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Lunar neutrinos

Cosmic rays bombard the lunar surface producing mesons, which attenuate inside the regolith. They get slower and decay weakly into mostly sub-GeV neutrinos leaving the surface. Thus the Moon shines in neutrinos. Here we calculate spectra of low energy neutrinos, which exhibit bright features potentially recognisable above isotropic neutrino background in the direction towards the Moon. Their observation, though a very challenging task for future neutrino large volume experiments, would make the Moon the nearest astrophysical source for which the concept of multimessenger astronomy works truly. Remarkably, some features of the lunar neutrino flux are sensitive to the surface mass density of the Moon.


Introduction
So far we have observed only a single extra-terrestrial permanent neutrino source, that is the Sun. Neutrinos can emerge upon hadron production in astrophysical sources, see e.g. [1,2], however the powerful ones are at quite large distances, so the individual positional identification requires an angular resolution far beyond the capabilities of the operating neutrino telescopes. Naturally, on this way it is tempting to start with a close source with obvious production mechanism to have both theoretical prediction robust and positional identification reliable.
A promising source of this kind is the Moon. Cosmic rays (CR) hit its surface freely, provided the absence of atmosphere. They initiate hadronic cascades developing inside the regolith, and numerous mesons weakly decay producing neutrinos. The cosmic ray spectrum degrades with energy and the scattering off regolith slows down the mesons, so the largest neutrino flux is expected from stopped mesons and muons. Hence the Moon becomes a source of neutrinos in sub-GeV energy range to be observed on the sky by neutrino telescopes. Semianalytic calculations of high energy (above 10 GeV) neutrino energy part of the spectrum were performed in the study [3] (see also [4,5] for earlier studies). It showed a suppression of the lunar neutrino flux as compared to the atmospheric one calculated over the Moon's solid angle by factor of 10 −2 − 10 −4 depending on neutrino energy. Remarkably, both lunar and atmospheric neutrinos have the same origin in cosmic rays.
In this letter we calculate the low energy part (from 10 MeV to about 10 GeV) of lunar neutrino spectrum and compare the lunar neutrino flux at the Earth with isotropic background from atmospheric and supernova neutrinos. We numerically simulate interactions of cosmic rays with the regolith and count neutrinos produced in meson and muon decays. We observe that although the ratio of total lunar to atmospheric neutrino fluxes in the low energy range is close to unity (contrary to the more energetic part) their spectra are quite different. This observation makes it potentially possible to distinguish between neutrinos of different origin, although the small Moon's solid angle with respect to the whole sky (∼ 5 × 10 −6 ) makes the task of lunar neutrino detection a great challenge even for near future neutrino experiments at the Earth.
Note that previous studies of the impact of cosmic ray bombarding the Moon include in particular simulations of the production of cosmogenic nuclides [6] and gamma-ray albedo [7]. Induced by the cosmic rays, γ-ray emission from the Moon was measured by FERMI [8,9]. A Moon shadow in high energy cosmic rays was measured in numerous experiments, most recently by ANTARES [10,11] and IceCube [12,13]. Possible implications of interactions of astrophysical neutrinos in the Moon were discussed in [14].

Simulation details and results
Absence of atmosphere makes the Moon a very effective cosmic ray dump. Namely, dominant sources of neutrinos, i.e. pions and kaons, which emerged in collisions of cosmic rays with the Moon soil, stop before decay, hence producing monoenergetic neutrinos.
To simulate neutrino flux from the Moon we exploit GEANT toolkit [15]. We involve the cosmic ray spectra dominated by protons and 4 He components. We fit the corresponding plot from PDG [16] by a power-law function, see Fig. 1  He measured fluxes (the latter is scaled by 10 −2 factor as in PDG) and numerical fits to them used in our analysis.
where E kin is kinetic energy of the nuclei, m p and m He are masses of proton and 4 He nuclei respectively. In simulations we set the threshold kinetic energy E th kin to 0.4 GeV for protons and to 0.8 GeV for 4 He. We checked that cosmic rays of lower energies produce rather small amount of neutrinos.
The Moon chemical composition is taken as shown in Table 1 Table 1: Chemical composition of lunar soil, adopted from Ref. [17].
compositions of Maria and Highlands parts of the Moon surface, see e.g. [17]. For the lunar soil density we take ρ ≈ 1.5 g/cm 3 which corresponds to an average density of the upper layer of regolith. We assume the isotropic arrival directions of the cosmic rays. We track the secondary particles produced by the collisions of cosmic rays with the lunar surface using GEANT (we exploit FTFP BERT physics model). Within the simulation we count all electron and muon neutrinos and antineutrinos leaving the Moon. Neutrino energy allows for constructing spectra. The fluxes of τ -neutrinos and antineutrinos at production are negligible as compared to those emerged due to oscillations, which we discuss below.
In Fig. 2 we show the obtained spectra of electron (left panel) and muon (right panel) neutrinos and antineutrinos N ν (E ν ) normalized to a single cosmic ray hitting the Moon; we simulated about 1.5 × 10 7 CR events and resulting neutrino events are distributed over 320 uniform logarithmic energy bins. No oscillations are taken into account at this stage. Most part of the neutrinos come from charged pion and kaon decays as well as from decays of secondary muons. Initially, pions π + and π − are produced in close fractions. They loose their energies in the course of elastic collisions with nuclei, interact and stop in the lunar soil. Most part of π − get captured by nuclei via Coulomb attraction [18] and do not produce neutrinos within discussed energy interval 10 MeV -10 GeV. On the other hand, stopped π + produce a monochromatic line of ν µ at energy E ν ≈ 29.8 MeV and anitmuons µ + which also stop in media and undergo the decay µ + → e + ν eνµ yielding neutrinos of energies below the threshold at E ν ≈ 52.8 MeV. Similar picture is valid for charged kaons, which produce a monochromatic ν µ line at E ν ≈ 235.6 MeV. On the right panel of Fig. 2 two peaks at E ν ∼ 30 MeV and 236 MeV in the red histogram correspond to those monochromatic neutrinos. The widths of 30 MeV and 236 MeV lines on this plot are chosen to be equal to the bin width. Their actual widths are considerably narrower, and hence the actual heights are considerably higher. We clarify their contribution numerically in due course. There is also a small bump at energies somewhat below the muon mass, E ν ≈ 100 MeV. It appears from the processes of muon capture by nuclei with subsequent conversion to neutrino, i.e. µ − + p → n + ν µ , see e.g. [19]. Spectra ofν µ and ν e at E ν < ∼ 53 MeV correspond to neutrinos from decays of antimuons µ + , most of which decay at rest. Note in passing that neutrinos from stopped pions and kaons were studied previously in different contexts (and in particular in searches for a signal from dark matter annihilations in the Sun) in Refs. [20][21][22][23][24].
Now we turn to calculation of lunar neutrino flux at the Earth. The lunar regolith emits neutrinos in all directions 1 . Introducing total integrated over energy isotropic flux Φ CR of cosmic rays used in the simulations we can find the number of cosmic ray particles bombarding the Moon per second where R Moon is the Moon radius. Then, a neutrino ν α flux at the Earth can be written as where L Moon is an average distance from the Earth to the Moon. Let us note that given Moon's orbit perigee about 362600 km and apogee about 405400 km one expects monthly variations of the lunar neutrino flux Φ να with the amplitude about 12%. In Figs. 3 and 4 we present differential neutrino fluxes multiplied by energy squared, i.e. E 2 ν dΦν dEν (in units of m −2 s −1 GeV), for muon and electron (anti)neutrinos (red lines) in comparison with atmospheric neutrino flux (blue lines) taken from [25,26] and diffuse supernova neutrino background (DSNB) flux (dashed black lines) taken from [27] in the direction towards the Moon, which are obtained 2 with multiplying the overall neutrino fluxes (no oscillations) by a factor equal to the fraction of the celestial sphere occupied by the Moon, ∆Ω 4π = πR 2 Moon 4πL 2 Moon ≈ 5 × 10 −6 . Comparing, for instance, lunar and atmospheric neutrinos in the energy interval from 10 MeV to 1 GeV we find that ratio of their energy integrated fluxes within the same solid angle is close to unity (about 1.4-1.5). At the same time, shapes of their energy spectra are drastically different due to prominent features related to peculiarities of neutrino production in the Moon. In particular, for an idealised neutrino detector having energy resolution of 10% the fluxes of   As we discussed above interactions of parent mesons and muons in the media have a great impact on spectra of neutrinos produced in collisions of cosmic rays with the Moon surface. To illustrate this point we perform the same numerical simulation taking somewhat larger values of the regolith density, ρ = 1.95 g/cm 3 . In Fig. 5 we show the ratios of neutrino fluxes with densities 1.5 and 1.95 g/cm 3 , respectively, for all neutrino flavors at production. We see the change of density has the most prominent impact on ν µ flux. In particular, in the matter of lower   Figure 5: Ratio of lunar neutrino flux at production calculated for different regolith densities, 1.5 and 1.95 g/cm 3 , respectively.
density the flux of ν µ is higher away from peaks at 29.7 MeV and 236 MeV. This corresponds to an increase of probability for mesons π + and K + (and subsequent µ + ) to decay in flight. At the same time, ν µ flux at the resonant energies, corresponding to decays of mesons at rest, decreases. Also we observe an increase of ν µ flux at the region around 100 MeV where these neutrinos appear from µ − − ν µ conversion on nuclei. Fluxes ofν e are also higher at energies corresponding to neutrino production from stopped muons µ − . It is related to lower probability for π − to be captured by nuclei in less dense media. At the same time fluxes of ν e andν µ which come from decays of µ + at this energy range are almost unchanged. Note that we omit the part of the plot at larger energies where small statistics do not allow to see clearly the density impact. It is worth to note that most neutrinos in the interesting energy range are produced in the surface upper layer of < ∼ 1 m depth. One can also expect dependence of lunar neutrino flux as on the density as well as on the composition of the regolith. Hence, the measurement of the neutrino spectra provides with a tool to investigate the lunar surface density. Now let us study impact of neutrino oscillations on lunar neutrinos. As most of them are produced in the surface region of the Moon, the corresponding neutrino oscillation probabilities should be averaged over the production point. The oscillation lengths for neutrinos of 10 MeV -1 GeV is considerably smaller than the baseline which is about L Moon and for lower part of this energy range is smaller than even radius of the Moon. This results in almost incoherent neutrino flavour transitions for the softest part of lunar neutrino spectra. However, matter effects in the Moon and the Earth may be important 3 . To model neutrino oscillation in the Moon we use profile of lunar density from Ref. [28] and we adopt PREM model [29] for the Earth structure to describe neutrino oscillations in the Earth. We numerically solve the Schrodinger equation for neutrino wave-function. For the PMNS matrix elements we take the best fit to neutrino oscillation experiments from Ref. [30] for normal neutrino mass hierarchy as an example. and the Earth. In the latter case we consider neutrino propagating through the center of the Earth, which implies in a real experiment that we account for the neutrino signal only when the Moon is in Nadir. In all three cases we average over the production point on the Moon surface and took L M oon = 384000 km as a distance between the Earth and the Moon. We checked that variation of L M oon due to ellipticity of the Moon orbit results only in minor corrections to the oscillation probabilities. Also we average over neutrino energy inside each of 320 energy bins. Second, in Fig. 7 we show dependence of oscillation probabilities for electron (anti)neutrinos on zenith angle θ M of the Moon (the case cos θ M > 0 corresponds to the Moon above the horizon, and so no matter effect in the Earth). In Fig. 8 we show the same probabilities but to muon (anti)neutrinos. Impacts from matter effect in the Moon and the Earth results in deviations from averaged neutrino oscillation probabilities and are clearly visible in those Figures. However, see Sec. 3, to make these studies feasible, one needs to collect a sufficient amount of lunar neutrinos from each position of the Moon, which implies unrealistically large operation time of experiment. In this case one may sum lunar neutrinos along the Moon trajectories, which average the matter effect in the Earth, but keep intact the matter effect in the Moon. In Fig. 9 we show resulting spectra of lunar neutrinos and antineutrinos at the Earth calculated for two cases: 1) assuming Moon to be above the horizon, which implies no matter effect in the Earth; 2) averaging over position of the Moon on the full sky. In realistic experimental setup additional averaging of oscillations comes from finite energy and directional resolution. Detailed study of this question is beyond the scope of this paper and generically depends on the experimental parameters. We will comment on that in the next Section. However, already from the plots of Fig. 9 one concludes that though the difference is recognisable by eye, in a realistic experiment the matter effect in the Earth has no observable effect at all. Let us note in passing that although here we study the oscillations numerically, one can use an analytical approach to describe evolution of low energy neutrinos, see e.g. [31,32]. Matter effect of neutrino propagation in the Moon may be potentially used to obtain an information about its inner structure similar to the studies for the Earth, see e.g. Refs. [33][34][35].

Discussion and conclusions
In the letter we calculated flux of neutrinos originated from cosmic rays hitting the Moon surface.
Apposed to the earlier study [3] we concentrated on low energy ( < ∼ 10 GeV) part of neutrino energy spectrum. Although lunar neutrinos have the same origin as atmospheric ones, absence of atmosphere at the Moon makes spectrum of the former to be quite peculiar and sensitive to the mass density of the surface layer. In particular, it appears to be shifted to lower energies where neutrinos originate from decays of mesons stopped in the regolith. Without neutrino oscillations the very low energy part (less than about 52.8 MeV) of the lunar neutrino spectrum at the Earth exceeds the atmospheric neutrino flux from the Moon's solid angle by up to an order of magnitude for ν e andν µ while the spectrum of ν µ exhibits two narrow peaks at energies E ν ≈ 29.8 MeV and 235.6 MeV corresponding to neutrinos from π + and K + decays. Oscillations of neutrinos mix those effects between all neutrino flavours. At energies lower than about 20-50 MeV the lunar neutrino flux is comparable with that of DSNB within Moon's solid angle. Contrary to atmospheric and diffuse neutrinos the lunar neutrino flux is expected to endue a periodic dependence with amplitude about 12% from small ellipticity of the Moon orbit. These  peculiarities, including explicit binding to the direction of the Moon, can be used to search for lunar neutrinos in future neutrino experiments and to distinguish them from atmospheric neutrino and diffuse supernova neutrino background. At present, neutrino detectors used or planned for study of sub-GeV neutrinos include water Cerenkov (WC) detectors such as Hyper-Kamiokande [36,37] and its predecessor Super-Kamiokande, liquid scintillator (LS) and liquid argon time projection chambers (LArTPC). The latter representative examples are JUNO [38] and DUNE [39], respectively. WC detectors can be more easily realized in large volume which is crucial requirement for detection of the pretty low neutrino flux from the Moon. However, in neutrino-nucleon scattering correlation between direction of charged lepton and incoming neutrino is weak for neutrinos from sub-GeV energy range. Moreover, for such neutrino energies an additional background in WC detectors comes from invisible muons. LS and LArTPC detectors are free of this type of the background. Still, typical detection channels used in LS detectors do not provide very large information about neutrino direction in the considered energy range. LArTPC detectors on the other hand can provide not only energy but also a directional information (see e.g. [24]) as for them it is possible to reconstruct positions of charged lepton and nucleon [40]. Still, even if very large exposures can be realized here, at present an angular resolution of these detectors is still quite poor as compared to the angular size of the Moon on the sky. We conclude that detection of lunar neutrinos would require not large but huge neutrino detectors with (so far) exceptional energy and, especially, angular resolution -the task which may one day be feasible.