Neutron Skin in CsI and Low-Energy Effective Weak Mixing Angle from COHERENT Data

Both the neutron skin thickness $\Delta R_{np}$ of atomic nuclei and the low-energy neutrino-nucleon ($\nu N$) interactions are of fundamental importance in nuclear and particle physics, astrophysics as well as new physics beyond the standard model (SM) but largely uncertain currently, and the coherent elastic neutrino-nucleus scattering (CE$\nu$NS) provides a clean way to extract their information. New physics beyond the SM may cause effectively a shift of the SM weak mixing angle $\theta_W$ in low-energy $\nu N$ interactions, leading to an effective weak mixing angle $\theta^*_W$. By analyzing the CE$\nu$NS data of the COHERENT experiment, we find that while a one-parameter fit to the COHERENT data by varying $\Delta R_{np}$ produces an unrealistically large central value of $\Delta R^{\rm{CsI}}_{np} \simeq 0.68$ fm for CsI with $\sin^2 \theta^*_W$ fixed at the low-energy SM value of $\sin^2\theta_W^{\rm{SM}} = 0.23857$, a two-dimensional fit by varying $\Delta R_{np}$ and $\sin^2 \theta^*_W$ gives significantly smaller central values of $\Delta R^{\rm{CsI}}_{np} \simeq 0.24$ fm and $\sin^2 \theta^*_W \simeq 0.21$, although their uncertainties are large. The implication of the substantial deviation of $\sin^2 \theta^*_W$ from $\sin^2\theta_W^{\rm{SM}}$ on new physics beyond the SM is discussed.

Both the neutron skin thickness ∆Rnp of atomic nuclei and the low-energy neutrino-nucleon (νN ) interactions are of fundamental importance in nuclear and particle physics, astrophysics as well as new physics beyond the standard model (SM) but largely uncertain currently, and the coherent elastic neutrino-nucleus scattering (CEνNS) provides a clean way to extract their information. New physics beyond the SM may cause effectively a shift of the SM weak mixing angle θW in low-energy νN interactions, leading to an effective weak mixing angle θ * W . By analyzing the CEνNS data of the COHERENT experiment, we find that while a one-parameter fit to the COHERENT data by varying ∆Rnp produces an unrealistically large central value of ∆R CsI np ≃ 0.68 fm for CsI with sin 2 θ * W fixed at the low-energy SM value of sin 2 θ SM W = 0.23857, a two-dimensional fit by varying ∆Rnp and sin 2 θ * W gives significantly smaller central values of ∆R CsI np ≃ 0.24 fm and sin 2 θ * W ≃ 0.21, although their uncertainties are large. The implication of the substantial deviation of sin 2 θ * W from sin 2 θ SM W on new physics beyond the SM is discussed.
Introduction.-The neutron skin thickness of atomic nuclei, defined as ∆R np = R n − R p where R n(p) = r 2 is the neutron (proton) rms radius of the nucleus, provides a good probe of the equation of state (EOS) for isospin asymmetric nuclear matter [1][2][3][4][5][6][7][8][9][10], which is critically important due to its multifaceted roles in nuclear physics and astrophysics [11][12][13][14] as well as some issues of new physics beyond the standard model (SM) [15][16][17][18][19]. While the R p can be measured precisely from electromagnetic processes (see, e.g., Refs. [20][21][22]), the R n is largely uncertain since it is usually determined from strong processes, which is generally model dependent due to the complicated nonperturbative effects. This provides a strong motivation for the Lead Radius Experiment (PREX) being performed at the Jefferson Laboratory to determine the R n of 208 Pb to about 1% accuracy by measuring the parity-violating electroweak asymmetry in the elastic scattering of polarized electrons from 208 Pb [23]. The PREX Collaboration reported the first result of the parity violating weak neutral interaction measurement of the ∆R np for 208 Pb, i.e., ∆R 208 np = 0.33 +0.16 −0.18 fm [24] (see, also, Ref. [25]). The central value of 0.33 fm means a surprisingly large neutron skin thickness in 208 Pb although there is no compelling reason to rule out a such large value [26].
Recently, the COHERENT Collaboration [27] reported the first observation of the coherent elastic neutrino-nucleus scattering (CEνNS) [28,29]. In Ref. [30], a value of the averaged ∆R np of 133 55 Cs and 127 53 I, i.e., ∆R CsI np ≃ 0.7 +0.9 −1.1 fm, is extracted from analyzing the COHERENT data. The extracted central value of ∆R CsI np ≃ 0.7 fm is unrealistically large. To the best of our knowledge, ∆R CsI np ≃ 0.7 fm is actually much larger than all the predictions of current nuclear models. Moreover, since 208 Pb is much more neutron-rich than 133 55 Cs and 127 53 I, the ∆R CsI np is expected to be smaller than the ∆R 208 np according to the neutron skin systematics [31,32], and thus * Corresponding author: lwchen@sjtu.edu.cn ∆R CsI np ≃ 0.7 fm is inconsistent with the PREX result. The inconsistency could be a hint of new physics in neutrino physics and this provides the main motivation of the present work.
We note that in Ref. [30], the ∆R CsI np is extracted from a one-parameter fit to the COHERENT data by varying ∆R CsI np with the low-energy weak mixing angle θ W fixed at the SM value sin 2 θ SM W = 0.23865 obtained in the modified minimal subtraction (MS) renormalization scheme at near zero momentum transfer Q = 0 [33] (the newest value is sin 2 θ SM W = 0.23857(5) [34]). Experimentally, the precise determination of sin 2 θ W at low Q 2 is an ongoing issue [35], and the atomic parity violation (APV) experiments offer the most precise results to date. For example, by measuring the 6s 1/2 − 7s 1/2 electric dipole transition in 133 Cs atom, a value of sin 2 θ W = 0.2356 (20) at Q ≃ 2.4 MeV is obtained [36][37][38], which is smaller than sin 2 θ SM W by about 1.5σ. In the mid-energy regime, the Qweak Collaboration reported the recent measurement on proton's weak charge and obtained sin 2 θ W = 0.2383(11) at Q = 0.158 GeV [39], agreeing well with the SM prediction. On the other hand, the low-energy neutrinonucleon (νN ) interactions could involve new physics beyond the SM [35,[40][41][42][43][44][45][46], which may cause effectively a shift of the SM weak mixing angle θ W in the νN interactions, leading to a low-energy effective weak mixing angle θ * W . Any experimental constraints on θ * W would provide useful information on new physics beyond the SM.
In this work, we extract the values of ∆R CsI np and sin 2 θ * W using a two-dimensional (2D) fit to the COHERENT data by varying ∆R CsI np and sin 2 θ * W . Compared to the results using one-parameter fit with sin 2 θ * W fixed at sin 2 θ SM W , we find significantly smaller central values of ∆R CsI np ≃ 0.24 fm and sin 2 θ * W ≃ 0.21 at Q ≃ 0.05 GeV (corresponding to the energy scale of COHERENT experiment), although their uncertainties are large.
CEνNS in the COHERENT experiment.-The differential cross section for coherent elastic neutrino-nucleus scattering has a straightforward SM prediction in the case with differ-ent proton and neutron distributions (form factors) in the nucleus. By neglecting the radiative corrections and axial contributions, the cross section can be expressed as [41,[47][48][49]: where G F is the Fermi coupling constant, M is the nucleus mass, E ν and T are neutrino energy and nuclear recoil kinetic energy, respectively. For a given E ν , the corresponding T varies from 0 to T max = 2E 2 ν /(M + 2E ν ). The proton and neutron neutral current vector couplings are defined, respectively, as g p V = 1 2 − 2 sin 2 θ W and g n V = − 1 2 . The form factor F n(p) (q 2 ) encapsulate the neutron (proton) number density distribution in nuclei, where the momentum transfer q is given by In the case of the COHERENT experiment, the measurement is performed using a CsI detector which is dominantly composed of 133 55 Cs and 127 53 I. The mass of a nucleus with N (Z) neutrons (protons) is determined by its corresponding total binding energy (E B ) from M = N ×m n +Z ×m p −E B where m n(p) is the rest mass of neutrons (protons). The binding energies per nucleon are 8.40998 MeV and 8.44549 MeV [50] for isotopes 133 Cs and 127 I, respectively. As for their density distributions, in order to test the model dependence, two analytic nuclear form factors are adopted, namely, the symmetrized Fermi (SF) form factor and the Helm form factor [30,51,52]. Both form factors are characterized by two parameters related to the nuclear radius and the surface thickness, respectively.
The SF form factor has the form [51] and the corresponding rms radius is expressed as where c is the half-density radius and a quantifies the surface thickness t = 4a ln 3. Experimentally, the proton distribution has been determined precisely, and we take the same parameters for proton distribution as in Ref. [30], which are obtained by fitting the proton structure data of 133 Cs and 127 I measured in muonic atom spectroscopy, namely, t p = 2.30 fm, c p,Cs = 5.6710 ± 0.0001 fm and c p,I = 5.5931 ± 0.0001 fm.
The corresponding proton rms radii for 133 Cs and 127 I are given by R Cs p = 4.804 fm and R I p = 4.749 fm, respectively. The Helm form factor is expressed as [52] F Helm (q 2 ) = 3 where j 1 (x) is the spherical Bessel function of order one, i.e., j 1 (x) = sin(x)/x 2 − cos(x)/x. The rms radius is simply given by Here, R 0 is the box radius and s quantifies the surface thickness. Again, for the proton distributions in 133 Cs and 127 I, we use s p = 0.9 fm following Ref. [30], which was determined for the proton form factor of similar nuclei [53], and the R 0,p is determined by the corresponding R p . For the parameters of the neutron distributions in 133 Cs and 127 I, they are essentially unknown. In these neutron-rich nuclei, in principle, the neutron distributions should be different from the proton distributions because of the charge difference, which means that the neutron distributions could have different radius parameters (c n and R 0,n ) and surface thickness parameters (t n and s n ) compared to the proton distributions. We will examine these effects in the following.
In the COHERENT experiment, the potoelectrons are counted to monitor the scattering events and extract the nuclear recoil energy, with approximately 1.17 photoelectrons expected per keV of nuclear recoil energy, denoted as ζ = 1.17 keV −1 [27]. The number of event counts in a nuclear recoil energy bin [T i , T i+1 ] can be obtained as where N CsI is the number of CsI in the detector and is given by N A m det /M CsI with N A being the Avogadro constant, m det = 14.57 kg the detector mass and M CsI = 259.8 g/mol the molar mass of CsI. The acceptance efficiency function A(x) is decribed by [54] A x < 5, 0.5 5 ≤ x < 6, 1 x ≥ 6.
The value of E min ν depends on T , and the E max ν is related to the neutrino source. At the Spallation Neutron Source, the neutrino flux is generated from the stopped pion decays π + → µ + + ν µ as well as the subsequent muon decays µ + → e + + ν µ + ν e . The neutrino population has the following energy distributions [30,42] with E max ν ≤ m µ /2. The normalization factor η is defined as η = rN POT /(4πL 2 ), where r = 0.08 is the averaged production rate of the decay-at-rest (DAR) neutrinos for each flavor per proton on target, N POT = 1.76 × 10 23 is the total number of protons delivered to the target and L = 19.3 m is the distance between the neutrino source and the CsI detector [27].
To evaluate the fitting quality on the COHERENT data in Fig. 3A of Ref. [27], following Ref. [30], we apply the the following least-squares function with only the 12 energy bins from i = 4 to i = 15, i.e., Here for each energy bin, the experimental number of events, denoted as N exp i , is generated from the C-AC differences, and B i is the estimated beam-on background with only prompt neutrons included [27]. The σ i = N exp i + 2B ss i + B i is the statistical uncertainty where B ss i is the estimated steadystate background determined with AC data [27]. The α and β are the systematic parameters corresponding to the uncertainties on the signal rate and the beam-on background rate, respectively. The fractional uncertainties corresponding to 1-σ variation are σ α = 0.28 and σ β = 0.25 [27]. All the experimental data are taken from the COHERENT release [54].
Results and discussions.-In the present work for CEνNS calculations, we replace the θ W in Eq. (1) by θ * W to effectively consider the possible effects of new physics in νN interactions. We first assume that the neutron and proton distributions have the same surface thickness parameters (i.e., t n = t p and s n = s p ) and the value of sin 2 θ * W is fixed at the SM value of sin 2 θ SM W = 0.23857, and then perform a one-parameter fit to the COHERENT data by varying R n to extract the neutron rms radius R CsI n of CsI ( 133 55 Cs and 127 53 I are assumed to have equal R n ). Our calculations lead to R n = 5.46 +0.91 −1.13 fm with the Helm form factor and R n = 5.47 +0.91 −1.13 fm with the SF form factor. Our results thus nicely confirm the modelindependent value of R n = 5.5 +0.9 −1.1 fm extracted in Ref. [30] with the same assumptions.
Furthermore, we examine the effects of the neutron surface thickness parameters. To this end, we perform a 2D fit to the COHERENT data by varying R CsI n and the surface thickness parameter while the effective weak mixing angle is fixed at sin 2 θ * W = sin 2 θ SM W . The results indicate that a variation of ±0.02 fm for ∆R CsI np arises when s n changes from 0.63 to 1.17 fm (corresponding to a variation of ±30% for s n = 0.9 fm) in the Helm form factor. The same conclusion is obtained when the SF form factor is used. Therefore, compared to the obtained neutron skin thickness of ∆R CsI np ≃ 0.68 +0.91 −1.13 fm, the effects of the neutron surface thickness parameters are indeed quite small, consistent with the statement in Ref. [30]. Now we turn to examining the effects of the low-energy effective weak mixing angle. The potential non-standard running of sin 2 θ * W in low-energy regime is expected to influence the extraction of the neutron distribution from the low-energy CEνNS experiments. The simultaneous precise determination of the neutron distribution and the low-energy sin 2 θ * W through CEνNS experiments can (in)validate our knowledge of nuclear physics and neutrino physics. Hence, we perform a 2D fit to the COHERENT data by varying R n and sin 2 θ * W using the Helm form factor with s n = s p . The resulting number of CEνNS event counts as a function of the number of photoelectrons is shown in Fig. 1 while the corresponding χ 2 contours are displayed in Fig. 2.
For comparison, we also include in Fig. 1 the corresponding results from the COHERENT data, the similar 2D fit by using the SF form factor with t n = t p , and the one-parameter fit by varying R n with fixed sin 2 θ * W = sin 2 θ SM W using both the Helm and SF form factors. It is seen from Fig. 1 that for both one-parameter and 2D fits, the SF and Helm form factors produce almost identical results, indicating the model independence of our results on the form of nuclear form factors. Furthermore, Fig. 1 indicates that compared to the oneparameter fit, the 2D fit predicts a fewer event counts in the region of 7 ∼ 15 for the photoelectron number, leading to the number of total event counts decreases by ∼ 3.2%. From Fig. 2, one sees that there exhibits a positive correlation between R n and sin 2 θ * W . Particularly interesting is that there exists favored center values for R n and sin 2 θ * W , i.e., although the uncertainty is large. We note that very similar results are obtained when the SF form factor is used. With the averaged rms radii of protons and neutrons in 133 Cs and 127 I, we then obtain the averaged neutron skin thickness of CsI as ∆R CsI np ≃ 0.24 +2.30 −2.03 fm.
The favored central value ∆R CsI np ≃ 0.24 fm is significantly smaller than ∆R CsI np ≃ 0.68 fm extracted from the oneparameter fit to the COHERENT data with fixed sin 2 θ * W = sin 2 θ SM W , indicating the importance of the sin 2 θ * W in the extraction of ∆R CsI np from CEνNS. Furthermore, we examine the effects of neutron surface thickness parameters using the 2D fit to the COHERENT data by varying R n and sin 2 θ * W with s n and t n fixed at various values. Our results indicate that the ∆R CsI np varies by ±0.03 fm (the corresponding R n varies from 4.99 fm to 5.05 fm) when the value of s n in the Helm form factor changes from 0.63 fm to 1.17 fm (corresponding to a variation of ±30% for s n = 0.9 fm). Similarly, we find the ∆R CsI np varies by ±0.04 fm (the corresponding R n varies from 4.99 fm to 5.07 fm) when the value of t n in the SF form factor changes from 1.61 fm to 2.99 fm (corresponding to a variation of ±30% for t n = 2.3 fm). The variation of ±(0.03 ∼ 0.04) fm is appreciable compared to the value of ∆R CsI np ≃ 0.24 fm, implying that one can extract useful information on the neutron surface thickness (diffuseness) parameters in atomic nuclei from the future precise measurements of CEνNS. Nevertheless, the extracted central value of ∆R CsI np ≃ 0.24 fm with an uncertainty of ±(0.03 ∼ 0.04) fm obtained in the present work is consistent with some carefully calibrated nuclear models (see, e.g., Refs. [26,30]).
On the other hand, a substantial deviation of sin 2 θ * W from sin 2 θ SM W , i.e., ∆ sin 2 θ * W = −0.02857, is obtained in the present work. This anomaly could be a hint of new physics beyond SM in neutrino physics. For example, one new physics scenario is to introduce the nonstandard interactions (NSIs) in the SM interactions, which has been widely discussed [40][41][42][43]. To make a rough estimate on the parameters in NSIs, we introduce an ad hoc nonstandard charge G NSI V to replace the G V in Eq. (2), i.e., where δ NSI = ǫ uV αα = ǫ dV αα (α = e, µ, τ represents the neutrino flavor) denotes the NSI parameters. Eq. (14) can be obtained from the more general NSIs (see, e.g., Refs. [40,41,43]) by neglecting the flavor-changing couplings ǫ qV αβ (α = β) and assuming that the new flavor-preserving couplings (ǫ qV αα ) are flavor symmetric for neutrinos and the first-generation quarks (q = u, d). Then one can estimate the value of δ NSI as by assuming F p (q 2 ) ≃ F n (q 2 ). These results indicate that the NSI contribution into the proton and neutron neutral current vector couplings is 3δ NSI = 0.024, which is even larger than the SM proton coupling g p V = 1 2 − 2 sin 2 θ SM W = 0.02286. Moreover, we would like to point out that the deviation of sin 2 θ * W from sin 2 θ SM W in neutrino physics can also potentially arise from the neutrino electromagnetic properties, e.g., the neutrino charge radius r 2 ν [43][44][45]. Furthermore, the deviation could be as well from the dark parity violation [35,46]. All these scenarios beyond the SM can effectively shift down the low-energy weak mixing angle in νN interactions and worthy of further investigation with forthcoming more precise CEνNS data in future. It will be also very interesting to check the similar effects in other weak neutral interaction measurements, e.g., APV and PREX.
Summary.-We have demonstrated that the low-energy effective weak mixing angle θ * W plays an important role in the extraction of neutron skin thickness of atomic nuclei from the CEνNS experiments. By analyzing the CEνNS data of the COHERENT experiment, we have found that while a one-parameter fit to the COHERENT data produces an unrealistically large central value of ∆R CsI np ≃ 0.68 fm with sin 2 θ * W fixed at sin 2 θ SM W = 0.23857, a 2D fit gives significantly smaller central values of ∆R CsI np ≃ 0.24 fm and sin 2 θ * W ≃ 0.21, although their uncertainties are large. While ∆R CsI np ≃ 0.24 seems to be reasonable, the substantial deviation of sin 2 θ * W from sin 2 θ SM W could give a hint on new physics in ν-nucleon interactions. Future high precision CEνNS measurements are extremely important to put more stringent constraints on ∆R CsI np and the low-energy effective sin 2 θ * W . It will be also interesting to explore the similar effects in other weak neutral interaction measurements.