Constraints on light vector mediators through coherent elastic neutrino nucleus scattering data from COHERENT

We present new constraints on three different models, the so-called universal, $B-L$ and $L_\mu-L_\tau$ models, involving a yet to be observed light vector $Z'$ mediator, by exploiting the recent observation of coherent elastic neutrino-nucleus scattering (CE$\nu$NS) in argon and cesium-iodide performed by the COHERENT Collaboration. We compare the results obtained from a combination of the above data sets with the limits derived from searches in fixed target, accelerator, solar neutrino and reactor CE$\nu$NS experiments, and with the parameter region that could explain the anomalous magnetic moment of the muon. We show that for the universal and the $B-L$ models, the COHERENT data allow us to put stringent limits in the light vector mediator mass, $M_{Z'}$, and coupling, $g_{Z'}$, parameter space.


Introduction
Since the first observation of the coherent elastic neutrino-nucleus scattering (CEνNS) process in cesium-iodide (CsI) by the COHERENT experiment [1,2], many intriguing physics results have been derived by a large community of physicists [3][4][5][6][7][8][9][10][11][12][13][14][15][16] on very diverse physical sectors. With the recent detection of CEνNS in a single-phase 24 kg liquidargon scintillation detector [17] the COHERENT Collaboration has started to probe the CEνNS cross-section dependence on the square of the number of neutrons, N 2 , and unrevealed a new way to test the standard model (SM). Indeed, this new measurement allowed to gain additional and complementary information to that provided by the CsI dataset on nuclear physics, neutrino properties, physics beyond the SM, and electroweak interactions [18].
CEνNS is a neutral current process induced by the exchange of a Z boson. It thus represents also a sensitive probe for non standard interactions (NSI) that are not included in the SM [19], induced by yet to be discovered neutral vector bosons [6], particularly if they are light. Indeed, for sufficiently light vector mediator masses, the scattering rate grows as 1/| q| 2 as one goes to lower energies, so the low momentum transfer of CEνNS experiments makes them ideal laboratories for such searches. In fact, CEνNS can occur when the three-momentum transfer | q| √ 2MT nr during neutrino scattering off a nucleus is smaller than the inverse of the nuclear radius R, which is of the order of few fm, where M is the nuclear mass, and T nr is the energy of the nuclear recoil of a few keV. This means that on average the momentum transfer is of the order of few tens of MeV, making CEνNS the perfect place to study scenarios including a new light vector mediator. The nature of the latter depends on the details of the specific model assumed.
In this paper, we present new constraints on different models involving a light vector Z mediator, obtained analyzing the new COHERENT argon (Ar) and CsI data, as well as those obtained with a combined analysis of these two datasets, using for CsI the same inputs described in Ref. [16]. In particular, we consider three models with different interactions of the light vector Z mediator. The first one is the so-called universal model, in which the mediator couples universally to all the SM fermions [6,20]. The second one is referred to as the B − L model [20,21], where the coupling of the mediator is different between quarks and leptons. Finally, the third one is the so-called L µ − L τ model [22], in which the mediator only couples with SM particles of the muonic or tauonic flavour. All these models are theoretically well motivated to provide a coherent explanation to a series of emerging discrepancies in precision studies of low-energy observables. Among those, the anomalous measurement of the magnetic moment of the muon, referred to as (g − 2) µ , performed by the E821 experiment at BNL [23] represents since almost two-decades an intriguing puzzle. Indeed, the experimental value differs from the SM prediction by about 3.7σ [24]. Thus, new constraints on this kind of models that incorporate new physics in the leptonic sector are very much awaited. In this context, the new COHERENT results provide a timely and stimulating opportunity to probe some of these models and improve the existing limits.
The plan of the paper is as follows. In Section 2 we describe briefly the CEνNS formalism used to simulate the CEνNS signal in the SM as well as the experimental inputs. In Section 3 we summarise the models of interest to incorporate the effect of a light vector Z mediator on the CEνNS cross section while in Section 4 we describe our method to analyse the COHERENT data. In Section 5 we derive the constraints using the results of the analysis of the COHERENT Ar and CsI data as well as the combined limits and we compare them with the existing limits. Finally, in Section 6 we summarise the results of the paper.

CEνNS signal prediction in the standard model
We follow closely the same formalism developed in Ref. [18] to analyse CEνNS Ar data and in Ref. [16] for CsI data from the COHERENT Collaboration, to which the reader is referred for the details. The SM weak-interaction differential cross section as a function of the nuclear kinetic recoil energy T nr of CEνNS processes with a spin-zero nucleus N with Z protons and N neutrons is given by [25][26][27] where G F is the Fermi constant, = e, µ, τ denotes the neutrino flavour, E is the neutrino energy and Q 2 , For the neutrino-proton, g p V , and the neutrino-neutron, g n V , couplings we consider the more accurate values that take into account radiative corrections in the minimal subtraction, MS, scheme [18,28], that correspond to These values are different from the tree-level values g p V = 0.0229 and g n V = −0.5, in particular for those of g p V (ν e ) and g p V (ν µ ). In Eq. (2.1) F Z (| q| 2 ) and F N (| q| 2 ) are, respectively, the form factors of the proton and neutron distributions in the nucleus. They are given by the Fourier transform of the corresponding nucleon distribution in the nucleus and describe the loss of coherence for | q|R p 1 and | q|R n 1, where R p and R n are, respectively, the rms radii of the proton and neutron distributions. For the form factors of the proton and neutron distributions we employ the Helm parameterisation [29], that is practically equivalent to the other two commonly-used symmetrized Fermi [30] and Klein-Nystrand [31] parameterisations. The description of these parameterisations can be found in several papers, for example in Refs. [3,13,14,18,30]. For the proton rms radii, in our calculations we use [32,33] R p (Cs) = 4.804 fm, R p (I) = 4.749 fm, R p (Ar) = 3.448 fm. (2.5) The value of the neutron rms radius is only poorly known experimentally both in CsI [3,4,10,13,14,16,34] and Ar [18]. For CsI, we adopt the values R n (Cs) = 5.01 fm and R n (I) = 4.94 fm obtained with the relativistic mean field (RMF) NL-Z2 [35] nuclear model calculation in Ref. [3], that are in good agreement with the most precise experimental value determined in combination with atomic parity violation (APV) experimental results in Ref. [16]. For argon, it is obtained starting from the experimental value of the proton rms radius and considering the average value of the so-called neutron skin, that corresponds to the difference among the two radii, predicted by different models with diverse nuclear interactions [36][37][38][39][40][41][42][43][44][45][46]. The neutron skin is predicted to be between 0.06 and 0.11 fm. Considering a neutron skin of 0.1 fm that is predicted by most of the models, we get R n (Ar) = 3.55 fm. The CEνNS event rate in the COHERENT experiment [1,17] depends on the neutrino flux produced from the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratories. The total differential neutrino flux is given by the sum of the three neutrino components coming from the pion decay (π + → µ + + ν µ ) and the subsequent muon decay (µ + → e + + ν e +ν µ ) Here m π and m µ are the pion and muon masses, and the normalization factor is η = rN POT /4πL 2 , where r is the number of neutrinos per flavour that are produced for each proton-on-target (POT), N POT is the number of proton on target, and L is the distance between the source and the detector. For the dataset collected by the COHERENT Ar detector, called CENNS-10 [1], we use r = (9 ± 0.9) × 10 −2 , N POT = 13.7 × 10 22 that corresponds to a total integrated beam power of 6.12 GW·hr and L = 27.5 m. The same numbers for CsI are r = 0.08, N POT = 17.6 × 10 22 and L = 19.3 m. The pions decay at rest producing ν µ 's which arrive at the COHERENT detector as a prompt signal within about 1.5 µs after protons-on-target. The decay at rest of µ + produces a delayed component of ν µ 's and ν e 's, since they arrive at the detector in a relatively longer time interval of about 10 µs. The theoretical CEνNS event-number N CEνNS i in each nuclear-recoil energy-bin i is given by 1 where A(T nr ) is the energy-dependent reconstruction efficiency, E min = √ MT nr /2, E max = m µ /2 ∼ 52.8 MeV, and dN ν /dE is the neutrino flux integrated over the experiment lifetime. The electron-equivalent recoil energy T ee [keV ee ], is transformed into the nuclear recoil energy T nr [keV nr ] thanks to the relation where f Q is the quenching factor, which is the ratio between the scintillation light emitted in nuclear and electron recoils. In CsI, the quenching factor is taken from Fig. 1 in Ref. [47]. This represents the most precise determination of the quenching factor in CsI up to date. For Ar, following Ref. [17], the quenching factor is parameterised as f Q (T nr ) = (0.246 ± 0.006 keV nr ) + ((7.8 ± 0.9) × 10 −4 )T nr up to 125 keV nr , and kept constant for larger values. Finally, the value of N(CsI) is given by N A M det /M CsI , where N A is the Avogadro number, M det is the detector active mass equal to 14.6 kg and M CsI = 259.8 g/mol is the molar mass of CsI. Similarly, the value of N(Ar) is obtained using M det = 24 kg and M Ar = 39.96 g/mol for 40 Ar. Here, the contribution of 36 Ar and 38 Ar has been neglected.

Light vector Z mediator in CEνNS
The SM cross section presented in Section 2 is modified by the presence of a new massive vector mediator which couples to SM leptons and quarks. Considering a vector neutralcurrent neutrino non-standard interaction [48] and assuming that the neutrino does not change flavour, it is generically described by the effective four-fermion interaction Lagrangian (see Ref. [49] and references therein) The parameters ε f V , where f = u, d stands for the flavour of the quark and = e, µ is the neutrino flavour 2 , describe the size of non-standard interactions relative to standard neutral-current weak interactions. The full cross section comes from coherently summing the contributions from the exchange of the SM and NSI mediators, which may interfere.
In particular, the NSI mediator effectively induces an energy-dependent modification of the SM factor Q 2 , SM in Eq. 2.1. Indeed, considering the NSI scenario described before it becomes In this paper, we focus on three simple models in which the vector neutral-current neutrino NSI is induced by a gauge boson Z with mass M Z and coupling g Z associated with a new U(1) symmetry. In this scenario, the CEνNS cross section can be determined by writing the parameter ε f V in terms of the light Z propagator as where Q are the charges under the new gauge symmetry.
The first model that we consider describes a Z boson which couples universally to all SM fermions [6,20]. We set the charges to be Q ≡ Q f = 1, and the coupling becomes the same for all the fermions. Under this model, the cross section in Eq. 2.1 becomes [6] dσ dT nr The second model that we consider is the so-called B − L (baryon number minus lepton number) extension of the SM [20,21]. In this case the gauge charges are determined by imposing that the theory is anomaly free. In particular, in this model the boson couples universally to the quarks, as well as to the neutrinos, but with different charges, namely Q = Q f . In particular, in the B − L model the gauge charges are such that Q = 1 and Q f = −Q /3. Under these assumptions, the cross section in Eq. 2.1 becomes (3.5) 2 We consider only the first generation of quarks since they are the only ones contained in nuclei and only electronic and muonic neutrinos since they are the only species present in the flux at the SNS.
The last scenario that we consider is a model with gauged L µ − L τ symmetry [22]. In this case, the new Z boson can couple directly only to muonic or tauonic flavour and there is no tree-level coupling to the quark sector. Thus, this model can be studied through the CEνNS process by considering the interaction between the new boson and quarks via kinetic loops of muons and tauons involving photons. Under these assumptions, the cross section for this process becomes [22] dσ dT nr where α EM is the electromagnetic fine-structure constant and m τ is the tau lepton mass.
As visible, only protons interact with the new boson due to the presence of the photon in the loop, and only the proton coupling is modified with a term that is proportional to the electric charge. As far as the CEνNS process in COHERENT is concerned, this last model modifies only the muonic neutrino and antineutrino cross sections, leaving the electronic neutrino cross section unchanged, since there is no direct coupling to the electronic flavour. It is worth to specify that in the case of antineutrinos, the term induced by the introduction of the light vector mediator changes sign as well as the other vector couplings.

COHERENT data analysis
Concerning the analysis of the CsI COHERENT dataset, we considered the least-squares function For each energy bin i, N exp i is the experimental event number in Ref. [2], N CEνNS i is the theoretical event number that is calculated as explained in Sections 2, B i is the estimated number of background events, and σ i is the statistical uncertainty, both taken from Ref. [2]. As explained in Ref. [16], we employ only the 12 energy bins from i = 4 to i = 15 of the COHERENT spectrum, because they cover the recoil kinetic energy of the more recent Chicago-3 quenching factor measurement [47] that we use in this work. In Eq. (4.2), α c and β c are nuisance parameters which quantify, respectively, the systematic uncertainty of the signal rate and the systematic uncertainty of the background rate, with the corresponding standard deviations σ α c = 0.112 and σ β c = 0.25 [1]. The uncertainty on the quenching factor is taken into account through a normalization factor η with σ η = 0.051, that contributes to the least-squares function.
Concerning the analysis of the Ar dataset, in the COHERENT paper [17] two independent analyses are presented, labeled as A and B, that differ mainly for the selection and the treatment of the background. In the following, we will use the data coming from the analysis A, whose range of interest of the nuclear recoil energy is [0, 120] keV ee (corresponding to roughly [0, 350] keV nr ), with 12 energy bins of size equal to 10 keV ee . In our analysis we considered the same least-squares function employed in Ref. [18], namely where PBRN stands for Prompt Beam-Related Background, LBRN for Late Beam-Related Neutron Background and with

Constraints on light vector mediator models
In this section we derive the constraints that can be obtained using CsI and Ar COHER-ENT data on the mass, M Z , and the coupling, g Z , of a light vector mediator that couples with the SM particles according to the models described in Section 3. Some of these constraints have been already derived in CsI for the universal model, see Ref. [6,8,13,21,50] and for the L µ − L τ model [51]. Here in our paper we derive the limits for CsI using the more recent Chicago-3 quenching factor measurement [47], we derive the same limits in the recently released Ar dataset and we perform for the first time their combination.
First of all, in Figure 1 we compare the SM differential CEνNS event rate for Ar (a) and CsI (b) to those obtained for the universal, B − L and L µ − L τ light vector mediator models. For illustrative purposes, we fixed the mass of the light vector mediator to M Z = 10 MeV and we use different values of the couplings g Z depending on the model. The latter have been chosen close to the current lower limits determined by other experiments. It is possible to see that all the light vector mediator models cause an increment of the differential event rate for small recoil energies, i.e., less than about 40 keV nr for Ar and less than about 15 keV nr for CsI, respectively. Interestingly, the universal model also shows a dip in the event rate due to the fact that in Eq. 3.4 the contribution of the light vector mediator enters with opposite sign with respect to the SM one. In CsI the cancellation between the two contributions is not perfect due to the presence of two different atomic species that compensate slightly each other.
When the experimental COHERENT efficiency is taken into account for the recoil energy bins described in Section 4, the theoretical event rates transform into the exper-     imental spectral distributions shown in Figure 2 for Ar (a) and CsI (b). The same mass but slightly different couplings of the light vector mediator for the different models as in the previous figure have been assumed for illustrative purposes.
In Figure 3(a) we report the 90% C.L. limits obtained using Ar and CsI COHER-ENT data for the universal Z model, where the grey shaded area represents the region excluded by the combination of the Ar and CsI datasets. The limits obtained from the separate Ar and CsI analyses are similar, with the Ar limit being slightly more stringent for M Z 100 MeV. Note that there is a thin diagonal strip that is allowed because it corresponds to values of g Z and M Z for which the universal Z cross section in Eq. (3.4) is almost degenerate with the SM cross section in Eq. (2.1) that fits well the data. This happens for values of M Z much larger than the typical momentum transfer of the order of few tens of MeV. Neglecting the | q| 2 in the denominator of the Z contribution, the small proton contribution to the SM cross section, and all the form factors, and considering g n V −1/2, we obtain the approximate degeneracy condition Taking into account that N/(Z + N) is about 0.55 for 40 Ar and about 0.58 for 127 I and 133 Cs, we can write the approximate degeneracy condition as One can easily see that the thin diagonal allowed strip in Figure 3(a) corresponds to this approximate relation. In Figure 3(b) the combined Ar and CsI result is shown at 2σ C.L. to allow a better comparison with the constraints at 95% C.L. obtained in the CONNIE reactor CEνNS experiment [52] using the Lindhard quenching factor and with the (g − 2) µ 2σ region in order to explain in this model the anomalous magnetic moment of the muon [23,24]. As one can see, the CONNIE bound is the most stringent in the very low mass region of the mediator, namely for M Z 5 MeV. In the region above, our limits obtained from the COHERENT data improve the CONNIE limits and allow us to exclude with much higher confidence level the (g − 2) µ region (the χ 2 difference with the minimum is more than 120, with two degrees of freedom, that means a practical certainty of exclusion). The high sensitivity to this model is due to the fact that the CEνNS cross section is strongly modified with respect to the SM one by the introduction of the universal Z mediator with sufficiently large coupling, as visible in Figure 1.
Let us also compare the limits that we obtain for CsI with those obtained in Ref. [13], that have been derived using the same quenching factor as in this work, but considering only the total number of events in the COHERENT CsI experiment. Comparing our Figure 3(a) with Fig. 8 of Ref. [13], one can see the impact of performing a spectral analysis of the COHERENT data instead of using only the total event number. Indeed, in Fig. 8 of Ref. [13] the degeneracy region extends down to very low Z masses, while the spectral information allows us to restrict it to a very narrow island only present for values of M Z larger than about 100 MeV. Moreover, also the overall limit becomes more stringent by including the spectral information. A comparable behaviour was also found in Ref. [6], that similarly to us used the spectral information, but employed the constant quenching factor in the COHERENT publication [1], that has a larger uncertainty than that in Ref. [47] used by ourselves. Comparing our Figure 3(b) to Fig. 2 of Ref. [6], one can see that our limits are much more stringent (for example, for M Z 10 MeV, we obtain g Z 3 × 10 −5 at 2σ, whereas the 2σ limit in Ref. [6] is g Z 6 × 10 −5 ).
In Figure 4(a) we show the 90% C.L. limits obtained using the Ar and CsI COHER-ENT data for the B − L Z model, together with their combination. In this case, the CsI limit is more stringent than that obtained with Ar for M Z 10 MeV and dominates the combined bound. The combination of the two data sets produces only a slight improvement of the separate limits for M Z 10 MeV. In Figure 4(b) the combined CsI and Ar bound is shown together with the regions excluded at 90% C.L. obtained by interpreting the BaBar [53] and LHCb [54] dark photon constraints in terms of the B − L Z model [69]. Moreover, the regions excluded at 90% C.L. by KLOE [55], Phenix [56], NA48/2 [57], Mainz [58,59] are also indicated. The areas excluded at 90% C.L. by fixed target experiments [60] (E774 [61], E141 [62], Orsay [63], U70 [64], E137 [65], CHARM [66] and LSND [67]) are also shown. For completeness, also the (g − 2) µ 90% C.L. region needed to explain the anomalous magnetic moment of the muon [23,24] and the (g − 2) e 90% C.L. limit [68] from the searches of an anomalous magnetic moment of the electron are depicted. The COHERENT limit obtained in this work allows us to improve the coverage between accelerator and fixed target experiment limits.
Our limits for the B − L Z model are similar to those obtained in Ref. [70], that have been derived using only the total number of events. In particular, it is difficult to compare our analysis of the COHERENT Ar dataset with that in Ref. [70], that used the fitted number of CEνNS events derived by the COHERENT Collaboration [17] instead of the experimental data.
Finally, in Figure 5(a) we report the the 90% C.L. limits obtained using the Ar and CsI COHERENT data for the L µ − L τ Z model and the combined bound. In this framework, as in the B − L Z case, the CsI limit is more stringent than that obtained with Ar for M Z 10 MeV and dominates the combined bound. However, the combination of the two data sets allows us to improve the separate limits for M Z 10 MeV slightly more than in the B − L Z case. In Figure 5(b) the combined CsI and Ar bound is shown together with the constraints at 95% C.L. obtained with the CCFR measurement [71] of the neutrino trident cross section, with the search of SM Z boson decay to four leptons in ATLAS [72,73] and CMS [74] reinterpreted under the hypothesis Z → Z µµ and with the search for e + e − → µ + µ − Z , Z → µ + µ − from BaBar [75]. The constraints at 90% C.L. for the Borexino [22,60,76,77] experiment and the (g − 2) µ 2σ band, related to the anomalous magnetic moment of the muon [23,24], are also shown. For the L µ − L τ model, the present COHERENT dataset is unfortunately unable to improve the current existing limits and it is not sensitive to the (g − 2) µ band, most of which is already excluded by other data, except for 10 M Z 400 MeV. However, the shape of the COHERENT combined CsI and Ar bound shows good potentiality to extend the sensitivity in the    [54] (green region), KLOE [55] (brown region), Phenix [56] (dark orange region), NA48/2 [57] (yellow region), A1/MAMI in Mainz [58,59] (pink region) and different fixed target [60] (orange regions) experiments (E774 [61], E141 [62], Orsay [63], U70 [64], E137 [65], CHARM [66] and LSND [67]). The (g − 2) µ 90% C.L. region needed to explain the anomalous magnetic moment of the muon [23,24] (purple region) and the (g − 2) e 90% C.L. limit [68] (cyan line) from the searches of an anomalous magnetic moment of the electron are also shown. (b) Figure 5.
(a) Excluded regions in the M Z − g Z plane for the L µ − L τ model at 90% C.L. using COHERENT Ar (above the dotted blue line) and CsI (above the dashed red) data sets, and their combination (solid black-grey shaded area). (b) Comparison of the combined COHERENT CsI and Ar bound at 2σ with the existing experimental bounds at 95% C.L. from the CCFR experiment [71] (green area) and from LHC searches in ATLAS [72,73] (yellow area) and CMS [74] (orange area), and the rescaled bounds at 95% C.L. from the BaBar [75] (purple area) and the Borexino [22,60,76,77] (red area) experiments. The (g − 2) µ 2σ region needed to explain the anomalous magnetic moment of the muon [23,24] (blue region) is also shown. region 10 M Z 400 MeV when further data now being collected by the COHERENT experiment will be released.
Our results for the L µ − L τ Z model are more stringent than those obtained recently in Ref. [78] by fitting the number of CEνNS events obtained by the COHERENT Collaboration from the fit of the data [17]. Since this is not a real fit of the COHERENT data, the result is questionable.

Conclusions
In this paper we discussed the limits on different light vector mediator models that can be obtained from the analysis of the recent CEνNS data on argon [17] and cesium iodide [1] of the COHERENT experiment. We also presented the results obtained by combining the analysis of the CsI and Ar data sets. We considered three models with a light Z mediator: one in which the Z couples universally to all SM fermions, another corresponding to a B − L extension of the SM, and a third one with a gauged L µ − L τ symmetry.
We compared the results obtained from the combination of the COHERENT CsI and Ar data sets with the limits derived from searches in fixed target, accelerator, solar neutrino and reactor CEνNS experiments. We showed that for the universal and the B − L Z models, the COHERENT data allow us to put stringent limits on the light vector mediator mass, M Z , and coupling, g Z , parameter space. In particular, in the framework of the universal Z model we improved significantly the limits obtained in the CONNIE reactor CEνNS experiment [52] and we excluded, with very high confidence level, the possibility of explaining the anomalous magnetic moment of the muon [23,24] assuming this model. Considering the B − L Z model, we have shown that the COHERENT data allow us to improve the coverage between accelerator and fixed target experiment limits. Finally, our results for the L µ − L τ Z model do not improve the current existing limits, but show good potentiality of future COHERENT data to extend the sensitivity in the region 10 M Z 400 MeV and g Z 8 × 10 −4 where the explanation with this model of the anomalous magnetic moment of the muon is still unconstrained.