Proceedings of the Third Meeting on CPT and Lorentz Symmetry: Bloomington, USA 4-7 August 2004

Proceedings of the

Third Meeting on CPT and Lorentz Symmetry

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Proceedings of the

Third Meeting on CPT and Lorentz Symmetry Bloomington, USA

4-7 August 2004

Editor

V. Alan KosteleckJi Indiana University, USA

N E W JERSEY

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LONDON

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r pWorld Scientific SINGAPORE

BElJlNG * S H A N G H A I

HONG KONG

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TAIPEI

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CHENNAI

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224

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CPT AND LORENTZ SYMMETRY Proceedings of the Third Meeting Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, withour written permission from the Publisher.

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PREFACE

The Third Meeting on CPT and Lorentz Symmetry was hosted by the Physics Department at Indiana University, Bloomington during the fourday period from Wednesday August 4 to Saturday August 7, 2004. Following the format of the previous meetings in this series, the talks spanned both experimental studies and theoretical topics in the field. Results presented on the experimental front included current sensitivities and future plans involving tests with the resonant-cavity and interferometric behavior of photons, the oscillations of neutrinos and neutral mesons, clock-comparison measurements on the Earth and in space, astrophysical observations, macroscopic matter, the spectroscopy of hydrogen and antihydrogen, various properties of fundamental particles, and gravitational phenomena. The theoretically oriented talks considered physical effects at the level of the Standard Model and beyond, possible sources and mechanisms for CPT and Lorentz violation, and associated classical and quantum topics in particle physics, field theory, gravity, and string theory. This proceedings volume begins with invited papers and follows with contributed ones, ordered according to their scheduling at the meeting. My thanks to all these authors for the timely preparation of their manuscripts. The efforts of many people were essential to the success of the event. Assistance in operational matters was provided by Brett Altschul, Quentin Bailey, Matt Mewes, Ali Picking, Samuel Santana, and Jay Tasson, among many others. Special thanks go in particular to Jordan Tillett, Robert Bluhm, and Neil Russell, whose organizational help was crucial to the meeting’s occurrence.

Alan Kostelecky October 2004

V

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CONTENTS Preface

............................................................ ......

Some Anomalies Related to Spontaneous Symmetry Breaking

v 1

Y. Nambu Tests of CPT and Lorentz Symmetry using Hydrogen and Noble-Gas Masers ............................ R.L. Walsworth

11

New Tests of Lorentz Invariance in the Photon Sector using Precision Oscillators and Interferometers ....................... M.E. Tobar et al.

20

..................................

29

................................

38

Lorentz Violation and Neutrinos M. Mewes ATHENA - First Production of Cold Antihydrogen and Beyond A. Kellerbauer et al.

The Physics of Generalized Maxwell Equations C. Lammerzahl and H. Miiller

...................

Operation of the K-3He Self-compensating Co-Magnetometer for a Test of Lorentz Symmetry ......................... T. W. Kornack and M. V. Romalis Lorentz Violation and Gravity V.A. Kostelecky’

:. .....

....................................

Short-Range Tests of the Gravitational Inverse-Square Law E. G. Adelberger Testing CPT Conservation using Atmospheric Neutrinos M.D. Messier New Tests of Lorentz Invariance using Optical Resonators A. Peters et al. vii

48

57

71

....... 80

..........

84

........

92

viii

Atomic Clocks on Earth and in Space for Tests of Fundamental Physics and Navigation

................. 101

K. Gibble Spacetime Symmetries and Varying Scalars

.....................

105

......................

115

R. Lehnert Gravity Probe B: Launch and Initialization G.M. Keiser et al.

Asymptotically Free Lorentz-Violating Field Theories B. Altschul Torsion-Balance Test of Lorentz-Symmetry Violation

........... ...........

124

133

B.R. Heckel &ED Tests of Lorentz Symmetry

..............................

141

R. Bluhm Lorentz and CPT Violation with LSND

.........................

150

T. Katori and R. Tayloe Neutrino Oscillations as Probes of New Physics

.................

159

C. Penla Garay

....................

165

.......................

175

BaBar Tests of Lorentz and CPT Symmetry

F. Martinez- Vidal Ultra- Sensitive Speedometer using Nonlinear Effects of Optical Pumping

B. T.H. Varcoe An Improved Test of Relativistic Time Dilation with Fast Stored Ions ............................................. G. Gwinner et al.

184

Prospects for Improved Lorentz-Violation Measurements using Cryogenic Resonators ......................................... J . A . Nzssen et al.

193

ix

Quantum Gravity Induced Granularity of Spacetime and Lorentz Invariance Violation ..................................

201

D. Sudarsky Vacuum Cerenkov Radiation in Maxwell-Chern-Simons Electrodynamics R. Lehnert and R. Potting

......................

211

.........................

220

Electrophobic Lorentz Invariance Violation for Neutrinos and the See-saw Mechanism .................................. S.F. King

228

The Bounds on Lorentz and CPT Violating Parameters ........................................... in the Higgs Sector I. Turan

235

............................

244

Tests of Lorentz Invariance and CPT Conservation using MINOS ................................................. B.J. Rebel and S.F. Mufson

251

Weighing the Antiproton using Antiprotonic Helium Atoms and Ions R.S. Hayano

Macroscopic Matter in Lorentz Tests H. Muller and C. Lammerzahl

An SME Analysis of Doppler-Effect Experiments C.D. Lane

................ 258

Nonrelativistic Ideal Gases and Lorentz Violations D. Colladay and P. McDonald

............... 264

A Laboratory Free-Fall Test of the ............................... Equivalence Principle - POEM R.D. Reasenberg and J.D. Phillips

270

.....................

277

Composite Mediators and Lorentz Violation A . Jenkins

X

Lorentz-Violating Electromagnetostatics

.........................

282

Q.G. Bailey Lorentz Violation in Supersymmetric Field Theories

.............

288

M.S. Berger Searching for CPT Violation and Missing Energy in Positronium Annihilation .....................................

294

P.A. Vetter Lorentz-Violating Vector Fields and the Rest of the Universe

.... 300

E.A. Lim CPT Test in Neutron- Antineutron Transitions

...................

306

Y.A. Kamyshkov Deformed Instantons

...........................................

312

D. Colladay and P. McDonald Proposal to Measure the Speed of Mu-Type Neutrinos to Two Parts in lo6 ............................................. T. Bergfeld et al. Particles and Propagators in Lorentz-Violating Supergravity R.E. Allen and S. Yokoo Testing Lorentz Symmetry in Space N . Russell

.....

.............................

318

324

330

SOME ANOMALIES RELATED TO SPONTANEOUS SYMMETRY BREAKING

YOICHIRO NAMBU University of Chicago, Illinois, USA E-mail: [email protected]

I discuss what I call Nielsen, Lorentz, and Newton anomalies accompanying some cases of spontaneous symmtry breaking (SSB). The first refers to an anomaly in the number and the dispersion law of the Nambu-Goldstone modes in the SSB of Lie algebras; the second to those in the case of Lorentz symmetry breaking, in a medium or in vacuo; and the third to the peculiar behavior of quasiparticle and other familiar excitations in a medium when they are regarded as classical particles. Speculations are made in regard to an SSB of Lorentz symmetry in mcuo.

1. Nielsen anomaly

In general the number of the Nambu-Goldstone (NG) bosons associated with a spontaneous symmetry breaking (SSB) G + H is equal to the number of symmetry generators Q, in the coset G I H . In the absence of a gauge field, their energy R goes as a power kY of wave number. In a relativistic theory, y = 1 necessarily unless Lorentz invariance is broken. There are, however, exceptions to the above ‘theorem.’’ Recently a few new examples have been found to occur in connection with color superconductivity in high density quark matter. I would like to give a general explanation to the phenomenon.2 Suppose a set of local operators 0 = (0,) forms a representation of a symmetry group G, under which the 0, transform into each other. The generators { Q,} of G themselves are an example. If one member of the elements, say 0 0 , develops a vacuum expectation value < 00>= C, then the other elements o k will generate the zero modes in their low energy part. If two of them are such that their commutator closes on 00: [Oj,O k ] N ZOO, or < , o k ] >NiC.Then their low-energy components behave like canonical conjugates of each other, therefore they belong to the same dynamical degree of freedom, and the number of NG bosons is thereby reduced to one

[o,

1

2

per each such pair. The set of generators {Qz} are precisely of this nature. The dispersion law y = 2 is obtained by a more detailed analysis. Basically one constructs a low-energy effective quadratic Hamiltonian in terms of the deviations (6%) and their canonical conjugate {Cz} by expanding around the expectation value < 00>= C:

H,jf = aCfi:

+ b6: + ck2C6:

(1)

The zero modes are (0,,j # 0) and have no b terms since they are in the flat directions from < 0 0 >. So from the a and c terms one gets a phononlike spectrum with y = 1. This is the normal case. However, if 0, and o k are canonical conjugates, then n k n, N o k , hence the a terms for them should also be missing, and their c terms directly lead to a dispersion law y = 2 . (The Higgs mode is 60,with bo # 0, and this gives a finite gap with k2 corrections.) Ferromagnetism and antiferromagnetism present the simplest examples to illustrate the normal and anomalous cases. Their relevant operators are the total spin (density) and the graded (staggered) sum N

(0,)= c,

o,,

(S,) = 0.

S and 0 together form an SO(4) algebra. The zero modes are expected to be Sz,Sy and O,,O, respectively. But since < [S,,S,] >=< [O,,O,] >= i < S, >, the ferromagnetic case has only one zero mode, whereas the antiferromagnetic case has two. In the former, the modes S, f is, N S, f in, serve respectively as the annihilation and creation operators of a single chiral mode behaving as exp(Fzwt), but in the latter 0, f in, and 0, f in, generate two nonchiral modes. I will mention two other examples of the anomalous case. They have been found in connection with color superconductivity. In one of them,4 the SSB of a complex scalar multiplet (Higgs field) q$ is induced by a chemical potential instead of a negative mass term: its potential term is of the form

v = -x4(Cp . $)2

- Zp(7.r q5 - (bt7rt) 1

(3)

The chemical potential destabilizes one of the $i away from zero. Since # 7r7 both 4 and 7r acquire nonzero expectation values, and this leads to noncommutativity among different zero modes. In still another example,6 one deals with a set of multicomponent Cooper pairs {&&}

4

3

{$j$:},

and some combination of which will condense. Since the commutators [ $ i $ j , $:$!] of the expected zero modes yield the bilinears {$$’+}, they may not be independent if one of the latter quantities has a nonzero expectation value because of a chemical potential.

2. Breaking of Lorentz Symmetries

The above considerations about the anomalous zero modes had to do with internal symmetries, except for the case of ferromagnetism since the spin is part of the Lorentz group. So the next natural problem would be to examine the spacetime symmetries. Can spacetime symmetries be spontaneously broken in vacuo? What would be the properties of the ensuing zero modes? The first question has been taken up by many people in the past, so I will concentrate here on the second one, in particular, the case of rotations and boosts. Here, one immediately realizes that the generators of the Lorentz group, namely L,, = x,a, - x,d, Cp,, are not homogeneous, i.e., their orbital part depends on the coordinates explicitly. Hence if a set of field operators transforming under L,, develops a vacuum expectation value, one cannot necessarily apply the usual argument and claim that the rest of the members would represent the zero modes. The spin part C,, of L,, is free of the problem, but it alone does not generate Lorentz invariance. To illustrate the point, let us consider a relativistic analogue of the ferromagnet in the vacuum. The spin density of a fermion field is the spatial part of the chiral current j 5 , = $757,$ = $ t ( p l , n z ) $ , and it is also a part of the angular momentum density Mozk satisfying the 4 2 ) (current) algebra relations.” Consider as a model the following nonlinear

+

leading to an SSB by an axial vector constant V, =< j 5 , >. Here a Dirac mass is assumed but the result is similar for a Majorana mass. In general

aIt is interesting to note that the relations among the spatial components of axial (ai) and vector (piai)currents (the latter corresponding to imaginary Lorentz boosts), stand in direct correspondence to those for ferro- and antiferromagnetism.

4

one may consider three possible cases.

I. g > 0, spacelike: V

(O,O,O,v), Z = (ZtZ,, Z,,-), vacuum has net spin in the z direction, T-C’P’, =

w 2 =p”,+P2,+(&%2kgv)2. 11. g < 0, timelike: V = (v,O,O,O), Z = (-,Zz,Z,,Zz), vacuum has net chiral charge, T+C+P-, w2 = m2

+ (Ipl =t gv)2.

111. lightlike: V

=

(wF gv)2 = pp + p i

(k’v, 0, 0, v),

Z = ( ~ ’ Z OZ,, , Z,,ZO),

+ ( p , * =t’gvl2 + m2.

(5)

Case I seems the more natural one since it corresponds to an exchange of

5 Figure 1.

The group velocity does not exceed 1, but the branch with the negative sign has various peculiarities, such as spacelike states, w < k, and negative group velocity relative to the momentum. I will come to them later, but our concern now is the properties of the associated zero modes 2,which follow from the loop integral < >. For example, the effective Lagrangian the present dynamical model, a stable solution exists if gv

>m

for Case I.

5

C and the dispersion law for Case I are found to be of the form

c = L,,Z~ZV,

+ BP,Pv + Cgp”P2, + Dgz,gz”P2 + E g z , g z v p ,2 x + F ( g z , p z p v + gzvpzp,) + iGEzpvxP

L,” = 4 , ” P

2

-M2g,v

( M +~

(6)

-AM2gz,gz”,

AM^ - p 2 ) 2

( M -~p 2 ) 2 - g 2 p 2

- g 2 p 2 = 0,

= 0.

(7)

The points to be noted are: (1)as was anticipated, the would-be zero modes are actually massive, and (2) there is a Chern-Simons term in the subspace ( t ,5 , Y).5c 3. Quasiparticles as classical objectsd

As we have seen above, the dispersion law for the fermion after an SSB can have a shape which leads to a group velocity opposite to the momentum. But this is not a new phenomenon. A similar behavior can be found in some familiar examples. One of them is the Laudau roton-phonon in superfluid 4He. Another is the Bogoliubov-Valatin (BV) quasiparticle in superconductors, which is a superposition of an electron and a hole:

@(PI

= a$T ( P )

+ p$l (-PI

(8)

having the dispersion law

w2 = ( p -

+A2,

(9)

where p~ is the Fermi momentum. My goal is now to study the implications of these peculiar dispersion laws. The relation between a dispersion law w ( k ) and the group velocity v = & / d p is equivalent to a Hamiltonian equation of motion: w H , k p, dx/dt = v = a H / a p . So let us regard a quasiparticle as a classical point particle and study its motion in an external field, assuming that the medium is of sufficiently large extent and the acceleration is sufficiently small to N

N

=The Chern-Simons term has been discussed in connection with Lorentz violation in electromagnetism. ’33 dFirst reported at a symposium in honor of P. Ramond, University of Florida, Jan. 2003, and at another in honor of O.W. Greenberg, University of Maryland, May 2003.

6

avoid the constraints of the uncertainty principle. Consider for example the following system:

+ +

H = Ho(p) V ( Z ) , Ho d x l d t = 3ap3 2bp,

= ap4

+ bp2 +

C,

dpldt = - d V / d x .

(10)

Qualitatively HO is similar to Eq. (9) if a > 0, b < 0. To have a clear spacetime picture, one would like to go back to the corresponding Lagrangian L = p i - H by solving p in terms of i.But 5 is not a single-valued function of p , and it has in general three real solutions. In other words, to a Hamiltonian given in the coordinates and the momenta does not correspond a unique Lagrangian in the coordinates and the velocities. This is at odds with the usual Newtonian causality, according to which the position and the velocity of a particle at a given time uniquely determines its motion at all times. I will call this a Newton anomalv. ~

~~~ ~~

-

~

o = ((p2 - ~ ~ ~ ) 2 / 4 rAn2)"2 *~+

I -4

VCO

-2

jo

- pF v>o I

2 p

VCO

pF v>o

Figure 2.

The behavior (9) of the BV quasiparticle spectrum w is shown in Fig. 2. There are four regions, I through IV, depending on the signs of the pair ( p , v ) . Regions I and IV correspond to the outside of the Fermi surface where the quasiparticle is mostly in the electron state $ ( p ) (la1 > lpl) ... . . .. . -- ___ with positive klnetlc mass, whereas regions 11 and 111 correspond to the inside where it is mostly in the hole state $ ( - p ) t (la1 < IpI) with negative kinetic mass. There is a degeneracy of energy in the range between A and

JA2

+~ ; / 4 r n * ~ .

7

Figure 3.

z(t) under constant force -g z = c - o(p), p = - g t

IF----

‘WP) = WI

+

Wll

Figure 4.

+

Wlll

+

WlV

8

It is interesting to follow the motion of a BV particle as a classical partice in the field of weak uniform gravity.e Given the kinetic part of the Hamiltonian Ho, the equivalence principle tells us that V = Hogz/c2 for nonrelativistic systems. The qualitative behavior, however, may be gleaned by taking V to be of the simple form V = Xz, with some constant A. The momentum p , keeps decreasing with time. Starting from a sufficiently large positive value outside of the Fermi surface, the particle component $ ( p ) will go through the Fermi sphere smoothly downwards, whereas the hole component $ ( - p ) t will move upwards (Fig. 3). In real space the motion is more interesting (Fig. 4). If one tosses the BV particle upwards with sufficiently large velocity, it rises up to a maximum height zmaX= z , where p = p~ but v = 0 , then starts falling down, stops once at zmin when p = 0 and v = 0 , rises up again to z, where p = - p ~ ,2, = 0 , and thereafter falls steadily. The violation of Newtonian causality is seen by noting that if the initial upward velocity is within the values between z,in and zm,, one cannot be sure if it is on the first upward swing or the second one.f In reality, the gravity is too weak to be able to exhibit the whole trajectory for a BV quasiparticle. But going back to quantum mechanics, the wave spatial function $ ( z ) between z,, and Zmin will consist of components from four regions, two corresponding to states below the Fermi surface, and two from above. 4. Speculative Remarks

In Section 3 a model of Lorentz symmetry violation in the vacuum by an axial vector was considered in analogy with ferromagnetism. In the real world I speculate that this could be a possibility, for example, for the neutrinos since they are the least known sector of the Standard Model. The violation occurs in the space component (in a certain Lorentz frame), and the vacuum has a spontaneous ‘magnetization’ of non-electromagnetic nature (C-even), but it could be also in the time component, in which case the vacuum has a net (non-spontaneous) chirality, perhaps due to the presence of a chemical potential. eGravity is considered because of its universal nature. One has t o assume that the medium is held against gravity, but the quasiparticles carry extra energy and should be able to respond to gravity freely. In real systems there will be complications such as the change of the density of the medium with the height. These effects are ignored. fOne could say that in the case of the BV paticle the nonuniqueness is due to the neglect of the spin degree of freedom in the classical description, but it is difficult to apply a similar argument in the case of the Landau roton.

9

hypothetical beta decay spectrum near the end point under an external chiral charge

40-

',\

\ .'.\..

....

--....

^

................

.......

. . I

..............

... .

-

.........

.

....

.

.,. . . , .

electron energy Emax- E Figure 5.

hypothetical reaction pattern

Figure 6.

One of the obvious manifestations of the effects would be in the beta decay spectrum. The presence of the peculiar dispersion law for a neutrino would be reflected in the electron energy spectrum near the end point. As is illustrated in Fig. 5 , the degeneracy of the spectrum is responsible for an

10

enchancement towards the end. The fact that the momentum and velocity can be in opposite directions causes some paradoxical phenomena if the particle in question is involved in scattering or decay (Fig. 6 ) . There is an apparent lack of momentum conservation when an anomalous particle (A) is produced, or absorbed.

Acknowledgment.

This work was supported by the University of

Chicago.

References 1. H.B. Nielsen and S. Chadha, Nucl. Phys. 8105, 445 (1976). 2. Y. Nambu, J. Statist. Phys. 115, 7 (2004), and references therein. 3. D. Colladay and V.A. Kostelecki, Phys. Rev. D 55,6760 (1997); Phys. Rev. D 58,116002 (1998); V.A. Kosteleck9 and R. Lehnert, Phys. Rev. D 63,065008 (2001). 4. V.A. Miransky and LA. Shovkovy, hep-ph/0108178; Phys. Rev. Lett. 88, 11601 (2002). 5. D. Ebert, V.Ch. Zhukovsky, and A.S. Razmovsky, hep-ph/0401241 v l . 6. D. Blaschke, D. Ebert, K.G. Klimenko, M.K. Volkov, and V.L. Yudichev, hep-ph/0403151. 7. S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990).

TESTS OF CPT AND LORENTZ SYMMETRY USING HYDROGEN AND NOBLE-GAS MASERS

RONALD L. WALSWORTH Haward-Smithsonian Center for Astrophysics Cambridge, M A 02138, U.S.A. We discuss two recent measurements constraining CPT and Lorentz violation using the 129Xe/3He Zeeman maser and atomic hydrogen masers. Experimental investigations of CPT and Lorentz symmetry provide important tests of the framework of the Standard Model of particle physics and theories of gravity. The two-species 129Xe/3He Zeeman maser sets stringent limit on rotation- and boost-dependent Lorentz and C P T violation involving the neutron, consistent respectively with no GeV. Measurements with atomic hyGeV and effect at the level of drogen masers provide a clean limit of rotation-violation of the proton at the GeV level.

1. Introduction Lorentz symmetry is a fundamental feature of modern descriptions of nature. Lorentz transformations include both spatial rotations and boosts. Therefore, experimental investigations of rotation and boost symmetry provide important tests of the framework of the Standard Model of particle physics and single-metric theories of gravity. Clock comparisons' provide sensitive tests of these symmetries by bounding the frequency variation of a given clock as its orientation changes, e.g., with respect to the fixed stars. In practice, the most precise limits are obtained by comparing the frequencies of two co-located clocks as they rotate with the Earth and as they revolve with the Earth around the Sun. Atomic clocks are typically used, involving the electromagnetic signals emitted or absorbed on hyperfine or Zeeman transitions. Here we discuss results from two recent atomic clock tests of CPT and Lorentz symmetry: (1) Using a two-species 12gXe/3HeZeeman maser we placed a limit on rotation-dependent Lorentz and CPT violation involving the neutron of GeV,2 improving by more than an order of magni11

12

tude on the best previous m e a ~ u r e m e n t . ~ With ? ~ the same device we performed the first clean test for the fermion sector of the symmetry of spacetime under boost transformations, placing a limit on boost-dependent Lorentz and CPT violation involving the neutron of GeV.5 (2) We employed atomic hydrogen masers to set an improved clean limit on rotation-violation of the proton, at the level of nearly GeV.6 2. Motivation

Our atomic clock comparisons are motivated by the Standard-Model Extension (SME) developed by Kosteleckf and other^.^ The SME parametrizes arbitrary coordinate-independent Lorentz violation. Since violation of CPT symmetry (the product of Charge conjugation, Parity inversion, and Time reversal) must come with Lorentz violation,8 the SME also parametrizes general CPT violation. Observable Lorentz and CPT violation could be a remnant of Planck-scale physics. One attractive origin is spontaneous Lorentz breaking in a fundamental t h e ~ r ybut , ~ other sources are possible." The SME provides a widely-accepted formalism for the interpretation and comparison of experimental measurements of Lorentz and CPT violation, and has been applied to many systems, including mesons, photons, and leptons. The atomic-clock comparisons presented here provide some of the most stringent tests of rotation and boost invariance, and hence of and the proton.6 Lorentz and CPT symmetry for the In particular, the Standard-Model Extension admits Lorentz-violating couplings of noble gas nuclei and hydrogen atoms to expectation values of tensor fields. (Some of these couplings also violate CPT.) Each of the tensor fields may have an unknown magnitude and orientation in space, to be limited by experiment.

3. 129Xe/3Hemaser test of CPT and Lorentz symmetry We provide here a brief review of the design and operation of the two-species 12gXe/3Hemaser. (See the schematic in Fig. 1.) Co-located ensembles of lZ9Xeand 3He atoms at pressures of hundreds of mbar are held in a doublechamber glass cell placed in a homogeneous magnetic field of 1.5 G. Both species have spin-1/2 nuclei and the same sign nuclear magnetic dipole moment, but no higher-order electric or magnetic nuclear multipole moments. In one chamber of the glass cell, the noble gas atoms are nuclear-spin-

-

13 Nested Magnetic Shtelds--;;r

H Maser Reference

Electric Reld Plates Enernal Resonator (T = 40 C)

Figure 1. Schematic of the 129Xe/3HeZeeman maser

polarized by spin-exchange collisions with optically-pumped Rb vapor." The noble gas atoms diffuse into the second chamber, which is surrounded by an inductive circuit resonant both at the 3He and 12'Xe Zeeman frequencies (4.9 kHz and 1.7 kHz, respectively). For a sufficiently high flux of population-inverted nuclear magnetization, active maser oscillation of both species can be maintained indefinitely. Due to the generally weak interactions of noble gas atoms with the walls and during atomic collisions, the 3He and 12'Xe ensembles can have long Zeeman coherence (7'2) times of hundreds of seconds. It is possible to achieve excellent absolute frequency stability with one of the noble-gas masers by using the second maser as a co-magnetometer. For example, Zeeman frequency measurements with sensitivity of 100 nHz are possible with averaging intervals of about an hour. This two-species noble gas maser can also serve as a sensitive NMR gyroscope:12the above quoted frequency stability implies a rotation sensitivity of 0.13 degree/hour. In the context of the SME, the neutron and hence the noble-gas masers are sensitive to Lorentz and CPT violation controlled by the coefficients b ~ , dAc, HAC, and g ~ c rof the SME.l We assume that these coefficients are static and spatially uniform in the Sun frame, at least over the course of a solar year. Thus, the frequencies of the noble-gas masers acquire a time dependence as a consequence of the Earth's rotation and its revolution around the Sun. In the Lorentz-symmetry test, the 12'Xe maser was phase-locked to a signal derived from a hydrogen maser in order to stabilize the magnetic field, which was oriented along the east-west direction. The leading LorentzN

14

violating frequency variation of the free-running 3He maser was given by: 6

u =~SUX~ sin weTe

+ Suy

cos weTe,

(1)

where

+,&(Ass sin!&T + AsccosS2~T)), buy = k (A, + ,Be(A,, sinReT A,, c o s R ~ T ) ) .

SUX

= k (A,

+

(2)

Here A, A,, A,,, A,,, ... are combinations of Sun-frame SME coefficients nHz/GeV.l mentioned above5 and k = -8.46 We note that Eqs. (1)and (2) cleanly distinguish the effects of rotation alone (terms proportional to A, and A,) from the effects of boosts due to the Earth's motion (terms proportional to A,,, A,,, A,,, A,,). These equations also indicate that the sensitivity of our experiment to violations of boost-symmetry is reduced by a factor of ,& 2 loe4 with respect to the sensitivity to rotation-symmetry violation.

Run mean date Figure 2. Time course of the mean values of bvx and bvy. For each plot the dashed line is the best fit obtained from Eq. (2), using the fit parameters Xc, As, Acc, A,,, As,, Ass. Dotted lines indicate the 10 confidence bands for the fit model.

As discussed in Refs. 2 and 5 we acquired noble-gas maser data in four different runs spread over about 13 months (see Fig. 2). Each run lasted about 20 days, and we reversed the direction of the magnetic field after the first 10 days in each run to help distinguish possible Lorentz-violating effects from diurnal systematic variations. We fit this data to Eq. (1). Figure 2 shows, for each run, the mean values we determined for dux and buy , N

15 Table 1. Bounds on 17 SME coefficients among the 44 coefficients describing possible leading-order Lorentz- and CPT-violating coupling of the neutron.

SME coefficients

GeV

the amplitudes of sidereal-day modulations of the 3He-maser frequency due to Lorentz-violating coefficients in the X and Y directions (Sun-centered frame). For each run, bux and buy correspond to a very good approximation to a single high-precision measurement of the X and Y components of 6 u performed ~ ~ at the run's mean time. Next, we fit the experimental values of bvx, buy to Eq. (2)) thus obtaining the fit shown graphically in Fig. 2, and the corresponding bounds on the SME coefficients of Table 1. We treated all fit parameters as independent and we extracted energy bounds for SME coefficients disregarding the possibility of accidental mutual cancellations. This analysis yielded no significant violation of rotation invariance with a limit of about 70 nHz on the magnitude of the daily sidereal variation in the 3He-maser frequency and no significant violation of boost invariance, with a limit of about 150 nHz on the magnitude of an annual modulation of the daily sidereal variation. We expect about an order of magnitude improvement in sensitivity to Lorentz/CPT violation of the neutron using a reengineered version of our '29Xe/3He maser. The new device has been designed to improve the medium term stability of the gas masers which limits the current sensitivity. Improved temperature control of the pump and maser regions, better co-magnetometry and the use of a narrow spectrum laser for optical pumping should help achieve this goal. Further improvements in senitivity may be possible with a 21Ne/3He Zeeman maser,I3 with masers located on a rotating table, or with space-based c10cks.l~ 4. Hydrogen maser test of CPT and Lorentz symmetry

Hydrogen masers operate on the A F = 1, AmF = 0 hyperfine transition in the ground state of atomic hydrogen.15 Hydrogen molecules are dissociated into atoms in an RF discharge, and the atoms are state selected via a hexapole magnet (Fig. 3). The high field seeking states, ( F = 1, m F = +1,0) are focused into a Teflon coated cell which resides in a microwave

16

cavity resonant with the AF = 1 transition at 1420 MHz. The F = 1, m F = 0 atoms are stimulated to make a transition to the F = 0 state by the field of the cavity. A static magnetic field of 1 milligauss is applied to maintain the quantization axis of the H atoms. N

Figure 3.

Schematic of the H maser in its ambient field stabilization loop.

The hydrogen transitions most sensitive to potential CPT and Lorentz violations are the F = 1, AmF = f l Zeeman transitions. In the 0.6 mG static field applied for these measurements, the Zeeman frequency is uz x 850 Hz. We utilize a double resonance technique to measure this frequency with a precision of 1 ~ H z . ' We ~ ? apply ~ ~ a weak magnetic field perpendicular to the static field and oscillating at a frequency close to the Zeeman transition. This audio-frequency driving field couples the three sublevels of the F = 1 manifold of the H atoms. Provided a population difference exists between the mF = f l states, the energy of the mF = 0 state is altered by this coupling, thus shifting the measured maser frequency in a carefully analyzed mannerl6?l7described by a dispersive shape (Fig. 4a). Importantly, the maser frequency is unchanged when the driving field is exactly equal to the Zeeman frequency. Therefore, we determine the Zeeman frequency by measuring the driving field frequency at which the maser frequency in the presence of the driving field is equal to the unperturbed maser frequency. N

17

We employ an active stabilization system to cancel external magnetic field fluctuations (Fig. 3). A fluxgate magnetometer placed within the maser’s outer magnetic shield controls large (2.4 m dia.) Helmholtz coils surrounding the maser via a feedback loop to maintain a constant ambient field. This feedback loop reduces the fluctuations at the sidereal frequency to below the equivalent of 1 pHz on the Zeeman frequency at the location of the magnetometer. The Zeeman frequency of a hydrogen maser was measured for 32 days. During data taking, the maser remained in a closed, temperature controlled room to reduce potential systematics from thermal drifts which might be expected to have 24 hour periodicities. The feedback system also maintained a constant ambient magnetic field. Each Zeeman measurement took approximately 20 minutes to acquire and was subsequently fit to extract a Zeeman frequency (Fig. 4a). Also monitored were maser amplitude, residual magnetic field fluctuation, ambient temperature, and current through the solenoidal coil which determines the Zeeman frequency (Fig. 3).

N

I

6m

E -2 854

PN

856

a58

860

Zeeman drive frequency (Hz)

0.10

0.00

-0.10

4%

96 144 192 240

time (hours)

Figure 4. (a) An example of a double resonance measurement of the F = 1, AmF = f l Zeeman frequency in the hydrogen maser. The change from the unperturbed maser frequency is plotted versus the driving field frequency. (b) Zeeman frequency data from 11 days of the Lorentz/CPT test using the H maser.

The data were then fit to extract the sidereal-period sinusoidal variation of the Zeeman frequency. (See Fig. 4b for an example of 11days of data.) In addition to the sinusoid, piecewise linear terms (whose slopes were allowed to vary independently for each day) were used to model the slow remnant drift of the Zeeman frequency. No significant sidereal-day-period variation of the hydrogen F = 1,Amp = f l Zeeman frequency was observed, setting a bound on the magnitude of such a variation of S v g 5 0.37 mHz (onesigma level).

18

In the context of the SME, the H maser measurement constrains CPT and Lorentz violations of the proton parameter &$ 5 2 GeV at the one sigma level. We expect that the sensitivity of the H maser Lorentz/CPT test can be improved by more than an order of magnitude through technical upgrades t o the maser’s thermal and magnetic field systems, better environmental control of the room housing the maser, and a longer period of data acquisition.

5. Conclusions Precision comparisons of atomic clocks provide sensitive tests of CPT and Lorentz symmetry, thereby probing extensions to the Standard Model in which these symmetries can be spontaneously broken. Measurements using the two-species 12gXe/3HeZeeman maser constrain rotation-violation of the GeV level and boost-dependent violations at the neutron at the GeV level. Measurements with atomic hydrogen masers provide clean tests of rotation-violation of the proton at the GeV level. Improvements in both experiments are being pursued.

Acknowledgments

I gratefully acknowledge my collaborators on the work described above: David Bear, Federico Can& Marc Humphrey, Alan Kosteleck?, Charles Lane, Edward Mattison, Matthew Rosen, David Phillips, Chris Smallwood, Richard Stoner, and Robert Vessot. Support for the Lorentz/CPT violation tests was provided by NASA and the Smithsonian Institution. References 1. V. A. Kostelecki and C. D. Lane, Phys. Rev. D 60, 116010/1-17 (1999). 2. D. Bear, R. E. Stoner, R. L. Walsworth, V. A. Kostelecki, and C. D. Lane, Phys. Rev. Lett. 85, 5038-5041 (2000); ibid., 89, 209902 (2002). 3. C. J. Berglund, L. R. Hunter, D. Krause, Jr., E. 0. Prigge, M. S. Ronfeldt, and S. K. Lamoreaux, Phys. Rev. Lett. 75, 1879-1882 (1995). 4. L. R. Hunter, C. J. Berglund, M. S. Ronfeldt, E. 0. Prigge, D. Krause, Jr., and S. K. Lamoreaux, “A Test of Local Lorentz Invariance Using Hg and Cs Magnetometers,” in CPT and Lorentz Symmetry, edited by V. A. Kostelecki, World Scientific, Singapore, 1999, pp. 180-186. 5. F. Can$, D. Bear, D.F. Phillips, M.S. Rosen, C.L. Smallwood, R.E. Stoner and R.L. Walsworth, submitted to PRL (physics/O309070). 6. D. F. Phillips, M. A. Humphrey, E. M. Mattison, R. E. Stoner, R. F. C. Vessot, and R. L. Walsworth, Phys. Rev. D 63,111101-111104 (2001).

19

7. D. Colladay and V.A. Kosteleckf, Phys. Rev. D 55,6760 (1997); 58,116002 (1998); V.A. Kosteleckf and R. Lehnert, Phys. Rev. D 63, 065008 (2001); V.A. Kosteleckf, Phys. Rev. D 69,105009 (2004). 8. O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002); Phys. Lett. B 567, 179 (2003). 9. V.A. Kosteleckf and S. Samuel, Phys. Rev. D 39, 683 (1989); Phys. Rev. Lett. 63,224 (1989); Phys. Rev. D 40,1886 (1989). V.A. Kosteleckf and R. Potting, Nucl. Phys. B 359,545 (1991); Phys. Rev. D 51,3923 (1995). 10. For reviews of approaches to Lorentz and CPT violation, see, for example, V.A. Kosteleckf, ed., CPT and Lorentz SymmetnJ I, 11, World Scientific, Singapore, 1999, 2002. 11. T.E. Chupp et al., Phys. Rev. A 38,3998 (1988); G.D. Cates et al., Phys. Rev. A 45,4631 (1992). 12. K.F. Woodman et al., J. Navig. 40,366 (1987). 13. R.E. Stoner and R.L. Walsworth, Phys. Rev. A 66,032704 (2002). 14. R. Bluhm et al., Phys. Rev. Lett. 88, 090801 (2002). 15. D. Kleppner, H. M. Goldenberg, and N. F. Ramsey, Phys. Rev. 126,603-615 (1962); D. Kleppner, H. C. Berg, S. B. Crampton, N. F. Ramsey, R. F. C. Vessot, H. E. Peters, and J. Vanier, Phys. Rev. 138,A972-983 (1965). 16. H. G. Andresen, Z. Physik, 210, 113-141 (1968). 17. M. A. Humphrey, D. F. Phillips, and R. L. Walsworth, Phys. Rev. A 62, 063405-063405 (2000).

NEW TESTS OF LORENTZ INVARIANCE IN THE PHOTON SECTOR USING PRECISION OSCILLATORS AND INTERFEROMETERS

M.E. TOBAR,l P. WOLF,2i3P.L. STANWIX,l A. FOWLER,l S. BIZE,2 A. CLAIRON,~J.G. HART NETT,^ E.N. IVANOV,~F. VAN KANN,' G. SANTARELLI,' M. SUSLI,~J. WINTERFLOOD' University of Western Australia, School of Physics 35 Stirling Hwy., Crawley 6009 W A , Australia E-mail: [email protected]. edu.au 2BNM-SYRTE, Observatoire de Paris 61 Av. de lObservatoire, '75014 Paris, France E-mail: [email protected] Bureau International des Poids et Mesures Pavillon de Breteuil, 92312 St?vres Cedex, France. High precision microwave oscillators and interferometers offer a very sensitive means to test the photon sector of the Standard Model of Physics. This work summarizes ongoing and new experiments at various stages of development, which are analyzed within the general Lorentz-violating extension of the Standard Model of particle physics (SME). Best present limits of seven parameters are reported, and we outline how we will improve on these measurements and set upper limits on more parameters in future experiments.

1. Introduction

In the photon sector of the SME numerous experiments have already set limits on 1 7 of the 19 possible Lorentz-violating coefficients. Of these, the 10 coefficients that depend on the polarization have had upper limits set at parts in lo3' by astrophysical tests.'?' Cavity experiment^^^^^^^^ have set upper limits on 4 of the 5 polarization-independent even-parity coefficients, ii:-y, it:--, ii:--, (ii:-x - iiF-y), at parts in and the 3 polarization-independent odd-parity coefficients, ii?:, I;::, ii::, at parts in lo1' (reduced due to boost suppression of order In this work we summarize the results of experiments based on Cryo20

21

genic Sapphire Oscillators (CSO) developed in collaboration by the University of Western Australia (UWA) and the Bureau National de Mktrologie - SystBmes de Rkfkrence Temps Espace (BNM-SYRTE) at the Paris Ob~ervatory.~ Then we describe the initial operation and results of a rotating dual-cavity CSO developed at UWA, which has the potential to improve on the upper limits of the above seven parameters by up to several orders of m a g n i t ~ d e ,and ~ ? ~provide the possibility of setting the first upper limit to one of the parameters not yet tested (2:: = i5-x izPy). Finally we summarize recent work that is dedicated to measuring the odd-parity coefficients directly (with no boost suppression).lo The same experiments allow the first determination of an upper limit of the scalar coefficient 2tT, with boost suppression. Of the classic special relativity experiments the Ives-Stilwell (IS) experiments are shown to exhibit this property, and we have shown that the best experiment sets a limit of order parts in lo5. To improve on the limit by several orders of magnitude we propose a magnetic asymmetric Mach-Zehnder (MZ) interferometer.

+

2. Ongoing experiments with cryogenic sapphire oscillators at the Paris Observatory Table 1. Results for the components of the SME Lorentz v i e lation parameters A,- (in and ko+ (in

from this work

e+y e+"

14(14) -1.8(1.5)

-1.2(2.6) -1.4(2.3)

Cf

0.1(2.7) 2.7(2.2)

During June 2000 a CSO from UWA was transported to BNM-SYRTE at the Paris Observatory, and has been essentially operating continuously since November 2001.7 This has allowed the current best test of the constancy of the speed of light (Kennedy-Thorndike e ~ p e r i m e n t )and ~ . ~seven of the nineteen parameters of the SME.6 The experiment is achieved by the comparison of a CSO and a hydrogen maser, and relies on the velocity and rotation of the earth around the sun to modify the orientation and velocity of the experiment at the sidereal ( w e ) and the annual (0,) frequencies respectively. During the initial stage of the experiment uncertanties were dominated by temperature variations of the local environment, but since Sept. 2002 a temperature control system has made the systematics insignif-

22

icant. The results of a recent publication that describe the analysis of this experiment in the framework of the SME6 are summarized in Table 1. The data were taken between Sept. 2002 and Jan. 2004 with differing lengths of 5 to 20 days (222 days in total). The sampling time for all data sets was 100 s. We note that our results for ii$-y and ii$-z are significant at about 2c1 while those of Ref. 4 are significant at about the same level for x-:i( -kF-y). The two experiments gave comparable results for ii$-- (within the la uncertainties) but not for the other two parameters, so the measured values of those are unlikely to come from a common source. Another indication of a non-genuine effect comes from analysis of the phase and nearby frequencies to those of interest, in which no prominant features occur. Details of the analysis are found in Ref. 6. In conclusion, we have not seen any Lorentz-violating effects in the general framework of the SME, and we have set limits on 7 parameters of the SME photon sector (cf. Table 1). These limits are up to an order of magnitude more stringent than those obtained from previous experiment^.^ Two of the parameters are significant (at NN 20). We believe that this is most likely a statistical coincidence or a neglected systematic effect. To verify this, our experiment is continuing and new, more precise experiments are ~ n d e r w a y .Their ~ ? ~ progress is summarized in the following sections.

3. New high precision rotating experiment at UWA

Further order of magnitude improvements are not likely for the experiments referred to in the previous section. This is due to the already long data set and systematic error limit.6 Significant improvements in the near future are more likely to come from new proposals. We are constructing new experiments on a rotating platform. The first experiment began initial operation in June 2004 and uses two orthogonal cylindrical Whispering Gallery resonators. Improvement of several orders of magnitude is possible, as the relevant time variations are now at the rotation frequency ( w 0.01 - 0.1 Hz) which is the range in which such resonators are the most stable (- 100 fold better stability). Ultimately, it has been proposedll to conduct these tests on board an Earth orbiting satellite, with a potential gain of several orders of magnitudes over current limits. In the following we summarize the initial operation of the rotating e ~ p e r i m e n t . ~ The sapphire cylinders each have a diameter of 31.60 mm and height of 30.05 mm. They have a spindle at either end and are supported by both spindles within superconducting niobium cavities of internal diameter

23

49.80 mm and height 48.90 mm. The resonators are oriented on their sides orthogonal to each other perpendicular to the normal to the Earth’s surface. Both resonators are excited in the whispering. gallery WGHs,o,o mode by separate oscillators with frequencies of approximately 10.000224 GHz and 9.999998 GHz at 4.2 K. Straight probes mounted in each lid couple to the resonators. The difference frequency between the two resonators of approximately 226 kHz corresponds to a diameter mismatch between the crystals of less than 1 pm. A schematic of the experiment is shown in Fig. 1. The resonators are placed inside a small stainless steel vacuum cylinder,

Figure 1.

Schematic of the rotating expriment

which is sealed with Mylar gaskets. The air is pumped out and sealed off. The resonator is coupled to the loop oscillator (at room temperature) with stainless steel coaxial lines. The small vacuum cylinder is sealed inside a large stainless steel cylinder that is supported by the insert inside the dewar. The double vacuum provides thermal insulation to the sapphire loaded cavities. The cavities are thermally connected to the liquid helium bath via a copper post that joins a stainless steel post connected to a triangular plate. This plate is bolted to three copper heat sinks which protrude into the liquid helium bath. The stainless steel post provides some thermal filtering of bath temperature fluctuations. A foil heater and a germanium

24

temperature sensor are attached to the copper post just above the cavity between the two vacuum cylinders to control the set point of operation. The temperature controller used is a Neocera LTC-21, with nominal mK stability. The cryogenic dewar containing the insert is suspended within a ring bearing and hangs below the floor level of the laboratory into a concrete pit. To avoid flexing of cables in the oscillator loop, the oscillator circuits are mounted on the suspended dewar, along with the control electronics. A detailed description of oscillator circuit operation and control systems can be found in Ref. 12. The suspension arrangement used is multiple ‘V’ shaped loops of elastic shock cord (‘bungee’ cord). The reason for using ‘V’ shaped loops over simple vertically stretched cord is to avoid high Qfactor pendulum modes by ensuring that the cord has to stretch and shrink (providing damping losses) for horizontal motion as well as vertical. A table mounted on the bearing carries peripheral equipment such as the temperature controller and power supply. The rotation system is driven by a microprocessor controlled stepper motor, with the rotation controlled by a computer. A rotating connector located at the bottom of the dewar allows the boil off from the liquid helium bath to be collected and re-liquified. This connector is mounted at the bottom of the pit on a scissor action bracket, allowing the connector to move vertically with the dewar as the helium boils off. A combination of two rotating electrical connectors is located above the experiment. A commercial 18 conductor slip ring connector with a hollow through bore transfers electrical power to the rotating experiment. Systematic effects including the temperature of the resonators, liquid helium bath level, ambient room temperature, oscillator control signals and tilt are monitored and transferred to the stationary data acquisition system via the same connector. The second mercury based coaxial rotating connector is mounted inside the larger slip ring connector in the same axis of rotation. The sole purpose of the connector is to transmit the beat signal to the stationary frequency counter and data acquisition system. Frequency stability results of the first experimental operation are shown in Fig. 2. Systematic effects associated with the rotation of the experiment can be seen up to 20 seconds. To evaluate the intrinsic stability the systematics were removed from the data using a least squares approach by fitting cosines at the rotation frequency and its harmonics. Comparison with the raw data suggests this is the source of the short term instabilities. Currently the stability is limited by temperature fluctuations. It has been difficult to maintain the operation of the resonators at the beat frequency

25

2-

1oo

1o2 Tau (s)

10'

1o3

1o4

Figure 2. Square Root Allen Variance (SRAV) frequency stability measurement. The hump at short integration times is due to systematic effects associated with the rotation of the experiment. These have been removed and the SRAV replotted for comparison.

temperature turning point due variations in the microwave power incident on the resonators and the high dependence of the individual resonators turning points on power. Currently, power is monitored at room temperature, which does not account for additional power variations caused by varying cable losses as the level of the liquid helium bath changes. These power variations result in a change in the temperature turning point of the beat frequency and hence the optimal temperature set point that minimizes frequency instabilities due to temperature fluctuations. This will be overcome by placing the power detectors for the power control system in the cryogenic environment, as close as possible to the resonator. The data from the experiment is essentially a list of time stamped beat frequency measurements. A Lorentz-violating signal is expected to show up as a periodic variation in the beat frequency as the experiment rotates. The amplitude and phase is determined by simultaneously fitting the parameters of Eq. (1) to the data. A and B determine the frequency offset and drift, while Ci and Si are the amplitude of a cosine and sine at frequency wi respectively. These parameters are calculated using the ordinary least squares technique.

Af = A + B.t + c ( C i cos(wit) + Si sin(wit)) -

f

(1)

i

First operation of the experiment produced a data set approximately 16

26

I---

Harmonics

3 8 -0

.-=) Y

F2

6

1

0'

I

I. ' 1

I

-7m

0.2

c

I

. Ic

I

0.4

. .- -"

r. I_ .,,p

I

..,.

0.6

-

r-

,

,,

I

0.8

Frequency (radk) Figure 3. Spectrum of amplitudes calculated using least squares, showing the level of systematics and noise floor from initial 16 hr data set.

hrs long. This data was analyzed using the approach outlined above. A spectrum of resulting amplitudes is shown in Fig. 3. A spike at the rotation frequency can clearly be seen, as well as its first and second harmonics. These are systematic effects associated with the rotation of the experiment and most likely due to magnetic field and tilt variations. It is possible to circumvent these systematics by looking for the sidereal and twice sidereal sidebands about twice the rotation frequency. The average amplitude of these four sidebands was calculated using weighted least squares to be 5.52(f5.32)x indicating that this experiment is able to provide a better order of resolution as Ref. 6 with one day of data. Improvements will be realized with longer data sets and improved frequency stability. 4. Proposals to improve limits on odd parity and scalar

parameters of the SME Recently we have focused on improving the limits of the odd parity coefficients by investigating experiments that have direct sensitivity and are not suppressed by the boost dependence.1° Also, we have shown that the same experiments allow the scalar coefficient to be determined via boost dependence. Of the experiments undertaken to date, IS experiments have the required properties, and by analysing the best experiment we have provided a first upper limit of parts in lo5 for the scalar coefficient." Furthermore, we have shown that a magnetically asymmetric MZ interferometer (mi-

27

crowave or optical) may provide a null experiment that is sensitive to the same SME parameters as the IS experiment. We have proposed recycling techniques to further enhance the sensitivity and have shown that the respective sensitivity to the odd parity and scalar coefficients are possible at parts in 1015 and 10l1 with current technology (Magnetostatic experiments have also been proposed to achieve a similar ~ensitivity.'~) The basic MZ interferometer with power and resonant recycling is shown in Fig 4. If the

-L-

Figure 4. Schematic of a MZ interferometer with two arms of permeability p r a and Resonant recycling with a travelling wave resonator is shown in arm a , and powerrecycling is demonstrated by feeding back the Bright Port (BP) of the interferometer to the input. The Dark Port (DP) remains the phase sensitive output.

prb.

interferometer arms labeled b, as shown in Fig. 4 contain vacuum, then the sensitivity of an odd parity Lorentz violation (with signal to noise ratio of one) will be

where N , is the number (or fraction) of cycles in one second, A, is the wavelength in free space, T& is the observation time, R is the power recycling factor, N is the resonant recycling factor and is the square root spectral density of phase noise, which is conservatively of order lo-' rads/&.14i15 For a rotating 10 GHz interferometer of order one meter long with a recycling factor of N 1 = 100 and R 1 = 100, the estimated sensitivity from (2) is of the order 2x for N , = 0.05 (20 second rotation period). Thus, a sensitivity of order is possible with only

+

+

28

450 seconds of data. For a non-rotating experiment N , = 1 . 1 5 7 ~ 1 0 -(one ~ day rotation period), a sensitivity of order is possible with 22.5 days and Rt,. of of data. This translates to sensitivity to R,+ of order order A further benefit of rotating over non-rotating experiments is direct sensitivity to all three odd-parity polarization-independent coefficients, as non-rotating experiments only allow two of the three coefficients to be tested directly.1° These ideas have also been extended to asymmetric resonant structures and possible resonator designs have also been proposed.1° The only way to accurately calculate the sensitivity of such experiments is through numerical simulation, and we will pursue this path in the future. Future work will concentrate on studying the detailed experimental feasability of each interferometer and resonator proposals, with the aim of realising such an experiment within the next few years.

Acknowledgments Helpful discussions with Alan Kosteleck$ and partial funding by the Australian Research Council are gratefully acknowledged.

References 1. V.A. Kostelecki and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001). V.A. Kosteleckjr and M. Mewes, Phys. Rev. D66, 056005 (2002). J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003). H. Miiller et al., Phys. Rev. Lett. 91, 020401 (2003). P. Wolf et al., Gen. Rel. Grav. 36, 2351 (2004). P. Wolf et al., Phys. Rev. D70, 051902 (2004); hep-ph/0407232. P. Wolf et al., Phys. Rev. Lett. 90, 060402, (2003). M.E. Tobar, J.G Hartnett, J.D. Anstie, Phys. Lett. A300, 33, (2002).

2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

14. 15.

P.L. Stanwix et al., Proc. IEEE Int. UFFC 50th Anniversary Conf., August 23-27, Montreal (2004). M.E. Tobar et al., submitted to Phys. Rev. D (2004); hep-ph/0408006. C. Lammerzahl et al., Class. Quantum Grav. 18, 2499 (2001). C.R. Locke, et al., Proc. Joint Meeting of IEEEIFCS/EFTF, 350, (2003). Q.G. Bailey and V.A. Kosteleckjr, submitted to Phys. Rev. D (2004); hepph/0407252. E.N. lvanov, M.E. Tobar and R.A. Woode IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 45, 1526 (1998). E.N. Ivanov and M.E. Tobar IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 49, 1160 (2003).

LORENTZ VIOLATION AND NEUTRINOS

MATTHEW MEWES Department of Physics and Astronomy, Carleton College, One North College Street, Northfield, MN 55057 Neutrino oscillations provide an opportunity for sensitive tests of Lorentz invariance. This talk reviews some aspects of Lorentz violation in neutrinos and the prospect of testing Lorentz invariance in neutrino-oscillation experiments. A general Lorentz-violating theory for neutrinos is discussed, and some signals of Lorentz violation are identified.

1. Introduction Neutrinos offer a promising avenue for the detection of new physics. Evidence for neutrino oscillations already indicates that the minimal Standard Model (SM) of particle physics needs modification.' The experiments providing this evidence are in an excellent position to detect tiny violations of Lorentz invariance2 that may exist as the low-energy the remnants of Planck-scale physic^.^ Here we discuss a general theoretical framework describing the free propagation of neutrinos in the presence of Lorentz violation. We examine the effects of Lorentz violation on neutrino oscillations and identify unconventional behavior and experimental signals. At attainable energies, violations of Lorentz invariance are described by a framework called the Standard-Model Extension (SME).4 While the SME was originally motivated by string t h e ~ r yit, ~also encompasses other origins for Lorentz violation such as spacetime varying coupling^.^ The SME provides the basis for a large number of experiments.6 In neutrinos, it gives a consistent theoretical framework for the study of Lorentz violation in oscillations and other phenomena. Neutrino-oscillation experiments provide sensitivity to Lorentz-violating that rival the best tests in any other sector of the SME.I1>I2 Remarkably, the current evidence for neutrino oscillations lies at levels where Planck-suppressed effects might be expected to appear. Furthermore, the possibility remains that Lorentz violation may be responsible at least in part for the observed oscillation^.^^^^^ Further analysis and experimentation 29

30

is needed to determine the extent to which Lorentz violation may play a role in neutrino oscillations. 2. Framework

In the SME, the propagation of neutrinos is governed by a modified multigeneration Dirac equation:2

(irfkB8v - M A B ) v B = 0

(1)

j

where three neutrino fields and their charge conjugates are included in order to allow for general Dirac- and Majorana-type terms; V A = {ve,vp,vT,v,", ,:v v,"}. Each of the quantities &J and MAB are 4 x 4 constant matrices in spinor space. Here we have included all terms arising from operators of renormalizable dimension, but in general, higher derivative terms can occur13 and may be i r n ~ 0 r t a n t . lIt ~ is also straightforward to include additional generations in order to accommodate sterile neutrinos. Common Lorentz-conserving scenarios exist as subsets of the general case. The matrices I'zB and MAB can be decomposed using the basis of y matrices. Following standard convention^,^^^^ we define J

Y ~ A+ cyByp B +d y B ~ 5 + ~ p efk+ ~ i.f;~~+ 5 igzaxp ,

~ :=B

MAB := m A B

+ i m 5 A B y 5 + a/"AB^(CL+ b2B'%Y5Y@ + i H Y B f f p W

.

(2)

In these equations, the masses m and m5 are Lorentz and CPT conserving. The coefficients c, d , H are CPT conserving but Lorentz violating, while a, b, e, f, g are both CPT and Lorentz violating. Requiring hermiticity of the theory imposes the conditions r f k B = y0(rLA)+y0and MAB = y O ( M ~ ~ ) twhich y O , implies all coefficients are hermitian in generation space. Equation (1) provides a basis for a general Lorentz- and CPT-violating relativistic quantum mechanics for freely propagating neutrinos. Construction of the relativistic hamiltonian is complicated by the unconventional time-derivative term, but this difficulty may be overcome in a manner similar to that employed in the QED e~tensi0n.l~ The result is

'FI =

x0- +(yo6r03-Io+ ~ o y o 6-ryo(i6rQj o) -6

~ ,)

(3)

where Ho = -yo(iyj8j - M o ) is the general Lorentz-conserving hamiltonian, Mo is the Lorentz-conservingpart of M , and 6I', bM are the Lorentzviolating parts of I', M .

31

A general treatment is possible but rather cumbersome and beyond the intended scope of this work. Therefore, we consider a simple physically reasonable case where oscillation between left- and right-handed neutrinos is highly suppressed. The resulting theory describes oscillations between three flavors of left-handed neutrinos due to mass or coefficients Lorentz violation. Within this restriction, a calculation gives a 6 x 6 effective hamiltonian describing the time evolution of active neutrinos and antineutrinos with momentum @:2

where v, and pa represent active neutrino (negative helicity) and antineutrino (positive helicity) states, and indices a,b range over { e , p , T } . The effective hamiltonian is given by

+’(

[(aL)’pp - (CL)pVP&]ab -id%p(f+)v[(gpVupu

id&,(f+)~[(gpuupu

+ H’””)C];b

[-(aL)’lp,

-

-

H

(cL)”vp,pV]~b

(5) This result assumes relativistic neutrinos with momentum 14 much larger than both mass and Lorentz-violating contributions. At leading order, the four momentum p , may be taken as p , = (Id;-3, and a suitable choice for ( E + ) ~ is ( E + ) ~ = & ( O ; P l iiz), where P I , i z are real and {g/lfl,Pl,i2} form a right-handed orthonormal triad. The above hamiltonian is consistent with the standard seesaw mechanism, where the right-handed Majorana masses are much larger than Dirac or left-handed Majorana masses. However, the above equations apply to any situation where only left-handed neutrinos are allowed to propagate or intermix. Only the first term in Eq. ( 5 ) arises from the minimal Standard Model. The second term corresponds to the usual massive-neutrino case without sterile neutrinos. The leading-order Lorentz-violating contributions are given by the last term. Lorentz-violating v H v mixing is controlled by the coefficient combinations (cL):; = (c+d)$’ and ( a ~ ) 5: ~(a+b):b. The remaining coefficients, (gPVuC),band (HpvC)ab,arise from gauge-violating Majorana-like couplings and generate Lorentz-violating v ++ p mixing resulting in lepton-number violations. Note that some combinations of coefficients are unobservable, either because of symmetries or because they can

+

32

be removed through field redefinition^.^^^^^^^^^^ Although this theory is observer independent and therefore independent of choice of coordinates, it is important to specify a frame for reporting experimental results. By convention this frame is taken as a Sun-centered celestial equatorial frame with coordinates { T ,X , Y,Z}.12117

3. Features A complete analysis of this construction is hampered by its generality and lies outside our present scope. Two lines of attack have been initiated in order to understand the theoretical and experimental implications of Lorentz violation.2 The first involves the construction of simple models that illustrate the various unconventional features and their potential to explain experimental data. Some possibilities are considered in the next section. An alternative strategy is to search for ‘smoking-gun’signals that are indicators of Lorentz violation. The many coefficients for Lorentz violation that appear in the effective hamiltonian (5) introduce a plethora of new effects, including unusual energy dependence, dynamics dependent on the direction of propagation, and neutrino-antineutrino mixing. Below we list six classes of model-independent features that represent characteristic signals of Lorentz violation in neutrino-oscillation experiments. A positive signal in any one of these classes would suggest the presence of Lorentz violation. Spectral anomalies. Each of the coefficients for Lorentz violation introduces energy dependence differing from the usual mass case. In the conventional massive-neutrino case, oscillations of neutrinos in the vacuum are determined by the energy-independent mixing angles 812, 613, 823 , phase 6, and mass-squared differences bm, Am. In this case, energy dependence enters the oscillation probabilities through the oscillation lengths LO 0; E/bm2,E / A m 2 . In contrast, coefficients for Lorentz violation can cause oscillation lengths that are either constant or decrease linearly with energy. For example, a simple model with only C L coefficients has much of the same structure as the mass case except that it has oscillation lengths Lo 0; ( E b c ~ ) -( ~E ,A c L ) - ~Combinations . of coefficients with different dimension can lead to very complex energy dependence in both the oscillation lengths and the mixing angles. Detection of a vacuum oscillation length that differs from the usual 0: E dependence or of energy dependence in the vacuum mixing angles would constitute a clear signal of Lorentz violation. L-E conflicts. This class of signal refers to a set of null and positive

33

measurements that conflict in any scenarios based on mass-squared differences. In the usual case, baseline and energy dependence enter through the ratio L/Lo o< L I E . So experiments that measure the same oscillation mode at similar ranges in L I E will have comparable sensitivity to neutrino oscillations. Because of the unusual energy dependence, in Lorentz-violating scenarios this may no longer be the case. If oscillations are caused by coefficients for Lorentz violation, it is possible that experiments operating in the same region of L I E space could see drastically different oscillation probabilities. A measurement of this effect would indicate physics beyond the simple mass case and would constitute a possible signal of Lorentz violation. Periodic variations. This signal indicates a violation of rotation invariance and would commonly manifest itself as either sidereal or annual variations in neutrino flux. The appearance of @inthe effective hamiltonian (5) implies that oscillations can depend on the direction of the propagation. In terrestrial experiments, where both the detector and the source are fixed relative to the Earth, the direction of the neutrino propagation changes as the Earth rotates. This can lead to periodic variations at the sidereal frequency w e N 27r/(23 h 56 min). For solar neutrinos, the variation in propagation of the detected neutrinos is due to the orbital motion of the Earth and can cause annual variations. Compass asymmetries. This class includes time-independent effects of rotation-invariance violations. They consist of unexplained directional asymmetries in the observed neutrino flux. For terrestrial experiments, averaging over time eliminates any sidereal variations, but may leave a dependence on the direction of propagation as seen from the laboratory. This can result in asymmetries between the compass directions north, south, east, and west. Neutrino-antineutrino mixing. This class includes any measurement that can be traced to u ij oscillations. This would indicate leptonnumber violation that could be due to g and H coefficients. All of these coefficientsintroduce rotation violation, so this signal may be accompanied by direction-dependent signals. Classic C P T test. This is the traditional test of CPT involving searches for violations of the relationship Pvb+va( t ) = Pfia.+Qb(t). This equation holds provided CPT is unbroken. An additional result holds in the event of lepton-number violation: Pvbzfia ( t )= P v a z f i(bt ) ,if CPT is unbroken. A measurement that contradicts either of these relations is a signal of CPT violation and would therefore imply Lorentz violation.

-

34

4. Illustrative models

In this section, we discuss some simple subsets of the general case ( 5 ) that exhibit some of the unconventional effects. While in most cases these models are not expected to agree with all existing data, they do provide useful insight into the novel behavior that Lorentz violation can introduce. An interesting open challenge is to identify general classes of realistic models that could be compared to experiment. The bicycle model7 and its variants offer possibilities that have no mass-squared differences and few degrees of freedom.

4.1. Fried-chicken models One simple class of models are those dubbed ‘fried-chicken’ (FC) models. The idea behind these is to restrict attention to direction-independent behavior by only considering isotropic coefficients. This restriction reduces the effective hamiltonian to (heE)zF = diag[(fi2/(2E)

+ ( a L ) T - $(‘LlTTE)&

(fi2/(2E)

-

(uL)T

-

,

$(cL)TTE):b]

*

(6)

A majority of the Lorentz-violating models considered in the literature are subsets of this general FC model.’ The differences in energy dependence between the various types of coefficients and mass is apparent in Eq. (6). FC models provide a workable context for studying the unconventional energy dependence without the complication of direction-dependent effects. However, it should be noted that Eq. ( 6 ) is a highly frame dependent. Isotropy in a given frame necessarily implies anisotropy in other frames boosted with respect to the isotropic one. While it may be appealing to impose isotropy in a frame such as the cosmic-microwave-background frame, it is difficult to motivate theoretically. 4.2. Vector models

In contrast to FC models, vector models are designed to study the effects of rotation-symmetry violation. These models contain coefficients that can be viewed as three-dimensional vectors that point in given directions. They are particularly useful in determining the types of signals that a given experiment might expect to see if rotation symmetry is violated. As an example consider a model where only the coefficients ( a ~ ) : ; , (UL);;, ( C L T) X ~ ;, and ( C L ) ; ~are nonzero. Each of these can be viewed as

35

vectors lying in the Earth’s equatorial plane. They are chosen to illustrate the periodic signals discussed in the previous section. With the above choice, we would see maximal mixing between up ++uT and Vp * VTl which are relevant oscillation modes for atmospheric neutrinos. So, this simple special case may serve as a test model for searches for sidereal variations in atmospheric neutrinos. The vacuum oscillation probability for a terrestrial experiment is

Pv,++v, = sin2 L (As)pT sinweTB

+

(AC)pT

C O S W ~ T, ~ )

(7)

where

( A ~= )- f i~x ((aL)ET ~ -2 ~ ( c L ) r ~ )f i y ( ( a L ) i T - 2 ~ ( c L ) r.~(9) ) Here N X and N Y are factors that are determined by the direction of the neutrino propagation as seen in the laboratory. In this example, both the unusual energy dependence and the sidereal variations are readily apparent. The dependence on beam direction through N X and N Y implies that a time average in this model also gives rise to compass asymmetries. These could be sought in atmospheric experiments and other experiments where neutrinos originate from different compass directions.

4.3. The bicycle model One class of interesting special cases are those that involve a Lorentzviolating seesaw mechanism. The resulting dynamics can be dramatically different than what is naively expected from the effective hamiltonian (5). One such model is the bicycle model.7 This model is also interesting because it crudely matches the basic features seen in solar and atmospheric neutrinos using only two degrees of freedom. The bicycle model consists of an isotropic C L with nonzero element ~ ( c L ) : : = 2; > 0 and an anisotropic aL with degenerate nonzero real elements (aL)& = = i=i/fi. The vacuum oscillation probabilities are

8 sin2(A31L/2) , = p”,-”, = 2sin2 8cos2 8 ~ i n ~ ( A 3 ~ L /,2 )

P,,+,, Pv,-vr

= 1 - 4 sin2 6 cos’

P”,+”,

= p”,-”, = 1 - sin2 Bsin2(A2,L/2) -

sin28cos2Bsin2(A31L/2) - cos2Bsin2(A32L/2)

,

36

Pvrcv,

= sin’

Osin2(A~lL/2) -

sin2Ocos’ Osin2(A31L/2)

+ cos’ Osin2(A32L/2)

, (10)

where A21 = J(EE)’

+ (6cos @)’ + EE

A31 = 2d(EE)’ a32

,

+ (6 cos Q)’ ,

+

= J ( E E ) ~ (iicoso)2 - EE

+

,

sin’ 8 = 11- E E / J ( E E ) ~(n cos 0 ) 2 ]

,

(11)

and where 0 is defined as the angle between the celestial north pole and the direction of propagation. These probabilities also hold for antineutrinos, which implies that it is possible to violate CPT and not produce the last signal discussed in Sec. 3. An important feature of this model is that at high energies, E >> 161/E, a seesaw mechanism takes effect and oscillations reduce to two-generation mixing with Pv,cv, N sin2(A32L/2), A32 II 6’ cos’ @/2EE. The energy dependence in this regime mimics exactly that of the usual mass case. However, the quantity that takes the place of mass, the pseudomass Am: = 6’ cos2@ / E , is dependent on the direction of propagation. So it is possible to construct models with conventional energy dependence but unconventional direction dependence. 5. Short baseline experiments Some circumstances are amenable to more general analyses. One case where this is true is when the baseline of an experiment is short compared to the oscillation lengths given by the hamiltonian (5).8 In this situation, the transition amplitudes can be linearized, which results in leading order probabilities given by

This approximation allows direct access to the coefficients for Lorentz violation without the complication of diagonalizing the hamiltonian. This makes an analysis of the general hamiltonian (5) more practical. This type of analysis may be relevant for the LSND experiment, which is consistent with a small oscillation probability PD,+p,N 0.26 over a short baseline of about 30 m.I8 This result is of particular interest because it

37

does not seem to fit into the simple three-generation solution to solar and atmospheric data. The possibility exists that Lorentz violation may provide a solution. References 1. For a recent review, see, for example R.D. McKeown and P. Vogel, Phys.

Rep. 394, 315 (2004). 2. V.A. Kostelecki and M. Mewes, Phys. Rev. D 69, 016005 (2004). 3. V.A. Kostelecki and S. Samuel, Phys. Rev. D 39, 683 (1989); Phys. Rev. D 40, 1886 (1989); Phys. Rev. Lett. 63, 224 (1989); Phys. Rev. Lett. 66, 1811 (1991); V.A. Kostelecki and R. Potting, Nucl. Phys. B 359, 545 (1991); Phys. Lett. B 381, 89 (1996); Phys. Rev. D 63, 046007 (2001); V.A. Kosteleckf, M. Perry, and R. Potting, Phys. Rev. Lett. 84, 4541 (2000). 4. D. Colladay and V.A. Kosteleckf, Phys. Rev. D 55, 6760 (1997); Phys. Rev. D 58, 116002 (1998); V.A. Kosteleckf, Phys. Rev. D 69, 105009 (2004). 5. V.A. Kostelecki, M. Perry, and R. Lehnert, Phys. Rev. D 68, 123511 (2003). 6. For recent overviews, see, for example, these proceedings and V.A. Kostelecki, ed., CPT and L o r e n t z S y m m e t r y 11,World Scientific, Singapore, 2002; 7. V.A. Kosteleckf and M. Mewes, Phys. Rev. D 70, 031902(R) (2004). 8. V.A. Kostelecki and M. Mewes, Phys. Rev. D, in press (hep-ph/0406255). 9. R. Foot e t al., Phys. Lett. B 443, 185 (1998); S. Coleman and S. L. Glashow, Phys. Rev. D 59, 116008 (1999); G.L. Fogli et al., Phys. Rev. D 60, 053006 (1999); P. Lipari and M. Lusignoli, Phys. Rev. D 60, 013003 (1999); V. Barger e t al., Phys. Rev. Lett. 85, 5055 (2000); J.N. Bahcall, V. Barger, and D. Marfatia, Phys. Lett. B 534, 114 (2002); A. de GouvGa, Phys. Rev. D 66, 076005 (2002); I. Mocioiu and M. Pospelov, Phys. Lett. B 537, 114 (2002). 10. S. Choubey and S.F. King, Phys. Lett. B 586, 353 (2004). 11. D. Bear et al., Phys. Rev. Lett. 85, 5038 (2000); F. Cane et al., physics/0309070; S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990); V.A. Kostelecki and M. Mewes, Phys. Rev. Lett. 87,251304 (2001). 12. V.A. Kosteleckf and M. Mewes, Phys. Rev. D 66, 056005 (2002). 13. See, for example, R. Brustein, D. Eichler, and S. Foffa, Phys. Rev. D 65, 105006 (2002). 14. V.A. Kostelecki and R. Lehnert, Phys. Rev. D 63, 065008 (2001). 15. D. Colladay and P. McDonald, J. Math. Phys. 43, 3554 (2002). 16. M.S. Berger and V.A. Kostelecki, Phys. Rev. D 65, 091701(R) (2002). 17. R. Bluhm e t al., Phys. Rev. D 68, 125008 (2003). 18. LSND Collaboration, C. Athanassopoulos et al., Phys. Rev. Lett. 81, 1774 (1998); LSND Collaboration, A. Aguilar et al., Phys. Rev. D 64, 112007 (2001).

ATHENA - FIRST PRODUCTION OF COLD ANTIHYDROGEN ANDBEYOND

A. KELLERBAUERl, M. AMORETT1273, C. AMSLER4, G. BONOMI', P. D. BOWE5, C. CANAL12,3, C. CARRARO'f, C. L. CESAR', M. CHARLTON5, M. DOSERl, A. FONTANA7,8, M. C. FUJIWARA', R. FUNAKOSHIl', P. GENOVA7?8, J. S. HANGST'l, R. S. HAYANOl', I. JOHNSON4, L. V. J0RGENSEN5, V. LAGOMARSIN02'3, R. LANDUAl, E. LODI RIZZIN112,7, M. MACRi2,3, N. MADSEN'l, G. MANUZ102,3, D. MITCHARD5, P. MONTAGNA7,8, H. PRUYS4, C. REGENFUS4, A. ROTOND1738, G. TESTERA213, A. VARIOLA5, L. VENTURELL112,7, D. P. VAN DER WERF5, Y. YAMAZAKI', AND N. ZURL0'2,7 Department of Physics, CERN, 1211 Genive 23, Switzerland Dipartimento d i Fisica, Universitci d i Genova, 16146 Genova, Italy 31NFN Sezione d i Genova, 16146 Genova, Italy Physik-Institut, University of Zurich, 8057 Zurich, Switzerland 5Department of Physics, University of Wales Swansea, Swansea SA2 8PP, UK 61nstituto d i Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21945-9'70, Brazil Dipartimento d i Fisica Nucleare e Teorica, Universith d i Pavia, 2'7100 Pavia, Italy 81NFN Sezione da Pavia, 27100 Pavia, Italy Atomic Physics Laboratory, RIKEN, Saitama 351-0198, Japan loDepartment of Physics, University of Tokyo, Tokyo 113-0033, Japan l1Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark 12Dipartimento d i Chimica e Fisica per l'lngegneria e per i Materiali, Universitd d i Brescia, 251 23 Brescia, Italy

'

(ATHENA Collaboration) Atomic systems of antiparticles are the laboratories of choice for tests of C P T symmetry with antimatter. The ATHENA experiment was the first to report the production of copious amounts of cold antihydrogen in 2002. This article reviews some of the insights that have since been gained concerning the antihydrogen production process as well as the external and internal properties of the produced anti-atoms. Furthermore, the implications of those results on future prospects of symmetry tests with antimatter are discussed.

38

39

1. Introduction According to the CPT theorem,l all physical laws are invariant under the combined operations of charge conjugation, parity (reversal of the spatial configuration), and time reversal. Since CPT transforms an elementary particle into its antiparticle, their fundamental properties such as mass, charge, and magnetic moment are either exactly equal or exactly opposed. This predestines antimatter for tests of CPT symmetry. Due to the fact that atomic spectroscopy on the transition between the ground and first relative excited states (1s-2s) of hydrogen has been carried out to precisioq2 this transition is also being targeted for CPT tests with hydrogen and antihydrogen @). In addition to atomic spectroscopy, antimatter gravity tests are being considered. These could be carried out by gravity interferometry3 or in atomic fountains, in analogy with current studies on ordinary matter in such setup^.^ Two dedicated experiments, ATHENA5 and ATRAP,6 have been set up at CERN’s Antiproton Decelerator’ (AD) since 1998 with the goal of producing sufficient amounts of antihydrogen to ultimately allow precision atomic spectroscopy and a comparison of its atomic spectrum with that of hydrogen. A production of large amounts of E was first demonstrated by ATHENA8 and later by ATRAP,’ using very similar schemes for antihydrogen production but different detection techniques. In the time since these independent proofs of principle, the main challenges have been to investigate the parameters that govern efficient production and the internal and external properties of the produced antihydrogen. 2. Setup and principle The ATHENA apparatus5 consists of three main components, shown in Fig. 1: the antiproton ( p ) capture trap, the mixing trap, and the positron ( e + ) source and accumulator. The former two are located in the 3-T field of a superconducting magnet whose bore is kept at liquid-nitrogen temperature. A liquid-helium cryostat, whose cold nose protrudes into the magnet bore and encloses the trap, reduces the temperature of the trap region further to about 15 K. The bunch of about 2-3 x lo7 antiprotons that is extracted from the AD after every deceleration and cooling cycle undergoes a final deceleration step in a thin (x 50 pm) degrader foil. The foil’s thickness is chosen in order to optimize the fraction of 17 that can be trapped by the capture trap’s high-voltage electrodes (5 kV potential). In the capture trap, the

40 Capturelmixingtrap

Positron accumulator

=Na source

Figure 1. Overview of the ATHENA apparatus. Shown on the left is the superconducting 3-T solenoid magnet which houses the capture trap, the mixing trap, and the antihydrogen annihilation detector. On the right, the radioactive sodium source for the positron production and the 0.14-T positron accumulation Penning trap.

confined antiprotons cool in Coulomb collisions with an electron plasma that was loaded prior to the p capture and allowed to cool by emission of synchrotron radiation. Typically, two p spills from the AD are stacked in the capture trap, resulting in about lo4 p ready for mixing. Simultaneously, positrons produced in the ,B decay of the radionuclide 22Naare moderated, then cooled in collisions with nitrogen buffer gas and accumulated in a low-field Penning trap at room temperature. After the independent p stacking and e+ accumulation phases, the axial potential in the mixing trap is brought into a so-called nested configuration." Figure 2(a) shows how this potential shape allows both positively and negatively charged particles to be simultaneously confined. The central well of this nested trap is then first filled with the NN 5 x lo' accumulated positrons. Just like the electrons in the capture trap, these cool to the ambient temperature of about 15 K by emitting synchrotron radiation. The antiprotons are transferred into a small lateral well and then launched into the mixing region with a relative energy of 30 eV. They oscillate between the lateral confines of the nested well, repeatedly traverse the e+ plasma, and rapidly cool in Coulomb collisions with the positrons. After some tens of ms, antihydrogen production spontaneously sets in with initial rates of several 100 Hz. The neutral atoms thus produced in the center of the mixing trap are no longer affected, to first order, by the electrical and magnetic fields used for the charged-particle confinement. They leave the interaction region with a momentum that is essentially equal to that of the p just before formation. When these anti-atoms impinge upon the Penning trap electrodes,

41

Mixina trap electrodes

z c

s! c

-100

-60

-40

-20

0

20

40

60

axial position (mm)

(a)

511-keV y

(outer) -

(b)

.__,

511-keV y

Figure 2. (a) Detailed sketch of the mixing trap, which is operated in a nested-trap configuration. The graph shows the axial trap potential before (dashed line) and after (solid line) the antiproton injection. (b) Sketch of the antihydrogen annihilation detector. With its highly granular silicon strip and CsI crystal modules, it allows a direct and unambiguous detection of production.

their constituents immediately annihilate with ordinary matter. The signal of these destructive events is recorded with the antihydrogen annihilation detector that surrounds the mixing trap. A sketch of this detector is shown in Fig. 2(b). It consists of 8192 silicon strips in two layers for the detection of the charged pions created in the p annihilation with a proton or a neutron and 192 cesium iodide crystals that record the (mainly back-to-back) y rays from the e+-e- annihilation. Despite its extremely compact dimensions, it allows a three-dimensional.reconstruction of the charged-particle vertex with a resolution of 4 mm and a spatial and temporal correlation of the p and e+ signals for an unambiguous identification of formation. As an example for event reconstruction, Fig. 3 illustrates the signal of the first production of cold antihydrogen' in 2002. In Fig. 3(a), the az-

42

E E 20 v g 10

a

s

Y

.c.

ln

c

0

’

0 0

c

120

Q)

3

c 0 0

C

240 ln

160

2

120

0

80

0

0

200

c I=

9

c

80

C

40 n -1

40

v

(b)

C

horizontal position (rnrn)

200

$

ln

c

-30

horizontal position (mrn)

160

Y

0. -10 8 e -20

.4-

$

i

2

0

‘P

53

(a)

-

30

?

-0.5

0 cos e,

0.5

1

n ”

-1

-0.5

0 cos e,

0.5

1

Figure 3. Signal of the first production of cold antihydrogen with ATHENA.8 (a) Charged-pion vertex distribution a s a function of the azimuthal coordinates. (b) Opening-angle distribution of the photons recorded in coincidence with the chargedparticle hits, as seen from the charged-particle vertex.

imuthal distribution of reconstructed vertices from the p annihilation is shown. In the left panel, where the positrons are in thermal equilibrium with the trap at 15 K (“cold mixing”), the largest numbers of events are recorded in a ring located at the position of the electrodes (25 mm diameter). When the positron plasma is heated by means of a radiofrequency (RF) excitation of the axial plasma modesll (“hot mixing”), production is suppressed and a smaller number of events, due to p annihilations with residual gas, is recorded in the center of the trap (right panel). Figure 3(b) shows the distribution of the opening angle of the two 511-keV y rays recorded in time coincidence with the charged-particle hits, as seen from the charged-particle vertex. In the left panel, a clear excess at an opening angle of 180’ is present for cold positrons, while it is suppressed when the positron plasma is heated. The right panel shows the good agreement of the data with Monte-Carlo simulations. The 131(22) fully reconstructed events that constitute the peak in the left panel of Fig. 3(b) correspond to a total number of about 50 000 produced atoms for this partial dataset

43

of 2002. A complete analysis of the 2002 data, together with more detailed Monte Carlo simulations, showed that the instantaneous trigger rate from the silicon detector is a good proxy for antihydrogen production, with 65% of all triggers over the entire mixing cycle due to annihilating antihydrogen atoms.” 3. Recent results

For precise antimatter studies, it is not sufficient to merely produce large numbers of antihydrogen. A fair knowledge of the temperature and kineticenergy distributions of the produced is required in order to estimate the fraction of anti-atoms which can be trapped. The atoms must also be produced in a well-defined internal quantum state, if possible the ground state. In this section, some results on the latter question, based on an investigation of the formation process, are presented.

n

n

n

3.1. Antihydrogen production 2002/2003

As a prerequisite for any quantitative studies on antihydrogen formation, the offline data analysis must allow a precise determination of the number of produced anti-atoms. In order to achieve this, one or several observables, such as the radial vertex distribution or the 27 opening angle distribution, can be considered as a linear combination of a pure signal (Monte-Carlo simulation of annihilations on the trap electrodes) and background. Since the background is expected to be mainly due t o p annihilations with residual gas, it can be represented by the signal obtained from runs in which the e+ were heated to several 1000 K, thereby inhibiting production. The total antihydrogen production of 2002 and 2003 obtained in this way is summarized in Tab. 1. It shows that ATHENA has produced more than lo6 H anti-atoms since its start of operations and that the production efficiency in terms of captured antiprotons from the AD is between 10 and 20%.

n

n

3.2. Recombination process

The formation of antihydrogen by direct capture of a positron onto an atomic orbit around an antiproton does not simultaneously conserve energy and momentum. The involvement of a third particle is needed in order to respect these conservation laws. That particle can either be a photon in the case of (spontaneous) radiative recombination (SRR)13 or a second positron in three-body recombination (TBR).14 These two processes are

44 Table 1. Comparative summary of ATHENA's antihydrogen production in 2002 and 2003. Cold mixing 2002

Cold mixing 2003

Total no. of cycles

341

416

Cycle duration

180 s

70 s

Total mixing time

17.1 h

8.1 h

~ _ _ _ _ _

~

Injected p Produced

2.92 x

n

Production efficiency Avg.

n production rate

H fraction of signal

lo6

5.07 x

lo6

4.94 x 105

7.04 x 105

16.9%

13.9%

8.0(4) Hz

24.2(1.3) Hz

65(5)%

74(3)%

predicted to have vastly different cross-sections and recombination rates, with TBR expected to be the dominant process at ATHENA's experimental conditions. The most important difference with a view to precision studies lies in the fact that SRR populates low-lying states ( n < 10) and TBR highly excited Rydberg states ( n >> 10). The two mechanisms also exhibit different dependencies on the positron temperature (SRR: 0; T-0.63;TBR: c( T-4.5) and density (SRR: 0; n,+; TBR: c( n:,), which can allow to distinguish between them. In order to determine the temperature dependence of production, we have performed mixing cycles with RF heating at various amplitudes applied to the positron ~ 1 a s m a . The l ~ positron temperature increase was measured with ATHENA's plasma diagnostics system'' by resonant excitation and detection of the axial plasma modes. In Fig. 4,the backgroundcorrected integrated number of triggers (left) and peak trigger rate (right) as possible proxies for production are shown as a function of the positron temperature, assuming an equilibrium temperature of 15 K. Neither of these plots shows the characteristics of a simple power law (a straight line in these logarithmic plots), but a best fit to the data yields a behavior of the form c( T-0.7(2), close to that expected from radiative recombination. However, the observed event rates are between 1 and 2 orders of magnitude higher than expected for this recombination process. The second access to the recombination process is via the positron density dependence. For this purpose, we have analyzed the 2003 data with a view to varying positron plasma density.16 Standard cold mixing runs with positron plasma densities between 3 x 108/cm3 and 1.5 x 10g/cm3

45

t 0

1 10 Id e+ plasma temperature (mev)

1 10 102 e+ plasma temperature (mev)

Figure 4. Dependence of the background-corrected integrated total number of chargedparticle triggers per mixing cycle (left) and the peak trigger rate (left) on the positron plasma t e m p e r a t ~ r e . 'The ~ number of triggers and trigger rate have been normalized to the signal for an e+ temperature of 1 meV. Note the logarithmic scale.

have been identified. However, this analysis is complicated by the fact that under ATHENA's typical experimental conditions, the f j cloud has a much larger radial extent than the e+ plasma. This means that the number of interacting antiprotons strongly depends on their radial density distribution. Measurements of this distribution are therefore required to extract the positron density dependence of B production. 4. Antihydrogen spectroscopy within the framework of the Standard-Model Extension Any measured difference in the hydrogen and antihydrogen atomic spectra would be a clear and unambiguous signal for CPT violation. On the other hand, a theoretical framework for such symmetry breaking can indicate which transitions are particularly suited for an experimental search. The Standard-Model Extension1' incorporates spontaneous CPT and Lorentz breaking at a fundamental level. It is an extension of the Standard Model that preserves energy and momentum conservation, gauge invariance, renormalizability, and microcausality. Within this framework, the sensitivity to CPT- and Lorentz-violating terms of spectroscopic experiments on H and confined in a magnetic trap can be predicted. Consider the energy states of (anti)hydrogen with zero angular momentum ( I = 0 ) , confined in a magnetic trap with axial solenoidal field and radial multipole magnetic fields. These states are subject to hyperfine as well as Zeeman splitting, as shown in Fig. 5. Before the excitation, only the low-field-seeking states Ic)l and Id)l are confined in a magnetic trap. It has been shown1* that the CPT-violating term bs shifts

46

]

A

E

0

0.1 0.2 0.3 0.4 0.5 0.6

0.7

8

low-field-seeking states

(T)

Figure 5. Hyperfine and Zeeman splitting of (anti)hydrogen confined in a magnetic field for states with zero angular momentum.

-

all states by the same amount, both in H and in H, and the CPTviolating effect is thus suppressed by a factor a2/87rin the Jd)l 142 transition. The transition I c ) ~ between the mixed-spin states does potentially produce an unsuppressed frequency shift due to the n dependence of the hyperfine splitting. This shift is different in H and in and leads both to diurnal variations in the frequency difference and to a nonzero instantaneous difference. However, this transition is field-dependent and thus subject to Zeeman broadening in inhomogeneous magnetic fields. As an alternative, it was therefore suggestedl8 to consider a transition between hyperfine levels of the ground state (n = 1)at the field-independent transition point B x 0.65 T. The transition Jd)l I c ) ~ in H and in is then subject to potential diurnal variations, and the instantaneous difference Av,d between these transitions in hydrogen and antihydrogen is directly proportional to the CPT-violating term bg. Based on these considerations, future antihydrogen spectroscopy experiments will include comparisons of the hydrogen and antihydrogen hyperfine structure.lg

Ic)~

-

n

-

n

5. Conclusions and outlook With the first production of copious amounts of cold antihydrogen, many of the challenges on the way to high-precision CPT tests with antimatter have been surmounted, but many more still remain. Future high-precision spectroscopic and interferometric measurements on antimatter atoms are contingent upon the ability to confine neutral atoms and to cool them with Lyman-a lasers. Our results on the temperature dependence of production suggest on the one hand that an appreciable fraction of the

47

antihydrogen may be produced in low-lying states accessible to precision atomic spectroscopy. On the other hand, recombination possibly sets in before complete thermalization of the antiprotons, thereby reducing the fraction of produced antihydrogen that can be confined in a magnetic trap. Further studies on antihydrogen production in a nested Penning trap are required to clarify these points. In parallel, tests with ordinary matter on the simultaneous confinement of charged and neutral particles in electromagnetic traps are being carried out in order to establish parameters for the efficient preparation of trapped antihydrogen for symmetry tests.

Acknowledgments This work was supported by the funding agencies INFN (Italy), CNPq (Brazil), MEXT (Japan), SNF (Switzerland), SNF (Denmark), and EPSRC (United Kingdom).

References W. Pauli, I1 Nuovo Cimento 6 (1957) 6. M. Niering et al., Phys. Rev. Lett. 84 (2000) 5496. T. J. Philips, Hyp. Int. 109 (1997) 357. A. Peters et al., Nature 400 (1999) 849. M. Amoretti et al., Nucl. Instrum. Methods A 518 (2004) 679. G. Gabrielse et al., Phys. Lett. B 455 (1999) 311. J . Y. H6mery and S. Maury, Nucl. Phys. A 655 (1999) 345c. M. Amoretti e t al., Nature 419 (2002) 456. G. Gabrielse et al., Phys. Rev. Lett. 89 (2002) 213401. G. Gabrielse et al., Phys. Lett. A 129 (1988) 38. M. Amoretti et al., Phys. Rev. Lett. 91 (2003) 055001. M. Amoretti et al., Phys. Lett. B 578 (2004) 23. J. Stevefelt et al., Phys. Rev. A 12 (1975) 1246. M. E. Glinsky et al., Phys. Fluids B 3 (1991) 1279. M. Amoretti et al., Phys. Lett. B 583 (2004) 59. G. Bonomi et al., t o be published. D. Colladay and V. A. Kostelecki, Phys. Rev. D 55 (1997) 6760. R. Bluhm, V. A. Kostelecki, and N. Russell, Phys. Rev. Lett. 82 (1999) 2254. 19. R. Hayano et al., these proceedings.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

THE PHYSICS OF GENERALIZED MAXWELL EQUATIONS

c. LAMMERZAHL Center f o r Applied Space Technology and Microgravity ( Z A R M ) , University of Bremen, A m Fallturm, 28359 Bremen, Germany E-mail: [email protected]

H. MULLER Physics Department, Stanford University, Stanford, C A 94305-4060 E-mail: [email protected] After some general remarks concerning the schemes for generalizations of standard equations for a phenomenological description of possible effects resulting from various quantum gravity approaches a new generalization of the feld equation governing dynamics of the electromagnetic fields is presented. This new field equation is capable of describing charge non-conservation. It also contains more Lorentz invariance violating parameters than the generalization given within the StandardModel Extension. As a consequence, observations and experiments related to electromagnetic radiation (birefringence and anisotropy experiments) alone are not enough to establish the ordinary Maxwell equations and, thus, Lorentz invariance. Also electromagnetostatic experiments are needed. New experiments are proposed for testing these new parameters.

1. Introduction One of the biggest goals of today’s theoretical physics is to resolve the fundamental inconsistency between quantum theory and gravity. As a solution to this problem, these two theories should emerge into a new universal theory called quantum gravity, covering both the quantum and the gravity regime in a new way. Presently, most promising aproaches are string theory, loop quantum gravity and nop-commutative geometry. It is an astonishing feature that all these approaches seem to lead to violations of Lorentz invariance (and also of the Universality of Free Fall and the Universality of the Gravitational Redshift, all encoded in Einstein’s Equivalence Principle, see, e.g., Ref. 1). Since the specific violations of Lorentz invariance are dif48

49

ficult to predict and since different aproaches predict different violations, a phenomenological scheme to treat all these effects in a unified way might be appropriate. Here we will discuss one way of obtaining a frame which is capable of describing a wide range of effects violating Lorentz invariance in the electromagnetic sector. Our generalized Maxwell equations describe violations of Lorentz invariance for radiation effects as well as effects which can be probed by electromagnetostatic setups only. Our aproach also can be used for a unified description of tests for charge conservation which is one of the basic principles in the construction of physical theories.

1.1. Methods of phenomenological generalizations of dynamical equations The current phenomenological generalizations of dynamical equations like the Dirac or the Maxwell equation have the status of a test theory and provide a link between the full quantum gravity formalism and the language which is appropriate to describe experiments. The structure of these test theories can be obtained by calculating some low energy approximation of the full quantum gravity scenario. This has been done for string theory2y3 as well as for loop quantum g r a ~ i t y .In ~ ?any ~ case, the ordinary equations of motion like the Dirac or Maxwell equations will turn out to be modified. The actual form of the generalization depends on the approximation used. Since the approximation schemes are not really settled, the results have to be considered not as exact predictions but, instead, as roadmaps to open up hypothetical possibilities of having new terms in the ordinary equations of motion. Therefore, these kinds of approaches to generalized equations of motion have been complemented by constructing test theories by its own. There are many generalizations of basic equations like the Maxwell or the Dirac equation. One method which has been applied intensively is the Standard-Model Extension (SME).6>778*9 In this approach one starts from the most general Lagrangian which is still quadratic in the field strengths or in the fermionic fields and requires further building principles like conservation of energy-momentum, Lorentz-covariance,conventional quantization, Hermiticity, microcausality, positivity of energy, gauge invariance, and power-counting renormalizability. The main advantage of this approach is the mathematical consistency and physical interpretability of this new theory in conventional terms and that this is most conservative (and, thus, the most probable) modification of estabished theories. The parameters of

50

the SME are just additional interactions with constant fields. This kind of generalizations in the photonic sector have been introduced and discussed earlier by Nilo and Haugan and Kauffmann." Another method introduces the generalization on the level of the dynamical equations. This is more general than the Lagrangian approach, see Refs. 12, 13 for examples dealing with generalized Dirac and Maxwell equations. In the case of the Dirac equation'' one is led to effects like the violation of the Universality of Free Fall, of the Universality of the Gravitational Redshift and of Local Lorentz Invariance (the same parametrization for describing effects due to violations of Local Lorentz Invariance have been also obtained later on in Ref. 8). More effects than in the SME are encountered for the generalized Maxwell equation13 which we are going to describe here. Charge conservation, which automatically comes out from the Lagrangian approach, can be violated in models generalizing the field e q ~ a t i 0 n s .In l ~ addition, in Ref. 13 also more Lorentz Invariance violating parameters than in the SME have been found. However, particular care must be taken for the mathematical consistency of the formalism if one "by hand" generalizes the dynamical equations. This will be automatically secured if one employs a constructive axiomatic scheme in order to derive such equations, see, e.g., Refs. 14, 15, 16 for a derivation of the generalized Dirac equation in terms of fundamental properties of the dynamics of fields.

1.2. Comparison with kinematical test theories for Lorentz

invariance Beside approaches for generalizing equations of motion in order to study violations of Lorentz invariance, there is a kinematical scheme introduced by Robertson17 and Mansouri and Sexl,18where a modification of Lorentztransformations are the basis for, e.g., anomalous effects of light propagation which, in turn, can be tested experimentally. Compared with the dynamical approach the kinematical approach is more powerful since it is independent of the specific particle model under consideration: it discusses the transformation properties of observed quantities for linear transformations between inertial systems. However, there are serious disadvantages of this model. (i) It assumes that in one frame light propagates isotropic. This requires singling out a preferred frame. This preferred frame usually is identified with the frame given by the cosmological microwave backround. However, if it will turn out that, e.g., a stochastic gravitational wave back-

51

ground exists which is different from the microwave background, then one has to decide which one will choose. This choice will influence the interpretation of the experiments. (ii) It is not justified why light should propagate isotropically in this preferred frame. This is violated in Finslerian spacetime models, for example. (iii) It is not possible to describe violations of Lorentz invariance due to birefringence effects. As a consequence, dynamical test theories like the SME or our more general approach definitely are the preferred way in order to describe tests of Lorentz invariance.

2. The generalized Maxwell equations

We assume that the homogeneous Maxwell equations dF = 0 are still true. In our approach13 the principles to formulate generalized inhomogenous Maxwell equations are: 0 0

0

linearity in the field strength, first order in the differentiation, and small deviation from the standard Maxwell equations.

Therefore, the generalized Maxwell equation which we are going to discuss and to confront with experiments is

The requirement that this equation describes a small deviation from the xP”Pa where rlP” is the standard theory leads to XP”pa = @’rf]” Minkowski metric diag(+ - - -) and, in that frame, all components of x P v P a and x P P a are small compared to unity. Therefore, all effects are calculated to first order in these quantities only. In our approach, the constitutive tensor XP”P“ and, thus, the tensor x P v P a are assumed to possess the symmetry P ” P a = X ~ ” [ P “ ] only (in the SME the constitutive tensor possesses the symmetry of the Riemann tensor.lg) The decomposition of this constitutive tensor and of x P P a reads (see Ref. 13 for the definition of the various irreducible parts)

+

52

A 3+1 decomposition of the generalized Maxwell equations gives 4rp = V E ~ T J "=

+ xoooaEa+ xoo23S83 + xoaPp"dFP" + xop"FPfl 2

+

+

(4)

Ea - (VX B)' + xaoo'E3+ X20'kBjk XaJP"djFpo Xap"Fpg,(5)

where E, = Fo, and B, = ;cykF3k. Using the homogeneous Maxwell equations, the time derivative of B can be replaced by a spatial derivative of the electric field. The appearance of the term xoooamakes both equations to dynamical equations for the electric field rendering them to be an overdetermined system. Therefore we have to require the vanishing of the coefficient xoooa.Since this should be true for any chosen frame of reference, we have to require

X(PVP)"= 0 ,

(6)

what is identical to (l)Z(apPp, = 0, 5," = 0, and APv = -z.ZPv. Due to the vanishing of these irreducible parts we get dP(Xfi"p"da,Fp,)= 0 so that only xPP" may lead to charge non-conservation:

+

47~8,= ~ ~xaPvdaFPv= ( l ) ~ a P " d Q F P ,f Y v d a F P u .

(7)

The validity of the homogenous Maxwell equations allows to require the vanishing of XP['p"] without any physical consequences. This leads to = 0 , Z P " = WE", and T P " = I Q P " 2 .

x

3. Radiation effects In order to discuss the effects due to the anomalous terms in radiation phenomena, we derive the wave equation

O = E, - A E ,

+ (V(V.E ) ) 2+ x""'da,dVEj + 2xZ0'E, + xak'dkEJ

(8)

and make a plane wave ansatz E = Eoezkpzp= Eoe-z(k'e-wt)which results in equations for the amplidude and two for the derivative of the amplitude:

0 = ( ( w 2 - lc2)6,,

+ k,k3 + 2xzPJ"kPk,)E ; .

(9)

53

and

1 p(n)= --W”npnv 2 2

(13)

1

n ( n )= - ~ , y 7 1 p ~ ~ a ~ u ~ ~ g ~ a ,~ n , n , n p n ,(14) 2 with n p = k,/w = (1,k / l k l ) . This generalizes results in Ref. 20, 19. The f sign in front of the square root indicates a hypothetical birefringence. If no birefringence is observed, then we can conclude (l)Wp”P“ = 0 and XDp” = 0. It has been shownlg that from astrophysical observations The remaining p ( n ) birefringence can be excluded at the level of in the dispersion relation leads to an asisotropic speed of light which has been excluded in laboratory experiments2’ at the level. Vanishing anisotropy implies W‘” = 0. Furthermore, from (10) a propagation equation for the amplitude can be derived V p dp Eo a =

-

(2wxZo3- kkXZk3)Ej” ,

(15)

where V P is the group velocity of the light ray. This is directly related to the charge non-conservation paremeter x p P O . Since no propagation of the polarization has been inferred from astrophysical observation22 the tensor x p p ‘ vanishes at the order GeV. From all these requirements, the remaining generalized Maxwell equations are 4 r j “ = (1- i W ) &Fp” --rfpZVpdpFp, 1 2

+ Z3 ~ p p Z ” U d p F p+, $apZpvdpFpv.

(16)

In the SME approach the Zp” are absent. Therefore, in our approach it is not possible to establish Lorentz invariance by radiation experiments only. Further experiments are needed.

54

4. Electromagnetostatics

The 3+1 decomposition of the above equations gives (we set aside the factor W since this can be absorbed into a redefinition of the electric charge and current) -=

-V . E

-

1. (Vx E ) - C C .

(Vx B )

(17)

€0

1 . c2

poj = - E -

1 1 1-c x 8 - ( & v ) B +xvB - -v(c.E ) + ; c ( v . E ) , C c2

(18) where we used SI units and defined Cz := qZoZand & := $ c z J k Z J k . The generalized Maxwell equations for a point charge at the origin are given by (17,18) with p = q6(r) and j = 0. Since we have a static problem, we neglect the time derivatives. We furthermore chose E = V+ and B = V x A and the gauge V . A = 0. Then the generalized Maxwell equations yield to first order in the perturbations

+=-- 1

9 4ncO r

A = - .4 t 4ncocr

(19)

This gives a magnetic field

which is beyond the SME scheme. Therefore, our model includes the feature that even in the case that all radiation effects respect Lorentz invariance, a point charge also creates a magnetic field. This field is different from the field of a magnetic moment. For a point charge located at the origin and a coordinate system with the e , base pointing in C direction the magnetic field lines are circles in the 2-y-plane, similar to the magnetic field lines around a wire. The strength, however, varies with l/r2 where r is the distance from the origin. If we take a charged line with linedensity X in direction n, then the magnetic field is

where p is the distance from the charged line, ep the radial unit vector orthogonal to n , and eV the unit tangent vector of a circle around that line. If the source of the Maxwell equations is a magnetic moment m localized at the origin, then the Maxwell equations are (17,18) with p = 0 and

55

j = m x V d ( r ) . We assume again a static situation and get to first order

A = -P-u o m x r 47r r3 ’

q $ = - POC (C x m) . r

(22)

47r r3 A magnetic moment also creates an electric field of an electrical dipole with dipole moment d = pocq,C x m. This feature is “dual” to the previous case. For the new Lorentz invariance violating parameters C and there seem to exist no experimental results. However, we expect strong bounds on the CZ components from measurements with SQUIDs and from atomic specT can troscopy. With SQUIDs weak magnetic fields of down to be measured. If we assume that a measurement with SQUIDs of a magnetic field from a point charge does not lead to any magnetic field larger T for a line than the SQUID sensitivity, then, from IX
If we choose the z-axis in direction of shift AEnlrn = ($nlrn I Hc I $nlm) is

C, then the corresponding energy

AEnim = This does not vanish for

$210

= R21Y1o where R21 =

A p 1 z e--r/(2a) T

7

Y ~=o G c o s 1 9 where a in the Bohr radius. In this case we get

With p z = eA/m, this yields AE2lo = C,1.8.10-2 eV. The stateof the art of high precision measurements of energy levels is of the order A E I E M Since the measured energy levels are still well described within the standard theory one gets for energies of about 10 eV at best an estimate I<, 1 5 which, however, is not as good as a direct measurement discussed above might yield. The parameter gives rise to small deviations from the unperturbed quantities only and, thus, cannot be measured so precisely.

1

56

References 1. C. Lammerzahl, in D. Giulini, C. Kiefer, and C. Lammerzahl (eds), Quantum Gravity - From Theory to Experimental Search (Springer Verlag, Berlin, 2003), p. 367. 2. J . Ellis, N.E. Mavromatos, and D.V. Nanopoulos, Gen. Rel. Grav. 32, 127 (2000). 3. J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, and G. Volkov, Gen. Rel. Grav. 32, 1777 (2000). 4. R. Gambini and J. Pullin, Phys. Rev. D 59,124021 (1999). 5. J. Alfaro, H.A. Morales-Tecotl, and L.F. Urrutia, Phys. Rev. Lett. 84,2318 (2000); Phys. Rev. D 65,103509 (2002); Phys. Rev. D 66,124006 (2002). 6. S. Coleman and S.L. Glashow, Phys. Lett. B 405,249 (1997). 7. D. Colladay and V.A. Kosteleck9, Phys. Rev. D 55, 6760 (1997). 8. D. Colladay and V.A. Kostelecki, Phys. Rev. D 58, 116002 (1998). 9. S. Coleman and S.L. Glashow, Phys. Rev. D 59, 116008 (1999). 10. W.-T. Ni, Phys. Rev. Lett. 38,301 (1977). 11. M.P. Haugan and T.F. Kauffmann, Phys. Rev. D 52,3168 (1995). 12. C. Lammerzahl, Class. Quantum Grav. 14,13 (1998). 13. C.Lammerzahl, A. Macias, and H. Muller. Lorentz invariance violation and charge (non-)conservation: A general theoretical frame for extensions of the Maxwell equations. submitted. 14. C. Lammerzahl, in V. deSabbata and J. Audretsch (eds), Quantum Mechanics in Curved Space-Time, N A T O A S I Series, Series B: Physics, volume 230 (Plenum Press, New York, 1990), p. 23. 15. J . Audretsch and C. Lammerzahl, in U. Majer U. and H.-J. Schmidt (eds), Semantical Aspects of Space-Time Geometry (BI Verlag, Mannheim, 1993), p. 21. 16. J. Audretsch, F.W. Hehl, and C. Lammerzahl, in J. Ehlers and G. Schafer (eds), Relativistic Gravity Research. With Emphasis on Experiments and Observations, Lecture Notes in Physics 410 (Springer-Verlag, Berlin, 1992), p. 368. 17. H.P. Robertson, Rev. Mod. Phys. 21,378 (1949). 18. R. Mansouri and R.U. Sexl, Gen. Rel. Grav. 8,497 (1977). 19. V.A. Kostelecki and M. Mewes, Phys. Rev. D 66,056005 (2002). 20. W.-T. Ni, Bull. Am. Phys. SOC.19,655 (1974). 21. H. Muller, S. Herrmann, C. Braxmaier, S. Schiller, and A. Peters, Phys. Rev. Lett. 91,020401 (2003); J.A. Lipa, J.A. Nissen, S. Whang, D.A. Stricker, and D. Avaloff, Phys. Rev. Lett. 90,060403 (2003); P. Wolf, M.E. Tobar, S. Bize, A. Clairon, A.N. Luiten, and G. Santarelli, Gen. Rel. Grav. 36 (2004), to appear. 22. S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41,1231 (1990).

OPERATION OF THE K-3HE SELF-COMPENSATING CO-MAGNETOMETER FOR A TEST OF LORENTZ SYMMETRY

T.W. KORNACK AND M.V. ROMALIS Department of Physics, Princeton University, "Princeton, N J 08550 USA E-mail: romalis@princeton. edu We report on progress towards a new test of Lorentz and CPT symmetry using a K 3 H e co-magnetometer. The K vapor forms a spin-exchange relaxation-free (SERF) magnetometer that has a sensitivity of about 1 ff/&. The polarized 3He cancels the magnetic field drifts and the Johnson noise generated by the magnetic shields. Together, the K-3He co-magnetometer retains sensitivity to anomalous, Lorentz-violating fields that couple to electron and nuclear spins differently than by their magnetic moments. We present the methods we have developed for running the co-magnetometer optimally, including techniques for zeroing the lightshifts and ambient magnetic fields, and for suppressing various systematic effects. Preliminary long-term data are also presented.

1. Introduction High precision tests of CPT and Lorentz symmetry are becoming increasingly important in the search for new physics beyond the Standard Model. Quantum gravity and string theory can break Lorentz invariance and thereby generate CPT- and Lorentz-violating effects. The theoretical framework presented by Alan Kosteleck? and colleagues for parameterizing CPT and Lorentz violating effects in the Standard-Model Extension has provided a concrete and comprehensive set of goals for experiments to pursue.' In the non-relativistic limit, CPT-violating effects can appear as magnetic-like anomalous fields that are assumed to couple to spins not in proportion to their magnetic moment.2 These anomalous fields point in a certain direction in spacetime that is constant on the scale of our solar system. An experiment can search for these fields by looking for sidereal or yearly signals as the Earth rotates and moves around the Sun. We have built a co-magnetometer using K and 3He spins that is sen57

58

sitive to the differential coupling of an anomalous, CPT-violating field to electron and nuclear spins. The K magnetometer is operated at sufficiently high density and low field that it becomes spin-exchange relaxation-free (SERF) with a significantly lower fundamental noise level than traditional alkali metal magnetometer^.^ Spin-exchange collisions between K and 3He polarize the 3He and enhance the dipolar interactions between K and 3He. This strong spin-exchange coupling allows the magnetometer to be operated in a self-compensating mode where the effects of magnetic fields are passively cancelled. Anomalous fields, however, are not cancelled and generate a signal. 2. The K-3He co-magnetometer The K-3He co-magnetometer can be fully described in two parts: first, as a sensitive, spin-exchange relaxation-free K magnetometer and second as a system of interacting K and 3He spin ensembles. Greater depth on these two topics can be found in Allred et aL3 and Kornack and R ~ m a l i s , ~ respectively. 2.1. The K magnetometer

The sensitivity of atomic magnetometers is limited by the shot-noise of the atomic vapor: 1

bB =

y J m '

This is the sensitivity obtained in a time t from atoms contained in a volume V with a density n and transverse spin relaxation time T2. In our co-magnetometer, we use K vapor in 50 torr N 2 to quench the K optical transitions and -7 atm 3He buffer gas to decrease diffusion rates and wall relaxation. The K spin relaxation is given by:

The first term is the relaxation due to spin-exchange collisions, the following 3 terms describe relaxation by spin-destructions collisions with 3He, N 2 and other K atoms, and the last term describes relaxation due to diffusion of atoms to the walls. The relaxation rate of typical alkali magnetometers is dominated by spin exchange collisions, but at sufficiently high density spin exchange relaxation can be suppressed in a low magnetic field:3

59

where wo is the spin precession frequency in low magnetic field:

If wo << TSEthen the transverse relaxation time due to spin-exchange is much longer than the time between spin-exchange collisions TSE.In low magnetic field the K spins do not precess continuously but rather achieve a certain average angle defined by competition between the precession frequency and relaxation dominated by the pumping rate. We have demonstrated spin-exchange relaxation-free operation of the K magnetometer, obtaining a 1 Hz resonance linewidth with a spin-exchange collision frequency of 20 kHz. For our 2.5 cm diameter cell, the sensitivity limit according to Equation (1)is 2 aT/&. We have obtained experimentally a sensitivity of 500 aT/& using a first order gradiometer, limited only by the Johnson noise from the magnetic shield^.^ 2.2. Coupled s p i n ensembles The dynamics of the KP3He co-magnetometer depend significantly on spinexchange interactions. The co-magnetometer cell contains ensembles of polarized K and 3He spins that are coupled to one another by both their spin-exchange contact interaction as well as by dipolar field interactions. The total interaction field experienced by each spin species can be expressed as B = ( 8 n ~ o / 3 ) Mwhere , KO is a spin-exchange enhancement factor due to the overlap of the K electron wavefunction and the 3He nucleus. For the cell temperature of 170°C KO = 5.9 for K-3He spin-exchange.6 Since the magnetometer cell is nearly spherical, the spin ensembles have negligible self-interaction and can be robustly approximated by a simple set of Bloch equations that couple the K magnetization, Me, with the 3He magnetization, M":

We include distinct transverse and longitudinal relaxation terms, Tzn and Tin, for the 3He, whereas a single relaxation relaxation term Te suffices for potassium. In the absence of interactions, the spin ensembles precess according to their gyromagnetic ratios: yn = p ~ s ~ ~=/ 2n f i x 3244 Hz/G and y,/S(P) = gpg/fiS(P) = 2n x 2.8/S(/3) MHz/G where S(p) is the slowingdown factor due to the K electron sharing angular momentum with the K

60

3He cancels the external field B, r

Figure 1. The co-magnetometer is insensitive to applied magnetic fields. (a) The 3He magnetization is cancelled by an externally applied B,. (b) In response t o a slowly changing transverse field B,, the 3He spins follow adiabatically the total field and to first order the 3He magnetization cancels the applied field.

nucleus, which ranges from S(p) = q = 6 for low K spin polarization to S(p) = (21+1) = 4 for high p~larization.~ We have also included the effects of pump and probe beam lightshifts, L, and L,, in the K system. Lightshifts can be treated like an effective magnetic field that only the potassium experiences; they are proportional to the product of the laser frequency detuning away from resonance and the degree of circular polarization. We set the pump beam lightshift to zero by tuning its frequency exactly onto the D1 resonance of K and the probe beam lightshift to zero by canceling any circular polarization. Finally, we have included anomalous field couplings be and b, to the K and 3He spins. The long-term systematic effects of drifting magnetic fields and lightshifts are the primary concerns in a test of CPT symmetry. Solving the Bloch equations for the potassium magnetization in steady state and keeping leading order terms for each magnetic field and lightshift, we obtain the following expression:

S=

MzYeTe

+

1+ [(Bc Lz)reTeI2

(6)

+

+

Here Bc = B, AMP AM: and is near zero when the external field B, cancels the 3He and (much smaller) K polarization. In the case that all

61

quantities B,, B,, B,,

L, and L, are zero, Equation ( 6 ) simplifies to: S = MzyeTe(b; - b i ) .

(7)

Hence, the co-magnetometer signal is proportional to the difference in field coupling to the two spin species. For a regular magnetic field b i = bi and the co-magnetometer signal vanishes. Importantly, the signal due to small changes in each quantity is suppressed by at least one small factor. Figure 1 illustrates how the 3He cancels an applied transverse field, leaving K in zero field. This behavior is useful for suppressing low-frequency Johnson noise from the magnetic shields or other drift in the ambient magnetic fields as well as maintaining the low field requirement of the spin-exchange relaxation-free operation of the K magnetometer. 3. Experimental implementation

We search for a sidereal signal from an anomalous field b interacting with the co-magnetometer as the Earth rotates on its axis and a yearly signal as the Earth rotates around the Sun. These long-term measurements place particularly stringent demands on the control of systematic effects and have a strong influence on the design of the experiment. 3.1. Co-magnetometer setup

The co-magnetometer is created using an optical pump-probe setup shown in Figure 2. K vapor is held in a 2.5 cm diameter GE180 aluminosilicate glass cell with thin walls. Also in the cell are 7 atm 3He and 50 torr Nz. A double-wall oven heats the cell to create 1013 cmP3 density K vapor by flowing 175°C hot air between the walls that surround the cell. The oven is insulated and placed in a water-cooled box inside five-layer magnetic shields. A 770 nm, 1W broad area diode laser optically pumps the D1 transition of the K vapor. The pump laser has an external cavity grating feedback and its wavelength is additionally stabilized by locking to a grating spectrometer. Intensity stabilization for the pump laser is achieved using feedback to a variable waveplate. A probe beam, created by a single frequency diode laser with a tapered amplifier, passes perpendicular to the pump beam through the cell. A Fabry-Perot interferometer stabilizes the frequency of the probe beam by modulating the cavity length and locking the frequency. Intensity stabilization is accomplished by employing a feedback to the tapered amplifier current.

62

Translating Lens

Figure 2. Schematic of the experiment indicating: pump and probe lasers with surrounding enclosures, frequency and intensity feedbacks, beam steering optics, the doublewalled oven and magnetic shielding.

The probe beam measures the component of K polarization parallel to the beam using optical rotation. To obtain a very high precision measurement of the probe beam polarization angle, we modulate the probe polarization angle so that the signal is separated from l/f noise. A faraday modulator using a Tb-doped glass faraday rod modulates the angle of polarization by (u = 5" at w, = 27r x 4.8 kHz before passing through the cell. The probe beam picks up an angle qh due to optical rotation in the cell before passing through a calcite polarizer set to extinction. The measured signal after the polarizer is given by:

A lock-in amplifier measures the first harmonic component of the dominantly second harmonic signal; if there is no optical rotation 4 then the first harmonic has zero amplitude. Using this modulation technique, we have been able to achieve sensitivity down to lo-* rad/&, which is sufficient to measure below 1 t T / a in this experiment.

63

Figure 3. Procedure for zeroing B,. A square wave modulation of &, creates a modulation of the co-magnetometer signal. The steady-state response is indicated by the difference between the dashed lines for B, # 0. Note that the signal changes sign as B, passes through zero, allowing the application of a zerefinding procedure. In case (b), when the applied B, exactly cancels the 3He and K magnetizations, the signal does not show any modulation. This illustrates the insensitivity of the co-magnetometer to regular magnetic field drift.

3.2. Zeroing Fields and Lightshifis

It is critical to the operation of the co-magnetometer and the validity of Equation 7 that the magnetic fields B,, B,, B, and lightshifts L be maintained at zero. This is particularly challenging because the comagnetometer is insensitive to these quantities when they are all near zero. We have developed a modulation technique where an appropriate choice of modulation allows the zeroing of each field. The modulation frequency is less than 1 Hz, far enough away from the 7 Hz resonance of 3He to validate the steady state solution in Equation 6. First consider modulating By and measuring the signal response. Equation (6) simplifies to: N

Zero B,:

dS

-c( B, + 0.

dBY

(9)

Here the signal response is proportional to B, and no other fields or lightshifts. It is possible, therefore, to employ a zero-finding routine that adjusts B, until the signal response and B, are exactly zero. At that zero point, the applied B, exactly cancels all the magnetizations of the atoms in the system as well as any ambient field generated by the magnetic shields. This procedure is illustrated in Figures 3 and 4. We must use a square wave modulation of B, that is sufficiently slow to allow the spin precession transients to decay before a measurement of the signal can be made. The

64

Bc (arb.) Figure 4. Summary of the data collected in Figure 3 including data that extends beyond the linear regime described by Equation (9). A zero-finding routine samples this curve to find the point where the signal response to B, modulation vanishes so that B, = 0

square wave modulation is also not perfectly sharp to reduce excitation of spin precession. Once the B, zero has been found, the remaining fields can be zeroed. By applying a B, modulation around its newfound zero point (modulating B, around the compensation field), we obtain the following response:

as

Zero By: - 0: (b; aBc

+ B y ) / B ,+ L,y,T,

4

0.

Here the quantity we are zeroing is the sum of terms with B y , Lx and the anomalous field b;. Note that the B,B, term is suppressed by being the product of two small factors. The L, term is zeroed independently at a later stage and the whole procedure is iterated several times. By zeroing (b; B y ) we do not render the magnetometer insensitive to anomalous fields because it is sensitive to the difference in anomalous field coupling as shown in Equation 7. To isolate the co-magnetometer response to B,, one must measure the signal response to the second derivative of Be:

+

Zero B,:

d2S

aB,z 0: Bx

-+

0.

We accomplish this by modulating Be between zero and a non-zero value. (If we were to modulate B, symmetrically around zero as we do with all the other fields we should expect to see the more complicated response 8.9 0: Bx(2Bc +

65

With the magnetic fields terms zeroed, the pump and probe beam lightshifts can be zeroed straightforwardly by modulating one while zeroing the other:

dS

Zero Lx: -c( Lx 4 0 , Zero L,:

dL, dS - 0: L , dLX

4

0.

In the experiment we alter the pump lightshift by tuning the angle of the feedback grating with a piezo stack. The probe lightshift is zeroed using a Pockel cell to control the degree of circular polarization and cancel the birefringence in the beam path before the cell. Now that the lightshift Lx is properly zeroed, one must return to re-zero the B, term in Equation (10). The five zeroing procedures described above are repeated several times and the zeroing results averaged to reduce noise. The 3He polarization is very sensitive to experimental conditions and is subject to slow drift. Since B, multiplies most of the terms in Equation ( 6 ) , we must re-zero B, frequently, especially before zeroing other terms. Currently the zeroing procedure is set to zero in this order: B, By B, Bx Lx Bg Bc Lx By Bc L z Bc Bx Lz Bc Bx. The zeroing procedure for all fields is performed approximately every 5 hours while zeroing of B, is performed every few minutes. During recording of the co-magnetometer signal we periodically block the pump beam to perform a measurement of the background probe rotation. By varying the fraction of the time the pump beam is on we can control the equilibrium 3He magnetization. A PID feedback system takes measurements of the 3He polarization from the B, zeroing procedure and makes adjustments to the background measurement time, thereby fixing the 3He polarization to a desired value. 3.3. Calibrating the co-magnetometer

We define the co-magnetometer signal to be the strength of an anomalous field in Tesla that couples exclusively to either K or 3He. One would obtain the same signal from a magnetic field of the same strength measured by an identical magnetometer with the 3He replaced by 4He. We define a calibration constant K that converts the K spin response expressed in Equation (7) into the magnetic field quantity b as follows:

b r KS = bna, - be,

where

K

= (M:reTe)-l.

(14)

66

250 I

1662

1663

1664

1665 1666 Time (Sidereal Days)

1667

1668

Figure 5 . Position of the probe beam (dashed line) is strongly correlated with the signal (solid line). If properly calibrated, the probe beam position measurement can be subtracted from the signal.

Since we cannot directly apply an anomalous field to calibrate the experiment, we determine IE by measuring the slope of the line in Figure 4 while zeroing B,. Combined with the known amplitudes of B, and the applied AB, modulation, the calibration constant can be expressed as:

In practice, the signal S that we measure is in volts from the lock-in amplifier and the calibration accounts for the optical rotation and conversions through the photodiode amplifier and lock-in amplifier.

3.4. Suppressing systematic eflects The primary sources of systematic noise in this experiment involve motion of the pump or probe beams. (1) Slight changes in the birefringence of the optics along the probe beam path, as temperature or beam position drift, can cause significant systematic error. Shuttering the pump beam every 10 s to measure the background probe signal and subtracting this background from the signal is effective in suppressing these effects. (2) If the angle between the pump and the probe beam changes then the K polarization projection onto the probe beam would change accordingly, an effect which is hard to distinguish from precession under the influence of an anomalous field. The orientation of the pump and probe beams is measured by four segment photodiodes as depicted in Figure 2. This correlation can be seen clearly in the long-term data shown in Figure 5. The

67

Sweet Spot

Probe Beam Figure 6 . An off-axis probe beam passing through the cell experiences linear dichroism depending on the angle at which it hits the cell wall, thereby rotating the angle of polarization of the light that passes through.

probe beam motion measured by these position sensors can be calibrated and subtracted directly from the signal. (3) If the probe beam passes through the cell off-center, motion of the probe beam can translate into noise. As depicted in Figure 6 , the transmitted probe beam experiences some linear dichroism due to different amounts of reflection at the interface depending on the angle between the plane of reflection and the incident polarization. As such, the probe beam motion across the cell, caused, for example, by air turbulence, translates into angular noise. However, if the probe beam passes exactly through the center of the cell, this effect vanishes to first order in the probe beam motion. We employ a set of optics before the cell, depicted in Figure 2, that allow us to independently translate and rotate the probe beam on the cell. These optics make it easier to find the ‘sweet spot’ where the probe beam passes through the center of the cell and experiences no dichroism due to small movements. Work is ongoing to discover and eliminate other sources of systematic effects and noise. We are actively pursuing the effects of pump beam motion, temperature fluctuations, optical table deformation, intermittent changes in the cooling water flow and changes in the magnetic field of the room. 4. Gyroscopic effects

It is possible to simulate CPT-violating terms in our experiment for diagnostic purposes by either introducing a lightshift or rotating the table. Both techniques couple to the spins differently than a magnetic field would and thereby qualify as ersatz CPT-violating fields. The K-3He comagnetometer could constitute a compelling gyroscope because of the rela-

68

Figure 7. Left, a diagram of the gyroscope experiment showing the non-contact position sensors and piezo driver. Right, angular velocity data from position sensors (dashed line) and cc-magnetometer signal (solid line) are plotted with no free parameters.

tively small magnetic moment of 3He and the insensitivity to applied magnetic fields and gradients. Signal due to rotation can be expressed as an effective anomalous coupling as follows:

Magnetic fields of magnitude bn and be would cause 3He and K spins, respectively, to both precess at w,. According to Equation (14), the magnetometer signal should be:

where if we have the proper calibration then K M Z ~ ~=T 1. , Solving for wT obtains a simple expression for the rotation signal:

With ye >> yn, the magnetometer is dominantly described by the motion of the 3He nucleus wT P ynb. We excite rotation of the optical table with the co-magnetometer cell at its center using a piezo stack as shown in Figure 7. S i x non-contact position sensors measure the orientation of the table as it rotates. The rotation signal from the position sensors and the rotation signal from the co-magnetometer agree to within the noise of the co-magnetometer. This

69

Frequency (sidereal days-') Figure 8. Lomb periodogram of 6 days of continuous data recording as a function of frequency. The CPT signal is expected to appear at the frequency of 1 (sidereal day)-' shown by the vertical line. Peaks at higher frequency correspond to periodic zeroing of the magnetometer fields.

rotation measurement provides a robust, independent verification of the calibration constant measured during the B, zeroing process. 5. Long term data analysis

Long term data acquisition is ongoing for the purpose of diagnosing sources of long-term drift and collecting CPT data. Because the data contain gaps for background measurements and zeroing of the magnetic fields, it is appropriate to analyze the results using a Lomb periodogram.' It gives the amplitude of a non-linear fit to a sine wave as a function of frequency. The results for a run containing about 6 days of data are shown in Figure 8. At low frequency the noise is dominated by large l/f drifts which we are currently working on eliminating. The existing limit on b, coupling8 corresponds to an effective magnetic field of about 1 fI'. 6. Conclusions

We have described in detail the operation and performance of the K-3He comagnetometer. It has high short term sensitivity, assisted by the magnetic

70

field compensation behavior of the 3He. Studies of long term drifts and systematic effects are the ongoing focus of this experiment. If these drifts can be reduced, we expect to achieve sensitivities to anomalous fields on GeV GeV for the neutron coupling and be the order of b, for the electron coupling. N

N

Acknowledgements We thank Rajat Ghosh, Tom Jackson, Igor Savukov, Charles Sule and Saee Paliwal for assistance in the lab, This work was supported by NASA, NSF, a NIST Precision Measurement grant, and The Packard Foundation.

References 1. D. Colladay and V.A. Kosteleck?, Phys. Rev. D 58, 116002 (1998). 2. V.A. Kosteleck?, C. Lane, Phys. Rev. D 60, 116010 (1999). 3. J.C. Allred, R.N. Lyman, T.W. Kornack, M.V. Romalis, Phys. Rev. Lett. 89, 130801 (2002). 4. T.W. Kornack and M.V. Romalis, Phys. Rev. Lett. 89, 253002 (2002). 5. I.K. Kominis, T.W. Kornack, J.C. Allred and M.V. Romalis, Nature, 422, 596 (2003). 6. A.B. Baranga et al., Phys. Rev. Lett. 80, 2801 (1998). 7. Numerical Recipes in C: the art of scientific computing, W.H.Press et al. (Cambridge University Press, Cambridge, 1992), 2nd ed. 8. D. Bear, R.E. Stoner, R.L. Walsworth, V.A. Kosteleck9, and C.D. Lane, Phys. Rev. Lett. 85, 5038 (2000), Erratum: Phys. Rev. Lett. 89, 209902(E) (2002).

L O R E N T Z VIOLATION A N D GRAVITY

v. ALAN KOSTELECKY Physics Department, Indiana University Bloomington, IN 4 7405, U.S.A . Lorentz symmetry lies at the heart of relativity and is a feature of low-energy descriptions of nature. Minuscule Lorentz-violating effects arising in theories of quantum gravity offer a promising candidate signal for new physics at the Planck scale. A framework is presented for incorporating Lorentz violation into general relativity and other theories of gravity. Applying this framework yields a proof that explicit Lorentz symmetry breaking is incompatible with generic Riemann-Cartan geometries. The framework also enables the construction of all possible terms in the effective low-energy action for the underlying quantum gravity. These terms form the gravitationally coupled Standard-Model Extension (SME), which offers a comprehensive guide to searches for observable phenomena. The dominant and sub-dominant Lorentz-violating terms in the gravitational and QED sectors of the SME are discussed.

1. Introduction Reconciling gravity with quantum mechanics to form a consistent theory of quantum gravity remains a major outstanding problem in theoretical physics. The difficulty of the problem is exacerbated by the lack of experimental guidance. Progress in physics is often made through the combination of theory and experiment working in tandem, but the natural scale for quantum gravity is the Planck scale, which lies some 17 orders of magnitude above our presently attainable energy scales. At first sight, this appears an insuperable barrier to the acquisition of experimental information about quantum gravity. Remarkably, under suitable circumstances, some experimental information about quantum gravity can nonetheless be obtained. The point is that minuscule effects emerging from the underlying quantum gravity might be detected in sufficiently sensitive experiments. To be identified as definitive signals from the Planck scale, such effects would need to violate some established principle of low-energy physics. One promising class of potential effects is relativity violations, arising from breaking the Lorentz symmetry 71

72

that lies at the heart of relativity.' Recent proposals suggest these effects could emerge from strings, loop quantum gravity, noncommutative field theories, or numerous other sources at the Planck scale.' Whatever the nature of the underlying quantum gravity, effective field theory is an appropriate tool for the general description of low-energy signals of Lorentz ~ i o l a t i o n To . ~ be realistic, a theory of this type must reproduce established physics. In Minkowski spacetime, nongravitational phenomena involving the basic particles and forces down to the quantum level are successfully described by the Standard Model (SM) of particle physics. Adding gravitational couplings and the Einstein-Hilbert action for general relativity yields the gravitationally coupled SM, which encompasses all known fundamental physics. This combined theory must therefore be a basic component of any realistic effective field theory. In Minkowski spacetime, relativity violations can be incorporated as additional terms in the SM action describing arbitrary coordinate-independent Lorentz violation, and all dominant contributions at low energies are explicitly known.4 However, the inclusion of Lorentz violation in an effective field theory containing also the Einstein-Hilbert action and the gravitationally coupled SM is more challenging. The study of relativity violations in the corresponding spacetimes requires a framework allowing violations of local Lorentz invariance while preserving general coordinate invariance. Also, since physical matter is formed from leptons and quarks, the framework must be sufficiently supple to incorporate spinors. This talk summarizes a suitable framework that meets all the above criteria, along with some key associated results. The framework described enables the construction of the general low-energy effective field theory, the Standard-Model Extension (SME), which serves as a comprehensive basis for theoretical and experimental studies of Lorentz violation in all gravitational and SM sectors. The talk is based on a selection of results obtained in Ref. 5, to which the reader is referred for more details.

2. Framework

The framework summarized here, appropriate for the comprehensive description of Lorentz violation, is founded on Riemann-Cartan geometry and the vierbein formalism.6 This formalism naturally distinguishes local Lorentz and general coordinate transformations and also allows the treatment of spinors. The basic gravitational fields are the vierbein epa and the spin connection wPab, and the action of the local Lorentz group at

73

each spacetime point allows three rotations and three boosts independent of general coordinate transformations. In this context, Lorentz violation appears in a local Lorentz frame when a nonzero vacuum value exists for one or more quantities carrying local Lorentz indices, called coefficients for Lorentz violation. As an illustrative example for the basic ideas of the framework, suppose a nonzero timelike coefficient k, = ( k ,0, 0,O) exists in a certain local Lorentz frame at some point P. Whenever particles (or localized fields) have observable interactions with k,, physical Lorentz violation occurs. The corresponding Lorentz transformations, called local particle Lorentz transformations, act to boost or rotate particles in the fked local frame at P , leaving k, and any other background quantities unaffected. Note, however, that the local Lorentz frame itself can be changed by local observer Lorentz transformations, under which k, behaves covariantly as a four-vector. Note also that the physics is covariant under general coordinate transformations, as desired, because a change of the observer’s spacetime coordinates x , induces a conventional general coordinate transformation on k , = ePak,. The breaking of Lorentz symmetry is called explicit if k,(x) is specified as a predetermined external quantity, while it is spontaneous if instead k, ( x ) is determined through a dynamical procedure such as the development of a vacuum value. In general, the Lorentz-violating piece SLV of the action for the effective field theory contains terms of the form

SLv 2 J d 4 x e k x J x , where k , is a coefficient for Lorentz violation in the covariant x representation of the observer Lorentz group. Also, J x lies in the corresponding contravariant representation and is a general-coordinate invariant formed from the vierbein, spin connection, and SM fields. The form (1) of terms in the effective action is independent of the origin of the Lorentz violation in the underlying quantum gravity, including whether the violation is spontaneous or explicit. 3. Spontaneous and Explicit Lorentz Violation

With this framework established, various issues concerning observable Lorentz violation can be addressed. One result is that explicit and spontaneous Lorentz violations have distinct implications for the energymomentum tensor. To see this, first separate the action of the full effective

74

theory into a piece Sgravity involving only the vierbein and spin connection and the remainder, Smatter = Smatter,O Smatter,LV, where Smatter,LV contains all Lorentz violations involving matter. Any term in the latter therefore has the general form (l),where the operator J" is now taken to be formed from matter fields fY and their covariant derivatives. For explicit Lorentz violation, consider a particular variation of Smatter for which all fields and backgrounds are allowed to vary, including the coefficients for explicit Lorentz violation, but in which the dynamical fields f Y satisfy equations of motion:

+

GSmatter

s

1

+

+

d42 e(TpYeUa6epa !jSpabbW,ab eJ"Slc,).

(2)

This expression defines the energy-momentum tensor T P " and the spindensity tensor Spa,, as usual. For infinitesimal local Lorentz transformations SePa,dwpab,Sk,, the variation (2) yields the condition

TP' - T'P - ( D ,

-

TDPa)S"Pu = -ePa eub k ~ ( x [ a b ] ) ~ y J ~(3)

on the symmetry of the energy-momentum tensor T P ' , where D, is the covariant derivative, TAPuis the torsion, and (X[ab])",is the representation for the local Lorentz generators. When instead the special variation (2) is induced by a diffeomorphism, the variations bepa, bwpab, dk, are Lie derivatives, yielding the covariant conservation law

(D, - TAA,)T", + TX,,TpA

+ iRabpUSpab = J"D,k,,

(4)

where RabPu is the curvature. In the limit of conventional general relativity, these equations reduce to the familiar expressions TPU= T u pand D,T'", = 0. In the Minkowski-spacetime limit with Lorentz violation, known results4 also emerge. For spontaneous Lorentz violation, the derivation can be adapted to obtain equations similar to (3) and (4),but with the terms involving lc, replaced by zero. This is because all coefficients arising from spontaneous breaking are vacuum field values and therefore must obey equations of motion, so the variation blc, in Eq. (2) is absent. The result can also be understood geometrically. The spacetime geometry implies a set of identities, the Bianchi identities, that are tied to the equations of motion and hence imply certain conditions on the matter sources. However, for sources involving explicit Lorentz violation, these conditions are generically incompatible with covariant conservation laws for the matter. For example, in general relativity the Bianchi identities are D,GPu = 0, the Einstein equations are (2,"" = 8.rrGNTpu,and substitution yields the condition D,TPU = 0, which

75

in the presence of explicit Lorentz violation is incompatible with the result (4). In contrast, spontaneous Lorentz violation yields consistent results, essentially because in this case the coefficients for Lorentz violation form an intrinsic part of the geometrical structure rather than being externally imposed. 4. Low-Energy Effective Action

A wide-ranging application of the general framework summarized here is the construction of all possible dominant terms in the low-energy effective action, independent of the structure of the underlying quantum gravity theory. The full SME effective action at low energies is a sum of partial actions, &ME

= Sgravity

+ SSM+ SLV+... .

(5)

Here, the term Sgravity represents the pure-gravity sector, involving the vierbein and the spin connection and including any Lorentz violation. The term SSMis the SM action with gravitational couplings. The term SLVcontains all Lorentz-violating terms that involve matter fields and dominate at low energies, including minimal gravitational couplings. The ellipsis represents low-energy terms of higher suppression order, including operators of mass dimension greater than four, some of which violate Lorentz symmetry. The pure-gravity action can be written

where the Lorentz-invariant piece C& and the Lorentz-violating piece Lt: involve only epa and wpab. The ellipsis represents possible dependence on nonminimal dynamical gravitational fields, such as the recently proposed cosmologically varying scalar fields that can lead to Lorentz ~ i o l a t i o nThe .~ Lorentz-invariant lagrangian L& can be expanded as usual,

C&

= eR - 2eA

while the Lorentz-violating lagrangian

C t :

+.

,

.,

(7)

has the form

+

+e(kTT)QPYXpuTQ,pyTX@Y. . . .

(8)

The Lorentz-violating matter action SLVcan also be constructed as a series of terms involving both SM and gravitational fields. For illustrative purposes, attention here is restricted to the special limit of single-fermion

76

gravitationally coupled quantum electrodynamics (QED), for which only the dominant and minimally coupled terms are considered. A discussion of the full theory can be found in Ref. 5. In this limit, the relevant U(1)-invariant action is a sum of partial actions for the Dirac fermion $ and the photon A,. The fermion partial action for the QED extension can be written as

where the symbols Pa and M are defined by

ra = y a - c,,,euaeClbyb - dPveua e 6 7 5 7b fi

-e,epa M

-

if,e'lay5 - 3gX,vevaeXbe~cgbc,

= m + im5y5 + a,ePaya

+ b,epay5ya + +H,,ePaeYboab.

(10) (11)

The first term of Eq. (10) and the first two terms of Eq. (11) are conventional, while the others involve Lorentz violation controlled by the coefficients a,, b,, cPv, d,,, e,, f,, gX,,, H,,, which typically vary with position. The covariant derivative D, in Eq. (9) is a combination of the spacetime covariant derivative and the usual U( 1)covariant derivative:

In the photon sector, the partial action is

+

SA = J d 4 x ( L ~ L A ) ,

(13)

where

l F= - I e F 4

P,

FP""- 1 4 e(kF)nXpuFnXFILU 7

L A = ~e(kAF)ncn~,,,AXFpu - e(kA),A".

(14) (15)

The electromagnetic field strength F,, is defined by the locally U(1)invariant form

F,,

-- D,A,

- D,A,

+ T X , , A ~ d , A , - &A,.

(16)

The Lorentz violation in this sector is controlled by the coefficients ( ~ F ) ~ E (X~ ~A ,F, ) , ,and ( k A ) , . In Minkowski spacetime, the coefficients for Lorentz violation in the SME predict a plethora of experimental signals for relativity violations, even when attention is limited to spacetime-constant coefficients for operators of mass dimension four or less. Experimental tests in this limit to date

77

include ones with photon^,^^^ electrons,10T11T12protons and mesons,15 muons,16 neutrinos,l’~~~ and the Higgs.lg In the full SME effective action including the gravitational couplings, the Lorentz-violating terms create spacetime anisotropies and spacetimedependent rescalings of the coupling constants in the field equations, which in turn induce further potentially significant physical effects. The Lorentzviolating behaviors of gravity modes and fundamental particles vary with momentum magnitude and orientation, spin magnitude and orientation, and particle species. Established results for post-newtonian physics, gravitational waves, black holes, cosmologies, and other standard scenarios typically acquire corrections depending on coefficients for Lorentz violation. In the gravitational sector, substantial deviations from conventional physics due to Lorentz violation are likely only in regions of large gravitational fields, such as near black holes or in the early Universe. Nonetheless, observable effects may emerge under suitable circumstances. For example, searches for Lorentz violation are feasible in laboratory and spacebased experiments studying post-newtonian gravitational physics,20including the classic tests of gravitational physics, of the inverse square law, and of gravitomagnetic effects. Similarly, spacetime anisotropies in the equations for gravitational waves21 can be sought in Earth- or space-based experiments. Comparisons of the speeds of neutrinos, light, and gravitational waves which can differ in the presence of Lorentz violation, may also eventually be feasible by observing certain violent astrophysical processes. On a larger scale, anisotropic Lorentz-violating corrections generated for the conventional homogeneous FRW cosmologies have the potential to generate a realistic anisotropic cosmology with detectable effects. One possible class of Lorentz-violating cosmological signals would be alignment anomalies on large angular scales, which have been reported in the WMAP data22but are absent in standard cosm~logies.~~ Certain coefficients for Lorentz violation can also contribute to an effective cosmological constant, dark matter, and dark energy. For instance, the small nonzero cosmological constant may be partially or entirely tied to small Lorentz violation and may also vary with spacetime position.

5. Summary

The gravitationally coupled SME discussed in this talk is the full low-energy effective field theory for gravitation and other fundamental interaction^.^ It offers a comprehensive basis for the study and analysis of experimental

78

tests of Lorentz symmetry, independent of the underlying quantum gravity. T h e detailed exploration of t h e associated theoretical and experimental implications is an open challenge of considerable interest, with the potential to uncover experimental signals from the underlying Planck-scale theory.

Acknowledgments This work was supported in part by NASA grants NAG8-1770 a n d NAGS2194 and by DOE grant DE-FG02-91ER40661.

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3. 4.

5. 6.

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9.

10.

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11.

12. 13.

14. 15.

16. 17.

18.

19. 20. 21. 22. 23.

Phys. Rev. Lett. 83, 2116 (1999); G. Gabrielse et al., Phys. Rev. Lett. 82, 3198 (1999); R. Bluhm et al., Phys. Rev. Lett. 82, 2254 (1999); Phys. Rev. Lett. 79, 1432 (1997); Phys. Rev. D 57, 3932 (1998); D. Colladay and V.A. Kostelecki, Phys. Lett. B 511, 209 (2001); B. Altschul, Phys. Rev. D 70, 056005 (2004). B. Heckel, in C P T and Lorentz Symmetry 11, Ref. 2; L.-S. Hou, W.-T. Ni, and Y.-C.M. Li, Phys. Rev. Lett. 90, 201101 (2003); R. Bluhm and V.A. Kostelecki, Phys. Rev. Lett. 84, 1381 (2000). H. Muller et al., Phys. Rev. D 68, 116006 (2003). L.R. Hunter et al., in C P T and Lorentz Symmetry, Ref. 2; D. Bear et al., Phys. Rev. Lett. 85, 5038 (2000); D.F. Phillips et al., Phys. Rev. D 63, 111101 (2001); M.A. Humphrey et al., Phys. Rev. A 68, 063807 (2003); Phys. Rev. A 62, 063405 (2000); F. Can6 et al., Phys. Rev. Lett. 93, 230801 (2004); V.A. Kostelecki and C.D. Lane, Phys. Rev. D 60, 116010 (1999); J. Math. Phys. 40, 6245 (1999). R. Bluhm et al., Phys. Rev. Lett. 88,090801 (2002); Phys. Rev. D 68, 125008 (2003). KTeV Collaboration, H. Nguyen, in C P T and Lorentz Symmetry II, Ref. 2; OPAL Collaboration, R. Ackerstaff et al., Z. Phys. C 76,401 (1997); DELPHI Collaboration, M. Feindt et al., preprint DELPHI 97-98 CONF 80 (1997); BELLE Collaboration, K. Abe et al., Phys. Rev. Lett. 86,3228 (2001); BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. 92, 142002 (2004); FOCUS Collaboration, J.M. Link et al., Phys. Lett. B 556, 7 (2003); V.A. Kostelecki and R. Potting, Phys. Rev. D 51, 3923 (1995); V.A. Kostelecki, Phys. Rev. Lett. 80, 1818 (1998); Phys. Rev. D 61, 016002 (2000); Phys. Rev. D 64, 076001 (2001). V.W. Hughes et al., Phys. Rev. Lett. 87, 111804 (2001); R. Bluhm et al., Phys. Rev. Lett. 84, 1098 (2000). Recent experimental studies of Lorentz and CPT violation with neutrinos are summarized elsewhere in these proceedings: M.D. Messier (SK); T. Katori and R. Tayloe (LSND); and B.J. Rebel and S.F. Mufson (MINOS). S. Coleman and S.L. Glashow, Phys. Rev. D 59, 116008 (1999); V. Barger et al., Phys. Rev. Lett. 85, 5055 (2000); J.N. Bahcall, V. Barger, and D. Marfatia, Phys. Lett. B 534, 114 (2002); I. Mocioiu and M. Pospelov, Phys. Lett. B 534, 114 (2002); A. de Gouvsa, Phys. Rev. D 66,076005 (2002); G. Lambiase, Phys. Lett. B 560, 1 (2003); V.A. Kosteleckf and M. Mewes, Phys. Rev. D 69, 016005 (2004); Phys. Rev. D 70, 031902(R) (2004); Phys. Rev. D 70, 076002 (2004); S. Choubey and S.F. King, Phys. Lett. B 586, 353 (2004); A. Datta et al., Phys. Lett. B 597, 356 (2004). D.L. Anderson, M. Sher, and I. Turan, Phys. Rev. D 70,016001 (2004); E.O. Iltan, Mod. Phys. Lett. A 49, 327 (2004). A detailed discussion is in C.M. Will, Theory and Experiment in Gravitational Physics, Cambridge University Press, Cambridge, England, 1993. V.A. Kostelecki and S. Samuel, Phys. Rev. D 40, 1886 (1989). G. Hinshaw et al., Astrophys. J. Suppl. 148, 135 (2003). See, for example, A. de Oliveira-Costa et al., Phys. Rev. D 69, 063516 (2004).

SHORT-RANGE TESTS OF THE GRAVITATIONAL INVERSE SQUARE LAW

E.G. ADELBERGER Center for Experimental Nuclear Physics and Astrophysics Box 354290, University of Washington Seattle, W A 98195-4290, U.S.A. E-mail: [email protected] Recent tests of the gravitational inverse square law are briefly reviewed and some upcoming results from the Eot-Wash Group are mentioned.

1. Motivations The motivations for testing the gravitational inverse-squarelaw at all length scales were recently reviewed' with particular emphasis on the possibilities for new physics at submillimeter length scales. The reader is referred to this reference. 2. Results

Reference 1 also gives a complete summary of the experimental constraints as of February 2003. The only new results that I am aware of since the publication of Ref. 1 are: (1) a thorough discussion and reanalysis of the Eot-Wash 10-hole experiments including previously unpublished data.2 (2) a new analysis by Nordtvedt of lunar laser-ranging (LLR) signatures of possible spatial gradients in GN or a. Nordtvedt4 points out that if such gradients exist, fixed in inertial space, the lunar orbit will be polarized in a definite sidereal direction. Because such a polarization of the orbit is so very close to being in resonance with the natural eccentric perturbation of the orbit (the frequency difference is 27r/(9 years)), this test is extremely sensitive. He concludes, for example, that the fine structure constant probably cannot vary across our 80

81

galaxy by more than a part in lolo, and cannot vary across a spaceslice of the entire universe by more than a part in lo4.

Figure 1. Schematic diagram of the pendulum and attractor used in the 6th generation experiment. The 3 small spheres near the top of the pendulum are used, together with 3 larger spheres which rotate continuously outside the vacuum chamber (not shown), to provide a gravitational octupole calibration. The octupole calibration supresses noise from fluctuating external gradients. The rectangular structure 2/3 of the way up the central column of the pendulum consists of 4 mirrors that form part of the twist-readout system.

3. Upcoming Eot-Wash Group results Daniel J. Kapner is completing his thesis on our 6th generation of inversesquare law tests using a much refined version of the same basic idea employed our original work.3 Compared to that work, his experiment employs a pendulum and attractor with smaller dimensions, 21-fold (rather than 10-fold) rotational symmetry, and much better sensitivity and calibration accuracy. Figure 1 shows the heart of his instrument. Figure 2 shows the

82

10’O

1o8 1o6

1oo 1o-2 10-4

I

I

I 1 1 1 1 1

10-6

10-5

I

I

I I 1 1 1 1

10-4

I

I

I I

IIIII

10-3

I

*

I

I I II I

1o-2

X [ml Figure 2. 95% confidence level contraints on new short-range Yukawa interactions. Heavy solid lines show published limits; the heavy dashed line shows the expected sensitivity of Kapner’s experiment. The light lines show theoretical expectations. See Fig. 34 of Ref. 2 for references to the experimental results and theoretical predictions.

expected sensitivity of his experiment to a new Yukawa interaction of the form V ( r )= -aGml m2 exp(-r/X)/r. Results should be available in early 2005.

Acknowledgments It is a great pleasure to acknowledge the skill and energy of my colleagues in the Eot-Wash Group. We are grateful for support from the NSF (Grant PHY-0355012), the DOE and NASA.

83

References 1. E.G. Adelberger, B.R. Heckel and A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003). 2. C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, U. Schmidt, and H.E. Swanson, Phys. Rev. D 70 042004 (2004). 3. C.D. Hoyle, U. Schmidt, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, D.J.Kapner, and H.E. Swanson, Phys. Rev. Lett. 86, 1418 (2001). 4. K. Nordtvedt, private communication (2004).

TESTING CPT CONSERVATION USING ATMOSPHERIC NEUTRINOS

M.D. MESSIER Department of Physics Indiana University Bloomington, IN, 47405 USA E-mail: messierQindiana.edu Recently there has been considerable interest in the possibility that neutrino OScillations are due to the effects of CPT or Lorentz violation. The interactions of cosmic-rays in the atmosphere are a natural source of v,, De,up, and Dp which span flight distances L ranging from 10 km to 12,000 km and energies E from 100 MeV t o 10 TeV. The Super-Kamiokande collaboration has completed an analysis of a 91.7 kiloton-year exposure of the detector to atmospheric neutrinos. These data have been found to be consistent with neutrino oscillations of v p -* vT with parameters 1.5 x lop3 < Am2 < 3.4 x 10W3eV2and sin2 28 > 0.92. In this report, I will present preliminary limits on CPT violating effects derived from these data. This includes tests of whether neutrinos and antineutrinos oscillate in the same way, and searches for deviations from the expected L I E behavior of “conventional” neutrino oscillations.

1. Atmospheric Neutrinos

Atmospheric neutrinos are produced as the decay products of hadron cascades resulting from the interactions of cosmic rays with nuclei in the upper atmosphere. The production of ue, up, and their antiparticles is dominated by the decay chain of pions: 7r+ -+ p+ up, p+ + e+ Cp Yel plus the charge-conjugates. This decay chain produces neutrinos roughly in the ratio (up Vp)/(ue+ f i e ) cv 2 at low energies. Calculations of the atmospheric neutrino fluxes are able to predict this ratio to 5% uncertainty over a broad range of energies, from 0.1 GeV to 10 GeV.13233Based solely on geometrical considerations, the atmospheric neutrino fluxes are expected to be symmetric about the horizon at energies where the Earth’s magnetic field has only a small effect on the propagation of charged particles (> 5 GeV). Atmospheric neutrinos are detected in Super-Kamiokande and other underground detectors via their charged-current interactions with nuclei, u N 4 1 X . The electroweak flavor of the incident neutrino is deduced

+

+

+

+

84

+ +

85

from the flavor of the final state lepton and the neutrino incident direction and energy are reconstructed from the total energy and momentum of the final state particles. 2. The Super-Kamiokande detector The details of the Super-Kamiokande (SK) detector are covered el~ewhere.~ The detector uses a 22.5 kilo-ton fiducial mass of ultra pure water which is viewed by 11,146 20” photomultiplier tubes (PMT). This volume is surrounded by a veto region used to tag entering and exiting particles. Final state particles resulting from neutrino-nucleon interactions are reconstructed based on the time and charge pattern formed by PMT illuminated by Cherenkov radiation. 3. Atmospheric neutrinos in Super-Kamiokande

The SK atmospheric neutrino analysis divides the data sample into four main categories based on the response of the detector. Events classified as fully contained (FC) are neutrino interactions where the final state particles are completely contained inside the inner detector volume. Partially contained (PC) interactions begin in the inner detector volume, but one or more particles (typically a muon) penetrate into the outer veto region surrounding the detector. Neutrino interactions on the rock outside the detector are also observed as upward-going muons and can be separated from downward-going cosmic ray muons based on the reconstructed muon direction. These events are classified as stopping and through-going depending on whether the muon stops inside the inner detector volume, or exits the detector. Typical energy ranges for these events range from 0.110 GeV for FC events, 1-100 GeV for PC and upward stopping events, and 10-104 GeV for upward through-going events. The FC sample is further divided based on the visible energy in the event and the number of visible Cherenkov rings. The data presented here were taken during SK’s first run from April 1996 through November 2001. Counts of events for this period of data taking are shown in Table 1 along with the estimated purity of each sample. The data recorded by SK-I exhibit a significant deviation in both the relative numbers of v p to u, events as well as upward-going to downwardgoing. In the sub-GeV sample we find R = ( p / e ) D A T A / ( p / e ) M C = 0.658 f O.O16(stat.) f O.O32(sys.). In the multi-GeV sample this double ratio R is 0 . 7 0 2 + ~ : ~ ~ ~ ( sf t a0.099(syst.). t.) The ratio of the number of up-

86 Table 1. Event rates in SK-I Fully contained sub-GeV 1-ring p-like multi-ring p-like 1-ring e-like multi-GeV 1-ring p-like multi-ring- .b-like 1-ring e-like Partiallv contained II Upward-going muons stopping through-going

Data

Monte Carlo

CC purity

3227 208 3353

4212.8 322.6 2978.8

94.5% 90.5% 88.0%

651 439 746 647

899.9 711.9 680.5 1034.5

99.4% 95.0% 82.6% 97.3%

417.7 1841.6

721.4 1684.4

N

N

100% 100%

going events to down-going events is U / D = 0.55+::$';(stat.) fO.O05(syst.). These ratios differ significantly from expectations (1.00) but can be explained by introducing neutrino oscillations of v p 4 v, according to the oscillation probability:

P,,-+y,= sin228sin2(1.27Am2[eV2]-L [kml ) , E[GeV] where Am2 is a difference of squared neutrino masses, and 6 is the mixing angle relating the neutrino mass eigenstates to the electroweak states. The neutrino flight distance and energy are L and E respectively. Introducing this oscillation probability into the simulations of atmospheric neutrinos allows us to bring the data and Monte Carlo event rates binned by energy and zenith angle into excellent agreement over the entire energy range.5 The best fit parameters occur at sin226 = 1.02, Am2 = 2.1 x lop3, x 2 = 174.9/177 dof allowing a ranges between 1 . 5l ~ o p 3 < Am2 < 3 . 4 l~o p 3 eV2 sin228 > 0.92 at 90% confidence level. The best fit results are shown in Fig. 1.

4. Fits to CPT violating models 4.1. Allowing diflerent parameters f o r neutrinos and antineutrinos

Phenomenologically,the simplest way to introduce CPT violation is to allow neutrinos and antineutrino oscillations to proceed with different values of sin2 28 and Am2.637,879 Although the SK detector cannot separate neutrino from antineutrino

87

multi-GeV p+PC

multi-GeV e

sub-GeV e

400

100

200

50 -1 t--JO

0

1

coso multi-ring 1

multi-ring NC

-1

coso

1

upstop p

Figure 1. Zenith angle rates for data (points), for up oscillations (boxes).

-+

0

-1

coso UPTh P

1

v, oscillations (line), and no

interactions, there is considerable sensitivity in the up/down ratio to variations in the neutrino and antineutrino oscillations. Any variation in the oscillation parameters away from the preferred values will cause the up/down symmetry to get closer to 1 either by suppressing oscillations of neutrinos from below the detector or by introducing oscillations into neutrinos from above the detector. As an extreme example to illustrate the point, Fig. 2 shows the expected zenith angle rates assuming that oscillations of neutrinos have proceeded with the best fit parameters, but that oscillations of antineutrinos have been switched off. The disagreement in the upward directions is apparent. For this analysis the values of Am2 and Am2 are scanned between lop4 and 10-1 eV2 while the mixing angles 6 and are varied t o minimize x2. The results of the fit are shown in Fig. 3. The resulting allowed region is consistent with Am2 = Am2 and 6 = 8, and hence CPT conservation at

88

multi-GeV b+PC

multi-GeV e

sub-GeV e

00

400

200

50 1

0 coso multi-ring

-1

coso multi-ring NC

coso

I

upstop p

150

100 50 0

-1

coso

Figure 2. Zenith angle rates for data (points), vp 4v, oscillations assuming no antineutrino oscillations (line) and no oscillations (boxes).

1-sigma. 4.2. Deviations

from L I E behavior

One generic feature of many CPT and Lorentz violating models may be a deviation from the (‘standard” L I E dependence of the oscillation probability.10~11912~13J4~15~16 SK has evidence that atmospheric neutrino oscillations follow an L I E pattern,17 so any deviations are expected to be small. Deviations from the conventional oscillation L I E dependence are introduced by modifying the standard oscillation formula in two ways:18,19

P”, +V r

L

= sin2 26sin2(l.27Am2E

f bL),

(2)

and,

L P”, -+vr = sin2 28sin2(1.27Arn2- f cLE), E

(3)

89

10 10 -3

10 -2

10

neutrino hz (ev2)

-’

Figure 3. Allowed region for oscillations allowing different mass splittings for neutrinos and antineutrinos.

+

where the sign is taken for neutrinos and the - sign is taken for antineutrinos. At 1-sigma, SK’s atmospheric neutrinos are consistent with the values b = 0 and c = 0. Preliminary 90% CL limits on b and c are:

< 8 x lop5 km-I (1.6 x c < 7 x lo-’ GeV-lkm-’ b

GeV), (1.4 x

(4)

4.3. Bicycle Model

As an example of one CPT violating model, we have performed a fit to the “bicycle” model described in Refs. 14, 15, 16. Again, since the deviation from the standard L I E dependence is expected to be small, we have performed a fit to the bicycle model using the parameters:

= ii2/ g, Ecrit= ti/ g .

Am&

Using these parameters the standard form for oscillations,

Pup-+,, = sin2 Osin2(1.27Am&L/E),

90

is recovered at neutrino energies below the critical energy Ecrit. Results for this test are shown in Fig 4. The best fit is obtained at a critical energy &it = 138 MeV. At the 1-sigma level the data are consistent with Ecrit = 0, and so an upper limit is set at Ecrit < 5 GeV at 90% CL.

68%CL 90% CL - 99%CL -----

.

10 -l

Best fit (138 MeV, 0.0018 eV2) x2 = 196/197 DOF Ec,,(GeV)

lo

lo2

Figure 4. Allowed region for oscillations in the “bicycle” model.

The bicycle model predicts a sidereal variation in the oscillation probability. At the edge of the 90% CL region found above, this variation is expected to be roughly 15%. Using the current data sample, we expect to have a 2-sigma sensitivity to this variation which is not enough to add significantly to the above analysis. 5 . Summary

In these proceedings I have presented preliminary limits for CPT and Lorentz violation using the Super-Kamiokande atmospheric neutrino data sample. At this time there is no evidence for CPT or Lorentz violating effects and limits on CPT and Lorentz violating parameters introduced in various models of neutrino oscillations have been set.

91

Acknowledgments

I would like t o thank Alan Kosteleckjr for encouragement and for an excellent workshop. I would also like to thank Matthew Mewes for many useful discussions. References 1. G. Barr et al., Phys. Rev. D39,3532 (1989); V. Agrawal, e t al., Phys. Rev.

D53, 1313 (1996); T.K. Gaisser and T. Stanev, Proc. 24th Int. Cosmic Ray Conf. (Rome) Vol. 1694 (1995). 2. M. Honda e t al., Phys. Lett. B248,193 (1990); M. Honda et al., Phys. Lett. D52, 4985 (1995). 3. T. K. Gaisser e t al., Phys. Rev. D54,5578 (1996) 4. Y. Fukuda et al., Nucl. Instrum. Meth. A 501,418 (2003). 5. Y. Fukuda et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 81, 1562 (1998) [arXiv:hep-ex/9807003]. 6. G. Barenboim, L. Borissov and J. Lykken, Phys. Lett. B 534, 106 (2002) [arXiv:hep-ph/0201080]. 7. G. Barenboim, J. F. Beacom, L. Borissov and B. Kayser, Phys. Lett. B 537, 227 (2002) [arXiv:hep-ph/0203261]. 8. G. Barenboim and J. Lykken, Phys. Lett. B 554, 73 (2003) [arXiv:hepph/0210411]. 9. G. Barenboim, L. Borissov and J. Lykken, arXiv:hep-ph/0212116. 10. S. R. Coleman and S. L. Glashow, Phys. Lett. B 405,249 (1997) [arXiv:hepph/9703240]. 11. S. L. Glashow, A. Halprin, P. I. Krastev, C. N. Leung and J. Pantaleone, Phys. Rev. D 56,2433 (1997) [arXiv:hep-ph/9703454]. 12. S. L. Glashow, Nucl. Phys. Proc. Suppl. 77,313 (1999). 13. S. L. Glashow, arXiv:hep-ph/0407087. 14. V. A. Kostelecki and M. Mewes, Phys. Rev. D 70, 031902 (2004) [arXiv:hepph/0308300]. 15. V. A. Kostelecki and M. Mewes, Phys. Rev. D 69,016005 (2004) [arXiv:hepph/0309025]. 16. V. A. Kostelecki, arXiv:hep-ph/0403088. 17. Y. Ashie e t al. [Super-Kamiokande Collaboration], arXiv:hep-ex/0404034. 18. D. Colladay and V. A. Kostelecki, Phys. Rev. D 55,6760 (1997) [arXiv:hepph/9703464]. 19. D. Colladay and V. A. Kostelecki, Phys. Rev. D 58, 116002 (1998) [arXiv:hep-ph/9809521].

NEW TESTS OF LORENTZ INVARIANCE USING OPTICAL RESONATORS

A. PETERS, s. HERRMANN, E.V. KOVALCHUK, AND H. MULLER Institut fur Physik, Humboldt- Universitat zu Berlin, Hausvogteiplatz 5-7, 101 17 Berlin, Germany E-mail: [email protected] We present a modern Michelson-Morley experiment testing Lorentz invariance by comparing the resonance frequencies of two orthogonal cryogenic optical resonators subject to Earth’s rotation over N 1 year. For a possible anisotropy of the speed of Within the general extension of the light c, we obtain Aec/co = ( 2 . 6 f 1.7). standard model of particle physics, we extract limits on 7 parameters at accuracies down to We also discuss prospects for further improving these limits and present a new experimental setup using a high-performance turntable.

1. Introduction

Michelson-Morley (MM) experiments, which test the isotropy of the speed of light, are a sensitive probe for Lorentz violation in electrodynamics. They have a long and fascinating history (see Fig.l), with the classic experiments’ even predating the formulation of Special Relativity. The basic principle of MM experiments is to compare the velocities c, and cy of light propagating in two orthogonal directions. While in the classical setup one compares the speed of light c in two interferometer arms by observing the interference fringes, modern techniques2 employ lasers whose frequencies are stabilized (“locked”) to the resonance frequencies u , , ~= rncZ,,/(2L) of orthogonal optical cavities, where L is the resonator length and m the integer mode number. A variation of u, - uy induced by a rotation of the setup then implies an anisotropy of the speed of light c and thus a violation of Lorentz-invarinace. Here, we present a modern variant of the Michelson-Morley experiment .314

2. Experimental setup In our experimental setup (Fig. 2, 3) we use cryogenic optical resonators (CORES) made from crystalline sapphire, which feature a remarkable long 92

93

1E7 1E9

lEll 4 c c

1E13 1E15 1E17 1880

1900

1920

1940

1960

1980

2000

Figure 1. Accuracy of tests of the isotropy of electromagnetic wave propagation. ’ the established upper bound to the speed of light anisotropy. Ac = c(@) - c ( @ ~ / 2 ) is Experiments until 1930 were performed using optical interferometers, later experiments using electromagnetic cavities.

+

term stability (upper limits on the drift rates are < 2 kHz/ 6 months and 20 Hz/day), making them a valuable tool for high precision measurements. For the MM experiment, it allows us to use solely Earth’s rotation, while previous experiments had no choice but to use relatively fast rotation on a turntable to overcome the large drift rates ( w 10 kHz/day) of room temperature cavities. Two such COREs are located inside a 4 K cryostat, with the light of two Nd:YAG lasers operating at 1064nm coupled to the COREs via windows. A highly optimized laser frequency stabilization scheme uses automatic beam positioning to compensate for cryostat movements caused by refills of coolants and an automatic offset compensation system5 to find the middle of the 100kHz wide resonator lines to about 1Hz accuracy. The frequency difference v, - vv of the lasers is measured by overlapping the beams on a high-speed photodetector and analyzing the heterodyne beat signal. Except for a 10 day break around New Year 2002, the COREs were operated continuously at 4.2 K for more than a year. Usable data (discounting data taken during adjustments or LHe refills, or data sets shorter than 12 hours) started on June 19, 2001 and was taken over 390days until July 13, 2002. A total of 146 data intervals of 12 h to 109 h in length, totaling 3461 h are available. 49 intervals (almost equally distributed in time) are longer N

94

Figure 2. Cavity arrangement for the MM-experiment. The two COREs are closely spaced and mounted inside a copper block to provide common-mode rejection of the influence of temperature variation: similar cavity length changes do not affect the difference of the cavity frequencies.

Figure 3. Setup. Inside the 4K cryostat the two orthogonal COREs are mounted in a copper block (see photo above) to provide common-mode rejection of thermal effects. Laser beams are coupled to the COREs via windows, with polarizers P and lock detectors P D inside the cryostat. For active beam positioning, beams pass through galvanometer (G) mounted glass plates. The horizontal and vertical displacements are adjusted to maximize coupling into the cavities, as measured using the Zfm signal from the detector in reflection.

then 24 h. Fig. 4 shows an example of one such dataset, together with the corresponding Allan-standard deviation.

95

-400

I

I

I

,

I

774.2

774.4 774.6 time since Jan 01,2000[days]

integration time T [s]

Figure 4. Left: Typical data set fitted with a 12 h sinewave amplitude. Linear drift and a constant offset have been removed. Peaks occur every few hours due to automatic LN2 refills. Right: Root Allan variance calculated from similar data (upper curve), and from a quiet part between two LN2-refills (lower curve).

3. Data analysis and results The data from the experiment were analyzed in the framework of the Standard-Model Extension (SME),6i7where the Lagrangian of the Standard Model is generalized by adding all observer Lorentz scalars that can be formed from known particles and Lorentz tensors. The part of the Lagrangian concerned with the electromagnetic field only is

Here, F,, denotes the electromagnetic field, and AX the vector potential. E ~ x , , is the total antisymmetric tensor. The second and third terms vanish if SR is valid. The parameters (I C A F ) ~ are strongly bounded by astrophysical measurements and can be assumed to be zero here. The remaining ( I C F ) ~ X ~ , has 19 independent components. All of these can, in principle, be tested with cavity experiments. However, 10 components describe the dependence of light propagation on the polarization, and are restricted to < 2 . lop3' by astrophysical observations of polarization of distant light sources.6 They are therefore also assumed to be zero here. The remaining 9 components describe boost and rotation invariance of c; they can be arranged into two traceless 3 x 3-matrices ( k e - ) A B (symmetric) and ( k ~ +(antisymmetric), ) ~ ~ plus one additional parameter. The phase velocity of light, and thus the resonance frequency of a cavity, can be expressed in terms of the matrix elements, the orientation of the cavity, and its velocity. From a measurement

96 according to the SME and fit results. Table 1. Signal amplitudes A”A? w @ M 2n/(23h56min) and 0~ = 2 n / l year denote the angular frequencies of Earth’s sidereal rotation and orbit; For each wi, we include only the largest term, in effect dropping all terms proportional to PL. A fit result that has been used to extract a parameter of the SME is set in bold face. The unused fit results lead to additional (but weaker) limits on the elements of k,+. Fit (Hz)

Wi

1.82f 1.91 -0.51 f 1.26 -0.43 f 1.68 -0.01 f 0.57

0.37f0.56

0.50f0.55

0.67 f 1.58

0.89f 1.74 1.83 f 1.83

0.25f 0.55 0.97f 0.53 0.75 f0.58

of the cavity frequency as a function of orientation and velocity, bounds for the 9 parameters can in principle be deduced. To perform this analysis, we divide the available data into blocks of 12 h (or 24 h for fitting the 24 h-period signals) length for the purpose of fitting violation signals. A simultaneous least-squares fit of a constant offset, a linear drift and the amplitude of a sinusoidal signal at a fixed frequency and phase as suggested by the test theory was performed for each data block; Fig. 4 shows an example. In total, 199 fits of 12 h data blocks and 48 fits of 24 h data blocks are obtained. The individual fit results are combined to the final result by vector averaging. In the SME, the hypothetical Lorentz violation signal (vz vy)/V = AS sin wiTe +A? coswiTe, where V = (vz vy)/2, has Fourier components at 6 frequencies wi (see Table 1). This assumes that the cavity length L is not substantially affected by the hypothetical Lorentz violation.8 As defined by Kostelecki et aZ.,6 Te = 0 on March 20, 2001, 11:31 UT.

xi

+

97

Because our data extends over more than one year, in the vector average we can resolve the 6 signal frequencies and extract a set of bounds on 7 components of ke- and Pio+:

U

1.7 f 2.6

1.7 f 2.6 -6.3 f 12.4

b

3.6 f 9.0

-6.3 f 12.4 3.6 f 9.0 -(a

ko+ =

+ b)

0

14 f 14

-1.2 f 2.6

-14 f 14

0

0.1 f 2.7

1.2 f 2.6 -0.1 f 2 . 7

0

with a - b = 8.9f4.9. These limits are about two orders of magnitude lower than the ones from previous mea~urements.~ Furthermore, while previously only linear combinations of the parameters entering Pio+ were known, our experiment allows a separate determination due to the > 1year span of our data. Meanwhile, these results have been confirmed and partially improved upon in later experiments.lO~l’ 4. Next generation experiments Future version of the experiment will employ various modifications (see Table 2) to increase accuracy by up to three orders of magnitude. The most significant improvement will be the use of a precison turntable to rotate the experimental setup at an optimal rate. While the low drift of COREs does allow us to perform measurements relying solely on Earth’s rotation (which would hardly be possible with room temperature cavities), the stability data in Fig.4 indicates that rotation rates (rotation period = 27) corresponding to the minimum of the Allan-variance at 30 - 300 seconds should result in much higher sensitivity. At a rate of 0.2/min the 7 . frequency stability is more than 10 times better than on the 12 h time scale used so far. Accumulating 500 measurements per day (2 per turn), one should thus be able to reach the 1O-l’ level of accuracy. Further improvements will include the use of a new generation of COREs, which by a combination of greater length and increased finesse

-

N

N

98

Table 2.

Comparison between current and next generation experiment

Rotation Resonators

Cryostat

Optical bench

Previous setup

Advanced setup

Earth’s rotation

Precision turntable

1 turn/day

1000 turns/day

Finesse = 100000

Finesse = 250000

AU = 50 kHz

AU = 5 kHz

2 days LHe

28 days LHe

3 hours LN2

28 days LN2

no inner vacuum chamber

inner vacuum chamber

attached to 4.2 K tank

flexible link to 4.2 K vibration-isolated

Optical coupling

windows

optical fiber

Vibration control

passive

active

/ windows

would offer a 10-fold decrease in linewidth. This should directly lead to a corresponding improvement in lock stability and - if used in conjunction with high performance vibration isolation - measurement sensitivity. To avoid disturbances from the refilling of cryogenic liquids, the resonators would also be mounted in a new large volume dewar with a hold-time of several weeks. All together, a 100 - 1000-fold improvement in measurement sensitivity should thus be relatively easy to reach. To utilize fully the increased measurement sensitivity in obtaining new limits on Lorentz-violation, it is essential to minimize systematic effects due to active rotation. First of all, this necessitates the use of a highperformance turntable (in our case using airbearings) with minimal deviations from perfect circular motion and precisely controlled angular velocity. Furthermore, parameters such as tilt and center-of-mass location have to be precisely controlled, while environmental effects such as temperature gradients and electromagnetic stray-fields have to be shielded as well as possible. It is important to note, though, that systematic effects don’t have to be controlled to level of the targeted accuracy: as the experiment is still performed on the rotating Earth, Lorentz-violation signals should generally not arise at the table-rotation frequency or its harmonics, but rather be shifted by multiples of the Earth’s rotation frequency. Thus, for measurement durations of more than 24 hours it should be possible to distinguish

99

Figure 5 .

Michelson-Morley experiment on rotary table.

rotation-table related disturbances from Lorentz-violation signals. However, as this strictly only holds as long as the systematic effects themselves are not modulated by Earth’s rotation (e.g., by day/night temperature fluctuations), it is still essential to minimize the underlying systematic effects.

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As a first step to a new measurement combining all envisioned improvements, we already implemented a precursor experiment set up on a highprecision air-bearing turntable (see Fig. s), but otherwise using existing equipment (i.e., cryostat, resonators, lock-electronics). During the first few days of operation with rotation periods of -100 seconds, some rotation related systematic effects were still visible. Still, 24 hours of these data were already sufficient to obtain limits on Lorentz violating coefficients comparable to those from the previous 1 year long measurement without active rotation. Thus, an improvement by 1order of magnitude for this setup, and 2-3 orders of magnitude for a next generation experiment using improved resonators in a new cryostat seem obtainable.

Acknowledgments This work has been supported by the Deutsche Forschungsgemeinschaft and the Optik-Zentrum Konstanz. We also thank Gerhard Ertl at the FritzHaber-Institute of the Max-Planck-Society, Claus Braxmaier at EADS Astrium, and Jiirgen Mlynek at the Humboldt-University of Berlin for making this project possible.

References 1. A.A. Michelson, Am. J. Sci.22, 120 (1881); A.A. Michelson and E.W. Morley, Am. J . Sci 34, 333 (1887). 2. A. Brillet and J.L. Hall, Phys. Rev. Lett. 42, 549 (1979). 3. H. Miiller et al., Phys. Rev. Lett. 91, 020401 (2003). 4. H. Muller et al., Appl. Phys. B 77,719 (2003). 5. H. Muller et al., Opt. Lett. 28, 2186 (2003). 6. V.A. Kosteleckf and M. Mewes, Phys. Rev. D 66, 056005 (2002). 7. D. Colladay and V.A. Kosteleckf, Phys. Rev. D 57, 6760 (1997). 8. H. Miiller et al., Phys. Rev. D 67,056006 (2003). 9. J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003). 10. P. Wolf et al., Phys. Rev. Lett. 90, 60402 (2003). 11. P. Wolf et al., Gen. Rel. Grav. 36, 2351 (2004); Phys. Rev. D, in press

(hep-ph/0407232).

ATOMIC CLOCKS ON EARTH AND IN SPACE FOR TESTS OF FUNDAMENTAL PHYSICS AND NAVIGATION

K. GIBBLE T h e Pennsylvania State University, 10.4 Davey Laboratory 232, University Park, PA 16802, U.S.A. E-mail: [email protected] I review the performance objectives for RACE, the Rubidium Atomic Clock Experiment that was to fly on the International Space Station. RACE’s performance could, in addition to significantly advancing clock tests of general relativity and Lorentz invariance, enable precise interplanetary navigation. I will also describe a juggling clock experiment that can sensitively probe interatomic forces at Angstrom ranges.

1. Introduction

The accuracies and stabilities of laboratory atomic clocks have surpassed 1 part in A number of clock experiments, PARCS, RACE, and ACES, were slated to fly on the ISS. All three aimed for high accuracy and stability. RACE was designed for ultra-high stability by taking advantage of the small Rb clock shift and the juggling techniques we have demonstrated. RACE’s stability goal is 3 x for one second of averaging and the accuracy Measuring the frequency differences of goal could approach 1 part in RACE and a rubidium fountain clock on Earth would dramatically improve the classic clock tests of general relativity. We have described a number of measurements and the RACE design previously in Ref. 1. In addition, new or better bounds can be put on a host of Lorentz violating terms by preparing the atoms in different spin states where the frequency differences are independent of magnetic field variations.2 Early in 2004, NASA announced a significant change in its priorities which emphasizes exploration of the Moon and interplanetary missions. Clocks such as ACES, PARCS, and RACE in an Earth orbit can support precise interplanetary navigation. Because of the clock’s high accuracy, interplanetary navigation may have a precision comparable to that of GPS 101

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navigation on Earth. Here I will describe a possible system, some of the considerations, and the requirements for the clocks. I will also describe a laboratory experiment which juggles atoms in an atomic fountain to demonstrate a new type of scattering measurement. It will directly measure the difference of scattering phase shifts to yield a very sensitive probe of atom-atom interactions at short distance scales.

2. Precise Interplanetary Navigation

One can imagine a variety of schemes to navigate to other planets that take advantage of atomic clocks. One limit is to construct a GPS type system around the planet of i n t e r e ~ tFor . ~ such a system, because the clock ranges to the planet are short, the clock performances can be modest and therefore the clocks can be relatively small. Another limit of the spectrum is to construct a system around the Earth. For such a system, the clock performance must be much better due to the long range to the planet of interest. While these clocks are also larger and more massive, they must only be launched to an orbit around the Earth. We lay out the possible performance of such a system of very good clocks orbiting the Earth which is one realization of very-long baseline interferometry. We consider clocks such as RACE in nearly geosynchronous orbits around the Earth. With a baseline equal to the diameter of a geosynchronous orbit, a clock stability of 1 x 1014 implies a transverse position uncertainty at Mars of 10 meters, comparable to GPS on Earth. The range from the clock to the receiver can be known to centimeter precision using the delay in the two-way communication time. To be most useful, a navigation system should have rapid position updates with little time lag. It is therefore essential that the clocks transmit their position and time, as in GPS, so that the receiver can calculate its position. This is essentially the reverse of the current operation of the NASA Deep Space Network (DSN) in which the DSN receives signals from a vehicle, calculates its position, and then sends new navigation instructions to the vehicle. For the range, after an initial two-way link, GPS carrier phase techniques can be used for range updates without data latency. The high-accuracy clocks are intrinsically stable and a receiver on the planet of interest, such as Mars, can be used to measure the clock offsets at a known position. To measure precisely small time differences, the atomic clocks will have to be in orbit to avoid uncertainties due to propagation delays in the Earth’s atmosphere. This is an important error for the DSN which

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uses large Earth-base radio-frequency antenna. One source of systematic position error is propagation delays in the Martian atmosphere. Because both clocks have an angular separation of hundreds of microradians, the propagation delay from both clocks will be largely common. For signal-tonoise and power and weight considerations, radio-frequency links will need to be replaced with laser links where radio-frequency signals are encoded on the laser light. While such a system has the advantage that it can be used for navigation to any planet, it is important to note that it needs to be augmented with inertial navigation to allow navigation when a vehicle is on the occluded side of a planet. Microgravity allows a significant improvement in clock performance. The high short-term stability of clocks like RACE may allow rapid and accurate interplanetary navigation. We hope that a navigation purpose would also allow such clocks to be used for the ultra-precise tests of fundamental physics for which they were conceived.

3. Direct measurement of scattering phase shifts

In our laboratory we are currently building a juggling Cs fountain experiment to perform a new type of scattering experiment. The experiment will accurately measure atomic scattering phase shifts using atomic clock techniques. We juggle by launching two balls of ultra-cold Cs atoms with a short time delay.4 The first juggled ball will be prepared in one of the Cs clock states. The microwaves excite and probe the atoms during successive passages through the cavity. The second ball is prepared in an internal energy eigenstate. When the two balls collide above the microwave cavity, the atoms in the first ball are in a coherent superposition of the two clock states and each of these states scatters off of the atoms in the second ball, experiencing an s-wave phase shift. In a clock, the unscattered as well as the scattered components of each atom are detected, resulting in a density dependent frequency shift. Here, however, we exclude the unscattered part of each clock atom in the first ball and detect only the scattered fraction after it returns through the microwave cavity. Because the scattering differentially shifts the phases of the two internal states of the clock atoms, the microwaves convert this phase difference into a population difference on the return passage. By detecting the populations, the difference in the s-wave scattering phase shifts can be directly measured as a phase shift of a Ramsey fringe pattern. This phase shift does not depend on the atomic density (in the single collision limit); the number of scattered atoms, the ampli-

104

tude of the Ftamsey fringe, is proportional to the density. This technique may enable extremely precise measurements of the differences of scattering phase shifts for a variety of spin states in a range of well-defined collision energies. Because the scattering is s-wave, it is sensitive to interatomic forces at ranges as close a few Angstroms.

Acknowledgments We acknowledge financial support from the NASA Microgravity program, the NSF, the Office of Naval Research, and Penn State University.

References 1. See C. Fertig, I. Rees, K. Gibble, J. Prestage, B. Klipstein, and R. Thompson, in CPT and Lorentz Symmetry II, V.A. Kosteleckg, ed., World Scientific, Singapore, 2002. 2. R. Bluhm, V.A. Kosteleckf, C.D. Lane, and N. Russell, Phys. Rev. Lett. 88, 090801 (2001); R. Bluhm, V.A. Kosteleckf, C.D. Lane, and N. Russell, Phys. Rev. D 68, 125008 (2003). 3. See S. Lichten, Lunar/Mars Navigation and Telecom Systems: GPS and GPS-like Systems, and the Role of Precise Clocks, in the Proceedings of the 2004 NASA/JPL Workshop on Physics for Planetary Exploration (in press). 4. R. Legere and K. Gibble, Phys. Rev. Lett. 81,5780 (1998).

SPACETIME SYMMETRIES AND VARYING SCALARS

RALF LEHNERT Department of Physics and Astronomy Vanderbilt University, Nashville, Tennessee, 37235 E-mail: ralf.lehnertOvanderbilt.edu This talk discusses the relation between spacetimedependent scalars, such as couplings or fields, and the violation of Lorentz symmetry. A specific cosmological supergravity model demonstrates how scalar fields can acquire timedependent expectation values. Within this cosmological background, excitations of these scalars are governed by a Lorentz-breaking dispersion relation. The model also contains couplings of the scalars to the electrodynamics sector leading to the time dependence of both the fine-structure parameter 0: and the 0 angle. Through these couplings, the variation of the scalars is also associated with Lorentz- and CPTviolating effects in electromagnetism.

1. Introduction

Despite its phenomenological success, the Standard Model of particle physics leaves unresolved a variety of theoretical issues. Substantial experimental and theoretical efforts are therefore directed toward the search for a more fundamental theory that includes a quantum description of gravity. However, most quantum-gravity effects in virtually all leading candidate models are expected to be minuscule due to Planck-scale suppression. Recently, minute violations of Lorentz and CPT symmetry have been identified as promising Planck-scale signals.' The idea is that these symmetries hold exactly in established physics, are accessible to ultrahighprecision tests, and can be broken in various quantum-gravity candidates. As examples, we mention strings,2 spacetime f ~ a m ,nontrivial ~,~ spacetime t ~ p o l o g yloop , ~ quantum gravity,6 and noncommutative g e ~ m e t r y . ~ The emerging low-energy effects of Lorentz and CPT breaking are described by the Standard-Model Extension (SME).8 The SME is a fieldtheory framework at the level of the usual Standard Model and general relativity. Its flat-spacetime limit has provided the basis for numerous experimental and theoretical studies of Lorentz and CPT violation involving mesons,9~10i11~12 b a ~ y o n s electron^,^^^^^^^^ , ~ ~ ~ ~ ~ ~photons,lg ~ ~ muons,2o and 105

106

the Higgs sector.21 We remark that neutrino-oscillation experiments offer the potential for d i s ~ o v e r y . ~ > ~ ~ * ~ ~ Varying scalars are another feature of many approaches to fundamental physics. Effective couplings, for instance, typically acquire time dependencies in models with extra dimension^.^^ Another class of models contains scalar fields, which can acquire time-dependent expectation values driven by the expansion of the universe. For example, in modern approaches to cosmology, such as quinte~sence,~~ k essence,26or inflation,27 scalar fields are frequently invoked to explain certain observations. In the present talk, it is demonstrated that the above potential quantumgravity features are interconnected. In particular, spacetime-dependent scalars are typically associated with Lorentz and possibly CPT violation. In Sec. 2, general arguments in favor of this claim are given. For further illustrations and some specific results, a toy model is introduced in Sec. 3. Lorentz violating effects within the scalar sector of our toy cosmology are discussed in Sec. 4. Section 5 discusses the Lorentz and CPT breaking in the electrodynamics sector of the model. Section 6 contains a brief summary. 2. General arguments

A spacetime-dependent scalar, regardless of the mechanism driving the variation, typically implies the breaking of spacetime-translation invariance. Since translations and Lorentz transformation are closely linked in the Poincare group, it is reasonable to expect that the translation-symmetry violation also affects Lorentz invariance. Consider, for instance, the angular-momentum tensor J P ” , which is the generator for Lorentz transformations:

J

P’-l -

d3x (9°P”z” - Bo”sP).

(1)

Note that this definition contains the energy-momentum tensor P”, which is not conserved when translation invariance is broken. In general, JP”” will possess a nontrivial dependence on time, so that the usual time-independent Lorentz-transformation generators do not exist. As a result, Lorentz and CPT symmetry are no longer assured. Intuitively, the violation.of Lorentz invariance in the presence of a varying scalar can be understood as follows. The 4-gradient of the scalar must be nonzero in some regions of spacetime. Such a gradient then selects a preferred direction in this region. Consider, for example, a particle that interacts with the scalar. Its propagation features might be different in the

107

directions parallel and perpendicular to the gradient. Physically inequivalent directions imply the violation of rotation symmetry. Since rotations are contained in the Lorentz group, Lorentz invariance must be violated. Lorentz violation induced by varying scalars can also be established at the Lagrangian level. Consider, for instance, a system with varying coupling [(x) and scalar fields 4 and @, such that the Lagrangian C contains a term [(x) 8’q58,@. The action for this system can be integrated by parts (e.g., with respect to the first partial derivative in the above term) without affecting the equations of motion. An equivalent Lagrangian Cf would then obey

L’

3

-K’”q50,@,

(2)

where KP = a,[ is an external nondynamical 4-vector, which clearly violates Lorentz symmetry. We remark that for variations of [ on cosmological scales, KP is constant to an excellent approximation locally-say on solarsystem scales. 3. Specific cosmological model

In the remainder of this talk, we illustrate the result from the previous section within a specific supergravity model. This model generates the variation of two scalars A and B in a cosmological context. It leads to a varying fine-structure parameter Q and a varying electromagnetic 8 angle. The starting point is pure N = 4 supergravity in four spacetime dimensions. Although unrealistic in its details, it can give qualitative insights into candidate fundamental physics because it is a limit of N = 1 supergravity in eleven dimensions, which is contained in M-theory. When only one graviphoton FPv is excited, the bosonic part of pure N = 4 supergravity reads28i29

Here,

FPv

B(A2

+ B2 + 1)

+ + +

A(A2 B2 - 1) (4) (1 A2 B 2 ) 2- 4A2 = ECL~P~F p ( T / 2 denotes the dual field-strength tensor, and g = M=

+ + B 2 ) 2- 4A2 ’

(1 A2

N=

-det(g,,). We remark that the redefinition FPv -+ the explicit appearance of the gravitational coupling of motion.

F p v / f i removes K.

in the equations

108

As a further ingredient, we gauge the internal SO(4) symmetry of the full N = 4 supergravity Lagrangian, which supports the interpretation of F”” as the electromagnetic field-strength tensor. This leads to a potential for the scalars A and B that is unbounded from below.30 At this point, we take a phenomenological approach and assume that in a realistic situation stability must be ensured by additional fields and interactions. At first order, we can then model the potential for the scalars with the following mass-type terms

which we add to CSgin Eq. (3). The full N = 4 supergravity Lagrangian also contains fermionic matter.28 In the present cosmological context, we can effectively represent the fermions by the energy-momentum tensor Tp, of dust describing galaxies and other matter:

As usual, p is the energy density of the matter and up is a unit timelike vector orthogonal to the spatial hypersurfaces. We are now ready to search for cosmological solutions of our supergravity model. We proceed under the usual assumption of an isotropic homogeneous flat (k = 0) Friedmann-Robertson-Walker universe with the conventional line element ds2 = dt2 - a 2 ( t )(dz2

+ dy2 + dz2) .

(7)

Here, a(t) denotes the scale factor and t the comoving time. Since isotropy requires F,” = 0 on large scales, our cosmology is governed by the Einstein equations and the equations of motion for the scalars A and B. Note that the fermionic matter is uncoupled from the scalars at tree level, so that we can take T,, as covariantly conserved separately. It then follows that p ( t ) = cn/a3(t),where c, is an integration constant. Although analytical solutions within this cosmological model can be found in special numerical integration is necessary in general. A particular solution is depicted in Figs. 1 and 2, where the following priors

109

have been used:31 mA = 2.7688

, GeV ,

x lop4’ GeV

m~ = 3.9765 x

cn = 2.2790 x 10-84GeV2 , a(tn) = 1 1 A(tn) = 1.0220426 , A(tn) = -8.06401 x 10-46GeV , B(t,) = 0.016598,

B(tn)= -2.89477

x 10-45GeV.

(8)

Here, the dot denotes differentiation with respect to the comoving time, and the subscript n indicates the present value of the quantity. For our present purposes, the details of this solution are less interesting. Note, however, that the scalars A and B have acquired a dependence on the comoving time t , so that they vary on cosmological scales.

1.o

0.8 0.6

0.4 0.2

0

0.2

0.4

0.6

0.8

1.o

fractional comoving time t/t, Figure 1. Scale factor a@) versus fractional comoving time t / t , . It turns out that the expansion history of this model matches closely the observed one.

Such a spacetime variation of A and B has various implications for the scalars themselves and (due to the coupling of A and B to P”) for electro-

110

1.2

Q W

8

P

-9 2

1.o

0.8 0.6

0.4 0.2

0

0.01

0.02

0.03

0.04

0.05

fractional comoving time t / t , Figure 2. Time dependence of the scalars A and B. At late times, the scalars approach constant values.

dynamics. These implications are discussed in Sec. 4 and 5, respectively. 4. Effects in the scalar sector

To gain insight into how the time-dependent cosmological background solutions Ab and Bb affect the scalars themselves, we investigate the propagation properties of small localized excitations 6A and SB of the cosmological background Ab and Bb. For such a study, it is appropriate to work in local coordinates, which we take to be anchored at the spacetime point XO. Substituting the ansatz

A ( s )= & ( Z ) B(X) = & ( X )

+ 6A(z) + bB(z)

(9)

for the scalars into the equations of motion for A and B determines the dynamics of the perturbations &Aand 6B. One obtains the following linearized equations:

+ 0 = [2APaP]6A + [O

0 = [O- 2BPaP 2miBzI6A - [2APa,- 2APBP- 4m;AbBb]6B1

+

+

2BPdP 6m;Bz - APAP B P B P ] 6 B . (10) Here, Ab and Bb as well as ACC = B;lap"Ab, and BP = B-'aPBb b are evaluated at z = zo.

-

111

Equation (10) determines the propagation features of 6A and 6 B in the varying cosmological background. In the context of this equation, /Ip and B p are external nondynamical vectors selecting a preferred direction in the local inertial frame. It follows that the propagation of 6A and 6B in the varying cosmological background fails to obey Lorentz symmetry. This result carries over to quantum theory: the traveling disturbances 6A and 6B would be seen as the effective particles corresponding to the scalars A and B, so that such particles would violate Lorentz invariance. We remark that the usual ansatz with e x p ( 4 p . z ) yields the plane-wave dispersion relation. As expected, this dispersion relation contains the fixed vectors Ap and B p contracted with the momentum pp implying Lorentz breaking. In addition, this dispersion relation exhibits imaginary terms leading to decaying solutions. This is consistent with the nonconservation of 4-momentum due to the violation of translational invariance. We also point out that the equations of motion are coupled, so that a plane-wave is a linear combination of 6A and 6B. More details can be found in Ref. 31. 5. Effects in the scalar-coupled sector Instead of excitations 6A and 6 B of the scalar fields, we now consider excitations of Fpw in our background cosmological solution Ab and B b . Again, it is appropriate to work in a local inertial frame. Then, the effective Lagrangian C ,,,, for localized Fpw fields follows from Eq. (3)

L,,,,

= -$bfbFpwFpw- $NbFpwfipw,

(11)

where i&, and N b are determined by the time-dependent cosmological solutions Ab and &, for the scalars. The physics content of L,,,, is best extracted by comparison to the conventional electrodynamics Lagrangian C,, given by

This shows that e2 = 1/Mb and 8 = 4n2Nb. Since Mb and N b are functions of the varying scalar background Ab and B b , the electromagnetic couplings e and 8 are no longer constant in general. In other words, in our supergravity cosmology the fine-structure parameter a! and the electromagnetic 6 angle acquire related spacetime dependences. The evolution of Q in the present model with initial conditions (8) is depicted in Fig. 3. Also shown is the recently reported Webb d a t a ~ e obtained t~~ from measurements of high-redshift spectra.

112

0

0.5

1.0

1.5

2.0

2.5

3.0

redshift z Figure 3. Relative change in the finestructure parameter a versus redshift. For comparison, the Webb data has been included into the the plot. Although disagreeing in detail, the data and the model exhibit variations of a of roughly the same order of magnitude.

Lorentz violation in our effective electrodynamics can be clearly established by inspection of the modified Maxwell equations resulting from the Lagrangian (11): 1 2 1 -dpFpy - - ( d p e ) F p y -+v)F,, =0 . e2 e3 47r In our supergravity cosmology, the gradients of e and 6 appearing in Eq. (13) are nonzero, approximately constant in local inertial frames, and act as a nondynamical external background. This vectorial background selects a preferred direction in the local inertial frame violating Lorentz symmetry. We remark that the term exhibiting the gradient of 0 can be identified with a Chern-Simons-type contribution to the modified Maxwell equations. Such a term, which is contained in the minimal SME, has received a lot attention recently.33 For example, it typically leads to vacuum Cerenkov radiation.34

+

6. Summary

In this talk, it has been demonstrated that the violation of spacetimetranslation invariance is closely intertwined with the breaking of Lorentz

113

symmetry. More specifically, a varying scalar-regardless of the mechanism driving the variation-is associated with a nonzero gradient, which selects a preferred direction in spacetime. This mechanism for Lorentz violation is interesting because varying scalars appear in many cosmological contexts.

Acknowledgments Funding by the Fundaqiio para a Ciencia e a Tecnologia (Portugal) under Grant No. POCTI/FNU/49529/2002 and by the Centro Multidisciplinar de Astrofisica (CENTRA) is gratefully acknowledged.

References 1. For an overview see, e.g., CPT and Lorentz Symmetry 11, edited by V.A. Kostelecki (World Scientific, Singapore, 2002). 2. V.A. Kostelecki and S. Samuel, Phys. Rev. D 39, 683 (1989); Phys. Rev. Lett. 63,224 (1989); 66,1811 (1991); V.A. Kosteleck? and R. Potting, Nucl. Phys. B 359, 545 (1991); Phys. Lett. B 381, 89 (1996); Phys. Rev. D 63, 046007 (2001); V.A. Kostelecky et al., Phys. Rev. Lett. 84,4541 (2000). 3. G. Amelino-Camelia et al., Nature (London) 393, 763 (1998). 4. D. Sudarsky et al., gr-qc/0211101; Phys. Rev. D 68,024010 (2003). 5. F.R. Klinkhamer, Nucl. Phys. B 578,277 (2000). 6. J. Alfaro et al., Phys. Rev. Lett. 84,2318 (2000); Phys. Rev. D 65,103509 (2002). 7. S.M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001); Z.Guralnik et al., Phys. Lett. B 517,450 (2001); A. Anisimov et al., Phys. Rev. D 65,085032 (2002); C.E. Carlson et al., Phys. Lett. B 518,201 (2001). 8. D. Colladay and V.A. Kosteleckf, Phys. Rev. D 55,6760 (1997); 58,116002 (1998); V.A. Kostelecki and R. Lehnert, Phys. Rev. D 63, 065008 (2001); V.A. Kosteleckf, Phys. Rev. D 69,105009 (2004). 9. KTeV Collaboration, H. Nguyen, in Ref. 1; OPAL Collaboration, R. Ackerstaff et al., Z. Phys. C 76,401 (1997); DELPHI Collaboration, M. Feindt et al., preprint DELPHI 97-98 CONF 80 (1997); BELLE Collaboration, K. Abe et al., Phys. Rev. Lett. 86,3228 (2001); BaBar Collaboration, B. Aubert et al., hep-ex/0303043; FOCUS Collaboration, J.M. Link et al., Phys. Lett. B 556,7 (2003). 10. V.A. Kosteleckf and R. Potting, Phys. Rev. D 51,3923 (1995). 11. D. Colladay and V.A. Kostelecki, Phys. Lett. B 344,259 (1995); Phys. Rev. D 52,6224 (1995); V.A. Kostelecki and R. Van Kooten, Phys. Rev. D 54, 5585 (1996); 0.Bertolami et al., Phys. Lett. B 395, 178 (1997); N. Isgur et al., Phys. Lett. B 515,333 (2001). 12. V.A. Kostelecki, Phys. Rev. Lett. 80,1818 (1998); Phys. Rev. D 61,016002 (2000); Phys. Rev. D 64,076001 (2001). 13. D. Bear et al., Phys. Rev. Lett. 85, 5038 (2000); D.F. Phillips et al., Phys. Rev. D 63,111101 (2001); M.A. Humphrey et al., Phys. Rev. A 68,063807

~

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14. 15. 16.

17. 18.

19.

20. 21. 22.

23. 24.

25. 26. 27. 28. 29. 30. 31. 32. 33.

34.

(2003); Phys. Rev. A 62, 063405 (2000); V.A. Kosteleckf and C.D. Lane, Phys. Rev. D 60,116010 (1999); J. Math. Phys. 40,6245 (1999). R. Bluhm et al., Phys. Rev. Lett. 88, 090801 (2002). F. Can& e t al., physics/O309070. H. Dehmelt et al., Phys. Rev. Lett. 83, 4694 (1999); R. Mittleman et al., Phys. Rev. Lett. 83,2116 (1999); G. Gabrielse et al., Phys. Rev. Lett. 82, 3198 (1999); R. Bluhm et al., Phys. Rev. Lett. 82, 2254 (1999); Phys. Rev. Lett. 79, 1432 (1997); Phys. Rev. D 57,3932 (1998). B. Heckel, in Ref. 1; L.-S. Hou et al., Phys. Rev. Lett. 90,201101 (2003); R. Bluhm and V.A. Kosteleckf, Phys. Rev. Lett. 84, 1381 (2000). H. Muller et al., Phys. Rev. D 68,116006 (2003); R. Lehnert, Phys. Rev. D 68,085003 (2003); J. Math. Phys. 45,3399 (2004). S.M. Carroll et al., Phys. Rev. D 41, 1231 (1990); V.A. Kosteleckf and M. Mewes, Phys. Rev. Lett. 87,251304 (2001); Phys. Rev. D 66,056005 (2002); J. Lipa et al., Phys. Rev. Lett. 90,060403 (2003); H.Muller e t al., Phys. Rev. Lett. 91,020401 (2003); Phys. Rev. D 67,056006 (2003); V.A. Kosteleckf and A.G.M. Pickering, Phys. Rev. Lett. 91,031801 (2003); G.M. Shore, Contemp. Phys. 44, 503 (2003); P. Wolf et al., Gen. Rel. Grav. 36, 2352 (2004); Q.Bailey and V.A. Kosteleckf, hep-ph/0407252. V.W. Hughes et al., Phys. Rev. Lett. 87, 111804 (2001); R. Bluhm et al., Phys. Rev. Lett. 84,1098 (2000); E.O. Iltan, JHEP 0306,016 (2003). E.O. Iltan, Mod. Phys. Lett. A 19,327 (2004); D.L. Anderson et al., Phys. Rev. D 70, 016001 (2004). S. Coleman and S.L. Glashow, Phys. Rev. D 59, 116008 (1999); V. Barger et al., Phys. Rev. Lett. 85,5055 (2000); J.N. Bahcall et al., Phys. Lett. B 534, 114 (2002); V.A. Kosteleckf and M. Mewes, hep-ph/0308300. V.A. Kosteleckf and M. Mewes, Phys. Rev. D 69,016005 (2004). E. Cremmer and J. Scherk, Nucl. Phys. B 118,61 (1977); P. Forgacs and Z. Horvath, Gen. Rel. Grav. 11,205 (1979); A. Chodos and S. Detweiler, Phys. Rev. D 21,2167 (1980); W.J. Marciano, Phys. Rev. Lett. 52, 489 (1984); T. Damour and A.M. Polyakov, Nucl. Phys. B 423,532 (1994). R.R. Caldwell et al., Phys. Rev. Lett. 80, 1582 (1998); A. Masiero et al., Phys. Rev. D 61,023504 (2000); Y.Fujii, Phys. Rev. D 62,044011 (2000). C. Armendkiz-Pic6n et al., Phys. Lett. B 458,209 (1999). A.H. Guth, Phys. Rev. D 23,347 (1981). E. Cremmer and B. Julia, Nucl. Phys. B 159,141 (1979). V.A. Kostelecki et al., Phys. Rev. D 68,123511 (2003). A. Das et al., Phys. Rev. D 16,3427 (1977). 0. Bertolami et al., Phys. Rev. D 69,083513 (2004). J.K. Webb et al., Phys. Rev. Lett. 82,884 (1999); 87,091301 (2001). R. Jackiw and V.A. Kosteleckf, Phys. Rev. Lett. 82,3572 (1999); C. Adam and F.R. Klinkhamer, Nucl. Phys. B 657,214 (2003); H.Belich et al., Phys. Rev. D 68,025005 (2003); M.B. Cantcheff et al., Phys. Rev. D 68,065025 (2003); B. Altschul, Phys. Rev. D 69,125009 (2004). R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004); hepph/0408285; see also these proceedings.

GRAVITY PROBE B: LAUNCH AND INITIALIZATION

G.M. KEISER, W.J. BENCZE, R.W. BRUMLEY, S. BUCHMAN, B. CLARKE, D. DEBRA, C.W.F. EVERITT, G. GREEN, M.I. HEIFETZ, D.N. HIPKINS, T. HOLMES, J. LI, J. MESTER, B. MUHLFELDER, D. MURRAY, Y. OHSHIMA, B. W. PARKINSON, M. SALOMON, D. SANTIAGO, P. SHESTOPLE, A.S. SILBERGLEIT, V. SOLOMONIK, M. TABER, J.P. TURNEAURE W .W.Hansen Experimental Physics Laboratories Stanford University, Stanford, CA 94305-4085, U.S. A . E-mail: [email protected] The scientific instrument and the major subsystems of the Gravity Probe B satellite are described. Following launch, the initial on-orbit operations were designed to check the operations of each of these major subsystems, provide an initial on-orbit calibration of the scientific instrument, set up the instrument in its operational mode, and spin up and align each of the four gyroscopes.

1. Introduction

The Gravity Probe B satellite was launched from Vandenburg Airforce Base at 9:57 am PDT on April 20, 2004. The Boeing Delta I1 7920-10 expendable launch vehicle carried it south over the Pacific Ocean. Six of the nine 40 inch diameter solid rocket motors were ignited at lift-off to supplement the Rocketdyne RS 27-A main engine, and the remaining three solid rocket motors were ignited in flight. Four minutes after lift-off, the second stage ignited. The second stage cutoff occurred shortly after crossing the equator as the satellite traveled south. After passing over the South Pole, a brief second stage burn occurred to circularize the orbit at an altitude of 642 km. Cameras attached to the second stage verified that all four solar arrays deployed before the second stage separation, which occurred when the satellite was about to pass over the North Pole. The Gravity Probe B satellite is designed to measure the precession of the spin axes of four electrostatically-supported, cryogenic, mechanical gyroscopes relative to the guide star, IM Pegasi or HR 8703. The proper 115

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motion of this guide star relative to extragalactic reference sources is independently being measured by a group at the Harvard-Smithsonian Center for Astrophysics and York University.' When these two measurements are combined, the drift rate of the gyroscope relative to the extragalactic reference sources will be known and may be compared to the values predicted by the general theory of relativity. In 1960, S ~ h i @and ~ Pugh4 independently pointed out that a gyroscope will precess due to two general relativistic effects. The geodetic effect, due to the gravitational interaction between the gyroscope and its orbital motion, will cause its spin axis to precess about a direction normal to the orbit plane. The frame-dragging effect will cause the gyroscope to precess about the direction of the Earth's rotation axis. At an altitude of 640 km, the predicted magnitude of the geodetic effect is 6.6 arc-sec/year (as/yr), and the predicted magnitude of the framedragging effect is N 42 milli-arc-sec/year (mas/yr). Accurate values for these effects as predicted by general relativity5 will be calculated based on the orbital data. The precession rates due to expected classical torques on each gyroscopes are less than 0.14 mas/yr, and statistical and systematic error in the measurement of the gyroscope drift rate, based on prelaunch tests, are expected be less than 0.17 mas/yr. When combined with the uncertainty in the proper motion of the guide star, the overall error in the drift rate of each of the gyroscopes relative to the extragalactic reference sources is expected to be less than 0.23 mas/yr.6 N

2. Gravity Probe B Payload and Satellite The rotor of each of the four gyroscopes is a fused quartz sphere, 1.9 cm in radius, which has been polished to a sphericity of better than 25 nm and coated with a 1.25 pm thick coating of niobium. Electron backscattering measurements verified that the uniformity of the coating was better than 2%. The niobium coating not only provides a conducting surface for the electrostatic suspension system but also is the superconducting surface for the London moment readout described below. Six electrodes lying along three perpendicular axes are used to both measure the position of the rotor with a 34.1 KHz capacitance bridge and also provide the electric field necessary to maintain the rotor at a centered position 32 pm from each of the electrodes. The rotors are spun up to full spin speed with helium gas at 6 K, which flows through the channel cut into one side of the housing. A four-turn superconducting pickup loop lies on the parting plane between the two halves of the housing. A superconducting cable connects the pickup

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loop to a SQUID magnetometer. The interior surfaces of the housing have been coated with a conducting ground plane to minimize the effects of electrostatic charges on the interior surface of the housing. On orbit, the rotor potential is expected to increase at a rate of 0.15 mV per day due to ionizing radiation. This rotor potential may be reduced by using fiber optic cables to illuminate the surface of the rotor and the housing with ultraviolet light. The direction of flow of the emitted photoelectrons is controlled by a special (dedicated) electrode so that the rotor potential relative to the ground plane may either be increased or decreased. The rotor potential is measured by applying a sinusoidally varying voltages with opposite sign on opposite electrodes and measuring the control effort necessary to maintain the rotor at its centered p o ~ i t i o n . ~ The superconducting coating of the spinning rotor produces a London magnetic dipole moment aligned with the instantaneous spin axis of the rotor. The orientation of the spin axis is designed to lie nearly in the plane of the readout loop. As the satellite rolls on its axis about the direction to the guide star, the magnetic flux through this pickup loop is modulated at the satellite roll rate. The orientation of the gyroscope spin axis may be determined from the magnitude and phase of the change in the magnetic flux through the pickup loop as measured by the SQUID readout system. Each of the four gyroscope housings are rigidly mounted within a fused quartz block which serves as a metrology reference frame. A Cassegrainian telescope, designed to operate at cryogenic temperatures, is bonded to the quartz block. On each of the two telescope axes, the light is focused on a roof prism which divides the light from the guide star. Silicon photodiodes measure the relative intensity of the light from each side of both roof prisms. When the telescope axis is aligned with the direction to the guide star, the light intensities on each side of the roof prism is equal. When the axis moves away, and the giude star stays within the linear range of the telescope, the pointing error is proportional to the difference between the photodiode currents for that axis. The four gyroscopes, the quartz block, and the telescope comprise the Science Instrument Assembly which is attached to the cryogenic probe. A vacuum can encloses the entire assembly. A sintered titanium vacuum cryopump within the vacuum probe provides the necessary surface area to adsorb the residual helium within the probe. After the gas spin up the residual gas within the probe is vented to space, and the cryogenic region of the vacuum probe and the cryopump are heated from a temperature of 2 K to 6 K. This temperature increase drives most of the adsorbed helium

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off the interior surfaces of the probe. With the exhaust valves open to vent the probe to space, the pressure on the interior of the probe is torr. When the valves are closed and the temperature of the interior surfaces are reduced to 2 K, most of the residual gas is adsorbed onto the interior surfaces. Ground based tests of this low temperature bakeout in the flight torr. probe showed that the residual pressures was less than 2 x The vacuum probe is inserted into the well of the liquid helium dewar, which also contains a superconducting magnetic shield. The residual magnetic field within this shield has been measured to be less than 3 pgauss, and the attenuation of the external magnetic field has been demonstrated to be better than 2 x Four magnetometers are mounted near the open end of the cylindrical shield to measure the magnetic field. At launch, the dewar holds 2317 liters of superfluid helium at 1.8 K, which is designed to hold the science instrument assembly at cryogenic temperatures for more than 16 months. A sunshade is attached to the warm end of the probe to prevent sunlight from entering the probe and interfering with the measurement of the light from the guide star. This entire assembly is inserted into the spacecraft framework. A Forward Equipment Enclosure provides a passive thermal shield for the sensitive electronics, which are mounted near the cable connectors at the top of the probe. Global Positioning System antennas receive the signals which are used to determine the position and velocity of the spacecraft. In addition, a retroreflector array, mounted on the aft end of the spacecraft, allows laser range measurements to be made. Two proton monitor telescopes are used to monitor protons incident from the side and the aft of the spacecraft. Two sets of rate gyroscopes and star trackers are used to determine the orientation of the spacecraft when the guide star is not in the field of view of the cryogenic telescope. In addition, these instruments measure the roll phase of the spacecraft. Proportional helium thrusters, which use the boiloff gas from the liquid helium dewar, control the attitude and translation of the spacecraft. At launch the four gallium arsenide solar arrays are folded to fit within the shroud of the Delta I1 rocket. N

3. Initial On-Orbit Operations

The initial phase of the on-orbit operations of the satellite began immediately after launch and continued through the end of August, 2004. These operations included the initial checkout of the major subsystems of the satellite, the acquisition and verification of the guide star, adjustments of

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the attitude and translation control system, the initial on-orbit calibration of science instrument, gyroscope operations, and the low temperature bakeout of the vacuum probe. Prior to launch, plans had been made to make the final adjustments to the orbit using the spacecraft thrusters to nudge the satellite into its final orbit. This orbit had been carefully selected to minimize the forces and torques acting on the gyroscopes due to the gradient in the Earths gravitational field. However, these plans for adjusting the final orbit were found to be unnecessary because of the very high accuracy of the initial orbit provided by the Delta I1 rocket. At the North and South Poles, this orbit differed by less than 100 meters from the planned orbit. The guide star, HR 8703, was acquired using the satellites star trackers and rate gyroscopes. Although the current from the photodiodes agreed well with prelaunch estimates, the identity of the guide star was confirmed by commanding the spacecraft to point toward two nearby stars, HR Pegasi and HD 216636. The second star is also used as a comparison star for ground-based photometry measurements by G. Henry at Tennessee State University. Both the brightness and location of these nearby stars confirmed that HR 8703 was correctly identified. Subsequent measurements by G. Henry8 of the variation with time in the brightness of HR 8703 agreed well with the brightness as measured by the Gravity Probe B telescope. The satellites attitude and translation control system has the demanding requirement to keep the telescope pointed to within 200 mas of the centroid of the star image while the guide star is within the field of view. Because of Gravity Probe Bs polar orbit, the guide star is eclipsed by the Earth for roughly 40% of each orbit. During this time, the attitude control system relies on the rate gyroscope and star trackers. Then, the attitude control system must reacquire the guide star and orient the spacecraft so that the telescope is pointed within its linear range. Ionizing radiation, particularly during passage through the South Atlantic Anomaly in the Earths magnetic field, decreased. by a small amount, the fraction of time that the telescope output can be used as a sensor for the attitude control system. Adjustments to the attitude control system and software to limit the impact of particle hits in the telescope photodiodes allowed the attitude control system to acquire the guide star within a few minutes and to use the telescope as a sensor during a significant portion of the passage through the South Atlantic Anomaly. In addition, the attitude control uses the rate gyroscopes and star trackers to maintain the satellite roll rate at 0.7742 rpm. Initial on-orbit calibrations included measurement of the SQUID and

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telescope noise, verification of the telescope and gyroscope readout scale factors, measurements of the temperature sensitivity of various electronics boards, and verification of the magnetic shielding provided by the Cryoperm and superconducting magnetic shields. In addition, while the gyroscopes were spinning slowly, rehearsals were performed for calibrations that are planned for the end of the mission when the gyroscopes will be spinning at full speed. These calibrations are designed to deliberately enhance the classical torques on the gyroscope to place tight limits on potential systematic experimental errors. The initial levitation of the gyroscopes with the electrostatic suspension systemg showed that primary digital suspension system and the backup analog suspension system were working reliably and the transition between these modes could easily be made. During these tests the average voltage applied to the electrodes was 10 volts, which was subsequently reduced to 0.2 volts. The differential control effort required to keep the four gyroscopes centered in the housing agreed well with the expected force from the gradient of the Earths gravitation field. While the gyroscopes are being spun up to the full spin speed, the force of the spinup gas on one side of the housing requires a higher voltage to keep the rotor centered. Tests of this digital spinup mode as well as its analog backup showed that it was performing reliably and that transitions between these various operating modes occurred smoothly. When each of the four gyroscopes were initially levitated, the rotor potential was found to be several tenths of a volt as expected. Since this rotor potential is larger than the planned control voltage for the electrostatic suspension system, the ultraviolet charge control system was used to reduce the rotor potential to within the required 10 millivolts of zero. Initial measurements of the variation of the magnetic flux through each of the four pickup loops indicated that the magnetic flux trapped in the superconducting coating of some of the four gyroscopes was higher than the required value. This increase in the magnetic trapped flux had been anticipated because similar results were found during the prelaunch acoustic tests of the spacecraft. To reduce this magnetic trapped flux, each of the four gyroscopes and the entire quartz block were heated above their superconducting transition temperature and then allowed to cool slowly through the transition temperature. Because the magnetic field within the superconducting shield is less than several microgauss, the magnetic field trapped in the rotors after cooling was expected to be at about the same level. This was confirmed with measurements of the slowly spinning rotors

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after their temperatures were cycled about the superconducting transition temperature. To reduce the force required to maintain each gyroscope at the tenter of its housing, and thereby reduce the torques acting on each of the gyroscopes, the satellite has the capability of using one of the gyroscopes as a drag-free sensor to control the translation of the spacecraft. There are two different operation modes of the drag-free control system. In the standard drag-free control, the spacecrafts translation control system uses the output of the gyroscopes position sensing capacitance bridge to control the acceleration of the spacecraft. In this operating mode the gyroscope rotor is maintained at the center of its housing by controlling the satellite to chase the gyroscope which has been designated as the drag-free sensor. There are no electrostatic forces which need to be applied to the rotor, and the gyroscope torques are thereby reduced. To the extent that the dragfree control system reduces the spacecraft acceleration due to the residual atmosphere and the solar radiation pressure, the forces and torques on the other gyroscopes will also be reduced. In the second operating mode of the drag-free control system, the electrostatic suspension system maintains control of the drag-free sensor, but the spacecraft translation control system minimizes the force required by the electrostatic suspension system to keep the rotor centered. During the initialization phase, tests were made of both of these operating modes, and the second mode was selected for the science data collection period because of its greater reliability. The spinup of each of the four gyroscopes occurred in three steps. Initial tests were made with the gyroscopes spinning at several tenths of a Hz. At this spin speed, extensive tests of the various operations planned for the gyroscopes at the higher spin speeds were made. After these tests were completed, the gyroscopes were spun for 90 seconds using the maximum flow rate of approximately 700 sccm, which brought each of the gyroscopes to a spin speed of approximately 3 Hz. After confirming that each of the four gyroscopes had a spindown rate commensurate with the pressure in the vacuum probe, all the four gyroscopes were spun to their final spin speed. As each subsequent gyroscope was spun up, the spindown rate of the other three gyroscopes increased because of the increase of the gas pressure due to leakage of gas from the spinup channel. Following the spin up of each of the four gyroscopes, the interior of the probe and the cryopump were heated to 6 K as described above and the gas was evacuated to space. The vent was closed and the temperatures were reduced to their values of approximately 2.5 K. The final spin speeds of the four gyroscopes are 79.4, 61.8, 82.1, and

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64.9 Hz. The measured spindown rates of three of the four gyroscopes are less than 1 pHz/hr while the fourth gyroscope is slightly over this value. This spindown rate corresponds to a spin down time constant of 15,000 years and a residual gas pressure of less than 1.5 x 10-l' torr. To reduce the gyroscope torques and minimize potential systematic readout errors, the spin axes of each of the four gyroscopes were aligned so that their expected average position over the course of a year would lie within 10 arc-seconds(as) of the average direction to the guide star. Since these gyroscopes had been deliberately designed to minimize the potential classical torques which could change the orientation of the spin axis, very different operating conditions were necessary to nudge the spin axis into its final alignment." An electrostatic torque due to the equatorial bulge of the spinning gyroscopes combined with a deliberate imbalance in the average electric field on two electrode axes which lie at 45" to the gyroscope spin axis was used for the final alignment. The voltages on the electrode were increased from 0.2 to 40 V and the average electric fields on two axes were deliberately modulated by 30% at the satellite roll frequency. The direction of the gyroscope drift under these conditions is determined by the phase of the modulation. After spinning up the gyroscopes with the gas spinup system, each of the spin axes was oriented from 100 to 200 as from the direction to the guide star. By deliberately applying this electrostatic torque to each of the four gyroscopes, their spin axes were gradually aligned in the desired direction at a rate of several arc-seconds per day. Because the aberration of starlight from the guide star due to the Earths motion about the Sun has a magnitude of 20 as in an East-West direction in late August, each of the four gyroscopes were aligned 20 as from the measured direction to the guide star, so that over the course of one year their average direction would lie within 10 as of the satellite roll axis. After the spin up and final alignment of the spin axes of each gyroscope, the electrode voltages were reduced to 0.2 V volts. The gyroscopes are expected to remain in this condition for most of the remainder of the mission. Data from the satellite are collected and reviewed every day and the analysis of the data to determine the drift rate of the gyroscopes is underway. The analysis of this data will also more tightly constrain potential systematic experimental errors. Several weeks before the liquid helium is exhausted, a number of operations are planned which will deliberately enhance potential systematic errors in an effort to place stringent limits on these errors. The latest measurements indicate that the liquid helium will last through July 2005. N

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Acknowledgments Many people contributed to the design, construction, launch, and operations of the Gravity Probe B satellite, and it is not possible to acknowledge the contributions that everyone made. Over 100 people at Stanford University worked on the satellite and 150 people at Lockheed-Martin Corporation. The authors would like to particularly acknowledge the valuable contributions of employees of the Lockheed Martin Corporation and NASAs Marshall Spaceflight Center. Very valuable contributions were made by R. Farley, J. Goebel, J. Kirshenbaum, K. Galal, J. Kolodziejczak, and R. Schultz. Papers describing these contributions in more detail will be forthcoming.

References 1. M.I. Ratner et al., in R. Ruffini, R.T. Jantzen, G.M. Keiser, eds., Proceed-

2. 3. 4.

5. 6. 7. 8. 9. 10.

ings of the Seventh Marcel Grossman Meeting on General Relativity, World Scientific, Singapore, 1996. L.I. Schiff, Proc. Nat. Acad. Sci. 46, 871 (1960). L.I. Schiff, Phys. Rev. Lett. 4, 215 (1960). G. E. Pugh, Proposal f o r a Satellite Test of the Coriolis Prediction of General Relativity, 111, Department of Defense, 1959. R.J. Adler, A.S. Silbergleit, Int. Journ. Theor. Phys. 39, 1291 (2000). A S . Silbergleit et al., in J.A. Miralles, J.A. Font, J.A. Pons, eds., Gravitational Radiation, University of Alicante, Alicante (Spain), 2004. S. Buchman et al., Rev. Sci. Instr. 66, 120 (1995). G. Henry, private communication, 2004. W.J. Bencze et al., in SICE Annual Conference 2003, Fukui, Japan, 2004. W.J. Bencze et al., in 1996 IEEE Conference on Decision and Control, Kobe, Japan, 1996.

ASYMPTOTICALLY FREE LORENTZ-VIOLATING FIELD THEORIES

B. ALTSCHUL Department of Physics Indiana University Bloomington, IN 4 7405, U.S. A . E-mail: [email protected] We examine Lorentz- and CPT-violating field theories, beyond those contained in the superficially renormalizable minimal Standard-Model Extension. In particular, we study a new class of scalar field self-interactions which are nonpolynomial in form, involving arbitrarily high powers of the field. Many of these interactions correspond to nontrivial asymptotically free theories. These theories are stable if rotation invariance remains unbroken. These results indicate that certain forms of Lorentz violation, if they exist, may naturally be quite strong.

Lorentz invariance is very well tested, in many areas of physics. So although it remains quite possible that some Lorentz-violating corrections to known physics could exist, those corrections must be small. This is not a problem if we expect that the coefficients governing the Lorentz violation should be suppressed by some power of the Planck mass. However, this is not necessarily always a reasonable expectation. While there has been a great deal of theoretical work on Lorentz violation, most systematic analyses of corrections to the Standard Model have focused only on superficially renormalizable (or almost superficially renormalizable) effective field t h e o r i e ~ However, . ~ ~ ~ ~there ~ ~ ~are other interactions that might also be important. In Lorentz-invariant scalar field theories, there are known to exist interactions, which are nonpolynomial in form (and thus superficially nonrenormalizable), yet which actually are renormalizable when the coupling constants' natural cutoff dependences are taken into a c c o ~ n t .There ~ > ~ are directions in the parameter space of interactions in which the bee field fixed point is ultraviolet (UV) stable, and these correspond to nontrivial asymptotically free theories. Moreover, as we shall see, there are also Lorentz-violating analogues of these nonpolynomial theories. 124

125

The key to finding these asymptotically free theories is an understanding of the renormalization group (RG) flow for Lorentz-violating scalar field theories. Since Lorentz violation in nature is a small effect, we shall restrict our attention to the linearized form of the RG transformation, in which only those terms that are first-order in the Lorentz-violating interaction are retained. We shall use the Wilsonian formulation of the RG,' in which the theory is considered with a momentum cutoff. Since violations of Lorentz symmetry may arise in a low-energy effective field theory as remnants of larger violations appearing in a fundamental theory at high energies, it is very natural to study Lorentz violation in the context of an effective theory with a cutoff. To determine the normal modes of the RG flow, we shall use the techniques developed in Refs. 8 and 9. The basic idea is to derive a differential equation relating the form of a particular interaction with that interaction's RG behavior. This strategy was first introduced by Periwal," who used more sophisticated exact RG techniques'' to generalize the Lorentz invariant results of Refs. 5 and 6 . We shall be considering complex scalar field theories in which the current operator j p = i [b*(8'4) - (Pq5*) 41 is contracted with a fixed vector up. The simplest theory of this sort has Lagrange density

C = ( P b * )(apb)+ upj,

-

m24*b,

(1)

where up is an externally prescribed vector and is the source of the Lorentz and CPT violation. The action in this theory is a bilinear function of q5* and 4, and so the theory is free. In particular, a field redefinition

4

~

eza.x+, f

.--)

e--za.x

b*

(2)

converts the Lagrange density into

C' = ( ~ 4 * (ap4) ) - (m2+ u2)4*4.

(3) We see that the theory possesses a physical spectrum only if a2 2 -m2; if u2 < -m2, then the energy is not bounded below. When we consider more general Lorentz-violating theories, the question of whether there is a positive definite energy will continue to be very important, and we shall find that in the interacting theories, much more stringent requirements on up will be needed in order to ensure stability. To study the stability of a theory, we must work in Minkowski spacetime, in which the energy is distinguished from the other components of the four-momentum. However, it will be simpler for us to perform our RG calculations in Euclidean space. We may then transform our results back into

126

Minkowski spacetime and determine whether or not the energy is bounded below. The gauge symmetries of the Standard Model restrict the allowed forms of any Lorentz-violating Higgs couplings.’*l2There is only one such gauge invariant, superficially renormalizable, CPT-violating interaction-a generalization of that given in Eq. (l),with a single value of up for both field components. Any nonpolynomial interactions are similarly restricted, so, while we shall work with only a single complex scalar field, a generalization of our results should be relevant to the study of the physical Higgs sector. We shall use bare perturbation theory to calculate the lowest-order radiative corrections to the effective action. We shall consider only the case of four spacetime dimensions, although our method is applicable in any number of dimensions d > 2. The Euclidean action for the bare theory is

s=

J

d42 { (8j4*) (8j4)- m24*4+ iaj [4* (W)- (3j4*)41

v (4*4>}

*

(4)

The function V parameterizing the interaction must be representable as a power series in 1412. When V(4’4) is not constant, the interaction term cannot be eliminated by means of a field redefinition. We shall determine the RG flow for this theory by finding the value of each effective n-point vertex generated by the interaction. To do this, we must sum up contributions from infinite numbers of diagrams. If we neglect the diagrams that are nonlinear in V , we find that all the contributions to the effective n-particle amplitude have a distinct form; each involves a single bare ( n 2Ic)-point vertex and Ic loops. Figure 1 shows the four lowest-order diagrams contributing to the four-particle amplitude. We shall regulate the theory with a momentum cutoff A, which will provide the only intrinsic scale in the theory. The coupling constants’ classical dependences on A will therefore be determined entirely by their dimension. A coupling 9~ with dimension (mass)dK will be associated with a dimensionless coupling constant C K according to

+

gK = CKhdK.

(5)

According to this scaling scheme, the mass parameter should have the form m2 = p 2 h 2 . The factors of A in Eq. ( 5 ) ensure the renormalizability of all interactions, essentially because any superficially nonrenormalizable couplings will vanish as A + m. However, the dimensionless couplings C K may remain finite, and it is the evolution of the C K under the action of the RG that is important. The inclusion of the extra factors of A in Eq.

127

Figure 1. Graphs contributing to the four-particle amplitude.

(5) will also allow us to write down a A-independent differential equation describing the normal modes of the RG flow. An explicit representation of V($*$)in terms of dimensionless couplings will be given below, in Eqs. (10) and (15). To determine the RG flow, we shall only need to consider the first term in the interaction: iaj [$* (8j$)]V ($*$). The contributions from the other term are only trivially different. A particular term in the power series expansion of V contributes Ln = iaj [4* (aj$)]($*$)"

(6)

to the Lagrange density. When the Feynman rules for the theory are worked out, L, gives rise to a vertex with n+ 1incoming particle lines (corresponding to $) and an equal number of outgoing (qY) lines. Moreover, in addition to the various combinatorial factors associated with the vertex, there is a

128

factor of a'. $, where p' is the momentum on one of the incoming legs; we must sum over all such incoming legs to which this momentum may be assigned. The linearized RG flow is generated by diagrams with a single vertex of the form generated by C,, with k outgoing and k incoming particle lines connected to form k tadpole loops. This generates an effective vertex, with a corresponding effective Lagrange density of the form c n - k . Determining the combinatorial factors associated with the effective vertex is a relatively simple matter. If we begin with a &-type diagram and contract one pair of lines into a loop, we get a factor of n 1 arising from the choice of which outgoing line is to be used and a factor of n from the choice of the incoming line. The difference between the two factors arises from the fact that we may only choose one of the incoming lines without the extra factor of a'. p'. If we did choose the leg with the momentum factor attached, we would obtain a loop integral whose integrand was an odd function of the momentum. Hence, the contribution from this contraction would vanish. This argument also guarantees that there can be no 0 (4*4) contribution O1 to the effective action generated by the tadpole loops; there must be at least one external leg on each nonvanishing diagram, to which the momentum factor may be attached. When acting on (@4)", the operator

+

[

+

generates n(n 1)(q5*4)n-1. So this differential operator will produce the necessary combinatorial factors accompanying a loop, when it acts on V (4*@).Each loop is also associated with a factor of DF(O),the Feynman propagator for the complex scalar field at zero spatial separation. Moreover, a diagram with k loops has a symmetry factor of k ! , because we are free to interchange the loops. The value of DF(O)is

We see that DF(O)has the form CA2, with 1

C = -16r2 [1- P2 log (1 + p-"1

.

(9)

The factors of A appearing in the zero-separation propagator are the source of the RG flow.

129

In order to study this RG flow, we must express the interactions in the nondimensionalized form described previously. Recalling that we are neglecting the (dq5*)4 term, we therefore write the interaction part of the effective action as

sint=

/

d42 iaj

[4* (aj$)] u ( ~ - 2 4 * 4,)

(10)

Since the action is dimensionless and iaj [4* (dj4)] has dimension mas^)^, U is a dimensionless function. In addition to the explicit A-dependence in this expression, iaj [q5* (dj4)] scales as A4, and U may also have a parametric dependence upon A; this parametric dependence will describe the RG flow of the effective potential. If only the tadpole diagrams contribute, then we may calculate U (A-24*q5) directly from V (4*4). Accounting for the combinatorial factors described above, the result is that

= exp

[-CA2D]V (4*4) .

(12)

Acting with the operator A& on Eq. (12) results in the differential equation

dU A- 2 (AP2q5*4) U’ (AP2q5*q5) = -2CA2DU (llP24*4) , (13) dll where the prime denotes differentiation of U with respect to its argument. The left-hand side of Eq. (13) describes the classical scaling behavior of U , while the right-hand side contains the effects of quantum corrections. If U is to describe a normal mode of the RG flow near the free-field fixed point, then it should display a power-law dependence on A. This is, we should have A% = -XU, for some constant A. In this case, the partial differential equation (Eq. (13)) is transformed into the ordinary differential equation

XU(Y) + 2YU’(Y) - 2c [YU”(Y) + U’(Y)l = 0, where y = A-2q5*~is the argument of U . To solve Eq. (14), we set M

n=O

and this leads to the recurrence relation

(14)

130

So the solution is

where g is some coupling constant, and M ( a ;p; z ) is the confluent hypergeometric (Kummer) function13

M ( a ; D ; z )= 1

a + -+ a ( a + 1 ) 2- 2+ . . . p l ! p(p+ 1) 2! 2

g

We see that U(y) has a polynomial form exactly if is a nonpositive integer. The full renormalized Sint corresponding to this normal mode of the RG flow (including the thus far omitted (aq5*)4term) is

X is the anomalous scaling dimension of the U factor in Sint. The operator iaj [4* (aj4 - (aj4*)4)]has dimension mas^)^, so the anomalous dimension of the entire interaction is X 4. This scaling dimension describes how the interaction scales with changes in the cutoff A; it need not describe the scaling of any correlation functions with respect to their external momenta. The calculation of such correlation functions involves the same sorts of complexities as are associated with similar calculations in the presence of Lorentz-invariant nonpolynomial potentials.1°*14 If X > 0, the potential is nonpolynomial (with U(y) behaving as yX/2-2ey/C for large y) and asymptotically free, with power-law coupling constant flow. The free field Gaussian fixed point is UV stable along the associated trajectories, so if the renormalized coupling is held fixed, then the bare coupling goes to zero as A 4 m. If X < 0, then the fixed point is infrared stable, and the interaction is irrelevant; this includes all interactions with polynomial U . The marginal case, X = 0, corresponds to the free theory discussed earlier. We must now determine whether the relevant interactions we have found lead to stable Minkowski-space theories. In fact, the canonical Hamiltonian density associated with the Lagrange density

+

131

is

All the terms in 3-t are manifestly positive except for the last one. If any spatial components of up are nonvanishing and V grows more rapidly than a constant, then this term can render 3-t arbitrarily negative. Hence we conclude that the theory must be quantized in a reference frame in which up is purely timelike (i.e. aj = 0 for j = 1 , 2 , 3 ) ;such a frame obviously can exist only if u2 2 0. In this special quantization frame, rotation invariance remains an unbroken symmetry, and the momentum regulator is symmetric. If the theory is boosted into a different frame, then a boosted regulator will be required. It is well established that consistency requirements in quantum field theories may constrain the values of Lorentz-violating coefficient^.^ Moreover, a purely timelike ap is appealing from a physical standpoint, because the universe displays a very high degree of isotropy in the reference frame of the cosmic microwave background, and this limits the possibilities for spacelike Lorentz-violating effects. However, the existence of strongly relevant directions in the parameter space of Lorentz-violating interactions does present a hierarchy problem. A generic field theory containing a scalar boson sector and Lorentz-violating coefficients with the same discrete symmetries as u0 (C-odd, P- and T-even) will generate, through radiative corrections, scalar self-interactions of the sort we have considered here. Since some of these interactions are asymptotically free, we expect them to be generated fairly strongly. The strong scalar field interactions will, in turn, generate Lorentz-violating interactions in other sectors of the theory. However, since Lorentz violation in nature is a weak effect, we must conclude that either there is some additional symmetry that prevents the generation of strong uO-typeeffects in the observable sectors of the theory, that the bare Lorentz-violating couplings in question are all extraordinarily small, or that nonlinear effects (possibly involving other interactions) become important even for very small values of g. To study this problem further, it would be desirable if the calculation of the RG flow could be extended beyond the linearized regime, but that could be a very difficult undertaking. We have seen that if Lorentz- and CPT-violating corrections to the Standard Model do exist, then there is good reason to believe that they may not be small, since the Higgs sector can support asymptotically free

132

Lorentz-violating interactions. These asymptotically free theories involve peculiar nonpolynomial potentials, and in order for the theories to be stable, the coefficients governing the Lorentz violation must be purely timelike in the frame in which the theory is quantized. The net result of these investigations is that we have obtained important theoretical insights and constraints on the forms to be taken by any possible Lorentz-violating interactions.

References 1. D. Colladay, V. A. Kosteleckf, Phys. Rev. D 55, 6760 (1997). D. Colladay, V. A. Kosteleckf, Phys. Rev. D 58, 116002 (1998).

2. 3. 4. 5.

V. A. Kosteleckf, R. Lehnert, Phys. Rev. D 63,065008 (2001). R. C. Myers, M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003). K. Halpern, K. Huang, Phys. Rev. Lett. 74, 3526 (1995). 6. K. Halpern, K. Huang, Phys. Rev. D 53,3252 (1996). 7. K. G. Wilson, K. Kogut, Phys. Rep. 12C,75 (1974). 8. B. Altschul, hepth/0403093.

9. 10. 11. 12. 13.

B. Altschul, hep-th/0407173.

V. Periwal, Mod. Phys. Lett. A 11,2915 (1996). J . Polchinski, Nucl. Phys. B 231,269 (1984). V. A. Kosteleckf, Phys. Rev. D 69,105009 (2004). M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover,

New York, 1977), p. 504-505. 14. K. Halpern, Phys. Rev. D 57, 6337 (1998).

TORSION BALANCE TEST OF LORENTZ SYMMETRY VIOLATION

B.R. HECKEL, FOR THE EOTWASH GROUP Physics Box 351 560, University of Washington, Seattle, WA 98195, U.S.A. E-mail: heckel@phys. washington. edu The torque produced on an electron spin polarized test mass of a torsion pendulum provides a sensitive probe for searching for rotational symmetry violation and new weak spin coupled forces. A second generation spin pendulum has been developed and is being used to search for Lorentz symmetry violation. Initial results indicate that the torsion balance is sensitive to certain Standard-Model Extension GeV, representing a factor of 20 improvement parameters at the level of over previous spin pendulum measurements.

1. Introduction

Torsion balances provide a sensitive experimental tool to detect weak forces that act on macroscopic objects. Unpolarized test bodies on torsion balances have recently been used to measure various aspects of gravity including the universality of free fal1,l the inverse square law at short distances,2 and Newton’s gravitational ~ o n s t a n t as , ~ well as to search for new forces weaker than g r a ~ i t y A . ~spin polarized torsion pendulum allows the possibility for searching for pseudoscalar particles such as the for testing Lorentz and CPT syrnmetrie~,6>~ and for detecting new spin coupled forces.8 In this paper, we present the status of our spin pendulum test of rotational invariance. In the context of the Standard-Model Extension, we are testing the electron sector of the theory, whose Lorentz-violating lagrangian terms are given

Le = -uZ@ypQ

- bL4757c”Q - iHE,&p”Q

+ 4 i ~ ; , & p ~ ~ Q / 2+ ;idE,&ypD”Q,

tt

(1)

where u: and b: are CPT-odd terms and the others CPT-even. For nonrelativistic electrons, the lagrangian in Eq. (1) gives rise to a coupling to 133

134

the electron spin given by:'

-

where bj" = bj" - medjeO- e j k l H i l / 2 . The Lorentz and CPT symmetry violation appears as a pseudo-magnetic field, b;, that couples to spin, along an axis fixed in space. At the Second Meeting on CPT and Lorentz Symmetry, we presented our initial spin pendulum limits the be parameters above:1°

at 95% C.L. Since that time, we have constructed an improved spin pendulum and have made improvements to the torsion balance apparatus, allowing us to collect data with a 20-fold increase in sensitivity. 2. The Spin Pendulum

We produce a pendulum test mass with a net spin dipole moment and minimal magnetic moments by assembling octagons out of Alnico and SmCo permanent magnets. The basic principle is illustrated in Figure 1.

Figure 1. One of four magnet rings of the spin pendulum. The shaded sections are SmCo magnets and the unshaded sections are Alnico. The solid arrows on top of the ring show the direction of magnetization. The open arrows on the sides of the ring show the electron spin polarization in each section.

The magnetization in Alnico V comes primarily from electron spin while the magnetization in SmCoS is produced both by the orbital angular momentum of the Sm ions and electron spin. After assembling the octagon

135

and magnetizing the Alnico to the same degree as the SmCo5, the magnetization runs toroidally within the ring, leaving a net spin excess within the Alnico side. We stack four such rings in an ABBA pattern, with their spin axes aligned, to form the spin pendulum, as shown in Figure 2.

Figure 2. The spin pendulum. The figure on the right shows the stack of 4 magnet rings. The darker sections are Alnico with small plates added to give them the same mass as the SmCo sections. The figure on the left shows the magnetic shield that contains the magnet rings, with mirrors and compensation screws.

The new spin pendulum, shown in Figure 2, differs from the original pendulum in several ways. The new version uses SmCos, rather than Sm2C017, to increase the fraction of magnetization from orbital angular momentum. Unlike the original spin pendulum that had soft iron corner pieces between the rectangular permanent magnets (shown in Fig. l),the new pendulum magnet pieces were cut to form an octagon without corner pieces. The thicker magnet sections of the new pendulum reduce the outer dimensions of the stack of rings (outer diameter of 2.6 cm), producing smaller gravitational multipole moments and allowing us to use a smaller (less massive) magnetic shield. We estimate that the 64 g of magnets provide a net spin dipole of (8 f 1) x electron spins that point perpendicular to the central axis of the magnet stack.ll A project is underway in our lab to measure the angular momentum associated with the spin excess of one of the magnet rings to verify the magnitude of the spin dipole. The magnet ring stack is mounted within a small mu-metal magnetic shield to reduce the leakage of magnetic fields from imperfections in the magnets. The shield supports four symmetrically placed mirrors (which we

136

label 1-4), allowing us to collect data with the spin dipole moment pointing in four different directions within the rotating torsion balance apparatus. The shield also supports eight small copper screws that we use to cancel residual gravitational moments of the spin pendulum. To search for new spin dependent interactions, it is essential to eliminate (or understand very well) any residual magnetic couplings. Figure 3 shows scans of the radial component of the magnetic field outside of the spin pendulum, taken at several radial distances from the pendulum axis.

Figure 3. Magnetic field scans outside of the spin pendulum, as the pendulum is rotated. The radial distance is measured from the center of the pendulum, in inches.

137

The bottom panel in Figure 2 is at a radial distance of 2.25”, which is the distance to the wall of the innermost magnetic shield of the torsion balance instrument. The dipole component of the magnetic field of the pendulum at the innermost shield wall is less than 0.1 mG. 3. Torsion Balance Apparatus

The spin pendulum is mounted within a rotating torsion balance apparatus, described in detail elsewhere.’ The pendulum is suspended from a small copper bellows (to damp other pendulum modes) followed by a 76 cm long, 30 pm diameter tungsten fiber, having a torsion constant of 0.18 erg/rad, and centered within four layers of high permeability magnetic shields. The innermost shield is gold-coated to minimize electrostatic coupling. The pendulum and shields are located within a vacuum vessel that is held at approximately lop6 torr by an ion pump. The vacuum vessel is mounted on a turntable that rotates at a constant rate of approximately four rev/hr. A feedback loop locks the output of a precise rotary encoder attached to the rotating vacuum vessel to the frequency of a crystal oscillator to ensure a constant rotation rate. Other important components of the apparatus include tilt sensors that allow the rotation axis to be monitored and aligned vertically, constant temperature water-cooled Cu shields that provided thermal isolation, and nearby machined Pb compensator masses used to cancel the Y31, Y41, and Y44 spherical multipole components of the local gravY21, YZZ, itational field. A set of three-axis Helmholtz coils surround the apparatus and are adjusted to cancel the local magnetic field to 2%. The most significant improvement to the apparatus since the origianl spin pendulum measurements has been the installation of a feedback system that holds the rotation axis to within 10 nrad of vertical (the lab floor tilt drifts by p a d s daily). As the vacuum vessel and pendulum within it rotate relative to the laboratory at an angle 4 = w t , an external potential, V, that couples to spin will produce a torque, r = aV/aq5, causing the spin pendulum to twist by an angle, 8,relative to the rotating vessel:

@($) = VHsin(40 - 4)/f% (3) where K. is the torsion constant of the fiber, VH is the horizontal component of the coupling to spin, and 40is the angle in the horizontal plane at which the coupling is largest. Diode laser light is doubly reflected from one of the four mirrors mounted on the pendulum and the reflected beam is focused

138

onto a linear position sensitive photodiode to monitor the angular position of the pendulum, 0. A rotary stage at the top of the torsion fiber allows the light to be centered onto any one of the four symmetrically placed mirrors. As different mirrors are used, 4 0 in Eq. (3) is altered by go", 180', or 270'.

4. Data and Analysis

We choose the turntable rotation period to be a half-integral multiple (typically 6.5) of the free torsion period of the pendulum (207 s) to avoid exciting the free torsion amplitude via turntable irregularities. O ( 4 ) and other sensor values are recorded every 10 s over 3 - 4 days (typically spanning a weekend when seismic noise from traffic is reduced). The long data run is divided into 'cuts,' each of two complete turntable revolutions duration; data points one half of a torsional cycle apart in time are averaged to filter out the free torsion amplitude; and the time sequence of filtered O(q5) values in each cut is fit to a Fourier series out to the eighth harmonic of the turntable frequency in addition to an offset and linear drift term.' An external coupling to spin will appear in the first harmonic of the Fourier series. The higher harmonics (due to imperfections in the turntable rotation rate and gravity gradient couplings) are monitored for stability. The results from a cut are rejected if the x2 of the Fourier fit exceeds a nominal value. Seismic disturbances can cause abrupt changes in the torsion oscillator amplitude and phase and are the dominant cause for cuts with large x2. The only correction applied to the O ( 4 ) Fourier amplitudes is an attenuation correction that accounts for the effects of the pendulum inertia, electronic time constants, and filtering. Between data sets, additional measurements are made to examine sources for systematic errors: temperature effects, magnetic coupling to the pendulum, and gravitational gradients. In each case, the driving term is greatly exaggerated. The temperature of the apparatus, normally constant to 0.1 mK, is made to oscillate by 1 K with a 24 hr period. The current in the Helmholtz coils is reversed, increasing the static magnetic field at the apparatus by a factor of 100. The machined Pb gravity gradient compensators are rotated to add to the local gradients rather than cancel them, increasing the gradients by a factor of typically 100. At present, there is no evidence for systematic error from these effects. In previous measurements, a drifting tilt of the rotation axis was the dominant source of systematic error. This error has been reduced to negligible level by the feedback system that holds the rotation axis constant.

139

5. Results for Lorentz Symmetry Violation The cutting and fitting process described above results in a time sequence of values for V H and $0 as defined in Eq. (3). A pseudo-magnetic field that violates Lorentz symmetry, as described in Eq. (2), would modulate VH and 40 as the Earth rotates. To search for such a pseudo-magnetic field, we follow the convention of Bluhm and K0steleck9~to define the nonrotating coordinate axes of The E axis is taken to lie along the rotational north pole of the Earth: a coupling of the spin pendulum to will produce a signal that does not vary as the Earth rotates. The 2 axis points from the Earth towards the Sun at the vernal equinox. For each cut within a data set, we compute the angular coordinates of our spin pendulum relative to the celestial axes. We then fit all of the first harmonic signals from the cuts to a function that describes the amplitude and phase of a coupling to any of the celestial axes.' The results of the fits for first four data sets are shown in Table 1.

&.

&

Table 1. Preliminary spin pendulum results for Lorentz violation

9/8/04

Averaged Results

-2.5 f 2.3 -0.4 f 2.0 4.3 f 2.7 0.4 f 1.1

-0.1 f 2.3 2.9 f 2.0 -1.5 f 2.7 0.2 f 1.1

The results above are preliminary; they are meant to illustrate the level of sensitivity that is being achieved. Additional data will be taken for one year. Investigations of systematic errors are still being pursued. The dominant error apprears to be vibrational excitation of pendulum modes due to traffic, which has daily period. After one year of data collection, a sensitivity to b& of % 3 x lop3' GeV may be achieved, as well as the potential to set limits on boost invariance violation in the electron sector.

Acknowledgments The group that contributed to the results presented here are: E.G. Adelberger, S. BaeBler, T. Cook, J.H. Gundlach, M.G. Harris, U. Schmidt, H.E. Swanson, and M. White. We gratefully acknowledge the financial support for this work provided by the NSF (Grant PHY-0355012) and the DOE through its support of the Center for Nuclear Physics and Astrophysics.

140

References 1. Y. Su, E.G. Adelberger, B.R. Heckel, J.H. Gundlach, G. Smith, M. Harris, and E. Swanson, Phys. Rev. D 50, 3614 (1994). 2. C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, U. Schmidt, and H.E. Swanson, Phys. Rev. D 70, 042004 (2004). 3. J.H. Gundlach and S.M. Merkowitz, Phys. Rev. Lett. 85, 2869-2872 (2000). 4. E.G. Adelberger, B.R. Heckel, and A.E. Nelson, Ann. Rev. Nucl. and Part. Sci. 53, 77 (2003). 5. J.E. Moody and F. Wilczek, Phys. Rev. D 30, 130 (1984). 6. D. Colladay and V.A. Kosteleckjr, Phys. Rev. D 55, 6760 (1997). 7. D. Colladay and V.A. Kosteleckjr, Phys. Rev. D 58, 1 (1998). 8. N. Arkani-Hamed, H.-T. Cheng, M. Luty, and J. Thaler, arXiv:hepph/0407034 v l (2004). 9. R. Bluhm and V.A. Kostelecki, Phys. Rev. Lett. 84, 1381 (2000). 10. B.R. Heckel, in V.A. Kosteleck9, ed., CPT and Lorentz Symmetry 11,World Scientific, Singapore, 173 (2002). 11. M.G. Harris, Ph.D. thesis, Univ. of Washington, unpublished (1998).

QED TESTS OF LORENTZ SYMMETRY

R. BLUHM Physics Department Colby College Waterville, M E 04901, U.S.A. E-mail: rtbluhmOcolby.edu A status report is given of some recent theoretical and experimental investigations looking for signals of Lorentz violation in QED. Experiments with light, charged particles, and atoms have exceptional sensitivity to small shifts in energy caused by Lorentz violation, including effects that could originate from new physics at the Planck scale.

1. Introduction

Lorentz symmetry is a fundamental feature of relativity theory. In special relativity, it is a global symmetry relating the laws of physics in different inertial frames under boosts and rotations. It is also linked by a general theorem to the combined discrete symmetry CPT formed from the product of charge conjugation C, parity P, and time reversal T. In general relativity, Lorentz symmetry becomes a local symmetry that relates the physics in different freely falling frames in a gravitational field. The Standard Model (SM) of particle physics does not include gravity as a fundamental interaction at the quantum level. It is therefore expected that the SM and gravity will merge in the context of a fundamental unified theory. The relevant energy scale for quantum gravity is the Planck scale Mp1 = N lOI9 GeV. Much current work in theoretical high-energy physics is aimed at finding a unified fundamental theory that describes physical interactions at the Planck scale. Promising candidates include string theory, D-branes, and theories of quantum gravity. Many of these include effects that violate assumptions of the SM, including higher dimensions of spacetime, unusual geometries, nonpointlike interactions, and new forms of symmetry breaking. In particular, it is possible that small violations of Lorentz symmetry might occur in theories of quantum gravity. For example, it is known that 141

142

there are mechanisms in string theory that can lead to spontaneous violations of Lorentz and CPT symmetry.l This is due to certain types of interactions in string theory among Lorentz-tensor fields that can destabilize the vacuum and generate nonzero vacuum expectation values for Lorentz tensors. These vacuum expectation values fill the true vacuum and cause spontaneous Lorentz violation. It is also known that geometries with noncommutative coordinates can arise naturally in string theory and that Lorentz violation is intrinsic to noncommutative field theories2 One method of searching for signals of Planck-scale physics is to look for highly suppressed effects involving inverse powers of the Planck scale. In this approach, Lorentz violation becomes an ideal signal since all of the interactions in the SM preserve Lorentz symmetry and therefore no conventional signal could mimic the effects of Lorentz violation. To observe a signal of Lorentz violation experimentally, one needs to perform experiments with exceptional sensitivity. Experiments in &ED systems provide many of the best oppportunities for testing Lorentz symmetry. One example is provided by measurements of photons that have traveled over cosmological distances. Any small phase effect would be amplified during the long transit time. Other examples with photons include highprecision laboratory-based experiments with resonance cavities. Experiments in atomic physics can also be performed at low energy with extremely high precision. For example, some atomic experiments are routinely sensitive to small frequency shifts at the level of 1 mHz or less. Interpreting this as being due to an energy shift expressed in GeV, it would correspond to a sensitivity of approximately 4 x GeV. Such a value is well within the range of energy one might associate with suppression factors originating from the Planck scale. The main focus of this work is to investigate tests of Lorentz and CPT symmetry performed in QED systems. The general goals are to analyze the sensitivity of QED systems to possible Lorentz and CPT violation, to uncover possible new signals that can be tested in experiments, and to express experimental sensitivities in the context of a common framework that permits comparisons across different experiments. To this end, we use the Standard-Model Extension (SME) as our theoretical f r a m e ~ o r k . ~ The SME permits detailed investigations of Lorentz and CPT tests in all particle sectors of the SM. Our analysis here focuses on the QED sector of the SME. This is presented in the following section and is then used to examine a number of experiments involving photons, trapped particles, atomic clocks, muons, and a spin-polarized pendulum. Additional details

143

about many of these experiments can be found as well in several of the other articles in this volume. 2. QED Sector of the SME The subset of the SME lagrangian relevant to experiments in QED systems can be written as

LQED= LO+ Lint

.

(1)

The lagrangian Lo contains the usual Lorentz-invariant terms in QED that describe photons, massive charged fermions, and their conventional couplings. If we restrict our investigation to the remormalizable and gaugeinvariant terms in the SME in flat spacetime, then the Lorentz-violating part of the lagrangian is given by4 Lint = -a,&p$

- b,4757’$

+ ic,,&yPD”$

+ i d p v 4 7 5 ~ ’ D V-~?~H,,I+W~$

-z(kF),+”FnAFp” 1

+ i(kAF)K.EKAp”AXFp”.

(2)

Here, natural units with h = c = 1 are used, and i D , = ia, - qA,. The terms with coefficients a,, b, and ( k A F ) , violate CPT, while those with H,,, cPv,d,,, and (kF),+, preserve CPT. All seven terms break Lorentz symmetry. Sensitivities to Lorentz and CPT violation can be expressed in terms of the SME coefficients. This provides a straightforward way of making comparisons across different types of experiments. Each different particle sector in the QED extension has an independent set of Lorentz-violating coefficients. These are distinguished using superscript labels. A thorough investigation of Lorentz and CPT violation requires looking at as many different particle sectors as possible. 3. Photon Experiments

The relevant part of the lagrangian for a freely propagating photon in the presence of Lorentz violation is given by

where F,, is the field strength, FPv = a,A, - &A,. The CPT-odd term with coefficient k A F has been investigated extensively both theoretically and e~perimentally.~?~ It is found theoretically that

144

this term leads to negative-energy contributions and is a potential source of instability. One solution is to set k A F M 0, which is consistent with radiative corrections in the SME. Stringent experimental constraints consistent with ~ A Fx 0 have also been determined by studying the polarization of radiation from distant radio g a l a x i e ~ .In ~ the following, we will therefore ignore the effects of k A F . The CPT-even term with coefficients k F have been investigated more r e ~ e n t l yIt . ~provides positive-energy contributions. The set of coefficients k F has 19 independent components. It is useful to make a decomposiand, ktr. Here, tion of these in terms of a new set: ke+, ke-, k,+, i,K,+, K,- , and ito- are 3 x 3 traceless symmetric matrices (with 5 independent components each), while k,+ is a 3 x 3 antisymmetric matrix (with 3 independent components), and the remaining coefficient ktr is the only rotationally invariant component. The lagrangian in terms of this decomposition becomes -

I

c = i[(l+ktr)E2 - (1 - k t r ) 9 ] + ; E . (ke+ + k e - ) . E

+-.

_ -: g . (ke+ - k e - ) . B + E . (k,+

+ ko-) .B

+

.

(4)

Here, 2 and are the usual electric and magnetic fields. The equations of motion following from this lagrangian give rise to modifications of Maxwell’sequations, which have been explored in several recent astrophysical and laboratory experiments. The ten coefficients ke+ and i,lead to birefrigence of light. Spectropolarimetry of light from distant galaxies leads to bounds on these parameters of order 2 x 10-32.7 Seven of the eight coefficients ke- and k,+ are bounded in experiments using optical and microwave cavities.’ Sensitivities on the order of ke- <, and KO+ - <, have been attained, and it is expected that future experiments in boosted frames will be sensitive to the remaining two parameters as well. 4. Atomic Experiments

In recent years, a number of atomic experiments have been performed which have very sharp sensitivity to Lorentz and CPT violation. Bounds from these experiments can be expressed in terms of the coefficients a,, b,, c P v , d,”, and H,” in the QED sector of the SME. Comparisons across different types of experiments can then be made which avoid the problems that can arise when different physical quantities (g factors, charge-to-mass ratios, masses, frequencies, etc.) are used in different experiments. In the

145

following, a number of atomic experiments involving the proton, neutron, electron. and muon are examined.

4.1. P e n n i n g - m a p Experiments Two recent sets of experiments with electrons and positrons in Penning traps provide sharp tests of Lorentz and CPT ~ y m m e t r y .Both ~ involve measurements of the anomaly frequency w, and the cyclotron frequency WC

.

The first consists of a reanalysis by Dehmelt’s group of existing data for electrons and positrons in a Penning trap.l0 The signal involves looking for an instantaneous difference in the anomaly frequencies of electrons and positrons, which can be nonzero when Lorentz and CPT symmetry are broken. In contrast the instantaneous cyclotron frequencies remain approximately equal at leading order in the Lorentz-violation corrections. Dehmelt’s original measurements of g - 2 did not involve looking for possible instantaneous variations in w,. Instead, the ratio W a I w , was computed using averaged values. It is important to realize that the Lorentz-violating corrections to the anomaly frequency w, can occur even though the g factor remains unchanged. The new analysis looks for an instantaneous difference in the electron and positron anomaly frequencies. A bound on this difference can be expressed in terms of the parameter bz, which is the component of b; along the quantization axis in the laboratory frame. It is given as lbzl <, 3 x GeV. A second signal for Lorentz and CPT violation in the electron sector has been obtained using data for the electron alone.’’ In this case, the idea is that the Lorentz-violating interactions depend on the orientation of the quantization axis in the laboratory frame, which changes as the Earth turns on its axis. As a result, both the cyclotron and anomaly frequencies have small corrections which cause them to exhibit sidereal time variations. These variations can be measured using electrons alone, eliminating the need for comparison with positrons. The bounds in this case are given with respect to a nonrotating coordinate frame such as celestial equatorial coordinates. The interactions involve a combination of laboratory-frame components that couple to the spin of the electron. This combination is denoted using tildes as bg = b$ - mds, - H t 2 . When expressed in terms of components X , Y , Z in the nonrotating frame, the obtained bound is <, 5 x 10-25GeV for J = X , Y .

161

146

4.2. Clock- Comparison Experiments

The Hughes-Drever experiments are classic tests of Lorentz i n v a r i a n ~ e . ’ ~ ? ~ ~ There have been a number of different types of these experiments performed over the years, with steady improvements in their sensitivity. They all involve making high-precision comparisons of atomic clock signals as the Earth rotates. The clock frequencies are typically hyperfine or Zeeman transitions. Many of the sharpest Lorentz bounds for the proton, neutron, and electron stem from atomic clock-comparison experiments. For example, Bear et al. use a two-species noble-gas maser to test for Lorentz and CPT violation in the neutron sector.14 They obtain a bound I@/ <, 10-31GeV for J = X , Y , which is currently the best bound for the neutron sector. Note that these Earth-based laboratory experiments are not sensitive to Lorentz-violation coefficients along the J = Z direction parallel to Earth’s rotation axis. They also neglect the velocity effects due to Earth’s motion around the sun, which would lead to bounds on the timelike components along J = T . These limitations can be overcome by performing experiments in space15 or by using a rotation platform. The earth’s motion can also be taken into account. A recent boosted-frame analysis of the dual noble-gas maser experiment yields bounds on the order of lop2’ GeV on many boost-dependent SME coefficients for the neutron that were previously unbounded.16 It should also be pointed out that certain assumptions about the nuclear configurations must be made to obtain bounds in clock-comparison experiments. For this reason, these bounds should be viewed as good to within about an order of magnitude. To obtain cleaner bounds it is necessary to consider simpler atoms or to perform more sophisticated nuclear modeling. 4.3. Hydrogen-Antihydrogen Experiments

The simplest atom in the periodic table is hydrogen, and the simplest antiatom is antihydrogen. There are three experiments underway at CERN that can perform high-precision Lorentz and CPT tests in antihydrogen. Two of the experiments (ATRAP and ATHENA) intend to make highprecision spectroscopic measurements of the 1s-2stransitions in hydrogen and antihydrogen. These are forbidden (two-photon) transitions that have a relative linewidth of approximately The ultimate goal is to measure the line center of this transition to a part in lo3 yielding a frequency comparison between hydrogen and antihydrogen at a level of 10-l’. An analysis of the 1s-2s transition in the context of the SME shows that the magnetic

147

field plays an important role in the attainable sensitivity to Lorentz and C P T vi01ation.l~For instance, in free hydrogen in the absence of a magnetic field, the 1s and 2s levels are shifted by equal amounts at leading order. As a result, in free H or H there are no leading-order corrections to the 1S-2s transition frequency. In a magnetic trap, however, there are fields that can mix the spin states in the four different hyperfine levels. Since the Lorentz-violating interactions depend on the spin orientation, there will be leading-order sensitivity to Lorentz and C P T violation in comparisons of 1S-2s transitions in trapped hydrogen and antihydrogen. At the same time, however, these transitions are field-dependent, which creates additional experimental challenges that would need to be overcome. An alternative to 1s-2s transitions is to consider the sensitivity to Lorentz violation in ground-state Zeeman hyperfine transitions. It is found that there are leading-order corrections in these levels in both hydrogen and a n t i h ~ d r 0 g e n . lThe ~ ASACUSA group a t CERN is planning to measure the Zeeman hyperfine transitions in antihydrogen. Such measurements will provide a direct C P T test. Experiments with hydrogen alone have been performed using a maser.18 They attain exceptionally sharp sensitivity to Lorentz and C P T violation in the electron and proton sectors of the SME. These experiments use a double-resonance technique that does not depend on there being a field-independent point for the transistion. The sensitivity for the proton attained in these experiments is lbP,l <, GeV. Due to the simplicity of hydrogen, this is an extremely clean bound and is currently the most stringent test of Lorentz and C P T violation for the proton.

4.4. Spin-Polarized Matter

Experiments a t the University of Washington using a spin-polarized torsion pendulumlg are able to achieve very high sensitivity to Lorentz violation in the electron sector.20 The sensitivity arises because the pendulum has a huge number of aligned electron spins but a negligible magnetic field. The pendulum is built out of a stack of toroidal magnets, which in one version of the experiment achieved a net electron spin S 2 8 x lo2’. The apparatus is suspended on a rotating turntable and the time variations of the twisting pendulum are measured. An analysis of this system shows that in addition t o a signal having the period of the rotating turntable, the effects due to Lorentz and C P T violation also cause additional time variations with a sidereal period caused by the rotation of the Earth. The group a t the University of Washington has analyzed their data and find that thay have

148

161

sensitivity to the electron coefficients at the levels of <, lop2’ GeV for J = X , Y and <, GeV.Ig These are currently the best Lorentz and CPT bounds for the electron. More recently, a new pendulum has been built, and it is expected that improved sensitivities will be attained.

161

4.5. Muon Experiments

Muons are second-generation leptons. Lorentz tests performed with muons are therefore independent of the tests involving electrons. There are two main classes of experiments with muons that have been conducted recently. These are experiments with muonium21 and g - 2 experiments with muons at Brookhaven. In muonium, measurements of the frequencies of groundstate Zeeman hyperfine transitions in a strong magnetic field have the greatest sensitivity to Lorentz and CPT violation. An analysis searching for sidereal time variations in these transitions was able to attain sensitivity to Lorentz violation at the level of GeV. At Brookhaven, 52x relativistic g - 2 experiments with positive muons have been conducted using muons with boost parameter 6 = 29.3. An analysis of the obtained data as a test of Lorentz symmetry is still forthcoming. We estimate that a sensitivity to Lorentz violation is possible in these experiments at a level of lopz5 GeV.22 5 . Conclusions

Experiments in &ED systems continue to provide many of the sharpest tests of Lorentz and CPT symmetry. In recent years, a number of new astrophysical and laboratory tests have been performed that have lead to substantially improved sensitivities for the photon. Similarly, atomic experimentalists continue to find ways of improving the sensitivity to Lorentz violation in many of the matter sectors of the SME. In particular, experiments in boosted frames are providing sensitivity to many of the previously unprobed SME coefficients. All of the bounds obtained are within the range of sensitivity associated with suppression factors arising from the Planck scale. The coming years are likely to remain productive. QED experiments will continue to provide increasingly sharp new tests of Lorentz and CPT symmetry.

References 1. V.A. Kosteleck? and S. Samuel, Phys. Rev. D 39 (1989) 683; ibid. 40 (1989) 1886; Phys. Rev. Lett. 63 (1989) 224; ibid. 66 (1991) 1811; V.A. KosteleckS;

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and R. Potting, Nucl. Phys. B 359 (1991) 545; Phys. Lett. B 381 (1996) 89. 2. S.M. Carroll, J.A. Harvey, V.A. Kosteleck?, C.D. Lane, and T. Okamoto, Phys. Rev. Lett. 87,141601 (2001). 3. V.A. Kosteleck? and R. Potting, Phys. Rev. D 51, 3923 (1995); D. Colladay and V.A. Kosteleck?, Phys. Rev. D 55,6760 (1997); 58, 116002 (1998); V.A. Kosteleck? and R. Lehnert, Phys. Rev. D 63,065008 (2001); V.A. Kosteleck?, Phys. Rev. D 69, 105009 (2004). 4. For simplicity, the additional terms involving parameters e p , fp, and gxpv are ignored. Gauge invariance excludes these terms in the SME; however, they can arise effectively due to strong binding in a nucleus. A more general discussion includes an investigation of these terms.12 5. S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41 (1990) 1231. 6. R. Jackiw and V.A. Kosteleck?, Phys. Rev. Lett. 82, 3572 (1999); 7. V.A. Kosteleck? and M. Mewes, Phys. Rev. Lett. 87,251304 (2001); Phys. Rev. D 66, 056005 (2002). 8. P. Wolf et al., Phys. Rev. Lett. 90, 060402 (2003); J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003); H. Miiller et al., Phys. Rev. Lett. 91, 020401 (2003); H. Miiller et al., Phys. Rev. D 68, 116006 (2003); P. Wolf et al., Phys. Rev. D 70,051902 (2004). 9. R. Bluhm, V.A. Kosteleck? and N. Russell, Phys. Rev. Lett. 79 (1997) 1432; Phys. Rev. D 57 (1998) 3932. 10. H.G. Dehmelt et al., Phys. Rev. Lett. 83 (1999) 4694. 11. R. Mittleman et al., Phys. Rev. Lett. 83 (1999) 2116. 12. V.A. Kosteleck? and C.D. Lane, Phys. Rev. D 60, 116010 (1999). 13. V.W. Hughes, H.G. Robinson, and V. Beltran-Lopez, Phys. Rev. Lett. 4 (1960) 342; R.W.P. Drever, Philos. Mag. 6 (1961) 683; J.D. Prestage e t al., Phys. Rev. Lett. 54 (1985) 2387; S.K. Lamoreaux et al., Phys. Rev. A 39 (1989) 1082; T.E. Chupp et al., Phys. Rev. Lett. 63 (1989) 1541; C.J. Berglund et al., Phys. Rev. Lett. 75 (1995) 1879. 14. D. Bear et al., Phys. Rev. Lett. 85 (2000) 5038. 15. R. Bluhm et al., Phys. Rev. Lett. 88,090801 (2002); Phys. Rev. D 68, 125008 (2003). 16. F. Can6 et al., Phys. Rev. Lett. in press, physics/O309070. 17. R. Bluhm, V.A. Kosteleck9 and N. Russell, Phys. Rev. Lett. 82 (1999) 2254. 18. D.F. Phillips et al., Phys. Rev. D 63,111101 (2001). 19. E.G. Adelberger, B. Heckel et al., to appear. 20. R. Bluhm and V.A. Kosteleck?, Phys. Rev. Lett. 84 (2000) 1381. 21. W. Liu et al., Phys. Rev. Lett. 82 (1999) 711. 22. R. Bluhm, V.A. Kosteleck? and C.D. Lane, Phys. Rev. Lett. 84 (2000) 1098.

L O R E N T Z AND CPT VIOLATION WITH LSND

T. KATORI AND R. TAYLOE FOR THE LSND COLLABORATION^ Indiana University Cyclotron Facility 2401 Milo B. Sampson Lane Bloomington, IN 47408, U.S.A. E-mail: [email protected]. edu

Lorentz and CPT violation is one of the predicted signals of Planck scale physics. A recently developed Standard-Model Extension (SME) for neutrino oscillations2 is used to analyze the sidereal time variation of the neutrino event excess observed in . ~is found that there is no statistically significant variation the LSND e ~ p e r i r n e n tIt in the sidereal time distribution. From a fit to this data, the allowed values for the coupled SME parameters values are extracted.

1. The L S N D signal for N e u t r i n o Oscillations

The Liquid Scintillator Neutrino Detector (LSND) e ~ p e r i m e n tperformed ,~ at the Los Alamos National Laboratory (LANL), observed an excess of Ve in a beam of 0, created from p+ decay at rest. The data analysis used the sample of detected pep + e+n events with positron energy 20 MeV < Ee+ < 60 MeV. This De excess implies a two-neutrino oscillation probability of (0.264 f 0.067 & 0.045)%. Here the first error is statistical and the second error is systematic (neutrino flux, particle detection efficiency). Despite the series of evidence for neutrino oscillations from solar, atmospheric, accelerator and reactor neutrinos, the oscillation signal observed a t LSND remains an unsolved puzzle. Since neutrino physics is one of the best places to search for new physics, the LSND anomaly is often discussed using unconventional models like a mass-difference C P T violating model6 or a sterile neutrino model (2 2 , 3 1 models’). The MiniBooNE experiment’ at Fermilab is currently taking data to test the LSND signal. Perhaps LSND is seeing the first signal of Planck-scale physics. In order t o describe neutrino oscillations including Lorentz and C P T violation, a recently developed formalism of neutrino oscillation^^^^ using the SME formalism5 is employed t o describe the LSND data sample, including any

+

150

+

151

sidereal time variations. 2. Short-Baseline Approximation

If the baseline of neutrino beam is short compared with the neutrino oscillation length, the neutrino transition probability can be expanded with an effective Hamiltonian and expressed to leading order in h e ~ , 4

Since p p in the effective Hamiltonian contains information about the propagation direction of the neutrino, this transition probability depends on the propagation direction. It is more convenient to fix the coordinate system to write down a phenomenological expression of transition probabi1ity.l' The standard choice is a Sun-centered system which is, to a good approximation, an inertial frame for the experiment. With this choice of coordinates, the neutrino oscillation probability becomes,

L2 P F ~ E+ -1 ~ ~ (C)QZ

+

sin waTe

+ (dc)ep cos weTB

+(BS)Ep sin 2 w e T e + ( B c ) e p cos 2 w e T e 12,

(2)

may depend on the sidereal time. Here, w e is the sidereal and frequency (=2~/23h56m4.ls)and Te is the sidereal time which is measured from a standard origin. , (dc)eo, (Bs)Ep, and (Bc)eQdepend on the The parameters ( C ) E ~(.As)<@, SME coefficients ( a ~ ) and " ( C L )and ~ ~ the neutrino propagation direction factors Nx,N Y , and Nz in the Sun-centered system. The direction factors depend on the colatitude x of LANL in the Earth-centered system and the zenith and azimuthal angles 8 and 4 in the LANL local coordinate system:

+

cos x sin 8 cos 4 sin x cos 8 sin 8 sin 4 - sin x sin8 cos 4 cos x cos 6'

+

(3)

For neutrinos from Los Alamos Neutron Science Center (LANSCE) beam to the LSND detector, x = 54.1", 8 = 99.0" and 4 = 82.6".11 The sidereal time is defined to be zero (Te = 0) at LANL local midnight at the autumnal equinox. At that time, the y-axis of the Earth-centered system coincides with the Y axis of the Sun-centered system. The estimated error

152

using this definition is 3 minutes which is sufficiently small compared to the time scale sensitivity of this analysis. The detailed expressions of the parameters ( C ) ~ p , ( d ~ ) , p , ( d ~ ) ~ p , ( B ~ ) ~ p and (Bc)Ep for the LSND experiment are,4 (Ckp =

(Ti%*):.

2E

+ [-(%)&

+ E [-0.17(~~)::

-

2 0.19(aL),p]

- 0.39(~~)::

+ 0.56(~~):,],

(4)

(ds)qj= [ - 0 . 9 8 ( ~ ~ ) $- 0 . 0 5 3 ( ~ ~ ) & ]

+ E [-1.96(~~)::

(dc)Ep

=

-

O.ll(c~ TY ) ,~ 0.38(~~)$:

+ + 1.96(~~)::

-

0 . 0 2 1 ( ~ ~ ) y(5) -~] w

7

[-0.053(~~)$ 0 . 9 8 ( ~ ~ ) & ]

+ E [-O.ll(c~):;

-

+

O.O21(c~)g 0.38(~~):,Z],(6)

+0.96(~~)$~],

( B s ) ~ p= E [ - 0 . 0 5 2 ( ( ~ ~ ) $ ~(CL):~) ( B c ) ~= p E [ 0 . 4 8 ( ( ~ ~) $( ~C L YY ) , ~) - 0

In the expression, (

u L )and ~ ( c ~ ) f are i~

. 1 0 ( ~X Y~ 1.) , ~

(7) (8)

the SME coefficients of Ref. 5.

3. Sidereal Time Distribution of the LSND Data

In the analysis of the final LSND data set,3 205 neutrino oscillation candidate events with an identified neutron-capture photon (R, > 1) were reported. The number of beam-off and v background events in this sample are 106.8f 2 . 5 and 39.2 f 3 . 1 respectively. These events were collected from runs in 1993 through 1998. There were 6, approximately contiguous, sets of runs, one in each of these years. The GPS time stamp, necessary for this analysis, was not included into the LSND data stream until midway through the 1994 run period. Because of this, only 186 of the 205 oscillation candidate events could be used in this analysis. The expected numbers of beam-off and v backgrounds in this smaller sample are 94.0 f 2.2 and 35.6 f 2.8 respectively. Ideally, an experiment to search for sidereal variations in a signal would run continuously throughout the calendar year so that one particular sidereal time bin would be drawn from the entire range of local time. LSND did not achieve this completely but did cover the space of local time vs sidereal time with reasonable completeness. This can be seen in Figure 1. The local time can be determined from Greenwich Mean (GM) time with local Los Alamos time equal to GM time - 6 (7) hours in the summer (winter). The experiment shows no sidereal variations in the beam-off data. This indicates that environmental or “day-night” effects are not substantial. Also, the beam-on “efficiency” of the experiment was reasonably flat in

153

4000

6000

5000

mn number

. / “ , * I

0,’

’

. , . , 40000 , ,

20000

,

, _ ,

t

60000

,

,

,

I

,1

80000

GM time (secs)

Figure 1. Distribution of beam-on neutrino candidate events in a) run number vs. solar day (day 1= January 1) and b) GM time vs sidereal time. The year of each run set is indicated in a).

sidereal time. These qualities can be checked by examining the sidereal time distributions of beam-off data that are shown in Figure 2. The distributions of the beam-on data in sidereal and GM time are shown in Figure 3. To determine the statistical significance of any time variation the Pearson’s x2 statistic12 was employed. The values obtained for this statistic from the sidereal and GM time distributions for a range of binning are summarized in Table 1. These values indicate that, while the sidereal time distributions have a higher x2 value for each binning choice, there is not a statistically significant deviation from a flat sidereal time distribution. For N = 37 (chosen to maximize the number of bins while keeping vi > 5) the cumulative x2 value is 0.15 (a “1.4a” deviation from zero).

4. Fits to Determine Allowed Values of SME Parameters

In general, the SME parameters are complex. The special case is considered here where these parameters are real. Also, the values extracted are an an effective average over the energy range 20-60 MeV. Using Eq. (2) to describe

154

al

a) &ndf = 21.1/23

20

2oooo

1

60000 SOOM) sidereal time (secs)

4oooo

2-

b)

Figure 2. Sidereal time distributions of beam-off data: a) unweighted and b) weighted with the beam-on duty ratio. b) corresponds to beam-on "efficiency" of the experiment. Table 1. Summary of Pearson's x2 test statistics for the sidereal and GM time distributions of the 186 oscillation candidate events. N is the number of time bins used and vi is the expected number of events per bin with the hypothesis of no time variation. P ( x 2 )is the x2 cumulative distribution. sidereal time

GM time

Vi

x2

37

5.03

44.75

0.150

27.65

0.840

36

5.17

43.16

0.162

30.00

0.708

24

7.75

24.84

0.359

21.74

0.536

12

15.50

12.97

0.295

8.19

0.696

9

20.67

12.00

0.151

7.84

0.449

6

31.00

9.23

0.100

4.77

0.444

N

P(X2)

x2

P(x2)

the sidereal time behavior of the LSND oscillation sample described and shown above, a maximum likelihood method was used to extract allowed values of the SME parameters. This was performed using an unbinned likelihood function using the 186 candidate oscillation events for 3 different parameter combinations.

155

GM time (secs)

Figure 3. Sidereal a) and GM b) time distributions of the 186 LSND beam-on oscillation candidate events with 24 bins. The value of vi is indicated with the solid line.

(1-parameter) ( C ) e p # 0; ( d s ) e p , ( A c ) e p ,(a,),,,( & k i p = 0 This model corresponds to the “rotationally invariant” case, which has been investigated p r e v i o ~ s l y . ~ (%parameter) (C), (As),,,(dc )ap # 0; (&),p, (&)ED = 0 This model especially corresponds to the case where there are only CPT-odd background fields. (5-parameter) (&p, ( d s ) e p , (.Ac)*,, (as),,,( B C ) e p# 0 This is the full minimal-SME model. The transition probability depends on both CPT-odd and CPT-even terms.

Using each of these three parameter sets, the log likelihood, L , was calculated as each of the parameters in the set was varied in a range around zero. The data along with the solution that maximized L are shown in Figure 4. The best-fit solutions are summarized below. The ( l a ) errors were calculated by determining the parameter ranges where the C decreased by 0.5 (1-parameter), 1.77 (%parameters), or 3.0 (5-parameters). 0

0

(1-parameter) (C), = 3.3 f 0.4 f 0.3 (3-parameter) There are 2 solutions within the la likelihood re-

156

~

~

'20000 ~

"

"

40000 "

"

"

60000 " ' ~ ' 80000 sidereal time (secs)

'

d 20000

E

40000

80000 sidereal timetsecs)

6oooO

10

d '

"

0; o

o

"

'

"

20000 i '

"

l

'

I

"

40000 ' l

' "

'

80000 sidereal time (secs)

l6oooO '

Figure 4. Sidereal time distributions of the LSND oscillation sample together with the best-fit results (solid lines) for a) 1, b) 3, and c) 5 parameter fits. The dotted lines indicate the estimated background distribution.

gion (see Fig. 5): Solution 1 (best-fit): ( C ) c p = -0.2 f 0.9 f 0.1, ( d S ) c p = 4.0 f 1.3 f 0.3,

( d c ) e p = 1.9 f 1.9 f 0.1

Solution 2: ( C ) e p = 3.3 f 0.8 f 0.3 ( d s ) e p = 0.1 f 1.1f 0.2 ( d c ) e p = -0.5

f 1.0 f 0.2

(5-parameter) There are multiple (connected) solutions possible in the 5-parameter case making a numerical extraction of errors impossible. The best-fit solution is: ( C ) E=~ -0.7, ( d s ) s p = 3.7, ( d c ) E p = 2.3, (Bs)q= 0.9, ( B c ) ~ p= -0.6.

All parameters have units of lop1' GeV and the errors quoted above are in the form f(statistical) f(systematic).

157

U

2 0

-2

4 -5

0

5 Ac GeV)

Figure 5. Results from the 3-parameter fit to the LSND sidereal time distribution: a) (A8)*, vs (C),p, b) ( d c ) e p vs ( C ) E ~c), ( d c ) e p vs (A,),,, and d) best-fit solution superimposed on the sidereal time distribution. The contours in a)-c) indicate the 1-0 allowed regions (L> Lm, - 1.77) and the dots indicate the best-fit parameter values.

In all of these results, there is an equivalent solution with opposite signs for all of the parameters. Note that a solution with all parameters M 0 is highly disfavored. This is equivalent to the statement the LSND oscillation excess is statistically significant. A more highly constrained value for the combination of parameters was extracted from the 3-parameter fit, l(C)cp12

+ i \ ( d s ) c p \ ' + $\(Ac)epJ2 = 9.9 f2.3 f 1.2 (10-19GeV)2,

(9)

and from the 5-parameter fit,

l(c)~pI'+ $ I ( d s ) e p I 2 + $l(dc)ep12 + $l(Bs)cp12 +!jI(&)epl'

= 10.7f 2.2

f 2.3 (10-19GeV)2.

(10)

These were obtained by examining the likelihood region with C > Lmm-0.5 (1-parameter region). This result is in agreement with Ref. 4. It should be noted that the values for the SME parameters extracted here would have significant implications in other neutrino experiments. A simple interpretation of these non-zero SME parameters within this model

158

of Lorentz and CPT violating neutrino oscillations2 would induce timeaveraged oscillations in other (reactor and long-baseline) neutrino oscillation experiments. However, there is no theoretical motivation that there is necessarily a simple solution for neutrino oscillations. A general solution of neutrino oscillations with Lorentz and CPT violation, obtained through a diagonalization of the effective Hamiltonian, may be able to accommodate all of the neutrino oscillation data. 5 . Conclusions

The recently developed neutrino oscillation formalism including Lorentz and CPT violation2i4has been used to fit the sidereal time distribution of the LSND oscillation data. There is no statistically significant modulation in sidereal time of the LSND that would definitely indicate a new theory of oscillations such as Lorentz and CPT violation. However, the LSND data is consistent with this model and a maximum likelihood method is used to extract allowed values for the SME parameters. With the LSND for the neutrino energy of x 0.01 GeV, these values imply a value FZ dimensionless expression of SME coefficients. In is interesting and perhaps telling, that this value is on the order naively expected from Planck scale physics (xE w / M p ) . References 1. The LSND collaboration consists of scientists from the following institutions: U. of California, Riverside; U. of California, San Diego; U. of California, Santa Barbara; Embry-Riddle Aeronautical U.; Los Alamos National Laboratory; Louisiana State U.; Southern U.; Temple U. 2. V.A. Kosteleckf and M. Mewes, Phys. Rev. D 69, 016005 (2004). 3. A.A. Aguilar et al. [LSND Collaboration], Phys. Rev. D 64, 112007 (2001). 4. V.A. Kosteleckf and M. Mewes, [arXiv:hep-ph/0406255]. 5 . D. Colladay and V.A. Kosteleckf, Phys. Rev. D 55, 6760 (1997). 6. H. Murayama and T. Yanagida, Phys. Lett. B 520, 263 (2001). 7. See, for example, M. Maltoni et al. [arXiv:hep-ph/0305312]. 8. M. Shaevitz, [arXiv:hep-ex/0407027]. 9. See, for example, S. Coleman and S.L. Glashow, Phys. Rev. D 59, 116008 (1999). 10. V.A. Kosteleckf and M. Mewes, Phys. Rev. D 66, 056005 (2002). 11. http://ww.terraserver.com. 12. A. Frodesen, 0. Skjeggestad, and H. Tofte, Probability and Statistics in Particle Physics, (Universitetsforlaget, Bergen, 1979).

NEUTRINO OSCILLATIONS AS PROBES OF NEW PHYSICS

C . PENA GARAY Institute for Advanced Study Einstein Drive Princeton, N J 08540, U.S.A. E-mail: penyaQias. edu Neutrino oscillations driven by neutrino masses have been confirmed by reactor and accelerator experiments as the main mechanism to explain solar and atmospheric neutrino data. Stringent tests of new physics are accessible to neutrino experiments, because tiny violations can be reflected in the mixing angles and/or in the phase of the oscillations. Neutrinos are particularly sensitive to CPT violation. Constraints on non-standard neutrino interactions and on CPT-violating fermion bilinears are discussed.

1. Three-Neutrino Oscillations Neutrino oscillations are entering in a new era in which the observations from underground experiments obtained with neutrino beams provided to us by Nature - either from the Sun or from the interactions of cosmic rays in the upper atmosphere- are being confirmed by experiments using “man-made” neutrinos from accelerators and nuclear reactors. Super-Kamiokande (SK) high statistics data established that the observed deficit in the p-like atmospheric events is due to the neutrinos arriving in the detector at large zenith angles, strongly suggestive of the v p oscillation hypothesis. This evidence was also confirmed by other atmospheric experiments such as MACRO and Soudan 2. Two reactor neutrino experiments, CHOOZ and Palo Verde, measured the flux of 0, from reactors with an energy of N MeV located at typical distance of 1 km. The data, consistent with no oscillations, established that the observed deficit in the p-like atmospheric events could not be dominantly due to v p + v, oscillations. On a different line of probing neutrino oscillations, the SNO results in combination with the SK data on the zenith angle dependence and

-

159

160

recoil energy spectrum of solar neutrinos and the Homestake, SAGE, GALLEX+GNO, and Kamiokande experiments, put on a firm observational basis the long-standing problem of solar neutrinos, establishing the need for u, conversions. The KEK to Kamioka long-baseline neutrino oscillation experiment (K2K) uses an accelerator-produced neutrino beam mostly consisting of up with a mean energy of 1.3 GeV and a neutrino flight distance of 250 km to probe the same oscillations that were explored with atmospheric neutrinos. K2K data show that both the number of observed neutrino events and the observed energy spectrum are consistent with neutrino oscillations with parameters in agreement with the ones derived by atmospheric neutrino data. The KamLAND experiment measures the flux of ge from nuclear reactors with a energy of MeV located at typical distance of 180 km with the aim of exploring with a terrestrial beam the region of neutrino parameters that is relevant for the oscillation interpretation of the solar data. KamLAND data show that both the total number of events and their energy spectrum are consistent with neutrino oscillations with parameters in agreement with the ones derived by solar neutrino data. Summarizing, solar, atmospheric, reactor, and accelerator neutrino data require that all three known neutrinos participate in oscillations due to their different mass eigenvalues and their non-parallel mass and flavor eigenstates. Six parameters characterize the phenomenology of neutrino oscillations: a) three angles, and one phase (named the CP phase) that parametrize the elements of the unitary matrix that relate mass and flavor states,

-

N

u

=

(

c12c13 -s12c23

s12c13

s13epi6

- c12s23s13ei6 c12c23 - s 1 2 s 2 3 s 1 3 e i 6 S23c13 -C12s23 - s12c23s13ei6 c23c13

s 1 2 s 2 3 - c12c23s13ei6

)

,

sij = sin B i j and Cij = cos 9ij; and b) two mass squared differences, Am$ = m: - m;, present in the differences of the eigenvalues of the evo-

where

lution operator. The physics of neutrino oscillations depends upon different scales: the distance between neutrino production and detection, L; the neutrino oscillation lengths in vacuum, l : ! = 4.rrE/Am$,where E is the neutrino energy; and, if ue is involved in the oscillations, the neutrino refraction length, Zmatt = 2.rr/&%~ ' N, where GF is the Fermi constant and N, is the electron density in the medium.

161

If lmatt >> lzoj”” for any i and j or v, does not participate in the oscillations, flavor conversion is driven by neutrino oscillations in vacuum, a function of the elements of the unitary matrix and the oscillation lengths in vacuum. This is the case of atmospheric neutrino oscillations with energies around a GeV. Running accelerator and reactor experiments were designed to be in this regime. If, moreover, l:? << L , the oscillations are averaged out and the energy and Am2 dependence is lost. This is the case of solar and atmospheric neutrinos at their lower energies. If lmatt 5 lzoj”” for a particular i and j and u, does participate in the oscillations, matter effects are important. In practice, the refraction length varies with the position of the neutrino in the medium, which leads to different evolution depending on the energy of the neutrino and the mixing angles. At lower energies (5 MeV), solar neutrinos oscillate in vacuum. At higher energies, solar neutrinos change flavor by adiabatic transitions: produced neutrinos follow adiabatically the varying hamiltonian. A combined analysis of all neutrino data (not including LSND results because they cannot be explained in this framework) leads to the following 3a ranges of five oscillation neutrino parameters (the present data are insensitive to the CP phase and to the sign of the higher mass squared difference): 7.0 5 1.6 5

Am& < 9.0 10-5 eV2 -

&Am:, 10-3 eV2

5 3.6,

0.28 5 tan2 812 5 0.60, 0.5 5 tan2823 I 2.1, sin2 Ql3 5 0.041.

These results can be translated into our present knowledge of the moduli of the mixing matrix U :

IU( =

0.79-0.88 0.19-0.52 0.20-0.53

(

0.47-0.61 0.42-0.73 0.44-0.74

< 0.20 0.58-0.82 0.56-0.8 1

1

.

(2)

Neutrino data also imply that matter matters in neutrino oscillations. Most neutrino experiments are not sensitive to matter interactions, which modify neutrino propagation due to coherent forward scattering in the medium, because the effects are small compared to the precision of the experiments. That’s not the case in solar neutrino experiments sensitive only to the higher energies, SNO and SK. Data are consistent with adiabatic transitions driven by a varying hamiltonian from the higher density

162

-

of the neutrino production region to the lower density outwards. Vacuum oscillations in this regime are excluded at 6 0 confidence level.

2. Constraints on non-standard interactions

Low-energy neutrino interactions can be described by four-fermion interaction vertices, L 3 C The vertices affecting neutrino evolution in matter are those containing two neutrino lines (2v). In the SM, these vertices receive contributions from neutral current (NC) processes and, if the initial state contains a charged lepton, also charged current (CC) processes. The NC processes are predicted to be flavor-preserving and universal. The strength of the possible non-standard interactions (NSI), both flavor-preserving and flavor-changing, normalized to G F ,between the neutrinos Y of flavors a and ,B and the left-handed (right-handed) components of the fermions f and f is parametrised by NSI can modify both the neutrino detection and neutrino propagation (oscillation) processes. Best bounds on the epsilons come from acceleratorbased experiments dedicated to studying neutrino interactions, constraining the vertices involving the muon neutrino, 5 - lop2. At the same time, bounds on NSI involving electron and tau neutrinos are rather loose.4 Bounds obtained by SU(2) symmetry can be avoided if, for example, the corresponding operators contain Higgs doublets. The propagation effects of NSI are sensitive to the vector component of cL$ when f = f. NSI modify propagation due to the change of the effective hamiltonian describing the interactions with matter. Therefore, the propagation will be sensitive to NSI in those regimes where matter matters. Present reactor and accelerator experiments are not sensitive to NSI. On the contrary, solar and atmospheric neutrinos can be sensitive to NSI matter effects. In fact, present-day loose bounds on some of the neutrino interaction parameters introduce uncertainties in the determinations of the oscillation neutrino parameters. In particular, the mass squared difference inferred by solar and KamLAND data could be wrong by a factor of 5-6 if the NSI couplings are 15 % of the Fermi c o n ~ t a n t These .~ uncertainties might be eliminated in the next several years, as more data are collected by KamLAND and future low energy solar neutrino experiments. In that case, strong bounds on tau neutrino interactions will be available from neutrino oscillation experiments.

t+&)q+.

fir.

N

163

3. Constraints on CPT Violation

3.1. Direct Constraints on CPT Violation CPT conservation was assumed in the discussion presented in the previous section. The CPT theorem rests on three assumptions: locality, hermicity of the hamiltonian, and Lorentz invariance. If CPT is found violated, at least one of the three assumptions has to be ~ i o l a t e dPhenomenologically, .~ the neutral kaon system puts a stringent limit on CPT violation; the mass difference between KO and I?’ is smaller than 5 x lo-’’ eV at 95% CL. Murayama? pointed out that, in the absence of a concrete theory of CPT violation, the kaon constraint may be looked as a mass squared difference instead of as a mass difference. In fact, mass squared is the natural parameter for bosons in lagrangian field theories. If reinterpreted as a limit of the mass squared,

lm2(Ko)- rn2(Ko)I< 0.25 eV2.

(3) Neutrino oscillations provide a stronger test, by constraining the mass squared differences between neutrinos and antineutrinos. Both neutrinos and antineutrinos contribute to the atmospheric signal and the analysis of the data leads to lArn$,(v)- Arn$,(g)l < lop2 eV2.6 Solar (neutrino) data and reactor (antineutrino) data provide us with a much stronger constraint. The current bound on (Am$,(u)- Arn$,(D)\ is, at the 3a CL18

lArngl(u) - Arng,(Q)l < 1.1 x lo-* eV2.

(4) On the other hand, bounds on the difference between mixing angles in the neutrino and the antineutrino sector, 1 sin2B,j - sin2&3)1, are 0.6, 0.6, and 0.9 at the three sigma confidence level, where ij is 12, 23, and 13, respectively. These bounds will not be significantly improved soon. 3.2. Constraints on Lorentz Violation The experimental upper limits on CPT violation can be used to test arbitrary future conjectures of CPT violation. Let’s consider an effective interaction which has been discussed by Coleman and Glashow, and by Colladay and K~stelecky,~ that violates both Lorentz invariance and CPT invariance. The interaction is of the form

C(ACPT))= D ~ b ~ ’ y , r / f , (5) where a , ,B are flavor indices, L indicates that the neutrinos are left-handed, and b is a Hermitian matrix. We discuss the special case with rotational invariance in the preferred frame in which the cosmic microwave background

164

(CMB) is isotropic. In the two-neutrino limit, atmospheric neutrino oscillation corresponds to conversion between mu and tau (anti)neutrinos driven by two terms: a vacuum term due to the squared mass difference term and a CPT-odd term given by the bo hermitian matrix. Atmospheric neutrinos have been measured by Super-Kamiokande in a huge range of energies expanded by three orders of magnitude. Therefore, this is the most sensitive experiment, at present, to test deviations of pure vacuum oscillations. The resulting bounds on violation of CPT invariance in terms of the difference of the eigenvalues of bo is?

Sbo < 2(5) x 10-23GeV

at 1u (3a).

(6)

Solar and KamLAND data also constrain the Lorentz violation interaction in Eq. (5). In some particular cases, the survival probabilities take a simple form. In the case of reactor antineutrinos, neutrinos oscillate in vacuum (matter effects are negligible in the range of parameters allowed by KamLAND data). In the case of solar neutrinos, sensitivity to bb in the oscillation phase is lost because of the fast oscillations due to the Am2 term. The sensitive to bb is in the mixing angle in matter, the projection of the flavor state to the eigenstate of the hamiltonian in the production point in the Sun. The bound obtained is two orders of magnitude weaker than the one obtained in the atmospheric sector.2 This bound could be improved by one order of magnitude. Matter effects should be known better if we want to go beyond that limit. Acknowledgments

CPG was supported by W.M. Keck Foundation and the National Science Foundation, PHY-0070928. References 1. M. C. Gonzalez-Garcia and C. Pena-Garay, Phys. Rev. D 68, 093003 (2003). 2. J.N. Bahcall et al., JHEP 0408, 016 (2004). 3. V.A. Kosteleckjr, ed., CPT and Lorentz Symmetry 11,World Scientific, 2002. 4. S. Davidson et al., JHEP 0303, 011 (2003). 5. A. Friedland e t al., Phys. Lett. B 594, 347 (2004). 6. M. C. Gonzalez-Garcia and M. Maltoni, Phys. Rev. D 70, 033010,2004. 7. H. Murayama, Phys. Lett. B 597, 73 (2004). 8. A. de Gouvea, and C. Pena-Garay, Phys. Rev. D in press, hep-ph/0406301.

BAEUR TESTS OF LORENTZ AND CPT SYMMETRY

F. MARTlXEZ-VIDAL FOR THE BABAR COLLABORATION IFIC, Instituto de Fisica Corpuscular, Universitat de Valbncia and CSIC, E-46071 Valencia, Spain; and Universitci d i Pisa, Dipartamento d i Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy. E-mail: [email protected] Tests of C P T and T symmetries and a limit on the difference between the decay rates of the two mass eigenstates in Bo-meson oscillations are reported. The reconstructed B decays, comprising both CP and flavor eigenstates, are obtained from T(4S) + BE decays collected by the BABAR detector at the PEP-I1 asymmetricenergy B Factory at SLAC. Sensitivity projections for sidereal-time modulation of the CPT-violating parameter based on an explicit and general CPT-breaking Standard-Model Extension are also discussed.

1. Introduction The original observation of CP violation in the Bo-meson system to states like J/$JK: and the current and foreseen high integrated luminosities at B factories have opened the window for more detailed investigations in Bmeson decays. Among them, those which are receiving today a great deal of experimental and theoretical attention are CP asymmetries and other Unitarity Triangle related measurements, using the large number of different B-decay channels (with branching ratios of the order of accessible in these experiments. Nevertheless, these are not the only studies required to understand fully the CP violation mechanism, and more generally the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix, of the Standard Model (SM). Moreover, although the mass difference between the Bo-mass eigenstates is known with high pre~ision,~ our knowledge of other aspects of the reach phenomenology of Bo-meson oscillations is meager. In particular, the violation of CP symmetry may include the possibility of CPT violation in oscillation, as well as violation of the T symmetry and a difference between the lifetimes of the mass eigenstates. In addition, an explicit and general CPT-breaking Standard-Model Extension (SME) prel l 2

165

166

dicts that the CPT-violating effects, if present, should reveal a dependence with the magnitude and orientation of the B-meson momentum and a corresponding variation with sidereal time. All these CPT-violating effects could provide clear signature for GUT-scale p h y ~ i c s . ~ 2. Quark-flavor oscillations in the Bo system

The Bo-meson system is usually described by the effective Hamiltonian H = M - Z / 2 , where M and r are two-by-two Hermitian matrices describing, respectively, the mass and decay-rate components. Under CP or CPT symmetry, M11 = M22 and rll = l?22.a In the limit of CP or T invariance, r12/M12 = F21/M21 = l?;2/M;2, so r12/M12 is real. The masses m H , L and decay rates r H , L of the two eigenstates of H form the complex eigenvahes W H , L = m H , L - $ r H , L . The light and heavy B-meson mass eigenstates are superpositions of Bo and

+

, ),

IBL)

=p G ( B O ) q G l B O )

IBH)

=p G I B o )- q G I B o

(1)

where 2 3

M12

-

g r12

'

(2)

with the definitions Sm = - M22, Sr = rll - r 2 2 , Am = m H - m L and A r = r H - r L . The CPT-violating and phase-convention-independent quantity5b z either vanishes by the CPT theorem, or it depends on the 4momentum of the B meson,4 as discussed in Sec. 3. A non-zero value of either Sm or S r is only possible if both CP and CPT are violated. Assuming CPT invariance (Sm = 0,Sr = 0), and anticipating lArl << Am, Am NN 21M121 and A r NN 21M121Re(I'12/M12). Detailed SM calculations6 find values for the ratio AI'/Am in the range -0.2% to -0.3%. A state that is initially Bo (Bo)will develop a Bo ( B o )component over time, whose amplitude is proportional to the complex factor q / p ( p / q ) ,

aThe index 1 indicates Bo and 2 indicates Bo. bThe parameter z is equivalent to cose in the D E @ formalism, and to 6 (to leading order) in the c, 6 (kaon system) formalism. See, for example, Ref. 4.

167

where g*(t) = (e--iwHt f e-iWLt)/2. Invariance under CP or under T requires that I(B0IEophys(t))I= I(~oIBophys(t))I, i.e., 1q/pJ = 1. Since the magnitude of (q/pI2M 1-1m is very nearly unity in the SM, this factor is usually assumed to be a pure phase. A detailed SM calculation yields 1q/pI - 1 = (2.5 - 6.5) x At the T(4S) resonance, Bo mesons are produced in coherent p-wave pairs. If Al,2 and 2 1 , 2 are the amplitudes for the decay of Bo and Bo, respectively, to the states f1 (at time t l ) and f 2 (at some later time tz), then the overall amplitude is given by A = a+g+(At) a-g-(At), where At = t2 - t l is the difference in proper decay times and

&

+

,

a+

=

-A122 +1I1A2

a-

=

d D [?A1A2 4

-

"-1

-A1A2 P

+Z

[A1Z2+Z1A2]

.

(4)

The decay rate then is5

+ (Ay)

Re(a;a-) sinh -

+

1

Im(a;a-) sin(AmAt) , (5)

where r = i(r11 l722). Note that if the decay f1 occurs second, we would need to redefine At, a+ and a- by interchanging the labels 1 and 2, which would leave Eq. (5) unaffected. Thus, we can retain the definitions _ _ At = t2 - t l and those of Eq. (4). Terms proportional to (q/p)AlAz and (p/q)A1A2 (A1712 and x1A2) are associated with decays with net (no net) oscillation. Thus, the phase-convention-independent quantity -

can be associated to each final state f . This is the usual parameter used to characterize CP violation which involves interference between decays with and without net o~cillation.~ If one of the B mesons decays to a CP eigenstate f c p , Xcp = (q/p)(&p/ACp), where ACP (&p) is the amplitude for Bo 4 f c p (Bo+ fcp). CP violation is in this case characterized by Xcp # ~ c p where , ~ c = p f l is the final state's CP eigenvalue. This is the type of CP violation observed in decays like B + J/$JKZ.'>~ Note that the other possible sources of CP violation contained in Xcp are T violation ((q/pl # 1) and I?Icp/AcpI # 1 (direct CPviolation). The combined time dependence of B mesons decaying either to flavor (Bfl,) or CP (Bw)eigenstates provides sensitivity to the full set of physical

168

parameters, since they are determined either from different samples, or from different proper-time dependen~e.~ The sensitivity to (Re X e / l X ~ p l )Rez and I m X e / l X e I is provided by B e decays, for which the At dependence is even for the former (cosh(AFAt/2) x 1 coefficient) and odd (sin(AmAt) coefficient) for the latter. Bfl, decays lack explicit dependence on ImXe/IXGpI and the dependence on Rez is scaled by the sinh (Art/2) term, which is small for small Ar.. In contrast, the parameters 1q/pI and Imz (and Am) are determined by the Bflavdecays due to its relative abundance compared to B e decays, where the former is associated with a Ateven distribution and the latter with a At-odd (sin(ArnAt) coefficient) distribution. The sensitivity to 1q/pI comes mostly from the Bflavsample since violation of CP and T in mixing leads to a difference between the Bo -+ Bo and Bo + Bo oscillation amplitudes proportional to 14/pI4 - 1.8For small values of AI’/I’, the determination of A r / r is dominated by the B e sample, since the leading dependence on A r is linear for B e decays, while it is quadratic for Bfl, states. Moreover, the contribution of sinh(Art/2) do not depend on whether the other B meson (.€?tag)is identified (“tagged”) as Bo or Do,so untagged data can be included to improve the sensitivity to A r . We cannot determine A r / r and Rez directly because both occur multiplied by Re Xcp in their dominant contributions to the decay rate. 3. Sidereal-time modulation of CPT-violating effects

Using the SME, a perturbative calculation to leading order returns 6m,6r x ,WAa, ,

(7)

~(1,p)

where pfi = is the four-velocity of the B-meson state in the laboratory frame, and Au, are the CPT- and Lorentz-violating coupling coefficient^.^ Equation (7) reveals that the CPT-parameter z will be affected by the meson-momentum magnitude and orientation in the laboratory frame. As a consequence, since laboratory frame rotates with Earth, if CPT-violating effects exist, z will exhibit a sidereal-time dependence after converting to a non-rotating frame. Defining the non-rotating frame to be compatible with celestial equatorial coordinates with Z-axis aligned along the Earth’s rotation axis, and taking the laboratory z-axis aligned with the direction of the colliding beams, the expression for z = z ( f , p 3 is found to ‘Note that a non-zero value of either bmd or b r d is only possible if both CPand CPTare violated.

169

be4 Z %

’

Am - i A r

[Aao + PA, cos x + P sin x (Aay sin 0i-t Aax cos

0q] ,

(8) where t denotes the sidereal time, 0 the Earth’s sidereal frequency and x the angle between the laboratory (beam) z-axis and the Earth’s rotation axis (cos = zZ = cos 0 cos 4, 8 and 4 being the polar and azimuthal coordinates of the detector with respect to the non-rotating frame). Therefore, in addition to the proper-time dependence, Eq. ( 5 ) also contains siderealtime and momentum dependence from z(i,p3. In deriving Eq. (8) it is assumed that the B mesons move along the z-axis with equal momenta, which is a good approximation at B factories operating close the T(4S) resonance. Since the meson decays occur quickly on the sidereal-time scale, we can treat !L as a parameter independent of At, and therefore it is appropriate to take z as independent of At but varying with t. Note that any measurement that ignores the dependence on t and meson momentum eliminates sensitivity to Aax and Aay components, being only sensitive to an average value of the first two terms of Eq. (8). Thus direct comparison of CPT results must be done carefully since the results are experiment dependent.

x

4. The experimental setup The PEP-I1 machine consists of two rings (2.2 km circumference), one of 9.0 GeV ( e - ) and one of 3.1 GeV ( e + ) , housed in the former PEP tunnel, with a single collision point. The machine uses the SLAC linac as injector. Table 1 summarizes some of the most relevant design and achieved parameters of the machine. The high luminosities (instantaneous and integrated) are achieved through strong focusing, high currents, large number of bunches and continuous ( “trickle”) injection. The asymmetric energy of the beams results in a boost /%y for the B mesons along the e- direction (z-axis) of about 0.56 in laboratory frame. The collider has delivered since the November 1999 until July 2004 a total of 254 fl-’ (about 266 million T(4S) -, B E decays), of which 244 f t - I (256 million) have been recorded by BABAR. Current luminosity projections estimate a total integrated luminosity of 1 - 2 ab-I by the end of the decade. At the single PEP-I1 collision point is placed the BABAR detector, which is described in detail el~ewhere.~ Figure 1 shows an schematic view of the detector and its components together with their main performances.

170 Table 1.

PEP-I1 design and reached main parameters. Design

Reached

E[GeV] e - / e I [ m A ]e - / e + Number of bunches 710.5(24-05-04)

ElectroMagnetic Calorimeter 6580 Cslfll) crystals brenkov Detector 144 quartz bars I 1000 PMTs

----.

40 stereo layers

Instrumented Flux Return ironIRPCs (rnuonlneutral hadrons)

SVT: 97% efficiency, 15 pm z hit resolution (inner layers, Itracks) SVT+DCH: 0(pT)/pT=0.13 % x pT+ 0.45 % DIRC: K-n: separation 4.20 Q 3.0 GeVk + >3.00 Q 4.0 GeVk EMC: GJE = 2.3% x E" CB 1.9 %

Figure 1. Schematic view of the BABAR detector and its subsystems, together with some of the main performances.

5. Results with fully reconstructed B decays The measurement of AI'/F, and CP, T, and CPT violation in mixing has been performed by analyzing T(4S) decays in which one neutral-B meson is fully reconstructed and the other is identified as being either Bo or Bo on the basis of the charges of leptons and kaons, and other indicators of f l a ~ o rassuming ,~ constant CPT-violating parameters. We reconstruct the flavor statesd Bflav= D(*)-.rr+(p+, u:) and J/+K*O(-+K + F ) and the CP eigenstates BCP =J/$K:, +(2S)K:, XclK:, and J/+K!. The data sample consists of 88 million T(4S) + B E decays (only about one third of the total dCharge conjugation is implied throughout this letter, unless explicitly stated otherwise.

171

recorded statistics), which corresponds to approximately 31,000 Bflavand 2,600 CP eigenstates. The time interval At between the two B decays is calculated from the measured separation AZ between the decay vertices of BreCand Btag along the boost direction ( z - a ~ i s )At ,~ trec - ttag M A./(Prc). The measurement requires a precision analysis since competing physics and detector effects that can mimic the behavior we seek must be i n ~ l u d e d . ~ First, the resolution for At is comparable to the B lifetime and is asymmetric in At. This asymmetry must be well understood lest it be mistaken for a fundamental asymmetry. Second, tagging assigns flavor incorrectly some fraction of the time. Third, interference between weak decays favored by the CKM quark-mixing matrix ( b 4 ctid, e.g., Bo + D-T+) and those doubly-Cabibbo-suppressed (DCS) ( b + ucd, e.g., Bo 4 D-T+), roughly proportional to jVu,V,.,/~bVu,l = 0.02, has to be explicitly taken into account. Fourth, direct CP violation in the B e sample could mimic CP violation in mixing and must be parametrized appropriately. Finally, we have to account for possible asymmetries induced by the differing response of the detector to positively and negatively charged particles. To disentangle all these issues we rely mainly on data, making use of a simultaneous maximum-likelihood fit to the time distributions of tagged and untagged, flavor and CP eigenstates. Backgrounds are primarily due to misreconstructed candidates and are studied in data using sidebands. A total of 58 free parameters (32 modeling the signal and 26 the background) are determined with the likelihood fit.5 The results are sgn(Re Xc*p)AI'/I' = -0.008 f 0.037(stat.) f 0.018(syst.)[-0.084,0.068], 1q/pI =

(dileptons)

1.029 f O.O13(stat.) f O.Oll(syst.)[ 1.001,1.057] , 0.998 f O.OOG(stat.) f 0.007(syst.)[ 0.983,1.013] ,

(ReXcp/lXeI) Rez = 0.014 f 0.035(stat.) f 0.034(syst.)[-O.072,0.101], Imz = 0.038 f 0.029(stat.) f 0.025(syst.)[-0.028,0.104]. The values in square brackets indicate the 90% confidence-level (CL) intervals. The measurements of sgn(ReXcp)AI'/I' and 1q/pI in the limit when CPT conservation is assumed are unchanged. The largest statistical correlation among the physics parameters appears between Im X c p / l X e l and Imz, which amounts to 17.4%. The second result of 1q/pI corresponds to the measurement performed when the two B mesons are reconstructed using high momentum leptons from semileptonic decays (dilepton events) .8 This measurement constrains 1q/pI without assumptions on the value of z. Fig-

172

ure 2 (left) shows the results in the ( l q / p l - l ,1). plane, compared to the SM expectations. If we express the CPT limits as ratios of the CPT-violating to the CPT-conserving terms we have

m

,

< 1.0 x

-

dr 0.156 < - < 0.042

r

at the 90% CL. The parameters ImXcp/lXcpI and Am are consistent with recent B-factory result^.^^^^^ The value of the CP- and T-violating parameter ImXcp/IXcpl increases by +0.011 when CPT violation is allowed, which is consistent with the correlations observed in the fit with CPT violation. The value of Am remains unchanged between the two fits. Most contributions to the systematic uncertainties are determined with data and will decrease with additional statistics. The largest single source of uncertainty is the DCS contribution to (ReXcp/IXcpI) Rez (0.032). This contribution will decrease since with additional statistics a better (and less conservative) treatment of DCS effects will be p ~ s s i b l e .The ~ sgn(Re Xcp)Ar/r and 1q/pI measurements can be used to set constraints on the complex ratio r12/M12, as shown in Fig. 2 (right). Solid contours show the results assuming Re X c p > 0 (as expected in the SM based on other experimental constraints), while dashed contours are for ReXcp < 0. Inner (outer) contours represent 68% (90%) CL regions for two degrees of freedom. The black region shows the predictions of SM calculations when all available experimental inputs are used.5

404

4 02

0 02

0.04

-0.1

-0.05

0

0.05

wr,#,J

-

0.1 ArlAm

Figure 2. (Left) Favored regions at 68% CL in the (Iq/pl - 1,121) plane, compared to SM expectation. The axis labels reflect the requirements that both CP and T be violated if Iq/pl # 1 and that both CP and CPT be violated if IzI # 0. (Right) Constraints at 68% and 90% CL on I112/1M12 as determined from the sgn(ReAm).Ar/r and Iq/p( measurements, compared to SM calculations.

173

6. Sensitivity to CPT-breaking sidereal-time modulation

As discussed in Sec. 3 and explicitly shown in Eq. (8), the CPT-violating parameter z, if non-zero, depends on center-of-mass boost (Pr),latitude of the collider (8) and azimuthal orientation of boost direction (4). The approximate values of these parameters for B B A R (Belle) are estimated to be 0.554, 37.42"N, and S35"E (0.425, 36.15"N, and S45"E), respectively, from which we estimate cos x = -0.65( -0.57) and sin x = 0.76(0.82). This implies that Belle's boost direction benefits sidereal-time varying terms, but BBAR's boost is larger, enhancing slightly all terms of Eq. (8), as shown in Table 2. In order to estimate the sensitivity to CPT-breaking sidereal Table 2. Sidereal-time dependence coefficients of Eq.(8)vv at B factories.

Term Aao

Coefficient

BABAR

Y

Aaz AaxlAaY

PY cos x

1.14 -0.36 0.42

B r sin Y

Belle 1.09 -0.24 0.35

modulation a CPT asymmetry sensitive to z can be defined as4

A:~(A~,E)

=

r f f ( A t > O,?) - r,j(At < 0 , f ) r f f ( A t > 0 , f )+rf,-(At < 0,E)

N

N

-2ImzsinArnAt 1 +cosAmAt ,(9)

only valid to first order in z and A r , and where f(f) denotes here a flavor eigenstate and its CP conjugate. Figure 3 shows this asymmetry for dimuon (e+e- -+ T(4S) 4 p+p-) events as a function of sidereal and Pacific time, using a data sample of about 100 ft-'. These events provide a At = 0 (AcPT = 0) benchmark, which can be used to evaluate not only tf the sensitivity but also detector biases and systematic effects to be used for correcting the signal data. A sensitivity on A:r at level translates into a sensitivity on z of about lop2. With full BABAR statistics at the end of the decade, the sensitivity will reach

7. Summary Using one third of the data already recorded by BABAR, we have performed a simultaneous measurement of the difference A r / r between the decay rates, and of CP, T and CPT violation in the Bo-meson system. The limits on A r / r and T violation in mixing have reached a precision at the level of 8% and 1% (90% CL), respectively, largely improving previous results." The CPT measurements, 16rnllrn < 1.0 x and -0.156 < 6r/r < 0.042,

174

Figure 3.

C P T asymmetry A C r T ( A t , i )as defined in Eq. (9) for e + e -

ff

+

T(4S) +

p+p- events as a function of sidereal (left) and Pacific (right) time, from about 100 fb-l.

represent the strongest and more general CPT invariance test in the Bo system to date." Previous mixing and CP BABAR measurements performed neglecting these effects are unaffected at this level of precision. We have also discussed within the framework of a general extension of the SM the sensitivity of the CPT-violating parameter to sidereal-time modulation. The magnitude and direction of the boost of the PEP-I1 machine and the latitude of the BABAR detector makes the experiment sensitive to sidereal modulation. The large data sample already recorded and the projections for the forthcoming years will provide the opportunity to perform high precision measurements in BE oscillations which may bring surprises.

References 1. BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 89,201802 (2002). 2. Belle Collaboration, K. Abe et al., Phys. Rev. D 66,071102(R) (2002). 3. Particle Data Group, S. Eidelman et al., Phys. Lett. B 592, 1 (2004). 4. V.A. Kostelecki, Phys. Rev. D 64,076001 (2001), and references therein. 5. BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 92,181801 (2004); Phys. Rev. D 70,012007 (2004). 6. M. Beneke et al., Phys. Lett. B 576,173 (2003). M. Ciuchini et al., JHEP 0308, 031 (2003). S. Laplace e t al., Phys. Rev. D 65,094040 (2002). 7. BABAR Collaboration, B. Aubert et al., Phys. Rev. D 66,032003 (2002). 8. BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 88, 231801 (2002). 9. BABAR Collaboration, B. Aubert et al., Nucl. Instr. Meth. A 479,1 (2002). 10. DELPHI Collaboration, J. Abdallah et al., Eur. Phys. Jour. C 28,155 (2003). 11. Belle Collaboration, N:C. Hastings et al., Phys. Rev. D 67,052004 (2003). OPAL Collaboration, R. AckerstafF e t al., Z.Phys. C 76,401 (1997).

ULTRA-SENSITIVE SPEEDOMETER USING NON LINEAR EFFECTS OF OPTICAL PUMPING

BENJAMIN T.H. VARCOE

Physics and Astronomy, University of Sussex, Falmer, Brighton BNl 9QH, UK Ernail: B. VarcoeQsussex.ac.uk Violation of Lorentz invariance in the form of a frame dependence of the speed of light is forbidden by most of modern physics, however this law is increasingly coming under pressure from theories such as string theory and theories of quantum gravity which predict violations of Lorentz invariance at some level. This paper presents a concept for a Lorentz invariance test using electromagnetically induced transparency which, in combination with an effect known as F’resnel drag, can amplify the effect of variations of the speed of light. It is shown how a model apparatus could in principle be used to search for frame dependences in the speed of light to a sensitivity of a part in

1. Introduction Testing Lorentz invariance is important both because, as a component of special relativity, it is one of the pillars of modern physics and also because varying degrees of violation of Lorentz invariance are predicted by theories such as string theory, string loop theory and quantum theories of gravity. The goal of this project is the construction of a new type of Michelson interferometer that makes use of a recently demonstrated technique from quantum optics called “Slow Light.”’ A Slow Light Assisted Michelson (SLAM) interferometer could potentially provide a test of Lorentz invariance several orders of magnitude better than is possible with current techniques. Violations of Lorentz invariance may arise when light becomes coupled to a field against which its velocity can be measured. This could occur either by the introduction of a new field or an as yet unknown coupling to an existing field. The hypothesized existence of new physics beyond the Standard Model is therefore a strong motivation for this type of experiment. To simplify the discussion of the new experimental method, an assumption is made that any potential violations will have the properties of a refractive index. This is quite a reasonable assumption and fills all of the requirements that a Lorentz violation may have, namely, that the speed 175

176

of light may not be the relativistic constant c and that we would be able to detect motion through this field via the frame dragging effect known as Fresnel drag. 2. Fresnel Drag Fresnel drag occurs when light moves through a medium with a refractive index (n # 1) that is moving with respect to the external or ‘measurement’ frame.2 In the measurement frame it appears that the light is ‘dragged’ by the medium. The effective velocity of light in the measurement frame is therefore given by ceff = c/n @ u (where @ represents the Lorentz velocity addition). To first order we can rewrite this as C

which is known as Fresnel drag. The second term in Eq. (1) is known as the Fresnel drag coefficient, cy=(l-$)u*

The result is that light, when traveling in a frame that is moving though a medium having refractive index n # 1, acquires a velocity component due entirely to this medium. But Eq. ( 2 ) is not the whole story; we also have to include dispersion to account properly for the relative difference in velocity between the source and the medium. In this case Eq. ( 2 ) becomes

which in terms of the group velocity in a medium in which n cy = c/vg

- 1.

N

1 becomes (4)

The conclusion is, if we can make vg very small cy can be made very large, the effective velocity change of the light as seen by atoms in the cell can become very large, or amplified.

3. Slow Light

A phase coherent ensemble of atoms represents a novel state of matter in which the susceptibility of the medium is modified. This can lead to a very steep dispersion and therefore a subsequently slow group velocity. This effect has been demonstrated by several groups who have demonstrated

177

very low group velocities,' recently reaching speeds as slow as 8 m/s in a hot gas cell.4 One method of producing slow light (there are now several) is when the medium exhibits Electromagnetically Induced Transparency (EIT). A set-up for producing slow light is shown in Fig. 1. Two lasers are used; one prepares the atomic ensemble in a phase coherent superposition (the drive laser) and the other produces the light that will be slowed by the medium (the probe laser). The probe laser is the laser that we will use to construct the SLAM interferometer.

Probe Laser

I

L-1'

/

Drive Laser

Figure 1. The experimental schematic for slow light. Assume for the moment that the atoms have no Doppler width in the plane of the lasers. This could be either a dense atomic beam or an ultra cold sample of atoms. It is shown later that the actual arrangement is however somewhat flexible.

In an EIT medium where the laser frequencies and polarizations are such that the arrangement of levels is that of Fig. 2, the susceptibility is given by3

where up and U D are the drive and probe laser frequencies, v = 3X3N/8r2, X is the probe laser wavelength, N is the atomic density, r 3 2 = 7 3 2 i[b

+ +

178

Figure 2. The laser transitions for creating slow light.

(kp - kd)u],7 3 is the radiative decay rate of level 3 to level 2 , 7 3 2 is the coherence decay rate of the two lower levels, y is the homogeneous half width of the transition, A p = ~ 3 -2 u p , AD = ~ 3 1 -u p , and 6 = A p - AD. flp, kp and RD, kD are the Rabi frequencies and wave numbers, respectively, of the drive and probe transitions. The susceptibility of a medium is divided into real and imaginary parts (i.e., x = x’ i f ) , where x’ is related to the refractive index via n = ( ~ ’ + l ) l and / ~ x’’ represents the loss of laser power per unit wavelength. The group velocity in such a medium is given by vg = c / [ 1 ( u / 2 )( d x ’ / d u ) ] . In a normal gas the dispersion dramatically increases as the frequency approaches resonance, and is normally accompanied by strong absorption (Fig. 3a, 3b). In EIT, the same medium is rendered transparent (for a very narrow range of frequencies) by the drive laser, which creates a superposition of states, decoupling the probe laser from the atoms and rendering the medium transparent (Fig. 3c, 3d). Plotting the susceptibility as a function of the probe laser frequency both with and without a drive laser reveals the dramatic effect of adding coherence to the atomic ensemble (Fig. 3). Plots 3a and 3b show the standard dispersion and absorption curves for excitation of a two level atom. Figures 3c and 3d show the change that occurs when the drive laser is added. The absorption at line center disappears and the dispersion curve changes, adding a strongly dispersive section to the line center. It is this strong dispersion that reduces the group velocity creating slow light and it also

+

+

179

Figure 3 . Plots a) and b) show the absorptive and dispersive properties of the atomic ensemble for the transition between states 12) and 13) without the drive laser. Plots c) and d) show the same properties for the 12) + 13) transition, with a drive laser coupling the 11) + 13) transition. The drive laser introduces a steep dispersion on resonance while the absorption goes to zero. The steepness the dispersion is inversely related to the power of the drive laser.

what gives the SLAM interferometer the high sensitivity to small variations in c. Note that it is not actually the slow group velocity, but the steepness of the dispersion that is important. The attenuation and group velocity I,,,/Ii, = exp(-2aL) and vg = c/(l n g ) (where n g = (v/2)(dx’/dv)) can now be obtained3 via the parameters ng and a,

+

and

180

where AWDis the Doppler width, which is a temperature dependent parameter. However if a'$>> y32(y AWD)the temperature of the atoms is factored out. Thus for a high enough drive intensity effects become independent of both velocity width and temperature, thus relaxing the requirement of stationary atoms in the phase coherent ensemble.

+

Figure 4. The group velocity and absorption of the gas cell as a function of the intensity of the drive laser. Even under conditions of EIT the medium becomes absorptive for low drive laser intensities. The probe laser has a finite width and as the dispersive part of the distribution narrows, more of the laser frequency width overlaps with regions of strong absorption. As mentioned in the text, the absorption can be reduced by detuning both lasers from resonance with state 13).

Figure 4 shows the absorption and group velocity as a function of the Rabi frequency of the drive laser (the Rabi frequency will be a property of the atom used in the medium and the square root intensity of the laser). Absorptive losses occur in the medium because the probe laser has a finite width which overlaps with the absorptive parts of the curve in Fig. 3d. Naturally the losses increase as the dispersion increases because the frequency .width becomes smaller. However the losses can be practically eliminated by detuning the laser from resonance thereby reducing the likelihood of loss by spontaneous emission. Slow light with reduced losses has been recently demonstrated by M. Kozuma et al. (2002).5 They report losses up to an order of magnitude smaller than the resonant scheme for the same group velocity. They also report that losses can be nearly eliminated using this scheme with some reduction of the dispersion causing an increase in group velocity by up to a factor of 10.

181

4. The Experiment

Date Acquisition

Figure 5 . A simplified experimental schematic to measure frame dependent parameters in a slow light experiment. The arrangement of the lasers is stylistic to clearly separate the roles, however this does relate to the arrangement one would have if an atomic beam rather than a cell was used.

A highly simplified schematic of the apparatus is shown in Fig. 5. A Mach-Zender apparatus is the archetypical method of measuring refractive index of a medium, thus it is the ideal starting point for this hypothetical tour of the experiment. In this case however the refractive index due to the medium is a property created by the drive laser and and measured by the probe laser. The refractive index is therefore a property of two influences, the frequency of the probe laser and the intensity of the drive laser. If both lasers are resonant with their respective atomic transitions, we expect to

182

find no refractive index shift due to the medium (the atomic medium is a disperse gas with a refractive index very close to 1). While the MachZender configuration is chosen here for clarity of operation, the common mode rejection offered by an appropriate modification of the traditional Michelson interferometer would be the preferred experimental arrangement. Imagine now that the apparatus is immersed in a Lorentz violating field through which it is moving (e.g., via the Earth’s motion relative to the hypothesized global rest frame). Fresnel drag by the Lorentz violating field causes the laser to acquire an additional velocity component to that of the test apparatus. This means that cell will have acquired a motion with respect to the light and we have to apply the Fresnel drag formula again to the cell to calculate its effect on the phase of the light. The cell and the global rest frame are totally separate effects here so each time there is a relative motion, Fresnel drag is required to perform the calculation. The light is therefore Doppler shifted with respect to the cell. To calculate what effect this might have on the phase we have to return to the full calculation of Fresnel drag (Eq. (2)) which tells us that the light will suffer a phase shift determined by the refractive index and dispersive properties of the cell. As the cell has strong dispersive properties, the effective speed can be multiplied by up to lo7 (see Eq. (4)). For a slow light group velocity of 30 m/s (i.e., ng = lo7) and assuming that, in an appropriate setup, frequency shifts of 1 mHz - which correspond to a part in 1017 - can be detected, sensitivity to bc of a part in is possible. In the short term a more reasonable target would be to detect shifts in the region of 1 kHz for group velocities of km per second, leading to sensitivities of parts in 1016 matching the current start of the art in cavity based experiments.6

5 . Conclusion

This is potentially a highly promising experiment as using a SLAM interferometer we can achieve a significantly enhanced sensitivity to 6c of a part in This is 9 orders of magnitude above the current state of the art for Fabry-Perot experiments. In addition there are several features of this experiment to recommend it for further investigation. Firstly, it is possible to mount the drive laser in a collinear arrangement, thus the interferometer can therefore be stabilized to the drive field component with no loss of sensitivity and a significantly increased stability. Secondly, the sensitivity of the apparatus is tunable via the intensity of the drive laser, thus helping

183

to eliminate systematic effects in candidate signals. Thirdly, the interferometer is small and self-contained and is therefore relatively easy to mount on an ultra stable platform to maintain its stability and hence sensitivity. Finally, the experiment proposed here also eliminates the need to rotate the apparatus through 360’ to achieve maximum sensitivity. This is because modulating the dispersion signal can modulate the sensitivity of the experiment; thus any frequency shift also displaying the same modulation is a real effect. The sensitivity of the arms can be independently modulated thus eliminating the need to interchange their roles by rotating the apparatus as in the traditional Michelson interferometer. Moreover if crossed cavities were used in a Michelson-like experiment this fits in well with the Standard-Model Extension7 when used in comparison to more conventional crossed cavity experiments (i.e., it does not probe the interaction of light with matter but is purely in the photon sector).

Acknowledgment This work is supported by PPARC’s Particle Physics Peer Review Panel.

References 1. A. Kasapi et al., Phys. Rev. Lett. 74,2447 (1995); 0.Schmidt et al.. Phys. Rev. A 53,R27 (1996); L.V. Hau et al., Nature (London) 397,594 (1999). 2. I. Lerche, Am. J. Phys. 45, 1154 (1977); G. Barton, “Introduction to the relativity principle,” Wiley Publishers, Chichester, 1999. 3. M.M. Kash et al., Phys. Rev. Lett. 8 2 , 5229 (1999). 4. D. Budker et al., Phys. Rev. Lett. 83, 1767 (1999). 5. M. Kozuma et al., Phys. Rev. A 66,031801(R) (2002). 6. J.A. Lipa et al., Phys. Rev. Lett. 90,060403 (2003); H.Miiller et al., Phys. Rev. Lett. 91,020401 (2003); P. Wolf et al., Phys. Rev. Lett. 90, 060402 (2003). 7. V.A. Kosteleckj. and M. Mewes, Phys. Rev. D 66,056005 (2002).

AN IMPROVED TEST OF RELATIVISTIC TIME DILATION WITH FAST, STORED IONS

G. GWINNER Department of Physics and Astronomy University of Manitoba Winnipeg, M B RST 2N2, Canada E-mail: [email protected]. ca S. REINHARDT, G. SAATHOFF, D. SCHWALM, AND A. WOLF Max-Planck-Institut fur Kernphysik 69029 Heidelberg, Germany G. HUBER, S. KARPUK, AND C. NOVOTNY Institut fur Physik Universitat M a i m 55099 Mainz, Germany A precise, laser-spectroscopic test of time dilation in special relativity has been performed with fast ions stored in the heavy-ion storage ring TSR in Heidelberg. At a velocity of 6.4% of the speed of light, the Doppler-shifted frequencies of a transition in 7Li+ have been measured in forward and backward direction. First for deviations from results confirm relativity and set an improved limit of 2.2 x the Lorentz factor 7 s =~1/J1 - v2/c2. Ongoing improvements will tighten the limit further. Preliminary investigations indicate that Doppler-type experiments can have unique sensitivity for testing Lorentz/CPT-violating extensions of the Standard Model.

1. Tests of Special Relativity and Kinematic Test Theories In his 1949 paper, Robertson’ concludes that the three landmark experiments of special relativity (SR) carried out by then, namely MichelsonMorley (MM),2 Kennedy-Thorndike (KT),3and Ives-Stilwell (IS)4 are sufficient to ‘enable us to replace the greater part of Einstein’s postulates with findings drawn inductively from observations.’ Any deviations from the laws of SR would violate Einstein’s relativity principle and lead to reference-frame-dependent effects. To quantify such deviations, a kinematic 184

185

test theory was devised by Robertson and was later reformulated by Mansouri and S e ~ lthey ; ~ consider generalized Lorentz transformations between a hypothetical preferred frame C ( T ,2)and a frame S ( t ,2) moving relative along X , and the speed of light c is assumed to be to C at a velocity constant and isotropic in C only. Using Einstein synchronization, these transformations read

d v ;

where = 1/ h( V 2 ) ,6( V 2 ) d( , V 2 )are velocity-dependent test functions describing modified time dilation, modified Lorentz contraction, and transverse Lorentz contraction, respectively. The latter is absent from SR, of course. For h = 6 = d = 1, SR is recovered. For convenience, these functions are usually expanded in powers of ( V / C ) i.e., ~, h = 1 &(V/C)'+ O ( C - ~ etc.; ) any non-zero value for any of the three resulting test parameters &,,8, and 8 indicates a breakdown of SR. The IS experiments determine h, i.e., test time dilation. MM, on the other hand, is sensitive to - 81, the difference between modified Lorentz contraction and transverse Lorentz contraction, and K T determines I& -,8I, the difference between the modifications of time dilation and Lorentz contraction. A possible candidate for a preferred frame C is the one in which the cosmic microwave background (CMB) is isotropic. Earth is moving with a velocity of V 350 km/s with respect to the CMB frame. It is customary (if arbitrary) to quote the sensitivity of SR tests based on this choice of preferred frame. The latest round of experiment^^%^ has yielded the limits Ip - 61 5 1.5 x lo-' and I& - ,bI 5 6.9 x The best constraint on time dilation prior to the storage-ring method presented in this paper was I&\ < 1.4 x

+

lb

A

,

.

2. Time Dilation via the Relativistic Doppler Effect

In SR, the radiation from a moving source is shifted according to VO

='-)'SR(1-,k?COS8)d,

(2)

where vo is the frequency in the rest-frame of the source, p' = V'/c the velocity of the source, 8 the angle between p' and the line of sight to the observer stationary in the lab, who measures a frequency u'. At 0 = n/2, the first

186

order (classical) Doppler effect vanishes, and Einstein already proposed in 1907 to use the resulting, purely relativistic, transverse Doppler effect to observe time dilation. However, at 8 = n / 2 , Eq. ( 2 ) is most sensitive to angular misalignment, the dependence on small deviations from perfect orthogonality is linear. On the other hand, at 8 = 0 and 8 = 7r, small angular misalignments enter only quadratically, making this the preferable geometry chosen by Ives and Stilwell 30 years later4. They used hydrogen atoms in canal rays moving at P = 0.005 and measured the Doppler-shifted frequencies up and u, of the n = 4 + n = 2 transition in parallel (8, = 0) and antiparallel (0, = T ) direction with respect to Within SR the respective Doppler shifts are given by

8.

vo = Y S R ( ~- Pcosep,a)up,a.

(3)

Multiplying the two equations for parallel and antiparallel observation, we yield a velocity-independent relation uo” = u,up,

(4)

if SR is correct. In the context of the Mansouri-Sex1test theory, a nonvanishing test parameter ti would modify this relation as

where @lab is the velocity of the lab with respect to the prefered frame C.8 Note that this result gives rise to two experimental methods. For ,8 >> it is advantageous to compare absolute frequencies; for ,8 < one can resort to observing sidereal variations in u,up. Essentially all IStype experiments using fast atomic beams have superior sensitivity to the P2 term. The original Ives-Stilwell experiment obtained an upper bound Great progress came along with the advent of laser of & < 1 x spectroscopy. Two-photon spectroscopy on a neon beam ( p = 0.0036) set bounds of ti < 2.3 x (P2 term),g and ti < 1.4 x l o p 6 from a sidereal variation analysis.’O 3. The Heidelberg Storage Ring Experiment

The advent of heavy-ion storage rings equipped with electron-cooler devices opened up the possibility of performing high-precision laser spectroscopy on significantly faster ion beams. Our experiment at the storage ring TSR at the Max-Planck-Institute for Nuclear Physics in Heidelberg uses He-like ’Li+ ions stored at P = 0.064. Laser spectroscopy is performed on the

187

2s 3S1+ 2p 3Pz transition at 548 nm. The 7Li nucleus has spin I = 312, leading to a hyperfine splitting. The F = 512 4 F’ = 712 transition is effectively a closed two-level system, due to the A F = 0 , f l selection rule for electric dipole transitions and the fact that the separation from neighbouring hyperfine levels is more than 10 GHz - much larger than the Doppler broadening ( m 2.5 GHz) and the widths of the lasers (less than 1 MHz). Negative Li ions are produced in an ion source and injected into a tandem Van de Graaff accelerator, where they are gas-stripped to Lif and accelerated to 13.3 MeV. This process leaves about 10% of the ions in the metastable 3S1 state required for the spectroscopy. About lo8 ions are injected into the TSR, and kept on a closed orbit of 55 m circumference. Collision with the background molecules (= 5 x mbar rest gas pressure) reduces the lifetime of the metastable beam fraction to about 13 s. For the first 5 s of storage, electron cooling is applied to the beam to reduce the beam diameter to = 500 pm and the divergence to M 50 p a d . The velocity of the beam is controlled by radio-frequency bunching. After the electron-cooling period, the beam is available for the Doppler spectroscopy. The longitudinal momentum spread of Aplp M 4 x lop5 leads to a Doppler width of the transition of about 2.5 GHz (FWHM) - about 5000 times larger than the accuracy with which the laser frequencies have to be measured for ti < To overcome this problem, the method of saturation spectroscopy is applied. The ion beam velocity is adjusted such that one of the lasers (a fixed-frequency argon-ion laser at 514 nm, providing u p ) is in resonance with a narrow group of ions in the center of the velocity distribution. The second, counter-propagating, tunable dye-laser at 585 nm (va) is then scanned across the Doppler profile of the transition. The ions’ fluorescence is recorded by photomultipliers as a function of the dye-laser frequency. The crucial point of saturation spectroscopy is that the intensities of the lasers are chosen sufficiently high to ‘saturate’ the transition. In this regime, the fluorescence yield (due to spontaneous emission) is no longer proportional to the intensity of the laser, as stimulated emission (which does not contribute to the fluorescence) becomes increasingly dominant. Generally, the two lasers talk to different velocity classes and their fluorescence yield simply adds up. However, at one point in the scan, both lasers are resonant with the same class. The fluorescence yield is then reduced due to the saturation, leading to the famous Lamb dip, which can be almost as narrow as the natural linewidth of the transition (3.8 MHz in our case). The dye-laser frequency u, at which the dip occurs can then be

188

used to test Eq. (4).

data from TSR

p

I ioncurrent

- mirror

L

'

I

Figure 1. The setup of the TSR experiment.

The experimental setup is shown in Fig. 1. To keep its frequency stable, the Ar-ion laser is locked to a high-finesse Fabry-Perot interferometer for short-term stability. Long-term stability is provided by a lock to a resonance in molecular iodine, whose frequency is well known. The dye-laser is typically scanned over a range of 200 MHz. Its absolute frequency is determined by recording simultaneously a saturated absorption spectrum of molecular iodine. The two laser beams are merged in a dichroic beamsplitter and transported in a single-mode fiber to the storage ring area, where the beams are expanded in an achromatic telescope and focused through one section of TSR onto a plane mirror, which in turn is adjusted to reflect the beam back onto itself. Proper alignment of the two laser beams with respect to each other and the ion beam is of utmost importance to the experiment. Using computer-controlled translation and rotation stages for the telescope and the retro-reflector, we reliably achieve alignment of better than 70 p a d for both the laser-laser and laser-ion angle (see Table 1 for the resulting uncertainties).

189

5

3000

...................... .................... .............. ........ ........................ ....... .................... ............. ...

LL

-400 -500},

i

1 -100

-50

0

50

100

Frequency (MHz)

Figure 2. Typical fluorescence signal observed with PMT3. The iodine signal is the derivative of the absorption profile.

The added result from 82 separate laser scans (each taking about 20 s) is shown in Fig. 2. To remove the influence of varying ion currents and laser power fluctuations on the data, we use a scheme where the laser light is sent to the experiment in four different configurations: (i) both lasers are on, (ii) only the dye-laser, (iii) only Ar-ion, (iv) no laser. Each period lasts for 200 ps. A spectrum containing data from period (i) will display a Lamb dip as shown in trace (b), whereas a spectrum assembled from the sum of (ii) and (iii) will have no dip (trace (a)),but is otherwise identical. Subtracting these spectra from each other, we obtain a Lamb dip with a good Lorentzian line-shape. Its frequency is compared to that of an iodine line (which is calibrated against a nearby iodine line measured at the PTB frequency standards lab). The measured line width of 15 MHz includes the effect of saturation broadening. Extrapolation to zero laser intensity gives a width of x 11 MHz, about 7 MHz in excess of the natural width. The laser frequency modulation required for our locking scheme and residual magnetic fields can account for this. As we use linearly polarized light, the stray magnetic fields (at most a few Gauss) should cause no net first-order Zeeman shift, only a broadening. The largest systematic uncertainty in this measurement turns out to be the laser-intensity dependence of the Lamb dip frequency, as shown in Fig. 3 (upper trace). We attribute this effect to local changes in the velocity distribution due to laser forces. Changes happening faster than the 200 ps

190 4 I

N

s

3

0

%

.-c u

.-

2 1

U

:.

0

.-

Q -0

n -1

;

2

o

2

4

6

a

10

12

14

16

Ar-ion laser power (mW)

Figure 3.

Dependence of the Lamb dip frequency on the laser power.

laser switching time cannot be cancelled by this method. Hence, we carry out an extrapolation to zero light intensity. Considering the systematics discussed above, an experimentally measured Lamb dip frequency vzxP is derived in Table l (columns l and 2). From the frequency vp of the Ar-ion laser and the rest-frequency vo (measured by Riis et al."), a SR prediction u,"" can be computed. Our result is in agreement with special relativity and we set a new limit12 &

2.2 x

(6)

The uncertainty in the rest-frequency is now the dominating error (note that vo enters quadratically).

3.1. New Developments Several improvements have been made to the experiment recently. It was established that a higher switching frequency for the lasers (20 ps per period instead of 200 ps) dramatically reduces the dependence of the Lamb dip frequency on the laser intensity (see Fig. 3, lower trace). Column 3 in Table 1 shows that the uncertainty associated with the intensity extrapolation is reduced from 460 kHz to 150 kHz. Including other improvements, the error on v z x P can be reduced from 517 kHz to 224 kHz. However, this would be meaningless without a more accurate value for the rest-frequency. For this reason, TSR measurements at p = 0.03 have been carried out over the last year. While they do not really provide a 'rest-frequency,' the results at different velocities can be compared in a meaningful way. Preliminary analysis of the data indicates that this measurement can provide a substitute rest-frequency for the high$ experiment

191

Table 1. Accuracy budget of the saturation specttion spectroscopy: errors are quoted as1 all values in kHz.

Next generation

Result by Saathoff et al.lz

p

p = 0.06 Iodine reference line (dye)

= 0.06

/3 = 0.03

Error estimates

Frequency

Error

512 671 028 023

152

100

100

50

100

100

-

-

Frequency calibration AOM shift

414 000

Lamb dip offset to iodine line

1550

460

150

150

Wavefront correction (dye)

-665

160

70

35

Ion beam divergence

50

50

25

Laser-laser angle

40

40

20

Laser-ion beam angle

10

10

5

224

212

99

122

Total vzxp

512 671 442 908

517

582 490 203 442

SR prediction v:“ -

= u,”/uF”

I

546 466 918 790

400

512 671 443 186

755

200

-278

915

300

SR va

100

with an uncertainty of only 100 kHz (column 4 in Table 1). Ultimately, we estimate that a limit & < 7 x lo-’ can be reached at TSR.

3.2. The Future: Faster I o n s Our experiments use the highest possible velocity at which Li+ can be stored in the TSR. The limit is dictated by the strength of the ring’s dipole magnets. At the ESR facility at GSI in Darmstadt, Li+ velocities up to P = 0.45 are attainable. A particularly intriguing situation arises at ,O = 1/3, where up = 2ua. In this case, the light can be generated by a single laser at 776 nm and a frequency-doubling unit. The absolute determination of the frequency can be accomplished with the new frequency-comb technique developed by the Hansch g r 0 ~ p . Naively, l~ the sensitivity for & is 25 times larger than at TSR.However, several systematic error sources also increase with velocity. We believe that an order of magnitude improvement is possible, pushing the limit for & into the range.

192

4. New Theoretical Developments: Sensitivity to the

Lorentz Violating Standard-Model Extension Lorentz invariance and CPT invariance (linked via the CPT theorem and a recently discussed ‘reverse’ CPT theorem14) can be violated, e.g., through spontaneous symmetry breaking, in unifying theories such as string theories. This should lead to observable effects in SR tests. Preliminary results by C. Lane15 indicate that Doppler-effect experiments have unique sensitivities to numerous parameters of the Standard-Model Extension of Kosteleckf et a1.,I6 in the nucleon sector as well for electrons. Some parameters can be bounded by our result. Others could be accessed by slightly modified versions of our measurement, where specific IF, m ~ -+) IF’, mh) transitions are excited. Our current technique averages over rn-substates (kinematic test theories are oblivious to the symmetry of the atomic states). Clearly, much more experimental and theoretical progress is possible.

Acknowledgments We would like to thank H. Buhr, L.A. Carlson, U. Eisenbarth, M. Grieser, H. Krieger, S. Krohn, and R. Muiioz Horta for their participation in the experiments. Helpful discussions with A. Kosteleckf, C. Lammerzahl, and C. Lane are acknowledged.

References 1. 2. 3. 4. 5.

6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16.

H. P. Robertson, Rev. Mod. Phys. 21,378 (1949). A. A. Michelson and E. W. Morley, Am. J. Sci. 34,333 (1887). R. J. Kennedy and E. M. Thorndike, Phys. Rev. 42,400 (1932). H. E. Ives and G. R. Stilwell, J . Opt. SOC.Am. 28,215 (1938). R. Mansouri and R. U. Sexl, Gen. Relativ. Gravit. 8, 497 (1977); 8, 515 (1977); 8,809 (1977). H. Muller et al., Phys. Rev. Lett. 91,020401 (2003). P. Wolf et al., Phys. Rev. Lett. 90,060402 (2003). M. Kretzschmar, Z. Phys. A 342,463 (1992). R. W. McGowan et al., Phys. Rev. Lett. 70, 251 (1993). E. Riis et al., Phys. Rev. Lett. 60,81 (1988). E. Riis et al., Phys. Rev. A 49,207 (1994). G. Saathoff et al., Phys. Rev. Lett. 91,190403 (2003). T. Udem et al., Nature 416,233 (2002). 0. W. Greenberg, Phys. Rev. Lett. 89,231602 (2002). C. Lane, these proceedings and private communication. D. Colladay and V. A. Kostelecki, Phys. Rev. D 58,116002 (1998).

PROSPECTS FOR IMPROVED LORENTZ VIOLATION MEASUREMENTS USING CRYOGENIC RESONATORS

J.A. NISSEN, J.A. LIPA, S. WANG, K. LUNA, D.A. STRICKER, AND D. AVALOFF Stanford University Stanford, CA 94205-4080 E-mail: [email protected] A new generation of clock experiments has pushed the search for Lorentz violation to new limits, while at the same time theoretical advancements in the StandardModel Extension (SME) have revealed possible Lorentz violating terms that were previously unrecognized. We present prospects for improved Lorentz symmetry tests both in Earth orbit and in ground based laboratories using clocks based on superconducting cavity oscillators.

1. Introduction Recently, Kosteleckq and co-workers1I2have developed a general framework that considers all possible forms of Lorentz violation that could occur within the Standard Model. This goes well beyond the special cases considered by Mansouri and S e ~ lencompassing ,~ other forms of spatial anisotropy and effects in all types of matter. The number of parameters introduced is large, well over 150, with many constraints from existing experiments. A number of important constraints in the fermion sector come from atomic clock experiments. In this framework microwave cavity experiments contribute primarily to the bounds on parameters in the photon sector, with small corrections from fermion effects in the cavity material. It is generally argued that Lorentz violations originating at the Plank scale could manifest themselves as fractional frequency variations at the level in the absence of suppression factors. In this paper we discuss some of our early work in this area and the prospects for the future. 2. Background

The oscillator we have developed tracks the TMolo (radial mode) of a niobium cavity operating under very high vacuum at a temperature of 1.2 K. 193

194

The cavities are fabricated from high purity niobium and are carefully etched and annealed to maximize their quality factor, Q. They are evacuated at room temperature, baked, and sealed off permanently to avoid contamination and excessive oxidation of the internal surface. Q values exceeding lo1' have been obtained with careful attention to processing and operation at very low temperatures. The resulting resonant line width is < 0.1 Hz, allowing exceptionally good frequency discrimination in a short measurement time. The cavity is driven by a 8.6 GHz voltage-controlled oscillator (VCO) which is locked to the cavity resonance. For Lorentz violation measurements, two such cavities are mounted in a helium cryostat and the outputs of the two VCOs are beat against each other to provide a low frequency differential signal. A low operating temperature is also needed to reduce the temperature coefficient of the resonant frequency. Experiments have shown that this effect is reduced substantially as a temperature of 1 K is approached. The main contributors to the temperature dependence of the cavity frequency are the lattice expansion and the variation with temperature of the skin depth. At 1.2 K we estimate that the temperature coefficient of the cavity resonant frequency is 2 x lO-'/K. Paramagnetic salt thermometers with resolutions below 10-l' K are used to obtain thermal control in the nanokelvin range. A four-stage thermal isolation system with this level of performance was flown successfully on the Space Shuttle in 1997 as part of the CHEX p r ~ g r a m It . ~ appears to be possible to reduce temperature effects on the fractional frequency stability to the range. The cavities used at the beginning of the program were supported at their ends, leading to a significant acceleration sensitivity. We have developed a center-mount version that significantly improves this situation. The primary adverse effect of residual accelerations is to cause a change in stored energy due to distortion of the cavity from mechanical stress. The sensitivity to variations in acceleration for a cavity suspended from one end is typically Af/f = 6 x lO-'/g. If a cavity is supported at its mid-plane the effect can be reduced substantially. Finite element structural analysis has shown that a sensitivity reduction of a factor of 100 relative to an end support is readily possible, and a factor of 1000 is achievable with tuning. So far, we have demonstrated an improvement factor of 50, yielding an implied frequency stability A f / f < for an acceleration change of lo-' g. This would be achievable in spacecraft with an orbit altitude of 1000 km, and at 500 km with a modest level of drag-free control using the helium boil-off gas. This technique has been demonstrated on GP-B.'

195

Variations in the electromagnetic energy stored in the cavity contribute to frequency shifts via variations in electromagnetic radiation pressure and changes in the non-linear surface reactance of the cavity. A typical frequency sensitivity is: A f / f = -1.7 x 10-6/Joule. Stein et a1.6 used a total stored energy of about 6 x lo-* J. Thus for a frequency stability of M a power stability of about one part in lo5 would be needed. With the use of lower noise electronics we expect to operate at two orders of magnitude lower powers, relaxing this requirement by perhaps a factor of 100. We have tested a low temperature power monitor based on germanium resistance thermometers and a sensing capability to one part in lo5 has been demonstrated. Power adjustment is performed at room temperature using an analog voltage-controlled attenuator and an integrating servo system. 3. Ground Based Measurements

In 2002 under NASA funding we compared the frequencies of two superconducting microwave cavities at irregular intervals over a 98 day period. One cavity had its symmetry axis along the local vertical while the other cavity had its symmetry axis oriented horizontally in the E-W direction. The cavities were operated at about 1.4 K in conventional helium cryostats. Microwave synthesizers were locked to the 8.6 GHz fundamental modes of the cavities using Pound frequency discrimination systems, and the beat note is mixed with an intermediate frequency oscillator to produce a signal in the 20-30 Hz range. We fitted nine data records, each of > 24 hr. duration, from May 30 to September 4 2002,’ with the function

Au _ - vo + u1Te + A s sin(wBTB)+ Bs sin(2wBTe) U

+

+ A c COS(WBTB)Bc C O S ( ~ W ~ T B ) ,

(1)

where uo and u1 were additional free parameters, TB is the laboratory time and W B is the angular velocity of the Earth’s rotation relative to the stars, i.e., the sidereal period. Six SME parameters as well as two combinations of parameters can then be determined from fitting the time variation of A s , A c , Bs and Bc as the Earth revolves around the Sun. The predicted variation2 in A s is: 1 4

1

As = - s i n 2 ~ ( k , , ) ~-’ ;,Be s i n 2 x [ s i n R ~ T ( k , ,-~ (k,,,)” )~~

+

sin 77 cos ~ e ~ ( k , , ~ cos ) ~ 77* cos R@T(~,,, IYx] 1 --PL(sin2 2 ~ ( i i , , ! ) ~-’ c o s 2 x ( ~ o t ) z y ,

-

(2)

196

where x is the colattitude of the laboratory (52.57’ for Stanford), r] is the inclination of the Earth’s rotational axis to the celestrial equitorial plane, Re is the angular frequency of the Earth’s orbit, T is time measured in the Sun-centered coordinate system, and PL = rewe sin x FZ lop6 is the speed of the laboratory due to the rotation of the Earth. For a TMolo cavity we have the relationships:

+(ie-)JK

( k e y = 3(ke+)JK ( k , l ) J K = 3(k,-)JK

+ (k,+)JK,

and from astrophysical tests8 we can set ie+ = KO- = 0. Ignoring the terms for the purposes of this paper Eq. (2) can then be reduced to:

As

=

(3)

PL

1 1 sin 2x( k e - ) y z - -,&sin 2x cos ReT 4 4

-

Similar considerations lead to:

BC =

1 -(I +sin2X)((ie-)xx - ( i e - ) y y ) - ; P ~ ( I+sin2X) 8 x [sinReT(io+)YZ- c o s r ] ~ ~ ~ R ~ T ( i , + ) ~ ~ ] . (7)

These equations are linear in sin RBT and cos RBT and can be utilized to extract the elements of the ieand i,+matrices. The results we obtained are listed in Table 1. Since these measurements were performed, other groups have obtained tighter bounds on the Re- and It,+ mat rice^.^^^^ We have also made improvements in our electronics to achieve similar performance. The next step in improving the Lorentz violation measurements would be to place the cavities on a rotating table. In order to take full advantage of the lowest noise performance of the resonators, the rotation should match the time period in which the lowest noise performance is obtained, approximately a 1000 second period. At this rotation rate FZ 100 measurements could be made per day, increasing the resolution by an or-

197

Table 1. Coefficients from best fits to the raw data. Parameter

Value

Uncertainty

(Ee_)YZ

7.21 x 10-14

2.6 x 1 0 - l ~

(Ee-)XZ

-1.24 x 1 0 - l ~

2.1 x 1 0 - l ~

(Re-)XY

-6.18 x

4.1 x

-4.66 x

(E,+)XY

2.3 x

5.59 x 1 0 - ~

2.4 x 1 0 - ~

7.05 x 1 0 - ~ (Ke- )xx - ( E . , - ) ~ ~ -2.04 x 10-13

2.6 x 1 0 - ~ 9.7 x

(Z.o+)ZY

der of magnitude from averaging. Further gains are expected from operating with the optimal noise performance. It appears that potential Lorentz violating terms could be measured to 1 part in l O I 7 in a few days, two orders of magnitude more precise than the best measurements to date. For the case of two cavities rotating together at angular frequency w,, one cavity oriented with its symmetry axis vertical on the rotating table and the second cavity oriented with its symmetry axis horizontal, a Lorentz violation signal would be expected to produce a beat frequency:

Au

- --

U

+

+ Bs, sin 2w,T@ + Ac cos weTe +Bc cos 2 w e T e + BcT cos ~ W T T @ + Ds+sin 2 (w@T@ + w,T@) +Dc+ cos 2 ( w @ T e+ w,T@) + Ds-sin 2 (w,T@ - w T T @ ) +Dc- cos 2 ( w @ T e w,T@) + Es+ sin (w@T@+ 2w,T@)

As sin w@T@ Bs sin 2weT@

-

+

+Ec+ cos ( w B T ~2W,T@)

+ Es- sin (w@T@2wTT@) -

+

+Ec- cos (w@T@ - 2wTT@) C.

(8)

Notice that the terms involving the rotation of the table are proportional to 2wT which greatly helps in eliminating systematic errors associated with the rotation mechanism which would typically be synchronous with w,. Because the rotation axis of the table is not aligned with the axis of the Sun-centered coordinate system the rotating experiment has access to the additional parameter (kel)zz not attainable from an apparatus fixed on the Earth’s surface. This can be seen from the form of the coeffi-

198

cient 3 sin2 x (&) zz Bc,. = -

16 1 +-Pe[sin2 X(sinvcosaaT (iio,>xy - ( i i o / ) y x ) 8 - cos r] cos R ~ ( T y i o / ) x z - sin R@T ( k o t ) y z )

(

+

-2 sin2 ~ ( C O Sr] cos ReT ( & _ o ’ ) ~ sin ~ ReT ( i i o ~ ) z y ) ] + W L )

(9)

To make full use of Eq. (8) it is clearly important to collect sufficient data to resolve the sideband amplitudes around the frequency 2w,.

4. Space Based Measurements

Due to cutbacks in NASA funding for basic research on the International Space Station (ISS), it no longer seems likely that clock comparison experiments will be performed there. We are now studying the possibilities presented by a dedicated free-flier mission. A basic free-flier mission would have a low inclination orbit at an intermediate altitude, in the neighborhood of 1000 km. The experiment would be capable of setting new bounds on the set of coefficients of Lorentz violation in the SME. Because of the lower drag in a high orbit and the possibility of rolling the spacecraft, this mission could set bounds more than an order of magnitude tighter than for a mission on the ISS or on the ground. The mission could be enhanced in a number of ways. By adding a high quality atomic clock and setting the eccentricity to about 0.05, the Kennedy-Thorndyke experiment and a differential red-shift test of General Relativity could be performed. The addition of a high time-resolution ground link to stations equipped with comparable atomic clocks would enable an improved absolute red-shift experiment. The lifetime of the basic mission should be the maximum possible, up to 6 months. There are two reasons for this. First, the potential violations of relativity that we seek to set bounds on are expected to be dependent on the orientation of the orbit relative to the Sun and to inertial space. A number of potential Lorentz violation signals would be modulated at the precessional period of the orbit, which is likely to be around 70 days. It would be desirable to observe at least 2 cycles of this behavior if an effect was seen. Also a 180-day mission would also allow significant noise reduction by integration of the signal if systematic effects are controlled.

199

The signals of interest are obtained with the cavity axes at right angles, with one axis at 45" to the plane of the orbit and the other parallel to the velocity vector. This configuration is designed to optimize the sensitivity to the parameters in the theory. Signals would be expected at orbital and twice orbital period. With roll, of course, additional modulation would occur. Other configurations are of course possible, but would be of reduced sensitivity. In this model, Lorentz violation terms exist which are independent of boost velocity, first order in boost and second order. The coefficients are quite cumbersome functions of the velocity and the parameters of interest, but are readily analyzed using numerical techniques. For example in the non-rolling configuration, Kostelecki and Mewes (KM) have shown that

As

1 4

= - c o ~ 2 [ [ s i n a ( i i ~, )~~o~ s

1 ( u ( i i ~ ) )-~sin2c[(1+ ~] sin20)(iie~)XX 8

+

Here [ and (u are slowly varying angles describing the geometry of the orbit. A complete set of expressions to first order in velocity was given by KM. By fitting data of the form in Eq. (10) with the detailed expressions for the coefficients, bounds can be set on up to eight parameters of interest. A ninth parameter is extractable with a sufficiently long piece of data. Using the technology described below, our goal is to reach a frequency discrimination level of zz 1x for a satellite roll period of about 1000 sec in a single measurement. Averaging over 6 months of data would then allow discrimination to the level. The potential gains over existing ground m e a s ~ r e m e n twould ~~*~ then ~ ~be as much as 750 for the zero-th order terms in p, lo4 for first order terms in p and lo5 for second order terms in p. We note that a similar mission named OPTISl' is being considered by the European Space Agency.

Acknowledgments We wish to thank the NASA Office of Life and Microgravity Sciences and Applications for its support with grants No. NAG3-1940 and NAG8-1439 and JPL for its support with contract JPL 1203716. We also thank G. J. Dick for many helpful comments.

200

References 1. D. Colladay and V.A. Kostelecki, Phys. Rev. D, 58, 116002 (1998). 2. V.A. Kostelecki and M. Mewes, Phys. Rev. D 66 , 056005 (2002). 3. R. Mansouri and R.U. Sexl, Gen. Rel. Grav. 8 , 497 (1977). 4. J.A. Lipa, D.R. Swanson, J.A. Nissen, Z.K. Geng, P.R. Williamson, D.A. Stricker, T.C. P. Chui, U.E. Israelsson and M. Larson, Phys. Rev. Lett. 84, 4894 (2000) 5. J.P. Turneaure et al., Adv. Space Res. 32, 1387, (2003) 6. S. R. Stein and J. P. Turneaure, IEEE Proceedings Letters, 1249 (Aug. 1975). 7. J. Lipa, J. A. Nissen, S. Wang, D. A. Stricker and D. Avaloff, Phys. Rev. Lett. 90, 060403 (2003). 8. V.A. Kostelecki and M. Mewes, Phys. Rev. Lett. 87, 2 51304 (2001). 9. H. Muller et al., Phys. Rev. Lett. 91,020401 (2003). 10. P. Wolf et al., Gen. Rel. Grav. 36, 2351 (2004); Phys. Rev. D, in press (hepph/0407232). 11. C. Lammerzahl, H. Dittus, A. Peters and S. Schiller , Class. Quantum Grav. 18,2499 (2001).

QUANTUM GRAVITY INDUCED GRANULARITY OF SPACETIME AND LORENTZ INVARIANCE VIOLATION

DANIEL SUDARSKY Instituto de Ciencias Nucleares Universidad Nacaonal Autdnoma de Mkxico A . Postal 70-543, Mkxico D.F. 04510, Mkxaco e-mail: [email protected] x We give short review of the way in which Lorentz invariance violation might arise in Loop Quantum Gravity as a result of a spacetime granularity associated with a preferential frame. We then discuss what would be its dominant manifestation: the effect of radiative corrections.

1. Introduction The history of the quest for a quantum theory of gravitation, one of the most daunting challenges of modern physics, has, for a long time, been overshadowed by the apparently clear theoretical expectation that no clues were to be expected to come from the empirical realm. Recently, motivated in part by the progress in different theoretical approaches to the subject, it has been suggested that quantum gravity could become manifest through slight deviations from Lorentz invariance.' On the other hand, and quite for some time, there has existed a robust program to test local Lorentz invariance and other symmetries usually considered to be exact symmetries of nature, such as CPT, mainly within the framework known as the Standard-Model Extension2 simply because they lie at the foundation of our current understanding and as such deserve to be tested to the highest precision possible. Quite remarkably the precision that has been achieved in many of these experimental programs, already calls into question the specific form and magnitude of the sought for quantum gravity effects. In fact these studies have resulted in very tight bounds on parameters that, although expected to be of order 1, can not be evaluated exactly. This drawback is a reflection of the fact that, despite their increasing sophistication, the current theoretical frameworks must still be regarded as heuristic constructs rather than detailed theoretical predictions. 201

202

There is however, one common aspect in many of these ideas: the occurrence of a preferred reference frame associated with some sort of granularity of spacetime resulting from the quantum gravity phenomena. The preferential frame has been identified with the only such object that seems to be globally singled out in our Universe: the one selected by the cosmic microwave background. The search for such breakdown of Lorentz invariance can thus be identified with the search for a dependence of the laws of physics with the state of motion of the system in question with respect to that frame, in analogy to the XIX century quest for the ether’s frame. However in contrast to what was needed to search for the ether, the current idea for experimental tests of these questions often calls for the study of particles with extremely high energies; cosmic rays, high energy gamma ray burst of cosmic origin, etc., because of the fact that the energy scale of ordinary physical phenomena is many orders of magnitude smaller than the Planck scale, the natural scale that would determine the situations in which those effects would become large. In a recent work done in collaboration with J. Collins, A. Perez, L. Urrutia and H. Vucetich3 we have shown how the consideration of radiative corrections would in this context lead to the emergence of Lorentz violating effects that are quite large. They can be thought to be the result of the fact that, although very high energy particles are in principle hard to come by, they do occur as virtual entities, contributing to all ordinary processes. In fact quantum field theory (QFT) teaches us that any process that is experimentally observed has contributions in which the intermediate, unobserved situations involve particles of all energies and momenta, including energies arbitrarily higher than those of the particles present in initial and final stages. In particular it is noted that if ordinary theories are to be regarded as effective theories valid €or particles that have energies and momenta, that in the preferred frame, are small compared with the Planck scale, one needs, for the sake of consistency, to demand a cut-off of the momenta relative to the preferential rest frame of the virtual particles appearing in any process. What we have found is that such frame specific cut-off would result in effects that can not be absorbed in the original terms of low energy theory. Moreover as could be expected these terms would be associated with a violation of Lorentz invariance, which as we shall see, would have an intensity such that its observable consequences are ruled out even by low precision and quite old experiments. In this article I will give a very brief overview of Loop Quantum Gravity4 in section 11, and in Section 111, I will describe how the ideas regarding

203

the possibility that the spacetime granularity which is intimately tied with this theory, might if it is also connected with a preferential frame, lead to observable effects in the propagation of free particles. Finally in section IV, I will discuss, how the considerations of interactions, treated within the field theory theoretical scheme, known to be an excellent description of the particles and forces that make up our universe, lead to the conclusion that the Preferred Frame Granularity of Spacetime would result in the very large effects mentioned above. References to other works have been kept to a bare minimum due to space limitations. I apologize to all colleagues whose work has not been cited and which should have. 2.

Overview of Loop Quantum Gravity

Loop Quantum Gravity (LQG) is a program that seeks to achieve a fully satisfactory and mathematically rigorous canonical quantization of General Relativity (GR), maintaining the background independence of the formulation. Recall that in GR the spacetime metric, represented by a tensor field g a b , determines the motion of free particles: the geodesics of the spacetime - such equation being the analogues of Newton’s laws of motion in the gravitational context - while at the same time the spacetime metric is itself influenced by the matter content, as described by an equation which is the analogous of the Poisson equation in Newtonian gravity: Einstein’s equation:

Here G a b is related to the curvature of the spacetime metric g a b , and Tab is the energy momentum tensor of the matter fields. One starts with the reformulation of this equation in terms of canonical variables within a Hamiltonian formalism. This is achieved by considering the foliation of spacetime by a sequence of spacelike hypersurfaces Ct which are the appropriate generalizations of the “space at a fixed time.” Now, given such a hypersurface, the spacetime metric gab determines the induced metric on such hypersurface hat, and the extrinsic curvature K a b which indicates how the hypersurface is embedded in the spacetime. Now, in considering the “time” evolution of such system we need to compare the metric at a given point on the hypersurface Ct with the metric at the “same position” but at a “time t dt.” This is achieved by fixing a lapse function N , which indicates the normal separation between the hypersurfaces and a shift vector Na indicating the tangent displacement on the hypersurfaces, which fixes the correspondence of points.

+

204

In this way we end up with a Hamiltonian formulation in which the canonical variables are: the spatial metric h a b , and its conjugate momentum nabwhich is a quantity associated with the extrinsic curvature K a b . In this context Einstein’s equations are replaced by equivalent set of equations which are divided in two classes: the constraints, and the evolution equations. The constraint equations are: 1) the Hamiltonian constraint CO(h a b , nab)= 8np, corresponding essentially to the time-time component of Einstein’s equation, where the “energy density” p stands for the appropriate component of the energy momentum tensor of the matter fields, and 2) the momentum constraints corresponding to the time-space components of Einstein’s equation: C b ( h a b , nab)= 87rPb where again P b represents the appropriate components of the energy momentum tensor of the matter fields. The rest of Einstein’s equations correspond to the canonical equations of motion for h a b and 7rab resulting from an appropriate Hamiltonian. The first step in the formalism is to rewrite the (3 1) decomposition described above, in terms of the so called Ashtekar-Barber0 variables (E,”,A:). The first one corresponds to a triad of vectors (the index a being the vector index, and the upper index i running from 1 to 3 labeling the specific vector) which play the role of “square roots” of the inverse spatial metric hab = E,”E;P, while the connection variable A: is related to the momenta nabas well as the metric h a b . In this way we obtain again a Hamiltonian description of general relativity, which is completely equivalent to the standard one. The process of quantization requires the description of the system in terms of suitable wave functions of one of the canonical variables, say, in our case the connection variable: Q[A:(rc)].Then one defines the connection operators kato act multiplicatively on these wave functionals and the triad operators E,” to be associated with functional derivatives i&. The physical states must satisfy the operator version of all the constraints. The crucial step in the LQG quantization program is the replacement of the local variables ( E f ,A:) by a very large set of integrated variables: the “holonomies” and “fluxes.” One considers the collection of all curves and associates with each such curve y, a certain line integral of A: known as “the holonomy,” stands for “path ordered” integral and similarly for all open 2surfaces one considers surface integrals of the triads E,”. The quantization then proceeds by considering wave functions Q associated with a curve (or collection of curves). These functions are, by construction, functions of the holonomies along the associated curves, and since the latter are

+

205

integrals of the connection, the wave functions are in fact, functional on the connections, as anticipated. Thus the wave function is determined by the curves on whose holonomies it depends, the representations ( or spin weights) of such holonomies that it is based on, and the way they are combined (specified by quantities known as the “intertwiners”). The resulting formulation then fixes a wave function through the specific curves and weights (and intertwiners) an object that is known as a “spin network.” Thus the wave functions are labeled by the corresponding spin network. When this is done all but the Hamiltonian constraints are satisfied automatically. Needless is to say that a lot of work goes into making all this construction mathematically rigorous. One outcome of this formulation is that all geometrical quantities such as areas, and volumes, have nonzero expectation values only if the region under consideration is intersected by the spin network that corresponds to the wave function which is describing the geometrical variables. Thus a wave function that might correspond to a classical spacetime in some appropriate limit would have to correspond to a spin network that endows every sufficiently large region with the appropriate areas, volumes, etc. Thus the spin network has to be highly convoluted and complex so as to pass sufficiently “close to all points” within very “large region.” The wave functions corresponding to such highly complex and convoluted spin networks are called weave states. 3. Effects of Preferential Frame Granularity on the

Propagation of Free Matter Fields It is now very interesting to consider how would a spacetime that is described at the basic level by such a quantum geometry be perceived by observers that use material probes such as light or ordinary matter. The starting point of such a n a l y s i ~ is ~ ?the ~ construction of the Hamiltonian operator for the matter fields evolving in the underlying geometry. This is achieved by focusing attention on the matter contribution to the Hamiltonian constraint, that as we mentioned above can be identified with the energy density of the matter fields. Such object depends on both the geometrical variables ( E ? ( z )A:($)), , and the matter field variables (cp(z),~ ( z ) ) represented here generically as configuration and momentum variables for a generic field. One then constructs the Hamiltonian as (2)

206

where the double hat reflects the fact that this operators depends both on the matter and the geometrical degrees of freedom. To take into account the specific geometry one is considering we take the expectation value of this operator in the weave state \weave > representing the geometry. In this way one obtains the effective Hamiltonian for the matter degrees of freedom propagating in the underlying geometry.

This object is then expanded in terms of the classical geometry that the state is designed to approximate, plus corrections to order l p l a n c k / X 1 where X is the wave-length of the matter field that is probing the underlying geometry. The first application of these ideas to the case of the Maxwell field6 result in an effective Hamiltonian given by &ff

=

/

+ + J1planck[E.V x E + B . V x B ] )

d 3 z ( E 2 B2

(4)

The first two terms correspond to the standard EM Hamiltonian, while the last term correspond to effect of the quantum gravitational fluctuations of the geometry. The parameter is an undetermined parameter that depends on the detailed properties of the weave state in question, and is expected a prior2 to be a number of order one. Note that this Hamiltonian violates Lorentz invariance as well as P and CPT. The dispersion relation for photons propagating in vacuum is then given by E 2 = IP12k<E3/Mplanckwhere E and P’ are the energy and momentum of the photon in the preferential frame, and the f correspond to the two different helicities of the photon. We can write this in a covariant looking way by introducing the four velocity of the preferential frame Wfi leading to PpPp = f((W’Pp)3/M~lanck. Similar analysis has been carried out for the case of f e r m i ~ n sleading ,~ to analogous results. As mentioned before these effects are suppressed by the ratio of the photon energy to the Planck mass, and therefore attention has focused on searches for extremely small violations of Lorentz invariance. Despite their very small values the precision available in some experiments and the availability of very distant sources, for which the effects would accumulate so as to become observable, has allowed the setting of very strong bounds. These bounds already call into question the scenarios contemplated in this way.

<

207

4. Effects of interactions In all previous work on the field one consider that is natural to assume that Lorentz violation at the Planck scale would only cause extremely small macroscopic effects. This seems natural and in agreement with our experience, where we find that the details of physical phenomena on one distance scale do not directly manifest themselves in physics on much larger scales. However, in quantum field theories like the Standard Model, such decoupling of short-distance from long-distance phenomena is quite non-trivial.8 The propagation of an isolated particle has contributions from Feynman diagrams containing “virtual particles,” and where all energies scales contribute entirely without suppression. When we work with the Standard Model by itself, i.e., ignoring gravity, as in almost all successful Standard Model phenomenology, there are indeed large contributions by high-energy virtual particles to low-energy processes, which however, do not give directly observable effects? the large effects of short-distance physics are equivalent to a “ren~rmali~ation” of the parameters of the Standard Model. When considering the type of situation we have been discussing, we must note that in such case, these virtual particles would explore all the possible energies and thus become affected by the spacetime granularity (as well as by other possible quantum gravity effects) and would transfer these effects to the low energy particles that are the subject of direct observations. What would be the magnitude of these effects? We addressed this issue by considering the simple case of Yukawa theory, the theory of a massive fermion field 9 interacting with a scalar field 4 via the Yukawa term. This theory has the joint advantages of simplicity and of representing one sector within the Standard Model. The Lagrangian for this theory is

L

= (1/2)ap4ay47’Ly - m;q52

+q i y a ,

M o p +go@*,

(5) where rno and MO are the bare masses of the scalar and fermion field respectively, and go is the bare coupling constant. When considering the effects of spacetime granularity on the propagation of free fields we need to take into account two facts: 1) the dispersion relations would be changed, and 2) the particles whose wave length, as measured in the preferential frame associated with the granular structure, is shorter than the granularity scale, would not exist. These effects would be represented by the following changes in the propagator of the fermions -

208

where A codifies the change in the dispersion relations and the cutoff function f eliminates the particles with transplanckian momenta. The cutoff scale A would normally be thought to be close to the Planck mass scale, and the magnitude of the tree momenta is evaluated in the preferred frame, so lpl = J(qPu - W,Wu)pppu).The cut-off function f(z)is required to go to 1 as 2 -+ 0 to recover low energy phenomenology, and to vanish as z 4 00 to cut-off transplanckian particles. There are similar changes in the bare scalar propagator. However we are now interested in the effects of virtual processes, i.e., the radiative corrections. We concentrate then on the scalar self energy II(p). This quantity is the sum of all graphs with two external scalar lines and which cannot be made disconnected by cutting a single internal line. The lowest order contribution corresponds to the virtual decay of the scalar into two fermions followed by their recombination into a scalar particle. We computed this quantity and following standard theorems which ensure that the would be divergences (in the absence of a cut-off) are confined to degree two polynomial in the momenta, we expanded the result about p = 0 to obtain

The interpretation of this result is the following: A and B are the standard mass and wave function renormalization constants, n L ’ ( p ) is a standard cut-off independent, and Lorentz invariant standard self energy term. The interesting term for us the term (ppW,)2E which corresponds to the generated term in the Lagrangian (E/2)apqGu4WpWu, and thus to a renormalization of the metric qpu -+ g g n = qp’ ZWpW”. The noteworthy fact is that E is independent of the cut-off scale A. In fact a direct calculation yields (after renormalization):

+

E = (g2/6.rr2)[1+ 2

Irn

f’(~)~zdz].

&,

We note that this quantity is positive definite, bounded from below by and dependent on the coupling constant. Thus different particles would effectively propagate on different renormalized spacetime metrics. The known widely different, couplings of different elementary fields imply that different particles have different values of c (the limiting velocity), with fractional differences of the order of typical one-loop corrections in the Standard Model, around 0.1% to 10%. This is completely incompatible with the observed limits, giving c = constant, at a fractional level well below

209

One could hope that other effects, arising also from the underlying quantum gravitational nature of spacetime, would cancel the effect we are considering, and in the absence of a precise theoretical description of such object it is impossible to prove that this might not happen, however barring a yet undiscovered principle, such cancellation would require a precise and unnatural fine tuning, i.e., a seemingly miraculous and unexplained coincidence. Furthermore, it is worth pointing out that, as there is no non-compact Lorentz invariant subset of the set of four momenta of the virtual particles, the introduction of any physical cut-off, such as one associated with spacetime granularity, would result in in a Lorentz violating boundary of the momentum space of the virtual particles, and “such boundary would be observable from any point,” i.e., the effects would permeate to the observable processes at all energies. This leads, as we have seen, to a Lorentz violating effects that are independent of the value of the cut-off which in our context would be the Planck mass or something close to it. Thus the effect is unsuppressed and would be so large as to become a blatant violation of Lorentz invariance. The cut-off independence of the effect is rather counterintuitive, and can be thought as the momentum space analog of Obler’s paradox regarding the brightness of the night’s sky: a finite static universe, no matter how large, would be different from an infinite universe now matter how sparsely populated by stars. The presence of the boundary makes a qualitative difference that could be perceived at any point no matter how distant it is from such boundary. Finally we should note that the existence of a minimum measurable length or spacetime granularity does not by itself imply the violation of local Lorentz in~ariance.~ The added element we have used is its association with a preferential reference frame.

5 . Conclusion

We have seen that the naive ideas that lead to the notion that spacetime granularity might be associated with a violation of Lorentz invariance are in conflict with our current understanding of nature in terms of quantum field theory together with the bounds that can be extracted from even very low accuracy experiments. Of course, it is possible that Lorentz invariance is violated and as with any accepted principle of science, it should be tested to the best of our abilities, however we must conclude that at this point there is no reliable

210

theoretical framework to estimate the magnitude of such effects. The assumption that these should be suppressed by powers of E/Mplanck on the basis of arguments relying on quantum gravitational ideas about the nature of spacetime are not compatible with our current understanding of physics. An optimistic point of view should be stressed: a branch of theoretical physics that has been considered for a long time to suffer from detachment from experimental guidance now finds itself in the opposite situation. Because of mechanisms intimately tied to the known ultra-violet divergences in conventional quantum field theories, certain kinds of Planck-scale phenomena, like a preferred frame, manifest themselves suppressed only by two powers of Standard Model couplings. Lorentz invariance continues to play the powerful role it has played throughout the twentieth century of imposing stringent requirements on our theories about nature.

Acknowledgments

I want to thank the organizers for the hospitality at this meeting and to acknowledge support from the project DGAPA-IN108103-3 of the UNAM. References 1. Amelino-Camelia, G., Ellis, J.R., Mavromatos, N.E., Nanopoulos, D.V. and Sarkar, S., Nature 393, 763 (1998). 2. Colladay, D. and Kosteleck?, V.A., Phys. Rev. D 58, 116002 (1998). 3. Collins, J., Perez, A., Sudarsky, D., Urrutia, L., and Vucetich, H., gr-

qc/0403053 4. For a real review see for instance: Rovelli, C., Livings Reviews, 1, 1 (1998), URL http://relativity.livingreviews.org/Articles;or Ashtekar, A. and Lewandoski, J., gr-qc/0404018. 5. Coleman, S.R. and Glashow, S.L., Phys. Rev. D 59, 116008 (1999). 6. Gambini, R. and J. Pullin, J., Phys. Rev. D 59, 124021 (1999). 7. Alfaro, J., Morales-Tbcotl, H.A. and Urrutia, L.F., Phys. Rev. Lett. 84, 2318 (2000). For a detailed and general treatment see also Sahlmann, H. and Thiemann, T., gr-qc/0207030, gr-qc/0207031. 8. Weinberg, S., The Quantum Theory of Fields, Vol. II, Cambridge Univ. Press, 1996. 9. Rovelli, C. and Speziale, S., Phys. Rev. D 67,064019 (2003); Dowker, F. and Sorkin, R., arXiv:gr-qc/0311055.

VACUUM CERENKOV RADIATION IN MAXWELL-CHERN-SIMONS ELECTRODYNAMICS

R. LEHNERT AND R. POTTING CENTRA, Departamento de Fisica, FCT Universidade do Algarue, Campus de Gambelas, 8000 Faro, Portugal We study the Cerenkov effect in the context of the Maxwell-Chern-Simons (MCS) limit of the Standard-Model Extension. We present a method to determine the exact radiation rate for a point charge.

1. The Cerenkov effect in the MCS model In recent years the so-called Standard-Model Extension (SME)’ has provided a convenient framework for studying minute Lorentz and C P T violations that may be low-energy signatures for Planck-scale physics.’ In this work we will study a subsector of the SME describing pure electrodynamics, where Maxwell theory has been modified with Chern-Simons-like term in the Lagrangian parametrized by dimensionful parameter (k A F ) p :

The Chern-Simons term explicitly violates Lorentz invariance, as well as PT and CPT invariance. (For an explicit mechanism generating it see Ref. 3.) We have explicitly included a coupling to an external current j p , which we take to satisfy a p j p = 0. As will become clear below, the inclusion of the ( k A F ) p term results in a modification of the photon dispersion relation, with the possibility of phase speeds smaller than the conventional speed of light in vacuum c. If realized in Nature, this opens up the possibility that ordinary charged matter could move with a velocity exceeding the phase velocity of radiation, and thus should emit Cerenkov radiation in vacuum. This effect is well established experimentally and theoretically in conventional macroscopic media.4 Recently, some unexpected features have been encountered in observations involving lead ions5 and in exotic condensed-matter systems.6 211

212

Some of these issues have been studied the~retically.~ In this talk, we will present recent work by the present authors in which vacuum Cerenkov radiation was investigated in detail.8 Our approach provides a new conceptual perspective on Cerenkov radiation, exploiting the fact that we have a fully relativistic Lagrangian, that allows arbitrary observer Lorentz transformations. In particular, going to the charge’s rest frame turns out to simplify the analysis. The dispersion relation that follows from (1) is given by: D(p”) = p4

+ 4p2k2

-

4(p. k)’

= 0,

(2)

where p” = (w,$ corresponds to the photon 4-momentum and k” 3 (kAF)”. Generally, this dispersion relation includes time-like as well as spacelike solutions for p”. In Fig. 1 the case of spacelike k” is depicted. It can be shown that the spacelike and timelike branches of the dispersion

Figure 1. Sample solution of the plane-wave dispersion relation. The solid lines correspond to the exact roots. The first-order solutions are shown as broken lines. The shaded region represents the interior of the @’-space lightcone.

relation correspond to deformed elliptical polarizations. At high momenta, they become left- and right circular polarizations. In order to determine the rate of emission of Cerenkov radiation, it will be necessary to determine the solution of the equations of motion in the

213

presence of a charge, that is, with nonzero four-current. The solution of the equation of motion that follows from lagrangian (1) is:

where A t ( % )is any solution to the free equations of motion (with j p = 0 ) , is the Fourier transform of the current, while the momentum space Green’s function equals

where

can be ignored as it yields a total derivative upon contraction with a conserved current, thus giving rise to a gauge artifact. The integration contour C, has to, be chosen judiciously to insure retarded boundary conditions. 2. Conditions for the emission of Cerenkov radiation We will now determine the rate of emission of Cerenkov radiation by a pointlike charge. As it turns out, the calculation is simplest in the rest frame of the charge. As the current is time-independent in that frame, we have for its Fourier transform

3” = 2 7 r S ( W ) j ” ( p 3 ,

(6)

where j”(p3 is the Fourier transform in 3-space. It follows that

with

Npu(p’)E p’2qp”- 2 i ~ ” ” ~ ~ k , p4k”k”. ,

(8)

As the source is independent of time, the resulting electromagnetic fields are expected to be stationary as well. Only spatial oscillations of the fields can occur. This time independence suggests that the radiated energy should be zero in the rest frame of the charge. Evaluating (7), it is advantageous, as usual, to extend the lp’l integral to the complex plane, and use residue calculus. It follows then directly that this integral yields a factor exp(ip’0 . q,

(9)

214

where @O satisfies the dispersion relation:

D(0,p’o)= 0.

(10)

We conclude that a nonzero imaginary part of po implies exponential decay of the fields with increasing r , while a nonzero real part corresponds an oscillatory behavior. As transport of energy-momentum to infinity can only occur in the presence of long-range fields, it follows that we can expect vacuum Cerenkov radiation only if there are real four-momenta pj’ = (0,g satisfying the plane wave dispersion relation in the +charge’s rest frame. In a general frame, where charge’s velocity is p’, the four-momentum p” = ( 0 , g is transformed into ($’.fl,fl), wherej? = @+(-y-l)(@@’)$’/l$’12. It follows that the phase velocity equals CLh =

lp” .$l/l$l 5 IPI,

(11)

so that the velocity of the particle must exceed the phase velocity of the waves. This corresponds exactly to the conventional condition for emission of Cerenkov radiation. It is useful to consider the analogue of a boat in still water. If the boat is in motion relative to the water, a v-shaped wavefront appears. For an observer on the boat, the wave pattern is stationary, while for a general observer on the shore it oscillates with decaying frequency (after the boat has passed). Figure 2 depicts a quantity related to the potential as a function of position, which clearly shows the nontrivial directional dependence of the emitted waves. Note that the MCS lagrangian implies a nontrivial dispersion relation (10). Consequently, the direction of the Cerenkov waves is frequency dependent, resulting in the absence of a sharp shock-wave. 3. Calculation of the emission rate

The usual way to determining Cerenkov rate involves integration of the r F 2 piece of Poynting vector over the boundary surface of space at infinity. However, this procedure is intractable in the present case, because determination of the asymptotic fields turns out to be difficult. An alternative approach has been developed in Ref. 8. We start with the following expression for the energy-momentum tensor

which obeys the conservation condition

6’,OPu = jpFj’”

215

Figure 2. General field pattern of a point charge resting at the origin. The function IFIIosc(F)is shown for F i n the xz plane with k along the z direction. This function was evaluated by an analytical Ip’l-type integration followed by numerical angular integrations. Uninteresting nonoscillatory pieces I,,, have been subtracted for clarity, so that only the oscillatory part I,,, I - I,,, contributes to this plot. The wave pattern is resernblant to that caused by a boat moving in water.

=

Integrating this equation over 3-volume yields

We now take the static point charge source J”(F‘) = (qS(F),o). It follows from Eq. (14) that

for the Poynting vector 810 3 Sl = -S1, so the net radiated energy is always zero in the charge’s rest frame, as anticipated. There is, however, a nonzero rate of radiation of 3-momentum:

216

which becomes 4

P=

d3r'Jp"d,.

(17)

Using the explicit (retarded) solution (7) obtained for A, one can calculate

fi by regularizing the delta-function defining the source, and performing the Fourier integral. It follows that P

4

q2 k: = -sgn(ko)--ek.

-

4lr #i$

Note that, as a consequence, P = 0 if ko = 0, that is, there is no radiation in the rest frame unless ko is nonzero. Transforming to general frame in which the charge has an arbitrary velocity generally yields non-zero components for all components of P, that depend on both p' and $. Figure 3 indicates the polarization of the radiation as a function of the direction of the wave vector p' in relation to p' and $.

Figure 3. Dependence of the polarization on direction. For vectors p' pointing in the clear (shaded) direction, the associated waves are right (left) polarized. The radiation exhibits linear polarization only when p'lies on one of the dashed lines. Vacuum cerenkov radiation may not be emitted into all directions. The wave 4-vector pfi = (p.p',p') is further constrained by the dispersion relation.

A natural question that presents itself is whether vacuum Cerenkov

217

radiation might be observed. As it turns out, in the laboratory frame the components of k” are observationally constrained by O(k”)5 GeV.’ The smallness of this bound implies that deviations of the photon phase speed from c are expected to be extremely small. Taking this bound to be saturated, it can be shown8 that a proton at the end of the observed cosmicray spectrum (1020eV) will only emit radiation of wavelengths larger than 1.2 x lo5 m. Conceivably, such radiation might be observable in high-energy astrophysical jets emitted in the direction of sight. 4. Back reaction on the charge Denoting the charge’s 4-momentum by Q”, momentum conservation yields

Q”

=

-P”(p’,.

(19)

It is possible to continue to use consistently the usual definition Q ’ = mu” so that one obtains the differential equtaion -Pp@)

= mu”@),

(20)

where P p ( p ’ , has been determined in the previous section (transforming formula (18) to the appropriate frame). For the important case of spacelike kpl this equation can be integrated explicitly in the laboratory frame in which ko = 0 , yielding the charge’s velocity as a function of time.’ One can show that: the component ,Blnormal to iis always constant in time; the charge is always slowed down by Cerenkov radiation; the characteristic time scale governing the time dependence is given by = 4rrn/q2i2d=; the trajectory is generally curved, with a characteristic scale size TPI.

One might speculate whether the slow-down effect of high-energy charges might lead to an effective cut-off in the cosmic-ray spectrum for primary particles carrying an electric charge. This idea has been raised in the literature to place bounds on Lorentz breaking. In the present model, however, the energy-loss rate is suppressed by two powers of the (experimentally tightly bounded) Lorentz-violating coefficient k p .

5. Phase space estimate While a full quantum field theory extension of the classical results obtained above is beyond the scope of the current work, we will here present a phase

218

space estimate of the radiation rate and show it is consistent with Eq. (18). We start with the decay rate of a particle a into two particles b and c in quantum field theory:

dr

=

' M a ~ b ~ c ' 2 ( 2 7 1 ) 4 6 ( 4- p) (f p ~-pf)d&d&. 2Ea

(21)

Here drIi (i = a , b, c) denote the phase-space elements. We take p: = pg = m2, corresponding to a mass m particle with a conventional Lorentz invariant dispersion relation, while c denotes photons with the MCS dispersion relation. It is possible to show that for light-like Ic"

is observer-invariant for the spacelike branches of the photon dispersion relation. For the amplitude we can take Ma-b,c

= 4EaM

(23)

as the generic form of the amplitude, with A4 a dimensionless function of external momenta and the Lorentz-violating parameters. An order-of-magnitude estimate for expression (21) can be worked out in the m + 00 limit, yielding for the decay rate:

-

Here IMI2 denotes a suitable angular average of [MI2. This result is in correspondence with classical result (18). 6. Conclusions

We considered the possiblity of Cerenkov radiation in the Maxwell-ChernSimons model, a particular limit of the SME. We showed how the Lorentzviolating modification of the plane-wave dispersion relation leads to the emission of radiation by moving charges. Our novel approach exploited the fact that observer Lorentz invariance always allows one to transform to the rest frame of the charge, where the calculations are less complicated. We investigated various properties of this radiation, and obtained the exact (classical) rate of emission of radiation by a point charge. The possibility of detection of vacuum Cerenkov radiation in astrophysical context

219

was considered, with the conclusion that the tight observational bounds on the ( k ~ F ) parameter p render any possible effect highly suppressed. We note that it would be interesting to consider the dimensionless k F term in the SME: some of its components are currently only bounded at the lo-' level," and a dynamical study paralleling the present one could yield less suppressed rates. We expect our methodology to have applicability in more general cases including macroscopic media.

References 1. D. Colladay and V.A. Kosteleck?, Phys. Rev. D 55,6760 (1997); Phys. Rev. D 58, 116002 (1998); V.A.Kostelecki and R. Lehnert, Phys. Rev. D 63, 065008 (2001); V.A. Kosteleck?, Phys. Rev. D 69,105009 (2004). 2. See, e.g., CPT and Lorentz Symmetry II, edited by V.A. Kosteleckf (World Scientific, Singapore, 2002). 3. V.A. Kosteleck?, R. Lehnert, and M.J. Perry, Phys. Rev. D 68, 123511 (2003); 0.Bertolami, R. Lehnert, R. Potting, and A. Ribeiro, Phys. Rev. D 69,083513 (2004). 4. 0. Heaviside, Phil. Mag. 27, 324 (1889); P.A. Cerenkov, Dokl. Akad. Nauk SSSR 2,451 (1934); S.I. Vavilov, Dokl. Akad. Nauk SSSR 2,457 (1934); I.E. Tamm and I.M. Frank, Dokl. Akad. Nauk SSSR 14, 107 (1937); E.Fermi, Phys. Rev. 57, 485 (1940); V.L. Ginzburg, J. Phys. USSR 2, 441 (1940); J.M. Jauch and K.M. Watson, Phys. Rev. 74, 950 (1948); Phys. Rev. 74, 1485 (1948); Phys. Rev. 75, 1249 (1949). 5. V.P. Zrelov, J. RuiiEka, and A.A. Tyapkin, JINR Rapid Commun. 1-87,23 (1998). 6. T.E. Stevens et al., Science 291,627 (2001); C. Luo et al., Science 299,368 (2003). 7. See, e.g., G.N. Afanasiev, V.G. Kartavenko, and E.N. Magar, Physica B 269, 95 (1999); I. Carusotto et al., Phys. Rev. Lett. 87,064801 (2001). 8. R. Lehnert and R. Potting, Phys. Rev. Lett. 93,110402 (2004); R. Lehnert and R. Potting, hep-ph/0408285. 9. H.B. Belich, M.M. Ferreira, J.A. Helayel-Neto, and M.T.D. Orlando, Phys. Rev. D 68, 025005 (2003). H.J. Belich, J.L. Boldo, L.P. Colatto, J.A. HelayelNeto, and A.L.M.A. Nogueira, Phys. Rev. D 68,065030 (2003). 10. J. Lipa et al., Phys. Rev. Lett. 90,060403 (2003); H. Miiller et al., Phys. Rev. Lett. 91,020401 (2003); P. Wolf et al., Gen. Rel. Grav. 36,2352 (2004).

WEIGHING THE ANTIPROTON USING ANTIPROTONIC HELIUM ATOMS AND IONS

R.S. HAYANO Department of Physics, University of Tokyo 7-3-1 Hongo, Bunkyo-ku Tokyo 119-0033, Japan E-mail: [email protected]

We discuss how the laser spectroscopy of antiprotonic helium atom, a neutral three-body Coulomb system consisting of an antiproton, a helium nucleus and an electron, is used to ‘weigh’ the antiproton. The best limit reached so far is lo-* in terms of Imp - mpl /mp (best baryonic C P T test), but this relied heavily on the results of (difficult) 3-body QED calculations, which have errors similar to those of the measured laser transition frequencies. Recently, we have succeeded to produce long-lived (T 2 100 ns) antiprotonic helium ions (two-body system: PHe++). Implications of this new discovery and future prospects are discussed.

1. Introduction Testing C P T invariance to highest-possible precision using low energy antiprotons is the common goal of the three ongoing experiments (ATHENA, ATRAP and ASACUSA) at the antiproton decelerator (AD) of CERN. The first two collaborations concentrate on antihydrogen production and spectroscopy, while our collaboration, ASACUSA, has so far concentrated on the study of antiprotonic helium atoms. Antihydrogens can now be routinely produced’>’ and the next milestone is to carry out spectroscopy, but there are still high hurdles to be cleared (e.g., produce enough number of cold antihydrogen in the ground state, trap them, and let them interact with stabilized laser beam(s) long enough, etc.), hence it is likely to take some time before precision laser-spectroscopic techniques can be applied t o a n t i h y d r ~ g e nMeanwhile, .~ high-precision laser spectroscopy of antiprotonic helium has shown a steady progress over the last ten year^,^^^^^^' and now provides the best baryonic C P T limits of 10-8.8 Although Imp - mpl/ m p may not be a suitable measure of C P T test in the Standard-Model Extension (SME) f r a r n e ~ o r kthis , ~ is how the Par220

221

ticle Data Group quotes the present baryonic CPT-test limit;' the best limit of lo-' now comes from the laser spectroscopy of antiprotonic helium (metastable three-body atom consisting of an antiproton, an electron and a helium nucleus, hereafter denoted pHe+).7

-3.2 W

9

-3.4

-3.6

-3.8

Nuclear Absorption

I

= 30

31

32

33

34

35

36

37

38

39

Figure 1. Level diagram of pHe+ in relation to that of pHe++. The solid and wavy

bars stand for metastable and short-lived states, respectively, and the dotted lines are for l-degenerate ionized states.

In the PHe+ laser spectroscopy (see Fig. l), laser in the visible-light region is used to induce resonant transitions between different antiprotonic orbits (not electronic states as in ordinary atomic spectroscopy). Here, the principal quantum number is typically n 39 and orbital quantum number is C 35, and the transition is either (n,C) -+ ( n - 1,C - 1) (downward or favored E l transitions) or (n,!) * ( n lC - 2) (upward or unfawored El transitions). As will be explained later, successful resonant transition can be detected by a sharp increase in the antiproton annihilation rate, occurring in response to to the applied laser pulse. The transition energy is proportional to m&Z'&e2Qg, where mg and Q p are the reduced mass and charge of p , respectively. The effective (shielded) nuclear charge Zeff (simply 2 in the case of pHe++) is (n,l ) dependent, and must be calculated for each state by using the state-of-the-art three-body QED t heories.l09 N

N

+

222

In the theoretical calculations, the proton mass measured in atomic units (current precision 4.6 x 10-lo)l2 and the helium-to-proton mass ratio (current precision 1.3 x for 4He and 1.9 x lo-' for 3He)12are used. The fact that the measured and calculated frequencies agree within error bars demonstrated the reliability of the 3-body QED calculations, as well as the equality of proton and antiproton masses. Our experimental results (K me&;) were then combined with the (more precise) antiproton chargeto-mass ratio (Qp/mp) measured to 9 x by the TRAP group at CERN LEAR13 to yield the present limit of quoted by the Particle Data Group. This is what is meant by 'weighing the antiproton.' 2. Antiprotonic helium atoms The antiprotonic helium atom @Hef) is a naturally-occurring antiproton trap which has the following remarkable features:14

(1) The atom can 'store' an antiproton for more than a microsecond. This longevity occurs when the antiproton occupies a near-circular orbit having a large n (- 38) and also large l ( 2 35). (2) Unlike antihydrogen, it is not at all difficult to make pHe+. Just stop antiprotons in a helium target. Then, about 3% of them automatically become trapped in the metastable states. (3) We usually use low-temperature (7' 10 K) helium gas as the target. The produced PHe+ atoms collide with the surrounding helium atoms and are thermalized. Therefore, the antiprotonic helium atoms are already cold, making them suitable for high-precision spectroscopy. (4)Very conveniently, the An = 1 energy difference around the region of metastability of PHef is in the visible-light region. This makes it possible to perform high-precision laser spectroscopy.

-

3. Laser spectroscopy

Let us use Fig. 1 to explain how the laser spectroscopy of PHe' works. In Fig. 1, the levels indicated by the solid lines have metastable ( 2 1ps) lifetimes and deexcite radiatively, while the levels shown in wavy lines are short lived ( ,< 10 ns) and deexcite by Auger transitions to antiprotonic helium ion states (shown in dotted lines). Since the ionic states are hydrogenic, Stark collisions quickly induce antiproton annihilation on the helium nucleus, as indicated in the figure.

223

Note that there is a boundary between metastable states and short-lived states. For example, (n, = (39,35) is metastable, while (n, l ) = (38,341, which can be reached from (39,35) by a favored El transition, is short lived. Thus, if we use a laser (A = 597 nm in this particular case) to induce a transition from (39,35) to (38,34), and if an antiproton happens to be occupying the (39,35) level at the time of laser ignition, the antiproton is deexcited to the short-lived state, which then Auger-decay to an ionic (ni,&)= (32,31) state within ;S 10 ns. The ionic state is then quickly (usually within ps) destroyed by Stark collisions. Hence, when a pulsed laser is used to induce the transition, a sharp increase in the p annihilation rate occurs in coincidence with the laser pulse. We measure the intensity of the laser-induced annihilation spike as a function of laser detuning, and compare the resonance-peak centroid vex, with the results of three-body QED calculations Vth (calculated assuming mp = mp and Q p = - Q p ) . So far, no statistically-significant deviation has been found. Our first experiment carried out in 19934 had a precision of about 50 ppm (5 x lop5). Theoretical predictions on the other hand scattered within about 1000 ppm, but were soon greatly improved to some 50 ppm (Korobov's non-relativistic calculation) and then to some 0.5 ppm (including relativistic corrections) by 1996. The precisions of theoretical calculations continued to improve, and they have now reached ,< 10-s.lO~ll In competition, experimental error bars also continued to decrease. After the first success, we soon found out that although pHe+ atoms are fairly stable against frequent collisions with helium atoms, the collisions induce frequency shift and broadening of the resonance lines.5 Soon after CERN AD was commissioned, by measuring the resonance centroids at different helium densities and by extrapolating to zero density, we reached a precision of 60 ppb in 2001.6 In our most recent measurements, we used a radio-frequency quadrupole decelerator (RFQD)15with which we decelerated the 5.3 MeV antiprotons extracted from AD to some 50 keV. A schematic drawing of the experimental setup (not to scale) is shown in Fig. 2. This made it possible to stop antiprotons in a very low density gas target ( N 1016-18 atoms/cm3), eliminating the need for the zero-density extrapolation. Fig. 3 shows the present status of the experiment-theory comparison for seven transitions in p4He+ (left) and six transitions in p3Hef (right). Experimental errors include the absolute-frequency calibration uncertainties, and are f l O O ppb. The theoretical predictions of Korobov (squares)lo and Kin0 (triang1es)ll are

e)

N

N

224 RF buncher

Solenoid magnets

Quadrupole magnets Cryogenic helium target

Figure 2. The 5.3 MeV antiprotons ejected from AD was decelerated using the RFQD to 50 keV and were stopped in a very low density target. Laser spectroscopy of pHe+ performed in the low-density environment made it possible to achieve higher precision, and also lead t o the discovery of metastable antiprotonic helium ions. N

mostly in the experimental error bars, but there are sometimes discrepancies of 100 ppb between the two. These differences do not yet affect the final CPT limits, but as the experimental precisions are improved, they may eventually become the dominant error source in the CPT limits deduced from the pHe+ spectroscopy. N

Figure 3. Commparisons between experimental results vexp (filled circles with errors)7 and theoretical predictions vth obtained assuming mp = mp and Qp = -Qp (squares10 and triangles.11)

225

4. Discovery of metastable antiprotonic helium ions

Let us go back again to Fig. 1 and consider the fate of pHe++ ions at very low target densities. The destruction of the j!jHe++ states usually take place in a matter of pic0 seconds. This is due to the Stark collisions. If the pHe++ ion is isolated in a vacuum, there are no collisions, and hence the pHe++ states should become metastable (the radiative lifetimes of the circular states around ni 30 is several hundred ns). We therefore expect that the prolongation of pHe++ lifetimes to occur at very low target densities. This is exactly what we recently observed. In the left panel of Fig. 4, we show the annihilation spike produced by inducing the P4He+ transition (n,a) = (39,35) -+ (38,34) measured by using the RFQD-decelerated beam at a low target density of 2 x 10l8 atom/cm3. At this density, the decay time constant of the laser spike is still consistent with the Auger lifetime of the (38,34) level. However, as shown in the right panel, the shape of the laser-induced spike changes drastically in an ultra-low target density of 3 x 10l6 atoms/crn3,l6 a clear indication of the metastability of pHe++. N

2.1

2.2

2.3

2.4

2.5

2.6

2.7 2.8

2.9

3

Elapsed time (ks)

Figure 4. Annihilation spike produced by inducing the FHe+ transition ( n , ! ) = (39,35) + (38,34), measured at a high target density (a). A prolongation of the tail is observed at ultra-low densities (b), indicating the formation of long-lived pHe++ ions.

5. Future prospects

All CPT-test experiments carried out until now by the ASACUSA collaboration dealt with the pHe+ atom, relying heavily on the results of 3-body QED calculations, which have errors similar to those of the measured laser transition frequencies. The long-lived antiprotonic helium ions $He++ and p3He++ are quite

226

interesting in this respect, since these are two-body systems and hence are practically free from theoretical errors. This motivates us to perform the laser spectroscopy of pHe++. In principle, this appears possible, since we found that there is up to 50% lifetime difference between the pHe++ levels when the principle quantum number of the ion ni is changed by one unit.16 Hence, if we use a laser to produce an ionic state, and then use another laser (in the UV region) to induce transitions between ni and nifl, we should be able to observe a slight change in the decay time constant of the laser spike tail (such as in Fig. 4(b)). This is by no means an easy measurement, but is nevertheless an important one. At the same time, we will continue to improve the precision of pHe+ (three-body) spectroscopy, by using a pulse-amplified CW laser system, in which the CW laser is stabilized and locked to an optical frequency comb. This way, we should be able to reach sub-ppb precision soon, thereby improving the CPT limits on proton-antiproton mass and charge comparison (the present goal is to reach the precision of proton mass measured in atomic units,12 4.6 x In order to achieve this, we need continuing efforts of the theory community, so that the existing differences of some 100 ppb in the transition-frequency calculations are diminished. Of course, our future goal includes high-precision antihydrogen spectroscopy. Our collaboration intends to do this differently from the other two (ATHENA and ATRAP) collaborations. Namely, we plan to measure the ground-state hyperfine splitting of antihydrogen using a beam geometry, without anti-atom trapping.'? Such a measurement enable us to directly probe the b parameter in SME,' while the achievable precision is likely to be modest (- lo-' in terms of ~ Y / Y ) . N

Acknowledgments This work has been carried out working closely with the members of ASACUSA collaboration, in particular, T. Yamazaki, J. Eades, E. Widmann, W. Pirkl, D. Horvath, H.A. Torii and M. Hori. Thanks are also due to V. I. Korobov and Y. Kin0 for theoretical inputs. This work was supported by the Grant-in-Aid for Specially Promoted Research (15002005) of MEXT Japan, AOARD, the Hungarian Scientific Research Fund (OTKA TO33079 and TeT-Jap-4/98), and the Japan Society for the Promotion of Science.

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References M. Amoretti et al., Nature 419,456 (2002). G. Gabrielse et al., Phys. Rev. Lett. 89,213401 (2002). A. Kellerbauer (for ATHENA), these proceedings. N. Morita e t al., Phys. Rev. Lett. 72, 1180 (1994). H.A. Torii et al., Phys. Rev. A 59,223 (1999). M. Hori et al., Phys. Rev. Lett. 87 093401 (2001). M. Hori et al., Phys. Rev. Lett. 91 123401 (2003). S. Eidelman et al., Phys. Lett. B 592,1 (2004). For example, D. Colladay, V.A. Kosteleckf, Phys. Rev. D 55, 6760 (1997); R. Bluhm, V.A. Kostelecki and N. Russell, Phys. Rev. Lett. 79,1432 (1997); Phys. Rev. D 57,3932 (1998); Phys. Rev. Lett. 82,2254 (1999). 10. V.I. Korobov, Phys. Rev. A 67,026501 (2003). 11. Y. Kin0 et al., Nucl. Instr. Methods B 214,84 (2004). 12. P. J. Mohr and B. N. Taylor, “The 2002 CODATA Recommended Values of the Fundamental Physical Constants, Web Version 4.0,” available at physics.nist .gov/constants. 13. G. Gabrielse et al., Phys. Rev. Lett. 82,3198 (1999). 14. T. Yamazaki et al., Phys. Rep. 366, 183 (2002). 15. A. M. Lombardi et al., in Proceedings of the 2001 Particle Accelerator Conference, Chicago, 2001 (IEEE, Piscataway, NJ, 200l), pp. 585-587. 16. M. Hori et al., submitted to Phys. Rev. Lett. 17. R.S. Hayano et al., Letter of intent submitted to CERN SPSC (CERN SPSC1-226). 1. 2. 3. 4. 5. 6. 7. 8. 9.

ELECTROPHOBIC LORENTZ INVARIANCE VIOLATION FOR NEUTRINOS AND THE SEE-SAW MECHANISM

S.F. KING Department of Physics and Astronomy, University of Southampton Highfield, Southampton SO1 7 1 BJ, U.K. E-mail: sj7cQhep.phys.soton. ac. uk

In this talk we show how Lorentz invariance violation (LIV) can occur for Majorana neutrinos, without inducing LIV in the charged leptons via radiative corrections. Such “electrophobic” LIV is due to the Majorana nature of the LIV operator together with electric charge conservation. Being free from the strong constraints coming from the charged lepton sector, electrophobic LIV can in principle be as large as current neutrino experiments permit. On the other hand electrophobic LIV could be naturally small if it originates from LIV in some singlet “right-handed neutrino” sector, and is felt in the physical left-handed neutrinos via a see-saw mechanism.

1. Introduction

In this talk we discuss a LIV scenario discussed in Ref. 1 with two desirable features: (i) natural explanation of smallness of LIV; (ii) protection of LIV in the neutrino sector from the bounds coming from the charged lepton sector. We satisfy (i) by supposing that such effects originate in the “righthanded neutrino” singlet sector, and are only fed down to the left-handed neutrino sector via the see-saw mechanism, thereby giving naturally small LIV in the left-handed neutrino sector. We satisfy (ii) by proposing a LIV operator which violates lepton number by two units - forbidden by electric charge conservation for charged fermions: “electrophobic LIV.” The motivation for LIV in the right-handed neutrino sector is as follows. 0

It is theoretically attractive since “right-handed neutrinos” could represent any singlet sector, and need not be associated with ordinary,quarks and leptons, except via their Yukawa couplings to 228

229

left-handed neutrinos. The fact that LIV is associated only with such a singlet sector could provide a natural explanation for why LIV appears to be a good symmetry for charged fermions, while being potentially badly broken in the neutrino sector.

0

2. CPT violation in the right-handed neutrino sector Suppose that CPT violation (CPTV) originates solely from the righthanded sector due to the operator:

Ng B'$prW{. < H>

Figure 1.

< H>

< H>

(1) < H>

See-saw mechanism with CPT violation in the right-handed neutrino sector.

The see-saw mechanism depicted in Fig. 1 leads to a naturally suppressed CPT violating operator in the left-handed neutrino sector:2

Mocioiu and Pospelov2 noted the following problem, namely that CPT violation is generated in the charged lepton sector via one-loop radiative corrections as shown in Fig. 2. The operator which is generated from Fig. 2 is given by:

LEbioop$prpL!, LL = (VL e ~ ) ~ . The CPT violating coefficient from Eq. (3) is given by: belectron

N

bfoop

N

10-2bp.

(3) (4)

The electron CPTV limit in this coefficient is given by belectron < GeV which implies that b < GeV. Is such a small amount of CPTV observable in the neutrino sector? To answer this question, consider the constraints arising from the CPTV operator V -a LIJ ~ ~ ~ Y ~ V L ~ .

(5)

230

L

N

Figure 2. One-loop contribution of CPT violation in the right-handed neutrino sector to CPT violation in the charged lepton sector.

It is conventional to consider the time component only of this operator: ~

bB . ~

~

p

~

o

~

~

(6)

The resulting two neutrino flavour equation of motion in the presence of CPTV is: - cos 28 sin 20 sin28 cos20

-

cos 286 sin 286

where

This results in the oscillation probability that an electron neutrino remains an electron neutrino given by:

Pee = 1 -

+

C2O20 2

sin2 (~+02 L) ,

(9)

where

c = Acos20 + Bcos28b,

D = Asin28

-

+ Bsin28b.

(10)

Neutrino oscillations are sensitive to b 10-” GeV. We therefore conclude that the electron CPTV limit belectron < GeV above renders any CPT violation in the neutrino sector unobservable.

3. Electrophobic LIV in the Right-Handed Neutrino Sector In order to overcome this problem we suggested the following LIV operator in the right-handed neutrino sector:’

H’$((NRc)a~py(iVR)~, AL = 2.

(11)

The see-saw mechanism depicted in Fig. 3 then leads to naturally sup-

231



Figure 3.

< H >

See-saw contribution of LIV operator in the right-handed neutrino sector.

pressed LIV in the left-handed neutrino sectox+

Note that both operators in Eqs. (ll),(12) are Majorana operators. They can never lead to LIV in the charged lepton sector to all orders of perturbation theory due to electric charge conservation! Expanding the electrophobic LIV operator in Eq. (12) gives:

where H* = (h23+hol)fi(h13+ho2).Eq. (13) shows that electrophobic LIV allows v, -+ Fa, whereas the CPT considered previously forbids v, 4 pa. We now consider constraints on the coefficient which controls electrophobic LIV:

-

aThis operator is reminiscent of the magnetic moment operator pap(u~),mp,(v~)pFpV. The main physical difference is that our operator is independent of any physical magnetic fields, and can in principle be arbitarily large.

232

The two neutrino equation of motion is: -Acos28 0 Asin28 B 0 -Acos28 -B Asin28) Asin28 -B Acos28 0 B Asin28 0 Acos28

($&) VaL

, (15)

where

A=-

Am2

4E '

B = Hap.

This leads to the two-flavour oscillation probabilities: = A2 A2sin2 + B2 28 sin2

p - = aP

(Jm L) ,

B2

A 2 + B 2 sin2 (JA2t_BzL )

(18)

Pacy=Pfifi=l-P 4 - P a p- ,

(19)

Pa@= 0.

(20)

We now summarise the experimental constraints on electrophobic LIV from different experiments. Constraints from CHOOZ/Palo Verde: CHOOZ and Palo Verde short baseline reactor experiments are consistent with no observed oscillation of ~7, at baseline L 1 km. This nonobservation of any oscillations can be used to constrain Hep ;S lo-'' GeV. [Hep (= Hep due to CPT invariance) is the LIV coefficient responsible for v,(Y,) + vp(v0) transition.] Constraints from the KamLAND experiment: KamLAND observes the electron antineutrinos produced in nuclear reactors from all over Japan and Korea. KamLAND results show a deficit of the antineutrino flux and are consistent with oscillations with Am2 and mixing given by LMA solar solution. KamLAND being a disappearance experiment is insensitive to whether the te oscillate into vp due to mass and mixing or pp due to LIV. However LIV driven oscillations are inconsistent with the KamLAND energy distortion data leading to He@< 7.2 x GeV. Constraints from the atmospheric neutrino data: The atmospheric neutrino experiments observe a deficit of the v p and tptype neutrinos, while the observed v, and t, are almost consistent with the atmospheric flux predictions. N

233

The LIV term would convert vp(Dp) into DT(vT),while flavor oscillations convert vp(Dp) to v,(D,). Since the experiments are insensitive to either v, or D,, they will be unable to distinguish between the two cases. LIV case is independent of the neutrino energy (same predicted suppression for the sub-GeV, multi-GeV, and the upward muon data). Therefore pure LIV term fails to explain the data but can exist as subdominant effect along with mass driven flavor oscillations, leading to the limit: HpT6 lo-’’ GeV. Constraints from the future long baseline experiments: Better constraints on LIV coefficient require experiments with longer baselines. MINOS and CERN to Gran Sasso (CNGS) experiments, ICARUS and OPERA, have a baseline of about 732 km, though the energy of the up beam in MINOS will be different from the energy of the CERN v p beam. However, since the LIV driven probability is independent of the neutrino energy, all these experiment would be expected to constrain Hpp 6 lop2’ GeV. JPARC has shorter baseline of about 300 km only, while the NuMI off-axis experiment is expected to have a baseline not very different from that in MINOS and CNGS experiments. The best constraints in terrestrial experiments would come from the proposed neutrino factory experiments, using very high intensity neutrino beams propagating over very large distances. Severe constraints, up to Hpp 6 GeV could be imposed for baselines of 10,000 km. Constraints from solar neutrinos: Neutrinos coming from the Sun travel over very long baselines 1.5 x 108 km. So one could put stringent constraints on Hep from the solar neutrino data. However the situation for solar neutrinos is complicated due to the presence of large matter effects in the Sun. Constraints from supernova neutrinos: Supernovae are one of the largest sources of astrophysical neutrinos, releasing about 3 x ergs of energy in neutrinos. The neutrinos observed from SN1987A, in the Large Magellanic Cloud, had traveled 50 kpc to reach the Earth. Neutrinos from a supernova in our own galactic center would travel distances 10 kpc. These would produce large number of events in the terrestrial detectors like the Super-Kamiokande. The observed flux and the energy distribution of the signal can then be used to constrain the LIV coefficient.

-

N

-

-

234

Constraints using the time of flight delay technique: The violation of Lorentz invariance could also change the speed of the neutrinos and hence cause delay in their time of flight. The idea is to find the dispersion relation for the neutrinos in the presence of LIV and extract their velocity v = d E / d p , where E is the energy and p the momentum of the neutrino beam. Then by comparing the time of flight of the LIV neutrinos, with particles conserving Lorentz invariance, one could in principle constrain the LIV coefficient. The presence of the LIV term in the Lagrangian gives a see-saw suppressed correction to the mass term. Therefore

vxl-

m2

+ miIv

E2 ’ where m is the usual mass of the neutrino concerned and m i I v is the LIV correction. 4. Conclusion

0

LIV may be introduced into a “right-handed neutrino” sector at some high scale, resulting in suppressed LIV in the left-handed neutrino sector via the see-saw mechanism. The AL = 2 lepton number violating operators induce LIV into the left-handed Majorana neutrino sector, while protecting LIV in the charged lepton sector to all orders of perturbation theory due to electric charge conservation.

References 1. S. Choubey and S. F. King, Phys. Lett. B 586 (2004) 353 [arXiv:hepph/0311326]. 2. I. Mocioiu and M. Pospelov, Phys. Lett. B 534 (2002) 114 [arXiv:hepph/0202160].

THE BOUNDS ON LORENTZ AND CPT VIOLATING PARAMETERS IN THE HIGGS SECTOR

ISMAIL TURAN Physics Department, Concordia University 7141 Sherbrooke West, Loyola Campus Montreal, Qc., H4B 1R6 C A N A D A E-mail: [email protected] In this talk, I discuss possible bounds on the Lorentz and CPT violating parameters in the Higgs sector of the so called minimal Standard-Model Extension (SME). The main motivation for this study is coming from the fact that unlike the parameters in the fermion and gauge sector, there are no published bounds on the parameters in the Higgs sector. From the one-loop contributions to the photon propagator the bounds on the CPT-even asymmetric coefficients are obtained and the cPv coefficients in the fermion sector determine the bound on the CPT-even symmetric coefficients. The CPT-odd coefficient is bounded from the non-zero vacuum expectation value of the Z-boson.

1. Introduction

Lorentz and CPT symmetries are assumed to be exact in nature within the framework of the Standard Model and this fact is in very good agreement to high precision with present-day experimental findings. However, it is widely believed that the Standard Model is nothing but a low energy version of some more complete (fundamental) theory, presumably valid at the Planck scale of 10’’ GeV, such as noncommutative field theory’ or string theory.2 It is then reasonable to search for some induced “new physics’’ effects at levels attainable by high precision experiments. The violation of Lorentz and CPT symmetries can be considered one of such effects. There is an explicit example from string theory in which non locality of the string leads to modification of the Lorentz properties of the vacuum. Among mechanisms to describe Lorentz and CPT violation, the most elegant way is to consider these symmetries exact at the scale of the fundamental theory and spontaneously broken at low energies due to the existence of nonvanishing expectation value of some background tensor fields. The 4-dimensional effective interactions between the background tensor 235

236

fields T and matter can be written as3

where all possible Lorentz indices are suppressed. For k = 0,1, the first two factors of the right hand side of Eq. (1)represent most of the CPT-violating terms in the fermion sector. At this point it is better to explain the difference between the observer Lorentz invariance and the particle Lorentz invariance, which are essential for understanding the minimal SME that I will describe briefly in the next section. The former involves transformations under rotations and boosts of coordinate system but the latter involves boosts on particle or localized fields but not on the background fields. Therefore, while, in the right hand side of Eq. (l),(T) and qria.1C, are both changing under observer Lorentz transformation such that their contraction stays invariant, particle Lorentz transformation leaves (2”) term unaffected which leads to a (particle) Lorentz violating effect when it is contracted with the matter term. The following example from conventional electrodynamics3 can be given to give further clarification. Let us consider a charged particle entering a region perpendicular to a uniform background magnetic field. Its path is circular. Suppose without changing the observer frame, one gives an instantaneous particle boost to the charged particle without affecting its direction. Then it will still keep moving on a circular path but with a bigger or smaller radius depending on the direction of the given boost. This boost leaves the background magnetic field unaffected (here, the background magnetic field is analogous to the field T ) . Let us now consider another observer frame which is obtained from our original frame by making a Lorentz transformation of coordinates. In that frame, the particle no longer makes a circular motion but a spiral motion (drift motion) due to the existence of induced electric field in addition to the magnetic field. The background field is obviously not a pure magnetic field at all. The important point is that the background field is changing to preserve the observer invariance, i.e., FpyFp” term is invariant. This means that any Lorentz indices in each term of Eq. (1) must be contracted. The outline of the talk, which is based on work done with David L. Anderson and Marc Sher4, is as follows. In Sec. 2, I will very briefly describe the minimal Standard-Model Extension by especially emphasing its fermion, photon and Higgs sector. The purpose of our study is to explore the bounds on the parameters appearing in the Higgs sector of the minimal SME. So, in Sec. 3, I consider the bounds on the CPT-even antisymmetric

237

and symmetric coefficients of the Higgs sector. A careful analysis of the COordinate and field redefinition issue will be done. The bounds on CPT-odd coefficient in the same sector are discussed in Sec. 4. 2. The minimal Standard-Model Extension

A framework for studying Lorentz and CPT violation has been constructed by Colladay and Ko~teleckjr,~ known as the minimal SME. It is a model based on the Standard Model but which relaxes the Lorentz and CPT invariance. The additional induced terms representing such violation are still invariant under SU(3) x S U ( 2 ) x U ( 1 ) gauge group of the Standard Model. As explained earlier, they preserve the observer Lorentz invariance but not the particle Lorentz invariance. The parameters in the minimal SME are assumed to be constant over spacetime and this is the reason why we call it “minimal”. An extension of the model by including gravity in the context of some non-Minkowski spacetimes has been recently discussed by Kostelecki‘ and the parameters become spacetime dependent. As an example, for simplicity, the QED sector of the minimal SME which involves the electron and photon sectors is given here.

where P and M denote

P = y ~ + I ’ ~M,= m + M l ,

p MI

=

+ dp’y5yp + + ifpy5 + -1g x u p a x v , 2 1 + bpy5yC”+ -2H , v a ~ u .

cw yu

= a,yp

ep

Here all constants a , b, ..,g and H represent expectation values of some background tensor fields and break the particle Lorentz invariance. The photon sector is given as

where the Lorentz violation is represented by k F and k A F terms. The parameters have some properties. Let us quote some of them here. All terms in M I and k A F have dimension of mass while all terms in :?I and k F are dimensionless. ( k F ) n x , v is antisymmetric with respect to first two and last two indices separately and it satisfies the double-trace condition, ( k F ) p v p v = 0, to be sure that the photon field is normalized properly. Only

238

b p , c p and ( k F ) & X p v will be relevant to our discussion here and there are many experimental and theoretical talks about them in this meeting. The Higgs sector is

1

--kpv 2

atWpv@ ,

$Jw

LHiggs rCPT-odd

+

= Zk;@+Dp@ H.c.

,

where k+4 has real symmetric and imaginary antisymmetric parts, which are separated as above and k 4 and ~ k4w have only real symmetric parts. All are CPT preserving (CPT-even) but Lorentz violating and dimensionless. The only CPT-odd and mass dimension coefficient is k4.

3. The CPT-even coefficients 3.1. The CPT-even antisymmetric coeficients Direct detection of these coefficients would necessitate producing large numbers of Higgs bosons, and the resulting bounds would be quite weak. However, there are extremely stringent bounds on Lorentz violation at low energies, and thus searching for the effects of these new interactions through loop effects will provide the strongest bounds. The most promising of these effects will be on the photon propagator. In this section, we will consider the bounds on the CPT-even antisymmetric coefficients, kg4, k4g and k$w. We first look at the most general CPT-even photon propagator, and then relate the k$+ coefficients to the Lorentz-violating terms in the photon propagator. Then, the experimental constraints on such terms lead directly to stringent bounds on the k& coefficients. We then consider the k+g and kbw coefficients. The modified photon propagator from the Lagrangian Eq. (2) is"

Ma6( p ) = g"6p2

- pap6

-

2( kF)QP+ppp,

.

(3)

The propagator is clearly gauge invariant (recall that k F is antisymmetric under exchange of the first or last two indices). Note that while the gp"p2 "We set IcAF-term to zero, since it is very tightly constrained from astrophysical ob~ervations.~

239

ppp” structure is mandated by gauge invariance, the k F term is separately

gauge invariant and may differ order by order in perturbation theory. For simplicity, we look at the divergent parts of the one loop diagrams only. Consideration of higher orders and finite parts will give similar, although not necessarily identical, results. We can consider each of the possible terms independently by assuming that there is no high-precision cancellations. Let us start with k$+. The k$+-term leads to photon-Goldstone boson-W boson and photonGoldstone boson-Goldstone boson type interactions which are absent in the conventional Standard Model. As we do in the Standard Model, it is possible to fix the gauge to simplify the calculations. The Standard Model gauge fixing removes the mixing between W r boson and the charged Goldstone boson A similar situation happens in the minimal SME if one modifies the gauge fixing functions by adding a i(k$+),,dpAY term to the S U ( 2 ) functions and a similar i(k$+),,dfiB” to the U(1) f ~ n c t i o n . ~ However, such generalization also leads to an unwanted mixing between the gauge boson Z, and the derivative of the Higgs field, &$I, which is contracted with (k$+)p”, as well as substantially complicating the photon propagator. An easier way is to use a mixed propagator of the form

+*.

Another feature of the k&,-term is the modification of the W-boson propagator. Up to the second order in k$+, the propagator in the ’t HooftFeynman gauge takes the form

The one-loop contributions to the photon vacuum polarization are given in Fig. 1. Here we only include diagrams with second order k$+, since one can show that all diagrams with one kg4 inclusion vanish. There is only one surviving structure, (k$+),x ( k & ) p v p x p x ’ , which is gauge invariant (it is clear when we contract any of two external momenta of the photon). Calculating the one-loop diagrams, and comparing with Eq. (3), we find that the components of k F can simply be expressed in terms of k$+ as ( k ~ ) , x x t= ~ i(k$+),x(k$+)yv. We now turn to the experimental bounds on the k F . Many speakers in this meeting have talked about the ke+ and by ko- which are 3 x 3 matrices defined from the components of k~ and represent 10 of 19 elements of kF.7 The strongest bound is coming

240

(d

(h)

(i)

Figure 1. One-loop contributions to the photon vacuum polarization involving Lorentzviolating interactions to second order. These diagrams are for k& case but similar diagrams exist for the other antisymmetric coefficients. Here the wavy (dashed) line circulating in the loop represents W boson (charged Goldstone boson). Each blob in vertices, W-propagator or W+ mixed propagator represents a single Lorentz-violating coefficient insertion. The rest of the diagrams can be obtained by permutations.

from birefringence constraints’ and is given by 3 x I should note that for any single or possible combination of non-zero elements of (k&)pv it is impossible for both ke+ and k0- to be null matrices, and thus the birefringence constraints apply.4 Therefore the upper bound of the (k&,)pv coefficients can be obtained as 3 x The discussion of k g and k$& is very parallel to the k& case. The k$ term does not induce a W-Goldstone mixing but leads to photon-Higgs scalar mixing instead. The k $ b term has very similar features to the k$+ case except for the photon-Higgs-boson mixing. The (kF)pxx’v = cos2Bw(k+B)px(k+B)xru and (kF)pxx/v = - &Z sin2Ow(k + w ) p x( k+w)xrv equalities hold, which sets the bound as 0.9 x and 1.7 x respectively. It is seen that the current bound on all three Lorentz violating coefficients is of the order of and can easily be updated as the bound on kF is updated.

&

3.2. Redefinitions and the symmetric coefficients

We consider bounds on the k$+ coefficients. In this case, the strongest bounds come from relating, through field redefinitions, these coefficients to other Lorentz violating coefficients in the fermion sector, and then using

241

previously determined bounds on those coefficients. Therefore, coordinate and field redefinitions need to be discussed carefully. Once any model is extended by relaxing some its symmetry properties, not all of the new parameters representing an apparent violation of these symmetries may be physical. That is, the model might have some redundant parameters. Therefore, the extended model should be carefully analyzed to check for redundant parameters. This analysis may yield several Lagrangians which are equivalent to each other by some coordinate and field redefinitions and r e s c a l i n g ~ The . ~ ~same ~ ~ ~situation ~ ~ ~ applies to the minimal SME case. A simple observation from the fermion sector is that $,rPDP$ - a P ~ , r ~ I4 + &PD,$ !J under $ + exp(-iapx,)$. Thus, a, is redundant unless gravity is included. A similar conclusion can be drawn for some components of k& in the Higgs sector under certain circumstances. Consider a case7l1O with only two Lorentz-violating parameters k44 and kF in the scalar and photon sectors, respectively. The Lagrangian is L = ,XFXIV FP” [gPv (k44),v] (Dp@)tD”@- m2@t@ - LF 4 P” 4(kF)pXX’vF 7 where D, = 8, iqA, and k44 is real and symmetric. Let us assume that only one component of k44, (k44)oo = k2 - 1, is n ~ n z e r oand ~~~~ that kF is taken as zero. The transformations t 4 k t , x -+ x and the field redefinitions A0 -+ Ao, A -+ k A with rescaling of the electric charge q 4 q / k move the Lorentz violation into the photon sector (Lphoton= (D,@)+D,@ - m2@t@ ;(E2 - k 2 B 2 ) , where E ( B ) is the electric(magnetic) field). A similar transformation works for the spatial diagonal components of k44 when they are assumed to be equal. However, for the other components of k44, there are no such obvious transformations. Another observation is from the electron sector of the extended QED.8 The free electron Lagrangian with explicit Lorentz violation L: ($(x)) transforms into Lof(x(x)) ic,,Xypa”x = L?(x(x’)) under the transformation I+!J(x)= (1 ~ , ~ ~ ~ ~ ”(i.e., ) x xfi ( x+ ) x’P = xP cfx”). Note that cPv is redundant unless fermion-photon interactions are included. Similarly the field redefinition of the Higgs doublet @(x) = [1+;(k&)PvxPau](p(x)eliminates the explicit Lorentz violation in the Higgs sector but the (/$&)-term reappears as a c-term in the photon sector. Thus, the redundancy of the parameters in the minimal SME is a matter of convention. Assuming a conventional fermion sector (and the photon sector in the case of including fermion-photon interactions) makes the ( kg4)+, physical. Otherwise, there is mixing among k&,cPv, and nine unbounded k p coefficients. In this study, we only concentrate on the Lorentz and CPT violation in the scalar sector of the SME, hence we assume that the theory has a conven-

+

+

+

+

+

+

242

tional fermion sector, which means that bounds on cPv will lead to effective bounds on k&. The best current bounds on the components of cPu are summarized in Table 1 as direct bounds on the components of (k&,)fiu. In general, we prefer using the measured cleaner bounds, if available, to some projected tighter bounds estimated from some planned experiments. Table 1. Estimated upper bounds for the Lorentz and CPT violating coefficients in the Higgs sector of the SME. Parameters (k,A)PV

( k 4 B ) ~ ~

(k4w )PV (@,) I I

Sources k,+, kEo3 x 10-16 0.9 x 10-16 1.7 x

Q TI TI

(~++)TT

CPV

Comments b, (GeV)

a b

4 x 10-l~ 10-25

C

( k & ) X Z , @&)YZ

(k&)XY

( k 4 ) X >( k 4 ) Y (k4)Zv (k4)T

2.8 x 10-27

d d e f

Note: a) Obtained from c F t r o n with the assumption that Lorentz violation is not i s ~ t r o p i c . ~ ~If ,it~ is ~ ,isotropic,the ’~ bound on ( ~ & ) T Ta ~ p 1 i e s . l ~ b) Obtained from the comparison of the anti-proton’s frequency with the hydrogen ion’s frequency.14 c) Estimated value based on the sensitivity calculations of some planned space experirnent~.”,~5,16,17 d) Obtained from the e) From with the use of a two-species noble-gas maser. From b y r o n , a weaker but cleaner bound of 1.2 x lowz5can be obtained. f ) This bound is from the spatial isotropy test of polarized electrons.

bPutron

4. The CPT-odd coefficient

One interesting effect of the CPT-odd k+-term is the modification of the conventional electroweak SU(2) x U( 1)symmetry breaking. Minimization of the static potential yields a nonzero expectation value for 2, boson field of the form (2,)o = e R e ( k 4 ) , . b The nonzero expectation value for the Q 2 will, when plugged into the conventional fermion-fermion-2 interaction, term.‘ Then the relation b, = +Re(k4), holds. The best yield a b,&fiLy5$ bound on b, for its X and Y components comes from the neutron with the bHere we have assumed all the other Lorentz-violating coefficients zero. ‘Alternatively, one can look at the one-loop effects on the photon propagator, however this will yield much weaker bounds.

243

use of a two-species noble-gas maserlg and it is of the order of b>,y 5 lop3' GeV. Details of the experiment and some new improvements can be found in these proceedings. The best bound on the Z component of b, comes from testing of cosmic spatial isotropy for polarized electrons" and it is of the order of b& 5 7.1 x lo-'' GeV in the Sun-centered frame. The bound on the time component of b, is around b$ 5 lopz7 GeV." The complete list of all bounds on the Lorentz and CPT violating parameters of the Higgs sector is given in Table 1.

Acknowledgments

I thank my collaborators Marc Sher and David L. Anderson. I am grateful to V. Alan Kosteleckq for many discussions and encouragement. I am also thankful to Mariana Frank who made this presentation possible. References 1. S. M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001). 2. See for example, V. A. Kosteleckjr and S. Samuel, Phys. Rev. D 39, 683 (1989); 40,1886 (1989); Phys. Rev. Lett. 63,224 (1989); 66,1811 (1991). 3. D. Colladay and V. A. Kosteleckjr, Phys. Rev. D 55,6760 (1997). 4. D. L. Anderson, M. Sher and I. Turan, Phys. Rev. D 70,016001 (2004). 5. D. Colladay and V. A. Kostelecki, Phys. Rev. D 58, 116002 (1998). 6. V. A. Kosteleckjr, Phys. Rev. D 69,105009 (2004). 7. V. A. Kostelecki and M. Mewes, Phys. Rev. D 66,056005 (2002). 8. D. Colladay and P. McDonald, J. Math. Phys. 43,3554 (2002). 9. D. Colladay, AIP Conf. Proc. 672,65 (2003). 10. H. Muller et al., Phys. Rev. D 68,116006 (2003). 11. S.K. Lamoreaux et al., Phys. Rev. Lett. 57,3125 (1986); Phys. Rev. A 39, 1082 (1989); T.E. Chupp et al., Phys. Rev. Lett. 63,1541 (1989). 12. R. Bluhm et al., Phys. Rev. D 68,125008 (2003). 13. V. A. Kostelecki and C. D. Lane, Phys. Rev. D 60,116010 (1999). 14. G. Gabrielse et al., in V.A. Kosteleckjr, ed., CPT and Lorentz Symmetry, World Scientific, Singapore, 1999. 15. V. A. Kosteleckjr and M. Mewes, Phys. Rev. D 69,016005 (2004). 16. V. A. Kosteleckg and M. Mewes, hep-ph/0308300. 17. A. Datta et al., Phys. Lett. B 597,356 (2004). 18. J.D. Prestage et al., Phys. Rev. Lett. 54,2387 (1985). 19. D. Bear et al., Phys. Rev. Lett. 85,5038 (2000) [erratum 89,209902 (2002)l. 20. L. -S. Hou, W. -T. Ni, and Y. -C. M. Li, Phys. Rev. Lett. 90,201101 (2003). 21. F. Cane et al., physics/O309070.

MACROSCOPIC MATTER IN LORENTZ TESTS

H. MULLER Physics Department, Stanford University, Stanford, C A 94305-4060 E-mail: [email protected]

c. LAMMERZAHL Center for Applied Space Technology and Microgravity ( Z A R M ) , University of Bremen, A m Fallturm, 28359 Bremen, Germany E-mail: [email protected] Lorentz violation affects the properties of solids. This modifies the sensitivity of Michelson-Morley type experiments and makes possible new tests of Lorentz violation, e.g., for the electron sector.

1. Introduction The Michelson-Morley experiment, originally performed 1881 in Potsdam,’ was the first experiment to test Lorentz symmetry. A light ray is split into two, run along two interferometer arms, and interfered thereafter; a differential change in the times required t o transverse the arms connected t o a rotation in space would move the interference fringes and indicate Lorentz violation. Very early, it was noted that such a change might be caused not only by a shift in the speed of light c, but also in the arm length L , maybe depending on the material used for the interferometer. For example, Morley and Miller2 used an interferometer out of sandstone to test whether Michelson’s famous null result was just a cancellation between simultaneous shifts in c and L for Michelson’s choice of material. Indeed, solids are bonded by electromagnetic forces, so Lorentz violation in electrodynamics should change the length of a crystaL3 Furthermore, the solid consists of electrons and protons (the uncharged neutrons play a minor role here), so Lorentz violation in the fermionic equations of motion will likely distort crystal^.^?^ These effects modify the sensitivity of interferometer or cavity experiments. Moreover, they allow new tests for fermionic Lorentz violation. 244

245

The Standard-Model Extension6 (SME) allows us to treat both the change in c and all matter effects on a common basis. Suitably fixing the definitions of coordinates and fields eliminates the proton terms and makes electron and photon terms independently meaningf~l.~ This brief article reviews these influences on crystals and, thus, cavity experiments. 2. Lorentz violation in the photon sector

Within the electrodynamic sector of the SME, Lorentz violation leads to a modified velocity of light c = co Sc and also a modified Coulomb potential of a point charge e, @(Z)= e2/(47r1Z1) V, where6

+

+

v = -e2 Z * K D E . Z 87r

(1)

)q3

The influence on solids can be treated for ionic crystal^,^ which in the simplest case (e.g., for NaCI) consists of a lattice of ions with opposite charges. The lattice is formed by the balance between attractive Coulomb forces and a quantum mechanical repulsion due to the overlap of the ionic orbitals. Perturbative calculations show that the change in the repulsive potential due to Lorentz violation is negligible.3 For estimating the influence of the modification of the Couomb potential V, we consider a cube with side lengths Li = LO SLi (i = 1 , 2 , 3 ) , where LO is the length for V = 0 and SLa are small Lorentz violating corrections due to V. The total energy of the cube can be written as

+

E ( L + S L ~=) const

+

+dLi

where the term proportional to the Young modulus E y is the elastic energy associated with a distortion of the crystal (for simplicity, here we restrict to isotropic elasticity). C V is the sum of V over all pairs of ions. Since L is a linear combination of the ion-ion distances as given by the primitive translations of the lattice, C V will depend on the geometry of the lattice. Whereas L minimizes E ( L ) without Lorentz violation, L SLZ minimizes the total energy including V. By minimizing E ( L SL), we obtain

+

+

with CJ and T being constants obtained from summing the Coulomb potential in analogy to the Madelung constants. ( K D E ) ~= , ( K D E ) ~ (~K ,D E ) I =

246 Table 1. Length change coefficients. Material NaCl sapphire

all

a1

-0.28 -0.03

0.10 0.01

Material LiF quartz

all

a1

-1.06 -0.11

0.37 0.04

01 and 212 are the number of valence charges for the atoms. The contributions of the length change and the change in the speed of light give the total frequency change of a cavity filled with vacuum due to Lorentz violation in electrodynamics

(5) Here, all = A(2a - 3711)and a, = -3A71. This has been simplified by noting that astrophysical tests lead to ( K H B ) = - ( K D E ) . For practical materials sapphire and quartz the length change effect is negligible (see Table 1). For future experiments using resonators made of other materials, however, the influence might be stronger and enhances the sensitivity.

Non-Lagrangian t e r m s The most general inhomogenous Maxwell equations that are linear and first order in the derivatives also contain terms that cannot be derived from a Lagrangian,? ($p$‘u

+x

~

~

+

~ xpPu ~ )Fpu~= 4.irj’ , F. ~

~

(6)

Here, x p ” P O is not necessarily symmetric in the first two indices and thus generalizes the ( I C F ) & X ~ ~term of the SME; xpp‘ generalizes the ICAF term. Some Lorentz-violating non-SME terms that cannot be ruled out by polarization or isotropy experiments enter the static field of a point charge. They are given by a 3-vector f and do not lead to an additional modification of the Coulumb potential, but to a magnetic vector potential

Now consider electrons moving in a lattice of point-like, charged atomic cores. The cores will generate a small vector potential with a corresponding magnetic field B’ (additional to any magnetic fields that may be

A’

247

conventionally present in crystals). Due to that field, the Hamiltonian $/(2m) for a electron having the momentum p’becomes

Since the term proportional to the electron momentum pi has zero expectation value in the rest frame of the crystal, we conclude that these non-SME terms do not additionally modify the geometry of crystals. This conclusion also holds for the influence of the magnetic field generated by a point charge within the SME.6 3. Lorentz violation in the electron sector

+

Here, the starting point is the nonrelativistic hamiltonian h = ho Sh of a free electron in the SME, where ho is the usual free-particle hamiltonian, and bh a Lorentz-violating correction. The only term that doesn’t drop out for non-spin-polarized materials (spin-polarized materials can also be treated4) corresponds to a modification of the kinetic energy of the electron:

where rn is the mass of the electron, p, are the momentum components, and Eik = -cjk - fcoobjk is given by the SME Lorentz tensor cpv for the electron. Since the electrons inside crystals have a nonzero expectation value (papj), which is a function of the geometry of the lattice, Lorentz violation will cause a geometry change (‘strain’) of the crystal. Strain is conventionally expressed by the strain tensor e t 3 . For i = j, it represents the relative change of length in 2,-direction, and for 2 # j , it represents the change of the right angle between lines originally pointing in x, and xj direction. A general linear relationship between the Lorentz violating E;k and strain is given by e d c = B d c p j ELj 7

(10)

with a ‘sensitivity tensor’ &pj that has to be determined from a model of the crystal. &pj can be taken as symmetric in the first and last index pair; symmetry under exchange of these pairs will hold only for some simple crystal geometries, like cubic. Thus, the tensor has at most 36 independent elements. To calculate the sensitivity tensor, the electronic states are described by Bloch wave functions to determine (papj);the corresponding strain is

248

calculated using elasticity theory. As a result, the sensitivity tensor &jp PdcmpKmj P d c m j K m p , where

+

=

can be calculated. Ne+ is the number of valence electrons per unit cell, Idet(Z,j)I is the volume of the unit cell expressed by the determinant of the matrix of the primitive direct lattice vectors, kml is the matrix containing the primitive reciprocal lattice vectors, and P d c m p the elastic compliance constants. The symmetric 3 x 3 matrix f l k is given by the Fourier coefficients of the Bloch wave functions. Its six parameters are unknown at this stage and can, e.g., be determined from a simple model that leads to f l k dlk. To eliminate these unknowns, an alternative method is used to calculate the strain for the simple case of isotropic Lorentz violation E i k d j k , and the result is compared to Eq. This yields six equations from which K a b (that depends solely on material properties) can be determined and re-inserted into Eq. (10):

-

N

Bdcjp

= PdcmpAaamj

+ PdcmjAaamp

'

(12)

Note that the theory now needs no assumptions that go beyond the use of Bloch states. Thus, it is very generally applicable and accurate. For convenience, we arrange the independent elements of eab into a sixvectors e = (e,,, eYY, ezz,eyz,ezz, ezy) and express Eq. (10) as e = B . E', where B is a 6 x 6 'sensitivity m a t r i ~ . For ' ~ cubical and trigonal crystals, respectively, it has the structure

'Bll

0 0 0 0 0 0 B12 B12 Bll 0 0 0 0 0 OB440 0 0 0 0 OB440 \ 0 0 0 0 OB44 B12 B12

B12 ,1311 a12

where B66 = ( & I - B12)/4. Table 2 gives values for the materials presently used for cavities. Among these, niobium has the highest sensitivity coef-

249 Table 2. Mat. Au A1203 Nb fused quartz

811

24.13 3.58 6.80 2.64

&2

-11.06 -1.05 -2.40 -0.32

Sensitivity coefficients. 813

814

831

833

841

844

12.34 -0.53

0.014

-0.57

3.14

0.004

5.08

17.9 3.95

ficients. Gold is included as an example for a material with exceptional sensitivity. The influence of the electron terms cclvon hydrogen molecules have also been ~alculated.~ Here, an explicit wave-function can be obtained from first principles, and Lorentz-violating changes in the frequencies of electronic and (ro-) vibrational transitions, as well as the bond length, have been obtained. This allows new tests of Lorentz symmetry that use molecules.

4. Applications and Outlook

An important conclusion is obvious from these calculations: Matter effects do not cancel the sensitivity in interferometer or cavity tests of Lorentz invariance. Instead, they enhance the sensitivity for Lorentz violation in electrodynamics, but only slightly for cavity materials presently in use. Moreover, the theory concerning the Dirac sector4 allowed to derive the first experimental limits on some of the electron coefficients cPv,at a level of 10-14. As a third remark, since the proton terms relevant here can be set to zero by fixing the coordinate and field definitions (this also makes the electron and photon parameters separately meaningful), the theories summarized here together with the change in light propagation6 constitute a complete description of all SME effects that influence experiment using vacuum-filled cavities. These experiments we thus particularly clean tests of Lorentz invariance. For also analyzing the recent and proposed experiments that use matter-filled cavities, the influence of Lorentz violation on the index of refraction can be derived, as will described el~ewhere.~

Acknowledgments H.M. likes to thank Sven Herrmann and Achim Peters (Berlin). A fellowship of the Alexander von Humboldt foundation is gratefully acknowledged.

250

References 1. A.A. Michelson, Am. J. Sci. 22,120 (1881); A.A. Michelson and E.W. Morley, ibid. 34, 333 (1887); Phil. Mag. 24, 449 (1897). 2. E.W. Morley and D.C. Miller, Phil. Mag. 8 , 753 (1904); ibid. 9, 680 (1905). 3. H. Muller e t al., Phys. Rev. D 67,056006 (2003). 4. H. Muller e t al., Phys. Rev. D 68, 116006 (2003); H. Muller, in preparation (2004). 5. H. Miiller e t al., Phys. Rev. D, in press (hep-ph/0405177) (2004). 6. V.A. Kostelecky and M. Mewes, Phys. Rev. D 66,056005 (2002); Q.G. Bailey and V.A. Kostelecky, hep-ph/0407252 (2004). 7. C. Lammerzahl, A. Macias, and H. Miiller, Phys. Rev. D, to be published (2004); C. Lammerzahl and H. Miiller, these proceedings.

TESTS OF LORENTZ INVARIANCE AND CPT CONSERVATION USING MINOS

BRIAN J. REBEL AND STUART L. MUFSON FOR THE MINOS COLLABORATION Department of Astronomy Indiana University Swain West 319, Bloomington, IN 47405, USA E-mail: [email protected] We present a potential analysis to use MINOS data to search for Lorentz invariance violation and C P T violation in neutrino oscillations using a model of KosteleckS; and Mewes.l MINOS will use both atmospheric neutrinos, specifically neutrinoinduced muon events, and beam neutrinos from Fermilab to test this model. A 5 live-year exposure will provide enough data to be sensitive to possible Lorentz invariance violation suggested by this model in both the atmospheric and beam neutrinos. The atmospheric neutrinos cannot conclusively show evidence for C P T violation in this model, but beam neutrinos would definitely be sensitive to possible C P T violation suggested by this model after a 5 live-year exposure.

1. Introduction

The Main Injector Neutrino Oscillation Search (MINOS) is a long baseline neutrino oscillation experiment designed to make a precision measurement of the neutrino oscillation parameters for vP + v, oscillations. The flavor change is said to be the result of the neutrino flavor states being linear combinations of neutrino mass states. The probability for vP -+ v, oscillations is given by

Pup-,,,

= sin2(28)sin2(1.27Am2L/E),

(1)

where 8 is the leptonic mixing angle that represents the rotation from the electroweak flavor basis into the mass basis, Am2 is the difference in the squared masses of the mass states, L is the distance traveled by the neutrino, known as its baseline, and E is the energy of the neutrino. In Eq. (1)Am2 has units of eV2, L is in km and E is in GeV. The Super-K,’ K2K,3 and MACRO4 results suggest that Am2 2.5 x 10-3eV2 and sin’(28) = 1. N

251

252

MINOS is an accelerator-based experiment consisting of a neutrino source and two detectors. The source is a beam of up generated by accelerating protons in the Fermilab Main Injector to high energies which then strike a carbon target. The kaons and pions produced in the collisions decay into muons and u p . One detector (“near detector”) is located at Fermilab, about 1 km away from the target; the second detector (“far detector”) is located 735 km downstream in the Soudan Underground Mine, Soudan, Minnesota, USA. Both detectors have toroidal magnetic fields making it possible to distinguish neutrino events from anti-neutrino events. Construction of the far detector was completed in July 2003. Since its completion the MINOS far detector has taken cosmic ray and atmospheric neutrino data. The near detector was completed in August 2004. It is expected that the beam will turn on in December 2004. 2. Model for Oscillations due to Lorentz Invariance

Violation and CPT Violation The model for neutrino oscillations from Kosteleckf and Mewes’ suggests that neutrino oscillations are due to Lorentz invariance violation and/or CPT violation rather than neutrino mass. In this model the probability for the vM + v, transition is

where L is given in meters, or GeV-l in natural units, E is in GeV, and ( A ) p T = fi’[(aL),”, - ~E(cL);?] - f i x [ ( a ~Y) M -T ~E(CL);:], (AC)MT

= - fiX[(UL)fT -2 E ( C L ) 3

- NY[(aL);T - 2 E ( C L ) 3 . ( 3 )

The ( a ~ ) ; , , where i = X , Y , are the CPT-violating factors. For the case of OM + V , oscillations, the ( a ~ ) ; ,are replaced by -(uL);,. The Lorentz invariance violating terms are the (c~)::. The sidereal frequency is w e = 2~123.9447and T, is the local sidereal time at which the neutrino is observed. The incoming direction of the neutrino is contained in the fi terms,

Nx = cos x sin 8, cos 4 + sin x cose,, N y = sine, sin$,

(4)

where x is the colatitude, defined as (90’ - 1 ) with 1 being the latitude of the detector. The nadir angle of the neutrino, en, measures the angle of the incoming neutrino direction where 9, = 0 for neutrinos coming from

253

directly below the detector. The ‘pseudo’-azimuthal angle of the neutrino is q5, where events from due south have q5 = 0 and those from due east have q5 = 7r/2. As seen from Eq. (2) and Eq. (3), the oscillation signal will be most apparent when the products (UL);,L and (cL)T:LE are near unity. One of the observable signals for this model is a variation in the neutrino flux as a function of sidereal time. If the CPT violating terms are nonzero, then the variation will be different for neutrinos versus antineutrinos.

3. Tests of the Model 3.1. Atmospheric Neutrinos One method of observing atmospheric up is to observe the muons produced by u,, interactions in the rock below and slightly above the detector’s horizon. These muons are called “neutrino-induced muons.” The neutrinos producing these muons have energies ranging from 1 GeV - lo4 GeV, with the distribution peaking near 10 GeV. Neutrinos coming from directions where 0.6 < cos0, 5 1 have baselines of order lo4 km. Those from directions with 0.2 < cos0, 5 0.6 have baselines ranging from lo3 - lo4 km and those with -0.1 5 COSO, 5 0.2 have baselines ranging from lo2 - lo3 km. The neutrino-induced muon analysis could probe for values of (cL);: as small as in this case, the oscillations will be most apparent for events coming from near the nadir. Larger values of (CL);; will cause the product of (CL);:LE to be much larger than unity for all directions making the oscillations very rapid, which results in the observed number of events being about 1/2 the expectation over all nadir angles. MINOS recorded a 231 live-day exposure between July 2003 and April 2004. A total of 41 neutrino-induced muons were found in the data set5 and the sidereal time distribution of the events is shown in Fig. 1. The errors shown are statistical errors and the number of events in each bin has been weighted to account for varying detector live-time in each bin. The data are consistent with Lorentz invariance. MINOS will take atmospheric neutrino data concurrently with beam data taking. The expected cos0, distribution for a 5 live-year exposure is shown in Fig. 2. The dashed lines indicate the boundaries for the different baseline regimes. In the following figures the distributions for the null oscillation scenario are shown by the solid line and the oscillated distributions are shown by the points with statistical uncertainties. The oscillated distribution in Fig. 2 assumes that the mixing is maximal with sin2(20) = 1, (CL);? = (cL);: = lopz4,and the CPT violating factors are set to 0. The N

254

x'

I ndf Prob

16-.,, I

" I

' ' I 1 ' ' ' 1

'

I

'

1 ' 1 . 1

" ' 1 '

P0

4.421 1 5 0.4906 6.943 1.254

+

14

-

12:

-

10L '

-

8i 6:

-

4-

-

2-

-

,.

I . . . I , . . I , . . I . . . I . . ~ I . . . I I I I I . . .I , . . I , ,

Figure 1. Sidereal time distribution of neutrino-induced muons in the 231 live-day exposure of the MINOS far detector.

-

No oscillations

cose,

Figure 2. Expected cos 0%distribution of neutrino-induced muons for a 5 live-year exposure of the MINOS far detector and the model of Eq. (2). The unoscillated distribution is shown by the solid line and the points with statistical uncertainties are for the oscillated distribution.

shape of the cos9, distribution shows that the model of Eq. (2) causes a small deficit versus the no oscillation case for events from above the horizon and that the deficit increases to 50% near the nadir. Comparing Fig. 3, which shows the cos en.distributions for p- (left) and p+ (right) events and ( u t ) t T= to Fig. 2 shows that the statistical uncertainties makes it impossible to distinguish between the combination of CPT violation and Lorentz invariance violation versus Lorentz invariance violation alone using neutrino-induced muons.

255

case,

case,

Figure 3. Expected cos distributions for a 5 live-year exposure at MINOS assuming the model of Eq. (2). The distribution for vp is shown on the left and for Dp is on the right.

3.2. Beam Neutrinos

The neutrino beam simplifies the process of probing for Lorentz invariance violation and CPT violation because the beam neutrinos have a fixed baseline of 735 km and they all come from the same direction. Moreover, the energy spectrum of the beam neutrinos is much narrower than the atmospheric neutrinos, peaking at 3 GeV with a high energy tail out to 40 GeV. MINOS should be sensitive to values of (CL):; between and and values of ( a ~ ) of; order ~ We first examined the case where the Lorentz invariance violating factors are nonzero and the CPT violating terms are set to 0. The predicted local sidereal time distribution for a 5 live-year exposure assuming maximal mixing and (CL):; = is shown in Fig. 4. This figure shows that the beam neutrinos will show a clear signal for Lorentz invariance violation in this model. The points at 6.5 and 18.5 hours do not show oscillations because weTe = 7r/2 for these times causing the Lorentz invariance violating terms in Eq. (2) to cancel. Next we let both the Lorentz invariance violation and CPT violating factors be nonzero, choosing ( u L ) = ~ ~ (UL);~ = and (CL):? = ( C L ) : ~= The local sidereal time distributions are shown in Fig. 5. This figure represents a 5 live-year exposure for both up, shown on the left, and Dp, shown on the right. Again, a clear signal will be seen if the model is correct. Additionally, the distribution for the neutrinos has a different shape than the antineutrino distribution. Moreover, it is easy to distinguish between Lorentz invariance violation alone and Lorentz

256

Figure 4. Expected local sidereal time distribution of beam neutrinos for a 5 liveyear exposure of the MINOS far detector and the model of Eq. (2) with maximal mixing and the C P T violating factors set to zero. The unoscillated expectation is shown by the solid line and the points with statistical uncertainties are the oscillated expectation.

invariance violation combined with CPT violation based on the shape of the distributions.

300 250 200

0'

2 4 6 8 10121416182022 Local Sidereal Time (h)

0'

o Oscillations

2 4 6 8 10121416182022 Local Sidereal Time (h)

Figure 5. Expected local sidereal time distribution of beam neutrinos for a 5 live-year exposures of the MINOS far detector to up (left) and Cp (right) using the model of Eq. (2). The unoscillated expectations are shown by the solid lines and the points with statistical uncertainties are the oscillated expectations.

257

4. Conclusions

We have presented two potential analyses for testing Lorentz invariance and/or CPT symmetry according to the neutrino oscillation model of Kosteleckf and Mewed using MINOS data. The first method involves using atmospheric up through the neutrino-induced muon events. This method is sensitive mainly to Lorentz invariance violation through the shape of the cosf3, distribution of the events for values of the Lorentz violating factors (CL);: of order The second method uses the beam neutrinos produced at Fermilab. It is sensitive to both Lorentz invariance violation and CPT violation for values of (c~):; of order and CPT violating factors ( c L L ) ; , of order For both methods, a 5 live-year exposure should be adequate to determine whether the Lorentz invariance violation and/or CPT violation is observed.

Acknowledgments We gratefully acknowledge the strong support of the MINOS collaboration in the preparation of this paper. We also thank A. Kosteleckf and M. Mewes for bringing this model to our attention.

References 1. V.A. Kostelecki and M. Mewes, hep-ph/0406255. 2. Y . Hayato, presented a t NOON’03 (2003). 3. M. H. Ahn et al., Phys. Rev. Lett. 90, 041801 (2003).

4. M. Ambrosio et al., Phys. Lett. B434,451 (1998). 5. B.J. Rebel, Ph.D. Dissertation, Indiana University (2004).

AN SME ANALYSIS OF DOPPLER-EFFECT EXPERIMENTS

CHARLES D. LANE Berry College Physics Department 2277 Martha Bemy Hwy Mount Berry, G A 30149-5004 E-mail: [email protected] We analyze a class of experiments that probe the Doppler shift of frequency measurements within the context of ‘the Lorentz-violating Standard-Model Extension (SME). It is found that these experiments are capable of probing many SME coefficients that are unprobed by any experiments to date. We use a particular recent experiment to place bounds on some SME coefficients.

1. Introduction Special relativity predicts specific relations between measurements of transition frequencies of atoms in relative motion. If Lorentz symmetry is violated,’ however, the conventional relations may not be precisely correct. In this talk, we present analysis of sensitive Doppler-shift experiments2 within the framework of the Lorentz-violating Standard-Model Extension3

(SME).

2. Basics Doppler-shift experiments involve the following ingredients. (1) Two groups of atoms, one group at rest in the lab frame and the -+ in the lab frame. The other forming a beam with velocity same transition should be studied in each group. (2) Two coordinate frames: S, the lab frame, and S, the moving atoms’ rest frame. (3) Three transition-frequency measurements: z&b= frequency associated with the atoms at rest in the lab frame, measured in the lab frame (i.e., in their own rest frame).

258

259

a

heam= frequency associated with the beam atoms, measured

a

in the beam frame (i.e., in their own rest frame). heam= frequency associated with the beam atoms, measured in the lab frame (not their own rest frame).

General considerations of Lorentz symmetry give qualitative insight into relationships within each of two pairs of the measurements. First, consider 'beam and h e a m . These are measurements of the same transition in the same atoms, but measured in different frames, so they are related by an observer Lorentz transformation. Since the SME preserves observer Lorentz symmetry, ubeam and fibeam obey their conventional Doppler relationship. Next, consider 'lab and h e a m . These are not measurements of the same quantity. However, since they are measurements of the same type of thing (a particular transition in a particular type of atom), they are related by a particle Lorentz transformation. Since the SME breaks particle Lorentz symmetry, they do not obey their conventional relationship, namely, equality. Doppler-effect experiments effectively probe the ratio of these two frequency measurements, bounding the amount by which it differs from unity. In practice, 'beam is not a single frequency measurement. Two different beam-atom transition frequencies are measured in the lab frame: up, the frequency of light emitted parallel to the beam's velocity in the lab frame, and v,, the frequency of light emitted antiparallel to the beam's velocity in the lab frame. In conventional physics, a nice identity relates these to the beam transition frequency as measured in the beam frame: upva = fizeam. Derivation of this relation uses only observer Lorentz transformations, so it holds also in the SME. The key experimental quantity is the ratio u p u , / ~ f a b . Due to the identity mentioned above, this can be expressed as the square of the ratio of two transition frequencies as measured in their own rest frames, heam and 'lab. As discussed above, these frequencies are equal in conventional physics, but may be different whenever particle Lorentz symmetry is violated. Since particle Lorentz symmetry is at least approximately valid, the experimental quantity of interest is nearly one: -2

'beam

'a'P-

2

Vlab

where E << 1.

-

2

'lab

= 1+

260

The most stringent bound on E to date results from a recent Dopplereffect experiment at the heavy-ion storage ring TSR in Heidelberg,2 with result

lEl

5 2 x 1 0 - ~.

(2)

3. SME Predictions In the SME, new interactions cause shifts in atomic energy levels with respect to conventional values. In turn, this generally induces transitionfrequency shifts of the form4 v=VSM+bu

bu

=

,

(...)bE+(...)(c;, + c ; ~ - ~ c ; ~ )

+ ( - . - ) b g + ( . . . ) ( ~ ;+cF2 ~ -2~;~)+... , (3) where u denotes any transition frequency measured in the atom’s own rest frame, U S M is the conventional (i.e., according-to-the-Standard-Model) frequency, and 6u is the shift induced by the SME. The quantities bz and c3ek denote tensor components associated with Lorentz violation in the electron while b: and cyk denote similar components for the neutron. A full expression of the frequency shift also involves proton-associated terms and many additional terms for each particle type. The coefficients of these components, represented by dots in parentheses, are combinations of atomic and nuclear expectation values such as spin and orbital-angular-momentum quantum numbers. This expression holds for both Vlab and heam, with the tensor components evaluated in the lab and beam frames, respectively. Since tensor components generally depend on the frame in which they’re evaluated, it usually pertains that b3 # b3, C i I # c11, etc. Thus, since they are measured in different frames, &&earn # bUlab and hence h e a m # Vlab. The experimental quantity of interest (1) can be expressed in terms of the conventional Standard-Model value of each frequency and the Lorentzviolation-induced frequency shifts as follows:

=

(.. . )

(@ -@)

+ (...) (c;-c;)

+ ... .

(4)

[Note: The quantity E may also receive contributions from the photon ~ e c t o r In . ~ the current analysis, we neglect all Lorentz-violating photon interactions, focusing entirely on the fermion sector.]

26 1

Since the lab and beam frames are attached to the rotating Earth, tensor components evaluated in them vary with time. Thus, each frequency shift varies with the Earth's rotation: Sheam-bulab

=const.+ ( * * * ) c o s R t(+. * . ) s i n a t +(. . . ) cos 2Rt (. ) sin 2Rt ,

+

+

(5)

where R is Earth's sidereal frequency. The unstated quantities in parentheses are composed of Lorentz-violation coefficients, atomic expectation values, geometric factors, and powers of the beam speed @beam. Data from a Doppler-effect experiment in which timing information is recorded could be fit to this function, yielding measurements of various combinations of Lorentz-violating coefficients. Even without timing information, interesting results may be achieved. If we assume that all frequency data is time-averaged, i.e., (cosRt) = (sinRt) = (cos2Rt) = (sin2Rt) = 0 , then the difference in frequency shifts is determined solely by the constant contribution. For the each transition studied in the heavy-ion experiment,

-

In this expression, the Lorentz-violation coefficients d % y , fi+x , etc. are evaluated in a Sun-centered nonrotating frame.6 Terms similar to these involving neutron- and proton-associated coefficientsalso make contributions, as do terms of higher order in @beam. Herein may be seen a large contrast with clock-comparison e x p e r i m e n t ~ , ~where , ~ , ~ a difference in transition-frequency shifts similar to Eq. (5) is studied. In clock-comparison experiments, the difference is nonzero (but constant) in conventional physics, so timing data is required to be sensitive to Lorentz-violating effects. In Doppler-effect experiments, the difference is identically zero in conventional physics, so timing data is not necessary. In the heavy-ion experiment12multiple transitions were studied and their measurements effectively averaged.8 As a result, all dipole-type frequency shifts from different transitions cancel out, while quadrupole-type frequency shifts are uncancelled. The time- and transition-averaged difference in frequency shifts then reduces to the following:

262

The beam speed /3beam M 0.064 in this experiment. The coefficient of E; amounts to a suppression factor of about lop5, while the proton and neutron contributions are each suppressed by about lop2. The results of the experiment imply that ( b h e a m - bl/lab)avg

<, lO-’vs~

x 500 kHz M 3 x

lop1’ GeV .

(8) If we assume that the contributions from different fermions don’t cancel each other, then the following bounds may be extracted:

IGI <,

GeV,

I E Z l <,

GeV,

1E;I <, lov1’ GeV. (9) The bounds nicely complement the bounds from clock-comparison4~7and optical cavityg experiments.

4. Comments on dansitions A few comments about choice of transitions studied in atomic Lorentzviolation experiments are in order. (1) All clock-comparison experiments to date are insensitive to quadrupole-type shifts involving electrons, and thus SME coefficients cjk. The heavy-ion experiment gains sensitivity to these by studying states where the electron’s orbital angular momentum 12 1. (2) Sensitivity to a large group of Lorentz-violating coefficients is lost upon averaging over many different transitions. A probe of a single transition could yield bounds on many so-far-unbounded electron coefficients. It may not be feasible, however, to probe single transitions with similar precision. Transitions involving spin-flipped electrons generically include Lorentz-violating contributions unsuppressed by the l o p 5 factor in Eq. (7). Again, however, it may not be feasible to probe such transitions with similar precision. (3) Standard clock transitions m F = 0 + m> = 0 are insensitive to Lorentz violation. 5. Summary Doppler-effect experiments provide a good example of the differences between observer and particle transformations. Already completed exper-

263

iments provide bounds on Lorentz-violating SME coefficients t h a t nicely complement t h e results of other types of atomic experiments.

References 1. For recent reviews of various theoretical and experimental aspects of Lorentz violation, see, for example, V.A. Kosteleckf, ed., CPT and Lorentz Symmetry 11, World Scientific, Singapore, 2002. 2. G. Saathoff et al., Phys. Rev. Lett. 91, 190403 (2003); G. Gwinner, these proceedings. 3. D. Colladay and V.A. Kostelecky, Phys. Rev. D 55,6760 (1997); 58,116002 (1998). 4. V.A. Kostelecky and C.D. Lane, Phys. Rev. D 60,116010 (1999); R. Bluhm, these proceedings. 5. M.E. Tobar et al., hep-ph/0408006. 6. R. Bluhm et al., Phys. Rev. Lett. 88, 090801 (2002); Phys. Rev. D, 68, 125008 (2003). 7. V.W. Hughes, H.G. Robinson, and V. Beltran-Lopez, Phys. Rev. Lett. 4 (1960) 342; R.W.P. Drever, Philos. Mag. 6 (1961) 683; J.D. Prestage et al., Phys. Rev. Lett. 54 (1985) 2387; S.K. Lamoreaux et al., Phys. Rev. A 39 (1989) 1082; T.E. Chupp et al., Phys. Rev. Lett. 63 (1989) 1541; C.J. Berglund et al., Phys. Rev. Lett. 75 (1995) 1879; D. Bear et al., Phys. Rev. Lett. 85 (2000) 5038; F.Can6 et al., physics/O309070; R. Walsworth, these proceedings. 8. G. Gwinner, personal communication. 9. H. Muller, Phys. Rev. D 68,116006 (2003); these proceedings.

NONRELATIVISITIC IDEAL GASES AND LORENTZ VIOLATIONS

D. COLLADAY AND P. MCDONALD Division of Natural Sciences New College of Florida Sarasota, FL 34243, USA E-mail: [email protected] We develop statistical mechanics for a nonrelativisitic ideal gas in the presence of Lorentz violating background fields. The analysis is performed using the StandardModel Extension (SME). We derive the corresponding laws of thermodynamics and find that, to lowest order in Lorentz violation, the scalar thermodynamic variables are corrected by a rotationally invariant combination of the Lorentz terms which can be interpreted in terms of a (frame dependent) effective mass. We find that spin couplings can induce a temperature independent polarization in the gas that is not present in the conventional case.

1. Introduction The Standard-Model Extension (SME) provides a convenient framework for studying the effects of spontaneous Lorentz and CPT symmetry breaking within the context of conventional quantum field theory.' In this report we develop a statistical mechanics formalism for calculations involving the SME. Our approach and our results are quite general and include a complete analysis of the effects of all Lorentz violating terms on a nonrelativistic ideal gas. Complete details of the results announced here will appear elsewhere.2 2. Notation and Framework We adopt the viewpoint of J a y n e ~In . ~this approach to statistical inference, one assumes a collection of states, and a finite collection of real valued functions on the collection of states, Given a distribution of states, qi = q ( $ i ) , we denote by brackets the corresponding expectations; (fj) = f j ( $ i ) q i . Given observations of the mean values { (fj)}&l, Jaynes argues that the most likely distribution for the given data is obtained by maximizing the (information) entropy, S = -k qi In ( q i ) , subject to

{$i}zl,

{fj}iZ1.

xi

xi

264

265

the constraints given by the observations (here k is a positive constant). A formal argument via variational calculus then leads immediately to a solution of the form

where the Lagrange multipliers, Xj, are real constants and Z is the partition function, Z(X) = C,e-c:=1 '~f3(+%). The same formalism permits the observables fj to depend on a finite number of parameters. When this formalism is applied to study the statistical mechanics of an ideal gas, one immediately identifies k with the Boltzmann constant and the Lagrange multipliers with the usual thermodynamic quantities (i.e., scaled inverse temperature, chemical potential, etc.). The central features of thermodynamics become consequences of formal computation. As was emphasized by Jaynes, the method provides accurate thermodynamic properties of a system assuming empirically accurate observations of the mean values and the correct laws of motion embedded in the hamiltonian. To use this framework to study ideal gases with Lorentz violation all that is required is the appropriate hamiltonian. For nonrelativistic spinfermions of mass m this has been worked out by Kosteleckf and Lane.4 The result to second order in 2, p momentum, is

a

where A , B j , Cj, Djk, Fjk and Gjkl are real parameters which can be given explicitly in terms of the standard collection of parameters defining the Lagrangian of the minimal SME.4>2 3. Single particle systems

We first consider a system consisting of a single free spin-; particle governed by the hamiltonian H appearing in (2), constrained to a cube of side length

L. Denote by $$; the standard unperturbed solutions for the hamiltonian where n = (n1,1zzln3)is a triple of positive integers and s E (1, -1) denotes a sign. Let EL:! denote the corresponding unperturbed energies. The first order correction to the energy levels due to the Lorentz-violating terms are

266

found using standard degenerate perturbation theory as:

where the vector G(n) is defined with components

Using the perturbed energy expression (3) and standard approximations, the partition function is

( :

Z(l) 2 2e-PAnQV 1 - - n ( F )

)

,

(5)

where V is the volume of the box, nQ = is the quantum concentration, and T r ( F ) = C Fii. It follows that only the A term corrects the energy and corresponds to a constant shift in all of the energy levels. The correction to the partition function due to the F term can be incorporated into an effective mass for the fermion

The expectation value of the spin can be calculated similarly: 1 (~('1) pv -PB - -Tr(G) 2 where the vector T r ( G ) is defined by ( n ( G ) ) kf Giik.

(7)

xi

4. Classical gas

The grand partition function for the classical gas system can be written in terms of the single-particle partition function and a chemical potential. This gives expressions for the expected particle number and energy. From these expressions it follows that there is no change in the ideal gas law. It is possible to solve for the chemical potential p(c) p(c) = -kT (ln

(s)

-5 1T r ( F ) )

where TLQ is the quantum concentration and n(c)= (N(c))/V is the concentration of the classical gas. One can also solve for the entropy, and there is a modification of the Sackur-Tetrode equation:

[; :

dC) = ( N ( C ) ) k - - -Tr(F)

I)$(

+ In

.

(9)

267

Finally, the expectation of the spin is

5. Quantum Gas - Fermions With notation from previous sections, and using zero subscripts to represent unperturbed quantities, the partition function for the grand canonical ensemble associated to a Fermi gas is given by

where A = h/(27rmkT)i is the thermal wavelength and f v ( e - " ) is the appropriate Fermi-Dirac integral.5 As in the classical case, first order corrections to the partition function occur only for Lorentz violating terms of type F. The corresponding partition function can be written as

As above, the perturbation can be absorbed as an effective mass. Calculations then give (13)

as well as the ideal gas law

where n(Q)is the concentration of the quantum gas. Since the map a 4 fq(e-") is invertible, we have a formal expression for the chemical potential in terms of the inverse F p(Q) 2 - k T 3

(

X3(m;)

ntu,)

*

(16)

Equation (16) can be used to obtain expressions for relevant thermodynamic quantities; for example, the Fermi energy and associated perturbations of the chemical potential at low temperature, low temperature expressions for specific heat and entropy, and low temperature perturbations of the ideal gas law.2

268

The expectation value for the spin can be calculated in the quantum regime using the fractional occupancies: 1

+ zTr(G)] .

( s ( Q )2 ) -(N(&))

At low temperatures, the contribution from the B term can be written as

where &F is the Fermi energy. 6. Quantum gas

- Bosons

It is possible to generate a model for a free spin-0 boson gas by combining two fermions into a singlet representation of the spin group. The resulting hamiltonian is given by

Choosing the ground state energy to be zero, employing the notation of the previous sections and making the standard approximation, the associated grand partition function for the unperturbed case can be written as 1 ln(Z(QB)(ao)) = -Vg;(e-*o) A3

- In(1- e-*O)

,

(20)

where A is the thermal wavelength, g;(e-*) is the appropriate BoseEinstein integral, and the ground state has been separated out. The only nontrivial leading order perturbation in (19) arises from the F term. A calculation which follows that done for the case of fermions gives

- In(1 - e-&)

It follows that for the perturbed case we have

As in the Fermi case, the chemical potential can be expressed as a function of the number of particles in excited states. Because only T r ( F ) enters

269

into the grand partition function, it is possible to use the concept of effective mass to absorb the effect as before. Standard results of Bose-Einstein condensation therefore hold in a given laboratory frame.

7. Conclusion Using Jaynesian formalism, we have developed a framework for statistical mechanics in the presence of symmetry violation which parallels the conventional case. We find that the laws of thermodynamics are the same as in the conventional case, with specific expectation values of thermodynamic quantities modified by the Lorentz-violating terms. For an ideal gas in the absence of any external applied fields, expectation values for scalar thermodynamic quantities such as energy and particle number were unaltered except for an overall scaling factor T r ( F ) . This correction can incorporated into theory as an effective mass m* = (1 i T r ( F ) ) r nin the hamiltonian, although the effective mass defined in this way depends on the observer’s Lorentz frame. Focusing on spin, we find nontrivial changes in the net spin expectation value arising from the terms that couple to the spin. The pure-spin coupling Bj mimics a constant background magnetic field and induces a corresponding magnetic moment per unit volume in the gas. The derivative-spin coupling G i j k generates a fundamentally new type of effect that induces a temperature-independent polarization in the classical gas that is proportional to Tr(G).

References 1. For a summary of recent theoretical models and experimental tests see, for

2. 3. 4. 5.

example, V.A. Kostelecki, ed., C P T and Lorentz Symmetry, World Scientific, Singapore, 1999; C P T and Lorentz Symmetry ZZ, World Scientific, Singapore, 2002. D. Colladay and P. McDonald, submitted, hep-ph/0407354. E. T. Jaynes, Phys. Rev. 106,620 (1957). V.A. Kosteleck? and C.D. Lane, J. Math. Phys. 40, 6245 (1999). R.K. Pathria, Statistical Mechanics, Oxford Press, Boston, 1996.

A LABORATORY FREE-FALL TEST OF THE EQUIVALENCE PRINCIPLE - POEM

ROBERT D. REASENBERG AND JAMES D. PHILLIPS Smithsonian Astrophysical Observatory Harvard-Smithsonian Center for Astrophysics 60 Garden Street, Cambridge, MA 02138, USA To test the Equivalence Principle (EP) to an accuracy of at least Ag/g = 5 x we are developing a modern version of the famous experiment historically attributed to Galileo Galilei. In our principle of equivalence measurement (POEM), we directly examine the relative motion of two test masses that are freely falling. Such an experiment tests both for a possible violation of the Weak Equivalence Principle (WEP), and for new forces that might mimic a WEP violation.

Why would one attempt a Galilean test when the best EP tests for the past century have been based on torsion pendula? Adelberger et al. have reached Ag/g = 5 x and further improvement is expected with their torsion balance, whereas Niebauer and Faller, who did the best Galilean test,' reached Ag/g = 5 x The answer is in three parts. First, the Galilean approach is sensitive to the full gravitational acceleration of Earth but the torsion pendulum is sensitive to the horizontal component or the solar term, each three orders smaller. Thus, to the extent that an EP test is limited by small effects (magnetic contamination, outgassing reaction forces, etc.), the Galilean approach is advantageous. Second, a Galilean test, but not a pendulum test, can evolve into a space-based experiment like STEP. Third, there has been limited interest in the Galilean approach, but evolving technology has created new opportunities. Only by exploiting them can we determine their potential. There are three key quantities in an EP test: the coherence time, Q, the total observing time, T, and the distance measurement sensitivity, ~ o ( T ) , which depends on the single-observationintegration time, r. A useful figure of merit is

270

271

where R < 1 is the fraction of the test-mass assembly (TMA) made of the test substance, and K is a factor that depends on the type of motion that would be caused by the EP violation. For constant acceleration, K = 12& M 27 (assuming the initial position and velocity are estimated along with the acceleration, as they must be); for sinusoidal motion with period 2Q, K = 7r2& M 14. In all cases, T must be much smaller than &. Tossing the test masses upward can increase Q by up to a factor of two. It also increases the fractional running time, a = Q / E 5 1, where E is the interval before a repetition of a Q-length observation sequence would be possible. For a torsion balance experiment, which can be run continuously, a = 1. For a drop-tower experiment, there is a gap between drops that makes a small (say to lop4). For POEM, a x 0.7. What are the limits on Q? For guided motion with precision roller bearings (or ball bushings, etc.), the maximum speed on the track is in the range of 3 to 5 m/s. Unguided motion is possible in an evacuated dropping tower. The tallest have a free-fall time of about 5 s. For longer unguided motion the free-falling system can be dropped inside an evacuated tube that is itself falling. For a 3 m tube dropped from 40 km, the free-fall time is about 30 s . Finally, ~ note that a spacecraft in low Earth orbit has a period of 1.5 hours. However, the STEP team plan to rotate the instrument so as to reduce the period of the EP signal and thus avoid the increased noise encountered at low freq~ency.~ On the other hand, were even longer free-fall periods desired, one could consider nearly Earth-synchronous rotation or spacecraft in solar orbit. For POEM, two pairs of TMA will be in free fall in a comoving vacuum chamber for about 0.8 s per “toss,” i.e., motion both up and down along a vertical path of about 90 cm. An additional vertical path of about 40 cm is used for the chamber to slow and reverse before transitioning to free fall (upward). This reversal of the moving vacuum chamber will take place in about 0.4 s using a “bouncer,” and the measurement will be repeated every 1.3 s. In this approach, there is no need of mechanisms inside the chamber to drive the motion of the TMA and the TMA-observing devices during each toss. However, it entails moving tens of kg at speeds approaching 5 m/s, which implies large forces and significant vibrational energy. The TMA of each pair inside the chamber are separated vertically by about 0.5 m, and the separation monitored by an SAO Tracking Frequency laser Gauge (TFG).4i5That gauge is expected to yield 10 pm for 7 = 10 ms (equivalent to 1 pm in 1s). The TFG‘ offers four advantages, all of which apply to POEM. First, it is free of the cyclic bias of the better-known

272

heterodyne gauges. That bias, which is of nm scale, has been reduced in some development projects, but yields a laser gauge that cannot be read out quickly (cf. the TFG that can read out at 100 kHz). Second, it can operate in a cavity and gain in precision in proportion to the finesse. Third, it can suppress polarization errors, which are also of nm scale. And fourth, the TFG can provide absolute distance at little additional cost or complexity. The POEM distance measurement uses a low finesse cavity formed by two solid glass retroreflectors and fed by an intra-cavity beamsplitter. With the expected precision, the derived relative acceleration from a single toss of a pair of TMA has a precision of 3 x 10-l' g. A one-hour run will have g on test g (on TMA; 2.1 x an acceleration error of 0.64 x substances for R = 0.3) due to the laser gauge. Thus, an experiment of the intended accuracy will be quick, and we will be able to test several (compositionally) different sample pairs. Among those substances being considered are Al, Be, Cu, U, and W. With rapid measurement, one can also investigate systematic errors and mitigate some kinds by repeating the experiment with altered conditions. Thus, we plan to: (a) Interchange the TMA horizontally, by robotic means, on a time scale probably about ten minutes. This will cancel the effect of the horizontal gravity gradient to within the precision of the interchanging of the locations of the TMA CM. (b) Interchange the TMA vertically in successive runs. This, along with the absolute laser gauging, will calibrate the effect of the dominant (vertical) gravity gradient. (c) Interchange payloads between TMA. This will address possible subtle differences in the optical and surface properties of the TMA. (d) Introduce temperature offsets (and gradients) and gravity gradients to demonstrate that their effects are acceptably small. (e) Clean components used inside the vacuum chamber and rerun to show that residual contamination does not introduce systematic error. The horizontal and vertical interchange of pairs of TMA address the largest single source of error, gravity gradient. In the left-right difference of acceleration of the two pairs of TMA, most of the effect of gradient cancels. -What remains is due to bias in the difference of the separations of their CM. The TFG will determine the absolute separation of the Optical Reference Points (ORP) of the TMA to 30 nm accuracy. The CM-ORP offset in each TMA will be adjusted, using TMA calibration tosses prior to a run, to be less than 2 pm. The difference of residual offsets comprises a bias in the absolute separation of CM. To remove this bias, the top and bottom TMA are swapped, and the difference of accelerations in the two orienta-

273

tions taken. Several effects limit the effectiveness of this cancellation, for example, the precision of the absolute distance measurements, and changes during the swap of gradient and of CM-ORP offset.

Modulated Light Entering Chamber (from beam launcher) Figure 1. Present POEM motion system (to left), and vacuum chamber (to right) with TMA and optical system

Figure 1 shows the principle components of POEM. A vacuum chamber is mounted to a cart that rides along a vertical track. Inside the track assembly is a linear motor (fixed magnets, moving coils mounted to the cart, and a position sensor) driven by a control unit that includes the power amplifier and a pair of computers. Conditioned laser light entering at the lower right reaches the beamsplitter, illuminating the optical cavity and then reaching the detector (upper right). The retroreflectors inside the TMA form the ends of the cavity and the compensator plate makes it possible to align the cavity in the presence of imperfect retroreflectors. The motion system performs three functions. First, the track ensures

274

that the chamber motion is linear to within a few 10s of pm, and with leveling of the track, that the motion is vertical. Second, during the free-fall portion of the motion cycle, the linear motor follows a frictionless trajectory. Third, following the free-fall portion of the motion, the "bouncer" reverses the chamber motion. In designing the bouncer, we balance track length and time required against the stress caused by large acceleration. Further, the change of acceleration needs to be smooth enough for the motor and its servo controller to follow without losing lock. The use of a bouncer addresses two related issues: what to do with the kinetic energy of the falling chamber, and how to supply the energy needed to send the chamber back up. Similarly, it reduces significantly the externally supplied power that would be dissipated close to the precision measurement components, and the size (and cost) of the motor and controller. Finally, this approach also is likely the fastest practical way of recycling the equipment between tosses.

SO cm

Figure 2. Original POEM bouncer, with vacuum chamber at left and ton of lead at right. Wire rope is the dotted line from Support at left, running over pulleys 1, 2, 3, 4, and the eccentric pulley t o the Support which is t o right of center.

The original bouncer used steel cables, pulleys, a 5:l lever, and a ton of lead as shown in Fig. 2. When the pulley mounted under the chamber first contacts the horizontal cable, there is no upward force, and only a

275

negligible mass must come quickly to the chamber’s speed ( ~ m/s.) 5 AS the moving chamber deflects the cable, the force grows. The Principle difficulty with this system is that the cable running over a (frictionless) pulley is dissipative; the cable strands rub against each other as the cable flexes. We found about 30% of the energy being lost in the bounce, which exceeded the motor capability. However, we did demonstrate the ability to launch a mass during the upward motion of the chamber. We are in the process of replacing the initial bouncer with one based on torsion bars. The new system maintains the horizontal cable and its associated smooth onset of force. The cable is reduced to a short section running between the ends of a pair of lever arms on the torsion bar ends. Excessive vibration comes from the wheels and their bearings. The slide (cart plus track assembly) is a commercial system that uses high quality precision ball bearings. A substantial reduction of vibration is not possible through a minor redesign of the cart. Therefore, we have moved up the long planned introduction of an air-bearing system to guide the motion of the chamber. The air bearing system is based on a pair of precision ground and plated hollow shafts (3” OD, 2” ID) supported on a steel structure. Riding on one shaft is a pair of cylindrical graphite air bushings. Together, they constrain four degrees of chamber-motion freedom: horizontal translations and rotations around horizontal axes. The remaining rotation is constrained by a third air bushing connected to the first two through a flexure that prevents overconstraint. The last degree of freedom - vertical motion - is constrained by connecting the chamber to the original cart. The connection is made using a “stinger” and contains a mechanical low-pass filter to reduce the transmission of vertical vibration. We are developing POEM in four stages. In Gen-I, a working freefall system will be completed, measurements taken on identical TMA with precision a(Ag)/g M 10-l’. The noise will be studied, but no science will result. In Gen-11, a differential system will be built (two pairs of TMA in free fall, and a second TFG), masses of different composition will be used, and the first science results will be obtained at an accuracy of a(Ag)/g < In Gen-111, automatic exchange of test masses will be implemented and errors reduced further to yield a(Ag)/g < 5 x The fourth stage is a spaceborne experiment with an accuracy goal of Ag/g = We expect to obtain the three order improvement by increasing the free-fall time from 1 s to between 10 and 30 s. Similarly, we will increase the laser gauge precision from 1 pm to between 0.1 and 0.01 pm (T = 1 s). Thus, these changes would increase the precision of the

276

experiment by a factor between lo3 and lo5. However, the real problem is systematic error, not sensitivity. Our preliminary design calls for as many as six pairs of TMA. As in the ground-based system, a series of interchanges is used to cancel systematic error. An improvement on the manual top-bottom interchange is obtained by rotating the chamber 180” about the along-track direction. Similarly, the equivalent of the robotic lateral interchanges is obtained by rotating the chamber around the Earth-pointing axis. Finally, these interchanges will be performed fast compared to the orbital period so as to reduce the contribution of “ l / f noise.” In operation, we would employ a five-step sequence corresponding to a single free fall of the terrestrial version: 1) Unclamp the TMA. 2) Use a capacitance gauge to sense the initial TMA state (position, velocity, orientation and rotation rate) and electrostatically force to precisely set the state. 3) Free fall, taking TFG data only. 4) Measure the final TMA state with the capacitance gauge. 5) Clamp the TMA. Acknowledgments We thank undergraduate students Brett Altschul (MIT), Giovanni DeSanti (Boston University), Dennis Feehan (Harvard), Jennifer Hoffman (Harvard), Alejandro Jenkins (Harvard), Naibi Marinas (U. Florida), and Brandon McKenna (Harvard) for skillful laboratory work. We thank Jim Faller (JILA), Robert Kimberk (SAO, Central Engineering), Tim Niebauer (Micro-g Corp.), and Doug Robertson (NGS/NOAA) for helpful discussions. This work was supported by the Smithsonian directly, through the SAO IR&D program, and through the SI Scholarly Studies Program. References T.M. Niebauer e t al., Metrologia 32, 159 (1995). E.C. Lorenzini et al., Nuovo Cimento 109B,1195 (1994). P. Worden, private communication (2003). M.C. Noecker et al., Proceedings of the SPIE Conference # 1947 on Spaceborne Interferometry (Orlando, FL, April 14-16, 1993), Vol. 1947, p. 174 (1993). 5. R.D. Reasenberg et al., “High Precision Interferometric Distance Gauge”, U S . Patent 5,412,474, issued 2 May 1995. 6. J.D. Phillips and R.D. Reasenberg, Proceedings of the SPIE Conference # 5495 on Astronomical Telescopes and Instrumentation (Glasgow, 21-25 June, 2004), in press; J.D. Phillips and R.D. Reasenberg, Review of Scientific Instruments, in press.

1. 2. 3. 4.

COMPOSITE MEDIATORS AND LORENTZ VIOLATION

ALEJANDRO JENKINS California Institute of Technology Pasadena, CA 91125, U.S.A. E-mail: [email protected]. edu We briefly review the history and current status of models of particle interactions in which massless mediators are given, not by fundamental gauge fields as in the Standard Model, but by composite degrees of freedom of fermionic systems. Such models generally require the breaking of Lorentz invariance. We describe schemes in which the photon and the graviton emerge as Goldstone bosons from the breaking of Lorentz invariance, as well as generalizations of the quantum Hall effect in which composite excitations yield massless particles of all integer spins. While these schemes are of limited interest for the photon (spin l), in the case of the graviton (spin 2) they offer a possible solution to the long-standing UV problem in quantum linear gravity.

1. Why local gauge invariance? The Dirac Lagrangian for a free fermion, C = q(zg-m)$, is invariant under the global U(1) gauge transformation $ H exp(ict)$. In the established model of quantum electrodynamics, this Lagrangian is transformed into an interacting theory by making the gauge symmetry local: the phase ct is allowed to be a function of the space-time point P.This requires the introduction of a gauge field A, with the the transformation property A, H A, d,a, and the use of a “covariant derivative” D, = a, - iA, instead of the usual derivative a,. The generalization to non-abelian gauge groups is well known, as is the Higgs mechanism to spontaneously break the gauge invariance and give the field A, a mass. A deeper insight into the physical meaning of local gauge invariance comes from realizing that a massless spin 1 particle, having no rest frame, cannot have its spin point along any axis other than that of its motion. Therefore, it has only two polarizations. By describing it as Lorentz vector A , (which has three polarizations) a mathematical redundacy is introduced. This redundancy is local gauge invariance. Something like it must appear in any Lorentz invariant theory of a massless spin 1 field coupled to matter. In general relativity, the graviton is a massless particle with two polariza-

+

277

278

tions, but it is described by a spin 2 field, which would ordinarily have five polarizations. This redundancy leads to diffeomorphism invariance, a symmetry analogous to local gauge invariance in the spin 1 case. (See, for instance, chapter 5.9 in Ref. 1 and chapter 111.3 in Ref. 2.) Modern particle theory is based on local gauge invariance, and it has been shown that gauge theories have the very attractive feature that they are always ren~rmalizable.~ But there is no clearly compelling a priori reason to impose local gauge invariance as an axiom. Also, it might appear unsatisfactory that our mathematical description of physical reality should be inherently redundant: local gauge invariance, unlike a true physical symmetry, does not mean that different physical configurations have the same behavior. Rather, it means that different field configurations represent exactly the same physics.2 Finally, local gauge invariance as a guarantee of renormalizability works only for spin 1. It is well known that quantizing h,, in linear gravity does not produce a perturbatively renormalizable field theory. 2. Goldstone photons

Before quantum chromodynamics (QCD), an S U ( 3 ) gauge theory, was accepted as a model for the strong nuclear force, Nambu and Jona-Lasinio (NJL) proposed a scheme in which protons and neutrons in nuclei would interact strongly by exchanging composite massless particles associated with the spontaneous breaking of the chiral symmetry $J H exp ( i ~ u y ~ )That $J.~ is, in their model, the pions were composite Goldstone bosons in a theory whose only fundamental fields were fermions. Shortly after the NJL model was published, Bjorken proposed using a similar idea to account for QED without postulating U(1) local gauge i n ~ a r i a n c eHe . ~ suggested ~~ that a theory with only self-interacting fermions might spontaneously break Lorentz invariance, yielding composite Goldstone bosons that could act as the mediators of the electromagnetic force. Conceptually, a useful way of understanding Bjorken’s proposal is to think of it as as a resurrection of the lumineferous “empty” space is no longer really empty.” Instead, the theory has a non-vanishing vacuum expectation value (VEV) for the current j , = Gyp$. This VEV, in turn, leads to a massive background gauge field A, 0: j,, as in the well-known aTaylor and Wheeler declare in Ref. 7 that one can think of Einstein’s special relativity (and therefore Lorentz invariance) simply as the statement that empty space is really empty.

279

London equations for the theory of superconductors.b Such a background spontaneously breaks Lorentz invariance and produces three massless excitations of A, (the Goldstone bosons) proportional to the changes Sj, associated with the three broken Lorentz transformations. Two of these Goldstone bosons can be interpreted as the usual transverse photons. The meaning of the third photon remains problematic. Bjorken originally interpreted it as the longitudinal photon in the temporalgauge QED, which becomes identified with the Coulomb force (see also Ref. 8). More recently, Kraus and Tomboulis have argued that the extra photon has an exotic dispersion relation and that its coupling to matter should be suppressed."

3. Goldstone gravitons The problem of the nonrenormalizability of linear gravity in the usual quantum field theory has been one of the major motivations for research in string theory, quantum loop gravity and other proposed theories of quantum gravity currently at the forefront of fundamental theoretical particle physics. An early suggestion for solving the problem of linear gravity in the UV was to make the graviton a composite degree of freedom of the low-energy regime. Weinberg and Witten, however, put an end to much of the speculation in this direction by using a strikingly simple argument to show that Lorentz invariant field theories with a Lorentz covariant energy-momentum tensor T,,, do not admit massless degrees of freedom, either fundamental or composite, with spin greater than 1.l' (GR is allowed to have a massless spin 2 particle, the graviton, because T,,, involves the field h,,,, which is not a Lorentz tensor.) It is, however, possible to invoke a mechanism similar to the one described by Bjorken in order to obtain composite gravitons as the Goldstone bosons of spontaneously broken Lorentz invariance. If a field h,, acquires a non-zero VEV, then the SO(3,l) Lorentz symmetry would be broken to nothing, generating six Goldstone bosons. The VEV of h,, can be thought of as proportional to some non-vanishing tensor bilinear in the background, -+

such as qC(7, d , -7,

+

a,,)$.''

The question of how to obtain such a VEV

Bjorken's work, A,, is just an auxiliary or interpolating field. Dirac had discussed somewhat similar ideas in an earlier paper,1° but, amusingly, he was trying to write a theory of electromagnetism with only a gauge field and no fundamental electrons. In both the work of Bjorken and the work of Dirac, the proportionality between A,, and j , is crucial.

280

remains pr0b1ematic.l~ Of the five massless excitations from the breaking of Lorentz invariance, two could be identified with the helicities of the graviton, while the other four should presumably have their interactions with matter suppressed. 4. Generalization of the quantum Hall effect In 1983, Laughlin explained the observed fractional quantum Hall effect in two-dimensional electronic systems by showing how such a system could form an incompressible quantum fluid whose excitations have charge e/3.14 That is, the low-energy theory of the interacting electrons in two spatial dimensions has composite degrees of freedom whose charge is a fraction of that of the electrons themselves. In 2001, Zhang and Hu used techniques similar to Laughlin’s to study the composite excitations of a higher-dimensional system.15 They imagined a four dimensional sphere in space, filled with fermions that interact via an S U ( 2 ) gauge field. In the limit where the dimensionality of the representation of S U ( 2 ) is taken to be very large, such a theory exhibits composite massless excitations of integer spin 1, 2 and higher. Like other theories from solid state physics, Zhang and Hu’s proposal falls outside the scope of Weinberg and Witten’s theorem because the proposed theory is not Lorentz invariant: the vacuum of the theory is not empty and has a preferred rest-frame (the rest frame of the fermions). However, the authors argued that in the three-dimensional boundary of the fourdimensional sphere, a relativistic dispersion relation will hold. One might then imagine that the relativistic, three-dimensional world we inhabit might be the edge of a four-dimensional sphere filled with fermions. Photons and gravitons would be composite low-energy degrees of freedom, and the problems currently associated with gravity in the UV would be avoided. The authors also argue that massless bosons with spin 3 and higher might naturally decouple from other matter, thus explaining why they are not observed in nature. 5. Outlook The model of the strong interactions proposed by Nambu and Jona-Lasinio was eventually superseded by QCD, and Bjorken’s proposal for composite photons was similarly overtaken by the steady rise of local gauge invariance as a sacred principle of theoretical particle physics. But NJL survived, in modified form, as the basis of chiral perturbation theory, and it is possible that Bjorken’s model might make a comeback, in the way that the old

281

Kaluza-Klein model was resurrected by modern theories with extra dimensions, such as string theory. A theory with composite gravitons currently seems especially appealing, given the interest in addressing the UV problems of gravity in the the quantum field theory context. Much work remains to be done in this area. One interesting issue is whether models of composite massless mediators might be associated with observable violations of Lorentz invariance. References 10, 5 , and 8 claimed that the proposed Lorentz violation was purely formal and had no observable consequences, since it appeared only as a VEV of A,, which could be gauged away. However, it has recently become apparent that in many such models the Lorentz violation is p h y ~ i c a l ~ and *" associated with the presence of an "iether" given by a non-empty Dirac sea of fermions in the background that also introduces a chemical p ~ t e n t i a l . ~ > l ~ On the other hand, the model proposed in Ref. 15 rescues special relativity in the boundary region in which they imagine our universe is located.

Acknowledgments The author thanks J.D. Bjorken and Y. Nambu for fruitful exchanges. This work was supported in part by an R.A. Millikan graduate fellowship from the California Institute of Technology.

References 1. S. Weinberg, The Quantum Theory of Fields, Vol. 1, Cambridge University Press, Cambridge, England, 1996. 2. A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, Princeton, NJ, 2003. 3. G. 't Hooft, Nucl. Phys. B33, 173 (1971); 35, 167 (1971). 4. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122,345 (1961); 124,246 (1961). 5. J.D. Bjorken, Ann. Phys. (N.Y.) 24, 174 (1963). 6. J.D. Bjorken, hep-th/Olll196. 7. E.F. Taylor and J.A. Wheeler, Spacetime Physics, 2nd ed., W.H. Freeman, New York, 1992. 8. Y. Nambu, Prog. Theor. Phys. extra num., 190 (1968). 9. Y. Nambu, in CPT and Lorentz Symmetry 11, ed. V.A. Kosteleck9, World Scientific, Singapore, 2002; J. Statis. Phys. 115, 7 (2004). 10. P.A.M. Dirac, Proc. Roy. SOC.,209, 291 (1951). 11. P. Kraus and E.T. Tomboulis, Phys. Rev. D 66, 045015 (2002). 12. S. Weinberg and E. Witten, Phys. Lett. 96B, 59 (1980). 13. A. Jenkins, Phys. Rev. D 69, 105007 (2004). 14. R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 15. S.C. Zhang and J. Hu, Science 294, 823 (2001).

LORENTZ-VIOLATING ELECTROMAGNETOSTATICS

QUENTIN G. BAILEY Physics Department Indiana University Bloomington, IN 47405, U.S.A. E-mail: [email protected] In this talk, the stationary limit of Lorentz-violating electrodynamics is discussed. As illustrated by some simple examples, the general solution includes unconventional mixing of electrostatic and magnetostatic effects. I discuss a high-sensitivity null-type measurement, exploiting Lorentz-violating electromagnetostatic effects, that could improve existing limits on parity-odd coefficients for Lorentz violation in the photon sector.

1. Introduction Experiments to date have shown that Lorentz symmetry is an exact symmetry of all known forces in nature. However, many ongoing experiments are searching for small violations of Lorentz symmetry that could arise in the low-energy limit of a unified theory of nature a t the Planck scale.' Much of the analysis of these experiments is performed within a theoretical framework called the Standard-Model Extension (SME).2 The SME is an effective field theory that extends the Standard Model (SM) and general relativity to include small violations of particle Lorentz and C P T symmetry while preserving observer Lorentz symmetry and the coordinate invariance of physics. The C P T and Lorentz-violating terms in the SME lagrangian have coupling coefficients with Lorentz indices which control the Lorentz violation, and can be viewed as low-energy remnants of the underlying physics at the Planck scale.3 Tests of this theory include ones with photon^,^^^^^^^^^ electron^,^ protons and neutrons,1° mesons,ll muons,12 neutrinos,13 and the Higgs.14 In the photon sector of the minimal SME, recent Lorentz symmetry tests have focused on the properties of electromagnetic waves in resonant cavities and propagating in vacuo. I show in this talk, however, that there are unconventional effects associated with the stationary, non-propagating limit 282

283

of the photon sector. I also discuss experimental possibilities based on these effects in high-sensitivity null-type measurements. A detailed discussion of this topic is contained in Ref. 16.a 2. Framework The lagrangian density for the photon sector of the minimal SME can be written as , f “ - - F1 F P - 14 (kF)nXpvFKXFp” 4 -jpA,. (1) +~(kAF)KEKXpvAXFpu

f)

In this equation, f’ = ( p , is the 4-vector current source that couples to the electromagnetic 4-potential A,, and F,” = a,A, - &A, is the electromagnetic field strength. From this definition the conventional homogeneous Maxwell equations are automatically satisfied. The coefficients (~ F ) & x , ” and ( k A F ) & are assumed constant and control the CPT and Lorentz violation. The current j , is taken to be conventional, thus assuming Lorentz violation is only present in the photon sector. The CPT-odd coefficients ( I C A F ) ~ are stringently bounded by cosmological observations and are set to zero in this a n a l y ~ i sThe . ~ lagrangian (1) yields the inhomogeneous equations of motion

,FPa

+ (kF),ap7aaF’Y + j ,

= 0.

(2)

These+ equations can be written as Maxwell equations in terms of 6 ,I?, E’ and B by defining appropriate vacuum constituency relations.6*16Equation ( 2 ) can be generalized to include regions of isotropic matter. The usual linear response of matter to applied fields is modified by Lorentz violation and additional matter coefficients appear in the constituency relations. l6 3. Electromagnetostatics

The stationary solutions of the modifed Maxwell equations (2) in will satisfy the time-independent equation of motion

where the coefficients ,@PkV are defined by

aSee Ref. 15 for theoretical literature on the photon sector of the SME.

DQCUO

284

From the homogeneous Maxwell equations the electrostatic and magnetostatic fields can be written in terms of the 4-potential A” = ( @ , A J )as E’ = -+@ and 2 = 9 x 2.The metric terms in Eq. (4) are the conventional terms that split (3) into separate equations for the scalar potential from charge density and the vector potential from current density. The presence of the ( I c ~ ) J fterm i ~ ~ implies that a static charge density generates a small vector potential and a modified scalar potential and similarly a steady-state current density generates a small scalar potential and a modified vector potential. Electrostatics and magnetostatics, while distinct in the conventional case, become convoluted in the presence of Lorentz violation. Discussing the static limit of Lorentz-violating electrodynamics therefore requires the simultaneous treatment of both electric and magnetic phenomena. l7 To obtain a general solution for the potentials @ and A’ I introduce Green functions Gpa(.’,.‘’) that solve Eq. (3) for a point source. Once a suitable Green theorem that incorporates the differential operator in Eq. (3) is found, the formal solution can be constructed for a spatial region V in terms of the Green functions, the 4-current density and the values of the potential on the boundary S. The general solution is

- ~ ~ ( i ? ) i F ” ~ ~ d f e ~ .’)I. ,~(i?,

(5)

Manipulation of Eq. (5) reveals four classes of boundary conditions that establish unique solutions for the electric and magnetic fields: (@,fi x A), (@,fix (fi.6, Ax (fi.6, fi x I?). With each of these sets of boundary conditions there are corresponding constraints on the Green functions.16 The mixing of @ and A’ in the boundary conditions comes from the unconventional definitions of the fields 3 and The solution (5) can be generalized to regions of isotropic matter using a modified version of Eq.

I?),

A),

I?.6316

(4).16

4. Applications

As a first application of Eq. (5) I consider the case of boundary conditions at infinity in which the surface terms are dropped. Imposing the Coulomb gauge, the explicit form for the Green functions can be extracted from

285

fourier decomposition in momentum space.16 For the case of a point charge at rest at the origin the scalar potential18 and vector potential are given by

Equation (6) shows explicitly that a point charge at rest produces a magnetic field in the presence of Lorentz violation, which is obtained from Bj = EjkldkA1.16 Consider now an example motivated by a possible experimental application. I seek the fields from a magnetic source surrounded by a conducting shell. In the idealized solution presented the magnetic source is a sphere of radius a and uniform magnetization 2 surrounded by a grounded conducting shell of radius R > a. The fields for this configuration can be obtained set of boundary conditions and treating the from (5) using the (@,fi x magnetic source as a current density J'= d x M . The leading order solution for the scalar potential @ in the region u < T < R, where T is the radial coordinate from the center of the sphere, is given by

A)

i . R,,

@(Z)=

47r

6

($-$),

(7)

where 6 = 47ra3M/3. Here we have made use of the zero-birefringence p qan anti-symmetric matrk6yl6 approximation that (Ro+)jk = ( k ~ ) O j P Q c ~ is The solution (7) becomes modified in the more realistic scenario with the magnet consisting of matter obeying Lorentz-violating matter constituency relations. l6 5. Experiment

Recent experiments in the photon sector are least sensitive to Ro+ and R.tr = - $ ( k ~ ) O j ' j . This is due to the parity-odd nature of the corresponding Lorentz-violating effects from Ro+ and the scalar nature of Xt, to which recent experiments are only indirectly sensitive. The setup of the second example in Sec. 4 is designed to be directly sensitive to parity-odd effects. It can be seen directly from (7) that the scalar potential, if taken to be the observable, is proportional to R,+ .I9 A suitable experiment would measure the potential from Eq. (7) in the space between the magnet and outer shell ( a < T < R). The outer conducting shell then serves to sheild the apparatus from external electric fields. For an estimate of the sensitivity that might

286

be attainable I assume the source is a ferromagnet with strength lo-’ T near its surface and the voltage sensitivity is at the level of nV. A null measurement could then achieve a sensitivity k,+ <, This represents an improvement by lo4 over the best existing sen~itivities.~ Equation (7) is written in the laboratory frame. Since this frame is fixed to the earth it is not inertial on the time scale of the earth’s rotation and revolution. The resultant time dependence of the signal can be obtained by transforming the laboratory-frame coefficients (k,,+){:,, to a Sun-centered inertial frame following Ref. 6 . Thus, with upper-case letters denoting Suncentered coordinates,

where T J k J K= RjJRkKand T!kJK= RjPRkJfKPQPQ are tensors containing the time dependence from the rotations RjJ and the boost ,@. One can also consider rotating the entire apparatus to produce a signal with a shorter time variation, which may increase sensitivity and reduce systematics. With these considerations one can attain time-dependent sensitivity to all three independent components of it,+ and time-dependent sensitivity to kt, suppressed by a single power of lp’l N_

References 1. For summaries of recent Lorentz tests, see, for example, V.A. Kostelecki, ed., CPT and Lorentr Symmetry I l , World Scientific, Singapore, 2002. 2. D. Colladay and V.A. Kostelecki, Phys. Rev. D 55,6760 (1997); Phys. Rev. D 58, 116002 (1998); V.A. Kostelecki, Phys. Rev. D 69,105009 (2004). 3. V.A. Kosteleckf and S. Samuel, Phys. Rev. D 39,683 (1989); V.A. Kosteleckf and R. Potting, Nucl. Phys. B 359,545 (1991). 4. J. Lipa et al., Phys. Rev. Lett. 90,060403 (2003). 5. H. Miiller et al., Phys. Rev. Lett. 91,020401 (2003); P. Wolf et al., Gen. Rel. Grav. 36,2351 (2004); Phys. Rev. D, Rapid Communications, in press (hep-ph/0407232). 6. V.A. Kostelecki and M. Mewes, Phys. Rev. D 66,056005 (2002). 7. S.M. Carroll et al., Phys. Rev. D 41, 1231 (1990); M.P. Haugan and T.F. Kauffmann, Phys. Rev. D 52,3168 (1995); 8. V.A. Kostelecki and M. Mewes, Phys. Rev. Lett. 87,251304 (2001). 9. H. Dehmelt et al., Phys. Rev. Lett. 83, 4694 (1999); R. Mittleman et al., Phys. Rev. Lett. 83,2116 (1999); G. Gabrielse et al., Phys. Rev. Lett. 82, 3198 (1999); R. Bluhm et al., Phys. Rev. Lett. 82, 2254 (1999); Phys. Rev. Lett. 79,1432 (1997); Phys. Rev. D 57,3932 (1998); D.Colladay and V.A. Kosteleckf, Phys. Lett. B 511, 209 (2001); B. Heckel, in Ref. 1; L.-S. Hou

287

et al., Phys. Rev. Lett. 90,201101 (2003); R. Bluhm and V.A. Kostelecki, Phys. Rev. Lett. 84,1381 (2000); H. Miiller et al., Phys. Rev. D 68,116006 (2003); B. Altschul, hep-ph/0405084; 10. L.R. Hunter et al., in V.A. Kosteleckf, ed., CPT and Lorentz Symmetry, World Scientific, Singapore, 1999; D. Bear et al., Phys. Rev. Lett. 85,5038 (2000); D.F. Phillips et al., Phys. Rev. D 63,111101 (2001); M.A. Humphrey et al., Phys. Rev. A 68,063807 (2003); Phys. Rev. A 62,063405 (2000); F. Can& et al., physics/O309070; V.A. Kostelecki and C.D. Lane, Phys. Rev. D 60,116010 (1999); J. Math. Phys. 40, 6245 (1999); R. Bluhm et al., Phys. Rev. Lett. 88, 090801 (2002); Phys. Rev. D 68,125008 (2003). 11. KTeV Collaboration, in Ref. 1; OPAL Collaboration, Z. Phys. C 76, 401 (1997); DELPHI Collaboration, preprint DELPHI 97-98 CONF 80 (1997); BELLE Collaboration, Phys. Rev. Lett. 86,3228 (2001); BaBar Collaboration, hep-ex/0303043; FOCUS Collaboration, Phys. Lett. B 556, 7 (2003); V.A. Kostelecki and R. Potting, Phys. Rev. D 51,3923 (1995); V.A. Kostelecki, Phys. Rev. Lett. 80, 1818 (1998); Phys. Rev. D 61,016002 (2000); Phys. Rev. D 64,076001 (2001). 12. V.W. Hughes et al., Phys. Rev. Lett. 87, 111804 (2001); R. Bluhm et al., Phys. Rev. Lett. 84,1098 (2000). 13. S. Coleman and S.L. Glashow, Phys. Rev. D 59,116008 (1999); V. Barger et al., Phys. Rev. Lett. 85,5055 (2000); J.N. Bahcall et al., Phys. Lett. B 534, 114 (2002); I. Mocioiu and M. Pospelov, Phys. Lett. B 537,114 (2002); A. de Gouvka, Phys. Rev. D 66,076005 (2002); G. Lambiase, Phys. Lett. B 560, 1 (2003); V.A. Kostelecki and M. Mewes, Phys. Rev. D 69,016005 (2004); Phys. Rev. D, in press (hep-ph/0308300); hep-ph/0406255; S. Choubey and S.F. King, Phys. Lett. B 586,353 (2004); A. Datta et al., hep-ph/0312027. 14. D.L. Anderson et al., hep-ph/0403116; E.O. Iltan, hep-ph/0405119. 15. R. Jackiw and V.A. Kostelecki, Phys. Rev. Lett. 82, 3572 (1999); M. PiezVictoria, JHEP 0104, 032 (2001); V.A. Kosteleckf et al., Phys. Rev. D 65,056006 (2002); C.Adam and F.R. Klinkhamer, Nucl. Phys. B 657,214 (2003); V.A. Kosteleckjr et al., Phys. Rev. D 68, 123511 (2003); H. Miiller et al., Phys. Rev. D 67,056006 (2003); T. Jacobson et al., Phys. Rev. D 67, 124011 (2003); V.A. Kostelecki and A.G.M. Pickering, Phys. Rev. Lett. 91, 031801 (2003); R. Lehnert, Phys. Rev. D 68, 085003 (2003); G.M. Shore, Contemp. Phys. 44,503 2003; B. Altschul, Phys. Rev. D 69,125009 (2004); hep-t h/0402036; hep-t h/047172; hep-t h/0403093; R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004); hep-ph/0408285; R. Lehnert, J. Math. Phys. 45,3399 (2004). 16. Q.G. Bailey and V.A. Kostelecki, Phys. Rev. D, in press (hep-ph/0407252). 17. These effects occur in CNC Lorentz violation. See C. Lammerzahl, CPT04 proceedings. 18. The scalar potential was first obtained in Ref. 6. 19. For an alternate method see P. Wolf et al., hep-ph/0408006; M. Tobar, CPT04 proceedings.

LORENTZ VIOLATION IN SUPERSYMMETRIC FIELD THEORIES

M.S. BERGER Indiana University Physics Department Bloomington, I N 47405, U.S.A. E-mail: bergerOindiana.edu Broken spacetime symmetries might emerge from a fundamental physical theory. The effective low-energy theory might be expected to exhibit violations of supersymmetry and Lorentz invariance. Some illustrative models which combine supersymmetry and Lorentz violation are described, and a superspace formulation is given.

1. Introduction There has been an increasing realization in recent years that the Lorentz and Poincarh symmetries assumed almost universally in models of particle physics might in fact be approximate symmetries that emerge from some more fundamental theory of quantum gravity. Other potential spacetime symmetries such as supersymmetry have yet to be uncovered and would have to be broken symmetries if we are to reconcile them with experimental physics. The interesting questions then involve how is the scale of the symmetry breaking determined in each case. Viewed from the Planck scale, the scale of Standard Model symmetry breaking and electroweak-scale supersymmetry are’very small. If the Lorentz symmetry is indeed broken, one of the most pressing issues would be to understand why the size of the physical effects are so incredibly tiny to have escaped all efforts to observe them experimentally. There seem to be at least some parallels in the violations of these spacetime symmetries, which has motivated some preliminary investigations to understand any possible connection between them. Of course, since a fully solvable theory of quantum gravity is not available, the issue cannot be addressed directly. Rather one must take a more phenomenological approach, allowing for all possible effects that are consistent with the remaining symmetries of the theory which might be 288

289

either exact or spontaneously broken. The approach taken in Ref. 1 is similar in spirit to the Minimal Supersymmetric Standard Model (MSSM) where supersymmetry breaking terms are added to the Standard Model where all particle fields have been expanded to include supermultiplets. Adding terms to a supersymmetric model that break the Lorentz symmetry while preserving the supersymmetry can be accomplished by modifying (deforming) the supersymmetric algebra and the supersymmetric transformation, or less generally one can leave the supersymmetric transformation unmodified. In the extensions to the Wess-Zumino model described below the supersymmetric algebra is modified, so the important issues are whether the algebra closes for the supersymmetric transformations when they are applied to the fields in the model. The approach is also in the spirit of the Standard-Model Extension (SME)2>3where Lorentz (and CPT) violating terms are introduced into the Standard Model Lagrangian. When one adds the requirement of supersymmetry, there emerge relationships between the Lorentz-violating coefficients in a fashion similar to how masses and couplings become related in a conventional case (MSSM, for example). In Refs. 1 all possible Lorentz-violating terms were added to the Wess-Zumino model which is a theory involving only a single chiral supermultiplet. These simple models do admit a superspace formulation: and this motivates future systematic studies in more realistic and interesting supersymmetric models. When supersymmetric particles are discovered at colliders, is it possible that Lorentz-violating effects could be experimentally interesting? If the effects are as suppressed as they appear to be for the observed particle content of the Standard Model, then it will be impossible to observe any new effects. In principle the Lorentz-violating effects could arise from terms in the Lagrangian involving so far unobserved superpartners to the Standard Model particles. From the point of view of phenomenology, this would mean that there are terms in the low-energy Lagrangian that violate both supersymmetry and the Lorentz ~ymrnetry.~ These terms could be less suppressed than the analogous terms in the SME. Presumably physical effects will appear radiatively in Standard Model physics, and bounds can be derived using existing bounds. It is well-known that the assumption of Lorentz invariance is needed in quantum field theory to avoid problems with microcausality. Field theories with Lorentz-violating terms should be regarded as effective theories and the issues involving microcausality will be addressed when the full character of the underlying fundamental theory emerges at the Planck scale.6

290

While the supersymmetric theories described here should be regarded as toy models, the experimental implications of Lorentz and CPT violation parameterized in this manner have been explored extensively in recent years.7 2. Superspace

Lorentz violation has been studied using superfields defined on superspace. Superspace is defined in terms of spacetime and superspace coordinates*

,P= (xp,e", &) ,

(1)

where 8" and gb each form two-component anticommuting Weyl spinors. A superfield @(z,e,e) is then a function of the commuting spacetime coordinates xp and of four anticommuting coordinates 8" and 8 b . A chiral superfield is a function of yp = x p i8a@ and 8. Since the expansion in powers of 8 eventually terminates this can be expanded as follows

+

@b, 878) = 4(Y) + JZe+(Y)+ ( @ ) F ( Y ) 1 = 4 ( ~+ ) ie&%p4(s) - ,(ee)(e6)o4(x) 7

+ iJZe&ea,+(x) + (ee)3(x). The chiral superfield can be described in terms of a differential operator which is defined as

uxZE ,ax ,

(2)

Ux (3)

where

x = (e&)a,. Then an expansion of

(4)

Uxyields

U,

=

1

+ i ( e a p L e ) a p - i(ee)(ee)o 4 .

(5)

This operator effects a shift x p 4 yp. Since the chiral superfield @(x, 8,e) is a function of yp and 8 only, the only dependence on 8 is in yp, so it must then be of the form @(s, 8, = U z q ( x ,8) for some function D! which depends only on x p and 8. The first supersymmetric model with Lorentz and CPT violation involved extending the Wess-Zumino model.g The Wess-Zumino Lagrangian can be derived from the superspace integral

e)

291

where the conjugate superfield is

@*(x, 8, e) = +*(z)+ JZeq(z)

+ (@)F(Z)

,

(7)

s

where z p = yp* = x p - i8d‘e. The superspace integral over d48 projects out the (&I)(@) component of the @*asuperfield while the s d 2 0 projects out the 88 component of the superpotential. The result

Lwz

=

a,dJ*av + ;[(a,!b).”ll+ (a,4)8”+]+ F*F

[w+ 4*F*- $+$- f44] +9 [d2F+ 4*2F*- 4(lcl!b) - 4*(dll)], +m

(8)

is a Lagrangian which transforms into itself plus a total derivative under a supersymmetric transformation. The procedure just outlined is well-known and forms a basis for constructing Lorentz-violating models involving chiral superfields.

3. Lorentz Violation Two Lorentz-violating extensions to the Wess-Zumino model were found,’ and these two models admit a superspace forrn~lation.~ Define new operators that can act on superfields as

where

The expansions are

Uy= 1+ ik,U(8d‘8)dv

1

-

-k,,k’Lp(e8)(~O)av~p 4 ,

(13)

k2

Tk = 1 - k,(Bd‘B)

+ -((ee)(ee) 4 .

(14)

Here k,, and k, are Lorentz-violating coefficients that transform under observer Lorentz transformations but do not transform (or transform as a scalar) under particle Lorentz transformations. They therefore represent possible descriptions of physically relevant effects. Since Y ,like X , is a derivative operator, the action of U, on a superfield S is a coordinate shift.

292

The appearance of terms of order O ( k 2 )in the Lagrangians is easily understood in both cases in terms of these operators. Furthermore we have Ui = while T i = T k and not its inverse. The supersymmetric models with Lorentz-violating terms can be expressed in terms of new superfields,

u;’

@&, 8,e) = u,uxxq~, 6) , a ; ( $ , e,e) = u,-~u;~Q*(Ic,

(15)

e) .

(16)

Applying U, to the chiral and antichiral superfields merely effects the substitution 8, -, a, lcPva”.Since U, involves a derivative operator just as U,, the derivation of the chiral superfield a, is a function of the variables xP+ - x P ieaP8 ikPV8a,8 and 8 analogous to how, in the conventional

+

+ +

case, @ is a function of the variables yP and 8. The Lagrangian is given by

=/d

+

1

+ h.c.

4 8 [Ui@*][Uy@]

For the CPT-violating model the superfields have the form

.

(17)

It is helpful to note that the transformation U, acts on D ! and its inverse U;’ acts on Q*, while the same transformation T k acts on both Q and @* (since 7’;= T k ) . A consequence of this fact is that the supersymmetry transformation will act differently on the components of the chiral superfield and its conjugate. Specifically the chiral superfield @ k is the same as @ with the substitution 8, -, a, ik, whereas the antichiral superfield @; is the same as @ * with the substitution 4 - ikP. The CPT-violating model can then be represented in the following way as a superspace integral:

+

1

a,

d48@;@k =

a,

d48@*C2K@

(20)

Unlike the CPT-conserving model, the (e8)(@) component of @*@ no longer transforms into a total derivative. A specific combination of components of @*@ does transform into a total derivative, and this combination is in fact the (ee)(ee)component of @ ; @ k . The Lagrangians for the two models in terms of the component fields can be found in Refs. 1, 4.

293

4. Conclusions

Lorentz-violating extensions of supersymmetric theories model can be understood in terms of analogous transformations on modified superfields and projections arising from superspace integrals. Such superspace formulations should allow efficient investigations into possible Lorentz violation in more complicated theories. Acknowledgments

This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-91ER40661. References 1. 2. 3. 4. 5.

6. 7. 8. 9.

M. S. Berger and V . A. Kostelecky, Phys. Rev. D 6 5 , 091701 (2002). D. Colladay and V. A. Kostelecki, Phys. Rev. D 55, 6760 (1997). D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998). M. S. Berger, Phys. Rev. D 68, 115005 (2003). H. Belich, J. L. Boldo, L. P. Colatto, 3. A. Helayel-Net0 and A. L. Nogueira, Phys. Rev. D 6 5 , 065030 (2003). V . A. Kostelecki and R. Lehnert, Phys. Rev. D 63, 065008 (2001). See, for example, V.A. Kostelecki, ed., CPT and Lorentz Symmetry 11,World Scientific, Singapore, 2002. A. Salam and J. Strathdee, Nucl. Phys. B 7 6 , 477 (1974). J. Wess and B. Zumino, Nucl. Phys. B 70, 39 (1974).

SEARCHING FOR CPT VIOLATION AND MISSING ENERGY IN POSITRONIUM ANNIHILATION

P.A. VETTER Nuclear Science Division, Lawrence Berkeley National Laboratory One Cyclotron Road, MS 88R0192 Berkeley, California 94 720 E-mail: pavetterO1bl.gov Many experiments on the annihilation of positronium have been searched for violations of Lorentz invariance and C P T symmetry. These experiments have not yet been analyzed within the framework of the Standard-Model Extension, although such calculations should be straightforward. Several observables in the annihilation process of Ps are sensitive to violations of CPT. There are current proposals for experiments t o measure the very rare decay mode of Ps, which would be sensitive to Randall-Sundrum extra dimensions.

Positronium, the (e+e-) atom, is arguably the “most symmetric” bound system. Bound by a central potential, Ps is an eigenstate of the parity operator, P. As a particle/antiparticle system, Ps is an eigenstate of the charge conjugation operator, C. Ps is therefore a unique laboratory for testing the discrete fundamental symmetries C, P, and T. In positronium, the C eigenvalue is Cp, = (-l)L+s, where L is the orbital and S the total spin angular momentum. The photon is intrinsically C-odd, so the eigenvalue for an n-photon state is C, = (-l)n. If charge conjugation symmetry is respected in the QED process of Ps annihilation, then positronium must decay to an even or odd number of photons such that (-l)L+s = (-l)n. The ground state of positronium is the spin 0 singlet IS0 (para-positronium), while the first excited state is the “metastable” 3S1 (ortho-positronium). In experiments, the two states can be distinguished since para-Ps has a lifetime against two-photon annihilation of 125 ps, while ortho-Ps, which annihilates to three photons, has a lifetime of 142 ns. A timing start signal can be generated at the emission of a positron, and the detection time of annihilation gamma rays allows separation of annihilation events into two populations - short timed p-Ps events and long timed 0-Ps events (with a correction from pick-off spin conversion of 0-Ps to p-Ps). 294

295

There have been many measurements to search for C violation in positronium annihilation by seeking decays of ortho-Ps and para-Ps to the “wrong number” of final state gamma rays.’ Theories of Beyond Standard Model physics provide new motivation for further searches for C-violating decay modes. Low-energy limits of string and brane theories can imply non-commutative effective field theories. Non-commutivity is intimately related to local Lorentz invariance violation. In non-commutative extensions of QED, direct photon-photon vertex couplings are allowed which do not preserve photon number. This destroys the usual odd-even selection rule for Ps decay, and would result (in lowest order) in the decay mode (p-Ps + 3y), with a three-photon energy spectrum distinct from the Ore-Powell 0-PS gamma ray energy spectrum.2 The most recent C violation test was performed using the Gammasphere detector array at Lawrence Berkeley National L a b ~ r a t o r y Gammasphere .~ is a highly segmented array of shielded high-purity germanium detectors. It has a high detection efficiency for high-multiplicity gamma ray events, and is an ideal tool to search for decays of Ps to three, four, or five photons. The experiment set a limit for the branching ratio of ortho-Ps to four photons: R$‘ rO-PS+4y r)o-ps--r37)) < 3.7 x lop6. This experiment also set the smallest ~

limit for any exotic decay of Ps, searching for (p-Ps + 5y), finding Rf < 2.7 x This is the smallest limit for an exotic branching ratio of Ps, but it may be specious. It seems unlikely that any tenable theory would predict a C-odd decay of (p-Ps 4 57) while not allowing other C-violating behavior (evading other limits), by virtue of cancellations at lower orders of Q. The decay of ortho-Ps has been extensively studied and has spawned useful fields of applied physics. When such well-developed experimental technique exists, attention often turns to tests of Lorentz invariance (regardless of theoretical motivation) by searching for shifts in observed signals as a function of time or spatial orientation as the laboratory moves with respect to the fixed stars. There has been one search for Lorentz invariance violation in (0-Ps + 3y), motivated by the ortho-positronium lifetime p u ~ z l e This . ~ was a relatively simple experiment which could be improved, particularly by using a rotating platform. Further theoretical work would be useful for suggesting ranges of sensitivity to Standard-Model Extension parameters. However, it seems unlikely that this experiment (at a relatively modest level of sensitivity of lop3 of the decay rate of 0-Ps for the size of a sidereal variation of conversion of 0-Ps to p-Ps through a spin-coupling to a Lorentz violating vector field) would be competetive in sensitivity to

296

SME parameters with other atom-based tests of Lorentz invariance. Since positronium is an eigenstate of CP, it can provide a decay system test of the operator CPT, with the usual caveat that this is the naive T operator which does not exchange initial and final states, but simply reverses momentum and spin directions. Theoretical implications of the CPT-odd triple correlation in the decay of polarized 0-Ps: s’.

(Gx G)

were studied in Ref. 5, and three experiments (Refs. 6 , 8, 9) have searched for such a correlation. However, a reappraisal of positronium observables within the new Standard-Model Extension formalism seems desirable. Interest in CPT violating theories motivated the Berkeley Gammasphere experiment on CPTg and a new effort on CP-violation by a positronium group in Z ~ r i c h . ~ In Eq. (l),S’is the spin vector of the ortho-positronium, and and are the momenta of the two most energetic annihilation photons. The quantity kl x k2 defines a vector normal to the decay plane. The CPTodd correlation is an up/down asymmetry of decay planes with respect to the spin direction of the 0-Ps. Two early experiments searched for this decay correlation at a level of about 2% of the total decay rate by searching for an up-down asymmetry in planar arrays of sodium iodide decay photon detectors as the spin direction of an 0-Ps source was reversed.‘Y8 These experiments searched for an amplitude C Aof the decay observable S: kl x k z in the otherwise symmetric distribution of annihilation. The experiments detected the most energetic photon y1 in one detector, and searched for an asymmetry between two detectors in the count rate of the second most energetic photon 72.Labelling the detectors up and down, these experiments measured the asymmetry A = (Nup- Ndown)/ (Nup N d o w n ) . If the average polarization of the 0-Ps was ( P ) ,then the angular correlation between spin and decay plane (CA)is derived from the measured count asymmetry by CA = A / ( P ) . The detector geometry of these experiments determines the sensitivity to the correlation (1). In the Gammasphere experiment (Ref. 9), all three annihilation photons y l , 72, and y3 were detected, and the decay plane reconstructed to calculate its orientation with respect to the initial spin axis. Because any of the detectors in the array could detect any of the three photons, and because it could detect decay planes at any orientation with respect to the spin (rather than merely parallel or antiparallel), this experiment was less sensitive to geometric asymmetries of the

6

(-

‘1

(-

+

’>

297

counter arrangement or Ps source, or to unequal detector efficiencies. Reference 9 improved the limit on the correlation in Eqn. 1 to CA < 3 x (la). This experiment used a large data set and long acquisition time, but the data were not explicitly analyzed for sidereal variation. Another observable suggested in Ref. 5 is CP-odd:

( 3 . k;) ( 3 . k; x

&) .

To search for this decay mode, aligned 0-Ps must be used. An experiment at University of Michigan used a Ps source in a magnetic field and an array of three NaI detectors, setting a limit for a CP-odd decay amplitude Ccp < 1.5% (1g).lo In a proposal by a group at Zurich to search for the correlation of Eq. 2, the apparatus described in Ref. 11 (previously used to search for “missing energy” Ps decay modes to invisible particles) will be used, and the sensitivity could be as much as one hundred times better.7 The improvement would primarily be due to using a segmented 47r array. The experiments performed to date to search for violations of Lorentz invariance, C and CPT violation have not been analyzed within the framework of the Standard-Model Extension.” Reference 13 pointed out that few predictions of the SME effective QED model for scattering or decay processes (such as Ps) have been evaluated, since most low-energy searches for CPT or Lorentz violating observables focus on comparative matter/antimatter properties and high precision tests of Lorentz invariance in QED. Reference 13 treated relativistic (e+ e- + 2 7 ) scattering, concluding that under the Standard-Model Extension, new terms are introduced in the cross section. These terms have angular dependence on the momentum variables in the scattering plane, and acquire the expected siderial time dependence. In this process, the new terms are sensitive to the czj components, from the new term yc’, in the momentum term in the SME lagrangian. Reference 13 averaged over spin states of the electron, positron, and photons. The experiment to search for Lorentz violation in 0-Ps annihlation essentially searched for an anomalous conversion of 0-Ps to p-Ps and thus a time variation of the yield of two photon annihlations from a positronium source. The coupling would be to both m = f l states of 3S1 ortho-Ps, and would seem to involve the coefficients gx,, and H,, in the SME. The search for the CPT-odd triple correlation, as a spin-dependent interaction, should be sensitive to the SME parameters b,, gx,,, and f,. Dependences on other terms would seem to be ruled out by the spin dependence or CPT proper tie^.'^ A complete treatment of these observables, evaluating the cross section for (e+e-) as in Ref. 13 in the limit of zero rel-

+

298

ative velocity would yield the full dependence on SME parameters. There may be unique sensitivities to the SME parameters in Ps annihilation experiments as compared to other &ED tests on stable atomic systems. Finally, there is current interest in Beyond Standard Model physics involving extra dimensions. Extra dimensions are required for consistent string theories, and address the gauge hierarchy problem. Two broad topological classes are possible: compact (Calabi-Yau) or large (RandallSundrum) extra dimensions. One consequence of Randall-Sundrum dimensions would be “missing energy” signals, in which mass-energy couples to gravitons which propagate in all dimensions, not merely the conventional three spatial dimensions. If the gravitons propagate completely on the Randall-Sundrum dimensions, energy can be lost in our dimensions. Such processes are strongly constrained by gauge symmetries and conservation laws which limit the loss of baryon or lepton number or charge. The Ps system as an uncharged particle/antiparticle bound state can evade these restrictions. A virtual annihilation photon with energy 2m, would couple to a graviton, and the expected annihilation radiation would be absent in an experiment. This process is estimated in Ref. 15 to occur with a branching ratio of roughly 10-(9-10) of the usual o-Ps decay rate. To detect this process, a group at Lawrence Berkeley and Lawrence Livermore National Laboratories is performing design studies for a large liquid scintillator detector, and a group at ETH Zurich has proposed a slow positron beam stopping in a large array of the high-Z scintillator bismuth germanate.16 Substantial technical problems must be solved to perform an experiment to detect such a tiny invisible branching ratio. But the tightly constrained theoretical range of the effect suggests that this could be a definitive experiment for extra-dimension physics.

Acknowledgments This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract NO. DE-AC-0376SF00098.

References 1. 2. 3. 4.

P.A. Vetter, Mod. Phys. Lett. A 19, 871 (2004). M. Caravati, A. Devoto, and W.W. Repko, Phys. Lett. B 556, 123 (2003). P.A. Vetter and S.J. Freedman, Phys. Rev. A 66, 052505 (2002). A.P. Mills, Jr. and D.M. Zuckerman, Phys. Rev. Lett. 64, 2637 (1989).

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5. W. Bernreuther, U. Low, J.P. Ma, and 0. Nachtmann, Z. Phys. C 41, 143 (1988); W. Bernreuther, and 0. Nachtmann, Z. Phys. C 11, 235 (1981). 6. B.K. Arbic, S. Hatamian, M. Skalsey, J. Van House, and W. Zheng, Phys. Rev. A 37,3189 (1988). 7. M. Felcini, Int. J. Mod. Phys. A, submitted for publication (2003). Proceedings of the Workshop on Positronium Physics, Zurich, 2003. 8. S.K. Andrukhovich, N. Antovich, and A.V. Berestov, Inst. and Exp. Tech. 43,453 (2000). 9. P.A. Vetter and S.J. Freedman, Phys. Rev. Lett. 91, 263401 (2003). 10. M. Skalsey and J. Van House, Phys. Rev. Lett. 67,1993 (1991). 11. A. Badertscher et al., Phys. Lett. B 542,29 (2002). 12. D. Colladay and V.A. Kostelecki, Phys. Rev. D 55,6760 (1997); Phys. Rev. D 58,116002 (1998). 13. D. Colladay and V.A. Kostelecki, Phys. Lett. B 511,209 (2001). 14. A.G.M. Pickering, in V.A. Kostelecki, ed., CPT and Lorentz Symmetry II, World Scientific, Singapore, 2002. 15. S.N. Ginenko, N.V. Krasnikov, and A. Rubbia, Phys. Rev. D 67, 075012 (2003); S.N. Gninenko, N.V. Krasnikov, and A. Rubbia, Mod. Phys. Lett. A 17,1713 (2002). 16. A. Badertscher et al., hep-ph/0311031.

LORENTZ-VIOLATING VECTOR FIELDS AND THE REST OF THE UNIVERSE

EUGENE A. LIM University of Chicago Enrico Fermi Institute, Kavli Center for Cosmological Physics 5640 South Ellis Ave. Chicago, 60637 IL, U.S.A. E-mail: elimOoddjob.uchicago.edu

We consider the gravitational effects of a single, fixed-norm, Lorentz-violating timelike vector field. In a cosmological background, such a vector field acts to rescale the effective value of Newton’s constant. This vector field similarly rescales Newton’s constant in the Newtonian limit, although by a different factor. We put constraints on the parameters of the theory using the predictions of primordial nucleosynthesis, demonstrating that the norm of the vector field should be less than the Planck scale by an order of magnitude or more.

1. Introduction Lorentz invariance is a fundamental requirement of the Standard Model of particle physics, verified to high precision by many tests.’ Indeed, many of the latest results from these tests are presented in this conference. A straightforward method of implementing local Lorentz violation in a gravitational setting is to imagine the existence of a tensor field with a non-vanishing expectation value, and then couple this tensor to gravity or matter fields. The simplest example of this approach is to consider a single timelike vector field with fixed norm. A special case of this theory was first introduced as a mechanism for Lorentz-violation by Kosteleck? and Samuel in Ref. 2. In this paper, we will consider the more general theory suggested by Jacobson and M a t t i n g l ~ . ~ ? ~ This vector field picks out a preferred frame at each point in spacetime, and any matter fields coupled to it will experience a violation of local Lorentz i n ~ a r i a n c eSince . ~ ~ ~many of the current tests of Lorentz violation are done on Earth-bound laboratories, it suffices to take the vector field to be a fixed element of a background flat spacetime and ignore gravity al300

301

together. In curved spacetime, however, there is no natural generalization of the notion of a constant vector field (since Vpu”= 0 generically has no solutions); we must therefore allow the vector to have dynamics. To ensure that this vector field has a non-zero vacuum expectation value, we fix its norm by choosing an appropriate action for the field. It follows that in the presence of gravity, these vector fields are no longer ad hoc fields coupled to the Standard Model as measures of Lorentz violation, since they contribute stress-energy to the Einstein equation. In this proceeding, we investigate the gravitational effects of such vector fields, especially in the context of cosmology. We find a non-trivial impact on the evolution of the universe, namely to decrease the value of Newton’s constant relative to that measured in the Solar System, resulting in a slowing down of the expansion rate for any fixed matter content in the universe. We note here that we do not consider any coupling of the vector field to the rest of the matter fields (presumably the Standard Model Lagrangian, but our analysis is general to any form of matter fields provided that they are not coupled to the vector field except minimally through gravity). In this proceeding, we will summarize the work done in papers’?’ that have been submitted for publication. 2. The Equations of Motion

The theory we consider consists of a vector field up minimally coupled to gravity, with an action of the form S=

/

d42J-g

-

(16:G,

R + Lu

+

Lm)

*

The parameter G, is given an asterisk subscript to emphasize that it will not be equal to the value of Newton’s constant that we would measure either in the Solar System or in cosmology. L, is the vector field Lagrange density while Lm denotes the Lagrange density for all other matter fields. The Lagrange density for the vector consists of terms quadratic in the field and its derivatives:

L,

=

- P ~ V ~ U ~-VP ~~ U( V, ~ U P~ ~) V~

+

+ m2). (2)

~ U T ~ U~ ~( u p u ,

Here the Pi’s are dimensionless parameters of our theory, and X is a Lagrange multiplier field; the vector field itself has a dimension of mass. This is a slight simplification of the theory introduced by Jacobson and M a t t i n g l ~ , ~ as we have neglected a quartic self-interaction term (uPVPu~)(uuVuup).

302

Note that we have not included any couplings between the vector field Lagrangian L, and the matter field Lagrangian L,. To observe particle Lorentz-violation effects, in the case where L, is the Standard Model Lagrangian these couplings would take the form of the Standard-Model EXtension introduced by Colladay and K0steleck9.~In this work, we focus on the large scale gravitational effects of the vector field and therefore we drop these couplings for the sake of simplicity. Varying the action with respect to the Lagrange multiplier enforces the fixed norm constraint upup = -m

2

.

(3)

Choosing m2 > 0 ensures that the vector will be timelike. Since the vector field is always non-vanishing, one can say that its vacuum has spontaneously broken Lorentz symmetry. In general, the vector field is allowed to have both timelike and spacelike components though in the cases we will consider below it turns out that it will only have a timelike component. The equation of motion for the vector field is

upu,VpJ p u + m2VpJ p p = 0 ,

(4)

where we have defined the current tensor Jpu

= -Plgpugup

-

P26:6,”

-

P36p:vuUp.

(5)

Using (4) and (5), the stress-energy tensor of the vector field has the form

TLE) = ~ P ~ ( V , U ~ V - ,VU P~~ , V , ~ , ) - 2 [ v P ( ~ ( , J p v )+ ) v , ( ~ p J ( p u ) ) )- v , ( ~ ( , J u ) p ) I

+

-2m-2uuVpJpuupu, gp,,L,.

(6)

3. Slowing Down the Universe

Our goal in this section is to find solutions to the Einstein equation

G,,

= 8xG, (Ti,

+ TG),

(7) where TZ denotes the stress-energy tensor for all other matter fields in the universe. We are ultimately interested in the potentially observable effects of the vector field on the expansion of the universe, which we will find to be a rescaling of the effective value of Newton’s constant. Such an effect would not truly be observable, however, if the same rescaling affected our measurements in the Solar System. We therefore begin with an examination of the Newtonian limit, in which fields and sources are taken to be both

303

static (no time derivatives) and weak (so that we may neglect terms beyond linear order). This limit suffices to describe any laboratory measurements of the effective value of Newton's constant, which we will denote G N . The metric in the Newtonian limit takes the form

+

+

ds2 = -(1+ 2CP(Z))dt2 (1- 2S(Z))(d2 dy2

+dz2).

(8)

We will look for solutions in which the dynamical vector field u p is parallel to the timelike Killing vector 5'" = (1,0,0,0), so the only nonvanishing component will be uo. Given the fixed-norm condition, at linear order we have uo = (1- @)m.

(9)

We model the matter stress-energy tensor by a static distribution of dust which at lowest order, the only non-vanishing component is the energy density

Ti?)= #om(.').

(10)

This might describe the Earth, the Sun, or laboratory sources. Assuming boundary conditions such that both CP and S vanish at spatial infinity, we find that the Newtonian potential obeys

v2@= 4 " r G ~ p ~ ,

(11)

where we have defined

GN =

G, 1 - 8nG,Plm2'

Equation (11)is just the usual Poisson equation where GN is the effective value of Newton's constant that we would actually measure via experiments within the Solar System. We that there is a unique rescaling of the strength of the gravitational force G,; since G, is not directly measurable, however, we have no constraint on the theory parameters Pi and m2. To obtain a constraint, we move beyond the Newtonian limit and consider cosmology, where our homogenous and spatially flat and isotropic universe is described by the Robertson-Walker metric ds2 = -dt2

+ u 2 ( t )(dr2 + r2dR2) ,

(13)

where u(t) is the cosmological scale factor. The expansion rate of the universe is given by the Hubble parameter

H=-.daldt U

304

For such a metric to solve Einstein’s equation in the presence of a fixednorm vector field, the vector must respect spatial isotropy, at least in the background. Thus the only component that the vector can possess is the timelike component. Using the fixed norm constraint (3), the components of the vector field are simply u p = (m,0, 0,O). We model the stress-energy tensor for the matter as a perfect fluid with energy density pm and pressure p,, = (pm

+ Pm)npnv+ pmgpv,

(15)

where np is a unit timelike vector field representing the fluid four-velocity. Although we refer to “matter,” this fluid may consist of any isotropic source; in particular, it can be a combination of different components with distinct equations of state. Using the Einstein equation, we find that the evolution of the cosmological scale factor a ( t ) proceeds as

where the effective Newton’s constant is G, = G,/(1+8nG,(P1+3Pz+P3)). These equations are simply the usual Friedmann equations except that the Newton’s constant has been rescaled. For a given energy density from all other matter fields, we would think that the expansion rate of the universe has been slowed down if

To show that this is the case, one need to move beyond the background theory and consider the behaviour of the quantum excitations of the vector field above its non-zero vacuum value. As we have shown,s insisting that the quantum degrees of freedom propagate subluminally and have nonghostlike behaviour imply that > 0, PI ,f32 P3 2 0, and p 2 2 0. Using these constraints, it is easy to show that the equality in (18) holds only for the trivial case of m = 0. Newton’s constant enters cosmological observations in different ways, including measurements of the expansion rate in the present universe, the formation of late time large-scale structure and the properties of perturbations in the Cosmic Microwave Background. However, the most straightforward test comes from the predictions of primordial abundances from Big Bang nucleosynthesis (BBN).” Decreasing the rate of cosmic expansion

+ +

305

during BBN results in a lower 4He to hydrogen H mass ratio, allowing US to put a bound on the theory parameters

Roughly speaking, if the dimensionless parameters pi are of order unity, the norm m of the vector field must be less than l0ls GeV, an order of magnitude below the Planck scale. In particular, Planck-scale vector fields are ruled out. 4. Conclusion

We have found solutions to Einstein’s equation in the presence of other matter fields for a class of Lorentz-violating, fixed-norm vector field theories, and find that they act to rescale the value of Newton’s constant. By comparing these rescalings in the Newtonian regime to those in cosmology, we find an observable deviation from ordinary general relativity. Following this, we use the predictions of BBN to place constraints on the value of the norm of this vector field. 5. Acknowledgments We would like to thank Christian Armendariz-Picon, Jim Chisholm, Christopher Eling, Wayne Hu, Ted Jacobson, Alan Kosteleck?, David Mattingly, Eduardo Rozo, Charles Shapiro, Susan Tolwinski and especially Sean Carroll for useful discussions. We would also like to thank the organizers of CPT’04 for their hospitality. This work was supported in part by the National Science Foundation (Kavli Institute for Cosmological Physics) , the Department of Energy, and the David and Lucile Packard Foundation.

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

V.A. Kosteleckj., hep-ph/0104227. V.A. Kosteleckj. and S. Samuel, Phys. Rev. D 40, 1886 (1989). T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001). V.A. Kosteleckj., Phys. Rev. D 69, 105009 (2004). D. Colladay and V.A. Kosteleckj., Phys. Rev. D 58, 116002 (1998). S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990). S.M. Carroll and E.A. Lim, hep-th/0407149. E.A. Lim, astro-ph/0407437. T. Jacobson and D. Mattingly, Phys. Rev. D 70, 024003 (2004). Y.I. Izotov and T.X. Thuan, Astrophys. J . 500, 188 (1998).

CPT TEST IN NEUTRON-ANTINEUTRON TRANSITIONS

Y.A. KAMYSHKOV Department of Physics and Astronomy University of Tennessee Knomille, TN 37996, U.S.A. E-mail: kamyshkovOutk. edu If neutron + antineutron transitions are discovered they will be a sensitive instrument for CPT tests.

1. Introduction

Transitions n 4 A1 would violate B (baryon) and B - L (baryon minus lepton) numbers by two units. Although direct violations of baryon or lepton number are not observed experimentally, they are assumed' to play an important role in the early Universe resulting in the existing Baryon Asymmetry of the Universe (BAU). If the asymmetry between matter and antimatter in the early Universe were created by B - L conserving interactions (e.g., like in the decay modes of proton p + T O e+ or p -+ K+ V ) then BAU would be wiped out at electroweak energy scale by Standard Model interaction^.^ Thus, a new interaction violating B - L at energy scale above lo3 GeV is required for an explanation of BAU. Also, heavy Majorana leptons required for the explanation of neutrino masses in the see-saw mechanism4 would violate L and B - L numbers by two units. Transitions n + fi could be a natural manifestation of a new physics5 below the Plank scale. Present status and importance of new experimental n 4 fi search are discussed elsewhere.6 With existing experimental facilities the sensitivity of new n 3 fi search experiments can be improved by factor above 1,000. If n 4 A transitions are discovered in future experiments, they can be used as a sensitive test of CPT as was first suggested by Abov, Djeparov, and Okun.' The purpose of this paper is to discuss the sensitivity of new n + fi search experiments to CPT violation effects. Violation of B or B - L global numbers is less fundamental than the violation of CPT with the natural scale for the latter to be above the Planck scale. Particle

+

306

+

307

antiparticle mass differences resulting from CPT violation can be probed in n -+ ii transitions with sensitivity Am/m < m n u c l e o n / m p l a n k . 2. n + iz transitions

The n + A transition can be described by the mixing amplitude o determined by a new interaction at a high-energy scale. The Q mixes the components of two-component neutron-antineutron wave function. The time evolution of the wave function is determined by the Hamiltonian

where En and EA are non-relativistic energy operators:

Several assumptions are made in the solution of the Hamiltonian:

(1) a(n + i i ) = ~ ( f+ i n ) = a)i.e., T invariance; ( 2 ) it is possible to find a reference frame for the neutron-antineutron system where momentum p = 0; (3) m, # mA if CPT is violated. Define Am = m, - m,; (4) gravipotential for n and f i is the same: AU,,,, = U, - U, = 0; (5) for magnetic moments of n and f i one assumes p ( i i ) = - p ( n ) as is required by CPT invariance. However, the magnetic moment of antineutron is not yet experimentally established! (6) magnetic field of the Earth interacting with magnetic moments of n and f i can be screened down to the negligible level of few nT. Using these assumptions for the particular Hamiltonian

(3)

H = ( m nQ+ v m A - V

the solution for probability of n + f i transitions can be found as a function of observation time tobs:

Pn-m(t) =

a2

a2

+ (V + Am/2)2sin2

(do2+

(V; Am/2)2 tobs)

9

(4)

where the potential V is due to interaction of the neutron or antineutron magnetic moment with the external magnetic field, positive for neutron and negative for antineutron.

308

If the external magnetic field is shielded or compensated and Am = 0, i.e., CPT is not violated, then the transition probability has a simple form:

Pn+,(t)

=

(tiQ tabs) 2 =

(">

2

7

(5)

Tnii

where rnn = h/a is a characteristic transition (or oscillation) time. From eV. existing experimental data the parameter a has a value Q < 3. Past and future n

3

? experiments i

The n fi transitions have been previously searched in two types of experiments (a complete list of experiments can be found in Ref. 8): ---$

(1) With free cold neutrons from research reactors. The sensitivity here is proportional to the neutron flux times the square of observation time. Best limit was obtained in a state-of-art reactor experiment at ILL/Grenobleg where no events were observed in the detector with zero background.

Present limit with free neutrons:

~

> 0.86 f i x lo8 s.

~

(2) In intranuclear transitions with bound neutrons. These transitions would be strongly suppressed" by the nuclear potential V, which is different for n and fi. Such a suppression for nucleus A would result in a usual exponential decay with a lifetime TA. This lifetime is related to rn, of vacuum transition as TA = R x T:,, where R is s-l) that can be predicted a nuclear suppression factor ( R theoreticallylo with an uncertainty of factor of 2. Intranuclear n 4 fi transitions can be searched in the large underground detectors. The best limit here was established with iron nuclei by Soudan-I1 Collaboration" in the detector where sensitivity was limited by atmospheric neutrino background. The measured limit of Soudan-I1 r p e > 7.2 x 1031 years. With the theoretical suppression factor R this results in the present best limit for vacuum n -+ fi transition time T~~ obtained with bound neutrons. N

N

Present limit with bound neutrons:

> 1.3 x lo* s.

T ~ ,

Thus, the limits reached by both types of experiments are presently quite similar. Although the sensitivity of the future intranuclear experiments will be limited by atmospheric neutrino background it is expected

309

that SNO and Super-K Collaborations in the analysis of their data could reach12 sensitivity corresponding to the limits TD(SNO)- 4.8 x years years. This sensitivity corresponds to the and T0(Super-K)" 7.5 x vacuum transition time limit: Future limit with bound neutrons:

-

T,~

3.3 x lo8 S.

fi search in the future can be Probably the best sensitivity for n reached by the experiments with free neutrons13 where it is possible to increase the sensitivity (level of appearance probability) by the factor up to lo4. This corresponds to the vacuum transition time:

Future limit with free neutrons:

T , ~N

1O'O s.

4. CPT violation effect in n -+ A

If Am # 0 , i.e., CPT is violated, the transitions for free neutrons (V = 0) are suppressed when Am > f i / t o b s , as was pointed out in the original paper.7 This can be seen from Eq. (4) and Fig. 1. Any conclusion about possible presence of Am suppression can only be made if Q # 0, i.e., if n -+ f i transitions are discovered. Fig. 1 shows the effect of transition suppression (increase of Q corresponding to a given fixed level of probability of n + f i observation) due to possible n and f i mass difference. The direct experimental limit on the difference of neutron-antineutron mass' is rather poor: (m, -mii)/m, = ( 9 f 5 ) x lop5 . Measurements with neutral kaons8 provide best existing limit for the difference of particle-antiparticle mass (mKo - mRO)/mKo < 1x Even for such small Am/m free n -+ fi transitions are strongly suppressed. The Am suppression effect disappears only for Am/m of several orders of magnitude lower. Suppression starts at Am > h / t o b s (where tabs is an average neutron flight time) and with larger values of Am goes to the oscillatory regime (assuming monochromatic neutrons, only the two first waves are shown in Fig. 1 complemented with lower suppression envelopes shown as dashed lines). The velocity spectrum of practical cold neutron sources will result in a spectrum of neutron flight times that averages the suppression factor. For bound-neutron transitions, the Am suppression in Eq. (4) is negligible compared to the intranuclear potential difference that is of the order of 10 MeV. For this reason, as was pointed out in the original paper17 the intranuclear transitions are not sensitive to Am CPT-violation effect. Present and future limits for intranuclear transitions are shown in Fig. 1 by horizontal lines.

310 -22

5:

2

....

-23

.....

.

c)

-0 f

*

1e

a

5

-24

____

.

1.-

50

.-E X

4

Y

C -25 ED

2i

__

-26

-30

-25

-20

-15

Log10 [AdmI Figure 1. The n -+ f i mixing amplitude a vs Am/m. Solid lines correspond to the appearance probability limit of the present experiments. Dotted lines correspond to expected experimental limits of the future experiments. Dash-dotted lines correspond to the envelope of the lowest suppression in the oscillating solution (see text). The circle indicates the upper limit on Am/m that can be established with the discovery of n + f i transitions in a future experiment.

If future experiments with free neutrons discover n 4 ii transitions at a probability level lo4 times below the present experimental limit, this result will indicate a x 7 x eV and a new established limit for Amlm < shown by the circle in Fig. 1. With further experiments, by applying weak magnetic field in the neutron flight volume and using Eq. (4), one can disentangle the suppressing effect of Am from the controlled suppression effect due to magnetic field and thus set a more stringent limit for Am/m at the level M Since the suppression depends on the combination of (V Am/2), the observation of n 4 ii transitions also sets a limit for the difference of gravitational potential14 between neutron and antineutron at the level of to GeV. If measurement with controlled magnetic field favors Am # 0 (or equivalent difference in gravitational potential) it is possible, using polarized neutrons and varying the polarity of magnetic field, to determine the sign of the mass difference.

+

311

Acknowledgments I am grateful to Mikhail B. Voloshin and William M. Bugg for useful discussions. This work was supported by the Neutron Science Consortium of the University of Tennessee. References 1. V. Kuzmin, J E T P Lett. 12, 228 (1970); R. Mohapatra and R. Marshak, Phys. Lett. 91B,222 (1980), Phys. Rev. Lett. 44, 1316 (1980) 2. A. Sakharov, JETP Lett. 5,24 (1967). 3. V. Kuzmin, V. Rubakov, and M. Shaposhnikov, Phys. Lett. 155B,36 (1985); V. Rubakov and M. Shaposhnikov, Sov. Phys. Usp. 39 461 (1996). 4. M. Gell-Mann, P. Ramond, and P. Slansky, in Supergravity, eds. P. van Nieuwenhuizen and D. Freedman, North Holland, 315 (1979); T. Yanagida, in Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, eds. 0. Sawada and A. Sugamoto, KEK Report No. 79-18, Tsukuba, 95 (1979); R. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980). 5. G. Dvali and G. Gabadadze, Phys. Lett. B 460, 47 (1999); K. Babu and R. Mohapatra, Phys. Lett. B 518,269 (2001); S.Nussinov and R. Shrock, Phys. Rev. Lett. 88,171601 (2002). 6. Y. Kamyshkov, arXiv:hep-ex/O211006. 7. Y. Abov, F. Djeparov, L. Okun, JETP Lett. 39 493 1984). 493-494; L. Okun, arXiv:hep-ph/9612247. 8. S. Eidelman et al., Phys. Lett. B592, 1 (2004). 9. M. Balelo-Ceolin et al., Z. Phys. C 63, 409 (1994). 10. C. Dover, A. Gal and J. Richard, Phys. Rev. D 27,1090 ,1983); W. Alberico, A. De Pace, and M. Pignone, Nucl. Phys. A 523,488 (1991); J. Hufner and B. Kopeliovich, Mod. Phys. Lett. A 13,2385 (1998). 11. J. Chung et al., Phys. Rev. D 66, 032004 (2002). 12. W.A. Mann, talk at International Workshop on n + ii Transition Search with Ultracold Neutrons, Indiana University, Bloomington, 13-14 September, 2002; www.iucf.indiana.edu/Seminars/NNBAR/workshop.shtml. 13. International Workshop on n -+ ? Transition i Search with Ultracold Neutrons, Indiana University Bloomington, 13-14 September 2002. 14. S. Lamoreaux, R. Golub and J. Pendlebury, Europhys. Lett. 14, 503 (1991).

DEFORMED INSTANTONS

DON COLLADAY AND PATRICK MCDONALD New College of Florida 5700 N Tamiami Trail Samsota, FL 34234, U S A . E-mail: colladayOncf.edu In this talk, instantons are discussed in the presence of Lorentz violation. Conventional topological arguments are applied t o classify the modified solutions to the Yang-Mills equations according to the topological charge. Explicit perturbations to the instantons are calculated in detail for the case of unit topological charge.

1. Introduction

Yang-Mills theories are typically constructed using a compact Lie group G with Lie algebra L(G ), and a Lie algebra valued vector field A p ( z ) . The action of the group on the vector field is defined by

where U is a group element. The field strength tensor is then defined as

+

where the covariant derivative is taken as Dp = W igAp. With this definition, the field strength transforms in the simple way

Fp” 4 U(z)FpYU-’(z) .

(3)

The standard gauge invariant action is constructed by forming the integrated trace over the Lorentz invariant square of the field tensor

Extremization of this action with respect to A yields the equations of motion

[D,, FP”] = 0 . 312

(5)

313

The Bianchi identity follows from the definition of the field strength and yields a further set of equations

[D,,FP””]= 0 , (6) where FP” = ;~P”(+flF(+p is the dual of F . Note that nonabelian groups yield nonlinear differential equations due to the nonvanishing field commutators. It is possible that more fundamental theories of nature may contain small Lorentz-breaking effects arising from new physics at higher energy scales.’ The Standard-Model Extension (SME) provides a general framework within which to study Yang-Mills theory in the presence of Lorentz v i ~ l a t i o nThis . ~ ~ type ~ of gauge theory has also been extended to include the gravitational sector .4 Including only gauge invariant and power-counting renormalizable corrections to the Yang-Mills sector yields the action

1

+ ( ~ A F ) ~ c ~ x 2igAXAPAv) ~ ~ ( A ~ F ~, ~ 3 where k F and kAF are constant background fields that parametrize the Lorentz violation. The ~ A F terms present theoretical difficulties involving negative energy issues5 even in the abelian case and are therefore neglected in the following analysis. The specific results discussed in this proceedings are derived in more detail elsewhere.6 2. Conventional Instantons In the standard case, static solutions to the Yang-Mills equations with nontrivial, finite action (called instantons7) only occur when there are four spatial dimensions.8 Therefore, it is convenient to transform the standard action to four-dimensional Euclidean space to perform the analysis:

&(A) =

f I d 4 . T r [ F P V F P ” ],

(7)

+

where FP” = P A ” - P A P ig[AP,A”] is the explicit form of the field tensor. It is convenient to define a quantity called the topological charge q as

where FP” P X P with

= %PuaflFafl 2 is

the dual of F . Using the identity i T r F F =

XP = I 4 ~ P u X n T ~ ( A uF $igA”AXA“) Xn

(9)

314

converts the integral to a surface integral. The net result is that Q must be an integer that represents the winding number of the group on the Euclidean three-sphere at infinity. Note that this argument is independent of the explicit form of the action and only depends on the asymptotic behavior of the fields. The Euclidean version of the equations of motion (5) and the Bianchi identity ( 6 ) yield a set of nonlinear coupled differential equations for A P . A clever argument for solving these equationsg has been developed. The key identity in obtaining the instanton solutions is

This can be rearranged to yield the condition

The inequality is saturated when the field strength satisfies the duality condition F = &p.This means that if a self-dual (or anti-self-dual) field strength can be found, it will automatically extremize the action and provide a solution to the equations of motion. A theorem by Bott’O states that any mapping of the Euclidean threesphere into the group can be continuously deformed into a mapping onto an SU(2) subgroup. It is therefore sufficient to consider SU(2) subgroups of the full Lie group G. An explicit example of a self-dual solution for q = 1 is given by

with associated field strength

where 7 0 i = oa and rij = e i j k a k are expressed in terms of the conventional Pauli matrices. The anti-self-dual solution is the parity transform of the above solution. Subsequent to this, all minimal action solutions have been classified’’ and formally constructed.

3. Deformed Instantons When Lorentz violation is present, the Euclidean action is modified as

315

Standard arguments demonstrate that instantons only exist in four Euclidean dimensions, as in the conventional case. The Lorentz violation is assumed small, therefore only leading order contributions from k~ are retained in the following analysis. As mentioned in the previous section, the topological charge q is an integer, regardless of the form of the action. A modified bound on the action can be derived as

where F =~ 4 ~ ~ u XFn k X n p ois @a‘ ugeneralized a~ dual to kF. This expression indicates the natural decomposition kF = k$ CB kF into its self-dual and anti-self-dual parts.

-

3.1. Case 1: kF = -kF This condition implies that k~ takes the form k$”uap = A k[ p [ a 6 ” ]where ~] A’L” = IkQpL””depends only on the trace components of kF. The action can be minimized using the modified duality condition

F ’ E ~ z, ~

+

(16)

where F‘p” = Fp” ~ k ~ ” ; v a P F aThe f l . explicit solutions can be constructed using the skewed coordinates 5Y = xp AiVxV and writing AF(x) rz Af”,,(5)+A~”A”,(x)in terms of the modified coordinates. These solutions therefore correspond to conventional instantons in a skewed coordinate system. Note that this is a result of the existence of field redefinitions that can be used to transform physical effects of this type between the fermion and Yang-Mills sectors.12

+

3.2. Case 2: kF = i~

In this case, kF is trace free and there is no obvious modified duality condition on F because the lower bound on the action given in Eq.(15) varies with small fluctuations 6 F , therefore the equations of motion must be solved directly. To find the deformation of the conventional instanton solution, the potential is expanded as A = A ~ +Ak, D where A ~ isDthe conventional selfdual solution and Ak is the unknown perturbation. The linearized equation of motion for Ak is [%’D,

[OgD,

+ 2ig[F&,

= j:

,

(17)

316

where j [ = F i g ] is a set of known functions. This gives a set of linear second-order elliptic differential equations that can be formally solved using propagator techniques:

/~y‘“’[Dfs.~,

where G is the appropriate Green’s function. As an explicit example, consider the deformation of a q = 1 instanton in SU(2). In this case, the direct Green’s function approach is unwieldy, therefore the following procedure was eventually adopted? 0

0

0

0

Perform a gauge transformation to the singular gauge using V ( x )= - i x . r t / x so that the fields become quadratic in the instanton size. Work to lowest order in the instanton size p using the approximate Green’s function G-’ N 47r2(x- Y ) ~ ,and integrate to find the potential. Use the tensorial structure of the resulting solution as an ansatz for general values of p:

Remarkably, this gives a differential equation for the unknown function f ( x 2 ) . Solve the differential equation for f to determine the perturbation to all orders in p.

This solution explicitly preserves the topological charge since the asymptotic fields at infinity and at the origin are unmodified. The structure of the instantons is only perturbed in the intermediate region. 4. Summary

Instantons in the presence of Lorentz violation retain their topological properties, but the detailed solutions are deformed. Explicit solutions have been presented here for the case of unit topological charge. The deformations fall into two cases, one for which a simple redefinition of coordinates provides solutions, and another that requires an explicit solution to a set of linear second-order elliptic differential equations. The explicit solution demonstrates that the instanton is unaltered at the boundaries, but is deformed in the intermediate regions.

317

References 1. V.A. Kostelecki and S. Samuel, Phys. Rev. D 39, 683 (1989); ibid. 40, 1886 (1989); Phys. Rev. Lett. 63, 224 (1989); ibid. 66, 1811 (1991); V.A. Kosteleck? and R. Potting, Nucl. Phys. B 359, 545 (1991); Phys. Lett. B 381,89 (1996); Phys. Rev. D 63,046007 (2001); V.A. Kostelecki, M. Perry, and R. Potting, Phys. Rev. Lett. 84,4541 (2000). 2. D. Colladay and V.A. Kostelecki, Phys. Rev. D 55,6760 (1997); Phys. Rev. D 58, 116002 (1998). 3. For a summary of recent experimental tests and theoretical progress, see, for example V.A. Kostelecki, ed., CPT and Lorentz Symmetry 11, World Scientific, Singapore, 2002; and these proceedings. 4. V. A. Kostelecki, Phys. Rev. D 69,105009 (2004). 5. S. M. Carroll, G . B. Field, and R. Jackiw, Phys. Rev. D 41,1231 (1990). 6. D. Colladay and P. McDonald, J. Math. Phys. 45,3228 (2004). 7. For reviews, see, for example Instantons in Gauge Theories, ed. M. Shifman, World Scientific, Singapore (1994); D. Freed and K. Uhlenbeck, Instantons and four-manifolds, New York, Springer-Verlag (1991). 8. S. Deser, Phys. Lett. 64B,463 (1976). 9. A. Belavin, A. Polyakov, A. Schwartz, and Y. Tyupkin, Phys. Lett. 59B,85 (1975). 10. R. Bott, Bull. SOC.Math. France 84,251 (1956). 11. M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, Phys. Lett. 65A 285 (1978). 12. D. Colladay and P. McDonald, J. Math. Phys. 43,3554 (2002); V. A. Kosteleck? and M. Mewes, Phys. Rev. D 66,056005 (2002).

PROPOSAL TO MEASURE THE SPEED OF MU-TYPE NEUTRINOS TO TWO PARTS IN lo6

T. BERGFELD, A. GODLEY, S.R. MISHRA, AND C. ROSENFELD Department of Physics and Astronomy University of South Carolina Columbia, SC 29208 We propose to measure the propagation time of muon-type neutrinos from the NuMI source at Fermilab to the MINOS detector in northern Minnesota, a distance of 735.34 km. The proposed timing instrumentation will provide an accuracy of f 2 ns in the v, propagation time. With an accuracy in the distance of f0.7 m, we expect to show that the speed of a neutrino differs from the speed of light by no more than two parts in lo6. The time-of-flight instrumentation will also enable a search for slow-moving weakly-interacting particles.

1. Introduction and Motivation

One of the neutrino beams constructed thirty years ago for the debut of Fermilab was the site of the first measurement of the speed of a neutrino.' The experiment showed for p-type neutrinos that Ivv/c - 11 < 40 x 10-6.2 An idea from that era that partially motivated the measurement was that neutrinos moving with a speed different from c might help to explain CP violation in K-meson decay.3 Information on the speed of e-type neutrinos comes exclusively from the observation of neutrinos from SN1987A. The supernova showed that ( v v / c- 11 < 2 x 10-9.4 We propose to undertake a measurement of the up speed with precision considerably higher than achieved previously. The MINOS experiment, designed for the study of neutrino oscillations, offers an incomparable opportunity for this measurement with its 735 km baseline. With additional instrumentation to measure the propagation time, we can determine the up speed to two parts in lo6. Still better accuracy may eventually become possible. The measurement that we propose, as good as it is, is not good enough to compete with conventional methods of constraining the up mass (analysis of charged pion decay gives the up mass < 190 keV,5 our speed measurement 318

319

would give no less than 5 MeV). There are however interesting theoretical ideas that can be tested by a speed measurement. Chodos, Hauser, and Kosteleckf6i7 proposed that the neutrino might be a tachyon, a fasterthan-light object whose speed is inversely related to its energy. Hughes and Stephenson’ criticized this idea. More recently several papers have proposed a limiting speed for neutrinos that is less than c.’ The time of flight measurement could test for Lorentz invariance violation.” Tests of CPT violation” could also be made in the neutrino sector and compared to parameter limits from the photon sector. While we view these theoretical speculations as interesting, they are not essential motivation for the project. If neutrinos indeed adhere to some unorthodox theory, it is just as likely a theory that has yet to be considered. The Main Injector RF imposes a high-contrast microstructure on the NuMI beam, and this feature is essential to the speed measurement. We expect neutrinos to arrive at Soudan in phase with the RF buckets. As proposed by Shrock” the voids between buckets offer the complementary opportunity of a search for weakly-interacting particles that are massive, in our case 2 10 MeV. Gallas et al. conducted this kind of search within the confines of Fermilab.13 They were sensitive to anomalous particles no lighter than 500 MeV. In the balance of this document we discuss methodology in the context of the uw speed measurement, but the identical technology and techniques enable the anomalous particle search. The two objectives will have equal claim on our interest and analysis effort. Our present estimate of the capital cost of this project is about $318,000. With a more beneficial satellite airtime scheme the capital costs could be as little as $282,000. Proposals to the NSF and NIST were not granted. We continue to search for this funding. 2. Methodology

2.1. Overview In a conventional time-of-flight measurement signals from separated sources are assembled at a single location where the delay is measured. The conventional approach becomes unattractive, if not infeasible, when the source separation grows to hundreds of kilometers. Timekeeping with atomic clocks offers a viable alternative, with one atomic clock (AC) at Fermilab and another at Soudan. The time of an arbitrary “event” at Fermilab is established by reference to the local AC and similarly for an event at Soudan. The delay between related events at the two locations may be determined

320

by comparing the recorded clock times “offline.” The bunching of the proton beam imposed by the accelerator RF is the essential feature of the experiment that allows events at the two locations to be correlated. If we assume that the pions produced in the proton target travel at precisely c and that the daughter neutrinos do likewise, then the neutrinos arriving at Soudan will faithfully preserve the microstructure of the primary protons. This structure consists of pulses of width 3 ns spaced by 19 ns. Although pions actually propagate down the decay pipe at a speed a bit less than c, the delay induced is typically only 300 ps for the neutrinos that we will catch at Soudan. We will measure the time of protons on target against the AC at Fermilab and the arrival time of a neutrino at Soudan against the AC in the cavern. Although the periodic bunching of the proton beam is essential to the speed measurement, clearly it leaves some ambiguity unresolved. We can not know from which bunch a neutrino was produced, and therefore we can know the transit time to Soudan only modulo the 19 ns period of the beam microstructure. Because we will measure the arrival time with precision much better than 19 ns, this ambiguity is of no real consequence. If, as we tend to believe, neutrinos propagate only with speed c, then all will arrive “in phase” with the microstructure. The alternative is a spectrum of arrival times with mean slightly lagging (bradyon) or slightly leading (tachyon) the in-phase arrival time. The 19 ns phase ambiguity will not confound the null hypothesis with these alternatives. A second factor that distinguishes this measurement from a typical timeof-flight measurement is the low interaction rate of neutrinos. Acquisition of an adequate number of events necessitates that all of the massive and voluminous MINOS detector participate in the measurement.

2.2. Timekeeping and T i m e Transfer A valid measurement of time of flight requires that we establish synchronization of the two clocks and maintain it while the neutrinos are in transit, an interval of about 2.5 ms. Even inexpensive AC’s routinely maintain synchronization to better than 1.0 ns over intervals of 5000 s. The challenge then is to resynchronize the AC’s on a schedule that holds the drift to less than 1.0 ns. Based on the literature14 and discussions with experts we believe that even after heroic efforts GPS would achieve synchronization no better than 5.0 ns, which is marginal at best for our purposes. The synchronization technology we prefer is two-way satellite

321

time transfer (TWSTT). In this technique a pulse generated at one clock propagates to the other via a geostationary satellite link. The readings of the clocks at the pulse time are recorded and exchanged either through the satellite link or the Internet. The clock comparison is repeated with the pulse generated first at one end of the link, then at the other end. Provided that the propagation delay of the link is independent of direction (to a part in lo8), the AT between the clocks can be determined. In practice the procedure is somewhat more elaborate. Most of the equipment, however, is available as a turn-key system.15 The Agilent 5071A is a commercially available AC that would unquestionably serve the objectives of this project. With the high-performance beam tube option the drift of this clock does not exceed 1.3 ns in 6 hr. Thus TWSTT twice per day would be adequate.

2.3. Temporal Calibration of the Far Detector

Atomic clocks generate a digital pulse at precisely 1.0 Hz. We need to determine the time of scintillations in the MINOS Far Detector (FD) with respect to the “ticks” of the AC. The FD electronics, however, can not assimilate an asynchronous 1 Hz signal, and independent clock electronics can not assimilate the 23,000 channels of the FD. To bridge the gap we will introduce an intermediate reference pulse (IRP) at roughly 4.0 ps, i.e., half way into the beam spill. We will insert the IRP into the FD electronics, which will treat it like a pulse from a PMT, and into electronics acquired from Timing Solutions Corp. (TSC), which will measure with 100 ps precision the delay between the IRP and neighboring ticks of the AC. Using only the FD electronics then, we will determine the delay between the reference pulse and all of the PMT signals that arrive during the beam spill. A “timing model” is used to compile the delays of all the scintillator strips, parametrized by track coordinates, to infer the instant that the neutrino impinged on the upstream end of the detector. We can generate a sufficiently accurate TM from the abundant cosmic ray data. There remain two uncalibrated delays, one between the instant at which the FD electronics registers the IRP and when the AC electronics registers the IRP, and the one global delay in the TM. To fix this we require a set of auxiliary calibration detectors.

322

2.4. Auxiliary Calibration Detectors

These detectors consist of a thick scintillator viewed by high-speed PMT’s and an adjacent hodoscope. We expect these detectors to achieve 300 ps resolution over a 1 m x 1 m area. A minimal system would require two of these detectors. We will deploy the “alpha” auxiliary in the MINOS near detector hall. It will detect neutrino-induced muons and will thus establish the phase of the neutrino bunches. Initially we will deploy the “beta” auxiliary at Fermilab also, along with all of the AC and TWSTT paraphernalia. It will sit one to three meters from the alpha detector so that a substantial flux of muons traverses both detectors. We will insert the signals from the alpha and beta detectors into the independent chassis of TSC clock electronics and thus measure the time of flight of muons between the nearby detectors. This exercise will determine the relevant electronic delay in this system. Next we will transfer the beta detector and its associated AC, clock electronics, and TWSTT system to Soudan. If every neutrino event would generate a pulse in the beta detector, the calibration task would already be complete, instead we must relatively time the beta detector and the FD. Muons that traverse both detectors will provide the requisite data, and cosmic rays are suited for this role just as they are for the timing model. 2.5. Clock Trips An independent assessment of the systematics of the TWSTT procedure would be highly desirable. The only robust method that has come to our attention is a clock trip (CT). This technique utilizes an additional Agilent 5071A to be transported from Fermilab to Soudan and back. The synchronization error for a CT grows like fi where R is the round-trip time. In order to minimize R we plan to transport the itinerant clock in a small aircraft. We expect to hold R to less than 12 hr, which would yield a single-trip error of about 1.3 ns. Environmental effects might degrade this error to 2.0 ns. To check TWSTT at the 1.0 11s level will require a set of five to ten CT’s. An annual repetition of the CT exercise should produce adequate confidence in the TWSTT method. 2.6. Error Considerations

We expect eventually to log arrival times for lo4 up charged current events. When we average over our final sample, the statistical component of the error will shrink to less than 50 ps, and the systematic error will dominate.

323

A signal from the Main Injector RF will provide the time of protons on target, and the auxiliary detector at Fermilab will provide the calibration for this signal. We expect this systematic error to be well under 1 ns. The error of the Fermilab-Soudan distance is currently 0.7 m, corresponding to 2 ns. This error likely can be reduced a factor of ten or more by improving the translation the surface benchmark to the MINOS cavern. We expect a major portion of the clock synchronization error to be statistical, with the single-event error less than 2 ns. The systematic part will surely be less than 1 ns, with the use of clock trips to constrain this. To constrain the systematic error from the Timing Model we will divide the Far Detector into four zones and difference the arrival times obtained for single events that traverse multiple zones, the means of these distributions giving an estimate of this error, expected at less than 1 ns. A portion of the distance from the proton target at Fermilab to the Far Detector is travelled by the parent n+. Our rough calculation of the average delay incurred gives 300 ps. By simulation we will greatly refine this estimate, and only the residual error in the computation will remain as a systematic error (a300 ps). In summary, we anticipate that none of the sources of systematic error will exceed 1 ns with the possible exception of the distance, and we will have methods for controlling all of these uncertainties. References 1. J . Alspector et al., Phys. Rev. Lett. 36,837-840 (1976). 2. G.R. Kalbfleisch et al., Phys. Rev. Lett. 43,1361-1364 (1979). 3. G.R. Kalbfleisch, BNL-20227 (1975). 4. M.J. Longo, Phys. Rev. D 36,3276-3277 (1987). L. Stodolsky, Phys. Lett. B 201,353-354 (1988). 5. K. Assamagan et al., Phys. Rev. D 53,6065-6077 (1996). 6. A. Chodos et al., Phys. Lett. B 150,431-435 (1985). 7. A. Chodos et al., Mod. Phys. Lett. A 7,467 (1992). 8. R.J. Hughes and G.J. Stephenson Jr., Phys. Lett. B 244,95-100 (1990). 9. S.C. Coleman and S.L. Glashow, Phys. Rev. D 59, 116008 (1999). V. Ammosov and G. Volkov, hep-ph/0008032. G.S. Asanov, hep-ph/0009305 (2000). 10. S. Choubey and S. King, hep-ph/0311326 (2003). 11. D. Colladay and V.A. Kostelecki, Phys. Rev. D 55, 6760 (1997). 12. R.E. Schrock, Phys. Rev. Lett. 40,1688-1691 (1978). 13. E. Gallas et al. (the FMMF Collaboration), Phys. Rev. D 52,6-14 (1995). 14. V.S. Zhang et al., Proceedings of the IEEE/EIA International Frequency Control Symposium, 2000, pp. 598-606. 15. Timing Solutions Corporation, www.timingsolutions.com.

PARTICLES AND PROPAGATORS IN LORENTZ-VIOLATING SUPERGRAVITY

ROLAND E. ALLEN AND SEIICHIROU YOKOO Physics Department, Texas A&M University College Station, T X 77843, U.S.A. email: allenOtamu.edu We obtain the propagators for spin 1/2 fermions and sfermions in Lorentz-violating supergravity.

Any violation of Lorentz invariance must be extremely small for ordinary matter under ordinary conditions.’ However, in Lorentz-violating supergravity2 there is Lorentz violation for both Standard Model particles at very high energy3 and their supersymmetric partners at even relatively low energy.4 Here we obtain the propagators for the fermions and sfermions of this theory, with the prefix “s” standing for “supersymmetric partner” rather than “scalar” in the present context, since these particles are spin 1/2 rather than spin zero b o ~ o n s . ~ In the present paper we also extend our previous work by considering left-handed as well as right-handed sfermion fields. Initially all fermion fields are right-handed, but one can transform half of them to left-handed field^,^ obtaining, e.g., the full 4-component field $ for the electron. The same transformation can be employed for the spin 1/2 sfermions of the present theory, with only one change: in the fourth step leading up to (11) in Ref. 3, bosonic fields commute rather than anticommute, so the final Lagrangian for massless particles has the form

where the upper sign holds for fermions and the lower for bosons. Here $L is a 2-component left-handed spinor, with Ck = -uk as usual, and r]pv = diag(-1, 1,1,1). The total Lagrangian has the following form, with leftand right-handed fields combined in a 4-component spinor $ (in the Weyl 324

325

representation, and coupled by a Dirac mass m in the case of fermions):

,c+

= m-14t724

+ ($+t?l4+ ~

+ L;,

c . )

(2)

where 51 and 5 2 are diagonal 4 x 4 matrices (with elements f l )inserted to cover all the sign possibilities. We treat both Standard Model fermions and bosonic sfermions together, with the following conventions: (1) the canonical momenta conjugate to $ and are respectively called T + and ii;(2) in defining these momenta, the derivative is taken from the right. The momenta are then

++

The equation of motion has the form

and we quantize by requiring that

[$,(3,.0)

,T;(3’,2)]* =d(3-Z’)6,p

[+: (31zo) ’ i i p ( z ’ ~ xO)]* = id (Z - z’) sap. The retarded Green’s function is defined by 2G:p

(x,).’

( I[

= 0 (t - t’) 0

+a

).(

1

+; (4*I 0)

and we can show that it is in fact a Green’s function by using

to obtain

(5)

(6)

(7)

326

where 0 (ap1) represents a term which will become negligibly small at energies low compared to m, after extremely high energy terms have been discarded from the representation of GR. (See the discussion below (30).) The causal Green’s function is defined by iGap (z, z’)

=

(&).(

( 0 (T

= Gfl (z, 5’)

I

(d))0 )

$J

F(0

I!$

(z’) ,$a

(10)

1

0)

(11)

so it satisfies the equation

= 26(4) (.

-

d)+ o (m-l) ,

(12)

where a 4 x 4 identity matrix implicitly multiplies 6(4) (z - z’) and the factor of 2 will be explained below. Let Gn and q ! ~respectively ~ represent the positive-frequency and negative-frequency solutions to (4). With b i = am, the field can be represented as in (4.17), (4.45)’ and (4.55) of Ref. 4: II,=

C n

+ C bL+m

(13)

m

with

qn (z) = Ax (p’)ux (p’) exp ( - 2 . c ~ +m (z) = A ,

(p’) 21, (p’)exp ( + Z E ,

(@’) t )exp (i@’ ’ 2) (g’)t )exp (-@” . 2)

(14) (15)

so that n cf g, X and m ++-$‘, ti. Here u and v are 4-component spinors, and the normalization is the same as in (4.35) of Ref. 4:

A: (p’) Ax (p’) = (1 + 2 ~ (5‘) x /a)-’ V-’ A: (p’)A, (p’) = (1 - 2 ~ (g’) , /m)-’ V-l. To obtain the Green’s function we need

(16)

(17)

327

The Fourier transform of the causal Green’s function

can be found by using

e(t-t/)=

s

dw’ exp (iw’ (t - t’))

-

2ni

w’ - i€

in each of the two terms:

with ux ( p )ul ( p ) and w, ( - p ) w: ( - p ) on the energy shell, in the sense that po = E (p3 in the first term and -po = E (p3 in the second. However, G (x - z’) can be equally well represented by exp (-ipo (t - t‘)) c e x p (ip’. ( 3 - 2 ‘ ) )G ( p ) ( 2 6 ) 6

328

(27) with po unrestricted, since, when the residue is evaluated at one of the poles, po is forced to equal f ( E (8 -k). Let us define a modified Green’s function for a fermion by

sf

Sf ( 2 7 4 = ( 0 IT (11,

4(2’))10)

7

4 = ++yo

(28)

so that

.sr ( P )

= iG (PI Y O UA

(29) 1

( P ) CA ( P )

i

Vn(-P)G(-P)

1

(30) In the remainder of this paper we limit attention to energies that are low compared to f i . In this case, and for massless right-handed fermions, it can be seen in (4.18)-(4.21) of Ref. 4 that there is only one value of A, and it corresponds to the normal branch with E X (8= Id. On the other hand, there are three values of n: one corresponds to the normal branch with E , (3= 14, and two to extremely high energy branches with E, (3= f i f Id. When a Dirac mass is introduced, these last two branches are hardly perturbed, and they will still give extremely large denominators in the expression above for They can then be neglected in calculations at normal energies, and at the same time 2p0/m can be neglected. With the high energy branches omitted, we have the relevant low energy propagator

sf.

+

where E (3= (g2 m2)1’2. In the sums, X and n are now each limited to the 2 usual values for a 4-component Dirac spinor, rather than the total of 8 values that one would have if the 4 extremely high energy solutions were retained. We should note, however, that the high-energy solutions give a contribution in the equation of motion (9) for the Green’s function that is equal to that of the low-energy solutions, because the derivatives bring down large energies which cancel those in the denominator. This accounts for the factor of 2 in (9) and (12), and at normal energies we obtain

($ - m )ZSf (2,d )= 6(4) (2 - 2’),

(32)

329

where$ = -r”aclwith our metric tensor vPv = diag (-1,1, 1,1).The usual Dirac spinors uf and v,” have the completeness relation

and they are normalized to 2p0, whereas our ux and v, are normalized to unity. We then have

and the standard expression for the Feynman propagator is regained. For sfermions, however, one must make a distinction even at low energy between the mathematical Green’s function G ( p ) , in which negative-norm solutions have been included, and the physical propagator sb ( p ) , which can contain only positive-norm solutions. It is a fundamental requirement of quantum mechanics that physical operators are allowed to connect only states in a positivenorm Hilbert space. The physical field operator &hys, and other physical operators, should therefore contain only creation and destruction operators for positive-norm states. To obtain the physical propagator, one can repeat the above treatment with only positivenorm solutions retained. In the present theory, it fortunately turns out that one still has a complete set of functions and q ! ~ as ~ ,is required to provide a proper representation of the original classical field and satisfy the quantization condition (5). On the other hand, one finds that the restriction to positivenorm solutions permits only one value each for X and K. in (25) or (27) at low energy, corresponding to sfermions and anti-sfermions which are righthanded before a mass is introduced. The physical implications of this, and the issue of sfermion masses, will be discussed elsewhere. References 1. V.A. Kosteleckf, ed., CPT and Lorentz Symmetry, World Scientific, Singapore, 1999; CPT and Lorentz Symmetry 11, World Scientific, Singapore, 2002. 2. R.E. Allen, in Proceedings of Beyond the Desert 8003,edited by H.V. Klapdor-

Kleingrothaus, (IOP, London, 2004), hep-th/0310039. 3. R.E. Allen and S. Yokoo, Nuclear Physics B Suppl. (in press), hep-th/0402154. 4. R.E. Allen, in Proceedings of Beyond the Desert 2UU.2, edited by H.V. KlapdorKleingrothaus, (IOP, London, 2003), hep-th/0008032.

TESTING LORENTZ SYMMETRY IN SPACE

NEIL RUSSELL Physics Department, Northern Michigan University 1401 Presque Isle Avenue, Marquette, MI 49855, U.S.A. E-mail: nrussellOnmu.edu Atomic clocks, masers, and other precision oscillators are likely t o be placed on the International Space Station and other satellites in the future. These instruments will have the potential to measure Lorentz-violation coefficients, and in particular may provide access to parts of the Lorentz-violation coefficient space at levels not accessible with Earth-based experiments. The basic issues are outlined in this talk.

1. Lorentz-Violating Standard-Model Extension (SME)

The Standard-Model Extension (SME) is essentially the conventional Standard-Model lagrangian of particle physics plus all possible coordinateindependent Lorentz- and CPT-violating terms constructed from the conventional fields of particle physics.1>2The additional terms could arise in a more fundamental theory, for example string t h e ~ r y Since . ~ the symmetryviolating effects are known to be small, perturbative methods can be adopted to calculate the effects in any experimental context. Calculations or measurements for the SME in various systems include investigations of meson^,^ neutrino oscillation^,^ spin-polarized matter,6 hydrogen and antihydr~gen,~ Penning traps,8 muons,g cosmological birefringence," electromagnetic cavities," electromagnetostatics, l2 and eerenkov radiation.13 Various other issues, including the SME in curved spacetime,14 have been examined in the literature.15 An SME analysis of clock comparison experiments16provides a comprehensive framework for relating various tests.17 These experiments search for signals that are due .to rotations and accelerations of the laboratory relative to an inertial reference frame. It is therefore natural to consider clock-comparison experiments performed in space since the laboratory motion offers various advantages. These proceedings provide an overview of the basic results of this a n a l y ~ i s . ~ ~ ~ ~ ~ 330

331

2. General Clock-comparison Experiments

An atomic clock is a device that provides a stable transition frequency in a particular type of atomic system. For most atoms of interest, the total atomic angular momentum and its projection along the quantization axis are conserved to a high precision, so the quantum states can be labeled as IF,mF). The shift in the energy levels due to the SME is found using a perturbation calculation giving

bE(F,m F ) = 6

F

x(&&+

bwiy

+ &a;)

W

+ G F ~ ( T W Z +:

xwg,")

(1)

*

W

The constants 6~and G F are ratios of Clebsch-Gordan coefficients given by h

mF:=-

m F

F

,

-

mF:'

3m$-F(F+1) 3F2-F(F+l)

*

In Eq. (I), the five tilde quantities are specific combinations of the SME coefficients for Lorentz violation. In the case of @, the definition is

@ := m,d;;j

+ imwdyo- i H g

.

(3)

Similar definitions apply for the remaining four tilde coefficients.16 Noting that mw is the mass of particle w, all five tilde coefficients have dimensions of mass. The index w is to be replaced with p for proton, e for electron, or n for neutron. The numerical subscripts refer to the laboratory-frame coordinate system, in which the third coordinate is the quantization axis by convention. Interestingly, these five tilde combinations are the only SME parameter combinations that can be bounded in clock-comparison experiments with ordinary matter. The aim of this work is to consider ways that atomic clock transition frequencies may be used to detect these tilde quantities. The five Greek-letter coefficientspw, T ~ b,, , K,, A, appearing in Eq. (1)are linear combinations of expectation values calculated for the state IF, F ) of particular operators in the nonrelativistic hamiltonian for the particle w. For example, in the case of b,, the expression is:

where p j are the momentum operators and 01 are the three Pauli matrices. These quantities are calculated for each particle of type w in a specific atom

332

and the index N labels each of the N , particles of that type; for example, in 133Cs,N p = N , = 55 and N , = 78. To calculate the values of , S and the other similar coefficients would require a detailed understanding of the many-body nuclear physics. However, reasonable approximations can be made within specific nuclear models. Dimensional arguments indicate that ,&, is of order unity, and the other quantities are suppressed by factors of about K p M K , N and K, 2 The frequency output f ( B 3 ) of a typical atomic clock is determined by the difference between two energy levels and in general depends on the magnetic field projected on the quantization axis, B3. Including the Lorentz-violating effects bw,the output frequency w is expressed as

+

w = f ( B 3 ) SW.

(5)

The transition frequency w is affected by both of the levels in the transition ( F ,m ~--+ )( F ' , mk), so Sw is determined from

6w = 6E(F,m F ) - SE(F',mk).

(6)

3. Standard Inertial Reference Frame

The Lorentz-violating effects in equation (1)are contained in the SME tilde quantities, which are tensors under observer transformations. Thus, their components in one inertial reference frame are related to those in another by the corresponding rotation or boost between observers. However, unlike the energy-momentum tensor, for example, they are not integrated from controllable experimental source configurations. They are instead fixed in space. In conventional physics, results are independent of the orientation or velocity of the laboratory, but this is no longer true since the interaction of the experiment with this fixed Lorentz-violating background introduces time-dependent effects. A measurement of for example, is timedependent since the third component in the laboratory frame is changing its orientation as the Earth rotates. The time dependence is determined by the laboratory motion relative to a standard reference frame. By convention, this frame is centered on the Sun with Z axis parallel to the rotation axis of the Earth, and with X axis pointing at the vernal equinox on the celestial sphere. The time T is measured from the vernal equinox in the year 2000. Measurement of the SME coefficients in the standard frame is done using the laboratory trajectory through a sequence of linear transformations. For the case of a

a$',

333

satellite, the motion is a combination of the circular motion of the Earth around the Sun and the circular motion of the satellite around the Earth. As an example of one of the laboratory-frame quantities expressed in terms of the inertial frame, the expression for 23 is

23 = cosw,T,{

[cix(-sinacosC) +&(cosacosC) +dz(sinC)]

+ ,&[seasonal Sun-frame tilde terms11

+ sinw,Ts{

p ~ ( - c o s a )+&(-sina)

I

+ cos2w T @, [constant Sun-frame tilde termsI > + sin2w T ,&[constant Sun-frame tilde termsI > + { ps [constant Sun-frame tilde termsI > .

4 4

I}

+ ,&[seasonal Sun-frame tilde terms

(7)

Here, the z or 3 direction in the lab is oriented along the velocity vector of the satellite relative to the Earth, while the 2 direction points towards the center of the Earth. The satellite time T, is related to the Sun-based time by T = T, To, where T = To is the time of a selected ascending node of the satellite. Other satellite orbital elements in the expression are the right ascension a of the ascending node and the inclination C between the orbital axis and the Earth’s axis. For the International Space Station, w, x 2 ~ / ( 9 2min) and ,Bs= 3 x lop5 are the orbital frequency and speed relative to the Earth. The speed of the Earth is ,& x 1.0 x and the seasonal terms refer to cyclic variations with angular frequency 27r/(one sidereal year). In Eq. (7), only the Sun-frame tilde components ax, &, and 2~ appear explicitly. Others appear in the seasonal and constant Sun-frame expressions, which are given in full and in tabular form in Ref. 19. Both single and double frequencies appear in the expressions, and can be understood as arising from single- and double-index coefficients in the SME. An advantage of using a satellite is the relatively high frequency uswhich reduces the limitation of clock stability over time. Use of a turntable in a ground-based laboratory, as is being done in some experiments, offers a similar stability payoff although the velocity factor ,f3, is reduced 16-fold to the value PL M 1.6 x The coefficients that a particular clock-comparison experiment could detect in principle depend on the atoms of the clock and the transition

+

334

used. An analysis has been done for rubidium clocks, cesium clocks, and hydrogen masers.lg Similar techniques can be applied to other systems. 4. Discussion

There are 120 coefficients that in principle clock-comparison experiments can detect at leading order, consisting of 40 for each of the three basic subatomic particles. About half of these coefficients are suppressed by a factor of ps, indicating that detection of these coefficients may be enhanced in a satellite moving at high ps. A number of coefficients have been probed with Earth-based experiments, even though the lab speed relative to the Earth is an order of magnitude less than in orbit. If experiments were done today with cesium and rubidium atomic clocks in space, several dozen unmeasured coefficients would be accessed. Others would be accessible with different clocks, and in principle, all 120 coefficients are accessible from space-based clock-comparison experiments.

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