Constraints on Neutrino Electric Millicharge from Experiments of Elastic Neutrino-Electron Interaction and Future Experimental Proposals Involving Coherent Elastic Neutrino-Nucleus Scattering

In several extensions of the Standard Model of Particle Physics (SMPP), the neutrinos acquire electromagnetic properties such as the electric millicharge. Theoretical and experimental bounds have been reported in the literature for this parameter. In this work, we first carried out a statistical analysis by using data from reactor neutrino experiments, which include elastic neutrino-electron scattering (ENES) processes, in order to obtain both individual and combined limits on the neutrino electric millicharge (NEM). Then, we performed a similar calculation to show an estimate of the sensitivity of future experiments of reactor neutrinos to the NEM, by involving coherent elastic neutrino-nucleus scattering (CENNS). In the first case, the constraints achieved from the combination of several experiments are −1:1 × 10−12e < qν < 9:3 × 10−13e (90% C.L.), and in the second scenario, we obtained the bounds −1:8 × 10−14e < qν < 1:8 × 10−14e (90% C.L.). As we will show here, these combined analyses of different experimental data can lead to stronger constraints than those based on individual analysis, where CENNS interactions would stand out as an important alternative to improve the current limits on NEM.


Introduction
In the SMPP, the neutrinos are massless, electrically neutral, and only interact weakly with leptons and quarks. Nevertheless, the neutrino oscillation experiments show that neutrinos have mass and are also mixed [1][2][3][4]. Hence, the idea of extending the SMPP so as to explain the origin of neutrino mass. Different extensions of SMPP allow the neutrino to have properties such as magnetic and electric dipole moments as well as anapole moment and electric millicharge [5][6][7]. Even in the Standard Model, it is well-known that the neutrinos also can have nonzero charge radius, as shown in reference [8,9]. Among these properties, the neutrino magnetic moment (NMM) has been quite studied in several research works, where different experimental constraints to this parameter were obtained, for instance, from reactor neutrino experiments [10][11][12][13][14], solar neutrinos [15,16], and astrophysical measurements [17,18]. The limits achieved for the NMM are around 10 −11 μ B , while the prediction of the simplest extension of the Standard Model, by including right-handed neutrinos, is 3:2 × 10 −19 μ B [19]. Furthermore, considering the representation of three active neutrinos, the magnetic moment is described by a 3 × 3 matrix whose components are the diagonal and transition magnetic moments. A complete analysis by considering the NMM matrix and using data from solar, reactor, and accelerator experiments was presented in reference [20,21]. In addition to NMM, the study of the remainder form factors is also important as they are a tool to probe new physics. Among them, the NEM has also been under consideration in the literature, and several constraints have been found mainly from reactor experiments and astrophysical measurements. The most restrictive bound on NEM so far, q ν ≲ 3:0 × 10 −21 e, was obtained in [18] based on the neutrality of matter. A limit of q ν ≲ 1:3 × 10 −19 e was achieved by considering the impact of the Neutrino Star Turning mechanism on a supernova (SN) explosion [22,23]. Some other astrophysical bounds come from analysis of SN 1987A neutrinos, in which effects of the galactic and extragalactic magnetic fields on the path of the neutrinos, such as q ν ≲ 2 × 10 −15 e in [24]. Regarding reactor neutrino experiments with ENES interactions, a bound of q ν ≲ 3:7 × 10 −12 e was calculated in reference [25] via data from TEXONO experiment, while one of the most strong limits was obtained by using data from GEMMA experiment and corresponds to q ν ≲ 1:5 × 10 −12 e at 90% C.L. [26,27]. A new interaction channel was considered in the reference [28], where electron antineutrinos ( ν e ) from reactors were studied via atomic ionizations with germanium detectors, reporting a combined limit on the neutrino charge fraction of |δq | <1 × 10 −12 e (90% C.L.). Most limits on the NEM from reactor experiments have been accomplished in the framework of ENES processes; however, the CENNS interactions also can be an important tool to constraint this parameter. The CENNS process had evaded the experimental detection for around forty years; nevertheless, its recent confirmation by the COHERENT collaboration renewed the interest for the study of different topics about physics beyond the standard model and CENNS [29]. Some of them are NSI interactions [30,31], searches for sterile neutrinos [32,33], measurements of precision on weak mixing angle [34], probes of nuclear density distributions [35], studies on neutrino magnetic moment [36,37], bounds on the neutrino charge radii from analysis of COHERENT experiment data [38,39], and neutrino couplings to light scalars [40]. Concerning the neutrino electric charge, recent limits of the order of 10 −7 e by including neutrino flavor change were obtained for the first time from the analysis of COHERENT experiment data [41]. These bounds are not competitive with those achieved for the effective charge from reactor neutrino experiment data, as mentioned in the same reference. Similar constraints on the transition charges were reported in [42] based on data from the COHERENT experiment with argon as detection material. At the present work, we will explore the sensitivity of CENNS future experiments of reactor neutrinos to neutrino electric millicharge.

The Neutrino Electric Millicharge
The electromagnetic interaction between a neutrino and a lepton or a quark can be expressed through a matrix element related to the electromagnetic current, which is given by the expression: where p i , p f , s i , and s f represent the initial and final neutrino momentum and spin projections, respectively [7,43]. The vertex function Γ μ is described through four form factors, Here, F D ðq 2 Þ, G D ðq 2 Þ, M D ðq 2 Þ, and E D ðq 2 Þ correspond to the charge, anapole, magnetic, and electric form factors, respectively. In the most general case, the form factors are represented by matrices including the neutrino mass eigenstates [44]. If we consider coupling with a real photon, q 2 = 0, then being q ν , a, μ ν , and d the neutrino millicharge, anapole moment, magnetic moment, and electric dipole moment, respectively [45]. On the other hand, in the Standard Model SUð2Þ L × Uð1Þ Y of electroweak interactions, the fact that the neutrino will be an electrically chargeless particle is a consequence of the quantization of the electric charge [6]. However, if we take into consideration the minimally extended Standard Model, which introduces a right-handed singlet neutrino ν R , the triangle anomaly can be removed from the U B−L symmetry of the SM. The introduction of an anomaly-free symmetry produces a dequantization of the electric charge; as a result, the neutrinos can be millicharged particles, whose value allowed by the measurements of nonneutrality of matter [46] and neutron charge [47] would be Q νL = Q νR = ε, being ε ≲ ð−0:6 ± 3:2Þ10 −21 e. Detailed information concerning the theoretical aspects of neutrino electric millicharge can be found in references [6,7] and citations therein. In the next sections, we will show the phenomenological analyses employed to obtain bounds on the neutrino electric millicharge from data of ENES experiments and from CENNS future proposals of reactor neutrinos.

Bounds on NEM from ENES Experiments of Reactor Neutrinos
3.1. χ 2 Statistical Analysis. The study of the electromagnetic properties of neutrinos is usually addressed within the framework of elastic neutrino-electron scattering. In this interaction, low-energy neutrinos and antineutrinos are scattered throughout the collision with electrons in solar, accelerator, and reactor experiments. In this analysis, we take into account data from reactor experiments. The total antineutrino-electron cross section will be composed by the Standard Model (SM) contribution, the electromagnetic (EM) term, and the interference term between SM and EM neutrino interactions, as follows: where being G F the Fermi coupling constant, T e the electron recoil energy, and E ν the incident neutrino energy; while the 2 Advances in High Energy Physics standard coupling constants are g L = sin 2 θ W and g R = sin 2 θ W + 1/2 [5]. Likewise, the term related to the electromagnetic component is given by where α is the fine structure constant and q ν stands for the neutrino electric millicharge measured in units of electron charge [6]. Finally, the interference term was calculated based on reference [8,9] and corresponds to the following: In order to calculate the total number of events (in the ith bin), we will use the following expression: The above integral is evaluated in the real recoil energy T e , the detected recoil energy of the electron T ′ e , and the neutrino energy E ν . λðE ν Þ represents the antineutrino spectrum from the nuclear reactor. For energies above 2 MeV, we used the λðE ν Þ parametrization given by reference [48], while for E ν < 2 MeV, we employ the spectrum of electronic reactor antineutrinos showed in [49]. Additionally, κ = N targ Φ ν e t tot corresponds to the product of the total number of targets times the total antineutrino flux times the total exposure time of the different experimental runs. Furthermore, we have used the resolution function given by the experiments, where σ accounts for the error in the measurement of the kinetic energy of the electrons. For some experiments studied in this work, there is no information available related to the resolution function; accordingly, we have considered it as a delta function, namely, RðT e , T ′ e Þ = δðT e − T ′ e Þ. Finally, we will use the χ 2 function to carry out the statistical analysis, Here, N SM represents the number of events measured by each experiment according to the Standard Model, N th stands for the theoretical prediction as a function of the neutrino millicharge, and Δ i is the statistical error in each energy bin. ENES interactions have not been measured by the GEMMA experiment; the constraint reported in [26,27] was obtained from estimates of the backgrounds; therefore, this bound could not be derived using equation (10). The method used in our analysis of the GEMMA experiment to obtain the respective χ 2 profile will be shown in Section 3.3. Moreover, if an experiment reports a single energy bin, we will use the following expression: where σ stat represents the statistical error.

Reactor Neutrino Experiments.
Regarding the NEM limits from neutrino-electron elastic processes, we used data reported by the following experiments: Rovno, Krasnoyarsk, MUNU, TEXONO, and GEMMA. And we will describe them briefly below.
(i) Rovno was an experiment located at the Rovno Nuclear Power Plant in Ukraine, which contained 75 kg of a silicon multidetector placed at 15 m from the center of the core of the reactor. The cross section ( ν e e) measured in the interval 0.6-2.0 MeV is σ = ð12:6 ± 0:62Þ × 10 −45 cm 2 /fission [10] (ii) The Krasnoyarsk experiment was developed at the nuclear power plant of Krasnoyarsk in Russia. A 103 kg detector of a fluoroorganic scintillator was used. The cross section for the ν e e scattering measured in the interval 3.15-5.175 MeV corresponds to ð4:5 ± 2:4Þ × 10 −46 cm 2 /fission [11] (iii) The MUNU collaboration has studied the existence of a magnetic moment of the electron antineutrino through ENES at low energy. The detector was located at 18 m from the core of a 2.75 GW th commercial nuclear reactor in the Bugey region in eastern France, being the main element of this detector a chamber filled with CF 4 . The measurements were taken considering 66.6 days live time of reactor-on data for electron recoil energies between 0.7 and 2.0 MeV, where 1:07 ± 0:34 events per day were observed [12] (iv) The TEXONO collaboration employed a CsI(Tl) scintillating crystal array as detector with a total mass of 187 kg, placed at 28 m from core 1 of the Kuo-Sheng Nuclear Power Station in Taiwan. In the interval 3.0-8.0 MeV, a total of event numbers of 414 ± 80 ± 61 were registered in 624.9 days of reactor ON and 156.1 days of rector OFF [13] (v) The GEMMA experiment is operating at the Kalinin Nuclear Power Plant (KNPP) in Russia. The distance between the detector of high-purity germanium (with a mass of 1.5 kg) and the 3 GW th reactor core is 13.9 m [14] Moreover, currently the development of low-energy detectors is one of the main purposes of the reactor neutrino experiments. However, for increasingly lower detection 3 Advances in High Energy Physics threshold energies (T < 10 keV), the atomic effects become relevant. According to the results showed in [28], for ENES interactions, the free electron approximation (FEA) (depicted in the equation (6)) would not be enough and the effects of atomic ionization should be included. In the reference [28], the atomic effects were taken into account through a new interaction channel, ν e + A → ν e + A + + e, where the cross section was calculated by using the multiconfiguration relativistic random-phase approximation (MCRRPA) [50]. The results display an enhancement of an order of magnitude in the differential cross section ( ν e − Aðδ q Þ) over FEA, when atomic ionization effects are considered. The analysis allowed to obtain a neutrino charge fraction of |δq | <1 × 10 −12 (90% C.L.) by including combined data from GEMMA and TEX-ONO experiments [28]. As a first approximation, in our calculations, we will not include atomic effects, taking into account that our priority is to initially show the advantages of CENNS interactions over ENES processes in constraining of the neutrino electric millicharge.

Limits on Neutrino Electric Millicharge.
In the different calculations corresponding to the bounds on NEM, we will use the χ 2 analysis given by equation (10) or (11). However, as we mentioned above, in the case of GEMMA experiment, we will use a different method to obtain the χ 2 profile. As shown in [26,27], the ratio between the electromagnetic cross sections for the NEM and NMM contributions is given by where it has been considered that NEM (q ν ) is smaller than NMM (μ ν ); μ B and e represent the Bohr magnet and the fundamental charge, respectively. From equation (12), it follows that Finally, the limit on NEM was computed by replacing the values: μ ν < 2:9 × 10 −11 μ B and T e = 2:8 keV in the equation (13), as presented in [26,27]. Nevertheless, we need the χ 2 profile as a function of q ν from the GEMMA experiment for combining it with the results from the rest of the reactor neutrino experiments under review. Therefore, and considering that the calculation to get the bound on NEM (q ν ≲ 1:5 × 10 −12 e) was based on [14], we will employ the parametrization presented there in order to express the electromagnetic cross section for NMM (equation (3.3) of [14]), being C a constant and X = ðμ ν /10 −11 μ B Þ 2 . Furthermore, the probability density function is given by where σ = 6:9, and X 0 = −5:79. Making use of the last expression, we found the values for the confidence level as function of the X parameter. The result can be seen in Figure 1(a), which corresponds to the right panel of Figure 8 from the reference [14]. As we saw above, the NMM is a function of X (μ ν = ffiffiffi ffi X p 10 −11 μ B ); then, the limit on the neutrino millicharge is given by On the other hand, the values of χ 2 and the X parameter are related through the composite function: χ 2 ðCLðXÞÞ, and the relation between the function χ 2 and the CL can be found following a recomendation of the Particle Data Group [51]. Considering the goodness-of-fit statistics is described by a χ 2 probability distribution function (p.d.f) [51], the p value corresponds to where the number of degrees of freedom is n d = 1, and f ðz, Figure 1(b) shows the result for the confidence level as a function of χ 2 , obtained by using the abovemethod from PDG. Finally, from χ 2 ðCLðXÞÞ and the equation (16), we reproduced the limit q ν ≲ 1:5 × 10 −12 e (90% C.L.) obtained in [26,27] based on the use of GEMMA experiment data. The corresponding χ 2 profile will be included in our combined analysis.
The results for the Δχ 2 sensitive profile (in terms of NEM) for each experiment, together with the combined analysis, are displayed in Figure 2. The Table 1 shows the values for the bounds on the NEM, and the limits from the combination of all experiments are −1:1 × 10 −12 e < q ν < 9:3 × 1 0 −13 e (90% C.L.). In this result, the upper constraint is lower than each of the individual bounds obtained from the other experiments studied here, including the limit achieved in [26,27] based on the analysis of GEMMA experiment data (q ν < 1:5 × 10 −12 e). According to the results in Table 1, the most important contributions to the combined limit are given by GEMMA, TEXONO, and MUNU experiments, respectively. 4 Advances in High Energy Physics

CENNS Interactions.
In 1974, Daniel Freedman suggested the possibility of neutrinos interact coherently with nucleus, based on the discovery of the weak neutral current in CERN and NAL [52]. This coherence condition takes place as long as the momentum transfer Q ≲ 1/R (R is the nuclear radius), which is satisfied by neutrinos with energies up to~50 MeV. The discovery of CENNS was reported by the COHERENT collaboration in 2017, and the observation corresponds to a signal at a 6:7σ confidence level by using a 14.6 kg CsI[Na] target as a detector of neutrinos from the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory [29]. In this case, the total cross section of the coherent elastic neutrino-nucleus scattering is given by where the contribution of the Standard Model is as follows: Here, G F is the Fermi coupling constant, and M and T represent the mass and the recoil energy of the nucleus, respectively; Z and N correspond to the number of protons and neutrons contained in the nucleus [30]. Likewise, FðQ 2 Þ = F Z ðQ 2 Þ = F N ðQ 2 Þ stands for the nuclear form factor, and the vector couplings of the neutral current between neutrinos and protons (neutrons) are described by  Figure 1: χ 2 analysis for the GEMMA experiment. (a) Confidence level as a function of X parameter; the result is in agreement with the right panel of Figure 8 from the reference [14]. (b) Relation between the confidence level and the χ 2 function; this result was obtained by following the Particle Data Group [51].

Advances in High Energy Physics
The radiative corrections in the last expressions are taken from the PDG [51]. As mentioned above, the transfer momentum Q 2 must be sufficiently small in order to satisfy the coherence condition. Thus, in the limit Q 2 → 0, we have the approximation FðQ 2 Þ ≃ 1, which means a full coherence [53]. Based on [54], the electromagnetic contribution and the interference term are given by respectively; where C V = g p V Z + g n V N and q ν represent the neutrino charge in units of the electron charge, q ν = e ν e. For the purpose of calculating the event number in a reactor neutrino detector, we used an expression similar to equation (8), taking into consideration the corresponding total CENNS cross section. Likewise, to obtain the constraints to the neutrino millicharge, we carried out an χ 2 analysis like the one described in equation (10), by including the respective expression for the number of events in CENNS interactions.

Future Neutrino Reactor Experiments with CENNS.
Several experimental proposals of reactor neutrinos that include CENNS interactions are being developed around the world. Each detector involves different features in terms of technology, configuration, and system detection. In our analysis, we have considered some experiments whose expectation is to achieve results in the near future, such as TEXONO, CON-NIE, CONUS, RED100, and MINER. A short description of each of these experiments is shown below.
(i) The TEXONO collaboration. The experiment is located at the Kuo-Sheng Neutrino Laboratory (Taiwan), where a distance of 28 m separates the detector from the 2.9 GW reactor core. Among the current goals of this collaboration is the development of germanium detectors with target mass of 1 kg, with sensitivities of up 100 eV and low-background specifications [55]. For the purposes of our study, we will consider a quenching factor Q f = 1 and an antineutrino flux of Φ ν e = 1 × 10 13 cm −2 s −1 ; the estimation for the expected number of events is 27962 [56].
(ii) The CONNIE experiment. This detector is located at a distance of 30 m from the core of the 3.8 GW th nuclear reactor in the Almirante Alvaro Alberto Nuclear Power Plant in Rio de Janeiro, Brazil. The core of the detector contains an array of Charged-Coupled Devices (CCDs), with silicon as detection material. The neutrino flux in the detector is Φ ν e = 7:8 × 10 12 cm −1 s −1 , and the expected rate of events would be 16.1 evt kg -1 day -1 [57]. In an exposition of 1 kg year, the total number of expected events corresponds to 5877, considering threshold energies up to 28 eV [58].    CONNIE −5:9 < q ν < 5:9 −12 < q ν < 12

RED100
−120 < q ν < 120 −370 < q ν < 340 Combined -3:8 < q ν < 3:8 −9:0 < q ν < 8:8 6 Advances in High Energy Physics (iii) The CONUS experiment is being developed in a nuclear power plant in Brokdorf, Germany, where the detector is placed 17 m from the core of the nuclear reactor. In the first measurements, a mass of 4 kg Ge was considered, and data collection is estimated to last five years. The total antineutrino flux at the detector is Φ ν e = 2:5 × 10 13 cm −2 s −1 , and the threshold energy is 0.1 keV. The expected number of the events under these conditions is 31200 evt year -1 [40,59] (iv) The RED100 experiment. This detector is located at the Kalinin Nuclear Power Plant at 19 m from the reactor core. The expectation is taking data by using a fiducial mass of 100 kg of 136-Xe. The antineutrino flux estimated at the detector location is Φ ν e = 1:35 × 10 13 cm −2 s −1 . In our analysis, we included the most conservative estimation for the number of the expected events (1020 events day -1 ), and the threshold energy corresponds to 500 eV [60,61].
(v) The MINER experiment. The proposal of the Mitchell Institute Neutrino Experiments at Reactor experiment is to use a combination of 72 Ge and 28 Si detectors at 2 : 1 mass ratio, with a total mass of 30 kg. The mean distance between the detector and the reactor core is 1 m. The antineutrino flux estimated at this location is Φ ν e = 2:5 × 10 13 cm −2 s −1 . The configuration allows for reaching a threshold energy of 10 eV, and 5-20 events kg -1 day -1 are expected at the detector. In the calculation, we have taken an event rate of 5 events kg -1 day -1 [62,63].

χ 2 Statistical
Analysis and Limits on NEM. As we mentioned above, in the framework of CENNS interactions, our main purpose is to determine the responsiveness of future experiments of reactor neutrinos to the NEM. In a first approximation, we assume only future statistical errors in the event spectrum at the detector; nevertheless, we will also include possible future systematic uncertainties in a second assessment. In this last case, we will choose the parametrization given by σ syst = pN th , where p represents the percentage of the systematic uncertainty. Then, the expression for the χ 2 calculation is as follows: where N SM corresponds to the event number that each experiment expects to measure according to the SM, and N th represents the theoretical number of events as a function of NEM, while σ stat denotes the statistical error. The results from the first scenario (only statistical uncertainties) are shown in Table 2 and Figure 3. In this case, the limits −1:8 × 10 −14 e < q ν < 1:8 × 10 −14 e (90% C.L.) were obtained for the combination of the experiments under study. As we can see, these last constraints are roughly two orders of magnitude lower than the combined bounds derived from ENES experiments, which would represent an outstanding result, taking into account that it will come from terrestrial experiments.
Among the lower constraints, the bounds obtained from the MINER, CONNIE, and CONUS experimental proposals stand up from the rest of the experiments studied here. As we mentioned above, in the case of the MINER proposal that produces the strongest individual limits, the detection material will be composed by two different elements 72 Ge and 28 Si, which could indicate that a combination of different 7 Advances in High Energy Physics materials in the detector might be a favorable option to improve the sensitivity to the neutrino electric millicharge in CENNS experiments. The constraints achieved by considering systematic errors are displayed in Table 3 and Figure 4. In this calculation, we included two values for the systematic uncertainty (p = 1% and p = 3%) in the theoretical number of events. It is worth mentioning that even for a systematic error of the order of 3% of the number of theoretical events, the combined limits are still of the order of 10 −14 e. This result could indicate that as the experiments reduce the systematic uncertainties, lower and lower constraints can be obtained on the neutrino electric millicharge.

Conclusions
There is an increasing interest in knowing more about the electromagnetic neutrino properties, as the neutrino electric millicharge. Different research works have been focused on obtaining limits for this parameter, including experimental and theoretical considerations. As regards to ENES processes, one of the most important limits, q ν ≲ 1:5 × 10 −12 e (90% C.L.), was reported in [26,27] from the analysis of GEMMA experiment data. In relation to CENNS interactions, limits of the order of 10 −7 e on the neutrino transition charges were obtained in [41,42] by using data for the first time from the COHERENT experiment. In this work, first we performed statistical analyses by considering data from past and current ENES neutrino experiments to restrict NEM. Then, and taking advantage of the great potential of coherent elastic neutrinonucleus scattering, we studied the possibilities of some CENNS future experiments of reactor neutrinos for setting bounds on the neutrino electric millicharge. The combined analyses were made by including data and estimations from different experimental collaborations. In the context of ENES experiments, we obtained the combined limits −1:1 × 10 −12 e < q ν < 9:3 × 1 0 −13 e, being this upper limit lower than the bound reported in [26,27] from the use of GEMMA experiment data. This indicates that the combination of data from different experiments gives rise to a more restrictive constraint than each limit achieved from the individual experiments. As regards CENNS future proposals, in a first approximation and considering only possible statistical errors in the calculations, the limits achieved here are −1:8 × 10 −14 e < q ν < 1:8 × 10 −14 e at 90% C.L. Additionally, we computed the constraints by including probable systematic uncertainties corresponding to the 1% and 3% in the event spectrum at the detector, obtaining the limits: −3:8 × 10 −14 e < q ν < 3:8 × 10 −14 e and −9:0 × 10 −14 e < q ν < 8:8 × 10 −14 e at 90% C.L., respectively. According to these results, CENNS combined constraints are approximately two orders of magnitude lower than those derived from ENES experiments. Which could indicate that in the near future, the CENNS experiments of reactor neutrinos will be one of the most important option to improve the limits on the neutrino electric millicharge, as the sensitivity increases at the detectors. In spite of that the bounds on NEM from CENNS proposals are larger than those coming from astrophysical measurements [18], this difference could be further reduced with the technological advancements in CENNS detectors.

Data Availability
The data used to support the findings of this study are included within the submitted article. The data corresponding to the elastic neutrino-electron scattering experiments are shown in Section 3.2, and the data from the coherent elastic neutrino-nucleus scattering experiments were included in Section 4.2. These data are publicly available for each of the experiments considered in this work. The references and other details are given in the manuscript.

Conflicts of Interest
The authors declare that they have no conflicts of interest.