Neutrino Oscillations, Lorentz=CPT Violation, and Dark Energy

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provided by Caltech Authors PHYSICAL REVIEW D 80, 123522 (2009) Neutrino oscillations, Lorentz=CPT violation, and dark energy

Shin’ichiro Ando,1 Marc Kamionkowski,1 and Irina Mocioiu2 1California Institute of Technology, Mail Code 350-17, Pasadena, California 91125, USA 2Pennsylvania State University, 104 Davey Lab, University Park, Pennsylvania 16802, USA (Received 22 October 2009; published 17 December 2009) If dark energy (DE) couples to neutrinos, then there may be apparent violations of Lorentz=CPT invariance in neutrino oscillations. The DE-induced Lorentz=CPT violation takes a speciﬁc form that introduces neutrino oscillations that are energy independent, differ for particles and antiparticles, and can lead to novel effects for neutrinos propagating through matter. We show that ultra-high-energy neutrinos may provide one avenue to seek this type of Lorentz=CPT violation in - oscillations, improving the current sensitivity to such effects by 7 orders of magnitude. Lorentz=CPT violation in electron-neutrino oscillations may be probed with the zenith-angle dependence for high-energy atmospheric neutrinos. More compelling evidence for a DE-neutrino coupling would be provided by a dependence of neutrino oscillations on the direction of the neutrino momentum relative to our peculiar velocity with respect to the cosmic microwave background rest frame. While the amplitude of this directional dependence is expected to be small, it may nevertheless be worth seeking in current data and may be a target for future neutrino experiments.

DOI: 10.1103/PhysRevD.80.123522 PACS numbers: 98.80.k, 14.60.Pq, 95.36.+x, 95.85.Ry

I. INTRODUCTION energy is highly restricted under fairly general assump- tions.1 The coupling engenders an additional source for The accelerated cosmic expansion [1] poses difficult neutrino mixing (e.g., Ref. [10]), resulting in neutrino questions for theoretical physics [2–4]. Is it simply due oscillations with a different energy dependence than vac- to a cosmological constant? Is some new negative-pressure uum oscillations and different oscillation probabilities for dark energy (DE) required? Is general relativity modified at neutrinos and antineutrinos. While similar large distance scales? The major thrust of the empirical Lorentz=CPT-violating oscillations have been considered assault on these questions has been to determine whether before [11–13], we emphasize here that cosmic accelera- the expansion history and growth of large-scale structure tion dictates a specific form for such effects. are consistent with a cosmological constant or require Data from Super-Kamiokande and K2K [14] and something more exotic [5]. AMANDA/IceCube [15] already tightly constrain However, it may be profitable to explore whether there CPT-violating parameters for - mixing, and those are other experimental consequences of the new physics— from solar-neutrino experiments and KamLAND [16]do which we collectively refer to as DE, although it may so for e- mixing. However, the effects of DE-induced involve a modification of gravity rather than the introduc- CPT violation become more significant at higher energies tion of some new substance—responsible for accelerated [17]. Here, we show that next-generation measurements of expansion. If cosmic acceleration is due to a cosmological ultra-high-energy neutrinos produced by spallation of constant (i.e., if general relativity is valid and the equation- ultra-high-energy cosmic rays will increase the sensitivity of-state parameter is w ¼1), then the vacuum is Lorentz to CPT-violating - oscillations by 7 orders of magni- invariant. If, however, something else is going on, then the CPT ‘‘vacuum’’ has a preferred frame: the rest frame of the tude. We also show that these -violating couplings cosmic microwave background (CMB). If, moreover, dark may lead to novel effects in the zenith-angle dependence energy couples somehow to standard-model particles, then for atmospheric neutrinos in the 100 GeV range. While such CPT-violating effects, if detected, could be there may be testable (apparent) violations of Lorentz CPT invariance. For example, if DE is coupled to the pseudo- attributed simply to intrinsic violation in fundamental scalar FF~ of electromagnetism [6], there may be a ‘‘cos- physics, not related to DE, a DE-neutrino coupling further mological birefringence’’ that rotates the linear predicts a directional effect: the neutrino-mixing parame- polarization of cosmological photons; CMB searches for ters depend on the neutrino-propagation direction relative such a rotation [7] constrain this rotation to be less than a to our peculiar velocity with respect to the CMB rest frame.

few degrees [8]. 1 =CPT The coupling of neutrinos to dark energy has also been Here, we explore DE-induced Lorentz -violating considered in the context of ‘‘mass-varying neutrinos’’ [9], but effects in the neutrino sector. We show that the form of a that implementation of the DE-neutrino coupling does not lead Lorentz-violating coupling between neutrinos and dark to the type of Lorentz=CPT-violating effects we discuss here.

1550-7998=2009=80(12)=123522(9) 123522-1 Ó 2009 The American Physical Society ANDO, KAMIONKOWSKI, AND MOCIOIU PHYSICAL REVIEW D 80, 123522 (2009) m m y m im P = While this signature will likely remain elusive even to where R ¼ð LÞ ¼ þ 5, L ¼ð1 5Þ 2, and P = N N m next-generation experiments, it would, if detected, be R ¼ð1 þ 5Þ 2. The 2 2 mass matrix R is written strong support for DE beyond a cosmological constant. It in terms of N N matrices L, R, and D, through is therefore worth considering as a long-range target for LD m : future neutrino experiments. It may also be worthwhile to R ¼ DT R (3) search current data in case an implementation of DE- neutrino coupling different from that we consider here Here, R and L are the right- and left-handed Majorana leads to a different energy dependence for these directional neutrino masses (L ¼ 0 is required if electroweak gauge effects. We therefore work out explicitly the directional invariance is preserved), and D is the Dirac-mass matrix. dependence to aid experimentalists who may wish to look The R and L matrices are required to be symmetric, and R, for such correlations in current data. L, and D can most generally be complex. Below, we first derive in Sec. II the form of the Lorentz=CPT violation allowed by a DE-neutrino cou- B. Dark-energy–induced Lorentz violation pling and discuss the resulting neutrino-oscillation physics. Lorentz violation in Eq. (2) is parametrized by the four- In Sec. III we apply the formalism to cosmogenic ultra- high-energy neutrinos, and obtain projected sensitivities of vectors a , b , and the antisymmetric tensor H . The a b CPT future detectors to these effects in - oscillations. In parameters and are both and Lorentz violating, H CPT Sec. IV we discuss matter-induced effects for e oscilla- while is Lorentz violating but conserving. While tions in high-energy atmospheric neutrinos in the presence these parameters are nonzero for the most general =CPT of Lorentz-invariance–violating mixings. Concrete formu- Lorentz -violating Dirac equation [13], the allowable a b H las for the directional dependence on oscillation probabil- forms for , , and are highly restricted if the =CPT ities are given in Sec. V. Finally, we discuss some Lorentz violation is induced by coupling to dark theoretical implications in Sec. VI and summarize and energy. conclude in Sec. VII. The smallness of the CMB quadrupole demands that the three-dimensional hypersurfaces of constant DE density II. THE DARK-ENERGY–NEUTRINO COUPLING must be closely aligned with those of constant CMB temperature [18]. The preferred frame associated with the A. General formalism cosmic expansion is then parametrized by a unit four- Following Ref. [13], the neutrino fields are denoted by vector l , which is orthogonal to surfaces of constant l Dirac spinors fe;;; gand their charge conjugates CMB temperature; i.e., in the CMB rest frame, it is ¼ c T ; ; ; by fec ;c ;c ; g, where xc x Cx is the ð1 0 0 0Þ. The symmetry of the problem thus dictates that a l b l H charge-conjugated spinor, and C is the charge-conjugation / and / . The tensor is antisymmetric, matrix. The 2N fields (where N is the number of flavors) and there is no way to construct an antisymmetric tensor H H ¼ and their conjugates are arranged in a single object A, from a single four-vector; we thus expect 0 for where A ranges over e; ; ; , ec;c;c; . DE-neutrino coupling. Furthermore, since neutrinos are produced and interact With a canonical kinetic term in the neutrino =CPT in weak eigenstates, it is only the combination ðaLÞab Lagrangian, the most general Lorentz -violating 2 a b Dirac equation is ð þ Þab that is relevant for neutrino phenomenology. Thus, the Lorentz/CPT violation induced in neutrino phys- ði @ MABÞB ¼ 0; (1) ics can be parametrized entirely by a single four-vector- where valued ðaLÞab / l matrix in the flavor space. M m im a b AB AB þ 5AB 5 þ AB þ AB 5 C. Neutrino oscillations 1 þ H : (2) 2 AB The propagation of the flavor eigenstates is then de- The usual mass terms are m þ im55 mLPL þ mRPR, scribed by an effective Hamiltonian ! p m2 = p a p =p h ab þð~ Þab 2 þð LÞab 0 ; ð effÞab ¼ 2 (4) 0 pab þðm~ Þab=2p ðaLÞab p=p

2 y where the flavor indices a and b run over the flavor jpj, with p the neutrino momentum, and m~ mlml is the c c c 1 T eigenstates e, , and and e , , and . Here, p usual mass matrix, with ml ¼ L DR D . Equation (4) has several implications: (i) Since the 2Additional possibilities arise with a noncanonical kinetic matrix is block-diagonal, there is no mixing between neu- term; we comment briefly on possible consequences below. trinos and antineutrinos (as may arise in more general

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Lorentz-violating scenarios [13]). (ii) Since aL appears neutrino oscillations are independent of the neutrino direc- with opposite sign in the neutrino and antineutrino entries tion. However, the Solar System moves with respect to the 1 in the Hamiltonian, a nonzero aL implies (apparent) CPT CMB rest frame with a velocity v ’ 370 km s . DE- violation—i.e., the propagation of neutrinos and antineu- induced neutrino oscillations will therefore depend on trinos is not the same. Thus, for example, if the anomalous ðaLÞ p / Eð1 v p^Þ, where p^ is the neutrino- LSND results had stood, the CPT-violating explanations propagation direction, and v is our peculiar velocity with (e.g., Ref. [19]) for them [20] may have implied DE- respect to the CMB rest frame. There will thus be an annual neutrino coupling. (iii) The mixing induced by DE- modulation in solar-neutrino oscillations, a diurnal modu- neutrino coupling is energy independent (like in the lation in laboratory neutrino-mixing experiments, and a Mikheev-Smirnov-Wolfenstein, or MSW, effect [21]), as direction dependence in oscillations of cosmogenic opposed to vacuum mixing, which declines as E1. Thus, neutrinos. these effects will become increasingly visible at higher Since neutrino mixing arises only as a consequence of energies. The detailed form of CPT violation implied by the traceless part of the propagation Hamiltonian, the DE- this effect is also thus different than that obtained with neutrino coupling must (like the vacuum mass matrix) be different m2 for neutrinos and antineutrinos. (iv) There flavor-violating if neutrino oscillations are to be affected. may also be novel effects for neutrinos propagating through matter, an effect we discuss further in Sec. IV D. Two-flavor oscillations below. The evolution equation for DE-induced two-flavor mix- Finally, (v) the neutrino oscillations induced by DE- ing is of the form neutrino coupling are frame dependent. If the observer is in the rest frame of the CMB, then ðaLÞ p / E, and ! d 1 m2 m ð v p^Þ m2 þ m ð v p^Þ ei i a 2E cos2 v eff 1 cos2 d 2E sin2 v eff 1 sin2 d a ; ¼ m2 i m2 (5) dt b m v p e m v p b 2 2E sin2 v þ effð1 ^Þ sin2 d 2E cos2 v þ effð1 ^Þ cos2 d m where eff is an effective mass parameter, and d and are Note that this time sin2 does indeed depend on ,asitis a mixing angle and relative phase in the DE-neutrino not the overall phase, but the relative one, that cannot be coupling matrix, respectively. There is also the usual vac- rotated away by redefinition of wave functions. The oscil- m2 L M1 uum mass difference (squared) and the vacuum- lation length is then osc ¼ . In the absence of DE- mixing angle v. The analogous propagation equations neutrino coupling, we recover the standard oscillation L E= m2 for antineutrinos are the same as Eq. (5) with the replace- length osc ¼ 4 and mixing angle ¼ v.If m m m m2= E L ments eff ! eff and ! , the changes in sign a eff 2 , then the oscillation length is osc ¼ CPT m 1 v p 1 manifestation of violation. 2 eff ð1 ^Þ . Recall that if the propagation Hamiltonian is of the form In general, m can be either positive or negative. eff cos2 sin2 However, from the symmetry of the Hamiltonian, the h ¼ M ; (6) relevant parameter space can be limited to m 0, 0 sin2 cos2 eff d =4, and 0 [14]. then the probability for one species of neutrino to convert Thus far, no deviations from standard three-flavor neu- to a different neutrino after a distance L is trino oscillations have been discovered in experimental 2 2 data (except LSND [20]), and this yields constraints on Pða ! bÞ¼sin 2sin ðMLÞ: (7) CPT m -violating parameters, especially for eff. By analyz- Here, we have neglected the CP-violating phase in Eq. (6) ing solar-neutrino and KamLAND data, Ref. [16] obtained m < : 20 because it does not affect the oscillation probability. The an upper limit of eff 3 1 10 GeV for e- mix- propagation Hamiltonian in Eq. (5) can be written in the ing. Atmospheric and accelerator data provide an upper m < 23 form of Eq. (6) with the following relations [22]: limit for - mixing of eff 5 10 GeV [14]. m2 2 m2 ð1 v p^Þ2 m2 M2 ¼ þ eff þ m ð1 v p^Þ III. ULTRA-HIGH-ENERGY NEUTRINOS 4E 4 4E eff A. Prediction ðcos2v cos2d þ sin2v sin2d cosÞ; (8) Given that DE-induced neutrino mixing becomes in- m2 2 m2 v p 2 creasingly important, relative to vacuum mixing, at high 2 1 2 effð1 ^Þ 2 sin 2 ¼ sin 2v þ sin 2d M2 4E 4 energies, the DE-neutrino coupling can be probed with ultra-high-energy cosmogenic neutrinos. These neutrinos m2 m v p : are produced by the interaction of ultra-high-energy þ effð1 ^Þ sin2 v sin2 d cos (9) 4E cosmic-ray protons with CMB photons [23]:

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þ þ þ p ! n ! n ! ne e: (10) where ðEÞ is the detector exposure to neutrinos in units of cm2 ssr, and it generally depends on neutrino energy. Here, The fact that the Greisen-Zatsepin-Kuzmin cutoff [24] has tot 0 KE2 we assume ¼ ¼ with a normalization con- now been observed by the HiRes [25] and Auger [26] K E 17 E 19 stant , min ¼ 2 10 eV, and max ¼ 2 10 eV. Collaborations implies that this interaction must be occur- This provides a good approximation for the spectrum of ring. And if so, there must be a population of cosmogenic 17 20 cosmogenic neutrinos (e.g., Ref. [29]). For simplicity we neutrinos with energies 10 –10 eV [23,27–29]. further assume that the detector exposure is independent of The characteristic distance between the source of these energy; the Auger exposure indeed depends on neutrino cH1 neutrinos and the Earth is the Hubble distance 0 , which energy only weakly [30]. Therefore, the total number of is much longer than the oscillation length—i.e., neutrino events is given by cH1 M1 m H 42 0 —as long as eff 0 ¼ 10 GeV,as is always the case here. Therefore, any oscillatory features K Ntot ¼ ; in neutrino mixing will be washed out; the probability for E (14) min conversion of a cosmogenic neutrino from its production flavor to another flavor en route from the source is then and the number of events is given by 2 = simply sin 2 2. Cosmogenic neutrinos mostly originate Z E max 1 from pion decays, with the characteristic flavor ratio N dE 2 0 E ¼ sin 2 ð Þ E e:: ¼ 1:2:0. The result of standard vacuum mixing min 2 Z E would be a flavor ratio at the Earth of e:: ¼ 1:1:1. K max ¼ dEE2sin22; While possible corrections to this flavor ratio can be in- (15) 2 Emin duced by small three-flavour oscillation effects or other tot Z E N E max new physics, here we concentrate on exploring the con- ¼ min dEE2sin22; E sequences of the DE-induced mixing. 2 min For the sake of simplicity, we focus on - mixing where we used Eq. (14) in the last equality. (and their antiparticles). In the absence of a DE-neutrino To investigate the sensitivity of a given experiment, we interaction, these two flavors are maximally mixed; i.e., assume a null detection of new physics; i.e., the result of v ¼ =4, and thus even if only are produced at the tot N is consistent with the standard expectation N =2 source, an equal number of and is generated by within statistical errors (we do not take systematic uncer- mixing. However, this can be altered if there is a DE- tainties into account). This will reject a certain range of neutrino interaction. The flux of and at the detector m 2 parameter space for ( eff, sin 2 d). More specifically, to is related to the flux at the source through, obtain 95% C.L. (2) limits for these parameters, we solve 1 2 0 sffiffiffiffiffiffiffiffi ¼ð1 sin 2Þ ; (11) 2 Ntot Ntot N > ; 2 (16) 1 2 0 ; 2 2 ¼ 2sin 2 (12) m m for eff and , using Eq. (15) for the left-hand side. In and will in general differ from v if eff 0. Fig. 1, we show the sensitivity of detectors that are expected to collect 12 and 100 neutrino events3 (total) and B. Proposed measurement that also have a -identification capability. If the true m 2 Here, we investigate the possibility of measuring using values of eff and sin 2 d are above these curves, then current or future ultra-high-energy-neutrino experiments we will see an anomalously suppressed flux compared such as Auger [30] and ANITA [31]. It is in principle with the standard mixing scenario. We also show the m possible to discriminate from by separately measur- current upper limit on eff obtained from the combined ing the Earth-skimming events () and almost horizontal analysis of Super-K and K2K data performed in Ref. [14]. events originating in air (). One can see from this Figure that by detecting cosmogenic tot neutrinos and by studying their flavor content, one can We assume that a given experiment detects N neutrino m events. This quantity is for the total neutrino and antineu- largely improve the current sensitivity to eff and d, tot quantifying further the suggestion of Ref. [17]. We also trino flux; i.e., N ¼ N þ N (here represents both note that a weaker sensitivity, albeit still much better than neutrinos and antineutrinos). If the flavor democracy expected from vacuum mixing is realized, then one expects tot 3 16 2 N ¼ N ¼ N =2. The number of neutrino events at the The current Auger exposure is 10 cm ssr[30], and an optimistic estimate for the flux of cosmogenic neutrinos is close detector is related to the flux through 2 to the Waxman-Bahcall bound [28], E ðEÞ Z E 108 GeV cm2 s1 sr1 [29]. Therefore, from Eq. (14), we max tot N ¼ dEðEÞðEÞ; (13) expect N & 1, which is still consistent with nondetection by E min Auger.

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To see when DE-induced mixing may bep significant,ffiffiffi recall that the value of the matter potential is 2GFNe ¼ 14 3 7:6 10 Yeð=gcm Þ eV. The Earth core has average : 3 Ycore density core ¼ 11 83 g cm and electron fraction e ¼ : 0 466, while the mantle has average density mantle ¼ 3 mantle 4:66 g cm and Ye ¼ 0:494, with the surface layer of the Earth having density as low as 2:6gcm3. The matter potential is thus about 1013 eV, so the effects of m DE-induced mixing may be manifest for eff around 1022 GeV, well below current upper limits. In the absence of matter, as discussed in Ref. [11], it is possible to obtain a resonance when all mixing angles involved are maximal m2 m : cos2 v þ eff cos2 d ¼ 0 (18) 2E Here, in the presence of matter and at high energies, a 2 FIG. 1 (color online). Sensitivity on (meff, sin 2d) plane of resonance can occur for a small mixing angle d when tot future experiments that would yield N ¼ 12 and 100 total pffiffiffi m G N : neutrino events. eff cos2 d ¼ 2 F e (19) The presence of a resonance is thus entirely determined by the current sensitivity, may be achieved with neutrinos of the densities encountered along the path and the DE cou- slightly lower energies [32]. pling parameters, with no (or very weak) energy dependence at high energies. To illustrate the possibilities, we integrate the neutrino- IV. MATTER EFFECTS IN ATMOSPHERIC propagation equation (including the small vacuum-mixing NEUTRINO OSCILLATIONS IN THE PRESENCE term) to calculate the -to-e transition probability as a OF A DARK-ENERGY COUPLING function of (cosine of) the zenith angle for atmospheric We now turn our attention to Lorentz=CPT-violating neutrinos propagating through the Earth. We use the den- effects in electron-neutrino oscillations, showing here sity profile of the Earth as given by the preliminary refer- that novel effects may arise with DE-neutrino coupling as ence earth model [33]. Figure 2 shows the results for two- m neutrinos propagate through the Earth. These effects may flavor oscillations forpffiffiffi different values of eff for d ¼ = m G N allow us to access with atmospheric neutrinos regions of 4. When eff 2 F e, the oscillation probability the DE-neutrino-coupling parameter space significantly is determined almost entirely by the DE term; there are below those currently probed. In this section, we consider regular large-amplitude variations of the oscillation proba- m two-flavor and three-flavor oscillations. bility as a function ofp zenithffiffiffi angle. As eff decreases to As neutrinos travel through matter, there is an additional values comparable to 2GFNe, the oscillation probability pcontributionffiffiffi to oscillations from the matter potential G N G N 1 2 F e (where F and e are, respectively, the Fermi -21 meff=10 GeV constant and electron density) relevant if electron neutrinos -22 m =10 GeV * eff are involved. Recalling that the matter potential is -23 m =5 x10 GeV 1022 GeV, the vacuum-mixing term m2=2E is small 0.8 eff for neutrino energies * 10 GeV. The mixing matrix Eq. (5) then becomes for e- mixing (neglecting the overall 0.6 ) e factor of 1=2, the directional dependence, and the phase ), −− ν µ ν pffiffiffi !P( 0.4 m G N m eff cos2 d þ 2 F e eff sin2pdffiffiffi : m m G N eff sin2 d eff cos2 d 2 F e 0.2 (17)

0 Note that here, both the DE term and the matter potential -1 -0.8 -0.6 -0.4 -0.2 0 θ change sign for antineutrinos, unlike the usual MSWeffect, cos in which the vacuum term does not change sign. Unlike FIG. 2 (color online). Oscillation probability as a function of MSW mixing, there is essentially no energy dependence, at zenith angle for atmospheric neutrinos of E ¼ 50 GeV, obtained sufﬁciently high energies, in this mixing matrix. with d ¼ =4.

123522-5 ANDO, KAMIONKOWSKI, AND MOCIOIU PHYSICAL REVIEW D 80, 123522 (2009) 0.3 neutrino coupling that is manifest in ways different than we θ =π/4 θ =0 12 and 13 have foreseen here. For example, if DE somehow produces θ =0 θ =π/4 0.25 12 and 13 θ =π/4 θ =π/4 Lorentz violation through a modiﬁcation of the kinetic 12 and 13 term in the Dirac equation, the energy dependence of the 0.2 mixing induced by Lorentz=CPT violation could be differ- ) e ent [13]. We therefore work out in this section expressions

−−ν 0.15 µ v p ν for the factor ^ to aid experimentalists who may wish to P( look for direction-dependent effects in their neutrino (or 0.1 other) data. To proceed, we first set our coordinate system. We set 0.05 the origin at the center of Earth and align the z axis along the rotational axis of Earth, so that the North Pole has 0 positive z coordinate. We set the x axis along the direction -1 -0.8 -0.6 -0.4 -0.2 0 cosθ to the Sun at vernal equinox. Since the Sun moves east- bound, its position at summer solstice aligns with the y FIG. 3 (color online). Oscillation probability as a function of axis. We can thus represent the seasonal shift by an azimu- zenith angle for atmospheric neutrinos of E ¼ 50 GeV, obtained thal angle ’, where ’ ¼ 0, =2, , and 3=4 for vernal d d for three-flavor oscillations with various values of 12 and 13 equinox, summer solstice, autumn equinox, and winter m 23 and eff ¼ 5 10 GeV. solstice, respectively. Note also that the orbital plane of x y : the Sun is inclined from the - plane by inc ¼ 23 5 [34]. decreases, and the oscillation length is seen to differ for The Sun is moving with respect to the CMB rest frame 1 trajectories that do ( cos & 0:8) and do not ( cos * with a speed of v ¼ 369 km s towards the direction 0:8) pass through the core. ¼ 168, ¼7:22 [35], where is right ascension m In Fig. 3 we show the oscillation probabilities for eff ¼ and is declination of the celestial coordinates [34]. In our 23 5 10 GeV for three-flavor mixing in the Lorentz- coordinates, the velocity of the Sun is v ¼ vðcoscos; d = d cossin;sinÞ¼ð358;76:1;46:4Þ kms1. The Earth violating sector. The case where 13 ¼ 4 and 12 ¼ 0 also corresponds to an effective two-flavor scenario, just is moving around the Sun with average orbital speed of like the previous results. It leads, however, to a very differ- V ¼ 29:8kms1. Thus, the velocity of the Earth with ent behavior due to the different contribution of the stan- respect to the CMB rest frame is d d = dard neutrino oscillations. The case where 13 ¼ 12 ¼ 4 corresponds to a full three-flavor oscillation scenario. We 0 1 sin’ have also studied the effects for other values of the mixing v ¼ v þ V@ ’ A angles and the same features remain present. A nonzero cos cos inc ’ v cos sin inc value of 13 for standard neutrino oscillations leads to 0 1 similar features in the zenith-angle distribution. However, 358 þ29:8 sin’ the effects are extremely small at the high energies con- ¼ @ 76:1 27:3 cos’ A km s1: (20) sidered here, orders of magnitude below those coming 46:4 11:9 cos’ from the Lorentz-invariance–violating terms.

V. DIRECTIONAL DEPENDENCE We neglect the contribution from the rotation of Earth ( & 0:5 km sec1) to our velocity with respect to the While detection of CPT=Lorentz-violating effects CMB rest frame. would be spectacular—it would imply new physics regard- Now we evaluate the direction of the neutrino beam p^. less of whether it is DE-related or not—more compelling We suppose that the beam runs from some point A on the evidence for a DE effect would be the directional depen- Earth’s surface to another point B on its surface (or vice dence, /ð1 v p^Þ, of neutrino-oscillation parameters. versa). It should be straightforward to generalize the argu- Given that our peculiar velocity with respect to the CMB ments below so that extraterrestrial neutrino production rest frame is 103 times the speed of light, the magnitude can be taken into account. We set the origin of time of this effect is going to be suppressed relative to the other coordinate T to ‘‘noon’’ (i.e., when the Sun reaches high- effects, discussed above, of a DE-neutrino interaction. est) at the point A. Therefore, the positions of A and B in Statistics well beyond the reach of current and forthcoming our coordinate are neutrino experiments will be required to detect this effect. Still, it is worth keeping in mind for future generations of 0 1 !T ’ experiments. cos A cosð A þ Þ x R @ !T ’ A; It may also be worth searching for such a directional A ¼ cos A sinð A þ Þ (21) dependence in current data, just in case there is a DE- sin A

123522-6 NEUTRINO OSCILLATIONS, Lorentz=CPT... PHYSICAL REVIEW D 80, 123522 (2009) 0 1 !T ’ x cos B cosð A þ þ Þ geometric longitude. The quantity appears in B be- x R @ !T ’ A; B ¼ cos B sinð A þ þ Þ (22) cause we measure the time (for both A and B) with respect sin B to noon of the point A, so the time difference is given by R ! the longitude difference (note also that the longitude in- where is the radius of the Earth, is the rotational p = T creases to the west). The direction of the neutrino beam ^ frequency (2 day), A is the time at the position A x x is then proportional to B A with proper normalization relative to noon, A;B is the geometric latitude of the points as A and B, and ¼ A B is the difference of the 0 1 !T ’ !T ’ cos B cosð A þ þ Þcos A cosð A þ Þ p 1 @ !T ’ !T ’ A: ^ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos B sinð A þ þ Þcos A sinð A þ Þ (23) 2ð1 cos cos cos sin sin Þ A B A B sin B sin A

Therefore, by combining Eqs. (20) and (23), we obtain The current best limits to the DE-neutrino coupling we the directional factor 1 v p^. Since it is a scalar quan- considered are obtained from atmospheric- and tity, the final result does not depend on the choice of the accelerator-neutrino experiments for - mixing, and coordinate system. from solar and reactor experiments for e- mixing. However, the higher the neutrino energy, the more promi- VI. THEORETICAL IMPLICATIONS nent the effect of the DE-neutrino interaction. We therefore Before closing, we discuss, for illustration, the implica- considered in this paper cosmogenic ultra-high-energy 17 19 tions of a measurement of a particular value of the Lorentz- (energies of 10 –10 eV) neutrinos produced by the in- m teraction of ultra-high-energy cosmic rays with CMB pho- invariance–violating effective mass parameter eff in terms of a specific model of DE-neutrino coupling. tons. We showed that future experiments targeting these Perhaps the simplest interaction of this kind has the form neutrinos will improve the sensitivity to a DE-neutrino m @ interaction by 7 orders of magnitude, down to eff 30 m & L ¼ ð Þ ; 10 GeV compared with the current upper bound eff int M 1 5 (24) 5 1023 GeV (Fig. 1). This corresponds to a sensitivity 6 where is a quintessence field, is a coupling-constant to an energy scale as large as 10 GeV for the DE- neutrino interaction. We then showed that the interplay of matrix, and M is some mass scale. Thus, aL _ t l=M m _ t =M DE- and matter-induced neutrino mixing could induce a ð Þ , and eff ð Þ , where is the novel zenith-angle dependence for e oscillations in at- difference between eigenvalues of the matrix. For quin- _ M H w 1=2 mospheric neutrinos. This effect may extend the sensitivity tessence, one expects Pl 0ð1 þ Þ (e.g., =CPT M to Lorentz -violating parameters in the e by roughly Ref. [3]), where Pl is the Planck energy scale. In this M m 3 orders of magnitude. case, the mass scale corresponding to a given eff is More compelling evidence of a DE-neutrino interaction 1 þ w 1=2 m (as opposed to some other origin for Lorentz=CPT viola- M ’ 106ðÞ eff GeV; (25) 0:01 1030 GeV tion) would be a directional dependence of the oscillation probabilities. The notion that Lorentz violation may give to the mass scale that controls the DE-neutrino interaction. rise to a directional dependence is not new (e.g., Ref. [36]) The ultra-high-energy - oscillation effects we have and searches for directional dependence in neutrino experi- M 6 discussed thus probe up to mass scales 10 GeV. ments have already been carried out (e.g., Ref. [37]), but The e- oscillations induced by the matter effects we prior work has considered Lorentz-violating parameters M discussed probe up to mass scales 100 MeV. introduced in an ad hoc manner and/or tested for direction-dependent effects in a Sun-centered inertial VII. CONCLUSIONS frame. We emphasize here that cosmic acceleration sug- We studied the implications of an interaction between gests that we seek a specific form of Lorentz violation, that dark energy and neutrinos for neutrino oscillations. The where the preferred frame is aligned with the CMB rest most general Lorentz=CPT-violating term induced by DE frame. Even though such a signal is expected to be small, it a a takes the form ð LÞ ð1 5Þ , where ð LÞ is a four- is still worth seeking in existing and future experimental vector normal to the CMB rest frame. This introduces a data. new source for neutrino oscillations that are energy inde- We have not discussed specific models for a DE- pendent and different for neutrinos and antineutrinos. neutrino interaction, beyond an illustrative toy model, but Furthermore, the motion of the Earth with respect to the it may be interesting to do so (see also Ref. [38]). The cosmic rest frame induces a directional dependence in the theoretical motivation to expect such a coupling may ad- oscillation probabilities. mittedly be slim. However, we are at square one in our

123522-7 ANDO, KAMIONKOWSKI, AND MOCIOIU PHYSICAL REVIEW D 80, 123522 (2009) understanding of DE, and such a coupling is no less likely ACKNOWLEDGMENTS to be expected, perhaps, than any of the many other man- We thank A. Friedland and S. Alexander for useful ifestations of new cosmic-acceleration physics that have discussions, and we acknowledge the hospitality of the been considered. Discovery of Lorentz=CPT-violating ef- Aspen Center for Physics. This work was supported by fects would be extremely important, even if not attributable the Sherman Fairchild Foundation (S. A.), DoE Contract directly to dark energy. A directional dependence, if dis- No. DE-FG03-92-ER40701 (M. K.), and NSF Grant covered, would be absolutely remarkable, as it would No. PHY-0555368 (I. M.). provide moreover clear evidence that there is more to cosmic acceleration than simply a cosmological constant.

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