aainstitutetext: School of Physics, Sun Yat-Sen University, Guangzhou 510275,
China bbinstitutetext: School of Physics, Nankai University, Tianjin 300071,
China

# Constraints on general neutrino interactions with exotic fermion from
neutrino-electron scattering experiments

Zikang Chen b Tong Li a Jiajun Liao<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The couplings between the neutrinos and exotic fermion can be probed in both
neutrino scattering experiments and dark matter direct detection experiments.
We present a detailed analysis of the general neutrino interactions with an
exotic fermion and electrons at neutrino-electron scattering experiments. We
obtain the constraints on the coupling coefficients of the scalar,
pseudoscalar, vector, axialvector, tensor and electromagnetic dipole
interactions from the CHARM-II, TEXONO and Borexino experiments. For the
flavor-universal interactions, we find that the Borexino experiment sets the
strongest bounds in the low mass region for the electromagnetic dipole
interactions, and the CHARM-II experiment dominates the bounds for other
scenarios. If the interactions are flavor dependent, the bounds from the
CHARM-II or TEXONO experiment can be avoided, and there are correlations
between the flavored coupling coefficients for the Borexino experiment. We
also discuss the detection of sub-MeV DM absorbed by bound electron targets
and illustrate that the vector coefficients preferred by XENON1T data are
allowed by the neutrino-electron scattering experiments.

###### Keywords:

Beyond Standard Model, Neutrino Physics

††arxiv: arXiv:2102.09784

##  1 Introduction

The phenomenon of neutrino oscillations has been well confirmed by various
neutrino experiments in the last two decades Zyla:2020zbs . Since the
explanation of neutrino oscillations requires nonvanishing neutrino masses,
which cannot be accounted for by the Standard Model (SM), the observation of
neutrino oscillations provides a strong motivation to search for new physics
beyond the SM that are associated with neutrinos. Moreover, the existence of
dark matter (DM) through abundant cosmological and astrophysical observations
is one of the most plausible evidences of new physics beyond the SM. The DM
direct detection experiments have pushed the limit on the cross section of DM
scattering off nucleus close to the neutrino floor for weak scale DM. The
couplings between the neutrinos and sub-GeV DM through the scattering off
nucleus have been studied in Ref. Brdar:2018qqj ; Chang:2019sel ; Dror:2019onn
; Dror:2019dib ; Chang:2020jwl ; Li:2020lba ; Hurtado:2020vlj ; Li:2020pfy .
There are also plenty of novel models and signatures proposed to search for
sub-GeV DM through the scattering off electrons Ge:2020jfn ; Shoemaker:2020kji
; Shakeri:2020wvk ; Dror:2020czw ; Brdar:2020quo ; AristizabalSierra:2020zod .

Recently, the XENON collaboration reported an excess of electronic recoil
events with the energy around 2-3 keV Aprile:2020tmw  and the event
distribution has a broad spectrum for the excess. They collected low energy
electron recoil data from the XENON1T experiment with an exposure of 0.65
tonne-years and analyzed various backgrounds for the excess events. Although a
small tritium background fits the excess data well, the solar axion
explanation or the solar neutrinos with magnetic moment can also provide a
plausible source for the peak-like excess. However, both of the two scenarios
have tension with stellar cooling constraints Viaux:2013lha ; Diaz:2019kim ;
DiLuzio:2020jjp ; Gao:2020wer ; Dent:2020jhf ; Brdar:2020quo . Some studies
instead proposed to explain the XENON1T excess through the electron recoil by
solar neutrinos with the sterile neutrino DM in the final states of inelastic
scattering Ge:2020jfn ; Shoemaker:2020kji . On the other hand, the inverse
process in which the incoming fermionic DM is absorbed by bound electron
targets and emits a neutrino is sensitive to the DM with mass below MeV
Shakeri:2020wvk ; Dror:2020czw . The two kinds of signals are governed by the
same interactions between SM neutrino and the exotic fermion. The relevant
interactions are inevitably constrained by the precision measurements in
neutrino experiments Brdar:2020quo ; AristizabalSierra:2020zod . In this work
we study the constraints on general neutrino interactions with sub-GeV exotic
fermion from neutrino-electron scattering experiments.

The large volume detectors enable precise measurements of neutrino properties.
The large neutrino detectors like Borexino can be used to place constraints on
general neutrino interactions. The Borexino experiment, located at the
Laboratori Nazionali del Gran Sasso, was built with a primary goal of
measuring solar neutrinos from the $pp$ chain Agostini:2017ixy . We employ the
Borexino measurements of low energy solar neutrinos to set limits on the
neutrino-electron scattering with an outgoing fermion $\chi$

$\displaystyle\nu e\to\chi e\;,$ (1)

where $\chi$ could be sterile neutrino or other possible exotic fermions. The
results apply for the exotic fermion $\chi$ being either DM candidate or not.
We restrict the general neutrino interactions categorized by dimension-5
dipole operators and dimension-6 four-fermion operators. They respect Lorentz
invariance and the gauge symmetries $SU(3)_{c}\times U(1)_{\rm em}$. The
scattering cross section from the magnetic and electric dipole operators is
inversely proportional to the recoil energy and thus the experiments with low
energy threshold are sensitive to them. For the four-fermion interactions, all
Lorentz-invariant operators (scalar, vector, pseudoscalar, axialvector and
tensor) will be explored in the neutrino-electron scattering. As the produced
solar electron neutrinos oscillate into muon and tau neutrinos, we can also
place limits on the general interactions of all neutrino flavors. In addition,
accelerator neutrinos with the energy being several tens of GeV can be used to
exploit large $\chi$ mass region. We thus take into account the constraints
from the CHARM-II experiment Vilain:1993kd ; Vilain:1994qy  as well as reactor
neutrino using TEXONO Deniz:2009mu  data. The relevant data were used to study
the electromagnetic interactions or the nonstandard vector-type interactions
of a neutrino via the transition $\nu e\to\nu e$ Khan:2014zwa ; Khan:2016uon ;
Khan:2017oxw ; Khan:2017djo ; Lindner:2018kjo ; Khan:2019jvr . The heavy
neutral leptons as well as DM scattering off electrons can be also searched in
high-energy neutrino beam-dump experiment or far-forward detector at the LHC
Jodlowski:2020vhr ; Batell:2021blf .

This paper is organized as follows. In Sec. 2, we discuss the effective
Lagrangian of an exotic fermion interacting with neutrino and electron. Then
we display the amplitudes and differential cross sections of neutrino-electron
scattering with the outgoing exotic fermion. In Sec. 3, we consider three
neutrino-electron scattering experiments: CHARM-II, TEXONO and Borexino, and
show the constraints on the general neutrino interactions. The detection of
sub-MeV DM absorbed by bound electron targets is then discussed in Sec. 5.
Finally, we summarize our conclusions in Sec. 6.

##  2 The general neutrino interactions with exotic fermion via neutrino-
electron scattering

We consider a Dirac fermion $\chi$ and its general interactions with neutrino
and electron. The effective Lagrangian including both dim-5 dipole operators
and dim-6 four-fermion operators reads as

$\displaystyle\mathcal{L}\supset{G_{F}\over\sqrt{2}}\left[\sum_{a}\bar{\chi}\Gamma^{a}\nu_{\alpha}~{}\bar{e}\Gamma^{a}(\epsilon^{a}_{\alpha}+\tilde{\epsilon}^{a}_{\alpha}\gamma^{5})e+{v_{H}\over\sqrt{2}}\bar{\chi}\sigma^{\mu\nu}(\epsilon^{M}_{\alpha}+\epsilon^{E}_{\alpha}\gamma_{5})\nu_{\alpha}F_{\mu\nu}\right]+h.c.\;,$
(2)

where $\alpha\equiv\\{e,\mu,\tau\\}$, $v_{H}\simeq 246$ GeV is the vacuum
expectation value of the SM Higgs, $F_{\mu\nu}$ is the electromagnetic field
strength tensor and
$\Gamma^{a}\equiv\\{I,i\gamma^{5},\gamma^{\mu},\gamma^{\mu}\gamma^{5},\sigma^{\mu\nu}\equiv{i\over
2}[\gamma^{\mu},\gamma^{\nu}]\\}$ correspond to the scalar ($S$), pseudoscalar
($P$), vector ($V$), axialvector ($A$) and tensor ($T$) operator,
respectively. The four-fermion operators are analogous to those in Ref.
Khan:2019jvr . Here the dimensionless parameters $\epsilon_{\alpha}^{M}$,
$\epsilon_{\alpha}^{E}$, $\epsilon_{\alpha}^{a}$ and
$\tilde{\epsilon}_{\alpha}^{a}$ are in general complex. The presence of new
interactions of Eq. (2) will give rise to the tree-level neutrino-electron
scattering, as shown in Fig.1.

Figure 1: The tree-level Feynman diagrams for the
$\nu_{\alpha}+e^{-}\rightarrow\chi+e^{-}$ process, where the circular bulb
(square) represents the effective dim-5 dipole (dim-6 four-fermion)
interaction.

In the SM, the neutrino-electron scattering is governed by both the weak
neutral current (NC) and charged current (CC). The effective Lagrangian for
the SM NC is given by

$\displaystyle\mathcal{L}_{\rm
NC}={G_{F}\over\sqrt{2}}\bar{\nu}\gamma^{\mu}(1-\gamma_{5})\nu\bar{e}\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})e\;,$
(3)

where $g_{V}=-{1\over 2}+2\sin^{2}\theta_{W}$ and $g_{A}=-{1\over 2}$. The CC
Lagrangian can be transmitted as

$\displaystyle\mathcal{L}_{\rm
CC}={G_{F}\over\sqrt{2}}\bar{\nu}\gamma^{\mu}(1-\gamma_{5})\nu\bar{e}\gamma_{\mu}(1-\gamma_{5})e\;.$
(4)

The CC current only contributes to the scattering of $\nu_{e}$. The
differential cross section of neutrino-electron scattering in the SM is
Khan:2019jvr

$\displaystyle\frac{d\sigma^{\rm SM}_{\alpha\beta}}{dE_{R}}$ $\displaystyle=$
$\displaystyle{G_{F}^{2}m_{e}\over
2\pi}\Big{[}\Big{(}g^{L}_{\alpha\beta}\Big{)}^{2}+\Big{(}g^{R}_{\alpha\beta}\Big{)}^{2}\Big{(}1-{E_{R}\over
E_{\nu}}\Big{)}^{2}-g^{L}_{\alpha\beta}g^{R}_{\alpha\beta}{m_{e}E_{R}\over
E_{\nu}^{2}}\Big{]}\;,$ (5)

where $\alpha$ ($\beta$) denotes the flavor of the neutrino in the initial
(final) states, $E_{\nu}$ is the neutrino energy, $E_{R}$ is the electron
recoil energy and

$\displaystyle\Big{(}g^{L}_{\alpha\beta},g^{R}_{\alpha\beta}\Big{)}=\left\\{\begin{array}[]{ll}(2\sin^{2}\theta_{W}+1,2\sin^{2}\theta_{W}),&\hbox{$\alpha=\beta=e$;}\\\
(2\sin^{2}\theta_{W}-1,2\sin^{2}\theta_{W}),&\hbox{$\alpha=\beta=\mu,\tau$;}\\\
0,&\hbox{$\alpha\neq\beta$.}\end{array}\right.$ (9)

We then calculate the differential cross section of
$\nu_{\alpha}+e\rightarrow\chi+e$ by following the procedure given in the
appendix. Here we show the differential cross sections for different
operators:

$\displaystyle\frac{d\sigma^{S}_{\nu_{\alpha}}}{dE_{R}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{e}}{8\pi}\left[|\epsilon_{\alpha}^{S}|^{2}\left(1+\frac{E_{R}}{2m_{e}}\right)+|\tilde{\epsilon}_{\alpha}^{S}|^{2}\frac{E_{R}}{2m_{e}}\right]\left(\frac{m_{e}E_{R}}{E_{\nu}^{2}}+\frac{m_{\chi}^{2}}{2E_{\nu}^{2}}\right)\;,$
(10) $\displaystyle\frac{d\sigma^{P}_{\nu_{\alpha}}}{dE_{R}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{e}}{8\pi}\left[|\epsilon_{\alpha}^{P}|^{2}\frac{E_{R}}{2m_{e}}+|\tilde{\epsilon}_{\alpha}^{P}|^{2}\left(1+\frac{E_{R}}{2m_{e}}\right)\right]\left(\frac{m_{e}E_{R}}{E_{\nu}^{2}}+\frac{m_{\chi}^{2}}{2E_{\nu}^{2}}\right)\;,$
(11) $\displaystyle\frac{d\sigma_{\nu_{\alpha}}^{V}}{dE_{R}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{e}}{4\pi}\left[\left(|\epsilon_{\alpha}^{V}|^{2}+|\tilde{\epsilon}_{\alpha}^{V}|^{2}\right)\left(1-{E_{R}\over
E_{\nu}}+{E_{R}^{2}\over 2E_{\nu}^{2}}-{m_{\chi}^{2}\over
2E_{\nu}m_{e}}+{m_{\chi}^{2}E_{R}\over 4E_{\nu}^{2}m_{e}}\right)\right.$ (12)
$\displaystyle\left.-\left(|\epsilon_{\alpha}^{V}|^{2}-|\tilde{\epsilon}_{\alpha}^{V}|^{2}\right)\left({E_{R}m_{e}\over
2E_{\nu}^{2}}+{m_{\chi}^{2}\over
4E_{\nu}^{2}}\right)-2\text{Re}[\epsilon_{\alpha}^{V}(\tilde{\epsilon}_{\alpha}^{V})^{*}]{E_{R}\over
E_{\nu}}\left(1-{E_{R}\over 2E_{\nu}}-{m_{\chi}^{2}\over
4E_{\nu}m_{e}}\right)\right]\;,$
$\displaystyle\frac{d\sigma_{\nu_{\alpha}}^{A}}{dE_{R}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{e}}{4\pi}\left[\left(|\epsilon_{\alpha}^{A}|^{2}+|\tilde{\epsilon}_{\alpha}^{A}|^{2}\right)\left(1-{E_{R}\over
E_{\nu}}+{E_{R}^{2}\over 2E_{\nu}^{2}}-{m_{\chi}^{2}\over
2E_{\nu}m_{e}}+{m_{\chi}^{2}E_{R}\over 4E_{\nu}^{2}m_{e}}\right)\right.$ (13)
$\displaystyle\left.+\left(|\epsilon_{\alpha}^{A}|^{2}-|\tilde{\epsilon}_{\alpha}^{A}|^{2}\right)\left({E_{R}m_{e}\over
2E_{\nu}^{2}}+{m_{\chi}^{2}\over
4E_{\nu}^{2}}\right)-2\text{Re}[\epsilon_{\alpha}^{A}(\tilde{\epsilon}_{\alpha}^{A})^{*}]{E_{R}\over
E_{\nu}}\left(1-{E_{R}\over 2E_{\nu}}-{m_{\chi}^{2}\over
4E_{\nu}m_{e}}\right)\right]\;,$
$\displaystyle\frac{d\sigma_{\nu_{\alpha}}^{T}}{dE_{R}}$ $\displaystyle=$
$\displaystyle\frac{2G_{F}^{2}m_{e}|\epsilon_{\alpha}^{T}-\tilde{\epsilon}_{\alpha}^{T}|^{2}}{\pi}\left[1-{E_{R}\over
E_{\nu}}+{E_{R}^{2}\over 4E_{\nu}^{2}}-{E_{R}m_{e}\over
4E_{\nu}^{2}}-{m_{\chi}^{2}\over 4E_{\nu}^{2}}\left({1\over 2}+{2E_{\nu}\over
m_{e}}-{E_{R}\over 2m_{e}}\right)\right]\;,$ (14)
$\displaystyle\frac{d\sigma_{\nu_{\alpha}}^{\text{EM}}}{dE_{R}}$
$\displaystyle=$
$\displaystyle{2\sqrt{2}\alpha_{\text{EM}}G_{F}|\epsilon_{\alpha}^{E}-\epsilon_{\alpha}^{M}|^{2}\over
m_{e}}\left[{m_{e}\over E_{R}}-{m_{e}\over E_{\nu}}-{m_{\chi}^{2}\over
E_{\nu}E_{R}}\left({1\over 2}-{E_{R}\over 4E_{\nu}}+{m_{e}\over
E_{\nu}}\right)-{m_{\chi}^{4}\over 8E_{\nu}^{2}E_{R}^{2}}\left(1-{E_{R}\over
m_{e}}\right)\right]\;,$ (15)

where $\alpha_{\text{EM}}\simeq 1/137$ is the electromagnetic fine structure
constant. Here we assume each of the scalar, pseudoscalar, vector,
axialvector, tensor and electromagnetic dipole operators dominates at a time.
The differential cross sections of
$\bar{\nu}_{\alpha}+e\rightarrow\bar{\chi}+e$ are the same as those of
$\nu_{\alpha}+e\rightarrow\chi+e$ except for the cross term of the axialvector
operator in Eq. (13) changed sign. From the above equations, we realize that
the the bounds on
$\\{\tilde{\epsilon}_{\alpha}^{S},\tilde{\epsilon}_{\alpha}^{P},\tilde{\epsilon}_{\alpha}^{V},\tilde{\epsilon}_{\alpha}^{A},\tilde{\epsilon}_{\alpha}^{T},\epsilon_{\alpha}^{E}\\}$
will be the same as
$\\{\epsilon_{\alpha}^{P},\epsilon_{\alpha}^{S},\epsilon_{\alpha}^{A},\epsilon_{\alpha}^{V},\epsilon_{\alpha}^{T},\epsilon_{\alpha}^{M}\\}$
if we consider only one $\epsilon$ parameter at a time.

##  3 Neutrino-electron scattering experiments

The parameter space of the general neutrino interactions with electrons can be
constrained by various neutrino-electron scattering experiments such as CHARM-
II Vilain:1993kd ; Vilain:1994qy , LAMPF Allen:1992qe , LSND Auerbach:2001wg ,
TEXONO Deniz:2009mu  and MINER$\nu$A Park:2015eqa ; Valencia:2019mkf . The
precision of these neutrino-electron scattering experiments can be inferred
from their measurement of $\sin^{2}\theta_{W}$ Rodejohann:2017vup , i.e.,
$\sin^{2}\theta_{W}=0.2324\pm 0.0083$ at CHARM-II, $0.249\pm 0.063$ at LAMPF
Allen:1992qe , $0.248\pm 0.051$ at LSND Auerbach:2001wg , and $0.251\pm 0.039$
at TEXONO Deniz:2009mu . Hence, we consider CHARM-II and TEXONO in our
analysis since they have the strongest sensitivity to
$\sin^{2}\theta_{W}$111The neutrino-electron scattering data at MINER$\nu$A
can be also used to impose competitive bounds on new interactions as CHARM-II,
see e.g. Arguelles:2018mtc . However, the full analysis of the measurement at
MINER$\nu$A requires a dedicated Monte Carlo simulation of the sideband
events, which is not available to us.. We also consider the Borexino
experiment since it measures the solar neutrinos which have much lower
energies than other experiments.

###  3.1 CHARM-II

The CHARM-II experiments measured the high energy $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ beam from the Super Proton Synchrotron (SPS) at CERN
Vilain:1993kd ; Vilain:1994qy . The mean neutrino energies of the $\nu_{\mu}$
and $\bar{\nu}_{\mu}$ beam are 23.7 GeV and 19.1 GeV, respectively. The
unfolded differential cross sections from the measurement have been given in
Ref. Vilain:1993kd , and the data points are shown in Fig. 2. Our SM
predictions are consistent with those given in the Ref. Vilain:1993kd .
Therefore, we consider the following $\chi^{2}$ function in our analysis for
new physics:

$\displaystyle\chi^{2}_{\text{CHARM-
II}}=\sum_{i}\frac{(d\sigma/dE_{R})_{i}-s_{i}^{0})^{2}}{\sigma_{i}^{2}}+(\nu_{\mu}\rightarrow\bar{\nu}_{\mu})\,,$
(16)

where $s_{i}^{0}$ and $\sigma_{i}$ are the measured differential cross section
and its corresponding uncertainties taken from Ref. Vilain:1993kd .

Figure 2: The differential cross sections of the $\nu_{\mu}$ (left) and
$\bar{\nu}_{\mu}$ (right) scattering on electrons at CHARM-II. The black data
points and the blue dashed SM prediction curve are taken from Ref.
Vilain:1993kd . The red curve corresponds to our best-fit prediction in the
SM.

###  3.2 TEXONO

The cross section of $\bar{\nu}_{e}$ scattering on electrons has been measured
by the TEXONO experiment utilizing electron antineutrinos produced by the Kuo-
Sheng Nuclear Power Reactor with a CsI(Tl) scintillating crystal detector
Deniz:2009mu . The detector is placed at a distance of 28 m from the 2.9 GW
reactor core. The range of recoil energy used in the analysis is from 3 MeV
and 8 MeV, respectively. The measured event rates and uncertainties have been
given in Ref. Deniz:2009mu , which are shown in Fig. 3. As seen from the red
and blue dashed curves in Fig. 3, our SM predictions agree quite well with
those given in the Ref. Deniz:2009mu . Therefore, we consider the following
$\chi^{2}$ function in our analysis for new physics:

$\displaystyle\chi^{2}_{\text{TEXONO}}=\sum_{i}\frac{(R_{i}^{0}-R_{i}(1+\alpha))^{2}}{\sigma_{R,i}^{2}}+\left(\frac{\alpha}{\sigma_{\alpha}}\right)^{2}\,,$
(17)

where $R_{i}$ ($R_{i}^{0}$) and $\sigma_{R,i}$ are the predicted (measured)
event rates and corresponding uncertainties in the $i$th recoil energy bin.
Here $\sigma_{\alpha}$ is the normalization uncertainty, and we take it to be
$5\%$ for conservation. Both the measured event rates and uncertainties are
taken from Ref. Deniz:2009mu . The predicted event rate in the $i$th recoil
energy bin is calculated by

$\displaystyle R_{i}=N_{e}\int
dE_{\nu}\phi_{\bar{\nu}_{e}}(E_{\nu})\int_{i}dE_{R}\frac{d\sigma_{\bar{\nu}_{e}}}{dE_{R}}\eta(E_{R})\,,$
(18)

where $N_{e}=2.5\times 10^{26}$ is the number of target electrons in the CsI
detector per kilogram, $\phi_{\bar{\nu}_{e}}(E_{\nu})$ is the reactor
antineutrino flux, and $\eta(E_{R})$ is the detection efficiency. Here we take
it to be 100%, which yields a good agreement for the SM predictions shown in
Fig. 16 of Ref. Deniz:2009mu .

Figure 3: The event rates of $\bar{\nu}_{e}-e^{-}$ scattering in the TEXONO
experiment. The black data points and the blue dashed SM prediction curve are
taken from Ref. Deniz:2009mu . The red curve corresponds to our best-fit
prediction in the SM.

###  3.3 Borexino

The Borexino experiment measured solar neutrinos at the Laboratori Nazionali
del Gran Sasso. Only $\nu_{e}$ are produced in the core of the Sun. However,
after adiabatic propagation in the sun, the solar neutrinos arrive at the the
Earth contain all three flavors: $\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$. The
survival probability of solar neutrinos is given by Liao:2017awz

$\displaystyle P_{ee}\approx
s_{13}^{4}+c_{13}^{4}(c_{12}^{2}\cos^{2}\theta_{12}^{m}+s_{12}^{2}\sin\theta_{12}^{m})\,,$
(19)

where $s_{12}$ ($c_{12}$) denotes $\sin\theta_{12}$ ($\cos_{12}$) ,
$\theta_{12}$ is the vacuum mixing angles in the PMNS matrix. Here
$\theta_{12}^{m}$ is the effective mixing angle at the production point in the
Sun, which is given by

$\displaystyle\theta_{12}^{m}=\frac{1}{2}\arctan\frac{\sin 2\theta_{12}}{\cos
2\theta_{12}-\hat{A}c_{13}^{2}}\,,$ (20)

with $\hat{A}\equiv 2\sqrt{2}G_{F}N_{e}^{S}E_{\nu}/\delta m_{21}^{2}$ and
$N_{e}^{S}$ being the number density of electron at the production point in
the Sun. Here we ignore the small corrections due to the day-night asymmetry
in the Borexino measurement Bellini:2011yj . We consider the pp, ${}^{7}Be$
and pep spectra measured in the Borexino phase-I Bellini:2011rx ;
Collaboration:2011nga ; Bellini:2013lnn  and phase-II Bellini:2014uqa ;
Agostini:2017ixy , and the ${}^{8}B$ data collected between January 2008 and
December 2016 Agostini:2017cav . The expected event rate at Borexino is given
by

$\displaystyle R^{i}_{\text{pre}}=N_{e}\int
dE_{\nu}\Phi^{i}(E_{\nu})\left[P_{ee}^{i}\sigma_{e}(E_{\nu})+(1-P_{ee}^{i})\sigma_{\mu}(E_{\nu})\right]\,,$
(21)

where $N_{e}=3.307\times 10^{31}/100$ ton is the density of target electrons
in the Borexino detector Agostini:2017ixy , and $i$ indicates solar neutrino
sources pp, ${}^{7}Be$, pep and ${}^{8}B$. $\Phi^{i}(E_{\nu})$ is the
corresponding solar neutrino flux taken from the standard solar model
(B16-GS98-HZ) Vinyoles:2016djt . $P_{ee}^{i}$ is the survival probability
given in Eq. (19). The cross section in Eq. (21) is calculated by

$\displaystyle\sigma_{\alpha}=\int
dE_{R}\frac{d\sigma_{\alpha}}{dE_{R}}\eta(E_{R})\,,$ (22)

where $\alpha=e,\mu$, $\frac{d\sigma_{\alpha}}{dE_{R}}$ is the differential
cross section, and $\eta(E_{R})$ is the detection efficiency. The detection
efficiency for ${}^{8}B$ is extracted from Fig. 2 in Ref. Agostini:2017cav
and we take $\eta(E_{R})$ to be 100% for other solar neutrino sources
Khan:2019jvr . The measured event rates and our predicted event rates in the
SM are given in Table 1. We see that our SM predictions agree with the
measured event rates. For new physics analysis, we employ the following
$\chi^{2}$ function Khan:2019jvr :

$\displaystyle\chi^{2}_{\text{Borexino}}=\sum_{i}\frac{\left[R^{i}_{\text{exp}}-R^{i}_{\text{pre}}(1+\alpha_{i})\right]^{2}}{(\sigma^{i}_{stat})^{2}}+\left(\frac{\alpha^{i}}{\sigma_{\text{th}}^{i}}\right)^{2}\,,$
(23)

where $R^{i}_{\text{exp}}$ ($\sigma^{i}_{exp}$) are the central values
(statistical uncertainties) of the $i$th measurement given in Table 1,
$R^{i}_{\text{pre}}$ is the predicted event rates calculated in Eq. (21), and
$\sigma^{i}_{th}$ is the theoretical uncertainties given in the last column in
Table 1.

Source | Measurement (cpd/100 t) | SM prediction (cpd/100 t) | Percentage error
---|---|---|---
pp | $134\pm 10^{+6}_{-10}$ | $136.0\pm 1.6$ | 1.2%
${}^{7}Be$ |  $46\pm 1.5^{+1.5}_{-1.6}$(phase I), $48.3\pm 1.1^{+0.4}_{-0.7}$(phase II) | $47.6\pm 2.9$ | 6.1%
pep |  $3.1\pm 0.6\pm 0.3$ (phase I), $2.43\pm 0.36^{+0.15}_{-0.22}$ (phase II) | $2.76\pm 0.04$ | 1.3%
${}^{8}B$ | $0.223^{+0.015}_{-0.016}\pm 0.006$ | $0.209\pm 0.025$ | 12.0%
Table 1: The measured event rates at Borexino and our predicted event rates in
the SM. The theoretical percentage uncertainties are given in the last column.

##  4 Constraints from the experimental data

In this section, we present our results of the constraints on the coupling
coefficients of the general neutrino interactions with $\chi$ and electrons
using the neutrino-electron scattering data from the CHARM-II, TEXONO and
Borexino experiments.

###  4.1 Flavor-universal bounds

We firstly consider the flavor-universal couplings, i.e. by setting
$\epsilon_{\alpha}^{i}\equiv\epsilon^{i}$ in Eq. (2), where $i$ indicates the
scalar (S), pseudoscalar (P), vector(V), axialvector (A), tensor (T) and
electromagnetic (E or M) dipole operators. Here we also assume only one
$\epsilon^{i}$ ($\tilde{\epsilon}^{i}$) exists at a time. As mentioned before,
the bounds on
$\\{\tilde{\epsilon}_{\alpha}^{S},\tilde{\epsilon}_{\alpha}^{P},\tilde{\epsilon}_{\alpha}^{V},\tilde{\epsilon}_{\alpha}^{A},\tilde{\epsilon}_{\alpha}^{T},\epsilon_{\alpha}^{E}\\}$
will be the same as
$\\{\epsilon_{\alpha}^{P},\epsilon_{\alpha}^{S},\epsilon_{\alpha}^{A},\epsilon_{\alpha}^{V},\epsilon_{\alpha}^{T},\epsilon_{\alpha}^{M}\\}$
in this case. The 90% CL upper bounds on the magnitude of the coefficients
$\epsilon^{i}$ ($\tilde{\epsilon}^{i}$) as a function of $m_{\chi}$ are shown
in Fig. 4. From Fig. 4, we see that the CHARM-II experiment yields the
strongest bounds for the scalar, pseudoscalar, vector, axialvector and tensor
interactions. For the electromagnetic dipole interaction, the Borexino
experiment has the best sensitivity for $m_{\chi}$ below 1 MeV. There is an
upper limit on $m_{\chi}$ for the CHARM-II bounds due to the kinematic
constraint. This can be explained by Eq. (32), from which we get
$m_{\chi}\leq\sqrt{(2E_{\nu}+m_{e})m_{e}}-m_{e}\simeq 155$ MeV for
$E_{\nu}=23.7$ GeV. Also, the upper limits on $m_{\chi}$ from the TEXONO and
Borexino experiments are much smaller than the CHARM-II experiment due to low
neutrino energies used in these two experiments. In addition, we see that the
bounds become flat at small $m_{\chi}$, which can be understood from Eqs.
(10), (11), (12), (13), (14) and (15) since the differential cross sections
are insensitive to $m_{\chi}$ as $m_{\chi}\ll E_{\nu}$. From Fig. 4, we find
that for $m_{\chi}\lesssim 50$ MeV, the strongest bounds on the magnitude of
$\epsilon^{S,P}$ ($\tilde{\epsilon}^{S,P}$), $\epsilon^{V,A}$
($\tilde{\epsilon}^{V,A}$) and $\epsilon^{T}$ ($\tilde{\epsilon}^{T}$) can
reach 1.0, 0.5 and 0.2, respectively. The strongest bounds on the magnitude of
$\epsilon^{M,E}$ can reach $1.3\times 10^{-6}$ for $m_{\chi}\lesssim 0.1$ MeV.

Figure 4: The 90% CL upper bounds on the magnitude of coupling coefficients as
a function of $m_{\chi}$ for the scalar, pseudoscalar, vector, axialvector,
tensor and electromagnetic dipole interactions. We assume that the couplings
are flavor universal and only one $\epsilon^{i}$ ($\tilde{\epsilon}^{i}$)
exists at a time. The gray, blue and red shaded regions are excluded by the
CHARM-II, TEXONO and Borexino experiment, respectively. The bounds on
$|\tilde{\epsilon}_{\alpha}^{S}|,|\tilde{\epsilon}_{\alpha}^{P}|,|\tilde{\epsilon}_{\alpha}^{V}|,|\tilde{\epsilon}_{\alpha}^{A}|,|\tilde{\epsilon}_{\alpha}^{T}|$,
and $|\epsilon_{\alpha}^{E}|$ are the same as
$|\epsilon_{\alpha}^{P}|,|\epsilon_{\alpha}^{S}|,|\epsilon_{\alpha}^{A}|,|\epsilon_{\alpha}^{V}|,|\epsilon_{\alpha}^{T}|$,
and $|\epsilon_{\alpha}^{M}|$, respectively.

###  4.2 Flavor-dependent bounds

Since only $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) are measured at the CHARM-II
experiment and only $\bar{\nu}_{e}$ are measured at the TEXONO experiment, the
bounds from these two experiments can be avoided if the coupling coefficients
are flavor non-universal. To illustrate the flavor dependence of these bounds,
we show the 90% CL allowed regions in the ($\epsilon_{e}$, $\epsilon_{\mu}$)
plane for the scalar and vector interactions in Fig. 5. Here we fixed
$m_{\chi}=1$ MeV, and assume the coupling coefficients are real for
simplicity. We also assume $\epsilon_{\tau}=\epsilon_{\mu}$ for the Borexino
experiments. As seen from Fig. 5, the CHARM-II experiment is not sensitive to
$\epsilon_{e}$, and the TEXONO experiment has no sensitivity to
$\epsilon_{\mu}$. The solar neutrino experiment at Borexino can impose
constraints on both $\epsilon_{e}$ and $\epsilon_{\mu}$ due to the flavor
transition in the Sun. We also show the allowed regions of the combined data
from these three experiments as the gray shaded regions in Fig. 5. From Fig.
5, one can see that the sensitivity of the combined data mainly comes from the
CHARM-II and TEXONO experiment.

Figure 5: The 90% CL allowed regions in the ($\epsilon_{e}$, $\epsilon_{\mu}$)
plane for the scalar (left panel) and vector (right panel) interactions. Here
we assume $m_{\chi}=1$ MeV and the coupling coefficients are real with
$\epsilon_{\tau}=\epsilon_{\mu}$. The region enclosed by the black, blue and
red curves correspond to the CHARM-II, TEXONO and Borexino experiments,
respectively. The gray shaded regions correspond to the allowed regions of the
combined data from these three experiments.

##  5 Prospects at DM experiments

The exotic fermion $\chi$ can also serve as a DM particle if the correct DM
abundance is satisfied. The relic abundance of DM is model dependent and
changes based on the specific production mechanism and dynamics in the early
Universe. The DM is either in thermal equilibrium with the SM bath and freezes
out below some temperature, or is produced non-thermally. The thermal
production of DM is a well-studied mechanism to obtain the correct abundance
of DM in the early Universe. In our case, DM starts to keep in thermal
equilibrium with the SM bath as a result of both the forward and backward
processes $\bar{\chi}\chi\leftrightarrow\bar{\nu}\nu$ happening in the early
Universe. As the Universe expands, the temperature drops below the DM mass and
the DM falls out of thermal equilibrium. As a result, only the forward process
of the annihilation of DM particles into neutrinos occurs and the abundance of
DM is quickly suppressed so as to reach the current equilibrium abundance.
This is the so-called freeze-out process for the production of DM. The DM in
thermal equilibrium should not significantly modifies the standard predictions
of the ratios of light elemental abundances by the big bang nucleosynthesis
(BBN). Also, the thermal DM species must not significantly alter the
temperature ratio of photons and neutrinos at recombination. It turns out that
thermal DM lighter than MeV scale will contribute to the radiation energy
density and be ruled out by $N_{eff}$ constraint Boehm:2013jpa . For the DM
mass scale smaller than MeV, the above $\chi-\nu$ operators can induce the
non-thermal production of DM after the SM neutrinos decouple and before the
electrons leave the bath. In the case of non-thermal production, if the
coupling to electron is large, the DM would suffer the constraint from
overproduction near BBN. The calculation of the final DM abundance replies on
the details of the UV completion. The annihilation cross section depends on
the $\chi-\nu$ mixing and the nature of mediator such as a dark photon for
vector interaction or a scalar for scalar interaction in a UV completion. We
refer the UV models to Ref. Dror:2020czw  and one can see that the XENON1T
excess preferred parameter space can avoid overproducing the DM abundance.

Here we briefly discuss the detection of DM hypothesis and the XENON1T excess.
If we reverse the above process and interpret the exotic fermion $\chi$ as DM
particle, the incoming DM $\chi$ can be absorbed by bound electron targets and
emit a neutrino

$\displaystyle\chi e\to\nu e\;.$ (24)

For the DM elastic scattering off the electron, to explain the XENON1T excess,
the key point is how to produce abundant DM particles with high velocity
$v_{\rm DM}\gtrsim 0.1$ Kannike:2020agf . In contrast, Ref. Dror:2020czw
proposed the above DM absorption scenario in which a DM particle deposits its
mass energy rather than kinetic energy and it is sensitive to sub-MeV
fermionic DM. For the DM absorption with free electrons $\chi e\to\nu e$,
analogous to the case with nucleus absorbing the DM Dror:2019onn ;
Dror:2019dib , the total event rate is naively given by

$\displaystyle R={\rho_{\chi}\over
m_{\chi}}\sigma_{e}N_{T}\Theta(E_{R}^{0}-E_{\rm th})\;,$ (25)

where $N_{T}$ is the number of target nuclei per detector mass, the local DM
density is $\rho_{\chi}\simeq 0.4~{}{\rm GeV}/{\rm cm}^{3}$, $\Theta$ is the
Heaviside theta function, $E_{\rm th}$ is the experimental threshold,
$E_{R}^{0}=m_{\chi}^{2}/2m_{e}$ for a free electron absorbing the DM, and
$\sigma_{e}$ is the absorption cross section per electron. For the XENON1T
experiment, we have $N_{T}\simeq 4\times 10^{27}/{\rm tonne}$ and
$E_{R}^{0}\simeq 2$ keV giving $m_{\chi}\simeq 45$ keV. With the total
exposure being 0.65 tonne$\cdot$years, XENON1T observed 285 events and the
expected event number is $232\pm 15$. This gives the total scattering cross
section $\sigma_{e}\simeq 2\times 10^{-48}~{}{\rm cm}^{2}$. In fact, the
absorbing electron in a shell with a binding energy would be ionized with
recoil energy Essig:2011nj ; Essig:2015cda . In Eq. (25) there should also
appear the ionization form factor of an electron in a certain shell and the
total differential ionization rate is obtained by summing over all possible
shells of the absorbing target electrons. The recoil energy of a free electron
absorbing the DM $E_{R}^{0}$ is then shifted. The best fit to the XENON1T data
was found to be $(m_{\chi}=56.5~{}{\rm keV},\sigma_{e}=1\times 10^{-49}~{}{\rm
cm^{2}})$ Dror:2020czw . We take the vector interaction for illustration. The
total scattering cross section $\sigma_{e}$ for vector operators is

$\displaystyle\sigma_{e}^{V}$ $\displaystyle=$
$\displaystyle{G_{F}^{2}m_{\chi}^{2}(m_{\chi}+2m_{e})^{2}\over
32\pi(m_{e}+m_{\chi})^{4}}\Big{[}|\epsilon^{V}|^{2}(2m_{e}^{2}+4m_{e}m_{\chi}+3m_{\chi}^{2})+|\tilde{\epsilon}^{V}|^{2}(6m_{e}^{2}+8m_{e}m_{\chi}+3m_{\chi}^{2})\Big{]}\;.$

By transforming the above best fit to the parameterization in our context, one
essentially obtains $|\epsilon^{V}|,|\tilde{\epsilon}^{V}|\simeq 0.1$ which is
allowed by the neutrino scattering experiments. Note that the LUX-ZEPLIN (LZ)
experiment at the Sanford Underground Research Facility (SURF) in South Dakota
Akerib:2019fml  will start to take data soon and is able to probe orders of
magnitude more parameter space than XENON1T. Such next generation detector
will help to test the XENON1T favored region.

The decaying DM scenario usually faces the requirement of stability. The
corresponding lifetime of $\chi$ should be longer than the age of the
Universe, i.e. $t_{\rm Universe}=4.4\times 10^{17}~{}{\rm sec}$ Ade:2015xua .
Requiring the DM being stable at the Universe time scale would set a very
stringent bound on the coupling and/or the DM mass. For the vector interaction
in our effective framework, the leading decay process is $\chi\to\nu\gamma$
with one photon radiated from the closed electron loop. The constraint would
be quite stringent if only electron is involved in the calculation of the
above decay width. In a realistic UV model in Ref. Dror:2020czw  with $\chi$
only coupled to right-handed neutrino, the decay $\chi\to\nu\gamma$ is
suppressed by the insertion of neutrino mass. The decay of $\chi$ into
$\nu\gamma\gamma$ is forbidden and the leading decay becomes
$\chi\to\nu\gamma\gamma\gamma$ which leads to a quite weak constraint. We
refer the detailed discussion of the sub-MeV DM absorption by electrons to
Ref. Dror:2020czw  and future studies. On the other hand, more stringent
constraints can also be placed on decaying DM from the observations of the
galactic and extra-galactic diffuse X-ray or gamma-ray background. For several
tens of keV DM, strong constraints have been obtained using INTEGRAL
Yuksel:2007xh ; Boyarsky:2007ge ; Essig:2013goa  and NuSTAR Perez:2016tcq ;
Roach:2019ctw . The bound on the lifetime of decaying DM depends on the
specific model parameters and decay topologies.

##  6 Conclusion

In this work we study the constraints on general neutrino interactions with
sub-GeV exotic fermion $\chi$ from neutrino-electron scattering experiments.
The general neutrino interactions are composed of dimension-5 dipole operators
and dimension-6 four-fermion operators. We employ the measurements of CHARM-
II, TEXONO and Borexino experiments to set limits on the neutrino-electron
scattering with an outgoing fermion $\chi$. We find that the bounds are
dominated by the CHARM-II experiment in most of the parameter space for the
flavor-universal interactions and $m_{\chi}$ below 155 MeV, while the Borexino
experiment sets the strongest bounds in the low mass region for the
electromagnetic dipole interactions. The limits are found to be
$|\epsilon^{S,P}|(|\tilde{\epsilon}^{S,P}|)<1$,
$|\epsilon^{V,A}|(|\tilde{\epsilon}^{V,A}|)<0.5$,
$|\epsilon^{T}|(|\tilde{\epsilon}^{T}|)<0.2$ for $m_{\chi}\lesssim 50$ MeV and
$|\epsilon^{M,E}|<1.3\times 10^{-6}$ for $m_{\chi}\lesssim 0.1$ MeV. If the
coupling coefficients are flavor non-universal, the bounds on $\epsilon_{e}$
($\epsilon_{\mu}$) can be avoided for the CHARM-II (TEXONO) experiment, and
there are correlations between the bounds on the coupling coefficients from
the Borexino experiment. Finally, as an example, we discuss the detection of
sub-MeV DM absorbed by bound electron targets. By transforming the best fit to
the XENON1T data in our parameterization, we obtain the preferred coefficients
for vector interactions as $|\epsilon^{V}|,|\tilde{\epsilon}^{V}|\simeq 0.1$
which is allowed by the neutrino experiments.

###### Acknowledgements.

TL is supported by the National Natural Science Foundation of China (Grant No.
12035008, 11975129) and “the Fundamental Research Funds for the Central
Universities”, Nankai University (Grant No. 63196013). JL 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.

##  Appendix A Calculation of the differential cross section

The amplitude for $\nu(p_{1})e(k_{1})\to\chi(p_{2})e(k_{2})$ is given by

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{u}_{\chi}(p_{2})P_{L}u_{\nu}(p_{1})~{}\bar{u}_{e}(k_{2})(\epsilon^{S}+\tilde{\epsilon}^{S}\gamma_{5})u_{e}(k_{1})$
(27) $\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{u}_{\chi}(p_{2})i\gamma_{5}P_{L}u_{\nu}(p_{1})~{}\bar{u}_{e}(k_{2})i\gamma_{5}(\epsilon^{P}+\tilde{\epsilon}^{P}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{u}_{\chi}(p_{2})\gamma_{\mu}P_{L}u_{\nu}(p_{1})~{}\bar{u}_{e}(k_{2})\gamma^{\mu}(\epsilon^{V}+\tilde{\epsilon}^{V}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{u}_{\chi}(p_{2})\gamma_{\mu}\gamma_{5}P_{L}u_{\nu}(p_{1})~{}\bar{u}_{e}(k_{2})\gamma^{\mu}\gamma_{5}(\epsilon^{A}+\tilde{\epsilon}^{A}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{u}_{\chi}(p_{2})\sigma_{\mu\nu}P_{L}u_{\nu}(p_{1})~{}\bar{u}_{e}(k_{2})\sigma^{\mu\nu}(\epsilon^{T}+\tilde{\epsilon}^{T}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$ $\displaystyle{iG_{F}v_{H}eQ_{e}\over
t^{2}}\bar{u}_{\chi}(p_{2})\sigma_{\mu\nu}(\epsilon^{M}+\epsilon^{E}\gamma_{5})P_{L}u_{\nu}(p_{1})~{}\bar{u}_{e}(k_{2})\gamma_{\mu}t_{\nu}u_{e}(k_{1})\;,$

where the projector $P_{L}=(1-\gamma_{5})/2$ is inserted to force the incoming
neutrinos to be left-handed and $t=p_{1}-p_{2}$. The amplitude for
$\overline{\nu}(p_{1})e(k_{1})\to\overline{\chi}(p_{2})e(k_{2})$ is given by

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{v}_{\nu}(p_{1})P_{R}v_{\chi}(p_{2})~{}\bar{u}_{e}(k_{2})(\epsilon^{S\ast}-\tilde{\epsilon}^{S\ast}\gamma_{5})u_{e}(k_{1})$
(28) $\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{v}_{\nu}(p_{1})P_{R}i\gamma_{5}v_{\chi}(p_{2})~{}\bar{u}_{e}(k_{2})i\gamma_{5}(\epsilon^{P\ast}-\tilde{\epsilon}^{P\ast}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{v}_{\nu}(p_{1})P_{R}\gamma_{\mu}v_{\chi}(p_{2})~{}\bar{u}_{e}(k_{2})\gamma^{\mu}(\epsilon^{V\ast}+\tilde{\epsilon}^{V\ast}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{v}_{\nu}(p_{1})P_{R}\gamma_{\mu}\gamma_{5}v_{\chi}(p_{2})~{}\bar{u}_{e}(k_{2})\gamma^{\mu}\gamma_{5}(\epsilon^{A\ast}+\tilde{\epsilon}^{A\ast}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$
$\displaystyle{G_{F}\over\sqrt{2}}\bar{v}_{\nu}(p_{1})P_{R}\sigma_{\mu\nu}v_{\chi}(p_{2})~{}\bar{u}_{e}(k_{2})\sigma^{\mu\nu}(\epsilon^{T\ast}-\tilde{\epsilon}^{T\ast}\gamma_{5})u_{e}(k_{1})$
$\displaystyle+$ $\displaystyle{iG_{F}v_{H}eQ_{e}\over
t^{2}}\bar{v}_{\nu}(p_{1})P_{R}\sigma_{\mu\nu}(\epsilon^{M\ast}-\epsilon^{E\ast}\gamma_{5})v_{\chi}(p_{2})~{}\bar{u}_{e}(k_{2})\gamma_{\mu}t_{\nu}u_{e}(k_{1})\;,$

where the projector $P_{R}=(1+\gamma_{5})/2$ is inserted to force the incoming
anti-neutrinos to be right-handed. The differential cross section of neutrino-
electron scattering $\nu(\bar{\nu})+e\to\chi(\bar{\chi})+e$ is

$\displaystyle{d\sigma(\nu e)\over dE_{R}}={1\over 32\pi
m_{e}E_{\nu}^{2}}\overline{|\mathcal{M}|^{2}}\;,$ (29)

where $\overline{|\mathcal{M}|^{2}}$ is the spin-averaged amplitude square.
The scattering angle is

$\displaystyle\cos\theta={E_{R}(E_{\nu}+m_{e})+m_{\chi}^{2}/2\over
E_{\nu}\sqrt{E_{R}^{2}+2m_{e}E_{R}}}\;.$ (30)

By requiring $\cos\theta\leq 1$, we can get the bounds on $E_{R}$ as

$\displaystyle E_{R}^{\rm
min(max)}={2m_{e}E_{\nu}^{2}-m_{\chi}^{2}(m_{e}+E_{\nu})\mp
E_{\nu}\sqrt{(2m_{e}E_{\nu}-m_{\chi}^{2})^{2}-4m_{e}^{2}m_{\chi}^{2}}\over
2m_{e}(m_{e}+2E_{\nu})}\;,$ (31)

and the minimal energy to generate the elastic scattering is

$\displaystyle E_{\nu}^{\rm min}=m_{\chi}+{m_{\chi}^{2}\over 2m_{e}}\;.$ (32)

## References

  * (1) Particle Data Group collaboration, P. A. Zyla et al., _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * (2) V. Brdar, W. Rodejohann and X.-J. Xu, _Producing a new Fermion in Coherent Elastic Neutrino-Nucleus Scattering: from Neutrino Mass to Dark Matter_ , _JHEP_ 12 (2018) 024, [1810.03626].
  * (3) W.-F. Chang and J. N. Ng, _KeV scale new fermion from a hidden sector_ , _Phys. Rev. D_ 101 (2020) 035028, [1903.12545].
  * (4) J. A. Dror, G. Elor and R. Mcgehee, _Directly Detecting Signals from Absorption of Fermionic Dark Matter_ , _Phys. Rev. Lett._ 124 (2020) 18, [1905.12635].
  * (5) J. A. Dror, G. Elor and R. Mcgehee, _Absorption of Fermionic Dark Matter by Nuclear Targets_ , _JHEP_ 02 (2020) 134, [1908.10861].
  * (6) W.-F. Chang and J. Liao, _Constraints on light singlet fermion interactions from coherent elastic neutrino-nucleus scattering_ , _Phys. Rev. D_ 102 (2020) 075004, [2002.10275].
  * (7) T. Li, X.-D. Ma and M. A. Schmidt, _General neutrino interactions with sterile neutrinos in light of coherent neutrino-nucleus scattering and meson invisible decays_ , _JHEP_ 07 (2020) 152, [2005.01543].
  * (8) N. Hurtado, H. Mir, I. M. Shoemaker, E. Welch and J. Wyenberg, _Dark Matter-Neutrino Interconversion at COHERENT, Direct Detection, and the Early Universe_ , _Phys. Rev. D_ 102 (2020) 015006, [2005.13384].
  * (9) T. Li and J. Liao, _Loop effect in the coherent neutrino-nucleus scattering_ , _JHEP_ 02 (2021) 099, [2008.00743].
  * (10) S.-F. Ge, P. Pasquini and J. Sheng, _Solar neutrino scattering with electron into massive sterile neutrino_ , _Phys. Lett. B_ 810 (2020) 135787, [2006.16069].
  * (11) I. M. Shoemaker, Y.-D. Tsai and J. Wyenberg, _An Active-to-Sterile Neutrino Transition Dipole Moment and the XENON1T Excess_ , 2007.05513.
  * (12) S. Shakeri, F. Hajkarim and S.-S. Xue, _Shedding New Light on Sterile Neutrinos from XENON1T Experiment_ , 2008.05029.
  * (13) J. A. Dror, G. Elor, R. McGehee and T.-T. Yu, _Absorption of sub-MeV fermionic dark matter by electron targets_ , _Phys. Rev. D_ 103 (2021) 035001, [2011.01940].
  * (14) V. Brdar, A. Greljo, J. Kopp and T. Opferkuch, _The Neutrino Magnetic Moment Portal: Cosmology, Astrophysics, and Direct Detection_ , _JCAP_ 01 (2021) 039, [2007.15563].
  * (15) D. Aristizabal Sierra, R. Branada, O. G. Miranda and G. Sanchez Garcia, _Sensitivity of direct detection experiments to neutrino magnetic dipole moments_ , _JHEP_ 12 (2020) 178, [2008.05080].
  * (16) XENON collaboration, E. Aprile et al., _Excess electronic recoil events in XENON1T_ , _Phys. Rev. D_ 102 (2020) 072004, [2006.09721].
  * (17) N. Viaux, M. Catelan, P. B. Stetson, G. Raffelt, J. Redondo, A. A. R. Valcarce et al., _Neutrino and axion bounds from the globular cluster M5 (NGC 5904)_ , _Phys. Rev. Lett._ 111 (2013) 231301, [1311.1669].
  * (18) S. A. Díaz, K.-P. Schröder, K. Zuber, D. Jack and E. E. B. Barrios, _Constraint on the axion-electron coupling constant and the neutrino magnetic dipole moment by using the tip-RGB luminosity of fifty globular clusters_ , 1910.10568.
  * (19) L. Di Luzio, M. Fedele, M. Giannotti, F. Mescia and E. Nardi, _Solar axions cannot explain the XENON1T excess_ , _Phys. Rev. Lett._ 125 (2020) 131804, [2006.12487].
  * (20) C. Gao, J. Liu, L.-T. Wang, X.-P. Wang, W. Xue and Y.-M. Zhong, _Reexamining the Solar Axion Explanation for the XENON1T Excess_ , _Phys. Rev. Lett._ 125 (2020) 131806, [2006.14598].
  * (21) J. B. Dent, B. Dutta, J. L. Newstead and A. Thompson, _Inverse Primakoff Scattering as a Probe of Solar Axions at Liquid Xenon Direct Detection Experiments_ , _Phys. Rev. Lett._ 125 (2020) 131805, [2006.15118].
  * (22) Borexino collaboration, M. Agostini et al., _First Simultaneous Precision Spectroscopy of $pp$, 7Be, and $pep$ Solar Neutrinos with Borexino Phase-II_, _Phys. Rev. D_ 100 (2019) 082004, [1707.09279].
  * (23) CHARM-II collaboration, P. Vilain et al., _Measurement of differential cross-sections for muon-neutrino electron scattering_ , _Phys. Lett. B_ 302 (1993) 351–355.
  * (24) CHARM-II collaboration, P. Vilain et al., _Precision measurement of electroweak parameters from the scattering of muon-neutrinos on electrons_ , _Phys. Lett. B_ 335 (1994) 246–252.
  * (25) TEXONO collaboration, M. Deniz et al., _Measurement of Nu(e)-bar -Electron Scattering Cross-Section with a CsI(Tl) Scintillating Crystal Array at the Kuo-Sheng Nuclear Power Reactor_ , _Phys. Rev. D_ 81 (2010) 072001, [0911.1597].
  * (26) A. N. Khan, D. W. McKay and F. Tahir, _Short baseline reactor $\bar{\nu}-e$ scattering experiments and nonstandard neutrino interactions at source and detector_, _Phys. Rev. D_ 90 (2014) 053008, [1407.4263].
  * (27) A. N. Khan, _Global analysis of the source and detector nonstandard interactions using the short baseline $\nu$-e and $\nu$¯-e scattering data_, _Phys. Rev. D_ 93 (2016) 093019, [1605.09284].
  * (28) A. N. Khan and D. W. McKay, _$\sin^{2}(\theta)w$ estimate and bounds on nonstandard interactions at source and detector in the solar neutrino low-energy regime_, _JHEP_ 07 (2017) 143, [1704.06222].
  * (29) A. N. Khan, _$\sin^{2}\theta_{W}$ Estimate and Neutrino Electromagnetic Properties from Low-Energy Solar Data_, _J. Phys. G_ 46 (2019) 035005, [1709.02930].
  * (30) M. Lindner, F. S. Queiroz, W. Rodejohann and X.-J. Xu, _Neutrino-electron scattering: general constraints on Z ${}^{{}^{\prime}}$ and dark photon models_, _JHEP_ 05 (2018) 098, [1803.00060].
  * (31) A. N. Khan, W. Rodejohann and X.-J. Xu, _Borexino and general neutrino interactions_ , _Phys. Rev. D_ 101 (2020) 055047, [1906.12102].
  * (32) Lowski and S. Trojanowski, _Neutrino beam-dump experiment with FASER at the LHC_ , 2011.04751.
  * (33) B. Batell, J. L. Feng and S. Trojanowski, _Detecting Dark Matter with Far-Forward Emulsion and Liquid Argon Detectors at the LHC_ , 2101.10338.
  * (34) R. C. Allen et al., _Study of electron-neutrino electron elastic scattering at LAMPF_ , _Phys. Rev. D_ 47 (1993) 11–28.
  * (35) LSND collaboration, L. B. Auerbach et al., _Measurement of electron - neutrino - electron elastic scattering_ , _Phys. Rev. D_ 63 (2001) 112001, [hep-ex/0101039].
  * (36) MINERvA collaboration, J. Park et al., _Measurement of Neutrino Flux from Neutrino-Electron Elastic Scattering_ , _Phys. Rev. D_ 93 (2016) 112007, [1512.07699].
  * (37) MINERvA collaboration, E. Valencia et al., _Constraint of the MINER $\nu$A medium energy neutrino flux using neutrino-electron elastic scattering_, _Phys. Rev. D_ 100 (2019) 092001, [1906.00111].
  * (38) W. Rodejohann, X.-J. Xu and C. E. Yaguna, _Distinguishing between Dirac and Majorana neutrinos in the presence of general interactions_ , _JHEP_ 05 (2017) 024, [1702.05721].
  * (39) C. A. Argüelles, M. Hostert and Y.-D. Tsai, _Testing New Physics Explanations of the MiniBooNE Anomaly at Neutrino Scattering Experiments_ , _Phys. Rev. Lett._ 123 (2019) 261801, [1812.08768].
  * (40) J. Liao, D. Marfatia and K. Whisnant, _Nonstandard interactions in solar neutrino oscillations with Hyper-Kamiokande and JUNO_ , _Phys. Lett. B_ 771 (2017) 247–253, [1704.04711].
  * (41) Borexino collaboration, G. Bellini et al., _Absence of day–night asymmetry of 862 keV 7Be solar neutrino rate in Borexino and MSW oscillation parameters_, _Phys. Lett. B_ 707 (2012) 22–26, [1104.2150].
  * (42) G. Bellini et al., _Precision measurement of the 7Be solar neutrino interaction rate in Borexino_ , _Phys. Rev. Lett._ 107 (2011) 141302, [1104.1816].
  * (43) Borexino collaboration, G. Bellini et al., _First evidence of pep solar neutrinos by direct detection in Borexino_ , _Phys. Rev. Lett._ 108 (2012) 051302, [1110.3230].
  * (44) Borexino collaboration, G. Bellini et al., _Final results of Borexino Phase-I on low energy solar neutrino spectroscopy_ , _Phys. Rev. D_ 89 (2014) 112007, [1308.0443].
  * (45) BOREXINO collaboration, G. Bellini et al., _Neutrinos from the primary proton–proton fusion process in the Sun_ , _Nature_ 512 (2014) 383–386.
  * (46) Borexino collaboration, M. Agostini et al., _Improved measurement of 8B solar neutrinos with $1.5kt·y$ of Borexino exposure_, _Phys. Rev. D_ 101 (2020) 062001, [1709.00756].
  * (47) N. Vinyoles, A. M. Serenelli, F. L. Villante, S. Basu, J. Bergström, M. Gonzalez-Garcia et al., _A new Generation of Standard Solar Models_ , _Astrophys. J._ 835 (2017) 202, [1611.09867].
  * (48) C. Boehm, M. J. Dolan and C. McCabe, _A Lower Bound on the Mass of Cold Thermal Dark Matter from Planck_ , _JCAP_ 08 (2013) 041, [1303.6270].
  * (49) K. Kannike, M. Raidal, H. Veermae, A. Strumia and D. Teresi, _Dark Matter and the XENON1T electron recoil excess_ , 2006.10735.
  * (50) R. Essig, J. Mardon and T. Volansky, _Direct Detection of Sub-GeV Dark Matter_ , _Phys. Rev. D_ 85 (2012) 076007, [1108.5383].
  * (51) R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto, T. Volansky and T.-T. Yu, _Direct Detection of sub-GeV Dark Matter with Semiconductor Targets_ , _JHEP_ 05 (2016) 046, [1509.01598].
  * (52) LZ collaboration, D. S. Akerib et al., _The LUX-ZEPLIN (LZ) Experiment_ , _Nucl. Instrum. Meth. A_ 953 (2020) 163047, [1910.09124].
  * (53) Planck collaboration, P. Ade et al., _Planck 2015 results. XIII. Cosmological parameters_ , _Astron. Astrophys._ 594 (2016) A13, [1502.01589].
  * (54) H. Yuksel, J. F. Beacom and C. R. Watson, _Strong Upper Limits on Sterile Neutrino Warm Dark Matter_ , _Phys. Rev. Lett._ 101 (2008) 121301, [0706.4084].
  * (55) A. Boyarsky, D. Malyshev, A. Neronov and O. Ruchayskiy, _Constraining DM properties with SPI_ , _Mon. Not. Roy. Astron. Soc._ 387 (2008) 1345, [0710.4922].
  * (56) R. Essig, E. Kuflik, S. D. McDermott, T. Volansky and K. M. Zurek, _Constraining Light Dark Matter with Diffuse X-Ray and Gamma-Ray Observations_ , _JHEP_ 11 (2013) 193, [1309.4091].
  * (57) K. Perez, K. C. Y. Ng, J. F. Beacom, C. Hersh, S. Horiuchi and R. Krivonos, _Almost closing the $\nu$MSM sterile neutrino dark matter window with NuSTAR_, _Phys. Rev. D_ 95 (2017) 123002, [1609.00667].
  * (58) B. M. Roach, K. C. Y. Ng, K. Perez, J. F. Beacom, S. Horiuchi, R. Krivonos et al., _NuSTAR Tests of Sterile-Neutrino Dark Matter: New Galactic Bulge Observations and Combined Impact_ , _Phys. Rev. D_ 101 (2020) 103011, [1908.09037].