CP-Violating Neutrino Non-Standard Interactions in Long-Baseline-Accelerator Data

Neutrino oscillations in matter provide a unique probe of new physics. Leveraging the advent of neutrino appearance data from NOvA and T2K in recent years, we investigate the presence of CP-violating neutrino non-standard interactions in the oscillation data. We first show how to very simply approximate the expected NSI parameters to resolve differences between two long-baseline appearance experiments analytically. Then, by combining recent NOvA and T2K data, we find a tantalizing hint of CP-violating NSI preferring a new complex phase that is close to maximal: $\phi_{e\mu}$ or $\phi_{e\tau}\approx3\pi/2$ with $|\epsilon_{e\mu}|$ or $|\epsilon_{e\tau}|\sim0.2$. We then compare the results from long-baseline data to constraints from IceCube and COHERENT.


I. INTRODUCTION
Neutrino oscillations have provided the only particle physics evidence for new physics beyond the standard model (BSM) to date [1,2], making it an excellent place to probe new physics scenarios. The phenomenology of neutrino oscillations is fairly unique, as it provides an opportunity to observe the accumulation of a relative phase over macroscopic distances, making neutrino oscillations one of the purest probes of quantum mechanics available. During propagation, the environment may also modify the phases due to an interaction. Such an interaction exists in the standard model (SM) and is called the Wolfenstein matter effect [3], wherein a neutrino in the electron state of the flavor basis experiences a potential with the background electrons via a charged-current (CC) interaction.
In the same paper that pointed out the SM matter effect, Wolfenstein also suggested the possibility of a new interaction that provides a matter effect, so-called neutrino non-standard interactions (NSI) [3][4][5]. Since then, there has been an explosion of interest to probe these new interactions. Numerous UV complete models have been developed [6][7][8][9][10][11] and the phenomenology has been generalized beyond vector currents [12][13][14]. In addition, several NSI parameters introduce various interesting degeneracies in oscillation or scattering experiments [4,, which demonstrates the importance of complementary measurements of the NSI parameters.
One of the most complete ways to probe neutrino oscillations is through long-baseline accelerator experiments with electron (anti)neutrino appearance. While these measurements are extremely challenging experimentally, they provide a wealth of information, as they are sensitive to many oscillation parameters, including those that are the least constrained, like the CP-violating phase δ from the leptonic mass mixing matrix. In addition, ap- * pdenton@bnl.gov; 0000-0002-5209-872X † jgehrlein@bnl.gov; 0000-0002-1235-0505 ‡ rebhawk8@vt.edu; 0000-0002-9634-1664 pearance measurements provide a crucial probe of certain NSI parameters.
The two state-of-the-art long-baseline neutrino experiments are NOvA and T2K [41,42]. Both are off-axis; therefore, each detects a flux of neutrinos with a relatively narrow energy distribution. The latest results from both experiments [43,44] show a slight tension at the ∼ 2σ level, depending on how exactly it is quantified. Both experiments prefer the normal mass ordering, but T2K prefers δ ∼ 3π/2 while NOvA does not have much preference and is generally around δ ∼ π. While this is not yet significant, it provides an interesting test case for new physics should it persist, as both experiments plan to accumulate additional data.
In this paper, we review NSI and show how to approximate the NSI parameters that describe the NOvA and T2K data in section II. We then describe our treatment of the NOvA and T2K data and show results in the standard oscillation picture in sections III and IV. Then, we show in section V that the NOvA and T2K data can be resolved by the inclusion of NSI with complex CP-violating (CPV) phases with a preference for CPV values over CP-conserving values. Finally, we discuss our results in a broader picture of other neutrino measurements and present some possible plans to improve these results, and we conclude in section VI. All the relevant data files are available at peterdenton.github.io/Data/NOvA+T2K NSI/index.html.

II. NSI OVERVIEW
NSI in oscillations provides an additional contribution to the matter potential of the neutrino oscillation Hamiltonian in the weak basis where E is the neutrino energy, U ≡ R 23 (θ 23 )U 13 (θ 13 , δ)R 12 (θ 12 ) is the PMNS mixing matrix [45,46] that is parameterized in the usual arXiv:2008.01110v1 [hep-ph] 3 Aug 2020 way [47], M 2 ≡ diag(0, ∆m 2 21 , ∆m 2 31 ) is the diagonal mass-squared matrix, a ≡ 2 √ 2G F N e E is the matter potential, and N e is the electron density. The αβ terms parameterize the size of the new interaction relative to the weak interaction and typically arise from effective Lagrangians of the form For simplicity, we only consider NSI with vector mediators. The Lagrangian level NSI parameters in eq. 2 are related to the Hamiltonian level terms in eq. 1 by where N f is the number density of fermion f . In the context of oscillations, it isn't possible to identify which matter particles (electrons, up quarks, or down quarks) the new physics is coupled to without comparing neutrino trajectories through materials with different neutron fractions, such as the Earth and the sun. Within the context of long-baseline trajectories through the crust, the neutron fraction is close to one. While the NSI parameters are often taken to be real for simplicity, we consider complex NSI, where αβ = | αβ |e iφ αβ for α = β, which violate CP [16,48], see ref. [49] for more on complex NSI.
In the presence of CPV NSI, one could interpret T2K's preference for δ T2K ∼ 3π/2 as approximately the true parameter, since the matter effect at T2K is comparably small. Thus, NOvA's measurements would be a function of the same δ T2K ∼ 3π/2 with a correction from NSI such that NOvA infers their best fit value of δ NOvA ∼ π (although NOvA has a broad allowed region in δ NOvA ). NSI phase reduction is possible to a good approximation for eµ and eτ . That is, at leading order the complex phases only appear as δ + φ eµ and δ + φ eτ . Thus, under the assumption that T2K experiences no matter effect and NOvA experiences a sizable matter effect, we find the following relation: for β = µ, τ (note that the phase reduction does not apply for µτ ). Therefore, we anticipate we will find that φ eβ ∼ 3π/2 will reduce the tension between the experiments. In addition, one can estimate the magnitude of the NSI parameter that would resolve different measurements of δ in experiments experiencing distinct matter potentials. We find that if two experiments at two different matter potentials measure two disparate values of δ due to eβ NSI for β ∈ {µ, τ }, the magnitude of the NSI in the NO is approximately given by where w β = s 23 or c 23 for β = µ or τ respectively 1 . The preferred value of eτ is larger than that for eµ since T2K prefers the upper octant and T2K is less affected by NSI than NOvA. The difference between eµ and eτ makes sense since long-baseline oscillations are dominated by ν 3 , which contains more ν µ in the upper octant, and thus, not as much NSI affecting ν µ is required to produce a given effect. We also note that the approximations presented here are quite consistent with our numerical results discussed below and shown in fig. 2 and table I. For more on the approximate derivations in this section, see appendix A.

III. ANALYSIS DETAILS
The appearance channels at NOvA and T2K can be approximated by counting experiments, while for the disappearance channels, the energy distribution of the events is important. This approximation ignores several potentially problematic issues: the energy distributions aren't exactly delta distributions, there are correlated systematics between the different channels, and the cross section systematics may well be related even between the different experiments. Nonetheless, we find an acceptable reproduction of the results with the simple treatment described below.
NOvA measures neutrinos with E ∼ 1.9 GeV after traveling 810 km through the Earth with density ρ = 2.84 g/cc, while T2K measures neutrinos with E = 0.6 GeV after traveling 295 km through the Earth with average density ρ = 2.3 g/cc. For the appearance channels, we find that the number of events can be expressed as a constant normalization term and a constant factor which multiplies the oscillation probability in matter (see also [50] for a similar approach). These constant factors can be derived from the provided bi-event plots in [43,44,51]. As wrong sign leptons contribute to the flux, especially in antineutrino mode, we parameterize the predicted numbers of events as and similarly for the antineutrino channel. For NOvA, a good fit is obtained for the neutrino channel without including the wrong sign leptons, so we find n(ν e ) NOvA = 13.97 + 472.60 × P (ν µ →ν e ) while for T2K, we find n(ν e ) T2K = 5.77 + 231.95 × P (ν µ →ν e ) The disappearance channel cannot be treated as a counting experiment, since the energy distribution of the events is important. At leading order, the oscillation probability for neutrinos and antineutrinos is the same in this channel. However, this changes in the presence of NSI. In the following, we will assume that the results in the disappearance channel are dominated by the neutrino sample, which provides higher statistics than the antineutrino sample. We adapt the results from [50] for the disappearance channel at NOvA, where they found as best fit |∆m 2 32 | = (2.41 ± 0.07) × 10 −3 eV 2 and 4|U µ3 | 2 (1−|U µ3 | 2 ) = 0.99±0.02. For T2K, we obtain the test statistic for θ 23 and ∆m 2 32 from the 1D distributions of the test statistics provided by the experiment [43] For the appearance channel, incorporating the effect of NSIs as described in eq. 1 is straightforward. For the disappearance channels, we calculate the effective vacuum mixing parameters by solving where A ≡ diag(a, 0, 0) and the N matrix contains the 's and is proportional to the matter potential a. Then, by diagonalizing U † M 2 U + N , one finds the vacuum parameters that a long-baseline accelerator experiment would extract in the presence of NSI. Various approximate techniques for the diagonalization of matrices in the context of neutrino oscillations in matter have been explored in [52][53][54][55][56][57][58][59]. The approach presented in eq. 10 is exact in the case of constant matter density; it does not apply to solar or atmospheric neutrinos, and additional care is necessary there. Finally, one can compare the effective vacuum mixing parameters extracted from M 2 and U to the measured oscillation parameters.
To analyze the data, we construct a test statistic using a log likelihood ratio with Poisson statistics for the appearance data and simple χ 2 pulls for the disappearance constraints.

IV. STANDARD OSCILLATION RESULTS
Before we address new physics in the neutrino sector, we show the preferred regions in the standard oscillation picture in fig. 1. Contours are drawn relative to the best fit point at ∆χ 2 = χ 2 − χ 2 bf = 4.61. Note that combining the data sets raises the minimum χ 2 by ∼ 5.5 over either experiment individually; this tension can be somewhat alleviated by switching the mass ordering [50,60]. We show the preferred regions of θ 23 and the Jarlskog invariant where J = s 12 c 12 s 13 c 2 13 s 23 c 23 sin δ is the Jarlskog [61], which is a parameterization-independent quantification of CPV in the leptonic mass matrix [62]. Note that the maximum value of the Jarlskog is 1/6 √ 3 ≈ 0.096; we are already quite far from maximal CPV in the leptonic sector due primarily to the fact that θ 13 is fairly small.
For fig. 1 we include a minimization over the four other standard oscillation parameters and the sign of cos δ for the Jarlskog panel. We include priors from KamLAND We find that the best fit parameters are at J = −0.0128, δ/π = 1.12, and sin 2 θ 23 = 0.556 in the NO. In the IO the best fit parameters are J = −0.0328, δ/π = 1.52, and sin 2 θ 23 = 0.56. These are compatible at the < 1σ level with the latest global fit to all oscillation experiments [60].
We see that in the normal mass ordering (NO), while T2K has some significance to disfavor J = 0, the inclusion of NOvA data weakens this, making CPV in the standard oscillation picture an important goal for NOvA and T2K [65,66] in coming years, as well as upcoming long-baseline accelerator neutrino experiments such as DUNE and T2HK [67,68]. This weakening of the significance in the NO when the experiments are combined emphasizes the slight tension between the experiments.
Similarly to refs. [50,60], we also find that while NOvA and T2K both individually prefer the NO, the combination shows a slight preference for the inverted mass ordering (IO) at χ 2 NO −χ 2 IO = 2.7. When combined with Super-KamiokaNDE (SK) atmospheric data [69,70], the best fit mass ordering (MO) remains normal [50,60] 3 . This MO question is of crucial significance beyond just measuring parameters in the SM. It may provide guidance about the structure of neutrino mass [71] and is a key input for many experimental measurements of neutrinos, including cosmological measurements of neutrino properties, kinematic measurements of neutrinos, and neutrinolessdouble-beta decay measurements should neutrinos have a Majorana mass term, see e.g. [72].
In the next section, we find that in the presence of NSI, the long-baseline data is better described by the NO than the IO, so we assume the NO unless otherwise specified. The MO can be confirmed independently of the presence of NSI via JUNO [73]. 3 SK preferred the NO at χ 2 IO − χ 2 NO > 5, but with their latest data release, the significance dropped to ∼ 3.2, although it is still enough to prefer the NO in total.

V. NSI RESULTS
We analyze one complex NSI parameter at a time, using the appearance and disappearance data from NOvA and T2K and assuming the NO. In fig. 2, we present the allowed parameter regions in the | αβ |-φ αβ plane for eµ and eτ . The results for µτ can be found in appendix B. For simplicity, we fix θ 13 , θ 12 , and ∆m 2 21 to the best fit values from Daya Bay and KamLAND as described above and marginalize over ∆m 2 31 , δ, and θ 23 , including the pull on ∆m 2 31 from Daya Bay. We have verified that including the pulls associated with θ 13 , θ 12 , and ∆m 2 21 do not significantly affect our results. The best fit values for the parameters for each case of eµ , eτ , and µτ in both MOs are given in table I. Note that while the combination of both experiments raises the χ 2 by about 5.5 as mentioned in the previous section, that can be nearly completely alleviated with the addition of eµ which provides an improvement in the test statistic of 4.7 (compare this to switching to the IO which only improves the test statistic by 2.7 and is in tension with SK data). In the presence of NSI, we still find that the upper octant is preferred with sin 2 θ 23 = 0.56 for all three NSI parameters and both MOs.
Consistent with our analytic estimates, we find moderate evidence for CP-violating NSI. The best solution is with the eµ parameter with maximal CP-violating phases for both the standard CP phase and the new NSI CP phase.
The constraints on complex NSI parameters from IceCube [74] slightly disfavor the preferred region for eµ , although it is possible to get an improved fit to the NOvA and T2K data while not being in too strong of tension with the IceCube data. In fact, the best fit point to the IceCube data for eµ is at | eµ | = 0.07 and φ eµ /π = 1.91, close to the relevant numbers for NOvA and T2K. It is also interesting to note that IceCube slightly disfavors | eµ | = 0 at just over 1σ.
We show the constraints from IceCube on complex NSI from [74] on figs. 2 and 3, which only slightly disfavors this NSI explanation of long-baseline data with eµ . The IceCube constraints are comparable to other constraints in the literature on real NSI from oscillation experiments [4,36,75].
COHERENT's measurement of the coherent elastic neutrino nucleus scattering (CEvNS) process [76] provides constraints [13,36,39,[77][78][79][80][81][82][83] on the NSI parameter space that is also an explanation of the NOvA and T2K data. While the parameters relevant for NOvA and T2K are are not strongly ruled out by COHERENT yet, they can be probed by COHERENT in coming years. It should be noted, however, that the NSI constraint derived from COHERENT only applies to NSI governed by mediators heavier than ∼ 10 MeV [36,84]. Constraints for lower mediators masses down to ∼ 1 MeV can by placed with upcoming low-threshold CEvNS experiments at nuclear reactors. Meanwhile, early universe measurements constrain mediators lighter than ∼ 5 MeV [85,86]. Thus we anticipate that COHERENT or future reactor CEvNS experiments will be able to probe the NSI parameters that could explain the NOvA and T2K data in coming years.

VI. CONCLUSIONS
Measuring and understanding CP violation is of the utmost importance in particle physics. Somewhat confusingly, the weak interaction violates CP while the strong interaction seems to conserve CP. Meanwhile, the quark mass mixing matrix has relatively small CP violation. To better understand the important role that CPV plays in particle physics, we must measure it and understand it in the leptonic sector.
In this manuscript, we have analyzed a new physics explanation for the slight tension in the recent NOvA and T2K data. We performed a fit to the data and showed that this tension can be resolved when introducing complex CP-violating NSI parameters. As an example, we analyzed non-zero eµ , eτ , µτ one at a time and found that the best fit points for the new complex phases of αβ prefers not only maximal CPV in the new interaction around 3π/2 for α = e, but also large CPV in the leptonic mass matrix. These NSI parameters are best constrained (not counting long-baseline experiments) by atmospheric oscillation measurements by Super-KamiokaNDE and IceCube. These measurements rule out the favored parameter region for µτ , whereas the atmospheric constraints only partially disfavor the preferred regions of eµ and eτ . We anticipate that improvements from Super-KamiokaNDE and IceCube can further test this hypothesis in the future. Furthermore, experiments that probe coherent elastic neutrino nucleus scattering will provide strong constraints on NSI parameters of a similar order of magnitude, though they only apply to mediators heavier than the ∼ 10 MeV scale. In addition, while without new physics, the IO is slightly preferred by NOvA and T2K, the inclusion of NSI shifts the preference back to the NO. JUNO's measurement of the MO, which has almost no dependence on the matter effect, will determine the MO independent of NSI.
We can see clearly from e.g. eq. 10 that in order to measure NSI with long-baseline neutrinos, one needs to either compare two different experiments or use a broad band beam such as that which DUNE will have [67].
To summarize, we have shown that the tension of the recent NOvA and T2K data can be resolved in a BSM scenario with the introduction of CP-violating NSI parameters, which can be further probed with near-future experiments. It would be interesting to see if other new physics models could also explain the discrepancy, such as the presence of sterile neutrinos, decoherence, or neutrino decay. FIG. 3. The preferred parameter region for µτ using the newest appearance and disappearance data from NOvA and T2K and assuming the NO (left) or the IO (right). The gray region is disfavored compared to the SM and the blue star shows the best fit point. The orange contours are drawn at integer values of ∆χ 2 . See table I for the best parameters. IceCube disfavors the region to the right of the black dotted curve at 90% [74].