Measuring the sterile neutrino CP phase at DUNE and T2HK

The European Physical Journal C, Apr 2018

The CP phases associated with the sterile neutrino cannot be measured in the dedicated short-baseline experiments being built to test the sterile neutrino hypothesis. On the other hand, these phases can be measured in long-baseline experiments, even though the main goal of these experiments is not to test or measure sterile neutrino parameters. In particular, the sterile neutrino phase \(\delta _{24}\) affects the charged-current electron appearance data in long-baseline experiment. In this paper we show how well the sterile neutrino phase \(\delta _{24}\) can be measured by the next-generation long-baseline experiments DUNE, T2HK (and T2HKK). We also show the expected precision with which this sterile phase can be measured by combining the DUNE data with data from T2HK or T2HKK. The T2HK experiment is seen to be able to measure the sterile phase \(\delta _{24}\) to a reasonable precision. We also present the sensitivity of these experiments to the sterile mixing angles, both by themselves, as well as when DUNE is combined with T2HK or T2HKK.

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Measuring the sterile neutrino CP phase at DUNE and T2HK

Eur. Phys. J. C Measuring the sterile neutrino CP phase at DUNE and T2HK Sandhya Choubey 0 1 Debajyoti Dutta 1 Dipyaman Pramanik 1 0 Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, AlbaNova University Center , 106 91 Stockholm , Sweden 1 Harish-Chandra Research Institute , HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019 , India The CP phases associated with the sterile neutrino cannot be measured in the dedicated short-baseline experiments being built to test the sterile neutrino hypothesis. On the other hand, these phases can be measured in longbaseline experiments, even though the main goal of these experiments is not to test or measure sterile neutrino parameters. In particular, the sterile neutrino phase δ24 affects the charged-current electron appearance data in long-baseline experiment. In this paper we show how well the sterile neutrino phase δ24 can be measured by the next-generation longbaseline experiments DUNE, T2HK (and T2HKK). We also show the expected precision with which this sterile phase can be measured by combining the DUNE data with data from T2HK or T2HKK. The T2HK experiment is seen to be able to measure the sterile phase δ24 to a reasonable precision. We also present the sensitivity of these experiments to the sterile mixing angles, both by themselves, as well as when DUNE is combined with T2HK or T2HKK. 1 Introduction Neutrino oscillation physics has reached precision era. Observation of neutrino oscillations by the solar [ 1 ] and atmospheric [ 2 ] neutrinos have been confirmed independently at accelerator [ 3,4 ] and reactor [5] experiments. The two mass squared differences and all three mixing angles of the three-generation neutrino oscillation theory have been well determined and the three-generation paradigm well established [ 6 ]. The only remaining questions which are still to be answered are CP violation in the leptonic sector, neutrino mass hierarchy (that is, whether ν3 is the lightest or the heaviest) and whether θ23 lies in the lower or in the higher octant. Future experiments like DUNE [ 7–10 ] and T2HK [ 11,12 ] are going to explore these questions and are expected to come with definite answers. Even though neutrino oscillation with three-generations is well established, there are some hints of neutrino oscillations at a higher frequency corresponding to a mass-squared difference m2 ∼ 1 eV2 [ 13 ]. LSND experiment [ 14,15 ] in Los Alamos, USA, first showed evidence for such oscillations, where a ν¯μ beam was sent to a detector and the observations showed a 3.8σ excess in the positrons, which could be explained in terms of ν¯μ → ν¯e oscillations. For the L and E applicable for the LSND experiment, this oscillation corresponds to a mass-squared difference of m2 ∼ 1 eV2. Experiments like KARMEN [16] and MiniBooNE [ 17–19 ] tested the claim. While KARMEN data did not show any evidence for ν¯μ → ν¯e oscillations, it could not rule out the entire allowed region from LSND. More recently, the MiniBooNE experiment ran in both neutrino as well as antineutrino mode. MiniBooNE did not find any significant excess in their neutrino mode, however they reported some excess in the antineutrino mode consistent with the LSND result. Apart from these, MiniBooNE also reported some excess in the low energy bins for both neutrino and antineutrino appearance channels, but these cannot be explained in terms of neutrino flavour oscillations. The one additional sterile neutrino can be fitted along with the three active neutrinos in the so-called 3 + 1 [ 20 ] type neutrino mass spectrum. The global fit of all the relevant short-baseline data shows severe tension between the appearance and disappearance data sets and the overall goodness of fit is only 31 %. However, if one consider appearance and disappearance channels separately, the goodness of fit improves slightly and it becomes 50 % and 35 %, respectively [ 21 ]. In the 3 + 1 scenario we have 3 mass squared differences1 m241, m231 and m221, 6 mixing angles and 3 phases. ln addition to the mixing angles θ12, θ23 and θ13 which appear 1 We define mi2j = mi2 − m2j . also in the three-generation sector we have 3 additional angles θ14, θ24 and θ34 involving the fourth generation. Also, there are two new CP phases δ24 and δ34 in addition to the standard CP phase δ13. At short-baseline experiments, the oscillation probabilities Pee, Pμμ and Pμe in the 3 + 1 scenario can be written in an effective two-generation framework which depends only on m241 and an effective mixing term given as a combination of the sterile mixing angles θ14, θ24 and θ34. As a result, the short-baseline experiments are completely insensitive to the sterile CP phases δ24 and δ34. On the other hand, it is now well known that even though the m241-driven oscillations get averaged out, these phases show up in the oscillation probabilities at the long-baseline experiments. A lot of effort in the last couple of years has gone into estimating the impact of the sterile neutrino mixing angles and phases on the measurement of standard oscillation parameters at longbaseline experiments [ 22–31 ].2 These papers showed that in presence of sterile neutrino mixing the sensitivity to the measurement of CP violation, mass hierarchy, as well as octant of θ23 becomes a band, where the width of the band comes from the uncertainty on both the values of the sterile mixing angles as well as the sterile phases. While the sterile neutrino mixing angles are constrained by the global short-baseline data, there are no constraints on the sterile phases. In the future, bounds on the sterile neutrino mixing angles are expected to improve by the data from forthcoming short-baseline experiments [54–56]. Studies have shown that the long-baseline experiments could also give constraints on the sterile neutrino mixing at their near [57] and far [24,58] detectors. The sterile phases on the other hand, can be constrained only in the longbaseline experiments. A short discussion on the study of sterile phases were done at T2K+reactor [ 29 ] and T2K+NOvA [ 28 ] and the sensitivity was shown to be poor. In this paper, we study how well the next-generation experiments DUNE and T2HK will be able to measure the sterile phases. We give the expected sensitivity of DUNE and T2HK alone as well as by combining data from the two experiments. To the best of our knowledge, this is the first time such a complete study is being performed. While the authors in [ 27 ] did attempt to present the expected precision on the sterile phase (which in their parametrisation was δ14) in the DUNE experiment, the analysis they performed has several short-comings. In their analysis, the authors of [ 27 ] keep the sterile mixing angles θ14, θ24 and θ34 fixed in the fit at their assumed true values. We allow these angles to vary freely in the fit we perform in this paper. This allows the uncertainty due to both the mixing angles as well as the phases to impact out final results. We also keep our sterile neutrino mixing angles within the currently allowed limits, which have been updated following the results from the NEOS, MINOS and MINOS+ experiments 2 Some recent studies on other new physics scenarios in the context of DUNE and T2HK can be found here [32–53]. [ 21 ]. Expected precision on the sterile phase from T2HK has never been studied before and we present them for the first time. We will also show the combined expected sensitivity of DUNE and T2HK to constrain the sterile mixing angles θ24 and θ14, both when the 3 + 1 scenario is true as well as when there is no positive evidence for sterile neutrino oscillations. The paper is organised as follows: In Sect. 2 we briefly describe the sterile neutrino hypothesis and the simulation procedure. In the same section we provide the details of the DUNE, T2HK and T2HKK experiments. In Sect. 3 we give our main results on how well the future long-baseline experiments will constrain the sterile phase δ24 if the 3 + 1 scenario is indeed true. In Sect. 4 we present our results on the expected constraints from future long-baseline experiments if the experiments do not see any positive signal for sterile neutrino oscillations. Finally, we conclude in Sect. 5. 2 Sterile neutrino mixing and simulation As discussed above, extra light neutrino state(s) with mass ∼ O(1eV2) have been proposed to explain the LSND results. In this article we have considered one additional sterile neutrino within the so-called 3 + 1 scenario, where the three active neutrinos are separated from the sterile neutrinos by a mass gap of ∼ 1 eV. In this scenario the mixing matrix is 4 × 4 and hence is defined in terms of six mixing angles and three phases. The mixing matrix can be parametrised in the following way: U 3+1 P M N S = O(θ34, δ34)O(θ24, δ24)R(θ14)R(θ23)O(θ13, δ13)R(θ12). (1) Here O(θi j , δi j ) are 4×4 orthogonal matrices with associated phase δi j in the i j sector, and R(θi j ) are the rotation matrices in the i j sector. There are three mass-squared differences in the 3 + 1 scenario – the solar mass-squared difference m221 7.5×10−5 eV2, the atmospheric mass-squared difference m231 2.5 × 10−3 eV2, and the LSND masssquared difference m241 1 eV2. One can of course write the most general oscillation probabilities in terms of these three mass-squared differences, six mixing angles and three phases. However, since oscillation driven by a given masssquared difference depends on the L/E of the experiment concerned, the expression for the oscillation probabilities simplify accordingly. In particular, the short-baseline experiments have a very small L/E such that sin2( mi2j L/4E ) ∼ 0 for m221 and m231 and only the terms for m241 survive. This is the one-mass-scale-dominance case, where only one oscillation frequency due to one mass scale survives. As a result the short-baseline experiments depend on only “effective” sterile mixing angles, which are combinations of the mixing angles θi j in Eq. (1). More importantly, since they have only one oscillation frequency, they do not depend on any CP violation phase. Hence, short-baseline experiments are completely insensitive to the sterile phases for the 3 + 1 scenario.3 In long-baseline experiments such as T2HK and DUNE, the oscillations driven by m231 dominate while those driven by m221 are sub-dominant, while the very fast oscillations driven by m241 ∼ O(1eV2) get averaged out. The transition probability Pμe in the limit sin2( m241 L/4E ) ∼ 1/2 and neglecting earth matter effect is [ 22 ]: Pμ4eν = P1 + P2(δ13) + P3(δ24) + P4(δ13 + δ24). (2) Here P1 is the term independent of any phase, P2(δ13) depends only on δ13, P3(δ24) depends only on δ24 and P4(δ13 + δ24) depends on the combination (δ13 + δ24). The full expression of the different terms in Eq. (2) are as follows: 1 P1 = 2 sin2 2θμ4νe P2(δ13) = ba2 sin 2θμ3νe cos(δ13) cos 2θ12 sin2 P3(δ24) = ba sin 2θμ4νe cos(δ24) + 1 a2 sin2 2θμ3νe − 4 sin2 2θ13 sin2 2θμ4νe ×(cos2 θ12 sin2 31 + sin2 θ12 sin2 + b2a2 − 41 a2 sin2 2θ12 sin2 2θμ3νe 1 − 4 cos4 θ13 sin2 2θ12 sin2 2θμ4νe sin2 + sin2 31 − sin2 32 1 − 2 sin(δ13) sin 2 21 − sin 2 31 + sin 2 32 , × cos 2θ12 cos2 θ13 sin2 − sin2 θ13(sin2 31 − sin2 32 1 + 2 sin(δ24) cos2 θ13 sin 2 21 21 + sin2 θ13(sin 2 31 − sin 2 32) , × 1 − 2 sin2 2θ12 cos2 θ13 sin2 + cos 2θ13(cos2 θ12 sin2 31 + sin2 θ12 sin2 32) 21 32) 21, (3) 21 (4) (5) P4(δ13 + δ24) = a sin 2θμ3νe sin 2θμ4νe cos(δ13 + δ24) 3 In the 3+2 mass spectrum case, there are two sterile neutrinos and two mass squared difference that affect the oscillations at very short baselines. In this case therefore, the short baseline experiments are sensitive to the sterile CP phases. 1 + 2 sin(δ13 + δ24) cos2 θ12 sin 2 31 We can see from Eq. (2) that even though the m241-driven oscillations are averaged out, the CP phases associated with the sterile sector still appear in the neutrino oscillation probability Pμe. This dependence comes in term P3(δ24) that depend only on the sterile phase δ24 as well as in term P4(δ13 + δ24) which depends on combination of δ13 and δ24. Hence, we can expect the next-generation long-baseline experiments to be sensitive to the sterile phases. We will see the anti-correlation between δ13 and δ24 manifest in our results on measurement of these phases in the long-baseline experiments. In fact, as has been pointed above, the sterile CP phases cannot be measured in the short-baseline experiments which are dedicated to measuring the sterile neutrino mixing. Hence, experiments like DUNE and T2HK are the only place where δ24 can be measured in the 3 + 1 scenario. Note that in Eq. (2) the probability Pμe does not depend on the mixing angle θ34, hence the corresponding phase associated with this angle δ34 also does not appear. Once earth matter effects are taken into account the probability Pμe picks up a θ34 dependence and hence depends on δ34 as well. However, for DUNE and T2HK experiments earth matter effects are rather weak and hence their corresponding sensitivity to δ34 cannot be expected to be strong. Therefore, as we will see in the Results section, these experiments are mainly able to put constraints on δ24. Prior to proceeding, we briefly discuss our simulation procedure as well as the present statues of the neutrino oscillation parameters. For our analysis we have used GLoBES (Global Long Baseline Experiment Simulator) [59,60] along with the additional codes [61,62] for calculating probabilities in the 3 + 1 scenario. We have used constant matter density for all the cases. Throughout the analysis we choose the true values4 of the standard oscillation parameters as: θ12 = 33.56◦, θ13 = 8.46◦, θ23 = 45◦, m221 = 7.5 × 10−5 eV2, m231 = 2.5 × 10−5 eV2 and δ13 = − 90◦ unless specified otherwise. This choice of parameters are consistent with the current limits [63]. Although one should marginalize over all the free parameters whenever one introduces some new physics, but new physics scenarios often give large number 4 Throughout this paper we refer to the oscillation parameter values at which the “data” is generated as the “true value” and values in the fit as “test values”. of parameters. Marginalisation over these large number of parameters is computationally challenging, so one has to do some approximation. For our analysis we have checked that the effect of marginalisation over the standard three neutrino parameters other than δ13 have no significant effect. So to save computational time we did not marginalise over these not so relevant set of parameters. For the sterile neutrino mixing, we have considered two scenarios. We first start by assuming that active-sterile neutrino oscillations indeed exist and find the expected constraints on the sterile neutrino mixing angles and phase δ24 assuming non-zero sterile neutrino mixing angles in the “data”. For this case we generate the “data” at true values m241 ≈ 1.7 eV2, θ14 ≈ 8.13◦, θ24 ≈ 7.14◦, θ34 = 0◦, which are the current best-fit values taken from [ 21 ]. The true values of the sterile phases will be specified. We marginalise our χ 2 over all sterile mixing parameters except m241 in the fit. The χ 2 is marginalised by varying θ14, θ24 and θ34 in the range [5◦, 10.5◦], [4◦, 9.5◦], and [0◦, 12◦], respectively [ 21 ] without any Gaussian prior, while the phases δ24 and δ34 are varied in their full range ∈ [−180◦, 180◦]. We next assume a scenario where the sterile neutrino mixing does not exist in nature and we show how well would then the long-baseline experiments DUNE and T2HK constrain the sterile neutrino mixing angles. For this case the data of course corresponds to true sterile mixing angles zero. The marginalisation of the χ 2 is done over all the three sterile mixing angles and the three phases. Mixing angles θ14, θ24 and θ34 are marginalised in the range [0◦, 10◦], [0◦, 10◦], and [0◦, 12◦], respectively, while the phases are allowed to vary in their full range ∈ [−180◦, 180◦]. 2.1 DUNE DUNE (Deep Underground Neutrino Experiment) [ 7–10 ] is a future long baseline experiment proposed in US. Purpose of DUNE is to address all the three unknowns of the neutrino oscillation sector – the leptonic CP violation, mass-hierarchy and the octant of the θ23. DUNE will consist of a source facility at Fermilab and a far detector at Sanford Underground Research facility in South Dakota at a distance of 1300 km from the source. Hence, the baseline of the experiment is 1300 km. The accelerator facility at Fermilab will give a proton beam of energy 80–120 GeV at 1.2–2.4 MW which will eventually give a wide-band neutrino beam of energy range 0.5–8.0 GeV. The far site will consist of 4 identical detector of 10 kt each which will give fiducial mass of 34 kt. All the detectors will be LArTPC (Liquid Argon Time Projection Chamber). In this work we have considered a Liquid Argon detector of fiducial mass 34 kt at a baseline of 1300 km. The neutrino flux is given by the 120 GeV, 1.2 MW proton beam. Here we have considered 5 years of neutrino and 5 years of antineutrino. Appearance and disappearance channels are combined for the analysis. The energy resolutions for the μ and e are taken to be 20%/√E and 15%/√E , respectively. The signal efficiency is taken to be 85%. The backgrounds are taken from [ 8 ]. In the neutrino (antineutrino) mode, the signal normalization error is 2% (5%), the background normalization error is 10% (10%) and the energy calibration error is 5% (5%). This choice of systematics is conservative compared to the projected systematics in [ 8 ].5 2.2 T2HK The Hyper-Kamiokande (HK) [ 11,12 ] is the upgradation of the Super-kamiokande (SK) [65] program in Japan, where the detector mass is projected to be increased by about twenty times the fiducial mass of SK. HK will consist of two 187 kt water Cherenkov detector modules, to be place near the current SK site about 295 km away from source. The detector will be 2.5◦ off-axis from the J-PARC beam which is currently being used by the T2K experiment [66]. T2HK has similar physics goals as DUNE, but since it will employ a narrow-band beam, it can be complimentary to the DUNE experiment. For our analysis we take a beam power of 1.3 MW and the 2.5◦ off-axis flux. We consider a baseline of 295 km and the total fiducial mass of 374 kt (two tank each of which is 187 kt). We consider 2.5 years of neutrino and 7.5 years of anti-neutrino runs in both appearance and disappearance channels. The energy resolution is taken to be 15%/√E . The number of events are matched with the Tables 2 and 3 of [ 12 ]. The signal normalization error in νe(ν¯e) appearance and νμ(ν¯μ) disappearance channel are 3.2% (3.6%) and 3.9% (3.6%), respectively. The background and energy calibration errors in all channels are 10 and 5%, respectively. 2.3 T2HKK In [ 12 ], the collaboration has also discussed the possibility of shifting one of the water tanks to a different location in Korea at a distance of about 1100 km from the source. This proposed configuration will consist of the same neutrino source but with one detector of fiducial mass 187 kt at a baseline of 295 km and another similar detector of fiducial mass 187 kt at a baseline of 1100 km. The second oscillation maximum takes place near Eν = 0.6 GeV at the second detector. Both detectors are taken 2.5◦ off-axis in our study. 5 The experimental specifications for DUNE has been updated in [64]. However, we have explicitly checked that the physics results do not differ much for the two specifications. The older version was for 5 + 5 years of run while the newer one is for 3.5 + 3.5 years of run and yet inspite of lower statistics the newer version is able to achieve similar physics goal because of the optimised fluxes, detector response and systematics. Since the flux peaks at the same energy for both detector locations, the Japan detector sees the flux at the first oscillation maximum while the Korea detector sees it at the second oscillation maximum. This whole setup is called T2HKK. In our analysis, we have considered signal normalization error of 3.2% (3.6%) in νe (ν¯e) appearance channel and 3.9% (3.6%) in νμ (ν¯μ) disappearance channel, respectively. The background and energy calibration errors are 10% and 5% in all the channels, respectively. 3 Measurement of the sterile phases In this section we discuss the ability of the long baseline experiments to constrain the sterile phases. The “data” is generated for the 3 + 1 scenario for the values of mixing parameters discussed in Sect. 2. In particular, for the sterile neutrino parameters we take the following values: m241(true) = 1.7 eV2, θ14(true) = 8.13◦, θ24(true) = 7.14◦, θ34(true) = 0◦. The true values of the phases δ24 will be taken at some benchmark values and will be mentioned whenever needed. The true values of standard oscillation parameters are taken at their current best-fit values, mentioned in Sect. 2. The χ 2 is marginalised over the relevant oscillation parameters in the 3 + 1 scenario, as discussed in Sect. 2, where the parameters are allowed to vary within their current 3σ ranges. Although there are three phases in the 3 + 1 scenario, the role of the phase δ34 is weak. As was discussed in the previous section, the mixing angle θ34 affects the oscillation probability Pμe only when matter effects become important. For DUNE and T2HKK earth matter effects are not very strong while for T2HK the effect of earth matter is even weaker. Since the impact of the phase δ34 on Pμe is proportional to the mixing angle θ34, the phase δ34 is also less important for Pμe for the same reason. Moreover, the current global best-fit for the angle θ34 turns out to be zero [ 21 ]. Therefore, in this work we set θ34(true) = 0◦ in the data. As a result the phase δ34 is not expected to be very crucial in our analysis and hence we take and δ34(true) = 0◦ in the data and show our results only in the δ13–δ24 plane. We reiterate that the χ 2 is marginalised over the mixing angle θ34 and phase δ34 in the fit, where the mixing angle is allowed to vary between [0◦, 12◦] [ 21 ]. The Fig. 1 shows the capability of DUNE, T2HK and T2HKK to measure the phase δ24. We show the plots of χ 2 as a function of δ24(test) for T2HK (dotted black lines), T2HKK (dash-dotted red lines) and DUNE (dashed blue lines) for δ24(true) of 0◦ (top-left panel), 90◦ (top-right panel), − 90◦ (bottom-left panel) and 180◦ (bottom-right panel). The true values of all other parameters are taken as detailed in Sect. 2 and the previous paragraph. The χ 2 plot has been marginalised over all relevant parameters as discussed before. The green solid horizontal lines show the χ 2 corresponding to 2σ CL. Table 1 shows that T2HK can better constrain the phase δ24 as compared to DUNE, while T2HKK is expected to perform better than DUNE but worse than T2HK. Note that the sensitivity of DUNE and T2HKK is marginally better for δ24(true) = 90◦ than for δ24(true) = − 90◦ while the reverse is true in case of T2HK (see Table 1). In order to understand why the measurement of δ24 is expected to be better at T2HK than DUNE, we show in Fig. 2 the expected electron events at DUNE (top left panel) and T2HK (top right panel). The four lines in each panel show the expected events for four values of δ24 = 0◦ (solid red lines), 90◦ (dashed blue lines), − 90◦ (dotted green lines) and 180◦ (dash-dotted dark red lines). The upper panels of the figure reveal that the two experiments behave in almost the same way as far as the dependence of the probability Pμe to δ24 is concerned. However, there is a clear difference between the two when it comes to the overall statistics. T2HK expects to see nearly 14 times more events than DUNE due to its bigger detector size. Hence the corresponding χ 2 for T2HK is also expected to be higher. Of course the systematic uncertainty for DUNE is considerably less than for T2HK and that compensates the effect of the lower statistics, however, the effect of statistic shows up in a non-trivial way for δ24 measurement at the long baseline experiments and T2HK with its bigger detector emerges as a better option in this regard. The lower panels of Fig. 2 show the event rate at the oscillation maximum for DUNE (lower left panel) and T2HK (lower right panel) as a function of δ24. The black solid curves show the expected number of events whereas the green and yellow bands show the 1σ and 3σ statistical deviation. The black short-dashed straight lines show the event rate at oscillation maxima for the four benchmark values of δ24 = 0◦, 180◦, 90◦ and − 90◦. The Fig. 1 had revealed that the χ 2 corresponding to δ24(true) = 0◦ and δ24(true) = 180◦ are much lower compared to that for δ24(true) = ± 90◦. This can be understood from the lower panels of Fig. 2 as follows. Figure 2 shows that the predicted number of events at oscillation maximum for δ24(true) = 0◦ and 180◦ lie between the predicted events for δ24(true) = ± 90◦. Therefore, for the cases where data is generated for δ24(true) = 0◦ and 180◦, it is easier for other δ24 values to fit the data and give a smaller χ 2. However, data corresponding to δ24(true) = ± 90◦ takes a more extreme value and the difference between the data and fit for other values of δ24 for these cases becomes larger, giving larger χ 2. Another interesting feature visible in the event plots in Fig. 2 is that the maxima and minima of the events are not at δ24 = 90◦ or −90◦. Rather they are slightly shifted towards the right. For the same reason the χ 2 plots in Fig. 1 are also asymmetric about the true value of δ24. One can explain this using Eq. (2). By inspecting the probability one can see that the correlation between m221 and δ24 is negligible. Also, we have taken δ13 = − 90◦ everywhere. Hence, for m221 = 0 and δ13 = − 90◦, the Eq. (2) can be rearranged as, 1 Pμe = A + B cos 2θ13 sin δ24 − 2 B cos δ24 , (8) where A and B are independent of δ24. In the absence of the last term, the probability would be a sine function shifted by the constant A. However, the presence of the cosine term shifts the curve and the shift is towards right because of the minus sign in front of the cosine term. In particular, the extrema of the probability in Eq. (8) is given by the condition, 339 appearance event rates at the oscillation maximum as a function of δ24 for DUNE (left) and T2HK (right). While the black curves give the expected number of events, the green and yellow bands show the 1σ and 3σ statistical uncertainties 1 cos δ24 = − 2 sin δ24 , which corresponds to minimum at δ24 = −63.4◦ and maximum at δ24 = 116.6◦. This agrees very well with the event plots in the lower panels of Fig. 2 which is obtained using the exact numerical probability. We next show in Fig. 3 the expected 95% CL allowed areas in the δ13(test)−δ24(test) plane, expected to be measured by the next generation long-baseline experiments T2HK (or T2HKK) and DUNE, and by combining them. The four panels of Fig. 3 have been generated for four different choices of δ24(true). The value of δ13(true)= −90◦ in all the panels. In each panel the benchmark point where the data is (9) generated is shown by the black star. The four panels correspond to δ24(true)= 0◦ (top left), 90◦ (top right), −90◦ (bottom left) and 180◦ (bottom right). In all the four cases we have considered 3+1 scenario both in the ‘data’ and in the ‘fit’ or ‘theory’. The χ 2 thus generated is then marginalised over the sterile mixing angles θ14, θ24, θ34 and δ34, as discussed before. The black dotted, red dash-dotted and blue dashed contours are for T2HK, T2HKK and DUNE, respectively, while the grey and magenta solid contours are for DUNE+T2HK and DUNE+T2HKK. As in Fig. 1 we note that T2HK can constrain the phase δ24 much better than DUNE, while T2HKK performs better than DUNE but worse than T2HK. We also see, as before, that for DUNE the precision δ24 = − 90◦ (δ24 = 180◦). The black dotted curve is for T2HK, the red dash-dotted curve is for T2HKK, the blue dashed curve is for DUNE, the grey solid curve is for DUNE+T2HK and the magenta solid curve is for DUNE+T2HKK on δ24 is expected to be better for δ24(true)= ±90◦ compared to when δ24(true)= 0◦ or 180◦. For T2HK this dependence of precision on δ24 measurement on δ24(true) is less pronounced. The effect of θ34 on the measurement of δ24 is also minimal. Finally, note that there is an anti-correlation between δ13 and δ24. This comes from the term P4(δ13 + δ24) of Eq. (2). The Fig. 3 also shows how the measurements of δ24 and δ13 improve as we combine DUNE with either T2HK or T2HKK. We see that combining DUNE with T2HKK improves the precision considerably, with the combined precision of DUNE and T2HKK becoming slightly better than the precision expected from T2HK alone. Combining DUNE with T2HK improves the precision even further, albeit only marginally, since T2HK alone can measure the phases rather precisely. The question on how the measurement of the standard CP phase δ13 gets affected by the sterile mixing angle phases in the 3 + 1 scenario is another pertinent question that one can ask. The Fig. 4 shows how the expected precision on δ13 changes in presence of sterile neutrinos. The left panel is for T2HK, middle panel is for T2HKK and right panel is for DUNE. The blue dashed curves are for the standard 3+0 case with no sterile neutrinos while the red dash-dotted curves are for the 3 + 1 case with δ24 = − 90◦ in data. The other standard and sterile neutrino oscillation parameters are taken in data as described above and the fit performed as before. The Fig. 4 shows that the expected precision on δ13 worsens when the sterile neutrino is present. From Eq. (6) one can see that there is an anti-correlation between δ13 and δ24 which makes the δ13 precision worse. For DUNE the effect is more compared to T2HK and T2HKK. For T2HK the δ13 measurement is seen to be nearly unaffected. DUNE measures δ24 worse than T2HK and T2HKK and hence the corresponding measurement of δ13 worsens due to the anti-correlation mentioned above. Table 2 summarises the expected precision on δ13 for the 3 + 0 and 3 + 1 scenario for four benchmark values of δ24(true). We see that DUNE’s measurement of δ13 gets affected for all δ24 while effect on T2HK’s measurement of δ13 is negligible. 4 Measurement of the mixing angles Prospects of measuring the sterile neutrino mixing angles at long-baseline experiments DUNE [ 24 ] and T2HK [58] has been studied before. Here we study how well the sterile mixing can be constrained by combining data from these experiments. We also present the sensitivity of the individual experiment DUNE, T2HK and T2HKK. Here we consider two complementary approaches. We first assume that sterile neutrino mixing does exist (as in the last section) and see how precisely the data from long-baseline experiments can measure and constrain the angles θ14 and θ24.6 We next consider 6 We do not study the mixing angle θ34 in this work. As discussed before, this affects Pμe and Pμμ only mildly through matter effects. To constrain this angle, we need to consider the neutral current data, which has been done in [67–69]. and the red dashed curves are for 3 + 1 case in both theory and data. The curves are δ24(true) = −90◦ the alternate situation where the active-sterile oscillations do not really exist and then we see how well DUNE, T2HK and T2HKK, as well as their combination, could put upper bounds on the sterile neutrino mixing angles θ14 and θ24. 4.1 Measuring the sterile mixing angles when 3 + 1 is true In this section we assume that the 3 + 1 scenario is indeed true in nature and the mixing angles θ14 and θ24 are indeed non-zero. We perform a χ 2 analysis with prospective data generated in the 3 + 1 scenario and fitted within the 3 + 1 scenario and give expected allowed CL regions in the sterile neutrino parameter spaces. As before, we take the true sterile oscillation parameters at the following benchmark values: m241(true) = 1.7 eV2, θ14(true) = 8.13◦, θ24(true) = 7.14◦, θ34(true)= 0◦ which are consistent with [ 21 ]. The standard oscillation parameters are taken and treated as discussed before. The χ 2 is marginalised over all relevant parameters in the fit and no Gaussian priors are included. In Fig. 5, we show the contours in sin2 θ14(test)– m241(test) plane (left panel) and sin2 θ24(test)– m241(test) plane (right panel). The colour code is same as Fig. 3. The point where the data is generated is shown by the black star in the two panels. The results show that in both panels, T2HK gives better results than both DUNE and T2HKK. Again, T2HKK is better than DUNE. Combining T2HK/T2HKK and DUNE experiments improves the results and the expected allowed ranges for the sterile neutrino mixing parameters shrink. The precision expected from the combined data from T2HK and DUNE is nearly the same as that from T2HKK and DUNE, which for the former is only marginally better. Figure 6 shows the contours in the sin2 θ14(test)–sin2 θ24(test). The colour code for the different data sets considered is the same as Fig. 3. We note that the expected upper limit on sin2 θ24 is the same for all the three individual experiments for 0.001 < sin2 θ14 < 0.03. Also, the expected upper bound on sin2 θ14 is seen to be better for T2HK than DUNE. Combining the experiments can improve the measurement of the sterile neutrino mixing angles as seen from the solid contours in Fig. 6. In particular, we now see a lower bound on sin2 θ24. 4.2 Excluding the sterile hypothesis when 3 + 1 is not true If the sterile neutrino hypothesis was wrong and there was no mixing between the active and sterile neutrinos the nextgeneration experiments would falsify it. There are a series of new short-baseline experiments planned which will be testing this hypothesis [54, 55, 57]. Even the near detector of planned long-baseline experiments are well-suited to check the sterile neutrino mixing as their baseline and energy match well to correspond to the maximum of m241-driven oscillations [57]. In the same vein it is pertinent to ask how well the next-generation long-baseline experiments could constrain this hypothesis, since the oscillation probabilities for long-baseline experiments also depend on the sterile neucorrespond to standard three-generation oscillation scenario with no sterile mixing while the fit is done in the 3 + 1 framework to obtain the exclusion contours. The colour code is same as Fig. 3 trino mixing and phases even though the m241-driven oscillations themselves average out. While some work in this direction has already been done in the literature [ 24, 58 ], we will present here, for the first time, the sensitivity of T2HKK set-up to the sterile neutrino mixing angles θ24 and θ14. We will also present the expected sensitivity from the combined prospective data-sets of T2HK (or T2HKK) and DUNE, which has not been studied before. In Fig. 7 we show the exclusion curves for the 3 + 1 hypothesis expected from the next-generation long-baseline experiments. The left panel of Fig. 7 shows the 95% CL exclusion plots in the m241(test)-sin2 θ14(test) plane while the right panel shows the results in m241(test)-sin2 θ24(test) plane. Here we generate the data assuming the standard three-generation neutrino scenario and then fit it with the 3 + 1 scenario. The blue dashed, black dotted and red dashdotted curves show the exclusion plots for DUNE, T2HK and T2HKK, while the magenta and grey solid curves show the expected exclusion sensitivity for DUNE+T2HKK and DUNE+T2HK, respectively. If only three neutrinos exist in the nature, then the parameter region in the top-right of the plots are excluded at 95% CL. Again, as in the previous results, T2HK constrains θ14 better than both T2HKK and DUNE for all values of m241 in the range 10−3 eV2 to 10 eV2. Combining DUNE with T2HK and T2HKK can improve the constraint on sin2 θ14 compared to the individual experiments. Since for higher values of m241 the oscillations average out, the experiments become almost insensitive to the value of m241. For 0.1 eV2 < m241 < 10.0 eV2, DUNE, T2HKK and T2HK can exclude sin2 θ14 ∼> 0.4, sin2 θ14 ∼> 0.27 and sin2 θ14 ∼> 0.21, respectively, at 95% CL. For the same range of values of m241, DUNE+T2HK and DUNE+T2HKK could put slightly tighter constrain on sin2 θ14 and the excluded regions are expected to be sin2 θ14 ∼> 0.165 and sin2 θ14 ∼> 0.18 at 95% CL, respectively. The results presented in the right panel show the capability of these experiments to constrain sin2 θ24. If m241 is small and lies in the range 10−3 eV2 < m241 < 0.01 eV2, T2HK gives better constraint on sin2 θ24 than both DUNE and T2HKK. But for higher values of m241, in the range of 0.014 eV2 < m241 < 0.1 eV2, the performance of DUNE is better than both T2HK and T2HKK. For 0.1 eV2 < m241 < 10.0 eV2, performance of T2HKK is almost similar to that of DUNE. Similar behaviour can be seen in the combined case. In the lower m241 region, DUNE+T2HK could constrain sin2 θ24 slightly better than DUNE+T2HKK. But in the higher m241 region, DUNE+T2HKK is expected to perform better than DUNE+T2HK. The expected exclusion sensitivity for DUNE, T2HKK and T2HK in the range 0.1 eV2 < m241 < 10.0 eV2 are given as sin2 θ24 ∼> 0.026, sin2 θ24 ∼> 0.026 and sin2 θ24 ∼> 0.03, at 95% CL Similarly, the expected exclusion limit for DUNE+T2HKK and DUNE+T2HK at 95% CL are sin2 θ24 ∼> 0.017 and sin2 θ24 ∼> 0.019, respectively, for 0.1 eV2 < m241 < 10.0 eV2. In Fig. 8 we show the expected exclusion contour in sin2 θ14(test)-sin2 θ24(test) plane. Here, the region outside the contour is excluded at 95% CL. The figure represents the slice at m241 = 1.7 eV2 of the contour in the sin214, sin2 θ24, m241 space. The colour code is the same as in Fig. 7. Here also, we observe better capability of T2HK to constrain sin2 θ14-sin2 θ24 parameter space than DUNE and T2HKK in most regions of the parameter space. The plot also shows that constraint on sin2 θ24 is complicated. We see that T2HK is better than DUNE and T2HKK in constraining sin2 θ24 for sin2 θ14 ∼> 10−2. However, for sin2 θ14 ∼< 10−2 DUNE and T2HKK perform better than T2HK in constraining sin2 θ24. Combining the data-sets improves the expected sensitivity on both the sterile mixing angles. 5 Conclusions There are a number of observational hints that support the existence of neutrino oscillations at short baselines. Since the m2 needed for these frequencies is inconsistent with the m2 needed to explain the solar and atmospheric neutrino anomalies – both of which have been confirmed by earth-based experiments – it has been postulated that there are additional light neutrino states which are mixed with the three standard neutrino states. Since the Z -decay width restricts the number of light neutrino states coupled to the Z boson to 3, the additional neutrino states should be “sterile”. In this paper we considered one extra such sterile neutrino in the so-called 3 + 1 mass spectrum. In the 3 + 1 scenario the neutrino oscillation parameter space is extended by one new mass squared difference m241, three new active-sterile mixing angles θ14, θ24 and θ34 and two new CP phases δ24 and δ34. We work within a parametrisation of the mixing matrix such that the phase δ24 is associated with the mixing angle θ24 and δ34 is associated with θ34. It is now well known that even though the m241-driven oscillations are averaged out in the longbaseline experiments, the active-sterile mixing angles and the additional phases appear in the oscillation probabilities and modify it. The sensitivity of the long-baseline experiments to the active-sterile mixing angles has been studied before. The impact of the sterile neutrino parameters on the physics reach of these experiments for standard parameter measurement such as CP violation, mass hierarchy measurement and octant of θ23 measurement has been investigated in details before. In this work, for the first time, we looked at the prospects of measuring the sterile CP phase δ24 in the long-baseline experiment T2HK (and T2HKK) and DUNE as well as when data from them is combined. Dedicated short-baseline experiments are being built to test the active-sterile neutrino oscillation hypothesis. However, these experiments are sensitive to oscillation probabilities that have one-mass-scale-dominance. In other words, these experiments mainly work within effective twogeneration scenarios and are sensitive to m241 and effective two-generation mixing angles, that can be written as combination of the mixing angles θ14 and θ24. Therefore, they are completely insensitive to the CP phases δ24 and δ34. On the other hand, these phases do affect the oscillation probabilities of the long-baseline experiments even though they are not sensitive to m241 since oscillations corresponding to this frequency averages out. In particular, the probability Pμe that affects the electron appearance data in long-baseline experiments depends on δ24. We exploited this dependence to show that the phase δ24 can be measured at DUNE, T2HK and T2HKK and estimated the expected precision on this parameter. Since Pμe depends on θ34 only through earth matter effects, the effect of δ34 on the long-baseline data is expected to be very small and we cannot constrain it easily. Hence, in this paper we concentrated only on δ24. We performed a χ 2 analysis of the prospective data at these experiments and presented the expected precision on δ24 expected at these experiments. We showed that with (7.5 + 2.5) years of running in (neutrino, antineutrino) mode, T2HK could constrain δ24 to within [−63.0◦,47.88◦], [30.10◦, 147.9◦] and [−145.95◦, −31.93◦] if the true value of δ24(true) is 0◦, 90◦ and − 90◦. For δ24(true) = 180◦, all test δ24 such that δ24 ≤ −128.43◦ and δ24 ≥ 136.22◦ are allowed in T2HK. The corresponding constraints from T2HKK and DUNE were seen to be weaker, with DUNE measurement on δ24 expected to be the weakest. We explained why T2HK is expected to perform better than DUNE, with the main reason being the higher statistics in T2HK. We also showed the expected allowed areas in the δ13(test)-δ24(test) plane from prospective data from DUNE, T2HK and T2HKK. We again reiterated that the expected constraints on δ24 were seen to be strongest from the T2HK experiment, while DUNE was seen to be weakest. We also presented the allowed areas in this plane expected from combined prospective data of DUNE and T2HK (or T2HKK). We showed that the combined data set could constrain the CP parameters better. Expected constraints from DUNE+T2HK was seen to be better than constraints from DUNE+T2HKK. We also discussed the impact of the sterile neutrino mixing angles and phases on the measurement of the standard CP phase δ13. We showed that for DUNE the expected δ13 precision worsens more than for T2HK. We also presented constraints on the mixing angles θ14 and θ24. Again, we did not consider θ34 since the appearance and disappearance data in long-baseline experiments depend only mildly on this angle, with the dependence coming solely from matter effects. We took two complementary approaches in this study. First we assumed the 3 + 1 scenario to be correct and generated data assuming non-zero values of the sterile neutrino oscillation parameters. This was used to present allowed areas in the sterile neutrino parameter space expected from full run of DUNE, T2HK, T2HKK and combinations of DUNE+T2HK and DUNE+T2HKK. We made a comparison between the expected precision reach of the different experiments. We next took the complementary approach where we assumed that the 3 + 1 scenario was not true. The data in this case was generated for no sterile mixing and fitted with the 3 + 1 hypothesis to yield expected exclusion limits on the sterile neutrino mixing parameters. Again, we did this analysis for DUNE, T2HK, T2HKK and combinations of DUNE+T2HK and DUNE+T2HKK and made a comparative analysis of the different data-sets. In conclusion, the sterile neutrino phases can be measured only in experiments that are sensitive to more than one oscillation frequency other than m241. The long-baseline experiments therefore are the best place to measure δ24. We showed that the sterile phase can be measured to reasonable precision in the next-generation long-baseline experiments. The T2HK set-up is better suited to measure δ24 compared to DUNE due to larger statistics. Acknowledgements We acknowledge the HRI cluster computing facility (http://cluster.hri.res.in). The authors would like to thank the Department of Atomic Energy (DAE) Neutrino Project of HarishChandra Research Institute. 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Sandhya Choubey, Debajyoti Dutta, Dipyaman Pramanik. Measuring the sterile neutrino CP phase at DUNE and T2HK, The European Physical Journal C, 2018, 339, DOI: 10.1140/epjc/s10052-018-5816-y