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
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CHENNAI
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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 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:? 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
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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
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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.
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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
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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
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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.
’ 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 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,+ 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).
299
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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