Nonstandard interactions in solar neutrino oscillations with Hyper-Kamiokande and JUNO

Measurements of the solar neutrino mass-squared difference from KamLAND and solar neutrino data are somewhat discrepant, perhaps due to nonstandard neutrino interactions in matter. We show that the zenith angle distribution of solar neutrinos at Hyper-Kamiokande and the energy spectrum of reactor antineutrinos at JUNO can conclusively confirm the discrepancy and detect new neutrino interactions.

matter NSI. Matter NSI can be described by dimension-six four-fermion operators of the form [8,9] L NSI = −2 √ 2G F f C αβ [ν α γ ρ P L ν β ] f γ ρ P C f + h.c. , (1) where α, β = e, µ, τ , C = L, R, f = u, d, e, and f C αβ specifies the strength of the new interaction in units of G F . The JUNO experiment will measure δm 2 21 to the percent level [6], but it is not sensitive to the NSI parameters due to its short baseline and low neutrino energy. The HK solar neutrino oscillation probabilities are strongly dependent on the NSI parameters due to the MSW effect [8,10], but HK will not precisely measure δm 2 21 due to systematic uncertainties. A combination of the two experiments could provide direct evidence for the existence of new physics if the δm 2 21 discrepancy persists in HK and JUNO data.
The paper is organized as follows. In Section II, we analyze the day-night asymmetry with NSI. In Section III, we describe our simulations of HK and JUNO. We discuss our results in Section IV, and sum up in Section V.

A. Formalism
The Hamiltonian in the three neutrino framework for neutrino propagation in the presence of matter NSI can be written in the flavor basis as where U is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [11], where R ij represents a real rotation by an angle θ ij in the ij plane, Γ δ = diag(1, 1, e iδ ), and s ij and c ij denote sin θ ij and cos θ ij respectively. The potential V originating from interactions of neutrinos in matter is where , and N f is the number density of fermion f at a given location. Following Ref. [12], we work in the new basis |ν = U † |ν α , withŨ = R 23 Γ δ R 13 . The Hamiltonian in the new basis becomes Ṽ ij , the third mass eigenstate decouples from the other mass eigenstates, and the evolution is governed by an effective 2 × 2 submatrix. After subtracting a constant diagonal matrix from the Hamiltonian, the effective matrix can be written as − cos 2θ 12 + 2Â(c 2 13 − D ) sin 2θ 12 + 2Â N sin 2θ 12 + 2Â * N cos 2θ 12 + 2Â D ,  Solar neutrinos produced in the core of the Sun arrive at the surface of the Earth as an incoherent sum of the three mass-eigenstates ν 1 , ν 2 and ν 3 . During the day, the neutrinos only travel a few km in the Earth, and the ν e survival probability from the source to the detector can be written as [12] where the superscript V represents neutrinos propagating in vacuum, and we have The superscript S represents neutrinos traveling in the Sun. We checked that the neutrino propagation in the Sun with NSI can be treated as adiabatic for the parameters we consider in this paper. For adiabatic propagation, we have . During the night, the neutrinos travel a large distance through the Earth, and the survival probability becomes where the superscript E represents neutrino propagation in the Earth. Since the third mass eigenstate decouples from the other mass eigenstates, P E 3e = s 2 13 . Also, from probability conservation, P E 1e = c 2 13 − P E 2e . However, the calculation of P E 2e in the presence of NSI is nontrivial. Here we derive a simple expression of P E 2e for a constant density profile.
For neutrino evolution in Earth matter, due to the decoupling of the third eigenstate, the effective Hamiltonian has the same form of Eq. (6) withÂ and X replaced byÂ where N E f is the number density of fermion f in the Earth. Then the effective Hamiltonian can be diagonalized by [14] where tan 2θ = | sin 2θ 12 and Then the evolution matrix in the new basis can be written as [12] For a constant density profile, The evolution matrix in the neutrino flavor basis becomes and we have = c 2 13 s 2 12 |α| 2 + c 2 12 |β| 2 + sin 2θ 12 Re(α * β ) .
Plugging the expressions in Eq. (18) into the above equation, we get cos 2θ 12 sin 2θ − sin 2θ 12 cos 2θ cos φ − c 2 13 sin 2θ 12 sin 2θ sin φ sin ωL cos ωL + c 2 13 s 2 12 . For earth matter,Â E 1, so we expand the above equation to leading order inÂ E and find Then from Eqs. (9) and (13), the day-night symmetry is  Since the day-night asymmetry is generally measured by integrating over the zenith angle and the oscillations in the above equation are averaged out, we obtain We have checked that the above equation is consistent with Eq. (13) in Ref. [15] for the two-flavor case.

B. Numerical analysis
Using Eq. (25), we estimate the day-night asymmetry for different SM and NSI parameters. In Fig 2, we show iso-A DN contours in the sin 2 θ 12 − δm 2 21 plane in the SM. We fix sin 2 θ 13 = 0.023, E ν = 7.0 MeV and Earth density ρ E = 3.0 g/cm 3 . The day-night asymmetry depends strongly on δm 2 21 , and its size decreases as sin 2 θ 12 increases in the second octant. In particular, a smaller value of δm 2 21 yields a larger | A DN |. Now we study the dependence of the day-night asymmetry on the NSI parameters. For the NSI parameters, we assume there are no nonstandard couplings to electrons since they would affect the electron-neutrino scattering cross section, yielding NSI at the SK and HK detectors. We also assume the NSI couplings to the up and down quarks are the same for simplicity.
We first examine the dependence on the diagonal NSI parameters. We consider the case in which only u ee = d ee is nonzero. From Eqs. (7) and (8), we see that E N = 0 and E D is linearly dependent on u ee . We show the iso-A DN contours in the space of δm 2 21 and u ee in Fig. 3. We find that for fixed δm , and E D is proportional to − u ee , as u ee increases, | A DN | becomes larger. This yields a degeneracy between δm 2 21 and u ee in the measurement of the day-night asymmetry, i.e., a daynight asymmetry that is consistent with δm 2 21 = 4.8×10 −5 eV 2 in the SM can also be obtained with δm 2 21 = 7.5×10 −5 eV 2 and u ee = d ee ∼ 0.1.
We also checked the dependence of the day-night asymmetry on the off-diagonal NSI parameters. Here we consider the case in which only u eτ = d eτ is nonzero. As can be seen from Eqs. (7) and (8), E D is suppressed by sin θ 13 and E N is proportional to − u eτ in this case. We first assume δ = 0 and u eτ is real for simplicity. In Fig. 4, we show the iso-A DN contours in the space of δm 2 21 and u eτ . The results in Fig. 4 can be understood from Eq. (25). As u eτ approaches 0.2, the factor 2 cos 2θ 12 Re( E N ) + sin 2θ 12 (c 2 13 − 2 E D ) approaches zero. We also checked the complex case by varying δ and the phase of u eτ , and find that for δm 2 21 =7.5 × 10 −5 eV 2 , the day-night asymmetry is always smaller than 2% for | u eτ | < 0.4. Since from Eq. (8) we know that the dominant contribution to f N comes from f eµ and f eτ , and the global-fit constraints on f eµ are stronger than on f eτ [13], an off-diagonal NSI parameter always gives a small daynight asymmetry for δm 2 21 =7.5×10 −5 eV 2 . We henceforth focus on the diagonal NSI parameters.

A. Hyper-Kamiokande
Solar neutrino experiments like HK detect neutrinos via the elastic scattering reaction, The expected event rate for the reconstructed electron kinetic energy of T is [16] where N is the overall normalization that gives the expected event rate in the absence of oscillations, Φ B (Φ hep ) is the normalized 8 B (hep) neutrino flux, and the factor 1.462 × 10 −3 is the relative total flux of hep to 8 B neutrinos in the standard solar model (SSM) (B16-GS98) [17]. The effective cross section is with where i = e, µ, T is the true electron kinetic energy, dσi dT (E ν , T ) is the differential scattering cross section with radiative corrections taken from Ref. [18], and the energy resolution g(T, T ) is given by Since an energy resolution of 10% at 10 MeV is achievable at the HK experiment [19], we choose the energy resolution function, Due to Earth matter effects, the electron neutrino survival probability at night is zenith angle dependent. Given a particular zenith angle, the relative amount of time that the detector is exposed to the Sun is determined by the latitude of the detector site. We use the exposure function at Kamiokande from Ref. [20], and weight each zenith angle by the exposure function. To obtain the survival probabilities, we adopt the average value of the production-point densities of the electron, up-quark and down-quark in the Sun from Ref. [21], and use the GLoBES software [22] with the new physics tools developed in Ref. [23] to calculate P E 2e numerically. We first simulate the detector with a fiducial volume of 0.56 Mt and the electron kinetic energy threshold of 7.0 MeV from the old HK design [24]. We normalize the number of events in our simulation to 200 events per day [24]. Since the HK collaboration has updated their design with a new two-tank configuration for the detector [2], we change the fiducial volume to 0.187 Mt per tank and assume the threshold energy is 5.0 MeV. Ergo, we expect 152 events per day per tank for the new design. The 2TankHK-staged configuration [2] has one tank taking data for 6 years and a second tank is added for another 4 years. We checked that our sensitivity to the day-night asymmetry is consistent with Fig. 134 in Ref. [2] for a 6.5 MeV energy threshold.
In our simulation of the HK experiment, we also consider two Earth density profiles: the Preliminary Reference Earth Model (PREM) [25] and the FLATCORE model [26], in which the density of the core is a constant, as shown in Fig. 5. Note that the FLATCORE model does not match the Earth's mass, and we only use it as an example to study the effects of the Earth's density profile on our results.

B. JUNO
The 20-kt liquid scintillator JUNO experiment will detect reactor antineutrinos from two reactor complexes with a total power of 36 GW via the inverse beta-decay reaction,ν e + p → e + + n . (32) Besides the primary goal of determining the neutrino mass hierarchy, JUNO will also provide a precise measurement of the solar neutrino oscillation parameters. We simulate the JUNO experiment using the GLoBES software with the tools developed in Refs. [23,27]. The baseline of the experiment is 52.5 km and we take the detector energy resolution to be 3%/ E(MeV). With 6 years running, the detector will collect a total of 1.52 × 10 5 events. An overall normalization error of 5% and a linear energy scale uncertainty of 3% is implemented in our simulation [28]. We consider 200 bins from 1.8 MeV to 8.0 MeV, and checked that the spectrum produced from our simulation is in good agreement with that in Fig. 2-15 of Ref. [6].

IV. RESULTS
A. Resolving the tension in δm 2

21
Since solar data are not sensitive to parameters related to m 3 or the CP phase for the case we are considering, we fix sin 2 θ 13 = 0.023, sin 2 θ 23 = 0.43 and δ = 0. We simulate HK and JUNO data with δm 2 21 = 7.5 × 10 −5 eV 2 , sin 2 θ 12 = 0.31, δm 2 31 = 2.43 × 10 −3 eV 2 for the normal mass hierarchy (NH), and u ee = d ee = 0.1, which gives a prediction for the day-night asymmetry that agrees with the current measurement at SK. We first perform a fit to only the SM parameters for each experiment separately to show how parameter degeneracies can occur with nonzero NSI, then perform a fit to the NSI parameters for the two experiments combined to study their ability to reject the SM. We always marginalize over the normal and inverted mass hierarchy (IH).

Day-night asymmetry
As an example, we first only use the day-night asymmetry in the HK analysis. The experimentally measured day-night asymmetry is defined as where N D (N N ) denotes the total number of events detected in the day (night) time. We fit the SM to the simulated data with NSI. From Fig. 6, we see an allowed region for HK around δm 2 21 = 4.8 × 10 −5 eV 2 . Note that the dependence of the HK allowed regions on sin 2 θ 12 is consistent with the prediction of Eq. (25) shown in Fig. 3. The two sets of allowed regions for JUNO (shown for comparison) around sin 2 θ 12 = 0.31 and 0.69 are a consequence of the generalized mass-hierarchy degeneracy [29]. Although the exact generalized mass-hierarchy degeneracy requires ee → − ee − 2, since JUNO is not sensitive to the NSI parameters, an approximate degeneracy holds.

Zenith-angle distribution
We now consider one bin with daytime data and six equisized (in the cosine of the zenith angle) nighttime bins, and define where σ α = 12% is the flux uncertainty in the SSM (B16-GS98) [17]. The results of a SM parameter space scan are shown in Fig. 7. The allowed regions around δm 2 21 = 4.8 × 10 −5 eV 2 persist. Compared to Fig. 6, we see that the new analysis gives a better constraint on sin 2 θ 12 ; however, the allowed regions in δm 2 21 are similar to those obtained from the day-night asymmetry. Hence the sensitivity to δm 2 21 at HK mainly comes from the day-night asymmetry, and HK alone cannot distinguish between the SM and NSI scenarios.
We also simulated data using the FLATCORE model for the Earth density profile. Then we fit the SM assuming the PREM model for the Earth density profile. The bestfit χ 2 to the HK data is 2.3, indicating that the PREM model provides a good fit to data simulated with the FLATCORE model. The allowed regions shown in Fig. 8 are similar to those in Fig. 7.

HK and JUNO combined analysis
Although HK data alone cannot distinguish between the SM and NSI scenarios, when combined with reactor data, measurements of the NSI parameters may be possible. We combine the data from JUNO and HK, and study their sensitivities to the NSI parameters. For the HK analysis, we use the zenith-angle distribution. Using simulated data with u ee = d ee = 0.1 at JUNO and HK, we scan over the range of u ee that is consistent with the global fit in Ref. [13]. After marginalizing over δm 2 21 , θ 12 and the mass hierarchy, we plot ∆χ 2 as a function of u ee for the JUNO and HK combined analysis. The solid curves in Fig. 9 show that the SM (with u ee = 0) is ruled out at 7.6σ, but large negative values of u ee are allowed at less than 3σ due to the generalized mass-hierarchy degeneracy.
In order to test the effect of the Earth density profile on our results, we also simulate the data with the FLAT-CORE model, and fit the data assuming the PREM model. The results are shown in Fig. 9 as the dashed curves. As expected, the sensitivity is reduced if the Earth density profile employed is inaccurate. However, the SM is still excluded at 6.2σ.

B. Detecting NSI
We now study the significance with which u ee = d ee = 0 can be established by ruling out the SM. We simulate data with δm 2 21 = 7.5 × 10 −5 eV 2 , NH, sin 2 θ 12 = 0.31 (sin 2 θ 12 = 0.7), and values of u ee that are roughly consistent with the global fit in Ref. [13] for the first (second) octant of θ 12 . For each value of u ee we calculate the sensitivity to reject the SM allowing for both mass hierarchies. for a range of u ee values. 2 The kink on the left side of the curve in Fig. 10 arises because the second octant of θ 12 and IH provides a better fit than the first octant for u ee ∼ −0.05. If sin 2 θ 12 = 0.7, Fig. 11 shows that for u ee ∼ −0.4, the SM is allowed at less than 3σ as a result of the generalized mass-hierarchy degeneracy.

V. SUMMARY
We explored the discrepancy in the current measurements of δm 2 21 from the SK solar neutrino and KamLAND reactor antineutrino experiments in the framework of NSI. Since the discrepancy mainly stems from the measurement of the day-night asymmetry, we first derived an analytic formula for the day-night asymmetry in the presence of NSI in the three-neutrino framework. We studied the dependence of the day-night asymmetry on both the diagonal and off-diagonal NSI parameters using the formula. We find that a diagonal NSI parameter could yield a large day-night asymmetry. In particular, for δm 2 21 = 7.5×10 −5 eV 2 , the value preferred by KamLAND, u ee = d ee = 0.1 could give a day-night asymmetry that agrees with the current measurement at SK. We also find that an offdiagonal NSI parameter always yields a small day-night asymmetry for δm 2 21 = 7.5 × 10 −5 eV 2 .
Since current SK solar and KamLAND reactor experiments cannot resolve the tension we studied the potential of the future solar neutrino experiment at HK and the future reactor antineutrino experiment at JUNO to provide a resolution. We find that by combining HK and JUNO data, the SM scenario can be rejected at 7.6σ if u ee = d ee = 0.1. Due to the generalized mass-hierarchy degeneracy, larger negative values of u ee are also allowed at less than 3σ. We find our conclusions to be robust under reasonable variations of the Earth density profile. Further, we demonstrated that by combining HK and JUNO data, the SM can be excluded at high confidence for a range of u ee values.