Nonstandard neutrino interactions at DUNE, T2HK and T2HKK

We study the matter effect caused by nonstandard neutrino interactions (NSI) in the next generation long-baseline neutrino experiments, DUNE, T2HK and T2HKK. If multiple NSI parameters are nonzero, the potential of these experiments to detect CP violation, determine the mass hierarchy and constrain NSI is severely impaired by degeneracies between the NSI parameters and by the generalized mass hierarchy degeneracy. In particular, a cancellation between leading order terms in the appearance channels when $\epsilon_{e\tau} = \cot\theta_{23} \epsilon_{e\mu}$, strongly affects the sensitivities to these two NSI parameters at T2HK and T2HKK. We also study the dependence of the sensitivities on the true CP phase $\delta$ and the true mass hierarchy, and find that overall DUNE has the best sensitivity to the magnitude of the NSI parameters, while T2HKK has the best sensitivity to CP violation whether or not there are NSI. Furthermore, for T2HKK a smaller off-axis angle for the Korean detector is better overall. We find that due to the structure of the leading order terms in the appearance channel probabilities, the NSI sensitivities in a given experiment are similar for both mass hierarchies, modulo the phase change $\delta \to \delta + 180^\circ$.


Introduction
The success of neutrino oscillation experiments in the last few decades is a significant triumph in modern physics, and the masses and mixing angles of neutrinos have been incorporated into the standard model (SM) [1]. The data from a plethora of neutrino experiments using solar, atmospheric, reactor, and accelerator neutrinos can be explained in the framework of three neutrino mixing, in which the three known neutrino flavor eigenstates (ν e , ν µ , ν τ ) are quantum superpositions of three mass eigenstates (ν 1 , ν 2 , ν 3 ). In the SM with three massive neutrinos, the neutrino oscillations probabilities are determined by six oscillation parameters: two mass-squared differences (δm 2 21 , δm 2 31 ), three mixing angles (θ 12 , θ 13 , θ 23 ) and one Dirac CP phase δ. Currently, the first five oscillation parameters have been well determined (up to the sign of δm 2 31 ) to the few percent level, and the main physics goals of current and future neutrino experiments are to measure the Dirac CP phase and to determine the neutrino mass hierarchy (MH), i.e., the sign of δm 2 31 , and the octant of θ 23 , i.e., whether θ 23 is larger or smaller than 45 • . Future neutrino oscillation experiments will reach the sensitivity to do precision tests of the three neutrino oscillation paradigm and probe new physics beyond the SM.
A model-independent way of studying new physics in neutrino oscillation experiments is provided by the framework of nonstandard interactions (NSI); for recent reviews see Ref. [2].
In this framework, new physics is parametrized as NSI at production, detection and in propagation according to their effects on the experiments. Since model-independent bounds on the production and detection NSI are generally an order of magnitude stronger than the matter NSI [3], we neglect production and detection NSI in this work, and focus on matter NSI, which can be described by dimension-six four-fermion operators of the form [4] where α, β = e, µ, τ , C = L, R, f = u, d, e, and fC αβ are dimensionless parameters that quantify the strength of the new interaction in units of G F .
The Hamiltonian for neutrino propagation in the presence of matter NSI can be written as where U is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [1] U = and V represents the potential from interactions of neutrinos in matter, Here c jk ≡ cos θ jk , s jk ≡ sin θ jk , A ≡ 2 √ 2G F N e E, N e is the number density of electrons, the unit contribution to V ee arises from the standard charged-current interaction. The effective NSI parameters are given by where N f is the number density for fermion f. In the earth, N u N d 3N e . The diagonal terms in V are real, and since the neutrino oscillation probabilities are not affected by a subtraction of a term proportional to the identity matrix, one of the diagonal terms can be chosen to be 0. The off-diagonal terms are in general complex.
Since neutral-current interactions affect neutrino propagation coherently, long-baseline neutrino experiments with a well-understood beam and trajectory are an ideal place to probe matter NSI. Studies of matter NSI effects in the MINOS experiment have been performed in Ref. [5] and by the MINOS collaboration [6]. NSI analyses related to the currently running T2K [7] and NOνA [8] experiments can be found in Refs. [9,10]. However, due to large systematic uncertainties and limited statistics, these experiments cannot make a definitive measurement of the matter NSI.

Experimental configurations
The main features of the three next generation long-baseline neutrino experiments we consider are summarized in Table 1 and details are described below.

DUNE:
The DUNE experiment sends neutrinos from Fermilab to the Homestake mine in South Dakota with a baseline of 1300 km. We followed the DUNE CDR [11] that uses a 40 kton liquid argon (LAr) detector sitting on axis with respect to the beam direction.
There is a range of beam design options and here we choose the optimized design, which provides a better sensitivity in the appearance channel than the reference design. The

Simulation details
We simulate the experiments using the GLoBES software [20]. We use the official GLoBES simulation files released by the DUNE collaboration [21]   in our simulation. We assume a normalization uncertainty of 2.5% for the signal rates, and 5% (20%) for the appearance (disappearance) background rates. Using the central values and uncertainties from a global fit in the SM scenario [22], we show the expected CP violation and mass hierarchy sensitivity of DUNE, T2HK, and T2HKK as a function of δ for both true normal and true inverted hierarchies in Figs. 1 and 2, respectively. From Fig. 1, we see that the expected CP violation sensitivities in our simulation are consistent with those in the DUNE [11] and Hyper-K [12] design reports.
For the NSI scenario, we use the new physics tools developed in Refs. [23,24]. In our simulation, we use the Preliminary Reference Earth Model density profile [25] with a 5% uncertainty for the matter density. 2 The central values and uncertainties for the mixing angles and mass-squared differences are adopted from the global fit with NSI in Ref. [26], which are For the NSI parameters, we scan over the following ranges suggested by the analysis of Ref. [26], and marginalize over all the NSI phases in our simulation.
2 Note that in the DUNE CDR [21] a 2% uncertainty is used for the matter density, while a 6% uncertainty is used in the T2HK [12] and T2HKK [13] reports.
From Eq. (8), we see that µµ , µτ and τ τ do not appear in the appearance probability up to second order in . Hence, they mainly affect the disappearance channel. Taking ee , eµ and eτ equal to zero, the disappearance probability can be written as Our result agrees with Ref. [29] for the SM terms (in the first two lines) and with Ref. [28] for the NSI terms up to second order after making the assumption that terms of order cos 2θ 23 2 can be ignored. Our result disagrees with Ref. [15] in the second-order terms in . We can see in Eq. (10) that µµ and τ τ appear in the form of their difference up to second order in . We therefore choose µµ = 0.
3.2 NSI in the appearance channels ( eµ , eτ and ee ) We only consider eµ , eτ and ee in this section because µµ , µτ and τ τ do not appear in the appearance probabilities up to second order in .

A single nonzero NSI parameter
For a single L/E, data consistent with the SM can be also described by a model with NSI if P SM (ν µ → ν e ) = P NSI (ν µ → ν e ) and P SM (ν µ →ν e ) = P NSI (ν µ →ν e ). Since the three mixing angles, δm 2 21 and |δm 2 31 | are well-measured by other experiments, if only one off-diagonal NSI ( eµ or eτ ) is nonzero, there exists a continuous four-fold degeneracy as a result of the unknown mass hierarchy and θ 23 octant [10].
The continuous degeneracy can be understood as follows. If only one off-diagonal NSI is nonzero, there are three unknowns to be determined in the NSI scenario: δ (the Dirac CP phase in P NSI ), the NSI magnitude and the NSI phase φ. Since a single measurement of P and P for a fixed L and E gives only two constraints, for each value of δ in the SM, a solution for and φ will exist for any value of δ . This leads to continuous degeneracies throughout the two-dimensional δ-δ space. An additional measurement at a different L and/or E can be made to reduce the degeneracies to lines in δ-δ space, i.e., for each value of δ there will only be one δ that will be degenerate. If there are multiple δ solutions, then a second additional measurement at a different L and/or E should in principle remove the degeneracies.
If only ee is nonzero, since it is real, an experiment that measures P and P at a single L/E should be able to fix the SM value of δ and the NSI values of δ and ee . If a nontrivial solution exists, then there is a simple two-fold degeneracy between the SM and NSI, and at least one additional measurement is needed to break the degeneracy between the SM and NSI with ee . Note however that the nonlinearity (in ee ) of the equations may yield several solutions with nonzero ee and δ = δ.
Since DUNE, T2HK and T2HKK effectively measure probabilities at a variety of energies, in principle these experiments can not only resolve the degeneracies with NSI solutions, but also put severe restrictions on the NSI parameters. If only one NSI parameter is nonzero, the expected allowed regions in the δ − ee , − eµ and − eτ planes are shown in Figs. 3, 4 and 5, respectively. We assume the data are consistent with the SM with δ = 0 and the NH.
The results are obtained after scanning over both mass hierarchies. From Fig. 3, we see that there is always an allowed region near ee = −2 and δ = 180 • . This degeneracy at DUNE was first shown in Ref. [10], and can be explained by the generalized MH degeneracy [18,30], which states that under the transformation, the Hamiltonian transforms as H → −H * , and the oscillation probabilities are unchanged [18]. From Fig. 4, we see that if only eµ is nonzero, the mass hierarchy degeneracy is resolved at DUNE and T2HKK. DUNE puts severe constraints on ( < ∼ 0.15 at 3σ) while T2HKK-1.5 places better constraints on |δ | ( < ∼ 30 • at 3σ). However, T2HK cannot resolve the mass hierarchy in this case; the IH is still allowed for δ ∼ 215 • . Also, there is a 2σ allowed region around eµ ∼ 0.5 arising from the θ 23 octant degeneracy. Around this region, the second octant of θ 23 for the IH has a smaller χ 2 than the first octant.
If only eτ is nonzero, from Fig. 5 we see that the mass hierarchy is not resolved for any of the experiments, although the IH is not allowed at the 1σ CL at DUNE. This could lead to a wrong determination of the Dirac CP phase in all the experiments.

Three nonzero NSI parameters
If eµ , eτ and ee are all nonzero, then there are six free NSI parameters: δ , ee , two magnitudes, and two phases. Even P and P measurements at three different L and E combinations (six equations and six unknowns) could at most reduce the degeneracy to a single point in NSI parameter space (or perhaps a finite number of points). Therefore, an experiment that measures probabilities at a large variety of energies and/or distances is needed to resolve the degeneracies in the presence of multiple NSI.
The expected allowed regions in the δ − ee , − eµ and − eτ planes are shown in Figs. 6, 7 and 8, respectively. For each , we scan over both NH and IH, and marginalize over all the other NSI parameters. As expected, constraints on the NSI parameters become much worse. In particular, from Fig. 7, we see that the constraint on eµ is much weaker at T2HK and T2HKK than at DUNE. This coincides with a strong degeneracy between eµ and eτ at T2HK and T2HKK (see Fig. 9), and can be explained by examining the appearance probability in Eq. (8).
A similar equation holds for the IH. As can be seen from Eq. (13), if the difference between the NSI and SM appearance probabilities is strongly suppressed in both the neutrino and antineutrino modes. Consequently, the constraint on eµ is very weak at T2HK and T2HKK if eτ cot θ 23 eµ . Since neutino energies at DUNE are much higher than at T2HK and T2HKK (e.g., E peak ∼ 3 GeV for whichÂ ∼ 0.28), higher order terms in Eq. (8) cannot be neglected, and the degeneracy between eµ and eτ can be resolved. Also, comparing the lower panels of Fig. 9 we see that T2HKK-1.5 starts to break the degeneracy between eµ and eτ for eµ < ∼ 0.5 since T2HKK-1.5 has a higher peak energy than T2HKK-

2.5.
We also find strong correlations between eτ and ee in all experiments, which can be seen in Fig. 10. The allowed regions are symmetric around ee = −1 due to the generalized MH degeneracy; the vertex of the V-shaped NH region is at ee = 0 and vertex of the V-shaped IH region is at ee = −2.

Dependence of the sensitivity on δ
Since both the Dirac CP phase δ and the mass hierarchy are unknown, the experimental performance may be affected by the true parameters in nature. In this section, we examine how the sensitivity changes with the true value of δ. In the next section we study the sensitivity if the true hierarchy is inverted. Although the mass hierarchy will not be measured in neutrino oscillation experiments because of the generalized MH degeneracy, future neutrinoless double beta decay experiments may determine the mass hierarchy if neutrinos are Majorana particles. We therefore entertain the possibilities that the MH is known and that it is unknown.
We assume that the data are consistent with the SM and the NH, and plot the constraints on δ as a function of δ if all three 's are nonzero; see Figs. 11 and 12 for the case when mass hierarchy is known and unknown, respectively. We see that if the mass hierarchy is known, since δ = δ always holds when = 0, the diagonal line in the δ versus δ plot is always allowed at less than 1σ. If the mass hierarchy is unknown, when the SM and NSI have the opposite mass hierarchy, there is a strong correlation between δ and δ (which can be described by δ = 180 − δ) as a result of the generalized MH degeneracy. We also see that T2HKK has a better performance than T2HK and DUNE in measuring δ. In fact, if the mass hierarchy is unknown, only T2HKK can measure δ at the 3σ CL when three 's are nonzero.
We also plot the minimum value of for which the NSI scenario can be discriminated from the SM at the 2σ CL. If there is only one nonzero , the expected sensitivities are shown in Figs. 13 and 14 if the MH is known and unknown, respectively. As expected, the sensitivity is always weaker if the MH is unknown than if the MH is known. Note that the minimum value of | ee | that is detectable is always larger than 2 if the MH is unknown due to the generalized MH degeneracy. From Fig. 13 we see that the sensitivity to | ee | and | eτ | at DUNE and T2HK improves for δ 90 and 240 • , while this is not the case for T2HKK. From Fig. 14 we see that there is a sharp improvement in the sensitivity to eτ at T2HKK-1.5 for δ 180 • because the IH is not allowed at the 2σ CL in this case.
In Fig. 15 we show the expected sensitivities at 2σ if ee , eµ and eτ are all nonzero and the MH is unknown. We find that the sensitivities to all three 's at T2HK and the sensitivity to eµ at T2HKK are outside the ranges that we scanned. Hence they are not shown in Fig. 15. If ee , eµ and eτ are all nonzero, knowledge of the mass hierarchy does not affect the sensitivity to eµ and eτ because we marginalize over the 's thereby covering the regime of the generalized MH degeneracy even if the MH is known. Furthermore, we see that the dependence on δ becomes much weaker if all three 's are nonzero, and that DUNE has the best sensitivity to the magnitude of the NSI parameters overall. An examination of our figures shows that T2HKK-1.5 has better sensitivities than T2HKK-2.5 in both the SM and NSI scenarios.

Sensitivity when the true mass hierarchy is inverted
We now study the scenario in which the data are consistent with the SM with the IH. We find that there is a similarity between the allowed regions for when the data are consistent with the IH and the allowed regions for when the data are consistent with the NH after a phase transformation in δ, i.e., An example of this similarity can be seen in Fig. 16 This similarity can be understood as follows. In order to fit the SM data with an NSI scenario, the two main constraints from the appearance channel in both the neutrino and antineutrino modes are where i, j = NH, IH.
Using Eq. (8) at leading order in , we have for the NSI scenario with the NH and the SM scenario with the IH. Switching the mass hierarchy of the SM and NSI scenarios (via ∆ → −∆, f ↔ −f , and g → −g), and applying the phase transformation of Eq. (15), leads to an interchange of Eq. (18) and Eq. (19) so that we obtain the same two constraints on the NSI parameters. Since the phase transformation does not depend on L and E, the allowed regions at DUNE, T2HK and T2HKK are similar for both hierarchies. However, note that if we take into account the higher order terms in Eq. (8), in particular the third and fifth lines in Eq. (8), the phase transformation does not leave the two constraints unchanged, which explains the small difference between the allowed regions in the two scenarios. There is a similar correspondence for any combination of hierarchies between the SM and NSI scenarios.
Because of the correspondence discussed above, we expect the NSI sensitivities to be similar whether the data are consistent with the SM in the NH or the SM in the IH. This can seen by comparing Fig. 15 with Fig. 17. In sum, the NSI sensitivities in a given experiment will be similar regardless of the true hierarchy, modulo the transformation δ → δ + 180 • .

NSI in the disappearance channels ( µτ and τ τ )
We find that the sensitivity of these experiments to µτ is outside the range of our scan.
After marginalizing over µτ and the mass hierarchy, we show the 2σ sensitivities to τ τ at DUNE, T2HK and T2HKK in Fig. 18. We see that T2HKK-1.5 has better sensitivity than T2HKK-2.5 because of its higher energy spectrum and larger statistics. We also see that the 2σ sensitivity at DUNE becomes quite weak at some δ values. This is due to a degenerate region near the boundary of the ranges that we have scanned, which was first noticed in Ref. [15] and can be seen in Fig. 19. Since this degenerate region, which occurs because of a correlation between τ τ and the deviation of θ 23 from maximal mixing, is at the boundary, we also show the 90% CL sensitivity curve at DUNE in Fig. 18 to emphasize that the sensitivity is uniform if the degenerate region is resolved by future atmospheric data.

Summary
We studied the sensitivities to NSI in the proposed next generation long-baseline neutrino ex- parameters, and between NSI parameters. As a specific realization of the latter, we find that a cancellation between terms at leading order in the appearance channel probabilities when eτ = cot θ 23 eµ strongly affects the sensitivities to these two NSI parameters at T2HK and T2HKK. Also, the sensitivities at all three experiments are worsened by the generalized mass hierarchy degeneracy in the NSI scenario. Because the generalized mass hierarchy degeneracy occurs at the Hamiltonian level, atmospheric neutrino and reactor neutrino experiments will not be able to resolve it.
We also studied the dependence of the sensitivities on the true CP phase δ and the true mass hierarchy. We find that the sensitivities are much weaker for all values of δ when multiple NSI are nonzero. Also, we find that due to leading order effects in the appearance channel probabilities, there is a similarity of the allowed regions for the NSI parameters between the case in which the data are consistent with the normal hierarchy and the case in which the data are consistent with the inverted hierarchy. Thus the sensitivities are similar whether nature has chosen the NH or the IH, modulo the transformation δ → δ + 180 • .
Overall DUNE has the best sensitivity to the magnitude of the NSI parameters, while T2HKK has the best sensitivity to CP violation whether or not there are NSI, and overall T2HKK-1.5 does better than T2HKK-2.5.
We further studied the sensitivities to µτ and τ τ that mainly come from the disappearance channel. We find that the sensitivities to µτ are limited compared to atmospheric experiments, and we obtained the sensitivity to τ τ at these three experiments. from a global fit in the SM scenario [22], and a 5% uncertainty for the matter density is assumed.           ∆ deg T2HKK 2.5 T2HK Figure 14: Same as Fig. 13, except that the mass hierarchy is unknown. The data are consistent with the SM and the NH. We assume ee , eµ and eτ are all nonzero, and the mass hierarchy is unknown. For each curve, all the other parameters have been marginalized over. All sensitivities for T2HK and the eµ sensitivity for T2HKK are outside the scan range and are therefore not shown. If the mass hierarchy is known, the sensitivities are unchanged for eµ and eτ and similar for | ee |. . We fit the data assuming only one of ee , eµ or eτ is nonzero. We scan over both mass hierarchies and marginalize over the NSI phases.  We also show the 90% CL sensitivity curve for DUNE to emphasize that the sensitivity is uniform if degenerate regions close to the boundaries of the scanned NSI parameter range are excluded by atmospheric data; see Fig. 19 for an illustration of the degenerate region. Figure 19: 1σ, 2σ and 3σ allowed regions in the τ τ versus δ plane at DUNE. The data are consistent with the SM with δ = 0 and the NH. We assume that only µτ and τ τ are nonzero and all the parameters not shown have been marginalized over.