Lorentz and CPT violation in neutrinos
V. Alan Kostelecky
´
and Matthew Mewes
Physics Department, Indiana University, Bloomington, Indiana 47405, USA
共Received 2 September 2003; published 30 January 2004兲
A general formalism is presented for violations of Lorentz and CPT symmetry in the neutrino sector. The
effective Hamiltonian for neutrino propagation in the presence of Lorentz and CPT violation is derived, and its
properties are studied. Possible definitive signals in existing and future neutrino-oscillation experiments are
discussed. Among the predictions are direction-dependent effects, including neutrino-antineutrino mixing, si-
dereal and annual variations, and compass asymmetries. Other consequences of Lorentz and CPT violation
involve unconventional energy dependences in oscillation lengths and mixing angles. A variety of simple
models both with and without neutrino masses are developed to illustrate key physical effects. The attainable
sensitivities to coefficients for Lorentz violation in the Standard-Model Extension are estimated for various
types of experiments. Many experiments have potential sensitivity to Planck-suppressed effects, comparable to
the best tests in other sectors. The lack of existing experimental constraints, the wide range of available
coefficient space, and the variety of novel effects imply that some or perhaps even all of the existing data on
neutrino oscillations might be due to Lorentz and CPT violation.
DOI: 10.1103/PhysRevD.69.016005 PACS number共s兲: 11.30.Cp, 11.30.Er, 14.60.Pq
I. INTRODUCTION
The minimal Standard Model 共SM兲 of particle physics
offers a successful description of most processes in nature
but leaves unresolved several experimental and theoretical
issues. On the experimental front, observations of neutrino
oscillations have accumulated convincing evidence that the
description of physical properties of neutrinos requires modi-
fication of the neutrino sector in the minimal SM. Most ex-
perimental results to date can be described theoretically by
adding neutrino masses to the minimal SM, but a complete
understanding of the existing data awaits further experimen-
tation. On the theoretical front, the SM is expected to be the
low-energy limit of a more fundamental theory that unifies
quantum physics and gravity at the Planck scale m
P
⯝10
19
GeV. Direct measurements at this energy scale are impracti-
cal, but suppressed low-energy signatures from the antici-
pated new physics might be detectable in sensitive existing
experiments.
In this work, we address both these topics by studying
effects on the neutrino sector of relativity violations, a prom-
ising class of Planck-scale signals. These violations might
arise through the breaking of Lorentz symmetry and perhaps
also the breaking of CPT symmetry 关1兴. Since the SM is
known to provide a successful description of most physics at
low energies compared to the Planck scale, any such signals
must appear at low energies in the form of an effective quan-
tum field theory containing the SM. The general effective
quantum field theory constructed from the SM and allowing
arbitrary coordinate-independent Lorentz violation is called
the Standard-Model Extension 共SME兲关2兴. It provides a link
to the Planck scale through operators of nonrenormalizable
dimension 关3,4兴. Since CPT violation implies Lorentz viola-
tion 关5兴, this theory also allows for general CPT breaking.
The SME therefore provides a realistic theoretical basis for
studies of Lorentz violation, with or without CPT breaking.
The Lagrangian of the SME consists of the usual SM
Lagrangian supplemented by all possible terms that can be
constructed with SM fields and that introduce violations of
Lorentz symmetry. The additional terms have the form of
Lorentz-violating operators coupled to coefficients with Lor-
entz indices, and they could arise in a variety of ways. One
generic and elegant mechanism is spontaneous Lorentz vio-
lation, proposed first in string theory and field theories with
gravity 关6兴 and then generalized to include CPT violation 关7兴.
Another popular framework for Lorentz violation is noncom-
mutative field theory, in which realistic models form a subset
of the SME involving operators of nonrenormalizable dimen-
sion 关8兴. Other proposed sources of Lorentz and CPT viola-
tion include various nonstring approaches to quantum gravity
关9兴, random dynamics 关10兴, and multiverses 关11兴. Planck-
scale sensitivity to the coefficients for Lorentz violation in
the SME has been achieved in various experiments, includ-
ing ones with mesons 关3,12,13兴, baryons 关14–16兴, electrons
关17,18兴, photons 关19–22兴, and muons 关23兴. However, no ex-
periments to date have measured neutrino-sector coefficients
for Lorentz violation.
Here, we explore neutrino behavior in the presence of
Lorentz and CPT violation using the SME framework. The
original proposal for Lorentz and CPT violation in neutrinos
关2兴 has since been followed by several theoretical investiga-
tions within the context of the SME 关24–29兴, most of which
have chosen to restrict attention to a small number of coef-
ficients. A comprehensive theoretical study of Lorentz and
CPT violation in neutrinos has been lacking. The present
work partially fills this gap by applying the ideas of the SME
to a general neutrino sector with all possible couplings of
left- and right-handed neutrinos and with sterile neutrinos.
We concentrate mostly on Lorentz-violating operators of
renormalizable dimension, which dominate the low-energy
physics in typical theories, but some generic consequences of
Lorentz-violating operators of nonrenormalizable dimension
are also considered 关3,4,30兴. The effective Hamiltonian de-
scribing free neutrino propagation is obtained, and its impli-
cations are studied. The formalism presented in this work
thereby provides a general theoretical basis for future studies
PHYSICAL REVIEW D 69, 016005 共2004兲
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of Lorentz and CPT violation in neutrinos. We also illustrate
various key physical ideas of Lorentz and CPT violation
through simple models, and we discuss experimental signals.
Our primary focus here is on oscillation data 关31兴, but the
formalism is applicable also to other types of experiments
including direct mass searches 关32兴, neutrinoless double-beta
decay 关33兴, and supernova neutrinos 关34兴.
Several features of Lorentz and CPT violation that we
uncover are common to other sectors of the SME, including
unconventional energy dependence and dependence on the
direction of propagation. We also find that Lorentz-violating
neutrino-antineutrino mixing with lepton-number violation
naturally arises from Majorana-like couplings. These fea-
tures lead to several unique signals for Lorentz and CPT
violation. For example, the direction dependence potentially
generates sidereal variations in terrestrial experiments as the
Earth rotates, annual variations in solar-neutrino properties,
and intrinsic differences in neutrino flux from different
points on the compass or different angular heights at the
location of the detector. The unconventional energy depen-
dence produces a variety of interesting potential signals, in-
cluding resonances in the vacuum 关25,29兴 as well as the
usual Mikheyev-Smirnov-Wolfenstein 共MSW兲 resonances in
matter 关35兴.
Experiments producing evidence for neutrino oscillations
to date include atmospheric-neutrino experiments 关36兴, solar-
neutrino experiments 关37–42兴, reactor experiments 关43兴, and
accelerator-based experiments 关44,45兴. Most current data are
consistent with the introduction of three massive-neutrino
states, usually attributed to Grand-Unified-Theory 共GUT兲
scale physics. However, as we demonstrate in this work, the
possibility remains that the observed neutrino oscillations
may be due at least in part and conceivably even entirely to
Lorentz and CPT violation from the Planck scale. In any
event, experiments designed to test neutrino mass are also
well suited for tests of Lorentz and CPT invariance, and they
have the potential to produce the first measurements of vio-
lations of these fundamental symmetries, signaling possible
Planck-scale physics.
The organization of this paper is as follows. Section II
presents the basic theory and definitions, obtaining the effec-
tive Hamiltonian for neutrino propagation and discussing its
properties. Issues of experimental sensitivities and possible
constraints from experiments in other sectors are considered
in Sec. III. Certain key features of neutrino behavior in the
presence of Lorentz and CPT violation are illustrated in the
sample models of Sec. IV. Some remarks about both generic
and experiment-specific predictions are provided in Sec. V.
Throughout, we follow the notation and conventions of Refs.
关2,4兴.
II. THEORY
A. Basics
Our starting point is a general theory describing N neu-
trino species. The theory is assumed to include all possible
Majorana- and Dirac-type couplings of left- and right-handed
neutrinos, including Lorentz- and CPT-violating ones. The
neutrino sector of the minimal SME is therefore included,
along with other terms such as those involving right-handed
neutrinos.
We denote the neutrino fields by the set of Dirac spinors
兵
e
,
,
,...
其
and their charge conjugates by
兵
e
C
⬅
e
C
,
C⬅
C
,
C⬅
C
,... 其, where charge conjugation
of a Dirac spinor is defined as usual:
a
C
⬅C
¯
a
T
. By defini-
tion, active neutrinos are detected via weak interactions with
left-handed components of
兵
e
,
,
其
. Complications may
arise in the full SME, where Lorentz-violating terms alter
these interactions and can modify the detection process.
However, such modifications are expected to be tiny and
well beyond the sensitivity of current experiments. In con-
trast, propagation effects can become appreciable for large
baselines. We therefore focus in this work on solutions to the
Lorentz-violating equations of motion that describe free
propagation of the N neutrino species.
It is convenient to place all the fields and their conjugates
into a single object
A
, where the index A ranges over the
2N possibilities
兵
e,
,
,...,e
C
,
C
,
C
,...
其
. This setup al-
lows us to write the equations of motion in a form analogous
to the Lorentz-violating QED extension 关2,4兴, and it can
readily accommodate Dirac, Majorana, or more general types
of neutrinos. Our explicit analysis in this section is per-
formed under the assumption that Lorentz-violating opera-
tors of renormalizable dimension dominate the low-energy
physics. Then, the general equations of motion for free
propagation can be written as a first-order differential opera-
tor acting on the object
A
:
共
i⌫
AB
⫺ M
AB
兲
B
⫽ 0. 共1兲
Here, each constant quantity ⌫
AB
, M
AB
is also a 4⫻4 ma-
trix in spinor space. Note that the usual equations of motion
for Dirac and Majorana neutrinos are special cases of this
equation.
The matrices ⌫
AB
and M
AB
can be decomposed using the
basis of
␥
matrices. We define
⌫
AB
⬅
␥
␦
AB
⫹ c
AB
␥
⫹ d
AB
␥
5
␥
⫹ e
AB
⫹ if
AB
␥
5
⫹
1
2
g
AB
,
M
AB
⬅m
AB
⫹ im
5AB
␥
5
⫹ a
AB
␥
⫹ b
AB
␥
5
␥
⫹
1
2
H
AB
. 共2兲
In these equations, the masses m and m
5
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 im-
poses the conditions ⌫
AB
⫽
␥
0
(⌫
BA
)
†
␥
0
and M
AB
⫽
␥
0
(M
BA
)
†
␥
0
, which implies all coefficients are hermitian
in generation space.
The above construction carries some redundancies that
stem from the interdependence of
and
C
. This implies
certain symmetries for ⌫
and M. Note first that charge con-
V. ALAN KOSTELECKY AND MATTHEW MEWES PHYSICAL REVIEW D 69, 016005 共2004兲
016005-2
jugation can be written as a linear transformation on
A
:
A
C
⫽ C
AB
B
, where C is the symmetric matrix with nonzero
elements C
ee
C⫽ C
C⫽ C
C⫽ •••⫽ 1. Then, in terms of C
and the spinor matrix C, the interdependence of
and
C
implies the relations
⌫
AB
⫽⫺C
AC
C
BD
C
共
⌫
DC
兲
T
C
⫺ 1
,
M
AB
⫽ C
AC
C
BD
C
共
M
DC
兲
T
C
⫺ 1
, 共3兲
where the transpose T acts in spinor space. Suppressing gen-
eration indices, this translates to
c
⫽ C
共
c
兲
T
C, m⫽ C
共
m
兲
T
C,
d
⫽⫺C
共
d
兲
T
C, m
5
⫽ C
共
m
5
兲
T
C,
e
⫽⫺C
共
e
兲
T
C, a
⫽⫺C
共
a
兲
T
C,
f
⫽⫺C
共
f
兲
T
C, b
⫽ C
共
b
兲
T
C,
g
⫽ C
共
g
兲
T
C, H
⫽⫺C
共
H
兲
T
C, 共4兲
where now the transpose T acts in generation space. Note
that the overall signs in the above equations are chosen to
match their derivation within the conventional lagrangian
formalism involving anticommuting fermion fields.
Equation 共1兲 provides a basis for a general Lorentz- and
CPT-violating relativistic quantum mechanics of freely
propagating neutrinos. However, the unconventional time-
derivative term complicates the construction of the corre-
sponding Hamiltonian. This difficulty also arises in the mini-
mal QED extension, but it may be overcome 关4兴 if there
exists a nonsingular matrix A satisfying the relationship
A
†
␥
0
⌫
0
A⫽ 1. The field redefinition
A
⫽ A
AB
B
then allows
the equations of motion 共1兲 to be written as (i
␦
AB
0
⫺ H
AB
)
B
⫽ 0, where the Hamiltonian is given by H
⫽⫺A
†
␥
0
(i⌫
j
j
⫺ M)A.
Denoting
␦
⌫
and
␦
M as the Lorentz-violating portions
of ⌫
and M, and under the reasonable assumption that
兩
␦
⌫
0
兩
⬍ 1, a satisfactory field redefinition is given by the
power series A⫽ (1⫹
␥
0
␦
⌫
0
)
⫺ 1/2
⫽ 1⫺
1
2
␥
0
␦
⌫
0
⫹ •••. Sepa-
rating the Hamiltonian H into a Lorentz-conserving part H
0
and a Lorentz-violating part
␦
H, which we assume is small
relative to H
0
, we can use the above expression for A to
obtain an expansion of
␦
H in terms of H
0
and coefficients
for Lorentz violation. Explicitly, at leading order in coeffi-
cients for Lorentz violation, we obtain
␦
H⫽⫺
1
2
共
␥
0
␦
⌫
0
H
0
⫹ H
0
␥
0
␦
⌫
0
兲
⫺
␥
0
共
i
␦
⌫
j
j
⫺
␦
M
兲
.
共5兲
This expression is therefore the basis for a general study of
leading-order Lorentz and CPT violation in the neutrino sec-
tor.
At this stage, prior to beginning our study of Eq. 共5兲,itis
useful to review the properties of the Lorentz-conserving
Hamiltonian 关46,47兴
H
0
⫽⫺
␥
0
共
i
␥
j
j
⫺ M
0
兲
. 共6兲
The Lorentz-conserving dynamics is completely determined
by the mass matrix M
0
, which in its general form can be
written
M
0
⫽ m⫹ im
5
␥
5
⫽ m
L
P
L
⫹ m
R
P
R
共7兲
with m
R
⫽ (m
L
)
†
⫽ m⫹ im
5
and P
L
⫽
1
2
(1⫺
␥
5
),P
R
⫽
1
2
(1
⫹
␥
5
). The components of the matrix m
R
⫽ m
L
†
can be iden-
tified with Dirac- or Majorana-type masses by separating m
R
into four N⫻N submatrices. It is often encountered in the
form of the symmetric matrix
m
R
C⫽
冉
LD
D
T
R
冊
. 共8兲
The matrices R and L are the right- and left-handed
Majorana-mass matrices, while D is the Dirac-mass matrix.
In general, R, L and D are complex matrices restricted only
by the requirement that R and L are symmetric. Note that a
left-handed Majorana coupling is incompatible with
electroweak-gauge invariance. In contrast, Dirac and right-
handed Majorana couplings can preserve the usual gauge in-
variance.
It is always possible to find a basis in which the mass
matrix M
0
is diagonal. Labeling the fields in this basis by
A
⬘
, where A
⬘
⫽ 1,...,2N, then the unitary transformation
relating the two bases can be written as
U
A
⬘
A
⫽ V
A
⬘
A
P
L
⫹
共
VC
兲
A
⬘
A
*
P
R
, 共9兲
where V isa2N⫻2N unitary matrix. Here, it is understood
that U
A
⬘
A
carries spinor indices that have been suppressed.
In the new basis, the mass matrix m
LA
⬘
B
⬘
⫽ m
RA
⬘
B
⬘
⫽ m
(A
⬘
)
␦
A
⬘
B
⬘
is diagonal with real non-negative entries. The
neutrinos
A
⬘
⫽
A
⬘
C
⫽ V
A
⬘
A
P
L
A
⫹ V
A
⬘
A
*
P
R
A
C
are Majorana
particles, regardless of the form of M
0
.
B. Effective Hamiltonian
The discussion above applies to an arbitrary number of
neutrino species and an arbitrary mass spectrum. Since a
general treatment is rather cumbersome, we restrict attention
in what follows to the minimal physically reasonable exten-
sion with N⫽3. For definiteness, we also assume a standard
seesaw mechanism 关48兴 with the components of R much
larger than those of D or L. This mechanism suppresses the
propagation of right-handed neutrinos, so the analysis below
also contains other Lorentz- and CPT-violating scenarios
dominated by light or massless left-handed neutrinos, includ-
ing the minimal SME.
Ordering the masses m
(A
⬘
)
from smallest to largest, we
assume that m
(1)
,m
(2)
,m
(3)
are small compared to the neu-
trino energies and possibly zero, and that the remaining
masses m
(4)
,m
(5)
,m
(6)
are large with the corresponding en-
ergy eigenstates kinematically forbidden. In this situation the
submatrix V
a
⬘
a
, where a⫽ e,
,
and a
⬘
⫽ 1,2,3, is approxi-
mately unitary.
To aid in solving the equations of motion, we define
LORENTZ AND CPT VIOLATION IN NEUTRINOS PHYSICAL REVIEW D 69, 016005 共2004兲
016005-3
A
共
t;x
ជ
兲
⫽
冕
d
3
p
共
2
兲
3
A
共
t;p
ជ
兲
e
ip•
ជ
x
ជ
,
A
共
t;p
ជ
兲
⫽ b
A
共
t;p
ជ
兲
u
L
共
p
ជ
兲
⫹
共
Cd
兲
A
共
t;p
ជ
兲
u
R
共
p
ជ
兲
⫹
共
Cb
兲
A
*
共
t;⫺ p
ជ
兲
v
R
共
⫺ p
ជ
兲
⫹ d
A
*
共
t;⫺ p
ជ
兲
v
L
共
⫺ p
ជ
兲
. 共10兲
This is chosen to satisfy explicitly the charge-conjugation
condition
A
C
⫽ C
AB
B
. The spinor basis
兵
u
L
(p
ជ
),u
R
(p
ជ
),
v
R
(⫺ p
ជ
),
v
L
(⫺ p
ជ
)
其
obeys the usual relations for massless fer-
mions, with
v
R,L
(p
ជ
)⫽ Cu
¯
L,R
T
(p
ជ
). It has eigenvalues of the
helicity operator
␥
5
␥
0
␥
ជ
•p
ជ
/
兩
p
ជ
兩
given by
兵
⫺ ,⫹ ,⫺ ,⫹
其
and
eigenvalues of the chirality operator
␥
5
given by
兵
⫺ ,⫹ ,⫹ ,
⫺
其
. For simplicity, we normalize with u
␣
†
u

⫽
v
␣
†
v

⫽
␦
␣
for
␣
,

⫽ L,R. The definition 共10兲 implies that the ampli-
tudes b
e,
,
may be approximately identified with active neu-
trinos and d
e,
,
with active antineutrinos. The remaining
amplitudes b
e
C
,
C
,
C and d
e
C
,
C
,
C cover the space of sterile
right-handed neutrinos, but a simple identification with fla-
vor neutrinos and antineutrinos would be inappropriate in
view of their large mass.
In the mass-diagonal Majorana basis, we restrict attention
to the propagating states consisting of the light neutrinos.
Taking the Hamiltonian in this basis
H
a
⬘
b
⬘
共
p
ជ
兲
⫽
␥
0
共
␥
ជ
•p
ជ
⫹ m
(a
⬘
)
兲
␦
a
⬘
b
⬘
⫹
␦
H
a
⬘
b
⬘
共
p
ជ
兲
, 共11兲
and applying it to
b
⬘
(t;p
ជ
)⫽ U
b
⬘
B
B
(t;p
ជ
) yields the equa-
tions of motion in terms of the amplitudes b and d. The result
takes the form of the matrix equation
关
i
␦
a
⬘
b
⬘
0
⫺ H
a
⬘
b
⬘
共
p
ជ
兲
兴
冉
b
b
⬘
共
t;p
ជ
兲
d
b
⬘
共
t;p
ជ
兲
b
b
⬘
*
共
t;⫺ p
ជ
兲
d
b
⬘
*
共
t;⫺ p
ជ
兲
冊
⫽ 0, 共12兲
where for convenience we have defined b
b
⬘
⫽ V
b
⬘
B
b
B
and
d
b
⬘
⫽ V
b
⬘
B
*
d
B
, and where H
a
⬘
b
⬘
is the spinor-decomposed
form of H
a
⬘
b
⬘
.
The propagation of kinematically allowed states is com-
pletely determined by the amplitudes b
a
⬘
and d
a
⬘
. However,
for purposes of comparison with experiment it is convenient
to express the result using the amplitudes associated with
active neutrinos, b
e,
,
and d
e,
,
. The relevant calculation
is somewhat lengthy and is deferred to Appendix A. It as-
sumes that the submatrix V
a
⬘
a
is unitary, and it neglects
terms that enter as small masses m
(a
⬘
)
multiplied by coeffi-
cients for Lorentz violation, since these are typically sup-
pressed. The calculation reveals that the time evolution of the
active-neutrino amplitudes is given by the equation
冉
b
a
共
t;p
ជ
兲
d
a
共
t;p
ជ
兲
冊
⫽ exp
共
⫺ ih
eff
t
兲
ab
冉
b
b
共
0;p
ជ
兲
d
b
共
0;p
ជ
兲
冊
, 共13兲
where h
eff
is the effective Hamiltonian describing flavor neu-
trino propagation. To leading order, it is given by
共
h
eff
兲
ab
⫽
兩
p
ជ
兩
␦
ab
冉
10
01
冊
⫹
1
2
兩
p
ជ
兩
冉
共
m
˜
2
兲
ab
0
0
共
m
˜
2
兲
ab
*
冊
⫹
1
兩
p
ជ
兩
冉
关共
a
L
兲
p
⫺
共
c
L
兲
p
p
兴
ab
⫺ i
冑
2p
共
⑀
⫹
兲
关共
g
p
⫺ H
兲
C
兴
ab
i
冑
2p
共
⑀
⫹
兲
*
关共
g
p
⫹ H
兲
C
兴
ab
*
关
⫺
共
a
L
兲
p
⫺
共
c
L
兲
p
p
兴
ab
*
冊
, 共14兲
where we have defined (c
L
)
ab
⬅(c⫹ d)
ab
and (a
L
)
ab
⬅(a
⫹ b)
ab
for reasons explained below. The approximate four
momentum p
may be taken as p
⫽ (
兩
p
ជ
兩
;⫺ p
ជ
) at leading
order. The Lorentz-conserving mass term results from the
usual seesaw mechanism with m
˜
2
⬅m
l
m
l
†
, where m
l
is the
light-mass matrix m
l
⫽ L⫺ DR
⫺ 1
D
T
. The complex vector
(
⑀
⫹
)
satisfies the conditions
p
共
⑀
⫹
兲
⫺ p
共
⑀
⫹
兲
⫽ i
⑀
p
共
⑀
⫹
兲
,
共
⑀
⫹
兲
共
⑀
⫹
兲
*
⫽⫺1. 共15兲
A suitable choice is (
⑀
⫹
)
⫽ (1/
冑
2)(0;
⑀
ˆ
1
⫹ i
⑀
ˆ
2
), where
⑀
ˆ
1
,
⑀
ˆ
2
are real and
兵
p
ជ
/
兩
p
ជ
兩
,
⑀
ˆ
1
,
⑀
ˆ
2
其
form a right-handed ortho-
normal triad. Note that (
⑀
⫹
)
and (
⑀
⫺
)
⬅(
⑀
⫹
)
*
is analo-
gous to the usual photon helicity basis. The appearance of
these vectors reflects the near-definite helicity of active neu-
trinos. The vectors
⑀
ˆ
1
and
⑀
ˆ
2
can be arbitrarily set by rota-
tions or equivalently by multiplying (
⑀
⫹
)
by a phase, which
turns out to be equivalent to changing the relative phase
between the basis spinors u
L
and u
R
.
Only the diagonal kinetic term in h
eff
arises in the mini-
mal SM. The term involving (m
˜
2
)
ab
encompasses the usual
massive-neutrino case without sterile neutrinos. The leading-
order Lorentz-violating contributions to neutrino-neutrino
mixing are controlled by the coefficient combinations (a
⫹ b)
ab
and (c⫹ d)
ab
. These combinations conserve the
usual SU(3)⫻ SU(2)⫻ U(1) gauge symmetry and corre-
spond to the coefficients (a
L
)
ab
and (c
L
)
ab
in the minimal
SME. Note that the orthogonal combinations (a⫺ b)
ab
and
V. ALAN KOSTELECKY AND MATTHEW MEWES PHYSICAL REVIEW D 69, 016005 共2004兲
016005-4
(c⫺ d)
ab
also conserve the usual gauge symmetry, but they
correspond to self-couplings of right-handed neutrinos and
are therefore irrelevant for leading-order processes involving
active neutrinos. The remaining coefficients (g
C)
ab
and
(H
C)
ab
appear in h
eff
through Majorana-like couplings that
violate SU(3)⫻ SU(2)⫻ U(1) gauge invariance and lepton-
number conservation. They generate Lorentz-violating
neutrino-antineutrino mixing.
Some combinations of coefficients may be unobservable,
either due to symmetries or because they can be removed
through field redefinitions 关2,4,49,50兴. For example, the trace
component
(c
L
)
is Lorentz invariant and can be ab-
sorbed into the usual kinetic term, so it may be assumed zero
for convenience. In fact, even if this combination is initially
nonzero, it remains absent from the leading-order effective
Hamiltonian because the trace of p
p
vanishes. Other ex-
amples of unobservable coefficients include certain combina-
tions of g
and H
. The antisymmetry properties g
⫽⫺g
, H
⫽⫺H
and the properties of (
⑀
⫹
)
can be
combined to prove that the physically significant combina-
tions of g
and H
are given by the relations
p
共
⑀
⫹
兲
g
⫽
兩
p
ជ
兩
共
⑀
⫹
兲
g
˜
,
p
共
⑀
⫹
兲
H
⫽
兩
p
ជ
兩
共
⑀
⫹
兲
H
˜
, 共16兲
where we have defined
g
˜
⬅g
0
⫹
i
2
⑀
0
␥
g
␥
,
H
˜
⬅H
0
⫹
i
2
⑀
0
␥
H
␥
. 共17兲
Only these combinations appear in h
eff
and are relevant to
neutrino oscillations.
In deriving Eq. 共14兲, we have focused on operators of
renormalizable dimension, which involve linear derivatives
in the equations of motion and a single power of momentum
in the Hamiltonian. Operators of nonrenormalizable mass di-
mension n⬎4 are also of potential importance 关3,4兴. They
appear as higher-derivative terms in the action, along with
corresponding complications in the equations of motion and
in the construction of the Hamiltonian. An operator of di-
mension n is associated with a term in the action involving
d⫽ n⫺ 3 derivatives, and the associated terms in the effec-
tive Hamiltonian involve d powers of the momentum. The
corresponding coefficient for Lorentz violation carries d⫹ 2
or fewer Lorentz indices, depending on the spinor structure
of the coupling and the number of momentum contractions
occurring. For the case n⬎ 4, we generically denote the co-
efficients by (k
d
)
•••
. These coefficients have mass dimen-
sion 1⫺d. Note that, depending on the theory considered,
the mechanism for Lorentz and CPT violation can cause
them to be suppressed by d-dependent powers of the Planck
scale 关3,4兴. Some effects of operators with d⫽ 2 have been
considered in the context of quantum gravity in Ref. 关30兴.
The mixing described by Eq. 共14兲 or its generalization to
operators of dimension n⬎4 can be strongly energy depen-
dent. For example, any nonzero mass-squared differences
dominate the Hamiltonian at some low-energy scale. How-
ever, while mass effects decrease with energy, Lorentz-
violating effects involving operators of renormalizable di-
mension remain constant or grow linearly with energy E and
so always dominate at high energies. For instance, the con-
tributions from a mass of 0.1 eV and a dimensionless coef-
ficient of 10
⫺ 17
are roughly comparable at an energy deter-
mined by E
2
⬃(0.1 eV)
2
/(10
⫺ 17
), or E⬃30 MeV. Below
this energy the mass term dominates, while above it the
Lorentz-violating term does. Similarly, a dimension-one co-
efficient of 10
⫺ 15
GeV has a transition energy E⬃10 keV.
More generally, effects controlled by the coefficients
(k
d
)
...
for Lorentz violation involving operators of dimen-
sion n⫽ d⫹ 3 grow as E
d
.
Although the perturbative diagonalization leading to Eq.
共14兲 is valid for dimensionless coefficients much smaller
than one and for energies much greater than any masses or
coefficients of dimension one, at sufficiently high energies
issues of stability and causality may require the inclusion of
Lorentz-violating terms of nonrenormalizable dimension in
the theory. In the context of the single-fermion QED exten-
sion, for example, a dimensionless c
00
coefficient can lead to
issues with causality and stability at energies ⬃m
fermion
/
冑
c
00
unless the effects of operators of nonrenormalizable dimen-
sion are incorporated 关4兴. A complete resolution of this issue
would be of interest but lies beyond our present scope. It is
likely to depend on the underlying mechanisms leading to
mass and Lorentz violation, and it may be complicated fur-
ther by the presence of multiple generations and the sterile
neutrino sector. We limit our remarks here to noting that the
values of the coefficients for Lorentz violation considered in
all the models in this work are sufficiently small that issues
of stability and causality can be arranged to appear only be-
yond experimentally relevant energies. In any case, the
renormalizable sector provides a solid foundation for the ba-
sic treatment of Lorentz and CPT violation in neutrinos.
C. Neutrinos in matter
In many situations, neutrinos traverse a significant volume
of ordinary matter before detection. The resulting forward
scattering with electrons, protons, and neutrons can have dra-
matic consequences on neutrino oscillations 关51兴. These mat-
ter interactions can readily be incorporated into our general
formalism. Since the effective Lagrangian in normal
matter is given by ⌬L
matter
⫽⫺
冑
2G
F
n
e
¯
e
␥
0
P
L
e
⫹ (G
F
n
n
/
冑
2)
¯
a
␥
0
P
L
a
, matter effects are equivalent to
contributions from CPT-odd coefficients
共
a
L,eff
兲
ee
0
⫽ G
F
共
2n
e
⫺ n
n
兲
/
冑
2,
共
a
L,eff
兲
0
⫽
共
a
L,eff
兲
0
⫽⫺G
F
n
n
/
冑
2, 共18兲
where n
e
and n
n
are the number densities of electrons and
neutrons. Adding these terms to the effective Hamiltonian
共14兲 therefore incorporates the effects of matter.
For some of the analyses of Lorentz violation below, it is
useful to review the treatment of matter effects in solar and
LORENTZ AND CPT VIOLATION IN NEUTRINOS PHYSICAL REVIEW D 69, 016005 共2004兲
016005-5
