Neutrino Oscillations at low energy long baseline experiments in the presence of nonstandard interactions and parameter degeneracy

We discuss the analytical expression of the oscillation probabilities at low energy long baseline experiments, such as T2HK and T2HKK in the presence of nonstandard interactions (NSIs). We show that these experiments are advantageous to explore the NSI parameters ($\epsilon_D$, $\epsilon_N$), which were suggested to be nonvanishing to account for the discrepancy between the solar neutrino and KamLAND data. We also show that, when the NSI parameters are small, parameter degeneracy in the CP phase $\delta$, $\epsilon_D$ and $\epsilon_N$ can be resolved by combining data of the T2HK and T2HKK experiments.


Introduction
In the last two decades we have been successful in determination of the oscillation parameters in the standard three flavor framework [1]. The three flavor neutrino oscillation is described by the mixing matrix where the following notations are adopted: c jk ≡ cos θ jk , s jk ≡ sin θ jk and θ jk ((j, k) = (1, 2), (1,3), (2,3)) are the three mixing angles and δ is the CP phase. The mixing angles θ 12 , θ 13 and the two mass squared differences ∆m 2 21 , |∆m 2 31 | have been measured with good precision [2,3,4], while the uncertainty in θ 23 and δ is still large. Furthermore, the mass hierarchy (whether the mass pattern is given by normal hierarchy or inverted hierarchy) and the octant of θ 23 (whether θ 23 is larger than π/4 or not) is not known, although the normal hierarchy and the higher octant θ 23 > π/4 are favored to some extent [2,3,4]. The uncertainties in these oscillation parameters are expected to be much reduced in the future long baseline experiments, T2HK [5] at L=295km, T2HKK [6] at L=1100km and DUNE [7] at L=1300km.
On the other hand, there have been a few experimental results which do not seem to be explained by the standard three flavor framework. One of them is the tension between the mass squared difference from the solar neutrino experiments and the KamLAND data. It has been pointed out that this tension can be removed by introducing either a nonstandard interaction (NSI) in the neutrino propagation [8,9] or sterile neutrinos with mass squared difference of O(10 −5 ) eV 2 [10]. 1 To know whether Nature is described by the NSI scenario discussed in Ref. [8], it is important to investigate how to check it. In the analysis of the long-baseline experiments and the atmospheric neutrino experiments, the dominant oscillation comes from the larger mass squared difference ∆m 2 31 and the oscillation probabilities are expressed in terms of ǫ αβ , which will be defined in Eq. (3) below, in addition to the standard oscillation parameters. While the results in Ref. [8] may suggest the existence of the NSI, the parametrization for the NSI parameters (ǫ D , ǫ N ), which will be defined in Eq. (7) below, is different from the one with ǫ αβ and it is not clear how the allowed region in Ref. [8] will be tested or excluded by the future experiments. In the past there were a couple of attempts to estimate the sensitivity of the future experiments to (ǫ D , ǫ N ). In Ref. [14], assuming the standard oscillation scenario, the excluded region in the (ǫ D , ǫ N )-plane by the atmospheric neutrino measurements at Hyper-Kamiokande was given. Ref. [15] estimated the sensitivity of future long baseline experiments in testing the current best fit point suggested by solar neutrino data.
In this paper we discuss the analytical expression of the oscillation probabilities in the presence of the NSI at low energy neutrino measurements ( < ∼ 1GeV), such as T2HK and T2HKK, and show that low energy neutrino measurements are advantageous because the oscillation probabilities involve the fewer NSI parameters including ǫ D , ǫ N . The oscillation probabilities at low energy in the presence of the NSI was discussed in Ref. [16] from a different point of view. The oscillation probabilities at higher energy experiments, such as DUNE, involve more NSI parameters and discussions at higher energy are left as a future work. We also show how parameter degeneracy can be resolved by combining data at different baseline length and different energy in the T2HK and T2HKK system. Parameter degeneracy in the presence of the NSI is a complicated problem and has been discussed by many people [17,18,19,20,21,22,23,24,16,25,26,27]. The situation of parameter degeneracy in low energy long baseline experiments is better than that at high energy, because the oscillation probabilities at low energy involve fewer numbers of the NSI parameters.

Nonstandard interactions in propagation
Suppose that we have a flavor-dependent neutral current NSI [28,29,30,31]: where f P and f ′ P are the fermions with chirality P = (1 ± γ 5 )/2, ǫ f f ′ P αβ is a dimensionless constant normalized in terms of the Fermi coupling constant G F . Then, the matter potential in the flavor basis is modified as where A ≡ √ 2G F N e , the new NSI parameters is defined as ǫ f αβ ≡ ǫ f f L αβ + ǫ f f R αβ , since the matter effect is sensitive only to the coherent scattering, and only to the vector part in the interaction, and N f stands for number density of fermion f , where f is assumed to be u or d quarks or electrons.
In the case of solar neutrino analysis [8,9], since the ratio of the density of protons to that of neutrons varies along the neutrino path, the case with ǫ u αβ = 0, the one with ǫ d αβ = 0, or the one with both must be analyzed separately. 2 On the other hand, in the case of atmospheric neutrinos or accelerator-based long baseline neutrinos, which go through the Earth, we can assume approximately that the numbers of density for electrons, protons and neutrons are almost equal, N e ≃ N p ≃ N n . So in this case, the matter potential (1) can be written as where the new parameter ǫ αβ is defined as While the constraints on ǫ f αβ by various experiments except neutrino oscillations were given in Refs. [32,33], the updated bounds on ǫ αβ by global analysis of oscillation experiments are given in Ref. [9]. The allowed region for ǫ αβ at 90% CL can be 2 The case with ǫ e αβ = 0 was not considered in Refs. [8,9] because of the complication in which the NSI ǫ e αβ would also affect the rate of the interactions between neutrinos and electrons at detection. read off from Fig. 9 in Ref. [9] as follows: 3 Oscillation probabilities at low energy

Solar neutrino flavor basis
At low energy E < ∼ 1GeV, the condition is satisfied and the ratio of the two scales is approximately given by ∆m 2 21 /|∆m 2 31 | ≃ 1/30. So the oscillation probability can be expressed analytically by a perturbation method with respect to this ratio.

4E
− cos 2θ 12 sin 2θ 12 sin 2θ 12 cos 2θ 12 + Ac 2 where ǫ f D and ǫ f N are related to the components of A ′ : It has been pointed out that the value of ∆m 2 21 inferred from the solar neutrino data and that from the KamLAND experiment have a tension at 2σ, and the results of Refs. [8,9] show that a nonvanishing value of (ǫ f D , ǫ f N ) solves this tension. This gives a motivation to take NSI in propagation seriously.
The oscillation probabilities are given by (See Appendix A for details.) Notice that Eqs. (12) and (13) are exact and the quantitiesŨ µj | 2 can be exactly obtained by the formalism by Kimura, Takamura and Yokomakura (KTY) [34,35] in the case with constant density of matter, as long as we know the energy eigenvaluesẼ (∓) j exactly. In reality, however, in order to obtainẼ (∓) j , we have to use a perturbation method with respect to ∆m 2 21 /|∆m 2 31 |. It should be emphasized that this approximation to obtainẼ (∓) j is independent of the baseline length L, so even with this approximation, Eqs. (12) and (13) are valid for arbitrary baseline length L. As described in Appendix B, applying the KTY formalism, we obtainŨ where ∆Ẽ 21 is defined by 3 In the standard parametrization [1] of the mixing matrix U αj , U µ3 is real. In the KTY formalism, however, the biliner formŨ in matter is expressed in terms of the same one U αj U * βj in vacuum, so we leave the notation of complex conjugate for U µ3 here to keep generality in the parametrization of U αj . and ǫ I , ǫ D and ǫ N are defined as From Eqs. (14) - (19) we see that the appearance probabilities involve only ǫ D and ǫ N while the disappearance probabilities also contain ǫ I , in addition to ǫ D and ǫ N . At low energy long baseline experiments on the Earth, therefore, all the oscillation probabilities involves only ǫ I , ǫ D and ǫ N and not ǫ ′ j3 (j = 1, 2, 3). Thus they are advantageous in determining ǫ D and ǫ N since there are less NSI parameters which appear in the oscillation probabilities compared with the experiments at higher energy (E > ∼ 1GeV).
4 Parameter degeneracy in δ, ǫ I , ǫ D and ǫ N In the standard three flavor framework, it has been known [36,37,38,39] that, even if we know exactly the appearance and disappearance probabilities for neutrinos and antineutrinos for a given neutrino energy and a given baseline length, there are in general eight-fold degeneracy in determination of δ, and this is called parameter degeneracy in neutrino oscillation. Here we discuss whether parameter degeneracy can be resolved at low energy long baseline experiments in the presence of the NSI. Our treatment here is based on analytical expressions of the oscillation probabilities and the experimental errors are not taken into account. However, such discussions give us an insight into the problem of parameter degeneracy in the presence of the NSI, like Refs. [36,37,38,39] did in the standard case.
Since the oscillation probabilities (14) - (19) are complicated functions of the NSI parameters, we make the following assumptions: (i) All the NSI parameters ǫ I , ǫ D and ǫ N are of order s 13 ≃ 0.15 or smaller than s 13 , and if the ratio of the next leading term to the leading one is of order s 13 , then the contribution of the next leading term is negligible. (ii) The following expansion is a good approximation: sin ∆Ẽ The assumption (i) may be almost justified from the constraints (4). On the other hand, in the energy region of the T2HK and T2HKK experiments (0.3GeV < ∼ E < ∼ 1GeV), we have ∆Ẽ (∓) 21 L < ∼ 0.54, and the error of the approximation |(sin x − x)/x| for the range 0 < x < 0.54 is less than 0.05. So in the present approximation the assumption (ii) is also justified. From the assumption (ii), we can expand the argument of the second term (solar term) in Eqs. (12) for both T2HK (L=295km) and T2HKK (L=1100km): First, let us discuss the disappearance probabilities at the T2HK experiment. In the case of T2HK (L=295km, E ≃ 0.6GeV), the term ∆Ẽ From this, we can determine the value of sin 2 2θ 23 in the present approximation. Next, let us discuss the appearance probabilities of T2HK. Since the second and third terms on the right hand side of Eq. (24) are multiplied by small quantities such as U e3 = e −iδ s 13 and ǫ N , the only surviving term on the right hand side of Eq. (24) is the first one U e2 U * µ2 ∆E 21 L/2. Thus, in the present approximation in which terms higher than s 13 etc. are ignored, the problem of determination of δ at T2HK is reduced to the same problem as that in the standard three flavor framework. Since the baseline length of T2HK satisfies |∆E 31 |L/2 ≃ π/2 and the mass hierarchy has a ratio ∆m 2 21 /|∆m 2 31 | ≃ 1/30, we have Notice that the appearance probabilities (27) and (28) at T2HK are independent not only of the NSI parameters but also of the mass hierarchy (sign(∆m 2 31 )) in the present approximation. This implies that there is no way to determine the mass hierarchy from the T2HK appearance channel, as is well known. The T2HK experiment as well as T2K [40] is performed at the oscillation maximum (|∆E 31 |L/2 ≃ π/2), and it is known [39] that the so-called intrinsic degeneracy becomes the ambiguity in the sign of cos δ in this case. This ambiguity cannot be removed by the T2HK alone, and as we will see below, we need the T2HKK data to remove this ambiguity. On the other hand, the appearance probabilities have some dependence on the octant of θ 23 , and we can resolve the octant degeneracy. Here, for concreteness, we take the true values as δ true = 5π/4 and θ true 23 = 16π/60 (with (s true 23 ) 2 = 0.552) which are almost the best fit values at present [1], respectively. The problem of determining δ from the two equations P ν µ → ν e ; δ, θ 23 = π 4 ± π 60 = P ν µ → ν e ; δ true = 5 4 π, θ true 23 = 16 60 π and P ν µ →ν e ; δ, θ 23 = π 4 ± π 60 = P ν µ →ν e ; δ true = 5 4 π, θ true 23 = 16 60 π (30) can be solved by looking for the intersection between the two circles in the complex plane of the variable z ≡ 2 exp(−iδ)s 13 s 23 as in Fig.1. Eq. (29) ((30)) tells us that the distance between the points 2 exp(−iδ)s 13 s 23 and −i(π/120)c 23 sin 2θ 12 (i(π/120)c 23 sin 2θ 12 ) in the complex plane is the same as that between the points 2 exp(−5iπ/4)s 13 s 23 and i(π/120)c 23 sin 2θ 12 (i(π/120)c 23 sin 2θ 12 ), respectively. If our hypothesis on the octant of θ 23 is correct (in the present case it is in the higher octant (θ true 23 = 16π/60 > π/4)), then we have two solutions corresponding to cos δ = ±| cos δ|, as is shown in Fig. 1 (a). On the other hand, if our hypothesis on the octant of θ 23 is wrong, then the absolute value of the intersection points is not equal to 2s 13 s 23 ( Fig. 1 (b) where a fit with θ 23 = 14π/60 < π/4 is attempted for the true value θ true 23 = 16π/60) or ( Fig. 1 (c) where a fit with θ 23 = 16π/60 > π/4 is attempted for the true value θ true 23 = 14π/60), and we can reject the wrong hypotheses on the assumption that difference between the true and fake points is large enough compared with the experimental errors. Note that the precise value of θ 13 , which was determined by the reactor experiments [1], is crucial to resolve the octant degeneracy because it uniquely specifies the radius of the thin circle in Fig. 1.
To summarize so far, we have the following results from the T2HK data: • For the sign degeneracy and the NSI parameters, we do not get any information.
• For the intrinsic degeneracy, we can determine the value of sin δ but we still have ambiguity in the sign of cos δ.
• For the octant degeneracy, we can resolve it, on the assumption that deviation |π/4 − θ 23 | is large enough compared with the experimental errors.
Let us now turn to the appearance probabilities at T2HKK (L=1100km, 0.3GeV < ∼ E < ∼ 1.1GeV). Since the T2HK appearance channel enables us to determine the value of sin δ and the octant of θ 23 , we assume in the following discussions that we know the value of sin δ and θ 23 , and the unknown are sign(cos δ), sign(∆m 2 31 ) and the NSI parameters. In the case of T2HKK, while AL/2 (≃ 1/4) and ∆E 21 L (∼ 0.2 (0.6GeV/E)) can no longer be treated as small quantity, the term U e3 U * µ3 ǫ D in Eq. (24) is of order s 2 13 from our assumption, so it can be ignored. Eq. (24) contains the factor ∆Ẽ 21 also gives a contribution to the appearance probabilities, and we have In the last equation in Eq. (34), the first line, which is assumed to be known up to the sign of cos δ, is the contribution of the standard three flavor framework and the second line is the NSI contribution. Assuming that δ is already known from the T2HK data (up to the sign of cos δ), the two equations and P (ν µ →ν e ; ǫ N ) = P (ν µ →ν e ; ǫ true N ) (37) give us a condition on ǫ N . A remark is in order. As was emphasized in Ref. [16], the reason that information on ǫ N can be still obtained after expanding a sine function with a small argument as sin(∆Ẽ 21 L/2 is because this is the case where a so-called vacuum mimicking phenomenon [28,41,42,43,44,45,46,47] does not occur. In the standard three flavor framework, if the argument of a sine function is small and expanded as sin x ≃ x, then the oscillation probability in matter is reduced to the one in vacuum, and this is call a vacuum mimicking phenomenon. In the present case, however, even after the approximation sin(∆Ẽ 21 L/2 is used, the term with ǫ N remains. This is an advantage of a long baseline experiment (L > ∼ 1000km) at low energy (E < ∼ 1GeV), such as T2HKK, since the other NSI parameters do not appear in the appearance probability to the leading order at low energy.
As in the case of T2HK, Eqs. (36) and (37) represent two circles in the complex plane of z ≡ AL U τ 3 ǫ N , and in general there are two intersections. To reject the fake solutions, we need more information. We therefore consider the appearance probabilities at different three energy regions, e.g., E=0.3 GeV, E=0.7 GeV and E=1.1 GeV. Here we take ǫ true N = 0 as the true value for simplicity. As we see in Fig. 2, there are four possible cases with right/wrong sign of cos δ and right/wrong sign of ∆m 2 31 . By demanding that there be a common intersection point among the three pairs of circles, we can resolve degeneracy of sign(sin δ) and that of sign(∆m 2 31 ), and we can determine both Re(ǫ N ) and Im(ǫ N ), on the assumption that the difference between the true and fake points is large enough compared with the experimental errors. So far we have taken ǫ true N = 0 as the true value for simplicity. If the true value ǫ true N is nonzero, then the same argument can be applied, since all the positions of the circles and ǫ true N in the complex plane are shifted by ǫ true N ( = 0). Hence even for ǫ true N = 0, we can determine ǫ N from the appearance probabilities of T2HKK and all the information from T2HK. Finally, let us discuss determination of ǫ D . In our approximation, ǫ D does not appear in the appearance probabilities to the leading order. So far we have already determined δ, θ 23 and ǫ N , so we assume in the following discussions that we already know the value of these parameters. To get information on ǫ D , let us discuss the disappearance probabilities at T2HKK. They are given by (See Appendix C for details.) where f (∓) = O(1) and g (∓) = O(1) are defined by From the discussions on the T2HK data and the appearance channel of T2HKK, we already know the values of δ and ǫ N . Assuming that the true values of the NSI parameters ǫ true N , ǫ true I and ǫ true D are zero for simplicity, the following equations give us information on ǫ I and ǫ D : P (ν µ →ν µ ; ǫ I , ǫ D ) = P (ν µ →ν µ ; 0, 0) Unlike in the case of the appearance probabilities, where the contributions from the atmospheric oscillation and from the solar one are both small, the NSI contributions in Eq. (58) are small compared with the one from atmospheric oscillation. So we can expand the disappearance probabilities in term of the small parameters ǫ I and ǫ D .
From these two equations we can determine ǫ I and ǫ D . Here we have assumed that the true value of the NSI parameters are zero for simplicity, but even for a nonvanishing value of the NSI parameters, the same argument can be applied.
To summarize, we have seen that, because the T2HK experiment has a relatively short baseline length, the oscillation probabilities at T2HK are approximately independent of the NSI parameters and T2HK can determine the value of sin δ and it can resolve the octant degeneracy, on the assumption that the difference between the true and fake points is large enough compared with the experimental errors. Furthermore, the T2HKK experiment can resolve degeneracy of the sign of ∆m 2 31 as well as the ambiguity of cos δ. By combining the appearance and disappearance probabilities at T2HK and T2HKK, we can determine the NSI parameters ǫ N , ǫ I and ǫ D .

Conclusions
At low energy (E < ∼ 1GeV), the description in the solar flavor basis is useful. In particular, in the presence of the nonstandard interactions in propagation of neutrinos, assuming that the NSI parameters are at most of order O(s 13 ), the appearance probabilities at low energy depend approximately only on one (ǫ N ) of the NSI parameters, while the disappearance ones do on three (Re(ǫ N ), ǫ I and ǫ D ). Furthermore, assuming that the experimental errors are small enough to justify the analytical discussions on the oscillation probabilities, we discussed how parameter degeneracy can be resolved by combining the T2HK and T2HKK experiments. These two low energy long baseline experiments are complementary to each other, because T2HK has little sensitivity to the matter effect and can therefore determine sin δ and the octant of θ 23 without being disturbed by the existence of the NSI whereas T2HKK has sensitivity to the matter effect and can give us information on the NSI parameters as well as sign(sin δ) and sign(∆m 2 31 ). Our treatment in this work is qualitative in the sense that the experimental errors are not taken into account, and quantitative estimation of the experimental errors is beyond the scope of this work. Nevertheless, we hope that the present work sheds light on the advantage of low energy long baseline experiments to investigate the NSI which is suggested by the tension between the solar neutrino data and that from the KamLAND experiment.
is the diagonal matrix with the energy eigenvalue of each mass eigenstate where the identity matrix times E 1 was subtracted without affecting the oscillation probability, andẼ is the diagonal matrix with the energy eigenvalue in matter. From Eq. (46) one can obtain the oscillation probability where we have used the unitarity property 3 j=1Ũ (∓) βjŨ (∓) * αj = δ αβ in the third line and we have defined Thus we can obtain the analytic expression if we get the bilinear formŨ Putting Eqs. (51)-(53) together, we have which can be easily solved by inverting the Vandermonde matrix: The eigenvalues can be obtained from the eigenequation Here the matrix can be expressed as where we have defined in Eq. (57) is given to the next leading order in ∆m 2 21 /|∆m 2 31 | because it is necessary to obtainŨ later. For simplicity, we obtain the biliner formŨ To perform perturbation calculations, it is convenient to rescale ∆E 21 → ǫ∆E 21 and A αβ → ǫA αβ . Then we have With these quantities, we have Eq. (14) to the leading order in O(ǫ): In the case of antineutrinos, we have to replace U αj by U * αj , and we have Eq. (15) ǫ I , ǫ D and ǫ N are defined in Eqs. (21), (22) and (23). In the case of antineutrinos, we have to replace U αj by U * αj and A by −A, and we have Eq.
As for the disappearance channel, on the other hand, we have the following: The bilinear form |Ũ (−) µ3 | 2 is thus given by Eq. (18) where Eq. (25) was used in the third step above. Here introducing the notation Thus we get the expressions (39) for f (∓) and (40) for g (∓) .