Effect of non-unitary mixing on the mass hierarchy and CP violation determination at the Protvino to Orca experiment

In this paper, we have estimated the neutrino mass ordering and the CP violation sensitivity of the proposed Protvino to Orca (P2O) experiment after 6 years of data-taking. Both unitary and non-unitary $3\times 3$ neutrino mass mixing have been considered in the simulations. A forecast analysis deriving possible future constraints on non-unitary parameters at P2O have been performed.


I. INTRODUCTION
Neutrino oscillation is parameterised by the the mixing of three mass eigen states into three flavour states by the following mixing scheme: where the vector on the left (right) hand side denotes the neutrino flavour (mass) eigen states.U is the 3 × 3 unitary PMNS mixing matrix parameterised as where c ij = cos θ ij and s ij = sin θ ij .Thus neutrino oscillation probabilities depend on three mixing angles θ 12 , θ 13 , and θ 23 , one CP violating phase δ CP , and two independent mass squared differences ∆ 21 = m 2 2 − m 2 1 and ∆ 31 = m 2 3 − m 2 1 .m i is the mass of the mass eigen state ν i .Among these parameters, θ 12 and ∆ 21 have been measured in the solar neutrino experiments [3,4], |∆ 31 | and sin 2 2θ 23 have been measured in atmospheric neutrino experiments [5] and long base line accelerator neutrino experiments like MINOS [6].The reactor neutrino experiments measured θ 13 [7][8][9].In table I, we have noted down the present global best-fit values of neutrino oscillation parameters.The current unknowns are the sign of ∆ 31 , δ CP and the octant of θ 23 .Depending on the sign of ∆ 31 , two scenarios are possible: a.Normal Hierarchy (NH): m 3 >> m 2 > m 1 and b.Inverted Hierarchy (IH): m 2 > m 1 >> m 3 .The long baseline accelerator neutrino experiments NOνA [10] and T2K [11] are currently taking data in both neutrino and anti-neutrino mode to observe both muon disappearances and electron appearances in the detector and are expected to measure the unknowns.However, their recent results [12,13] show tension between the allowed regions in the sin 2 θ 23 − δ CP plane, with NOνA excluding T2K best-fit point at 90% confidence level and T2K excluding NOνA best-fit point at 3 σ.If this trend continues with the future data, upcoming long baseline accelerator experiments like DUNE [14] and T2HK [15] will play vital role in resolving the tension.
Apart from the tension in the sin 2 θ 23 − δ CP plane, the latest data from NOνA and T2K also fail to make any distinction between the two hierarchies [16].Therefore, it's really important for the future experiments to distinguish between the two hierarchies.Atmospheric neutrino experiments with multi-megaton ice/water Cherenkov detectors, like PINGU [17] and KM3NeT [18], can provide large statistics, however their mass hierarchy sensitivity is limited by the uncertainties in the flux.Careful analysis to minimise these uncertainties using dedicated detector specific simulations have been done and it has been shown that it is possible to achieve 3 σ hierarchy sensitivity after 3-4 years of data taking [19].However the CP sensitivity is rather low unless future upgrade is possible [20].
Combining multi megaton water/ice Cherenkov detector with accelerator neutrino source over a very long baseline to explore the hierarchy and CP sensitivity has been considered in literature [21][22][23][24][25][26].In ref. [27], the hierarchy and CP sensitivity of a possible future experiment with beam from Fermilab towards the ORCA detector of KM3NeT collaboration was discussed.Ref. [28] discussed the physics potential of an experiment with beam from Protvino to Orca (P2O).In 2019, the letter of intent (LOI) of the P2O experiment came out [29].The fluxes used in the LOI was less optimistic than the ones used in ref. [28].Therefore it is important to explore the physics potential of the P2O with the updated beam.
Apart from measuring the standard oscillation parameter, future long baseline accelerator neutrino experiments will explore beyond Standard Model (BSM) physics [30].Recently, there have been studies where BSM physics has been used to resolve the tension between the NOνA and T2K.In ref. [31,32] non-standard neutral current interaction during propa-gation has been used to resolve this tension, while in ref. [33] same has been done with CPT violating Lorentz invariance violation.Ref. [34] tried to probe non-unitary mixing with the recent NOνA and T2K data.They found out that the tension between the two experiments can be reduced with non-unitarity and that both the experiments prefer non-unitary mixing.They also showed that the signature of non-unitarity in these two experiments have grown stronger over the years.However, a recent paper [35] showed that the short baseline experiments NOMAD and NuTeV prefer unitary mixing and put strong constraints on nonunitary parameters.This means that the short baseline experiments data have a tension with the long baseline data from the NOνA and T2K.Thus it is important to explore the effects of non-unitary mixing with different baselines and energy.
Earlier, effects of non-unitary parameters on the determination of unknown neutrino oscillation parameters in long baseline experiments like NOνA, T2K, DUNE, T2HK and TNT2K have been discussed in ref. [36][37][38][39][40]. Ref. [41] explored the role of neutral current measurements in measuring the bounds on non-unitary parameters.In this paper, we have studied the effect of non-unitary mixing on the determination of hierarchy and CP violation in P2O experiment.In section II, we have discussed the detail of the P2O experiment.
Neutrino oscillation probabilities with unitary and non-unitary mixing have been discussed in section III.The physics sensitivities to determine hierarchy both with unitary and nonuniray oscillation have been talked about in section IV.Section V talks about the physics potential to establish CP violation in the P2O experiment with unitary and non-unitary mixing schemes.The future constraints on non-unitary parameters from P2O experiment have been discussed in section VI.The conclusion has been drawn in section VII.

II. EXPERIMENTAL DETAILS
ORCA (Oscillation Research with Cosmics in the Abyss) is the low energy component of the KM3NeT Consortium [18], housing two detectors in the Mediterranean sea.It is located about 40 km off the coast of Toulon, France, at a depth of ∼ 2450 m.Upon completion ORCA will house 115 strings with an effective volume of ∼ 5 MT.The water Cherenkov detector is optimized for the study of atmospheric neutrino oscillations in the energy range of 3 to 100 GeV with the primary goal to determine the neutrino mass ordering.
A 3 σ sensitivity to mass hierarchy is expected after 3 years of full-ORCA data taking [29].
ORCA will provide a complimentary measurement of ∆ 32 and θ 23 , measure the ν τ flux normalisation, and probe a variety of new physics scenarios namely, sterile neutrinos, nonstandard interactions, neutrino decay and so on.Currently, 6 ORCA lines are live and taking data since an year.
The Protvino accelerator complex, situated at 100 km south of Moscow, sits a U-70 accelerator.The baseline between Protvino and ORCA is roughly 2595 km.A beam power of 90 kW, corresponding to 0.8 × 10 20 protons on target (POT) per year [29] is expected.
However, there is a proposal to upgrade the beam power to 450 kW [29].
In ref. [28], the analyses have been done assuming an optimistic beam power of 450 kW.
In this work, we have taken a conservative approach followed in the P2O LOI [29].To do so, we have first matched the expected event numbers with the LOI [29].To calculate event numbers, the GLoBES [42,43] simulation package has been used.For energy smearing, GLoBES incorporates a Gaussian resolution function: where E is the true neutrino energy and E is the reconstructed neutrino energy.The energy resolution is parameterised by: Values of α = 0, β = 0, and γ = 0.4 are assumed in this work.
Depending on the Cherenkov signatures of the outgoing lepton from the ν e and ν µ CC and NC interactions, two distinct event topologies are observed at the detector: track-like and shower-like events.ν µ CC and those ν τ CC interactions with muonic τ decays mostly account for track-like topology.The shower-like topology has events from ν e CC, ν τ CC interactions with non-muonic τ decays and NC interactions of all flavours.
For the disappearance channels, the signal comes from ν µ CC events, and the backgrounds come from NC and tau events.For appearance channels, the signal comes from ν e CC events and the backgrounds come from ν τ CC events, NC events and also the mis-identification.In our analysis, we have modified GLoBES to match the expected event rates given in ref. [29].
The energy distributions of the expected neutrino events across various channels after 3 years of running in the neutrino mode (corresponding to 2.4 × 10 20 POT) are shown in Figure 1.

MIXING
The standard unitary neutrino oscillation probability with matter effect for uniform matter density can be written as [44] P µe sin 2 2θ 13 sin 2 θ 23 sin where A is the Wolfenstein matter term [45], given by , where E is the neutrino beam energy and L is the length of the baseline.
The hierarchy-δ CP degeneracy is important for experiments like NOνA and T2K, making it impossible to determine the hierarchy when NH (IH) and 0 < δ CP < 180 • (−180 But for P μē , the separation between these two hierarchy-δ CP combinations is maximum.
Thus, in case of P2O addition have anti-neutrino run will have better hierarchy sensitivity than only neutrino run for NH-δ CP in UHP, and IH-δ CP in LHP.In the case of non-unitary mixing, the mixing matrix is defined as [61] The mixing between flavour states and mass eigenstates can be written as where β denotes the flavour states and i denotes the mass eigenstates.The evolution of neutrino mass eigenstates during the propagation, can be written as where H vac is the Hamiltonian in vacuum and it is defined as The non-unitary neutrino oscillation probability in vacuum can be written as If written explicitly neglecting the cubic products of α 10 , sin θ 13 , and ∆ 21 , the oscillation probability takes the form [ where I 012 = −dcp = φ 10 − φ 20 + φ 21 , and I N P = φ 10 − Arg(α 10 ).φ ij s are the phases associated with α ij = |α ij |e iφ ij .It should be noted that non-unitary parameters α 00 , α 11 , and α 10 have most significant effects on P NU µe (vac).While, propagating through matters, the neutrinos undergo forward scattering and the neutrino oscillation probability gets modified due to interaction potential between neutrino and matter.In case of non-unitary mixing, the CC and NC interaction Lagangian is given as where V CC = √ 2G F N e , and V NC = −G F N n / √ 2 are the potentials for CC and NC interactions respectively.Therefore, the effective Hamiltonian becomes The non-unitary neutrino oscillation probability after neutrinos travel through a distance L is given as A detailed description of non-unitary neutrino oscillation probability in presence of matter effect has been discussed in ref. [34,[62][63][64].
Probability plots for P2O with the standard and with non-unitary oscillations have been plotted for [φ 10 =0, +90 and -90 degree] for different values of δ cP under Normal hierarchy and Inverted hierarchy for neutrinos and anti-neutrinos separately.Oscillation parameters used for probability plots are given in table II.Please note that apart from α 00 , α 10 , and α 11 , all other non-unitary parameters have been fixed at their unitary values.This is because α 00 , α 10 , and α 11 have maximum effects on the oscillation probability, and in this paper, we will concentrate on the effects of these three parameters on the physics potential of P2O.We have used the software GLoBES [42,43] to calculate both unitary and non-unitary oscillation probabilities.For non-unitary oscillation probabilities, GLoBES has been modified aptly to include non-unitary mixing.Figure 4, 5 and 6 shows the neutrino and anti neutrino probability plots corresponding to φ 10 =0, +90 and -90 degree respectively.

IV. MASS HIERARCHY SENSITIVITY
In this section, we will discuss the mass hierarchy sensitivity of the P2O experiment for different neutrino and anti-neutrino run times.From now on, x years of neutrino run and y years of anti-neutrino run will be defined as x + y years.

A. Mass hierarchy sensitivity with standard unitary oscillation
First we have considered that NH is our true hierarchy and IH is test hierarchy.For the true data set, values of δ cp has been varied in the range [−180 • : 180 • ].Other standard unitary oscillation parameter values have been fixed at the values given in ref. [1,2].We have assumed unitary mixing for both true and test event rates.In the test data set ∆ 31 , and sin 2 θ 23 have been varied in their possible 3 σ range given in ref. [1,2], δ CP has been varied in its complete range [−180 • : 180 • ].The χ 2 between true and test event rates have been calculated using GLoBES [42,43].Automatic bin based energy smearing for generated test events has been implemented in the same way as described in the GLoBES manual [42,43].
We used 5% normalisation and 3% energy calibration systematics uncertainty for e and µ like events [29].Implementing systematics uncertainty has been discussed in details in GLoBES manual [42,43].
For simulations, χ 2 and ∆χ 2 are always equivalent.σ = (χ 2 ) has been calculated as a function of true values of δ cp by marginalising it over the test parameter values.The same procedure has been repeated with the considerations that IH is true and NH is test hierarchy.In this study, both disappearance and appearance channels have been considered and combined χ 2 for ν and ν has been shown.MH sensitivity plots have been generated assuming 10% muon mis identification in P2O detector and are shown in Figure 10 and with 5% muon misidentification has been shown in Figure 11.
From fig. 10, it is obvious that for 10% muon mis-identification factor, changes in neutrino and anti-neutrino run time for a total run of 6 years do not affect hierarchy sensitivity when NH is the true hierarchy.However, for IH, changes in run time have significant effect on hierarchy sensitivity and 3 + 3 years run has the maximum hierarchy sensitivity.This is because the true value of θ 23 is in HO, and as we have already shown in section III with fig. 3 that the octant-hierarchy combination HO-IH has the maximum separation from another combination LO-NH for P μē .Hence, addition of anti-neutrino run improves the hierarchy sensitivity for IH.Another important feature is that for NH δ CP = −90 • has the maximum hierarchy sensitivity for the 6 + 0 run, where as δ CP = 90 • has the same for 3 + 3 run.This  In case of IH being the true hierarchy (fig.13), in case of α 00 , hierarchy can be determined at 6 σ, after 6 years of only neutrino run.Anti-neutrino run can improve the sensitivity up to 8 σ.However, for α 11 , the hierarchy can be established at ∼ 8 σ after 6 years, and antineutrino run does not have significant effect on hierarchy sensitivity.In case of |α 10 | , after 6 years of neutrino run, hierarchy can be established at 2 σ C.L. Anti-neutrino run can improve the sensitivity up to 8 σ.

V. ESTABLISHING CP VIOLATION
In this section, we discuss about the CP violation sensitivity, i.e. the sensitivity to rule out CP conserving δ CP values, of P2O, when the data are analysed with both unitary and non-unitary mixing hypothesis.To do so, the true δ CP values have been varied in the complete range.Other standard unitary oscillation parameter values have been fixed at the values given in Table II.

A. CP violation sensitivity with standard unitary oscillations
At first, the test event rates and the χ 2 between true and test event rates have been ).These constraints are less stringent than the present constraints from the global analysis [65], but better than the constraints put on by the present NOνA and T2K experimental data [34].

FIG. 1 :
FIG. 1: Energy distribution of the expected number of beam neutrino events that would be detected by ORCA after 3 years of running for 4 different values of the CP phase for the case of normal (left panel) and inverted (right panel) neutrino mass ordering.θ 23 = 45 o is assumed.The x-axis shows the true neutrino energy.

FIG. 2 :
FIG. 2: (Anti-) Neutrino appearance probability as a function of energy in the upper (lower) panel assuming NH and IH with standard unitary oscillation parameters given in Table I for δ cp = −90 • and δ cp = +90 •

FIG. 4 :FIG. 6 :FIG. 8 :
FIG. 4: Neutrino and Anti-neutrino appearance (top row) and disappearance (bottom row) probability plots assuming NH with standard oscillation parameters and with non-unitary parameters as given in Table II for φ 10 = 0 o for different true values of δ cp is because P µe for NH and δ CP = −90 • is well separated from that for IH for all δ CP values, whereas NH and δ CP = 90 • has the maximum separation from IH and δ CP = −90 • for P μē .This has already been explained in section III with fig.2.It can also be noted that wrong hierarchy can be ruled out for all the δ CP values and all the run times at more than 10 σ (8 σ) confidence level (C.L.), when NH (IH) is the true hierarchy.When muon misidentification factor is reduced to 5%, mass hierarchy sensitivity of the experiment increases.Now the wrong hierarchy can be ruled out for all δ CP values and all the different run times at more than 11 σ (9 σ) C.L. when NH (IH) is the true hierarchy.Other features remain same.

FIG. 10 :FIG. 13 :
FIG. 10: MH sensitivity as a function of true values of δ cp assuming 10% muon misidentification and for different run times for 6 years.True hierarchy has been assumed to be normal (inverted) for the upper (lower) panel.
12 s 12 c 13 s 13 e −iδ CP −s 12 c 23 − c 12 s 23 s 13 e iδ CP c 12 c 23 − s 12 s 23 s 13 e iδ CP s 23 c 13 s 12 s 23 − s 13 c 12 c 23 e iδ CP −c 12 s 23 − s 13 c 23 s 12 e iδ CP c 23 c 13 [59] 0 lower half plane (LHP) of δ CP .In eq. 5, NH (IH) enhances (suppresses) P µe , while δ CP in UHP (LHP) suppresses (enhances) P µe .Thus the value of probabilities for NH and δ CP in UHP, and IH and δ CP in LHP are close to each other, making these two combinations as the unfavourable hierarchy-δ CP combinations.On the other hand, for the combinations NH-δ CP in LHP, and IH-δ CP in UHP, the values of P µe are well separated.These two are the favourable hierarchy-δ CP combinations.Hierarch-δ CP degeneracy exists for P μē as well.However, this degeneracy can be broken in the case of bi-magic baseline 2540 km, as shown in ref.[59].The P2O baseline 2595 km is very close to the bi-magic baseline, and hence the special effects enhancing the hierarchy sensitivity of bi-magic baseline are also present in the case of P2O as well.In fig.2, we have shown the oscillation probabilities P µe and P • < δ CP < 0) is the true hierarchy-δ CP combination chosen by nature.We call 0 < δ CP < 180 • upper half plane (UHP), and −180 • < δ μē as a function of energy for P2O and for different hierarchy and δ CP values as mentioned in the plot.The other oscillation parameters have been fixed at values mentioned in table I.We have only plotted probabilities pertaining to δ CP values −90 • and 90 • .All other δ CP values fall in between.It can be easily noticed that the hierarchy-δ CP degeneracy gets broken and hierarchy can be determined even for the unfavourable hierarchy-δ CP combinations.However, for P µe , the separation between NH-δ CP = +90 • , and IH-δ CP = −90 • is least compared to other hierarchy-δ CP combinations.