Nonmaximal neutrino mixing at NO$\nu$A from nonstandard interactions

Muon neutrino disappearance measurements at NO$\nu$A suggest that maximal $\theta_{23}$ is excluded at the 2.6$\sigma$ CL. This is in mild tension with T2K data which prefer maximal mixing. Considering that NO$\nu$A has a much longer baseline than T2K, we point out that the apparent departure from maximal mixing in NO$\nu$A may be a consequence of nonstandard neutrino propagation in matter.

Recently, NOνA released a new measurement of θ 23 from the ν µ disappearance channel which indicates that θ 23 = π/4 is excluded at the 2.6σ CL [1]. T2K measurements in the same channel prefer θ 23 = π/4 [2]. Neutrinos in both the NOνA and T2K experiments travel through a long distance in matter, so that their oscillation probabilities are modified by the interactions with matter via the MSW effect [3,4]. Since the NOνA baseline (810 km) and neutrino energy (∼ 2 GeV) are greater than those for T2K (295 km and ∼ 0.6 GeV), the matter effect in the NOνA experiment is much larger than in T2K. However, the standard weak interactions with matter have a negligible effect on the ν µ survival probabilities. In this Letter, we study the matter effects induced by nonstandard interactions (NSI) on the ν µ survival probabilities, and show that they reconcile the discrepancy between the NOνA and T2K measurements of θ 23 .
NSI are motivated by physics beyond the standard model (SM), and provide a model-independent way to study subdominant effects in neutrino oscillation experiments; for recent reviews, see Ref. [5,6]. NSI can in general affect neutrino production, detection, and propagation in matter. Here we focus on the matter NSI, which can be described in an effective theory by the dimensionsix operators [3] where α, β = e, µ, τ , C = L, R, f = u, d, e, and f C αβ are dimensionless parameters that quantify the strength of the new interaction in units of the Fermi constant G F . The Hamiltonian that describes neutrino propagation in matter with NSI is where U is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [7], ∆m 2 ij = m 2 i − m 2 j , and Here, Ne gives the effective strength of NSI relative to the SM chargedcurrent interaction in matter, and N f is the number density of fermion f .
The effects of matter NSI on neutrino oscillations at NOνA have been analyzed at the probability level in Ref. [8]. We recently showed that matter NSI could lead to wrong determinations of the Dirac CP phase, the mass hierarchy, and θ 23 octant at NOνA and T2K [9]. For example, the current hint of δ CP = −π/2 from T2K [10] could be due to a nonzero eτ [11]. Previous analyses of NOνA have focused on the ν e appearance channel, in which the NSI terms related to ee , eµ and eτ are dominant [9]. However, in the ν µ disappearance channel, the leading NSI contributions come from the µ − τ sector.
The Super-Kamiokande experiment has obtained the strong 90% CL constraints, | µτ | < 0.011 and | τ τ − µµ | < 0.049 [12], using a two-flavor analysis of its atmospheric neutrino data. However, it has been shown that the two-flavor framework is not adequate to constrain NSI parameters using atmospheric neutrino experiments [13,14]. Note that the Super-Kamiokande collaboration also performed a three-flavor analysis in Ref. [12] with the standard oscillation parameters fixed. As shown in Ref. [13], constraints on τ τ − µµ are significantly weaker when the standard oscillation parameters are marginalized over. This is confirmed in a global three-flavor analysis of neutrino oscillation data including matter NSI, which yields the approximate 3σ CL bounds, (taking e αβ = 0, and | αβ | , | µτ | 0.10, | eτ | 1.34 and −0.68 τ τ − µµ 0.66; these are the limits from the SNO-DATA variant of the solar analysis in Ref. [15]. 1 Since Ref. [15] only considered NSI with one flavor f = e, f = u or f = d at a time in the analysis of solar data, we consider these bounds to be representative. Nonmaximal mixing from NSI. To understand the dependence of the survival probabilities on the NSI parameters at NOνA and T2K, we first consider the two-flavor framework. The Hamiltonian that describes nonstandard neutrino propagation in matter induced by NSI in the 1 If instead, we assume uncorrelated errors and take the separate bounds on | u αβ | and | d αβ | in quadrature, i.e., , and c ij (s ij ) denotes cos θ ij (sin θ ij ). The matter density is constant for the relevant baselines, and the ν µ survival probability in the two-flavor framework can be written in the form [16] where As can be seen from the above equations, even with maximal mixing in vacuum, i.e., θ 23 = π/4, the NSI terms can generate nonmaximal mixing in matter. Also, because cos 2θ 23 sin 2θ 23 , the diagonal parameter τ τ − µµ has a larger effect on the deviation from maximal mixing than the off-diagonal parameter µτ . This, coupled with the fact that µτ is more tightly constrained than τ τ − µµ , leads us to fix µτ = 0.
In the three-flavor framework, we ignore the solar masssquared difference (since ∆m 2 21 /|∆m 2 32 | ≈ 0.03), take the NSI parameters to be real, and the CP phase to be vanishing. Henceforth, we set µµ = 0, as the oscillation probabilities are not affected by subtracting an overall diagonal term in the Hamiltonian. We only consider nonzero τ τ and eτ for simplicity.
where R ij is a real rotation by an angle θ ij in the ij plane. If we assume the terms in the square bracket of Eq. (6) are diagonalized by U α = R α 23 R α 13 R α 12 , where R α ij is a real rotation by an angle α ij in the ij plane, then the mixing matrix that diagonalizes the Hamiltonian in matter is U m = R 23 U α . SinceÂ ≈ 0.17 Eν 2 GeV at NOνA, 2 In anticipation of our numerical results, we mention that our NOνA data analysis is insensitive to the O(1) values of ee allowed by the global fit of Ref. [15]. Consequently, our conclusions are unaffected by an O(1) ee invoked to satisfy the relation, | eτ | 2 τ τ (1 + ee), required by high energy atmospheric data [13].
we find [17] α 23 = − s 23 c 23 τ τÂ c 2 13 + cos 2θ 23 τ τÂ Then the ν µ disappearance probability can be written in the form of Eq. 5, with the oscillation amplitude replaced by where θ m 23 = θ 23 + α 23 and θ m 13 = α 13 . Data analysis. To analyze NOνA's ν µ disappearance results we extract the unoscillated spectrum, backgrounds, and data from Ref. [1]. Since the data above 2.5 GeV are noisy and have an insignificant effect on the parameter fit [1], we only include 7 bins in the energy range [0.75 GeV, 2.5 GeV] in our analysis. The expected number of events per bin N th i is calculated as where N unosc i is the expected number of events without oscillations, N bkg i is the expected background, and P (ν µ → ν µ ) i is the average survival probability in each bin. We calculate the survival probabilities in the threeflavor framework using the GLoBES software [18] supplemented with the results of Ref. [19]. We choose θ 12 , ∆m 2 21 to be the global best-fit values [7], vary |∆m 2 32 | between (2.0 − 3.5) × 10 −3 eV 2 and set δ CP = 0 because the CP phase has a negligible effect on our analysis.
To evaluate the significance of each scenario, we define where N obs i is the observed number of events in each bin, and σ i is obtained by summing the statistical and systematic uncertainties in quadrature. Both sets of (asymmetric) uncertainties are extracted from Ref. [1].
Results. For the SM we find that the best fit value of sin 2 θ 23 is 0.41 in the first octant and 0.63 in the second octant. Defining we find that in the SM case ( τ τ = eτ = 0) maximal mixing is excluded at the 2.2σ CL for the normal hierarchy, which is close to the NOνA result of 2.6σ CL [1]. In Fig. 1, we display the confidence level at which maximal mixing is excluded as a function of τ τ after marginalizing over | eτ | < 1.2 for both the normal and inverted hierarchy, and for two possible values of θ 13 . We choose sin 2 θ 13 = 0.022, which is a weighted average of recent reactor neutrino results, and sin 2 θ 13 = 0.030, which is at the edge of the 3σ range obtained in Ref. [15]. We checked that varying θ 13 within the 3σ allowed range has very little effect on the exclusion of maximal mixing in the SM analysis. The exclusion weakens significantly for some large values of | τ τ |. For the normal hierarchy, an NSI scenario with τ τ = 0.6 and eτ = 1.2 is perfectly consistent with θ 23 = π/4. (Incidentally, χ 2 min (0.6, 1.2) = 4.39 which represents a very good fit to the NOνA data.) Inserting τ τ = 0.6, eτ = 1.2, µµ = µτ = 0, θ 23 = π/4, sin 2 θ 13 = 0.030 andÂ(E ν = 1.625 GeV) ≈ 0.14 into Eqs. (7)(8)(9)(10) gives sin 2 θ = 0.41, which is close to the best-fit obtained by NOνA.
In Fig. 2 we plot the event distributions and survival probabilities for three different scenarios with a normal mass hierarchy. The SM(a) scenario has parameters close to the best-fit values from the T2K experiment [2] with maximal mixing. The SM(b) scenario corresponds to the best-fit values from the recent NOνA measurement [1]. For the NSI scenario, we choose θ 23 = π/4 with τ τ = 0.6 and eτ = 1.2 and all other NSI parameters set to zero. In both panels we see that the NSI scenario with maximal mixing is substantially similar to the SM(b) scenario with nonmaximal mixing. The NOνA measurement of nonmaximal mixing in the standard scenario could be interpreted as a hint for NSI with maximal mixing. The T2K curves in the right panel are almost overlapping for the NSI and SM(a) scenarios because T2K has a relatively short baseline. The small difference between them is due to the different values of ∆m 2 32 . It is noteworthy that with three years of data in the neutrino mode and three years in the antineutrino mode, NOνA will differentiate between the NSI scenario with maximal mixing depicted in Fig. 2 and the SM scenario with nonmaximal mixing at about the 3σ CL.
Summary. We analyzed the recent NOνA ν µ disappearance data in the framework of nonstandard neutrino interactions. We find that if the NSI parameters | eτ | and | τ τ | are O(1), the recent NOνA data are consistent with θ 23 = π/4, a value preferred by the T2K data. O(1) values for these NSI parameters have a negligible effect on the T2K measurement. This means that the value of θ 23 measured by T2K is close to the vacuum value, while the nonmaximal mixing detected by NOνA could be a hint of matter NSI.
We consider our study to be a proof of principle that demonstrates that if this anomaly blossoms into something more significant, then currently running experiments may lead us to NSI, and that we may not have to wait more than a decade for experiments like DUNE and T2HK to discover the existence of NSI [9,20]. In fact, the drastically altered mission of future long-baseline experiments would be to corroborate this discovery.