Implications of first LZ and XENONnT results: A comparative study of neutrino properties and light mediators

Next generation direct dark matter detection experiments are favorable facilities to probe neutrino properties and light mediators beyond the Standard Model. We explore the implications of the recent data reported by LUX-ZEPLIN (LZ) and XENONnT collaborations on electromagnetic neutrino interactions and neutrino generalized interactions (NGIs). We show that XENONnT places the most stringent upper limits on the effective and transition neutrino magnetic moment (of the order of few $\times 10^{-12}~\mu_B$) as well as stringent constraints to neutrino millicharge (of the order of $\sim 10^{-13}~e$)--competitive to LZ--and improved by about one order of magnitude in comparison to existing constraints coming from Borexino and TEXONO. We furthermore explore the XENONnT and LZ sensitivities to simplified models with light NGIs and find improved constraints in comparison to those extracted from Borexino-Phase II data.

WIMP-nucleus or background event. The first WIMP search of LZ is consistent with the null hypothesis (background-only). The main background sources in their reconstructed ER region of interest (ROI) arise from β-decay events and elastic neutrino-electron scattering (EνES) due to the pp and 7 Be 0.861 components of the solar neutrino spectrum. Another important background source may potentially come due to 37 Ar events originating from xenon exposure to cosmic rays before filling up the TPC detector and getting transferred underground. However, for the case of XENONnT this component has a negligible contribution to the background model.
Apart from being state-of-the-art direct dark matter detection experiments-by analyzing the first LZ and XENONnT data-we show that they have also reached a better sensitivity on low-energy neutrino physics, surpassing dedicated neutrino experiments by up to an order of magnitude.
Prompted by the lack of WIMP-induced events in the ROI, in this work we are motivated to explore potential deviations from the Standard Model (SM) EνES cross section with the new data available.
Indeed, as recently pointed out in Ref. [4] the new LZ data can be used to set the stringent limits on effective neutrino magnetic moments, further constraining previous limits [5] from the analysis of Borexino Phase-II data [6] by about a factor 2.5. These results are competitive, though slightly less stringent, to those obtained in Ref. [7] using the recent XENONnT data. In addition to the latter works, here we also provide the corresponding constraints on the fundamental transition magnetic moments (TMMs) [8], improving previous constraints obtained from laboratory based experiments reported in Ref. [9]. We furthermore note that TMMs are more interesting since they have the advantage of being directly comparable with existing constraints from different laboratory experiments [10] and astrophysics [11]. We then demonstrate that the XENONnT and LZ data can be exploited to probe additional electromagnetic (EM) neutrino properties such as the neutrino millicharge and charge radius [12]. Regarding LZ data, here for the first time we show that they can be used for obtaining the most severe upper limits on neutrino millicharges, which we found to be of the order of 10 −13 e. 1 We furthermore show that these sensitivities are somewhat less severe compared to those extracted from the analysis of XENONnT data, in a good agreement with Ref. [7]. On the other hand, we stress that LZ and XENONnT are placing weak sensitivities on the neutrino charge radii. We point out for the first time that the new LZ data can be used to probe neutrino generalized interactions (NGIs) due to light mediators, improving previous constraints set by Borexino. Similarly to the case of electromagnetic properties, here we confirm previous results regarding light mediator scenarios from a similar analysis performed in Ref. [7] and again we find slightly improved sensitivities from the analysis of XENONnT data. Let us finally note that here we furthermore consider the case of a light tensor mediator which was neglected in Ref. [7].
The remainder of the paper is organized as follows: In Sec. 2 we discuss the EνES cross sections within and beyond the SM and we present the simulated signals expected at LZ and XENONnT.
Sec. 3 presents the statistical analysis we have adopted and the discussion of our results. We finally summarize our concluding remarks in Sec. 4.

EνES in the SM
Within the framework of the SM the tree-level differential EνES cross section with respect to the electron recoil energy E er , takes the form [13] dσ να dE er where m e is the electron mass, E ν the incoming neutrino energy, G F the Fermi constant, while the +(−) sign accounts for neutrino (antineutrino) scattering. The vector and axial vector couplings are given by with the Kronecker delta δ αe term accounting for the charged-current contributions to the cross section, present only for ν e -e − andν e -e − scattering.

Electromagnetic neutrino properties
The existence of nonzero neutrino mass, established by the observation of neutrino oscillations in propagation [14,15] stands up as the best motivation for exploring nontrivial EM neutrino properties [16][17][18][19]. The most general EM neutrino vertex is expressed in terms of the EM neutrino form factors F q (q), F µ (q), F ε (q) and F a (q) (for a detailed review see Ref. [12]). The observables at a low energy scattering experiment are the charge, magnetic moment, electric moment and anapole moment, respectively, which coincide with the aforementioned EM form factors evaluated at zero momentum transfer 2 q = 0.
The helicity-flipping neutrino magnetic moment contribution to the EνES cross section adds incoherently to the SM and reads [20] dσ να dE er with a EM denoting the fine structure constant. Note that the so-called effective magnetic moment is expressed in terms of the fundamental neutrino magnetic (µ) and electric (ε) dipole moments, which for solar neutrinos takes the form µ eff να = k | j U * αk λ jk | 2 , where λ jk = µ jk − iε jk represent the TMMs [8,10]. For Majorana neutrinos, the latter is an antisymmetric matrix which in the mass basis takes the form [21] where for simplicity the definition Λ i = ijk λ jk has been introduced. Then, the most general effective neutrino magnetic moment taking into account also the effect of neutrino oscillations in propagation reads [22] with U αi being the entries of the lepton mixing matrix, ∆m 2 ij the neutrino mass splittings and L the distance between the neutrino source and detection points. For the case of solar neutrinos we are interested in this work, the exponential in Eq.(5) can be safely neglected and the effective neutrino magnetic moment takes the form [23] with |Λ| 2 = |Λ 1 | 2 + |Λ 2 | 2 + |Λ 3 | 2 , c 13 = cos θ 13 and P 2ν e1 = 0.667 ± 0.017 which corresponds to the average probability value for pp neutrinos [24]. The latter expression is used to map between the effective neutrino magnetic moments and the fundamental TMMs.
On the other hand the helicity-preserving EM contributions for millicharge (q να ), anapole moment (a να ) and neutrino charge radius 3 ( r 2 να ) are taken via the substitution [12]: where e is the electric charge of electron.

Light mediators
Sensitive experiments with extremely low-energy threshold capabilities such as LZ and XENONnT constitute excellent probes of new physics interactions that involve spectral features induced in the presence of novel mediators [25,26]. Many such beyond the SM physics scenarios can be accommodated in the context of model independent NGIs [27,28]. Let us note that in this framework all Lorentz invariant forms X = {S, P, V, A, T } employing Wilson coefficients of dimension-six effective operators can be incorporated [29]. Here, we consider the EνES contributions of light scalar (S), pseudoscalar (P ), vector (V ), axial vector (A) and tensor (T ) bosons with mass m X and coupling g X = √ g νX g eX , and explore how well they can be constrained in the light of the recent data. For X = {V, A} interactions, the corresponding differential cross sections can be obtained from Eq.(1) and the replacements [30] For the case of X = {S, P, T } interaction there is no interference with the SM cross section and the relevant contributions read [31] dσ να dE er

Simulated event rates at LZ and XENONnT
At LZ and XENONnT the differential EνES event rate for the different interactions ξ = {SM, EM, NGI} is calculated, as [32] dR dE er where E and N T = Z eff m det N A /m Xe denote the exposure and number of electron targets respectively, with m det being the fiducial mass of the detector, N A the Avogadro number and m Xe the molar mass of 131 Xe. Due to atomic binding, Z eff (E er ) accounts for the number of electrons that can be ionized for an energy deposition E er . The latter is approximated through a series of step functions that depend on the single particle binding energy of the ith electron, evaluated from Hartree-Fock calculations [33]. In the ROI of LZ and XENONnT experiments, EνES populations are mainly due to pp neutrinos with a subdominant contribution coming from 7 Be 0.861 neutrinos, while the rest fluxes of the solar neutrino spectrum [34], (dΦ ν i /dE ν ), contribute negligibly. Since solar neutrinos undergo oscillations in propagation before reaching the Earth, the cross section in 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  Eq. (12) is weighted with the relevant oscillation probability and reads where P ee = cos 4 θ 13 P eff +sin 4 θ 13 is the solar neutrino survival probability calculated in the two-flavor approximation following the standard procedure detailed in Ref. [32], while P eµ = (1 − P ee ) cos 2 θ 23 and P eτ = (1 − P ee ) sin 2 θ 23 are the corresponding transition probabilities with the atmospheric mixing angle θ 23 taken from [24]. Here, P eff = (1 + cos 2θ M cos 2θ 12 ) /2 which depends also on the neutrino propagation path and accounts for matter effects [35] with neutrino production distribution functions and neutrino fluxes (pp and CNO). For evaluating P eff , the required oscillation parameters ∆m 2 12 , θ 12 , θ 13 are all taken from Ref. [24] assuming their central values for normal ordering. The minimum neutrino energy required to induce an electronic recoil E er is trivially obtained from the kinematics of the process, as E min ν = E er + E 2 er + 2m e E er /2. In order to accurately simulate the LZ signal, the true differential event rate of Eq.(12) is then smeared with a normalized Gaussian resolution function with width σ(E reco er ) = K · E reco er [36] (E reco er is the reconstructed recoil energy and K = 0.323 ± 0.001 (keV ee ) 1/2 ) and converted to a reconstructed spectrum. Finally, the reconstructed spectrum is weighted by the efficiency function, A(E reco er ) taken from the Ref. [4] 4 , where the authors used the NEST 2.3.7 [37] software and the LZ data release [1] for accurately extracting the efficiency in terms of ER, since in the data release that latter was provided in units of nuclear recoil energy. For the case of XENONnT, a similar procedure is followed, while the efficiency is taken from Ref. [2] and the resolution function from Ref. [3]. In Fig. 1 we present a comparison of the experimental data with the expected signal at LZ (left panel) and XENONnT (right panel) for various new physics scenarios involving EM neutrino properties and NGIs. We have furthermore checked that our integrated SM EνES spectra in the ROI of LZ and XENONnT agree well with those reported by the two collaborations.

RESULTS AND DISCUSSION
In order to explore the new physics parameter(s) of interest S with the LZ data, our statistical analysis is based on the Poissonian χ 2 function [38] where R i exp stands for the experimental differential events in ith bin reported in [1], while the predicted differential spectrum-which contains EνES and background components-is taken as Here, the nuisance parameters α, β and δ are introduced to incorporate the uncertainty on background, flux normalization and 37 Ar components with σ α = 13%, σ β = 7% and σ δ = 100% (see Refs. [1,4]). Let us note that while we do not vary the oscillation parameters in our fitting procedure, their uncertainty is effectively accounted for in the large flux normalization uncertainty. Following Ref. [4], the R i bkg spectrum is taken by subtracting the SM and 37 Ar contributions from the total background reported in [1] by normalizing the integrated spectrum of 37 Ar to its nominal value given by the LZ collaboration, i.e. 97 events. For the case of XENONnT we employ a Gaussian χ 2 function where R i pred (S, β) = (1 + β) R i EνES (S) + B i 0 . Here, B 0 represents the modeled background reported in [2] from which we have subtracted the SM EνES contribution.
In Fig. 2 we show the one-dimensional ∆χ 2 profiles corresponding to the effective neutrino magnetic moment, obtained from the analysis of LZ and XENONnT data. Differently from Ref. [4] where a universal effective neutrino magnetic moment has been considered, here we present the individual limits on the flavored effective magnetic moments according to Eq. (13). At 90% C.L. we find the upper limits: Phase-II data [6] carried out in Ref. [5] for µ eff νe , µ eff νµ and µ eff ντ as well as the TEXONO [39] and GEMMA [40] limits on µ eff νe . Assuming an effective neutrino magnetic moment that is universal over all flavors we find the upper limits: µ eff ν = 10.1 (6.3) × 10 −12 µ B for LZ (XENONnT). Notice, that the latter limit is in excellent agreement with the one reported by XENONnT [2]. Going one step further, for the first time we derive the corresponding constraints on the fundamental magnetic moments λ ij (see Section 2 and Refs. [10,21] for details). Using the definition Λ i = ikj λ jk we find the limits: at 90% C.L. from the analysis of LZ (XENONnT) data. The latter limits are directly comparable and competitive with astrophysical limits derived from plasmon decay: µ eff ν = i |Λ i | 2 = 4.5×10 −12 µ B (95% C.L.) [11]. In the upper panel of Fig. 3 we show the 90% C.L. allowed regions in the (µ eff να , µ eff ν β ) plain assuming the third effective magnetic moment to be vanishing, while in the lower panel we demonstrate the corresponding 90% C.L. limits in the TMM parameter space (Λ i , Λ j ) by marginalizing over Λ k .
In the left (right) panel of Fig. 4 we present the corresponding sensitivities on the neutrino millicharge (charge radius). As for the case of the neutrino magnetic moment, the extracted constraints refer to the different flavors and indicate that the LZ and XENONnT data are very sensitive to this EM parameter. For each flavor we find the limits from the XENONnT at 90% C.L. from TEXONO [41] as well as from those extracted in Ref. [43] through a combined analysis of the recent coherent elastic neutrino-nucleus scattering (CEνNS) data by COHERENT [48,49] and Dresden-II [50]. Notice, that the XENONnT limits are by a factor 1.2 more stringent in comparison    Table I summarizes the 90% C.L. limits on EM neutrino properties extracted in the present work from the analysis of the LZ and XENONnT data. Also listed are the corresponding limits on the neutrino charge radii, for which as expected the new data are not placing a competitive sensitivity.
This is due to the absence of signal enhancement at low momentum transfer, unlike the cases of neutrino magnetic moment and millicharge. For the sake of comparison, also shown in the Table   are the most stringent existing limits placed from the different experimental data mentioned above.
At this point we are interested to perform a combined analysis allowing two nonzero neutrino parameters to vary at a time, assuming vanishing contribution from the third. First, for the case of neutrino millicharges the allowed regions at 90% C.L. are illustrated in the upper panel of Fig. 5.
Then, assuming two nonzero charge radii at a time, the 90% C.L. allowed regions in the parameter space of ( r 2 να , r 2 ν β ) are depicted in the lower panel of Fig. 5. Before closing this discussion, we wish to emphasize that LZ and XENONnT data can place only weak limits on the neutrino charge radii.
We finally turn our attention on simplified NGIs with light X = {S, P, V, A, T } mediators. The corresponding allowed regions by the LZ and XENONnT data in the (g X , m X ) plain are illustrated at 90% C.L. in Fig. 6. We stress that NGI limits from the analysis of LZ data are presented for the first time in this work. As can be seen, for the case of tensor (pseudoscalar) interaction the most (least) stringent bounds are found, in agreement with the projected sensitivities explored in Ref. [51]. In Fig. 7, we first reproduce the limits from  allowed regions in the charge radius parameter plains ( r 2 να , r 2 ν β ). In both cases, the results are presented at 90% C.L. and correspond to the analysis of LZ (blue) and XENONnT (red) data. Two nonvanishing parameters are allowed to vary at a time, while the third is set to zero. data from: COHERENT [52] (See also Ref. [53]), CONNIE [54] and CONUS [55], and through the EνES channel at Borexino [5]. Compared to Borexino Phase-II limits extracted in Ref. [5] the present analysis leads to improved sensitivities, with the XENONnT data being slightly more constraining compared to LZ.  In particular, we have analyzed the recent data reported by the two collaborations focusing on potential EνES contributions in the presence of EM neutrino properties and light NGI mediators.
We find that in all cases the XENONnT data are competitive with LZ, though yielding slightly improved constraints. Since the two datasets lead to essentially similar sensitivities for the BSM scenarios considered, one would not expect a notable improvement from a combined analysis. Projects to support Post-Doctoral Researchers" (Project Number: 7036). The work of RS has been supported by the SERB, Government of India grant SRG/2020/002303.

COMMENT ON THE ANAPOLE MOMENT
As pointed out in Ref. [12] the charge radius and the anapole moment cannot be distinguished in the SM. Indeed, the anapole moment is related to the neutrino charge radius according to a να = − r 2 να /6. For completeness, in Fig. 8 we present the respective constraints on a να .