Probing new neutral gauge bosons with CEνNS and neutrino-electron scattering

The potential for probing extra neutral gauge boson mediators (Z ′) from lowenergy measurements is comprehensively explored. Our study mainly focuses on Z ′ mediators present in string-inspired E6 models and Left-Right symmetry. We estimate the sensitivities of coherent-elastic neutrino-nucleus scattering (CEνNS) and neutrino-electron scattering experiments. Our results indicate that such low-energy high-intensity measurements can provide a valuable probe, complementary to highenergy collider searches and electroweak precision measurements.


INTRODUCTION
Despite its amazing success [1] it is well-accepted that the Standard Model (SM) can not be the whole truth. Although the SM seems to capture the most essential features concerning the gauge description of fundamental interactions, it leaves many in the open. Indeed, many are the theoretical motivations for having an extended gauge structure. The latter include the desire of incorporating a dynamical seesaw mechanism that can naturally account for small neutrino masses [2,3] in such a way that these are linked to the origin of parity violation in the weak interaction [4]. Embedability into a simple unified structure at high energies [5] also motivates the existence of new gauge bosons.
Searches for heavy intermediate vector bosons have been extensively performed using high energy accelerators such as the LHC [6]. Their existence could also have important implications for electroweak precision tests [7][8][9] and induce charged lepton flavor violation [10,11]. Building up on early work [12,13] here we will examine the sensitivity of a number of experimental low-energy setups to the existence of heavy electrically neutral intermediate vector bosons Z . These are expected in theories with gauged B-L [14,15], in extended electroweak models predicting the number of families [16,17], in models with dynamical symmetry breaking [18][19][20], in string-inspired extensions of the SM [21], as well as in ambitious "comprehensive unification" scenarios with extra dimensions [22]. It has been shown that a neutral Z can have masses at the TeV scale in a way consistent with neutrino mass generation as well as gauge coupling unification in SO (10) [23]. In this work we focus on scenarios where a Z boson has mass around the few TeV scale.
In the present paper, we further explore the complementarity of the high-intensity, lowenergy approach as a tool to search for new physics. In the same spirit as Refs. [12,13], here we consider the sensitivity of various low-energy experimental setups involving CEνNS and neutrino-electron scattering to the existence of extra neutral gauge bosons arising from wellmotivated Left-Right (LR) symmetric and E 6 -based theories. Regarding SNS neutrinos, in addition to the "first light" CEνNS measurement with a CsI[Na] detector, we also explore the new physics potential at the future Ge, Liquid Argon (LAr) and NaI[Tl] detector subsystems of COHERENT [48]. In addition, we test the corresponding capabilities at various proposed reactor-based CEνNS facilities such as CONUS [49], CONNIE [50], MINER [51], TEXONO [52], RED100 [53], RICOCHET [54], NUCLEUS [55]. We also explore the potential for probing these vector mediators through ν e − e − scattering using a Liquid Xenon (LXe) detector exposed to neutrinos from a 51 Cr 1 source [58].
Our paper is organized as follows. In Sec. 2 we introduce the formalism for CEνNS and neutrino-electron scattering in Left-Right and E 6 theories. The various experimental setups using both SNS and reactor neutrinos for the case of CEνNS , as well as 51 Cr neutrinos for the case of neutrino-electron scattering are described in Sec. 3. Finally, in Sec. 4 we present our numerical results for the expected sensitivities to the mass of Z gauge bosons in the context of the models discussed.

CEνNS AND NEUTRINO-ELECTRON SCATTERING WITHIN Z MODELS
In this section we first introduce the notation relevant for the description of CEνNS and ν e − e − cross sections in the SM. We provide the new couplings in the neutrino-quark and neutrino-lepton sectors present in extended electroweak models based on Left-Right or E 6 gauge symmetries. Next, we discuss their subleading effect on the dominant Standard Model cross sections.
We start from the neutral-current interaction cross-section of a neutrino with energy E ν scattering off a nucleus with Z protons, N = A − Z neutrons (A is the mass number) and mass m A . In the framework of Standard Model interactions only and for sufficiently low momentum transfer, the CEνNS channel dominates the cross-section, provided that the coherence condition q ≤ 1/R (R is the nuclear radius) is satisfied. Assuming a fourfermion contact interaction, the relevant CEνNS cross-section can be expressed in terms of the nuclear recoil energy T A as [47,59] dσ where G F denotes the Fermi constant and Q V is the vector weak charge written in the form For later convenience, Q V is expressed in terms of the left-and right-handed couplings of the quark q = {u, d} to the Z-boson, as with the weak mixing angle taken in the MS scheme, i.e.ŝ 2 Z = 0.2312. The radiative corrections from the PDG: ρ N C νN = 1.0082,κ νN = 0.9972, λ L u = −0.0031, λ L d = −0.0025 and λ R d = 2λ R u = 3.7 × 10 −5 are also included. In the present study, important corrections due to the finite nuclear size are incorporated through the momentum variation of the nuclear form factors F (Q 2 ). These lead to a suppression of the expected CEνNS event rate. A comprehensive analysis of the form factor effects has been recently conducted in Refs. [30,31] using the first COHERENT data. Here we consider the symmetrized Fermi (SF) approximation [60] with where c and a represent the half-density radius and diffuseness, respectively.
Within the SM, the differential cross-section describing ν e − e − scattering arises from both neutral-and charged-current interactions and reads [13] dσ where the QED corrections have been neglected and the chiral couplings take the form with the radiative corrections ρ νe = 1.0128 andκ νe = 0.9963.

Left-Right Symmetry
There are various Left-Right-symmetric models using the gauge group SU (2) L ⊗SU (2) R ⊗ U (1) B−L , restoring the parity symmetry at high energies [4]. These models give an interesting phenomenology, associated to the existence of additional charged and neutral gauge bosons [7][8][9][10][11]. Here we consider models where the Z arises from Left-Right symmetrical extensions of the SM. In contrast to the charged intermediate vector bosons, it has been shown that the neutral one, Z , can have masses at the TeV scale consistent with neutrino mass generation and gauge coupling unification in SO(10) [23]. In what follows, we will focus on the phenomenology coming from such Z boson.

CEνNS
In the framework of the Left-Right symmetric model, the relevant parameters describing CEνNS are modified as follows.
with the definitions and

Neutrino-electron scattering
Turning to the case of neutrino-electron scattering in the Left-Right symmetric model, the relevant couplings are trivially obtained as where the dependence on the Z mass is incorporated through the parameters A and B defined as in the case of CEνNS in Eq. (9).

E 6 models
New neutral gauge bosons also appear in the primordial E 6 gauge symmetry [7][8][9]. Since it is a rank-six group, E 6 in general yields two neutral gauge bosons beyond those present in the SM. These gauge bosons couple to two new hypercharges, χ and ψ that correspond to the U (1) symmetries present in E 6 /SO(10) and in SO(10)/SU (5). The corresponding hypercharge quantum numbers are given in Table I. We assume that, at low-energies, there is only one U (1) symmetry, written as the combination of the symmetries U (1) χ and U (1) ψ . This defines a one-parameter family of models with hypercharge given as whereas the charge operator takes the usual form Q = T 3 +Y . Within this framework, we can write the expressions for the low-energy effective Lagrangian and compute the corresponding corrections to the SM couplings.

Coherent elastic neutrino-nucleus scattering
In the context of the E 6 model, the new couplings read where the ε P q contributions are written as [61] ε with the abbreviations c β = cos β and s β = sin β. Three different E 6 models are considered here, namely the (χ, ψ, η) models corresponding to cos β = (1, 0, 3/8). Note that, for cos β = (− 5/32, 0), the new physics contributions vanish and, therefore, there is no sensitivity to Z , i.e. the ψ model can not be probed in CEνNS studies.

Neutrino-electron scattering
For this case the relevant couplings read with the new contributions written as and γ defined as previously in the CEνNS case. Here, it is interesting to note that, for cos β = − 5/32, the coupling constants are equal to zero, so there is no sensitivity to new physics in this case [61].

EXPERIMENTAL SETUPS
We now examine a number of conceivable experimental setups that may be used to probe for the existence of new neutral gauge bosons. In particular, we consider CEνNS experiments employing both SNS and reactor neutrinos with various possible targets, as well as a future ν e − e − scattering experiment using a 51 Cr source.

CEνNS from accelerator and reactor neutrinos
For the case of CEνNS experiments, the total number of events between the threshold T th and the maximum nuclear recoil energy allowed by the kinematics, T max As indicated, the sum is taken over the detector isotopes, x, and the neutrino flavors, α. In this expression, F x = N x targ Φ ν denotes the corresponding luminosity on the detector. This depends on the neutrino flux at the detector, Φ ν (L), and the number of target nuclei, N x targ , (see Table II). The efficiency function E(T A ) for each given experiment is taken according to Table II   For the pion decay at rest (π-DAR) neutrinos, relevant for the COHERENT experiment, the neutrino energy distributions are adequately described by the Michel spectrum [62] For reactor-based neutrino experiments, we consider the corresponding antineutrino energy distribution λν e (E ν ) resulting from the fission products 2 of 235 U, 238 U, 239 Pu and 241 Pu [63], while for Eν e < 2 MeV we rely on the theoretical spectrum given in Ref. [64].

Neutrino-electron scattering from a 51 Cr source
Another experimental configuration that we have studied uses neutrinos from an artificial neutrino source, as suggested in [56]. Following Ref. [58], we consider for this case a cylindrical LXe detector with height h = 1.38 m, diameter d = 1.38 m, located at L = 1 m above a 1 MCi radioactive 51 Cr source with flux φ 0 = 2.94 × 10 15 ν/(MCi m 2 s). The emitted neutrino spectra consist of two monochromatic beams with energies E ν 1 = 430 keV and E ν 2 = 750 keV with relative strength α 1 = 10% and α 2 = 90%, respectively. Due to the exponentially decaying nature of the source within a time interval ∆t, we take the time-averaged activity [58] where R 0 Cr51 denotes the initial radioactivity and τ = 39.96 days is the mean lifetime of 51 Cr. The number of neutrino-electron scattering events within a bin i with recoil energy in the range [T e,i , T e,i + δT e ] and maximum recoil energy T max e = 2E 2 ν /(2E ν + m e ) is given by with F = Φ 51 Cr avg V n e ∆t. Here, V represents the detector fiducial volume, n e is the electron density of the target material, while the factor Φ 51 Cr avg = φ 0 1m 2 r 2 avg R Cr51 represents the average neutrino flux. Finally, due to its cylindrical geometry, the average distance r avg between the source and the detector is written as [67] As a test case, in our calculations we assume the three configurations assumed in Ref.

NUMERICAL RESULTS
We now perform a statistical analysis of the different experimental configurations discussed in the previous sections. Our present study is based on a χ 2 fit of the measured (COHERENT with CsI detector) or expected (other CEνNS experiments) number of events. We minimize over the nuisance parameters and probe the Z mass by computing ∆χ 2 (S) = χ 2 (S) − χ 2 min (S) with S ≡ {M Z , β}. Our statistical analysis of the COHERENT data relies on the χ 2 function [24] with N meas = 142 (measured number of events), σ a 1 = 0.28 (normalization uncertainty on the signal events) and σ a 2 = 0.25 (normalization uncertainty on the background events).
N theor denotes the calculated number of events in the Left-Right or E 6 model. The statistical uncertainty is calculated as σ stat = √ N meas + B 0n + 2B ss with B 0n = 6 and B ss = 405 being the prompt-neutron and steady-state background events respectively (see Refs. [24,65] for more details). For the analysis of future CEνNS data expected at the Ge, LAr and NaI detector subsystems at COHERENT, as well as at the different reactor-based experiments, we consider a single nuisance parameter a and assign conservative values for the statistical and systematic uncertainties, namely σ stat = σ sys = 0.2. Although quite simplified, we think this analysis is justified at present. The χ 2 function in this case reads where N SM represents the number of events assuming purely Standard Model interactions and, as previously, N theor is the calculated number of events in the presence of LR and E 6 interactions.
As a first step, we obtain the sensitivity on M Z in the framework of the LR-symmetric model from the available data of COHERENT in terms of a χ 2 fit as described above. In a similar manner, we estimate the projected sensitivities at the future SNS and reactor experiments looking for CEνNS events. The results are presented in the upper-left (upperright) panel of Fig. 1 for the case of SNS (reactor) experiments. From this analysis, it becomes evident that for all the setups the sensitivities are rather poor compared to current bounds from the LHC [6], i.e. we find that M Z 125 GeV at 90% C.L.
We now turn to the Z models obtained in the context of E 6 symmetry. In particular, we explore the corresponding sensitivity on M Z in the (χ, ψ, η) realizations of E 6 by fixing cos β = (1, 0, 3/8). For each model we extract the sensitivity on M Z assuming the available CEνNS data as well as the data expected in the future. The results obtained are presented in the lower-panel of Fig. 1, where the red (gray) band corresponds to the χ (η) model. Note that, in comparison with the upper plots in Fig. 1, here the color labels corresponding to each experiment are dropped. The band width illustrates the sensitivity range considering all the experiments: for the COHERENT experiment, the least (most) constraining detector is the CsI (NaI), as can be seen in the upper-panel Fig. 1, while for reactor-based facilities one finds that essentially all experiments have the same sensitivity. In both cases, the sensitivity to the Z mass at 90% C.L. is slightly below (above) 200 GeV for the η (χ) model. A summary of the obtained limits at 90% C.L. is listed in Table III. Notice that CEνNS is not sensitive to the ψ model, since the ε V p = 2ε V u + ε V d and ε V n = ε V u + 2ε V d couplings 3 are vanishing, see Eq. (13). 3 Note that ε V q = ε L q + ε R q with q = {u, d}.

Improving the sensitivities on the Z mass with future CEνNS experiments
We now explore to what extent the control of uncertainties will offer improved sensitivities on the vector mediator mass. To this purpose, we perform a χ 2 analysis assuming different values for the statistical and systematic uncertainties, under the approximation σ sys = σ stat , while keeping all other detector specifications fixed according to Table II Table II and the text). The values of the reported limits are given in GeV units. We also present results for the proposed 51 Cr-LXe experiment, corresponding to the scenarios (A,B,C).
factor measurements with a better understanding of the nuclear form factors and neutrino fluxes, as well as from the expected substantial improvements on detector technologies aimed at the future CEνNS experiments. Our results are presented in the upper and lower panel of Fig. 2 for the LR symmetric and E 6 models, respectively. As previously, the left and right panels show the sensitivities of SNS and reactor CEνNS experiments. Focusing on the LR symmetric model, it can be seen that, for very low uncertainty, the LAr and NaI detectors perform better while, for larger uncertainty, the CsI detector is optimal. Similarly, for the case of reactor-based CEνNS experiments, the Xenon-based RED100 (Ge-based TEXONO) appears to have the best (worst) performance. The same conclusions are drawn for the case of E 6 models where, for convenience, only the bands are displayed. In both cases, one sees the improvement with respect to Fig. 1. At this point, we turn to the impact of neutrino luminosities on improving the attainable sensitivities on the Z mass at future CEνNS experiments. We do this by scaling up the number of events, assuming a correspondingly larger detector mass and running period. This information is encoded in the future detector luminosity factor, that we denote here as F . We have checked that, with the chosen values of statistical and systematic uncertainties [see Eq. (22)], the sensitivity shown in Fig. 1 remains practically unaffected by an increase in the exposure. Indeed, the sensitivity on the Z mass is dominated by the systematic uncertainty at all experiments. Figure 3 illustrates the projected sensitivity on M Z at 90% C.L. as a function of the ratio F /F, where F corresponds to the current or proposed luminosity of each experiment in Table II and σ stat = σ sys = 5%. This level of systematic uncertainty can only be reached through a substantial improvement on the quenching factor uncertainty 4 , an improved determination of the nuclear form factors and a better understanding of the neutrino energy distribution. We therefore conclude that higher intensity CEνNS experiments will offer only slightly improved results with respect to those expected from the current experimental setups. Moreover, one sees that the expected sensitivity for the χ model is better than that expected in the η or LR symmetric models.

Improved Z sensitivities with future neutrino-electron scattering experiments
We now expand our analysis by including also information coming from neutrino-electron scattering, which involves both neutral and charged currents. As a concrete example, we focus on new interesting proposals that aim to measure neutrino-electron scattering events by employing a LXe detector exposed to neutrino emissions from a radioactive 51 Cr source [58]. Our statistical analysis in this case is based on the χ 2 function with δN i SM = N i SM . Since we are dealing with a large number of events, here we have binned the sample with δT e = 5 keV, assuming 120 bins in the range [0, T max e ].
The sensitivities on M Z for the LR symmetric and E 6 models is shown in the left and right panel of Fig. 4, respectively. As previously, the χ model (dashed lines) is more sensitive to M Z compared to the η model (solid lines), while the ψ model (dotted lines) is the least sensitive. Our results indicate that neutrino-electron scattering within 50-100 days will reach a sensitivity of the order of 200-600 GeV, i.e. more competitive with respect to the one extracted from the purely neutral-current CEνNS.
We can obtain the sensitivity contours in the (cos β, M Z ) plane, where β is the parameter defining a one-parameter family of E 6 theories, as shown in Fig. 5. The left panel illustrates the allowed regions obtained from CEνNS at the SNS and at reactor experiments, indicated by the shaded blue and green bands, respectively. The right panel shows the corresponding regions from a neutrino-electron scattering experiment using a 51 Cr source, for the different configurations assumed (A, B and C). The following conclusions can be extracted from the figure: concerning CEνNS, the SNS facilities are less sensitive compared to the reactor-based ones, while neutrino-electron scattering at LXe with a 51 Cr source is the optimum choice, since it can exclude a larger region of the parameter space. Notice as well the presence of two special β values for which CEνNS experiments have no sensitivity to M Z , since the Z couplings vanish in this case. Likewise, one sees that neutrino-electron scattering presents only one such special β value with no Z sensitivity. These results are in agreement with our discussion in Sec. 2 2.2.
One sees that the potential for probing a new Z mediator from low-energy measurements of neutrino-electron scattering or CEνNS seems to lie well below the sensitivity reached by direct searches at the LHC, i.e. the search for high-mass dilepton resonances produced a la Drell-Yan [6]. However, our analysis has been very conservative, as it relies only on the proposed experimental configurations of the first generation of CEνNS experiments. Moreover, we have presented the sensitivity range expected from the various experiments of each type, considering one experiment at a time.

Combined expected sensitivities on the Z mass
So far our strategy has been to explore the potential in probing Z physics within a given low-energy experiment. However, one may explore the phenomenological potential of high intensity low-energy experiments by performing a combined analysis of CEνNS and ν e − e − scattering experiments. Due to the lack of experimental data, however, only the COHERENT-CsI, CONNIE and 51 Cr-LXe experiments are taken into account. While considering only three experiments for this particular analysis, we however note that different neutrino sources and detector materials are assumed, minimizing the impact of correlation effects of the present analysis. The corresponding results are illustrated in Fig. 6 for the χ and η models of E 6 , assuming different choices of systematic or statistical uncertainties ranging from 20% to zero. Notice that different experimental uncertainties are assumed for the case of CEνNS experiments, while for neutrino-electron scattering the C configuration is assumed. As before, one can also present the sensitivities to the various gauge bosons of E 6 models associated to different values of β. One sees that future neutrino-electron scattering and high-intensity CEνNS data with a better control of uncertainties has promising prospects for reaching few TeV scale. This is the scale currently probed at the high energy frontier experiments, such as the LHC 5 . It follows that low-energy measurements may offer new probes of Z parameters, complementary to the high-energy frontier approach.

CONCLUSIONS AND OUTLOOK
In this work we have quantified the attainable sensitivities on extra neutral gauge bosons Z that can be reached at high-intensity, low-energy facilities. We focused on existing and next generation coherent-elastic neutrino-nucleus scattering (CEνNS) as well as neutrinoelectron scattering experiments. As neutrino sources, we have discussed the spallation neu- tron source as well as reactor neutrinos and neutrinos from radioactive sources. We have concentrated on compelling models that predict the existence of a new neutral vector boson mediator, such as string-inspired E 6 schemes and models with Left-Right symmetry. The Z contributions to the CEνNS and neutrino-electron scattering in this class of theories were studied. Current Z limits are obtained from Fig. 1 and given in Table III. Future Z sensitivities from individual CEνNS at SNS and reactor experiments are given in Figs. 2 and 3. A comparison with the expected sensitivity from a future neutrino-electron scattering using a 51 Cr source and a ton-scale Liquid Xenon detector is also given in Fig. 4. Expected sensitivities for an arbitrary E 6 model are presented in Fig. 5. Finally, combined global sensitivities were presented in Fig. 6. The high-intensity, low-energy approach is not only complementary to the high-energy frontier measurements at colliders, but could also become competitive in the long run, as shown in Fig. 6.