Probing neutrino magnetic moment at the Jinping neutrino experiment

Neutrino magnetic moment ($\nu$MM) is an important property of massive neutrinos. The recent anomalous excess at few keV electronic recoils observed by the Xenon1T collaboration might indicate a $\sim 2.2\times10^{-11} \mu_B$ effective neutrino magnetic moment ($\mu_\nu^{eff}$) from solar neutrinos. Therefore, it is essential to carry out the $\nu$MM searches at a different experiment to confirm or exclude such hypothesis. We study the feasibility of doing $\nu$MM measurement with 4 kton active mass at Jinping neutrino experiment using electron recoil data from both natural and artificial neutrino sources. The sensitivity of $\mu_\nu^{eff}$ can reach $1.2\times10^{-11}\mu_B$ at 90\% C.L. with 10-year data taking of solar neutrinos. Besides the intrinsic low energy background $^{14}$C in the liquid scintillator, we find the sensitivity to $\nu$MM is highly correlated with the systematic uncertainties of $pp$ and $^{85}$Kr. Reducing systematic uncertainties ($pp$ and $^{85}$Kr) and the intrinsic background ($^{14}$C and $^{85}$Kr) can help to improve sensitivities below these levels and reach the region of astrophysical interest. With a 3 mega-Curie (MCi) artificial neutrino source $^{51}$Cr installed at Jinping neutrino detector for 55 days, it could give us a sensitivity to the electron neutrino magnetic moment ($\mu_{\nu_e}$) with $1.1\times10^{-11} \mu_B$ at 90\% C.L.. With the combination of those two measurements, the flavor structure of the neutrino magnetic moment can be also probed at Jinping.

A high precision measurement of solar neutrinos has been proposed by the Jinping collaboration in China, aiming to obtain high precision of solar neutrinos at the sub-percentage level [31]. We explore the possibility of carrying the measurement of νMM at Jinping neutrino experiment via solar neutrinos. We also consider a MCi-scale electron capture neutrino source like mostly 51 Cr [32][33][34][35][36], which can release sub-MeV neutrinos as well for this study, as proposed by [37,38].
The paper is organized as follows. Section 2 depicts the study on νMM measurement with the natural neutrino source. Section 3 presents the research on νMM with a specific artificial neutrino source. Conclusions are drawn in section 4.
2 νMM measurement with solar neutrinos

Jinping neutrino experiment
The Jinping neutrino experiment (Jinping) [31] aims to study MeV-scale neutrinos, including solar neutrinos, geoneutrinos and supernova neutrinos. It is located in one of the deepest underground laboratories in the world with 2400 m vertical rock-overburden shielding, leading to a much small cosmic-ray muon background. The target material is the water-based liquid scintillator (LS), whose Cherenkov light can indicate the direction of charged particles and scintillation light can be used for the precise energy reconstruction of the particles. The nominal energy resolution of this kind of material is nominally 500 PE/MeV. In this study we assume a 4 kton fiducial target mass with 5 kton total mass. Comparing with Borexino, Jinping has a smaller cosmic-ray and a bigger detector mass. Therefore, Jinping could obtain more remarkable results.
νMM is detected though the neutrino elastic scattering (νES) from solar neutrinos or artificial neutrino source. The cross section of νES with νMM can be expressed as = sin 2 ϑ W 0.23. For νMM cross section, µ ef f ν is the effective νMM in µ B units. The νMM cross section from neutrino magnetic moment is proportional to (1/T e − 1/E ν ) , which leads that the measurement of νMM significantly depends on the capacity of the detection at low energy.
In general, the prediction of the electron scattering signal from solar neutrinos can be counted by N e is the total electron number of the fiducial volume counted as N e = V ρ LS ρ e N A = 1.35×10 33 with the total volume V , T is the exposure time, the LS density ρ LS , the electron density per gram ρ e (mol/g) and the Avogadro constant N A . i is the i th solar neutrino source. φ i is the corresponding neutrino flux. S i is the normalized energy spectrum of such neutrino. P i eα (E ν ) is the oscillation probability, which is weighted by the different neutrino distributions in the sun, with e − α flavor transition from the sun to the earth. The i th P i eα (E ν ) can be approximately given as where R is the radius of the sun, F i (r) is the normalized i th neutrino distribution [40] as a function of r, which is the distance to the center of the sun, and P eα (r, E ν ) is the oscillation probability of neutrino produced at r with an energy E ν and detected at the earth. As a good approximation, the matter effect from the earth is neglected in the night due to the very low neutrino energy of pp and 7 Be (E max ν <1 MeV). Therefore, P eα (r, E ν ) takes no account of the day-night effect. σ α is the cross section of νES with α-flavor neutrinos.
In this study, we assume 100% detection efficiency in the fiducial volume.    The SM ES signal is also background for νMM. The yellow band represents the region with the relative large νMM signal-to-background ratio shown in the subplot for all neutrinos, assuming µ ef f ν = 2.2 × 10 −11 µ B , which is the best fit of νMM hypothesis in Xenon1T [25].
shows the predictions of each components including signal and background. We generate the solar neutrino signal by using the parameters in PDG 2020 [41]. In order to measure νMM precisely, it is crucial to evaluate accurately all components which are overwhelming the most sensitive νMM detection region, the region of interest (ROI) plotted with the yellow band in this figure. For the solar neutrinos, the SM νES part of pp could mimic νMM ES part of itself because of the similar spectra of them. Therefore, the external constraint of the flux of pp takes an important role. the flux of pp neutrino can be mainly limited by the radiochemical constraints with 5% from gallium experiments [42]. The electron recoil with 7 Be neutrinos rises an outstanding "shoulder" structure up from about 500 to 700 keV, providing a precise measurement of 7 Be neutrino flux. The fluxes of CNO and pep could be counted from about 900 to 1300 keV, where νMM is negligible due to the higher neutrino energy and lower fluxes than pp and 7 Be.

Background
In general, three types background should be considered: cosmic-ray muon induced background, internal radioactive background and external radioactive background. However, for νMM study, all SM νES components are also kinds of background to νMM. The rate of cosmic-ray muon induced background can be about 200 times lower than those in Borexino due to the depth of the Jinping laboratory [31]. The main low-energy background from cosmic-ray is 11 C at low energy. However, it does not affect the measurement of νMM due to its high energy (> 1.5 MeV). Other cosmogenic isotopes will not be considered in this study. We assume the internal radioactive background can be reduced to the same level as Borexino Phase-II [43] after purifications. The intrinsic 14 C of LS is 2.7 × 10 −18 g/g in Borexino. The external γ-rays can be predicted with a exponential scale factor according to the distance from the edge of the fiducial volume to the detector surface. However, it is also insignificant because it dominates at high energy range (> 1.5 MeV). For νMM study, 14 C, 210 Bi, 85 Kr and 210 Po are the main background shown with the rate in table 1. The pile-up of 14 C-14 C has been considered for Jinping. We assume a naive 200 ns signal window (δt) to coarsely estimate the rate as where M F V is the fiducial volume mass, M T V is the total volume mass and δt is the signal integral window. The spectrum of it is generated through the convolution with the spectrum of 14 C. Other pile-up events, mainly 14 C with external γ-rays and 210 Po with external γ-rays [44], are negligible.
In general, 14 C could be determined independently from the main analysis as Borexino suggested [45]. 14 C shuts down the feasibility of any signal detection at low energy due to its huge abundance below 150 keV. Thanks to the better resolution at Jinping, 14 C could not bury pp and 7 Be neutrinos severely above 150 keV. However, the measurement of pp is still challenging because of the pile-up of 14 C-14 C. Optimistically, we neglect the shape uncertainty of 14 C [46]. 210 Po, overwhelming in the region of 350-550 keV, can be fitted clearly with a gaussian distribution. In addition, α particle from 210 Po could also be discriminated with scintillation pulse shape from e ± . Conservatively, 210 Po is still considered in this study. 210 Bi appears as a "shoulder" structure in the region from about 700 to 1000 keV with a relatively big event number. Therefore, 210 Bi could be fitted well.
Thanks to the energy resolution of Jinping experiment, there is a wide ROI, the yellow band in figure 1, between 14 C and 210 Po spectra to measure the νMM. However, 85 Kr hides under all other components in ROI, resulting in a difficult measurement. In addition, It could almost freely mimic the shape and rate of νMM component, especially νMM from 7 Be, in ROI of νMM. That is to say, the residual 85 Kr of the detector material can significantly influence the measurement of νMM. Therefore, the good purification and the independent measurement of 85 Kr could accordingly improve the capability of νMM measurement for Jinping experiment. In general, it is uncertain the the purification of 85 Kr improves 1 or 2 order, according to the analysis in Borexino Phase-II [43]. Fortunately, 85 Kr has a rather small branch of β decay (0.4%) with a coincidence signal, which could be selected with 18% efficiency proposed by Borexino Phase-II [43], giving a 4% bound by 10-year measurement at Jinping.

Sensitivity
For the sensitivity study, we build a χ 2 function as where N i pre and N i obs are the event number in the i th bin of the prediction and the observation with visible energy from 150 to 1500 keV, and ( δα σα ) 2 is the penalty term to constrain solar neutrino oscillation parameters (i.e. θ 12 and ∆m 2 21 ), solar neutrino fluxes and background. In ( δα σα ) 2 , δ α means the difference between the value of fitting parameter α and the center value of its prior and σ α is 1 σ error.
Because there are multiple components, which are overlapping with each other at different visible energy regions, a simultaneous analysis method is necessary to take out all information for each part. Therefore, Markov chain Monte Carlo (MCMC) technique is used to study the correlations among multi parameters, especially the relations between νMM and any other parameter, and to obtain the individual posterior distribution of each component simultaneously. MCMC is based on a pythonic package named emcee [47], in which a likelihood function is needed. Therefore, we convert the χ 2 function into likelihood through L = exp − 1 2 χ 2 for MCMC. Figure 2 presents a multi-parameter scatter plots with the full parameters by MCMC after a 10-year data taking, showing the correlations of each two parameters and the distributions of full parameters. Solar neutrino parameters are relative to the HZ fluxes: R φ = φ /φ truth . Background parameters are the relative differences to the background truth rates in table 1: δBG=(BG-BG truth )/BG truth . This MCMC assumes that sin 2 θ 12 and ∆m   cases. The sensitivities to νMM from The MCMC method and χ 2 minimizer are consistent with each other. We find that fixing sin 2 θ 12 and ∆m 2 21 leads no differences about the sensitivity of νMM in fitter, compared with the case that they are constrained by JUNO. The standard case assumes that sin 2 θ 12 , ∆m 2 21 , CNO and pep are fixed, pp is bounded by Gallium experiments and other parameters are all free. In figure 3, the standard case can reach 1.2 × 10 −11 µ B , which is much better than other natural neutrino experiments. The 4% 85 Kr bound can boost the 90% upper limit to 1 × 10 −11 µ B as well as 3%/ E(MeV), high light yield LS. Moreover, HZ flux bound, particularly pp and 7 Be fluxes, can also improve a lot, leading to the νMM bound down to 0.8 × 10 −11 µ B . The most ambitious sensitivity one is the statistics-only case with 3.9 × 10 −12 µ B level at 90% C.L. for the case of such background level in table 1. If the background level could reduced, the result will get better.
With naive background reductions, we calculate the sensitivity to νMM with different individual reductions of 14 C, 85 Kr, 210 Po and 210 Bi. We find that the reductions of 14 C and 85 Kr could significantly improve the sensitivity to νMM. It would take a more than 10,000-fold reduction in 14 C background to reach 1.0 × 10 −12 µ B level. A more than 1000fold reduction in 85 Kr background could at most reach 6.0 × 10 −12 µ B level. More reduction in 85 Kr could not improve any sensitivity. Any reduction of 210 Po and 210 Bi could hardly improve the sensitivity to νMM. We also find that Jinping has more than 5 σ (3 σ) to confirm or exclude the νMM hypothesis about the recent excess in Xenon1T in 10 years (4 years).
3 νMM with artificial neutrino source 3.1 51 Cr neutrino signals As a specific example, a 3 MCi initialized 51 Cr source [38] is assumed to be placed outside with 1 meter away from the edge of the fiducial volume or inside at the center of the detector with shielding. Figure 4 shows the cartoon sketch of the proposed source positions. The decay of 51 Cr is 51 Cr + e − −→ 51 V + ν e , with a 27.7-day half-life. The monoenergetic neutrino energies are 752 keV (9%), 747 keV (81%), 432 keV (1%) and 427 keV (9%) respectively.

R [m]
Outside Figure 5: The normalized event profile with respect to R from the source.
We calculate event vertex distribution as the function of R, which is the distance from the event position to the artificial source for both cases and the event number based on appendix A. Figure 5 presents the event profile in the different slices as a function of R, assuming an ideal reconstruction performance. The position information can provide a strong capability to separate the signal of 51 Cr and the almost uniform solar neutrinos and background in the fiducial volume. We obtain the effective neutrino fluxes relative to the solar neutrino with 55-day 51 Cr source for both cases as Considering both cases, we simulate the signal and background as shown in figure 6. For solar neutrino, the event contains the νMM contribution from ν e , ν µ and ν τ . However, the source can only contribute the ν e MM. Therefore, the artificial neutrino source signal can significantly break the structure of ν e , ν µ and ν τ magnetic moment in solar neutrinos, which has been presented in [24,49], when combined with solar neutrino signal. In this case, the probing of ν e MM will be robuster than the others. In figure 6, 51 Cr shows a outstanding "shoulder" structure between 500 and 600 keV, where the flux of 51 Cr could be counted precisely. Moreover, the amount of ν e MM shown as dash line in blue color from 51 Cr source is almost equal to the summation of ν e MM, ν µ MM and ν τ MM shown as dash line in fuchsia color from solar neutrinos for both cases. That is to say, the sensitivity to ν e MM should be better than other neutrino flavor magnetic moment. The ROI of νMM with 51 Cr source is the same as the solar-only case in figure 1.

Sensitivity
We build a similar χ 2 function as eq. (2.5) to study µ νe , µ νµ , µ ντ separately. We find that T e − R (visible energy and event vertex by R) two dimensional fit is almost equal T e − R − T (visible energy, event vertex by R and time) three dimensional fit. However, T e (visible energy) one dimensional fit gets much worse sensitivity than others. Therefore, we adopt the 2 dimensional χ 2 function in the following study as where i is the i th T e bin from 150 to 1500 keV, j is the j th reconstructed R bin. Compared with section 2, the fitter has an extra parameter, the flux of 51 Cr ν e . We bound it within 1% in the following analyses. And other parameters are consistent with the standard case in section 2. Figure 7 presents the individual results about νMM with different flavors. Obviously, µ νe gets the most stringent constraint among all neutrino flavors due to the strong ν e source.  When assuming the inside case, µ νe gets more tightly bounded than the outside case on account of more electron recoil event from the source. However, the bounds on µ ντ and µ ντ from solar neutrinos become a little weaker because 51 Cr ν e affects as a kind of background for µ ντ and µ ντ . With 55-day data taking, we obtain the results with 90% C.L. upper limits shown in table 2. If we reduce the systematic uncertainties (pp and 85 Kr) and the intrinsic background ( 14 C and 85 Kr) and enrich the strength of 51 Cr source, we will obtain better results.

Combined analyses with 55-day artificial neutrino source and 10-year solar neutrinos
10-year data taking of solar neutrinos could give a stringent bound on µ ef f ν with the combination of µ νe , µ νµ and µ ντ . And 55-day data taking of 51 Cr could make a tighter limit on µ νe . To reach a better sensitivity, we combine them with 10-year solar neutrinos and 55-day 51 Cr source neutrinos. We define a mixing of µ νµ and µ ντ , i.e. µ ef f νµτ , which presents the mixing of ν µ and ν τ in solar neutrinos. An approximate mixing µ ef f  We find the sensitivity of µ ef f ν can reach 1.2 × 10 −11 µ B level at 90% C.L. with 10year solar neutrino data taking at Jinping, which can validate the νMM hypothesis in Xenon1T by more than 5 σ. A 4% bound on 85 Kr or 3% energy resolution could improve the sensitivity to 1.0 × 10 −11 µ B . The HZ model flux bound on pp and 7 Be could lead to 0.8 × 10 −11 µ B at 90% C.L.. We find that more than 10,000-fold reduction of 14  In the end, we present the current results of νMM and the sensitivities of both solar neutrino and artificial neutrino sources at Jinping in figure 9. The left segment of figure 9 shows µ ef f ν from this work compared to terrestrial experiments: Borexino [24], Xenon1T [25] and PandaX-II [26], also with the astrophysical observations: cooling of globular clusters [29], white dwarfs [30] and red clump stars [50]. The right segment presents the electronic neutrino magnetic moment from this work with 51 Cr source compared to Gemma experiment [23]. Jinping could validate the νMM that is suggested by Xenon1T in the future. It could reach the region of astrophysical interest by reducing the systematic uncertainties and the intrinsic background or by enriching the strength of 51 Cr source. During preparing this paper, we notice an independent and similar study [arXiv:2103.11771] is also proposed and studied by Z. Ye et al. simultaneously.
where N e (R) is the total electron number density as a function of R in the shell, ψ(R, t, E i ) is the neutrino flux as a function of radius R, time t and the i th neutrino branch with different monoenergetic E i . This formula can also be used for the light sterile neutrino study such as ref. [51]. In our case, there is almost no neutrino oscillation. Therefore, it can reduce to where, N e (R) = S(R)ρ LS ρ e N A with the area of the shell S(R) and ψ(R, t, is the fraction of the i th branch and R51 Cr (t) is the decay rate initialized with 3 MCi. The area of the shell S(R) is expressed by S(R) = 2π 1 − is the radius of the fiducial volume sphere and x (1 m) is the shortest distance from the source to the edge of the fiducial volume. So far, we obtain the total event number as a function of time t and T e with the integral by R from x to x + 2R 0 Te − 1 Eν . Moreover, remove the SM and ν e MM contributions from the event number, resulting in where µ ef f ντ can be split into µ νµ and µ ντ . In this study, 7 Be (862 keV) dominates the proportions of µ νµ and µ ντ in µ ef f νµτ at ROI, the yellow band in figure 1. Therefore, (B.2) can reduce to P Be eα approximates P eα (r = 0.06R , E = 862 keV), the probability from the densest point of 7 Be production in the sun to the earth. Therefore, we get µ ef f νµτ 2 0.49µ 2 νµ + 0.51µ 2 ντ with the neutrino oscillation parameters in ref. [41].