# Physics implications of recent Dresden-II reactor data

Anirban Majumdar<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science Education and Research - Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 462066, India  Dimitrios K. Papoulias
<EMAIL_ADDRESS>Department of Physics, National and Kapodistrian
University of Athens, Zografou Campus GR-15772 Athens, Greece  Rahul
Srivastava<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Science Education and Research - Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 462066, India  José W. F. Valle
<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular – CSIC/Universitat
de València
C/Catedrático José Beltrán, 2 E-46980 Paterna, Spain

###### Abstract

Prompted by the recent Dresden-II reactor data we examine its implications for
the determination of the weak mixing angle, paying attention to the effect of
the quenching function. We also determine the resulting constraints on the
unitarity of the neutrino mixing matrix, as well as on the most general type
of nonstandard neutral-current neutrino interactions.

##  1 Introduction

The neutral-current observation of neutrino events opens new windows that may
reveal novel aspects of neutrino physics. Coherent elastic neutrino nucleus
scattering (CE$\nu$NS) is a low-energy process, proposed by Freedman [1],
where the entire nucleus is scattered off elastically by the neutrinos.
Despite the large predicted CE$\nu$NS cross section, its detection faces
experimental challenges mainly because of the very tiny nuclear recoil signals
produced in the aftermath of a CE$\nu$NS event. CE$\nu$NS was first observed
at the Oak Ridge National Laboratory by the COHERENT Collaboration which
exploited the $\pi$-DAR (pion-decay-at-rest) neutrino beam coming from the
Spallation Neutron Source facility (SNS) using a CsI detector [2, 3] and
subsequently using a liquid argon detector [4]. Nuclear power plants
constitute another important experimental probe for CE$\nu$NS studies, with
several experiments currently underway such as CONNIE [5], CONUS [6, 7] and
$\nu$GEN [8] or in preparation (e.g. MINER [9], RICOCHET [10], $\nu$-cleus
[11], TEXONO [12], $\nu$IOLETA [13] and Scintillating Bubble Chamber (SBC)
[14]). These facilities have the advantage of producing an extremely intense
beam of low-energy antineutrinos ($E_{\nu}<10$ MeV). Thus, because of the very
low energy-momentum transfer involved, the expected CE$\nu$NS signal at
reactors does not suffer from nuclear physics uncertainties, in contrast to
$\pi$-DAR based experiments [15, 16].

On the other hand, reactor experiments are subject to large neutron
backgrounds, which together with the large quenching factor (QF) uncertainties
are the main limiting factors for CE$\nu$NS detection at reactor facilities.

Very recently, the Dresden-II Collaboration [17, 18] has reported a suggestive
piece of evidence pointing to the first ever observation of CE$\nu$NS with
reactor antineutrinos, using a 3 kg germanium detector, namely NCC-1701
(Neutrino Coherent Coupling-$1701$). The experiment is located $10.39$ meters
away from the Dresden-II boiling water reactor and collected beam-on data for
a total time of $96.4$ days, operating with a very low threshold of $0.2\text{
keV}_{ee}$. The new Dresden-II data have prompted phenomenological analyses
that resulted in complementary constraints on various parameters within and
beyond the Standard Model (SM). The relevant works focused on the
determination of the weak mixing angle [19, 20, 21] as well as on popular SM
extensions such as those involving electromagnetic neutrino interactions [19,
20, 21] and light mediators [19, 22, 23]. Nonstandard interactions (NSIs) were
also explored in Refs. [23, 24]. They are a sideshow of the physics associated
with low-scale neutrino mass generation [25]. Indeed, neutrino oscillation
experiments have provided robust evidence for massive neutrinos, prompting
many efforts for underpinning the origin of their masses. A remarkable example
is the low-scale seesaw mechanism, such as the inverse [26, 27] and the linear
seesaw [28, 29, 30]. In both of these, we have massless neutrinos in the limit
of lepton-number conservation. They lead to small, symmetry-protected neutrino
masses, mediated by heavy quasi-Dirac neutrino exchange. These schemes can be
tested directly at high energies [31, 32, 33, 34, 35], a possibility taken up
by the ATLAS and CMS collaborations at the LHC [36, 37, 38], or in future
proposals—see, e.g., Refs. [39, 40]. Here we focus on their associated signals
at low-energy precision studies of neutrino properties. An example is
unitarity violation in neutrino mixing, as expected in a low-scale seesaw
mechanism.

Building upon previous work [19], in this paper we revisit the impact of
Dresden-II data on the determination of the weak mixing angle. In so doing, we
adopt an improved analysis taking into account the full Dresden-II data [18].
Concerning new physics scenarios, we first explore unitarity violation in the
lepton mixing matrix, a characteristic feature of low-scale seesaw mechanisms.
Moreover, we consider neutrino generalized interactions (NGI) with heavy
mediators by taking into consideration two different quenching factor models
namely, the photoneutron (YBe) and iron-filter (Fef) QF reported by the
Dresden-II Collaboration.

The remainder of the paper is organized as follows: In Sec. 2 we provide the
necessary theoretical framework regarding CE$\nu$NS processes. We begin by
discussing CE$\nu$NS within the SM followed by a brief review of the unitarity
violation and NGIs. Section 3 contains the details of our Dresden-II simulated
signal and the statistical analysis we have adopted as well as discussion of
our results. We finally summarize our concluding remarks in Sec. 4.

##  2 Theoretical Framework

In this section we discuss the CE$\nu$NS differential cross section within the
SM framework. We also describe the necessary modifications for incorporating
unitarity violation effects in active neutrino mixing, NSIs or the most
general NGIs. For the latter cases, we only consider the effective new physics
interactions resulting from integrating out the heavy mediators.

###  2.1 CE$\nu$NS within the SM

The tree-level SM differential CE$\nu$NS cross section with respect to the
nuclear recoil energy $E_{nr}$ is expressed as111It should be emphasized that
Eq. (1) holds for adequately low momentum transfer such that the coherence
criterion $\mathfrak{q}\leq 1/R_{A}$ [41] holds, where $R_{A}$ is the nuclear
radius and $\mathfrak{q}^{2}=2m_{N}E_{nr}$ is the three-momentum transfer.
[42]

$\left.\frac{d\sigma}{dE_{nr}}\right|_{\text{SM}}=\frac{G_{F}^{2}m_{N}}{\pi}Q_{W}^{2}\left(1-\frac{m_{N}E_{nr}}{2E_{\nu}^{2}}-\frac{E_{nr}}{E_{\nu}}\right)\,,$
(1)

where $G_{F}$ represents the Fermi constant and $m_{N}$ the mass of the target
nucleus. Here the SM vector weak charge $Q_{W}$ is explicitly written as [43],

$Q_{W}=g_{V}^{p}ZF_{Z}(\mathfrak{q}^{2})+g_{V}^{n}NF_{N}(\mathfrak{q}^{2})\,,$
(2)

where $Z$ and $N$ denote the number of protons and neutrons of the target
nucleus, while $g_{V}^{p}~{}=1/2-2\sin^{2}{\theta_{W}}$ and $g_{V}^{n}=-1/2$
denote the corresponding SM tree-level vector couplings for protons and
neutrons, with $\theta_{W}$ representing the weak mixing angle. We have
verified that the radiative corrections employed in [21] lead to less than 1%
difference in the expected event rates. Equation (2) also incorporates the
nuclear proton and neutron form factors $F_{Z}(\mathfrak{q}^{2})$ and
$F_{N}(\mathfrak{q}^{2})$, which are the Fourier transforms of the
corresponding nuclear charge density distribution and reflect the loss of
coherence for sizable momentum transfer [44, 45]. The nuclear form factors are
essential in the analysis of COHERENT data which utilized $\pi$-DAR neutrino
beams. Note, however, that for the recoil energy region of interest (ROI) at
the Dresden-II experiment $F_{Z}(\mathfrak{q}^{2})$ and
$F_{N}(\mathfrak{q}^{2})$ are roughly equal to unity.

#### Implications of weak mixing angle

The weak mixing angle is an important input of the Salam-Weinberg electroweak
theory [46]. The theoretically estimated value of the weak mixing angle at the
$Z$ pole is
$\left.\sin^{2}\theta_{W}\right|_{\overline{\text{MS}}}(\mu=m_{Z^{0}})=0.23121\pm
0.00004$. For low energies, relevant to the CE$\nu$NS process, i.e.
$\mathfrak{q}\approx 0$, its value has been obtained by RGE extrapolation in
the minimal subtraction ($\overline{\text{MS}}$) renormalization scheme:
$\sin^{2}\theta_{W}(\mathfrak{q}=0)=0.23857\pm 0.00005$ [47, 48]. A precise
measurement of this parameter at low energies is an important test for the
validation of the SM. Although many efforts have been conducted so far [47],
measurements of the weak mixing angle at low energy still come up with
significant uncertainties. CE$\nu$NS is a low-energy process that can be used
for the determination of the weak mixing angle in the low-energy regime.
Indeed, Ref. [49] pointed out that the uncertainty in the measurement of
$\sin^{2}\theta_{W}$ obtained from the combined fit of the full COHERENT-CsI +
APV 222APV stands for atomic parity violation in cesium atoms; it is measured
in Refs. [50, 51].  datasets is still rather large compared to other
determinations of this parameter at low momentum transfer. Note that a
limiting factor behind the poor COHERENT sensitivity on the weak mixing angle
relates to the additional nuclear physics uncertainties involved in the
CE$\nu$NS signal [49]. On the other hand, being free from nuclear physics
uncertainties, the Dresden-II reactor data offer a new opportunity for an
improved determination of the weak mixing angle.

###  2.2 Violation of lepton unitarity

Neutrinos in the SM are massless and have no isosinglet “right-handed”
partners. A simple way to give them mass is the (high-scale) seesaw mechanism
in which one postulates the existence of a right-handed neutrino associated
with each left-handed one, following in a conventional “(3,3)” structure.
Given that right-handed partners are gauge singlets one can add any number of
them [52, 53]. The low-scale seesaw mechanisms assume a common “(3,6)” seesaw
template: instead of having a single heavy isosinglet right-handed neutrino
associated with each family, it contains two heavy singlets in each family,
making up three sequential Dirac leptons, in the limit of lepton-number
conservation. These schemes include both the inverse [26, 27] and the linear
seesaw mechanism [28, 29, 30]. In both of them, a massless neutrino setup is
the common template for building a genuine low-scale seesaw mechanism, in
which the small symmetry-protected neutrino masses are mediated by heavy
quasi-Dirac neutrino exchange. These schemes can be probed directly at high
energies [31, 32, 33, 34, 35], as in the ATLAS and CMS collaborations at the
LHC [36, 37, 38], or in future neutral heavy lepton search proposals—see,
e.g., Refs. [39, 40].

Low-scale seesaw schemes imply signals at low-energy precision studies of
neutrino properties, such as unitarity violation in neutrino mixing. Indeed, a
characteristic feature of these models is, of course, the existence of new
neutral leptons. The latter are assumed to be heavy and participate in weak
interaction processes only through mixing with the SM (active) neutrinos. The
admixture of heavy leptons in the charged current implies that the weak
interaction mixing matrix needs to be rectangular [52], leading to effective
violation of lepton unitarity in many processes.

Here we consider the possibility of constraining the nonunitarity (NU)
parameters in the light of the Dresden-II reactor data. The generalized
charged-current weak interaction mixing matrix characterizing unitarity
violation can be written in the form [54]

$N=N^{\text{NP}}U^{3\times 3}\,,$ (3)

where $U^{3\times 3}$ is the standard $3\times 3$ lepton mixing matrix, while
$N^{\text{NP}}$ denotes the new physics matrix describing unitarity violation.
The matrix $N^{\text{NP}}$ can be conveniently parametrized as follows [54]

$N^{\text{NP}}\equiv\begin{pmatrix}\alpha_{11}&0&0\\\
\alpha_{12}&\alpha_{22}&0\\\
\alpha_{13}&\alpha_{23}&\alpha_{33}\end{pmatrix}\,,$ (4)

where the diagonal entries ($\alpha_{ii}$) are real, while the off-diagonal
ones ($\alpha_{ij}$) are small but in general complex.

Within this context, the initial neutrino flux generated at the source,
undergoes oscillation in propagation and reaches the detector as a modified
flux according to [55]

$\begin{pmatrix}\frac{d\Phi^{\text{NU}}_{\nu_{e}}}{dE_{\nu}}+\frac{d\Phi^{\text{NU}}_{\bar{\nu}_{e}}}{dE_{\nu}}\\\
\frac{d\Phi^{\text{NU}}_{\nu_{\mu}}}{dE_{\nu}}+\frac{d\Phi^{\text{NU}}_{\bar{\nu}_{\mu}}}{dE_{\nu}}\\\
\frac{d\Phi^{\text{NU}}_{\nu_{\tau}}}{dE_{\nu}}+\frac{d\Phi^{\text{NU}}_{\bar{\nu}_{\tau}}}{dE_{\nu}}\\\
\end{pmatrix}=\begin{pmatrix}P_{ee}&P_{e\mu}&P_{e\tau}\\\
P_{e\mu}&P_{\mu\mu}&P_{\mu\tau}\\\
P_{e\tau}&P_{\mu\tau}&P_{\tau\tau}\end{pmatrix}\begin{pmatrix}\frac{d\Phi_{\nu_{e}}}{dE_{\nu}}+\frac{d\Phi_{\bar{\nu}_{e}}}{dE_{\nu}}\\\
\frac{d\Phi_{\nu_{\mu}}}{dE_{\nu}}+\frac{d\Phi_{\bar{\nu}_{\mu}}}{dE_{\nu}}\\\
\frac{d\Phi_{\nu_{\tau}}}{dE_{\nu}}+\frac{d\Phi_{\bar{\nu}_{\tau}}}{dE_{\nu}}\\\
\end{pmatrix}\,,$ (5)

where $P_{\alpha\beta}$ represents the probability of transition from $\alpha$
to $\beta$ neutrino or antineutrino flavor
($P_{\alpha\beta}=P[\nu_{\alpha}\to\nu_{\beta}]=P[\bar{\nu}_{\alpha}\to\bar{\nu}_{\beta}]$)
and satisfies the condition $P_{\alpha\beta}=P_{\beta\alpha}$. The oscillation
probabilities in the presence of NU are expressed as [54]

$\displaystyle P_{ee}=\alpha_{11}^{4}P_{ee}^{3\times 3}\,,$ (6) $\displaystyle
P_{e\mu}=\left(\alpha_{11}\alpha_{22}\right)^{2}P_{e\mu}^{3\times
3}+\alpha_{11}^{2}\alpha_{22}\left|\alpha_{21}\right|P_{e\mu}^{I}+\alpha_{11}^{2}\left|\alpha_{12}\right|^{2}\,,$
$\displaystyle
P_{e\tau}=\left(\alpha_{11}\alpha_{33}\right)^{2}P_{e\tau}^{3\times
3}+\alpha_{11}^{2}\alpha_{33}\left|\alpha_{31}\right|P_{e\tau}^{I}+\alpha_{11}^{2}\left|\alpha_{13}\right|^{2}\,.$

where $P^{3\times 3}_{ee}$ is the standard survival probability, and
$P^{3\times 3}_{e\mu}$ and $P^{3\times 3}_{e\tau}$ are the standard transition
probabilities, while $P_{e\mu}^{I}$ and $P_{e\tau}^{I}$ contain CP-violating
terms as explained in Ref. [54]. For very short baseline experiments such as
Dresden-II, the standard neutrino oscillation probability can be well
approximated as $P_{\alpha\beta}^{3\times 3}(L\approx
0)=\delta_{\alpha\beta}$, with $\delta_{\alpha\beta}$ denoting the Kronecker
delta (zero-distance effect). It follows that the expected number of events
can be cast in the simple form

$R_{\text{NU}}=\alpha_{11}^{2}\left(\alpha_{11}^{2}+|\alpha_{12}|^{2}+|\alpha_{13}|^{2}\right)R_{\text{SM}}\,,$
(7)

where $R_{\text{SM}}$ represents the unoscillated number of CE$\nu$NS events
in the SM. Hence, for the case of reactor $\bar{\nu}_{e}$, the only relevant
NU parameters affecting the propagation of neutrinos are $a_{11}$,
$|\alpha_{12}|$, and $|\alpha_{13}|$. Note that, in contrast to oscillation
experiments, the expected neutrino signal in short-baseline reactor neutrino
experiments is directly proportional to the NU parameters, independently of
CP-violating terms and standard oscillation phenomena. To our knowledge, up to
now, only projected NU sensitivities for the CE$\nu$NS channel exist in the
literature—e.g., using future data expected at COHERENT [55] or at the planned
SBC experiment [56]. In the present work, we instead consider the actual data
from the Dresden-II experiment.

###  2.3 Neutrino nonstandard interactions

Neutrino nonstandard interactions (or NSIs, for short) [57, 58] arise in most
neutrino mass generation schemes, and their expected magnitude can be sizeable
in the case of low-scale models [25]. Prominent theories of neutrino mass
generation borrow the chiral structure of the Standard Model and, as a result,
imply a restricted Lorentz structure for the resulting NSIs. These have been
extensively studied in various contexts using Wilson coefficients of
nonrenormalizable dimension-6 operators [59, 60, 61, 62]. NSIs have been
previously searched for using the COHERENT data [2, 3, 4] (see Refs. [63, 64,
65, 66, 67, 68, 69, 70]) and the CONUS data [7]. Studies have also been
performed simulating future data expected at the European Spallation Source
[71] and at the SBC [56].

Because of the basic gauge structure NSIs are restricted to be vector or axial
vector-type interactions (see below the most general case of the NGIs). The
relevant nonstandard-interaction neutral-current Lagrangian reads [43]

$\mathscr{L}^{\text{NSI}}_{\text{eff}}\;=\;-2\sqrt{2}\,G_{F}\,\varepsilon_{\alpha\beta}^{qC}\,\bigl{(}\overline{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta}\bigr{)}\bigl{(}\overline{q}\gamma_{\mu}P_{C}q\bigr{)}\;,$
(8)

where $\alpha,\beta=e,\mu,\tau$ is the neutrino flavor, $q=u,d$ stands for the
quark fields, and $C=L,R$ denotes the chirality projections. The strength of
the new interaction is taken with respect to the Fermi constant, through the
NSI parameters $\varepsilon_{\alpha\beta}^{qC}$, which can be either flavor
preserving ($\alpha=\beta$) or flavor changing ($\alpha\neq\beta$). The
former, flavor-preserving NSIs, are usually referred to as nonuniversal. It is
often convenient to express the latter in vector and axial-vector form as
$\varepsilon_{\alpha\beta}^{qV}\equiv\varepsilon_{\alpha\beta}^{qR}+\varepsilon_{\alpha\beta}^{qL}$
and
$\varepsilon_{\alpha\beta}^{qA}\equiv\varepsilon_{\alpha\beta}^{qR}-\varepsilon_{\alpha\beta}^{qL}$,
respectively.

For the case of CE$\nu$NS on intermediate or heavy target nuclei, axial-vector
NSIs are usually ignored, since they are expected to contribute
negligibly333Axial-vector CE$\nu$NS interactions are suppressed by a factor
$\sim 1/A$ (see Ref. [43]). with respect to their vector counterpart. As a
result, for an incoming (anti)neutrino of flavor $\alpha$, the NSI CE$\nu$NS
cross section is given from Eq.(1) with the substitution $Q_{W}\to
Q_{\text{NSI}}^{V}$, where the NSI charge reads

$\displaystyle Q_{\text{NSI}}^{V}$
$\displaystyle=\left(g_{V}^{p}+2\varepsilon_{\alpha\alpha}^{uV}+\varepsilon_{\alpha\alpha}^{dV}\right)ZF_{Z}(\mathfrak{q}^{2})+\left(g_{V}^{n}+\varepsilon_{\alpha\alpha}^{uV}+2\varepsilon_{\alpha\alpha}^{dV}\right)NF_{N}(\mathfrak{q}^{2})$
(9)
$\displaystyle+\sum\limits_{\beta\neq\alpha}\left(2\varepsilon_{\alpha\beta}^{uV}+\varepsilon_{\alpha\beta}^{dV}\right)ZF_{Z}(\mathfrak{q}^{2})+\left(\varepsilon_{\alpha\beta}^{uV}+2\varepsilon_{\alpha\beta}^{dV}\right)NF_{N}(\mathfrak{q^{2}})\,,$

with $g_{V}^{p}$ and $g_{V}^{n}$ denoting the corresponding SM vector
couplings for protons and neutrons [see Eq.(2)].

###  2.4 Neutrino generalized interactions

Generalized neutrino interactions provide a model-independent framework for a
variety of new physics scenarios. They incorporate all possible Lorentz-
invariant interactions up to dimension $6$ between neutrinos and other SM
fermions—i.e., scalar, pseudoscalar, vector, axial vector, and tensor
[72]—hence, extending the conventional NSIs discussed above.

Effective field theories (EFTs) resulting from new physics [73] provide a
suitable framework for describing NGIs in a model-independent manner below the
electroweak scale. They can be described through the four-fermion effective
interaction Lagrangian [74]

$\mathscr{L}^{\text{NGI}}_{\text{eff}}=\frac{G_{F}}{\sqrt{2}}\sum_{X=S,P,V,A,T}\left[\bar{\nu}\Gamma^{X}\nu\right]\left[\bar{q}\Gamma_{X}\left(C_{X}^{q}+i\gamma_{5}D_{X}^{q}\right)q\right]\,,$
(10)

with
$\Gamma_{X}=\left\\{\mathbb{1},\gamma_{5},\gamma_{\mu},\gamma_{\mu}\gamma_{5},\sigma_{\mu\nu}\right\\}$
corresponding to $X=\\{S,P,V,A,T\\}$ interactions.

The relative strength of the new physics interaction $X$ is specified by the
real dimensionless coefficients $C_{X}^{q}$ and $D_{X}^{q}$ which are of the
order $(\sqrt{2}/G_{F})(g_{X}^{\nu}g_{X}^{q}/m_{X}^{2})$. Here, $m_{X}$
denotes the mediator mass $(\mathfrak{q}\ll m_{X})$, and $g_{X}^{\nu}$
($g_{X}^{q}$) denotes the coupling between the mediator and neutrino (quark).
Note that we have neglected pseudoscalar and axial vector interactions, since
they are expected to be suppressed by the nuclear spin; we choose
$D_{X}^{q}=0$.

The transition from quark-level to nuclear-level interactions is achieved by
assuming that the nucleonic matrix element of the quark current at
$\mathfrak{q}=0$ is proportional to that of the corresponding nucleon
current—i.e.,
$\left<N_{f}|\bar{q}\Gamma^{X}q|N_{i}\right>=\mathcal{F}^{X}\left<N_{f}|\bar{N}\Gamma^{X}N|N_{i}\right>$
($\mathcal{F}^{X}$ is the hadronic form factor) as argued in Refs. [74, 75,
76]. The neutrino-nucleus effective Lagrangian reads [74]

$\mathscr{L}_{\nu-N}\sim\sum_{X=S,V,T}C_{X}\left[\bar{\nu}\Gamma^{X}\nu\right]\left[\bar{N}\Gamma_{X}N\right]\,,$
(11)

where the neutrino-nucleus effective couplings $C_{X}$ are expressed in terms
of $C_{X}^{q}$ as [74]

$\displaystyle\begin{aligned}
C_{S}=&Z\,F_{Z}(\mathfrak{q})\,\sum_{q}C_{S}^{q}\frac{m_{p}}{m_{q}}f^{p}_{q}+N\,F_{N}(\mathfrak{q})\sum_{q}C_{S}^{q}\frac{m_{n}}{m_{q}}f^{n}_{q}\,,\\\
C_{V}=&Z\,F_{Z}(\mathfrak{q})\left(2C_{V}^{u}+C_{V}^{d}\right)+N\,F_{N}(\mathfrak{q})\left(C_{V}^{u}+2C_{V}^{d}\right)\,,\\\
C_{T}=&Z\,F_{Z}(\mathfrak{q})\,\sum_{q}C_{T}^{q}\delta^{p}_{q}+N\,F_{N}(\mathfrak{q})\,\sum_{q}C^{q}_{T}\delta^{n}_{q}\,.\end{aligned}$
(12)

The hadronic form factors for the scalar case ($S$) are $f_{u}^{p}=0.0208$,
$f_{u}^{n}=0.0189$, $f_{d}^{p}=0.0411$, and $f_{d}^{n}=0.0451$ [77], and those
for the tensor case ($T$) are $\delta^{p}_{u}=\delta^{n}_{d}=0.54$ and
$\delta^{p}_{d}=\delta^{n}_{u}=-0.23$ [77].

Neglecting the higher-order terms of $\mathcal{O}(E_{nr}^{2}/E_{\nu}^{2})$ and
taking into account the simultaneous presence of all these interactions the
differential cross section corresponding to the Lagrangian density in Eq.(11)
takes the form [74]

$\displaystyle\left.\frac{d\sigma}{dE_{nr}}\right|_{\text{NGI}}=\frac{G_{F}^{2}m_{N}}{\pi}\Bigl{\\{}$
$\displaystyle
C_{S}^{2}\frac{m_{N}E_{nr}}{8E_{\nu}^{2}}+\left(\frac{C_{V}}{2}+Q_{W}\right)^{2}\left(1-\frac{m_{N}E_{nr}}{2E_{\nu}^{2}}-\frac{E_{nr}}{E_{\nu}}\right)$
(13)
$\displaystyle+2C_{T}^{2}\left(1-\frac{m_{N}E_{nr}}{4E_{\nu}^{2}}-\frac{E_{nr}}{E_{\nu}}\right)\pm\mathcal{R}\frac{E_{nr}}{E_{\nu}}\Bigr{\\}}\,.$

In the above expression, one notices the interference between the SM and
vector NGIs, as well as the interference between the scalar and tensor terms
given as $\mathcal{R}=C_{S}C_{T}/2$. The plus (minus) sign accounts for
coherent elastic antineutrino (neutrino) scattering off nuclei (see also Ref.
[78]). Notice that only flavor-diagonal NGI couplings have been considered in
Eq.(13), while nondiagonal NGI coefficients are neglected. It is worth
mentioning also that the vector NSI contributions discussed in Sec. 2.3 are
accounted for in the vector part of the generic NGI cross section, since
$C_{V}^{u}\equiv 2\varepsilon^{uV}_{\alpha\alpha}$ and $C_{V}^{d}\equiv
2\varepsilon^{dV}_{\alpha\alpha}$.

NGIs have been previously searched for through CE$\nu$NS using the COHERENT
CsI measurement data [74], while projected sensitivities at CONUS [72] and the
planned SBC experiment have also been reported [78]. Here, for the first time
we explore the NGIs parameter space using actual reactor data, from the
Dresden-II experiment. Notice also the complementarity of our present work
with Ref. [19], which analyzed the Dresden-II data assuming light mediators.

##  3 Results

In this section we briefly review the specifications of the Dresden-II
experiment [17, 18], and provide the technical details required to
successfully simulate the CE$\nu$NS and background signals expected at the
NCC-1701 detector. We also describe the procedure of the statistical analysis
adopted in this work and discuss our results.

###  3.1 Simulation of the Dresden-II signal

In its current configuration, the Dresden-II reactor experiment used a $2.924$
kg ultralow-noise p-type point-contact germanium detector. It was exposed to
the intense electron antineutrino flux coming from the Dresden-II boiling
water reactor. The experiment has collected data for 96.4 days (Rx-ON) during
which the reactor operated with its full nominal power of $P=2.96$ GW. The
antineutrino energy flux used here is taken from Appendix A of Ref. [79],

$\frac{d\Phi_{\bar{\nu}_{e}}}{dE_{\nu}}=\frac{P}{4\pi
d^{2}\epsilon}\,\sum_{i}\frac{dN^{i}_{\bar{\nu}_{e}}}{dE_{\nu}}\,,$ (14)

where $\epsilon=205.24$ MeV is the average energy released per fission and the
index $i$ runs over all the fissible isotopes $\mathrm{{}^{235}U}$,
$\mathrm{{}^{238}U}$, $\mathrm{{}^{239}Pu}$, $\mathrm{{}^{241}Pu}$ and
$\mathrm{{}^{238}U(n,\gamma)^{239}U}$. Given the baseline of the experiment
$d=10.39$ m, the neutrino flux normalization reaching the NCC-1702 detector
can be estimated to be $\mathscr{N}=4.8\times 10^{13}/\text{cm}^{2}/\text{s}$.

The theoretical CE$\nu$NS events in the $i$th bin can be estimated as444The
nuclear recoil energy spectra has a lower cutoff ($E_{nr}^{\text{min}}$)
corresponding to electron equivalent ionization energy, $E_{er,\text{
min}}^{\text{true}}\simeq 2.98\text{ eV}_{ee}$, necessary to produce a
electron-hole pair in Ge at $77$ K [80, 81]. [19]

$\left[R_{i}\right]_{\xi}^{\text{CE}\nu\text{NS}}=t_{\text{run}}N_{\text{target}}\int_{E_{er}^{i}}^{E_{er}^{i+1}}dE_{er}^{\text{reco}}\int_{E_{nr}^{\text{min}}}^{E_{nr}^{\text{max}}}dE_{nr}\,G(E_{er}^{\text{reco}},E_{er}^{\text{true}})\int_{E_{\nu}^{\text{min}}}^{E_{\nu}^{\text{max}}}dE_{\nu}\frac{d\Phi_{\bar{\nu}_{e}}}{dE_{\nu}}\left.\frac{d\sigma}{dE_{nr}}\right|_{\xi}\,,$
(15)

where $\xi=\\{\text{SM, NU, NGI}\\}$ denotes the type of CE$\nu$NS
interaction555Note that, by switching off the scalar ($C_{S}$) and tensor
($C_{T}$) terms in Eq.(13), the NGI cross section reduces to that for the
conventional NSI CE$\nu$NS case. From here on, we give the NSI results as a
particular case of the NGI ones.. Here, $t_{\text{run}}$ represents the
exposure time, and $N_{\text{target}}=m_{\text{det}}N_{A}/m_{\text{Ge}}$
denotes the number of target nuclei. $N_{A}$ is the Avogadro number,
$m_{\text{Ge}}$ the molar mass of the Ge isotope, and $m_{\text{det}}=2.924$
kg the mass of NCC-1701. We consider all the stable isotopes of Ge with mass
number $\\{70,72,73,74,76\\}$ along with their corresponding relative
abundances $\\{0.2057,0.2745,0.0775,0.3650,0.0773\\}$ taken from [82]. In Eq.
(15), $E_{\nu}^{\text{max}}$ ($E_{\nu}^{\text{min}}$) denotes the maximum
(minimum) electron antineutrino energy, while the kinematics of the process is
used to determine the $E_{\nu}^{\text{min}}$ required to produce a nuclear
recoil with energy $E_{nr}$, as

$E_{\nu}^{\text{min}}=\frac{1}{2}\left[E_{nr}+\sqrt{E_{nr}^{2}+2m_{N}E_{nr}}\right]\approx\sqrt{\frac{m_{N}E_{nr}}{2}}\,.$
(16)

The “electron equivalent” ionization energy, $E_{er}^{\text{true}}$, is
related to the nuclear recoil energy, $E_{nr}$, through the QF which
quantifies the fraction of energy loss to heat, as
$E_{er}^{\text{true}}(E_{nr})=\mathcal{Q}_{f}(E_{nr})\cdot E_{nr}$. For the
latter, we take into consideration the two QF models described in the Dresden-
II data release [18]. The first one (YBe) was obtained from observations of
photoneutron sources [83, 84], while the second one (Fef) was determined from
iron-filtered monochromatic neutrons. A truncated Gaussian distribution is
used to define the detector’s energy-resolution function as [23]

$G(E_{er}^{\text{reco}},E_{er}^{\text{true}})=\left(\frac{2}{1+\text{Erf}\left(\frac{E_{er}^{\text{true}}}{\sqrt{2}\sigma_{e}}\right)}\right)\frac{1}{\sqrt{2\pi}\sigma_{e}}e^{-{\left(\frac{{E_{er}^{\text{reco}}-E_{er}^{\text{true}}}}{\sqrt{2}\sigma_{e}}\right)}^{2}}\,,$
(17)

with $E_{er}^{\text{reco}}$ being the measured (or “reconstructed”) ionization
energy. The energy resolution power of the detector has been determined to be
$\sigma_{e}(E_{er}^{\text{true}})=\sqrt{\sigma_{n}^{2}+\eta\,F_{f}\,E_{er}^{\text{true}}}$.
Here, $\sigma_{n}=68.5~{}\mathrm{eV_{ee}}$ is the intrinsic electronic noise
of the detector (from Rx-ON data), and $\eta=2.96~{}\mathrm{eV_{ee}}$ denotes
the average energy for electron-hole formation in Ge, while $F_{f}=0.105$ is
the average value of the germanium Fano factor [18].

In the ROI of Dresden-II: $0.2~{}\text{keV}_{ee}\leq E_{er}^{\text{reco}}\leq
1.5\text{ keV}_{ee}$, the background model given in Ref. [18] depends on four
individual components, coming from epithermal neutrons and three electron
capture (EC) peaks and can be fully characterized by seven free parameters
(see below). Explicitly, the background model can be described as a constant
term, a decaying exponential, and three Gaussian probability density functions
(each depending on three free parameters: amplitude, centroid, and standard
deviation) [18]:

$R_{\text{bkg}}(\mathbf{\upbeta})=N_{\text{epith}}+A_{\text{epith}}\,e^{-\left(\frac{E_{er}^{\text{reco}}-E_{\text{epith}}}{\tau_{\text{epith}}}\right)}+\sum_{i=L_{1},L_{2},M}A_{i}\,e^{-{\left(\frac{E_{er}^{\text{reco}}-E_{i}}{\sqrt{2}\sigma_{i}}\right)}^{2}}\,.$
(18)

The elastic scattering of epithermal neutrons contributes at the same recoil
energy region where the CE$\nu$NS signal amounts to a significant number of
events. This background component comprises a free constant term
$N_{\text{epith}}$ plus a free decaying exponential function with amplitude
$A_{\text{epith}}$, centroid $E_{\text{epith}}=0.2~{}\mathrm{keV_{ee}}$, and a
decay constant $\tau_{\text{epith}}$. The remaining background contribution is
due to the EC peaks in ${}^{71}\text{Ge}$, namely the $L_{1}$-, $L_{2}$-, and
$M$-shell peaks. The $L_{1}$-shell peak depends on its own amplitude,
centroid, and standard deviation $(A_{L_{1}}$, $E_{L_{1}}$, $\sigma_{L_{1}}$),
while the $L_{2}$\- and $M$-shell peaks are expressed in terms of the
$L_{1}$-shell peak parameters as given in Table 1. The ratio of the number of
counts under the $M$\- and $L_{1}$-shell peaks, $\beta_{M/L_{1}}$, has been
experimentally determined to be $0.16\pm 0.03$ [85, 86]; thus, the amplitude
of the $M$-shell can be expressed as
$A_{M}=\beta_{M/L_{1}}\times\frac{\sigma_{L_{1}}}{\sigma_{M}}\times
A_{L_{1}}$, while its centroid is fixed to the nominal value
$E_{M}=0.158~{}\mathrm{keV_{ee}}$ as it can not be determined by the data.
Finally its uncertainty is equal to the electronic noise
$\sigma_{M}\equiv\sigma_{n}=68.5~{}\mathrm{eV_{ee}}$ [18]. For the $L_{2}$
peak, the theoretical amplitude $A_{L_{2}}=0.008\times A_{L_{1}}$ and the
centroid $E_{L_{2}}=1.142~{}\mathrm{keV_{ee}}$ are used, while its width is
taken to be equal to that of the $L_{1}$-shell, i.e.,
$\sigma_{L_{2}}=\sigma_{L_{1}}$ [18]. In what follows, each analysis requires
fitting of the seven free background parameters, which are represented by the
vector
$\mathbf{\upbeta}=\\{N_{\text{epith}},A_{\text{epith}},\tau_{\text{epith}},A_{L_{1}},E_{L_{1}},\sigma_{L_{1}},\beta_{M/L_{1}}\\}$.
More details on the background treatment are given in the Appendix A.

Parameters |  $L_{2}$-shell |  $M$-shell
---|---|---
Amplitude ($A_{i}$) | $0.008\times A_{L_{1}}$ | $\beta_{M/L_{1}}\times\frac{\sigma_{L_{1}}}{\sigma_{M}}\times A_{L_{1}}$
Centroid ($E_{i}$) | $1.142~{}\mathrm{keV_{ee}}$ | $0.158~{}\mathrm{keV_{ee}}$
Standard deviation ($\sigma_{i}$) | $\sigma_{L_{1}}$ | $68.5~{}\mathrm{eV_{ee}}$
Table 1: Parameter details of $L_{2}$\- and $M$-shell peaks. $\beta_{M/L_{1}}$
is the ratio of the number of counts under $M$\- and $L_{1}$-shell peaks (see
the main text and Ref. [18] for details).

Figure 1:  Estimated number of events within the SM and various new physics
scenarios considered here, for given benchmark values for the new physics
parameters. We also show a comparison of the Rx-ON data reported by the
Dresden-II Collaboration. In each panel, the low-lying spectra with thin lines
show the predicted signal from “CE$\nu$NS only,” while those with thick lines
give the estimated “CE$\nu$NS + fitted background.” Different colors
correspond to two QF models. Figure 2: Dresden-II sensitivity on the weak
mixing angle, assuming the Fef and YBe QF models. Figure 3: RGE running of the
weak mixing angle and experimental measurements across different energies. The
inset plot shows our current result obtained from the analysis of Dresden-II
data for the different QF models. A comparison with the COHERENT data [3] is
also shown.

###  3.2 Statistical analysis

The statistical analysis in our present work is based on the Gaussian
$\chi^{2}$ function [23]

$\chi^{2}(\overrightarrow{S},a,\upbeta)=\sum_{i=1}^{130}{\left[\frac{(1+a)R^{i}_{\text{CE$\nu$NS}}(\overrightarrow{S})+R^{i}_{\text{bkg}}(\mathbf{\upbeta})-R^{i}_{\text{exp}}}{\sigma^{i}_{\text{exp}}}\right]}^{2}+{\left(\frac{a}{\sigma_{a}}\right)}^{2}+{\left(\frac{\beta_{M/L_{1}}-0.16}{\sigma_{\beta_{M/L_{1}}}}\right)}^{2}\,,$
(19)

where $R^{i}_{\text{CE$\nu$NS}}$ stands for the theoretically estimated
CE$\nu$NS events in the $i$th bin, and $\overrightarrow{S}$ denotes the set of
new physics parameters, while $R^{i}_{\text{exp}}$ and
$\sigma^{i}_{\text{exp}}$ are the experimental number of events and the
corresponding uncertainty in the $i$th bin, all taken from data release [18].
The neutrino flux normalization uncertainty is taken into consideration
through the nuisance parameter $a$ with $\sigma_{a}=2$%. Following Ref. [18],
the uncertainty of $\beta_{M/L_{1}}$ is taken to be
$\sigma_{\beta_{M/L_{1}}}=0.03$, and the prior $0.16$ is assigned. In what
follows, for a given parameter of interest from the set $\overrightarrow{S}$,
our analysis involves minimization of the $\chi^{2}$ function over each
component of $\upbeta$ and $a$.

As a first step, we simulate the event spectra expected at Dresden-II, a
procedure which has also served as a validation of our calculation. Figure 1
depicts the efficiency-corrected event rates within and beyond the SM for the
different new physics scenarios of interest here. In each panel, we show both
the “CE$\nu$NS only” signal and “CE$\nu$NS + fitted background” signal, while
the event rates are zoomed in on the recoil energy range
$[0.2,0.5]~{}\text{keV}_{ee}$, where CE$\nu$NS has a significant contribution
(for the full spectra in the ROI, see Fig. 7 in the Appendix A). Assuming that
the experimental data are populated by backgrounds only, we find the best-fit
value $\chi^{2}_{\text{min}}=110.69/\text{d.o.f.}$ (d.o.f. stands for the
total number of degrees of freedom). On the other hand, by including the
predicted SM CE$\nu$NS signal in our analysis, we find the best-fit values
$\chi^{2}_{\text{min}}=\\{102.66,105.95\\}/\text{d.o.f.}$ for {Fef, YBe}
QF—i.e., a clear improvement with respect to the background-only hypothesis.

Based on the $\chi^{2}$ function given in Eq. (19) and the Dresden-II data
release, we proceed by presenting our results—namely, the parameter
determination sensitivities within and beyond the SM for the various physics
scenarios described in Sec. 2.

We begin by discussing our new restrictions imposed on the SM weak mixing
angle. The corresponding $\Delta\chi^{2}$ profiles of $\sin^{2}{\theta_{W}}$
for the Fef and YBe QFs are shown in Fig. 2. The extracted 1$\sigma$
determinations of the weak mixing angle for our two QFs read

$\begin{split}\text{Photoneutron QF:}&\text{ }\text{
}\sin^{2}{\theta_{W}}=0.321^{+0.075}_{-0.094}\,,\\\ \text{Iron filter
QF:}&\text{ }\text{
}\sin^{2}{\theta_{W}}=0.236^{+0.048}_{-0.058}\,.\end{split}$

One sees that there is a strong dependence on the QF model used. Moreover, our
results are in good agreement with those of Ref. [21], though our best-fit
point lies closer to the RGE prediction, especially for the case of Fef QF. We
can also compare our results with those obtained in Refs. [19, 20], based on
the background-subtracted data in Ref. [18]. Exploiting the full Dresden-II
data in the present analysis, we find a better agreement with the RGE
prediction, but with a higher uncertainty compared to the latter studies. In
Fig. 3, we compare our results extracted from the analysis of Dresden-II data
with determinations from other probes across a wide range of energies. Let us
finally emphasize that even though the Dresden-II data constrain the weak
mixing angle with lower significance than the measurement reported by the
COHERENT Collaboration [3], it is interesting to notice the complementarity
between the two measurements. Clearly, the Dresden-II data fill a gap in the
regime of low momentum transfer—e.g., within $10^{-2}-10^{-1}~{}\mathrm{GeV}$.

We now examine the possibility of using the Dresden-II data to extract
information on unitarity-violation effects in the neutrino mixing matrix in
Eqs. (3) and (4). In Fig. 4 we present the $\Delta\chi^{2}$ profiles of
$\alpha_{11}$ and $|\alpha_{12}|$ or $|\alpha_{13}|$, by varying one parameter
at a time. As can be seen from Eq.(7), the expected number of events has the
same dependence on the off-diagonal NU parameters $|\alpha_{12}|$ and
$|\alpha_{13}|$. As a result, they can be constrained by the Dresden-II data
with exactly the same sensitivity. We present our results for the two QF
models, Fef and YBe. As expected, the Fef-driven results are more
constraining, since the pure SM event rate evaluated with the YBe QF is more
suppressed, compared to the Fef spectrum. For the case of YBe QF, the
predicted event rate contains much fewer events than the spectrum calculated
with the Fef QF (see Fig. 1), and hence the NU parameters $|\alpha_{21}|$,
$|\alpha_{31}|$ would reach values larger than 1 in order to describe the
Dresden-II data, i.e. they would become unphysical.

The 90% C.L. sensitivities on the NU parameters of interest using Dresden-II
data are summarized in Table 2. They are compared with those derived from
neutrino oscillation global fits [87]. As expected, due to low statistics,
large uncertainties on the background model of Dresden-II, as well as poor QF
determination, the sensitivities extracted in this work are not competitive
with those achievable at the large-scale oscillation experiments. We note,
however, that the upcoming sensitivities at the next-generation CE$\nu$NS
experiments are rather promising, and they might become competitive with
oscillation experiments—see, e.g., Ref. [55].

NU parameter Oscillations [87] YBe QF Fef QF $\alpha_{11}$ $>0.98$ $>0.758$
$>0.805$ $\left|\alpha_{12}\right|$ $<10^{-2}$ – $<0.747$
$\left|\alpha_{13}\right|$ $<4.2\times 10^{-2}$ – $<0.747$

Table 2: 90% C.L. sensitivities on NU parameters from the Dresden-II data for
YBe and Fef QF. We also give, for comparison, those derived from neutrino
oscillation data in Ref. [87].

Figure 4:  Sensitivity of Dresden-II on NU parameters, $\alpha_{11}$ (left)
and $|\alpha_{12}|$ or $|\alpha_{13}|$ (right), for the Fef and YBe QF. Also
shown are the corresponding 90% C.L. constraints extracted from neutrino
oscillation global fits [87].

Figure 5: 90% C.L. allowed regions in the vector (top), scalar (lower-left)
and tensor (lower-right) NGI parameters ($C_{X}^{u},C_{X}^{d}$). We assume one
nonvanishing interaction $X=S,V,T$ at a time and nonuniversal couplings for
$u$ and $d$ quarks, for the Fef (in blue) and YBe (in red) quenching factors.
For vector interactions, the COHERENT results [3] are superimposed for
comparison.

Figure 6: Same as Fig. 5, but for two nonvanishing interactions $X=S,V,T$ at a
time, using universal quark couplings.

Now, we finally come to our discussion of the results of our phenomenological
analysis of NGIs that includes, as a special case, the conventional NSIs. As
discussed in Sec. 2.3, these are expected in low-scale models of neutrino mass
generation; see Ref. [25] for a more extended discussion. If NGIs exist, the
Dresden-II reactor experiment will be sensitive to a particular combination of
the $C_{X}^{q}$ coefficients, specified in Eq. (12). In Fig. 5, we show the
corresponding $90\%$ C.L. allowed regions for the NGI coefficients in the
$(C_{X}^{u},C_{X}^{d}$) plane. These are obtained by switching on a single
interaction $X$ on top of the SM, taking the other two to vanish. As
previously, the results are given for both QFs, with the Fef constraints more
stringent than the YBe ones. Concerning the NSI case of the vector
interaction, our ($C_{V}^{u},C_{V}^{d}$) sensitivity contours (see top graph
in Fig. 5) agree very well with those in Refs. [23, 24], and with existing
constraints from COHERENT [3]. However, the NGI framework allows us to explore
the sensitivity of Dresden-II data on scalar and tensor interactions as well
[72]. The corresponding results are given in the lower-left (lower-right)
graph of Fig. 5 for scalar (tensor) interactions. As expected, in both cases,
there is no destructive interference and the sensitivity contours appear as a
single band, unlike the vector case 666For vector interaction, the new physics
part interferes either destructively or constructively with the SM, leading to
the white region between the two colored stripes.. Interestingly, by comparing
the three contours, one can see that the least (most) constrained NGI
parameter is the vector (scalar) one. Indeed, we find the best-fit values:
$\chi^{2}_{\text{min}}=\\{104.53,105.16,104.82\\}/\text{d.o.f.}$ using YBe QF
and $\\{102.59,102.63,102.62\\}/\text{d.o.f.}$ using Fef QF for the
$(C_{S}^{u},C_{S}^{d})$, $(C_{V}^{u},C_{V}^{d})$,
$(C_{T}^{u},C_{T}^{d})$—i.e., the presence of NGIs implies a slight
improvement in the fit with respect to the SM result, of no statistical
significance.

Within the NGI framework, in principle, all interactions $X=S,V,T$ can be
simultaneously active with a total number of six free NGI parameters. Notice
that the different kinematic terms corresponding to $X=S,V,T$ interactions in
the NGI cross section of Eq.(13) affect the shape of the CE$\nu$NS event rate
distribution. In order to reduce the degrees of freedom, while still being
able to come out with intuitive results, we will make further reasonable
simplifications. First, we assume universal couplings for $u$ and $d$
quarks—i.e., $C_{X}^{u}=C_{X}^{d}=C_{X}^{q}$—thus reducing the degrees of
freedom by half without losing information on the spectral shape of the
signal. Second, we allow two different interactions to float simultaneously,
while the third is set to zero. This way, we can perform a more general
analysis, with two different interaction types taken into account
simultaneously in our fit. Figure 6 shows the corresponding 90% C.L.
sensitivity contours in the $(C_{V}^{q},C_{S}^{q})$, $(C_{V}^{q},C_{T}^{q})$,
and $(C_{S}^{q},C_{T}^{q})$ planes for the two QFs. One sees that, in the
presence of the vector NGI there is a destructive interference, leaving a
“hole” in the $(C_{V}^{q},C_{S}^{q})$, $(C_{V}^{q},C_{T}^{q})$ planes. The
best-fit values are:
$\chi^{2}_{\text{min}}=\\{103.81,104.39,104.54\\}/\text{d.o.f.}$ using YBe QF,
and $\\{102.01,102.05,102.59\\}/\text{d.o.f.}$ using Fef QF for
$(C_{V}^{q},C_{S}^{q})$, $(C_{V}^{q},C_{T}^{q})$, and $(C_{S}^{q},C_{T}^{q})$,
respectively. Thus, we conclude that a combined analysis with two nonzero NGIs
at a time leads to a slight improvement in the fit compared to the pure SM
case, or a single nonvanishing NGI type.

##  4 Conclusions

We have analyzed the new data released by the Dresden-II Collaboration which
reported, for the first time, a suggestive evidence of CE$\nu$NS observation
using reactor antineutrinos. We performed a careful statistical analysis of
the event spectral distributions, estimating the expected background sources
according to the prescription given by the Dresden-II Collaboration: see Fig.
1. Using the available data, we explored different physics schemes within and
beyond the SM. We first examined the new determination of the weak mixing
angle in the low-energy regime, given in Fig. 2. Although our Dresden-II
determination is not as competitive as the existing one from the COHERENT
data, the best-fit value is in better agreement with the theoretical
prediction from RGE extrapolations (Fig. 3). However, one should note that the
best-fit value differs dramatically for the two QF models we have adopted.

Concerning new physics scenarios, we focused on two different examples namely,
the potential violation of unitarity in the neutrino mixing matrix, as well as
the presence of new neutrino interactions due to heavy mediators. As seen in
Fig. 4, current Dresden-II data do not have sufficient statistics to place a
competitive constraint on unitarity violation parameters. However, short
baseline reactor CE$\nu$NS experiments may provide a promising unitarity
violation probe in the long run. Finally, we examined the Dresden-II limits on
the most general neutrino interactions due to heavy mediators, given in Figs.
5 and 6. These are competitive with those obtained at COHERENT for the case of
vector interaction. We have also obtained the corresponding sensitivities for
scalar and tensor CE$\nu$NS interactions. Though not significant, their
interplay may improve the description of the Dresden-II data.

###### Acknowledgements.

We thank Prof. Juan I. Collar for providing us the updated version of Ref.
[18] containing ranges of the free background parameters for the Dresden-II
analysis as well as their $1~{}\sigma$ limit. This work has been supported by
the SERB, Government of India grant SRG/2020/002303, by the Spanish Agencia
Estatal de Investigacion under grant N. PID2020-113775GB-I00
(AEI/10.13039/501100011033) and by Generalitat Valenciana under Prometeo grant
N. CIPROM/2021/054.

##  Appendix A Details of the data fitting

In this section, we briefly discuss the details of the determination of the
expected background events for the Dresden-II data used in this work. As
already mentioned in Sec. 3, the background model characterized by seven free
parameters is reported in Ref. [18]. Note also that every analysis requires
simultaneous fitting of the free background parameters in addition to the
physics parameters of interest within or beyond the SM. Hence, in this work we
paid special attention in adopting a careful fitting procedure. In Fig. 7, we
demonstrate the SM “CE$\nu$NS only” signal and the “CE$\nu$NS + fitted
background” signal in the full ROI of the Dresden-II experiment. In contrast
to Refs. [23, 21], which employed a differential spectrum, here we make
different background assumptions. Namely, the background model of Eq.(18) is
taken to be an integrated spectrum for which the parameters
$N_{\text{epith}}$, $A_{\text{epith}}$, and $A_{L_{1},L_{2},L_{M}}$ are given
in units of $\mathrm{counts/\left(10~{}eV_{ee}\cdot 3~{}kg\cdot day\right)}$,
in line with the approach followed by the Dresden-II Collaboration. Although
this may not lead to significant differences, we stress that in our analysis
the background parameters are varied within their allowed ranges, unspecified
in Refs. [23, 21]. The most notable difference with those works is that here
the exponential decay function characterizing the epithermal neutron
background component is taken to be centered at
$E_{\text{epith}}=0.2~{}\mathrm{keV_{ee}}$ instead of zero. The allowed ranges
and the best-fit values of free background parameters are listed in Table 3.
For each case, the best-fit values we obtain are in agreement with the
corresponding $1~{}\sigma$ limit reported in the ancillary file of Ref. [18].

Background Parameters ($\upbeta$) |  Allowed range |  YBe QF |  Fef QF
---|---|---|---
$N_{\text{epith}}$ | $[0,~{}25]$ | $14.082$ | $13.779$
$A_{\text{epith}}$ | $[0,~{}150]$ | $66.505$ | $62.947$
$\tau_{\text{epith}}\text{ (keV)}$ | $[0,~{}2]$ | $0.249$ | $0.262$
$A_{L_{1}}$ | $[70,~{}250]$ | $84.577$ | $84.63$
$E_{L_{1}}\text{ (keV)}$ | $[1.2,~{}1.4]$ | $1.29228$ | $1.29237$
$\sigma_{L_{1}}\text{ (keV)}$ | $[0.04,~{}0.1]$ | $0.0694$ | $0.06963$
$\beta_{M/L_{1}}$ | $[0,~{}0.3]$ | $0.17$ | $0.158$
Fitted events (SM $+$ Background) |  | $5080.29$ | $5103.4$
$\chi^{2}_{\text{min}}$ |  | $105.95/\text{d.o.f.}$ | $102.66/\text{d.o.f.}$
Table 3: Best-fit values for the background model parameters and total number
of signal events. The values of $N_{\text{epith}}$, $A_{\text{epith}}$,
$A_{L_{1}}$, as well as the fitted events are all given in units of
$\mathrm{[counts/\left(10~{}eV_{ee}\cdot 3~{}kg\cdot day\right)]}$. Figure 7:
SM “CE$\nu$NS only” (thin lines) and “CE$\nu$NS + fitted background” (thick
lines) signal in the full ROI of the Dresden-II measurement.

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