Sensitivity of reactor experiments to nonstandard neutrino interactions in beta decay rates

The Standard Model has been one of the most powerful tools for studying particle physics. Yet, there remain many open questions about phenomena that seemingly cannot be explained within its conventional framework. Such physics is beyond the Standard Model (i.e., BSM physics). For example, the discovery of neutrino oscillations super-kamiokande_collaboration_evidence_1998 arising from mismatching flavor and mass eigenstates highlighted the need for BSM searches—or at least for revising the Standard Model to accommodate for new physics. As research has concentrated on this area, one expanding body of work has included studies of “nonstandard neutrino interactions" (NSI). Operating within this framework has been a popular way of approaching BSM neutrino physics.

Toward the beginning of the 2000’s, an important study of general NSI bounds came with Ref. biggio_general_2009 along with an updated consideration of these authors’ loop bounds with Ref. biggio_loop_2009 . Charged-current (CC) NSI in these studies were constrained below $\mathcal{O}(10^{-1})$ using measurements of the Fermi constant $G_{F}$ and of the Cabibbo-Kobayashi-Maskawa (CKM) quark matrix. Another important study for CC NSI a few years later looked at low-energy and collider experiment bounds cirigliano_non-standard_2013 . A summary ohlsson_status_2013 of phenomenological bounds—including a review of NSI—reported direct bounds on matter NSI, direct bounds on production and detection NSI, bounds on NSI in neutrino cross-sections, and bounds on NSI using accelerators. One of the most recent status reports on NSI analyzed bounds from several more sources dev_neutrino_2019 . Indeed, this research has expanded to touch much of neutrino physics and astrophysics.

This NSI framework has been applied to neutrino experiments at all stages of development and operation. Ref. liao_nonstandard_2017 studied NSI at the future Hyper-Kamiokande experiment and Jiangmen Underground Neutrino Observatory (JUNO). NSI at the Deep Underground Neutrino Experiment (DUNE), the Tokai-to-Hyper-Kamiokande (T2HK) experiment, and the Tokai-to-Hyper-Kamiokande-and-Korea (T2HKK) experiment were studied in Ref. liao_nonstandard_2017-1 . Refs. de_gouvea_non-standard_2016 and bakhti_sensitivities_2017 focused solely on NSI at DUNE. Finally, Ref. papoulias_recent_2019 considered the effect of recent and future experiments on both standard and nonstandard physics in nuclei.

The NSI can be manifested in a diverse set of observable quantities in experiments cirigliano_non-standard_2013 ; biggio_general_2009 ; severijns_tests_2006 ; bakhti_sensitivities_2017 ; blennow_non-unitarity_2017 ; ohlsson_status_2013 ; dev_neutrino_2019 ; papoulias_recent_2019 ; liao_nonstandard_2017-1 ; de_gouvea_non-standard_2016 ; liao_nonstandard_2017 ; biggio_loop_2009 . Among these have been particle decay rates cirigliano_non-standard_2013 ; biggio_general_2009 ; severijns_tests_2006 , neutrino mixing parameters bakhti_sensitivities_2017 ; blennow_non-unitarity_2017 ; dev_neutrino_2019 , neutrino CP-violating phases bakhti_sensitivities_2017 ; blennow_non-unitarity_2017 ; dev_neutrino_2019 , scattering cross-sections ohlsson_status_2013 ; dev_neutrino_2019 , and matter-affected oscillation probabilities ohlsson_status_2013 ; dev_neutrino_2019 . This body of research has constrained NSI (both charged-current and neutral-current) to be anywhere on the order of $\mathcal{O}(10^{-4})$ to $\mathcal{O}(10^{-1})$, depending on the data utilized for the analysis and on the NSI model chosen. These constraints tend to be more strict with CC NSI.

Our work here is motivated by another potential observable for NSI. We focus on reported perturbations in nuclear decay rates previously unexplored specifically in the NSI framework. Evidence suggestive of BSM physics is included in several reports, and some of the principal studies are presented in Refs. jenkins_evidence_2009 ; jenkins_additional_2012 ; fischbach_time-dependent_2009 ; sturrock_analysis_2014 ; jenkins_perturbation_2009 ; fischbach_indications_2018 . Table 1 in Ref. nistor_phenomenology_2014 summarizes the majority of the reported decay rate perturbations. These perturbations exhibit periodic fluctuations in the daily, yearly, and $\sim 12$-year ranges. There are additionally reports of “single-event" decay rate perturbations associated with solar storms mohsinally_evidence_2016 and a binary neutron star inspiral fischbach_indications_2018 .

At first glance, one feasible explanation for the decay rate perturbations—especially in their annual periodicity—is the effect of instrumentation sensitivity to the environment. For example, it would make sense that temperature-dependent effects would be manifested in annual perturbations in the detector counting rate. While this could explain some of the decay rate deviations from the expected behavior, Ref. jenkins_analysis_2010 argued that external influences such as temperature could not sufficiently explain the entirety of the fluctuations. However, there are still criticisms of the possibility that something external to instrumentation errors could influence decay rates. For example, Ref. pomme_evidence_2016 argues against the influence of the Sun on decay rates.

Regardless, it is intriguing that most of the decay rate perturbations have been associated with beta-minus decays specifically, as evident in Table 1 of Ref. nistor_phenomenology_2014 . Furthermore, the decay rate perturbations appear at the $\mathcal{O}(10^{-3})$ level for a variety of independent experiments. Finally, these perturbations appear to be correlated with a varying neutrino flux through the decaying sample. One reported connection between periodic fluctuations and a varying neutrino flux was presented in Ref. sturrock_comparative_2016 in which the authors reference both annual neutrino flux variations at the Super-Kamiokande experiment as well as longer (e.g., $12.5$ year^-1) variations suggestive of periodic internal solar processes (e.g., the rotation of the radiative zone).

With evidence suggestive of decay rate perturbations correlated with varying neutrino fluxes, we ask ourselves the following question: What can the NSI framework say about beta-minus decay rate perturbations? In answering this question, we aim to construct a more complete mathematical framework to model the decay perturbations. Our goal is to construct a framework for which there would be experimental sensitivity to the model parameters. To accomplish this, we will extend our analysis of decay perturbations beyond a dependence solely on local neutrino flux to include parameters such as neutrino energy and flavor. Finally, we will explore what NSI constraints can be determined using current and future data from a nuclear reactor experiments.

In this study, we are focused on beta-minus decay rate NSI. For this reason, we will concentrate on the area of quark-neutrino dominating charged-current (CC) NSI. To model these NSI, we introduce a relative strength to the quark-neutrino dominating interaction of beta-minus decays. This is achieved with a modification $\mathcal{L}_{\text{NSI}}^{q}$ to the Standard Model Lagrangian, given by

In this model, there is a relative CC NSI strength $\varepsilon_{\alpha\beta}^{qq^{\prime}P}\in\mathbb{C}$ to an electroweak process involving quark flavors $q,q^{\prime}$ and lepton generations $\alpha,\beta$. The two quark flavors of interest in beta-minus decay are up-type ($q=u$) and down-type ($q^{\prime}=d$), respectively generated and annihilated. This leaves $V_{qq^{\prime}}=V_{ud}$ as one element of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix. Finally, we are interested in the generation of an electron $\alpha=e$ and the introduction of an electron anti-neutrino $\beta=e$.

It is important to distinguish between the expression for vector ($\varepsilon_{\alpha\beta}^{udR}+\varepsilon_{\alpha\beta}^{udL})$ and axial-vector ($\varepsilon_{\alpha\beta}^{udR}-\varepsilon_{\alpha\beta}^{udL}$) structures for $P=L,R$. For the beta-minus process here, we will make use of the vector structure

From this framework, we know that the beta decay rate $\Gamma$ is related to $G_{F}$ and $V_{ud}$ through

We will take two approaches to constrain the NSI. First, we will take into account only the uncertainty $\delta\Gamma$ in beta-minus decay rates. This is accomplished with the following form:

To constrain $\varepsilon_{ee}^{udV}$ and the other $\varepsilon_{e\alpha}^{udV}$, it is standard practice to only allow one nonzero $\varepsilon$ at a time in the analysis. This is done to avoid cancellations from the sum of negative and positive values that can come from $\text{Re}(\varepsilon_{ee}^{udV})$ and the $|\varepsilon_{e\alpha}^{udV}|^{2}$. Using a Markov chain Monte Carlo (MCMC) simulation gelman_bayesian_2014 , we apply the Metropolis-Hastings algorithm for skewed jumping rules to constrain the $|\varepsilon_{e\alpha}^{udV}|$ to be non-negative. Then, the following bounds (95% confidence interval) can be placed:

In the second approach, we will still consider the uncertainty in beta-minus decay rates as from above, but we will also include uncertainties in the Fermi constant and the CKM matrix element $|V_{ud}|$. This is done to determine how sensitive the NSI constraints are to all possible contributions to their strength. We then write down

Next, we aim to further parameterize the CC NSI. Previous studies have explored a perturbed decay rate of the form $\Gamma=\Gamma_{0}(1+\Delta)$ in beta decays (e.g., in Section 4 in Ref. barnes_upper_2019 ). This modified decay rate has been generally defined as a function of neutrino flux through a sample, i.e., $\Delta=\Delta(F_{\nu})$. However—as we alluded to before—there has not been in-depth consideration of dependence on other parameters. Namely, this includes neutrino flavor, type (antimatter or matter), energy, and source.

One way in which we could expect to constrain our model parameters would be with reactor experiments at a short range. In this scenario, the electron neutrino survival probability $P_{ee}$ is arbitrarily close to 1, i.e., $P_{e\mu}\approx P_{e\tau}\approx 0$ such that $F_{\mu},F_{\tau}\ll F_{e}$. This would lead to a measurement of $\varepsilon\rightarrow\varepsilon_{e}$ in which the effects of $\nu_{\mu}$ and $\nu_{\tau}$ are negligible.

To illustrate this, we will assume that the $\nu_{e}$ energy and flux contribute independently to $f_{e}$ around the solar neutrino flux and energy. This will allow us to break down the function $f_{e}$ into manageable pieces for the energy contribution we will call $X_{e}$ and the flux contribution we will call $Y_{e}$:

Electron neutrino fluxes at the centers of nuclear reactors—for the sake of neutrino physics experiments—can be at least $F_{e}(L=0)=10^{14}$ cm^-2 s^-1 (e.g., see Ref. texono_collaboration_production_2005 ). For $E\approx E_{\text{solar}}$, the $\nu_{e}$ survival probability is rather large at short distances, since oscillations are generally relevant only for $L/E>100$ m MeV^-1. The required distance to reproduce the solar neutrino flux would be on the order of $L_{0}\sim 10$ m, assuming spherically emitted neutrinos, i.e., $F_{e}(L)=F_{e}(L=0)/4\pi L^{2}$. For a practical example, the experiment in Ref. barnes_upper_2019 intercepted electron anti-neutrinos around $5$ m from the nuclear reactor at which the anti-neutrino flux was $\sim 50$ times the solar neutrino flux.

Since we already know that $X_{e,0}=1$ from the normalization in Section 4, we also know that $Y_{e}=1$ when $F_{e}$ is at the solar neutrino flux. For small perturbations around $Y_{e}$, then, we could expect a first-order effect:

In summary, we used the CC NSI framework to model time-varying beta decay rates by assuming a neutrino-quark dominating effect. First, we related the coherent and incoherent contributions of the NSI $\varepsilon_{e\alpha}^{udV}$ to the beta decay rate with $\Gamma_{\text{obs}}\propto 1+2\text{Re}(\varepsilon_{ee}^{udV})+\sum_{\alpha}|\varepsilon_{e\alpha}^{udV}|^{2}$. Using data on beta-minus decays, the Fermi coupling constant $G_{F}$, and the CKM matrix element $V_{ud}$, we determined that the $\varepsilon$ were constrained at $\mathcal{O}(10^{-4})$ to $\mathcal{O}(10^{-2})$, which is consistent with bounds placed in existing literature. Next, we constructed a model based on potential parameters governing the new CC NSI. This model considered the NSI strength $\varepsilon$ to be a function of neutrino flavor, energy, and local flux. We allowed the energy and flux of each flavor $\alpha$ to contribute nominally to the NSI with strength $f_{\alpha}$, and then these nominal strengths were each weighted by a factor $\beta_{\alpha}\in\mathbb{C}$ to obtain $\varepsilon=\varepsilon_{ee}^{udV}\sum_{\alpha}\beta_{\alpha}f_{\alpha}$. We determined that current technology for nuclear reactors could produce neutrino experiments sensitive to the $\beta_{\alpha}$ and to $y_{e}$. These experiments would study electron neutrino production at short distances such that other flavor contributions would be negligible to the NSI.

From this work, we found that it is possible to parameterize CC NSI in order to model beta-minus decay rate perturbations as functions of several neutrino properties. These parameters could be constrained by existing data and potentially by new experiments using current reactor technology. However, there remain several open questions. For example, it would be instructive for future studies to consider whether the origin or oscillatory nature of neutrinos had any additional effect on the NSI framework presented here. Since solar neutrino oscillations are much different from terrestrial oscillations produced in experiments, the solar neutrino flux could engage in different NSI that we did not take into account. Regardless, the feasibility for NSI frameworks to complete our understanding of neutrino physics is a promising area for further research.