Developing the MeV Potential of DUNE: Detailed Considerations of Muon-Induced Spallation Backgrounds and More

The Deep Underground Neutrino Experiment (DUNE) could be revolutionary for MeV neutrino astrophysics, due to its huge detector volume, unique event reconstruction capabilities, and excellent sensitivity to the $\nu_e$ flavor. However, its backgrounds are not yet known. A major background is expected due to muon spallation of argon, which produces unstable isotopes that later beta decay. We present the first comprehensive study of MeV spallation backgrounds in argon, detailing isotope production mechanisms and decay properties, analyzing beta energy and time distributions, and proposing experimental cuts. We show that above a nominal detection threshold of 5 MeV, the most important backgrounds are --- surprisingly --- due to low-A isotopes, such as Li, Be, and B, even though high-A isotopes near argon are abundantly produced. We show that spallation backgrounds can be powerfully rejected by simple cuts, with clear paths for improvements. We compare these background rates to rates of possible MeV astrophysical neutrino signals in DUNE, including solar neutrinos (detailed in a companion paper arXiv:1808.08232), supernova burst neutrinos, and the diffuse supernova neutrino background. Further, to aid trigger strategies, we quantify the rates of single and multiple MeV events due to spallation, radiogenic neutron capture, and other backgrounds, including through pileup. Our overall conclusion is that DUNE has high potential for MeV neutrino astrophysics, but reaching this potential requires new experimental initiatives.


I. INTRODUCTION
Astrophysical neutrinos are uniquely penetrating probes of their sources, whose extreme physical conditions in turn allow new tests of neutrino properties. In the MeV energy range, there are three important targets: solar neutrinos [1][2][3][4], supernova burst neutrinos [5][6][7][8][9], and the diffuse supernova neutrino background (DSNB) [10][11][12]. Despite great achievements in solar neutrino studies, opportunities remain for detailed tests of astrophysics (e.g., the first detection of the hep flux) and particle physics (e.g., resolving the discrepancy between reactor and solar mixing parameters). The next Galactic core-collapse supernova will enable multi-flavor neutrino measurements, revealing details of explosion physics and testing neutrino mixing at high densities. Meanwhile, the DSNB could be detected as a steady source, which would probe the core-collapse history and test black-hole versus neutron-star formation. With new experiments, exciting progress on these and other topics could be made. DUNE, the leading next-generation neutrino experiment in the United States [13][14][15][16][17][18][19], offers such opportunities. Its principal science goals are to measure CP violation and the mass ordering, to search for nucleon decay, and to detect the next galactic supernova burst. For MeV neutrino astrophysics, DUNE has three key advantages.
First, its far detector will be huge, 20 kton in total for two modules (eventually twice that) of liquid argon (LAr), comparable to the current largest MeV neutrino detector, Super-Kamiokande. Second, with the Liquid Argon Time-Projection Chamber (LArTPC) technology, DUNE will have excellent capabilities in event reconstruction, enabling the separation of different types of events (e.g., electrons vs. gammas). Third, the charged-current detection channel in DUNE (ν e + 40 Ar → e − + 40 K * ) isolates the ν e flavor. Compared to elastic scattering (ν e,µ,τ + e − → ν e,µ,τ + e − ), the main ν e detection channel in current experiments, the charged-current channel has a much larger cross section and a much sharper correlation between neutrino energy and electron energy. Therefore, new results from DUNE should powerfully complement results from previous and ongoing experiments.
Understanding the detector backgrounds is an essential step for successful MeV neutrino programs in DUNE. Above 5 MeV, the nominal detection threshold, a significant background rate is expected from muon-induced spallation. (Another, from radiative neutron captures in the detector due to neutrons produced by radioactivities in the rock, given some consideration below, is detailed in Ref. [1].) When cosmic-ray muons pass through the detector, they produce secondary particles, which then occasionally break argon nuclei and make other isotopes. Unstable daughter isotopes decay later, emitting betas, which can mimic neutrino signals. Prior spallation studies on LAr were incomplete. Barker et al. [20] focused on only high-A isotopes (near argon) and Gehman et al. [21] considered only isotopes produced by high-energy neutrons. More recently, Franco et al. [22] produced a thorough study of spallation backgrounds for low-energy (≤ 1.3 MeV) solar neutrinos, but did not provide the details needed for the higher energies (5-20 MeV) of DUNE.
Building on the work of Li and Beacom [23][24][25], our goals for this paper are to calculate the spallation backgrounds for DUNE in detail, understand their physical mechanisms, and use this understanding to develop cuts to reject backgrounds. We compare these background rates to the signal rates for solar neutrinos, supernova burst neutrinos, and the DSNB, finding that spallation backgrounds can be well controlled. We aim for a factor of ∼ 2 precision on isotope yields, which is appropriate given the hadronic uncertainties. This is adequate to guide development of DUNE as a detector for MeV neutrino astrophysics. Once there are measurements, which could begin soon with surface-based detectors, the uncertainties can be reduced via empirical calibrations.
This paper is organized as follows.
In Sec. II, we overview the physics of spallation, using Super-Kamiokande as a concrete example. Our main results are in Sec. III, where we calculate the isotope yields, describe the production mechanisms, show the component background energy spectra and time distributions, and detail our proposed background-rejection methods. Using the projected post-cut background levels, we discuss the possible MeV neutrino programs in DUNE in Sec. IV, along with new results to aid trigger development. Finally, we present our conclusions in Sec. V.

II. OVERVIEW OF SPALLATION
In this section, we review the physics of spallation isotope production by cosmic-ray muons, which is now understood, due to Refs. [23][24][25]. Though those papers focus on the water-based detector Super-Kamiokande, the results can be widely applied with modifications, including to the argon-based DUNE. The most important concepts are as follows: • Almost all isotopes are made by muon secondaries, not directly by muons. (This point was known earlier [26,27].) • Almost all of these secondaries are made in showers, which are relatively rare along muon tracks.
• Almost all the isotope-producing secondaries are made in hadronic showers, which are even rarer. ( 11 C, a dominant background isotope in oil, is made in electromagnetic showers.) • The positions of decaying isotopes produced by spallation can be constrained by localizing the preceding showers. (This point was found empirically in Ref. [28], though the physical reason was unknown.) One could go further by identifying the showers that are hadronic.
We now detail spallation processes, beginning with cosmic-ray muon energy loss [23][24][25]29]. Muons lose energy in two ways: ionization when interacting with atomic electrons, and radiative processes when interacting with atomic nuclei. The ionization losses have a typical value of ∼ 2 MeV g −1 cm 2 , moderately depending on muon energy and the medium material. These losses can be separated into a restricted energy loss from soft collisions and delta-ray production from hard collisions, where delta rays can be almost as energetic as the parent muon. Radiative losses produce most secondary particles, and the rate rises with muon energy. For muons up to several hundred GeV, the dominant radiative processes are pair production and bremsstrahlung. Photonuclear interactions, a low-Q 2 analog to deep inelastic scattering, are less frequent. The first-generation particle production is closely associated with muon energy loss. The most abundantly produced particles are electrons from delta-ray production, followed by electrons and positrons from pair production. There are also some gamma rays, made mostly through bremsstrahlung. Even though muons mainly lose energy electromagnetically, a small number of hadrons are made through photonuclear interactions. The dominant hadrons are pions, which are almost equally distributed among the three charges. In Super-Kamiokande, there are 3.6, 0.4, 0.04, and 0.003 daughter particles above 0.1, 1, 10, and 100 GeV per vertical muon (of track length 32.2 m) [24], showing that shower frequencies are small.
Isotopes are born primarily in showers induced by secondaries. There are two types of showers. Electrons, positrons, and gammas make electromagnetic showers, which typically have no hadronic components, aside from some low-energy neutrons made through (γ, n) interactions. Pions induce hadronic showers, producing roughly equal numbers of π + , π − , π 0 in each generation, in analogy to e + , e − , γ for electromagnetic showers. Hadronic showers always have large electromagnetic components, because neutral pions decay promptly to gammas. For Super-Kamiokande solar neutrino analyses, the most dangerous background isotopes are produced in the less frequent hadronic showers, where pions and neutrons are very efficient at making isotopes, due to the strong interactions. There are also many unstable isotopes made from the more frequent electromagnetic showers, but those tend to have harmless decays, e.g., 15 O, produced by (γ, n), decays by electron capture. Overall, isotope production in Super-Kamiokande is a rare process, with the most abundantly produced background isotope, 16 N, having a yield of 0.006 per muon [23].
A small fraction (e.g., ∼ 7% in Super-Kamiokande [23]) of isotopes are made by stopping muons. Once µ + are brought to rest, they simply decay. However, once µ − are brought to rest, nearly all undergo atomic capture, an electromagnetic process in which electrons are ejected, because muons are bound more tightly by a factor ∼ m µ /m e [30][31][32]. Of muons in atomic orbits, an appreciable fraction undergo nuclear capture, a weak pro-cess that converts µ − +p → ν µ +n, often removing several low-energy nucleons from the nucleus [30][31][32]. Because stopping muons enter the detector with only a few GeV, they have vanishing radiative losses and hence do not produce isotopes along their tracks. Therefore, powerful cuts on isotopes produced by stopping muons can be made by concentrating cuts at the ends of their tracks.
The precision of predicting isotope yields, which is mostly limited by the uncertainties in hadronic processes, is typically a factor of ∼ 2. For example, in Super-Kamiokande [33], the FLUKA-predicted yields of some isotopes agree with measurements within a few tens of percent; some are off by a factor ∼ 2-3. In Borexino [34], FLUKA predictions also agree well with experimental measurements. A few tens of percent agreement is found for some isotopes, but a factor of 2-4 for some others. As for the predicted yields from GEANT4, a factor of ∼ 2 agreement with data is observed for some isotopes, while a few differ by a factor of ∼ 10. Overall, a factor of ∼ 2 precision is adequate as isotope yields usually differ by orders of magnitude. Because the decay time profiles and energy spectra are known from laboratory data, all that needed is the yield constants. Theory is needed to get the predicted yields right enough to identify the key physical processes and to develop cuts, and then these predictions can be refined with experimental measurements.
There are multiple ways to cut the background betas from unstable-isotope decays. The basic concepts of spallation cuts can be explained in a simplified picture. While neutrino signals are uniform in the detector, spallation backgrounds are highly correlated with muons. One strategy is a cylinder cut, where one discards all events inside a cylinder of radius R surrounding each muon track in a time window of duration ∆t. The values of R and ∆t are chosen such that both cut efficiency and the resulting deadtime are acceptable. For example, in Super-Kamiokande solar neutrino analyses, 90% of the isotopes can be rejected with 20% deadtime via a likelihood-based version of this approach. Separately, one can also cut on shower energy, because isotope yields rise with increasing muon energy losses. In Super-Kamiokande, about 60% of the spallation yields could be cut by rejecting the 2% of muons with the highest energy losses [24]. Finally, more advanced cuts use reconstructions of the shower profiles [25,28]. Instead of cutting a whole cylinder for each muon, one could only cut where isotopes are made -the rare shower regions. This technique is under development for solar neutrino detection in Super-Kamiokande.

III. SPALLATION BACKGROUNDS IN DUNE
We calculate muon-induced spallation backgrounds for MeV astrophysics studies in DUNE. Under some reasonable assumptions about the detector properties, we present our simulation inputs, the calculated isotope production rates, and the component background spectra.
We show that the spallation backgrounds are low after a simple two-step cut we propose, and could be improved.

A. Basic Facts of DUNE
Located 4850 ft (4300 m.w.e.) underground in the Homestake mine, the DUNE far detector will have two 10-kton (fiducial) modules of LArTPC deployed by 2024 [13], and two more modules later.
The LArTPC technique that DUNE will use is superb in tracking and calorimetry performance [13][14][15][16][17][18][19]35]. Charged particles cause ionization and excitation of argon atoms. The ionization electrons then drift to wire planes at a speed of 1.6 mm µs −1 under an applied electric field of 500 V cm −1 and form a 2-D particle track [16]. Combined with the timing information from prompt scintillation light emitted by argon excimer states, one can reconstruct the 3-D image.
For all three astrophysical sources we consider, the neutrinos either elastically scatter off electrons or have charged-current interactions with argon nuclei. In the elastic-scattering channel, the final state is an electron.
In the charged-current channel, there would be one electron plus multiple gammas from 40 K * de-excitations. These gammas do not typically overlap with the outgoing electron in space, because the 14-cm radiation length [29] is much larger than the position resolution in DUNE ( 0.5 cm [16]). The ability to detect these gamma rays has recently been demonstrated for ArgoNeuT [36]. A precise gamma-ray energy reconstruction would aid neutrino energy reconstruction, and help signal and background separation. To be conservative, we assume no ability to separate charged-current events and elasticscattering events, and take the signal for both channels to be one outgoing electron, following Ref. [1].
Given such neutrino signals, we take the backgrounds to be just the betas from spallation isotope decays. We especially focus on the energy range above 5 MeV, which could be a reasonable choice (see Ref. [16]), although the energy threshold could vary in different analysis programs. We also remark on the background rates below 5 MeV to help with trigger design. In addition, we expect good energy reconstructions for the spallation betas, because their energies ( 20 MeV) are well below the electron critical energy of 45 MeV [37], and hence radiative losses are minimal [38]. When smearing the background spectra in Sec. III E, we use a 7% energy resolution [16], of which the specific value has little effect on the continuum spectra, except for the tails. In the Appendix, we show results for 20% energy resolution.

B. Setup of the Calculation
We use the Monte Carlo code FLUKA [39,40] (version 2011.2x-1) to simulate muon propagation in liquid argon. FLUKA is a well-known package for simulating par-ticle transport and interactions in matter on an eventby-event basis. For hadron-nucleus interactions, FLUKA uses its own PEANUT model [41,42]. It describes target nuclei with a local Fermi gas model, and hadronic inelastic interactions in a Generalized IntraNuclear Cascade (GINC) approach, where the cross sections used are a mixture of tabulated data and parametrized fits. The GINC step continues until all nucleons are below 30-100 MeV and all non-nucleons (typically pions) are decayed or absorbed. Then the preequilibrium stage takes over, which is mostly based on Geometry Dependent Hybrid Model. At the end of the preequilibrium stage, a compound nucleus (Z, N) with known momentum and excitation energy is left, starting from which the evaporation/fission/fragmentation stage is modeled. When the excitation energy of the residual nucleus is below the particle emission threshold, the remaining energy is released through gamma emission.
In our simulations, the PRECISIOn card is used. All relevant electromagnetic processes and hadronic processes are included, such as ionization, bremsstrahlung, pair production, photonuclear, Compton scattering, pion production and transport, photo-disintegration, low-energy neutron interactions, etc. We switch on EVAPORAT and COALESCE through the PHYSICS cards to enable accurate residual nuclei scoring. When we calculate isotope production yields, the RADDECAY card is switched off to make sure that only the isotopes produced by muon spallation are counted, i.e., not including daughter nuclei from spallation isotope decays (such as 39 Ar from 39 Cl decay). It is later switched on when we simulate radioactive decays.
The first main input is the detector setup. Each module is a box of liquid argon (active volume), with dimensions 58 m (l) × 14.5 m (w) × 12 m (h). To match the 10kton fiducial mass, for which our results are calculated, we assume that the fiducial volume has dimensions 56 m (l) × 12.5 m (w) × 10 m (h). The detector material is natural argon, consisting of 99.6% 40 Ar, 0.3% 36 Ar, and 0.1% 38 Ar. Chemical impurities (water, air, etc.) and radioactive impurities ( 39 Ar, 42 Ar) have tiny abundances, so we ignore them as possible targets for muons. Outside of the active volume, we include 2 m of rock to induce full showers but not significantly affect the muon spectrum. The rock chemical composition follows that given in Ref. [43]. In reality, there is ∼ 1 m of LAr cryostat layer outside of the active volume. We have verified that our calculated isotope yields in the fiducial volume are not affected by more than a few tens of percent if we included that extra LAr layer in the simulation. This is expected, because the 2 m of rock has enabled full shower development, and the production of muon secondaries is nearly material-independent. Figure 1 shows the other main input, the simulated cosmic-ray muon spectrum, averaged over zenith angles, at the DUNE underground site, based on the simulations of Kudryavtsev et al. [44][45][46]. In their calculations, the sea-level muon flux follows Gaisser's formula [47], modified for large zenith angles and prompt muon flux with the best fit to the LVD data [48]; muon propagation throughout the rock is then carefully modeled in MUSIC/MUSUN [45,49]. The good agreement with the measured muon flux by the Davis experiment [50] and the Majorana Demonstrator [51] shows that this simulated muon spectrum should be reliable. We plot the spectrum as E dΦ/dE = 2.3 −1 dΦ/d log 10 E, so that the relative heights at different energy decades correctly represent their relative contributions to the total flux of Φ µ = 5.66 × 10 −9 cm −2 s −1 . The muon rate in four modules of DUNE will be 0.2 Hz, roughly 10 times lower than that for Super-Kamiokande. The muons have an average energy of 283 GeV. The minimum kinetic energy required for a muon to vertically pass through DUNE's fiducial volume is 2 GeV, so 2% of the muons would stop in the detector. The muon critical energy is 484 GeV in argon [29]; muons at higher energy dominantly lose energy through radiative processes, and hence are more likely to make showers and produce isotopes.
There are two main simplifications we adopt for the primary muons. One is that we only simulate µ − , because the energy losses and hence isotope production rates of µ + are almost identical. The only difference comes for stopping muons (details in Sec. II), for which we could correct the relevant isotope yields with the expected µ + /µ − ratio of 1.38 [44][45][46], but we choose not to because of negligible differences (details in Sec. III C).
The other simplification is about muon injection. We inject single muons above the rock, vertically downward at the center of the detector. (Muons that miss the detector will be discussed separately.) In reality, muons arrive with a variety of angles and positions, which can be easily measured. Once the muon track is localized, there is no difference in the analysis procedure compared to a vertically throughgoing muon. All that matters is the muon track length, because isotope production is a Poisson process. According to MUSIC/MUSUN [44][45][46], most muons are downward going. They have a mean zenith angle of cos θ z 0.9, resulting an average path length of h/ cos θ z 11 m, very close to the fiducial-volume height (h = 10 m). Besides the single-muon case, there can be two muons appearing in a readout time window ( 5.4 ms [16]), either due to muon bundles from cosmicray showers or an accidental coincidence ( 1 per day per module). If these muons are far from each other, then they can be treated separately. If they are close, such that their possible showers could overlap, then one could apply a wider cylinder cut. The results are not appreciably affected by our simplifications.
With the specified inputs, we expect three main outputs from FLUKA: the isotope yields, the energy spectra of the isotope decay products, and the time and spatial correlations between the decay secondaries and the muons. We record the first from the RESNUCLEi card, which directly returns per-muon isotope yields, and the latter two from a modified mgdraw.f subroutine.
In addition to the RESNUCLEi card, isotope yields can be recorded from subroutine usrrnc.f and mgdraw.f as well. In usrrnc.f, isotopes are identified with the arguments IZ (atomic numbers) and IA (mass numbers). In mgdraw.f, we note that isotopes are divided into two categories, both of which should be accounted for, to get the correct yields. Some isotopes are characterized with the arguments ICRES (residual nucleus atomic number) and IBRES (mass number). Some isotopes, mostly with small mass numbers, are treated as heavy ions, whose charge and mass number are stored in the arguments ICHEAV and IBHEAV. As expected, these three counting methods return the same values of isotope yields.

C. Predicted Isotope Yields
The isotope yields reveal key features of spallation production processes in argon. We find that there are about 100 different isotopes produced in DUNE, but only about 10 isotopes contribute significantly to the background rate above 5 MeV. The underlying physics can be explained by a two-group structure. Below, we summarize our results, highlighting the differences in three characteristics between high-A (Ar, Cl, S, etc.) isotopes and low-A (Li, Be, B, etc.) isotopes, with corresponding details given in Table I. The first difference concerns decay properties. In general, beta-decay Q values and half-lives follow Sargent's rule, t 1/2 ∝ Q −5 [30], although nuclear-structure differences cause deviations. In DUNE, high-A isotopes mostly have small Q values ( 2-3 MeV) and long half-lives (minutes to years). Only a few isotopes from the high-A group can decay to betas above 5 MeV, among which 40 Cl is the one with the largest Q value (7.48 MeV). In contrast, low-A isotopes often have large Q values (∼ 10-20 MeV) and short half-lives (milliseconds to seconds). The largest Q value in DUNE is 21 MeV, for 14 B and 11 Li, which determines the end of the background energy range. We show in Sec. III E that these two isotopes and a few others with large Q values are completely rejected after a 250-ms cut, due to their short half-lives.
The third difference is production yields. High-A isotopes typically have large yields. The most abundantly produced one is 41 Ar, which has a yield of 0.47 per muon, corresponding to 1600 per day in one module. Next is 39 Ar, with a yield of 0.35 per muon. These two together comprise 82% of the unstable isotope yields. In contrast, low-A isotopes are made much more rarely. The highest yield from the low-A group is 1.4×10 −3 per muon ( 5 per day) from 8 Li, while a typical yield is even smaller, of order 10 −4 or 10 −5 per muon. Table I shows the decay properties and production yields of selected isotopes in DUNE. The average production yield of all isotopes is 1.5 per vertical muon, of which 1 per muon are beta-unstable, although the production rate of troublesome isotopes is much lower, as described below. The statistical variation of this average isotope yield is negligible, given the huge number ( 4 × 10 9 ) of primary muons we use in the simulation. In the top block, we list isotopes relevant to the backgrounds above 5 MeV. Its upper part contains the isotopes that make at least 1 background event per year, among which each of the 10 isotopes in bold individually makes more than 1% of the total background. Its lower part contains the isotopes that will be decimated by the 250-ms cut (details below). In the bottom block, we list selected isotopes that either are stable or have small Q values. Overall, we see that most backgrounds above 5 MeV are from low-A isotopes (near oxygen), while abundantly produced high-A isotopes (near argon) are more important for the energy range below 5 MeV.
The production yields shown in Table I are dominantly due to throughgoing muons, with only a small fraction from stopping µ − . Once µ − are stopped in DUNE, 76% of the time they make isotopes though nuclear capture. For most isotopes, the yields from µ − capture are orders of magnitude lower than those due to throughgoing  53]. The yields are from simulation. The conversion factor from the 5th to 6th column is 3423. Yields per muon above 0.01 are quoted with three digits after the decimal; smaller yields are expressed in scientific notation. Top block (upper part): Isotopes that survive from the 250-ms cut (discussed below) and are relevant for the backgrounds above 5 MeV, sorted by Q values. All isotopes that make at least 1 event yr −1 (10 kton) −1 above 5 MeV are listed. Individual isotopes that make more than 1% of the backgrounds are shown in bold. Top block (lower part): All isotopes eliminated by the 250-ms cut (making 1 event (10 yr) −1 (10 kton) − 40 Cl and 38 Cl would differ by ∼ 20% and ∼ 5%, respectively, which is below the precision of our calculation, so we neglect the muon charge ratio correction in our subsequent discussions. We compare our predicted isotope yields with those of previous papers. Franco et al. [22] used FLUKA to simulate the spallation backgrounds in a 100-ton fiducial LArTPC, which is inside a scintillator sphere within a water tank. Their dark-matter-detection-style detector and DUNE have very different strengths. The detector in Ref. [22] has a very low energy threshold, and excellent intrinsic radio-purity. Although this kind of detector drifts charge, the main detection strategy is collecting light. In contrast, DUNE is a "tracking" detector, primarily collecting charge, and the much larger volume ensures much larger statistics, although the energy threshold in DUNE is higher. Nevertheless, the detection material in both detectors is LAr, so we could compare our calculated spallation yields with the 75 isotope decay rates listed in their Table A. For the relative yields, there is good agreement for most isotopes, although several less important isotopes show discrepancies that can be explained. Some of those ( 9 C, 32 Al, 30 S, etc.) have large yield uncertainties in Franco et al.'s simulation, due to their low statistics. Others are extremely long-lived isotopes ( 14 C, 32 Si, 39 Ar, etc.), so the comparison between our production rates and their decay rates is not valid, because these isotopes will not be in equilibrium. For the absolute yields, if we take their muon track length as the detector height, then our yields are a factor of 2-3 higher. Our results should agree when normalized by volume or muon track length, as the muon spectrum is similar, the target is similar (the LAr is the same; while the presence of surrounding material is important, its composition is not), and we both use FLUKA (which can have some changes over time).
To investigate this discrepancy further, we have done our own simulations for the detector setup of Ref. [22], and find yields per volume or muon track length comparable to ours for DUNE, i.e., higher than those of Ref. [22] by the factor of 2-3 noted above. However, we can recover their results if we ignore the surrounding material, i.e., if we start the muons just outside the fiducial volume instead of outside the full detector. The physical reason for the difference would then be that showers are not fully developed. We have discussed this with the lead author of Ref. [22], who agrees this may be a possible explanation for their results being different, and this is being investigated further.
We also compare our isotope yields with previous results that are from a different simulation package. Barker et al. [20] used GEANT4 and listed about 20 high-A isotope production rates in their Table VI. For half of them, including some important ones such as 40 Cl, 38 Cl, 34 P, etc., we find a good agreement, within a factor of ∼ 2. For some rarely produced isotopes ( 37 P, 33 Cl, 35 Ar, etc.), our yields are a factor ∼ 0.1 of theirs. However, even with Barker et al.'s yields, those isotopes would still be unimportant, due to other much more frequently produced isotopes that have similar decay Q values. For 39 Ar and 41 Ar, when we remove the rock layer and count in the active volume, as in Ref. [20], our yields are a factor ∼ 10 of theirs. We note the correlations of their production channels, i.e., 40 Ar(n, γ) 41 Ar, 40 Ar(γ, n) 39 Ar. Nevertheless, these two isotopes are not particularly problematic due to their low Q values. We think it is possible that the nuclear breakup models in FLUKA and GEANT4 differ enough to cause the factor of ∼ 10 differences for certain isotope yields, as has been seen in Ref. [34]. Gehman et al. [21] took muon-induced neutrons coming from the rock as the primary particles, so their yields are subdominant to ours (details in Sec. III E).
In summary, we believe our spallation predictions for LAr to be the most complete and accurate available.

D. Overview of Spallation Spectrum in DUNE
Given the predicted isotope yields and the beta-decay properties of each isotope, we calculate the all-isotope background energy spectra and time distributions from FLUKA. We hold off on applying energy resolution until the next subsection, to provide a useful starting point in case the DUNE energy resolution turns out to be better than assumed below.
Getting the individual beta spectra right is important. For most decays, the spectrum is specified completely by the Q value. For some isotopes, extra care is needed. For example, 36 P has a Q value of 10 MeV, but the probability to decay to the branch with the highest end-point energy (7 MeV) is only 1% [52]. Both 8 Li and 8 B have special spectral shapes, because they decay to the 8 Be 2 + continuum state, which has an excitation energy of 3 MeV and a width of 1.5 MeV [54][55][56]. FLUKA simulates their decays correctly. We find two problems when cross checking the decay spectra from FLUKA with the analytic spectra given the nuclear data from Refs. [52,53]. One is 11 Li, for which the endpoint energy in FLUKA is incorrect. The other is 11 Be, which FLUKA does not simulate. For these, we use analytic spectra.
In addition, some decays are more complex than just beta decays, producing also gamma rays or neutrons. Gammas produce electrons through Compton scattering or pair production, with radiation length 14 cm [29]. Neutrons typically travel many meters and eventually get captured on 40 Ar, making gammas, which then Compton-scatter or pair-produce electrons. Unlike in water or scintillator, here the original beta and the γ (or n) induced electrons from the same decay can be well separated, due to the good position resolution in DUNE ( 0.5 cm). Thus, we only focus on the betas from spallation isotope decays, and ignore the energy deposited by the accompanying gammas or neutrons, if any. spectrum in DUNE, with several components shown individually to explain the breakpoints near 2, 5, 15, and 20 MeV. The rate scales corresponding to these breakpoints are roughly 10 4 , 10 1 , 10 −1 , and 10 −4 , spanning almost 10 orders of magnitude, showing the large variations among the isotope yields. From the individual components, we see that high-A isotopes (near argon) have large yields and dominate the spectrum at low energy, while low-A isotopes (near oxygen) have small yields and dominate the spectrum at high energy. The clear separation of high-A and low-A isotopes shown here makes visual the points made above about Table I. Only 0.2% of the spallation betas appear above 5 MeV, corresponding to 0.002 per muon or 7 events per day in each module. Figure 2 (right panel) shows the spallation isotope decay time distributions relative to the cosmic-ray muons. Because we use a log time axis, we plot the decay profile for each isotope with half-life t 1/2 as t dN/dt = 2.3 −1 dN/d log 10 t ∝ t/t 1/2 × 2 −t/t 1/2 , which shows the number of decays per log time. We see this generic shape for different isotopes, while the decay time scales vary greatly. High-A isotopes are typically long-lived. Even though they are less important for higher energies in DUNE, they could form a steady-state background once production and decay reach equilibrium, which may affect the trigger rate. For example, 41 Ar (Q = 2.5 MeV) saturates after half a day of exposure, resulting a steady decay rate of 1600 events per day in each module of DUNE. Another high-A isotope, 39 Ar (Q = 0.6 MeV), has a similar production rate, but its half-life is extremely long (268 yrs). We expect that 39 Ar made from spallation has a decay rate of only 3 per day after one year of operation, while it would increase to 30 per day once DUNE has run for 10 years, which is still insignificant compared to the rate (∼ 10 MHz) due to the pre-existing 39 Ar in the atmospheric argon that DUNE will use [35]. Low-A isotopes, the dominant source for the backgrounds at higher energies, are typically short-lived. This is a crucial point for successful background cuts.

E. Spallation Backgrounds After Cuts
Even though the exact detector performance and analysis procedures of DUNE in the MeV energy range are not available yet, it would be valuable to estimate how much the backgrounds can be reduced. In this subsection, we propose a two-step spallation cut as an example. We smear the energy spectra with 7% resolution, and choose 5 MeV as the energy threshold (i.e., the aftersmearing background events below 5 MeV are automatically rejected).
The 1st-step cut is a time cut, discarding events with t < 250 ms for each whole module with a muon, assuming an energy threshold of 5 MeV. The time cut mainly rejects those short-lived low-A isotopes that dominate at higher energies. We choose 250 ms as long enough to reject those backgrounds and short enough to not introduce significant deadtime. In addition, this means that time resolution becomes irrelevant as the 250 ms is much longer than the few-ms readout time. Because the muon rate is low ( 0.05 Hz per module), the deadtime from this cut is only 1%. With only the time cut, longlived high-A isotopes that dominate the isotope production yields still exist, as shown in Figure 2 (right panel). However, having a 5-MeV threshold rejects almost all of them, because those high-A isotope decays are dominantly at lower energies, as shown in Figure 2 (left panel). Thus, after the 1st-step cut, one should have a softer energy spectrum and a much smaller background rate. The concepts behind this simple cut proposed here could work well for detectors besides DUNE. Figure 3 (left panel) shows the spallation background energy spectrum after the 1st-step cut. Note that these spectra now have energy resolution included. Above 5 MeV, 48% of the backgrounds are rejected, which is entirely due to the 250-ms cut. Its more important effect is to lower the endpoint of the background spectrum. Precut, the spectrum at high energies ( 18 MeV) is dominated by 11 Li and 14 B, both of which have a Q value of 21 MeV and a half-life of 10 ms. Post-cut, only a fraction ∼ 10 −7 of them remain, corresponding to a negligible decay rate of ∼ 10 −6 per year. Now, the residual backgrounds at high energies are mostly due to 8 Li, with 8 B becoming comparable near the endpoint; at low energies, a few high-A isotopes are important. Even though some of them (such as 38 Cl and 34 P) barely decay to betas above 5 MeV, they could be visible due to large production yields and energy smearing effect. The 2nd-step cut is a cylinder cut for throughgoing muons, discarding events with R < 2.5 m and t < 40 s, and a sphere cut for stopping muons, discarding events with R < 3 m and t < 10 min. As noted, in DUNE it will be easy to determine the muon positions. Through the correlations between the background events and the parent muons, we can cut the backgrounds further. A key variable is the perpendicular distance of the backgrounds to the muon track. Figure 4 shows the cumulative distance distribution of spallation betas relative to the muon track. We find that, on average, 99% of the decays happen within 2.5 m. Isotopes decay nearly at the same places as where they are born, so the distribution shown here also reveals the isotope parent particle (γ, n, π) absorption distances. Neutron behavior in LAr is highly energy dependent. Highenergy neutrons efficiently make isotopes. They typically die within a few meters, as shown in the figure, because of large inelastic hadronic cross sections. Low-energy neutrons ( 10 MeV) do not make isotopes except for 41 Ar. They have to travel much longer distances to lose enough energy through elastic scattering, and eventually get captured via 40 Ar(n, γ) 41 Ar or escape. Once they are captured, the emitted gammas could cause backgrounds (details in Sec. IV D 3).
The measured isotope decay distances relative to the muon track could differ from that shown in Figure 4, due to the potential movements of the isotopes prior to their decays. One cause could be fluid motion in LAr, which is expected to have a speed 3 cm s −1 [18]. In addition, spallation isotope ions could drift under the electric field. Those isotopes could be created in a fully ionized state, but would quickly catch electrons from argon until they are singly or doubly ionized, and drift with a speed typically 1 cm s −1 , about 5 orders of magnitude lower than that of electrons [57]. With such speeds, most background isotopes in Table I would only drift for negligible  distances, 10 cm, before decay. Thus, our proposed cuts below would not be affected at all. Two exceptions are for 40 Cl (t 1/2 = 81 s) and 38 Cl (t 1/2 = 2234 s). Due to their long half-lives, the nominal drift distances would be of order 1 and 10 m, respectively. In principle, one could develop likelihood-based techniques that take into account the isotope drift and the maximum drift distances allowed in the detector to reject these two isotopes. Our simple cuts do little to reject these two long-lived isotopes, which means even though they drift out the cylinder, our post-cut backgrounds would not be appreciably affected. Therefore, the simulated distances shown in Figure 4 should fulfill our purpose.
The cylinder cut is especially useful for rejecting shortlived isotopes. The cut efficiency and resulting detection deadtime can be estimated with the predicted spatial and time distributions of the backgrounds to the parent muons. The remaining background flux after an (R, ∆t) cylinder cut is where i denotes a sum over all background isotopes. For each isotope, P i (D > R) represents the fraction of decays outside the cylinder of radius R. The averaged value over all background isotopes in DUNE can be extracted from Figure 4. Similarly, for each isotope with half-life t 1/2 , P i (t > ∆t) = 2 −∆t/t 1/2 represents the fraction of the decays outside the time window of duration ∆t. The resulting deadtime given the muon flux and the detector geometry described in Sec. III B is accordingly In DUNE, the (2.5 m, 40 s) cylinder cut in the 2nd step would cause a 4% deadtime. The stopping-muon cut can better reject those relatively long-lived isotopes, as one can afford a larger time window than for the cylinder cut. The reason is that, unlike throughgoing muons, stopping muons have much lower rates. In DUNE, the per-module stopping muon rate is only 0.001 Hz. With the (3 m, 10 min) stoppingmuon cut in the 2nd step, all the 36% of 40 Cl made through 40 Ar(µ, ν µ ) 40 Cl could be removed, and the deadtime is 1%. Figure 5 (left panel) shows the spallation energy spectrum after the 2nd-step cut. Under this simple cut, only 3% of the backgrounds above 5 MeV remain. The post-cut flux in the low energy end is dominantly from 38 Cl and 40 Cl. In the more important energy range above 8 MeV, 8 Li and 11 Be make 86% of the remaining backgrounds. This cut efficiency is basically limited by the cylinder time window ∆t for high-A isotopes (low-energy spectrum) and by the cylinder radius R for low-A isotopes (high-energy spectrum). This can be understood from Eq. (1). For all isotopes, the decay fraction outside of the cylinder, P (D > 2.5 m), is typically 1%, as shown in Figure 4. While the decay fraction outside of the time window, P (t > 40 s), could be as big as 76% for high-A isotopes with half-lives of order 100 s, and as small as 10 −12 for low-A isotopes with half-lives of order 1 s (and 6% for 10 s). This suggests that a set of cylinders might be useful if one wants to have a good cut efficiency in the entire energy range. of 40 Cl. Now, the residual spectrum seems to tell us that the high-A isotopes, 38 Cl and 40 Cl, are the most important ones. However, note that the majority of the energy range (8-15 MeV) is still covered by the low-A isotopes, as shown in the left panel.
This two-step spallation cut is very encouraging, and improvements are possible. Under this simple cut, the backgrounds are already rejected by a factor of 1.7 × 10 4 , resulting an acceptable deadtime of 5%, with 1% from the 1st-step cut and 4% from the 2nd-step cut. One improvement would be using a shower cut. Isotopes are dominantly born in hadronic showers, which are rare and have special characteristics (greater fluctuations, transverse size, and muon and neutron counts [58]), and hence should be easily identified in LAr. Those showers usually extend ∼ 5 m (less than the detector height) and, more importantly, not every muon showers. Compared to the cylinder cut used here, cutting the shower region for rare hadronic showers would enable a smaller cut volume and a smaller cut frequency, and hence allow stronger cuts.
So far, our focus is on the muons coming into the detector, but there can be muons that miss the detector but send in secondary particles. Neutrons are especially dangerous, as they could enter the detector invisibly. Low-energy neutrons can get captured and emit gammas in the fiducial volume. However, their rate is much lower than that from radioactivities in the rock (details in Sec. IV D 3). High-energy neutrons can make isotopes, but they must be from the muons that are close to the detector edge, because the isotope production probability drops significantly when its distance to the muon track gets larger, as shown in Figure 4. Taking the 1 m of active LAr shielding that is outside of the fiducial volume into account, the isotope yield in the fiducial volume per muon-in-rock would be 7% of the yield per muonin-detector. From simulation, we find that the isotope production yields in the fiducial volume from the muoninduced neutrons in the rock are typically 3-4 orders of magnitude lower than those from throughgoing muons. They could be cut further by recognizing the electrons and gammas accompanying the incoming neutrons. If we cut incoming electromagnetic showers with deposited energies larger than 50 MeV, we find that only 20% of the isotopes remain. Therefore, we ignore isotope production due to muons that miss the detector.
Interestingly, the isotopes that matter most in LAr are many of the same isotopes that matter most in water-or scintillator-based detectors: low-A isotopes of Li, Be, B, etc. (and for water, also 16 N). These isotopes are outliers in the chart of nuclides in that they have both high Q values and long half-lives, which makes them survive cuts. Isotope production is closely associated with muon secondaries. The rates of those secondary particles are largely material-independent, but the rates of specific isotopes do strongly depend on the target, due to the different nuclear processes. For example, for the low-A isotopes that matter most in LAr, their yields are suppressed by at least an order of magnitude compared to water.
One essential step forward is to understand detector backgrounds. Below, we focus on spallation backgrounds, both pre-and post-cut, as well as neutroncapture backgrounds and other backgrounds, including through pileup. We evaluate their impact on MeV program of DUNE from two aspects: the absolute background rates compared to the signal rates and how backgrounds would affect trigger design.

A. Solar Neutrinos
We first summarize the predicted solar neutrino signals from Ref. [1], where 100 kton-year exposure, 5-MeV energy threshold, and 7% energy resolution are assumed. The two signal channels are elastic-scattering interaction ν e,µ,τ + e − → ν e,µ,τ + e − , and charged-current interaction ν e + 40 Ar → e − + 40 K * . They can be well separated by a forward-cone angular cut, because elasticscattering events are forward-peaked whereas chargedcurrent events are nearly isotropic. For 8 B signals, inside the cone, the elastic-scattering channel dominates (∼ 10 4 events); outside the cone, the charged-current channel dominates (∼ 10 5 events). For hep signals, the sensitivity is largely from the charged-current events above 11 MeV that are outside the cone (∼ 150 events). Given the background rates we explain below, DUNE would measure sin 2 θ 12 and ∆m 2 21 with a factor of 1.5 and 3 better precision, respectively, than all combined solar experiments to date, a factor of 1.6 better precision on 8 B flux than from SNO, and make the first detection of the hep flux, with a precision of 11%. Figure 6 shows the solar neutrino signals and the spallation backgrounds. For illustration purposes, we show only the combined signal rate from the two detection channels (details in Ref. [1]). Pre-cut, spallation backgrounds are subdominant but important. There are 2.4 × 10 4 background events above 5 MeV in 100 ktonyear. After the two-step cut we propose in Sec. III E, the backgrounds are reduced to 700 events above 5 MeV, as shown in the figure. These post-cut backgrounds are  p o s tc u t s p a l l a t i o n   FIG. 6. Solar neutrino signal rates [1] and spallation background rates in DUNE, with nominal 7% energy resolution. (Other backgrounds, due especially to neutron capture, which dominates at low energy, are not shown.) The deadtime imposed by the spallation cut is not accounted for the signal rates. Orange solid line: total 8 B signal rates from charged-current channel and elastic-scattering channel. Orange dashed line: total hep signal rates from the two channels. Light blue shaded: pre-cut spallation spectrum. Dark blue shaded: post-cut spallation spectrum (same as Fig. 5).
negligible compared to both 8 B and hep event rates, and the imposed deadtime is only 5%. For this and the next figure, we show in the Appendix versions where we adopt 20% energy resolution.

B. Supernova Neutrinos
The expected counts (mostly ν e ) from a supernova at 10 kpc are large, ∼ 800 in 10 s in each module [14].
If we have independent information on when a supernova is happening, spallation backgrounds are negligible. The total background rate is 0.4 in 10 s in each module, and it reduces to 0.001 if only the backgrounds above 5 MeV are counted. On top of that, the backgrounds could be rejected further if cuts (e.g., as proposed in Sec. III E) are applied. The harder case, where one waits to trigger on a supernova, is discussed below.

C. Diffuse Supernova Neutrino Background
The DSNB is the flux of neutrinos emitted by all core-collapse supernovae throughout the universe. While being a unique probe for both stellar astrophysics and neutrino physics, the DSNB has not been detected. Currently, the strongestν e flux limit is set by Super-Kamiokande [28,69,70]. Future progress could be made by the joint efforts from next-generation experiments.
Three ingredients are needed to calculate the DSNB event rates [11]. The first is the supernova neutrino emission spectrum. For each flavor, the neutrino spectrum can be approximated by a Fermi-Dirac distribution, where the two parameters, total energy E ν,tot and temperature T , should be determined from experiments, although there are oft-quoted estimates. The second is the cosmic supernova rate. This is closely related to the star-formation rate, which has been measured. The third is the neutrino interaction cross sections with argon. For the charged-current interaction that dominates the DSNB signal rates, the uncertainty on the cross section is 10% [1]. Following Ref. [71], which provides the star-formation rate from Ref. [72], we calculate the DSNB ν e flux for neutrinos at T = 4, 5, and 6 MeV, and then calculate the DSNB signal rates with the cross section from Refs. [73,74]. Figure 7 shows our calculated DSNB ν e signal rates, together with the spallation backgrounds and solar neutrinos, which are treated as backgrounds in this case. Another background, due to atmospheric neutrinos, arising at ∼ 40 MeV [66], is not shown here. Pre-cut, the spallation backgrounds and the hep events are comparable above 15 MeV, resulting a low-energy threshold for the DSNB signals at 17 MeV. After the two-step cut, the spallation backgrounds above 15 MeV are completely rejected. Unfortunately, it would not lower the energy threshold for the DSNB because of the hep events. In principle, the hep elastic-scattering rates can be reduced by ∼ 80% with a forward cone cut. However, it would not significantly help the DSNB search, due to the remaining much larger hep charged-current rates. Taking 17 MeV as the threshold, the event rates of DSNB in an exposure of 100 kton-year would be about 1, 2, and 5 events, for neutrinos at T = 4, 5, and 6 MeV. The event rates might be higher, ∼ 10 in 100 kton-year, if one considers neutrino oscillations, such as in Ref. [66], where the after-mixing neutrino spectrum is equivalent to assuming T 8 MeV, which is unrealistically high.

D. Trigger Considerations
Now that we have discussed potential signals and backgrounds, it is important to also consider the trigger requirements to collect the data. Here, in an effort to aid DUNE trigger design, we provide new results on rates, spectra, and multiplicities of low-energy events. Though the details are unknown, it is expected that there will be one kind of trigger for single low-energy events, e.g., solar neutrinos, and another for a burst of events, e.g., a supernova, which may allow a lower energy threshold per event. For burst-event trigger, a limiting factor will likely be storage of signal events, as it is expected that all data will be recorded for say minutes in event of a possible supernova burst, and that backgrounds will be negligible during a real burst. However, backgrounds could lead to a considerable amount of fake triggers, due to the long waiting period for a burst.
The supernova trigger design is ongoing, so we make some reasonable assumptions. First, we assume a conservative energy resolution of 20%, which may be more realistic at the trigger level. Second, we assume an energy threshold of 5 MeV. This means that any event below 5 MeV is invisible at the trigger level, unless there is pileup, which is discussed in detail in Sec. IV D 5.
Under such assumptions, we consider two types of supernova triggers, based on the burst characteristics. For a supernova at 10 kpc, the expected counts are ∼ 1-2 per 5.4-ms readout window in the first ∼ 1 s of the supernova, and falling thereafter. Type I triggers on nearcontinuous tracks, i.e., individual electron tracks above 5 MeV, in several readout windows. Type II triggers on many tracks in a single readout window, which is probably more suitable for the start of a supernova at a smaller distance. For both types, determining the value of n, i.e., the number of 5.4-ms bins or the number of tracks per bin, is not trivial, because if n is too small, there would be too many fake triggers due to backgrounds; if n is too large, one may miss the first tens of milliseconds of supernova events. Below, we detail how backgrounds would affect these two types of supernova neutrino trigger de- signs. We report the calculated fake trigger rates in units of month −1 (10 kton) −1 , as each module is independent in terms of trigger, and once per month is likely the scale to determine whether a fake rate is acceptable.
In short, we find that muon-generated backgrounds are unimportant, but those due to radiogenic neutrons from the rock are important.

Muon Activity in the Detector
One potential fake trigger scenario is the aftermath of a cosmic-ray muon. Throughgoing muons easily trigger the detector, due to the typical GeV-range energy deposition. In the first 5.4-ms readout period, the detector will be very active. In its first 2.7 ms (one drift period), the charges due to shower particles drift to the wire planes; in the second 2.7 ms, any remaining associated charge depositions are due to neutron captures and spallation isotope decays. We ignore this first 5.4ms bin in our following discussions, as it is already triggered by a muon and it is likely to be discarded to clear the charges. Starting from the next 5.4 ms, any associated charge depositions are purely from spallation isotope decays. The neutron capture rate beyond 5.4 ms is vanishing, because most neutron captures happen with τ 0.35 ms ( 30% of captures are at energies above 0.01 MeV and hence sooner). The detector module will become relatively quiet until another muon comes in, which is typically 20 s later. Understanding the decay multiplicities in the time period t > 5.4 ms is im-   Figure 8 shows the spallation decay multiplicity distributions due to throughgoing muons. The multiplicity does not follow a Poisson distribution, because isotopes are made in showers sampled from a broad energy range. For the decays below 5 MeV, even though their multiplicities are high, we find that pileup is negligible, following our calculations in Sec. IV D 5. Thus, only decays above 5 MeV are relevant. To approximately take into account the energy smearing effect, we use 4 MeV as the presmearing threshold in our analysis, which is conservative even for a 20% energy resolution.
Table II, in its first three rows, lists the rates of selected decay multiplicities for different energy thresholds. To quantify their impact on the triggers, we further look into the per-readout multiplicities. For a 5-MeV threshold, in the fourth row of Table II, we list the rates of decay multiplicities from 0 to 5 in the 5.4-10.8 ms bin, which has the maximum decay yield for t > 5.4 ms. The rate to have 1 decay from a muon in that bin is 9 month −1 , corresponding to a probability of 10 −4 for each muon. We note that for any trigger, it must have more than 1 event. For a Type I trigger, the rate to have a following decay in the next 5.4 ms is negligible, 10 −3 month −1 , so the spallation background is not a concern. For a Type II trigger, it is not a problem either, because the rate to have more than 1 decay per readout is tiny, 0.3 month −1 , as shown in the table. For a 10-MeV threshold, the trigger rates are even smaller. Overall, even under conservative assumptions, spallation decays from throughgoing muons would not affect the supernova trigger.

Muon Activity in the Rock
Similar to the throughgoing-muon case, there can be muons that pass through rock near the detector. Their spallation betas (entering the detector) are also not a concern for the supernova triggers. Note that our results for throughgoing muons are calculated for the precut backgrounds, so recognizing the muon track or not does not matter. In that sense, the spallation multiplic- ities in the detector due to the muons in the rock would have a similar distribution as shown in Figure 8 and Table II, but have a much smaller normalization, due to the much lower yields of isotope in the detector (details in Sec. III E). The neutron capture backgrounds due to these muons in the rock would not be a concern either, because their rates are ∼ 10 −5 of those due to radioactivities in the rock, as shown in Figure A11, and one can further cut these neutrons from the near-detector muons by recognizing the accompanying electromagnetic showers that enter the detector.

Radioactivity Neutrons
The dominant source of neutron backgrounds is due to radioactivities in the rock, primarily 238 U and 232 Th. The neutrons from muons in the rock or from radioactivities in the detector (orders of magnitude lower U/Th concentrations) are much fewer. The most relevant energy range is at or below a few MeV, as neutrons at these energies can easily reach all parts of the detector and get captured inside the detector. (Neutrons between a few MeV and about 10 MeV can travel long distances and may escape; neutrons at higher energies have short atten- 0.01 --uation lengths due to inelastic interactions, and cannot travel far into the detector.) For neutrons at or below a few MeV, their mean free path due to elastic scattering is λ ∼ 15 cm. Because the elastic-scattering energy loss on 40 Ar is very inefficient, a 1-MeV neutron needs to scatter n ∼ 400 times to be thermalized. Thus, the diameter of random walk before it gets captured is 2 √ n λ ∼ 6 m, comparable to the fiducial volume height or width. Once a neutron is captured on 40 Ar, a total energy of 6.1 MeV will be released in gamma rays. These gamma rays will mostly Compton scatter, producing electrons that can be backgrounds for supernova neutrinos.
The background rate due to neutron captures is determined by two factors: the neutron capture rate and the number of electrons per capture that are above the energy threshold. Given the radioisotope concentration in the rock, we find that the neutron capture rate is 81 s −1 in each module. With a 5-MeV energy threshold and a 20% energy resolution, the background electron rate is 5 s −1 . As we show below, such a high background rate can be a concern for both types of trigger.
Tables III and IV show our calculated neutron background multiplicity distributions relevant to the Type I and II triggers, respectively. Because these backgrounds follow a Poisson distribution, the rates shown in the ta-bles are determined by µ, the expected counts in a 5.4ms readout window. For example, the background electron rate of 5 s −1 corresponds to a µ 0.03. For a Type I trigger, if the definition is to have n tracks in a row of 5.4-ms bins, then the fake trigger rate would be (µ n ×0.0054 −1 ) s −1 . For a Type II trigger, if it requires n tracks in a single 5.4-ms bin, then the fake rate would be (µ n /n! × 0.0054 −1 ) s −1 . As shown in the tables, the fake trigger rates in both cases could be as large as hundreds per month per module for some choices of n > 2.
To lower the fake trigger rate, one approach is to enforce a higher energy threshold. When smeared with a 20% energy resolution, the number of electrons per capture that are above E thr = 5, 6, 7, 8, 9, and 10 MeV are about 0.066, 0.013, 0.002, 10 −4 , 8.4×10 −6 , and 2.6×10 −7 , respectively. The background electron rates are accordingly about 5, 1, 0.2, 0.01, 7×10 −4 , and 2×10 −5 s −1 , respectively. With an 8 or 9 MeV threshold, the fake rate could be reduced significantly, as shown in the tables. However, great physics opportunities might be lost, e.g., detecting the beginning of the infall period ( E νe 8 MeV [75]), and detecting the large fraction of all supernova events at low electron energies.
Alternatively, to avoid sacrifice on the energy threshold, we propose to add passive water (/oil/plastic) shielding that would greatly reduce the neutron capture rate. With no shielding, 20-cm shielding, and 40-cm shielding, the neutron capture rates in each module are about 81, 0.7, and 0.02 s −1 , respectively. Assuming a 5-MeV threshold, the background electron rates are accordingly about 5, 0.05, and 0.001 s −1 , respectively. With shielding, the fake trigger rate would be negligible, even for a threshold at 5 MeV, as shown in the tables. This strategy will benefit not only the supernova detection, but also the solar neutrino program proposed in Ref. [1].

Radioactivities in the Detector
For intrinsic radioactivities in the detector, relevant beta-decay sources could be 39 Ar and 42 Ar in the atmospheric argon used in DUNE, 42 ) has not yet been measured at DUNE. Conservatively, we assume a 222 Rn activity of ∼ 10 mBq/m 3 , mainly from the detector materials, corresponding to ∼ 10 2 Hz per module. (In Super-Kamiokande, after the detector was sealed and the initial radon decayed away, the 222 Rn activity was ∼ 20 mBq/m 3 until water-purification procedures lowered it to a few mBq/m 3 [79].) Thus, 214 Bi should be unimportant compared to 42 K in DUNE.
With the above arguments, the most important in-trinsic radioactive isotopes are 39 Ar (Q 0.6 MeV, R ∼ 10 7 Hz) and 42 K (Q 3.5 MeV, R ∼ 10 3 Hz). They are abundant, but due to their low Q values, they cannot trigger the detector unless there is pileup.

Pileup
We first present a framework to calculate pileup rates due to sub-threshold events. With specific numbers and conservative assumptions, we show that the expected pileup rates would be negligible for a nominal threshold near 5 MeV in electron energy.
In general, the rate of n-event pileup is R × P (µ, n − 1), where R is the single-event rate, and P (µ, n − 1) is the Poisson probability for another (n − 1) events that satisfy the pileup conditions of coincidence within a space volume ∆x × ∆y × ∆z and a time window ∆t.
We first consider how pileup would work at the reconstruction level, due to greater simplicity, though that is not the whole story. At the reconstruction level, the resolution in the drift direction, which we call z, is 0.8 mm, determined by the t 0 resolution and the drift speed v, while the resolution in x and y is 5 mm, limited by the wire spacing. A 5-MeV electron's track length is 2.5 cm long, crossing 5 wires. For simplicity, we conservatively choose ∆x = ∆y = ∆z = 5 cm. The time window ∆t should be decided by the charge pulse width on the wire planes, i.e., δt 50 µs [80], conservatively. For n events to appear as a single event, the maximum time separation between the first and the last pulse peak is ∆t = (n − 1) δt, which is very conservative given the δt value we use.
At the trigger level, t 0 might not be extractable, and one would have a higher pileup rate. We start with the pileup of two events, with single-event rates R i and R j , respectively. We use the ∆x and ∆y defined above, and derive ∆z. Suppose the first event happens at (z 0 , t 0 = 0), where z 0 ∈ [0, z max ], i.e., anywhere along the drift direction. If a second event, originated at (z , t > 0), reaches the wire plane within δt of the first event, these two events would appear as one. Thus, the pileup condition is: for any z 0 , all (z , t ) that satisfies | z − (z 0 + vt ) | ≤ v δt, which is a band of area f (z 0 ) = 2 (z max − z 0 ) δt on a z − t plot. The Poisson expectation would accordingly be This is equivalent to setting ∆z = z max and ∆t = δt. The length scales we use are x max = 58 m, y max = 12 m, z max = 14.5 m (maximum total drift distance, to be conservative). Given R i and µ j , the two-event pileup rate is R i ×P (µ j , 1) = R i µ j . For n-event pileup at the trigger level, the above derivation can be simply generalized by requiring ∆z = z max and ∆t = (n − 1) δt. For example, the pileup rate due to four 39 Ar decays is 0.3 Hz per module, consistent with the 1 Hz from Ref. [80].
In this framework, we now calculate the pileup rates at the trigger level with conservative assumptions. We assume that 39 Ar and 42 K always decay to betas with the endpoint energies. For the sub-threshold electrons due to neutron captures, we assume that there are two 3-MeV electrons produced per capture [81][82][83], with a rate of ∼ 200 Hz per module without shielding. Therefore, neutron capture would have a rate ∼ 5 times lower than that of 42 K decay. Thus, there are only two sources that could dominate the pileup rate: 0.6-MeV betas from 39 Ar, with a rate of ∼ 10 7 Hz, and 3.5-MeV betas from 42 K, with a rate of ∼ 10 3 Hz.
For pileups that appear as a single 5-MeV event, there are two types. The first type is among multiple 39 Ar or 42 K events themselves. There could be 10 decays of 39 Ar, with a per-module rate of ∼ 10 −15 s −1 , or 2 decays of 42 K, with a per-module rate of ∼ 10 −4 s −1 . The second type is between 39 Ar and 42 K events, where the only scenario would be 3 decays of 39 Ar with 1 decay of 42 K, with a per-module rate of ∼ 10 −5 s −1 .
Even such pileup rates at the trigger level would be negligible compared to neutron-capture background rate above 5 MeV, which is 5 s −1 without shielding and 10 −3 s −1 with 40-cm shielding (details in Sec. IV D 3). At the reconstruction level, or with a higher energy threshold, the pileup rates would be even smaller than above. We conclude that pileup is irrelevant for E > 5 MeV. Even for slightly lower thresholds, the results would hold, as we have unrealistically assumed that every decay beta has its full Q value.

V. CONCLUSIONS
DUNE could provide a precious opportunity for MeV neutrino astrophysics. A key step to probe DUNE's MeV potential is to understand the detection backgrounds.
We calculate the muon-induced spallation background in DUNE. Using the Monte Carlo code FLUKA and theoretical insights, we detail the physical mechanism of isotope production, calculate the isotope yields, evaluate the isotope decay energy and time profiles, and develop cuts to reduce the backgrounds. Complementary to previous work on scintillation detectors and water-Cherenkov detectors, we provide a thorough understanding of the spallation backgrounds in argon.
The uncertainty of our simulation is likely to be around a factor of 2, mostly due to the uncertainties of hadronic processes. This is good enough in the sense that isotope yields vary by orders of magnitude. The calibration could be done in situ, and the isotope yields in argon could be checked with detectors such as MicroBooNE.
We are the first to show explicitly that the essential difference between argon and oil or water is revealed by the two-group isotope production mechanism. In DUNE, high-A isotopes (e.g., Ar, Cl, S) are abundantly produced. However, these isotopes typically have small decay Q values (2)(3), and hence dominantly produce betas well below the expected energy threshold. Low-A isotopes (e.g., Li, Be, B), despite their small yields, are important background sources at higher energies, due to their large decay Q values (∼ 10-20 MeV). This two-group production might reveal a hidden universality, suggesting that maybe the most important isotopes for all target nuclei will be the same ones, such as Li, Be, B, and N. Future work is needed to check this.
The decay properties of those low-A isotopes are the key for us to design the cuts, so that the backgrounds become controllable. While there are many unknowns for the future MeV programs in DUNE, we propose a two-step spallation cut, based on reasonable assumptions of the detector, showing that the spallation backgrounds can be greatly reduced. Our 1st-step cut is discarding all events with t < 250 ms relative to each muon, assuming a 5-MeV energy threshold. After this cut, the endpoint of the background is lowered down from 20 MeV to 15 MeV, and the per-module deadtime is only 1%. The 2nd-step cut is a cylinder cut for throughgoing muons, discarding events with R < 2.5 m and t < 40 s, and using a sphere cut for stopping muons. After the 2nd step, the background rates above 5 MeV are reduced from 7 per day per module to 0.2 per day per module, and the total deadtime is only 5%.
The background cuts can be further improved. We briefly note some possibilities. In principle, one could cut muons with large energy loss to reduce the deadtime, because isotope yields are roughly proportional with muon energy losses. In addition, one could use a shower profile cut. Li and Beacom [23][24][25] show that most isotopes are made in rare hadronic showers, which should be easily recognized in DUNE. In this approach, the deadtime would be reduced at least by a factor of a few.
As a last step, we evaluate how the backgrounds would affect MeV programs in DUNE. One aspect is understanding the absolute background rates compared to the signal rates. For solar neutrinos, the pre-cut spallation backgrounds have comparable rates to the signals. However, a simple two-step cut could make the spallation backgrounds totally subdominant [1]. For supernova neutrinos, the spallation background has a negligible rate compared to the intense burst. For the DSNB, hep solar neutrinos turn out to be the limiting background, the pre-cut spallation backgrounds are of comparable importance though.
Another aspect is to aid trigger design for supernova neutrino detection -the primary MeV program in DUNE. In this regard, the most significant background is not spallation isotope decays, but rather neutron captures on argon, where the neutrons are produced by U/Th decays in the rock. This neutron capture background could also affect the solar-neutrino program proposed in Ref. [1], but we showed that it could be avoided by setting a high electron-energy threshold of 7 MeV (for 7% energy resolution; higher if worse). This would still allow a strong solar-neutrino program, but it would take somewhat longer to accumulate statistics. For the supernova-neutrino program, to have an acceptable trigger rate, the threshold would need to be at least 8 MeV, which would significantly cut into the detectable supernova spectrum. As shown in Ref. [1], both of these problems could be solved by even modest passive shielding. If there is no shielding, the deleterious effects can be reduced if the energy resolution is good, which depends on a robust light-detection system [13][14][15][16][17][18][19]. To fully realize its potential, DUNE must take new steps to ensure robust detector capabilities at MeV energies, as detailed above and in Ref.  Second, we compare the neutron-capture backgrounds due to radiogenic [1] and cosmogenic sources. Figure A11 shows the neutron-capture rates in the DUNE fiducial volume with different assumed thicknesses of passive water/oil/plastic shielding. With zero or modest shielding, the dominant neutron source is radioactivities (assumed 3.43 ppm 238 U and 7.11 ppm 232 Th) in rock, which produce MeV-range neutrons at high rates. These can be stopped efficiently by the shielding. In argon, the neutron capture rate is ∝ e −x/λ , where x is the shielding thickness, and the length scale is λ 5 cm. Muons in the rock, unseen by the detector, produce GeV-range neutrons with low rates. Because more shielding is needed to stop those energetic neutrons, the capture rate in argon is still ∝ e −x/λ , but with λ 70 cm. As a high-energy neutron propagates, it can make many low-energy neutrons that capture more easily. This process, which is not very efficient, has been taken into account in our simulation. We conclude that only in detectors with significant shielding are the cosmogenic neutrons more important than the radiogenic neutrons.