Searching for BSM neutrino interactions in dark matter detectors

Neutrino interactions beyond the Standard Model (BSM) are theoretically well motivated and have an important impact on the future precision measurement of neutrino oscillation. In this work, we study the sensitivity of a multi-ton-scale liquid Xenon dark matter detector equipped with an intense radiative neutrino source to various BSM neutrino-electron interactions. We consider the conventional Non-Standard Interactions (NSIs), other more generalized four-fermion interactions including scalar and tensor forms, and light-boson mediated interactions. The work shows that with realistic experimental setups, one can achieve unprecedented sensitivity to these BSM neutrino-electron interactions.

(Dated: March 26, 2019) Neutrino interactions beyond the Standard Model (BSM) are theoretically well motivated and have important impact on future precision measurement of neutrino oscillation. In this work, we study the sensitivity of a multi-ton-scale liquid Xenon dark matter detector equipped with an intense radiative neutrino source to various BSM neutrino-electron interactions. We consider the conventional Non-Standard Interactions (NSIs), other more generalized four-fermion interactions including scalar and tensor forms, and light-boson mediated interactions. The work shows that with realistic experimental setups, one can achieve unprecedented sensitivity to these BSM neutrino-electron interactions.

I. INTRODUCTION
The existence of small neutrino masses is calling for new physics. It is fairly reasonable to speculate that the interactions of neutrinos may also go beyond the Standard Model (BSM). Searching for BSM neutrino interactions is of increasing importance since they could make a considerable impact on future precision oscillation measurements. For example, it is known that the Non-Standard Interactions [1][2][3][4] of neutrinos may affect the measurement of δ CP in long-baseline oscillation experiments [5][6][7][8][9][10][11][12]. Moreover, due to various well-known problems of the SM, theorists have for decades been looking for more satisfactory theories. As neutrino masses and mixing have become established, precision measurement of neutrino interactions are increasingly important in directing theoretical exploration.
In this paper, we study the potential of 51 Cr-LXe experiments for probing BSM neutrino interactions. Our previous study [49] showed that the combination of a 5 MCi 51 Cr source and a 6 ton LXe detector could generate the tightest terrestrial constraint on neutrino magnetic moments, and competitive sensitivity to sterile neutrino oscillation. Therefore, we expect similar setups would have excellent sensitivity to a wide range of BSM neutrino interactions, including Non-Standard Interactions (NSIs), general four-fermion effective interactions (e.g., including tensor and scalar forms), and light and weakly coupled mediators (e.g. dark photons).
The paper is organized as follows. In Sec. II, we introduce the basic experimental setup and evaluate the event rates of signals and backgrounds for several configurations. In Sec. III, we study the precision measurement of the SM parameters relevant to neutrino interactions. In Sec. IV, we include sensitivity studies for several types of BSM neutrino interactions which are widely discussed in the literature. Finally, we conclude and summarize the results in Sec. V.

II. EVENT RATES
Let us consider a general 51 Cr-LXe experiment and evaluate the event numbers of ν-e scattering. In the 51 Cr-LXe experiment, the number of ν-e scattering events appearing in the infinitesimal volume dV around position r ≡ (x, y, z) in the detector, during the time interval t to t + dt, with the recoil energy from T to T + dT can be evaluated as where n e , φ, dσ dT , and E ν denote the electron number density, the neutrino flux, the differential cross section, and the neutrino energy respectively.
For a point-like radioactive source, φ has the following spatial and temporal dependence: where is the neutrino flux at 1 meter from a 1 MCi radioactive source; τ = 39.96 days is the mean lifetime of 51 Cr; R 0 Cr51 is the initial (t = 0) radioactivity; and f (E ν ) describes the spectral shape, normalized by f (E ν )dE ν = 1. The neutrino spectrum of 51 Cr simply consists of mono-energetic neutrino emissions at 750 keV (90%) and 430 keV (10%): Since the source decays exponentially, it is useful to define a time-averaged activity, relative to the initial activity, R 0 Cr51 , and the exposure time, ∆t. Likewise, an average distance between the detector and the source, r avg , can be defined as: With these averaged values, it is equivalent use the following replacement in our analyses: while collapsing the time and space integrals to the ∆t and the detector fiducial volume, V . Hence the number of events in the recoil energy bin [T i , T i + ∆T ] can be written as with where φ avg (E ν ) has been defined in Eq. (7). For a detector with a cylindrical fiducial region, r avg can be computed analytically (see Appendix A). We take the same geometrical profile as Ref. [49], i.e., the height (h) and diameter (d) of the cylinder are h = d = 1.38 m, and the source is L = 1 m below the bottom. Using Eq. (A5), this profile has r avg = 1.63 m and contains 6 tons of LXe, which corresponds to V n e =1.5 × 10 30 electrons.
Plugging in the definition of φ avg , the integral in Eq. (9) works out as In practical calculations, we also need the maximal recoil energy, T max , which is defined as the maximal recoil of electron that can be generated by a certain E ν : For T > T max the corresponding dσ dT in Eq. (10) is set to zero. In Tab. I, we summarize useful quantities for the configurations considered in this work. The event numbers with the SM ν-e elastic scatting cross section [54] are evaluated using Eq. (8) and presented in Fig. 1. Regarding backgrounds, we assume a "normal" level [55], which includes the solar neutrino background and decays of embedded radioactive isotopes ( 136 Xe, 85 Kr, 222 Rn, etc.), for configuration A; and a " 136 Xe depleted" level, which consists of the normal solar background with only 10% of the normal level from radioactive decays, for configurations B and C. In configurations B and C, the background rate is further decreased by halving the exposure time, ∆t. Finally, in configuration C, the signal-to-noise ratio is further increased by the use of a more intense source.

III. MEASUREMENT OF THE SM PARAMETERS
The study of neutrino-electron scattering has played an important role in the precision electroweak tests of the SM -see Chapter 10 of PDG [56]. For instance, the CHARM II experiment, which has performed the most precise measurement of ν µ -e and ν µ -e scattering thus far, has determined the electroweak mixing angle, sin 2 θ W , to be: with about 3% precision [43]. In comparison (−) ν e scattering experiments have not yet achieved such precision. The best measurements of ν e -e and ν e -e scattering come from LSND [44] and TEXONO [30] respectively: We will show that a 51 Cr-LXe experiment could provide a superior precision measurement of sin 2 θ W , which not only exceeds that of these aforementioned experiments, but also reverses the current situation in which ν e scattering is less precise than (−) ν µ scattering. In the SM, the low energy interactions of neutrinos and electrons can be described by the effective Lagrangian where, for ν e -e scattering, g e V and g e A are given at the tree level by which includes both the neutral current (NC) and the charged current (CC) contributions. At the tree level, the differential cross section for ν-e scattering is given by [54]: where T is the electron's recoil energy and The experimental value of sin 2 θ W is well determined at the Z-pole and found to be 0.23122 (defined in the Modified Minimal Subtraction (MS) scheme [56]). At low energies ( 0.1 GeV), its value can be extrapolated theoretically using renormalization group equation (RGE) running [57]: Including the radiative corrections, the expressions of g V and g A should also be modified [58]: Taking sin 2 θ W as a varying parameter, one can measure it in the 51 Cr-LXe experiment, as a low-energy precision test of the SM. The simulated data is generated assuming the numerical values in Eqs. (18) and (19), and including the backgrounds shown in Fig. 1. Then we perform a χ 2 -fit on the data to obtain the 1σ confidence level (CL) of sin 2 θ W . The results are presented in Fig. 2 and Tab. II. Comparing these results with the current state-of-the-art from CHARM II, LSND, and TEXONO, we see that the 51 Cr-LXe experiment is able to reach unprecedented precision on sin 2 θ W as measured by ν-e scattering, even in the least sensitive configuration that we have considered.
In Fig. 2, we show the role of 51 Cr-LXe measurements in probing the RGE running of sin 2 θ W . So far, the most precise measurements of sin 2 θ W are from colliders, including LEP, LHC, SLC, and Tevatron. All the colliders essentially measure the value of sin 2 θ W only at the Z-pole scale. Other measurements, including the weak charge of protons (Q weak ), Atomic Parity Violation (APV), and electron deep-inelastic scattering (eDIS), are located at various lower energy scales. This plot shows that a precision probe of the RGE running of sin 2 θ W , is still needed at the lowest energies, which a 51 Cr-LXe measurement (marked in red as LXe-A, B and C) would provide.
The 51 Cr-LXe experiment would also provide a high precision measurement of the weak coupling parameters g e V and g e A . We performed a simultaneous χ 2 -fit of g e V and g e A based on Eqs. (16) and (17). Fig. 3 shows the fit result in the (g e V , g e A ) plane, and compares it with the measurements from TEXONO, LSND, and CHARM II, where for ν µ -e scattering measurements of CHARM II, the weak coupling constants are converted using g e V = g µ V +1 and g e A = g µ A +1. Due to low statistics, TEXONO and LSND appear as long bands, which are mainly determined by the total event numbers with little impact from spectral information. In other words, TEXONO and LSND effectively measured only the total cross section since their low-statistics measurements were not sensitive to spectral distortions from variations of g V and g A . The CHARM II measurement, which has much higher statistics, gives  and g µ A = −0.503±0.017 (see the overlap of two orange bands). However, since the neutrino energy is much higher than the electron mass in CHARM II, it also has some discrete parameter degeneracy-the cross section is approximately invariant when . For the 51 Cr-LXe experiment, depending on the signal/background ratio, similar situation may appear as well. This can be understood by reformatting the cross section in powers of T /E ν : which implies that for low recoil energies (T /E ν 1) we essentially measure only g 2 1 + g 2 2 , while higher energy recoils are also sensitive to 2g 2 2 + g 1 g 2 me Eν . With these two factors determined, (g 1 , g 2 ) would be approximately known up to some discrete ambiguities. This explains why configurations A and B have two separate low-∆χ 2 regions, while for configuration C, this degeneracy is broken by the higher signal/background measurement, which is also sensitive to the quadratic term in T /E ν .

IV. BSM NEUTRINO INTERACTIONS
In many BSM theories, neutrinos have new interactions, which can be categorized into two types, (i) interactions mediated by heavy particles, and (ii) interactions mediated by light particles.
For light mediators, the interactions can be much more multifarious. As a demonstration of light mediator sensitivity, we will study a light Z with a very weak gauge coupling. Such a scenario is particular important for dark matter studies [69][70][71][72][73][74][75][76][77][78][79][80] since additional U (1) symmetries have been widely used to stabilize the dark mater candidates.

A. Non-Standard Interactions
NSIs in ν-e scatting are usually formulated by the following Lagrangian: where ψ e is the Dirac spinor of the electron, L αβ and R αβ are flavor-dependent constants, and P L/R ≡ (1 ∓ γ 5 )/2. This Lagrangian originates in some models with heavy vector bosons that interact with neutrinos and electrons (see, e.g., [81]). In addition, purely scalar interactions can also generate NSIs at the loop level, and these loop-induced NSI could potentially be very large in the presence of some secret neutrino-scalar interactions [82].
Including NSIs, the cross section of ν e -e scattering can be derived by summing the cross sections of the three processes ν e + e → ν α + e, where α = e, µ or τ , which is  [56], and the CHARM II constraint comes from Ref. [43].
Note that there is no interference between the three processes because they have different flavor neutrinos in the final state. But for α = e, there is interference between the NSI and the SM interactions, which significantly enhances the sensitivity to L ee and R ee . In Fig. 4, we present the result of a χ 2 -fit analysis on the NSI sensitivity. The χ 2 -fit is performed each time for a pair of ( L eα , R eα ) with the other ε's set to zero. Because the cross section is completely symmetric under the µ-τ exchange, ν e -e scattering should have exactly the same sensitivities to ( L eµ , R eµ ) and to ( L eτ , R eτ ). So they are shown in the same plot in the right panel. As we can see, ( L ee , R ee ) would be stringently constrained by the 51 Cr-LXe experiment while ( L eµ , R eµ ) and to ( L eτ , R eτ ) would be less constrained due to the lack of interference. The 51 Cr-LXe bounds are compared to the current known bounds on NSIs. In general, direct measurements of neutrino-electron scattering and measurements of neutrino oscillation are sensitive to the NSIs considered here. In particular, there have been bounds from LSND and TEXONO that can be readily superposed on our results, as shown in Fig. 4. As for other bounds, we refer to Ref. [3] for the recently updated summary, which have been included in Tab. III. We conclude that for most NSIs parameters, the 51 Cr-LXe sensitivity to NSIs would generally exceed the current known bounds.

B. SPVAT interactions
More generally one can adopt an effective field theory (EFT) approach to study BSM neutrino interactions and write down all the possible Lorentz invariant operators as follows: with Here C a and D a are real constants if i a = i for a = S, P and T ; and i a = 1 for a = V and A. It should be noted that this reduces to the SM Lagrangian when and all other C a and D a are zero. Regarding theoretical motivations, the first four types (a = S, P, V and A) could originate from integrating out some heavy scalar or vector mediators. The tensor interactions could be generated by integrating out heavy charged scalar mediators following necessary the Fierz transformations [84]. For Dirac neutrinos, all the SPVAT interactions could exist with 10 free parameters (C a , D a ) in Eq. (26). For Majorana neutrinos, there are further constraints on the SPVAT interactions such that the allowed parameter space is actually smaller, containing only 6 free parameters [32]. Such a difference could be used to distinguish between Dirac and Majorana neutrinos [85]. In this work, we will ignore the additional constraints of Majorana neutrinos and simply take the full parameter spaces containing 10 parameters into consideration.
In this model space, the cross section of ν e -e scattering is given as follows [32]: where In our analysis we are interested in extracting the non-SM part, so in our χ 2 -fit we define and make a 2-parameter fit for each pair of (δC a , δD a ). The result is presented in Fig. 5. For a = S or P , the 51 Cr-LXe experiment would reach sensitivity around 0.5 ∼ 2. For the other cases, the measurement could be even more precise. Note that for a = V or A, there is parameter degeneracy which would be removed in configuration C. The situation is similar to Fig. 3, with the reason explained at the end of Sec. III.
The Lagrangian of light Z that concerns ν-e scattering is generally formulated as follows where g Z is the gauge coupling and is a charged lepton. Depending on the models, there can be different charge assignments. In addition, many models also have both mass mixing and kinetic mixing between a Z and the SM Z 0 boson. All these variations have additional impacts on ν-e scattering which are model-dependent. A complete analysis (see, e.g., [33]) is outside the scope of this work. Here we simply assume the charge assignments are the same as the U (1) B−L model (i.e., all leptons have the same charge) and the mixing between Z and Z 0 is negligibly small. The contribution of Z to the ν e -e cross section can be included by adding an energy-dependent term to (g 1 , g 2 ) in Eq. (16): In Fig. 6, we present the 51 Cr-LXe sensitivity to g Z and compare it with other known bounds taken from Refs. [33,46,86]. As shown in Fig. 6, in general neutrino-electron scattering experiments (CHARM II, TEXONO, and GEMMA) provide the leading constraints on light Z prior to some constraints obtained from atomic physics and measurements of muon and electron anomalous magnetic moments, (g −2) e and (g −2) µ . They are also complementary to some other constraints from fixed target experiments, astrophysical observations (energy loss in the sun and globular clusters), and the B-Factories. Due to its low threshold and high statistics, the 51 Cr-LXe experiment would significantly improve upon existing constraints from neutrino-electron scattering. In the low mass limit, the 51 Cr-LXe experiment has sensitivity to small g Z down to 10 −7 . For heavy Z , it would exceed the CHARM II experiment which is based on high-energy neutrino beams.  Figure 6. Constraints on light Z . The constraints of TEXONO, CHARM II, and GEMMA are taken from Ref. [33], other constraints are taken from Refs. [46,86]. The green dashed curve is for 51 Cr-LXe with configuration A, and dot-dashed for configuration C. Configuration B is not shown here for simplicity, which should be between the dashed and dot-dashed curves.

V. CONCLUSION
In this work, we have studied the potential for ν e -e scattering in a LXe dark matter detector using 51 Cr as a radioactive neutrino source ( 51 Cr-LXe). Assuming a 5 MCi 51 Cr source located 1 m below a 6-ton LXe detector with the current state-of-the-art background and running for 100 days (configuration A), one can already achieve a high statistics (1.2 × 10 4 signal events) measurement of ν e -e scattering. However, only the low energy part (T 100 keV) of the recoil spectrum can be efficiently measured while the higher energy signal would be submerged by the background, as shown in Fig. 1. To enhance the signal/background ratio at high T so that the full recoil spectrum (0 < T 559 keV) can be measured, the background must be reduced by, for example, using 136 Xe-depleted LXe and a shorter exposure time. For this purpose, we also considered configurations B and C (see Tab. I) for which the signal is greater than the background over the whole spectrum (see Fig. 1).
Based on these three configurations, we study low energy precision measurement of the SM parameters sin 2 θ W , g V and g A . For sin 2 θ W , we find that 1 ∼ 2% precision can be attained, as presented in Fig. 2, and Tab. II. As for g V and g A ), such an experiment would provide the most precise measurement among (−) ν e scattering experiments (see Fig. 3), and be comparable to the CHARM II measurements based on (−) ν µ scattering. Due to this high precision, we expect that the 51 Cr-LXe experiment would be excellent at constraining BSM neutrino interactions. Indeed, according to our analyses, it would generate leading or complementary constraints on NSIs, SPVAT interactions, and light Z , as presented in Figs. 4, 5, and 6 respectively. For example, take NSIs, the most extensively studied scenario: the 51 Cr-LXe experiment is sensitive to six of the NSI parameters, namely L ee , R ee , L eµ , R eµ , L eτ , and R eτ . In Tab. III and Fig. 4 we show that L eµ , R eµ , L eτ , and R eτ would be constrained to the limit around 0.1 ∼ 0.2, and L ee and R ee can be more stringently constrained due to the interference with the SM signal, sometimes even down to 6 × 10 −3 . These constraints would generally exceed the existing bounds, which would be important for future long baseline experiments such as DUNE and T2HK. In addition, probing SPVAT interactions and light Z in ν e -e scattering would shed light on the fundamental theories of both neutrinos and dark matter.
where the volume is Here the Cartesian coordinate x-y-z is set in such a way that z is the cylindrical axis the x-y is the horizontal plane. By defining we can compute the integral in the cylindrical coordinate system: