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4.3 Symmetry and relevant/irrelevant Lorentz violating operators

The above Section 4.2 illustrates a crucial issue in searches for Lorentz violation that are motivated by quantum gravity: Why is Lorentz invariance such a good approximate symmetry at low energies? To illustrate the problem, let us consider the standard assumption made in much of the work on Lorentz violation in astrophysics - that there exist corrections to particle dispersion relations of the form (n) n n- 2 f p EPl with n > 3 and (n) f of order one. Without any protective symmetry, radiative corrections involving this term will generate dispersion terms of the form f (n)p2 + EPlf (n)p. These terms are obviously ruled out by low energy experiment.10 Accordingly, the first place to look for Lorentz violation is in terrestrial experiments using the standard model extension rather than astrophysics with higher dimension operators. However, no evidence for such violation has been found. The absence of lower dimension operators implies that either there is a fine tuning in the Lorentz violating sector [91Jump To The Next Citation Point], some other symmetry is present that protects the lower dimension operators, or Lorentz invariance is an exact symmetry. It is always possible that Lorentz violation is finely tuned - there are other currently unexplained fine-tuning problems (such as the cosmological constant) in particle physics. However, it would be far preferable if there was some symmetry or partial symmetry that could naturally suppress/forbid lower dimension operators. For rotation invariance, a discrete remnant of the original symmetry is enough. For example, hypercubic symmetry on a lattice is enough to forbid dimension four rotation breaking operators for scalars.11 No physically meaningful equivalent construction exists for the full Lorentz group, however (see [223] for a further discussion of this point). A discrete symmetry that can forbid some of the possible lower dimension operators is CPT. A number of the most observationally constrained operators in the mSME are CPT violating, so imposing CPT symmetry would explain why those operators are absent. However, the CPT even operators in the mSME are also very tightly bounded, so CPT cannot completely resolve the naturalness problem either. Supersymmetry is currently the only known symmetry (other than Lorentz symmetry itself) that can protect Lorentz violating operators of dimension four or less [137Jump To The Next Citation Point15965Jump To The Next Citation Point], much as SUSY protects some lower dimension operators in non-commutative field theory [216]. If one imposes exact SUSY on a Lorentz violating theory, the first allowed operators are of dimension five [137]. These dimension five operators do not induce n = 3 type dispersion like the operators (40View Equation). Instead, in a rotationally invariant setting they produce dispersion relations of the form
( ) 2 2 2 (1) E-- E = p + m 1 + f EPl + ... . (45)
Such modifications are completely unobservable in astrophysical processes, although high precision terrestrial experiments can still probe them. Dimension 6 SUSY operators in SQED also yield dispersion relations that are untestable by high energy astrophysics [65Jump To The Next Citation Point].

Fortunately, we do not live in a SUSY world, so it may be that upon SUSY breaking appropriate sized operators at each mass dimension are generated. This question has recently been explored in [65Jump To The Next Citation Point]. For CPT violating dimension five SUSY operators in SQED, the authors find that SUSY breaking yields dimension three operators of the form am2s/M, where ms is the SUSY breaking scale, M is the scale of Lorentz violation, and a is an O(1) coefficient. For ms as light as it could be (around 100 GeV), spin polarized torsion balances (see Section 5.4) are able to place limits on M between 5 10 10 - 10 EPl. It therefore is probable that these operators are observationally unacceptable. However, dimension five SUSY operators are CPT violating, so a combination of CPT invariance and SUSY would forbid Lorentz violating operators below dimension six. The low energy dimension four operators induced by SUSY breaking in the presence of dimension six operators would then presumably be suppressed by 2 2 m s/M. This is enough suppression to be compatible with current experiment if M is at the Planck scale and ms < 1 TeV.

Another method by which Lorentz violation can occur but might have small dimension < 4 matter operators is via extra dimension scenarios. For example, in [70] a braneworld scenario was considered where four-dimensional Lorentz invariance was preserved on the brane but broken in the bulk. The only particle which can then directly see Lorentz violation is the graviton - the matter fields, being trapped on the brane, can only feel the bulk Lorentz violation through graviton loops. The induced dimension < 4 operators can be quite small, depending on the exact extra-dimension scenario considered. Note though that this approach has been criticized in [91], whose authors argue that significant Lorentz violation in the infrared would still occur.

In summary, the current status of Lorentz violation in EFT is mildly disconcerting for a phenomenologist (if one really wants to believe in Lorentz violation). From an EFT point of view, without custodial symmetries one would expect that we would have seen signs of Lorentz violation by now. Imposing SUSY + CPT or a braneworld scenario may fix this problem, but then we are left with a model with more theoretical assumptions. Furthermore a SUSY + CPT model is unlikely to ever be testable with astrophysics experiments and requires significant improvement in terrestrial experiments to be seen [65]. Fortunately, since this is a phenomenological review we can blithely ignore the above considerations and simply classify and constrain all possible operators at each mass dimension. This is also the safest approach. After all, we are searching for a possible signal from the mysterious realm of quantum gravity and so must be careful about overly restricting our models.

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