theory, where the k+ = 0 singularity of the instantaneous interaction is defined by a simple infrared regularization, one obtains the correct spectrum of QCD(1 + 1) without any need for vacuum-related contributions. In the case of QCD(3+1), the momentum-independent four-gluon nonAbelian interaction in principle can couple the perturbative vacuum to a state with four collinear gluons in which all of the gluons have all components fcf = 0, thus hinting at role for zero modes in theories with massless quanta. In fact, zero modes of auxiliary fields are necessary to distinguish the thetavacua of massless QED(1+1), 28 ' 29 ' 30 or to represent a theory in the presence of static external boundary conditions or other constraints. Zero-modes provide the light-front representation of spontaneous symmetry breaking in scalar theories. 31 There are a number of other simplifications of the light-front formalism: 1. The light-front wavefunctions describe quanta which have positive energy, positive norm, and physical polarization. The formalism is thus physical, and unitary. No ghosts fields appear explicitly, even in non-Abelian theory. The wavefunctions are only functions of three rather than four physical momentum variables: the light-front momentum fractions Xj and transverse momenta k±. The quarks and gluons each have two physical polarization states. 2. The set of light-front wavefunctions provide a frame-independent, quantum-mechanical description of hadrons at the amplitude level capable of encoding multi-quark and gluon momentum, helicity, and flavor correlations in the form of universal process-independent hadron wavefunctions. Matrix elements of spacelike currents such as the spacelike electromagnetic form factors have an exact representation in terms of simple overlaps of the light-front wavefunctions in momentum space with the same Xi and unchanged parton number. 32 ' 33 ' 34 In the case of timelike decays, such as those determined by semileptonic B decay, one needs to include contributions in which the parton number An = 2. 35 The leading-twist off-forward parton distributions measured in deeply virtual Compton scattering have a similar light-front wavefunction representation. 36,37 3. The high x —• 1 and high A;_L limits of the hadron wavefunctions control processes and reactions in which the hadron wavefunctions are highly stressed. Such configurations involve far-off-shell intermediate states and can be systematically treated in perturbation theory. 38 ' 1 4. The leading-twist structure functions qi(x,Q) and g{x,Q) measured

11

in deep inelastic scattering can be computed from the absolute squares of the light-front wavefunctions, integrated over the transverse momentum up to the resolution scale Q. All helicity distributions are thus encoded in terms of the light-front wavefunctions. The DGLAP evolution of the structure functions can be derived from the high fcj_ properties of the light-front wavefunctions. Thus given the light-front wavefunctions, one can compute 1 all of the leading twist helicity and transversity distributions measured in polarized deep inelastic lepton scattering. For example, the helicity-specific quark distributions at resolution A correspond to

«A,/A, M

d

-^i

=E / f [ "i9o

x 16TT3* U~it

.7=1

ElV^fcH^)]2

(6)

A;

<*i\ &{2) ( E M

*(* " *«MA,A, 6(A2 - M2n) ,

where the sum is over all quarks qa which match the quantum numbers, lightfront momentum fraction x, and helicity of the struck quark. Similarly, the transversity distributions and off-diagonal helicity convolutions are defined as a density matrix of the light-front wavefunctions. This defines the LC factorization scheme1 where the invariant mass squared .M 2 = YA=\ (^li + ml)/xi of the n partons of the light-front wavefunctions is limited to M\ < A2 5. The distribution of spectator particles in the final state in the proton fragmentation region in deep inelastic scattering at an electron-proton collider are encoded in the light-front wavefunctions of the target proton. Conversely, the light-front wavefunctions can be used to describe the coalescence of comoving quarks into final state hadrons. 6. The light-front wavefunctions also specify the multi-quark and gluon correlations of the hadron. Despite the many sources of power-law corrections to the deep inelastic cross section, certain types of dynamical contributions will stand out at large x\,j since they arise from compact, highly-correlated fluctuations of the proton wavefunction. In particular, there are particularly interesting dynamical 0(1/Q2) corrections which are due to the interference of quark currents; i.e., contributions which involve leptons scattering amplitudes from two different quarks of the target nucleon.39 7. The higher Fock states of the light hadrons describe the sea quark structure of the deep inelastic structure functions, including "intrinsic" strangeness and charm fluctuations specific to the hadron's structure rather than gluon substructure. 40 ' 41 Ladder relations connecting state of different particle number follow from the QCD equation of motion and lead to Regge behavior of the quark and gluon distributions at x ->• 0. 42

12

8. The gauge- and process-independent meson and baryon valence-quark distribution amplitudes (J>M(X, Q), and (J>B(XI,Q) which control exclusive processes involving a hard scale Q, including heavy quark decays, are given by the valence light-front Fock state wavefunctions integrated over the transverse momentum up to the resolution scale Q. The evolution equations for distribution amplitudes follow from the perturbative high transverse momentum behavior of the light-front wavefunctions.23 9. The line-integrals needed to defining distribution amplitudes and structure functions as gauge invariant matrix elements of operator products vanish in light-front gauge. 10. Proofs of factorization theorems in hard exclusive and inclusive reactions are greatly simplified since the propagating gluons in light-cone gauge couple only to transverse currents; collinear divergences are thus automatically suppressed. 11. At high energies each light-front Fock state interacts distinctly; e.g., Fock states with small particle number and small impact separation have small color dipole moments and can traverse a nucleus with minimal interactions. This is the basis for the predictions for "color transparency" in hard quasiexclusive 43 ' 44 and diffractive reactions 3,4,5 . 12. The Fock state wavefunctions of hadron can be resolved by a high energy diffractive interaction, producing forward jets with momenta which follow the light-front momenta of the wavefunction. 3,4,5 . 13. The deuteron form factor at high Q2 is sensitive to wavefunction configurations where all six quarks overlap within an impact separation bu < 0(1/Q). The leading power-law fall off predicted by QCD is Fd(Q2) = f(ag(Q2))/(Q2)5, where, asymptotically, f(as(Q2)) oc a s (Q 2 ) 5 + 2 ^. 4 5 , 4 6 In general, the six-quark wavefunction of a deuteron is a mixture of five different color-singlet states. The dominant color configuration at large distances corresponds to the usual proton-neutron bound state. However at small impact space separation, all five Fock color-singlet components eventually evolve to a state with equal weight, i.e., the deuteron wavefunction evolves to 80% "hidden color." 46 The relatively large normalization of the deuteron form factor observed at large Q2 hints at sizable hidden-color contributions. 47 Hidden color components can also play a predominant role in the reaction jd —• J/ippn at threshold if it is dominated by the multi-fusion process ^gg ->• J/if).48 Hard exclusive nuclear processes can also be analyzed in terms of "reduced amplitudes" which remove the effects of nucleon substructure. Light-front wavefunctions are thus the frame-independent interpolating functions between hadron and quark and gluon degrees of freedom. Hadron amplitudes are computed from the convolution of the light-front wavefunctions

13

with irreducible quark-gluon amplitudes. More generally, all multi-quark and gluon correlations in the bound state are represented by the light-front wavefunctions. The light-front Fock representation is thus a representation of the underlying quantum field theory. I will discuss progress in computing lightfront wavefunctions directly from QCD in sections 9 and 10. 2

Other Theoretical Tools

In addition to the light-front Fock expansion, a number of other useful theoretical tools are available to eliminate theoretical ambiguities in QCD predictions: (1) Conformal symmetry provides a template for QCD predictions, 49 leading to relations between observables which are present even in a theory which is not scale invariant. For example, the natural representation of distribution amplitudes is in terms of an expansion of orthonormal conformal functions multiplied by anomalous dimensions determined by QCD evolution equations. 50,51,52 Thus an important guide in QCD analyses is to identify the underlying conformal relations of QCD which are manifest if we drop quark masses and effects due to the running of the QCD couplings. In fact, if QCD has an infrared fixed point (vanishing of the Gell Mann-Low function at low momenta), the theory will closely resemble a scale-free conformally symmetric theory in many applications. (2) Commensurate scale relations 53,54 are perturbative QCD predictions which relate observable to observable at fixed relative scale, such as the "generalized Crewther relation" 55 , which connects the Bjorken and Gross-Llewellyn Smith deep inelastic scattering sum rules to measurements of the e + e~ annihilation cross section. Such relations have no renormalization scale or scheme ambiguity. The coefficients in the perturbative series for commensurate scale relations are identical to those of conformal QCD; thus no infrared renormalons are present. 49 One can identify the required conformal coefficients at any finite order by expanding the coefficients of the usual PQCD expansion around a formal infrared fixed point, as in the Banks-Zak method. 26 All nonconformal effects are absorbed by fixing the ratio of the respective momentum transfer and energy scales. In the case of fixed-point theories, commensurate scale relations relate both the ratio of couplings and the ratio of scales as the fixed point is approached. 49 (3) ay and Skeleton Schemes. A physically natural scheme for defining the QCD coupling in exclusive and other processes is the av{Q2) scheme defined from the potential of static heavy quarks. Heavy-quark lattice gauge theory can provide highly precise values for the coupling. All vacuum polariza-

14

tion corrections due to fermion pairs are then automatically and analytically incorporated into the Gell Mann-Low function, thus avoiding the problem of explicitly computing and resumming quark mass corrections related to the running of the coupling.56 The use of a finite effective charge such as a y as the expansion parameter also provides a basis for regulating the infrared nonperturbative domain of the QCD coupling. A similar coupling and scheme can be based on an assumed skeleton expansion of the theory. 25 ' 26 (4) The Abelian Correspondence Principle. One can consider QCD predictions as analytic functions of the number of colors Nc and flavors Np. In particular, one can show at all orders of perturbation theory that PQCD predictions reduce to those of an Abelian theory at Nc —• 0 with a — CFOL8 and Np = 2NF/CF held fixed.57 There is thus a deep connection between QCD processes and their corresponding QED analogs. 3

Applications of light-front Wavefunctions to Current Matrix Elements

The light-front Fock representation of current matrix elements has a number of simplifying properties. The space-like local operators for the coupling of photons, gravitons and the deep inelastic structure functions can all be expressed as overlaps of light-front wavefunctions with the same number of Fock constituents. This is possible since one can choose the special frame q+ = 0 32 ' 33 for space-like momentum transfer and take matrix elements of "plus" components of currents such as J+ and T++. No contributions to the current matrix elements from vacuum fluctuations occur. Similarly, given the local operators for the energy-momentum tensor TliV(x) and the angular momentum tensor M M " A (x), one can directly compute momentum fractions, spin properties, and the form factors A(q2) and B(q2) appearing in the coupling of gravitons to composite systems. 21 In the case of a spin-| composite system, the Dirac and Pauli form factors 2 Fi(q ) and i^fa 2 ) are defined by = u(P') [ f i f o r V +F2(q2)^Ma»aqa]u(P)

,

(7)

where gM = (P' - P) M and u(P) is the bound state spinor. In the light-front formalism it is convenient to identify the Dirac and Pauli form factors from the helicity-conserving and helicity-fiip vector current matrix elements of the J+ current 34 : P + QA

J+(0) PA)=F1tf), 2P+

(8)

15

J+(0) p , \

P + qA 2P+

,„1

• 2^2(g2)

(9)

The magnetic moment of a composite system is one of its most basic properties. The magnetic moment is defined at the q2 —> 0 limit, H=^[Fi(0)

+ F2(0)},

(10)

where e is the charge and M is the mass of the composite system. We use the standard light-front frame {q± = q° ±q3):

P = (P+,P-,P±) = ( P + , ^ , f f x ) ,

(11)

where q2 = —2P q = —q\ is 4-momentum square transferred by the photon. The Pauli form factor and the anomalous magnetic moment K = ^Fi (0) can then be calculated from the expression

(12) where the summation is over all contributing Fock states a and struck constituent charges ej. The arguments of the final-state light-front wavefunction are * l i = leu + (1 - Zi)tfi

(13)

for the struck constituent and k'±i = k±i - xtq±

(14)

for each spectator. Notice that the magnetic moment must be calculated from the spin-flip non-forward matrix element of the current. In the ultrarelativistic limit where the radius of the system is small compared to its Compton scale 1/M, the anomalous magnetic moment must vanish. 34 The light-front formalism is consistent with this theorem. The form factors of the energy-momentum tensor for a spin-| composite are denned by (P'\T^(0)\P)

= u(P') [A{q2)^Pv)

+B{q2)^P{lia^qa

+C(q2)^(q^-g^q2)]u(P),

(15)

16 where q" = ( P ' - P ) " , P M = \{P' + py, a("&") = I ( a " 6 " + avb"), and u(P) is the spinor of the system. One can also readily obtain the light-front representation of the A(q2) and B(q2) form factors of the energy-tensor Eq. (15). 21 In the interaction picture, only the non-interacting parts of the energy momentum tensor T + + ( 0 ) need to be computed: P + qA

P + qA

T++(0) PA) 2(P+)2

= A{qi)

T++(0) PA^-iq1 2(P+) 2

V)

(16)

(17)

2M

The A(q2) and B{q2) form factors Eqs. (16) and (17) are similar to the Fi(q2) and F2(q2) form factors Eqs. (8) and (9) with an additional factor of the light-front momentum fraction x = k+/P+ of the struck constituent in the integrand. The B(q2) form factor is obtained from the non-forward spin-flip amplitude. The value of -B(O) is obtained in the q2 —> 0 limit. The angular momentum projection of a state is given by (J*) = ±eijk

[d3x(T0kxj-T°ixk)

= A(0) (L*) + [A(0) + B(0)}u(P)-aiu(P)

.

(18)

This result is derived using a wave-packet description of the state. The (L') term is the orbital angular momentum of the center of mass motion with respect to an arbitrary origin and can be dropped. The coefficient of the (L l ) term must be 1; A(0) = 1 also follows when we evaluate the four-momentum expectation value (P M ). Thus the total intrinsic angular momentum Jz of a nucleon can be identified with the values of the form factors A{q2) and B(q2) at q2 = 0 : (J') = (1
.

(19)

The anomalous moment coupling P>(0) to a graviton is shown to vanish for any composite system. This remarkable result, first derived by Okun and Kobzarev, 59 ' 60 ' 61 ' 62 ' 63 is shown to follow directly from the Lorentz boost properties of the light-front Fock representation. 21 Dae Sung Hwang, Bo-Qiang Ma, Ivan Schmidt, and I 21 have recently shown that the light-front wavefunctions generated by the radiative corrections to the electron in QED provides a simple system for understanding the

17

spin and angular momentum decomposition of relativistic systems. This perturbative model also illustrates the interconnections between Fock states of different number. The model is patterned after the quantum structure which occurs in the one-loop Schwinger a/2-K correction to the electron magnetic moment. 34 In effect, we can represent a spin-| system as a composite of a spin-1 fermion and spin-one vector boson with arbitrary masses. A similar model has been used to illustrate the matrix elements and evolution of lightfront helicity and orbital angular momentum operators. 64 This representation of a composite system is particularly useful because it is based on two constituents but yet is totally relativistic. We can then explicitly compute the form factors F\{q2) and Fi(q2) of the electromagnetic current, and the various contributions to the form factors A(q2) and B(q2) of the energy-momentum tensor. Another remarkable advantage of the light-front formalism is that exclusive semileptonic B-decay amplitudes such as B -* A£u can also be evaluated exactly. 35 The time-like decay matrix elements require the computation of the diagonal matrix element n —>• n where parton number is conserved, and the off-diagonal n + 1 ->• n — 1 convolution where the current operator annihilates a qq' pair in the initial B wavefunction. See Fig. 1. This term is a consequence of the fact that the time-like decay q2 = (pe +Pv)2 > 0 requires a positive light-front momentum fraction q+ > 0. Conversely for space-like currents, one can choose q+ = 0, as in the Drell-Yan-West representation of the space-like electromagnetic form factors. However, as can be seen from the explicit analysis of the form factor in a perturbative model, the off-diagonal convolution can yield a nonzero q+/q+ limiting form as q+ —• 0. This extra term appears specifically in the case of "bad" currents such as J~ in which the coupling to qq fluctuations in the light-front wavefunctions are favored. In effect, the q+ —• 0 limit generates 5(x) contributions as residues of the n + 1 —> n — 1 contributions. The necessity for such "zero mode" S(x) terms has been noted by Chang, Root and Yan 65 , Burkardt 66 , and Ji and Choi. 67 The off-diagonal n + 1 —¥ n — 1 contributions give a new perspective for the physics of B-decays. A semileptonic decay involves not only matrix elements where a quark changes flavor, but also a contribution where the leptonic pair is created from the annihilation of a qq' pair within the Fock states of the initial B wavefunction. The semileptonic decay thus can occur from the annihilation of a nonvalence quark-antiquark pair in the initial hadron. This feature will carry over to exclusive hadronic Z?-decays, such a s B ° - > n~D+. In this case the pion can be produced from the coalescence of a du pair emerging from the initial higher particle number Fock wavefunction of the B. The D meson

18

A,q±

1-A,-q_L Xn.k

rv Mn

A,q±

1-A,-q±

Figure 1. Exact representation of electroweak decays and time-like form factors in the lightfront Fock representation.

is then formed from the remaining quarks after the internal exchange of a W boson. In principle, a precise evaluation of the hadronic matrix elements needed for B-decays and other exclusive electroweak decay amplitudes requires knowledge of all of the light-front Fock wavefunctions of the initial and final state hadrons. In the case of model gauge theories such as QCD(1+1) 68 or collinear QCD 69 in one-space and one-time dimensions, the complete evaluation of the light-front wavefunction is possible for each baryon or meson bound-state using the DLCQ method. It would be interesting to use such solutions as a model for physical B-decays.

19

4

Light-front Representation of Deeply Virtual Compton Scattering

The virtual Compton scattering process ff (l*P -> 7P) for large initial photon virtuality q2 = — Q2 has extraordinary sensitivity to fundamental features of the proton's structure. Even though the final state photon is on-shell, the deeply virtual process probes the elementary quark structure of the proton near the light cone as an effective local current. In contrast to deep inelastic scattering, which measures only the absorptive part of the forward virtual Compton amplitude i>n7^«p-+T»p, deeply virtual Compton scattering allows the measurement of the phase and spin structure of proton matrix elements for general momentum transfer squared t. In addition, the interference of the virtual Compton amplitude and Bethe-Heitler wide angle scattering Bremsstrahlung amplitude where the photon is emitted from the lepton line leads to an electron-positron asymmetry in the e±p -> e^yp cross section which is proportional to the real part of the Compton amplitude. 70 ' 71 ' 72 The deeply virtual Compton amplitude -y*p —• •yp is related by crossing to another important process 7*7 -> hadron pairs at fixed invariant mass which can be measured in electron-photon collisions.73 To leading order in 1/Q, the deeply virtual Compton scattering amplitude 7*(Q)P(P) -* 1{Q')P(P') factorizes as the convolution in x of the amplitude V1" for hard Compton scattering on a quark line with the generalized Compton form factors H(x,t, £), E(x, t, Q, H(x,t,Q, and E(x,t,Q of the target proton. 60 ' 61 ' 74 ' 75 ' 78 ' 79 ' 77 - 80 ' 81 ' 82 - 83 - 84 Here x is the light-front momentum fraction of the struck quark, and C = Q 2 / 2 P • q plays the role of the Bjorken variable. The square of the four-momentum transfer from the proton is given by t = A 2 = 2 P • A = - ^ " f f j f l ^ , where A is the difference of initial and final momenta of the proton (P = P' + A). We will be interested in deeply virtual Compton scattering where q2 is large compared to the masses and t. Then, to leading order in 1/Q2, ^jh^ = C • Thus £ plays the role of the Bjorken variable in deeply virtual Compton scattering. For a fixed value of —t, the allowed range of £ is given by

The form factor H(x,t,Q describes the proton response when the helicity of the proton is unchanged, and E(x, t, C) is for the case when the proton helicity is flipped. Two additional functions H(x, t, £)> and E(x,t,Q appear, corresponding to the dependence of the Compton amplitude on quark helicity.

20

5-2000 8530A6

Figure 2. light-cone time-ordered contributions to deeply virtual Compton scattering. Only the contributions of leading twist in l/q2 are illustrated. These contributions illustrate the factorization property of the leading twist amplitude.

Recently, Markus Diehl, Dae Sung Hwang and I 36 have shown how the deeply virtual Compton amplitude can be evaluated explicitly in the Fock state representation using the matrix elements of the currents and the boost properties of the light-front wavefunctions. For the n -* n diagonal term (An = 0), the arguments of the final-state hadron wavefunction are y £ ^ , fc_Li — l~*} Aj_ for the struck quark and j ^ , k±t + j ^ A i for the n — 1 spectators. We thus obtain formulae for the diagonal (parton-number-conserving) contribution to the generalized form factors for deeply virtual Compton amplitude in the domain 82 ' 81,85 £ < X\ < 1:

21

x ^ t K ^ l i , ^ ) ^(x^fcxnA^Cx/f^C)1-",

^

^

(

^

2(1=0 )

^

T

^

^

(21)

^

O

=E n jf«fe«<*u j ^ ^ - t ^ ^ (z ^ x^n)(^,^,Ai) ^(^.fcxi.A^Cv^^C)1-",

(22)

where a;7! = ^ ^ ,

ATQ

x.\ — f±f ,

k'±i = fcj.j + i ^ A j ,

= k_ii —

Y^F-A±

for the struck quark,

(23)

for the (n - 1) spectators.

A sum over all possible helicities A* is understood. If quark masses are neglected, the currents conserve helicity. We also can check that Y17=i x'i = 1> For the n + 1 -> n - 1 off-diagonal term (An = — 2), consider the case where partons 1 and n + 1 of the initial wavefunction annihilate into the current leaving n — 1 spectators. Then x„+i = C _ xii £in+i = Aj_ — k±\. The remaining n — 1 partons have total momentum ((1 - C)-P+> ~ AJL). The final wavefunction then has arguments x\ = ^ v and k'±i = k±i + ^^A±. We thus obtain the formulae for the off-diagonal matrix element of the Compton amplitude in the domain 0 < X\ < £:

(n+l-+n-l)(xlit,()

C2 -

-£y===f2(n+i->n-l)(xi>t,0

=Eyo *^y ap^y -^orgi, d^y 2W)

3

22 n+1

x

/n+1

\

V'[ n li)(*j,*l<,Ai) ^ ^ ( { i i . i j . i n + i = C - a ; i } , {fc_Li,£_u,fc_L„+i = A ± -fcj_i},{Ai,Ai,A n+ i = -Ai}),

r ^

-/ H

+

C

X^-iA

2 ( 1 - 0 /

2M

2

), h(n+i^n-i){xut,0

= Ey o * ^ y 2(2^7 ^ ^ n y o ^ y

( X

n+1

\

(24)

/n+1

w

\

i-E^j ^» E^' [ « ' -

^(n-l)( X i>£j-i> A *) ^ ( n + ^ C i ^ l ^ i ^ n + l = C - * l } ,

{fc±i,fcxi,fc±n+i = Aj_ - fcxi},{Ai,Aj, A n + i = -Ai}), (25) where i = 2,3, • • •, n label the n — 1 spectator partons which appear in the final-state hadron wavefunction with

*i = T?^>

*±.i = Z±i + Y^Z±.

(26)

We can again check that the arguments of the final-state wavefunction satisfy The above representation is the general form for the generalized form factors of the deeply virtual Compton amplitude for any composite system. Thus given the light-front Fock state wavefunctions of the eigensolutions of the light-front Hamiltonian, we can compute the amplitude for virtual Compton scattering including all spin correlations. The formulae are accurate to leading order in 1/Q 2 . Radiative corrections to the quark Compton amplitude of order as(Q2) from diagrams in which a hard gluon interacts between the two photons have also been neglected.

23

5

Applications of QCD Factorization to Hard Q C D Processes

Factorization theorems for hard exclusive, semi-exclusive, and diffractive processes allow the separation of soft non-perturbative dynamics of the bound state hadrons from the hard dynamics of a perturbatively-calculable quarkgluon scattering amplitude. The factorization of inclusive reactions is reviewed in ref. For reviews and bibliography of exclusive process calculations in QCD see ref.23-88 The light-front formalism provides a physical factorization scheme which conveniently separates and factorizes soft non-perturbative physics from hard perturbative dynamics in both exclusive and inclusive reactions. 1 ' 86 In hard inclusive reactions all intermediate states are divided according to M\ < A2 and .M 2 > A2 domains. The lower mass regime is associated with the quark and gluon distributions defined from the absolute squares of the LC wavefunctions in the light cone factorization scheme. In the high invariant mass regime, intrinsic transverse momenta can be ignored, so that the structure of the process at leading power has the form of hard scattering on collinear quark and gluon constituents, as in the parton model. The attachment of gluons from the LC wavefunction to a propagator in a hard subprocess is power-law suppressed in LC gauge, so that the minimal quark-gluon particle-number subprocesses dominate. It is then straightforward to derive the DGLAP equations from the evolution of the distributions with log A 2 . The anomaly contribution to singlet helicity structure function g\{x,Q) can be explicitly identified in the LC factorization scheme as due to the j*g —> qq fusion process. The anomaly contribution would be zero if the gluon is on shell. However, if the off-shellness of the state is larger than the quark pair mass, one obtains the usual anomaly contribution. 89 In exclusive amplitudes, the LC wavefunctions are the interpolating amplitudes connecting the quark and gluons to the hadronic states. In an exclusive amplitude involving a hard scale Q2 all intermediate states can be divided according to M„ < A2 < Q 2 and M„ < A2 invariant mass domains. The high invariant mass contributions to the amplitude has the structure of a hard scattering process TH in which the hadrons are replaced by their respective (collinear) quarks and gluons. In light-cone gauge only the minimal Fock states contribute to the leading power-law fall-off of the exclusive amplitude. The wavefunctions in the lower invariant mass domain can be integrated up to an arbitrary intermediate invariant mass cutoff A. The invariant mass domain beyond this cutoff is included in the hard scattering amplitude TH- The TH satisfy dimensional counting rules. 90 Final-state and initial state corrections from gluon attachments to lines connected to the color-singlet distribution

24

amplitudes cancel at leading twist. Explicit examples of perturbative QCD factorization will be discussed in more detail in the next section. The key non-perturbative input for exclusive processes is thus the gauge and frame independent hadron distribution amplitude 86 ' 1 defined as the integral of the valence (lowest particle number) Fock wavefunction; e.g. for the pion M a * , A) = y ^ k i . V $ , r f o , * ± i , A )

(27)

where the global cutoff A is identified with the resolution Q. The distribution amplitude controls leading-twist exclusive amplitudes at high momentum transfer, and it can be related to the gauge-invariant Bethe-Salpeter wavefunction at equal light-cone time. The logarithmic evolution of hadron distribution amplitudes <j>H(%i, Q) can be derived from the perturbatively-computable tail of the valence light-front wavefunction in the high transverse momentum regime. 86 ' 1 The conformal basis for the evolution of the three-quark distribution amplitudes for the baryons 91 has recently been obtained by V. Braun et a/.52 The existence of an exact formalism provides a basis for systematic approximations and a control over neglected terms. For example, one can analyze exclusive semi-leptonic B-decays which involve hard internal momentum transfer using a perturbative QCD formalism 6 ' 7 ' 8 ' 9 ' 10 ' 92 patterned after the perturbative analysis of form factors at large momentum transfer. The hardscattering analysis again proceeds by writing each hadronic wavefunction as a sum of soft and hard contributions

i>n = C f t ( K < A2) + V£ard(M2„ > A2),

(28)

where M„ is the invariant mass of the partons in the n-particle Fock state and A is the separation scale. The high internal momentum contributions to the wavefunction t/>Jjard can be calculated systematically from QCD perturbation theory by iterating the gluon exchange kernel. The contributions from high momentum transfer exchange to the B-decay amplitude can then be written as a convolution of a hard-scattering quark-gluon scattering amplitude TJJ with the distribution amplitudes <j>(xi,A), the valence wavefunctions obtained by integrating the constituent momenta up to the separation scale Mn < A < Q. Furthermore in processes such as B —• TTD where the pion is effectively produced as a rapidly-moving small Fock state with a small color-dipole interactions, final state interactions are suppressed by color transparency. This is the basis for the perturbative hard-scattering analyses. 6 ' 8,9 ' 10 ' 92 In a systematic analysis, one can identify the hard PQCD contribution as well as the soft con-

25

tribution from the convolution of the light-front wavefunctions. Furthermore, the hard-scattering contribution can be systematically improved. Given the solution for the hadronic wavefunctions tpn ' with M2n < A 2 , one can construct the wavefunction in the hard regime with M2n > A2 using projection operator techniques. The construction can be done perturbatively in QCD since only high invariant mass, far off-shell matrix elements are involved. One can use this method to derive the physical properties of the LC wavefunctions and their matrix elements at high invariant mass. Since M\ = X)iLi ( 1 +xm l ' t n * s m e t n ° d also allows the derivation of the asymptotic behavior of light-front wavefunctions at large k±, which in turn leads to predictions for the fall-off of form factors and other exclusive matrix elements at large momentum transfer, such as the quark counting rules for predicting the nominal power-law fall-off of two-body scattering amplitudes at fixed 6Cm-90 and helicity selection rules. 93 The phenomenological successes of these rules can be understood within QCD if the coupling ay(Q) freezes in a range of relatively small momentum transfer. 94 6

Two-Photon Processes

The simplest and perhaps the most elegant illustration of an exclusive reaction in QCD is the evaluation of the photon-to-pion transition form factor -F-y-^Q 2 ) 1 , 9 5 which is measurable in single-tagged two-photon ee —• een0 reactions. The form factor is defined via the invariant amplitude TM = —ie2FKJ(Q2)e,i''f"rp*epq(T . As in inclusive reactions, one must specify a factorization scheme which divides the integration regions of the loop integrals into hard and soft momenta, compared to the resolution scale Q. At leading twist, the transition form factor then factorizes as a convolution of the 7*7 -» qq amplitude (where the quarks are collinear with the final state pion) with the valence light-front wavefunction of the pion: F7M(Q2)

4

= ^

J

r1

dx
(29)

The hard scattering amplitude for 77* -> qq is T"M(x,Q2) = [(1 — x)Q2]~1 (1 + 0(a8)). The leading QCD corrections have been computed by Braaten 96 . The evaluation of the next-to-leading corrections in the physical ay scheme is given in Ref.94. For the asymptotic distribution amplitude 2 2 \ ^ P ) 0 as y m p t ( a ; ) = 73/^(1 _ x) o n e predicts Q F T7r (Q ) = 2 / * ( l where Q* = e~3/2Q is the BLM scale for the pion form factor. The PQCD predictions have been tested in measurements of e-y ->• eir° by the CLEO

26

collaboration 97 . See Fig. 4 (b). The observed flat scaling of the Q2F~fn(Q2) data from Q2 = 2 to Q2 = 8 GeV2 provides an important confirmation of the applicability of leading twist QCD to this process. The magnitude of Q2FTK{Q2) is remarkably consistent with the predicted form, assuming the asymptotic distribution amplitude and including the LO QCD radiative correction with av{e~3/2Q)/n ~ 0.12. One could allow for some broadening of the distribution amplitude with a corresponding increase in the value of a y at small scales. Radyushkin 101 , Ong 102 , and Kroll 103 have also noted that the scaling and normalization of the photon-to-pion transition form factor tends to favor the asymptotic form for the pion distribution amplitude and rules out broader distributions such as the two-humped form suggested by QCD sum rules 104 . The two-photon annihilation process 7*7 —> hadrons, which is measurable in single-tagged e + e~ —> e + e _ hadrons events, provides a semi-local probe of C = + hadron systems 7r°,7j°, 77', jj c ,7r + 7r _ , etc. The 7*7 -> TT+TT~ hadronpair process is related to virtual Compton scattering on a pion target by crossing. The leading twist amplitude is sensitive to the l / x — 1/(1 — x) moment of the two-pion distribution amplitude coupled to two valence quarks 85,73 . Two-photon reactions, 77 —> HH at large s = (fci + fo)2 and fixed 0cm, provide a particularly important laboratory for testing QCD since these crosschannel "Compton" processes are the simplest calculable large-angle exclusive hadronic scattering reactions. The helicity structure, and often even the absolute normalization can be rigorously computed for each two-photon channel 95 . In the case of meson pairs, dimensional counting predicts that for large s, sida/dt(iyy -+ MM scales at fixed t/s or 9c.m. up to factors of lns/A 2 . The angular dependence of the 77 —• HH amplitudes can be used to determine the shape of the process-independent distribution amplitudes, <J>H{X,Q). An important feature of the 77 —> MM amplitude for meson pairs is that the contributions of Landshoff pitch singularities are power-law suppressed at the Born level - even before taking into account Sudakov form factor suppression. There are also no anomalous contributions from the x -> 1 endpoint integration region. Thus, as in the calculation of the meson form factors, each fixed-angle helicity amplitude can be written to leading order in 1/Q in the factorized form [Q2 =p\. = tu/s; Qx = min(xQ, (I - x)Q)]\ M

-rr^M-M=

dx

dt/]jj(j/,0y)7,//(a;,2/,s,
(30)

where TH is the hard-scattering amplitude 77 ->• {qq){qq) for the production of the valence quarks collinear with each meson, and 4>M(%, Q) is the amplitude for finding the valence q and q with light-front fractions of the meson's

27

momentum, integrated over transverse momenta k± < Q. The contribution of non-valence Fock states are power-law suppressed. Furthermore, the helicityselection rules 93 of perturbative QCD predict that vector mesons are produced with opposite helicities to leading order in 1/Q and all orders in as. The dependence in x and y of several terms in 7 \ y is quite similar to that appearing in the meson's electromagnetic form factor. Thus much of the dependence on n+ p.~ amplitudes at large s and fixed 6CM is nearly insensitive to the running coupling and the shape of the pion distribution amplitude: f(77->7r+0

4\FAs)\2 2

^ ( 7 7 ->• n+n~) ~ 1 - cos ec.m.'

, l

The comparison of the PQCD prediction for the sum of 7r+7r~ plus K+K~ channels with recent CLEO d a t a " is shown in Fig. 3. The CLEO data for charged pion and kaon pairs show a clear transition to the scaling and angular distribution predicted by PQCD 9 5 for W = ^ 7 7 > 2 GeV. See Fig. 3. It is clearly important to measure the magnitude and angular dependence of the two-photon production of neutral pions and p+p~ cross sections in view of the strong sensitivity of these channels to the shape of meson distribution amplitudes. QCD also predicts that the production cross section for charged p-pairs (with any helicity) is much larger that for that of neutral p pairs, particularly at large 9c.m. angles. Similar predictions are possible for other helicity-zero mesons. The cross sections for Compton scattering on protons and the crossed reaction 77 -» pp at high momentum transfer have also been evaluated, 98,100 providing important tests of the proton distribution amplitude. It is particularly compelling to see a transition in angular dependence between the low energy chiral and PQCD regimes. The success of leadingtwist perturbative QCD scaling for exclusive processes at presently experimentally accessible momentum transfer can be understood if the effective coupling &v{Q*) is approximately constant at the relatively small scales Q* relevant to the hard scattering amplitudes 94 . The evolution of the quark distribution amplitudes In the low-Q* domain at also needs to be minimal. Sudakov suppression of the endpoint contributions is also strengthened if the coupling is frozen because of the exponentiation of a double logarithmic series. A debate has continued 105 ' 106 ' 107 ' 108 on whether processes such as the pion and proton form factors and elastic Compton scattering ^p —• 7^ might be dominated by higher-twist mechanisms until very large momentum transfer. If one assumes that the light-front wavefunction of the pion has the form VWtfofcjJ = Aexp(—b3,/1f >), then the Feynman endpoint contribution to

28

1-2001 8SG1A2G

COS © I

Figure 3. Comparison of the sum of 7 7 —• 7r+7r~ and 7 7 —>• K+K~ meson pair production cross sections with the scaling and angular distribution of the perturbative QCD 96 prediction . The data are from the CLEO collaboration".

the overlap integral at small k± and x ~ 1 will dominate the form factor compared to the hard-scattering contribution until very large Q2. However, this ansatz for ipSOft{x,kj_) has no suppression at k± = 0 for any x; i.e., the wavefunction in the hadron rest frame does not fall-off at all for fcj_ = 0 and kz ->• —00. Thus such wavefunctions do not represent well soft QCD contributions. Endpoint contributions are also suppressed by the QCD Sudakov form factor, reflecting the fact that a near-on-shell quark must radiate if it absorbs large momentum. One can show1 that the leading power dependence of the two-particle light-front Fock wavefunction in the endpoint region is 1 — x, giving a meson structure function which falls as (1 — a;)2 and thus by duality a non-leading contribution to the meson form factor F(Q2) oc 1/Q 3 . Thus the dominant contribution to meson form factors comes from the hard-scattering regime. Radyushkin 106 has argued that the Compton amplitude is dominated by soft end-point contributions of the proton wavefunctions where the two photons both interact on a quark line carrying nearly all of the proton's momentum. This description appears to agree with the Compton data at least at forward angles where -t < 10 GeV 2 . From this viewpoint, the dominance of the factorizable PQCD leading twist contributions requires momentum transfers much higher than those currently available. However, the endpoint model cannot explain the empirical success of the perturbative QCD fixed 0c.m. scaling s7da/dt('jp -+ 7r + n) ~ const at relatively low momentum transfer in pion photoproduction 109 . Clearly much more experimental input on hadron wavefunctions is needed, particularly from measurements of two-photon exclusive reactions into meson

29 and baryon pairs at the high luminosity B factories. For example, the ratio TIT(77 ~* 7 r ° 7 r °)/^"(77 ~* 7r+?r~) is particularly sensitive to the shape of pion distribution amplitude. Baryon pair production in two-photon reactions at threshold may reveal physics associated with the soliton structure of baryons in QCD 1 1 0 , 1 1 1 . In addition, fixed target experiments can provide much more information on fundamental QCD processes such as deeply virtual Compton scattering and large angle Compton scattering. 7

Self-Resolved Diffractive Reactions and Light Cone Wavefunctions

Diffractive multi-jet production in heavy nuclei provides a novel way to measure the shape of the LC Fock state wavefunctions and test color transparency. For example, consider the reaction 3 ' 4 ' 5 TTA -»• Jeti + Jet 2 + A' at high energy where the nucleus A' is left intact in its ground state. The transverse momenta of the jets balance so that k±t + &X2 = q± < -R -1 A • The light-front longitudinal momentum fractions also need to add to x\ + x% ~ 1 so that Api, < R^1. The process can then occur coherently in the nucleus. Because of color transparency, the valence wavefunction of the pion with small impact separation, will penetrate the nucleus with minimal interactions, diffracting into jet pairs. 3 The X\ = x, Xi = 1 — x dependence of the di-jet distributions will thus reflect the shape of the pion valence light-front wavefunction in x; similarly, the fc_u —fcj.2relative transverse momenta of the jets gives key information on the derivative of the underlying shape of the valence pion wavefunction. 4 ' 5 ' 114 The diffractive nuclear amplitude extrapolated to t = 0 should be linear in nuclear number A if color transparency is correct. The integrated diffractive rate should then scale as A2/R\ ~ A4/3. Preliminary results on a diffractive dissociation experiment of this type E791 at Fermilab using 500 GeV incident pions on nuclear targets. 112 appear to be consistent with color transparency. The measured longitudinal momentum distribution of the jets 1 1 3 is consistent with a pion light-cone wavefunction of the pion with the shape of the asymptotic distribution amplitude, <j>fympt(x) = %/3/ffx(l - a;). Data from CLEO 97 for the 77* -> ir° transition form factor also favor a form for the pion distribution amplitude close to the asymptotic solution to the perturbative QCD evolution equation. 86 ' 1 The diffractive dissociation of a hadron or nucleus can also occur via the Coulomb dissociation of a beam particle on an electron beam (e.g. at HERA or eRHIC) or on the strong Coulomb field of a heavy nucleus (e.g. at RHIC or nuclear collisions at the LHC). 114 The amplitude for Coulomb exchange at small momentum transfer is proportional to the first derivative ^ et-^-tp

30

of the light-front wavefunction, summed over the charged constituents. The Coulomb exchange reactions fall off less fast at high transverse momentum compared to pomeron exchange reactions since the light-front wavefunction is effective differentiated twice in two-gluon exchange reactions. It will also be interesting to study diffractive tri-jet production using proton beams pA ->• Jeti + Jet2 + Jet 3 + A' to determine the fundamental shape of the 3-quark structure of the valence light-front wavefunction of the nucleon at small transverse separation. 4 For example, consider the Coulomb dissociation of a high energy proton at HERA. The proton can dissociate into three jets corresponding to the three-quark structure of the valence light-front wavefunction. We can demand that the produced hadrons all fall outside an opening angle 0 in the proton's fragmentation region. Effectively all of the light-front momentum 53 • Xj ~ 1 of the proton's fragments will thus be produced outside an "exclusion cone". This then limits the invariant mass of the contributing Fock state M\ > A2 = P+2 sin2 0/4 from below, so that perturbative QCD counting rules can predict the fall-off in the jet system invariant mass M. At large invariant mass one expects the three-quark valence Fock state of the proton to dominate. The segmentation of the forward detector in azimuthal angle <j> can be used to identify structure and correlations associated with the three-quark light-front wavefunction.114 An interesting possibility is that the distribution amplitude of the A(1232) for Jz = 1/2,3/2 is close to the asymptotic form X1X2X3, but that the proton distribution amplitude is more complex. This ansatz can also be motivated by assuming a quark-diquark structure of the baryon wavefunctions. The differences in shapes of the distribution amplitudes could explain why the p —• A transition form factor appears to fall faster at large Q2 than the elastic p -» p and the other p -» N* transition form factors. 115 One can use also measure the dijet structure of real and virtual photons beams j*A —> Jeti + Jet2 + A' to measure the shape of the light-front wavefunction for transversely-polarized and longitudinally-polarized virtual photons. Such experiments will open up a direct window on the amplitude structure of hadrons at short distances. The light-front formalism is also applicable to the description of nuclei in terms of their nucleonic and mesonic degrees of freedom. 116 ' 117 Self-resolving diffractive jet reactions in high energy electron-nucleus collisions and hadron-nucleus collisions at moderate momentum transfers can thus be used to resolve the light-front wavefunctions of nuclei.

31

8

Higher Fock States and the Intrinsic Sea

One can identify two contributions to the heavy quark sea, the "extrinsic" contributions which correspond to ordinary gluon splitting, and the "intrinsic" sea which is multi-connected via gluons to the valence quarks. The leading 1/rriQ contributions to the intrinsic sea of the proton in the heavy quark expansion are proton matrix elements of the operator 118 ,rf'rfGaV.GpvGaP which in light-cone gauge rfA^ = A+ = 0 corresponds to three or four gluon exchange between the heavy-quark loop and the proton constituents in the forward virtual Compton amplitude. The intrinsic sea is thus sensitive to the hadronic bound-state structure. 119,40 The maximal contribution of the intrinsic heavy quark occurs at XQ ~ TTIXQI J2i m±- where m±_ = yjm2 + k\; i.e. at large XQ, since this minimizes the invariant mass .M 2 . The measurements of the charm structure function by the EMC experiment are consistent with intrinsic charm at large a; in the nucleon with a probability of order 0.6 ± 0.3%. 41 which is consistent with the recent estimates based on instanton fluctuations. 118 Chang and Hou 120 have recently discussed the consequences of intrinsic charm in heavy quark states such as the B, AB, and T, such as an anomalous momentum distribution for B —> J/ipX. The characteristic momenta characterizing the B meson is most likely higher by a factor of 2 compared to the momentum scale of light mesons, This effect is analogous to the higher momentum scale of muonium p,+e~ versus that of positronium e+e~ in atomic physics because of the larger reduced mass. Thus one can expect a higher probability for intrinsic charm in heavy hadrons compared to light hadrons. One can also distinguish "intrinsic gluons" 121 which are associated with multi-quark interactions and extrinsic gluon contributions associated with quark substructure. One can also use this framework to isolate the physics of the anomaly contribution to the Ellis-Jaffe sum rule. 89 Thus neither gluons nor sea quarks are solely generated by DGLAP evolution, and one cannot define a resolution scale Qo where the sea or gluon degrees of freedom can be neglected. It is usually assumed that a heavy quarkonium state such as the J/tp always decays to light hadrons via the annihilation of its heavy quark constituents to gluons. However, as Karliner and I 122 have shown, the transition J/t/j —>• pir can also occur by the rearrangement of the cc from the J/ip into the | qqcc) intrinsic charm Fock state of the p or TT. On the other hand, the overlap rearrangement integral in the decay ip' -> pir will be suppressed since the intrinsic charm Fock state radial wavefunction of the light hadrons will evidently not have nodes in its radial wavefunction. This observation provides a natural explanation of the long-standing puzzle 123 why the J/ip decays

32

prominently to two-body pseudoscalar-vector final states, breaking hadron helicity conservation, 93 whereas the ip' does not. The higher Fock state of the proton | uudss) should resemble a | Kk) intermediate state, since this minimizes its invariant mass M. In such a state, the strange quark has a higher mean momentum fraction x than the s. 12 > n ,i3 Similarly, the helicity of the intrinsic strange quark in this configuration will be anti-aligned with the helicity of the nucleon. 12 ' 13 This Q «+ Q asymmetry is a striking feature of the intrinsic heavy-quark sea. 9

Non-Perturbative Solutions of Light-Front Quantized Q C D

Is there any hope of computing light-front wavefunctions from first principles? The solution of the light-front Hamiltonian equation H^Q10] * ) = M2\ Vt) is an eigenvalue problem which in principle determines the masses squared of the entire bound and continuum spectrum of QCD. If one introduces periodic or anti-periodic boundary conditions, the eigenvalue problem is reduced to the diagonalization of a discrete Hermitian matrix representation of H£CD. The light-front momenta satisfy x+ = ^ n * and P+ = ^-K, where £ \ rij = K. The number of quanta in the contributing Fock states is restricted by the choice of harmonic resolution. A cutoff on the invariant mass of the Fock states truncates the size of the matrix representation in the transverse momenta. This is the essence of the DLCQ method, 124 which has now become a standard tool for solving both the spectrum and light-front wavefunctions of one-space one-time theories - virtually any 1 + 1 quantum field theory, including "reduced QCD" (which has both quark and gluonic degrees of freedom) can be completely solved using DLCQ. 125,69 The method yields not only the bound-state and continuum spectrum, but also the light-front wavefunction for each eigensolution. 126 ' 127 In the case of theories in 3+1 dimensions, Hiller, McCartor, and 1128>129 have recently shown that the use of covariant Pauli-Villars regularization with DLCQ allows one to obtain the spectrum and light-front wavefunctions of simplified theories, such as (3+1) Yukawa theory. Dalley et al. have shown how one can use DLCQ in one space-one time, with a transverse lattice to solve mesonic and gluonic states in 3 + 1 QCD. 130 The spectrum obtained for gluonium states is in remarkable agreement with lattice gauge theory results, but with a huge reduction of numerical effort. Hiller and I 131 have shown how one can use DLCQ to compute the electron magnetic moment in QED without resort to perturbation theory. One can also formulate DLCQ so that supersymmetry is exactly preserved in the discrete approximation, thus combining the power of DLCQ with the

33

beauty of supersymmetry. 132 ' 133 ' 134 The "SDLCQ" method has been applied to several interesting supersymmetric theories, to the analysis of zero modes, vacuum degeneracy, massless states, mass gaps, and theories in higher dimensions, and even tests of the Maldacena conjecture. 132 Broken supersymmetry is interesting in DLCQ, since it may serve as a method for regulating nonAbelian theories. 129 There are also many possibilities for obtaining approximate solutions of light-front wavefunctions in QCD. QCD sum rules, lattice gauge theory moments, and QCD inspired models such as the bag model, chiral theories, provide important constraints. Guides to the exact behavior of LC wavefunctions in QCD can also be obtained from analytic or DLCQ solutions to toy models such as "reduced" QCD(1 + 1). The light-front and many-body Schrodinger theory formalisms must match In the nonrelativistic limit. It would be interesting to see if light-front wavefunctions can incorporate chiral constraints such as soliton (Skyrmion) behavior for baryons and other consequences of the chiral limit in the soft momentum regime. Solvable theories such as QCD(1 + 1) are also useful for understanding such phenomena. It has been shown that the anomaly contribution for the 7r° —• 77 decay amplitude is satisfied by the light-front Fock formalism in the limit where the mass of the pion is light compared to its size. 135 10

Non-Perturbative Calculations of the Pion Distribution Amplitude

The distribution amplitude 4>(x,Q) can be computed from the integral over transverse momenta of the renormalized hadron valence wavefunction in the light-cone gauge at fixed light-cone time 23 :

J>(x,Q) = Jd2k±e (Q2 -

*x_ J ipW&kl),

(32)

where a global cutoff in invariant mass is identified with the resolution Q. The distribution amplitude (x,Q) dependence in the ultraviolet logQ scale are the same as those which appear in the DGLAP evolution of structure functions 50 . The decay IT -» (iv normalizes the wave function at the origin: a 0 /6 = £ dx4>(x, Q) = A/(2 V / 3). One can also compute the distribution amplitude from the gauge invariant Bethe-Salpeter wavefunction at equal light-

34

cone time. This also allows contact with both QCD sum rules and lattice gauge theory; for example, moments of the pion distribution amplitudes have been computed in lattice gauge theory 14,136,137 . Dalley 138 has recently calculated the pion distribution amplitude from QCD using a combination of the discretized DLCQ method for the x~ and x+ light-cone coordinates with the transverse lattice method 139>140 in the transverse directions, A finite lattice spacing a can be used by choosing the parameters of the effective theory in a region of renormalization group stability to respect the required gauge, Poincare, chiral, and continuum symmetries. The overall normalization gives fn = 101 MeV compared with the experimental value of 93 MeV. Figure 4 (a) compares the resulting DLCQ/transverse lattice pion wavefunction with the best fit to the diffractive di-jet data (see the next section) after corrections for hadronization and experimental acceptance 2 . The theoretical curve is somewhat broader than the experimental result. However, there are experimental uncertainties from hadronization and theoretical errors introduced from finite DLCQ resolution, using a nearly massless pion, ambiguities in setting the factorization scale Q2, as well as errors in the evolution of the distribution amplitude from 1 to 10 GeV 2 . Instanton models also predict a pion distribution amplitude close to the asymptotic form 141 . In contrast, recent lattice results from Del Debbio et al.137 predict a much narrower shape for the pion distribution amplitude than the distribution predicted by the transverse lattice. A new result for the proton distribution amplitude treating nucleons as chiral solitons has recently been derived by Diakonov and Petrov 142 . Dyson-Schwinger models 15 of hadronic Bethe-Salpeter wavefunctions can also be used to predict light-cone wavefunctions and hadron distribution amplitudes by integrating over the relative k~ momentum. There is also the possibility of deriving Bethe-Salpeter wavefunctions within light-cone gauge quantized QCD 24 in order to properly match to the light-cone gauge Fock state decomposition. 11

Conclusions

In these lectures I have shown how the universal, process-independent and frame-independent light-front Fock-state wavefunctions encode the properties of a hadron in terms of its fundamental quark and gluon degrees of freedom. Knowledge of such wavefunctions will be critical for progress in understanding exclusive B decays. I have shown how, given the proton's light-front wavefunctions, one can compute not only the moments of the quark and gluon distributions measured in deep inelastic lepton-proton scattering, but also the multi-parton

35 1

T

_i

1

i

4 Q 2 (GeV2)

r

i

i

8

Figure 4. (a) Preliminary transverse lattice results for the pion distribution amplitude at Q 2 ~ 10 GeV 2 . The solid curve is the theoretical prediction from the ctfmbined DLCQ/transverse lattice method 1 3 8 ; the chain line is the experimental result obtained from jet diffractive dissociation 2 . Both are normalized to the same area for comparison. (b) Scaling of the transition photon to pion transition form factor Q2F o (Q2)- The dotted and solid theoretical curves are the perturbative QCD prediction at leading and next-toleading order, respectively, assuming the asymptotic pion distribution The data are from the CLEO collaboration 9 7 .

correlations which control the distribution of particles in the proton fragmentation region and dynamical higher twist effects. Light-front wavefunctions also provide a systematic framework for evaluating exclusive hadronic matrix elements, including time-like heavy hadron decay amplitudes, form factors, and the generalized form factors that appear in deeply virtual Compton scattering. The light-front Hamiltonian formalism also provides a physical factorization scheme for separating hard and soft contributions in both exclusive and inclusive hard processes. The leading-twist QCD predictions for exclusive two-photon processes such as the photon-to-pion transition form factor and 77 —> hadron pairs are based on rigorous factorization theorems. The recent data from the CLEO collaboration on F 7 „.(Q 2 ) and the sum of 77 -> TT+TT~ and 77 -> K+K~ channels are in excellent agreement with the QCD predictions. It is particularly compelling to see a transition in angular dependence between the low energy chiral and PQCD regimes. The success of leading-twist perturbative QCD scaling for exclusive processes at presently experimentally accessible momentum transfer can be understood if the effective coupling ay(Q*) is approximately constant at the relatively small scales Q* relevant to the hard scattering amplitudes 94 . The evolution of the quark distribution amplitudes In the low-Q* domain at also needs to be minimal. Sudakov suppression of the endpoint contributions is also strengthened if the coupling is frozen because

36

of the exponentiation of a double logarithmic series. One of the formidable challenges in QCD is the calculation of nonperturbative wavefunctions of hadrons from first principles. The recent calculation of the pion distribution amplitude by Dalley 138 using light-cone and transverse lattice methods is particularly encouraging. The predicted form of (j)^ (x, Q) is somewhat broader than but not inconsistent with the asymptotic form favored by the measured normalization of Q2F77ro (Q2) and the pion wavefunction inferred from diffractive di-jet production. Clearly much more experimental input on hadron wavefunctions is needed, particularly from measurements of two-photon exclusive reactions into meson and baryon pairs at the high luminosity B factories. For example, the ratio %{ll ~• 7r ° 7r °)/^f (77 ~~^ 7r+7r~) is particularly sensitive to the shape of pion distribution amplitude. Baryon pair production in two-photon reactions at threshold may reveal physics associated with the soliton structure of baryons in QCD 1 1 0 . In addition, fixed target experiments can provide much more information on fundamental QCD processes such as deeply virtual Compton scattering and large angle Compton scattering. A remarkable new type of diffractive jet production reaction, "selfresolving diffractive interactions" can provide direct empirical information on the light-front wavefunctions of hadrons. The recent E791 experiment at Fermilab has not only determined the main features of the pion wavefunction, but has also confirmed color transparency, a fundamental test of the gauge properties of QCD. Analogous reaction involving nuclear projectiles can resolve the light-front wavefunctions of nuclei in terms of their nucleon and mesonic degrees of freedom. It is also possible to measure the light-front wavefunctions of atoms through high energy Coulomb dissociation. There has been notable progress in computing light-front wavefunctions directly from the QCD light-front Hamiltonian, using DLCQ and transverse lattice methods. Even without full non-perturbative solutions of QCD, one can envision a program to construct the light-front wavefunctions using measured moments constraints from QCD sum rules, lattice gauge theory, and data from hard exclusive and inclusive processes. One can also be guided by theoretical constraints from perturbation theory which dictate the asymptotic form of the wavefunctions at large invariant mass, x -¥ 1, and high fcj_One can also use ladder relations which connect Fock states of different particle number; perturbatively-motivated numerator spin structures; conformal symmetry, guidance from toy models such as "reduced" QCD(1 + 1); and the correspondence to Abelian theory for Nc -+ 0, as well as many-body Schrodinger theory in the nonrelativistic domain.

37

Acknowledgments Work supported by the Department of Energy under contract number DEAC03-76SF00515. I wish to thank the organizers of this meeting, Hsiang-nan Li and Wei-Min Zhang, for their outstanding hospitality in Taiwan. Much of this work is based on collaborations, particularly with Markus Diehl, Paul Hoyer, Dae Sung Hwang, Peter Lepage, Bo-Qiang Ma, Hans Christian Pauli, Johan Rathsman, Ivan Schmidt, and Prem Srivastava. References 1. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980). 2. D. Ashery [E791 Collaboration], hep-ex/9910024. 3. G. Bertsch, S. J. Brodsky, A. S. Goldhaber and J. F. Gunion, Phys. Rev. Lett. 47, 297 (1981). 4. L. Frankfurt, G. A. Miller and M. Strikman, Phys. Lett. B304, 1 (1993) [hep-ph/9305228]. 5. L. Frankfurt, G. A. Miller and M. Strikman, Found. Phys. 30, 533 (2000) [hep-ph/9907214]. 6. A. Szczepaniak, E. M. Henley and S. J. Brodsky, Phys. Lett. B243, 287 (1990). 7. A. Szczepaniak, Phys. Rev. D 54, 1167 (1996). 8. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999) [hep-ph/9905312]. 9. Y. Keum, H. Li and A. I. Sanda, hep-ph/0004004. 10. Y. Y. Keum, H. Li and A. I. Sanda, hep-ph/0004173. 11. A. I. Signal and A. W. Thomas, Phys. Lett. B191, 205 (1987). 12. M. Burkardt and B. Warr, Phys. Rev. D 45, 958 (1992). 13. S. J. Brodsky and B. Ma, Phys. Lett. B381, 317 (1996) [hep-ph/9604393]. 14. G. Martinelli and C. T. Sachrajda, Amplitude," Phys. Lett. B190, 151 (1987). 15. M. B. Hecbt, C. D. Roberts and S. M. Schmidt, nucl-th/0008049. 16. D. G. Robertson, E. S. Swanson, A. P. Szczepaniak, C. R. Ji and S. R. Cotanch, Phys. Rev. D 59, 074019 (1999) [hep-ph/9811224]. 17. S. J. Brodsky, H. Pauli and S. S. Pinsky, Phys. Rept. 301, 299 (1998) [hep-ph/9705477]. 18. P. A. Dirac, Rev. Mod. Phys. 21, 392 (1949). 19. S. Weinberg, Phys. Rev. 150, 1313 (1966). 20. S. J. Brodsky, R. Roskies and R. Suaya, Momentum Frame," Phys. Rev. D 8, 4574 (1973).

38

21. S. J. Brodsky, D. S. Hwang, B. Ma and I. Schmidt, Nucl. Phys. B593, 311 (2001) [hep-th/0003082]. 22. P. P. Srivastava and S. J. Brodsky, hep-ph/0011372. 23. S. J. Brodsky and G. P. Lepage, in Perturbative Quantum Chromodynamics, A. H. Mueller, Ed. (World Scientific, 1989). 24. P. P. Srivastava and S. J. Brodsky, Phys. Rev. D 6 1 , 025013 (2000), hep-ph/9906423, and SLAC-PUB 8543, in preparation. 25. J. M. Cornwall and J. Papavassiliou, Phys. Rev. D 40, 3474 (1989). 26. S. J. Brodsky, E. Gardi, G. Grunberg and J. Rathsman, hep-ph/0002065. 27. A. Bassetto, L. Griguolo and F. Vian, hep-th/9911036. 28. K. Yamawaki, hep-th/9802037. 29. G. McCartor, hep-th/0004139. 30. P. P. Srivastava, Phys. Lett. B448, 68 (1999) [hep-th/9811225]. 31. S. S. Pinsky and B. van de Sande, Phys. Rev. D49, 2001 (1994), hepph/9310330. 32. S. D. Drell and T. Yan, Phys. Rev. Lett. 24, 181 (1970). 33. G. B. West, Phys. Rev. Lett. 24, 1206 (1970). 34. S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236 (1980). 35. S. J. Brodsky and D. S. Hwang, Nucl. Phys. B543, 239 (1999) [hepph/9806358]. 36. S. J. Brodsky, M. Diehl and D. S. Hwang, hep-ph/0009254. 37. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, hep-ph/0009255. 38. S. J. Brodsky, M. Burkardt and I. Schmidt, distributions," Nucl. Phys. B441, 197 (1995) [hep-ph/9401328]. 39. S. J. Brodsky, hep-ph/0006310. 40. S. J. Brodsky, P. Hoyer, C. Peterson and N. Sakai, Phys. Lett. B93, 451 (1980). 41. B. W. Harris, J. Smith and R. Vogt, Nucl. Phys. B461, 181 (1996) [hepph/9508403]. 42. F. Antonuccio, S. J. Brodsky and S. Dalley, Phys. Lett. B412,104 (1997) [hep-ph/9705413]. 43. S. J. Brodsky and A. H. Mueller, Phys. Lett. B206, 685 (1988). 44. L. L. Frankfurt and M. I. Strikman, Phys. Rept. 160, 235 (1988). 45. S. J. Brodsky and B. T. Chertok, Phys. Rev. D 14, 3003 (1976). 46. S. J. Brodsky, C. Ji and G. P. Lepage, Phys. Rev. Lett. 51, 83 (1983). 47. G. R. Farrar, K. Huleihel and H. Zhang, Phys. Rev. Lett. 74, 650 (1995). 48. S. J. Brodsky, E. Chudakov, P. Hoyer and J. M. Laget, hep-ph/0010343. 49. S. J. Brodsky and J. Rathsman, hep-ph/9906339. 50. S. J. Brodsky, Y. Frishman, G. P. Lepage and C. Sachrajda, Phys. Lett.

39

91B, 239 (1980). 51. D. Miiller, Phys. Rev. D49, 2525 (1994). 52. V. M. Braun, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, Nucl. Phys. B553, 355 (1999), hep-ph/9902375. 53. S". J. Brodsky and H. J. Lu, Phys. Rev. D51, 3652 (1995), hepph/9405218. 54. S. J. Brodsky, J. R. Pelaez and N. Toumbas, Phys. Rev. D60, 037501 (1999), hep-ph/9810424. 55. S. J. Brodsky, G. T. Gabadadze, A. L. Kataev and H. J. Lu, Phys. Lett. B372, 133 (1996) hep-ph/9512367. 56. S. J. Brodsky, M. S. Gill, M. Melles and J. Rathsman, Phys. Rev. D 58, 116006 (1998) [hep-ph/9801330]. 57. S. J. Brodsky and P. Huet, Phys. Lett. B417, 145 (1998), hepph/9707543. 58. C. R. Ji and H. M. Choi, Fizika B8, 321 (1999). 59. L. Okun and I. Yu. Kobzarev, ZhETF, 43 1904 (1962) ( English translation : J E T P 16 1343 (1963)); L. Okun, in proceedings of the International Conference on Elementary Particles, 4th, Heidelberg, Germany (1967). Edited by H. Filthuth. North-Holland, (1968). 60. X. Ji, hep-ph/9610369. 61. X. Ji, Phys. Rev. Lett. 78, 610 (1997), hep-ph/9603249. 62. X. Ji, Phys. Rev. D55, 7114 (1997), hep-ph/9609381. 63. O. V. Teryaev, hep-ph/9904376. 64. A. Harindranath and R. Kundu, Phys. Rev. D59, 116013 (1999), hep-ph/9802406. 65. S. Chang, R. G. Root and T. Yan, Phys. Rev. D 7, 1133 (1973). 66. M. Burkardt, Nucl. Phys. A504, 762 (1989). 67. H.-M. Choi and C.-R. Ji, Phys. Rev. D 58, 071901 (1998). 68. K. Hornbostel, S. J. Brodsky and H. C. Pauli, Phys. Rev. D 41, 3814 (1990). 69. F. Antonuccio and S. Dalley, Phys. Lett. B348, 55 (1995) [hepth/9411204]. 70. S. J. Brodsky, F. E. Close and J. F. Gunion, Phys. Rev. D 5, 1384 (1972). 71. S. J. Brodsky, F. E. Close and J. F. Gunion, Phys. Rev. D 6, 177 (1972). 72. S. J. Brodsky, F. E. Close and J. F. Gunion, Phys. Rev. D 8, 3678 (1973). 73. M. Diehl, T. Gousset and B. Pire, Phys. Rev. D 62, 073014 (2000) [hepph/0003233]. 74. A. V. Radyushkin, Phys. Lett. B380, 417 (1996) [hep-ph/9604317]. 75. X. Ji and J. Osborne, Phys. Rev. D 58, 094018 (1998) [hep-ph/9801260]. 76. A. V. Radyushkin, Phys. Rev. D 56, 5524 (1997) [hep-ph/9704207].

40

77. A. V. Radyushkin, Phys. Rev. D 59, 014030 (1999) [hep-ph/9805342]. 78. P. A. Guichon and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 41, 125 (1998) [hep-ph/9806305]. 79. M. Vanderhaeghen, P. A. Guichon and M. Guidal, Phys. Rev. Lett. 80, 5064 (1998). 80. J. C. Collins and A. Freund, Phys. Rev. D 59, 074009 (1999) [hepph/9801262]. 81. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Phys. Lett. B460, 204 (1999) [hep-ph/9903268]. 82. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur. Phys. J. C8, 409 (1999) [hep-ph/9811253]. 83. J. Blumlein and D. Robaschik, Nucl. Phys. B 5 8 1 , 449 (2000) [hepph/0002071]. 84. M. Penttinen, M. V. Polyakov, A. G. Shuvaev and M. Strikman, Phys. Lett. B491, 96 (2000) [hep-ph/0006321]. 85. D. Muller, D. Robaschik, B. Geyer, F. M. Dittes and J. Horejsi, Fortsch. Phys. 42, 101 (1994) [hep-ph/9812448]. 86. G. P. Lepage and S. J. Brodsky, Phys. Lett. B 87, 359 (1979). 87. J. C. Collins, D. E. Soper and G. Sterman, ITP-SB-89-31 in Perturbative Quantum Chromodynamics, A. H. Mueller, Ed. (World Scientific, 1989). 88. S. J. Brodsky, representation," SLAC-PUB-8649. 89. S. D. Bass, S. J. Brodsky and I. Schmidt, Phys. Rev. D 60, 034010 (1999) [hep-ph/9901244]. 90. S. J. Brodsky and G. R. Farrar, Phys. Rev. D 11, 1309 (1975). 91. G. P. Lepage and S. J. Brodsky, Phys. Rev. Lett. 43, 545 (1979). 92. H. Li, gamma (pi) 1 anti-nu," hep-ph/0012140. 93. S. J. Brodsky and G. P. Lepage, Phys. Rev. D 24, 2848 (1981). 94. S. J. Brodsky, C. Ji, A. Pang and D. G. Robertson, Phys. Rev. D 57, 245 (1998) [hep-ph/9705221]. 95. S. J. Brodsky and G. P. Lepage, in C81-04-06.1.4 Phys. Rev. D 24, 1808 (1981). 96. E. Braaten and S. Tse, Phys. Rev. D 35, 2255 (1987). 97. J. Gronberg et al. [CLEO Collaboration], Phys. Rev. D57, 33 (1998), hep-ex/9707031; and H. Paar, presented at PHOTON 2000: International Workshop on Structure and Interactions of the Photon Ambleside, Lake District, England, 26-31 Aug 2000. 98. G. R. Farrar and H. Zhang, Phys. Rev. Lett. 65, 1721 (1990). 99. Paar, H., et al. CLEO collaboration (to be published); See also Boyer, J. et al., Phys. Rev. Lett. 56, 207 (1980); TPC/Two Gamma Collaboration (H. Aihara et al.), Phys. Rev. Lett. 57,404 (1986).

41

100. T. C. Brooks and L. Dixon, Phys. Rev. D 62, 114021 (2000) [hepph/0004143]. 101. A. V. Radyushkin, Acta Phys. Polon. B26, 2067 (1995) [hepph/9511272]. 102. S. Ong, Phys. Rev. D 52, 3111 (1995). 103. P. Kroll and M. Raulfs, Phys. Lett. B387, 848 (1996) [hep-ph/9605264]. 104. V. L. Chemyak and A. R. Zhitnitsky, Phys. Rept. 112, 173 (1984). 105. N. Isgur and C. H. Llewellyn Smith, Phys. Lett. B217, 535 (1989). 106. A. V. Radyushkin, Phys. Rev. D 58, 114008 (1998) [hep-ph/9803316]. 107. J. Bolz and P. Kroll, Z. Phys. A356, 327 (1996) [hep-ph/9603289]. 108. C. Vogt, hep-ph/0010040. 109. R. L. Anderson et al, Phys. Rev. Lett. 30, 627 (1973). 110. H. M. Sommermann, R. Seki, S. Larson and S. E. Koonin, Phys. Rev. D 45, 4303 (1992). 111. S. Brodsky and M. Karliner, in preparation. 112. E. M. Aitala et al. [E791 Collaboration], hep-ex/0010044. 113. E. M. Aitala et al. [E791 Collaboration], pion light-cone wave function squared," hep-ex/0010043. 114. S. Brodsky, M. Diehl, P. Hoyer, and S. Peigne, in preparation. 115. P. Stoler, Few Body Syst. Suppl. 11, 124 (1999). 116. G. A. Miller, nucl-th/9910053. 117. G. A. Miller, S. J. Brodsky and M. Karliner, Phys. Lett. B481, 245 (2000) [hep-ph/0002156]. 118. M. Franz, M. V. Polyakov and K. Goeke, Phys. Rev. D 62, 074024 (2000) [hep-ph/0002240]. 119. S. J. Brodsky, C. Peterson and N. Sakai, Phys. Rev. D 23, 2745 (1981). 120. C. V. Chang and W. Hou, in B Meson," hep-ph/0101162. 121. S. J. Brodsky and I. A. Schmidt, Phys. Lett. B234, 144 (1990). 122. S. J. Brodsky and M. Karliner, Phys. Rev. Lett. 78, 4682 (1997) [hepph/9704379]. 123. S. J. Brodsky, G. P. Lepage and S. F. Tuan, Phys. Rev. Lett. 59, 621 (1987). 124. H. C. Pauli and S. J. Brodsky, Phys. Rev. D 32, 2001 (1985). 125. S. Dalley and I. R. Klebanov, Phys. Rev. D 47, 2517 (1993) [hepth/9209049]. 126. F. Antonuccio and S. Dalley, Phys. Lett. B376, 154 (1996) [hepth/9512106]. 127. F. Antonuccio and S. Dalley, Nucl. Phys. B461, 275 (1996) [hepph/9506456]. 128. S. J. Brodsky, J. R. Hiller and G. McCartor, Phys. Rev. D 58, 025005

42

(1998) [hep-th/9802120]. 129. S. J. Brodsky, J. R. Hiller and G. McCartor, Phys. Rev. D 60, 054506 (1999) [hep-ph/9903388]. 130. S. Dalley and B. van de Sande, Phys. Rev. D 62, 014507 (2000) [heplat/9911035]. 131. J. R. Hiller and S. J. Brodsky, Phys. Rev. D 59, 016006 (1999) [hepph/9806541]. 132. F. Antonuccio, I. Filippov, P. Haney, O. Lunin, S. Pinsky, U. Trittmann and J. Hiller [SDLCQ Collaboration], hep-th/9910012. Brodsky: 1999xj 133. O. Lunin and S. Pinsky, hep-th/9910222. 134. P. Haney, J. R. Hiller, O. Lunin, S. Pinsky and U. Trittmann, Phys. Rev. D 62, 075002 (2000) [hep-th/9911243]. 135. G. P. Lepage, S. J. Brodsky, T. Huang and P. B. Mackenzie, CLNS82/522, published in Banff Summer Inst.l981:0083 (QCD161:B23:1981); S. J. Brodsky, T. Huang and G. P. Lepage, In *Banff 1981, Proceedings, Particles and Fields 2*, 143-199. 136. D. Daniel, R. Gupta and D. G. Richards, Phys. Rev. D 43, 3715 (1991). 137. L. Del Debbio, M. Di Pierro, A. Dougall and C. Sachrajda [UKQCD collaboration], Nucl. Phys. Proc. Suppl. 83, 235 (2000) [hep-lat/9909147]. 138. S. Dalley, hep-ph/0007081. 139. W. A. Bardeen and R. B. Pearson, Phys. Rev. D 14, 547 (1976). 140. M. Burkardt, Phys. Rev. D 54, 2913 (1996) [hep-ph/9601289]. 141. V. Y. Petrov, M. V. Polyakov, R. Ruskov, C. Weiss and K. Goeke, Phys. Rev. D 59, 114018 (1999) [hep-ph/9807229]. 142. D. Diakonov and V. Y. Petrov, hep-ph/0009006.

LECTURES O N T H E T H E O R Y OF N O N - L E P T O N I C B DECAYS MATTHIAS NEUBERT Newman Laboratory of Nuclear Studies, Cornell University Ithaca, NY 14853, USA E-mail: [email protected] These notes provide a pedagogical introduction to the theory of non-leptonic heavymeson decays recently proposed by Beneke, Buchalla, Sachrajda and myself. We provide a rigorous basis for factorization for a large class of non-leptonic two-body B-meson decays in the heavy-quark limit. The resulting factorization formula incorporates elements of the naive factorization approach and the hard-scattering approach, and allows us to compute systematically radiative ("non-factorizable") corrections to naive factorization for decays such as B —• DTT and B —• 7T7T.

1

Introduction

Non-leptonic two-body decays of B mesons, although simple as far as the underlying weak decay of the b quark is concerned, are complicated on account of strong-interaction effects. If these effects could be computed, this would enhance tremendously our ability to uncover the origin of CP violation in weak interactions from data on a variety of such decays being collected at the B factories. In these lecture, I review recent progress towards a systematic analysis of weak heavy-meson decays into two energetic mesons based on the factorization properties of decay amplitudes in QCD l i 2 . My discussion will follow very closely the detailed account of this approach given in 2 . (We have worked so hard on this paper that any attempt to improve on it were bound to fail and leave the author in despair.) Much of the credit for these notes belongs to my collaborators Martin Beneke, Gerhard Buchalla, and Chris Sachrajda. As in the classic analysis of semi-leptonic B ->• D transitions 3 ' 4 , our arguments make extensive use of the fact that the b quark is heavy compared to the intrinsic scale of strong interactions. This allows us to deduce that nonleptonic decay amplitudes in the heavy-quark limit have a simple structure. The arguments to reach this conclusion, however, are quite different from those used for semi-leptonic decays, since for non-leptonic decays a large momentum is transferred to at least one of the final-state mesons. The results of our work justify naive factorization of four fermion operators for many, but not all, nonleptonic decays and imply that corrections termed "non-factorizable", which up to now have been thought to be intractable, can be calculated rigorously if 43

44

the mass of the decaying quark is large enough. This leads to a large number of predictions for CP-violating B decays in the heavy-quark limit, for which measurements will soon become available. Weak decays of heavy mesons involve three fundamental scales, the weakinteraction scale M\y, the 6-quark mass m&, and the QCD scale AQCD, which are strongly ordered: Mw > mj, » AQCD- The underlying weak decay being computable, all theoretical work concerns strong-interaction corrections. QCD effects involving virtualities above the scale mt are well understood. They renormalize the coefficients of local operators 0, in the effective weak Hamiltonian 5 , so that the amplitude for the decay B ->• M\M2 is given by

A(B -+ MXM2) = ^

J2 A« C*M (M1M2\Oi(^)\B),

(1)

where each term in the sum is the product of a Cabibbo-Kobayashi-Maskawa (CKM) factor A,, a coefficient function Cj(/u), which incorporates stronginteraction effects above the scale p ~ mi, and a matrix element of an operator Oj. The difficult theoretical problem is to compute these matrix elements or, at least, to reduce them to simpler non-perturbative objects. A variety of treatments of this problem exist, which rely on assumptions of some sort. Here we identify two somewhat contrary lines of approach. The first one, which we shall call "naive factorization", replaces the matrix element of a four-fermion operator in a heavy-quark decay by the product of the matrix elements of two currents 6 , 7 ; e.g. (D+Tr-\(cb)v-A(du)V-A\Bd)

-»• (Tr-\(du)V-A\0)

(D+\(cb)V-A\Bd)

•

(2)

This assumes that the exchange of "non-factorizable" gluons between the n~ and the (B^ D+) system can be neglected if the virtuality of the gluons is below fj, ~ mi,. The non-leptonic decay amplitude then reduces to the product of a form factor and a decay constant. This assumption is in general not justified, except in the limit of a large number of colours in some cases. It deprives the amplitude of any physical mechanism that could account for rescattering in the final state. "Non-factorizable" radiative corrections must also exist, because the scale dependence of the two sides of (2) is different. Since such corrections at scales larger than fj, are taken into account in deriving the effective weak Hamiltonian, it appears rather arbitrary to leave them out below the scale fi. Various generalizations of the naive factorization approach have been proposed, which include new parameters that account for non-factorizable corrections. In their most general form, these generalizations have nothing to do with the original "factorization" ansatz, but amount to

45

a general parameterization of the matrix elements. Such general parameterizations are exact, but at the price of introducing many unknown parameters and eliminating any theoretical input on strong-interaction dynamics. The second method used to study non-leptonic decays is the hard-scattering approach, which assumes the dominance of hard gluon exchange. The decay amplitude is then expressed as a convolution of a hard-scattering factor with light-cone wave functions of the participating mesons, in analogy with more familiar applications of this method to hard exclusive reactions involving only light hadrons 8 ' 9 . In many cases, the hard-scattering contribution represents the leading term in an expansion in powers of AQCD/<3, where Q denotes the hard scale. However, the short-distance dominance of hard exclusive processes is not enforced kinematically and relies crucially on the properties of hadronic wave functions. There is an important difference between light mesons and heavy mesons in this regard, because the light quark in a heavy meson at rest naturally has a small momentum of order AQCD, while for fast light mesons a configuration with a soft quark is suppressed by the endpoint behaviour of the meson wave function. As a consequence, the soft (or Feynman) mechanism is power suppressed for hard exclusive processes involving light mesons, but it is of leading power for heavy-meson decays. It is clear from this discussion that a satisfactory treatment should take into account soft contributions, but also allow us to compute corrections to naive factorization in a systematic way. It is not at all obvious that such a treatment would result in a predictive framework. We will show that this does indeed happen for most non-leptonic two-body B decays. Our main conclusion is that "non-factorizable" corrections are dominated by hard gluon exchange, while the soft effects that survive in the heavy-quark limit are confined to the (BMi) system, where M% denotes the meson that picks up the spectator quark in the B meson. This result is expressed as a factorization formula, which is valid up to corrections suppressed by powers of AQCD/'mb- At leading power, non-perturbative contributions are parameterized by the physical form factors for the B —» M\ transition and leading-twist light-cone distribution amplitudes of the mesons. Hard perturbative corrections can be computed systematically in a way similar to the hard-scattering approach. On the other hand, because the B -> Mx transition is parameterized by a form factor, we recover the result of naive factorization at lowest order in as. An important implication of the factorization formula is that strong rescattering phases are either perturbative or power suppressed in AQCD/W;,. It is worth emphasizing that the decoupling of M 2 occurs in the presence of soft interactions in the {BMX) system. In other words, while strong-interaction effects in the B —» Mi transition are not confined to small transverse dis-

46

tances, the other meson M 2 is predominantly produced as a compact object with small transverse extension. The decoupling of soft effects then follows from "colour transparency". The colour-transparency argument for exclusive B decays has already been noted in the literature 1 0 ' u , but it has never been developed into a factorization formula that could be used to obtain quantitative predictions. The approach described in 1,:2 is general and applies to decays into a heavy and a light meson (such as B —> Dir) as well as to decays into two light mesons (such as B -* -KIT). Factorization does not hold, however, for decays such as B —> irD and B —• DD, in which the meson that does not pick up the spectator quark in the B meson is heavy. For the main part in these lectures, we will focus on the case of B -»• D^L decays (with L a light meson), for which the factorization formula takes its simplest form, and power counting will be relatively straightforward. Occasionally, we will point out what changes when we consider more complicated decays such as B —> 7T7r. A detailed treatment of these processes can be found in 12 . The outline of these notes is as follows: In Sect. 2 we state the factorization formula in its general form. In Sect. 3 we collect the physical arguments that lead to factorization and introduce our power-counting scheme. We show how light-cone distribution amplitudes enter, discuss the heavy-quark scaling of the B —> D form factor, and explain the cancellation of soft and collinear contributions in "non-factorizable" vertex corrections to non-leptonic decay amplitudes. We also comment on the implications of our results for finalstate interactions in hadronic B decays. The cancellation of long-distance singularities is demonstrated in more detail in Sect. 4, where we present the calculation of the hard-scattering functions at one-loop order for decays into a heavy and a light meson. Various sources of power-suppressed effects, which give corrections to the factorization formula, are discussed in Sect. 5. They include hard-scattering contributions, weak annihilation, and contributions from multi-particle Fock states. We then point out some limitations of the factorization approach. In Sect. 7 we consider the phenomenology of B -¥ D^L decays on the basis of the factorization formula and discuss various tests of our theoretical framework. We also examine to what extent a charm meson should be considered as heavy or light. Section 8 contains the conclusion. 2

Statement of the factorization formula

In this section we summarize the factorization formula for non-leptonic B decays. We introduce relevant terminology and provide definitions of the hadronic quantities that enter the factorization formula as input parameters.

47

2.1

The idea of factorization

In the context of non-leptonic decays the term "factorization" is usually applied to the approximation of the matrix element of a four-fermion operator by the product of a form factor and a decay constant, as illustrated in (2). Corrections to this approximation are called "non-factorizable". We will refer to this approximation as "naive factorization" and use quotes on "non-factorizable" to avoid confusion with the (much less trivial) meaning of factorization in the context of hard processes in QCD. In the latter case, factorization refers to the separation of long-distance contributions to the process from a shortdistance part that depends only on the large scale m&. The short-distance part can be computed in an expansion in the strong coupling as(mb). The long-distance contributions must be computed non-perturbatively or determined experimentally. The advantage is that these non-perturbative parameters are often simpler in structure than the original quantity, or they are process independent. For example, factorization applied to hard processes in inclusive hadron-hadron collisions requires only parton distributions as non-perturbative inputs. Parton distributions are much simpler objects than the original matrix element with two hadrons in the initial state. On the other hand, factorization applied to the B —> D form factor leads to a nonperturbative object (the "Isgur-Wise function"), which is still a function of the momentum transfer. However, the benefit here is that symmetries relate this function to other form factors. In the case of non-leptonic B decays, the simplification is primarily of the first kind (simpler structure). We call those effects non-factorizable (without quotes) which depend on the long-distance properties of the B meson and both final-state mesons combined. The factorization properties of non-leptonic decay amplitudes depend on the two-meson final state. We call a meson "light" if its mass m remains finite in the heavy-quark limit. A meson is called "heavy" if its mass scales with m\, in the heavy-quark limit, such that m/mb stays fixed. In principle, we could still have m >• AQCD for a light meson. Charm mesons could be considered as light in this sense. However, unless otherwise mentioned, we assume that m is of order AQCD for a light meson, and we consider charm mesons as heavy. In evaluating the scaling behaviour of the decay amplitudes, we assume that the energies of both final-state mesons (in the -B-meson rest frame) scale with mi, in the heavy-quark limit. 2.2

The factorization

formula

We consider a generic weak decay B —• M\Mi in the heavy-quark limit and differentiate between decays into final states containing a heavy and a light

48

Figure 1. Graphical representation of the factorization formula. Only one of the two formfactor terms in (3) is shown for simplicity.

meson or two light mesons. Our goal is to show that, up to power corrections of order AQCD/WJ,, the transition matrix element of an operator 0\ in the effective weak Hamiltonian can be written as

{M1M2\Oi\B) = TF^^imDfM,

f dul$(u) **,(«) J

3

°

if Mi is heavy and M 2 is light,

(M1M2\Oi\B) = Y^Ff^imDfM,

f CIUT^QMAU)

3

+ fisfMjM2

+ (Mi ^ M2)

J

°

/ d^dudvTf'i^^v)

Jo

<M£)$ M l (v)$M 2 (U)

if Mi and M 2 are both light. M

2

(3) 2

2

Here i ^ " ^ ( m ) denotes a B - > M form factor evaluated at q = rri , mi, 2 are the light meson masses, and $ x (u) is the light-cone distribution amplitude for the quark-antiquark Fock state of the meson X. These non-perturbative quantities will be defined below. Ty(u) and T''{^,u,v) are hard-scattering functions, which are perturbatively calculable. The factorization formula in its general form is represented graphically in Fig. 1. The second equation in (3) applies to decays into two light mesons, for which the spectator quark in the B meson (in the following simply referred to as the "spectator quark") can go to either of the final-state mesons. An example is the decay B~ -> ir°K~. If the spectator quark can go only to

49

one of the final-state mesons, as for example in B^ —• ir+K~, we call this meson Mi, and the second form-factor term on the right-hand side of (3) is absent. The formula simplifies when the spectator quark goes to a heavy meson (first equation in (3)), such as in B^ —> D+n~. Then the second term in Fig. 1, which accounts for hard interactions with the spectator quark, can be dropped because it is power suppressed in the heavy-quark limit. In the opposite situation that the spectator quark goes to a light meson but the other meson is heavy, factorization does not hold, because the heavy meson is neither fast nor small and cannot be factorized from the B —> Mi transition. Finally, notice that annihilation topologies do not appear in the factorization formula, since they do not contribute at leading order in the heavy-quark expansion. Any hard interaction costs a power of as. As a consequence, the hardspectator term in the second formula in (3) is absent at order a ° . Since at this order the functions Tfj(u) are independent of u, the convolution integral results in the normalization of the meson distribution amplitude, and (3) reproduces naive factorization. The factorization formula allows us to compute radiative corrections to this result to all orders in aB. Further corrections are suppressed by powers of AQCD/"1|, in the heavy-quark limit. The significance and usefulness of the factorization formula stems from the fact that the non-perturbative quantities appearing on the right-hand side of the two equations in (3) are much simpler than the original non-leptonic matrix elements on the left-hand side. This is because they either reflect universal properties of a single meson (light-cone distribution amplitudes) or refer only t o a B - > meson transition matrix element of a local current (form factors). While it is extremely difficult, if not impossible 13 , to compute the original matrix element (MiM 2 |0;|.B) in lattice QCD, form factors and lightcone distribution amplitudes are already being computed in this way, although with significant systematic errors at present. Alternatively, form factors can be obtained using data on semi-leptonic decays, and light-cone distribution amplitudes by comparison with other hard exclusive processes. After having presented the most general form of the factorization formula, we will from now on restrict ourselves to the case of heavy-light final states. Then the (simpler) first formula in (3) applies, and only the first term shown in Fig. 1 is present at leading power. 2.3

Definition of non-perturbative parameters

The form factors Ff~*M(q2) in (3) arise in the decomposition of current matrix elements of the form {M($f)\qTb\B(p)), where T can be any irreducible

50

Dirac matrix that appears after contraction of the hard subgraph to a local vertex with respect to the B -» M transition. We will often refer to the matrix element of the vector current evaluated between a B meson and a pseudoscalar meson P, which is conventionally parameterized as (Ptf)\rfb\B{p))

= F^p(q2)

(p" + p"*)

+ [F0B^P(q>) - F^p(q2)}

"^

~™p q» ,

(4)

where q = p—p', and F ^ ^ p ( 0 ) = F^p(0) at zero momentum transfer. Note that we write (3) in terms of physical form factors. In principle, Fig. 1 could be looked upon in two different ways. We could suppose that the region represented by Fj accounts only for the soft contributions to the B —>• M\ form factor. The hard contributions to the form factor would then be considered as part of T?j (or as part of the second diagram). Performing this split-up would require that one understands the factorization of hard and soft contributions to the form factor. If Mi is heavy, this amounts to matching the form factor onto a form factor defined in heavy-quark effective theory 14 . However, for a light meson Mi the factorization of hard and soft contributions to the form factor is not yet completely understood. We bypass this problem by interpreting Fj as the physical form factor, including hard and soft contributions. This avoids the above problem, and in addition has the advantage that the physical form factors are directly related to measurable quantities. Light-cone distribution amplitudes play the same role for hard exclusive processes that parton distributions play for inclusive processes. As in the latter case, the leading-twist distribution amplitudes, which are the ones we need at leading power in the 1/m.b expansion, are given by two-particle operators with a certain helicity structure. The helicity structure is determined by the angular momentum of the meson and the fact that the spinor of an energetic quark has only two large components. The leading-twist light-cone distribution amplitudes for pseudoscalar mesons (P) and longitudinally polarized vector mesons (Vj|) with flavour content (qq1) are defined as (P(q)\q(y)aq'(x)0\O)

= *-£ (jfrsW f

{V\\{q)m)a
= -\if>a

due^"x+u^

fdue^o*^

<Mu,/z), $||(u,M),

(5)

where (x - y)2 = 0. We have suppressed the path-ordered exponentials that connect the two quark fields at different positions and make the light-cone

51

operators gauge invariant. The equality sign is to be understood as "equal up to higher-twist terms". It is also understood that the operators on the lefthand side are colour singlets. When convenient, we use the "bar"-notation u = 1 — u. The parameter fj, is the renormalization scale of the light-cone operators.on the left-hand side. The distribution amplitudes are normalized as J 0 du $ x (u, fj) = 1 with X = P, Vj|. One defines the asymptotic distribution amplitude as the limit in which the renormalization scale is sent to infinity. In this case ^(«,/i)"=°°Ml-n).

(6)

The use of light-cone distribution amplitudes in non-leptonic B decays requires justification, which we will provide in Sects. 3 and 4. The decay amplitude for a B decay into a heavy-light final state is then calculated by assigning momenta uq and uq to the quark and antiquark in the outgoing light meson (with momentum q), writing down the on-shell amplitude in momentum space, and performing the replacement f1

if Uaa(uq)

T(u,.

. .)a/3,abVl3b{uq)

->• ~ -

/

du $ P ( u ) ( # y 5 ) / 3 a T ( u , . . .)a0taa

(7)

for pseudoscalars and, with obvious modifications, for vector mesons. (Even when working with light-cone distribution amplitudes it is not always justified to perform the collinear approximation on the external quark and antiquark lines right away. One may have to keep the transverse components of the quark and antiquark momenta until after some operations on the amplitude have been carried out. However, these subtleties do not concern calculations at leading-twist order.) 3

Arguments for factorization

In this section we provide the basic power-counting arguments that lead to the factorized structure shown in (3). We do so by analyzing qualitatively the hard, soft and collinear contributions to the simplest Feynman diagrams. 3.1

Preliminaries and power counting

For concreteness, we label the charm meson which picks up the spectator quark by Mx = D+ and assign momentum p' to it. The light meson is labeled Mi = n~ and assigned momentum q = En+, where E is the pion energy in the B rest frame, and n± = (1,0,0, ±1) are four-vectors on the light-cone. At leading power, we neglect the mass of the light meson.

52

The simplest diagrams that we can draw for a non-leptonic decay amplitude assign a quark and antiquark to each meson. We choose the quark and antiquark momenta in the pion as

f2 lq=uq

+ l± + 4 ^ n - ,

f2

_ l9 = a q - l

±

+ ^ n - .

(8)

2

Note that q^ lq + lq, but the off-shellness (lq + lq) is of the same order as the light meson mass, which we can neglect at leading power. A similar decomposition (with longitudinal momentum fraction v and transverse momentum l'±) is used for the charm meson. To prove the factorization formula (3) for the case of heavy-light final states, one has to show that: i) There is no leading (in powers of AQCD/WJ,) contribution to the amplitude from the endpoint regions u ~ AQCD/WIJ, and u ~ AQCD/WI6ii) One can set l± = 0 in the amplitude (more generally, expand the amplitude in powers of l±) after collinear subtractions, which can be absorbed into the pion wave function. This, together with i), guarantees that the amplitude is legitimately expressed in terms of the light-cone distribution amplitudes of pion. iii) The leading contribution comes from v ~ A Q C D / " 1 6 (the region where the spectator quark enters the charm meson as a soft parton), which guarantees the absence of a hard spectator interaction term. iv) After subtraction of infrared contributions corresponding to the lightcone distribution amplitude and the form factor, the leading contributions to the amplitude come only from internal lines with virtuality that scales with rribv) Non-valence Fock states are non-leading. The requirement that after subtractions virtualities should be large is obvious to guarantee the infrared finiteness of the hard-scattering functions Ty. Let us comment on setting transverse momenta in the wave functions to zero and on endpoint contributions. Neglecting transverse momenta requires that we count them as order AQCD when comparing terms of different magnitude in the scattering amplitude. This conforms to our intuition and the assumption of the parton model, that intrinsic transverse momenta are limited to hadronic scales. However, in QCD transverse momenta are not limited, but logarithmically distributed up to the hard scale. The important point is that

53

contributions that violate the starting assumption of limited transverse momentum can be absorbed into the universal light-cone distribution amplitudes. The statement that transverse momenta can be counted of order AQCD is to be understood after these subtractions have been performed. The second comment concerns endpoint contributions in the convolution integrals over longitudinal momentum fractions. These contributions are dangerous, because we may be able to demonstrate the infrared safety of the hardscattering amplitude under assumption of generic u and independent of the shape of the meson distribution amplitude, but for u —> 0 or u —> 1 a propagator that was assumed to be off-shell approaches the mass-shell. If such a contribution were of leading power, we would not expect the perturbative calculation of the hard-scattering functions to be reliable. Estimating endpoint contributions requires knowledge of the endpoint behaviour of the light-cone distribution amplitude. Since it enters the factorization formula at a renormalization scale of order mj,, we can use the asymptotic form (6) to estimate the endpoint contribution. (More generally, we only have to assume that the distribution amplitude at a given scale has the same endpoint behaviour as the asymptotic amplitude. This is generally the case, unless there is a conspiracy of terms in the Gegenbauer expansion of the distribution amplitude. If such a conspiracy existed at some scale, it would be destroyed by evolving the distribution amplitude to a different scale.) We count a light-meson distribution amplitude as order AQCD/"^& in the endpoint region (defined as the region the quark or antiquark momentum is of order AQCD)I and order 1 away from the endpoint, i.e. (for X = P, Vy) x. r \

1;

I

generics,

[ AQCD/TO6 ;

U, U ~ AQCD/m;,.

Note that the endpoint region has a size of order A Q C D / T I J , SO that the endpoint suppression is ~ (AQCD/rot) 2 . This suppression has to be weighted against potential enhancements of the partonic amplitude when one of the propagators approaches the mass shell. The counting for B mesons, or heavy mesons in general, is different. Naturally, the heavy quark carries almost all of the meson momentum, and hence we count *B(0~(m*/AQCD;

[

0;

f-AQCD/m*.

(1Q)

f ~ 1.

The zero probability for a light spectator with momentum of order m^ must be understood as a boundary condition for the wave function renormalized at a scale much below m&. There is a small probability for hard fluctuations that

54

o o o

(a)

(b)

Figure 2. Leading contributions to the B -> D form factor in the hard-scattering approach. The dashed line represents the weak current. The two lines to the left belong to the B meson, the ones to the right to the recoiling charm meson.

transfer large momentum to the spectator. This "hard tail" is generated by evolution of the wave function from a hadronic scale to a scale of order mj,. If we assume that the initial distribution at the hadronic scale falls sufficiently rapidly for £ » AQCD/TTH, this remains true after evolution. We shall assume a sufficiently fast fall-off, so that, for the purposes of power counting, the probability that the spectator-quark momentum is of order mj, can be set to zero. The same counting applies to the D meson. (Despite the fact that the charm meson has momentum of order mi,, we do not need to distinguish the rest frames of B and D for the purpose of power counting, because the two frames are not connected by a parametrically large boost. In other words, the components of the spectator quark in the D meson are still of order AQCD-) 3.2

The B -+ D form factor

We now demonstrate that the B -» D form factor receives a leading contribution from soft gluon exchange. This implies that a non-leptonic decay cannot be treated completely in the hard-scattering picture, and so the form factor should enter the factorization formula as a non-perturbative quantity. Consider the diagrams shown in Fig. 2. When the exchanged gluon is hard the spectator quark in the final state has momentum of order m\,. But according to the counting rule (10) this configuration has no overlap with the D-meson wave function. On the other hand, there is no suppression for soft gluons in Fig. 2. It follows that the dominant behaviour of the B -» D form factor in the heavy-quark limit is given by soft processes. Because of this argument, we can exploit the heavy-quark symmetries to determine how the form factor scales in the heavy-quark limit. The wellknown result is that the form factor scales like a constant (modulo logarithms), since it is equal to one at zero velocity transfer and independent of mj as long

55

as the Lorentz boost that connects the B and D rest frames is of order 1. The same conclusion follows from the power-counting rules for light-cone wave functions. To see this, we represent the form factor by an overlap integral of wave functions (not integrated over transverse momentum),

* V ( 0 ) ~ / ^ ^ ffntt,*x) ^(^(O.fcj.),

(11)

where £' (£) is fixed by kinematics, and we have set q2 = 0 for simplicity. The probability of finding the B meson in its valence Fock state is of order 1 in the heavy-quark limit, i.e. / ^ | * B , ^ , * X ) |

2

~ 1 .

(12)

Counting k\_ ~ AQCD and d£ ~ AQCD/"1(„ we deduce that V&B(£, A;J_) ~ m l M Q C D - F r o m (11)) w e then obtain the scaling law F+j?D(Q) ~ 1, in agreement with the prediction of heavy-quark symmetry. The representation (11) of the form factor as an overlap of wave functions for the two-particle Fock state of the heavy meson is not rigorous, because there is no reason to assume that the contribution from higher Fock states with additional soft gluons is suppressed. The consistency with the estimate based on heavy-quark symmetry shows that these additional contributions are not larger than the two-particle contribution. 3.3

Non-leptonic decay amplitudes

We now turn to a qualitative discussion of the lowest-order and one-gluon exchange diagrams that could contribute to the hard-scattering kernels T^ (u) in (3). In the figures below, the two lines directed upwards represent TT - , the lines on the left represent Bd, and the lines on the right represent D+. Lowest-order diagram There is a single diagram with no hard gluon interactions shown in Fig. 3. According to (10) the spectator quark is soft, and since it does not undergo a hard interaction it is absorbed as a soft quark by the recoiling meson. This is evidently a contribution to the left-hand diagram of Fig. 1, involving the B -» D form factor. The hard subprocess in Fig. 3 is just given by the insertion of a four-fermion operator, and hence it does not depend on the longitudinal momentum fraction u of the two quarks that form the emitted TT - . Consequently, the lowest-order contribution to T^{u) in (3) is independent of u, and the u-integral reduces to the normalization condition for the pion

56

A/ Figure 3. Leading-order contribution to the hard-scattering kernels T?Au). The weak decay of the b quark through a four-fermion operator is represented by the black square.

distribution amplitude. The result is, not surprisingly, that the factorization formula reproduces the result of naive factorization if we neglect gluon exchange. Note that the physical picture underlying this lowest-order process is that the spectator quark (which is part of the B -»• D form factor) is soft. If this is the case, the hard-scattering approach misses the leading contribution to the non-leptonic decay amplitude. Putting together all factors relevant to power counting, we find that in the heavy-quark limit the decay amplitude for a decay into a heavy-light final state (in which the spectator quark is absorbed by the heavy meson) scales as A(Bd -> D+7T-) ~ GFm2b FB^D(0)

/„ ~ GFm2b A Q C D .

(13)

Other contributions must be compared with this scaling rule. Factorizable diagrams In order to justify naive factorization as the leading term in an expansion in a g and AQCD/W&, we must show that radiative corrections are either suppressed in one of these two parameters, or already contained in the definition of the form factor and the pion decay constant. Consider the graphs shown in Fig. 4. The first three diagrams are part of the form factor and do not contribute to the hard-scattering kernels. Since the first and third diagrams contain leading contributions from the region in which the gluon is soft, they should not be considered as corrections to Fig. 3. However, this is of no consequence since these soft contributions are absorbed into the physical form factor. The fourth diagram in Fig. 4 is also factorizable. In general, this graph would split into a hard contribution and a contribution to the evolution of the pion distribution amplitude. However, as the leading-order diagram in Fig. 3 involves only the normalization integral of the pion distribution amplitude, the sum of the fourth diagram in Fig. 4 and the wave-function renormalization of the quarks in the emitted pion vanishes. In other words, these diagrams would renormalize the (ud) light-quark current, which however is conserved.

57

X>OflS^

Figure 4. Diagrams at order aa that need not be calculated.

"Non-factorizable" vertex corrections We now begin the analysis of "non-factorizable" diagrams, i.e. diagrams containing gluon exchanges that cannot be associated with the B -> D form factor or the pion decay constant. At order a g , these diagrams can be divided into three groups: vertex corrections, hard spectator interactions, and annihilation diagrams. The vertex corrections shown in Fig. 5 violate the naive factorization ansatz (2). One of the key observations made in l i 2 is that these diagrams are calculable nonetheless. Let us summarize the argument here, postponing the explicit evaluation of these diagrams to Sect. 4. The statement is that the vertex-correction diagrams form an order-a s contribution to the hardscattering kernels Tfj{u). To demonstrate this, we have to show that: i) The transverse momentum of the quarks that form the pion can be neglected at leading power, i.e. the two momenta in (8) can be approximated by uq and uq, respectively. This guarantees that only a convolution in the longitudinal momentum fraction u appears in the factorization formula, ii) The contribution from the soft-gluon region and gluons collinear to the direction of the pion is power suppressed. In practice, this means that the sum of these diagrams cannot contain any infrared divergences at leading power in A Q C D / " H . Neither of the two conditions holds true for any of the four diagrams individually, as each of them separately contains collinear and infrared divergences. As will be shown in detail later, the infrared divergences cancel when one sums over the gluon attachments to the two quarks comprising the emission pion ((a+b), (c+d) in Fig. 5). This cancellation is a technical manifestation of Bjorken's colour-transparency argument 1 0 , stating that soft gluon interactions with the emitted colour-singlet (ud) pair are suppressed because they interact with the colour dipole moment of the compact light-quark pair. Collinear divergences cancel after summing over gluon attachments to the b and c quark lines ((a+c), (b+d) in Fig. 5). Thus the sum of the four diagrams (a-d) involves only hard gluon exchange at leading power. Because the hard gluons transfer large momentum to the quarks that form the emission pion,

58

JV (a)

W (b)

V^ (c)

\A (d)

Figure 5. "Non-factorizable" vertex corrections.

the hard-scattering factor now results in a non-trivial convolution with the pion distribution amplitude. "Non-factorizable" contributions are therefore non-universal, i.e. they depend on the quantum numbers of the final-state mesons. Note that the colour-transparency argument, and hence the cancellation of soft gluon effects, applies only if the (ud) pair is compact. This is not the case if the emitted pion is formed in a very asymmetric configuration, in which one of the quarks carries almost all of the pion momentum. Since the probability for forming a pion in such an endpoint configuration is of order ( A Q C D / W J , ) 2 , they could become important only if the hard-scattering amplitude favoured the production of these asymmetric pairs, i.e. if T^- ~ 1/u2 for u -» 0 (or T^ ~ 1/u2 for u -t 1). However, we will see that such strong endpoint singularities in the hard-scattering amplitude do not occur. To complete the argument, we have to show that all other types of contributions to the non-leptonic decay amplitudes are power suppressed in the heavy-quark limit. This includes interactions with the spectator quark, weak annihilation graphs, and contributions from higher Fock components of the meson wave functions. This will be done in Sect. 5. In summary, then, for hadronic B decays into a light emitted and a heavy recoiling meson the first factorization formula in (3) holds. At order as, the hard-scattering kernels TlAu) are computed from the diagrams shown in Figs. 3 and 5. Naive factorization follows when one neglects all corrections of order AQCD/"I& and as. The factorization formula allows us to compute systematically corrections to higher order in as, but still neglects power corrections. 3.4

Remarks on final-state interactions

Some of the loop diagrams entering the calculation of the hard-scattering kernels have imaginary parts, which contribute to the strong rescattering phases. It follows from our discussion that these imaginary parts are of order a8 or A Q C D / « V This demonstrates that strong phases vanish in the heavy-quark

59 limit (unless the real parts of the amplitudes are also suppressed). Since this statement goes against the folklore that prevails from the present understanding of this issue, and since the subject of final-state interactions (and of strong-interaction phases in particular) is of paramount importance for the interpretation of CP-violating observables, a few additional remarks are in order. Final-state interactions are usually discussed in terms of intermediate hadronic states. This is suggested by the unitarity relation (taking B -> TTTT for definiteness)

Im AB^,™ ~ Yl *4B-m -4JU™ >

( 14 )

n

where n runs over all hadronic intermediate states. We can also interpret the sum in (14) as extending over intermediate states of partons. The partonic interpretation is justified by the dominance of hard rescattering in the heavy-quark limit. In this limit, the number of physical intermediate states is arbitrarily large. We may then argue on the grounds of parton-hadron duality that their average is described well enough (up to AQCD/W& corrections, say) by a partonic calculation. This is the picture implied by (3). The hadronic language is in principle exact. However, the large number of intermediate states makes it intractable to observe systematic cancellations, which usually occur in an inclusive sum over hadronic intermediate states. A particular contribution to the right-hand side of (14) is elastic rescattering (n = 7T7r). The energy dependence of the total elastic 7T7r-scattering cross section is governed by soft pomeron behaviour. Hence the strong-interaction phase of the B -t mr amplitude due to elastic rescattering alone increases slowly in the heavy-quark limit 1 5 . On general grounds, it is rather improbable that elastic rescattering gives an appropriate representation of the imaginary part of the decay amplitude in the heavy-quark limit. This expectation is also borne out in the framework of Regge behaviour, as discussed in 15 , where the importance (in fact, dominance) of inelastic rescattering was emphasized. However, this discussion left open the possibility of soft rescattering phases that do not vanish in the heavy-quark limit, as well as the possibility of systematic cancellations, for which the Regge approach does not provide an appropriate theoretical framework. Eq. (3) implies that such systematic cancellations do occur in the sum over all intermediate states n. It is worth recalling that similar cancellations are not uncommon for hard processes. Consider the example of e+e~ —»• hadrons at large energy q. While the production of any hadronic final state occurs on a time scale of order 1/AQCD (and would lead to infrared divergences if we

60

attempted to describe it using perturbation theory), the inclusive cross section given by the sum over all hadronic final states is described very well by a (qq) pair that lives over a short time scale of order 1/q. In close analogy, while each particular hadronic intermediate state n in (14) cannot be described partonically, the sum over all intermediate states is accurately represented by a (qq) fluctuation of small transverse size of order 1/m;,. Because the (qq) pair is small, the physical picture of rescattering is very different from elastic 7T7r scattering. In perturbation theory, the pomeron is associated with two-gluon exchange. The analysis of two-loop contributions to the non-leptonic decay amplitude in 2 shows that the soft and collinear cancellations that guarantee the partonic interpretation of rescattering extend to two-gluon exchange. Hence, the soft final-state interactions are again subleading as required by the validity of (3). As far as the hard rescattering contributions are concerned, two-gluon exchange plus ladder graphs between a compact (qq) pair with energy of order m^ and transverse size of order 1/mt and the other pion does not lead to large logarithms, and hence there is no possibility to construct the (hard) pomeron. Note the difference with elastic vector-meson production through a virtual photon, which also involves a compact (qq) pair. However, in this case one considers s » Q2, where y/s is the photon-proton center-ofmass energy and Q the virtuality of the photon. This implies that the (qq) fluctuation is born long before it hits the proton. It is this difference of time scales, non-existent in non-leptonic B decays, that permits pomeron exchange in elastic vector-meson production in j*p collisions. 4

B —>• DTT: Factorization at one-loop order

We now present a more detailed treatment of the exclusive decays £<* -¥ £)(*)+L~, where L is a light meson. We illustrate explicitly how factorization emerges at one-loop order and compute the hard-scattering kernels 7y(u) in the factorization formula (3). For each final state / , we express the decay amplitudes in terms of parameters Oi (/) defined in analogy with similar parameters used in the literature on naive factorization. 4-1

Effective Hamiltonian and decay topologies

The effective Hamiltonian for B ->• D-K is Heft = % V:dVcb (C0O0 + C808).

(15)

61

We choose to write the two independent four-quark operators in the singletoctet basis O0 = c 7 A t ( l - 7 5 ) 6 d 7 M ( l - 7 5 ) u , 08 = c y ( l -l5)TAbd^(l

-l5)TAu,

(16)

rather than in the more conventional basis of 0\ and Oi. The Wilson coefficients C 0 and Cs describe the exchange of hard gluons with virtualities between the high-energy matching scale Mw and a renormalization scale n of order m&. (These coefficients are related to the ones of the standard basis by C0 =Ci + C 2 /3 and C 8 = 2C 2 .) They are known at next-to-leading order in renormalization-group improved perturbation theory and are given by 5 C0 = ^ ± l c 2JVC

+T

+ ^ ^ C - , 2NC

CS = C+-C-,

(17)

B± = ±^1B,

(is)

where

^=(1

+ ^ ) ^ ^= ) , cv / 4TT

=J

^'

=

2NC

and C±{y) =

aa(Mw) ««(/«)

d±

r

1 +

a8(Mw) - ag(n) 4^

5±

(19)

For 7Ve = 3 and / = 5, we have d+ = ^ and d_ = — 1 | , as well as S+ = IIH and S- = — | | | i . The scheme dependence of the Wilson coefficients at next-to-leading order is parameterized by the coefficient B in (18). We note that B N D R = 11 in the naive dimensional regularization (NDR) scheme with anticommuting 75, and .BHV = 7 in the 't Hooft-Veltman (HV) scheme. We will demonstrate below that the scale and scheme dependence of the Wilson coefficients is canceled by a corresponding scale and scheme dependence of the hadronic matrix elements of the operators OQ and OsBefore continuing with a discussion of these matrix elements, it is useful to consider the flavour structure for the various contributions to B -> Dw decays. The possible quark-level topologies are depicted in Fig. 6. In the terminology generally adopted for two-body non-leptonic decays, the decays Ba ->• D+n~, Ba ->• £>°7r° and B~ -> D°n~ are referred to as class-I, class-II and class-Ill, respectively 16 . In Bd ->• D+ir~ and B~ ->• D°ir~ decays the pion can be directly created from the weak current. We call this a class-I contribution, following the above terminology. In addition, in the case of Bd —* D+TT~ there is a contribution from weak annihilation, and a classII amplitude contributes to B~ -> D°it~. The important point is that the

62 d

u

b—M— (a>

c

c

u

6 — M — < (b)

(c)

Figure 6. Basic quark-level topologies for B - • DTT decays (q = u, d): (a) class-I, (b) classII, (c) weak annihilation. Bd -+ D+ir~ receives contributions from (a) and (c), Bd -> D°7r° from (b) and (c), and B~ -+ D°TV~ from (a) and (b). Only (a) contributes in the heavyquark limit.

spectator quark goes into the light meson in the case of the class-II amplitude. This amplitude is suppressed in the heavy-quark limit, as is the annihilation amplitude. It follows that the amplitude for Bd ->• D°ir°, receiving only classII and annihilation contributions, is subleading compared with Bd -> D+ir~ and B~ ->• D°n~, which are dominated by the class-I topology. We shall use the one-loop analysis for Bd —• D+TT~ as a concrete example to illustrate explicitly the various steps involved in establishing the factorization formula. Most of the arguments given below are standard from the theory of hard exclusive processes involving light hadrons 8 . However, it is instructive to repeat these arguments in the context of B decays. 4-2

Soft and collinear cancellations at one-loop order

In order to demonstrate the property of factorization for the decay -B<j —>• D+w~, we now analyze the "non-factorizable" one-gluon exchange contributions shown in Fig. 5 in some detail. We consider the leading, valence Fock state of the emitted pion. This is justified since higher Fock components only give power-suppressed contributions to the decay amplitude in the heavyquark limit (as demonstrated later). For the purpose of our discussion, the valence Fock state of the pion can be written as

k(?)> =J^i^^wc

(4W6tto)-°tto)b\to)) 1°)*(u'k)'

(20) where a\ (6j) denotes the creation operator for a quark (antiquark) in a state with spin s = | or s =4-, and we have suppressed colour indices. The wave function ^(u,l±) is defined as the amplitude for the pion to be composed of two on-shell quarks, characterized by longitudinal momentum fraction u and

63

transverse momentum l±. The on-shell momenta of the quark and antiquark are chosen as in (8). For the purpose of power counting, l± ~ AQCD
- ^ = * * ( « , / ! ) (75 *)«/» eil<* ,

= Jdu ^

(21)

which appears as an ingredient of the B —> D-K matrix element. It is now straightforward to obtain the one-gluon exchange contribution to the B -¥ D-K matrix element of the operator Og- For the sum of the four diagrams in Fig. 5, we find -gluon —

%.

2CF

1 f d4k , „D+ _, &lx **(ti,/1) t rr + l _ A , J6N B r l f i v 1 f du

£ ~ t j (27)4 < Mi(*0 I <*> jfca J0

ie^ ^ f

(22)

,, „ , ,,,

^ 4M(i„k,k)],

where Ai(fc) = A2{lg,lg,k)-

7 A ( ^ c - U + mc)T 2pc-kfc2 2l_ fe +

fe2

2lq.k

T(jib+ )i + m 6 ) 7 A 2p6 • k + k2 + k2-

(23)

Here T = 7 M (1 — 75), and pt, pc are the momenta of the b- and c-quark, respectively. There is no correction to the matrix element of O0 at order ag, because in this case the (du) pair is necessarily in a colour-octet configuration and cannot form a pion. In (22) the pion wave function ^(u,l±) appears separated from the B -* D transition. This is merely a reflection of the fact that we have represented the pion state in the form shown in (20). It does not, by itself, imply factorization, since the right-hand side of (22) still involves non-trivial integrations over l± and the gluon momentum fc, and long- and short-distance contributions are not yet disentangled. In order to prove factorization, we

64

need to show that the integral over k receives only subdominant contributions from the region of small k2. This is equivalent to showing that the integral over k does not contain infrared divergences at leading power in AQCD/'716To demonstrate infrared finiteness of the one-loop integral J=

fdik^A1{k)®A2(lq,lq-,k)

(24)

at leading power, the heavy-quark limit and the corresponding large lightcone momentum of the pion are again essential. First note that when k is of order mj, J ~ 1 by dimensional analysis. Potential infrared divergences could arise when k is soft or collinear to the pion momentum q. We need to show that the contributions from these regions are power suppressed. (Note that we do not need to show that J is infrared finite. It is enough that logarithmic divergences have coefficients that are power suppressed.) We treat the soft region first. Here all components of k become small simultaneously, which we describe by scaling k ~ A. Counting powers of A (cPk ~ A4, 1/fc2 ~ A - 2 , 1/p • k ~ A - 1 ) reveals that each of the four diagrams in Fig. 5, corresponding to the four terms in the product in (24), is logarithmically divergent. However, because k is small the integrand can be simplified. For instance, the second term in A2 can be approximated as 7 A(&+

jpr

2/,-fc +

=

fc

2

7A(^ 4+V± + £% 2uq-k + 2l±-k+j^sn--k

*-+ *0 r

-. g* r ,

+k

2

(25)

qk

'

where we used that 4 to the extreme left or right of an expression gives zero due to the on-shell condition for the external quark lines. We get exactly the same expression but with an opposite sign from the other term in A2, and hence the soft divergence cancels out. More precisely, we find that the integral is infrared finite in the soft region when l± is neglected. When l± is not neglected, there is a divergence from soft k which is proportional to ' i / m 6 ~ A Q C D / T O 6 - I n either case, the soft contribution to J is of order A Q C D / T I J or smaller and hence suppressed relative to the hard contribution. This corresponds to the standard soft cancellation mechanism, which is a technical manifestation of colour transparency. Each of the four terms in (24) is also divergent when k becomes collinear with the light-cone momentum q. This implies the scaling k+ ~ A0, k± ~ A, and jfe- ~ A2. Then d4k ~ dk+dk~d2k± ~ A4, and q • k = q+k~ ~ A2, fe2 = 2k+k~ + k\ ~ A2. The divergence is again logarithmic, and it is thus sufficient to consider the leading behaviour in the collinear limit. Writing

65

k = aq + ... we can now simplify the second term of A? as 7 ^ + ?)r^2(. +

a)rj

(26)

No simplification occurs in the denominator (in particular, l± cannot be neglected), but the important point is that the leading contribution is proportional to q\. Therefore, substituting k = aq into A\ and using q2 — 0, we obtain

gAAi^c

+ me)r_r(^ + m ^

2apc •q

= 0)

2apb •q

employing the equations of motion for the heavy quarks. Hence the collinear divergence cancels by virtue of the standard Ward identity. This completes the proof of the absence of infrared divergences at leading power in the hard-scattering kernel for JS^ —• D+n~ to one-loop order. Similar cancellations are observed at higher orders. A complete proof of factorization at two-loop order can be found in 2 . Having established that the "non-factorizable" diagrams of Fig. 5 are dominated by hard gluon exchange (i.e. that the leading contribution to J arises from k of order rrn,), we may now use the fact that \l±\ -C -B to expand A^ in powers of \l±\/E. To leading order the expansion simply reduces to neglecting l± altogether, which implies lq = uq and lq — uq in (8). As a consequence, we may perform the l± integration in (22) over the pion distribution amplitude. Defining

/ the matrix element of 0$ in (22) becomes {D+7r-\08\Bd)1

~92s ^

(29)

-gluon —

/ ( 0 r (D+lcAiikWB*) 1 UJ

du^ujtrfos

^A2(uq,uq,k)}.

On the other hand, putting y on the light-cone in (21) and comparing with (5), we see that the Zj_-integrated wave function $,r(u) in (28) is precisely the light-cone distribution amplitude of the pion. This demonstrates the relevance of the light-cone wave function to the factorization formula. Note that the collinear approximation for the quark and antiquark momenta emerges automatically in the heavy-quark limit.

66

After the k integral is performed, the expression (29) can be cast into the form (D+*-\Oa\Bd)1-glm,n~FB->D(0)

f

duT8(u,z)$w(u),

(30)

Jo

where z = mc/mb, Ts(u,z) is the hard-scattering kernel, and FB~*D(Q) the form factor that parameterizes the (£>+|c[.. .]b\Bd) matrix element. Because of the absence of soft and collinear infrared divergences in the gluon exchange between the (cb) and (du) currents, the hard-scattering kernel Ts is calculable in QCD perturbation theory. 4-3

Matrix elements at next-to-leading order

We now compute these hard-scattering kernels explicitly to order as. effective Hamiltonian (15) can be written as neS

-k^C {n) C+ + %v;,v«[ [2N ^ C

= ^kv:dVcb{

+

+

The

+ -h^C-Ui) C

2NC -M+ + ^^-^rBCsifj.) 4. 2JV 0 5 C 'H

C8(fi)08\,

O0

(31)

where the scheme-dependent term in the coefficient of the operator 0$ has been written explicitly. Because the light-quark pair has to be in a colour singlet to produce the pion in the leading Fock state, only OQ gives a contribution to zeroth order in as. Similarly, to first order in a8 only 0$ can contribute. The result of evaluating the diagrams in Fig. 5 with an insertion of Og can be presented in a form that holds simultaneously for a heavy meson H = D,D* and a light meson L = w,p, using only that the (ud) pair is a colour singlet and that the external quarks can be taken on-shell. We obtain (z = mc/mb) (H(p')L(q)\08\Bd(p)) x \-(6\TI^

= ^^JLifL

j\u*L(u)

+ B]({JV)-(JA))+F(U,Z)(JV)-F(U,-Z)(JA)

(32) ,

where {Jv) = (H(p')\c4b\Bd(p)),

(JA) = {H(pl)\cfab\Bd(p)).

(33)

It is worth noting that even after computing the one-loop correction the (ud) pair retains its V-A structure. This, together with (5), implies that the form

67

of (32) is identical for pions and longitudinally polarized p mesons. (The production of transversely polarized p mesons is power suppressed in AQCD/"I&-) The function F(u, z) appearing in (32) is given by F(u, z) = (3 + 2 In 3 ) In z2 - 7 + f(u, z) + f{u,

1/z),

(34)

where f(u,z)

=

u(l - z2)[3(l - u(l - z2)) + z] ln[u(l - z2)] [\-u{l-z2)Y<

l-u(l-z2)

' ln[u(l z2)] - l n > ( l - z2)} - Li a [l - u(l - z2)} - {u -> u] + 2 l-u(l-z2)

(35)

and Li2(a;) is the dilogarithm. The contribution of f(u,z) in (34) comes from the first two diagrams in Fig. 5 with the gluon coupling to the b quark, whereas f(u,l/z) arises from the last two diagrams with the gluon coupling to the charm quark. Note that the terms in the large square brackets in the definition of the function f(u, z) vanish for a symmetric light-cone distribution amplitude. These terms can be dropped if the light final-state meson is a pion o r a p meson, but they are relevant, e.g., for the discussion of Cabibbosuppressed decays such as B^ —> D^+K~ and £4 —> D^+K*~. The discontinuity of the amplitude, which is responsible for the occurrence of the strong rescattering phase, arises from f(u, 1/z) and can be obtained by recalling that z2 is z2 — it with e > 0 infinitesimal. We find z2)[3(l-u(l-z2)) 7T

[1

+ z]

•u(l-*2)]2

ln[l-u(l-z2)] + 2 1 n u + -

u(l-z2)

{u ->• u}

(36)

As mentioned above, (32) is applicable to all decays of the type Bd -»• where L is a light hadron such as a pion or a (longitudinally polarized) p meson. Only the operator Jv contributes to Bd -> D+L~~, and only J A contributes to Bd -> D*+L~. Our result can therefore be written as D(*'+L~,

(D+L-\O0tS\Bd)

= (D+\&y»(l-l5)b\Bd)-ifLqii

[ duTQ,s(u,z)L(u), ./o

(37)

where L = TT, p, and the hard-scattering kernels are T0(u,z) = l + O(a2a), Ts(u,z)

=

Os

Cf_

4TT

2AT,

-6 In

-B

+ F{u,z) +

0(a2s).

(38)

68

When the D meson is replaced b y a D * meson, the result is identical except that F(u,z) must be replaced with F(u, -z). Since no order-a g corrections exist for Oo, the matrix element retains its leading-order factorized form (D+L-\O0\Bd)

= ifLqiM < D + | g y ( l - -y5)b\Bd)

(39)

to this accuracy. From (35) it follows that Tg (u, z) tends to a constant as u approaches the endpoints (u -» 0, 1). (This is strictly true for the part of Ts(u,z) that is symmetric i n « H u ; the asymmetric part diverges logarithmically as u —• 0, which however does not affect the power behaviour and the convergence properties in the endpoint region.) Therefore the contribution to (37) from the endpoint region is suppressed, both by phase space and by the endpoint suppression intrinsic to $x,(u). Consequently, the emitted light meson is indeed dominated by energetic constituents, as required for the self-consistency of the factorization formula. The final result for the class-I, non-leptonic Bd —> D^+L~ decay amplitudes, in the heavy-quark limit and at next-to-leading order in a8, can be compactly expressed in terms of the matrix elements of a "transition operator"

r

GF ,

=7T^

a1(DL)Qv-a1(D*L)QA],

(40)

where Qv = cry^b d^^l

- -yh)u,

QA = cj^^b

® dj^l

- j5)u,

(41)

and hadronic matrix elements of Qv,A are understood to be evaluated in factorized form, i.e. {DL\h ®h\B)

= (D\h\B) (L\j2\0).

(42)

Eq. (40) defines the quantities ai(D^L), which include the leading "nonfactorizable" corrections, in a renormalization-scale and -scheme independent way. To leading power in AQCD /m,b these quantities should not be interpreted as phenomenological parameters (as is usually done), because they are dominated by hard gluon exchange and thus calculable in QCD. At next-to-leading order we get

r

u2

r1

-61n-S-+ / m b Jo

duF(u,z)$L(u)

69 ai(D*L)

= ^±IC (M) + 2N ^r^-M 2N + C

a8 CF

C

u2

r1

-61n-^-+ / duF{u,-z)
T(Bd -> D+TT-) T(Bd

-+

D*+TT~)

C 8 (M)

(D+\c4{l-l5)b\Bd) {D*+\c4{l-1,)b\Bd)

ai(Dir) ai(£)*7r)

(44)

where for simplicity we neglect the light meson masses as well as the mass difference between D and D* in the phase-space for the two decays. At nextto-leading order ax [DTT) ai(D*n)

= 1+

^%%Re!odU[F[U'Z)

~

F(U

' ~Z)] *"{U) •

(45)

Our result for the symmetric part of the kernel agrees with that found in 5

17

.

Power-suppressed contributions

Up to this point we have presented arguments in favour of factorization of non-leptonic U-decay amplitudes in the heavy-quark limit, and have explored in detail how the factorization formula works at one-loop order for the decays Bd —t D(*)+L~. It is now time to show that other contributions not

70

-V- -M o o

o o

Figure 7. "Non-factorizable" spectator interactions.

considered so far are indeed power suppressed. This is necessary to fully establish the factorization formula. Besides, it will also provide some numerical estimates of the corrections to the heavy-quark limit. We start by discussing interactions involving the spectator quark and weak annihilation contributions, before turning to the more delicate question of the importance of non-valence Fock states. 5.1

Interactions with the spectator quark

Clearly, the diagrams shown in Fig. 7 cannot be associated with the formfactor term in the factorization formula (3). We will now show that for JB decays into a heavy-light final state their contribution is power suppressed in the heavy-quark limit. (This suppression does not occur for decays into two light mesons, where hard spectator interactions contribute at leading power. In this case, they contribute to the kernels T?1 in the factorization formula (second term in Fig. 1).) In general, "non-factorizable" diagrams involving an interaction with the spectator quark would impede factorization if there existed a soft contribution at leading power. While such terms are present in each of the two diagrams separately, they cancel in the sum over the two gluon attachments to the (ud) pair by virtue of the same colour-transparency argument that was applied to the "non-factorizable" vertex corrections. Focusing again on decays into a heavy and a light meson, such as Bd -» D+n~, we still need to show that the contribution remaining after the soft cancellation is power suppressed relative to the leading-order contribution (13). A straightforward calculation leads to the (simplified) result A(Bd ->• D + 7r~) s p e c ~ GF UfDfB

Jo £

as

Jo V

~ GF a8 mb A Q C D .

Jo u (46)

71

>

^

>

(a)

^ (b)

>

^ (c)

>

^

(d)

Figure 8. Annihilation diagrams.

This is indeed power suppressed relative to (13). Note that the gluon virtuality is of order £77 m^ ~ A Q C D and so, strictly speaking, the calculation in terms of light-cone distribution amplitudes cannot be justified. Nevertheless, we use (46) to deduce the scaling behaviour of the soft contribution, as we did for the heavy-light form factor in Sect. 3.2.

5.2

Annihilation topologies

Our next concern are the annihilation diagrams shown in Fig. 8, which also contribute to the decay Bd -> D+n~. The hard part of these diagrams could, in principle, be absorbed into hard-scattering kernels of the type T*1. The soft part, if unsuppressed, would violate factorization. However, we will see that the hard part as well as the soft part are suppressed by at least one power of A Q C D/m 6 . The argument goes as follows. We write the annihilation amplitude as A{Bd -> D+7T-) ann

x f d^dr1du^B(O^D(ri)^^)Tann(^v,u), Jo

(47)

where the dimensionless function T ann (£,?j,u) is a product of propagators and vertices. The product of decay constants scales as Aq C D /mi,. Since d£$B(Q scales as 1 and so does dr)$D{v), while du^^iu) is never larger than 1, the amplitude can only compete with the leading-order result (13) if Tann(t;,r],u) can be made of order (mi,/AQCU)3 or larger. Since T a n n (£, r/,u) contains only two propagators, this can be achieved only if both quarks the gluon splits into are soft, in which case T ann (£,r), u) ~ (m&/^AQCD)4- But then du$„(u) ~ ( A Q C D / T I 6 ) 2 , so that this contribution is power suppressed.

72

Figure 9. Diagram that contributes to the hard-scattering kernel involving a quarkantiquark-gluon distribution amplitude of the B meson and the emitted light meson.

5.3

Non-leading Fock states

Our discussion so far concentrated on contributions related to the quarkantiquark components of the meson wave functions. We now present qualitative arguments that justify this restriction to the valence-quark Fock components. Some of these arguments are standard 8 ' 9 . We will argue that higher Fock states yield only subleading contributions in the heavy-quark limit.

Additional hard partons An example of a diagram that would contribute to a hard-scattering function involving quark-antiquark-gluon components of the emitted meson and the B meson is shown in Fig. 9. For light mesons, higher Fock components are related to higher-order terms in the collineax expansion, including the effects of intrinsic transverse momentum and off-shellness of the partons by gauge invariance. The assumption is that the additional partons are collinear and carry a finite fraction of the meson momentum in the heavy-quark limit. Under this assumption, it is easy to see that adding additional partons to the Fock state increases the number of off-shell propagators in a given diagram (compare Fig. 9 to Fig. 3). This implies power suppression in the heavy-quark expansion. Additional partons in the B-meson wave function are always soft, as is the spectator quark. Nevertheless, when these partons are connected to the hard-scattering amplitudes the virtuality of the additional propagators is still of order m&AQCD, which is sufficient to guarantee power suppression. Let us study in more detail how the power suppression arises for the simplest non-trivial example, where the pion is composed of a quark, an antiquark, and an additional gluon. The contribution of this 3-particle Fock state to the B - • D-K decay amplitude is shown in Fig. 10. It is convenient to use the Fock-Schwinger gauge, which allows us to express the gluon field A\ in

73

Figure 10. The contribution of the qqg Fock state to the Bj, -> D+ir decay amplitude.

terms of the field-strength tensor Gp\ via Ax(x) = [ dv vxp GpX(vx). (48) Jo Up to twist-4 level, there are three quark-antiquark-gluon matrix elements that could potentially contribute to the diagrams shown in Fig. 10. Due to the V — A structure of the weak-interaction vertex, the only relevant three-particle light-cone wave function has twist-4 and is given by 18,1Q (7r(g)|d(0)7M75 g.Gap(vx)

u(0)|0)

Qa90^Jvu±(ui)eivU39X

= U(Qfi9a» ~

+ U " ^ (QaXfi - qpxa)Jvu

(±(ui) + 0||(«0) e l w » « - .

(49)

Here J Vu = JQ du\ du?. du% 6(1 —Ui — U2—U3), with u\, u 2 and 113 the fractions of the pion momentum carried by the quark, antiquark and gluon, respectively. Evaluating the diagrams in Fig. 10, and neglecting the charm-quark mass for simplicity, we find (D+n-\08\Bd)m

= iU (D+\c4(l

- ^)b\Bd)

f Vu ^ l l . J

(50)

W3 771^

Since <j>\\ ~ A Q C D , the suppression by two powers of AQCD/WJ, compared to the leading-order matrix element is obvious. Note that due to G-parity 4>\\ is antisymmetric in ui «•> u-i for a pion, so that (50) vanishes in this case. Additional soft partons A more precarious situation may arise when the additional Fock components carry only a small fraction of the meson momentum, contrary to the assumption made above. It is usually argued 8,Q that these configurations are

74

(a)

(b)

Figure 11. (a) Soft overlap contribution which is part of the B -> D form factor, (b) Soft overlap with the pion which would violate factorization, if it were unsuppressed.

suppressed, because they occupy only a small fraction of the available phase space (since / dui ~ AQCD /m,b when the parton that carries momentum fraction m is soft). This argument does not apply when the process involves heavy mesons. Consider, for example, the diagram shown in Fig. 11 (a) for the decay B —> DTT. Its contribution involves the overlap of the B-meson wave function involving additional soft gluons with the wave function of the D meson, also containing soft gluons. There is no reason to suppose that this overlap is suppressed relative to the soft overlap of the valence-quark wave functions. It represents (part of) the overlap of the "soft cloud" around the b quark with (part of) the "soft cloud" around the c quark after the weak decay. The partonic decomposition of this cloud is unrestricted up to global quantum numbers. (In the case where the B meson decays into two light mesons, there is a form-factor suppression ~ (AQCD/"I&) 3 / ' 2 for the overlap of the valence-quark wave functions, but once this price is paid there is again no reason for further suppression of additional soft gluons in the overlap of the B-meson wave function and the wave function of the recoiling meson.) The previous paragraph essentially repeated our earlier argument against the hard-scattering approach, and in favour of using the B —> D form factor as an input to the factorization formula. However, given the presence of additional soft partons in the B -> D transition, we must now argue that it is unlikely that the emitted pion drags with it one of these soft partons, for instance a soft gluon that goes into the pion wave function, as shown in Fig. 11 (b). Notice that if the (qq) pair is produced in a colour-octet state, at least one gluon (or a further (qq) pair) must be pulled into the emitted meson if the decay is to result in a two-body final state. What suppresses the process shown in Fig. 11 (b) relative to the one in Fig. 11 (a) even if the emitted (qq) pair is in a colour-octet state? It is once more colour transparency that saves us. The dominant config-

75

K

+

f\

Figure 12. Quark-antiquark-gluon distribution amplitude in the gluon endpoint region.

uration has both quarks carry a large fraction of the pion momentum, and only the gluon might be soft. In this situation we can apply a non-local "operator product expansion" to determine the coupling of the soft gluon to the small (qq) pair 2 . The gluon endpoint behaviour of the qqg wave function is then determined by the sum of the two diagrams shown on the right-hand side in Fig. 12. The leading term (for small gluon momentum) cancels in the sum of the two diagrams, because the meson (represented by the black bar) is a colour singlet. This cancellation, which is exactly the same cancellation needed to demonstrate that "non-factorizable" vertex corrections are dominated by hard gluons, provides one factor of AQCD/TMJ, needed to show that Fig. 11 (b) is power suppressed relative to Fig. 11 (a). In summary, we have (qualitatively) covered all possibilities for nonvalence contributions to the decay amplitude and find that they are all power suppressed in the heavy-quark limit. 6

Limitations of the factorization approach

The factorization formula (3) holds in the heavy-quark limit m\, -> oo. Corrections to the asymptotic limit are power-suppressed in the ratio AQCD/TTI6 and, generally speaking, do not assume a factorized form. Since m;, is fixed to about 5GeV in the real world, one may worry about the magnitude of power corrections to hadronic J5-decay amplitudes. Naive dimensional analysis would suggest that these corrections should be of order 10% or so. We now discuss several reasons why some power corrections could turn out to be numerically larger than suggested by the parametric suppression factor A-Qcn/mb- Most of these "dangerous" corrections occur in more complicated, rare hadronic B decays into two light mesons, but are absent in decays such as B - • DTT. 6.1

Several small parameters

Large non-factorizable power corrections may arise if the leading-power, factorizable term is somehow suppressed. There are several possibilities for such

76

a suppression, given a variety of small parameters that may enter into the non-leptonic decay amplitudes. i) The hard, "non-factorizable" effects computed using the factorization formula occur at order as. Some other interesting effects such as final-state interactions appear first at this order. For instance, strong-interaction phases due to hard interactions are of order as, while soft rescattering phases are of order AQCD/W;,. Since for realistic B mesons aa is not particularly large compared to AQCD /m&, we should not expect that these phases can be calculated with great precision. In practice, however, it is probably more important to know that the strong-interaction phases are parametrically suppressed in the heavy-quark limit and thus should be small. (This does not apply if the real part of the decay amplitudes is suppressed for some reason; see below.) ii) If the leading, lowest-order (in as) contribution to the decay amplitude is colour suppressed, as occurs for the class-II decay Bd -> ir°n°, then perturbative and power corrections can be sizeable. In such a case even the hard strong-interaction phase of the amplitude can be large 1'2. But at the same time soft contributions could be potentially important, so that in some cases only an order-of-magnitude estimate of the amplitude may be possible. iii) The effective Hamiltonian (1) contains many Wilson coefficients Cj that are small relative to C\ m 1. There are decays for which the entire leading-power contribution is suppressed by small Wilson coefficients, but some power-suppressed effects are not. An example of this type is B~ -> K~K°. This decay proceeds through a penguin operator b -¥ dss at leading power. But the annihilation contribution, which is power suppressed, can occur through the current-current operator with large Wilson coefficient C\. Our approach does not apply to such (presumably) annihilation-dominated decays, unless a systematic treatment of annihilation amplitudes can be found. iv) Some amplitudes may be suppressed by a combination of small CKM matrix elements. For example, B -» •KK decays receive large penguin contributions despite their small Wilson coefficients, because the so-called tree amplitude is CKM suppressed. This is not a problem for factorization, since it applies to the penguin and the tree amplitudes. We are not aware of any case (for ordinary B mesons) in which a purely power-suppressed term is CKM enhanced and which would therefore dominate the decay

77

amplitude. (But this situation could occur for Bc -t D°K , where the QCD dynamics is similar if we consider the charm quark as a light quark.) 6.2

Power corrections enhanced by small quark masses

There is another enhancement of power-suppressed effects for some decays into two light mesons, connected with the curious numerical fact that 2

/

^ - ? ^ -

= -*M«3GeV

(51)

is much larger than its naive scaling estimate AQCD- (Here (qq) = (0|tm|0) = (0|dd|0) is the quark condensate.) Consider the contribution of the penguin operator 0 6 = (dibj)V-A(ujUi)v+A to the Bd -»• n+n~ decay amplitude. The leading-order graph of Fig. 3 results in the expression (n+ir-lidib^v^iujuJv+AlBd)

= im% F*-+*(0) U x — ,

(52)

771 f>

which is formally a AQCD/WI;, power correction compared to the corresponding matrix element of a product of two left-handed currents, but numerically large due to (51). We would not have to worry about such terms if they could all be identified and the factorization formula (3) applied to them, since in this case higher-order perturbative corrections would not contain non-factorizing infrared logarithms. However, this is not the case. After including radiative corrections, the matrix element on the left-hand side of (52) is expressed as a non-trivial convolution with pion light-cone distribution amplitudes. The terms involving ^ can be related to two-particle twist-3 (rather than leading twist-2) distribution amplitudes, conventionally called $ p (u) and $ a ( u ) . We find that the radiative corrections to the matrix element in (52) do indeed factorize. However, at the same order there appear twist-3 corrections to the hard spectator interaction shown in Fig. 7, and these contributions contain an endpoint divergence (related to the fact that the distribution amplitudes $ p (u) and $'a(u) do not vanish at the endpoints). In other words, the twist-3 "corrections" to the hard spectator term in the second factorization formula in (3) relative to the "leading" twist-2 contributions are of the form a s x logarithmic divergence, which we interpret as being of order 1. The non-factorizing character of the "chirally-enhanced" power corrections can introduce a substantial uncertainty in some decay modes 12 . As in the related situation for the pion form factor 2 0 , one may argue that the endpoint divergence is suppressed by a Sudakov form- factor. However, it is likely that when mi, is not large enough to suppress these chirally-enhanced terms, then it is also not large enough to make Sudakov suppression effective.

78

We stress that the chirally-enhanced terms do not appear in decays into a heavy and a light meson such as B ->• DTT, because these decays have no penguin contribution and no contribution from the hard spectator interaction. Hence, the twist-3 light-cone distribution amplitudes responsible for chirally-enhanced power corrections do not enter in the evaluation of the decay amplitudes. 6.3

Non-leptonic decays when M 2 is not light

The analysis of non-leptonic decay amplitudes in Sect. 3.3 referred to decays where the emission particle M 2 is a light meson. We now briefly discuss the case where M 2 is heavy. Suppose that M 2 is a D meson, whereas the meson that picks up the spectator quark can be heavy or light. Examples of this type are the decays Bd —> TT°D0 and B^ —• D+D~. It is intuitively clear that factorization must be problematic in these cases, because the heavy D meson has a large overlap with the BTT or BD systems, which are dominated by soft processes. In more detail, we consider the coupling of a gluon to the two quarks that form the emitted D meson, i.e. the pairs of diagrams in Fig. 5 (a+b), (c+d) and Fig. 7. Denoting the gluon momentum by A;, the quark momenta by lq and lq, and the .D-meson momentum by q, we find that the gluon couples to the "current" A =

7A(%+* + ™„)r _ r(ft+ Vhx;

(53)

where Y is part of the weak decay vertex. When k is soft (all components of order AQCD), each of the two terms scales as 1/AQCD- Taking into account the complete amplitude as done explicitly in Sect. 4.2, we can see that the decoupling of soft gluons requires that the two terms in (53) cancel, leaving a remainder of order l/m;,. This cancellation does indeed occur when M 2 is a light meson, since in this case lq and lq are dominated by their longitudinal components. When M 2 is heavy, the momenta lq and lq are asymmetric, with all components of the light antiquark momentum lq of order AQCD in the Bor D-meson rest frames, while the zero-component of lq is of order mt- Hence the current can be approximated by _ SX0T * ~ ~1

J

K0

T(j/g+ #) 7 A 1^7 , , , 2 ~ 2lq • k + k*

1 A AQCD

.

(54)

and the soft cancellation does not occur. (The on-shell condition for the charm quark has been used to arrive at this equation.) It follows that the emitted D meson does not factorize from the rest of the process, and that a factorization formula analogous to (3) does not apply

79 to decays such as Bd -> 7r°D° and B^ -> D+D~. An important implication of this statement is that one should also not expect naive factorization to work in these cases. In other words, we expect that "non-factorizable" corrections modify the factorized decay amplitudes by terms of order 1. 6.4

Difficulties with charm

There are decay modes, such as B~ ->• D°ir~, in which the spectator quark can go to either of the two final-state mesons. The factorization formula (3) applies to the contribution that arises when the spectator quark goes to the D meson, but not when the spectator quark goes to the pion. However, even in the latter case we may use naive factorization to estimate the power behaviour of the decay amplitude. Adapting (13) to the decay B~ —> D°ir~, we find that the non-factorizing (class-II) amplitude is suppressed compared to the factorizing (class-I) amplitude by A(B~ -» Z?°7r-) class -„ _ FB^(m2D)fD

_ ( AQCD ^ 2

(5g)

Here we use that FB~>n(q2) ~ (AQCD/wf,)3/2 even for q2 ~ m 2 , as long as qmax—q2 is also of order m 2 . (It follows from our definition of heavy final-state mesons that these conditions are fulfilled.) As a consequence, strictly speaking factorization does hold for B~ —• D°n~ decays in the sense that the class-II contribution is power suppressed with respect to the class-I contribution. Unfortunately, the scaling behaviour for real B and D mesons is far from the estimate (55) valid in the heavy-quark limit. Based on the dominance of the class-I amplitude we would expect that R=*!T±gq„l

(56)

in the heavy-quark limit. This contradicts existing data which yield R = 1.89 ± 0.35, despite the additional colour suppression of the class-II amplitude. One reason for the failure of power counting lies in the departure of the decay constants and form factors from naive power counting. The following compares the power counting to the actual numbers (square brackets): SD „ / A Q C D \ 1 / 2

F^(m2D)

(AQCu\3/2

, . _,

....

However, it is unclear whether the failure of power counting can be attributed to the form factors and decay constants alone.

80

Note that for the purposes of power counting we treated the charm quark as heavy, taking the heavy-quark limit for fixed mc/mb. This simplified the discussion, since we did not have to introduce mc as a separate scale. However, in reality charm is somewhat intermediate between a heavy and a light quark, since mc is not particularly large compared to AQCD- In this context it is worth noting that the first hard-scattering kernel in (3) cannot have A Q C D / W C corrections, since there is a smooth transition to the case of two light mesons. The situation is different with the hard spectator interaction term, which we argued to be power suppressed for decays into a D meson and a light meson. We shall come back to this in Sect. 7.5, where we estimate the magnitude of this term for the D-K final state, relaxing the assumption that the D meson is heavy. 7

P h e n o m e n o l o g y of B —> D^L

decays

The matrix elements we have computed in Sect. 4.3 provide the theoretical basis for a model-independent calculation of the class-I non-leptonic decay amplitudes for decays of the type B -¥ D^ L, where L is a light meson, to leading power in AQCD /"i& and at next-to-leading order in renormalizationgroup improved perturbation theory. In this section we discuss phenomenological applications of this formalism and confront our numerical results with experiment. We also provide some numerical estimates of power-suppressed corrections to the factorization formula. 1.1

Non-leptonic decay amplitudes

The results for the class-I decay amplitudes for B -¥ D^L are obtained by evaluating the (factorized) hadronic matrix elements of the transition operator T defined in (40). They are written in terms of products of CKM matrix elements, light-meson decay constants, B -» D^ transition form factors, and the QCD parameters a\ (Z)W L). The decay constants can be determined experimentally using data on the weak leptonic decays P~ -> l~Di{~f), hadronic T~ -> M~vT decays, and the electromagnetic decays V° -> e + e - . Following 16 , we use fn = 131 MeV, fK = 160 MeV, fp = 210 MeV, fK* = 214 MeV, and fai = 229 MeV. (Here ai is the pseudovector meson with mass mai ~ 1230 MeV.) The non-leptonic B D^+L~ decay amplitudes for L = ir, p can be expressed as A(Bd - • D + O = i ~

V:dVcb oi {D-K) / , F 0 (mJ) {m% -

m2D),

81

A(Bd -*

D*+TT-)

A(Bd^D+p-)

= - &

v:dVcbai(D*7r)

UMO

2™»« e*-P,

= -i^V:dVcba1(Dp)fpF+(m2p)2mpr1*-p,

(58)

where p (p7) is the momentum of the I? (charm) meson, e and r\ are polarization vectors, and the form factors FQ, F+ and Ao are defined in the usual way 16 . The decay mode Bd -» D*+p~ has a richer structure than the decays with at least one pseudoscalar in the final state. The most general Lorentz-invariant decomposition of the corresponding decay amplitude can be written as A(Bd -> D*+p-) = » ^ | V:dVcb e* V " (St g„v - S 2 q^v

+ iS3 e^p

p V ) ,

(59) where the quantities Si can be expressed in terms of semi-leptonic form factors. To leading power in A Q C D / ^ ; , , we obtain Si = oi (D*p) mpfp (mB + mD. )AX (m2p), S 2 = ax(D*p)mpfp

2A2 m

( J

.

(60)

The contribution proportional to S3 in (59) is associated with transversely polarized p mesons and thus leads to power-suppressed effects, which we do not consider here. The various B —> D^ form factors entering the expressions for the decay amplitudes can be determined by combining experimental data on semileptonic decays with theoretical relations derived using heavy-quark effective theory 3>16. Since we work to leading order in A Q C D / T I 6 , it is consistent to set the light meson masses to zero and evaluate these form factors at q2 = 0. In this case the kinematic relations F o (0) = F+(0),

( m B + m i > . ) A i ( 0 ) - (mB - mD.)A2(0)

=

2mD.A0(0) (61) allow us to express the two Bd -> D+L rates in terms of i*+(0), and the two Bd -+ D*+L~ rates in terms of Ao(0). Heavy-quark symmetry implies that these two form factors are equal to within a few percent 14 . Below we adopt the common value i*+(0) = Ao(0) = 0.6. All our predictions for decay rates will be proportional to the square of this number.

82 Table 1. Numerical values for the integrals J diiF(ti,z)$i(u) (upper portion) and J 0 du F(u, —z) $£, (u) (lower portion) obtained including the first two Gegenbauer moments.

z 0.25 0.30 0.35 0.25 0.30 0.35

7.2

Leading term -8.41-9.51i -8.79 - 9.09i -9.13 - 8.59i -8.45 - 6.56i -8.37 - 5.99i -8.24 - 5.44i

Coefficient of a f 5.92 - 12.19i 5.78 - 12.71i 5.60 - 13.21i 6.72 - 10.73i 6.83 - 11.49i 6.81 - 12.29i

Coefficient of a\ -1.33 + 0.36i -1.19 + 0.58i -1.00 + 0.73* -0.38 + 0.93* -0.21 + 0.85* -0.08 + 0.75*

Meson distribution amplitudes and predictions for a\

Let us now discuss in more detail the ingredients required for the numerical analysis of the coefficients ai{D^L). The Wilson coefficients C* in the effective weak Hamiltonian depend on the choice of the scale \i as well as on the value of the strong coupling a s , for which we take ag(mz) = 0.118 and two-loop evolution down to a scale /x ~ m\,. To study the residual scale dependence of the results, which remains because the perturbation series are truncated at next-to-leading order, we vary n between mt,/2 and 2m(,. The hard-scattering kernels depend on the ratio of the heavy-quark masses, for which we take z = mc/mb = 0.30 ± 0.05. Hadronic uncertainties enter the analysis also through the parameterizations used for the meson light-cone distribution amplitudes. It is convenient and conventional to expand the distribution amplitudes in Gegenbauer polynomials as $L(U)

= 6u(l — u)

l + ga^C^O-l)

(62)

n=l

where c[3/2){x) = 3z, c£/2)(x) = |(5a; 2 - 1), etc. The Gegenbauer moments a„(iJ.) are multiplicatively renormalized. The scale dependence of these quantities would, however, enter the results for the coefficients only at order a , , which is beyond the accuracy of our calculation. We assume that the leading-twist distribution amplitudes are close to their asymptotic form and thus truncate the expansion at n = 2. However, it would be straightforward to account for higher-order terms if desired. For the asymptotic form of the

83 Table 2. The QCD coefficients ai(D^L) at next-to-leading order for three different values of the renormalization scale /t. The leading-order values are shown for comparison.

H — mb/2 fj, = 2rrib £t = mb 1.074 + 0.037i 1.055 + 0.020i 1.038 + 0.011i -(0.024 - 0.0520 af - ( 0 . 0 1 3 - 0.028i) af -(0.007-0.0150 af 1.072 + 0.024i 1.054 + 0.013i 1.037 + 0.007? -(0.028 - 0.0470 af -(0.015-0.025i) af -(0.008-0.0140 af 1.049 1.025 1.011

ai(DL) ai{D*L)

ct0

distribution amplitude, $ L ( U ) = 6u(l - u), the integral in (43) yields pi

/ duF(u,z)$L(u) Jo

+

=

3]nz*-7 3(2 - 3z + 2z2 + z3) ln(l - z2) {l-z)(l + z)2

6z(l - 1z) fir' ^r-U {z2) 2 3 (l-z) (l+z) V 6 2

+

4 - 17z + 20z2 + 5z3 2(l-z)(l + z)2

+{Z

•1/z}

(63)

and the corresponding result with the function F(u, —z) is obtained by replacing z —p — z. More generally, a numerical integration with a distribution amplitude expanded in Gegenbauer polynomials yields the results collected in Table 1. We observe that the first two Gegenbauer polynomials in the expansion of the light-cone distribution amplitudes give contributions of similar magnitude, whereas the second moment gives rise to much smaller effects. This tendency persists in higher orders. For our numerical discussion it is a safe approximation to truncate the expansion after the first non-trivial moment. The dependence of the results on the value of the quark mass ratio z = mc/rrib is mild and can be neglected for all practical purposes. We also note that the difference of the convolutions with the kernels for a pseudoscalar D and vector D* meson are numerically very small. This observation is, however, specific to the case of B -p D^L decays and should not be generalized to other decays. Next we evaluate the complete results for the parameters a,i at next-toleading order, and to leading power in AQCD/W&. We set z = mc/mb = 0.3. Varying z between 0.25 and 0.35 would change the results by less than 0.5%.

84

The results are shown in Table 2. The contributions proportional to the second Gegenbauer moment a\ have coefficients of order 0.2% or less and can safely be neglected. The contributions associated with a\ are present only for the strange mesons K and K*, but not for n and p. Moreover, the imaginary parts of the coefficients contribute to their modulus only at order a , , which is beyond the accuracy of our analysis. To summarize, we thus obtain \ai(DL)\

= 1.055ig;gl? " (0.013±g-°£)af ,

\ai(D*L)\

= 1.054tg;°i? - ( O . O l S t ^ K ,

(64)

where the quoted errors reflect the perturbative uncertainty due to the scale ambiguity (and the negligible dependence on the value of the ratio of quark masses and higher Gegenbauer moments), but not the effects of power-suppressed corrections. These will be estimated later. It is evident that within theoretical uncertainties there is no significant difference between the two a\ parameters, and there is only a very small sensitivity to the differences between strange and non-strange mesons (assuming that \a^ \ < 1). In our numerical analysis below we thus take |ai| = 1.05 for all decay modes. 7.3

Tests of factorization

The main lesson from the previous discussion is that corrections to naive factorization in the class-I decays Bd —> D^+L~ are very small. The reason is that these effects are governed by a small Wilson coefficient and, moreover, are colour suppressed by a factor l/A^. For these decays, the most important implications of the QCD factorization formula are to restore the renormalization-group invariance of the theoretical predictions, and to provide a theoretical justification for why naive factorization works so well. On the other hand, given the theoretical uncertainties arising, e.g., from unknown power-suppressed corrections, there is little hope to confront the extremely small predictions for non-universal (process-dependent) "non-factorizable" corrections with experimental data. Rather, what we may do is ask whether data supports the prediction of a quasi-universal parameter \a,i\ ~ 1.05 in these decays. If this is indeed the case, it would support the usefulness of the heavy-quark limit in analyzing non-leptonic decay amplitudes. If, on the other hand, we were to find large non-universal effects, this would point towards the existence of sizeable power corrections to our predictions. We will see that within present experimental errors the data are in good agreement with our prediction of a quasi universal ai parameter. However, a reduction of the experimental uncertainties to the percent level would be very desirable for obtaining a more conclusive picture.

85

We start by considering ratios of non-leptonic decay rates that are related to each other by the replacement of a pseudoscalar meson by a vector meson. In the comparison of B -> Dir and B -> D*n decays one is sensitive to the difference of the values of the two ai parameters in (64) evaluated for a f = 0. This difference is at most few times 1 0 - 3 . Likewise, in the comparison of B -> D-K and B -> Dp decays one is sensitive to the difference in the lightcone distribution amplitudes of the pion and the p meson, which start at the second Gegenbauer moment a^. These effects are suppressed even more strongly. Prom the explicit expressions for the decay amplitudes in (58) it follows that T(Bd -» D+n-) T{Bd

-»•

D*+TT~)

T(Bd -» D+p-) T{Bd -> D+7T-)

_ {m\ - mlf\q | D i r ( F0{m%) 4m||g| 3 9 » 7

MO

3

=

2

4m||
F+{ml) F0(ml)

ai(Dir) Ol(£>*7T)

ai(Dp) cn(Dn)

(65)

Using the experimental values for the branching ratios reported by the CLEO Collaboration 21 we find (taking into account a correlation between some systematic errors in the second case) ai(£»7r) Oi(£>*7T)

Oi(Dp) a^Dir)

Fo(ml)

MO F+(mp Mml) =

1.00 ± 0 . 1 1 , 1.16 ± 0 . 1 1 .

(66)

Within errors, there is no evidence for any deviations from naive factorization. Our next-to-leading order results for the quantities ai(D^L) allow us to make theoretical predictions which are not restricted to ratios of hadronic decay rates. A particularly clean test of these predictions, which is essentially free of hadronic uncertainties, is obtained by relating the Bd —> D^+L~ decay rates to the differential semi-leptonic Bd —¥ Z)W + l~v decay rate evaluated at q2 = ml- In this way the parameters |ai| can be measured directly 10 . One obtains

R? =

dT(Bd -> D(.*)+l-v)/dq2

(67) where Xp = X* = 1 for a vector meson (because the production of the lepton pair via a V — A current in semi-leptonic decays is kinematically equivalent to that of a vector meson with momentum q), whereas X^ and X* deviate from

86

1 only by (calculable) terms of order m^/m2B, which numerically are below the 1% level 16 . We emphasize that with our results for ai given in (43) the above relation becomes a prediction based on first principles of QCD. This is to be contrasted with the usual interpretation of this formula, where a\ plays the role of a phenomenological parameter that is fitted from data. The most accurate tests of factorization employ the class-I processes Bd —• D*+L~, because the differential semi-leptonic decay rate in B —i D* transitions has been measured as a function of q2 with good accuracy. The results of such an analysis, performed using CLEO data, have been reported in 2 3 . One finds Rl = (1.13 ± 0.15) GeV2

=>-

\cn {D*ir)\ = 1.08 ± 0.07,

R* = (2.94 ± 0.54) GeV2

=»

\ai (D*p)\ = 1.09 ± 0.10,

R*ai = (3.45 ± 0.69) GeV2

=>•

\ai{D*ai)\

= 1.08± 0.11.

(68)

This is consistent with our theoretical result in (43). In particular, the data show no evidence for large power corrections to our predictions obtained at leading order in AQCD/"I&- However, a further improvement in the experimental accuracy would be desirable in order to become sensitive to processdependent, non-factorizable effects. 7-4 Predictions for class-I decay amplitudes We now consider a larger set of class-I decays of the form B^ —> D^+L~, all of which are governed by the transition operator (40). In Table 3 we compare the QCD factorization predictions with experimental data. As previously we work in the heavy-quark limit, i.e. our predictions are model independent up to corrections suppressed by at least one power of AQCD/W;,. The results show good agreement with experiment within errors, which are still rather large. (Note that we have not attempted to adjust the semi-leptonic form factors F+(0) and Ao(0) so as to obtain a best fit to the data.) We take the observation that the experimental data on class-I decays into heavy-light final states show good agreement with our predictions obtained in the heavy-quark limit as evidence that in these decays there are no unexpectedly large power corrections. We will now address the important question of the size of power corrections theoretically. To this end we provide rough estimates of two sources of power-suppressed effects: weak annihilation and spectator interactions. We stress that, at present, a complete account of power corrections to the heavy-quark limit cannot be performed in a systematic way, since these effects are not dominated by hard gluon exchange. In other words,

87 Table 3. Model-independent predictions for the branching ratios (in units of 10 - 3 ) of classI, non-leptonic Bd -> £>(*)+£,- decays in the heavy-quark limit. All predictions are in units of (|ai |/1.05) 2 . The last two columns show the experimental results reported by the CLEO Collaboration 2 1 , and by the Particle Data Group 2 4 .

Decay mode Bd -» -D+7J-

Bd Bd Bd Bd

-> D+K-> D+p-> D+K*-* D+a.1

Bd Bd Bd Bd

-• -> -• -•

D*+irD*+K~ £>*+pD*+J£T*-

Theory (HQL) 3.27 0.25 7.64 0.39 7.76 x[F + (0)/0.6] 2 3.05 0.22 7.59 0.40 8.53 x[4,(0)/0.6] 2

CLEO data 2.50 ±0.40

PDG98 3.0 ± 0 . 4

7.89 ±1.39

7.9 ± 1 . 4

8.34 ±1.66

6.0 ± 3 . 3

2.34 ±0.32

2.8 ± 0 . 2

7.34 ±1.00

6.7 ± 3 . 3

11.57 ±2.02

13.0 ± 2 . 7

factorization breaks down beyond leading power, and there are other sources of power corrections, such as contributions from higher Fock states, which we will not address here. We believe that the estimates presented below are nevertheless instructive. To obtain an estimate of power corrections we adopt the following, heuristic procedure. We treat the charm quark as light compared to the large scale provided by the mass of the decaying b quark (m c ^ mj, and mc fixed as mi -> oo) and use a light-cone projection similar to that of the pion also for the D meson. In addition, we assume that m c is still large compared to AQCD- We implement this by using a highly asymmetric D-meson wave function, which is strongly peaked at a light-quark momentum fraction of order A Q C D / W D . This guarantees correct power counting for the heavy-light final states we are interested in. As discussed in Sect. 5.2, there are four annihilation diagrams with a single gluon exchange (see Fig. 8 (a)-(d)). The first two diagrams are "factorizable" and their contributions vanish because of current conservation in the limit m c —> 0. For non-zero m c they therefore carry an additional suppression factor m^/mfg PS 0.1. Moreover, their contributions to the decay amplitude are suppressed by small Wilson coefficients. Diagrams (a) and (b) can therefore safely be neglected. From the non-factorizable dia-

88

grams (c) and (d) in Fig. 8, the one with the gluon attached to the b quark turns out to be strongly suppressed numerically, giving a contribution of less than 1% of the leading class-I amplitude. We are thus left with diagram (d), in which the gluon couples to the light quark in the B meson. This mechanism gives the dominant annihilation contribution. (Note that by deforming the light spectator-quark line one can redraw this diagram in such a way that it can be interpreted as a final-state rescattering process.) Adopting a common notation, we parameterize the annihilation contribution to the B
Ax Ufofsfdu^

J do*^*3UfDfB

f*,*$±.

(69)

The B-meson wave function simply integrates to / B , and the integral over the pion distribution amplitude can be performed using the asymptotic form of the wave function. We take $D(V) in the form of (62) with the coefficients a f = 0.8 and a§ = 0.4 (af = 0, i > 2). With this ansatz $D(v) is strongly peaked at v ~ AQCD/"1£>- The integral over $D(V) in (69) is divergent at v = 1, and we regulate it by introducing a cut-off such that v < l - A / m g with A « 0.3 GeV. Then J dv $D(V)/V2 m 34. Evidently, the proper value of A is largely unknown, and our estimate will be correspondingly uncertain. Nevertheless, this exercise will give us an idea of the magnitude of the effect. For the ratio of the annihilation amplitude to the leading, factorizable contribution we obtain A T~

2-nas C+ + C3 2C++C-

fDfB f, * (v) 1 * * D^ « 0.04. Fo(0)m%

(70)

We have evaluated the Wilson coefficients at fj, = mi, and used fo = 0.2 GeV, j B = 0.18 GeV, .Fo(O) = 0.6, and a8 = 0.4. This value of the strong coupling constant reflects that the typical virtuality of the gluon propagator in the annihilation graph is of order AQCD"^B- We conclude that the annihilation contribution is a correction of a few percent, which is what one would expect for a generic power correction to the heavy-quark limit. Taking into account that j B ~ AQ C D(A Q cD/m B ) 1 / 2 , Fo{0) ~ (A Q C D/m B ) 3 / 2 and fD ~ AQCD, we observe that in the heavy-quark limit the ratio A/T indeed scales as A Q C D / T I 6 , exhibiting the expected linear power suppression, (Recall that we consider the D meson as a light meson for this heuristic-analysis of power corrections.)

89 Using the same approach, we may also derive a numerical estimate for the non-factorizable spectator interaction in Bd -> D+ir~ decays, discussed in Sect. 5.1. We find T 5pec ^ 27ras C+ - CfDfB Tlead3 2C++C-F0(0)m%\B

mB

f J

*D{v) _ v ~

^

((l)

where the hadronic parameter XB = O ( A Q C D ) is defined as fQ (d£/£) $.B(£) = mB/XB. For the numerical estimate we have assumed that \B « 0.3 GeV. With the same model for §D{V) as above we have J dv $D(V)/V « 6.6, where the integral is now convergent. The result (71) exhibits again the expected power suppression in the heavy-quark limit, and the numerical size of the effect is at the few percent level. We conclude from this discussion that the typical size of power corrections to the heavy-quark limit in class-I decays of B mesons into heavy-light final states is at the level of 10% or less, and thus our prediction for the near universality of the parameters a\ governing these decay modes appears robust.

7.5 Remarks on class-II and class-Ill decay amplitudes In the class-I decays Bd -> D^+L~, the flavour quantum numbers of the final-state mesons ensure that only the light meson L can be produced by the (du) current contained in the operators of the effective weak Hamiltonian in (15). The QCD factorization formula then predicts that the corresponding decay amplitudes are factorizable in the heavy-quark limit. The formula also predicts that other topologies, in which the heavy charm meson would be created by a (cu) current, are power suppressed. To study these topologies we now consider decays with a neutral charm meson in the final state. In the class-II decays Bd —> D^°L° the only possible topology is to have the charm meson as the emission particle, whereas for the class-Ill decays B~ -> D(*)°L~ both final-state mesons can be the emission particle. The factorization formula predicts that in the heavy-quark limit class-II decay amplitudes are power suppressed with respect to the corresponding class-I amplitudes, whereas class-Ill amplitudes should be equal to the corresponding class-I amplitudes up to power corrections. It is convenient to introduce two common parameterizations of the decay amplitudes, one in terms of isospin amplitudes Ai/2 and A3/2 referring to the isospin of the final-state particles, and one in terms of flavour topologies (T for "tree topology", C for "colour suppressed tree topology", and A for

90 Table 4. CLEO data 2 1 , 2 2 on the branching ratios for the decays B -> D^L in units of 10 - 3 . Upper limits are at 90% confidence level. See text for the definition of the quantities <5 and U.

Class-I ( D W + L - ) Class-II (DW°L°) Class-Ill (£>W°L-) 6

n

B -y Dn

B -y Dp

B -> £>*TT

B -> D*p

2.50 ±0.40 <0.12 4.73 ±0.44 < 22° 1.34 ±0.13

7.89 ±1.39 <0.39 9.20 ±1.11 < 30° 1.05 ±0.12

2.34 ±0.32 <0.44 3.92 ±0.63 < 57° 1.26 ±0.14

7.34 ±1.00 <0.56 12.77 ±1.94 < 31° 1.28 ± 0 . 1 3

"annihilation topology"). Taking the decays B -> DTT as an example, we have A(Bd -> D+7T-) = \J\A3,2

V2A(Bd

A(B-

+ y | ^ i / 2 = T + A,

-»• £>°7T0) = ^ 3 / 2 - \J\AI,2

D°TT-)

=

C-A,

(72)

= VaiAj/a = T + C.

A similar decomposition holds for the other B -¥ D^L decay modes. Isospin symmetry of the strong interactions implies that the class-Ill amplitude is a linear combination of the class-I and class-II amplitudes. In other words, there are only two independent amplitudes, which can be taken to be A1/2 and A3/2, or (T + A) and (C — A). These amplitudes are complex due to stronginteraction phases from final-state interactions. Only the relative phase of the two independent amplitudes is an observable. We define 5 to be the relative phase of Ax/2 and A 3 / 2 , and 6TC the relative phase of (T + A) and (C - A). The QCD factorization formula implies that Al/2 y/2A3/2 C-A T + A

= 1 + 0{AQCv/mb), 0(AQ C D/m&).

5=

STC

0(AQCD/mb),

= 0(1).

(73)

In the remainder of this section, we will explore to what extent these predic tions are supported by data.

91

In Table 4 we show the experimental results for the various B -> D^L branching ratios reported by the CLEO Collaboration 21>22. We first note that no evidence has been seen for any of the class-II decays, in accordance with our prediction that these decays are suppressed with respect to the class-I modes. Below we will investigate in more detail how this suppression is realized. The fourth line in the table shows upper limits on the strong-interaction phase difference S between the two isospin amplitudes, which follow from the relation 16 . 2 9 T(B-) Bv{Bd -> £>°7T0) (74) Sm 2 r(Bd) B r ( B - - • D°ir~) The strongest bound arises in the decays B -> Dn, where the stronginteraction phase is bound to be less than 22°. This confirms our prediction that the phase S is suppressed in the heavy-quark limit. Let us now study the suppression of the class-II amplitudes in more detail. We have already mentioned in Sect. 6.4 that the observed smallness of these amplitudes is more a reflection of colour suppression than power suppression. This is already apparent in the naive factorization approximation, because the appropriate ratios of meson decay constants and semi-leptonic form factors exhibit large deviations from their expected scaling laws in the heavy-quark limit, as shown in (57). Indeed, it is obvious from Table 4 that there are significant differences between the class-I and class-Ill amplitudes, indicating that some power-suppressed contributions are not negligible. In the last row of the table we show the experimental values of the quantity

n=

A{B~ - • £>M°L-) A{Bd -> D W + L - )

r{Bd)

B r ( B - - • £>(*)«£-)

Y T(B-) Br(Bd -> £>(•)+£-) '

(75)

which parameterizes the magnitude of power-suppressed effects at the level of the decay amplitudes. If we ignore the decays B -> D*p with two vector mesons in the final state, which are more complicated because of the presence of different helicity amplitudes, then the ratio 1Z is given by

n= i

+

C-A T + A

1+x

02

(76)

where a\ are the QCD parameters entering the transition operator in (40), and Nc + 1 „ Nc 1 C- + "non-factorizable corrections" CT4 (77) a-i = 2NC ~ 2NC are the corresponding parameters describing the deviations from naive factorization in the class-II decays 16 . All quantities in (76) depend on the nature

92 of the final-state mesons In particular, the parameters x(£>7r) : x(Dp) --

(ml-ml)fDF0B^(m2D) _ (ml-ml)UFB-+°(ml)~U-y> -/pFf^(m^)~U-5'

x(D*ir) -account for the ratios of decay constants and form factors entering in the naive factorization approximation. For the numerical estimates we have assumed that the ratios of heavy-to-light over heavy-to-heavy form factors are approximately equal to 0.5, and we have taken fr> — 0.2 GeV and fo* = 0.23 GeV for the charm meson decay constants. Note that in (76) it is the quantities x that are formally power suppressed ~ (AQCD /m,b)2 m the heavy-quark limit, not the ratios 02/01. For the final states containing a pion the power suppression is clearly not operative, mainly due to the fact that the pion decay constant fv is much smaller than the quantity (fD^/fnB)2^3 « 0.42 GeV. To reproduce the experimental values of the ratios 1Z shown in Table 4 requires values of 02/01 of order 0.1-0.4 (with large uncertainties), which is consistent with the fact that these ratios are of order 1/NC in the large-iVc limit, i.e. they are colour suppressed. The QCD factorization formula (3) allows us to compute the coefficients oi in the heavy-quark limit, but it does not allow us to compute the corresponding parameters 02 in class-II decays. The reason is that in class-II decays the emission particle is a heavy charm meson, and hence the mechanism of colour transparency, which was essential for the proof of factorization, is not operative. For a rough estimate of 02 in B —¥ TTD decays we consider as previously the limit in which the charm meson is treated as a light meson (m c « fflj), however with a highly asymmetric distribution amplitude. In this limit we can adapt our results for the class-II amplitude in B -¥ in: decays derived in x , with the only modification that the hard-scattering kernels must be generalized to the case where the leading-twist light-cone distribution amplitude of the emission meson is not symmetric. We find that

a2

Nc + 1 „

c+(M)

Ne-1~

^^vr "^vr c - (M) + £ wc [C+ ^ + c- (M)] ("6 ln ^+fl + fn)'

(79)

93 where // = / Jo

dv$D(v) \n2v + In v + —- - 6 + in {2 In v - 3) + 0(v) O

f

" = -W F^Hml)ml-XB-J^—-

m

The contribution from / / / describes the hard, non-factorizable spectator interaction. Note that this term involves f CIV$D{V)/V, which can be sizeable but remains constant in the heavy-quark limit implied here (mj —> oo with mc constant). Using the same numerical inputs as previously, we find that / / / « 13 and / / « - 1 - 19i. In writing the hard-scattering kernel for / / we have only kept the leading terms in v, which is justified because of the strongly asymmetric shape of &D(V)- Note the large imaginary part arising from the "non-factorizable" vertex corrections with a gluon exchange between the final-state quarks. Combining all contributions, and taking \i — nib for the renormalization scale, we find a2«0.25e-i41°,

(81)

which is significantly larger in magnitude than the leading-order result a%° m 0.12 corresponding to naive factorization. We hasten to add that our estimate (81) should not be taken too seriously, since it is most likely not a good approximation to treat the charm meson as a light meson. Nevertheless, it is remarkable that in this idealized limit one obtains indeed a very significant correction to naive factorization, which gives the right order of magnitude for the modulus of a-i and, at the same time, a large strong-interaction phase. For completeness, we note that the value for a-i in (81) would imply a stronginteraction phase difference 6 « 10° between the two isospin amplitudes A1/2 and A3/2 in B -> Dn decays, and hence is not in conflict with the experimental upper bound on the phase 5 given in Table 4. The phase STC, on the other hand, is to leading order simply given by the phase of a-z and is indeed large, in accordance with (73). 8

Conclusion

With the recent commissioning of the B factories and the planned emphasis on heavy-flavour physics in future collider experiments, the role of B decays in providing fundamental tests of the Standard Model and potential signatures of new physics will continue to grow. In many cases the principal source of systematic uncertainty is a theoretical one, namely our inability to quantify

94

the non-perturbative QCD effects present in these decays. This is true, in particular, for almost all measurements of CP violation at the B factories. In these lectures, I have reviewed a rigorous framework for the evaluation of strong-interaction effects for a large class of exclusive, two-body nonleptonic decays of B mesons. The main result is contained in the factorization formula (3), which expresses the amplitudes for these decays in terms of experimentally measurable semi-leptonic form factors, light-cone distribution amplitudes, and hard-scattering functions that are calculable in perturbative QCD. For the first time, therefore, we have a well founded field-theoretic basis for phenomenological studies of exclusive hadronic B decays, and a formal justification for the ideas of factorization. For simplicity, I have mainly focused o n B - > Dn decays here. A detailed discussion of B decays into two light mesons will be presented in a forthcoming paper 12 . We hope that the factorization formula (3) will form the basis for future studies of non-leptonic two-body decays of B mesons. Before, however, a fair amount of conceptual work remains to be completed. In particular, it will be important to investigate better the limitations on the numerical precision of the factorization formula, which is valid in the formal heavy-quark limit. We have discussed some preliminary estimates of power-suppressed effects in the present work, but a more complete analysis would be desirable. In particular, for rare B decays into two light mesons it will be important to understand the role of chirally-enhanced power corrections and weak annihilation contributions 12 ' 25 . For these decays, there are also still large uncertainties associated with the description of the hard spectator interactions. Theoretical investigations along these lines should be pursued with vigor. We are confident that, ultimately, this research will result in a theory of nonleptonic B decays, which should be as useful for this area of heavy-flavour physics as the large-mi, limit and heavy-quark effective theory were for the phenomenology of semi-leptonic decays. Acknowledgements I would like to thank the organizers of the workshop, in particular Hsiangnan Li, Wei-Min Zhang and Xiao-Gang He, for the invitation to present these lectures and for their hospitality and support. I am grateful to the students for attending the lectures and contributing with questions and discussions. I also wish to express my gratitude to Wei-Shu Hou for his hospitality during my stay in Taipei. Finally, I am indebted to my collaborators Martin Beneke, Gerhard Buchalla and Chris Sachrajda, who deserve much credit for these notes. This work was supported in part by the National Science Foundation.

95

References 1. M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999). 2. M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Nucl. Phys. B 591, 313 (2000). 3. N. Isgur and M.B. Wise, Phys. Lett B 232, 113 (1989); ibid. 237, 527 (1990). 4. M.A. Shifman and M.B. Voloshin, Sov. J. Nucl. Phys. 45, 292 (1987) [Yad. Fiz. 45, 463 (1987)]; ibid. 47, 511 (1988) [47 (1988) 801]. 5. For a review, see: G. Buchalla, A.J. Buras and M.E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). 6. D. Fakirov and B. Stech, Nucl. Phys. B 133, 315 (1978). 7. N. Cabibbo and L. Maiani, Phys. Lett. B 73, 418 (1978); ibid. 76, 663 (1978) (E). 8. G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22, 2157 (1980). 9. A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94, 245 (1980). 10. J.D. Bjorken, Nucl. Phys. (Proc. Suppl.) B 11, 325 (1989). 11. M.J. Dugan and B. Grinstein, Phys. Lett. B 255, 583 (1991). 12. M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization for B -¥ -KK decays, Preprint hep-ph/0007256, to appear in the Proceedings of the 30th International Conference on High-Energy Physics (ICHEP 2000), Osaka, Japan, 27 July-2 August 2000, and paper in preparation. 13. L. Maiani and M. Testa, Phys. Lett. B 245, 585 (1990). 14. For a review, see: M. Neubert, Phys. Rep. 245, 259 (1994). 15. J.F. Donoghue, E. Golowich, A.A. Petrov and J.M. Soares, Phys. Rev. Lett. 77, 2178 (1996). 16. For a review, see: M. Neubert and B. Stech, in: Heavy Flavours II, ed. A.J. Buras and M. Lindner (World Scientific, Singapore, 1998) pp. 294 [hep-ph/9705292]; M. Neubert, Nucl. Phys. (Proc. Suppl.) B 64, 474 (1998). 17. H.D. Politzer and M.B. Wise, Phys. Lett. B 257, 399 (1991). 18. A. Khodjamirian and R. Riickl, in: Heavy Flavours II, ed. A.J. Buras and M. Lindner (World Scientific, Singapore, 1998) pp. 345 [hep-ph/9801443]. 19. V.M. Braun and I.E. Filyanov, Z. Phys. C 4 8 , 239 (1990). 20. B.V. Geshkenbein and M.V. Terentev, Phys. Lett. B 117, 243 (1982); Sov. J. Nucl. Phys. 39, 554 (1984) [Yad. Fiz. 39, 873 (1984)]. 21. B. Barish et a l , CLEO Collaboration, Conference report CLEO CONF 97-01 (EPS 97-339).

96 22. B. Nemati et al., CLEO Collaboration, Phys. Rev. D 57, 5363 (1998). 23. J.L. Rodriguez, in: Proceedings of the 2nd International Conference on B Physics and CP Violation, Honolulu, Hawaii, March 1997, ed. T.E. Browder et al. (World Scientific, Singapore, 1998) pp. 124 [hep-ex/9801028]. 24. C. Caso et al., Particle Data Group, Eur. Phys. J. C 3, 1 (1998). 25. Y.Y. Keum, H.-N. Li and A.I. Sanda, Fat penguins and imaginary penguins in perturbative QCD, Preprint hep-ph/0004004.

NEUTRINO PHYSICS PETR VOGEL Physics Department, Caltech, Pasadena, CA 91125 USA E-mail: [email protected] Selected topics in the emerging field of the neutrino mass and oscillation searches are reviewed. In the first chapter the overall theoretical framework is sketched. The second chapter is devoted to the neutrinoless double beta decay. It is shown that the observation of the 0i//8/9 decay would be complementary to the oscillation results, and would establish the absolute scale of neutrino masses. The third chapter is devoted to the oscillation searches using nuclear reactors, in particular to the tests of the atmospheric and solar neutrino oscillations.

1

Lecture 1: Generalities

In this lecture various issues encountered in the study of neutrino intrinsic properties will be briefly discussed, telegramm style. 1.1

Introductory remarks

We have entered a new era of neutrino physics. In the last few years we have witnessed several remarkable discoveries: • Oscillations of atmospheric neutrinos • Solar neutrino deficit • K2K oscillation hints • LSND oscillation observations These discoveries, some of them more firmly established than others, offer perhaps a first glimpse of the "Physics Beyond the Standard Model". In order to appreciate the significance of this development we shall concentrate in these lectures on the intrinsic neutrino properties, and the ways one can study them. The things we would like to know are: • Are neutrinos massless as the standard model assumes, or do they have some, albeit small, mass? • Why are the masses so small? Do they follow any pattern? 97

98 • If neutrinos are massive, are the neutrinos produced in weak decays particles with a definite mass (so-called mass eigenstates), or are they mixtures of the mass eigenstates? • Is the total lepton number L conserved? In other words, are massive neutrinos Dirac particles (with distinct antiparticles) are are they Majorana particles? • Are massive neutrinos stable? If not, what are the decay modes and/or decay rates? • Do neutrinos couple to the electromagnetic field? Do they have nonvanishing magnetic and/or electric dipole moments? • Which experiments can one do to answer these questions? And what are the experiments that have been done already telling us? 1.2

How can one determine neutrino mass?

The most straightforward way is to use the energy and momentum conservation and detect charged particles created together with neutrinos in weak decays instead of the difficult detection of neutrinos. For example, in the 7r+ -¥ /J,+ + fM decay, muon momentum is related to the pion mass mv, muon mass mM, and neutrino mass m„ by ml = m\ + m\-

2m^^{pfl + mj) .

(1)

The trouble is that the neutrino mass enters as a difference of two large numbers. Present limit (after resolving troubles with an incorrect determination of the pion mass) is mV/i < 190 keV 1, not very good considering the cosmological bound. For electron antineutrinos the best mass limit is obtained from the study of three-body decays, such as the nuclear beta decay, (Z, A) -+ (Z + 1, A) + e~ + ve . In them the electron spectrum has the shape ^ = canst x pE(E0 - E)[(E0 - Ef - mlfl2F{Z +1,E), (2) aE where E,p is the electron energy and momentum, E0 is the endpoint energy and F(Z + 1, E) is the Fermi function descibing the Coulomb effect. Obviously, appreciable sensitivity to neutrino mass exists only near the endpoint EQ, where the spectrum terminates abruptly (with an infinite slope), manifestly distinguishing it from the spectrum for the case of massless neutrino which has a vanishing slope. In an experiment, the spectrum thus needs to

99

be studied near E0, so that E0 - E = AE ~ mv. However, the number of decays, AN /AE ~ (AE/Q)3, becomes extremely small there. Recently, various difficulties in the analysis of the tritium beta decay which resulted in the unphysical negative m 2 fitted value were overcome; the best present limit is mv < 2.2 eV (95% CL) from the Mainz experiment 2 . Another, conceptually simple, but in practice difficult, method uses the neutrino time-of-flight. Clearly, if one can determine both the velocity and the energy of a particle, it is easy to deduce its mass. Indeed, from

V =

p E

=

(E> - m 2 ) 1 / 2 E *

1 _

m2 2 ^ '

(3)

for an ultrarelativistic particle travelling a distance d, the time delay caused by a nonvanishing rest mass equals Ai(ms) = 5.15(d/10 kpc)(m/leV) 2 (10 MeV/E)2

,

(4)

where the time in milliseconds, the mass in eV, the energy in units of 10 MeV, and the distance in units of 10 kpc, which is approximately the distance to the galactic center. Obviously, only Supernova neutrinos can be used for this purpose. However, the mass determination is complicated by the rather wide (and unknown) duration of the pulse, finite statistics and associated statistical fluctuations, and the difficulties of separating the signal of different flavors. Last but not least, the SN frequency is at best only a few per century. However, when the next supernova is detected by its neutrinos, we will have a chance to determine mVll and m„T down to ~ 20 eV and m„, to an accuracy comparable to the present limit of a few eV (see Ref. 3,4 ). Thus, these direct mass determinations are unlikely to reach sensitivities (i.e., upper limits) below the ~ 1 eV range any time soon. On the other hand, the oscillation experiments give lower limits on the neutrino masses which are typically much smaller. I will argue in the next lecture that among the various possibilities, the neutrinoless double beta decay appears to be the method which offers at present the best hope of reaching the mass sensivity suggested by the oscillation results. 1.3

Mass terms; see-saw mechanism

The two Lorentz invariant and hermitian neutrino mass terms are: ip'ip and •ij}ctpc (Dirac) as well as ^ij)0 and frip (Majorana). (In Dirac-Pauli representation charge conjugation is tpc ~ 72V'*-) The most general mass Lagrangian for one 'flavor' therefore depends on

100

three real parameters mD,ml,m2

(TOM = m\ + 17712),

The parameters mo and TUM can be connected to other known masses by the see-saw mechanism introduced by Gell-Mann, Ramond & Slansky and by Yanagida in 1979. This mechanism is based on the Majorana mass and is therefore unique to neutrinos. To see, schematically, how it works, rewrite LM in the basis of chiral projections $ L , ^ R ,

M' = (mKm°)

,

(6 )

where rriR = m\ + |m 2 | , rriL =m\ — |m 2 |Now make the assignments m i ~ 0, rriR ~ MQUT — 10 14 - 10 16 GeV, mo — Tnfermion , where mfermion represents either charged lepton or quark masses. There are two eigenvalues of M', a very heavy (mostly VR) Majorana neutrino with m ~ MGUT, and a light one (mostly * L ) m ~ rn2ermion/MGUTThus, one might expect that m„e : m„„ : mVr = m2e : m£ : m2.,

(7)

mv, : mVii : m„T = ml : m2c : m2.

(8)

or,

While we do not know if this works, it would explain the smallness of neutrino masses, and suggest that they are Majorana particles. 1.4

Oscillation phenomenology

Assume for simplicity that only two neutrinos flavors exist. Diagonalization of L M then gives \ve) = c o s % 1 ) + s i n 0 | i / 2 ) I^M) = ~sin9\vi)

(9)

+ cos6\i/2) ,

where the mass eigenstates v\ and vi evolve as \ML) > = e - i p i i |i/i(0)>; M L ) > = e~ip>L h ( 0 ) > .

(10)

Consequently, a beam that began as ue evolves as \ve(L) >=cos9e-ipiL\u1(0))

+sm8e-ip*L\v2(0))

•

(11)

101

Projecting again on 'weak eigenstates' we find that \ve(L) > = ( e - i p i i c o s 2 ^ + e- i p 2 Z 'sin 2 ^|i/ e ) + c o s e s i n e ( e - i p 2 i ' - e - i p i Z ' ) | ^ ) . (12) Thus, such a beam now contains a component of fM. Therefore, the probability of the flavor oscillation is P(„ e - •

Vfl)

= | ( ^ | ^ e ( i ) ) | 2 = \ sin2 26 (l - cos ^ ' ^ L J

,

(13)

thus revealing the oscillatory character of the signal at L. There are two basic approaches to the experimental study of neutrino oscillations. If one begins with a beam of a definite flavor, say ve as above, one can look for the appearance of the 'wrong' flavor (lepton flavor number nonconservation) described by Eq. (13). Naturally, the beam energy in this case must be high enough that the charged current reactions, revealing the neutrino flavor, are energetically possible. Or one can look for the disappearance, i.e. deviation of the beam intensity from the simple geometrical scaling. Such a test can be performed at all energies, and involve a sum over all possible final flavors. The absolute flux measurement, however, must be performed in this case. The characteristic length is 2EV 2.48x£„(MeV) or L L 0 8 C = 27T,—J\m\-m\\ 2i »' ° r Losc(meters) »°c(™terS)- = j—^— | m 2 _m2mi^„uB 2|(eV)2 •

(14)

Note that Losc is proportional to the neutrino energy E„, and inversely proportional to the mass difference Am 2 = |m 2 — m\\. In other words, for a given Am 2 the oscillation pattern depends on the ratio L/Ev. Generally, the transformation between the flavor neutrino eigenstates \v{) and neutrino states of the definite mass \vi) is

k<> = I > u k > ,

(is)

i

where Uu is a unitary N x N matrix, recently often denoted as the MakiNakagawa-Sakata (MNS) matrix, and |m?-m?|£

Pi* -> «*) = £ \UnU+\2 + X^YtUuUZ,UlJU+et-t*rJi

i

j^i

.

(i 6)

102

1.5

How many parameters we have to determine?

In an oscillation experiment, one gains information on the mass differences Am 2 from the behaviour of the signal as the function of L/Ev, and information on the mixing matrix from the amplitude of the oscillations. How many experiments do we have to perform in order to completely determine the mixing matrix and all Am 2 values5? The mixing matrix is a TV x TV unitary matrix with TV2 parameters. (The matrix itself has 27V2 parameters, subject to TV normalization and TV (TV - 1) orthogonality conditions.) If T-invariance is valid the mixing matrix is an orthogonal matrix and contains only TV(TV — l ) / 2 parameters. In generality, the mixing matrix therefore contains TV(TV — l ) / 2 mixing angles and TV (TV + l ) / 2 phases. However, TV phases of charged leptons can be chosen arbitrarily and hence are unphysical. For Dirac neutrinos additional (TV — 1) phases can be eliminated as well. This cannot be done for Majorana neutrinos, for which there are consequently TV - 1 undetermined CP phases. From this follows that there are (TV — 1)(TV — 2)/2 true CP violating phases, just like in the case of quarks. In practice, for TV = 2 we have 1 mixing angle, no CP violation, and 1 CP phase in 0.0 decay. For TV = 3 there are 3 mixing angles, 1 CP violating phase (like in the CKM matrix) and 2 CP phases governing lepton number violation. 1.6

CP and T violation in neutrino oscillations

What are the signals of the CP and T violation 6,7 ? If v

t = YjUuVi'then

b yC P T s

ymmetry

Q

u

*ipi •

t =S

i

(1T)

i

Generally, as stated earlier, a beam which began as vy will at time t have the probability amplitude for vi AW (t) = (vi(t)\vt>) = £ UuUfaexpl-iEit]

.

(18)

i

From CPT it follows that A*w(-t) = Au< (t), while \AW (-*)| 2 = \AW (*)|2 (T invariance), and \Aw{t)\2 = |j4««(t)| 2 {CP invariance). Therefore, in order to establish violation of T or CP one would have to show that P(ve - • vi) = \Au>(i)|2 ± P{pv -* vt) = \Au, {t)\2 ,

(19)

i.e., that for example the probability of v^ oscillating into ve must be different from the probability of PM oscillating into ve. It immediately follows that there

103

are no T or CP violating effects observable in 'diagonal', i.e., disappearance oscillation experiments (e.g., solar and reactor). For the usual case of three neutrino flavors, one can parametrize the mixing matrix in terms of the three angles 9i, #2, and 93 and the CP violating phase 5. ve\

(

C1C3 -C2S3 - Sis2c3eiS S2S3 - sic2c3eiS

C1S3 sie~'s\ I vx lS c 2 c 3 - SiS2S3e cis 2 V2 -S2C3 - siC2S3ei6 cic 2 / \etSu3t

(One often sees a different notation, not more intuitive, namely 81 = 0\3, 62 = #23, and 03 = 0 i 2 ) . After some algebra, it turns out that the T or CP violation is P(p, -> e) - P(p -* e) = -[P(p, -> f) - P(/* -> T)] = P(e -» r) - P(e -> f !J2&) -4ciS 1 C2S2C 3 s 3 sin5[sinAi2 +sinA 2 3 + s i n A 3 1 ] , where Ay = (m? - m | ) x X/E. Thus, we see that: a) The size of the effect is the same in all three channels, b) CP violation is observable only if all three masses are different (i.e. nondegenerate), c) and all three angles are nonvanishing, d) Naturally, the CP violating phase must be 6 / 0, e) Finally, when L/E is large ( » 1/Am 2 ) the effect averages to zero. 1.7

Dirac and Majorana neutrinos

For free fermions of mass m one can rewrite the four-component Dirac equation as a coupled system of two two-component equations:

»(£"a„)** - m*L = °

( 21 )

t(a"5M)*L - m*fl = 0 , where $ # , ^1 are the chiral projections (eigenstates of 75 with eigenvalues ±1) and or" = (a 0 , -3) , £" = {a0, a). The chiral projections $ « , $ £ are Lorentz invariant; they coincide with the states of helicity ± 1 only if m = 0. The helicity of a state is not, generally, Lorentz invariant. In contrast, Majorana equation of a particle with mass m is a twocomponent equation, which for the field ^R has the form i^d^R

- me*R = 0 ,

(22)

104

while %>L can have, generally, a different mass m' t(<7"0M)¥i+m'e*i=Ol

(23)

where in the usual representation e = -iav. Note that (up to a possible phase) multiplication by e plus the complex conjugation are the operations associated with charge conjugation. In other words, the L component of the neutrino field and the R component of the antineutrino field obey one Majorana equation ( if <&L = C * R ) , and similarly the R component of the neutrino field and the L component of the antineutrino field obey another Majorana equation (if $ f i = -e^*L). Since neutrinos interact only weakly, and weak interactions affect only one of the two chiral components, all effects related to the difference between the Dirac and Majorana neutrinos scale as m/Ev. This is the essence of the so-called practical confusion theorem which states that the differences between the two possibilities disappear for m —> 0. Nevertheless, once it is established (as we believe at has been established now) that neutrinos are massive, the problem becomes the central issue of neutrino physics (Pontecorvo). So how can one tell? • Any process which violates the total lepton number, e.g. K~ —> 7r+fj,"ii~, v -> v oscillation, • • • is a signature of massive Majorana neutrinos. Among such processes the Oi//3/3 decay appears to be the most promising avenue (see Lecture 2). • Some processes that do not violate lepton number nevertheless can, in principle, but unlikely in practice, make the distinction. Consider, e.g. a radiative decay v^ —> ui + 7. The ratio of right and left polarized photons is 1 for Majorana neutrinos, and (mi/rrih)2 for Dirac neutrinos. As we saw earlier the difference between Dirac and Majorana neutrinos will also affect the transformation properties of the mixing matrices. The gauge transformations h^ei0'lh

iax

Vl^e

vi

(24)

are both allowed for the Dirac neutrinos, but only the first one is allowed for the Majorana neutrinos. Consequently, as stated earlier, for Majorana neutrinos there are additional (N - 1) CP phases that must be determined from the lepton number violating processes, and affect only such processes.

105

1.8

Neutrino Oscillations in Matter

Vacuum oscillations are an interference effect caused by the phase v{t) = v(0)e^x'E^

ss u(0)e-u^

.

(25)

However, when neutrinos propagate through matter, an additional phase difference arises, as shown first by Wolfenstein8 and subsequently applied by Mikheyev and Smirnov 9 . This is the so-called MSW, or matter oscillation effect. In matter, ipx -» ipnx, where n is the index of refraction . IITN

n^l+fHf/,(())

,

(26)

P One can understand schematically what happens, by remembering that the ve propagating in matter interact by the W exchange with electrons, and other flavors do not. Thus Eeff*p+T^

+ {eu\Heff\eu)

w p + — ± V2GFNe

,

(27)

where ± is for ve respectively De. Therefore, in matter there is an additional phase, now only for electron neutrinos, ue{x) = ve(0)eipnx

= ve{Q)e-iy/^GpN'x

,

(28)

and a corresponding matter oscillation length 1.7 x 10 7 L0 = —p= « — —IT m . (29) V2GFNe p(gcm-3)f The effective mixing angle 6m in matter depends on the vacuum mixing angle, and on the vacuum and matter oscillation lengths Losc and L0 through the relation T

2TT

tan 20 m = tan 29 (1 - - ~ sec 20 J

(30)

One can see easily the possibility of resonance as well as the dependence on the sign of Losc and through it on the sign of Am 2 . Of particular relevance are matter oscillations for neutrinos propagating through a medium with a changing density, such as the Sun, Supernovae, and to some extent Earth. Analogous matter oscillations may be used to identify the difference between the sterile (i.e. noninteracting by the neutral currents) and standard weakly interacting neutrinos.

106

1.9

Neutrino decay

A heavy massive neutrino uH, with mass > 2me can decay into a lighter one and e~ 4- e + -+ e +e+

VH

(31)

+ut

The decay is described by the left graph below for a Dirac neutrino, while both graphs contribute for a Majorana neutrino.

Uei

Ue

The decay rate is easy to calculate -iCM _

G\

mz

1927T3 miH\UeH\Wh(-^)mt

.

(32)

Here h(a) —> 0 for a —> 4 and h(a) —• 1 for a —• 0 as expected (assuming that mVi is negligible. If the e~e+ pairs are observed in a detector of length Ldet the probability of the decay is p

'^JLTCMLdet

=

(33)

Naturally, the number of observable pairs also depends on the probability of producing VH which contains factors \Uen|2 or \U^H|2 depending on the neutrino source. Neutrinos can also decay radiatively VH

-+l + fi •

(34)

The decay rate can be calculated in the standard model. For a Dirac neutrinos of negligible mass mVi „n2

pCM _ ^ GFa

m5

16 128TT 4 mlVH

J2UiHUu

mi

Mw

(35)

107

In the Standard model radiative decay lifetimes are much longer than the age of the universe due to the presence of the {mi/Mw)i 'GIM' suppression factor. However, various ways to speed up the process, i.e. to enhance the coupling to the electromagnetic field has been proposed. They include e.g. left-right symmetric models, or models with additional particles such as heavy leptons etc. For neutrinos propagating in matter there is another mode involving the W exchange 10 . This decay mode does not involve the GIM suppression, but it is still very slow. 1.10

Neutrino magnetic moment

Massive Dirac neutrinos will not only 7 decay in the Standard Model but also, by the same mechanism, acquire a tiny magnetic moment Vv

3eGF

v HBohr : 3 X 1 0 ~ 1 9m—

=m„

8TT2X/2

(36)

lev In extensions of the SM larger /j,v can occur, even without large neutrino masses. How can one measure fxv? In i/-electron scattering the weak and electromagnetic parts contribute incoherently: da dT

2Glme

2

,

9L+9R

2

1-

T T^-

-

9L9R

El

2

+

T/Ev 2 7ra 1 ^ m? T

(37)

The last term is of a 'universal form'. The only quantity that depends on the beam composition is the effective magnetic moment \iv. This moment, in principle, depends on the oscillation parameters. So, e.g. in a beam which was initially a pure ue, the effective magnetic moment is Me

2.

£"« fee-iB L k

Mjfc

(38)

i However, in many cases of practical relevance, the magnetic moments 'do not oscillate', i.e. the effective /z„ does not depend on distance or beam energy n . One can separate the weak and electromagnetic scattering by the recoil electron shape (through the 1/T dependence). The existing limits are close B

MBoft.r-

108

2

Lecture 2: Neutrinoless double beta decay

Observation of neutrino oscillations allows us to determine the neutrino mass differences Am 2 = \m\ - m\\ and hence the lower limit of the mass of one of the two neutrinos involved, say m i , since mi > \ / A m 2 . We cannot, however, distinguish between the case where m 2 ~ 0 and hence mi ~ v^Am2 on one hand, and the case of almost degenerate neutrinos where mi ~ m2 > \ / A m 2 on the other hand. Among the various possibilities to fix the absolute neutrino mass scale, neutrinoless double beta decay offers at present the best hope. 2.1

What is {3/3 decay?

In double beta decay two neutrons bound in a nucleus are simultaneously transformed into two protons, like e.g. in the transitions 32Ge76 —• 34Se76 or 5 4Xe 136 -» 5 6 Ba 1 3 6 . In order to conserve the electric charge two electrons must be emitted at the same time, and perhaps one or more neutral particles as well. If only the two electrons and nothing else are emitted, the process, called neutrinoless /3/3 decay or 0v(3(3 , violates the lepton number conservation. Its existence implies that the electron neutrino have a massive Majorana component. The rate of the 0u{3/3 decay is equal to (for more details see, e.g. Ref. 12 ) l/T??2=G(EtoUZ)M2X2

,

(39)

where G(Etot,Z) is the calculable phase space factor, M. is a nuclear matrix element, calculable with difficulties, and X is an unknown fundamental parameter (e.g. the effective neutrino Majorana mass). Until now we did not know what value of X to expect, we even did not know what physics determines the X. By determining a lower limit on the Oi//3/3 lifetime, we were able to constrain simultaneously a whole array of possible sources of X. (It is reasonable to assume that the various possible mechanism do not cancel each other.) In fact, a variety of Feynman graphs can contribute to the Of/3/? decay rate. Among them is a) the exchange of a light massive Majorana neutrino with left-handed weak interaction, b) the exchange of a light massive Majorana neutrino combined with the righthanded weak interaction in one of the verteces, c) the exchange of a very heavy Majorana neutrino, or d) the exchange of various supersymmetric particles, to name just a few. Each of these possibilities contains some 'new physics' so far unobserved anywhere else. However, neutrino mass has been observed, and most theorists

109 believe that the massive neutrinos are Majorana particles. Thus, the mechanism a) seems to be the least exotic possibility. We will in the following discuss only that mechanism. Note in passing that the /?/? decay in which two neutrinos and two electrons are emitted together, so called 2v/3f3 decay is the second order weak process that does not violate any conservation law (again, more details can be found in 13 ). That process is just slow, but has been observed in many cases now. We will mention it only as a the testing case for the evaluation of the nuclear matrix elements, and as an unwanted source of background in the search for the lepton number violating Oi>00 decay. 2.2

Neutrino mass pattern

Here we shall use the results of the searches for neutrino oscillations and attempt to derive the pattern of neutrino masses, and from it the effective Majorana mass that governs the rate of Qv/3/3 decay. Let us assume that the atmospheric neutrino observations really imply that v^ and vT mix maximally, with Am 2 ~ 3 x 10 _ 3 eV 2 , and that the solar neutrino observations are explained by the large mixing angle (LMA) solution with Am 2 ~ (1 - 10) x 10 _ 5 eV 2 . These solutions are preffered by the most recent data 14 ' 15 and suggest that the neutrino mixing matrix has the bi-maximal form 1/V2 1/V2 0\ fuA - 1 / 2 1/2 1/^2 U2 • 1/2 - 1 / 2 1/^2 J \f3/ Neutrino masses can be arranged in one of the two patterns depicted below, with A m ^ ~ Amf,2 ~ 3 x 10~ 3 eV 2 and A m ^ ~ 1 - 10 x 10~ 5 eV 2 in both of them. However, it is impossible to decide which one is correct, and to decide where the zero mass value is. Normal hierarchy

Inverse hierarchy 7713

JTli

mi mi

m3

110

Now, in double beta decay with an exchange of a light Majorana neutrino, the quantity X is the effective neutrino Majorana mass defined as

X = < m ^ >= X > e i i | W i ,

(40)

i

where e* = ± 1 is the CP phase of the mass eigenstate i. For the bimaximal mixing only i = 1,2 matter and there is only one phase factor e, thus < meff

>= 2 ( m i +

e m

2)

(41)

Considering the neutrino mass patterns depicted above, it is easy to see that for e = + 1 the magnitude < m^f, > > 10 meV is a reasonable bet. What would, on the other hand, e = — 1 mean? It would mean that two mass eigenstates, 1 and 2, are almost degenerate, and have opposite CP phases. Thus, they form a 'quasi-Dirac' particle, since as pointed out above, Dirac neutrino is equivalent to two exactly degenerate Majorana neutrinos with opposite CP phases. It remains to be seen whether it is 'natural' to encounter such a situation in reality. In any case, the above crude analysis suggests that a search for the Qv0ft decay with sensitivity well below 100 meV has the potential of fixing the neutrino mass scale. 2.3

Complications

What if the mixing matrix slightly deviates from the bi-maximal form? This can happen if the third mixing angle #1,3 is small, so that sin0i ]3 = 8. The mixing matrix is then, to the first order in 8 (vt\ K \Ur)

(

=

1/V2 l/>/2 8 \ -1/2(1 + *) 1/2(1-*) 1/V2" \ 1 / 2 ( 1 - 5 ) - 1 / ( 1 + 5)2 1 / V 2 /

M\ U, \Vs)

From the Chooz and Palo Verde experiments (see Lecture 3) we know that sin 2 (20i ]3 ) < 0.2 and therefore 82 < 0.05 is indeed small.. The formula for the effective neutrino mass is then modified to < m"ef} >= - (mi + e m 2 + 282e'm3)

,

(42)

where e' is the additional CP phase of m 3 with respect to m i . In most considered cases that additional term changes nothing. Nevertheless, any

111

improvement in the determination of the third mixing angle, i.e. of S2, would be very welcome. What if we do not disregard the LSND result as we have done so far? We must postulate then the existence of a fourth sterile neutrino va and also of a fourth massive neutrino VQ. The pattern of masses consists in such a case of two pairs, separated by the rather large interval Am = 0.1 — 1.4 eV dictated by the LSND result. The inverse hierarchy is already (or soon will be) excluded by the Of/?/? results. The normal hierarchy will be constrained only when sensitivities near 10 meV range are reached. Thus, allowing #i | 3 ^ 0 or including the LSND result does not change the conclusions about the significance of the Of/3/3 decay. (For a detailed quantitative analysis, considering all possible solution of the solar neutrino proble, see also Ref. i e .) 2.4

Experimental

situation

The search for the Of/?/? decay has been going on for a long time. Briefly, in the experiment one uses certain amount of the material containing the parent nuclei, and tries to identify, using various detection methods, the peak in the sum electron spectrum corresponding to the Q value of the decay. The presence of the sharp peak at Q is the distinguishing feature of the Of/?/? decay; in contrast, the allowed 2v/?/3 decay is characterized by the contiuous spectrum which vanishes at Q and has a broad maximum below Q/2. Experimentally, the best cases involve materials that can be used at the same time as a detection medium. The longest halflife limit of 1.9xl0 25 years has been achieved in 76 Ge with an exposure of more than 20 kg-y, Ref. 17 (the limit quoted here is based on a later communication by the authors). Quite large exposures were also achieved in the case of 136 Xe, 8 kg-y, Ref. 18 , T 1 / 2 > 4.4 x 1023y, and 130 Te, 0.85 kg-y, Ref. 19 , T 1 / 2 > 1.4 x 1023y. In order to translate the lower halflife limits into the upper effective neutrino mass limits we have to know the nuclear matrix elements. In order to evaluate such matrix elements, one needs to know the many-body wave function of the initial and final nuclear states. This represents a nontrivial problem which can be solved only in various approximations. Moreover, it is difficult to judge the degree of validity of these approximations. There are two basic widely used methods. The Quasiparticle Random Phase Approximation (QRPA) which treats a class of graphs (or configurations) to all orders, but neglects other terms, and the Nuclear Shell Model (NSM) which treats exactly all states of a limited number of nucleons. The convergence with respect to the number of allowed states is the basic issue in

112 Table 1. Ov/3,8 halflives corresponding to the < m^ff > = 100 meV for various candidate nuclei, and the QR.PA and NSM nuclear matrix elements (columns 2 and 3). The source mass needed for 1 decay/year (columns 4 and 5).

Nucleus

™Ge

100

Mo Cd

116

130Te 136

Xe

QRPA xl0 2 6 y 2.3 1.3 0.49 0.49 2.2

NSM xl0 2 6 y 17 12

QRPA kg 42 31 14 15 72

NSM kg 310 392

the NSM. The Table below illustrates the range of halflives one obtains for a representee < mveff > = 100 meV. The table also gives the source mass (in kg) needed for 1 decay event per year. The QRPA results are based on the matrix elements of Staudt et al. 20 and the NSM on the matrix elements of Caurier et al. 21 Clearly, it is important to better understand the origin of the difference between the two methods (and to some extend between the results obtained by the same method but by different authors). But it is equally clear that in order to move below the 100 meV range, the exposure must be increased to hundreds of kg-y, or better yet to tons-y. 2.5

Experimental future

We argued above that the < mveff > range of 10 - 50 meV is the range where we expect to actually see the Qv/3/3 decay. In order to reach the corresponding sensitivity, two challenges must be overcome. First, one needs very large fiducial mass (preferably ~ tons). This implies that an unprecendented isotope enrichment program must be used. Such program is not only technically challenging, but also potentially very costly. Second, new ways to reduce the backgrounds must be implemented. The present experiments are already dominated by background. Note that the sensitivity varies as ~ l/{Nt)1/2 if there is no background but as ~ l/(iVi) 1 / 4 if the background scales with the exposure Nt. Thus, while the increase in Nt is essential, without the corresponding reduction in background it is unproductive. Here is a very brief list of the running, under construction, and

113

planned experiments. References are given only sparingly, the reader can usually find them on the Neutrino Oscillation Industry Web page at http://hepunx.rl.ac.uk/neutrino-industry/. a) First, the running experiments. These are of multi-kg size and will presumably continue for several more years: • Heidelberg-Moscow, last published 24.2 kg-yr, present halflife limit 1.9xl0 25 years (see the comment above). • IGEX, last published 5.7 kg-yr, present halflife limit 0.8 xlO 2 5 years. b) Under construction (or planned) experiments with ~ 10 kg of source material, which either have the enriched material already, or do not need it, or have a good chance of acquiring it: • NEMOS, tracking detector + calorimeter in the Frejus tunnel plans to use ~ 10 kg of 100 Mo. Operational, at least in part, at the present time. • Cuoricino, cryogenic detector in Gran Sasso plans to use 56 crystals of natural T e 0 2 (~ 12 kg of 130 Te). Results with 20 smaller crystals published. Approved and funded. • CAMEO II, so far mostly an idea, will use the modified CTF of Borexino, plans to have 65 kg of enriched 1 1 6 CdW0 3 scintillating crystals, estimated sensitivity ~ 1026y. c) Proposed ton-size experiments (in alphabetical order of the corresponding acronyms): CUORE, EXO, GENIUS, MAJORANA, MOON. • CUORE 2 2 : Cryogenic set-up of 1020 crystals of natural Te0 2 (34% abundance of 130 Te), represents 210 kg of 130 Te. It would be housed in a single specially constructed dilution refrigerator in Gran Sasso. A tower with 20 Te0 2 crystals cooled to 8 mK has reached a resolution of 8 keV at 2.6 MeV and run for 0.66 kg-y of 130 Te, with 0.5 c/(fceV-kg-y) background. • EXO 23 : Assumes that ton quantities of enriched 136 Xe will be available (negotiations are in progress). Lasers will be used to individually detect the daughter 1 3 6 Ba + ions resulting from the /?/? decay of 136 Xe. The detector will be either a large high pressure TPC or (this is currently the more favored scenario) a liquid Xe detector from which the B a + ions will be extracted and identified in low pressure environment. Projected

sensitivity for one ton is T1/2 ~ 1027y , going to T 1 / 2 ~ 1028y for a 10 tons ultimate experiment. Presented to SAGENAP, some R&D funds approved. • GENIUS 24 : Proposed to operate 300 'naked' detectors of enriched 76 Ge ( 1 ton) in a large (12 m diameter) liquid nitrogen shielding. Projected sensitivity is < rn"ff > ~ 10 meV. Located either at Gran Sasso or WIPP. A prototype detector with 'naked' Ge crystals has been tested. Requires enrichment of 1 ton of 76 Ge. • MAJORANA 25 : Uses the successful observation of the 2f/5/3 decay of 100 Mo to the excited 0 + state in 100 Ru. This is accomplished above ground by observing in coincidence the two 7-rays de-exciting the 0 + state. Similar transition very likely exists in 7 e Ge and offers efficient background suppression. Moreover, transitions to the 76 Se ground state will also have a distinct signature. It is proposed to use 500 kg of 76 Ge in the form of 210 crystals. The enrichment could be done in Krasnoyarsk, the experiment will be located at the WIPP site, Carlsbad, NM. Projected sensitivity is < mve^ > ~ 20 meV. Requires enrichment of 500 kgof76Ge. • MOON2*: Natural Mo (9.6% of 100 Mo) in thin foils (0.05 g/cm 2 ) interleaved with scintillators, 34 tons, good localization (10 cm 2 ) is a must. Projected sensitivity is < mvef* > ~ 30 meV or 7 \ / 2 ~ 1027y. Possible used of the enriched 100 Mo is being considered. 2.6

00 decay conclusions

Thus, we see that based on the existing neutrino oscillation evidence, there is a good possibility that the neutrinoless 00 decay corresponding to < m | ^ > > 10 meV really exists. To prove that it indeed does exist, i.e. to actually observe it, is of extreme physics interest. This can be achieved only with much larger and thus more challenging and costly experiments than those presently running. There is a wealth of proposals how to do it, which have to be carefully scrutinized and, if found feasible, the international physics community should wholeheartedly support them (or at least the best of them). Hand in hand with the development of the novel detection schemes for Ot/00 decay, the community should also support the attempts to utilize the large scale isotope enrichment program in the former SU.

115

3

Lecture 3: Oscillation searches with terrestrial low energy neutrino sources - reactors

These are not the most exciting of the v oscillation experiments, at least as of now. However, they are potentially decisive for the existence of sterile neutrinos and/or resolution of the 'solar neutrino puzzle'. At the same time they are of moderate size, allowing to obtain important results with relatively modest investments in manpower and resources. Reactors produce only ve of low energies (< 10 MeV), strictly isotropic; the spectrum and intensity is well understood. Clearly, with reactors one can perform only the disappearance oscillation tests. In them, a detector is placed at the distance L and one compares the signal, corresponding to the neutrino energy Ev, to the expectation for no oscillation. Since the reactor energies are low, the values of L/Ev are high, and the reactor experiments are sensitive to low values of Am 2 . In fact, these are the lowest Am 2 values accessible with terrestrial neutrino sources. The interpretation is based on the premise that the 'no oscillation' signal is accurately known. 3.1

Reactor Pe spectrum

Reactors are powerful sources of electron antineutrinos, with the source strength of about 6 x l O 2 0 ^ s - 1 for a typical 3 GWtherm power reactor. These Pe are produced by the @~ decay of the fission fragments, with approximately 6 ue per fission. It is easy to understand this number. Take, as an example, the most common 235 U fission, which produces two unequal fragments, and typically two new neutrons that sustain the chain reaction, 235

U + n->X1+X2

+ 2n.

(43)

The mass distribution of the fragments (so called fission yields) is shown in Fig. 1. The lighter fragments have, on average, A ~ 94 and the heavier ones A ~ 140. The stable nucleus with A — 94 is 40 Zr 94 and the stable A = 140 nucleus is 53 Ce 140 . These two nuclei have together 98 protons and 136 neutrons, while the initial fragments, as seen from the equation above, have 92 protons and 142 neutrons. To reach stability, therefore, on average 6 neutrons bound in the fragments must /? decay, emitting the required 6 ve. While the total number of ve is straightforward to estimate, and can be accurately calculated given the known fission yields, their energy spectrum, which is of primary interest for the oscillation searches discussed here, requires

116 10.000

0.100

"p

0.010

0.001

70

80 90 100 110 120 130 140 150 160 Mass number A

F i g u r e 1. Yields for 2 3 6 thermal neutron fission (normalized to 2 for the two fragments)

more care. In fact, the most commonly used neutrino detection reaction, the inverse neutron beta decay, has 1.8 MeV threshold. Only about 1.5 i/e/fission (i.e. ~ 25%) of the total are above that threshold and hence can be detected. The conceptually most straightforward method of determining the spectra of the ve is the summation of the contributions from all individual 0~ decays. Thus diV (44) fpr = ^Yn{Z,A,t)Y,Ki(K)PA^,E^,Z) , dEp where Yn (Z, A, t) is the number of p decays per unit time of the fragment Z, A after the fissioning material has been exposed to neutrons for a time t and the label n characterizes each fragment. For t larger than the /? decay lifetime of the fragment Z, A the Yn converges toward the so-called cumulative yield and becomes independent of t. In practical reactor experiments the yields saturate after about a day of exposure. Naturally, each fission fuel is characterized by a different set of yields Yn. The &„,*(£„) are the branching ratios for the zth branch with the maximal electron energy (endpoint energy) EQ. Finally, the function Pp(Ep,El, Z) is the allowed normalized /3 decay spectrum shape, Eq. (2). The weakness of this method is the incomplete information on the endpoint distribution and branching ratios of some fission fragments, in particular those with very short lifetimes and therefore higher decay energies. These

117

'unknown' decays contribute as much as 25% of the ue at energies above 4 MeV. In practice, nuclear models are used to supplement the missing data. Examples of calculations based on this method are 27>28. An example of an extension to lower De energies, where the neutron activation of the reactor materials plays a role is in 2 9 . For the ue associated with 2 3 5 U, 2 3 9 Pu, and 2 4 1 Pu, which undergo thermal neutron fission, it is customary to use a less model dependent hybrid method, based on the conversion of the carefully measured electron spectra 30 31 ' associated with the thermal neutron induced fission into the De spectra. Obviously, the electron and ve originate from the same beta decay and share the available endpoint energy. For a single branch the conversion is therefore trivial. However, in general we have many branches, and many nuclei with different charges. In order to perform the conversion, the electron spectrum is expressed as a finite sum of 30 hypothetical beta decay branches with branching ratios b, and endpoints E^, Y(Ee) = J2bik(El,Z)8(Ee,Et0)peEe(Et0

- Ee)2F(Ee,Z

+ 1) ,

(45)

i

where S(Ee,E^) describes the small outer radiative corrections and Z is the average fragment charge. One can now begin with the largest value of E^ (only one branch) and determine the corresponding branching ratio bi, and continue in this fashion step by step until the smallest EQ is reached. Possible variations in the number and distribution of the endpoints E^ affects the resulting De spectrum not more than at 1% level (see 3 0 ) . Having determined the set bi and EQ it is trivial to obtain

Y(EPJ = Y^bik(E^Z)El(E0-Ep,)[(Eo-E^)2-ml}1/2F(Eo-EPe,Z+l)

,

(46) where the irrelevant radiative corrections were omitted. This is the procedure used in deriving the neutrino spectra associated with fission of 2 3 5 U, 2 3 9 Pu, and 2 4 1 Pu which account for about 90% of the reactor ve. The ultimate check of the accuracy of the prediction outlined above consists in comparing the results, in terms of the ue energy spectrum, with the measurements performed in short baseline reactor oscillation experiments. Since such experiment have not reported the observation of oscillations we can assume that their measurement represents the direct observation of the reactor spectrum at production. This assumption is strengthened by the fact that some of the short baseline experiments such as Gosgen 32 or Bugey 3 3 ' 3 4 involved measurements with different baselines observing no difference between the spectra and neither between their normalization. In particular the

118

U . a) 1.1 1.0 0.9--

•-«--

•4-.... 1

i

i

i

5

6

0

1

2

3

4

7

0

1

2

3

4 5 6 7 Positron energy (MeV)

Figure 2. Ratio between Bugey 3 measurements and different predictions. In a) the measurements are compared to the a-priori calculations from Ref. 2 8 . In b) Bugey 3 data is compared to the prediction obtained using the /9 spectra measurements of Ref. 3 l , 3 ° (and the calculation mentioned for 238 U). The dashed envelopes are estimates of the overall systematics.

Bugey 3 measurements were performed at 15—40 m distance from the core and recorded very high statistics (some 1.5 x 105 ve events). The corresponding comparison is shown in Fig. 2. 3.2

Detection reaction

Inverse neutron beta decay, ve + p -> e + + n, is the reaction of choice for the detection of reactor antineutrinos. It is also, by far, the reaction giving the largest yield for the detection of supernova neutrinos. The LSND 3 5 and KARMEN 3e experiments use PM antineutrinos from n+ decay at rest to search for the oscillation appearance of ue events, also detected by this reaction. There are also searches for ve antineutrinos from the Sun. The total cross section for this reaction, neglecting terms of order E/M, is given by the standard formula

a£l=o-o(f + tf)EePe = 0.0952 where Ee,pe

/ EePe EePe \ -42 xlO cm 22 MeV ,/ VlMeV

(47)

are the positron energy and momentum. The cross section can

119

be expressed in terms of the neutron lifetime and the phase space factor / « , . = 1.7152 37 a$ = ^

2

«

>

.

(48)

Jp.sTn

The cross section normalization was measured in Ref. 39 and found to be in agreement with the expectation from the neutron lifetime at the 3% level. This measurement thus tests, at the same time, the overall normalization of the reactor ue spectrum to the same accuracy. The (small) energy-dependent outer radiative corrections to atot are given in Refs. 38>40. The corrections to the cross section of order E/M and the angular distribution of the positrons and electrons are described in Ref. 41 . In Fig. 3 we show the total cross section as well as the quantity (cos0) which characterizes the positron angular distribution (see 4 1 ). The high energy extension of the total and differential cross section has been discussed already in the classic paper 4 2 . Near threshold, however, that treatment must be modified as shown in Fig. 3. 10 8 )

6

2 4

0 -0.01

$" -0.02 o u ""

-0.03

-0.04 0

1

2

3

4 5 6 Ev [MeV]

7

8

9 10

Figure 3. Upper panel: total cross section for ve + p -+ e+ + n; bottom panel: (cos#); as a function of the antineutrino energy. The solid line is the O(lfM) result and the shortdashed line is the O ( l ) result. The long-dashed line is the result of Eq.(3.18) of Ref. 4 2 , and the dot-dashed line contains the threshold modifications to the same. The solid and dot-dashed lines are not distinguishable in this figure.

Once the cross section and ue spectra are known, the corresponding

120

positron yield is easily evaluated as the simple product of these two functions. (For simplicity we neglect possible energy dependent detection efficiency.) In reactor experiments, the neutron recoil is quite small (10-50 keV) and thus the positron energy is simply related to the incoming De energy, Ee+ = EPe - A M , where A M = Mn - Mp ~ 1.3 MeV. The corresponding threshold is Ethr = A M + me ~ 1.8 MeV. In Fig. 4 we show the shape of the positron spectra assuming that there are no neutrino oscillations.

0

1

2

3

4

5

6

7

8

Positron kinetic energy (MeV) Figure 4. The i>e spectrum (dashed), the reaction cross section (dot-and dashed) and their product (full), i.e. the positron yield.

The effect of oscillations is easily included as illustrated in Fig. 5. There we used the oscillation parameters relevant for the atmospheric neutrinos, even though the flavors involved, as we know by now, are different. The distance of 800 m corresponds to the distance actually used in the Palo Verde experiment. The effect would be clearly visible, both in the total yield, and in the spectrum shape. 3.3

Testing the atmospheric anomaly

In order to test the atmospheric neutrino anomaly, which suggests that Am 2 ~ 1 0 - 3 eV 2 , with reactor ve of a few MeV energy, a baseline of the order of 1 km must be used. That, in turn, requires a target (= detector) mass in excess of a few tons. Two experiments have been built to test the hypothesis that ue play a role in the atmospheric neutrino oscillations. The two experiments,

121 25 1

0

.

1

.

1

2

1

.

1

3

.

r

4

5

6

7

8

Position kinetic energy (MeV) Figure 5. Positron yield without oscillations (full curve) and with the indicated oscillation parameters (dashed curve).

Chooz43 and Palo Verde44 are completed now. The Chooz detector used a pre-existing underground cavity under a ~ 100 m rock overburden (~ 300 m.w.e). This shielding allowed the use of a homogeneous detector with large efficiency so that a 5 ton active mass was sufficient for the experiment. The Palo Verde detector, on the other hand, was located in a shallow underground bunker at the Palo Verde Nuclear Generating Station, ~ 80 km west of Phoenix, in the Arizona desert. The power station has three identical reactors, two 890 m from the detector, and the third one at 750 m. The rather large cosmic muon flux produced relatively many secondary neutrons in the detector so that a segmented detector was needed to take full advantage of the triple coincidence given by the e + ionization and subsequent 7's from its annihilation. This more elaborate topological signature reduced the detector efficiency and pushed the fiducial mass to 12 ton. The Chooz detector was built at a distance of ~ 1000 m from the two reactors of the new Chooz power plant of Electricity de France in the Ardenne region of France. The detector was operational during the commissioning phase of the reactors, and was able to record data as a function of the reactor power. The resulting plot, Fig. 6, allows direct determination of the background corresponding to zero reactor power. Both the measured rate and the positron energy distribution agree perfectly with the expectation without

122

•8

> Q

15

10

0 0

2

4

6

8

10

Reactor Power (GW)

Figure 6. Chooz data during the reactor commissioning. This allows to determine the background of 1.1 ± 0.25 events/day.

neutrino oscillations. Thus, one can construct the exclusion plot shown in Fig. 7 below, which clearly excludes the possibility that v^ -¥ ve oscillations are responsible for the atmospheric anomaly. The Palo Verde dector was segmented for active background rejection and used, like Chooz, 0.1% Gd-loaded scintillator to shorten the neutron capture time and shift the neutron capture energy upwards. The reaction signal thus consisted of a triple coincidence (e + and the two annihilation 7), followed by the delayed neutron capture on Gd signal. Again, no deviation from the expected rate was observed. In both experiments the final result can be expressed as the (integrated over energies ) ratio between De detected and expected R = 1.01 ± 2.8%(stat) ± 2.7%(syst)

Chooz

(49)

PaloVerde

(50)

and R = 1.04 ± 3%(stat) ± 8%(syst) in both cases consistent with 1. 3.4

Three flavor analysis

The neutrino oscillation phenomenology was discussed above in subsection 1.4. In particular, the general formula for the oscillation probability was

123

introduced in Eq. (16). That formula can be substantially simplified if one makes the assumption that one of the masses, say m^, differs substantially from the other two (see subsection 2.2). In other words when AmJ ]3 ~ Am2,^ = Am2 » Amf)2 .

(51)

2

2

This means that for L/E ~ 1/Am the terms containing Am 2 can be neglected. The oscillation probabilities can be now expressed in terms of only two mixing angles and one mass difference. P{ue -> i/M) = sin2 2013 sin2 923 sin2 (

™£ J

P(ye -> vT) = sin2 20 13 cos2 d23 sin2 (

™£ J

P{v* "> VT) = cos4 0 13 sin2 2023 sin2 ( ^ ^ j

•

(52)

In particular, in the reactor experiments, where one measures the ue disappearance, P(ve->vx)

= sin2 2 0 i 3 s i n 2 f A £ | ^

.

(53)

The reactor experiments, which do not indicate any need for oscillations, thus constrain the mixing angle #13 as shown in Fig. 7. The figure at the same time indicates that the study of atmospheric neutrinos is relatively insensitive to that mixing angle. Clearly, the angle #13 is small, or vanishing. Thus, the bimaximal form of the MNS mixing matrix, as discussed in subsection 2.2, finds strong support in the reactor experiments. The question, however, remains. How small # 13 really is? Or maybe it simply vanishes? The answer has important consequences. If #13 = 0 exactly, then as shown in subsection 1.6 no CP or T violation effects are observable. Unfortunately, it is unlikely that the reactor experiments can reduce the upper limit on #13 substantially. It remains to be seen how much improvement can be achieved in the planned long baseline appearance experiments. Until now we have neglected, essentially troughout, the evidence for the 9^ —• ve and i/M —• ve oscillations with small oscillation probability of 0.25 ± 0.06 ± 0.04 % of the LSND experiment 45 ' 46 . If that result is indeed attributed to neutrino oscillations, the most favored region of Am 2 is a band from about 0.2 to 2.0 eV 2 . That is clearly incompatible with the existence of only three neutrinos on one hand and the implications of the atmospheric

124

Palo Verde (excluded)

— CHOOZ (excluded) SK 90% CL (allowed)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Figure 7. The allowed region of 813 and Am2 based on the SuperKamiokande preliminary analysis (the region inside the dotted curve). The region excluded by the reactor experiments are to the right of the corresponding curves.

neutrino oscillations which suggest that one of the Am ~ 3 x 10~ 3 eV , and the solar neutrino deficit that suggests that another Am 2 < 1 0 - 4 eV 2 , on the other hand. If, or when, the LSND result is confirmed, the oscillation phenomenolgy would have to be modified substantially. 3.5

Testing the solar v deficit with reactors - KamLAND

The solar neutrino deficit has been very convincingly established in several experiments with different thresholds. Nevertheless, the 'smoking gun' proof, i.e. the demonstration of the oscillatory behaviour as the function of L/Ev, is still missing. Moreover, the exploration of the corresponding L/Ev range is particularly challenging, due to the huge L/Ev values required. The very low-energy of reactor neutrinos makes an oscillation experiment able to reach at least one of the solutions, the Large Mixing Angle (LMA) solution. It should be remembered that, unless CPT symmetry is violated, a reactor experiment (ue disapperance) is an exact replica of the astrophysical experiment (ye disapperance), only built on earth. In order to completely explore the LMA solution one needs a Am 2 sensitivity of at least 1 0 - 5 eV2 at large mixing angles. This, in turn, implies that a » 100 km baseline is needed and that consequently power x fiducial-mass

125

product between 107 and 109 MW t h x tons is required. Such an experiment could only be placed near a concentration of nuclear reactors, which means in Europe, Eastern United States, or Japan. The Kamioka site has the necessary requirements. There is an anti-neutrino flux of c± 3 x 10 6 cm _ 2 s _ 1 from reactors at a distance between 140 km and 210 km there. This flux gives the non-oscillation rate of ~750 kton _ 1 year _ 1 for a C„H2n+2 target.

^

o •o

0.O1 / v

*

3

5 3 o d ^ a

/ 0.008

\

/

/

0.006

X''

v

s :

. v *

;-..:

X"'%" /Am,=2»10'*(?V*

/

/\\

I / W l / \ V ^k.

0.004

0.002

0

\

/

Cou

"

•-«•

\

/

c

Mf <-.*,,«pvrffnft* \

•

/ 1

^ ^ w ^ 2

3

4

5

6

7

8

9

e* energy / MeV

Figure 8. Positron spectra expected at KamLAND for no oscillations and with the oscillation parameters corresponding to the LMA solar solution.

The KamLAND detector is being built in the cavity originally excavated for the Kamiokande detector under the summit of Mt. Ikenoyama in the Japanese Alps. The fiducial volume consists of a sphere containing 1000 tons of liquid scintillator. Simulations indicate that sufficient signal-to-noise ratio will be achieved even without the Gd loading of the scintillator. The predicted positron spectra for the KamLAND detector and different oscillation parameters coverning the LMA solar solution are shown in Fig. 8. After three years of data taking, and the expected 10/1 signal to noise ratio, it is projected that the oscillation parameters Am 2 and sin2 26 should be determined to ~ 10% and ~ 20%, respectively. The detector will begin data taking in the late Summer of 2001. If the anticipated radiopurity is achieved

126

and other background suppression efforts are successful, the KamLAND detector in the next phase will be able to detect also the low energy 7 Be solar neutrinos by their scattering on electrons. In that way, the whole spectrum of the solar oscillation solutions will be accessible. 4

Conclusion

In these lectures I concentrated on few issues in the fast growing field of the quest for neutrino mass and oscillations. Even for these rather special topics I discuss, by necessity, only the main features. For more details an inquisitive reader must consult one or more of the quoted references. I would like to thank the organizers of PPP2000 for inviting me, and for the splendid organization of the meeting. Part of the work reported here was supported by the US Department of Energy. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

D. E. Groom et al., Eur. Phys. J C 1 5 , 1 (2000). J. Bonn et al., talk at Neutrino 2000, to be published. J. F. Beacom and P. Vogel, Phys. Rev. D 58, 053010 (1998). J. F. Beacom and P. Vogel, Phys. Rev. D 58, 093012 (1998). I. Yu. Kobzarev et al, Sov. J. Nucl. Phys. 32, 823, (1981). N. Cabibbo, Phys. Lett. 72B, 333, (1978). V. Barger, K. Whisnant, and R. J. N. Phillips Phys. Rev. Lett. 45, 2084 (1980). L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). S. P. Mikheyev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 1441, (1985). J. Nieves and P. Pal, Phys. Rev. D 40, 1693 (1989); J. C. D'Olivo, J. Nieves and P. Pal, Phys. Rev. D 40, 3679 (1989). J. F. Beacom and P. Vogel, Phys. Rev. Lett. 83, 5222 (1999). F. Boehm and P. Vogel, Physics of Massive Neutrinos, 2nd ed., (Cambridge University Press, Cambridge, 1992). M. Moe and P. Vogel, Annu. Rev. Nucl. Part. Sci. 44, 247 (1994). H. Sobel, talk at Neutrino 2000, to be published. Y. Suzuki, talk at Neutrino 2000, to be published. M. Czakon, J. Studnik and M. Zralek, hep-ph/0006339. L. Baudis et al., Phys. Rev. Lett. 83, 41 (1999). L. Luescher et al., Phys. Lett. 434B, 407, (1998). A. Alessandrello, Phys. Lett. 486B, 13, (2000).

127

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

A. Staudt et a l , Europhysics. Lett. 13, 31 (1990). E. Caurier et a l , Phys. Rev. Lett. 77, 1954 (1996). E. Fiorini, Phys. Rep. 307, 309 (1998). M. Danilov et al., Phys. Lett. B480, 12 (2000). H. V. Klapdor-Kleingrothaus et al., hep-ph/9910205. L. de Braeckeleer, to be published. H. Ejiri et al., nucl-ex/9911008. B. R. Davis et al. Phys. Rev. C19, 2259 (1979); P. Vogel et al. Phys. Rev. C24, 1543 (1981). H-V. Klapdor and J. Metzinger, Phys. Rev. Lett. 48, 127 (1982); HV. Klapdor and J. Metzinger, Phys. Lett. B112, 22 (1982. V. I. Kopeikin, L. A. Mikaelyan, and V. V. Sinev, Phys. Atom. Nucl. 60, 172 (1997). K. Schreckenbach et al., Phys. Lett. B160, 325 (1985. A.A. Hahn et al., Phys. Lett. B218, 365 (1989). G. Zacek et al., Phys. Rev. D34, 2621 (1986). B. Achkar et al., Nucl. Phys. B434, 503 (1995). B. Achkar et al., Phys. Lett. B374, 243 (1996). C. Athanassopoulos et al., Phys. Rev. C 54, 2685 (1996); C. Athanassopoulos et a l , Phys. Rev. Lett. 77, 3082 (1996). B. Armbruster et a\.,Phys. Rev. C57, 3414 (1998). D.H. Wilkinson, Nucl. Phys. A377, 474 (1982). P. Vogel, Phys. Rev. D29, 1918 (1984). Y. Declais et al, Phys. Lett. B338, 383 (1994). S. A. Fayans, Sov. J. Nucl. Phys. 42, 590 (1985). P. Vogel and J. F. Beacom, Phys. Rev. D60, 053003 (1999). C. H. Llewellyn-Smith, Phys. Rep. 3, 261 (1972). M. Apollonio et al., Phys. Lett. B420, 397 (1998); M. Apollonio it et al., Phys. Lett. B466, 415 (1999). F. Boehm et al., Phys. Rev. Lett. 84, 3764 (2000); F. Boehm et al., Phys. Rev. D62, 072002 (2000). C. Athanassopoulos eta\., Phys. Rev. Lett. 77, 3082 (1996); C. Athanassopoulos et al., Phys. Rev. C54, 2685 (1996). C. Athanassopoulos et al., Phys. Rev. Lett. 81,1774 (1998); C. Athanassopoulos et al., Phys. Rev. C58, 2489 (1998).

Invited Talks

R E C E N T RESULTS F R O M LATTICE QCD O N CP-PACS SINYA AOKI Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan E-mail: [email protected] I report the recent results from lattice QCD, obtained on the massively parallel computer CP-PACS at the center for computational physics, University of Tsukuba. I focus my attention on two topics, the dynamical quark effects in QCD and the calculation of the K meson weak matrix elements.

1

Introduction

QCD is the fundamental theory for the strong interaction, where the standard perturbative description of the quantum field theory fails to describe nonperturbative phenomena such as a confinement. The lattice QCD is the most promising non-perturbative description of QCD and numerical simulations open a possibility for solving QCD numerically. In this talk I will summarize the recent status of lattice QCD simulations, focusing my attention on results from the CP-PACS collaboration. The CP-PACS, which stands for Computational Physics by Parallel Array Computer System, is a massively parallel computer with 2048 processing units, 614 GFLOPS peak speed and 128 GBytes total memory, at the Center for Computational Physics, University of Tsukuba, Japan. It started to operate in the spring of 19961. The CP-PACS collaboration is a research group for lattice QCD calculations using the CP-PACS. Current members are A. Ali Khan, S. Aoki, Y. Aoki, R. Burkhalter, S. Ejiri, M. Pukugita, S. Hashimoto, N. Ishizuka, Y. Iwasaki, T. Izubuchi, K. Kanaya, T. Kaneko, Y. Kuramashi, T. Manke, K. Nagai, J. Noaki, M. Okamoto, M. Okawa, H. P. Shanahan, Y. Taniguchi, A. Ukawa, T. Yoshie. Projects of lattice QCD calculations carried out by this collaboration so far include the light hadron spectrum in quenched QCD 2 , Nf = 2 full QCD simulations 3 , the dynamical quark effects on B meson decay constants 4 , the dynamical quark effect on bottomonium 5 , a test for the quenched domainwall QCD(DWQCD) 6 , the calculation of K meson B parameters in quenched DWQCD 7 ' 8 , the equation of states of QCD at finite temperature 9 and more. Form these topics I will mainly discuss the Nf — 2 full QCD simulation and K meson B parameters in the quenched DWQCD.

131

132

2 2.1

Light hadron spectrum in full QCD Quenched approximation

Since the calculation of the fermion determinant det£) is the most timeconsuming part in lattice QCD simulations, the quenched approximation, by setting d e t D = 1, has been often used in the past simulations. Physically this approximation prohibits quark-antiquark pair creations from vacuum.

Figure 1. Quenched light hadron spectrum in the continuum limit.

Recently the CP-PACS collaboration has obtained the light hadron spectrum very precisely within this approximation 2 . All systematic uncertainties except the one associated with the quenched approximation are well-controlled within a few %. The quenched light hadron spectrum in the continuum limit are given in Fig. 1. Here the experimental value of rho meson mass mp is used to fix the scale and the pion mass is used for the light quark mass(m u = m<j in the lattice simulations), while the strange quark mass is determined by either K meson mass(solid circles; .ftT-input) or (j> meson mass(open circles; -input). In the figure horizontal lines correspond to experimental values. This figure shows that the overall pattern of light hadron spectrum is reproduced by the quenched lattice QCD at a 5-10% level. At the same time, however, the systematic deviation between the quenched spectrum and the experiment becomes manifest beyond the error of 2-3% in strange hadrons. In particular, the hyperfine splitting between
133

2.2

Nf = 2 full QCD simulation

Since the quencning error becomes manifest in the light hadron spectrum, the effect of dynamical quarks must be included into simulations, in order to reproduce the correct hadron spectrum within a few % accuracy. The CP-PACS collaboration has performed the full QCD simulation with Nf = 2 dynamical fermions, which correspond to up and down quarks with m u = m<j, to investigate the dynamical quark effect on the light hadron spectrum. In order to take the continuum limit, the simulation has been carried out at three values of the lattice spacing, a ~ 0.22 fm, 0.16 fm and 0.11 fm, while keeping the lattice volume roughly constant (~ (2.4 fm) 3 ), which seems large enough for mesons but may be a little smaller for baryons. To reduce the expected large scaling violation at our coarsest lattice such as a ~ 0.2 fm, the improved actions for both gauge and fermion field have been employed: the renormalization group(RG) improved one for gluons 10 and the clover action for quarks 11 . Several values of the dynamical quark mass, corresponding to m^jmp ~ 0.6-0.8, are employed for the chiral extrapolation to the physical u, d quark mass. 2.3

Meson mass

0 1.00

*

•

O experiment • full QCD D quenched QCD

K-input -

—

—

~~~——•-_ §

J--B-H—-a-

"

•

*

•

"

"

•

—

-

«

.

_

_

.

0.95

0.90

K*

* }___--$.-*-—-«_ - ~~*-

-•—

0.85 0.10

0.15

a[fm]

Figure 2. <£ and K meson masses from K"-input as a function of a for Nf = 2 full QCD(solid circles) as well as the quenched QCD(open squares), together with experimental values(open diamonds).

I now discuss how the simulation with dynamical quarks removes the quenching error in the meson mass. In Fig. 2 <j> and K* meson masses from

134

the If-input are plotted as the function of the lattice spacing for the Nf — 2 full QCD simulation 3 as well as the quenched one 2 . The figure shows that the quenching error for <j> and K meson masses are almost removed by the Nf = 2 full QCD simulation after the continuum (a -» 0) limit is taken. The effect of dynamical quarks is really important for reproducing the correct spectrum. Thanks to the action improvement employed, the scaling violation is reasonably small even at a > 0.2 fm. Small deviations of masses from experimental values could be a quenching effect of the strange quark. 2.4

Quark mass

Quark masses are fundamental parameters of the Standard Model, and important for phenomenology of the elementary particle physics. Because of quark confinement, they can be determined indirectly from a comparison of experimental hadron masses and their theoretical prediction in terms of quark mass parameters. The hadron mass calculation in lattice QCD can therefore gives also light quark masses(up, down, strange).

I* ¥

160 ,''

Q>

.r

quench fy VWipq^

1 . 140 > •

at CD w

120

^

£*-<.'. .-> quench K

li,-«=-

v

0.00

0.05

.-*

^*TulU

"

full K

0.10 0.15 a[fm]

Figure 3. Strange quark mass as a function of a for Nf quenched QCD(open)

AWi

0.20

0.25

: 2 full QCD(solid) as well as the

In lattice QCD calculation, the quark mass can be determined directly form the relation between hadron masses and the quark mass parameters in the action, or from the axial Ward-Takahashi identity (AWI), 2 m ^ / = t hm

(p(0)pw>

.

(1)

135 Table 1. Light quark masses in the MS scheme at n = 2 GeV.

m„d[MeV] 4.57±0.18 Nf = 2

o 44+0.14 o, **-0.22

ms [MeV] AT-input 0-input 116±3 144±6 88±^ 901^

where A± is the axial current and P is the pseudo-scalar density. The former is called as the vector Ward-Takahashi identity (VWI) quark mass and denoted mywi while the latter is the axial Ward-Takahashi identity(AWI) quark mass, JTIAWI- For the strange quark mass, either if-input or 0-input is used, as was done for the spectrum. In order to convert the lattice quark mass to the one in the MS scheme, 1-loop renormalization has been employed for the Z factor, = Z(JM) • m l a t t i c e (l/a) a2 Z{JM) = 1 + ^ [dm log(/io) + zm]

m^in)

(2) (3)

where dm is the anomalous dimension of the quark mass and zm is the finite part of the renormalization constant. In Fig. 3 the strange quark mass is plotted as a function of the lattice spacing a, for both quenched QCD and Nf = 2 full QCD. One first notices that mvwi and JTIAWI agree in the continuum limit for both quenched and full QCD, though they are very different at non-zero a. This agreement between two definitions indicates that not only the continuum extrapolation but also the 1-loop renormalization work reasonably well in the calculations. One find, however, that the strange quark mass from -ftT-input(circles) and 0-input(squares) disagree even in the continuum limit for the quenched QCD. This is one of the manifestations of the quenching error, already observed for meson masses in the previous subsection. On the other hand, the different between two inputs almost disappears in the continuum limit for the full QCD. This reflects the fact, discussed already, that masses of strange mesons agree well with experimental values in the Nf = 2 full QCD. Values of mU(j and ms in the MS scheme at fj, = 2 GeV are given in tablel. The large dynamical quark effect is observed on quark masses: it reduces quark masses by 20~ 30 %. In particular, the strange quark mass, mMS(2 GeV) ~ 90(10) MeV, is much smaller than the quenched one, and is very close to the lower bound from QCD sum rules 12 . The value of the strange quark mass might be reduced further by the inclusion of the dynamical strange quark.

136

3

Domain-wall QCD and K meson weak matrix elements

The Wilson and the Kogut-Susskind(KS) fermions, popularly used in numerical simulations, have some disadvantages: The former breaks the chiral symmetry explicitly, while the latter breaks the flavor symmetry. The domain-wall fermion formulation, recently proposed 13 , has good flavor and chiral symmetries even at non-zero o in the Ns -t oo limit, where N„ is a parameter of this formulation, while the cost of numerical simulations is Ns times larger than the cost for the Wilson fermion. 3.1

Domain-wall QCD

The domain-wall fermion (DWF) is the 5 dimensional massive Wilson fermion with a free boundary condition in 5-th direction interacting with the 4 dimensional gauge field. Since one can interpret the 5-th dimensional coordinate as "flavors", the lattice action becomes

+ i>n,s [MPR + M^PL] n;™
(4)

where n,m are the 4 dimensional lattice sites, the "flavor" indices s,t run from 1 to Ns, Un,n is the link variable for the 4-dimensional gauge field, and M. is the flavor mixing mass matrix. For the weak enough gauge coupling, this action contains the light-handed zero mode near s = 1 and the left-handed one near s = Ns as Ns —• oo, so that these two zero modes are combined into one massless Dirac fermion, which is defined as qn = P R ^ „ , I + PLtpn,N,

Qn = '/'n.l-PL + i>n,N,PR-

(5)

In the presence of the fermion mass term, given by —mfqnqn, the pole mass of the fermion becomes m p o l e = M(2 - M)[mf + (1 - M)N']

(6)

for 0 < M < 2, and it vanishes in the limit that Ns -¥ oo and m / -> 0: the chiral symmetry is restored. We call the lattice QCD with DMF as the domain-wall QCD(DWQCD). This good property of the chiral symmetry for DWQCD has been investigated in the quenched numerical simulations 6 : While the chiral symmetry breaking seems to remain non-zero in the Ns ->• oo limit at coarse lattice(l/a ~ 1.0 GeV), much better chiral behaviour is observed at finer

137

lattice(l/o ~ 2.0 GeV). In particular it is actually observed in the case of the RG improved gauge action at 1/a ~ 2.0 GeV that the chiral symmetry breaking vanishes exponentially in N„. Therefore the CP-PACS collaboration has attempted to apply the DWQCD formulation to the calculation of weak matrix elements of K meson decays, the quark mass dependence of which is strongly restricted by the chiral symmetry. These subjects will be discussed in the next two subsections. 3.2

Neutral K meson mixing parameter BK BK(NDR, 2GeV) vs. mpa q =1/a, 3-loop coupling, 5 points

O KS (non-invariant) O KS (invariant) KDWF

mpa Figure 4. B J C ( N D R , 2 GeV) as a function mpa in the quenched QCD with the domainwall fermion(solid symbols) and the KS fermion(open symbols). Two different four quark operators are employed for the latter fermion.

The indirect CP-violation in K meson decays, parameterized by e, is related to the CKM matrix elements as *?[(! • p)A + B}BK

(7)

= %

where p and fj are the (modified) Wolfenstein parameters, and A, B, and C are numerical constants. The KQ-KQ mixing parameter BK is defined as the K meson matrix element of the AS = 2 four quark operator: BK

=

(K°\s^(l

-

7 5 )d

• g 7M (l -

l(K°\slfll5d\0)(d\sWd\K°)

l5)d\K°)

(8)

138

The result of BK calculated by the quenched DWQCD 7 with the RG improved gauge action is plotted as the function of the lattice spacing in Fig. 4, together with the previous high-precision result with the KS quark action 14 . While the large scaling violation(a dependence) is observed in the case of the KS fermion, the result from DWQCD is almost a independent and a little smaller but consistent with the value from the KS fermion in the continuum limit. This proves that DWQCD works well in practice. 3.3

K -t TTTT decays

Recent experiments indicate the non-zero value of e', the parameter of the direct CP violation in K -» mr decays: - = (19.2 ± 2.5) x l ( T 4 ,

(9)

while an approximated formula predicts that

where B6'8 ' ' is the B parameter of A7 = 1/2,3/2 K —• mr decay induced by the operator Q6,sQe = (sadb)V-A

Y^tfv'W+A,

QS = ^(sadb)V-AJ2eg(Qbqa)v+A.

(11)

It has been found that the direct calculation of the K -> ITK matrix elements is very difficult, and so far chiral perturbation theory is used to relate them to simpler ones as follows: {w+n-\Qi\K°)

fK ~ "*' (12) iPn-PK)f where the coefficient a, is determined by the condition that 0 = (0|Q, — ttiQsub\Ko), with the subtracted two quark operator Qsub = (™d + rns)sd + (ma — ms)sj5d. Again the chiral symmetry is important for this relation to hold. Recently the CP-PACS collaboration has started to evaluate these matrix elements, using the quenched DWQCD with both plaquette and RG improved gauge actions at 1/a ~ 2.0 GeV 8 . In Fig. 5 the quark mass dependence of (7r+7r_|Q6° l^ 0 ) an(^ i t s breakup into (7r+|Q^0)|A:+) and -a6('K+\Qsub\K+) is plotted in the case of the RG =

(TT+IQ,

-

+

aiQsub\K

)

139 breakup of , RG improved

$—*—at—x—

:

• total # •subtraction term

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

m,

Figure 5. The quark mass dependence of (7r+7r~|Qg |K"°)(squares), together +

0)

+

< T IQa l-K" >(circles) and

with

-as{Tr+\Q3ub\K+)(aia.monds).

gauge action, where superscript (0) indicates that the operator has AI

=

1/2(7 = 0 for the final TTTT ).

While good signals are obtained, a huge cancellation between (Tr+\Q^'\K+) and -ae(n+\Qsub\K+) is observed, so that the total contribution (n+n-lQ^lK0) becomes very small. As shown in Fig. 6, however, one can still get enough signals to estimate the total matrix element except at the lightest quark mass, m / = 0.01, where the finite size effect seems non-negligible. In order to get the reliable estimate of the K —• mr matrix element, the finite size effect as well as the scaling violation must be carefully investigated.

4

Conclusion

I show that the dynamical quark effect can be seen on meson masses and quark masses at the present accuracy of the lattice QCD simulation performed on the CP-PACS. The domain-wall QCD is proven to work well for BK, and the preliminary results for K —> mr matrix element are presented.

140 , RG improved 0.3

1

0.2 0.1

-

j

1

_ ' I J-i *" 1

-0.3 -0.4

-•£

-:

1

-0.2

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 6. The quark mass dependence of (TT+TV |Qg |K"0)(squares).

Acknowledgments I would like to thank the organizers of this workshop for their kind hospitality. This work is supported in part by Grants-in-Aid of the Ministry of Education(Nos. 12014202, 12640253). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

Y. Iwasaki, Nucl. Phys. B (Proc.Suppl.) 60A, 246 (1998). S. Aoki, et al, Phys. Rev. Lett. 84, 238 (2000). A. Ali Khan, et al, Phys. Rev. Lett. 85, 4674 (2000); hep-lat/0010078. A. Ali Khan, et al, hep-lat/0010009. T. Manke, et al, Phys. Rev. D 62, 114508 (2000). A. Ali Khan, et al, Nucl. Phys. B (Proc. Suppl.) 83-84, 591 (2000); hep-lat/0007014. A. Ali Khan, et al, hep-lat/0010079. A. Ali Khan, et al, hep-lat/0011007. M. Okamoto, et al, Phys. Rev. D D60, 094510 (1999); A. Ali Khan, et al, hep-lat/0008011. Y. Iwasaki, preprint UTHEP-118(Dec. 1983), unpublished. B. Sheikholeslami and R. Wohlert, Nucl Phys. B 259, 527 (1985). L. Lellouch, E. de Rafael, J. Taron, Phys. Lett. B 414, 195 (1997). D. Kaplan, Phys. Lett. B 288, 342 (1992); Y. Shamir, Nucl Phys. B 406, 90 (1993); V. Purman and Y. Shamir, Nucl Phys. B 439, 54 (1995). S. Aoki, et. al, Phys. Rev. Lett. 80, 5271 (1998).

QCD O N A T R A N S V E R S E LATTICE MATTHIAS BURKARDT AND SUDIP SEAL Department of Physics, New Mexico State University, Las Graces, NM 88003-0001, USA We present results from a transverse lattice study of low lying mesons. Special emphasis is put on the issue of Lorentz invariant energy-momentum dispersion relations for these mesons. The light-cone wave function for the 7r obtained in this framework is very close t o its asymptotic shape.

1

Introduction

The transverse lattice 1 is an attempt to combine advantages of the light-front (LF) and lattice formulations of QCD. In this approach to QCD the time and one space direction (say x3) are kept continuous, while the two 'transverse' directions XJL = (x 1 ,x2) are discretized. Keeping the time and a;3 directions continuous has the advantage of preserving manifest boost invariance for boosts in the a;3 direction. Furthermore, since x^ = x° ± x3 also remain continuous, this formulation still allows a canonical LF Hamiltonian approach. On the other hand, working on a position space lattice in the transverse direction allows one to introduce a gauge invariant cutoff on _L momenta — in a similar fashion as is done in Euclidean or Hamiltonian lattice gauge theory. In summary, the LF formulation has the advantage of utilizing degrees of freedom that are very physical since many high-energy scattering observables (such as deep-inelastic scattering cross sections) have very simple and intuitive interpretations as equal LF-time (x+) correlation functions. Using a gauge invariant (position space-) lattice cutoff in the J_ direction within the Lf framework has the advantage of being able to avoid the notorious l/k+ divergences from the gauge field in LF-gauge which plague many other Hamiltonian LF approaches to QCD 2 . The hybrid treatment (continuous versus discrete) of the long./J_ directions implies an analogous hybrid treatment of the long, versus J. gauge field: the long, gauge field degrees of freedom are the non-compact A1*, while the _L gauge degrees of freedom are compact link-fields. Each of these degrees of freedom depend on two continuous (a^) and two discrete (nj_) space-time variables, i.e. from a formal point of view the canonical J_ lattice formulation is equivalent to a large number of coupled 1 + 1 dimensional gauge theories (the long, gauge fields at each n i ) coupled to nonlinear a model degrees of freedom (the link fields). 141

142

2

The color dielectric formulation

For a variety of reasons it is advantageous to work with J_ gauge degrees of freedom that are general matrix fields rather than U € SU(Nc). First of all, we would like to work at a cutoff scale which is small (in momentum scale) since only then do we have a chance to find low lying hadrons that are simple (i.e. contain only few constituents). If one wants to work on a very coarse lattice, it is useful to imagine introducing smeared or averaged degrees of freedom. Upon averaging over neighboring 'chains' of SU(Nc) fields one obtains degrees of freedom that while they still transform in the same way as the original SU(Nc) degrees of freedom under gauge transformations, they are general matrix degrees of freedom which no longer obey U^U — 1 and det(U) = 1. The price that one has to pay for introducing these smeared degrees of freedom are more complicated interactions. The advantage is that low lying hadrons can be described in a Fock expansion (this has been confirmed by calculations of the static quark-antiquark potential 3 and glueball spectra 4 ) . Another important advantage of this 'color-dielectric' approach is that it is much easier to construct a Fock expansion of states out of general linear matrix fields than out of fields that are subject to non-linear SU(Nc) constraints. In the color-dielectric approach the complexity is shifted from the states to the Hamiltonian: In principle, there exists an exact prescription for the transformation from one set of degrees of freedom (here C/'s) to blocked degrees of freedom M = £ „ „ I 1 « ^ e-s.„.(iiO =

f [dU] e-s„..(u)5 (M _ J2 JJ U^\ . •*

V

av

i

(1)

/

The problem with this prescription is that 5 e / / . is not only very difficult to determine directly, but in general also contains arbitrarily complicated interactions. A much more practical approach towards determining the effective interaction among the link fields nonperturbatively is the use of Lorentz invariance. This strategy has been used in a systematic study of glueball masses in Ref. 4 , where more details can be found regarding the effective interaction. One starts by making the most general ansatz for the effective interaction which is invariant under those symmetries of QCD that are not broken by the ± lattice. This still leaves an infinite number of possible terms and for practical reasons, only terms up to fourth order in the fields and only local (in the ± direction) terms have been included in the Ref. 4 . The coefficients of the remaining terms are then fitted to maximize Lorentz covariance for physical

143

observables, such as the QQ potential (rotational invariance!) and covariance of the glueball dispersion relation. It should be emphasized that these are first principle calculations in the sense that the only phenomenological input parameter is the overall mass scale (which can for example be taken to be the lowest ghieball mass or the string tension). The only other input that is used is the requirement of Lorentz invariance. The numerical results from Refs. 4 ' 3 within this approach are very encouraging: « with only a few parameters, approximate Lorentz invariance could be achieved for relatively large number of glueball dispersion relations simultaneously 4 as well as for the QQ potential • the glueball spectrum that was obtained numerically on the _L lattice for Nc -» oo is consistent with Euclidean Monte Carlo lattice gauge theory calculations performed at finite Nc and extrapolated to Nc -> oo . For further details on these very interesting results, the reader is referred to Refs. 4 ' 3 and references therein. 3

Fermions on the J. lattice

Encouraged by the very successful calculations within the pure glue sector, we proceeded to conduct numerical studies that include fermions 5 . In this framework, states that have meson quantum numbers consist either of a q and a q on the same transverse site with no link fields required or of a q and a q at an arbitrary _L separation with a chain of link fields in the J. direction connecting them. Hopping of the quarks in the J_ direction is accompanied by emission or absorption of link field quanta on the link across which the quarks hop. For each transverse site there is a longitudinal gauge field interaction (very similar to the interaction in QCDi+i) which couples to q and q on that site as well as to link fields adjacent to that site. In the color dielectric approach the link fields are also subject to the effective interaction discussed above. Similar to other lattice field theories with fermions, species doubling also occurs for the J_ lattice action. Of course, one main difference to the Euclidean formulation is that the naive _l_ lattice action for fermions on the J. lattice exhibits only 2 2 = 4 fold species doubling, since only two directions are discretized. Nevertheless, although species doubling is a less extreme phenomenon here, it is a problem that needs to be addressed.

144

At this point one has several options to proceed. One obvious possibility is to add a Wilson r-term of the form 5Cr = ar^d\ij}

(2)

to the -L lattice action. Obviously, such a term violates chiral symmetry for finite lattice spacing, but this is just a consequence of the well known NielsonNinomya theorem, which states that any local and hermitian action for lattice fermions which is chirally symmetric does necessarily exhibit species doubling. Within the LF framework there seems to be an alternative way to eliminate 'doublers': The crucial observation is that it is possible to write down a fermion (kinetic) mass term within this framework which is chirally invariant (but nonlocal) 6Cm = 8m2i>^--i>.

(3)

Since it is possible to write down a chirally invariant mass term, it is also possible to write down a chirally invariant r-term to remove the doublers. a In Ref. 5 we investigated the differences between adding a conventional rterm and such a modified chirally invariant r-term to the ± lattice action. The main problem is that in the canonical LF approach, where half the fermion degrees of freedom are eliminated using a constraint equation, the usual chiral transformations become dynamical operations and therefore the meaning of the usual chiral symmetry becomes obscure. For further details on this issue see Ref. 5 . Both approaches to fermions on the _L lattice give rise to two kind of hopping terms for the fermions: one that has the Dirac structure of a vector coupling and which flips the helicity (hereafter referred to as spin-flip hopping) and one which has a scalar Dirac structure and does not flip the helicity (hereafter referred to as r-term). The difference between conventional and modified r-term is how coefficients in the LF Hamiltonian are related to coefficients in the Lagrangian. In the spirit of the color-dielectric approach we regard both coefficients in the Hamiltonian as free parameters. For numerical reasons, we limited the Fock space to states where q and q are on the same J_ lattice site (no link field, 2 particle Fock component) and states where q and q are separated by one link (3 particle Fock component), i.e the femtoworm approximation 5 . "Of course, it is non-local, which is why this does not contradict the Nielson-Ninomya theorem.

145

4

Fit of p a r a m e t e r s

The parameters in the Hamiltonian are: • longitudinal gauge coupling • r-term coupling (hopping without spin flip) • spin-flip coupling (hopping with spin flip) • kinetic mass (2 particle sector) • kinetic mass (3 particle sector). The observables that we studied showed little dependence on this parameter, so we kept it fixed at a constituent mass. • link field mass. Similar to kinetic mass in 3 particle sector. Furthermore, demanding Lorentz invariance in the pure glue sector, one finds a renormalized trajectory. This makes sense since the _L lattice scale is unphysical. We keep the link field mass fixed at a value which yields relatively small ± lattice spacing. The observables that we studied, including Lorentz invariance, show rather little sensitivity b to the precise values of the 3-particle sector masses, which we thus keep at a value corresponding to a constituent mass (about half the p mass). This leaves us with 4 free parameters. As input parameter, we use the physical value of the p mass, chiral symmetry in the sense that we demand a small n mass and the physical string tension. As explained in the appendix, demanding only Lorentz invariance would drive <7 2 / m p to infinity and would thus give rise to string tensions that are inconsistent with phenomenology. Although the dependence of physical parameters on the input parameters is in general rather complex and non-perturbative, one can understand at a qualitative level how the input parameters influence the relevant physical scales: The physical string tension in the longitudinal direction determines G 2 in physical units. Modulo mass renormalization due to the Yukawa couplings, the quark mass in the 2-particle Fock component and the gauge coupling are strongly constrained by fitting the physical string tension (which determines G2 and the center of mass of the n-p system. This leaves us with only the r-term and the helicity flip hopping term couplings as parameters to vary. As explained in the 6

As long as we kept the other parameters floating!

146

\

"

Figure 1. Light-cone wave function for the n obtained on the _l_ lattice.

appendix, the helicity flip term is not only responsible for ir-p splitting within our approximations but also for violations of Lorentz invariance (different _L lattice spacings in physical units for different mesons). Using a value of G2 = g 2 ^ g ta 0.4 GeV2 in physical units, as determined from the string tension in Ref. 4 (which is larger than the one previously used 5 ' 7 ) , we were able to produce the physical n-p splitting with only a relatively small spin-flip coupling. This allows us to chose the r-term large enough so that r-term hopping dominates over spin-flip hopping and therefore violations of Lorentz invariance (as measured by comparing quadratic terms in the dispersion relation of mesons in the ir-p sector) are only on the order of 20%. The average _L lattice spacing (from the dependence of the energy on Pj_) is found to be a± m 0.5/m. For the 7r wave function, we find a shape that is very close to the asymptotic shape {4>1„m{x) <* ^(1 _ x))- This is surprising if one consider that the lattice spacing is still relatively large and hence the momentum scale is still very low. We should also point out that the shape of our it wave function disagrees with the results from Ref. 6 (the 7T wave function obtained in Ref. 6 is much more flat than ours). Since the Hamiltonian and Fock space truncation in both works are the same, the only real difference are the basis functions used to cast the Hamiltonian into matrix form: we used continuous basis functions, while Ref. 6 uses DLCQ. However, it is not clear that this difference alone can explain the different shape of 7r-wave functions.

Acknowledgments M.B. would like to thank H.N.-Li and W.-M. Zhang for the invitation to this interesting workshop and S. Dalley for many interesting and clarifying

147

discussions about the ± lattice. This work was supported by a grant from DOE (FG03-95ER40965) and through Jefferson Lab by contract DE-AC0584ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility. Appendix: perturbative analysis of n-p splitting on the _L lattice In order to gain a qualitative understanding about the interplay between different parameters in the J. lattice Hamiltonian, it is instructive to study a simple model, where one treats the admixture of the 3-particle Fock component to the 7r and p as a perturbation. To 0th order, i.e. when the coupling between 2 and 3 particle Fock component is turned off, there is no spin dependence of the interactions and the •K and p are degenerate. Likewise, there is no 'hopping' (i.e. J_ propagation) of mesons and thus energies are independent of kj_ giving rise to an infinite J. lattice spacing (in physical units). In the next order we treat the coupling between 2 and 3 particle Fock components as a perturbation (note that interactions which are diagonal in the particle number, such as the confining interaction in the longitudinal direction are still treated non-perturbatively). There are two interactions that mix Fock sectors: hopping due to the r-term (without helicity flip) and hopping due to the vector coupling (with helicity flip). In order to understand the effect of these two types of hopping, it is very useful to point out that there is no 'mixing' between these two hopping terms in the femtoworm approximation, i.e. there is a complete cancellation among the various hopping terms where the 2 to 3 transition is caused by say the r-term and the subsequent 3 to 2 transition is caused by the w-term (and the other way round). For kj_ = 0 this follows trivially from the fact that the _L lattice Hamiltonian is invariant under rotations around the z — axis by multiples of ir/2, giving rise to conservation of total angular momentum modulo 4. Since Jz = Sz in the 2 particle sector, and Sz can only assume the values —1, 0, and 1, Sz in the 2 particle sector is conserved. Since 'mixed' hopping would change Sz by one unit, this means that the sum of all contributions from mixed hopping must add up to zero. For kj_ 7^ 0 the argument is a little more complicated, since rotations also change the direction of kj_. However, both the J. lattice Hamiltonian as well as kj_ are invariant under the a sequence consisting of a rotation by -K around the z-axis followed by a _L reflection on the x-axis and then a _L reflection on the y axis PvPxRi80- As a result, Jz is conserved modulo 2, which still rules out mixing between the r-term and the w-term. Starting from a basis of 't Hooft eigenstates which are plane waves in the

148

± direction and where the qq in the 2 particle Fock component carry spins I t t ) , I t l ) , I I t ) , and | 4-4-) respectively, one thus finds for the energy in second order perturbation theory 0

Cx ~r Cy

0 0

H = M2 - M\

CX

0 0

V o + Kv

0 \ cv

Cx

cx

0 0

Cx "i Cy

0 0

I °0

0 0

\ Cy

0

\J

0

CX +CyJ

vJ

Cy

Cx \

CX +Cy

0

0 0

0 0

> Cy

\

(4)

/

Here M 2 r and M2V are some second order perturbation theory expressions involving matrix elements between 2 and 3 particle states that are eigenstates of the diagonal parts of the Hamiltonian (kinetic + Coulomb), and c; = cos fcj. Several general and important features can be read off from this result. First of all, and most importantly, Eq. (4) Shows that the r-term gives rise to a dispersion relation with the same _L speed of light for the TT and the p's, while the vector interaction breaks that symmetry. This observation already indicates that it may be desirable to keep the r-term much larger than the spin-flip term. We will elaborate on this point below. At kx = ky = 0, the eigenstates of the above Hamiltonian are the p±i i.e. | t t ) and | 4-1), with M 2 = M\x = M02 - Mfr, the \po) = \U + I t ) , with M 2 = M ^ + M 2 „ and the |TT> = | U - I t ) , with M 2 = M\x - M 2 „ For nonzero J_ momenta, there will in general be mixing among the p+i and the p_i, but not among the other states since helicity in the 2-particle Fock sector is still conserved modulo 2. Expanding around k± = 0, and denoting M2 = MQ — M 2 r one finds to 0(k 2 L ) the following eigenstates and eigenvalues state

M 2 (0)

tl-lt

M2 - Mlv

M2(k2)-M2(0) M

l,r

2

^ ml,v

2

tl + ltM

2 2 m + M 2 „ MMl , r ^ I2± ^ - M *,v *2 ^

tt-II

M2

lvl

tt +II

M2

M2 M

2

l,r

l,r

2 ".T-i, 2

m

l,« 2

, JU-2 l,v

+ M

k k *- y 2

(5)

149 Eq. (5) illustrates a fundamental dilemma that hampers any attempt to fully restore Lorentz invariance within the femtoworm approximation: Mf v not only governs the splitting between the n and the (h=0) p but is also responsible for violations of Lorentz invariance among the different helicity states: If one determines the J_ lattice spacing in physical units for each meson separately, by demanding that the ± speed of light equals 1, one finds for example

Mttr + Mt>v Kr ~ Mlv

a\

(6)

i.e. increasing the 7r — p splitting is typically accompanied by an increase in Lorentz invariance violation 1

1

Ml-Ml

(7)

For the p±i the breaking is of a similar scale, plus one also observes an anisotropy in the dispersion relation on the same scale. Therefore, in order to avoid a large breaking of Lorentz invariance, it will be necessary that M 2 X » M2po - Ml

(8)

If one keeps the -K — p splitting fixed at its physical value then there are two ways to achieve this condition. One possibility is to simply increase the Yukawa coupling that appears in the r-term. This increase of the r-term tends to decrease the ± lattice spacing for both -K and p's and in order to achieve satisfactory Lorentz invariance (in the sense of uniform _L lattice spacings) one needs to make the lattice spacing smaller than the Compton wavelength of the p meson. However, one cannot make the r-term coupling arbitrarily large because at some point there occurs an instability (tachyonic M 2 !). Such instabilities for large coupling are common in the LF formulation of models with Yukawa coupling and might be related to a phase transition (similar to the phase transition in 4 theory that occurs as the coupling is increased). Fortunately, there exists another possibility to make these matrix elements large, without increasing the Yukawa couplings. This derives from the fact that the hopping interactions are proportional to {\ ± h) -Z-^T, where x \ 3*

31 ' 3* 3*

(x1) are the momenta of the active quark before(after) the hopping. Because of the singularity as x, x' -> 0, matrix elements of the hopping terms are greatly enhanced if the unperturbed wave functions are large near x = 0 and

150

x = 1. Since the unperturbed wave functions in the 2 particle Fock component vanish like x@ near x = 0, where /3 oc ^ - , matrix elements of the hopping interaction become very large when one makes —^ very large. Q

Therefore, the larger one chooses ^5-, the more one restores Lorentz invariance of the n and p dispersion relations because one can keep the n-p splitting fixed while decreasing the coupling of the spin flip interaction. At the same time, keeping the r-term interaction fixed one increases the dominance of the r-term contribution in \ and thus not only reduces the lattice spacing in physical units, but also obtains dispersion relations for the n and the p's that look more and more similar — as demanded by Lorentz invariance. Unfortunately, we are not completely free to pick whatever value of Q? we like because m^ and G2 are largely fixed by the center of mass in the ir-p system as well as by fitting the physical string tension in the pure glue sector. References 1. W.A. Bardeen, R.B. Pearson, and E. Rabinovici, Phys. Rev. D 2 1 , 1037 (1980). 2. M. Burkardt, invited talk given at "11th International Light-Cone School and Workshop", Eds. C. Ji and D.-P. Min, hep-th/9908195. 3. M. Burkardt and B. Klindworth, Phys. Rev. D 55, 1001 (1997); hepph/9809283. 4. S. Dalley and B. vande Sande, Phys. Rev. D 59, 065008 (1999); Phys. Rev. Lett. 82, 1088 (1999); Nucl. Phys. B(Proc. Suppl.) 83,116 (2000); Phys. Rev. D 62, 014507 (2000). 5. M. Burkardt and H. El-Khozondar, Phys. Rev. D 60, 054504 (1999). 6. S. Dalley, Nucl. Phys. B(Proc. Suppl.) 90, 227 (2000). 7. S.K. Seal and M. Burkardt, Nucl. Phys. B(Proc. Suppl.) 90,233(2000).

QCD AT T H E TEVATRON A N D LHC JOEY HUSTON Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA In this talk, I review QCD results from the Tevatron and QCD phenomenology relating both to the Tevatron and to the LHC. 1

Introduction

In this talk, I will briefly review some of the QCD physics results from Run 1 at the Fermilab Tevatron Collider. In the (Run 1) Collider, 900 GeV protons collided with 900 GeV antiprotons, leading to a center-of-mass energy of 1.8 TeV, the highest energy then accessible. The Tevatron Collider completed a very successful Run 1 in 1996, with each of the main experiments (CDF and DO) accumulating data samples on the order of 100 pb_1. Most analyses have already been published or are nearing publication. The main theme of my talk will be the success with which perturbative QCD has been applied to the data from the Collider. There are enough mysteries left, however, to make life interesting (and to provoke the need for larger data samples), with several of the mysteries involving the remaining uncertainties in the gluon distribution. DGLAP-based perturbative QCD predictions remain very successful and the search for convincing evidence for BFKL effects continue. These results serve not only as a venue for precision tests of QCD but also as a much-needed testing ground for LHC calculations. I will also discuss recent work on determining the uncertainties for parton distribution functions (pdf's). I will conclude with an outlook/preview of the upcoming Run 2. 2

Q C D Results from the Tevatron

As discussed above, the Tevatron Collider serves as an arena for precision tests of QCD involving photons, W/Z bosons, jets, leptons and heavy quarks in the final state. These measurements involve the highest Q2 scales currently achievable, which is the natural place to search for new physics involving small distance scales. There is a sensitivity to parton distributions over a wide kinematic range, which then allows these measurements to be used in parton distribution fits over this range. There are many two-scale problems, which 151

152

allows the testing of the effects of soft gluon resummation. In addition, there are large cross sections for the diffractive production of jets, W/Z bosons and heavy flavors. The data are compared to next-to-leading (NLO), resummed, leading log Monte Carlo and fixed order perturbative QCD predictions. Due to the limitations of space, I will concentrate in this writeup on measurements of the inclusive jet cross section by the two experiments and their implications. There have been many recent reviews of QCD physics at the Tevatron to which I will refer for more detail than provided in this account 1>2>3-4'5. 2.1

Inclusive Jet Production at the Tevatron

The inclusive jet cross section in the central rapidity region has been measured by both the CDF and DO experiments, while DO, in addition, has measured the inclusive cross section in the forward rapidity region. Jets are defined primarily using an iterative fixed cone algorithm with a radius R (i/A?j 2 + A0 2 ) of 0.7, although some analyses have used other cone sizes or have used the kr algorithm for jet definition. These same jet algorithms are also applied to the partons in the theory; the details of the jet algorithm are critical for precision comparisons to higher order QCD predictions 2 . The inclusive measurement spans the transverse energy range from 15 GeV/c to of the order of 500 GeV/c; in this range the jet cross section drops by over 9 orders of magnitude. The highest ET jet events probe the smallest distance scales (10~ 17 cm) currently accessible. Any new physics that might exist at these distance scales, such as compositeness, might manifest itself in the jet cross section. The CDF Collaboration published the inclusive jet cross section from Run 1A (19.1 pb~x) for the rapidity region (0.1 < |r/| < 0.7) 6 . Good agreement with the NLO prediction was observed (using the CTEQ4M parton distribution functions), except at the highest values of transverse energy. The CTEQ collaboration performed a global fit using the CDF Run 1A jet data, giving a large emphasis in the fit to the high ET data. The resulting fit (CTEQ4HJ) contains a gluon distribution substantially greater at high x than in conventional pdf's, leading to a better agreement with the CDF Run 1A data 7 . Both CDF and DO have published jet cross section results using the higher statistics obtained in Run IB 8,Q. The inclusive cross sections compared to NLO QCD predictions using the CTEQ5M and CTEQ5HJ pdf's are shown in Figures 1 and 2 10 . The CTEQ5M pdf's were determined using the CDF and DO Run IB jet cross sections in the global pdf fit (with normal weight), while in the CTEQ5HJ fit, an increased statistical weight was given to the higher

153

Ratio: Prel. data / NLO QCD (CTEQ5M | CTEQ5HJ)

cor

1.4

1.2

K

-"•—IS

,

V*-""' V » •

1

Data/CTEQ5M CTEQ5HJ/CTEQ5M CDF Data (Prel.) CTEQ5HJ CTEQ5M

0.8 Incl. Jet : pf * da/dp t 0.6

0.4

0.2

0 50

100

150

200

250

300

350

400

PT

Figure 1. The CDF jet cross section from Run I B compared to NLO QCD predictions using the C T E Q 5 M and CTEQ5HJ p d f s.

ET data points (for both the CDF and DO measurements). The CTEQ5HJ pdf's (and in particular the gluon distribution) are very similar to those in the CTEQ4HJ set. From these two figures, it can be observed that the CDF and DO jet cross sections are in good agreement with each other, and with the predictions using the CTEQ5HJ pdf's. Note that the normalizations of the jet cross sections in these plots have been slighly adjusted. There is a small normalization difference between the CDF and DO measurements, which is expected since the two experiments use different values of the proton-antiproton total cross sections to normalize their luminosities. It is partially because of this luminosity uncertainty that a proposal has been made to normalize all Tevatron (and LHC) physics cross sections to the cross section for W production. As mentioned previously, DO has also measured the inclusive jet cross section as a function of rapidity n . The wider kinematic range and greater statistical power leads to a more powerful discrimination among pdf's. A detailed quantitative comparison shows that NLO QCD predictions using CTEQ4HJ provide the best agreement with the DO data.

154

1.4

Ratio: Data / NLO QCD (CTEQ5M | CTEQ5HJ) CTEQ5M: CTEQ5HJ:*

2

24/24 = 25C4

, "°

.

r m fad0r:

1.2

tLJJtil*'<"i's('*ii

DO

1.04 1.08

,r,"T

1-1•

1 Incl. Jet : p? * da/dp t

•

0.6

Dala/CTEQ5M CTEQ5HJ/CTEQ5M DO data CTEQ5HJ OTEQ5M

Error bass: siailslical only 0.2

3% < Corr. Sys. Err. < 30%

200

300

400

PT

Figure 2. The DO jet cross section from Run I B compared to NLO QCD predictions using the CTEQ5M and CTEQ5HJ pdf s.

3 3.1

P D F Determination and Uncertainties PDF

Determination

The calculation of the production cross sections at the Tevatron and LHC, for both interesting physics processes and their backgrounds, relies upon a knowledge of the distribution of the momentum fraction x of the partons in a proton in the relevant kinematic range. These parton distribution functions (pdf's) are determined by global fits to data from deep inelastic scattering (DIS), Drell-Yan (DY), and jet and direct photon production at current energy ranges. There are several groups that provide semi-regular updates to the parton distributions when new data and/or theoretical developments become available 10>12'13 The pdf's obtained are by definition the best fit to the data sets. Of great interest also is the uncertainty to the fit and thus the uncertainty on the pdf's. Lepton-lepton, lepton-hadron and hadron-hadron interactions probe complementary aspects of perturbative QCD (pQCD). Lepton-lepton processes provide clean measurements of as(Q2) and of the fragmentation functions of partons into hadrons. Measurements of deep-inelastic scattering (DIS) struc-

155

ture functions (^2,-^3) in lepton-hadron scattering and of lepton pair production cross sections in hadron-hadron collisions provide the main source on quark distributions fa(x, Q) inside hadrons. At leading order, the gluon distribution function g(x,Q) enters directly in hadron-hadron scattering processes with direct photon and jet final states. Modern global parton distribution fits are carried out to next-to-leading (NLO) order which allows as(Q2),qa(x,Q) and g(x, Q) to all mix and contribute in the theoretical formulae for all processes. Nevertheless, the broad picture described above still holds to some degree in global pdf analyses. The data from DIS, DY, direct photon and jet processes utilized in pdf fits cover a wide range in x and Q. The kinematic 'map' in the (x,Q) plane of the data points used in a recent parton distribution function analyses is shown in Figure 3. The HERA data (Hl+ZEUS) are predominantly at low x, while the fixed target DIS and DY data are at higher x. There is considerable overlap, however, with the degree of overlap increasing with time as the statistics of the HERA experiments increase. DGLAP-based NLO pQCD should provide an accurate description of the data (and of the evolution of the parton distributions) over the entire kinematic range shown. At very low x and Q, DGLAP evolution is believed to be no longer applicable and a BFKL description must be used. No clear evidence of BFKL physics is seen in the current range of data; thus all global analyses use conventional DGLAP evolution of pdf's. There is a remarkable consistency between the data in the pdf fits and the NLO QCD theory fit to them. Over 1300 data points are shown in Figure 3 and the % 2 /DOF for the fit of theory to data is on the order of 1. 3.2

PDF Uncertainty

In addition to having the best estimates for the values of the pdf's in a given kinematic range, it is also important to understand the allowed range of variation of the pdf's, i.e. their uncertainties. The conventional method of estimating parton distribution uncertainties is to compare different published parton distributions. This is unreliable since most published sets of parton distributions (for example from CTEQ and MRST) adopt similar assumptions and the differences between the sets do not fully explore the uncertainties that actually exist. Quantifying the uncertainties in a global QCD analysis is far from being a straightforward exercise in statistics. There are non-gaussian sources of uncertainty from perturbation theory (e.g., higher order and power law corrections), from choices of the non-pertubative input (i.e. intial parton distributions at a low energy scale), from uncertain nuclear corrections to experiments performed on nuclear targets, and from normal experimental

156

DIS (fixed target) !•• HERA (94) s DY W-asymmetry 0 Olrect-Y ' Jets

to2

:

* >

«„ -. *

10°

101

10 2

I03

104

1/X

Figure 3. The kinematic map in the (x,Q) plane of data points used in the CTEQ5 analysis.

statistical and systematic errors. Several approaches to this problem have been proposed, with rather different emphases on the rigor of the statistical method, scope of experimental input, and attention to various practical complications ".is.ie.iT.is.io. In particular, there are two recent results from the Michigan State University group, of which the author is a member 18 ' 19 . In the first paper, the Lagrange Multiplier method is used to explore the multi-dimensional parameter space, using an effective x 2 function that conveniently combines the global experimental, theoretical and phenomenological inputs to give a quantitative measure of the goodness-of-fit for a given set of pdf parameters. The method probes directly the variation of the effective x2 along a specific direction in pdf parameter space-that of the maximum variation of a specified physical variable. The result is a robust set of optimized sample PDF's from which the uncertainty of the physical variable can be assessed quantitatively without the approximations inherent in the traditional error matrix approach. From this analysis, the uncertainty of the prediction for the W and Z cross sections at the Tevatron was determined to be approximately ± 3 — 4% and approximately ± 8 - 10% at the LHC. Given the possible use of W production as a normalization for all physics processes, it is important to understand both

157

W production at the Tevatron

1350 1320 •a

11290 * 1260 1230 1200 21

21.5 22 0w (nb)

22.5

Figure 4. The minimum global x2 versus aw at the Tevatron. The points were determined by the Lagrange Multiplier Method. The curve is a polynomial to the points.

this and other sources of theoretical uncertaintiy for the W cross section. A plot of the global x 2 versus W cross section at the Tevatron is shown in Figure 4. In the second paper, a general method is developed to quantify the uncertainties of parton distribution functions and their physical interpretations, using the Hessian formalism to study an effective chi-squared function that quantifies the fit between theory and experiment. The result is a set of 2d Eigenvector Basis parton distributions (where d=16 is the number of parton parameters) from which the uncertaintiy on any physical quantity due to the uncertainty in parton distributions can be calculated. As an example, the uncertainty for the gluon distribution resulting from this analysis is shown in Figure 5. The uncertainty on the gluon is relatively small for lower values of x, but as discussed earlier, the gluon distribution is relatively unconstrained for a; values above 0.25. It is in this region than an increased gluon distribution in the CTEQ4HJ/5HJ pdf's provides a better description of the CDF and DO jet data.

158

Figure 5. The fractional uncertainty (at a Q value of 10 GeV) for the gluon distribution in the proton. The solid curves represent two extreme pdf's within the uncertainty band.

4

Outlook/Conclusions

Both CDF and DO are nearing the end of $110M upgrade projects. CDF has recently finished a commissioning run in November of 2000 and Run 2 will begin in earnest for both experiments in March of 2001. In Run 2A (20012003), both experiments will accumulate on the order of 2 fb~l of data. This will lead, for example, to 2 million W events, 6 thousand jets with transverse energies above 300 GeV/c and with the jet cross section measured precisely to 500 GeV/c. In addition to the increased statistics in Run 2A, the jet yield at high ET is greatly aided by the increase in center-of-mass energy from 1.8 TeV to close to 2.0 TeV. Hopefully, the greatly increased statistics will allow a determination of whether an excess in the inclusive jet cross section really exists at high ET and if so, whether the cause is a larger gluon distribution. An increased luminosity in Run 2B (2003-2007) may lead to an accumulation of the order of 30 fb"1 of data, leading to a discovery potential for a relatively light Higgs.

159

Acknowledgments This work was supported in part by the National Science Foundation under grant PHY-9901946. The author wishes to thank the organizers for this workshop for their hospitality. References 1. J. Huston, QCD at High Energy, invited talk at the ICHEP98, Vol. 1, 283-302, hep-ph/9901352. 2. See, for example, the writeups from the recent Workshop and QCD and Weak Boson Physics, available at http://wwwtheory.fnal.gov/people/ellis/QCDWB/QCDWB.html. 3. See, for example, the writeup from the 1999 Les Houches Workshop on Physics at TeV Colliders, J. Huston et al., hep-ph/0005114. 4. G. Blazey, A QCD Survey: 0 < Q2 < 105 GeV 2 , invited talk at DPFOO, hep-ex/0011078. 5. G. Blazey and B. Flaugher, Ann. Rev. Nucl. Part. Sci. 49, 633 (1999). 6. F. Abe et al., (CDF Collaboration), Phys. Rev. Lett. 77, 438 (1996), hep-ex/9601008. 7. J. Huston et al., Phys. Rev. Lett. 77, 444 (1996), hep-ph/9511386. 8. T. Affolder et al., (CDF Collaboration), submitted to Phys. Rev. D, hep-ph/0102074. 9. B. Abbott et al., (DO Collaboration), submitted to Phys. Rev. Lett, hep-ex/0011036. 10. H. L. Lai, J. Huston et a l , Eur. Phys. J. C 12, 375 (2000), hepph/9903282. 11. See for example, L. Babukhadia, hep-ex/0005026. 12. A.D. Martin, R.G. Roberts, W.J. Stirling and R. Thome, Eur. Phys. J C4, 463 (1998), hep-ph/9803445, and earlier references cited therein. 13. M. Gluck, E. Reya, A. Vogt, Eur. Phys. J. C5, 461 (1998). 14. V. Barone, C. Pascaud and F. Zomer, Eur. Phys. J. C12, 243 (2000), hep-ph/9907512; C. Pascaud and F. Zomer, LAL-95-05. 15. S. Alekhin, Eur. Phys. J. C10, 395 (1999), hep-ph/9611213; contribution to Proceedings of Standard Model Physics (and more) at the LHC, 1999; and S. I. Alekhin, hep-ph/0011002. 16. W.T. Giele and S. Keller, Phys. Rev. D58, 094023 (1998), hepph/9803393; W. T. Giele, S. Keller and D. Kosower, Proceedings of the Fermilab Run II Workshop (2000). 17. M. Botje, Eur. Phys. J C14, 285 (2000), hep-ph/9912439.

160

18. R. Brock, D. Casey, J. Huston, J. Kalk, J. Pumplin, D. Stump, W.K. Tung, hep-ph/0101051. 19. J. Pumplin, R. Brock, D. Casey, J. Huston, J. Kalk, H.L. Lai, D. Stump, W. K. Tung, hep-ph/0101032.

R A R E B P H Y S I C S RESULTS F R O M BELLE C.H. W A N G National

Lien-Ho

Institute Belle

of Technology, Collaboration

Miao-Li,

Taiwan

With 5.27 Million BB events recorded on the T(45) resonance at K E K B , the preliminary results on the search of rare B decays has been reported. The branching ratios and the upper limit will be given in the charmless B decay modes, which include strong and electro-weak penguin decays: B —• K*(p)i, B —• <j>K, B —• h+h~ (TT°). The results on the Cabibbo-suppressed modes B —>• D^K and J/rpKi will also be presented here.

1

Introduction

According to the Standard Model (SM), penguin diagrams play the key role in the CP violation and the charmless Rare B meson decays provide a rich information for us to understand the penguin phenomenon 1. Up to now, dozen charmless B decays have been reported by CLEO 2 and need to be confirmed by results from KEK or SLAC B-factory. Penguin diagrams dominate most of those decays, some are mix-up between tree and penguin diagrams. Penguin diagram is one of the major mechanisms of the CP violation. With more charmless B decays being measured, we can have more understandings on penguin diagrams and more information to possibly extract CP phase. New Physics may also arise from some unexpected suppressed or large decay branching ratios of these charmless B decays. On the other hand, Cabibbo-suppressed modes B -> DK can provides a theoretically clean method to determine 03 (also known as 7). Ratio of the Cabiboo suppressed B -> D^K to the Cabiboo allowed B -t D^n will be reported here, which will give us information on Cabiboo mixing angle, pion and kaon decay constants assuming factorization and SU(3) symmetry 3 . The first measurement of B -¥ J/ipKx will also be presented here. In this paper, the KEKB storage ring, Belle detector and its performance will not be described here. The complete description can be found in reference elsewhere4. This paper will be organized as the follows. The brief analysis techniques will be described in the section 2. Following that section will be the preliminary results of selected decay modes, where the more detailed information can be found in http://bsunsrvl.kek.jp/conferences/papers/.

161

162

2

General Features of the Data Analysis

B candidates are identified through the beam constrained mass MB = V-Vbeam - PB* and AE = Ebeam - EB, where EB, PB* are the B energy and momentum in CM frame and Ebeam = ECMS/2Because of the two body decay features of the charmless decay modes we have studied, the daughter particles' momentum will generally fall between 2 and 3 GeV in CM frame which is very rare in general b to c decay. Therefore, the major dominant background will come from continuum events which is from ee -* 7* - • qq. Due to the production diagram, the continuum event has a two-jet structure and is very different from T(45) —• BB where B decay almost at rest (spherical shape) in T(45) rest(CM) frame. To discriminate between jetty and spherical event topologies, modified Fox-Wolfram5 is used, which is defined as

where ps is the momentum of one signal particle, p° is the momentum of one of the other particles in that event that does not belong to signal B candidate, and Pi(cos9) is the Legendre Polynominal. Through Fisher Discriminant, those moments are then combined to form one Super Fox Wolfram variable (SFW): SFW = 24=1,4(0, V " + Pith0'0) where a and P are Fisher coefficients. The content of SFW may differ slightly depending on the analysis mode and the way of calculating hi3'" may also vary. But the basic structure is the same. People should refer the Belle conference web site for the exact definitions of the variables used in different analysis. Other variable usually being used to suppress continuum background is cosdth, where 9th is the angle between the thrust of the B candidate and the thrust of the rest of the event. The cosdth distribution will be flat for spherical event shape and peak around -1 and + 1 for jetty background. There are basically two approach to use SFW to extract signal. The first one is the slice cuts method. People just make a straight cuts on the shape variables SFW ,cosdthrust or combination ot both for each event, then fit MB and AE to get the signal yields. Another way is to combine SFW, cosOthrust-, MB and AE into a likelihood ratio function, yields are extracted by maximized that function (Maximum Likelihood ratio fit).

163

(a)

(b)

(c)

Figure 1. (a.)AE and (b)Mfl distributions for B -> K*y The upper one is K*° mode and the lower one is K*+ mode, (c) The energy spectrum of b —• 37 before (top) and after (bottom) background subtraction. The solid histogram is the signal Monte Carlo. The open circle in the top are estimated from SFW side-band and the dash hitogram are from continuum Monte Carlod data.

3 3.1

Radiative B decays B -¥ K*j, B -> py

K*(892) is reconstructed though four different K-K modes with I M R ^ I < 75MeV/c?. Loose particle identification has been applied to that kaon track. The 7 is required to be within Calorimeter Barrel region, 33° < 9lab < 128° and energy(lab) between 1.8 GeV and 3.3 GeV. Calorimeter shower shape and 7T° and r\ veto are also applied here, p is reconstructed through nn mode with |M ffff | < IhQMeV/c?. Tight particle identification is applied on those pions and K*j feed down will also be removed by requiring \MK-K — M(K*)\ > 50MeV/(?. SFW is used to suppress the continuum background. Signal yields are obtained through the fit on the MB distribution with AE cut in the signal region. The distribution is in Fig. 1. No excess is seen for fry mode which set the limit for the ratio B -*• fry/B -» K*i < 0.28 (90% C.L.). The results are in Table 1. 3.2

b -»• S7

For inclusive b -> s'y decays, we combined one high energy 7 (1?7 > 2.1GeV), one K± or Ks and no more than 4 pions (at most one 7r°) to form the B candidate. The best candidate is cosen based on the best of AE information.

164 Table 1. Belle preliminary results for radiative B decays. The first error is statistical, the second error is systematic, and the third error is the theoretical. Upper limits are all stated in 90% C.L. The branching ratio and the upper limit are in units of 1 0 - 8 .

Mode b -> S7 B° -> K*° K*+j B° -> p ° 7

Signal Yield 92 ±14 33.7 ±6.9 8.7 ±4.2

B+ ->• p+7

UglCtV)

(a)

AEtCeV)

(b)

BR 33.4±5.0±l;*+a'; 4.94±0.93ig"g 2.87±1.20ig-g <0.56 <2.27

M(GeV)

(c)

Figure 2. AE distributions for (a) B -¥ K+ir~, (b)7r+7r_ and (c) K°n+ modes. The smaller portion of the fits are the n+(K+)ir~ feed-down for K+(w+)ir~ mode.

7 selection is similar to the previous one and additional cuts are made based on MB, the SFW, the invariant mass of kn7r(m(X s )). Signal yields are extracted through the 7 energy distribution in the CM frame. The 7 spectrum from SFW side-band are treated as the backgrounds contribution. The results are shown in Fig. 1 and listed in Table 1. 4 4.1

Hadronic charmless B decays B-t

Kir,inr,KK

The B -¥ PP modes include h+h~, Kgh*, h^-K0 and Kg7r°. For modes without 7T°, likelihood ratio function is formed using signal and background SFW shape and the yields are extracted from simultaneous fit on MB and AE distribution after cuts on likelihood ratio function. For modes with w°, Maximum likelihood ratio fit method is applied which combine SFW, Mg and

165 Table 2. Belle preliminary results for charmless B —• PP decays. The first error is statistical, the second error is systematic. Upper limits are all stated in 90% C.L. T h e branching ratio and upper limit are in units of 10~ 6 .

Mode B° -> K+-KB+ -»• K+n° B+ -+ K°ir+

Signal Yield

B° -> K°n°

25.6+^ 32.3±l\ 5.712:$ 10.8l44:u

B° -> TT+TT-

9-3±|:?

B+ -> TT+TT0

B° -> K+KB+ -> If+if 0

(A)

5.4^. 4

o.8ig:S8 o.o±S: 0

Sig. 4.4 5.0 2.4 3.9 1.9 1.3

BR

UL

1.741S:« ±0.34 1.88iX;S±0.23 u -, fifi+0.98+ -^ i .O D -0.78_o.24

9

<3.4

in+o.93+u.ab

^•iu-0.78_o.23

0 . 6 3 + ^ ±0.16 0.33tS:2? ± °-

07

< 1.65 < 1.01 <0.6 <0.51

(B)

Figure 3. (A) Projections of A E and MB distributions for hw0: (top)B —• K'+7r°,(middle)7r + 7r 0 and (bottom) ft"°7r° modes. The dash portion in mode 7r+7r° are the K+n° feed-down. (B) MB distribution and fits for B -* (102Q)K+ mode.

AE all together. The MB and AE distribution are in Fig. 2 and Fig. 3. The results are listed in Table 2. 4.2

B -» 0(1020)^

Due to the excellent performance of the Belle particle identification system, (j> can be easily reconstructed with relatively small background. The <j> candi-

166

A

•

"

•:•,«.;

ftp (a)

(b)

Figure 4. AE distributions for (a) B D—K+ and (c) B~ ->• D*°K- modes.

(c)

D°7i-(upper) and D°K~ (lower), (b)B°

Table 3. Belle preliminary results for Cabiboo suppressed B —»• D^*'if decays. The Cabiboo allowed mode B —> D^n is also listed. The ratio if the ratio of Cabiboo suppressed vs Cabiboo allowed. The first error is statistical, the second error is systematic.

Mode B~-> D»K-(«-) B~

-+D*°K-(TT-)

B°^D*+K-(ir-)

Yields for K 48.7 ± 8 . 4

13.333 6.7±*;*

Yields for n 832 ± 34 155.4 ±14.4 164.8 ± 13.6

Ratio of BR 0.081 ±0.014 ±0.011 0 . 1 3 4 i ^ | ±0.015 0 - 0 6 3 + S ± 0.013

dates are reconstructed through two kaon with cuts IMKK—M,/,] < 10MeV/c2. Both tracks have been applied loose particle identification and have to form a good vertex. Continuum background are suppressed by cuts on cos6thrust, B flight direction and helicity angle. Signal yields are obtained by fitting on MB distribution. The results(Fig. 3) give 9.21^9 signals for K+, which correspond to a branching fraction of (1.72JIo".i4 ± 0.18) x 1 0 - 5 . No excess is seen in <j>Ks mode. 5

Cabiboo Suppressed B —»• D^K

decays

Also thanks to the good particle identification system in Belle, we can easily reduce the Cabiboo favored B —• D^n feed-down and extract the B -> D^K. The D*+ and D*° candidate are reconstructed through D*° -> £>°7r°, D*+ -»• D°Tr+andD+Tr° with D° -¥ K~ir+,K- -Tr+Tr°,K-*{**) and D+ -» K^-K+-K+,K8-K+,KS-K+'K0,KS-K+-K+'K~.

The invariant mass of D

is required to be within 4
167

S300

-l?20

mrJ(

,ii\,ri,„n,,

0 00 b| AE (GeV) all channels

0 20

- - - -.^t'Sr^lvr.-':'-'.- -J ..t^tiE.Y1:.-;:;-;1,.'^

100

1 5D

ZOO

100

150

3 00

5250

Ai-rffl

,§Z,7,5

oaa

(a)

(b)

Figure 5. (a)mr vs Kmr scatter plots and projection in MB — AE signal (right) and sideband (left) region, (b) MB and AE for various if i mode after K\ mass cuts.

of the branching ratio R = Br(B -> D^K)/Br{B ->• D^n) to reduce the systematic uncertainty. The results are in Table 3. The AE distribution for Fig. 4 6

B-> J/rl>Ki(1270)

In this decay modes, ifi(1270) is reconstructed through modes: K+TT+TT~, + + 0 K TT~TT and Ksir n~. The feed-down from tp(2s) -t J/ipmr are also excluded here. The signal is confirmed by looking at the mass scatter plot Mn7r vs M^TTTT in B MB and AE signal region with B -» J/ipKirir. A cluster in the area near Mnv ~ Q.75GeV/c2, MK-K-K ~ l-3GeV/c2 which indicate the ifi(1270) resonance(Fig. 5). The yields are extracted through simultaneous fits on MB and AE of B candidate while cutting Kirir in the Jfi(1270) mass range. The results are shown in Fig. 5 and the numbers are listed in Table 4. Acknowledgments We gratefully acknowledge the effort of the KEK-B staff in providing us with excellent luminosity. We acknowledge the support of all the participating institutions within Belle. We also acknowledge the support from National Science Council and Education Ministry of Taiwan.

168 Table 4. Belle preliminary results for B -> J/T)>KI(1270) cal, the second error is systematic.

decays. The first error is statisti-

mode

yields or ratio 25.2_ 6 ; 3

J/^iVT+Tr+TTJ/ipK+Tr°Tr-

12.9+J-;

J/ipK°ir+Tr-

J/rpK+ Br(B° -+ J/i/;K1v)/Br(B+ Br(B+ -> J/ipKi+)/Br(B+

-+ J/ipK+) -»• J/i})K+)

238.0 ±15.9 1.4 ± 0 . 4 ± 0 . 4 1.5 ± 0 . 4 ± 0 . 3

References 1. For discussions of theory of rare B meson, please see: M. Neubert, hep-ph/0001334(2000); Y.Y. Kuem, H.N. Li, A.I. Sanda, hepph/0004004(2000); W.S. Hou and K.C. Yang Phys. Rev. D 6 1 , 73014 (2000). 2. For various CLEO's charmless B decays mode, please see CLEO conference papers:CLEO-CONF-99-12,99-13,99-14 (1999). 3. D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78, 3257 (1997). 4. The Belle Collaboration, KEK Progress Report 2000-4. 5. G. Fox and S. Wolfram, Phys. Rev. Lett. 4 1 , 1581 (1978).

R E C E N T B C P P R O G R E S S I N TAIWAN HSIANG-NAN LI Department of Physics, National Gheng-Kung University, Tainan, Taiwan 701, Republic of China Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300, Republic of China I review theoretical progresses on B physics and CP violation which were made in Taiwan recently. I concentrate on the approaches to exclusive B meson decays based on factorization assumption, SU(3) symmetry, perturbative QCD factorization theorem, QCD factorization, and light-front QCD formalism. 1

Introduction

The collaboration on B physics and CP violation (BCP) is one of the most active groups in Taiwan. In this talk I will briefly review theoretical progresses on BCP, which were made in Taiwan recently. For experimental review, please refer to Dr. Wang's talk in this workshop. I will concentrate on five approaches to exclusive B meson decays based on factorization assumption (FA), SU(S) symmetry, perturbative QCD (PQCD) factorization theorem, QCD factorization (QCDF), and light-front QCD (LFQCD) formalism. Abundant results of exclusive B meson decays have been produced and important dynamics has been explored. The effective Hamiltonian for b quark decays, for example, the b —• s transition, is given by 1 10

GF

9) #eff = -k > . Vq d(M)Oi (M) + C2{v)0<$\n) + Y^CiMOiin) , ( 1 ) V2 q=u,c i=3

with the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements Vq — V*sVqb. Using the unitarity condition, the CKM matrix elements for the penguin operators O3-O10 can also be expressed as Vu + Vc — —V*. The definitions of the operators Oi are referred to 1 . According to the Wolfenstein parametrization, the CKM matrix upto 0(A 3 ) is written as 'Vud Vcd

Vtd

Vus Vub\ VC8 Vcb\

Vu

VtbJ

( = [

1-V _A

\AX3{l-p-ir}) 169

A 1- £

-Ay

AX3(p-ir])\ AA2

1

/

,

(2)

170

with the parameters 2 A = 0.2196 ± 0.0023, A = 0.819 ± 0.035, and Rb = y/p2 + V2 = 0.41 ± 0.07. The unitarity angle fa is denned via Vub = \Vub\exp(-ifa)

.

(3)

One of the important missions of B fatories is to determine the angles fa, fa and fa. The angle fa can be extractedfrom the CP asymmetry in the B ->• J/ipKs decays, which arises from the B-B mixing. Due to similar mechanism of CP asymmetry, the decays B° - • n+7r~ are appropriate for the extraction of the angle fa. However, these modes contain penguin contributions such that the extraction suffers large uncertainty. It has been proposed that the angle fa can be determined from the decays B -> Kir, 7r7r3'4'5'6. Contributions to these modes involve interference between penguin and tree amplitudes, and their analyses rely heavily on theoretical formulations. 2

Factorization A s s u m p t i o n

The conventional approach to exclusive nonleptonic B meson decays is based on FA 7 , in which nonfactorizable and annihilation contributions are neglected and final-state-interaction (FSI) effects are assumed to be absent. Factorizable contributions are expressed as products of Wilson coefficients and hadronic form factors, which are then parametrized by models. With these approximations, the FA approach is simple and provides qualitative estimation of various decay branching ratios. To extract the angle fa, we consider the ratios R and R^ defined by

BijB'-HC***) U

~~ B r ( 5 ± -»• KH±)

Br(i%->***?) '

*

Br(B° -»• 7r±7rT) '

w

where Bv(Bd -» if±7T=F) represents the CP average of the branching ratios Br(B9 ->• K+ir-) and Br(J§2 -> K~TT+), and the definition of B r ( S ± -»• K°n^) is similar. It has been shown that the data R ~ 1 imply fa ~ 90° 8 . To explain the data of R„ ~ 4, a large angle fa ~ 130° must be postulated 8 . It is easy to observe from Eqs. (2) and (??) that the products of the CKM matrix elements V*sVub and V*dVub have the same weak phase, and that the real parts of V£Vtb and V^dVub are opposite in sign. That is, the tree-penguin interference in the decays B —t Kw and B —> ww is anticorrelated. A fa > 90° then leads to constructive interference between the tree and penguin contributions in B —• Kn, and a large R^. The determination fa ~ 114° from the global fit to charmless B meson decays 8 , located between the two extreme cases 90° and 130°, is then understood. On the other hand, a large B -> p form factor AQP ~ 0.48 has been extracted, which

171

accounts for the large B —• pir branching ratios. In the modified FA approach with an effective number of colors N%s, a large unitarity angle 3 ~ 105° is also concluded 9 . The above
SU(3) Symmetry

The model-dependent determination of the angle fa from FA seems not to be satisfactory. A more model-independent approach based on SU(3) symmetry has been proposed 11 , in which the light quarks u, d and s form a SU(3) triplet, while the heavy quarks c, 6, and t form 5C/(3) singlets. According to the above assumption, the B mesons Bu, B^, and Bs form a SU(3) triplet at the hadronic level. Pseudo-scalar mesons P and vector mesons V also possess definite St/(3) structures, V2

V6

K~

V3

M) =

\/2

Ve

y/3

K°

K+

K°

\ (5)

_2aa _|_ m^ V6 ^ V^

K*~\

v? =

P+ K*+

K*°

K*°

•

(6)


Similarly, the four-fermion operators in the weak Hamiltonian can be decomposed into operators with definite SU(3) structures. For example, the penguin operators O3-6 are labelled as 3 states, qb(uu + dd + ss) +* 3 ,

(7)

since qb forms a triplet 3 and uu + dd+ss froms a singlet. The operators 0\t2 are written as quub 0 3 x 3 x 3 = 3 + 3 + 6 + 1 5 .

(8)

Following Eqs. (7) and (8), the effective Hamiltonian in Eq. (1) is decomposed into operators carrying different SC/(3) structures, such as H(3), H(6) and

172

H(15), whose coefficients are the linear combinations of the Wilson coefficients. Employing the above results, we formulate various decay amlitudes of the tree and annihilation topologies for B -»• PP modes. For example, the parameter Cg associated with the contraction BiMjJMJ*.H'(3)-7 represents a tree amplitude. The parameter A% associated with the contraction BiH(2>)lMfMlk represents an annihilation amplitude. Collecting all expressions for the branching ratios and the CP asymmetries of the B -» nw, Kn and KK modes, there are totally 13 free parameters. This number is too big for a global analysis of currently available data. As an approximation, annihilation contributions are neglected. 8 parameters, the absolute values of C j \ C^, Cj and C^5, the phases 5£, 8$, and 6^5, and the CKM phase fa, are then left, where T (P) denotes the tree operators Oi,2 (the penguin operators O3-10).

The best fit to data gives 03 = 70°, ,9 = 0.17, 7? = 0.37, C j = 0.28, Cf = 0.14, Cf = 0.33, # = 12°, # = 6 ° , tf6 = 74°.

Cf5 = 0.14, (9)

If the SU(3) symmetry breaking effect from fx/ f-K ^ 1 is taken into account, IK (/TT) being the kaon (pion) decay constant, the results are shifted only a bit. Hence, there is no indication that
Perturbative QCD

It has been shown that the decay amplitudes and strong phases discussed in the previous sections can be evaluated in the PQCD framework, and that it is possible to extract z from the B ->• Kir data 1 2 . According to PQCD factorization theorem, a B meson decay amplitude is expressed as convolution of a hard b quark decay amplitude with meson wave functions. A meson wave function, absorbing nonperturbative dynamics of a QCD process, is not calculable, while a hard amplitude is. In perturbation theory nonperturbative dynamics is reflected by infrared divergences in radiative corrections. It has been proved to all orders that these infrared divergences can be separated and absorbed into meson wave functions 13 . A formal definition of wave functions as matrix elements of nonlocal operators has been constructed, which, if evaluated perturbatively, reproduces the infrared divergences. The gauge invariance of the above factorization has been proved in 14 . A meson wave function must be determined by

173

nonperturbative means, such as lattice gauge theory and QCD sum rules, or extracted from experimental data. A salient feature of PQCD factorization theorem is the universality of nonperturbative wave functions. Because of universality, a B meson wave function extracted from some decay modes can be employed to make predictions for other modes. This is the reason PQCD factorization theorem possesses a predictive power. In the practical calculation small parton transverse momenta kr are included 15 , which are essential for smearing the end-point singularities from small momentum fractions 12 . Because of the inclusion of kr, double logarithms In (Pb) are generated from the overlap of collinear and soft enhancements in radiative corrections to meson wave functions, where P denotes the dominant light-cone component of a meson momentum and 6 is a variable conjugate to kr- The resummation of these double logarithms leads to a Sudakov form factor exp[—s(P, b)] 16 ' 17 , which suppresses the long-distance contributions in the large b region, and vanishes as b = 1/A, A = AQCD being the QCD scale. This suppression guarantees the applicability of PQCD to exclusive decays around the energy scale of the b quark mass 18 . The hard amplitude contains all possible Feynman diagrams 19 ' 20 , such as factorizable diagrams, where hard gluons attach the valence quarks in the same meson, and nonfactorizable diagrams, where hard gluons attach the valence quarks in different mesons. The annihilation topology is also included, and classified into factorizable or nonfactorizable one. Therefore, FA for twobody B meson decays is not necessary. It has been shown that factorizable annihilation contributions are in fact important, and give large strong phases in PQCD 1 2 . We emphasize that the hard amplitude is characterized by the virtuality of internal particles, t ~ \/\MB ~ 1.5 GeV, A = MB—mb- The RG evolution of the Wilson coefficients C±$(t) dramatically increase as t < MB/2, such that penguin contributions are enhanced 12 ' 21 . With this penguin enhancement, the observed branching ratios of the B -> Kw and B -» mr decays can be explained in PQCD using a smaller angle fa = 90°. That is, the data of R^ do not demand large fa. Such a dynamical enhancement of penguin contributions does not exist in the FA approach. Our predictions for the branching ratio of each Kir mode corresponding to fa = 90° 12 , Br(B+ -> JF£T07r+) = 21.72 x 1(T 6 , Br(fi- -> KQ-K~) = 21.25 x 10~ 6 , Bv{B°d -> K+TT-) = 24.19 x 1(T 6 , Br(BJ} -> K~ir+) = 16.84 x 1(T 6 , Br(B+ -» K+TT0) = 14.44 x 1(T 6 , B r ( £ - -> K'n0) = 10.65 x 1(T 6 , Br(£°. -> K°7r°) = 11.23 x 1(T 6 , Bv(B°d - • K°n°) = 11.84 x 1(T 6 , (10)

174

are consistent with the CLEO data 2 2 , Bv{B± -s- JftT°7r±) = (18.2±|;g ± 1.6) x 10~ 6 , Br(B° -> K*^)

= (17.2l 2 j ± 1.2) x 1(T 6 ,

Br(B± -» JftT±7r°) = (11.6±|;?±i;|) x 10" 6 , Br(B° -> K°7T0) = (U.6tlit23A3)

x 1(T 8 ,

ACp{B°d -» K*1**) = -0.04 ± 0.16 , A C p( J B ± -s. J ft: 0 7r ± )=0.17±0.24. r±

(11)

T

In the above expressions B(Bd -> J< 7r )_ represents the CP average of the branching ratios B(B°d -» K+TT') and B(B° - • #-71-+). 5

Q C D Factorization

Recently, Beneke, Buchalla, Neubert, and Sachrajda proposed the QCDF formalism for two-body nonleptonic B meson decays 23 . They claimed that factorizable contributions, for example, the form factor FBn in the B -> TTTT decays, are not calculable in PQCD, but nonfactorzable contributions are in the heavy quark limit. Hence, the former are treated in the same way as FA, and expressed as products of Wilson coefficients and FBv. The latter, calculated as in the PQCD approach, are written as the convolutions of hard amplitudes with three (B, n, n) meson wave functions. Annihilation diagrams are neglected as in FA, but can be included as 1/Mjg correction. Values of form factors at maximal recoil and nonperturbative meson wave functions are treated as free parameters. Here I mention some essential differences between the QCDF and PQCD approaches. For more detailed comparisons, refer to 2 4 . Because of the neglect of annihilation diagrams in QCDF, strong phases and CP asymmetries are much smaller than those predicted in PQCD. In QCDF the leading-order diagrams are those that contain vertex corrections to the four-fermion operators. These diagrams, however, appear at the next-to-leading order in PQCD. This difference implies different characteristic scales in the two approaches: the former is characterized by the b quark mass m&, while the latter is characterized by the virtuality t of internal particles, which leads to the penguin enhancement emphasized above. Without penguin enhancement, a large fe is still necessary to account for the large ratio R.„25. The B ->• (f>K decays have been analyzed in the QCDF formalism 26 ' 27 , and branching ratios much smaller than experimental data have been obtained. Since these modes are dominated by penguin contributions, the penguin en-

175

hancement may be crucial for explaining the data. 6

Light-front QCD

The evaluation of a form factor is simple in the LFQCD formalism, which is written as an overlap integral of initial- and final-state meson wave functions 28 . Various form factors have been computed, such as the B ->• n, p, K and K* form factors 29 and the Aj —> A form factors 30 . The results have been employed to predict the decay spectra of the B -> KMJL(TT) and At -»• AjU^(rr) modes. This formalism has been also applied to the radiative leptonic B meson decays B —• Z+Z~731. These predictions can be compared with data in the future. 7

Conclusion

In this talk I have briefly summarized the theoretical progresses on exclusive B meson decays, which were made by Taiwan BCP community recently. With the active collaboration, more progresses are expected in the near future. Acknowledgments This work was supported in part by the National Science Council of R.O.C. under the Grant No. NSC-89-2112-M-006-033, and in part by Grant-in Aid for Special Project Research (Physics of CP Violation) and by Grant-in Aid for Scientific Exchange from Ministry of Education, Science and Culture of Japan. References 1. G. Buchalla, A. J. Buras and M. E. Lautenbacher, Review of Modern Physics 68, 1125 (1996). 2. Review of Particle Physics, Eur. Phys. J. C 3, 1 (1998). 3. M. Gronau, J.L. Rosner, and D. London, Phys. Rev. Lett. 73, 21 (1994); R. Fleischer, Phys. Lett. B 365, 399 (1996). 4. R. Fleischer and T. Mannel, Phys. Rev. D 57, 2752 (1998). 5. M. Neubert and J. Rosner, Phys. Lett. B 441, 403 (1998); M. Neubert, J. High Energy Phys. 02, 14 (1999). 6. A.J. Buras and R. Fleischer, Eur. Phys. J. C 11, 93 (1999). 7. M. Bauer, B. Stech, M. Wirbel, Z. Phys. C 34, 103 (1987); Z. Phys. C 29, 637 (1985).

176

8. N.G. Deshpande, X.G. He, W.S. Hou and, S. Pakvasa, Phys. Rev. Lett. 82, 2240 (1999); W.S. Hou, J.G. Smith, and F. Wurthwein, hepex/9910014. 9. H.Y. Cheng, hep-ph/9912372; H.Y. Cheng and K.C. Yang, hepph/9910291. 10. W.S. Hou and K.C. Yang, Phys. Rev. Lett. 84, 4806 (2000). 11. X.G. He, Eur. Phys. J. C 9, 443 (1999); N.G. Deshpande, X.G. He, and J.Q. Shi, Phys. Rev. D 62, 034018 (2000). 12. Y.Y. Keum, H-n. Li, and A.I. Sanda, hep-ph/0004004; hep-ph/0004173. 13. H-n. Li, hep-ph/0012140. 14. H.Y. Cheng, H-n. Li, and K.C. Yang, Phys. Rev. D 60, 094005 (1999). 15. H-n. Li and G. Sternum Nucl. Phys. B381, 129 (1992). 16. J.C. Collins and D.E. Soper, Nucl. Phys. B193, 381 (1981). 17. J. Botts and G. Sterman, Nucl. Phys. B325, 62 (1989). 18. H-n. Li and H.L. Yu, Phys. Rev. Lett. 74, 4388 (1995); Phys. Lett. B 353, 301 (1995); Phys. Rev. D 53, 2480 (1996). 19. C.H. Chang and H-n. Li, Phys. Rev. D 55, 5577 (1997). 20. T.W. Yeh and H-n. Li, Phys. Rev. D 56, 1615 (1997). 21. C. D. Lii, K. Ukai, and M. Z. Yang, hep-ph/0004213. 22. CLEO Coll., Y. Kwon et al, hep-ex/9908039. 23. M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); hep-ph/0006124. 24. Y.Y. Keum and H-n. Li, hep-ph/0006001. 25. D. Du, D. Yang, and G. Zhu, hep-ph/0005006; T. Muta, A. Sugamoto, M.Z. Yang, and Y.D. Yang, hep-ph/0006022. 26. H.Y. Cheng and K.C. Yang, hep-ph/0012152. 27. X.G. He, J.P. Ma, and C.Y. Wu, hep-ph/0008159. 28. H.Y. Cheng, C.Y. Cheung, C.W. Hwang, and W.M. Zhang, Phys. Rev. D 57, 5598 (1998). 29. C.Y. Cheung, C.W. Hwang, W.M. Zhang, Z. Phys. C 75, 657 (1997); H.Y. Cheng, C.Y. Cheung, and C.W. Hwang, Phys. Rev. D 55, 1559 (1997). 30. C.H. Chen and C.Q. Geng, hep-ph/0012003. 31. C.Q. Geng, C.C. Lih, and W.M. Zhang, Phys. Rev. D 62, 074017 (2000).

Q C D - I M P R O V E D FACTORIZATION IN N O N L E P T O N I C B DECAYS JUNEGONE CHAY Department of Physics, Korea University, Seoul 136-701, Korea E-mail: [email protected] I consider nonleptonic decays of B mesons into two light mesons using the lightcone wave functions for the mesons. In the heavy quark limit, nonfactorizable contributions are calculable from first principles in some decay modes. I review the idea of the QCD-improved factorization method and discuss the implications in phenomenology.

1

Introduction

Nonleptonic decays of B mesons have attracted a lot of attention recently since they were observed experimentally in CLEO, BaBar and BELLE. These decays are important in extracting the information on the Cabibbo-KobayashiMaskawa (CKM) matrix elements and CP violation. On the theoretical side, it is the least understood area. Since nonleptonic decays involve nonperturbative effects such as final-state interactions, it is difficult to obtain a quantitative theoretical prediction. However, it has been recently found that nonperturbative effects such as the nonfactorizable contribution in nonleptonic decays could be systematically understood in the heavy quark limit with mj —> oo. Here I will review the current status of understanding on nonleptonic B decays very schematically. I will focus on the underlying ideas omitting technical complication. I hope that this talk will give a clear sketch of what is being studied currently. It has been found that nonfactorizable contributions in nonleptonic B decays into two light mesons could be calculated using perturbation theory in the heavy quark limit mj -¥ oo. I will explain in detail how we treat nonleptonic B decays in the heavy quark limit and discuss some phenomenological aspects. The theoretical framework for B decays is to use the effective weak Hamiltonian at the renormalization scale \i » mt,, which is schematically given as H

*« = % E ^ ^ M O J M ,

(i)

where Fy are the CKM matrix elements, Cj are the Wilson coefficients and 0\ are the effective four-quark operators. The Wilson coefficients are calculable order by order in perturbation theory, and the main issue in analyzing 177

178

nonleptonic decays is how tp_ evaluate the matrix elements of the four-quark operators. For example, if B decays into the final two mesons Mi and M 2 , how do we evaluate ( M I M 2 | 0 ? | J B ) ?

First we can use the naive factorization in which we neglect the strong interaction effects between mesons and separate the operators into a currentcurrent form, and calculate their matrix elements 1 . Schematically this process can be expressed as (M1M2\Oi\B)

» (M 1 |J M |0>(M 2 |J"|B>.

(2)

Here JM is the current operator in the effective Hamiltonian. Each matrix element is parameterized by a decay constant or a form factor, which describe intrinsic nonperturbative effects. The decay constant and the form factors can be obtained from either experiment or other theoretical techniques such as QCD sum rules 2 . Unfortunately this naive factorization is unsatisfactory. First of all, there is no justification in neglecting the final-state interactions between mesons. The argument of color transparency 3 can be applied in the case of two final light mesons, but it should be proved explicitly if the matrix elements can be truly factorized including final-state interactions. Secondly, theoretically the naive factorization gives an unphysical result. Decay constants and form factors are independent of the renormalization scale fi, and the decay amplitude has an arbitrary ju dependence through the Wilson coefficients Ci(fi). This is because we replace the matrix elements, which depend on fj,, by the decay constants and the form factors, which are independent of fi. Therefore the resulting amplitude is unphysical. As a remedy to this problem, Ali and Greub 4 suggested to calculate the radiative corrections of the operators before taking the matrix elements. Following this procedure, we can write the matrix element as {M1M2\Oi\B)

» F( A i l a.)[(M 1 |J M |0>(M 2 |J"|B>] t r e e .

(3)

It turns out that the n dependence in F(ti,ag) arising from the radiative corrections cancel the /x dependence in the corresponding Wilson coefficients at any given order, hence the decay amplitude does not depend on the renormalization scale. I call this procedure the "improved factorization". The unphysical fj, dependence is absent in the improved factorization, but it has other significant problems. First of all, in order to avoid infrared divergence, F(n,a„) is calculated with off-shell external quarks 5 . The external momenta p 2 play a role of the infrared cutoff. However, because of this, the decay amplitudes depend on the choice of the gauge, and the employed reg-

179

ularization schemes 6 . Therefore the decay amplitude depends on arbitrary gauges and the schemes, hence also unphysical. If we put external quarks on their mass shell, the gauge dependence and the scheme dependence go away, but, in this case, the amplitude becomes infrared divergent and gives also an unphysical result. Therefore we need a consistent scheme which solves all the problems I mentioned above. Before 1 explain the main idea of the QCD-improved factorization, note that the Feynman diagrams from which infrared divergence appears are those with vertex corrections of the weak currents. 2

QCD-improved factorization

The source of infrared divergence in the radiative corrections of the effective weak Hamiltonian, as mentioned above, suggests an interesting idea. The Feynman diagrams which cause infrared divergence are the radiative corrections of vertices. In other words, the infrared divergence comes from the radiative corrections for the decay constant of a light meson and form factors for B -> Mi. It reminds us of the hadron-hadron scattering, in which the infrared divergence of the scattering amplitude is absorbed in the redefinition of the parton distribution functions. The remaining hard scattering amplitude and the parton distribution functions are factorized. We can apply the same idea to B decays. The infrared divergences can be attributed to the renormalization of the decay constant and the form factors, and other radiative corrections constitute nonfactorizable contributions. That is, the infrared divergence is absorbed in the definition of the light-cone meson wave function or the form factors. This is first observed in Ref. 7 Recently Beneke et al. 8 formulated this problem in the heavy quark limit and extensively studied nonleptonic B decays into two final-state mesons. The idea can be summarized as follows: We first take the heavy quark limit rrib —» oo. In this limit we can calculate nonfactorizable contributions systematically in perturbative QCD. Also we can obtain corrections to the perturbative results by expanding in powers of AQCD/WI;,. The next step is to arrange external quarks using Fierz transformation such that the quark-antiquark pair, which forms a meson, is included in a single current. And we use the light-cone wave function for the mesons. If we can use the operators in the effective Hamiltonian as they are in arranging quarks, we call that configuration of quarks as "charge-retention configuration". If we have to switch some quarks using Fierz transformation, we call that configuration as "charge-changing configuration". This arrangement is important

180

since it determines which radiative corrections correspond to the corrections to the decay constants or the form factor. Since the decay amplitude in each process depends on how we arrange quarks, the scattering amplitude becomes process-dependent. The infrared divergence is attributed to the renormalization of the wave function or the form factors. And we calculate all the nonfactorizable contributions along with the spectator contribution and the annihilation channels. This procedure is called the ''QCD-improved factorization". If all the nonfactorizable contributions are infrared finite, and suppressed as m\, goes to infinity, these processes can be treated in the QCD-improved factorization. In this case, we have a theoretical method to analyze nonleptonic B decays from first principles. In the QCD-improved factorization method, we can formally write the matrix element as (MiMalOilB) « (M 1 |J M |0)(M 2 |J' i |B)[l + O(a 8 ) + 0 ( A Q c D / m 6 ) ] .

(4)

If we neglect the radiative corrections and the l/m;, corrections, we restore the result obtained in the naive factorization. The matrix element can be written as a product of a decay constant and a form factor. And we can systematically calculate the corrections to the result in the naive factorization. Before we discuss the details of the QCD-improved factorization, I would like to explain another approach to study nonleptonic B decays. Keum et al. 9 have also considered nonleptonic B decays using light-cone wave functions, but they concentrated on the calculation of the form factors instead of nonfactorizable contributions. They calculated a single gluon exchange which is responsible for the correction to the form factor. There also appears infrared divergence in the form factor, but they introduce the Sudakov factor which smears the endpoint region such that there is no infrared divergence. And the origin of imaginary parts in their calculation comes from the modification of the propagator with transverse momentum, which is totally different from the source of the imaginary part in the QCD-improved factorization. They do not consider nonfactorizable contributions at the moment, and hence the leading-order Wilson coefficients are employed. Because what is calculated is totally different in the two approaches, we have to be careful when we try to compare the results in both approaches. 3

Nonfactorizable contribution

Let us go into the detail of how to calculate nonfactorizable contributions. When we use the light-cone meson wave functions for exclusive decays, the

181

transition amplitude of an operator 0$ in the weak effective Hamiltonian is given by

(M1M2|Oi|B) = J2Ff^M2 f

dxT/jix^M^x)

+ / d^dxduT('(^,x,u)4>B(0^Mi(x)^M,(u), Jo

(5)

where p^M2 axe the form factors for B -¥ M2, and (J>M{ (X) is the light-cone wave function for the meson Mt. T^(x) and T 77 (£,a;,u) are hard-scattering amplitudes, which are perturbatively calculable. The second term in Eq. (5) represents spectator contributions.

Figure 1. Feynman diagrams for nonfactorizable contribution at order a3. The dots represents the operators Oj.

The relevant Feynman diagrams for Tf- are shown in Fig. 1. Each Feynman diagram has an infrared divergence. But if we sum over all the Feynman diagrams and symmetrize with respect to x «-» 1 — x, where x is the momentum fraction of the outgoing meson, we have an infrared-finite result. Another feature of the nonfactorizable contribution is that there appears an imaginary part due to the final-state interactions. This plays an important role in studying CP violation in nonleptonic decays. The strong phase is calculable in perturbation theory. And since we are working at next-to-leading order accuracy, the dependence on fj, becomes very mild. There are other nonfactorizable contributions such as the spectator contributions. The Feynman diagrams for Ty7 are shown in Fig. 2. In calculating nonfactorizable contributions, we use the light-cone wave functions. The projection of the quark bilinears to each pseudoscalar light meson wave function to the order of twist three can be written as

(P(j>)\qa(yWp(x)\0) = ~

f

dtiei<'*-''+<1-">''">

182

Figure 2. Feynman diagrams for the spectator contribution.

^{MM-vp^M-v^iy-xY—^-)}-

(6)

For the B meson, we can write the projection as

{Q\qMB) = - ^ M O { ( * B +mBh5}0a

(7)

Here is the leading-twist wave function, and <j>p and a axe twist-three wave functions for the pseudoscalar and the tensor currents respectively. For the B wave function, we take the leading-twist wave function only. Since we expand in powers of 1/mt, at leading order 0 B ( £ ) ~ £(£ — A Q C D / « I ; , ) . In calculating the spectator contribution, we have the integral of the form

[\+2&„™ JO

A

?

(8) QCD

which is enhanced. We can consider corrections of order 0 ( A Q C D / " ^ 6 ) - The most important contribution comes from the term proportional to pp/nib which is given by

„p =

m

P

(9)

^ mi + m 2 where m,\ and mi axe the masses of the quarks which form a meson with mass mp. Compared to other O ( A Q C D ) terms, this is numerically large. For the case of ir+, for instance, it is /xT+ ~ 1.4 GeV. Therefore it has been of great interest to calculate higher-twist effects proportional to \s,p. However, the spectator contribution from 0\ and Oi with higher-twist wave functions is infrared divergent 10 . It has been suggested that we introduce some parameters to regulate the infrared divergence and regard the parameter as a theoretical uncertainty. But these contributions axe numerically large and especially the extraction of the strong phase becomes too ambiguous to say anything quantitatively.

183

The annihilation topology poses another problem. When we calculate the spectator contribution with the operator 0 5 , the amplitude is also infrared divergent. Therefore the annihilation topology can give a significant power correction to the decay amplitude. The analysis on the annihilation channel with radiative correction is in progress. One way to look at the infrared divergence is that it is not a serious problem. The divergence comes from the endpoint and it actually gives the logarithmic enhancement as /

— «In

.

(10)

Since the cutoff AQCD is actually arbitrary, we can parameterize this contribution and regard this as a theoretical uncertainty. But its magnitude is numerically large, and thus it enlarges theoretical uncertainty. Another view is a more conservative one. If there appears an infrared divergence in the hard scattering amplitude, the effect of soft gluon exchange is really significant and the QCD-improved factorization breaks down at this order. It remains to be seen if the QCD-improved factorization really breaks down, or there are some other contributions which cancel the infrared divergence rendering the final result infrared finite. 4

Conclusion

The understanding of nonleptonic B decays into two mesons has acquired a new sophisticated level. In the heavy quark limit, nonfactorizable contributions are calculable using perturbative QCD for light final-state mesons. When one of the final-state meson is heavy, we can still use the QCD-improved factorization for the case in which the spectator quark in the B meson goes to the heavy meson in the final state. If the spectator quark goes to a light meson, as in class II decays, the nonfactorizable contribution is infrared divergent, and the effect of soft gluon exchange is significant. But the analysis of higher-twist effects is yet far from satisfactory. For example, the spectator contribution which is proportional to the twist-three contribution of the meson wave function is infrared divergent. And the annihilation topology also has the infrared divergence. The status of the QCDimproved factorization method for nonleptonic B decays into two mesons is not complete until we disentangle the infrared divergence to give a quantitative prediction. It will be an interesting project to combine the QCD-improved factorization with the calculation of the form factors using the light-cone wave

184

functions. Currently, in the QCD-improved factorization, we use the form factors extracted from experiment, as in semileptonic B decays. On the other hand, in Ref. 9 they only consider the calculation of form factors. It will be interesting to see whether we can give a consistent theoretical description of nonleptonic B decays combining these two approaches. Acknowledgments I would like to thank Pyungwon Ko for the collaboration on the subject. I am also grateful to the organizers of the conference for their kind support. This research is supported in part by the Ministry of Education grants KRF-99-042D00034 D2002, the Center for Higher Energy Physics, Kyungbook National University, and Korea University. References 1. M. Bauer, B.Stech, and M. Wirbel, Z. Phys. C 34, 103 (1987). 2. See, for example, A. Khodjamirian, R. Riickl, S. Weinzierl, C. W. Winhart, and O. Yakovlev, Phys. Rev. D 62, 114002 (2000). 3. J. D. Bjorken, Nucl. Phys. (Proc. Suppl), B l l 325 (1989). 4. A. Ali and C. Greub, Phys. Rev. D 57, 2996 (1998). 5. G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). 6. A. J. Buras, and L. Silvestrini, Nucl. Phys. B 548, 293 (1999). 7. H. D. Politzer and M. B. Wise, Phys. Lett. B 257, 399 (1991). 8. M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B 591, 313 (2000); hep-ph/0009328. 9. Y.-Y. Keum, H.-n. Li and A. I. Sanda, hep-ph/0004173. 10. M. Beneke, hep-ph/0009328.

R A R E R A D I A T I V E B DECAYS IN P E R T U R B A T I V E Q C D DAN PIRJOL Department of Physics, University of California at San Diego 9500 Gilman Drive, La Jolla, CA 92093 We report on recent progress on perturbative QCD calculations of certain exclusive rare weak B meson decays involving hard photons. In the limit of a photon energy E~f much larger than KQCD-, the amplitudes for such processes can be analyzed in a twist expansion in powers of A/E-,. The leading twist amplitude is given by the convolution of a hard scattering amplitude with the B meson light-cone wavefunction. This approach is applied to a calculation of the leptonic radiative B —> "fli>i formfactors and to an estimate of the weak annihilation contribution to the penguin decays B -»• p(w)7. As an application we discuss a few methods for constraining the unitarity triangle with exclusive radiative B decays.

1

Introduction

Rare radiative decays of B hadrons have been extensively studied as a possible way of probing possible new physics effects. Most effort has been concentrated on the theoretically cleaner inclusive decays, which can be analyzed in an 1/mt expansion with the help of an operator product expansion. On the other hand, exclusive rare B decays are sensitive to long-distance QCD effects and the relevant amplitudes depend on details of the hadronic bound states. This makes their theoretical description considerably more model-dependent, and a full description is still lacking. In the following we describe a systematic treatment of exclusive decays involving one very energetic on-shell photon emitted from an internal quark line. In this situation the internal light quark moves very close to the light cone, and an expansion in powers of the small parameter A/J5 T becomes useful1. This is analogous to the twist expansion for exclusive processes involving only light quarks 3 . The simplest process which can be studied along these lines is the leptonic radiative decay4 B -> 'ylui, which is discussed in Sec. 2. A basic ingredient of the method is the light-cone description of a heavy-light meson, which is introduced in Sec. 2.1. In Sec. 2.2 we argue that the general structure of the leading-twist B -> -ylvi form factors is given by a convolution of the B light-cone wave function with a hard scattering amplitude Tula Sec. 3 we describe an application of this method to the calculation 5 of the weak annihilation contribution to the weak radiative decay B —> pj. Taking into account all possible long distance amplitudes compatible with 185

186

SU(3) symmetry, we present in Sec. 3.1 a few methods for constraining the unitarity triangle using isospin violating effects in B - • p(uj)j decays 6 . 2 2.1

Leading twist calculation of the B ->• jivi

decay

Heavy mesons light-cone wavefunctions

The most general form for the Bethe-Salpeter wave function of a heavy-light meson B(v) can be written in terms of 4 scalar functions V>» (v-z, z2) (z = x—y) i>ap{z,v) = (O\Tba(x)q0(y)\B(v))

= { ( ^ - ± ( # - tffifo + #

+ #/>*)}

3

•

Tl) In the heavy quark limit the structure of the wavefunction is further constrained by the condition t/ijj = tp. This reduces the number of independent structures to 2, which can be taken as tpit2 V>«0 (*, V) = | ^ # !

+ [ / - V(V • Z)]^2h5 }

•

(2)

The momentum space B meson wavefunction is defined as ipa0(k) = Jd4zeik-zt/;a0(v,z).

(3)

In the physical applications to be discussed in the following, this wavefunction is convoluted with a scattering amplitude Ttf(fc) which does not depend on one light-cone component of the relative momentum fcM in the bound state (e.g. k-). Therefore, this component can be integrated over, which effectively puts z on the light cone (z+ = 0). We denned here light-cone coordinates as k± = k° ± k3, which can be projected out by dotting into the light-cone basis vectors n = (1,0,0,1) and n = (1,0,0, - 1 ) . This gives fc+ = n • k, fc_ = n • k. It is convenient to write the wavefunction (2) as a linear combination of light-cone projectors in spinor space A + — \4i>, A_ = \ijlijl, ^a0(z+

= 0,z.,z±=0)

= j i y ^ A + l M * - ) + A-iMz-Jhs}

•

(4)

with ip±(z-) = ipiT \z~i>2- [When used to compute a physical amplitude, only one of these functions will contribute to leading twist (e.g. if>+)-] This gives the most general expression for the B wavefunction in the heavy quark limit, which is often found in the literature 7 ' 8 ^ ^ • ^

2

) = ^{W'

+

-^W'

+

-V'-))75}

a

•

(5)

187

However, in the following we will use the form (4) which has a simple interpretation in the parton model. With the kinematics adopted above, the wavefunction ip+(k+) gives the probability to find the light quark in the B meson carrying momentum k+. Its moments are related to matrix elements of local operators r°° (jfe£) = / <**+*+lM*+) = (0\qA+Min Jo

• D)Nhv\B(v)).

(6)

The first two moments can be expressed in terms of known B hadronic parameters: (fc°_) = / B " I B and (A;+) = f A / s m s , with A = m j - mj, ~ 350 MeV the binding energy of the 6 quark in a B meson. The average of &+1 will play an important role in the following. Although it cannot be related to the matrix element of a local operator, it is possible to give a model-independent lower bound on its magnitude. Using the normalization conditions for N = 0,1 and the positivity of the distribution function ip+ one finds4

For more specific (but less model-independent) predictions, some model has to be adopted for the distribution function ip+(k+). We used an ansatz inspired by a quark model with harmonic oscillator potential tp+(k+) = Nk+ exp (— 2^5-(fc+ — a) 2 ). The width parameter u is varied as OJ = 0.1 — 0.3 MeV and a is constrained from the normalization conditions. Using A = 0.3 - 0.4 MeV gives a = 0.05 - 0.5 MeV. 2.2

Leading twist form factors in B -t j£i/f

The simplest hard photon process which can be analyzed using an expansion in l/Ej is the radiative leptonic decay B -t jive. This proceeds through the weak annihilation of the spectator quark in the B meson, as depicted in the quark diagrams in Fig. 1. Interest in this decay was sparked by the observation9 that its branching ratio is enhanced relative to that for the leptonic decay B -t Iv^ with £ = e, /i. Although adding one photon introduces a factor of a = ^ in the rate, it also removes the helicity suppression factor {mi/niB)2; the overall effect is an enhancement in the rate of the leptonic radiative mode. Later work 10 investigated this decay using a variety of approaches, both in connection with the prospect of constraining fB and/or |V u j| and in the context of the weak radiative decays B -> p(w)7.

188

(a)

(b)

Figure 1. Tree level contributions to the hard scattering amplitude for B —• ilvi decay. The cross denotes the weak current JM = g7(x(l — 75)6. The dominant contribution comes from the photon being emitted from the light quark line (a) while the diagram (b) is suppressed by the inverse heavy quark mass 1/m;,.

The amplitude for B -> •yii/f is parameterized by two formfactors fv,A(E^) defined as i^a0s£*avpqdfv(E7)

<7(9.e)kfTM&|B(t;)> = 1 -7==(7(q,s)\qj^b\B(v))

(8)

= [q^v • e*) - e* (W • q)] fA(Ey).

(9)

Computing the contribution of the two diagrams in Fig. 1 with the B wavefunction (4) one finds for the formfactors (8), (9) fBTTlB

fv(Ej)Ji) — iA\^-t) fA{E-y) — = -gij— \ v ? J t '7 \

_

~m ; + 0(A. /Ey) bJ

(10)

where Qq,Qb are the spectator- and heavy-quark electric charges and the hadronic parameter R describes the photon coupling to the light quark. This is given by a convolution of the leading twist B meson wavefunction with a hard-scattering amplitude TH(E~,,

k+)

00

R{E.

•> =

/

Jo

dk+-

V>+(*+) TH{E-y,k+) k+

The hard scattering amplitude TH(Ej,k+)

(11)

is given to one-loop order by 4

189

Using the model distribution function described in Sec. 2.1 one finds at tree level R(EJ = 2 - 4 G e V - 1 , while the lower bound (7) predicts R > 2.1 G e V - 1 (corresponding to A = 350 MeV). We briefly discuss in the following a few interesting consequences of these results. a) To leading twist the formfactors fv,A(E7) scale like 1/.ET, with a proportionality coefficient which is independent on the heavy quark flavor (up to l/mft corrections). This property can be used to determine the CKM matrix element | V^t,| from a comparison of the B —• ^ivi and D —¥ j£u( photon spectra 4 . b) The form factors of the vector and axial current are equal to leading twist. This is in contrast to their behaviour in the low E1 region, where they receive contributions from intermediate states with different quantum numbers 9 (Jp = l - for fy and Jp = 1+ for /A)The equality of the form factors (10) is a particular case of a symmetry relation analogous to those discussed for the soft components of semileptonic formfactors in 11 . The gluon couplings of a quark moving close to the light cone possess a higher symmetry 12 . This can be formalized by going over to an effective theory, which should include in addition to the soft gluon modes (LEET 12 ), also collinear gluons. A complete discussion including collinear gluons has been given only recently 13 . There is however an important difference between the status of the symmetry relation fy = /A among the leptonic radiative formfactors, and the analogous symmetry relations for the soft semileptonic formfactors11. While the latter receive symmetry breaking corrections from hard gluon exchange (which have been computed recently 8 ), the former relation is not changed by such effects, as checked by explicit calculation to one-loop order 4 . Hadronic amplitudes similar to R in (11) appear in many physical quantities involving long-distance effects induced by hard photon or gluon emission from weak annihilation topologies 14 . [When computed in the quark model, this quantity appears as the inverse of the constituent quark mass R -> l/mq.] In the following section we study such an important application, to longdistance effects in exclusive penguin induced decays b -» cfy. 3

Long-distance contributions to the B -> pj decay

An important class of weak radiative decays are those mediated by the penguin mechanism b —• s(d),y Hes = ^LvtbV^C7^F"vStTia>PRb

+ nlA.,

(13)

190

Figure 2. Long distance contributions to B -> Vj weak radiative decays. The crosses denote the possible attachments of the photon to the internal quark lines. The quark diagrams are denoted in text as a) weak annihilation (^4); b) VT-exchange (E); c) penguin ; >(0') and c-quark (P<>tfh amplitudes containing u-quark (P^ ) loops. We distinguish between amplitudes with the photon attaching to the spectator quark (i = 1) and to the quark in the loop (i = 2). d) annihilation penguin amplitudes (PAU c ); e) gluon penguin amplitudes (AfW).

with small long-distance contributions expected from time-ordered products of the weak nonleptonic Hamiltonian with the minimal electromagnetic coupling (denoted as H\.d.)We show in Fig. 2 the different quark diagrams responsible for longdistance contributions to a typical weak radiative decay B —• V7. In the SU(3) limit any such amplitude can be written as a linear combination 5 of the amplitudes in Fig. 2 with CKM factors VJ (for each photon helicity A)

A(B -+ Vlx) = J^ VqbV*8 q=u,c,t

E

Ci,qMiiq\ .

(14)

Mi,q=A,E,Pq,PA„

This is analogous to a similar decomposition of nonleptonic B decay amplitudes into graphic amplitudes 15 , and is equivalent to a more usual SU(3) analysis in terms of reduced matrix elements. The long-distance amplitudes appearing in (14) are notoriously dimcult to calculate. They have been estimated using various methods such as QCD sum rules 16 , vector meson dominance 17 , perturbative QCD 18 and Regge methods 19 . A few sample results are tabulated in Table 1, separately for the two helicities of the photon A = L, R. These calculations show that the dominant long-distance amplitude in 6 -> d-y decays comes from the weak

191 Table 1. Estimates of the short-distance and long-distance amplitudes in B -¥ P7 decays (in units of 10~ 6 GeV). The estimates of the WA and W-exchange amplitudes Ax and E\ used R = 2.5 GeV - 1 . The penguin type amplitudes Pu and Pc have been estimated using vector meson dominance.

Photon helicity X= L \ = R

\Ptx\ 1.8 0

1*5*1

I^«A|

0.16 0.04

0.03 0.007

\Ax\ 0.6 0.07

\Ex\ 0.05 0.007

annihilation graph A in Fig. 2(a). It is therefore rather fortunate that the weak annihilation amplitude can be computed in an esentially model-independent way. In the factorization approximation, one finds5 AL,R

= -^(C2

+ | K / ,

UB

+ E^fv

± /A))

(15)

with fv,A(E7) the radiative leptonic form factors defined in (8), (9), and Ci(mj,) = —0.29, Ci(mj) = 1.13 are Wilson coefficients in the weak nonleptonic Hamiltonian. Nonfactorizable corrections to this result arise from hard gluons connecting the initial and final state quarks, but appear only at higher twist 5 . Furthermore, to leading order of a twist expansion for the radiative leptonic formfactors fv,A (10), the weak annihilation amplitude couples predominantly to left-handed photons. A similar suppression of the right-handed helicity amplitudes is noted for all other long-distance contributions (see Table 1), which has implications for proposals to search for new physics through photon helicity effects in b -> sj decays 20 . The main motivation for measuring exclusive weak B radiative decays is connected with the possibility of extracting the CKM matrix element \Vtd\The weak annihilation amplitude A introduces the most significant theoretical uncertainty in such a determination. Keeping only the leading long-distance amplitude, one finds for the ratio of charge-averaged rates 2

Hd

B(J3± -> !f*±7)

vtts

R •51/(3)

('-

SA

vubv:d vtdvti

cos a cos A •0(£))

(16) with eAe%4,A = AL/PtL and RSu(s) = 0.76±0.22 an SU(3) breaking parameter in the short-distance amplitude 21 . Using the estimates of Table 1 one finds EA — 0.3, which gives an uncertainty of about 15% in this determination of jVtdl from long-distance effects. The amplitude A induces also a direct CP asymmetry in charged B decay ACP — 2 VubV,:d e,4 sin a sin 0,1, which can v«v t i

192

be as large as 30% for optimal values of the weak and strong phases a, (J>A3.1

Constraining the CKM matrix with exclusive weak radiative B decays

Interference between the e.m. penguin and long-distance amplitudes can produce isospin breaking effects in B± ->• pj decays. Some of the latter contribute with a different weak phase than the former, which led to the suggestion to use isospin breaking in these decays in order to extract information about CKM parameters 22 . A more careful treatment 6 shows that such an approach could receive contaminations from additional long-distance effects with the same weak phase as the short-distance amplitude. These amplitudes can be related by SU(3) symmetry to isospin-breaking effects in B —• K*-y decays. Experimental data on these modes have been recently reported by the CLEO 23 , BaBar 24 and Belle25 collaborations, which find B{B± -> JSf ± 7) = (3.76±g;|l ± 0.28) x 10~ 5 (2.87 ± 1.20lg;^) x 10~ 34

B(B° -t i T ° 7 ) = (4.55tS;S ± °- ) *

5

10_5

(4.94 ± 0.93±g;||) x 10~ 5 (5.2 ± 0.82 ± 0.47) x 1 0 - 5

(CLEO)

(17)

{BELLE) {CLEO)

(18)

{BELLE) {BABAR)

The long-distance amplitudes responsible for the difference in rate between charged and neutral B -»• K*-y modes are shown in Figs. 2(c), (e), and contain a charm loop or a gluon penguin in which the photon attaches to the spectator quark. The preliminary data (17), (18) indicate that these effects could be significant. It has been proposed6 therefore to eliminate them by combining B -> py with B -¥ K*j data, by forming the combined ratio B{B± -» p ± 7 ) B{B° -» iC*°7) 2B{B° -> p° 7 ) ' B{B± -> K*±^) _ l - 2 e ^ - 2 ^

—

cos a cos <J>A + C(e 2 ) •

We expanded here to linear order in the ratios of long-/short-distance amplitudes Si. The residual contamination from the (OZI-suppressed) penguinannihilation amplitude epAc Fig. 2(d) can be expected to be very small. An upper bound on its size can be given in terms of experimental data on Bs decays as \ePAc.\2 < 2T{BS -> p 0 7 ) / r ( B ± -»• K*^). Any measurement of the ratio (19) different from 1 can be translated into a constraint on the CKM factors in the last term. The cleanest approach

193

11

1

0.8 0.6

\

—

0.4 y

0.1' i

:

' X^^^% :

i

i

i

i

i

0 0.2 0.4 0.6 0.8 1.0

^

p

Figure 3. Illustrative example of a constraint on the CKM parameters from isospin breaking in B —> p(u)y decays. The shaded area is the 68% CL contour obtained from a global fit of the unitarity triangle 26 . The region contained between the dotted lines is the exclusion region corresponding to \RP — 1\/SA = 0 . 1 .

involves extracting the factor v»iv;d EA from a combination of data in B -> vuv*, jivi and B -t K*j decays6 This results into a bound on the weak phase a which excludes values around a = 90°. A simpler (although less model-independent) approach would combine theoretical calculations of £A using (15), in order to constrain the combination of CKM parameters cos a. A typical region in the (p, rj) plane which can be excluded in this way is shown in Fig. 3. This largely overlaps with the area presently favored by global fits of the unitarity triangle, and can therefore be expected to be useful in further constraining it, as more data on weak radiative B decays become available. Acknowledgments It has been a pleasure collaborating with Ben Grinstein, Yuval Grossman, G. Korchemsky and Tung-Mow Yan on the issues discussed here. I am grateful to the organizers of the P P P workshop for the invitation to give a talk and

194

to the Center for Theoretical Sciences, R. O. C. for financial support. This research was supported by the DOE grant DOE-FG03-97ER40546. References 1. Another class of decays which can be described systematically corresponds to diagrams with one far offshell photon q2 >• A2 emitted from an internal quark line. The internal quark propagator is contracted to a point and the decay amplitudes are expanded in powers of 1/q2 with the help of an OPE 2 . 2. B. Grinstein and R. F. Lebed, Phys. Rev. D 60, 031302 (1999); D. Evans, B. Grinstein and D. R. Nolte, Phys. Rev. D 60, 057301 (1999); Phys. Rev. Lett. 83, 4947 (1999); Nucl. Phys. B 577, 240 (2000); B. Grinstein, D. R. Nolte and I. Rothstein, Phys. Rev. Lett. 84, 4545 (2000). 3. S. J. Brodsky and G. P. Lepage, Phys. Rev. D 22, 2157 (1980). 4. G. Korchemsky, D. Pirjol and T. M. Yan, Phys. Rev. D 61, 114510 (2000). 5. B. Grinstein and D. Pirjol, Phys. Rev. D 62, 093002 (2000). 6. D. Pirjol, Phys. Lett. B 487, 306 (2000). 7. M. Beneke, G. Buchalla, M. Neubert and C. Sachrajda, Nucl. Phys. B 591, 313 (2000). 8. M. Beneke and T. Feldmann, Nucl. Phys. B 592, 3 (2000). 9. G. Burdman, T. Goldman and D. Wyler, Phys. Rev. D 51, 111 (1995). 10. A. Khodjamirian, G. Stoll and D. Wyler, Phys. Lett. B 358, 129 (1995); P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B 372, 331 (1996) and Phys. Lett. B 386, 328 (1996); D. Atwood, G. Eilam and A. Soni, Mod. Phys. Lett. A 11, 1061 (1996); G. Eilam, I. Halperin and R. R. Mendel, Phys. Lett. B 361, 137 (1995); C. Q. Geng, C. C. Lih and Wei-Min Zhang, Phys. Rev. D 57, 5697 (1998). 11. J. Charles et al, Phys. Rev. D 60, 014001 (1999). 12. M. J. Dugan and B. Grinstein, Phys. Lett. B 255, 583 (1991). 13. C. Bauer, S. Fleming and M. Luke, hep-ph/0005275; C. Bauer, S. Fleming, D. Pirjol and I. Stewart, hep-ph/0011336. 14. M. Bander, D. Silverman and A. Soni, Phys. Rev. D 44, 7 (1980); 44, 962(E) (1980). 15. D. Zeppenfeld, Z. Phys. C 8, 77 (1981); M. Gronau, O. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 52, 6374 (1995). 16. A. Khodjamirian, G. Stoll and D. Wyler, Phys. Lett. B 358, 129 (1995); A. Ali and V. M. Braun, Phys. Lett. B 359, 223 (1995); A. Khodjamirian, R. Riickl, G. Stoll and D. Wyler, Phys. Lett. B 402, 167 (1997).

195

17. E. Golowich and S. Pakwasa, Phys. Lett. B 205, 393 (1988); Phys. Rev. D 5 1 , 1215 (1995); H. Y. Cheng, Phys. Rev. D 5 1 , 6228 (1995). 18. C. E. Carlson and J. Milana, Phys. Rev. D 5 1 , 4950 (1995). 19. J. F. Donoghue, E. Golowich, A. V. Petrov and J. M. Soares, Phys. Rev. Lett. 77, 2178 (1996); J. F. Donoghue, E. Golowich and A. V. Petrov, Phys. Rev. D 55, 2657 (1997). 20. D. Atwood, M. Gronau and A. Soni, Phys. Rev. Lett. 79, 185 (1997); Y. Grossman and D. Pirjol, J H E P 0006: 029 (2000). 21. P. Ball and V. M. Braun, Phys. Rev. D 58, 094016 (1998). 22. A. Ali and V. M. Braun, Phys. Lett. B 359, 223 (1995); A. Ali, L. T. Handoko and D. London, DESY report DESY 00-088, hep-ph/0006175. 23. CLEO Collaboration, T. E. Coan et al, Phys. Rev. Lett. 84, 5283 (2000). 24. C. Jessop et al, BABAR-PROC-00/15, hep-ex/0011054. 25. A. Abashian et al, BELLE-CONF-00-03. 26. F. Caravaglios at al, hep-ph/0002171.

N E U T R I N O E X P E R I M E N T S : HIGHLIGHTS HENRY TSZ-KING WONG Institute of Physics, Academic Sinica, Nankang 11529, Taipei, Taiwan. E-mail: [email protected] This article consists of two parts. The first section presents the highlights on the goals of neutrino physics, status of the current neutrino experiments and future directions and program. The second section describes the theme, program and research efforts for the TEXONO Collaboration among scientists from Taiwan and China.

1

Introduction

With the strong evidence presented by the Super-Kamiokande experiment on neutrino oscillations *, there are intense world-wide efforts to pursue the next-generation of neutrino experiments. The aims of the first section of this article is to "set the stage" for students and researchers not in the field by summarizing the key ingredients and highlights of the goals, status and future directions in neutrino physics experiments. It is not meant to be a comprehensive lecture or detailed review article. Interested readers can pursue the details from the listed references on textbook accounts 2 , latest status 3 and Web-links 4 . The second part of this article presents an account of the research program of the TEXONO Collaboration. 2 2.1

Neutrino Physics Experiments Why Neutrino Physics

Neutrino exists — and exists in large quantities in the Universe, comparable in number density to the photon. It is known that there are three flavors of light neutrino coupled via weak interaction to the Z gauge boson. Yet the fundamental properties : (1) masses, denoted by m; for mass eigenstate vu and (2) mixings, denoted by XJu for mixing matrix elements between flavor eigenstate vi and mass eigenstate vx (alternatively by fly for mixing angles between mass eigenstates v\ and Vj), remains unknown or at least not accurately known enough. In field theory language, this translates to the crucial question on the structure (or even the possible existence) of a "neutrino mass term" L(i/mass) in the total Lagrangian. Standard Model sets this to be identically 196

197 io» I

Iff* liaV

i

Iff 3 itnV

1 aV

•

i

Vfi luv

10» MaV

10s 3aV

•

'

i

10" TeV

10" PaV

10" E»v

Neutrino Energy (aV)

Figure 1. The expected neutrino spectra from various celestial and terrestrial sources. Neutrinos from man-made accelerators, typically at the range of 10 MeV to 100 GeV, are not shown here since accelerator parameters differ.

zero, but without any compelling reasons - in contrast to the massless-ness of the photons being dictated by gauge invariance. The detailed structures and values of this term can reveal much about the Grand Unified Theories. At the large length-scale frontier, neutrino mass is related to the composition and structural evolution of the Universe. Neutrino has been a candidate of Dark Matter 3 : in fact it is the only candidate within the list that is proven to exist. Experimentally, the probing of L(i/-mass) is carried out by studying various processes related to neutrino masses and mixings, such as direct mass measurement through the distortion of /J-spectra, neutrinoless double beta decays, neutrino oscillations, neutrino magnetic moments, neutrino decays and so on. These investigations are realized by a wide spectrum of experimental techniques spanned over several decades of energy scale with different neutrino sources. The expected neutrino spectrum due to terrestrial and astrophysical sources are shown in Figure 1. In addition, neutrino has been used as probe (as "beam" from accelerators and reactors and even astrophysical sources) to study electroweak physics, QCD, structure function physics, nuclear physics, and to provide otherwise

198

inaccessible information on the interior of stars and supernovae. Therefore, the study of neutrino physics and the implications of the results connect many disciplines together, from particle physics to nuclear physics to astrophysics to cosmology. 2.2

Current Status and Interpretations

Neutrino interactions are characterized by cross-sections at the weak scale (100 fb at 100 GeV to < 1 0 - 4 fb at 1 MeV). As an illustration, the mean free path in water for i7e from power reactors at the typical energy of 2 MeV is 250 light years! The central challenge to neutrino experiments is therefore on how to beat this small cross section and/or slow decay rate. Usually massive detectors are necessary to compensate by their large target size. Then the issue becomes how to keep the cost and background low. After half a century of ingenious experiments since the experimental discovery of the neutrinos by Cowan and Reines, there are several results which may indicate the existence of neutrino masses, and hence physics beyond the Standard Model. All these results are based on experimental searches of neutrino oscillation, a quantum-mechanical effect which allows neutrino to transform from one flavor eigenstate to another as it traverses in space. This process depends on the mass difference (Am 2 = |m 2 — m 2 |) rather than the absolute mass, resulting in enhanced sensitivities. A simplified summary of the results of neutrino oscillation experiments is shown in Figure 2. The "allowed regions" are due to anomalous results from experiments in: 1. Atmospheric Neutrinos: Data from the Super-Kamiokande experiments 1, supported by other experiments, indicates a smaller (fM + i^)/(i/ e + ve) ratio than would be expected from propagation models of cosmic-ray showering. The "smoking gun" for new physics is that the deficit has a statistically strong dependence with the zenith angle, meaning the effect depends on the propagation distance of the neutrinos from the production point to the detector. The combined fit supports a scenario of v^ -> vr oscillation. 2. Solar Neutrinos: All solar neutrinos to date reported an observed solar neutrino flux less than the predictions of Standard Solar Model. The deficit is different among the experiments, suggesting an energy dependence which is difficult to be explained by standard solar models. However, within an individual experiment, the potential "smoking gun" effects (day-night variation, seasonal variation, spectral distortion) are absent or statis-

199 1& 10" ID 1 lO"

•ff" 10"

to" -S. 10^

i f

ior10 r 10*

m " 10

a a v iyanium\ 4 m*

itf*

ia*

10"

1D1

sin*afl

Figure 2. A summary of results of neutrino oscillation experiments in (Am2,sin220) parameter space. The allowed regions are shaded.

tically weak. The combined data favors neutrino oscillations of ve to another species, active or sterile: i>e —> i/x, either in vacuum or due to matter-enhanced oscillation (the "MSW" Effect) in the Sun. 3. LSND Anomaly: The LSND experiment with accelerator neutrinos reported unexpected excess of ue and ve in a v^ + v^ beam, which can be explained by v^ ->• fe oscillation. The results are yet to be reproduced (or totally excluded) by other experiments. If all experimental results are correct and to be explained by neutrino oscillations, one must incorporate a fourth-generation sterile neutrino. If one takes the conservative approach that LSND results must be reproduced by an independent experiment before they are incorporated into the theoretical framework, then the favored scenario would be a three-family scheme with: • Neutrino M a s s Differences: Atmospheric neutrino oscillation is driven by A m ^ ~ 1 0 - 2 — 1 0 - 3 eV2 which is much bigger than the scale which drives solar neutrino oscillation at Amf2 ~ 1 0 - 4 — 1 0 - 5 eV2 due to matter-enhanced oscillation, as depicted in Figure 3. The sign of A m | 3 remains undetermined.

200 m^

.2 ••

_^ * *•

Ani-^t,,,

,

m s2 m12

^m2si>|

m22 Scheme A: Am 2 32 > 0

Scheme B: Am 2 3 j < 0

Figure 3. The favored hierarchical structures of the neutrino mass matrix to explain atmospheric and solar neutrino results.

• Mixing Angles: Surprisingly when compared to the quark sector, the atmospheric neutrino data supports strongly large mixing angles (order of 1) for sin 2 2023 • Combining all data from solar neutrinos also statistically favors large sin2 2#i2. The 1-km baseline reactor neutrino experiments set limits on sin 2 20 13 < 0.1. 2.3

Future Experimental Program

Among the various neutrino sources depicted in Figure 1, only a relatively small window from ~ 1 MeV to ~100 GeV is detectable by present techniques. The future of neutrino experiments will therefore evolve along various directions: (I) Further Exploration of the Measured Window • Long Baseline (LBL) Neutrino Oscillation Experiments: There is a running experiment with accelerator neutrinos produced by "proton-on-target" (PoT) from KEK to the Super-Kamiokande detector (250 km), while two others (both 730 km) in construction: Fermilab to MINOS experiment at Soudan, and CERN to ICANOE and OPERA experiments at Gran Sasso. The goals will be to reproduce and improve on the i/ atm parameters, to observe the E/L oscillation pattern, and to detect vT appearance explicit by observing vT charged-current interactions leading to the production of r-lepton. In addition, there are two experiments, KamLAND and BOREXINO, with capabilities of performing LBL experiments with reactor neutrinos

201

from power plants ~100 km away. Their goals will be the probe the "Large Mixing Angle" MSW solution to the solar neutrino data. • Detection of Weak/Rare Signals: Various big underground experiments are sensitive to neutrinos from supernovae, with the hope of detecting thousands of events from the next supernova, as compared to the 20 events from SN 1987a. The LBL reactor neutrino experiments will also try to observe "terrestrial neutrinos" produced by the radioactive (mainly 2 3 2 Th and 238 U series) from the Earth's crust. • Further Double Beta Decay Searches: Neutrinoless Double Beta Decay is sensitive to the absolute scale of mf rather then Am?. Many R&D projects are underway to try to achieve the interesting range of 1113 ~ 0.1 — 0.01 which is a scenario suggested by atmospheric neutrino results. • Neutrino Factories from Muon Storage Rings: There are intense pilot efforts to study the feasibilities of performing LBL experiments with a "Neutrino Factory" where the neutrinos are produced by muons decay in a muon storage ring. Unlike conventional PoT neutrino beam, /i-decay gives beams which are selectable in ve, ue u^ and v^, and with well-known spectra and compositions. Coupled to the high luminosity and small beam size, they can be powerful tools to study neutrino physics. The goals for the LBL program, where the baseline will have to be the 1000-10000 km range, would be to do precision measurements on the i/ atm parameters, to probe sin 2 20 13 to O(10~ 4 ) (from ve -¥ v^), to determine the signs of A m ^ (from the asymmetric ve/ve matter effects), and to probe CP-violation in the lepton sector (comparing ue -> v^ & ^e -> ^ ) - There are many technical challenges which must be addressed towards the realization of such projects. (II) Extension of Detection Capabilities: • High Energy Frontiers: There are several "neutrino telescope" projects (Lake Baikal, AMANDA, NESTOR, ANTARES) whose objective is towards the construction of an eventual "km 3 " detector. The scientific goals are (a) to identify and understand the astrophysics of high-energy (TeV to PeV) neutrino sources from active galactic nuclei, gamma-ray bursts, neutron stars and other astrophysical objects, and (b) to use these high-energy neutrinos for neutrino physics like very long baseline studies.

• Low Energy Frontiers: There are a host of new solar neutrino experiments: SNO is taking data while Borexino and KamLAND are under construction, as well as many R&D ideas and projects. Their goals are to measure the solar neutrino spectrum and particularly those from the dominant sub-MeV pp and 7 Be neutrinos, and to study the details of spectral shape, day-night and seasonal variations, as well as the neutral-current to charged-current ratios, so as to identify a unique solutioin to "solar neutrino problem", and to measure the (Am 2 ,sin 2 26) parameters. The low-energy solar neutrino spectrum also provides a probe to the other inaccessible physics at the interior of the Sun. Weakly Interacting Massive Particles (WIMP) are candidates for Dark Matter. Their experimental searches also employ the techniques of lowbackground low-energy experiments. Many experiments are running or being planned, based on crystal scintillators, cryogenic detectors and other techniques. Finally, the relic "Big Bang" neutrino, the counterpart to the 2.7 K cosmic microwave photon background (CMB), has large number density (110 c m - 3 for Majorana neutrinos, comparing to 411 c m - 3 for CMB) but extremely small cross sections due to the meV energy scale at an effective temperature of 1.9 K. The relic neutrinos decouples from matter at a much earlier time (1 s) than the CMB (3xl0 5 years), and hence are, in principle, better probes to the early Universe. A demonstration of its existence and a measurement of its density is a subject of extraordinary importance. Though there is no realistic proposals on how to detect them, it follows the traditions of offering a highly rewarding challenge to and pushing the ingenuity of neutrino experimentalists. 3

Research Program of the TEXONO Collaboration

Since 1997, the TEXONO" Collaboration has been built up to initiate and pursue an experimental program in Neutrino and Astroparticle Physics 5 . By the end of 2000, the Collaboration comprises more than 40 research scientists from major institutes/universities in Taiwan (Academia Sinica*, ChungKuo Institute of Technology, Institute of Nuclear Energy Research, National Taiwan University, National Tsing Hua University, and Kuo-Sheng Nuclear Power Station), Mainland China (Institute of High Energy Physics*, Institute of Atomic Energy*, Institute of Radiation Protection, Nanjing University) and "Taiwan Experiment On NeutrinO

203

the United States (University of Maryland), with AS, IHEP and IAE (with t) being the leading groups. It is the first research collaboration of this size and magnitude, among Taiwanese and Mainland Chinese scientists The research program 6 is based on the the unexplored and unexploited theme of adopting the scintillating crystal detector techniques for low-energy low-background experiments in Neutrino and Astroparticle Physics 7 . The "Flagship" experiment 10 is based on CsI(Tl) crystals placed near the core of the Kuo-Sheng Nuclear Power Station (KSNPS) at the northern shore of Taiwan to study low energy neutrino interactions. It is the first particle physics experiment performed in Taiwan where local scientists are taking up major roles and responsibilities in all aspects of its operation: conception, formulation, design, prototype studies, construction, commissioning, as well as data taking and analysis. In parallel to the flagship reactor experiment, various R&D efforts coherent with the theme are initiated and pursued. 3.1

Scintillating Crystal Detector for Low-Energy

Experiments

One of the major directions and experimental challenges in neutrino physics 3 is to extend the measurement capabilities to the sub-MeV range for the detection of the p-p and 7 Be solar neutrinos, Dark Matter searches and other topics. For instance, while high energy (GeV) neutrino beams from accelerators have been very productive in the investigations of electroweak, QCD and structure function physics, the use of low energy (MeV) neutrino as a probe to study particle and nuclear physics has not been well explored. Nuclear power reactors are abundant source of electron anti-neutrinos (i/e) at the MeV range and therefore provide a convenient laboratory for these studies. On the detector technology fronts, large water Cerenkov and liquid scintillator detectors have been successfully used in neutrino and astro-particle physics experiments. New detector technology must be explored to open new windows of opportunities. Crystal scintillators may be well-suited to be adopted for low background experiments at the keV-MeV range 7 . They have been widely used as electromagnetic calorimeters in high energy physics 8 , as well as in medical and security imaging and in the oil-extraction industry. There are matured experience in constructing and operating scintillating crystal detectors to the mass range of 100 tons. This technique offer many potential merits for low-energy low-background experiments. In particular, they are usually made of high-Z materials which provide strong attenuation for 7's of energy less than 500 keV, as indicated in the photon attenuation plot in Figure 4. As an illustration, 10 cm of CsI(Tl)

204

P h o t o n Energy ( MeV )

Figure 4. The attenuation length, as defined by the interactions that lead to a loss of energy in the media, for photons at different energies, for CsI(Tl), water, and liquid scintillator.

has the same attenuating power as 5.6 m of liquid scintillators at 7-energy of 100 keV. Consequently, it is possible to realize a compact detector design with minimal passive materials equipped with efficient active veto and passive shielding. Externally originated photons in this energy range from ambient radioactivity or from surrounding equipment cannot penetrate into the fiducial volume. Therefore, the dominating contribution to the experimental sensitivities is expected to be from the internal background in the crystal itself, either due to intrinsic radioactivity or cosmic-induced long-lived isotopes, both of which can be identified and measured such that the associated background can be subtracted off accordingly. The experimental challenges are focussed on the understanding and control of the internal background. Pioneering efforts have already been made with Nal(Tl) crystals for Dark Matter searches 9 . 3.2

Flagship Experiment with Reactor Neutrinos

An experiment towards a 500 kg CsI(Tl) scintillating crystal detector to be placed at a distance of 28 m from a core in KSNPS is under construction to study various neutrino interactions at the keV-MeV range 10 , and to establish and explore the general techniques for other low-energy low-background applications. The layout of the experimental site is shown in Figure 5 The physics objectives of the experiment are to improve on or to explore the subjects of: (1) Neutrino-electron scatterings, which is a fundamental interaction providing information on the electro-weak parameters

205 Kuo-sheng Nuclear Power Station: Reactor Building

Figure 5. Schematic side view, not drawn to scale, of the Kuo-sheng Nuclear Power Station Reactor Building, indicating the experimental site. The reactor core-detector distance is about 28 m.

(gv, gA> and sin 2 0\y), and are sensitive to small neutrino magnetic moments (fjtv) and the mean square charge radius, as well as the W-Z interference effects; (2) Neutrino interactions on nuclei such as 133 Cs and 1 2 7 I, and in particular the neutral current excitation (NCEX) channels where the signatures are gamma-lines at the characteristic energies — their cross-sections are related to the physics of nucleon structure, and are essential to explore the scenario of applying these interaction channels in the detection of low energy solar and supernova neutrinos; (3) Matter effects on neutrinos, since this is the first big high-Z detector built for reactor neutrino studies. To fully exploit the advantageous features of the scintillating crystal approach in low-energy low-background experiments, the experimental configuration should enable the definition of a fiducial volume with a surrounding active 47r-veto, and minimal passive materials. This is realized by a design shown schematically in Figure 6. One CsI(Tl) crystal unit consists of a hexagonalshaped cross-section with 2 cm side and a length 20 cm, giving a mass of 0.94 kg. Two such units are glued optically at one end to form a module. The modules will be installed in stages towards an eventual 17 x 15 matrix configuration. The light output are read out at both ends by custom-designed 29 mm

206 Csl(TI)

Longitudinal View

Cross-Sectional view

Figure 6. Schematic drawings of the CsI(Tl) target configuration.

diameter photo-multipliers (PMTs) with low-activity glass, whose signals will pass through amplifiers and shapers to be digitized by 20 MHz FADCs n . The sum and difference of the PMT signals gives information on the energy and the longitudinal position of the events, respectively. The passive shieldings consist of, from inside out, 5 cm of copper, 25 cm of boron-loaded polyethylene, 5 cm of steel, 15 cm of lead and finally plastic scintillators as cosmic-ray veto. The target is housed in a nitrogen environment to prevent background events due to the diffusion of the radioactive radon gas. Extensive measurements on the crystal prototype modules have been performed 12 . The response is depicted in Figure 7, showing the variation of collected light for Qi, Q2 and Q to t as a function of position within one crystal module. The error bars denote the FWHM width of the 137 Cs photo-peaks. The discontinuity at L=20 cm is due to the optical mis-match between the glue (n=1.5) and the CsI(Tl) crystal (n=1.8). It can be seen that Q to t is only weakly dependent of the position and a 10% FWHM energy resolution is achieved at 660 keV. The detection threshold (where signals are measured at both PMTs) is <20 keV. The longitudinal position can be obtained by considering the variation of the ratio R = (Qi - Qa)/(Qi + Q2) along the crystal. Resolutions of 2 cm and 3.5 cm at 660 keV and 200 keV, respectively, have been demonstrated. In addition, CsI(Tl) provides powerful pulse shape discrimination capabilities to differentiate 7/e from a events, with an excellent separation of >99% above 500 keV. The light output for a's in CsI(Tl) is quenched less than that in liquid scintillators. The absence of multiple a-peaks above 3 MeV in

207

•

1


\m

Qi

i 1

i

t

*

*

Mill

'

(lilt » T

•

•

•+ » • +

» t t

:,,,,,,,,,,,,,,,,,,, ,, 0

5

10

15

, . , ,

Z0

1 . , , . 1 , . , , 1 25 3a 35

H

40

position (cm)

Figure 7. The measured variations of Qi, Q2 and Qtot = Qi + Q2 along the longitudinal position of the crystal module. The charge unit is normalized to unity for both Qi and Q2 at their respective ends.

the prototype measurements suggests that a 238 U and 2 3 2 Th concentration (assuming equilibrium) of < 1 0 - 1 2 g/g can be achieved. By early 2001, after three and a half years of preparatory efforts, the "Kuo-Sheng Neutrino Laboratory", equipped with flexibly-designed shieldings, cosmic-ray 4-7r active veto, complete electronics which allow full digitization of multi-channel detectors for a 10 ms long duration, data acquisition and monitoring systems, as well as remote assess capabilities, was formulated, designed, constructed and commissioned. Prototype CsI(Tl) detector has been operating in the home-base laboratory. The first Reactor ON/OFF data taking period will be based on a 1 kg low-background germanium detector with 100 kg of CsI(Tl) running in conjunction. 3.3

R&D Projects

Various projects with stronger R&D flavors are proceeding in parallel to the flagship reactor experiment. A feasibility study of using boron-loaded liquid scintillator for the detection of ve has completed 13 . The cases of using of GSO 14 and Lil(Eu) 1 5 as well as Yb-based scintillating crystals for low energy solar neutrinos are explored. The adaptations of CsI(Tl) crystal scintillator for Dark Matter WIMP searches are studied, which includes a neutron beam measurement to study the response due to nuclear recoils 16 . A generic and convenient technique to measure trace concentration of radio-isotopes in sample materials is of great importance to the advance of low-

208

energy low-background experiments. A R&D program is initiated to adapt the techniques of Accelerator Mass Spectrometry(AMS) with the established facilities at the 13 MV TANDEM accelerator at IAE 17 . The goals are to device methods to measure 2 3 8 U, 2 3 2 Th, 8 7 Rb, 40 K in liquid and crystal scintillators beyond the present capabilities 18 . Complementary to these physics-oriented program are detector R&D efforts. Techniques to grow CsI(Tl) mono-crystal of length 40 cm, the longest in the world, have been developed and are deployed in the production for future batches. 4

Outlook

Neutrino physics and astrophysics will remain a central subject in experimental particle physics in the coming decade and beyond. There are room for ground-breaking technical innovations - as well as potentials for surprises in the scientific results. A Taiwan, Mainland China and U.S.A. collaboration has been built up with the goal of playing a major role in this field. It is the first generation collaborative efforts in large-scale basic research between scientists from Taiwan and Mainland China. The technical strength and scientific connections of the Collaboration are expanding and consolidating. The flagship experiment is to perform the first-ever particle physics experiment in Taiwan. Many R&D projects are being pursued. The importance of the implications and outcomes of the experiment and experience will lie besides, if not beyond, neutrino physics. Acknowledgments The author is grateful to the members, technical staff and industrial partners of TEXONO Collaboration, as well as the concerned colleagues for the many contributions which "make things happen" in such a short period of time. Funding are provided by the National Science Council, Taiwan and the National Science Foundation, China, as well as from the operational funds of the collaborating institutes. References 1. Y. Fukuda et al., Phys. Rev. Lett. 8 1 , 1562 (1998); 82, 1810 (1999); 82, 2430 (1998); 85, 3999 (2000).

209

2. For textbook descriptions, see, for example: "Physics of Massive Neutrinos", 2nd Edition, F. Boehm and P. Vogel, Cambridge University Press (1992); "Neutrino Physics", ed. K. Winter, Cambridge University Press (1991). 3. For the overview of present status, see, for example: "Neutrino 1998 Conf. P r o c " , eds. Y. Suzuki and Y. Totsuka, Nucl. Phys. B (Procs. Suppl.) 77 (1999); "TAUP 1999 Conf. P r o c " , eds. J. Dumarchez, M. Froissart, and D. Vignaud, Nucl. Phys. B (Procs. Suppl.) 87 (2000); "Neutrino 2000 Conf. P r o c " , ed. A. McDonald, Nucl. Phys. B (Procs. Suppl.) in press (2001), and references therein. 4. For Neutrino-related Web-site, click, for example: "The Neutrino Oscillation Industry", http://www.hep.anl.gov/ndk/hypertext/nuindustry.html; "The Ultimate Neutrino Page", http://cupp.oulu.fi/neutrino/; "History of the Neutrinos", http://wwwlapp.in2p3.fr/neutrinos/aneut.html, and links therein. 5. C.Y. Chang, S.C. Lee and H.T. Wong, Nucl. Phys. B (Procs. Suppl.) 66, 419 (1998). 6. Henry T. Wong and Jin Li, Mod. Phys. Lett. A 15, 2011 (2000). 7. H.T. Wong et al., Astropart. Phys. 14, 141 (2000). 8. For a recent review, see, for example, G. Gratta, H. Newman, and R.Y. Zhu, Ann. Rev. Nucl. Part. Sci. 44, 453 (1994). 9. See, for example: R. Bernabei et al., Phys. Lett. B 389, 757 (1996); B 450, 448 (1999). 10. H.B. Li etal.,TEXONO Coll., hep-ex/0001001, Nucl. Instrum. Methods A, in press (2001). 11. W.P. Lai et al., TEXONO Coll., hep-ex/0010021, Nucl. Instrum. Methods A, in press (2001). 12. C.P. Chen et al., in preparation for Nucl. Instrum. Methods (2001). 13. S.C. Wang et al., Nucl. Instrum. Methods A 432, 111 (1999). 14. R.S. Raghavan, Phys. Rev. Lett. 78, 3618 (1997); S.C. Wang, H.T. Wong, and M. Fujiwara, submitted to Nucl. Instrum. Methods A (2000). 15. C.C. Chang, C.Y. Chang, and G. Collins, Nucl. Phys. (Proc. Suppl.) B 35, 464 (1994). 16. M.Z. Wang et al., in preparation for Phys. Rev. C (2001). 17. S. Jiang et al., Nucl. Instrum. Methods B 52, 285 (1990); B 92, 61 (1994). 18. D. Elmore and F.M. Phillips, Science 346, 543 (1987).

N E U T R I N O S A N D COSMOLOGY SANDIP PAKVASA Department of Physics and Astronomy University of Hawaii Honolulu, HI 96822 USA E-mail: [email protected] This is an elementary review of some aspects of neutrinos vis-a-vis cosmology. I review the standard discussion of neutrino energy density and neutrino counting in cosmology. The possibility of detection of relic neutrinos is discussed. Finally, the possibility of using neutrinos from distant sources to deduce cosmological parameters is briefly reviewed.

1

Introduction

Until recently most of observed quantitative information cosmology could be summarised by a few numbers: (i) the Hubble expansion parameter which is usually quoted as H=100h km/sec/MPc with h now thought to be about 0.7; (ii) the temperature of microwave background radiation T T « 2.7°.ftT = 2.510~4eV; (iii) the baryon density which is about ( 1 0 - 1 0 —10~9) times photon density (and the fact that anti-baryon density is essentially zero). Now, with the observations of type 1A supernovae 1 and the anisotropy measurements 2 of BOOMERANG and MAXIMA there is a large amount of data becoming available, and much more is on the way. 2

Neutrino Number and Energy Density

We first estimate the neutrino energy density, number density and temperature. Let us repeat a simple argument to deduce the critical energy density 3 . In a homogenous uniform universe one can choose a centre anywhere, then the energy of a (point-like) galaxy at radius R is E = K + U=

)-mR2

H2

H

8 p

~3^G

(1)

using Newtonian mechanics and R = HR. Now LHS = 0 corresponds to the critical point between eternal expansion and eventual collapse and hence the critical density pc is p e = JL ^ 1 « tf(i.05) x 104 eV/cc

210

(2)

211

This result holds relativistically as well. We define flj (where \=v, b, DM (Dark Matter) or A for cosmological constant term) by fij = Pi/pc- Both theoretical prejudice as well as current observational evidence 4 point to fltot — ^ A + UDM + Clb+Qv ml. Using Bose distribution for photons, the energy density in photons is given by pa = aT* = 0.25eV/cc

(3)

for Tj « 2.7°K. The photon number density is -

T

~

(2 4)

'

2

7T (/IC)

3

(fcTT)3 » 4 0 0 / c c

(4)

T

For relativistic fermions, using Fermi distribution, 7 aT* />/ = o o-TJ 8 3 nf = - aT-4f

(5)

As the temperature dropped past the point where electron pairs could no longer be created, the present temperature of the photon gas was effectively boosted by e+e~ annihilation (neutrinos had already decoupled somewhat earlier). Since entropy is conserved (5 = ^pR3/T). 4 7?3 S> = - — {l + 2x7/8}aT*

(6)

Hence Tv = T> = ( £ ) 1 / 3 T< = ( £ ) 1 / 3 T 7 « 1.9°tf. Since v and P each have only one helicity, v and v together count as one fermion and for (no degeneracy) n„ + np = 3/4(4/ll)n T « 110/cc

(7)

per flavor. Furthermore, if N (< 3) families have mass mv then the total neutrino energy density is Pv

andn„« (^)eVycc.

= Nmv (110) eV/cc

(8)

212

For example, for 3 degenerate v's of mass ~ 0(eV),n„ ~ 0.06. From the Super-K results on atmospheric neutrinos 5 , Qv > 0.0015. Progressively stronger constraints on neutrino masses have been derived starting from the conservative bound H„ < 1 which yields NMV < 47eV or mv < l§eV for 3 degenerate neutrinos. From the recent observations of red shifts of type 1A Supernova and of CMB anisotropy (BOOMERANG and MAXIMA) a consensus about Q, is emerging 4 ; namely QA ~ 0.7, QDM ~ 0.25 and Clb ~ 0.04 to 0.06. From these, one may bound ilu < €IDM leading to total mass in neutrinos bounded by 5.5 eV. Furthermore, the Sloan Digital Sky Survey (SDSS) 6 should measure the power spectrum of Large Scale Structure to a 1% accuracy. The presence of massive neutrinos tends to suppress the power at "smaller" scales and a bound can be obtained 7 :

m„ - 0.33 ( ^ H

8

eV.

(9)

In the near future with the MAP and PLANCK data, neutrino masses as small as a fraction of eV (~ 0.25 eV) can be probed. This compares very favorably with Laboratory experiments 8 which hope to probe masses at a level of leV (or less) in Tritium beta-decay and 0.05 eV in neutrino-less double beta decay (for Majorana neutrinos).

3

Neutrino Counting

At temperatures sufficiently high (>• 1 MeV) the numbers of neutrons and protons must have been equal since reactions such as e~p «-> ni/ e and uep <-* e+n had equal rates in both directions. During expansion when the temperature drops enough (to T*) so that e~p -*• nve and vep -» e+n rates get suppressed, then the n/p ratio remains fixed at exp(—(mn — mp)/T*). T* is fixed by the condition that the expansion rate (which goes ^/g T2, with g being the number of light degrees of freedom) is equal to the reaction rate for the above reactions (Tw ~ G%T&). This yields T* ~ g*G*Gp2 ~ 0.66_MeV. The neutron to proton ratio is then n/p ~ \. Assuming all neutrons form helium-4, the number of helium-4 nuclei is \n; and the number of surviving protons is (p-n). The fractional number of 4He is

af

= jV(^e)/(p-n) = ^

= 1 ^

= 1/12

213

The mass fraction is y

MjAHe) MQ-HJ + M^H)

=

Ax l + 4z

=

'

A more detailed treatment yields Y of about 0.24 ± 0.04. This T* and the resulting Y was calculated with three flavors of light neutrinos {Nv = 3). If Nv (or other light degrees of freedom) is increased, the effective g above increases and in turn the the expansion rate increases. This raises T*, and hence n and hence Y. Roughly the change in Y, 6Y, goes as 0.025NV. This allows one to place a bound on the number of light neutrinos beyond the three known ones 9 . The exact bound is not unanimous but 5NV is surely less than 2. 4

Detection of Relic Neutrinos

The average momentum of relic neutrinos is 3.2T„ ~ 5.2 x 10~ 4 eV/c. The neutrino current density is cn„ ~ 10 1 3 cm~ 2 s - 1 for massless neutrinos and 5.109 c m ~ 2 s - 1 for a mass of 0(eV). The effective interaction Hamiltonian for neutrinos with neutral matter is proportional t o o e = (3Z — A) for ve and aM = (A — Z) for v^ and vT. The i/ a -scattering cross section (at very low energies) then goes as aa ~ a" <% ml

(10)

Many early proposals to detect relic neutrinos by reflection or coherent effects turned out to be incorrect. There are three methods which some day may prove to be practical. The first is a 1975 proposal due to Stodolsky 10 . The idea needs neutrino degeneracy i.e. excess of v (or v) over v (or v) to work. Then a polarized electron moving in a background of CMB neutrinos can change its polarization due to the axial vector parity violating interaction. The effective neutrino density (for nv » rip) goes as p2f/6ir2 where p / is the Fermi momemtum. The effective interaction goes as Heff ~—T=-v*-vnv

(11)

With v ~ 300A;m/sec and pf ~ 0(eV) this leads to a rotation of the polarization of about 0.02" in a year. Can such small spin rotations can be detected? Certainly not at present, but technology may someday allow this.

214

The second method is one suggested by Zeldovich and collaborators n . The idea is to take advantage of momentum transfer in neutrino-nucleus scattering. Consider an object made up of small spheres of radius a « A (neutrino wavelength) packed loosely with pore sizes also of the same size, (to avoid destructive interference). If the number of atoms in the target is NA, then the effective coherent cross-section is a = aQN\ where aa is as given in Eq. transfer is

(12)

(10). Assuming total reflection, momentum Ap s 2m„ vv

and the force / = jvaAp

(13)

is given by / = 2nvaa NAmv

v2v

(14)

The most optimistic estimates are obtained by assuming some clustering (n„ ~ 10 7 /c.c.),m v ~ 0(eV), vv ~ lOFcm/s, p ~ 20gm/cc; leading to a=^-=

—-

W-23(aa/A)2

cm.s-2

(15)

Such accelerations are at least ten orders of magnitude removed from current sensibility and possible detection remains far in future 12 . The third possibility is the one proposed by Weiler in 1984 13 . The basic idea is as follows. If neutrinos have masses in the eV range and there are sources of very high energy neutrinos at large distances, then the H.E. v can annihilate on the C.B.R. v and make a Z° on-shell at resonance creating an absorption dip in the neutrino spectrum. The threshold for Z production would be at E ~ m | / 2 m „ which is about 4.10 21 eV. This seemed like an unlikely possibility, since it required large neutrino fluxes at very high energies to see the neutrino spectrum and then the absorption dip. But all this changed dramatically recently with the clear signal of cosmic rays beyond the GZK cut-off 14 . The GZK cut-off is the energy at which cosmic ray protons pass the threshold for pion production off the CMB photons. This is at an energy E ~ ravrapjE1 ~ 6.1019 eV. Above this energy, the mean free path of protons is less than lOMPc and hence these protons have to be "local". The flux should then decrease dramatically since we believe the cosmic rays are not produced locally. Recently, what used to be hints of the cosmic ray signal extending beyond this cut-off, has become a clear signal 15 . The events are most likely due to primary protons. Then an explanation is called for. One intriguing proposal 1 6 is that these events are nothing but a signal for the Z's

215 produced by the vv -» Z process with the protons coming from the subsequent Z decay! Of course, the original problem of needing sources of high energy neutrinos remains. If this explanation is valid, we have already seen (indirect) evidence for the existence of relic neutrinos. In principle, this proposal can be tested: (i) the events should point back out at the neutrino sources; (ii) there is an eventual cut-off when the energy reaches the threshold energy for Z production, E ~ 4.10 21 (jjJ-J eV; (iii) j/p ratio should be large near threshold and (iv) the large i/-flux should be eventually seen directly in large i/-telescopes. 5

Cosmology with neutrinos

We know from supernova studies that there are several effects of neutrino masses and mixings on the observation of neutrino bursts. A pulse spreads in time due to dispersion of velocities (from non-zero mass); a pulse separates into several pulses due to a neutrino of a given flavor being mixture of different masses and flavors conversion changing the initial flavor composition. One can apply these considerations to neutrino pulses from sources which are at cosmological distances. Then the effects come to depend on cosmological parameters. For example, the time difference 17 between two mass eigen-states which left at the same time is given by At » z/H \ - ^ z . . :

[Ef

Eh

(16)

where Et are the energies observed at earth and z, H and q have the usual meanings. The spreading of a pulse of a given mass neutrino is given by 17 At « z/H 1 -

(3 + 9)

Mi-i}

(17

>

Finally, the conversion probability for an emitted flavor a to become /? at detection is given by Pap = sin220 sin2<j>/2 where the phase is

18

(18)

216

The basic flight time factors are rather small, for eV neutrino masses and GeV energies, At ~ 50 milliseconds at 1000 MPc. These time spreads and separation may be shorter than the times involved in the production process thus making them difficult to observe. But if heavier neutrinos exist and mix with the three light flavors or if heavy particle (WIMP?) emission accompanies neutrino emission then the times become accessible. As for flavor conversion, emitted v'^s can get converted into v'Ts and thus produce a significant incoming flux of v'Ts (which is essentially absent initially in most neutrino production scenarios). How the flavor mix of the incoming beam can be determined has been delineated by Learned and Pakvasa 1 9 , then Pa0 and hence the phase (ft can be deduced by comparing to expected initial relative fluxes. Provided the phase 0/2 is not too large (such that sin 2 <j>/2 averages to 1/2) one has sensitivity to the parameters z, q, and H. With such measurements of At and <j>, one can potentially meassure the Hubble expansion parameters, z and q. This would be the first time that the red-shift or any other cosmological parameters are measured for anything other than light. There is another advantage of using neutrinos. This is the fact that the initial flavor mixing only depends on microphysics and so the comparison is free from problems such as evolution or worries about standard cadles etc. Conclusion The study of neutrinos and cosmology is reaching a very interesting and exciting stage and the next few years both will bear watching closely. Acknowledgements I thank the organizers Hsiang-nan Li and Wei-Min Zhang for their hospitality and conducting a most stimulating meeting. This work was supported in part by U.S.D.O.E under Grant DE-FG03-94ER40833. References 1. S. Perlmutter et al, Ap. J. 517, 565 (1999); A.G. Reiss et al, Ap. J. 507, 46 (1998). 2. P. De Bernardis et al, Nature, 404, 955 (2000); S. Hanany, et al, Ap. J. 545, L55 (2000). 3. S. Weinberg, The First Three Minutes, Basic Books, Inc. N.Y. (1977); E.W. Kolb and M. Turner, The Early Universe, Addison Wesley, N.Y.

217

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

(1990). See e.g. C.H. Lineweaver, astro-ph/0011448. Y. Fukuda et al, Phys. Rev. Lett. 8 1 , 1562 (1998). The Sloan Digital Sky Survey (SDSS) homepage is http://ssds.nasa.gov/ W. Hu, D.J. Eisenstein and M. Tegmark, Phys. Rev. Lett. 280, 5255 (1998). See talk by E. Fiorini, V.M. Lobashev and C. Weinheimer at Neutrino 2000, to be published in the Proceedings. S. Buries, K.M. Nollett and M.S. Turner, astro-ph/0008495. L. Stodosky, Phys. Rev. Lett. 39, 110(1975); B.A. Campbell and P.J.O'Donnell,P/»j/s. Rev. D 6, 1487 (1982). B.F. Shvartsman et al. JETP Lett. 36, 277 (1982). See however the proposal by C. Hagmann, astro-ph/9905258. T.J. Weiler, Ap.J. 285, 495 (1984). K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. Zatsepin and V. Kuzmin, JETP Lett. 4, 78 (1966). M. Takeda et al, Phys. Rev. Lett. 8 1 , 1163 (1998); M. Are et al, Phys. Rev. Lett. 85, 2244 (2000). T.J. Weiler, Astropart. Phys. 11, 303 (1999). L. Stodolsky, Phys. Lett. B 473, 61 (2000). T.J. Weiler, W. Simmons, J.G. Learned, and S. Pakvasa, hepph/9411432. J.G. Learned and S. Pakvasa, Astropart. Phys. 3, 267 (1995).

E M B E D ZEE N E U T R I N O M A S S MODEL I N T O S U S Y KINGMAN CHEUNG National Center for Theoretical Science, National Tsing Hua Hsinchu, Taiwan R.O.C. E-mail: cheungQphys.cts.nthu.edu.tw

University,

In this talk, I summarize a work done in collaboration x with Otto Kong on the Zee neutrino mass model. We show that the MSSM with explicit fi-parity violation actually contains the Zee model with the right-handed sleptons as the Zee singlet. We determine the conditions on the parameter space such that the neutrino mass matrix provides a viable texture that explains the atmospheric and solar data.

1

Introduction

We have seen substantial amount of experimental evidences from solar and atmospheric neutrino experiments that neutrinos in fact have masses. Among the experiments, SuperKamiokande 2 provided the strongest evidence for the atmospheric neutrino deficit, especially the impressive zenith angle distribution. The neutrino oscillation of v^ —> vT provides the best explanation for the atmospheric neutrino deficit. On the other hand, the solar neutrino deficit is best explained by ue —> Vy,,vT. So where do we stand if neutrinos do in fact oscillate? 1. Neutrino oscillation necessarily implies neutrinos have masses and of different masses. 2. However, we do not know the absolute values of the masses. We only know the mass differences. The mass difference required to explain the atmospheric neutrino deficit is 3 Amltm

~ 3 • 10- 3 eV2 ,

while a few solutions to the solar neutrino deficit exist. For example, the LMA solution requires a mass difference of 3 Am 2 oIar ~ 10- 5 eV2

(MSW) .

3. Though we do not know the absolute mass scale of the neutrinos, we have indirect constraints from various sources. The cosmological constraint f&hot ~ 0.1 implies mv < 3 eV, assuming neutrinos make up the hot dark matter. The end point of Tritium decay also constrains mv<. < 2.2 eV. Nevertheless, the best constraint comes from the neutrinoless double 218

219

beta (Oi^/?)decay. The absence of Ov(30 decay put an upper bound on the effective neutrino mass, as < m „ > e = 5 ^ mViV£ < 0 . 2 eV . i

We know of two widely separated mass scales in neutrinos: Am1tm and Amg 0lar . Two possibilities of arranging the three neutrino masses exist: (1) mi <§: m 2 ~ m 3 or (2) mi ~ m 2 <S m 3 , assuming mi < m 2 < m 3 :

f

~}AmLr

Am

*

2

*

L _

Am

sclar { IIIIIIZIZ.

•

Types of neutrino mass

There are three types of neutrino mass according to the structure of the mass term. (i) Dirac neutrino mass: ijJLM.DXR + h.c, in which \R ls the right-handed neutrino field. This is analogous to the Dirac mass term for charged leptons. However, this term is not allowed in the SM, because the bare mass term is forbidden by gauge invariance and the SM does not have the right-handed neutrino field. Even in the case of charged leptons, the Dirac mass term must be derived from the Yukawa term with a Higgs field or equivalent, in order that gauge invariance is fulfilled before the symmetry breaking, followed by spontaneous symmetry breaking that the Higgs field develops a VEV. (ii) Left-handed marjorana neutrino mass: tp'[C~1MLipL, where C is the charge conjugation operator. Again, this bare mass term is not allowed in the SM due to gauge invariance. Therefore, it must be derived from a Yukawa term with a Higgs field or equivalent. However, in this case a / = 1, Y = 2 Higgs field is required to generate such a mass term. SM does not have such a Higgs field. (iii) Right-handed marjorana mass: X # C - 1 MRXLIn the SM, there is no right-handed neutrino field. Therefore, to generate nonzero neutrino mass one has to include new physics beyond the SM. In both (i) and (iii) a right-handed field has to be

220

introduced while the case (ii) does not necessarily require a right-handed field. The hierarchy between the small neutrino mass and the charged lepton mass tells us something special about the mechanism that generates the neutrino mass, otherwise a fine tuning of the small Yukawa coupling for neutrinos is needed. A natural way to generate small neutrino mass is the see-saw mechanism, making use of a very large mass scale. Suppose there exist heavy right-handed neutrino fields Xfl's that couple to the left-handed neutrino fields via the usual Yukawa coupling. After electroweak symmetry breaking, £ = W, (MD)^ XRi + XTHi C~l (Mn)^ XRi + h.c. ,

(1)

where the first term is the Dirac mass term for the neutrinos and the last term is the majorana mass for the right-handed fields. We can then write the mass matrix as

2 ^ **H Ml MR ) UJ + "- C -

(2)

After diagonalizing the mass matrix, the mass matrix of the light neutrinos is given by Mv = -MDM^Ml

,

(3)

where M^1 is the inverse of the majorana mass matrix. If MR is sufficiently large, it naturally obtains small neutrino mass. Or equivalently, in terms of a dim-5 operator: C=^-(LiH2)(LjH2). To explain the observed neutrino mass the scale of MR ~ 1 0 1 0 - 1 3 GeV for a typical Yukawa coupling. Such an intermediate scale arouses a lot of theoretical speculations and interests. Should the \R related to SUSY breaking or early unification (a prediction of the Type I string theory is that the string scale is around 10 11 GeV.) Another natural way to generate small neutrino masses is to make use of loop suppression. This need not introduce right-handed neutrino fields, though new physics is still needed to generate the neutrino mass. One nice example is the Zee model 4 . 3

Zee mass model

Zee model 4 provides an economical way to generate small neutrino masses with a favorable texture 4 ' 5 ' 6 . The model consists of a charged gauge singlet

221

PiN. O

i l

m

(hZee ) - ' ' '

-5s-

-i

V,

w

h

2

M-* m, tan p ^

N

h1

<

X

<—lj

1

m,

7

^~ '/

VLi

Figure 1. A Feynman diagram for the Zee model, embedded in the R P V SUSY framework.

scalar h~, which couples to lepton doublets ipLj via the interaction

fij {WM,) ^ h- ,

(4)

where a,/? are the SU(2) indices, i,j are the generation indices, C is the charge-conjugation matrix, and / y ' are Yukawa couplings antisymmetric in i and j . Another ingredient of the model is an extra Higgs doublet (in addition to the one that gives masses to charged leptons) that develops a VEV and thus provides mixing between the charged Higgs boson and the Zee singlet. The one-loop diagram for the Zee model is depcited in Fig. 1. The Zee model can provide a mass matrix of the following texture 5 , e 0 meM meT meii 0 e meT € 0

(5)

where e is small compared with m e / i and meT, which is able to provide a compatible mass pattern that explains the atmospheric and solar neutrino data. Diagonal elements are guaranteed to vanish while the mM1- entry, denoted by e, has to be suppressed by some means. Moreover, meM ~ meT is required to give the maximal mixing solution for the atmospheric neutrinos. First, take e = 0 the matrix can be diagonalized by

222

with the eigenvalues m , - m , 0 for vLl,^,vL3,

respectively, and m

1ii+ %rThe atmospheric mass-squared difference Am ~ 3 x 4' 10~ eV , is to be identified with m = m + m . The transition probam

m

2

tm

3

2

2

bilities for v.

2 M

2

r

are

^Vi... —*VT.„ VIJU-JTVL

U •

metlmeT \ 2 . 2 ( {m2eiM + Pv,...^v,._ - 4 —=—• TT sin

m^ + m^ J "'

m2eT)L

4£

\

If meiM ~ mer, then sin 20 a t m ~ 1. This provides the maximal mixing solution for the atmospheric neutrino anomaly. If we choose a nonzero e, but keep e <S m e ^, e r . Then after diagonalizing the matrix we have the following eigenvalues = \ mi -4

£

mvl

n e\i

mv2 = -\

-L- rn-<3

+ m* _+L e*:

mj

+ miT + e

„

mV3 = - 2 e

~

2

er

*—_-

TfleUrffleT

™eM +

m

er

The mass-square difference between m ^ and m 2 2 can be fitted to the solar neutrino mass. If one takes the LMA solution and requires 4e

m

«"m"-

= Am 2 0l ~ 2 x 1(T 5 eV2 ,

giving (we have used mefl ~ meT) e m, efi 4

5 x 1(T 3

Neutrino mass in S U S Y

The original Zee model was not embedded into any grand unified theories or supersymmetric models. It would be very interesting if the Zee model naturally exists in some GUT or SUSY theories. In fact, the minimal supersymmetric standard model (MSSM) with a minimal extension, namely, the i?-parity violation, contains the Zee model. The right-handed sleptons in SUSY have the right quantum numbers to play the role of the charged Zee singlet. The E-parity-violating (RPV) A-type couplings could provide the

223

terms in Eq.(4). It is also easy to see that the RPV bilinear M"type couplings (/j,iLH2) would allow the second Higgs doublet H2 in SUSY to be the second ingredient of the Zee model. However, in RPV SUSY framework, there are three other sources for neutrino masses, in addition to the Zee model contribution. They are (i) the tree-level mixing with the higgsinos and gauginos, (ii) the one-loop diagram that involves the usual mass mixing between the left-handed and right-handed sleptons proportional to mt (Af —/itan/3), and (iii) the one-loop diagram that again involves the mixing between the left-handed and right-handed sleptons but this time via the A and jii couplings. They may deviate from the texture of the Zee mass matrix of Eq. (5). The tree-level mixing among the higgsinos, gauginos, and neutrinos gives rise to a 7 x 7 neutral fermion mass matrix MM under SVP 7 : 0 g'v2/2 MN

= 0 0 0

0 M2 -gv2/2 gvr/2 0 0 0

g'v2/2 -gv2/2 0 -M -Mi -Mi -Mi

-g'v,/2 gvJ2 -M 0 0 0 0

0 0

0 0

0 0

-Mi

-M2

"Ma

\

0 0 0 (m^x ( m ^ ( m ^ {ml\ «U K),3 (m°)31 (171%, {mil, J

(7)

whose basis is (—iB, —iW,h2,h°,vLc,vLii,vL^). In the above 7 x 7 matrix, the whole lower-right 3 x 3 block (m°) is zero at tree level. They are induced via one-loop contributions. We can write the mass matrix in the form of block submatrices: MM

-(MW\

(8)

ATR7 '

where M is the upper-left 4 x 4 neutralino mass matrix, £ is the 3 x 4 block, and m° is the lower-right 3 x 3 neutrino block in the 7 x 7 matrix. The resulting neutrino mass matrix after block diagonalization is given by

(m„) = -ZM'1? + (m°v)

(9)

The first term here corresponds to the tree level contributions, which are see-saw suppressed. Through this gaugino-higgsino mixing, nonzero /XJ'S give tree-level see-saw type contributions to (mv)ij proportional to HiHj, given by (mv)ij

= -

v2cos2^ (g2M1+g2M2) HiHj 2 2/j, [2nM,M2 - v sin/3 cos/3 (g2M, + g'2M2)]

(10)

224

A diagonal (mv)kk term is always present for a nonzero Hk- To eliminate these tree-level terms requires either very stringent constraints on the parameter space or extra Higgs superfields beyond the MSSM spectrum. This is a major difficulty of the present MSSM formulation of supersymmetric Zee model. Zee mechanism. The Feynman diagram is shown in Fig. 1. The lRh is the charged Zee singlet. To complete the diagram the charged Higgs boson K[ from the Higgs doublet H^ is on the other side of the loop and a IRk -h\ mixing at the top of the loop is provided by a F term of Lk: nkrriikK[l^ {h°)/(h°), where h% takes on its VEV, for a nonzero fik. Thus, the neutrino mass term (m®)ij has a Hkrrkh\ijk(m?. - n$)

(11)

dependence, where m,. 's are the charged lepton masses. LR slepton mass mixing comes from the one-loop diagram with two Acoupling vertices and the usual (AB — /j, tan /3)-type LR slepton mixing. Neglecting the off-diagonal entries in AE, the contribution to (m°)f,- with the pair Xuk and Xjki is proportional to [ (A% - /itan/3) + (1 - tfw)(^? - fiUmP) ] mtkmtlXiikXjki

.

(12)

LR slepton mass mixing via RPV couplings comes from a F term of L,: ^iXijkt-L-f-a (h°), where h° takes on the VEV. This is similar to the £Rk-h~ mixing in the Zee model, except that this time we have a A-type coupling in place of the Yukawa coupling. With a specific choice of a set of nonzero /Vs and A's, this type of mixing gives rise to the off-diagonal (m°)jj terms only and, therefore, of particular interest to our perspectives of Zee model. Taking the pair Xuk and Xjhi for the fermion vertices and a F term of Lg providing a coupling for the scalar vertex in the presence of a \ig and a Xghk, a ( m °)ij term is generated and proportional to Hgrrit^ghkXukXjhi • When we allow only one nonzero from Xijj but not from those with Xijj and Hj, there is a contribution dependence. We conclude that a minimal zeroth order Zee matrix is {

A12* ,

(13)

A at a time, the only contribution comes distinct indices. Suppose we have nonzero to the off-diagonal (m°)y with a Hjmtj Af^set of RPV couplings needed to give the A13fc ,

nk

}.

As at least one of the two A's has the form Xikk (= -Xkik), all types of contributions that have been discussed above are there. We want to make the

225

contribution from the Zee mechanism dominate over the others, or at least to suppress the diagonal entries in (m„). This necessarily requires suppression of the contributions from the tree-level see-saw mechanism and from the (AE — /ztan/3)-type LR slepton mixing. So, it is the Zee mechanism and the LR mixing via RPV couplings are required to be the dominant ones. 5

Scenarios and conditions to maintain Zee Texture

Because of space limitation we only show the best scenario: ^A^g, A ^ , ana H3}. The resulting neutrino mass matrix is given by

(

C 4 mT A,33

C 2 mT m M /LJ3A123 + Cs vn,T fiaX-aa^-aa 0

C 2 mT /H3A133 + Cb mT (13^133 \ 0 Ci»l

(14)J

where _ _

v2cos20(g2M1+g'2M2) 2/x [2pMxM* - v2 sin/3 cos/3 (g2M, + g'2M2)] '

Isi — ~;

C: = ~ ~

c

(A* - Mtan/3) f(M?L,M?R)

,

' = -iS?^<-^)'

(15)

where f(x,y) = ^ \og(y/x). In the above, we have neglected terms suppressed by me/m^ or me/mT. In order to maintain the zeroth order Zee texture, we need m e / i and meT to dominate over the other entries. Moreover, we need m e / i ~ meT ~ V / ^ C ( ~ 5 x 10- 1 1 GeV). Requiring the tree-level gaugino-higgsino mixing contribution to be well below meM gives H\ cos2/3 < fM,

(1 x 1(T 14 GeV" 1 ) .

(16)

For the (AE — n tan/3) LR slepton mixing contribution to be much smaller than m eM , we have

226

This corresponds to mee. It tells us that A^ can hardly be much larger than 1 0 - 3 . On the other hand, A^ is constrained differently because it does not contribute to this type of neutrino mass term. From the tree-level Zee-scalar mediated fj, decay, the constraint is A,„ 123

< lO" 8 GeV~ 2 ,

MlTR ~

(18)

which tells us that \-a3 can be as large as order of 0.01 for scalar masses of order of 0(100) GeV. Both meii and meT have two terms. Let us look at meii first. For the first term in mefl (the one with a C'2 dependence) in Eq. (14) to give the required value of atmospheric neutrino mass, we need meii ~ ?^g

(7 x 10" 7 GeV 2 ) ~ (5 x lO" 1 1 GeV) (19)

\

cos2/3 max(Mf_,M| R ) or (/x3 cos/3) AE3 ~ cos3/3 max(M 2 ., M | J (7 x 10~ 5 GeV - 1 ) . (20) This result looks relatively promising. If we take cos/3 = 0.02, all the involved scalar masses at 100 GeV and AK3 at the corresponding limiting 0.01 value, /j,3 cos/3 has to be at 5.6 x 1 0 - 4 GeV to fit the requirement. This means pushing for larger Mx (and M 2 ) and \i values but may not be ruled out. The corresponding first term in m e r has a A,33 dependence in the place of A^jj with an extra enhancement of m^./m'^, in comparison to m eM . That is to say, requiring meM « meT gives, in this case, AU3 « ^

A™ .

(21)

This gives a small A^a easily satisfying Eq. (17). The small A,33 also suppresses the second terms in both meM and m e r , the C5 dependent terms in Eq. (14). To produce the neutrino mass matrix beyond the zeroth order Zee texture, the subdominating first-order contributions are required to be substantially smaller in order to fit the solar neutrino data. Here, it is obvious that it is difficult to further suppress the tree level gaugino-higgsino mixing contribution to mTT, which makes it even more difficult to get the scenario to work. Explicitly, the requirement for the solar neutrino is l4 cos2/3 ~ n2M1 (1 x 10- 1 6 GeV" 1 ) .

(22)

227

6

A general version of supersymmetric Zee model

The conditions for maintaining the Zee neutrino mass matrix texture is extremely stringent, if not impossible, mainly because of the tree-level mixings via the bilinear RPV couplings. An alternative without the bilinear RPV couplings is to introduce an additional pair of Higgs doublet superfields. Denoting them by H3 and Hi, bearing the same quantum numbers as H^ and i? 2 , respectively, RPV terms of the form can be introduced. With a trivial extension of notations we obtain a Zee diagram contribution to {mv)ij through Ay* as follows :

1^2 Hy K - <) *M A"^ f(MZ- ,Mlj.

(23)

Here the slepton IHk keeps the role of the Zee singlet. Notice that the second Higgs doublet of the Zee model, corresponding to H3 here, is assumed not to have couplings of the form LiH3E?. The condition for the LR slepton mixing contribution to be below the required rnefi would be the same as discussed in the last section. However, there is a new contribution to {mv)kk given by

M)2^/K-,MJR). I^jg <-i^At-i,

(24)

This is a consequence of the fact that the term X^H^HfE^ provides new mass mixings for the charged Higgsinos and the charged leptons. The essential difference here is that unlike the /Uj terms the X'f.H^HfEl term does not contribute to the mixings between neutrinos and the gauginos and higgsinos on tree level. Similar to the above we are interested in only the minimal set of couplings {Ai2 k j A13 k , A" } with a specific k. For expression (23) to give the right value to m eM , we need ^kX"k

R

^

"

| g

(7x 10-7GeV-x) ,

(25)

and similarly for meT, it requires A13fc = (m^/m^)A12fc . This condition is easy to satisfy when we take {h3)/(h°} = 0.1. For Eq. (23) to dominate over Eq. (24), it requires A12fc » Afc

(ft?)

^

.

*»* » \

m

m2

•

W

228

The most favorable scenario is then the k = 1 case, where mt is just the m e . The above requirements are then easily satisfied. Also, the requirement for suppression of the LR slepton mixing is the same as before, and we also have Eq. (18) from the tree-level Zee-scalar induced muon decay. All these constraints can now be easily satisfied. Hence, such a supersymmetric Zee model looks very feasible. 7

Conclusions

Zee model provides a viable texture that explains the data. The minimal extension of MSSM with i?-parity violation actually contains the Zee model, with the right-handed sleptons £R as the charged singlet, A ^ couplings providing lepton-number violation, and Hu providing the mixing. However, there are other sources of neutrino mass in RPV SUSY, some of which wipe away the favorable texture. In order for the Zee contribution to dominate over the others we pick the best minimal scheme {A12fc , A13fc , fj,k}, k = 3, and determine the requirements on the parameter space, which turns out quite stringent but still possible. Finally, we offered a further consideration that abandons the bilinear RPV couplings but introduces two additional Higgs doublets. This model turns out quite feasible. I would like to thank Otto Kong for the pleasant collaboration on the work presented here. References 1. K. Cheung and O. Kong, Phys. Rev. D61, 113012 (2000). 2. Y. Fukuda et al. (Super-K Coll.), Phys. Rev. Lett. 8 1 , 1562 (1998); Phys. Lett. B467, 185 (1999). 3. see e.g. M. Gonzalez-Garcia, M. Maltoni, C. Pena-Garay, and J.W.F. Valle, Phys. Rev. D63, 033005 (2001). 4. A. Zee, Phys. Lett. 93B, 389 (1980). 5. P. Frampton and S. Glashow, Phys. Lett. B461,95 (1999). 6. C. Jarlskog, M. Matsuda, S. Skadhauge, and M. Tanimoto, Phys. Lett. B449, 240 (1999). 7. M. Bisset, O. Kong, C. Macesanu, and L. Orr, Phys. Lett. B430, 274 (1998); Phys. Rev. D62, 035001 (2000).

ELECTROWEAK SUDAKOV CORRECTIONS AT 2 L O O P L E V E L HIROYUKI KAWAMURA Department of Physics, Hiroshima University Higashi-Hiroshima 739-8526, JAPAN [email protected]. sci.hiroshima-u.ac.jp In processes at the energy much higher than electroweak scale, weak boson mass act as the infrared cutoff in weak boson loops and resulting Sudakov log corrections can be as large as 10%. Since electroweak theory is off-diagonally broken gauge theory, its IR structure is quite different from that of QCD. We briefly review recent developments on electroweak Sudakov and discuss on the exponentiation of Sudakov double logs and explicit 2 loop calculations in Feynman gauge.

1

Introduction

High energy experiments in TeV region are planned in the near future to obtain the hints to the fundamental problems in particle physics such as the mechanism of electroweak symmetry breaking, the gauge hierarchy problem and so on. The precision measurements in this region are expected to give us the important information to construct the scenario favorable up to Plank scale.* In order to extract these information from experimental data, it is crucial to carry out the theoretical calculations at least in the same level of accuracy in the standard model and in other possible models. It is very important also for the estimation of the background of the production of the new particle. Recently Ciafaloni and Comelli 1,a pointed out that in processes with energy much higher than electroweak scale, the "heavy particle masses" Mw — Mz(= M) acts as IR cutoff in weak boson loop integrals and resulting IR log corrections give large contributions to cross sections, comparable or larger than QCD corrections. For example, 1 loop log corrections to e+e~ —> n+n~ are 3 , a"=o*"x\\

+

a { o . 6 l n ^ + 9 . 4 1 n ^2 - 1 . 4 1 n 2 g U2 . 47rsur0w L M M MJ

UV log

IR log

Sudakov log

"The possibility of the large extra dimension is not considered in this work. 229

230

The second and the third terms which vanish in LEP energy region give the dominant corrections in TeV region. Especially Sudakov double logarisms 4 which come from the overlap of the soft and the collinear region of the loop integral give the dominant radiative corrections in the asymptotic regime (Their typical value at VS ~ ITeV is j ^ r ^ l o g 2 ^ ~ 6.6%). Since factors like o / M o g 2 " ^ (LL), a n l o g 2 n _ 1 j^r (NLO), • • •, may spoil the perturbation expansion, it is necessary to control or resum these contributions to obtain a reliable and precise calculation in extremely high energy processes. The infrared structure of gauge theory has been extensively investigated and it is well-known that Sudakov logs exponentiate for the form factor in QED 4 ' 5 and QCD 6 and that they can be resummed in various physical processes 7 . The exponentiation of Sudakov logs in these cases results from the gauge symmetry and Lorentz symmetry in the factorization formula of cross sections 8 . In case of the electroweak theory, the situation is quite different. Since the gauge symmetry is broken and its breaking pattern is off-diagonal, the exponentiation of Sudakov logs in this case is a non-trivial matter. Several discussions on the all order behavior of Sudakov terms have been presented with different results 9>10>n and the explicit 2 loop calculations 13,14,15 imply the exponentiation of Sudakov logs. From these discussions, we can see that the methods valid in QCD can not be applied straightforwardly to electroweak case and adequate modifications are needed The most striking feature of electroweak Sudakov is that they appear not only in the exclusive processes but also in the inclusive processes since non-abelian charge is not confined 2 . In non-abelian gauge theory, it is known that Bloch-Nordsieck cancellations occur in leading power when both the sum of the final degenerate states and the initial color average are taken 1 6 . In QCD prcesses where initial color averages are always taken due to the color confinement, Sudakov double logs appear only in the end-point region of the phase space where the soft cancellation fails due to the suppression of the real emissions. On the other hand, initial state in electroweak processes are color (isospin) non-singlet and Sudakov logs remain even when the number of weak boson is not counted in the final state. Therefore, it can be said that the effects of symmetry breaking remain even we go up to the extremely high energy in which we usually consider that the gauge symmetry is restored. In the following sections, we introduce the discussions on all order behavior of Sudakov logs and explain 2-loop calculation of the form factor in detail and discuss on other recent development.

231

2

Discussion on all order behavior of Sudakov logarisms

Discussion on all order behavior of Sudakov logarisms have been presented by several authors 1 0 ' 9 > n . The author of 10 calculated all order In 2 " -fife terms in the self-energy in Axial gauge including only the weak boson loop and showed that Sudakov terms does not exponentiate due to the mixing effect 10 . Ciafaloni and Comelli 9 estimated the leading log terms of the form factor in Feynman gauge using soft insertion formula which is the formalism developed in QCD 17 . Soft insertion formula means that we can obtain leading log terms by inserting the eikonal current at fermion legs, and estimating the diagrams with energy ordering. Their result was that the QED effects and electroweak effects factorize and Sudakov logs exponentiate in operator form but does not exponentiate numerically. Fadin et.ai. n gave a general argument using the infrared evolution equation 18 , which is the differencial equation with respect to the infrared cutoff derived from Gribov's Bremsstrahlung theorem. They derive two equations, one is for the region where \i < Mw with QED kernel and another is for JJ, > Mw with electroweak kernel. The exponentiation as the numerical value is concluded naturally from their recursive form. 3

Explicit calculation

Explicit 2 loop calculations of leading logarisms for the form factor are helpful to resolve the controversy discussed above and to construct the rigorous all order proof of exponentiation of electroweak Sudakov. We describe the similarity and the difference between this case and the QCD case concerning the cancellation of the non-exponential factors. 3.1

QCD case

2 loop calculations of the leading singularities of the form factor were accomplished many years ago in QED 20 and QCD 21 . QCD double log corrections for the form factor come from vertex corrections in Feynman gauge. Using the mass regularization for infrared divergences 1-loop result is, e2Q2 , 16TT2

2

S A2

'

where CF is the SU(3) Casimir operator for the fundamental represen-

232

tation and S is the momentum transfer. At 2 loop level, the leading double logs appear from the ladder diagram, the crossed ladder diagram, and the diagrams including triple gluon coupling as follows,

(4^24ln

X

X C F

'

$)^**W-?C*C') X2

'

= -(IrVl^A^^

Then the sum of these contributions are exactly a half of the 1-loop contribution squared.The non-exponential terms which appear in the crossed diagram and the diagram with the triple gluon coupling due to the non-abelian nature cancel each other, and the result is consistent with the exponentiation of the Sudakov logarisms. 3.2

Electroweak theory

2-loop expricit calculations has been done for self energy in axial gauge by Beenakker and Werthenbach and for the form factor of right-handed fermion in Feynman gauge by Melles14. Here we concentrate on the calculations of the form factor for left-handed fermions 15 . The situation become more complicated in electroweak theory than QCD since the gauge symmetry is spontaneously broken and the pattern of the symmetry breaking is not diagonal. In physical processes photon must be treated in semi-inclusive way since it is uncountable, on the other hand, weak bosons can be treated both inclusive and exclusive ways. The process we consider now is the fermion pair production from the SU(2)(g>U(l) singlet source and the infrared divergences by photon loops are regularzed by the fictitious photon mass A. We must treat gauge bosons with different mass ("mass gap"). Within the leading log approximations, the fermion chirality is conserved and W and Z boson mass can be approximated to be equal and the field of unbroken phase i.e., W and Z can be used though S » M™ = M.

233

The mixing effect comes from

P^.QJ^O, ±

where T is the SU(2) generator and Q = T3 + Y is the charge of fermion. For the right-handed fermion, the gauge group of electroweak interaction is reduced to U(1)Q ® U(1)Y-Q and the exponentiation become a trivial matter. It also should be noted that if we take M2 — A2, the mixing effects disappear and the calculation becomes same with that for unbroken S£/(2)E/(l) gauge theory with regularization mass M and the exponentiation holds as QCD case. The group factors for each SM boson exchange become, ' 7 exchange : e2Q2 W exchange : g2 £ 0 = 1 , 2 T Q r a a2 Z exchange : , „ (T 3 - sin2 6WQ)(T3 - sin2 6WQ) COS^ Vw

= g2T3T3 + g'2Y2 - e2Q2 Then, 1 loop result turn out to be,

r<

fa2&+aPr

' -eV)

" - > - jk

ln2

& - (ST8"?2 ln' i

Here CF denotes the SU(2) Casimir operator for the fundamental representation. At the 2 loop level, there appear 16 diagrams which are classified in 3 groups. That is, ladder diagrams, crossed ladder diagrams (Fig.l) and the diagrams including 3 gauge boson couplings (Fig.2). Here we use the following definitions for brevity, e2Q2 = 1 -

7

{g2CF + g'2Y2 - e2Q2) = (W + Z) ,

, 1

,

S

' = —T=- 1 H T 7 ,

2V^TT

A2

l

T -

1

S

L •= — - = - ln —— ,

2V27T

M2

for the group factors and the loop factors respectively. After the several cancellations among different diagrams, the 2 loop results for ladder diagrams, crossed ladder diagrams, and the diagrams including 3 point coupling turn out to be, ladder = (la.) + (lc.) + (Id.) + (1/.) = 7 2 ^ I" + 2j(W + Z) \ l2I? + (W + Zf + y(W

+

l i

4

Z)\^L4-~L3l

234

(lb.)

(le.)

(14)

dg.)

Figure 1. The ladder and crossed ladder diagrams. The dashed (wavy) line represents the photon (W and/or Z) with the mass A (M).

(2a.)

(2b.)

(2c.)

(2d.)

Figure 2. The diagrams which have the triple point couplings. The meaning of lines is the same as in Fig.l.

crossed = (16.) + (le.) x 2 + (Ig.)

+7(W + Z)

iL'+lL3>

3 point = [(2a.) + (26.) + (2c.) + (2d.)]

2„2/TT3

+ 2gze*QT

6

3

235

94lcvCF~L"

+ 2g^QT3

i

L4 + L31

l

Here the underlined terms are the extra terms compared to QCD case which come from the mixing effect. After these terms cancels each other, the sum of these contributions becomes,

T(2)

= 1 - ik* ^ 2(i?) + *'2y2 " e 2 Q 2 ) l n 2 W - I^ g2Q2 ln2 I • 1 iR)+9 2y2 2! is? ^ ' - ^ln21? + i e 2 Q 2 ln2 £ 2

+

which imply the exponentiation of Sudakov form factor. We can see from this result that (1) 7 must be included to extract the correct coefficient of In4

-^.

(2) Non-exponential factor appear also in ladder diagrams. The first point comes from the fact that the singularity from the photon which propagates inside the W and/or Z loop in Figs.Id and 2c is regulated by the W and/or Z mass 14 . The gauge invariant set including photon contributions must be included to extract the correct In2 -£p terms. The second is the important difference from QCD. In QCD case, the contributions of the ladder diagrams and similar part of the crossed ladder diagrams remain in final results and those of "3-point" diagrams act only as the counter term against the nonexponential term, while the sum of former two contributions can be extracted from the ordered ladder diagrams. Therefore, the approach by soft insetion formula which is valid for QCD can not be applied straightforwardly to this case and modifications seems to be necessary. 4

Discussions

Electroweak log corrections become very important in the processes in TeV region. Since they give large corrections to the cross sections, It is crucial to control them in perturbation series to reduce the uncertainty of theoretical predictions in precision measurements and in the backgroud estimations for the signals of the new physics. Exponentiation of Sudakov double logarisms seems to be valid from the discussion using the infrared evolution equation and the explicit 2loop calculations. The resummation of next to leading logarisms were

236

accomplished by Ktihn, Penin and Smirnov 12 for fermion external line and by Melles for general external line and for the top Yukawa enhanced part 19>22. As for the phenomenology, several works have been done. 1-loop ? , and 2-loop 12 Sudakov-type log corrections to e + e > ff and 1-loop effects in MSSM 24 were calculated. Denner and Pozzorini 25 presented the complete 1-loop electroweak log corrections for the general processes. Baur 26 discussed on the impact of electroweak Sudakov on the W boson mass measurement at LHC. Above calculations indicate that NLO Sudakov effects are comparable with LO Sudakov effects at the energy around 1 TeV. Therefore, it can be said that we can not yet control the electroweak log corrections within the 1% level and much have to be done from now.

Acknowledgements The author would like to thank M.Hori and J.Kodaira for useful discussions through the collaboration. He is also grateful to H-n.Li, and W.M.Zhang and all organizers of PPP2000 for the kind invitation to a fruitful, enjoyable workshop. The work of H.K was supported by the Monbusyo Grant-in-Aid for Scientific Research No. 10000504.

References 1. P. Ciafaloni and D. Comelli, Phys. Lett. B446 (1999) 278. 2. M. Ciafaloni, P. Ciafaloni and D. Comelli, Phys. Rev. Lett.84 (2000) 4810; Nucl. Phys. B589 (2000) 359; and hep-ph/0007096. 3. M. Beccaria, P.Ciafaloni, D. Comelli, F. Renard and C. Verzegnassi, Phys. Rev. D61 (2000) 073005. 4. V. V. Sudakov, Sov. Phys. JETP 3 (1956) 65. 5. D. Yennie, S. Frautschi and H. Suura, Ann. Phys.13 (1961) 379. 6. See e.g. J. C. Collins, in Perturbative Quantum Chromodynamics ed. A. H. Mueller (World Scientific, Singapore,1989) P. 573; and references therein. 7. See e.g. G. Sterman, in QCD and Beyond ed. D. Soper (World Scientific,

237

Singapore, 1996)P. 327; and references therein ; hep-ph/9606312. 8. H. Contopanagos, E. Laenen and G.Sterman, Nucl. Phys. B484 (1997) 303. 9. P. Ciafaloni and D. Comelli, Phys. Lett. B476 (2000) 49. 10. J. H. Kiihn and A. A. Penin, hep-ph/9906545. 11. V. S. Fadin, L. N. Lipatov, A. D. Martin and M. Melles, Phys. Rev. D 6 1 (2000) 094002. 12. J. H. Kiihn, A. A. Penin and V. A. Smirnov, Eur. Phys. J. C17 (2000) 97. 13. W. Beenakker and A. Werthenbach, Phys. Lett.B489 (2000) 148. 14. M. Melles, Phys. Lett.B495 (2000) 81. 15. M. Hori, H. Kawamura and J. Kodaira, Phys. Lett.B491 (2000) 275. 16. See e.g. M. Ciafaloni, in Perturbative Quantum Chromodynamics ed. A. H. Mueller (World Scientific, Singapore,1989) P. 491. 17. A. Bassetto, M. Chiafaloni and G. Marchesini, Phys. Rept 100 (1983) 201 ; S. Catani and G.Marchesini, JVucL Phys. B249 (1985) 301. 18. V. N. Gribov, Yad. Fiz. 5 (1967) 399 [Sov. J. Nucl. Phys. 5 (1967) 280]; R. Kirschner and L. N. Lipatov, JETP 56 (1982) 266; Phys. Rev. D 2 6 (1982) 1202 ; L. N. Lipatov, Nucl. Phys. B307 (1988) 705; V. Del. Duca, Nucl. Phys. B345 (1990) 369. 19. M. Melles, Phys. Rev. D 6 3 (2001) 034003. 20. R. Jakiew, Ann. Phys. 48 292; P. M. Fishbane and J. D. Sullivan, Phys. Rev. D 4 (1971) 458; T. Appelquist and J. R. Primack Phys.Rev. D 4 (1971) 2454 etc. 21. J. Carrazone, E. Poggio and H. Quinn, Phys. Rev. D l l 2286. J. M. Cornwall and G. Tiktopoulos, Phys. Rev. D 1 3 (1976) 3370; J. Frenkel, M. -L. Frenkel and J. C. Taylor, JVuci. Phys. B124 (1977) 268; and references therein. 22. M. Melles, hep-ph/0012157. 23. M. Beccaria, P.Ciafaloni, D. Comelli, F. Renard and C. Verzegnassi, Phys. Rev. D61 (2000) 011301. 24. M. Beccaria, F. Renard and C. Verzegnassi, hep-ph/0007224. 25. A. Denner and S. Pozzorini, hep-ph/0010201. 26. U. Baur, hep-ph/0007287.

List of Participants S. Aoki A. Arhrib S. Baek S. Brodsky M. Burkardt J. Chay W.F. Chang A. Chen C.H. Chen C.P. Chen C.C. Cheng C.Y. Cheung K. Cheung T.W. Chiu C.K. Chua M. Chung C.Q. Geng J.H. He I.L. He X.G. He M. Hori W.S. Hou Y.K. Hsiao R.C. Hsieh H.C. Hu M.H. Huang J. Huston C.W. Hwang W.F. Kao H. Kawamura Y.Y Keum P. Ko 0 . Kong H.L. Lai F.F. Lee H.P. Lee

Tsukuba U., Japan NTU NTU SLAC, USA NMSU, USA Korea U., Korea NTHU NCU NCKU AS NTHU AS NCTS NTU NTU NCTS NTHU NCKU NTHU NTU Hiroshima U., Japan NTU NTU NCKU NCKU AS MSU, USA NTHU NCTU Hiroshima U., Japan AS KAIST AS MHIT NCTU AS

240

J.Y. Leon H-n. Li H.S. Liao C.C. Lih G.L. Lin L. Lin S.Y. Lin T.F. Lin W.L. Lin C.C. Liu T.Liu C.L. Lu C.S. Luo M. Neubert S. Pakvasa D. Pirjol W. Sam J.Q. Shi S.Y. Tsai Y.Y. Tsao J.J. Tseng D. Tsou P. Vogel C.H. Wang Y.C. Wang T.K. Wong M.C. Wu M.H. Wu T.H. Wu H.S. Yang K.C. Yang T.W. Yeh W.M. Zhang B.A. Zhuang

NTU NCKU NTHU NCKU NCTU NCHU AS NTTTC NTNU NTHU AS NCKU AS Cornell U., USA Hawaii U., USA UCSD, USA AS NTU NTU NCTU NCTU NCTS Caltech, USA NLIT NCKU AS NCTU NCKU NTHU NTU CYCU NCTU NCKU AS

Hllfflsini

WiiwMp^ ..:;.,:.'!•.»

.:!-i,

•

:»!-•••.••

^HWK'nlmnwIir™ Ir*Xti

• -

•1 •

•

•

•

•

| •

•

-

'

•

•

• ' I

m

r^

- Lm I

I m IHH ISBN 981-02-4645-5

wiw. worldscientific. com 4723 he