Non-standard interactions in SMEFT confronted with terrestrial neutrino experiments

The discovery of neutrino oscillations more than two decades ago Fukuda:1998mi ; Ahmad:2002jz began an era where the understanding of the neutrino properties has improved at an accelerating pace. Not only have the majority of the neutrino oscillation parameters describing the oscillation of $\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$ been measured to a relatively high precision, but the recent observations from the long-baseline (LBL) accelerator experiments have also hinted that these oscillations may indicate CP violation in the leptonic sector (see Ref. Esteban:2020cvm for a review). The standard theory of neutrino oscillations has been tested in a variety of neutrino oscillation experiments, which have provided a wealth of data using different neutrino sources and measurement techniques. The present generation of on-going neutrino oscillation experiments is already starting to show the level of precision at which probes for non-standard neutrino physics can be started using the novel neutrino oscillation data.

There have been several attempts to look for signals of the physics beyond the Standard Model (BSM) in the neutrino sector Davidson:2003ha ; Barranco:2005ps ; Barranco:2007ej ; Forero:2011zz ; Adhikari:2012vc ; Girardi:2014kca ; DiIura:2014csa ; Tang:2017khg ; Babu:2019mfe . While many studies have interpreted the consequences of UV-complete models in the neutrino oscillations through the mixing with the heavy right-handed neutrinos, a great deal of discussion has been placed on the possibility of non-standard interactions (NSIs) influencing neutrino oscillations Ohlsson:2012kf ; Dev:2019anc ; Esteban:2020cvm . The NSI of neutrinos is typically discussed in the context of the ad-hoc parameterization of effective couplings $\epsilon^{s}$, $\epsilon^{d}$ Grossman:1995wx ; GonzalezGarcia:2001mp ; Ohlsson:2008gx ; Farzan:2017xzy and $\epsilon^{m}$ Wolfenstein:1977ue , which describe the non-standard effects in the source, detection and propagation, respectively. The majority of the literature focused on finding the constraints on each parameter in neutrino oscillation experiments, but new physics has so far been mainly treated as uncorrelated energy-independent parameters. The efforts to connect the knowledge into a full UV-complete model have altogether been fairly limited. There is now a growing demand for a systematic treatment that connects the low-energy BSM phenomenology of the neutrino NSI to the high-scale regime.

The Standard Model Effective Fields Theory (SMEFT) provides a systematic and model-independent way to parameterize the effects of some heavy degrees of freedom in the UV theory into Wilson coefficients of a set of higher dimensional operators. These operators are constructed by the SM fields and respect the SM gauge symmetries. Therefore, the SMEFT is very robust to study the effects of some UV physics on neutrino NSI parameters. A schematic workflow to find relations between the parameters in the UV model and the neutrino NSI parameters is shown in figure 1. Generally, in the top-down point of view, the Wilson coefficients of the SMEFT operators can be derived as a function of parameters in the UV theory by integrating out the heavy fields at the UV scale $\Lambda$. This procedure of deriving the relations between the Wilson coefficients and the couplings in the UV model is called matching, and is indicated by the first row of figure 1. Note that the Wilson coefficients after this matching are defined at the UV scale $\Lambda$, while current experiments are performed at a much lower scale. For example, observables from collider experiments are usually defined at the weak scale $m_{W}$, thus the renormalization group equation (RGE) running of these Wilson coefficients needs to be considered to correctly find contributions to these observables from the UV theory. This step is represented by the arrow connecting the purple and the green in figure 1.

On the other hand, neutrino oscillation experiments are performed at an even lower scale, and theoretically, the low energy effective field theory (LEFT) around $2\rm\,GeV$ stands as an ideal framework for the study. Obviously, there exists an energy gap between the SMEFT after the aforementioned RGE running and this LEFT. Therefore, extra matching and running need to be appropriately done before making a correct and meaningful prediction for the neutrino NSI parameters from the UV theory. These extra steps are indicated by the last two rows of figure 1, where one firstly translates the SMEFT at the weak scale into the LEFT through matching as shown by the arrow connecting the two greens. This second matching is achieved by breaking the electroweak symmetry and integrating out the the top quark, the Higgs boson and the massive gauge bosons. Note that all the Wilson coefficients in the LEFT after this step are defined at the weak scale $m_{W}$. Then secondly, one applies the RGE and runs these Wilson coefficients down to the 2 GeV scale for neutrino oscillation experiments. This is indicated by the arrow connecting the green and the yellow in figure 1. Now the conventional neutrino NSI parameters in the quantum field theory (QFT) description are encoded in the Wilson coefficients $\epsilon_{L,R,S,P,T}$ of the several charged current operators in LEFT as shown in figure 1 and eq. (3) at the 2 GeV scale. From there on, we leave out the scale parameter $\mu$ and unless otherwise specified, $\epsilon_{L,R,S,P,T}$ shall all be understood as parameters defined at the 2 GeV scale. Finally, these Wilson coefficients can be matched to the NSI parameters $\epsilon^{s,d}$ in the quantum mechanic (QM) formalism. A faithful matching of these two types of neutrino NSI parameters only exist when the event rates calculated in the QFT formalism are expanded to linear order in the NSI parameters. This point was recently presented in Ref. Falkowski:2019kfn and is represented by the arrow connecting the two yellows in our schematic plot figure 1.

In the present work, we make an effort to systematically study the connection between terrestrial neutrino oscillation experiments and the high-energy phenomenology. We study the physical potential of the neutrino oscillation experiments by reflecting on the recent findings of the LBL accelerator experiments Tokai-to-Kamioka (T2K) Abe:2019vii ; Abe:2011ks and NuMI Off-axis neutrino Appearance (NO$\nu$A) Ayres:2007tu ; Acero:2019ksn and the reactor experiments Daya Bay An:2012bu ; An:2013zwz ; Adey:2018zwh , Double Chooz Ardellier:2006mn ; DoubleChooz:2019qbj and Reactor Experiment for Neutrino Oscillation (RENO) Ahn:2012nd ; Bak:2018ydk . Practically, we use the Wilson package Aebischer:2018bkb to take care of the running from the scale $\Lambda$ down to 2 GeV, as well as the second matching between SMEFT and the LEFT at $\mu=m_{W}$ shown in figure 1.¹¹1For the matching and running between SMEFT and LEFT for charge-current interactions, see Refs. Cirigliano:2009wk ; Gonzalez-Alonso:2017iyc . The first matching is model dependent and is thus usually worked out manually model by model. We will discuss more about it in section 2 and provide an example in section 5 for illustration. The last matching is universal and has been studied in Ref. Falkowski:2019kfn recently. The NSI parameters acquired this way serve as our input to the General Long-Baseline Experiment Simulator (GLoBES) Huber:2004ka ; Huber:2007ji through its new physics extension NuPhys .

This article is decomposed into the following sections: in section 2 we review the standard parameterization of the NSIs in neutrino oscillations and summarize the matching between the QFT and the QM formalisms relevant for our study. Section 3 focuses on the connection between the SMEFT and the neutrino NSI parameters. We describe the neutrino oscillation data and the related analysis methods used in this work in section 4. Then we present a UV example, i.e., the simplified leptoquark model in section 5 to illustrate our approach following the workflow of figure 1. We finally present a systematical analysis on the dominant dimension-6 SMEFT operators by providing constraints on the UV scale $\Lambda$ and the Wilson coefficients from neutrino experiments in section 6. The implications on the CP violation in the leptonic sector are briefly discussed in section 7. The results are then summarized in section 8.

Since neutrino oscillations are low-energy experiments, the Quantum Mechanical (QM) formalism is often used to compute the oscillation probability $P_{\alpha\beta}=|\langle\nu_{\alpha}^{d}|e^{-iHL}|\nu_{\beta}^{s}\rangle|^{2}$, which describes the probability of a source neutrino of flavor $\beta$ being detected as a neutrino of flavor $\alpha$. The meaning of “flavor” here is modified according to the inclusion of the non-standard interactions and is different from the original definition of the flavor that is associated with the lepton doublets diagonalizing the charged lepton mass matrix. The relation between these two flavor eigenstates are related by the QM production and detection NSI parameters $\epsilon^{s}$ and $\epsilon^{d}$ Grossman:1995wx ; GonzalezGarcia:2001mp ; Ohlsson:2008gx ; Farzan:2017xzy :

$\displaystyle|\nu_{\alpha}^{s}\rangle=\frac{(1+\epsilon^{s})_{\alpha\gamma}}{N^{s}_{\alpha}}|\nu_{\gamma}\rangle,\ \langle\nu_{\beta}^{d}|=\langle\nu_{\gamma}|\frac{(1+\epsilon^{d})_{\gamma\beta}}{N^{d}_{\beta}},$

(1)

where $N_{\alpha}^{s}=\sqrt{[(1+\epsilon^{s})(1+\epsilon^{s\dagger})]_{\alpha\alpha}}$ and $N_{\beta}^{d}=\sqrt{[(1+\epsilon^{d{\dagger}})(1+\epsilon^{d})]_{\beta\beta}}$. The repeated indices $\alpha$ and $\beta$ on the right hand side of equations refer to the diagonal elements, and are not to be confused with indices contraction. The information about matter effects affecting the propagation of the neutrino oscillation in media is encoded in the Hamiltonian $H$:

$$H=\frac{1}{2E_{\nu}}\left[U\left(\begin{array}[]{ccc}0&0&0\\ 0&\Delta m_{21}^{2}&0\\ 0&0&\Delta m_{31}^{2}\end{array}\right)U^{\dagger}+A\left(\begin{array}[]{ccc}1+\varepsilon_{ee}^{m}&\epsilon_{e\mu}^{m}&\,\,\,\epsilon_{e\tau}^{m}\\ \epsilon_{e\mu}^{m*}&\epsilon_{\mu\mu}^{m}&\,\,\,\epsilon_{\mu\tau}^{m}\\ \epsilon_{e\tau}^{m*}&\epsilon_{\mu\tau}^{m*}&\,\,\,\epsilon_{\tau\tau}^{m}\end{array}\right)\right],$$

(2)

where $U$ is the Pontecorvo-Maki-Nakagawa-Sakata matrix Pontecorvo:1957cp ; Pontecorvo:1957qd ; Maki:1960ut ; Maki:1962mu ; Pontecorvo:1967fh and $A=\sqrt{2}G_{F}N_{e}$, with $G_{F}$ denoting the Fermi constant and $N_{e}$ the electron number density in the medium, defines the standard matter potential Wolfenstein:1977ue ; Mikheev:1986gs . The matrix elements $\epsilon^{m}_{\alpha\beta}$ ($\alpha$, $\beta=e$, $\mu$, $\tau$) are the matter NSI parameters. A schematic illustration of the source NSI is presented in figure  2, where the pion decay process is influenced by non-standard neutrino interactions.

The NSI parameters defined in the QM formalism have been widely used in the study of neutrino oscillations, and their concepts are rather phenomenological. However, to study the effects originating from some UV physics, it is better to parameterize the NSI parameters in the QFT framework, i.e., in terms of the Wilson coefficients in the Lagrangian of the LEFT since these Wilson coefficients are directly related to the parameters in the UV theory as depicted in figure 1. On the other hand, it was recently pointed out in Ref. Falkowski:2019kfn that matching the differential rate $R_{\alpha\beta}$, defined as the number of oscillation events observed per second per neutrino energy Antusch:2006vwa ; Falkowski:2019kfn , in both formalisms to obtain the connection between the NSI parameters and the Wilson coefficients in LEFT is highly non-trivial. Furthermore, this matching procedure is not guaranteed to give a meaningful match of the two sets of parameters unless one works in the linear approximation in terms of $\epsilon^{s,d}$. In what follows, in light of the smallness of the neutrino NSI parameters, we will adopt the matching procedure described in Ref. Falkowski:2019kfn to the linear order and study effects from these charge-current NSI parameters.²²2For the neutral-current neutrino NSIs, see, for example, Ref. Dev:2019anc for a recent review and Refs. Choudhury:2018xsm ; Masud:2018pig ; Friedland:2011za ; Pandey:2019apj ; Babu:2020nna ; Liu:2020emq ; Dey:2020fbx for their recent studies. In this work, however, we will only focus on the charged-current NSI parameters.

To start, we write down the most general charged-current (CC) neutrino NSIs in LEFT asFalkowski:2019kfn :³³3For this CC NSIs, we adopt the convention used in Ref. Falkowski:2019kfn , where the (V+A), scalar, pseudoscalar and tensor-type interactions are also introduced. In the case of general neutrino interactions, we refer the reader to Ref. Bischer:2019ttk .

	$\displaystyle\mathcal{L}_{\rm CC}\supset$	$\displaystyle-\frac{2V_{ud}}{v^{2}}\left\{\left[\mathbf{1}+\epsilon_{L}\right]^{ij}_{\alpha\beta}\left(\bar{u}_{i}\gamma^{\mu}P_{L}d_{j}\right)\left(\bar{\ell}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)+\left[\epsilon_{R}\right]^{ij}_{\alpha\beta}\left(\bar{u}_{i}\gamma^{\mu}P_{R}d_{j}\right)\left(\bar{\ell}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)\right.$
		$\displaystyle+\frac{1}{2}\left[\epsilon_{S}\right]^{ij}_{\alpha\beta}(\bar{u}_{i}d_{j})\left(\bar{\ell}_{\alpha}P_{L}\nu_{\beta}\right)-\frac{1}{2}\left[\epsilon_{P}\right]^{ij}_{\alpha\beta}\left(\bar{u}_{i}\gamma_{5}d_{j}\right)\left(\bar{\ell}_{\alpha}P_{L}\nu_{\beta}\right)$
		$\displaystyle+\left.\frac{1}{4}\left[\epsilon_{T}\right]^{ij}_{\alpha\beta}\left(\bar{u}_{i}\sigma^{\mu\nu}P_{L}d_{j}\right)\left(\bar{\ell}_{\alpha}\sigma_{\mu\nu}P_{L}\nu_{\beta}\right)+\mathrm{h.c.}\right\},$		(3)

where from now on we use the Roman letters $i,j=\{1,2,3\}$ to represent the flavor of quarks, and the Greek letters $\alpha,\beta=\{e,\mu,\tau\}$ to that of charged leptons and neutrinos, $P=P_{L,R}$ are the chiral projection operators for the left- and the right-handed state of Dirac spinors. Since the energy scale of neutrino oscillation experiments are oftentimes very small, the first generation of quarks is to be understood when the $i,j$ indices are omitted for the NSI parameters $\epsilon_{L,R,S,P,T}$.

Starting from eq. (3) and following the matching procedure of Ref. Falkowski:2019kfn , we reproduce the following matching formulas originally derived in Ref. Falkowski:2019kfn to the linear order of $\epsilon^{s,d}_{\alpha\beta}$’s:⁴⁴4Here, we use the same notation for the nucleon tensor form factor $g_{T}$ as in Ref. Falkowski:2019kfn in eqs. (4-5).

	$\displaystyle\epsilon_{e\beta}^{s}=$	$\displaystyle\left[\epsilon_{L}-\epsilon_{R}-\frac{g_{T}}{g_{A}}\frac{m_{e}}{f_{T}\left(E_{\nu}\right)}\epsilon_{T}\right]_{e\beta}^{*},\quad{\text{~{}($\beta$ decay)}}$		(4)
	$\displaystyle\epsilon_{\beta e}^{d}=$	$\displaystyle\left[\epsilon_{L}+\frac{1-3g_{A}^{2}}{1+3g_{A}^{2}}\epsilon_{R}-\frac{m_{e}}{E_{\nu}-\Delta}\left(\frac{g_{S}}{1+3g_{A}^{2}}\epsilon_{S}-\frac{3g_{A}g_{T}}{1+3g_{A}^{2}}\epsilon_{T}\right)\right]_{e\beta},{\text{~{}(inverse $\beta$ decay)}}$		(5)
	$\displaystyle\epsilon_{\mu\beta}^{s}=$	$\displaystyle\left[\epsilon_{L}-\epsilon_{R}-\frac{m_{\pi}^{2}}{m_{\mu}\left(m_{u}+m_{d}\right)}\epsilon_{P}\right]_{\mu\beta}^{*},\quad{\text{~{}(pion decay)}}$		(6)

where $g_{S}=1.022(10)$, $g_{A}=1.251(33)$ and $g_{T}=0.987(55)$ are the scalar, axial-vector and tensor charges of the nucleon Chang:2018uxx ; Gupta:2018qil ; Aoki:2019cca ; Gonzalez-Alonso:2013ura , $\Delta\equiv m_{n}-m_{p}\simeq 1.3\rm~{}MeV$, $E_{\nu}$ is the neutrino energy, and $f_{T}(E_{\nu})$ is the nucleon form factor resulting from the tensor-type NSI in eq. (3), for which we use the same parameterization as that in Ref. Falkowski:2019xoe . Note that for the inverse $\beta$ decay, the formula above only includes the Gamow-Teller type interactions since reactor beta transitions like Daya Bay, Double Chooz and RENO discussed above, are mostly of this type Hayes:2016qnu . On the other hand, one may also expect contributions to $\epsilon^{s}_{e\beta}$ from muon decay, however, as we will discuss in the next section, NSI effects from muon decay will be absorbed into the definition of the Fermi constant in our setup.

Before discussing the effects from the CC NSIs, we would like to emphasize that whenever matching is involved, one shall note that all the parameters in eq. (3) are to be understood as parameters defined at the $2\rm\,GeV$ scale. Therefore, when one studies the effects of some new physics that lives at a much higher scale, the running effects, as illustrated in figure 1, shall be included as they could contribute significantly to these parameters due to the large QCD coupling. A complete study of the running effects is provided in Ref. Jenkins:2017dyc . In our numerical analysis, we count on the Wilson package Aebischer:2018bkb to take care of these effects.

Since no new particles have been observed at the LHC since 2012 after the discovery of the Higgs, SMEFT has become a popular toolset to systematically study potential deviations from SM predictions due to new physics residing in an energy scale $\Lambda$ much higher than the weak scale. Hence, it is thus natural to study the effects on neutrino NSIs from this new physics in the framework of SMEFT. To be specific, we shall derive constraints on the SMEFT Wilson coefficients from neutrino oscillation experiments following the workflow in figure 1 and using eqs. (4-6) for the matching that essentially relates the UV parameters to $\epsilon^{s,d}$. Similar work focusing on the neutral-current source NSI can be found in Ref. Terol-Calvo:2019vck , though no source and detection NSI were taken into account. Ref. Falkowski:2017pss studied charge-current NSIs, but they only considered the lepton flavor conserving part of the NSI parameters in eq. (3). In this section, we will describe the matching framework we use to derive our results. Our study follows the results in Ref. Jenkins:2017jig and is implemented in the Python package Wilson.

In general, the SMEFT can be organized by the dimension of the operators:

$${\cal L}_{\text{SMEFT}}=\sum_{D}\sum_{i_{D}}\frac{c_{i_{D}}}{\Lambda^{D-4}}{\cal O}_{i}^{D},$$

(7)

where $D$ denotes the dimensions of the operators starting from 5, $i_{D}$ labels the effective operators for a given dimension $D$, $c_{i_{D}}$ is the dimensionless Wilson coefficient, $\Lambda$ represents the characteristic UV scale. In this work, we will focus on the dimension-6 operators only and assume that neutrino masses are generated by the dimension-5 Weinberg operator Weinberg:1979sa .

Starting from a certain UV theory around scale $\Lambda$, its contributions to SMEFT operators can be readily obtained by integrating out the heavy fields with the covariant derivative expansion (CDE) techniques Gaillard:1985uh ; Cheyette:1987qz ; Henning:2014wua . Alternatively, one can calculate the amplitudes in both the UV theory and the SMEFT, and then obtain their relations through amplitudes matching. See, for example, Ref. Skiba:2010xn for a detailed discussion. This is a paradigm of the top-down EFT approach in studying the phenomenology of new physics, depicted by the first row of figure 1. We will show a concrete example of this approach to study constraints from neutrino oscillations on a simplified scalar leptoquark model in section 5. On the other hand, in the bottom-up EFT approach, one can view the SMEFT at an arbitrary new physics scale as the starting point and treat all the Wilson coefficients as independent parameters. This approach is more general while the correlation among the Wilson coefficients is lost. In section 6, through a careful study on the correlation of dimension-6 SMEFT operators for neutrino oscillation experiments, we will demonstrate this correlation is in general important.

In both the EFT approaches described above, the Wilson coefficients of the SMEFT operators are all defined at the new physics scale $\Lambda$, while a natural matching between SMEFT and LEFT is around $m_{W}$, where one integrates out $t$ quark, the Higgs boson, and the $W,Z$ bosons in the SM as implied by the second row of figure 1. Therefore, before the matching procedure is done, one needs to run the SMEFT Wilson coefficients from the scale $\Lambda$ down to $m_{W}$, i.e., from the first to the second row of figure 1. This running of the SMEFT Wilson coefficients were thoroughly studied in Ref. Alonso:2013hga ; Jenkins:2013zja ; Jenkins:2013wua . Due to the running effects, neutral current interactions in the UV model can finally generate charged current NSIs. Taking the minimal $Z^{\prime}$ model as an example, we assume that the SM gauge group is extended by a $U(1)_{z}$ associated with the new gauge boson $Z^{\prime}$ and a complex scalar $\phi$ which is responsible for the spontaneous breaking of the $U(1)_{z}$ giving mass to $Z^{\prime}$. With no new fermion fields introduced, the gauge anomaly cancellation restricts the $U(1)_{z}$ charges of the SM fields to be proportional to their $U(1)_{Y}$ hypercharges or trivially to equal to zero. In this case we find that a tree level matching generates flavor conserving neutral current $C_{\begin{subarray}{c}ll\\ ee\mu\mu\end{subarray}}$, which generates $C_{\begin{subarray}{c}ll\\ e\mu\mu e\end{subarray}}$ through the RGE running and further induces the $\epsilon_{L}$ via the calibration of $G_{F}$ as we will discuss below.

The matching between SMEFT and the LEFT is studied in detail in Ref. Jenkins:2017jig . At tree level, we classify the SMEFT operators that contribute to the neutrino NSI operators in the LEFT after matching into two types as listed below, where the type-A ones generates the operators in the LEFT directly via expansions of the left-handed lepton and the quark doublets, and the type-B ones contribute by modifying the interactions between the gauge bosons and fermions after setting the Higgs field to its vacuum expectation value $v_{T}$ and integrating out the $W$ bosons and setting.

Type-A	${\cal O}^{(1)}_{\begin{subarray}{c}lequ\end{subarray}}=(\bar{l}_{\alpha}^{j}e_{\beta})\epsilon_{jk}(\bar{q}_{s}^{k}u_{t})\quad{\cal O}^{(3)}_{\begin{subarray}{c}lequ\end{subarray}}=(\bar{l}_{\alpha}^{j}\sigma_{\mu\nu}e_{\beta})\epsilon_{jk}(\bar{q}_{s}^{k}\sigma^{\mu\nu}u_{t})$
	${\cal O}_{\begin{subarray}{c}ledq\end{subarray}}=(\bar{l}_{\alpha}^{j}e_{\beta})(\bar{d}_{s}q_{tj})\quad{\cal O}^{(3)}_{\begin{subarray}{c}lq\end{subarray}}=(\bar{l}_{\alpha}\gamma^{\mu}\tau^{I}l_{\beta})(\bar{q}_{s}\gamma_{\mu}\tau^{I}q_{t})$
Type-B	${\cal O}^{(3)}_{\begin{subarray}{c}Hl\end{subarray}}=(H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H)(\bar{l}_{\alpha}\tau^{I}\gamma^{\mu}l_{\beta})\quad{\cal O}^{(3)}_{\begin{subarray}{c}Hq\end{subarray}}=(H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H)(\bar{q}_{p}\tau^{I}\gamma^{\mu}q_{r})$
	$\quad{\cal O}_{\begin{subarray}{c}Hud\end{subarray}}=(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{u}_{p}\gamma^{\mu}d_{r})\quad{\cal O}_{\begin{subarray}{c}ll\\ 2112\end{subarray}}=(\bar{l}_{2}\gamma^{\mu}l_{1})(\bar{l}_{1}\gamma_{\mu}l_{2})$
	${\cal O}_{\begin{subarray}{c}ll\\ 1221\end{subarray}}=(\bar{l}_{1}\gamma^{\mu}l_{2})(\bar{l}_{2}\gamma_{\mu}l_{1})\quad{\cal O}^{(3)}_{\begin{subarray}{c}Hl\\ 22\end{subarray}}=(H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H)(\bar{l}_{2}\tau^{I}\gamma^{\mu}l_{2})$
	${\cal O}^{(3)}_{\begin{subarray}{c}Hl\\ 11\end{subarray}}=(H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H)(\bar{l}_{1}\tau^{I}\gamma^{\mu}l_{1})$

One example of the type-A operators is:

$${\cal O}^{(1)}_{lequ}=\bar{l}^{j}_{\alpha}e_{\beta}\epsilon_{jk}\bar{q}^{k}_{s}u_{t},$$

(8)

where $l_{\alpha}=(\nu_{L,\alpha},\ell_{L,\alpha})^{T}$ and $q_{s}=(u_{L,s},d_{L,s})^{T}$ are the $SU(2)_{L}$ lepton and quark doublets. Expanding the conjugate of this operator, one can find that it contains the following CC neutrino NSI relevant operators:

$$\bar{l}^{j}_{\alpha}e_{\beta}\epsilon_{jk}\bar{q}^{k}_{s}u_{t}\supset(\bar{u}_{t}P_{L}d_{s})(\bar{\ell}_{\alpha}P_{L}\nu_{\beta})=\frac{1}{2}(\bar{u}_{t}d_{s})(\bar{\ell}_{\alpha}P_{L}\nu_{\beta})-\frac{1}{2}(\bar{u}_{t}\gamma^{5}d_{s})(\bar{\ell}_{\alpha}P_{L}\nu_{\beta}).$$

(9)

Hence, if the corresponding dimensionless Wilson coefficients of the operators in eq. (8) is $C^{(1)}_{\begin{subarray}{c}lequ\\ prst\end{subarray}}$, it can then be related to $\epsilon_{S}$ and $\epsilon_{P}$ defined in eq. (3) as:

$$(\epsilon_{S})^{ts}_{\alpha\beta}=(\epsilon_{P})^{ts}_{\alpha\beta}=(C^{(1)}_{\begin{subarray}{c}lequ\\ \alpha\beta st\end{subarray}})^{*}.$$

(10)

For the Type-B operators, one can parameterize their effects in the following Lagrangian Jenkins:2017jig :

$\displaystyle{\cal L}_{\rm gauge}=-\frac{\bar{g}_{2}}{\sqrt{2}}\{\mathcal{W}^{+}_{\mu}j^{\mu}_{\mathcal{W}}+h.c.\}-\bar{g}_{Z}Z_{\mu}j^{\mu}_{Z},$

(11)

where once again we only focus on the CC $j^{\mu}_{W}$:

$\displaystyle j_{\mathcal{W}}^{\mu}=\left[W_{l}\right]_{\alpha\beta}\bar{\nu}_{L\alpha}\gamma^{\mu}e_{L\beta}+\left[W_{q}\right]_{pr}\bar{u}_{Lp}\gamma^{\mu}d_{Lr}+\left[W_{R}\right]_{pr}\bar{u}_{Rp}\gamma^{\mu}d_{Rr},$

(12)

with $W_{l,q,R}$ parameterizing the modification of the $W$ boson coupling to the corresponding fermion currents

$\displaystyle\left[W_{l}\right]_{\alpha\beta}=\left[\delta_{\alpha\beta}+v_{T}^{2}C_{\begin{subarray}{c}Hl\\ \alpha\beta\end{subarray}}^{(3)}\right],\quad\left[W_{q}\right]_{pr}=\left[\delta_{pr}+v_{T}^{2}C_{\begin{subarray}{c}Hq\\ pr\end{subarray}}^{(3)}\right],\quad\left[W_{R}\right]_{pr}=\left[\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}Hud\\ pr\end{subarray}}\right]$

(13)

The $W$ boson mass is $M_{\mathcal{W}}=\bar{g}_{2}^{2}v_{T}^{2}/4$, with $\bar{g}_{2}$ the modified gauge coupling of the SU(2) group when normalization of the gauge fields due to the presence of ${\cal O}_{HW}=|H|^{2}W^{I}_{\mu\nu}W^{I\mu\nu}$ Jenkins:2017jig . However, as we shall see immediately, ${\cal O}_{HW}$ is not relevant for our tree-level matching. After integrating out the $W$ boson at tree level, the effective Lagrangian can be written as:

$\displaystyle{\cal L}_{\rm C}=-\frac{\bar{g}^{2}_{2}}{2M_{\mathcal{W}}}j^{\mu}_{\mathcal{W}}j_{\mathcal{W}{\mu}}=-\frac{2}{v_{T}^{2}}j^{\mu}_{\mathcal{W}}j_{\mathcal{W}{\mu}},$

(14)

note that $\bar{g}_{2}$ cancels in the numerator and the denominator. Upon expanding $j^{\mu}_{\mathcal{W}}$, the above equation will then generate all the terms in eq. (3). However, one needs to pay particular attention to the fact that $v_{T}$ above is different from $v$ in eq. (3). The latter is defined through the Fermi constant as $G_{F}=1/(\sqrt{2}v^{2})$, and the relation between $v_{T}$ and $G_{F}$, the Fermi constant $G_{\mu}$ obtained from muon decay in our setup, is as follows:

$\displaystyle\frac{4G_{F}}{\sqrt{2}}=\frac{4G_{\mu}}{\sqrt{2}}=\frac{2}{v_{T}^{2}}-C_{\begin{subarray}{c}ll\\ \mu ee\mu\end{subarray}}-C_{\begin{subarray}{c}ll\\ e\mu\mu e\end{subarray}}+C^{(3)}_{\begin{subarray}{c}Hl\\ \mu\mu\end{subarray}}+C^{(3)}_{\begin{subarray}{c}Hl\\ ee\end{subarray}}$

(15)

The term on the right hand side is essentially the Wilson coefficients of the LEFT operators $\left(\bar{\nu}_{L\mu}\gamma^{\mu}\nu_{Le}\right)\left(\bar{e}_{L}\gamma_{\mu}\mu_{L}\right)$ after the tree level matching. In principle, there could be a non-vanishing Wilson coefficients of $\left(\bar{\nu}_{L\mu}\gamma^{\mu}\nu_{Le}\right)\left(\bar{e}_{R}\gamma_{\mu}\mu_{R}\right)$ from the new physics effects contributing to the muon decay process, while its contribution is higher order or suppressed by the $m_{e}/m_{\mu}$ in the interference terms due to the opposite chirality of the electron in the two operators Jenkins:2017jig . Since there are two neutrinos involved in muon decay, the matching between the QFT and the QM formalisms are less trivial compared with pion decay, beta and inverse beta decay discussed in section 2. On the other hand, for different setups with respect to $G_{F}$ as discussed above, it would become necessary to have the matching formulae for muon decay. For this reason, we present these matching formulae in Appendix A. Our results agree with those in Ref. Falkowski:2019kfn for both neutrinos in the final states after Taylor expansion around small electron mass $m_{e}$. A comparison between our results and those in Ref. Falkowski:2019kfn is also discussed in Appendix A.

The last subtlety of this framework is related to the definition of quark flavors in operators. At tree level, the mass matrices $\left[M_{\psi}\right]_{rs}$ with SMEFT dimension-6 operators are:

$\displaystyle\left[M_{\psi}\right]_{rs}=\frac{v_{T}}{\sqrt{2}}\left(\left[Y_{\psi}\right]_{rs}-\frac{1}{2}v^{2}C_{\psi H}^{*}\right),\quad\psi=u,d.$

(16)

Depending on whether the up-type quark or the down-type quark mass matrix is diagonal, there are two types of bases at the definition scale, i.e., the “Warsaw up” and the “Warsaw down” bases:

	$\displaystyle\text{Warsaw up}:M_{d}\rightarrow\operatorname{diag}\left(m_{d},m_{s},m_{b}\right)V^{\dagger},\quad M_{u}\rightarrow\operatorname{diag}\left(m_{u},m_{c},m_{t}\right),$		(17)
	$\displaystyle\text{Warsaw down}:M_{d}\rightarrow\operatorname{diag}\left(m_{d},m_{s},m_{b}\right),\quad M_{u}\rightarrow V^{\dagger}\operatorname{diag}\left(m_{u},m_{c},m_{t}\right).$		(18)

For the lepton sector, in either the “Warsaw up” or the “Warsaw down” basis, we always choose the bases of the lepton doublets and the right-handed charged lepton fields such that the mass matrix of the charged leptons is diagonal. We would like to emphasize here that both bases are specifically defined at a fixed scale such that when running effects are included, off-diagonal elements in the mass matrices are likely to reappear at a different scale. Therefore, whenever a running is performed, one needs to rotate the basis of fermion fields to change the operators back into the “Warsaw up” or the “Warsaw down” basis.

The focus of our work is in the present generation of neutrino oscillation experiments and to investigate the effects of dimension-6 SMEFT operators on neutrino oscillation. To this end, we simulate the neutrino oscillation corresponding to the recently collected data in the LBL experiments T2K and NO$\nu$A and the reactor experiments Daya Bay, Double Chooz and RENO. We consider the data sets published in the Neutrino 2020 conference Neutrino2020 and reproduce them with GLoBES. In this section, we briefly describe the nature of the simulated data and numerical methods used in this work.

To illustrate how the matching and the running work, as well as how the numerical analysis of neutrino oscillation experiments discussed in last section can be used to impose constraints on the UV models, we adopt the simplified scalar leptoquark model in this section. We will first set up this simple UV model and then illustrate how neutrino experiments are used to constrain this model. As shown in figure 1, since this simplified model is defined at a UV scale much above that at which neutrino experiments are carried out, the running effects need to be considered. The Wilson package is used for the running and the matching at the weak scale, after which a comparison between theoretical prediction from this simplified UV model and experimental bounds on the NSI parameters is done to obtain constraints on this model. As we will see later in this section, though neutrino experiments are performed at a very low energy scale, they put much more stringent constraints than those from high-energy experiments at colliders.

In the last section, the robustness of using neutrino NSI parameters to study new physics at high energies is illustrated by the simplified scalar leptoquark model. In this section, following figure 1, we generalize our study to all dimension-6 SMEFT operators and present our final results as lower constraints on the UV scale $\Lambda$ as well as upper constraints on the Wilson coefficients pertaining to individual operators. Due to the large number of dimension-6 SMEFT operators, we only present the results for operators that are within the reach of the neutrino experiments considered in this work. To that end, we first take the bottom-up EFT approach where one assumes all the dimension-6 SMEFT operators are independent at the UV scale $\Lambda$ and then derives the experimental constraints on $\Lambda$ and the Wilson coefficient of each operator. The results are presented in subsection 6.1 and achieved by simulating the neutrino oscillation data in the LBL neutrino experiments T2K and NO$\nu$A and the reactor antineutrino experiments Daya Bay, Double Chooz and RENO.

On the other hand, as discussed in section 3, the correlation among different operators will in general get lost in the bottom-up EFT approach. However, as already can be seen in the simplified scalar leptoquark model case presented in section 5, effects due to the correlation among the dimension-6 SMEFT operators already show up at the UV scale $\Lambda$. As a result, neutrino NSI parameters at the 2 GeV scale will also be affected by the correlation. We present our results demonstrating the correlation among different operators in subsection 6.2.