# Electron-target experiment constraints on light dark matter produced in
primordial black hole evaporation

Tong Li<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin
300071, China  Jiajun Liao<EMAIL_ADDRESS>School of Physics, Sun
Yat-Sen University, Guangzhou 510275, China

###### Abstract

Light sub-GeV dark matter (DM) particles in the Milky Way or macroscopic
objects such as primordial black holes (PBHs) become attractive DM candidates
due to null results of WIMP from direct detection experiments. We explore the
possibility in which the present PBHs play as a novel source to produce light
boosted DM and confine light PBHs with current and future terrestrial
facilities. We study the electron elastic scattering data and obtain the
current constraints from Super-Kamiokande and XENON1T on the boosted DM from
PBH evaporation. The prospective bounds on the sub-GeV DM-electron scattering
cross section and the fraction of DM composed of PBHs $f_{\rm PBH}$ are also
imposed for future Xenon experiments.

##  I Introduction

Astrophysical evidence suggests that 84% of the matter in the Universe is
constituted by dark matter (DM). However, the constitution and the properties
of DM are still unknown. One longstanding candidate of thermal DM is the
weakly interacting massive particle (WIMP) with weak scale interaction. Due to
null conclusive evidence of WIMP search at DM direct detection (DD)
experiments Schumann (2019), however, much more efforts have been recently
paid to the hypotheses beyond the WIMP in both theoretical and experimental
aspects. The explorations of alternative DM candidates motivated a
hypothetical light particle with sub-GeV mass Griest and Seckel (1991); Essig
_et al._ (2012, 2017); Elor  _et al._ (2021) or macroscopic objects such as
primordial black holes (PBHs) Zel’dovich (1967); Carr and Hawking (1974); Carr
(1975) (see Ref. Carr and Kuhnel (2021) for a recent review).

The DM in the Milky Way (MW) halo can be in part composed of a non-thermal
light DM which gets boosted to a semi-relativistic velocity by cosmic rays
Cappiello  _et al._ (2019); Yin (2019); Bringmann and Pospelov (2019); Ema
_et al._ (2019), the DM semi-annihilation D’Eramo and Thaler (2010) or the
annihilation of heavier DM component Agashe  _et al._ (2014). The boosted DM
(BDM) was proposed in many well-motivated DM models and can induce high-energy
recoil events by upscattering on electrons in the terrestrial experiments
Kannike  _et al._ (2020); Fornal  _et al._ (2020); Su  _et al._ (2020); Cao
_et al._ (2021); Lei  _et al._ (2020); Bloch  _et al._ (2021); Kannike  _et
al._ (2020). On the other hand, it is well known that PBHs with the mass $\sim
10^{15}$ g would evaporate due to Hawking radiation Hawking (1974) and cannot
provide all the observed abundance of DM Barrau  _et al._ (2004). The emitted
particles such as gamma-rays and $e^{\pm}$ in the evaporation process suffer
from a variety of cosmological constraints Carr  _et al._ (2020); Laha (2019);
Laha  _et al._ (2020); Saha and Laha (2022); Ray  _et al._ (2021). The
neutrinos emitted from PBHs were also studied to a large extent Dasgupta  _et
al._ (2020); Wang  _et al._ (2021); Calabrese  _et al._ (2021a); De Romeri
_et al._ (2021); Ghosh  _et al._ (2021); Capanema  _et al._ (2021); Chao  _et
al._ (2021); Bernal  _et al._ (2022). The emitted particles can acquire
energies to a few hundred MeV from the evaporation of PBHs with the mass $\sim
10^{15}$ g. The light particles can gain enough kinetic energy to travel
through the Earth and reach the terrestrial detectors. Besides the Standard
Model (SM) particles, there is a possibility that the present PBHs play as a
novel source to produce and boost light DM Khlopov  _et al._ (2006). It is
thus interesting to confine the PBHs through the boosted DM scattering with
current and future terrestrial facilities.

In this work we explore the boosted DM from the present PBH evaporation
(denoted by “PBHBDM” below) so as to connect the macroscopic PBHs to sub-GeV
DM particle. The PBHs with mass $\sim 10^{14}-10^{16}$ g evaporate DM flux in
the present Universe. The light components of DM are boosted to
(semi-)relativistic velocities. Then PBHBDM travels through the Earth and
scatters with the electrons in the underground detector targets of the
terrestrial experiments 111The scattering of boosted DM evaporated from PBHs
and nucleons was studied in Ref. Calabrese  _et al._ (2021b).. The electrons
then induce photoelectron signatures in the time-projection chambers (TPCs).
Such “S2” signature can be sensitive to smaller DM mass region compared to the
DM-nucleon scattering.

Both neutrino and DM scattering experiments can search for the PBHBDM in light
of the electron elastic scattering events. Super-Kamiokande (Super-K) with
161.9 kiloton-years exposure Kachulis  _et al._ (2018) provide a strong
constraint on sub-GeV BDM Ema  _et al._ (2019). The kinetic energy of the
recoiled electron is above 100 MeV at Super-K. The Xenon collaboration
recently collected low-energy electron recoil data from the XENON1T experiment
with an exposure of 0.65 tonne-years and reported the event distribution with
a broad spectrum Aprile  _et al._ (2020a). If using the BDM event rate as well
as the XENON1T data, one can construct a conservative method to give a general
exclusion limit on the DM-electron scattering. This limit then becomes a
strong existing constraint on DM lighter than a few MeV. We will obtain the
current constraints from Super-K and XENON1T on PBHBDM, and compute the
prospects in future Xenon experiments. Moreover, these bounds can also be
converted into the upper limit on the fraction of DM composed of PBHs $f_{\rm
PBH}$. Compared with the current evaporation constraints from extragalactic
gamma-rays, we expect the bound on $f_{\rm PBH}$ can be improved from
terrestrial facilities.

This paper is organized as follows. In Sec. II we evaluate the BDM spectrum
from PBH evaporation. The DM-electron scattering and the scattering rate in
the terrestrial facilities are then calculated in Sec. III. We obtain the
current constraints on PBHs from Super-K and XENON1T, and give the future
prospects in XENONnT in Sec. IV. Our conclusions are drawn in Sec. V.

##  II The DM flux from PBH evaporation

The PBHs have quantum properties and thermally radiate with a temperature
Hawking (1975); Page (1976a, 1977); MacGibbon and Webber (1990); MacGibbon
(1991)

$\displaystyle T_{\rm PBH}={\hbar c^{3}\over 8\pi GM_{\rm PBH}k_{\rm
B}}\approx 10^{-7}\Big{(}{M_{\rm PBH}\over M_{\odot}}\Big{)}^{-1}{\rm K}\;,$
(1)

where $G$ is the Newtonian constant of gravitation, $M_{\rm PBH}$ denotes the
PBH mass and $k_{\rm B}$ is the Boltzmann constant. We use the public code
BlackHawk v2.0 Arbey and Auffinger (2019, 2021) 222BlackHawk v2.0 provides the
additional emission of massive DM and we directly apply it to our massive
Dirac fermion DM case. to calculate the differential number of DM per unit
time emitted by PBHs Hawking (1971); Page (1976a, b)

$\displaystyle{d^{2}N_{\chi}\over dT_{\chi}dt}={g_{\chi}\over
2\pi}{\Gamma(T_{\chi},M_{\rm PBH})\over{\rm exp}((T_{\chi}+m_{\chi})/k_{\rm
B}T_{\rm PBH})+1}\;,$ (2)

where $\Gamma$ is the function of greybody factor which encodes the
probability of an elementary spin-1/2 DM $\chi$ to escape the PBH
gravitational well, $T_{\chi}$ is the emitted DM kinetic energy and
$g_{\chi}=4$ denotes the degrees of freedom for the emitted DM as Dirac
fermions. We next ignore the spin of PBH for simplicity 333It is customary in
the literatures to only consider non-spinning PBHs (in Refs. Wang  _et al._
(2021); Calabrese  _et al._ (2021a, b) and etc.). In principle, spinning PBHs
evaporate faster than the non-spinning PBHs and thus can contribute to the DM
content nowadays for $M_{\rm PBH}\gtrsim 7\times 10^{15}$ g Dasgupta  _et al._
(2020). In this work, we focus on $M_{\rm PBH}\lesssim 1\times 10^{16}$ g and
only consider non-spinning PBHs..

For the DM flux from PBH evaporation, we consider the contributions of both
the PBHs in galactic halo and the extragalactic PBHs Wang  _et al._ (2021);
Calabrese  _et al._ (2021a); De Romeri  _et al._ (2021)

$\displaystyle{d^{2}\phi_{\chi}\over dT_{\chi}d\Omega}={d^{2}\phi_{\chi}^{\rm
MW}\over dT_{\chi}d\Omega}+{d^{2}\phi_{\chi}^{\rm EG}\over
dT_{\chi}d\Omega}\;,$ (3)

where $\phi_{\chi}^{\rm MW}$ and $\phi_{\chi}^{\rm EG}$ correspond to the DM
flux from PBHs in Milky Way (MW) and extragalactic (EG) PBHs, respectively,
and $\Omega$ is the solid angle. The differential galactic DM flux is given by

$\displaystyle{d^{2}\phi_{\chi}^{\rm MW}\over dT_{\chi}d\Omega}={f_{\rm
PBH}\over 4\pi M_{\rm PBH}}{d^{2}N_{\chi}\over
dT_{\chi}dt}\int{d\Omega_{s}\over 4\pi}\int dl\rho_{\rm MW}[r(l,\psi)]\;,$ (4)

where $f_{\rm PBH}$ denotes the fraction of DM composed of PBHs, $\rho_{\rm
MW}[r(l,\psi)]$ is the DM density of the MW halo,
$r(l,\psi)=\sqrt{r_{\odot}^{2}+l^{2}-2lr_{\odot}\cos\psi}$ is the
galactocentric distance with $r_{\odot}$ the solar distance from the galactic
center, $l$ the line-of-sight distance to the PBH and $\psi$ the angle between
these two directions. The maximal limit of distance $l$ is taken to be
$(r_{h}^{2}-r_{\odot}^{2}\sin^{2}\psi)^{1/2}+r_{\odot}\cos\psi$ with the halo
radius being $r_{h}=200$ kpc Wang  _et al._ (2021). The angular integration is
defined as $\int d\Omega_{s}=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\psi\sin\psi$
with the azimuthal angle $\phi$. For illustration, we employ the generalized
Navarro-Frenk-White (NFW) DM profile Navarro  _et al._ (1997)

$\displaystyle\rho_{\rm MW}(r)=\rho_{\odot}\Big{(}{r\over
r_{\odot}}\Big{)}^{-\gamma}\Big{(}{1+r_{\odot}/r_{s}\over
1+r/r_{s}}\Big{)}^{3-\gamma}\;,$ (5)

where $\rho_{\odot}=0.4~{}{\rm GeV}/{\rm cm}^{3}$ is the local DM density,
$r_{\odot}=8.5$ kpc, $r_{s}=20$ kpc is the radius of the galactic diffusion
disk, and the inner slope of the NFW halo profile is fixed as $\gamma=1$. For
the extragalactic contribution, the corresponding differential BDM flux over
the full sky is

$\displaystyle{d^{2}\phi_{\chi}^{\rm EG}\over dT_{\chi}d\Omega}={f_{\rm
PBH}\rho_{\rm DM}\over 4\pi M_{\rm PBH}}\int^{t_{\rm max}}_{t_{\rm
min}}dt[1+z(t)]{d^{2}N_{\chi}\over dT_{\chi}dt}\Big{|}_{E^{s}_{\chi}}\;,$ (6)

where the average DM density of the Universe at the present epoch determined
by Planck Aghanim  _et al._ (2020) is $\rho_{\rm DM}=2.35\times
10^{-30}~{}{\rm g}/{\rm cm}^{3}$,
$E^{s}_{\chi}=\sqrt{(E_{\chi}^{2}-m_{\chi}^{2})(1+z(t))^{2}+m_{\chi}^{2}}$
denotes the energy at the source which is related to the energy $E_{\chi}$ in
the observer’s frame by the redshift $z(t)$. The DM emitted from the PBHs in
the very early Universe is sufficiently redshifted. As a result, their
energies are very low today and cannot be detected. Thus, for the integral
limits, we fix $t_{\rm min}=10^{11}$ s being close to the era of matter-
radiation equality Wang  _et al._ (2021) and $t_{\rm max}=\tau_{0}$ with
$\tau_{0}$ as the age of Universe. Actually, it turns out that changing the
lower limit has no impact on the results De Romeri  _et al._ (2021). In Fig. 1
we show the differential BDM flux from PBHs $f_{\rm
PBH}^{-1}d^{2}\phi_{\chi}/dT_{\chi}d\Omega$ for three benchmark values of
$M_{\rm PBH}$. As seen in Eqs. (4) and (6), smaller PBHs exhibit harder
spectra of the evaporated DM. The PBHBDMs have maximal kinetic energies of
order $T_{\chi}\sim\mathcal{O}(100)$ MeV for $M_{\rm PBH}\lesssim 10^{15}$ g.
The spectrum generally peaks at lower energies for each PBH mass. The energy
where the spectrum peaks at is higher for less massive PBH.

Figure 1: The differential BDM flux from PBHs $f_{\rm
PBH}^{-1}d^{2}\phi_{\chi}/dT_{\chi}d\Omega$ as a function of kinetic energy
$T_{\chi}$ for three benchmark values of $M_{\rm PBH}=5\times 10^{14}$ g
(blue), $1\times 10^{15}$ g (red), and $8\times 10^{15}$ g (black). The EG and
MW contributions are denoted by solid and dashed lines, respectively. We fix
$m_{\chi}=1$ MeV.

##  III PBHBDM-electron scattering in the terrestrial facilities

The produced BDM $\chi$ can travel a distance underground and scatter with the
electrons in the detector of the terrestrial facilities. After traveling a
distance of $z$, the actual DM flux reaching the detector is given by
Calabrese  _et al._ (2021b)

$\displaystyle{d^{2}\phi_{\chi}^{d}\over
dT_{\chi}^{d}d\Omega}\approx{4m_{\chi}^{2}e^{\tau}\over(2m_{\chi}+T_{\chi}^{d}-T_{\chi}^{d}e^{\tau})^{2}}{d^{2}\phi_{\chi}\over
dT_{\chi}d\Omega}\Big{|}_{T^{0}_{\chi}}\;,$ (7)

where $T_{\chi}^{0}$ and $T_{\chi}^{d}$ are defined as the DM kinetic energies
at Earth’s surface and at the detector, respectively. They are related as

$\displaystyle T^{0}_{\chi}(T^{d}_{\chi})={2m_{\chi}T^{d}_{\chi}e^{\tau}\over
2m_{\chi}+T^{d}_{\chi}-T^{d}_{\chi}e^{\tau}}\;,$ (8)

where $\tau=z/\ell_{\rm int}^{\oplus}$ , with

$\displaystyle
z=-(R-d)\cos\theta_{z}+\sqrt{R^{2}-(R-d)^{2}\sin^{2}\theta_{z}}\,,$ (9)

and

$\displaystyle\ell_{\rm int}^{\oplus}=\Big{[}n^{\oplus}_{e}\sigma_{\chi
e}{2m_{e}m_{\chi}\over(m_{e}+m_{\chi})^{2}}\Big{]}^{-1}\;.$ (10)

Here $R$ is the radius of the Earth, $\theta_{z}$ is the zenith angle of the
detector, $d$ is the depth of the detector from the Earth surface at the zero
zenith angle, $\sigma_{\chi e}$ is the total cross section of BDM-electron
scattering, and $n^{\oplus}_{e}$ is the electron number density of the Earth.
For simplicity, we take the average value over the Earth, i.e.,
$n^{\oplus}_{e}=8\times 10^{23}~{}\text{cm}^{-3}$ Ema  _et al._ (2019). Note
that the BDM can also be stopped by the scattering in the atmosphere Emken and
Kouvaris (2018); Bringmann and Pospelov (2019); Ema  _et al._ (2019);
Cappiello and Beacom (2019) and there would be an additional $z/\ell_{\rm
int}^{\rm atm}$ due to atmosphere attenuation in quantity $\tau$. To estimate
the electron number density in the atmosphere, we consider the density profile
of the US Standard Atmosphere model with an altitude of $d^{\rm atm}=86$ km
Emken and Kouvaris (2018). The atmosphere was modeled as 4 layers of falling
mass density as a function of the altitude. We take a constant density
$\rho=1.1~{}{\rm kg}~{}{\rm m}^{-3}$ in the first layer and the most abundant
elements in the atmosphere (nitrogen and oxygen) to obtain the most optimistic
number density of electron as $n_{e}^{\rm atm}=3.2\times 10^{20}~{}{\rm
cm}^{-3}$. For zero zenith angle, we find that the value of $d^{\rm atm}n^{\rm
atm}_{e}$ in $\tau$ is 30 and 40 times smaller than $dn^{\oplus}_{e}$ of
Super-K and XENON1T detectors, respectively. Thus, the atmosphere attenuation
is negligible and we neglect it in the following analysis.

After combining the above DM flux, the number of scattered DM events per unit
time per solid angle per recoil energy $T_{e}$ of the electron, is then given
by

$\displaystyle{d^{3}N_{\chi}\over dtd\Omega dT_{e}}=N_{e}\int
dT_{\chi}^{d}\sigma_{\chi
e}D^{e}_{\chi}(T_{e},T_{\chi}^{d}){d^{2}\phi_{\chi}^{d}\over
dT_{\chi}^{d}d\Omega}\;,$ (11)

where $N_{e}$ is the total number of free electrons in targets for a detector.
We define a transfer function which encodes the electron energy spectrum
induced by the scattering with BDM $\chi$ as

$\displaystyle D^{e}_{\chi}(T_{e},T_{\chi})={1\over T_{e}^{\rm
max}(T_{\chi})}\Theta(T_{e}^{\rm max}(T_{\chi})-T_{e})\;,$ (12)

where the final kinetic energy of electrons is

$\displaystyle T_{e}=T_{e}^{\rm max}{1-\cos\theta\over 2}\;,$ (13)

with

$\displaystyle T_{e}^{\rm
max}={2m_{e}(T_{\chi}^{2}+2m_{\chi}T_{\chi})\over(m_{\chi}+m_{e})^{2}+2m_{e}T_{\chi}}\;.$
(14)

Here $\theta$ is the scattering anle in the center of mass (CM) frame, and we
assume the scattering is isotropic in the CM frame.

Figure 2: Top: the predicted event spectrum (blue dashed) and the estimated
backgrounds (red solid) at Super-K. For the predicted events, we set
$\alpha_{1}=-0.35$, $\alpha_{2}=0$ and $\alpha_{3}=0$. Bottom: the electron
recoil spectrum in XENON1T induced by PBHBDM with (blue dashed) the background
model spectrum $B_{0}$ (red solid).

###  III.1 Super-Kamiokande

Energy bin | Data | Background | Uncertainty
---|---|---|---
$0.1\text{ GeV}<T_{e}<1.33\text{ GeV}$ | 4042 | 3992.9 | 19%
$1.33\text{ GeV}<T_{e}<20\text{ GeV}$ | 658 | 772.6 | 19%
$T_{e}>20\text{ GeV}$ | 3 | 7.4 | 23%
Table 1: The number of measured events, estimated backgrounds and systematic
uncertainties for each electron recoil energy bin at Super-K Kachulis  _et
al._ (2018).

To set bounds on the PBHBDM parameter space from Super-K, we use the “electron
elastic scatter-like” events from Ref. Kachulis  _et al._ (2018), and define
the $\chi^{2}$ as

$\displaystyle\chi^{2}=\sum_{i=1}^{3}\min_{\alpha_{i}}\left[2\left(N_{\text{th}}^{i}-N_{\text{obs}}^{i}+N_{\text{obs}}^{i}\ln\frac{N_{\text{obs}}^{i}}{N_{\text{th}}^{i}}\right)+\left(\frac{\alpha_{i}}{\sigma_{i}}\right)^{2}\right]\,,$

where the predicted number of events per bin
$N_{\text{th}}^{i}=N_{\text{bkg}}^{i}(1+\alpha_{i})+N_{\text{sig}}^{i}$. Here
$N_{\text{bkg}}^{i}$ ($N_{\text{obs}}^{i}$) [$N_{\text{sig}}^{i}$] is the
number of background (observed) [signal] events in each bin, $\alpha_{i}$ is
the background normalization factor that is minimized over, and $\sigma_{i}$
is the uncertainty for each bin. The main background at Super-K is due to
atmospheric neutrinos Kachulis  _et al._ (2018). Follow Ref. Kachulis  _et
al._ (2018), we take a conservative value of $19\%$ for the uncertainties in
the two lower energy bins, and 23% for the uncertainty in the highest energy
bin. The number of measured event, estimated background and the uncertainty
for each energy bin are listed in Table 1. For the signal events, we take into
account the Earth attenuation effect in the calculation of the predicted
number of events from PBHBDM. Since the distance between the detector and the
surface of the Earth depends on the direction of observation, we also
integrate over the solid angle for the Super-K detector. The predicted number
of events is obtained by integrating Eq. (11) over the solid angle with the
data-taking time of 2628.1 days and $N_{e}^{\text{SK}}=7.5\times 10^{33}$, and
we also apply the signal efficiency extracted from Fig. 1 in Ref. Kachulis
_et al._ (2018). For illustration, a predicted event spectrum for $M_{\rm
PBH}=10^{15}$ g, $f_{\rm PBH}=3.9\times 10^{-7}$, $\sigma_{\chi
e}=10^{-33.5}~{}{\rm cm}^{2}$, $m_{\chi}=10^{-3}$ GeV with $\alpha_{1}=-0.35$,
$\alpha_{2}=0$ and $\alpha_{3}=0$ at Super-K is shown in the upper panel of
Fig. 2. The number of PBHBDM events drops very fast as the electron recoil
energy increases, and there are almost no PBHBDM events at the higher energy
bins. The background-only hypothesis yields a $\chi^{2}_{0}=2.7$, and the
2$\sigma$ exclusion bound is set with
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{0}=4.0$ for one degree of freedom.

###  III.2 XENON1T and prospects

We also use the XENON1T electron recoil data measured in Ref. Aprile  _et al._
(2020a) to set bounds on PBHBDM. After taking account of the Earth attenuation
effect, the differential number of events from PBHBDM is given by

$\displaystyle{dN_{\chi}\over
dT_{e}}=Z_{\text{eff}}^{\text{Xe}}(T_{e}){M_{\text{det}}t\over
m_{\text{Xe}}}\int d\Omega dT_{\chi}^{d}\sigma_{\chi
e}D^{e}_{\chi}(T_{e},T_{\chi}^{d}){d^{2}\phi^{d}_{\chi}\over
dT^{d}_{\chi}d\Omega}\;,$

where $M_{\text{det}}$ is the fiducial mass of the detector, $t$ is the
exposure time, $m_{\text{Xe}}$ is the atomic mass of Xenon, and
$Z_{\text{eff}}^{\text{Xe}}$ is the effective electron charge of Xenon that
depends on the electron recoil energy $T_{e}$. Here we take the values of
$Z_{\text{eff}}^{\text{Xe}}(T_{e})$ from the Appendix A of Ref. Atzori Corona
_et al._ (2022), which are obtained by using the edge energies extracted from
the photoabsorption data Chen  _et al._ (2017); Henke  _et al._ (1993). To
compare with the measured data in Fig. 4 of Ref. Aprile  _et al._ (2020a), we
take $M_{\text{det}}t=1$ ton$\cdot$year, and convolve the differential number
of events with a Gaussian detector response function for the electron recoil
spectrum

$\displaystyle\sigma(E)=a\sqrt{E}+bE\;,$ (17)

where $a=0.31~{}\sqrt{\rm keV}$ and $b=0.0037$ Aprile  _et al._ (2020a). We
further apply the detector efficiency and add the background $B_{0}$ from Ref.
Aprile  _et al._ (2020a) to the PBHBDM contribution. The background components
at XENON1T come from solar neutrinos, $\gamma$ emissions from radioimpurities
in detector materials, $\beta$ decay of 214Pb, 85Kr, and 133Xe, internal
conversion of 131mXe and 83mKr, and electron capture of 125I and 124Xe Aprile
_et al._ (2020a). To place the bound on PBHDM parameter space, we define a
simple test statistic as follows

$\displaystyle\chi^{2}=\sum_{i}\frac{\left[\frac{dN_{\chi+B_{0},i}}{dT_{\text{rec}}}-\frac{dN_{obs,i}}{dT_{\text{rec}}}\right]^{2}}{\sigma_{i}^{2}}\,,$
(18)

where $T_{\text{rec}}$ is the reconstructed energy after the smearing,
$\frac{dN_{obs,i}}{dT_{\text{rec}}}$ and $\sigma_{i}$ are the observed event
rate and uncertainty in the $i^{\text{th}}$ bin extracted from Fig. 4 in Ref.
Aprile  _et al._ (2020a), respectively. We find the background-only hypothesis
yields a $\chi^{2}_{B_{0}}=46.4$. The 2$\sigma$ bound is estimated with
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{B_{0}}=4.0$ assuming the test statistic
follows a chi-squared distribution with one degree of freedom.

For the projected sensitivity at the future XENONnT experiment Aprile  _et
al._ (2020b), we consider a detector with 20 ton$\cdot$year exposure. We use
the same binning as the measured spectrum at XENON1T and perform a $\chi^{2}$
analysis with

$\displaystyle\chi^{2}=\sum_{i}\frac{N_{\chi,i}^{2}}{N_{B,i}}\;,$ (19)

where $N_{\chi,i}$ ($N_{B,i}$) is the expected number of signal (background)
events in the $i^{\text{th}}$ bin at XENONnT. The detection efficiency and the
energy resolution are assumed to be the same as those in the current XENON1T
detector, and the number of background events per ton$\cdot$year are reduced
by a factor of 6 at XENONnT Aprile  _et al._ (2020a, b).

##  IV Results

We first show the bounds on the $\sigma_{\chi e}$ versus $m_{\chi}$ parameter
space from Super-K and XENON1T in Fig. 3 assuming $M_{\rm PBH}=1\times
10^{15}$ g and $f_{\rm PBH}=3.9\times 10^{-7}$ allowed by the current
constraints from extragalactic gamma-rays Carr  _et al._ (2020). From Fig. 3,
one can see that the DM-electron cross section between $5\times 10^{-34}$
($4\times 10^{-34}$) and $3\times 10^{-32}~{}{\rm cm}^{2}$ ($1\times
10^{-32}~{}{\rm cm}^{2}$) is excluded by XENON1T (Super-K) for light
$m_{\chi}\lesssim 1$ MeV. Above this region, the cross section $\sigma_{\chi
e}$ is too large to make the BDM reach the detector. We also estimate the
projected sensitivity on $\sigma_{\chi e}$ as a function of $m_{\chi}$ at
future XENONnT experiment with 20 ton$\cdot$year exposure Aprile  _et al._
(2020b). As we see from Fig. 3, the future XENONnT experiment can improve the
current lower bounds by about a factor of 3. The constraints on the
$\sigma_{\chi e}-m_{\chi}$ parameter space from various DM direct detection
experiments are also added for comparison, and we see that the regions to the
right of the curves have been already excluded by existing DM direct detection
experiments.

These bounds can also be translated into the upper limit on $f_{\rm PBH}$ as a
function of $M_{\rm PBH}$, which is shown in Fig. 4 for different DM-electron
cross sections and $m_{\chi}=10^{-3}$ GeV. As we see from Fig. 4, the bounds
from Super-K and XENON experiments are complementary to each other. For
illustration, assuming $\sigma_{\chi e}=10^{-33}~{}{\rm cm}^{2}$, Super-K sets
a stronger bound than XENON1T (XENONnT) for $M_{\text{PBH}}\lesssim 10^{15}$ g
($M_{\text{PBH}}\lesssim 7\times 10^{14}$ g). Compared with the current
evaporation constraint from extragalactic gamma-rays, an overall improvement
of $f_{\rm PBH}$ bound can be achievable for Super-K and XENON1T combined.

Figure 3: $2\sigma$ bounds from Super-K (shaded blue) and XENON1T (shaded
gray) and projected sensitivity at XENONnT (black dashed line) on
$\sigma_{\chi e}$ as a function of $m_{\chi}$ for $M_{\rm PBH}=10^{15}$ g and
$f_{\rm PBH}=3.9\times 10^{-7}$. The constraints from DM direct detection
experiments are also shown, including SENSEI Barak  _et al._ (2020) (cyan),
DAMIC Aguilar-Arevalo  _et al._ (2019) (red), EDELWEISS Arnaud  _et al._
(2020) (yellow), CDMS Agnese  _et al._ (2018) (magenta), DarkSide50 Agnes  _et
al._ (2018) (green), XENON10/100 Essig  _et al._ (2017) (orange), XENON1T
Aprile  _et al._ (2019) (pink) and PandaX-II Cheng  _et al._ (2021) (brown).
Figure 4: $2\sigma$ bounds on $f_{\text{PBH}}$ as a function of
$M_{\text{PBH}}$ for different DM-electron cross sections $\sigma_{\chi
e}=10^{-33}~{}{\rm cm}^{2}$ (blue) and $\sigma_{\chi e}=10^{-34}~{}{\rm
cm}^{2}$ (red), and $m_{\chi}=10^{-3}$ GeV. The bounds and prospect are from
Super-K (dashed), XENON1T (solid) and XENONnT (dotted). The current
evaporation constraints from extragalactic gamma-rays Carr  _et al._ (2020)
are also shown for comparsion.

##  V Conclusions

The macroscopic PBH and the light sub-GeV DM particle are two interesting DM
candidates beyond the WIMP. The boosted DM from PBH evaporation can be
searched at both neutrino and DM direct detection experiments. Using the
current data from Super-K and XENON1T, we place bounds on the DM-electron
cross section for sub-GeV DM. We find that for light $m_{\chi}\lesssim 1$ MeV,
assuming $f_{\rm PBH}=3.9\times 10^{-7}$ and $M_{\rm PBH}=10^{15}$ g, the DM-
electron cross section between $5\times 10^{-34}$ ($4\times 10^{-34}$) and
$3\times 10^{-32}~{}{\rm cm}^{2}$ ($1\times 10^{-32}~{}{\rm cm}^{2}$) can be
excluded by current XENON1T (Super-K) data. Future XENONnT experiment can
improve the current lower bounds by about a factor of 3. We also find the
bounds from Super-K and XENON on the $f_{\text{PBH}}$ versus $M_{\text{PBH}}$
parameter space are complementary to each other. Compared with the current
evaporation constraints from extragalactic gamma-rays, the bound on $f_{\rm
PBH}$ from Super-K and XENON1T/XENONnT can be substantially improved with a
certain range of DM-electron cross section for light DM particles. Note that
these bounds on $f_{\rm PBH}$ and $\sigma_{\chi e}$ are obtained
simultaneously and are fully correlated.

###### Acknowledgements.

T.L. would like to thank Roberta Calabrese, Wei Chao and Xin-He Meng for
useful discussions. T.L. is supported by the National Natural Science
Foundation of China (Grant No. 11975129, 12035008) and “the Fundamental
Research Funds for the Central Universities”, Nankai University (Grant No.
63196013). J.L. is supported by the National Natural Science Foundation of
China (Grant No. 11905299), Guangdong Basic and Applied Basic Research
Foundation (Grant No. 2020A1515011479), the Fundamental Research Funds for the
Central Universities, and the Sun Yat-Sen University Science Foundation. Note
added. After the completion of this work, a separate work on constraints on
light dark matter produced in primordial black hole evaporation from electron-
target experiments has appeared Calabrese  _et al._ (2022). Their bounds for a
fixed $\sigma_{\chi e}$ are similar to our Fig. 4.

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$\begin{split}D_{3}^{\chi\chi\hat{h}}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}=\frac{Gk}{(1-2\lambda)\pi}&\Bigg{\\{}\frac{15}{256}+\frac{1}{1-2\lambda}\big{[}\frac{7}{16}-\frac{15}{128}\pi\big{]}+(\frac{28}{32}-\frac{27}{128}\pi)\beta\\\
&+\alpha\,\Big{[}-\frac{119}{48}+\frac{55}{64}\pi+\frac{1}{1-2\lambda}\big{[}\frac{13}{20}-\frac{3}{16}\pi\big{]}+\frac{1}{4}\beta\Big{]}\\\
&+\alpha^{2}\Big{[}-\frac{173}{24}+\frac{65}{32}\pi-\frac{17}{60}\frac{1}{1-2\lambda}+(-\frac{45}{8}+\frac{45}{32})\beta\Big{]}-\frac{9}{20}\alpha^{3}\Big{]}\Bigg{\\}}.\end{split}$
(111)

For completeness we also give the expressions for the tadpole diagrams shown
in Fig. 16, which are evaluated rather straightforwardly, and read

$\begin{split}&D_{\rm
Tad}^{h}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}=\frac{Gk}{(1-2\lambda)^{2}\pi}\,\bigg{\\{}\alpha\,\Big{[}\frac{1}{8}-\frac{1}{48}\eta_{N}\Big{]}\bigg{\\}},\\\
&D_{\rm
Tad}^{\hat{h}}\Big{|}_{R\,\bar{\theta}\gamma^{5}\theta}=\frac{Gk}{(1-2\lambda)^{2}\pi}\,\bigg{\\{}\alpha\,\Big{[}\frac{27}{8}-\frac{9}{16}\eta_{N}+\frac{1}{1-2\lambda}\big{[}-2+\frac{1}{4}\eta_{N}\big{]}\Big{]}\bigg{\\}}.\end{split}$
(112)

Together, the contributions (102), (105), (110), (111), and (112) give rise to
the beta function (29). This closes our discussion on evaluating the
functional renormalization group equation (1) including fermions in a curved
background.

###### Acknowledgements.

This article has been prepared for the Special Issue “Asymptotic Safety in
Quantum Gravity” (Universe) edited by A. D. Pereira. We thank A. D. Pereira
and M. Reuter for interesting discussions. J. W. acknowledges the China
Scholarship Council (CSC) for financial support.

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