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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 < 4 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 < 4 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 [90Jump To The Next Citation Point] 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 SU (3) × SU (2) × U (1) 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

- (a ) L--gmL - (a ) R--gmR , (23) L mAB A B R mAB A B
where LA is the left-handed lepton doublet LA = (nA lA)L,RA is the right singlet (lA)R, and A and B are flavor indices. The coefficients (aL,R)mAB are constant vectors that can mix flavor generations.6 For quarks we have similarly
-- -- -- - (aQ)mAB QAgmQB - (aU )mAB U AgmUB - (aD)mAB DAgmDB, (24)
where QA = (uA dA)L, UA = (uA)R, and DA = (dA)R. In the gauge sector we have
(k ) Bk + (k ) ekcmnB B 0k 1 k ( c mn ) kcmn 2- +(k2)ke Tr WcWmn + 3 igWcWmWn ( ) +(k3)kekcmn Tr GcGmn + 2ig3GcGmGn . (25) 3
Here Bm, Wm, and Gm are the U (1), SU (2), and SU (3) gauge fields, and Bmn, Wmn, and Gmn are their respective field strengths. The k0 term in Equation (25View Equation) is usually required to vanish as it makes the theory unstable. The remaining a,k coefficients have mass dimension one. The CPT even operators for leptons in the mSME are
-- <--> -- <--> 1i(cL)mnAB LAgm Dn LB + 1i(cR)mnABRAgm Dn RB, (26) 2 2
while we have for quarks
1- -- m <-->n 1- -- m <--> n 1- -- m <-->n 2i(cQ)mnABQAg D QB + 2i(cU)mnAB UAg D UB + 2 i(cD)mnAB DAg D DB. (27)
For gauge fields the CPT even operators are
- 1(kB)kcmnBkcBmn - 1(kW )kcmn Tr(W kcW mn) - 1-(kG)kcmn Tr(GkcGmn). (28) 4 2 2

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 A, B 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

1[ -- mn -- * mn -- mn ] - 2 (HL)mnAB LAfs RB + (HU )mnABQAf s UB + (HD)mnAB QAfs DB + h.c. (29)
Finally, there are also additional terms for the Higgs field alone. The CPT odd term is
i(kf)mf†Dmf + h.c., (30)
while the CPT even terms are
[ ] 1- (kff)mn (Dmf) †Dnf - (kfB)mn f†fBmn - (kfW )mn f†Wmnf + h.c. (31) 2
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 SU (2) 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

- a ygmy - b yg- gmy (32) m m 5
and the CPT even terms
-- -- <--> -- <--> - 1Hmn ysmny + 1icmnygm Dn y + 1-idmnyg5gm Dn y, (33) 2 2 2
where the fermion spinor is denoted by y. Each possible particle species has its own set of coefficients. For a single particle the a m term can be absorbed by making a field redefinition y ---> e-ia.xy. However, in multi-particle theories involving fermion interactions one cannot remove am for all fermions [89]. However, one can always eliminate one of the am, i.e. only the differences between am for various particles are actually observable.

As an aside, we note that there are additional dimension < 4 U (1) invariant terms for fermions that could be added to the mSME once SU (2) gauge invariance is broken. These operators are

1 --<--> 1 -- <--> i -- <--> -en y Dn y - -fnyg5 Dn y + -gcmnyscm Dn y. (34) 2 2 4
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.7 Current tests of Lorentz invariance for gauge bosons directly constrain only the electromagnetic sector. The Lorentz violating terms for electromagnetism are
1- kc mn 1- k c mn - 4(kF )kcmnF F + 2(kAF ) ekcmnA F , (35)
where the kF term is CPT even and the kAF term is CPT odd. The kAF 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 (32View Equation, 33View Equation, 35View Equation). The dispersion relation for photons will be useful when deriving birefringence constraints on kF. If kF /= 0, spacetime acts as a anisotropic medium, and different photon polarizations propagate at different speeds. The two photon polarizations, labelled e±, have the dispersion relation [179Jump To The Next Citation Point]

| | E = (1 + r ± s)|--->p |, (36) | |
where r = 1~ka 2 a, s2 = 1(~k )2- r2 2 ab, ~k = (k ) p^g ^pd ab F agbd, and p^a = pa/|--->p |. Strong limits can be placed on this birefringent effect from astrophysical sources [179Jump To The Next Citation Point], 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 ua, that describes the preferred frame. We will normalize ua to have components (1,0,0,0) in the preferred frame, placing constraints on the coefficients instead. The rotationally invariant extra terms are

-- m 1- --m <-->n 1- -- m <--> n - bumyg5g y + 2icumun yg D y + 2 idumun yg5g D y (37)
for electrons and
- 1(k )u j u F kcFmn (38) 4 F k cm n
for photons. The high energy (EPl » E » m) dispersion relations for the mSME will be necessary later. To lowest order in the Lorentz violating coefficients they are
2 2 2 (1) (2) 2 E = m + p + fe p + fe p , 2 ( (2)) 2 (39) E = 1 + fg p ,
where, if s = ± 1 is the helicity state of the electron, (1) (2) fe = - 2bs,fe = - (c - ds), and (2) fg = kF/2. The positron dispersion relation is the same as Equation (39View Equation) with the replacement p-- > - p, which will change only the f (1e) term.

In the QED sector dimension five operators that give rise to n = 3 type dispersion have also been investigated by [230Jump To The Next Citation Point] with the assumption of rotational symmetry:

-q--umFma(u .@)(un ~Fna) + -1--umygm(jLPL + jRPR)(u .@)2y, (40) EPl EPl
where P = 1/2(1 ± g5) L,R are the usual left and right projection operators and ~F ab = 1eabgdF 2 gd is the dual of Fab. 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 uaubyDaDby. These other operators, however, do not give rise to n = 3 dispersion as they are CPT even.

The birefringent dispersion relation for photons that results from Equation (40View Equation) is

fg(3)E3 E2 = p2 ± ------- (41) EPl
for right (+) and left (-) circularly polarized photons, where f(g3)= 2q. Similarly, the high energy electron dispersion is
f (3) E3 E2 = m2 + p2 + -e(R,L)---, (42) EPl
where (3) fe(R,L) = 2jR,L.8 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 2j R and 2j L, the corresponding coefficients for a positron’s positive and negative helicity states are - 2jL and - 2jR. This will be crucially important when deriving constraints on these operators from photon decay.
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