Constraining the Mass Scale of a Lorentz-Violating Hamiltonian with the Measurement of Astrophysical Neutrino-Flavor Composition
We study Lorentz violation effects on flavor transitions of high energy astrophysical neutrinos. It is shown that the appearance of Lorentz violating Hamiltonian can drastically change the flavor transition probabilities of astrophysical neutrinos. Predictions of Lorentz violation effects on flavor compositions of astrophysical neutrinos arriving on Earth are compared with IceCube flavor composition measurement which analyzes astrophysical neutrino events in the energy range between and . Such a comparison indicates that the future IceCube-Gen2 will be able to place stringent constraints on Lorentz violating Hamiltonian in the neutrino sector. We work out the expected sensitivities by IceCube-Gen2 on dimension- CPT-odd and dimension- CPT-even operators in Lorentz violating Hamiltonian. The expected sensitivities can improve on the current constraints obtained from other types of experiments by more than two orders of magnitudes for certain range of the parameter space.
PACS numbers: 95.85.Ry, 14.60.Pq, 95.55.Vj
Although physical laws are believed to be invariant under Lorentz transformation, violations of Lorentz symmetry might arise in string theory as discussed in Kostelecky:1988zi (); Kostelecky:1991ak (). It is possible to incorporate Lorentz violation (LV) effects in an observer-independent effective field theory, the so-called Standard-Model Extension (SME) Colladay:1996iz (); Colladay:1998fq (), which encompasses all the features of standard model particle physics and general relativity plus all possible LV operators Kostelecky:2003fs (); AmelinoCamelia:2005qa (); Bluhm:2005uj (). While LV signatures are suppressed by the ratio with the electroweak energy scale and the Planck scale, experimental techniques have been developed for probing such signatures Kostelecky:2008ts (); Mattingly:2005re (). The effects of LV on neutrino oscillations were pointed out in Kostelecky:2003cr (); Kostelecky:2003xn (); Kostelecky:2004hg (). One can categorize LV effects to neutrino flavor transitions into three aspects: the modifications to energy dependencies of neutrino oscillation probabilities, the directional dependencies of oscillation probabilities, and the modifications to neutrino mixing angles and phases. In the standard vacuum oscillations of neutrinos, the oscillatory behavior of flavor transition probability is determined by the dimensionless variable with the neutrino mass-squared difference, the neutrino propagation distance, and the neutrino energy. This dependence results from the Hamiltonian with . The extra terms in Lorentz violating Hamiltonian introduces and dependencies into the oscillation probability, in addition to the standard dependence. The directional dependence of oscillation probability is due to the violation of rotation symmetry in . The coefficients of LV operators change periodically as the Earth rotates daily about its axis. This induces temporal variations of neutrino oscillation probability at multiples of sidereal frequency . Finally the full Hamiltonian is diagonalized by the unitary matrix which differs from due to the appearance of . Hence the values of neutrino mixing angles and phases associated with deviate from those associated with . Such deviations increase with neutrino energies since is while contains and terms.
Experimentally, effects of Lorentz violation on neutrino oscillations have been investigated in short-baseline neutrino beams Auerbach:2005tq (); Adamson:2008aa (); AguilarArevalo:2011yi (); Adamson:2012hp (), in long-baseline neutrino beams Adamson:2010rn (); Rebel:2013vc (), in reactor neutrinos at Double Chooz Abe:2012gw (); Diaz:2013iba (), and in atmospheric neutrinos at IceCube Abbasi:2010kx () and Super-Kamiokande Abe:2014wla (). These experiments probe either the spectral anomalies of the oscillated neutrino flux or the sidereal variations of neutrino oscillation probabilities. In this paper, we shall focus on LV effects to neutrino mixing angles and phases. As mentioned before, these effects grow with neutrino energies. Thus it is ideal to probe such effects through the flavor transitions of high energy astrophysical neutrinos dispersion (). For simplicity, we only consider isotropic LV effects.
The observation of high energy astrophysical neutrinos by IceCube Aartsen:2013bka (); Aartsen:2013jdh (); Aartsen:2013eka (); Aartsen:2014gkd () is a significant progress in neutrino astronomy and provides new possibilities for testing neutrino properties. The first result by IceCube on the flavor composition of observed astrophysical neutrinos has been published in Aartsen:2015ivb (), and was updated in Aartsen:2015knd () by a combined-likelihood analysis taking into account more statistics. Meanwhile, independent efforts have been made to determine neutrino flavor compositions from IceCube data Mena:2014sja (); Winter:2014pya (); Chen:2014gxa (); Palomares-Ruiz:2015mka (); Palladino:2015zua (). As we shall see in latter sections, the flavor measurement in Aartsen:2015knd () is not yet able to constrain more stringently than the previous experiments. Fortunately there is an active plan for extending the current IceCube detector to a larger volume, which is referred to as IceCube-Gen2 Aartsen:2014njl (); Aartsen:2015dkp (). This extension shall increase the effective area of the current 86-string detector up to a factor of 5. The expected improvement on neutrino flavor discrimination by IceCube-Gen2 has been studied in Shoemaker:2015qul (). Using this result, we shall study sensitivities of IceCube-Gen2 to the parameters of .
Astrophysical neutrinos are commonly produced by either or collisions at astrophysical sources. For sufficiently high energies, collisions produce equal number of and , which decay to neutrinos through and . This leads approximately to the flux ratio for both neutrinos and anti-neutrinos. Here denotes generically the flux of neutrino or anti-neutrino of flavor . This type of source is referred to as the pion source. A more detailed study on the neutrino flavor fraction with the consideration of neutrino spectral index is given in Lipari:2007su (). For an spectrum, the neutrino flavor fraction at the source is , where . However, for the purpose of this work, it suffices to take . We note that the secondary muons in some astrophysical objects can lose energy quickly by synchrotron cooling in magnetic fields or interactions with matter before their decays. Hence the neutrino flavor fraction at the source becomes . This type of source is referred to as the muon-damped source Kashti:2005qa (); Kachelriess:2007tr (); Hummer:2010ai (). In fact, there are also cases that the flavor fraction of astrophysical neutrinos at the source is energy dependent. For example, the flavor fraction of neutrinos can gradually changes from at lower energies to at high energies. Such a phenomenon has been discussed in Kashti:2005qa (); Kachelriess:2007tr () and investigated systematically in Hummer:2010ai (). The latter work also discusses sources with flavor fractions different from those of the pion source and muon-damped source. While a general study should consider the energy dependence of neutrino flavor fraction and variations of neutrino flavor fractions among different sources, we shall only focus on the simplified scenario that all sources of astrophysical neutrinos arising from collisions possess an energy independent flavor fraction for neutrinos at .
The production mechanism of astrophysical neutrinos with collisions is more complicated. The leading process of this category is which gives rise to the flavor fraction for neutrinos and for anti-neutrinos. The sub-leading process is which is non-negligible when the spectral index of the target photon is harder than Murase:2005hy (); Baerwald:2010fk (). This process produces equal number of neutrinos and anti-neutrinos with a common flavor fraction . Since the flavor fraction of neutrinos produced by collisions is relatively uncertain, we will not consider astrophysical neutrinos produced by such a mechanism.
We note that effects of new-physics Hamiltonian (with Lorentz violation as a special case), parametrized as , on the flavor transitions of astrophysical neutrinos were discussed in Arguelles:2015dca (); Katori:2016eni () for and (similar discussions were also given in Barenboim:2003jm (); Hooper:2005jp (); Bustamante:2010nq (); Bustamante:2015waa ()), and comparisons with earlier IceCube flavor measurement Aartsen:2015ivb () were made. The authors scan all possible structures of the mixing matrix for given new-physics scales and and determine the allowed range of astrophysical neutrino flavor fractions on Earth resulting from the full Hamiltonian . In our work, we shall focus on LV effects which are parameterized in a different form from the above new-physics Hamiltonian. We shall discuss current and future constraints on LV effects by comparing the predicted neutrino flavor fraction with the range of flavor fraction measured by the current IceCube detector Aartsen:2015knd () and that expected Shoemaker:2015qul () in the future IceCube-Gen2 detector. Our results can be directly compared with the previously most stringent constraints obtained by Super-Kamiokande Abe:2014wla ().
This paper is organized as follows. In Sec. II, we incorporate LV effects into the full neutrino Hamiltonian in the framework of SME. We then study analytically the flavor transition of astrophysical neutrinos assuming the dominance of over . As stated before such a dominance is possible for high energy astrophysical neutrinos. We discuss constraints on LV effects by the current IceCube flavor measurement. Such discussions pave the way for detailed numerical studies in the next section. In Sec. III, we study the flavor transitions of astrophysical neutrinos with the full Hamiltonian . The expected sensitivities of IceCube-Gen2 to are studied. We conclude in Sec. IV.
Ii Lorentz Violation in Neutrino Oscillations
LV effects in neutrino oscillations are incorporated by introducing an additional Lorentz violating term to the full Hamiltonian of the neutrino. Hence
where is the standard model neutrino Hamiltonian in vacuum with the neutrino mass matrix
and the PMNS matrix. Here we do not consider matter effects due to neutrino propagations inside the Earth. This is because we only focus on neutrino events with energies higher than few tens of TeV. In this case the Earth regeneration effect to the neutrino flavor transition is negligible. For neutrinos, the general form of LV Hamiltonian is given by
Since we shall only consider isotropic LV effects, we have the simplified form for given by Kostelecky:2003cr ()
where is the time component of Sun-centered celestial equatorial coordinate . For anti-neutrinos, we have
The two terms on the right hand side of are distinguished by their CPT transformation properties and dimensionality of the operators they are originated from. The first term is CPT-odd and originated from dimension- operator while the second term is CPT-even and originated from dimension- operator. Diagonalizing the full Hamiltonian in Eq. (1) yields a new mass-flavor mixing matrix . The neutrino flavor transition probability is then given by
where is the difference between the energy eigenvalues. For high-energy astrophysical neutrinos, is so large that the rapid oscillating terms are averaged out so that
Since depends only on the elements of , the neutrino flavor composition observed on the Earth for a given astrophysical neutrino source is affected by LV parameters. Therefore, the measurement of neutrino flavor fraction by neutrino telescopes such as IceCube is useful for constraining LV parameters. For convenience in discussions, we shall first concentrate on constraints on by setting . The constraints on will be commented later.
Recently, Super-Kamiokande Abe:2014wla () has set upper limits for , which are of the order GeV. With of this energy scale, it is interesting to note that is smaller than by more than 3 orders of magnitude for neutrino energies beyond a few tens of TeV. Hence for neutrino events analyzed in IceCube flavor measurement Aartsen:2015knd (), the LV term dominates over the standard model Hamiltonian if any term is set at the SK limit, . Therefore, IceCube measurements of flavor ratios should be useful for constraining the LV mass scale.
To illustrate the current IceCube capability of constraining LV parameters, we calculate the accessible ranges of neutrino flavor fractions on Earth resulting from the full Hamiltonian and the astrophysical pion source for neutrinos with the flavor fraction . For an illustrative purpose, we consider special scenarios for where only one pair of matrix elements in LV Hamiltonian, for instance, and its complex conjugate , are non-vanishing. We classify these special scenarios as , , , and , respectively. For the last scenario we take . In each special scenario for , the magnitude of the relevant matrix element is varied from zero to the current Super-Kamiokande C. L. limit, the phase of is varied from to , and the neutrino mixing parameters in are taken to be their best-fit values Gonzalez-Garcia:2015qrr (). The predicted ranges of flavor fractions on Earth by the full Hamiltonian for all considered scenarios of LV Hamiltonian are shown in Fig. 1. We stress that dictates the neutrino flavor fraction when is taken at the current SK limit in each special scenario. For comparison, the standard-model predicted neutrino flavor fractions with neutrino mixing angles and CP phase in varied over range Bustamante:2015waa () is also shown as the green area comment_SM () in Fig. 1. It is clear that, except for a tiny piece of area, the predicted ranges of flavor fractions of neutrinos by the full Hamiltonian are all within the current IceCube contour. Therefore a stringent constraint to requires IceCube-Gen2, which is the main target of our study in the next session.
Iii The sensitivity of IceCube-Gen2 to the LV parameters
In this section, we apply the projected flavor discrimination sensitivity of IceCube-Gen2 Shoemaker:2015qul () to estimate the future constraints on LV parameters. In the above projected sensitivity, only the pion source produced by collisions is considered. Therefore we shall only consider this type of source in the following discussions.
Before studying constraints to the most general flavor structure of , it is useful to summarize our analysis in the previous section. Let us take as the neutrino flavor fraction on the Earth. Since we shall focus on the pion source caused by collisions, there are equal numbers of neutrinos and anti-neutrinos produced with the flavor fraction at the source for both neutrinos and anti-neutrinos. Therefore we have . Since still holds with the addition of LV Hamiltonian, we thus have . Hence . Similarly we can show that , and . Clearly for astrophysical neutrinos arising from the pion source, the deviation of their flavor fraction on Earth to is due to symmetry breaking effects in the transition probability matrix. For the standard model Hamiltonian , the symmetry breaking effects are small. To leading orders in and , one has , with (taking ) Lai:2010tj () where is the CP violation phase. Hence LV effects can be detectable provided they introduce sizable symmetry breaking effects in the neutrino flavor transition probability matrix.
In the case that only and are non-vanishing in , symmetry is clearly broken. If dominates over , the flavor transition probability is determined by LV Hamiltonian and we find and in this limit. Consequently, the flavor fraction of astrophysical neutrinos arriving on Earth deviates significantly from . This corresponds to the tip of purple area in Fig. 1, which represents the flavor fraction . Similarly, large symmetry breaking occurs in the scenarios and (). On the other hand, symmetry is preserved in the scenario .
We have just seen that the symmetry breaking effect in can be probed with the pion source produced by collisions. Since we have assumed that all astrophysical neutrinos come from the pion source, it is essential to quantify the symmetry breaking effect in . To do that, it is useful to write with
Similar decomposition can be applied to .
We note that the simplified structure has been considered as the LV coupling between dark energy and neutrinos and the measurement of astrophysical and event difference was proposed to constrain in the future Ando:2009ts (). Here we shall begin with simplified scenarios that and . We then proceed to discuss the general case with . We shall study the sensitivities of IceCube-Gen2 to these Hamiltonians.
For , we can write
The first term of is proportional to the identity matrix and does not affect the neutrino flavor transition probability. One can ignore this term and rewrite as
where , , , , and is the phase of . Since is positive by definition, varies between and . The Hamiltonian can be inferred from by the replacements and . Taking into account the total Hamiltonian, , one can predict the neutrino flavor fraction on Earth assuming the initial neutrino flavor fraction at the source to be . We note that the neutrino energy appearing in should in principle follow the distribution with the threshold at TeV according to Ref. Shoemaker:2015qul (). However, for simplicity, we fix TeV. This is a conservative choice that makes less suppressed in comparison to the dominant .
Given the IceCube-Gen2 sensitivity shown in Fig. 1, we obtain the expected constraints on the LV mass scale as a function of mixing angle with the phase varied between and and the ratio of the order of unity. The expected constraints on are shown in that part of Fig. 2 labeled by . To derive the expected constraints on , we first fix the symmetry breaking parameter while allow the parameters and to vary. We then identify the critical value of such that the resulting neutrino flavor fraction on the Earth reaches to the boundary of IceCube-Gen2 C.L. contour. In this way we obtain an expected constraint on for a specific . We repeat the above procedure for different values of so that the entire sensitivity curve is obtained. The parameter range above the sensitivity curve will be ruled out at if no deviation to the standard neutrino flavor transition mechanism is observed.
We note that the symmetry limit in corresponds to while the maximum breaking corresponds to . This can be seen from the matrix structure given by Eq. (11) or the neutrino flavor transition probabilities resulting from the Hamiltonian . For the latter we found , , and . It is clear that indeed determines the above symmetry breaking effects in neutrino flavor transition probabilities. For , the sensitivity of IceCube-Gen2 to is about . The sensitivity to diminishes for (). In our numerical studies, the neutrino mixing parameters in are taken as the best fit values given in Gonzalez-Garcia:2015qrr (). This will be our choice for neutrino mixing parameters throughout the rest of the paper. We also vary each neutrino mixing parameter over range to see the effect. No appreciable effect in the sensitivity to is found. We note that the current SK C.L. limits on the related matrix elements are GeV and GeV Abe:2014wla (). It is clear that the expected bounds by IceCube-Gen2 shall improve the current bounds by more than two orders of magnitudes provided . Particularly the IceCube Gen2 sensitivity presented here is at C.L.
For , we can write
where , , , and are phases of and , respectively. The Hamiltonian can be inferred from by the replacements , , and . Since both and are positive by definition, the angle is between and . Taking into account the total Hamiltonian, , one can predict the neutrino flavor fraction on Earth assuming the initial neutrino flavor fraction at the source is .
Given the IceCube-Gen2 sensitivity shown in Fig. 1, we obtain the expected constraints on the LV mass scale as a function of mixing angle with the phases and varied between and . The sensitivity to is shown in that part of Fig. 2 labeled by . We have varied each neutrino mixing parameter over range and no appreciable effect on the sensitivity to is found. The parameter that characterizes the degree of symmetry breaking in is . The symmetry limit corresponds to , i.e., . On the other hand, the maximum breaking corresponds to , i.e., or . This is seen from the matrix structure given by Eq. (12) or the neutrino flavor transition probabilities resulting from the Hamiltonian . For the latter one can show that the neutrino flavor transition probabilities depend on both and . Hence a specific value of corresponds to two different neutrino flavor transition probabilities distinguished by the sign of . In principle there are two sensitivity points for each but we have chosen the more conservative one to plot the sensitivity curve.
For , the sensitivity of IceCube-Gen2 to varies slowly from to . In comparison, the current SK C.L. limits on related matrix elements are GeV, GeV, GeV and GeV Abe:2014wla (). One can see that the expected bounds by IceCube-Gen2 shall improve the current bounds by more than two orders of magnitudes provided . The sensitivity to diminishes for ().
For the general case with , the mass scales and of and , respectively, are independent parameters. These two scales can be comparable or one of the scales is suppressed in comparison to the other. Since the latter scenario has already been discussed, we only focus on the former case. To simplify our discussions, we take . The sensitivity of IceCube-Gen2 to is shown in that part of Fig. 2 labeled by . The parameter that characterizes the degree of symmetry breaking is . For , one must have both and equal to unity, i.e., the symmetry is respected in both and . For , either or (or both) breaks symmetry maximally. The sensitivity of IceCube-Gen2 to is GeV for . The sensitivity becomes GeV for . All these sensitivities improve significantly from the current SK bounds. The sensitivity of IceCube-Gen2 to diminishes for . We also vary the neutrino mixing parameter in range and no appreciable effect on the sensitivity to is found.
iii.4 Sensitivities to
So far we have only discussed IceCube-Gen2 sensitivities to . One can also study the sensitivities to parameters by turning off . Clearly replaces when the latter is turned off. It should however be noted that, for the anti-neutrino case, is changed into while is turned into .
Following the previous treatment, one can also decompose the dimension-, CPT-even LV Hamiltonian into two terms such that
where we have used to denote CPT-even LV Hamiltonian. The LV Hamiltonian for anti-neutrinos can be obtained by taking complex conjugates. Analogous to our definitions of and from , we can define dimensionless parameters and , respectively. Let us consider the full LV Hamiltonian and takes . The sensitivity of IceCube-Gen2 to is shown in Fig. 3.
We have taken as the parameter to characterize the degree of symmetry breaking, with and . Furthermore we also take TeV in for simplicity. The sensitivity of IceCube-Gen2 to is about for . Such a sensitivity shall improve significantly from the current SK C.L. limits, , , , , and much less stringent constraints on . The sensitivity curve rises up immediately for . This behavior is quite distinct from the behavior of sensitivity curve in Fig. 2 which rises mildly in the range before its sharp rise at . We attribute the shape difference between two sensitivity curves to the sign difference between and terms. To see this we change the sign of () terms in the neutrino sector while keeping the sign of () in the anti-neutrino sector unchanged. It is found that the shape of sensitivity curve in Fig. 3 is completely identical to the shape of sensitivity curve in Fig. 2 as it should be according to Eqs. (4) and (5).
Iv Discussions and Conclusions
In this paper, we discuss the sensitivities of future IceCube-Gen2 to Lorentz violation parameters in the neutrino sector. We consider the effects of Lorentz violating Hamiltonian on the flavor transitions of astrophysical neutrinos coming from the pion source produced by collisions. In such a case, there are equal numbers of neutrinos and anti-neutrinos produced with the flavor fraction at the source for both neutrinos and anti-neutrinos. We have shown that the flavor fraction of such neutrinos as they arrive at the Earth is if the neutrino Hamiltonian respects symmetry. The deviation to such a flavor fraction is therefore controlled by the breaking of symmetry in the neutrino Hamiltonian. For both CPT-odd and CPT-even LV Hamiltonian, we decompose the LV Hamiltonian into two matrix structures as shown in Eqs. (8), (9), (13), and (14). For each matrix structure we define the parameter that characterizes the degree of symmetry breaking and the scale of the matrix to be probed by the measurement of astrophysical neutrino flavor fractions.
Since the neutrino Hamiltonian in the Standard Model is approximately symmetric, the effect from the new physics Hamiltonian is important only when this Hamiltonian significantly breaks the symmetry. Taking Fig. 2 as an example, the LV Hamiltonian breaks the symmetry significantly for () such that the expected constraint to the LV mass scale by IceCube-Gen2 is stringent. It is of interest to see how restricted the parameter range is. Without specific preference to the detailed structure of , one can assume the angle to be uniformly distributed from to for a fixed LV mass scale . The condition requires either or . Such a range for occupies of the total parameter space for . For , the LV mass scale is testable for the parameter range . Assuming is uniformly distributed between and , the range for required by the above condition occupies about of the total parameter space for . Finally for the case of full LV Hamiltonian with , the LV mass scale is testable in the parameter range . This is of the total parameter space of and evaluated by a simple Monte Carlo. In the case of CPT-even LV Hamiltonian, the dimensionless LV scale () of is testable for . Clearly the percentage of total parameter space of and that satisfies this condition is also around .
In summary, we have taken a phenomenological approach that incorporate all LV effects in the neutrino sector with a set of local operators Colladay:1996iz (); Colladay:1998fq (); Kostelecky:2003fs (); AmelinoCamelia:2005qa (); Bluhm:2005uj (). We only focus on the isotropic LV effects Kostelecky:2003cr () so that the structure of LV Hamiltonian is given by Eqs. (4) and (5). We have worked out the sensitivities of IceCube-Gen2 to CPT-odd LV parameter originated from dimension- operator and the CPT-even LV parameters originated from dimension- operators. We have shown that the expected IceCube-Gen2 sensitivities to LV mass scales can improve the current SK bounds Abe:2014wla () by at least two orders of magnitudes for sufficiently large symmetry breaking effects in LV Hamiltonian. We reiterate again that our results are based upon the assumption that all sources of astrophysical neutrinos have an energy independent flavor fraction for neutrinos at . It is worthwhile to pursue further studies with both the energy dependence of neutrino flavor fraction and the variations of neutrino flavor fractions among different sources taken into account.
We thank M. Bustamante for useful comments. This work is supported by Ministry of Science and Technology, Taiwan under Grant Nos. 106-2112-M-182-001 and 105-2112-M-009 -014.
Note added.—As we were revising this paper, we became aware of the newest IceCube analysis on Lorentz violation effects in neutrino sector using atmospheric neutrino data Aartsen:2017ibm (), which sets C.L. bounds on and at GeV and C.L. bounds on and at .
- (1) V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).
- (2) V. A. Kostelecky and R. Potting, Nucl. Phys. B 359, 545 (1991).
- (3) D. Colladay and V. A. Kostelecky, Phys. Rev. D 55, 6760 (1997).
- (4) D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998).
- (5) V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004).
- (6) G. Amelino-Camelia, C. Lammerzahl, A. Macias and H. Muller, AIP Conf. Proc. 758, 30 (2005).
- (7) R. Bluhm, Lect. Notes Phys. 702, 191 (2006).
- (8) V. A. Kostelecky and N. Russell, Rev. Mod. Phys. 83, 11 (2011).
- (9) D. Mattingly, Living Rev. Rel. 8, 5 (2005).
- (10) V. A. Kostelecky and M. Mewes, Phys. Rev. D 69, 016005 (2004).
- (11) V. A. Kostelecky and M. Mewes, Phys. Rev. D 70, 031902 (2004).
- (12) V. A. Kostelecky and M. Mewes, Phys. Rev. D 70, 076002 (2004).
- (13) L. B. Auerbach et al. [LSND Collaboration], Phys. Rev. D 72, 076004 (2005).
- (14) P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. 101, 151601 (2008).
- (15) A. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. Lett. B 718, 1303 (2013).
- (16) P. Adamson et al. [MINOS Collaboration], Phys. Rev. D 85, 031101 (2012).
- (17) P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. 105, 151601 (2010).
- (18) B. Rebel and S. Mufson, Astropart. Phys. 48, 78 (2013).
- (19) Y. Abe et al. [Double Chooz Collaboration], Phys. Rev. D 86, 112009 (2012).
- (20) J. S. Diaz, T. Katori, J. Spitz and J. M. Conrad, Phys. Lett. B 727, 412 (2013).
- (21) R. Abbasi et al. [IceCube Collaboration], Phys. Rev. D 82, 112003 (2010).
- (22) K. Abe et al. [Super-Kamiokande Collaboration], Phys. Rev. D 91, no. 5, 052003 (2015).
- (23) We note that the neutrino dispersion relation can be modified by Lorentz violation. Thus it is possible that neutrino loses energy via Cherenkov radiation during its propagation. For astrophysical neutrinos, such an energy loss mechanism can also lead to stringent bounds on Lorentz violation as pointed out in J. S. Diaz, A. Kostelecky and M. Mewes, Phys. Rev. D 89, no. 4, 043005 (2014).
- (24) M. G. Aartsen et al. [IceCube Collaboration], Phys. Rev. Lett. 111, 021103 (2013).