Lorentz-violating extension of the standard model
D. Colladay and V. Alan Kostelecky
´
Physics Department, Indiana University, Bloomington, Indiana 47405
~Received 24 June 1998; published 26 October 1998!
In the context of conventional quantum field theory, we present a general Lorentz-violating extension of the
minimal SU~3!3SU~2!3U~1! standard model including CPT-even and CPT-odd terms. It can be viewed as
the low-energy limit of a physically relevant fundamental theory with Lorentz-covariant dynamics in which
spontaneous Lorentz violation occurs. The extension has gauge invariance, energy-momentum conservation,
and covariance under observer rotations and boosts, while covariance under particle rotations and boosts is
broken. The quantized theory is Hermitian and power-counting renormalizable, and other desirable features
such as microcausality, positivity of the energy, and the usual anomaly cancellation are expected. Spontaneous
symmetry breaking to the electromagnetic U~1! is maintained, although the Higgs expectation is shifted by a
small amount relative to its usual value and the Z
0
field acquires a small expectation. A general Lorentz-
breaking extension of quantum electrodynamics is extracted from the theory, and some experimental tests are
considered. In particular, we study modifications to photon behavior. One possible effect is vacuum birefrin-
gence, which could be bounded from cosmological observations by experiments using existing techniques.
Radiative corrections to the photon propagator are examined. They are compatible with spontaneous Lorentz
and CPT violation in the fermion sector at levels suggested by Planck-scale physics and accessible to other
terrestrial laboratory experiments. @S0556-2821~99!01601-X#
PACS number~s!: 11.30.Er, 12.60.2i, 12.20.Fv, 41.20.Jb
I. INTRODUCTION
The minimal SU~3!3SU~2!3U~1! standard model, al-
though phenomenologically successful, leaves unresolved a
variety of issues. It is believed to be the low-energy limit of
a fundamental theory that also provides a quantum descrip-
tion of gravitation. An interesting question is whether any
aspects of this underlying theory could be revealed through
definite experimental signals accessible with present tech-
niques.
The natural scale for a fundamental theory including grav-
ity is governed by the Planck mass M
P
, which is about 17
orders of magnitude greater than the electroweak scale m
W
associated with the standard model. This suggests that ob-
servable experimental signals from a fundamental theory
might be expected to be suppressed by some power of the
ratio r'm
W
/M
P
.10
2 17
. Detection of these minuscule ef-
fects at present energy scales would be likely to require ex-
periments of exceptional sensitivity, preferably ones seeking
to observe a signal forbidden in conventional renormalizable
gauge theories.
To identify signals of this type, one approach is to exam-
ine proposed fundamental theories for effects that are quali-
tatively different from standard-model physics. For example,
at present the most promising framework for a fundamental
theory is string (M) theory. The qualitative difference be-
tween particles and strings means that qualitatively new
physics is expected at the Planck scale. An interesting chal-
lenge would be to determine whether this could lead to ob-
servable low-energy effects.
In the present work, we consider the possibility that the
new physics involves a violation of Lorentz symmetry. It has
been shown that spontaneous Lorentz breaking may occur in
the context of string theories with Lorentz-covariant dynam-
ics @1#. Unlike the conventional standard model, string theo-
ries typically involve interactions that could destabilize the
naive vacuum and trigger the generation of nonzero expec-
tation values for Lorentz tensors. Note that some kind of
spontaneous breaking of the higher-dimensional Lorentz
symmetry is expected in any realistic Lorentz-covariant fun-
damental theory involving more than four spacetime dimen-
sions. If the breaking extends into the four macroscopic
spacetime dimensions, apparent Lorentz violation could oc-
cur at the level of the standard model. This would represent
a possible observable effect from the fundamental theory,
originating outside the structure of conventional renormaliz-
able gauge models.
A framework has been developed for treating the effects
of spontaneous Lorentz breaking in the context of a low-
energy effective theory @2#, where certain terms can be in-
duced that appear to violate Lorentz invariance explicitly. It
turns out that, from a theoretical perspective, the resulting
effects are comparatively minimal.
An important point is that Lorentz symmetry remains a
property of the underlying fundamental theory because the
breaking is spontaneous. This implies that various attractive
features of conventional theories, including microcausality
and positivity of the energy, are expected to hold in the low-
energy effective theory. Also, energy and momentum are
conserved as usual, provided the tensor expectation values in
the fundamental theory are spacetime-position independent.
Moreover, standard quantization methods are unaffected, so
a relativistic Dirac equation and a nonrelativistic Schro
¨
dinger
equation emerge in the appropriate limits.
Another important aspect of the spontaneous breaking is
that both the fundamental theory and the effective low-
energy theory remain invariant under observer Lorentz trans-
formations, i.e., rotations or boosts of an observer’s inertial
frame @2#. The presence of nonzero tensor expectation values
in the vacuum affects only invariance properties under par-
PHYSICAL REVIEW D, VOLUME 58, 116002
0556-2821/98/58~11!/116002~23!/$15.00 ©1998 The American Physical Society58 116002-1
ticle Lorentz transformations, i.e., rotations or boosts of a
localized particle or field that leave unchanged the back-
ground expectation values.
This framework for treating spontaneous Lorentz viola-
tion has been used to obtain a general extension of the mini-
mal SU~3!3SU~2!3U~1! standard model that violates both
Lorentz invariance and CPT @2#. In addition to the desirable
features of energy-momentum conservation, observer Lor-
entz invariance, conventional quantization, hermiticity, and
the expected microcausality and positivity of the energy, this
standard-model extension maintains gauge invariance and
power-counting renormalizability. It would emerge from any
fundamental theory ~not necessarily string theory! that gen-
erates the standard model and contains spontaneous Lorentz
and CPT violation.
The present work continues our previous theoretical in-
vestigations of spontaneous Lorentz and CPT breaking.
Working first at the level of the standard model, we provide
explicitly in Sec. II the full Lorentz-violating extension, in-
cluding the CPT-even Lorentz-breaking terms described im-
plicitly in Ref. @2#. We also give some details of the modifi-
cations to the usual electroweak symmetry breaking.
Since many sensitive measures of Lorentz and CPT sym-
metry involve tests of quantum electrodynamics ~QED!,itis
useful to extract from the standard-model extension a gener-
alized QED that allows for possible Lorentz and CPT vio-
lations. This extended QED, given in Sec. III, involves modi-
fications of the usual QED in both the fermion and the
photon sectors. Some comments are also given in Sec. III
about the implications of this theory for experimental tests
with electrons and positrons.
In the remainder of this paper, we focus primarily on the
photon sector of the extended QED, presenting a study of the
theoretical and experimental implications of the modifica-
tions to photon properties arising from the possible Lorentz
and CPT violations. Section IV discusses changes in the
basic theory, including the modified Maxwell equations and
properties of their solutions. One possible effect is vacuum
photon birefringence, and some associated features are de-
scribed. We show that feasible measurements limiting bire-
fringence on cosmological scales could tightly constrain the
Lorentz-violating terms. In Sec. V, some important consis-
tency checks on the theory at the level of radiative correc-
tions are presented, largely at the one-loop level. The types
of Lorentz violation that can be affected by radiative correc-
tions are identified, and explicit calculations are given. We
show that the effects are compatible with spontaneous Lor-
entz and CPT violation in the fermion sector at levels acces-
sible to other QED experiments.
Since the standard-model extension provides a quantita-
tive microscopic theory of Lorentz and CPT violation, it is
feasible to identify potentially observable signals and to es-
tablish bounds from various experiments other than ones in
the photon sector. Numerous tests of Lorentz invariance and
CPT exist. The present theory provides a single coherent
framework at the level of the standard model and QED that
can be used as a basis for the analysis and comparison of
these tests. Although many experiments are insensitive to the
suppressed effects motivating our investigation, certain high-
precision ones might have observable signals within this
framework. In particular, the results in the present paper
have been used to examine possible bounds on CPT and
Lorentz violations from measurements of neutral-meson os-
cillations @3–6#, from tests of QED in Penning traps @7,8#,
and from baryogenesis @9#. Several other investigations are
underway, including a study @10# of possible Lorentz and
CPT effects on hydrogen and antihydrogen spectroscopy
@11# and another @12# of limits attainable in clock-
comparison experiments @13#.
The analyses of the standard-model and QED extensions
performed in the present work leave unaddressed a number
of significant theoretical issues arising at scales between the
electroweak mass and the Planck mass. These include the
‘‘dimension problem’’ of establishing whether spontaneous
Lorentz breaking in the fundamental theory near the Planck
scale indeed extends to the four physical spacetime dimen-
sions and, if so, the mechanism for its suppression or, if not,
why exactly four spacetime dimensions are spared. Other
issues include the effects of mode fluctuations around the
tensor expectation values and possible constraints and effects
arising from a nonminimal standard model or ~super!unifica-
tion below the Planck scale.
Another potentially important topic is the implication of
spontaneous Lorentz violation for gravity at observable en-
ergies. Like the usual standard model, the standard-model
extension considered here disregards gravitational effects.
The particle Lorentz symmetry that is broken in this theory is
therefore effectively a global symmetry, and so one might
expect Nambu-Goldstone modes. Since gravity is associated
with local Lorentz invariance, it is natural to ask about the
role of these modes in a version of the standard-model ex-
tension that includes gravity. In a gauge theory, when a suit-
able scalar acquires a nonzero expectation value, the Higgs
mechanism occurs: the propagator for the gauge boson is
modified, and a mass is generated. Similarly, in a theory with
gravitational couplings, when a Lorentz tensor acquires a
nonzero expectation value, the graviton propagator can be
modified. However, no mass for the graviton is induced be-
cause the gravitational connection is related to the derivative
of the metric rather than to the metric itself @1#. In this sense,
there is no gravitational Higgs effect.
The theory described here appears at present to be the sole
candidate for a consistent extension of the standard model
providing a microscopic theory of Lorentz violation. A com-
plete review of alternative approaches to possible Lorentz
and CPT violation lies beyond the scope of this paper.
Works known to us of relevance in the present context are
referenced in the body of the text below. Among other ideas
in the literature are several distinctive ones developed from
perspectives very different from ours. Following early work
by Dirac and Heisenberg, several authors have considered an
unphysical spontaneous Lorentz breaking in an effort to in-
terpret the photon as a Nambu-Goldstone boson @14#.
Nielsen and his colleagues have suggested the converse of
the philosophy in the present work: that the observed Lorentz
symmetry in nature might be a low-energy manifestation of a
fundamental theory without Lorentz invariance. A discussion
of this idea and a brief review of the literature on Lorentz
D. COLLADAY AND V. ALAN KOSTELECKY
´
PHYSICAL REVIEW D 58 116002
116002-2
breaking prior to the establishment of the usual minimal
standard model may be found in Ref. @15#. Hawking has
suggested @16# the possibility that conventional quantum me-
chanics is invalidated by gravitational effects and that this
might lead to CPT violation, among other effects. The im-
plications for experiments in the kaon system @17# are known
to be entirely different from those arising in the present
standard-model extension, which is based on conventional
quantum theory. There is also a body of literature pertaining
to unconventional theories of gravity ~without standard-
model physics!, among which are some models containing
various possible sources of local Lorentz violation @18#.
II. STANDARD-MODEL EXTENSION
In this section, we extend the minimal standard model by
adding all possible Lorentz-violating terms that could arise
from spontaneous symmetry breaking at a fundamental level
but that preserve SU~3!3SU~2!3U~1! gauge invariance and
power-counting renormalizability. Terms that are odd under
CPT are explicitly given in Ref. @2# but are also included
here for completeness.
The general form of a Lorentz-violating term involves a
part that acts as a coupling coefficient and a part constructed
from the basic fields in the standard model. The requirements
of the derivation impose various limitations on the possible
structures of both parts. Taken together, these requirements
place significant constraints on the form of terms in the
standard-model extension.
The part acting as a coupling coefficient carries spacetime
indices reflecting the properties under observer Lorentz
transformations of the relevant nonzero expectation values
from the fundamental theory. The coupling coefficient may
be complex, but it is constrained by the requirement that the
Lagrangian be Hermitian. For a coupling coefficient with an
even number of spacetime indices, the pure trace component
is irrelevant for present purposes because it maintains Lor-
entz invariance. A coupling coefficient of this type can there-
fore be taken traceless.
The field part may involve covariant derivatives and, if
fermions are involved,
g
matrices. Gauge invariance requires
that the field part be a singlet under SU~3!3SU~2!3U~1!,
while power-counting renormalizability implies that it must
have mass dimension no greater than four. The requirement
that the standard-model extension originates from spontane-
ous Lorentz breaking in a covariant fundamental theory im-
plies the whole Lorentz-violating term must be a singlet un-
der observer Lorentz transformations, so the field part must
have indices matching those of the coupling coefficient.
Following the discussion in the introduction, all coupling
coefficients are assumed to be heavily suppressed by some
power of the ratio r of the light scale to the Planck scale. In
the absence of a satisfactory explanation of the suppression
mechanism, it would seem premature to attempt specific de-
tailed predictions about the relative sizes of different cou-
pling coefficients. As a possible working hypothesis, one
might attribute comparable suppression factors to all terms at
the level of the standard-model extension. Note that a term
with the field part having mass dimension n must have a
coupling coefficient with mass dimension 42 n, and the rel-
evant scale for these effects is roughly the Planck mass. The
hypothesis would therefore suggest that in the low-energy
theory a term with the field part of mass dimension n1 1
would have a coupling coefficient suppressed by an addi-
tional power of r relative to the coefficient of a field term of
mass dimension n. This scheme would be compatible with
interpreting the standard model as an effective field theory,
in which each additional derivative coupling would involve
an additional suppression factor in the coupling coefficient. It
would imply a distinct hierarchy among the coupling coeffi-
cients introduced below, and would suggest that certain de-
rivative couplings could be neglected relative to comparable
nonderivative ones. However, since this hypothesis presently
has no basis in a detailed theory, in what follows we have
chosen to retain on an equal footing all renormalizable terms
compatible with the gauge symmetries of the standard model
and with an origin in spontaneous Lorentz breaking.
In what follows, we denote the left- and right-handed lep-
ton and quark multiplets by
L
A
5
S
n
A
l
A
D
L
, R
A
5
~
l
A
!
R
,
Q
A
5
S
u
A
d
A
D
L
, U
A
5
~
u
A
!
R
, D
A
5
~
d
A
!
R
, ~1!
where
c
L
[
1
2
~
12
g
5
!
c
,
c
R
[
1
2
~
11
g
5
!
c
, ~2!
as usual, and where A5 1,2,3 labels the flavor: l
A
[(e,
m
,
t
),
n
A
[(
n
e
,
n
m
,
n
t
), u
A
[(u,c,t), d
A
[(d,s,b).
We denote the Higgs doublet by
f
, and in unitary gauge we
represent it as
f
5
1
&
S
0
r
f
D
. ~3!
The conjugate doublet is
f
c
. The SU~3!,SU~2!, and U~1!
gauge fields are denoted by G
m
, W
m
, and B
m
, respectively.
The corresponding field strengths are G
m
n
, W
m
n
, and B
m
n
,
with the first two understood to be Hermitian adjoint matri-
ces while B
m
n
is a Hermitian singlet. The corresponding cou-
plings are g
3
, g, and g
8
. The electromagnetic U~1! charge q
and the angle
u
W
are defined through q5 g sin
u
W
5g
8
cos
u
W
, as usual. The covariant derivative is denoted by
D
m
, and A
]
J
m
B[A
]
m
B2 (
]
m
A)B. The Yukawa couplings
are G
L
, G
U
, G
D
. Throughout most of this work we use
natural units, which could be obtained from the SI system by
redefining \5 c5
e
0
5 1, and we adopt the Minkowski metric
h
m
n
with
h
00
511.
The complete Lagrangian for the Lorentz-breaking
standard-model extension can be separated into a sum of
terms. For completeness, we first provide the Lagrangian
terms in the usual SU~3!3SU~2!3U~1! minimal standard
model:
L
lepton
5
1
2
iL
¯
A
g
m
D
J
m
L
A
1
1
2
iR
¯
A
g
m
D
J
m
R
A
, ~4!
LORENTZ-VIOLATING EXTENSION OF THE STANDARD MODEL PHYSICAL REVIEW D 58 116002
116002-3
L
quark
5
1
2
iQ
¯
A
g
m
D
J
m
Q
A
1
1
2
iU
¯
A
g
m
D
J
m
U
A
1
1
2
iD
¯
A
g
m
D
J
m
D
A
,
~5!
L
Yukawa
52
@~
G
L
!
AB
L
¯
A
f
R
B
1
~
G
U
!
AB
Q
¯
A
f
c
U
B
1
~
G
D
!
AB
Q
¯
A
f
D
B
#
1H.c., ~6!
L
Higgs
5
~
D
m
f
!
†
D
m
f
1
m
2
f
†
f
2
l
3!
~
f
†
f
!
2
, ~7!
L
gauge
52
1
2
Tr
~
G
m
n
G
m
n
!
2
1
2
Tr
~
W
m
n
W
m
n
!
2
1
4
B
m
n
B
m
n
. ~8!
The usual
u
terms have been omitted, and possible analogous
total-derivative terms that break Lorentz symmetry are dis-
regarded in this work.
In the fermion sector of the standard-model extension, the
contribution to the Lagrangian can be divided into four parts
according to whether the term is CPT even or odd and
whether it involves leptons or quarks:
L
lepton
CPT-even
5
1
2
i
~
c
L
!
m
n
AB
L
¯
A
g
m
D
J
n
L
B
1
1
2
i
~
c
R
!
m
n
AB
R
¯
A
g
m
D
J
n
R
B
, ~9!
L
lepton
CPT-odd
52
~
a
L
!
m
AB
L
¯
A
g
m
L
B
2
~
a
R
!
m
AB
R
¯
A
g
m
R
B
, ~10!
L
quark
CPT-even
5
1
2
i
~
c
Q
!
m
n
AB
Q
¯
A
g
m
D
J
n
Q
B
1
1
2
i
~
c
U
!
m
n
AB
U
¯
A
g
m
D
J
n
U
B
1
1
2
i
~
c
D
!
m
n
AB
D
¯
A
g
m
D
J
n
D
B
, ~11!
L
quark
CPT-odd
52
~
a
Q
!
m
AB
Q
¯
A
g
m
Q
B
2
~
a
U
!
m
AB
U
¯
A
g
m
U
B
2
~
a
D
!
m
AB
D
¯
A
g
m
D
B
.
~12!
In these equations, the various coupling coefficients c
m
n
and
a
m
are understood to be Hermitian in generation space. The
coefficients a
m
have dimensions of mass. The dimensionless
coefficients c
m
n
can have both symmetric and antisymmetric
parts but can be assumed traceless. A nonzero trace would
not contribute to Lorentz violation and in any case can be
absorbed by a conventional field normalization ensuring the
usual kinetic operator for the matter fields.
The standard-model extension also contains Lorentz-
violating couplings between the fermions and the Higgs
field, having the gauge structure of the usual Yukawa cou-
plings but involving nontrivial
g
matrices. These terms are
all CPT even:
L
Yukawa
CPT-even
52
1
2
@~
H
L
!
m
n
AB
L
¯
A
f
s
m
n
R
B
1
~
H
U
!
m
n
AB
Q
¯
A
f
c
s
m
n
U
B
1
~
H
D
!
m
n
AB
Q
¯
A
f
s
m
n
D
B
#
1H.c. ~13!
The coefficients H
m
n
are dimensionless and antisymmetric,
but like the Yukawa couplings G
L,U,D
they are not necessar-
ily Hermitian in generation space.
The possible contributions in the Higgs sector can be
CPT even or CPT odd:
L
Higgs
CPT-even
5
1
2
~
k
ff
!
m
n
~
D
m
f
!
†
D
n
f
1H.c.
2
1
2
~
k
f
B
!
m
n
f
†
f
B
m
n
2
1
2
~
k
f
W
!
m
n
f
†
W
m
n
f
,
~14!
L
Higgs
CPT-odd
5 i
~
k
f
!
m
f
†
D
m
f
1 H.c. ~15!
In Eq. ~14!, the dimensionless coefficient k
ff
can have sym-
metric real and antisymmetric imaginary parts. The other co-
efficients in Eq. ~14! have dimensions of mass and must be
real antisymmetric. The coefficient k
f
for the CPT-odd term
~15! also has dimensions of mass and can be an arbitrary
complex number.
The gauge sector has both CPT-even and CPT-odd con-
tributions. The CPT-even ones are
L
gauge
CPT-even
52
1
2
~
k
G
!
k
l
m
n
Tr
~
G
k
l
G
m
n
!
2
1
2
~
k
W
!
k
l
m
n
Tr
~
W
k
l
W
m
n
!
2
1
4
~
k
B
!
k
l
m
n
B
k
l
B
m
n
. ~16!
In this equation, the dimensionless coefficients k
G,W,B
are
real. They must have the symmetries of the Riemann tensor
and a vanishing double trace. The point is that any totally
antisymmetric part involves only a total derivative in the
Lagrangian density, while a nonzero double trace can be ab-
sorbed into a redefinition of the normalization of the corre-
sponding kinetic term ~8!.
The CPT-odd gauge terms are given by the following
expression @19#:
L
gauge
CPT-odd
5
~
k
3
!
k
e
k
l
m
n
Tr
~
G
l
G
m
n
1
2
3
ig
3
G
l
G
m
G
n
!
1
~
k
2
!
k
e
k
l
m
n
Tr
~
W
l
W
m
n
1
2
3
igW
l
W
m
W
n
!
1
~
k
1
!
k
e
k
l
m
n
B
l
B
m
n
1
~
k
0
!
k
B
k
. ~17!
The coefficients k
1,2,3
are real and have dimensions of mass,
while k
0
is also real and has dimensions of mass cubed. It
turns out that, if any of these CPT-odd terms do indeed
appear, they would generate instabilities in the minimal
theory. They are all associated with negative contributions to
the energy, and in addition the term with k
0
would directly
generate a linear instability in the potential. It might there-
fore seem desirable that all the coefficients k
0,1,2,3
vanish.
While this could be imposed at the classical level, radiative
quantum corrections from, say, the fermion sector might a
priori be expected to generate nonzero values. Remarkably,
the structure of the standard-model extension appears to be
such that no corrections arise, at least to one loop. These
issues are discussed further in what follows, in particular in
Secs. IV A and V.
It is known that some apparently CPT- and Lorentz-
violating terms can be eliminated from the action via field
D. COLLADAY AND V. ALAN KOSTELECKY
´
PHYSICAL REVIEW D 58 116002
116002-4
redefinitions @2#. Several types of redefinition can be consid-
ered. In the context of the present standard-model extension,
we have investigated a variety of possibilities for each field.
As a general rule, the more complex the theoretical structure
becomes, the less likely it is that a useful field redefinition
exists. For instance, the presence of Lorentz-violating
CPT-even derivative couplings in the standard-model exten-
sion complicates the analysis for CPT-odd terms provided in
Ref. @2#, although it turns out that the conclusions still hold.
Here, we summarize a few methodological results and de-
scribe some special cases of particular interest.
To eliminate a Lorentz-breaking term, a field redefinition
must involve the associated coupling coefficient. When de-
rivative couplings play a role, the field redefinition may also
involve spacetime-position variables. The assumption that
the coupling coefficients are small can be helpful, in some
cases directly assisting in derivations and in others leading to
a set of approximate field redefinitions. Under the latter, a
theory with first-order Lorentz-breaking effects may be rede-
fined into one with effects appearing only at second or higher
orders. Alternatively, some first-order Lorentz-breaking
terms may be absorbed into others. A partial constraint on
allowable redefinitions is provided by the transformation
properties of the various Lorentz-violating terms under the
discrete symmetries C, P, T. Only terms with identical
discrete-symmetry properties can be absorbed into one an-
other by first-order redefinitions.
Two types of redefinition that we have found of particular
value are linear phase redefinitions and linear normalization
redefinitions. For example, some terms involving the coeffi-
cients a
L,R,Q,U,D
can be eliminated by position-dependent
field-phase redefinitions, as described in Ref. @2#. Another
example is provided by terms involving the coefficients
H
L,U,D
, some of which can absorb through field-
normalization redefinitions certain other terms involving the
coefficients c
L,R,Q,U,D
. These examples have specific inter-
esting implications for the quantum-electrodynamics limit of
the standard-model extension, and their explicit forms for
that case are given in Sec. III below. Useful nonlinear field
redefinitions might also exist in principle, but these are typi-
cally more difficult to implement meaningfully because they
may represent ~noncanonical! transformations between dif-
ferent physical systems rather than reinterpretations of the
same physics.
We next consider the issue of electroweak SU~2!3U~1!
symmetry breaking. The static potential for the gauge and
Higgs fields can be extracted from the Lagrangian terms
given above for the standard-model extension. It is possible
to work in the unitary gauge as usual, since the Lorentz-
breaking terms do not affect the gauge structure of the
theory. The analysis is somewhat more complicated than the
conventional case, as it involves additional terms depending
on the coupling coefficients k
ff
, k
f
W
, k
f
, k
W
, k
2
, and k
0
.
In principle, there are also contributions from the SU~3! sec-
tor, but these decouple from the Higgs field and so the gluon
expectation values can be taken to be zero as usual. As men-
tioned above, the terms k
2
and k
0
are expected to vanish for
consistency of the minimal theory, and so we assume this in
what follows. In fact, a nonzero k
2
would have no effect on
the vacuum values of the fields, but the linear instability that
would be introduced by a nonzero k
0
would exclude a stable
vacuum in the absence of other ~nonlinear! effects.
Extremizing the static potential produces five simulta-
neous equations. Three of these are satisfied if the expecta-
tion values of W
m
6
and the photon A
m
vanish. The other two
equations can be solved algebraically for the expectation val-
ues of the Higgs and Z
m
0
fields. In the general case, both of
these are nonzero and are given by
^
Z
m
0
&
5
1
q
sin 2
u
W
~
Re k
ˆ
ff
!
m
n
21
k
f
n
, ~18!
^
r
f
&
5 a
S
12
1
m
2
~
Re k
ˆ
ff
!
m
n
21
k
f
m
k
f
n
D
1/2
, ~19!
where k
ˆ
ff
m
n
[
h
m
n
1k
ff
m
n
and a[
A
6
m
2
/l. Note that the quan-
tity (Re k
ˆ
ff
)
m
n
21
always exists when the Lorentz violation is
small,
u
(k
ff
)
m
n
u
!1. Note also that
^
r
f
&
is a scalar under
both particle and observer Lorentz transformations, so quan-
tities such as the fermion mass parameters remain scalars
despite the presence of Lorentz breaking.
As might be anticipated, the above pattern of expectation
values leaves unbroken the electromagnetic U~1! symmetry,
and it can be shown that fluctuations about the extremum are
stable. When substituted into the Lagrangian for the
standard-model extension, the unconventional nonzero ex-
pectation value for the field Z
m
0
generates some additional
CPT- and Lorentz-violating contributions. However, these
are all of the same form as other CPT- and Lorentz-violating
terms already present in the theory, so they can be absorbed
into existing coupling coefficients.
Some analyses of experimental tests of the standard-
model extension involving flavor-changing oscillations in
neutral mesons have been performed in Refs. @3,5,6#. Tests at
the level of quantum electrodynamics are mentioned below.
Note that some bounds on both the fermion and the gauge
sectors might be obtained from available experimental infor-
mation about the Z
m
0
and perhaps the W
m
6
. Such limits would
be of interest in their own right, although it seems likely that
they would be much weaker than required to detect sup-
pressed Lorentz violation at the levels estimated in this work.
III. EXTENDED QUANTUM ELECTRODYNAMICS
In much the same way that conventional quantum electro-
dynamics ~QED! can be obtained from the usual standard
model, a generalized quantum electrodynamics incorporating
Lorentz-breaking terms can be extracted from the standard-
model extension given in Sec. II. This is of particular interest
because QED has been tested to high precision in a variety of
experiments, some of which may tightly constrain the cou-
pling coefficients of the possible Lorentz-violating terms.
A straightforward way to obtain the extended QED is as
follows. After the SU~2!3U~1! symmetry breaking, set to
zero the fields G
m
for the gluons, W
m
6
,Z
m
0
for the weak
bosons, and the physical Higgs field ~but not the expectation
value of the Higgs doublet, which generates fermion
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