Luminous solar neutrinos I: Dipole portals

Solar neutrinos upscattering inside the Earth can source unstable particles that can decay inside terrestrial detectors. Contrary to naive expectations we show that when the decay length is much shorter than the radius of the \emph{Earth} (rather than the detector), the event rate is independent of the decay length. In this paper we study a transition dipole operator (neutrino dipole portal) and show that Borexino's existing data probes previously untouched parameter space in the 0.5--20 MeV regime, complementing recent cosmological and supernova bounds. We briefly comment on similarities and differences with luminous dark matter and comment on future prospects for analogous signals stemming from atmospheric neutrinos.


I. INTRODUCTION
The Earth is constantly pelted with neutrinos from the Sun (among other sources), and dark matter from our local galaxy; both can serve as a resource with which to search for new physics. The canonical strategy, as it pertains to dark matter, is to detect particles directly via elastic or inelastic scattering within a detector, however this strategy applies equally well to new physics coupled via some neutrino-portal. For example, electron recoil data can be used to constrain both elastic and transition neutrino dipole moments, both of which have attracted recent interest as potential explanations of the XENON1T excess [1][2][3][4][5] (see also [6] for a review) . More broadly speaking, a transition dipole operator can serve as "dipole portal" to a dark sector [7][8][9] and is described by the interaction Lagrangian 1 where ν L is the Standard Model left-handed neutrino, and F µν is the electromagnetic field strength. The index a = e, µ, τ denotes the neutrino flavour and N R is a (possibly Dirac or Majorana) sterile neutrino. The dipole operator, d a , can generically be flavour dependent however since the solar neutrino flux contains all three flavors the sensitivity derived here is only midly flavor dependent. This model was proposed as a phenomenological explanation of the MiniBooNE and LSND anomalies [10], and subsequently studied in the context of radiative muon capture [11], supernovae cooling [8], collider searches [8], double bangs at ICECUBE [12], fixed-target facilities [8,[13][14][15], low-recoil dark matter detectors [16] and cosmology of the early universe [9]. This paper is the Upscattering of a solar neutrino inside the Earth's mantle (or core, or crust) leads to a "heavy" O(MeV) sterile neutrino, N . In the weak coupling limit N is long-lived, and can propagate long distances eventually decaying inside a large-scale detector. For a neutrino-dipole portal this leads to a deposited photon with an energy of order ∼ 100 keV -10 MeV.
first in a series of two that investigates the physics potential for detecting upscattered unstable particles sourced by solar neutrinos inside the Earth with a neutrino dipole portal serving as a useful benchmark example. In [17] we consider a mass-mixing portal whose upscattering and decay phenomenology is sufficiently different to warrant an isolated treatment. The parametric sensitivity to the dipole operator, d, of a search strategy that uses a flux of incident neutrinos directly is straightforward. The rate of scattering events inside a detector is proportional to the flux times the cross section Φ × σ multiplied by the number of targets inside the detector's fiducial volume. The flux of incident neutrinos is independent of d, while the detection cross section scales quadratically with the dipole operator, σ ∼ d 2 . The signal rate, R, therefore scales as R ∼ d 2 . Another possibility is to make use of astrophysical fluxes indirectly by leveraging their ability to produce a flux of new physics particles. For instance, cosmic rays interacting in the upper atmosphere can produce longlived particles [18] in much the same way that dedicated fixed target facilities (e.g. NA62 [19], SHiP [20] etc.) can. In both cases pp and/or pA collisions source mesons, which promptly decay, ultimately producing a flux of light, long-lived, new physics particles e.g. heavy neutral leptons [18], light dark matter [21], or millicharged particles [22]. Such an indirect production mechanism costs an extra two powers of the coupling constant, such that for a neutrino dipole-portal the flux itself scales as Φ ∼ d 2 leading to an event rate that scales as R ∼ d 4 . These indirect fluxes are therefore expected to provide meaningful sensitivity to new physics only at moderately small couplings.
A seemingly innocuous twist on this latter scenario is to consider an indirect flux of new particles that decay within a detector rather than scattering against its constituents. Consider a detector with a characteristic length scale , and a particle with a decay length, λ, that satisfied λ . A long-lived particle is unlikely to decay inside a finite sized detector, since the probabilitty of decay within a distance is, P dec = 1 − exp[− /λ] and for λ this scales as P dec ≈ /λ ∼ d 2 . Naively, therefore, the signal rate in this scenario scales as R ∼ d 4 .
The parametric dependence of the signal changes dramatically, however, if the indirect flux is sourced by upscattering inside the Earth and the decay length satisfies λ R ⊕ with R ⊕ the radius of the Earth. In this case, the upscattered flux that arrives at the detector is proportional to λ such that Φ × P dec is independent of λ at leading order in /λ. This can be understood as the effective column density of targets growing with λ in such a way as to precisely cancel the 1/λ penalty arising from the rarity of decays within the detector. This effect persists until it is saturated by the boundaries of the Earth after which, rather than being suppressed by a factor that is O( /λ) the rate is instead suppressed by a factor that is O(R ⊕ /λ).
We can make our discussion more concrete by considering a flux of incident particles on a thick slab of material of length L slab which terminates in a detector as depicted in Fig. 1. If we consider an infinitesimally thin slice of the slab (thickness dz) then the flux of long-lived particles, N , arriving at the front detector is dΦ N = Φ ν n A σ ν→N e −z/λ dz, where z is the distance from the slice to the detector, n A is the number density of upscattering targets and σ ν→N is the upscattering cross section for νA → N A. Integrating over z we find the flux at the detector is given by where the quantity in the square braces can be interpreted as the effective column density of scatterers along the line of sight. The rate of decays within the detector will be proportional to the product of this flux, the area of the detector, and the probability of decaying within it where we have assumed L slab λ . This can be compared to the rate of quasi-elastic scattering νX → Constraints on a muon-only dipole coupling dµ, from cosmology (BBN and CMB) [9], SN-1987A [8], Borexino eν scattering data [9,23,24], CHARM-II eν scattering data [12,25], MiniBooNE [8,26], and NOMAD [13,27]; the last viable parameter space [9] to explain the XENON1T excess is indicated with a star. This work (ν → N → νγ) is shown in green with a solid (dashed) line for α = −1 ((α = 0) corresponding to a maximally CP-violating Dirac (Majorana) N . Constraints were obtained by multiplying the solar neutrino flux by Peµ(Eν ) [28][29][30].
Xν signal events from the direct flux of neutrinos We have included the label X, because for scattering events to be visible inside the detector, their energy deposition must be observable as X-recoil energy. Low-energy nuclear recoils are difficult to observe as compared to electron recoils, which means that νe → N e scattering often provides better sensitivity. Upscattering off of electrons, however, has a much smaller cross section than νA → N A. In upscatter-decay scenarios, the nuclear recoil of a target inside the Earth does not need to be detected, and so νA → N A scattering is an ever-present production mechanism which dominates over νe → N e. Therefore, despite the upscattered flux being indirect (i.e. sourced by scattering) the event rate is parametrically identical to direct detection (scaling as R ∼ d 2 rather than R ∼ d 4 ). Furthermore, one can clearly see a number of avenues via which event rates from upscattered long-lived particles can supersede those of direct elastic recoil: 1. While both event rates are proportional to the volume of the detector, R dec scales with the density of upscattering targets inside the Earth. The interior of the Earth tends to be 3-12 times more dense than detector material.
2. The cross sections entering the two expressions are different, and the upscattering cross section may be much larger. For example νe → N e (detection) has a much smaller cross section than νA → N A (upscattering).
Even when νX → N X scattering is kinematically allowed, points 1. and 2. can make the decay event rate much larger than the elastic scattering event rate. The rest of this paper is dedicated to investigating the sensitivity of large-scale detectors to a dipole portal for m N ≤ 18.8 MeV, the cut-off of the solar-neutrino flux. Our results are summarized and shown in context with other constraints for a muon-only dipole portal in Fig. 2; constraints derived in this paper for d e and d τ are broadly similar, while constraints based on fixed-target facilities either weaken or disappear entirely for d e and d τ because the flux considered is predominantly composed of muon neutrinos. Solar neutrinos offer a robust probe of lowmass sterile neutrinos independent of neutrino flavor.
The rest of the article is organized as follows: In Section II we discuss the upscattering of solar neutrinos into unstable right-handed neutrinos via the dipole portal. Next, in Section III we discuss decay properties including the difference between Majorana and Dirac decays [34][35][36]. Then, in Section IV, we discuss how these two pieces of phenomenology conspire to produce photon spectra at large-scale detectors on Earth, the details of which are determined by the geometry of the Earth relative to the Sun, and the latitude of the detector. In Section V we use the photon spectra derived in Section IV to set lim-its on a dipole portal couplings using Borexino [23,24] and Super-Kamiokande data [37]. We conclude with Section VI where we summarize our main results, emphasize the overlap between luminous dark matter searches and luminous solar neutrino searches, and comment on qualitative effects that are important for future dedicated analysis.

II. DIPOLE-PORTAL UPSCATTERING
Solar neutrinos never exceed 20 MeV in energy, and this means that their (hypothetical) electromagnetic interactions with nuclei fall firmly in the coherent regime F (Q 2 ) ≈ 1. Moreover, the four-momentum transfer is limited by E ν , and so the recoil energy of a nucleus, A, never exceeds T max Therefore, neutrino-up-scattering on nuclei can be reliably approximated by considering the nucleus as a fixed and static charge density sourcing a Coulomb field. Within this approximation E N = E ν by energy conservation and the matrix element for up-scattering is given by where flavour indices have been suppressed for brevity's sake. Using dσ = 1 16π 2 |M| 2 dΩ (appropriate for potential scattering), noting that dt = 2E 2 ν d cos θ, and integrating over the azimuthal angle we have This is logarithmically enhanced to prefer forward scattering such that it is a good approximation to treat the resultant "beam" of steriles produced via solar neutrino upscattering to be parallel with the solar neutrino flux. We restrict upscattering to "forward" angles satisfying cos θ > 0 such that (for Q 2 = −t) and clearly in the E ν m N limit this give the result We can then calculate the rate of "forward" scattering (denoted by the superscript ) as The terms in the top line have been organized based on their scaling with respect to = m N /E ν . At logarithmic accuracy in , we therefore have σ ∼ 16Z 2 αd 2 log(2E ν /m N ) with corrections being of order ∼ few % for < 1/2. The Z 2 enhancement means that scattering from nuclei will always dominate over scattering from electrons. Keeping only the leading-logarithm and taking d = 1.97 · 10 −9 MeV −1 [a convenient choice for decay-length purposes c.f. Eq. (20)] For E ν m N (near threshold) one must use the full expression in the first line of Eq. (9) and the resultant flux is shown for a few benchmark masses in Fig. 3.
The flux of steriles Φ N emerging from an infinitesimal slab of thickness dz with target density n A is then given by This flux, unsculpted by the energy dependent decay length, is plotted in Fig. 3. For flavor dependent couplings oscillation effects should be included in the solar neutrino spectrum and the result should be summed over flavors appropriately weighted by d 2 a . This leads to, at most, an O(1) modification of the constraints and we neglect it in the discussion hereafter, but include it in our flavor-dependent results Fig. 2.

III. RADIATIVE STERILE DECAY
Once the flux of N 's has been determined, the resultant spectral shape of the photons from N → νγ can be calculated. The shape is somewhat model dependent, being determined by the Majorana vs Dirac nature of N and the level of CP-violation which is present [36,38], which determines the angular dependence of the differential decay rate in the rest frame of N A Majorana N has α = 0, whereas a Dirac N (naively favoured to suppress dipole contributions to neutrino textures [8]) can have α = 0 for a combination of magnetic and electric dipole moments [34][35][36]38]. For illustration we include α = 0 and α = −1. In the lab frame, for an N with energy E N , this leads to where tri The photon energy spectrum is then obtained by integrating over E N weighted by the flux of N . For example for the α = −1 case we have Taking the density to be constant, n A (z) = n A , integrating over z, treating λ, and multiplying by the size of the detector we arrive at This form is applicable to a spatially varying density profile if we make the replacement n A (1 − e −L slab /λ ) → Generally the closer a detector is to the equator the more advantageous its sensitivity will be, however this effect is relatively small provided we restrict possible detector sites to lie between the Arctic Circle and the Antarctic Circle. Omnipresent in our current discussion is the decay length λ whose relative size determines whether our signal scales as R ∼ d 2 or R ∼ d 4 . The lifetime of N is given by and its decay length, λ = γβτ is therefore where d eff = a d 2 a . It turns out that the values of d and m N that lead to a roughly Earth-scale decay length are right at the boundary of currently unexplored parameter space.

IV. SIGNALS AT LARGE SCALE DETECTORS
Unlike the simple picture presented in the introduction, the "slab" of Earth that is traversed by solar neutrinos in transit to a terrestrial detector is time dependent. This is obvious in that during the day the solar neutrinos pass through the Earth's crust whereas at night the majority of the line-of-sight density they encounter is the Earth's mantle. This can be visualized by working in a coordinate system where the solar flux is incident from thex direction. In this frame, the Earth rotates about its axis daily, and precesses about theẑ axis yearly as depicted in Fig. 4.
Let us highlight a number of qualitative effects • There is a stark day-night asymmetry with almost no signal during the day and all of the signal coming at night. The seasonal variation is less extreme being an O(1) effect.
• The neutrinos only pass through the core of the Earth 2 , which is composed of high-density and high-Z material, during winter for detectors in the northern hemisphere. Since Φ N ∝ Z 2 n A , and the core is a high-density, high-Z material this also introduces an important seasonal modulation.
• The direction of the photons will be highly correlated with the zenith angle for highly boosted N 's (most of the spectrum).
Many of these issues are discussed in detail in [33] for inelastic dark matter, which also has mostly-forward upscattering. For our present purposes we will focus on the time-averaged rate of photon deposition in a given experiment, and we therefore ignore details of seasonal and daily signal modulation. Furthermore, we consider a simplified model of the Earth that is designed to give a conservative estimate of the photon yield, while remaining analytically tractable.
We take the Earth to be a uniform sphere composed of one atomic species with Z = Z eff and number density n A , designed to mimic the average density of the Earth's mantle as shown in Appendix B. We assume that λ few km, such that the detector is effectively on the surface. Each day the detector traces out the path labeled by the frequency Ω in Fig. 4. The length of the slab through which the neutrinos pass (shown as a dashed circle in Fig. 4) changes in time such that the factor of (1 − e −L slab /λ ) in Eq. (18) is replaced by where · i labels an average over the i th day of the year and we have introduced the dimensionless variable ζ = R ⊕ /λ. The function I i (ζ) can be evaluated numerically, but also has simple analytic limits. For λ R ⊕ we have the intuitive result that such that the rate of photons is independent of the decay length. Averaging over a full year's exposure would then yield a factor of 1/2 because of the absence of signal during the day. In the opposite limit, where λ R ⊕ , the function is simply related to the average column density seen by the neutrinos Making use of the seasonal modulation in signal could serve as a powerful tool for characterizing backgrounds and substantially improve the sensitivity of large terrestrial detectors as discussed in [33] for the case of inelastic dark matter.
For simplicity, we ignore seasonal variations and focus here on the total integrated rate, or equivalently the time averaged rate taken over a full year of run time. This has the advantage of being directly comparable to publicly released data from Borexino and SK. For this purpose we can define the average rate for the full live-time of an experiment as where R i (E γ ) is defined the same way as in Eq. (18) but with the replacement (1 − e −L slab /λ ) → I i (ζ). This leads immediately to the function where we have introduced ζ(E N , m N , d) = R ⊕ /λ. The quantity I(ζ) has a mild lattitude dependence as illustrated in Fig. 5. With this function defined we can express the year-averaged differential rate of photon deposition, R(E γ ) (per unit time per unit energy) as In the limit where λ R ⊕ , but λ h (with h the overburden) we can replace I(ζ) → 1 2 (half of the exposure is day-time where there is no signal 3 ). If instead λ R ⊕ (i.e. ζ → 0) then we can replace I(ζ) → L(t) /λ such that for the two limits we have where Φ ν = Φ ν (E ν = E N ), and we have introduced a reference cross section, 3 If λ h then production in the Earth's crust (rather than the mantle) and there will be signal both day and night. that is independent of d and Z, but depends on m N and E ν as described in Eq. (9).
The function dR /dE γ is proportional to the differential rate of photons produced per target nucleus A, and is left unsculpted by the dependence of λ on E N . In contrast, the function dR /dE γ is sculpted by λ(E N ) ∝ 1/P N , because the rate depends on the ratio L(t) /λ; up-scattered N 's with more energy are less likely to decay inside the detector.
The most important feature of Eq. (27) is the transition between R ∼ d 2 and R ∼ d 4 scaling. The inter-polation between these two limits is given by Eq. (26), which can be safely used over the entire m N − d plane of parameter space. It is, however, instructive to consider experimental sensitivity in the two limits shown in Eq. (27) Future dedicated analyses at large scale detectors should include core of the Earth, the radial dependence of the mantle density, photon spectral information, and make use of time-modulation to discriminate backgrounds from signal. At this level of detail it becomes mandatory to use Eq. (26) rather than Eq. (27) (especially in the cross-over region between the two limits). All results in this paper make use of Eq. (26).

V. NEW CONSTRAINTS ON DIPOLE PORTALS
We consider data from Borexino and Super-Kamiokande to set constraints on a neutrino dipole portal. We perform a conservative analysis on the Borexino low-energy region (LER) data set from [23] (available online at [24]), binning their data into a region with E vis < 0.8 keV (LER-I) and a region with E vis > 0.8 keV (LER-II). Since the efficiency of Borexino is so high and photons are reconstructed nearly identically to electrons we neglect efficiencies in our estimates (setting them equal to one). Our signal's photon spectrum is very broad and so we do not include the effects of the O(50  [23,24]. The collaboratioin's best fit to the data is shown overtop the raw data, with the black circles representing the residuals (i.e. data-fit). The vertical band shows our binning into low-energy region I and low-energy region II, and the horizontal black lines correspond to the average rate used in each region to set our exclusions. The rate chosen in LER-I lies beneath well understood radioactive backgrounds, but above the data's residuals and the 7 Be neutrino event rate (not shown). In LER-II the rate is conservatively chosen to lie above the full observed signal. Photon spectral shapes (Dirac N with α = −1) are shown for three different masses. . keV) broadening from the detector resolution as this will be a negligible effect.
Our analysis is (up to a coarse re-binning) a rate-only analysis and so is insensitive to spectral shape details. In LER-I the Borexino residuals are small relative to the event rate, with eight different radioactive backgrounds contributing to the expected event rate. We use an event rate of 0.1 events per 100 t−day per keV in LER-I. This corresponds roughly to the full measured event rate in LER-I after subtracting off the 210 Po background and is larger than the 7 Be solar neutrino signal (which Borexino was able to measure with percent level precision). This is important, because the spectrum tends to be rather flat as can be seen in Fig. 6, which qualitatively resembles the 7 Be signal. A quoted statistical significance would therefore require some prior on the 7 Be flux normalization uncertainty while simultaneously accounting for the tight correlation between the 7 Be signal from νe scattering and the ν → N → νγ signal from the solar neutrinos. We do not attempt this here, and instead set limits whenever the rate from N → νγ would be rougly three times as large as the expected Standard Model νe scattering signal with nominal expectations for the solar neutrino flux. For LER-II we do not have to contend with large radioactive backgrounds, and choose to set constraints by demanding that the N decays do yield more than 0.01 events per 100 t-day. As can be seen in Fig. 6 this would completely overwhelm the signal observed in the detector and is clearly incompatible with the measured data. A more sophisticated analysis would include a full spectral shape fit, and an accounting for the daily and seasonal modulation of the signal. For Super-Kamiokande we use recent results from SK-IV, taking cuts were applied. Photon events are difficult to distinguish from electron or positron events since the radiation length of an MeV photon is tens of centimeters, and so pair produces rapidly in medium. The cuts applied on the SK-IV data involved external event vetoes and a tight fiducial volume cut [37]. Since the decay length of N is four to five orders of magnitude larger than the Super-Kamiokande detector, we would expect the photons to be uniformly distributed throughout the detector volume and so these cuts do not modify the expected rate normalized to fiducial volume. To set limits on d eff we require that the N → νγ do not produce a signal in excess of the observed rate above 4 MeV, which we calculate to be 4.62 events per kton-day. We do not make use of any spectral shape information, day-night asymmetry, or zenith angle dependence all of which could be leveraged for improved sensitivity in a dedicated analysis. Finally, we emphasize that the limits set here are conservative and systematically underestimate the reach of both experiments. It is not entirely fair to compare the sensitivities of Borexino and Super-Kamiokande based on the current analysis since each experiment's own detailed spectral shape and day-night asymmetry may significantly impact their ultimate sensitivity; this comparison warrants further study.

VI. CONCLUSIONS
The presence of solar neutrinos and coherent upscattering with the high-density, and moderate-Z material of the Earth's mantle can lead to a dramatic flux of sterile neutrinos passing through terrestrial detectors. Theses fluxes are so large that they compensate for the relatively low probability of decay within the detector volume re-  Fig. 2 of [23] (available online at [24]), and Fig. 14 of [37]. The excluded rate is a highly conservative choice c.f. Fig. 6. A proper statistical analysis, spectral shape information, and day-night asymmetry could easily improve sensitivity.
sulting in rates that dominate over elastic scattering by orders of magnitude.
As we explore in [17], similar phenomenology can be used to study other upscattering processes such as ν → N via a mass-mixing portal, followed by N → νe + e − . The phenomenology differs qualitatively from the dipolescattering example, however, because the upscattering process is quasi-isotropic in the lab frame. This eliminates day-night asymmetry to a first approximation, and requires a slightly re-worked treatment. We refer the interested reader to [17] where these details are discussed and similar bounds are worked out from the nonobservation of energetic e + e − pairs in Borexino.
Unsurprisingly, luminous neutrino signals share many of the same qualitative features with luminous dark matter. The signal rate is independent of detector composition depending only on the volume of the detector, scattering is preferentially forward, and the signal modulates in time. While similarities exist, there are also important differences. Luminous dark matter is expected to be heavy m χ ∼ 1 TeV, with a mass-splitting of O(100 keV) and a non-relativisitic velocity [33]. This means that the resultant photons are monochromatic with an energy set by the mass-splitting, and that threshold effects are important 4 . In contrast the model considered in this paper leads to moderately relativistic sterile neutrinos, whose lab-frame photon distribution is therefore broadened. The solar neutrino flux is broad, and since neutrinos are much lighter than nuclei, the only threshold effect that manifests itself in this work is the requirement that E ν ≥ m N . Looking forward it is important to emphasize that any astrophysical source of neutrinos (e.g. atmospherics) could produce similar phenomenology, albeit with the potential mandatory inclusion of neutrino oscillation effects. Luminous signals of inelastic transitions within the Earth have many common ingredients, but their signatures inside detectors can be somewhat model dependent. Our study further motivates a dedicated program at future large scale detectors to search for upscattered new physics within the Earth. This includes inelastic dark matter, but also any neutrinophilic inelastic transition.
Our study has demonstrated the discovery potential using solar neutrinos, but heavier sterile neutrinos, N , can be produced from atmospheric neutrinos and it would be interesting to study their impact on dipole portals and other inelastic upscattering operators. Atmospheric neutrinos have a much smaller flux, but could probe heavier m N , where constraints are lacking as shown in Fig. 2.
Future experiments such as JUNO and Hyper-Kamiokande can serve as sensitive probes of upscattered particles, as can any other large-volume detector capable of detecting MeV scale photons (this statement is independent of detector density). Searches could leverage time modulation, which needs to be studied in more detail as has been done for luminous dark matter [33], zenith-dependence, and spectral shape characteristics to distinguish new physics from backgrounds. To calculate the decay-weighted average path length I(ζ) we take our detector to be described by a point defined in the reference frame of the Earth (ẑ parallel to the magnetic pole). We then rotate the Earth's pole via a the rotation matrix where θ tilt = 23.5 degrees. This gives us an Earth that is titled and spinning about its own axis, but is static relative to the Sun. In reality the Earth precesses relative to the Sun. We therefore apply a final rotation R prec. (ωt) for rotation about theẑ axis (perpendicular to the Sun) ultimately arriving at the depiction shown in Fig. 4 p (t) = R prec (ωt)R tilt p(Ωt) .
Next, we draw a straight line (ray) from p to the Sun, which in the solar-beam frame is equivalent to adding −Lx to p . We then solve for the value of L for which which is the distance to the point of intersection of the ray with the surface of the Earth closest to the Sun. The value of L is then the time-dependent slab length For a given day of the year the day-averaged rate one should make the replacement Ωt → ωt + Ωt (accounting for the slow-drift in position from the Earth's precession relative to the Sun) is given by with ωt = 2π × (day i /365). For the year-averaged quantity used for limit setting in this paper we have Finally, in a crude two-region model of the Earth's interior, where there is a sharp boundary between the core and mantle at some radius r < R ⊕ , the same basic machinery can be used to estimate the time-averaged value of n A and Z eff (which, as discussed above, will generally be time dependent for a heterogeneous model of the Earth). Let us take p (t) − Lx again, but now search for solutions of the form  Table 3. of [39] and a density of ρ = 4 g/cm 3 . Elements with ni ≤ 7 · 10 19 cm −3 are omitted.
for which there will generically be two solutions L ± (t) corresponding to when the ray goes enters and exits the core. Taking the difference of these two quantities then yields the path-length that resides within the core, which we will denote L core . The mantle length can then be found from the full slab-length as L mantle = L slab − L core . As we will discuss in the next section, for the λ R ⊕ limit this information is sufficient to estimate the added flux due to the high-density, high-Z material of the Earth's core.

Appendix B: Effective composition of the Earth
Our choice of parameters for the uniform-density model of the Earth was driven by conservative considerations. First, the core of the Earth will always yield a higher-rate of ν → N production than an equivalent volume filled with mantle, and so treating the core as if it were made of mantle will always underestimate the rate of N production. The average density of Earth is roughly 5.5 g/cm 3 ; we conservatively use ρ = 4 g/cm 3 in setting our limits since for certain regions of parameter space we consider only that outer mantle contributes significantly to upscattering.
For shorter decay lengths probed by SK (namely λ 1 km) one may worry that ρ = 4 g/cm 3 is not a conservative choice since the crust has a slightly lower density of roughly 2.7 g/cm 3 . This concern turns out to be unwarranted, however, because in this limit both the crust beneath and above the experiment (i.e the overburden) would contribute to scattering roughly doubling the signal over the "night only" estimates presented here. We therefore conclude that ρ = 4 g/cm 3 is indeed conservative over the full range of parameter space we consider.
The elemental composition of the Earth is given in Table 3. of [39] and we use the DMM column, which gives the mass percentage by molecular compound. We are ultimately interested in where the bracketed sum is adding up Z 2 within the atoms of each molecular compound. This can be reexpressed in terms of the mantle density via where f m is the mass fraction given in Table 3. of [39] (Bulk DMM). We can define Z 2 eff via which is a density independent definition. The effective number density is then defined as As we have shown above, for m N ≤ 5 − 10 MeV, the combined sensitivity of Borexino and Super-Kamiokande probes parameter space where λ R ⊕ . For internal consistency, we do not make use of the high-density, high-Z nature of the Earth's core in the main text, because for the higher mass regions m N ≥ 1 − 10 MeV (depending on the experiment) the parameter space being probed satisfies λ R ⊕ . In this limit, any upscattered particles from the Earth's core would decay prior to reaching the detector and so, for limit setting purposes, including the Earth's core in the definition of the average column density is unrealistically aggressive.
For the low-mass region m N 2 MeV, however, it is easy to include the effect of the Earth's core, since in this region λ R ⊕ , most all of the upscattered particles will survive to the detector, and the core just serves to enhance the effective column density. The tricky part to include is the geometry of the core, which only appears along the line-of-sight to the Sun in the winter months the details of which are detector latitude dependent (see e.g. Fig. 3 of [33]). Using the procedure outlined at the end of