# Nuclear neutron radius and weak mixing angle measurements from latest
COHERENT CsI and atomic parity violation Cs data

M. Atzori Corona<EMAIL_ADDRESS>Dipartimento di Fisica,
Università degli Studi di Cagliari, Complesso Universitario di Monserrato -
S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  Istituto
Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Complesso
Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato
(Cagliari), Italy  M. Cadeddu<EMAIL_ADDRESS>Istituto Nazionale di
Fisica Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di
Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  N.
Cargioli<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli
Studi di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  Istituto Nazionale di Fisica
Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di Monserrato -
S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  F. Dordei
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione
di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700,
09042 Monserrato (Cagliari), Italy  C. Giunti<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria
1, I–10125 Torino, Italy  G. Masia<EMAIL_ADDRESS>Dipartimento di
Fisica, Università degli Studi di Cagliari, Complesso Universitario di
Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy

( , )

###### Abstract

The COHERENT collaboration observed coherent elastic neutrino nucleus
scattering using a 14.6 kg cesium-iodide (CsI) detector in 2017 and recently
published the updated results before decommissioning the detector. Here, we
present the legacy determination of the weak mixing angle and of the average
neutron rms radius of ${}^{133}\mathrm{Cs}$ and ${}^{127}\mathrm{I}$ obtained
with the full CsI dataset, also exploiting the combination with the atomic
parity violation (APV) experimental result, that allows us to achieve a
precision as low as $\sim$4.5% and to disentangle the contributions of the
${}^{133}\mathrm{Cs}$ and ${}^{127}\mathrm{I}$ nuclei. Interestingly, we show
that the COHERENT CsI data show a 6$\sigma$ evidence of the nuclear structure
suppression of the full coherence. Moreover, we derive a data-driven
APV+COHERENT measurement of the low-energy weak mixing angle with a percent
uncertainty, independent of the value of the average neutron rms radius of
${}^{133}\mathrm{Cs}$ and ${}^{127}\mathrm{I}$, that is allowed to vary freely
in the fit. Additionally, we extensively discuss the impact of using two
different determinations of the theoretical parity non-conserving amplitude in
the APV fit. Our findings show that the particular choice can make a
significant difference, up to 6.5% on $R_{n}$(Cs) and 11% on the weak mixing
angle. Finally, in light of the recent announcement of a future deployment of
a 10 kg and a $\sim$700 kg cryogenic CsI detectors, we provide future
prospects for these measurements, comparing them with other competitive
experiments that are foreseen in the near future.

##  I Introduction

Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) is a purely weak-
neutral-current process, predicted by Freedmann in 1974 Freedman (1974), which
happens at low-momentum transfer when the de Broglie wavelength of the
exchanged Z boson mediator is greater than the nucleus radius. In this
process, the neutrino interacts with the nucleus as a whole, i.e. coherently,
making the cross section roughly proportional to the square of the number of
neutrons of the target nucleus. For this reason, it can become some orders of
magnitude bigger than that of other low-energy processes which involve
neutrinos. Nonetheless, CE$\nu$NS has evaded experimental observation for
about 43 years since its first theoretical postulation, due to the difficulty
in detecting such a low-energy (few keVs) nuclear recoil produced as the
single outcome of the interaction. In fact, the first observation of CE$\nu$NS
was reported by the COHERENT collaboration only in 2017 Akimov  _et al._
(2017, 2018), thanks to 14.6 kg of cesium-iodide (CsI) scintillating crystals
exposed to the neutrino flux produced by pion-decays-at-rest at the Spallation
Neutron Source (SNS), situated in the Oak Ridge Nation Laboratory. In the
subsequent years, the COHERENT collaboration reported a second measurement of
CE$\nu$NS with a liquid argon (LAr) detector Akimov  _et al._ (2021, 2020),
and later on updated the results with the same CsI detector Akimov  _et al._
(2022a). Furthermore, tantalizing evidence of CE$\nu$NS using antineutrinos
from the Dresden-II boiling water reactor has been recently reported in Ref.
Colaresi  _et al._ (2022) using an ultra-low noise 2.924 kg p-type point-
contact germanium detector.

In recent years, CE$\nu$NS has proven to be one of the most profitable tools
to study a plethora of diverse physical phenomena, ranging from neutrino non-
standard interactions and light mediators Liao and Marfatia (2017); Denton
_et al._ (2018); Lindner  _et al._ (2017); Flores  _et al._ (2020); Coloma
_et al._ (2017); Giunti (2020); Coloma  _et al._ (2022); Cadeddu  _et al._
(2021a); Atzori Corona  _et al._ (2022a); Banerjee  _et al._ (2021); Bertuzzo
_et al._ (2022) to neutrino electromagnetic properties Kosmas  _et al._
(2015); Billard  _et al._ (2018); Coloma  _et al._ (2022); Cadeddu  _et al._
(2018a); Atzori Corona  _et al._ (2022b), as well as to perform intriguing
tests of nuclear physics and electroweak interactions Cadeddu  _et al._
(2020a); Miranda  _et al._ (2020); Cadeddu  _et al._ (2021b, 2018b); Papoulias
_et al._ (2020); Papoulias and Kosmas (2018); Cadeddu  _et al._ (2020b);
Papoulias (2020); Khan and Rodejohann (2019); Dutta  _et al._ (2019);
Aristizabal Sierra  _et al._ (2018); Cadeddu and Dordei (2019); Dutta  _et
al._ (2020); Abdullah  _et al._ (2018); Ge and Shoemaker (2018); Miranda  _et
al._ (2021); Flores  _et al._ (2020); De Romeri  _et al._ (2022). Indeed, in
previous works Cadeddu  _et al._ (2018b, 2020b); Papoulias (2020); Miranda
_et al._ (2020); Huang and Chen (2019); Cadeddu  _et al._ (2021b), it has been
shown that CE$\nu$NS can be exploited to extract information on the neutron
root-mean-square (rms) radius distribution inside atomic nuclei, $R_{n}$. Such
a quantity is of extreme interest in nuclear physics and astrophysics since it
provides valuable information on the equation of state (EOS) of neutron-rich
matter, which is at the basis of the formation, structure, and stability of
atomic nuclei, whether stars collapse into neutron stars or black holes, and
also the structure of neutron stars as well as the understanding of
gravitational wave events Shen  _et al._ (2020); Horowitz (2019); Wei  _et
al._ (2020); Thiel  _et al._ (2019); Adhikari  _et al._ (2021). Despite its
importance, $R_{n}$ is experimentally very poorly known and it has been
usually determined using strong probes, which are known to be model dependent
and affected by non-perturbative effects Thiel  _et al._ (2019). This lack of
knowledge has been behind the Lead Radius Experiment (PREX) construction at
the Jefferson Laboratory in order to precisely determine $R_{n}$ of lead,
208Pb, by measuring the parity-violating electroweak asymmetry in the elastic
scattering of polarized electrons from 208Pb Abrahamyan  _et al._ (2012);
Adhikari  _et al._ (2021). Notably, the PREX-II collaboration has demonstrated
the feasibility of measuring the neutron rms radius of 208Pb at the percent
level Adhikari  _et al._ (2021), being able to obtain the current most
accurate determination of the neutron skin, i.e. the difference between
$R_{n}$ and the proton rms radius distribution $R_{p}$, of a heavy and
neutron-rich nucleus like lead. More recently, the CREX experiment, the twin
experiment of PREX that uses calcium, has reported a very precise
determination of the neutron radius of ${{}^{48}\textrm{Ca}}$ Adhikari  _et
al._ (2022). These achievements strengthen the importance of further
exploiting electroweak probes to obtain direct, reliable and precise
determinations of the neutron distribution of neutron-rich nuclei. Luckily,
CE$\nu$NS provides a promising and long-lasting tool, especially as the
community is putting a lot of effort into developing the future CE$\nu$NS
program Akimov  _et al._ (2022b); Baxter  _et al._ (2020); Abdullah  _et al._
(2022); Angloher  _et al._ (2019); Strauss  _et al._ (2017).

The CE$\nu$NS process is also sensitive to the so-called weak mixing angle
$\textrm{sin}^{2}\vartheta_{W}$, also known as the Weinberg angle, which is a
key parameter of the electroweak theory as it describes the mixing of the
$\textrm{SU}(2)_{L}\otimes\textrm{U}(1)_{Y}$ gauge boson fields Workman  _et
al._ (2022). Many experiments measured its value very precisely at
intermediate or high energies scales Workman  _et al._ (2022), while its low-
energy value still lacks of a precise experimental determination. Since the
weak mixing angle can be significantly modified in some beyond the standard
model (SM) scenarios Safronova  _et al._ (2018); Cadeddu  _et al._ (2021c),
such as in presence of dark $Z$ bosons Coloma  _et al._ (2022); De Romeri  _et
al._ (2022); Atzori Corona  _et al._ (2022a); Young  _et al._ (2007); Cadeddu
_et al._ (2021c), and it may have an impact also for nuclear physics
measurements Corona  _et al._ (2022), it is of major importance to exploit the
most recent available data to obtain its most accurate state-of-the-art
determination using electroweak probes. At the moment, the most precise
determination of the weak mixing angle available in the low-energy sector
comes from another low-energy electroweak probe, so-called Atomic Parity
Violation (APV) experiments, also known as parity non-conservation (PNC)
experiments, on cesium atoms Wood  _et al._ (1997); Guena  _et al._ (2005). In
particular, a PNC interaction mixes S and P eigenstates, and, in cesium, this
feature permits for the 6S$\rightarrow$7S transition between Zeeman sub-levels
Gilbert  _et al._ (1985), that would be otherwise forbidden. In Ref. Cadeddu
and Dordei (2019); Cadeddu  _et al._ (2021b), it has been demonstrated for the
first time the complementarity of CE$\nu$NS and APV experiments to
simultaneously extract information on the weak mixing angle and the nuclear
physics parameters, making it interesting to combine these results.

The goal of this paper is therefore to use the latest CsI data-release
provided by the COHERENT collaboration in 2021 with a refined quenching factor
(QF) determination and more statistics Akimov  _et al._ (2022a, c), both alone
and in combination with the information content coming from APV on cesium
atoms, in order to extract the most precise information on the weak mixing
angle and nuclear physics parameters. This paper is organized as follows. In
Sec. II we give the necessary theoretical tools for the determination of the
physics parameters of interest, while in Sec. III we provide the experimental
details and statistical procedures used in this analysis as well as describe
and comment the results obtained analysing COHERENT and APV data. Before to
summarise our conclusions in Sec. V, in Sec. IV we review future perspectives
for these measurements.

##  II Nuclear and electroweak physics using CE$\nu$NS and APV processes

As stated in the introduction, CE$\nu$NS and atomic parity violation represent
two of the most promising electroweak probes to access the neutron nuclear
distribution. The proton and neutron nuclear form factors,
$F_{Z}\left(|\vec{q}|^{2}\right)$ and $F_{N}\left(|\vec{q}|^{2}\right)$,
account respectively for the spatial distribution of protons and neutrons
inside the target nucleus. Their measurement can be extracted analysing
CE$\nu$NS data corresponding to a neutrino $\nu_{\ell}$ ($\ell=e,\mu,\tau$)
scattering off a nucleus $\mathcal{N}$ with $Z$ protons and $N$ neutrons (in
this analysis, we use $(Z,N)_{\mathrm{Cs}}=(55,78)$, and
$(Z,N)_{\mathrm{I}}=(53,74)$). They depend on the size of the nucleus and in
these kind of processes they parameterize the loss of coherence as the
momentum transfer $|\vec{q}|$ grows. They represent a crucial ingredient of
the CE$\nu$NS SM differential cross section which, as a function of the
nuclear recoil energy $T_{\mathrm{nr}}$ and for a spin-zero nucleus, is given
by Freedman (1974); Drukier and Stodolsky (1984); Barranco  _et al._ (2005);
Patton  _et al._ (2012)

$\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\mathrm{nr}}}(E,T_{\mathrm{nr}})=\dfrac{G_{\text{F}}^{2}M}{\pi}\left(1-\dfrac{MT_{\mathrm{nr}}}{2E^{2}}\right)\left[g_{V}^{p}\left(\nu_{\ell}\right)ZF_{Z}\left(|\vec{q}|^{2}\right)+g_{V}^{n}NF_{N}\left(|\vec{q}|^{2}\right)\right]^{2},$
(1)

where $G_{\text{F}}$ is the Fermi constant, $E$ is the neutrino energy, $M$ is
the nuclear mass, and $g_{V}^{p}$ and $g_{V}^{n}$ are the neutrino couplings
to protons and neutrons, respectively. Their tree-level values are
$g_{V}^{p}=1/2-2\;\textrm{sin}^{2}\;\vartheta_{\text{W}}$, which encapsulates
the dependence on the weak mixing angle such that it could be determined from
CE$\nu$NS data, and $g_{V}^{n}=-1/2$. In the SM, the value of the weak mixing
angle $\vartheta_{W}$ evaluated at zero-momentum transfer is equal to
$\textrm{sin}^{2}\vartheta_{\text{W}}(q^{2}=0)=\hat{s}^{2}_{0}=0.23863$, as
reported in the Particle Data Group (PDG) Workman  _et al._ (2022). A precise
determination of these couplings is of fundamental importance to retrieve
correctly $F_{Z}\left(|\vec{q}|^{2}\right)$ and
$F_{N}\left(|\vec{q}|^{2}\right)$. Thus, in this paper, we calculated the
couplings taking into account the radiative corrections in the
$\overline{\mathrm{MS}}$ scheme following Refs. Erler and Su (2013); Workman
_et al._ (2022)

$\\!\begin{aligned}
g_{V}^{\nu_{\ell}\,p}&=\rho\left(\frac{1}{2}-2\;\textrm{sin}^{2}\vartheta_{\text{W}}\right)+2\;{\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}+\Box_{WW}-2\diameter_{\nu_{\ell}W}+\rho(2\boxtimes_{ZZ}^{uL}+\boxtimes_{ZZ}^{dL}-2\boxtimes_{ZZ}^{uR}-\boxtimes_{ZZ}^{dR})\,,\\\
g_{V}^{\nu_{\ell}\,n}&=-\frac{\rho}{2}+2\Box_{WW}+{\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}+\rho(2\boxtimes_{ZZ}^{dL}+\boxtimes_{ZZ}^{uL}-2\boxtimes_{ZZ}^{dR}-\boxtimes_{ZZ}^{uR}).\end{aligned}$
(2)

The quantities in Eq. (2), $\Box_{WW}$,
${\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}$ and
$\boxtimes_{ZZ}^{fX}$, with $f\in\\{u,d\\}$ and $X\in\\{L,R\\},$ are the
radiative corrections associated with the $WW$ box diagram, the $WW$ crossed-
box and the $ZZ$ box respectively, while $\rho=1.00063$ is a parameter of
electroweak interactions. Moreover, $\diameter_{\nu_{\ell}W}$ describes the
neutrino charge radius contribution and introduces a dependence on the
neutrino flavour $\ell$ (see Ref. Erler and Su (2013) or the appendix B of
Ref. Atzori Corona  _et al._ (2023) for further information on such
quantities). Numerically, the values of these couplings correspond to
$g_{V}^{p}(\nu_{e})=0.0382$, $g_{V}^{p}(\nu_{\mu})=0.0300$, and
$g_{V}^{n}=-0.5117$.

The form factors $F_{N}\left(|\vec{q}|^{2}\right)$ and
$F_{Z}\left(|\vec{q}|^{2}\right)$ can be parametrized using a symmetrized
Fermi Cadeddu  _et al._ (2018b) distribution which is given by the analytic
expression

$F_{Z}^{\text{SF}}(q^{2})=\dfrac{3}{qc\left[(qc)^{2}+(\pi
qa)^{2}\right]}\left[\dfrac{\pi qa}{\sinh(\pi qa)}\right]\left[\dfrac{\pi
qa\sin(qc)}{\tanh(\pi qa)}-qc\cos(qc)\right].$ (3)

The parameter $a$ is the so-called diffuseness, which is related to the
surface thickness $t$ through $t=4\;a\;\textrm{ln}(3)$ and is commonly fixed
to the value $t=2.30$ fm both for the neutron and proton form factor, as most
of theoretical nuclear models predict roughly the same density drop between
proton and neutron distributions. The rms radius is related to $a$ through
$R^{2}=3/5\,c^{2}+7/5\,(\pi a)^{2}$ and, for the proton distribution inside
the nucleus, we use the proton rms radii obtained from the muonic atom
spectroscopy data Fricke  _et al._ (1995); Angeli and Marinova (2013) as
explained in Ref. Cadeddu  _et al._ (2020a), namely
$R_{p}(\mathrm{Cs})=4.821(5)~{}\mathrm{fm}$ and
$R_{p}(\mathrm{I})=4.766(8)~{}\mathrm{fm}$. Since only little knowledge on the
neutron rms radii of ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$ is available
using electroweak probes Hoferichter  _et al._ (2020); Cadeddu  _et al._
(2018b); Papoulias  _et al._ (2020); Cadeddu  _et al._ (2018a); Huang and Chen
(2019); Papoulias (2020); Khan and Rodejohann (2019); Cadeddu  _et al._
(2020a, b), we take as a reference the values of the neutron rms radii
extracted from the theoretical nuclear shell model (NSM) calculations of the
corresponding neutron skins $\Delta R_{\rm{np}}$ Hoferichter  _et al._ (2020)
which are

$\displaystyle R^{\rm NSM}_{n}(^{133}\mathrm{Cs})\simeq
5.09~{}\mathrm{fm},\quad\quad R^{\rm NSM}_{n}(^{127}\mathrm{I})\simeq
5.03~{}\mathrm{fm}.$ (4)

In the calculations which assume these values of the neutron rms radii, we
take into account the effect of their uncertainties by considering a 3.4%
uncertainty for the CsI CE$\nu$NS rates Akimov  _et al._ (2022a).

An alternative description is provided by the Helm parameterization Helm
(1956), which practically gives the same results as the SF. The Helm
parameterization is given by

$F^{\textrm{Helm}}\left(|\vec{q}|^{2}\right)=3\frac{j_{1}(qR_{0})}{qR_{0}}e^{-|\vec{q}|^{2}s^{2}/2},$
(5)

where $j_{1}(x)=\textrm{sin}(x)/x^{2}-\textrm{cos}(x)$ is the order-one
spherical Bessel function, while $R_{0}$ is the box (or diffraction) radius.
The rms radius of the corresponding nucleon distribution is given by
$R^{2}=3/5R_{0}^{2}+3s^{2}$, where the parameter $s$ quantifies the so-called
surface thickness. We consider a value of $s=0.9$ fm, which is the typical
value determined for the proton form factor for this type of nuclei Friedrich
and Voegler (1982).

The low-energy measurement of the nuclear weak charge, $Q_{W}$, of
${}^{133}\text{Cs}$ from APV experiments also depends on the nuclear physics
parameters and the weak mixing angle, making it interesting to simultaneously
constrain these parameters exploiting COHERENT CsI data. In particular, the
nuclear weak charge depends on the weak mixing angle according to the relation
Workman  _et al._ (2022)

$\displaystyle Q_{W}^{\mathrm{th}}(\sin^{2}\vartheta_{W})=$
$\displaystyle-2[Z(g_{AV}^{ep}(\sin^{2}\vartheta_{W})+0.00005)+N(g_{AV}^{en}+0.00006)]\left(1-\dfrac{\alpha}{2\pi}\right),$
(6)

where $\alpha$ is the fine-structure constant, while $g_{AV}^{ep}$ and
$g_{AV}^{en}$ are the couplings of electrons to protons and neutrons,
respectively, including radiative corrections in the $\overline{\mathrm{MS}}$
scheme Workman  _et al._ (2022); Erler and Su (2013) (see also appendix A of
Ref. Cadeddu  _et al._ (2021b)). The small numerical corrections to the
couplings are discussed in Ref. Erler and Su (2013) and include the
calculation of the $\gamma Z$-box correction from Ref. Blunden  _et al._
(2012). The values of the couplings correspond to

$\displaystyle\hbox{}g_{AV,\text{SM}}^{ep}=2g_{AV,\text{SM}}^{eu}+g_{AV,\text{SM}}^{ed}=-0.0357,$
(7)
$\displaystyle\hbox{}g_{AV,\text{SM}}^{en}=g_{AV,\text{SM}}^{eu}+2g_{AV,\text{SM}}^{ed}=0.495,$
(8)

where $g_{AV,\text{SM}}^{eu}=-0.1888$ and $g_{AV,\text{SM}}^{ed}=0.3419$.
These values give $Q_{{W}}^{\text{SM}}=-73.23\pm 0.01$.

Experimentally, the weak charge of a nucleus is extracted from the ratio of
the parity violating amplitude, $E_{\text{PNC}}$, to the Stark vector
transition polarizability, $\beta$, and by calculating theoretically $E_{\rm
PNC}$ in terms of $Q_{W}$

$Q_{W}=N\left(\dfrac{{\rm Im}\,E_{\rm PNC}}{\beta}\right)_{\rm
exp.}\left(\dfrac{Q_{W}}{N\,{\rm Im}\,E_{\rm PNC}}\right)_{\rm th.}\beta_{\rm
exp.+th.}\,,$ (9)

where $\beta_{\rm exp.+th.}$ and $(\mathrm{Im}\,E_{\rm PNC})_{\rm th.}$ are
determined from atomic theory, and Im stands for imaginary part Workman  _et
al._ (2022). We use $({\rm Im}\,E_{\rm PNC}/{\beta})_{\rm exp}=(-3.0967\pm
0.0107)\times 10^{-13}|e|/a_{B}^{2}$ Workman  _et al._ (2022), where $a_{B}$
is the Bohr radius and $|e|$ is the absolute value of the electric charge, and
$\beta_{\rm exp.+th.}=(27.064\pm 0.033)\,a_{B}^{3}$ Workman  _et al._ (2022).
In order to extract information on the neutron distribution inside cesium and
to directly evaluate $R_{n}$ from a combined fit with the COHERENT data, one
has to subtract to $({\rm Im}\,E_{\rm PNC})_{\rm th.}$ the so-called “neutron
skin” correction $\delta E^{\mathrm{n.s.}}_{\mathrm{PNC}}(R_{n})$ in order to
obtain $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}$ Cadeddu  _et al._
(2021b). This correction has been introduced in Ref. Derevianko (2001) to take
into account the fact that the difference between $R_{n}$ and $R_{p}$ is not
considered in the nominal atomic theory derivation. Hence, the neutron skin
corrected value of the weak charge, $Q_{W}^{\mathrm{n.s.}}(R_{n})$, is
retrieved by summing to $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}$ the
correction term $\delta
E^{\mathrm{n.s.}}_{\mathrm{PNC}}(R_{n})=\left[(\mathrm{N}/Q_{W})\left(1-(q_{n}(R_{n})/q_{p})\right)E_{\mathrm{PNC}}^{\mathrm{w.n.s.}}\right]$
Derevianko (2001); Cadeddu  _et al._ (2020b); Cadeddu and Dordei (2019). In
this way, the dependence of $Q_{W}$ on the neutron rms radius becomes also
explicit. Indeed, the factors $q_{p}$ and $q_{n}$ incorporate the radial
dependence of the electron axial transition matrix element by considering the
proton and the neutron spatial distribution, respectively Pollock  _et al._
(1992); Viatkina  _et al._ (2019); Pollock  _et al._ (1992); Derevianko
(2001). For the calculation of $q_{p}$ and $q_{n}$ we refer to appendix B of
Ref. Cadeddu  _et al._ (2021b). In this work, we will rely on the theoretical
value of the PNC amplitude with the neutron skin correction used in Ref.
Cadeddu  _et al._ (2021b) $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm
w.n.s.}=(0.8995\pm 0.0040)\times 10^{-11}|e|a_{B}\frac{Q_{W}}{N}$ Dzuba  _et
al._ (2012), but we will also discuss the implications to consider the result
of the more recent calculation reported in Ref. Sahoo  _et al._ (2021) which
yields to a smaller value, namely $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm
w.n.s.}=(0.8930\pm 0.0027)\times 10^{-11}|e|a_{B}\frac{Q_{W}}{N}$.

##  III Results

In this section, we present the new measurements of the weak mixing angle,
performed at the energy scale probed by the experiments, i.e.
$|\vec{q}|^{2}\simeq(50\ {\rm{MeV}})^{2}$ and $|\vec{q}|^{2}\simeq(2.4\
{\rm{MeV}})^{2}$ for COHERENT and APV, respectively, and of the cesium nuclear
parameters, both obtained using the latest COHERENT CsI data-set Akimov  _et
al._ (2022a) alone and in combination with APV.

Concerning the treatment of the COHERENT data, we largely follow the same
procedure described in details in Ref. Atzori Corona  _et al._ (2022a), to
which we refer the reader. The physical parameters of interest are extracted
using the following Poissonian least-squares function Baker and Cousins
(1984); Workman  _et al._ (2022)

$\displaystyle\chi^{2}_{\mathrm{CsI}}=\hbox{}$
$\displaystyle\hbox{}2\sum_{i=1}^{9}\sum_{j=1}^{11}\left[\sum_{z=1}^{4}(1+\eta_{z})N_{ij}^{z}-N_{ij}^{\text{exp}}+N_{ij}^{\text{exp}}\ln\left(\frac{N_{ij}^{\text{exp}}}{\sum_{z=1}^{4}(1+\eta_{z})N_{ij}^{z}}\right)\right]+\sum_{z=1}^{4}\left(\dfrac{\eta_{z}}{\sigma_{z}}\right)^{2},$
(10)

where the indices $i,j$ represent the nuclear-recoil energy and arrival time
bin, respectively, while the indices $z=1,2,3,4$ for $N_{ij}^{z}$ stand,
respectively, for CE$\nu$NS,
($N_{ij}^{1}=N_{ij}^{\mathrm{CE}\nu\mathrm{NS}}$), beam-related neutron
($N_{ij}^{2}=N_{ij}^{\text{BRN}}$), neutrino-induced neutron
($N_{ij}^{3}=N_{ij}^{\text{NIN}}$) and steady-state
($N_{ij}^{4}=N_{ij}^{\text{SS}}$) backgrounds obtained from the anti-
coincidence data. In our notation, $N_{ij}^{\text{exp}}$ is the experimental
event number obtained from coincidence data and $N_{ij}^{\text{CE$\nu$NS}}$ is
the predicted number of CE$\nu$NS events that depends on the physics model
under consideration, according to the cross-section in Eq. (1), as well as on
the neutrino flux, energy resolution, detector efficiency, number of target
atoms and the CsI quenching factor Atzori Corona  _et al._ (2022a). We take
into account the systematic uncertainties with the nuisance parameters
$\eta_{z}$ and the corresponding uncertainties
$\sigma_{\text{CE}\nu\text{NS}}=0.12$, $\sigma_{\text{BRN}}=0.25$,
$\sigma_{\text{NIN}}=0.35$ and $\sigma_{\text{SS}}=0.021$ as explained in
Refs. Atzori Corona  _et al._ (2022a); Akimov  _et al._ (2022a).

When performing the analysis of the APV data, we use the least-squares
function given by

$\displaystyle\chi^{2}_{\mathrm{APV}}=\left(\dfrac{Q_{W}^{\rm
Cs\,n.s.}(R_{n})-Q_{W}^{\mathrm{th}}(\sin^{2}\vartheta_{W})}{\sigma_{\rm
APV}(R_{n},\sin^{2}\vartheta_{W})}\right)^{2}\,,$ (11)

where $\sigma_{\rm APV}$ is the total uncertainty. Finally, when performing a
combined analysis of the COHERENT data with APV, we use the least-squares
function given by

$\displaystyle\chi^{2}=\chi^{2}_{\mathrm{CsI}}+\chi^{2}_{\mathrm{APV}}.$ (12)

###  III.1 Determination of $\mathbf{\sin^{2}\vartheta_{W}}$ and the average
CsI neutron radius

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Figure 1:  Constraints on the weak mixing angle (a) and on the average rms CsI neutron radius (b) at different confidence levels (CL). The different curves refer to the COHERENT CsI data (CsI), the APV data using the PNC amplitude of Ref. Dzuba  _et al._ (2012) (APV-PDG) and that recently calculated in Ref. Sahoo  _et al._ (2021) (APV-2021), as well as their combination (APV-PDG+CsI and APV-2021+CsI). In (a) the combined curves are practically indistinguishable from the APV only fit. The green lines represent (a) the low-energy SM value of the weak mixing angle and (b) the average rms CsI neutron radius from the nuclear shell model prediction in Eq. (4). In the inset in the top left of (a), a zoom of x-axes is shown to better appreciate the APV only determinations.  | $\sin^{2}\vartheta_{W}$ | $R_{n}(\rm{CsI})[fm]$
---|---|---
| $\textrm{best-fit}^{+1\sigma+90\%\rm{CL}+2\sigma}_{-1\sigma-90\%\rm{CL}-2\sigma}$ | $\chi^{2}_{\rm{min}}$ | $\textrm{best-fit}^{+1\sigma+90\%\rm{CL}+2\sigma}_{-1\sigma-90\%\rm{CL}-2\sigma}$ | $\chi^{2}_{\rm{min}}$
COH-CsI | $0.231^{+0.027+0.046+0.058}_{-0.024-0.039-0.047}$ | 86.0 | $5.47^{+0.38+0.63+0.76}_{-0.38-0.72-0.89}$ | 85.2
APV PDG | $0.2375^{+0.0019+0.0031+0.0038}_{-0.0019-0.0031-0.0038}$ | - | $5.29^{+0.33+0.55+0.66}_{-0.34-0.56-0.68}$ | -
APV 2021 | $0.2399^{+0.0016+0.0026+0.0032}_{-0.0016-0.0026-0.0032}$ | - | $4.86^{+0.28+0.46+0.56}_{-0.29-0.48-0.58}$ | -
APV PDG + CsI | $0.2374^{+0.0020+0.0032+0.0039}_{-0.0018-0.0031-0.0037}$ | 86.0 | $5.35^{+0.25+0.41+0.50}_{-0.26-0.43-0.53}$ | 85.3
APV 2021 + CsI | $0.2398^{+0.0016+0.0026+0.0032}_{-0.0015-0.0026-0.0031}$ | 86.0 | $5.04^{+0.23+0.38+0.46}_{-0.24-0.40-0.48}$ | 86.6
Table 1: Summary of the constraints obtained in this work on the weak mixing
angle $\sin^{2}{\vartheta_{\text{W}}}$ and on the average rms CsI neutron
radius $R_{n}(\rm{CsI})$ at different confidence levels. The different labels
refer to the COHERENT CsI data (COH- CsI), the APV data using the PNC
amplitudes of Ref. Dzuba  _et al._ (2012) (APV-PDG) and Ref. Sahoo  _et al._
(2021) (APV-2021), as well as their combination (APV-PDG+CsI and
APV-2021+CsI). For each constrain we also report the minimum value of the
least square function provided by the fit.

In this section, we report the limits obtained on
$\sin^{2}{\vartheta_{\text{W}}}$ and $R_{n}(\rm{CsI})$ separately, i.e., by
fitting for only one parameter at a time and fixing the other to the
theoretical prediction. To this end, we exploit the COHERENT CsI and APV
dataset separately but also their combination.

In Sec. I we introduced the weak mixing angle $\vartheta_{\text{W}}$, whose
experimental determination might provide a powerful tool to test the SM of
electroweak theory and to investigate possible new physics extensions of it.
With this aim in mind, we updated the previous COHERENT CsI analysis performed
in Ref. Cadeddu  _et al._ (2021b) using a refined quenching factor Akimov  _et
al._ (2022a, c) and an improved fitting procedure which exploits the neutrino
time arrival information and uses a poissonian $\chi^{2}$ function, instead of
the canonical gaussian definition, which provides a more correct and reliable
fit to the data when the number of events in each bin is small. Relying on the
theoretical prediction for the neutron radius, we can fix the average CsI
radius to $R_{n}^{\rm{NSM}}(\rm{CsI})\simeq 5.06\;\rm{fm}$, according to Eq.
(4), to perform a fit to extract the weak mixing angle using the least-square
function in Eq. (10). The result is shown graphically in Fig. 1 and at the
$1\sigma$, 90$\%$ and $2\sigma$ confidence level (CL) we find

$\sin^{2}{\vartheta_{\text{W}}}(\mathrm{COH-
CsI})=0.231^{+0.027}_{-0.024}(1\sigma)^{+0.046}_{-0.039}(90\%\textrm{CL})^{+0.058}_{-0.047}(2\sigma),$
(13)

which is in agreement with the theoretical SM prediction $\hat{s}^{2}_{0}$ and
the result recently presented in Ref. De Romeri  _et al._ (2022) when fitting
the COHERENT CsI data with a different approach. Another derivation performed
by the COHERENT collaboration Akimov  _et al._ (2022a) reports
$\sin^{2}{\vartheta_{\text{W}}}(\mathrm{CsI})=0.220^{+0.028}_{-0.026}$, which
agrees rather well with our result although some small differences are
expected due to the different description of the nuclear structure, i.e.
different choices of the reference values for the neutron nuclear radii, and a
different approach to radiative corrections for neutrino-nucleus scattering.
Moreover, we checked the impact of using a different quenching factor, by
comparing our nominal results obtained using Refs. Akimov  _et al._ (2022a, c)
and the derivation in Ref. Lewis and Collar (2021). The latter lower QF
decreases the total number of CE$\nu$NS events resulting in a larger
$\sin^{2}{\vartheta_{\text{W}}}$ by about 10%.

This result can be compared with those obtained using the APV experiment on Cs
and the least-squares function in Eq. (11), namely

$\displaystyle\sin^{2}{\vartheta_{\text{W}}}(\mathrm{APV\,PDG})=0.2375\pm
0.0019\,(1\sigma)\pm 0.0031\,(90\%\textrm{CL})\pm 0.0038\,(2\sigma),$ (14)
$\displaystyle\sin^{2}{\vartheta_{\text{W}}}(\mathrm{APV\,2021})=0.2399\pm
0.0016\,(1\sigma)\pm 0.0026\,(90\%\textrm{CL})\pm 0.0032\,(2\sigma),$ (15)

that we derived exploiting the experimental value of $Q_{W}$ obtained with the
theoretical prediction of the PNC amplitude of Ref. Dzuba  _et al._ (2012),
referred to as APV PDG, and that recently calculated in Ref. Sahoo  _et al._
(2021), referred to as APV 2021. It is possible to see that the APV dataset
allows us to achieve a factor of more than 10 better precision in the
determination of $\sin^{2}{\vartheta_{\text{W}}}$. As shown in the insert of
Fig. 1, the two PNC amplitudes point to a value of the weak mixing angle that
is below and above the theoretical prediction, respectively, by less than
1$\sigma$. From the combination of the APV and COHERENT CsI dataset we obtain

$\displaystyle\sin^{2}{\vartheta_{\text{W}}}(\mathrm{APV\,PDG+COH-
CsI})=0.2374^{+0.0020}_{-0.0018}(1\sigma)^{+0.0032}_{-0.0031}(90\%\textrm{CL})^{+0.0039}_{-0.0037}(2\sigma),$
(16) $\displaystyle\sin^{2}{\vartheta_{\text{W}}}(\mathrm{APV\,2021+COH-
CsI})=0.2398^{+0.0016}_{-0.0015}(1\sigma)^{+0.0026}_{-0.0026}(90\%\textrm{CL})^{+0.0032}_{-0.0031}(2\sigma).$
(17)

Clearly, the combination is vastly dominated by the APV result, with the PNC
amplitude from Ref. Sahoo  _et al._ (2021) being slightly more precise. All
the results shown are summarised in Table 1.

Moving now to the poorly-known neutron distribution inside the nuclei, in
order to obtain information on it we fix the weak mixing angle to the SM low-
energy value, that is currently calculated with very high precision Workman
_et al._ (2022), to let the average CsI neutron distribution radius
$R_{n}(\rm{CsI})$ free to vary in the fit. Indeed, given the fact that the
difference between the rms neutron radii of Cs and I is expected to be small
compared to the current precision of experimental data, the choice to fit for
an average value is a fair approximation. Clearly, in this case, the
contribution due to the neutron form factor to the total systematic
uncertainty on $N_{ij}^{\mathrm{CE}\nu\mathrm{NS}}$ is removed in the least-
square function evaluation. The result of the fit is shown in Fig. 1 and at
the $1\sigma$, 90$\%$ and $2\sigma$ CL we find

$R_{n}(\mathrm{COH-
CsI})=5.47^{+0.38}_{-0.38}(1\sigma)^{+0.63}_{-0.72}(90\%\textrm{CL})^{+0.76}_{-0.89}(2\sigma)\;{\rm{fm}},$
(18)

which is in agreement, within the uncertainty, with the NSM expected value for
$R_{n}^{\rm{NSM}}(\rm{CsI})$, despite the central value pointing toward a
large neutron skin. Moreover, this result is almost $10\%$ more precise than
the previous determination of Ref. Cadeddu  _et al._ (2021b). To better
appreciate the sensitivity of CE$\nu$NS to $R_{n}$, in Fig. 2 we show the
impact of the nuclear structure to the theoretical prediction of the CE$\nu$NS
event rates. In particular, we show the COHERENT excess counts, namely the
background subtracted COHERENT data, as a function of both the photoelectrons
(PE) and the corresponding nuclear recoil energy ($T_{\rm{nr}}$) and we
compare them with the prediction obtained in case of full coherence, i.e.,
setting all nuclear form factors equal to unity, and with the best fit
obtained leaving $R_{n}$ free to vary, as described in this section. We find
that COHERENT data shows a 6$\sigma$ evidence of the nuclear structure
suppression of the full coherence, making it an extremely powerful probe to
determine nuclear parameters.

Figure 2: COHERENT CE$\nu$NS only data versus the number of photoelectrons
(PE) and the nuclear kinetic recoil energy ($T_{\mathrm{nr}}$). The histograms
represent the theoretical prediction in the case of full coherence (blue dash-
dotted line) and the best fit obtained leaving $R_{n}$ free to vary (red
dashed line). The red shadowed area represents the $\pm 1\sigma$ variation in
the $R_{n}$ value.

Also APV data is sensitive to $R_{n}$ and using the least-squares function in
Eq. (11) we get

$\displaystyle
R_{n}(\mathrm{\mathrm{APV\,PDG}})=5.29^{+0.33}_{-0.34}(1\sigma)^{+0.55}_{-0.56}(90\%\textrm{CL})^{+0.66}_{-0.68}(2\sigma)\;{\rm{fm}},$
(19) $\displaystyle
R_{n}(\mathrm{\mathrm{APV\,2021}})=4.86^{+0.28}_{-0.29}(1\sigma)^{+0.46}_{-0.48}(90\%\textrm{CL})^{+0.56}_{-0.58}(2\sigma)\;{\rm{fm}}.$
(20)

Differently from the case of the weak mixing angle, the precision achieved in
this case is only slightly better than that achieved with COHERENT, such that
the constraints improve significantly by performing a combination of these two
experiments. The $\chi^{2}$-curves we obtain are summarised in Fig. 1 and the
numerical values we find are

$\displaystyle R_{n}(\mathrm{\mathrm{APV\,PDG+COH-
CsI}})=5.35^{+0.25}_{-0.26}(1\sigma)^{+0.41}_{-0.43}(90\%\textrm{CL})^{+0.50}_{-0.53}(2\sigma)\;{\rm{fm}},$
(21) $\displaystyle R_{n}(\mathrm{\mathrm{APV\,2021+COH-
CsI}})=5.04^{+0.23}_{-0.24}(1\sigma)^{+0.38}_{-0.40}(90\%\textrm{CL})^{+0.46}_{-0.48}(2\sigma)\;{\rm{fm}}.$
(22)

It is possible to see that the combination obtained using the 2021 PNC
amplitude of Ref. Sahoo  _et al._ (2021) returns a neutron distribution rms
radius that is very well in agreement with the theoretical prediction, while
both COHERENT and the PDG PNC amplitude of Ref. Dzuba  _et al._ (2012) suggest
a larger neutron skin. Given that in the latter case the two dataset point
toward a similar value, we also get a smaller value for the minimum
$\chi^{2}$, as shown in Table 1. In both scenarios, a precision of less than
5% is obtained in the determination of $R_{n}$.

###  III.2 Simultaneous determination of $\mathbf{\sin^{2}\vartheta_{W}}$ vs
$\mathbf{R_{n}}$(CsI) and combined analysis with APV

|
---|---

Figure 3:  Constraints obtained fitting the COHERENT CsI data on the weak
mixing angle and the average rms CsI neutron radius (a) and on the plane of
$R_{n}(^{133}{\rm{Cs}})$ and $R_{n}(^{127}{\rm{I}})$ (b) together with their
marginalizations, at different CLs. The green lines indicate the theoretical
low-energy value of the weak mixing angle and the NSM prediction for the
corresponding rms neutron distribution radius.

|
---|---

Figure 4:  Constraints on the weak mixing angle and the average rms CsI
neutron radius together with their marginalizations, at different CLs obtained
fitting the COHERENT CsI data in combination with APV data, using the value
for the neutron skin corrections of Ref. Dzuba  _et al._ (2012) (a) and Ref.
Sahoo  _et al._ (2021) (b). The green lines indicate the theoretical low-
energy value of the weak mixing angle and the NSM prediction for the average
rms CsI neutron radius.

In this section, we extend the analysis presented in Sec. III.1 by letting
both the weak mixing angle and the average CsI neutron radius to vary freely
in the fit such that correlation between these two observables is properly
taken into account. This allows one to obtain simultaneous information on both
parameters, taking into account their degeneracy and obtaining thus a more
reliable result. We also perform this fit in combination with APV data using
Eq. (12), in order to exploit the different dependence of the two experiments
on these two parameters. This allows us to get a more precise and solid
determination of both quantities that uses the overall constraining power of
two different electroweak probes. The contours at different CLs of the allowed
regions in the plane of the weak mixing angle and the average CsI neutron
radius are reported in Fig. 3, using COHERENT CsI data alone. At the $1\sigma$
CL we obtain

${\rm{COH-
CsI:}}\;\;\;\sin^{2}\vartheta_{W}=0.31^{+0.08}_{-0.07},\;\;R_{n}(\mathrm{CsI})=6.6^{+1.4}_{-1.1}\;{\rm{fm}}.$
(23)

The fit tends to prefer large values for both parameters, with the theoretical
value of the weak mixing angle and the rms average neutron radius of CsI that
lie respectively at $\sim 1\sigma$ and $\sim 1.3\sigma$ outside the
marginalized allowed region, despite the large uncertainties. Indeed, the
dataset is better fitted considering unusually large values for the weak
mixing angle and the average rms neutron radius. In fact, it is interesting to
notice that the fit of the COHERENT-CsI data improves noticeably with respect
to the case in which both parameters are fixed to their theoretical value with
a $\Delta\chi^{2}$ (i.e. the difference between the $\chi^{2}_{\rm{min}}$ in
this fit and the $\chi^{2}$ obtained fixing the parameters to their
theoretical value) which is $\Delta\chi^{2}=-2.2$.

These constraints have been further tested by performing a combined fit with
the APV experiment on cesium. The results of the combined analyses are
reported in Fig. 4 and Fig. 4, using the experimental value of $Q_{W}$
obtained with the theoretical prediction of the PNC amplitude of Ref. Dzuba
_et al._ (2012) and that recently calculated in Ref. Sahoo  _et al._ (2021),
respectively. Using $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}$ from Ref.
Dzuba  _et al._ (2012) we obtain

${\rm{APV\,PDG+COH-
CsI:}}\;\;\;\sin^{2}\vartheta_{W}=0.2397^{+0.0033}_{-0.0032},\;\;R_{n}(\mathrm{CsI})=5.4^{+0.5}_{-0.4}\;{\rm{fm}}.$
(24)

The impact of including the APV data is noticeable, both in the uncertainty of
the parameters, that is improved by more that one order of magnitude for the
weak mixing angle and by a factor of $\sim 3$ for $R_{n}(\mathrm{CsI})$, as
well as in their central values, that are moved towards the expected values,
especially for $\textrm{sin}^{2}\vartheta_{W}$.
Using $({\rm Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}$ from Ref. Sahoo  _et
al._ (2021), at the $1\sigma$ level we obtain

${\rm{APV\,2021+COH-
CsI:}}\;\;\;\sin^{2}\vartheta_{W}=0.2423^{+0.0032}_{-0.0029},\;\;R_{n}(\mathrm{CsI})=5.5^{+0.4}_{-0.4}\;{\rm{fm}}.$
(25)

These results are depicted by the red data points in Fig. 5, where a summary
of the weak mixing angle measurements as a function of the energy scale Q is
shown along with the SM predicted running calculated in the
$\overline{\textrm{MS}}$ scheme. They represent an alternative derivation of
the weak mixing angle from APV that is fully data-driven and that keeps into
account the correlation with the value of $R_{n}$ determined simultaneously
using two electroweak probes, that are known to be practically model
independent. Indeed, the nominal derivation of the weak mixing angle from APV
data, that is reported in the PDG Workman  _et al._ (2022) and is depicted by
the grey point in Fig. 5, uses a value of $R_{n}$ that is extrapolated from
hadronic experiments using antiprotonic atoms, which are known to be affected
by considerable model dependencies. By comparing the two new determinations
reported in this work, it is possible to see that the weak mixing angle is
especially affected by the particular choice of the PNC amplitude, underlying
thus the importance for the future to clarify the discrepancies between the
two different approaches used in Refs. Sahoo  _et al._ (2021); Dzuba  _et al._
(2012).

Figure 5: Running of the weak mixing angle in the SM (green line) as a
function of the energy scale $Q$. The black experimental determination
represent the status of the art of the measurements at different energy scales
Workman  _et al._ (2022); Dzuba  _et al._ (2012); Anthony  _et al._ (2005);
Wang  _et al._ (2014); Zeller  _et al._ (2002); Androic  _et al._ (2018). The
red points show the determinations from the combined analysis of APV(Cs) and
COHERENT-CsI measurements retrieved in this work, which supersedes the nominal
APV determination depicted in grey Wood  _et al._ (1997).

###  III.3 Simultaneous determination of $\mathbf{R_{n}}$(I) vs
$\mathbf{R_{n}}$(Cs) and combined analysis with APV

|
---|---

Figure 6:  Constraints on the plane of $R_{n}(^{133}{\rm{Cs}})$ and
$R_{n}(^{127}{\rm{I}})$ together with their marginalizations, at different CLs
obtained fitting the COHERENT CsI data in combination with APV data, using the
value for the neutron skin corrections of Ref. Dzuba  _et al._ (2012) (a) and
Ref. Sahoo  _et al._ (2021) (b). The green lines indicate the corresponding
NSM prediction for the average rms neutron radius of Cs and I.

In this section, we describe the study of the correlation between
$R_{n}({\rm{Cs}})-R_{n}({\rm{I}})$ using the latest COHERENT CsI data alone
and combined with APV Cs to obtain the up-to-date and more accurate
constraints on both these quantities. In fact, since APV depends only on the
Cs neutron radius, while the COHERENT CsI result depends on both
$R_{n}{(^{133}{\rm{Cs}})}$ and $R_{n}{(^{127}{\rm{I}})}$, we are able to break
their degeneracy by fixing the weak mixing angle to its SM value. The result
of the COHERENT CsI analysis is reported in Fig. 3, where we show the contours
at different CLs in the plane of $R_{n}({\rm{Cs}})$ and $R_{n}({\rm{I}})$.
Namely, we get

${\rm{COH-
CsI:}}\;\;\;R_{n}({\rm{Cs}})=5.3^{+1.3}_{-1.2}\;{\rm{fm}},\;\;R_{n}(\mathrm{I})=5.6^{+1.6}_{-1.2}\;{\rm{fm}}.$
(26)

As expected, COHERENT CsI data alone does not allow to disentangle the two
contributions, motivating the need to perform a combined analysis with APV
data. The results of the combined analysis are reported in Fig. 6 and Fig. 6,
using the experimental value of $Q_{W}$ obtained with the theoretical
predictions of the PNC amplitude considered in this work. In these two
scenarios we obtain

$\displaystyle{\rm{APV\,PDG+COH-CsI:}}\;\;\;R_{n}(\mathrm{Cs})$
$\displaystyle=$ $\displaystyle
5.29^{+0.31}_{-0.34}\;{\rm{fm}},\;\;R_{n}(\mathrm{I})=5.6^{+1.0}_{-0.8}\;{\rm{fm}},$
(27) $\displaystyle{\rm{APV\,2021+COH-CsI:}}\;\;\;R_{n}(\mathrm{Cs})$
$\displaystyle=$ $\displaystyle
4.85^{+0.30}_{-0.25}\;{\rm{fm}},\;\;R_{n}(\mathrm{I})=6.0^{+0.9}_{-0.9}\;{\rm{fm}}.$
(28)

The corresponding neutron skins are, respectively, $\Delta
R_{\rm{np}}({\rm{Cs}})=0.20^{+0.31}_{-0.34}\;{\rm{fm}}$-$\Delta
R_{\rm{np}}({\rm{I}})=0.57^{+1.0}_{-0.8}\;{\rm{fm}}$ and $\Delta
R_{\rm{np}}({\rm{Cs}})=-0.24^{+0.30}_{-0.25}\;{\rm{fm}}$-$\Delta
R_{\rm{np}}({\rm{I}})=1.0^{+0.9}_{-0.9}\;{\rm{fm}}$. Also in this case the
usage of the different PNC amplitudes play a major role and with the second
analysis, the slightly larger value of the iodium rms neutron radius is
compensated by a significantly smaller value of $R_{n}(\textrm{Cs})$, that
translates in an almost-zero neutron skin for cesium, with smaller
uncertainties than those of the first analysis. Moreover, in all the
scenarios, the central values suggest that
$R_{n}(\textrm{I})>R_{n}(\textrm{Cs})$, while all theoretical models (see e.g.
Table I of Ref. Cadeddu  _et al._ (2021b)) predicts the opposite. We thus
redetermine these measurements after imposing the well-motivated constraint
$R_{n}(\textrm{I})\leq R_{n}(\textrm{Cs})$. In this case the measurements
performed in this section become

$\displaystyle{\rm{COH-CsI}}\,[R_{n}(\textrm{I})\leq R_{n}(\textrm{Cs})]$
$\displaystyle:$
$\displaystyle\;\;\;R_{n}({\rm{Cs}})=5.5^{+1.1}_{-0.4}\;{\rm{fm}},\;\;R_{n}(\mathrm{I})=5.4^{+0.4}_{-1.0}\;{\rm{fm}},$
(29) $\displaystyle{\rm{APV\,PDG+COH-CsI}}\,[R_{n}(\textrm{I})\leq
R_{n}(\textrm{Cs})]$ $\displaystyle:$
$\displaystyle\;\;\;R_{n}(\mathrm{Cs})=5.32^{+0.30}_{-0.23}\;{\rm{fm}},\;\;R_{n}(\mathrm{I})=5.30^{+0.30}_{-0.6}\;{\rm{fm}},$
(30) $\displaystyle{\rm{APV\,2021+COH-CsI}}\,[R_{n}(\textrm{I})\leq
R_{n}(\textrm{Cs})]$ $\displaystyle:$
$\displaystyle\;\;\;R_{n}(\mathrm{Cs})=5.07^{+0.21}_{-0.26}\;{\rm{fm}},\;\;R_{n}(\mathrm{I})=5.06^{+0.22}_{-0.4}\;{\rm{fm}}.$
(31)

Imposing this constraint, we achieve an uncertainty as low as 4% on
$R_{n}(\textrm{Cs})$. The corresponding constraints on the plane of
$R_{n}(^{133}{\rm{Cs}})$ and $R_{n}(^{127}{\rm{I}})$ together with their
marginalizations, at different CLs can be found in Appendix A.

###  III.4 Overview of the results on $\mathbf{R_{n}}$

Given the vast amount of measurements of the neutron rms radius distribution
presented in this work under different hypotheses, we summarised all of them
in Fig. 7(a) and (b) when using APV with the PNC amplitude from Ref. Dzuba
_et al._ (2012) or from Ref. Sahoo  _et al._ (2021), respectively. Despite the
different fit configurations used to extract the values of
$R_{n}(\mathrm{CsI})$, $R_{n}(\mathrm{Cs})$ and $R_{n}(\mathrm{I})$, a
coherent picture emerges with an overall agreement between the COHERENT and
APV results and the theoretical predictions. However, we would like to note
that using APV PDG we obtain on average larger values on the radii, even if
still compatible within uncertainties. On the contrary, APV 2021 shifts
downwards the measured radii towards the predictions, but in the simultaneous
2D fit with $\sin^{2}\vartheta_{\text{W}}$ where the correlation with the
latter increases the extracted central value of $R_{n}(\mathrm{CsI})$.
Moreover, we checked the impact of using a different quenching factor, by
comparing our nominal results obtained using Refs. Akimov  _et al._ (2022a, c)
and the derivation in Ref. Lewis and Collar (2021). The latter lower QF
decreases the total number of CE$\nu$NS events resulting in a smaller
$R_{n}(\mathrm{CsI})$ by about 10%.

---

---

Figure 7: Overview of the different results presented in this work obtained
using CE$\nu$NS CsI COHERENT (COH) data and APV with the PNC amplitude (a)
from Ref. Dzuba  _et al._ (2012) (APV PDG) or (b) from Ref. Sahoo  _et al._
(2021) (APV 2021) as well as their combination on $R_{n}(\mathrm{CsI})$ and
$R_{n}(\mathrm{Cs})$ when fixing the weak mixing angle to $\hat{s}^{2}_{0}$
(black circles), on $R_{n}(\mathrm{CsI})$ when fitting simultaneously for
$R_{n}$ and $\sin^{2}\vartheta_{\text{W}}$ (red up triangles), on
$R_{n}(\mathrm{Cs})$ and $R_{n}(\mathrm{I})$ when fixing the weak mixing angle
to $\hat{s}^{2}_{0}$ (blue down triangles) as well as when imposing the
constraint $R_{n}(\mathrm{Cs})\geq R_{n}(\mathrm{I})$ (green squares). The
yellow and orange areas represent the regions where the theoretical
predictions of $R_{n}(\mathrm{Cs})$ and $R_{n}(\mathrm{I})$ can be found, as
taken from Table I of Ref. Cadeddu  _et al._ (2021b) and from NSM calculations
in Ref. Hoferichter  _et al._ (2020).

##  IV Future perspectives

In this section, we describe a sensitivity study that we performed to outline
the potentialities of CE$\nu$NS to measure the CsI neutron radius and the weak
mixing angle at the SNS, in the context of the COHERENT experimental program.
As shown in the previous sections and in several previous papers (see, e.g.,
the review in Ref. Abdullah  _et al._ (2022)), CE$\nu$NS is a very powerful
tool, being a process which turned out to be very versatile in putting
constraints on a variety of parameters. Nevertheless, the current level of
accuracy of CE$\nu$NS in the determination of the neutron radius, both the CsI
one reported in this work as well as the argon (Ar) one reported in Ref.
Cadeddu  _et al._ (2020a), is still lower with respect to that obtained using
parity-violating electron scattering on similar nuclei. This is visible in
Fig. 8 where we show the current status for different neutron distribution
radii measured via diverse electroweak probes. As shown, the precision
achieved by the PREX Abrahamyan  _et al._ (2012); Adhikari  _et al._ (2021);
Becker  _et al._ (2018) and CREX Adhikari  _et al._ (2022) experiments is
indeed greater than that obtained through CE$\nu$NS for CsI and Ar Cadeddu
_et al._ (2018b, 2020b, 2021b, 2020a); Akimov  _et al._ (2022b).

Luckily, this is not the end of the story. The COHERENT collaboration has
additional existing and planned near-future deployments in the Neutrino Alley
at the SNS with exciting physics potential. In particular, the experimental
program under development includes a tonne-scale liquid argon time-projection
chamber detector as well as a large scale CsI cryogenic detector. These new
detectors, together with planned upgrades to the SNS proton beam, will further
broaden and deepen the physics reach of the COHERENT experiment. Moreover, the
European Spallantion Source (ESS) is currently under construction in Lund,
Sweden Abele  _et al._ (2022). At design specifications, the ESS will operate
at 5 MW using a proton linac with a beam energy of 2 GeV. In addition to
providing the most intense neutron beams, the ESS also provides a large
neutrino flux and so it will be exploited to study CE$\nu$NS, using in
particular a 31.5 kg CsI target kept at 80 K Abele  _et al._ (2022); Baxter
_et al._ (2020). Finally, a new CE$\nu$NS detection experiment is under
construction in China, where undoped CsI crystals coupled with two photon
multiplier tubes each, will be cooled down to 77 K and placed at the China
Spallation Neutron Source (CSNS) Su  _et al._ (2023).

In this work, we performed a sensitivity study using COHERENT plans as
described in Ref. Akimov  _et al._ (2022b). However, similar conclusions and
prospects can also be drawn for the already mentioned CsI detectors expected
at the ESS and the CSNS as they foresee similar technologies. The aim of this
study is to assess to which extent CE$\nu$NS will be competitive in the
future. To better appreciate the reach of the CE$\nu$NS program, we compare
our COHERENT projections with other competitive measurements that are foreseen
for the neutron radius of different nuclei and for the weak mixing angle. In
this way, we obtain a complete and comprehensive review of the current and
future status of knowledge on these fundamental quantities. The first upgrade
of the SNS is planned for 2025, where the proton beam energy $(E_{p})$ will be
increased up to $1.3\;\rm{GeV}$ with respect to the current $0.984\;\rm{GeV}$.
Moreover, the beam power $P_{\rm{beam}}$ will increase to $2\;\rm{MW}$,
compared to the current $1.4\;\rm{MW}$ so that the number of neutrinos per
flavor produced for each proton-on-target (POT) will increase to a value of
0.12. A second target station is planned in the 2030s, for a final power of
$2.8\;\rm{MW}$. Using this information, we evaluate the number of proton-on-
target ($N_{\rm{POT}}$) expected, that describe the intensity of the neutrino
flux (see e.g. Ref. Atzori Corona  _et al._ (2022a)), which is given by

$N_{\rm{POT}}=t_{\rm{exp}}\frac{P_{\rm{beam}}}{E_{p}},$ (32)

$t_{\rm{exp}}$ being the running time of the experiment. This means that the
future SNS upgrade will be able to get a much higher neutrino flux for each
neutrino flavor with respect to the current configuration.

Figure 8: Current status and future projections for neutron distribution radii
of different nuclei measured via electroweak probes. The upper plots show the
current and foreseen measurements of $R_{n}(\mathrm{Cs}$ from CE$\nu$NS with
CsI crystal detectors Cadeddu  _et al._ (2018b, 2020b, 2021b) (left) and of
$R_{n}(\mathrm{Ar}$ liquid Ar detectors Cadeddu  _et al._ (2020a); Akimov  _et
al._ (2022b) (right), compared to the lower plots where the current and
foreseen measurements from parity-violating electron scattering are shown, for
the case of Pb Abrahamyan  _et al._ (2012); Adhikari  _et al._ (2021) (left)
and Ca Adhikari  _et al._ (2022) (right). For CsI, similar uncertainties are
expected to be achieved thanks to the detectors planned at the ESS Abele  _et
al._ (2022) and at the CSNS Su  _et al._ (2023).

See the text for more details.

The so-called COH-CryoCsI-I experiment, scheduled for 2025, will have a mass
of about 10 kg and will exploit an undoped CsI crystal at cryogenic
temperature $(\sim 40\rm{K})$, which would permit to use SiPM arrays instead
of PMTs in order to remove the Cherenkov radiation background emitted by the
latter. Moreover, the undoped CsI crystals at cryogenic temperature have
approximately twice the light yield of the CsI[Na] crystal at $300~{}\rm{K}$.
The following upgrade will be the COH-CryoCsI-II experiment, planned in the
2030s, that will operate in similar conditions with a 700 kg undoped CsI
detector. Both the COH-CryoCsI-I and COH-CryoCsI-II detectors will be able to
lower the energy threshold, which is a fundamental requirement for CE$\nu$NS
precision physics. Following Ref. Akimov  _et al._ (2022b), we considered a
threshold of $1.4\;\rm{keV}_{nr}$ while keeping the shape of the energy
efficiency unchanged. In addition, the systematic uncertainty on the neutrino
flux will be strongly reduced thanks to the planned $\rm{D}_{2}O$ detector and
will approach 4.7(2)$\%$ statistical uncertainty after 2(5) SNS-years of
operation. In this sensitivity study, we will use the value of 4.7$\%$ for the
statistical uncertainty on the neutrino flux for the Cryo-CsI-I detector,
while we use 2$\%$ for the Cryo-CsI-II detector. Another fundamental
ingredient is the quenching factor of the CsI crystal, whose behavior at
cryogenic temperature is currently under investigation. It will be directly
measured by the COHERENT collaboration, but for this sensitivity study, we
will consider a quenching factor of 5$\%$ as reported in Ref. Akimov  _et al._
(2022b) and as it has already been established in Ref. Lewis and Collar
(2021). Given the conservative value used, we do not consider a systematic
error on the quenching factor. Thus, the total systematic uncertainty
considered for the CE$\nu$NS prediction is $\sigma_{\rm{CE}\nu\rm{NS}}=0.062$
for the Cryo-CsI-I detector and $\sigma_{\rm{CE}\nu\rm{NS}}=0.046$ for the
Cryo-CsI-II detector. Assuming three years of data taking, $\sim 2\cdot
10^{3}$ and $\sim 2\cdot 10^{5}$ events are expected for COH-CryoCsI-I and
COH-CryoCsI-II111For comparison, at the ESS a compact 31.5 kg cryogenic CsI
detector is expected to measure $\sim 1.2\cdot 10^{4}$ CE$\nu$NS events per
year Abele  _et al._ (2022)., respectively, to be compared with the current
CsI available statistics of $\sim 300$ events. Given the high statistics of
these detectors, we find that the choice of using a poissonian or a gaussian
definition for the least square function leads to almost identical results.
Moreover, in order to keep into account the timing information in the
sensitivity study, we used the least-square function definition in Eq. (10),
using the same time binning of the data and time efficiency of the latest CsI
data release analyzed in this paper. In this sensitivity study, we considered
the SS background obtained by rescaling the one measured by the current CsI
detector for the exposure time and the mass of COH-CryoCsI-I and COH-CryoCsI-
II, respectively. Given that the SS background is unknown for energies smaller
than the actual COHERENT energy threshold, we simply extended the background
currently measured by the CsI COHERENT detector also to the first energy bin.

In Fig. 8, we report the projections on the neutron radius of CsI as obtained
from this sensitivity study, along with a comprehensive review of the current
and future status of different neutron radius measurements on various nuclei
using other weak probes. We find that COH-CryoCsI-I will be able to measure
the neutron rms CsI radius with a precision of
$\sigma(R_{n}\textrm{(CsI)})=0.19\,\rm{fm}$ corresponding to a relative
accuracy of about $4\%$222For these studies, we consider a value of
$R_{n}\textrm{(CsI)}=5.06\;\rm{fm}$ as provided by the NSM calculations. See
Sec. II for more information.. Similarly, for the COH-CryoCsI-II scenario we
obtain a sensitivity projection for $R_{n}\rm{(CsI)}$ that corresponds to
$\sigma(R_{n}\textrm{(CsI)})=0.023\,\rm{fm}$, meaning that COH-CryoCsI-II will
be able to reach a per-mille accuracy level, i.e., about $0.5\%$. It is worth
noticing that, in this regime, the projected uncertainty on the neutron radius
will become smaller than the difference between the Cs and I radius, expected
to be $\sim 0.06\,\rm{fm}$. Thus, it will be of paramount importance to keep
into account the different contributions of Cs and I by performing a
simultaneous fit on these two quantities as we did in Sec. III.3. This
precision, which is also expected to be achieved at the ESS with a similar
amount of foreseen CE$\nu$NS events, will represent an unprecedented window
into nuclear physics, making CE$\nu$NS very competitive with respect to the
other weak probes. Specifically, the other available and currently world-
leading measurements on the neutron radius of heavy and neutron-rich nuclei
come from parity-violating electron scattering as shown in the lower left
panel of Fig. 8, for the case of ${}^{208}\rm{Pb}$ Abrahamyan  _et al._
(2012); Adhikari  _et al._ (2021) as measured by PREX-I and PREX-II,
respectively. It is worth noticing that the MREX experiment Becker  _et al._
(2018) also plans to measure the ${}^{208}\rm{Pb}$ neutron radius with an
accuracy of about $0.5\%$. Talking about lighter nuclei, the currently
available measurements come from COHERENT for ${}^{40}\rm{Ar}$ Akimov  _et
al._ (2021) and for ${}^{48}\rm{Ca}$ Adhikari  _et al._ (2022) from the CREX
experiment exploiting again parity-violating electron scattering. The COHERENT
collaboration foresees to measure $R_{n}(\rm{Ar})$ to 4.6$\%$ with the
upgraded tonne scale argon detector (see also Ref. Sierra (2023)), so-called
COH-Ar-750 Akimov  _et al._ (2022b), as shown in the upper right panel of Fig.
8.

In Fig. 9, we report the projections on the weak mixing angle as obtained from
this sensitivity study, along with a comprehensive review of the current and
future measurements that are known for an energy scale below $100\
\mathrm{MeV}$. In particular we depicted the evolution of the APV
determination in the last decade Amsler  _et al._ (2008); Nakamura  _et al._
(2010); Olive  _et al._ (2014); Zyla  _et al._ (2020), that moved
significantly due to different theoretical re-evaluations, the value of the
weak mixing angle extracted from the CsI COHERENT in Ref. Cadeddu  _et al._
(2021b) and in this work as well as the values obtained from the combination
with APV. As it can be seen, many measurements of
$\sin^{2}\vartheta_{\text{W}}$ are expected in the near future in the low
energy sector, as those coming from the P2 Becker  _et al._ (2018); Dev  _et
al._ (2021) and MOLLER Benesch  _et al._ (2014) experiments, and from future
CE$\nu$NS experiments (CO$\nu$US Lindner  _et al._ (2017), TEXONO Wong  _et
al._ (2007), CONNIE Aguilar-Arevalo  _et al._ (2016), and MI$\nu$ER Agnolet
_et al._ (2017)) that will be really powerful for further constraining such a
quantity. It is worth noticing that CE$\nu$NS from reactor antineutrinos
already proved to be able to provide a determination of the weak mixing angle.
Indeed, this has been shown in Ref. Atzori Corona  _et al._ (2022b);
Aristizabal Sierra  _et al._ (2022), even if the uncertainty so far is still
too large to be depicted in Fig. 9. In this scenario, CE$\nu$NS determinations
with CsI will help with both the Cryo-CsI-I detector that will reach a
precision of about $\sigma(\sin^{2}\vartheta_{W})=0.012$ and in particular
with COH-CryoCsI-II, where a precision of about
$\sigma(\sin^{2}\vartheta_{W})=0.007$ will be achieved. Similar precision are
also expected to be achieved by the other large cryogenic CsI targets
measuring CE$\nu$NS as highlighted in this section. The sensitivity projection
on the weak mixing angle for the Cryo-CsI-I detector has been reported also in
Ref. Akimov  _et al._ (2022b) by the COHERENT collaboration, where a slightly
better precision corresponding to $\sigma(\sin^{2}\vartheta_{W})\sim 0.009$
has been found. The different result can be explained considering that the
sensitivity to the weak mixing angle depends strongly on the values of
$R_{n}(\rm{Cs})$ and $R_{n}(\rm{I})$ used to describe the loss of coherence
for increasing recoil energies. The values from the NSM calculations adopted
in our work differ from the significantly larger value used in the
aforementioned work, which seems to be $R_{n}(\mathrm{CsI})\sim 6\;$fm. We
verified that we are able to obtain a better agreement with their projections
using the latter value for the nuclear radius. When more data will become
available, it will be therefore essential to perform a simultaneous
determination of these parameters, as investigated in this work.

Figure 9: Current status and future projections for weak mixing angle
measurements below $Q\lesssim 100\ \mathrm{MeV}$. The gray points show the
measurements from APV on cesium atoms during the years Amsler  _et al._
(2008); Nakamura  _et al._ (2010); Olive  _et al._ (2014); Zyla  _et al._
(2020); Workman  _et al._ (2022). The brown measurements refer to the COHERENT
only and the combination between COHERENT and APV as determined in 2020 in our
previous work (see Ref. Cadeddu  _et al._ (2021b)), while the dark blue and
red points refer to the updated measurements reported in this work. The
projections for future CE$\nu$NS experiments (CO$\nu$US Lindner  _et al._
(2017), TEXONO Wong  _et al._ (2007), CONNIE Aguilar-Arevalo  _et al._ (2016),
and MI$\nu$ER Agnolet  _et al._ (2017)), shown by the light blue triangles and
dashed error bars, are taken from Ref. Cañas  _et al._ (2018). The purple
triangles are the projections for the future electron scattering experiment
MOLLER Benesch  _et al._ (2014) and P2@MESA Becker  _et al._ (2018); Dev  _et
al._ (2021). The dark blue triangles with dashed error bars are the
projections for the future CryoCsI-I and CryoCsI-II determinations as derived
in this work. Similar uncertainties are expected to be achieved thanks to the
detectors planned at the ESS Abele  _et al._ (2022) and at the CSNS Su  _et
al._ (2023).

In the inset in the top left, a zoom of the y axes is shown to better
appreciate the statistical uncertainties of the reported measurements and
projections, removing the measurements from COHERENT-only which suffer from
larger uncertainties.

##  V Conclusions

Motivated by the recent update of the observation of CE$\nu$NS using a 14.6 kg
CsI detector by the COHERENT collaboration, we provide in this manuscript a
complete and in-depth legacy determination of the average neutron rms radius
of ${}^{133}\mathrm{Cs}$ and ${}^{127}\mathrm{I}$ and of the weak mixing
angle. To do so, we take advantage of the knowledge developed in the latest
years, employing the most up-to-date and accurate description of the CE$\nu$NS
data, profiting from a precise determination of the radiative corrections, the
inclusion of the neutrino arrival time information, a refined quenching factor
derivation, and an appropriate least-square function definition.
Interestingly, we show that the COHERENT CsI data show a 6$\sigma$ evidence of
the nuclear structure suppression of the full coherence. Moreover, we also
perform the combination with the APV experimental result, that allows us to
disentangle the contributions of the ${}^{133}\mathrm{Cs}$ and
${}^{127}\mathrm{I}$ nuclei. A precision as low as $\sim$5(4)$\%$ is obtained
on $R_{n}(\mathrm{Cs})$ leaving it free to vary in the fit (imposing the
constraint $R_{n}(\mathrm{Cs})\geq R_{n}(\mathrm{I})$). The combination of
APV+COHERENT impacts in particular the determination of the weak mixing angle
and allows us to obtain a data-driven measurement of the low-energy weak
mixing angle with a percent uncertainty, independent of the value of the
average neutron rms radius of ${}^{133}\mathrm{Cs}$ and ${}^{127}\mathrm{I}$,
that is allowed to vary freely in the fit. In all the APV-related results, we
exploit two different derivations of the theoretical PNC amplitude, showing
that the particular choice can make a difference as large as 6.5% on
$R_{n}(\mathrm{Cs})$ and 11% on the weak mixing angle, underlying thus the
importance for the future to clarify the discrepancies between the two
different approaches used in Refs. Sahoo  _et al._ (2021) and Dzuba  _et al._
(2012). At the time of writing, the CsI COHERENT detector has been dismantled,
but a new data taking with an upgraded experiment is foreseen in 5 years.
Thus, in light of the recent announcement of a future deployment of a 10 kg
and a $\sim$700 kg cryogenic CsI detectors, we provide future prospects for
these measurements thanks to a sensitive study that we performed considering
all the foreseen improvements. We compared the uncertainties forecasted for
$R_{n}(\mathrm{CsI})$ and $\sin^{2}\vartheta_{W}$, which are
$\sigma(R_{n}\textrm{(CsI)})=0.023\,\rm{fm}$ and
$\sigma(\sin^{2}\vartheta_{W})=0.007$ for the $\sim$700 kg configuration, with
different future measurements of the same quantities using weak probes,
highlighting thus the impact that future CE$\nu$NS measurements will provide.

##  Appendix A Contours and marginalizations of $R_{n}(^{133}{\rm{Cs}})$ and
$R_{n}(^{127}{\rm{I}})$ with the constraint $R_{n}({\rm{Cs}})\geq
R_{n}({\rm{I}})$

In this appendix, we list the constraints on the plane of
$R_{n}(^{133}{\rm{Cs}})$ and $R_{n}(^{127}{\rm{I}})$ together with their
marginalizations, at different CLs, obtained fitting the COHERENT CsI data
alone, Fig. 10, and in combination with APV data, Figs. 10 and 10, imposing
the constraint $R_{n}({\rm{Cs}})\geq R_{n}({\rm{I}})$. The results in
combination with APV have been obtained using the value for the neutron skin
correction of Ref. Dzuba  _et al._ (2012) and the value recently calculated in
Ref. Sahoo  _et al._ (2021), respectively.

|
---|---

---

Figure 10:  Constraints on the plane of $R_{n}(^{133}{\rm{Cs}})$ and
$R_{n}(^{127}{\rm{I}})$ together with their marginalizations, at different CLs
obtained fitting the COHERENT CsI data alone (a) and in combination with APV
data (b) and (c), using the value for the neutron skin corrections of Ref.
Dzuba  _et al._ (2012) (b) and Ref. Sahoo  _et al._ (2021) (c). In all cases,
we impose the constraint $R_{n}({\rm{Cs}})\geq R_{n}({\rm{I}})$. The green
lines indicate the corresponding NSM prediction for the average rms neutron
radius of Cs and I.

## References

  * Freedman (1974) Daniel Z. Freedman, “Coherent effects of a weak neutral current,” Phys. Rev. D 9, 1389–1392 (1974).
  * Akimov  _et al._ (2017) D. Akimov _et al._ (COHERENT), “Observation of Coherent Elastic Neutrino-Nucleus Scattering,” Science 357, 1123–1126 (2017), arXiv:1708.01294 [nucl-ex] .
  * Akimov  _et al._ (2018) D. Akimov _et al._ (COHERENT), “COHERENT Collaboration data release from the first observation of coherent elastic neutrino-nucleus scattering,”  (2018), 10.5281/zenodo.1228631, arXiv:1804.09459 [nucl-ex] .
  * Akimov  _et al._ (2021) D. Akimov _et al._ (COHERENT), “First Measurement of Coherent Elastic Neutrino-Nucleus Scattering on Argon,” Phys. Rev. Lett. 126, 012002 (2021), arXiv:2003.10630 [nucl-ex] .
  * Akimov  _et al._ (2020) D. Akimov _et al._ (COHERENT), “COHERENT Collaboration data release from the first detection of coherent elastic neutrino-nucleus scattering on argon,”  (2020), 10.5281/zenodo.3903810, arXiv:2006.12659 [nucl-ex] .
  * Akimov  _et al._ (2022a) D. Akimov _et al._ (COHERENT), “Measurement of the Coherent Elastic Neutrino-Nucleus Scattering Cross Section on CsI by COHERENT,” Phys. Rev. Lett. 129, 081801 (2022a), arXiv:2110.07730 [hep-ex] .
  * Colaresi  _et al._ (2022) J. Colaresi, J. I. Collar, T. W. Hossbach, C. M. Lewis, and K. M. Yocum, “Measurement of Coherent Elastic Neutrino-Nucleus Scattering from Reactor Antineutrinos,” Phys. Rev. Lett. 129, 211802 (2022), arXiv:2202.09672 [hep-ex] .
  * Liao and Marfatia (2017) Jiajun Liao and Danny Marfatia, “COHERENT constraints on nonstandard neutrino interactions,” Phys. Lett. B 775, 54–57 (2017), arXiv:1708.04255 [hep-ph] .
  * Denton  _et al._ (2018) Peter B. Denton, Yasaman Farzan, and Ian M. Shoemaker, “Testing large non-standard neutrino interactions with arbitrary mediator mass after COHERENT data,” JHEP 07, 037 (2018), arXiv:1804.03660 [hep-ph] .
  * Lindner  _et al._ (2017) Manfred Lindner, Werner Rodejohann, and Xun-Jie Xu, “Coherent Neutrino-Nucleus Scattering and new Neutrino Interactions,” JHEP 03, 097 (2017), arXiv:1612.04150 [hep-ph] .
  * Flores  _et al._ (2020) L. J. Flores, Newton Nath, and Eduardo Peinado, “Non-standard neutrino interactions in U(1)’ model after COHERENT data,” JHEP 06, 045 (2020), arXiv:2002.12342 [hep-ph] .
  * Coloma  _et al._ (2017) Pilar Coloma, M. C. Gonzalez-Garcia, Michele Maltoni, and Thomas Schwetz, “COHERENT Enlightenment of the Neutrino Dark Side,” Phys. Rev. D 96, 115007 (2017), arXiv:1708.02899 [hep-ph] .
  * Giunti (2020) C. Giunti, “General COHERENT constraints on neutrino nonstandard interactions,” Phys. Rev. D 101, 035039 (2020), arXiv:1909.00466 [hep-ph] .
  * Coloma  _et al._ (2022) Pilar Coloma, Ivan Esteban, M. C. Gonzalez-Garcia, Leire Larizgoitia, Francesc Monrabal, and Sergio Palomares-Ruiz, “Bounds on new physics with data of the Dresden-II reactor experiment and COHERENT,” JHEP 05, 037 (2022), arXiv:2202.10829 [hep-ph] .
  * Cadeddu  _et al._ (2021a) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, and Y. Y. Zhang, “Constraints on light vector mediators through coherent elastic neutrino nucleus scattering data from COHERENT,” JHEP 01, 116 (2021a), arXiv:2008.05022 [hep-ph] .
  * Atzori Corona  _et al._ (2022a) M. Atzori Corona, M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, C. A. Ternes, and Y. Y. Zhang, “Probing light mediators and ($(g-2)_{\mu}$ through detection of coherent elastic neutrino nucleus scattering at COHERENT,” JHEP 05, 109 (2022a), arXiv:2202.11002 [hep-ph] .
  * Banerjee  _et al._ (2021) Heerak Banerjee, Bhaskar Dutta, and Sourov Roy, “Probing L$\mu$-L$\tau$ models with CE$\nu$NS: A new look at the combined COHERENT CsI and Ar data,” Phys. Rev. D 104, 015015 (2021), arXiv:2103.10196 [hep-ph] .
  * Bertuzzo  _et al._ (2022) Enrico Bertuzzo, Giovanni Grilli di Cortona, and Lucas Magno D. Ramos, “Probing light vector mediators with coherent scattering at future facilities,” JHEP 06, 075 (2022), arXiv:2112.04020 [hep-ph] .
  * Kosmas  _et al._ (2015) T. S. Kosmas, O. G. Miranda, D. K. Papoulias, M. Tortola, and J. W. F. Valle, “Probing neutrino magnetic moments at the Spallation Neutron Source facility,” Phys. Rev. D 92, 013011 (2015), arXiv:1505.03202 [hep-ph] .
  * Billard  _et al._ (2018) Julien Billard, Joseph Johnston, and Bradley J. Kavanagh, “Prospects for exploring New Physics in Coherent Elastic Neutrino-Nucleus Scattering,” JCAP 11, 016 (2018), arXiv:1805.01798 [hep-ph] .
  * Cadeddu  _et al._ (2018a) Matteo Cadeddu, Carlo Giunti, Konstantin A. Kouzakov, Yu-Feng Li, Yi-Yu Zhang, and Alexander I. Studenikin, “Neutrino Charge Radii From Coherent Elastic Neutrino-nucleus Scattering,” Phys. Rev. D 98, 113010 (2018a), [Erratum: Phys.Rev.D 101, 059902 (2020)], arXiv:1810.05606 [hep-ph] .
  * Atzori Corona  _et al._ (2022b) M. Atzori Corona, M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, C. A. Ternes, and Y. Y. Zhang, “Impact of the Dresden-II and COHERENT neutrino scattering data on neutrino electromagnetic properties and electroweak physics,” JHEP 09, 164 (2022b), arXiv:2205.09484 [hep-ph] .
  * Cadeddu  _et al._ (2020a) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, and Y. Y. Zhang, “Physics results from the first COHERENT observation of coherent elastic neutrino-nucleus scattering in argon and their combination with cesium-iodide data,” Phys. Rev. D 102, 015030 (2020a), arXiv:2005.01645 [hep-ph] .
  * Miranda  _et al._ (2020) O. G. Miranda, D. K. Papoulias, G. Sanchez Garcia, O. Sanders, M. Tórtola, and J. W. F. Valle, “Implications of the first detection of coherent elastic neutrino-nucleus scattering (CEvNS) with Liquid Argon,” JHEP 05, 130 (2020), [Erratum: JHEP 01, 067 (2021)], arXiv:2003.12050 [hep-ph] .
  * Cadeddu  _et al._ (2021b) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, C. A. Ternes, and Y. Y. Zhang, “New insights into nuclear physics and weak mixing angle using electroweak probes,” Phys. Rev. C 104, 065502 (2021b), arXiv:2102.06153 [hep-ph] .
  * Cadeddu  _et al._ (2018b) M. Cadeddu, C. Giunti, Y. F. Li, and Y. Y. Zhang, “Average CsI neutron density distribution from COHERENT data,” Phys. Rev. Lett. 120, 072501 (2018b), arXiv:1710.02730 [hep-ph] .
  * Papoulias  _et al._ (2020) D. K. Papoulias, T. S. Kosmas, R. Sahu, V. K. B. Kota, and M. Hota, “Constraining nuclear physics parameters with current and future COHERENT data,” Phys. Lett. B 800, 135133 (2020), arXiv:1903.03722 [hep-ph] .
  * Papoulias and Kosmas (2018) D. K. Papoulias and T. S. Kosmas, “COHERENT constraints to conventional and exotic neutrino physics,” Phys. Rev. D 97, 033003 (2018), arXiv:1711.09773 [hep-ph] .
  * Cadeddu  _et al._ (2020b) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, and Y. Y. Zhang, “Neutrino, electroweak, and nuclear physics from COHERENT elastic neutrino-nucleus scattering with refined quenching factor,” Phys. Rev. D 101, 033004 (2020b), arXiv:1908.06045 [hep-ph] .
  * Papoulias (2020) Dimitrios K. Papoulias, “COHERENT constraints after the COHERENT-2020 quenching factor measurement,” Phys. Rev. D 102, 113004 (2020), arXiv:1907.11644 [hep-ph] .
  * Khan and Rodejohann (2019) Amir N. Khan and Werner Rodejohann, “New physics from COHERENT data with an improved quenching factor,” Phys. Rev. D 100, 113003 (2019), arXiv:1907.12444 [hep-ph] .
  * Dutta  _et al._ (2019) Bhaskar Dutta, Shu Liao, Samiran Sinha, and Louis E. Strigari, “Searching for Beyond the Standard Model Physics with COHERENT Energy and Timing Data,” Phys. Rev. Lett. 123, 061801 (2019), arXiv:1903.10666 [hep-ph] .
  * Aristizabal Sierra  _et al._ (2018) D. Aristizabal Sierra, Valentina De Romeri, and N. Rojas, “COHERENT analysis of neutrino generalized interactions,” Phys. Rev. D 98, 075018 (2018), arXiv:1806.07424 [hep-ph] .
  * Cadeddu and Dordei (2019) Matteo Cadeddu and Francesca Dordei, “Reinterpreting the weak mixing angle from atomic parity violation in view of the Cs neutron rms radius measurement from COHERENT,” Phys. Rev. D 99, 033010 (2019), arXiv:1808.10202 [hep-ph] .
  * Dutta  _et al._ (2020) Bhaskar Dutta, Doojin Kim, Shu Liao, Jong-Chul Park, Seodong Shin, and Louis E. Strigari, “Dark matter signals from timing spectra at neutrino experiments,” Phys. Rev. Lett. 124, 121802 (2020), arXiv:1906.10745 [hep-ph] .
  * Abdullah  _et al._ (2018) Mohammad Abdullah, James B. Dent, Bhaskar Dutta, Gordon L. Kane, Shu Liao, and Louis E. Strigari, “Coherent elastic neutrino nucleus scattering as a probe of a Z’ through kinetic and mass mixing effects,” Phys. Rev. D 98, 015005 (2018), arXiv:1803.01224 [hep-ph] .
  * Ge and Shoemaker (2018) Shao-Feng Ge and Ian M. Shoemaker, “Constraining Photon Portal Dark Matter with Texono and Coherent Data,” JHEP 11, 066 (2018), arXiv:1710.10889 [hep-ph] .
  * Miranda  _et al._ (2021) O. G. Miranda, D. K. Papoulias, O. Sanders, M. Tórtola, and J. W. F. Valle, “Low-energy probes of sterile neutrino transition magnetic moments,” JHEP 12, 191 (2021), arXiv:2109.09545 [hep-ph] .
  * De Romeri  _et al._ (2022) V. De Romeri, O. G. Miranda, D. K. Papoulias, G. Sanchez Garcia, M. Tórtola, and J. W. F. Valle, “Physics implications of a combined analysis of COHERENT CsI and LAr data,” (2022), arXiv:2211.11905 [hep-ph] .
  * Huang and Chen (2019) Xu-Run Huang and Lie-Wen Chen, “Neutron Skin in CsI and Low-Energy Effective Weak Mixing Angle from COHERENT Data,” Phys. Rev. D 100, 071301 (2019), arXiv:1902.07625 [hep-ph] .
  * Shen  _et al._ (2020) Hong Shen, Fan Ji, Jinniu Hu, and Kohsuke Sumiyoshi, “Effects of symmetry energy on equation of state for simulations of core-collapse supernovae and neutron-star mergers,” Astrophys. J. 891, 148 (2020), arXiv:2001.10143 [nucl-th] .
  * Horowitz (2019) C. J. Horowitz, “Neutron rich matter in the laboratory and in the heavens after GW170817,” Annals Phys. 411, 167992 (2019), arXiv:1911.00411 [astro-ph.HE] .
  * Wei  _et al._ (2020) Jin-Biao Wei, Jia-Jing Lu, G. F. Burgio, Z. H. Li, and H. J Schulze, “Are nuclear matter properties correlated to neutron star observables?” Eur. Phys. J. A 56, 63 (2020), arXiv:1907.08761 [nucl-th] .
  * Thiel  _et al._ (2019) M. Thiel, C. Sfienti, J. Piekarewicz, C. J. Horowitz, and M. Vanderhaeghen, “Neutron skins of atomic nuclei: per aspera ad astra,” J. Phys. G 46, 093003 (2019), arXiv:1904.12269 [nucl-ex] .
  * Adhikari  _et al._ (2021) D. Adhikari _et al._ (PREX), “Accurate Determination of the Neutron Skin Thickness of 208Pb through Parity-Violation in Electron Scattering,” Phys. Rev. Lett. 126, 172502 (2021), arXiv:2102.10767 [nucl-ex] .
  * Abrahamyan  _et al._ (2012) S. Abrahamyan _et al._ , “Measurement of the Neutron Radius of 208Pb Through Parity-Violation in Electron Scattering,” Phys. Rev. Lett. 108, 112502 (2012), arXiv:1201.2568 [nucl-ex] .
  * Adhikari  _et al._ (2022) D. Adhikari _et al._ (CREX), “Precision Determination of the Neutral Weak Form Factor of Ca48,” Phys. Rev. Lett. 129, 042501 (2022), arXiv:2205.11593 [nucl-ex] .
  * Akimov  _et al._ (2022b) D. Akimov _et al._ , “The COHERENT Experimental Program,” in  _2022 Snowmass Summer Study_ (2022) arXiv:2204.04575 [hep-ex] .
  * Baxter  _et al._ (2020) D. Baxter _et al._ , “Coherent Elastic Neutrino-Nucleus Scattering at the European Spallation Source,” JHEP 02, 123 (2020), arXiv:1911.00762 [physics.ins-det] .
  * Abdullah  _et al._ (2022) M. Abdullah _et al._ , “Coherent elastic neutrino-nucleus scattering: Terrestrial and astrophysical applications,” (2022), arXiv:2203.07361 [hep-ph] .
  * Angloher  _et al._ (2019) G. Angloher _et al._ (NUCLEUS), “Exploring $\hbox{CE}\nu\hbox{NS}$ with NUCLEUS at the Chooz nuclear power plant,” Eur. Phys. J. C 79, 1018 (2019), arXiv:1905.10258 [physics.ins-det] .
  * Strauss  _et al._ (2017) R. Strauss _et al._ , “The $\nu$-cleus experiment: A gram-scale fiducial-volume cryogenic detector for the first detection of coherent neutrino-nucleus scattering,” Eur. Phys. J. C 77, 506 (2017), arXiv:1704.04320 [physics.ins-det] .
  * Workman  _et al._ (2022) R. L. Workman _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2022, 083C01 (2022).
  * Safronova  _et al._ (2018) M. S. Safronova, D. Budker, D. DeMille, Derek F. Jackson Kimball, A. Derevianko, and C. W. Clark, “Search for New Physics with Atoms and Molecules,” Rev. Mod. Phys. 90, 025008 (2018), arXiv:1710.01833 [physics.atom-ph] .
  * Cadeddu  _et al._ (2021c) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, and E. Picciau, “Muon and electron g-2 and proton and cesium weak charges implications on dark Zd models,” Phys. Rev. D 104, 011701 (2021c), arXiv:2104.03280 [hep-ph] .
  * Young  _et al._ (2007) Ross Daniel Young, Roger D. Carlini, Anthony William Thomas, and Julie Roche, “Testing the standard model by precision measurement of the weak charges of quarks,” Phys. Rev. Lett. 99, 122003 (2007), arXiv:0704.2618 [hep-ph] .
  * Corona  _et al._ (2022) Mattia Atzori Corona, Matteo Cadeddu, Nicola Cargioli, Paolo Finelli, and Matteo Vorabbi, “Incorporating the weak mixing angle dependence to reconcile the neutron skin measurement on Pb208 by PREX-II,” Phys. Rev. C 105, 055503 (2022), arXiv:2112.09717 [hep-ph] .
  * Wood  _et al._ (1997) C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner, and Carl E. Wieman, “Measurement of parity nonconservation and an anapole moment in cesium,” Science 275, 1759–1763 (1997).
  * Guena  _et al._ (2005) J. Guena, M. Lintz, and M. A. Bouchiat, “Measurement of the parity violating 6S-7S transition amplitude in cesium achieved within 2 x 10(-13) atomic-unit accuracy by stimulated-emission detection,” Phys. Rev. A 71, 042108 (2005), arXiv:physics/0412017 .
  * Gilbert  _et al._ (1985) S. L. Gilbert, M. C. Noecker, R. N. Watts, and Carl E. Wieman, “MEASUREMENT OF PARITY NONCONSERVATION IN ATOMIC CESIUM,” Phys. Rev. Lett. 55, 2680–2683 (1985).
  * Akimov  _et al._ (2022c) D. Akimov _et al._ (COHERENT), “Measurement of scintillation response of CsI[Na] to low-energy nuclear recoils by COHERENT,” JINST 17, P10034 (2022c), arXiv:2111.02477 [physics.ins-det] .
  * Drukier and Stodolsky (1984) A. Drukier and Leo Stodolsky, “Principles and Applications of a Neutral Current Detector for Neutrino Physics and Astronomy,” Phys. Rev. D 30, 2295 (1984).
  * Barranco  _et al._ (2005) J. Barranco, O. G. Miranda, and T. I. Rashba, “Probing new physics with coherent neutrino scattering off nuclei,” JHEP 12, 021 (2005), arXiv:hep-ph/0508299 .
  * Patton  _et al._ (2012) Kelly Patton, Jonathan Engel, Gail C. McLaughlin, and Nicolas Schunck, “Neutrino-nucleus coherent scattering as a probe of neutron density distributions,” Phys. Rev. C 86, 024612 (2012), arXiv:1207.0693 [nucl-th] .
  * Erler and Su (2013) Jens Erler and Shufang Su, “The Weak Neutral Current,” Prog. Part. Nucl. Phys. 71, 119–149 (2013), arXiv:1303.5522 [hep-ph] .
  * Atzori Corona  _et al._ (2023) M. Atzori Corona, W. M. Bonivento, M. Cadeddu, N. Cargioli, and F. Dordei, “New constraint on neutrino magnetic moment and neutrino millicharge from LUX-ZEPLIN dark matter search results,” Phys. Rev. D 107, 053001 (2023), arXiv:2207.05036 [hep-ph] .
  * Fricke  _et al._ (1995) G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera, and C. W. de Jager, “Nuclear Ground State Charge Radii from Electromagnetic Interactions,” Atom. Data Nucl. Data Tabl. 60, 177–285 (1995).
  * Angeli and Marinova (2013) I. Angeli and K. P. Marinova, “Table of experimental nuclear ground state charge radii: An update,” Atom. Data Nucl. Data Tabl. 99, 69–95 (2013).
  * Hoferichter  _et al._ (2020) Martin Hoferichter, Javier Menéndez, and Achim Schwenk, “Coherent elastic neutrino-nucleus scattering: EFT analysis and nuclear responses,” Phys. Rev. D 102, 074018 (2020), arXiv:2007.08529 [hep-ph] .
  * Helm (1956) Richard H. Helm, “Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei,” Phys. Rev. 104, 1466–1475 (1956).
  * Friedrich and Voegler (1982) J. Friedrich and N. Voegler, “The salient features of charge density distributions of medium and heavy even-even nuclei determined from a systematic analysis of elastic electron scattering form factors,” Nucl. Phys. A 373, 192–224 (1982).
  * Blunden  _et al._ (2012) P. G. Blunden, W. Melnitchouk, and A. W. Thomas, “$\gamma-Z$ box corrections to weak charges of heavy nuclei in atomic parity violation,” Phys. Rev. Lett. 109, 262301 (2012), arXiv:1208.4310 [hep-ph] .
  * Derevianko (2001) Andrei Derevianko, “Correlated many-body treatment of the breit interaction with application to cesium atomic properties and parity violation,” Phys. Rev. A 65, 012106 (2001).
  * Pollock  _et al._ (1992) S. J. Pollock, E. N. Fortson, and L. Wilets, “Atomic parity nonconservation: Electroweak parameters and nuclear structure,” Phys. Rev. C 46, 2587–2600 (1992).
  * Viatkina  _et al._ (2019) A. V. Viatkina, D. Antypas, M. G. Kozlov, D. Budker, and V. V. Flambaum, “Dependence of atomic parity-violation effects on neutron skins and new physics,” Phys. Rev. C 100, 034318 (2019).
  * Dzuba  _et al._ (2012) V. A. Dzuba, J. C. Berengut, V. V. Flambaum, and B. Roberts, “Revisiting parity non-conservation in cesium,” Phys. Rev. Lett. 109, 203003 (2012), arXiv:1207.5864 [hep-ph] .
  * Sahoo  _et al._ (2021) B. K. Sahoo, B. P. Das, and H. Spiesberger, “New physics constraints from atomic parity violation in Cs133,” Phys. Rev. D 103, L111303 (2021), arXiv:2101.10095 [hep-ph] .
  * Baker and Cousins (1984) Steve Baker and Robert D. Cousins, “Clarification of the Use of Chi Square and Likelihood Functions in Fits to Histograms,” Nucl. Instrum. Meth. 221, 437–442 (1984).
  * Lewis and Collar (2021) C. M. Lewis and J. I. Collar, “Response of undoped cryogenic CsI to low-energy nuclear recoils,” Phys. Rev. C 104, 014612 (2021), arXiv:2101.03264 [physics.ins-det] .
  * Anthony  _et al._ (2005) P. L. Anthony _et al._ (SLAC E158), “Precision measurement of the weak mixing angle in Moller scattering,” Phys. Rev. Lett. 95, 081601 (2005), hep-ex/0504049 [hep-ex] .
  * Wang  _et al._ (2014) D. Wang _et al._ (PVDIS), “Measurement of parity violation in electron–quark scattering,” Nature 506, 67–70 (2014).
  * Zeller  _et al._ (2002) G. P. Zeller _et al._ (NuTeV), “A precise determination of electroweak parameters in neutrino nucleon scattering,” Phys. Rev. Lett. 88, 091802 (2002), hep-ex/0110059 .
  * Androic  _et al._ (2018) D. Androic _et al._ (Qweak), “Precision measurement of the weak charge of the proton,” Nature 557, 207–211 (2018).
  * Becker  _et al._ (2018) Dominik Becker _et al._ , “The P2 experiment,” Eur. Phys. J. A 54, 208 (2018), arXiv:1802.04759 [nucl-ex] .
  * Abele  _et al._ (2022) H. Abele _et al._ , “Particle Physics at the European Spallation Source,” (2022), arXiv:2211.10396 [physics.ins-det] .
  * Su  _et al._ (2023) Chenguang Su, Qian Liu, and Tianjiao Liang (CE$\nu$NS@CSNS), “CE$\nu$NS Experiment Proposal at CSNS,” (2023), arXiv:2303.13423 [physics.ins-det] .
  * Sierra (2023) D. Aristizabal Sierra, “Extraction of neutron density distributions from high-statistics coherent elastic neutrino-nucleus scattering data,” (2023), arXiv:2301.13249 [hep-ph] .
  * Amsler  _et al._ (2008) Claude Amsler _et al._ (Particle Data Group), “Review of Particle Physics,” Phys. Lett. B 667, 1–1340 (2008).
  * Nakamura  _et al._ (2010) K. Nakamura _et al._ (Particle Data Group), “Review of particle physics,” J. Phys. G 37, 075021 (2010).
  * Olive  _et al._ (2014) K. A. Olive _et al._ (Particle Data Group), “Review of Particle Physics,” Chin. Phys. C 38, 090001 (2014).
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * Dev  _et al._ (2021) P. S. Bhupal Dev, Werner Rodejohann, Xun-Jie Xu, and Yongchao Zhang, “Searching for $Z^{\prime}$ bosons at the P2 experiment,” (2021), arXiv:2103.09067 [hep-ph] .
  * Benesch  _et al._ (2014) J. Benesch _et al._ (MOLLER), “The MOLLER Experiment: An Ultra-Precise Measurement of the Weak Mixing Angle Using Moller Scattering,” (2014), arXiv:1411.4088 [nucl-ex] .
  * Wong  _et al._ (2007) H. T. Wong _et al._ (TEXONO), “A Search of Neutrino Magnetic Moments with a High-Purity Germanium Detector at the Kuo-Sheng Nuclear Power Station,” Phys. Rev. D 75, 012001 (2007), arXiv:hep-ex/0605006 .
  * Aguilar-Arevalo  _et al._ (2016) A. Aguilar-Arevalo _et al._ (CONNIE), “Results of the Engineering Run of the Coherent Neutrino Nucleus Interaction Experiment (CONNIE),” JINST 11, P07024 (2016), arXiv:1604.01343 [physics.ins-det] .
  * Agnolet  _et al._ (2017) G. Agnolet _et al._ (MINER), “Background Studies for the MINER Coherent Neutrino Scattering Reactor Experiment,” Nucl. Instrum. Meth. A 853, 53–60 (2017), arXiv:1609.02066 [physics.ins-det] .
  * Aristizabal Sierra  _et al._ (2022) D. Aristizabal Sierra, V. De Romeri, and D. K. Papoulias, “Consequences of the Dresden-II reactor data for the weak mixing angle and new physics,” JHEP 09, 076 (2022), arXiv:2203.02414 [hep-ph] .
  * Cañas  _et al._ (2018) B. C. Cañas, E. A. Garcés, O. G. Miranda, and A. Parada, “Future perspectives for a weak mixing angle measurement in coherent elastic neutrino nucleus scattering experiments,” Phys. Lett. B 784, 159–162 (2018), arXiv:1806.01310 [hep-ph] .