Signals of CPT violation and non-locality in future neutrino oscillation experiments Academic research paper on "Physical sciences"

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Signals of CPT violation and non-locality in future neutrino oscillation experiments

S. Antusch, E. Fernandez-Martinez *

Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany

ARTICLE INFO ABSTRACT

We investigate the sensitivities of future neutrino oscillation experiments for measuring the neutrino mass squared differences and leptonic mixing angles independently with neutrinos and anti-neutrinos. We update the expected sensitivities of Neutrino Factories to the "atmospheric" (anti-)neutrino parameters using an optimized setup. A dedicated ß-Beam facility, in combination with a SPMIN reactor experiment, could give excellent sensitivities also to the "solar" parameters, for neutrinos and antineutrinos respectively. A signal of a different mass matrix for neutrinos and anti-neutrinos would imply CPT violation and non-locality of the underlying particle theory.

© 2008 Elsevier B.V. All rights reserved.

Article history:

Received 25 April 2008

Received in revised form 30 May 2008

Accepted 2 June 2008

Available online 12 June 2008

Editor: A. Ringwald

1. Introduction

With the expected high sensitivities of envisioned neutrino facilities like Neutrino Factories [1-3] or /)-Beams [4], neutrino physics could enter a new era of precision [5]. One main goal of these experiments is the measurement of the yet unknown low energy parameters of the lepton sector accessible to oscillation physics, i.e., the Dirac CP phase 8, the mixing angle 813 and the sign of the "atmospheric" mass squared difference (corresponding to whether the mass spectrum is normally ordered or inverted). However, in addition to the determination of these parameters, future precision experiments may also find interesting signals of physics beyond the SM or even signals of violation of fundamental principles such as CPT invariance, Lorentz invariance or locality [6]. One interesting signal of this type would be a difference between the masses and mixing angles measured by neutrino oscillation experiments operating with neutrinos and anti-neutrinos.

The CPT theorem [7] states that any local Quantum Field Theory (QFT) which is Lorentz invariant and has a Hermitean Hamiltonian must have CPT symmetry. From the derivation of the CPT theorem it also follows that CPT violation implies violation of Lorentz invariance. Bounds on CPT and Lorentz invariance violation from processes involving quarks and charged leptons are quite strong, for example from the K0-K0 system [8]. On the other hand, CPT and Lorentz invariance are tested to a much less precision in the neutrino sector by the current experiments, as we will review below in Section 2.1. In addition to the less tight constraints, another

* Corresponding author.

E-mail addresses: antusch@mppmu.mpg.de (S. Antusch), enfmarti@mppmu.mpg.de (E. Fernandez-Martinez).

0370-2693/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2008.06.005

motivation to look for signals of CPT and Lorentz invariance violation in the neutrino sector is the potentially different mechanism for generating the small neutrino masses compared to the masses of quarks and charged leptons, which might be especially sensitive to new physics.

One direct consequence of CPT invariance is that neutrinos and anti-neutrinos have the same masses and mixing angles. It has been pointed out in Ref. [9] that a signal for the violation of this prediction, i.e., different masses and mixing angles for neutrinos and anti-neutrinos, would allow to draw the additional conclusion that in addition to CPT and Lorentz invariance also locality must be violated. More precisely, it has been shown that a difference between the masses for particles and anti-particles implies non-locality in the sense that field (anti-)commutators do not vanish for space-like distances and furthermore that the fields enter terms in the Lagrangian at different space-time points [9]. Non-locality is in general predicted by extensions of the SM towards a unified theory with gravity. Regarding CPT violation, it has been argued in Ref. [10] that non-local interactions in string theory might generate CPT violation close to present bounds. Specific scenarios of CPT breaking can also be realised in noncommutative geometries [11]. We would like to note, however, that if a signal of CPT violation via different masses of neutrinos and anti-neutrinos would be observed, it could be challenging to distinguish the possible intrinsic CPT violation and non-locality from "fake" signals, caused, e.g., by non-standard matter effects. In any case, if the experimental data would point to different masses and/or mixing angles for neutrinos and anti-neutrinos, it would be an intriguing signal of physics beyond the SM.

Various aspects regarding CPT violation in the neutrino sector have been analysed in previous studies. For example, Ref. [12] has discussed the potential to test CPT by combining solar neu-

trino data and data from the KamLAND reactor experiment. The sensitivity of experiments with atmospheric neutrinos has been discussed in [13] and the prospects of the MINOS experiment has been studied in [14]. For the "atmospheric" parameters, high sensitivity to CPT violation could be achieved in Neutrino Factories [15,16]. Additional tests might be possible with the data from supernova neutrinos [17] and with neutrinoless double beta decay [18]. One motivation for the study of models of CPT violation for neutrino physics has been the LSND anomaly, where CPT violation in the neutrino sector was proposed as a solution [19]. The case studied by most authors in this context is that of a CPT violating background which leads to CPT and Lorentz invariance violating local effective operators at low energy [20]. Recently, a parameterisation for a three family oscillation analysis in the presence of specific types of CPT violation has been studied in [21].

In order to search for a signal of CPT violation and non-locality in neutrino oscillation experiments a precise determination of the neutrino mass splittings and/or leptonic mixing angles, independently with neutrinos and anti-neutrinos, is desirable. Recently, significant progress has been made in the design of possible future neutrino oscillation facilities, for example regarding Neutrino Factories [5]. Furthermore, /)-Beam facilities have been proposed which, in some aspects, could even allow more precise measurements than a Neutrino Factory. In addition, new reactor experiments (for (13 [22] as well as for precision measurements of (12 [23-25]) have been envisioned. In this Letter, we therefore investigate the combined sensitivity of such future neutrino oscillation experiments for measuring the "solar" and "atmospheric" mass squared differences and leptonic mixing angles independently with neutrinos and anti-neutrinos. We update the expected sensitivities for the "atmospheric" mass squared difference and mixing angle with Neutrino Factories using an optimized setup and analyse how a dedicated /)-Beam facility, in combination with a SPMIN reactor experiment [23-25], could improve the sensitivities for the "solar" mass splitting and mixing angle.

2. Potential signals of CPT violation and non-locality

In the following we denote parameters for anti-neutrinos with bars and parameters for neutrinos without bars. Using standard PDG parameterisation of the PMNS matrix [8], CPT invariance implies mj = m, and 8ij = (j. As discussed above, a violation of these equalities, i.e., mi = mi for one neutrino mass eigenvalue or (ij = (¡ij for one of the mixing angles, if not confused with a "fake" signal of different new physics, would signal violation of CPT invariance (and Lorentz invariance) as well as non-locality of the underlying particle theory. To analyse the expected sensitivities to such signals of new physics beyond the SM, we study how well neutrino oscillation experiments can determine the neutrino masses and leptonic mixing angles for neutrinos and antineutrinos, i.e., using only data from oscillation of neutrinos and anti-neutrinos, respectively. In our analysis we treat the parameters Qij^ij and mi, mi as independent quantities and assume the standard three family oscillation formulae for neutrinos va and anti-neutrinos va, thereby testing the consistency of the CPT symmetric description. Bounds on CPT violation are presented as the constraints on differences | sin2 dij — sin2 (ij | for the mixing angles and |Am2j — Am2 | for the mass squared differences (defined as Amfj = m2 — m2 and Am2 = m2 — m2) which the considered experiments can impose. Before we turn to the discussion of the expected sensitivities of future neutrino oscillation facilities in this respect, we review the bounds on the difference between neutrino and anti-neutrino parameters from the present data. A graphical summary of the present bounds can be found in Fig. 55 of [26].

2.1. Present bounds (anti-)neutrino parameters

Regarding the "solar" neutrino parameters 012, (12 and Am^1, Am221, the most relevant data stems from experiments with solar neutrinos [27-32] and from the KamLAND experiment [33] observing anti-neutrinos from nuclear reactors. If CPT invariance is assumed, the complementarity between these two data sets allows for a quite good determination of the "solar" mixing angle as well as of the "solar" mass squared difference. However, separating into neutrino and anti-neutrino parameters shows that the bounds on their difference are comparatively weak. While solar neutrino data allows for a quite precise measurement of sin2 012, there is very little information on Am^1. The reason for the good sensitivity on sin2 012 is that solar matter effects adiabatically convert the ve produced in the solar core into v2 when leaving the sun, and thus detectors on earth can extract |Ue2|2 = sin2 012 by comparing the measured ve flux to the theoretically expected flux without oscillations or to the total neutrino flux measured through neutral current interactions. The sensitivity of the solar neutrino experiments to the mass splitting Am221 is very poor because the distance between earth and sun is so large that the oscillatory behaviour of the transition probability is averaged out. KamLAND, on the other hand, has a much shorter average baseline of ~ 100 km. The (almost) vacuum oscillation signal of the reactor ve and the measurement of the distortion of the anti-neutrino energy spectrum allows a precise measurement of Am21. However, the overall normalization of the neutrino flux is rather difficult due to the fact that anti-neutrinos from many reactors (as well as from other sources) are detected by KamLAND. Consequently the constraints on sin2 (12 from KamLAND is comparatively weak. Furthermore, the near vacuum oscillations observed by KamLAND depend on the quantity sin22(12, which implies that the octant of (12 cannot be determined from the present data on anti-neutrinos. In summary, assuming that (12 lies in the first octant, the present bounds on CPT violation in the "solar" sector are at 3a

I sin2 e12 — sin2 (121 < 0.3, (1)

|Am21 — Am21|< 1.1 x 10—4 eV2. (2)

Regarding the "atmospheric" parameters (23, (23 and Am321, Am321, the most important experimental data stems from atmospheric neutrinos observed by SuperKamiokande [34,35] and from the accelerator experiments sensitive to the "atmospheric" sector, K2K [36] and MINOS [37]. K2K and MINOS operate with neutrinos and the atmospheric signal is dominated by the neutrino event rates due to the larger cross section. Therefore, the measurements of sin2 (23 and in particular of Am321 are much less precise than the ones of sin2 (23 and Am321 (see Fig. 55 of [26]). The present bounds on CPT violation for the "atmospheric" sector are at 3a [5,38]:

|sin2 e23 — sin2 (23| < 0.45, (3)

IAm21 — Am211 < 1 x 10—2 eV2. (4)

Finally the bounds on the unknown 1-3 mixing angles 013 and (13 for neutrinos and anti-neutrinos, respectively, are similar. The bound sin2 (13 < 0.1 (at 3a) is obtained from the CHOOZ experiment [39] and a bound of sin2 d13 < 0.3 can be derived from the combined data of experiments on solar, atmospheric and accelerator neutrino oscillations [26]. From the present data, without a measurement of neither 013 nor (13, the bound on the difference (at 3a) is given by

|sin2 e13 — sin2 (131 < 0.3. (5)

We will now discuss how these bounds (or equivalently the sensitivities for discovering a signal) might be improved in future neutrino oscillation facilities.

2.2. Strategies for improving the sensitivities

As has been discussed in previous studies [16], the bounds on | sin2d23 — sin2§23\ and |Am21 — Am211 can be strongly improved at a Neutrino Factory facility [1-3]. Excellent sensitivities to | sin2 d13 — sin2 d13 \ are also to be expected since the determination of 613 is one of the main goals of the Neutrino Factory. In the context of the IDS, International Design Study of the Neutrino Factory, a baseline for the Neutrino Factory acceleration complex and detection systems has recently been defined [5,40]. This baseline setup would store 25 GeV muons, whose (anti-)neutrino fluxes aim at two 50 kton Magnetized Iron Neutrino Detectors (MIND) located at L ~ 4000 km and L ~ 7500 km from the source, respectively. The goal luminosity for such a facility is 5 x 1020 useful muon decays per year per polarity per baseline. The efficiencies, backgrounds and energy resolution of the MIND detector when exposed to such a beam have been studied in [41]. We will present the up-dated bounds with these specifications in Section 2.3. In comparison, little work has been done so far with respect to the improvement of the bounds on the "solar" parameters, and we will therefore focus on this issue in the remainder of this section.

As discussed in Section 2.1, the present uncertainty on | sin2 d12 — sin2 d12 \ is dominated by the comparatively low precision on sin2 612, while the error on |Am21 — Am211 stems from the weak constraints on Am21. Improving the former is much easier than the latter. A dedicated reactor experiment placed at the minimum of the Ve survival probability (SPMIN) could provide an excellent measurement of sin2 612 [23-25]. A precise determination of Am221, however, is much more challenging. The observation of the oscillations driven by the small "solar" splitting requires large values of L/E. The neutrino charged current (CC) interaction, however, is very suppressed at low energies since the neutrons with which they can interact are confined in nuclei and a minimum threshold energy is normally required for the interaction to occur. Moreover, while nuclear reactors provide a convenient source of electron anti-neutrinos, it is challenging to produce low energy electron neutrinos in large amounts on earth.

One possibility to improve the determination of Am21 would be an intense ve beam of Ev ~ 0.4 GeV from the /) decay of 18Ne ions accelerated to y = 100 at a /)-Beam facility [4,42-44]. However, to optimize the determination of Am21 and 612 with neutrinos, we consider also a very long baseline of 4000 km to a Mton class water Cerenkov detector [45-47]. Somewhat less sensitivity can be reached with a more conventional 750 km baseline. We have also studied the possible measurements achievable with the ve that this facility could also provide from the decays of 6He. However we found that the performance of the anti-neutrino beam was significantly worse. This asymmetry is caused by the different matter effects for neutrinos and anti-neutrinos. In particular, the electron (anti-)neutrino survival probabilities for 613 = 0 are given by (see, e.g., [48]):

Pee = 1 —

sin2 2012

i(Amh L, I 4E '

s = ysin2 ie12 + (cos ie12 ± A)2,

with A = VmE and where the ± signs apply to the probabilities of anti-neutrinos and neutrinos, respectively. Thus, matter effects reduce the amplitude of the oscillation probability and increase the oscillation frequency. Due to the + sign in Eq. (7) for

Fig. 1. ve (solid) and ve (dashed) oscillation probability for E = 0.4 GeV as a function of the baseline.

anti-neutrinos the effect is enhanced, the oscillation amplitude is smaller and a less precise reconstruction of the parameters follows. This situation is illustrated in Fig. 1. The solid and dashed lines correspond to the ve and ve survival probabilities respectively. Consequently, present bounds on Am21 and sin2 612 will not be improved by this setup. On the other hand, Eq. (7) is no longer symmetric under the octant of 612. Changing the octant has the same effect as changing the sign of A or, equivalently, interchanging the neutrino and anti-neutrino curves in Fig. 1. This means that the ve from the 6He decay could distinguish the octant of 612 which would not be possible even at a dedicated SPMIN experiment for a precise measurement of sin2 2612 due to the small matter effects.

The /)-Beam setup with 4000 km baseline is thus optimized for the measurement of "solar" parameters through ve disappearance, however it is unrealistic to expect that such a dedicated facility would ever be built. As already mentioned above, we have therefore also considered a setup with a shorter, more conventional baseline of 750 km. This is the baseline usually chosen for the Y = 350 version of the /)-Beam, which could provide excellent sensitivities to CP-violation [49] even outperforming the Neutrino Factory for some regions of the parameter space. A y = 100 run of the /)-Beam at this baseline, such as the one we consider here to improve the measurement on Am221, can improve the sensitivity to the mass hierarchy and allow to solve degeneracies when combined with the higher Y run [50].

2.3. Expected sensitivities of future facilities

For the experimental setup we consider the "IDS baseline" Neutrino Factory [40], which provides excellent sensitivities to the "atmospheric" parameters and 813 with both neutrino and anti-neutrino beams. In this facility intense (anti-)neutrino beams would be produced from the decay of 25 GeV (anti-)muons with a luminosity of 5 x 1020 muon decays per year per muon polarity. This beams illuminate two identical 50 Kton iron calorimeters located at 4000 km and 7000 km. We consider 5 years of data taking with each muon polarity. We did not consider the Opera-like detector to observe the silver channel [51] since it did not improve significantly any bound.

In order to improve the present bounds on the "solar" sector we also considered a /)-Beam facility producing electron (anti-)neu-trinos from the decay of 18Ne (6He) ions accelerated to Y = 100. We considered 5 years of data taking with luminosities of 1019 ion decays per year [52] for both ions. These neutrino beams would be detected at a Mton class water Cerenkov detector at a 750 or 4000 km baseline. To describe the detector efficiencies and backgrounds when exposed to these beams we followed Ref. [44].

Fig. 2. 90%, 95%, 99% and 3a contours for the measurements of sin2 012, sin2 023, sin2 013, with either neutrino data (vertical axes) or anti-neutrino data (horizontal axes) alone. The left (right) panels show the constraints achievable with the 4000 km (750 km) baseline for the /¡-Beam.

Finally a reactor experiment with an exposure of 60 GWktonyr to a detector located at 60 km distance was considered to improve the measurement of the "solar" parameters with anti-neutrinos. We considered a 5% systematic error in the normalization of the signal and the energy resolution of the detector was set to a = 0.05 x E. The signal was distributed in 10 energy bins in the range between 0.0018 and 0.008 GeV.

For the numerical analysis we independently combined the neutrino and anti-neutrino data from the Neutrino Factory, SPMIN and /-Beam setups to derive measurements on the neutrino and anti-neutrino parameters. Since the combination of the three facilities allows to constrain all the oscillation parameters no prior information on any of them was assumed except for the measure-

ment that will not be improved by this setup, namely 012 from solar neutrino oscillations, for which we assumed a 1a uncertainty of 10%. A 5% uncertainty in the PREM density profile was also assumed. The Globes 3.0 [53] software was used to perform the numerical analysis. The following (CPT conserving) input values were assumed for the oscillation parameters: sin2 012 = 0.3, sin2 023 = 0.5, sin2 013 = 0, Am21 = 7.6 x 10—5 eV2, Am21 = 2.5 x 10—3 eV2.

Figs. 2 and 3 show the 90%, 95%, 99% and 3a contours for the constraints on sin2 012, sin2 023, sin2 013, Am21 and Am21 with either neutrino data (vertical axes) or anti-neutrino data (horizontal axes) alone. The left (right) panels contain the constraints achievable with the 4000 km (750 km) baseline for the /-Beam. Comparing it with Fig. 55 of [26] the dramatic improvement on

Fig. 3. 90%, 95%, 99% and 3a contours for the measurements of Am^1 and Am^1 with either neutrino data (vertical axes) or anti-neutrino data (horizontal axes) alone. The left (right) panels show the constraints achievable with the 4000 km (750 km) baseline for the ¡-Beam.

the constraints on the "atmospheric" parameters with the Neutrino Factory is manifest. The constraints for sin2 023 and sin2 023 shrink by about one order of magnitude, while the improvement in Am321 and Am21 is about two and three orders of magnitude respectively. The bounds on CPT violation that could be derived from this precision measurements including the 750 km baseline /-Beam would be:

Table 1

Present bounds and expected future experimental sensitivities to differences between masses and mixing angles for neutrinos and anti-neutrinos at 3a. The considered future facilities are explained in the main text

Quantity

Present bound Future 4000 km) Future 750 km)

| sin2 012 — sin2 012 | | Sin2 013 — Sin2 013 |

| sin2 023 — sin2 0231

|sin2 023 — sin2 023 I < 0.043, I Amf1 — Am211 < 3.3 x 10—5 eV2.

(8) (9)

- Am2-

0.3 0.14

0.3 5.7 x 10—4

0.45 0.043

1.1 x 10—4eV2 1.3 x 10—5 eV2

1 x 10—2 eV2 2.6 x 10—5 eV2

5.7 x 10—4 0.044

2.2 x 10—5 eV2

3.3 x 10—5 eV2

With such a Neutrino Factory setup, the constraints on both for the 4000 km baseline p-Beam or

sin2 013 and sin2 013 could also be significantly improved by almost three orders of magnitude to

I sin2 013 — sin2 013 I

< 5.7 x 10—

I sin2 012 — sin2 0121 < 0.14, | Am^ — Aim^I < 2.2 x 10—5 eV2,

In the "solar" sector the improvements are more modest, especially for the shorter /-Beam baseline. For the 4000 km baseline matter effects allow to measure the octant of 023, but at 750 km their strength is not sufficient for this task, as can be seen in the top panels of Fig. 2. With neutrinos an improvement of Am221 by an order of magnitude could be accomplished with the /-Beam and some improvement in the lower bound of sin2 012 could also be achieved. These improved measurements could test CPT invariance to the level of

I sin2 012 — sin2 0121 < 0.14, IAm2i — Am2J < 1.3 x 10—5 eV2,

(11) (12)

for the 750 km baseline /-Beam. In the latter case the octant of 012 is not measured. 012 has been assumed to lie in the first octant in order to compare with Eq. (2). A summary of present and possible future bounds is presented in Table 1.

3. Discussion and conclusions

In this study we have investigated the sensitivity of future neutrino oscillation experiments for measuring the "solar" and "atmospheric" mass squared differences and leptonic mixing angles independently with neutrinos and anti-neutrinos. If a difference between the parameters for neutrinos and anti-neutrinos would be

Am321 — Am321

established, it would imply CPT (and Lorentz invariance) violation as well as non-locality.

To improve the present bounds on this form of CPT violation we have considered three types of possible future neutrino oscillation experiments: an optimized Neutrino Factory, a y = 100/-Beam (with 750 km or 4000 km baseline) pointing to a Mton water Cerenkov detector and a SPMIN reactor experiment at the minimum of the ve survival probability. The possible improvements with respect to the present bounds are summarised in Table 1.

Regarding the "atmospheric" parameters and 013 (vs. 013), a dramatic improvement could be accomplished with a Neutrino Factory operating with neutrinos as well as with anti-neutrinos. The sensitivity on | sin2 013 — sin2 0131 could be improved almost three orders of magnitude, the sensitivity on | sin2 023 — sin2 0231 by one order of magnitude and the sensitivity on |Am21 — Am211 by more than two orders of magnitude.

To improve the sensitivity with respect to the "solar" parameters, the combination of the /-Beam with the SPMIN reactor experiment could also be very successful. While the /-Beam experiment could improve the measurements of the parameters with neutrinos (mainly Am21) and determine the octant of 012, the SP-MIN experiment could further improve the measurement of 012. While the gain in sensitivity on | sin2 012 — sin2 0121 amounts about 50%, the sensitivity on |Am21 — Am211 could be improved by about one order of magnitude.

In summary, future neutrino oscillation experiments have the potential to measure neutrino mass squared differences and lep-tonic mixing angles separately with neutrinos and anti-neutrinos to high precision and may detect possible differences between neutrino and anti-neutrino parameters. Such a difference would be a clear signal of new physics beyond the Standard Model of particle physics. If other types of new physics which may lead to "fake" signals, e.g., non-standard matter effects, could be excluded it would signal CPT (and Lorentz invariance) violation and also a non-local nature of the underlying particle theory.

Acknowledgements

We would like to thank Andrea Donini for helpful discussions and Patrick Huber for kindly providing the description of the IDS baseline for the Neutrino Factory. This work was partially supported by The Cluster of Excellence for Fundamental Physics "Origin and Structure of the Universe" (Garching and Munich).

References

[1] S. Geer, Phys. Rev. D 57 (1998) 6989, hep-ph/9712290; S. Geer, Phys. Rev. D 59 (1999) 039903, Erratum;

A. De Rujula, M.B. Gavela, P. Hernandez, Nucl. Phys. B 547 (1999) 21, hep-ph/ 9811390.

[2] A. Cervera, A. Donini, M.B. Gavela, J.J. Gomez Cadenas, P. Hernandez, O. Mena, S. Rigolin, Nucl. Phys. B 579 (2000) 17, hep-ph/0002108;

A. Cervera, A. Donini, M.B. Gavela, J.J. Gomez Cadenas, P. Hernandez, O. Mena, S. Rigolin, Nucl. Phys. B 593 (2001) 731, Erratum.

[3] M. Apollonio, et al., hep-ph/0210192.

[4] P. Zucchelli, Phys. Lett. B 532 (2002) 166.

[5] A. Bandyopadhyay, et al., ISS Physics Working Group, arXiv: 0710.4947 [hep-ph].

[6] See, e.g.: V.A. Kostelecky, M. Mewes, Phys. Rev. D 69 (2004) 016005, hep-ph/ 0309025, and references therein.

[7] J. Schwinger, Phys. Rev. 82 (1951) 914;

G. Luders, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd. 28 (5) (1954); W. Pauli, in: W. Pauli, L. Rosenfeld, V. Weisskopf (Eds.), Niels Bohr and the Development of Physics, Pergamon, London, 1955, p. 30; J.S. Bell, Proc. R. Soc. London A 231 (1955) 479.

[8] W.M. Yao, et al., Particle Data Group Collaboration, Rev. Part. Phys. J. Phys. G 33 (2006) 1.

[9] O.W. Greenberg, Phys. Rev. Lett. 89 (2002) 231602, hep-ph/0201258. [10] V.A. Kostelecky, R. Potting, Nucl. Phys. B 359 (1991) 545;

J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Int. J. Mod. Phys. A 11 (1996) 1489, hep-ph/9212057;

J.R. Ellis, J.L. Lopez, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 53 (1996) 3846, hep-ph/9505340.

[11] G. Amelino-Camelia, S. Majid, Int. J. Mod. Phys. A 15 (2000) 4301, hep-th/ 9907110;

I. Mocioiu, M. Pospelov, R. Roiban, Phys. Rev. D 65 (2002) 107702, hep-ph/ 0108136.

[12] J.N. Bahcall, V. Barger, D. Marfatia, Phys. Lett. B 534 (2002) 120, hep-ph/ 0201211.

[13] A. Datta, R. Gandhi, P. Mehta, S. Uma Sankar, Phys. Lett. B 597 (2004) 356, hep-ph/0312027.

[14] B.J. Rebel, S.L. Mufson, For MINOS Collaboration, arXiv: 0802.3785 [hep-ph].

[15] V.D. Barger, S. Pakvasa, T.J. Weiler, K. Whisnant, Phys. Rev. Lett. 85 (2000) 5055, hep-ph/0005197.

[16] S.M. Bilenky, M. Freund, M. Lindner, T. Ohlsson, W. Winter, Phys. Rev. D 65 (2002) 073024, hep-ph/0112226.

[17] H. Minakata, S. Uchinami, Phys. Rev. D 72 (2005) 105007, hep-ph/0505133.

[18] G. Barenboim, J.F. Beacom, L. Borissov, B. Kayser, Phys. Lett. B 537 (2002) 227, hep-ph/0203261.

[19] H. Murayama, T. Yanagida, Phys. Lett. B 520 (2001) 263, hep-ph/0010178.

[20] D. Colladay, V.A. Kostelecky, Phys. Rev. D 55 (1997) 6760, hep-ph/9703464; S.R. Coleman, S.L. Glashow, Phys. Rev. D 59 (1999) 116008, hep-ph/9812418; For limits on explicit models, see also: I. Mocioiu, M. Pospelov, Phys. Lett. B 534 (2002) 114, hep-ph/0202160.

[21] A. Dighe, S. Ray, arXiv: 0802.0121 [hep-ph].

[22] F. Ardellier, et al., Double Chooz Collaboration, hep-ex/0606025.

[23] A. Bandyopadhyay, S. Choubey, S. Goswami, Phys. Rev. D 67 (2003) 113011, hep-ph/0302243.

[24] H. Minakata, H. Nunokawa, W.J.C. Teves, R. Zukanovich Funchal, Phys. Rev. D 71 (2005) 013005, hep-ph/0407326.

[25] A. Bandyopadhyay, S. Choubey, S. Goswami, S.T. Petcov, Phys. Rev. D 72 (2005) 033013, hep-ph/0410283.

[26] M.C. Gonzalez-Garcia, M. Maltoni, arXiv: 0704.1800 [hep-ph].

[27] B.T. Cleveland, et al., Astrophys. J. 496 (1998) 505.

[28] J.N. Abdurashitov, et al., SAGE Collaboration, Phys. Rev. C 60 (1999) 055801, astro-ph/9907113.

[29] W. Hampel, et al., GALLEX Collaboration, Phys. Lett. B 447 (1999) 127.

[30] S. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 86 (2001) 5651, hep-ex/0103032.

[31] Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 87 (2001) 071301, nucl-ex/0106015.

[32] S.N. Ahmed, et al., SNO Collaboration, Phys. Rev. Lett. 92 (2004) 181301, nucl-ex/0309004.

[33] K. Eguchi, et al., KamLAND Collaboration, Phys. Rev. Lett. 90 (2003) 021802, hep-ex/0212021.

[34] Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 81 (1998) 1562, hep-ex/9807003.

[35] M. Ambrosio, et al., MACRO Collaboration, Phys. Lett. B 517 (2001) 59, hep-ex/ 0106049.

[36] E. Aliu, et al., K2K Collaboration, Phys. Rev. Lett. 94 (2005) 081802, hep-ex/ 0411038.

[37] D.G. Michael, et al., MINOS Collaboration, Phys. Rev. Lett. 97 (2006) 191801, hep-ex/0607088.

[38] M.C. Gonzalez-Garcia, M. Maltoni, Phys. Rev. D 70 (2004) 033010, hep-ph/ 0404085.

[39] M. Apollonio, et al., CHOOZ Collaboration, Eur. Phys. J. C 27 (2003) 331, hep-ex/ 0301017.

[40] See http://www.hep.ph.ic.ac.uk/ids/docs/index.html.

[41] T. Abe, et al., arXiv: 0712.4129 [physics.ins-det].

[42] J. Bouchez, M. Lindroos, M. Mezzetto, AIP Conf. Proc. 721 (2004 ) 37, hep-ex/ 0310059;

M. Mezzetto, J. Phys. G 29 (2003) 1771, hep-ex/0302007;

M. Mezzetto, Nucl. Phys. B (Proc. Suppl.) 143 (2005) 309, hep-ex/0410083.

[43] A. Donini, E. Fernández-Martínez, P. Migliozzi, S. Rigolin, L. Scotto Lavina, Nucl. Phys. B 710 (2005) 402, hep-ph/0406132;

A. Donini, E. Fernández-Martínez, S. Rigolin, Phys. Lett. B 621 (2005) 276, hep-ph/0411402.

[44] J.E. Campagne, M. Maltoni, M. Mezzetto, T. Schwetz, JHEP 0704 (2007) 003, hep-ph/0603172.

[45] Y. Itow, et al., T2K Collaboration, hep-ex/0106019.

[46] A. de Bellefon, et al., hep-ex/0607026.

[47] M. Diwan, et al., hep-ex/0608023.

[48] E.K. Akhmedov, R. Johansson, M. Lindner, T. Ohlsson, T. Schwetz, JHEP 0404 (2004) 078, hep-ph/0402175.

[49] J. Burguet-Castell, D. Casper, J.J. Gomez-Cadenas, P. Hernandez, F. Sanchez, Nucl. Phys. B 695 (2004) 217, hep-ph/0312068;

J. Burguet-Castell, D. Casper, E. Couce, J.J. Gomez-Cadenas, P. Hernandez, Nucl. Phys. B 725 (2005) 306, hep-ph/0503021.

[50] A. Donini, E. Fernandez-Martinez, Phys. Lett. B 641 (2006) 432, hep-ph/ 0603261;

E. Fernandez-Martinez, hep-ph/0605101.

[51] A. Donini, D. Meloni, P. Migliozzi, Nucl. Phys. B 646 (2002) 321, hep-ph/ 0206034;

A. Donini, D. Meloni, P. Migliozzi, J. Phys. G 29 (2003) 1865, hep-ph/0209240;

D. Autiero, et al., Eur. Phys. J. C 33 (2004) 243, hep-ph/0305185.

[52] See the talk by M. Lindroos at the second plenary IDS meeting, http://www. hep.ph.ic.ac.uk/ids/communication/RAL-2008-01-16/agenda.html.

[53] P. Huber, J. Kopp, M. Lindner, M. Rolinec, W. Winter, Comput. Phys. Commun. 177 (2007) 432, hep-ph/0701187.