Physics results from the first COHERENT observation of CE$\nu$NS in argon and their combination with cesium-iodide data

We present the results on the radius of the neutron distribution in $^{40}\text{Ar}$, on the low-energy value of the weak mixing angle, and on the electromagnetic properties of neutrinos obtained from the analysis of the coherent neutrino-nucleus elastic scattering data in argon recently published by the COHERENT collaboration, taking into account proper radiative corrections. We present also the results of the combined analysis of the COHERENT argon and cesium-iodide data for the determination of the low-energy value of the weak mixing angle and the electromagnetic properties of neutrinos. In particular, the COHERENT argon data allow us to improve significantly the only existing laboratory bounds on the electric charge $q_{\mu\mu}$ of the muon neutrino and on the transition electric charge $q_{\mu\tau}$.


I. INTRODUCTION
The observation of coherent elastic neutrino-nucleus scattering (CEνNS) in cesium-iodide performed in 2017 by the COHERENT experiment [1,2] unlocked an innovative and powerful tool to study many and diverse physical phenomena [3][4][5][6][7][8][9][10][11][12]. Recently, the COHERENT collaboration observed CEνNS for the first time also in argon [13], using a single-phase 24 kg liquid-argon (LAr) scintillation detector, with two independent analyses that prefer CEνNS over the background-only null hypothesis with greater than 3 σ significance. The experimental challenge behind this analysis is the need to observe nuclear recoils with a very small kinetic energy T nr of a few keV, and thus the need of a low nuclear-recoil-energy threshold, in presence of a larger background, when compared to the cesium-iodide case. This requirement is necessary for the coherent recoil of the nucleus which occurs for | q|R 1 [14], where | q| √ 2M T nr is the three-momentum transfer, R is the nuclear radius of a few fm, and M is the nuclear mass, of about 40 GeV for argon nuclei. The observation in argon, which is the lightest nucleus for which CEνNS process has been measured, allows to demonstrate the CEνNS cross-section dependence on the square of the number of neutrons N 2 , but it can also provide valuable information on nuclear physics, neutrino properties, physics beyond the standard model (SM), and electroweak (EW) interactions.
In this paper, we present the bounds on different parameters of the EW interaction and neutrino electromagnetic properties obtained analyzing the new COHERENT Ar data and those obtained with a combined analysis of the COHERENT CsI and Ar data, using the results of the analysis of the CsI data in Ref. [15]. During the completion of this work, another analysis of this type appeared on arXiv [16], but the results are not comparable with ours because we fit the COHERENT Ar data, whereas the analysis of Ref [16] is not a fit of the COHERENT Ar data, but a fit of the number of CEνNS events obtained by the COHERENT collaboration from the fit of the data [13]. Such an indirect analysis underestimates the systematic uncertainties, especially those due to the background that are not taken into account.
The plan of the paper is as follows. In Section II we describe briefly the CEνNS formalism used in our calculation as well as the inputs employed for simulating the signal spectra. In Sections III and IV we derive, respectively, the results on the average rms radius of the neutron distributions in Ar and on the weak mixing angle. In Sections V, VI, and VII we present, respectively, the constraints on the neutrino charge radii, neutrino electric charges and magnetic moments. Finally, in Section VIII we summarize the results of the paper.

II. FORMALISM AND SIGNAL PREDICTIONS
The SM weak-interaction differential cross section as a function of the nuclear kinetic recoil energy T nr of CEνNS processes with a spin-zero nucleus N with Z protons and N neutrons is given by [17][18][19] where G F is the Fermi constant, = e, µ, τ is the neutrino flavor, E is the neutrino energy. The well-known tree-level values of g p V and g n V are where ϑ W is the weak mixing angle, also known as the Weinberg angle. In this paper we consider the following more accurate values that take into account radiative corrections in the MS scheme [20]: where sin 2 ϑ W = 0.23857 ± 0.00005 [21] is the low-energy value of the weak mixing angle, often denoted withŝ 2 0 [20,21], and ρ = 1.00058 [21], (6) s 2 Z = 0.23122 ± 0.00003 [21], α −1 Z = 127.955 ± 0.010 [21], (8) α s (m W ) = 0.123 ± 0.018 ± 0.017 [22] (9) are, respectively, the ρ parameter of electroweak interactions, the value of sin 2 ϑ W at the scale of the Z-boson mass, the value of the electromagnetic fine-structure constant at the scale of the Z-boson mass, and the value of the strong constant at the scale of the W -boson mass. The value ofα s (m W ) in Eq. (9) is the only measured one that we found in the literature. It is in agreement with the PDG summary in Figure 9.5 of Ref. [21]. In any case, a precise value of α s (m W ) is not needed, because its contribution is practically negligible. The terms in Eqs. (3) and (4) proportional toα Z /ŝ 2 Z , which in turn is proportional to the square of the chargedcurrent weak coupling constant, are due to box diagrams with W -boson propagators. The last term in Eq. (3) depends on the flavor of the interacting neutrino ν through the corresponding charged lepton mass m . This term can be interpreted as the contribution of the neutrino charge radius and is consistent with the expression of the neutrino charge radius calculated in Refs. [23][24][25], that we will discuss in Section V.
Numerically, neglecting the small uncertainties, we obtain These values are different from the tree-level values g p V = 0.0229 and g n V = −0.5 obtained with Eq. (2), especially those of g p V (ν e ) and g p V (ν µ ). In Eq. (1) F Z (| q| 2 ) and F N (| q| 2 ) are, respectively, the form factors of the proton and neutron distributions in the nucleus. They are given by the Fourier transform of the corresponding nucleon distribution in the nucleus and describe the loss of coherence for | q|R p 1 and | q|R n 1, where R p and R n are, respectively, the rms radii of the proton and neutron distributions. For the two form factors one can use different parameterizations. The three most popular ones are the symmetrized Fermi [26], Helm [27], and Klein-Nystrand [28] parameterizations that give practically identical results, as we have verified (see Figure 2). Here, we briefly describe only the Helm parameterization (descriptions of the other parameterizations can be found in several papers, for example in Refs. [3,26,29,30]), that is given by where j 1 (x) = sin(x)/x 2 − cos(x)/x is the spherical Bessel function of order one and R 0 is the box (or diffraction) radius. The rms radius R of the corresponding nucleon distribution is given by For the parameter s, that quantifies the so-called surface thickness, we consider the value s = 0.9 fm which was determined for the proton form factor of similar nuclei [31]. We determined the value of the rms proton distribution radius R p from the value of the 40 Ar charge radius measured precisely in electromagnetic experiments [32,33]: The charge radius R c is given by [34,35] 1 where R point p is the point-proton distribution radius, r 2 p 1/2 = 0.8414 ± 0.0019 fm [36] is the charge radius of the proton and r 2 n c = −0.1161 ± 0.0022 fm 2 is the squared charge radius of the neutron [21]. Since the proton form factor F Z (| q| 2 ) in the cross section in Eq. (1) describes only the interaction of the protons in the nucleus, the corresponding proton distribution radius R p is given by From the experimental value of R c in Eq. (15), we obtain This is the value of the rms radius R p that we used in our calculations. Let us now consider the neutron distribution radius R n that determines the neutron form factor F N (| q| 2 ) in the cross section in Eq. (1). Experimentally, the value of R n is not known and we can get information on it from the fit of the COHERENT data, as discussed in Section III. However, in our analysis it would be unphysical to consider R n as a completely free parameter, because it is very plausible that the neutron distribution radius is larger than the proton distribution radius R p in Eq. (18), since the 40 Ar nucleus has 22 neutrons and only 18 protons. In order to check if this hypothesis is supported by the nuclear theory, we have calculated the proton and neutron radii with two publicly available numerical codes: the Sky3D code [47] of nonrelativistic nuclear mean-field models based on Skyrme forces, and the DIRHB code [48] of relativistic self-consistent mean-field models. Table I  in all the nuclear models that we have considered and the excess is between 0.08 and 0.11 fm. Since where r 2 n 1/2 r 2 p 1/2 is the radius of the neutron (this approximation is supported by the measured value of the neutron magnetic radius r 2 n 1/2 mag = 0.864 +0.009 −0.008 fm [21], that is close to the measured value of the proton charge radius r 2 p 1/2 = 0.8414 ± 0.0019 fm [36]). Hence, from the nuclear model prediction R point n R point p + 0.1 fm we obtain the approximate relation Therefore, in our analyses of the COHERENT Argon data we consider two cases: Fixed R n : where R n is given by Eq. (20) with the value in Eq. (18) for R p : Free R n : where R n is considered as a free parameter between R p and 4 fm: The CEνNS event rate in the COHERENT experiment [13] depends on the neutrino flux dN ν /dE produced from the Spallation Neutron Source (SNS) at Oak Ridge Spallation Neutron Source. It is given by the sum of with the normalization factor η = rN POT /4πL 2 , where r = (9±0.9)×10 −2 is the number of neutrinos per flavor that are produced for each proton-on-target (POT), N POT = 13.7 × 10 22 is the number of proton on target corresponding to a total integrated beam power of 6.12 GW·hr and L = 27.5 m is the distance between the source and the COHERENT Ar detector, called CENNS-10 [1]. The pions decay at rest (π + → µ + + ν µ ) producing ν µ 's which arrive at the COHERENT detector as a prompt signal within about 1.5 µs after protons-on-targets. The decay at rest of µ + (µ + → e + + ν e +ν µ ) produces a delayed component ofν µ 's and ν e 's, since they arrive at the detector in a relatively longer time interval of about 10 µs. In order to extract the physical parameter of interest, the first step is to simulate the CEνNS signal at CENNS-10 as a function of the nuclear recoil energy. The theoretical CEνNS event number N CEνNS i in each nuclear recoil energy bin i is given by where A(T nr ) is the energy-dependent reconstruction efficiency given in Fig. 3 in Ref. [13], E min = M T nr /2 and E max = m µ /2 ∼ 52.8 MeV, m µ being the muon mass, N (Ar) is the number of Ar atoms in the detector, and dNν dE is the neutrino flux integrated over the experiment lifetime. Concerning the former element, we digitalise the efficiency as a function of the electron-equivalent recoil energy T ee [keV ee ], which is subsequently transformed as a function of the nuclear recoil energy T nr [keV nr ] thanks to the relation Here, f Q is the quenching factor, which is the ratio between the scintillation light emitted in nuclear and electron recoils and determines the relation between the number of detected photoelectrons and the nuclear recoil kinetic energy. Following Ref. [13], the quenching factor is parameterized as f Q (T nr ) = (0.246 ± 0.006 keV nr ) + ((7.8 ± 0.9) × 10 −4 )T nr up to 125 keV nr , and kept constant for larger values. The value of N (Ar) is given by N A M det /M Ar , where N A is the Avogadro number, M det is the detector active mass equal to 24 kg and M Ar = 39.96 g/mol is the molar mass of 40 Ar. Actually, one should consider that atmospheric argon is contaminated by a small percentage of 36 Ar and 38 Ar, namely F ( 36 Ar) = 0.334% and F ( 38 Ar) = 0.063%. However, since the amount of 36 Ar and 38 Ar is very small and the uncertainties are large, in practice one gets the same results considering F ( 40 Ar) = 100% and F ( 36 Ar) = F ( 38 Ar) = 0. In Ref. [13] two independent analyses, labeled A and B, are described, that differ mainly for the selection and the treatment of the background. In the following, we will use the data coming from the analysis A, whose range of interest of the nuclear recoil energy is [0, 120] keV ee (corresponding to roughly [0, 350] keV nr ), with 12 energy bins of size equal to 10 keV ee . We have also performed the analyses of the data corresponding to analysis B of the COHERENT collaboration [13] described in appendix A, where we considered only the determination of the radius of the nuclear neutron distribution and of the weak mixing angle.
In our analysis corresponding to analysis A of the COHERENT collaboration [13], we considered the least-squares function where PBRN stands for Prompt Beam-Related Background, LBRN for Late Beam-Related Neutron Background and with σ CEνNS = 13.4% for fixed R n , or 13.2% for free R n , are the estimated number of PBRN and LBRN background events, and σ i is the total signal uncertainty. The Beam Related Neutron Energy Shape (BRNES) 5.8% uncertainty (σ BRNES ) is taken into account by distributing it over the 12 bins in an uncorrelated way. All the numbers are taken from Ref. [13].
In Eq. (29), η CEνNS , η PBRN and η LBRN are nuisance parameters which quantify, respectively, the systematic uncertainty of the signal rate and the systematic uncertainty of the PBRN and LBRN background rate, with corresponding standard deviations σ CEνNS , σ PBRN and σ LBRN .
The COHERENT spectral data are shown in Figure 1(a) together with the histogram obtained with the CEνNS cross section of Eq. (1) and the neutron form factor corresponding to the minimal neutron distribution radius R n = 3.45 fm, that gives the larger CEνNS cross section for R n in the range in Eq. (22). We obtained (χ 2 S ) min = 9.9 with 11 degrees of freedom, corresponding to a goodness of fit of 54%. This is an acceptable goodness of fit, but one can already see from Figure 1(a) that the theoretical histogram in the low-energy bins, where the CEνNS contribution is mostly relevant, is lower than the data. We tested also the case of full coherence, i.e. without the suppression of the neutron and proton form factors, as done in the case of the COHERENT CsI data [3,15]. In this case, a larger CEνNS cross section is obtained, as shown in Figure 1(b). Indeed the full coherence (χ 2 S ) min is 7.5, that is smaller than the 9.9 obtained with the minimal neutron distribution radius. However, we will not consider the full coherence in the rest of the paper, because we are not aware of any physical mechanism that can justify the absence of the form-factor suppression corresponding to the physical nucleon distributions in the nucleus.

III. RADIUS OF THE NUCLEAR NEUTRON DISTRIBUTION
The observation of CEνNS scattering in argon can be used to probe the nuclear neutron distribution [3,4,19,49]. We fitted the COHERENT data in order to determine the neutron rms radius R n of Ar, considering for R n the lower bound in Eq. (22), without an upper bound.   Figure 2 shows the comparison of ∆χ 2 = χ 2 S − (χ 2 S ) min as a function of the rms neutron distribution radius R n of 40 Ar using the three most popular form factor parameterizations: symmetrized Fermi [26], Helm [27], and Klein-Nystrand [28]. One can see that the three form factor parameterizations give practically the same result and the best fit is obtained for the minimal allowed value R n = 3.45 fm. Therefore, from the analysis of the COHERENT data we can only put the following upper bounds on the value of 40 Ar neutron distribution radius: These bounds are in agreement with the nuclear model predictions in Table I, but unfortunately they are too weak to allow us a selection of the models.

IV. WEAK MIXING ANGLE
The weak mixing angle is a fundamental parameter in the theory of the EW interactions and its experimental determination provides a direct probe of physics phenomena not included in the SM, usually referred to as new physics. In particular, low-energy determinations of ϑ W offer a unique role, complementary to those at high-energy, being highly sensitive to extra Z (Z ) bosons predicted in grand unified theories, technicolor models, supersymmetry and string theories [50]. This underscores the need for improved experimental determinations of ϑ W in the low-energy regime. We fitted the COHERENT CENNS-10 data in order to determine the value of sin 2 ϑ W in Ar, considering R n either fixed or free. The result for the weak mixing angle is independent on the assumption used for R n and in both cases we get: which is about 1.7σ above the SM prediction, sin 2 ϑ SM W = 0.23857(5) [21]. The reason of this small discrepancy is that a larger weak mixing angle increases the CEνNS cross section and it allows a better fit of the low-energy bins of the Ar data. Given the independence of sin 2 ϑ W on the value of R n , in the following we will consider only the case with R n fixed. Figure 3 shows the comparison of ∆χ 2 = χ 2 S − (χ 2 S ) min as a function of sin 2 ϑ W using the Helm parameterization for the neutron form factor.
Following the approach used in Ref. [15], where we improved the bounds on several physical quantities from the analysis of the COHERENT CsI data [1] considering the improved quenching factor in Ref. [51], we derive here the result for the weak mixing angle also exploiting the COHERENT CsI dataset. Fixing R n (Cs) and R n (I) to 5.01 fm and 4.94 fm [52], respectively, we get sin 2 ϑ W (CsI) = 0.24 ± 0.04 (1σ), ±0.09 (2σ), +0. 13 −0.14 (3σ),  [56] (e 2 H PVDIS) and from neutrino-nucleus scattering [57] (NuTeV) and the new result from the proton's weak charge at q = 0.158 GeV [58] (Q weak ). In green it is shown the result derived in this paper, obtained fitting the Ar and CsI COHERENT dataset. For clarity we displayed the Tevatron and LHC points horizontally to the left and to the right, respectively.
in very good agreement with the SM prediction. The corresponding ∆χ 2 is also shown in Figure 3. Finally, we performed a combined fit of the CsI and Ar data. The value found for the weak mixing angle is sin 2 ϑ W (CsI + Ar) = 0.27 +0.04 −0.03 (1σ), ±0.07 (2σ), ±0.11 (3σ), which is slightly more precise than the CsI result alone and in agreement within slightly more than 1σ with the SM prediction. Unfortunately, as it is possible to see in Figure 4, the uncertainty obtained for the weak mixing angle from COHERENT is still very large when compared to the other determinations at low-momentum transfer.

V. NEUTRINO CHARGE RADII
The neutrino charge radii are the only electromagnetic properties of neutrinos that are nonzero in the Standard Model of electroweak interactions. They are induced by radiative corrections, with the predicted values [23][24][25] where m W and m are the W boson and charged lepton masses ( = e, µ, τ ), and we use the conventions in Refs. [10,15,59]. The Standard Model charge radii of neutrinos are diagonal in the flavor basis, because in the Standard Model the generation lepton numbers are conserved. Numerically, the predicted values of r 2 νe SM and r 2 νµ SM , that can be probed with the data of the COHERENT experiment, are The current 90% CL experimental bounds for r 2 νe and r 2 νµ obtained in laboratory experiments that do not involve CEνNS are listed in Table I of Ref. [10]. Since they are only about one order of magnitude larger than the Standard Model predictions, they may be the first neutrino electromagnetic properties measured by new experiments in a near future.
As discussed in Section II the contribution of the Standard Model charge radius of ν is taken into account by the last term in the expression (3) of g p V (ν ). Here, we want to study the effects of the neutrino charge radii in the CEνNS data of the COHERENT experiment independently of the origin of the charge radii, that can have contributions both from the Standard Model and from physics beyond the Standard Model. Therefore we consider the differential cross section whereg p V = 0.0204 is given by Eq. (3) without the last term that contains the contribution of the Standard Model charge radius. The effects of the charge radii r 2 ν in the cross section are expressed through [60] We consider the general case in which neutrinos can have both diagonal and off-diagonal charge radii in the flavor basis. The off-diagonal charge radii, as well as part of the diagonal charge radii, can be generated by physics beyond the Standard Model. The effects of the charge radii in the cross section are sometimes expressed through [61,62] that is considered equivalent toQ in Eq. (42) [63] through the well-known relations G F / √ 2 = g 2 /8m 2 W and g 2 sin 2 ϑ W = e 2 = 4πα, where g is the weak charged-current coupling constant and e is the elementary electric charge (see, for example, Ref. [64]). The problem is that the equivalence holds only at tree level and radiative corrections induce a significant difference. Indeed, using the PDG values of all quantities [21] we obtain, neglecting the uncertainties, √ 2πα/3G F = 2.38 × 10 30 cm −2 and 2m 2 W sin 2 ϑ W /3 = 2.64 × 10 30 cm −2 , that differ by about 10%. Therefore, the form in Eq. (43) overestimates the effect of the charge radius by about 10% with respect to the form in Eq. (42), that is the correct one for low-energy interactions because it depends only on measured lowenergy quantities. Moreover, one can notice that the electromagnetic interaction due to the charge radius must be proportional to the electromagnetic fine-structure constant α and must be independent of the Fermi weak interaction constant G F . Indeed, the G F in the denominator of Eq. (42) cancels the G F in the cross section (41).
The diagonal charge radii of flavor neutrinos contribute to the cross section coherently with the neutrino-proton neutral current interaction, generating an effective shift of sin 2 ϑ W . In the case ofν -N scattering, we have g p,n V → −g p,n V and r ν → rν = − r ν . Therefore, the charge radii of flavor neutrinos and antineutrinos contribute with the same sign to the shift of sin 2 ϑ W in the CEνNS cross section.
There are five charge radii that can be determined with the COHERENT CEνNS data: the two diagonal charge radii r 2 νee and r 2 νµµ , that sometimes are denoted with the simpler notation r 2 νe and r 2 νµ in connection to the Standard Model charge radii in Eqs. (38)- (40), and the absolute values of the three off-diagonal charge radii r 2 νeµ = r 2 νµe * , r 2 νeτ , and r 2 νµτ . In Ref. [10] we obtained the bounds on the neutrino charge radii from the analysis of the COHERENT CsI data [1]. In Ref. [15] we improved these bounds considering the improved quenching factor in Ref. [51]. Here we present the bounds on the neutrino charge radii that we obtained from the analysis of the spectral Ar data of the COHERENT experiment [13] and those obtained with a combined fit of the CsI and Ar data. We also revise the CsI limits on the charge radii presented in Ref. [15] because they have been obtained through Eq. (43), that overestimates their contribution by about 10%, as discussed above.
The results of our fits for fixed and free R n are given in Table II. One can see that the bounds obtained with fixed and free R n are similar. Therefore, our results are practically independent from the unknown value of R n , and in the following, for simplicity, we discuss only the case of fixed R n .
The bounds in Table II obtained from the COHERENT Ar data are compatible, but less stringent than those obtained from the CsI data, and the bounds of the combined fit are similar to those obtained with the CsI data only. This is illustrated by Figure 5, that depicts the allowed regions in different planes of the parameter space of the neutrino charge radii. It is interesting, however, that the contribution of the argon data shrinks the allowed region in the vicinity of the Standard Model values of the diagonal charge radii given in Eqs. (39) and (40) and shown by the white cross near the origin in Figure 5(d). In the combined fit, the point corresponding to the Standard Model values of the diagonal charge radii lies at the edge of the 1σ allowed region. This is due to the bad fit of the low-energy bins in Figure 6 due to the small value of the CEνNS cross section for the small Standard Model values of the diagonal charge radii. Figure 6 shows that the best fit of the COHERENT Ar data is obtained for relatively large values of the charge radii (shown by the blue crosses in Figure 5) that enhance the CEνNS cross section in order to fit better the low-energy bins. The best-fit large values of r 2 νe and r 2 νµ are, however, completely excluded by the bounds obtained by other experiments (see Table I of Ref. [10]). The black rectangle near the origin in Figure 5(d) shows the most stringent 90% bounds on r 2 νe and r 2 νµ obtained, respectively in the TEXONO [65] and BNL-E734 [66] experiments. Unfortunately the CEνNS data still do not allow us to limit the neutrino charge radii with such small precision, but it is interesting to see that they tend to favor negative values of the charge radii.
We considered also the case of absence of the neutrino transition charge radii, that is motivated by the attempt to probe the values of the neutrino charge radii in the Standard Model, where only the diagonal charge radii with the values in Eqs. (39) and (40) exist. It is also possible that the physics beyond the Standard Model generates off-diagonal neutrino charge radii that are much smaller than the diagonal charge radii and can be neglected in a first approximation. Figure 7 shows the allowed regions in the ( r 2 νe , r 2 νµ ) plane. One can see that the contribution of the Ar data leads to a restriction of the allowed regions. Although the combined fit tends to favor the allowed island at large negative values of r 2 νµ , we cannot consider it as possible, because it lies well outside the black rectangle near the origin that shows the 90% bounds of the TEXONO [65] and BNL-E734 [66] experiments. The allowed island of the combined CsI and Ar analysis for values of r 2 νµ around zero is compatible at about 2σ with these bounds, as well as with the Standard Model values of the neutrino charge radii.

VI. NEUTRINO ELECTRIC CHARGES
As discussed in Ref. [10], the CEνNS process is sensitive not only to the neutrino charge radii, but also to the neutrino electric charges. Usually neutrinos are considered as exactly neutral particles, but in theories beyond the SM they can have small electric charges (often called millicharges). This possibility was considered in many experimental and theoretical studies (see the review in Ref. [59]).
The differential CEνNS cross section that takes into account the contribution of the neutrino electric charges in addition to Standard Model neutral-current weak interactions is with g p V and g n V given, respectively, by Eqs. (3) and (4) Rn obtained from the analysis of COHERENT CsI data (red lines), from the analysis of COHERENT Ar data in this paper (blue lines), and from the combined fit (shaded green-yellow regions). The crosses with the corresponding colors indicate the best fit points. The white cross near the origin in panel (d) indicates the Standard Model values in Eqs. (39) and (40). The black rectangle near the origin shows the 90% bounds on r 2 νe and r 2 νµ obtained, respectively in the TEXONO [65] and BNL-E734 [66] experiments. electric charges q ν contribute through [59,60] where q 2 = −2M T nr is the squared four-momentum transfer. Although the electric charges of neutrinos and antineutrinos are opposite, neutrinos and antineutrinos contribute with the same sign to the shift of sin 2 ϑ W , as in the case of the charge radii, because also the weak neutral current couplings change sign from neutrinos to antineutrinos.
In this Section, we present the bounds on the neutrino electric charges that we obtained from the analysis of the spectral Ar data of the COHERENT experiment [13] and those obtained with a combined fit of the CsI and Ar data. We also revise the CsI limits on the electric charges presented in Ref. [15] because they have been obtained through an expression similar to that in Eq. (43) (see Eq. (30) of Ref. [15]), that overestimates their contribution by about 10%, as discussed in Section V for the charge radii.
There are five electric charges that can be determined with the COHERENT CEνNS data: the two diagonal electric charges q νee and q νµµ , and the absolute values of the three transition electric charges q νeµ = q * νµe , q νeτ , and q νµτ . Contours of the allowed regions in the ( r 2 νe , r 2 νµ ) plane obtained with fixed Rn obtained from the analysis of COHERENT CsI data (red lines), from the analysis of COHERENT Ar data in this paper (blue lines), and from the combined fit (shaded green-yellow regions), assuming the absence of transition charge radii. The crosses with the corresponding colors indicate the best fit points. The white cross near the origin indicates the Standard Model values in Eqs. (39) and (40). The black rectangle near the origin shows the 90% bounds on r 2 νe and r 2 νµ obtained, respectively in the TEXONO [65] and BNL-E734 [66] experiments.
The results of our fits for fixed and free R n are given in Table III. Since the bounds are similar in the two cases, in  From Table III and Figure 8 one can see that the COHERENT Ar data allow us to put slightly more stringent limits on the neutrino electric charges than the CsI data, in spite of the larger uncertainties. The larger sensitivity of the Ar data to the electric charges is in contrast with the smaller sensitivity to the charge radii discussed in Section V. It follows from the enhancement of the neutrino electric charge effect in CEνNS at low q 2 , because of the denominator in Eq. (45). Since q 2 = −2M T nr , light nuclei are more sensitive than heavier ones at the neutrino electric charges for similar nuclear recoil kinetic energies T nr . The acceptance functions of both the CsI and Ar experiments have a threshold of about 5 keV nr . Since M ( 40 Ar) 37 GeV, M ( 133 Cs) 123 GeV, and M ( 127 I) 118 GeV, the minimum value of |q 2 | can be about 3.2 times smaller in the Ar experiment than in the CsI experiment. However, this enhancement of a factor as large as 3.2 of the neutrino electric charge effect for nuclear recoil kinetic energies above the experimental threshold is mitigated by the different sizes of the energy bins: in the Ar experiment the first bin includes energies from the threshold to about 36 keV nr , whereas the CsI energy bins have a size of about 1.7 keV nr . Therefore, the enhancement of the electric charge effect occurs only in the first energy bin of the Ar experiment. Nevertheless, this enhancement is sufficient for achieving a slightly better performance of the Ar data in constraining the neutrino electric charges in spite of the larger uncertainties, as can be seen in Table III and Figure 8.
The combined fit of the COHERENT CsI and Ar data leads to a significant restriction of the allowed values of the neutrino electric charges, especially the diagonal ones, because of the incomplete overlap of the CsI and Ar allowed regions that can be seen in Figure 8(d). Although the best-fit values of q νee and q νµµ are visibly different from zero, the deviation is not significant, because the 1σ allowed region includes well the point q νee = q νµµ = 0. From Figures 8(a), 8(b), and 8(c), one can see that the best-fit values of the off-diagonal electric charges are close to zero and the values of the off-diagonal electric charges are well constrained.
As already noted in Ref. [15], the bounds of the order of 10 −7 e that we obtained are not competitive with the bounds on the electron neutrino electric charges obtained in reactor neutrino experiments, that are at the level of 10 −12 e [21,59,67,68]. These limits are given in the literature for the diagonal electron neutrino charge q νee , because the contribution of the off-diagonal charges was not considered. However, since the off-diagonal charges contribute to the cross section in a quantitatively comparable way, we can consider them to be bounded at the same order of magnitude level of 10 −12 e. Therefore our bounds are not competitive with the reactor bounds for q νee , q νeµ , and q νeτ . On the other hand, they are the only existing laboratory bounds for q νµµ and q νµτ .   8. Contours of the allowed regions in different planes of the neutrino electric charge parameter space obtained with fixed Rn obtained from the analysis of COHERENT CsI data (red lines), from the analysis of COHERENT Ar data in this paper (blue lines), and from the combined fit (shaded green-yellow regions). The crosses with the corresponding colors indicate the best fit points.

VII. NEUTRINO MAGNETIC MOMENTS
The neutrino magnetic moment is the electromagnetic neutrino property that is most studied and searched experimentally. The reason is that its existence is predicted by many models beyond the Standard Model, especially those that include right-handed neutrinos. It is also phenomenologically important for astrophysics because neutrinos with a magnetic moment can interact with astrophysical magnetic fields leading to several important effects (see the reviews in Refs. [59,69]).
The CEνNS process is sensitive to neutrino magnetic moments [7,15,29,30,70,71]. In this Section, we present the bounds on the neutrino magnetic moments that we obtained from the analysis of the COHERENT Ar data and those that we obtained from the combined fit of the COHERENT CsI and Ar data.
For the analysis of the COHERENT data we used the least-squares function in Eq. (29), with the theoretical predictions N CEνNS i calculated by adding to the Standard Model weak cross section in Eq. (1) the magnetic moment interaction cross section  TABLE IV. Limits at 1σ, 2σ, and 3σ for the neutrino magnetic moments in units of 10 −10 µB, obtained from the analysis of COHERENT CsI data in Ref. [15], from the analysis of COHERENT Ar data in this paper, and from the combined fit.   where m e is the electron neutrino mass, µ ν is the effective magnetic moment of the flavor neutrino ν in elastic scattering (see Ref. [59]), and µ B is the Bohr magneton. The results of the fits for fixed and free R n are given in Table IV. Again, one can see that the bounds are robust with respect to our lack of knowledge of the value of R n , because the bounds are similar for fixed and free R n . For simplicity, in Figure 9(a) we show only the allowed regions in the (|µ νe |, |µ νµ |) plane obtained with fixed R n .
From Figure 9(a) one can see that the best fit of the Ar data is obtained for relatively large values of the neutrino magnetic moments and the case of zero magnetic moments is out of the 1σ allowed region. The reason is similar to that discussed in Section V for the neutrino charge radii: the enhancement of the CEνNS cross section with a sizable neutrino magnetic moment contribution fits better the low-energy bins of the Ar data set than the Standard model cross section, as illustrated in Figure 9(b). In the combined CsI and Ar analysis we find the best fit for |µ νe | = 0, but a best-fit value of |µ νµ | that is relatively large. However, we cannot consider this as a valid indication in favor of a nonzero |µ νµ | because the best-fit value is much larger than the bounds obtained in accelerator experiments with ν µ − e scattering (see Table IV of Ref. [59]). The most stringent of those bounds is the LSND bound |µ νµ | < 6.8 × 10 −10 µ B at 90% CL [72] shown in Figure 9(a). Nevertheless, the 1σ allowed region of the combined fit is compatible with this bound, as well as with the stringent bounds on |µ νe | established in reactor neutrino experiments (the currently best one, |µ νe | < 2.9 × 10 −11 µ B [21,59,73], is not shown in Figure 9(a) because it would not be distinguishable from the y axis).

VIII. CONCLUSIONS
In this paper we discussed the information on nuclear physics, on the low-energy electroweak mixing angle and on the electromagnetic properties of neutrinos that can be obtained from the analysis of the recent CEνNS data on argon of the COHERENT experiment [13]. We also presented the results obtained by combining the analysis of the COHERENT Ar data with the analysis of the COHERENT CsI data [1] performed in Ref. [15].
The information on nuclear physics provided by CEνNS measurements concerns the radius of the neutron distribution in the target nucleus. In Section III we calculated the bounds on the radius of the neutron distribution in 40 Ar. These bounds are in agreement with the nuclear model predictions in Table I, but are rather weak, because the data have large uncertainties. Therefore, they do not allow us to discriminate the different nuclear models.
For the low-energy weak mixing angle, from the analysis of the COHERENT Ar data we obtained a relatively large value which, however, is compatible with that predicted by the Standard Model at about 1.7σ. Including in the analysis the COHERENT CsI data, we found a value that is still larger than that predicted by the Standard Model, but compatible at about 1σ.
The analysis of the COHERENT Ar data allows us to constrain the neutrino charge radii and magnetic moments, but not as well as the analysis of the COHERENT CsI data. Therefore, the combined fits are dominated by the CsI data, with small changes due to the Ar data with respect to the results obtained in Ref. [15]. On the other hand, the Ar data are more sensitive to the neutrino electric charges than the CsI data because of the lower nuclear mass, as discussed in Section VI. Therefore, the Ar data allowed us to improve the constraints on the neutrino electric charges that can be obtained with CEνNS. In particular, we improved the only existing laboratory bounds on the electric charge q µµ of the muon neutrino and on the transition electric charge q µτ .
In conclusion, we would like to emphasize the importance of the results of the COHERENT experiment, that opened the way for CEνNS measurements, first with the CsI detector [1] and then with the LAr detector [13]. Even if the first CEνNS data on argon have large uncertainties, they give us useful physical information. We believe that future experimental improvements will lead to far-reaching results.