# Meson-exchange currents in quasielastic electron scattering in a generalized
superscaling approach

Paloma R. Casale  Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.  J.E. Amaro<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.  M.B. Barbaro
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Torino, 10125 Torino,
Italy  INFN Sezione di Torino, 10125 Torino, Italy

###### Abstract

We present a model that incorporates the effect of two-body currents in
quasielastic electron-nucleus scattering within the framework of a consistent
superscaling formalism. This is achieved by defining an averaged single-
nucleon hadronic tensor based on the 1p1h matrix element of the one-body
current plus meson-exchange currents (MEC). The consistent treatment of one-
and two-body currents in our model enables the calculation of exchange current
effects in the kinematical region where the Fermi gas response is zero, but
not the scaling function. The effect of MEC is consistently taken into account
when extracting the phenomenological scaling function from electron scattering
data. With this model, we investigate the effect of MEC on the response
functions taking into account the effective mass of the nucleon, and examine
the consequences it has on the inclusive $(e,e^{\prime})$ cross section. We
find that 1p1h MEC deplete the quasielastic transverse response, while they
not alter significantly the scaling behavior of (e,e’) data.

Quasielastic electron scattering, meson-exchange currents, superscaling

##  I Introduction

The pursuit of neutrino oscillation experiments represents a significant
scientific endeavor, encompassing both experimental and theoretical efforts
Kami98 ; Nova16 ; Agu10 ; Abe18 ; Acc14 ; Ace19 . In particular, theoretical
nuclear physics has been propelled to analyze neutrino-induced nuclear
reactions in these experiments Alv14 ; Mos16 ; Kat17 ; Ank17 ; Ben17 ; Alv18 ;
Ama05 ; Ama20 ; Ama21 ; Ank22 . The ultimate goal is to minimize uncertainties
arising from nuclear effects, which are a primary source of systematic errors
when determining the neutrino interactions in detectors. Simultaneously, there
has been a renewed interest in electron scattering studies Ama10 ; Meg16 ;
Ama17 ; Don23 ; Don23b , as theoretical models can be calibrated using (e,e’)
data and subsequently extended to neutrinos by incorporating the contribution
of the axial current.

At typical energies around 1 GeV in many neutrino experiments, a significant
contribution arises from quasielastic nucleon emission, which dominates at
transferred energies around $\omega=|Q^{2}|/2m^{*}_{N}$, where $\omega$ is the
energy transfer, $Q^{2}=\omega^{2}-q^{2}<0$, and $q$ is the momentum transfer
to a nucleon with relativistic effective mass $m_{N}^{*}$ Ros80 ; Ser86 ;
Dre89 ; Weh93 ; Bar98 . It is crucial to take into account that the
transferred energies involved in neutrino experiments necessitate a
relativistic treatment of the reaction. This requirement introduces
significant challenges in constructing appropriate models for these
interactions Giu15 ; Pan15 ; Mar16 ; Gon19 ; Lov20 .

In this article, we focus on the study of one-particle one hole (1p1h)
transverse and longitudinal responses in the QE peak Ama02 ; Ama03 ; Ama09 ,
at intermediate and high momentum transfer, including the effect of meson
exchange currents (MEC) for electron scattering. The MEC are two-body currents
that involve the exchange of mesons between nucleons and virtual excitation of
nuclear resonances. This can have a significant impact on the scattering cross
section and on the distribution of energy and momentum transferred during the
interaction. The emission of two particles (2p2h), stemming from MEC and
short-range correlations, has emerged as a focal point in studies on lepton-
nucleus scattering.

Extensive research has been dedicated to understanding its effects on the
cross-section of both electron and neutrino interactions Alb84 ; Van80 ; Dek94
; Mar10 ; Lal12 ; Nie13 ; Cuy17 ; Pac03 ; Rui17 ; Mar21 ; Mar21b . However, it
is often overlooked that MEC also contribute to the emission of a single
particle (1p1h), thereby introducing interference effects with the one-body
current. Notably, calculations have shown a reduction in the quasielastic
transverse response compared to the impulse approximation when employing
nuclear shell or Fermi gas models Koh81 ; Alb90 ; Ama94 ; Ama03 ; Meu02 ;
Ama10 ; Cuy17 . This reduction is mainly due to the exchange part of the 1p1h
matrix element of the $\Delta$ current.

In this work, we aim also to incorporate the effect of Meson Exchange Currents
(MEC) consistently into the quasielastic peak within the framework of the
relativistic effective mass Superscaling (SuSAM*) model Mar17 ; Ama18 . This
is an extension of SuSA model based on the approximation of factorizing the
nuclear response into a single nucleon response multiplied by a superscaling
function Don99 . The phenomenological superscaling function accounts for
nuclear structure and reaction effects, as it is fitted to experimental data.
The motivation behind scaling models arises from the observation that
inclusive data, when divided by an appropriate single nucleon prefactor,
approximately scale when plotted against a suitable scaling variable, $\psi$,
extracted from the Relativistic Fermi Gas (RFG) model Don99 . The SuSA model
Ama05 , along with its improved version SuSAv2, and SuSAM*, has been
extensively utilized to analyze inclusive electron and neutrino scattering
data Gon14 ; Meg16 ; Meg16b ; Rui18 ; Ama21 . These efforts represent
important strides in understanding and predicting neutrino-nucleus
interactions. By establishing a phenomenological scaling function that
successfully describes (e,e’) data, these models provide a valuable foundation
for extrapolating to neutrino cross-sections.

The SuSAM* model builds upon the SuSA framework but incorporates the effective
mass dependence from the Relativistic Mean Field (RMF) theory. A notable
feature of the RMF model of nuclear matter (such as the Walecka or
$\sigma-\omega$ model Ser86 ) is that it reproduces the (e,e’) cross-section
better than the RFG model when an appropriate value for the effective mass M*
is chosen Ros80 ; Weh93 ; Ama15 . Motivated by this the SuSAM* model employs
the RMF model’s scaling variable, $\psi^{*}$, and single nucleon prefactor
dependent on the effective mass, with the aim to capture the essential
dynamics associated with the interaction process more accurately. This
approach capitalizes on the reasonable dynamical aspects embedded in the RMF
model and offers an alternative description of the scaling behavior observed
electron scattering cross section. It provides a comprehensive framework that
combines the strengths of the RMF model and the superscaling formalism,
leading to an improved understanding and interpretation of experimental data.

Until now, a unified model that incorporates 1p1h Meson Exchange Currents in
the superscaling function had not been proposed. This was primarily due to the
violation of scaling properties by MEC, even at the Fermi gas level Ama02 .
Additionally, the 1p1h matrix element of MEC is not easily extrapolated to the
$|\psi|>1$ region outside the range where the Fermi gas response is zero, as
nucleons are constrained by the Fermi momentum. In this work, we address both
of these challenges in a unified manner by modifying the scaling model to
account for the contribution of MEC within the single nucleon prefactor.
Furthermore, we take the opportunity to enhance the recently improved
superscaling model by eliminating the extrapolation of single-nucleon
responses averaged over the Fermi gas to the region $|\psi|>1$ Cas23 . Instead
of extrapolation, we introduce a new approach where the single nucleon
response is averaged with a smeared momentum distribution around the Fermi
surface. As a result, the averaged single nucleon responses are well defined
for all the values of $\psi$.

In the modified superscaling framework proposed in this work, the single
nucleon response incorporates the contribution of MEC to the effective one-
body current operator. This modification allows us to define a new prefactor
that already includes the effects of MEC, enabling a novel scaling analysis of
the data. Importantly, it should be noted that the Fermi gas now exhibits
exact scaling behavior when utilizing the new single nucleon response: scaling
violations associated to the MEC are exactly canceled by the dividing factor
used to construct the scaling function. By incorporating these modifications,
we overcome the limitations of previous models and provide a comprehensive
framework that encompasses both MEC and modified superscaling effects. By
consistently integrating 1p1h MEC within the SuSAM* model, we aim at refining
our understanding of the underlying nuclear dynamics in the quasielastic peak.
This comprehensive approach allows us to account for both the scaling behavior
observed in inclusive data and the contributions from meson exchange currents,
leading to a more accurate and comprehensive description of the reaction.

The article is structured as follows. In Sect. 2, we introduce the formalism
of quasielastic electron scattering within the framework of the Relativistic
Mean Field (RMF) model of nuclear matter, incorporating Meson Exchange
Currents (MEC). In Sect. 3, we present our unified scaling model that
incorporates MEC effects. We describe the modifications made to the
conventional scaling approach to account for the contribution of MEC within
the single nucleon prefactor. In Sect. 4 we present the results of our
calculations and analyses based on the unified scaling model with MEC. Finally
in Sect. 5 we present the conclusions drawn from our study.

##  II Formalism

###  II.1 Response functions

We start with the inclusive electron scattering cross section in plane-wave
Born approximation with one photon-exchange. The exchanged photon transfers an
energy $\omega$ and a momentum ${\bf q}$ to the nucleus. The initial electron
energy is $\epsilon$, the scattering angle is $\theta$, and the final electron
energy is $\epsilon^{\prime}=\epsilon-\omega$. The double-differential cross
section is written in terms of the longitudinal and transverse response
functions, $R_{L}(q,\omega)$ and $R_{T}(q,\omega)$,

$\frac{d\sigma}{d\Omega d\epsilon^{\prime}}=\sigma_{\rm
Mott}\left(v_{L}R_{L}(q,\omega)+v_{T}R_{T}(q,\omega)\right),$ (1)

where $\sigma_{\rm Mott}$ is the Mott cross section and $v_{L}$ and $v_{T}$
are the kinematic coefficients defined as

$\displaystyle v_{L}$ $\displaystyle=$ $\displaystyle\frac{Q^{4}}{q^{4}}$ (2)
$\displaystyle v_{T}$ $\displaystyle=$
$\displaystyle\tan\frac{Q^{4}}{q^{4}}-\frac{Q^{2}}{2q^{2}}$ (3)

with $Q^{2}=\omega^{2}-q^{2}<0$ the four-momentum transfer. The nuclear
response functions are the following combinations of the hadronic tensor

$R_{L}=W^{00},\kern 28.45274ptR_{T}=W^{11}+W^{22}.$ (4)

The inclusive hadronic tensor is constructed from the matrix elements of the
electromagnetic current operator $\hat{J}^{\mu}({\bf q})$ between the initial
and final hadronic states:

$\displaystyle W^{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{f}\overline{\sum_{i}}\left\langle
f\right|\hat{J}^{\mu}({\bf q})|\left.i\right\rangle^{*}\left\langle
f\right|\hat{J}^{\nu}({\bf q})|\left.i\right\rangle$ (5) $\displaystyle\times$
$\displaystyle\delta(E_{f}-E_{i}-\omega),$

where the sum is performed over the undetected final nuclear states
$|f\rangle$ and the average over the initial ground state $|i\rangle$ spin
components.

In this work, our approach aims at exploting the scaling symmetry of
quasielastic data. This scaling symmetry states that the scaling function,
that is, the cross-section divided by an appropriately averaged single-nucleon
cross-section and multiplied by a kinematic factor, only depends on a single
kinematic variable, $\psi$, rather than on the three variables
$(\epsilon,q,\omega)$. The scaling function is approximately the same for all
nuclei Ama18 . The starting point for the scaling analysis is the relativistic
Fermi gas (RFG) model, where this symmetry holds exactly. In the case of real
nuclei, it is only approximately fulfilled, but it proves to be very useful
for analyzing experimental data and performing calculations and predictions.

###  II.2 1p1h hadronic tensor

In independent particle models, the main contribution to the hadronic tensor
in the quasielastic peak comes from the one-particle one-hole (1p1h) final
states. As the transferred energy increases, there are contributions from two-
particle two-hole (2p2h) emission, the inelastic contribution of pion emission
above the pion mass threshold, and the deep inelastic scattering at higher
energies. Therefore, the hadronic tensor can be generally decomposed as the
sum of the 1p1h contribution and other contributions:

$W^{\mu\nu}=W^{\mu\nu}_{1p1h}+W^{\mu\nu}_{2p2h}+\ldots$ (6)

In this work we focus on the 1p1h response which, in the RFG model, reads

$\displaystyle W^{\mu\nu}_{1p1h}$ $\displaystyle=$
$\displaystyle\sum_{ph}\left\langle
ph^{-1}\right|\hat{J}^{\mu}|\left.F\right\rangle^{*}\left\langle
ph^{-1}\right|\hat{J}^{\nu}|\left.F\right\rangle$ (7) $\displaystyle\times$
$\displaystyle\delta(E_{p}-E_{h}-\omega)\theta(p-k_{F})\theta(k_{F}-h)$

where $|p\rangle\equiv|{\bf p}s_{p}t_{p}\rangle$ and $|h\rangle\equiv|{\bf
h}s_{h}t_{h}\rangle$ are plane wave states for particles and holes,
respectively, and $|F\rangle$ is the RFG ground state with all momenta
occupied below the Fermi momentum $k_{F}$. The novelty compared to previous
works on scaling is that we start from a current operator that is a sum of
one-body and two-body operators. This approach allows us to consider the
contributions of both the usual electromagnetic current of the nucleon and the
meson-exchange currents (MEC) to the 1p1h response:

$\hat{J}^{\mu}=\hat{J}^{\mu}_{1}+\hat{J}^{\mu}_{2},$ (8)

where $\hat{J}_{1}$ represents the one-body (OB) electromagnetic current of
the nucleon, while $\hat{J}_{2}$ is the two-body MEC. Both currents can
generate non-zero matrix elements for 1p1h excitation. MEC are two-body
operators and they can induce 1p1h excitation due to the interaction of the
hit nucleon with a second nucleon acting as a spectator. The many-body matrix
elements of these operators are given by

$\left\langle
ph^{-1}\right|\hat{J}_{1}^{\mu}|\left.F\right\rangle=\left\langle
p\right|\hat{J}_{1}^{\mu}|\left.h\right\rangle$ (9)

for the OB current and

$\left\langle
ph^{-1}\right|\hat{J}_{2}^{\mu}|\left.F\right\rangle=\sum_{k<k_{F}}\left[\left\langle
pk\right|\hat{J}_{2}^{\mu}|\left.hk\right\rangle-\left\langle
pk\right|\hat{J}_{2}^{\mu}|\left.kh\right\rangle\right]$ (10)

for the two-body current, where the sum over spectator states $(k)$ is
performed over the occupied states in the Fermi gas, considering both the
direct and exchange matrix elements. Due to momentum conservation, the matrix
element of the OB current between plane waves can be written as

$\langle p|\hat{J}_{1}^{\mu}|h\rangle=\frac{(2\pi)^{3}}{V}\delta^{3}({\bf
q}+{\bf h}-{\bf p})\frac{m_{N}}{\sqrt{E_{p}E_{h}}}j_{1}^{\mu}({\bf p},{\bf
h}),$ (11)

where $V$ is the volume of the system, $m_{N}$ is the nucleon mass,
$E_{p}=\sqrt{p^{2}+m_{N}^{2}}$ and $E_{h}=\sqrt{h^{2}+m_{N}^{2}}$ are the on-
shell energies of the nucleons involved in the process, and $j_{1}^{\mu}({\bf
p},{\bf h})$ is the OB current (spin-isospin) matrix

$j^{\mu}_{1}({\bf p},{\bf h})=\bar{u}({\bf
p})\left(F_{1}\gamma^{\mu}+i\frac{F_{2}}{2m_{N}}\sigma^{\mu\nu}Q_{\nu}\right)u({\bf
h}),$ (12)

being $F_{1}$ and $F_{2}$ the Dirac and Pauli form factors of the nucleon. In
the case of the two-body current, the elementary matrix element can be written
in a similar form:

$\displaystyle\langle
p^{\prime}_{1}p^{\prime}_{2}|\hat{J}_{2}^{\mu}|p_{1}p_{2}\rangle$
$\displaystyle=$ $\displaystyle\frac{(2\pi)^{3}}{V^{2}}\delta^{3}({\bf
p}_{1}+{\bf p}_{2}+{\bf q}-{\bf p}^{\prime}_{1}-{\bf p}^{\prime}_{2})$ (13)
$\displaystyle\kern-28.45274pt{}\times\frac{m_{N}^{2}}{\sqrt{E^{\prime}_{1}E^{\prime}_{2}E_{1}E_{2}}}j_{2}^{\mu}({\bf
p}^{\prime}_{1},{\bf p}^{\prime}_{2},{\bf p}_{1},{\bf p}_{2}).$

Here $j_{2}^{\mu}({\bf p}^{\prime}_{1},{\bf p}^{\prime}_{2},{\bf p}_{1},{\bf
p}_{2})$ is a spin-isospin matrix and it depends on the momenta of the two
nucleons in the initial and final state. The two-body current contains the sum
of the diagrams shown in Figure 1, including the seagull, pionic, and $\Delta$
isobar currents. The specific form of the two-body current function will be
given later when we discuss the MEC model. By inserting (13) into Eq. (10) we
obtain an expression similar to (11) that resembles the matrix element of an
effective one-body (OB) current for the MEC:

$\left\langle
ph^{-1}\right|\hat{J}_{2}^{\mu}|\left.F\right\rangle=\frac{(2\pi)^{3}}{V}\delta^{3}({\bf
q}+{\bf h}-{\bf p})\frac{m_{N}}{\sqrt{E_{p}E_{h}}}j_{2}^{\mu}({\bf p},{\bf
h}).$ (14)

Here the effective OB current generated by the MEC involves a sum over the
spectator nucleons and is defined by

$\displaystyle j_{2}^{\mu}({\bf p},{\bf h})$ $\displaystyle\equiv$ (15)
$\displaystyle\kern-28.45274pt\sum_{k<k_{F}}\frac{m_{N}}{VE_{k}}\left[j_{2}^{\mu}({\bf
p},{\bf k},{\bf h},{\bf k})-j_{2}^{\mu}({\bf p},{\bf k},{\bf k},{\bf
h})\right].$

Note that in the thermodynamic limit $V\rightarrow\infty$ the above sum will
be transformed into an integral over the momenta occupied in the Fermi gas:

$\frac{1}{V}\sum_{k<k_{F}}\rightarrow\sum_{s_{k}t_{k}}\int\frac{d^{3}k}{(2\pi)^{3}}\theta(k_{F}-k).$
(16)

Finally, we can write the transition matrix element of the total current
between the ground state and the 1p1h state as

$\left\langle
ph^{-1}\right|\hat{J}^{\mu}|\left.F\right\rangle=\frac{(2\pi)^{3}}{V}\delta^{3}({\bf
q}+{\bf h}-{\bf p})\frac{m_{N}}{\sqrt{E_{p}E_{h}}}j^{\mu}({\bf p},{\bf h}),$
(17)

where the effective total current for the 1p1h excitation includes
contributions from both the one-body current and MEC:

$j^{\mu}({\bf p},{\bf h})=j_{1}^{\mu}({\bf p},{\bf h})+j_{2}^{\mu}({\bf
p},{\bf h}).$ (18)

By inserting (17) into Eq. (7) and taking the thermodynamic limit, we obtain
the following expression for the hadronic tensor:

$\displaystyle W^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{V}{(2\pi)^{3}}\int{d^{3}h\delta(E_{p}-E_{h}-\omega)\frac{m_{N}^{2}}{E_{p}E_{h}}}2w^{\mu\nu}({\bf
p},{\bf h})$ (19) $\displaystyle\times$
$\displaystyle\theta(p-k_{F})\theta(k_{F}-h),$

where ${\bf p}={\bf h}+{\bf q}$ by momentum conservation after integration
over ${\bf p}$. The function $w^{\mu\nu}$ is the effective single-nucleon
hadronic tensor in the transition

$\displaystyle w^{\mu\nu}({\bf p},{\bf
h})=\frac{1}{2}\sum_{s_{p}s_{h}}j^{\mu}({\bf p},{\bf h})^{*}j^{\nu}({\bf
p},{\bf h}).$ (20)

In this equation, we did not include the sum over isospin $t_{p}=t_{h}$.
Therefore, $w^{\mu\nu}$ refers to the tensor of either proton or neutron
emission, and the total tensor would be the sum of the two contributions. Note
that the effective single-nucleon tensor $w^{\mu\nu}$ includes the
contribution of MEC, thus encompassing an interference between the one-body
and two-body currents. Indeed, the relevant diagonal components of the
effective single-nucleon hadronic tensor for the longitudinal and transverse
responses (4) can be expanded as

$\displaystyle w^{\mu\mu}({\bf p},{\bf h})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s_{p}s_{h}}|j^{\mu}_{1}+j^{\mu}_{2}|^{2}$ (21)
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum|j^{\mu}_{1}|^{2}+\mbox{Re}\sum(j^{\mu}_{1})^{*}j^{\mu}_{2}+\frac{1}{2}\sum|j^{\mu}_{2}|^{2}$
$\displaystyle\equiv$ $\displaystyle
w^{\mu\mu}_{1}+w^{\mu\mu}_{12}+w^{\mu\mu}_{2}$

where $w^{\mu\mu}_{1}$ is the tensor corresponding to the one-body current,
$w^{\mu\mu}_{12}$ represents the interference between the one-body and two-
body currents, and $w^{\mu\mu}_{2}$ corresponds to the contribution of the
two-body current alone. The one-body part is the leading contribution in the
quasielastic peak, while the dominant contribution of the MEC corresponds to
the interference with the one-body current Ama94 ; Ama03 , being the pure
contribution of the two-body current generally smaller.

###  II.3 Responses in the relativistic mean field approach

Going beyond the Relativistic Fermi Gas (RFG) model, the Relativistic Mean
Field (RMF) approach for nuclear matter allows for the inclusion of dynamic
relativistic effects. The simplest approximation in this framework is to
introduce constant mean scalar and vector potentials with which the nucleons
interact Ros80 ; Ser86 ; Weh93 ; Bar98 . The scalar potential is attractive,
while the vector potential is repulsive. The single-particle wave functions
still exhibit plane-wave behavior with momentum $p$ in nuclear matter, but
with an on-shell energy given by

$E=\sqrt{m_{N}^{*2}+p^{2}},$ (22)

where $m_{N}^{*}$ is the relativistic effective mass of the nucleon, defined
as

$m_{N}^{*}=m_{N}-g_{s}\phi_{0}=M^{*}m_{N}.$ (23)

Here $\phi_{0}$ is the scalar potential energy of the RMF and $g_{s}$ the
corresponding coupling constant Ser86 , and $M^{*}=0.8$ for 12C, the nucleus
considered in this work Mar21 . To account for the interaction with the vector
potential, a positive energy term needs to be added to the on-shell energy.
Therefore, the total energy of the nucleon can be expressed as:

$E_{RMF}=E+E_{v}.$ (24)

In this work we use the value $E_{v}=141$ MeV, obtained in Ref. Mar21  for
12C. Note that in observables that only depend on the energy differences
between initial and final particles, the vector energy cancels out, and only
the on-shell energy appears. This cancellation happens, as we will see, in the
response associated to the one-body current. However, in the case of the two-
body current, the vector energy needs to be taken into account in the $\Delta$
current, as we will see in the next section.

In the present RMF approach of nuclear matter, the evaluation of the hadronic
tensor is done similarly to the RFG, with the difference that the spinors
$u(p)$ now correspond to the solutions of the Dirac equation with the
relativistic effective mass $m_{N}^{*}$. From Eq. (19) the 1p1h nuclear
response functions are then given by

$\displaystyle R_{K}(q,\omega)$ $\displaystyle=$
$\displaystyle\frac{V}{(2\pi)^{3}}\int
d^{3}h\frac{(m^{*}_{N})^{2}}{E_{p}E_{h}}2w_{K}({\bf p},{\bf h})$ (25)
$\displaystyle\kern-28.45274pt\times\theta(p-k_{F})\theta(k_{F}-h)\delta(E_{p}-E_{h}-\omega),$

where $E_{p},E_{h}$ are the on-shell energies with effective mass $m_{N}^{*}$,
and $w_{K}$ are the single-nucleon responses for the 1p1h excitation

$\displaystyle w_{L}=w^{00},\kern 28.45274ptw_{T}=w^{11}+w^{22}.$ (26)

The effective single-nucleon tensor $w^{\mu\nu}$ is constructed as in Eq.
(20), but the current is obtained from matrix elements using spinors with the
relativistic effective mass $m_{N}^{*}$ instead of the normal nucleon mass.
This prescription is also followed when evaluating the 1p1h matrix elements of
the MEC (as discussed in the next section).

To compute the integral (25), we change to the variables $(E_{h},E_{p},\phi)$,
using $h^{2}dhd\cos\theta=(E_{h}E_{p}/q)dE_{h}dE_{p}$. Then the integral over
$E_{p}$ can be performed using the Dirac delta. This fixes the angle
$\theta_{h}$ between ${\bf q}$ and ${\bf h}$

$\cos\theta_{h}=\frac{2E_{h}\omega+Q^{2}}{2hq}.$ (27)

The integration over the angle $\phi$ gives $2\pi$ by symmetry of the
responses when ${\bf q}$ is on the $z$-axis Ama20 . The result is an integral
over the initial nucleon energy

$R_{K}(q,\omega)=\frac{V}{(2\pi)^{3}}\frac{2\pi
m_{N}^{*3}}{q}\int_{\epsilon_{0}}^{\infty}d\epsilon\,n(\epsilon)\,2w_{K}(\epsilon,q,\omega),$
(28)

where we have defined the adimensional energies $\epsilon=E_{h}/m^{*}_{N}$ and
$\epsilon_{F}=E_{F}/m_{N}^{*}$. Moreover we have introduced the energy
distribution of the Fermi gas $n(\epsilon)=\theta(\epsilon_{F}-\epsilon)$. The
lower limit of the integral (28), $\epsilon_{0}$, represents the minimum
energy that an on-shell nucleon can have when it absorbs energy $\omega$ and
momentum $q$ Ama20

$\epsilon_{0}={\rm
Max}\left\\{\kappa\sqrt{1+\frac{1}{\tau}}-\lambda,\epsilon_{F}-2\lambda\right\\},$
(29)

where we have defined the dimensionless variables

$\lambda=\frac{\omega}{2m_{N}^{*}},\kern
28.45274pt\kappa=\frac{q}{2m_{N}^{*}},\kern
28.45274pt\tau=\kappa^{2}-\lambda^{2}.$ (30)

From Eq. (28) the nucleons that contribute to the response function
$R{{}_{K}}(q,\omega)$ are those with energy ranging from $\epsilon_{0}$ to
$\epsilon_{F}$. For fixed values of $\phi,q,\omega$, the integral over energy
$\epsilon$ in Eq. (28) corresponds to integrating the single nucleon response
over a path in the momentum space of the hole ${\bf h}$, weighted with the
momentum distribution. The angle between ${\bf h}$ and ${\bf q}$ for each
energy is given by Eq. (27). The minimum momentum $h_{0}$ correspond to the
minimum energy $\epsilon_{0}$. Indeed, for a specific value of $\omega$, the
lower limit of the integral becomes $h=0$ or $\epsilon_{0}=1$, which
corresponds to the center of the quasielastic peak. Using Eq. (29), it is
straightforward to verify that this point corresponds to $\lambda=\tau$ in the
regime without Pauli blocking.

###  II.4 Scaling

Scaling is based on the approximated factorization of an averaged single-
nucleon response from the nuclear cross section. This factorization is exact
in the RMF model with the OB current. In previous works, analytical
expressions were obtained from the RFG and RMF models by explicit integration
of the one-body responses, Eq. (28). However in this case, it is not possible
to perform the integration (28) analytically because now $w_{K}$ includes also
the matrix elements of the two-body operator. Nevertheless, we can still
define averaged single-nucleon responses as

$\displaystyle\overline{w}_{K}(q,\omega)=\frac{\int^{\infty}_{\epsilon_{0}}d\epsilon\,n(\epsilon)w_{K}(\epsilon,q,\omega)}{\int_{\epsilon_{0}}^{\infty}d\epsilon\,n(\epsilon)}$
(31)

and we can rewrite Eq. (28) in the form

$R_{K}(q,\omega)=\frac{V}{(2\pi)^{3}}\frac{2\pi
m_{N}^{*3}}{q}2\overline{w}_{K}(q,\omega)\int_{\epsilon_{0}}^{\infty}d\epsilon\,n(\epsilon).$
(32)

The averaged single-nucleon responses, $\overline{w}_{K}(q,\omega)$, include
the combined effect of both the OB current and the MEC in all the 1p1h
excitations compatible with given values of $(q,\omega)$. Eq. (32) shows that
in the RMF model (or the RFG model for effective mass $M^{*}=1$) the nuclear
responses factorize as the product of the averaged single-nucleon response
(including MEC) and the scaling function. In fact a superscaling function can
be defined as

$f^{*}(\psi^{*})=\frac{3}{4}\frac{1}{\epsilon_{F}-1}\int_{\epsilon_{0}}^{\infty}n(\epsilon)d\epsilon,$
(33)

where $\epsilon_{F}-1$ is the kinetic Fermi energy in units of $m_{N}^{*}$ and
the $\psi^{*}$-scaling variable is related to the minimum nucleon energy,
$\epsilon_{0}$, as

$\psi^{*}=\sqrt{\frac{\epsilon_{0}-1}{\epsilon_{F}-1}}{\rm
sgn}(\lambda-\tau).$ (34)

The scaling variable, $\psi^{*}$, is negative (positive) for $\lambda<\tau$
($\lambda>\tau$). In the RMF the scaling function is easily evaluated from Eq.
(33), giving

$f^{*}(\psi^{*})=\frac{3}{4}(1-\psi^{*}{}^{2})\theta(1-\psi^{*}{}^{2}).$ (35)

Note that the scaling function of nuclear matter is zero for
$\epsilon_{0}>\epsilon_{F}$, and this is equivalent to $|\psi^{*}|>1$. This is
a consequence of the maximum momentum $k_{F}$ for the nucleons in nuclear
matter, which implies that $\epsilon_{0}<\epsilon_{F}$.

Using $V/(2\pi)^{3}=N/(\frac{8}{3}\pi k_{F}^{3})$ for nuclear matter we can
write the response functions (32) as

$R_{K}(q,\omega)=\frac{\epsilon_{F}-1}{m_{N}^{*}\eta_{F}^{3}\kappa}(Z\overline{w}^{p}_{K}(q,\omega)+N\overline{w}^{n}_{K}(q,\omega))f^{*}(\psi^{*}),$
(36)

where we have added the contribution of $Z$ protons and $N$ neutrons to the
response functions, and $\eta_{F}=k_{F}/m_{N}^{*}$.

###  II.5 SuSAM*

The expression given by Eq (36) for the response function is formally the same
as the response in the RMF, the only difference being that the averaged
single-nucleon response now includes the contribution of MEC to the 1p1h
excitation. This equation, valid for the RMF, serves as the starting point for
performing the superscaling analysis with relativistic effective mass (SuSAM*)
using electron scattering data, extending the formula to the region
$\epsilon_{0}>\epsilon_{F}$ or $|\psi^{*}|>1$. We will follow the procedure
suggested by Casale et al. Cas23 .

In the Fermi gas, it is not possible to extend the averaging formula for
$\epsilon_{0}>\epsilon_{F}$ because the momentum distribution is zero and the
denominator in (31) vanishes. Therefore, what we do is slightly modify the
Fermi gas distribution by allowing a smeared Fermi surface, so that the
distribution is not exactly zero above $k_{F}$, allowing for the averaging
procedure. By substituting the Fermi distribution with a distribution that is
not significantly different from the original one for $h<k_{F}$, the average
of the single-nucleon response will not change significantly in the Fermi gas
region $|\psi^{*}|<1$.

By this method, the extension of the single-nucleon average is done smoothly
and continuously to the region $|\psi^{*}|>1$, with the added meaning that, in
this way, we are taking into account, at least partially, the high-momentum
distribution. This is because it is primarily the nucleons with momenta
greater than $k_{F}$ that contribute to this region. A possible distribution
that can be used to extend the averaging procedure is the Fermi distribution:

$n(h)=\frac{a}{1+{\rm e}^{(h-k_{F})/b}}.$ (37)

Using this distribution, the integrals in the numerator and denominator of Eq
(31) extend to infinity and are well-defined for $\epsilon_{0}>\epsilon_{F}$
or $|\psi^{*}|>1$. An appropriate value for the smearing parameter is $b=50$
MeV/c, used in ref. Cas23 , where the averaged single-nucleon responses were
evaluated for the one-body current, and it was found to yield practically the
same results as the analytically calculated responses in the strict Fermi gas
region. The averaged responses were also found to be very similar to the
traditionally extrapolated responses outside this region. This proposed method
provides a simple approach that allows for the definition of generalized
scaling, including the MEC, consistently, and also takes into account that the
nucleons are not limited by a maximum Fermi momentum.

Several approaches exist to obtain a phenomenological scaling function.
Different methods are based on different assumptions for the scaling function
or the single-nucleon response, but all are ultimately adjusted to
experimental data. The original SuSA model, based on the RFG, was fitted to
the scaling data of the longitudinal response, to obtain a longitudinal
scaling function, $f_{L}$, while in the extended SuSA-v2 approach, the RMF
model for finite nuclei was used to obtain a transverse scaling function,
$f_{T}$ . The SuSAM* model, based on the nuclear matter RMF with effective
mass, directly fitted the quasielastic data of the cross section after
discarding the non-scaling data points, to obtain a single phenomenological
scaling function valid for both the L and T channels Ama17 .

In the generalized SuSAM* model proposed here, we will follow the same
procedure as described in references [42, 43]. First, we subtract the
calculated inclusive cross section for two-particle emission in the RMF with a
relativistic MEC model from the (e,e’) data. This subtraction aims to
partially remove the contribution of 2p2h processes present in the data, in
order to isolate the purely quasielastic data as much as possible. Next, we
will scale each residual data point by dividing it by the contribution of the
single nucleon to the cross section, as given by Eq. (36),

$f_{exp}^{*}=\frac{\displaystyle\left(\frac{d\sigma}{d\Omega
d\omega}\right)_{exp}-\left(\frac{d\sigma}{d\Omega
d\omega}\right)_{2p2h}}{\sigma_{M}(v_{L}r_{L}+v_{T}r_{T})},$ (38)

where the single nucleon cross section includes the averaged single-nucleon
responses including MEC

$r_{K}=\frac{\epsilon_{F}-1}{m_{N}^{*}\eta_{F}^{3}\kappa}(Z\overline{w}^{p}_{K}(q,\omega)+N\overline{w}^{n}_{K}(q,\omega)).$
(39)

In the results section, we will proceed with the scaling analysis for the
obtained $f_{exp}^{*}$ data, by a plot as a function of $\psi^{*}$, calculated
using Eq. (34). This analysis includes a selection process to identify the
data points that are most likely to be quasielastic (which exhibit approximate
scaling behavior) and discarding the remaining data points (mainly non-scaling
inelastic processes). Finally, we will fit a phenomenological scaling function
to the surviving data points, aiming to describe the global scaling behavior
of the quasielastic region.

Figure 1: Feynman diagrams for the 2p2h MEC model used in this work.

###  II.6 Meson-exchange currents

In this work, we use the relativistic meson exchange currents (MEC) model
described in Ref. Rui17 . The Feynman diagrams shown in Fig. 1 illustrate the
different components of the MEC model. Diagrams (a) and (b) correspond to the
seagull current, diagram (c) represents the pion-in-flight current, and
diagrams (d,e) and (f,g) depict the forward- and backward- $\Delta(1232)$
currents, respectively. The specific treatment of the $\Delta$ current is
model-dependent, and various versions exist with possible corrections to the
off-shell relativistic interaction of the $\Delta$. Other widely used models
for MEC include those described in Refs. Pac03 ; Ama03 ; Pas95 .

While these different models may exhibit slight variations and corrections to
the $\Delta$ off-shell interaction, they generally yield similar results for
the dominant transverse response at the quasielastic peak. In particular, in
the results section, we will compare our findings with the model presented in
Refs. Pas95 ; Ama03 , which we previously employed to assess the impact of MEC
on the 1p1h response.

In our model the MEC functions defined in Eq. (13) correspond to the sum of
diagrams of Fig. 1

$j_{2}^{\mu}({\bf p}^{\prime}_{1},{\bf p}^{\prime}_{2},{\bf p}_{1},{\bf
p}_{2})=j^{\mu}_{sea}+j^{\mu}_{\pi}+j^{\mu}_{\Delta},$ (40)

where the $\Delta$ current is the sum of forward and backward terms

$j^{\mu}_{\Delta}=j^{\mu}_{\Delta F}+j^{\mu}_{\Delta B}.$ (41)

These functions are defined by

$\displaystyle j^{\mu}_{sea}$ $\displaystyle=$
$\displaystyle[I_{z}]_{t^{\prime}_{1}t^{\prime}_{2},t_{1}t_{2}}\frac{f^{2}}{m_{\pi}^{2}}V^{s^{\prime}_{1}s_{1}}_{\pi
NN}(p^{\prime}_{1},p_{1})F_{\pi
NN}(k_{1}^{2})\bar{u}_{s^{\prime}_{2}}(p_{2}^{{}^{\prime}})F_{1}^{V}\gamma^{5}\gamma^{\mu}u_{s_{2}}(p_{2})+(1\leftrightarrow
2)$ (42) $\displaystyle j^{\mu}_{\pi}$ $\displaystyle=$
$\displaystyle[I_{z}]_{t^{\prime}_{1}t^{\prime}_{2},t_{1}t_{2}}\frac{f^{2}}{m_{\pi}^{2}}F_{1}^{V}V^{s^{\prime}_{1}s_{1}}_{\pi
NN}(p^{\prime}_{1},p_{1})V^{s^{\prime}_{2}s_{2}}_{\pi
NN}(p^{\prime}_{2},p_{2})(k_{1}^{\mu}-k_{2}^{\mu})$ (43) $\displaystyle
j^{\mu}_{\Delta F}$ $\displaystyle=$
$\displaystyle[U_{z}^{F}]_{t^{\prime}_{1}t^{\prime}_{2},t_{1}t_{2}}\frac{ff^{*}}{m_{\pi}^{2}}V^{s^{\prime}_{2}s_{2}}_{\pi
NN}(p^{\prime}_{2},p_{2})F_{\pi
N\Delta}(k_{2}^{2})\bar{u}_{s^{\prime}_{1}}(p^{\prime}_{1})k_{2}^{\alpha}G_{\alpha\beta}(p_{1}+Q)\Gamma^{\beta\mu}(Q)u_{s_{1}}(p_{1})+(1\leftrightarrow
2)$ (44) $\displaystyle j^{\mu}_{\Delta B}$ $\displaystyle=$
$\displaystyle[U_{z}^{B}]_{t^{\prime}_{1}t^{\prime}_{2},t_{1}t_{2}}\frac{f^{2}f^{*}}{m_{\pi}^{2}}V^{s^{\prime}_{2}s_{2}}_{\pi
NN}(p^{\prime}_{2},p_{2})F_{\pi
N\Delta}(k_{2}^{2})\bar{u}_{s^{\prime}_{1}}(p^{\prime}_{1})k_{2}^{\beta}\hat{\Gamma}^{\mu\alpha}(Q)G_{\alpha\beta}(p^{\prime}_{1}-Q)u_{s_{1}}(p_{1})+(1\leftrightarrow
2)$ (45)

We will evaluate these matrix elements in the framework of the RMF model,
where the spinors $u(p)$ are the solutions of the Dirac equation with
relativistic effective mass $m_{N}^{*}$. The four-vectors
$k_{i}^{\mu}=p^{\prime}_{i}{}^{\mu}-p_{i}^{\mu}$ with $i=1,2$ are the momenta
transferred to the nucleons 1,2. We have defined the following function that
includes the $\pi NN$ vertex, a form factor, and the pion propagator

$V^{s^{\prime}_{1}s_{1}}_{\pi NN}(p^{\prime}_{1},p_{1})=F_{\pi
NN}(k_{1}^{2})\frac{\bar{u}(p^{\prime}_{1})\gamma^{5}\not{k}_{1}u(p_{1})}{k_{1}^{2}-m_{\pi}^{2}}.$
(46)

We apply strong form factors at the pion absorption/emission vertices given by
Alb84 ; Som78

$F_{\pi NN}(k)=F_{\pi
N\Delta}(k)=\frac{\Lambda^{2}-m_{\pi}^{2}}{\Lambda^{2}-k^{2}}.$ (47)

The charge structure of the MEC involves the isospin matrix element of the
operators

$\displaystyle I_{z}$ $\displaystyle=$ $\displaystyle
i[\tau(1)\times\tau(2)]_{z},$ (48) $\displaystyle U^{F}_{z}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}\sum_{i=1}^{3}{(T_{i}T_{z}^{\dagger})\otimes\tau_{i}},$
(49) $\displaystyle U^{B}_{z}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}\sum_{i}^{3}{(T_{z}T_{i}^{\dagger})\otimes\tau_{i}},$
(50)

where we denote by $T^{\dagger}_{i}$ the Cartesian coordinates of the
$\frac{1}{2}\rightarrow\frac{3}{2}$ transition isospin operator, defined by
its matrix elements Eri88

$\textstyle\langle\frac{3}{2}t_{\Delta}|T^{\dagger}_{\mu}|\frac{1}{2}t\rangle=\langle\frac{1}{2}t1\mu|\frac{3}{2}t_{\Delta}\rangle$
(51)

$T^{\dagger}_{\mu}$ being the spherical components of the vector
$\vec{T}^{\dagger}$. With the aid of the expression
$T_{i}T^{\dagger}_{j}=(2/3)\delta_{ij}-\frac{i}{3}\tau_{i}\tau_{j}$ and making
the summation, we can rewrite the isospin operators in the forward and
backward $\Delta$ current as

$\displaystyle U^{F}_{z}$
$\displaystyle=\sqrt{\frac{3}{2}}\left(\frac{2}{3}\tau_{z}^{(2)}-\frac{i}{3}\left[\tau^{(1)}\times\tau^{(2)}\right]_{z}\right)$
(52) $\displaystyle U^{B}_{z}$
$\displaystyle=\sqrt{\frac{3}{2}}\left(\frac{2}{3}\tau_{z}^{(2)}+\frac{i}{3}\left[\tau^{(1)}\times\tau^{(2)}\right]_{z}\right).$
(53)

The $\gamma N\rightarrow\Delta$ transition vertex in the forward $\Delta$
current is defined as Lle72 ; Her07

$\Gamma^{\beta\mu}(Q)=\frac{C_{3}^{V}}{m_{N}}(g^{\beta\mu}\not{Q}-Q^{\beta}\gamma^{\mu})\gamma_{5}$
(54)

while for the backward $\Delta$ current

$\hat{\Gamma}^{\mu\alpha}(Q)=\gamma^{0}[\Gamma^{\alpha\mu}(-Q)]^{\dagger}\gamma^{0}.$
(55)

In this vertex we have neglected contributions of order $O(1/m_{N}^{2})$. Note
that the $\Gamma^{\beta\mu}$ operator is a spin matrix and depends on the
vector form factor $C_{3}^{V}$. In this paper, we use the vector form factor
in $\Delta$ current from Refs. Nie13 ; Her07 :

$C_{3}^{V}(Q^{2})=\frac{2.13}{(1-\frac{Q^{2}}{M_{V}^{2}})^{2}}\frac{1}{1-\frac{Q^{2}}{4M_{V}^{2}}}.$
(56)

Various alternative approximations to the propagator have been proposed Qui17
. However, in the case of the quasielastic peak, the typical kinematics are of
the order of 1 GeV, and these issues are not expected to be relevant. They are
overshadowed by other more significant nuclear effects that dominate in this
energy regime. Here we use the $\Delta$ propagator commonly used for the
spin-3/2 field

$G_{\alpha\beta}(P)=\frac{{\cal
P}_{\alpha\beta}(P)}{P^{2}-M_{\Delta}^{2}+iM_{\Delta}\Gamma(P^{2})+\frac{\Gamma(P^{2})^{2}}{4}}$
(57)

where $M_{\Delta}$ and $\Gamma$ are the $\Delta$ mass and width respectively.
The projector ${\cal P}_{\alpha\beta}(P)$ over spin-3/2 on-shell particles is
given by

$\displaystyle{\cal P}_{\alpha\beta}(P)$ $\displaystyle=$
$\displaystyle-(\not{P}+M_{\Delta})$ (58)
$\displaystyle\kern-28.45274pt\times\left[g_{\alpha\beta}-\frac{\gamma_{\alpha}\gamma_{\beta}}{3}-\frac{2P_{\alpha}P_{\beta}}{3M_{\Delta}^{2}}+\frac{P_{\alpha}\gamma_{\beta}-P_{\beta}\gamma_{\alpha}}{3M_{\Delta}}\right].$

Finally, the $\Delta$ width $\Gamma(P^{2})$ is given by

$\Gamma(P^{2})=\Gamma_{0}\frac{m_{\Delta}}{\sqrt{P^{2}}}\left(\frac{p_{\pi}}{p^{res}_{\pi}}\right)^{3}.$
(59)

In the above equation, $p_{\pi}$ is the momentum of the final pion resulting
from the $\Delta$ decay an $p^{res}_{\pi}$ is its value at resonance
($P^{2}=m_{\Delta}^{2}$), and $\Gamma_{0}=120$ MeV is the width at rest. The
width (59) corresponds to the $\Delta$ in vacuum, and it is expected to be
slightly different in the medium depending on the kinematics. One could
investigate the dependence of the results on the choice of the width. However,
in this work, we do not delve into this issue because, as we will see, the
effect of the MEC on the 1p1h response is generally small, and corrections due
to fine-tuning of the model are unlikely to substantially alter the results.

Figure 2: Diagrams for the 1p1h MEC matrix elements

In the relativistic mean field description used in this work, we consider that
the $\Delta$ is also interacting with scalar and vector fields, acquiring an
effective mass and vector energy. To treat this case, we make the following
substitutions in the $\Delta$ propagator for the $\Delta$ mass and momentum
Weh93 ; Kim96 :

$M_{\Delta}\rightarrow M_{\Delta}^{*},\kern
28.45274ptP^{*\mu}=P^{\mu}-\delta_{\mu 0}E_{v}^{\Delta}.$ (60)

We use the value $M^{*}_{\Delta}=1042$ MeV, taken from Mar21b , and the
universal vector coupling $E_{v}^{\Delta}=E_{v}$.

With the MEC current defined in Eqs. (24-27), the effective one-body current
$j_{2}({\bf p},{\bf h})$ is generated by summing over the spin, isospin and
momentum of the spectator nucleon, as in Eq. (15). First, it can be observed
that due to the sum over isospin $t_{k}$, the direct term is zero (see Ref.
Ama03  for details). Therefore, the many-body diagrams that contribute to the
1p1h MEC are those shown in Figure 2. Furthermore, it can be verified that
diagrams e and f are also zero. Therefore, only diagrams a, b, c, and d
survive and contribute to the 1p1h MEC matrix elements.

##  III Results

In this section, we present results for the effects of MEC on the 1p1h
response functions using several models: the relativistic Fermi gas, the
relativistic mean field, and the generalized SuSAM* model. By employing these
different models, we take into account relativistic kinematics and we can
analyze the impact of including the relativistic effective mass of the nucleon
and the $\Delta$ resonance appearing in the MEC. The scaling analysis
described in the previous Section will allow us to study the influence of MEC
on the generalized scaling function also in the region $|\psi^{*}|>1$ where
the RFG and RMF responses are zero. Moreover, we can investigate how the
inclusion of MEC affects the scaling function and compare it with the
predictions of the RFG and RMF models.

Figure 3: Interference OB-MEC in the transverse response of 40Ca for two
values of the momentum transfer, with $k_{F}=237$ MeV/c. In the graph, the
curve labeled $\Delta_{1}$ corresponds to using the $\Delta$ current of the
present work in RFG. The curve $\Delta_{2}$ corresponds to the calculation
from reference Ama03 .  Figure 4: Comparison between relativistic and non
relativistic MEC transverse responses in 12C. Black lines: RFG. Red lines:
non-relativistic Fermi gas. Left panels show the interference OB-$\pi$, and
left panels the interference OB-$\Delta$. In these calculations the strong
form factors in the pion vertices are set to one.

Unless stated otherwise, we present the results for 12C with a Fermi momentum
of $k_{F}=225$ MeV/c. We use an effective mass of $M^{*}=0.8$, following the
same choice of parameters as in reference Mar21 ; Mar21b . The calculation of
1p1h responses involves evaluating the 1p1h matrix element of the MEC, as
given by Eq (15). This requires performing a numerical three-dimensional
integration to account for the momentum dependence. Subsequently, a one-
dimensional integration is carried out to calculate the averaged single-
nucleon responses, as described in Eq (31).

First, since this work is an extension of the MEC model from Ref. Ama03  to
the superscaling formalism, we will compare with the OB-MEC interference
responses presented in Ama03  within the framework of the RFG. It should be
noted that in Ama03  a different version of the $\Delta$ current was used. The
$\Delta$ current was obtained from the $\gamma N\Delta$ Lagrangian proposed by
Pascalutsa Pas95

${\cal L}_{\gamma
N\Delta}=ie\frac{G_{1}}{2m_{N}}\overline{\psi}^{\alpha}\Theta_{\alpha\mu}\gamma_{\nu}\gamma_{5}T_{3}^{\dagger}NF^{\nu\mu}+\mbox{h.c.},$
(61)

plus $O(1/m_{N}^{2})$ terms that give negligible contribution in the
quasielastic energy region. The tensor $\Theta_{\alpha\mu}$ may contain an
off-shell parameter and another arbitrary parameter related to the contact
invariance of the Lagrangian. In this work we use the simplest form

$\Theta_{\alpha\mu}=g_{\alpha\mu}-\frac{1}{4}\gamma_{\alpha}\gamma_{\mu}.$
(62)

The coupling constant $G_{1}$ was determined in Pas95  by fitting Compton
scattering on the nucleon. However, there is a detail that needs to be
clarified: the isospin operator used by Pascalutsa is normalized differently
from the standard convention. That is,
$T_{i}^{\text{Pascalutsa}}=\sqrt{\frac{3}{2}}T_{i}$, where $T_{i}$ is the
operator used in our calculation. This means that if we use the standard
$T_{i}$ in the Lagrangian (61), it should be multiplied by
$\sqrt{\frac{3}{2}}$. This is equivalent to multiplying Pascalutsa’s coupling
constant $G_{1}=4.2$ by the factor $\sqrt{\frac{3}{2}}$. In reference Ama03 ,
this detail went unnoticed, and the $\sqrt{3/2}$ factor was not included in
the calculations.

Using the Lagrangian given by Eq. (61), the following $\Delta$ current is
obtained:

$\displaystyle j^{\mu}_{\Delta F}$ $\displaystyle=$
$\displaystyle[(T_{i}T_{3}^{\dagger})\otimes\tau_{i}]_{t^{\prime}_{1}t^{\prime}_{2},t_{1}t_{2}}\frac{ff^{*}}{m_{\pi}^{2}}F_{\Delta}(Q^{2})V^{s^{\prime}_{2}s_{2}}_{\pi
NN}(p^{\prime}_{2},p_{2})F_{\pi N\Delta}(k_{2}^{2})$ (63)
$\displaystyle\bar{u}_{s^{\prime}_{1}}(p^{\prime}_{1})k_{2}^{\alpha}\left[\Theta^{\alpha\beta}G_{\beta\rho}(p_{1}+Q)\frac{G_{1}}{2m_{N}}[\Theta^{\rho\mu}\gamma^{\nu}-\Theta^{\rho\nu}\gamma^{\mu}]\gamma_{5}Q_{\nu}\right]u_{s_{1}}(p_{1})+(1\leftrightarrow
2)$

and a similar expression for the $\Delta$ backward current. This current was
used in Ref. Ama03  to compute the OB-MEC interference with the following form
factor

$F_{\Delta}(Q^{2})=G^{p}_{E}(Q^{2})\left(1-\frac{Q^{2}}{3.5(GeV/c)^{2}}\right)^{-1/2}$
(64)

where $G_{E}^{p}$ is the electric form factor of the proton.

Figure 5: Averaged single nucleon responses computed with and without MEC, for
several values of the momentum transfer as a function of the scaling variable
$\psi^{*}$.

In Figure 3, we present the interference between the OB and $\Delta$ currents
in the transverse response of 40Ca. We compare our results with the model of
reference Ama03  in RFG, where the Lagrangian of Pascalutsa was used. The
results of Ama03  have been corrected with the factor of $\sqrt{\frac{3}{2}}$
mentioned earlier. For $q=500$ MeV/c, there is little difference between the
two models. However, for $q=1$ GeV/c, the difference becomes more noticeable.

The results of Fig. 3 show that the $\Delta$ current model used in this work
does not differ significantly from the model in reference Ama03 , providing
similar results. The small differences observed can be attributed to the
different form factor and coupling constants, and can be understood as a model
dependence in these results. From here on, all the results refer to the
$\Delta$ current model described in the equations (44,45).

It is expected that any relativistic model should reproduce the results of the
well-established non-relativistic model for small values of energy and
momentum in the non-relativistic limit Hok73 . As a check in this regard, in
Fig. 4 we compare the present model with the non-relativistic Fermi gas model
from ref. Ama94 . The non relativistic $\Delta$ current used is taken from
Ama03 . To perform this comparison the same form factors and coupling
constants are used in the relativistic and non relativistic models. To take
this limit in Fig. 4, we follow the procedure as follows: $q$ is small and
$k_{F}=q/2$. We show the comparison between the two models for various values
of $q$ ranging from 100 to 500 MeV/c. In the left panels, we present the
contribution of the transverse response stemming from the interference
OB-$\pi$ between the pure pionic MEC (diagrams a-c in Fig.2) and in the right
panels we show the OB-$\Delta$ interference (diagrams d-g in Fig.2) for the
same values of $q$. As expected, we observe that for $q=100$ MeV/c, the
relativistic and non-relativistic models practically coincide, demonstrating
the consistency between the two models in the non-relativistic limit.

In Fig.4 one can also observe that for low values of $q$ the dominant
contributions to the MEC are from the seagull and pion-in-flight diagrams,
with the seagull diagram playing a particularly important role. These diagrams
contribute positively to the MEC, enhancing the overall response. On the other
hand, the contribution from the $\Delta$ resonance is negative. As $q$
increases, the influence of the $\Delta$ resonance becomes more significant,
and it starts to dominate the MEC contribution for $q$ values around 400
MeV/c.

Before performing the scaling analysis, we examine the averaged single-nucleon
responses that will be used to scale the data. In Figure 5, we display the
longitudinal and transverse single-nucleon responses for various values of $q$
as a function of the scaling variable. The calculated responses are shown
separately for the OB current and the total responses including the MEC and
taking into account the sum of protons and neutrons. The total response, which
we have defined in equation (36), comes from the product of the single nucleon
with the phenomenological scaling function obtained from the $(e,e^{\prime})$
data as shown below. We have used a Fermi distribution, Eq.(37), with a
smearing parameter $b=50MeV/c$, although the single nucleon responses do not
depend much on this specific value. It is observed that the effect of the MEC
is negligible in the longitudinal response, as the curves for the OB current
and total response overlap. However, in the transverse response, the effect of
the MEC becomes appreciable, resulting in a reduction of the $w_{T}$ response
compared to the OB current. This reduction can be attributed to the
interference between the one-body and two-body currents, which leads to a
modified transverse response. The comparison between the OB current and the
total response including the MEC provides insights into the contributions of
the MEC to the single-nucleon responses and sets the stage for the subsequent
scaling analysis.

Figure 6: Scaling analysis of 12C data including MEC and relativistic
effective mass $M^{*}=0.8$. The Fermi momentum is $k_{F}=225$ MeV/c. In the
top panel, we show the data points after scaling, representing the overall
distribution. In the middle panel, we display the selected data points, which
have been chosen after eliminating those that do not exhibit clear scaling
behavior. In the bottom panel, we present the phenomenological scaling
function, which has been fitted to the selected data points, compared to the
scaling function obtained in a similar analysis without MEC. Experimental data
are taken from Refs. archive ; archive2 .

Note that the center of the quasielastic peak corresponds to $\psi^{*}=0$,
where the energy and momentum can be transferred to a nucleon at rest. We see
that MEC have a larger impact in the region $\psi^{*}>0$, that is, the right-
hand side of the peak, corresponding to higher energy transfers.

In Figure 6, we present the scaling analysis of the 12C data. In the top
panel, the experimental data, $f^{*}_{exp}$, are plotted against $\psi^{*}$ in
the interval $-2<\psi^{*}<2$. Experimental data are from Refs. archive ;
archive2  and cover a wide electron energy range, from 160 MeV up to 5.8 GeV.
We observe a significant dispersion of many data points, indicating a wide
range of inelastic scattering events. However, we also notice that a portion
of the data points cluster together and collapse into a thick band. These data
points can be considered as associated to quasielastic (1p1h) events. To
select these quasielastic data, we apply a density criterion. For each point,
we count the number of points, $n$, within a neighborhood of radius $r=0.1$,
and eliminate the point if $n$ is less than 25. Points that have been
disregarded are likely to correspond to inelastic excitations and low energy
processes that violate scaling and cannot be considered within quasielastic
processes. We observe that the remaining selected points, about half of the
total, shown in the middle panel of Fig 6, form a distinct thick band. These
points represent the ones that best describe the quasielastic region and
approximately exhibit scaling behavior. The red curve represents the
phenomenological quasielastic function $f^{*}(\psi^{*})$, that provides the
best fit to the selected data using a sum of two Gaussian functions:

$f^{*}(\psi^{*})=a_{3}e^{-(\psi^{*}-a_{1})^{2}/(2a_{2}^{2})}+b_{3}e^{-(\psi^{*}-b_{1})^{2}/(2b_{2}^{2})}.$
(65)

The parameters found are shown in table 1.

$a_{1}$ | $a_{2}$ | $a_{3}$ | $b_{1}$ | $b_{2}$ | $b_{3}$
---|---|---|---|---|---
$-0.01015$ | 0.46499 | 0.69118 | 0.86952 | 1.16236 | 0.17921
Table 1: Table of fitted parameters of the scaling function.

In the bottom panel of Fig. 6 we compare the scaling function obtained in our
analysis with the scaling function obtained without including the MEC
contributions. When including the MEC, the scaling function appears slightly
higher since the single-nucleon response with MEC is slightly smaller than
without them. However, both analyses provide a similarly acceptable
description of the data. This suggests that while the MEC do have an impact on
the scaling behavior, their effect is relatively small and does not
significantly alter the overall scaling pattern observed in the data.

Figure 7: Interference OB-MEC responses separated in seagull, pion-in-flight,
and $\Delta$ contributions for 12C and $q=500$ MeV/c. In each panel we compare
the results of RFG (with $M^{*}=1$, dot-dash), with the RMF (with $M^{*}=0.8$)
and the SuSAM* model.  Figure 8:  The same as Fig. 7 for $q=700$ MeV/c.
Figure 9: The same as Fig. 7 for $q=1000$ MeV/c.  Figure 10: The same as Fig.
7 for $q=1500$ MeV/c.

Now that we have obtained the phenomenological scaling function through the
scaling analysis, we can utilize this function to calculate the response
functions of the model beyond the RMF. By multiplying the scaling function by
the averaged single nucleon responses, as stated in Eq (36), we can extend our
calculations to different kinematic regimes and explore the behavior of the
responses beyond the relativistic mean field description. This allows us to
investigate the influence of various factors, such as the MEC and relativistic
effects, on the response functions and cross sections.

In Figures 7-10, we present the interferences of the OB-MEC in the response
functions for different values of $q$ (500, 700, 1000, and 1500 MeV/c). We
separate the interferences into OB-seagull, OB-pionic, and OB-$\Delta$
contributions for both the longitudinal and transverse responses as functions
of $\omega$. Each panel displays three curves corresponding to the free RFG
(with effective mass $M^{*}=1$), the RMF (with effective mass $M^{*}=0.8$),
and the present SuSAM* model. These figures allow us to analyze the relative
contributions of the different OB-MEC interferences in the response functions
at various kinematic regimes. By comparing the results obtained from the RFG,
RMF, and SuSAM* models, we can observe the effects of including the
relativistic interaction through the effective mass and the scaling function
on the interferences.

First is observed that the introduction of the effective mass $M^{*}=0.8$
shifts the responses to the right, towards higher energy values. The effective
mass takes into account the binding of the nucleon in the nucleus, which
causes the quasielastic peak to approximately coincide with the maximum of the
experimental cross section. In the RFG, this is traditionally taken into
account by subtracting a binding energy of approximately 20 MeV from $\omega$
to account for the average separation energy of the nucleons. In the RMF, this
is automatically included by considering the effective mass of the nucleon,
$M^{*}=0.8$, which was adjusted for 12C precisely to achieve this effect.

In the transition from the RMF to the SuSAM* model, we replace the scaling
function of the RFG with the phenomenological scaling function that we have
adjusted. This new scaling function extends beyond the region of
$-1<\psi^{*}<1$, where the RFG scaling function is zero. As a result, we
observe in figures 7-11 that the interferences acquire a tail towards high
energies, similar to the behavior of the scaling function.

The tail effect is more pronounced in the longitudinal responses because the
single-nucleon longitudinal response, as shown in Figure 5, increases with
$\omega$. This amplifies the tail when multiplied by the scaling function.
However, it is important to note that the contribution of the MEC to the
longitudinal response is relatively small compared to the dominant transverse
response. Therefore, while the tail effect is observed in the longitudinal
responses, its impact on the cross section is not as significant as in the
transverse channel, if not negligible.

In the dominant transverse response, the seagull contribution from the MEC is
positive, leading to an enhancement of the response, while the pionic and
$\Delta$ contributions are negative, causing a reduction in the overall
response when including the MEC. This is in line with pioneering calculations
by Kohno and Otsuka Koh81  and by Alberico et al. Alb90  in the non-
relativistic Fermi gas. Also in shell model calculations, similar results have
been obtained Ama94 , showing that the MEC contributions also lead to a tail
and extension of the response functions to higher values of $\omega$, as in
the SuSAM* approach. It is worth noting that the relative importance of these
contributions can depend on the momentum transfer $q$ and the energy transfer
$\omega$. For the values considered in Figures 7-11, the $\Delta$ current is
found to be the dominant contribution, leading to a net negative effect from
the MEC.

The observation in Fig. 10 of a sign change and a small bump in the
OB-$\Delta$ transverse response for high values of $\omega$ is indeed
interesting. The change of sign is already observed for $q$=1 GeV/c in Fig. 9.
This connects with the findings in reference Ama10 , where a pronounced bump
and sign change were reported in a semi-relativistic shell model calculation
based on the Dirac equation with a relativistic energy-dependent potential. In
the present calculation the bump is observed but it is very small compared to
the results of Ref, Ama10 . It is important to note that, in the present work,
the fully relativistic SuSAM* approach is employed, which takes into account
the dynamical properties of both nucleons and the $\Delta$, as well as the
scaling function. This differs from the approach in reference Ama10 , where a
static propagator for the $\Delta$ was used. To definitively clarify the
difference with the present results, a fully relativistic calculation in
finite nuclei, considering the dynamical properties of the $\Delta$ would be
necessary.

Figure 11: Comparison of OB-MEC interference in the transverse response (black
lines) with the pure MEC transverse response (red lines) for several values of
$q$ in the RMF model. Figure 12: The same as Fig. 11 in the SuSAM* model.

The comparison of the OB-MEC interference with the MEC contribution alone
(represented by $w_{12}^{\mu\nu}$ and $w_{2}^{\mu\nu}$, respectively in Eq,
(21)) in the transverse response is shown in Figs. 11 and 12. We observe that
the MEC contribution alone represents a small and almost negligible
contribution to the transverse response. This justifies the previous
calculations that focused only on the OB-MEC interference (e.g., the semi-
analytical calculations in references Ama94a ; Ama94  for the non-relativistic
Fermi gas), as it provides an excellent approximation. This observation holds
true for both the RMF model in Fig. 11 and the SuSAM* model in Fig. 12. It
highlights the fact that the dominant contribution to the transverse response
arises from the interference between the OB and MEC, while the pure MEC
contribution is relatively small. It is also worth stressing that while the
pure MEC contribution is, of course, positive, the interference contribution
is negative.

Figure 13:  Response functions calculated in the generalized SuSAM* model
(black curves). The red curves do not include the MEC.

In Fig. 13, we present the total responses of 12C computed using the
generalized SuSAM* model. These responses are obtained by multiplying the
phenomenological scaling function by the averaged single-nucleon response and
summing over protons and neutrons, as given by Eq. (36). The responses are
shown for different values of $q$ as a function of $\omega$. In the same
figure, we also show the results without including the MEC contributions,
which corresponds to setting the terms $w_{12}+w_{2}$ associated with the two-
body current (Eq. (21)) to zero.

Comparing the results with and without MEC, we observe that the impact of MEC
is more significant in the transverse response compared to the longitudinal
response. This is expected since the corrections due to MEC in the
longitudinal response are higher-order effects in a non-relativistic expansion
in powers of $v/c$, as known from previous studies Ris89 . Therefore, the MEC
contributions to the longitudinal response are minimal and only start to
become noticeable for $q>$1 GeV in the high-energy region. However, this high-
energy region is dominated and overshadowed by pion emission and inelastic
processes, making it difficult to isolate the 1p1h longitudinal response.

The inclusion of MEC in the single-nucleon leads to a reduction of the
transverse response by around 10% or even more for all studied values of $q$.
This is consistent with previous calculations in RFG and the shell model Ama94
; Fab97 ; Ama03 ; Ama03b ; Ama10 . These calculations have consistently shown
that MEC in the 1p1h channel tend to decrease the transverse response compared
to the contribution from the one-body current. It is important to note that
this reduction in the transverse response is a direct consequence of the
destructive interference between the one-body current and MEC. The
contribution of MEC to the transverse response is negative because the direct
two-body matrix element is zero (in symmetric nuclear matter, $N=Z$) or almost
zero (in asymmetric nuclear matter, $N\neq Z$, or in finite nuclei) after
summing over isospin.

Figure 14: Comparison of the transverse interference OB-$\Delta$ computed in
the generalized SuSAM* model with and without relativistic effective mass and
vector energy for the $\Delta$.

The treatment of the $\Delta$ resonance in the medium is subject to various
ambiguities and uncertainties. In our generalized SuSAM* model, we have
assumed that the $\Delta$ resonance acquires an effective mass
$M_{\Delta}^{*}$ and vector energy $E_{v}^{\Delta}$ due to its interaction
with the RMF. This requires modifying the propagator according to the
formalism proposed in references Weh93 ; Kim96 . To estimate the effect of
this treatment, in Fig. 14 we compare the transverse response for the
OB-$\Delta$ interference calculated assuming that the $\Delta$ remains
unchanged in the medium, i.e., setting $M_{\Delta}^{*}=M_{\Delta}$ and
$E_{v}^{\Delta}=0$. The response with the free $\Delta$ without medium
modifications is slightly smaller in absolute value, around 10% depending on
the momentum transfer. This can be seen as an estimation of the uncertainty
associated with the $\Delta$ interaction in the medium.

Another related issue is the modification of the $\Delta$ width in the medium,
which we have not considered here assuming the free width (59). This effect
can also influence the results, but it is expected to be of the same order as
the observed effect in Fig. 14. It is important to note that the treatment of
the $\Delta$ resonance in the medium is a complex topic, and further
investigations and refinements are needed to fully understand its effects and
uncertainties.

Figure 15: Total transverse responses for 12C including MEC in the RMF model
with $M^{*}=0.8$ compared to the generalized SuSAM* model.

In Fig. 15, we compare the total transverse response calculated in the RMF
model with an effective mass of $M^{*}=0.8$ to the results obtained in the
generalized SuSAM* approach for various momentum transfers, ranging from
$q=300$ MeV/c to $q=1500$ MeV/c. Both calculations include the effects of MEC.
One notable difference between the two approaches is the presence of a
pronounced tail at high energy transfer rates in the SuSAM* results. This tail
extends well beyond the upper limit of the RFG responses, reflecting the
effect of the phenomenological scaling function used in the SuSAM* approach.
Similar effects are found in the longitudinal response. Additionally, it is
worth noting that the peak height of the transverse response in the SuSAM*
approach is generally higher compared to the RMF model. Overall, the
comparison in Fig. 15 highlights the improvements and additional physics
captured by the SuSAM* approach, by extending the scaling function of the RFG
to describe the transverse response in a wider energy transfer range.

Figure 16: Cross section of 12C for several kinematics computed with the
generalized SuSAM* model, including MEC, compared with the same calculation
without MEC. Experimental data are from Refs. archive ; archive2 .

Finally, in Fig. 16, we present the results for the (e,e’) double differential
cross section of 12C calculated with the generalized SuSAM* model including
MEC, compared to experimental data for selected kinematics. We also compare
with the same model but assuming that only the single-nucleon contribution is
present, i.e., setting the MEC to zero. We observe that the inclusion of MEC
in this model leads to a small reduction in the cross section compared to the
case without MEC. This reduction is a consequence of the decrease in the
transverse response due to the presence of MEC. The generalized scaling
approach, including the inclusion of MEC, provides a global description of the
cross section that is comparable to other previous analyses, such as the
SuSAM* model with the one-body current only, or the SuSAv2 model, which
factorize different definitions of the single nucleon (without effective mass
and with extrapolation of the Fermi gas single nucleon in the case of SuSAv2).
All of these approaches reasonably describe the quasielastic cross section
because the scaling function has been properly adjusted to reproduce the
global scaling data. The generalized scaling approach, like any
parametrization, is a phenomenological framework that aims to capture the
essential physics of the reaction. It provides a functional form for the cross
section that incorporates the known ingredients and leaves the unknowns to be
determined by the scaling function. The scaling function encapsulates the
effects of various dynamical and correlation effects, allowing for a global
description of the data.

##  IV Discussion and concluding remarks

From the results seen in the previous section we have observed that, in all
the models considered, the transverse response decreases when including meson
exchange currents in the 1p1h channel. This result is consistent with previous
independent calculations performed in the relativistic and non-relativistic
Fermi gas models as well as in the non-relativistic and semi-relativistic
shell models. The result is a consequence of the fact that the main
contribution arises from the interference of the OB and $\Delta$ currents, in
particular through the exchange diagram, carrying a minus sign. The
contribution from the direct part of the MEC matrix element is zero in the
Fermi gas, and this is the reason for the negative contribution.

It is worth mentioning the existence of some calculations that disagree with
this result and suggest a different effect of MEC on the transverse response.
We would like to comment in particular on two notable model calculations: the
Green Function Monte Carlo (GFMC) model from reference Lov16  and the
Correlated Basis Function (CBF) calculation by Fabrocini Fab97 , both
including meson exchange currents in the 1p1h sector. In both approaches, the
effect of MEC is positive in the quasielastic peak and quite significant,
around 20%, in the transverse response. This substantial effect is attributed
to the simultaneous effect of tensor correlations in the wave function and
MEC. In fact, in Fabrocini’s calculation, the origin of this effect was found
to be the tensor-isospin correlation contribution in the direct matrix element
of the $\Delta$ current, which is non-zero when summing over isospin for
correlated wave functions. This effect can also be understood in terms of
presence of short-range correlations in the nuclear wave function. The direct
matrix element of MEC, when a proton is emitted, involves the interaction of
the proton with protons as well as with neutrons, i.e., the MEC matrix element
involves PN and PP pairs. The high-momentum component of these pairs is
significantly different because PN pairs contain the ${}^{3}S_{1}$ and
${}^{3}D_{1}$ deuteron-like waves, while PP pairs do not. Therefore, when
summing over isospin, there is no cancellation between PP and PN pairs in the
high-momentum part of the wave function, resulting in a non-zero direct matrix
element. This is in agreement with the conclusion of Fabrocini, as the tensor-
isospin term precisely generates this significant difference between PP and PN
pairs. An alternative way to investigate this hypothesis would be to perform
calculations in the independent particle model by solving the Bethe-Goldstone
equation Cas23b  for PP and PN pairs and using a correlation current similar
to the one proposed in Mar23 . Such calculations could provide further
insights into the effect of short-range correlations on the MEC contributions
to the transverse response.

On the other hand the results of Fabrocini reproduce the well-known effect
that MEC has a negative impact on the transverse response when the
correlations functions are set to zero, consistent with the results from
uncorrelated models. Since in the present work we started with an uncorrelated
model, the relativistic mean field, the effects of correlations in the
transverse current are expected to be included phenomenologically in the
scaling function. This and other mechanisms, such as final state interactions,
contribute to the violation of scaling observed in the data.

To summarize, this work presents a generalized scaling analysis of the (e,e’)
cross section of 12C, including the MEC consistently in the formalism. To
achieve this, we have introduced a new definition of the single nucleon tensor
in the factorization of the model. The average per particle of the hadronic
tensor for 1p1h emission has been defined by considering the sum of the one-
body and two-body currents, without modifying the definition of the scaling
function, which remains the same as in the one-body current case in the Fermi
gas. This averaging definition has been extended beyond the scaling region
$-1<\psi^{*}<1$ of the Fermi gas by slightly modifying the momentum
distribution with a smeared Fermi distribution that allows the evaluation of
MEC for any value of the scaling variable.

By incorporating the MEC and using the phenomenological scaling function, we
have calculated the 1p1h response functions in the RFG, RMF, and SuSAM*
models. The results show the impact of the MEC on the response functions,
particularly in the transverse sector. The MEC reduce the transverse response
while the longitudinal response is found to be hardly affected by the MEC.
Furthermore, the analysis of the OB-MEC interference and the comparison
between the SuSAM* and RFG models highlight the role of the effective mass and
the $\Delta$ resonance in the response functions.

Overall, the generalized scaling analysis with the inclusion of MEC provides a
consistent framework for studying quasielastic electron scattering in nuclei
accounting for relativistic dynamical effects through the effective mass. The
approach adopted in this work differs from other scaling analyses, such as the
original SuSAM* model, in the definition of the single-nucleon dividing
factor, which now incorporated the effect of MEC in the 1p1h channel. However,
the ultimate results are compatible between different models because the
improvement in scaling symmetry is not significant when modifying the single
nucleon in this manner. This means that both formalisms will describe the
experimental cross section data similarly, as they have been adjusted
accordingly. The difference between various approaches lies in how the scaling
function is adapted and rectified based on the chosen prefactor of the single
nucleon. The equivalence between these models and others, such as SuSAv2,
indicates the flexibility of the scaling approach to adapt to the
circumstances of the emphasized model. Scaling is only an approximate symmetry
of quasielastic data, and the degree of violation of this symmetry should be
attributed to all effects that break the factorization of the cross section in
a many-body system with complex interactions and correlations between
particles.

In conclusion, this work presents the first comprehensive study of
quasielastic electron scattering in nuclei that includes the 1p1h meson
exchange currents (MEC) consistently in a generalized scaling approach,
extending previous work where this contribution was evaluated in the
relativistic Fermi gas (RFG) framework. Looking ahead, this work opens the
door to future developments and applications, including the extension of the
model to study neutrino-nucleus scattering.

##  V Acknowledgments

Work supported by: Grant PID2020-114767GB-I00 funded by
MCIN/AEI/10.13039/501100011033; FEDER/Junta de Andalucia-Consejeria de
Transformacion Economica, Industria, Conocimiento y
Universidades/A-FQM-390-UGR20; Junta de Andalucia (Grant No. FQM-225); INFN
under Project NUCSYS; and University of Turin under Project BARM-RILO-23-01.

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