Searching for Afterglow: Light Dark Matter Boosted by Supernova Neutrinos

A novel analysis is performed, incorporating time-of-flight (TOF) information to study the interactions of dark matter (DM) with standard model particles. After supernova (SN) explosions, DM with mass $m_\chi\lesssim\mathcal{O}({\rm MeV})$ in the halo can be boosted by SN neutrinos (SN$\nu$) to relativistic speed. The SN$\nu$ boosted DM (BDM) arrives on Earth with TOF which depends only on $m_\chi$ and is independent of the cross section. These BDMs can interact with detector targets in low-background experiments and manifest as afterglow events after the arrival of SN$\nu$. The characteristic TOF spectra of the BDM events can lead to large background suppression and unique determination of $m_\chi$. New cross section constraints on $\sqrt{\sigma_{\chi e} \sigma_{\chi\nu}}$ are derived from SN1987a in the Large Magellanic Cloud with data from the Kamiokande and Super-Kamiokande experiments. Potential sensitivities for the next galactic SN with Hyper-Kamiokande are projected. This analysis extends the existing bounds on $\sqrt{\sigma_{\chi e}\sigma_{\chi \nu}}$ over a broad range of $r_\chi=\sigma_{\chi \nu}/\sigma_{\chi e}$. In particular, the improvement is by 1-3 orders of magnitude for $m_\chi<\mathcal{O}(100\,{\rm keV})$ for $\sigma_{\chi e}\sim\sigma_{\chi \nu}$. Prospects of exploiting TOF information in other astrophysical systems to probe exotic physics with other DM candidates are discussed.

Introduction.-Althoughthere is compelling evidence on the existence of dark matter (DM) as an additional gravity source, its properties and interactions remain unknown [1,2].Experimental searches of DM are intensely pursued worldwide [3][4][5][6][7][8][9][10][11][12].Direct detection (DD) experiments focus on the weakly interacting massive particle (WIMP) scenario of DM mass m χ ≳ O(GeV), with sensitivities approaching the neutrino floor [2].The search for lighter WIMPs is an active area of research.One scenario with rapidly expanding interest is where light DM is upscattered or boosted by known cosmic particles including baryons, electrons, and neutrinos .The boosted DM (BDM) then carries kinetic energy T χ much larger than when it is nonrelativistic (with velocity v χ ∼ 10 −3 ) according to the Halo model.Nuclear and electron recoil events from BDM interaction with the detector targets will therefore have increased energy deposition, making DM with m χ ≲ O(GeV) experimentally accessible.Time-of-flight (TOF) techniques are matured laboratory tools for differentiation or measurement of particle masses.This technique, however, has not been well exploited to probe exotic physics in astrophysical systems.One notable exception is the neutrino mass constraints derived from the timing distributions of supernova neutrinos (SNνs) from SN1987a [42,43].We explore in this Letter a novel scenario of BDM with kinetic energy injected by SNν interactions, and in particular where the prompt SNν burst is also detected, providing a time-zero definition in terrestrial experiments.The prompt SNν events will be followed by time-evolving BDM afterglow events where energy and time can be measured.The delay time between BDM and SNν is a distinctive "smoking-gun" signature and provides unique information to infer m χ , independent of the interaction cross section. Secifically, a delay time of ∆t ≃ 10 days×[R/(8 kpc)][m χ /(10 keV)] 2 [T χ /(10 MeV)] −2 for SNνBDM traveling an astronomical distance R before reaching the Earth highlights that although BDM has v χ ∼ c, the delay can be substantial but measurable in a duration post the arrival of SNν.In contrast, most proposed BDM scenarios rely on steady sources, e.g., cosmic rays [13-17, 19-29, 32, 35-37], stellar ν [18,30], diffuse SNν [31,34], etc, for which the BDM flux is constant with time and lacks any time-dependent feature.
We explore the signatures of SNνBDM with SN1987a in the Large Magellanic Cloud (LMC) and a future supernova (SN) in the Galactic Center (GC) to derive the fluxes and the associated electron-recoil event rates via σ χe in multikiloton water Cherenkov detectors, including Kamiokande, Super-Kamiokande (Super-K), and Hyper-Kamiokande.The scenario of SNνBDM depends on finite DM cross section with ν (σ χν ), which may originate from an effective Lagrangian χΓχ li Γℓ i /Λ where χ and ℓ i = (ν i , i) are the DM and SM fields with i = e, µ, τ .The vertex Γ denotes the interaction type and Λ indicates certain cutoff scale.Possible interactions between χ with ν is a subject of intense recent interest [44][45][46][47][48][49][50][51].They can naturally arise in many particle physics models such as the extensively studied B − L and L µ − L τ , where the new gauge bosons can kinematically mix with the standard model photon.Further constraints will be provided by this work.
DM boosted by SNν.-Assuming a SN explodes near the center of a galaxy (location O in Fig. 1), it emits a large amount of O(10) MeV neutrinos within τ ≈ 10 s carrying total luminosity L ν,tot ≈ 3 × 10 52 erg s −1 .We approximate these SNν by an expanding thin spherical shell with a radius r away from O and a thickness d ≈ cτ (see Fig. 1).The radially propagating SNν within the shell has a number density of where L νi = L ν,tot /6 is the luminosity of each flavor (ν e , ν µ , ν τ and their antineutrinos).We take the average energy ⟨E νe ⟩, ⟨E νe ⟩, and ⟨E νx ⟩ (ν x ∈ {ν µ , ν τ , νµ , ντ }) to be 11, 16, 25 MeV, respectively [52].The energy distribution follows a Fermi-Dirac distribution f νi with a pinch parameter η νi ≡ µ νi /T νi = 3, such that T νi ≈ ⟨E νi ⟩/3.99.With a nonvanishing DM-ν interaction, these neutrinos can upscatter DM in the halo [with number density n χ (r)] when they propagate outward.The BDM from location A can reach the Earth at B (with a distance R away from the center) with a scattering angle α after traveling a length ℓ.At neutrino energy E ν much larger than the typical DM kinetic energy in the halo, DM can be approximated as at rest, and the BDM kinetic energy is given by where θ c ∈ [0, π] is the scattering angle in the center-ofmass (c.m.) frame.One can relate θ c to the lab frame Assuming σ χν is independent of θ c in the c.m. frame, the normalized BDM angular distribution in the lab frame is given by such that dΩ α f χ (α, E ν ) = 1 for any given E ν , where dΩ α = 2π sin αdα.In Fig. 2, we plot 2π sin αf χ (α) for a fixed T χ = 10 MeV (corresponding to different E ν ) with different m χ .It shows that for BDM with m χ /T χ ≪ 1, they are confined within a small scattering angle relative to the direction of SNν.The BDM emissivity j χ at location A can be written as where the BDM velocity v χ /c = T χ (2m χ + T χ )/(m χ + T χ ), and can be evaluated using Eqs.( 1) to (3).Time-dependent BDM flux at Earth.-To obtain the BDM flux (number of BDM per unit time per unit energy per solid angle) at Earth dΦ χ /(dT χ dΩ) (location B in Fig. 1), we shall integrate all j χ along the line of sight ℓ, where dΩ = 2π sin θdθ is viewed from B. The Heaviside functions limit j χ to being nonzero only within the spherical shell of width d where SNν are present.The arrival time of BDM, t ′ , relative to the time of SN explosion, includes the propagation time of SNν from O to A (r/c) and the traveling time of BDM from A to B (ℓ/v χ ).Integrating Eq. ( 5) over dΩ and approximating where appears due to the change of variable dℓ = J dt ′ .Note that for a given (t ′ , θ), one can find a unique solution of (r, ℓ, α) and compute the integration.BDM flux from SN in the GC and LMC.-We now compute the BDM fluxes at the Earth from SN1987a in LMC and from a SN in the GC.We characterize n χ in the Milky Way (MW) and LMC by Navarro-Frenk-White (NFW) and Hernquist profiles respectively.Both share the same expression with (n, ρ s , r s ) = (2, 184 MeV cm −3 , 24.4 kpc) for MW [53] and (n, ρ s , r s ) = (3, 68 MeV cm −3 , 31.9 kpc) for LMC [54].
The distances R for the two are (R GC , R LMC = (8.5, 50) kpc.We neglect the contribution from r < 10 −5 kpc since the profile in the inner region is highly uncertain and the adopted profile diverges when r → 0.
Fig. 3 shows dΦ χ /dT χ versus t > τ for T χ = 10 MeV with different m χ for SN in the GC (solid) and in LMC (dot-dashed), assuming σ χν = 10 −35 cm 2 .Note that we define a shifted time coordinate t = t ′ − R/c as the delayed arrival time for BDM relative to SNν.For m χ = 1 keV and 1 MeV, the most prominent feature is that the BDM fluxes contain a rising part and peak at t p ≈ R(1/v χ − 1/c).This is mainly due to the increase of n χ ∝ r −1 toward the halo center.The postpeak tails are due to BDM contributions with larger scattering angles.For m χ = 1 eV, t p ≈ 0.004 s is too short and overlaps with the 10 s duration of SNν to be shown in Fig. 3. Comparing BDM fluxes coming from the GC to LMC, the LMC cases have smaller fluxes and larger t p due to larger R and smaller halo density.
Fig. 3 also shows another important feature-the BDM flux for a given T χ and m χ vanishes after some time post t p , which is related to the sharp cutoff of f χ shown in Fig. 2.This allows us to consider a reduced duration for BDM searches after the arrival of SNν.Practically, a detector that can probe BDM has a threshold energy T th , below which the detector is insensitive to BDM.Thus, for a given m χ , one can define the latest possible arrival time of BDM with T χ = T th as the vanishing time t van to analyze the data.We stress that all these time-dependent features only depend on m χ but not σ χν .Consequently, if such BDM is detected, analyzing the time profile of the signal will allow direct measurements of m χ .
Events in Kamiokande and Super-K.-ForBDM that also interact with electrons with a cross section σ χe , they can produce signals in neutrino or DM experiments.The total event number N χ induced by BDM with T th ≤ T χ ≤ T max within an exposure time t 0 ≤ t ≤ t exp is given by with N e the total target number of electrons and ϵ the signal efficiency.We consider the water Cherenkov experiments, Kamiokande and Super-K, to calculate N χ for BDM from LMC (by SN1987a) and from a SN in GC.They have N e = (M T /m H2O )N A n e with M T the fiducial detector mass, m H2O the water molar mass, N A the Avogadro constant and n e the electron number per water molecule.We take M T = 2.2 and 22.5 kton for Kamiokande and Super-K, respectively [55,56], and set (T th , T max ) = (5, 100) MeV for both.We make a conservative choice of taking ϵ = 50%, lower than the energy-dependent efficiency roughly ranging from 50% to 75% reported for solar ν detection in Super-K [56].For signal duration, we consider t 0 = 10 s to approximately exclude events produced by SNν, and let t exp = min(t van , t cut = 35 yrs) depending on m χ .For the LMC case, the considered duration thus includes the running time of Kamiokande from 1987 to 1996 and Super-K after 1996 for heavier m χ .For the GC case, we consider Super-K only.The main background for both comes from the solar and atmospheric neutrinos for T χ ≲ 20 MeV and T χ ≳ 20 MeV, respectively.We adopt values in Table XIV of Ref. [56] and the FLUKA simulation result in Ref. [57] to estimate the background.Fig. 4 shows N χ vs. m χ resulting from the GC and LMC given σ χν = σ χe = 10 −35 cm 2 .We first discuss the GC case where only Super-K is considered.The red-solid dots show that N χ ∝ m −1 χ perfectly for m χ ≤ 25 keV, which corresponds to having t exp = t van ≤ t cut .This is because for smaller m χ , all BDM arrive at the detector before t cut so that N χ is proportional to the amount of DM in the halo.For heavier m χ , however, a larger part of BDM flux only arrives after t cut (see Fig. 3), leading For LMC, events in Kamiokande (hollow-green squares) and Super-K (hollow-orange triangles) are shown separately.For the GC, only Super-K (red dots) is considered.Background counts (dashed lines) are also shown for both cases.
to a faster decrease of N χ with increasing m χ .For the same reason, the background counts (red-dashed curve) N b ≃ 526M T t exp for m χ ≤ 25 keV due to a constant background rate of ∼ 526 events per kton per year [56].For m χ > 25 keV where t exp = t cut is applied, N b stays constant.
For BDM associated with SN1987a in LMC, we plot N χ in Kamiokande (1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996) and Super-K (after 1996) by hollow green squares and orange triangles separately.The behavior of N χ (m χ ) in Kamiokande is similar to that of the GC case, but falls off faster for large m χ due to the maximal exposure time of 9 yr only.The difference at small m χ is mainly due to different detector fiducial mass M T , geometric dilution factor 1/R 2 , and the characteristic density ρ s of DM profiles.A simple estimate gives 103 ) consistent with Fig. 4. Super-K here only starts to accumulate events for m χ ≳ 1.1 keV whose t van > 9 yr, and eventually dominates the contribution to N χ more than that from Kamiokande for larger m χ .For comparison, we also plot the combined background numbers N b from both detectors.
SN1987a in LMC, and the projected sensitivity at n σ = 2.0 for a SN in the GC.In order to compare s with existing constraints based exclusively on σ χe [9-12, 15, 58], a model-dependent choice relating σ χν and σ χe has to be made 1 .Under a generic description of r χ = σ χν /σ χe , the specific case of r χ = 1 was selected as illustration, with which the resulting bounds are superimposed in Fig. 5.The Super-K constraints are derived from the average background rates [56] and statistical uncertainties.Time stability can be inferred from the absence of anomalous time variations in the solar ν annual modulation analysis [59,60].Limits derived with BDM from SN1987a in LMC leads to orders of magnitude improvement over existing bounds for m χ < 2 keV over a large range of r χ .For instance, more stringent limits are derived at m χ ∼ 10 −6 MeV for r χ > 10 −6 .Moreover, a future SN in the GC can improve the sensitivity by a factor of ∼ 30 with Super-K, since s GC χ /s LMC simply due to N χ ∝ m −1 χ (see Fig. 4).On the other hand, the sensitivities for m χ ≳ 100 keV weaken considerably due to the reduced BDM that can arrive at the detector within 35 years.Finally, we include an additional projection with Hyper-K for the GC case (red-dashed curve).The analysis is similar to Super-K, with fiducial mass and background rate scaled up by a factor of 10, which then leads to another improvement of ∼ 2-3 over the Super-K result.
Summary and prospects.-Wehave examined the scenario of halo DM being boosted by prompt SNν, and extracted a wealth of information from its TOF measurements.The BDM events on Earth are characterized by unique timing distributions, which vanish beyond m χdependent end points and are independent of the interaction cross sections, while their peak positions provide information on the SN locations and m χ .
A new constraint was derived on s = √ σ χν σ χe using Kamiokande and Super-K data on the SN1987a in LMC.
Our results probe and exclude new parameter space over a large range of r χ and in particular improve over the existing cosmic-ray BDM bounds for m χ < 100 keV by 1-3 orders of magnitude at σ χe ∼ σ χν .A future SN in the GC can provide improved sensitivity by another factor of 30-100 with Super-K or Hyper-K.The improvement over other probes [9-12, 15, 30, 31, 58] in the sub-MeV mass range originates from the transient BDM flux arriving in a short duration that can be calibrated by the detection of SNν, thereby minimizing the background counts.The constraint and sensitivity of this work were derived by a conservative analysis which stands on the BDM rates not being larger than those of background.A detailed analysis that optimally exploits the m χ -dependent TOF temporal profile or combines multiple detectors is beyond the scope of this work but will further enhance the sensitivities.Furthermore, most BDM arriving on Earth are within a small solid angle relative to the SN direction for m χ ≲ O(MeV) (see Fig. 2).Coupled with the good pointing capability for galactic SN [61][62][63][64], the angular information can be exploited to greatly reduce the background.
Other effects such as the distortion of SNν spectra, the recently proposed SNν echo [65], and the impact of χ − ν interaction on SNν emission have been neglected here.Estimations suggest that SNν spectra be minimally affected for the parameter space examined.These effects may be combined with the TOF profiles of SNνBDM to provide severe constraints on specific phenomenological models relating σ χe and σ χν .With all the rich information, the next galactic SN will offer new insights to the nature of DM.Furthermore, TOF analysis following SN or other transient astrophysical events can be applied in a similar vein to studies of other exotic physics interactions.A broad range of interesting scenario will be explored in our future research.
Code availability.-Weprovide a Python package snorer [66], which can fully reproduce the results in this Letter.The package is available on both GitHub and PyPI.It offers numerous new features, such as including DM spikes, user-specified SN locations in arbitrary distant galaxies, and an implementation of any particle physics model.See its official page for further details.

FIG. 1 .
FIG. 1. Schematic plot of DM boosted by SNν within an expanding spherical shell with width d at radius r.The SN occurs at O. BDM from A arrives B with an scattering angle α.
FIG. 3. The BDM flux at Earth vs. t with different mχ for Tχ = 10 MeV and σχν = 10 −35 cm 2 .Fluxes resulting from a SN in the GC and from SN1987a in LMC are shown with solid and dash-dotted lines.The black dashed line indicates the maximum exposure time texp = 35 years (see text for details).
FIG.4.BDM events in water Cherenkov detectors Nχ as a function of mχ for both the GC and LMC (SN1987a) cases.For LMC, events in Kamiokande (hollow-green squares) and Super-K (hollow-orange triangles) are shown separately.For the GC, only Super-K (red dots) is considered.Background counts (dashed lines) are also shown for both cases.