NT@UW-24-04

#  Electromagnetic form factors for nucleons in short-range correlations and
the EMC effect

Dmitriy N. Kim1, Or Hen2, Gerald A. Miller1, E. Piasetzky3, M. Strikman4 and
L. Weinstein5 1 Department of Physics, University of Washington, Seattle, WA
98195-1560, USA  2 Department of Physics, Massachusetts Institute of
Technology, Cambridge Massachusetts 02139, USA  3 School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 69978, Israel  4 Department of
Physics, The Pennsylvania State University, State College, PA 16802 , USA  5
Department of Physics, Old Dominion University, Norfolk, Virginia, 23529, USA

###### Abstract

The relationship between medium modifications of nucleon electromagnetic form
factors and nucleon structure functions is examined using a model motivated by
Light-Front Holographic QCD (LFHQCD). These modifications are closely
connected with the influence of short-ranged correlations. The size of the
modifications to nucleon form factors is shown to be about the same as the
modifications to the structure functions. Thus, small limits on form factors
modifications do not rule out an explanation of the EMC effect motivated by
the influence of short range correlations, as claimed by a recent paper.

Recent experimental studies Weinstein  _et al._ (2011); Hen  _et al._ (2017);
Duer  _et al._ (2019); Schmookler  _et al._ (2019a) seem to support models in
which the EMC effect (medium modification of nucleon structure functions at
$0.3<x<0.7$) is mainly due to the influence of nucleon-nucleon short-range
correlations (SRCs) (high-relative momentum nucleon-nucleon correlations).
This hypothesis requires that the structure functions for nucleons involved in
SRCs be heavily suppressed compared to that of free nucleons.

Ref. Xing  _et al._ (2023) claimed that medium modifications of structure
functions motivated by the influence of short range correlations causes a very
large strong suppression of elastic electromagnetic form factors of bound
nucleons at values of momentum transfer, $Q^{2}>0.25\,\rm GeV^{2}$. Since this
strong suppression seems unlikely, Ref. Xing  _et al._ (2023) casts doubt on
the causal nature of the EMC-SRC correlation.

The expansion of a nucleon’s radius in the medium was one of the early
proposed hypotheses for the explanation of the EMC effect, see the reviews
Geesaman  _et al._ (1995); Norton (2003); Hen  _et al._ (2013); Rith (2015);
Malace  _et al._ (2014); Hen  _et al._ (2017). Thus the suggestion of Ref.
Xing  _et al._ (2023) encourages us to seek a deeper understanding of the
connections between the medium modifications of nucleon structure functions
and elastic form factors. It turns out that in doing so we shall refute the
claims of Ref. Xing  _et al._ (2023).

There are strong constraints on medium modifications of form factors —they
have been shown to be small Sick (1985), with an upper limit between 3 and 6%
for values of $Q^{2}$ between 1 and 4 GeV2, for values of missing momentum
($p$ of Fig. 1) less than about 150 MeV/c. Therefore it is necessary to verify
that any model of the EMC effect does not violate such constraints.
Constraints for larger values of the missing momentum do not seem to exist.

The purpose of this is paper is to study the connection between medium
modifications of nucleon structure functions and elastic electromagnetic form
factors, as raised in the early papers, Refs. Frankfurt and Strikman (1985);
Ciofi degli Atti  _et al._ (2007).

We begin with a necessarily somewhat lengthy summary of previous work related
to the current effort. The early papers showed that the nuclear modification
of the nucleon wave function depends on the virtuality,
$\mathcal{V}=p^{2}-M^{2}$, of the nucleon, where $p$ is the four-momentum of
the bound nucleon and $M$ is the nucleon mass. The relevance of virtuality can
be seen from the simple Feynman diagram shown in Fig. 1. If the energy of the
$A-1$ residual nucleus is given by $M_{A}-M+\Delta E$, the use of four-
momentum conservation tells us that $p^{2}-M^{2}=-2M\Delta E+\Delta E^{2}$. If
one is examining a regime in which the scattering is from a correlated pair of
nucleons (quasi-deuteron model) then $\mathcal{V}\approx-2{\bf p}^{2}$. Small
values of $\Delta E$ correspond to typical mean-field effects, and larger
values generally arise from the effects of SRCs. The structure of the nucleon
depends on the value of $p^{2}$ that is necessarily different than $M^{2}$.

Figure 1:  Deep inelastic scattering event leaving a residual nucleus
$(A-1)^{*}$ in an excited state.

It was argued that larger values of the magnitude of the virtuality led to
larger suppression of the point-like configurations (PLCs) Frankfurt and
Strikman (1985), quark configurations that dominate the high-$x$ behavior of
nucleon structure functions. Thus larger virtuality causes a large EMC effect
and presumably large modifications of the electromagnetic form factors. Ref.
Ciofi degli Atti  _et al._ (2007) computed ratios of nuclear structure
functions to free nucleon structure functions at a value of Bjorken $x=1/2$,
where the effects of Fermi motion are small. It was also suggested that deep
inelastic processes might be more effective than quasi-elastic processes for
determining the medium modifications of form factors. This was because in the
former higher momentum components are probed.

The idea of PLC suppression was taken up in Ref. Kim and Miller (2022) in a
calculation that used an AdS/QCD motivated light-front quark-diquark model to
compute the nuclear structure functions. The related modifications of nucleon
radii were also examined. The present paper presents a more detailed study of
medium-modified form factors at all values of momentum transfer.

We summarize the work of Ref. Kim and Miller (2022). Motivated by the the QCD
physics of color transparency Frankfurt and Strikman (1985); Jennings and
Miller (1993), Ref. Kim and Miller (2022) treated the infinite number of
quark-gluon configurations of the nucleon as two configurations: a large-
sized, blob-like configuration (BLC), consisting of complicated configurations
of many quarks and gluons, and a small-sized, PLC consisting of 3 quarks. The
BLC can be thought of as an object larger in average size than a nucleon, and
the PLC is meant to represent a three-quark system of small size that is
responsible for the high-$x$ behavior of the distribution function. This is
because the smaller the number of quarks, the more likely one can carry a
large momentum fraction. Recent support for the idea that configurations of a
proton with large values of $x$ have smaller than average size comes from an
an analysis hadron production at backward rapidities and from an analysis of
jet production data in proton- and deuteron-nucleus collisions at RHIC and the
LHC Alvioli  _et al._ (2016, 2018).

When placed in a nucleus, the BLC feels the usual nuclear attraction, thus its
energy decreases. The PLC feels far less nuclear-attraction by virtue of color
screening Frankfurt  _et al._ (1994), in which the effects of gluons emitted
by small-sized configurations are cancelled in low-momentum transfer
processes. The nuclear attraction increases the energy difference between the
BLC and the PLC, therefore reducing the PLC probability Frankfurt and Strikman
(1985). Reducing the PLC probability in the nucleus reduces the quark momenta,
in qualitative agreement with the EMC effect. Working out the consequences of
the BLC-PLC model enables the connection between the EMC effect and virtuality
to be clarified.

The Hamiltonian for a free nucleon in the two-component model is given by

$H_{0}=\left[\begin{array}[]{cc}E_{B}&V\\\ V&E_{P}\end{array}\right],$ (1)

where subscripts $P$ and $B$ represent the PLC and BLC respectively. We define
the energy difference between the PLC and the BLC to be $\Delta=E_{P}-E_{B}$.
The hard-interaction potential, $V$, connects the two components, causing the
eigenstates of $H_{0}$ to be $|N\rangle$ and $|X\rangle$ rather than
$|B\rangle$ and $|P\rangle$. The normalized eigenstates are given by

$\displaystyle|N\rangle=\frac{1}{\sqrt{1+\epsilon^{2}}}(|B\rangle+\epsilon|P\rangle),$
$\displaystyle|X\rangle=\frac{1}{\sqrt{1+\epsilon^{2}}}(\epsilon|B\rangle-|P\rangle),$
(2)

where $\epsilon=-2V/(\Delta+\overline{\Delta})$ and
$\overline{\Delta}\equiv\sqrt{\Delta^{2}+4V^{2}}$. The notation $|X\rangle$ is
used to denote the orthogonal excited state. The probability of the PLC,
$P_{PLC}$, is then $P_{PLC}=\frac{\epsilon^{2}}{1+\epsilon^{2}}.$

Now suppose the nucleon is bound to a nucleus. The nucleon feels an
attractive, flavor-dependent, nuclear potential acting only on the BLC, so the
Hamiltonian is given by

$\displaystyle H=\left[\begin{array}[]{cc}E_{B}-|U_{N}|&V\\\
V&E_{P}\end{array}\right],$ (5)

where $U_{N}$ is the attractive nuclear potential and the subscript $N$ is the
flavor of the nucleon, proton or neutron. These interactions increase the
energy difference between the bare BLC and PLC states, and thereby decrease
the PLC probability.

The medium-modified nucleon wave function $|\tilde{N}\rangle$ is

$|\tilde{N}\rangle=\frac{1}{\sqrt{1+\tilde{\epsilon}^{2}_{N}}}(|B\rangle+\tilde{\epsilon}_{N}|P\rangle),$
(6)

with $\tilde{\epsilon}_{N}\approx\epsilon(1-{|U_{N}|\over\overline{\Delta}})$
and the probability of the modified PLC for the nucleon, $\tilde{P}_{PLC}$, is
given by

$\tilde{P}_{PLC}=P_{PLC}\left(1-\frac{2|U_{N}|}{\overline{\Delta}(1+\epsilon^{2})}\right).$
(7)

.

The next step is to relate $U_{N}$ to the fractional virtuality ${\cal
V}_{N}={\cal V}/M^{2}$, which is done in Ref. Ciofi degli Atti  _et al._
(2007). The result is

$\displaystyle{\cal V}_{N}={-2|U_{N}|\over M},$ (8)

in which $U$ and ${\cal V}_{n}$ are average quantities.

Ref. Kim and Miller (2022) introduced a model of the EMC effect using LFHQCD
nucleon form factors Sufian  _et al._ (2017) as a starting point. In that work
the SU(6) spin-flavor symmetric quark model was used to calculate the
effective charges of positive and negative helicity protons and neutrons.
These effective charges were then used to obtain expressions for the nucleon
form factors. The key idea is that the nuclear medium will affect the
probabilities of finding a spin up or down quark $q$ in a proton or neutron,
breaking SU(6) symmetry; thus, the effective charges are ultimately modified
as well.

We next consider free nucleon form factors and PDFs from LFHQCD using
Ref.Sufian  _et al._ (2017). In their work the form factor for an arbitrary
twist-$\tau$ (number of partons) hadron is expressed as

$F_{\tau}(Q^{2})={1\over(1+{Q^{2}\over M^{2}_{0}})(1+{Q^{2}\over
M^{2}_{1}})\cdots(1+{Q^{2}\over M^{2}_{\tau-2}})},$ (9)

a product of poles along the vector meson Regge radial trajectory in terms of
the $\rho$ vector meson mass, $M_{\rho}$, and its radial excitations of mass,
$M_{n}$, with $M_{n}^{2}=2M_{\rho}^{2}(n+1/2)$.

The use of SU(6) symmetry lead Ref. Sufian  _et al._ (2017) to obtain

$F_{1}^{p}(Q^{2})=F_{\tau=3}(Q^{2})$ (10)

$F_{1}^{n}(Q^{2})=-\frac{1}{2}F_{\tau=3}(Q^{2})+\frac{1}{2}F_{\tau=4}(Q^{2}).$
(11)

One can obtain the up ($u$) and down ($d$) valence PDFs of the free proton and
neutron by using a flavor decomposition of nucleon form factors Cates  _et
al._ (2011), charge (isospin) symmetry Miller  _et al._ (2006), and by writing
the form factors for quark flavor $q$, $F^{q}$, in terms of the valence GPD
$H^{q}_{v}(x,t)$,

$(F_{1}^{u})^{p}=2F_{1}^{p}+F_{1}^{n},\,\,\
(F_{1}^{d})^{p}=F_{1}^{p}+2F_{1}^{n}.$ (12)

Then de Teramond  _et al._ (2018),

$(F^{q}_{1})^{N}=\int_{0}^{1}dx\,H^{q}_{v}(x,t)=\int_{0}^{1}dx\,q_{v}^{N}(x)e^{tf(x)},$
(13)

where $(F_{1}^{q})^{N}$ is the $F_{1}$ flavor form factor for quark $q$ in
nucleon $N$, $q_{v}^{N}(x)$ is the valence PDF for quark flavor $q$ in nucleon
$N$, and $f(x)$ is the profile function.

Furthermore, Ref. de Teramond and Brodsky (2010) recast Eq. (9) in terms of an
Euler Beta Function and determined what the PDF is for
$(F^{q}_{1})^{N}=F_{\tau}$. The corresponding PDF for $F_{\tau}$, referred to
as $q_{\tau}(x)$, is normalized to unity and is given by,

$\displaystyle
q_{\tau}(x)=\frac{\Gamma\left(\tau-\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(\tau-1)}\big{(}1-w(x)\big{)}^{\tau-2}\,w(x)^{-{\frac{1}{2}}}\,w^{\prime}(x),\,\,\,\,\,\,$
(14)

with $w(x)=x^{1-x}e^{-a(1-x)^{2}}$, where the flavor-independent parameter
$a=0.531\pm 0.037$. Using Eqs. (10, 11, 12, 13) one obtains de Teramond  _et
al._ (2018) the valence $u$ and $d$ proton quark distributions at the matching
scale between LFHQCD and pQCD, $\mu_{0}=1.06\pm 0.15$ GeV. The size of the EMC
effect is approximately independent of the factorization scale Geesaman  _et
al._ (1995); Norton (2003); Hen  _et al._ (2013); Rith (2015); Malace  _et
al._ (2014); Hen  _et al._ (2017).

$\displaystyle u_{v}^{p}(x)={3\over 2}q_{3}(x)+{1\over
2}q_{4}(x),\,\,d_{v}^{p}(x)=q_{4}(x).$ (15)

The neutron valence PDFs are obtained by using isospin symmetry. The square of
the proton and neutron wave functions are characterized by,
$\Psi_{p}^{2}\sim\frac{3}{2}q_{3}(x)+\frac{3}{2}q_{4}(x)$ and
$\Psi_{n}^{2}\sim\frac{3}{2}q_{3}(x)+\frac{3}{2}q_{4}(x)$, which are related
to the elastic form factor $F_{1}^{N}$.

The elastic form factors in the LFHQCD model, Eq. (9), fall asymptotically as
$1/Q^{2\tau}$, and the slope of form factors as $Q^{2}=0$ increases with
increasing values of $\tau$. Thus an increase in the value of $\tau$
corresponds to an increase in effective size. Because $\tau$ represents the
number of constituents in a given Fock component of the hadron, the function
$q_{3}$ is naturally associated with the a three quark PLC system and $q_{4}$
with the BLC. This association is also consistent with the discussion
regarding the PLC dominating at high-$x$ as can be seen by analyzing
$g(x)\equiv q_{3}/q_{4}$, which approaches infinity as $x$ nears 1. See Ref.
Kim and Miller (2022) for a plot. The exact expression for $g(x)$ is obtained
from Eq. (14). A qualitatively accurate representation of $g(x)$ for values of
$x$ between 0.3 and 0.7 is given by $g(x)\approx 1/(1-x)^{4/3}.$ Lastly, the
expressions for the wave function tell us that the probability of the PLC,
$P_{PLC}$, for both nucleons is equal to 1/2, thus $\epsilon^{2}=1$.

Ref. Kim and Miller (2022) used holographic QCD to obtain modified nucleon
PDFs by using the consequences of nuclear medium breaking of SU(6) symmetry.
Motivated by neutron-proton dominance in SRCs, we expect the nuclear potential
to depend on whether one introduces a proton or neutron into the nucleus. For
example, we expect the proton to feel a stronger attraction to a nucleus if
there is an abundance of neutrons and vice versa. We apply medium effects by
introducing two free positively-valued parameters that depend on mass and
atomic numbers $A$ and $Z$, $\delta r_{p}(A,Z)$ and $\delta r_{n}(A,Z)$. The
$A$ and $Z$ dependencies in $\delta r_{p}$ and $\delta r_{p}$ are dropped to
simplify the notation, but are implied. The signs in front of $\delta r_{p}$
and $\delta r_{n}$ are motivated by the suppression of the PLC from the two-
component model. The above parameterization leads to a suppression of the PLC,
$q_{3}(x)$. The net result was

$\tilde{F}_{1}^{p}(Q^{2})=(1-\delta r_{p})F_{\tau=3}(Q^{2})+\delta
r_{p}F_{\tau=4}(Q^{2}),$ (16)

and

$\tilde{F}_{1}^{n}(Q^{2})=-\left(\frac{1}{2}+\delta
r_{n}\right)(F_{\tau=3}(Q^{2})-F_{\tau=4}(Q^{2})).$ (17)

The Pauli form factor $F_{2}$ is not modified in the model of Kim and Miller
(2022).

The essence of our procedure is the dynamical connection between the medium-
modifications, $\delta r_{p,n}$ and the underlying dynamics of the nucleon in
the medium as represented by the virtuality. The same model that describes the
EMC effect is used to compute the medium modifications of the form factors.
This is different than the work of Xing  _et al._ (2023) which obtained huge
suppression of the form factors. That work did not present a model of the
nucleon wave function in the medium. Instead they utilized the SRC model from
Ref. Schmookler  _et al._ (2019b), providing them with the ratio of SRC
nucleons to deuteron structure functions, and a light-front quark-diquark
AdS/QCD model Maji and Chakrabarti (2016) that is different from LFHQCD, which
gives them formulae connecting form factors and structure functions. They
multiplied the ratio obtained from the SRC model by free proton and neutron
structure functions from their AdS/QCD model. This gave them their results for
the structure functions of SRC nucleons. Then, they modified the parameters of
their AdS/QCD model to match the shape of the SRC structure functions, and
used those parameters to approximate the SRC form factors in the AdS/QCD
model.

Following the lengthy introduction, we now are in position to study form
factor modifications over a wide range of values of $Q^{2}$. The medium
modification ratios can now be obtained using Eq. (16) and Eq. (10), with the
result:

$\displaystyle{\tilde{F}_{1}^{p}(Q^{2})\over{F}_{1}^{p}(Q^{2})}=(1-\delta
r_{p})+{\delta r_{p}\over 1+{Q^{2}\over 5M_{\rho}^{2}}}$ (18)
$\displaystyle{\tilde{F}_{1}^{n}(Q^{2})\over{F}_{1}^{n}(Q^{2})}={1+2\delta
r_{n}}$ (19)

The values of $\delta r_{N}$ were obtained by describing EMC ratios and are
given in Table I. These values are to be considered as averages over the
nucleus. The medium modifications grow with increasing virtuality, which
corresponds to increasing values of missing momentum.

The relation between $\delta r_{N}$ and the virtuality is given by $\delta
r_{N}=\frac{1}{4}\frac{|U_{N}|}{\overline{\Delta}}=-\frac{M}{8}\frac{{\cal
V}_{N}}{\overline{\Delta}}$from Eq. (8). The quantity $\delta r_{N}$ is
proportional to the nuclear potential $|U_{N}|$, which is also proportional to
the virtuality. We can determine $|U_{N}|$ from our fitted $\delta r_{N}$
values, and noting that the energy difference between the ground state and the
first excited state in this model is $2V$ which is at least 500 MeV, the
energy of the Roper Resonance. For $A>4$, the values of $\delta r_{N}$ range
from 0.044 to 0.078, thus $80\,\,\mbox{MeV}\leq|U_{N}|\leq 156\,\,\mbox{MeV}$.
We decompose the nuclear potential into mean field and SRC contributions. This
is much larger than the shell model potential at the center of the nucleus
which is about 50 MeV for the absolute value of the mean field Lilley (2009).
These values of $|U_{N}|$ are consistent with the finding that both mean-field
and SRC effects contribute to the average virtuality, roughly in the ratio
$1:2$ Ciofi degli Atti  _et al._ (2007). Indeed both effects must be involved,
as noticed long ago Hen  _et al._ (2013). The fundamental forces that bind
nuclei involve two and three nucleons. The mean-field is obtained by averaging
the fundamental forces over densities. This is only an approximation. There
must be residual effects of the basic forces, those that cause short-range-
correlations.

It is immediately apparent from the small values of $\delta r_{N}$ in Table I
that the effects of the modifications of the elastic electromagentic form
factors are only a few percent. This is in contrast with the results of Xing
_et al._ (2023) which showed huge suppression factors. At this time, there are
no experimental limits on modification of neutron radii, although a method has
been suggested Cloët  _et al._ (2009). Thus we focus on the medium
modifications of the proton form factor, examining the limits at small and
large values of $Q^{2}$. We find that the ratio of the medium-modified mean
square radii to the free radii, $\tilde{R}_{p}^{2}/R_{p}^{2}$, from the
$F_{1}$ form factor is

$\displaystyle\tilde{R}_{p}^{2}/R_{p}^{2}=1+{3\over 20}\delta r_{p},$ (20)

a truly tiny effect. The limit of large values of $Q^{2}$ is simply

$\displaystyle{\tilde{F}_{1}^{p}(Q^{2})\over{F}_{1}^{p}(Q^{2})}=(1-\delta
r_{p}),$ (21)

which is no more than a few percent.

Nucleus | Ref. Kim and Miller (2022)
---|---
|  $\delta r_{p}$ |  $\delta r_{n}$
2H |  0.010 $\pm$ 0.003 |  0.010 $\pm$ 0.003
3He |  0.031 $\pm$ 0.003 |  0.061 $\pm$ 0.006
3H |  0.032 $\pm$ 0.006 |  0.016 $\pm$ 0.003
4He |  0.040 $\pm$ 0.004 |  0.040 $\pm$ 0.004
9Be |  0.044 $\pm$ 0.004 |  0.035 $\pm$ 0.003
12C |  0.049 $\pm$ 0.003 |  0.049 $\pm$ 0.003
27Al |  0.057 $\pm$ 0.003 |  0.053 $\pm$ 0.003
56Fe |  0.074 $\pm$ 0.003 |  0.064 $\pm$ 0.003
63Cu |  0.052 $\pm$ 0.003 |  0.044 $\pm$ 0.003
197Au |  0.072 $\pm$ 0.004 |  0.048 $\pm$ 0.003
208Pb |  0.078 $\pm$ 0.005 |  0.051 $\pm$ 0.003
Table 1: The $\delta r_{p}$ and $\delta r_{n}$ medium modifications from Ref.
Kim and Miller (2022).

The free and medium-modified elastic form factors $F_{1}$, as determined in
Kim and Miller (2022) are shown in Fig. 2 for a Pb nucleus where the EMC
effect is largest, providing an upper limit within the model. The influence of
the nuclear medium is shown to be very modest indeed, in accord with Eq. (18)
and produces effects that do not violate the results of any existing
measurements. This is because the probability of the excited state components
are proportional to the square of the size of the EMC effect. Since the EMC
effect size is of order 10%, the probability of the excited state components
is only of order 1%.

Figure 2: (color online) (top) Results for the $F_{1}$ elastic form factor of
a free (blue) and medium modified (orange) proton in a Pb nucleus from Ref.
Kim and Miller (2022). The bottom plot presents the ratio of the medium
modified $F_{1}$ form factor to the free one. Figure 3: (color online) (top)
The ratio ${\tilde{F}^{p}_{1}\over F^{p}_{1}}$ for a fixed value of
$Q^{2}=5M_{\rho}^{2}=3$ GeV2 as a function of virtuality in units of $M^{2}$.
If the scattering is from a correlated pair, ${\cal V}\approx-2{\bf p}^{2}$.

We also note that the size of the medium modification increases with
virtuality, as shown in Fig. 3 which shows the ratio of medium-modified to
free form factors is as a function of virtuality for a fixed value of
$Q^{2}=5M_{\rho}^{2}=3$ GeV2. Next turn to the medium modifications of the
nucleon structure functions. Ref. Kim and Miller (2022) found the EMC ratio,
$R^{EMC}(A,Z)$, for nuclear structure function relative to deuterium, for a
nucleus of mass and atomic numbers A and Z to be

$\displaystyle R^{EMC}(A,Z)=\frac{2}{A}\frac{F_{2}^{A}}{F_{2}^{d}}$
$\displaystyle=\frac{2}{A}\frac{ZF_{2}^{p}+NF_{2}^{n}+\frac{5x}{3}(Z\delta
r_{p}+N\delta r_{n})(q_{4}-q_{3})}{F_{2}^{p}+F_{2}^{n}+\frac{10x}{3}\delta
r(^{2}H)(q_{4}-q_{3})}.$ (22)

We make a simple analysis to bring out the essential features of this result.
Take $N=Z$ so that $\delta r_{p}=\delta r_{n}=\delta r$ and also neglect the
small effects of $\delta r(^{2}H)$. Then, with a bit of algebra, one finds
that

$\displaystyle R^{EMC}(A,Z)=1+{5\over 9}\delta r{q_{4}(x)-q_{3}(x)\over
q_{4}(x)+q_{3}(x)}$ $\displaystyle\approx 1+{5\over 9}\delta r(0.395-1.53x),$
(23)

with the last expression being an accurate representation of the ratio for
values $0.3\leq x\leq 0.7.$ Comparing Eq. (20) and Eq. (21) with Eq. (23) it
is evident that medium modification effects on elastic electromagnetic form
factors and quark distribution functions are of about the same size. Thus,
large modifications of the electromagnetic form factors are not required or
even implied by an SRC-caused, virtuality-dependent EMC effect.

This paper shows that one can construct a model in which the medium
modification of elastic electromagnetic form factors is about the same size as
the effects that cause the EMC effect even though SRC are the origin. This
contradicts the claims of Ref. Xing  _et al._ (2023), so that the causal
connection between SRC and the EMC effect is alive and well. In any case, it
would be very useful to obtain a direct experimental measurement of medium-
modified form factors. It would be especially useful for such measurements to
be made at kinematics where the nucleon virtuality is large, because all
current measurements correspond to small values of the virtuality. A
discussion of different experimental approaches is contained in Klimenko  _et
al._ (2006); Wang  _et al._ (2019), see also Melnitchouk  _et al._ (1997).

Acknowledgments DNK and GAM would like to thank Guy F. de Téramond for
pointing out an error in a previous version of this paper. The work of DNK and
GAM was supported by the U.S. Department of Energy Office of Science, Office
of Nuclear Physics under Award No. DE- FG02-97ER41014. The research of MS was
supported by the US Department of Energy Office of Science, Office of Nuclear
Physics under Award No. DE-FG02-93ER40771. The work of LW was supported by the
U.S. Department of Energy Office of Science, Office of Nuclear Physics under
Award No. DE- FG02-96ER40960.

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