Constraints on neutrino non-standard interactions from COHERENT and PandaX-4T

In the Standard Model (SM), neutrinos interact with ordinary matter through the exchange of $W$ and $Z$ bosons. Neutrino non-standard interactions (NSIs) in the charged and neutral currents beyond the SM (BSM) were initially formulated by Lee-Yang Lee and Yang (1956) and Wolfenstein Wolfenstein (1978), respectively. While the detection of neutrinos in neutrino oscillation experiments only involves charged-current interactions, the coherent elastic neutrino-nucleus scattering (CE$\nu$NS) Freedman (1974) serves as a unique probe of the neutral-current neutrino NSIs, potentially arising from new mediators, such as $Z^{\prime}$ boson Barranco et al. (2005).

Despite the coherent enhancement of the CE$\nu$NS cross section, this process is difficult to detect due to small deposited energy. It was first observed in the COHERENT experiment using the CsI[Na] scintillation detector Akimov et al. (2017) and later argon Akimov et al. (2021) and germanium Adamski et al. (2024) detectors with neutrinos produced from the spallation neutron source (SNS). These results as well as the follow-up detection with a larger exposure of CsI[Na] Akimov et al. (2022a) and other experimental efforts Aguilar-Arevalo et al. (2019); Bonet et al. (2021); Alekseev et al. (2022); Colaresi et al. (2022); Su et al. (2023); Augier et al. (2023); Ackermann et al. (2024); Yang et al. (2024); Cai et al. (2024); Xiao (2024) have motivated diverse phenomenological studies Coloma et al. (2017a); Liao and Marfatia (2017); Cadeddu et al. (2018); Ge and Shoemaker (2018); Aristizabal Sierra et al. (2018a); Ciuffoli et al. (2018); Farzan et al. (2018); Billard et al. (2018); Aristizabal Sierra et al. (2018b); Cadeddu and Dordei (2019); Altmannshofer et al. (2019); Miranda et al. (2019); Aristizabal Sierra et al. (2019); Papoulias (2020); Giunti (2020); Canas et al. (2020); Hoferichter et al. (2020); Skiba and Xia (2022); Cadeddu et al. (2021); Du et al. (2022); Dasgupta et al. (2021); Atzori Corona et al. (2022); De Romeri et al. (2023); Li et al. (2024), see also Refs. Lindner et al. (2017); Dent et al. (2017); Coloma et al. (2017b) for earlier studies and Ref. Abdullah et al. (2022) for a recent review.

On the other hand, with the tremendous progress in the sensitivity of dark matter (DM) direct detection, it is anticipated that the DM experiments can detect neutrinos from astrophysical sources. These neutrinos exhibit nuclear recoil signatures resembling those of DM, which pose as irreducible backgrounds in DM direct detection and are often referred to as the “neutrino frog” Billard et al. (2014); O’Hare (2021); Tang and Zhang (2023). This is not unexpected since the idea of detecting DM that scatters off nuclei Goodman and Witten (1985) was inspired by the proposal to detect MeV-range neutrinos via CE$\nu$NS Drukier and Stodolsky (1984).

Recently, the solar ⁸B neutrino was measured through CE$\nu$NS in the PandaX-4T Bo et al. (2024) and XENONnT Aprile et al. (2024) experiments with corresponding statistical significance of $2.64\sigma$ and $2.73\sigma$, which signify the first step into the neutrino frog experimentally. Assuming the SM cross section of CE$\nu$NS, the signals are interpreted as the measurements of solar ⁸B neutrino flux of $(8.4\pm 3.1)\times 10^{6}\mathrm{\leavevmode\nobreak\ cm}^{-2}\mathrm{\leavevmode\nobreak\ s}^{-1}$ and $\left(4.7_{-2.3}^{+3.6}\right)\times 10^{6}\mathrm{\leavevmode\nobreak\ cm}^{-2}\mathrm{\leavevmode\nobreak\ s}^{-1}$, respectively, both of which are consistent with the standard solar model predictions Vinyoles et al. (2017); Bahcall et al. (1996, 2006) and the results from dedicated solar neutrino experiments Aharmim et al. (2013); Abe et al. (2011, 2016); Agostini et al. (2020).

In the SM, empirical nuclear form factors Helm (1956); Klein and Nystrand (1999) that parameterize the nuclear response are usually adopted to calculate the cross section of CE$\nu$NS. However, an improved treatment is necessary if neutrino NSIs are present, which is feasible in the effective field theory (EFT) approach Altmannshofer et al. (2019); Skiba and Xia (2022); Hoferichter et al. (2020)¹¹1For vector and axial-vector NSIs, one can also refine the CE$\nu$NS cross section by modifying the weak charge Barranco et al. (2005); Abdullah et al. (2022). , analogous to the situation of DM-nucleus scattering Cirigliano et al. (2012); Menendez et al. (2012); Klos et al. (2013); Vietze et al. (2015); Hoferichter et al. (2015); Fitzpatrick et al. (2013); Anand et al. (2014); Bishara et al. (2017a, b).

An end-to-end EFT framework Altmannshofer et al. (2019) was developed and utilized to describe the CE$\nu$NS process from the new physics scale to the nuclear scale, which takes advantages of the heavy baryon chiral perturbation theory (HBChPT) Jenkins and Manohar (1991) and multipole expansions for nuclear responses Walecka (1995)²²2Indeed, the multipole analysis was applied to neutrino-nucleus scattering processes in the SM weak charged currents slightly before the proposal of CE$\nu$NS O’ Connell et al. (1972). The effect of SM weak neutral currents in neutrino scattering off nuclei was later investigated in Ref. Donnelly and Peccei (1979). . Compared to the nuclear form factor approach, the EFTs enable controlled uncertainties in systematic power countings, which allows for potential theoretical improvements in nuclear shell-model calculations Hoferichter et al. (2020) and beyond Abdel Khaleq et al. (2024).

In this work, we will investigate the sensitivities to neutrino NSIs in the EFT framework, including matching between several EFTs. We consider relevant dimension-6 operators in the low-energy effective field (LEFT) and QCD chiral Lagrangian with external sources and heavy baryon expansion, and finally match to nuclear response function to obtain the most stringent constraints from the measurement conducted with the CsI[Na] detector in the COHERENT experiment Akimov et al. (2022a) and the first result of new physics using the measurement of solar ⁸B neutrinos by PandaX-4T Bo et al. (2024).

The remainder of the paper is organized as follows. In Sec. II, we discuss the neutrino NSIs from quark level to the nucleon level using the LEFT and HBChPT. In Sec. III, we derive the CE$\nu$NS cross section using the multipole expansions for nuclear responses. In Sec. IV, we evaluate the event rates of the CE$\nu$NS signals in the COHERENT and PandaX-4T experiments (considering the matter effects in the neutrino propagation by using package PEANUTS), and obtain the constraints on the Wilson coefficients for specific neutrino flavors, which are interpreted as lower bounds on the NSI energy scale using the $\chi^{2}$ analysis. Two-dimensional constraint on the NSI parameters is also obtained for the comparison of these two CE$\nu$NS experiments. We have an assessment of the sensitivities of future measurements of solar ⁸B neutrinos via CE$\nu$NS in DM detectors. We conclude in Sec. V. In the appendix, we provide details of the detector resolution and efficiency in COHERENT CsI measurement.

The neutral-current (NC) neutrino-quark interactions can be parameterized as Wolfenstein (1978); Scholberg (2006); Barranco et al. (2005); Davidson et al. (2003); Du et al. (2022)

where $P_{L/R}=(1\mp\gamma_{5})/2$, $\alpha,\beta$ denote the flavors of neutrinos, and $q=u,d$, $G_{F}$ is the Fermi constant.

In the LEFT, the relevant effective Lagrangian is

where the dimension-6 operators are defined as Jenkins et al. (2018); Altmannshofer et al. (2019)

The correspondence between the coefficients $\epsilon_{\alpha\beta}^{qL(R)}$ and the Wilson coefficients $\hat{\cal C}_{1,q}^{(6)}$ and $\hat{\cal C}_{2,q}^{(6)}$ is

The SM contributions to the Wilson coefficients after integrating out the $Z$ boson are Altmannshofer et al. (2019); Abdullah et al. (2022)

where $\theta_{W}$ is the weak mixing angle with $\sin^{2}\theta_{W}=0.2312$ Workman et al. (2022). We consider contributions from neutrino NSIs, so that

The dimensionful Wilson coefficient can also be expressed as

where $\left.\mathcal{C}_{i}^{(6)}\right|_{\rm NSI}$ is dimensionless, and the energy scale $\Lambda_{\rm NSI}$ is determined by the mass of mediator that is responsible for the neutrino NSIs³³3For light mediator, there is additional momentum dependence from its propagator..

To obtain the neutrino-nucleus cross section of CE$\nu$NS, we need to consider the matching in two steps Altmannshofer et al. (2019); Hoferichter et al. (2020): (1) from the quark level to the nucleon level; (2) from the nucleon level to the nucleus level. In the first step, the nucleons in the target are considered non-relativistic due to the small momentum exchange $q$ compared to the nucleon mass. The interaction Lagrangian for the neutrinos and non-relativistic nucleons is

where $N=p,n$, and $d$ denotes the number of derivatives in the operator.

By using the HBChPT, the following neutrino-nucleon operators at leading order are obtained:

where $N_{v}$ denotes the large component of the nucleon field, and $v^{\mu}$ is the nucleon velocity, the spin operator $S^{\mu}=\gamma_{5}\gamma_{\perp}^{\mu}/2$ with $\gamma_{\perp}^{\mu}\equiv\gamma^{\mu}-v^{\mu}/v$. In the lab frame, $v^{\mu}=(1,\vec{0})$, and $S^{\mu}=(0,\vec{\sigma}/2)$ with $\vec{\sigma}$ being the Pauli matrices. Denoting $p_{1}$ and $p_{2}$ ($k_{1}$ and $k_{2}$) as the momenta of income and outcome neutrinos (nucleons), respectively, we can define the momentum transfer as $q=p_{1}-p_{2}$.

From the quark-level interactions to the nucleon-level interactions, the matching conditions are expressed as⁴⁴4Note that we use the symbol $q$ to represent both the momentum transfer and the quark in the conventional manner.

where $F_{i}$ denote the momentum-dependent nucleon form factors describing the hadronization of quark currents. We use the values of $F_{i}$ evaluated at $q^{2}\to 0$ Bishara et al. (2017b), which are accurate enough for our purpose,

In the second step, the nuclear response to neutrino scattering needs to be considered at the nuclear level, which is described similarly to DM detection Fitzpatrick et al. (2013); Anand et al. (2014). In this framework, the many-body nuclear matrix elements are expanded using the multipole expansions Walecka (1995) in the harmonic oscillator basis, and can be calculated in the nuclear shell model Haxton and Lunardini (2008); Hoferichter et al. (2020).

To this end, we classify the Lagrangian terms according to charge operator ($1_{N}$) and nuclear spin operator ($\vec{S}$) Altmannshofer et al. (2019):

where the Dirac structures are given by

and $\vec{l}_{5}$ in Eq. (14) is the spatial three-vector components of $l_{5}^{\mu}$.

The differential cross section in the rest frame of the target nucleus is Altmannshofer et al. (2019)⁵⁵5We have corrected a missing factor of $1/8$ in Eq.(3.23) of Ref. Altmannshofer et al. (2019), and have verified it by comparing the SM result with the calculation using the nuclear form factor Lindner et al. (2017); Abdullah et al. (2022).

where $M$ is the target nucleus mass, $E_{\nu}$ is the initial neutrino energy, $T_{\rm nr}$ is the nuclear recoil energy, and the spin-averaged amplitude square is expressed as

Here, $J_{A}$ is the spin of the target nucleus, $W_{i}$ denotes the nucleus response functions Fitzpatrick et al. (2013); Anand et al. (2014), and the kinematic factors $R_{i}$ are given by Altmannshofer et al. (2019)

where $\tau,\tau^{\prime}$ are the isospin indices, $\hat{q}\equiv\vec{q}/|\vec{q}|$, and $i,j=1,2,3$. In the isospin basis, $l_{0,\tau}=\left[l_{0,p}+(-1)^{\tau}l_{0,n}\right]/2$ and $l_{5,\tau}^{\mu}=\left[l_{5,p}^{\mu}+(-1)^{\tau}l_{5,n}^{\mu}\right]/2$.

In the CE$\nu$NS experiments, neutrinos from the source will interact with detector target nuclei, causing nucleus recoils. The resulting signal can be translated into the event rate. In the following, we will investigate constraints on neutrino NSIs from the measurements by the COHERENT and PandaX-4T experiments.

In this work, we have studied the constraints on the neutrino non-standard interactions (NSIs) using the latest CE$\nu$NS measurements in the COHERENT and PandaX-4T experiments. The cross section of CE$\nu$NS is calculated within an end-to-end effective field theory framework. In this approach, the dimension-6 operators in the low-energy EFT are matched to hadronic operators in the heavy baryon chiral perturbation theory, while the nuclear response of target nuclei is described using multipole expansions.

We have performed the $\chi^{2}$ analyses of the CE$\nu$NS events observed in the CsI[Na] detector of COHERENT and PandaX-4T, and have derived the one-parameter-a-time lower bounds on the NSI scale for specific neutrino flavors. We have found that the constraints for the $e$ and $\mu$ flavors from COHERENT are more stringent than those from PandaX-4T, while the latter provides unique sensitivities to the neutrino NSIs for the $\tau$ flavor due to the oscillation of solar ⁸B neutrinos propagating from the Sun to the Earth (considering the matter effects in the neutrino propagation by using package PEANUTS). Besides, we have compared the two-dimensional constraints on the NSI parameters for the $e$ flavor from COHERENT and PandaX-4T, and have obtained that the sensitivity of PandaX-4T is limited by the central value of measured CE$\nu$NS counts.

Moreover, we have assessed the measurements of the solar ⁸B neutrinos via CE$\nu$NS in dark matter detectors, leveraging the current results from PandaX-4T as a reference. Assuming that the central value of measured CE$\nu$NS counts aligns with the SM prediction, the sensitivity is significantly improved for the exposure of $6\leavevmode\nobreak\ {\rm tonne}\cdot{\rm year}$, and is comparable to that imposed by the COHERENT CsI measurement if the systematic uncertainties are further disregarded.

Note added: After this paper was finished, another paper Aristizabal Sierra et al. (2024) appeared, which has some overlap of our work. However, we use an end-to-end EFT framework to investigate the neutrino NSIs sensitivities, and we emphasize that it is important to include systematic uncertainties in the analysis.