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  Colladay and Kostelecky derived just such a theory in flat space - the so-called (minimal) Standard Model Extension (mSME).5 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.
We deal with CPT odd terms first. The additional Lorentz violating CPT odd operators for leptons are6 For quarks we have similarly
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
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. 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 are7 Current tests of Lorentz invariance for gauge bosons directly constrain only the electromagnetic sector. The Lorentz violating terms for electromagnetism are
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 , 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
In the QED sector dimension five operators that give rise to type dispersion have also been investigated by  with the assumption of rotational symmetry:
The birefringent dispersion relation for photons that results from Equation (40) is8 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.
© Max Planck Society and the author(s)