A view of Coherent Elastic Neutrino-Nucleus Scattering

We review the physics of coherent elastic neutrino-nucleus scattering and the results and perspectives for the measurements of the radius of the neutron distribution of the nucleus, of the weak mixing angle, and of new neutrino interactions due to physics beyond the Standard Model.

Introduction. -Coherent elastic neutrino-nucleus scattering (CEνNS) is a peculiar interaction process which was predicted theoretically in 1973 [1] (see also [2,3]), but it was observed experimentally for the first time only in 2017 in the COHERENT experiment [4], with neutrinos produced by the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory. The cross-section of CEνNS is larger than the cross-sections of other neutrino interactions at the same neutrino energy. Thus, one may wonder why it took 44 years to observe CEνNS after its theoretical prediction. The reason is that the only observable signature of CEνNS is the recoil of the nucleus with an extremely small kinetic energy which is very difficult to observe. The recent development of new very sensitive detectors allowed this magnificent achievement.
The main characteristic of CEνNS is that the interactions of the neutrino with the nucleons must be coherent, such that the nucleus recoils as a whole, without any change in its internal structure. In this case, the predicted Standard Model (SM) weak-interaction cross-section is approximately proportional to the square of the number of neutrons N in the nucleus. Therefore, the probability of CEνNS interactions is approximately N times larger than the sum of the probabilities of neutrino interaction with every single nucleon in the same nucleus, which is only approximately proportional to N . The reason why the number of neutrons N in the nucleus is more relevant than the number of protons is that, as explained later, the crosssection of neutrino-proton interaction is much smaller than the cross-section of neutrino-neutron interaction.
CEνNS is a process that is not only interesting per se. It gives useful information on the nuclear structure, in particular on the neutron density distribution in the nucleus, which surprisingly is still unknown for most of the nuclei.
It gives also information on the precise value of the neutrino neutral-current interaction at low energy, which is quantified by the so-called "weak mixing angle". Moreover, CEνNS is sensitive to effects of physics Beyond the Standard Model (BSM) such as neutrino electromagnetic interactions, neutrino interactions mediated by new neutral BSM bosons, and, in general, neutrino neutral-current nonstandard interactions (NSI).
So far, all the results obtained from CEνNS experiments are compatible with the SM prediction. After the first observation by the COHERENT Collaboration of CEνNS in 2017 using a cesium-iodide (CsI) detector [4], a larger dataset of scattering events have been collected with the same experiment [5] and the same process has also been observed with a liquid argon detector [6]. Recently, the experimental collaboration working at the Dresden-II experiment reported the observation of CEνNS produced by electron antineutrinos from a nuclear power reactor [7]. Other experiments (CONUS [8] and CONNIE [9]) have been able to obtain upper limits on CEνNS, which are compatible with the SM prediction, and are still working towards their first observation. Several other CEνNS experiments are planned or under construction: new CO-HERENT detectors [10], MINER [11], NEON [12], NU-CLEUS [13], RED-100 [14], Ricochet [15], a project at the European Spallation Source [16], and others [17]. Therefore, we expect many new experimental results in the next years, which will bring very interesting information on nuclear physics, weak neutral-current neutrino interactions and BSM physics.
CEνNS in the Standard Model. -CEνNS is the neutral-current process  where ν α represents a neutrino with flavor α = e, µ, or τ , and A Z N denotes a nucleus with A nucleons, of which Z are protons and N = A − Z are neutrons. Neutral-current processes are characterized by the absence of an exchange of electric charge between the interacting particles. Hence, they are mediated by the exchange of a neutral boson. In the SM the mediator is the Z 0 neutral boson, as illustrated in Fig. 1(a). Figure 2 illustrates three types of interactions of a neutrino ν α with a nucleus, which depend on the wavelength λ Z 0 = h/| q| of the Z 0 neutral vector boson which mediates SM neutral-current weak interactions (h is Planck's constant and q is the Z 0 three-momentum). When λ Z 0 2R, where R is the radius of the nucleus, the Z 0 has a high probability to interact with a single nucleon in the nucleus (a neutron in the illustration), ejecting it. When λ Z 0 2R the Z 0 has a high probability to interact with a group of nucleons in the nucleons, exciting the latter to the state N * . Elastic coherent neutrino-nucleus scattering can happen only when λ Z 0 2R. In this case, the wavelength of the Z 0 covers all the nucleus, allowing a coherent nuclear recoil without any internal change of the nucleus.
For CEνNS, the absolute value | q| of the Z 0 threemomentum is related to the measurable value of the kinetic energy of the recoiling nucleus T by | q| √ 2 M T , where M is the mass of the nucleus. Therefore, the coherency condition is satisfied only for very small values of T , which are difficult to measure. For example, let us consider a heavy nucleus with A ≈ 100 nucleons, for which the nuclear radius can be approximately estimated with the well-known semi-empirical formula R ≈ 1.2 A 1/3 fm ≈ 5 fm. Therefore, CEνNS can happen for | q| 40 MeV, which implies T 10 keV. This is a very low value of the recoil energy of the nucleus, which could not be measured before the recent technical developments of very sensitive detectors employed in the search of dark matter in the form of weakly interacting massive particles (WIMPs).
Indeed, WIMPs with masses of O(10 GeV/c 2 ) induce nuclear recoils in the keV range which are very similar to those produced by CEνNS processes.
Moreover, since the average momentum transfer depends on the neutrino energy, it is only possible to observe CEνNS generated by low-energy neutrinos (E ν 30 MeV). There are various natural sources of low-energy neutrinos: solar neutrinos, geoneutrinos, supernova neutrinos, and the low-energy tail of atmospheric neutrinos [17]. Several experiments are sensitive to CEνNS generated by these natural low-energy neutrino fluxes (especially experiments on the search of nuclear recoils produced by dark matter [18]), but so far these neutrino fluxes have been observed only through other interactions.
CEνNS has been observed only in two laboratory experiments: 1. The COHERENT experiment [4][5][6], with neutrinos produced by the Oak Ridge SNS. In this experiment, the neutrinos are generated by the decay at rest of positive pions (π + ) and muons (µ + ). The positive pions are created by the interaction of a pulsed beam of protons with an energy of about 1 GeV with a mercury target (the produced negative pions are rapidly absorbed by the positive nuclei of the target before decay). They stop in the target and decay at rest with a lifetime of about 26 ns with the process π + → µ + +ν µ , generating a prompt flux of muon neutrinos (ν µ ). The produced positive muons also stop in the target and decay at rest with a longer lifetime of about 2.2 µs with the process µ + → e + +ν µ + ν e , generating delayed fluxes of muon antineutrinos (ν µ ) and electron neutrinos (ν e ). Since the energies of these neutrino fluxes are below about 53 MeV, they generate observable rates of CEνNS in the COHERENT CsI [4,5] and LAr [6] detectors, where the total neutrino flux is about 4.3 × 10 7 ν cm −2 s −1 . It is remarkable that the detectors are very small in comparison with typical neutrino detectors, with mass scales of the order of multiples of tons. The CsI detector weighs only 14.6 kg and the LAr detector weighs 29 kg. The detection of CEνNS with such small detectors is possible because of the large CEνNS cross-section.
2. The Dresden-II experiment [7] with electron antineutrinos (ν e ) generated by the Dresden-II nuclear power reactor of the Dresden Generating Station located near Morris, Illinois, USA. The flux of electron antineutrinos at the very sensitive germanium detector is huge, about 4.8 × 10 13ν e cm −2 s −1 , with energy below about 10 MeV. It is well suited to generate an observable rate of CEνNS in the extraordinarily small germanium detector weighing only 3 kg. We note that there is, however, some criticism of this result related to the anomalously large quenching factor, the reactor-related background and the incompatibility with the recent CONUS results [19] which must be clarified [20].
Illustration of three types of interactions of a neutrino να with a nucleus: inelastic incoherent scattering when λ Z 0 2R, elastic incoherent scattering when λ Z 0 2R, and elastic coherent scattering (CEνNS) when λ Z 0 2R. Here R is the radius of the nucleus and λ Z 0 = h/| q| is the wavelength of the Z 0 neutral vector boson, where q is its three-momentum and h is Planck's constant.
The measurable CEνNS differential cross-section of a neutrino with a spin-zero nucleus N is where E ν is the neutrino energy, and G F is the Fermi constant. The interaction is characterized by the so-called "weak charge of the nucleus" Q N W (| q|), a function of the three-momentum q transferred from the neutrino to the nucleus that depends on the interaction type. In the SM The functions F N N (| q|) and F N Z (| q|) are, respectively, the neutron and proton form factors of the nucleus N , which characterize the amount of coherency of the interaction and depend on the neutron and proton distributions in the nucleus. The coefficients g n V and g p V quantify the weak neutral-current interactions of neutrons and protons, respectively. In the SM they are approximately given by 1 where ϑ W is the weak mixing angle, with sin 2 ϑ W 0.239. Therefore, the neutron contribution is much larger than the proton contribution and the CEνNS cross section is approximately proportional to N 2 .
Nuclear Physics. -The approximate increase of the total CEνNS cross-section with the square of the number of neutrons in the nucleus is shown by the black curve in Fig. 3. However, one can see that the result of the CO-HERENT measurement of the total CEνNS cross-section with the CsI detector lies below the black line and is compatible with the green line which takes into account a small amount of incoherency quantified by a realistic form factor F N N (| q|). Indeed, the COHERENT CsI data show a 6σ evidence of the nuclear structure suppression of the full coherence [21]. 1 The more accurate numerical values which incorporate the socalled radiative corrections can be found in [21]. The dependence of the cross-section on the neutron form factor F N N (| q|) is a powerful tool for obtaining information on the neutron distribution in the nucleus, which is not well-known. Indeed, while the proton distribution of most nuclei is known from electromagnetic measurements [22], the nuclear neutron distribution can be probed only through measurements which employ the strong or weak forces. The interpretation of the results of experiments with hadron probes which explore the nuclear neutron distribution through the strong force is difficult, since the effects of strong-force interactions cannot be calculated with sufficient approximation and the interpretation can be done only by assuming a strong-interaction model with all its limitations [23]. On the other hand, the effects of the weak neutral-current interactions, embodied by Q N W (| q|) in Eq. (3), are known with good approximation. Therefore, weak neutral-current processes as CEνNS are ideal for probing the nuclear neutron distribup-3 tion. Besides CEνNS, the nuclear neutron distribution has been probed with neutral-current weak interactions through parity-violating electron scattering for only two nuclei: 208 Pb in the PREX experiment [24] and 48 Ca in the CREX experiment [25]. The quantity which in practice is measured is the radius R n of the neutron distribution in the nucleus. Since in heavy nuclei there are more neutrons than protons, all nuclear models predict that R n should be larger than the radius R p of the proton distribution in the same nucleus. The excess ∆R np = R n − R p is called "neutron skin". It is a quantity of great interest for nuclear physics and astrophysics, because it 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 [26]. It gives information on the nuclear equation of state, which determines the size of neutron stars [27].
For CEνNS, the neutron form factor F N N (| q|), being the Fourier transform of the neutron distribution in the nucleus, depends on R n . The CEνNS measurements of the COHERENT collaboration with the CsI detector [4,5] give information on the average radius R CsI n of the neutron distributions in 133 Cs and 127 I [21, 28-34]: R CsI n = 5.47 ± 0.38 fm [21]. This value is in agreement, within the uncertainty, with the recent Nuclear Shell Model (NSM) prediction R CsI n (NSM) 5.06 fm [35,36]. The CEνNS measurement of the COHERENT collaboration with the LAr detector [6] has larger uncertainties and leads only to an upper bound for the radius R Ar n of the neutron distributions in 40 Ar [37,38]: R Ar n < 4.2 fm [38] at 1σ (i.e. 68% confidence level). This bound is compatible with the recent NSM prediction R Ar n 3.6 fm [35]. Further improvements in the determination of R CsI n and R Ar n are expected when new measurements of the COHERENT collaboration will be available [10], as illustrated in Fig. 4.
An important improvement in the determination of R CsI n can be achieved by combining the COHERENT CsI results with those of Atomic Parity Violation (APV) experiments with Cs atoms [39,40], which are sensitive to R Cs n [21,36,41]. This method permits to disentangle the measurements of the radii of the neutron distributions of 133 Cs and 127 I: R Cs n = 5.32 +0. 30 −0.23 fm and R I n = 5.30 +0.3 −0.6 fm [21]. Obviously R Cs n is better determined than R I n . Both are compatible with the NSM predictions R Cs n (NSM) 5.09 fm and R I n (NSM) 5.03 fm [35,36]. Note that the results of reactor neutrino CEνNS experiments, as those of the Dresden-II experiment [7], do not give information on the neutron distribution of the target nucleus, because the neutrino energy is low, of a few MeV, and there is an observable CEνNS signal only for very small momentum transfers. Therefore, the interaction is fully coherent and the form factors in Eq. (3) are practically equal to one. On the other hand, in these experiments there is the advantage that the extraction from the data of other physical quantities does not depend on the poorly known neutron distribution of the nucleus.
Electroweak Physics. -The CEνNS cross section depends on the value of sin 2 ϑ W through the proton coupling g p V in Eq. (4). Although the proton contribution to the CEνNS cross section is subdominant with respect to the neutron one, it must be taken into account when performing accurate calculations. In quantum field theory, the value of sin 2 ϑ W (as well as other quantities) is not a constant, but depends on the energy scale at which it is measured, as illustrated in Fig. 5. This figure shows the SM prediction for sin 2 ϑ W obtained from high-precision measurements at a scale of about 100 GeV in experiments at the Tevatron, LEP, SLC and LHC high-energy colliders [42]. The figure shows also that the low-energy value of sin 2 ϑ W has been probed in several experiments with various results which have different degrees of compatibility with the SM prediction. The main interest in measuring the low-energy value of sin 2 ϑ W lies in the possibility of discovering a deviation from the SM due to new physics contributions [48,49], such as the mediation of the interaction by dark Z bosons [49][50][51][52][53]. The determination of the low-energy value of the weak mixing angle has been discussed in several studies [21,28,[30][31][32][36][37][38]41,51,[54][55][56][57][58][59][60][61][62], with the recent updated result sin 2 ϑ W = 0.231 +0.027 −0.024 [21], obtained from the latest COHERENT CsI data [5]. As in the case of the radius of the neutron distribution in the nucleus, the measurement of sin 2 ϑ W can be improved with a combined analysis of the CEνNS COHER-ENT CsI data and those of APV experiments with Cs atoms [39,40], which are also sensitive to the weak mixing angle: sin 2 ϑ W = 0.2398 +0.0016 −0.0015 [21] 2 . This measure-  [39,[42][43][44][45][46][47]. The result obtained in [21] from the combination of COHERENT CEνNS data and APV data is shown in red. ment becomes sin 2 ϑ W = 0.2423 +0.0032 −0.0029 [21], as reported in Fig. 5, when the neutron distribution is simultaneously determined. Clearly, the current combination is dominated by the APV result, but future measurements of the CO-HERENT collaboration [10] are expected to reduce significantly the uncertainty of the CEνNS determination of the weak mixing angle, as illustrated in Fig. 6. As it can be seen, more sin 2 ϑ W measurements are also expected from other future CEνNS experiments (COνUS [64], TEX-ONO [65], CONNIE [66], and MINER [11]) that will be really powerful for further constraining such a quantity.
CEνNS beyond Standard Model. -CEνNS measurements are powerful probes of physics beyond the SM which can induce neutral current interactions as those depicted in Fig. 1(b), which involve the mediation of a photon γ, or a new massive vector boson Z , or a new massive scalar boson Φ. Note that in these non-standard interactions the neutrino flavor can change: ν α + A Z N → ν β + A Z N , with a final neutrino flavor β which can be equal or different of the initial neutrino flavor α. This is different from the SM CEνNS process in Fig. 1(a), where the initial and final neutrino flavors are equal, because the SM neutralcurrent weak interaction conserves flavor. In models BSM there can be flavor-changing neutral-current interactions. Unfortunately, the change of neutrino flavor is not observable in the current and foreseeable experiments, because the final neutrino cannot be detected (the only observable signal of CEνNS is the recoil of the nucleus, as explained above). However, specific flavor-changing NSI can be probed by their contribution to CEνNS.
The possible mediators γ, Z , and Φ in Fig. 1(b) correspond to different scenarios of physics beyond the SM: I) The photon mediates electromagnetic interactions. In the SM and in the common belief neutrinos are exactly neutral and do not interact electromagnetically. However, in models BSM neutrinos can have small electromagcan make a difference as large as 11% [21].  Fig. 6: Current status and future projections for low-energy weak mixing angle measurements. The gray points show the measurements from APV on Cs atoms during the years [67][68][69][70][71]. The brown measurements refer to the COHERENT only and the combination between COHERENT and APV as determined in 2020 [36], while the dark blue and red points refer to the updated measurements reported in [21]. The projections for future CEνNS experiments (CONUS [64], TEXONO [65], CONNIE [66], and MINER [11]), shown by the light blue triangles and dashed error bars, are taken from [72]. The purple triangles are the projections for the future electron scattering experiment MOLLER [73] and P2@MESA [74,75]. The dark blue triangles with dashed error bars are the projections for the future CryoCsI-I and CryoCsI-II determinations [21].
netic interactions [76] which contribute to CEνNS [17,77], due to three non-standard neutrino electromagnetic properties: a very small electric charge (often called "millicharge") [78], a small magnetic moment, and a small anomalous charge radius, which is the only neutrino electromagnetic property that is non-zero already in the SM. So far there is no experimental indication in favour of BSM neutrino electromagnetic interactions, but the search is very active [77]. Several studies have probed neutrino electromagnetic interactions through the analysis of CEνNS data [29,30,32,37,38,50,51,54,55,79,80], finding competitive upper limits. In particular, CEνNS processes permit to improve the limit on the ν e charge radius and represent the only existing laboratory bounds on the muonic and tauonic neutrino electric charges [79]. Thanks to the lower energy of reactor antineutrinos and the low energy threshold of semiconductor detectors, the data from Dresden-II provide complementary information with respect to CEνNS processes observed with ν's produced at SNSs, with negligible dependence on the neutron distribution even if with larger backgrounds and larger dependence on the so-called quenching factor that is not wellconstrained at low energies.
II) The existence of a new massive Z vector boson is predicted in several BSM scenarios [81]. Its mass can be very high, much larger than the so-called "electroweak scale" of about 100 GeV, or even as low as about 100 MeV. The search for the possible effects of these bosons is very active in different types of experiments [82], including p-5 CEνNS [32, 50-52, 54, 55, 62, 79, 80, 83-87]. Existing data permit to put limits on a variety of vector boson mediator models: the so-called universal, the B − L and other anomaly-free U (1) gauge models with direct couplings of the new vector boson with neutrinos and quarks, and the anomaly-free L e − L µ , L e − L τ , and L µ − L τ gauge models where the coupling of the new vector boson with the quarks is generated by kinetic mixing with the photon at the one-loop level. The constraints related to models where the muon couples to the new boson mediator can also be compared with the values that can explain the muon g − 2 anomaly [88] and further data will be needed to fully exclude these interpretations of the anomaly [52].
III) The existence of a new massive Φ scalar boson is more exotic, but we cannot miss the chance to probe it in CEνNS. Interesting constraints have already been put using existing data [32, 50-52, 54, 55, 79, 80, 85].
Conclusions. -The CEνNS era started with the first measurement of such a process by the COHERENT experiment in 2017 [4], 44 years after its theoretical prediction in 1973 [1]. Since then there is great interest in this field, with many experiments already operating or planned [8][9][10][11][12][13][14][15][16][17] that will provide complementary information. The aim is to observe CEνNS in different nuclei, confirming the theoretical prediction, and to obtain from the data information on the radius of the neutron distribution of the nucleus, on the weak mixing angle, and on new neutrino interactions. The search for effects of physics beyond the Standard Model is the main goal of current theoretical and experimental research in particle physics, because we know that the Standard Model, albeit very successful, cannot be the final theory of everything: for example, it cannot explain the smallness of neutrino masses, the existence of Dark Matter and Dark Energy, the matter-antimatter asymmetry in the Universe, and does not include gravity in a consistent theory [93]. Moreover, CEνNS is also important for the search for Dark Matter, because CEνNS events produced by solar and atmospheric neutrinos are expected to be detected in future experiments with high sensitivity. This detection would be very interesting for the study of neutrino properties, but it constitutes a background for Dark Matter searches that can limit their sensitivity, the so-called "neutrino floor/fog" [94,95], making thus of utmost importance to precisely study CEνNS in direct detection dedicated experiments. For all these reasons, we expect interesting CEνNS news in the next years and we look forward to the future. * * * The work of C. Giunti is supported by the research grant "The Dark Universe: A Synergic Multimessenger Approach" number 2017X7X85K under the program "PRIN 2017" funded by the Italian Ministero dell'Istruzione, Università e della Ricerca (MIUR).