††institutetext:  Max-Planck-Institut für Kernphysik, Postfach 103980, D-69029
Heidelberg, Germany

#  $\sin^{2}\theta_{W}$ and neutrino electromagnetic interactions in
CE$\overline{\nu}_{e}$NS with different quenching factors

Amir N. Khan,111Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recently, evidence for the observation of about 2 keV and below nuclear
recoils from the coherent scattering of reactor anti-neutrinos off the
germanium nuclei has been reported. We analyze the observed data to estimate
the value of the weak mixing angle and constrain the neutrino millicharge,
magnetic moment, charge radius and anapole moment contributing to the coherent
scattering process. Currently, there is no definite model available for the
quenching factor at such low energies. To this end, we consider various models
of the quenching factor and show how it affects the interpretation of the
obtained results. We find that the bounds obtained are stronger in some cases
while comparable or weaker in other cases which show a strong dependence on
the choice and accuracy of a particular quenching factor model. The results
are the first at such low-energy nuclear recoils. We present an exhaustive
list of analytical functions for the different quenching factors corresponding
to the existing models and to the data from various experiments. Such
functions will be useful for any new physics study using the nuclear recoils
due to the reactor neutrinos.

##  1 Introduction

##  2 Introduction

The first evidence for the observation of coherent elastic anti-neutrino
nucleus scattering (CE$\overline{\nu}_{e}$NS) with Dresden-II boiling water
reactor neutrinos and p-type point-contact germanium, also known as NCC-1701
detector, has been suggested in refs. Colaresi  _et al._ (2022); Collar and
Lewis (2022), following from the earlier work Colaresi  _et al._ (2021). The
result also suffers from large experimental uncertainties in the region of
interest (sub-keV). On top of that, there is no unique theoretical prediction
of the standard event spectrum possible because of our limited understanding
of the quenching factor (QF) at such low energies. This leads to a large
disparity in new physics predictions. The maximum nuclear recoil produced by
reactor neutrinos is up to $2\ \rm keV$ and the corresponding ionization
energy is below one keV, while the detection threshold is 0.2 keV ionization
energy. Other competitive experiments looking for nuclear recoils with reactor
neutrinos such as MINERAgnolet  _et al._ (2017), RED-100 Akimov  _et al._
(2017a), $\nu$-cleus Strauss  _et al._ (2017), CONUS Hakenmüller  _et al._
(2019); Bonet  _et al._ (2021a, b); Bonhomme  _et al._ (2022); Bonet  _et al._
(2022), CONNIE Aguilar-Arevalo  _et al._ (2019, 2020, 2021), TEXONO Sharma
_et al._ (2021), vIOLETA Fernandez-Moroni  _et al._ (2021) and SCB Flores  _et
al._ (2021) will provide further concrete information about this observation.

The coherent elastic neutrino-nucleus scattering process was predicted long
ago Freedman (1974); Freedman  _et al._ (1977); Tubbs and Schramm (1975);
Drukier and Stodolsky (1984), however, it was first observed only a few years
ago by the COHERENT experiment with nuclear recoils of 10 keV and above and
with neutrino energies of a few tens of MeV produced from spallation neutron
source Akimov  _et al._ (2017b, 2018, 2021). Several standard model (SM) and
nuclear physics parameters were studied Barranco  _et al._ (2005); Cadeddu
_et al._ (2018a); Tomalak  _et al._ (2021) and new physics interactions and
models were explored using the coherent scattering process Anderson  _et al._
(2012); deNiverville  _et al._ (2015); Lindner  _et al._ (2017); Coloma  _et
al._ (2017); Bauer  _et al._ (2018); Billard  _et al._ (2018); Denton  _et
al._ (2018); Dutta  _et al._ (2019); Khan and Rodejohann (2019); Khan  _et
al._ (2021); Khan (2022); Coloma  _et al._ (2022); Liao  _et al._ (2022);
Aristizabal Sierra  _et al._ (2022); Corona  _et al._ (2022); Abdullah  _et
al._ (2022); de la Vega  _et al._ (2021); Denton and Gehrlein (2021); Suliga
and Tamborra (2021); Du and Yu (2021); Cerdeño  _et al._ (2021); Alikhanov and
Paschos (2021); Fernandez-Moroni  _et al._ (2022). With reactor neutrinos, no
signal of the CE$\overline{{\nu}_{e}}$NS process was observed yet until
recently when its observation evidence was reported with Dresden-II reactor
neutrinos and NCC-1701 detector Colaresi  _et al._ (2022).

Here, we analyze the data of the observed measurement with the 3 kg detector
mass, 2.96 Giga-Watt reactor and data-taking period of 96.4 days Colaresi  _et
al._ (2022). We mainly focus on estimating the weak mixing angle
($\sin^{2}{\theta_{W}}$) and derive bounds on the electromagnetic
interactions; neutrino millicharges, magnetic moment, charge radius and
anapole moment. At such low energies, the detection of nuclear recoils suffers
from the lack of accurate knowledge about the quenching factor and from large
uncertainties about it. Therefore, we use different models for the quenching
factor in our analysis and compare our results from them. Two of them are
theoretical models; one is the famous Linhard model Lindhard  _et al._ (1963)
while the other is an ansatz introduced by Sarkis et al Sarkis  _et al._
(2020) which is also based on a modification to the Linhard model and an
improvement of recent work Sorensen (2015). In the third case, we will use the
experimental data of the QF obtained from the NCC-1701 detector calibration,
called Iron-filtered (Fef) data. In all three cases, we evaluate the quenching
factor as a function of the observed ionization energy.

We will show how the three quenching factors give different results for
$\sin^{2}{\theta_{W}}$ and for electromagnetic properties of neutrinos. At
such low energies, neutrino millicharge is more sensitive than the other
electromagnetic interactions due to their interference with the standard weak
interactions and their dependence on the inverse power of the recoil energy
and on the target mass Khan (2020, 2021); Aprile  _et al._ (2020); Khan
(2022). Further, we will derive constraints on all electromagnetic properties
of neutrinos with the observed data.

The paper is organized as follows. In the next section, we discuss different
quenching factors that we will use for our analysis. In Sec. 4, we discuss the
differential cross-section of the CE$\nu$NS in the SM and all necessary
notations. In Sec. 5, we discuss reactor neutrino fluxes and calculate the
event energy spectrum. In Sec. 7, we introduce the electromagnetic properties
of neutrinos and discuss our results in view of the three QF used here.
Finally, we summarize and conclude in Sec. 8.

##  3 Models and data for the quenching factors

For ionization detectors, the visible nuclear recoil energy, called the
ionization energy ($E_{I}$) is always less than the actual nuclear recoil
energy ($E_{nr}$) due to the energy loss in exciting the binding atoms. This
effect is even stronger at lower energy nuclear recoils, particularly for the
nuclear recoils due to reactor neutrinos. The ratio between the two energies
is often called the quenching factor, denoted here by $Q$. This could be
conveniently taken as a function of $E_{nr}$ or $E_{I}$. We define it as a
function of $E_{I}$ in the following,

$Q(E_{I})=\frac{E_{I}}{E_{nr}}\,.$ (1)

The commonly used theoretical model for the quenching factor is the so-called
Linhard model Lindhard  _et al._ (1963). Although this model is successful at
nuclear recoils above a few keV, it fails to describe the low energy
phenomena. For example, this model predicts that for one keV true nuclear
recoil energy for a Germanium nucleus is reduced by about 80%. This reduction
is even more for the nuclear recoils below one keV. This limitation is caused
by several approximations made in the formulation of the model of the atomic
binding energy at low energies. To account for properly treating the atomic
binding energy at lower energy recoils some attempts were made in refs.
Sorensen (2015) and Sarkis  _et al._ (2020). The authors of refs. Sorensen
(2015); Sarkis  _et al._ (2020) have revisited the original Linhard model and
modified it for the low energy recoils by relaxing the binding energy
approximations which were made in the formulation of the Linhard model. Ref.
Sorensen (2015) includes new kinematic effects and shows how they affect the
low energy cut-off for the atomic binding energy. This approach has recently
been improved further by Sarkis $et\ al$ Sarkis  _et al._ (2020) by
considering a semi-hard sphere interaction model and by solving the original
Linhard integral equation and bringing the cut-off on the binding energy up to
$\sim$ 200 eV. Another choice for the quenching factor is to use the
experimental data obtained directly for the calibration of the particular
detector with photon sources from the neutron scattering measurements.

For our analysis here we will consider three different quenching factors: $i)$
Linhard Model $ii)$ Sarkis $et\ al$ model $iii)$ Iron-filtered (Fef) data
obtained from the calibration of the Germanium-based NCC-1701 detector. We
have calculated the quenching factor as a function of the ionization energy
for each case, as discussed in the following.

Linhard Model: The Linhard model Lindhard  _et al._ (1963) depends on the
element-specific parameter ‘$k$’ which is a function of atomic mass and atomic
number of the relevant atom $(k=0.133\times Z^{2/3}\times A^{-1/2})$. For
Germanium $\rm(^{72}Ge)$ with $k=0.158$, we derive the following quenching
factor as a function of the ionization energy,

$Q(E_{I})=E_{I}\sum_{i=0}^{6}\left[\frac{a_{i}E_{I}^{i}}{b_{i}E_{I}^{i}}\right]\
\ \ \ (\rm Linhard\ Model)\,,$ (2)

where $a_{i}$ and $b_{i}$ are the evaluated fitting parameters. The numerical
values of the fit parameters are given in the Ist and 2nd columns of Table 1.
The resultant quenching factor as a function of $E_{I}$ is shown in Fig. 1 in
blue.

Bonhomme data: The most recent direct measurement of the quenching factor has
been reported in Bonhomme  _et al._ (2022). Four data sets were obtained at
different energies in this measurement and the data were fitted with the
Linhard model. The best-fit value of the parameter ‘$k=0.133$’ was obtained
Bonhomme  _et al._ (2022). We use the Linhard model with this value and derive
the quenching factor in terms of the ionization energy. We obtain the
following fit function for the quenching factor as a function of the
ionization energy

$Q(E_{I})=E_{I}\sum_{i=0}^{6}\left[\frac{a_{i}E_{I}^{i}}{b_{i}E_{I}^{i}}\right]\
\ \ \ (\rm Bonhomme\ \textit{et al}\ \ data)\,,$ (3)

where $a_{i}$ and $b_{i}$ are the evaluated fitting parameters. The numerical
values of the fit parameters are given in the 3rd and 4th columns of Table 1.
The resultant quenching factor as a function of $E_{I}$ is shown in Fig. 1 in
magenta.

Sarkis $et\ al$ Model: Another theoretical model, which is a refined form of
the Linhard model, is the Sarkis $et\ al$ Sarkis  _et al._ (2020) model. This
model also takes into account the internal energy of the system at the low
energy recoils. We use the numerical solution of the model for germanium as
given in ref. Sarkis  _et al._ (2020) and we derive the following quenching
factor as a function of $E_{I}$,

$Q(E_{I})=E_{I}\sum_{i=0}^{7}\left[\frac{a_{i}E_{I}^{i}}{b_{i}E_{I}^{i}}\right]\
\ \ \ (\rm Sarkis\ \textit{et al}\ \ Model)\,$ (4)

where $a_{i}$ and $b_{i}$ are the evaluated fit parameters and their numerical
values are given in the 5th and 6th columns of Table 1. The resultant
quenching factor as a function of $E_{I}$ is shown in Fig. 1 in green. There
is about 3% increase in the ionization efficiency in the Sarkis $et\ al$ model
compared with the Linhard model as clear from the figure.

YBe data: Another recent direct measurement of the quenching factor for
germanium is based on photo-neutron (YBe) data ref. Colaresi  _et al._ (2022)
which was taken for calibrating the NCC-1701 detector to detect the
CE$\overline{\nu}_{e}$NS of the Dresden-II reactor neutrinos. We calculate the
following fitting function for the YBe data

$Q(E_{I})=E_{I}\sum_{i=0}^{6}\left[\frac{a_{i}E_{I}^{i}}{b_{i}E_{I}^{i}}\right]\
\ \ \ (\rm YBe\ data)\,,$ (5)

where $a_{i}$ and $b_{i}$ are the evaluated fitting parameters. The numerical
values of the fit parameters are given in the 7th and 8th columns of Table 1.
The related quenching factor as a function of $E_{I}$ is shown in Fig. 1 in
purple. Note that above 0.3 keV, no real data of Fef was used, the fit was
extrapolated using the Linhard model expectations Colaresi  _et al._ (2022).

Jones $et\ al$ data: We also use the old measurement of the ionization
efficiency by Jones et al Jones and Kraner (1975) because of this overlap with
the ionization energy scale relevant for the nuclear recoils due to reactor
neutrinos. Since the interpolation of this data results in the multi-valued
quenching factor function, therefore we derive an approximate fit function for
this data. The best-fit quenching factor as a function of ionization was
obtained as in the following

$Q(E_{I})=\sum_{i=0}^{6}\left[\frac{E_{I}}{a_{i}E_{I}^{i}}\right]\ \ \ \ (\rm
Jones\ \ \textit{et al}\ \ data)\,,$ (6)

where $a_{i}$ are the evaluated fitting parameters. The numerical values of
the fit parameters are given in the 9th column of Table 1. The corresponding
quenching factor as a function of $E_{I}$ is shown in Fig. 1 in orange.

Fef data: Finally, we use the iron-filtered calibration data provided with the
data release file of ref. Colaresi  _et al._ (2022) and find the quenching
factor as a function of the ionization energy, $E_{I}$, for it. The obtained
fit function is given in the following

$Q(E_{I})=E_{I}\left[\frac{\sum_{i=0}^{8}a_{i}E_{I}^{i}}{\sum_{i=0}^{7}b_{i}E_{I}^{i}}\right]\
\ \ \ (\rm FeF\ data)\,,$ (7)

where $a_{i}$ and $b_{i}$ are the evaluated fit parameters. The numerical
values of the fit parameters are given in the 11th and 12th columns of Table
1. The corresponding quenching factor as a function of $E_{I}$ is shown in
Fig. 1 in red. There is about 15% increase relative to the Linhard model in
the range between the ionization threshold and 0.3 keV. Note that above 0.3
keV, no real data of Fef was used, the fit was extrapolated using the Linhard
model expectations Colaresi  _et al._ (2022).

$i$ | Linhard | Bonhomme | Sarkis | YBe | Jones | Fef
---|---|---|---|---|---|---
ai | bi | ai | bi | ai | bi | ai | bi | ai | bi | ai | bi
0 | 0.01143 | 0.00001 | 0.00069 | 0.00001 | 0.00005 | 0.00001 | -0.60425 | 11.84055 | 1.09756 | - | 0.03200 | -0.00793
1 | 2.52766 | 0.11801 | 0.15017 | 0.00700 | -0.0210 | -0.00050 | 55.9154 | -141.433 | -4.73588 | - | -0.70192 | 0.38355
2 | 60.7988 | 19.2150 | 3.54089 | 1.11711 | 4.90730 | 0.02390 | -668.480 | 780.109 | 38.0178 | - | 6.06352 | -6.54642
3 | 248.556 | 370.305 | 14.1941 | 21.1202 | 70.4665 | 36.0068 | 4017.06 | -1843.22 | -126.10 | - | -23.4161 | 55.5917
4 | 196.196 | 1246.77 | 10.9871 | 69.7824 | 84.2566 | 365.219 | -10316.7 | 2011.00 | 268.504 | - | 16.2113 | -253.096
5 | 25.3242 | 804.766 | 1.39028 | 44.2150 | 12.7504 | 344.445 | 12684.2 | -583.964 | -276.87 | - | 180.122 | 551.519
6 | 0.13558 | 78.2333 | 0.00725 | 4.22059 | 0.21177 | 39.3241 | -5674.99 | -213.231 | 104.875 | - | -590.858 | -99.4481
7 | - | - | - | - | 0.00008 | 0.39971 | - | - | - | - | 580.171 | -1779.01
8 | - | - | - | - | - | - | - | - | - | - | - | 2326.26

Table 1: Fit parameters for the three models. Each parameter has unit of the
inverse power of keV where the powers of ‘$\rm a_{i}$’ correspond to the index
‘$i+1$’ and powers of ‘$\rm b_{i}$’ correspond to ‘$i$’. Figure 1: The
quenching factor models and iron-filtered (Fef) data as a function of the true
ionization energy. The dashed vertical line corresponds to the Germanium based
NCC-1701 detector threshold of 0.2 keV. Figure 2: Observed event energy
spectrum versus predictions as a function of the true ionization energy for
three quenching factors considered here.

Now we discuss the coherent elastic anti-neutrino nucleus scattering in the SM
in addition to the form factor and introduce our notation.

##  4 Coherent elastic anti-neutrino nucleus scattering

At the tree level in SM, the differential cross-section of the reactor
electron anti-neutrino scattering off the spin-0 nucleus of Germanium
$\rm(^{72}Ge)$ with proton number ‘$\rm Z$’ and neutron number ‘$\rm N$’ is
given by Freedman (1974); Freedman  _et al._ (1977); Tubbs and Schramm (1975);
Lindner  _et al._ (2017),

$\frac{d\sigma_{{\overline{\nu}_{e}}\mathcal{N}}}{dE_{nr}}(E_{\overline{\nu}_{e}},E_{nr})=\frac{G_{F}^{2}M}{\pi}\left[Zg_{p}^{V}+Ng_{n}^{V})\right]^{2}\left(1-\frac{E_{nr}}{E_{\nu}}-\frac{ME_{nr}}{2E_{{\overline{\nu}_{e}}}^{2}}\right)F^{2}(q^{2})\,,$
(8)

where ‘$G_{F}$’ is the Fermi constant, ‘$E_{\overline{\nu}_{e}}$’ is the
energy of the incoming neutrinos, ‘$E_{nr}$’ is the nuclear recoil energy,
$q^{2}=-2ME_{nr}$ is the squared momentum transfer, ‘$M$’ is the mass of the
target nucleus. Here, $g_{p}^{V}=(2g_{u}^{V}+g_{d}^{V})$ and
$g_{n}^{V}=(g_{u}^{V}+2g_{d}^{V})$, where $g_{u}^{V}$ and $g_{d}^{V}$ are the
neutral current coupling constants for the ‘up’ and ‘down’ quarks which, in
terms of the weak mixing angle ‘$\theta_{W}$’ are given by

$\displaystyle g_{u}^{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{4}{3}\sin^{2}\theta_{W}\,,$ $\displaystyle
g_{d}^{V}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{2}{3}\sin^{2}\theta_{W}\,.$ (9)

To include all the radiative corrections, we will use $\rm
sin^{2}\theta_{W}=0.23857\pm 0.00005$, the low energy value evaluated in
$\overline{\rm MS}$ scheme Marciano and Sirlin (1980); Erler and Ramsey-Musolf
(2005); Zyla  _et al._ (2020). In eq. (8), $F(q^{2})$ is nuclear form factor,
where we use the Klein-Nystrand form factor Klein and Nystrand (2000) as given
in the following

$F(q^{2})=\frac{4\pi\rho_{0}}{Aq^{3}}[\sin(qR_{A})-qR_{A}\cos(qR_{A})]\left[\frac{1}{1+a^{2}q^{2}}\right],$
(10)

where $\rho_{0}$ is the normalized nuclear number density, $A$ is the atomic
number of $\rm{}^{72}Ge$, $R_{A}=1.2A^{1/3}\,\mathrm{fm}$ is the nuclear
radius, and $a=0.7\,\mathrm{fm}$ is the range of the Yukawa potential. At such
low energy recoils, the form factor effects are less than $3\%$.

##  5 Predicted spectrum and the statistical analysis

The Germanium based NCC-170 detector of total mass 3 kg receives electron
anti-neutrinos from Dresden-II boiling water reactor of thermal power of 2.96
$\rm GW_{th}$. The detector is located at distance $d=8$ m from the reactor
source. For this experimental setup, the total number of events of the nuclear
recoil in a given bin ‘$i$’ of the ionization energy reads,

$N^{i}=n.t.\int_{E_{I}^{\prime\rm i}}^{E_{I}^{\prime\rm
i+1}}dE_{I}^{\prime}\int_{0}^{E_{I}^{\rm
max}}dE_{I}\int_{E_{\overline{\nu}_{e}}^{\rm
min}}^{E_{\overline{\nu}_{e}}^{\rm
max}}dE_{\overline{\nu}_{e}}\frac{d\sigma_{{\overline{\nu}_{e}}\mathcal{N}}}{dE_{I}}(E_{\nu},E_{I})\frac{d\phi_{\overline{\nu}_{e}}({E_{\overline{\nu}_{e}}})}{dE_{\nu}}G(E_{I}^{\prime},E_{I})\mathcal{E}\\\
(E^{\prime}),$ (11)

where $n=2.43\times 10^{25}$ is the number of target nuclei corresponding to
the 2.924 kg fiducial mass of $\rm{}^{72}Ge$, t = 96.4 days is the data taking
time, $E_{I}$ denotes the actual ionization energy and $E_{I}^{\prime}$
denotes the visible ionization energy. The integration range of the
$E_{I}^{\prime}$ was taken 10 eV according to the observed spectrum. Here,
$E_{I}^{\rm max}=2E_{\overline{\nu}_{e}}^{2}/(2E_{\overline{\nu}_{e}}+M)$ is
maximum true ionization energy, $E_{\overline{\nu}_{e}}^{\rm
min}=E_{nr}+\sqrt{E_{nr}^{2}+2ME_{nr}}/2$ is the minimum anti-neutrino energy
that produces a nuclear recoil, $E_{nr}$, while the
$E_{\overline{\nu}_{e}}^{\rm max}$ is the endpoint of the reactor neutrino
spectrum, which we take as 9 MeV. Note that $E_{\overline{\nu}_{e}}^{\rm min}$
is expressed as a function of $E_{I}$ using eq. (1). Here,
$\frac{d\sigma_{{\overline{\nu}_{e}}\mathcal{N}}}{dE_{I}}$ is the differential
cross-section as a function of the true ionization energy that can be obtained
from eq. (1) and (8) using the derivative chain rule. The result can be
written as,

$\frac{d\sigma_{{\overline{\nu}_{e}}\mathcal{N}}}{dE_{I}}(E_{\nu},E_{I})=\left(\frac{Q(E_{I})-E_{I}\frac{dQ(E_{I})}{dE_{I}}}{Q^{2}(E_{I})}\right)\left(\frac{d\sigma_{{\overline{\nu}_{e}}\mathcal{N}}}{dE_{nr}}(E_{\nu},E_{nr})\Bigr{|}_{\begin{subarray}{c}E_{nr}=\frac{E_{I}}{Q(E_{I})}\end{subarray}}\right).$
(12)

The reactor anti-neutrino energy spectrum
($\frac{d\phi_{\overline{\nu}_{e}}}{{dE_{\nu}}}(E_{\nu})$) for energies above
$\sim$ 2 MeV is given by Kopeikin  _et al._ (2004),

$\frac{d\phi_{\overline{\nu}_{e}}}{{dE_{\nu}}}(E_{\nu})=\frac{n_{f}}{4\,\pi\,d^{2}}\left(\sum_{i}f_{i}\exp{\left[a_{0i}+\left(\sum_{j=1}^{5}a_{j}E_{\nu}^{j}\right)_{i}\right]}\right)\,\text{,}$
(13)

where $n_{f}=9\times 10^{19}$ fissions per second corresponding to 2.96 Giga-
watt thermal power and an average energy release of 205.25 MeV by all reactor
components, ‘$d$’ is the distance of the detector from the reactor core and
‘$i$’ sums over the fuel constituents
${{}^{235}\mathrm{U}},{{}^{238}\mathrm{U}}\
{{}^{239}\mathrm{Pu}},{{}^{241}\mathrm{Pu}}$. The coefficients, $f_{i}$, which
represent the fission rate of each component element and the coefficients of
the neutrino energy $E_{\nu}$ in the exponent (a0 and aj) were both taken from
Mueller  _et al._ (2011). The low energy part of the flux spectrum ($\lesssim$
2MeV) is mainly governed by the slow neutron capture by the
${{}^{238}\mathrm{U}}$. We use the numerical data for this part of the
spectrum, taken from ref. Wong  _et al._ (2007).

$G(T^{\prime},T)$ is the Gaussian distribution function which accounts for the
detector energy resolution as given in the following,

$G(E^{\prime}_{I},E_{I})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp{\left[-\frac{(E_{I}^{\prime}-E_{I})^{2}}{2\sigma^{2}}\right]}\,.$
(14)

Here, the Gaussian width ‘$\sigma$’ is given by
$\sigma=\sqrt{\sigma_{n}^{2}+E_{I}\eta F}$ where $\sigma_{n}=68.5$ eV is the
electronic noise, $\eta=2.96$ eV is the average energy of photons to create an
electron-hole pair in germanium, and $F=0.105$ is the Fano factor Colaresi
_et al._ (2022). Finally, $\mathcal{E}(E^{\prime})$ represents the signal
acceptance. The measured data has already been corrected for the signal
acceptance Colaresi  _et al._ (2022), therefore, we do not use it in
calculating our predicted spectrum. Note that we normalize the integrand of
the variable $E_{I}$ in eq. (11) with the integral of the Gaussian function of
eq. (14).

To fit our expected spectrum for $\sin^{2}{\theta_{W}}$ and electromagnetic
interaction parameters to the observed data, we use the following $\chi^{2}$
function,

$\chi^{2}=\underset{i=1}{\overset{50}{\sum}}\left(\frac{N_{\rm obs}^{i}-N_{\rm
exp}^{i}(1+\alpha)}{\sigma^{i}}\right)^{2}+\left(\frac{\alpha}{\sigma_{\alpha}}\right)^{2}\,,$
(15)

where $N_{\rm obs}^{i}$ is the background-subtracted observed events in the
$i-$th energy bin given in units of per 10 eV per 3 kg per day as shown in
Fig. 2, which was taken from Collar and Lewis (2022), $N_{\rm exp}^{i}$ is the
expected events in the corresponding energy bin, $\sigma_{i}$ is the
uncertainty in the data points as shown in error bars in Fig. 2, which include
the combination of the signal acceptance and statistical uncertainties. The
pull term in eq. (15) is added in to account for the theoretical
uncertainties, ‘$\alpha$’ is the pull parameter and
$\sigma_{\alpha}=\sqrt{\sigma_{f}^{2}+\sigma_{qf}^{2}}$ is the total
theoretical uncertainty, where $\sigma_{f}=5\%$ is reactor flux total
uncertainty and $\sigma_{qf}$ is the uncertainty in each QF considered here.
The average uncertainty in the Sarkis $et\ al$ model is $\sigma_{qf}=25\%$
uncertainty. We use the same value also for the Linhard model. On the other
hand, we use the average value for the uncertainty on Fef data, which is
$\sigma_{qf}=40\%$ as shown in Fig. (4) of ref. Colaresi  _et al._ (2022) and
likewise for the other quenching factors.

Next, we present and discuss our results. We will analyze the observed data
for the $\sin^{2}{\theta_{W}}$, neutrino millicharge, magnetic moment, charge
radius and neutrino anapole moment.

##  6 Weak mixing angle at keV nuclear recoils

With the statistical model introduced in the preceding section, we first
discuss the implications of the observed reactor neutrino coherent scattering
process for the weak mixing angle, namely $\sin^{2}{\theta_{W}}$, using the
three different quenching factors discussed in Sec. 3. We fit
$\sin^{2}{\theta_{W}}$ using the two theoretical models and the ion-filter
data for the QF. The results of the parameter fitting are shown in Fig. 3 in
the form of 1-dimensional $\Delta\chi^{2}$ distributions. The best-fit values
with 1 $\sigma$ uncertainties for the three cases are given in the following,

$\displaystyle\sin^{2}\theta_{W}=0.50^{+0.09}_{-0.12}$ $\displaystyle\ \ \ \
(\text{Lindhard Model}),\ $
$\displaystyle\sin^{2}\theta_{W}=0.48^{+0.14}_{-0.18}$ $\displaystyle\ \ \ \
(\text{Bonhomme {et al} data}),\ $
$\displaystyle\sin^{2}\theta_{W}=0.47^{+0.08}_{-0.11}$ $\displaystyle\ \ \ \
(\text{Sarkis $et\ al$ Model}),\ $
$\displaystyle\sin^{2}\theta_{W}=0.33^{+0.13}_{-0.18}$ $\displaystyle\ \ \ \
(\text{YBe data}),\ $ $\displaystyle\sin^{2}\theta_{W}=0.22^{+0.06}_{-0.11}$
$\displaystyle\ \ \ \ (\text{Jones data}),\ $
$\displaystyle\sin^{2}\theta_{W}=0.20^{+0.04}_{-0.05}$ $\displaystyle\ \ \ \
(\text{Fef data}).\ $ (16)

By comparing the three results, it is clear how sensitive the best-fit values
and the corresponding uncertainties are to QF. The theoretically predicted
value at the related energy scale in the $\overline{\rm MS}$ scheme is $\rm
sin^{2}\theta_{W}=0.23857$, Marciano and Sirlin (1980); Erler and Ramsey-
Musolf (2005); Zyla  _et al._ (2020). This gives, respectively, 52%, 49% and
14% discrepancy with the Linhard model, Sarkis $et\ al$ model and with the Fef
data. The rest accure in this range. However, notice that the percentage
change in the case of two models is increasing while in the case of Fef data
it is decreasing. This implies that the theoretical models for the QF
overestimate the true value of $\rm sin^{2}\theta_{W}$ while the Fef data
underestimate its value, therefore, the true value lies in between the two
types of extreme values of the fits. Notice that the Sarkis $et\ al$ model
improves the agreement by 3%.

This is the first determination of the $\rm sin^{2}\theta_{W}$ at the sub-keV
nuclear recoils using the CE$\overline{{\nu}_{e}}$NS data with the reactor
anti-neutrinos. Before it was determined using the COHERENT data Akimov  _et
al._ (2017b, 2018, 2021) with nuclear recoils above $\sim$ 5 keV Khan and
Rodejohann (2019); Cadeddu  _et al._ (2020a, b). The results obtained here
show that we need a better understanding for the quenching factor. More
importantly, the new data for the QF needs more precision. Also, its
consistency with theoretical modeling is essential.

Figure 3: $\Delta\chi^{2}$ distribution of $\sin^{2}{\theta_{W}}$ with
1$\sigma$ and $90\%$ C.L. projections using two theoretical models and the
iron-filtered data for the quenching factors. See text for discussion.

##  7 Electromagnetic interactions of neutrinos at keV nuclear recoils

###  7.1 Millicharge neutrinos

Electric charge quantization is assumed in the standard model, however,
several theories beyond the SM like those with magnetic monopoles Dirac
(1931), grand unified theories Georgi and Glashow (1974); Pati and Salam
(1974) and the extra dimension models Arkani-Hamed  _et al._ (2007) predict
the charge quantization. Other extensions predict new particles with
fractional charges Ignatiev  _et al._ (1979); Okun  _et al._ (1984); Holdom
(1986); Kors and Nath (2004); Batell and Gherghetta (2006), which can be
promising candidates for dark matter Goldberg and Hall (1986); Mohapatra and
Rothstein (1990); Kors and Nath (2005); Gies  _et al._ (2006); Cheung and Yuan
(2007); Feldman  _et al._ (2007); Berlin  _et al._ (2020); Berlin and Schutz
(2021). Among the SM particles, neutrinos are the most promising candidates
for such particles, often called milli-charged neutrinos Babu and Mohapatra
(1989, 1990); Foot  _et al._ (1990, 1993). The electric charge dequantization
is also related to the emergence of the gaugeable $U(1)$ symmetries,
$L_{e}-L_{\mu}$, $L_{\mu}-L_{\tau}$ and $L_{e}-L_{\tau}$ and $B-L$. Only one
of the three differences can be anomaly-free and the corresponding difference
is added to the hypercharge of the SM which leads to the fractional charges of
Dirac-type neutrinos Foot  _et al._ (1990, 1993); Babu and Mohapatra (1989).
In these simple extensions of the SM, the value of neutrino millicharges is
arbitrary and only experimental Davidson  _et al._ (1991); Babu  _et al._
(1994); Bressi  _et al._ (2011); Gninenko  _et al._ (2007); Chen  _et al._
(2014); Singh  _et al._ (2019); Khan and Rodejohann (2019); Cadeddu  _et al._
(2020a, b); Khan (2020) or observational Barbiellini and Cocconi (1987);
Raffelt (1999); Davidson  _et al._ (2000); Melchiorri  _et al._ (2007);
Studenikin and Tokarev (2014) limits are available.

The contribution of the neutrino millicharges to the SM weak interaction
process of the coherent neutrino-nucleus ($\overline{\nu}-N$) scattering is
parameterized in terms of $Q_{\overline{\nu}_{\alpha}}$ and the interaction
term is given by

$\mathcal{L}_{\alpha}^{em}=-ie\left(Q_{\overline{\nu}_{\alpha}}{\overline{\nu}_{\alpha}}\gamma_{\mu}{\nu}_{\alpha}+{\overline{N}}\gamma_{\mu}{N}\right)A^{\mu},$
(17)

where $A^{\mu}$ is mediating electromagnetic field and ‘$e$’ is the unit
electric charge. In principle, neutrinos with intrinsic electric charge should
be negatively charged while the antineutrinos should be positively charged,
however, since this is a direct way of probing the property, we, therefore,
make no distinction in the sign of neutrino and anti-neutrinos. Thus, we add
its contribution to the SM interactions. Further, the electromagnetic
interactions due to the intrinsic electric charge of neutrinos add up
coherently to the vector part of the SM weak interaction, therefore, its
effect on reactor anti-neutrinos can be included through the weak mixing angle
in eq. (9) accordingly as,

$\displaystyle\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}\left(1-\frac{\pi\alpha_{em}}{\sqrt{2}\sin^{2}\theta_{W}G_{F}ME_{nr}}Q_{{\overline{\nu}_{e}}}\right),$
(18)

where $\alpha_{em}$ is the fine structure constant.

Figure 4: $\Delta\chi^{2}$ distributions of neutrino millicharge (NMC) with
1$\sigma$ and $90\%$ C.L. projections using two theoretical models and the
iron-filtered data for the quenching factors. See text for details.

Notice that for the CE$\overline{\nu}_{e}$NS with reactor anti-neutrinos there
is only one parameter involved, namely, $Q_{\overline{\nu}_{e}}$. We determine
the constraints on neutrino millicharge by fitting, $Q_{\overline{\nu}_{e}}$
using the two theoretical models for the QF and the Fef data. The results of
the parameter fitting are shown in Fig. 4 in the form of 1-dimensional
$\Delta\chi^{2}$ distributions. At 90% C.L., we obtain the following
constraints,

$\displaystyle-2.43\times 10^{-8}<Q_{\overline{\nu}_{e}}/e<5.73\times
10^{-8}\,\ \ \ \ \ \ \ \ \ (\text{Lindhard Model})\ $
$\displaystyle-2.26\times 10^{-8}<Q_{\overline{\nu}_{e}}/e<5.45\times
10^{-8}\,\ \ (\text{Bonhomme {et al} data})\ $ $\displaystyle-2.11\times
10^{-8}<Q_{\overline{\nu}_{e}}/e<5.32\times 10^{-8}\,\ \ \ \ \ \ (\text{Sarkis
{et al} Model})\ $ $\displaystyle-1.36\times
10^{-8}<Q_{\overline{\nu}_{e}}/e<4.46\times 10^{-8}\,\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ (\text{YBe data})\ $ $\displaystyle-0.09\times
10^{-8}<Q_{\overline{\nu}_{e}}/e<4.15\times 10^{-8}\,\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ (\text{Jones data})\ $ $\displaystyle-0.16\times
10^{-8}<Q_{\overline{\nu}_{e}}/e<0.66\times 10^{-8}\,\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ (\text{Fef data})\ $ (19)

In the case of Linhard and Sarkis et al models, the bounds are comparable to
those obtained before from the coherent scattering using COHERENT data Khan
(2022) while from the Fef QF data they are two orders of magnitude stronger
than the previous bounds Khan (2022).

It is important to note that for the three quenching factors considered here,
there is a mild preference for the non-zero neutrino millicharges as shown in
Fig. 4. This preference can be attributed to the interference between the SM
and the neutrino millicharge term and its dependence on the inverse double
power of the nuclear recoil energy and on the target mass in the cross-section
(see eq. (18) in combination to eq. (8)). This is unlike the other
electromagnetic interactions. These aspects of the coherent scattering process
have been discussed in detail in ref. Khan (2022).

Stronger limits on millicharge neutrinos are available from the observational
studies Barbiellini and Cocconi (1987); Raffelt (1999); Davidson  _et al._
(2000); Melchiorri  _et al._ (2007); Studenikin and Tokarev (2014). The
strongest available upper limit is $Q_{\nu}\leq 2\times 10^{-15}e$ which comes
from the time arrival dispersion and the energy spread of neutrinos from
SN1987A Barbiellini and Cocconi (1987). The laboratory bounds from the $\nu-e$
are also several orders of magnitude from the limits derived here Davidson
_et al._ (1991); Babu  _et al._ (1994); Bressi  _et al._ (2011); Gninenko  _et
al._ (2007); Chen  _et al._ (2014); Singh  _et al._ (2019); Khan and
Rodejohann (2019); Cadeddu  _et al._ (2020a, b); Khan (2020). For instance,
the TEXONO experiment derives the limit, $Q_{\nu}\leq 2.1\times 10^{-12}e$.
However, this difference can be easily understood from the kinematical
considerations Khan (2022). It was shown in ref. Khan (2022) that robustness
of the bounds or preference of non-zero neutrino millicharges totally depends
on the experimental precision.

###  7.2 Neutrino magnetic moment

The neutrino magnetic moment in the coherent neutrino-nucleus scattering
process has been studied before and constraints with the COHERENT data were
derived before in ref. Khan and Rodejohann (2019); Cadeddu  _et al._ (2020a).
In the general coupling for Majorana ($M$) or Dirac ($D$) neutrinos to the
electromagnetic field strength ($F^{\mu\nu}$), the neutrino magnetic moment
interaction term can be written as (Fujikawa and Shrock, 1980; Shrock, 1982;
Vogel and Engel, 1989; Abak and Aydin, 1989; Grimus and Stockinger, 1998)

${\cal L}^{M}=-\frac{1}{4}\bar{\nu}_{\alpha
L}^{c}\,\lambda_{\alpha\beta}^{M}\,\sigma_{\mu\nu}\,\nu_{\beta
L}\,F^{\mu\nu}~{}\mbox{ or }~{}{\cal L}^{D}=-\frac{1}{2}\bar{\nu}_{\alpha
R}\,\lambda_{\alpha\beta}^{D}\,\sigma_{\mu\nu}\,\nu_{\beta L}\,F^{\mu\nu}\,,$
(20)

where $\lambda^{X}=\mu^{X}-i\epsilon^{X}$, which is hermitian for the Dirac
neutrinos and antisymmetric for Majorana neutrinos. For Majorana neutrinos,
only transition magnetic moments are possible while the flavor diagonal is
zero. Here, we consider the flavor diagonal neutrino magnetic moment of
reactor electron antineutrino ($\mu_{\overline{\nu}_{e}}$). Again, there is
only one parameter involved. The SM prediction of the non-zero neutrino
magnetic moment at a loop level can be quantified as below, (Fujikawa and
Shrock, 1980; Vogel and Engel, 1989)

$\mu_{\overline{\nu}_{e}}=\frac{3eG_{F}\rm
m_{\overline{\nu}_{e}}}{8\sqrt{2}\pi^{2}}\sim 3\times
10^{-19}\mu_{B}\left(\frac{\rm m_{\overline{\nu}_{e}}}{1\rm eV}\right).$ (21)

As clear from eq. (20), for Dirac neutrinos, the helicity of the final state
neutrino changes in interaction due to its magnetic moment, therefore, no
interference with the SM weak interaction can occur. The corresponding
contribution adds to the SM weak cross-section incoherently. We can write down
the differential cross-section for the neutrino magnetic moment (MM) of
electron anti-neutrinos scattering off a spin-0 nucleus of ${}^{72}\rm Ge$
with proton number (Z) as in the following Khan and Rodejohann (2019),

$\frac{d\sigma_{{\overline{\nu}_{e}}\mathcal{N}}^{MM}}{dE_{nr}}(E_{\overline{\nu}_{e}},E_{nr})=\left(\frac{\pi\alpha_{em}^{2}\,\mu_{\overline{\nu}_{e}}^{2}}{m_{e}^{2}}\right)\text{
}\left(\frac{1}{E_{nr}}-\frac{1}{E_{\overline{\nu}_{e}}}+\frac{E_{nr}}{4E_{\overline{\nu}_{e}}^{2}}\right)Z^{2}F^{2}(q^{2}),$
(22)

where $\mu_{\overline{\nu}_{e}}$ is expressed in units of Bohr’s magneton
($\mu_{B}$) and $m_{e}$ is the electron mass. One can notice that in
comparison to the millicharge neutrinos, as given in eq. (18) in combination
with eq. (8), the neutrino magnetic moment has no interference with the SM
weak interaction and the dependence on the inverse power of the nuclear recoil
is only linear in the leading term. Notice that eq. (22) can be written in
terms of the ionization energy $E_{I}$ in a similar fashion as described for
the SM cross-section around eq. (12).

Using eq. (22) in terms of the ionization energy in addition to eq. (12) and
our $\chi^{2}$ function defined in eq. (15), we fit $\mu_{\overline{\nu}_{e}}$
to the data and derive the constraints according to the three QFs. The results
are shown in Fig. 5 in the form $\Delta\chi^{2}$ profile. For guidance, we
also show the 1$\sigma$ and 90% C.L. projections in the figure. We obtain the
following bounds at 90% C.L.,

$\displaystyle-0.25\times 10^{-8}<\mu_{\overline{\nu}_{e}}/\mu_{B}<0.25\times
10^{-8}\,\ \ \ \ \ \ \ \ \ \ (\text{Lindhard Model})\ $
$\displaystyle-0.24\times 10^{-8}<\mu_{\overline{\nu}_{e}}/\mu_{B}<0.24\times
10^{-8}\,\ \ (\text{Bonhomme {et al} data})\ $ $\displaystyle-0.23\times
10^{-8}<\mu_{\overline{\nu}_{e}}/\mu_{B}<0.23\times 10^{-8}\,\ \ \ \ \ \
(\text{Sarkis {et al} Model})\ $ $\displaystyle-0.12\times
10^{-8}<\mu_{\overline{\nu}_{e}}/\mu_{B}<0.12\times 10^{-8}\,\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (\text{YBe data})\ $ $\displaystyle-0.07\times
10^{-8}<\mu_{\overline{\nu}_{e}}/\mu_{B}<0.07\times 10^{-8}\,\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ (\text{Jones data})\ $ $\displaystyle-0.06\times
10^{-8}<\mu_{\overline{\nu}_{e}}/\mu_{B}<0.06\times 10^{-8}\,\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ (\text{Fef data})\ $ (23)

In the case of Linhard and Sarkis et al models, the bounds are comparable to
those obtained before from the coherent scattering using COHERENT data Khan
(2022) while from the Fef QF data they are a factor of seven stronger than the
previous bounds Khan (2022).

Figure 5: $\Delta\chi^{2}$ distributions of neutrino magnetic moment (NMM)
with 1$\sigma$ and $90\%$ C.L. projections using two theoretical models and
the iron-filtered data for the quenching factors. See text for details.

###  7.3 Neutrino charge radius

In the SM, the neutrino charge radius for neutrinos is induced by radiative
corrections. Its relevance to CE$\nu$NS has been discussed before in ref.
Papavassiliou  _et al._ (2006). For the general effective electromagnetic
vertex of massive neutrinos, $\bar{\nu}\Lambda_{\mu}\nu A^{\mu}$, the neutrino
charge radius term is written as Bernabeu  _et al._ (2000, 2002, 2004);
Fujikawa and Shrock (2004),

$\Lambda_{\mu}(q)=\gamma_{\mu}F_{\nu}(q^{2})\simeq\gamma_{\mu}q^{2}\frac{\langle
r^{2}\rangle}{6}\,,$ (24)

where $q$ is the momentum transfer and $F_{\nu}(q^{2})$ is a form factor that
is related to the neutrino charge radius $\langle r_{\nu}^{2}\rangle$ via

$\langle
r_{\nu}^{2}\rangle=6\left.\frac{dF_{\nu}(q^{2})}{dq^{2}}\right|_{q^{2}=0}\,.$
(25)

Notice that choice for the sign in the definition of the charge radius in eq.
(25) is completely conventional. Here, we consider positive signs. The SM
prediction of the neutrino charge radius, therefore, is Bernabeu  _et al._
(2000, 2002, 2004); Fujikawa and Shrock (2004); Novales-Sanchez  _et al._
(2013); Cadeddu:2Sarkis018dux,

$\langle r_{\nu_{\alpha}}^{2}\rangle_{\rm
SM}=-\frac{G_{F}}{2\sqrt{2}\pi}\left[3-2\ln\left(\frac{m_{\alpha}^{2}}{m_{W}^{2}}\right)\right],$
(26)

where $m_{\alpha}$ is the mass of the charged lepton associated to
$\nu_{\alpha}$ and $m_{W}$ is the mass of the $W^{\pm}$ boson. The numerical
values for the electronic flavor in the SM therefore is Bernabeu  _et al._
(2000, 2002, 2004); Novales-Sanchez  _et al._ (2013); Cadeddu  _et al._
(2018b)

$\displaystyle\langle r_{\overline{\nu}_{e}}^{2}\rangle_{\rm SM}=-0.83\times
10^{-32}\ \ \rm cm^{2}.$ (27)

Like the neutrino millicharges, the neutrino charge radii contribute
coherently to the SM process Papavassiliou  _et al._ (2006) and its effect on
the CE$\overline{\nu}$NS process can be added to the weak mixing angle by
making the following replacement in eq. (9),

$\displaystyle\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}\left(1+\frac{\pi\alpha_{em}}{3\sqrt{2}\sin^{2}\theta_{W}G_{F}}\langle
r_{\nu_{\alpha}}^{2}\rangle\right).$ (28)

We note that, unlike the millicharge neutrinos in eq. (18), the charge radius
does not have dependence on the inverse power of the recoil energy and on the
target mass. Therefore, a weaker sensitivity compared to the millicharge
neutrinos is expected. This was also noted before in refs Khan (2020, 2019)
and recently for analysis with the COHERENT data Khan (2022). Here, the only
parameter that contribute is the $\langle r_{\overline{\nu}_{e}}^{2}\rangle$
and we fit this parameter using the observed data of the sub-keV nuclear
recoils. The obtained results for the three QFs are shown in Fig. 6 and the
constraints obtained at 90% C.L. are the following,

$\displaystyle-0.85\times 10^{-30}<\langle
r^{2}_{\overline{\nu}_{e}}\rangle/\rm cm^{2}<0.35\times 10^{-30}\,\ \ \ \ \ \
\ \ \ \ (\text{Lindhard Model})\ $ $\displaystyle-0.82\times 10^{-30}<\langle
r^{2}_{\overline{\nu}_{e}}\rangle/\rm cm^{2}<0.32\times 10^{-30}\,\ \
(\text{Bonhomme {et al} data})\ $ $\displaystyle-0.80\times 10^{-30}<\langle
r^{2}_{\overline{\nu}_{e}}\rangle/\rm cm^{2}<0.30\times 10^{-30}\,\ \ \ \ \ \
(\text{Sarkis {et al} Model})\ $ $\displaystyle-0.70\times 10^{-30}<\langle
r^{2}_{\overline{\nu}_{e}}\rangle/\rm cm^{2}<0.18\times 10^{-30}\,\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ (\text{YBe data})\ $ $\displaystyle-0.55\times
10^{-30}<\langle r^{2}_{\overline{\nu}_{e}}\rangle/\rm cm^{2}<0.04\times
10^{-30}\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{Jones data})\ $
$\displaystyle-0.50\times 10^{-30}<\langle
r^{2}_{\overline{\nu}_{e}}\rangle/\rm cm^{2}<0.03\times 10^{-30}\,\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ (\text{Fef data})\ $ (29)

In the case of Linhard and Sarkis et al models, the bounds are comparable to
those obtained before from the coherent scattering using COHERENT data Khan
(2022) while from the Fef QF data are a factor of five stronger than the
previous bounds Khan (2022).

Figure 6: $\Delta\chi^{2}$ distributions of neutrino charge radius (NCR) with
1$\sigma$ and $90\%$ C.L. projections using two theoretical models and the
iron-filtered data for the quenching factors. See text for details.

###  7.4 Neutrino anapole moment

If neutrino carries a non-zero charge radius, it can also have a non-zero
anapole moment which is induced from the gauge-invariant parity-odd axial
current Zel’dovich (1958); Zel’dovich and Perelomov (1960); Barroso  _et al._
(1985); Abak and Aydin (1987); Musolf and Holstein (1991); Dubovik and
Kuznetsov (1998); Rosado (2000); Novales-Sanchez  _et al._ (2013). Physically,
it determines the correlation between the spin and charge distributions of
neutrinos has the same dimensions as that of the charge radius. The anapole
term from the general vertex for electromagnetic interactions,
$\bar{\nu}\Lambda_{\mu}\nu A^{\mu}$, is defined by Barroso  _et al._ (1985);
Abak and Aydin (1987); Rosado (2000)

$\Lambda_{\mu}(q)=-\gamma_{\mu}\gamma_{5}F(q^{2})\simeq-\gamma_{\mu}\gamma_{5}q^{2}\textit{{a}}_{\nu}\,,$
(30)

where the form factor ‘$F(q^{2})$’ is related to the neutrino anapole moment
‘$\textit{{a}}_{\nu\alpha}$’ by the expression,

$\textit{{a}}_{\nu}=-\left.\frac{dF_{\nu}(q^{2})}{dq^{2}}\right|_{q^{2}=0}\,.$
(31)

By comparing with eq. (25), the SM prediction for the neutrino anapole moment
can be written in terms of the neutrino charge radius as in the following
Barroso  _et al._ (1985); Abak and Aydin (1987); Rosado (2000); Novales-
Sanchez  _et al._ (2013); Cadeddu  _et al._ (2018b),

$\textit{{a}}_{\nu_{\rm SM}}=-\frac{\langle r_{\nu}^{2}\rangle_{\rm SM}}{6},$
(32)

and numerical value for the electronic flavor accordingly is given by

$\displaystyle\textit{{a}}_{\overline{\nu}_{e\rm SM}}=4.98\times 10^{-32}\ \
\rm cm^{2}.$

For the CE$\overline{\nu}_{e}$NS with reactor anti-neutrinos, the anapole
moment contribution is added to the SM cross-section by replacing the
$\sin^{2}{\theta_{W}}$ with the effective weak mixing angle in eq. (9) in the
following way,

$\displaystyle\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}\left(1-\frac{\pi\alpha_{em}}{18\sqrt{2}\sin^{2}\theta_{W}G_{F}}\textit{{a}}_{\nu_{\alpha}}\right).$
(33)

Again, unlike neutrino millicharges, the anapole moment does not have a direct
dependence on the inverse nuclear recoil energy and on the target mass which
makes it relatively less sensitive at lower recoil. Now, we fit the parameter
$\textit{{a}}_{\overline{\nu}_{e\rm}}$ with the observed data for the three
QFs. The results obtained from this fitting analysis are shown in Fig. 7 while
the parameter bounds at 90% C.L. are given in the following

$\displaystyle-2.10\times 10^{-30}<\textit{{a}}_{\overline{\nu}_{e}}/\rm
cm^{2}<5.10\times 10^{-30}\,\ \ \ \ \ \ \ \ \ \ (\text{Lindhard Model})\ $
$\displaystyle-1.90\times 10^{-30}<\textit{{a}}_{\overline{\nu}_{e}}/\rm
cm^{2}<4.90\times 10^{-30}\,\ \ (\text{Bonhomme {et al} data})\ $
$\displaystyle-1.38\times 10^{-30}<\textit{{a}}_{\overline{\nu}_{e}}/\rm
cm^{2}<4.80\times 10^{-30}\,\ \ \ \ \ \ (\text{Sarkis {et al} Model})\ $
$\displaystyle-1.15\times 10^{-30}<\textit{{a}}_{\overline{\nu}_{e}}/\rm
cm^{2}<4.13\times 10^{-30}\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{YBe
data})\ $ $\displaystyle-0.28\times
10^{-30}<\textit{{a}}_{\overline{\nu}_{e}}/\rm cm^{2}<3.26\times 10^{-30}\,\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{Jones data})\ $ $\displaystyle-0.14\times
10^{-30}<\textit{{a}}_{\overline{\nu}_{e}}/\rm cm^{2}<3.15\times 10^{-30}\,\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{Fef data})\ $ (34)

Again, Linhard and Sarkis et al models give comparable bounds to those
obtained before from the coherent scattering using COHERENT data Khan (2022)
while from the Fef data they are a factor of five stronger than the previous
bounds Khan (2022).

Figure 7: $\Delta\chi^{2}$ distributions of neutrino anapole moment (NAM) with
1$\sigma$ and $90\%$ C.L. projections using two theoretical models and the
iron-filtered data for the quenching factors. See text for details.

##  8 Summary and Conclusions

We have analyzed data from the recently reported observation of the coherent
elastic antineutrino- nucleus scattering process with Dresden-II boiling water
nuclear reactor neutrinos using the germanium detector. The maximum nuclear
recoils produced with reactor antineutrinos can be about 2 keV or below. At
such low energy recoils, there is unprecedentedly large uncertainty in the
quenching factor. This leads to a large fluctuation in the theoretical
prediction of the event energy spectrum in the standard model. Any new physics
searches depend on the accuracy of the quenching factor. In this regard, we
used several theoretical models and data on the quenching factor from various
experiments. In the main part of this work, we have shown how different
choices of the quenching factor alter the standard model predictions or any
new physics sensitivities.

It was shown that the precise knowledge of the quenching factor is of central
importance for the observation of the true signal CE$\overline{\nu}_{e}$NS in
the SM and for any new physics sensitivity. We have included a complete list
of the analytical functions of the quenching factor and used them for
estimating the value of the weak mixing angle and the neutrino electromagnetic
interactions. It is important that these analytical functions could be used
for any other new physics models with CE$\overline{\nu}_{e}$NS from the
reactor antineutrinos. It should be noted that there also exist other
experimental data for the quenching factor of germanium element, but for
nuclear recoils due to the reactor energies, the quenching factors used here
are the only relevant ones. We have shown that the Linhard, Sirkis $et\ al$
model and Bonhomme $et\ al$ are in strong tension with the Jones, iron-filter
and photo-neutron data in the regime of nuclear recoils below about 2 keV and
a more realistic model is needed. It was consistently shown that the iron-
filtered data over-estimate the theoretical predictions of the event energy
spectrum while the Linhard model underestimates the predictions which leads to
the large disagreement for the value of weak mixing angle and on the bounds of
the neutrino electromagnetic parameters.

Using the two theoretical models, namely the Linhard model Lindhard  _et al._
(1963) and the Sarkis $et\ al$ model Sarkis  _et al._ (2020), and four
available data sets for the quenching factor, an older one by Jones $et\ al$
Jones and Kraner (1975) and the three recent ones with iron-filtered Colaresi
_et al._ (2022) and photo-neutrons and by Bonhomme $et\ al$ Bonhomme  _et al._
(2022), we have derived the weak mixing angle and constraints on the possible
electromagnetic interactions of neutrinos using the observed data. We have
shown how the best fits and the uncertainties on the weak mixing angle and
bounds on neutrino millicharge, neutrino magnetic moment, neutrino charge
radius and neutrino anapole moment depend on the choice of the quenching
factor. The iron-filtered data gives stronger constraints on all parameters
comparable to the two theoretical models. Our results for the weak mixing
angle and neutrino magnetic moment are with iron-filtered data and photo-
neutron data are in agreement with those obtained in ref. Coloma  _et al._
(2022); Liao  _et al._ (2022); Aristizabal Sierra  _et al._ (2022) while the
constraints on the other electromagnetic interactions obtained here are the
first ones.

In general, our study provides the first direct limits on electromagnetic
properties of neutrinos in addition to the weak mixing angle using the keV
scale nuclear recoils due to the reactor antineutrinos. The constraints on the
neutrino millicharges are one to two orders of magnitude stronger than the
previous bounds from the coherent scattering process using the COHERENT data,
while other bounds are comparable to the previous ones. As discussed before
these variations are due to the lack of precise knowledge of the quenching
factor for pure elements like germanium at low nuclear recoils. We note that
neutrino millicharges are more sensitive to the choice of the quenching factor
because of the interference of its amplitude with the standard model
interactions and its special dependence on the inverse power of the nuclear
recoil energy and on the target nuclear mass. We conclude that the precise
understanding of the quenching factor is essential for the new physics
searches with such low-energy nuclear recoils due to the
CE$\overline{\nu}_{e}$NS.

###### Acknowledgements.

I am thankful to professor Juan Collar for providing me with all the necessary
data and information about the experiment. I am also thankful to Dr. Y. Sarkis
and Dr. B. Cervantes for useful discussions. This work is financially
supported by the Alexander von Humboldt foundation.

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