# Nonstandard Interactions and Prospects for Studying Standard Parameter Degeneracies in DUNE and T2HKK

Authors: Surender Verma; Shankita Bhardwaj
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

The future long baseline experiments such as DUNE and T2HKK have promising prospects to determine the neutrino mass hierarchy and measuring standardCP phase δ. However, presence of possible nonstandard interactions of neutrinos with matter may intricate this picture and is the subject matter of the present work. We have studied the standard parameter degeneracies in presence of nonstandard interactions (NSI) with DUNE and T2HKK experiments. We examine the mass hierarchy degeneracy assuming (i) all NSI parameters to be nonzero and (ii) one NSI parameter (ϵeμ) and its corresponding CP phase (δeμ) to be nonzero. We find that the latter case is more appropriate to resolve mass hierarchy degeneracy with DUNE and T2HKK experiments due to relatively small uncertainties emanating from the NSI sector. We have, also, investigated the octant degeneracy with neutrino (νμ→νe) and antineutrino (ν¯μ→ν¯e) mode separately. We find that to resolve this degeneracy the long baseline experiment with combination of neutrino and antineutrino mode is essential. Furthermore, we have considered DUNE in conjunction with T2HKK experiment to study CP phase degeneracy due to standard (δ) and nonstandard (δeμ) CP phases. We find that DUNE and T2HKK, in conjunction, have more sensitivity for CP violation effects (10σ for true NH and 8.2σ for true IH).


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## Body

## 1. Introduction

The discovery of nonzero neutrino masses and lepton flavor mixing by the reactor [1], accelerator [2], atmospheric [3], and solar [4] neutrino oscillation experiments have revealed the values of oscillation parameters such as mass squared differences Δm212,Δm312 and mixing angles θ12,θ23,θ13 [5], to an unprecedented accuracy. At present, there are some unknown quantities in standard three-neutrino framework, namely, (i) sign of Δm232, (ii) the octant of θ23, and (iii) the CP-phase δ, the determination of which is the prime objective of current and future neutrino oscillation experiments. The difficulty in the determination of these unknowns is the existence of degeneracies in neutrino oscillation parameters. To overcome these degeneracies, one of the methods is to combine data from different neutrino oscillation experiments. Recently, this procedure has been adopted by various studies [6–8], where the synergy between current and future experiments has been considered. In principle, future neutrino oscillation experiments have sensitivity reach to perform precision test of standard neutrino oscillation paradigm and to probe new physics beyond standard model (SM). In neutrino oscillation experiments, one model-independent way to study new physics (NP) is given by the framework of nonstandard interactions (NSI) [9, 10].An alternative phenomenon to explain neutrino flavor transitions, on the basis of NSI, was first proposed by Wolfenstein [11]. Although we know that they will show their effect in neutrino oscillation experiments at subleading level, they are important, with the emergence of next generation experiments like Tokai-to-Hyper-Kamiokande (T2HK) [12], Deep Underground Neutrino Experiment (DUNE) [13], Tokai-to-Hyper-Kamiokande-and-Korea (T2HKK) [14], etc., where such type of interactions can be probed. In general, the NSIs may manifest itself in propagation of neutrino through matter and the processes involved in its creation and detection. The possible manifestations of NSIs have been widely studied in the literature and bounds on NSI parameters have been derived from various experiments [9, 10, 15, 16]. Furthermore, the model-independent bounds on NSI in production and detection regions are an order of magnitude stronger than the matter NSI [15]. In this work, we focus on matter NSI which can be defined by dimension-six four-fermion operators given by [11, 17](1)LNSI=22GFϵαβζKν¯αγρPLνβζ¯γρPKζ+h.c.,where α,β=e,μ,τ, K=L,R, ζ=u,d,e, and ϵαβζK are dimensionless parameters indicating the strength of the new interaction having units of GF. To probe matter NSI long baseline neutrino experiments (LBNE) are ideal and the neutral current (NC) interactions which affect the neutrino propagation coherently can also be studied at far detectors. The next generation LBNE such as DUNE, T2HK, and T2HKK may reach the sensitivity to reveal NSI in neutrino sector.In the leptonic sector, theCP violation can render leptogenesis mechanism which in turn may shed light on baryogenesis [19, 20]. It is very difficult to measure leptonic CP-violation in presence of NSI as it will get bewildered by the existence of possible CP-violation generated by NSI itself. Undoubtedly, the existence of NSI has opened an entirely new window to explore NP beyond standard model.Previously, the authors of [6] have explored the ability to disentangle the CP violating effects due to standard and nonstandard contributions under the assumption that only one NSI parameter ϵeμ or ϵeτ is present. In [7], the parameter degeneracies in LBNE originating from nonstandard interactions have been studied and [21] has focused on NSI at DUNE, T2HK, and T2HKK and has concluded that overall DUNE has the best sensitivity to the magnitude of the NSI parameters, while T2HKK has the best CP violation sensitivity with or without NSI. Furthermore, in [22], the authors have studied the impact of nonzero NSI on the CP precision of DUNE. The authors of [23–25] have explored the effects of NSI on CP violation sensitivity and hierarchy sensitivity at DUNE, respectively. In [8] the authors have studied the sensitivity to mass hierarchy, the octant of θ23, and CP phase δ in the future long baseline experiments T2HK and DUNE assuming standard interactions (SI) only. In general, earlier studies on standard parameters degeneracies with SI or matter NSI have mostly focused on DUNE [23–26]. Motivated by the long baseline of T2HKK experiment, it is imperative to study physics potential of T2HKK and DUNE+T2HKK, in resolving standard parameter degeneracies in presence of NSI. In the present work, we have investigated prospects for lifting mass hierarchy degeneracy (sign degeneracy), θ23-octant degeneracy, and CP-phase degeneracy in DUNE, T2HKK, and DUNE+T2HKK with matter NSIs.T2HKK is a long baseline experiment proposed to enhance the hierarchy sensitivity of T2HK by setting one of the two tanks of HK detector at a site in Korea. This multidetector setup is advantageous as it gives access to a longer baseline of 1100 km and simultaneously boosts the data at the T2HK with baseline of 295 km [27]. The neutrino oscillation probabilities are strongly affected by the matter effects in long baseline experiments. These matter effects can be beneficial in lifting up the standard parameter degeneracies. Therefore, we have considered the T2HKK setup with larger baseline of 1100 km in the analysis. In present work, we have studied the standard parameter degeneracies, i.e., mass hierarchy degeneracy and octant degeneracy in presence of matter NSI with DUNE and T2HKK experiment. We have assumed all NSI parameters to be nonzero in one case and only one off-diagonal NSI parameter ϵeμ to be nonzero, in another case. We find that the latter case is better at resolving standard parameter degeneracies in case of both DUNE and T2HKK experiments. Due to the larger baseline, T2HKK is found to have similar sensitivity as DUNE experiment to resolve standard parameter degeneracies including NSI. Furthermore, we have investigated the CP phase degeneracy occurring due to the contribution from standard and nonstandard CP phases. We observe that it is difficult to disentangle the CP effects due to SI phase δ from NSI phase δeμ at DUNE+T2HKK experiment as this conjunction is more sensitive to study CP violation effects [27].We organize the paper as follows: in Section2, we present the formalism to write oscillation probability in presence of matter NSIs. We discuss about the long baseline experiments DUNE, T2HKK, and corresponding simulation details in Section 3. In Section 4, we discuss the prospects to resolve standard parameter degeneracies in these LBNEs. We have presented our results and, subsequent, discussion in Section 5. Finally, we conclude in Section 6.


## 2. Formalism: Oscillation Probabilities

The Hamiltonian for the neutrino propagation in presence of matter NSI can be written as(2)H=12EUdiag0,Δm212,Δm312U†+V,where U is the PMNS mixing matrix containing three mixing angles(θij,i<j=1,2,3) and one CP phase δ, Δmji2≡mj2-mi2. V is the matter potential due to interaction of neutrino with matter, viz.,(3)V=A1+ϵeeϵeμeiδeμϵeτeiδeτϵeμe-iδeμϵμμϵμτeiδμτϵeτe-iδeτϵμτe-iδμτϵττ,where A≡22GFNe(r)E. The unit contribution in the first element of the matrix V is due to the matter term contribution from standard charged-current interactions. The diagonal element of V is real, i.e., δαβ=0 (where α,β=e,μ,τ) for α=β and ϵαβ≡∑ζ,KϵαβζK(Nζ/Ne). The oscillation probability for μ→e channel can be written as [7](4)Pνμ→νe=p2f2+2pqfgcosΔ+δ+q2g2+4A^ϵeμpfs232fcosδeμ+δ+c232gcosΔ+δ+δeμ+qgc232gcosδeμ+s232fcosΔ-δeμ+4A^ϵeτs23c23pffcosδeτ+δ-gcosΔ+δ+δeτ-qggcosδeτ-fcosΔ-δeτ+4A^2g2c232c23ϵeμ-s23ϵeτ2+4A^2f2s232s23ϵeμ+c23ϵeτ2+8A^2fgs23c23c23cos⁡Δs23ϵeμ2-ϵeτ2+2c23ϵeμϵeτcos⁡δeμ-δeτ-ϵeμϵeτcos⁡Δ-δeμ+δeτ+Os132ϵ,s13ϵ2,ϵ3,(5)p≡2s13s23,q≡2rs12c12c23,r=Δm212Δm312,f,f¯≡sin⁡Δ1∓A^1+ϵee1∓A^1+ϵee,(6)g≡sin⁡A^1+ϵeeΔA^1+ϵee,Δ≡Δm312L4E,A^≡AΔm312.where sij=sin⁡θij,cij=cos⁡θij,i<j, (i,j)=1,2,3. Similar expression can be obtained for inverted hierarchy (IH) by replacing Δm312→-Δm312 (i.e., Δ→-Δ,A^→-A^,f→-f¯,g→-g and q→-q). The expression for antineutrino oscillation probability can be written by replacing A^→-A^,δ→-δ and δαβ=-δαβ in (4).


## 3. Experimental Setups

Considering the sensitivity reach of the present and future long baseline neutrino oscillation experiments (for example, DUNE and T2HKK), it is very important to study the individual and collective effects of NSI parameters on parameter degeneracies. We have used GLoBES package [28, 29] with best-fit values and ranges of the standard neutrino mixing parameters, as given in [30] to simulate the DUNE and T2HKK. The current bounds on NSI parameters used in present analysis are ϵee<4.2,|ϵeμ|<0.33,|ϵeτ|<3.0,ϵμμ<0.068,|ϵμτ|<0.04,ϵττ<0.15 [15]. The CP phases δαβ of the off-diagonal NSI parameters are still unconstrained and can lie in the range δαβ∈[-π,+π].The experimental configurations, energy resolutions, and systematic uncertainties considered in the present work are as follows.


### 3.1. DUNE

The DUNE experiment [13], situated in the USA, is a globally synchronized endeavor of neutrino physicists around the world. Out of many others, the neutrino physics goals of the experiment are to unravel the sign of neutrino mass hierarchy (Δm312) and to measure the CP phase(s). The experiment is planned to direct neutrino beam from Fermilab to Homestake mine in South Dakota providing an optimum baseline of 1300 km for manifestation of matter effects in neutrino oscillations. Unlike Hyper-K, DUNE is an on-axis experiment. We have used DUNE CDR [13, 31] with 35 kt LAr far detector. The optimized beam design that employs 80 GeV beam of protons having 1.0 MW power has been used to simulate the experiment. We have considered 5(+5) years of run in neutrino (antineutrino) mode resulting in an exposure of 350 kt.MW.years. The appearance efficiency (ϵapp) and energy resolutions (ERe,ERμ) taken in the present analysis are 80% (0.15/E, 0.2/E), respectively. The normalization and energy calibration uncertainty for νe signal (NS,ES)/background (NB,EB) is taken to be NS=5%, ES=2%, NB=10%, and EB=10%. For νμ signal (NS,ES)/background (NB,EB) the values are NS=5%, ES=5%, NB=10%, and EB=10%.


### 3.2. T2HKK

The T2HKK experiment [14], an extension T2HK [12], is proposed to be stretched over Japan and Korea. The neutrino beam will be directed from J-PARC facility in Japan to two water-Cherenkov detectors: (i) first detector at Kamioka mine in Japan with a baseline of 295 km; (ii) second detector to be built in Korea providing a baseline of 1100 km. In the present work, we have considered 1100 km baseline (also, referred to as T2HKK), with detector at 1.5° off-axis with respect to the neutrino beam, where matter effects will be large. We choose 13 MW.years beam power which is similar to that of T2HK. The running time, in ratio 1:3 for neutrino and antineutrino mode, is 10 years amounting to total exposure of 2.7×1022 protons on target (POT). The appearance efficiency (ϵapp) and energy resolutions (ERe,ERμ) taken in the present analysis are 50% (0.085/E, 0.085/E), respectively. The normalization and energy calibration uncertainty for νe signal(NS,ES)/background(NB,EB) are taken to be NS=5%, ES=0.01%, NB=5%, and EB=0.01%. For νμ signal(NS,ES)/background(NB,EB) the values are NS=2.5%, ES=0.01%, NB=20%, and EB=0.01%.


## 3.1. DUNE

The DUNE experiment [13], situated in the USA, is a globally synchronized endeavor of neutrino physicists around the world. Out of many others, the neutrino physics goals of the experiment are to unravel the sign of neutrino mass hierarchy (Δm312) and to measure the CP phase(s). The experiment is planned to direct neutrino beam from Fermilab to Homestake mine in South Dakota providing an optimum baseline of 1300 km for manifestation of matter effects in neutrino oscillations. Unlike Hyper-K, DUNE is an on-axis experiment. We have used DUNE CDR [13, 31] with 35 kt LAr far detector. The optimized beam design that employs 80 GeV beam of protons having 1.0 MW power has been used to simulate the experiment. We have considered 5(+5) years of run in neutrino (antineutrino) mode resulting in an exposure of 350 kt.MW.years. The appearance efficiency (ϵapp) and energy resolutions (ERe,ERμ) taken in the present analysis are 80% (0.15/E, 0.2/E), respectively. The normalization and energy calibration uncertainty for νe signal (NS,ES)/background (NB,EB) is taken to be NS=5%, ES=2%, NB=10%, and EB=10%. For νμ signal (NS,ES)/background (NB,EB) the values are NS=5%, ES=5%, NB=10%, and EB=10%.


## 3.2. T2HKK

The T2HKK experiment [14], an extension T2HK [12], is proposed to be stretched over Japan and Korea. The neutrino beam will be directed from J-PARC facility in Japan to two water-Cherenkov detectors: (i) first detector at Kamioka mine in Japan with a baseline of 295 km; (ii) second detector to be built in Korea providing a baseline of 1100 km. In the present work, we have considered 1100 km baseline (also, referred to as T2HKK), with detector at 1.5° off-axis with respect to the neutrino beam, where matter effects will be large. We choose 13 MW.years beam power which is similar to that of T2HK. The running time, in ratio 1:3 for neutrino and antineutrino mode, is 10 years amounting to total exposure of 2.7×1022 protons on target (POT). The appearance efficiency (ϵapp) and energy resolutions (ERe,ERμ) taken in the present analysis are 50% (0.085/E, 0.085/E), respectively. The normalization and energy calibration uncertainty for νe signal(NS,ES)/background(NB,EB) are taken to be NS=5%, ES=0.01%, NB=5%, and EB=0.01%. For νμ signal(NS,ES)/background(NB,EB) the values are NS=2.5%, ES=0.01%, NB=20%, and EB=0.01%.


## 4. Parameter Degeneracies

In general, different set of oscillation parameters may give identical predictions for oscillation probability resulting in parameter degeneracies and making it difficult to uniquely determine these parameters. The biprobability plots are well known constructions to study parameter degeneracies in presence of NSI parameters andCP phases. In presence of off-diagonal NSI the neutrino/antineutrino oscillation probability exhibits degeneracy between SI and NSI phase when PSI(δ)=PNSI(ϕ,ϵeμ,δeμ) and P¯SI(δ)=P¯NSI(ϕ,ϵeμ,δeμ), where ϕ is Dirac CP phase in a model with NSI. If we presume that mixing angles and mass squared differences are known from some other experiment, then for every value of SI phase δ there will be three unknowns (ϕ, ϵeμ, and the phase δeμ) which generate an off-diagonal NSI degeneracy. Accordingly, a measurement of P and P¯ in an experiment provides two constraints; for each value of δ, a solution for ϵeμ and δeμ will exist for any value of ϕ resulting in parameter degeneracy. As a representative plot to depict standard parameter degeneracies we have shown, in Figure 1, mass hierarchy and octant degeneracy assuming NSI parameters ϵeμ=0.04,δeμ=π/2 and varying δ from 0 to 2π for DUNE experiment. All other NSI parameters are assumed to be zero. The solid and dashed ellipses correspond to higher octant (HO) and lower octant (LO) of θ23, respectively, for normal hierarchy (NH). Similarly, the dotted and dash-dotted ellipses correspond to higher octant (HO) and lower octant (LO) of θ23, respectively, for inverted hierarchy (IH). The significant overlapping of the ellipses shows fourfold θ23 octant and mass hierarchy degeneracy. For example, points of intersection of solid (dotted) and dashed (dash-dotted) ellipses exhibit θ23 octant degeneracy as neutrino and antineutrino oscillation probabilities are the same in both cases for normal (inverted) hierarchy.Figure 1
The biprobability plots withϵeμ=0.04 and CP-phase δeμ=π/2 for DUNE with baseline L=1300 km and E=3 GeV. The standard CP-phase δ is varied from 0 to 2π.


### 4.1. Mass Hierarchy Degeneracy

In the determination of unknown neutrino mixing parameters an ambiguity exists in correlated way betweenδ and sign of Δm312. The sign of Δm312 can be determined by measuring interference between the vacuum and the matter effects. The simultaneous determination of δ and sign of Δm312 can be done in long baseline experiments in both standard [8] and nonstandard cases [6]. Thus, we study the prospects to resolve this degeneracy with two future long baseline experiments DUNE and T2HKK involving NSIs. Throughout this work we have considered the νμ→νe channel to study various parameter degeneracies. We have obtained plots for DUNE and T2HKK for two cases assuming (i) all NSI parameters to be nonzero, i.e., ϵee=0.4,ϵeμ=0.04,ϵeτ=0.04,ϵμμ=0.03,ϵμτ=0.04,ϵττ=0.1 and NSI CP phases δαβ=[-π,π]; (ii) only ϵeμ,δeμ to be nonzero (ϵeμ=0.04,δeμ=[-π,π]) (Figure 2).Figure 2
The appearance probabilityP(νμ→νe) as function of neutrino beam energy E for DUNE (first row) and T2HKK (second row). The left (right) panel describes mass hierarchy assuming all (ϵeμ,δeμ) NSI parameters nonzero. θ23=42°(θ23=48°) for left panel (right panel). The band comes due to variation of δ,δeμ∈[-π,π] and boundaries correspond to δ=±π/2 for NH as well as IH.
(a) (b) (c) (d)We have, also, shown sensitivity plots for mass hierarchy for cases (i) and (ii) in Figures3(a) and 3(b) and Figures 3(c) and 3(d), respectively. We plot significance (σ=χ2) as a function of δ(true) to study hierarchy sensitivity of DUNE and T2HKK experiments for true NH(true IH) in left panel(right panel) of Figure 3. The true NH (true IH) sensitivity plot is obtained by considering NH (IH) in the true spectrum and IH (NH) in test spectrum. We have marginalized over δ,θ23 and ϵ by considering them in test spectrum. The statistical definition of χ2 for understanding the aspects of the mass hierarchy sensitivity plot in case of true NH is as follows:(7)χNH2≡minδ,θ23,ϵtest∑i=1x∑j,k=12NNHi,j,kδ,θ23,ϵtrue-NIHi,j,kδ,θ23,ϵtest2NNHi,j,kδ,θ23,ϵtrue, where NNHi,j,k and NIHi,j,k denote the number of true events and test events for NH and IH in the (i,j,k)th bin, respectively. The index i runs over 1 to x; x is the number of bins for particular experiment. For DUNE, x=39, of 250 MeV width in 0.5-10 GeV range and for T2HKK x=20, of 40 MeV width in 0.4-1.2 GeV range. The index j describes type of mode, i.e., neutrino or antineutrino. j=1(j=2) for neutrino (antineutrino) mode. The index k describes type of channel considered, i.e., appearance or disappearance. k=1(k=2) for appearance (disappearance) channel. The NSI parameter ϵ=ϵαβeiδαβ. We have minimized over the marginalization ranges of NSI parameters as given in Table 1.Table 1
The true values and marginalization ranges for all NSI parameters used in the analysis [15, 18].
NSI Parameter True value Marginalization range ϵee 0.4 [-0.4,0.4] ϵμμ 0.03 [-0.05,0.05] ϵττ 0.1 [-0.15,0.15] ϵeμ 0.04 [0,0.10] ϵeτ 0.04 [0,0.10] ϵμτ 0.04 [0,0.04] δ e μ [-π,π] [-π,π] δ e τ [-π,π] [-π,π] δ μ τ [-π,π] [-π,π]Figure 3
Sensitivity plots for mass hierarchy. The plots for significance (χ2) as a function of δ(true) with true NH (left panel) and true IH (right panel). The first (second) row describes mass hierarchy sensitivity assuming all (ϵeμ,δeμ) NSI parameters nonzero. Figures 3(e) and 3(f) show the sensitivity plots for SI case. The band comes due to variation of θ23 and boundaries correspond to θ23=42° and 48°.
(a) (b) (c) (d) (e) (f)


### 4.2. Octant Degeneracy

The oscillation probabilities for disappearance channels1-P(νμ→νe) (neutrino) and 1-P(νμ¯→νe¯) (antineutrino) show main contribution from sin22θ23. In case, θ23 is not maximal, we have two possibilities: either cos⁡2θ23>0 or cos⁡2θ23<0. This ambiguity creates two solutions (θ23,δ) and (90°-θ23,δ′). The resolution of uncertainty in θ23 due to octant degeneracy is important for precise measurement of θ23. We study this degeneracy in both neutrino and antineutrino mode with DUNE and T2HKK assuming NSI parameters ϵeμ=0.04 and δeμ=[-π,π] (Figure 4).Figure 4
The appearance probability for neutrino (antineutrino) mode as function of neutrino beam energyE for DUNE (first row) and T2HKK (second row) assuming NSI parameters ϵeμ=0.04 and δeμ=[-π,π]. The octant degeneracy is represented for neutrino mode (left panel) and for antineutrino mode (right panel). The band comes due to variation of δ,δeμ∈[-π,π] and the boundaries of the bands correspond to δ=±π/2 for LO and HO. The value of θ23=42° for LO and θ23=48° for HO.
(a) (b) (c) (d)To study theθ23 octant degeneracy, we have obtained octant sensitivity plot for true NH(true IH) in Figure 5. We plot significance (σ) as a function of θ23(true). One can define octant sensitivity by considering LO in true spectrum (HO in true spectrum) and HO in test spectrum(LO in test spectrum). The octant sensitivity tells us about the ability of an experiment to distinguish the θ23 lower octant from its higher octant. In order to obtain sensitivity plots for octant degeneracy, we have marginalized over sign (Δm312), δ, and NSI parameter ϵ and the χ2 in true NH case is(8)χNH2≡minδ,θ23,Δm312,ϵtest∑i=1x∑j,k=12NNHi,j,kδ,θ23,Δm312,ϵtrue-NNHi,j,kδ,θ23,Δm312,ϵtest2NNHi,j,kδ,θ23,Δm312,ϵtrue,Figure 5
Sensitivity plots forθ23 octant with true NH (left) and true IH (right), assuming NSI parameters ϵeμ=0.04 and δeμ=[-π,π].
(a) (b)


### 4.3.CP Phase Degeneracy

In presence of new physics, there may appear additional sources ofCP violation other than due to Dirac-type CP phase δ. The CP effects will, in general, include contributions from both standard and nonstandard CP phases. As the value of CP phases δ and δeμ is not known and possible existence of CP violation in nature, we look for all possible values of δ(true),δeμ(true) which are distinct from the CP conserving values of δ and δeμ. For this purpose, we define CP fraction [32] F(δ)(F(δeμ)) as the fraction of total permitted range of δ(true)(δeμ(true)), i.e., [-π,π] where CP violation effects corresponding to standard(nonstandard) CP phases can be explored. Also, we have excluded the CP conserving values by marginalizing over δ(test)(δeμ(test)) for {0,π} which implies that the value of δ(test)(δeμ(test)) get fixed to its CP conserving values 0 or π (Figure 6(b)). So, we can write CP fraction F(δ)(F(δeμ)) at 3σC.L.=Totalrangeofδ(true)(δeμ(true))valuesabove3σC.L./ Totalpermittedrangeofδ(true)(δeμ(true))([-π,π]) which will be discussed in detail, in Section 5.Figure 6
TheCP asymmetry as function of neutrino beam energy E for DUNE+T2HKK (left panel). The CP fractions corresponding to δ and δeμ for which significance ≥ 3σ (right panel).
(a) (b)To resolveCP phase degeneracy, we need to find out the CP sensitivity with which an experiment can distinguish between CP conserving cases and CP violating cases. In standard oscillations there is only one degree of freedom in χ2-function corresponding to standard CP phase δ. However, in case of neutrino oscillations with NSI there are two degrees of freedom due to standard and nonstandard CP phases (δ,δeμ). We define the χ2-function as(9)χ2≡minδ,ϵeμ,δeμ∑i=1x∑j2Ntruei,jδ,ϵeμ,δeμ-Ntesti,jδ,ϵeμ,δeμ2Ntruei,jδ,ϵeμ,δeμ, where Ntruei,j are the number of true events for (δ,δeμ) in the range [-π,π] and Ntesti,j are the number of test events in (i,j)th bin for (δ,δeμ) with {0,π}.To studyCP violation discovery, we obtain plot for χ2 as a function of δ(true) for true NH and true IH for DUNE, T2HKK, and DUNE+T2HKK experiments (Figure 7).Figure 7
Sensitivity plots forCP violation with true NH (left) and true IH (right) assuming NSI parameters ϵeμ=0.04 and δeμ(true)=π/2.
(a) (b)


## 4.1. Mass Hierarchy Degeneracy

In the determination of unknown neutrino mixing parameters an ambiguity exists in correlated way betweenδ and sign of Δm312. The sign of Δm312 can be determined by measuring interference between the vacuum and the matter effects. The simultaneous determination of δ and sign of Δm312 can be done in long baseline experiments in both standard [8] and nonstandard cases [6]. Thus, we study the prospects to resolve this degeneracy with two future long baseline experiments DUNE and T2HKK involving NSIs. Throughout this work we have considered the νμ→νe channel to study various parameter degeneracies. We have obtained plots for DUNE and T2HKK for two cases assuming (i) all NSI parameters to be nonzero, i.e., ϵee=0.4,ϵeμ=0.04,ϵeτ=0.04,ϵμμ=0.03,ϵμτ=0.04,ϵττ=0.1 and NSI CP phases δαβ=[-π,π]; (ii) only ϵeμ,δeμ to be nonzero (ϵeμ=0.04,δeμ=[-π,π]) (Figure 2).Figure 2
The appearance probabilityP(νμ→νe) as function of neutrino beam energy E for DUNE (first row) and T2HKK (second row). The left (right) panel describes mass hierarchy assuming all (ϵeμ,δeμ) NSI parameters nonzero. θ23=42°(θ23=48°) for left panel (right panel). The band comes due to variation of δ,δeμ∈[-π,π] and boundaries correspond to δ=±π/2 for NH as well as IH.
(a) (b) (c) (d)We have, also, shown sensitivity plots for mass hierarchy for cases (i) and (ii) in Figures3(a) and 3(b) and Figures 3(c) and 3(d), respectively. We plot significance (σ=χ2) as a function of δ(true) to study hierarchy sensitivity of DUNE and T2HKK experiments for true NH(true IH) in left panel(right panel) of Figure 3. The true NH (true IH) sensitivity plot is obtained by considering NH (IH) in the true spectrum and IH (NH) in test spectrum. We have marginalized over δ,θ23 and ϵ by considering them in test spectrum. The statistical definition of χ2 for understanding the aspects of the mass hierarchy sensitivity plot in case of true NH is as follows:(7)χNH2≡minδ,θ23,ϵtest∑i=1x∑j,k=12NNHi,j,kδ,θ23,ϵtrue-NIHi,j,kδ,θ23,ϵtest2NNHi,j,kδ,θ23,ϵtrue, where NNHi,j,k and NIHi,j,k denote the number of true events and test events for NH and IH in the (i,j,k)th bin, respectively. The index i runs over 1 to x; x is the number of bins for particular experiment. For DUNE, x=39, of 250 MeV width in 0.5-10 GeV range and for T2HKK x=20, of 40 MeV width in 0.4-1.2 GeV range. The index j describes type of mode, i.e., neutrino or antineutrino. j=1(j=2) for neutrino (antineutrino) mode. The index k describes type of channel considered, i.e., appearance or disappearance. k=1(k=2) for appearance (disappearance) channel. The NSI parameter ϵ=ϵαβeiδαβ. We have minimized over the marginalization ranges of NSI parameters as given in Table 1.Table 1
The true values and marginalization ranges for all NSI parameters used in the analysis [15, 18].
NSI Parameter True value Marginalization range ϵee 0.4 [-0.4,0.4] ϵμμ 0.03 [-0.05,0.05] ϵττ 0.1 [-0.15,0.15] ϵeμ 0.04 [0,0.10] ϵeτ 0.04 [0,0.10] ϵμτ 0.04 [0,0.04] δ e μ [-π,π] [-π,π] δ e τ [-π,π] [-π,π] δ μ τ [-π,π] [-π,π]Figure 3
Sensitivity plots for mass hierarchy. The plots for significance (χ2) as a function of δ(true) with true NH (left panel) and true IH (right panel). The first (second) row describes mass hierarchy sensitivity assuming all (ϵeμ,δeμ) NSI parameters nonzero. Figures 3(e) and 3(f) show the sensitivity plots for SI case. The band comes due to variation of θ23 and boundaries correspond to θ23=42° and 48°.
(a) (b) (c) (d) (e) (f)


## 4.2. Octant Degeneracy

The oscillation probabilities for disappearance channels1-P(νμ→νe) (neutrino) and 1-P(νμ¯→νe¯) (antineutrino) show main contribution from sin22θ23. In case, θ23 is not maximal, we have two possibilities: either cos⁡2θ23>0 or cos⁡2θ23<0. This ambiguity creates two solutions (θ23,δ) and (90°-θ23,δ′). The resolution of uncertainty in θ23 due to octant degeneracy is important for precise measurement of θ23. We study this degeneracy in both neutrino and antineutrino mode with DUNE and T2HKK assuming NSI parameters ϵeμ=0.04 and δeμ=[-π,π] (Figure 4).Figure 4
The appearance probability for neutrino (antineutrino) mode as function of neutrino beam energyE for DUNE (first row) and T2HKK (second row) assuming NSI parameters ϵeμ=0.04 and δeμ=[-π,π]. The octant degeneracy is represented for neutrino mode (left panel) and for antineutrino mode (right panel). The band comes due to variation of δ,δeμ∈[-π,π] and the boundaries of the bands correspond to δ=±π/2 for LO and HO. The value of θ23=42° for LO and θ23=48° for HO.
(a) (b) (c) (d)To study theθ23 octant degeneracy, we have obtained octant sensitivity plot for true NH(true IH) in Figure 5. We plot significance (σ) as a function of θ23(true). One can define octant sensitivity by considering LO in true spectrum (HO in true spectrum) and HO in test spectrum(LO in test spectrum). The octant sensitivity tells us about the ability of an experiment to distinguish the θ23 lower octant from its higher octant. In order to obtain sensitivity plots for octant degeneracy, we have marginalized over sign (Δm312), δ, and NSI parameter ϵ and the χ2 in true NH case is(8)χNH2≡minδ,θ23,Δm312,ϵtest∑i=1x∑j,k=12NNHi,j,kδ,θ23,Δm312,ϵtrue-NNHi,j,kδ,θ23,Δm312,ϵtest2NNHi,j,kδ,θ23,Δm312,ϵtrue,Figure 5
Sensitivity plots forθ23 octant with true NH (left) and true IH (right), assuming NSI parameters ϵeμ=0.04 and δeμ=[-π,π].
(a) (b)


## 4.3.CP Phase Degeneracy

In presence of new physics, there may appear additional sources ofCP violation other than due to Dirac-type CP phase δ. The CP effects will, in general, include contributions from both standard and nonstandard CP phases. As the value of CP phases δ and δeμ is not known and possible existence of CP violation in nature, we look for all possible values of δ(true),δeμ(true) which are distinct from the CP conserving values of δ and δeμ. For this purpose, we define CP fraction [32] F(δ)(F(δeμ)) as the fraction of total permitted range of δ(true)(δeμ(true)), i.e., [-π,π] where CP violation effects corresponding to standard(nonstandard) CP phases can be explored. Also, we have excluded the CP conserving values by marginalizing over δ(test)(δeμ(test)) for {0,π} which implies that the value of δ(test)(δeμ(test)) get fixed to its CP conserving values 0 or π (Figure 6(b)). So, we can write CP fraction F(δ)(F(δeμ)) at 3σC.L.=Totalrangeofδ(true)(δeμ(true))valuesabove3σC.L./ Totalpermittedrangeofδ(true)(δeμ(true))([-π,π]) which will be discussed in detail, in Section 5.Figure 6
TheCP asymmetry as function of neutrino beam energy E for DUNE+T2HKK (left panel). The CP fractions corresponding to δ and δeμ for which significance ≥ 3σ (right panel).
(a) (b)To resolveCP phase degeneracy, we need to find out the CP sensitivity with which an experiment can distinguish between CP conserving cases and CP violating cases. In standard oscillations there is only one degree of freedom in χ2-function corresponding to standard CP phase δ. However, in case of neutrino oscillations with NSI there are two degrees of freedom due to standard and nonstandard CP phases (δ,δeμ). We define the χ2-function as(9)χ2≡minδ,ϵeμ,δeμ∑i=1x∑j2Ntruei,jδ,ϵeμ,δeμ-Ntesti,jδ,ϵeμ,δeμ2Ntruei,jδ,ϵeμ,δeμ, where Ntruei,j are the number of true events for (δ,δeμ) in the range [-π,π] and Ntesti,j are the number of test events in (i,j)th bin for (δ,δeμ) with {0,π}.To studyCP violation discovery, we obtain plot for χ2 as a function of δ(true) for true NH and true IH for DUNE, T2HKK, and DUNE+T2HKK experiments (Figure 7).Figure 7
Sensitivity plots forCP violation with true NH (left) and true IH (right) assuming NSI parameters ϵeμ=0.04 and δeμ(true)=π/2.
(a) (b)


## 5. Results and Discussion

The degeneracy in sign ofΔm312 and δ can be resolved with the experiments which involve matter effects such as DUNE and T2HKK. In left (right) panel of Figure 2, we have shown the mass hierarchy degeneracy assuming all (ϵeμ,δeμ) NSI parameters, along with corresponding CP phases, to be nonzero for DUNE and T2HKK experiments. The meshed region between solid and dashed lines corresponds to the normal hierarchy (NH) whereas the forward-diagonal region between dash-dotted and dotted lines corresponds to the inverted hierarchy (IH) of neutrino masses. The normal and inverted hierarchy regions overlap for DUNE and T2HKK experiments due to contributions from all the nonzero NSI parameters (left panel) making it difficult to resolve the mass hierarchy degeneracy in these experiments. However, there is no overlap in the NH and IH regions for DUNE and T2HKK for energy range 1 to 4 GeV assuming only ϵeμ and corresponding phase to be nonzero (right panel). Furthermore, the future long baseline experiments such as DUNE and T2HKK with higher statistics and better energy resolution, focusing on neutrino beam energy between 1 and 4 GeV energy bracket, may provide better opportunity to resolve mass hierarchy degeneracy (Figures 2(b) and 2(d)). However, for the same energy range DUNE has brighter prospects to resolve the mass degeneracy than T2HKK.In Figure3, we have obtained the sensitivity plots for mass hierarchy in presence of all(ϵeμ,δeμ) NSI parameters for DUNE and T2HKK in first (second) row with true NH and true IH. Here, all NSI parameters referred to ϵee,ϵeμ,ϵeτ,ϵμμ,ϵμτ,ϵττ with corresponding CP phases and their true and test values have been taken from Table 1. For ready reference, we have, also, shown the sensitivity plots for the SI case in Figures 3(e) and 3(f). We find from Figures 3(a) and 3(b) that there is no distinction between true NH and true IH sensitivities for any value of δ(true) at nearly 20σ(3σ) C.L. for DUNE (T2HKK) experiments. The hierarchy sensitivity get enhanced when only one NSI parameter ϵeμ and its corresponding CP phase δeμ is present as shown in Figures 3(c) and 3(d). The DUNE experiment in case of true NH shows stronger hierarchy sensitivity for -180°<δ<0° as compared to true IH and is maximum at δ≈-60° (15σ C.L.), whereas for 0°<δ<180° the hierarchy sensitivity in true IH case is stronger and is maximum at δ≈40°(11σ C.L.). For T2HKK experiment the hierarchy sensitivity of true NH case is stronger than true IH case for the region -90°<δ<90° and lies nearly at 5σ C.L. It can be noted from Figures 3(a) and 3(b) that χ2 decreases as θ23 changes from lower to higher octant which is in contradistinction to the case with one off-diagonal NSI (Figures 3(c) and 3(d)) and SI case (Figures 3(e) and 3(f)). Consequent to the presence of multiple NSI parameters and long baselines of DUNE and T2HKK, θ23 measurement will be severely affected by the NSI-modified matter effects due to degeneracies between SI and NSI parameters and between NSI parameters. Also, the mass hierarchy sensitivity will be decreased as compared to SI case due to cancellation effects induced by the off-diagonal NSI parameters. In particular, for all NSI parameters, at lower value of θ23 the cancellation effects will be relatively small as compared to SI matter effects which are reflected as increase in the χ2 value. However, cancellation effects will be appreciable for larger value of θ23 resulting in low χ2.In Figure4, we have shown octant degeneracy in DUNE and T2HKK with NSI parameters ϵeμ=0.04 and δeμ=[-π,π]. It is evident from Figure 4 that octant degeneracy can be resolved with DUNE and T2HKK experiments using combination of neutrino and antineutrino oscillation modes. In Figure 4, the meshed region between solid and dashed lines represents the lower octant (LO) and the forward-diagonal region between dash-dotted and dotted lines represents the higher octant (HO). The left (right) panel in Figure 4 represents neutrino (antineutrino) mode of DUNE and T2HKK experiments. For LO(θ23<45°), with δ=π/2 and for HO(θ23>45°) with δ=-π/2, both DUNE and T2HKK can resolve the octant degeneracies with neutrino mode only. Moreover, for LO with δ=-π/2 and HO with δ=π/2, octant degeneracies can be resolved with both DUNE and T2HKK with antineutrino mode only. Thus, in general, the neutrino and antineutrino modes are exigent to resolve octant degeneracy in DUNE and T2HKK experiments with matter NSI. Also, we find that the neutrino (antineutrino) beam energy bracket of 1 to 4 GeV can, simultaneously, resolve the mass hierarchy and octant degeneracies in DUNE and T2HKK.In Figure5, we have obtained the sensitivity plots of θ23 octant with true NH and true IH for DUNE and T2HKK. It can be seen from Figure 5 that the octant degeneracy can be resolved for both DUNE and T2HKK experiments for true NH and true IH. The DUNE experiment shows strong sensitivity for LO in true NH (18σ C.L.) and for HO in true IH (19σ C.L.). The T2HKK experiment has weak sensitivity to resolve octant degeneracy in both true NH and true IH case. We have, also, checked that there is not much improvement in θ23 octant sensitivity (over DUNE case) if we take DUNE and T2HKK conjunctively. For comparison, we have, also, shown the SI sensitivity curves of θ23 octant with true NH and true IH for DUNE and T2HKK.The current and future neutrino oscillation experiments are diligently aiming at measuring neutrino mass hierarchy andCP violating phase δ. In presence of NSI (for example, assuming ϵeμ and δeμ nonzero) the situation becomes more complicated due to presence of additional sources of CP violation. The nature may intromit CP violation for wide range of δ(δeμ).In Figure6(a), we have shown CP asymmetry with neutrino beam energy E for both δ and δeμ for DUNE+T2HKK. In Figure 6(a), the solid (dashed) line represents δ=0,δeμ=π/2(δ=π/2,δeμ=0) case whereas in Figure 6(b), the solid (dashed) line represents the CP fraction corresponding to the δeμ(δ). In Figure 6(b), instead of focusing on measurement of CP phases (δ,δeμ) we have obtained all possible δ(true) and δeμ(true) values which are different from CP conserving values of δ and δeμ at 3σ C.L. for DUNE+T2HKK. We have shown the effect of real NSI parameter (ϵeμ) on the discovery reach of CP violation due to δ,δeμ in Figure 6(b).For real NSI case (ϵeμ∈[10-4,1.0],δeμ=0), we calculate CP fraction F(δ), i.e., fraction of δ(true)∈[-π,π] which is distinguishable from its CP conserving values, at 3σ C.L. The parameter space for which significance is less than 3σ C.L. has been excluded; thus, δ(test) has been fixed to its CP conserving values {0,π}. The χ2 has been minimized for δ(true) value from -π to π and the parameter space for which significance ≥3σ is considered. It is evident from Figure 6(b) that real NSI parameter (ϵeμ) has no effect on the discovery reach of CP violation due to δ(F(δ)), within the available bound on ϵeμ, at DUNE+T2HKK.For complex NSI (ϵeμ∈[10-4,1.0],δeμ≠0), with δ=0, i.e., CP violation is only due to NSI phase δeμ, we have calculated CP fraction F(δeμ) on similar lines as that for F(δ) in real NSI case (replace δ by δeμ). It can be seen from Figure 6(b) that F(δeμ)∈[0,0.27] for ϵeμ∈[10-4,1.0]. However, for longer baseline F(δeμ) may be larger even for small value(s) of ϵeμ. In this case, if CP violation is not observed at shorter baselines then the larger value of F(δeμ) implies CP violation due to matter NSI. Also, it is observed from Figure 6(b) that CP fraction F(δ)=0.57 implies that there exist certain range(s) of δ(true) for which CP violation is undetectable because significance is less than 3σ. For the case when CP violation is due to both δ and δeμ and δ(true) lies below 3σ significance, CP violation may be observed for certain range(s) of NSI parameters ϵeμ and δeμ. However, it will be difficult to disentangle the source of CP violation i.e., whether it is due to SI phase δ or NSI phase δeμ.Figure7 depicts the CP violation sensitivity plots for DUNE, T2HKK, and DUNE+T2HKK experiments with true NH and true IH. The CP violation sensitivity of T2HKK is stronger than DUNE in true NH irrespective of the value of δ(true) (Figure 7(a)). For true IH, CP violation sensitivity of T2HKK is stronger than DUNE except for 25° range of δ(true) in δ(true)<0 region and δ(true)>0 region for which CP violation sensitivity of DUNE is stronger than T2HKK. The CP violation sensitivity of DUNE experiment is stronger in true NH than in true IH and is maximum at δ=-115°(δ=115°), in true NH (IH). In combined analysis DUNE+T2HKK, the CP violation sensitivity increases to 10σ and 8.2σ for true NH and true IH, respectively. In Figure 7(b), the dip in the T2HKK sensitivity is due to the hierarchy-δ degeneracy. The δ(true) values at the dip correspond to wrong hierarchy solutions. SI CP violation sensitivity curves for DUNE, T2HKK, and DUNE + T2HKK experiments with true NH and true IH have, also, been shown.


## 6. Conclusions

In conclusion, we have investigated the sensitivities of DUNE and T2HKK experiments to resolve mass hierarchy and octant degeneracies in presence of matter NSI. We have, also, studied theCP phase degeneracy due to standard and nonstandard CP phases for DUNE+T2HKK. The results are in consonance with the earlier studies on DUNE. We have analyzed the standard parameter degeneracies in presence of matter NSI for T2HKK experiment. We find that the mass hierarchy degeneracy cannot be resolved in presence of all NSI parameters due to their large experimental uncertainties (Figures 2(a) and 2(c)). However, it can be resolved for neutrino beam energy range 1 to 4 GeV in case of one nonzero NSI parameter ϵeμ and corresponding NSI CP phase δeμ for DUNE and T2HKK experiments (Figures 2(b) and 2(d)).DUNE and T2HKK show poor sensitivity to resolve mass hierarchy degeneracy in presence of all NSI parameters (Figures3(a) and 3(b)). However, the sensitivity to mass hierarchy gets enhanced when only one NSI parameter ϵeμ and its corresponding CP phase δeμ are present (Figures 3(c) and 3(d)). DUNE shows stronger hierarchy sensitivity for -180°<δ<0° in true NH than true IH case with maximum sensitivity at δ≈-60°(15σ C.L.), whereas for 0°<δ<180° the hierarchy sensitivity in true IH case is stronger and is maximum at δ≈40°(11σ C.L.). T2HKK shows stronger hierarchy sensitivity in true NH case than true IH case for the region -90°<δ<90° and lies nearly at 5σ C.L.Furthermore, for LO (θ23<45°), with δ=π/2 and for HO (θ23>45°) with δ=-π/2, both DUNE and T2HKK can resolve the octant degeneracies with neutrino mode only. Moreover, for LO with δ=-π/2 and HO with δ=π/2, octant degeneracies can be resolved with both DUNE and T2HKK with antineutrino mode only. Thus, combination of neutrino and antineutrino mode of DUNE and T2HKK can resolve the octant degeneracy.The octant degeneracy can be resolved for both DUNE and T2HKK experiments for true NH and true IH (Figure5). The DUNE experiment shows stronger sensitivity for LO in true NH (18σ C.L.) and for HO in true IH (19σ C.L.). The T2HKK experiment has weak sensitivity to resolve octant degeneracy in both true NH and true IH case. We have, also, checked that there is not much improvement in θ23 octant sensitivity (over DUNE case) if we take DUNE and T2HKK conjunctively.TheCP asymmetry from nonstandard CP phase δeμ is more than the standard CP phase δ for neutrino beam energy 1.5 to 7 GeV (Figure 6(a)). From Figure 6(b) we observe that real NSI parameter (ϵeμ) has no effect on the discovery reach of CP violation due to δ(F(δ)=0.57), within the available bound on ϵeμ, at DUNE+T2HKK. Also, for complex NSI (ϵeμ∈[10-4,1],δeμ≠0) with δ=0, F(δeμ)∈[0,0.27] for ϵeμ∈[10-4,1]. It increases with increase in ϵeμ and is 0.27 when ϵeμ=1. Also, it is observed that F(δ)=0.57 implying that there exist certain range(s) of δ(true) for which CP violation is undetectable because significance is less than 3σ. For δ,δeμ≠0 and δ(true) lies below 3σ significance, then CP violation may be observed for certain range(s) of NSI parameters ϵeμ and δeμ. However, it will be difficult to disentangle the source of CP violation, i.e., whether it is due to SI phase δ or NSI phase δeμ.TheCP violation sensitivity of T2HKK is stronger than DUNE in true NH (Figure 7(a)). However, there exists a 25° range of δ(true) in case of true IH for which CP violation sensitivity of DUNE is stronger than T2HKK. In combined analysis DUNE+T2HKK, the CP violation sensitivity increases to 10σ and 8.2σ for true NH and true IH, respectively.



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*Source: 8464535-2019-05-13.xml*