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

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.

In previous works [3, 14-16, 19, 20, 22], it has been shown that the CEνNS 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 a matteo.cadeddu@ca.infn.it b nicola.cargioli@ca.infn.it c francesca.dordei@cern.ch d carlo.giunti@to.infn.it e liyufeng@ihep.ac.cn f emmanuele.picciau@ca.infn.it g ternes@to.infn.it h zhangyiyu@ihep.ac.cn 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 CEνNS process can also give information on the weak mixing angle, usually referred to as sin 2 ϑ W , a fundamental parameter of the electroweak theory of the SM. However, in the low-energy sector, the most precise sin 2 ϑ W measurement performed so far belongs to the socalled atomic parity violation (APV) experiment, using cesium atoms [30,31]. This latter measurement depends on the value of R n ( 133 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 Cs and 127 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 Cs and 127 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 mix-ing angle with a percent uncertainty, independent of the value of the average neutron rms radius of 133 Cs and 127 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 CEνNS 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 COHER-ENT CEνNS data; in Section IV we describe the APV data analysis; in Section V we present the results on the 133 Cs and 127 I neutron radii obtained from the combined analysis of COHERENT CEνNS and APV data; in Section VI we discuss the determination of the weak mixing angle from the combined analysis of COHERENT CEνNS and APV data; finally, in Section VII, we briefly summarize the results presented in the paper.

II. COHERENT CEνNS DATA ANALYSIS
The SM CEνNS differential cross section as a function of the true nuclear kinetic recoil energy T nr , considering a spin-zero nucleus N with Z protons and N neutrons, is given by [48][49][50] where G F is the Fermi constant, = e, µ, τ is the neutrino flavor, E is the neutrino energy and | q| 2M T 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 p V = 1/2 − 2 sin 2 ϑ SM W = 0.0229 and g n V = −1/2, where sin 2 ϑ SM W = 0.23857 ± 0.00005 [51] is the low-energy value of the weak mixing angle, in order to take into account radiative corrections in the MS scheme [52], namely g p V (ν e ) = 0.0401, g p V (ν µ ) = 0.0318, and g n V = −0.5094. In Eq. (1), F Z (| q| 2 ) and F N (| q| 2 ) are, respectively, the proton and neutron form factors. They describe the loss of coherence for | q|R p 1 and | q|R n 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 results 1 .
1 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 neutron distribution there is only poor knowledge of R n ( 133 Cs) and R n ( 127 I) obtained in the analyses of the COHERENT 2017 data [3,4,10,[14][15][16]22]. 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 fm and 0.26 fm [59], leading to  Table I, 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 point p,n given by the models adding in quadrature the contribution of the rms nucleon radius r 2 N 1/2 0.84 fm, that is considered to be approximately equal for the proton and the neutron, The analysis of COHERENT data is performed in each nuclear recoil energy bin i and time interval j with the least-squares function where z = 1, 2, 3 stands for CEνNS, Beam-Related Neutron (BRN) and Steady-State (SS) backgrounds. N exp ij is the experimental event number, N CEνNS ij is the predicted number of CEνNS events in Eq. (8), N BRN ij and N SS ij are the estimated number of BRN and SS background events, respectively, and σ ij is the statistical uncertainty, all taken from Ref. [21]. The uncertainties of the η z nuisance parameters, which quantify the systematic uncertainty of the signal rate, of the BRN and of the SS background rates, are σ CEνNS = 13%, σ BRN = 0.9% and σ SS = 3% [21].
We calculated the CEνNS event number N CEνNS i in each nuclear recoil energy bin i with where T nr is the reconstructed nuclear recoil energy, A(T nr ) is the energy-dependent detector efficiency, T max MeV, m µ being the muon mass, E min (T nr ) = M T nr /2, dN ν /dE is the neutrino flux integrated over the experiment lifetime and N CsI is the number of 133 Cs and 127 I atoms in the detector. The latter is given by N A M det /M CsI , where N A is the Avogadro number, M det is the detector active mass equal to 14.6 kg and M CsI is the molar mass of CsI. The neutrino flux from the spallation neutron source, dN ν /dE, is given by the sum of the prompt ν µ component and the delayed ν e andν µ components, considering 8.48×10 −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 × 10 23 and a distance of 19.3 m between the source and the COHERENT detector are used. The energy resolution function, R(T nr , T 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 PE /keV [60] and the quenching factor, f Q (T 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 CEνNS event number, N CEνNS ij , in each nuclear recoil energy bin i and time interval j with where P (νµ) j and P (νe,νµ) j are obtained by integrating the arrival time distributions in the corresponding time intervals with the time-dependent efficiency function [21,60].
Using the SM inputs, the experimental values of R p ( 133 Cs) and R p ( 127 I) in Eqs. (2) and (3), and the NSM values of R EFT n ( 133 Cs) and R EFT n ( 127 I) in Eq. (4), the total number of predicted events is found to be N CEνNS = 311.8.

III. AVERAGE CSI NEUTRON RADIUS
We fitted the COHERENT CsI data to get information on the average neutron rms radius of 133 Cs and 127 I, R n (CsI), obtaining 2 R n (CsI) = 5.55 ± 0.44 fm.
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 NSM n (CsI) 5.06 fm, is compatible with the determination in Eq. (9) at 1.1σ.
The SHF and RMF predictions in Table I  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 Cs in APV experiments, that is related to the weak mixing angle through the relation where α is the fine-structure constant and the couplings of electrons to nucleons, g ep AV and g en AV , using sin 2 ϑ SM W and taking into account radiative corrections in the SM [52,61,62], are given by (see Appendix A) g en AV,SM = g eu AV,SM + 2g ed AV,SM = 0.495, where g eu AV,SM = −0.1888 and g ed AV,SM = 0.3419. These values give Q SM W = −73.23 ± 0.01. Experimentally, the weak charge of a nucleus is extracted from the ratio of the parity violating amplitude, E PNC , to the Stark vector transition polarizability, β, and by calculating theoretically E PNC in terms of Q W where β exp.+th. and (Im E PNC ) th. are determined from atomic theory, and Im stands for imaginary part [61]. We use (Im 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 n.s. W (R n ), is thus retrieved by summing to (Im E PNC ) w.n.s.
PNC ] [12,14,64]. 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 [64][65][66][67][68]. Our calculation of q p and q n is described in Appendix B. We performed the combined APV and COHERENT analysis with the least-squares function where the first term is defined in Eq. (6) and the second term represents the contribution of the APV measurement, where σ APV is the total uncertainty.
Considering that the COHERENT data depend separately on R n ( 133 Cs) and R n ( 127 I), while APV depends only on R n ( 133 Cs), we disentangled for the first time the two nuclear contributions. Assuming sin 2 ϑ W = sin 2 ϑ SM W , we obtained R n ( 133 Cs) = 5.27 +0. 33 −0.33 fm , R n ( 127 I) = 5.9 +1.0 −0.9 fm. (15) The contours at different confidence levels (CL) of the allowed regions in the plane of R n ( 133 Cs) and R n ( 127 I) are shown in Figure 1, from which one can see that NSM expected values in Eq. (4) lie in the 1σ allowed region. Thanks to the combination with APV, R n ( 133 Cs) is well constrained and practically uncorrelated with R n ( 127 I).
The value in Eq. (15) represents the most precise determination of R n ( 133 Cs) and implies a value of the neutron skin ∆R np ( 133 Cs) = R n ( 133 Cs) − R p ( 133 Cs) = 0.45 +0. 33 −0.33 fm, that tends to be larger than the SHF and RMF nuclear model predictions in Table I. This value can be translated in terms of the proton and neutron point radii to allow a direct comparison with ∆R point np measured with parity-violating electron scattering on 208 Pb in the PREX experiments [69][70][71][72]. The comparison is shown in Figure 2, together with the neutron skin predictions given in Table I, 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 Pb and 133 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, CEνNS, and parity-violating electron scattering.
The combination of the precise PREX results with the unique determination of ∆R np ( 133 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 deriva- Point neutron skin predictions for 208 Pb and 133 Cs according to different models (blue squares and red circles, see text and Table I 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. tive of the symmetry energy with respect to density at saturation.
Theoretical calculations show a strong correlation [79][80][81][82] between ∆R np and L, namely larger neutron skins translate into larger values of L. Thus, an experimental measurement of ∆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 [24,25], 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 ∆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 [77,84,85]. 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 Pb [91,92], and nuclear masses [93]. All these constraints indicate an allowed region of L corresponding to 40 L 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 127 I, ∆R np ( 127 I) = 1.1 +1 −0.9 fm, even though with large uncertainty.

VI. WEAK MIXING ANGLE
Leaving sin 2 ϑ W free to vary in the χ 2 in Eq. (14) and assuming R n ( 133 Cs) R n ( 127 I) R n (CsI), it is possible to constrain simultaneously R n (CsI) and sin 2 ϑ W . In this analysis we assume that sin 2 ϑ W has the same value at the momentum transfer scales of COHERENT CEνNS 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 ϑ W from the SM prediction that is constant between about 2 and 100 MeV.
In this analysis we considered R n ( 133 Cs) R n ( 127 I) because the data do not allow us to obtain separate in-  formation 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 I), that is smaller than the uncertainties of the determinations in Eq. (15) of the two radii assuming the SM value of sin 2 ϑ W .
We obtained 3 R n (CsI) = 5.60 +0.47 −0.50 fm, sin 2 ϑ W = 0.2406 ± 0.0035. (17) The contours at different CL in the plane of R n (CsI) and sin 2 ϑ W are shown in Figure 3. One can see that the NSM expected value for R n (CsI) and the SM value of sin 2 ϑ W lie in the 1σ allowed region. The inclusion of the experimental input of R n (CsI) has the effect of shifting the measurement of sin 2 ϑ W towards larger values with respect to the Particle Data Group (PDG) APV value sin 2 ϑ PDG W = 0.2367 ± 0.0018 [61], while keeping the uncertainty at the percent level.
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 µ is shown along with the SM predicted running calculated in the MS scheme [51,101,102]. It is important to remark that, before this paper, the value of R n ( 133 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 ϑ 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 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 (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 ϑ 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 ϑ 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 Cs and 127 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.

ACKNOWLEDGMENTS
We would like to thank A. Konovalov  In order to determine the APV weak charge, Q th W , it is necessary to study in detail the calculation of the couplings, taking into account the radiative corrections. Following Refs. [52,62,115,116] the lepton-fermion couplings are In these relations for up and down quarks, ρ = 1.00063 represents a low-energy correction for neutralcurrent processes and Q f is the fermion charge. Hereŝ 2 0 = sin 2 ϑ SM W , which keeps the same value for µ < O(0.1 GeV). The other corrections inserted in equations (A1)-(A2) come from different contributions, such as electron charge radii (∅ eW ,∅ eZ ), EW box diagrams ( f ZZ , W W , W W ) and vacuum polarization of γZ diagrams ( f γZ ) [62]. They can be expressed as In the expressions above, indicates the lepton involved in the interaction (in our case = e), while f indicates the quarks (in our case f = u, d).
For the electromagnetic-running coupling we adopt the abbreviationα ij ≡α( m i M j ) andα Z ≡ α(M Z ). In particular,α f Z , that is present in the f γZ contribution in Eq. (A3d), 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 valuesα s (M Z ) = 0.1185 [61] and α s (M W ) = 0.123 [117].
Inside the correction diagrams in Eqs. (A3b), (A3c), (A3d), the neutral-current couplings enter at tree level and can be written as [52] Their products are defined as It is important to remark, as reported in Ref. [62], that for the EW box corrections (Eqs. (A3c), (A3e), (A3f)) the sine is evaluated at the value of the Z mass,ŝ 2 Z ≡ sin 2θ W (M Z ) = 0.23121 [61], while in the f γZ term (Eq. (A3d)) the sine is evaluated at scale µ = m p M Z . Finally, inside the ∅ Z term (Eq. (A3b)) the coupling g V A is obtained using the value sin 2 θ W ( √ m 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 g ep AV = 2g eu AV + g ed AV , (A7) However, as pointed out in Refs. [52,62], it is necessary to take into account also a correction relative to the f γZ contribution, and this is obtained by adding to the proton and neutron couplings some small constants such that g ep AV → g ep AV + 0.00005, (A9) g en AV → g en AV + 0.00006, 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. [64,67], where the interaction matrix is proportional to the electroweak couplings to protons and neutrons Here G F is the Fermi constant and This coupling depends on the integrals where ρ 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 (Zα), and for most of the atoms of interest, in particular for (Zα) up to ∼ 0.7, cutting off the series at (Zα) 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 (Zα) 2 , for any nucleus, f (r) is given by where V (r) represents the radial electric potential determined uniquely by the charge distribution ρ c (r) of the nucleus. One can obtain the potential through the Poisson equation (B10) Using the above results and Eq. (B3), one can calculate the proton and neutron integrals. It is worth to notice that in the case of constant density, the integrals in Eq. (B3) 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 q cd p,n = 1 − (Zα) 2 − Under the approximation R c R p and for R 2 n /R 2 p − 1 1, it is possible to obtain the typically used forms of q p,n q cd 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 CEνNS cross section. Therefore, we evaluated numerically the quantities in Eqs. (B3), (B4), and (B6). In practice, we used the Helm parameterization [54] with R c ( 133 Cs) = 4.8041 fm and R p ( 133 Cs) = 4.8212 fm which, for reference, give as a result q p ( 133 Cs) = 0.9567.