# Progress in understanding short-range structure in nuclei: an experimental
perspective

John Arrington  1 Nadia Fomin  2 and Axel Schmidt3 1Lawrence Berkeley National
Laboratory, Berkeley, CA94720; email<EMAIL_ADDRESS>2Department of
Physics and Astronomy, University of Tennessee, Knoxville, TN, 37996; email:
<EMAIL_ADDRESS>3Department of Physics, The George Washington University,
Washington, DC, 20052

###### Abstract

High-energy electron scattering is a clean and precise probe for measurements
of hadronic and nuclear structure, with a key role in understanding the role
of high-momentum nucleons (and quarks) in nuclei. Jefferson Lab has
dramatically expanded our understanding of the high-momentum nucleons
generated by short-range correlations, providing sufficient insight to model
much of their impact on nuclear structure in neutron stars, and in low- to
medium-energy scattering observables including neutrino oscillation
measurements. These short-range correlations also appear to be related to the
modification of the quark distributions in nuclei, and efforts to improve our
understanding of the internal structure of these short-distance and high-
momentum configurations in nuclei will provide important input on a wide range
of high-energy observables.

###### doi:

10.1146/annurev-nucl-102020-022253

###### keywords:

high-energy electron nucleus scattering, short-range nucleon-nucleon
correlations, deep inelastic electron scattering, nuclear structure functions,
super-fast quarks

###### Contents

  1. 1 Introduction
  2. 2 Measurements of high-momentum nucleons and SRCs
    1. 2.1 Challenges in isolating/measuring high-momentum nucleon
    2. 2.2 Experimental Signatures of SRCs
    3. 2.3 Nuclear Dependence of SRCs
  3. 3 Internal structure of SRCs
    1. 3.1 Measurements of center-of-mass motion of SRCs in nuclei
    2. 3.2 Isospin structure of SRCs
      1. 3.2.1 Two-nucleon knockout measurements
      2. 3.2.2 Single nucleon knockout
      3. 3.2.3 Inclusive studies on non-isoscalar nuclei
    3. 3.3 SRC internal momentum distributions
      1. 3.3.1 Generalized Contact Formalism
    4. 3.4 Impact of SRCs on other fields of physics
    5. 3.5 Structure of the nucleons within SRCs
      1. 3.5.1 EMC-SRC correlation
      2. 3.5.2 Potential implication for flavor dependence of the EMC effect
      3. 3.5.3 Impact on other physics
  4. 4 Three-nucleon short-range correlations
  5. 5 Key questions, future directions
    1. 5.1 Summary of our understanding of SRCs and remaining questions
    2. 5.2 Extensions of the current experimental program
      1. 5.2.1 2N-SRC and EMC effect measurements in new nuclei
      2. 5.2.2 3N-SRCs
      3. 5.2.3 2N-knockout
    3. 5.3 New directions: partonic structure and non-nucleonic degrees of freedom
    4. 5.4 New directions: Flavor-dependent EMC effect
    5. 5.5 New directions: Tagged measurements of SRCs
    6. 5.6 Summary and Conclusions

##  1 Introduction

The structure of atomic nuclei is well described by mean-field calculations,
which can reproduce a wide range of static properties and low-energy
scattering observables over the full range of the periodic table. For some
observables at higher energies scales, the contribution of nucleons with
momenta above the Fermi momentum, $k_{F}$, become more important and it has
been a long-time goal of higher energy nuclear structure experiments to
isolate and study these contributions. While the contribution from nucleons
above $k_{F}$ to the momentum distribution of nucleons in nuclei is low, the
fact that they are at large momenta and, as we will see below, are associated
with large excitation of the residual (A-1) nucleus, make them an important
piece of nuclear structure and give them a dominant role in certain high-
energy scattering observables.

These high-momentum nucleons are generated by the short-distance part of the
nucleon-nucleon (NN) potential - the strong tensor attractive component for
separations near 1 fm, and the repulsive core at shorter distances. Hard NN
interactions at short distances generate pairs of nucleons with large relative
momenta, but small total momentum, known as short-range correlations (SRCs).
This leads to a picture of the nucleus where nucleons with momenta below the
Fermi momenta are dominated by the mean-field structure of complex many-body
nuclei, while structure at larger momenta is dominated by the two-body physics
associated with the short-distance NN interaction.

Because of the underlying two-body physics, these contributions are similar in
scattering from heavy nuclei or scattering from the deuteron. In principle,
this makes is possible to combine information from a variety of different
observables in different nuclei to provide an improved understanding of the
nature of SRCs, and to map out their strength and structure in light and heavy
nuclei.

Electron scattering provides a clean and powerful probe of nuclear structure,
due to the combination of the point-like nature of the probe, the well
understood interaction via virtual photon exchange, and the fact that the
electromagnetic interaction that allows a uniform probe of the entire nuclear
volume. However, several complications arise when trying to isolate the high-
momentum components of the nuclear wave-function, and our understanding of the
structure of SRCs comes from a combination of different measurements, each
yielding unique but limited information, but together providing an ever
clearer picture of these important but poorly understood component of nuclear
structure.

For the early theoretical foundations for SRCs, see Refs. [1, 2]. Additional
examinations of the experimental considerations for probing SRCs are presented
in [3, 4]. More recent reviews include [4, 5, 6], but significant experimental
progress has been made since these works were published. In this review, we
will provide a brief overview of early studies, a more complete view of recent
experimental progress and efforts to tie together information from different
experiments. We will also highlight key remaining questions and future
experimental work.

##  2 Measurements of high-momentum nucleons and SRCs

One important consideration is the fact the the momentum distribution of
nucleons in a nucleus is not an experimental observable. Most experiments work
in the framework of the Plane-Wave Impulse Approximation (PWIA), where the
A(e,e’p) cross section is connected to the spectral function, which accounts
for both the momentum and energy distribution of the nucleons. In this case,
the spectral function (or momentum distribution) is directly related to the
cross section in the PWIA assumption, so direct comparison of the cross
section with calculations based on a momentum distribution of spectral
function are possible. This is the approximation we make when we discuss using
the cross section data to study the momentum distribution or spectral function
of nuclei.

We begin by highlighting early attempts to probe the distribution of nucleons
at large momenta using A(e,e’p) measurements and note some of the issues with
interpreting these data. We then summarize how SRCs were isolated in inclusive
scattering, with a handful of precision measurements mapping out the relative
contribution of SRCs in nuclei. The following section 3 will discuss studies
of the momentum distribution of SRCs in nuclei, as well s their internal
isospin and momentum structure.

###  2.1 Challenges in isolating/measuring high-momentum nucleon

In the Plane-Wave Impulse Approximations (PWIA), one can treat the A(e,e’p)
single-nucleon knockout reaction as billiard-ball scattering from a single
proton with (A-1) spectator nucleons. Knowing the electron beam energy and
detecting the scattered electron and knocked-out proton, there is enough
information to reconstruct the initial momentum (and energy) of the struck
proton, referred to as the missing momentum ($p_{m}$) and missing energy
($E_{m}$). In principle, this should allow extraction of the proton
distribution in the nucleus, but effects beyond the PWIA make this a
significant challenge.

The potential to have large contributions from meson-exchange currents (MEC)
and final-state interactions (FSIs) make probing the momentum distribution
difficult, especially for large values of missing momentum, even for light
nuclei [3, 4]. Proton knockout measurements from the deuteron, measured at
Jefferson Lab over a wide kinematic range [7, 8], have been used to examine
final-state interactions, with comparisons to the Generalized Eikonal
Approximation [9] showing that this formalism provides a reasonable
description of FSIs over a large kinematic range. However, recent measurements
have pushed such studies to higher four-momentum transfer, $Q^{2}$, and larger
$p_{m}$, focusing on regions of minimal FSIs, and observed that none of the
calculations reproduce the data above $p_{m}=700$ MeV/c. Because of this, and
the lack of high-precision systematic studies, it is not yet clear to what
degree the FSI corrections can be minimized or corrected for. Because of this,
SRC studies have generally moved away from A(e,e’N) measurements, with the
exception of comparisons of proton and neutron knockout, where these effects
are assumed to have significant cancellation. These will be discussed in more
detail in Section 3.2.1 where we discuss progress in exclusive studies.

Inclusive scattering has also been used to try to study the momentum
distribution of nucleons. The shape of the quasielastic (QE) is driven by the
e-p and e-n elastic cross sections smeared by the nucleon momentum
distributions [10], with scattering below the kinematic threshold for
stationary nucleon, allowing for clean separation of QE from inelastic
scattering. Under certain assumptions, $y$ scaling [11, 12, 10] provides a
connection between the QE cross section and the underlying momentum
distribution. However, this relies on the PWIA, and MECs and FSIs can yield
large corrections at low $Q^{2}$. It also assumes a final spectator (A-1)
nucleus in an unexcited state, which is a poor approximation for scattering
from an SRC within a nucleus. Model-dependent corrections (e.g. $y_{CW}$ or
$y*$ scaling [13, 14]) have been proposed to account for this, but it is
difficult to quantify the uncertainty in such an approach. We note, however,
that in QE scattering from the deuteron at high $Q^{2}$, the $y$-scaling
assumptions appear to be reasonable, and momentum distributions extracted from
such data are in good agreement with momentum distributions calculated from
realistic NN potentials, as shown in [15].

###  2.2 Experimental Signatures of SRCs

As noted in the previous section, the shape of the QE peak is driven by the
distribution of protons and neutron in the nucleus. One can decompose the
cross section into contributions based on the structure (in particular high-
momentum nucleons) in the breakup of the nucleus. In the naive SRC model [16],
the cross section is separated into scattering from single nucleons in the
nucleus with the spectator in an unexcited or minimally-excited states,
scattering from two-nucleon SRCs where the final state is dominated by the
high-momentum spectator and an (A-2) residual nucleus, and contributions from
multi-nucleon SRCs where multiple nucleons in the initial state have large
momenta. The bulk of the cross section comes from the first term, which
represents the shell structure of the nucleus. Contributions from SRCs are
assumed to be significantly smaller, with the contributions falling as the
number of high-momentum nucleons increases. Therefore, initial studies of SRCs
assumed that only 2N-SRCs would have significant contributions, and made
measurements focused on scattering from high-momentum nucleons ($k>k_{Fermi}$)
to suppress the contributions from the shell-model structure.

Figure 1: (Left) Minimum nucleon momentum for quasielastic scattering from a
nucleon as a function of $x$ and $Q^{2}$ (Right) Fe/D inclusive cross section
ratios (per nucleon) from SLAC over a range of $Q^{2}$ values; figure adapted
from Ref. [16].

Electron scattering from a stationary proton is kinematically forbidden for
energy transfer, $\nu$, below that for elastic e-p scattering, corresponding
to $x=Q^{2}/(2M\nu)=4$. In a nucleus, scattering at $x>1$ is can occur because
of the motion of the nucleon in the nucleus, with larger values of $x$
corresponding to larger initial nucleon momenta. Therefore, by selecting
scattering at $x>1$, one can set a minimum initial nucleon momentum that
depends on $x$ and $Q^{2}$, as shown in the left plot of Fig. 1. Measurements
at modest-to-large $Q^{2}$ values ($>$1-2 GeV2) will minimize final state
interactions, while the low energy transfer required to reach large $x$
suppresses inelastic scattering. Thus, it was predicted that it should be
possible to isolate scattering from the high-momentum nucleons in SRCs by
requiring that $x$ and $Q^{2}$ be large enough that scattering from nucleons
below the Fermi momentum is forbidden [1, 2].

In this region, the inclusive scattering from any nucleus is driven by
scattering from 2N-SRCs, generated by the same underlying two-body physics.
The scattering in this region should therefore show a universal behavior,
yielding a nuclear cross section ratio in the SRC-dominated region that is
independent of both $x$ and $Q^{2}$. The first extensive set of data to
examine this prediction was from SLAC [16], which found a plateau in the A/D
cross section ratio for $x>1.4$ and $Q^{2}>1.4$ GeV2 for 3He, 4He, 12C, 27Al,
56Fe, and 197Au. The results for the Fe/D ratios are shown for a range of
$Q^{2}$ values the right panel of Fig. 1. High-momentum nucleons
($k>k_{Fermi}$) are accessible at lower values of $x$ as $Q^{2}$ increases, as
seen in the right panel of Fig. 1, yielding a larger plateau region in the
high-$Q^{2}$ A/D ratios. These data confirmed the prediction of identical
behavior in light and heavy nuclei as a function of $x$ and $Q^{2}$ for
measurements made in the SRC-dominated region, supporting the naive SRC model.
Taking the cross section for scattering from SRCs in heavy nuclei to be
identical to scattering from a deuteron, the quantity $a_{2}$ \- the A/D ratio
in the plateau region - was taken as an approximate measure of the relative
contribution of SRCs relative to that in the deuteron.

Figure 2: 12C/D per-nucleon cross section ratios from [15] plotted as a
function of $x$ (left panel) and $\alpha_{2N}$ (right panel). $Q^{2}$ values
for $x=1$ for the different data sets are shown in the left plot; the data is
nearly independent of $Q^{2}$ as a function of $\alpha_{2N}$.

At very large $Q^{2}$, $x$ can be directly equated with a specific minimum
nucleon momentum, while going to lower $Q^{2}$ yields a reduction in the range
of nucleon momenta associated with the fixed value of $x$. Thus, the use of
$x$ as a proxy for the initial nucleon momentum an approximation at lower
$Q^{2}$ values, yielding a $Q^{2}$ dependence in the region where the
scattering is not dominated by SRCs. This $Q^{2}$ dependence can be largely
removed by using the light-cone variable $\alpha_{i}$, rather than $x$, as a
surrogate for nucleon momentum. The light-cone momentum fraction $\alpha_{i}$
is analogous, but is calculated for a bound nucleon in a nucleus, representing
fraction of the total nuclear momentum, carried by the nucleon on the light-
cone. The general form is $\alpha_{i}=\frac{p_{i\\_}}{p_{A\\_}/A}$, where
$p_{i}\\_$ is the longitudinal component of the momentum on the light-cone for
the bound nucleon, and $p_{A}\\_$ is the same for the nucleus. In the case of
a 2N-SRC, the expression becomes:

$\alpha_{2N}=2-\frac{q_{min}+2M}{2M}\left(1+\frac{\sqrt{W^{2}-4M^{2}}}{W}\right)~{}~{},$
(1)

where $q_{min}=\nu-|\vec{q}|$, $W^{2}=\sqrt{-q^{2}+4M\nu+4M^{2}}$, and $M$ is
the proton mass. When the 2N-SRC cross-section ratios are examined as a
function of $\alpha_{2N}$, rather than $x$, as shown in Fig. 2, the $Q^{2}$
dependence largely removed, and the onset of scaling is more consistent. Note
that for $x\approx 1$, the larger values of $Q^{2}$ show deviations from
scaling even when shown vs $\alpha_{2N}$, as the large inelastic contributions
begin to become significant in the cross section. Because the larger smearing
in heavy nuclei increases the contribution from the inelastic, the A/D ratio
increases for $x\approx 1$ and large $Q^{2}$, leading to the increase in the
ratio.

Once scaling of the cross-sections is observed, there are a couple of
lingering obstacles to address in attempting to quantify the contribution of
SRCs in these nuclei. A common concern is the effect of FSIs, which, despite
falling off quickly with increasing $Q^{2}$ can have a lingering effect. Most
calculations agree that the FSIs at large $Q^{2}$ are limited to interactions
between the nucleons in the SRC pair, and as such, cancel in the cross section
ratios at $x>1$ [1, 2, 16, 17, 18, 4], and this assumption is applied
universally in the interpretation of the inclusive ratios in terms of SRC
contributions. High-precision $x>1$ inclusive deuteron data from Hall C (Ref.
[15]) were used to extract the underlying momentum distribution at several
kinematic settings based on the $y$-scaling approach which assumes no FSI. The
results across settings were consistent, showing no $Q^{2}$ dependence in the
extracted momentum distributions, even in the SRC-dominated region. In the
presence of large FSIs for $x>1.4$, the momentum distribution would be
distorted, which is not observed. While this supports the idea the effect of
FSIs is small, it is difficult to set a precise limit given, among other
things, the uncertainty in the strength of the momentum distribution for
$k>k_{F}$.

Having established that the inclusive ratios can isolate and quantify the
presence of SRCs in nuclei, later Jefferson Lab experiments compared heavy
nuclei to 3He in the SRC region [19, 20], and made precision measurements of
A/D ratios in SRC kinematics for a range of light and heavy nuclei [15, 21].
In the naive SRC model, these measurements directly provide a relative measure
of the cross section in the SRC-dominated region, and allow for a comparison
of the strength of SRCs in various nuclei. However, there are corrections to
the naive SRC model which have to be addressed for a quantitative
interpretation of SRCs, as described in the next subsection.

###  2.3 Nuclear Dependence of SRCs

Scaling of inclusive A/D cross-section ratios at $x>1$ [16, 15, 21] was taken
to be the signature of the presence of high-momentum nucleons, born in SRCs.
This ratio, taken in the scaling region (which depends on kinematics), is
referred to as $a_{2}$. However, $a_{2}$ was also extracted from A/3He ratios
[19, 20], combined with ${}^{3}He/D$ data and/or calculations to extract the
contribution of SRCs relative to the deuteron. The left plot in Fig. 3 shows
the A/D ratios for a wide range of nuclei from Ref. [15]. One can see that the
scaling sets in earlier for light nuclei, while heavier nuclei require larger
$x$ to isolate nucleons with momenta above the larger Fermi momentum
associated with heavy nuclei.

The extracted values of $a_{2}$ from all inclusive measurements are shown in
the right panel of Fig. 3, with the A/3He ratios converted to A/2H using
3He/2H=2.12$\pm$0.06 - the average of refs. [16, 15]. While there are some
systematic disagreements between different data sets, they do not appear to be
significant given the sizes of the uncertainties, especially given that there
will be a scale uncertainty on all measurements from a given data set
associated with the use of common set of deuterium data (or 3He for [20].
Early works assumed that $a_{2}$ would scale with the average nuclear density
(or A-1/3 as an approximation to nuclear density), but the results of Ref.
[15] showed that 9Be was an outlier [22] from this model, and the the details
of the nuclear structure are important. For heavier, the ratio is
approximately consistent, consistent with the idea of an effect that saturates
in heavy nuclei with roughly 5 ’nearest neighbors’ able to generate SRCs with
any given nucleon (implying roughly 10 ’nearest neighbors’, only half of which
are available to form SRCs assuming np dominance). However, because the
heavier nuclei have significant neutron excess, it is difficult to cleanly
separate A dependence from isospin effects for medium-to-heavy mass nuclei.

Figure 3: (Left) Cross section ratios from JLab E02-019 [15] (Right) Extracted
values of $a_{2}$ vs A from all measurements; some values offset slightly in A
to make the results more visible. Note that smearing corrections have been
estimates to decrease the relative number of SRCs from the extracted $a_{2}$
value by about 10% for 3He, and 20% for A $\geq 12$.

Some inclusive SRC ratio measurements [19, 20] assumed equal probability for
all isospin SRC pairs (nn, np, pp). This meant that those data for non-
isoscalar nuclei were corrected for the $a_{2}$ determination. While heavier
nuclei also have pp- and nn-SRCs, their contributions are significantly
smaller, as discussed in Sec. 3.2. Therefore, the data presented in Fig. 3 are
shown without the isoscalar corrections to the cross sections that applied in
some workds to try to account for pp and nn contributions [19]. It is worth
noting that the momentum distribution for nucleons in np-SRCs is very
different from pp-SRCs [23], meaning that this becomes a more important issue
when trying to make quantitative comparisons of inclusive scattering, which
integrates over a range of initial nucleon momenta, and exclusive reactions
which measure the ratio as a function of initial momentum. This will be
discussed in more detail in Secs. 3.2.1 and 3.2.2.

There has been significant theory work trying to model the contribution of
2N-SRCs as a function of A, and trying to estimate the relative contributions
of np-SRC and pp(nn)-SRCs. A naive starting point is the total number of np
pairs ($N\times Z$) or pp pairs ($Z\times(Z-1)/2$), but the ratio in the high-
momentum tails does have to be proportional to the overall number of np and pp
pairs, as the tensor interaction generates hard interactions between nucleons
in isospin zero states [23, 24, 25, 26] (as discussed in detail Sec. 3.2).
Thus, associating SRCs with pairs of specific spin, isospin states provides a
better approximation to the data [27]. These simple scaling models can be
compared to values of $a_{2}$ extracted [23, 28, 29] based on comparing high-
momentum (or short-distance) parts of realistic nuclear structure calculations
[23, 30, 26, 27, 31, 32, 33]. Such approaches yield qualitatively consistent
results: a significant dominance of np-SRCs or pp- and nn-SRCs, and a weak
dependence of the total contribution and np/pp ratio in $A\geq 12$ Nuclei.
Note that while we focus on an interpretation of SRCs and their isospin
structure at the hadronic level, based on the NN interaction, a quark-level
model explains np dominance by associating SRCs with diquark correlations
between neighboring nucleons [34], with the ratio of pp- to np-SRCs depending
on the contributions of 3-quark and diquark-quark contributions in the
nucleon.

A more quantitative examination of the A dependence requires going beyond the
assumptions of the naive SRC model, as there are a number of effects that may
make the A/D cross-section ratios ($a_{2}$) differ from the relative number of
2N-SRC pairs. The general argument for inclusive measurements is that the
cross-section strength at $x>1$ is due to the presence of high-momentum
nucleons from short-range interactions. That means that the high-momentum tail
of the momentum distribution (which is probed at $x>1$) should look similar
across nuclei, with $A>2$ tails looking approximately like rescaled $A=2$
tails. This leads to the expectation that the signature of 2N SRCs will be a
plateau in the cross-section ratios. This is what is observed in the data, as
Fig. 3 shows. However, there are small deviations from a perfectly flat
plateau, such as a small increase as one approaches $x=2$, that are more
visible in heavier nuclei. There are a couple of mechanisms at play here. One
has to do with the motion of the SRC pair in the field of the other nucleons
(discussed below). Another possibility is that at larger $x$ there may begin
to be contributions 3N-SRCs, in the region where the 2N-SRC contributions are
dropping off. The 3N-SRC contribution, relative to 2N-SRCs, is expected to
grow with the size of the nucleus [35], leading to increasingly imperfect
scaling ratios.

A major correction comes from the center-of-mass (CM) motion of the 2N-SRC
pair in $A>2$ nuclei. This modifies the shape of the momentum distribution and
redistributes strength (from the quasielastic peak to the high-momentum tail).
The first data to be corrected for this effect were those of Ref. [15]. The
authors calculated the effect of this smearing using parametrizations of the
$F(y)$ scaling function convolved with realistic parametrizations for the CM
motion from [18]. The correction was on the order of 20$\%$ for most nuclei,
with a low value of 10$\%$ for 3He. A more recent work [36] examines the same
effect in relativistic and non-relativistic convolution models, using the
Generalized Contact Formalism (GCF) to provide the structure of the SRCs to be
smeared, and also examining the impact of having a larger excitation of the
residual (A-2) system. However, Ref. [36] does not provide direct comparisons
of calculations with and without SRC smearing or (A-2) binding, making it
difficult to determine the size of the individual effects.

For the rest of this work, we will typically refer to $a_{2}$ as a measure of
the relative contribution of SRCs, consistent with most of the literature,
although efforts are ongoing to better quantify the corrections between the
cross section ratios and the relative SRC contributions.

##  3 Internal structure of SRCs

In addition to mapping out the size of the SRC contributions in nuclei, a
variety of experimental studies have focused on obtaining a more detailed
understanding of the structure of SRCs in nuclei. This addresses some of the
questions raised by the initial inclusive measurements: what is the isospin
structure of SRCs, what is the momentum distribution of SRCs in nuclei, and
what is the distribution of relative momenta within SRCs? Finally, as SRCs
involve large energy scales arising from finite objects interaction strongly
at short distances, is the internal structure of nucleons modified by being in
these dense, energetic configurations?

While the inclusive measurements were able to confirm SRC contributions by
identifying the universal behavior in all nuclei in the $x,Q^{2}$ range
predicted by the SRC model, the earlier measurements were sensitive mainly to
the overall contribution from SRCs. They did not provide information on the
isospin structure of the SRCs - the relative contribution of pp-, np-, and nn-
SRCs - and did not measure the momentum of SRCs in nuclei. In both cases,
corrections had to be applied based on assumptions or measurements of the
isospin structure and momentum distribution of SRCs. Below, we discuss our
understanding of these effects and their experimental and theoretical support.

###  3.1 Measurements of center-of-mass motion of SRCs in nuclei

Theoretical predictions and experimental results suggest that typical momenta
of the center-of-mass (CM) of SRC pairs is small, i.e., on the order of mean-
field momenta [26]. Nevertheless, determining the center-of-mass motion
distribution of pairs has important implications. As noted in section 2.3, the
inclusive measurements have to apply a correction for the motion of the SRCs
inside of the nucleus to extract the relative SRC contributions in different
nuclei. Furthermore, the CM momentum distribution may shed some light on how
short-range correlations form. So far, experiments have aimed to quantify the
width of the CM momentum distribution, and have assumed a Gaussian shape,
which is broadly consistent with the data given the available statistics and
resolution. For what follows, we use the following convention: for two
nucleons in an SRC pair with momenta $\vec{p}_{1}$ and $\vec{p}_{2}$, the
center-of-mass momentum is defined by
$\vec{p}_{CM}\equiv\vec{p}_{1}+\vec{p}_{2}$, while the relative momentum is
$\vec{p}_{rel.}\equiv\frac{1}{2}(\vec{p}_{1}-\vec{p}_{2})$.

Experimental determinations have been made through two-nucleon knockout
reactions. One of the earliest such measurements was performed by colliding a
6–9 GeV proton beam from a fixed carbon target in the Brookhaven EVA
spectrometer and identifying SRC break-up events from the 12C(p,ppn) reaction
[37]. Guided by theoretical indications that the transverse directions would
be far more sensitive to FSIs, Tang et al., measured the longitudinal
component of the CM distribution to have a Gaussian width of $143\pm 17$
MeV$/c$, i.e., somewhat smaller than the Fermi momentum. Electron scattering
experiments using the spectrometers in Jefferson Lab’s Hall A measured a width
of $136\pm 20$ MeV$/c$ in the (e,e’pp) channel for carbon [38] and a smaller
width for helium-4 in the (e,e’pn) channel of $100\pm 20$ MeV$/c$ [39]. In
both cases, the small acceptances of the spectrometers required the CM width
be inferred from the measured distribution using a simulation in which the
electron scatters from a nucleon in a moving pair. The result was obtained by
matching the data and simulated distributions.

Measurements on a number of different nuclei have been made at CLAS, where the
large acceptance is advantageous for reconstructing a multi-nucleon final
state. Refs. [40, 41] reported measurements of 3He(e,pp)n and used this to
study both the relative and CM momentum of the nucleons in SRCs. Ref. [42]
made similar measurements for additional nuclei, finding that pp-SRCs could be
well described as having Gaussian CM momentum distributions with widths
ranging between $143\pm 7$ MeV/c for 12C to $157\pm 14$ MeV/c for 208Pb. These
results were generally consistent with the widths assumed based on mean-field
momenta of the nuclei. A summary of these results is shown in the left panel
of Fig. 4.

Figure 4: (Left) pp-SRC CM momentum distributions from Ref. [42] in the
direction transverse to $p_{m}$ in the x (red) and y (blue) directions, before
correcting for the CLAS acceptance. (Right) Extraction of the gaussian width
associated with SRC motion in nuclei from ref. [42] compared to calculations
from Refs. [18, 27, 43]; figure adapted from Ref. [42].

###  3.2 Isospin structure of SRCs

As we discuss in the following sections, a series of measurements have made it
clear that 2N-SRCs are dominated by np-SRCs, meaning that the isospin
structure of SRCs is nearly identical for all nuclei, and the scattering from
an SRC is roughly proportional to the sum of the e-p and e-n cross sections.

While early inclusive measurements neglected the isospin structure of SRCs,
the comparison of heavy nuclei to the deuteron will be impacted by the
presence of pp- and nn-SRCs, not possible in the deuteron. As noted in Sec.
2.3, some measurements applied corrections to the cross section per nucleon to
correct for the difference between the elastic e-p and e-n cross sections.

Multiple approaches have been used to study the isospin structure of SRCs. As
noted in sec. 2, A(e,e’p) measurements are challenging to interpret at high
missing momenta. The first measurements examining the isospin structure used
detection of both the struck nucleon from the SRC and it’s spectator partner.
While this approach has to worry about large FSI corrections, certain
contributions will cancel in the comparison of np and pp final states.
Additional studies using A(e,e’p)/A(e,e’n) at large and small missing momenta
and recent inclusive studies making use of the isospin structure of the target
support the picture of np dominance, with each having it’s own advantages and
limitations.

####  3.2.1 Two-nucleon knockout measurements

Information about the isospin structure of short-range correlations can be
directly accessed in measurements in which both correlated nucleons are
detected. Such measurements face a couple of experimental challenges. First,
since the detection of neutrons is accomplished differently than that of
protons, care must be taken to control the relative acceptance and efficiency
between the two particles. Second, as with all measurements with a nucleon in
the final state, final-state interactions are important. For these
measurements, the results are particularly sensitive to charge-exchange
reactions (where an outgoing proton knocks out a spectator neutron or vice
versa) or scattering from a low-momentum nucleon which rescatters from a
spectator nucleon, generating a final state that’s identical to scattering
from a pre-existing SRC. Different experiments have used a variety of
strategies to negotiate these challenges, and while these issues impact the
quantitative interpretation, the body of experimental data points a clear
picture of dominance of np-SRCs in the momentum range of $\approx$ 300–600
MeV/c. The evolution of isospin structure at higher momenta is an area of
active research.

The first estimate of the isospin structure of SRC pairs was made using
12C(p,ppn) and 12C(p,pp) data collected by the EVA Spectrometer at Brookhaven,
finding that correlated protons were accompanied by the emission of back-to-
back neutron $92_{-18}^{+8}\%$ of the time [44]. This was confirmed by an
electron-scattering experiment in Jefferson Lab Hall A, in which high
resolution spectrometers were used to detect the scattered electron as well as
a knocked-out proton in high $p_{\text{miss}}$ kinematics, while a third
spectrometer was used to look for recoiling protons and neutrons. The pp pair
fraction was inferred from both the 12C(e,e’pp)/12C(e,e’p) [38] and
12C(e,e’pp)/12C(e,e’pn) ratios [45]. The results of these measurements are
shown in the left panel of Fig. 5. A follow-up measurement confirmed np
dominance for the 4He nucleus [39].

An analysis from CLAS considered np-dominance in non-isoscalar nuclei [46].
Since CLAS had limited capabilities for neutron detection, inferences were
drawn from a measurement of the A(e,e’pp)/A(e,e’p) ratio. To compensate for
the number of undetected recoil protons, double ratios were formed relative to
carbon. The results showed that pp pairing is as prevalent in aluminum, iron,
and lead as it is in carbon, i.e., no significant decrease increasing neutron
excess. This suggests that in neutron-rich nuclei, protons have larger average
kinetic energy. A follow up analysis that in which $A(e,e^{\prime}np)$ events
were identified via neutron hits in the CLAS electromagnetic calorimeter had
consistent findings [47]. Note that the FSI corrections applied in this work
were substantially larger than applied in the initial measurement at JLab
[38], yielding a result $\sim$70% larger than the initial work. The right
panel of Fig. 5 shows the ratio of np-SRCs to pp-SRCs extracted from all
direct two-nucleon knockout measurements.

The measurements described above have looked at scattering from nucleons in
correlated pairs, with the possible additional detection of a spectator. By
contrast, an analysis of 3He(e,e’pp)n data at CLAS was performed in kinematics
where the spectator pair is correlated, while the struck nucleon is not [41].
Events with pp and np pairs were identified by relative energy transfer among
the detected protons and undetected neutrons. This analysis confirmed the
relative scarcity of pp pairs while also observing that the effect diminishes
for large pair CM momentum [41]. It is not clear whether this is a result of
the details of the $A=3$ wave function, caused by final-state effects, or
produced by some other phenomenon.

Figure 5: (Left) The pp-SRC and np-SRC fractions as a function of missing
momentum from Refs. [44, 45, 38, 47]; figure adapted from Ref. [45]. (Right)
np-SRC/pp-SRC ratio from direct measurements using two-nucleon knockout. The
blue band shows the one-sigma range from the the average of all measurements
excluding Ref. [45] (hollow circle) which significantly underestimated [47]
the FSI corrections.

####  3.2.2 Single nucleon knockout

One can also attempt to study the isospin structure of SRCs using single
nucleon knockout. One technique is to compare relative rates of proton and
neutron knockout in high $p_{\text{miss}}$ kinematics. A CLAS analysis of
(e,e’p) and (e,e’n) in both correlated and in mean-field kinematics was
undertaken to study the asymmetry dependence of pairing [48]. It was found
that while the (e,e’n)/(e,e’p) ratio in mean-field kinematics scales roughly
with N/Z (after correcting for the difference between the elastic e-p and e-n
cross sections), this ratio is approximately unity in SRC kinematics, a clear
sign of np dominance persisting in asymmetric nuclei. The relative abundance
of nucleons in correlated and mean-field states, i.e., (e,e’N)SRC/(e,e’N)MF,
was also studied for both protons and neutrons. Since the relative acceptance
for the two kinematics was very different, a double-ratio relative to carbon
was used. It was found that the fraction of correlated protons steadily
increases with N/Z, while the double ratio remains flat for neutrons. In a
simple np-dominance picture, the neutron ratio would have a slight decrease.

Isospin structure can also be tested in isospin mirror nuclei, i.e., where the
behavior of protons in one nucleus can provide information about the neutrons
in the other. An (e,e’p) experiment was conducted in Hall A on 3H and 3He pair
with a goal of mapping the relative momentum distribution for protons and
neutrons in the $A=3$ system. The measurement was conducted in anti-parallel,
$x>1$ kinematics, for $p_{\text{miss}}$ in the range of 50 to 500 MeV$/c$,
i.e., across the transition from the conventional mean-field to the correlated
regimes. The ratio 3He(e,e’p)/3H(e,e’p) [49] was measured to be above 2 at low
missing momentum, falling to approximately 1.4 at $p_{\text{miss}}=250$
MeV$/c$, consistent with both the onset of np dominance, and with PWIA
calculations based on three-body spectral functions [50]. However, for even
larger missing momentum, the ratio rose again, inconsistent with PWIA
predictions. An analysis of the absolute cross sections showed that while the
$x>1$ kinematics go a long way to reducing FSIs, the can still affect the
measured cross section ratio, particularly through charge-exchange reactions,
which can increase the helium (e,e’p) cross section and decrease the tritium
(e,e’p) cross section with a non-trivial $p_{\text{miss}}$-dependence [51].
While the general result is in agreement with the np-dominance picture, in
general, detailed final-state interaction calculations are necessary for
drawing firm quantitative conclusions about the nuclear ground state.

####  3.2.3 Inclusive studies on non-isoscalar nuclei

Because inclusive scattering is sensitive to both proton and neutron knockout,
the cross section is proportional to a combination on np-, pp-, and nn-SRCs.
In general, this means that there is little sensitivity to the isospin
structure of the SRCs, especially in scattering from isoscalar nuclei, and
deviations from simple models of the A dependence associated with neutron
excess do not provide sufficient sensitivity to look for isospin dependence.
This is in part because the effects are not large, but also because it is
difficult to separate the A dependence from the N/Z dependence given that the
deviation of N/Z tends to grow with mass in nuclei for which SRC measurements
exist. However, by comparing scattering from targets with similar mass but
different isospin structure one can attempt to isolate the isospin structure
of the SRCs.

The first such experiment compared 48Ca and 40Ca scattering in 2N-SRC
kinematics [50]. These nuclei have similar mass and should therefore equal
contribution from NN-SRCs, but 48Ca should have an excess of nn-SRCs relative
to pp-SRCs due to the additional neutrons. Because the e-n cross section is
lower than the e-p cross section, this would yield a decrease in the cross
section per nucleon in 2N-SRC kinematics, unless np- and pp-SRCs have
negligible contributions. Taking into account the small difference in the
expected SRC contribution for A=40 and A=48 and the ratio of the e-p to e-n
cross section, the 48Ca/40Ca per-nucleon cross section ratio was predicted to
yield 0.975 in the case of 100% np-SRCs and 0.930 for isospin-independent SRC
formation [50]. The measurement observed a ratio of 0.971(12), corresponding
to near-total np dominance, with a 2$\sigma$ (1$\sigma$) lower limit on the
np/pp enhancement factor of 2.9 (10.6). This confirmed the observation from
2N-knockout measurements while avoiding the need for large FSI corrections.
However, given the very small difference between the predictions for isospin
independence and np dominance, potential corrections to the assumptions used
to interpret the data could impact the quantitative interpretation.

More recently, a comparison of scattering from mirror nuclei 3H and 3He was
performed [52], which has several advantages over the measurement using
calcium isotopes. First, because 3H and 3He structure are nearly identical,
there is next to no correction for the difference in nuclear structure apart
from the isospin structure. Second, because the fractional change in N/Z is
much larger between the two nuclei, there is a much larger difference for the
prediction for the 3H/3He cross section ratio in the SRC regime: 0.74 for
isospin independent vs. 1 for np dominance. This suggests that the measurement
will be almost an order of magnitude more sensitive to the relative
contribution of pp-SRCs.

Figure 6: (Left) Measured pn/p and pp/p fractions compared to GCF calculation;
Figure reproduced from Ref. [53].

###  3.3 SRC internal momentum distributions

As discussed in the previous sections, the ratio of np/pp SRCs varies with
$p_{m}$ [39], suggesting a difference in the relative internal momentum
distribution for np and pp SRCs, which is also seen in ab initio calculations
of the np and pp pair distributions in nuclei [23, 24, 25, 26]. As such, a
more detailed understanding of the role of SRCs has to include study of the
internal structure of the SRCs, which generate the observed $p_{m}$
dependence. In addition, while many experiments talk about the ratio of SRCs
(between nuclei, or the ratio of np to pp pairs), the measurements are
sensitive to the cross section contribution of those pairs over the $p_{m}$
range of the measurement. As such, one must go beyond treating each
measurements of the relative number of pairs, especially when comparing data
from different reactions or covering different $p_{m}$ ranges.

In addition, measurements that include detection of one or more nucleons in
the final state need to account for final-state interactions. Even in a mean
field model where there are no SRCs in the initial state, simple rescattering
can generate a measurement with a leading struck nucleon and a second high-
momentum nucleon (above the Fermi momentum), which are indistinguishable from
scattering from an SRC pair in the initial state. In addition, charge-exchange
rescattering can be an exceptionally large correction, in particular when
trying to isolate the very small pp-SRC contributions from the much larger np-
SRC contributions.

####  3.3.1 Generalized Contact Formalism

One of the challenges of interpreting the results of two-nucleon knockout
measurements has been, until recently, the lack of theoretical cross section
calculations with which to compare results. While there are methods for
calculating the break-up of deuterium and $A=3$ nuclei, and single-nucleon
spectral functions for heavier nuclei, calculations of the full two-nucleon
knockout decay function for nucleons in short-range correlated pairs have
proven difficult. For that reason, several recent analyses have turned to
Generalized Contact Formalism (GCF), a factorized model of short-range
correlations within a nucleus, to calculate fully differential two-nucleon
knockout cross sections to compare to data. A theoretical derivation of GCF
can be found in Ref. [54], but the core concept is the assumption that at
asymptotically short-distance or high-momentum scales, the nuclear wave
function, $\Psi$, can be factorized,

$\Psi_{r_{ij}\rightarrow
0}\longrightarrow\sum_{\alpha}\varphi_{\alpha}(\mathbf{r}_{ij})A_{ij}^{\alpha}(\mathbf{R}_{ij},\\{\mathbf{r}_{k\neq
ij}\\}),$ (2)

where $\mathbf{r}_{ij}$ is the separation vector between nucleons $i$ and $j$,
$\mathbf{R}_{ij}$ is their center of mass coordinate, $\alpha$ represents the
set of possible quantum numbers of a pair of nucleons, $\varphi$ represents a
universal two-body wave-function that is independent of nucleus, and
$A_{ij}^{\alpha}$ represents the wave function of the remaining $A-2$
nucleons. A similar formulation can be written in momentum space. Over these
asymptotic scales, $A_{ij}^{\alpha}$ is approximately constant, and one can
define the contact matrix,

$C^{\alpha,\beta}=16\pi^{2}N_{ij}\langle
A_{ij}^{\alpha}|A_{ij}^{\beta}\rangle,$ (3)

where $N_{ij}$ is the number of possible pair combinations. There are a number
of selection rules that determine which off-diagonal elements can be non-zero,
and in practice one only considers the diagonal terms, referred to as the
“contacts,” $C^{\alpha}$, which can be interpreted as the abundance of short-
range correlated pairs with quantum numbers $\alpha$, in a nucleus.

The asymptotic behavior of the nucleons within a correlated pair is determined
by the universal functions $\varphi_{\alpha}(\mathbf{r}_{ij})$ in position
space, or $\tilde{\varphi}_{\alpha}(\mathbf{k}_{ij})$ in momentum space. These
functions can be estimated from a given model NN potential by solving for the
zero-energy solution to the 2-body Schrödinger Equation [55]. While different
model NN potentials lead to different universal functions, comparisons with ab
initio variational Monte Carlo (VMC) calculations confirm the asymptotic
behavior predicted by GCF as well as the universality of pair behavior across
different nuclei [29]. Comparison with VMC calculations is one of the ways
that values of the contacts, i.e., the SRC pair abundances, can be estimated
[55, 29].

In addition to modelling properties of pairs within nuclei, GCF can also be
used to predict the fully differential cross section for the knockout of SRC
nucleons. This is especially powerful for two-nucleon knockout, where the
differential cross section has three extra differential dimensions. In single
nucleon knockout, the cross section within the PWIA can be factorized

$d^{6}\sigma\sim K\sigma_{eN}S(E_{m},\mathbf{P}_{m}),$ (4)

where $K$ is the kinematic factor, $\sigma_{eN}$ is the cross section for the
electron to scatter elastically from an off-shell nucleon, and
$S(E_{m},\mathbf{p}_{m})$ is the spectral function, describing the probability
to find a nucleon within a nucleus with momentum equal to the missing
momentum, $\mathbf{p}_{m}$, and separation energy equal to the missing energy,
$E_{m}$. In two-nucleon knockout, the spectral function is supplanted by the
two-nucleon decay function [56]:

$d^{8}\sigma\sim K^{\prime}\sigma_{eN}D(E_{m},\mathbf{p}_{m},\mathbf{p}_{r}),$
(5)

where $K^{\prime}$ is a different kinematic factor, and
$D(E_{m},\mathbf{p}_{m},\mathbf{p}_{r})$ additionally describes the
probability for a second recoil nucleon to be ejected with momentum
$\mathbf{p}_{r}$. In GCF, the decay function can be written in a factorized
form:

$D(E_{m},\mathbf{p}_{m},\mathbf{p}_{r})=\sum\limits_{\alpha}C^{\alpha}\left|\tilde{\varphi}^{\alpha}\left(\frac{1}{2}(\mathbf{p}_{m}-\mathbf{p}_{r})\right)\right|^{2}P^{\alpha}_{CM}(\mathbf{p}_{m}+\mathbf{p}_{r})\delta(m_{A}-E_{1}-E_{r}-E_{A-2}),$
(6)

where $E_{1}\equiv E_{N}-\omega$. In the above equation,
$|\tilde{\varphi}^{\alpha}|^{2}$ represents a universal, i.e., nucleus-
independent, two-body momentum distribution for pairs with quantum numbers
$\alpha$. $P_{CM}^{\alpha}$ represents the center of mass momentum
distribution, typically modelled by a Gaussian. A similar equation can be
formulated in light-front coordinates [57].

The tunable parameters of Eq. 6 are the contact terms, $C_{\alpha}$,
parameters defining the CM momentum distribution (typically a single Gaussian
width, $\sigma_{CM}$), and finally, any parameters describing the excitation
energy of the $A-2$ system (typically denoted by a single average value,
$E^{*}$). These parameters can either be estimated from the results of ab
initio variational Monte Carlo calculations (e.g., [29]), or from data.

The usefulness of GCF for interpreting experiments is wide-ranging, and and
many new avenues are under active development. First and foremost, GCF
specifies a fully differential cross section for scattering from correlated
nucleons. The GCF cross sections can reproduce, in $x>1$ kinematics, the
multi-dimensional kinematic distributions of one- and two-nucleon knock-out
[58]. Such theoretical calculations have been lacking, or used simpler
approximations, in previous SRC studies. Another benefit is that GCF provides
a very simple connection to underlying nuclear properties within the model,
i.e., pair abundances, center-of-mass motion, etc. GCF can be used to study
the sensitivity measurements of those properties, whether through the
calculation of various correction factors [57], or by performing inference
from data directly [36]. Since GCF specifies a plane-wave cross section, it
can be used to identify the effects of final state interactions through
significant deviations from data. Lastly, since the model is factorized, it is
applicable to reactions other than electron scattering, for instance, proton
scattering in inverse kinematics [59], or photoproduction [60].

###  3.4 Impact of SRCs on other fields of physics

While 2N-SRCs are most easily studied in high-energy scattering measurements,
their impact on nuclear structure can be seen in neutron stars [61, 62],
$\nu$-A scattering [63, 64, 65] for nuclear structure studies and $\nu$
oscillation, and understanding the nuclear quark distributions (EMC effect)
through understanding the high-momentum part of spectral function [66].

The impact of 2N-SRCs in nuclei can modify e-A, $\nu$-A, and A-A interactions
in a way that is not captured in mean-field calculations of the nuclear
structure. While ab initio calculations of light nuclei and infinite nuclear
matter can include these contributions, including the observed np-dominance,
this is more challenging for medium- and heavy-mass nuclei. In addition, there
are many cases where mean-field calculations that do not include SRC
contributions in the scattering observables have been used, due to lack of
appropriate ab initio calculations in the past or where the mean-field
structure was assumed to be sufficient.

Our present understanding of SRCs provides guidance on the overall strength
and isospin structure of their contributions. In many cases, this is
sufficient to allow detailed calculations or at least realistic modeling of
the contribution from SRCs in other reactions. Where ab initio calculations
are not available, or where their input is not directly used in scattering
calculations, momentum distributions or spectral functions can be constructed
based on a combination of mean-field and SRC contributions [18, 67], although
there has been progress recently on extracting structure functions directly
from ab initio calculations [26, 68, 69]. The Generalized Contact Formalism
(GCF) discussed in Sec. 3.3.1 provides a relatively simple and flexible way of
implementing SRCs in structure and scattering calculations.

###  3.5 Structure of the nucleons within SRCs

The scale of nuclear binding (tens of MeV) is so much smaller than the GeV
energy scales associated with deep-inelastic scattering measurements which
probe the structure function and quark distributions of nucleons and nuclei.
Because of this scale separation, it was initially assumed that the quark
distributions in a nucleus would be the sum of its constituent proton and
neutron quark distributions, with percent level effects associated with
binding and Fermi momentum. The EMC collaboration first observed [70] that the
structure function from iron different significantly from that of deuteron,
taken as an approximation for the free proton and neutron distributions. This
demonstrated that the quark structure of the nucleus has significant
deviations from the proton and neutron distributions, and was dubbed the EMC
effect. Experiments at SLAC [71] expanded these measurements to additional
nuclei showing that this was a universal feature of finite nuclei, with the
deviation of the nucleus from a simple sum-of-nucleons picture grew with
nuclear density [72].

While many explanations have been proposed to explain the EMC effect, as
summarized in Refs. [73, 74], we do not yet have a clear understanding of the
underlying physics. Several works have examined in more detail the impact of
binding and Fermi motion of the nucleons, non-nucleonic contributions to the
nuclear quark distributions (e.g. those carried by virtual pions in the
nucleus), and modification of the nucleon structure when bound in a nucleus.
While there is a growing consensus that contributions beyond binding and Fermi
motion are required [75, 76], additional measurements are required to better
constrain attempts to explain nuclear quark distributions.

Several experiments have been proposed to try and elucidate the origin of the
EMC effect [73, 74, 77]. In the following section we will discuss recent ideas
that were motivated by Jefferson Lab studies of both SRCs and the EMC effect,
as well as the experiments that motivated these new ideas.

####  3.5.1 EMC-SRC correlation

As noted in Sec. 2.3, measurements of SRCs in light nuclei [15] showed that
the contribution of SRCs in light nuclei deviated significantly from the
simple scaling models used to describe data in heavier nuclei [72]. Models
where the SRC contribution scales with average nuclear density failed to
describe the anomalously large value of $a_{2}$ in 9Be, whose average density
is very low, while models that scale with the target mass could not explain
the large difference between $a_{2}$ in 3He and 4He [22].

The data on SRCs matched the unexpected behavior of the EMC effect [78, 79]
showed a non-trivial correlation between the EMC effect and presence of SRCs.
Prior to this, it was generally assumed that both effects scale with the
average nuclear density [72, 73] , while the unexpected behavior in light
nuclei showed that both the EMC effect and the presence of SRCs are sensitive
to the detailed nuclear structure, suggesting that both effects are driven by
the same underlying physics or that the EMC effect is driven by the presence
of SRCs in nuclei. This has been referred to as the EMC-SRC correlation, and
while a great deal of effort has gone into examining the correlation, it is as
yet unclear what underlying physics drives the connection between these two
observables.

This idea of a connection between the EMC effect and SRCs was first raised in
Ref. [80], which quantified the correlation between the quantities and
interpreted as a consequence of the density dependence of both effects.
Shortly after, Ref. [81] examined this correlation and speculated that the EMC
effect was associated with the high virtuality of nucleons in SRCs. At the
time, only 3He, 4He, 12C, and 56Fe had measurements of both the EMC effect and
SRC contributions, and it was already known that both effects scaled
approximately with nuclear density or A-1/3 [72, 71]. It wasn’t until the SRC
data on 9Be [15] that there was a common deviation from the simple dependence
on density for both the EMC effect and for SRCs.

Initially, the 9Be EMC effect was initially explained [78] by looking at the
local density, as seen by the struck nucleon, rather than the average density
of the nucleus. This, combined with the $\alpha$-cluster structure of 9Be,
yielded a large difference between the predication based on average density or
local density (which can also be interpreted as quantifying the amount of
overlap in nucleons). Ref. [81] described the EMC-SRC correlation as a
consequence of the high-momentum (large-virtuality) nucleons in SRCs. Because
the SRCs are generated by nucleons interacting at short distances, there is a
connection between the contribution of nucleons at short distances and the
number of high-momentum nucleons, making it difficult to determine whether the
EMC effect is driven by short-distance configurations, as in the local density
(LD) picture, or high-momentum SRCs in the high-virtuality (HV) picture. In
both cases, the cluster structure of 9Be yields short-distance configurations
which would explain the larger-than-expected contribution of SRCs. In the HV
approach, the enhanced SRC contribution causes the increased EMC effect, while
in the LD picture, the short-distance, high-density clusters yield an enhanced
EMC effect as well as an enhanced SRC contribution. Both pictures can explain
the correlation, but may have quantitative differences, e.g. because np-
dominance of SRCs would imply that only np pairs would generate the EMC effect
in the HV picture, while all short-distance NN pairs can contribute in the LD
model.

Ref. [82] reexamined the correlation including the data of [15] and found that
the observation of the EMC-SRC correlation was largely independent of what
corrections were applied to the data, e.g. to account for potential isospin
structure of the SRCs. The “robustness” of the correlation is a consequence of
the limited precision of the measurements, in particular for the EMC effect,
making it difficult to use the correlation to test different hypotheses of the
EMC-SRC correlation with the data available at the time. Even so, a
quantitative analysis comparing simple models of the correlation for the LD
and HV approaches showed a slightly better correlation in the LD picture, but
the difference was only at the two standard deviation level [22].

####  3.5.2 Potential implication for flavor dependence of the EMC effect

The observed EMC-SRC correlation, combined with the observation of np-
dominance in SRCs suggests the possibility that the EMC effect might have an
isospin dependence associated with the excess of high-momentum protons in
neutron-rich nuclei. In this case, the fraction of high-momentum protons
(neutrons) scales as N/A (Z/A), such that there are an equal number of high-
momentum protons and neutrons. This suggests that the EMC effect in protons
(neutrons) would increase (decrease) with neutron excess. This will introduce
a flavor dependence of the EMC effect, with enhanced modification of the
u-quark distributions in neutron rich nuclei. Note that that a flavor-
dependent EMC effect is also a natural consequence of looking individually at
the proton and neutron distributions, whether you assume that the EMC effect
is driven by high virtuality or local density or even different overlap
between protons and neutrons, e.g. in a neutron-rich nucleus with a neutron
skin [28].

Such a flavor dependence is difficult to observe in inclusive EMC effect
measurements, as the neutron excess tends to increase with the mass of the
nucleus, and the existing data make it very difficult to separate an A
dependence from an N/Z dependence in heavy nuclei. However, an extension of
the analysis mentioned above [22] was performed, assuming that only np-SRCs
contribute to the EMC effect, thus providing a specific flavor dependence for
the EMC effect and allowing for the extraction of a ‘universal’ EMC effect for
the np-SRCs [21]. They extracted their proposed universal modification
function, under the assumptions noted above, and found that it was largely
independent of A, concluding that was in fact a universal function with with a
flavor-dependent EMC effect driven by the isospin structure of np-SRC
dominance.

Figure 7:  Three versions of universal deuterium EMC effect. Top panels are
the High Virtuality (HV) and Local Density (LD) functions of Ref. [83], and
the bottom left panel is the Naive universal function with no explicit isospin
dependence (EMC effect for each nucleus scaled down by it’s measured value of
$a_{2}$). The bottom right panel compares the slope from each of the universal
functions for the data from Refs. [71, 78, 79].

Shortly thereafter, another analysis compared the universal function based on
np-SRC dominance (“HV”) with an equivalent function based on the local density
model (“LD”), where all NN pairs at short distance contribute (analogous to
the comparison of the EMC-SRC correlation assuming HV or LD in Ref. [22]). The
results are shown as the top left (HV) and top right (LD) plots of Fig. 7.
While the exact form of the isoscalar corrections applied to the data differed
slightly from Ref [21], the universal function based on the HV assumption (np-
dominance) was nearly identical to the result of [21]. The LD-based universal
function, assuming a flavor-independent EMC effect, showed a more consistent
EMC slope as a function of A. Examining the A dependence of the universal
functions (bottom right panel of Fig. 7, the LD model was more consistent with
the hypothesis of A-independence, but only at the two standard deviation
level.

The bottom left plot of Fig. 7 shows the slopes from a third version of a
universal modification function based simply on the observation of the EMC-SRC
correlation. In this case the universal function is obtained by scaling down
the nuclear EMC effect (deviation of the structure function ratio from unity)
by the factor $a_{2}$, with no assumptions about the underlying cause or
flavor dependence dependence. For isoscalar nuclei, this is equivalent to the
HV universal function from [21], while for the non-isoscalar nuclei, it has an
extremely small impact on the slope of the universal EMC function, as seen in
the bottom right panel of Fig. 7. As such, the data have very little direct
sensitivity to the explicit assumption of flavor dependence in the HV-based
universal function compared to the naive model of a flavor-independent EMC
effect that scales with $a_{2}$. The difference between the HV and LD
functions does not come directly from the explicit flavor-dependent effect,
instead it comes from the fact that the LD model takes $a_{2}$ as a relative
measure of the contribution of np-SRCs, and then scales from the number of np
pairs (NZ) in the nucleus to the total number of NN pairs (A(A-1)/2). This
scaling factor, A(A-1)/(2NZ), yields an A dependence even for isoscalar
nuclei, and gives a reduced A dependence in the universal EMC slope. However,
this makes the implicit assumption that all NN pairs contribute equally at
short distances, while the short-distance distributions for pp, np, and nn
pairs can differ. Thus, these comparisons are useful in examining the impact
of certain assumptions on the extracted size of the EMC effect in the
deuteron, but the data have limited sensitivity even in the extreme/simplified
cases assumed in such analysis. Future data [84, 85] will improve the
situation by measuring the contribution of SRCs and the EMC effect for a range
of light nuclei and for medium-to-heavy nuclei covering a range in N/Z,
providing a data set with a greater ability to separate the A dependence and
the N/Z dependence of the EMC effect.

####  3.5.3 Impact on other physics

The question of whether the EMC effect is flavor dependent and, if so, how
large the effect is, is an important question that relates to both the origin
of the EMC effect and the question of obtaining reliable quark distributions
for both non-isoscalar nuclei and the neutron. The bottom right plot in Fig. 7
shows a smaller universal EMC effect for the deuteron, by almost a factor of
two based on the average of all nuclei, and a factor of 1.5 if the behavior is
extrapolated to the deuteron using the fits shown. This difference is
significant in the extraction of the neutron structure function [86, 87] and
quark distributions [88, 89] from comparisons of proton and deuteron structure
functions. In addition, it was pointed out in [22] that a flavor dependence of
the EMC effect in non-isoscalar nuclei would impact the extraction of the
neutron structure function from comparisons of 3H and 3He [90], and initial
examinations of the impact of such a flavor dependence using the data from
[90] have already appeared [91, 92].

In addition, high-energy scattering or collider experiments (e-A, $\nu$-A, or
A-A) with heavy neutron-rich nuclei. A reliable analysis of such data will
require an understanding of the modification of up and down quarks in non-
isoscalar nuclei. In addition, polarized 3He is often used as an effective
polarized neutron target, as the spins of the protons cancel almost
completely. However, if used for DIS studies of the polarized pdfs of semi-
inclusive DIS, a flavor dependent EMC effect will yield an additional
correction currently neglected in such analyses. Beyond this, the fact that
the neutron polarization is largely associated with lower-momentum nucleons
[26] means that the flavor dependent EMC effect may differ from the
unpolarized EMC effect both in their $x$ dependence and in their sensitivity
to the part of the 3He wave function sampled in the experiment.

Thus, the question of whether only np-SRCs or all short-distance pairs
contribute to the EMC effect has a significant impact on nuclear pdfs, the
predicted EMC effect for the deuteron, and spin structure studies on 3He. Note
that the question of flavor-dependent vs flavor-independent effects is not the
same as the question of local density vs high virtuality. The LD hypothesis
does not require that the EMC effect be flavor independent (or flavor
dependent); this is just the assumption typically made for comparison as the
goal is to examine how sensitive the data is to different models of the flavor
dependence. In addition, other ideas have been proposed to make direct
measurements of the flavor dependence of the EMC effect in pion-induced Drell-
Yan [93], SIDIS [94, 95] and Parity-violating electron scattering [96, 97]. A
review of ideas and future plans to more fully understand the EMC effect can
be found in [77].

##  4 Three-nucleon short-range correlations

Figure 8: (Left) 4He/3He per-nucleon cross section ratio vs $x$; figure
adapted from Ref. [98]. (Right) Calculation of the mean field, 2N-SRC, and
3N-SRC contributions from Ref. [99].

So far, we have focused on 2N SRCs, where only two nucleons interact to
produce large relative but small total momenta. Short-range configurations
with more than two nucleons are not prohibited, but are expected to occur with
decreasing probability. Following searches for 2N SRCs, attempts were also
made to observe a second plateau in nuclear cross section ratios relative to
3He. The first observation claim was published by the CLAS collaboration [20],
showing a second plateau starting at $x\approx 2.25$, shown in Fig. 8. This
was surprising, as the $Q^{2}$ value was low (average of 1.7 GeV2) and the
onset of scaling was low in $x$. Other experiments were carried out [15, 98]
with high resolution spectrometers, with the aim of looking at the $Q^{2}$
dependence of the second plateau as well as looking at its nuclear dependence.
The data from the Hall C E02-019 experiment [15] are inconclusive, as the data
are consistent with a plateau but the uncertainties in the region of interest
are large. However, the ratio significantly higher than what was observed by
CLAS, and the would-be plateau starts later in $x$ (which aligns better with
expectations), contradicting the interpretation of the CLAS data in terms of
isolating 3N-SRCs. A follow-up Hall A experiment [98] performed a scan over
$Q^{2}$ including the CLAS kinematics, but failed to observe a 3N plateau, as
can be seen in Fig. 8. The plateau initially observed in Ref. [20] was later
shown to be an effect of bin migration, where all the $x>2.2$ bins came from a
single $E^{\prime}$ bin [100]. This means that we still have no definitive
experimental observations of 3N SRCs and the question of whether there is a
kinematic region where they dominate the scattering is still open.

Figure 9: Different potential momentum sharing and isospin configurations for
symmetric (A) and asymmetric (B, C) 3N-SRC configurations.

While the onset of the 2N SRC plateau has well defined kinematics, the picture
is less clear for 3N-SRCs due to multiple possible configurations, as shown in
Fig. 9. Three nucleons can share three units of momentum equally (symmetric
configuration), which would result in nucleons with high momenta comparable to
those in a 2N configuration. Additionally, a high-asymmetric configuration is
also possible. While we can calculate $p_{min}$ as a function of $x$ and
$Q^{2}$ for heavy nuclei, this value is dependent on the momentum distribution
of the spectators in the 3N-SRC, so a more realistic estimate of relevant
minimum momenta for 3N-SRCs is dependent for the symmetric and asymmetric
momentum configurations.

Another difficulty arises because the contribution from 2N-SRCs falls off more
slowly than the mean field contributions, and so predicting what nucleon
momentum is required for the 3N-SRCs to dominate (and result in a second
plateau) depends on the details of size and nature of the 3N-SRC
contributions. On top of this, the structure of the 3N-SRC is more complicated
and predictions for its contribution are highly model dependent. Results of
one such calculation are shown on the RHS of Fig. 8, assuming the star-
configuration for a 3N-SRC [99]. In this model, the 3N-SRC contribution begins
to have a significant contribution at $\alpha\approx 1.5$, but does not
dominate until $\alpha\approx 1.8$. While the transition from mean field to
2N-SRC dominance is clear, due to the rapid and well understood falloff of the
mean field momentum distribution, the transition to 3N-SRCs depends strongly
on both the 2N-SRC and 3N-SRC structure assumed in the calculations. Based on
this calculation it was argued that 3N-SRCs might be visible somewhere above
$\alpha=1.6$ [35], corresponding to $x=2.3$ for the E02-019 data [15], with
all other data sets stopping before $\alpha=1.6$. While the ratio is
consistent with a plateau for $x>2.3$, the statistics are limited and data
exist for only one $Q^{2}$, meaning that this data set is not sufficient to
identify universal 3-body behavior in 3He and 4He. Ref. [35] also predicts
that $a_{3}(A)$, the A/3He ratio in the 3N-SRC region, should go like
$a_{2}^{2}$ (neglecting isospin differences), and they find that the E02-019
data are consistent with this prediction. Thus, there are hints that
$\alpha=1.6$ might be sufficient to study 3N-SRCs, but new data at higher
$Q^{2}$ will be needed to confirm the presence of 3N-SRCs [84, 101].

The 3N-SRC landscape is further clouded by the possibility of multiple isospin
configurations. If we expect some analog with the 2N SRC, where there’s a
region of np dominance, there should be ppn and pnn dominance among 3N-SRC
configurations. Even if SRCs are dominated by ppn and pnn contributions (as
opposed to 3 of a kind, which could be less probable, as pp and nn are in
2N-SRCs), scattering from isoscalar nuclei should have identical contributions
from these configurations (plus possible configurations from ppp- or nnn-
SRCs). Previous and upcoming measurements have used the 3He nucleus for the
measurement in the denominator, as it is the smallest stable nucleus with
three nucleons. However, in 3He, only the ppn 3N-SRC configuration is
possible, meaning the $a_{3}$ ratio will be sensitive to differences in the
number of protons and neutrons at the largest momenta in 3N-SRCs.

In a symmetric configuration (left panel of Fig. 9), scattering from the
highest momentum nucleons in 3He will involve both protons and the neutron. In
a highly asymmetric configuration (middle and right panels of Fig. 9), the
high-momentum part of the distribution could be dominated by the singly-
occurring neutron, the double-occurring proton, or some combination. Thus, the
question of whether the highest-momentum nucleons are protons, neutrons, or a
mix of both, will modify the A/3He ratio. If the relative contributions of
protons and neutrons varies as a function of momentum, this could also distort
the $x$ dependence of the ratio. Future JLab 12 GeV measurements [84, 101]
should shed light on many of these questions.

##  5 Key questions, future directions

In this review, we have tried to highlight recent progress in our
understanding of the short-range structure of nuclei, highlighting key
discoveries and important insight from theory. Here, we summarize some of
these key points and discuss the remaining physics questions that will
complete our picture of short-range correlations and understand their impact
on nuclear structure at both the hadronic and partonic levels. Plans to expand
on previous measurements and new novel probes of SRCs aimed at addressing
these remaining questions will be presented.

###  5.1 Summary of our understanding of SRCs and remaining questions

Inclusive measurements identified kinematic regions of 2N-SRC dominance in
nuclei, confirming the predictions for the kinematic region where SRCs should
dominate and observing the expected universal two-body behavior through
scaling in $x$ and $Q^{2}$. Such measurements mapped out the A dependence of
the contribution of SRCs, identifying 9Be as an outlier from the expected
scaling with density and indicating that understanding SRCs requires a
microscopic picture of the nuclear structure.

They also showed an intriguing correlation with measurements of the EMC
effect, raising questions about the flavor dependence of the EMC effect.
However, only the 9Be data indicate that this correlation goes beyond simple
scaling of both effects with nuclear density, and the direct sensitivity of
both SRC and EMC measurements to isospin structure is limited, due to the
limited number of measurements on nuclei with neutron excess, and the fact
that the neutron excess tends to grow with mass in heavier nuclei, making it
difficult to separate mass- and isospin-dependent effects.

Triple-coincidence measurements have given us insight into the isospin
structure of 2N-SRCs, showing an enhancement of np-SRCs over pp- and nn-SRCs
at the $\sim$30:1 level in all nuclei measured. This observation has been
confirmed in a variety of measurements, with new results to come for 3H and
3He [29, 52].

While 2N-SRCs have been identified, quantified, and significant progress has
been made in understanding their momentum and isospin structure, we do not yet
know what if any role 3N-SRCs play in nuclear structure. While early
measurements suggested that they had been observed [20], later experiments
[15, 102] did not agree with the initial observation, which is now believed to
have been limited by the experimental momentum resolution [100]. Because each
of these experiments had one or more significant limitations - low $Q^{2}$
values, poor statistics, insufficient resolution - an argument can be made
that there has yet to be a meaningful test of the 3N-SRC hypothesis.

Based on the discussion above, we identify the following as key questions that
require further data and insight to be fully answered.

  * •

How can we improve direct, quantitative comparisons of different measurements
of 2N-SRCs. How well can we constrain calculations by combining the results of
these different experiments?

  * •

What is the contribution of 3N-SRCs, and what is their internal momentum and
isospin structure?

  * •

What is the origin of the EMC-SRC correlation and what does it imply for the
internal structure of nucleons in SRCs and the flavor dependence of the EMC
effect?

###  5.2 Extensions of the current experimental program

Several experiments are planned which will extend the type of SRC studies we
have discussed already. These include experiments that will make measurements
on new targets, extending the kinematic coverage to higher $x$ or $Q^{2}$, or
providing improved statistics. We summarize these experiments below, and in
the following sections discuss new and novel measurements will probe SRCs and
their internal structure in completely new ways, and are discussed in the
following sections.

####  5.2.1 2N-SRC and EMC effect measurements in new nuclei

Two experiments are scheduled to run in 2022 that will measure A/D ratios to
extract the contribution of SRCs [84] and the EMC effect [85] in a wide range
of nuclei. This includes several light nuclei, 3,4He, 6,7Li, 9Be, 10,11B, and
12C to look at the A dependence of both effects in these well understood
nuclei. In addition, they will study medium-to-heavy nuclei over a range of
N/Z to try and disentangle the A dependence and isospin dependence of SRCs and
the EMC effect. It will also provide a better data set to examine the impact
of isospin structure on the EMC-SRC correlation.

####  5.2.2 3N-SRCs

Experiment [85] will also perform the first significant search for 3N-SRCs,
providing the first data with large enough $Q^{2}$ values and high enough
precision to identify a plateau in $x$ at $x>2$, as discussed in Sec. 4.
Depending on what is observed, further measurements could be performed [101]
to confirm the $Q^{2}$ independence of the scattering and to map out the
A-dependence. In addition, measurements on 3H and 3He could provide a
sensitive probe of the momentum/isospin structure of 3N-SRCs, even if
inclusive scattering does not allow for a clean isolation of 3N-SRC
contributions [95, 101].

####  5.2.3 2N-knockout

The nuclear targets program at CLAS-12 [103], which began data collection in
2021, will leverage the larger acceptance, higher luminosity and data rate,
and increased neutron detection capabilities of CLAS-12 to pursue higher-
statistics studies of two-nucleon knockout. Combined with improved modeling of
reaction effects and final-state interactions this program will refine our
understanding of isospin structure, momentum structure, size, and asymmetry
dependencies of SRCs, with the additional goal of detecting 3N knockout
events.

A number of new programs will use different probes to test the degree to which
reaction effects factorize from the nuclear structure. A 2021 experiment using
the GlueX spectrometer aims to identify SRCs in 2N-knockout using
photoproduction [60]. A program of inverse kinematics measurements has begun
at the Joint Institute for Nuclear Research (JINR) in Dubna, using beams of
nuclei incident on proton targets. The advantage of this technique is that the
residual nucleus and correlated spectator nucleons have significant momentum
in the lab frame, making them easier to detect and momentum analyze. For
example, the detection of an intact boron nucleus in the break-up of SRC pairs
in carbon has been shown to reduce apparent final state effects [59]. This
technique could open the possibility of studying SRCs in unstable nuclei, for
instance, at the future FAIR Facility.

In addition to these measurements, which represent natural extensions of the
already-completed studies, there are new measurements that offer the
possibility of providing significant new information, allowing us to better
understand the microscopic structure of SRCs and the origin of the EMC effect.
While there are several possibilities to make new measurements that can help
identify the origin of the EMC effect [77], we provide three examples below
which are more directly connected with the connection between the EMC effect
and SRCs.

###  5.3 New directions: partonic structure and non-nucleonic degrees of
freedom

The existing inclusive studies have demonstrated that it is possible to
isolate scattering from SRCs by going to sufficiently large $x$ values at
modest-to-large $Q^{2}$ values. By extending to much higher $Q^{2}$, but
maintaining sufficiently large $x$, it should be possible to stay in the SRC-
dominated regime while makings DIS measurements to probe the target’s quark
distributions. In this way, we can study the nuclear pdfs at $x>1$ and extract
the distribution of so-called ‘super-fast quarks’ in nuclei [1, 3]. In this
region, where SRC contributions should be large, different models make very
different predictions for the super-fast quark distribution. This might also
provide a way to test quark-level descriptions of SRCs, e.g. where SRCs are
associated with diquark correlations between neighboring nucleons [34].

The first such proposed measurement should take data in 2022 [84], with
$Q^{2}$ values from 15-20 GeV2 for $x$ up to 1.4. Such a measurement should be
extremely sensitive to certain types of more exotic configurations within
nuclei. Early examinations of the EMC effect looked at the contribution of
6-quark bags as a model for exotic configurations for which the quark momentum
distributions would be significantly modified. A small (5%) contribution from
6q bags would have very little impact on the structure function in the DIS
region, at most 1-2% [84], as illustrated in the left panel of Fig. 10.
However, the free momentum sharing between 6 quarks carrying twice the
longitudinal momentum of a single nucleon yields a large (fractional)
enhancement for $x>1$, matching or even significantly exceeding the
contributions that come from Fermi smearing. Thus, what is a percent-level
modification for the deuteron pdf for $x<0.8$ could yield an order of
magnitude enhancement over the pdfs obtained from a simple convolution of a
proton and neutron, as illustrated in the right panel of Fig. 10. While this
estimate was made using a 6q bag model [104, 3, 84, 99], any enhancement of
momentum sharing between overlapping nucleons would be expected to yield a
qualitatively similar effect. In models where the EMC effect is driven by
highly virtual nucleons, models show a significant suppression of the
structure function for off-shell as opposed to on-shell nucleons at large $x$
and light-cone momentum $\alpha$ [105]. DIS at $x>1$ requires $\alpha>1$, as
shown in Fig. 35 of Ref. [5], allowing for a measurement that does not
directly tag $\alpha$ for the nucleon, but which is still sensitive to the
large-$x$ nucleon pdfs at $\alpha>1$, where significant suppression of the
free nucleon structure function is predicted by several models [106, 105, 5].

Figure 10: (Left) Plot showing the calculated valence deuteron structure
function based on convolution of proton plus neutron (red dashed line) vs. the
structure function (blue solid line) where the deuteron includes a 5%
component of 6q bag [107] (green dotted line). (Right) Same on a logarithmic
scale focused on $x>1$.

One critical aspect of such a measurement will be demonstrating sensitivity to
the pdfs at $x>1$, where the conventional kinematics criteria for DIS
scattering on a nucleon does not apply. The data taken at 6 GeV were limited
to $Q^{2}<10$ GeV2, and yet they were well reproduced by a QCD-inspired fit
[108]. The 12 GeV measurement [84] will extend these measurements to
$Q^{2}\approx 20$ GeV2, making them more appropriate for an analysis in terms
of the nuclear quark distributions. In addition, if non-nucleonic
contributions yield a very large deviation from the simple proton plus nucleon
convolution, the effect should be clear as long as the data is DIS-dominated,
even if it is not purely DIS.

One can also make measurements that are explicitly aimed at identifying non-
nucleonic degrees of freedom including hidden color configurations [109, 110,
111] through measurements of A(e,e’N$\Delta$) or A(e,e’$\Delta\Delta$). In the
presence of such non-nucleonic configurations at short-distance scales, there
would be a significant enhancement of SRC-like N-$\Delta$ or
$\Delta$-$\Delta$-like pairs [112]. Initial estimates suggested that such
measurements would be much more sensitive at 12 GeV or even higher energies.

###  5.4 New directions: Flavor-dependent EMC effect

As noted in section 3.5.2, the EMC-SRC connection raised the question of
whether the EMC effect is flavor dependent, in particular for non-isoscalar
nuclei. This will be examined by measurements of the EMC effect for non-
isoscalar nuclei, e.g. 40Ca and 48Ca, but as shown in Fig. 7, the flavor
dependence associated with the neutron excess in heavy nuclei has only a
modest impact on the EMC effect. While a direct measurement of sufficient
sensitivity does not appear to be possible, the systematic study of JLab
E12-10-008 [85] may allow for a model-dependent separation of A and isospin
dependence.

The use of SIDIS as a way of flavor tagging scattering on nuclei has been
proposed as a way to examine the flavor dependence of the EMC effect [94, 95].
However, measurements in heavy nuclei, where the expected effects are expected
to be relatively significant, have to deal with model-dependent corrections
and other possible nuclear effects which may make it difficult to cleanly
isolate the flavor dependence of the nuclear pdfs. These issues can be
addressed by going to light nuclei, in particular in a comparison of 3H and
3He, but it is not clear that realistic models of the flavor dependence will
be large enough to measure in such experiments.

A clean and sensitive measurement can be performed using parity-violating
electron scattering on non-isoscalar nuclei [96]. This involves looking for a
modest change in a parity-violating asymmetry of order 100 parts-per-million,
and while this is within the capabilities of modern parity-violating
measurements, it requires a high luminosity and large acceptance as well as
good control of false asymmetries. This appears to be within the capabilities
of the SoLID detector planned for parity-violating DIS measurements at
Jefferson Lab [113], and a proposal was submitted to perform such a
measurement on 48Ca [97] that can provide excellent sensitivity to estimates
of the flavor dependence based on simple scaling models [28] or the
calculations of Ref. [96].

###  5.5 New directions: Tagged measurements of SRCs

Another approach related to flavor dependence is the technique of spectator
tagging, especially in deuterium, where by the detection of a recoiling
spectator nucleon can identify the struck-nucleon, and provide information
about the deuteron’s initial configuration (i.e., short range vs. long range).
The requirement of nucleon detection makes the technique sensitive to final
state interactions, though theoretical estimates indicate the effects are
smallest when the spectator recoils in the backwards direction [114, 115].
This technique is used by the BoNUS experiment in CLAS, in which the detection
of a low-momentum spectator proton indicates that the accompanying DIS
reaction took place on a neutron. By extrapolating to stationary neutron
limit, BoNUS aims to determine the structure of the free neutron. First
results were published in 2011 from the 6 GeV program [116, 117, 118]. A
subsequent 12 GeV measurement is currently under analysis [119].

Whereas BoNUS is focused on slow-moving spectators, two other Jefferson Lab
experiments aim to detect high-momentum (300–600 MeV$/c$) spectators, as a way
of determining the structure modification in SRC-pairs, thus testing models of
the EMC effect. The BAND Experiment [120] took data in CLAS in 2019 with a
backward-angle detection system added to measure backwards recoiling spectator
neutrons. The Hall C LAD Experiment [121] will detect recoiling spectator
protons. Both experiments aim to determine how nucleon structure modification
changes with virtuality. Their combined results may give an indication of how
bound proton modification differs from that of the bound neutron. Such studies
can be extended to higher energy and larger spectator momenta at the proposed
Electron-Ion Collider [122].

###  5.6 Summary and Conclusions

Over the past few decades, extensive sets of data related to short-range
correlations have become available, mainly from the Jefferson Lab experimental
program. With this data in hand, we have mapped out the contribution of SRCs
over a range of nuclei, and measured their isospin and momentum structure.
Overall, enough has been learned that we are in a position to incorporate this
short-distance physics into nuclear structure calculations related to low
energy scattering, neutrino-nucleus scattering, and neutron star structure.
These new data have driven significant experimental progress in quantifying
SRCs and understanding their internal structure, although more work is
required to understand final-state interactions and the momentum structure of
np and pp SRCs well enough make detailed, quantitative comparisons of
different measurements that are sensitive to SRCs. Beyond this, the issue of
3N-SRCs remains unanswered: what is their isospin and momentum structure, and
can they be isolated experimentally? These questions are important enough that
studies SRCs and related high-energy nuclear structure measurements continue
to be a significant part of the future Jefferson Lab experimental program
[123].

Beyond the direct measurements of SRCs in nuclei, this experimental program
has also raised questions that touch on other aspects of high-energy nuclear
structure. The observation of the EMC-SRC correlation raises the question of
whether this represents a commonality of origin or a direct causal connection
where SRCs generated modified nucleon structure because of their short-
distance and/or high-momentum nature. The possibility of a causal connection,
combined with the strong np dominance of 2N-SRCs, suggests the possibility of
a flavor-dependent EMC effect in non-isoscalar nuclei. This motivates further
studies of the EMC effect, as well as a program of studies to probe the
internal partonic structure of SRCs, either through DIS measurements at
kinematics that isolate scattering from SRCs, or from tagged measurements
which look at the nucleon structure (effective form factors or pdfs) as a
function of nucleon virtuality.

Answering these questions will require not only the experimental program
described in this paper, but also improvements in the theory necessary to
interpret these measurements. Improved FSI corrections are needed for more
quantitative studies in one- and two-nucleon knockout reactions, and the
extraction of the modification of the nucleon structure through tagged
measurements of effective form factors or pdfs relies on careful comparison to
calculations which account for both the conventional nuclear effects and
modification of the nucleon structure within SRCs. Close collaboration has
driven the simultaneous advancement of both experimental data and theoretical
understanding, and will continue to open up new avenues for future study.

Acknowledgements: The authors wish to thank Wim Cosyn, Adam Freese, Shujie Li,
Augusto Machiavelli and Jerry Miller for useful discussions and other
contributions.

This work supported in part by the Department of Energy’s Office of Science,
Office of Nuclear Physics, under contracts DE-AC02-05CH11231, DE-SC0013615 and
DE-SC0016583.

## References

  * [1] Frankfurt LL, Strikman MI.  Phys. Rept. 76:215 (1981)
  * [2] Frankfurt LL, Strikman MI.  Phys. Rept. 160:235 (1988)
  * [3] Sargsian MM, et al.  J. Phys. G 29:R1 (2003)
  * [4] Arrington J, Higinbotham DW, Rosner G, Sargsian M.  Prog. Part. Nucl. Phys. 67:898 (2012)
  * [5] Hen O, Miller GA, Piasetzky E, Weinstein LB.  Rev. Mod. Phys. 89:045002 (2017)
  * [6] Fomin N, Higinbotham D, Sargsian M, Solvignon P.  Ann. Rev. Nucl. Part. Sci. 67:129 (2017)
  * [7] Egiyan KS, et al.  Phys. Rev. Lett. 98:262502 (2007)
  * [8] Boeglin WU, et al.  Phys. Rev. Lett. 107:262501 (2011)
  * [9] Sargsian MM.  Phys. Rev. C 82:014612 (2010)
  * [10] Benhar O, day D, Sick I.  Rev. Mod. Phys. 80:189–224 (2008)
  * [11] Day DB, et al.  Phys. Rev. Lett. 59:427–430 (1987)
  * [12] Arrington J, et al.  Phys. Rev. Lett. 82:2056 (1999)
  * [13] Ciofi degli Atti C, West GB.  Phys. Lett. B 458:447–453 (1999)
  * [14] Arrington J. 2003.  In 9th International Conference on the Structure of Baryons
  * [15] Fomin N, et al.  Phys. Rev. Lett. 108:092502 (2012)
  * [16] Frankfurt LL, Strikman MI, Day DB, Sargsian M.  Phys. Rev. C 48:2451 (1993)
  * [17] Ciofi degli Atti C, Simula S.  Phys. Lett. B 325:276 (1994)
  * [18] Ciofi degli Atti C, Simula S.  Phys. Rev. C 53:1689 (1996)
  * [19] Egiyan KS, et al.  Phys. Rev. C 68:014313 (2003)
  * [20] Egiyan KS, et al.  Phys. Rev. Lett. 96:082501 (2006)
  * [21] Schmookler B, et al.  Nature 566:354 (2019)
  * [22] Arrington J, Daniel A, Day D, Fomin N, Gaskell D, Solvignon P.  Phys. Rev. C 86:065204 (2012)
  * [23] Schiavilla R, Wiringa RB, Pieper SC, Carlson J.  Phys. Rev. Lett. 98:132501 (2007)
  * [24] Alvioli M, Ciofi degli Atti C, Morita H.  Phys. Rev. Lett. 100:162503 (2008)
  * [25] Wiringa RB, Schiavilla R, Pieper SC, Carlson J.  Phys. Rev. C 78:021001 (2008)
  * [26] Wiringa RB, Schiavilla R, Pieper SC, Carlson J.  Phys. Rev. C 89:024305 (2014)
  * [27] Colle C, Hen O, Cosyn W, Korover I, Piasetzky E, et al.  Phys. Rev. C 92:024604 (2015)
  * [28] Arrington J.  EPJ Web Conf. 113:01011 (2016)
  * [29] Cruz-Torres R, et al.  Nature Phys. 17:306 (2021)
  * [30] Feldmeier H, Horiuchi W, Neff T, Suzuki Y.  Phys. Rev. C 84:054003 (2011)
  * [31] Ryckebusch J, Cosyn W, Vanhalst M.  J. Phys. G 42:055104 (2015)
  * [32] Mosel U, Gallmeister K.  Phys. Rev. C 94:034610 (2016)
  * [33] Ryckebusch J, Cosyn W, Vieijra T, Casert C.  Phys. Rev. C 100:054620 (2019)
  * [34] Rittenhouse West J.  arXiv:2009.06968 (2020)
  * [35] Sargsian MM, Day DB, Frankfurt LL, Strikman MI.  Phys. Rev. C 100:044320 (2019)
  * [36] Weiss R, Denniston AW, Pybus JR, Hen O, Piasetzky E, et al.  Phys. Rev. C 103:L031301 (2021)
  * [37] Tang A, Watson JW, Aclander J, Alster J, Asryan G, et al.  Phys. Rev. Lett. 90(4):042301 (2003)
  * [38] Shneor R, et al.  Phys. Rev. Lett. 99:072501 (2007)
  * [39] Korover I, et al.  Phys. Rev. Lett. 113:022501 (2014)
  * [40] Niyazov RA, et al.  Phys. Rev. Lett. 92:052303 (2004), [Erratum: Phys.Rev.Lett. 92, 099903 (2004)]
  * [41] Baghdasaryan H, et al.  Phys. Rev. Lett. 105:222501 (2010)
  * [42] Cohen EO, et al.  Phys. Rev. Lett. 121:092501 (2018)
  * [43] Moniz EJ, Sick I, Whitney RR, Ficenec JR, Kephart RD, Trower WP.  Phys. Rev. Lett. 26:445–448 (1971)
  * [44] Piasetzky E, Sargsian M, Frankfurt L, Strikman M, Watson JW.  Phys. Rev. Lett. 97:162504 (2006)
  * [45] Subedi R, et al.  Science 320:1476 (2008)
  * [46] Hen O, et al.  Science 346:614–617 (2014)
  * [47] Duer M, et al.  Phys. Rev. Lett. 122:172502 (2019)
  * [48] Duer M, et al.  Nature 560:617 (2018)
  * [49] Cruz-Torres R, et al.  Phys. Lett. B 797:134890 (2019)
  * [50] Nguyen D, et al.  Phys. Rev. C 102:064004 (2020)
  * [51] Cruz-Torres R, et al.  Phys. Rev. Lett. 124:212501 (2020)
  * [52] Li S, et al.  The isospin structure of short-range correlations in the mirror nuclei 3H and 3He (2021), submitted to Phys. Rev. Lett.
  * [53] Korover I, et al.  Phys. Lett. B 820:136523 (2021)
  * [54] Weiss R, Bazak B, Barnea N.  Phys. Rev. C 92:054311 (2015)
  * [55] Weiss R, Cruz-Torres R, Barnea N, Piasetzky E, Hen O.  Phys. Lett. B780:211–215 (2018)
  * [56] Sargsian MM, Abrahamyan TV, Strikman MI, Frankfurt LL.  Phys. Rev. C 71:044615 (2005)
  * [57] Pybus JR, Korover I, Weiss R, Schmidt A, Barnea N, et al.  Phys. Lett. B 805:135429 (2020)
  * [58] Schmidt A, et al.  Nature 578:540 (2020)
  * [59] Patsyuk M, et al.  Nat. Phys. 17:693–699 (2021)
  * [60] Hen O, Dutta D, Gao H, Patsyuk M, Piasetzky E, et al.  E12-19-003: Studying Short-Range Correlations with Real Photon Beams at GlueX (2019)
  * [61] Higinbotham D, Piasetzky E, Strikman M.  CERN Cour. 49N1:22–24 (2009)
  * [62] Frankfurt L, Sargsian M, Strikman M.  Int. J. Mod. Phys. A 23:2991–3055 (2008)
  * [63] Kulagin SA, Petti R.  Phys. Rev. D 76:094023 (2007)
  * [64] Niewczas K, Sobczyk JT.  Phys. Rev. C 93:035502 (2016)
  * [65] Van Cuyck T, Jachowicz N, González-Jiménez R, Martini M, Pandey V, et al.  Phys. Rev. C 94:024611 (2016)
  * [66] Kulagin SA, Petti R.  Nucl. Phys. A 765:126–187 (2006)
  * [67] Ciofi degli Atti C, Mezzetti CB, Morita H.  Phys. Rev. C 95:044327 (2017)
  * [68] Carlson J, Gandolfi S, Pederiva F, Pieper SC, Schiavilla R, et al.  Rev. Mod. Phys. 87:1067 (2015)
  * [69] Sobczyk JE, Acharya B, Bacca S, Hagen G.  Phys. Rev. Lett. 127:072501 (2021)
  * [70] Aubert JJ, et al.  Phys. Lett. B 123:275 (1983)
  * [71] Gomez J, et al.  Phys. Rev. D 49:4348 (1994)
  * [72] Sick I, Day D.  Phys. Lett. B 274:16 (1992)
  * [73] Geesaman DF, Saito K, Thomas AW.  Ann. Rev. Nucl. Part. Sci. 45:337 (1995)
  * [74] Malace S, Gaskell D, Higinbotham DW, Cloet I.  Int. J. Mod. Phys. E 23:1430013 (2014)
  * [75] Miller GA, Smith JR.  Phys. Rev. C 65:015211 (2002), [Erratum: Phys.Rev.C 66, 049903 (2002)]
  * [76] Smith JR, Miller GA.  Phys. Rev. C 65:055206 (2002)
  * [77] Cloët IC, et al.  J. Phys. G 46:093001 (2019)
  * [78] Seely J, et al.  Phys. Rev. Lett. 103:202301 (2009)
  * [79] Arrington J, et al.  Phys. Rev. C 104:065203 (2021)
  * [80] Higinbotham DW, Gomez J, Piasetzky E.  arXiv:1003.4497 (2010)
  * [81] Weinstein LB, Piasetzky E, Higinbotham DW, Gomez J, Hen O, Shneor R.  Phys. Rev. Lett. 106:052301 (2011)
  * [82] Hen O, Piasetzky E, Weinstein LB.  Phys. Rev. C 85:047301 (2012)
  * [83] Arrington J, Fomin N.  Phys. Rev. Lett. 123:042501 (2019)
  * [84] Arrington J, Day D, Fomin N, Solvignon-Slifer P.  E12-06-105: Inclusive Scattering from Nuclei at $x>1$ in the quasielastic and deeply inelastic regimes (2006)
  * [85] Gaskell D, Arrington J, Daniel A, Fomin N.  E12-10-008: Detailed studies of the nuclear dependence of $F_{2}$ in light nuclei (2010)
  * [86] Arrington J, Coester F, Holt RJ, Lee TSH.  J. Phys. G 36:025005 (2009)
  * [87] Arrington J, Rubin JG, Melnitchouk W.  Phys. Rev. Lett. 108:252001 (2012)
  * [88] Accardi A, Christy ME, Keppel CE, Melnitchouk W, Monaghan P, et al.  Phys. Rev. D 81:034016 (2010)
  * [89] Accardi A, Melnitchouk W, Owens JF, Christy ME, Keppel CE, et al.  Phys. Rev. D 84:014008 (2011)
  * [90] Abrams D, et al.  arXiv:2104.05850 (2021)
  * [91] Cocuzza C, Keppel CE, Liu H, Melnitchouk W, Metz A, et al.  Phys. Rev. Lett. 127:242001 (2021)
  * [92] Segarra EP, et al.  arXiv:2104.07130 (2021)
  * [93] Dutta D, Peng JC, Cloet IC, Gaskell D.  Phys. Rev. C 83:042201 (2011)
  * [94] Dutta D, Gaskell D, Hafidi K, et al.  Precise Measurement of pi+/pi- Ratios in Semi-inclusive Deep Inelastic Scattering Part II: Unraveling the Flavor Dependence of the EMC Effect.  JLab Proposal PR12-09-004 (2009)
  * [95] Weinstein L, et al.  E12-21-004: Semi-Inclusive Deep Inelastic Scattering Measurement of A=3 Nuclei with CLAS12 in Hall B (2021)
  * [96] Cloet IC, Bentz W, Thomas AW.  Phys. Rev. Lett. 109:182301 (2012)
  * [97] Arrington J, Beminiwattha R, Gaskell D, Mammei J, Reimer P.  PR12-21-002: First Measurement of the Flavor Dependence of Nuclear PDF Modification Using Parity Violating Deep Inelastic Scattering (2021)
  * [98] Ye Z, et al.  Phys. Rev. C 97:065204 (2018)
  * [99] Freese AJ, Sargsian MM, Strikman MI.  Eur. Phys. J. C 75:534 (2015)
  * [100] Higinbotham DW, Hen O.  Phys. Rev. Lett. 114:169201 (2015)
  * [101] Arrington J, Fomin N, Li S.  LOI12-21-001: 3N Short Range (2021)
  * [102] Ye Z, Arrington J. 2018.  In 13th Conference on the Intersections of Particle and Nuclear Physics
  * [103] Hen O, Piasetzky E, Schmidt A, Stepanyan S, Szumila-Vance H, Weinstein L.  E12-17-006A: Exclusive Studies of Short Range Correlations in Nuclei using CLAS12 (2018)
  * [104] Bickerstaff RP, Birse MC, Miller GA.  Phys. Rev. Lett. 53:2532–2535 (1984)
  * [105] Melnitchouk W, Sargsian M, Strikman MI.  Z. Phys. A 359:99–109 (1997)
  * [106] Melnitchouk W, Schreiber AW, Thomas AW.  Phys. Rev. D 49:1183–1198 (1994)
  * [107] Mulders PJ, Thomas AW.  Phys. Rev. Lett. 52:1199 (1984)
  * [108] Fomin N, et al.  Phys. Rev. Lett. 105:212502 (2010)
  * [109] Brodsky SJ, Robertson DG. 1995.  In ELFE (European Laboratory for Electrons) Summer School on Confinement Physics
  * [110] Brodsky SJ. 2004.  In 58th Scottish Universities Summer School in Physics (SUSSP58): A NATO Advanced Study Institute and EU Hadron Physics 13 Summer Institute
  * [111] Brodsky SJ.  Eur. Phys. J. A 24S1:129–135 (2005)
  * [112] Ji CR, Brodsky SJ.  Phys. Rev. D 34:1460 (1986)
  * [113] Souder P, et al.  Precision Measurement of Parity-violation in Deep Inelastic Scattering Over a Broad Kinematic Range (2010)
  * [114] Cosyn W, Sargsian M.  Int. J. Mod. Phys. E 26:1730004 (2017)
  * [115] Strikman M, Weiss C.  Phys. Rev. C 97:035209 (2018)
  * [116] Baillie N, et al.  Phys. Rev. Lett. 108:142001 (2012), [Erratum: Phys.Rev.Lett. 108, 199902 (2012)]
  * [117] Tkachenko S, et al.  Phys. Rev. C 89:045206 (2014), [Addendum: Phys.Rev.C 90, 059901 (2014)]
  * [118] Griffioen KA, et al.  Phys. Rev. C 92:015211 (2015)
  * [119] Bueltmann S, Christy M, Fenker H, Griffioen K, Keppel C, et al.  E12-06-113: The structure of the free neutron at large x-bjorken (2006)
  * [120] Chen O, Hakobyan H, Piasetzky E, Weinstein L.  E12-11-003A: In Medium Proton Structure Functions, SRC, and the EMC effect (2015)
  * [121] Chen O, Gilad S, Weinstein L, Wood S.  E12-11-107: In Medium Nucleon Structure Functions, SRC, and the EMC effect (2011)
  * [122] Jentsch A, Tu Z, Weiss C.  Phys. Rev. C 104:065205 (2021)
  * [123] Arrington J, et al.  arXiv:2112.00060 (2021)