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6.8 Neutrinos

Neutrinos can provide excellent probes of Lorentz violation, as their mass is much smaller than any other known particle. To see this consider the modified dispersion framework. For an electron with n = 3 and n = 4 dispersion the energies at which Lorentz violation can become appreciable are at 10TeV and 5 10 TeV, respectively. However, for a neutrino with mass even at 1 eV the corresponding energies are only 1 GeV for n = 3 and 1 TeV for n = 4, well within the realm of accelerator physics. The most sensitive tests of Lorentz violation in the neutrino sector come from neutrino oscillation experiments, which we now describe. For a more comprehensive overview of neutrino mixing, see for example [102165].

6.8.1 Neutrino oscillations

Lorentz violating effects in the neutrino sector have been considered by many authors [86182Jump To The Next Citation Point181Jump To The Next Citation Point180Jump To The Next Citation Point83Jump To The Next Citation Point5518Jump To The Next Citation Point126Jump To The Next Citation Point19182]. To illustrate how Lorentz violation affects neutrino propagation, we consider the simplest case, where the limiting speeds for mass eigenstates of the neutrino are different, i.e. neutrinos have dispersions

( ) E2 = 1 + f(2) p2 + m2 , (104) ni i
where i denotes the energy eigenstate. In this case, the energy eigenstates are also the mass eigenstates (this is not necessarily the case with general Lorentz violation). This is a special case of the neutrino sector of the mSME if we assume that c00 n is flavor diagonal and is the only non-zero term. For relativistic neutrinos, we can expand the energy to be
( ) f(2ni)- m2i- E = 1 + 2 p + 2p . (105)
Now consider a neutrino produced via a particle reaction in a definite flavor eigenstate I with energy E. We denote the amplitude for this neutrino to be in a particular energy eigenstate i by the matrix UIi, where sum † iUJiUIi = dIJ. The amplitude for the neutrino to be observed in another flavor eigenstate J at some distance L,T from the source is then
sum sum [ ( (2) 2) ] A = U † exp [- i(ET - pL)] U ~~ U† exp - i fni-E + m-i L U (106) IJ Ji Ii Ji 2 2E Ii i i
for relativistic neutrinos. If we define an “effective mass” Ni as
2 2 (2) 2 Ni = m i + fni E , (107)
then the probability 2 PIJ = |AIJ| can be written as
( ) ( ) sum 2 dNi2jL dNi2jL PIJ = dIJ- 4FIJijsin ------ + 2GIJijsin ------ , (108) i,j>i 4E 2E
where dN = N 2- N 2 ij i j, and F IJij and G IJij are functions of the U matrices.

We can immediately see from Equation (108View Equation) that Lorentz violation can have a number of consequences for standard neutrino oscillation experiments. The first is simply that neutrino oscillation still occurs even if the mass is zero. In fact, some authors have proposed that Lorentz violation could be partly responsible for the observed oscillations [181Jump To The Next Citation Point]. Oscillations due to the type of Lorentz violation above vary as EL [181]. Current data support neutrino oscillations that vary as a function of L/E [35125], so it seems unlikely that Lorentz violation could be the sole source of neutrino oscillations. It is possible, however, that Lorentz violation may explain some of the current problems in neutrino physics by giving a contribution in addition to the mass term. For example it has been proposed in [182] that Lorentz violation might explain the LSND (Liquid Scintillator Neutrino Detector) anomaly [36]30, which is an excess of - - nm --> nm events that cannot be reconciled with other neutrino experiments [129]. We note that the above model for Lorentz violating effects in neutrino oscillations is perhaps the simplest case. In the neutrino sector of the mSME there can be more complicated energy dependence, directional dependence, and new oscillations that do not occur in the standard model. For a discussion of these various possibilities see [180]. The difference in speeds between electron and muon neutrinos was bounded in [88] to be |f(n2e)- fn(2m)|< 10 -22. Oscillation data from Super Kamiokande have improved this bound to O(10 - 24) [114]. Current neutrino oscillation experiments are projected to improve on this by three orders of magnitude, giving limits on maximal speed differences of order -25 10 [126]. For comparison, the time of flight measurements from supernova 1987A constrain (2) (2) |fni - fg |< 10 -8 [265]. Neutrino oscillations are sensitive enough to directly probe non-renormalizable Lorentz violating terms. In [69] current neutrino oscillation experiments are shown to yield bounds on dimension five operators stringent enough that the energy scale suppressing the operator must be a few orders of magnitude above the Planck energy. Such operators are therefore very unlikely in the neutrino sector. Ultra-high energy neutrinos, when observed, will provide further information about neutrino Lorentz violation. For example, flavor oscillations of ultra-high energy neutrinos at 1021 eV propagating over cosmic distances would be able to probe Lorentz violating dispersion suppressed by seven powers of EPl [83] (or more if the energies are even higher).

Additionally, neutrino Lorentz violation can modify the energy thresholds for reactions involving neutrinos, which can have consequences for the expected flux of ultra-high energy neutrinos for detectors such as ICECUBE. The expected flux of ultra-high energy neutrinos is bounded above by the Bahcall-Waxman bound [273] if the neutrinos are produced in active galactic nuclei or gamma ray bursters. It has been shown [18] that Lorentz violation can in fact raise (or lower) this bound significantly. A higher than expected ultra-high energy neutrino flux therefore could be a signal of Lorentz violation.

6.8.2 Neutrino Čerenkov effect

Finally, neutrinos can also undergo a vacuum Čerenkov effect. Even though a neutrino is neutral there is a non-zero matrix element for interaction with a photon as well as a graviton. Graviton emission is very strongly suppressed and unlikely to give any useful constraints. The matrix element for photon emission, while small, is still larger than that for graviton emission and hence the photon Čerenkov effect is more promising. The photon-neutrino matrix element can be split into two channels, a charge radius term and a magnetic moment term. The charge radius interaction is suppressed by the W mass, leading to a reaction rate too low for current neutrino observatories such as AMANDA to constrain n = 3,4 Lorentz violation. However, the rate from the charge radius interaction scales strongly with energy, and it has been estimated [154] that atmospheric PeV neutrinos may provide good constraints on n = 3 Lorentz violation. The magnetic moment interaction has not yet been conclusively analyzed, so possible constraints from the magnetic moment interaction are unknown. In Lorentz invariant physics, the magnetic moment term is suppressed by the small neutrino mass, so energy loss rates are likely small. However, it should be noted that some Lorentz violating terms in an effective field theory give rise to effective masses that scale with energy. These might be much larger than the usual neutrino mass at high energies, yielding a large neutrino magnetic moment.

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