aainstitutetext: Mitchell Institute for Fundamental Physics and Astronomy,
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
Texas 77843, USAbbinstitutetext: School of Physics, Korea Institute for
Advanced Study, Seoul 02455, Koreaccinstitutetext: CAS Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing, 100190, People’s Republic Chinaddinstitutetext: School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
People’s Republic China

# Non-standard neutrino interactions in light mediator models at reactor
experiments

Bhaskar Dutta a, b Sumit Ghosh c, d Tianjun Li a Adrian Thompson a and Ankur
Verma<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Compared to other neutrino sources, the huge anti-neutrino fluxes at nuclear
reactor based experiments empower us to derive stronger bounds on non-standard
interactions of neutrinos with electrons mediated by light scalar/vector
mediators. At neutrino energy around $200$ keV reactor anti-neutrino flux is
at least an order of magnitude larger compared to the solar flux. The atomic
and crystal form factors of the detector materials related to the details of
the atomic structure becomes relevant at this energy scale as the momentum
transfers would be small. Non-standard neutrino-electron interaction mediated
by light scalar/vector mediator arises naturally in many low-scale models. We
also propose one such new model with a light scalar mediator. Here, we
investigate the parameter space of such low-scale models in reactor based
neutrino experiments with low threshold Ge and Si detectors, and find the
prospect of probing/ruling out the relevant parameter space by finding the
projected sensitivity at $90\%$ confidence level by performing a
$\chi^{2}$-analysis. We find that a detector capable of discriminating between
electron recoil and nuclear recoil signal down to a very low threshold such as
$5$ eV placed in reactor based experiment would be able to probe a larger
region in parameter space compared to the previously explored region. A Ge
(Si) detector with $10$ kg-yr exposure and 1 MW reactor anti-neutrino flux
would be able to probe the scalar and vector mediators with masses below 1 keV
for coupling products $\sqrt{g_{\nu}g_{e}}$ $\sim$ $1\times
10^{-6}\leavevmode\nobreak\ (9.5\times 10^{-7})\leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ 1\times 10^{-7}\leavevmode\nobreak\ (8\times
10^{-8})$, respectively.

###### Keywords:

NSI, Reactor Neutrinos

††preprint: MI-HET-775, KIAS-P22043

##  1 Introduction

The solar, atmospheric, reactor, and accelerator neutrino experiments have
established that three neutrinos oscillate and at least two neutrinos in the
Standard Model (SM) are massive Super-Kamiokande:1998kpq ; SNO:2002tuh ;
KamLAND:2002uet ; T2K:2011ypd ; DoubleChooz:2011ymz ; DayaBay:2012fng ;
RENO:2012mkc . However, the origin of neutrino masses and the variety of
neutrino interactions still remain to be explained. Many models have been
proposed to explain the tiny non-zero masses of neutrinos with the scales of
new physics covering a wide range of values Minkowski:1977sc ; Yanagida:1979as
; Gell-Mann:1979vob ; Mohapatra:1979ia ; Mohapatra:1980yp ; Cheng:1980qt ;
Foot:1988aq ; Schechter:1980gr ; Chikashige:1980ui ; Ballesteros:2016euj ;
Barbieri:1979hc ; Grossman:1999ra ; Huber:2003sf ; Agashe:2015izu ;
Mohapatra:1986aw ; Mohapatra:1986bd ; Kersten:2007vk ; Zee:1980ai ;
Wolfenstein:1980sy ; Zee:1985rj ; Babu:1988ki ; Ma:1998dn ; Ma:2006km ;
Ma:2007yx ; Dev:2012sg ; Cao:2017xgk ; Dutta:2018qei ; Ibarra:2018dib . The
well-established phenomena of neutrino oscillations and non-zero masses
establish the fact that there exists new physics in the neutrino sector beyond
the SM. The non-standard interactions in the neutrino sector are a well-
motivated phenomenological approach to understand the new physics in the
neutrino sector Wolfenstein:1977ue ; Wolfenstein:1979ni ; Bergmann:1997mr ;
Valle:1987gv ; Roulet:1991sm ; Krastev:1997cp ; Miranda:2015dra . So far,
there is no direct experimental observation that shows new interactions of
neutrinos beyond the SM weak interactions. If they exist, they would show some
signals in the ongoing and upcoming neutrino experiments, and thus would
directly indicate the presence of new physics beyond the SM.

Recently, the low-scale neutrino models containing non-standard interaction
mediated by light mediators have been receiving a lot of attention. Such
models can naturally accommodate neutrino-electron interactions beyond the SM
weak interactions. There exist many vector models in the literature
Foot:1990mn ; He:1990pn ; He:1991qd ; Dutta:2019fxn ; Dutta:2022qvn , but
there is a dearth of light scalar mediator models. We propose one light scalar
model by extending the SM scalar sector by a doublet, a triplet and a singlet
scalar. We will use these models as the base models for our analysis. Many
ongoing beam-dump based neutrino experiments, e.g., COHERENT Akimov:2017ade ;
Akimov:2018vzs ; Akimov:2018ghi ; Akimov:2019xdj ; Akimov:2020pdx ;
COHERENT:2021pvd , CCM CCM1 ; CCM2 ; CCM:2021leg , JSNS2 Harada:2013yaa ;
Ajimura:2015yux ; Harada:2015tcp ; Ajimura:2017fld ; Rott:2020duk  etc., and
reactor based experiments, e.g., GEMMA Beda:2013mta , Borexino
BOREXINO:2018ohr , MINER MINER:2016igy , CONUS Buck:2020opf , CONNIE Aguilar-
Arevalo:2016khx  etc. are investigating these low mass mediator models
Cerdeno:2016sfi ; Dutta:2020vop ; Strauss:2017cuu ; Dent:2016wcr . Since the
neutrinos produce a background for the dark matter direct detection
experiments, these new interactions are being investigated at the dark matter
experiments, e.g., XENONnT XENON:2017lvq ; XENON:2022mpc , LZ Akerib:2019fml ,
SENSEI SENSEI:2020dpa , etc. Some of these experiments put stringent bounds on
the parameter space of these models. In addition to these, the light mediators
of these models can also be constrained from astrophysical and cosmological
data Babu:2019iml ; Harnik:2012ni ; Escudero:2019gvw . The neutrino
interactions with low mass mediators also appear to be resolving the so-called
Hubble tension Escudero:2019gzq ; Bernal:2016gxb ; Alcaniz:2019kah ;
Vagnozzi:2019ezj .

Among various sources of neutrinos available for neutrino experiments, reactor
neutrino flux can be larger compared to others for a particular neutrino
energy range for proper settings of the reactor. For example, at $\sim
200-300$ keV neutrino energy range a 1 MW class reactor with source-to-
detector distance of 1 m can give one order of magnitude larger flux compared
to solar flux. Therefore, one can expect that the reactor based neutrino
experiments can probe a much bigger parameter space compared to the solar
neutrino based analysis such as XENONnT and Borexino. This is one of the main
motivations of this paper.

In this paper, we investigate the parameter space of neutrino-electron
scattering mediated by light mediators in low scale neutrino models utilizing
the large anti-neutrino flux from reactor based experiments. The mediators can
be scalar, pseudo-scalar, vector, and axial-vector types, but we mainly focus
on scalar and vector cases. Similar analysis can be performed for pseudo-
scalar and axial-vector mediators as well. The neutrino-electron scattering
processes via light mediators have an interesting feature: the cross section
becomes larger at low recoil energies, hence getting an enhancement in the
rate. Therefore, the low threshold detectors are a good probe to the light
mediators. For our analysis purpose, we consider that the detector used in the
experiment would have a very low threshold such as 5 eV, and a very low
constant background over the SM weak current mediated neutrino-electron
scattering events. We further consider that the detector can distinguish
events based on whether it is coming from electronic recoil or nuclear recoil.
Such capability of the detector turns out to be very fruitful for the ranges
of parameters we are considering.

For light mediators with mass $\mathcal{O}(1)$ keV, the energy transferred in
the scattering falls in sub-keV range and the related distance scale becomes
comparable to the atomic scale. Therefore, a free electron approximation,
where we treat the atom as something with $Z$ number of free electrons, is not
a good choice for our analysis. For proper treatment, one needs to consider
the many body interaction picture inside the target atoms and consider the
ionization effects. Moreover, the target materials we are considering, i.e.
silicon and germanium, are semiconductor crystals. Below some energy scale the
energy band structure of the crystals becomes relevant. The dynamics of many
body ionization and the transition between different energy bands inside the
crystal can be encapsulated in atomic and crystal form factors, respectively.
We use a combination of ionization form factor and crystal form factor
depending on the energy scale. The crystal form factor accounts for low energy
$\lesssim$ 60 eV transitions from valence to conduction bands. On the other
hand, the atomic form factor is used for the transitions to energies $\gtrsim$
60 eV in the conduction bands from some initial electronic state in the
crystal. We use the DarkARC python tool catena_atomic_2020  to get the atomic
form factors, and compute the crystal form factors using the Fortran program
EXCEED-DM griffin_extended_2021 .

The rest of the paper is organized as follows: In section 2, we give a brief
review of neutrino-electron scattering mediated by light mediators. The
relevant ionization and crystal form factors are discussed in section 3. In
section 4, we propose a new light scalar model that can give rise to non-
standard interactions between neutrinos and electrons. Few light vector models
have been reviewed as well. We present our results in section 5. We conclude
in section 6.

##  2 Neutrino-Electron Scattering

In this section, we discuss the basic framework of NSI with electrons via
light mediators arising from some new physics scenarios. The simplest type of
NSI, which leads to neutrino-electron scattering at low energies, is through
the exchange of light scalar or vector particles.

In the SM, the low energy neutrinos and anti-neutrinos interact with electrons
through the elastic scattering process via the charged current (exchange of
$W$ boson) and neutral current ( exchange of $Z$ boson). The neutrino-electron
scattering cross section in SM is given by

$\displaystyle\frac{d\sigma_{e}^{\mathrm{SM}}}{dT}=\frac{G_{F}^{2}m_{e}}{2\pi}\left[(g_{v}+g_{a})^{2}+(g_{v}-g_{a})^{2}\left(1-\frac{T}{E_{\nu}}\right)^{2}-(g_{v}^{2}-g_{a}^{2})\frac{m_{e}T}{E_{\nu}^{2}}\right]\leavevmode\nobreak\
,\leavevmode\nobreak\ $ (1)

where $G_{F}$ is the Fermi constant; $m_{e}$ is the electron mass; $T$ is the
recoil energy of the outgoing electron; and $E_{\nu}$ is the incoming neutrino
energy. The couplings $g_{v}$ and $g_{a}$ depend on the flavor of the
neutrinos. This is due to the fact that the electron neutrino scattering with
electron receives contributions from both $W$-boson and $Z$-boson exchange
diagrams, whereas for muon and tao neutrino scattering with electron only the
$Z$-boson exchange diagram is possible. For $\nu_{e}$ and $\bar{\nu}_{e}$ we
have

$g_{v;e}=\frac{1}{2}+2\sin^{2}\theta_{W};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ g_{a;e}=+\frac{1}{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ $ (2)

whereas for $\nu_{\mu,\tau}$ and $\bar{\nu}_{\mu,\tau}$ we have

$g_{v;\mu,\tau}=-\frac{1}{2}+2\sin^{2}\theta_{W};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ g_{a;\mu,\tau}=-\frac{1}{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ $ (3)

where $\theta_{W}$ is the weak mixing angle with $\sin^{2}_{\theta_{W}}=0.231$
ParticleDataGroup:2020ssz . Note that, for the range of electron recoil energy
where, $T\ll E_{\nu}$, the SM differential cross section is constant.

The neutrino-electron scattering cross section deviates from Eq. 1 in the
presence of NSI. The new physics contributions will be different for light
scalar and light vector mediators. In the following, we summarize the new
physics contributions to the neutrino-electron scattering cross section for
light scalar and light vector particle exchange processes.

  * •

Light scalar mediator: We consider a light scalar particle, $\phi$, with mass
$m_{\phi}$ that couples to both electron and neutrino at tree level with
coupling constants $g_{e,\phi}$ and $g_{\nu,\phi}$, respectively. The
necessary terms of the low-energy Lagrangian below the electroweak (EW) scale
can be written as

$\mathcal{L}_{S}\supset
g_{e,\phi}\bar{e}e\phi+g_{\nu,\phi}\bar{\nu}\nu\phi\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (4)

The $\bar{\nu}\nu\phi$ interaction term can be realized in different ways. One
can simply add a right-handed neutrino and a Dirac type interaction term
$\bar{\nu}_{R}\nu_{L}\phi$. Another possibility is to generate a Majorana type
interaction term like $\bar{\nu}^{c}_{L}\nu_{L}\phi$ using a triplet scalar
field, which we will show in detail in subsection 4.1. The new contribution to
the differential cross section of neutrino-electron scattering is given by

$\frac{d\sigma_{e}}{dT}-\frac{d\sigma_{e}^{\mathrm{SM}}}{dT}=\frac{{g_{\nu,\phi}}^{2}g_{e,\phi}^{2}Tm_{e}^{2}}{4\pi
E_{\nu}^{2}(2Tm_{e}+m_{\phi}^{2})^{2}}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (5)

Note that, the new contribution to the differential cross section varies as
$1/T$ for the range $T>m_{\phi}$, whereas it decreases proportional to smaller
$T$, when $T$ is much smaller than $m_{\phi}$. Also note that there is no
interference term.

  * •

Light vector mediator: Here we consider a light vector mediator $A^{\prime}$
with mass $m_{A^{\prime}}$. The $A^{\prime}$ couples to both electron and
neutrino at tree level with respective coupling constants $g_{e,A^{\prime}}$
and $g_{\nu,A^{\prime}}$. The low-energy Lagrangian below EW scale is

$\mathcal{L}_{V}\supset
g_{e,A^{\prime}}\bar{e}\gamma^{\rho}eA^{\prime}_{\rho}+g_{\nu,A^{\prime}}\bar{\nu}_{L}\gamma^{\rho}\nu_{L}A^{\prime}_{\rho}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (6)

The new contribution to the differential cross section can be written as

$\frac{d\sigma_{e}}{dT}-\frac{d\sigma_{e}^{\mathrm{SM}}}{dT}=\frac{\sqrt{2}G_{F}m_{e}g_{v}g_{\nu,A^{\prime}}g_{e,A^{\prime}}}{\pi\left(2m_{e}T+m_{A^{\prime}}^{2}\right)}+\frac{m_{e}g_{\nu,A^{\prime}}^{2}g_{e,A^{\prime}}^{2}}{2\pi\left(2m_{e}T+m_{A^{\prime}}^{2}\right)^{2}}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (7)

The new light vector contribution to the differential cross section has an
interference term with SM. The cross section first increases by $1/T^{2}$ when
$m_{A^{\prime}}$ can be neglected compared to $T$, and becomes constant when
energy transfer is smaller than the mediator mass.

Figure 1:  Anti-neutrino spectrum produced in the nuclear fission process at
reactor based neutrino experiment. We use the spectrum corresponding to 1 MW
power reactor with core to detector distance 1 meter (solid line). We also
show that, by using a 1 GW power reactor with core to detector distance 10
meters, we can get one order of magnitude more flux (Dashed line).

For our analysis, we mostly consider the anti-neutrino flux generated from
nuclear fission in a reactor Schreckenbach:1985ep ; Kopeikin:2012zz .
Specifically, we use the anti-neutrino fission spectrum which is used for the
analysis in the MINER experiment at Texas A$\&$M University Dutta:2015vwa .
MINER is a reactor based experiment equipped with a low threshold ( $\sim$ 5
eV) cryogenic detector. The reactor is a megawatt (MW) class TRIGA research
reactor with movable core stocked with low-enriched ($\sim$ 20 $\%$)
${}^{235}U$. This high resolution detector with a movable core provides an
ideal setup to probe neutrino NSI where the recoil energy is very low. The
detector can be installed at a baseline separation of 1 m from the reactor
core. The thermal power in the reactor is generated by the nuclear fission
process. The intrinsic anti-neutrino flux can be obtained by combining the
reactor thermal power and the relevant fission fractions. At 1.00 MW reactor
power, the extrapolated fission rate is $3.1\times 10^{16}s^{-1}$ with an
intrinsic anti-neutrino production rate of $1.9\times 10^{17}s^{-1}$, and a
fission anti-neutrino flux of $\Phi_{\nu}=1.5\times 10^{12}cm^{-2}s^{-1}$ at a
mean distance from core of 1 m. Note that, these fluxes are proportional to
$1/r^{2}$, where $r$ is the mean distance between the detector and the reactor
core. Therefore the flux is reduced by a factor of 4 (9) for a mean distance
of 2 m (3 m). The anti-neutrino flux used for analysis is shown in figure 1.

The anti-neutrino flux from MW class reactors has a similar energy profile to
the solar neutrino flux SNO:2011hxd ; Super-Kamiokande:2016yck ;
BOREXINO:2018ohr  with characteristic neutrino energies $\leq 1$ MeV. Both
reactor anti-neutrino and solar neutrino flux are dominated by the neutrinos
with energy $\mathcal{O}(100)$ keV. At 1.00 MW reactor power with a core to
detector mean distance of 1 m, the anti-neutrino flux gets a peak between
200-300 keV neutrino energy, whereas the solar neutrino flux gets a peak
around 200 keV. One advantage of the reactor anti-neutrino flux is that at
this keV energy scale it gives one order of magnitude more flux compared to
the solar neutrino flux. Experiments like XENONnT and Borexino have already
provided constraints on the parameter space of the neutrino NSI using the
solar-$pp$ flux. Thus, using the larger anti-neutrino flux from the reactors
we expect to see even greater sensitivity of the parameter space. Note that,
with a gigawatt (GW) class nuclear reactor we can get 10 times more flux even
with a 10 m mean distance between the reactor core and the detector, such as
in vIOLETA experiment. Also note that at the typical neutrino energies $\leq
200$ keV, the atomic/crystal structure of the target material becomes
relevant. Therefore, one needs to incorporate these effects in the rate
calculation Chen:2014ypv , which we will discuss in detail in section 3.

In figure 2, we show the flux averaged differential cross-section for
different mediator masses of both scalar and vector particles. We have used
the reactor anti-neutrino flux from figure. 1. For the scalar case, we choose
a benchmark coupling $g_{\nu}=g_{e}=5\times 10^{-7}$. And for the vector
mediator, we take $g_{\nu}=g_{e}=1.5\times 10^{-7}$. The cross sections for
the scalar mediated case are first increasing as $1/T$ with decreasing $T$,
then attain a peak and further fall down linearly with $T$. On the other hand,
the vector mediated cross sections increase proportional to $1/T^{2}$ when
energy transfer is large compared to the mass of the mediator. The cross
section then becomes independent of $T$ and go parallel to the x-axis. Note
that, for scalar (vector) mediated cases, the curves attain a peak (become
constant) below $\mathcal{O}(1)$ keV for light mediator of mass
$\mathcal{O}(10)$ keV.

Figure 2:  Flux averaged differential cross section. We have used the reactor
anti-neutrino flux from figure 1. The left panel is for scalar, and the
couplings are: $g_{\nu}=g_{e}=5\times 10^{-7}$. The vector case is on the
right panel, and the benchmark couplings are: $g_{\nu}=g_{e}=1.5\times
10^{-7}$.

##  3 Atomic Ionization by Neutrino NSI

The conventional free electron approximation for isolated atoms is expected to
be a good approximation when the momentum transferred in the scattering
process is large so that the corresponding de Broglie wavelength is much
smaller than the typical distance between the electrons in a multi-electron
atom. In other words, when the typical scattering energy is high compared to
the atomic scales, the atomic effects can be neglected. This case would just
correspond to a factor of Z in the scattering rate accounting for the Z
electrons in the atom. However, as has been shown in some previous works
Chen:2013iud ; Chen:2014ypv ; Hsieh:2019hug  that for the incident neutrino
energy less than the typical binding momentum of electron $\sim Zm_{e}\alpha$
in a hydrogen like atom, the simple free electron approximation is not
accurate. This would correspond to neutrino energies of
$\mathcal{O}(100)\leavevmode\nobreak\ {\rm keV}$ which is quite typical of
neutrino energies we get from a MW class reactor flux.

For the NSI with light mediators the momentum transferred during an
interaction event is small and the differential cross section peaks at very
low recoil energies as can be seen in figure 2. As the kinematics in neutrino
scattering with sub-keV energy transfer starts to overlap with atomic scales,
the atomic binding effects modify the free scattering rate formula. Thus, for
getting a better understanding of detector responses at low recoil energies
one must deal with a proper treatment of many electron dynamics in atomic
ionization. The effect of this complication can be encoded inside an atomic
ionization form factor in the rate calculation. Furthermore, for crystal
targets like Si/Ge the atoms are not isolated which modifies the outer shell
electron wave functions. The outer-shell electronic wavefunctions are
delocalized forming a complicated band structure with an energy band gap
between the occupied valence bands and unoccupied conduction bands. This
modification becomes important at ultra-low energy transfers when most of the
transitions will be occurring from the valence band to the conduction band.
This is accounted for in the form of a crystal form factor which describes the
likelihood that a given momentum transfer will result in a transition from
some state in valence band to conduction band.

Therefore, the free scattering differential rate formula will be modified
after incorporating the form factors. The differential rate can be thought of
as a sum of two contributions: rate modified by atomic form factor and crystal
form factor. We briefly describe the atomic and crystal form factors and how
they modify the differential rate formula.

  * •

Atomic states to free state transitions for deposited energy $E_{e}>60$ eV
using atomic form factor.

For a scattering event with high energy transfer, we can safely ignore the
crystal structure in Si and Ge and treat the transitions to be occurring from
some atomic state to free electronic state. This approximation is indeed more
reasonable for high energy transitions from core states to free state since
the electronic states further away from the band gap are not substantially
modified by the crystal structure. There is a slight inaccuracy though when we
treat the valence states as atomic states (4s, 4p for Ge and 3s, 3p for Si).
But at large for high energy transfers we do not expect a sizeable deviation
from this approximation if one is to perform an exact valence to conduction
calculation instead of a semi-analytical valence atomic state to free state
calculation. In fact, we observe that the valence atomic state to free
calculation gives a decent continuation of the valence to conduction rate
spectra to energies above 60 eV.

We use the DarkARC code catena_atomic_2020 ; timon_emken_2019_3581334  to
evaluate the atomic form factors. DarkARC uses Roothan Hartree Fock (RHF)
approximation for the ground state wave functions and calculates the form
factor for transition to the free state. The RHF Slater Type Orbitals (STO)
coefficients for Si and Ge tabulated in bunge1993  are used as inputs to
DarkARC to get the atomic ionization form factors for Si and Ge. The
dimensionless atomic ionization form factor evaluated in DarkARC as described
in catena_atomic_2020  is expressed as

$\left|f_{\text{ion
}}^{n\ell}\left(k^{\prime},q\right)\right|^{2}=V\frac{4k^{\prime
3}}{(2\pi)^{3}}\sum_{\ell^{\prime}=0}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\left|\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\psi_{2}^{*}(\mathbf{k}+\mathbf{q})\psi_{1}(\mathbf{k})\right|^{2}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (8)

Incorporating the form factor the ionization rate can be expressed as

$\frac{d\mathcal{R}_{a}}{d\ln E_{e}}=\frac{N_{T}}{4}\int
dE_{\nu}\Phi(E_{\nu})\int
dq\left(\frac{d\sigma}{dq}\right)\left|f_{\text{ion}}^{n,l}(q,E_{e})\right|^{2}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (9)

  * •

Valence band to conduction band transtions for deposited Energy $E_{e}\leq 60$
eV using valance-to-conduction band crystal form factor.

To evaluate the rate more reliably at low energies ($\lesssim 60$ eV), the
alteration of atomic orbitals near the band gap must be taken into account.
These states are computed numerically using DFT methods and has been included
in calculations of previous works for Dark Matter-electron
scattering.Essig:2015cda ; griffin_extended_2021

For this work, we use a modification of the EXCEED-DM code
tanner_trickle_2022_6097642  to get the valence to conduction crystal form
factor binned in magnitude of momentum transfer $q$ and deposited energy
$E_{e}$ and employ it in our neutrino electron scattering rate calculations
for Si and Ge. Our definition of the form factor that we extract from EXCEED-
DM is defined as follows

$\displaystyle\left|f_{v\rightarrow c}\left(q,E_{e}\right)\right|^{2}$
$\displaystyle=$
$\displaystyle\frac{4\pi^{2}\Omega}{q^{3}}\sum_{i,f}\int_{\mathrm{BZ}}\frac{d^{3}k_{i}d^{3}k_{f}}{(2\pi)^{6}}E_{e}\delta\left(E_{e}-E_{f\mathbf{k}_{f}}+E_{i\mathbf{k}_{i}}\right)\sum_{\mathbf{G}}q\delta\left(q-\left|\mathbf{k}_{f}+\mathbf{G}-\mathbf{k}_{i}\right|\right)$
(10) $\displaystyle\times\left|\frac{1}{\Omega}\int_{\text{cell
}}d^{3}xe^{i\mathbf{G}\cdot\mathbf{x}}u_{f,\mathbf{k}_{f}}^{*}(\mathbf{x})u_{i,\mathbf{k}_{i}}(\mathbf{x})\right|^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ $

where $\Omega$ is the primitive cell volume; $i(f)$ are band numbers which
index the initial and final states ; $k_{i}$ and $k_{f}$ are the initial and
final Bloch momentum in the first Brillouin zone; $\mathbf{G}$ is a reciprocal
lattice vector

$\displaystyle\frac{d\mathcal{R}_{v\rightarrow c}}{d\ln E_{e}}=N_{cell}\int
dE_{\nu}\Phi\left(E_{\nu}\right)\int dq\frac{d\sigma_{e}}{dq}$
$\displaystyle\left|f_{v\rightarrow c}\left(q,E_{e}\right)\right|^{2}$ (11)

where $N_{cell}$ is the number of primitive unit cells in the crystal;
$\Phi\left(E_{\nu}\right)$ is the incoming neutrino flux;
$\frac{d\sigma_{e}}{dq}$ is the differential cross section as a function of
momentum transfer assuming that the initial electron was at rest.

The DFT-computed wavefunctions with all-electron reconstruction used in the
calculations are taken from sinead_m_griffin_2022_6015637 . The calculation
includes 4 valence bands for Si and 4 valence bands $+$ 10 outer core
bands(corresponding to 3d shell electrons) for Ge. All bands up to 60 eV above
the band gap are included in the conduction band. This choice therefore covers
all transitions with deposited energy below 60 eV. The lowest energy that can
be deposited is the energy band gap between occupied valence bands and
unoccupied conduction bands which is 1.11 eV for Si and 0.67 eV for Ge.

As a matter of fact, the EXCEED-DM code provides a robust idea for calculation
of scattering rate in semiconductor crystals by using a combination of Density
Functional Theory (DFT) and semi-analytic methods. The work divides the
electronic states in a (pure) crystal into four categories: core, valence,
conduction, and free. The valence and conduction electronic band structures
and wave functions are approximated by numerical calculations using DFT. The
states further away from the band gap (the core and free states) are modelled
using semi-analytic approximations. The total transition rate is then
expressed as the sum of four contributions: valence to conduction (v to c),
core to conduction(c to c), valence to free ( v to f) and core to free (c to
f) transition rates. Though the form factors obtained from EXCEED-DM works
well for dark matter scattering, we observe that the core to conduction (c to
c), valence to free (v to f), and core to free (c to f) calculations lead to
some unwarranted results for neutrino induced scattering via light mediators.
On closer inspection, we find that the modelling of the initial and final
states in EXCEED-DM leads to some fallacious behaviour in the form factor at
low momentum transfers. Since the scattering rates for neutrino electron
scattering via light mediator NSI are heavily dependent on contributions from
low momentum transfers, we therefore only choose to use valence to conduction
(v to c) form factors from EXCEED-DM for this work and instead approximate the
remaining rate spectra for transitions to states with energies above $60$ eV
using DarkARC. We find that this approximation using DarkARC in fact gives a
reasonable continuation to the valence to conduction rate above $E_{e}>60$ eV.
Also, the sensitivity calculations are largely insensitive to this choice of
approximating the rate using atomic form factor.

Figure 3:  Differential event rate as a function of the deposited energy
$E_{e}$ for neutrino scattering off silicon and germanium target for various
mediator masses. The appropriate atomic and crystal form factors have been
considered. The curves for the low mass mediators rise at low deposited
energy. The benchmark couplings taken are: $g_{\nu}=g_{e}=5\times 10^{-7}$ for
scalar mediator (Top Panel), and $g_{\nu}=g_{e}=1.5\times 10^{-7}$ for vector
mediator (Bottom Panel).

Recoil spectra for silicon and germanium detectors incorporating the above
form factors are shown in figure 3 for a few different mediator masses for
both scalar and vector cases. The $x$-axis in the plots corresponds to the
total energy deposited in the detector. The benchmark neutrino and electron
couplings have been taken to be $g_{\nu}=g_{e}=1.5\times 10^{-7}$ and
$g_{\nu}=g_{e}=5\times 10^{-7}$ for vector and scalar mediators, respectively.
Note that, the spectra corresponding to mediators with mass $\mathcal{O}(1)$
keV or below rise at low recoil energies. Therefore, more events would be
registered in such scenarios. Thus, the low threshold detectors with very
small background are ideal to probe light mediators with masses $\leq 1$ keV.

Note that we present the recoil spectra as a function of the deposited energy.
But in reality, the experiments do not measure the deposited energy directly.
Rather they measure the number of electron-hole pairs produced in a scattering
event, i.e., the ionization signal, $Q$. When an incoming neutrino scatters
off an electron inside the atom, it deposits energy $E_{e}$ to the electron,
causing the electron to jump from the valence bands or the core states to
conduction bands. The electron produced in the primary scattering then goes
through a chain of secondary scattering processes and rapidly redistributes
the initial deposited energy by creating many low energy electron-hole pairs,
until all the electrons and holes have energies below the pair creation
threshold. A realistic modelling of the secondary scattering processes is very
complicated. Therefore, a simple linear approximation can be adopted which
does a decent job. Under this approximation, the ionization signal Q is
related to the deposited energy $E_{e}$ by the expression

$Q=1+\left\lfloor\frac{E_{e}-E_{\mathrm{g}}}{\varepsilon}\right\rfloor\leavevmode\nobreak\
,\leavevmode\nobreak\ $ (12)

where $\lfloor\leavevmode\nobreak\ \rfloor$ is the greatest integer function;
$\varepsilon$ is the mean energy per electron-hole pair; and $E_{g}$ is the
bang-gap energy. The values are given by: $\varepsilon$ for silicon
(germanium) is 3.6 eV (2.9 eV); and $E_{g}$ is 1.1 eV (0.67 eV) for silicon
(germanium) 1968JAP….39.2029K .

##  4 Light Mediator Models

In this section we mention a few specific low-energy models with neutrino-
electron interactions through the exchange of light mediators, where our
analysis could be applicable. When considering the models, we confine
ourselves to those with light scalar and vector mediators only. The masses of
the mediators are in the sub-MeV range. Our analysis of the neutrino-electron
scattering can either put a tight bound on the parameter space of these models
or can predict their discovery potential in the future and upcoming
experiments.

###  4.1 Scalar mediator models

The SM scalar sector is the simplest possible scalar structure with only one
Higgs doublet Higgs:1964pj ; Higgs:1966ev ; Englert:1964et ; Guralnik:1964eu ;
Kibble:1967sv . Several well motivated extensions of the SM scalar sector have
been considered in the literature. Among them, the two Higgs doublet model
(2HDM) Lee:1973iz ; Branco:2011iw  is simple and has considerable
phenomenological potential. Recently, it was shown that one can generate a
physical scalar spectrum with one light scalar with mass $\mathcal{O}(100)$
MeV by extending the 2HDM with a singlet scalar field without any fine-tuning
Dutta:2020scq . The $\mathcal{O}(100)$ MeV scalar field can be useful to study
NSI and anomalous magnetic moment problems along with other well known
tensions in the experimental results (e.g. MiniBooNE anomalous events). But we
need even a lighter scalar with mass $\mathcal{O}$(keV) to do the analysis of
neutrino-electron scattering at the reactor neutrino experiments. One needs to
do some level of fine-tuning to get a scalar of mass $\mathcal{O}$(keV) from
SM symmetry only by extending the scalar sector. There are not any specific
models in the literature with such a light scalar particle that generates
considerable non-standard interactions of neutrinos. Therefore we propose one
such model below. We briefly describe the model here in the context of our
analysis. The details of the complete model and its phenomenology will appear
in a future work Dutta:2022future . Note that, one can avoid the fine-tuning
by considering a new $U(1)$ gauge symmetry or some other new physics.

In the following, we briefly mention our new model and explain how it can
generate a non-standard neutrino-scalar and electron-scalar interaction terms.
We extend the scalar sector of the SM by adding one additional doublet, one
scalar singlet, and one scalar triplet field. The scalar fields under
$SU(2)_{L}\times U(1)_{Y}$ symmetry are:

$\displaystyle H_{1}$ $\displaystyle\sim$
$\displaystyle(2,1/2),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
H_{2}\sim(2,1/2),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\phi$ $\displaystyle\sim$
$\displaystyle(1,0),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Delta\sim(3,1)\leavevmode\nobreak\ .\leavevmode\nobreak\ \,$ (13)

For simplicity, we consider that the scalar potential is CP-conserving. In
general, they can be CP-violating. Also, note that we do not impose any
discrete symmetry in the scalar sector. The most general renormalizable scalar
potential is then given by

$\displaystyle V$ $\displaystyle=$ $\displaystyle
m_{1}^{2}H_{1}^{\dagger}H_{1}+m_{2}^{2}H_{2}^{\dagger}H_{2}+m_{\phi}^{2}\phi^{\dagger}\phi+m_{\phi^{\prime}}^{2}(\phi^{2}+\phi^{\dagger
2})+m_{\Delta}^{2}\mbox{Tr}\leavevmode\nobreak\
\Delta^{\dagger}\Delta+m_{12}^{2}(H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1})$
(14)
$\displaystyle+m_{11\phi}H_{1}^{\dagger}H_{1}(\phi^{\dagger}+\phi)+m_{22\phi}H_{2}^{\dagger}H_{2}(\phi^{\dagger}+\phi)+m_{12\phi}(H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1})(\phi^{\dagger}+\phi)$
$\displaystyle+m_{\Delta\Delta\phi}(\phi^{\dagger}+\phi)\mbox{Tr}\leavevmode\nobreak\
\Delta^{\dagger}\Delta+m_{3\phi}(\phi^{3}+{\phi^{\dagger}}^{3})+m_{3\phi}^{\prime}(\phi^{2}\phi^{\dagger}+{\phi^{\dagger}}^{2}\phi)$
$\displaystyle+m_{11\Delta}(H_{1}^{T}i\sigma_{2}\Delta^{\dagger}H_{1}-H_{1}^{\dagger}\Delta
i\sigma_{2}H_{1}^{*})+m_{22\Delta}(H_{2}^{T}i\sigma_{2}\Delta^{\dagger}H_{2}-H_{2}^{\dagger}\Delta
i\sigma_{2}H_{2}^{*})$
$\displaystyle+m_{12\Delta}(H_{1}^{T}i\sigma_{2}\Delta^{\dagger}H_{2}-H_{2}^{\dagger}\Delta
i\sigma_{2}H_{1}^{*})+m_{21\Delta}(H_{2}^{T}i\sigma_{2}\Delta^{\dagger}H_{1}-H_{1}^{\dagger}\Delta
i\sigma_{2}H_{2}^{*})$
$\displaystyle+\frac{\lambda_{1}}{2}(H_{1}^{\dagger}H_{1})^{2}+\frac{\lambda_{2}}{2}(H_{2}^{\dagger}H_{2})^{2}+\frac{\lambda_{\phi}}{2}(\phi^{\dagger}\phi)^{2}+\frac{\lambda_{\phi}^{\prime}}{2}({\phi^{\dagger}}^{3}\phi+\phi^{3}\phi^{\dagger})+\frac{\lambda_{\phi}^{\prime\prime}}{2}({\phi^{\dagger}}^{4}+\phi^{4})+\frac{\lambda_{\Delta}}{2}(\mbox{Tr}\leavevmode\nobreak\
\Delta^{\dagger}\Delta)^{2}$
$\displaystyle+\frac{\lambda^{\prime}_{\Delta}}{2}\mbox{Tr}\leavevmode\nobreak\
(\Delta^{\dagger}\Delta)^{2}+\lambda_{3}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{4}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\frac{\lambda_{5}}{2}\left[(H_{1}^{\dagger}H_{2})^{2}+(H_{2}^{\dagger}H_{1})^{2}\right]$
$\displaystyle+\lambda_{6}\left[(H_{1}^{\dagger}H_{1})(H_{1}^{\dagger}H_{2})+(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{1})\right]+\lambda_{7}\left[(H_{2}^{\dagger}H_{2})(H_{1}^{\dagger}H_{2})+(H_{2}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})\right]$
$\displaystyle+\lambda_{8}(H_{1}^{\dagger}H_{1})(\phi^{\dagger}\phi)+\lambda_{8}^{\prime}(H_{1}^{\dagger}H_{1})({\phi^{\dagger}}^{2}+\phi^{2})+\lambda_{9}(H_{2}^{\dagger}H_{2})(\phi^{\dagger}\phi)+\lambda_{9}^{\prime}(H_{2}^{\dagger}H_{2})({\phi^{\dagger}}^{2}+\phi^{2})$
$\displaystyle+\lambda_{10}(H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1})\phi^{\dagger}\phi+\lambda_{10}^{\prime}(H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1})({\phi^{\dagger}}^{2}+\phi^{2})+\lambda_{11}H_{1}^{\dagger}\Delta\Delta^{\dagger}H_{1}$
$\displaystyle+\lambda_{12}(H_{1}^{\dagger}H_{1})\mbox{Tr}(\Delta^{\dagger}\Delta)+\lambda_{13}H_{2}^{\dagger}\Delta\Delta^{\dagger}H_{2}+\lambda_{14}(H_{2}^{\dagger}H_{2})\mbox{Tr}(\Delta^{\dagger}\Delta)+\lambda_{15}\left[H_{1}^{\dagger}\Delta\Delta^{\dagger}H_{2}+H_{2}^{\dagger}\Delta\Delta^{\dagger}H_{1}\right]$
$\displaystyle+\lambda_{16}\left[(H_{1}^{\dagger}H_{2})\mbox{Tr\leavevmode\nobreak\
}\Delta^{\dagger}\Delta+(H_{2}^{\dagger}H_{1})\mbox{Tr\leavevmode\nobreak\
}\Delta^{\dagger}\Delta\right]+\lambda_{17}(\phi^{\dagger}\phi)\mbox{Tr}(\Delta^{\dagger}\Delta)+\lambda_{17}^{\prime}({\phi^{\dagger}}^{2}+\phi^{2})\mbox{Tr}(\Delta^{\dagger}\Delta)$
$\displaystyle+\lambda_{18}(H_{1}^{T}i\sigma_{2}\Delta^{\dagger}H_{1}-H_{1}^{\dagger}\Delta
i\sigma_{2}H_{1}^{*})(\phi^{\dagger}+\phi)+\lambda_{19}(H_{2}^{T}i\sigma_{2}\Delta^{\dagger}H_{2}-H_{2}^{\dagger}\Delta
i\sigma_{2}H_{2}^{*})(\phi^{\dagger}+\phi)$
$\displaystyle+\lambda_{20}(H_{1}^{T}i\sigma_{2}\Delta^{\dagger}H_{2}-H_{2}^{\dagger}\Delta
i\sigma_{2}H_{1}^{*})(\phi^{\dagger}+\phi)+\lambda_{21}(H_{2}^{T}i\sigma_{2}\Delta^{\dagger}H_{1}-H_{1}^{\dagger}\Delta
i\sigma_{2}H_{2}^{*})(\phi^{\dagger}+\phi)\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$

By tuning the parameters of the scalar potential, we can achieve a scenario
where only one of the doublet $H_{1}$ obtains vacuum expectation value (VEV),
$\braket{H_{1}}=v_{1}/\sqrt{2}$, which completely controls the spontaneous
electroweak symmetry breaking. The triplet field $\Delta$ and the singlet
field $\phi$ also obtain VEVs in the process. The VEV of $\Delta$ is
constrained by the electroweak precision measurement of the $\rho$ parameter
Langacker:1980js ; ParticleDataGroup:2020ssz , given by
$v_{\Delta}<0.032v_{1}$, where $v_{1}=246$ GeV. The vev of the $\phi$ field is
unconstrained and we choose to work with $1$ GeV value.

After the spontaneous symmetry breaking, the scalars can be expressed as

$\displaystyle H_{1}$ $\displaystyle\sim\left(\begin{array}[]{c}{G}^{+}\\\
\frac{1}{\sqrt{2}}(v_{1}+\rho_{1}+iG_{0})\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
H_{2}\sim\left(\begin{array}[]{c}{\phi_{2}}^{+}\\\
\frac{1}{\sqrt{2}}(v_{2}+\rho_{2}+i\eta_{2})\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (18)
$\displaystyle\phi$
$\displaystyle\sim\frac{1}{\sqrt{2}}(v_{\phi}+\rho_{\phi}+i\eta_{\phi}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta\sim\left(\begin{array}[]{cc}\Phi^{+}&\sqrt{2}\Phi^{++}\\\
\frac{1}{\sqrt{2}}(v_{\Delta}+\rho_{t}+i\eta_{t})&-\Phi^{+}\end{array}\right)\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (21)

There are total 16 scalar degrees of freedom (dof) present in the model. Out
of that 3 dof ($G^{\pm}$ and $G^{0}$) become the longitudinal polarization
states of the vector gauge bosons $W^{\pm}$ and $Z$ and make them massive. The
rest of them become physical scalar particles. Two doubly charged physical
scalars $h_{3}^{\pm\pm}$ come from $\Phi^{\pm\pm}$ while 4 single charged
physical scalars, named $h_{1}^{\pm}$ and $h_{2}^{\pm}$ come from the mixing
of $\phi_{2}^{\pm}$ and $\Phi^{\pm}$. There are total 4 CP-even states
$\rho_{1}$, $\rho_{2}$, $\rho_{\phi}$ and $\rho_{t}$. They mix among
themselves and generate four physical neutral Higgs particles named as $h$,
$h_{1}$, $h_{2}$ and $h_{3}$. We identify $h$ as the SM Higgs particle having
mass $125.5$ GeV. Here, $h_{1}$ can be a very light scalar with mass
$\sim\mathcal{O}(1)$ keV. The three CP-odd states $\eta_{2},\eta_{\phi}$ and
$\eta_{t}$ mix and give three physical neutral pseudoscalars, we call them
$s_{1},s_{2}$ and $s_{3}$. All of them have masses that satisfy the LHC
bounds.

Parameters | Descriptions |  Benchmark values
---|---|---
$m_{1}^{2}$, $m_{2}^{2}$, $m_{\phi}^{2}$, $m_{\phi^{\prime}}^{2}$, $m_{\Delta}^{2}$ and $m_{12}^{2}$ |  these are the mass square parameters $\sim[\mathcal{O}$(100) GeV]2 |  $m_{1}^{2}=-(126)^{2}$ GeV2, $m_{2}^{2}=(843)^{2}$ GeV2, $m_{\phi}^{2}=m_{\phi}^{\prime 2}=-(54)^{2}$ GeV2, $m_{\Delta}^{2}=(703)^{2}$ GeV2, $m_{12}^{2}=-(100)^{2}$ GeV2.
$m_{11\phi}$, $m_{22\phi}$, $m_{12\phi}$, $m_{\Delta\Delta\phi}$, $m_{3\phi}$, $m_{3\phi}^{\prime}$, $m_{11\Delta}$, $m_{22\Delta}$, $m_{12\Delta}$ and $m_{21\Delta}$ |  the mass dimension of these parameters are one with values $\sim\mathcal{O}(0.1-10)$ GeV |  $m_{11\phi}=0.138$ GeV, $m_{11\Delta}=11.54$ GeV and rest of them are $-10$ GeV.
$\lambda_{1}$, $\lambda_{2}$, $\lambda_{\phi}$, $\lambda_{\phi}^{\prime}$, $\lambda_{\phi}^{\prime}$ $\lambda_{\Delta}$ $\lambda_{\Delta}^{\prime}$, $\lambda_{3}$, $\lambda_{4}$, $\lambda_{5}$, $\lambda_{6}$, $\lambda_{7}$, $\lambda_{8}$, $\lambda_{8}^{\prime}$, $\lambda_{9}$, $\lambda_{9}^{\prime}$, $\lambda_{10}$, $\lambda_{10}^{\prime}$, $\lambda_{11}$, $\lambda_{12}$, $\lambda_{13}$, $\lambda_{14}$, $\lambda_{15}$, $\lambda_{16}$, $\lambda_{17}$, $\lambda_{17}^{\prime}$, $\lambda_{18}$, $\lambda_{19}$, $\lambda_{20}$ and $\lambda_{21}$ |  these are the dimensionless couplings with values between $0.1-1.0$ |  $\lambda_{1}=0.53$, $\lambda_{6}=0.33$; rest of them are 0.1
Table 1:  Brief descriptions and the necessary range of values for all the
parameters that appear in the scalar potential in eq. 14. One particular
benchmark scenario is shown. We use this benchmark to generate a light scalar
of mass $\sim\mathcal{O}(1)$ keV along with other heavy physical scalars
consistent with the LHC bounds. The values of the vev’s are $v_{1}=246$ GeV,
$v_{2}=0$ GeV, $v_{\phi}=1$ GeV and $v_{\Delta}=1$ GeV.

The Yukawa sector Lagrangian in the interaction basis is given by

$\displaystyle-\mathcal{L}_{Yukawa}$ $\displaystyle=$
$\displaystyle\bar{q}^{\prime}_{L_{i}}(y^{\prime}_{1d})_{ij}d^{\prime}_{R_{j}}H_{1}+\bar{q}^{\prime}_{L_{i}}(y^{\prime}_{1u})_{ij}u^{\prime}_{R_{j}}\tilde{H_{1}}+\bar{l}^{\prime}_{L_{i}}(y^{\prime}_{1e})_{ij}e^{\prime}_{R_{j}}H_{1}+\bar{q}^{\prime}_{L_{i}}(y^{\prime}_{2d})_{ij}d^{\prime}_{R_{j}}H_{2}$
(22) $\displaystyle+$
$\displaystyle\bar{q}^{\prime}_{L_{i}}(y^{\prime}_{2u})_{ij}u^{\prime}_{R_{j}}\tilde{H_{2}}+\bar{l}^{\prime}_{L_{i}}(y^{\prime}_{2e})_{ij}e^{\prime}_{R_{j}}H_{2}+\bar{l}^{\prime
c}_{L_{i}}(y^{\prime})_{ij}i\sigma_{2}\Delta
l^{\prime}_{L_{j}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ $

where $i,j$ are the family indices with values $1,2$, and $3$; also note that
the primed fermions are the interaction states. The first three terms give
rise to the masses of the charged leptons and quarks upon diagonalization. The
next three terms give interaction with $H_{2}$ which is not constrained by the
fermion masses. Note that, these interactions do not respect the SM flavor
symmetries and thus can generate flavor changing neutral currents at tree
level through the interactions with the neutral components of $H_{2}$. The
fermions have no direct interactions with the scalar singlet, the only way
they can interact with the singlet is via the scalar mixing. Therefore, the
physical scalars, light and heavy, interact with all flavors of fermions,
diagonally and off-diagonally.

The last term of eq. 22 is the most interesting term as it is generating the
non-standard interactions of neutrinos and neutrino mass. It generates a
Majorana type interaction term $-\bar{\nu}^{\prime
c}_{L_{i}}i(y^{\prime})_{ij}\nu^{\prime}_{L_{j}}\rho_{t}$, where $\rho_{t}$ is
the CP-even component of the neutral component of $\Delta$ field. The
$\rho_{t}$ mixes with other CP-even states and generates a very light physical
scalar particle $h_{1}$. Therefore in the mass basis the term
$\bar{\nu^{c}}\nu h_{1}$ is giving the non-standard neutrino-scalar
interaction term. On the other side, the light scalar $h_{1}$ couples to both
leptons and quarks diagonally and off-diagonally. Therefore, in general, there
are all possible non-standard neutrino interactions with SM fermions mediated
by $h_{1}$. Also, note that these interactions are possible for all the other
heavy physical scalars and pseudo-scalars. Therefore, the physical basis
Lagrangian responsible for the non-standard neutrino interactions can be
written as follows

$-\mathcal{L}_{Yukawa}\supset\bar{{\nu}^{c}}_{i}(y_{\nu})_{ij}\Gamma\nu_{j}h_{k}+\bar{f}_{i}(y_{f})_{ij}\Gamma
f_{j}h_{k}\leavevmode\nobreak\ ,\leavevmode\nobreak\ $ (23)

where $i$ and $j$ are flavor indices, $i,j=1,2,3$; $f=d,u,e$;
$h_{k}=h,h_{1},h_{2},h_{3},s_{1},s_{2},s_{3}$; and $\Gamma=1$ for scalars
while $\Gamma=\gamma^{5}$ for the pseudo-scalars. As we are only interested in
the neutrino-electron interactions, we assume that only the electron diagonal
interactions are non-zero. In general, there is no strong reason to forbid the
other interactions. Actually it is very interesting, though computationally
very complex, to turn on all possible interactions. In that case, a global
analysis based on the neutrino oscillation and scattering data will affect the
parameter space. Previously such analysis was performed mostly for the heavy
mediator case (see for example Dutta:2020che ; Coloma:2022umy ;
Esteban:2018ppq ; Esteban:2019lfo ; Khan:2016uon ). A similar global analysis
for the light scalar case will be very crucial to get bounds for a realistic
light scalar model Dutta:2022future . From this point forward, the light
scalar $h_{1}$ will be identified as $\phi$. In this work, we only consider
the electron and $\nu_{e}$ couplings with the light mediators to investigate
them at the reactors.

Physical particles |  Mass values and descriptions
---|---
Neutral scalars: $h$, $h_{1}$, $h_{2}$ and $h_{3}$ |  $m_{h}=125.5$ GeV, this is the SM Higgs; $m_{h_{1}}=0.001$ GeV, very light scalar, neccessary for our analysis; $m_{h_{2}}=500$ GeV, $m_{h_{3}}=600$ GeV they satisfy the LHC bounds.
Neutral pseudo-scalars: $s_{1}$, $s_{2}$ and $s_{3}$ |  $m_{s_{1}}=500$ GeV, $m_{s_{2}}=500$ GeV, $m_{s_{3}}=600$ GeV, they satisfy LHC bounds.
Single charged scalars: $h_{1}^{\pm}$, $h_{2}^{\pm}$ |  $m_{h_{1}^{\pm}}=500$ GeV, $m_{h_{2}^{\pm}}=600$ GeV, in agreement with LHC bounds.
Double charged scalars: $h_{3}^{\pm\pm}$ |  $m_{h_{3}^{\pm\pm}}=500$ GeV, satisfying LHC limits.
Table 2:  The physical scalar spectrum generated using the benchmark values
from Table 1. The physical scalar $h$ is identified as the SM Higgs scallar.
The $h_{1}$ is the light scalar which will be identified as $\phi^{\prime}$ in
the rest of the paper. The scalar masses are consistent with the LHC bounds.

###  4.2 Vector Models

Here we mention a few well-studied models that can have neutrino-electron
interactions through a light vector boson mediator. The extension of the SM by
a $U(1)$ gauge symmetry can have the required interactions and has been well-
studied. We consider three gauge symmetries $U(1)_{L_{e}-L_{\mu}}$,
$U(1)_{L_{e}-L_{\tau}}$, and $U(1)_{L_{\mu}-L_{\tau}}$ Foot:1990mn ; He:1990pn
; He:1991qd  that are anomaly-free without any additional particles. Note
that, for the first two scenarios one can get neutrino-electron interactions
at tree level. The necessary terms of the Lagrangian that can give rise to
neutrino-electron couplings are,

$\displaystyle-\mathcal{L}=\bar{l}_{L_{i}}\gamma^{\mu}l_{L_{i}}A^{\prime}_{\mu}+\bar{e}_{R_{i}}\gamma^{\mu}e_{R_{i}}A^{\prime}_{\mu}-\bar{l}_{L_{j}}\gamma^{\mu}l_{L_{j}}A^{\prime}_{\mu}-\bar{e}_{R_{j}}\gamma^{\mu}e_{R_{j}}A^{\prime}_{\mu}\leavevmode\nobreak\
,\leavevmode\nobreak\ $ (24)

where $i$ and $j$ are not equal. In all three cases, the SM fermions are
charged under both the new $U(1)$ gauge symmetry as well as the SM hypercharge
symmetry, and thus a kinetic mixing term automatically arises between the new
gauge boson $A^{\prime}$ and the SM hypercharge boson at one loop level even
if we consider that the tree level kinetic mixing is zero. And this allows to
have the desired neutrino-electron coupling in the third scenario. The
coupling will be suppressed by the kinetic mixing factor which is roughly
$\epsilon_{\mu\tau}\sim eg_{\mu\tau}/4\pi^{2}$.

Another possible gauge group is the difference between baryon and lepton
number $U(1)_{B-L}$. This group can be anomaly-free if one adds right-handed
neutrinos. The kinetic mixing at one-loop level is unavoidable. We can get the
neutrino-electron coupling at tree level as follows

$\displaystyle-\mathcal{L}=-\bar{l}_{L}\gamma^{\mu}l_{L}A^{\prime}_{\mu}+\bar{e}_{R}\gamma^{\mu}e_{R}A^{\prime}_{\mu}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (25)

The neutrino-electron interaction is also possible in the low energy
incarnation of the gauge group $U(1)_{T3R}$ Dutta:2019fxn ; Dutta:2022qvn ,
where only the right handed SM particles are charged. Right handed neutrino
fields are needed to cancel the anomalies. The desired interaction is possible
either using active-sterile neutrino mixing or using the mixing of the new
gauge boson $A^{^{\prime}}$ with SM $Z$ gauge boson.

The details of these models are not necessary for our purpose. We briefly
mention various constraints on these models in the next subsection.

###  4.3 Constraints

One common feature of these models would be that they have neutrino-neutrino,
neutrino-electron and electron-electron interactions induced by the new light
mediators. Therefore, the parameter space of these models will be constrained
by various laboratory based experiments and/or astrophysical/cosmological
data.

Neutrino-electron scattering experiments such as Borexino BOREXINO:2018ohr
and Gemma Beda:2013mta  are quite sensitive to the modification of the
scattering cross section by the non-standard interactions of neutrinos with
electrons mediated by low mass mediators. The non-observation of any anomalous
events at these experiments can be converted into bounds on the parameter
space of coupling-mediator mass plane. We have translated the bounds into
constraints on the couplings of the light mediators (both scalar and vector)
and shown them in figure 4.

Different astrophysical processes such as energy loss of supernova 1987A,
solar cooling process, and cooling of stars in globular clusters can also put
bounds on the relevant parameter space for light mediator models. The light
mediator particle can be produced inside the core of the supernova and then
subsequently escape, resulting in energy loss, which can be compared to the
SN1987A data to find bounds on light mediator mass and couplings. Similarly,
they can be produced inside the sun or any other stars in a globular cluster,
and contribute to the cooling process as well. One can derive constraints from
these cooling processes by requiring that the luminosity due to the new
mediator particle be small compared to the luminosity due to the photon. Note
that, these astrophysical bounds can be avoided in light mediators models by
using the chameleon effect. In general, the masses of the mediators depend on
the background matter density. The environment in the core of a supernova or a
star is extremely dense. In such a highly dense environment, the effective
mass of the mediator particle can be larger than it is on Earth, and thus the
allowed parameter space can be increased. Astrophysical bounds for light
scalar mediator can be found in Ref. Babu:2019iml  and the relevant bounds for
vector boson were discussed in Ref. Harnik:2012ni .

The light mediators can be produced in the early Universe and can be in
equilibrium with the thermal bath. Therefore, they can contribute to the
$\Delta N_{eff}$ depending on their abundance during the time of
recombination. If $m_{\phi,A}<1$ MeV, then $\phi$ can decay into $\nu\nu$
final state at tree level. Note that, $\gamma\gamma$ final state is also
possible for $\phi$ in this case via a one-loop diagram consisting of an
electron inside the loop. The detailed analysis is out of the scope of this
paper, however, we refer Ref.Escudero:2019gvw  where a detailed analysis has
been performed for similar mass ranges of a light mediator.

Non-standard interactions of the neutrinos can have effects on the oscillation
and scattering experiments as well. Low threshold experiments based on
coherent elastic neutrino-nucleus scattering (CE$\nu$NS) can place constraints
on both scalar and vector non-standard interactions. Vector interaction
introduces a new flavor-dependent matter potential during the propagation
through matter that can affect neutrino oscillation. Current data from
COHERENT can put constraints for mediator mass $\geq 10$ MeV Denton:2018xmq .
Future reactor CE$\nu$NS experiments can probe very light mediators. The light
scalar mediators, on the other hand, do not give rise to a matter potential as
the scalar interaction cannot be converted into a vector current. Rather it
contributes as a correction to the neutrino mass matrix. The correction factor
is proportional to the inverse mass square of the mediator, therefore a small
mediator mass can produce large enough interactions Ge:2018uhz ; Babu:2019iml
.

##  5 Results

In this section, we describe our strategy for the numerical analysis to probe
the light scalar and vector mediated non-standard neutrino-electron
scattering. Then we present the results of our analyses for silicon and
germanium detectors. Our analyses are focused towards low threshold reactor
based neutrino experiments. We also give the expected total number of events
as a function of the detector threshold. First, we briefly describe the key
aspects of our analysis.

  * •

The differential cross sections of neutrino-electron scattering for both
scalar and vector cases increase at low recoil energies with decreasing
mediator masses, as evident from figure 2. This causes a significant
enhancement in the differential rate spectra at low recoils. It can be seen
explicitly in figure 3. In particular, for a light vector mediator with mass
$\sim 1$ keV almost 99$\%$ of the events recorded are expected to deposit less
than a keV of energy in the detector. Therefore, the experiments with low
threshold detectors are expected to be sensitive to light mediators with
masses $\leq\mathcal{O}(1)$ keV. For our analyses, we consider detectors with
threshold as low as 5 eV.

  * •

A major background to the neutrino-electron scattering events in a typical low
threshold detector (e.g. CCD detector) comes from CE$\nu$NS interactions
mostly below $\mathcal{O}(1)$ keV, since both $\nu-e$ and CE$\nu$NS have the
same experimental signature in these detectors. Most low threshold detector
technologies being utilized in current experiments can not discriminate
between nuclear recoil (NR) and electron recoil (ER) signals at very low
recoil energies due to signal to noise limitations. Therefore, the
experimental sensitivity to the non-standard neutrino-electron couplings in
reactor experiments will be dependent on the capability of ER and NR events
discrimination of the detector down to very low recoil energies. Recently it
was shown that a new hybrid detector at MINER incorporating both ZIP
(Z-sensitive Ionization and Phonon detector) and HV (High Voltage) technology
is capable of discriminating ER and NR up to a very low threshold ($\sim 1$
keV) and would be able to do so even at a lower threshold in near future
Neog:2020ily . We assume that the detector is capable to discriminate between
NR and ER signals down to the threshold of 5 eV. Note that, above
$\mathcal{O}(1)$ keV recoil energies there is no contribution from CE$\nu$NS
interactions, therefore the discrimination between ER and NR is not needed.
But in this region, the rate of non-standard neutrino-electron scattering is
very low and does not give any interesting result.

  * •

The main background for such analyses is the SM neutrino-electron scattering
events mediated by the weak interactions. In addition to that, there would be
more background events specific to the experiments. Here we assume that there
exists a constant background over the SM value. We consider two such cases:
0.1 DRU and 1.0 DRU (where [DRU] = [counts $\cdot$ kg-1 keV-1 day-1]). The
second case is achievable at the present facilities whereas the first case can
be achieved in near future MINER:2016igy . We perform our analyses for both
cases.

We perform a $\chi^{2}$ test over the distribution of deposited energy to find
the bound on the parameter space in the coupling-mass plane. We start by
assuming a set of independent background expectations $B_{i}$ and an expected
signal $S_{i}$, which depends on the coupling and mediator mass, across bins
$i=1,...,\nu$. If the experiment sees data consistent with the null hypothesis
(only background events), the median value of the $\chi^{2}$ distribution
tested against our $S+B$ model will exceed the $1-\alpha$ quantile of the
background only $\chi^{2}$ distribution. In the absence of real experimental
data, we generate a set of independent, normally distributed pseudo data
$d_{i}\sim N(B_{i},\sqrt{B_{i}})$ for each bin.

However, in the limit of a large number of pseudo-experiments, the $1-\alpha$
quantile of this distribution approaches the theoretical limit of the inverse
cumulative distribution function $F^{-1}(1-\alpha,\nu)\equiv\chi^{2}_{\alpha}$
which allows us to construct a simplifying heuristic for the separation
between the two $\chi^{2}$ distributions described above. The expectation
value of the $\chi^{2}$ tested against the $S+B$ model can be written as,

$E\left[\chi^{2}_{S+B}\right]=E\left[\sum_{i=1}^{\nu}\frac{(d_{i}-S_{i}-B_{i})^{2}}{B_{i}}\right]$
(26)

Using the linearity of expectation values and the assumption that $d_{i}$ are
independent, and $Var[d_{i}]=B_{i}$, we can simplify eq. 26,

$E\left[\chi^{2}_{S+B}\right]=\nu+\sum_{i=1}^{\nu}\frac{S^{2}_{i}}{B_{i}}$
(27)

Therefore, the test statistic can be defined as,

$t=\sum_{i=1}^{\nu}\frac{S^{2}_{i}}{B_{i}}=E\left[\chi^{2}_{S+B}\right]-\nu$
(28)

For relatively large degrees of freedom, Median$[\chi^{2}]$ $\simeq
E[\chi^{2}]$. On the other hand, for large number of pseudo experiment and
data consistent with background-only scenario, we have
$\text{Median}[\chi^{2}_{S+B}]>\chi^{2}_{\alpha}$. Therefore, we need to find
the smallest signal such that

$t\gtrsim\chi^{2}_{\alpha}-\nu$ (29)

By using Eq. 29, this method circumvents the need to generate pseudoexperiment
monte carlo as a matter of computational ease while sacrificing little in the
accuracy of the test statistic. The validity can be measured by generating
pseudoexperiment data with the assumptions above and checking that the
separation in the null-hypothesis $\chi^{2}$ distribution and the $S+B$
model’s $\chi^{2}$ distribution matches well with the heuristic measure in Eq.
29. In order to get the exclusion curve at 90$\%$ confidence level, we
consider $\alpha=0.1$ and take number of bins as $\nu=37$ in the log space of
the deposited energy.

Figure 4: Projected bounds on the coupling $\sqrt{g_{\nu}g_{e}}$ at $90\%$
confidence level as a function of the mediator masses. The bounds are given
for four different exposure values: 1 kg-yr, 10 kg-yr, 100 kg-yr, and 1000 kg-
yr for scalar mediators (top panel) for silicon (left) and germanium (right)
target. For the vector mediator (bottom panel) three different exposure have
been considered: 1 kg-yr, 10 kg-yr, and 100 kg-yr. The current excluded
regions from XENONnT (yellow region) and Gemma/Borexino (orange region) are
also shown. We have assumed a constant background of 0.1 DRU and the detector
threshold is 5 eV.

Figure 5: Projected bounds on the coupling $\sqrt{g_{\nu}g_{e}}$ at $90\%$
confidence level as a function of the mediator masses assuming a constant
background of 1 DRU and the detector threshold is 5 eV.. The bounds are given
for four different exposure values: 1 kg-yr, 10 kg-yr, 100 kg-yr, and 1000 kg-
yr for scalar mediators (top panel) for silicon (left) and germanium (right)
target. For the vector mediator (bottom panel) three different exposure have
been considered: 1 kg-yr, 10 kg-yr, and 100 kg-yr. The current excluded
regions from XENONnT (yellow region) and Gemma/Borexino (orange region) are
also shown.

In figure 4 we show the projected sensitivity at 90% confidence level for both
scalar and vector mediator models considering a constant background of 0.1 DRU
over the SM $\nu-e$ background. Along with this we also show the excluded
region of parameter space from GEMMA, Borexino, and XENONnT experiments. The
constraints from CONUS and CONNIE are subdominant. A similar plot for a
constant background of 1.0 DRU is shown in figure 5. As expected we see an
enhanced sensitivity to light mediators using a detector with low threshold.
The nature of the exclusion curve can be described as follows;

  * •

For heavier mediator masses the coupling sensitivity decreases since for
$q<<M_{\phi/A^{\prime}}$ the new-physics interaction becomes the Fermi like
contact interaction. This gives a line with a slope of 2 in the log-log space
of the coupling vs mediator mass plot.

  * •

For light mediators, the mediator mass may be neglected in the propagator when
it becomes small with respect to the momentum transfer
$q>>M_{\phi/A^{\prime}}$ which corresponds to the sensitivity becoming flat
for very light mediators.

We notice that the larger flux of the reactor experiment allows for a deeper
reach into the parameter space when compared to solar neutrino experiments.
Even a 1 year run of a 1 kg Si or Ge detector can probe larger area of the
parameter space compared to the Gemma, Borexino, and XENONnT experiments. This
shows that a reactor based experiment using low threshold detectors capable of
differentiating ER-NR signals like those intended for MINER can provide
prominent bounds on large regions of parameter space. Note that one can reach
an even greater sensitivity at a GW class reactor experiment but there one is
limited by the distance the detector can be placed from the reactor core which
is typically $\sim$10 m. Overall, the flux is still enhanced by an order of
magnitude at GW reactor experiments.

Figure 6:  Integrated yield of electron recoil events (per ton per year)
captured by silicon and germanium detectors as a function of detector
threshold $E_{Th}$. The coupling values are taken as $g_{\nu}=g_{e}=5\times
10^{-7}$ for scalar mediator (Top Panel), and $g_{\nu}=g_{e}=1.5\times
10^{-7}$ for vector mediator (Bottom Panel).

In figure 6 we show the projected total number of events per ton per year as a
function of experimental recoil threshold assuming the vector and scalar
couplings to be $g_{\nu}=g_{e}=1.5\times 10^{-7}$ and $5\times 10^{-7}$,
respectively. The electron recoil events for both Si and Ge detector targets
for various mediator masses are plotted. As expected, in all the cases shown,
lowering the threshold increases the reach of the experiment for light
mediator masses. We see that for scalar case a $\sim$100 eV threshold is good
enough to capture most of the events, though an even lower threshold can help
provide a better resolution of the light mediator mass. On the other hand, for
the vector case a $\sim$1 eV threshold detector can capture an order of
magnitude more events compared to a $\sim$1 keV threshold detector for
$\mathcal{O}(1)$ keV mediator mass . This implies that the valence to
conduction transition rate calculation used in this work is particularly
significant for these very light vector mediators.

##  6 Conclusions

The non-standard neutrino-electron interactions via light and weakly coupled
scalar or vector mediators occur in many extensions of the SM such as
$U(1)_{L_{i}-L_{j}}$, $U(1)_{B-L}$, and $U(1)_{T3R}$, etc. But not many such
scalar mediator models exist. Here we proposed one light scalar model by
extending only the scalar sector of the SM using doublet, triplet, and singlet
scalar fields. In this model, we have non-standard interactions associated
with different flavors of neutrinos for both heavy and light mediators.
Results of many previous experiments utilizing solar and reactor flux such as
XENONnT, Borexino, GEMMA, etc provide constraints on the parameter space of
such light mediator models associated with non-standard $\nu_{e}-e$
scattering.

Since MW class reactor experiments have an order of magnitude more neutrino
flux (at a mean distance of 1m from the core) compared to solar experiments,
reactor experiments are expected to provide greater sensitivity. Ongoing and
future reactor experiments such as MINER, CONUS, CONNIE, vIOLETTA, etc. can
provide a particularly important probe of such light mediator models if one is
to leverage ultra low threshold detectors such as Si and Ge. We discussed the
prospects of probing non-standard neutrino interactions with low energy
neutrino-electron elastic scattering in reactor based experiments. Since for
these neutrino-electron interactions via light mediators, the typical momentum
transfer is low, the energy deposited in the detector is in the sub-keV range.
At these sub keV energies, one cannot reliably use the free electron
approximation (FEA) and must include the atomic and crystal effects in the
calculation. Taking this into account via atomic form factor from DarkARC and
valence to conduction crystal form factor from EXCEED-DM we evaluated the
sensitivity of a typical MW-class reactor experiments to weakly coupled NSIs
in light mediator models. We showed that using ultra low threshold detectors
such as Si/Ge the reactor experiments can outdo the current constraints on
light scalar and vector mediators couplings to neutrinos and electrons. We
also find that a low threshold detector with very low background capable of
differentiating between ER and NR is very crucial for such deeper reach of the
parameter space. A germanium (silicon) detector with $10$ kg-yr exposure and 1
MW reactor anti-neutrino flux would be able to probe a scalar and vector
mediator with masses below 1 keV for coupling products $\sqrt{g_{\nu}g_{e}}$
$\sim$ $1\times 10^{-6}\leavevmode\nobreak\ (9.5\times
10^{-7})\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ 1\times
10^{-7}\leavevmode\nobreak\ (8\times 10^{-8})$, respectively.

###### Acknowledgements.

We are grateful to Timon Emken, C.-P. Liu, Rupak Mahapatra, Tanner Trickle, P.
S. Bhupal Dev and Miguel Escudero for useful discussions. The work of BD, SG,
AT and AV is supported in part by DOE grant DE-SC0010813. The work of SG is
also supported in part by National Research Foundation of Korea (NRF) Grant
No. NRF-2019R1A2C3005009 (SG). T.L. is supported in part by the National Key
Research and Development Program of China Grant No. 2020YFC2201504, by the
Projects No. 11875062, No. 11947302, No. 12047503, and No. 12275333 supported
by the National Natural Science Foundation of China, as well as by the Key
Research Program of the Chinese Academy of Sciences, Grant NO. XDPB15. We
acknowledge that portions of this research were conducted with the advanced
computing resources provided by Texas A&M High Performance Research Computing.

#### Note added.

We provide the files containing the codes for the numerical calculations here:
https://github.com/averma609/NSI-calc

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