# New insights into nuclear physics and weak mixing angle using electroweak
probes

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, and INFN, Sezione 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  Y.F. Li
<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  E. Picciau
<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di
Cagliari, and INFN, Sezione 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  C.A. Ternes<EMAIL_ADDRESS>Istituto Nazionale di Fisica
Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy
Y.Y. Zhang<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China

(11 September 2021)

###### Abstract

Using the new results on coherent elastic neutrino-nucleus scattering data in
cesium-iodide provided by the COHERENT experiment, we determine a new
measurement of the average neutron rms radius of ${}^{133}\text{Cs}$ and
${}^{127}\text{I}$. In combination with the atomic parity violation (APV)
experimental result, we derive the most precise measurement of the neutron rms
radii of ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$, disentangling for the
first time the contributions of the two nuclei. By exploiting these
measurements we determine the corresponding neutron skin values for
${}^{133}\text{Cs}$ and ${}^{127}\text{I}$. These results suggest a preference
for models which predict large neutron skin values, as corroborated by the
only other electroweak measurements of the neutron skin of ${}^{208}\text{Pb}$
performed by PREX experiments. Moreover, for the first time, we obtain 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 ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$, that is allowed to vary
freely in the fit. The value of the low-energy weak mixing angle that we found
is slightly larger than the standard model prediction.

##  I Introduction

The detection of coherent elastic neutrino-nucleus scattering (CE$\nu$NS) in
2017 in cesium-iodide (CsI) by the COHERENT experiment [1, 2] motivated a
burst of studies of diverse physical phenomena, with important implications
for high-energy physics, astrophysics, nuclear physics, and beyond [3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. After a fruitful discovery
period, recently enriched by the observation of CE$\nu$NS in argon [18, 19,
20], a new era of precision measurements has now begun, thanks to the new data
recorded by the COHERENT experiment using a CsI target [21]. Indeed, the
larger CE$\nu$NS statistics collected together with a refined quenching factor
determination allow us to perform stringent tests of the Standard Model (SM).

In previous works [3, 14, 16, 22, 15, 19, 20], it has been shown that the
CE$\nu$NS process gives model-independent information on the neutron nuclear
form factor, which is more difficult to obtain than the proton one. Form
factors represent the Fourier transform of the corresponding nucleon
distribution, necessary for obtaining in turn measurements of the neutron rms
radius, $R_{n}$, which is a crucial ingredient of the nuclear matter equation
of state (EOS). The latter plays an essential role in understanding nuclei in
laboratory experiments and several processes, like heavy ion collisions, and
the structure and evolution of compact astrophysical objects as neutron stars
[23, 24, 25, 26]. However, while the proton form factor is well known since it
can be measured through electromagnetic processes [27, 28], the same cannot be
said for the neutron one. Indeed, despite its importance, $R_{n}$ is still
unknown for many nuclei, especially in a model independent way, since the
interpretation of hadron scattering experiments depends on the model used to
describe nonperturbative strong interactions [29].

The CE$\nu$NS process can also give information on the weak mixing angle,
usually referred to as $\sin^{2}{\vartheta_{W}}$, a fundamental parameter of
the electroweak theory of the SM. However, in the low-energy sector, the most
precise $\sin^{2}{\vartheta_{W}}$ measurement performed so far belongs to the
so-called atomic parity violation (APV) experiment, using cesium atoms [30,
31]. This latter measurement depends on the value of $R_{n}(^{133}\text{Cs})$
that, at the time of Ref. [32], could only have been extrapolated from a
compilation of antiprotonic atom x-rays data [33]. A combination of COHERENT
and APV data is thus highly beneficial to determine simultaneously in a model-
independent way these two fundamental parameters, keeping their correlations
into account.

In this paper, we present improved measurements of the average neutron rms
radius of ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$ obtained analyzing the
updated COHERENT CsI data [21]. In combination with the APV experimental
result, we derive the most precise measurement of $R_{n}$ of
${}^{133}\text{Cs}$ and ${}^{127}\text{I}$, disentangling for the first time
the contributions of the two nuclei. Moreover, for the first time, we 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
${}^{133}\text{Cs}$ and ${}^{127}\text{I}$ (that is allowed to vary freely in
the analysis), from a simultaneous fit of the COHERENT and APV experimental
results.

The plan of the paper is as follows: in Section II we introduce the CE$\nu$NS
cross section and we describe the method of analysis of the COHERENT data; in
Section III we present the results on the average CsI neutron rms radius
obtained from the analysis of the COHERENT CE$\nu$NS data; in Section IV we
describe the APV data analysis; in Section V we present the results on the
${}^{133}\text{Cs}$ and ${}^{127}\text{I}$ neutron radii obtained from the
combined analysis of COHERENT CE$\nu$NS and APV data; in Section VI we discuss
the determination of the weak mixing angle from the combined analysis of
COHERENT CE$\nu$NS and APV data; finally, in Section VII, we briefly summarize
the results presented in the paper.

##  II COHERENT CE$\nu$NS data analysis

| ${}^{127}\text{I}$ | ${}^{133}\text{Cs}$ | ${}^{208}\text{Pb}$
---|---|---|---
Model | $R_{p}^{\text{point}}$ | $R_{p}$ | $R_{n}^{\text{point}}$ | $R_{n}$ | $\Delta R_{np}^{\text{point}}$ | $\Delta R_{np}$ | $R_{p}^{\text{point}}$ | $R_{p}$ | $R_{n}^{\text{point}}$ | $R_{n}$ | $\Delta R_{np}^{\text{point}}$ | $\Delta R_{np}$ | $R_{p}^{\text{point}}$ | $R_{p}$ | $R_{n}^{\text{point}}$ | $R_{n}$ | $\Delta R_{np}^{\text{point}}$ | $\Delta R_{np}$
SHF SkI3 [34] | 4.68 | 4.75 | 4.85 | 4.92 | 0.17 | 0.17 | 4.74 | 4.81 | 4.91 | 4.98 | 0.18 | 0.18 | 5.43 | 5.49 | 5.66 | 5.72 | 0.23 | 0.23
SHF SkI4 [34] | 4.67 | 4.74 | 4.81 | 4.88 | 0.14 | 0.14 | 4.73 | 4.80 | 4.88 | 4.95 | 0.15 | 0.14 | 5.43 | 5.49 | 5.61 | 5.67 | 0.18 | 0.18
SHF Sly4 [35] | 4.71 | 4.78 | 4.84 | 4.91 | 0.13 | 0.13 | 4.78 | 4.85 | 4.90 | 4.98 | 0.13 | 0.13 | 5.46 | 5.53 | 5.62 | 5.69 | 0.16 | 0.16
SHF Sly5 [35] | 4.70 | 4.77 | 4.83 | 4.90 | 0.13 | 0.13 | 4.77 | 4.84 | 4.90 | 4.97 | 0.13 | 0.13 | 5.45 | 5.52 | 5.62 | 5.68 | 0.16 | 0.16
SHF Sly6 [35] | 4.70 | 4.77 | 4.83 | 4.90 | 0.13 | 0.13 | 4.77 | 4.84 | 4.89 | 4.97 | 0.13 | 0.13 | 5.46 | 5.52 | 5.62 | 5.68 | 0.16 | 0.16
SHF Sly4d [36] | 4.71 | 4.79 | 4.84 | 4.91 | 0.13 | 0.12 | 4.78 | 4.85 | 4.90 | 4.97 | 0.12 | 0.12 | 5.48 | 5.54 | 5.65 | 5.71 | 0.17 | 0.17
SHF SV-bas [37] | 4.68 | 4.76 | 4.80 | 4.88 | 0.12 | 0.12 | 4.74 | 4.82 | 4.87 | 4.94 | 0.13 | 0.12 | 5.44 | 5.51 | 5.60 | 5.66 | 0.15 | 0.15
SHF UNEDF0 [38] | 4.69 | 4.76 | 4.83 | 4.91 | 0.14 | 0.14 | 4.76 | 4.83 | 4.92 | 4.99 | 0.16 | 0.15 | 5.46 | 5.52 | 5.65 | 5.71 | 0.19 | 0.19
SHF UNEDF1 [39] | 4.68 | 4.76 | 4.83 | 4.91 | 0.15 | 0.15 | 4.76 | 4.83 | 4.90 | 4.98 | 0.15 | 0.15 | 5.46 | 5.52 | 5.64 | 5.70 | 0.18 | 0.17
SHF SkM* [40] | 4.71 | 4.78 | 4.84 | 4.91 | 0.13 | 0.13 | 4.76 | 4.84 | 4.90 | 4.97 | 0.13 | 0.13 | 5.46 | 5.52 | 5.63 | 5.69 | 0.17 | 0.17
SHF SkP [41] | 4.72 | 4.80 | 4.84 | 4.91 | 0.12 | 0.12 | 4.79 | 4.86 | 4.91 | 4.98 | 0.12 | 0.12 | 5.48 | 5.54 | 5.62 | 5.68 | 0.15 | 0.14
RMF DD-ME2 [42] | 4.67 | 4.75 | 4.82 | 4.89 | 0.15 | 0.15 | 4.74 | 4.81 | 4.89 | 4.96 | 0.15 | 0.15 | 5.46 | 5.52 | 5.65 | 5.71 | 0.19 | 0.19
RMF DD-PC1 [43] | 4.68 | 4.75 | 4.83 | 4.90 | 0.15 | 0.15 | 4.74 | 4.82 | 4.90 | 4.97 | 0.16 | 0.15 | 5.45 | 5.52 | 5.65 | 5.71 | 0.20 | 0.20
RMF NL1 [44] | 4.70 | 4.78 | 4.94 | 5.01 | 0.23 | 0.23 | 4.76 | 4.84 | 5.01 | 5.08 | 0.25 | 0.24 | 5.48 | 5.55 | 5.80 | 5.86 | 0.32 | 0.31
RMF NL3 [45] | 4.69 | 4.77 | 4.89 | 4.96 | 0.20 | 0.19 | 4.75 | 4.82 | 4.95 | 5.03 | 0.21 | 0.20 | 5.47 | 5.53 | 5.74 | 5.80 | 0.28 | 0.27
RMF NL-Z2 [46] | 4.73 | 4.80 | 4.94 | 5.01 | 0.21 | 0.21 | 4.79 | 4.86 | 5.01 | 5.08 | 0.22 | 0.22 | 5.52 | 5.58 | 5.81 | 5.87 | 0.29 | 0.29
RMF NL-SH [47] | 4.68 | 4.75 | 4.86 | 4.94 | 0.19 | 0.18 | 4.74 | 4.81 | 4.93 | 5.00 | 0.19 | 0.19 | 5.45 | 5.52 | 5.72 | 5.78 | 0.26 | 0.26
Table 1:  Theoretical values in units of fermi of the rms proton and neutron
point and physical radii and the neutron skin of ${}^{127}\text{I}$,
${}^{133}\text{Cs}$, and ${}^{208}\text{Pb}$ obtained with nonrelativistic
Skyrme-Hartree-Fock (SHF) and relativistic mean field (RMF) nuclear models.

The SM CE$\nu$NS differential cross section as a function of the true nuclear
kinetic recoil energy $T^{\prime}_{\mathrm{nr}}$, considering a spin-zero
nucleus $\mathcal{N}$ with $Z$ protons and $N$ neutrons, is given by [48, 49,
50]

$\displaystyle\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT^{\prime}_{\mathrm{nr}}}(E,T^{\prime}_{\mathrm{nr}})\hbox{}$
$\displaystyle\hbox{}=\dfrac{G_{\text{F}}^{2}M}{\pi}\left(1-\dfrac{MT^{\prime}_{\mathrm{nr}}}{2E^{2}}\right)\times$
$\displaystyle\hbox{}\left[g_{V}^{p}ZF_{Z}(|\hskip-1.0pt\vec{\hskip
1.0ptq}|^{2})+g_{V}^{n}NF_{N}(|\hskip-1.0pt\vec{\hskip
1.0ptq}|^{2})\right]^{2},$ (1)

where $G_{\text{F}}$ is the Fermi constant, $\ell=e,\mu,\tau$ is the neutrino
flavor, $E$ is the neutrino energy and
$|\vec{q}|\simeq\sqrt{2MT^{\prime}_{\mathrm{nr}}}$ is the three-momentum
transfer, $M$ being the nuclear mass. As introduced in Ref. [19], we modify
the tree-level values of the vector couplings
$g_{V}^{p}=1/2-2\sin^{2}\\!\vartheta^{\mathrm{SM}}_{W}=0.0229$ and
$g_{V}^{n}=-1/2$, where $\sin^{2}\vartheta_{W}^{\mathrm{SM}}=0.23857\pm
0.00005\,\text{\cite[cite]{[\@@bibref{Number}{Tanabashi:2018oca}{}{}]}}$ is
the low-energy value of the weak mixing angle, in order to take into account
radiative corrections in the $\overline{\text{MS}}$ scheme [52], namely
$g_{V}^{p}(\nu_{e})=0.0401$, $g_{V}^{p}(\nu_{\mu})=0.0318$, and
$g_{V}^{n}=-0.5094$. In Eq. (1), $F_{Z}(|\hskip-1.0pt\vec{\hskip
1.0ptq}|^{2})$ and $F_{N}(|\hskip-1.0pt\vec{\hskip 1.0ptq}|^{2})$ are,
respectively, the proton and neutron form factors. They describe the loss of
coherence for $|\hskip-1.0pt\vec{\hskip 1.0ptq}|R_{p}\gtrsim 1$ and
$|\hskip-1.0pt\vec{\hskip 1.0ptq}|R_{n}\gtrsim 1$, where $R_{p}$ is the rms
radius of the proton distribution. The three most popular parameterizations of
the form factors are the symmetrized Fermi [53], Helm [54], and Klein-Nystrand
[55], that give practically identical results111 These parameterizations of
the form factors depend on two parameters: the rms radius and a parameter that
quantifies the nuclear surface thickness. For all the nuclear proton and
neutron form factors, we considered the standard surface thickness of 2.30 fm
[56], that is in agreement with the values extracted from measured charge
distributions of similar nuclei [57]. We verified that the results are
practically independent of small variations of the value of the surface
thickness..

For the values of $R_{p}$, we correct the charge radii determined
experimentally from muonic atom spectroscopy [56, 58] as in Ref. [19],
obtaining

$\displaystyle\hbox{}R_{p}(^{133}\text{Cs})=4.821\pm 0.005\,\text{fm},$ (2)
$\displaystyle\hbox{}R_{p}(^{127}\text{I})=4.766\pm 0.008\,\text{fm}.$ (3)

For the neutron distribution there is only poor knowledge of
$R_{n}(^{133}\text{Cs})$ and $R_{n}(^{127}\text{I})$ obtained in the analyses
of the COHERENT 2017 data [3, 4, 10, 22, 15, 16, 14]. Plausible theoretical
values can be obtained using the recent nuclear shell model (NSM) estimate of
the corresponding neutron skins, the differences between the neutron and the
proton rms radii, $0.27\,\text{fm}$ and $0.26\,\text{fm}$ [59], leading to

$R_{n}^{\text{NSM}}(^{133}\text{Cs})\simeq 5.09\,\text{fm},\quad
R_{n}^{\text{NSM}}(\mathrm{{}^{127}\text{I}})\simeq 5.03\,\text{fm}.$ (4)

These values are slightly larger than those in Table 1, that we obtained using
nonrelativistic Skyrme-Hartree-Fock (SHF) and relativistic mean-field (RMF)
nuclear models. We calculated the physical proton and neutron radii $R_{p,n}$
from the corresponding point-radii $R_{p,n}^{\text{point}}$ given by the
models adding in quadrature the contribution of the rms nucleon radius
$\langle r_{N}^{2}\rangle^{1/2}\simeq 0.84\,\text{fm}$, that is considered to
be approximately equal for the proton and the neutron,

$R_{p,n}^{2}=(R_{p,n}^{\text{point}})^{2}+\langle r_{N}^{2}\rangle.$ (5)

The analysis of COHERENT data is performed in each nuclear recoil energy bin
$i$ and time interval $j$ with the least-squares function

$\displaystyle\chi^{2}_{\mathrm{C}}=\hbox{}$
$\displaystyle\hbox{}\sum_{i=2}^{9}\sum_{j=1}^{11}\left(\dfrac{N_{ij}^{\text{exp}}-\sum_{z=1}^{3}(1+\eta_{z})N_{ij}^{z}}{\sigma_{ij}}\right)^{2}+\sum_{z=1}^{3}\left(\dfrac{\eta_{z}}{\sigma_{z}}\right)^{2},$
(6)

where $z=1,2,3$ stands for CE$\nu$NS, Beam-Related Neutron (BRN) and Steady-
State (SS) backgrounds. $N_{ij}^{\text{exp}}$ is the experimental event
number, $N_{ij}^{\text{CE$\nu$NS}}$ is the predicted number of CE$\nu$NS
events in Eq. (8), $N_{ij}^{\text{BRN}}$ and $N_{ij}^{\text{SS}}$ are the
estimated number of BRN and SS background events, respectively, and
$\sigma_{ij}$ is the statistical uncertainty, all taken from Ref. [21]. The
uncertainties of the $\eta_{z}$ nuisance parameters, which quantify the
systematic uncertainty of the signal rate, of the BRN and of the SS background
rates, are $\sigma_{\text{CE$\nu$NS}}=13\%$, $\sigma_{\text{BRN}}=0.9\%$ and
$\sigma_{\text{SS}}=3\%$ [21].

We calculated the CE$\nu$NS event number $N_{i}^{\text{CE$\nu$NS}}$ in each
nuclear recoil energy bin $i$ with

$\displaystyle N_{i}^{\text{CE$\nu$NS}}=\hbox{}$
$\displaystyle\hbox{}N_{\text{CsI}}\int_{T_{\text{nr}}^{i}}^{T_{\text{nr}}^{i+1}}dT_{\text{nr}}\,A(T_{\text{nr}})$
$\displaystyle\hbox{}\times\int_{0}^{T^{\prime\text{max}}_{\text{nr}}}dT^{\prime}_{\text{nr}}\,R(T_{\text{nr}},T^{\prime}_{\text{nr}})\int_{E_{\text{min}}(T^{\prime}_{\text{nr}})}^{E_{\text{max}}}dE$
$\displaystyle\hbox{}\times\sum_{\nu=\nu_{e},\nu_{\mu},\overline{\nu}_{\mu}}\frac{dN_{\nu}}{dE}(E)\,\dfrac{d\sigma_{\nu\text{-}\text{CsI}}}{dT^{\prime}_{\text{nr}}}(E,T^{\prime}_{\text{nr}}),$
(7)

where $T_{\text{nr}}$ is the reconstructed nuclear recoil energy,
$A(T_{\text{nr}})$ is the energy-dependent detector efficiency,
$T^{\prime\text{max}}_{\text{nr}}=2E_{\text{max}}^{2}/M$,
$E_{\text{max}}=m_{\mu}/2\sim 52.8$ MeV, $m_{\mu}$ being the muon mass,
$E_{\text{min}}(T^{\prime}_{\text{nr}})=\sqrt{MT^{\prime}_{\text{nr}}/2}$,
$dN_{\nu}/dE$ is the neutrino flux integrated over the experiment lifetime and
$N_{\text{CsI}}$ is the number of ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$
atoms in the detector. The latter is given by
$N_{\mathrm{A}}\,M_{\mathrm{det}}/M_{\mathrm{CsI}}$, where $N_{\mathrm{A}}$ is
the Avogadro number, $M_{\mathrm{det}}$ is the detector active mass equal to
14.6 kg and $M_{\mathrm{CsI}}$ is the molar mass of CsI. The neutrino flux
from the spallation neutron source, $dN_{\nu}/dE$, is given by the sum of the
prompt $\nu_{\mu}$ component and the delayed $\nu_{e}$ and $\bar{\nu}_{\mu}$
components, considering $8.48\times 10^{-2}$ as the number of neutrinos per
flavor that are produced for each proton-on-target (POT). A number of POT
equal to $3.20\times 10^{23}$ and a distance of $19.3\,\text{m}$ between the
source and the COHERENT detector are used. The energy resolution function,
$R(T_{\text{nr}},T^{\prime}_{\text{nr}})$, is parameterized in terms of the
number of photoelectrons (PE) following Ref. [60]. The number of PE is related
to the nuclear recoil kinetic energy thanks to the light yield
$13.348\,N_{\text{PE}}/\text{keV}$ [60] and the quenching factor,
$f_{\text{Q}}(T_{\text{nr}})$, that is parameterized as a fourth order
polynomial as in Ref. [60].

In order to exploit also the arrival time information, we calculated the
CE$\nu$NS event number, $N_{ij}^{\text{CE$\nu$NS}}$, in each nuclear recoil
energy bin $i$ and time interval $j$ with

$N_{ij}^{\text{CE{\hbox{\nu}}NS}}=(N_{i}^{\text{CE{\hbox{\nu}}NS}})_{\nu_{\mu}}P_{j}^{(\nu_{\mu})}+(N_{i}^{\text{CE{\hbox{\nu}}NS}})_{\nu_{e},\bar{\nu}_{\mu}}P_{j}^{(\nu_{e},\bar{\nu}_{\mu})},$
(8)

where $P_{j}^{(\nu_{\mu})}$ and $P_{j}^{(\nu_{e},\bar{\nu}_{\mu})}$ are
obtained by integrating the arrival time distributions in the corresponding
time intervals with the time-dependent efficiency function [60, 21].

Using the SM inputs, the experimental values of $R_{p}(^{133}\text{Cs})$ and
$R_{p}(^{127}\text{I})$ in Eqs. (2) and (3), and the NSM values of
$R_{n}^{\text{EFT}}(^{133}\text{Cs})$ and
$R_{n}^{\text{EFT}}(\mathrm{{}^{127}\text{I}})$ in Eq. (4), the total number
of predicted events is found to be $N^{\text{CE$\nu$NS}}=311.8$.

##  III Average CsI neutron radius

We fitted the COHERENT CsI data to get information on the average neutron rms
radius of ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$, $R_{n}(\mathrm{CsI})$,
obtaining222 We considered also a fit with equal ${}^{133}\text{Cs}$ and
${}^{127}\text{I}$ neutron skins, which gave the almost equivalent result
$R_{n}(\mathrm{Cs})=5.56{}^{+0.45}_{-0.43}\,\text{fm}$ and
$R_{n}(\mathrm{I})=5.51{}^{+0.45}_{-0.43}\,\text{fm}$.

$R_{n}(\mathrm{CsI})=5.55\pm 0.44\,\text{fm}.$ (9)

This result is almost a factor of 2.5 more precise than previous
determinations [3] using the 2017 COHERENT dataset [1, 2].

The average of the NSM expected values in Eq. (4),
$R_{n}^{\text{NSM}}(\text{CsI})\simeq 5.06\,\text{fm}$, is compatible with the
determination in Eq. (9) at $1.1\sigma$.

The SHF and RMF predictions in Table 1 give
$R_{n}^{\text{SHF}}(\text{CsI})\simeq 4.92-4.95\,\text{fm}$ and
$R_{n}^{\text{RMF}}(\text{CsI})\simeq 4.93-5.05\,\text{fm}$, that differ from
the value in Eq. (9) by about $1.4\sigma$ and $1.2\sigma$, respectively.

Therefore, the COHERENT CsI data still do not allow to exclude some nuclear
models, but tends to favor those that predict a relatively large value of
$R_{n}$, as the NSM, RMF NL1, and RMF NL-Z2.

##  IV APV data analysis

The COHERENT data do not allow us to disentangle the contributions of the two
nuclei, but only to constrain their average. A separation of the two
contributions can be achieved in combination with the low-energy measurement
of the weak charge, $Q_{W}$, of ${}^{133}\text{Cs}$ in APV experiments, that
is related to the weak mixing angle through the relation

$\displaystyle Q_{W}^{\mathrm{th}}(\sin^{2}\vartheta_{W})=$
$\displaystyle-2[Z(g_{AV}^{ep}(\sin^{2}\vartheta_{W})+0.00005)$
$\displaystyle+N(g_{AV}^{en}+0.00006)]\left(1-\dfrac{\alpha}{2\pi}\right),$
(10)

where $\alpha$ is the fine-structure constant and the couplings of electrons
to nucleons, $g_{AV}^{ep}$ and $g_{AV}^{en}$, using
$\sin^{2}\vartheta_{W}^{\mathrm{SM}}$ and taking into account radiative
corrections in the SM [61, 52, 62], are given by (see Appendix A)

$\displaystyle\hbox{}g_{AV,\text{SM}}^{ep}=2g_{AV,\text{SM}}^{eu}+g_{AV,\text{SM}}^{ed}=-0.0357,$
(11)
$\displaystyle\hbox{}g_{AV,\text{SM}}^{en}=g_{AV,\text{SM}}^{eu}+2g_{AV,\text{SM}}^{ed}=0.495,$
(12)

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 $Q_{W}$

$Q_{W}=N\left(\dfrac{{\rm Im}\,E_{\rm PNC}}{\beta}\right)_{\rm
exp.}\left(\dfrac{Q_{W}}{N\,{\rm Im}\,E_{\rm PNC}}\right)_{\rm th.}\beta_{\rm
exp.+th.}\,,$ (13)

where $\beta_{\rm exp.+th.}$ and $(\mathrm{Im}\,E_{\rm PNC})_{\rm th.}$ are
determined from atomic theory, and Im stands for imaginary part [61]. We use
$({\rm Im}\,E_{\rm PNC}/{\beta})_{\rm exp}=(-3.0967\pm 0.0107)\times
10^{-13}|e|/a_{B}^{2}$ [61], where $a_{B}$ is the Bohr radius and $|e|$ is the
absolute value of the electric charge, and $\beta_{\rm exp.+th.}=(27.064\pm
0.033)\,a_{B}^{3}$ [61].

For the imaginary part of $E_{\rm PNC}$ we use $({\rm Im}\,E_{\rm PNC})_{\rm
th.}^{\rm w.n.s.}=(0.8995\pm 0.0040)\times 10^{-11}|e|a_{B}\frac{Q_{W}}{N}$
[32], where we subtracted the correction called “neutron skin”, introduced in
Ref. [63] to take into account the difference between $R_{n}$ and $R_{p}$ that
is not considered in the nominal atomic theory derivation. Here we remove this
correction in order to be able to directly evaluate $R_{n}$ from a combined
fit with the COHERENT data. The neutron skin corrected value of the weak
charge, $Q_{W}^{\mathrm{n.s.}}(R_{n})$, is thus retrieved by summing to $({\rm
Im}\,E_{\rm PNC})_{\rm th.}^{\rm w.n.s.}$ the correcting term $\delta
E^{\mathrm{n.s.}}_{\mathrm{PNC}}(R_{n})=\left[(\mathrm{N}/Q_{W})\left(1-(q_{n}(R_{n})/q_{p})\right)E_{\mathrm{PNC}}^{\mathrm{w.n.s.}}\right]$
[64, 14, 12]. 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 [65, 66, 67, 68,
64]. Our calculation of $q_{p}$ and $q_{n}$ is described in Appendix B.

##  V ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$ neutron radii

We performed the combined APV and COHERENT analysis with the least-squares
function

$\displaystyle\chi^{2}=\chi^{2}_{\text{C}}+\left(\dfrac{Q_{W}^{\rm
Cs\,n.s.}(R_{n})-Q_{W}^{\mathrm{th}}(\sin^{2}\vartheta_{W})}{\sigma_{\rm
APV}(R_{n},\sin^{2}\vartheta_{W})}\right)^{2}\,,$ (14)

where the first term is defined in Eq. (6) and the second term represents the
contribution of the APV measurement, where $\sigma_{\rm APV}$ is the total
uncertainty.

Considering that the COHERENT data depend separately on
$R_{n}(^{133}\mathrm{Cs})$ and $R_{n}(^{127}\mathrm{I})$, while APV depends
only on $R_{n}(^{133}\mathrm{Cs})$, we disentangled for the first time the two
nuclear contributions. Assuming
$\sin^{2}\vartheta_{W}=\sin^{2}\vartheta_{W}^{\rm SM}$, we obtained

$R_{n}(^{133}\mathrm{Cs})=5.27{}^{+0.33}_{-0.33}\,\text{fm}\,,\,R_{n}(^{127}\mathrm{I})=5.9{}^{+1.0}_{-0.9}\,\text{fm}.$
(15)

Figure 1:  Contours at different CL of the allowed regions in the plane of
$R_{n}(^{133}\mathrm{Cs})$ and $R_{n}(^{127}\mathrm{I})$, together with their
marginalizations, obtained from the combined fit of the COHERENT and APV data.
The green lines indicate the NSM expected values
$R_{n}^{\text{NSM}}(^{133}\text{Cs})\simeq 5.09\,\text{fm}$ and
$R_{n}^{\text{NSM}}(\mathrm{{}^{127}\text{I}})\simeq 5.03\,\text{fm}$.

The contours at different confidence levels (CL) of the allowed regions in the
plane of $R_{n}(^{133}\mathrm{Cs})$ and $R_{n}(^{127}\mathrm{I})$ are shown in
Figure 1, from which one can see that NSM expected values in Eq. (4) lie in
the $1\sigma$ allowed region. Thanks to the combination with APV,
$R_{n}(^{133}\mathrm{Cs})$ is well constrained and practically uncorrelated
with $R_{n}(^{127}\mathrm{I})$.

The value in Eq. (15) represents the most precise determination of
$R_{n}(\rm^{133}Cs)$ and implies a value of the neutron skin

$\Delta
R_{np}(\mathrm{{}^{133}Cs})=R_{n}(\mathrm{{}^{133}Cs})-R_{p}(\rm^{133}Cs)=0.45{}^{+0.33}_{-0.33}\,\text{fm},$
(16)

that tends to be larger than the SHF and RMF nuclear model predictions in
Table 1.

Figure 2:  Point neutron skin predictions for ${}^{208}\text{Pb}$ and
${}^{133}\text{Cs}$ according to different models (blue squares and red
circles, see text and Table 1 for details). Constraints set by PREX-1 [69,
70], PREX-2 and their combination [71, 72] and the constraint derived in this
work using COHERENT+APV data are also shown by the blue, purple, green, and
light red bands, respectively.

This value can be translated in terms of the proton and neutron point radii to
allow a direct comparison with $\Delta R_{np}^{\mathrm{point}}$ measured with
parity-violating electron scattering on ${}^{208}\text{Pb}$ in the PREX
experiments [70, 71, 69, 72]. The comparison is shown in Figure 2, together
with the neutron skin predictions given in Table 1, that have been obtained
with nonrelativistic Skyrme-Hartree-Fock models (red circles) and relativistic
mean-field models (blue squares). A clear model-independent linear correlation
is present between the neutron skin of ${}^{208}\text{Pb}$ and
${}^{133}\text{Cs}$ within the nonrelativistic and relativistic models with
different interactions. This has been already discussed in the literature [73,
74, 75, 76, 77], but here for the first time we are able to compare different
experimental determinations of the neutron skin of two nuclei obtained
exploiting three electroweak processes, namely atomic parity violation,
CE$\nu$NS, and parity-violating electron scattering.

The combination of the precise PREX results with the unique determination of
$\Delta R_{np}(^{133}\text{Cs})$ from APV and COHERENT prefers models that
predict large neutron skins. The neutron skin of a neutron-rich nucleus is the
result of the competition between the Coulomb repulsion between the protons,
the surface tension, that decreases when the excess neutrons are pushed to the
surface, and the symmetry energy [78]. The latter reflects the variation in
binding energy of the nucleons as a function of the neutron to proton ratio.
Its density dependence, that is a fundamental ingredient of the EOS, is
expressed in terms of the slope parameter, $L$, that depends on the derivative
of the symmetry energy with respect to density at saturation.

Theoretical calculations show a strong correlation [79, 80, 81, 82] between
$\Delta R_{np}$ and $L$, namely larger neutron skins translate into larger
values of $L$. Thus, an experimental measurement of $\Delta R_{np}$ represents
the most reliable way to determine $L$, which in turn provides critical inputs
to a wide range of problems in physics. Among others, it would greatly improve
the modeling of matter inside the cores of neutron stars [25, 24], despite a
difference in size with the nucleus of 18 orders of magnitude. Specifically,
given that $L$ is directly proportional to the pressure of pure neutron matter
at saturation density, larger values of $\Delta R_{np}$ imply a larger size of
neutron stars [83]. In Figure 2 we indicated the lower limit for $L$ suggested
by the combined COHERENT and APV result, namely $L>38.5$ MeV. Interestingly,
these findings are not in contrast with laboratory experiments or
astrophysical observations [84, 85, 77]. Indeed, our bound is compatible with
the constraints on the slope parameter $L$ derived in Ref. [86, 87] from a
combined analysis of a variety of experimental and theoretical approaches,
comprising heavy ion collisions [88], neutron skin-thickness of tin isotopes
[89], giant dipole resonances [90], the dipole polarizability of
${}^{208}\text{Pb}$ [91, 92], and nuclear masses [93]. All these constraints
indicate an allowed region of $L$ corresponding to $40\lesssim L\lesssim 65$.
However, the central value of the averaged PREX result as well as of the
combined COHERENT and APV determination presented in this paper suggest rather
large neutron skin-thicknesses that would imply a fairly stiff EOS at the
typical densities found in atomic nuclei. This finding is in contrast with the
current understanding of the neutron star parameters coming from the
observation of gravitational waves from GW170817 [94, 95]. If both are
correct, it would imply the softening of the EOS at intermediate densities,
followed by a stiffening at higher densities [84], that may be indicative of a
phase transition in the stellar core [96].

For completeness, using the result in Eq. (15) we are also able to measure for
the first time the neutron skin of 127I, $\Delta
R_{np}(\rm^{127}I)={1.1}^{+1}_{-0.9}\,\text{fm}$, even though with large
uncertainty.

##  VI Weak mixing angle

Leaving $\sin^{2}\vartheta_{W}$ free to vary in the $\chi^{2}$ in Eq. (14) and
assuming $R_{n}(^{133}\mathrm{Cs})\simeq R_{n}(^{127}\mathrm{I})\simeq
R_{n}(\mathrm{CsI})$, it is possible to constrain simultaneously
$R_{n}(\mathrm{CsI})$ and $\sin^{2}\vartheta_{W}$. In this analysis we assume
that $\sin^{2}\vartheta_{W}$ has the same value at the momentum transfer
scales of COHERENT CE$\nu$NS data (about 100 MeV) and the APV data (about 2
MeV), as in the SM prediction. Therefore, our analysis probes new physics
beyond the SM that can generate a deviation of $\sin^{2}\vartheta_{W}$ from
the SM prediction that is constant between about 2 and 100 MeV.

In this analysis we considered $R_{n}(^{133}\mathrm{Cs})\simeq
R_{n}(^{127}\mathrm{I})$ because the data do not allow us to obtain separate
information on the two radii together with the weak mixing angle. This choice
is acceptable, since the two radii are expected to have values that differ by
less than 0.1 fm (see Eq. (4) and Table 1), that is smaller than the
uncertainties of the determinations in Eq. (15) of the two radii assuming the
SM value of $\sin^{2}\vartheta_{W}$.

We obtained333 Considering a fit with equal ${}^{133}\text{Cs}$ and
${}^{127}\text{I}$ neutron skins, we obtained the almost equivalent result
$R_{n}(\mathrm{Cs})=5.63{}^{+0.47}_{-0.50}\,\text{fm}$,
$R_{n}(\mathrm{I})=5.58{}^{+0.47}_{-0.50}\,\text{fm}$, and
$\sin^{2}{\vartheta_{W}}=0.2407\pm 0.0035$.

$R_{n}(\mathrm{CsI})=5.60{}^{+0.47}_{-0.50}\,\text{fm},\,\sin^{2}{\vartheta_{W}}=0.2406\pm
0.0035.$ (17)

The contours at different CL in the plane of $R_{n}(\text{CsI})$ and
$\sin^{2}{\vartheta_{W}}$ are shown in Figure 3.

Figure 3:  Contours at different CL of the allowed regions in the plane of
$R_{n}(^{133}\mathrm{Cs})$ and $\sin^{2}{\vartheta_{W}}$, together with their
marginalizations, obtained from the combined fit of the COHERENT and APV data.
The green lines indicate the average of the NSM expected values in Eq. (4),
$R_{n}^{\text{NSM}}(\text{CsI})\simeq 5.06\,\text{fm}$, and
$\sin^{2}\vartheta_{W}=\sin^{2}\vartheta_{W}^{\rm SM}\simeq 0.23857$.

One can see that the NSM expected value for $R_{n}(\text{CsI})$ and the SM
value of $\sin^{2}\vartheta_{W}$ lie in the $1\sigma$ allowed region. The
inclusion of the experimental input of $R_{n}(\text{CsI})$ has the effect of
shifting the measurement of $\sin^{2}{\vartheta_{W}}$ towards larger values
with respect to the Particle Data Group (PDG) APV value
$\sin^{2}{\vartheta}_{W}^{\mathrm{PDG}}=0.2367\pm 0.0018$ [61], while keeping
the uncertainty at the percent level.

Figure 4:  Variation of $\sin^{2}\vartheta_{\text{W}}$ with energy scale
$\mu$. The SM prediction is shown as the solid curve, together with
experimental determinations in black [51, 30, 32, 97, 97, 98, 99, 100]. The
result derived in this paper is shown in red.

Our result is depicted by the red data point in Figure 4, where a summary of
the weak mixing angle measurements as a function of the energy scale $\mu$ is
shown along with the SM predicted running calculated in the
$\overline{\text{MS}}$ scheme [51, 101, 102].

It is important to remark that, before this paper, the value of
$R_{n}(^{133}\mathrm{Cs})$ used in the APV result was extrapolated from
hadronic experiments using antiprotonic atoms [33], that are known to be
affected, unlike electroweak measurements, by considerable model dependencies
and uncontrolled approximations that may be underestimated in the nuclear
uncertainty [29]. Among others, antiprotonic atoms test the neutron
distribution in the nuclear periphery, where the density drops exponentially,
under the strong assumption that a two-parameter Fermi distribution can be
safely used to extrapolate the information on the nuclear interior. Thus, it
is legit to question if the uncertainty of the official APV result is
realistic. On the contrary, the measurement of $\sin^{2}\vartheta_{\text{W}}$
presented in this paper in Eq. (17) 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.

In this regard, the precise determination of $R_{n}$ for different nuclei from
electroweak measurements, as shown in this paper in Eq. (15) for
${}^{133}\mathrm{Cs}$, provides a valuable benchmark to calibrate the result
of experiments involving hadronic probes, that are fundamental to map the
large neutron skins of exotic nuclei. In the future, the COHERENT program
[103] will include more detectors, each based on a different material allowing
more determinations of $R_{n}$. Besides more data that will be available using
a single-phase liquid argon detector, that so far allowed a first constrain on
$R_{n}(\text{Ar})$ [19], there will be two future experiments that are still
being developed: a germanium detector, that is also the target used by the
CONUS experiment [104], and an array of NaI crystals.

It is also important to note that the central value of the
$\sin^{2}\vartheta_{\text{W}}$ measurement presented in Eq. (17) is slightly
larger with respect to the SM prediction. Combined with the other low-energy
measurements, it could be interpreted in terms of a presence of a new dark
boson [105, 106, 107, 108]. Further measurements of
$\sin^{2}\vartheta_{\text{W}}$ in the low energy sector should come from the
P2 [109] and MOLLER [110] experiments, from the near DUNE detector [111] and
the exploitation of coherent elastic neutrino scattering in atoms [112] and
nuclei [19, 113, 114].

##  VII Conclusions

In conclusion, in this paper we discussed the results on nuclear physics and
on the low-energy electroweak mixing angle obtained from the analysis of the
new COHERENT CsI data in combination with the atomic parity violation result
in cesium. We obtained the most precise measurement of the neutron rms radius
and neutron-skin values of ${}^{133}\text{Cs}$ and ${}^{127}\text{I}$,
disentangling for the first time the two nuclear contributions. Moreover, for
the first time, we derived a data-driven APV+COHERENT measurement of the low-
energy weak mixing angle with a percent uncertainty fully determined from
electroweak processes and independent of the average neutron rms radius of CsI
that was allowed to vary freely in the fit.

###### Acknowledgements.

We would like to thank A. Konovalov and D. Pershey for the useful information
provided for the analysis of the COHERENT data. The work of C. Giunti and C.A.
Ternes is supported by the research grant ”The Dark Universe: A Synergic
Multimessenger Approach” number 2017X7X85K under the program PRIN 2017 funded
by the Ministero dell’Istruzione, Università e della Ricerca (MIUR). The work
of Y.F. Li and Y.Y. Zhang is supported by the National Natural Science
Foundation of China under Grant No. 12075255 and No. 11835013, and by Beijing
Natural Science Foundation under Grant No. 1192019. Y.F. Li is also grateful
for the support by the CAS Center for Excellence in Particle Physics (CCEPP).

##  Appendix A APV Weak Charge Calculation

In order to determine the APV weak charge, $Q^{\mathrm{th}}_{W}$, it is
necessary to study in detail the calculation of the couplings, taking into
account the radiative corrections. Following Refs. [62, 52, 115, 116] the
lepton-fermion couplings are

$\displaystyle g_{AV}^{\ell f}$ $\displaystyle=$
$\displaystyle\rho\Big{[}-\frac{1}{2}+2Q_{f}\hat{s}_{0}^{2}-2Q_{f}\varnothing_{\ell
Z}+\boxtimes_{ZZ}^{\ell f}+\boxtimes_{\gamma Z}^{\ell f}\Big{]}$ (18)
$\displaystyle-$ $\displaystyle 2Q_{f}\varnothing_{\ell W}+\square_{WW},\ \ \
\ \mathrm{for}\ f=u,$ $\displaystyle g_{AV}^{\ell f}$ $\displaystyle=$
$\displaystyle\rho\Big{[}\frac{1}{2}+2Q_{f}\hat{s}_{0}^{2}-2Q_{f}\varnothing_{\ell
Z}+\boxtimes_{ZZ}^{\ell f}+\boxtimes_{\gamma Z}^{\ell f}\Big{]}$ (19)
$\displaystyle-$ $\displaystyle 2Q_{f}\varnothing_{\ell
W}+\mathbin{\;{\rotatebox{90.0}{$\small\bowtie$}}}_{WW},\ \ \ \ \mathrm{for}\
f=d.$

In these relations for up and down quarks, $\rho=1.00063$ represents a low-
energy correction for neutral-current processes and $Q_{f}$ is the fermion
charge. Here $\hat{s}_{0}^{2}=\sin^{2}{\vartheta}_{W}^{\mathrm{SM}}$, which
keeps the same value for $\mu<\mathcal{O}(0.1~{}\mathrm{GeV})$. The other
corrections inserted in equations (18)-(19) come from different contributions,
such as electron charge radii ($\varnothing_{eW}$,$\varnothing_{eZ}$), EW box
diagrams ($\boxtimes_{ZZ}^{\ell f}$, $\square_{WW}$,
$\mathbin{\;{\rotatebox{90.0}{$\small\bowtie$}}}_{WW}$) and vacuum
polarization of $\gamma Z$ diagrams ($\boxtimes_{\gamma Z}^{\ell f}$) [62].
They can be expressed as

$\displaystyle\varnothing_{\ell W}$ $\displaystyle=$
$\displaystyle\frac{2\alpha}{9\pi},$ (20a) $\displaystyle\varnothing_{\ell Z}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{6\pi}Q_{\ell}g_{VA}^{\ell\ell}\Big{(}\ln\frac{M_{Z}^{2}}{m_{\ell}^{2}}+\frac{1}{6}\Big{)},$
(20b) $\displaystyle\boxtimes_{ZZ}^{\ell f}$ $\displaystyle=$
$\displaystyle-\frac{3\hat{\alpha}_{Z}}{16\pi\hat{s}_{Z}^{2}\hat{c}_{Z}^{2}}\Big{(}g_{VA}^{\ell
f}g_{VV}^{\ell f}+g_{AV}^{\ell f}g_{AA}^{\ell f}\Big{)}\times$ (20c)
$\displaystyle\Big{[}1-\frac{\hat{\alpha}_{s}(M_{Z})}{\pi}\Big{]},$
$\displaystyle\boxtimes_{\gamma Z}^{\ell f}$ $\displaystyle=$
$\displaystyle\frac{3\hat{\alpha}_{fZ}}{2\pi}Q_{f}g_{VA}^{\ell
f}\Big{(}\ln\frac{M_{Z}^{2}}{m_{f}^{2}}+\frac{3}{2}\Big{)},$ (20d)
$\displaystyle\square_{WW}$ $\displaystyle=$
$\displaystyle-\frac{\hat{\alpha}_{Z}}{2\pi\hat{s}^{2}_{Z}}\Big{[}1-\frac{\hat{\alpha}_{s}(M_{W})}{2\pi}\Big{]},$
(20e) $\displaystyle\mathbin{\;{\rotatebox{90.0}{$\small\bowtie$}}}_{WW}$
$\displaystyle=$
$\displaystyle\frac{\hat{\alpha}_{Z}}{8\pi\hat{s}_{Z}^{2}}\Big{[}1+\frac{\hat{\alpha}_{s}(M_{W})}{\pi}\Big{]}.$
(20f)

In the expressions above, $\ell$ indicates the lepton involved in the
interaction (in our case $\ell=e$), while $f$ indicates the quarks (in our
case $f=u,d$).
For the electromagnetic-running coupling we adopt the abbreviation
$\hat{\alpha}_{ij}\equiv\hat{\alpha}(\sqrt{m_{i}M_{j}})$ and
$\hat{\alpha}_{Z}\equiv\alpha(M_{Z})$. In particular, $\hat{\alpha}_{fZ}$,
that is present in the $\boxtimes_{\gamma Z}^{\ell f}$ contribution in Eq.
(20d), is evaluated considering the quark masses equal to the proton one, and
inside the logarithmic term the same value ($m_{q}=m_{p}$) is used. For the
strong coupling, we use the values $\hat{\alpha}_{s}(M_{Z})=0.1185$ [61] and
$\hat{\alpha}_{s}(M_{W})=0.123$ [117].

Inside the correction diagrams in Eqs. (20b), (20c), (20d), the neutral-
current couplings enter at tree level and can be written as [52]

$\displaystyle g_{V}^{f}$ $\displaystyle\equiv$
$\displaystyle\sqrt{2}\frac{T_{f}^{3}-2Q_{f}\sin^{2}\vartheta_{W}(\mu)}{\cos\vartheta_{W}(\mu)},$
(21) $\displaystyle g_{A}^{f}$ $\displaystyle\equiv$
$\displaystyle\sqrt{2}\frac{T_{f}^{3}}{\cos\vartheta_{W}(\mu)}.$ (22)

Their products are defined as

$g_{\alpha\beta}^{\ell
f}=\cos^{2}\vartheta_{W}(\mu)g_{\alpha}^{\ell}g_{\beta}^{f}\ \ \ \
\mathrm{for}\ \ \alpha,\beta=V,A.$ (23)

It is important to remark, as reported in Ref. [62], that for the EW box
corrections (Eqs. (20c), (20e), (20f)) the sine is evaluated at the value of
the $Z$ mass, $\hat{s}_{Z}^{2}\equiv\sin^{2}\hat{\theta}_{W}(M_{Z})=0.23121$
[61], while in the $\boxtimes_{\gamma Z}^{\ell f}$ term (Eq. (20d)) the sine
is evaluated at scale $\mu=\sqrt{m_{p}M_{Z}}$. Finally, inside the
$\varnothing_{\ell Z}$ term (Eq. (20b)) the coupling $g_{VA}^{\ell\ell}$ is
obtained using the value $\sin^{2}\theta_{W}(\sqrt{m_{\ell}M_{Z}})$ as
discussed in Ref. [62].
In order to determine the couplings to the proton and to the neutron it is
sufficient to use the fact that

$\displaystyle g_{AV}^{ep}$ $\displaystyle=$ $\displaystyle
2g_{AV}^{eu}+g_{AV}^{ed},$ (24) $\displaystyle g_{AV}^{en}$ $\displaystyle=$
$\displaystyle g_{AV}^{eu}+2g_{AV}^{ed}.$ (25)

However, as pointed out in Refs. [52, 62], it is necessary to take into
account also a correction relative to the $\boxtimes_{\gamma Z}^{\ell f}$
contribution, and this is obtained by adding to the proton and neutron
couplings some small constants such that

$\displaystyle g_{AV}^{ep}\rightarrow g_{AV}^{ep}+0.00005,$ (26)
$\displaystyle g_{AV}^{en}\rightarrow g_{AV}^{en}+0.00006,$ (27)

obtaining the theoretical expression for the APV weak charge written in Eq.
(7).

##  Appendix B Nuclear integrals calculation

The approach used to model the nuclear size and shape of the nucleus in APV
experiments is based on Refs. [67, 64], where the interaction matrix is
proportional to the electroweak couplings to protons and neutrons

$\mathcal{M}\propto G_{\rm F}\widetilde{Q}_{W}.$ (28)

Here $G_{\rm F}$ is the Fermi constant and

$\widetilde{Q}_{W}\equiv Zq_{p}(1-4\sin^{2}\vartheta_{W})-Nq_{n}.$ (29)

This coupling depends on the integrals

$q_{p,n}=4\pi\int_{0}^{\infty}\rho_{p,n}(r)f(r)r^{2}\mathrm{d}r,$ (30)

where $\rho_{p,n}(r)$ are the proton and neutron densities in the nucleus as
functions of the radius $r$ and $f(r)$ is the matrix element of the electron
axial current between the atomic $s_{1/2}$ and $p_{1/2}$ wave functions inside
the nucleus normalized to $f(0)=1$. The function $f(r)$ can be expressed as a
series in power of $(Z\alpha)$, and for most of the atoms of interest, in
particular for $(Z\alpha)$ up to $\sim 0.7$, cutting off the series at
$(Z\alpha)^{2}$ is more than adequate to fulfil the requirements of precision
for the comparison with experimental observation. According to Eq. (13) of
Ref. [67], at order $(Z\alpha)^{2}$, for any nucleus, $f(r)$ is given by

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-2\int_{0}^{r}\frac{V(r^{\prime})}{r^{\prime
2}}\int_{0}^{r^{\prime}}V(r^{\prime\prime})r^{\prime\prime
2}\mathrm{d}r^{\prime\prime}\mathrm{d}r^{\prime}$
$\displaystyle+\left(\frac{1}{r}\int_{0}^{r}V(r^{\prime})r^{\prime
2}\mathrm{d}r^{\prime}\right)^{2},$ (31)

where $V(r)$ represents the radial electric potential determined uniquely by
the charge distribution $\rho_{c}(r)$ of the nucleus. One can obtain the
potential through the Poisson equation

$\frac{1}{r}\frac{d^{2}}{dr^{2}}[rV(r)]=-4\pi Z\alpha\rho_{c}(r),$ (32)

whose general solution is

$V(r)=4\pi Z\alpha\left[\frac{1}{r}\int_{0}^{r}\rho_{c}(r^{\prime})r^{\prime
2}\mathrm{d}r^{\prime}+\int_{r}^{\infty}\rho_{c}(r^{\prime})r^{\prime}\mathrm{d}r^{\prime}\right].$
(33)

At this point one has to choose how to parameterize the charge density in
order to perform the calculation. The easiest choice is to imagine the nucleus
as a sphere of radius $R_{c}$ and constant density

$\rho_{c}^{\mathrm{cd}}(r)=\frac{3}{4\pi R_{c}^{3}}\,\Theta(R_{c}-r),$ (34)

$\Theta(R_{c}-r)$ is the Heaviside function, and the potential, using Eq. (33)
turns out to be

$V^{\mathrm{cd}}(r)=\begin{cases}\frac{Z\alpha}{2R_{c}}\left(3-\frac{r^{2}}{R_{c}^{2}}\right)&\mbox{for}\
r<R_{c}\\\ \frac{Z\alpha}{r}&\mbox{for}\ r>R_{c}\end{cases}.$ (35)

By using Eq. (31), it is possible to derive the analytical form of
$f^{\mathrm{cd}}(r)$ for $r<R_{c}$

$\displaystyle f^{\mathrm{cd}}(r)$
$\displaystyle=1-\frac{(Z\alpha)^{2}}{2}\left(\frac{r^{2}}{R_{c}^{2}}-\frac{r^{4}}{5R_{c}^{4}}+\frac{r^{6}}{75R_{c}^{6}}\right),$
(36)

and for $r>R_{c}$

$\displaystyle f^{\mathrm{cd}}(r)$
$\displaystyle=1-\frac{(Z\alpha)^{2}}{2}\left(\frac{13}{30}+\frac{2R_{c}^{2}}{5r^{2}}-\frac{R_{c}^{4}}{50r^{4}}+2\ln\left(\frac{r}{R_{c}}\right)\right).$
(37)

Using the above results and Eq. (30), one can calculate the proton and neutron
integrals. It is worth to notice that in the case of constant density, the
integrals in Eq. (30) have a cut-off at the value of the proton distribution
radius $R_{p}$, and the neutron distribution radius $R_{n}$. Since both
$R_{p}$ and $R_{n}$ are larger than $R_{c}$, one has to use both forms for
$f(r)$, depending on the region of integration. These considerations lead to

$\displaystyle q_{p,n}^{\mathrm{cd}}=1-(Z\alpha)^{2}$
$\displaystyle\left(-\frac{7}{60}+\frac{3}{5}\frac{R_{c}^{2}}{R_{p,n}^{2}}-\frac{16}{63}\frac{R_{c}^{3}}{R_{p,n}^{3}}\right.$
$\displaystyle\left.+\frac{3}{100}\frac{R_{c}^{4}}{R_{p,n}^{4}}+\ln{\frac{R_{p,n}}{R_{c}}}\right).$
(38)

Under the approximation $R_{c}\simeq R_{p}$ and for $R_{n}^{2}/R_{p}^{2}-1\ll
1$, it is possible to obtain the typically used forms of $q_{p,n}$

$\displaystyle q_{p}^{\mathrm{cd}}$ $\displaystyle\simeq
1-\frac{817}{3150}(Z\alpha)^{2},$ (39) $\displaystyle q_{n}^{\mathrm{cd}}$
$\displaystyle\simeq
1-(Z\alpha)^{2}\left[\frac{817}{3150}+\frac{116}{525}\left(\frac{R_{n}^{2}}{R_{p}^{2}}-1\right)\right].$
(40)

In this manuscript we performed the calculations considering the more accurate
charge, proton and neutron distribution densities that correspond to the form
factors in the CE$\nu$NS cross section. Therefore, we evaluated numerically
the quantities in Eqs. (30), (31), and (33). In practice, we used the Helm
parameterization [54] with $R_{c}(^{133}\mathrm{Cs})=4.8041$ fm and
$R_{p}(^{133}\mathrm{Cs})=4.8212$ fm which, for reference, give as a result
$q_{p}(^{133}\mathrm{Cs})=0.9567$.

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