# DUNE potential as a New Physics probe

Adriano Cherchiglia  Instituto de Física Gleb Wataghin, Universidade Estadual
de Campinas, Rua Sérgio Buarque de Holanda, 777, 13083-859, Campinas, SP,
Brazil  Departamento de Física Teórica y del Cosmos, Universidad de Granada,
Campus de Fuentenueva, E-18071 Granada, Spain  José Santiago  Departamento de
Física Teórica y del Cosmos, Universidad de Granada, Campus de Fuentenueva,
E-18071 Granada, Spain

###### Abstract

Neutrino experiments, in the next years, aim to determine with precision all
the six parameters of the three-neutrino standard paradigm. The complete
success of the experimental program is, nevertheless, attached to the non-
existence (or at least smallness) of Non-Standard Interactions (NSI). In this
work, anticipating the data taken from long-baseline neutrino experiments, we
map all the weakly coupled theories that could induce sizable NSI, with the
potential to be determined in these experiments, in particular DUNE. Once
present constraints from other experiments are taken into account, in
particular charged-lepton flavor violation, we find that only models
containing leptoquarks (scalar or vector) and/or neutral isosinglet vector
bosons are viable. We provide the explicit matching formulas connecting weakly
coupled models and NSI, both in propagation and production. Departing from the
weakly coupled completion with masses at TeV scale, we also provide a global
fit on all NSI for DUNE, finding that NSI smaller than $10^{-2}$ cannot be
probed even in the best-case scenario.

##  1 Introduction

Neutrino experiments are entering a high precision era. Within the next decade
we expect to have new long-baseline experiments such as DUNE and T2HK, whose
main goal is to measure with high accuracy two of the parameters of the three-
neutrino paradigm, namely $\theta_{23}$ and $\delta_{CP}$. These measurements,
however, can be sensitive to the presence of Non-Standard Interactions (NSI),
which represent new sources for neutral (charged) currents apart from the
usual Z (W) mediated processes, see [1, 2, 3] for reviews. Given their nature,
NSI can affect neutrino production, detection and/or propagation through
matter [4, 5]. Interestingly, new effects such as violation of the unitarity
of the PMNS matrix may be seen [6, 7], which will definitely call for an
departure from the three-neutrino picture. These answers, nonetheless, will
only be known in the 2030s or later, after the data taken from these planned
experiments has been analyzed.

In the meantime, there are currently significant efforts towards the search
for new physics at colliders [8] and at low-energy experiments [9]. Even
though there are some hints of anomalies in experimental data, no clear
picture can be drawn yet. In view of this scenario, it is desirable to merge
the present knowledge in a unified front, such that any theoretical model can
be scrutinized by as many experimental data as currently available. This is
especially relevant if an anomaly is observed in a particular experiment, as
one can immediately rule out or confirm, based on other experiments, if a
particular model can be the origin of the anomaly, also making predictions of
unmeasured observables that could further confirm the nature of the anomaly.

In this work, anticipating the results from planned long-baseline experiments,
in particular DUNE, we give a comprehensive list of which weakly coupled
models, based only on the Standard Model (SM) gauge group, could be
responsible for any sizeable anomaly seen by the experiment. This can be
achieved under the assumption that new physics is heavy, as compared with the
electroweak scale, and weakly coupled, making the loop expansion meaningful.
Under these circumstances we can use the tree-level SM effective field theory
(SMEFT) dictionary [10] that includes the most general model with heavy
particles of spin 1 or less that can contribute to the SMEFT Lagrangian at
tree level and mass dimension 6 111A similar analysis, taking into account
operators of dimension 6 and 8 containing only leptons was performed in [11].
The complete tree-level dictionary has been recently extended up to dimension
7 in [12] and the calculation of the dimension 6, one-loop dictionary is
currently underway [13].. Given that the new particles are heavier than the
electroweak scale, we will use an effective approach based on the SMEFT. As
first noticed in [14], an immediate problem to be faced by any weakly coupled
model with sizable non-diagonal NSI, is to evade the strong bounds imposed by
the null observation of charged lepton flavor violation (CLFV) processes. In
references [11, 14], it is argued that, by a suitable cancellation of the
Wilson coefficients (WC), it is possible to generate large NSI at the same
time that CLFV processes are under control. Their analysis, previous to the
proper definition of the Warsaw basis [15] is, however, incomplete. In
particular, they consider as independent the operators
$(\bar{\ell}_{p}\gamma_{\mu}\ell_{s})(\bar{\ell}_{r}\gamma^{\mu}\ell_{t})$,
$(\bar{\ell}_{p}\gamma_{\mu}\tau^{I}\ell_{s})(\bar{\ell}_{r}\gamma^{\mu}\tau^{I}\ell_{t})$,
which are actually related by a Fierz transformation. Therefore, it is not
possible to obtain large non-diagonal NSI by considering only four-lepton
operators. By considering operators containing two leptons and two quarks, the
situation is even more dramatic given the very strong bounds coming from
$\mu-e$ conversion in nuclei, for instance.

Although the previous discussion was mainly related to NSI coming from
propagation, a similar reasoning can be applied for those coming from
production/detection. The main question is: how sensitive will future long-
baseline experiments (such as DUNE) be to non-diagonal NSI in comparison with
bounds from CLFV processes. By performing a global fit, we will obtain that
DUNE can be sensitive to NSI at most of order $10^{-2}$. However, CLFV
constraints from processes involving muon and electron are of order $10^{-6}$
(or lower) while those involving taus are of order $10^{-4}$ (or lower) [16,
17]. Therefore, it is clear that, if DUNE sees any anomaly related to non-
diagonal NSI, it will be very challenging to invoke a weakly coupled model at
the TeV scale as an explanation. Other possibilities such as considering
lighter mediators and/or enlarging the gauge group could be envisaged (see [1,
3] and references therein).

Analyses discussing the connection between NSI and EFT were presented, for
instance, in [18, 19, 20, 21, 22, 16, 23, 24, 25, 26, 17, 27, 28]. In [22], a
particular set of models is discussed focusing on neutrino mass mechanisms. It
is shown that, for some portions of the parameter space, it is possible to
generate sizeable diagonal NSI, evading the strong bounds from charged lepton
flavor violating processes. It is not clear, however, if there are other
weakly coupled models that could still be viable. This work intends to fill
this gap. In summary, this work goes beyond the current results in the
literature in the following.

  * •

We provide a complete list of tree-level weakly coupled models relevant for
long-baseline experiments, as well as their matching to the NSI.

  * •

We adopt the quantum field theory (QFT) formalism to treat NSI at production,
including indirect effects recently pointed out in [28]. We show that, if the
non-diagonal NSI in production are suppressed, the diagonal ones cannot be
probed. This implies that production can be treated as SM-like, allowing the
bounds extracted in [29] to be applied, even though they were derived under
the hypothesis of SM fluxes.

  * •

We go beyond the weakly coupled theories paradigm by performing a global fit
considering arbitrary NSI at production and propagation, in order to asses the
best-case scenario sensitivity of DUNE on such parameters.

The work is organized as follows. In Section 2 we review the connection
between NSI and the neutrino transition rate in the QFT formalism, while in
Section 3 we present the matching at tree-level between NSI and SMEFT WC. In
Section 4 we identify and analyze the weakly coupled models, with new
particles heavier than the electroweak scale, that can potentially generate
sizeable NSI to be probed by DUNE. In Section 5 we perform the numerical
analysis, and present our results. Section 6 is devoted to the summary of our
work while we collect relevant notation and the explicit matching result in
Appendices A and B, respectively.

##  2 Neutrino transition rate

In this work we will adopt the more general QFT framework to describe neutrino
oscillations in long-baseline experiments, following [21] (see also [30, 31,
32] for previous treatment in QFT not including NSI). The main point is to
consider the entire process (from neutrino production to its detection) as an
unique process in QFT. Thus, the differential event rate for neutrinos of
flavor $\beta$ with energy $E_{\nu}$ to be detected at a distance $L$ from the
source $S$, where they were produced with flavor $\alpha$, is given by

$R_{\alpha\beta}^{S}=\frac{N_{T}}{32\pi
L^{2}m_{S}m_{T}E_{\nu}}\sum_{k,l}e^{-i\frac{L\Delta m_{kl}^{2}}{2E_{\nu}}}\int
d\Pi_{P^{\prime}}\mathcal{M}^{P}_{\alpha k}\overline{\mathcal{M}}^{P}_{\alpha
l}\int d\Pi_{D}\mathcal{M}^{D}_{\beta k}\overline{\mathcal{M}}^{D}_{\beta
l}~{},$ (1)

where $N_{T}$ is the number of target particles, $m_{S,T}$ are the masses of
the source and target particles respectively, and we denote complex
conjugation with a bar. As usual, $\Delta m_{kl}^{2}\equiv
m_{k}^{2}-m_{l}^{2}$ denotes the difference of neutrino masses squared, while
the phase-space integrals are given by
$d\Pi\equiv\frac{d^{3}k_{1}}{(2\pi)^{3}2E_{1}}\dots\frac{d^{3}k_{n}}{(2\pi)^{3}2E_{n}}(2\pi)^{4}\delta^{4}(\sum
p_{n}-\sum k_{i})$. Here, we denote by $k_{i}$ and $E_{i}$ the four-momenta
and energies of the final states, while the sum $\sum p_{n}$ amounts to the
total four-momentum of the initial state. Since we are interested in the
differential number of events per incident neutrino energy $E_{\nu}$, we
actually define $d\Pi_{P}\equiv d\Pi_{P^{\prime}}dE_{\nu}$, as can be seen in
eq.(1). Notice also that we are considering oscillation in vacuum only. Matter
effects will be taken into account below.

As already stated, we will focus on long-baseline experiments, in which
neutrinos are mainly produced by pion decay. Moreover, we will only consider
beyond the SM (BSM) possible effects at neutrino production and/or propagation
(by interaction with the medium). This is justified by the complicated nature
of detection in long-baseline experiments. To properly include BSM effects, it
would require first a theoretical description of standard neutrino interaction
with nucleus, which is still not satisfactory [33]. Thus, regarding the
detection amplitude, $\mathcal{M}^{D}_{\beta k}$, it will be given by

$\mathcal{M}^{D}_{\beta k}=U_{\beta k}A^{D}\,,$ (2)

where $A^{D}$ is a reduced matrix element whose explicit form is not relevant,
and $U_{ij}$ is the PMNS matrix.

As recently pointed out [28], in the presence of BSM effects care must be
exercised when calculating the phase-space integral related to the production
amplitude. The amplitude concerning pion decay,
$\pi^{+}\to\ell_{\alpha}^{+}\nu_{k}$, is given by [28]

$\displaystyle{\cal M}^{P,\pi}_{\alpha
k}\equiv\mathcal{M}(\pi^{+}\to\ell_{\alpha}^{+}\nu_{k})=-i\,m_{\mu}f_{\pi^{\pm}}\frac{V_{ud}}{v^{2}}[{\cal
P}U]_{\alpha k}^{*}(\bar{u}_{\nu_{k}}P_{L}v_{\ell_{\alpha}})~{},$ (3)

where $f_{\pi^{\pm}}$ is the pion decay constant, and we are using the
notation

$\displaystyle[{\cal P}]_{\alpha\beta}$ $\displaystyle\equiv$
$\displaystyle\delta_{\alpha\sigma}+\bar{\epsilon}_{\alpha\sigma},\quad\mbox{where}\quad\bar{\epsilon}_{\alpha\sigma}=(\epsilon_{L})_{\alpha\sigma}-(\epsilon_{R})_{\alpha\sigma}-\frac{m_{\pi^{\pm}}^{2}}{m_{\ell_{\alpha}}(m_{u}+m_{d})}(\epsilon_{P})_{\alpha\sigma}\,.$
(4)

The NSI $\epsilon_{L},\epsilon_{R},\epsilon_{P}$ are defined in eq.(3.1). As
can be seen, $\bar{\epsilon}_{\alpha\sigma}$ actually encodes the deviation
from the PMNS matrix $U$ due to the presence of NSI.

At first sight, one could insert eq.(3) into eq.(1), perform the phase-space
integration, and obtain that BSM effects appear only in the numerator of
eq.(1). However, this reasoning is incorrect, since we are assuming the BSM
presence in all cases that the pion decay is involved, related (or not) to
experiments that aim to probe neutrinos properties. Therefore, we should re-
express eq.(3) in terms of the experimental measured decay width of the pion,
which was already affected by BSM effects as well. Proceeding this way, we
obtain [28]

$\displaystyle\int d\Pi_{P^{\prime}}\mathcal{M}_{\alpha
k}^{P,\pi}\bar{\mathcal{M}}_{\alpha
l}^{P,\pi}=2\,m_{\pi^{\pm}}\Gamma_{\pi\to\ell_{\alpha}\nu}\frac{[{\cal
P}U]_{\alpha l}[U^{\dagger}{\cal P}^{\dagger}]_{k\alpha}}{[{\cal P}{\cal
P}^{\dagger}]_{\alpha\alpha}}\,\delta\bigg{(}E_{\nu}-E_{\nu,\pi}\bigg{)}~{},$
(5)

where $E_{\nu,\pi}=(m^{2}_{\pi^{\pm}}-m_{\alpha}^{2})/(2m_{\pi^{\pm}})$ stands
for the energy of the neutrino emitted. Finally, we can write eq.(1) in the
form

$R_{\alpha\beta}^{S}=N_{T}\sigma_{\beta}^{\text{SM}}(E_{\nu})\Phi_{\alpha}^{\text{SM}}(E_{\nu})\sum_{k,l}e^{-i\frac{L\Delta
m_{kl}^{2}}{2E_{\nu}}}\frac{[{\cal P}U]_{\alpha l}[U^{\dagger}{\cal
P}^{\dagger}]_{k\alpha}}{[{\cal P}{\cal P}^{\dagger}]_{\alpha\alpha}}U_{\beta
k}U_{\beta l}^{*}~{},$ (6)

where $\Phi_{\alpha}^{\text{SM}}$ is the SM flux (with the decay width of the
pion, $\Gamma_{\pi\to\ell_{\alpha}\nu}$, as input), while
$\sigma_{\beta}^{\text{SM}}$ is the SM cross-section. Notice that, by
including the indirect effects (the denominator in $R_{\alpha\beta}^{S}$),
there is no sensitivity to diagonal-only NSI.

At this stage we would like to include matter effects. As is well-known, the
vector part of the neutral current (NC) NSI interactions introduced in
eq.(3.2) will introduce flavor dependent matter effects, which can be
parametrized in the Hamiltonian

$H=\frac{1}{2E}\left[U\left(\begin{array}[]{ccc}0&&\\\ &\Delta m_{21}^{2}&\\\
&&\Delta
m_{31}^{2}\end{array}\right)U^{\dagger}+a\left(\begin{array}[]{ccc}1+\epsilon_{ee}&\epsilon_{e\mu}&\epsilon_{e\tau}\\\
\epsilon_{e\mu}^{*}&\epsilon_{\mu\mu}&\epsilon_{\mu\tau}\\\
\epsilon_{e\tau}^{*}&\epsilon_{\mu\tau}^{*}&\epsilon_{\tau\tau}\end{array}\right)\right]\,.$
(7)

We introduced the parameter $a\equiv 2\sqrt{2}G_{F}N_{e}E$ which is known as
the Wolfstein matter potential, $N_{e}$ is the electron number density, and
$E$ is the neutrino energy. Notice that, in the absence of NSI effects, only
the first element of the matter matrix survives, which stands for the standard
charged current (CC) mediated by the W-boson. In the equation above we are
using the NSI notation in the Hamiltonian framework. The connection to the
parameters introduced at Lagrangian level, see eq.(3.2), is straightforward
[1]

$\epsilon_{\alpha\beta}=\sum_{f\in\left\\{e,u,d\right\\}}\left\langle\frac{N_{f}(x)}{N_{e}(x)}\right\rangle\left(\epsilon^{ff}_{V}\right)_{\alpha\beta}\,,$
(8)

where $N_{f}(x)$ is the number density of fermion of type $f$ at position $x$.
In Earth, we generally have $N_{u}/N_{e}\sim N_{d}/N_{e}\sim 3$ [34].
Moreover, in oscillation experiments one can, without loss of generality,
remove a free parameter from the diagonal. In general, $\epsilon_{\mu\mu}$ is
removed, which we will also adopt.

##  3 Matching NSI to the SMEFT

The connection between NSI and the dimension-six operators in the SMEFT was,
to the best of our knowledge, discussed first in [16]. Here we review the
formalism, and present a concise formula relating both quantities at the end
of each subsection.

###  3.1 Charged current: neutrino production

The Lagrangian relevant for neutrino production (mainly through pion decay) is
given by [35, 36]

$\displaystyle\mathcal{L}_{\rm WEFT}\supset-\frac{2\,V_{jk}}{v^{2}}\Big{\\{}$
$\displaystyle[1+\epsilon^{jk}_{L}]_{\alpha\beta}\left(\bar{u}^{j}\gamma^{\mu}P_{L}d^{k}\right)\left(\bar{\ell}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)+[\epsilon^{jk}_{R}]_{\alpha\beta}\left(\bar{u}^{j}\gamma^{\mu}P_{R}d^{k}\right)\left(\bar{\ell}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)$
$\displaystyle\left.+\frac{1}{2}\left[\epsilon_{S}^{jk}\right]_{\alpha\beta}\left(\bar{u}^{j}d^{k}\right)\left(\bar{\ell}_{\alpha}P_{L}\nu_{\beta}\right)-\frac{1}{2}\left[\epsilon_{P}^{jk}\right]_{\alpha\beta}\left(\bar{u}^{j}\gamma^{5}d^{k}\right)\left(\bar{\ell}_{\alpha}P_{L}\nu_{\beta}\right)\right.$
$\displaystyle+\frac{1}{4}\left[\epsilon_{T}^{jk}\right]_{\alpha\beta}\left(\bar{u}^{j}\sigma^{\mu\nu}P_{L}d^{k}\right)\left(\bar{\ell}_{\alpha}\sigma_{\mu\nu}P_{L}\nu_{\beta}\right)+\mathrm{h.c.}\Big{\\}}~{},$
(9)

We are considering the quark fields as well as the charged leptonic fields in
the mass eigenstate basis, $V$ is the CKM matrix, and $v\approx 246$ GeV is
the Higgs vacuum expectation value. This Lagrangian can be directly translated
to the Low Energy EFT (LEFT) one, which reads, following the notation in [36]

$\displaystyle
L_{r}^{V,LL}=-\frac{2\,V_{kj}^{*}}{v^{2}}\left[\delta_{\beta\alpha}+(\epsilon_{L}^{kj})_{\beta\alpha}\right]^{*}\,,\qquad
L_{r}^{V,LR}=-\frac{2\,V_{kj}^{*}}{v^{2}}(\epsilon_{R}^{kj})_{\beta\alpha}^{*},$
(10) $\displaystyle
L_{r}^{S,RR}+L_{r}^{S,RL}=-\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{S}^{kj})_{\beta\alpha}^{*}\,,\qquad
L_{r}^{S,RL}-L_{r}^{S,RR}=-\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{P}^{kj})_{\beta\alpha}^{*},$
(11) $\displaystyle
L_{r}^{T,RR}=-\frac{V_{kj}^{*}}{2v^{2}}(\epsilon_{T}^{kj})_{\beta\alpha}^{*},$
(12)

where $r=\begin{subarray}{c}\nu edu\\\ \alpha\beta jk\end{subarray}$ and we
are using an asterisk to denote complex conjugation here. The matching to the
SMEFT is also given in [36], which we reproduce below

$L_{\begin{subarray}{c}\nu edu\\\
prst\end{subarray}}^{V,LL}=\sum_{x}V^{*}_{xs}L_{\begin{subarray}{c}\nu edu\\\
prxt\end{subarray}}^{V,LL}=\sum_{x}V^{*}_{xs}\left(2C^{(3)}_{\begin{subarray}{c}lq\\\
prxt\end{subarray}}-\frac{{\overline{g}_{2}}^{2}}{2M_{W}^{2}}\left[W_{l}\right]_{pr}\left[W_{q}\right]_{tx}^{*}\right),$
(13) $L_{\begin{subarray}{c}\nu edu\\\
prst\end{subarray}}^{V,LR}=-\frac{{\overline{g}_{2}}^{2}}{2M_{W}^{2}}\left[W_{l}\right]_{pr}\left[W_{R}\right]_{ts}^{*},$
(14) $\displaystyle L_{\begin{subarray}{c}\nu edu\\\
prst\end{subarray}}^{S,RR}=\sum_{x}V_{xs}^{*}C^{(1)}_{\begin{subarray}{c}lequ\\\
prxt\end{subarray}},\quad L_{\begin{subarray}{c}\nu edu\\\
prst\end{subarray}}^{S,RL}=C_{\begin{subarray}{c}ledq\\\
prst\end{subarray}},\quad L_{\begin{subarray}{c}\nu edu\\\
prst\end{subarray}}^{T,RR}=\sum_{x}V_{xs}^{*}C^{(3)}_{\begin{subarray}{c}lequ\\\
prxt\end{subarray}},$ (15)

where

$\displaystyle[W_{l}]_{pr}=\left[\delta_{pr}+v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}\right],\quad[W_{q}]_{pr}=\left[\delta_{pr}+v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}\right],\quad[W_{R}]_{pr}=\left[\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}Hud\\\
pr\end{subarray}}\right].$ (16)

We are adopting the up basis where the up quark and charged lepton Yukawa
matrices are diagonal. In this case, flavor and mass eingenstates of the left-
handed down-quarks are connected by the CKM matrix.

Once all the pieces are known, it is straightforward to express the NSI
parameter in production in terms of the WC of the SMEFT as below

$\displaystyle\bar{\epsilon}_{\beta\alpha}=\frac{v^{2}}{2}\left[2C^{(3)}_{\begin{subarray}{c}Hl\\\
\alpha\beta\end{subarray}}+2\sum_{x}V_{x1}^{*}\left(\delta_{\alpha\beta}C^{(3)*}_{\begin{subarray}{c}Hq\\\
1x\end{subarray}}-C^{(3)}_{\begin{subarray}{c}lq\\\ \alpha\beta
1x\end{subarray}}\right)+\delta_{\alpha\beta}C^{*}_{\begin{subarray}{c}Hud\\\
11\end{subarray}}+p_{\beta}\left(C_{\begin{subarray}{c}ledq\\\ \alpha\beta
11\end{subarray}-}-\sum_{x}V_{x1}^{*}C^{(3)}_{\begin{subarray}{c}lequ\\\
\alpha\beta x1\end{subarray}}\right)\right]^{*}\,.$ (17)

where we used the notation
$p_{\alpha}=\frac{m_{\pi^{\pm}}^{2}}{m_{\ell_{\alpha}}(m_{u}+m_{d})}$.

###  3.2 Neutral current: neutrino propagation

Regarding neutrino propagation, we need to take into account the interaction
of neutrinos with the medium. This is given by the NC interactions between
neutrinos and the other fermions [37, 36]:

$\displaystyle\mathcal{L}_{\rm WEFT}^{\rm quarks}\supset-\frac{1}{v^{2}}$
$\displaystyle\left\\{[g_{V}^{qq}\,\mathbb{1}+\epsilon_{V}^{qq}]_{\alpha\beta}\left(\bar{q}\gamma^{\mu}q\right)\left(\bar{\nu}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)\right.$
$\displaystyle\left.+[g_{A}^{qq}\,\mathbb{1}+\epsilon_{A}^{qq}]_{\alpha\beta}\left(\bar{q}\gamma^{\mu}\gamma^{5}q\right)\left(\bar{\nu}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)\right\\},$
$\displaystyle\mathcal{L}_{\rm WEFT}^{\rm leptons}\supset-\frac{1}{v^{2}}$
$\displaystyle\left\\{[g_{V}^{ee}\,\mathbb{1}+\bar{g}_{V}^{ee}+\epsilon_{V}^{ee}]_{\alpha\beta}\left(\bar{e}\gamma^{\mu}e\right)\left(\bar{\nu}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)\right.$
$\displaystyle\left.+[g_{A}^{ee}\,\mathbb{1}+\bar{g}_{A}^{ee}+\epsilon_{A}^{ee}]_{\alpha\beta}\left(\bar{e}\gamma^{\mu}\gamma^{5}e\right)\left(\bar{\nu}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta}\right)\right\\}.$
(18)

Since we are considering neutrino interaction with a medium (Earth), only
first generation fermions are to be considered, allowing us to omit some
indices. Moreover, $g_{i}^{ff}$ are the SM-coefficients coming from Z-exchange
while $\bar{g}_{i}^{\ell\ell}$ are related to W-exchange. Their explicit forms
are:

$g_{V}^{ff}=T^{3}_{f}-2Q_{f}\sin^{2}\theta_{W},\quad
g_{A}^{ff}=-T^{3}_{f},\quad(\bar{g}_{V}^{\ell\ell})_{\alpha\beta}=-(\bar{g}_{A}^{\ell\ell})_{\alpha\beta}=\delta_{\alpha
e}\delta_{\beta e}\;,$ (19)

where $Q_{f}$ and $T^{3}_{f}$ are the electric charge and weak isospin of
fermion $f$, respectively.

The matching to the LEFT is straightforward

$\displaystyle L_{\begin{subarray}{c}\nu q\\\ \alpha\beta
11\end{subarray}}^{S,LL}$
$\displaystyle=-\frac{1}{v^{2}}\left[(g_{V}^{qq}-g_{A}^{qq})\delta_{\alpha\beta}+(\epsilon_{V}^{qq}-\epsilon_{A}^{qq})_{\alpha\beta}\right]\,,$
(20) $\displaystyle L_{\begin{subarray}{c}\nu q\\\ \alpha\beta
11\end{subarray}}^{S,LR}$
$\displaystyle=-\frac{1}{v^{2}}\left[(g_{V}^{qq}+g_{A}^{qq})\delta_{\alpha\beta}+(\epsilon_{V}^{qq}+\epsilon_{A}^{qq})_{\alpha\beta}\right]\,,$
(21) $\displaystyle L_{\begin{subarray}{c}\nu e\\\ \alpha\beta
11\end{subarray}}^{S,LL}$
$\displaystyle=-\frac{1}{v^{2}}\left[(g_{V}^{ee}-g_{A}^{ee})\delta_{\alpha\beta}+(\bar{g}_{V}^{ee}-\bar{g}_{A}^{ee})_{\alpha\beta}+(\epsilon_{V}^{ee}-\epsilon_{A}^{ee})_{\alpha\beta}\right]\,,$
(22) $\displaystyle L_{\begin{subarray}{c}\nu e\\\ \alpha\beta
11\end{subarray}}^{S,LR}$
$\displaystyle=-\frac{1}{v^{2}}\left[(g_{V}^{ee}+g_{A}^{ee})\delta_{\alpha\beta}+(\bar{g}_{V}^{ee}+\bar{g}_{A}^{ee})_{\alpha\beta}+(\epsilon_{V}^{ee}+\epsilon_{A}^{ee})_{\alpha\beta}\right]\,.$
(23)

Finally, the matching to the SMEFT is given by

$\displaystyle L_{\begin{subarray}{c}\nu e\\\ prst\end{subarray}}^{S,LL}$
$\displaystyle=C_{\begin{subarray}{c}ll\\\
prst\end{subarray}}+C_{\begin{subarray}{c}ll\\\
stpr\end{subarray}}-\frac{{\overline{g}_{2}}^{2}}{2M_{W}^{2}}\left[W_{l}\right]_{pt}\left[W_{l}\right]_{rs}^{*}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{\nu}\right]_{pr}\left[Z_{e_{L}}\right]_{st},$
(24) $\displaystyle L_{\begin{subarray}{c}\nu u\\\ prst\end{subarray}}^{S,LL}$
$\displaystyle=C^{(1)}_{\begin{subarray}{c}lq\\\
prst\end{subarray}}+C^{(3)}_{\begin{subarray}{c}lq\\\
prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{\nu}\right]_{pr}\left[Z_{u_{L}}\right]_{st},$
(25) $\displaystyle L_{\begin{subarray}{c}\nu d\\\ prst\end{subarray}}^{S,LL}$
$\displaystyle=\left(C^{(1)}_{\begin{subarray}{c}lq\\\
prxz\end{subarray}}-C^{(3)}_{\begin{subarray}{c}lq\\\
prxz\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{\nu}\right]_{pr}\left[Z_{d_{L}}\right]_{xz}\right)V_{tz}V_{sx}^{*},$
(26)

$\displaystyle L_{\begin{subarray}{c}\nu e\\\ prst\end{subarray}}^{S,LR}$
$\displaystyle=C_{\begin{subarray}{c}le\\\
prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{\nu}\right]_{pr}\left[Z_{e_{R}}\right]_{st},$
(27) $\displaystyle L_{\begin{subarray}{c}\nu u\\\ prst\end{subarray}}^{S,LR}$
$\displaystyle=C_{\begin{subarray}{c}lu\\\
prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{\nu}\right]_{pr}\left[Z_{u_{R}}\right]_{st},$
(28) $\displaystyle L_{\begin{subarray}{c}\nu d\\\ prst\end{subarray}}^{S,LR}$
$\displaystyle=C_{\begin{subarray}{c}ld\\\
prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{\nu}\right]_{pr}\left[Z_{d_{R}}\right]_{st},$
(29)

where

$\displaystyle[Z_{\nu}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(\frac{1}{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}+\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}\right],$ (30) $\displaystyle[Z_{e_{L}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(-\frac{1}{2}+{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}-\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}\right],$ $\displaystyle[Z_{e_{R}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(+{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}He\\\
pr\end{subarray}}\right],$ (31) $\displaystyle[Z_{u_{L}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(\frac{1}{2}-\frac{2}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}+\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}\right],$ $\displaystyle[Z_{u_{R}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(-\frac{2}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}Hu\\\
pr\end{subarray}}\right],$ (32) $\displaystyle[Z_{d_{L}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(-\frac{1}{2}+\frac{1}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}-\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}\right],$ $\displaystyle[Z_{d_{R}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(+\frac{1}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}Hd\\\
pr\end{subarray}}\right].$ (33)

The connection to NSI in propagation for Earth based long-baseline experiments
is

$\displaystyle\epsilon_{\alpha\beta}=\frac{v^{2}}{2}\Bigg{[}$ $\displaystyle
C^{(1)}_{\begin{subarray}{c}Hl\\\
\alpha\beta\end{subarray}}-C^{(3)}_{\begin{subarray}{c}Hl\\\
\alpha\beta\end{subarray}}+2\delta_{1\alpha}C^{(3)*}_{\begin{subarray}{c}Hl\\\
\beta 1\end{subarray}}+2\delta_{1\beta}C^{(3)}_{\begin{subarray}{c}Hl\\\
\alpha 1\end{subarray}}-C_{\begin{subarray}{c}le\\\ \alpha\beta
11\end{subarray}}-3C_{\begin{subarray}{c}ld\\\ \alpha\beta
11\end{subarray}}-3C_{\begin{subarray}{c}lu\\\ \alpha\beta
11\end{subarray}}-C_{\begin{subarray}{c}ll\\\ \alpha\beta
11\end{subarray}}-C_{\begin{subarray}{c}ll\\\ 11\alpha\beta\end{subarray}}$
$\displaystyle-3\left(C^{(1)}_{\begin{subarray}{c}lq\\\ \alpha\beta
11\end{subarray}}+C^{(3)}_{\begin{subarray}{c}lq\\\ \alpha\beta
11\end{subarray}}+\sum_{x,z}\Big{(}C^{(1)}_{\begin{subarray}{c}lq\\\
\alpha\beta xz\end{subarray}}-C^{(3)}_{\begin{subarray}{c}lq\\\ \alpha\beta
xz\end{subarray}}\Big{)}V_{1,z}V_{1x}^{*}\right)$
$\displaystyle-\delta_{\alpha\beta}\left(3C_{\begin{subarray}{c}Hd\\\
11\end{subarray}}+3C_{\begin{subarray}{c}Hu\\\
11\end{subarray}}+C_{\begin{subarray}{c}He\\\
11\end{subarray}}+C^{(1)}_{\begin{subarray}{c}Hl\\\
11\end{subarray}}+C^{(3)}_{\begin{subarray}{c}Hl\\\ 11\end{subarray}}\right)$
$\displaystyle-3\delta_{\alpha\beta}\left(C^{(1)}_{\begin{subarray}{c}Hq\\\
11\end{subarray}}+C^{(3)}_{\begin{subarray}{c}Hq\\\
11\end{subarray}}+\sum_{x,z}\Big{(}C^{(1)}_{\begin{subarray}{c}Hq\\\
xz\end{subarray}}++C^{(3)}_{\begin{subarray}{c}Hq\\\
xz\end{subarray}}\Big{)}V_{1,z}V_{1x}^{*}\right)\Bigg{]}\,.$ (34)

##  4 NSI probed by long-baseline experiments

In long-baseline experiments, the aim is to measure the conversion rate from a
muon neutrino (antineutrino) to an electron neutrino (antineutrino),
$R_{\mu\beta}$. The beam of muon neutrinos (anti-neutrinos) is generally
generated by pion decay. In the particular case of DUNE, it is expected that
92$\%$ (90.4$\%$) of the beam will correspond to muon neutrinos
(antineutrinos), wrong-sign contamination is as large as 7$\%$ (8.6$\%$) and
the background corresponding to electron neutrino and antineutrino is 1$\%$
(1$\%$) [38]. Thus, by allowing NSI interaction at production, we can probe
the combination, $\overline{\epsilon}_{\mu\sigma}$, where

$[{\cal
P}]_{\mu\sigma}=\delta_{\mu\sigma}+\bar{\epsilon}_{\mu\sigma},\quad\mbox{with}\quad\bar{\epsilon}_{\mu\sigma}=(\epsilon_{L})_{\mu\sigma}-(\epsilon_{R})_{\mu\sigma}-p_{\mu}(\epsilon_{P})_{\mu\sigma},$
(35)

and $p_{\mu}=\frac{m_{\pi^{\pm}}^{2}}{m_{\mu}(m_{u}+m_{d})}\sim 27$. This
stands for 3 free parameters in the real case. For illustration, consider the
connection between $\bar{\epsilon}_{\mu e}$ and the SMEFT operators (for
simplicity, we restrict ourselves to real coefficients and consider the CKM
matrix to be approximately diagonal):

$\displaystyle\bar{\epsilon}_{\mu
e}=\frac{v^{2}}{2}\left[2C^{(3)}_{\begin{subarray}{c}Hl\\\
12\end{subarray}}-2C^{(3)}_{\begin{subarray}{c}lq\\\
1211\end{subarray}}+p_{\mu}\left(C_{\begin{subarray}{c}ledq\\\
1211\end{subarray}}-C^{(3)}_{\begin{subarray}{c}lequ\\\
1211\end{subarray}}\right)\right]\,.$ (36)

In order to have values for $\bar{\epsilon}_{\mu\sigma}$ that can be probed at
DUNE (roughly of order $10^{-2}$), the WC has to be at least of order
$(\unit{TeV})^{-2}$. However, present constraints coming from charged lepton
flavour changing processes, in particular $\mu-e$ conversion in nuclei,
requires that these WC are several orders of magnitude lower [16], precluding
any chance that they can be seen in DUNE. Even more importantly, as we already
stated in the introduction, if an anomaly is seen in this experiment in non-
diagonal channels, it will definitely require new physics to be other than a
weakly coupled theory with masses above the electroweak scale. Even though we
focused on $\bar{\epsilon}_{\mu e}$, which is more heavily constrained, a
similar reasoning applies to $\bar{\epsilon}_{\mu\tau}$. The main message is:
it is extremely difficult to generate sizable non-diagonal NSI in production,
once a connection to the SMEFT is made.

On the other hand, diagonal NSI at production could, in principle, be
generated, with milder constraints (mainly from pion decay measurements).
However, as we show below, the inclusion of indirect effects recently pointed
out in [28], will remove all dependence of NSI at production from the neutrino
transition rate. Therefore, even though they could be sizable, they cannot be
observed in long-baseline experiments such as DUNE. Once the NSI is only-
diagonal, we obtain

$[{\cal P}U]_{\mu l}[U^{\dagger}{\cal
P}^{\dagger}]_{k\mu}=(1+\bar{\epsilon}_{\mu\mu})^{2}U_{\mu
l}U^{\dagger}_{k\mu}\,,$ (37)

while $[{\cal P}{\cal P}^{\dagger}]_{\mu\mu}=(1+\bar{\epsilon}_{\mu\mu})^{2}$.
Thus, by inspection of eq.(6), we see that the standard rate in the three-
neutrino paradigm is obtained. In summary, it is possible to consider the SM
prediction for the production fluxes even if diagonal NSI in production are
generated by a particular UV model.

We conclude by considering NSI in propagation. Recall that long-baseline
experiments actually probe the combination

$\epsilon_{\alpha\beta}=\sum_{f\in\left\\{e,u,d\right\\}}\left\langle\frac{N_{f}(x)}{N_{e}(x)}\right\rangle\left(\epsilon^{ff}_{V}\right)_{\alpha\beta}\,\sim\left(\epsilon^{ee}_{V}\right)_{\alpha\beta}+3\left(\epsilon^{uu}_{V}\right)_{\alpha\beta}+3\left(\epsilon^{dd}_{V}\right)_{\alpha\beta}\,.$
(38)

As before, we illustrate the role played by non-diagonal effects with
$\epsilon_{\mu e}$, which is related to the SMEFT operators below

$\displaystyle\epsilon_{\mu e}=\frac{v^{2}}{2}\Bigg{[}$ $\displaystyle
C^{(1)}_{\begin{subarray}{c}Hl\\\
21\end{subarray}}+C^{(3)}_{\begin{subarray}{c}Hl\\\
21\end{subarray}}-C_{\begin{subarray}{c}le\\\
2111\end{subarray}}-3C_{\begin{subarray}{c}ld\\\
2111\end{subarray}}-3C_{\begin{subarray}{c}lu\\\
2111\end{subarray}}-C_{\begin{subarray}{c}ll\\\
2111\end{subarray}}-C_{\begin{subarray}{c}ll\\\
1121\end{subarray}}-6C^{(1)}_{\begin{subarray}{c}lq\\\
2111\end{subarray}}\Bigg{]}\,.$ (39)

For simplicity, we consider only real coefficients and a diagonal CKM. In
order to have sizable NSI (of order $10^{-2}$), we need that the WC are not
smaller than roughly one $(\unit{TeV})^{-2}$. However, even if one can avoid
the stringent bounds from conversion in nuclei for some the operators above,
there are strong bounds from processes such as $\mu\rightarrow eee$ [16].
Therefore, we arrive at the same conclusion before: it is extremely
challenging to have sizable non-diagonal NSI in propagation, for weakly
coupled theories with masses at the electroweak scale (or above).
Nevertheless, diagonal NSI can still be probed, with sizable values generated
by well-motivated UV models. In the next subsection, we present all the weakly
coupled models that can generate NSI.

###  4.1 Connection to UV model

In [10], a general Lagrangian that maps all renormalizable weakly coupled
models up to spin 1 was presented. Given eqs. (17-3.2), it is straightforward
(yet tedious) to find the UV models that can generate the WC entering in these
equations, as well as, their matching conditions. This task can be automated
making use of the implementation of the tree level dictionary [10] in the
MatchingDB format (see https://gitlab.com/jccriado/matchingdb and Section 3.6
of [39]). In order to (re)check the calculation, the explicit tree-level
matching was performed for all the models relevant for our purposes. Also the
results from the dictionary implemented in MatchingDB were cross-checked
against the results of [10], showing perfect agreement. For completeness, we
have collected in Appendix A the Lagrangian containing all the relevant models
to us, organized in terms of the particle spin. Also, the explicit quantum
number for the fields is given. In Appendix B we present the explicit
connection between the UV models and the NSI.

We collect in table 1 the weakly coupled models that can generate diagonal NSI
at propagation and/or production, while in table 2 we show the explicit
connection between the WC in the WEFT, the WC in the SMEFT and the UV model
that can generate them. In order to make the table more compact, in each line
the UV models depicted can generate at least one of the SMEFT WC presented in
the same line. The only exception is for the vector boson models that can
generate $\epsilon_{V}$, $\epsilon_{A}$, which are all collected in one single
block.

NSI | Scalar | VL-fermion | Vector boson
---|---|---|---
propagation | $\mathcal{S}_{1},\varphi,\Xi_{1},\omega_{1},\Pi_{1},\Pi_{7},\zeta$ | $N,E,\Sigma,\Sigma_{1}$ | $\mathcal{B},\mathcal{W},\mathcal{L}_{1},\mathcal{L}_{3},\mathcal{U}_{2},\mathcal{Q}_{1},\mathcal{Q}_{5},\mathcal{X}$
production | $\varphi,\omega_{1},\Pi_{7},\zeta$ | $N,E,\Sigma,\Sigma_{1}$ | $\mathcal{B}_{1},\mathcal{W},\mathcal{L}_{1},\mathcal{U}_{2},\mathcal{Q}_{5},\mathcal{X}$
Table 1: weakly coupled models that can generate diagonal NSI at propagation and/or production. WC WEFT | SMEFT | Scalar UV | VL-fermion UV | Vector boson UV
---|---|---|---|---
$\epsilon_{L}$ | $\mathcal{O}_{lq}^{(3)}$ | $\omega_{1},\zeta$ | - | $\mathcal{W},\mathcal{U}_{2},\mathcal{X}$
| $\mathcal{O}^{(3)}_{\phi l},\mathcal{O}^{(3)}_{\phi q}$ | - | $N,E,\Sigma,\Sigma_{1},U,D,T_{1},T_{2}$ | $\mathcal{L}_{1}$
$\epsilon_{R}$ | $\mathcal{O}_{\phi ud}$ | - | $Q_{1}$ | $\mathcal{B}_{1}$
$\epsilon_{S}$, $\epsilon_{P}$ | $\mathcal{O}^{(1)}_{lequ},\mathcal{O}_{ledq}$ | $\varphi,\omega_{1},\Pi_{7}$ | - | $\mathcal{L}_{1},\mathcal{U}_{2},\mathcal{Q}_{5}$
$\epsilon_{T}$ | $\mathcal{O}^{(3)}_{lequ}$ | $\omega_{1},\Pi_{7}$ | - | -
| $\mathcal{O}_{ll},\mathcal{O}_{lq}^{(1)},\mathcal{O}_{lq}^{(3)}$ | $\mathcal{S}_{1},\Xi_{1},\omega_{1},\zeta$ | - |
$\epsilon_{V}$, $\epsilon_{A}$ | $\mathcal{O}^{(1)}_{\phi l},\mathcal{O}^{(3)}_{\phi l},\mathcal{O}^{(1)}_{\phi q},\mathcal{O}^{(3)}_{\phi q}$ | - | $N,E,\Sigma,\Sigma_{1},U,D,T_{1},T_{2}$ | $\mathcal{B},\mathcal{W},\mathcal{L}_{1},\mathcal{L}_{3},$
| $\mathcal{O}_{le},\mathcal{O}_{lu},\mathcal{O}_{ld}$ | $\varphi,\Pi_{1},\Pi_{7}$ | - | $\mathcal{U}_{2},\mathcal{Q}_{1},\mathcal{Q}_{5},\mathcal{X}$
| $\mathcal{O}_{\phi e},\mathcal{O}_{\phi u},\mathcal{O}_{\phi d}$ | - | $\Delta_{1},\Delta_{3},Q_{1},Q_{5},Q_{7}$ |
Table 2: Relation among WC in eqs.(3.1),(3.2), the SMEFT operators, and the UV
completion.

In order to have an idea of the impact that each WC can have on the NSI given
current bounds, we present in Fig.1 the maximum value that each WC can attain
according to the global fit performed in [40]. Since
$C^{(1)}_{\begin{subarray}{c}lq\\\ 3311\end{subarray}}$,
$C_{\begin{subarray}{c}ld\\\ 3311\end{subarray}}$,
$C_{\begin{subarray}{c}lu\\\ 3311\end{subarray}}$ were not present, we have
used smelli [41, 42, 43] to obtain the bounds on these coefficients. We have
also checked the results for all the other WC’s with smelli, and kept the most
stringent bounds. We are focusing only on diagonal NC-NSI, as already
explained, implying that only the operators that generate $\epsilon_{V}$ are
relevant. Some comments are in order:

  * •

WC related to quarks and taus can be much larger than
$\sim\mathcal{O}(\mathrm{TeV}^{-2})$. They are going to be constrained by
future experiments, such as DUNE, implying that it is still possible to have
sizable NSI in these channels222Very recently, an analysis performed at CMS
for leptoquarks models coupling the tau and first generation quarks was
performed [44]. It can be translated in bounds on $C_{\begin{subarray}{c}ld\\\
3311\end{subarray}}$ and $C_{\begin{subarray}{c}lu\\\ 3311\end{subarray}}$,
which we take into account into our numerical analysis. They can be generated
by models with scalar leptoquarks ($\omega_{1}$, $\Pi_{1}$, $\Pi_{7}$,
$\zeta$), neutral isosinglet vector bosons ($\mathcal{B}$), or vector
leptoquarks ($\mathcal{U}_{2}$, $\mathcal{Q}_{1}$, $\mathcal{Q}_{5}$,
$\mathcal{X}$).

  * •

With the exception of the tau channels, only $C_{\begin{subarray}{c}ld\\\
2211\end{subarray}}$ can be of order $(\unit{TeV})^{-2}$, inducing NSI of
order $10^{-2}$. It may be generated by models with scalar leptoquarks
($\Pi_{1}$), neutral isosinglet vector bosons ($\mathcal{B}$), or vector
leptoquarks ($\mathcal{Q}_{5}$).

  * •

In single field models, we generally have
$(\epsilon_{V})_{\alpha\beta}\sim\lambda_{\alpha}\lambda_{\beta}$, where
$\lambda$ is the generic coupling of the weakly coupled field under
consideration. Thus, to avoid flavor-violating NSI, only one of the couplings
can be non-zero. To generate sizable NSI in both diagonal directions probed by
long-baseline experiments, one needs to consider at least two fields, where
one only couples to the muon (chosen from the set
$\\{\Pi_{1}\;,\mathcal{B}\;,\mathcal{Q}_{5}\\}$), and the other only couples
to the tau (chosen from the set
$\\{\omega_{1}\;,\Pi_{1}\;,\Pi_{7}\;,\zeta\;,\mathcal{B}\;,\mathcal{U}_{2}\;,\mathcal{Q}_{1}\;,\mathcal{Q}_{5}\;,\mathcal{X}\\}$).

  * •

Models with vector-like fermions are extremely constrained by electroweak
precision data, and a sizable NSI cannot be induced.

Figure 1: Maximum absolute value of the WC that affect diagonal NC-NSI at
90$\%$ C.L. [40, 43]. The bounds were extracted considering only one non-null
WC. The bounds for $C_{\begin{subarray}{c}lq\\\ 3311\end{subarray}}$,
$C_{\begin{subarray}{c}ld\\\ 3311\end{subarray}}$, and
$C_{\begin{subarray}{c}lu\\\ 3311\end{subarray}}$ were multiplied by
$10^{-2}$.

In summary, the most promising combination to induce sizeable diagonal NC-NSI
is

$\displaystyle\epsilon_{ee}-\epsilon_{\mu\mu}=\frac{v^{2}}{2}\left(3C_{\begin{subarray}{c}ld\\\
2211\end{subarray}}\right)\;,\quad\quad\epsilon_{\tau\tau}-\epsilon_{\mu\mu}=\frac{v^{2}}{2}\left(-3C_{\begin{subarray}{c}ld\\\
3311\end{subarray}}-3C_{\begin{subarray}{c}lu\\\
3311\end{subarray}}-6C^{(1)}_{\begin{subarray}{c}lq\\\
3311\end{subarray}}+3C_{\begin{subarray}{c}ld\\\
2211\end{subarray}}\right)\;,$ (40)

where we are considering the CKM matrix to be approximately diagonal.

Regarding the connection between sizeable diagonal NSI and TeV mediators,
models containing scalar leptoquarks $\omega_{1}$, $\Pi_{1}$, $\Pi_{7}$,
$\zeta$ were considered previously in [22] (in their notation, fields $\chi$,
$\Omega$, $\delta$, $\rho$, respectively)333In Ref. [16], the set of
leptoquarks (scalar or vector) that can generate NSI is identified, which
agrees with our results. The explicit matching formulas are, however, not
provided, which prevents one to conclude how sizable the NSI can actually be
in these models.. To the best of our knowledge, models containing extra vector
bosons were considered only for light mediator, for instance in [1], or did
not take into account bounds from charged-lepton violation processes [45].
Finally, leptoquarks may also affect neutrino experiments other than long-
baseline ones [46, 47, 48]. See, for instance, [49, 50, 51, 52, 53, 54], for
recent updates.

##  5 Numerical results

###  5.1 Details of the simulation

To perform our simulation of long-baseline experiments, we will use the GLoBES
package [55, 56], together with MonteCUBES [57] for efficient sampling. In
this work, we are interested in the DUNE experiment for concreteness, however
the same approach could be applied to any other existing (or future)
experiment. In order to ensure that our results can be reproducible, we used
the public GLoBES files supplemented by the DUNE collaboration [58] which are
based on the Technical Design Report configuration [59]. We will be assuming
that the standard three-neutrino paradigm corresponds to our true hypothesis
in our simulation, which renders $\Delta\chi^{2}=\chi^{2}_{\rm
BSM}-\chi^{2}_{\rm SM}$. Moreover, we will consider the mixing angles,
$\delta_{\rm CP}$ and the mass squared difference given by the NuFIT444NuFIT
5.2 (2022), www.nu-fit.org collaboration [60], which we reproduce in table 3.

| Normal Ordering | Inverted Ordering
---|---|---
$\sin^{2}\theta_{12}$ | $0.303^{+0.012}_{-0.011}$ | $0.303^{+0.012}_{-0.011}$
$\sin^{2}\theta_{13}$ | $0.02203^{+0.00056}_{-0.00059}$ | $0.02219^{+0.00060}_{-0.00057}$
$\sin^{2}\theta_{23}$ | $0.572^{+0.018}_{-0.023}$ | $0.578^{+0.016}_{-0.021}$
$\delta_{\rm CP}$/° | $197^{+42}_{-25}$ | $286^{+27}_{-32}$
$\frac{\Delta m^{2}_{21}}{10^{-5}\unit{eV}^{2}}$ | $7.41^{+0.21}_{-0.20}$ | $7.41^{+0.21}_{-0.20}$
$\frac{\Delta m^{2}_{31}}{10^{-3}\unit{eV}^{2}}$ | $2.511^{+0.028}_{-0.027}$ | $-2.498^{+0.032}_{-0.025}$
Table 3: Best-fit values for three-flavor oscillation parameters taken from
[60].

Since long-baseline experiments are not sensitive to the solar mixing
parameters ($\theta_{12}$ and $\Delta m_{21}^{2}$), we will fix them to their
best-fit values. For simplicity, we will also only consider normal ordering.
The detailed results for inverted ordering will be different but the general
conclusions will be similar. When performing our fits, we will consider two
scenarios. The first one, denoted by optimistic, will consider that the
remaining parameters $\theta_{13}$, $\theta_{23}$, $\delta_{\rm CP}$, $\Delta
m_{31}^{2}$ were measured elsewhere, and DUNE will corroborate such values.
This case corresponds to fixing all oscillation parameters to their best-fit
values, which would allow DUNE to set stringent bounds on BSM signals. In the
second scenario, denoted by conservative, the remaining parameters will be
minimized when computing our $\Delta\chi^{2}$, with a prior assigned only to
the reactor mixing angle ($3.7\%$ uncertainty) since it is better constrained
by reactor experiments such as Daya Bay [61] and Reno [62]. In both scenarios,
we will adopt a $5\%$ uncertainty on the average matter density, whose value
of $\rho=2.848\;\unit{g}/\unit{cm}^{3}$ is chosen according to the PREM
profile [63].

###  5.2 Numerical results and current bounds

In this section we present the results of our simulation, following the
methodology described in the last section. As pointed out before, inspired by
weakly coupled theoretical models, only diagonal NSI in propagation can be
probed, which renders two free parameters. In fig.2 we show the 90% C.L.
limits on diagonal NC-NSI, considering the two scenarios: optimistic (left)
and conservative (right). We also present the most-up-to-date constraints [29]
as dashed lines in the plots. These bounds should be interpreted with care,
since we are strictly setting non-diagonal NSI to zero, while in [29] they
were marginalized over. Moreover, since not only oscillation data was
included, the bounds are obtained for individual diagonal NSI rather than
their differences. The lines we present are thus estimated from two-
dimensional exclusion plots on the diagonal NSI. Finally, we checked our own
implementation for NSI in GLoBES against previous results in the literature,
for instance [64], finding compatible results.

Figure 2: Expected sensitivity on diagonal NC-NSI by DUNE, where the red
(blue) region is for 68$\%$ (90$\%$) C.L. The lines represent the current
bounds at 90$\%$ C.L. extracted from the most recent global fit to oscillation
experiments [29].

As can be seen in fig.2, DUNE will provide a better sensitivity in both
directions, for the two scenarios we considered. As a by-product, we also
provide a semi-analytical formula for $\Delta\chi^{2}$ in the optimistic
scenario, in terms of $\epsilon_{ee}-\epsilon_{\mu\mu}$, and
$\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$

$\displaystyle\Delta\chi^{2}|_{\text{opt}}=$ $\displaystyle
115.1\left(\epsilon_{\text{ee}}-\epsilon_{\mu\mu}\right){}^{2}-197.1\left(\epsilon_{\text{ee}}-\epsilon_{\mu\mu}\right)\left(\epsilon_{\tau\tau}-\epsilon_{\mu\mu}\right)-0.1577\left(\epsilon_{\text{ee}}-\epsilon_{\mu\mu}\right)$
$\displaystyle+429.4\left(\epsilon_{\tau\tau}-\epsilon_{\mu\mu}\right){}^{2}+0.8418\left(\epsilon_{\tau\tau}-\epsilon_{\mu\mu}\right)+0.001713.$
(41)

By employing eq.(5.2) and the matching conditions of eq.(3.2), it is possible
to obtain the allowed region for any weakly coupled model of table 1. For
illustration, we consider two examples: models S and V. In model S, we have
two scalar leptoquarks, $\Pi_{1}$, $\omega_{1}$, where they only couple to
first-generation down-type quarks and second/third-generation leptons
respectively
($\big{(}y_{\Pi_{1}}\big{)}_{21},\big{(}y_{\omega_{1}}^{ql}\big{)}_{13}\neq
0$, see eq.(44) ). A similar model was studied by [22] in connection to
radiative neutrino mass generation models. The diagonal NSI induced at tree-
level by this model are

$\displaystyle\epsilon_{ee}-\epsilon_{\mu\mu}=\frac{3v^{2}}{2}\left(-\frac{\left|\big{(}y_{\Pi_{1}}\big{)}_{21}\right|^{2}}{2M^{2}_{\Pi_{1}}}\right)\;,\quad\quad\epsilon_{\tau\tau}-\epsilon_{\mu\mu}=\frac{3v^{2}}{2}\left(-\frac{\left|\big{(}y_{\omega_{1}}^{ql}\big{)}_{13}\right|^{2}}{2M^{2}_{\omega_{1}}}-\frac{\left|\big{(}y_{\Pi_{1}}\big{)}_{21}\right|^{2}}{2M^{2}_{\Pi_{1}}}\right)\;.$
(42)

It is immediate to observe that only negative values can be attained, and for
non-null values of $y_{\Pi_{1}}$ both directions have an absolute lower bound.
We choose as benchmark for the masses $M^{2}_{\Pi_{1}}=1250\;\unit{GeV}$,
$M^{2}_{\omega_{1}}=1000\;\unit{GeV}$ in accordance to the constraints derived
in [22]. For the Yukawas, we varied $\big{(}y_{\Pi_{1}}\big{)}_{21}$ in the
range allowed by perturbative unitarity $y<\sqrt{2\pi}$, while
$\big{(}y_{\omega_{1}}^{ql}\big{)}_{13}$ is bounded to be lower than 0.8 in
accordance with the new result from CMS [44]. The region allowed by present
constraints can be seen as the gray area in the left plot of fig. 3, while in
red (black) we show the 68$\%$ (90$\%$) C.L. allowed region from DUNE for the
model S.

For model V, we consider vector leptoquarks $\mathcal{Q}_{5}$, and
$\mathcal{U}_{2}$, where they only couples to first-generation down-type
quarks and second/third-generation leptons respectively
($\big{(}g_{\mathcal{Q}_{5}}^{dl}\big{)}_{12},\big{(}g_{\mathcal{U}_{2}}^{lq}\big{)}_{31}\neq
0$, see eq.(46)). Clearly it is a simplified model, whose UV completion is
beyond the scope of our work. In this case, the diagonal NSI are given by

$\displaystyle\epsilon_{ee}-\epsilon_{\mu\mu}=\frac{3v^{2}}{2}\left(\frac{\left|\big{(}g_{\mathcal{Q}_{5}}^{dl}\big{)}_{12}\right|^{2}}{M^{2}_{\mathcal{Q}_{5}}}\right)\;,\quad\quad\epsilon_{\tau\tau}-\epsilon_{\mu\mu}=\frac{3v^{2}}{2}\left(\frac{|\big{(}g_{\mathcal{U}_{2}}^{lq}\big{)}_{31}|^{2}}{M^{2}_{\mathcal{U}_{2}}}+\frac{\left|\big{(}g_{\mathcal{Q}_{5}}^{dl}\big{)}_{12}\right|^{2}}{M^{2}_{\mathcal{Q}_{5}}}\right)\;.$
(43)

Contrarily to model S, the NSI can only be positive, while, similarly to
before, there is an lower bound for their values if
$g_{\mathcal{Q}_{5}}^{dl}\neq 0$. Using as benchmark for the masses the values
$M^{2}_{\mathcal{Q}_{5}}=1250\;\unit{GeV}$,
$M^{2}_{\mathcal{U}_{2}}=1000\;\unit{GeV}$, and allowing the gauge couplings
to be as large as unit, we obtain the right plot of fig.3. The colours follow
the same pattern used for model S. Similar analysis can be performed for an
arbitrary number of fields collected in table 1 in a straightforward way. This
feature will be particularly important after data is collected from the DUNE
experiment. At that stage, only the $\chi^{2}$ function given in eq.(5.2) will
have to be updated, and, for instance, exclusion analyses regarding the weakly
coupled models of table 1 will follow automatically. Even more interestingly
is the case where the DUNE experiment reports an anomaly in any of the NSI
diagonal directions. In this scenario, all weakly coupled models that could
potentially explain the anomaly can be easily identified, and their
predictions can be straightforwardly obtained.

Figure 3: Expected sensitivity on diagonal NC-NSI by DUNE, for some
illustrative models containing leptoquarks. The left (right) plot is for the
scalar (vector) case, where the red (black) region is for 68$\%$ (90$\%$) C.L.
The gray area denote the presently allowed region given other constraints.

Finally, one can ask which are the best-possible bounds expected by DUNE on
NSI, for both propagation and production. A comprehensive analysis was
performed in [65], showing the appearance of degenerate solution for
$\theta_{23}$ and $\delta_{\rm CP}$, as well as for the NSI. Nevertheless,
this reference is not only previous to the 2020 update in the three-neutrino
paradigm parameters [60], but it does not consider the updated experimental
setup planned by the DUNE collaboration [38, 59, 58] neither the indirect
effects included recently in the treatment of NSI in production [28].
Therefore, it is justified to perform an updated analysis, which we discuss on
the remaining of this section.

We emphasize, once again, that this scenario does not represent any viable
weakly coupled theory with masses at the weak scale or above. However, from
the point of view of model-building, one could consider other alternatives
with light fields or extra symmetry groups. In this case, an immediate
question is how large a given NSI has to be in order to be probed by DUNE. The
best-case scenario stands for the optimistic scenario, where all the three-
neutrino standard paradigm parameters were measured with high precision
elsewhere, and none deviation was found. We emphasize that this approach is
different from the one adopted in [65] in a fundamental way, since our aim is
to provide an easy access to the capabilities of DUNE to constrain generic
models in the best-case scenario, rather than precisely derive the bounds on
NSI that are expected by the DUNE experiment.

In order to make the global fit feasible, we will adopt the optimistic
scenario which in practice corresponds to fix all the three-neutrino standard
paradigm parameters to their best-fit values. Regarding the NSI parameters, we
have now 11 free parameters, five from propagation and six from production (we
restrict ourselves to real coefficients). As already explained, if only
diagonal CC-NSI are included, they will cancel in the neutrino conversion
rate. This constitutes a flat direction in our plot. In order to break this
direction, we could add data from other experiments that aim to measure the
rate $\Gamma_{\pi\to e\nu}/\Gamma_{\pi\to\mu\nu}$, for instance. This
approach, however, requires that the likelihood is obtained independently from
GLoBES, with dedicated routines. Another possibility is to include priors on
all CC-NSI in the GLoBES engine, penalizing values that do not comply with the
experimental rate $\Gamma_{\pi\to e\nu}/\Gamma_{\pi\to\mu\nu}$ at 90$\%$ C.L.
We have opted for the latter approach, so the simulation to the DUNE
experiment already implemented in GLoBES can be used. Since our goal is to
obtain the best-case scenario for DUNE, we have adopted the most restrictive
possibility when implementing the constraints. It stands to consider only one
non-null CC-NSI at time, in similarity to the approach performed in [17]. We
will show these bounds as lines in the plot of fig.5.

After marginalization of the CC-NSI parameters, we present in fig.4, the
sensitivity for NC-NSI from DUNE.

Figure 4: Expected sensitivity on NC-NSI by DUNE, where the red (blue) region
is for 68$\%$ (90$\%$) C.L. The lines represent the current bounds at 90$\%$
C.L. extracted from the most recent global fit to oscillation experiments
[29].

As can be seen in the figure, the sensitivity on the diagonal NSI decreases
when compared to our previous analysis of fig. 2. This is expected since the
minimum chi squared in both cases were similar, however, non-diagonal NSI were
set to zero before, while they are marginalized over now. Moreover, the most
stringent bounds are expected for $\epsilon_{e\mu}$, and $\epsilon_{\mu\tau}$,
showing that NSI smaller than $10^{-2}$ cannot be probed by DUNE. Therefore,
since in weakly coupled theories the non-diagonal elements are already
constrained to be several orders of magnitude smaller [16], any possible
anomaly seen in these channels will require a new theoretical framework.

Finally, in fig.5 we present the sensitivity on CC-NSI after marginalization
of the NC-NSI parameters.

Figure 5: Expected sensitivity on CC-NSI by DUNE, where the red (blue) region
is for 68$\%$ (90$\%$) C.L. The lines represent the current bounds at 90$\%$
C.L. coming from $\Gamma_{\pi\to e\nu}/\Gamma_{\pi\to\mu\nu}$.

We recall that we have adopted priors coming from the experimental measurement
of the pion decay, which are quite stringent. Therefore, for most of the
parameters, DUNE, even in the best-case scenario, will not be able to improve
the bounds. The only exemption is for the last plot, considering the
parameters $\bar{\epsilon}_{\mu e}$, $\bar{\epsilon}_{\mu\tau}$. In these
channels, DUNE could (potentially) report an anomaly. This is reasonable since
these parameters will modify the neutrino fluxes for muon neutrino conversion,
which is the main channel for neutrino production at DUNE.

##  6 Summary

There is at present a very active experimental program in neutrino physics,
with many planned experiments in the near future. Their main goal is to
measure with precision three ($\theta_{23}$, $\delta_{CP}$, $\Delta
m^{2}_{31}$) of the six parameter of the three-neutrino standard paradigm. It
cannot be dismissed, however, that the standard scenario is incomplete, since
it does not introduce, per se, a mechanism for neutrino mass generation. The
simplest proposal, seesaw type I, is realized with very heavy sterile
neutrinos, implying that no deviation from the standard paradigm should be
seen in future experiments. If, however, the next-generation long-baseline
experiments report an anomaly, it may open a window not only to the neutrino
mass generation mechanism realized in Nature, but potentially to the presence
of new weakly coupled states.

In this work we aimed to identify all the weakly coupled states up to spin 1
that could potentially be probed by long-baseline neutrino experiments, in
particular DUNE. This was achieved by employing an effective field theory
framework, where the new states are assumed to be heavier than the electroweak
scale (SMEFT). We rederived the matching between neutrino non-standard
interactions (NSI) coefficients (defined at low-energy) and the WC in the
SMEFT at tree-level [16], as well as the matching from the SMEFT to the
relevant weakly coupled theories. The final formulas are organized in appendix
B, which can be employed not only for long-baseline neutrino experiments, but
for all cases that the description by NSI parameters is applicable. In
particular, for long-baseline experiments, only a small number of weakly
coupled fields can induce sizable NSI parameters, in view of present
constrains from collider, electroweak precision data and the non-observance of
charged lepton flavor violation. The latter strongly suppress non-diagonal
NSI, as noted long ago [14]. For diagonal NSI, we found that only models
containing leptoquarks (scalar or vector) or an extra neutral gauge boson are
relevant at tree-level matching. We illustrated the potential from DUNE to
unveil some of these models. For ease of the reader the full chain for the
matching, as well as the chi squared function in terms of SMEFT WC is provided
in https://gitlab.com/alcherchiglia/eft-neutrinos. Therefore, any combination
of fields can be studied, in similarity to the analysis that lead to figure 3.

Finally, by departing from the weakly coupled framework, we also performed a
global fit to assess DUNE sensitivity to NSI coming from propagation and
production simultaneously. Our approach is different from the usual one
adopted in the literature, since our main aim is to provide model builders
with easy access to the strongest bound that DUNE can set on each of the NSI
parameters. Our results can be seen in figs.4-5, where we assumed that the
three-neutrino standard paradigm parameters are going to be measured elsewhere
with good precision. In this best-case scenario, DUNE could potentially report
anomalies in non-diagonal NSI channels, which would necessarily indicate that
new physics is light (below the electroweak scale) and, potentially, the gauge
group of the SM need to be extended.

## Acknowledgments

We gratefully acknowledge Jorge Blas for valuable comments regarding global
fits, Juan Carlos Criado for providing the library of UV models, Pablo Olgoso
for his help with smelli and flavio, Pedro Pasquini for his help with GLoBES
codes, and Bruno Zamorano García for an enlightening discussion about the DUNE
experiment. A.C. acknowledges support from National Council for Scientific and
Technological Development – CNPq through projects 166523/2020-8 and
201013/2022-3, and FAPESP through Grant 2022/08954-2. This work has been
partially supported by MCIN/AEI (10.13039/501100011033) and ERDF under grants
PID2019-106087GB-C22 and PID2022-139466NB-C21 and by the Junta de Andalucía
grant FQM 101.

##  Appendix A Lagrangian

For convenience of the reader, we reproduce below the parts of the Lagrangian
defined in [10] relevant for our work. We also present the representation
under the SM gauge group for each of the fields. The complete set of
conventions used can be found in the original reference.

###  A.1 New Scalars

In table 4 we collect the new scalars that may contribute to NSI while in
eq.(44) we present their interacting Lagrangian.

Name | ${\cal S}_{1}$ | $\varphi$ | $\Xi_{1}$ | ${\omega}_{1}$ | $\Pi_{1}$ | $\Pi_{7}$ | $\zeta$ |
---|---|---|---|---|---|---|---|---
Irrep | $\left(1,1\right)_{1}$ | $\left(1,2\right)_{\frac{1}{2}}$ | $\left(1,3\right)_{1}$ | $\left(3,1\right)_{-\frac{1}{3}}$ | $\left(3,2\right)_{\frac{1}{6}}$ | $\left(3,2\right)_{\frac{7}{6}}$ | $\left(3,3\right)_{-\frac{1}{3}}$ |
Table 4: New scalars and their irrepresentation under the gauge group of the
SM $(\rm{SU}(3),\rm{SU}(2))_{\rm{U}(1)}$.

$\displaystyle-\mathcal{L}_{\mathrm{S}}=$
$\displaystyle\;+\left\\{(y_{\mathcal{S}_{1}})_{rij}\mathcal{S}_{1r}^{\dagger}\bar{l}_{Li}i\sigma_{2}l^{c}_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(y^{e}_{\varphi})_{rij}\varphi_{r}^{\dagger}\bar{e}_{Ri}l_{Lj}+(y^{d}_{\varphi})_{rij}\varphi_{r}^{\dagger}\bar{d}_{Ri}q_{Lj}+(y^{u}_{\varphi})_{rij}\varphi_{r}^{\dagger}i\sigma_{2}\bar{q}^{T}_{Li}u_{Rj}\right.$
$\displaystyle\quad\left.+(\lambda_{\varphi})_{r}\left(\varphi^{\dagger}_{r}\phi\right)\left(\phi^{\dagger}\phi\right)+\mathrm{h.c.}\right\\}$
$\displaystyle+\frac{1}{2}(\lambda_{\Xi_{1}})_{rs}\left(\Xi^{a\dagger}_{1r}\Xi^{a}_{1s}\right)\left(\phi^{\dagger}\phi\right)+\frac{1}{2}(\lambda^{\prime}_{\Xi_{1}})_{rs}f_{abc}\left(\Xi_{1r}^{a\dagger}\Xi_{1s}^{b}\right)\left(\phi^{\dagger}\sigma^{c}\phi\right)$
$\displaystyle+\left\\{(y_{\Xi_{1}})_{rij}\Xi^{a\dagger}_{1r}\bar{l}_{Li}\sigma^{a}i\sigma_{2}l^{c}_{Lj}+(\kappa_{\Xi_{1}})_{r}\Xi_{1r}^{a\dagger}\left(\tilde{\phi}^{\dagger}\sigma^{a}\phi\right)+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(y^{ql}_{\omega_{1}})_{rij}\omega_{1r}^{\dagger}\bar{q}^{c}_{Li}i\sigma_{2}l_{Lj}+(y^{qq}_{\omega_{1}})_{rij}\omega_{1r}^{A\dagger}\epsilon_{ABC}\bar{q}^{B}_{Li}i\sigma_{2}q^{c\,C}_{Lj}\right.$
$\displaystyle\quad\left.+(y^{eu}_{\omega_{1}})_{rij}\omega_{1r}^{\dagger}\bar{e}^{c}_{Ri}u_{Rj}+(y^{du}_{\omega_{1}})_{rij}\omega_{1r}^{A\dagger}\epsilon_{ABC}\bar{d}^{B}_{Ri}u^{c\,C}_{Rj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(y_{\Pi_{1}})_{rij}\Pi_{1r}^{\dagger}i\sigma_{2}\bar{l}^{T}_{Li}d_{Rj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(y^{lu}_{\Pi_{7}})_{rij}\Pi_{7r}^{\dagger}i\sigma_{2}\bar{l}^{T}_{Li}u_{Rj}+(y^{eq}_{\Pi_{7}})_{rij}\Pi_{7r}^{\dagger}\bar{e}_{Ri}q_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(y^{ql}_{\zeta})_{rij}\zeta_{r}^{a\dagger}\bar{q}^{c}_{Li}i\sigma_{2}\sigma^{a}l_{Lj}+(y^{qq}_{\zeta})_{rij}\zeta^{a\dagger}_{r}\epsilon_{ABC}\bar{q}^{B}_{Li}\sigma^{a}i\sigma_{2}q^{c\,C}_{Lj}+\mathrm{h.c.}\right\\}.$
(44)

###  A.2 New Fermions

In table 5 we collect the new vector-like fermions that may contribute to NSI
while in eq.(45) we present their interacting Lagrangian.

Name | $N$ | $E$ | $\Delta_{1}$ | $\Delta_{3}$ | $\Sigma$ | $\Sigma_{1}$ |
---|---|---|---|---|---|---|---
Irrep | $\left(1,1\right)_{0}$ | $\left(1,1\right)_{-1}$ | $\left(1,2\right)_{-\frac{1}{2}}$ | $\left(1,2\right)_{-\frac{3}{2}}$ | $\left(1,3\right)_{0}$ | $\left(1,3\right)_{-1}$ |
|  |  |  |  |  |  |
Name | $U$ | $D$ | $Q_{1}$ | $Q_{5}$ | $Q_{7}$ | $T_{1}$ | $T_{2}$
Irrep | $\left(3,1\right)_{\frac{2}{3}}$ | $\left(3,1\right)_{-\frac{1}{3}}$ | $\left(3,2\right)_{\frac{1}{6}}$ | $\left(3,2\right)_{-\frac{5}{6}}$ | $\left(3,2\right)_{\frac{7}{6}}$ | $\left(3,3\right)_{-\frac{1}{3}}$ | $\left(3,3\right)_{\frac{2}{3}}$
Table 5: New vector-like fermions and their irrepresentation under the gauge
group of the SM $(\rm{SU}(3),\rm{SU}(2))_{\rm{U}(1)}$.

$\displaystyle-\mathcal{L}_{\mathrm{fermions}}=$
$\displaystyle\;(\lambda_{N})_{ri}\bar{N}_{Rr}\tilde{\phi}^{\dagger}l_{Li}+(\lambda_{E})_{ri}\bar{E}_{Rr}\phi^{\dagger}l_{Li}+(\lambda_{\Delta_{1}})_{ri}\bar{\Delta}_{1Lr}\phi
e_{Ri}+(\lambda_{\Delta_{3}})_{ri}\bar{\Delta}_{3Lr}\tilde{\phi}e_{Ri}$
$\displaystyle+\frac{1}{2}(\lambda_{\Sigma})_{ri}\bar{\Sigma}^{a}_{Rr}\tilde{\phi}^{\dagger}\sigma^{a}l_{Li}+\frac{1}{2}(\lambda_{\Sigma_{1}})_{ri}\bar{\Sigma}^{a}_{1Rr}\phi^{\dagger}\sigma^{a}l_{Li}+(\lambda_{U})_{ri}\bar{U}_{Rr}\tilde{\phi}^{\dagger}q_{Li}+(\lambda_{D})_{ri}\bar{D}_{Rr}\phi^{\dagger}q_{Li}$
$\displaystyle+(\lambda^{u}_{Q_{1}})_{ri}\bar{Q}_{1Lr}\tilde{\phi}u_{Ri}+(\lambda^{d}_{Q_{1}})_{ri}\bar{Q}_{1Lr}\phi
d_{Ri}+(\lambda_{Q_{5}})_{ri}\bar{Q}_{5Lr}\tilde{\phi}d_{Ri}+(\lambda_{Q_{7}})_{ri}\bar{Q}_{7Lr}\phi
u_{Rj}$
$\displaystyle+\frac{1}{2}(\lambda_{T_{1}})_{ri}\bar{T}^{a}_{1Rr}\phi^{\dagger}\sigma^{a}q_{Li}+\frac{1}{2}(\lambda_{T_{2}})_{ri}\bar{T}^{a}_{2Rr}\tilde{\phi}^{\dagger}\sigma^{a}q_{Li}+\mathrm{h.c.}~{}.$
(45)

###  A.3 New Vectors

In table 6 we collect the new vector bosons that may contribute to NSI while
in eq.(46) we present their interacting Lagrangian.

Name | ${\cal B}$ | ${\cal B}_{1}$ | ${\cal W}$ | ${\cal L}_{1}$ | ${\cal L}_{3}$ | ${\cal U}_{2}$ | ${\cal Q}_{1}$ | ${\cal Q}_{5}$ | ${\cal X}$
---|---|---|---|---|---|---|---|---|---
Irrep | $\left(1,1\right)_{0}$ | $\left(1,1\right)_{1}$ | $\left(1,3\right)_{0}$ | $\left(1,2\right)_{\frac{1}{2}}$ | $\left(1,2\right)_{-\frac{3}{2}}$ | $\left(3,1\right)_{\frac{2}{3}}$ | $\left(3,2\right)_{\frac{1}{6}}$ | $\left(3,2\right)_{-\frac{5}{6}}$ | $\left(3,3\right)_{\frac{2}{3}}$
Table 6: New vector bosons and their irrepresentation under the gauge group of
the SM $(\rm{SU}(3),\rm{SU}(2))_{\rm{U}(1)}$.

$\displaystyle-\mathcal{L}_{\mathrm{V}}=$
$\displaystyle\;(g^{l}_{\mathcal{B}})_{rij}\mathcal{B}^{\mu}_{r}\bar{l}_{Li}\gamma_{\mu}l_{Lj}+(g^{q}_{\mathcal{B}})_{rij}\mathcal{B}^{\mu}_{r}\bar{q}_{Li}\gamma_{\mu}q_{Lj}+(g^{e}_{\mathcal{B}})_{rij}\mathcal{B}^{\mu}_{r}\bar{e}_{Li}\gamma_{\mu}e_{Lj}$
$\displaystyle+(g^{d}_{\mathcal{B}})_{rij}\mathcal{B}^{\mu}_{r}\bar{d}_{Li}\gamma_{\mu}d_{Lj}+(g^{u}_{\mathcal{B}})_{rij}\mathcal{B}^{\mu}_{r}\bar{u}_{Li}\gamma_{\mu}u_{Lj}+\left\\{(g^{\phi}_{\mathcal{B}})_{r}\mathcal{B}^{\mu}_{r}\phi^{\dagger}iD_{\mu}\phi+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(g^{du}_{\mathcal{B}_{1}})_{rij}\mathcal{B}^{\mu\dagger}_{1r}\bar{d}_{Ri}\gamma_{\mu}u_{Rj}+(g^{\phi}_{\mathcal{B}_{1}})_{r}\mathcal{B}^{\mu\dagger}_{1r}iD_{\mu}\phi^{T}i\sigma_{2}\phi+\mathrm{h.c.}\right\\}$
$\displaystyle+\frac{1}{2}(g^{l}_{\mathcal{W}})_{rij}\mathcal{W}^{\mu
a}_{r}\bar{l}_{Li}\sigma^{a}\gamma_{\mu}l_{Lj}+\frac{1}{2}(g^{q}_{\mathcal{W}})_{rij}\mathcal{W}^{\mu
a}_{r}\bar{q}_{Li}\sigma^{a}\gamma_{\mu}q_{Lj}$
$\displaystyle+\left\\{\frac{1}{2}(g^{\phi}_{\mathcal{W}})_{r}\mathcal{W}^{\mu
a}_{r}\phi^{\dagger}\sigma^{a}iD_{\mu}\phi+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(\gamma_{\mathcal{L}_{1}})_{r}\mathcal{L}_{1r\mu}^{\dagger}D^{\mu}\phi+\mathrm{h.c.}\right\\}$
$\displaystyle+i(g^{B}_{\mathcal{L}_{1}})_{rs}\mathcal{L}_{1r\mu}^{\dagger}\mathcal{L}_{1s\nu}B^{\mu\nu}+i(g^{W}_{\mathcal{L}_{1}})_{rs}\mathcal{L}_{1i\mu}^{\dagger}\sigma^{a}\mathcal{L}_{1j\nu}W^{a\,\mu\nu}$
$\displaystyle+i(g^{\tilde{B}}_{\mathcal{L}_{1}})_{rs}\mathcal{L}_{1r\mu}^{\dagger}\mathcal{L}_{1s\nu}\tilde{B}^{\mu\nu}+i(g^{\tilde{W}}_{\mathcal{L}_{1}})_{rs}\mathcal{L}_{1r\mu}^{\dagger}\sigma^{a}\mathcal{L}_{1s\nu}\tilde{W}^{a\,\mu\nu}$
$\displaystyle+(h^{(1)}_{\mathcal{L}_{1}})_{rs}\left(\mathcal{L}_{1r\mu}^{\dagger}\mathcal{L}^{\mu}_{1s}\right)\left(\phi^{\dagger}\phi\right)+(h^{(2)}_{\mathcal{L}_{1}})_{rs}\left(\mathcal{L}_{1r\mu}^{\dagger}\phi\right)\left(\phi^{\dagger}\mathcal{L}^{\mu}_{1s}\right)$
$\displaystyle+\left\\{(h^{(3)}_{\mathcal{L}_{1}})_{rs}\left(\mathcal{L}_{1r\mu}^{1\dagger}\phi\right)\left(\mathcal{L}^{\dagger\mu}_{1s}\phi\right)+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(g_{\mathcal{L}_{3}})_{rij}\mathcal{L}^{\mu\dagger}_{3r}\bar{e}^{c}_{Ri}\gamma_{\mu}l_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(g^{ed}_{\mathcal{U}_{2}})_{rij}\mathcal{U}^{\mu\dagger}_{2r}\bar{e}_{Ri}\gamma_{\mu}d_{Rj}+(g^{lq}_{\mathcal{U}_{2}})_{rij}\mathcal{U}^{\mu\dagger}_{2r}\bar{l}_{Li}\gamma_{\mu}q_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(g^{ul}_{\mathcal{Q}_{1}})_{rij}\mathcal{Q}^{\mu\dagger}_{1r}\bar{u}^{c}_{Ri}\gamma_{\mu}l_{Lj}+(g^{dq}_{\mathcal{Q}_{1}})_{rij}\mathcal{Q}^{A\mu\dagger}_{1r}\epsilon_{ABC}\bar{d}^{B}_{Ri}\gamma_{\mu}i\sigma_{2}q^{c\,C}_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{(g^{dl}_{\mathcal{Q}_{5}})_{rij}\mathcal{Q}^{\mu\dagger}_{5r}\bar{d}^{c}_{Ri}\gamma_{\mu}l_{Lj}+(g^{eq}_{\mathcal{Q}_{5}})_{rij}\mathcal{Q}^{\mu\dagger}_{5r}\bar{e}^{c}_{Ri}\gamma_{\mu}q_{Lj}\right.$
$\displaystyle\quad\left.+(g^{uq}_{\mathcal{Q}_{5}})_{rij}\mathcal{Q}^{A\mu\dagger}_{5r}\epsilon_{ABC}\bar{u}^{B}_{Ri}\gamma_{\mu}q^{c\,C}_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{\frac{1}{2}(g_{\mathcal{X}})_{rij}\mathcal{X}^{a\mu\dagger}_{r}\bar{l}_{Li}\gamma_{\mu}\sigma^{a}q_{Lj}+\mathrm{h.c.}\right\\}$
$\displaystyle+\Big{\\{}\frac{1}{f}\mathcal{L}^{\mu\dagger}_{1r}\bigg{[}(\tilde{\gamma}^{(1)}_{\mathcal{L}_{1}})_{r}\left(\phi^{\dagger}D_{\mu}\phi\right)\phi+(\tilde{\gamma}^{(2)}_{\mathcal{L}_{1}})_{r}\left(D_{\mu}\phi^{\dagger}\phi\right)\phi+(\tilde{\gamma}^{(3)}_{\mathcal{L}_{1}})_{r}\left(\phi^{\dagger}\phi\right)D_{\mu}\phi$
$\displaystyle\quad\qquad\qquad+(\tilde{\gamma}^{B}_{\mathcal{L}_{1}})_{r}B_{\mu\nu}D^{\nu}\phi+(\tilde{\gamma}^{\tilde{B}}_{\mathcal{L}_{1}})_{r}\tilde{B}_{\mu\nu}D^{\nu}\phi$
$\displaystyle\quad\qquad\qquad+(\tilde{\gamma}^{W}_{\mathcal{L}_{1}})_{r}W^{a}_{\mu\nu}\sigma^{a}D^{\nu}\phi+(\tilde{\gamma}^{\tilde{W}}_{\mathcal{L}_{1}})_{r}\tilde{W}^{a}_{\mu\nu}\sigma^{a}D^{\nu}\phi$
$\displaystyle\quad\qquad\qquad+(\tilde{g}^{eDl}_{\mathcal{L}_{1}})_{rij}\bar{e}_{Ri}D_{\mu}l_{Lj}+(\tilde{g}^{Del}_{\mathcal{L}_{1}})_{rij}D_{\mu}\bar{e}_{Ri}l_{Lj}+(\tilde{g}^{dDq}_{\mathcal{L}_{1}})_{rij}\bar{d}_{Ri}D_{\mu}q_{Lj}$
$\displaystyle\quad\qquad\qquad+(\tilde{g}^{Ddq}_{\mathcal{L}_{1}})_{rij}D_{\mu}\bar{d}_{Ri}q_{Lj}+(\tilde{g}^{qDu}_{\mathcal{L}_{1}})_{rij}i\sigma_{2}\bar{q}^{T}_{Li}D_{\mu}u_{Rj}+(\tilde{g}^{Dqu}_{\mathcal{L}_{1}})_{rij}i\sigma_{2}D_{\mu}\bar{q}^{T}_{Li}u_{Rj}$
$\displaystyle\quad\qquad\qquad+(\tilde{g}^{du}_{\mathcal{L}_{1}})_{rij}\tilde{\phi}\bar{d}_{Ri}\gamma_{\mu}u_{Rj}+(\tilde{g}^{e}_{\mathcal{L}_{1}})_{rij}\phi\bar{e}_{Ri}\gamma_{\mu}e_{Rj}+(\tilde{g}^{d}_{\mathcal{L}_{1}})_{rij}\phi\bar{d}_{Ri}\gamma_{\mu}d_{Rj}$
$\displaystyle\quad\qquad\qquad+(\tilde{g}^{u}_{\mathcal{L}_{1}})_{rij}\phi\bar{u}_{Ri}\gamma_{\mu}u_{Rj}+(\tilde{g}^{l}_{\mathcal{L}_{1}})_{rij}\phi\bar{l}_{Ri}\gamma_{\mu}l_{Lj}+(\tilde{g}^{l\prime}_{\mathcal{L}_{1}})_{rij}\left(\sigma^{a}\phi\right)\left(\bar{l}_{Li}\gamma_{\mu}\sigma^{a}l_{Lj}\right)$
$\displaystyle\quad\qquad\qquad+(\tilde{g}^{q}_{\mathcal{L}_{1}})_{rij}\phi\bar{q}_{Li}\gamma_{\mu}q_{Lj}+(\tilde{g}^{q\prime}_{\mathcal{L}_{1}})_{rij}\left(\sigma^{a}\phi\right)\left(\bar{q}_{Li}\gamma_{\mu}\sigma^{a}q_{Lj}\right)\bigg{]}+\mathrm{h.c.}~{}\\!\Big{\\}}.$
(46)

##  Appendix B Explicit matching results

We provide in the following explicit matching results for the UV models
identified in table 2.

###  B.1 Scalar models

$\varphi$ model
---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{S}^{kj})^{*}_{\alpha\beta}$ | $-\frac{1}{M_{\varphi}^{2}}\left[(y^{d}_{\varphi})_{jk}+\sum_{x}V_{xj}^{*}(y^{u}_{\varphi})_{xk}\right](y^{e}_{\varphi})^{*}_{\alpha\beta}$
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{P}^{kj})^{*}_{\alpha\beta}$ | $-\frac{1}{M_{\varphi}^{2}}\left[(y^{d}_{\varphi})_{jk}-\sum_{x}V_{xj}^{*}(y^{u}_{\varphi})_{xk}\right](y^{e}_{\varphi})^{*}_{\alpha\beta}$
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $\frac{(y^{e}_{\varphi})^{*}_{1\alpha}(y^{e}_{\varphi})_{1\beta}}{2M_{\varphi}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $\frac{(y^{e}_{\varphi})^{*}_{1\alpha}(y^{e}_{\varphi})_{1\beta}}{2M_{\varphi}^{2}}$
Table 7: Matching to $\varphi$ model. |  $\omega_{1}$ model |  $\Pi_{7}$ model
---|---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $\frac{\sum_{x}V_{xj}^{*}(y^{ql}_{\omega_{1}})_{x\beta}^{*}(y^{ql}_{\omega_{1}})_{k\alpha}}{2M_{\omega_{1}}^{2}}$ | -
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{S}^{kj})^{*}_{\alpha\beta}$ | $-\frac{\sum_{x}V_{xj}^{*}(y^{eu}_{\omega_{1}})_{\alpha k}(y^{ql}_{\omega_{1}})_{x\beta}^{*}}{2M_{\omega_{1}}^{2}}$ | $-\frac{\sum_{x}V_{xj}^{*}(y^{eq}_{\Pi_{7}})^{*}_{\alpha x}(y^{lu}_{\Pi_{7}})_{\beta k}}{2M_{\Pi_{7}}^{2}}$
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{P}^{kj})^{*}_{\alpha\beta}$ | $\frac{\sum_{x}V_{xj}^{*}(y^{eu}_{\omega_{1}})_{\alpha k}(y^{ql}_{\omega_{1}})_{x\beta}^{*}}{2M_{\omega_{1}}^{2}}$ | $\frac{\sum_{x}V_{xj}^{*}(y^{eq}_{\Pi_{7}})^{*}_{\alpha x}(y^{lu}_{\Pi_{7}})_{\beta k}}{2M_{\Pi_{7}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | - | $\frac{(y^{lu}_{\Pi_{7}})_{\beta 1}^{*}(y^{lu}_{\Pi_{7}})_{\alpha 1}}{2M_{\Pi_{7}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | - | $\frac{(y^{lu}_{\Pi_{7}})_{\beta 1}^{*}(y^{lu}_{\Pi_{7}})_{\alpha 1}}{2M_{\Pi_{7}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $-\frac{\sum_{x,z}V_{1z}V_{1x}^{*}(y^{ql}_{\omega_{1}})_{x\alpha}^{*}(y^{ql}_{\omega_{1}})_{z\beta}}{2M_{\omega_{1}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $\frac{\sum_{x,z}V_{1z}V_{1x}^{*}(y^{ql}_{\omega_{1}})_{x\alpha}^{*}(y^{ql}_{\omega_{1}})_{z\beta}}{2M_{\omega_{1}}^{2}}$ | -
Table 8: Matching to $\omega_{1}$, and $\Pi_{7}$ models. |  $\mathcal{S}_{1}$ model |  $\Xi_{1}$ model |  $\Pi_{1}$ model
---|---|---|---
WEFT WC | Matching
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $-\frac{(y_{\mathcal{S}_{1}})_{\beta 1}^{*}(y_{\mathcal{S}_{1}})_{\alpha 1}}{M_{\mathcal{S}_{1}}^{2}}-\frac{(y_{\mathcal{S}_{1}})_{1\beta}^{*}(y_{\mathcal{S}_{1}})_{1\alpha}}{M_{\mathcal{S}_{1}}^{2}}$ | $-\frac{(y_{\Xi_{1}})_{\beta 1}^{*}(y_{\Xi_{1}})_{\alpha 1}}{M_{\Xi_{1}}^{2}}-\frac{(y_{\Xi_{1}})_{1\beta}^{*}(y_{\Xi_{1}})_{1\alpha}}{M_{\Xi_{1}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $\frac{(y_{\mathcal{S}_{1}})_{\beta 1}^{*}(y_{\mathcal{S}_{1}})_{\alpha 1}}{M_{\mathcal{S}_{1}}^{2}}+\frac{(y_{\mathcal{S}_{1}})_{1\beta}^{*}(y_{\mathcal{S}_{1}})_{1\alpha}}{M_{\mathcal{S}_{1}}^{2}}$ | $\frac{(y_{\Xi_{1}})_{\beta 1}^{*}(y_{\Xi_{1}})_{\alpha 1}}{M_{\Xi_{1}}^{2}}+\frac{(y_{\Xi_{1}})_{1\beta}^{*}(y_{\Xi_{1}})_{1\alpha}}{M_{\Xi_{1}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | - | - | $\frac{(y_{\Pi_{1}})_{\beta 1}^{*}(y_{\Pi_{1}})_{\alpha 1}}{2M_{\Pi_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | - | - | $\frac{(y_{\Pi_{1}})_{\beta 1}^{*}(y_{\Pi_{1}})_{\alpha 1}}{2M_{\Pi_{1}}^{2}}$
Table 9: Matching to $\mathcal{S}_{1}$, $\Xi_{1}$, and $\Pi_{1}$ models. |  $\zeta$ model
---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $\frac{\sum_{x}V_{xj}^{*}(y^{ql}_{\zeta})_{x\beta}^{*}(y^{ql}_{\zeta})_{k\alpha}}{M_{\zeta}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $-\frac{\sum_{x,z}V_{1z}V_{1x}^{*}(y^{ql}_{\zeta})_{x\alpha}^{*}(y^{ql}_{\zeta})_{z\beta}}{2M_{\Xi_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $\frac{\sum_{x,z}V_{1z}V_{1x}^{*}(y^{ql}_{\zeta})_{x\alpha}^{*}(y^{ql}_{\zeta})_{z\beta}}{2M_{\Xi_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $-\frac{(y^{ql}_{\zeta})_{1\alpha}^{*}(y^{ql}_{\zeta})_{1\beta}}{M_{\zeta}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $\frac{(y^{ql}_{\zeta})_{1\alpha}^{*}(y^{ql}_{\zeta})_{1\beta}}{M_{\zeta}^{2}}$
Table 10: Matching to $\zeta$ model.

###  B.2 Vector-like models

|  $N$ model
---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $-\frac{(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}V_{kj}^{*}}{2M_{N}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $-\frac{(\lambda_{N})_{1}^{*}\delta_{1\alpha}(\lambda_{N})_{\beta}+(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{1}\delta_{1\beta}-(1-4s^{2})(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}}{2M_{N}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $\frac{(\lambda_{N})_{1}^{*}\delta_{1\alpha}(\lambda_{N})_{\beta}+(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{1}\delta_{1\beta}-(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}}{2M_{N}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(-3+8s^{2})(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}}{6M_{N}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $\frac{(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}}{2M_{N}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $\frac{(3-4s^{2})(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}}{6M_{N}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $-\frac{(\lambda_{N})_{\alpha}^{*}(\lambda_{N})_{\beta}}{2M_{N}^{2}}$
Table 11: Matching to $N$ model. |  $E$ model
---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $-\frac{(\lambda_{E})_{\alpha}^{*}(\lambda_{E})_{\beta}V_{kj}^{*}}{2M_{E}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $-\frac{(\lambda_{E})_{1}^{*}\delta_{1\alpha}(\lambda_{E})_{\beta}+(\lambda_{E})_{\alpha}^{*}(\lambda_{E})_{1}\delta_{1\beta}-(\lambda_{E})_{1}^{*}(\lambda_{E})_{1}\delta_{\alpha\beta}}{2M_{E}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $\frac{(\lambda_{E})_{1}^{*}\delta_{1\alpha}(\lambda_{E})_{\beta}+(\lambda_{E})_{\alpha}^{*}(\lambda_{E})_{1}\delta_{1\beta}-(\lambda_{E})_{1}^{*}(\lambda_{E})_{1}\delta_{\alpha\beta}}{2M_{E}^{2}}$
Table 12: Matching to $E$ model. |  $\Delta_{1}$ model |  $\Delta_{3}$ model
---|---|---
WEFT WC | Matching
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $-\frac{(\lambda_{\Delta_{1}})_{1}^{*}(\lambda_{\Delta_{1}})_{1}\delta_{\alpha\beta}}{2M_{\Delta_{1}}^{2}}$ | $\frac{(\lambda_{\Delta_{3}})_{1}^{*}(\lambda_{\Delta_{3}})_{1}\delta_{\alpha\beta}}{2M_{\Delta_{3}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $-\frac{(\lambda_{\Delta_{1}})_{1}^{*}(\lambda_{\Delta_{1}})_{1}\delta_{\alpha\beta}}{2M_{\Delta_{1}}^{2}}$ | $\frac{(\lambda_{\Delta_{3}})_{1}^{*}(\lambda_{\Delta_{3}})_{1}\delta_{\alpha\beta}}{2M_{\Delta_{3}}^{2}}$
Table 13: Matching to $\Delta_{1}$, and $\Delta_{3}$ models. |  $\Sigma$ model
---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $\frac{(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}V_{kj}^{*}}{8M_{\Sigma}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $\frac{(\lambda_{\Sigma})_{1}^{*}\delta_{1\alpha}(\lambda_{\Sigma})_{\beta}+(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{1}\delta_{1\beta}-2(\lambda_{\Sigma})_{1}^{*}(\lambda_{\Sigma})_{1}\delta_{\alpha\beta}+(1-4s^{2})(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}}{8M_{\Sigma}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $-\frac{(\lambda_{\Sigma})_{1}^{*}\delta_{1\alpha}(\lambda_{\Sigma})_{\beta}+(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{1}\delta_{1\beta}-2(\lambda_{\Sigma})_{1}^{*}(\lambda_{\Sigma})_{1}\delta_{\alpha\beta}+(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}}{8M_{\Sigma}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(-3+8s^{2})(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}}{24M_{\Sigma}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $\frac{(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}}{8M_{\Sigma}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $\frac{(3-4s^{2})(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}}{24M_{\Sigma}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $-\frac{(\lambda_{\Sigma})_{\alpha}^{*}(\lambda_{\Sigma})_{\beta}}{8M_{\Sigma}^{2}}$
Table 14: Matching to $\Sigma$ model. |  $\Sigma_{1}$ model
---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $\frac{(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}V_{kj}^{*}}{8M_{\Sigma_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $\frac{(\lambda_{\Sigma_{1}})_{1}^{*}\delta_{1\alpha}(\lambda_{\Sigma_{1}})_{\beta}+(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{1}\delta_{1\beta}+(\lambda_{\Sigma_{1}})_{1}^{*}(\lambda_{\Sigma_{1}})_{1}\delta_{\alpha\beta}-2(1-4s^{2})(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}}{8M_{\Sigma_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $-\frac{(\lambda_{\Sigma_{1}})_{1}^{*}\delta_{1\alpha}(\lambda_{\Sigma_{1}})_{\beta}+(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{1}\delta_{1\beta}+(\lambda_{\Sigma_{1}})_{1}^{*}(\lambda_{\Sigma_{1}})_{1}\delta_{\alpha\beta}-2(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}}{8M_{\Sigma_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(3-8s^{2})(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}}{12M_{\Sigma_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $-\frac{(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}}{4M_{\Sigma_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $\frac{(-3+4s^{2})(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}}{12M_{\Sigma_{1}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $\frac{(\lambda_{\Sigma_{1}})_{\alpha}^{*}(\lambda_{\Sigma_{1}})_{\beta}}{4M_{\Sigma_{1}}^{2}}$
Table 15: Matching to $\Sigma_{1}$ model. |  $U$ model |  $D$ model
---|---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $-\frac{\sum_{x}V_{xj}^{*}(\lambda_{U})_{x}^{*}(\lambda_{U})_{k}\delta_{\alpha\beta}}{2M_{U}^{2}}$ | $-\frac{\sum_{x}V_{xj}^{*}(\lambda_{D})_{x}^{*}(\lambda_{D})_{k}\delta_{\alpha\beta}}{2M_{D}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | - | $\frac{\sum_{x,z}V_{1x}^{*}V_{1z}(\lambda_{D})_{x}^{*}(\lambda_{D})_{z}\delta_{\alpha\beta}}{2M_{D}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | - | $-\frac{\sum_{x,z}V_{1x}^{*}V_{1z}(\lambda_{D})_{x}^{*}(\lambda_{D})_{z}\delta_{\alpha\beta}}{2M_{D}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $-\frac{(\lambda_{U})_{1}^{*}(\lambda_{U})_{1}\delta_{\alpha\beta}}{2M_{U}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $\frac{(\lambda_{U})_{1}^{*}(\lambda_{U})_{1}\delta_{\alpha\beta}}{2M_{U}^{2}}$ | -
Table 16: Matching to $U$, and $D$ models. |  $Q_{1}$ model |  $Q_{5}$ model |  $Q_{7}$ model
---|---|---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{R}^{kj})^{*}_{\alpha\beta}$ | $-\frac{(\lambda_{Q_{1}})_{j}^{*}(\lambda_{Q_{1}})_{k}\delta_{\alpha\beta}}{M_{Q_{1}}^{2}}$ | - | -
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $-\frac{(\lambda_{Q_{1}})_{1}^{*}(\lambda_{Q_{1}})_{1}\delta_{\alpha\beta}}{2M_{Q_{1}}^{2}}$ | $\frac{(\lambda_{Q_{5}})_{1}^{*}(\lambda_{Q_{5}})_{1}\delta_{\alpha\beta}}{2M_{Q_{5}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $-\frac{(\lambda_{Q_{1}})_{1}^{*}(\lambda_{Q_{1}})_{1}\delta_{\alpha\beta}}{2M_{Q_{1}}^{2}}$ | $\frac{(\lambda_{Q_{5}})_{1}^{*}(\lambda_{Q_{5}})_{1}\delta_{\alpha\beta}}{2M_{Q_{5}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(\lambda_{Q_{1}})_{1}^{*}(\lambda_{Q_{1}})_{1}\delta_{\alpha\beta}}{2M_{Q_{1}}^{2}}$ | - | $-\frac{(\lambda_{Q_{7}})_{1}^{*}(\lambda_{Q_{7}})_{1}\delta_{\alpha\beta}}{2M_{Q_{7}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $\frac{(\lambda_{Q_{1}})_{1}^{*}(\lambda_{Q_{1}})_{1}\delta_{\alpha\beta}}{2M_{Q_{1}}^{2}}$ | - | $-\frac{(\lambda_{Q_{7}})_{1}^{*}(\lambda_{Q_{7}})_{1}\delta_{\alpha\beta}}{2M_{Q_{7}}^{2}}$
Table 17: Matching to $Q_{1}$, $Q_{5}$ and $Q_{7}$ models. |  $T_{1}$ model |  $T_{2}$ model
---|---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $\frac{\sum_{x}V_{xj}^{*}(\lambda_{T_{1}})_{x}^{*}(\lambda_{T_{1}})_{k}\delta_{\alpha\beta}}{8M_{T_{1}}^{2}}$ | $\frac{\sum_{x}V_{xj}^{*}(\lambda_{T_{2}})_{x}^{*}(\lambda_{T_{2}})_{k}\delta_{\alpha\beta}}{8M_{T_{2}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $\frac{\sum_{x,z}V_{1x}^{*}V_{1z}(\lambda_{T_{1}})_{x}^{*}(\lambda_{T_{1}})_{z}\delta_{\alpha\beta}}{8M_{T_{1}}^{2}}$ | $-\frac{\sum_{x,z}V_{1x}^{*}V_{1z}(\lambda_{T_{2}})_{x}^{*}(\lambda_{T_{2}})_{z}\delta_{\alpha\beta}}{4M_{T_{2}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $-\frac{\sum_{x,z}V_{1x}^{*}V_{1z}(\lambda_{T_{1}})_{x}^{*}(\lambda_{T_{1}})_{z}\delta_{\alpha\beta}}{8M_{T_{1}}^{2}}$ | $\frac{\sum_{x,z}V_{1x}^{*}V_{1z}(\lambda_{T_{2}})_{x}^{*}(\lambda_{T_{2}})_{z}\delta_{\alpha\beta}}{4M_{T_{2}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(\lambda_{T_{1}})_{1}^{*}(\lambda_{T_{1}})_{1}\delta_{\alpha\beta}}{4M_{T_{1}}^{2}}$ | $-\frac{(\lambda_{T_{2}})_{1}^{*}(\lambda_{T_{2}})_{1}\delta_{\alpha\beta}}{8M_{T_{2}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $-\frac{(\lambda_{T_{1}})_{1}^{*}(\lambda_{T_{1}})_{1}\delta_{\alpha\beta}}{4M_{T_{1}}^{2}}$ | $\frac{(\lambda_{T_{2}})_{1}^{*}(\lambda_{T_{2}})_{1}\delta_{\alpha\beta}}{8M_{T_{2}}^{2}}$
Table 18: Matching to $T_{1}$, and $T_{2}$ models.

###  B.3 Gauge boson models

|  $\mathcal{B}$ model
---|---
WEFT WC | Matching
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{B}})_{\alpha\beta}\left((g^{e}_{\mathcal{B}})_{11}+(g^{l}_{\mathcal{B}})_{11}\right)}{M_{\mathcal{B}}^{2}}+\frac{g^{\phi}_{\mathcal{B}}\left[\left((g^{e}_{\mathcal{B}})_{11}+(g^{l}_{\mathcal{B}})_{11}\right)\delta_{\alpha\beta}-(1-4s^{2})(g^{l}_{\mathcal{B}})_{\alpha\beta}\right]}{2M_{\mathcal{B}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{B}})_{\alpha\beta}\left((g^{e}_{\mathcal{B}})_{11}-(g^{l}_{\mathcal{B}})_{11}\right)}{M_{\mathcal{B}}^{2}}+\frac{g^{\phi}_{\mathcal{B}}\left[\left((g^{e}_{\mathcal{B}})_{11}-(g^{l}_{\mathcal{B}})_{11}\right)\delta_{\alpha\beta}+(g^{l}_{\mathcal{B}})_{\alpha\beta}\right]}{2M_{\mathcal{B}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{B}})_{\alpha\beta}\left((g^{u}_{\mathcal{B}})_{11}+(g^{q}_{\mathcal{B}})_{11}\right)}{M_{\mathcal{B}}^{2}}+\frac{g^{\phi}_{\mathcal{B}}\left[3\left((g^{u}_{\mathcal{B}})_{11}+(g^{q}_{\mathcal{B}})_{11}\right)\delta_{\alpha\beta}+(3-8s^{2})(g^{l}_{\mathcal{B}})_{\alpha\beta}\right]}{6M_{\mathcal{B}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{B}})_{\alpha\beta}\left((g^{u}_{\mathcal{B}})_{11}-(g^{q}_{\mathcal{B}})_{11}\right)}{M_{\mathcal{B}}^{2}}+\frac{g^{\phi}_{\mathcal{B}}\left[3\left((g^{u}_{\mathcal{B}})_{11}-(g^{q}_{\mathcal{B}})_{11}\right)\delta_{\alpha\beta}-(g^{l}_{\mathcal{B}})_{\alpha\beta}\right]}{2M_{\mathcal{B}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{B}})_{\alpha\beta}\left((g^{d}_{\mathcal{B}})_{11}+\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{B}})_{xz}\right)}{M_{\mathcal{B}}^{2}}+\frac{g^{\phi}_{\mathcal{B}}\left[3\left((g^{d}_{\mathcal{B}})_{11}+\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{B}})_{xz}\right)\delta_{\alpha\beta}-(3-4s^{2})(g^{l}_{\mathcal{B}})_{\alpha\beta}\right]}{6M_{\mathcal{B}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{B}})_{\alpha\beta}\left((g^{d}_{\mathcal{B}})_{11}-\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{B}})_{xz}\right)}{M_{\mathcal{B}}^{2}}+\frac{g^{\phi}_{\mathcal{B}}\left[3\left((g^{d}_{\mathcal{B}})_{11}-\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{B}})_{xz}\right)\delta_{\alpha\beta}+(g^{l}_{\mathcal{B}})_{\alpha\beta}\right]}{2M_{\mathcal{B}}^{2}}$
Table 19: Matching to $\mathcal{B}$ model. |  $\mathcal{W}$ model
---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{W}})_{\beta\alpha}\sum_{x}V_{xj}^{*}(g^{q}_{\mathcal{W}})^{*}_{xk}}{2M_{\mathcal{W}}^{2}}-\frac{g^{\phi}_{\mathcal{W}}\left[\sum_{x}V_{xj}^{*}(g^{q}_{\mathcal{W}})_{kx}^{*}\delta_{\alpha\beta}+(g^{l}_{\mathcal{W}})_{\alpha\beta}V_{kj}^{*}\right]}{4M_{\mathcal{W}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{W}})_{\alpha 1}(g^{l}_{\mathcal{W}})_{1\beta}}{2M_{\mathcal{W}}^{2}}+\frac{g^{\phi}_{\mathcal{W}}\left[(g^{l}_{\mathcal{W}})_{11}\delta_{\alpha\beta}-2(g^{l}_{\mathcal{W}})_{\beta 1}^{*}\delta_{1\alpha}-2(g^{l}_{\mathcal{W}})_{\alpha 1}\delta_{1\beta}+(1-4s^{2})(g^{l}_{\mathcal{W}})_{\alpha\beta}\right]}{8M_{\mathcal{W}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{ee})_{\alpha\beta}$ | $-\frac{(g^{l}_{\mathcal{W}})_{\alpha 1}(g^{l}_{\mathcal{W}})_{1\beta}}{2M_{\mathcal{W}}^{2}}-\frac{g^{\phi}_{\mathcal{W}}\left[(g^{l}_{\mathcal{W}})_{11}\delta_{\alpha\beta}-2(g^{l}_{\mathcal{W}})_{\beta 1}^{*}\delta_{1\alpha}-2(g^{l}_{\mathcal{W}})_{\alpha 1}\delta_{1\beta}+(g^{l}_{\mathcal{W}})_{\alpha\beta}\right]}{8M_{\mathcal{W}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{W}})_{\alpha\beta}(g^{q}_{\mathcal{W}})_{11}}{4M_{\mathcal{W}}^{2}}-\frac{g^{\phi}_{\mathcal{W}}\left[3(g^{q}_{\mathcal{W}})_{11}\delta_{\alpha\beta}+(3-8s^{2})(g^{l}_{\mathcal{W}})_{\alpha\beta}\right]}{24M_{\mathcal{W}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $-\frac{(g^{l}_{\mathcal{W}})_{\alpha\beta}(g^{q}_{\mathcal{W}})_{11}}{4M_{\mathcal{W}}^{2}}+\frac{g^{\phi}_{\mathcal{W}}\left[(g^{q}_{\mathcal{W}})_{11}\delta_{\alpha\beta}+(g^{l}_{\mathcal{W}})_{\alpha\beta}\right]}{8M_{\mathcal{W}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | $-\frac{(g^{l}_{\mathcal{W}})_{\alpha\beta}\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{W}})_{xz}}{4M_{\mathcal{W}}^{2}}+\frac{g^{\phi}_{\mathcal{W}}\left[3\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{W}})_{xz}\delta_{\alpha\beta}+(3-4s^{2})(g^{l}_{\mathcal{W}})_{\alpha\beta}\right]}{24M_{\mathcal{W}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | $\frac{(g^{l}_{\mathcal{W}})_{\alpha\beta}\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{W}})_{xz}}{4M_{\mathcal{W}}^{2}}-\frac{g^{\phi}_{\mathcal{W}}\left[3\sum_{xz}V_{1z}V_{1x}^{*}(g^{q}_{\mathcal{W}})_{xz}\delta_{\alpha\beta}+(g^{l}_{\mathcal{W}})_{\alpha\beta}\right]}{8M_{\mathcal{W}}^{2}}$
Table 20: Matching to $\mathcal{W}$ model. |  $\mathcal{L}_{1}$ model |  $\mathcal{L}_{3}$ model |  $\mathcal{B}_{1}$ model
---|---|---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{R}^{kj})^{*}_{\alpha\beta}$ | $\frac{1}{f}\frac{\sum_{x}\gamma_{\mathcal{L}_{1}}(\tilde{y}_{u})^{*}_{kx}(\tilde{g}^{dDq}_{\mathcal{L}_{1}})_{jx}\delta_{\alpha\beta}}{2M_{\mathcal{L}_{1}}^{2}}$ | - | $\frac{g_{\mathcal{B}_{1}}^{\phi}(g_{\mathcal{B}_{1}}^{du})_{jk}\delta_{\alpha\beta}}{M_{\mathcal{B}_{1}}^{2}}$
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{P}^{kj})^{*}_{\alpha\beta}$ | $\frac{1}{f}\frac{\gamma_{\mathcal{L}_{1}}(\tilde{g}^{eDl}_{\mathcal{L}_{1}})_{\alpha\beta}\left[\sum_{x}V_{xj}(\tilde{y}_{u})^{*}_{kx}+(\tilde{y}_{d})^{*}_{jk}\right]}{2M_{\mathcal{L}_{1}}^{2}}$ | - | -
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{S}^{kj})^{*}_{\alpha\beta}$ | $-\frac{1}{f}\frac{\gamma_{\mathcal{L}_{1}}(\tilde{g}^{eDl}_{\mathcal{L}_{1}})_{\alpha\beta}\left[\sum_{x}V_{xj}(\tilde{y}_{u})^{*}_{kx}-(\tilde{y}_{d})^{*}_{jk}\right]}{2M_{\mathcal{L}_{1}}^{2}}$ | - | -
$\frac{2}{v^{2}}(\epsilon_{V}^{ee})_{\alpha\beta}$ | $-\frac{1}{f}\frac{\gamma_{\mathcal{L}_{1}}(\tilde{y}_{e})_{1\beta}(\tilde{g}^{eDL}_{\mathcal{L}_{1}})_{1\alpha}^{*}}{4M_{\mathcal{L}_{1}}^{2}}$ | $-\frac{(g_{\mathcal{L}_{3}})_{1\beta}(g_{\mathcal{L}_{3}})_{1\alpha}^{*}}{M_{\mathcal{L}_{3}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $-\frac{1}{f}\frac{\gamma_{\mathcal{L}_{1}}(\tilde{y}_{e})_{1\beta}(\tilde{g}^{eDL}_{\mathcal{L}_{1}})_{1\alpha}^{*}}{4M_{\mathcal{L}_{1}}^{2}}$ | $-\frac{(g_{\mathcal{L}_{3}})_{1\beta}(g_{\mathcal{L}_{3}})_{1\alpha}^{*}}{M_{\mathcal{L}_{3}}^{2}}$ | -
Table 21: Matching to $\mathcal{L}_{1}$, $\mathcal{L}_{3}$, and $\mathcal{B}_{1}$ models. |  $\mathcal{Q}_{1}$ model |  $\mathcal{Q}_{5}$ model |  $\mathcal{U}_{2}$ model |  $\mathcal{X}$ model
---|---|---|---|---
WEFT WC | Matching
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{L}^{kj})^{*}_{\alpha\beta}$ | - | - | $\frac{\sum_{x}V_{xj}^{*}(g^{lq}_{\mathcal{U}_{2}})^{*}_{\alpha x}(g^{lq}_{\mathcal{U}_{2}})_{\beta k}}{M_{\mathcal{U}_{2}}^{2}}$ | $-\frac{\sum_{x}V_{xj}^{*}(g_{\mathcal{X}})_{\alpha x}^{*}(g_{\mathcal{X}})_{\beta k}}{4M_{\mathcal{X}}^{2}}$
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{P}^{kj})^{*}_{\alpha\beta}$ | - | $\frac{2(g^{eq}_{\mathcal{Q}_{5}})_{\alpha k}(g^{dl}_{\mathcal{Q}_{5}})_{j\beta}^{*}}{M_{\mathcal{Q}_{5}}^{2}}$ | $-\frac{2(g^{ed}_{\mathcal{U}_{2}})_{\alpha j}^{*}(g^{lq}_{\mathcal{U}_{2}})_{\beta k}}{M_{\mathcal{U}_{2}}^{2}}$ | -
$\frac{2V_{kj}^{*}}{v^{2}}(\epsilon_{S}^{kj})^{*}_{\alpha\beta}$ | - | $\frac{2(g^{eq}_{\mathcal{Q}_{5}})_{\alpha k}(g^{dl}_{\mathcal{Q}_{5}})_{j\beta}^{*}}{M_{\mathcal{Q}_{5}}^{2}}$ | $-\frac{2(g^{ed}_{\mathcal{U}_{2}})_{\alpha j}^{*}(g^{lq}_{\mathcal{U}_{2}})_{\beta k}}{M_{\mathcal{U}_{2}}^{2}}$ | -
$\frac{2}{v^{2}}(\epsilon_{V}^{dd})_{\alpha\beta}$ | - | $-\frac{(g^{dl}_{\mathcal{Q}_{5}})_{1\alpha}^{*}(g^{dl}_{\mathcal{Q}_{5}})_{1\beta}}{M_{\mathcal{Q}_{5}}^{2}}$ | - | $\frac{\sum_{xz}V_{1x}V_{1z}^{*}(g_{\mathcal{X}})_{\alpha x}(g_{\mathcal{X}})_{\beta z}^{*}}{2M_{\mathcal{X}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{dd})_{\alpha\beta}$ | - | $-\frac{(g^{dl}_{\mathcal{Q}_{5}})_{1\alpha}^{*}(g^{dl}_{\mathcal{Q}_{5}})_{1\beta}}{M_{\mathcal{Q}_{5}}^{2}}$ | - | $-\frac{\sum_{xz}V_{1x}V_{1z}^{*}(g_{\mathcal{X}})_{\alpha x}(g_{\mathcal{X}})_{\beta z}^{*}}{2M_{\mathcal{X}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{V}^{uu})_{\alpha\beta}$ | $-\frac{(g^{ul}_{\mathcal{Q}_{1}})_{1\alpha}^{*}(g^{ul}_{\mathcal{Q}_{1}})_{1\beta}}{M_{\mathcal{Q}_{1}}^{2}}$ | - | $\frac{(g^{lq}_{\mathcal{U}_{2}})_{\alpha 1}(g^{lq}_{\mathcal{U}_{2}})_{\beta 1}^{*}}{M_{\mathcal{U}_{2}}^{2}}$ | $\frac{(g_{\mathcal{X}})_{\alpha 1}(g_{\mathcal{X}})_{\beta 1}^{*}}{4M_{\mathcal{X}}^{2}}$
$\frac{2}{v^{2}}(\epsilon_{A}^{uu})_{\alpha\beta}$ | $-\frac{(g^{ul}_{\mathcal{Q}_{1}})_{1\alpha}^{*}(g^{ul}_{\mathcal{Q}_{1}})_{1\beta}}{M_{\mathcal{Q}_{1}}^{2}}$ | - | $-\frac{(g^{lq}_{\mathcal{U}_{2}})_{\alpha 1}(g^{lq}_{\mathcal{U}_{2}})_{\beta 1}^{*}}{M_{\mathcal{U}_{2}}^{2}}$ | $-\frac{(g_{\mathcal{X}})_{\alpha 1}(g_{\mathcal{X}})_{\beta 1}^{*}}{4M_{\mathcal{X}}^{2}}$
Table 22: Matching to $\mathcal{Q}_{1}$, $\mathcal{Q}_{5}$, $\mathcal{U}_{2}$,
and $\mathcal{X}$ models.

## References

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