# Neutrino and antineutrino charged-current multi-nucleon cross sections
revisited

J. E. Sobczyk  Institut für Kernphysik and PRISMA+ Cluster of Excellence,
Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany  J. Nieves
Instituto de Física Corpuscular (centro mixto CSIC-UV), Institutos de
Investigación de Paterna, Apartado 22085, 46071, Valencia, Spain

###### Abstract

In this work we improve on several aspects of the computation of the
(anti-)neutrino charged-current multi-nucleon cross section carried out in
Phys.Rev.C 83 (2011) 045501 and Phys.Rev.C 102 (2020) 024601. Most
importantly, we implement a consistent treatment of the nucleon self-energy in
the $W^{\pm}N\to N^{\prime}\pi$ amplitude entering the definition of the two-
particle two-hole (2p2h) cross-section, and estimate the source of uncertainty
of our model due to a simplified treatment of the $\Delta$ self-energy. Our
new predictions are around 20-40% higher than previously. We show comparisons
for the inclusive lepton double-differential cross sections, with no pions in
the final state, measured by MiniBooNE on carbon and by T2K on carbon and
oxygen. In all cases, we find an excellent reproduction of the experiments,
and in particular, the neutrino MiniBooNE data is now well described without
requiring a global 90% re-scaling of the flux. In addition, we take the
opportunity of this revision to discuss in detail several important issues of
the calculation of the 2p2h cross section, delving into the microscopic
dynamics of the multi-nucleon mechanisms. The improved treatment presented in
this work provides realistic first-step emitted two-nucleon final state
momentum configurations, beyond the approximation of phase-space
distributions.

##  I Introduction

The ambitious experimental programs to precisely determine neutrino
properties, test the three-generation paradigm, establish the order of mass
eigenstates and investigate leptonic CP violation are underway Abi  _et al._
(2020); Abe  _et al._ (2015). The success of these endeavours will require an
excellent control over the systematic uncertainties, and therefore a solid
microscopic understanding of neutrino-nucleus interactions Alvarez-Ruso  _et
al._ (2018). Neutrino and antineutrino beams used in long-baseline oscillation
experiments are not monochromatic making the experiments sensitive to all
contributing mechanisms up to the energies of the order of 1 GeV. Therefore,
the reconstruction of the incident neutrino energy requires a consistent
framework in which all these relevant interaction channels are included, not
only at the level of inclusive cross-sections, but also giving access to
exclusive final states Filali  _et al._ (2024). This modelling is a
challenging task which encompasses a theoretical description of the so-called
primary vertex of interaction, in which hadrons are produced inside of a
nucleus, as well as the intra-nuclear cascade which can lead to a different
final distribution of particles.

As a prominent example, the CC0$\pi$ cross section, charge-current (CC) driven
(anti-)neutrino reaction with no pions in the final state, reported by
experiments, requires also accounting for real pion production in the first
step, after which the final pion gets absorbed in the nuclear medium. The so-
called CC quasielastic-like (CCQE-like) cross section can be obtained after
Monte Carlo correcting the CC$0\pi$ one for those events where CC pion
production was followed by pion absorption. However, it still does not
correspond to the pure quasielastic (QE) process since multinucleon
mechanisms, where the gauge boson $W^{\pm}$ is absorbed by a pair of
interacting nucleons, should be accounted for. Mismatching the signal coming
from these two reaction mechanisms would lead to a bias in the (anti-)neutrino
energy reconstruction Nieves  _et al._ (2012a).

In this paper we follow a theoretical framework which consistently describes
(anti-)neutrino QE, multi-nucleon knock-out and pion production processes
Nieves  _et al._ (2004, 2011). The same theoretical approach has been widely
used to model the pion propagation in the nuclear medium, accounting for pion
absorption, charge exchange and production mechanisms. Actually, the scheme is
an extension of the framework developed by the Valencia group, successfully
used for many different nuclear reactions at intermediate energies.
Importantly, these works include inclusive scattering of photons Carrasco and
Oset (1992), electrons Gil  _et al._ (1997) and pions from nuclei Salcedo  _et
al._ (1988); Garcia-Recio  _et al._ (1991); Nieves  _et al._ (1993a, b), where
multi-nucleon mechanisms played a crucial role. The fundamental object in the
application of this approach for (anti-)neutrinos is the imaginary part of the
$W$ boson self-energy inside of the nuclear medium. The scheme presented in
Ref. Nieves  _et al._ (2011) amounts to perform a many-body expansion in which
the relevant gauge boson absorption modes are systematically incorporated:
absorption by one nucleon (QE), or a pair of nucleons or even three nucleon
mechanisms, real and virtual meson ($\pi$, $\rho$) production, excitation of
$\Delta$ or higher-energy resonance degrees of freedom, etc. In addition,
nuclear effects such as RPA collective excitations or short range correlations
(SRC) are also taken into account. In particular, the nucleon-nucleon
interaction, when modeling the excitation of a two-particle two-hole (2p2h)
nuclear component, goes far beyond the simple pion exchange, commonly used in
the literature, and which is shown here to be residual for intermediate energy
and momentum transfers. It is used the so-called induced interaction Oset  _et
al._ (1982); Ericson and Weise (1988); Nieves (1996), which consists of
longitudinal and transverse channels opened by pion and rho exchanges,
respectively, modulated by SRC and RPA re-summed to account for collective
effects. This is one of the pillars of the model developed by the Valencia
group to study many different nuclear reactions at intermediate energies.

We will focus in this work on the 2p2h mechanism. It stirs a lot of attention,
since already for electron scattering, the so-called “dip” region is often
underestimated by theoretical modeling. Several approaches have been developed
to account for this nuclear process, like the SuSA model Megias  _et al._
(2016a, b, 2019); Ivanov  _et al._ (2019), relativistic mean-field González-
Jiménez  _et al._ (2019); Martinez-Consentino  _et al._ (2021); Martinez-
Consentino and Amaro (2023); Martinez-Consentino  _et al._ (2024), spectral
functions Lovato  _et al._ (2023) or short-time approximation Pastore  _et
al._ (2020). Currently, a lot of effort is devoted to deliver predictions at
the semi-exclusive level, i.e. accounting for the final nucleons’
distributions. As already mentioned, it is a crucial observable for neutrino
oscillation experiments. While some theoretical models have intrinsic problems
to predict directly exclusive distributions, in our framework this can be
achieved in a straightforward way as it was shown in Ref. Sobczyk  _et al._
(2020).

Here, we improve on several aspects of the computation of the (anti-)neutrino
CC multi-nucleon cross section carried out in Ref. Sobczyk  _et al._ (2020).
Most importantly, we implement a correct treatment of nucleon self-energy in
the $W^{\pm}N\to N^{\prime}\pi$ vertex which enters in the calculation. This
consistent description leads to 2p2h strengths around 40% higher than those
obtained in the previous works of Refs. Sobczyk  _et al._ (2020); Nieves  _et
al._ (2011). We have also identified the largest source of theoretical
uncertainty in the 2p2h cross sections presented in this work and have
displayed error bands in all our predictions. In addition, we take the
opportunity of this revision to discuss in detail several important issues of
the calculation of the 2p2h cross section, delving into the microscopic
dynamics of the multi-nucleon mechanisms.

This work is organized as follows. In Sect. II, we review the main elements of
the calculation of the CCQE-like cross section carried out in Refs. Nieves
_et al._ (2011); Sobczyk  _et al._ (2020). We first briefly describe the model
for the pion production on the nucleon, paying special attention to recent
developments not considered in Nieves  _et al._ (2011). Subsect. II.1 is
devoted to the induced spin-isospin $NN,N\Delta,\Delta\Delta$ interactions
inside of a nuclear medium. In Subsec. II.2, further physical effects are
presented, together with a correct nucleon self-energy treatment in the in-
medium $W^{\pm}NN$ vertex and the identification of the largest source of
theoretical uncertainty in this calculation of the 2p2h cross sections. Next,
in Sect. III, we show predictions for the MiniBooNE carbon CCQE-like and T2K
carbon and oxygen CC0$\pi$ lepton double-differential cross sections. We also
compare with some published distributions obtained within the Lyon (Martini
_et al._ (2009, 2011)) and previous Valencia (Sobczyk  _et al._ (2020); Nieves
_et al._ (2012b)) schemes. Finally, the main conclusions of this work are
collected in Sect. IV.

##  II Microscopic calculation of 2p2h cross section

The basic dynamical ingredient driving the 2p2h mechanism is the
(anti-)neutrino pion production off nucleons (see top panel of Fig. 1). The
$W^{\pm}N\to N^{\prime}\pi$ amplitude, represented by the green dot, is
comprised of seven contributions depicted at the bottom panel of the figure.
The model developed in Hernandez  _et al._ (2007) contains the excitation of
the $\Delta(1232)$ resonance and non-resonant background, required by chiral
symmetry. It was further improved in Ref. Alvarez-Ruso  _et al._ (2016) by
unitarizing the leading vector and axial multi-poles applying Watson’s
theorem.

Figure 1: Top: four diagrams contributing to 2p2h constructed from the $WN\to
N^{\prime}\pi$ amplitude (green circle-box): pion production from the first
(a) and second (b) of the initial nucleons and the corresponding anti-
symmetrized terms (c) and (d), which are obtained from the previous ones by
crossing the final nucleons. Bottom: Model for the $WN\to N^{\prime}\pi$
amplitude, comprised of seven Feynman diagrams: pion pole (PP), contact term
(CT), pion in flight (PF), Delta pole ($\Delta$P) and cross Delta pole
(C$\Delta$P), and nucleon pole (NP) and cross nucleon pole (CNP). The light
blue circle-boxes in the diagrams stand for the weak transitions and include
phenomenological form-factors.

The most recent version of the model was derived in Ref. Hernández and Nieves
(2017), where the $\Delta$ propagator was changed from the Rarita-Schwinger
form to the pure spin-$3/2$ projector operator and consistent couplings
Pascalutsa (2001) were employed.111In Ref. Hernández and Nieves (2017), there
is nevertheless a very small local contribution stemming from a possible
$1/2$-spin part in the off-shell $\Delta$ propagator that is fitted to the
$\nu_{\mu}n\to\mu^{-}n\pi^{+}$ cross section data. As a consequence, a
$p^{2}_{\Delta}/M^{2}_{\Delta}$ local factor, which is equal to one at the
$\Delta-$peak, appears in the $\Delta$ propagator, which now reads

$G_{\mu\nu}(p_{\Delta})=\frac{p^{2}_{\Delta}}{M^{2}_{\Delta}}\frac{P_{\mu\nu}^{\frac{3}{2}}(p_{\Delta})}{p_{\Delta}^{2}-M_{\Delta}^{2}+iM_{\Delta}\Gamma_{\Delta}}\
,\quad
P_{\mu\nu}^{\frac{3}{2}}(p_{\Delta})=-(\not{p}_{\Delta}+M_{\Delta})\bigg{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{1}{3p_{\Delta}^{2}}(\not{p}_{\Delta}\gamma_{\mu}p_{\Delta\nu}+p_{\Delta\mu}\gamma_{\nu}\not{p}_{\Delta})\bigg{]}$
(1)

These improvements lead to a substantially enhanced
$\nu_{\mu}n\to\mu^{-}n\pi^{+}$ cross section, which is now quite well
described222The factor $p^{2}_{\Delta}/M^{2}_{\Delta}$ in Eq. (1) reduces
considerably the contribution of the $\Delta-$cross term of the amplitude in
all channels, especially in the $\nu_{\mu}n\to\mu^{-}n\pi^{+}$ one. , together
with the rest of available data from the Argonne and Brookhaven bubble-chamber
experiments. Moreover, the resulting model leads to an accurate reproduction
of the Watson’s unitary theorem in the dominant $P_{33}-$multipoles without
including any further sizable complex phase Hernández and Nieves (2017), which
is quite reassuring from the theoretical perspective. Finally, the complete
model of Ref. Hernández and Nieves (2017) also includes the contribution from
the $N^{*}(1520)$ resonance which is the only resonance, apart from the
$\Delta(1232)$, that gives a significant contribution in the few GeV neutrino
energy region Leitner  _et al._ (2009). It is interesting to note that the
model was successfully confronted to experimental total and differential cross
section for pion production off nucleons induced by both real and and virtual
photons Hernández and Nieves (2017); Sobczyk  _et al._ (2018a). In the latter
case, the theoretical predictions for polarized-electron reactions were also
compared to data, as a further test of the vector content of the model.
Moreover, an exhaustive comparison of the pion angular distributions obtained
within the scheme of Ref. Hernández and Nieves (2017) and the Argonne-Osaka
DCC (dynamical coupled-channel) model Matsuyama  _et al._ (2007); Kamano  _et
al._ (2013); Nakamura  _et al._ (2015) was carried out in Sobczyk  _et al._
(2018a). Given the high degree of complexity of the DCC approach, it is really
remarkable that the bulk of its predictions for electroweak production of
pions in the $\Delta$ region could be reproduced, with a reasonable accuracy,
by the simpler model of Ref. Hernández and Nieves (2017).

For simplicity, in the 2p2h calculations below, we will neglect the weak
excitation of the $D_{13}$ $N^{*}(1520)$ resonance which is negligible for
MiniBooNE and T2K experiments, and so we employ only the diagrams depicted in
the lower panel of Fig. 1. Furthermore, we will strictly consider the
$\Delta(1232)$ propagator of Eq. (1), neglecting any spin $1/2$ off-shell term
and the ($W_{\pi N},q^{2}$)-dependent unitarity relative phases between
background and resonant terms deduced from Watson’s theorem since they become
very small once consistent $\Delta$ couplings are used (Fig. 5 of Ref.
Hernández and Nieves (2017)).

###  II.1  Nuclear 2p2h cross section: meson exchange, short range
correlations and RPA

In many-body quantum field theory Fetter and Walecka (1971); Mattuck (1976),
the diagrams in the upper panel of Fig. 1 represent the $WNN\to NN$ absorption
reaction inside the nuclear medium. They can be equivalently expressed as the
imaginary parts of the $W$ self-energy diagrams depicted in Fig. 2, where the
initial and outgoing nucleons are put on-shell by the Cutkosky cuts shown in
the figure. The sum of the modulus squared of each of the four 2p2h mechanisms
depicted in Fig. 1 leads to the many-body diagram (a) of Fig. 2, while the
interference between (direct and crossed) mechanisms where the pion is
produced in the first or in the second of the initial nucleons is accounted by
the many-body diagram (b). These are the contributions considered in our
approach. The (c) and (d) mechanisms of Fig. 2 account for the direct-exchange
interference Ruiz Simo  _et al._ (2017), i.e., contributions stemming from the
interference between the direct and the antisymmetrized diagrams of Fig. 1. We
will further comment on the direct-exchange interference contributions below.

Our approach is based on the local density approximation333We would like to
point out that the local Fermi gas provides nucleon momentum distributions
much more realistic than those obtained in the global Fermi gas model, as
illustrated in Figs. 3 and 4 of Refs. Bourguille  _et al._ (2021) and Alvarez-
Ruso  _et al._ (2014), respectively.  (LDA), following the density power
counting (number of hole lines). In Sec. IIA of Sobczyk  _et al._ (2020), we
explained in detail, within the adopted scheme, the many-body diagrams that
account for initial and final state correlations and meson exchange current
contributions. There we also discussed the power density counting of each of
the latter many-body mechanisms.

The key feature of the approach lies in employing a widely tested effective
interaction in the nuclear medium between $NN,N\Delta,\Delta\Delta$, i.e. in
the diagrams driven by the $\Delta$P, C$\Delta$P, NP and CNP vertices (green
dots) in Fig. 2. Such isovector baryon-baryon interaction accounts for both
the spin longitudinal and transverse channels, going beyond the one pion
exchange (OPE) potential, and includes short and long range correlations. We
note that this differentiates our approach from many calculations of the 2p2h
cross section Megias  _et al._ (2019); Ruiz Simo  _et al._ (2017); Megias  _et
al._ (2016a); Lovato  _et al._ (2023); Martinez-Consentino and Amaro (2023)
which are based only on OPE.

The longitudinal and transverse channels opened by $\pi$ and $\rho$ exchanges,
respectively, are firstly supplemented by the SRC $g_{l}^{\prime}(k)$,
$g_{t}^{\prime}(k)$ terms, which account for the dynamics at shorter distances
and effectively prevent nucleons from getting close to each other. The SRC
Landau-Migdal parameters $g^{\prime}_{l/t}(k)$ inside of the nuclear medium
have a smooth dependence on the carrier’s four-momentum $k$ Oset  _et al._
(1982); Oset and Salcedo (1987); Garcia-Recio  _et al._ (1988), which has been
recently re-derived in Ref. Sobczyk  _et al._ (2020). Thus in momentum space,
the effective interaction reads for the $NN\to NN$ case

$\begin{split}&V_{l}(k)=\frac{f_{\pi
NN}^{2}}{m_{\pi}^{2}}\left[F_{\pi}^{2}(k^{2})\,\frac{\vec{k}^{2}}{k^{2}-m_{\pi}^{2}+i\epsilon}+g^{\prime}_{l}(k)\right](\vec{\tau}_{1}\cdot\vec{\tau}_{2})\,\vec{\sigma}_{1}^{i}\vec{\sigma}_{2}^{j}\,\hat{k}^{i}\hat{k}^{j}\;,\\\
&V_{t}(k)=\frac{f_{\pi
NN}^{2}}{m_{\pi}^{2}}\left[C_{\rho}F_{\rho}^{2}(k^{2})\,\frac{\vec{k}^{2}}{k^{2}-m_{\rho}^{2}+i\epsilon}+g^{\prime}_{t}(k)\right](\vec{\tau}_{1}\cdot\vec{\tau}_{2})\,\vec{\sigma}_{1}^{i}\vec{\sigma}_{2}^{j}\,(\delta^{ij}-\hat{k}^{i}\hat{k}^{j}),\end{split}$
(2)

with $\hat{k}=\vec{k}/|\vec{k}\,|$ the unitary three momentum of the virtual
pion or rho mesons, $\sigma_{i}$ and $\tau_{i}$ ($i=1,2$), Pauli matrices
acting on the spin and isospin nucleon degrees of freedom of each nucleon,
respectively, $C_{\rho}=2$, $f^{2}_{\pi NN}/4\pi=0.08$ and the form factors
($m_{\pi}=139\ \text{MeV},m_{\rho}=770\ \text{MeV}$)

$F_{\pi/\rho}(k^{2})=\frac{\Lambda_{\pi/\rho}^{2}-m_{\pi/\rho}^{2}}{\Lambda_{\pi/\rho}^{2}-k^{2}}\,,\quad
k^{2}=(k^{0})^{2}-\vec{k}^{\,2}\,,\quad\Lambda_{\pi}=1200\
\text{MeV},\quad\Lambda_{\rho}=2500\ \text{MeV}$ (3)

The $N\Delta$ and $\Delta\Delta$ interactions are obtained by replacing
$\vec{\sigma}\to\vec{S}$, $\vec{\tau}\to\vec{T}$, where $\vec{S},\vec{T}$ are
the spin, isospin $N\Delta$ transition operators Oset  _et al._ (1982);
Ericson and Weise (1988) and $f_{\pi NN}\to f^{*}_{\pi N\Delta}=2.13~{}f_{\pi
NN}$, for any $\Delta$ which replaces a nucleon.

For moderate momenta, $V_{l}(k)$ is suppressed due to cancellations between
OPE and the SRC $g^{\prime}_{l}(k)$ term444The SRC term $g^{\prime}_{l}(k)$ is
not the same one that is fitted in the free space, since SRC are modified
inside of the nuclear medium. Values around 0.6-0.7, fitted for the Valencia
group in the 1980’s and 1990’s, led to successful data descriptions of
inclusive photo-, electron- and pion-nuclear reactions, of pionic atom levels,
and of many other processes sensitive to the (pseudoscalar & vector)-isovector
part of the $NN,N\Delta,\Delta\Delta$ interactions, which involved moderate
momenta around 300-400 MeV. This range of values for $g^{\prime}_{l}$ is also
consistent with the results of Ref. Speth  _et al._ (1980) from calculations
of nuclear electric and magnetic moments, transition probabilities, and giant
electric and magnetic multipole resonances. In this regime, we can see that
the SRC greatly cancels out the OPE interaction, since
$\vec{k}^{2}D_{\pi}(k^{2})\sim-1+[m_{\pi}^{2}-(k^{0})^{2}]/\vec{k}^{2}+\cdots$
(4) and as result, the longitudinal channel of the interaction turns out to be
largely suppressed as compared to the OPE (see left plot of Fig. 5 of Ref.
Sobczyk  _et al._ (2020)). and so the transverse channel of the isovector part
of the $NN$, $N\Delta$, $\Delta\Delta$ interaction plays a crucial role in
many nuclear reactions at intermediate energies. We note that the
(pseudoscalar and vector)-isovector part of the $NN$ interaction is of little
importance to describe the bulk of the structure of closed-shell isospin
symmetric nuclei, which is mostly given by scalar-isoscalar and vector-
isoscalar exchanges (e.g. Quantum hadrodynamics Serot and Walecka (1997)
begins with effective $\sigma$ and $\omega$ meson exchanges). However, this
piece of the interaction becomes extremely important when external probes
(pions, real and virtual photons or $W^{\pm}$ and $Z^{0}$ bosons) excite these
spin-isospin components of the nuclei, and it becomes really essential when
the transferred energies make it possible to excite the $\Delta$ resonance.
Thus, in these situations, the nuclear responses are in great extent
determined by these degrees of freedom that otherwise have little relevance in
the nuclear structure.

Figure 2: 2p2h many-body contributions to the $W$ self-energy driven only by
pion exchange. The green circle-boxes account for the seven $W^{\pm}N\to
N^{\prime}\pi$ Feynman diagrams shown in the two bottom rows of Fig. 1. The
horizontal dotted lines show the cuts that put on the mass-shell the 2p2h
excitations and that account for $WNN\to NN$ absorption inside of the nuclear
medium.  Figure 3: Diagrammatic representation of the RPA series of 1p1h and
$\Delta$h excitations, driven by the effective $\pi+\rho+{\rm SRC}$
interaction.

The $\pi+\rho+{\rm SRC}$ model for the $NN$, $N\Delta$, $\Delta\Delta$ medium
interactions is further improved by including nuclear collective polarization
corrections, as diagrammatically shown in Fig. 3. Hence, ultimately the
picture for the spin-isospin dependent part of the interaction accounts also
for the propagation of the mesons through the medium in the RPA sense,
producing indistinctly 1p1h and $\Delta$-hole ($\Delta$h) nuclear excitations
in the whole nucleus.555This has an obvious resemblance with the properties of
the instantaneous Coulomb interaction in an electron gas. One electron inside
of an electron gas polarizes the medium in such a way that, in a region around
it, the negative charges are slightly displaced away from the electron,
leaving behind the background charge of the positive ions. As a consequence of
this polarization, the photon acquires an effective mass, which leads to the
screening of the Coulomb interaction Fetter and Walecka (1971). Thus, one
finds two (longitudinal and transverse) geometrical series, which do not
interfere, and that can be easily summed up leading to the induced spin-
isospin interaction, which derivation can be found e.g. in Refs. Oset  _et
al._ (1982); Ericson and Weise (1988) or more recently in the Section 2.2 of
Ref. Nieves (1996). Thus in the many body diagrams of Fig. 2, one should
replace $V_{\pi}(k)$ (OPE) by the induced spin-isospin interaction in all
interaction lines that emerge from the direct and cross $\Delta(1232)$ and
nucleon pole terms ($\Delta$P, C$\Delta$P, NP and CNP). Pions in the PP, PF
and CT vertices should not be modified because those mechanisms require an
actual pion.

This induced spin-isospin interaction has been satisfactorily used for many
different nuclear reactions at intermediate energies by Prof. Oset’s Valencia
group Oset and Weise (1979); Oset  _et al._ (1982, 1985); Salcedo  _et al._
(1988); Oset and Salcedo (1987); Garcia-Recio  _et al._ (1988); Carrasco  _et
al._ (1993); Fernandez de Cordoba  _et al._ (1993, 1995); Garcia-Recio  _et
al._ (1992); Fernandez de Cordoba and Oset (1992); Gil  _et al._ (1997);
Carrasco and Oset (1992); Nieves  _et al._ (1993a, b), and that importantly
include inclusive scattering of photons Carrasco and Oset (1992), electrons
Gil  _et al._ (1997) and pions from nuclei Nieves  _et al._ (1993a, b), where
multinucleon mechanisms play a crucial role. In all these works, the many-body
exchange diagrams of panels (c) and (d) of Fig. 2 had never been considered.
They were discussed in the context of the calculation of the $\Delta$ self-
energy in the pioneering work of Ref. Oset and Salcedo (1987) (see text
related to Fig. 4 of that reference). Firstly, such contributions were found
to produce corrections of the order of $25\%$, which agrees with the claims
made in Ref. Ruiz Simo  _et al._ (2017) using only OPE. Doubts emerged,
however, due to to some undesirable cancellation between
$g^{\prime}$-contributions in the direct and exchange diagrams. It was argued
in Oset and Salcedo (1987) that $g’$ is associated to the repulsive forces
that arise as a consequence of the antisymmetry of the quarks when two bags
overlap, and hence it is the antisymmetry of the quarks what matters and one
should not play with the antisymmetry of the nucleons. In any case, within the
scheme adopted by the Valencia group, such exchange diagrams were always
neglected, and the effective parameters, like $g^{\prime}_{l}$,
$g^{\prime}_{t}$ or form-factors, of the in-medium interaction were
successfully fitted to pion-nuclear inclusive processes (pionic atom data,
elastic and inelastic pion-nucleus scattering and pion absorption in nuclei),
since the pion precisely selects the spin-isospin piece of the $NN$,
$N\Delta$, $\Delta\Delta$ interactions. The subsequent results for real
photon- and electro-nuclear processes or CC neutrino reactions are predictions
of the model. Including at any later point the exchange contribution, would
necessarily require a re-fit of the effective interactions, since these
contributions are implicitly (effectively) included.666See also the discussion
of Fig. 31 of Ref. Carrasco and Oset (1992), where the interaction of real
photons with nuclei from 100 MeV to 500 MeV is studied.

###  II.2 Further physical effects and considerations

Figure 4:  $W$ self-energy many-body diagrams obtained from the diagonal
$\Delta$P-$\Delta$P (left), diagonal NP-NP (middle) and off-diagonal
$\Delta$P-NP (right) terms of the $WN\to N^{\prime}\pi$ amplitude. In all
diagrams the OPE interaction has been replaced by the full induced interaction
of Fig. 3. Note that the NP-NP diagram can be also seen as a QE contribution
where the nucleon propagator of the particle state in the 1p1h excitation is
dressed. In all cases the 2p2h cut is depicted, and in the off-diagonal
$\Delta$P-NP and diagonal NP-NP terms, 1p1h cuts are also shown to illustrate
that even though the 2p2h excitation is on-shell, one can still put on the
mass shell the nucleons cut by the circular dotted line.

To compute the 2p2h cross sections we adopt in this paper the scheme of Ref.
Sobczyk  _et al._ (2020). The latter work improves on some aspects of the
previous work of Ref. Nieves  _et al._ (2011), the main motivation being to
retrieve the distribution of outgoing nucleons in the primary reaction, which
otherwise were distributed according to phase-space in Monte Carlo event
generators. To this end, we separated the two (2p2h) and three (3p3h) nucleon
contributions driven by the $\Delta$ excitation, which in Ref. Nieves  _et
al._ (2011) were treated together. Moreover, in the description of the in-
medium interactions between nucleons and $\Delta$’s, the energy- and momentum-
transfer dependence of the SRC $g^{\prime}_{l}$ and $g^{\prime}_{t}$ is
maintained. In addition, the treatment of the weak excitation of the $\Delta$
resonance was also refined, according to the latest model for the $WN\to
N^{\prime}\pi$ amplitude derived in Ref. Hernández and Nieves (2017), and it
includes changes in the $\Delta$ propagator (see discussion of Eq. (1)) and in
the value at $q^{2}=0$ of the dominant axial $(C_{5}^{A}(q^{2}))$ $WN\Delta$
form factor, which is in excellent agreement with the value derived from the
$\pi N\Delta$ coupling (off-diagonal Goldberger-Treiman relation) and around a
20% larger than that used in Ref. Nieves  _et al._ (2011). Finally, we did not
average in Sobczyk  _et al._ (2020) the initial nucleons’ momenta.777The
computation of the 2p2h contribution of the many-body diagrams involving NP
terms is subtle, since when placing the 2p2h excitation on-shell, the nucleon
propagator that is part of the $WN\to N^{\prime}\pi$ amplitude can be still
placed on-shell (circular cuts in Fig. 4) for a virtual $W$, and thus there
exists a single pole in the integration over the hole three momentum. The
correct treatment requires i) to dress the nucleon propagator with a complex
selfenergy $\Sigma_{N}(p)$, which moves the pole to the complex plane out of
the integration domain, and ii) to explicitly perform all the nucleon phase-
space integrals avoiding the use of any type of average over the kinematics of
the nucleons Gil  _et al._ (1997).

In what follows we enumerate some considerations and additional improvements
on the treatment of Ref. Sobczyk  _et al._ (2020) implemented in this work.

  * a)

$\Delta$ self-energy — As pointed out in Ref. Nieves  _et al._ (2011), the RPA
series of $\Delta$h excitations driven by the $\pi+\rho+{\rm SRC}$ spin-
isospin interaction888Actually, one should use the full induced spin-isospin
interaction sketched in Fig. 3, which amounts to account also for intermediate
1p1h excitations in the RPA series (see for instance Fig. 16 of Ref. Gil  _et
al._ (1997)). The discussion below is not affected by including these
corrections. gives rise to an irreducible $W$ self-energy piece and therefore
to a contribution to the hadron tensor. The latter depends on the particular
Lorentz components of the tensor which is being evaluated. Each of these
tensor components are the result of matrix elements of different operators
${\cal O}_{i}$, selecting either the longitudinal or transverse pieces of the
$\pi+\rho+{\rm SRC}$ interaction

$\sum_{\rm spins}\sum_{\rm isospins}\ \sum_{i,j\in WN\Delta}\langle N|{\cal
O}_{i}|\Delta\rangle\langle\Delta|{\cal O}_{j}|N\rangle$ (5)

which leads to the fact that each of the hadron tensor elements
$W^{\mu\nu}_{\Delta}$ could be modified by a different value of the real part
of the $\Delta$ self-energy ${\rm Re}\Sigma^{\rm
eff}_{\Delta}(p_{\Delta},q;\rho)$.

In the study of inclusive cross sections of real photons by nuclei, due to the
transverse nature of the $\gamma N\Delta$ vertex, only $V_{t}$ is relevant
which leads to Carrasco and Oset (1992)

${\rm Re}\Sigma^{\rm eff,t}_{\Delta}(p_{\Delta},q;\rho)={\rm
Re}\Sigma_{\Delta}(p_{\Delta};\rho)+\frac{4}{9}\left(\frac{f^{*}_{\pi
N\Delta}}{m_{\pi}}\right)^{2}V_{t}(q)\rho$ (6)

where ${\rm Re}\Sigma_{\Delta}\sim-50\rho/\rho_{0}$ MeV Oset and Salcedo
(1987); Ericson and Weise (1988) is referred to the potential felt by the
nucleons in the LFG model (bound by $\sim-k_{F}^{2}(r)/2M$)999It means that
for $\rho=\rho_{0}$, the real part of the $\Delta$-nucleus interaction is
around 50 MeV more attractive that the nucleon mean-field potential.. For
momentum transfers smaller than $m_{\rho}$, $V_{t}$ is dominated by the SRC
Landau-Migdal parameter $g^{\prime}_{t}\sim 0.6$ and hence ${\rm
Re}\Sigma^{\rm eff,t}_{\Delta}(\rho)\sim+30\rho/\rho_{0}$ MeV, at most. Thus
the RPA re-summation provides a natural explanation for why, although the
$\Delta-$peak is much wider because of the broadening of the $\Delta$ in the
medium, its position hardly changes in inclusive $\gamma A$ nuclear
reactions.101010This can be confirmed in the right panel of Fig. 8.17 of the
book T. Ericson and W. Weise ”Pions in Nuclei” Ericson and Weise (1988).
There, the experimental $\gamma^{12}$C total cross section as a function of
the laboratory photon energy is compared to $12\times\sigma(\gamma p)$. In the
left panel of the same figure is shown the $\pi^{12}$C total cross section,
for which the $\Delta$ peak is around 50-75 MeV shifted towards lower masses,
indicating that in that case the $\Delta$ feels an important attraction in the
nuclear medium, as we will see next.

On the other hand, the interaction of nuclei with external pions selects only
the spin longitudinal channel of the $\Delta$h-$\Delta$h interaction in the
RPA re-summation, leading to Oset  _et al._ (1982); Nieves  _et al._ (1993a,
b) the replacement

${\rm Re}\Sigma^{\rm eff,l}_{\Delta}(p_{\Delta},q;\rho)={\rm
Re}\Sigma_{\Delta}(p_{\Delta};\rho)+\frac{4}{9}\left(\frac{f^{*}_{\pi
N\Delta}}{m_{\pi}}\right)^{2}V_{l}(q)\rho$ (7)

where $V_{l}(q)<0$ and it is small because of the large cancellation between
the OPE piece and the SRC Landau-Migdal parameter $g^{\prime}_{l}$. Hence, the
effective longitudinal real part of the $\Delta$ self-energy remains more
attractive than the result of ${\rm
Re}\Sigma_{\Delta}(p_{\Delta};\rho)=-50\rho/\rho_{0}$ MeV found in Ref. Oset
and Salcedo (1987).

When considering the CC neutrino scattering, the weak $W^{\pm}$ gauge boson
will probe both longitudinal and transverse operators of the $WN\Delta$ vertex
and therefore each tensorial component of the hadron tensor will be dominated
by either the longitudinal or the transverse RPA re-summation. A correct
treatment requires a detailed non-trivial microscopic evaluation, as it was
done in Nieves  _et al._ (2004) for the QE contributions. For that reason in
the previous calculations of Refs. Nieves  _et al._ (2011) and Sobczyk  _et
al._ (2020), the real part of the position of the $\Delta$-peak was not
renormalized in the medium taking

${\rm Re}\Sigma^{\rm eff}_{\Delta}(p_{\Delta},q;\rho)=0$ (8)

Here, we will allow ${\rm Re}\Sigma^{\rm eff}_{\Delta}(p_{\Delta},q;\rho)$ to
vary in the range $[-50\rho/\rho_{0},+30\rho/\rho_{0}]$ MeV and the dispersion
of CCQE-like predictions will be taken as a systematic error of the
calculation.

  * b)

Removal energy — In the calculation of the 2p2h cross section, we introduce
two-nucleon removal energy corrections Bourguille  _et al._ (2021), and the
energy transferred to the hadronic system is reduced by the two-nucleon
separation energy,

$q^{0}=E_{\nu}-E_{\mu}\to q^{0}_{\rm
eff}=E_{\nu}-E_{\mu}-S_{N_{1}N_{2}},\qquad
S_{N_{1}N_{2}}=M^{\prime}_{A-2}-M_{A}+m_{N_{1}}+m_{N_{2}}$ (9)

with $M_{A}$ and $M^{\prime}_{A-2}$ the ground-state masses of the initial and
final nuclei, and $m_{N_{i}}$ the masses of the ejected nucleons. In the
calculations below, we take approximate average values for $S_{N_{1}N_{2}}$ of
around 27 and 22 MeV for carbon and oxygen respectively.

  * c)

Nucleon self-energy — In the previous computations of Refs. Nieves  _et al._
(2011); Sobczyk  _et al._ (2020) the incoming and outgoing nucleons of $WN\to
N^{\prime}\pi$ amplitude (entering the 2p2h diagrams) are considered to be
moving in a local mean-field potential

$E(\vec{p};\rho)=\sqrt{M^{2}+\vec{p}^{\,2}\,}-\frac{k_{F}^{2}(r)}{2M},\qquad
k_{F}(r)=\left[\frac{3\rho(r)}{2\pi^{2}}\right]^{\frac{1}{3}}$ (10)

The constant term, $-k_{F}^{2}(r)/2M$, which cancels out in the energy
conservation, was not included in the energy of the virtual nucleon that
appears in the NP and CNP terms of the $WN\to N^{\prime}\pi$ amplitude. This
is inconsistent because this virtual nucleon should feel the same attraction,
neglecting momentum dependent nuclear potential terms. In the case of virtual
$\Delta$ in $\Delta$P diagram, the term $-k_{F}^{2}(r)/2M$ amounted to include
effectively a repulsive interaction, since no other real part of the
$\Delta$-nucleus potential was included.

This inconsistent description induced smaller 2p2h cross sections. Here, we
decide to remove everywhere the constant term $-k_{F}^{2}(r)/2M$ from the
definition of nucleon energies. It does not affect the overall energy-balance
of the reaction (since both the initial and final nucleons are shifted by the
same constant term). This choice is also consistent with the calculation of
${\rm Re}\Sigma_{\Delta}$ in Ref. Oset and Salcedo (1987), since the numerical
values obtained in that work are referred to the potential felt by the
nucleons in the LFG model.

The 2p2h calculation of Ref. Nieves  _et al._ (2011), implemented in some
Monte Carlo event generators has the same theoretical inconsistency, which is
inherited by the derived works of Refs. Nieves  _et al._ (2012b) and Nieves
_et al._ (2013) on neutrino and antineutrino MiniBooNE double differential
cross sections. All the theoretical developments made in the work on the
reconstruction of the neutrino energy and the shape of the CCQE-type total
cross section of Ref. Nieves  _et al._ (2012a) are correct, although the
quantitative predictions are affected. Finally, the 2p2h cross sections shown
in Refs. Gran  _et al._ (2013); Bourguille  _et al._ (2021) for MINERvA
energies and in Ref. Bourguille  _et al._ (2021) for the T2K CC0$\pi$, were
also evaluated using the results of Ref. Nieves  _et al._ (2011). The
predictions which we will show in the next section for MiniBooNE and T2K
experiments supersede all previous calculations.

###  II.3 Quasi-elastic contribution

The description of CCQE-like cross section requires a consistent calculation
of the multinucleon and the QE contributions. The latter process corresponds
to a 1p1h nuclear excitation and here we use the results obtained from the LFG
model of Ref. Nieves  _et al._ (2004). This QE model accounts for a correct
energy balance using the experimental $Q$ values, the Coulomb distortion of
the outgoing charged lepton and medium polarization (RPA) effects. In the
spin-isospin channel, the RPA includes also $\Delta$h degrees of freedom and
it is driven by the $\pi+\rho+{\rm SRC}$ interaction, leading to the induced
interaction shown in Fig. 3. Notably, the RPA re-summation was significantly
improved in Nieves  _et al._ (2004) by a correct tensorial treatment of the
RPA response function and the inclusion of the scalar-isovector channel of the
nucleon-nucleon interaction. In the scheme of Ref. Nieves  _et al._ (2004),
there exists also a possibility of dressing the nucleon propagators in the
nuclear medium, which amounts to work with nucleon spectral functions (SF)
Benhar  _et al._ (2005); Benhar (2013); Vagnoni  _et al._ (2017).

Here, however, the QE cross section will be computed within the RPA model of
Ref. Nieves  _et al._ (2004) without including SF effects. The reason for it
is the following. The hole and particle SFs used in Nieves  _et al._ (2004);
Nieves and Sobczyk (2017); Sobczyk  _et al._ (2018b, 2019) are obtained from
the semi-phenomenological nucleon self-energies developed in Ref. Fernandez de
Cordoba and Oset (1992). The real part of self-energy (${\rm Re}\Sigma_{N}$)
acts as an energy-dependent effective potential which changes the in-medium
dispersion relation of the nucleons Nieves and Sobczyk (2017). It largely
cancels out between the hole and particle states and can be safely
neglected.111111Indeed, it provides a high energy tail in the QE
distributions, which was shown in Ref. Benhar  _et al._ (2010) to have little
effect in the reproduction of the MiniBooNE two-dimensional CCQE-like cross
section Aguilar-Arevalo  _et al._ (2010).  The imaginary part, on the other
hand, plays a much more important role. It accounts for $NN$ collisions and it
is much larger for particle than for hole states Fernandez de Cordoba and Oset
(1992), because of the enhanced available phase-space. The events induced by
the imaginary part of the nucleon self-energy account for various collision
channels (hadronic states) at the primary vertex. If we are interested in the
semi-exclusive cross sections, these events (driven by collisional broadening)
in the intranuclear cascade would need to be treated separately from the one
body (1p1h) ones in the primary reaction, which are obtained after neglecting
${\rm Im}\Sigma_{N}$. All in all, in the calculation of the 1p1h contribution,
we neglect both ${\rm Re}\Sigma_{N}$ and ${\rm Im}\Sigma_{N}$, while we
maintain RPA corrections and relativistic kinematics. Within this discussion,
we point out that the diagonal NP-NP diagram (middle one in Fig. 4) can be
cast as a 1p1h term, where the particle nucleon is dressed with a complex
self-energy $\Sigma_{N}$ obtained from a single 1p1h insertion. Hence, it
belongs to the QE calculation using SF and so in our calculations this term is
excluded from the sum implicit in the generic 2p2h contribution depicted in
the first diagram of Fig. 2.

##  III Results

Figure 5: MiniBooNE neutrino Aguilar-Arevalo  _et al._ (2010) (left) and
antineutrino Aguilar-Arevalo  _et al._ (2013) (right) CCQE-like muon angle and
kinetic energy distribution $d^{2}\sigma/d\cos\theta_{\mu}dT_{\mu}$ on carbon
compared to predictions obtained in this work. Experimental errors do not
include overall flux uncertainties. We average the theoretical cross sections
(integrate and divide by the length of the interval) for each of the
$\cos\theta_{\mu}$-bins. The uncertainty bands in the theoretical results show
the variation produced when ${\rm Re}\Sigma^{\rm
eff}_{\Delta}(p_{\Delta},q;\rho)$ change in the range
$[-50\rho/\rho_{0},+30\rho/\rho_{0}]$ MeV. The largest (smallest) cross
sections are obtained for ${\rm Re}\Sigma^{\rm
eff}_{\Delta}(p_{\Delta},q;\rho)=-50\rho/\rho_{0}$ MeV ($+30\rho/\rho_{0}$
MeV), while the central values have been calculated using ${\rm Re}\Sigma^{\rm
eff}_{\Delta}$=0.

Figure 6: Flux-unfolded MiniBooNE muon neutrino Aguilar-Arevalo  _et al._
(2010) (left) and antineutrino Aguilar-Arevalo  _et al._ (2013) (right) CCQE-
like cross section on carbon, divided by the number of active nucleons, as a
function of (anti-)neutrino energy compared to predictions obtained in this
work. The theoretical multi-nucleon curves have been computed from the 2p2h
cross section using $\sigma^{\rm 2p2h}_{\rm appx}$ defined in Eq. (19) of Ref.
Nieves  _et al._ (2012a). The uncertainty bands in the theoretical results
have been obtained as explained in Fig. 5.

Figure 7: T2K neutrino muon angle and momentum CC0$\pi$ double differential
distribution $d^{2}\sigma/d\cos\theta_{\mu}dp_{\mu}$ on carbon (left) and
oxygen (right) Abe  _et al._ (2020), divided by the number of nucleons. We
also show the predictions obtained in this work, supplemented by the
contribution “CC Res” taken from the NEUT results of Ref. Bourguille  _et al._
(2021) for carbon, and re-scaled by the number of nucleons for oxygen. We
average the theoretical cross sections (integrate and divide by the length of
the interval) for each of the $\cos\theta_{\mu}$-bins. The uncertainty bands
in the theoretical results have been obtained as explained in Fig. 5.

Firstly, we compare the predictions of our updated model with the MiniBooNE
CCQE-like neutrino and antineutrino $d^{2}\sigma/d\cos\theta_{\mu}dT_{\mu}$
differential cross-section of Refs. Aguilar-Arevalo  _et al._ (2010, 2013). We
find an excellent reproduction of neutrino data across a broad range of muon
kinetic energy and angular bins. In the earlier comparison of Ref. Nieves  _et
al._ (2012b), an agreement was found only when the neutrino flux was quenched
by $10\%$, which could be accommodated thanks to the global normalization
uncertainty of 10.7% admitted by the MiniBooNE Collaboration Aguilar-Arevalo
_et al._ (2010). The new 2p2h cross section is larger than the one computed in
Nieves  _et al._ (2012b), and allows for an excellent description of the
experiment without having to reduce the data-points by an overall factor of
0.9. Still, we support the findings of Ref. Nieves  _et al._ (2012b), that the
simultaneous consideration of RPA and multinucleon mechanisms makes the
present microscopic model more appropriate than a pure impulse approximation.
We also observe the significant size of the uncertainty on the theoretical
2p2h cross section induced by the variation of ${\rm Re}\Sigma^{\rm
eff}_{\Delta}(p_{\Delta},q;\rho)$ in the range
$[-50\rho/\rho_{0},+30\rho/\rho_{0}]$ MeV (see discussion of item a) in
Subsect. II.2). The theoretical description of the MiniBooNE antineutrino
CCQE-like double differential cross section (right panels of Fig. 5) on
carbon, though not as good as the neutrino one, is quite reasonable and
certainly better than the predictions shown in Nieves  _et al._ (2013). Here,
we also confirm that the combined RPA and multinucleon nuclear effects lead to
a robust microscopical description of the data. For all angular bins,
multinucleon mechanisms improve agreement with data. Their contribution tends
to concentrate at low muon energies so that more energy is transferred to the
nucleus than in genuine QE events. Comparing antineutrino and neutrino double
differential distributions, we observe a different pattern because of the more
forward peaked character of the antineutrino induced reaction and the lower
average energies of the MiniBooNE antineutrino flux. For reference, the 3p3h
contribution ($WNNN\to NNN$) is separately shown in the
$0.80<\cos\theta_{\mu}<0.90$ window for both neutrino and antineutrino
reactions. It is small in both cases because of the low energies of the
MiniBooNE fluxes, and in the antineutrino case, it is barely visible.

Next, in Fig. 6 we show our predictions for the flux-unfolded MiniBooNE muon
neutrino Aguilar-Arevalo  _et al._ (2010) (left) and antineutrino Aguilar-
Arevalo  _et al._ (2013) (right) CCQE-like cross section on carbon. The 2p2h
contribution $\sigma^{\rm 2p2h}_{\rm appx}$ is calculated following the QE
unfolding procedure explained in Nieves  _et al._ (2012a) (see Eq. (19) of
that work). The reproduction of the data for the neutrino reactions is
spectacular, as good as that exhibited in Ref. Nieves  _et al._ (2012a) using
the 2p2h contribution of Ref. Nieves  _et al._ (2011), but without relying on
a global reduction of the data-points by a factor 0.9. Since in both works the
genuine QE cross section is the same, the difference is only due to larger
multinucleon cross sections here. For instance at $E=1$ GeV we obtain now,
including the uncertainty due to the $\Delta$h RPA, values for $\sigma^{\rm
2p2h}_{\rm appx}/6$ (units of $10^{-38}$ cm2) in the range of 0.23-0.29, while
it was around 0.15 in Ref. Nieves  _et al._ (2012a). This quantitative
difference does not alter the conclusions of Ref. Nieves  _et al._ (2012a)
about the inadequacy of the genuine-QE algorithm to reconstruct the neutrino
energy. The MiniBooNE unfolded CCQE-like cross section exhibits an excess
(deficit) at low (high) neutrino energies, which is an artifact of an
unfolding procedure that ignores multinucleon mechanisms. This effect should
be now larger than that inferred from the solid green-curve and the pseudo
data-points in Fig. 5 of Ref. Nieves  _et al._ (2012a), since the 2p2h cross
section is now bigger. This systematic bias is relevant to neutrino
oscillation experiments that uses the CCQE-like samples to compute the
neutrino energy.

The agreement with antineutrino unfolded cross-section on the right panel of
Fig. 6 is good, actually it is slightly better than in Ref. Nieves  _et al._
(2013) where 2p2h results from Ref. Nieves  _et al._ (2011) were used. For
antineutrinos, we also find that the multinucleon cross section is now larger
in this revised and corrected calculation, but the increase is significantly
more moderate than in the case of the neutrino-induced reaction. Thus at $E=$1
GeV, $\sigma^{\rm 2p2h}_{\rm appx}/6$ takes values in the $(0.07-0.09)\times
10^{-38}$ cm2 range, while one can read off $0.06\times 10^{-38}$ cm2 from the
blue-circles of Fig. 1 of Ref. Nieves  _et al._ (2013). The error reported by
MiniBooNE for the total cross sections is already of the order of $0.04\times
10^{-38}$ cm2, comparable with the size of the unfolded multinucleon
contribution, which makes multinucleon effects less relevant than in the
neutrino driven process.

We finish this part of the discussion of results with Fig. 7, where we compare
our predictions with the T2K neutrino muon angle and momentum CC0$\pi$ double
differential distributions on carbon (left) and oxygen (right) Abe  _et al._
(2020). Since these data-samples include all events with no pions in the final
state, we have supplemented our QE+2p2h cross sections by the small
contribution labelled as “CC Res” in the plots and taken from the NEUT results
of Ref. Bourguille  _et al._ (2021) for carbon. The “CC Res” contribution for
oxygen has been obtained from carbon by re-scaling them by a factor $8/6$. It
includes primarily events originated from production of a real pion in the
initial step followed by its absorption before leaving the nuclear medium,
which were removed by the MiniBooNE Collaboration from its two-dimensional
distributions, as well as a very small deep inelastic scattering contribution
from the higher-energy tail of the T2K neutrino spectrum.121212We cannot
extend our theoretical calculation of the QE and 2p2h cross sections to all
T2K-flux neutrino energies and for these contributions we have neglected the
flux above 3 GeV. For both nuclei, we reproduce quite well the distributions
supporting the validity of the theoretical model discussed in this work.
Results for carbon, using the 2p2h cross sections of Ref. Nieves  _et al._
(2011), are shown in the left panels of Fig. 5 of Ref. Bourguille  _et al._
(2021), where it can be seen that a worse description of the data is found.

###  III.1 Comparison with other theoretical calculations

In Fig. 8, we compare the actual neutrino and antineutrino 2p2h cross sections
$\sigma^{\rm 2p2h}$ evaluated in this work and those derived in Ref. Sobczyk
_et al._ (2020). We clearly see the sizable effect produced by the
inconsistent treatment of the energy conservation between the hole-nucleon and
the virtual nucleon or $\Delta$ that enter in the $WN\to(N,\Delta)\pi$
vertices, which are part of the full $WN\to N^{\prime}\pi$ amplitude. At $E=1$
GeV and in units of $10^{-38}$ cm2, the new calculation provides for the
neutrino mode a band of values comprised between 1.69 [for ${\rm
Re}\Sigma^{\rm eff}_{\Delta}(p_{\Delta},q;\rho)=+30\rho/\rho_{0}$ MeV] and
2.39 [for ${\rm Re}\Sigma^{\rm
eff}_{\Delta}(p_{\Delta},q;\rho)=-50\rho/\rho_{0}$ MeV], while the calculation
carried out in Ref. Sobczyk  _et al._ (2020) gives around 1.38. These numbers
for the antineutrino cross sections are [0.50-0.62] for the uncertainty band
of the present evaluation and 0.31 for the calculation of Ref. Sobczyk  _et
al._ (2020).

Figure 8: Neutrino and antineutrino 2p2h (including three body absorption
contributions) cross section on 12C as a function of incoming neutrino energy
$E$. We show both the results obtained in this work and those in Ref. Sobczyk
_et al._ (2020) (PRC 102, 024601). In all cases, a cut $|\vec{q}\,|<1.2$ GeV
is implemented.

Figure 9: Left: MiniBooNE neutrino CCQE-like muon angle and kinetic energy
distribution $d^{2}\sigma/d\cos\theta_{\mu}dT_{\mu}$ on carbon for
$0.80<\cos\theta_{\mu}<0.90$ Aguilar-Arevalo  _et al._ (2010). We show
separately the total, QE-RPA and multi-nucleon theoretical distributions
obtained in this work, previously depicted in Fig. 5, and by the Arpajon-Lyon
group taken from Fig. 6 of Ref. Martini  _et al._ (2011) (reddish curves, PRC
84, 055502). We also show the multi-nucleon contribution (green dotted curve,
PLB 707, 72) from Fig. 3 of Ref. Nieves  _et al._ (2012b). The uncertainty
bands in the theoretical predictions of this work have been obtained in all
cases as explained in Fig. 5. Right: Various contributions to the CCQE-like
$\nu_{\mu}+^{12}$C differential cross section at $E=0.7$ GeV from this work,
as a function of the energy transferred to the nucleus. Quasi-elastic
calculations Nieves  _et al._ (2004) include RPA corrections and the
uncertainty band induced by the variation of ${\rm Re}\Sigma^{\rm
eff}_{\Delta}$ (see text of Fig. 5) is shown for the multi-nucleon
distribution. For this latter cross section, we also show the Lyon group
results (red curves) of Fig. 22 of Ref. Martini  _et al._ (2009) for the two
different extrapolations examined in that work. The one peaked at low values
of $\omega$ is based on the approach of J. Marteau of Ref. Marteau (1999),
while the another one exploits a microscopic evaluation of the 2p2h
contribution to the transverse magnetic response of $(e,e^{\prime})$
scattering Alberico  _et al._ (1984).

Lastly, we would like to compare our predictions with the Lyon group results
Martini  _et al._ (2009, 2011). Both models are based on the LDA, include RPA
effects and have clear theoretical similarities in the treatment of the
$\Delta$ in the medium. The RPA 1p1h-1p1h, 1p1h-$\Delta$h and
$\Delta$h-$\Delta$h forces are constructed qualitatively in a similar way,
namely from $\pi+\rho+{\rm SRC}$ interactions, with similar parameters as in
Nieves  _et al._ (2004), except the $\rho$-driven potential which is a factor
2/3 weaker in the Lyon group scheme. The original calculations of Ref. Martini
_et al._ (2009) are non-relativistic. In the later work of the Lyon group
Martini  _et al._ (2011), a relativistic modification of the nuclear response
is implemented as proposed in Refs. Barbaro  _et al._ (1996); De Pace (1998).
This amounts to work in a non-relativistic frame with the substitution
$\omega\to\omega(1+\omega/2M)$ in the nuclear responses, and by multiplying
the responses by $(1+\omega/M)$. As far as we understand, similar recipes are
also used for the multinucleon cross sections also given in Martini  _et al._
(2009).

The Lyon 2p2h predictions of Ref. Martini  _et al._ (2009) in a first step
relies on Marteau’s model of Ref. Marteau (1999). There, and attending to Fig.
1 of that work, the 2p2h cross section is computed from the first many-body
diagram of Fig. 2, considering only the NP, CNP, $\Delta$P and C$\Delta$P
terms of the $WN\to N^{\prime}\pi$ amplitude, and replacing the OPE by a more
realistic interaction. The direct $\Delta$P-$\Delta$P contribution to the 2p2h
cross section is considered as driven by a $\Delta$ self-energy diagram, which
is taken into account through the parametrization of the in-medium
$\Delta-$width of the Valencia group Oset and Salcedo (1987). The rest of
contributions that are not reducible to a modification of the $\Delta$ width
are accounted in two different ways, as we will describe below.

In the left panel of Fig. 9 we compare our new results with those previously
published in Refs. Martini  _et al._ (2011) and Nieves  _et al._ (2012b) for
the MiniBooNE neutrino CCQE-like muon angle and kinetic energy distribution
$d^{2}\sigma/d\cos\theta_{\mu}dT_{\mu}$ on carbon in the angular window
$0.80<\cos\theta_{\mu}<0.90$. QE cross sections, including RPA effects, appear
to agree very well as expected, except for the lowest muon kinetic energies,
where some deviations, although small, are visible. For the multinucleon cross
section, the differences are more visible. Our new predictions (dashed-dotted
line) are substantially larger than those of Ref. Nieves  _et al._ (2012b)
(dotted-line), based on the findings of Ref. Nieves  _et al._ (2011). The
uncertainties induced by the lack of a proper RPA re-summation of $\Delta$h
excitations (band depicted in the plot) is not sufficient to cover for the
discrepancies between both sets of predictions. Thus for instance, for the
muon kinetic energy $T_{\mu}$ of 0.6 GeV, the result from Ref. Nieves  _et
al._ (2012b) is $0.34\times 10^{-38}$cm2/GeV/$(A-Z)$, while the band stands
for the interval $0.49-0.67$ in the same units. Indeed, the present CCQE-like
results describe well the MiniBooNE data without requiring a global 90% re-
scaling down of the flux. As we have already pointed out, there exist several
improvements implemented here, including the correction for the inconsistent
treatment of the energy conservation between the hole-nucleon and the virtual
nucleon or $\Delta$ that enter in the $WN\to N\pi$ vertices, responsible for
this enhancement of the cross section.

On the other hand, our results are now closer to those obtained by the Lyon
group in Martini  _et al._ (2011) (red dashed-dotted line in the plot) and
they can be accommodated within the uncertainty band of our scheme. This is
reassuring, taking into account the theoretical similarities of both
calculations. The results presented in the left panel of Fig. 9 are flux-
folded, which prevents from a more in-depth comparison. Therefore, in the
right panel we show the QE and 2p2h contributions to the differential cross-
section $d\sigma/d\omega$ for a given neutrino energy. As mentioned, the Lyon
group performed two different parametrizations for the the multi-nucleon
distribution, in which the pieces that are not reducible to a modification of
the $\Delta$ width are accounted for in two different ways. The authors of
Marteau (1999) employ an extrapolation of the two-body pion absorption at
threshold carried out in Ref. Faessler  _et al._ (1980). Within this model, an
accumulation of the 2p2h strength at low energy is produced (dashed line,
“param 1”, in the right panel of Fig. 9). Alternatively, the authors of Ref.
Martini  _et al._ (2009) presented results in which they exploit a microscopic
evaluation of the 2p2h contribution to the transverse magnetic response of
$(e,e^{\prime})$ scattering Alberico  _et al._ (1984). It improves on the
previous parametrization which had no momentum dependence, and it agrees
better with our microscopic results (dashed-dotted line, “param 2”). The Lyon
group results shown in the left plot of Fig. 9 were obtained using this more
realistic model for the 2p2h contribution. Still, it becomes clear from the
above discussion that neither of the two 2p2h models discussed in Ref. Martini
_et al._ (2009) can be directly used to predict exclusive two-nucleon final
states Martini  _et al._ (2009, 2010, 2011); Martini and Ericson (2013).

##  IV Conclusions

The ambitious experimental program which is underway to precisely determine
neutrino properties, requires a solid microscopic understanding of neutrino-
nucleus interactions. The (anti-)neutrino CC multi-nucleon cross section
calculation presented in this work provides an adequate theoretical tool for
neutrino oscillation experiments at low and moderate energies. It is based on
a state of the art description of the neutrino pion production off the nucleon
and a vast experience in describing nuclear responses sensitive to the spin-
isospin channel of the $NN$, $N\Delta$ and $\Delta\Delta$ in medium
interactions.

We presented a revisited calculation of the CC multinucleon knockout process
improved on several aspects of the computation of the CCQE-like cross section
carried out in Ref. Sobczyk  _et al._ (2020). Importantly, we introduced a
consistent treatment of the nucleon self-energy in the $WN\to N^{\prime}\pi$
amplitude leading around $20-40\%$ enhancement in respect to the previous
results. We have taken the opportunity of this revision to discuss in detail
several important issues of the calculation of the 2p2h cross section, delving
into the microscopic dynamics of the multi-nucleon mechanisms. Within this
discussion, we identified the RPA series of $\Delta$h excitations driven by
the $\pi+\rho$+SRC spin-isospin interaction, as the largest source of
theoretical uncertainty on the 2p2h cross sections presented in this work.

Our current predictions agree very well at the inclusive level with the
(anti-)neutrino CCQE-like and neutrino CC0$\pi$ lepton double-differential
cross sections measured by MiniBooNE Aguilar-Arevalo  _et al._ (2010, 2013) on
carbon and by T2K Abe  _et al._ (2020) on carbon and oxygen, respectively. In
particular, the neutrino MiniBooNE data is now well described without
requiring a global 90% re-scaling down of the flux, as it was needed in Ref.
Nieves  _et al._ (2012b).

Moreover, as pointed out in Sobczyk  _et al._ (2020), the microscopic
calculation performed in our scheme provides the two-nucleon final states in
the primary vertex of the interaction. Work is currently underway to include
our predictions in Monte Carlo event generators. It will allow to account for
the exclusive distributions of the outgoing nucleons, modified on their way
leaving out the nucleus. Consequently, we will be able to make a comparison of
our predictions, including both the QE and multinucleon knockout mechanisms,
with semi-exclusive cross-sections measured with an increasing statistics by
present (e.g. T2K, MicroBooNE) and future (e.g. SBND) experiments.

## Acknowledgements

We warmly thank José Enrique Amaro, Eulogio Oset, Federico Sánchez and Jan
Sobczyk for useful discussions and clarifications. J.E.S. acknowledges the
support of the European Union’s Horizon 2020 research and innovation programme
under the Marie Skłodowska-Curie grant agreement No. 101026014. This research
has been supported by the Spanish Ministerio de Ciencia e Innovación (MICINN)
and the European Regional Development Fund (ERDF) under contract
PID2020-112777GB-I00, the EU STRONG-2020 project under the program
H2020-INFRAIA-2018-1, grant agreement no. 824093. It has also been funded by
Generalitat Valenciana under contract PROMETEO/2020/023 and the “Planes
Complementarios de I+D+i” program (Grant No. ASFAE/2022/022) by MICINN with
funding from the European Union NextGenerationEU and Generalitat Valenciana,
the Deutsche Forschungsgemeinschaft (DFG) through the Cluster of Excellence
“Precision Physics, Fundamental Interactions, and Structure of Matter”
(PRISMA+ EXC 2118/1) funded by the DFG within the German Excellence Strategy
(Project ID 39083149).

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