# One and Two-Body Current Contributions to Lepton-Nucleus Scattering

Alessandro Lovato  Physics Division, Argonne National Laboratory, Argonne,
Illinois 60439, USA  INFN-TIFPA Trento Institute of Fundamental Physics and
Applications, Via Sommarive, 14, 38123 Trento, Italy  Noemi Rocco  Theoretical
Physics Department, Fermi National Accelerator Laboratory, P.O. Box 500,
Batavia, IL 60410, USA  Noah Steinberg  Theoretical Physics Department, Fermi
National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60410, USA

###### Abstract

Modeling lepton-nucleus scattering with the accuracy required to extract
neutrino-oscillation parameters from long- and short-baseline experiments
necessitates retaining most quantum-mechanical effects. One such effect is the
interference between one- and two-body current operators in the transition
currents, which has been known to enhance the cross-sections, especially in
transverse kinematics. In this work, we incorporate such interference in the
spectral function formalism, which combines relativistic currents and
kinematics with an accurate description of the initial target state. Our
analysis of lepton-scattering off 12C demonstrates that interference effects
appreciably enhance the transverse electromagnetic response functions and the
flux-folded neutrino-nucleus cross section, in both cases, improving the
agreement with experimental data. We discuss the impact on the neutrino-
oscillation program and the determination of nucleon axial form factors.

††preprint: FERMILAB-PUB-23-458-CSAID-ND-T

##  I Introduction

Neutrino-oscillation experiments rely on precise theoretical calculations of
neutrino interactions with target nuclei to accurately extract oscillation
parameters. These calculations will become even more relevant for high-
statistics DUNE and Hyper-Kamiokande Acciarri  _et al._ (2015); Abe  _et al._
(2021a, 2018), where modeling neutrino-nucleus cross sections constitutes a
primary source of systematic uncertainty Benhar  _et al._ (2017); Katori and
Martini (2018); Alvarez-Ruso  _et al._ (2018); Ruso  _et al._ (2022). While
inclusive cross-sections are crucial for determining the energy distribution
of charged-current events based on the kinematics of the outgoing charged
lepton, models capable of providing accurate semi-exclusive predictions for
the final state become invaluable for correct calorimetric energy
reconstruction Ankowski  _et al._ (2015). The vast amount of available
electron scattering data, combined with the increasing body of measurements
from the accelerator neutrino program, has highlighted deficiencies in models
of lepton-nucleus cross sections routinely used in neutrino event generators
Carneiro  _et al._ (2020); Abe  _et al._ (2020); Khachatryan  _et al._ (2021);
Cai  _et al._ (2020); Ascencio  _et al._ (2022); Abe  _et al._ (2021b);
Abratenko  _et al._ (2023a, 2022); Ruterbories  _et al._ (2022); Abratenko
_et al._ (2023b); Kleykamp  _et al._ (2023). These limitations pose a major
obstacle to achieving the precision design goal of DUNE, which requires
percent-level accuracy in neutrino-nucleus cross sections Abi  _et al._
(2021).

This challenge has served as a catalyst for improving theoretical models of
neutrino-nucleus scattering. Accurate predictions at moderate energies and
momentum transfers have been achieved using “ab-initio” methods, starting from
consistent Hamiltonians and currents. Continuum Quantum Monte Carlo Carlson
_et al._ (2015); Gandolfi  _et al._ (2020) (QMC) and, specifically, the
Green’s Function Monte Carlo (GFMC) methods have been successfully employed to
compute electroweak response functions of nuclei up to 12C in the quasielastic
(QE) region Lovato  _et al._ (2020, 2018, 2016). These calculations retain
quantum many-body correlations in both the initial and final states of the
reaction that are consistent with one- and two-body current operators. More
recently, Coupled Cluster (CC) theory has been applied to carry out the first
ab-intio calculation of the electromagnetic response functions of 40Ca Sobczyk
_et al._ (2023), including one-body currents only. Despite this remarkable
accomplishment, due to the slow convergence of single-particle basis
expansion, CC must use relatively soft nuclear forces as input Ekström  _et
al._ (2015); Jiang  _et al._ (2020), limiting its applicability in modeling
the final state of the reaction to momentum transfers $q\lesssim 400$ MeV. In
addition, both the GFMC and CC are constrained by their non-relativistic
nature, although some relativistic effects can be controlled by performing
calculations in a specific reference frame Rocco  _et al._ (2018);
Nikolakopoulos  _et al._ (2023). Most importantly, they can only predict
inclusive observables, while liquid argon time projection chambers have
excellent calorimetric reconstruction Abratenko  _et al._ (2021), which is
enabled by its unique combination of size, position resolution precision, and
low energy thresholds Castiglioni  _et al._ (2020).

These novel detector capabilities and the high energies characterizing most
neutrino beams have motivated the development of theoretical models that can
accommodate fully-relativistic kinematics and currents and provide detailed
information on the kinematic variables associated with the hadronic final
states. A key role in this regard is played by methods relying on the
factorization of the final state of the reaction, such as the spectral-
function (SF) formalism Benhar and Pandharipande (1993); Benhar  _et al._
(2008); Rocco (2020) and the short-time approximation (STA) Pastore  _et al._
(2020), which retain the vast majority of correlation effects in both the
initial and final states of the reaction. Both the recently introduced QMC-SF
Korover  _et al._ (2023), constructed from spectroscopic overlaps and two-
nucleon momentum distributions computed within the variational Monte Carlo
method, and the STA adopt the same model of nuclear dynamics as the GFMC for
describing the initial target state. Therefore, benchmarking the corresponding
predictions for the inclusive electron-nucleus scattering cross-section, as in
Ref. Andreoli  _et al._ (2022) for $A=3$ nuclei, brings to light uncertainties
inherent in factorization schemes and relativistic effects.

A recent comparison on electron- and neutrino scattering off 12C between the
GFMC and QMC-SF Nikolakopoulos  _et al._ (2023) has demonstrated the accuracy
of the GFMC at intermediate momentum transfer, provided that GFMC calculations
are performed in a reference frame that minimizes nucleon momenta and the
kinematics of the reaction is corrected leveraging the “two-fragment” model
Efros  _et al._ (2010); Rocco  _et al._ (2018). Assessing the uncertainty
inherent in the factorization of the final state when two-body currents are
present is hindered by the fact that GFMC includes interference between one-
and two-body currents, generally neglected in the SF formalism. Refs. Benhar
_et al._ (2015); Rocco  _et al._ (2016) are notable exceptions; there the
interference leading to two-particle two-hole final states was considered.
However, it was found to peak in the so-called “dip” region, while in GFMC
interference effects enhance the transverse response function in the QE region
Lovato  _et al._ (2020, 2018, 2016).

Consistent with the GFMC, both within the STA and relativistic mean-field
approaches, the interference between one- and two-body contributions has been
found to enhance the transverse electromagnetic response function in the QE
region Pastore  _et al._ (2020); Franco-Munoz  _et al._ (2022, 2023). A
similar pattern emerges from correlated-basis functions calculations of
infinite nuclear matter, and it has been ascribed to tensor correlations in
the nuclear many-body wave function Fabrocini (1997). On the other hand,
relativistic Fermi Gas calculations report either similar constructive
interference in the transverse channel Umino and Udias (1995) or negligible
effects Amaro  _et al._ (2002a). In this work, we significantly improve the
accuracy of the QMC-SF formalism by including one-nucleon knockout processes
induced by the interference between one- and two-body currents. The relative
importance of these processes with respect to pure one-body and two-body
contributions is analyzed in detail.

This work is organized as follows. In Sec. II, we introduce the SF formalism
and the factorization scheme, reviewing the calculation of the “diagonal” one-
body one-body and two-body two-body current contributions. The extension of
this formalism to accommodate the interference term between one- and two-body
currents is discussed in Sec. III, while the explicit expression of the two-
body current operator is reported in Sec. II. In Sec. IV, we show the improved
agreement between theoretical calculations that account for the interference
contributions and both electron-nucleus and neutrino-nucleus scattering data.
Finally, in Sec. V, we draw our conclusions and outline future perspectives of
this work.

##  II Factorization Scheme with One- and Two-Body Currents

In the one-boson exchange approximation, the inclusive lepton-nucleus
scattering cross section can be expressed in terms of the response tensor,
which contains all information on nuclear dynamics

$R^{\mu\nu}=\sum_{f}\langle 0|J^{\mu\dagger}|f\rangle\langle
f|J^{\nu}|0\rangle\delta(E_{0}+\omega-E_{f})\,.$ (1)

In the above equation, $|0\rangle$ is the nuclear ground state, $|f\rangle$
denotes the hadronic final state, and $J^{\mu}$ is the electroweak current
operator. While factorization schemes can accommodate real-pion production
Rocco  _et al._ (2019a), in this work we restrict our analysis to final states
containing nucleons only. The latter are the most relevant ones for lower-
energy experiments, like T2K Abe  _et al._ (2023) and the SBN program Machado
_et al._ (2019), whose neutrino fluxes peak at or below $1$ GeV. In this
regime, the nuclear current operator consists of a sum of one- and two-body
contributions

$J^{\mu}=\sum_{i}j^{\mu}_{i}+\sum_{j>i}j^{\mu}_{ij}\,,$ (2)

In the QE region, the dominant reaction mechanism is single nucleon knockout.
For large values of momentum transfer, the hadronic final state can be
approximated by

$|f\rangle\simeq|p^{\prime}\rangle\otimes|f_{A-1}\rangle,$ (3)

where $|p^{\prime}\rangle$ is the plane-wave state describing the nucleon
produced at the interaction vertex, and $|f_{A-1}\rangle$ is the spectator
system, carrying momentum $\textbf{p}_{A-1}$.

Inserting this factorization ansatz as well as a single-nucleon completeness
relation leads to the following expression for the one-body matrix element

$\langle 0|J^{\mu}|f\rangle\rightarrow\sum_{h}\langle
0|[|h\rangle\otimes|f_{A-1}\rangle]\langle
h|\sum_{i}j^{\mu}_{i}|p^{\prime}\rangle\,.$ (4)

The first term in the above summation can safely be computed within
nonrelativistic nuclear many body theory, as it is independent of momentum
transfer. The second piece is the nucleon level matrix element, which is fully
specified by the relativistic one-body currents and single-particle plane wave
states. The incoherent one-body current contribution to the response tensor
can then be expressed as

$\displaystyle R_{1b}^{\mu\nu}(\textbf{q},\omega)=$
$\displaystyle\int\frac{d^{3}h}{(2\pi)^{3}}dEP_{h}(\textbf{h},E)\frac{m^{2}_{N}}{e(\textbf{h})e(\textbf{h}+\textbf{q})}$
(5) $\displaystyle\times\sum_{i}\langle h|j_{i}^{\mu\dagger}|h+q\rangle\langle
h+q|j_{i}^{\nu}|h\rangle$
$\displaystyle\times\delta(\tilde{\omega}+e(\textbf{h})-e(\textbf{h}+\textbf{q})),$

where we use the relativistic expression of the single-particle energy
$e(\mathbf{k})=\sqrt{m_{N}^{2}+\mathbf{k}^{2}}$. The factors
$m_{N}/e(\textbf{h})$ and $m_{N}/e(\textbf{h}+\textbf{q})$ are included to
account for the covariant normalization of the Dirac spinors in the matrix
elements of the relativistic current. The energy transfer is replaced by
$\tilde{\omega}=\omega+m_{N}-E-e(\textbf{h})$ to account for the fact that the
initial nucleon is bound. The expression for the response tensor given in Eq.
(5) is identical for electron and neutrino-nucleus scattering — the
corresponding one-body current electromagnetic and electroweak operators can
be found in Ref. Lovato  _et al._ (2023). Note that for the axial form factor
we consider both the dipole parameterization with $M_{A}=1.0$ GeV, as well as
recent results from Lattice Quantum Chromo Dynamics (LQCD) Park  _et al._
(2022); Djukanovic  _et al._ (2022); Bali  _et al._ (2020).

The single-nucleon spectral function $P_{h}(\textbf{h},E)$ provides the
probability of removing a nucleon with momentum h and leaving the residual
nucleus with an excitation energy $E$ Benhar  _et al._ (2008)

$P_{h}(\textbf{h},E)=\sum_{f}|[_{A-1}\langle f|\otimes\langle
k|]|0\rangle|^{2}\delta(E-E_{f}^{A-1}+E_{0})$ (6)

It can be expressed as a sum of two components

$P_{h}(\textbf{h},E)=P_{\rm MF}(\textbf{h},E)+P_{\rm corr}(\textbf{h},E)\,.$
(7)

The mean-field term corresponds to $|f_{A-1}\rangle$ remaining in a bound
state, while the correlation contribution involves at least nucleon in the
continuum. The one used in this work is based on QMC calculations of
spectroscopic overlaps and two-body momentum distributions to obtain
$P_{h}^{\rm MF}(\textbf{h},E)$ and $P_{h}^{\rm corr}(\textbf{h},E)$,
respectively Lovato  _et al._ (2023); Simons  _et al._ (2022); Korover  _et
al._ (2023), and it is normalized as

$\int\frac{d^{3}k}{(2\pi)^{3}}dEP_{h}(\mathbf{k},E)=1\,.$ (8)

In the results presented, we also include the contribution of two-body
currents commonly denoted as meson exchange currents (MEC), whose primary
effect is to induce two-nucleon knockout. This channel is subdominant with
respect to single-nucleon knockout in the QE region, but it has been shown to
provide important strength at intermediate values of energy transfer between
the QE peak and resonance production region Dekker  _et al._ (1994); De Pace
_et al._ (2003); Rocco  _et al._ (2016). Two-nucleon emission processes
induced by MEC play a significantly enhance the flux-folded neutrino-
scattering cross section, bringing theoretical predictions closer to
experimental measurements Rodrigues  _et al._ (2016); Acero  _et al._ (2020);
Ascencio  _et al._ (2022); Martini  _et al._ (2010, 2011); Nieves  _et al._
(2011, 2012); Megias  _et al._ (2015); Gonzaléz-Jiménez  _et al._ (2014);
Pandey  _et al._ (2015).

MEC leading to two-nucleon knockout have been incorporated in the
factorization scheme as discussed in Refs. Benhar  _et al._ (2015); Lovato
_et al._ (2023); Rocco  _et al._ (2016, 2019b); Simons  _et al._ (2022). Here,
we report the final expression for the pure two-body current component of the
response tensor, associated with two-nucleon emission

$\displaystyle R^{\mu\nu}_{\rm 2b}({\bf q},\omega)=\frac{V}{4}\int
dE\frac{d^{3}h}{(2\pi)^{3}}\frac{d^{3}h^{\prime}}{(2\pi)^{3}}\frac{d^{3}p}{(2\pi)^{3}}\frac{m_{N}^{4}}{e({\bf
h})e({\bf h^{\prime}})e({\bf p})e({\bf p^{\prime}})}$
$\displaystyle\qquad\times P_{h}^{\rm NM}({\bf h},{\bf
h}^{\prime},E)\sum_{ij}\,\langle
h\,h^{\prime}|{j_{ij}^{\mu}}^{\dagger}|p\,p^{\prime}\rangle_{a}\,{}_{a}\langle
p\,p^{\prime}|j_{ij}^{\nu}|h\,h^{\prime}\rangle$
$\displaystyle\qquad\times\delta(\omega-E+2m_{N}-e(\mathbf{p})-e(\mathbf{p}^{\prime}))\,.$
(9)

The normalization volume $V=\rho/A$ with $\rho=3\pi^{2}k_{F}^{3}/2$ depends on
the Fermi momentum of the nucleus, which for 12C is taken to be $k_{F}=225$
MeV. The factor $1/4$ accounts for the fact that we sum over indistinguishable
tow-particle two-hole final state. The two-nucleon spectral function in Eq.
(II) is computed using QMC techniques and retains correlations between the two
struck particles Simons  _et al._ (2022).

Figure 1: Feynman diagrams describing two-body currents contributions
associated to: pion in flight (a), seagull (b), pion-pole (c), and delta
excitations (d) processes. Solid, thick, and dashed lines correspond to
nucleons, deltas, pions, respectively. The wavy line represents the vector
boson.

The electroweak MEC used in this work are those of Ref. Ruiz Simo  _et al._
(2017), which have been derived by coupling to a second nucleon line the pion-
production amplitudes obtained within the non-linear $\sigma$ model in Ref.
Hernandez  _et al._ (2007). Their electromagnetic counterparts can be obtained
by setting the axial pieces of each current to zero and modifying their
isospin dependence. The MEC is the sum of four different contributions

$\displaystyle j^{\mu}_{\rm MEC}=j^{\mu}_{\pi}+j^{\mu}_{\rm sea}+j^{\mu}_{\rm
pole}+j^{\mu}_{\Delta}\,,$ (10)

whose corresponding Feynman diagrams are depicted in Fig. 1. We note that
these currents are symmetric under the exchange $1\leftrightarrow 2$.

Introducing the pion momenta $k_{1}=p-h$ and $k_{2}=p^{\prime}-h^{\prime}$,
the pion-in-flight current operator corresponding to diagram (a) of Fig. 1 is
written as

$\displaystyle j^{\mu}_{\pi}$ $\displaystyle=(I_{V})_{\pm}J^{\mu}_{\pi}\ ,$
$\displaystyle J^{\mu}_{\pi}$
$\displaystyle=(J^{\mu}_{\pi})_{V}+(J^{\mu}_{\pi})_{A}\ ,$
$\displaystyle(J^{\mu}_{\pi})_{V}$ $\displaystyle=\frac{f^{2}_{\pi
NN}}{m_{\pi}^{2}}F_{1}^{V}(q)F_{\pi NN}(k_{1})F_{\pi NN}(k_{2})$
$\displaystyle\times\Pi(k_{1})_{(1)}\Pi(k_{2})_{(2)}(k_{1}^{\mu}-k_{2}^{\mu})\,.$
(11)

while its axial component vanishes, $(J^{\mu}_{\pi})_{A}=0$. In the above
equation, $f^{2}_{\pi NN}/(4\pi)$=0.08 and the pion propagation and absorption
is described by

$\displaystyle\Pi(k)=\frac{\gamma_{5}\not{k}}{k^{2}-m_{\pi}^{2}}\,.$ (12)

The isospin raising-lowering operator is given by

$\displaystyle(I_{V})_{\pm}=(\tau^{(1)}\times\tau^{(2)})_{\pm}\ ,$ (13)

where $\pm\rightarrow x\pm iy$, where in the electromagnetic case we simply
take $\pm\rightarrow z$.

Consistent with the conserved vector current (CVC) hypothesis, the vector part
of the pion-in-flight current operator includes the electromagnetic form
factor

$\displaystyle F_{1}^{V}(q)=G_{E}^{p}(q)-G_{E}^{n}(q)\,.$ (14)

The $\pi NN$ coupling is described using a form factor that accounts for the
off-shellness of the pion

$\displaystyle F_{\pi
NN}(k)=\frac{\Lambda_{\pi}^{2}-m_{\pi}^{2}}{\Lambda_{\pi}^{2}-k^{2}}\ ,$ (15)

where $\Lambda_{\pi}$=1300 MeV.

The electroweak seagull current operator, given by the sum of diagram (b) of
Fig. 1 and the one obtained interchanging particles 1 and 2, reads

$\displaystyle j^{\mu}_{\rm sea}$ $\displaystyle=(I_{V})_{\pm}J^{\mu}_{\rm
sea}\ ,$ $\displaystyle J^{\mu}_{\rm sea}$ $\displaystyle=(J^{\mu}_{\rm
sea})_{V}+(J^{\mu}_{\rm sea})_{A}\ ,$ $\displaystyle(J^{\mu}_{\rm sea})_{V}$
$\displaystyle=\frac{f^{2}_{\pi NN}}{m_{\pi}^{2}}F_{1}^{V}(q)F^{2}_{\pi
NN}(k_{1})\Pi(k_{1})_{(1)}\big{(}\gamma_{5}\gamma^{\mu}\big{)}_{(2)}$
$\displaystyle-(1\leftrightarrow 2)\ ,$ $\displaystyle(J^{\mu}_{\rm sea})_{A}$
$\displaystyle=\frac{f^{2}_{\pi
NN}}{m_{\pi}^{2}}\frac{1}{g_{A}}F_{\rho}(k_{2})F^{2}_{\pi
NN}(k_{1})\Pi(k_{1})_{(1)}(\gamma^{\mu}\big{)}_{(2)}$
$\displaystyle-(1\leftrightarrow 2)\,.$ (16)

The form factor $F_{\rho}(k)$, included to account for the $\rho$ meson
dominance of the $\pi NN$ coupling, is given by Hernandez  _et al._ (2007)

$\displaystyle F_{\rho}(k)=\frac{1}{k^{2}-m^{2}_{\rho}},\ \ \ m_{\rho}=775.8\
{\rm MeV}$ (17)

The expression for the pion-pole current operator, represented by diagram (c)
of Fig. 1, is

$\displaystyle j^{\mu}_{\rm pole}$ $\displaystyle=(I_{V})_{\pm}J^{\mu}_{\rm
pole}\ ,$ (18) $\displaystyle J^{\mu}_{\rm pole}$ $\displaystyle=(J^{\mu}_{\rm
pole})_{V}+(J^{\mu}_{\rm pole})_{A}\ ,$ (19) $\displaystyle(J^{\mu}_{\rm
pole})_{V}$ $\displaystyle=0,$ (20) $\displaystyle(J^{\mu}_{\rm pole})_{A}$
$\displaystyle=\frac{f^{2}_{\pi
NN}}{m_{\pi}^{2}}\frac{1}{g_{A}}F_{\rho}(k_{1})F^{2}_{\pi
NN}(k_{2})\Pi(k_{2})_{(2)}$
$\displaystyle\times\Big{(}\frac{q^{\mu}\not{q}}{q^{2}-m_{\pi}^{2}}\Big{)}_{(1)}-(1\leftrightarrow
2)\ .$ (21)

Diagrams (d), as well as the corresponding two in which particles 1 and 2 are
interchanged, are associated with two-body current terms involving a
$\Delta$-resonance in the intermediate state. The expression of this operator
is largely model dependent, owing to the purely transverse nature of this
current,  i.e. the form of the vector part is not subject to current-
conservation constraints. We adopted the parametrization of Ref. Hernandez
_et al._ (2007)

$\displaystyle j^{\mu}_{\Delta}$ $\displaystyle=\frac{3}{2}\frac{f_{\pi
NN}f^{\ast}}{m^{2}_{\pi}}\bigg{\\{}\Pi(k_{2})_{(2)}\Big{[}\Big{(}-\frac{2}{3}\tau^{(2)}+\frac{I_{V}}{3}\Big{)}_{\pm}$
$\displaystyle\times F_{\pi NN}(k_{2})F_{\pi
N\Delta}(k_{2})(J^{\mu}_{a})_{(1)}-\Big{(}\frac{2}{3}\tau^{(2)}+\frac{I_{V}}{3}\Big{)}_{\pm}$
$\displaystyle\times F_{\pi NN}(k_{2})F_{\pi
N\Delta}(k_{2})(J^{\mu}_{b})_{(1)}\Big{]}+(1\leftrightarrow 2)\bigg{\\}}$ (22)

where $f^{\ast}$=2.14 and

$F_{\pi N\Delta}(k)=\frac{\Lambda^{2}_{\pi N\Delta}}{\Lambda^{2}_{\pi
N\Delta}-k^{2}}\ ,$ (23)

with $\Lambda_{\pi N\Delta}=1150$ MeV. The $N\rightarrow\Delta$ transition
vertices entering the left and right (d) diagrams, corresponding to
$J^{\mu}_{a}$ and $J^{\mu}_{b}$, respectively are expressed as

$\displaystyle J^{\mu}_{a}$
$\displaystyle=(J^{\mu}_{a})_{V}+(J^{\mu}_{a})_{A}\ ,$
$\displaystyle(J^{\mu}_{a})_{V}$
$\displaystyle=\frac{C_{3}^{V}}{M}\Big{[}k_{2}^{\alpha}G_{\alpha\beta}(h+q)\Big{(}g^{\beta\mu}\not{q}-q^{\beta}\gamma^{\mu}\Big{)}\Big{]}\gamma_{5}\
,$ $\displaystyle(J^{\mu}_{a})_{A}$
$\displaystyle=C_{5}^{A}\Big{[}k_{2}^{\alpha}G_{\alpha\beta}(h+q)g^{\beta\mu}\Big{]}$
(24)

and

$\displaystyle J^{\mu}_{b}$
$\displaystyle=(J^{\mu}_{b})_{V}+(J^{\mu}_{b})_{A}\ ,$
$\displaystyle(J^{\mu}_{b})_{V}$
$\displaystyle=\frac{C_{3}^{V}}{M}\gamma_{5}\Big{[}\Big{(}g^{\alpha\mu}\not{q}-q^{\alpha}\gamma^{\mu}\Big{)}G_{\alpha\beta}(p-q)k_{2}^{\beta}\Big{]},$
$\displaystyle(J^{\mu}_{b})_{A}$
$\displaystyle=C_{5}^{A}\Big{[}g^{\alpha\mu}G_{\alpha\beta}(p-q)k_{2}^{\beta}\Big{]}\,.$
(25)

Since the above $\Delta$ current is applied in the resonance region, the
standard Rarita-Schwinger propagator

$\displaystyle
G^{\alpha\beta}(p_{\Delta})=\frac{P^{\alpha\beta}(p_{\Delta})}{p^{2}_{\Delta}-M_{\Delta}^{2}}$
(26)

has to be modified to account for the possible $\Delta$ decay into a physical
$\pi N$ state. To this aim, following Refs. Dekker  _et al._ (1994); De Pace
_et al._ (2003), we replace the real resonance mass $M_{\Delta}$=1232 MeV by
$M_{\Delta}-i\Gamma(p_{\Delta})/2$. The energy-dependent decay width
$\Gamma(p_{\Delta})/2$ effectively accounts for the allowed phase space for
the pion produced in the physical decay process. It is given by

$\Gamma(p_{\Delta})=\frac{(4f_{\pi N\Delta})^{2}}{12\pi
m_{\pi}^{2}}\frac{|\mathbf{k}|^{3}}{\sqrt{s}}(m_{N}+E_{k})R(\mathbf{r}^{2})$
(27)

where $(4f_{\pi N\Delta})^{2}/(4\pi)=0.38$, $s=p_{\Delta}^{2}$ is the
invariant mass, $\mathbf{k}$ is the decay three-momentum in the $\pi N$ center
of mass frame, such that

$\mathbf{k}^{2}=\frac{1}{4s}[s-(m_{N}+m_{\pi})^{2}][s-(m_{N}-m_{\pi})^{2}]\,$
(28)

and $E_{k}=\sqrt{m_{N}^{2}+\mathbf{k}^{2}}$ is the associated energy. The
additional factor

$R(\mathbf{r}^{2})=\left(\frac{\Lambda_{R}^{2}}{\Lambda_{R}^{2}-\mathbf{r}^{2}}\right)$
(29)

depending on the $\pi N$ three-momentum $\mathbf{r}$, with
$\mathbf{r}^{2}=(E_{k}-\sqrt{m_{\pi}^{2}+\mathbf{k}^{2}})^{2}-4\mathbf{k}^{2}$
and $\Lambda_{R}^{2}=0.95\,m_{N}^{2}$, is needed to better reproduce the
experimental phase-shift $\delta_{33}$ Dekker  _et al._ (1994). The medium
effects on the $\Delta$ propagator are accounted for by modifying the decay
width as

$\displaystyle\Gamma_{\Delta}(p_{\Delta})\to\Gamma_{\Delta}(p_{\Delta})-2\rm{Im}[U_{\Delta}(p_{\Delta},\rho=\rho_{0})],$
(30)

where $U_{\Delta}$ is a density dependent potential obtained from a Bruckner-
Hartree-Fock calculation using a coupled-channel $NN\oplus N\Delta\oplus\pi
NN$ model Lee (1983, 1984); Lee and Matsuyama (1985, 1987) and we fixed the
density at the nuclear saturation value $\rho_{0}$ =0.16 fm3. A detailed
analysis of medium effects in the MEC contribution for electron-nucleus
scattering can be found in Ref. Rocco  _et al._ (2019a). The spin 3/2
projection operator reads

$\displaystyle P^{\alpha\beta}(p_{\Delta})$
$\displaystyle=(\not{p}_{\Delta}+M_{\Delta})\Big{[}g^{\alpha}\beta-\frac{1}{3}\gamma^{\alpha}\gamma^{\beta}-\frac{2}{3}\frac{p_{\Delta}^{\alpha}p_{\Delta}^{\beta}}{M_{\Delta}^{2}}$
$\displaystyle+\frac{1}{3}\frac{p_{\Delta}^{\alpha}\gamma^{\beta}-p_{\Delta}^{\beta}\gamma^{\alpha}}{M_{\Delta}}\Big{]}\
.$ (31)

The vector and axial form factors adopted in this work are those of Ref.
Hernandez  _et al._ (2007)

$\displaystyle C_{3}^{V}$ $\displaystyle=\frac{2.13}{(1-q^{2}/M_{V}^{2})^{2}}\
\frac{1}{1-q^{2}/(4M_{V}^{2})}\ ,$ (32) $\displaystyle C_{5}^{A}$
$\displaystyle=\frac{1.2}{(1-q^{2}/M^{2}_{A\Delta})^{2}}\
\frac{1}{1-q^{2}/(3M^{2}_{A\Delta})}\ ,$ (33)

where $M_{V}=0.84$ GeV and $M_{A\Delta}=1.05$ GeV. In this work we only keep
the leading vector and axial $N\rightarrow\Delta$ form factors, though
including the sub-leading terms in the current operator does not pose any
conceptual difficulties.

##  III Interference Contribution

A fully quantum mechanical treatment of the scattering process requires
accounting for the interference between one- and two-body current
contributions, potentially leading to both one and two-nucleon knockouts.
Within the factorization scheme, calculations based on realistic spectral
functions have shown that the contribution to the response tensor from
interference leading to two-nucleon knockout is relatively small Rocco  _et
al._ (2016). Here, we focus on the interference between one- and two-body
currents resulting in single-nucleon knockout (one-particle, one-hole) final
states. The corresponding response tensor is given by

$R^{\mu\nu}_{\rm{12b}}=\sum_{f}\langle 0|j_{2b}^{\mu\dagger}|f\rangle\langle
f|j_{1b}^{\nu}|0\rangle\delta(\omega+E_{0}-E_{f})\,.$ (34)

Let us initially focus on non-interacting, homogeneous, isospin-symmetric
nuclear matter — a Fermi gas of protons and neutrons. In this case, the two-
body current yields a one-particle, one-hole final state, represented by
$|ph^{-1}\rangle$, via the interaction of the struck nucleon with a second
spectator nucleon.

We start by explicitly writing the initial and final nuclear states as
properly normalized Slater determinants

$\displaystyle|ph^{-1}\rangle$
$\displaystyle=\mathcal{A}[\phi_{\tilde{n}_{1}}\phi_{\tilde{n}_{2}}\dots\phi_{\tilde{n}_{A}}]$
$\displaystyle|0\rangle$
$\displaystyle=\mathcal{A}[\phi_{n_{1}}\phi_{n_{2}}\dots\phi_{n_{A}}]\,.$ (35)

In nuclear matter, to respect translation invariance, the single-particle
orbitals are plane waves

$\langle\mathbf{r}_{i}|\phi_{n}\rangle\equiv\phi_{n}(\mathbf{r}_{i})=\frac{e^{i\mathbf{k}\cdot\mathbf{r}_{i}}}{\sqrt{V}}|\eta_{n}\rangle\,,$

where the spinor $\eta_{n}$ represents the spin and the isospin of state i.
The indexes $\tilde{n}_{i}$ and ${n}_{i}$ denote the single particle states of
the ground-state and the one-particle, one-hole final state, respectively.
Note that, $\tilde{n}_{i}=n_{i}$ except when $n_{i}=h$, as in that case
$\tilde{n}_{i}=p$.

Exploiting the orthogonality of single-particle orbitals, and the fact that
MEC are symmetric under the exchange $1\leftrightarrow 2$, the many-body
matrix element of the reduces to

$\displaystyle\langle ph^{-1}|j^{\mu}_{2b}|0\rangle=\sum_{n}\int
d^{3}r_{1}d^{3}r_{2}\phi^{*}_{p}(\mathbf{r}_{1})\phi^{*}_{n}(\mathbf{r}_{2})$
$\displaystyle\times
j_{12}(\mathbf{r}_{1},\mathbf{r}_{2})\mathcal{A}[\phi_{h}(\mathbf{r}_{1})\phi_{n}(\mathbf{r}_{2})]\,.$
(36)

As observed in Refs. Amaro  _et al._ (1998, 2002b), the direct term vanishes
for owing to both spin-isospin traces being zero in unpolarized isospin-
symmetric nuclear matter, and the presence of a pion carrying zero momentum,
thus

$\displaystyle\langle ph^{-1}|j^{\mu}_{2b}|0\rangle=-\sum_{n}\int
d^{3}r_{1}d^{3}r_{2}\phi^{*}_{p}(\mathbf{r}_{1})\phi^{*}_{n}(\mathbf{r}_{2})$
$\displaystyle\times
j_{12}^{\mu}(\mathbf{r}_{1},\mathbf{r}_{2})\phi_{n}(\mathbf{r}_{1})\phi_{h}(\mathbf{r}_{2})\,.$
(37)

The coordinate space expressions of the two-body current operators are
connected to the ones in momentum space provided in Section II by the
following Fourier transformation

$\displaystyle j^{\mu}_{12}(\mathbf{r}_{1},\mathbf{r})=$
$\displaystyle\int\frac{d^{3}k_{1}}{(2\pi)^{3}}\frac{d^{3}k_{2}}{(2\pi)^{3}}e^{i\mathbf{k}_{1}\cdot\mathbf{r}_{1}}e^{i\mathbf{k}_{2}\cdot\mathbf{r}_{2}}$
$\displaystyle(2\pi)^{3}\delta(\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{q})j^{\mu}_{12}(\mathbf{k}_{1},\mathbf{k}_{2})\,.$
(38)

We can now insert the latter expression into Eq. (37) and integrate over
$\mathbf{r}_{1}$ and $\mathbf{r}_{2}$. Recalling that

$\displaystyle\sum_{n}=V\int\frac{d^{3}k}{(2\pi)^{3}}\theta({k_{F}}-{\bf k})$
$\displaystyle\int
d^{3}re^{i\mathbf{k}\cdot\mathbf{r}}=(2\pi)^{3}\delta(\mathbf{k})\,,$ (39)

and integrating over $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$, we obtain

$\displaystyle\langle ph^{-1}|j^{\mu}_{2b}|0\rangle$
$\displaystyle=-\frac{1}{V}\sum_{\eta_{k}}\int d^{3}k\delta^{3}({\bf q}-{\bf
h}+{\bf p})$ $\displaystyle\times\theta({k_{F}}-{\bf
k})\langle\eta_{p}\eta_{k}|j^{\mu}_{12}(\mathbf{k}_{1},\mathbf{k}_{2})|\eta_{h}\eta_{p}\rangle\,.$
(40)

Following the same steps outlined above, the nuclear-matter matrix element of
the one-body current operator can be readily expressed as

$\displaystyle\langle ph^{-1}|j^{\mu}_{1b}|0\rangle$
$\displaystyle=\frac{1}{V}(2\pi)^{3}\delta^{3}({\bf q}-{\bf h}+{\bf
p})\langle\eta_{p}|j_{1}|\eta_{h}\rangle\,.$ (41)

Substituting the expressions just obtained for the one- and two-body current
matrix elements into Eq. (34), the interference contribution to the response
tensor for the nuclear matter reads

$\displaystyle R^{\mu\nu}_{12b}=$
$\displaystyle-V\sum_{\eta_{p},\eta_{h},\eta_{k}}\int\frac{d^{3}h}{(2\pi)^{3}}\frac{d^{3}k}{(2\pi)^{3}}\Big{[}\delta({\bf
q}-{\bf h}+{\bf p})$ $\displaystyle\times\theta({\bf
p}-k_{F})\theta({k_{F}}-{\bf h})\theta({k_{F}}-{\bf
k})\langle\eta_{p}|j_{1}|\eta_{h}\rangle$
$\displaystyle\times\langle\eta_{p}\eta_{k}|j_{12}|\eta_{h}\eta_{k}\rangle\delta(\omega-e(\mathbf{p})+e(\mathbf{h}))\Big{]}\,.$
(42)

To make contact with finite nuclei, we replace

$\theta(k_{F}-{\bf k})\rightarrow\tilde{n}_{\rm MF}({\bf k})$ (43)

and

$\displaystyle\theta(k_{F}-{\bf h})\delta(\omega+e({\bf h})-e({\bf
p}))\rightarrow$ $\displaystyle\qquad\qquad\int dE{\tilde{P}}_{\rm MF}({\bf
h},E)\delta(\tilde{\omega}+e({\bf h})-e({\bf p}))\,.$ (44)

To keep the normalization consistent with the infinite matter case, we defined
$\tilde{n}_{\rm MF}({\bf h})=k_{F}^{3}/(6\pi^{2})n_{\rm MF}({\bf h})$ and
$\tilde{P}_{\rm MF}({\bf h},E)=k_{F}^{3}/(6\pi^{2})P_{\rm MF}({\bf h},E)$. In
order to select final states in the reaction with only one-nucleon emission,
we consider just the mean-field component of the hole spectral function. The
corresponding momentum distribution is obtained as

$n_{\rm MF}(\mathbf{k})=\int dEP_{\rm MF}(\mathbf{k},E)\,.$ (45)

Note that, the normalization of the latter is less than $1$, thereby quenching
the interference between one- and two-body currents. The final expression for
latter that we adopt in our numerical calculation reads

$\displaystyle R^{\mu\nu}_{12b}=$
$\displaystyle-V\sum_{\eta_{p},\eta_{h},\eta_{k}}\int\frac{d^{3}h}{(2\pi)^{3}}\frac{d^{3}k}{(2\pi)^{3}}dE\Big{[}{\tilde{P}}_{\rm
MF}({\bf h},E)$ $\displaystyle\times{\tilde{n}}_{\rm MF}({\bf k})\delta({\bf
q}-{\bf h}+{\bf p})\theta({\bf p}-k_{F})\langle\eta_{p}|j_{1}|\eta_{h}\rangle$
$\displaystyle\times\langle\eta_{p}\eta_{k}|j_{12}|\eta_{h}\eta_{k}\rangle\delta(\tilde{\omega}-e(\mathbf{p})+e(\mathbf{h}))\Big{]}\,.$
(46)

Figure 2: Longitudinal (upper panel) and transverse (lower panel) response
functions from e-12C scattering at $|\mathbf{q}|=570$ MeV. Contributions are
separated into pure one-body (red), pure two-body (orange), interference
between one and two-body (purple), and total (blue).

##  IV Results

We begin our analysis by comparing against inclusive electron scattering data,
for which the Rosenbluth technique can be applied to separate the longitudinal
and transverse contributions. The upper and lower panels of Figure 2 display
theoretical calculations of the longitudinal and transverse response functions
of ${}^{12}\rm{C}$, respectively, at a momentum transfer of $|\mathbf{q}|=570$
MeV compared against the world data analysis conducted by Jourdan Jourdan
(1996). Within the extended factorization scheme, final state interaction
effects are included by convoluting the impulse-approximation results with a
folding function that both shifts and redistributes strength from the peak to
the tails Benhar  _et al._ (2008). Contributions from pure one- and two-body
currents, as well as their interference, are separately shown to better
identify their relative importance.

Consistent with the GFMC calculations in Ref. Lovato  _et al._ (2016), two-
body currents in the longitudinal channel appear to have a negligible effect.
The pure two-body component is nearly zero in the QE peak, while it brings
about a modest enhancement of the response in the dip region. On the other
hand, the interference contribution is small in magnitude and negative,
resulting in a minor depletion of the total response. The scenario is markedly
different in the transverse channel, where two-body effects play a crucial
role. This is not entirely surprising, given that the dominant contribution is
the Delta current, which is fully transverse. Both the pure two-body and
interference components contribute about the same additional strength to the
response beneath the QE peak, constituting almost 20% of the peak height. This
enhancement significantly improves the agreement between theoretical
calculations and experimental data. As highlighted in several previous works —
see for instance Refs. Amaro  _et al._ (2003); Rocco  _et al._ (2016) — the
pure two-body term becomes dominant in the dip region. It is important to note
that the missing strength compared to experimental data has to be ascribed to
resonance-production mechanisms Rocco  _et al._ (2019b), which have not been
accounted for in the present work.

Figure 3: One- and two-body interference contribution to the longitudinal
(upper panel) and transverse (lower panel) response functions from e-12C
scattering at $|\mathbf{q}|=500$ MeV separated into isospin pairs: pp (red),
np (orange), nn (blue), and total (black).

Thanks to our improved numerical implementation of the two-body currents, we
are now able to separately compute the contribution of each isospin pair to
the nuclear response. Figures 3 show the electromagnetic responses for
$|\mathbf{q}|=500$ MeV, categorized into $pp$, $pn$, and $nn$ pairs. In the
transverse response, the contribution of $pn$ pairs is about 10 times as large
as that of $pp$ and $nn$ pairs, consistent with the isospin structure observed
in SRCs in $(e,e^{\prime}NN)$ experiments on heavy nuclei Korover  _et al._
(2021); Duer  _et al._ (2019). The dependence of the pure two-body and
interference responses on the isospin of the initial and final state,
especially for two-nucleon knockout, is relevant for the MicroBooNE and SBND
experiments, as these will constitute a significant fraction of total events
Abratenko  _et al._ (2023a). These experiments aim to measure the outgoing
hadronic system with unprecedented accuracy, making semi-exclusive predictions
necessary for a correct interpretation of the data. In this context, we note
that MicroBooNE has already measured an exclusive sample of $pp$ final state
and found large disagreements between model predictions Abratenko  _et al._
(2022).

Figure 4: Inclusive electron cross sections on Carbon at several beam energies
and scattering angles. Contributions are separated into pure one-body (red),
pure two-body (orange), interference between one and two-body (purple), and
total (blue).

Next, we examine inclusive electron-12C cross sections at various beam
energies and scattering angles, as depicted in Fig.4. The inclusion of one-
and two-body interference (indicated by the purple line in this figure)
improves the agreement with experimental data by increasing the strength in
the QE region by approximately 10%. Interference effects in the inclusive
cross sections appear to be relatively modest, especially for smaller values
of $\theta_{e^{\prime}}$. This behavior can be attributed to the fact that to
obtain the inclusive cross section, the longitudinal and transverse responses
must be multiplied by their respective leptonic response factors, which depend
on the scattering angle. Consequently, two-body currents in general and
interference effects, in particular, are more pronounced at backward angles,
where the transverse response contributes a significant fraction of the
strength, as illustrated in Fig.4. Again, the missing strength in the
resonance-production region is due to resonance-production processes, which
are not the main focus of this work.

Moving to the electroweak sector, in Fig.5, we display the flux-folded
charged-current quasielastic (CCQE) $\nu_{\mu}-^{12}$C cross section from
MiniBooNE in selected kinematic binsAguilar-Arevalo  _et al._ (2010). We
present experimental data for both the CCQE-like and the CCQE measurements,
with the latter obtained using a model-dependent background subtraction to
account for pions created at the interaction vertex and subsequently
reabsorbed in the nuclear medium. The difference between the blue (total
contribution) and red (one-body currents only) lines highlights the sizable
additional strength brought about by two-body currents. As the cross sections
are convoluted with the broad MiniBooNE flux, the contribution from
interference effects is approximately the same in each bin of
$\cos\theta_{\mu}$, constituting about 10% of the cross section.

Figure 5: Flux-averaged $\nu_{\mu}$ differential cross sections on 12C for
MiniBooNE. Three bins of $\cos\theta_{\mu}$ are shown with the one-body
contributions in red, pure two-body contributions in orange, one- and two-body
interference in purple, and total in blue. The width of the error band
interpolates between the dipole axial form factor with $M_{A}=1$ GeV, and the
LQCD form factor of Ref. Bali  _et al._ (2020). The open circles are the cross
section to which the background reported in Ref. Aguilar-Arevalo  _et al._
(2010) is added

The theoretical calculations for the one-body, interference, and total cross
sections displayed in Figure 5 are accompanied by colored bands, whose widths
reflect the dependence of the inclusive cross section on different
parameterizations of the axial form factor. Specifically, we employ both the
dipole parameterization with $M_{A}=1$ GeV Bodek  _et al._ (2008), as
extracted from neutrino-deuteron scattering and pion electroproduction
experiments, and the LQCD form factor computed in Ref.Bali  _et al._ (2020).
As extensively discussed, the dipole parameterization is unphysical, and
several recent LQCD calculations and measurements of $\bar{\nu}_{\mu}-p$
scattering cross sections tend to favor a form factor with a shallower fall-
off for $Q^{2}>0.3,\rm{GeV}^{2}$Meyer  _et al._ (2016); Bali  _et al._ (2020);
Park  _et al._ (2022); Djukanovic  _et al._ (2022); Cai  _et al._ (2023);
Simons  _et al._ (2022).

Theoretical calculations using the LQCD form factor as input consistently
yield larger pure one-body cross sections, corresponding to the upper edge of
the band. This behavior is particularly evident at backward muon angles, where
the $Q^{2}$ dependence of the two axial form factor parameterizations differs
more significantly. In contrast, the interference contribution shows a milder
dependence on the choice of the axial form factor, increasing by up to 10% in
the most backward angle bin. It is crucial to emphasize that these bands
should not be interpreted as the uncertainty in the axial form factor
propagated forward to the uncertainty of the cross section but should be seen
as a way to interpolate between these two form factor parameterizations.

While the choice of the LQCD form factor seems to significantly improve the
agreement with data, the model dependent background subtraction method adopted
by the MiniBooNE collaboration as well as the lack of a prediction including
events with absorbed pions make quantitative comparisons difficult. We note
that in our factorization scheme the enhancement from the LQCD form factor
matches the enhancement seen in Green’s Function Monte Carlo (GFMC)
calculations of flux folded cross section using the same LQCD form factor
Simons  _et al._ (2022). As these are two completely different many body
methods, only linked by the same underlying nuclear Hamiltonian, the
sensitivity to the choice in axial form factor seems robust.

##  V Conclusions

Providing accurate theoretical predictions, accompanied by reliable
uncertainty quantification, for neutrino-nucleus scattering cross-sections in
the energy regime relevant to the neutrino-oscillation problem is highly non-
trivial. The primary challenges lie in combining a microscopic, quantum-
mechanical description of real-time nuclear dynamics with relativistic
kinematics and currents. In this regard, the extended factorization scheme,
based on realistic spectral functions obtained from Quantum Monte Carlo
calculations, offers an ideal framework for addressing this problem. This
scheme can accommodate fully relativistic kinematics while maintaining an
accurate description of the initial state and the hadronic final state.

To extract neutrino-oscillation parameters with high accuracy and capitalize
on the next-generation experiments, it is essential to account for as many
quantum-mechanical effects as possible. One such effect is the interference
between one- and two-body relativistic current operators, resulting in a final
state with one nucleon in the continuum. In this work, we have incorporated
this process into the extended factorization scheme in a consistent manner
with pure one- and two-body current transitions, ensuring the avoidance of
potential double counting.

We first assessed the impact of interference between one- and two-body
currents on the electromagnetic longitudinal and transverse response
functions. Consistent with GFMC and STA calculations Lovato  _et al._ (2016);
Pastore  _et al._ (2020); Andreoli  _et al._ (2022), we observed a substantial
enhancement of the transverse response in the QE region, while the
longitudinal response remained largely unchanged. By analyzing the individual
contributions of $pp$, $nn$, and $np$ pairs, we determined that the $np$ pairs
predominantly drive the transverse enhancement. This observation is relevant
for both electron-scattering and neutrino-oscillation experiments, as it
provides insights into the isospin of the emitted nucleon and, consequently,
that of the spectator system. Interference effects remain sizable in inclusive
electron and neutrino scattering of 12C, contributing to over 10% of the cross
section, depending on the kinematics. Overall, the inclusion of interference
effects in the extended factorization scheme improves the agreement with
experimental data for both electron and flux-folded neutrino-nucleus
scattering.

Quantifying the uncertainties introduced by approximating the solution of the
quantum many-body problem, such as assuming the factorization of the final
hadronic state, will become increasingly relevant as statistical and
experimental systemic errors diminish in short- and long-baseline experiments.
A promising approach to assess the intrinsic errors in the factorization
scheme involves comparisons with virtually exact — albeit non-relativistic —
GFMC calculations. By incorporating interference effects in the extended
factorization scheme and addressing relativistic effects in the GFMC through
the selection of an appropriate reference frame and the two-fragment model
Efros  _et al._ (2010); Rocco  _et al._ (2018); Nikolakopoulos  _et al._
(2023), we can perform more accurate comparisons between these two methods.

Interference effects also play a crucial role in analyzing the axial-vector
form factor. Currently, the majority of models rely on the dipole form factor
with an axial mass of approximately $1$ GeV, derived from fits to bubble
chamber data and pion electro-production Bodek  _et al._ (2008). However, this
parameterization is recognized to underestimate uncertainty, being both
unphysical and inconsistent with QCD sum rules Meyer  _et al._ (2016). In
contrast, LQCD computations of single nucleon form factors have advanced to
the point where the axial-vector form factor has been computed and accompanied
by credible uncertainty estimates. Employing the LQCD form factor of Ref.Bali
_et al._ (2020) in the extended factorization scheme significantly enhances
the QE neutrino-nucleus cross section, bringing it closer to experimental
data—although the model dependence on modeling pion reabsorption remains to be
fully determinedMeyer  _et al._ (2022).

Event generators typically use the axial mass as a fit parameter in their
tunes, extracting it from neutrino-nucleus scattering data. This approach
assumes that the underlying nuclear quantum many-body method, which takes the
axial form factor as input, is complete, meaning it accurately incorporates
all possible interaction channels. However, since most event generators do not
include interference effects between one and two-body currents — which
contribute additional strength in the QE region — these extractions may yield
incorrect values for the axial mass. It would be timely to investigate how the
inclusion of one- and two-body interference effects affects the extracted
value of the axial mass from high-precision neutrino scattering data.

##  VI Acknowledgements

This work was supported in part by the U.S. Department of Energy (DOE), Office
of Science, Office of Nuclear Physics under contract number DE-AC02-06CH11357
(A.L.), and by SciDAC-NeuCol (A.L., N.R.) projects. N.S. is also supported by
a Fermilab LDRD award. A.L. is also supported by DOE Early Career Research
Program awards. Numerical calculations were performed on the parallel
computers of the Laboratory Computing Resource Center, Argonne National
Laboratory, the computers of the Argonne Leadership Computing Facility via the
INCITE grant “Ab-initio Nuclear Structure and Nuclear Reactions”.

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