4.1 Effective field theory
The most conservative approach for a framework in which to test Lorentz violation from quantum
gravity is that of effective field theory (EFT). Both the standard model and relativity can be considered
EFT’s, and the EFT framework can easily incorporate Lorentz violation via the introduction of extra
tensors. Furthermore, in many systems where the fundamental degrees of freedom are qualitatively different
than the low energy degrees of freedom, EFT applies and gives correct results up to some high energy scale.
Hence following the usual guideline of starting with known physics, EFT is an obvious place to start looking
for Lorentz violation.
4.1.1 Renormalizable operators and the Standard Model Extension
The standard model is a renormalizable field theory containing only mass dimension
operators. If we
considered the standard model plus Lorentz violating terms then we would expect a tower of operators
with increasing mass dimension. However, without some custodial symmetry protecting the
theory from Lorentz violating dimension
operators, the lower dimension operators will be
more important than irrelevant higher mass dimension operators (see Section 4.3 for details).
Therefore the first place to look from an EFT perspective is all possible renormalizable Lorentz
violating terms that can be added to the standard model. In [90
] Colladay and Kostelecky
derived just such a theory in flat space - the so-called (minimal) Standard Model Extension
(mSME).
One can classify the mSME terms by whether or not they are CPT odd or even. We first will show the
terms with manifestly
gauge invariance. After that, we shall give the coefficients
in a more practical notation that exhibits broken gauge invariance.
4.1.2 Manifestly invariant form
We deal with CPT odd terms first. The additional Lorentz violating CPT odd operators for leptons are
where
is the left-handed lepton doublet
,
is the right singlet
, and
and
are flavor indices. The coefficients
are constant vectors that can mix flavor
generations.
For quarks we have similarly
where
,
, and
. In the gauge sector we have
Here
,
, and
are the
,
, and
gauge fields, and
,
, and
are their respective field strengths. The
term in Equation (25) is usually required to
vanish as it makes the theory unstable. The remaining
coefficients have mass dimension
one.
The CPT even operators for leptons in the mSME are
while we have for quarks
For gauge fields the CPT even operators are
The coefficients for all CPT even operators in the mSME are dimensionless. While the split of CPT even
and odd operators in the mSME correlates with even and odd mass dimension, we caution the reader that
this does not carry over to higher mass dimension operators. Finally, we will in general drop the subscripts
when discussing various coefficients. These terms without subscripts are understood to be the flavor
diagonal coefficients.
Besides the fermion and gauge field content, the mSME also has Yukawa couplings between the fermion
fields and the Higgs. These CPT even terms are
Finally, there are also additional terms for the Higgs field alone. The CPT odd term is
while the CPT even terms are
This concludes the description of the mSME terms with manifest gauge invariance.
4.1.3 Practical form
Tests of the mSME are done at low energies, where the
gauge invariance has been broken. It will
be more useful to work in a notation where individual fermions are broken out of the doublet with their own
Lorentz violating coefficients. With gauge breaking, the fermion Lorentz violating terms above give the
additional CPT odd terms
and the CPT even terms
where the fermion spinor is denoted by
. Each possible particle species has its own set of coefficients. For
a single particle the
term can be absorbed by making a field redefinition
. However, in
multi-particle theories involving fermion interactions one cannot remove
for all fermions [89]. However,
one can always eliminate one of the
, i.e. only the differences between
for various particles are
actually observable.
As an aside, we note that there are additional dimension
invariant terms for fermions that
could be added to the mSME once
gauge invariance is broken. These operators are
These terms do not arise from gauge breaking of the renormalizable mSME in the previous Section 4.1.2.
However, they might arise from non-renormalizable terms in an EFT expansion. As such, technically they
should be constrained along with everything else. However, since their origin can only be from higher
dimension operators they are expected to be much smaller than the terms that come directly from the
mSME.
Current tests of Lorentz invariance for gauge bosons directly constrain only the electromagnetic sector.
The Lorentz violating terms for electromagnetism are
where the
term is CPT even and the
term is CPT odd. The
term makes
the theory unstable, so we assume it is zero from here forward unless otherwise noted (see
Section 6.3). Now that we have the requisite notation to compare Lorentz violating effects
directly with observation we turn to the most common subset of the mSME, Lorentz violating
QED.
4.1.4 Lorentz violating QED
In many Lorentz violating tests, the relevant particles are photons and electrons, making Lorentz violating
QED the appropriate theory. The relevant Lorentz violating operators are given by Equation (32, 33, 35).
The dispersion relation for photons will be useful when deriving birefringence constraints on
. If
, spacetime acts as a anisotropic medium, and different photon polarizations propagate at
different speeds. The two photon polarizations, labelled
, have the dispersion relation [179
]
where
,
,
, and
. Strong limits can be
placed on this birefringent effect from astrophysical sources [179
], as detailed in Section 6.3.
A simplifying assumption that is often made is rotational symmetry. With rotational symmetry all the
Lorentz violating tensors must be reducible to products of a vector field, which we denote by
, that
describes the preferred frame. We will normalize
to have components
in the preferred
frame, placing constraints on the coefficients instead. The rotationally invariant extra terms are
for electrons and
for photons. The high energy (
) dispersion relations for the mSME will be necessary later.
To lowest order in the Lorentz violating coefficients they are
where, if
is the helicity state of the electron,
, and
.
The positron dispersion relation is the same as Equation (39) with the replacement
, which will
change only the
term.
In the QED sector dimension five operators that give rise to
type dispersion have also been
investigated by [230
] with the assumption of rotational symmetry:
where
are the usual left and right projection operators and
is the
dual of
. One should note that these operators violate CPT. Furthermore, they are not the only
dimension five operators, a mistake that has sometimes been made in the literature. For example, we could
have
. These other operators, however, do not give rise to
dispersion as they are
CPT even.
The birefringent dispersion relation for photons that results from Equation (40) is
for right (
) and left (
) circularly polarized photons, where
. Similarly, the high energy
electron dispersion is
where
.
We note that since the dimension five operators violate CPT, they give rise to different dispersions for
positrons than electrons. While the coefficients for the positive and negative helicity states of an electron are
and
, the corresponding coefficients for a positron’s positive and negative helicity states are
and
. This will be crucially important when deriving constraints on these operators from
photon decay.