# Electron scattering of light new particles from evaporating primordial black
holes

Roberta Calabrese  Dipartimento di Fisica “Ettore Pancini”, Università degli
studi di Napoli “Federico II”, Complesso Univ. Monte S. Angelo, I-80126
Napoli, Italy  INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo,
I-80126 Napoli, Italy  Marco Chianese  Dipartimento di Fisica “Ettore
Pancini”, Università degli studi di Napoli “Federico II”, Complesso Univ.
Monte S. Angelo, I-80126 Napoli, Italy  INFN - Sezione di Napoli, Complesso
Univ. Monte S. Angelo, I-80126 Napoli, Italy  Damiano F.G. Fiorillo
Dipartimento di Fisica “Ettore Pancini”, Università degli studi di Napoli
“Federico II”, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy  INFN -
Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy
Niels Bohr International Academy, Niels Bohr Institute, University of
Copenhagen, Copenhagen, Denmark  Ninetta Saviano  Scuola Superiore
Meridionale, Università degli studi di Napoli “Federico II”, Largo San
Marcellino 10, 80138 Napoli, Italy  INFN - Sezione di Napoli, Complesso Univ.
Monte S. Angelo, I-80126 Napoli, Italy

###### Abstract

Primordial black holes are a possible component of dark matter, and a most
promising way of investigating them is through the product of their Hawking
evaporation. As a result of this process, any species lighter than the Hawking
temperature is emitted, including possible new particles beyond the Standard
Model. These can then be detected in laboratory-based experiments via their
interaction with the Standard Model particles. In a previous work, we have
first proposed and studied this scenario in the presence of an interaction
between the light new species and nucleons. Here we extend this discussion to
include the case of interaction with electrons. We show that the simultaneous
presence of primordial black holes and species lighter than about $100$ MeV
can be constrained by the measurements of direct detection experiments, such
as XENON1T, and water Cherenkov neutrino detectors, such as Super-Kamiokande.
Our results provide a complementary and alternative way of investigation with
respect to cosmological and collider searches.

##  I Introduction

Primordial black holes (PBHs) are hypothetical black holes formed due to
gravitational collapse of over-density fluctuations in the very early Universe
Zel’dovich and Novikov (1967); Harrison (1970). Different from astrophysical
black holes, which must have sufficiently large masses in order to encounter
the instability leading to their formation, PBHs can be produced with any mass
larger than about 0.1 g Carr  _et al._ (2021), because of the different
equation of state involved. From the cosmological point of view, PBHs behave
as a component of dark matter (DM). The possibility of observing PBHs with
masses between $5\times 10^{14}$ and $10^{18}$ g is mainly connected with the
phenomenon of Hawking evaporation Hawking (1974, 1975); Zel’dovich (1976);
Carr (1976); Page (1976a, b, 1977); MacGibbon and Webber (1990); MacGibbon
(1991), namely the emission of particles induced by the PBHs gravitational
field. The nonobservation of any production that could be connected with
Hawking evaporation has allowed to severely constrain the contribution of PBHs
with masses $M_{\rm PBH}\lesssim 10^{17}$ g to the dark matter content of the
Universe Calabrese  _et al._ (2021); Coogan  _et al._ (2021); Dasgupta  _et
al._ (2021); Wang  _et al._ (2021); Iguaz  _et al._ (2021). In different mass
ranges, the amount of PBHs can further be constrained by different means, see
Ref. Carr  _et al._ (2021) for a review.

Hawking evaporation produces any species that is gravitationally interacting.
This includes also any elementary particle beyond the Standard Model with a
mass below the Hawking temperature. In particular, if DM is dominated by a
sufficiently light particle $\chi$ (with $m_{\chi}\lesssim 100$ MeV), it can
be emitted by PBHs. In Ref. Calabrese  _et al._ (2022), we proposed this
scenario as a possible source of experimental signatures in direct detection
experiments Agnes  _et al._ (2015); Aalseth  _et al._ (2018); Aprile  _et al._
(2012, 2019a); Meng  _et al._ (2021); Akerib  _et al._ (2020); Bueno (2006);
Fiorillo  _et al._ (2008); Marchionni  _et al._ (2011); Amaudruz  _et al._
(2016); Angloher  _et al._ (2005); Agnes  _et al._ (2015); Angloher  _et al._
(2016); Aalbers  _et al._ (2016); Agnese  _et al._ (2017); Aprile  _et al._
(2018); Akerib  _et al._ (2020); Aprile  _et al._ (2019b); Angloher  _et al._
(2017); Abdelhameed  _et al._ (2019); Aprile  _et al._ (2020a); Adhikari  _et
al._ (2021), due to the DM-nucleon interaction. We showed that the DM emitted
by PBHs would provide a signal qualitatively similar to the DM boosted by
cosmic-ray interactions Cappiello  _et al._ (2019); Bringmann and Pospelov
(2019); Ema  _et al._ (2019); Cappiello and Beacom (2019); Ema  _et al._
(2021); Andriamirado  _et al._ (2021). The nonobservation of such a signal was
used to constrain the combined parameter space of PBHs and light DM.

Given our ignorance of the interactions of DM with the Standard Model, a
natural question is what are the corresponding constraints if DM interacts
only with electrons Essig  _et al._ (2012a, b, 2017); Crisler  _et al._
(2018); Agnes  _et al._ (2018); Aguilar-Arevalo  _et al._ (2019); Andersson
_et al._ (2020). Here we aim to answer this question. Recently, the same
subject has been discussed in Ref. Li and Liao (2022), where the authors apply
our treatment discussed in Ref. Calabrese  _et al._ (2022) to the DM-electron
interaction and obtain constraints from Super-Kamiokande Kachulis  _et al._
(2018) and XENON1T Aprile  _et al._ (2020b) measurements. However, the authors
do not correctly account for the ionization of atoms that results from the DM-
electron scattering. Rather, they assume electrons to be free. While this may
be a reasonable approximation for Super-Kamiokande Kachulis  _et al._ (2018);
Ema  _et al._ (2019); Cappiello and Beacom (2019), it is not the correct
treatment for XENON1T Lee  _et al._ (2015). For the same reason, the
discussion of energy loss in the Earth is also not applicable to the DM-
electron interaction, since the DM particle does not just lose energy, but
rather performs a random walk traversing the Earth for sufficiently large
cross sections Cappiello and Beacom (2019). Finally, the authors of Ref. Li
and Liao (2022) model the interaction between $\chi$ and electrons with a
differential cross section flat in the electron recoil energy, which may not
be easily realized in a realistic particle model setting.

In the present work, which was developed independently at the same time as
Ref. Li and Liao (2022), we approach the determination of the constraints with
a correct treatment of the atom ionization for Xenon nuclei. We adopt an
effective field theory for the DM-electron interaction, and we use it to
deduce the constraints from Super-Kamiokande and XENON1T. Remarkably, the
parameter space of the model is also affected by additional constraints, such
as cosmological Henning and Murayama (2012); Sabti  _et al._ (2020, 2021) and
collider Liang  _et al._ (2021) ones, which are completely absent in Ref. Li
and Liao (2022). The constraints that we draw are not necessarily connected
with the assumption that the species emitted by PBHs is dark matter. Indeed,
any light species is emitted by Hawking evaporation and can subsequently be
detected if it has a coupling with electrons. We emphasize that, in order to
reach Earth from very far distances, such a species should be stable on
cosmological scales. Therefore, throughout the paper, we will refer to this
species as a generic light new particle $\chi$, and will derive the
constraints with no further assumptions.

We structure the discussion as follows. In Sec. II we compute the flux of the
new fermionic particles $\chi$ from the evaporation of the primordial black
holes. In Sec. III we discuss the possible detection of $\chi$ particles in
two different categories of experiments: XENON1T and Super-Kamiokande. Then,
we report in Sec. IV the constraints we obtain for the combined parameter
space of primordial black holes and light new species. Finally, we draw our
conclusions in Sec V.

##  II Flux of light particles from Primordial Black Holes

In this section we derive the flux of a fermionic species $\chi$ emitted from
evaporating PBHs. PBHs are fully characterized by mass, charge, and spin.
However, evaporating, they lose mass slower than they lose charge Page
(1976b); Zaumen (1974); Carter (1974); Gibbons (1975); Page (1976a, 1977). For
this reason, it is reasonable today to consider neutral PBHs that formed in
the very early Universe. Moreover, we consider spinless PBHs since rotating
PBHs evaporate faster causing a higher flux on Earth and leading to more
stringent constraints. The radiation emitted by a single PBH of a mass ${\rm
M}_{\rm PBH}$ is characterized by a thermal blackbodylike spectrum defined by
the Hawking temperature $\mathcal{T}_{\rm PBH}$. In the case of nonrotating
and neutral PBHs, it takes the following expression Hawking (1974, 1975)

$k_{B}\mathcal{T}_{\rm PBH}=\frac{\hbar c^{3}}{8\pi G_{\rm N}{\rm M}_{\rm
PBH}}\approx 1.06\left[\frac{10^{16}~{}{\rm g}}{{\rm M}_{\rm
PBH}}\right]\mathrm{MeV}\,,$ (1)

where $k_{B}$, $G_{N}$ and $\hbar$ are the Boltzmann, the gravitational and
the Planck constants, respectively and $c$ is the speed of light. The
differential spectrum of a single species $\chi$ from an evaporating PBH is
given by Hawking (1974, 1975)

$\frac{{\rm d}N}{{\rm d}t{\rm
d}T}=\frac{n_{\mathrm{dof}}^{\chi}\Gamma^{\chi}(T,\mathcal{T}_{\rm
PBH})}{2\pi(e^{(T+m_{\chi})/\mathcal{T}_{\rm PBH}}+1)},\,$ (2)

where $T$ is kinetic energy of $\chi$, $m_{\chi}$ is the mass and
$n_{\mathrm{dof}}^{\chi}=4$ is the number of degrees of freedom. The grey-body
factor $\Gamma^{\chi}$ takes into account the distortions in the black-body
spectrum and it is provided by the BlackHawk code Arbey and Auffinger (2019,
2021). This spectrum is peaked at an energy approximately corresponding to
$\sim 5$ times the Hawking temperature and is kinematically suppressed if its
mass is greater than such a value.

For clarity of exposition, we assume the PBH mass distribution to be
monochromatic. However, the method can be easily generalized to mass
distributions. We are able then to compute the $\chi$ flux from the spectrum
of primordial black holes. Two components contribute to the flux, one
resulting from the total emission of Extra-Galactic PBHs
($\Phi_{\mathrm{EG}}$) and the other from Galactic ($\Phi_{\mathrm{G}}$) ones.
The Extra-Galactic contribution can be written as

$\frac{{\rm d}\Phi_{\chi}^{\mathrm{EG}}}{{\rm
d}T}=\int_{t_{\mathrm{min}}}^{t_{\rm
max}}\differential{t}\left.\frac{\differential N}{\differential t\differential
T}\right|_{E_{s}}\frac{f_{\rm PBH}\,\Omega_{\rm DM}\rho_{\mathrm{cr}}}{{\rm
M}_{\rm PBH}}[1+z(t)],$ (3)

where $z(t)$ is the redshift and $E_{s}=\sqrt{(E^{2}_{\chi}-m_{\rm
DM}^{2})(1+z(t))^{2}+m_{\rm DM}^{2}}$ is the redshifted energy of $\chi$
particles. The quantity $\Omega_{\rm DM}$ is the cosmological dark matter
density and $\rho_{\mathrm{cr}}$ is the critical density. Indeed, since PBHs
are dark matter candidates, it is convenient to refer to the PBHs abundance as
the fraction of the dark matter content of the Universe:

$f_{\rm PBH}=\frac{\rho_{\rm PBH}}{\rho_{\mathrm{DM}}}=\frac{\Omega_{\rm
PBH}}{\Omega_{\mathrm{DM}}}\,,$ (4)

where $f_{\rm PBH}\leq 1$. The integral in Eq. (3) is performed from the time
of matter-radiation equality ($t_{\mathrm{min}}$) and the age of the Universe
($t_{\rm max}$) assuming that PBHs are not fully evaporated today. We
emphasize that considering times smaller than $t_{\rm min}$ would not affect
the flux in a appreciable way. The Extra-Galactic component is assumed to be
isotropic.

Concerning the Galactic component, this can be written as

$\frac{\differential\Phi_{\chi}^{\mathrm{G}}}{\differential
T}=\int\frac{\differential{\Omega}}{4\pi}\int_{0}^{l_{\rm
max}}\differential{l}\frac{\differential N}{\differential t\differential
T}\frac{f_{\rm PBH}\,\rho_{\rm NFW}(r(l,\phi))}{{\rm M}_{\rm PBH}},$ (5)

where $\rho_{\mathrm{NFW}}(r)$ is the standard Navarro-Frenk-White profile
defined as Navarro  _et al._ (1997)

$\rho_{\mathrm{NFW}}(r)=\rho_{\odot}\,\left(\frac{r_{\odot}}{r}\right)\left(\frac{1+r_{\odot}/r_{s}}{1+r/r_{s}}\right)^{2}$
(6)

where $\rho_{\odot}=0.4~{}{\rm GeV\,cm^{-3}}$ is the local DM density,
$r_{\odot}=8.5~{}{\rm kpc}$ is the distance between the Sun and the Milky Way
center, and $r_{s}=20~{}{\rm kpc}$ is the scale radius. The galactocentric
distance $r$ is

$r(l,\phi)=\sqrt{r_{\odot}^{2}-2lr_{\odot}\cos\phi+l^{2}}\,,\hskip
28.45274ptl_{\rm
max}=\sqrt{r_{h}^{2}-r_{\odot}^{2}\sin^{2}\phi}+r_{\odot}\cos\phi\,.$ (7)

where $r_{h}=200~{}{\rm kpc}$ is the halo radius. We mention that since our
analysis takes into account the $\chi$ flux from the whole sky (with a
sizeable contribution from Extra-Galactic PBHs), the exact choice of the
galactic profile is not so relevant.

According to the previous Eq.s (3) and (5), the total flux of $\chi$ particles
is proportional to $f_{\rm PBH}/{\rm M}_{\rm PBH}$. Therefore, the lower the
PBH mass, the higher the $\chi$ flux. However, at lower masses the fraction
$f_{\rm PBH}$ is typically constrained to a very small value ($<<1$) as given
in Ref. Carr  _et al._ (2021), thus suppressing the $\chi$ flux.

##  III Detection of light particles

The possibility of detecting the light particle $\chi$ is crucially dependent
on its interactions with the Standard Model. In Ref. Calabrese  _et al._
(2022) we assumed that $\chi$ interacts with nucleons, whereas here we focus
on its interaction with electrons. For the sake of definiteness, we consider
the coupling Lagrangian

$\mathcal{L}_{\chi}=\frac{1}{\Lambda^{2}}\bar{\chi}\chi\,\bar{\ell}\ell\,.$
(8)

This is a non-renormalizable effective field theory (EFT) coupling
characterized by the energy scale $\Lambda$, and $\ell$ refers to the Standard
Model lepton fields. Such an interaction arises at the effective level from a
scalar mediator with mass higher than the energies involved in the process,
which in our case is at most $\sim 10$ MeV. With this condition, the scalar
mediator can be integrated out, leaving the effective four-fermions
interaction of Eq. (8).

With this coupling, the $\chi$ particles emitted by PBHs can be looked for in
mainly two classes of experiments:

  * •

double phase dark matter direct detection experiments based on noble liquid
technology Agnes  _et al._ (2015); Aalseth  _et al._ (2018); Aprile  _et al._
(2012, 2019a); Meng  _et al._ (2021); Akerib  _et al._ (2020); Bueno (2006);
Fiorillo  _et al._ (2008); Marchionni  _et al._ (2011); Amaudruz  _et al._
(2016);

  * •

neutrino experiments based on water Cherenkov Aardsma  _et al._ (1987);
Aslanides  _et al._ (1999); Bazarko (2001); Ahrens  _et al._ (2002); Fukuda
_et al._ (2003).

In this paper, we consider XENON1T and Super-Kamiokande as representatives of
the two classes, respectively. For both of them, we use the measured data to
constrain the scenario of $\chi$ particles emitted from evaporating PBHs. In
this section, we first present the general framework for $\chi$ interacting
with the electrons bound to the atoms in the detector, and then specialize our
discussion to each of the two experiments separately.

The particles $\chi$ hit detector’s atoms interacting with electrons through
the effective interaction of Eq. (8). The $\chi$-$e$ scatterings would give
rise to atom ionization with an outgoing non-relativistic free electron as

$\chi+A\to\chi+A^{*}+e^{-},$ (9)

where $A$ denotes the atom, and $A^{*}$ is the ionized atom. The key quantity
characterizing the detection of the signal is the differential event rate
${\rm d}R_{\chi}/{\rm d}E_{r}$ per unit recoil energy $E_{r}$, which
corresponds to the free electron kinetic energy. It can be computed as:

$\frac{\differential R_{\chi}}{\differential
E_{r}}=n_{t}\,\eta(E_{r})\,F(E_{r})\,\int\differential{T}\,\frac{\differential\Phi_{\chi}}{\differential
T}\sum_{n,\,l}\frac{\differential\sigma^{n,\,l}}{\differential
E_{r}}(E_{r},m_{\chi},T).$ (10)

Here, the quantity $n_{t}$ is the number of detector’s targets per tonne and
$\eta$ is the detector’s efficiency. The quantity $F(E_{r})$ is the Fermi
factor that takes into account the distortion of the scattered electron
wavefunction by the presence of the atom. In the non-relativistic limit, we
have

$F(E_{r})=\frac{2\pi\nu}{1-e^{-2\pi\nu}},$ (11)

where $\nu=Z_{\rm eff}(\alpha m_{e}/\sqrt{2\,m_{e}\,E_{r}})$ and $Z_{\rm eff}$
is the effective charge that is felt by the scattered electron. In our
analysis, we conservatively set $Z_{\rm eff}=1$.111In general $Z_{\rm eff}$ is
greater than one since the shielding of the escaping electron by the remaining
bounded electrons is imperfect. In Ref. Lee  _et al._ (2014), it was pointed
out that $Z_{\rm eff}=1$ is a good approximation for outer-shell electrons.
Moreover, assuming $Z_{\rm eff}$ greater than the unity would enhance the
event rate. Therefore, our choice is good for outer-shell electrons and
conservative for the others. Finally, $\differential\Phi_{\chi}/\differential
T$ is the differential $\chi$ flux introduced in Sec. II, and
$\differential\sigma^{n,l}/\differential E_{r}$ is the differential cross
section for scattering of a $\chi$ particle on a bound electron with principal
quantum number $n$, orbital quantum number $l$, and recoil energy $E_{r}$.
Denoting by $E_{b}^{n,\,l}$ the electron binding energy of the atomic orbital
$(n,l)$, from energy conservation we have

$E_{\chi}-|E_{b}^{n,\,l}|=E_{\chi}^{\prime}+E_{r}$ (12)

where $E_{\chi}=T+m_{\chi}$ and $E_{\chi}^{\prime}$ are the initial and final
energy of $\chi$ particle, respectively.

The strength of the $\chi$-$e$ interaction can be parameterized in terms of
the coupling at the Lagrangian level. However, it is more common to express it
in terms of the cross section on a free electron at a fixed momentum transfer
$\alpha m_{e}$

$\overline{\sigma}_{\chi e}=\frac{\mu_{\chi
e}^{2}}{\pi\,\Lambda^{4}}\left(1+\frac{\alpha^{2}\,m_{e}^{2}}{4\,m_{\chi}^{2}}\right)\,$
(13)

with $\alpha$ being the fine-structure constant. This is the quantity that we
aim to constrain. Before discussing how the analysis proceeds for the two
experiments separately, it is worth noticing that the same $\chi$-$e$
interaction would cause an attenuation effect of the $\chi$ flux due to the
propagation in the atmosphere and the Earth Starkman  _et al._ (1990); Mack
_et al._ (2007); Kavanagh  _et al._ (2017); Emken and Kouvaris (2018).
However, as shown by Ref. Ema  _et al._ (2019); Cappiello and Beacom (2019)
the attenuation is negligible for $\overline{\sigma}_{\chi e}\lesssim
10^{-31}~{}{\rm cm^{2}}$. For this reason, we restrict our analysis to smaller
cross sections only.

###  III.1 XENON1T event rate

Figure 1: Xenon ionization function as a function of the momentum transfer
$\tilde{q}$ for orbitals and with $E_{r}=5$ keV and
$\tilde{k}^{\prime}=\sqrt{2\,m_{e}\,E_{r}}$.

XENON1T utilizes a dual-phase liquid Xe time projection chamber, used in most
of the experiments based on noble liquid, with a 2.0-tonne active target. Each
interaction inside the detector produces a prompt scintillation signal (signal
1) and a delayed electroluminescence signal (signal 2). The technology of this
kind of detector allows one to study both the elastic scattering between
$\chi$ particles and nuclei, and the ionization of electrons bound to atoms
due to $\chi$-$e$ interactions. Here, we use the measurements from XENON1T
Aprile  _et al._ (2020b) of the event rate for recoil energies $E_{r}$ in the
range $[1-30]$ keV to constrain the $\chi$-$e$ cross section in Eq. (13). For
this experiment the target density is $n_{t}=4.59\cdot 10^{27}\,{\rm
ton^{-1}}$ corresponding to the number of detector’s Xe nuclei, and we use the
detector efficiency $\eta(E_{r})$ reported in Ref. Aprile  _et al._ (2020b).

Assuming the outgoing electron to be non-relativistic, it is possible to show
that the differential cross section takes the following expression222In this
paper, given a four-momentum $k$ we denote $\tilde{k}=|\boldsymbol{k}|$.

$\frac{d\sigma^{n,\,l}}{dE_{r}}(E_{r},m_{\chi},T)=\frac{1}{8\pi\,\Lambda^{4}\,\tilde{p}^{2}\,m_{e}}\int_{\tilde{q}_{-}}^{\tilde{q}_{+}}\differential{\tilde{q}}\tilde{q}\frac{1}{2\,\tilde{k}^{\prime\,2}}\left|f^{n,\,l}_{\rm
ion}(\tilde{k}^{\prime},\tilde{q})\right|^{2}\left[(p\cdot(p-q)+m_{\chi}^{2})\,(k^{\prime}\cdot(k^{\prime}-q)+m_{e}^{2})\right]\,.$
(14)

where $p$ is the four-momentum of incoming $\chi$ particles, $k^{\prime}$ is
the one of the outgoing free electron, and $q$ is the momentum transfer. The
integration extremes $\tilde{q}_{\pm}$ are the minimum and maximum momentum
transfer kinematically allowed:

$\tilde{q}_{\pm}=\sqrt{(T+m_{\chi})^{2}-m_{\chi}^{2}}\pm\sqrt{\left(T+m_{\chi}-\varepsilon_{n,l}\right)^{2}-m_{\chi}^{2}}\,,$
(15)

where $\varepsilon_{n,l}=E_{r}+|E_{b}^{n,l}|$. Moreover, the quantities
$f^{n,l}_{\mathrm{ion}}$ are the ionization functions, which can be written as
Kopp  _et al._ (2009); Lee  _et al._ (2015); Catena  _et al._ (2020)

$|f^{n,\,l}_{\mathrm{ion}}(\tilde{k}^{\prime},\tilde{q})|^{2}=\frac{(2l+1)\tilde{k}^{\prime\,2}}{4\pi^{3}\tilde{q}}\int_{\tilde{k}^{\prime}-\tilde{q}}^{\tilde{k}^{\prime}+\tilde{q}}\differential{\tilde{k}}\,\tilde{k}|\widetilde{\mathcal{R}}_{n,\,l}(\tilde{k})|^{2}\,,$
(16)

where the integral is performed for all the allowed values for the module of
the electron initial momentum, $\tilde{k}$, and
$\widetilde{\mathcal{R}}_{n,\,l}(\tilde{k})$ is the Fourier transform of the
radial part of the bound electron wavefunction (see Kopp  _et al._ (2009) for
details)

$\widetilde{\mathcal{R}}_{n,\,l}(\tilde{k})=\sum_{\theta}\frac{C_{nl\theta}2^{-l+n_{l\theta}}}{\Gamma(3/2+l)}\left(\frac{2\pi
a_{0}}{Z_{l\theta}}\right)^{3/2}\left(\frac{i\tilde{k}a_{0}}{Z_{l\theta}}\right)^{l}\frac{(1+n_{l\theta}+l)!}{\sqrt{(2n_{l\theta})!}}\,_{2}F_{1}\left[\frac{(2+n_{l\theta}+l)}{2},\frac{(3+n_{l\theta}+l)}{2},\frac{3}{2}+l,-\left(\frac{\tilde{k}a_{0}}{Z_{l\theta}}\right)^{2}\right]$
(17)

where ${}_{2}F_{1}(a,b,c,x)$ is the hyper-geometric function. All the
coefficients appearing in this expression, $C_{nl\theta}$, $n_{l\theta}$ and
$Z_{l\theta}$, are tabulated in Ref. Bunge  _et al._ (1993), while $a_{0}$ is
the Bohr radius. In Fig. 1 we show some examples of the ionization functions
for Xenon orbitals as a function of the momentum transfer $\tilde{q}$ once the
electron recoil energy is fixed to 5 keV.

Expanding the terms in the last square bracket in Eq. (14) in the non-
relativistic regime for the electron and substituting the reference cross
section defined in Eq. (13), we obtain

$\frac{\differential\sigma^{n,\,l}}{\differential
E_{r}}(E_{r},m_{\chi},T)=\frac{\overline{\sigma}_{\chi e}\,m_{e}}{8\,\mu_{\chi
e}^{2}\tilde{k}^{\prime\,2}\,\tilde{p}^{2}\,\left(1+\frac{\alpha^{2}\,m_{e}^{2}}{4\,m_{\chi}^{2}}\right)}\int_{\tilde{q}_{-}}^{\tilde{q}_{+}}\differential{\tilde{q}}\tilde{q}\left|f_{ion}^{n,\,l}(\tilde{q},\tilde{k}^{\prime})\right|^{2}\left(2m_{\chi}^{2}+\frac{\tilde{q}^{2}-\varepsilon^{2}_{n,l}}{2}\right),$
(18)

Having determined the cross section, we proceed to obtain the event rate in
the XENON1T detector and compare it with the measurements from the
collaboration. The free parameters of the model are ${\rm M}_{\rm PBH}$,
$f_{\rm PBH}$, $m_{\chi}$, and $\overline{\sigma}_{\chi e}$. Since our aim is
to constrain the model, we adopt the statistical procedure suggested in Ref.
Cowan  _et al._ (2011) for setting upper limits. We define the chi-squared
variable

$\chi^{2}\left({\rm M}_{\rm PBH},\,m_{\chi},\,\overline{\sigma}_{\chi
e},\,f_{\rm PBH}\right):=\sum_{i}\frac{\left[\frac{dR_{\rm
obs}}{dE_{r}}-\left(\frac{dR_{\rm
BCK}}{dE_{r}}+\frac{dR_{\chi}}{dE_{r}}\right)\right]^{2}_{E_{r}=E_{r}^{i}}}{\sigma_{i}^{2}}\,.$
(19)

Here $dR_{\rm obs}/dE_{r}$ is the observed event rate, while $dR_{\rm
BCK}/dE_{r}$ is the estimated background event rate. For both of them we take
the results of Ref. Aprile  _et al._ (2020b). The sum is performed over all
the energy bins, and $\sigma_{i}$ are the uncertainties on $dR_{\rm
obs}/dE_{r}(E_{r}^{i})$. We then set the test statistic $\lambda$ for upper
limits as

$\lambda=\begin{cases}\chi^{2}\left({\rm M}_{\rm
PBH},\,m_{\chi},\,\overline{\sigma}_{\chi e},\,f_{\rm
PBH}\right)-\chi^{2}\left({\rm M}_{\rm
PBH},\,m_{\chi},\,\hat{\overline{\sigma}}_{\chi e},\,f_{\rm
PBH}\right),\qquad\overline{\sigma}_{\chi e}>\hat{\overline{\sigma}}_{\chi
e}\\\ 0,\qquad\overline{\sigma}_{\chi e}<\hat{\overline{\sigma}}_{\chi
e}\,,\end{cases}$ (20)

where $\hat{\overline{\sigma}}_{\chi e}$ is the value of the cross section
which minimizes the chi-squared. In this way, we provide the most conservative
upper limits on this quantity. We then exclude at $90\%$ confidence level the
region of the parameter space in which $\lambda>2.71$, following the
prescription in Ref. Cowan  _et al._ (2011).

###  III.2 Super-Kamiokande’s event rate

Super-Kamiokande is a water Cherenkov detector realized by a cylindrical tank
filled with 50kt of water. Charged particles in the water produce Cherenkov
radiation that is recorded by the photo-multipliers. The light particle $\chi$
is also expected to yield Cherenkov radiation when scattering inside the
detector. Super-Kamiokande has observed $N_{\rm SK}=4042$ events in the recoil
energy range between $[0.1-1.33]~{}{\rm GeV}$ Kachulis  _et al._ (2018).

In the case of Super-Kamiokande, previous analyses assume the electrons to be
free and at rest in the observer frame Necib  _et al._ (2017); Ema  _et al._
(2019); Cho  _et al._ (2020); Granelli  _et al._ (2022). In this work, we
follow the same approach and assume the number of targets to be
$n_{t}=3.34\times 10^{28}~{}{\rm ton^{-1}}$ and the detector efficiency to be
equal to 0.93 in the energy range considered Kachulis  _et al._ (2018). In the
limit of free electrons, we have $F(E_{r})=1$, and the differential cross
section is simply given by

$\frac{\differential\sigma}{\differential E_{r}}=\frac{\overline{\sigma}_{\chi
e}\,\Theta(E_{\rm max}-E_{r})}{8\,\mu^{2}_{\chi
e}\,\tilde{p}^{2}\left(1+\frac{\alpha^{2}\,m_{e}^{2}}{4\,m_{\chi}^{2}}\right)}(2m_{e}+E_{r})(2m_{\chi}^{2}+m_{e}E_{r})$
(21)

where $E_{\rm max}$ is the maximum allowed recoil energy equal to

$E_{\rm
max}=\frac{2\,m_{e}\,T\,(T+2\,m_{\chi})}{\left(\left(m_{e}+m_{\chi}\right)^{2}+2\,m_{e}\,T\right)}\,.$
(22)

Plugging this equation into Eq. (10) we can compute the expected number of
events in Super-Kamiokande provided by $\chi$ particles. Following Ref. Ema
_et al._ (2019), we conservatively obtain the constraints on
$\overline{\sigma}_{\chi e}$ by simply requiring that

$\mathcal{E}_{\rm SK}\times\int_{0.1{\rm GeV}}^{1.33{\rm
GeV}}\differential{E_{r}}\,\frac{\differential R_{\chi}}{\differential
E_{r}}<N_{\rm SK}\,,$ (23)

where $\mathcal{E}_{\rm SK}=161.9~{}{\rm kton~{}yr}$ is the Super-Kamiokande
exposure Kachulis  _et al._ (2018).

##  IV Results

Figure 2: Constraints on the reference $\chi$-$e$ cross section
$\overline{\sigma}_{\chi e}$ in Eq. (13) as a function of $\chi$ mass
$m_{\chi}$ in case of XENON1T (left panel) and Super-Kamiokande (right panel).
The color shaded regions represents our results for PBH masses once their
fraction $f_{\rm PBH}$ is fixed to the corresponding maximum value according
to current constraints (see footnote 3). The hatched region is excluded by Big
Bang Nucleosynthesis, while the thin dashed lines correspond to the limits
that apply only if $\chi$ particles are dark matter: (1) boosted dark matter
from cosmic-ray up-scatterings Cappiello and Beacom (2019); (2) Solar
reflection with XENON1T An  _et al._ (2018); (3) combined constraints from
dark matter direct detection experiments (see Ref. Cappiello and Beacom
(2019)).

The analysis at XENON1T Aprile  _et al._ (2020b) and Super-Kamiokande Kachulis
_et al._ (2018) allows to constrain the combined parameter space of the model,
which is expressed in terms of the PBH mass $M_{\mathrm{PBH}}$, the PBH
fraction of DM $f_{\mathrm{PBH}}$, the mass of the light particle $m_{\chi}$,
and the cross section $\overline{\sigma}_{\chi e}$ introduced above in Eq.
(13). The large dimensionality of the parameter space requires to show the
constraints as sections in the two-dimensional planes. In particular, in Fig.
2 we show the constraints in the $m_{\chi}-\overline{\sigma}_{\chi e}$ plane
for varying masses of the PBHs. For each mass of the PBHs, we have assumed
$f_{\mathrm{PBH}}$ to be as large as allowed by the present constraints on the
PBH parameter space alone. We present separately the constraints from XENON1T
(left panel) and Super-Kamiokande (right panel). We show as well the
constraints (dashed thin lines) which would apply if $\chi$ is identified as
the dominant component of dark matter (see the caption).

Due to the differences in the cuts in the recoil energies, the two experiments
are able to probe a different range of PBH masses. In particular, XENON1T can
look at somewhat heavier PBHs with a smaller Hawking temperature and
corresponding smaller recoil energies. We find that in both cases the
constraints that can be drawn, assuming that PBHs are as many as allowed,333In
this analysis, we considering the constraints on $f_{\rm PBH}$ derived from
Extra-Galactic gamma-ray data Carr  _et al._ (2021) and from isotropic X-ray
observations Iguaz  _et al._ (2021). The latter provides the stronger limit
for $\mathrm{M}_{\mathrm{PBH}}\gtrsim 10^{16}~{}{\rm g}$. Such constraints do
not depend on the choice of the dark matter galactic profile since they are
based on Extra-Galactic and isotropic measurements. are complementary to the
constraints on light DM. However, if the species $\chi$ is not identified as
DM, the constraints from the dashed lines do not apply. Nevertheless, for
$m_{\chi}\lesssim 1$ MeV any light species sufficiently strongly interacting
would be in serious tension with a successful Big Bang Nucleosynthesis. For
this reason, we consider this region as completely excluded, even though
specific models can be devised to evade these constraints Elor  _et al._
(2021).

Among the possible additional constraints, we can consider the ones from
collider experiments Liang  _et al._ (2021), which we do not show for purely
graphical reasons, since they would exclude the region
$\overline{\sigma}_{\chi e}\gtrsim 10^{-44}$ cm2, which is much lower than the
range of our figures. However, we have to stress that the collider constraints
are strongly sensitive to the mediator mass, as shown in Ref. Liang  _et al._
(2021). While the authors do not provide the constraints for a scalar mediator
with arbitrary mass, which we are studying in this work, for the vector
mediator model they clearly show that the constraints significantly weaken for
lighter mediators, already at a mass of $0.7$ GeV. Furthermore, the
constraints from collider are expected to be limited from above by a ceiling,
when the cross section becomes sufficiently large for the $\chi$ particle to
interact in the calorimeter. This has been explicitly studied for the case of
$\chi$-nucleon interaction Cappiello  _et al._ (2019), yet no such study
exists for $\chi$-electron interaction. Therefore, obtaining constraints which
are complementary to the collider ones is especially necessary. Finally, light
dark matter coupled to electrons can be emitted in dense astrophysical
environments, in particular in supernova for these DM masses. Indeed, the
emission of light particles from supernova would lead to an additional cooling
mechanism and change the duration of the neutrino burst. This would have led
to observable consequences for the case of SN1987A. The constraints from
SN1987A are discussed in the context of a specific interaction model with
vector mediator in Ref. Chang  _et al._ (2018). These results show that, at
$\overline{\sigma}_{\chi e}\gtrsim 10^{-39}$ cm2, SN1987A does not lead to
significant constraints because the $\chi$ particle would be trapped inside
the supernova without leading to an observable signal. Since we focus on a
scalar interaction, these results are not directly applicable. However, they
suggest that at the large cross sections we consider, the constraints from
SN1987A would not be applicable.

A crucial point is the comparison between our results and the results of Ref.
Li and Liao (2022), since they are representing the same constraints. For
Super-Kamiokande we find constraints which, for the same PBH mass, are about
two orders of magnitude stronger. This is mainly due to our use of a concrete
particle model for the $\chi$-electron interaction, whereas Ref. Li and Liao
(2022) simply assumes a differential cross section flat in the electron recoil
energy, which is not necessarily realizable in a realistic particle model. For
XENON1T the constraints we find are significantly weaker, by as much as three
orders of magnitude. We attribute this difference mostly to the complete
different treatment of the DM-electron interaction. In fact, whereas in Ref.
Li and Liao (2022) the electrons are treated as free, we correctly account for
the ionization of the xenon atoms in the detector.

Figure 3: Constraints shown in the plane $(f_{\rm
PBH}\cdot\overline{\sigma}_{\chi e})-{\rm M}_{\rm PBH}$ for different masses
$m_{\chi}$. In warm tones are shown the constraints obtained using Super-
Kamiokande, while in cold tones are shown the constraints obtained using
XENON1T. The dark grey band shows the region that cannot be probed in our
analysis due to existing constraints on $f_{\rm PBH}$ (see footnote 3) and
assuming $\overline{\sigma}_{\chi e}\lesssim 10^{-31}~{}{\rm cm^{2}}$ to
neglect attenuation effects. The light grey band is excluded by requiring
$\overline{\sigma}_{\chi e}\lesssim 10^{-44}~{}{\rm cm^{2}}$ according to
collider constraints Liang  _et al._ (2021).

A more compact, although less conventional, way of showing our results is to
exploit the proportionality of the signal in any detector to the product
$f_{\mathrm{PBH}}\cdot\overline{\sigma}_{\chi e}$. Therefore, the parameters
effectively determining the constraints are in reality three, namely
$m_{\chi}$, $\mathrm{M}_{\mathrm{PBH}}$, and
$f_{\mathrm{PBH}}\cdot\overline{\sigma}_{\chi e}$. We show the constraints in
the $(f_{\mathrm{PBH}}\cdot\overline{\sigma}_{\chi
e})-\mathrm{M}_{\mathrm{PBH}}$ plane in Fig. 3 both for XENON1T and Super-
Kamiokande. This figure again shows that the range of PBH masses probed by the
two experiments is different. Furthermore, for XENON1T this range potentially
extends up to $10^{18}$ g depending on the mass $m_{\chi}$, since the light
particle is only emitted if $m_{\chi}$ is smaller than the Hawking temperature
of the PBH.

##  V Conclusions

In this paper, we have extended the work Calabrese  _et al._ (2022) where we
proposed to test the potential existence of new light particles beyond the
Standard Model through the evaporation of primordial black holes. In
particular, we have focused on the possible interactions of such new particles
with the electrons through an effective coupling mediated by a heavy scalar.
Hence, we have investigated the consequent detection of $\chi$ particles from
evaporating primordial black holes in dark matter direct detection
experiments, such as XENON1T, and neutrino detectors, such as Super-
Kamiokande. The nonobservation of the expected signal has been used to set
constraints on the combined parameter space of primordial black holes and
light new particles. In the case of XENON1T, we have performed a binned
likelihood analysis comparing the observed event rate with the one predicted
in our scenario. We have properly taken into account the ionization process of
Xe atoms due to $\chi$-$e$ scattering. On the other hand, for Super-Kamiokande
we have performed a more conservative analysis based on the total number of
events detected. We have found that XENON1T (Super-Kamiokande) is able to
constrain the $\chi$-$e$ cross section down to $10^{-32}~{}{\rm cm^{2}}$
($10^{-35}~{}{\rm cm^{2}}$) for $m_{\chi}=1~{}{\rm MeV}$ in case of PBH with
masses from $5\times 10^{14}$ to $10^{18}$ g. These limits are complementary
to cosmological and collider constraints and do not require the $\chi$
particles to be dark matter.

### Acknowledgements

We thank Gennaro Miele and Stefano Morisi for useful comments and discussions.
This work was supported by the research grant number 2017W4HA7S “NAT-NET:
Neutrino and Astroparticle Theory Network” under the program PRIN 2017 funded
by the Italian Ministero dell’Università e della Ricerca (MUR) and by the
research project TAsP (Theoretical Astroparticle Physics) funded by the
Istituto Nazionale di Fisica Nucleare (INFN). RC acknowledges also financial
support from the agreement ASI-INAF n.2017-14-H.O.. The work of DFGF is
partially supported by the Villum Fonden under project no. 29388. This project
has received funding from the European Union’s Horizon 2020 research and
innovation program under the Marie Sklodowska-Curie grant agreement No. 847523
‘INTERACTIONS’.

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