aainstitutetext: Theoretical Physics Department, CERN, Geneva,
Switzerlandbbinstitutetext: PRISMA Cluster of Excellence & Mainz Institute for
Theoretical Physics,
Johannes Gutenberg University, Staudingerweg 7, 55099 Mainz,
Germanyccinstitutetext: Theoretical Physics Department, Fermi National
Accelerator Laboratory,
P.O. Box 500, Batavia, IL 60510, USAddinstitutetext: Northwestern University,
Department of Physics & Astronomy,
2145 Sheridan Road, Evanston, IL 60208, USA

# Unleashing the Power of EFT
in Neutrino–Nucleus Scattering

Joachim Kopp c Noemi Rocco d Zahra Tabrizi<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Neutrino physics is advancing into a precision era with the construction of
new experiments, particularly in the few GeV energy range. Within this energy
range, neutrinos exhibit diverse interactions with nucleons and nuclei. This
study delves in particular into neutrino–nucleus quasi-elastic cross sections,
taking into account both standard and, for the first time, non-standard
interactions, all within the framework of effective field theory (EFT). The
main uncertainties in these cross sections stem from uncertainties in the
nucleon-level form factors, and from the approximations necessary to solve the
nuclear many-body problem. We explore how these uncertainties influence the
potential of neutrino experiments to probe new physics introduced by left-
handed, right-handed, scalar, pseudoscalar, and tensor interactions. For some
of these interactions the cross section is enhanced, making long-baseline
experiments an excellent place to search for them. Our results, including
tabulated cross sections for all interaction types and all neutrino flavors,
can serve as the foundation for such searches.

##  1 Introduction

How do neutrinos interact with matter? While the answer to this question is
well understood at the qualitative level, making quantitative predictions for
neutrino–nucleus cross sections has proven to be a daunting task and is a
leading source of systematic uncertainty in present and upcoming precision
neutrino experiments. It may therefore seem foolish to even try to complicate
matters further by considering the possible impact of physics beyond the
Standard Model (SM). In this paper, we will nevertheless take a step forward
in precisely this direction. Working in the context of effective field theory
(EFT), we will compute neutrino–nucleus interaction cross sections including
new couplings with arbitrary Lorentz structure. These results can form the
cornerstone of future experimental searches for “new physics” in the neutrino
sector, as well as for phenomenological work relating extensions of the SM to
observables.

Our starting point is SM Effective Field Theory (SMEFT) Buchmuller:1985jz ;
Grzadkowski:2010es , which is defined by requiring the same gauge symmetries
and the same particle content as in the SM, but adding non-renormalizable
operators to parameterize the low-energy footprint of particles much heavier
than the electroweak scale. But while SMEFT is a suitable framework for new
physics searches at the electroweak scale – that is, at the LHC – for lower-
energy probes such as hadron/meson decay and accelerator-based neutrino
experiments, it is more appropriate to work with a different EFT, one in which
electroweak-scale degrees of freedom ($W$, $Z$, Higgs, top) have been
integrated out. This theory is called Weak Effective Field Theory (WEFT) or
Low-Energy Effective Field Theory (LEFT) in the literature. It can be ultra-
violet (UV) completed by SMEFT, and the two theories can be related to one
another by renormalization group running and parameter matching at the
electroweak scale Cirigliano:2012ab ; Falkowski:2017pss ; Jenkins:2017jig ;
Jenkins:2017dyc ; Falkowski:2018dmy ; Aebischer:2018bkb ; Descotes-
Genon:2018foz ; Bischer:2019ttk ; Dekens:2019ept ; Falkowski:2019kfn ;
Falkowski:2020pma ; Falkowski:2021vdg ; Falkowski:2021bkq . (This is of course
assuming that there are no new particles at the electroweak scale or below. If
new physics exists already at a scale lower than the $W$ boson mass, WEFT
would be consistent up to the scale of the new physics, and it would need to
be UV-completed by an extension of SMEFT that includes the new light degrees
of freedom.) At even lower energy scales, namely in the regime of quasi-
elastic (QE) scattering, the relevant hadronic degrees of freedom are nucleons
instead of quarks. QE scattering makes a significant and often dominant
contribution to the total detection rate in current and future accelerator-
based long-baseline experiments with GeV-scale beams.

Calculating neutrino cross-sections in the QE regime requires first describing
the interactions of the neutrino with individual nucleons, and then taking
into account nuclear effects. Accurate predictions at low and moderate
neutrino energies have been achieved through “ab-initio” methods like Green’s
Function Monte Carlo and the Coupled Cluster approaches Lovato:2020kba ;
Lovato:2017cux ; Lovato:2016gkq ; Sobczyk:2023sxh . For higher neutrino
energies, relevant for accelerator neutrino experiments, alternative
approaches utilize the factorization of the nuclear final state. The spectral
function (SF) formalism Benhar:1994hw ; Benhar:2006wy ; Benhar:2010nx ,
adopted in this work, incorporates relativistic effects in both kinematics and
the interaction vertex, while providing a precise depiction of nuclear
dynamics. While the findings presented here are specifically focused on the QE
region, the SF approach has been generalized to describe multiple reaction
mechanisms. Previous works Rocco:2018mwt ; Rocco:2020jlx demonstrate its
application to phenomena like meson exchange currents and pion production.
Additionally, alternative methods such as SuSA-v2 and the relativistic mean-
field approach also provide predictions of neutrino-nucleus cross sections up
to high neutrino energies, encompassing a diverse array of reaction mechanisms
Amaro:2019zos ; Gonzalez-Jimenez:2014eqa ; Megias:2016lke ; Amaro:2021sec .

The calculation of neutrino cross sections in the QE regime for arbitrary WEFT
interactions and including nuclear effects, as well as the discussion of the
associated uncertainties, are the main topics of this paper. The most
important novel challenge we face is that nucleon-level form factors have to
be precisely known for arbitrary WEFT inteactions.

In the past, physics beyond the SM in the neutrino sector has often been
discussed in the framework of “non-standard interactions”, which is similar to
WEFT: the relevant degrees of freedom are those present below the electroweak
scale (but above the QCD scale), and the new operators are dimension-six four-
fermion interactions Wolfenstein:1977ue ; Valle:1987gv ; Grossman:1995wx ;
Kopp:2007mi ; Kopp:2007ne ; Antusch:2008tz ; Gavela:2008ra ; Ohlsson:2012kf ;
Miranda:2015dra ; deGouvea:2015ndi ; Coloma:2023ixt . However, this widely
used formalism has several shortcomings (see e.g. the discussion in
Falkowski:2019kfn ): (i) it typically does not take into account
renormalization group effects and the necessity of an embedding into a theory
that respects electroweak symmetry. This means that correlations between
operators that would arise from such an embedding are typically neglected.
(ii) even more problematic, some authors treat operators affecting neutrino
production and those affecting neutrino detection separately, even though in
many cases, they are the same. First steps in resolving these shortcomings
have been taken in refs. Falkowski:2019xoe ; Falkowski:2019kfn , which
introduce a comprehensive EFT framework for neutrino oscillations, offering a
systematic approach applicable to experiments with extended baselines and
diverse neutrino production and detection processes. The practicality of this
formalism has been demonstrated in ref. Falkowski:2021bkq by applying it to
the FASER$\nu$ neutrino experiment at the LHC, which showcases the capability
of FASER$\nu$ to probe new physics scales up to
$\sim$10\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$ for certain operators.

In the following, we will first briefly introduce the effective field theory
that we work in (section 2.1) and then proceed to the calculation of first the
neutrino–nucleon interaction amplitudes for all EFT operators (section 2.2),
and then the interaction cross sections including nuclear effects (section
2.3). We will present our results in section 3 and discuss their
uncertainties, before summarizing and concluding in section 4.

Our main cross-section results in tabulated form are available from GitHub
github .

##  2 Formalism

###  2.1 Weak Effective Field Theory

In this section we introduce the effective field theory (EFT) formalism in
which we will calculate neutrino–nucleus cross section in the presence of
physics beyond the SM. In particular, we are interested in Weak Effective
Field Theory (WEFT), valid below the electroweak scale, with the electroweak
gauge bosons, the Higgs boson, and the top quark integrated out. The part of
the WEFT Lagrangian that is relevant to neutrino–nucleus interactions is given
by Falkowski:2021bkq :

$\displaystyle\mathcal{L}_{\rm WEFT}$
$\displaystyle\supset-\,\frac{2V_{ud}}{v^{2}}\Big{\\{}[{\bf
1}+\epsilon_{L}]_{\alpha\beta}(\bar{q}_{u}\gamma^{\mu}P_{L}q_{d})(\bar{\ell}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta})\,+\,[\epsilon_{R}]_{\alpha\beta}(\bar{q}_{u}\gamma^{\mu}P_{R}q_{d})(\bar{\ell}_{\alpha}\gamma_{\mu}P_{L}\nu_{\beta})$
$\displaystyle\quad+\,\frac{1}{2}[\epsilon_{S}]_{\alpha\beta}(\bar{q}_{u}q_{d})(\bar{\ell}_{\alpha}P_{L}\nu_{\beta})-\frac{1}{2}[\epsilon_{P}]_{\alpha\beta}(\bar{q}_{u}\gamma_{5}q_{d})(\bar{\ell}_{\alpha}P_{L}\nu_{\beta})$
$\displaystyle\quad+\,\frac{1}{4}[\epsilon_{T}]_{\alpha\beta}(\bar{q}_{u}\sigma^{\mu\nu}P_{L}q_{d})(\bar{\ell}_{\alpha}\sigma_{\mu\nu}P_{L}\nu_{\beta})+{\rm
h.c.}\Big{\\}}\,.$ (1)

Here,
$v\equiv(\sqrt{2}G_{F})^{-1/2}\approx$246\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$
is the vacuum expectation value (vev) of the Higgs field, $V_{ud}$ is the $ud$
element of the Cabibbo–Kobayashi–Maskawa (CKM) matrix,
$P_{L,R}=\frac{1}{2}(1\mp\gamma^{5})$ are chirality projection operators, and
$\sigma^{\mu\nu}=\tfrac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. The $q_{u}$ and
$q_{d}$ fields describe up and down quarks, respectively. The charged leptons
are labeled $\ell_{\alpha}$ (with $\alpha$ their flavor index), and the
neutrino fields in the flavor basis are $\nu_{\alpha}$. They can be connected
to the mass eigenstates $\nu_{k}$ using the Pontecorvo–Maki–Nakagawa–Sakata
(PMNS) mixing matrix: $\nu_{\alpha}=\sum_{k=1}^{3}U_{\alpha k}\nu_{k}$. The
first term of eq. 1 describes the SM charged current (CC) interaction between
neutrinos, charged leptons and quarks, while the subsequent terms add the new
left-handed ($L$), right-handed ($R$), scalar ($S$), pseudo-scalar ($P$) and
tensor ($T$) interactions to the Lagrangian. These terms are dimension-six,
and their dimensionless Wilson coefficients $[\epsilon_{X}]_{\alpha\beta}$
(with $X=L,R,S,P,T$) describe the interaction strengths of the new operators
relative to SM weak interactions.

###  2.2 Neutrino–Nucleon Scattering

We are interested in studying the effects that the new interactions introduced
in the Lagrangian of eq. 1 have on charged current quasi-elastic (CCQE)
neutrino interactions. At the nucleon level, the interactions we are
interested in are

$\displaystyle\nu_{\beta}+n\to\ell_{\alpha}^{-}+p^{+}\,$ (2)

for neutrinos, and

$\displaystyle\bar{\nu}_{\beta}+p^{+}\to\ell_{\alpha}^{+}+n\,$ (3)

for anti-neutrinos.The structure of the WEFT Lagrangian implies that the
amplitudes related to the neutrino–nucleon interactions take the form

$\displaystyle{\mathcal{M}}_{\alpha\beta}$
$\displaystyle=\delta_{\alpha\beta}A_{L,\alpha}+\sum_{X=L,R,S,P,T}[\epsilon_{X}]_{\alpha\beta}A_{X,\alpha}\,,$
(4)

where the $A_{X,\alpha}$ are the reduced matrix elements from which the WEFT
coefficients have been factored out. In other words, $A_{X,\alpha}$ gives the
amplitude for a neutrino to produce a charged lepton of flavor $\alpha$ via an
interaction of type $X$, assuming this interaction is as strong as SM weak
interactions. We discuss these amplitudes in detail below. Note that _no_
summation over $\alpha$ is implied in eq. 4. For anti-neutrinos eq. 4 holds
after changing $\epsilon\to\epsilon^{*}$, and replacing $A_{X,\alpha}$ by the
corresponding anti-neutrino amplitudes. The total neutrino–nucleon scattering
cross section is

$\displaystyle\sigma_{\alpha\beta}^{\text{tot}}$
$\displaystyle=\frac{1}{2\kappa}\sum_{\text{spins}}\int\\!d\Pi\,|{\mathcal{M}}_{\alpha\beta}|^{2}$
$\displaystyle=\hat{\sigma}_{LL,\alpha}\delta_{\alpha\beta}+\sum_{X}\,[\epsilon_{X}]_{\alpha\beta}\hat{\sigma}^{\text{int}}_{LX,\alpha}\delta_{\alpha\beta}+\sum_{X}\,[\epsilon_{X}]^{*}_{\alpha\beta}(\hat{\sigma}^{\text{int}}_{LX,\alpha})^{*}\delta_{\alpha\beta}+\sum_{X,Y}\,[\epsilon_{X}]_{\alpha\beta}[\epsilon_{Y}]^{*}_{\alpha\beta}\hat{\sigma}^{\text{NP}}_{XY,\alpha}\,,$
(5)

where the definition $\kappa=4E_{\nu}m_{n,p}$ has been introduced, with
$m_{n,p}$ the mass of the target nucleon. Moreover,
$\int\\!d\Pi=\int\\!\big{[}\prod_{j}d^{3}p_{j}/(2\pi)^{3}\big{]}\times(2\pi)^{4}\delta^{(4)}(p_{i}-p_{f})$
denotes the phase space integral. The integral runs over the three-momenta of
all final-state particles (indexed by $j$), and the $\delta$-function ensures
energy momentum conservation, with $p_{i}$ the total initial-state momentum
and $p_{f}$ the total final-state momentum. Note that on the right-hand side
of eq. 5, again no sum over flavor indices is implied. Setting
$\epsilon_{X}=0$ recovers the SM cross section,

$\displaystyle\hat{\sigma}_{LL,\alpha}\equiv\frac{1}{2\kappa}\sum_{\text{spins}}\int\\!d\Pi\,|A_{L,\alpha}|^{2}\,.$
(6)

The terms describing the interference between the new physics and the SM
amplitude are

$\displaystyle\hat{\sigma}^{\text{int}}_{LX,\alpha}\equiv\frac{1}{2\kappa}\sum_{\text{spins}}\int\\!d\Pi(A_{L,\alpha}A^{*}_{X,\alpha})\,,$
(7)

while the terms quadratic in the new physics interactions are

$\displaystyle\hat{\sigma}^{\text{NP}}_{XY,\alpha}\equiv\frac{1}{2\kappa}\sum_{\text{spins}}\int\\!d\Pi(A_{X,\alpha}A^{*}_{Y,\alpha})\,.$
(8)

In these relations the pre-factor of $\tfrac{1}{2}$ comes from averaging over
initial state spins..

Let us now compute the reduced matrix elements for the CCQE interaction
$\nu_{\beta}+n\to\ell_{\alpha}^{-}+p^{+}$. They can be expressed as

$\displaystyle\begin{split}A_{L,\alpha}&=-\frac{2V_{ud}}{v^{2}}\Big{[}\bar{u}_{\ell_{\alpha}}(p_{\ell_{\alpha}})\gamma^{\mu}P_{L}u_{\nu}(p_{\nu})\Big{]}\braket{p(p_{p})}{\bar{q}_{u}\gamma_{\mu}P_{L}q_{d}}{n(p_{n})}\,,\\\
A_{R,\alpha}&=-\frac{2V_{ud}}{v^{2}}\Big{[}\bar{u}_{\ell_{\alpha}}(p_{\ell_{\alpha}})\gamma^{\mu}P_{L}u_{\nu}(p_{\nu})\Big{]}\braket{p(p_{p})}{{\bar{q}}_{u}\gamma_{\mu}P_{R}q_{d}}{n(p_{n})}\,,\\\
A_{S,\alpha}&=-\frac{V_{ud}}{v^{2}}\Big{[}\bar{u}_{\ell_{\alpha}}(p_{\ell_{\alpha}})P_{L}u_{\nu}(p_{\nu})\Big{]}\braket{p(p_{p})}{{\bar{q}}_{u}q_{d}}{n(p_{n})}\,,\\\
A_{P,\alpha}&=\frac{V_{ud}}{v^{2}}\Big{[}\bar{u}_{\ell_{\alpha}}(p_{\ell_{\alpha}})P_{L}u_{\nu}(p_{\nu})\Big{]}\braket{p(p_{p})}{{\bar{q}}_{u}\gamma_{5}q_{d}}{n(p_{n})}\,,\\\
A_{T,\alpha}&=-\frac{V_{ud}}{2v^{2}}\Big{[}\bar{u}_{\ell_{\alpha}}(p_{\ell_{\alpha}})\sigma^{\mu\nu}P_{L}u_{\nu}(p_{\nu})\Big{]}\braket{p(p_{p})}{{\bar{q}}_{u}\sigma_{\mu\nu}q_{d}}{n(p_{n})}\,,\end{split}$
(9)

where the $u_{i}$ are the spinor wave functions of the leptons and hadrons
(not to be confused with the up-quark field, $u$). The reduced matrix elements
for the corresponding anti-neutrino interaction
$\bar{\nu}_{\beta}+p\to\ell_{\alpha}^{+}+n$, are obtained by replacing
$u_{\ell_{\alpha}}\to v_{\nu}$, $u_{\nu}\to v_{\ell_{\alpha}}$, and taking the
complex conjugate of the hadronic current as well as the CKM element $V_{ud}$.
Note that, as long as neutrino masses are neglected, these amplitudes do no
longer depend on the neutrino flavor as this dependence has been factored out
in eq. 4. The hadronic matrix elements can be parameterized in terms of a set
of Lorentz-invariant form factors, one corresponding to each possible quark
bilinear Weinberg:1958ut ; Cirigliano:2013xha :

$\displaystyle\braket{p(p_{p})}{{\bar{q}}_{u}\gamma_{\mu}q_{d}}{n(p_{n})}$
$\displaystyle=\bar{u}_{p}(p_{p})\bigg{[}G_{V}(Q^{2})\gamma_{\mu}+i\frac{\tilde{G}_{T(V)}(Q^{2})}{2M_{N}}\sigma_{\mu\nu}q^{\nu}-\frac{\tilde{G}_{S}(Q^{2})}{2M_{N}}q_{\mu}\bigg{]}u_{n}(p_{n})\,,$
(10)
$\displaystyle\braket{p(p_{p})}{{\bar{q}}_{u}\gamma_{\mu}\gamma_{5}q_{d}}{n(p_{n})}$
$\displaystyle=\bar{u}_{p}(p_{p})\bigg{[}G_{A}(Q^{2})\gamma_{\mu}\gamma_{5}+i\frac{\tilde{G}_{T(A)}(Q^{2})}{2M_{N}}\sigma_{\mu\nu}q^{\nu}\gamma_{5}-\frac{\tilde{G}_{P}(Q^{2})}{2M_{N}}q_{\mu}\gamma_{5}\bigg{]}u_{n}(p_{n})\,,$
(11) $\displaystyle\braket{p(p_{p})}{{\bar{q}}_{u}q_{d}}{n(p_{n})}$
$\displaystyle=G_{S}(Q^{2})\,\bar{u}_{p}(p_{p})u_{n}(p_{n})\,,$ (12)
$\displaystyle\braket{p(p_{p})}{{\bar{q}}_{u}\gamma_{5}q_{d}}{n(p_{n})}$
$\displaystyle=G_{P}(Q^{2})\,\bar{u}_{p}(p_{p})\gamma_{5}u_{n}(p_{n})\,,$ (13)
$\displaystyle\braket{p(p_{p})}{{\bar{q}}_{u}\sigma_{\mu\nu}q_{d}}{n(p_{n})}$
$\displaystyle=\bar{u}_{p}(p_{p})\bigg{[}G_{T}(Q^{2})\sigma_{\mu\nu}-\frac{i}{M_{N}}G_{T}^{(1)}(Q^{2})(q_{\mu}\gamma_{\nu}-q_{\nu}\gamma_{\mu})$
$\displaystyle-\frac{i}{M_{N}^{2}}G_{T}^{(2)}(Q^{2})(q_{\mu}P_{\nu}-q_{\nu}P_{\mu})-\frac{i}{M_{N}}G_{T}^{(3)}(Q^{2})(\gamma_{\mu}q\\!\\!\\!{\bf/}\gamma_{\nu}-\gamma_{\nu}q\\!\\!\\!{\bf/}\gamma_{\mu})\bigg{]}u_{n}(p_{n})\,,$
(14)

where $u_{p}$ and $u_{n}$ are the proton and neutron spinors, $q=p_{p}-p_{n}$
is the transferred momentum, $Q^{2}\equiv-q^{2}$, $P=p_{n}+p_{p}$, and
$M_{N}\equiv(m_{n}+m_{p})/2$ is the average nucleon mass.

The crucial task now is to obtain the form factors $G_{X}$.

  1. 1.

Vector current: In this case, there are three contributions, parameterized by
the vector form factor $G_{V}(Q^{2})$, the induced tensor-vector form factor
$\tilde{G}_{T(V)}(Q^{2})$, and the induced scalar form factor
$\tilde{G}_{S}(Q^{2})$, respectively. In the literature, $G_{V}(Q^{2})$ is
sometimes called the Dirac form factor, and $\tilde{G}_{T(V)}(Q^{2})$ is
sometimes called the Pauli form factor, anomalous magnetic moment form factor,
or weak magnetism form factor. Naively, all terms in eq. 10 could be expected
to be $\mathcal{O}(1)$. (The prefactors $\sim\mathcal{O}(q/M_{N})$
accompanying $\tilde{G}_{S}(Q^{2})$ and $\tilde{G}_{T(V)}(Q^{2})$ are
$\mathcal{O}(1)$ for GeV-scale neutrino scattering.) However, in the isospin-
symmetric limit (neglecting the differences between the up and down quark
masses), $\tilde{G}_{S}(Q^{2})$ vanishes, as can be seen by replacing
$p_{p}\leftrightarrow p_{n}$ and showing that this leads to a sign change in
the induced scalar current. Isospin symmetry thus requires
$\tilde{G}_{S}(Q^{2})=0$. (In the language of ref. Weinberg:1958ut , the
induced scalar current is a second-class current which vanishes in the
isospin-symmetric limit). Corrections to isospin symmetry are of
$\mathcal{O}(10^{-3})$ and can be safely neglected here in view of
uncertainties on other terms that are much larger.

The vector [$G_{V}(Q^{2})$] and induced tensor-vector
[$\tilde{G}_{T(V)}(Q^{2})$] form factors are related to the electromagnetic
form factors of the nucleon $N=p,n$ by Giunti:2007ry

$\displaystyle G_{V}(Q^{2})$
$\displaystyle=F_{1}^{p}(Q^{2})-F_{1}^{n}(Q^{2})\,,$ (15)
$\displaystyle\tilde{G}_{T(V)}(Q^{2})$
$\displaystyle=F_{2}^{p}(Q^{2})-F_{2}^{n}(Q^{2})\,.$ (16)

$F_{1}^{N}(Q^{2})]$ and $F_{2}^{N}(Q^{2})$, in turn can be related to the
Sachs electric and magnetic form factors:

$\displaystyle\begin{split}G_{E}^{p}(Q^{2})&=F_{1}^{p}(Q^{2})-\frac{Q^{2}}{4M_{N}^{2}}F_{2}^{p}=G_{D}(Q^{2})\,,\\\
G_{E}^{n}(Q^{2})&=F_{1}^{n}(Q^{2})-\frac{Q^{2}}{4M_{N}^{2}}F_{2}^{n}=0\,,\\\\[5.69046pt]
G_{M}^{p}(Q^{2})&=F_{1}^{p}(Q^{2})+F_{2}^{p}(Q^{2})=\mu_{p}G_{D}(Q^{2})\,,\\\\[14.22636pt]
G_{M}^{n}(Q^{2})&=F_{1}^{n}(Q^{2})+F_{2}^{n}(Q^{2})=\mu_{n}G_{D}(Q^{2})\,,\end{split}$
(17)

where $\mu_{p}=2.79284$ is the proton magnetic moment in units of the nuclear
magneton, and $\mu_{n}=-1.91304$ is the equivalent for neutrons
ParticleDataGroup:2016lqr . Finally, we can parameterize $G_{D}(Q^{2})$ as a
dipole function,

$\displaystyle
G_{D}(Q^{2})=\frac{1}{\big{(}1+\frac{Q^{2}}{M_{V}^{2}}\big{)}^{-2}}\,,$ (18)

with $M_{V}=$0.84\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

Precise knowledge of the nucleon vector form factors can be obtained through
high-statistics electron scattering experiments.

  2. 2.

Axial current: Of the three terms contributing to the axial current, one –
namely the induced tensor-axial form factor, $\tilde{G}_{T(A)}$ – vanishes in
the isospin-symmetric limit Weinberg:1958ut . The nucleon axial form factor,
$G_{A}$, which is not accessible in electron-scattering experiments,
introduces significant systematic uncertainties in neutrino–nucleus cross-
section calculations. Existing experimental constraints on $G_{A}$ primarily
rely on beta decay measurements, on neutrino scattering on nuclear targets,
and on pion electro-production. However, the accuracy of these measurements
remains relatively poor compared to measurements of the vector form factors.
Beta decay experiments are sensitive only to the absolute normalization of the
nucleon axial coupling, $g_{A}\equiv G_{A}(0)$, while neutrino scattering and
pion electro-production experiments face challenges due to limited statistics
and uncertainties in the nuclear modeling.

Historically, the most common parametrization used for the axial form factor
is the dipole form:

$\displaystyle
G_{A}(Q^{2})=\frac{g_{A}}{\big{(}1+\frac{Q^{2}}{m_{A}^{2}}\big{)}^{2}}\,,$
(19)

where the axial mass is $m_{A}\simeq 0.961$ GeV, used by GENIEv3 neutrino
generator for their nominal $10a_{0}2_{1}1a$ model Andreopoulos:2009rq ) and
$g_{A}=1.2728\pm 0.0017$ Gonzalez-Alonso:2018omy . This parametrization,
however, proves incapable of capturing the observed shape of the axial form
factor. The so-called $z$-expansion parametrization Simons:2022ltq ,

$\displaystyle G_{A}(Q^{2})=\sum_{k=0}^{\infty}a_{k}\,[z(Q^{2})]^{k}\,,$ (20)

substantially improves the accuracy of the fit to data Bhattacharya:2011ah ;
Meyer:2016oeg . Here, $z(q^{2})$ is an analytic function Hill:2006ub ;
Hill:2010yb ; Meyer:2016oeg motivated by QCD assumptions, while the
coefficients $a_{k}$ can be obtained from lattice QCD or by fitting to
neutrino–nucleus scattering and/or pion electro-production data.

The different parametrizations of the axial form factor – the dipole (dip)
form factor from eq. 19, the $z$-expansion fit to neutrino–deuteron scattering
data (D2), and the $z$-expansion fit to lattice QCD results – differ
substantially, which reflects the large uncertainty the axial form factor
introduces to the cross section. The top panel of fig. 1 compares the
different parametrizations and reveals discrepancies by almost an order of
magnitude at large momentum transfer. This illustrates how important it is to
quantitatively assess these systematic uncertainties. For the SM case such a
study has been carried out in Ref. Simons:2022ltq . In the present paper, we
analyze for the first time how form factor uncertainties affect the study of
new interactions beyond the SM.

Figure 1: The nucleon form factors appearing in eqs. 10, 11, 12, 13 and 14 as
a function of $Q^{2}=-q^{2}$. For the form factors affected by the uncertainty
in $G_{A}(Q^{2})$, the left panel shows results for three different
parametrizations, namely the dipole form factor with
$m_{A}=$0.961\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ (thin solid lines), the
$z$-expansion fitted to neutrino–deuterium scattering data (“D2”, shaded
bands) Meyer:2016oeg , and a $z$-expansion fit to lattice QCD calculations by
the RQCD Collaboration (hatched bands) RQCD:2019jai . In the latter two cases,
the width of the colored bands indicates the uncertainties quoted in the
respective references. The remaining form factors are shown in the right
panel. Note the sign change from negative to positive for $G_{T}^{(1)}$ and
from positive to negative for $G_{T}^{(2)}$.

  3. 3.

Pseudoscalar current: By using the partial conservation of the axial current
(PCAC) and applying it to the axial current matrix element of the nucleon, one
can find a relation between the axial, induced-pseudoscalar, and pseudoscalar
form factors Gonzalez-Alonso:2013ura :

$\displaystyle
G_{P}(Q^{2})=\frac{M_{N}}{m_{q}}G_{A}(Q^{2})+\frac{Q^{2}/2M_{N}}{2m_{q}}\tilde{G}_{P}(Q^{2})\,.$
(21)

Here, $m_{q}=$3.410(43)\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ is the
average light quark mass, taken from lattice calculations (2019 FLAG review,
$N_{f}=2+1+1$ FlavourLatticeAveragingGroup:2019iem ;
EuropeanTwistedMass:2014osg ; FermilabLattice:2018est ). Note the large
prefactors proportional to $M_{N}/m_{q}$ and $Q^{2}/(M_{N}m_{q})$, which are
both of order $10^{3}$. These prefactors lead to an enhancement of, e.g.,
$G_{P}(Q^{2})$ compared to other form factors.

We can moreover use the pion pole dominance (PPD) ansatz to derive an
approximate relation between the induced-pseudoscalar and axial form factors.
More precisely, we use the fact that $G_{P}(Q^{2})$ is dominated by the pion-
pole contribution because the pseudoscalar current has the same quantum
numbers as the pion. This suggests to write
$G_{P}(Q^{2})=G_{P}(0)\,m_{\pi}^{2}/(Q^{2}+m_{\pi}^{2})$, where $m_{\pi}$ is
the pion mass. Moreover, in the low-$Q^{2}$ limit, eq. 21 implies
$G_{P}(Q^{2})\simeq(M_{N}/m_{q})G_{A}(Q^{2})$. Plugging both relations into
eq. 21 leads to Sasaki:2007gw ; Falkowski:2021vdg ; Meyer:2022mix :

$\displaystyle\tilde{G}_{P}(Q^{2})=-\frac{4M_{N}^{2}}{Q^{2}+m_{\pi}^{2}}G_{A}(Q^{2})\,.$
(low-$Q^{2}$ limit) (22)

For higher accuracy, we will instead use the $z$-expansion parametrization
Simons:2022ltq also for the pseudoscalar form factors, and write:

$\displaystyle\tilde{G}_{P}(Q^{2})=-\frac{4M_{N}^{2}}{Q^{2}+m_{\pi}^{2}}\sum_{k=0}^{\infty}a^{\tilde{P}}_{k}\,z(Q^{2})^{k}\,,\,$
(23)

and

$\displaystyle
G_{P}(Q^{2})=\frac{M_{N}}{m_{q}}\frac{m_{\pi}^{2}}{Q^{2}+m_{\pi}^{2}}\sum_{k=0}^{\infty}a^{P}_{k}\,z(Q^{2})^{k}\,,\,$
(24)

where the $a^{\tilde{P}}_{k}$ and $a^{P}_{k}$ coefficients are found by
fitting to lattice QCD results. The first few coefficients in the expansion,
taken from ref. RQCD:2019jai , are shown in table 1. We have checked that
these lattice QCD results satisfy the PPD and PCAC relations.

$X$ | $a_{0}^{X}$ | $a_{1}^{X}$ | $a_{2}^{X}$ | $a_{3}^{X}$ | $a_{4}^{X}$ | $a_{5}^{X}$ | $a_{6}^{X}$
---|---|---|---|---|---|---|---
$A$ | $1.009$ | $-1.756$ | $-1.059$ | $1.621$ | $3.919$ | $-5.739$ | $2.005$
$\tilde{P}$ | $1.008$ | $-1.831$ | $-1.713$ | $4.994$ | $-1.522$ | $-1.984$ | $1.047$
$P$ | $1.066$ | $-1.461$ | $-1.053$ | $-2.504$ | $12.446$ | $-12.260$ | $3.766$
Table 1: The coefficients of the $z$-expansion parametrization of the axial
(first line) induced pseudoscalar (second line) and pseudoscalar (last line)
form factors, determined by the lattice QCD calculations from Ref.
RQCD:2019jai .

  4. 4.

Scalar current: We can find the scalar form factor by using the conservation
of the vector current (CVC), which relates the divergence of the vector
current in (eq. 10) to the scalar current (eq. 12) and leads to Gonzalez-
Alonso:2013ura

$\displaystyle G_{S}(Q^{2})=-\frac{\delta M_{N}^{QCD}}{\delta
m_{q}}G_{V}(Q^{2})+\frac{Q^{2}/2M_{N}}{\delta m_{q}}\tilde{G}_{S}(Q^{2})\,,$
(25)

where $\delta
M_{N}^{QCD}=m_{n}-m_{p}=$2.58(18)\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ is
the difference between the neutron and proton mass in pure QCD Gonzalez-
Alonso:2013ura ) and $\delta
m_{q}=m_{d}-m_{u}=$2.527(47)\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$
FlavourLatticeAveragingGroup:2019iem . Note that the induced tensor-vector
form factor, $\tilde{G}_{T(V)}$, does not appear in eq. 25 because the
divergence of the corresponding term in eq. 10 vanishes.

The induced scalar form factor $\tilde{G}_{S}(Q^{2})$ poses a problem as no
robust measurements or lattice calculations for it exist. This is not
surprising, given that, using Weinberg’s language from ref. Weinberg:1958ut ,
it corresponds to a second-class current (that is, a current that violates $G$
parity). Such currents vanish in the isospin-symmetric limit, so their
physical effects are always suppressed by small isospin-breaking factor such
as $\delta M_{N}$ or $\delta m_{q}$. In eq. 25, however,
$\tilde{G}_{S}(Q^{2})$ appears with a prefactor that is _enhanced_ by
$1/\delta m_{q}$, compensating for the smallness of the form factor.

In the absence of robust measurements or lattice calculations for
$\tilde{G}_{S}(Q^{2})$, we consider two approaches. The first is to simply set
$\tilde{G}_{S}(Q^{2})=0$. The second one is based on the constituent quark
model (CQM) which relates $\tilde{G}_{S}$ to the vector form factor, but
suffers from large theoretical uncertainties. The CQM prediction is
Sharma:2009hg

$\displaystyle\tilde{G}_{S}(Q^{2})=\frac{2M_{N}}{(m_{u}+m_{d})^{\text{CQM}}}\bigg{(}\frac{\delta
M_{N}^{QCD}}{2M_{N}}\,g_{A}-\frac{\delta
m_{q}}{(m_{u}+m_{d})^{\text{CQM}}}\bigg{)}\,G_{V}(Q^{2})\,,$ (26)

where in the constituent quark model one takes $m_{u}=m_{d}=m_{p}/3$.

  5. 5.

Tensor current: Unlike the vector, axial-vector, scalar, and pseudoscalar
currents, constraining the matrix elements of the antisymmetric tensor current
is much more challenging. While the relevant form factors are not easily
determined through experimental data, Ward identities, or low-energy theorems,
lattice QCD techniques and theoretical considerations can nevertheless provide
valuable insights. In the following, we draw from Ref. Hoferichter:2018zwu .
In the isospin-symmetric limit, which we adopt here, the $G^{(3)}_{T}(Q^{2})$
form factor vanishes, and the following relation between the charged and
neutral currents holds FlavourLatticeAveragingGroup:2019iem :

$\displaystyle\langle p(p_{p})|\bar{u}\sigma^{\mu\nu}d|n(p_{n})\rangle$
$\displaystyle=\langle
p(p_{p})|\bar{u}\sigma^{\mu\nu}u-\bar{d}\sigma^{\mu\nu}d|p(p_{p})\rangle$
$\displaystyle=\langle
n(p_{n})|\bar{d}\sigma^{\mu\nu}d-\bar{u}\sigma^{\mu\nu}u|n(p_{n})\rangle\,.$
(27)

The two equivalent matrix elements on the right-hand side of this equation can
be evaluated on the lattice. The results can be written as

$\displaystyle G_{T}(Q^{2})$
$\displaystyle=F^{u}_{1,T}(Q^{2})-F^{d}_{1,T}(Q^{2})\,,$ (28) $\displaystyle
G_{T}^{(1)}(Q^{2})$ $\displaystyle=F^{u}_{2,T}(Q^{2})-F^{d}_{2,T}(Q^{2})\,,$
(29) $\displaystyle G_{T}^{(2)}(Q^{2})$
$\displaystyle=F^{u}_{3,T}(Q^{2})-F^{d}_{3,T}(Q^{2})\,,$ (30)

with the $F_{i,T}^{q}(Q^{2})$ ($q=u,d$) functions parameterized by:

$\displaystyle F_{1,T}^{q}(Q^{2})$
$\displaystyle=\pm\frac{1}{2}\Big{[}F_{1,T}^{u}(0)-F_{1,T}^{d}(0)\Big{]}D_{b_{1}}(Q^{2})+\frac{1}{2}\Big{[}F_{1,T}^{u}(0)+F_{1,T}^{d}(0)\Big{]}D_{h_{1}}(Q^{2})\,,$
(31) $\displaystyle F_{2,T}^{u}(Q^{2})$
$\displaystyle=-\frac{M_{N}}{2m_{\pi}}B_{T}^{\pi,u}(0)\Big{[}2G^{p}_{M}(Q^{2})+G^{n}_{M}(Q^{2})\Big{]}+2\frac{M_{N}^{2}}{m^{2}_{b_{1}}}F_{1,T}^{u}(0)D_{b_{1}}(Q^{2})\,,$
(32) $\displaystyle F_{2,T}^{d}(Q^{2})$
$\displaystyle=-\frac{M_{N}}{2m_{\pi}}B_{T}^{\pi,u}(0)\Big{[}G^{p}_{M}(Q^{2})+2G^{n}_{M}(Q^{2})\Big{]}+2\frac{M_{N}^{2}}{m^{2}_{b_{1}}}F_{1,T}^{d}(0)D_{b_{1}}(Q^{2})\,,$
(33) $\displaystyle F_{3,T}^{u}(Q^{2})$
$\displaystyle=\frac{M_{N}}{4m_{\pi}}B_{T}^{\pi,u}(0)\Big{[}2F^{p}_{2}(Q^{2})+F^{n}_{2}(Q^{2})\Big{]}-\frac{M_{N}^{2}}{m^{2}_{b_{1}}}F_{1,T}^{u}(0)D_{b_{1}}(Q^{2})\,,$
(34) $\displaystyle F_{3,T}^{d}(Q^{2})$
$\displaystyle=\frac{M_{N}}{4m_{\pi}}B_{T}^{\pi,u}(0)\Big{[}F^{p}_{2}(Q^{2})+2F^{n}_{2}(Q^{2})\Big{]}-\frac{M_{N}^{2}}{m^{2}_{b_{1}}}F_{1,T}^{d}(0)D_{b_{1}}(Q^{2})\,.$
(35)

In the first of these relations, the plus sign corresponds to $q=u$ and the
minus sign to $q=d$. Numerically, Ref. Hoferichter:2018zwu gives
$F_{1,T}^{u}(0)=0.784$, $F_{1,T}^{d}(0)=-0.204$, and $B_{T}^{\pi,u}(0)=0.195$
for the tensor charges that determine the normalization. The functions
$G^{N}_{M}(Q^{2})$ and $F_{2}^{N}(Q^{2})$ are the Sachs magnetic and Pauli
form factors defined in eq. 17. Finally,

$\displaystyle
D_{b_{1}/h_{1}}(Q^{2})=\frac{m^{2}_{b_{1}/h_{1}}}{m^{2}_{b_{1}/h_{1}}+Q^{2}}$
(36)

are dipole functions with
$m_{h_{1}}=$1.166\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$m_{b_{1}}=$1.229\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

Note that several of the terms in eqs. 31, 32, 33, 34 and 35 benefit from an
enhancement factor $M_{N}/m_{\pi}\sim 10$, which facilitates detection of
tensor currents.

The above parametrizations of the form factors – especially the axial vector
and tensor ones – have been derived with momentum transfers
$\sqrt{Q^{2}}\lesssim$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ in mind.
However, given their weak dependence on $Q^{2}$ at
$\mathcal{O}($\mathrm{G}\mathrm{e}\mathrm{V}$)$ momentum transfers, which is
evident from fig. 1, we will use them also to extrapolate to larger momentum
transfers relevant for accelerator neutrino experiments.

We are now ready to express the spin-summed squared amplitudes for CCQE
neutrino–nucleon scattering ($\nu_{\beta}+n\to\ell_{\alpha}^{-}+p^{+})$ in
terms of the form factors and as functions of the Mandelstam variables:

$\displaystyle\frac{1}{2}\sum_{\text{spin}}A_{X,\alpha}A^{*}_{Y,\alpha}=C_{0}\bigg{[}B_{\alpha,XY}(Q^{2})+C_{\alpha,XY}(Q^{2})\frac{s-u}{M_{N}^{2}}+D_{\alpha,XY}(Q^{2})\frac{(s-u)^{2}}{M_{N}^{4}}\bigg{]}\,,$
(37)

where $X,Y={L,R,S,P,T}$ and $C_{0}\equiv 4|V_{ud}|^{2}M_{N}^{4}/v^{4}$. The
factor $\frac{1}{2}$ on the left hand side averages over spin orientations of
the initial nucleon. In the approximation that the nucleon is initially at
rest, kinematics dictates that
$s-u=4M_{N}E_{\nu}-Q^{2}-m_{\ell_{\alpha}}^{2}$. Assuming moreover that proton
and neutron masses are identical, the functions $B_{\alpha,XY}(Q^{2})$,
$C_{\alpha,XY}(Q^{2})$ and $D_{\alpha,XY}(Q^{2})$ are111The full expressions
without these approximation are given in appendix A. For our numerical
results, we will use the full expressions because nucleons bound in nuclei
cannot be assumed to be initially at rest.

$\displaystyle B_{\alpha,LL}$
$\displaystyle=\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\bigg{\\{}\Big{(}1+\frac{Q^{2}}{4M_{N}^{2}}\Big{)}G_{A}^{2}-\Big{(}1-\frac{Q^{2}}{4M_{N}^{2}}\Big{)}\Big{(}F_{1}^{2}-\frac{Q^{2}}{4M_{N}^{2}}F_{2}^{2}\Big{)}+\frac{Q^{2}}{M_{N}^{2}}F_{1}F_{2}$
$\displaystyle\quad-\frac{m_{\ell_{\alpha}}^{2}}{4M_{N}^{2}}\bigg{[}\Big{(}F_{1}+F_{2}\Big{)}^{2}+\Big{(}G_{A}-\tilde{G}_{P}\Big{)}^{2}-\Big{(}1+\frac{Q^{2}}{4M_{N}^{2}}\Big{)}\tilde{G}_{P}^{2}\bigg{]}\bigg{\\}}\,,$
(38) $\displaystyle C_{\alpha,LL}$
$\displaystyle=\frac{Q^{2}}{M_{N}^{2}}\Big{(}G_{A}(F_{1}+F_{2})\Big{)}\,,$
(39) $\displaystyle D_{\alpha,LL}$
$\displaystyle=\frac{1}{4}\Big{(}G_{A}^{2}+F_{1}^{2}+\frac{Q^{2}}{4M_{N}^{2}}F_{2}^{2}\Big{)}\,,$
(40) $\displaystyle B_{\alpha,RR}$ $\displaystyle=B_{\alpha,LL}\,,$ (41)
$\displaystyle C_{\alpha,RR}$ $\displaystyle=-C_{\alpha,LL}\,,$ (42)
$\displaystyle D_{\alpha,RR}$ $\displaystyle=D_{\alpha,LL}\,,$ (43)
$\displaystyle B_{\alpha,SS}$
$\displaystyle=\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\Big{(}1+\frac{Q^{2}}{4M_{N}^{2}}\Big{)}G_{S}^{2}\,,$
(44) $\displaystyle C_{\alpha,SS}$ $\displaystyle=0\,,$ (45) $\displaystyle
D_{\alpha,SS}$ $\displaystyle=0\,,$ (46) $\displaystyle B_{\alpha,PP}$
$\displaystyle=\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\frac{Q^{2}}{4M_{N}^{4}}G_{P}^{2}\,,$
(47) $\displaystyle C_{\alpha,PP}$ $\displaystyle=0\,,$ (48) $\displaystyle
D_{\alpha,PP}$ $\displaystyle=0\,,$ (49) $\displaystyle B_{\alpha,TT}$
$\displaystyle=-\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\bigg{\\{}G_{T}^{2}+\frac{Q^{2}}{4M_{N}^{2}}\bigg{[}+4G_{T}G_{T}^{(1)}-\frac{m_{\ell_{\alpha}}^{2}}{M_{N}^{2}}\Big{(}2G_{T}^{(2)}\big{(}G_{T}-4G_{T}^{(2)}\big{)}$
$\displaystyle\hskip
227.62204pt+\big{(}G_{T}^{(1)}\big{)}^{2}-4G_{T}^{(1)}G_{T}^{(2)}\Big{)}\bigg{]}$
$\displaystyle\quad+\frac{m_{\ell_{\alpha}}^{2}}{2M_{N}^{2}}\bigg{[}G_{T}^{2}-4G_{T}\Big{(}({G_{T}^{(1)}}+{G_{T}^{(2)}}\Big{)}+2\Big{(}G_{T}^{(1)}+2G_{T}^{(2)}\Big{)}^{2}\bigg{]}$
$\displaystyle\quad+4\bigg{(}\frac{Q^{2}}{4M_{N}^{2}}\bigg{)}^{2}\Big{(}\big{(}G_{T}^{(1)}\big{)}^{2}+\frac{m_{\ell_{\alpha}}^{2}}{M_{N}^{2}}\big{(}G_{T}^{(2)}\big{)}^{2}\Big{)}\bigg{\\}}\,,$
(50) $\displaystyle C_{\alpha,TT}$ $\displaystyle=0\,,$ (51) $\displaystyle
D_{\alpha,TT}$
$\displaystyle=\frac{1}{2}G_{T}^{2}+\frac{Q^{2}}{4M_{N}^{2}}\bigg{[}-2G_{T}G_{T}^{(2)}+\Big{(}G_{T}^{(1)}+2G_{T}^{(2)}\Big{)}^{2}\bigg{]}+4{G_{T}^{(2)}}^{2}\Big{(}\frac{Q^{2}}{4M_{N}^{2}}\Big{)}^{2}\,,$
(52) for $XX$-type interactions, and $\displaystyle B_{\alpha,LR}$
$\displaystyle=\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\bigg{\\{}-\Big{(}1+\frac{Q^{2}}{4M_{N}^{2}}\Big{)}G_{A}^{2}-\Big{(}1-\frac{Q^{2}}{4M_{N}^{2}}\Big{)}\Big{(}F_{1}^{2}-\frac{Q^{2}}{4M_{N}^{2}}F_{2}^{2}\Big{)}+\frac{Q^{2}}{M_{N}^{2}}F_{1}F_{2}$
$\displaystyle\quad-\frac{m_{\ell_{\alpha}}^{2}}{4M_{N}^{2}}\bigg{[}\Big{(}F_{1}+F_{2}\Big{)}^{2}-\Big{(}G_{A}-\tilde{G}_{P}\Big{)}^{2}+\Big{(}1+\frac{Q^{2}}{4M_{N}^{2}}\Big{)}\tilde{G}_{P}^{2}\bigg{]}\bigg{\\}}\,,$
(53) $\displaystyle C_{\alpha,LR}$ $\displaystyle=0\,,$ (54) $\displaystyle
D_{\alpha,LR}$
$\displaystyle=\frac{1}{4}\Big{(}-G_{A}^{2}+F_{1}^{2}+\frac{Q^{2}}{4M_{N}^{2}}F_{2}^{2}\Big{)}\,,$
(55) $\displaystyle B_{\alpha,LS}$ $\displaystyle=0\,,$ (56) $\displaystyle
C_{\alpha,LS}$
$\displaystyle=\frac{m_{\ell_{\alpha}}}{2M_{N}}\Big{(}F_{1}-\frac{Q^{2}}{4M_{N}^{2}}F_{2}\Big{)}G_{S}\,,$
(57) $\displaystyle D_{\alpha,LS}$ $\displaystyle=0\,,$ (58) $\displaystyle
B_{\alpha,LP}$
$\displaystyle=-\frac{m_{\ell_{\alpha}}}{2M_{N}}\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\Big{(}G_{A}+\frac{Q^{2}}{4M_{N}^{2}}G_{P}\Big{)}G_{P}\,,$
(59) $\displaystyle C_{\alpha,LP}$ $\displaystyle=0\,,$ (60) $\displaystyle
D_{\alpha,LP}$ $\displaystyle=0\,,$ (61) $\displaystyle B_{\alpha,LT}$
$\displaystyle=-\frac{m_{\ell_{\alpha}}^{2}+Q^{2}}{M_{N}^{2}}\frac{m_{\ell_{\alpha}}}{M_{N}}\bigg{\\{}F_{1}\Big{(}3G_{T}-2G_{T}^{(1)}-4G_{T}^{(2)}\Big{)}+2F_{2}G_{T}$
$\displaystyle\quad+\frac{Q^{2}}{4M_{N}^{2}}\Big{[}4F_{1}\big{(}G_{T}^{(1)}-G_{T}^{(2)}\big{)}+F_{2}\big{(}-G_{T}+6G_{T}^{(1)}+4G_{T}^{(2)}\big{)}\Big{]}+4F_{2}G_{T}^{(2)}\Big{(}\frac{Q^{2}}{4M_{N}^{2}}\Big{)}^{2}\bigg{\\}}\,,$
(62) $\displaystyle C_{\alpha,LT}$
$\displaystyle=-\frac{m_{\ell_{\alpha}}}{M_{N}}\Big{(}3G_{A}G_{T}+(\tilde{G}_{P}G_{T}-4G_{A}G_{T}^{(1)})\frac{Q^{2}}{4M_{N}^{2}}\Big{)}\,,$
(63) $\displaystyle D_{\alpha,LT}$ $\displaystyle=0\,.$ (64)

for the interference terms with the SM amplitude ($LX$-type interactions).

For anti-neutrinos interacting via
$\bar{\nu}_{\beta}+p^{+}\to\ell_{\alpha}^{+}+n$, the relevant functions for
the LL, RR, LS and LT interactions are
$({\bar{C}}_{\alpha,XY})=-(C_{\alpha,XY})$, for the LP interaction we have
$\bar{B}_{\alpha,LP}=-B_{\alpha,LP}$, and the remaining functions are
identical to their $\nu$ counterparts. Finally, for both neutrino and anti-
neutrino scatterings we always have
$\sum_{\text{spin}}A^{*}_{X,\alpha}A_{Y,\alpha}=\sum_{\text{spin}}{A}_{X,\alpha}A^{*}_{Y,\alpha}$.

Let us focus first on the interference ($LX$-type) terms which are typically
more important than the $XX$-type contributions because they will be
multiplied by only one power of the new physics couplings $\epsilon_{X}$. We
see that among the interference terms, the only ones that are not proportional
to $m_{\ell_{\alpha}}/M_{N}$ are of the $LR$ type (interference between the
left-handed SM current and a right-handed new physics current, eqs. 53, 54 and
55). Therefore, for electron (muon) neutrinos, for which
$m_{\ell_{\alpha}}/M_{N}\sim$5\text{\times}{10}^{-4}$~{}(0.1)$, one might
expect right-handed new physics currents parameterized by $\epsilon_{R}$ to be
the easiest to detect. However, it turns out that the $F_{1}$, $F_{2}$, and
$G_{S}$ form factors appearing in eqs. 53 and 55 are only $\mathcal{O}(1)$,
while tensor (eqs. 62 and 63) and pseudoscalar interactions (eq. 59) benefit
from form factors that are larger (see discussion below eq. 21 and below eq.
36), compensating the smallness of $m_{\ell_{\alpha}}/M_{N}$.

At quadratic order in the $\epsilon_{X}$’s we do not have any such
suppressions and all interactions contribute, especially the pseudoscalar and
tensor ones with their large form factors.

###  2.3 Neutrino–Nucleus Scattering

While interactions of neutrinos with nucleons are already highly non-trivial,
the fact that nucleons are embedded in nuclei adds another layer of
complexity. In this section we consider the scattering of (anti)neutrinos off
a target nucleus $A$. We will focus in particular on $A=16$ (oxygen, relevant
for water Čerenkov detectors), for which nuclear spectral functions are
available.222Our results should to a good approximation be applicable also to
scattering on carbon ($A=12$) and nitrogen ($A=14$), given that these nuclei
are similar in mass to oxygen. We plan to repeat our analysis for scattering
on 40Ar once the corresponding spectral functions – currently under
development – are published.

In the limit of moderate momentum transfer
($|{\mathbf{q}}|\gtrsim$400\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$), a
factorization scheme can be adopted to describe the hadronic final state
Benhar:2006wy . In the quasielastic region, the latter can be simplified as
the product of a single nucleon with momentum
$p^{\prime}=(E^{\prime},{{\mathbf{p}}^{\prime}})$ that is decoupled, and the
$A-1$-nucleon residual system:

$\displaystyle\ket{\Psi_{f}^{A}}=\ket{p^{\prime}}\otimes\ket{\Psi^{A-1}_{n},p_{A-1}}\,.$
(65)

Here, $\ket{p^{\prime}}$ is the final state nucleon produced at the primary
vertex, assumed to be in a plane wave state, and
$\ket{\Psi^{A-1}_{n},p_{A-1}}$ is the wave-function describing the remnant
nucleus with momentum $p_{A-1}$. The subscript $n$ labels the possible states
of the nuclear remnant, with $n=0$ corresponding to the ground state and $n>0$
to excited states. With this factorization ansatz, and inserting a complete
set of free single-nucleon states satisfying $\sum_{N}\int
d^{3}p_{N}\ket{N,p_{N}}\bra{p_{N},N}=1$ (where the sum runs over all nucleons
in the target nucleus and the integral runs over the nucleon 3-momentum), the
matrix element of the one-body current operator is

$\displaystyle\bra{\Psi_{f}^{A}}J^{\mu}\ket{\Psi^{A}_{0}}=\sum_{N}\int
d^{3}p_{N}\big{[}\\!\bra{\Psi^{A-1}_{n}}\otimes\bra{p_{N},N}\big{]}\ket{\Psi^{A}_{0}}\bra{p^{\prime}}\sum_{i}j^{\mu}_{i}\ket{p_{N},N}\,.$
(66)

In this expression $\ket{\Psi^{A}_{0}}$ is the ground state of the initial-
state nuclear system, $p^{\prime}=q+p_{N}$ is the final 4-momentum of a free
nucleon with initial 4-momentum $p_{N}$, and $J^{\mu}=\sum_{i}j^{\mu}_{i}$, is
the nuclear current operator, which we write as a sum of one-body currents,
each of them corresponding to one of the quark bilinear operators from eqs.
10, 11, 12, 13 and 14.

Exploiting momentum conservation at the single nucleon vertex we can rewrite
the squared spin-summed amplitudes in the nuclear case. Considering the system
where a nucleon with momentum $p_{N}$ is removed, leaving behind a residual
nucleus with excitation energy $E^{*}$,333In other words, $E^{*}$ is the
absolute value of the energy required to remove one nucleon from the nucleus.
we have

$\displaystyle\frac{1}{2}\sum_{\text{spin}}\mathcal{A}_{X,\alpha}\mathcal{A}^{*}_{Y,\alpha}$
$\displaystyle=\frac{1}{2}\int\frac{d^{3}p_{N}}{(2\pi)^{3}}P_{h}({{\mathbf{p}}_{N},E^{*}})\frac{m_{N}}{e({\mathbf{p}}_{N})}\frac{m_{N}}{e({\mathbf{q}}+{\mathbf{p}}_{N})}$
$\displaystyle\quad\times\sum_{\text{spin}}\sum_{N}A_{X,\alpha}A^{*}_{Y,\alpha}\,\delta\big{(}\tilde{\omega}+e({\mathbf{p}}_{N})-e({\mathbf{q}}+{{\mathbf{p}}_{N}})\big{)}\,,$
(67)

where $e({\mathbf{p}}_{N})\equiv({\mathbf{p}}_{N}^{2}+m_{N}^{2})^{1/2}$, and
$\tilde{\omega}$ is the effective energy transfer from the neutrino to the
nucleon. It is defined in terms of the physical energy transfer $\omega$ (the
zero component of the 4-vector $q\equiv(\omega,{\mathbf{q}})$) via

$\displaystyle\tilde{\omega}$
$\displaystyle=\omega+m_{N}-E^{*}-e({\mathbf{p}}_{N})\,.$ (68)

The correction terms on the right-hand-side of eq. 68 account for the fact
that the initial state nucleon is bound. The one-nucleon spectral function,
$P_{h}({\mathbf{p}}_{N},E)$, can be written as

$\displaystyle
P_{h}({\mathbf{p}}_{N},E^{*})=\sum_{n}\Big{|}\bra{\Psi^{A}_{0}}\big{[}\ket{p_{N}}\otimes\ket{\Psi^{A-1}_{n}}\big{]}\Big{|}^{2}\,\delta(E^{*}+E_{0}^{A}-E_{n}^{A-1}+m_{N})\,,$
(69)

where the energy of the nuclear ground state is $E_{0}^{A}$ and the one of the
remnant system, which can be found in any state $n$, is $E_{n}^{A-1}$. The
factors $m_{N}/e({\bf p_{N}})$ and $m_{N}/e(\textbf{q}+{\bf p_{N}})$ in eq. 67
are included to account for the fact that the four-spinors in the relativistic
matrix element are defined using the covariant normalization, whereas the
spinors appearing in the spectral function are normalized to unity.

Various methods can be employed to calculate the spectral function of finite
nuclei Dickhoff:2004xx ; Barbieri:2016uib ; Barbieri:2019ual ; Sobczyk:2023mey
; CLAS:2022odn ; Andreoli:2021cxo . In this study, we utilize the spectral
function of 16O from Ref. Benhar:1994hw ; Benhar:1989aw . This approach
employs a semi-phenomenological method, combining $(e,e^{\prime}p)$ scattering
data with nuclear matter calculations derived from the Correlated Basis
Function formalism.

##  3 Cross Section Results

Using the formalism outlined in the previous sections, we can now calculate
the CCQE neutrino–nucleus cross-section, including the full set of EFT
operators. In the laboratory frame, where the initial nucleus is at rest, the
differential neutrino–nucleus cross-section is given by

$\displaystyle\frac{d\sigma_{\alpha\beta}}{dQ^{2}}$
$\displaystyle=\frac{d\hat{\sigma}_{LL,\alpha}}{dQ^{2}}\delta_{\alpha\beta}+\sum_{X}\Big{(}[\epsilon_{X}]_{\alpha\beta}\frac{d\hat{\sigma}_{LX,\alpha}}{dQ^{2}}\delta_{\alpha\beta}+h.c.\Big{)}+\sum_{X,Y,\beta}[\epsilon_{X}]_{\alpha\beta}\,[\epsilon_{Y}]^{*}_{\alpha\beta}\frac{d\hat{\sigma}_{XY,\alpha}}{dQ^{2}}\,,$
(1)

with

$\displaystyle\frac{d\hat{\sigma}_{XY,\alpha}}{dQ^{2}}$
$\displaystyle=\frac{1}{64\pi}\frac{1}{(p_{\nu}\cdot
p_{N_{i}})^{2}}\bigg{[}\frac{1}{2}\sum_{\text{spin}}\mathcal{A}_{X,\alpha}\mathcal{A}^{*}_{Y,\alpha}\bigg{]}\,.$
(2)

Here, $N_{i}=n$ for neutrino interactions ($\nu n$ scattering) and $N_{i}=p$
for anti-neutrino interactions ($\bar{\nu}p$ scattering). For the scattering
on free nucleons, one simply has to replace
$\mathcal{A}_{X,\alpha}\mathcal{A}^{*}_{Y,\alpha}$ by
$A_{X,\alpha}A^{*}_{Y,\alpha}$. From kinematics we have $p_{\nu}\cdot
p_{N_{i}}=M_{N}E_{\nu}$. The total cross section (eq. 5) is obtained by
integrating eq. 1 over $Q^{2}$, with the lower and upper integration limits
ParticleDataGroup:2016lqr

$\displaystyle Q^{2}_{\text{min}}$
$\displaystyle=\frac{1}{2E_{\nu}+M_{N}}\bigg{[}2M_{N}E_{\nu}^{2}-m_{\ell_{\alpha}}^{2}(M_{N}+E_{\nu})-E_{\nu}\sqrt{-2M_{N}^{2}(s+m_{\ell_{\alpha}}^{2})+(s-m_{\ell_{\alpha}}^{2})^{2}+M_{N}^{4}}\bigg{]}\,,$
(3) $\displaystyle Q^{2}_{\text{max}}$
$\displaystyle=\frac{1}{2E_{\nu}+M_{N}}\bigg{[}2M_{N}E_{\nu}^{2}-m_{\ell_{\alpha}}^{2}(M_{N}+E_{\nu})+E_{\nu}\sqrt{-2M_{N}^{2}(s+m_{\ell_{\alpha}}^{2})+(s-m_{\ell_{\alpha}}^{2})^{2}+M_{N}^{4}}\bigg{]}\,.$
(4)

The squared center of mass energy is $s=M_{N}^{2}+2M_{N}E_{\nu}$.

The individual contribution, $\hat{\sigma}_{XY,\alpha}$, to the total cross
sections as a function of the neutrino energy are shown in figs. 2 and 3 for
scattering on oxygen. Remember that the $\hat{\sigma}_{XY,\alpha}$ can be
negative – only after multiplying with the $[\epsilon_{X}]_{\alpha\beta}$
coefficients and summing according to eq. 5, the result for the total cross
section needs to be positive. In fact, $\sigma_{LR}$ exhibits a sign change
(from negative at lower energies to positive at higher energies) at
$E_{\nu}\sim$0.8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

Figure 2: Contributions to the CCQE differential cross sections for muon
neutrinos scattering on an oxygen target, as a function of the neutrino
energy. Results for $\nu_{e}$ scattering are very similar. The different
colored curves correspond to operators with different Lorentz structures, with
the SM ($LL$) case shown in gray. For interactions depending on the axial form
factor, we compare different parameterizations of that form factor: the dipole
from eq. 19 (dotted), the $z$-expansion fitted to neutrino–deuteron scattering
data (dashed), and the $z$-expansion fitted to lattice QCD results (solid).
For comparisons, we also show results for neutrino scattering on free nucleons
(thinner dot-dashed lines). The content of this plot is available in tabulated
form from the Our main cross-section results in tabulated form are available
from GitHub github . Figure 3: Same as fig. 2, but for $\nu_{\tau}$
scattering.

|  | $\nu_{e}$ | $\nu_{\mu}$ | $\nu_{\tau}$
---|---|---|---|---
| $E_{\nu}~{}[{\rm{GeV}}]$ | $\leq 1$ | $1<\cdots\leq 4$ | $>4$ | $\leq 1$ | $1<\cdots\leq 4$ | $>4$ | $\leq 4$ | $>4$
| LL | 2.4 | 3.1 | 3.0 | 2.3 | 3.1 | 3.0 | 3.4 | 3.4
| RR | 1.4 | 2.1 | 2.6 | 1.4 | 2.1 | 2.6 | 0.2 | 2.8
D2 Difference ($\%$) | PP | 4.9 | 7.5 | 7.8 | 4.5 | 7.2 | 7.5 | 12.1 | 6.1
| LR | 10.4 | 23.5 | 19.6 | 9.1 | 23.5 | 19.6 | 8.0 | 11.7
| LP | 1.5 | 0.04 | 0.2 | 1.4 | 0.9 | 0.05 | 9.4 | 4.1
| LL | 18.3 | 34.9 | 35.7 | 17.8 | 34.9 | 35.7 | 70.0 | 44.5
| RR | 11.8 | 22.4 | 30.5 | 11.7 | 22.4 | 30.5 | 30.8 | 37.0
LQCD Difference ($\%$) | PP | 60.5 | 145.0 | 162.7 | 56.5 | 138.6 | 154.6 | 310.6 | 90.4
| LR | 76.8 | 167.5 | 176.3 | 76.3 | 168.1 | 176.5 | 154.1 | 152.3
| LP | 14.0 | 10.5 | 4.1 | 16.3 | 17.3 | 10.9 | 974.8 | 86.0

Table 2: Differences (in per cent) between the energy-integrated cross
sections evaluated using axial form factors based on neutrino–deuteron
scattering (top) or lattice QCD (bottom) compared to the simple dipole form
factor from eq. 19. Different columns correspond to different energy ranges
and neutrino flavors, while different rows are for interactions with different
Lorentz structures.

Our main findings from figs. 2 and 3 can be summarized as following:

  1. 1.

New interactions can benefit from significant cross-section enhancements,
aiding their detection. This is particularly true for pseudoscalar and tensor
interactions, which benefit from large nucleonic form factors. In the case of
pseudoscalar currents, this is related to the fact that the pion is a
pseudoscalar, and nucleons couple strongly to pions. For the tensor case, the
enhancement can similarly be related to the existence of two-pion intermediate
states Hoferichter:2018zwu .

  2. 2.

The axial form factor introduces significant systematic uncertainties to the
neutrino–nucleus cross-section, indicated by the colored bands in figs. 2 and
3 and also summarized in table 2. This affects the SM contribution as well as
many types of new interactions, notably those with right-handed and
pseudoscalar Lorentz structures. We see that using lattice QCD, which predicts
a larger form factor, as input leads to cross sections that are larger by
$\mathcal{O}(10\%)$ factor than those based on the dipole form factor or
neutrino–deuteron scattering. Discrepancies are largest in the multi-GeV
energy range which is most relevant to long-baseline experiments like DUNE.
Note, however, that the large uncertainties affect mostly the normalization of
the cross section, not the energy dependence. Therefore, the energy spectrum
of observed neutrino events can be used to distinguish different types of new
interactions.

  3. 3.

Nuclear effects are crucial even at multi-GeV energies, in contradiction to
the widespread assumption that neutrino scattering on free nucleons is a good
approximation in the calculation of the CCQE cross sections at large $q^{2}$.
This is particularly apparent for tensor interactions (green lines in figs. 2
and 3) at energies $\gtrsim$6\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. The
observed phenomenon can be attributed to the Fermi motion of nucleons within
the nuclear medium. This introduces additional terms that are negligible when
scattering occurs on a single nucleon at rest in the laboratory frame.

  4. 4.

For $\tau$ neutrinos, the initial-state momentum of the bound target nucleon
lowers the energy threshold for $\nu_{\tau}$ interactions from
$3.45\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ for scattering on nucleons to
$\sim$2.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ for scattering on oxygen
nuclei. The effect is expected to be even more significant in larger nuclei
like 40Ar thanks to the larger initial-state momenta. The shift of the
threshold is much larger than the typical experimental energy resolution and
should therefore be clearly visible in the event spectrum, given sufficient
statistics.

##  4 Summary and Conclusions

Accurate calculations of neutrino–nucleus interaction cross sections are
pivotal for precision neutrino physics. And while controlling hadronic
uncertainties within the Standard Model already presents a substantial
challenge, the task becomes even more daunting when potential new interactions
beyond the Standard Model are introduced

In this work, we have taken an important step forward by deriving, for the
first time, a comprehensive set of neutrino–nucleus cross sections that
incorporate all conceivable hadronic currents within the framework of
Effective Field Theory. We have carefully determined the relevant vector,
axial vector, scalar, pseudoscalar, and tensor form factors of the nucleon.
Furthermore, we have established the corresponding nucleon matrix elements and
integrated them with the nuclear spectral function to account for nuclear
effects. An emphasis was placed on understanding systematic uncertainties,
particularly those tied to the nucleon axial form factor.

Our findings show that specific Lorentz structures, especially pseudoscalar
and tensor interactions, exhibit cross sections notably enhanced compared to
those of the Standard Model. Nonetheless, there exists a considerable margin
of uncertainty. Lattice QCD predicts the axial form factor, and the
pseudoscalar one which is derived from it, to be larger by about 10% compared
to the simple dipole parameterization and the $z$-expansion approach based on
neutrino–deuteron scattering data. We also observed that nuclear effects hold
significance across all Lorentz structures, even at multi-GeV energies. For
charged current $\nu_{\tau}$ interactions, factoring in the initial state
momentum of the impacted nucleon smears out the energy threshold.

Our results can be used in future studies of new physics in the neutrino
sector to relate observed event rates to the coupling constants of an
underlying theoretical model. Further recommendations on how to use our
results in practice are given in appendix B.

Directions for future work include applying our methods to different target
nuclei, as well as updating form factors and spectral functions with new data
from neutrino–nucleus and electron–nucleus scattering experiments. Attention
must be paid to the fact that this data itself may be contaminated by physics
beyond the Standard Model.

###### Acknowledgements.

It is a pleasure to thank Salvador Urrea and Noah Steinberg for very useful
discussions and for collaboration during parts of this project. This
manuscript was managed by Fermi Research Alliance, LLC under Contract No. DE-
AC02- 07CH11359 with the U.S. Department of Energy, Office of Science, Office
of High Energy Physics and by the NeuCol SciDAC-5 program (NR). The work of ZT
is supported by the Neutrino Theory Network Program Grant #DE-AC02-07CHI11359
and the US Department of Energy under award #DE-SC0020250. She also
appreciates the hospitality of Aspen Physics Center where this work was
partially developed. JK would like to acknowledge support from a Fermilab
Neutrino Physics Center Fellowship.

##  Appendix A Complete Expressions for the Squared Amplitudes

In this appendix, we present the full expression for the squared amplitudes,
without the approximation of nucleons at rest that was used in eqs. 38 to 64.
With the definitions $D_{0}\equiv 2V_{ud}^{2}/v^{4}$ and
$\mathcal{P}_{ij}\equiv p_{i}\cdot p_{j}$, we find

$\displaystyle\frac{1}{2}\sum_{\text{spin}}|A_{L(R),\alpha}|^{2}$
$\displaystyle=D_{0}\Bigg{\\{}\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
p}+\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
n}\Big{)}\bigg{[}\big{(}2F_{1}+F_{2}\big{)}^{2}+\big{(}\tilde{G}_{P}-2G_{A}\big{)}^{2}+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\big{(}F_{2}^{2}-\tilde{G}_{P}^{2}\big{)}\bigg{]}$
$\displaystyle\hskip
19.91684pt-\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
n}+\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
p}\Big{)}\bigg{[}4F_{1}F_{2}+3F_{2}^{2}+\tilde{G}_{P}\big{(}\tilde{G}_{P}-4G_{A}\big{)}-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\big{(}F_{2}^{2}+\tilde{G}_{P}^{2}\big{)}\bigg{]}$
$\displaystyle\hskip 14.22636pt\mp
8\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
p}-\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
n}\Big{)}\bigg{[}G_{A}\big{(}F_{1}+F_{2}\big{)}\bigg{]}$ (1)
$\displaystyle\hskip 19.91684pt-
M_{N}^{2}\mathcal{P}_{\ell_{\alpha}\nu}\bigg{[}\Big{(}1-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\Big{(}4F_{1}F_{2}+F_{2}^{2}+\tilde{G}_{P}\big{(}4G_{A}-\tilde{G}_{P}\big{)}-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\big{(}F_{2}^{2}-\tilde{G}_{P}^{2}\big{)}\Big{)}$
$\displaystyle\hskip
284.52756pt+4\big{(}F_{1}^{2}-G_{A}^{2}\big{)}\bigg{]}\Bigg{\\}}\,,$
$\displaystyle\frac{1}{2}\sum_{\text{spin}}|A_{S,\alpha}|^{2}$
$\displaystyle=2D_{0}G_{S}^{2}\mathcal{P}_{\ell_{\alpha}\nu}\Big{(}\mathcal{P}_{np}+M_{N}^{2}\Big{)}\,,$
(2) $\displaystyle\frac{1}{2}\sum_{\text{spin}}|A_{P,\alpha}|^{2}$
$\displaystyle=2D_{0}G_{P}^{2}\mathcal{P}_{\ell_{\alpha}\nu}\Big{(}\mathcal{P}_{np}-M_{N}^{2}\Big{)}\,,$
(3) $\displaystyle\frac{1}{2}\sum_{\text{spin}}|A_{T,\alpha}|^{2}$
$\displaystyle=4D_{0}\Bigg{\\{}\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
n}+\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
p}\Big{)}\bigg{[}2(G_{T}^{(1)}+G_{T}^{(2)})\Big{(}G_{T}-G_{T}^{(1)}-G_{T}^{(2)}\Big{)}$
$\displaystyle\hskip
199.16928pt+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{(}{G_{T}^{(1)}}^{2}-2{G_{T}^{(2)}}^{2}\Big{)}\bigg{]}$
(4) $\displaystyle\qquad+\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
p}+\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
n}\Big{)}\bigg{[}2\frac{\mathcal{P}_{np}}{M_{N}^{2}}G_{T}^{(2)}\bigg{(}-G_{T}+2G_{T}^{(1)}+G_{T}^{(2)}\Big{(}1+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{)}$
$\displaystyle\hskip 236.15796pt+\big{(}G_{T}-G_{T}^{(1)}\big{)}^{2}\bigg{]}$
$\displaystyle\qquad+M_{N}^{2}\mathcal{P}_{\ell_{\alpha}\nu}\bigg{[}G_{T}^{2}\frac{\mathcal{P}_{np}}{M_{N}^{2}}+2\Big{(}1-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{(}G_{T}^{(1)}+G_{T}^{(2)}\Big{(}1+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{)}$
$\displaystyle\hskip
170.71652pt\times\bigg{(}G_{T}-G_{T}^{(1)}-G_{T}^{(2)}\Big{(}1+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{)}\bigg{]}\Bigg{\\}}\,,$

where in appendix A the $-$ and $+$ signs correspond to $|A_{L,\alpha}|^{2}$
and $|A_{R,\alpha}|^{2}$, respectively. For the interference terms with the SM
we find

$\displaystyle\frac{1}{2}\sum_{\text{spin}}A_{L,\alpha}A^{*}_{R,\alpha}$
$\displaystyle=D_{0}\Bigg{\\{}\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
p}+\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
n}\Big{)}\bigg{[}\big{(}2F_{1}+F_{2}\big{)}^{2}-\big{(}\tilde{G}_{P}-2G_{A}\big{)}^{2}+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\big{(}F_{2}^{2}+\tilde{G}_{P}^{2}\big{)}\bigg{]}$
$\displaystyle\quad-\Big{(}\mathcal{P}_{\ell_{\alpha}n}\mathcal{P}_{\nu
n}+\mathcal{P}_{\ell_{\alpha}p}\mathcal{P}_{\nu
p}\Big{)}\bigg{[}4F_{1}F_{2}+3F_{2}^{2}-\tilde{G}_{P}\big{(}\tilde{G}_{P}-4G_{A}\big{)}-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\big{(}F_{2}^{2}-\tilde{G}_{P}^{2}\big{)}\bigg{]}$
$\displaystyle\quad+M_{N}^{2}\mathcal{P}_{\ell_{\alpha}\nu}\bigg{[}\Big{(}1-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\Big{(}4F_{1}F_{2}+F_{2}^{2}-\tilde{G}_{P}\big{(}4G_{A}-\tilde{G}_{P}\big{)}-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\big{(}F_{2}^{2}+\tilde{G}_{P}^{2}\big{)}\Big{)}$
$\displaystyle\hskip
256.0748pt+4\big{(}F_{1}^{2}+G_{A}^{2}\big{)}\bigg{]}\Bigg{\\}}\,,$ (5)
$\displaystyle\frac{1}{2}\sum_{\text{spin}}A_{L,\alpha}A^{*}_{S,\alpha}$
$\displaystyle=D_{0}m_{\ell_{\alpha}}M_{N}G_{S}\Big{(}\mathcal{P}_{\nu
n}+\mathcal{P}_{\nu
p}\Big{)}\bigg{[}2F_{1}+F_{2}\Big{(}1-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{]}\,,$
(6) $\displaystyle\frac{1}{2}\sum_{\text{spin}}A_{L,\alpha}A^{*}_{P,\alpha}$
$\displaystyle=-D_{0}m_{\ell_{\alpha}}M_{N}G_{P}\Big{(}\mathcal{P}_{\nu
n}-\mathcal{P}_{\nu
p}\Big{)}\bigg{[}2G_{A}-\tilde{G}_{P}\Big{(}1-\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{]}\,,$
(7) $\displaystyle\frac{1}{2}\sum_{\text{spin}}A_{L,\alpha}A^{*}_{T,\alpha}$
$\displaystyle=-D_{0}m_{\ell_{\alpha}}M_{N}\Bigg{\\{}\Big{(}\mathcal{P}_{\nu
n}-\mathcal{P}_{\nu
p}\Big{)}\bigg{[}2F_{1}\Big{(}3G_{T}-4G_{T}^{(1)}-2G_{T}^{(2)}\Big{)}$ (8)
$\displaystyle\qquad\qquad+F_{2}\Big{(}5G_{T}-6G_{T}^{(1)}-2G_{T}^{(2)}\Big{)}$
$\displaystyle\qquad\qquad+\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{(}4F_{1}\big{(}G_{T}^{(1)}-G_{T}^{(12}\big{)}-F_{2}\big{(}G_{T}-6G_{T}^{(1)}\big{)}+2F_{2}G_{T}^{(2)}\frac{\mathcal{P}_{np}}{M_{N}^{2}}\Big{)}\bigg{]}$
$\displaystyle\qquad+\Big{(}\mathcal{P}_{\nu n}+\mathcal{P}_{\nu
p}\Big{)}\bigg{[}2G_{A}\big{(}3G_{T}-2G_{T}^{(1)}\big{)}-\tilde{G}_{P}G_{T}+\big{(}4G_{A}G_{T}^{(1)}+G_{T}\tilde{G}_{P}\big{)}\frac{\mathcal{P}_{np}}{M_{N}^{2}}\bigg{]}\Bigg{\\}}\,.$

##  Appendix B Application to Oscillation Experiments

In this appendix we explain how the results of this work can be used at
neutrino oscillation experiments. We will follow closely the formalism
introduced at Refs. Falkowski:2019xoe ; Falkowski:2019kfn . The first
important point to note is that the new interactions introduced in the
Lagrangian of eq. 1 will in general affect not only the neutrino detection
cross sections, but also the production rate. Let us assume neutrinos are
produced at a source $S$ in conjunction with a charged lepton $\ell_{\beta}$,
and that the experiment can have multiply “sources” (e.g. different meson
decays, $S=\pi^{\pm},K^{0}_{S},K^{0}_{L},K^{\pm}$, etc., in a neutrino beam
experiment). Neutrinos scatter off the target nucleon or nucleus and produce a
charged lepton $\ell_{\alpha}$. The differential event rate in a detector a
distance $L$ away from the source is Falkowski:2021bkq :

$\displaystyle\frac{dR_{\alpha}}{dE_{\nu}}$
$\displaystyle=N_{T}\,\sigma^{\text{SM}}_{\alpha}(E_{\nu})\,\sum_{\beta,S}\Phi^{S,\text{SM}}_{\beta}(E_{\nu})\,\tilde{P}^{S}_{\beta\alpha}(E_{\nu},L)\,.$
(1)

In this expression, $N_{T}$ is the number of target particles in the detector,
the SM neutrino flux for each source $S$ (in the absence of new physics) is
$\Phi^{S,\text{SM}}_{\beta}(E_{\nu})$, and the SM detection cross section is
$\sigma^{\text{SM}}_{\alpha}(E_{\nu})=\hat{\sigma}_{LL,\alpha}$. The pseudo-
probability $\tilde{P}^{S}_{\beta\alpha}(E_{\nu},L)$ is the oscillation
probability convoluted with terms describing the new physics in neutrino
production and detection. It is given by:

$\displaystyle\tilde{P}^{S}_{\beta\alpha}(E_{\nu},L)$
$\displaystyle=\sum_{n,m}e^{-i\Delta m_{nm}^{2}L/(2E_{\nu})}$
$\displaystyle\times\bigg{[}U_{\beta n}^{*}U_{\beta
m}+\\!\sum_{X,j,k}p_{XL,\beta}^{S,jk}[\epsilon_{X}^{jk}U]_{\beta
n}^{*}U_{\beta m}+\\!\\!\sum_{X,j,k}p_{XL,\beta}^{S,jk*}U_{\beta
n}^{*}[\epsilon_{X}^{jk}U]_{\beta
m}+\\!\\!\\!\sum_{X,Y,j,k}p_{XY,\beta}^{S,jk}[\epsilon_{X}^{jk}U]_{\beta
n}^{*}[\epsilon_{Y}^{jk}U]_{\beta m}\bigg{]}$
$\displaystyle\times\bigg{[}U_{\alpha n}U_{\alpha
m}^{*}+\\!\sum_{X,r,s}d_{XL,\alpha}^{rs}[\epsilon_{X}^{rs}U]_{\alpha
n}U_{\alpha m}^{*}+\\!\sum_{X,r,s}d_{XL,\alpha}^{rs*}U_{\alpha
n}[\epsilon_{X}^{rs}U]^{*}_{\alpha
m}+\\!\sum_{X,Y,r,s}d_{XY,\alpha}^{rs}[\epsilon_{X}^{rs}U]_{\alpha
n}[\epsilon_{Y}^{rs}U]^{*}_{\alpha m}\bigg{]}\,,$ (2)

where $\Delta m^{2}_{nm}\equiv m_{\nu_{n}}^{2}-m_{\nu_{m}}^{2}$ are the mass
squared differences, $U_{\alpha m}$ are the elements of the effective PMNS
matrix in matter,444It is important to keep in mind that new charged-current
interactions are typically accompanied by new neutral-current interactions
once the EFT defined by eq. 1 is embedded into an ultraviolet completion that
respects the SM $SU(2)$ symmetry. If the new neutral-current interactions have
an $L$ or $R$ Lorentz structure, they contribute to neutrino coherent forward
scattering, thereby modifying the effective PMNS matrix in matter. In
addition, new scalar ($S$) interactions lead to an effective modification of
the neutrino mass matrix in matter, which implies modifications to both the
PMNS matrix elements $U_{\alpha n}$ and to the mass-squared differences
$\Delta m_{nm}^{2}$. Precision studies should always take into account both
neutral-current and charged-current manifestations of the new physics. and the
sums run over neutrino mass eigenstates $n$, $m$, interactions $X=L,R,S,P,T$,
and quark flavor indices $j$, $k$ at the source or $r$, $s$ at the detector.
The impact of the new physics effects is parameterized by the production
coefficients, $p_{XY,\beta}^{S,jk}$, and the detection coefficients,
$d_{XY,\alpha}^{jk}$, which quantify how strongly a new interaction affects
the neutrino event rate. Roughly speaking, the $p_{XY,\beta}^{S,jk}$ give the
magnitude of the new physics contribution to the neutrino fluxes relative to
the SM flux. The coefficients with $X\neq Y$ describe the interference between
operators with different Lorentz structures, whereas the ones with $X=Y$
describe the non-interfering terms. Notably $p_{LL,\beta}^{S,jk}=1$ as $LL$ is
the Lorentz structure of SM weak interactions. Similarly, the
$d_{XY,\alpha}^{jk}$ are the ratios of the (partial) detection cross sections
with and without new physics. As long as only scattering on up and down quarks
is considered, they are given by

$\displaystyle
d_{XY,\alpha}^{ud}=\frac{\hat{\sigma}_{XY,\alpha}}{\sigma_{\alpha}^{\rm{SM}}}\,,$
(3)

where $\hat{\sigma}_{XY,\alpha}$ are the CCQE partial cross sections including
new physics from section 3, and $\sigma_{\alpha}^{\text{SM}}$ is the total SM
CCQE cross section.

For additional details on the production coefficients and the effect of new
physics on various neutrino fluxes we refer the reader to Refs.
Falkowski:2019xoe ; Falkowski:2019kfn ; Falkowski:2021bkq .

## References

  * (1) W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621–653.
  * (2) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, [arXiv:1008.4884].
  * (3) V. Cirigliano, M. Gonzalez-Alonso, and M. L. Graesser, Non-standard Charged Current Interactions: beta decays versus the LHC, JHEP 02 (2013) 046, [arXiv:1210.4553].
  * (4) A. Falkowski, M. González-Alonso, and K. Mimouni, Compilation of low-energy constraints on 4-fermion operators in the SMEFT, JHEP 08 (2017) 123, [arXiv:1706.03783].
  * (5) E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03 (2018) 016, [arXiv:1709.04486].
  * (6) E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions, JHEP 01 (2018) 084, [arXiv:1711.05270].
  * (7) A. Falkowski, G. Grilli di Cortona, and Z. Tabrizi, Future DUNE constraints on EFT, JHEP 04 (2018) 101, [arXiv:1802.08296].
  * (8) J. Aebischer, J. Kumar, and D. M. Straub, Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale, Eur. Phys. J. C 78 (2018), no. 12 1026, [arXiv:1804.05033].
  * (9) S. Descotes-Genon, A. Falkowski, M. Fedele, M. González-Alonso, and J. Virto, The CKM parameters in the SMEFT, JHEP 05 (2019) 172, [arXiv:1812.08163].
  * (10) I. Bischer and W. Rodejohann, General neutrino interactions from an effective field theory perspective, Nucl. Phys. B 947 (2019) 114746, [arXiv:1905.08699].
  * (11) W. Dekens and P. Stoffer, Low-energy effective field theory below the electroweak scale: matching at one loop, JHEP 10 (2019) 197, [arXiv:1908.05295]. [Erratum: JHEP 11, 148 (2022)].
  * (12) A. Falkowski, M. González-Alonso, and Z. Tabrizi, Consistent QFT description of non-standard neutrino interactions, JHEP 11 (2020) 048, [arXiv:1910.02971].
  * (13) A. Falkowski, M. González-Alonso, and O. Naviliat-Cuncic, Comprehensive analysis of beta decays within and beyond the Standard Model, JHEP 04 (2021) 126, [arXiv:2010.13797].
  * (14) A. Falkowski, M. González-Alonso, A. Palavrić, and A. Rodríguez-Sánchez, Constraints on subleading interactions in beta decay Lagrangian, arXiv:2112.07688.
  * (15) A. Falkowski, M. González-Alonso, J. Kopp, Y. Soreq, and Z. Tabrizi, EFT at FASER$\nu$, JHEP 10 (2021) 086, [arXiv:2105.12136].
  * (16) A. Lovato, J. Carlson, S. Gandolfi, N. Rocco, and R. Schiavilla, Ab initio study of $(\nu_{\ell},\ell^{-})$ and $(\overline{\nu}_{\ell},\ell^{+})$ inclusive scattering in 12C: confronting the MiniBooNE and T2K CCQE data, Phys. Rev. X 10 (2020), no. 3 031068, [arXiv:2003.07710].
  * (17) A. Lovato, S. Gandolfi, J. Carlson, E. Lusk, S. C. Pieper, and R. Schiavilla, Quantum Monte Carlo calculation of neutral-current $\nu-^{12}C$ inclusive quasielastic scattering, Phys. Rev. C 97 (2018), no. 2 022502, [arXiv:1711.02047].
  * (18) A. Lovato, S. Gandolfi, J. Carlson, S. C. Pieper, and R. Schiavilla, Electromagnetic response of 12C: A first-principles calculation, Phys. Rev. Lett. 117 (2016), no. 8 082501, [arXiv:1605.00248].
  * (19) J. E. Sobczyk, B. Acharya, S. Bacca, and G. Hagen, 40Ca transverse response function from coupled-cluster theory, arXiv:2310.03109.
  * (20) O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Spectral function of finite nuclei and scattering of GeV electrons, Nucl. Phys. A 579 (1994) 493–517.
  * (21) O. Benhar, D. day, and I. Sick, Inclusive quasi-elastic electron-nucleus scattering, Rev. Mod. Phys. 80 (2008) 189–224, [nucl-ex/0603029].
  * (22) O. Benhar, P. Coletti, and D. Meloni, Electroweak nuclear response in quasi-elastic regime, Phys. Rev. Lett. 105 (2010) 132301, [arXiv:1006.4783].
  * (23) N. Rocco, C. Barbieri, O. Benhar, A. De Pace, and A. Lovato, Neutrino-Nucleus Cross Section within the Extended Factorization Scheme, Phys. Rev. C 99 (2019), no. 2 025502, [arXiv:1810.07647].
  * (24) N. Rocco, Ab initio Calculations of Lepton-Nucleus Scattering, Front. in Phys. 8 (2020) 116.
  * (25) J. E. Amaro, M. B. Barbaro, J. A. Caballero, R. González-Jiménez, G. D. Megias, and I. Ruiz Simo, Electron- versus neutrino-nucleus scattering, J. Phys. G 47 (2020), no. 12 124001, [arXiv:1912.10612].
  * (26) R. Gonzaléz-Jiménez, G. D. Megias, M. B. Barbaro, J. A. Caballero, and T. W. Donnelly, Extensions of Superscaling from Relativistic Mean Field Theory: the SuSAv2 Model, Phys. Rev. C 90 (2014), no. 3 035501, [arXiv:1407.8346].
  * (27) G. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Caballero, and T. W. Donnelly, Inclusive electron scattering within the SuSAv2 meson-exchange current approach, Phys. Rev. D 94 (2016) 013012, [arXiv:1603.08396].
  * (28) J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, R. Gonzalez-Jimenez, G. D. Megias, and I. R. Simo, Neutrino-nucleus scattering in the SuSA model, Eur. Phys. J. ST 230 (2021), no. 24 4321–4338, [arXiv:2106.02857].
  * (29) L. Wolfenstein, Neutrino Oscillations in Matter, Phys. Rev. D 17 (1978) 2369–2374.
  * (30) J. W. F. Valle, Resonant Oscillations of Massless Neutrinos in Matter, Phys. Lett. B 199 (1987) 432–436.
  * (31) Y. Grossman, Nonstandard neutrino interactions and neutrino oscillation experiments, Phys. Lett. B 359 (1995) 141–147, [hep-ph/9507344].
  * (32) J. Kopp, M. Lindner, and T. Ota, Discovery reach for non-standard interactions in a neutrino factory, Phys. Rev. D 76 (2007) 013001, [hep-ph/0702269].
  * (33) J. Kopp, M. Lindner, T. Ota, and J. Sato, Non-standard neutrino interactions in reactor and superbeam experiments, Phys. Rev. D 77 (2008) 013007, [arXiv:0708.0152].
  * (34) S. Antusch, J. P. Baumann, and E. Fernandez-Martinez, Non-Standard Neutrino Interactions with Matter from Physics Beyond the Standard Model, Nucl. Phys. B 810 (2009) 369–388, [arXiv:0807.1003].
  * (35) M. B. Gavela, D. Hernandez, T. Ota, and W. Winter, Large gauge invariant non-standard neutrino interactions, Phys. Rev. D 79 (2009) 013007, [arXiv:0809.3451].
  * (36) T. Ohlsson, Status of non-standard neutrino interactions, Rept. Prog. Phys. 76 (2013) 044201, [arXiv:1209.2710].
  * (37) O. G. Miranda and H. Nunokawa, Non standard neutrino interactions: current status and future prospects, New J. Phys. 17 (2015), no. 9 095002, [arXiv:1505.06254].
  * (38) A. de Gouvêa and K. J. Kelly, Non-standard Neutrino Interactions at DUNE, Nucl. Phys. B 908 (2016) 318–335, [arXiv:1511.05562].
  * (39) P. Coloma, M. C. Gonzalez-Garcia, M. Maltoni, J. a. P. Pinheiro, and S. Urrea, Global constraints on non-standard neutrino interactions with quarks and electrons, JHEP 08 (2023) 032, [arXiv:2305.07698].
  * (40) A. Falkowski, M. González-Alonso, and Z. Tabrizi, Reactor neutrino oscillations as constraints on Effective Field Theory, JHEP 05 (2019) 173, [arXiv:1901.04553].
  * (41) J. Kopp, N. Rocco, and Z. Tabrizi, 2023.  https://github.com/ztabrizi/EFT-in-Neutrino-Nucleus-Scattering/.
  * (42) S. Weinberg, Charge symmetry of weak interactions, Phys. Rev. 112 (1958) 1375–1379.
  * (43) V. Cirigliano, S. Gardner, and B. Holstein, Beta Decays and Non-Standard Interactions in the LHC Era, Prog. Part. Nucl. Phys. 71 (2013) 93–118, [arXiv:1303.6953].
  * (44) C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics.  Oxford University Press, 2007.
  * (45) Particle Data Group Collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016), no. 10 100001.
  * (46) C. Andreopoulos et al., The GENIE Neutrino Monte Carlo Generator, Nucl. Instrum. Meth. A 614 (2010) 87–104, [arXiv:0905.2517].
  * (47) M. González-Alonso, O. Naviliat-Cuncic, and N. Severijns, New physics searches in nuclear and neutron $\beta$ decay, Prog. Part. Nucl. Phys. 104 (2019) 165–223, [arXiv:1803.08732].
  * (48) D. Simons, N. Steinberg, A. Lovato, Y. Meurice, N. Rocco, and M. Wagman, Form factor and model dependence in neutrino-nucleus cross section predictions, arXiv:2210.02455.
  * (49) B. Bhattacharya, R. J. Hill, and G. Paz, Model independent determination of the axial mass parameter in quasielastic neutrino-nucleon scattering, Phys. Rev. D 84 (2011) 073006, [arXiv:1108.0423].
  * (50) A. S. Meyer, M. Betancourt, R. Gran, and R. J. Hill, Deuterium target data for precision neutrino-nucleus cross sections, Phys. Rev. D 93 (2016), no. 11 113015, [arXiv:1603.03048].
  * (51) R. J. Hill, The Modern description of semileptonic meson form factors, eConf C060409 (2006) 027, [hep-ph/0606023].
  * (52) R. J. Hill and G. Paz, Model independent extraction of the proton charge radius from electron scattering, Phys. Rev. D 82 (2010) 113005, [arXiv:1008.4619].
  * (53) RQCD Collaboration, G. S. Bali, L. Barca, S. Collins, M. Gruber, M. Löffler, A. Schäfer, W. Söldner, P. Wein, S. Weishäupl, and T. Wurm, Nucleon axial structure from lattice QCD, JHEP 05 (2020) 126, [arXiv:1911.13150].
  * (54) M. González-Alonso and J. Martin Camalich, Isospin breaking in the nucleon mass and the sensitivity of $\beta$ decays to new physics, Phys. Rev. Lett. 112 (2014), no. 4 042501, [arXiv:1309.4434].
  * (55) Flavour Lattice Averaging Group Collaboration, S. Aoki et al., FLAG Review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80 (2020), no. 2 113, [arXiv:1902.08191].
  * (56) European Twisted Mass Collaboration, N. Carrasco et al., Up, down, strange and charm quark masses with Nf = 2+1+1 twisted mass lattice QCD, Nucl. Phys. B 887 (2014) 19–68, [arXiv:1403.4504].
  * (57) Fermilab Lattice, MILC, TUMQCD Collaboration, A. Bazavov et al., Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD, Phys. Rev. D 98 (2018), no. 5 054517, [arXiv:1802.04248].
  * (58) S. Sasaki and T. Yamazaki, Nucleon form factors from quenched lattice QCD with domain wall fermions, Phys. Rev. D 78 (2008) 014510, [arXiv:0709.3150].
  * (59) A. S. Meyer, A. Walker-Loud, and C. Wilkinson, Status of Lattice QCD Determination of Nucleon Form Factors and their Relevance for the Few-GeV Neutrino Program, Ann. Rev. Nucl. Part. Sci. 72 (2022) 205–232, [arXiv:2201.01839].
  * (60) N. Sharma, H. Dahiya, P. K. Chatley, and M. Gupta, Weak vector and axial-vector form factors in the chiral constituent quark model with configuration mixing, Phys. Rev. D 79 (2009) 077503, [arXiv:0904.2246].
  * (61) M. Hoferichter, B. Kubis, J. Ruiz de Elvira, and P. Stoffer, Nucleon Matrix Elements of the Antisymmetric Quark Tensor, Phys. Rev. Lett. 122 (2019), no. 12 122001, [arXiv:1811.11181]. [Erratum: Phys.Rev.Lett. 124, 199901 (2020)].
  * (62) W. H. Dickhoff and C. Barbieri, Selfconsistent Green’s function method for nuclei and nuclear matter, Prog. Part. Nucl. Phys. 52 (2004) 377–496, [nucl-th/0402034].
  * (63) C. Barbieri and A. Carbone, Self-consistent Green’s function approaches, Lect. Notes Phys. 936 (2017) 571–644, [arXiv:1611.03923].
  * (64) C. Barbieri, N. Rocco, and V. Somà, Lepton Scattering from 40Ar and Ti in the Quasielastic Peak Region, Phys. Rev. C 100 (2019), no. 6 062501, [arXiv:1907.01122].
  * (65) J. E. Sobczyk and S. Bacca, 16O spectral function from coupled-cluster theory: applications to lepton-nucleus scattering, arXiv:2309.00355.
  * (66) CLAS Collaboration, I. Korover et al., Observation of large missing-momentum (e,e’p) cross-section scaling and the onset of correlated-pair dominance in nuclei, Phys. Rev. C 107 (2023), no. 6 L061301, [arXiv:2209.01492].
  * (67) L. Andreoli, J. Carlson, A. Lovato, S. Pastore, N. Rocco, and R. B. Wiringa, Electron scattering on A=3 nuclei from quantum Monte Carlo based approaches, Phys. Rev. C 105 (2022), no. 1 014002, [arXiv:2108.10824].
  * (68) O. Benhar, A. Fabrocini, and S. Fantoni, The Nucleon Spectral Function in Nuclear Matter, Nucl. Phys. A 505 (1989) 267–299.