Invisible neutrino decays at the MOMENT experiment

We investigate invisible decays of the third neutrino mass eigenstate in future accelerator neutrino experiments using muon-decay beams such as MuOn-decay MEdium baseline NeuTrino beam experiment (MOMENT). MOMENT has outstanding potential to measure the deficit or excess in the spectra caused by neutrino decays, especially in νμ and ν¯μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{\nu}}_{\mu } $$\end{document} disappearance channels. Such an experiment will improve the constraints of the neutrino lifetime τ3. Compared with exclusion limits in the current accelerator neutrino experiments T2K and NOvA under the stable ν assumption, we expect that MOMENT gives the bound of τ3/m3 ≥ 10−11 s/eV at 3σ, which is better than their recent limits: τ3/m3 ≥ 7 × 10−13 s/eV in NOvA and τ3/m3 ≥ 1.41 × 10−12 s/eV in T2K. The non-decay scenario is expected to be excluded by MOMENT at a confidence level > 3σ, if the best fit results in T2K and NOvA are confirmed. We further find that reducing systematic uncertainties is more important than the running time. Finally, we find some impact of τ3/m3 on the precision measurement of other oscillation parameters.


Introduction
The oscillation pattern of three-flavour neutrino mixing has been established through solar, atmospheric, accelerator and reactor neutrino experiments [1][2][3][4]. In the standard threeflavour paradigm, neutrino oscillations are dominated by two mass-squared splittings (i.e., ∆m 2 31 , ∆m 2 21 ) and three mixing angles (i.e. θ 12 , θ 13 , θ 23 ) [5]. Up to now, most of the oscillation parameters have been measured well [6], except the Dirac CP phase δ and the neutrino mass ordering (normal mass hierarchy: ∆m 2 31 > 0; inverted mass hierarchy: ∆m 2 31 < 0). The precision of measuring θ 23 is not good enough to discriminate the octant degeneracies with a specific prediction θ 23 = 45 • . All these unknown parameters will be measured in the near future by medium baseline reactor experiments: JUNO [7] and RENO [8], and by the long-baseline accelerator neutrino experiments: T2K [9], NOvA [10], T2HK [9] and DUNE [11]. Recent results from T2K and NOvA incline to a normal mass hierarchy and indicate a hint of δ ≈ 270 • [12,13] only at a low confidence level. Therefore, we are looking forward to data provided by the next-generation experiments to attain a compelling conclusion. Since we are entering an era of precision measurements, it is natural to expect near future neutrino oscillation experiments to search for new physics beyond three-generation neutrino oscillations including sterile neutrinos, neutrino decays and non-standard neutrino interactions, and so on.

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focusing on the constraints on the ν 3 lifetime, compare it to the reach of current experiments, and investigate the impacts of the total running time, systematic uncertainty and energy resolution on this measurement, with the study on the expected exclusion level to the stable-neutrino assumption and their impacts on precision measurements of θ 23 and ∆m 2 31 . Finally, we summarize in section 5.

Neutrino oscillations with invisible neutrino decays
The latest results from MiniBooNE have an excess for reconstructed oscillation spectra [44], suggesting the existence of sterile neutrinos. We assume that the neutrino decay products are sterile neutrinos. In addition, we consider that the third mass eigenstate decays in the following channel: ν 3 → ν 4 + J, where normal mass hierarchy and a light sterile neutrino are considered (i.e. m 3 > m 2 > m 1 > m 4 ). The connection between flavour eigenstates and mass eigenstates can be given as: The Hamitonian of neutrino propagation in matter can be written as: where U is the PMNS mixing matrix [45,46], G F is the Fermi coupling constant, N e is the electron density, E is the neutrino energy and τ 3 is the lifetime of ν 3 . Obviously, the probabilities for neutrino and antineutrino modes remain invariant with a replacement of δ → −δ and N e → −N e , i.e. P να→ν β (E, L; δ, N e ) = Pν α→νβ (E, L; −δ, −N e ). Then we can calculate the numerical oscillation probabilities by diagonalizing the Hamitonian matrix. The diagonalization method can be found in ref. [47]. Our numerical tool to evaluate the probabilities with neutrino decays has been checked by comparing our result with those shown in ref. [31]. To cross check validity of our codes, we have reproduced the invisibleneutrino-decay result from ref. [32], highlighting the current measurement at T2K and NOvA. The probability for the antineutrino mode has been cross checked by a comparison with the neutrino mode taking the opposite sign of δ and N e .
The probabilities with ν decays in vacuum are given as follows,

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This can be further expanded as For the antineutrino mode, δ is replaced by −δ. Eq. (2.4) is consistent with eq. (A.2) in [29]. It is clear that through the final 4 terms of eq. (2.4) neutrino decays provide damping effects to the ∆m 2 31 and ∆m 2 32 oscillations. Further, the decays also cause an overall decrease via the third term. Both effects can be seen in the following.
We show the probability for four channels ν µ → ν µ (upper-left), ν e → ν e (upper-right), ν e → ν µ (lower-left), and ν µ → ν e (lower-right) of MOMENT in figure 1 (those for the antineutrino mode are in figure 2). For the case with neutrino decays, we consider those within 10 −12 s/eV < τ 3 /m 3 < 10 −9 s/eV (red band), and compare it with that for the case without neutrino decays (black curve). As we can see, the case with τ 3 /m 3 = 10 −9 s/eV is overlapping with the curves for the case without neutrino decays. For the other extreme case τ 3 /m 3 = 10 −12 s/eV, the probabilities are far from the black curves. In the following, we compare the case for τ 3 /m 3 = 10 −12 s/eV and that without neutrino decays. Except for the minima, in the ν µ → ν µ channel, we see significant deficits. Around the minima, we notice the fact that the probability with neutrino decays goes above or below the curve corresponding to the stable-neutrino assumption. This is because the suppression term dominates the damping ones. Moving to the smaller τ 3 /m 3 , U * α3 (δ)U β3 (δ)U α3 (δ)U * β3 (δ) exp(−2Γ 3 L) gets smaller earlier than the damping terms because of the factor of 2 in the exponential. When this effect does not dominate the damping one, the probability goes upper around the minima. The competition between these two effects is also seen in the ν e → ν µ andν µ →ν e channels. Therefore, the maxima in the ν µ andν µ disappearance channels could be useful for measuring the effect of neutrino decays. The damping effect in ν e andν e disappearance channels is obvious. Further, we see an overall decrease in P (ν e → ν µ ), while the impact of neutrino decays on P (ν µ → ν e ) is similar to that for e disappearance channels -it smoothens out the probability (damping effects). The amount of impact in P (ν µ → ν e ) is similar to that in P (ν e → ν µ ). We see similar results for the antineutrino mode, except for the opposite pattern in the appearance channels: P (ν µ → ν e ) ∼ P (ν e →ν µ ) and [GeV] Figure 1. The oscillation probabilities within 10 −12 s/eV < τ 3 /m 3 < 10 −9 s/eV (red band) for MOMENT. We especially present the probability with τ 3 /m 3 = 10 −12 s/eV (red-dotted curve), = 10 −9 s/eV (red-dashed curve) and = ∞ (black curve). Four channels are shown: P (ν e → ν µ ) ∼ P (ν µ →ν e ). Based on the size of variations, we reach the conclusion that the µ-flavour disappearance channel is the more important than the other channels in the measurement of τ 3 /m 3 .

Simulated spectra with neutrino decays in MOMENT
The simulation details for MOMENT are shown in table 1 with the neutrino sources, detector descriptions and running time [48,49]. MOMENT, as a medium muon decay accelerator neutrino experiment, is proposed as a future experiment to measure the leptonic CP-violating phase. The neutrino fluxes are kindly offered by the MOMENT working group [36]. Here we utilize eight oscillation channels: ν e → ν e , ν e → ν µ , ν µ → ν e , ν µ → ν µ and their CP-conjugate partners. We have to consider flavour and charge identifications to distinguish secondary particles by means of an advanced neutrino detector. The chargedcurrent interactions are used to identify neutrino signals: ν e + n → p + e − ,ν µ + p → n + µ + , ν e + p → n + e + , and ν µ + n → p + µ − . We consider the new technology using Gd-dopped water to separate both Cherenkov and coincident signals from the capture of thermal neutrons [50,51]. The major backgrounds are mostly from the atmospheric neutrinos, [GeV] Figure 2. The oscillation probabilities within 10 −12 s/eV < τ 3 /m 3 < 10 −9 s/eV (red band) for MOMENT. We especially present the probability with τ 3 /m 3 = 10 −12 s/eV (red-dotted curve), = 10 −9 s/eV (red-dashed curve) and = ∞ (black curve). Four channels are considered:ν µ → ν µ (upper-left),ν e →ν e (upper-right),ν e →ν µ (right-left), andν µ →ν e (lower-right). The following oscillation parameters are used: neutral current backgrounds and charge mis-identifications. They can be largely suppressed by the beam direction and proper modelling of background spectra within the beam-off period, which is to be extensively studied in detector simulations. In section 4.2, we will compare the physics capabilities under different assumptions, including a change of total running time.
Our simulation is carried out with the help of a GLoBES package [52,53]. The following central values and their uncertainties of the standard neutrino oscillation parameters are taken from the latest NuFit4.0 results [6]: In the following, we will assume the normal mass hierarchy, i.e. ∆m 2 31 > 0. We present the event spectra for each channels of MOMENT in figures 3 and 4. Similar to figures 1 and 2, the spectra for the case with τ 3 /m 3 = 10 −9 s/eV exactly overlap the spectra for the case without neutrino decays. The extreme case τ 3 /m 3 = 10 −12 s/eV is far from the black spectra, which are predicted assuming stable neutrinos. In the following, we focus on a comparison of results given different assumptions. We observe the advantage of the lower energy events, as the larger deviations from the spectra for the case without neutrino decays appear in the lower-energy bins. Comparing all panels in figures 3 and 4, we Channels    are reminded of the conclusion from section 2 that the muon-flavour disappearance channels are the most important ones for the measurement of τ 3 /m 3 , as the larger deviations from the black spectra are observed. In ν µ andν µ disappearance channels shown in figure 3, we see both suppression and damping effects. The event rate decreases all the way in energy because of neutrino decays. However, the degree of deficit becomes larger around the maximum, while it gets smaller at the minimum. The change in these ν µ andν µ disappearance channels can be a few hundred events per bin, and much larger than those in the other six channels, in which the deficit is a few tens of events per bin. The overall decrease is also seen in ν e → ν µ andν µ →ν e channels. However, the number of events decreases in the lower energy bin but increases in the higher energy bin because of neutrino decays in the ν e andν e disappearance channels. We see a reduction of event rates in most energy bins inν e →ν µ , ν µ → ν e ,ν e →ν µ andν µ →ν e , as shown in the lower panels of figures 3 and 4.
To sum up, it is clear that when we turn on neutrino decays with τ 3 /m 3 = 10 −12 s/eV, a distinct difference between the cases with and without decays can be easily measured -8 -

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by MOMENT. Invisible decays can wash out the extreme of neutrino oscillations. Therefore, the focus on the maximum or minimum can help us to detect the effect of neutrino decays. Furthermore, the differences in ν µ andν µ disappearance channels are larger than the other six channels. This implies that the ν µ → ν µ andν µ →ν µ channels will play an important role in the analysis. This could affect the precision measurement of neutrino mixing parameters such as θ 23 and ∆m 2 31 which are mostly involved in these channels. As a result, the other channels could help with a clarification of this bias induced by neutrino decays. We eventually come up with the conclusion that MOMENT is expected to have high-level sensitivities to the lifetime of ν 3 since they have multiple channels, and this exactly demonstrates the advantage of Gd-dopped water Cherenkov technology.

Results
Based on simulated event spectra with/without neutrino decays, we investigate the precision measurement on τ 3 /m 3 of MOMENT, and compare it with the reach by the current experiments. We also study the expected exclusion level to the stable neutrino hypothesis (τ 3 /m 3 = ∞) assuming various true values of τ 3 /m 3 . We further present our results on the impact of statistical error, systematic uncertainty and energy resolution. Finally, we study the contours at 3σ on the θ 23 − τ 3 /m 3 , ∆m 2 31 − τ 3 /m 3 and θ 23 − ∆m 2 31 planes.

Bound on the lifetime of ν 3
In figure 5, we show the constraint on τ 3 /m 3 for four different true values: τ 3 /m 3 = ∞ (black solid), 10 −11 (green dashed-dotted), 5.01 × 10 −12 (blue short-dashed), and 3.16 × 10 −12 s/eV (red dotted). The latter three values are the current results from NOvA, T2K and the combined analysis of these two. It is obvious that for larger neutrinodecay effects, the constraint becomes tighter. The appearance of the upper bound at 3σ, which does not show up in the current measurements, is notable. In the case of τ 3 /m 3 = 10 −11 s/eV, the lower (upper) bound at 3σ is at log 10 (τ 3 /m 3 ) ∼ −11.25 (−10.5).
With τ 3 /m 3 = 5.01 × 10 −12 s/eV, the 3σ constraint is about 10 −11.5 -10 −11.1 s/eV, while with τ 3 /m 3 = 3.16×10 −12 s/eV the 3σ uncertainty runs from ∼ 10 −11.65 to ∼ 10 −11.35 s/eV. The whole behaviour of ∆χ 2 is that starting from the true value, it climbs to infinity when τ 3 /m 3 gets smaller, while ∆χ 2 approaches to a certain value when τ 3 /m 3 → ∞. The behaviour can be understood in figures 3 and 4. When τ 3 /m 3 is larger enough, the spectra behave the same as those for the stable-neutrino case. Therefore, ∆χ 2 approaches to a certain value when τ 3 /m 3 → ∞. We note that the behaviour of ∆χ 2 looks symmetric for τ 3 /m 3 = 3.16 × 10 −12 [s/eV] in figure 5, but does not for the larger value of τ 3 /m 3 . It is because in the case with τ 3 /m 3 = 3.16 × 10 −12 [s/eV] ∆χ 2 is approaching to ∼ 120 when τ 3 /m 3 → ∞. The range of ∆χ 2 shown in figure 5 is near the bottom. Therefore, the behaviour of ∆χ 2 looks symmetric for τ 3 /m 3 = 3.16 × 10 −12 [s/eV]. In figure 6 we compare the result from MOMENT (black curve) with the current experiments (red short-dashed curves), which are taken from ref. [32]. The upper-left panel shows the constraint assuming the case without neutrino decays. As we can see, the bound at 3σ for τ 3 /m 3 is pushed up by about one order of magnitude from the bound at as the best fit of ref. [32] in this figure. The most striking feature of MOMENT we see in this figure is that it provides the upper bound for τ 3 /m 3 measurement at 3σ, while the lower bound is also greatly reduced. In the other words, instead of giving us a lower bound, MOMENT provides a complete range with the upper and lower limits at a considerable confidence level. The upper bound is important for excluding the case without neutrino decays, if the neutrino decay is confirmed.
From figure 5, it is natural to expect that these experiments have a great ability to exclude the stable-neutrino hypothesis τ 3 /m 3 = ∞. We therefore discuss while the true τ 3 /m 3 is not infinity, how much MOMENT can exclude the stable-neutrino hypothesis, and therefore find a hint of new physics. 1 We show our results in figure 7, in which the red curve is the exclusion ability for MOMENT. The statistical quantity we are studying is ∆χ 2 for the hypothesis m 3 /τ 3 = 0 assuming the various true values τ 3 /m 3 (x-axis). We also compare these results with the constraint on τ 3 /m 3 assuming the case with neutrino decays (black curves). We find that if in the nature log 10 (τ 3 /m 3 [s/eV]) ∼ −10.85, MOMENT can detect a "hint" at around 3σ. These τ 3 /m 3 values are larger than our current discovery from T2K and NOvA. This means MOMENT could be sensitive enough to claim a "hint" if the current results are confirmed. 1 We call the tension between the experimental result and the stable-neutrino prediction "hint".

Impact of the total running time, systematic uncertainty, and energy resolution
We are interested in studying the impact of the total running time (the short-dashed grey curve), the systematic uncertainty (the black solid curve) and the energy resolution (the red solid curves) in figure 8. We present the constraint power assuming the case without neutrino decays at the 3σ confidence level. 2 Going through the total running time (νmode +ν-mode) from 1 to 20 years, a 3σ bound can improve from about τ 3 /m 3 = 10 −11 to 10 −10.7 s/eV. It soars from log 10 (τ 3 /m 3 [s/eV]) = −11 to about ∼ −10.9 at the fourth year before a slow climb to −10.7 at the twentieth year. This means that once it runs for more than 4 years, it gets more difficult to improve the sensitivity by increasing the running time. Moving to the impact of systematic uncertainties, we vary the size of the 2 We also undergo the same study for the 3σ exclusion ability to the stable-neutrino hypothesis. The results are almost the same as those shown in figure 8. Energy resolution Figure 8. The constraint at 3σ on τ 3 /m 3 assuming the stable-neutrino case against the total running time (the short-dashed grey curve), the size of systematic uncertainty σ s (black), and the energy resolution σ res (red). We focus on the total running time from 1 to 20 years, while σ s and σ res vary in the range [1%, 20%].
We find an important result by comparing two curves, representing the impact of the total running time and σ s . Our default setting for MOMENT is the case with 10 years for the total running time and roughly the point for σ s = 5%; comparing to two curves, we can see improving σ s = 1% can improve better (log 10 (τ 3 /m 3 [s/eV]) = −10.6) than that by doubling the total running time (log 10 (τ 3 /m 3 [s/eV]) = −10.7). Then, we further conclude that improving our understanding of systematic uncertainties is more important than doubling the total running time. As we see in figure 1, the measurement of τ 3 /m 3 largely depends on the disappearance channel, which is sensitive to θ 23 and ∆m 2 31 . We are therefore interested in the performance of 3σ contours on the τ 3 /m 3 −θ 23 (upper-left), τ 3 /m 3 −∆m 2 31 (upper-right) and θ 23 −∆m 2

31
(lower) planes in figure 9. We assume three true values: τ 3 /m 3 = 10 −11 (dashed-dotted green), 5.01 × 10 −12 (short-dashed blue) and 3.16 × 10 −12 (red dotted) s/eV. Thanks to the high precision of the τ 3 /m 3 measurement, we see a complete contour, instead of a band as what we see in current fitting result, shown in ref. [32]. On average, the precision at 3σ of θ 23 is almost 3-3.5 • for MOMENT. We observe some impact from the true τ 3 /m 3 value on the θ 23 measurement. The 3σ uncertainty of ∆m 2 31 is about 0.05 × 10 −3 eV 2 . We further study the 3σ contour on the θ 23 − ∆m 2 31 plane (the lower panel). We also include results for the stable neutrino case. We consider two scenarios -τ 3 /m 3 fixed at ∞ (black) and let this parameter vary (grey). It is obvious that the impact of neutrino decays mainly worsens the measurement of θ 23 from ∼ 1.5 • to ∼ 3-3.5 • . In comparison, there is little impact on the measurement of ∆m 2 31 . We also see a little correlation, once we include τ 3 /m 3 into fitting.

Summary
In this paper we have considered the third neutrino mass eigenstate ν 3 decaying to invisible states in MOMENT, using eight channels of neutrino oscillation (ν e → ν e , ν e → ν µ , ν µ → ν e , ν µ → ν µ and their CP-conjugate partners) with the help of the following detection processes in a Gd-doped Cherenkov detector: ν e + n → p + e − ,ν µ + p → n + µ + ,ν e + p → n + e + , and ν µ + n → p + µ − . Neutrino decays cause suppression and damping effects on neutrino oscillation probabilities, and could be measured in the reconstructed energy spectra of MOMENT, especially in ν µ andν µ disappearance channels. And we have found that focusing on the maximum or minimum is a strategy to measure these effects. Events with lower neutrino energy do not only avoid the sizeable matter effect, but also enhance the effects caused by neutrino decays. We have simulated the MOMENT experiment and JHEP04 (2019) Figure 9. The exclusion contour at 3σ on the planes any two of log 10 (τ 3 /m 3 [s/eV]) and θ 23 (left) and ∆m 2 31 (right). We show for different true values: τ 3 /m 3 = 10 −11 (dashed-dotted green), 5.01 × 10 −12 (short-dashed blue) and 3.16 × 10 −12 (dotted red) s/eV. In the lower panel, we further consider two scenarios -τ 3 /m 3 fixed at ∞ (black) and let this parameter vary (grey).
found outstanding potential to constrain the τ 3 /m 3 parameter in figure 5. Given the bestfit values hinted by T2K and NOvA [32], we have found that MOMENT would improve the precision measurement of invisible neutrino decays. We reach an interesting conclusion that if the current best fit discovered in [32] is confirmed, the standard non-decay scenario can be excluded with a statistics level higher than 3σ. At 3σ confidence level, the projections of θ 23 − log 10 (τ 3 /m 3 ), ∆m 2 31 − log 10 (τ 3 /m 3 ) and θ 23 − ∆m 2 31 have demonstrated little correlations between θ 23 and ∆m 2 31 . The impact of neutrino decays mainly decrease the 3σ precision of θ 23 by 1-1.5 • .
We have further investigated the impact of statistical and systematic uncertainties by varying the total running time, changing the size of the normalisation uncertainty σ s and energy resolution respectively. We have demonstrated the 3σ constraint assuming the standard non-decay scenario. By increasing the total running time or reducing the systematic uncertainties, we will improve the sensitivity in invisible neutrino decays. A comparison of two methods has guided us to the conclusion that reducing systematic uncertainties is more important than increasing the total running time in the MOMENT experiment. We have also checked that there is no sizeable impact from improved energy resolution in the detector.

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As MOMENT has outstanding potential to measure neutrino decays, we also have to emphasize that future atmospheric and astrophysical neutrino experiments will significantly improve the current understanding of neutrino decays. They are complementary to each other, though.