Looking for Lorentz invariance violation (LIV) in the latest long baseline accelerator neutrino oscillation data

In this paper, we have analysed the latest data from NO$\nu$A and T2K with the Lorentz invariance violation along with the standard oscillation hypothesis. We have found that the NO$\nu$A data cannot distinguish between the two hypotheses at $1\, \sigma$ confidence level. T2K data and the combined data analysis excluded standard oscillation at $1\, \sigma$. All three cases do not have any hierarchy sensitivity when analysed with LIV. There is a mild tension between the two experiments, when analysed with LIV, as $\theta_{23}$ at \nova best-fit is at higher octant but the same for T2K is at lower octant. NO$\nu$A has a new degeneracy over $\sin^2 \theta_{23}$ value, when analysed with LIV.

The current unknowns are the sign of ∆ 31 , octant of θ 23 and the CP violating phase δ CP .There can be two possible mass orderings for the masses of the neutrino mass eigenstates, depending on the sign of ∆ 31 : normal hierarchy (NH), which implies m 3 >> m 2 > m 1 , and inverted hierarchy (IH), which implies m 2 > m 1 >> m 3 [9]. It is expected that the current 2 accelerator based long baseline neutrino oscillation experiments NOνA [10] and T2K [11] will measure these unknowns by measuring the ν µ → ν e oscillation probabilities in presence of matter effect. Recently both the experiments have published their latest analysis. The best-fit values for NOνA is sin 2 θ 23 = 0.57 +0.04 −0.03 and δ CP = 0.82π for NH [12]. For T2K, the best-fit values are sin 2 θ 23 = 0.53 +0.03 −0.04 for both mass hierarchies, and δ CP /π = −1.89 +0.70 −0.58 (−1.38 +0.48 −0.54 ) for normal (inverted) hierarchy [13]. Therefore, there is a moderate tension between the outcomes of the two experiments. The measured best-fit δ CP values of both the experiments are far apart. Moreover, there is no overlap between the allowed regions on sin 2 θ 23 − δ CP plane at 1 σ confidence level (C.L.). Although the individual experiments prefer NH over IH, their combined analysis has the best-fit point at IH [14].
Apart from the unknown standard oscillation parameters, these experiments will also investigate about the possibility of beyond standard model (BSM) physics.A large number of studies have been done about exploring BSM physics with long baseline neutrino oscillation experiments [15]. Recently non-unitary neutrino mixing [16] and non-standard neutrino interaction during propagation through matter [17,18] have been used to resolve the tension between NOνA and T2K. It is important to test other BSM physics models to resolve the tension as well.
Neutrino oscillation requires neutrinos to be massive albeit extremely light. This curious and interesting characteristic makes neutrino oscillation the first experimental signature of BSM physics. Without loss of any generality, SM can be considered as the low energy effective theory derived from a more general theory governed by Planck mass (M P 10 19 GeV). This more fundamental theory unifies gravitational interactions along with strong, weak and electro-magnetic interactions. There exists theoretical models [19][20][21][22][23] which include spontaneous Lorentz invariance violation (LIV) and CPT violations in that more complete framework at Planck scale. At the observable low energy, these violations can give rise to minimal extension of SM through perturbative terms suppressed by M P . Particles and anti-particles have same mass and lifetime due to CPT invariance. Any observed difference between masses or lifetimes of particles and anti-particles would be a signal for CPT violation. The present upper limit on CPT violation from kaon system is |m K 0 − mK 0 |/m K < 6 × 10 −18 [24]. Since, kaons are Bosons and the natural mass term appearing in the Lagrangian is mass squared term, the above constraints can be rewritten as |m 2 K 0 − m 2K 0 | < 0.25 eV 2 . Current neutrino oscillation data provide the bounds |∆ 21 −∆ 21 | < 5.9 × 10 −5 eV 2 and 3 |∆ 31 −∆ 31 | < 1.1 × 10 −3 eV 2 [25]. If these differences are non-zero and they are manifestation of some kind of CPT violating effects, they can induce changes in neutrino oscillation probabilities [26][27][28][29]. Various studies have been done about the LIV/CPT violation with neutrinos [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Several neutrino oscillation experiments have looked for LIV/CPT violations and put on constraints on the LIV/CPT violating parameters [46][47][48][49][50][51][52][53]. Ref. [54] includes the list of constraints on all the relevant LIV/CPT violating parameters. But till date no study has been made to look for LIV/CPT violation in the long baseline accelerator neutrino oscillation experiments data. In this paper, we will consider the minimal extension of SM that violates Lorentz invariance as well as CPT symmetry. We will test the model with the latest data from NOνA and T2K and try to see if there is any hint of CPT violating LIV in the individual as well as combined data set and whether the tension between the two experiments can be resolved with the help of this new physics hypothesis.
In section II, we will discuss the theoretical framework of LIV in neutrino oscillation and present the comparison between oscillation probabilities with and without LIV. In section III, we will discuss the details method of our analysis and in section IV we will present our results after analysing data from NOνA and T2K. The conclusion will be drawn in section V.

II. THEORETICAL FRAMEWORK
The Lorentz invariance violating neutrinos and anti-neutrinos can be described by the effective Lagrangian [26,55] is a 2N dimensional spinor containing ψ α(β) , which is a spinor field with α(β) ranging over N spinor flavours, and their charge conjugates given by ψ C α(β) = Cψ T α(β) . Therefore, Ψ A(B) can be expressed as Q in eq. II.2 is a generic Lorentz invariance violating operator. The first term in the right side of eq. II.2 is the kinetic term, the second term is the mass term involving the mass matrix M and the third term gives rise to the LIV effect.Q is small and perturbative in nature. 4 We will restrict ourselves only to the renormalizable Dirac couplings in the theory, i.e.
terms only with mass dimension ≤ 4 will be incorporated. Doing so, one can write the Lorentz invariance violating Lagrangian in the flavour basis as [26] where a µ αβ , b µ αβ , c µν αβ and d µν αβ Lorentz invariance violating parameters. Since, only left handed neutrinos are present in SM, the observable effects in the neutrino oscillation experiments can be parameterized as These are constant Hermitian matrices which can modify the standard Hamiltonian in vacuum. The first combination involves CPT violation, where as the second combination is the CPT conserving Lorentz invariance violating neutrinos. In this paper, we will consider only direction independent isotropic terms, and hence we will only consider the µ = ν = 0.
From now on, for simplicity, we will call a 0 αβ terms as a αβ and c 00 αβ term as c αβ . Taking into account only these isotropic LIV terms, the neutrino Hamiltonian with LIV effect becomes: G F is the Fermi coupling constant and N e is the electron density along the neutrino path.
The −4/3 in front of the second term arises due to non observability of the Minkowski trace of the CPT conserving LIV term c L which relates xx, yy, and zz component to the 00 component [26]. The effects of a αβ are proportional to the baseline L and those of c αβ are proportional to LE. In this paper, we will consider the effects of CPT violating LIV parameters a αβ only.
It is noteworthy that the Hamiltonian due to LIV is analogous to that with neutral current (NC) non standard interaction (NSI) during the propagation of neutrinos through αβ are the strength of NSI. Thus, a relation between CPT violating LIV and NSI can be found by following equation [56]: In this work, we will consider the effects of parameters a eµ = |a eµ |e iφeµ and a eτ = |a eτ |e iφeτ , because these two parameters have the highest influences on ν µ → ν e oscillation probability [45], which is responsible for octant, δ CP , and hierarchy sensitivity of long baseline accelerator neutrino like NOνA and T2K. Since, we are mostly concerned about determining these unknown standard oscillation parameters in the long baseline accelerator neutrino experiment, we fixed all other LIV parameters, except a eµ and a eτ , to zero. It implies that in this paper, the contribution from Lorez invariance violation in the NOνA and T2K experiments is coming only from the CPT violating Lorentz violation and mostly in the appearance channels. The ν µ andν µ disappearance channels still conserve Lorentz invariance. The current constraint on these parameters from Super-kamiokande experiment at 95% confidence level The ν µ → ν e oscillation probability in presence of LIV parameters a eµ and a eτ can be written in the similar way as the oscillation probability in presence of NSI parameters eµ and eτ [57][58][59]: The first term in eq. II.13 is the oscillation probability in the presence of standard matter 6 effect. It can be written as [60] P µe (SM)= sin 2 2θ 13 sin 2 θ 23 A is the Wolfenstein matter term [61], given by where E is the neutrino beam energy and L is the length of the baseline.
For the second and third terms in eq. II.13, describing the effects of a eµ and a eτ respectively, we follow the similar approach of NSI, described in references [57][58][59], and replace αβ terms by a αβ terms according to eq. II.11. Doing so, we can write where β = µ, τ ; and W eβ = cos θ 23 sin 2 θ 23 sin∆ From, eq. II.15, it can be seen that the LIV effects considered in this paper are matter independent.
The oscillation probability Pμē for anti-neutrino can be calculated from equations II.14 and II.15 by substituting ability. To do so, we modified the probability code of the software to include LIV. After that, GLoBES is capable of calculating the oscillation probability without the approximations required to derive equations II.13-II. 15 H is the Hamiltonian from eq. II.6. The oscillation probability of ν µ → ν e after travelling through a distance L can be written as In fig. 1 is very small compared to that of NOνA (810 km).
It would be interesting to note down the the ability to discriminate between the two models for different δ CP values and baselines. To do so, we first fixed the neutrino energy good discrimination capability between the two models at the oscillation probability level for all three cases and with both the hierarchies.
In the next step, we repeated the same exercise by fixing the energy E = 0.7 GeV at the T2K. We varied |∆ µµ | in its 3 σ range around the MINOS best-fit value 2.32 × 10 −3 eV 2 with 3% uncertainty [65]. ∆ µµ is related with ∆ 31 by the following relation [66] ∆ µµ = sin 2 θ 23 ∆ 31 + cos 2 θ 12 ∆ 32 + cos δ CP sin 2θ 12 sin θ 13 tan θ 12 ∆ 21 . We calculated the theoretical event rates and the χ 2 between data and theoretical event rates using GLoBES [62,63]. The data has been taken from [12,13]. To calculate the theoretical event rates, we fixed the signal and background efficiencies by matching with the Monte-Carlo simulations given by the collaborations [12,13]. Automatic bin based energy smearing for generated theoretical events has been implemented in the same way as described in the GLoBES manual [62,63]. For this purpose, we used a Gaussian smearing where E is the reconstructed energy. The energy resolution function is given by • 5% normalisation and 0.01% energy callibration systematics uncertainty for µ like events.
Implementing systematics uncertainty has been discussed in details in GLoBES manual [62,63]. During the calculations of χ 2 we added priors on sin 2 2θ 13 . After calculating χ 2 , we found out minimum of these χ 2 s and subtracted it from all the χ 2 s to calculate ∆χ 2 .

IV. RESULTS AND DISCUSSIONS
At first, we have analysed the data with the standard matter effect without any LIV hypothesis. The minimum χ 2 for NOνA (T2K) with 50 (88) bins is 48.65 (95.85) and it is at NH. For the combined analysis, the minimum χ 2 with 138 bins is 147.14 and it occurs at IH. In fig. 9, we have shown the analysis in the sin 2 θ 23 − δ CP plane. This plot is comparable to the ones presented by the collaborations in references [12,13]. It is evident that there is a tension between the two experiments in terms of the best-fit δ CP values. Moreover, there is no overlap between the 1 σ region of each experiment.
Once the standard analysis is done, we proceeded to analyse the data with LIV hypothesis. We found out that the minimum χ 2 for NOνA (T2K) with 50 (88) bins is 47.71 (93.14) and it is at NH. Both the experiments, however, have a degenerate solution at IH with ∆χ 2 = 0.1 For the combined analysis the minimum χ 2 is 145.09 at IH for 138 bins. The combined analysis has a degenerate solution at NH with ∆χ 2 = 0.1. Therefore, the present accelerator neutrino oscillation data has no hierarchy sensitivity when analysed with LIV.
In fig. 10, we have presented our result with LIV hypothesis on the sin 2 θ 23 − δ CP plane. T2K (NOνA) disfavours (includes) NOνA (T2K) best-fit points at 1 σ C.L. Now, there is a large overlap between the 1 σ allowed regions of the two experiments. Hence, one can conclude that the tension between the two experiments gets reduced when the data are analysed with LIV. However, there is a new mild tension in terms of the best-fit sin 2 θ 23 values. θ 23 at T2K best-fit point is at lower octant, while the same is at higher octant for NOνA. But NOνA has a nearly degenerate (∆χ 2 = 0.35) best-fit point at lower octant.
Similarly, T2K also cannot rule out higher octant at 1 σ C.L. Thus, both the experiments lose their octant sensitivity when analysed with LIV. In fig. 11    values have ∆χ 2 < 1. For the combined analysis, |a eµ | = 0 has a ∆χ 2 > 1, but |a eτ | = 0 has ∆χ 2 < 1. Therefore, it can be said that the present NOνA data do not favour any of the two hypotheses over the other. However T2K data and the combined analysis disfavour standard oscillation at 1 σ C.L. In tables II, III and IV, we have presented the best-fit values of the standard and non-standard unknown oscillation parameters.
To emphasize our result, in fig. 12, we have presented the expected electron and positron events rates for each energy bins for both standard oscillation and oscillation with LIV as a function of energy for both NOνA and T2K. The experimental event rates have also been plotted. It is obvious that for NOνA, there is not any significant difference between the expected event rates at best-fit points of the two models and both models give a good fit to the data. However, for T2K, there is a clear distinction at the expected event rates at the best-fit points of the two models. Also, LIV gives a better fit to the data especially at energies close to the flux peak energy. neutrino oscillation data lose hierarchy sensitivity when analysed with LIV. The 1 σ allowed regions from the two experiments have a large overlap with LIV, unlike the standard oscillation case. Therefore, one can comment that the tension between the two experiments is reduced when the data are analysed with LIV. However, there is a new mild tension between the best-fit values of sin 2 θ 23 . While θ 23 for NOνA best-fit is at HO, it is at LO for the T2K bst-fit point. But NOνA (T2K) has a nearly degenerate best-fit point at lower (higher) octant as well. Therefore, both NOνA and T2K data do not have octant and hierarchy determination capability when analysed with LIV.
In light of these, we recommend that the long baseline accelerator neutrino experiments to be analysed with LIV along with other BSM physics. If the future data continue to favour LIV over standard oscillation, it can be considered as a prominent signal for LIV.