# Simultaneous Extraction of the Weak Radius and the Weak Mixing Angle
from Parity-Violating Electron Scattering on ${}^{12}\mathrm{C}$

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  J. Erler  erler@uni-
mainz.de PRISMA+ Cluster of Excellence, Institute for Nuclear Physics,
Johannes Gutenberg-University, 55099 Mainz, Germany  M. Gorchtein
<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence, Institute for Nuclear
Physics, Johannes Gutenberg-University, 55099 Mainz, Germany  J. Piekarewicz
<EMAIL_ADDRESS>Department of Physics, Florida State University,
Tallahassee, FL 32306-4350, USA  Xavier Roca-Maza<EMAIL_ADDRESS>INFN, Sezione di Milano, 20133 Milano, Italy  Dipartimento di Fisica “Aldo
Pontremoli”, Università degli Studi di Milano, 20133 Milano, Italy
Departament de Física Quàntica i Astrofísica, Martí i Franqués, 1, 08028
Barcelona, Spain  Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franqués, 1, 08028 Barcelona, Spain  H. Spiesberger  spiesber@uni-
mainz.de PRISMA+ Cluster of Excellence, Institute for Nuclear Physics,
Johannes Gutenberg-University, 55099 Mainz, Germany

###### Abstract

We study the impact of nuclear structure uncertainties on a measurement of the
weak charge of ${}^{12}\mathrm{C}$ at the future MESA facility in Mainz.
Information from a large variety of nuclear models, accurately calibrated to
the ground-state properties of selected nuclei, suggest that a $0.3$ %
precision measurement of the parity-violating asymmetry at forward angles will
not be compromised by nuclear structure effects, thereby allowing a world-
leading determination of the weak charge of ${}^{12}\mathrm{C}$. Further, we
show that a combination of measurements of the parity-violating asymmetry at
forward and backward angles for the same electron beam energy can be used to
extract information on the nuclear weak charge distribution. We conclude that
a $0.34$% precision on the weak radius of ${}^{12}\mathrm{C}$ may be achieved
by performing a $3$% precision measurement of the parity-violating asymmetry
at backward angles.

##  I Introduction

The weak mixing angle $\theta_{\mathrm{W}}$ is the fundamental parameter of
the electroweak sector of the standard model (SM), which describes the mixing
between the SU(2) and U(1) gauge fields. Over the years, many measurements of
the weak mixing angle have been performed to test the SM and to search for
possible hints of beyond the SM effects. Its sine squared
$\sin^{2}\theta_{\mathrm{W}}$ enters the neutral current vector charge of SM
fermions and can be accessed through a variety of processes at very different
energy scales: from the “Z-pole” measurements performed at colliders Schael
_et al._ (2006); Aaltonen  _et al._ (2018); Aaij  _et al._ (2015);
Collaboration (2018); Sirunyan  _et al._ (2018), to deep inelastic scattering
with electrons Wang  _et al._ (2014); Prescott  _et al._ (1979), and neutrinos
Zeller  _et al._ (2002), to parity violation in atoms Wood  _et al._ (1997);
Guena  _et al._ (2005), and parity-violating electron scattering (PVES) off
protons Androić  _et al._ (2018) and electrons Anthony  _et al._ (2005). At
low energies, also the coherent elastic neutrino-nucleus scattering process
can provide a determination of the weak mixing angle Atzori Corona  _et al._
(2023). Although the current precision is still much worse than the other
available probes, it provides complementary information to atomic parity
violation and PVES measurements with nuclei Atzori Corona  _et al._ (2024). A
set of precise measurements at different energy scales are needed to test the
running of the weak mixing angle, which arises from loop-level contributions
Erler and Ramsey-Musolf (2005); Erler and Ferro-Hernández (2018). Our current
theoretical knowledge of this running is much more precise than the
corresponding experimental accuracy, especially at low energies. Future, more
precise low-energy measurements are thus necessary to further test the SM and
to disentangle beyond the SM scenarios. There are efforts to exploit atomic
parity violation to measure the weak charge of heavy nuclei at the per mille
precision Antypas  _et al._ (2019); Zhang  _et al._ (2016). On the PVES side,
the Qweak experiment has reached a 0.5% accuracy Androić  _et al._ (2018);
Androic  _et al._ (2022), and the upcoming P2@MESA Becker  _et al._ (2018) and
the MOLLER Benesch  _et al._ (2014) experiments aim at improving this
precision by factors of 4 and 6, respectively.
PVES on nuclei is not only useful for testing the SM, but it also allows one
to gain information on the structure of the target nucleus which has been
traditionally accessible only via strongly interacting probes that could carry
significant systematic unceratainties of the experimental analysis. In recent
years, PVES off neutron-rich nuclei has been used to determine a crucial
nuclear property known as the neutron skin thickness, defined as the
difference between the root mean square radii of neutron and proton nuclear
distributions. Intimately connected to this non-observable quantity is the so-
called weak skin, defined as the difference between root mean square radii of
the weak and the charge nuclear distributions. This quantity may indeed be
directly measured in the experiment. A strong motivation to study these
quantities is to constrain the equation of state (EOS) of neutron-rich matter
Horowitz and Piekarewicz (2001); Reed  _et al._ (2021), which is central to
our understanding of the structure of neutron stars Horowitz and Piekarewicz
(2001); Thiel  _et al._ (2019).
Challenging experimental programs to extract the neutron skin via PVES were
carried out successfully at Jefferson Lab. In particular, the PREX Abrahamyan
_et al._ (2012); Adhikari  _et al._ (2021) and CREX Adhikari  _et al._ (2022)
Collaborations measured the neutron skins of 208Pb and 48Ca, respectively.
Recently, also the Qweak Collaboration released a first neutron skin
measurement from PVES on ${}^{27}\mathrm{Al}$ Androic  _et al._ (2022). The
interpretation of these experiments in terms of the neutron skin has been made
possible thanks to a considerable input from the nuclear and particle theory
communities.
The interplay between the weak mixing angle and the neutron skin for a PVES
measurement was already studied in Ref. Corona  _et al._ (2022) for the case
of lead, while in a previous work Koshchii  _et al._ (2020) some of the
present authors addressed the feasibility of an intriguingly precise
simultaneous extraction of the weak charge and the neutron skin from a PVES
experiment off a light nucleus, such as 12C. Such an approach is highly
beneficial and complementary to the coherent elastic neutrino nucleus
scattering (CE$\nu$NS) campaign that is sensitive to the same observables
Akimov  _et al._ (2019); Yang  _et al._ (2019); Atzori Corona  _et al._
(2023); Cadeddu  _et al._ (2021a); Atzori Corona  _et al._ (2024).
In the present study, we build upon the latter work for 12C using a more
advanced treatment of the information provided by a broader set of theoretical
models. In particular, we include in our new study a collection of models
using non-relativistic Agrawal  _et al._ (2005); Steiner  _et al._ (2005);
Guichon  _et al._ (2006); Alex Brown (1998); Beiner  _et al._ (1975); Van Giai
and Sagawa (1981); Bartel  _et al._ (1982); Dobaczewski  _et al._ (1984);
Reinhard and Flocard (1995); Chabanat  _et al._ (1998); Reinhard  _et al._
(1999); Roca-Maza  _et al._ (2012, 2013); Agrawal  _et al._ (2003) and
covariant Chen and Piekarewicz (2014a, 2015); Fattoyev and Piekarewicz (2013)
energy density functionals.

##  II Parity violating electron scattering

The experimental observable of a PVES measurement is the parity-violating
asymmetry, $A^{\rm PV}$, which is defined as the difference between the
differential cross sections for elastic scattering of longitudinally polarized
electrons off a spin-zero target,

$A^{\rm PV}=\dfrac{\sigma_{R}-\sigma_{L}}{\sigma_{R}+\sigma_{L}},$ (1)

where $\sigma_{R(L)}$ represents the cross sections with right(left)-handed
electron polarization. The asymmetry arises from the interference between two
parts of the scattering amplitude: one due to the exchange of a virtual photon
and the other one from exchanging a virtual $Z^{0}$-boson. Adopting
momentarily the plane-wave Born (tree-level) approximation, the asymmetry
takes the form

$A^{\rm PV}_{\rm
PWBA}=-\dfrac{G_{F}Q^{2}}{4\sqrt{2}\pi\alpha}\dfrac{Q_{W}}{Z}\dfrac{F_{\rm
wk}(Q^{2})}{F_{\rm ch}(Q^{2})},$ (2)

with $G_{F}$ the Fermi constant, $\alpha$ the fine-structure constant, and
$Q^{2}$ the four-momentum transfer squared. Moreover, $Q_{W}$ denotes the
nuclear weak charge, $Z$ the nuclear electric charge, while $F_{\rm wk}/F_{\rm
ch}$ is the ratio of the weak and charge form factors that encode the nuclear
structure information. The formulation in Eq. (2) is very useful as it clearly
factorizes the dependence on fundamental SM parameters from nuclear effects
encoded in the form factor ratio. To further underscore this aspect we define
the normalization factor $A_{0}$ as follows:

$A_{0}=-\dfrac{G_{F}Q^{2}}{4\sqrt{2}\pi\alpha}\dfrac{Q_{W}}{Z},$ (3)

which incorporates only kinematical factors, SM parameters, and the nuclear
weak and electric charges. $A_{0}$ represents the value of the asymmetry for a
point-like nucleus, that is, with the form factors equal to unity independent
of the momentum transfer.
The weak charge of a nucleus composed of $Z$ protons and $N$ neutrons is given
by

$\displaystyle Q_{W}(Z,N)=ZQ_{W}^{p}+NQ_{W}^{n},\;\;{\rm where}$ (4a)
$\displaystyle Q_{W}^{p}=(1-4\sin^{2}\theta_{W})\;\;{\rm
and}\;\;Q_{W}^{n}=-1,$ (4b)

and the last equation denotes tree-level proton and neutron weak charges,
respectively. This expression highlights the dependence of the nuclear weak
charge on the weak mixing angle. Including radiative corrections, the
numerical value of the weak charge of ${}^{12}\mathrm{C}$ is $Q^{\rm
SM}_{W}(^{12}\mathrm{C})\\!=\\!-5.499$ Erler  _et al._ (2014).
The charge form factor in Eq. (2) has been extracted from elastic electron
scattering data. The best fit to data is obtained by using the sum-of-
Gaussians (SOG) parametrization of the density profile De Vries  _et al._
(1987). The slope of the charge form factor at zero momentum transfer is
related to the nuclear charge radius. For 12C, we use the reference value
obtained from scattering data, $R_{\rm ch}^{\rm
SOG}\\!=\\!2.469(5)\,\mathrm{fm}$. X-ray spectroscopy of muonic atoms is known
to usually give more precise values of nuclear charge radii; however,
oftentimes the two methods give different values for the radius De Vries  _et
al._ (1987); Angeli and Marinova (2013). The case of carbon is special in this
respect: not only are the respective charge radius extractions compatible,
$R_{\rm ch}^{\mu}\\!=\\!2.4702(22)\,\mathrm{fm}$ Angeli and Marinova (2013),
the value obtained from the electron scattering data is only slightly less
precise. We opt to use the SOG radius for a more conservative error analysis.
In turn, the weak form factor is only loosely constrained by hadronic
experiments that involve large and poorly understood systematic uncertainties
Thiel  _et al._ (2019).
In the Born approximation, Eq. (3), Coulomb corrections which generically
scale as $\pi\alpha Z$ ($\sim\\!0.1$ for 12C) are negelected. Given the
experimental precision goal of $0.3\%$, not only are these corrections
essential, but they have to be resummed to all orders. This is done within the
distorted wave Born approximation (DWBA) by numerically solving the Dirac
equation for an electron in the presence of the Coulomb potential generated by
the entire nuclear charge distribution. This requires to go beyond the Coulomb
potential of a point charge. In addition, in order to provide predictions for
the PV asymmetry, the Coulomb potential has to be supplemented with the weak
interaction potential which depends on the nuclear weak charge distribution
and acts on left- and right-handed electrons with the opposite sign Horowitz
(1998). We note that one can go beyond the tree level in DWBA calculations by
including higher-order QED corrections such as the vacuum polarization
(Uehling potential) and self-energy corrections Shabaev  _et al._ (2000).
These two contributions are expected to partially cancel each other Shabaev
_et al._ (2000); Roca-Maza (2018). We do not include these corrections in this
work since they are much smaller than the uncertainties associated with the
nuclear weak form factor. Finally, we note that for numerical calculations, we
employ the DREPHA code Klein  _et al._ (1986). We checked its consistency with
the ELSEPA package Salvat  _et al._ (2005) which was used in Ref. Koshchii
_et al._ (2020).

##  III Theoretical Framework

Given that the main nuclear inputs to the parity-violating asymmetry are
ground-state densities and their associated form factors, we start this
section by reviewing the theoretical framework underpinning our predictions.
Since such a theoretical approach assumes point nucleons, we then proceed to
discuss how single nucleon form factors are used to refine our predictions. We
end this section by quantifying the theoretical uncertainties that affect the
precision by which $\sin^{2}\theta_{\mathrm{W}}$ may be determined from a
measurement of the parity violating asymmetry with 12C.

###  III.1 Density Functional Theory

Predictions for the parity violating asymmetry for 12C will be made by a
collection of both non-relativistic and relativistic (or covariant) energy
density functionals Agrawal  _et al._ (2005); Steiner  _et al._ (2005);
Guichon  _et al._ (2006); Goriely  _et al._ (2009); Alex Brown (1998); Beiner
_et al._ (1975); Van Giai and Sagawa (1981); Bartel  _et al._ (1982);
Dobaczewski  _et al._ (1984); Reinhard and Flocard (1995); Chabanat  _et al._
(1998); Reinhard  _et al._ (1999); Roca-Maza  _et al._ (2012, 2013); Agrawal
_et al._ (2003); Chen and Piekarewicz (2014a, 2015); Fattoyev and Piekarewicz
(2013). The energy density functional (EDF) is the main building block behind
Density Functional Theory (DFT), a powerful technique developed by Kohn and
collaborators Hohenberg and Kohn (1964); Kohn and Sham (1965) to understand
the electronic structure of complex many-body systems, but since then extended
to many other areas of physics. Perhaps the most remarkable result of DFT is
that the exact ground-state energy of the complicated many-body system may be
obtained from minimizing a suitable EDF that only depends on the one-body
density. By doing so, DFT not only reduces drastically the complexity of the
problem, but also benefits from incorporating physical insights into the
construction of the functional. This is particularly relevant given that the
DFT formalism is formulated as an existence theorem that offers no guidance on
how to construct the appropriate EDF. In nuclear physics, one incorporates a
myriad of physical insights into the construction of the functional and leaves
the parameters of the model to be determined through a calibration procedure
informed by the ground-state properties of finite nuclei.

Inspired by the long history of Hartree-Fock theory in a variety of
disciplines, Kohn and Sham replaced the explicit construction of the energy
density functional by an equivalent system of non-interacting particles moving
in a suitably generated one-body potential Kohn and Sham (1965). It is
important to underscore that the equivalent Kohn-Sham potential must be
sophisticated enough to reproduce the exact one-body density of the
interacting system. This often demands the inclusion of one-body terms that
have a strong density dependence. The reformulation of the DFT problem in
terms of one-particle orbitals has several advantages, first and foremost that
self-consistent problems of this kind have a long history in nuclear physics.
The outcome of solving the mean-field-like Kohn-Sham equations is a single-
particle spectrum from which proton and neutron densities may be determined.
In turn, the ground state form factors may be computed by taking the Fourier
transform of the corresponding proton and neutron densities. Finally, one
accounts for the finite nucleon size by incorporating nucleon form factors as
described in the next section. The resulting charge and weak charge form
factors computed in this manner are the sole nuclear input to the parity
violating asymmetry given in Eq. (2).

###  III.2 Nucleon Form Factors

In order to obtain reliable theoretical predictions for the electromagnetic
and weak charge densities, one must fold the point-proton and -neutron
densities obtained from the nuclear structure models with single nucleon form
factors. For the electromagnetic single nucleon form factors we adopt the
following simple dipole parametrization Kelly (2002); Horowitz and Piekarewicz
(2012):

$G_{\rm E}^{\,p}(Q^{2})=\frac{G_{\rm M}^{\,p}(Q^{2})}{\mu_{p}}=\frac{G_{\rm
M}^{\,n}(Q^{2})}{\mu_{n}}=G_{\rm D}(Q^{2})\;.$ (5)

The form factors depend on the four-momentum transfer $Q^{2}\\!\\!=\\!{\bf
q}^{2}\\!-\\!\omega^{2}$ which is determined from the corresponding three-
momentum transfer ${\bf q}$ and the energy loss $\omega$. The dipole form
factor is given by

$G_{\rm D}(Q^{2})=\left(1+\frac{Q^{2}}{12}r_{p}^{2}\right)^{-2}\;,$ (6)

where $r_{p}^{2}\\!=\\!0.707\,{\rm fm}^{2}$ is the mean square proton radius.
Note that for the proton charge radius we have adopted the 2018 CODATA
recommended value of $r_{p}\\!=\\!0.841\,{\rm fm}$ Tiesinga  _et al._ (2021).
For the magnetic form factors we use $\mu_{p}\\!=\\!2.792\,847\,344\,63(82)$
and $\mu_{n}\\!=\\!-1.913\,042\,76(45)$ for the proton and neutron magnetic
moments (in units of the nuclear magneton), respectively Tiesinga  _et al._
(2021).
In turn, for the electric neutron form factor we use the Galster
parametrization Kelly (2002):

$G_{\rm
E}^{\,n}(Q^{2})=-\left(\frac{Q^{2}r_{n}^{2}/6}{1+Q^{2}/M^{2}}\right)G_{\rm
D}(Q^{2})\;,$ (7)

where $r_{n}^{2}\\!=\\!-0.116\,{\rm fm}^{2}$ is the mean square radius of the
neutron. All electromagnetic form factors are expressed exclusively in terms
of the experimentally determined mean square radii and magnetic moments.

Although one could invoke more sophisticated parametrizations that may be
extended to large values of $Q^{2}$ Ye  _et al._ (2018), for the modest values
of the momentum transfer probed in this work, $Q\sim 78-292\,\mathrm{MeV}$,
the two parametrizations are largely indistinguishable; in the relevant range
the differences for $G_{\rm E}^{\,p}(Q^{2})$ amount to less than 0.2%. We have
also tested that the errors in the form factor parameters are negligible for
our purposes.

We conclude this section by connecting the single nucleon electromagnetic form
factors to the corresponding weak form factors. Invoking isospin symmetry and
neglecting any strangeness contribution Horowitz and Piekarewicz (2012) we
obtain

$\displaystyle\widetilde{G}_{E}^{p}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{V}^{p}G_{E}^{p}(Q^{2})+g_{V}^{n}G_{E}^{n}(Q^{2}),$ (8a)
$\displaystyle\widetilde{G}_{E}^{n}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{V}^{n}G_{E}^{p}(Q^{2})+g_{V}^{p}G_{E}^{n}(Q^{2}),$ (8b)
$\displaystyle\widetilde{G}_{M}^{p}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{V}^{p}G_{M}^{p}(Q^{2})+g_{V}^{n}G_{M}^{n}(Q^{2}),$ (8c)
$\displaystyle\widetilde{G}_{M}^{n}(Q^{2})$ $\displaystyle=$ $\displaystyle
g_{V}^{n}G_{M}^{p}(Q^{2})+g_{V}^{p}G_{M}^{n}(Q^{2}),$ (8d)

where $g_{V}^{p}\\!=\\!0.0709(2)$ and $g_{V}^{n}\\!=\\!-0.9901(2)$ are the
proton and neutron weak vector charges, respectively, including radiative
corrections Workman  _et al._ (2022).

###  III.3 Spin-Orbit Currents

The importance of the so-called spin-orbit currents was studied in Ref. Ong
_et al._ (2010) and shortly after extended to the electroweak sector Horowitz
and Piekarewicz (2012). The nucleon matrix elements of both the
electromagnetic current as well as the vector part of the weak neutral current
may in general be written in terms of Lorentz vector and tensor components
multiplied by Dirac and Pauli (single-nucleon) form factors, respectively.
These single-nucleon form factors encode the quark substructure of the
nucleon. In particular, the spin-orbit contribution to the electroweak current
is associated with the tensor component of the nucleon electromagnetic
current.

Although in general small and often neglected, the spin-orbit current makes a
significant contribution to the form factor when one of the spin-orbit partner
states is not fully occupied. Indeed, in the CREX analysis Adhikari  _et al._
(2022), it was critical to include the spin-orbit contribution given that in
48Ca the neutron $f_{7/2}$ orbital is full while its $f_{5/2}$ spin-orbit
partner is empty. This may suggest that for 12C, with both proton and neutron
$p_{3/2}$ orbitals filled but its spin-orbit partner $p_{1/2}$ empty, spin-
orbit currents may play an important role. However, as we demonstrate in
Appendix B, this is not the case: its contribution to both $R_{\rm ch}$ and
$R_{\rm wk}$ does not exceed 0.05%.

###  III.4 Nuclear Radii

Using $R_{\rm ch}^{\rm SOG}\\!=\\!2.469(5)\,\mathrm{fm}$ and neglecting spin-
orbit effects (0.05%) in Eq. (22), one extracts an “experimental” point-proton
radius of $R_{p}=2.389(5)\,{\rm fm}$. There have been several attempts to
extract the point-neutron radius of 12C using hadronic probes. For example, an
elastic proton scattering experiment with 12C carried out at the Los Alamos
Meson Physics Facility (LAMPF) reported $R_{n}\\!=\\!2.4\pm\\!0.1\,{\rm fm}$
Blanpied  _et al._ (1978), a central value that is surprisingly larger than
the proton one, albeit with a large error. More recently, interaction,
reaction, and charge-exchange cross sections have been used to extract both
point-proton and matter radii Ozawa  _et al._ (2001); Kanungo  _et al._
(2016), from which one extracts a neutron radius of $R_{n}\\!=\\!2.35(4)\,{\rm
fm}$. Based on these results, one must conclude that the precise extraction of
the neutron radius of 12C using hadronic probes is challenging. Perhaps the
safest assumption is that proton and neutron radii are equal within errors,
given that it is a nucleus with $N\\!=\\!Z$. However, the repulsive Coulomb
interaction pushes the protons out relative to the neutrons suggesting a
negative neutron skin. Therefore, we now proceed to estimate the nuclear
structure uncertainties associated with the predictions for the point proton
and neutron densities.

The ground-state proton and neutron densities for 12C are computed using a
large set of nonrelativistic and covariant energy density functionals
calibrated to the binding energy and charge radii of a collection of magic and
semi-magic nuclei that do not include 12C. Folding these point-nucleon
densities with the single-nucleon form factors one obtains the charge and
weak-charge densities that serve as input for the distorted wave calculation
of the parity violating asymmetry. All the considered models fall within the
general framework of energy density functionals. To avoid any possible bias in
the analysis it would be useful to extend the study to ab-initio predictions
for the proton and neutron distributions in 12C, which unfortunately are not
available for such a light nucleus (see Appendix C for a tentavite discussion
on model consistency).

##  IV Analysis

To assess the nuclear-structure uncertainty, we display in Fig. 1 model
predictions for the weak (vertical axis) and charge (horizontal axis) radii of
12C. Also shown in the figure is the experimental charge radius $R_{\rm
ch}^{\rm exp}\\!=R_{\rm ch}^{\rm SOG}=\\!2.469(5)\,{\rm fm}$ De Vries  _et
al._ (1987). As anticipated, given that the major difference between the two
radii is due to the Coulomb repulsion, all models predict a small negative
weak skin of about $R_{\rm wk}\\!-\\!R_{\rm ch}\\!\approx\\!-0.015\,{\rm fm}$.

Figure 1: Scatter plot of the charge and weak radii obtained from a variety of
nuclear models. The gray band shows the linear fit to the data points,
obtained assuming a 1% uncertainty of the model prediction for the weak
radius. The black square is placed at the experimental value of the charge
radius, $R_{\rm ch}=2.469(5)$ fm, assuming a zero weak skin.

The new observation in Fig. 1 is a strong correlation between the model
predictions for the weak and charge radii, generally expected based on the
strong neutron-proton nuclear interaction. This strong correlation is shown by
the black line in Fig. 1 which was obtained by performing a linear fit to the
model predictions. To perform a statistical analysis and define a $\chi^{2}$
function, we assigned a conservative 1% uncertainty to each prediction Chen
and Piekarewicz (2014b); Roca-Maza  _et al._ (2015). This procedure returns
the following estimate for the weak radius of 12C:

$R_{\rm wk}=0.9942(15)\times R_{\rm ch}\,.$ (9)

The gray band in Fig. 1 indicates the corresponding $\pm 1\sigma$ region.

Figure 2: Asymmetry normalized to the pre-factor $A_{0}$ at the appropriate
value of $Q^{2}$, evaluated for all non-relativistic Agrawal  _et al._ (2005);
Steiner  _et al._ (2005); Guichon  _et al._ (2006); Goriely  _et al._ (2009);
Alex Brown (1998); Beiner  _et al._ (1975); Van Giai and Sagawa (1981); Bartel
_et al._ (1982); Dobaczewski  _et al._ (1984); Reinhard and Flocard (1995);
Chabanat  _et al._ (1998); Reinhard  _et al._ (1999); Roca-Maza  _et al._
(2012, 2013); Agrawal  _et al._ (2003) and covariant Chen and Piekarewicz
(2014a, 2015); Fattoyev and Piekarewicz (2013) models using only the nuclear
model densities. Each point is set at the corresponding prediction of the weak
skin. The left plot shows the results for forward kinematics ($\theta_{\rm
f}=29^{\circ}$), the right plot for backward kinematics ($\theta_{\rm
b}=145^{\circ}$). The black squared data points with error bars at the left
border show the expected precision for the asymmetry measurements centered at
an arbitrary value.

It is informative to note a discrepancy between two sets of models: while all
the Skyrme models systematically over-predict the charge radius, the
relativistic models do the opposite. The source for such a discrepancy may
have its origin in the competition between the saturation density and the
surface energy. Skyrme interactions tend to predict that symmetric nuclear
matter saturates at a higher density than the covariant models, typically
$0.16\,{\rm fm}^{-3}$ versus $0.15\,{\rm fm}^{-3}$. On its own, this would
suggest that Skyrme EDFs predict smaller nuclear radii than covariant EDFs.
However, both sets of functionals use charge radii of medium to heavy nuclei
as input for the calibration. This indicates that, in order for both sets of
functionals to reproduce the same charge radius of, e.g., 208Pb, the surface
energy in the Skyrme models must be softer than in the relativistic models.
However, 12C is relatively light to display any evidence of a saturated core,
so its charge radius is dominated by the surface energy—leading to
systematically larger radii for Skyrme EDFs than for covariant EDFs.

Returning to the discussion of Fig. 1, we observe that most of the models fail
to reproduce the experimental value of the charge radius $R_{\rm ch}^{\rm
exp}$ by as much as $\pm 5\%$. To circumvent this significant model
dependence, in Ref. Koshchii  _et al._ (2020), some of the authors of the
present work hand-picked just five nuclear mean-field models by requiring that
the charge radius be predicted within $\approx 1\%$ from the experimental one.
Consequently, the respective model predictions for the weak density (fitted to
a 2-parameter Fermi function form Piekarewicz  _et al._ (2016) to accelerate
the numerical evaluation) were used to quantify the nuclear theory uncertainty
for the weak skin extracted from a possible future measurement of the PV
asymmetry. In doing so, the charge density was fixed to the experimental one
(i.e., SOG). The conclusion of Ref. Koshchii  _et al._ (2020) was that the
extraction of the weak charge of 12C from a 0.3% measurement of the PV
asymmetry at low momentum transfer was contaminated by the nuclear theory
uncertainty, such that an additional measurement at a larger value of the
momentum transfer was necessary.

Here we argue that the strong correlation between the weak and charge radii
observed in Fig. 1 makes this arbitrary model selection unnecessary, allowing
for a much more reliable statistical analysis on a large sample of nuclear
models. Therefore, for consistency and accuracy, we prefer to directly use the
theoretical density predicted by the nuclear models instead of a two-parameter
Fermi function fitted to it. Finally, we take into account that nuclear models
predict point-proton and point-neutron densities, rather than the
experimentally measured weak and charge densities. Hence, to prevent the
introduction of additional systematic errors, we determine the weak and charge
densities by folding the theoretical predictions with the single-nucleon form
factors introduced in Sec.III.2.
To further illustrate the importance of maintaining the $R_{\rm
wk}\\!-\\!R_{\rm ch}$ correlation, we display predictions for the PV asymmetry
in Fig. 2. As mentioned earlier, for a realistic prediction for the PV
asymmetry one has to go beyond PWBA and account for Coulomb distortions. This
entails numerically solving the Dirac equation for an electron moving in the
Coulomb field of the nuclear charge distribution supplemented by a PV weak
interaction potential which is the source of the asymmetry. As such, the main
input into the DWBA code are the nuclear weak and charge densities which we
consistently take from each of the models adopted in this work.

As in Ref. Koshchii  _et al._ (2020), we explore two scenarios at a fixed
electron beam energy of $E_{e}=155\ \mathrm{MeV}$: A forward ($\theta_{\rm
f}=29^{\circ}$) and backward ($\theta_{\rm b}=145^{\circ}$) angle
configuration to highlight two different features of the PV asymmetry. Results
for the PV asymmetry are displayed in Fig. 2; left for the forward angle
measurement and right for the backward one. We find that in this new approach
the spread of both the weak skin and the PV asymmetry is dramatically reduced
relative to the uncertainty found in Ref. Koshchii  _et al._ (2020).

We observe on the left-hand panel of Fig. 2 that the predictions for the PV
asymmetry at the forward angle are fairly scattered without following a
particular trend as a function of the weak skin. However, whereas the spread
in the weak skin is relatively large for such a light nucleus, it does not
lead to any significant spread in $A^{\rm PV}$. Indeed, the range of the
predicted PV asymmetry for the forward measurement is within the $0.3\%$
precision goal, as indicated by the vertical error bar with the central value
(black squared point) placed arbitrarily around 1.034. We note that a measured
central value differing from 1.034 by more than a 0.3% would clearly imply a
flaw in current nuclear structure models based on Density Functional Theory.
The latter would be a very interesting although very unexpected result.

The right-hand panel of Fig. 2 contains the same information, but this time
for a backward scattering angle. The models display a systematic trend
indicating that the PV asymmetry is anti-correlated to the weak skin. Although
we have argued on the importance of Coulomb distortions, the PWBA provides a
simple explanation for this anti-correlation. Expanding both form factors in
Eq. (2) to first order in $Q^{2}$ one obtains

$\frac{A^{\rm PV}_{\rm PWBA}}{A_{0}}\simeq 1-\frac{Q^{2}}{3}R_{\rm ch}R_{\rm
wskin}$ (10)

where $R_{\rm wskin}\\!=\\!R_{\rm wk}\\!-\\!R_{\rm ch}$ is the weak skin.

Armed with these observations, we proceed to an analysis of the statistical
ensemble of model predictions for $A^{\rm PV}$ in the two experimental
scenarios, for forward and backward scattering angles.

We start with the forward measurement and notice that while the correlation
between $A^{\rm PV}$ and the weak skin in Fig. 2 is weak, a stronger
correlation emerges if the asymmetry is plotted versus the charge radius, as
shown in Fig. 3. In this figure, the black solid line indicates the result of
a linear fit. To define the $\chi^{2}$ of this fit, we assign to each
individual model prediction a conservative 1% uncertainty. The gray band in
the figure indicates the resulting $1\sigma$ region obtained in this manner.
The vertical red line is set to the experimental value of the charge radius of
12C. While its uncertainty is $\sim 0.2\%$, even an overly conservative $\pm
2\%$ range translates into a corresponding uncertainty for the PV asymmetry of
about $0.18\%$, as indicated by the orange box and lines. Given that this
theoretical uncertainty is below the anticipated precision goal of the
experiment, we conclude that the determination of the weak charge of 12C from
the forward measurement alone could be considered a clean measurement of the
weak mixing angle, without any significant contamination from nuclear
structure effects.

Figure 3: Linear fit to the predictions of the forward asymmetry normalized to
the pre-factor $A_{0}$ (black solid lines) as a function of the charge radius.
The $1\sigma$ region is shown by the gray band. The red solid line shows the
measured value of the charge radius, while the dashed vertical lines indicate
its $\pm 2\%$ uncertainty region. The orange box indicates the estimated
uncertainty on the asymmetry from the correlation of the nuclear models.

We next turn to the analysis of the backward measurement of $A^{\rm PV}$. Its
observed sensitivity to the weak skin motivates us to perform a linear fit of
the following form:

$A^{\rm PV}_{b}(R_{\rm wskin})=\mathcal{A}\cdot R_{\rm wskin}+\mathcal{B}\,.$
(11)

To perform the linear fit, we again assign to each individual $A^{\rm PV}$
prediction an uncertainty of $1\%$.
The linear fit returns

$\displaystyle\mathcal{A}$ $\displaystyle\\!=\\!$ $\displaystyle(-39\pm
4)\times 10^{-6}\,\mathrm{fm}^{-1}\,,$ (12) $\displaystyle\mathcal{B}$
$\displaystyle\\!=\\!$ $\displaystyle(7.37\pm 0.05)\times 10^{-6}$

for the two coefficients, and is shown in Fig. 4 by a black line with the
shaded gray band indicating the $1\sigma$ region. Different experimental
scenarios for the attainable precision of the backward measurement, denoted by
$\epsilon_{b}$, are indicated by colored horizontal bands: an orange band for
$\epsilon_{b}\\!=\\!3\%$, green for $\epsilon_{b}\\!=\\!5\%$, and blue for
$\epsilon_{b}\\!=\\!7\%$. Since the backward measurement retains sensitivity
to both the weak charge and weak skin, we must resort to a combined analysis
of the forward and backward angle results, which we will discuss in the next
section.

Figure 4: Linear fit to predictions of the backward asymmetry normalized to
the pre-factor $A_{0}$ as a function of the weak skin (black solid line). The
$\pm 1\sigma$ uncertainty range is shown by the gray band. The short-dashed
lines indicate reference values for the weak skin and the corresponding
asymmetry. The colored horizontal bands indicate the different precision
scenarios for a possible measurement: $\epsilon_{b}=3\%$ (orange),
$\epsilon_{b}=5\%$ (green) and $\epsilon_{b}=7\%$ (blue).

##  V Combined sensitivity analysis
for the weak mixing angle
and the weak skin

In this section we explore the impact of a combined analysis of the forward
and backward measurements of the PV asymmetry on the simultaneous
determination of the weak mixing angle and the weak skin of 12C. We have
established already that the forward measurement at the $0.3\%$ precision (or
worse) provides a clean determination of the nuclear weak charge. The backward
measurement retains sensitivity to both the weak charge and weak skin.

To analyze the backward measurement, we take as reference values $R^{\rm
ref}_{\rm wk}\\!=\\!2.455\,\mathrm{fm}$ for the weak radius, which has been
obtained by exploiting the correlation in Eq. (9) and $R_{\rm ch}=R_{\rm
ch}^{\rm SOG}$. The latter weak radius corresponds to a weak skin of $R^{\rm
ref}_{\rm wskin}\\!=\\!-0.0143\,\mathrm{fm}$, which falls about in the middle
of the predictions of the considered models sample. These choices imply a
corresponding reference value for the PV asymmetry at backward angles of
$A^{\rm PV,ref}_{b}\\!=\\!7.92$ ppm, obtained by using the definition in Eq.
(11) (and shown in Fig. 4 by the dashed black horizontal line). As the
asymmetry at forward angles is practically insensitive to the reference value
chosen for the weak skin, we consider as a reference prediction an average
value obtained from the considered nuclear models, which numerically
corresponds to $A^{\rm PV,ref}_{f}\\!=\\!0.517$ ppm (and is shown by the black
squared data point in Fig. 3). The optimization of the parameters is obtained
by minimizing the following $\chi^{2}$-function:

$\displaystyle\chi^{2}$ $\displaystyle=\left(\dfrac{A_{f}^{\rm
PV,ref}-\eta_{1}A^{\rm PV}_{f}(\sin^{2}\theta_{W})}{\epsilon_{f}}\right)^{2}$
$\displaystyle+\left(\dfrac{A_{b}^{\rm PV,ref}-A^{\rm
PV}_{b}(\eta_{2},\sin^{2}\theta_{W},R_{\rm wk})}{\sigma_{b}}\right)^{2}$
$\displaystyle+\left(\dfrac{\eta_{1}-1}{\sigma_{\eta_{1}}}\right)^{2}+\eta_{2}^{2}\,,$
(13)

which combines the information from the forward angle measurement with the
fixed precision goal of $\epsilon_{f}=0.3\%$ (first term), information from
the backward measurement with different precision scenarios for $\epsilon_{b}$
(second term), and information on the model uncertainty (last terms). The
adopted uncertainty in $\sigma_{b}$ is obtained by adding in quadratures the
anticipated experimental error $\epsilon_{b}$ with the uncertainty in the
$\mathcal{B}$ parameter associated with the linear fit given in Eq. (12),
$\sigma_{b}^{2}=\epsilon_{b}^{2}+\sigma_{\mathcal{B}}^{2}$. By writing the
asymmetry as a function $A_{f}^{\rm PV}(\sin^{2}\theta_{W})$, we explicitly
indicate its dependence on the weak mixing angle from its relation to the
nuclear weak charge, see Eq. (2). Further, $\eta_{1}$ and $\sigma_{\eta_{1}}$
are introduced as nuisance parameters to account for our estimate of the
nuclear uncertainty, i.e. $\sigma_{\eta_{1}}=0.18\%$. This allows the PV
asymmetry at the forward angle to vary withing the orange box indicated in
Fig. 3. Finally, to describe the uncertainty of the parameter $\mathcal{A}$ in
Eq. (12) we introduce the nuisance parameter $\eta_{2}$ and parametrize the
backward asymmetry as follows:

$\displaystyle A^{\rm PV}_{b}$
$\displaystyle(\eta_{2},\sin^{2}\theta_{W},R_{\rm
wk})=\dfrac{Q_{W}(\sin^{2}\theta_{W})}{Q_{W}^{\rm SM}}$ $\displaystyle\times$
$\displaystyle\left[\left(\mathcal{A}+\eta_{2}\,\sigma_{\mathcal{\\!A}}\right)R_{\rm
wskin}+\mathcal{B}\right]\,.$ (14)

The dependence on the weak mixing angle is restored by rescaling the weak
charge as in the case of the forward asymmetry. The second line in Eq. (14) is
introduced to describe the dependence on the nuclear models, as shown in Fig.
4 and Eq. (11). The quantity $\sigma_{\mathcal{A}}$ is the uncertainty of
$\mathcal{A}$, as reported in Eq. (12). $\eta_{2}$ is defined such that for
$\eta_{2}=\pm 1$ one retrieves the $\pm 1\sigma$ value for $\mathcal{A}$, i.e.
the slope of the prediction for the asymmetry goes from the lower to the
higher dashed black lines in Fig. 4.

Figure 5: Results of the sensitivity to the weak mixing angle and the weak
radius for the combined forward-backward measurement. The different
$\Delta\chi^{2}=1$ contours correspond to the different precision scenarios
for the backward measurement (orange for $\epsilon_{b}=3\%$, green for
$\epsilon_{b}=5\%$ and blue for $\epsilon_{b}=7\%$). The dashed lines indicate
the results of the independent analyses of the forward and backward
measurements.

The results of the combined fit to both the forward and backward measurements
are displayed in Fig. 5. The various contours, obtained by imposing the
condition $\Delta\chi^{2}\\!=\\!1$, correspond to the three different values
for $\epsilon_{b}$.

We can extract the individual sensitivities of a combination of the forward
and backward measurements to the weak mixing angle and the weak radius by
splitting the $\chi^{2}$-function in Eq. (13) and retaining only the relevant
contribution.
The horizontal dashed lines show the marginalization in terms of the weak
mixing angle: as already mentioned, its extraction is not contaminated by the
nuclear model dependence, nor does it depend on the precision of the backward
measurement. Retaining only the information from the backward measurement
results in nearly vertical bands in the parameter space.
The various colors for the dashed vertical bands correspond to different
choices for the precision of the backward measurement $\epsilon_{b}$, as
indicated in the legend of the figure. The fact that these bands are slightly
tilted indicates the residual (mild) dependence on the weak mixing angle.
Therefore, a combined analysis of the forward and backward measurements is
necessary to simultaneously extract both parameters.

$\epsilon_{f}$ | $\epsilon_{b}$ |  $R_{\rm wk}$ $[{\rm fm}]$ |  $\sigma_{R_{\rm wk}}$ $[\%]$ | $\sin^{2}\theta_{W}$
---|---|---|---|---
$0.3\%$ | $3\%$ | $2.455\pm 0.008$ | $0.34\%$ | $0.2386\pm 0.0008$
$5\%$ | $2.455\pm 0.012$ | $0.48\%$ | $0.2386\pm 0.0008$
$7\%$ | $2.455\pm 0.016$ | $0.63\%$ | $0.2386\pm 0.0008$

Table 1: Summary of the best fit and precision values corresponding to the
results shown in Fig. 5.

The numerical results obtained by marginalizing the contours shown in Fig. 5
are summarized in Tab. 1, where we collect the best fit values and the
relative uncertainties of the weak radius and the weak mixing angle. The
assumed precision of the forward measurement $\epsilon_{f}=\\!0.3\%$ directly
translates into that for the weak mixing angle. Similarly, depending on the
precision assumed for the backward measurement, the precision of the weak
radius ranges between $\sim\\!0.34\%$ and $\sim\\!0.63\%$.

We have also explored a wider set of values for the precision of a backward
measurement. Results are shown in Fig. 6, where $\epsilon_{b}$ is varied
between 1% and 8%. The figure shows the achievable precision for the weak
radius (both relative and absolute).

Figure 6: Precision for the weak radius as a function of the precision of the
backward asymmetry. The colored lines indicate the precision scenarios
considered in Ref. Koshchii  _et al._ (2020).

The impact of a sub-percent extraction of the weak mixing angle at low
energies is demonstrated in Fig. 7 where we display
$\sin^{2}\theta_{\mathrm{W}}$ along with existing measurements Navas  _et al._
(2024).
One can see that a $0.3\%$ measurement of the PV asymmetry with 12C at MESA
has the potential of providing one of the most precise measurements of the
weak mixing angle at low momentum transfers. This could allow one to obtain
new limits on models for physics beyond the SM. As an example we show in Fig.
7 how the running of the weak mixing angle could be modified if an additional
light vector mediator, such as a dark $Z$ boson Davoudiasl  _et al._ (2012a,
b, 2013, c); Arcadi  _et al._ (2020); Cadeddu  _et al._ (2021b), existed. In
such a model, the effect on the running is significant only at low scales,
depending on the coupling and the mass of the new boson. The dashed blue lines
shown in Fig. 7 correspond to a model with a $\sim\\!50\,\mathrm{MeV}$ dark
$Z$ boson, using the results obtained by the analysis in Ref. Cadeddu  _et
al._ (2021b). Having a sub-percent measurement of the weak mixing angle at
around $Q\sim\\!78\,\mathrm{MeV}$ will help constraining a hypothetical
deviation of the running weak mixing angle from the SM prediction.

Figure 7: Current status of $\sin^{2}\theta_{W}$ measurements as a function of
the energy scale $Q$ Wood  _et al._ (1997); Dzuba  _et al._ (2012); Androic
_et al._ (2022); Anthony  _et al._ (2005); Wang  _et al._ (2014); Navas  _et
al._ (2024). The blue solid curve represents the SM prediction for the running
weak mixing angle Workman  _et al._ (2022). The red data point shows our
result and is set at $Q\sim 78\ \mathrm{MeV}$. The light blue dashed curves
and the blue filled region indicate the modification of the running weak
mixing angle in the presence of a dark $Z$ boson, as discussed in Ref. Cadeddu
_et al._ (2021b).

As a final note we point out that the possibility to cleanly determine the
point-neutron radius and the neutron skin of the stable daughter nuclei
participating in the superallowed nuclear $\beta$ decays will have an impact
on the extraction of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix element
$V_{ud}$ Seng and Gorchtein (2023); Gorchtein and Seng (2024). Future
measurements of the PV asymmetry in PVES with nuclei such as 26Mg, 42Ca or
54Fe can be carried out at the MESA facility. A dedicated feasibility study
for each of these nuclei, similar to the 12C primer performed here, will be
necessary.

##  VI conclusions

In conclusion, we have revisited the physics reach of a measurement of the
parity violating asymmetry for ${}^{12}\mathrm{C}$. Improvements with respect
to the previous work reported in Ref. Koshchii  _et al._ (2020) include a much
larger sample of nuclear models analyzed, and a more robust procedure of
assessing the nuclear theory uncertainties. We observed a tight correlation
between the nuclear models’ predictions for the weak and charge radii. The
difference of the two, the weak skin, is expected to be small and negative due
to the Coulomb repulsion among the protons inside an $N\\!=\\!Z$ nucleus.
Predictions for the asymmetry at forward angles show no significant dependence
on the weak skin, and solely remain correlated with the charge radius which is
known experimentally with a sufficient precision. Thus, the forward
measurement of the PV assymetry of 12C planned at MESA will determine the weak
mixing angle at low momentum transfer, not contaminated by nuclear
uncertainties. The asymmetry in the backward kinematics is, instead, strongly
correlated with the weak skin. This correlation has been exploited to assess
the precision required to provide a clean electroweak determination of the
weak radius of 12C. We performed a combined analysis of the forward-backward
measurement and showed that assuming the planned $0.3\%$ precision at forward
angles, it directly translates into a $0.3\%$ determination of the weak mixing
angle; meanwhile, a $3\%$ measurement at backward angle will result in the
first electroweak determination of the weak radius of ${}^{12}\mathrm{C}$ with
a precision of $\sim\\!0.34\%$. More aggressive scenarios naturally result in
an even more precise determination of the weak radius. We discussed the
consequences of such an ambitious experimental program for precision tests of
the standard model and beyond with the running weak mixing angle.

###### Acknowledgements.

XRM acknowledges support by grants PID2020-118758GB-I00 funded by
MCIN/AEI/10.13039/5011 00011033; by the “Unit of Excellence María de Maeztu
2020-2023” award to the Institute of Cosmos Sciences, Grant CEX2019-000918-M
funded by MCIN/AEI/10.13039/501100011033; and by the Generalitat de Catalunya,
grant 2021SGR01095. JP acknowledges support from the U.S. Department of Energy
Office of Science, Office of Nuclear Physics under Award Number DE-
FG02-92ER40750. M.G. acknowledges support by EU Horizon 2020 research and
innovation programme, STRONG-2020 project under grant agreement No 824093, and
by the Deutsche Forschungsgemeinschaft (DFG) under the grant agreement GO
2604/3-1.

##  Appendix A Nuclear form factors for non-relativistic energy density
functionals

In Sec. III.2 we describe the formalism adopted to treat the nucleon form
factors. Those are at the basis of the construction of the nuclear form
factors. Here, we report the procedure employed in order to obtain the charge
and weak form factors for non-relativistic energy density functionals, which
follows closely Refs. Horowitz and Piekarewicz (2012); Reinhard and Nazarewicz
(2021). The charge density is calculated as

$\rho_{\rm ch}(\mathbf{r})=\dfrac{1}{(2\pi)^{3}}\int
d^{3}qe^{-i\mathbf{q}\cdot\mathbf{r}}F_{\rm ch}(\mathbf{q}),$ (15)

and the charge form factor as Reinhard and Nazarewicz (2021)

$\displaystyle F_{\rm ch}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\sum_{t=p,n}\Big{[}G^{t}_{E}(\mathbf{q})\Big{(}1-\frac{1}{2}\mathbf{q}^{2}\mathcal{D}\Big{)}F_{t}(\mathbf{q})$
(16) $\displaystyle-$
$\displaystyle\mathcal{D}\Big{(}2G_{M}^{t}(\mathbf{q})-G_{E}^{t}(\mathbf{q})\Big{)}F_{ls}^{t}(\mathbf{q})\Big{]},$

with $\mathcal{D}=\dfrac{\hbar^{2}}{(2mc)^{2}}$, and Reinhard and Nazarewicz
(2021)

$\displaystyle F_{t}(\mathbf{q})$ $\displaystyle=$ $\displaystyle\int
d^{3}re^{i\mathbf{q\cdot r}}\rho_{t}(\mathbf{r})\ ,$ (17) $\displaystyle
F_{ls,t}(\mathbf{q})$ $\displaystyle=$ $\displaystyle\int
d^{3}re^{i\mathbf{q\cdot r}}\nabla\cdot\mathbf{J}_{t}(\mathbf{r})\ .$ (18)

$F_{t}(\mathbf{q})$ is the point-proton (neutron) form factor, while
$F_{ls,t}(\mathbf{q})$ is a form factor related to the current density.
The weak charge density is the Fourier transform of the weak form factor which
is introduced in full analogy to Eq. (16),

$\displaystyle F_{\rm wk}(\mathbf{q})$
$\displaystyle=\sum_{t=p,n}\Big{[}\widetilde{G}^{t}_{E}(\mathbf{q})\Big{(}1-\frac{1}{2}\mathbf{q}^{2}\mathcal{D}\Big{)}F_{t}(\mathbf{q})$
$\displaystyle-\mathcal{D}\Big{(}2\widetilde{G}_{M}^{t}(\mathbf{q})-\widetilde{G}_{E}^{t}(\mathbf{q})\Big{)}F_{ls}^{t}(\mathbf{q})\Big{]}.$
(19)

The charge and weak radii are obtained from the respective charge densities as

$\displaystyle R^{2}_{ch}$ $\displaystyle=$ $\displaystyle\dfrac{1}{Z}\int
r^{2}\rho_{ch}(r)d^{3}r\ ,$ (20) $\displaystyle R^{2}_{\rm wk}$
$\displaystyle=$ $\displaystyle\dfrac{1}{Q_{\rm wk}}\int r^{2}\rho_{\rm
wk}(r)d^{3}r\ .$ (21)

##  Appendix B Spin-orbit currents

Given that the treatment of spin-orbit currents is different for relativistic
and non-relativistic EDFs, this could potentially introduce an additional
theoretical uncertainty. Here, we demonstrate that this is not the case.

The impact of the spin-orbit current on the charge radius of a nucleus with
$N\\!=\\!Z$ was shown to take the following simple form Horowitz and
Piekarewicz (2012):

$R_{\rm ch}^{2}=R_{p}^{2}+\Big{(}r_{p}^{2}+r_{n}^{2}\Big{)}+\Big{(}\langle
r_{p}^{2}\rangle_{\rm so}+\langle r_{n}^{2}\rangle_{\rm so}\Big{)},$ (22)

where in general, the spin-orbit contributions to $R_{\rm ch}$ can not be
evaluated in closed form. Yet, in Ref. Horowitz and Piekarewicz (2012) a good
estimate was obtained by assuming that the lower component of the Dirac spinor
may be derived from the upper component assuming a free space relation. In
this case one obtains

$\langle r_{p}^{2}\rangle_{\rm so}+\langle r_{n}^{2}\rangle_{\rm
so}=\frac{2}{3}\frac{(\kappa_{p}+\kappa_{n})}{M^{2}}\approx-0.004\,{\rm
fm}^{2},$ (23)

where $\kappa_{p}\\!=\\!+1.793$ and $\kappa_{n}\\!=\\!-1.913$ are the
anomalous magnetic moments of the proton and neutron (in units of the nuclear
magneton), respectively. Although the magnitude of the contribution from each
nucleon is about $4/3M^{2}\approx 0.06\,{\rm fm}^{2}$, their combined
contribution is more than ten times smaller because of the almost equal but
opposite values of the anomalous magnetic moments.

In the general case where the relation between the upper and lower components
is determined dynamically, the effect is enhanced because the effective
nucleon mass $M^{\star}$ is reduced in the nuclear medium. Yet, the large
cancellation induced by the anomalous magnetic moments persists, resulting in
a combined contribution of $\langle r_{p}^{2}\rangle_{\rm so}+\langle
r_{n}^{2}\rangle_{\rm so}\\!\approx\\!-0.007\,{\rm fm}^{2}$.

Given that the anomalous weak magnetic moments are simply related to the
corresponding electromagnetic ones, a similar cancellation occurs in the case
of the weak charge radius $R_{\rm wk}$. Thus, we conclude that the spin-orbit
contribution to both $R_{\rm ch}$ and $R_{\rm wk}$ is of the order of 0.05%.

##  Appendix C Comparison between EDF and ab-initio models

To test the validity of the sample of models considered in this analysis,
which is built within the general framework of energy density functionals, we
compared with predictions from ab-initio models. Although to our knowledge
there are no ab-initio predictions for the proton and neutron distributions in
12C, predictions are available for the proton and neutron radii of 40Ca both
in coupled-cluster Hagen  _et al._ (2016) and dispersive-optical models Pruitt
_et al._ (2020). Given that 40Ca is a heavier $N\\!=\\!Z$ nucleus than 12C,
one anticipates that the stronger Coulomb repulsion in 40Ca will generate a
slightly larger negative skin than in 12C. The predictions for the neutron
skin thickness of 40Ca are $-0.0461\\!\leq\\!R_{\rm skin}^{40}({\rm
fm})\\!\leq\\!-0.0400$ for the coupled-cluster model and $R_{\rm
skin}^{40}\\!=\\!(-0.051\pm 0.004)$ fm for the dispersive-optical model. These
values compare well with the predictions obtained by the same EDF models
employed in this work, namely $-0.05\\!\leq\\!R_{\rm skin}^{40}({\rm
fm})\\!\leq\\!-0.04$ for non-relativistic models and $-0.053\\!\leq\\!R_{\rm
skin}^{40}({\rm fm})\\!\leq\\!-0.046$ for the covariant ones. Although the
extrapolation from 40Ca to 12C may not be trivial, it is gratifying to see
that the set of energy density functionals used in this work are entirely
consistent with these values for the case of 40Ca, thereby lending credibility
to our approach. Neverthless, new predictions from ab-initio calculations for
nuclei as light as 12C would be benefitial for improving the current work in
the future.

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