1 Introduction

# The possibility to observe the non-standard interaction by the Hyperkamiokande atmospheric neutrino experiment

The possibility to observe the non-standard interaction by the Hyperkamiokande atmospheric neutrino experiment

Shinya Fukasawa and Osamu Yasuda

Department of Physics, Tokyo Metropolitan University,

Minami-Osawa, Hachioji, Tokyo 192-0397, Japan

Abstract

It was suggested that a tension between the mass-squared differences obtained from the solar neutrino and KamLAND experiments can be solved by introducing the non-standard flavor-dependent interaction in neutrino propagation. In this paper we discuss the possibility to test such a hypothesis by atmospheric neutrino observations at the future Hyper-Kamiokande experiment. Assuming that the mass hierarchy is known, we find that the best-fit value from the solar neutrino and KamLAND data can be tested at more than 8 , while the one from the global analysis can be examined at 5.0 (1.4 ) for the normal (inverted) mass hierarchy.

## 1 Introduction

It is well established by solar, atmospheric, reactor and accelerator neutrino experiments that neutrinos have masses and mixings [1]. In the standard three flavor neutrino oscillation framework, there are three mixing angles , , and two mass-squared differences , . Their approximate values are determined as eV,, eV,, . However we do not know the value of the Dirac CP phase , the sign of (the mass hierarchy) and the octant of (the sign of ). Future neutrino oscillation experiments with high statistics [2, 3] are planned to measure these undetermined neutrino oscillation parameters and we are entering an era of the precision measurements. With these precision measurements, we can probe the new physics by looking at the deviation from the standard three flavor neutrino mixing scenario.

Flavor-dependent neutrino NonStandard Interactions (NSI) have been studied as the new physics candidates which may be searched at the future neutrino experiments. There are two types of NSI. One is a neutral current nonstandard interaction [4, 5, 6] and the other is a charged current nonstandard interaction [7]. The neutral current NSI affects the neutrino propagation through the matter effect and hence experiments with a long baseline such as atmospheric neutrino and LBL experiments are expected to have the sensitivity to the neutral current NSI. On the other hand, the charged current NSI causes zero distance effects in neutrino oscillation. In this paper, we concentrate on the effects of neutral current NSI in neutrino propagation and study the sensitivity of the future atmospheric neutrino experiments Hyper-Kamiokande to NSI with a parametrization introduced to study solar neutrinos.

It was pointed out in Ref. [8] that there is a tension between the mass-squared difference deduced from the solar neutrino observations and the one from the KamLAND experiment, and that the tension can be resolved by introducing the flavor-dependent NSI in neutrino propagation. Such a hint for NSI gives us a strong motivation to study NSI in propagation in details.

In Ref. [9] it was shown that the atmospheric neutrino measurements at Hyper-Kamiokande has a very good sensitivity to the NSI, on the assumptions that (i) all the components of the NSI vanish and (ii) the (, ) component is expressed in terms of the other components as is suggested by the high energy atmospheric neutrino data. In this paper we discuss the sensitivity of the atmospheric neutrino measurements at Hyper-Kamiokande to NSI without the assumptions (i) and (ii) mentioned above. Since the parametrization which is used in Ref. [8] is different from the ordinary one in the three flavor basis, a non-trivial mapping is required to compare the results in these two parametrizations. Our analysis was performed by taking this non-trivial mapping into account.

Constraints on have been discussed by many people in the past. from atmospheric neutrinos [10, 11, 12, 13, 14], from colliders [15], from the compilation of various neutrino data [16], from solar neutrinos [17, 18, 19], from or scatterings [20, 21], from solar and reactor neutrinos [22], from solar, reactor and accelerator neutrinos [23]. The constraints on and from the atmospheric neutrino has been discussed in Ref. [24] along with those from the long-baseline experiments, in Ref. [25] by the Super-Kamiokande Collaboration, in Ref. [26, 27, 28] on the future extension of the IceCube experiment, in Ref. [29] on the future experiment with the iron calorimeter or liquid argon detectors, with the ansatz different from ours.

This paper is organized as follows. In Section 2, we describe the current knowledge and constraints on NSI in propagation from solar neutrinos and atmospheric neutrinos. In Section 3, we study the sensitivity of the future atmospheric neutrino experiment Hyper-Kamiokande to NSI. In Section 4, we draw our conclusions. In the appendix A, we derive the relation between the two different parametrizations of NSI.

## 2 Three flavor neutrino oscillation framework with NSI

### 2.1 Nonstandard interactions

Let us start with the effective flavor-dependent neutral current neutrino nonstandard interactions in propagation given by

 L\tiny{\rm NSI}\rm\scriptsize eff=−2√2ϵff′PαβGF(¯¯¯ναLγμνβL)(¯¯¯fPγμf′P), (1)

where and stand for fermions with chirality and is a dimensionless constant which is normalized by the Fermi coupling constant . The presence of NSI (1) modifies the MSW potential in the flavor basis:

 √2GFNe⎛⎜⎝100000000⎞⎟⎠→A, (2)

where

 A≡√2GFNe⎛⎜⎝1+ϵeeϵeμϵeτϵμeϵμμϵμτϵτeϵτμϵττ⎞⎟⎠, (3)

is defined by

 ϵαβ≡∑f=e,u,dNfNeϵfαβ, (4)

and stands for number densities of fermions . Here we defined the new NSI parameters as and since the matter effect is sensitive only to the coherent scattering and only to the vector part in the interaction. As can be seen from the definition of , the neutrino oscillation experiments on the Earth are sensitive only to the sum of . We call the most general parametrization (3) of NSI in the flavor basis the standard NSI parametrization in this paper. In the three flavor neutrino oscillation framework with NSI, the neutrino evolution is governed by the Dirac equation:

 iddx⎛⎜⎝νe(x)νμ(x)ντ(x)⎞⎟⎠=[Udiag(0,ΔE21,ΔE31)U−1+A]⎛⎜⎝νe(x)νμ(x)ντ(x)⎞⎟⎠, (5)

where is the leptonic mixing matrix defined by

 U ≡ ⎛⎜⎝c12c13s12c13s13e−iδCP−s12c23−c12s23s13eiδCPc12c23−s12s23s13eiδCPs23c13s12s23−c12c23s13eiδCP−c12s23−s12c23s13eiδCPc23c13⎞⎟⎠, (6)

and , , .

### 2.2 Solar neutrinos

In Refs. [8, 30] it was pointed out that there is a tension between the two mass squared differences extracted from the KamLAND and solar neutrino experiments. The mass squared difference () extracted from the solar neutrino data is smaller than that from the KamLAND data (). The authors of Refs. [8, 30] discussed the tension can be removed by introducing NSI in propagation.

To discuss the effect of NSI on solar neutrinos, we reduce the Hamiltonian in the Dirac equation Eq. (5) to an effective Hamiltonian to get the survival probability because solar neutrinos are approximately driven by one mass squared difference [8]. The survival probability can be written as

 P(νe→νe)=c413Peff+s413. (7)

can be calculated by using the effective Hamiltonian written as

 Heff=Δm2214E(−cos2θ12sin2θ12sin2θ12cos2θ12)+(c213A000)+A∑f=e,u,dNfNe(−ϵfDϵfNϵf∗NϵfD),

where and are linear combinations of the standard NSI parameters:

 ϵfD = c13s13Re[eiδCP(s23ϵfeμ+c23ϵfeτ)]−(1+s213)c23s23Re[ϵfμτ] −c2132(ϵfee−ϵfμμ)+s223−s213c2232(ϵfττ−ϵfμμ) ϵfN = c13(c23ϵfeμ−s23ϵfeτ)+s13e−iδCP[s223ϵfμτ−c223ϵf∗μτ+c23s23(ϵfττ−ϵfμμ)]. (8)

Ref. [8, 30] discussed the sensitivity of solar neutrino and KamLAND experiments to and real for one particular choice of or at a time. The best fit values from the solar neutrino and KamLAND data are and and that from the global analysis of the neutrino oscillation data are and . These results give us a hint for the existence of NSI. In addition to the above, Ref. [8, 30] also discussed the possibility of the dark-side solution ( and ) which requires NSI in the solar neutrino problem. The allowed regions for the dark-side solution are disconnected from that for the standard LMA solution in the plane and those for the dark-side solution within do not contain the standard scenario .

### 2.3 Atmospheric neutrinos

In this subsection, we describe the constraints on NSI from the atmospheric neutrino experiments and introduce a relation between , and and a matter angle . Atmospheric neutrinos go through the Earth and interact with electrons, up and down quarks. In the Earth, the number densities of electrons, protons and neutrons are approximately equal and hence those of up quarks and down quarks are approximately the same. From these, one can define as

 ϵαβ=ϵeαβ+3ϵuαβ+3ϵdαβ, (9)

and we have the following limits [31] on at 90% C.L.:

 ⎛⎜ ⎜⎝|ϵee|<4×100|ϵeμ|<3×10−1|ϵeτ|<3×100 |ϵμμ|<7×10−2|ϵμτ|<3×10−1|ϵττ|<2×101 ⎞⎟ ⎟⎠. (10)

To investigate the sensitivity of the atmospheric neutrino experiment to and , we have to convert and into because and are valid only in the solar neutrinos analysis. and are expressed in terms of as the following:

 |ϵfeτ|=sin(ϕfμτ)t13sin(δcp+ϕfeτ)s23|ϵfμτ|+sin(δcp+ϕfeμ)t23sin(δcp+ϕfeτ)|ϵfeμ| −sin(δcp+ψf)sin(δcp+ϕfeτ)s23c13|ϵfN|, +sin(ϕfeμ−ϕfeτ)s23t13sin(δcp+ϕfeτ)|ϵfeμ|−2sin(ψf−ϕfeτ)sin(δcp+ϕfeτ)s13sin2θ23|ϵfN|, ϵfee−ϵfμμ=2⎡⎢ ⎢⎣s223−s213c223c213⎧⎪ ⎪⎨⎪ ⎪⎩cosϕfμτtan2θ23+sinϕfμτtan(δcp+ϕfeτ)sin2θ23⎫⎪ ⎪⎬⎪ ⎪⎭ +t213t23⋅sinϕfμτtan(δcp+ϕfeτ)−1+s2132c213sin2θ23cosϕμτ⎤⎥ ⎥⎦|ϵfμτ| +2⎡⎢ ⎢⎣s223−s213c223s23sin2θ13⋅sin(ϕfeμ−ϕfeτ)sin(δcp+ϕfeτ)+t13s23cos(δcp+ϕfeμ) +t13c23t23⋅sin(δcp+ϕfeμ)tan(δcp+ϕfeτ)sin2θ23⎤⎥ ⎥⎦|ϵfeμ| −2⎡⎢ ⎢⎣s223−s213c223s13c213sin2θ23⋅sin(ψf−ϕfeτ)sin(δcp+ϕfeτ) +t13t23c13⋅sin(δcp+ψf)tan(δcp+ϕfeτ)⎤⎥ ⎥⎦|ϵfN|−2c213ϵfD, (11)

where , and . When we consider only one particular choice of or at a time as in Ref. [8], from the definition of (9), we cannot distinguish the case of from that of in the atmospheric neutrinos analysis. Therefore we concentrate on only one particular choice of in this paper and then we have

 ϵαβ =3ϵdαβ (12) ϕαβ ϵD =ϵdD ϵN =ϵdN ψ ≡arg(ϵdN).

#### The case with ϵαμ=0 (α=e,μ,τ)

It was pointed out in Refs. [32, 33] that if the components of are set to zero then the high-energy atmospheric neutrino data, where the matter effects are dominant, are consistent with NSI only when the following inequality is hold:

 min±(∣∣∣1+ϵee+ϵττ±√(1+ϵee−ϵττ)2+4|ϵeτ|2∣∣∣)\raisebox0.853583pt\em$<$\raisebox−3.983386pt\em$∼$0.4, (13)

where the arguments of the absolute value on the left hand side are the two nonzero eigenvalues of the matrix in the absence of component, and the () sign in is chosen when is negative (positive). Notice that in the limit of

 min±(∣∣∣1+ϵee+ϵττ±√(1+ϵee−ϵττ)2+4|ϵeτ|2∣∣∣)=0, (14)

and satisfy a parabolic relation

 ϵττ=|ϵeτ|21+ϵee (15)

and hence can be eliminated. In the limit of Eq. (15), the disappearance oscillation probability of the high-energy atmospheric neutrinos can be reduced to vacuum oscillation like two-flavor form ( is a mixture of and due to the presence of NSI) in spite of nonvanishing component in the matter potential. This means that the disappearance oscillation probability with NSI of the high-energy atmospheric neutrinos is proportional to

 1−P(νμ→νμ)=sin22θatmsin2(Δm2atmL4E)∝1E2 (16)

as in the case of the standard two flavor neutrino oscillation framework.

Next let us introduce the matter angle [32, 33] which determines the mixing between the standard flavor basis defined by the W-boson exchange interaction and the modified flavor basis due to the presence of NSI with components . It is convenient to take the modified flavor basis in the discussion on the sensitivity of atmospheric neutrino experiments to NSI. The matter angle is defined as

 tanβ≡|ϵeτ|1+ϵee. (17)

In the case of SK for 4438 days analysis, the constraint to from the energy rate analysis is given by [9]. If we rewrite the matter potential as

 A=√2GFNe⎛⎜⎝1+ϵee0ϵeτ000ϵ∗eτ0|ϵeτ|2/(1+ϵee)⎞⎟⎠,

then the allowed region which was obtained in Ref. [9] from the SK atmospheric neutrino data at is

 |ϵττ|=|ϵeτ|2|1+ϵee|\raisebox0.853583pt\em$<$\raisebox−3.983386pt\em$∼$2. (18)

Notice that the bound (18) on is much weaker than what is obtained from the two flavor analysis assuming only the transition [10, 11, 12, 13, 14, 25]. This is because in the two flavor analysis is assumed, and the parabolic relation (15) would imply in this case.1

It is instructive to discuss the relation between the standard parametrization and the set of the parametrizations (, ) in the simplest case. In the simplest case, we assume the parabolic relation (15) and set , , which is a good approximation to some extent. Then, introducing a new angle

 tanβ′≡tanβ√2, (19)

we can derive the following relation (See Appendix A for the derivation and the expression for a more general case.):

 |3ϵN|1/2−3ϵD=tan2β′. (20)

The region  , which is the area surrounded by the axis and the straight line with the gradient and the -intercept , is the allowed region in the (, ) plane by the atmospheric neutrino data under the assumption of the parabolic relation (15). The corresponding region in the (, ) plane is approximately given by the one surrounded by the axis and the straight line with the gradient and the -intercept .

#### The case with ϵαμ≠0 (α=e,μ,τ)

From here we take into consideration all the components of including the components, and lift the parabolic relation (15). Even in this case, because of the strong constraints (10) on the components, the three eigenvalues of the matter potential matrix are approximately 0 and . So most of the discussions in the previous subsubsection are approximately valid. In particular, the constraint from the high energy data of the atmospheric neutrinos can be approximately given by Eq. (13). We note that another derivation of the relation (15) was given in Ref. [39]. The high-energy behavior of the disappearance oscillation probability in the presence of NSI without switching off any can be written as

 1−P(νμ→νμ)≃c0+c1√2GFNeE+O(1E2). (21)

This expression requires and so that the presence of NSI is consistent with the high-energy atmospheric neutrino experiments data. The constraints on and imply and .

## 3 Analysis

In this section we discuss the sensitivity of the Hyper-Kamiokande (HK) atmospheric neutrino experiment whose data is assumed to be taken for 4438 days to and with the codes that were used in Ref. [36, 37, 38, 9]. We assume that the HK fiducial volumes are 0.56 Mton 2, and that the HK detector has the same detection efficiencies as those of Super-Kamiokande (SK) and that HK atmospheric neutrino data comprise the sub-GeV, multi-GeV and upward going events as in the case of SK. As HK is the future experiment, the number of events calculated with the standard three flavor oscillation scenario are used as the experimental data for fitting. The reference values of oscillation parameters used in the calculation of the experimental data are the following:

 Δ¯m231=2.5×10−3\rm eV2,sin2¯θ23=0.5,¯δCP=0, sin22¯θ12=0.86,sin22¯θ13=0.1,Δ¯m221=7.6×10−5\rm eV2, (22)

where the parameters with a bar denote those for the reference value of “the experimental data”. The information on the zenith angle bins for the sub-GeV, multi-GeV and upward going events are given in Ref. [35] while that on the energy bins is not. We analyze with the ten zenith angle bins as in Ref. [35]. As the experimental data is calculated by our codes, we can use any information on the energy spectrum of the number of events and analyze with any number of the energy bins.

The analysis was performed using -method and is defined as

 χ2=minθ23,|Δm232|,δ,ϵαβ(χ2sub−GeV+χ2multi−GeV+χ2upward+χ2prior), (23)

where

 χ2sub−GeV (24) = minαs,β′s,γ′s[β2s1σ2βs1+β2s2σ2βs2+γ2L1σ2γL1+γ2L2σ2γL2+γ2H1σ2γH1+γ2H2σ2γH2 +∑A=L,H10∑j=1⎧⎨⎩1nsAj(e)⎡⎣αs⎛⎝1−βs12+βs22+γjA12⎞⎠NsAj(νe→νe) +αs⎛⎝1+βs12+βs22+γjA12⎞⎠NsAj(νμ→νe) +αs⎛⎝1−βs12−βs22+γjA12⎞⎠NsAj(¯νe→¯νe) +αs⎛⎝1+βs12−βs22+γjA12⎞⎠NsAj(¯νμ→¯νe)−nsAj(e)⎤⎦2 +1nsAj(μ)⎡⎣αs⎛⎝1−βs12+βs22+γjA22⎞⎠NsAj(νe→νμ) +αs⎛⎝1+βs12+βs22+γjA22⎞⎠NsAj(νμ→νμ) +αs⎛⎝1−βs12−βs22+γjA22⎞⎠NsAj(¯νe→¯νμ) +αs⎛⎝1+βs12−βs22+γjA22⎞⎠NsAj(¯νμ→¯νμ)−nsAj(μ)⎤⎦2⎫⎪⎬⎪⎭⎤⎥⎦,
 χ2multi−GeV (25) = minαm,β′s,γ′s[β2m1σ2βm1+β2m2σ2βm2+γ21σ2γ1+γ22σ2γ2 +∑A=L,H10∑j=1{1nmAj(e)[αm(1−βm12+βm22+γj12)NmAj(νe→νe) +αm(1+βm12+βm22+γj12)NmAj(νμ→νe) +αm(1−βm12−βm22+γj12)NmAj(¯νe→¯νe) +αm(1+βm12−βm22+γj12)NmAj(¯νμ→¯νe)−nmAj(e)]2 +1nmAj(μ)[αm(1−βm12+βm22+γj22)NmAj(νe→νμ) +αm(1+βm12+βm22+γj22)NmAj(νμ→νμ) +αm(1−βm12−βm22+γj22)NmAj(¯νe→¯νμ) +αm(1+βm12−βm22+γj22)NmAj(¯νμ→¯νμ)−nmAj(μ)]2⎫⎬⎭⎤⎥⎦,
 χ2upward = minαu{α2uσ2α+10∑j=11nuj(μ)[(1+αu)Nuj(νe→νμ)+(1+αu)Nuj(νμ→νμ) (26) +(1+αu)Nuj(¯νe→¯νμ)+(1+αu)Nuj(¯νμ→¯νμ)−nuj(μ)]2},
 χ2prior = Δχ2prior|ϵfeμ|2|δϵfeμ|2+Δχ2prior|ϵfμτ|2|δϵfμτ|2. (27)

Where in stands for for CL with 1 d.o.f. and stand for constraint on corresponding NSI at CL, respectively. The summation on and run over the ten zenith angle bins and the two energy bins, respectively. The indices and stand for the lower () and higher () energy bins, respectively. For all the zenith angle bins, the threshold energy for the sub-GeV events is 0.5GeV and that for the multi-GeV events is 3.2GeV. The threshold energy is chosen so that the numbers of events for the lower and higher energy bins are approximately equal. The experimental data () stands for the sum of the number of neutrinos and antineutrinos events for the sub-GeV and multi-GeV events, and the experimental data stands for that for the upward going events. () stands for the prediction with our codes for the number of -like events () of the sub-GeV and multi-GeV events and () stands for that of the upward going events. stands for the uncertainty in the overall flux normalization for the sub-GeV, multi-GeV, and upward going events, () stands for the uncertainty in the relative normalization between - flux ( - flux) for the sub-GeV () and multi-GeV () events, respectively, and stand for the flavor and energy dependent relative normalization between the upward and downward bins for the sub-GeV and multi-GeV events:

 γjA1,2 = ⎧⎨⎩γA1,2(j≤j\rm% \scriptsize th;A=L,H)−γA1,2(j>j