XENONnT and LUX-ZEPLIN constraints on DSNB-boosted dark matter

We consider a scenario in which dark matter particles are accelerated to semi-relativistic velocities through their scattering with the Diffuse Supernova Neutrino Background. Such a subdominant, but more energetic dark matter component can be then detected via its scattering on the electrons and nucleons inside direct detection experiments. This opens up the possibility to probe the sub-GeV mass range, a region of parameter space that is usually not accessible at such facilities. We analyze current data from the XENONnT and LUX-ZEPLIN experiments and we obtain novel constraints on the scattering cross sections of sub-GeV boosted dark matter with both nucleons and electrons. We also highlight the importance of carefully taking into account Earth's attenuation effects as well as the finite nuclear size into the analysis. By comparing our results to other existing constraints, we show that these effects lead to improved and more robust constraints.


I. INTRODUCTION
It is estimated that 85% of the matter content of the Universe is in the form of a hypothetical kind of matter, dubbed dark matter (DM) [1].One of the biggest mysteries in contemporary physics and astronomy is to understand its microscopic nature.However, since DM does not interact with photons and interacts very "weakly" with ordinary matter, it proves challenging to detect it.On the other hand, DM gravitational effects on visible matter allow us to infer its existence despite its elusiveness.One of the most compelling solutions to the DM puzzle assumes it to be in the form of some unknown particle [2], thus calling for an extension of the Standard Model (SM).Several strategies, including direct and indirect detection experiments and collider searches, have been developed to try to detect it [3,4].Although a conclusive finding of DM has not been achieved yet, these searches have imposed very tight constraints on its potential properties.As part of the continuous effort to understand this enigmatic component, new experiments, and observations are being carried out.
The possibility that DM has been produced thermally in the early Universe and that its abundance is determined by thermal freeze-out has motivated numerous large direct detection (DD) experiments, which aim at observing the scattering of a DM particle off a target in a deep underground detector.These experiments have experienced an increasingly, decades-long progress which has brought them into the multi-ton era [5].Current most-sensitive constraints in the highmass regime include those set by liquid xenon (LXe) experiments like LUX-ZEPLIN (LZ) [6,7], XENONnT [8,9], XENON1T [10], PandaX-II [11] and LUX [12], together with measurements on liquid argon (LAr) detectors like DEAP-360 [13] and DarkSide-50 [14] and the solid-state cryogenic detector of SuperCDMS [15].
We are interested in the results recently released by two DD experiments, XENONnT [8] and LZ [7].Both experiments use state-of-the-art LXe detectors that aim at observing low-energy electron and nuclear recoils induced by DM scattering.Being one of the most sensitive DM DD experiments at present, XENONnT [16,17], installed at the Gran Sasso National Laboratories in Italy, is the upgrade phase of XENON1T [10].Thanks to its larger active target mass, superior photon detection mechanism, and extremely low background, XENONnT is an order of magnitude more sensitive to weakly-interacting DM particles than its predecessor.The recently released XENONnT data correspond to a total exposure of 1.16 tonne×years [8].The LZ experiment [18], located at the Sanford Underground Research Facility in South Dakota, is a detector centered on a dual-phase time projection chamber, also filled with LXe.The recently available LZ data correspond to an exposure of 5.5 tonne×60 days [7].The LZ collaboration has reported results from a blind search for DM particles and established the current strongest constraint for masses above 9 GeV, testing a cross section as small as 6 × 10 −48 cm 2 at a DM mass of 30 GeV.
In this work, we investigate the possibility that the DM in the Milky Way halo is boosted to semi-relativistic velocities, via its scattering on the DSNB [57,58].The DSNB is a cumulative and isotropic flux of MeV neutrinos of all flavors produced from core-collapse supernovae explosions along the whole history of the Universe.While not yet observed, the DSNB is an irreducible background, expected to be within the reach of near-future experiments.Even though less energetic than cosmic rays, it seems reasonable to assume possible interactions of local DM with this isotropic neutrino background.By employing XENONnT and LZ latest data releases [7,8], we derive stringent constraints on both DM-electron and DM-nucleon scattering cross sections in the sub-GeV range, thus providing complementary results to the standard analyses offered by the two collaborations [7,9].We highlight and pay special attention to the Earth's attenuation effects, that, as we will show, play an important role in the region of interest of the parameter space.Additionally, we also take into account nuclear effects which further improve the sensitivity and robustness of our analysis.DSNB-boosted DM had previously been considered in Ref. [45] as a possible explanation to an excess of electron recoil events in the low energy region, now disappeared, observed by XENON1T [59].Ref. [46] also set limits on DSNB-boosted DM scattering off electrons using XENON1T and Super-Kamiokande data.Here we improve upon these previous results by presenting for the first time constraints on DSNB-boosted DM, from the most recent XENONnT and LZ data, for both nuclear and electron scattering.
The remainder of this paper is organized as follows.Section II provides a discussion on theoretical predictions for the DSNB flux.Sec.III explains how sub-GeV non-relativistic DM particles in the Milky Way halo can attain semi-relativistic speeds due to interactions with DSNB neutrinos, and highlights the importance of Earth's attenuation effects as well as the nuclear form factors.In Sec.IV, we delve into the simulation of the DSNB-boosted DM-induced signal predicted for the XENONnT and LZ detectors.Our results are presented in Sec.V, while we finally provide our concluding remarks in Sec.VI.

II. THEORETICAL ESTIMATE OF THE DSNB FLUX
Right after the first star formation event, the Universe has been surrounded by an isotropic flux of MeV-energy neutrinos and antineutrinos of all flavors, produced from all supernovae events from the core-collapse explosions of huge stars throughout the Universe.The theoretical prediction for the differential DSNB flux, per neutrino flavor α, can be estimated as [57,[60][61][62] E s ν being the neutrino energy at the source.The integral is performed over the redshift parameter, z, and we take the maximum redshift at which star-formation occurs as z max ∼ 6.Moreover, H(z) is the Hubble function determined from the Friedmann equation as where H 0 = 67.45km s −1 Mpc −1 is the Hubble constant [1,63], Ω M = 0.315 ± 0.007 and Ω Λ = 0.685 ± 0.007 denote the matter and vacuum contributions to the present-Universe energy density, while the best current measurement for the equation-of-state parameter for the dark energy is w = −1.028±0.031[1].The DSNB flux further depends upon the rate of Core-Collapse Supernovae (CCSN), which reads [61] R CCSN (z) = ρ * (z) where ψ(M ) is the initial mass function (IMF) of stars, indicating the star density within a certain mass range.For our analysis we have assumed the IMF to be a power-law distribution, ψ(M ) ∝ M −2.35 according to [64].The redshift evolution of the co-moving cosmic star-formation rate, ρ * (z), can be modelled as [60,65] ρ * (z where the overall normalization factor is ρ0 = 0.0178 +0.0035 −0.0036 M ⊙ yr −1 Mpc −3 [61].The constants B, and C are expressed as [60,65]:  1).The bands illustrate a 40% error in the normalization uncertainty of the DSNB spectra [60].
where z 1 = 1, and z 2 = 4 represent the redshift breaks, while a, b and c denote the logarithmic slopes for the low, intermediate, and high redshift ranges.An analytical fit to data from different astronomical surveys [60,65] gives Finally, a non-degenerate Fermi-Dirac distribution is used to parametrize the flavor-dependent neutrino spectra released by a CCSN event [57,61] where E tot ν = 3 × 10 53 erg,1 represents the total amount of energy released as neutrinos [57], and T να denotes the temperature of each flavor of neutrinos.In our present study, we consider T νe = 6.6 MeV, T νe = 7 MeV, T νx = 10 MeV (ν x denotes either ν µ or ν τ or their antiparticles), satisfying the upper limit extracted from Super-Kamiokande [66].
We show in Fig. 1 the predicted DSNB fluxes, for the different neutrino flavors, as a function of the neutrino energy.In the following calculations we will assume an uncertainty of 40% in the normalization of the DSNB spectra, estimated from uncertainties in the cosmic star-formation rate [60].This uncertainty on the DSNB fluxes is illustrated by the shaded bands in Fig. 1.

III. THE DSNB-BOOSTED DARK MATTER FLUX
In this section, we discuss how the DM particles in the Milky Way halo get boosted to considerably greater velocities due to their scattering with DSNB neutrinos.We remain agnostic of the specific DM model 2 and for the sake of uniformity in comparing the final results we assume the DM to be made of one particle species χ that scatters with neutrinos and electrons (σ νχ = σ χe ) or with neutrinos and nucleons (σ νχ = σ χn ) with the same cross section, as the benchmark for our analysis.These assumptions can be naturally realized in flavor-dependent gauged U (1) extensions such as U (1) B i −3L i , i being generation index or U (1) B−3L i models [68][69][70] (for an overview of such a model, see Appendix E).Furthermore, scenarios deviating from this assumption can be easily accounted for by using the product √ σ νχ σ χe or √ σ νχ σ χn as applicable, see Sec.V for further discussion.Before entering into details, it is noteworthy to stress that the initial DM galactic escape velocity is irrelevant [19], as the scattering between χ and DSNB neutrinos accelerates the DM to significantly higher velocities.
The DSNB-boosted DM differential flux, induced by its scattering with the DSNB given in Eq. ( 1), can be estimated as [45] where T χ is the energy transferred to χ and dΦ DSNB ν dEν is the sum over all neutrino flavors of the DSNB flux given in Eq. ( 1).The neutrino-DM scattering cross section can be cast in the form where m χ denotes the DM mass, while σ νχ controls the strength of the interaction.The maximum transferred energy to which the DM can be boosted for a given neutrino energy E ν , is dictated by the kinematics of the process and is incorporated in the Heaviside step function3 : The maximum neutrino energy in our numerical calculations is taken to be E max ν = 100 MeV, while the lower integration limit in Eq. ( 7) can be obtained by inverting the expression for T max χ which gives the minimum neutrino energy required to boost the DM, i.e.
The D−factor (D halo ) in Eq. ( 7) encodes the DM density distribution within our galactic halo, and it is expressed as the integral of the density profile along the line of sight (l.o.s.) ℓ and over the solid angle Ω: Here, we assume a Navarro-Frenk-White (NFW) profile4 , defined as [72] ρ where the scale radius is r s = 20 kpc and the local DM density is ρ ⊙ = 0.4 GeV cm −3 .The galactocentric distance reads with r ⊙ = 8.5 kpc being the distance between the Earth and the galactic centre and ψ the angle of view defining the l.o.s..The upper limit of the l.o.s.integral is given by ℓ max = R 2 − r 2 ⊙ sin 2 ψ + r ⊙ cos ψ, with the galactic halo virial radius taken to be R = 200 kpc.Given these values, we hence obtain D halo = 2.22 × 10 25 MeV cm −2 over the whole galactic halo.

A. Attenuation effects
In this subsection, we will focus our attention on the modifications expected to occur in the energy profile of the DSNB-boosted DM flux during its propagation through the atmosphere and the Earth [19,26,[73][74][75][76].For sufficiently large interaction cross sections, dσ χi /dT i , the DM particles may lose a significant amount of energy due to their scattering on nuclei (i = N ) or electrons (i = e), resulting into a sizeable attenuation of the DM flux before reaching the detector.This effect can be accounted for via the energy loss equation [19,26] where T i denotes the energy lost by the boosted DM particle in a collision and n i is the number density of nucleus species or electrons.Here, z denotes the distance travelled from the location of the scattering point (inside the atmosphere or the Earth) to the detector.In the most general case, Eq. ( 12) relates the initial energy at the top of the atmosphere (z = 0), T 0 χ , with the average kinetic energy, T z χ , after travelling a distance z before reaching the underground detector.In our analysis, we neglect the impact of atmospheric attenuation as it is expected to be negligible compared to Earth's attenuation [26], for the cross sections under consideration.Hence, we take z = 0 at the Earth's surface.Then, the distance z can be expressed as (see Appendix A for more details) where R E stands for the radius of the Earth, θ z refers to the detector's zenith angle and h d indicates the depth of the detector's location from the Earth's surface, at the point where the zenith angle is zero.Moreover, for the sake of simplicity, we have adopted a mean average electron density n e of Earth's most abundant elements between the surface and depth z, n e = 8 × 10 23 cm −3 [20].In the case of attenuation due to χ scattering on nuclei, we have determined the nuclear number density at depth z through a weighted average of the most abundant elements found in the Earth's crust, mantle, and core, yielding n N = 3.44 × 10 22 cm −3 and A ≈ 33.3 (for details see Appendix B and Refs.[73,[77][78][79]).
The differential cross section for DM-electron or DM-nucleus scattering takes the form where the maximum recoil energy that can be lost by χ during the attenuation process, is obtained from the kinematics of the process and reads with m i indicating the nuclear (i = N ) or electron (i = e) mass.The solution of Eq. ( 12) gives the DM energy as a function of the distance and the initial DM energy, i.e.T z χ ≡ T z χ (T 0 χ , z) with z depending on the zenith angle and the detector depth as indicated in Eq. ( 13).The resulting attenuated DM flux reaching the detector after averaging over angles5 , dΦ z χ /dT z χ , is given by the expression where Ω is the solid angle.

Scattering with electrons
For the case of DM-electron scattering we have σ χi = σ χe in Eq. ( 14).In this case Eq. ( 12) can be solved analytically.The solution for T z χ at a given depth z can be expressed in terms of the DM energy at the surface, T 0 χ , as where l E represents the mean free path for energy loss, given by l −1 E = n e σ χe 2memχ (me+mχ) 2 .By inverting Eq. ( 17) we obtain the expression for T 0 χ as a function of T z χ and z, which reads As a consequence, the attenuated DM flux given in Eq. ( 16) that eventually reaches the detector can be simplified as follows Before closing this discussion let us stress that, as discussed before, our analysis is done for the benchmark σ νχ = σ χe .Note that the bounds that we will eventually obtain in Sec.V will mainly depend on the product √ σ νχ σ χe , which simplifies to σ χe under the assumption σ νχ = σ χe , plus corrections due to the dependence of l E (see Eq. 17) on σ χe .We will discuss this in more detail in Sec.V.

Scattering with nuclei: Effect of the finite nuclear size
For the case of DM-nucleus scattering we take σ χi = σ SI χN in Eq. ( 14), where the spin-independent (SI) DM-nucleus elastic scattering cross section is expressed as6 [80, 81] where A denotes the atomic mass number of the target nuclei, q = √ 2m N T N stands for the threemomentum transfer, σ χn is the DM-nucleon SI cross section and µ N (µ n ) represents the DM-nucleus (DM-nucleon) reduced mass.We again take the benchmark σ νχ = σ χn for our analysis, but, as mentioned above, our bounds are mainly dependent on √ σ νχ σ χn (see Sec. V).The SI cross section σ SI χN is a momentum-dependent quantity due to the presence of the nuclear form factor, F (q 2 ), which accounts for the finite nuclear size and has been parametrized by a Helm-type7 effective form factor [82].Notice that the energy dependence of the SI cross section prevents us from obtaining an analytical solution for Eq. ( 12), unlike the case of DM-electron scattering discussed above, hence we need to solve Eq. ( 12) numerically.The fact that the DM travels through Earth to reach the detectors, leads to attenuation of the flux.In Fig. 2 we present the angle-averaged DSNB-boosted DM flux for unattenuated and attenuated cases.The results are plotted for the benchmark parameters m χ = 300 MeV and σ νχ = 10 −29 cm 2 , assuming a depth of h d = 1.4 km which corresponds to the underground location of XENONnT.Note that the results remain essentially unchanged for the case of LZ (h d = 1.47 km).The solid blue line in Fig. 2 corresponds to the unattenuated flux.The dashed lines show the attenuated fluxes corresponding to attenuation due to DM-nucleus (orange, dashed) and DMelectron (green, dashed) scattering as a function of the DM energy.For the case of χ−N scattering, the effect of the finite nuclear size is illustrated by comparing the resulting fluxes for two cases: i) by incorporating the Helm form factor in the calculation (orange, dashed) and ii) by assuming F = 1 i.e. completely ignoring nuclear physics (orange, dotted).
As can be seen, Earth's attenuation effects shift the peak of the DSNB-boosted DM flux towards lower energies and reduce it up to a factor 2 (3.5) for χ − e (χ − N ) scattering.Furthermore, the high-energy endpoint of the differential DSNB-boosted DM flux spectra exhibits a faster decline when finite nuclear size is neglected, as opposed to the case where the finite nuclear size effects are taken into account.The Earth's attenuation and nuclear size effects play a crucial role in the results presented in the remainder of the work (see also Appendix C).
Before closing this discussion let us stress that in the present analysis, the calculated attenuated flux is not taking into account additional effects related to the direction of DNSB-boosted DM particles after each scattering process nor the possibility of multiple scatterings.These effects only become relevant when the DM mass and energy are significantly lower than the mass of the nucleus [26,73] and are important for probing diurnal effects [83].

IV. DARK MATTER SIGNAL AT UNDERGROUND DETECTORS
After reaching the underground detector, the DSNB-boosted DM can scatter off both the electrons and nuclei of the target material, thus inducing both electronic and nuclear recoils.The differential event rate with respect to the recoil energy T i can be written as [45] where t run and N target denote the exposure time and number of targets of the detector, respectively, while A represents the detection efficiency provided by each experiment.At this point we should clarify that the detection efficiency is provided either in terms of true A(T i ) or reconstructed A(T reco i ) recoil energy, hence its explicit dependence has been dropped in Eq. ( 21) to avoid confusion.Regarding DM-electron scattering, our calculations incorporate the detection efficiency provided by LZ [6] in terms of true recoil energy A(T e ), while for the case of XENONnT we consider the efficiency provided in Ref. [8] in terms of reconstructed recoil energy A(T reco e ).Regarding DM-nucleus scattering, we account for the detection efficiency A(T N ) provided in terms of true nuclear recoil energy as reported by both LZ [7] and XENONnT [9].In what follows, we will use the recent data released by XENONnT and LZ collaborations to put constraints on the DM mass and DM-electron/nucleon cross sections.
We first focus on DM-electron scattering, i.e. we calculate the differential event spectrum dR/dT e given in Eq. ( 21) for i = e.In this case, χ particles scatter off electrons in the underground detector with a cross section σ χe .The angle-averaged DSNB-boosted DM flux, accounting for the attenuation effects (see Sec. III A 1) is given in Eq. (19).Since very low energy scatterings occur, our calculations take into consideration atomic binding effects which lead to a slight cross section suppression at very low recoil energies.To this purpose, the number of target electrons in Eq. ( 21) is expressed as N e target = m det N A Mr × Z eff (T e ) where m det , M r and N A represent the detector mass, the molar mass of the target material and the Avogadro's number, respectively.The recoil energydependent quantity Z eff (T e ) denotes the effective charge of the atomic nucleus that is seen by DM for a given energy deposition T e .The latter can be approximated by a series of step functions that depend on the single particle binding energy of the ith electron, following the Hartree-Fock calculations of Ref. [84].
Turning our attention to the case of DM scattering with nuclei, we calculate the corresponding differential event spectrum dR/dT N that follows from Eq. ( 21) for i = N .For the sake of clarity let us note that in this case, although our calculated event rates refer to DM-nucleus scattering, our results will be always expressed in terms of the fundamental DM-nucleon cross section σ χn , instead of the SI DM-nucleus cross section σ SI χN [see e.g.Eq. ( 20)].In this case the angle-averaged attenuated boosted DM flux has been computed numerically as discussed in Sec.III A 2. Since both the LZ and XENONnT collaborations have reported their measured data in terms of electron-recoil spectra, we convert our calculated nuclear recoil spectrum dR/dT N into an "electron-equivalent" recoil spectrum, according to the expression where the quenching factor, Q f (T N ) = T e /T N , quantifies the energy loss to heat in the aftermath of a DM-nucleus scattering event.In the present analysis we adopt the standard Lindhard quenching factor [85].Notice also that for DM-nucleus scattering the effective charge Z eff is irrelevant and hence we take Z eff = 1.
By following the above method we have verified that our predictions are in excellent agreement with those reported by XENONnT and LZ for both electron and nuclear recoils.First, based on previous work [86], we have calculated the elastic neutrino-electron scattering spectra and found that a total of ∼30 events are expected for LZ and 76 events (300 events in the full region [1,140] keV ee ) for XENONnT, in agreement with Refs.[7,8].Second, regarding nuclear recoils we have calculated the corresponding coherent elastic neutrino-nucleus scattering (CEνNS) expected events induced by 8 B neutrinos and found 0.16 events for LZ and 0.24 events for XENONnT, in agreement with Refs.[7,9], respectively.Following these prescriptions, we show in Fig. 3 the simulated recoil spectra as a function of the electron-equivalent ionization energy, expected at both the LZ and XENONnT detectors.The black points depict the experimental data together with the error bars, when available 8 , as provided by the collaborations.The blue histograms indicate the background events, also given by the collaborations.The red and green histograms represent the sum of the background and of the simulated number of events, assuming m χ = 300 MeV and two different values of χ-nucleon or χ-electron scattering cross sections, as indicated in the legend.In the case of the LZ detector, the light brown histogram further represents the 37 Ar background, which originates from cosmogenic activation of the xenon prior to underground deployment, producing short-lived 37 Ar that decayed during the first run [7].

A. LZ analysis
For the analysis of LZ data, we have performed a spectral analysis using the following Poissonian where R i exp denotes the experimental differential events in the ith recoil energy bin, as reported in Ref. [7], while the predicted differential events contain the DSNB-boosted DM signal, as well as all background components: R i pred ( Ar .It is worth noting that, in accordance with Ref. [88], the R bkg spectrum is calculated by eliminating the 37 Ar contributions from the total background provided in Ref. [7].The nuisance parameters {α, β, δ} are introduced to incorporate the uncertainty on background, neutrino flux distribution9 and 37 Ar components with σ α = 13%, σ β = 40% [60] and σ δ = 100%.For each new physics parameter belonging to − → S (i.e.m χ or σ χi ), we have marginalized the χ 2 function over all nuisance parameters.

B. XENONnT analysis
The following Gaussian χ 2 function is used for the analysis of XENONnT data [87] Here, ) + B i 0 , with B 0 denoting the simulated background mentioned in Ref. [8].The rest of the details are similar to the LZ analysis.

V. RESULTS
We present in Fig. 4 the 90% C.L. exclusion regions on the DSNB-boosted DM, in the planes (m χ , σ χe ) and (m χ , σ χn ).We consider the case of DM scattering off electrons (left panel) and nuclei (right panel), and show both constraints obtained using LZ (blue) and XENONnT (red) experimental data.
In particular, Lyman-α data provide stringent constraints on σ νχ [117,118], although dependent on several assumptions, including massless neutrinos.A more recent analysis, accounting for neutrino masses, points toward a preference for a non-zero DM-neutrino interaction strength [119] thus providing further motivations for our work.Finally, we compare our results with recent bounds on sub-GeV cosmic-ray boosted DM (CRDM), also derived using LZ data [30] (light yellow).While many references in the literature have addressed cosmic-ray boosted DM, Ref. [30] allows for a direct comparison of our results given that we both analyze the same LZ data set.As can be seen, our constraints obtained assuming a boost from DSNB neutrinos are ruling out a larger region of the parameter space.This is understood since the local interstellar population of cosmic rays is about one order of magnitude less intense compared to the flux of DSNB neutrinos, the latter peaking at lower (∼ 10 MeV) energies, though.Notice also that Ref. [30] ignored nuclear-physics corrections, that are rather important for a CRDM-based analysis where a larger momentum transfer is involved (compared to our DSNB-based analysis).The correct inclusion of such effects would drastically modify the CRDM region shown in the plot.Although not shown here, nuclear effects in CRDM studies have been considered by incorporating Helm-type nuclear form factors, for example, in the analysis of XENON1T excess data in Refs.[26,29].At this point we should note that given that the initial CRDM flux peaks beyond 100 MeV and extends up to GeV energies, a large momentum transfer is involved in the process and it cannot be realistically accounted for through the inclusion of nuclear form factors.For an appropriate treatment of nuclear structure at such large momentum transfer see Ref. [28].
While a significant part of our constraints lie in a region of parameter space already probed by other searches, these results highlight the complementarity and significance of the LZ and XENONnT data in probing the sub-GeV DM parameter space.Also, it is worth mentioning that both these experiments have just started taking data and we are only using their very first data sets obtained with exposure time of only a few months, but still the bounds are already competitive with other bounds.As the statistics of these experiments will increase, their data will play a much more important role in constraining the DSNB-boosted DM parameter space.Moreover, and as mentioned in Sec.III, let us recall that we present the limits on σ χe and σ χn under the assumption of σ νχ = σ χe and σ νχ = σ χn , respectively.However, note that the lower limit of our closed regions is basically dependent only on √ σ χe σ νχ and √ σ χn σ νχ , respectively, so it can be easily recast into alternative scenarios in which the magnitude of the two cross sections is different.The upper limit of our closed contours though, has a stronger dependence on the attenuation effects and therefore depends on a more complicated combination of σ χe,n and σ νχ .As it can be noticed, our exclusion regions have a closed shape, due to the inclusion of attenuation effects (see the discussion below).Large scattering cross sections i.e. σ χe ≳ 2 × 10 −28 cm 2 (σ χn ≳ 8 × 10 −28 cm 2 ) for DM scattering off electrons (nucleons) and m χ = 0.1 MeV result into a strong attenuation during the propagation of the DM particles through the Earth and are therefore disfavored.To understand the shape of our bounds in more detail, we now explore the implications of considering attenuation effects and adopting a realistic nuclear form factor.We examine how these factors influence the exclusion limits on the scattering cross section and DM mass.The energy loss experienced by the DSNB-boosted DM due to its scattering with the Earth's material introduces a significant impact on the derived exclusion limits.Figure 5 illustrates the consequences of considering (blue, closed contour) or neglecting (red exclusion line) attenuation effects for χ−e scattering, while Fig. 6 depicts the same for χ−N scattering.Clearly, attenuation effects impose an upper bound on the exclusion region.Above a certain scattering cross section, energy loss becomes substantial such that the DSNB-boosted DM particles cannot be detected because of severe attenuation.This fact confirms the necessity of properly accounting for Earth's attenuation effects to ensure accurate and robust constraints on the DNSB-boosted DM parameters.Finally, for low DM mass, e.g.m χ ≲ 1 MeV, we find that the exclusion regions which incorporate attenuation effects (blue contours) are excluding lower cross sections (below σ χe = 10 −31 cm 2 ) compared to the case where attenuation effects are ignored (red contours) in Fig. 5.This is expected since in the low-mass region, for a fixed m χ the number of events corresponding to the attenuated case is larger compared to the unattenuated one.The reason behind this behavior is twofold: First for low m χ the DM flux reaching the detector is larger at low T χ when attenuation effects are taken into account in comparison to the unattenuated case.Second, the non-vanishing attenuated DM flux at lower T χ is triggering lower recoils T e in the detector, which due to their inversely proportional dependence on the differential cross section given in Eq. ( 14) are eventually leading to an enhancement of the event rates given by Eq. ( 21).The same reasoning applies for the case of nuclear recoils, discussed below, but due to kinematics this effect is not visible in Fig. 6.Further details are given in Appendix D. Furthermore, for the case of χ−N scattering, considering a realistic nuclear form factor [see e.g.Eq. ( 20)] introduces visible effects, as illustrated in Fig. 6.The inclusion of the Helm form factor effectively modifies the energy-loss dynamics compared to the F = 1 scenario.Indeed, the finite nuclear size reduces the differential DM-nucleus cross section given in Eq. ( 14), thus leading to a decrease in the energy-loss rate dT z χ /dz when a larger momentum transfer is involved i.e. for the high-energy tail of DSNB-boosted DM flux (for an illustration see Fig. 10 in Appendix C).This modification results in a shift of the upper bound of the exclusion region, allowing for slightly higher σ χn values before energy loss renders particles undetectable.
Such an exploration of the interplay between attenuation effects and nuclear-physics considerations leads to a more comprehensive and robust understanding of the complex dynamics governing DSNB-boosted DM scatterings.These insights emphasize the significance of accounting for the latter effects in the accurate interpretation of experimental results, providing insights into the implications of χ−N scattering.
Before closing our discussion we should stress that obtaining large, O(10 −30 ) cm 2 , DM-nucleon cross sections from conventional DM models is challenging, primarily due to the stringent bounds imposed on mediators coupled to nucleons, as pointed out in [67,120].A model-dependent study should be performed to thoroughly address the applicability of our limits.The interested reader is for instance referred to [120], where a sub-GeV DM candidate is presented, dubbed HYPER, that can accommodate large DM-nucleon cross sections.This example presents a promising pathway to reconcile large direct detection cross sections with cosmological and laboratory observations.

VI. CONCLUSIONS
In this work we have revisited the possibility that sub-GeV DM is boosted to semi-relativistic velocities through collisions with the DSNB.Such a very energetic component of the total DM flux, while subdominant, would be detectable at DM DD experiments thus amplifying their experimental reach.
We have analyzed the most recent data from two cutting-edge DM experiments, LZ and XENONnT, and we have obtained stringent constraints on the DSNB-boosted DM parameter space.For the first time, we have considered both electron and nuclear scatterings and obtained bounds on the relevant cross sections and DM mass.These new bounds extend the reach of typical DM DD searches to even lower DM mass ranges, and they are consistent with other searches for cosmic-ray boosted DM.In this regard, we have illustrated that due to the higher intensity of the DSNB flux in comparison to cosmic rays, the former allows to exclude a larger part of the available parameter space.We further point out that in obtaining our results for DM-nuclei scattering we reliably account for corrections due to the finite nuclear size by incorporating a Helm-type nuclear form factor.
Our results hence complement other existing searches for sub-GeV DM.Most of all, they show that even with their very first and limited exposure time data sets, the low-threshold XENONnT and LZ experiments dominate the terrestrial limits on DM-nucleus scattering at very low DM masses, with good complementarity to neutrino experiments like Super-Kamiokande and cosmological observations.Finally we have highlighted the importance of including Earth's attenuation effects in the analysis.In particular, we have demonstrated that they have a strong impact on the upper bound of our derived exclusion regions disfavoring large DM scattering cross sections, namely σ χe ≳ 2 × 10 −28 cm 2 and σ χn ≳ 8 × 10 −28 cm 2 for m χ = 0.1 MeV.
In summary our current analysis, by taking into account the Earth's attenuation and finite nuclear size effects, provides accurate and robust constraints on the parameter space of low mass DSNB-boosted DM, using the first data sets of LZ and XENONnT.configuration of a detector located at a depth h d below the Earth's surface, as depicted in Fig. 7 10 .Our primary concern is to determine the distance z that corresponds to the distance traveled by DM particles between a specific impact point on the Earth's surface (I) and the detector (D).This distance depends on the zenith angle, θ z , representing the angle between the vertical direction and the line connecting the detector to the chosen point on the Earth's surface.
For the purpose of computing the distance z we employ the law of cosines within the triangle △ODI (see Fig. 7), leading to the following expression: By means of this expression, we can precisely determine the distance z that characterizes the spatial relationship between the position of the detector and the Earth's surface point of interest for a given zenith angle.Understanding this geometric configuration is crucial for a more realistic simulation of the interactions between DM particles and the detector and in predicting potential signals in DM experiments.For the case of χ−N scattering, incorporating finite nuclear size effects through the nuclear form factor becomes particularly relevant.Figure 10 shows the impact of nuclear effects by considering two distinct scenarios in the calculation of the final kinetic energy of DM particles reaching the detector: (i) including a Helm-type nuclear form factor and (ii) completely neglecting nuclear effects, i.e.F = 1.Notably, the effect driven by nuclear physics becomes evident around z = 0.1 km.In particular for high-energy DSNB-boosted DM particles, the disparity between the two scenarios becomes substantial.All in all, incorporating nuclear physics effects through the Helm form factor leads to a reduction of the total cross section, resulting in a mitigated energy loss.Remarkably, DSNB-boosted DM particles with kinetic energies exceeding 100 MeV undergo such a marginal energy loss that their energy remains nearly constant at distances around z ≲ 100 km.

Appendix D: Behavior of the exclusion regions in the low DM mass regime
Here we provide further clarifications regarding the behavior of the exclusion regions observed in Fig. 5, where the sensitivity of XENONnT and LZ experiments to DM-electron scattering was presented assuming both attenuated and unattenuated DSNB-boosted DM fluxes.For relatively low DM masses m χ , the exclusion limit in Fig. 5 corresponding to DM-electron scattering when attenuation effects are taken into account (blue contours) is extending to lower cross section values compared to the unattenuated case (red contours).As we will explain, this is a direct consequence related to the shape of the energy distribution of DSNB-boosted DM particles reaching the terrestrial detector as well as to the recoil dependence of the differential event rates.
For low values of T χ the attenuated flux becomes significantly larger in comparison to the unattenuated one, a behaviour that is more pronounced when the value of m χ gets smaller.Moreover, a non-vanishing DM flux at lower values of T χ results into detectable signals of lower recoil energies T e .As explained in the main text, the number of detected events is inversely proportional to T e , see e.g.Eqs.(14,21), thus a signal enhancement is expected for lower recoils.As a consequence, the events originating from the attenuated flux are enhanced in comparison to those coming from the unattenuated flux, affecting the sensitivity in the parameter space of (m χ , σ χe ) accordingly.This is particularly relevant for small values of m χ (especially for m χ ≲ 1 MeV).The opposite behavior occurs for larger values of m χ (see the discussion below).
A detailed illustration of these effects is presented in Fig. 11.Each row from top to bottom corresponds to different values of DM mass m χ = 0.1, 1, 100 MeV, while left and right panels show the differential DSNB-boosted DM flux distribution and the respective differential event rates at XENONnT11 .For each case, a comparison is given between the attenuated and unattenuated cases, while the chosen cross section values are allowed by the exclusion limits in Fig. 5. Results in the left panels of Fig. 11 demonstrate that the lower the DM mass the larger the attenuated flux becomes in the low T χ regime.In the right panels, the corresponding event rates show that the lower the DM mass the larger the enhancement of the events coming from the attenuated flux becomes compared to the unattenuated case.Let us finally note that for m χ = 100 MeV, the unattenuated flux is larger and so is the respective number of events; consequently the exclusion region that corresponds to the unattenuated flux is slightly more constraining at this value of m χ .
Before closing, let us stress that the obtained results highlight the intricate nature of direct detection of sub-GeV DM and the importance of considering these factors when interpreting experimental results.(1, 2, −1) (3, 2, iQ j γ µ D µ Q j + iu j γ µ D µ u j + id j γ µ D µ d j + iL j γ µ D µ L j + il j γ µ D µ l j + iν j R γ µ D µ ν j R .

(E2)
■ Gauge Kinetic Sector: The gauge kinetic term is where W a µν , B µν , and Z ′ µν are the field strength tensors corresponding to the gauge groups SU (2) L , U (1) Y , and U (1) B 1 −3Le respectively.■ Yukawa Sector: The Lagrangian density in the Yukawa sector is given by R ν e R ϕ + h.c. .
The cross sections for DM scattering with electrons, nucleons, and neutrinos through Z ′ mediator can be expressed as follows: • DM-electron scattering (σ χe ): DM particles can scatter off electrons in direct-detection experiments through Z ′ exchange; the DM-electron cross section can be expressed in terms of the coupling [121] σ χe = 36x 2 µ 2 χe g 4 where µ χe denotes the DM-electron reduced mass and m Z ′ the mediator mass.
• Neutrino-DM scattering (σ νχ ): Finally, the neutrino-DM cross section can be expressed as [123,124] Thus, the B 1 − 3L e gauge symmetry simultaneously provides a framework for the DM matter extension of SM and naturally leads to our benchmark scenario.

FIG. 1 .
FIG.1.Predicted DSNB fluxes for various neutrino flavors (ν x denotes either ν µ or ν τ or their antiparticles), as a function of the neutrino energy, estimated from Eq. (1).The bands illustrate a 40% error in the normalization uncertainty of the DSNB spectra[60].

FIG. 2 .
FIG.2.The angle-averaged DNSB-boosted DM flux distribution as a function of the DM energy for m χ = 300 MeV, σ νχ = 10 −29 cm 2 and a detector depth of h d = 1.4 km for the benchmarks σ νχ = σ χe and σ νχ = σ χn .The unattenuated flux is shown by the solid blue line.The attenuated DM flux for the case of DM scattering with electrons (nuclei including Helm-type form factor) inside the Earth is displayed with a green (orange) dashed line.The effect of the nuclear form factor is illustrated by comparing the results assuming a Helm-type form factor (dashed line) and F = 1 (dash-dotted line).See main text for more details.

,
FIG. 3. Simulated signal (colored histograms) and experimental data (black points with error bars) as a function of the electron-equivalent ionization energy, for the LZ (left) and XENONnT (right) experiments.The DSNB-boosted DM events have been computed assuming m χ = 300 MeV and different values of scattering cross sections, see the legend.The blue and light brown histograms refer to the background events as provided by the collaborations.

FIG. 4 .
FIG. 4. 90% C.L. exclusion regions in the DSNB-boosted DM parameter space, obtained for scattering off electrons, for the benchmark σ νχ = σ χe (left panel) and nuclei, for the benchmark σ νχ = σ χn (right).We show results obtained with LZ (blue) and XENONnT (red) data.For comparison purposes, existing limits from other studies are also shown (see main text for details).

2 ) 2 )
FIG. 5. 90% C.L. exclusion limits on σ χe and m χ , obtained including (blue closed contour) or neglecting (red line) attenuation effects due to DM scattering off Earth's elements before reaching the detectors.Left panel shows bounds obtained with XENONnT data, while right panel refers to LZ data.

1 FIG. 6 .
FIG. 6. 90% C.L. exclusion limits on σ χn and m χ , obtained including (blue closed contour) or neglecting (red line) attenuation effects due to DM scattering off Earth's elements before reaching the detectors.Left panel shows bounds obtained with XENONnT data, while right panel refers to LZ data.Additionally, the impact of the nuclear form factor is shown, with solid lines representing the Helm form factor, and dashed lines obtained assuming F = 1 for both attenuated and unattenuated scenarios.

FIG. 7 .
FIG. 7. Geometry of the underground detector's location at a depth h d below Earth's surface, and distance z traveled by DM before detection.

FIG. 9 .
FIG. 9. Contour plot of values of T z χ in the (z, T 0 χ ) plane, for m χ = 300 MeV and σ χe (σ χn ) = 10 −29 cm 2 .The left panel refers to DM-electron scattering, while the right panel refers to DM-nuclei scattering.Both panels depict the variation of T z χ across different values of T 0 χ and z, providing insights into the interplay between these parameters.

1 FIG. 10
FIG. 10. T zχ as a function of the distance z, for different initial kinetic energies of DM particles at Earth's surface (T 0 χ ), focusing on χ − N scattering interactions.The DM particle mass and cross section are fixed to m χ = 300 MeV and σ χn = 10 −29 cm 2 .The depicted results are obtained considering both the Helm form factor (solid lines) and F = 1 (dotted lines).

σχe = 9 . 9 ×
FIG.11.Left: Differential DSNB-boosted DM flux distribution as a function of the DM energy (T χ ) for both attenuated and unattenuated scenarios.Right: Simulated differential events for the XENONnT experiment in response to DSNB-boosted DM interactions with electrons, corresponding to both attenuated and unattenuated scenarios.From top to bottom rows the DM mass is fixed at m χ = 0.1, 1 and 100 MeV, while the corresponding cross section values are obeying the exclusion limits of Fig.5and read σ χe = 6.6 × 10 −32 , 4.5 × 10 −31 and 9.9 × 10 −31 cm 2 , respectively.

TABLE I .
[77,78]ies of the most abundant elements in the Earth's geosphere.The table presents the main elements found within the Earth's crust, mantle, and core[77,78].For each element, the respective mass number (A), relative abundance, and nuclear number density (n N ) are provided.

TABLE II .
Matter content and charge assignment of the gauged B 1 − 3L e model.