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3.1 Systematic modified dispersion

Perhaps the simplest kinematic framework for Lorentz violation in particle based experiments is to propose modified dispersion relations for particles, while keeping the usual energy-momentum conservation laws. This was the approach taken in much of the work using astrophysical phenomena in the late 1990’s. In a given observer’s frame in flat space, this is done by postulating that the usual Lorentz invariant dispersion law 2 2 2 E = p + m is replaced by some function 2 E = F (p,m). In general the preferred frame is taken to coincide with the rest frame of the cosmic microwave background. Since we live in an almost Lorentz invariant world (and are nearly at rest with respect to the CMBR), in the preferred frame F (p,m) must reduce to the Lorentz invariant dispersion at small energies and momenta. Hence it is natural to expand F (p,m) about p = 0, which yields the expression
2 2 2 (1)i (2) i j (3) i j k E = m + p + F i p + Fij p p + F ijkp p p + ..., (13)
where the constant coefficients (n) Fij...n are dimensionful and arbitrary but presumably such that the modification is small. The order n of the first non-zero term in Equation (13View Equation) depends on the underlying model of quantum gravity taken. Since the underlying motivation for Lorentz violation is quantum gravity, it is useful to factor out the Planck energy in the coefficients (n) F and rewrite Equation (13View Equation) as
(3) E2 = m2 + p2 + E f (1)pi + f (2)pipj + fijkpipjpk + ..., (14) Pl i ij EPl
such that the coefficients (n) f are dimensionless.

In most of the literature a simplifying assumption is made that rotation invariance is preserved. In nature, we cannot have the rotation subgroup of the Lorentz group strongly broken while preserving boost invariance. Such a scenario leads immediately to broken rotation invariance at every energy which is unobserved.2 Hence, if there is strong rotation breaking there must also be a broken boost subgroup. However, it is possible to have a broken boost symmetry and unbroken rotation symmetry. Either way, the boost subgroup must be broken. Phenomenologically, it therefore makes sense to look first at boost Lorentz violation and neglect any violation of rotational symmetry. If we make this assumption then we have

(3) E2 = m2 + p2 + E f (1)|p|+ f(2)p2 + f--|p|3 + .... (15) Pl EPl
There is no a priori reason (from a phenomenological point of view) that the coefficients in Equation (15View Equation) are universal (and in fact one would expect the coefficients to be renormalized differently even if the fundamental Lorentz violation is universal [6]). We will therefore label each (n) f as (n) f A where A represent particle species.

Modified dispersion and effective field theory
Effective field theory (EFT) is not applicable if one wishes to stick to straight kinematics, however the EFT implications for modified dispersion are so significant that they must be considered. As will be shown in detail in Section 4.1, universal dispersion relations cannot be imposed for all n from an EFT standpoint. For example, rotationally invariant n = 1,3 type dispersion cannot be imposed universally on photons [90Jump To The Next Citation Point230Jump To The Next Citation Point]. The operators that give rise to n = 1,3 dispersion are CPT violating and induce birefringence (the dispersion modifications change sign based on the photon helicity). Since EFT requires different coefficients for particles with different properties and there is no underlying reason why all coefficients should be the same, it is phenomenologically safest when investigating modified dispersion to assume that each particle has a different dispersion relation. After this general analysis is complete the universal case can be treated with ease.

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