Search for boosted keV-MeV light dark matter particles from evaporating primordial black holes at the CDEX-10 experiment

We present novel constraints on boosted light dark matter particles (denoted as ``$\chi$'') from evaporating primordial black holes (PBHs) using 205.4 kg$\cdot$day data from the China Jinping Underground Laboratory's CDEX-10 p-type point contact germanium detector with a 160 eVee analysis threshold. $\chi$ from PBHs with masses ranging from 1$\times$10$^{15}$ g to 7$\times$10$^{16}$ g are searched in this work. In the presence of PBH abundance compatible with present bounds, our result excludes the $\chi$-nucleon elastic-scattering cross section region from 3.4$\times$10$^{-32}$ cm$^{2}$ to 2.3$\times$10$^{-29}$ cm$^{2}$ for $\chi$ of 1 keV to 24 MeV from PBHs with masses of 5$\times$10$^{15}$ g, as well as from 1.1$\times$10$^{-28}$ cm$^{2}$ to 7.6$\times$10$^{-28}$ cm$^{2}$ for $\chi$ of 1 keV to 0.6 MeV from PBHs with masses of 7$\times$10$^{16}$ g. If the $\chi$-nucleon elastic-scattering cross section can be determined in the future, the abundance of PBHs may be severely constrained by $\chi$ evaporation. With the lower threshold (160 eVee) of the CDEX-10 experiment compared to the previously used experiments, this work allows for a better reach at soft spectra produced by heavier PBHs, which demonstrates the vast potential of such a technical route to pursue $\chi$ from larger PBHs with a low threshold.

In this study, the constraints on keV-MeV boosted χ from evaporating PBHs are derived based on the 205.4 kg·day exposure data [23,84] from the CDEX-10 experiment, which aims at searching χ with p-type point contact germanium detectors [85] at the China Jinping Underground Laboratory (CJPL) [86], where the physics analysis threshold is 160 eVee (electron equivalent energy).

II. χ SPECTRA FROM EVAPORATING PBHS
The Hawking temperature of an evaporating PBH with mass M P BH is [87][88][89] where ℏ is the reduced Planck constant, c is the speed of light in vacuum, G is the Newtonian constant of gravitation, and k B is the Boltzmann constant. PBHs with larger masses have lower Hawking temperatures. The differential number of χ per unit time evaporated by PBHs is calculated by the public code Blackhawk-v2.1 [90,91] where E χ is the evaporated χ energy and Γ χ is the graybody factor. Here, we consider the case of Dirac fermions with four degrees of freedom (g χ =4) and chargeless and spinless PBHs.

IV. EARTH ATTENUATION
χ must travel through a few kilometers of rock before reaching the underground laboratory. At large scattering cross sections with a mean free path in rocks less than a few kilometers, χ will scatter with the nucleus and lose kinetic energy before reaching the detectors at underground locations. This phenomenon is called the Earth shielding effect. In this work, this process is modeled with a ballistic-trajectory approximation following Refs. [31,65], which is subjected to systematic uncertainties in the numerical calculations [23,94]. We note that different results are obtained by different numerical frameworks [23,32,95]. The subject of χ attenuation by the Earth remains an open research question and will be a theme of our future efforts.
The χ flux will be changed from where d represents the depth of rock through which incident χ passes, T 0 represents the initial kinetic energy of χ, and T d represents the kinetic energy of χ reaching the detectors after traveling a distance of d. The χ flux arriving at the CJPL can be evaluated by with where τ = d/l, d = 2400 m is the rock overburden depth of the CJPL [86] and l is the interaction length given by where n N is the number density of element N . The density and the elemental compositions of the rocks are provided in Ref. [94]. Omitting the nuclear form factor, the χ-nucleus elastic scattering cross section σ χN can be expressed with where A N is the mass number of nucleus N , σ SI χp is the SI χ-nucleon elastic scattering cross section, and m N and m p are the masses of nucleus N and proton, respectively. This approximation of omitting the form factor will result in an error of less than 1% because the nuclear recoil energy spectra are softer than the ones in Ref. [94].

V. EXPECTED EVENT RATE IN GE DETECTORS
The differential event rate of the χ-germanium nucleus scattering in the detector is calculated by where N Ge = 8.30×10 21 g −1 is the number of germanium nuclei per gram and Θ is the Heaviside function. E max r is the maximum value of the recoil energy E r expressed with .
The χ-germanium nucleus elastic scattering differential cross section σ χGe is given by where F (E r ) is the nuclear form factor, for which the conventional Helm form factor [96] is adopted. σ SI χp is the SI χ-nucleon elastic scattering cross section. m p is the mass of the proton. A Ge = 72.5 is the mass number of germanium, and m Ge = 6.77×10 4 MeV is the mass of germanium nuclei.
In a germanium semiconductor detector, the observed total deposit energy E det is different from the real nuclear-recoil energy E r and should be corrected by the quenching factor, E det = Q nr E r . The differential event rate per deposit energy E det can be obtained as follows: where the quenching factor Q nr is determined using the Lindhard formula [97], and κ = 0.16 (a typical value reported in the literature that closely matches recent measurements in the low-energy range [98]). Expected event rates at 0.16 keVee in the CDEX-10 detector for σ SI χp = 10 −28 cm 2 , 10 −29 cm 2 , and 10 −30 cm 2 . The quenching factor in germanium is determined by the Lindhard formula [97] and κ = 0.16 [98]. The energy resolution of CDEX-10 was considered, and its standard deviation is 35.8 + 16.6×E 1 2 (eV), where E is expressed in keV [19,23,99].
The expected event rates in a germanium detector convolved with energy resolution for different M P BH s are shown in Fig. 2(a) when m χ = 0.1 MeV and σ SI χp = 10 −29 cm 2 . PBHs with larger masses have softer expected energy spectra. As shown in Fig. 2(b), the expected event rate decreases at E det = 0.16 keVee, whereas the χ spectrum becomes softer as the PBH mass increases when M P BH ≳ 8 × 10 15 g. The search for χ from PBHs with larger masses is more sensitive to the threshold. Figure 3 shows the expected event rates for boosted χ from evaporating PBHs at the specified parameters (M P BH , σ SI χp , B). As M P BH becomes larger, the expected spectrum becomes significantly softer. This means that low-threshold detectors can perform well in the search for χ from PBHs with larger masses.

VI. EXCLUSION RESULT AND DISCUSSION
Data analysis for this work is based on a 205.4 kg·day dataset from CDEX-10. At the subkeVee energy range relevant to this analysis, background events are dominated by the Compton scattering of high-energy gamma rays and internal radioactivity from long-lived cosmogenic isotopes. Figure 3 shows the measured spectrum after subtracting the contributions from the L-and Mshell x-ray peaks derived from the corresponding K-shell x-ray intensities. Following our previous WIMP, dark photon, and χ-electron scattering analysis [18,84,99], a minimum-χ 2 analysis is applied to the residual spectrum in the range of 0.16−2.16 keVee: where n i and ∆ i denote the measured data and standard deviation with statistical and systematic components at the ith energy bin, respectively; S i denotes the predicted event rate; and B denotes the assumed flat-background contribution from the Compton scattering of high-energy gamma rays. Figure 4 shows the 90% confidence level (C.L.) exclusion regions (m χ , σ SI χp ) for M P BH = 0.5, 1, 2, 3 FIG. 4. Exclusion regions (mχ, σ SI χp ) with 90% confidence level for MP BH = (0.5, 1, 2, 3, 5, 7) ×10 16 g. The exclusion region from cosmology [101][102][103][104] and the published limits under the standard halo model scenario from CDEX-10 Migdal effect analysis [94], CRESST ν-cleus 2017 surface run [105][106][107], and EDELWEISS-surface [108] are superimposed. For visual clarity, the constraints from phenomenological interpretations of the XENON-1T data [65] are omitted from this figure. and 7×10 16 g, derived by the Feldman-Cousins unified approach with ∆χ 2 = 1.64 [100]. The constraints from the Migdal effect analysis of CDEX-10 data [94] and cosmology [101][102][103][104], CRESST ν-cleus 2017 surface run [105][106][107], and EDELWEISS-Surface [108] are shown for comparison. For visual clarity, the constraints from phenomenological interpretations of the XENON-1T data [65], obtained for different values of M P BH = (0.5, 1.0, 8.0) ×10 15 g, are omitted from the figure.
As Fig. 4 shows, both the upper and lower lines of the exclusion regions for χ with masses below 0.1 MeV are flat. We search the exclusion regions for M P BH from 1×10 15 g to 7×10 16 g choosing m χ = 0.1 MeV as a benchmark value for our analysis. The results are shown in Fig. 5. The constraints from phenomenological interpretations by Calabrese et al. of the XENON-1T data [65] for M P BH = 8×10 15 g are also plotted. Adoption of the latest f P BH values in Table I would give rise to weaker constraints as indicated by the red arrows, such that this work significantly extends the excluded parameter space.
The next generation of the CDEX experiment, CDEX-50, is currently under construction. A 50 kg germanium detector array will be run in a low-radioactive environment, and the radioactive background will be further reduced to ∼ 0.01 cpkkd in the sub-keV region [99]. As shown by the purple dashed line in Fig. 5, the lower constraints on σ SI χp from 50 kg·year data of CDEX-50 will be improved by up to approximately three orders of magnitude compared to those from CDEX-10. At the same time, χ from PBHs with a broader mass range can be  [65] for MP BH = 8×10 15 g are also plotted in horizontal lines with one-side shading. If the latest fP BH values in Table I are adopted, the excluded parameter space will be reduced, as indicated by the red arrows. effectively searched.
In other words, if the Earth shielding effect can be omitted; as f P BH decreases, σ SI χp decreases qualitatively. We fix σ SI χp = 10 −29 , 10 −30 and 10 −31 cm 2 and then search for f P BH with m χ = 0.1 MeV. Figure 6 shows the upper bounds with 90% C.L. on f P BH , obtained for different σ SI χp . According to the projection results (thin dotted dashed lines in Fig. 6), CDEX-50 can be well placed to search for χ from PBHs with a mass of M P BH ≳ 2×10 16 g. Some strong existing bounds from extragalactic γ-ray background (EGR) [75], EDGES 21 cm [71], Voyager-1 [77], and COMPTEL [76] observations are superimposed. The constraints from phenomenological interpretations by Calabrese et al. of the XENON-1T data [65] are also shown by dashed lines for comparison.
Finally, there is the degeneracy of σ SI χp and f P BH to consider. Following the determination of one of σ SI χp and f P BH in the future measure, the other one can be known. For example, if PBHs exist and f P BH is accurately measured in the observation of Hawking radiation, such as γ-rays and neutrinos, accurate constraints on σ SI χp can be derived. As the spectrum becomes softer, the CDEX experiment has vast potential as a technical route to pursue χ from larger PBHs with a low threshold. . Some strong existing bounds from extragalactic γ-ray background (EGR) [75], EDGES 21 cm [71], Voyager-1 [77], COMPTEL [76] observations are shown, and the strongest ones marked with black circles are selected for input fP BH listed in Table I. Projection results from CDEX-50 are shown by thin dotted-dashed lines. The numbers in the legend, 10 −29 , 10 −30 , 10 −31 and 10 −32 , are σ SI χp s in the unit of cm 2 . The constraints from phenomenological interpretations by Calabrese et al. of the XENON-1T data are also shown by dashed lines, where for σ SI χp = 10 −28 cm 2 , the Earth shielding effect is more evident, pushing the events to lower-recoil energies and weakening the XENON-1T's sensitivity [65].