# Maker-Breaker domination game played on corona products of graphs

Athira Divakarana, Email<EMAIL_ADDRESS>Tijo Jamesb, Email:
<EMAIL_ADDRESS>Sandi Klavžar c,d,e, Email<EMAIL_ADDRESS>Latha S Naira, Email<EMAIL_ADDRESS>
###### Abstract

In the Maker-Breaker domination game, Dominator and Staller play on a graph
$G$ by taking turns in which each player selects a not yet played vertex of
$G$. Dominator’s goal is to select all the vertices in a dominating set, while
Staller aims to prevent this from happening. In this paper, the game is
investigated on corona products of graphs. Its outcome is determined as a
function of the outcome of the game on the second factor. Staller-Maker-
Breaker domination numbers are determined for arbitrary corona products, while
Maker-Breaker domination numbers of corona products are bounded from both
sides. All the bounds presented are demonstrated to be sharp. Corona products
as well as general graphs with small (Staller-)Maker-Breaker domination
numbers are described.

a Department of Mathematics, Mar Athanasius College, Kothamangalam, India

b Department of Mathematics, Pavanatma College, Murickassery, India

c Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

d Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

e Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

Keywords: Maker-Breaker game; Maker-Breaker domination game; Maker-Breaker
domination number; corona product of graphs

AMS Subj. Class. (2010): 05C57, 05C69

##  1 Introduction

The Maker-Breaker game was introduced by Erdős and Selfridge in [11]. The game
is played on an arbitrary hypergraph by two players named Maker and Breaker.
The players alternately select an unplayed vertex during the game. Maker’s aim
is to occupy all the vertices of some hyperedge, the goal of Breaker is to
prevent him from doing it. The game has been extensively researched, both in
general and in specific cases, cf. the book [16], the recent paper [20], and
references therein.

In this paper we are interested in the domination version of the Maker-Breaker
game which was introduced in 2020 by Duchêne, Gledel, Parreau, and Renault
[9]. The Maker-Breaker domination game (MBD game for short) is a game played
on a graph $G=(V(G),E(G))$ by two players named Dominator and Staller. These
names were chosen so that the players are named in line with the previously
intensively researched domination game [2, 3]. Just as in the general case,
the two players alternately select unplayed vertices of $G$. The aim of
Dominator is to select all the vertices of some dominating set of $G$, while
Staller aims to select at least one vertex from every dominating set of $G$.
There are two variants of this game depending on which player has the first
move. A D-game is the MBD game in which Dominator has the first move and an
S-game is the MBD game in which Staller has the first move.

The following graph invariants are naturally associated to the MBD game [4,
15]. The Maker-Breaker domination number, $\gamma_{\rm MB}(G)$, is the minimum
number of moves of Dominator to win the D-game on $G$ when both players play
optimally. That is, $\gamma_{\rm MB}(G)$ is the minimum number of moves of
Dominator such that he wins in this number of moves no matter how Staler is
playing. If Dominator has no winning strategy in the D-game, then set
$\gamma_{\rm MB}(G)=\infty$. The Staller-Maker-Breaker domination number,
$\gamma_{\rm SMB}(G)$, is the minimum number of moves of Staller to win the
D-game on $G$ when both players play optimally, where $\gamma_{\rm
SMB}(G)=\infty$ if Staller has no winning strategy. In a similar manner,
$\gamma_{\rm MB}^{\prime}(G)$ and $\gamma_{\rm SMB}^{\prime}(G)$ are the two
parameters associated with the $S$-game. Briefly, we will call $\gamma_{\rm
MB}(G)$, $\gamma_{\rm MB}^{\prime}(G)$, $\gamma_{\rm SMB}(G)$, and
$\gamma_{\rm SMB}^{\prime}(G)$ the MBD numbers of $G$.

In the seminal paper [9] it was proved, among other things, that deciding the
winner of the MBD game can be solved efficiently on trees, but it is PSPACE-
complete in general. These results have been very recently strengthened in
[10] by proving that the game is PSPACE-complete on split and bipartite graphs
and by developing a linear time algorithm to solve this game in forests. The
paper [4] focuses on $\gamma_{\rm SMB}(G)$ and $\gamma_{\rm SMB}^{\prime}(G)$
and among other results establishes appealing exact formulas for $\gamma_{\rm
SMB}^{\prime}(G)$ where $G$ is a path. In [5], for every positive integer $k$,
trees $T$ with $\gamma_{\rm SMB}(T)=k$ are characterized and exact formulas
for $\gamma_{\rm SMB}(G)$ and $\gamma_{\rm SMB}^{\prime}(G)$ derived for
caterpillars. In the main result of [13], $\gamma_{\rm MB}$ and $\gamma_{\rm
MB}^{\prime}$ are determined for Cartesian products of $K_{2}$ by a path. In
[7], the MBD game is further studied on Cartesian products of paths, stars,
and complete bipartite graphs. The total version of the MBD game was
introduced in [14] and further investigated in [12].

The corona product of graphs is a graph operation that has already been
studied from many perspectives, [1, 8, 17, 18, 19, 21] is a selection of
recent relevant contributions. In this paper, we investigate the MBD game
played on these products and proceed as follows. In the rest of the
introduction definitions and known results needed are stated. In the next
section we determine the outcome of the MBD game on an arbitrary corona
product of graphs. In Section 3 we determine $\gamma_{\rm SMB}$ and
$\gamma_{\rm SMB}^{\prime}$, and bound $\gamma_{\rm MB}$ and $\gamma_{\rm
MB}^{\prime}$ of arbitrary corona products. We demonstrate that all the bounds
are sharp. In particular, $\gamma_{\rm MB}$ and $\gamma_{\rm MB}^{\prime}$ are
determined for corona products of arbitrary graphs by trees with perfect
matchings. In Section 4 we describe corona products with small MBD numbers and
along the way describe general graphs with small MBD numbers.

Let $G=(V(G),E(G))$ be a graph. The order of $G$ will be denoted by $n(G)$.
For a vertex $v\in V(G)$, its open neighbourhood is denoted by $N(v)$ and its
closed neighbourhood by $N[v]$. The degree of $v$ is $\deg(v)=|N(v)|$. The
minimum and the maximum degree of $G$ are denoted by $\delta(G)$ and
$\Delta(G)$. An isolated vertex is a vertex of degree $0$, a leaf is a vertex
of degree $1$. A support vertex is a vertex adjacent to a leaf, a strong
support vertex is a vertex that is adjacent to at least two leaves. A
dominating set of $G$ is a set $D\subseteq V(G)$ such that each vertex from
$V(G)\setminus D$ has a neighbour in $D$. The domination number $\gamma(G)$ of
$G$ is the minimum cardinality of a dominating set of $G$. A vertex of degree
$n(G)-1$ is a dominating vertex.

The corona $G\odot H$ of graphs $G$ and $H$ is obtained by taking one copy of
$G$ and $n(G)$ copies of $H$ by joining the $i^{{\rm th}}$ vertex of $G$ to
each vertex in the $i^{{\rm th}}$ copy of $H$. Clearly, $n(G\odot
H)=n(G)(n(H)+1)$ and $\delta(G\odot H)=\delta(H)+1$. It is also
straightforward to see that $\gamma(G\odot H)=n(G)$.

We next collect known results about the MBD game to be used later on.

######  Lemma 1.1 (No-Skip Lemma for Dominator [15])

In an optimal strategy of Dominator to achieve $\gamma_{\rm MB}(G)$ or
$\gamma_{\rm MB}^{\prime}(G)$ it is never an advantage for him to skip a move.
Moreover, if Staller skips a move it can never disadvantage Dominator.

######  Lemma 1.2 (No-Skip Lemma for Staller [4])

In an optimal strategy of Staller to achieve $\gamma_{\rm SMB}(G)$ or
$\gamma_{\rm SMB}^{\prime}(G)$ it is never an advantage for her to skip a
move. Moreover, if Dominator skips a move it can never disadvantage Staller.

######  Theorem 1.3 (Continuation Principle [15])

Let $G$ be a graph with $A,B\subseteq V(G)$. If $B\subseteq A$ then
$\gamma_{\rm MB}(G|A)\leq\gamma_{\rm MB}(G|B)$ and $\gamma_{\rm
MB}^{\prime}(G|A)\leq\gamma_{\rm MB}^{\prime}(G|B)$.

###### Theorem 1.4

[4] Let G be a graph.

  1. 1.

If $\gamma_{\rm SMB}^{\prime}(G)<\infty$, then $\gamma_{\rm
SMB}^{\prime}(G)\leq\left\lceil\frac{n(G)}{2}\right\rceil$.

  2. 2.

If $\gamma_{\rm SMB}(G)<\infty$, then $\gamma_{\rm
SMB}(G)\leq\left\lfloor\frac{n(G)}{2}\right\rfloor$.

Moreover, both bounds are sharp.

##  2 Outcome of the game on corona products

In this section we determine the winner of the MBD game played on corona
products. Before that we argue in the rest we can restrict our attention to
corona products in which the first factor is of order at least two.

Let $H$ be an arbitrary graph. Then it is straightforward to verify that

$\displaystyle\gamma_{\rm MB}(K_{1}\odot H)$ $\displaystyle=1\,,$
$\displaystyle\gamma_{\rm MB}^{\prime}(K_{1}\odot H)$
$\displaystyle=\gamma_{\rm MB}(H)\,,$ $\displaystyle\gamma_{\rm
SMB}(K_{1}\odot H)$ $\displaystyle=\infty\,,$ $\displaystyle\gamma_{\rm
SMB}^{\prime}(K_{1}\odot H)$ $\displaystyle=\gamma_{\rm SMB}(H)\,.$

Therefore, we may indeed restrict in the rest of the paper to corona products
$G\odot H$, where $G$ is a connected graph with $n(G)\geq 2$. In what follows,
we will also always assume that $V(G)=\\{v_{1},\ldots,v_{n(G)}\\}\subseteq
V(G\odot H)$, and denote by $H_{i}$ the copy of $H$ in $G\odot H$ assigned to
$v_{i}$, $i\in[n(G)]$.

It was proved in [9] that for the outcome $o(G)$ of the MBD game played on $G$
we have $o(G)\in\\{\mathcal{D},\mathcal{S},\mathcal{N}\\}$, with the following
meaning:

  * •

$o(G)=\mathcal{D}$: Dominator has a winning strategy no matter who starts the
game;

  * •

$o(G)=\mathcal{S}$: Staller has a winning strategy no matter who starts the
game;

  * •

$o(G)=\mathcal{N}$: the first player has a winning strategy.

We now prove that the winner in the MBD game played on a corona product
depends only on the winner of game played on the second factor.

###### Theorem 2.1

If $G$ is a connected graph with $n(G)\geq 2$ and $H$ is a graph, then

$o(G\odot
H)=\begin{cases}\mathcal{D};&o(H)\in\\{\mathcal{D},\mathcal{N}\\},\\\
\mathcal{S};&o(H)=\mathcal{S}.\end{cases}$

Proof. Suppose first that $o(H)\in\\{\mathcal{D},\mathcal{N}\\}$. Then in both
cases we have $\gamma_{\rm MB}(H)<\infty$. We describe a strategy for
Dominator by which he can win the MBD game played on $G\odot H$, no matter who
is the first player. First, if Staller selects a vertex $v_{i}$ as her first
move, then Dominator replies by a vertex in $H_{i}$ which is his optimal first
move when the game is played on $H$. During later stages of the game, if
Staller at some point selects another vertex of $H_{i}$, Dominator replies by
an optimal move (w.r.t. the game played in $H$) in $H_{i}$. Playing like that,
at the end of the game Dominator will dominate the vertices of $H_{i}$.
Second, if Staller first selects some vertex of $H_{i}$, then Dominator
replies by playing $v_{i}$ and then clearly at the end of the game he will
dominate the vertices of $H_{i}$. Following the described strategy of
Dominator we conclude that $o(G\odot H)=\mathcal{D}$ when
$o(H)\in\\{\mathcal{D},\mathcal{N}\\}$.

Suppose second that $o(H)=\mathcal{S}$. As $n(G)\geq 2$, Staller can select
some vertex $v_{i}$ as her first move, no matter who starts the game, such
that no vertex of $H_{i}$ has yet been played. In the continuation of the
game, Staller can isolate a vertex in $H_{i}$ because $o(H)=\mathcal{S}$, that
is, she follows her optimal strategy on $H_{i}$. We conclude that
$o(H)=\mathcal{S}$. $\square$

##  3 MBD numbers of corona products

In this section we consider the MBD numbers of corona products $G\odot H$. In
view of Theorem 2.1 we consider three cases depending on $o(H)$. We start with
the simplest case $o(H)=\mathcal{S}$ which enables an exact solution as
follows.

###### Theorem 3.1

If $o(H)=\mathcal{S}$ and $G$ is a connected graph with $n(G)\geq 2$, then

$\gamma_{\rm SMB}(G\odot H)=\gamma_{\rm SMB}^{\prime}(G\odot H)=1+\gamma_{\rm
SMB}(H).$

Proof. Using Theorem 1.3, a vertex $v_{i}$ of $G$ is an optimal first move for
Dominator. Then all the vertices in its closed neighbourhood in $G\odot H$ are
dominated. Staller responds by selecting a vertex $v_{j}$, where $j\neq i$.
Staller’s plan afterwards is to play only on $H_{j}$. If Dominator plays his
second move on $H_{j}$, then Staller can finish the game by selecting
$\gamma_{\rm SMB}(H)$ vertices of $H_{j}$ and if Staller is able to play first
in $H_{j}$, this can be done in $\gamma_{\rm SMB}^{\prime}(H)$ moves by
Staller. As $\gamma_{\rm SMB}(H)\geq\gamma_{\rm SMB}^{\prime}(H)$, see [4,
Proposition 2.7], Staller thus has a strategy to end the game in at most
$1+\gamma_{\rm SMB}(H)$ moves. On the other hand, Dominator can force this
number of moves by playing (optimally in the S-game on $H$) his second move in
$H_{j}$, hence $\gamma_{\rm SMB}(G\odot H)\leq 1+\gamma_{\rm SMB}(H)$.

The above argument is actually independent of who moves first since $n(G)\geq
2$ and Staller can select a vertex of $G$ as her first move. Therefore we also
have $\gamma_{\rm SMB}^{\prime}(G\odot H)=1+\gamma_{\rm SMB}(H)$. $\square$

The next case we consider is $o(H)=\mathcal{D}$, for which the following
holds.

###### Theorem 3.2

If $H\in\mathcal{D}$ and $G$ is a connected graph with $n(G)\geq 2$, then

$\left\lceil\frac{n(G)}{2}\right\rceil+\left\lfloor\frac{n(G)}{2}\right\rfloor\gamma_{\rm
MB}(H)\leq\gamma_{\rm MB}(G\odot
H)\leq\left\lceil\frac{n(G)}{2}\right\rceil+\left\lfloor\frac{n(G)}{2}\right\rfloor\gamma_{\rm
MB}^{\prime}(H)$

and

$\left\lfloor\frac{n(G)}{2}\right\rfloor+\left\lceil\frac{n(G)}{2}\right\rceil\gamma_{\rm
MB}(H)\leq\gamma_{\rm MB}^{\prime}(G\odot
H)\leq\left\lfloor\frac{n(G)}{2}\right\rfloor+\left\lceil\frac{n(G)}{2}\right\rceil\gamma_{\rm
MB}^{\prime}(H)\,.$

Proof. Consider the following strategy of Dominator in the D-game played on
$G\odot H$. His basic idea is to select as many vertices of $G$ as possible.
In particular, if at some point of the game Staller plays a vertex of $H_{i}$
and the vertex $v_{i}$ has not yet been played, then Dominator selects $v_{i}$
as his next move. On the other hand, if at some point of the game Staller
plays a vertex $w$ of $H_{i}$ and the vertex $v_{i}$ has been already selected
by Staller, then Dominator replies by a vertex in $H_{i}$ which is an optimal
reply for him in the game played on $H\cong H_{i}$ after the first move $w$ of
Staller. In the continuation of the game, whenever Staller plays in some
$H_{i}$, Dominator replies optimally there. It is also possible that Dominator
is the first to play in some $H_{i}$ in which no vertex has yet been played
(but $v_{i}$ was played by Staller), in which case Dominator selects a vertex
which is optimal in the D-game played on $H$.

Following the above strategy, Dominator will select at least
$\left\lceil\frac{n(G)}{2}\right\rceil$ vertices of $G$. If Staller was the
first to select a vertex in $H_{i}$, then Dominator will select at most at
most $\gamma_{\rm MB}^{\prime}(H)$ vertices of $H_{i}$. Similarly, If
Dominator was the first to select a vertex in $H_{i}$, then he will select at
most at most $\gamma_{\rm MB}(H)$ vertices of $H_{i}$. Since $\gamma_{\rm
MB}(H)\leq\gamma_{\rm MB}^{\prime}(H)$ (see [15, Theorem 3.1]), Dominator will
play at most $\gamma_{\rm MB}^{\prime}(H)$ vertices of each $H_{i}$, where
$v_{i}$ was selected by Staller. As there are at most
$\left\lfloor\frac{n(G)}{2}\right\rfloor$ such vertices, the upper bound for
$\gamma_{\rm MB}(G\odot H)$ follows.

To prove the lower bound for $\gamma_{\rm MB}(G\odot H)$, consider the
following strategy of Staller. During the first phase of the game, Staller
selects as many vertices of $G$ as possible, that is,
$\left\lfloor\frac{n(G)}{2}\right\rfloor$ vertices. After that, whenever
Dominator plays a vertex of $H_{i}$, Staller replies there by her optimal
strategy in the game played on $H$. In this way Staller forces to select at
least $\gamma_{\rm MB}(H)$ vertices in $H_{i}$ if Dominator was the first to
play in $H_{i}$ and at least $\gamma_{\rm MB}^{\prime}(H)$ vertices if Staller
was the first to play in $H_{i}$. Since $\gamma_{\rm
MB}^{\prime}(H)\geq\gamma_{\rm MB}(H)$, in each of the
$\left\lfloor\frac{n(G)}{2}\right\rfloor$ graphs $H_{i}$ for which $v_{i}$ was
played by Staller, at least $\gamma_{\rm MB}(H)$ vertices will be played by
Dominator. This proves the lower bound.

The arguments for the bounds for $\gamma_{\rm MB}^{\prime}(G\odot H)$ are
parallel and hence omitted. $\square$

The bounds of Theorem 3.2 are sharp as can be demonstrated by the following
two consequences of the theorem, where the first one directly follows from it.

###### Corollary 3.3

If $H\in\mathcal{D}$ with $\gamma_{\rm MB}(H)=\gamma_{\rm MB}^{\prime}(H)$,
and $G$ is a connected graph with $n(G)\geq 2$, then

$\gamma_{\rm MB}(G\odot
H)=\left\lceil\frac{n(G)}{2}\right\rceil+\left\lfloor\frac{n(G)}{2}\right\rfloor\gamma_{\rm
MB}(H)$

and

$\gamma_{\rm MB}^{\prime}(G\odot
H)=\left\lfloor\frac{n(G)}{2}\right\rfloor+\left\lceil\frac{n(G)}{2}\right\rceil\gamma_{\rm
MB}(H).$

###### Theorem 3.4

Let $G$ be a graph with $n(G)\geq 2$ and let $T$ be a tree with a perfect
matching. Then

$\gamma_{\rm MB}(G\odot
T)=\left\lceil\frac{n(G)}{2}\right\rceil+\left\lfloor\frac{n(G)}{2}\right\rfloor\frac{n(T)}{2}$

and

$\gamma_{\rm MB}^{\prime}(G\odot
T)=\left\lfloor\frac{n(G)}{2}\right\rfloor+\left\lceil\frac{n(G)}{2}\right\rceil\frac{n(T)}{2}\,.$

Proof. Since $T$ is a tree with a perfect matching, we know from [14] that
$o(T)=\mathcal{D}$ as well as that $\gamma_{\rm MB}(T)=\gamma_{\rm
MB}^{\prime}(T)=\frac{n(T)}{2}$. The conclusion then follows by Corollary 3.3.
$\square$

The last case to consider is $o(H)=\mathcal{N}$, for which the following
holds.

###### Theorem 3.5

If $o(H)=\mathcal{N}$ and $G$ is a graph with $n(G)\geq 2$, then

$1+\left\lfloor\frac{n(G)-1}{2}\right\rfloor\gamma(H)+\left\lfloor\frac{n(G)}{2}\right\rfloor\gamma_{\rm
MB}(H)\leq\gamma_{\rm MB}(G\odot H)\leq 1+(n(G)-1)\gamma_{\rm MB}(H)$

and

$\left\lfloor\frac{n(G)}{2}\right\rfloor\gamma(H)+\left\lceil\frac{n(G)}{2}\right\rceil\gamma_{\rm
MB}(H)\leq\gamma_{\rm MB}^{\prime}(G\odot H)\leq n(G)\gamma_{\rm MB}(H)\,.$

Proof. Consider the D-game on $G\odot H$.

To prove the lower bound, consider the following strategy of Staller. After
Dominator selects some vertex of $G$, say $v_{i}$, as his optimal first move
in view of Theorem 1.3, Staller selects as her first move another vertex of
$G$, say $v_{j}$. Because $O(H)=\mathcal{N}$, Dominator must reply by playing
(an optimal move with respect to the game played on $H$) in $H_{j}$. After
that Staller continues by selecting the vertices of $G$ until all are vertices
of $G$ but $v_{i}$ are played by Staller. For each such vertex $v_{k}$
Dominator was forced to select a vertex from $H_{k}$. Hence until this point
of the game Dominator played $n(G)$ vertices, the vertex $v_{i}$ and one
vertex in each $H_{j}$, $j\neq i$. In the rest of the game, Staller will be
able to play the second move in at least
$\left\lfloor\frac{n(G)}{2}\right\rfloor$ copies $H_{j}$ of $H$. Consequently,
she will be able to force at least $\gamma_{\rm MB}(H)$ moves of Dominator in
each of these copies. As in the remaining
$\left\lfloor\frac{n(G)-1}{2}\right\rfloor$ copies of $H$ at least $\gamma(H)$
moves must be done Dominator, the bound follows.

For the upper bound, we argue as follows. After Dominator selects a vertex
$v_{i}$ as his first move, Staller responds to this move by playing a vertex
$v_{j}$, $j\neq i$. Otherwise, if Staller selects a vertex in $H_{j}$ then
Dominator can dominate all the vertices of $H_{j}$ by selecting the vertex
$v_{j}$ in a single move. Since $\gamma_{\rm MB}(H)<\infty$, Dominator has an
optimal strategy in the D-game on $H$ to finish it in at most $\gamma_{\rm
MB}(H)$ moves. Dominator must play a vertex of $H_{j}$ as his response after
the Staller’s move on $v_{j}$ because $\gamma_{\rm MB}^{\prime}(H)=\infty$.
Now Dominator’s strategy is as follows. If Staller plays on $H_{j}$ then
Dominator uses his optimal strategy on $H_{j}$ for the remaining moves in
$H_{j}$. Otherwise, if Staller selects a vertex of $G$, say $v_{k}$, then
Dominator uses his optimal strategy in $H_{k}$ for the remaining moves in
$H_{k}$. In this strategy, Dominator can finish the game on $G\odot H$ in
$1+(n(G)-1)\gamma_{\rm MB}(H)$ moves. Here, Dominator may not play optimally
but Staller plays optimally in $G\odot H$. Therefore $\gamma_{\rm MB}(G\odot
H)\leq 1+(n(G)-1)\gamma_{\rm MB}(H)$.

Similar arguments also hold for the S-game on $G\odot H$. The only difference
is that Staller starts the game and hence she can select all the vertices of
$G$ in $G\odot H$. After that Dominator can ensure that he will play at most
$\gamma_{\rm MB}(H)$ vertices in each $H_{i}$ by replying to each move of
Staller in some $H_{j}$ by an optimal move in the same $H_{j}$. As for the
lower bound, Staller can now force Dominator to play at least $\gamma_{\rm
MB}(H)$ moves in $\left\lceil\frac{n(G)}{2}\right\rceil$ copies $H_{j}$ of
$H$. $\square$

All the bounds of Theorem 3.5 are sharp as follows from the next consequence
of Theorem 3.5.

###### Corollary 3.6

If $H$ is a graph with $o(H)=\mathcal{N}$ and $\gamma_{\rm MB}(H)=\gamma(H)$,
and $G$ is a graph with $n(G)\geq 2$, then

$\gamma_{\rm MB}(G\odot H)=1+(n(G)-1)\gamma_{\rm MB}(H)$

and

$\gamma_{\rm MB}^{\prime}(G\odot H)=n(G)\gamma_{\rm MB}(H)\,.$

In particular, if $\gamma_{\rm MB}(H)=2$, then $\gamma_{\rm MB}(G\odot
H)=2n(G)-1$ and $\gamma_{\rm MB}^{\prime}(G\odot H)=2n(G)$.

Proof. Since $\gamma_{\rm MB}(H)=\gamma(H)$ and
$\left\lfloor\frac{n(G)-1}{2}\right\rfloor+\left\lfloor\frac{n(G)}{2}\right\rfloor=n-1$,
the bounds for $\gamma_{\rm MB}(G\odot H)$ in Theorem 3.5 coincide, and since
$\left\lfloor\frac{n(G)}{2}\right\rfloor+\left\lceil\frac{n(G)}{2}\right\rceil=n$
the same conclusion holds for the bounds for $\gamma_{\rm MB}^{\prime}(G\odot
H)$.

If $\gamma_{\rm MB}(H)=2$, then $G$ clearly does not contain a dominating
vertex, hence $\gamma(H)\geq 2$ and therefore $\gamma(H)=2$. The last
assertion of the corollary now follows from the first one. $\square$

Let $H_{m}$, $m\geq 3$, be the graph obtained from $m$ disjoint copies of
$C_{4}$ by selecting a vertex in each of the copies and identifying these
vertices. (The identified vertex is of degree $2m$ in $H_{m}$.) Then it is not
hard to check that $o(H_{m})=\mathcal{N}$ and that $\gamma_{\rm
MB}(H_{m})=\gamma(H_{m})=m+1$. Hence the graphs $H_{m}$ fulfill the
assumptions of Corollary 3.6.

On the other hand, the bounds of Theorem 3.5 are not sharp in general. To see
this, consider the example from Fig. 1 in which $G=P_{3}$ and $H$ is clear
from the figure. To make the figure clearer, the edges between the vertices of
$G$ and $H$ are only indicated.

$g_{1}=s_{1}$$d_{2}$$s_{3}$$H_{1}$$g_{2}=d_{1}$$H_{2}$$g_{3}=s_{2}$$d_{3}$$d_{4}$
!$H_{3}$ Figure 1: An example in which it is useful for Dominator to skip a
move in some copy of $H$

Note first that $O(H)=\mathcal{N}$. in particular, the support vertex of $H$
is the optimal first move of Staller in the S-game. Moreover, the same vertex
is the optimal first move of Dominator in the D-game and we have $\gamma_{\rm
MB}(H)=3$. Consider now the D-game played on $P_{3}\odot H$, see Fig. 1 again.
In view of Theorem 1.3, Dominator begins the game by selecting $d_{1}=g_{2}$.
After Staller replies by $s_{1}=g_{1}$, Dominator is forced to play $d_{2}$ in
$H_{1}$, and after the move $s_{2}=g_{3}$ Dominator must play in $H_{3}$.
Assume without loss of generality that afterwards Staller plays a vertex of
$H_{1}$. Now it is useful for Dominator not to follow Staller in $H_{1}$ but
instead playing the vertex $d_{4}$ of $H_{3}$ as shown in the figure. In this
way, Dominator will finish the game in six moves, which is also optimal for
him, that is, $\gamma_{\rm MB}(P_{3}\odot H)=6$ which coincides with the lower
bound of the theorem. On the other hand, the upper bound of Theorem 3.5 yields
$\gamma_{\rm MB}(P_{3}\odot H)\leq 7$.

We conclude with another consequence of Theorem 3.5 which demonstrates the
sharpness of bounds.

###### Corollary 3.7

If $H\in\mathcal{D}$, then $\gamma_{\rm MB}(K_{2}\odot H)=1+\gamma_{\rm
MB}(H)$.

Proof. In this case the bounds of Theorem 3.5 coincide. $\square$

##  4 Corona products with small MBD numbers

In this section we characterize corona products whose MBD numbers are equal to
$2$ or $3$. For this purpose we describe along the way graphs whose MBD
numbers are equal to $2$. This fills a gap in the literature. In fact, graphs
$G$ with $\gamma_{\rm MB}(G)=2$ were already described as follows. For this
sake note that if $\gamma(G)=1$, then $\gamma_{\rm MB}(G)=1$, and if
$\gamma(G)\geq 3$, then $\gamma_{\rm MB}(G)\geq 3$.

###### Proposition 4.1

[14, Proposition 3.2] Let $G$ be a graph with $\gamma(G)=2$. Then $\gamma_{\rm
MB}(G)=\gamma(G)=2$ if and only if $G$ has a vertex that lies in at least two
$\gamma$-sets of $G$.

We next complement Proposition 4.1 with the following result.

###### Proposition 4.2

Let $G$ be a graph with $\gamma(G)\leq 2$. Then $\gamma_{\rm
MB}^{\prime}(G)=2$ if and only if $G$ has at most one dominating vertex and at
least two vertices that lie in two different dominating sets of $G$ of
cardinality $2$.

Proof. Assume first that $\gamma_{\rm MB}^{\prime}(G)=2$. Then $G$ has at most
one dominating vertex for otherwise Dominator can finish the game in his first
move. Suppose that $G$ contains at most one vertex that lies in two different
dominating sets of $G$. If $x$ is a such a vertex, Staller can select $x$ as
her first move. Then, no matter how Dominator plays, he needs at least three
moves to finish the game.

Conversely, assume that $G$ has at most one dominating vertex and at least two
vertices that lie in two different dominating sets of $G$ of cardinality $2$.
Then Staller can prevent Dominator from winning in one move by selecting first
a possible dominating vertex. However, no matter how Staller starts, after her
move there exists a not yet played vertex which lies in two dominating sets of
$G$ of cardinality $2$. By selecting such a vertex, Dominator will win the
game in his second move. $\square$

It is straightforward to check that in the case of corona products $G\odot H$,
each of Propositions 4.1 and 4.2 yields the same condition on $H$, that is, we
have:

###### Corollary 4.3

If $G$ is a graph with $n(G)\geq 2$, then $\gamma_{\rm MB}(G\odot
H)=\gamma_{\rm MB}^{\prime}(G\odot H)=2$ if and only if $n(G)=2$ and $H$ has a
dominating vertex.

Graphs $G$ with $\gamma_{\rm SMB}(G)=2$ and with $\gamma_{\rm
SMB}^{\prime}(G)=2$ can be respectively described as follows.

###### Proposition 4.4

Let $G$ be a graph with $\delta(G)\geq 1$. Then the following hold.

  1. (i)

$\gamma_{\rm SMB}(G)=2$ if and only if $G$ has at least two strong support
vertices.

  2. (ii)

$\gamma_{\rm SMB}^{\prime}(G)=2$ if and only if $G$ has a strong support
vertex.

Proof. (i) Let $G$ be a graph with $\delta(G)\geq 1$ and assume that
$\gamma_{\rm SMB}(G)=2$. Suppose that $G$ has at most one strong support
vertex. Dominator first selects this strong support vertex. Now, Staller
replies to this move with a vertex $v$. If at most one leaf is adjacent to
$v$, then Dominator selects this leaf as his next turn and Staller cannot
finish the game in her next move. Clearly, Staller can finish the game in her
next move only when either $G$ has isolated vertices or at least two leaves
adjacent to $v$. Since $\delta(G)\geq 1$, $G$ has no isolated vertices. Thus
$G$ has at least two strong support vertices.

Conversely, assume that $G$ has at least two strong support vertices. In the
first move Staller can select a strong support vertex such that none of its
leaves has been played by Dominator in his first move. Then Staller can
isolate one leaf in her second move, hence $\gamma_{\rm SMB}(G)\leq 2$. Since
$\delta(G)\geq 1$, we also have $\gamma_{\rm SMB}(G)\geq 2$. Therefore
$\gamma_{\rm SMB}(G)=2$.

(ii) Assume that $u$ us a strong support vertex of $G$. Since $\delta(G)\geq
1$, we have $\gamma_{\rm SMB}^{\prime}(G)\geq 1+\delta(G)\geq 2$. If Staller
first selects $u$ in the S-game, then she can win the game in her second move,
hence we also have $\gamma_{\rm SMB}^{\prime}(G)\leq 2$.

Conversely, assume that $\gamma_{\rm SMB}^{\prime}(G)=2$. Since
$1+\delta(G)\leq\gamma_{\rm SMB}^{\prime}(G)=2$, we see that $G$ has a leaf.
Staller first selects a vertex $u$ which is adjacent to a leaf for otherwise
$\gamma_{\rm SMB}^{\prime}(G)\geq 3$. If there is only one leaf adjacent to
$u$, then Dominator selects that leaf as his next move. Since $\gamma_{\rm
SMB}^{\prime}(G)=2$, Staller can finish the game in her next move. This is
possible only when $G$ has an isolated vertex or $u$ has an unplayed leaf.
Since $\delta(G)\geq 1$, $G$ has no isolated vertices. Therefore $u$ is a
strong support vertex. $\square$

Proposition 4.4 has been independently established by Bujtás and Dokyeesun and
is contained in [6].

It is again straightforward to check that in the case of corona products
$G\odot H$, each of the conditions (i) and (ii) of Propositions 4.4 and 4.2
yields the same condition on $H$ as follows.

###### Corollary 4.5

If $G$ is a graph with $n(G)\geq 2$, then $\gamma_{\rm SMB}(G\odot
H)=\gamma_{\rm SMB}^{\prime}(G\odot H)=2$ if and only if $H$ has at least two
isolated vertices.

To describe graphs with SMB numbers equal to three is more tricky.
Nevertheless we can do it for $\gamma_{\rm SMB}^{\prime}$ as follows.

###### Proposition 4.6

Let $G$ be a graph with $\delta(G)\geq 1$. Then $\gamma_{\rm
SMB}^{\prime}(G)=3$ if and only if $G$ satisfies the following conditions.

  1. (i)

$G$ has no strong support vertices.

  2. (ii)

There exists a vertex $u\in V(G)$ such that either $G-u$ has at least two
strong support vertices or $G-u$ has exactly one isolated vertex and a strong
support vertex.

Proof. Assume first that $\gamma_{\rm SMB}^{\prime}(G)=3$. Then $G$ has no
isolated vertices and no strong support vertices. Let $u$ be an optimal first
move of Staller in the S-game. After this move, the remaining game is a D-game
on $G-u$ and we have $\gamma_{\rm SMB}(G-u)=2$. In view of Proposition 4.4
this is possible only when either $G-u$ has at least two strong support
vertices or $G-u$ has exactly one isolated vertex and a strong support vertex.

Conversely, assume that $G$ is a graph which satisfies (i) and (ii). Let $u$
be a vertex that satisfies (ii) and let $u$ be the first move of Staller in
the S-game. If $G-u$ has an isolated vertex then Dominator selects this vertex
for otherwise Staller could claim it in the next move to win the game. After
that, Staller selects a strong support vertex and wins the game in her third
move. Similarly, if $G-u$ has no isolated vertex, then $G-u$ has two strong
support vertices and again Staller can win in her third move. We conclude that
$\gamma_{\rm SMB}^{\prime}(G)=3$. $\square$

For corona products Proposition 4.6 yields:

###### Corollary 4.7

Let $G$ be a graph with $n(G)\geq 2$. Then $\gamma_{\rm SMB}(G\odot
H)=\gamma_{\rm SMB}^{\prime}(G\odot H)=3$ if and only if either $H$ has at
least two strong support vertices, or $H$ has an isolated vertex and at least
one strong support vertex.

An alternative way to derive Corollary 4.7 is to use Theorem 3.1, that is,
$\gamma_{\rm SMB}(G\odot H)=\gamma_{\rm SMB}^{\prime}(G\odot H)=3$ if and only
if $\gamma_{\rm SMB}(H)=2$. If $\delta(H)\geq 1$, then Proposition 4.4(i)
implies that this is if and only if $H$ has at least two strong support
vertices. On the other hand, if $H$ has an isolated vertex, we infer that it
must at least one strong support vertex.

We conclude the section by a description of corona products with $\gamma_{\rm
MB}$ equal to three.

###### Proposition 4.8

Let $G$ be a graph with $n(G)\geq 2$ and let $H$ be graphs. Then $\gamma_{\rm
MB}(G\odot H)=3$ if and only if either $n(G)=2$ and $\gamma_{\rm MB}(H)=2$, or
$n(G)=3$ and $H$ has a dominating vertex.

Proof. If $\gamma_{\rm MB}(G\odot H)=3$, then clearly $n(G)\leq 3$. Assume
first $n(G)=2$. In their first two moves, Dominator and Staller will
respectively select the two vertices of $G$, no matter whether the D-game or
the S-game is played. After that, $\gamma_{\rm MB}(G\odot H)=3$ holds if and
only if $\gamma_{\rm MB}(H)=2$. Similarly, if $n(G)=3$, then Dominator can win
if and only if $H$ has a dominating vertex. $\square$

##  5 Concluding remarks

In this paper, we explore the Maker-Breaker domination game on corona products
of graphs. We have identified some general features, but several directions
remain open for further exploration. Let us point to some of them. In Theorem
3.4 we have determined $\gamma_{\rm MB}(G\odot T)$ and $\gamma_{\rm
MB}^{\prime}(G\odot T)$ for an arbitrary graph $G$ with with $n(G)\geq 2$ and
an arbitrary tree $T$ with a perfect matching. It would be interesting to
derive such a result also for arbitrary trees, that is, also for trees which
do not contain perfect matchings. With respect to small MBD numbers it in
particular remains to consider arbitrary graphs $G$ with $\gamma_{\rm
SMB}(G)=3$, or with $\gamma_{\rm MB}(G)=3$, or with $\gamma_{\rm
MB}^{\prime}(G)=3$, as well as corona products $G\odot H$ with $\gamma_{\rm
MB}^{\prime}(G\odot H)=3$.

## Acknowledgements

Sandi Klavžar was supported by the Slovenian Research Agency ARIS (research
core funding P1-0297 and projects J1-2452, N1-0285).

## Declaration of interests

The authors declare that they have no conflict of interest.

## Data availability

Our manuscript has no associated data.

## References

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2021

[1]Axel Schmidt

1]Department of Physics, The George Washington University, Washington, DC
20052, USA

# Studying the Impact of Virtuality-Dependent Nucleon Structure Modification
on Spectator-Tagged Deep Inelastic Scattering

Sara Ratliff<EMAIL_ADDRESS>[

###### Abstract

Measurements of deep inelastic scattering from nuclei have revealed that the
partonic structure of bound nucleons differs from that of free nucleons. One
hypothesis is that this structure modification primarily occurs in highly
virtual nucleons participating in short-range correlations, although
distinguishing this from other hypotheses is difficult with inclusive
measurements alone. Spectator-tagged deep inelastic scattering, on the other
hand, may offer a way to specifically probe the partonic structure of highly-
virtual nucleons by detecting the correlated emission of a spectator nucleon.
Here, we present a method for calculating a “spectator-tagged” structure
function for a nucleus by combining Generalized Contact Formalism’s
description of short-range correlations with light-cone convolution formalism
to determine the impact of nucleon motion on the structure function. We apply
this method to calculate predictions for helium-4, and find that differences
in the virtuality-dependence of nucleon structure modification can lead to
large measurable changes in the tagged structure function. The recent CLAS12
Short-Range Correlations Experiment, which collected electron scattering data
on helium-4 and other nuclear targets, may be able to constrain this
virtuality-dependence and help test whether correlations are the origin of the
modification of bound nucleon structure.

###### keywords:

EMC Effect, Deep Inelastic Scattering, Short-Range Correlations, Spectator
Tagging, CLAS12, Jefferson Lab

##  1 Introduction

Deep inelastic scattering (DIS) measurements on nuclei have revealed that the
partonic structure of nuclei differs from that of free nucleons in substantial
ways EuropeanMuon:1983wih ; Arnold:1983mw ; BCDMS:1987upi ; Gomez:1993ri ;
Seely:2009gt . Coherent destructive and constructive interference effects,
dubbed “shadowing” and “anti-shadowing” respectively, alter the nuclear
$F_{2}$ structure function below the valence region. In the valence region, a
universal reduction in $F_{2}$ has been observed, which is now called the “EMC
Effect.” The EMC Effect grows with nuclear size, saturating at about a 15%
reduction at a momentum fraction of approximately $0.7$ in the heaviest
nuclei. While conventional nuclear physics effects such as Fermi motion and
nuclear pions play a role, the magnitude of the effect requires a modification
of the intrinsic structure of bound nucleons. The underlying mechanism driving
the effect is not conclusively known. See Refs. Malace:2014uea ; Hen:2016kwk
for recent reviews of the topic.

One of the many hypotheses for the cause of the EMC Effect is that the
observed reduction in $F_{2}$ is driven primarily by a large modification to
the structure of nucleons participating in a short-range nucleon-nucleon
correlations. Short-range correlations (SRCs) consist of pairs of nucleons in
close proximity, whose strong interactions lead to large relative momenta,
often well above the nuclear Fermi momentum. As a result, correlated nucleons
tend to be the highest virtuality nucleons in the nucleus. Like the EMC
Effect, such correlations appear to be a universal feature of nuclear
structure, and have been experimentally observed in a wide range of nuclei
through a wide range of experimental techniques (e.g., Piasetzky:2006ai ;
Fomin:2011ng ; Terashima:2018bwq ). This universality is understood to emerge
from the short-range effective nucleon-nucleon interaction Frankfurt:1993sp ;
CiofidegliAtti:1995qe . See Ref. Arrington:2022sov  for more details on recent
experimental progress.

The EMC-SRC hypothesis is supported by several experimental and theoretical
findings. There is a striking correlation between the magnitude of the EMC
Effect and the prevalence of SRC pairs across all measured nuclei
Weinstein:2010rt ; Hen:2012fm ; Arrington:2012ax . Effective field theory
calculations suggest that this correlation is a natural consequence of scale
separation Chen:2016bde . Schmookler et al. have shown that nuclear DIS cross
section ratios are consistent with the universal modification of the nucleons
in SRCs CLAS:2019vsb . Universal modification would have consequences for
extractions of free neutron structure from nuclear DIS data Segarra:2019gbp ,
and these predictions are largely consistent with recent results from the
MARATHON Experiment JeffersonLabHallATritium:2021usd .

There are multiple possibilities for the mechanism driving structure
modification of SRCs. Due to their close proximity, nucleons in short-range
correlations experience significantly higher nuclear density than typical
nucleons. This may even allow quarks in the two nucleons to interact, an idea
considered in six-quark bag Bickerstaff:1984gut ; CiofidegliAtti:1994vf  and
hidden color models West:2020rlk ; West:2020tyo . Alternatively, correlated
nucleons have momenta exceeding the nuclear Fermi momentum and are highly
virtual, which could lead to modification CiofidegliAtti:2007ork . Arrington
and Fomin have attempted to study whether local-density dependence or high-
virtuality dependence are more consistent with experimental data, without
strong conclusions Arrington:2012ax ; Arrington:2019wky . An analysis of
quantum Monte Carlo calculations indicates that there may not be a distinction
between local-density and virtuality dependences Cruz-Torres:2019fum .

Efforts to understand the EMC Effect and the EMC-SRC hypothesis are eventually
run into the limits of what can be learned by DIS data alone. Inclusive
nuclear DIS measurements are sensitive to the average modification of all
nucleons. Efforts have been made to study how the EMC Effect changes across
light nuclei with different nuclear structures Seely:2009gt ;
Arrington:2021vuu , but these systematic efforts will always face difficulties
in establishing unambiguous conclusions. New experimental techniques that add
additional information beyond what inclusive DIS can provide are needed.

A new experimental technique which holds promise for testing the EMC-SRC
hypothesis is spectator-tagged DIS, in which a spectator nucleon is detected
in coincidence with the outgoing electron from DIS. The detection of a
spectator momentum, specifically high-momentum oriented anti-parallel to the
momentum transfer, is a strong indication that the electron scattered from a
nucleon in an SRC. Spectator-tagged DIS was measured on deuterium at Jefferson
Lab in the 6-GeV era CLAS:2005ekq ; CLAS:2011qvj ; CLAS:2014jvt , and two
dedicated experiments, BAND BAND:proposal  and LAD LAD:proposal  were proposed
to study the relationship between SRCs and structure modification in
deuterium. At the time of this writing, BAND has collected data and is under
analysis, while LAD is expected to run in a little more than one year. In
addition, the BAND Experiment necessitated the construction of the BAND
detector Segarra:2020txy , a Backward Angle Neutron Detector for detecting
spectator neutrons that is now part of the standard equipment of the CLAS-12
Spectrometer in Jefferson Lab Hall B Burkert:2020akg . Spectator neutron-
tagging analyses can be performed on data from a wide-variety of CLAS-12
experiments.

BAND and LAD targeted deuterium because the system of two nucleons is simpler
to study, even though any structure modification effects are expected to be
smaller. In this paper, we consider the possibility of studying spectator
tagged DIS on 4He; a slightly more complicated system, but one whose EMC
Effect is larger, and on which abundant data have been collected as part of a
recent CLAS-12 Experiment RGM:proposal . We have developed a formalism for
estimating the spectator-tagged spectral function for a nucleus, building on
the convolution framework developed in Ref. Segarra:2020plg , and modeling
short-range correlations using the light-cone formulation of Generalized
Contact Formalism, described in Ref. Pybus:2020itv . We have performed
calculations under several different assumptions for the virtuality-dependent
modification. We find that a reasonable range of assumptions, which are all
consistent with current data, can lead to a wide variability on the tagged-
structure function for 4He. This suggests that spectator-tagging measurements
on 4He can help shed light on the dependence of structure modification in
bound nucleus on virtuality, and on the role short-range correlations play in
the EMC Effect.

##  2 Formalism

We have developed a prescription for calculating a “spectator neutron-tagged
$F_{2}$ structure function,” i.e., the $F_{2}$ structure function for deep
inelastic electron scattering from a nucleus, in which there is also the
simultaneous emission of a spectator neutron. This prescription extends
formalisms described in Refs. Segarra:2020plg ; Pybus:2020itv . In particular,
we employ a convolution formalism with virtuality-dependent modification
developed in Ref. Segarra:2020plg  to define a structure function for
scattering from moving, off-shell nucleons with modified partonic structure.
We use the light-cone formulation of Generalized Contact Formalism (GCF)
Pybus:2020itv  to describe the motion of nucleons along with correlated
spectators participating in short-range correlations. In this section, we
detail how these formalisms can be combined to allow a calculation of the
spectator tagged structure function.

For what follows, we consider deep-inelastic electron scattering, in which the
scattered electron transfers energy $\nu$ and momentum $\vec{q}$ to a single
parton in a nucleus with $Z$ protons, and $N=A-Z$ neutrons. We define
$Q^{2}=\lvert\vec{q}\rvert^{2}-\nu^{2}$ as the negative squared four-momentum
transfer. We use the Bjorken $x_{B}$ parameter, which is given by
$Q^{2}/2m_{N}\nu$ in the fixed-target frame, where $m_{N}$ is the nucleon
mass. We will work in a light-cone coordinate system in which momentum four-
vectors are defined by $p^{+}=E+p_{z}$, $p^{-}=E-p_{z}$, and transverse
components $p^{\perp}=\langle p_{x},p_{y}\rangle$. We choose the $z$ axis to
point in the direction of $-\vec{q}$, i.e., in the direction opposite the
3-momentum transferred by the scattered electron. We use the term light-cone
momentum fraction, denoted by the symbol $\alpha$ to be the fraction of
$p^{+}$ momentum. Our goal is therefore to build a prescription for the
spectator-tagged structure function
$F_{2}^{A,\text{tag}}(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})$, which, in
addition to the familiar dependence on $x_{B}$ and $Q^{2}$, depends on the
light-cone momentum fraction of the spectator, $\alpha_{2}$, as well as its
transverse momentum $\vec{p}_{2}^{\perp}$. We will use the index ‘2’ to refer
to the spectator nucleon, and index ‘1’ to refer to the struck nucleon.

In convolution formalism, the inclusive structure function for scattering from
a nucleus, $F_{2}^{A}(x_{B},Q^{2})$ can be defined in terms of light-cone
density functions, $\rho_{1}(\alpha_{1},v_{1})$, which describe the
distribution for nucleons in the nucleus to have light-cone momentum fraction
$\alpha_{1}$ and virtuality
$v_{1}\equiv(E_{1}^{2}-p_{1}^{2}-m_{N}^{2})/m_{N}^{2}$ Segarra:2020plg . In
terms of a light-cone density for protons, $\rho_{1,p}(\alpha_{1},v_{1})$, and
$\rho_{1,n}(\alpha_{1},v_{1})$ for neutrons, the inclusive structure function
is given by:

$F_{2}^{A}(x_{B},Q^{2})=\frac{1}{A}\int_{x_{B}}^{A}\frac{d\alpha_{1}}{\alpha_{1}}\int_{-\infty}^{0}dv_{1}\\\
\left[Z\rho_{1,p}(\alpha_{1},v_{1})F_{2}^{p}(\tilde{x},Q^{2})+N\rho_{1,n}(\alpha_{1},v_{1})F_{2}^{n}(\tilde{x},Q^{2})\right],$
(1)

where $F_{2}^{p}$ is the free proton structure function, $F_{2}^{n}$ is the
free neutron structure function, and
$\tilde{x}\equiv\frac{Q^{2}}{2q_{\mu}p_{1}^{\mu}}$ is the fraction of the off-
shell nucleon’s momentum carried by the struck parton. In the Bjorken limit,
$\tilde{x}\rightarrow\frac{x_{B}m_{N}}{\alpha_{1}\bar{m}}$, which was used in
Ref. Segarra:2020plg , and which we will adopt here. This structure function
prescription can be extended to include a virtuality-dependent modification
term to get:

$F_{2}^{A}(x_{B},Q^{2})=\frac{1}{A}\int_{x_{B}}^{A}\frac{d\alpha_{1}}{\alpha_{1}}\int_{-\infty}^{0}dv_{1}\\\
\left[Z\rho_{1,p}(\alpha_{1},v_{1})F_{2}^{p}(\tilde{x},Q^{2})+N\rho_{1,n}(\alpha_{1},v_{1})F_{2}^{n}(\tilde{x},Q^{2})\right]\\\
\times\left(1+v_{1}f_{\text{off}}(\tilde{x})\right).$ (2)

where $f_{\text{off}}(\tilde{x})$ describes the $\tilde{x}$-dependence of this
modification. Ref. Segarra:2020plg  studied the effects of using different
assumptions for $f_{\text{off}}(\tilde{x})$, including constant and linear
models.

To determine the spectator-tagged structure function, we assume that the
emission of a spectator will occur when the struck nucleon and spectator
nucleon are in a short-range correlation together, and can thus be described
using GCF. In Ref. Pybus:2020itv , GCF is used to define the light-cone decay
function, i.e., the distribution of pairs of correlated nucleons in the
nucleus, given by

$\rho(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})=\frac{\alpha_{1}}{\alpha_{\text{CM}}}\rho_{\text{SRC}}(\alpha_{\text{rel.}},\vec{p}_{\text{rel.}}^{\perp})\rho_{\text{CM}}(\alpha_{\text{CM}},\vec{p}_{\text{CM}}^{\perp}),$
(3)

where $\alpha_{1}$,$\vec{p}_{1}^{\perp}$ and
$\alpha_{2}$,$\vec{p}_{2}^{\perp}$ combine according to

$\displaystyle\alpha_{\text{CM}}$ $\displaystyle=\alpha_{1}+\alpha_{2}$ (4)
$\displaystyle\alpha_{\text{rel.}}$
$\displaystyle=\frac{2\alpha_{2}}{\alpha_{\text{CM}}}$ (5)
$\displaystyle\vec{p}_{\text{CM}}^{\perp}$
$\displaystyle=\vec{p}_{1}^{\perp}+\vec{p}_{2}^{\perp}$ (6)
$\displaystyle\vec{p}_{\text{rel.}}^{\perp}$
$\displaystyle=\vec{p}_{2}^{\perp}-\frac{\alpha_{2}}{\alpha_{\text{CM}}}\vec{p}_{CM}^{\perp}$
(7)
$\displaystyle=\frac{\alpha_{1}\vec{p}_{2}^{\perp}-\alpha_{2}\vec{p}_{1}^{\perp}}{\alpha_{\text{CM}}},$
(8)

and $\rho_{\text{SRC}}$ and $\rho_{\text{CM}}$ refer to the relative and
center-of-mass distributions, respectively, of nucleons within SRC pairs.
Following Ref. Pybus:2020itv , we model $\rho_{\text{CM}}$ as a Gaussian:

$\rho_{\text{CM}}(\alpha_{\text{CM}},\vec{p}_{\text{CM}}^{\perp})=\frac{\bar{m}\alpha_{\text{CM}}}{(2\pi\sigma_{\text{CM}})^{3/2}}\exp\left\\{-\frac{\bar{m}^{2}(2-\alpha_{\text{CM}})^{2}+\lvert\vec{p}_{\text{CM}}^{\perp}\rvert^{2}}{2\sigma_{\text{CM}}^{2}}\right\\}$
(9)

where $\bar{m}\equiv\frac{m_{A}}{A}$ and $\sigma_{\text{CM}}$ represents the
width of the CM momentum distribution, $\approx 100$–$150$ MeV$/c$
Colle:2013nna ; LabHallA:2014wqo ; CLAS:2018qpc .

In Generalized Contact Formalism, the relative distribution takes the form of
sum over pairs of different relative quantum numbers, which we represent by
index $j$:

$\rho_{\text{SRC}}(\alpha_{\text{rel}},\vec{p}_{\text{rel}}^{\perp})=\sum_{j}C_{j}\frac{\sqrt{m_{N}^{2}+k^{2}}}{2-\alpha_{\text{rel}}}\frac{\lvert\tilde{\phi}_{j}(k)\rvert^{2}}{(2\pi)^{3}},$
(10)

where $\tilde{\phi}_{j}(k)$ is a universal (i.e., nucleus-independent) two-
body momentum space wave function, $C_{j}$ is the nucleus-dependent contact
term representing the abundance of SRC pairs with quantum numbers $j$, and $k$
is given by:

$k^{2}\equiv\frac{m_{N}^{2}+\lvert\vec{p}_{\text{rel}}^{\perp}\rvert^{2}}{\alpha_{\text{rel}}(2-\alpha_{\text{rel}})}-m_{N}^{2}.$
(11)

The decay function can be restricted based on the isospin projection of
nucleons 1 and 2, which determines which quantum numbers are considered in the
sum over $j$ in Eq. 10. We’ll use the notation $\rho_{pn}$ to refer to the
decay function where nucleon 1 is a proton and nucleon 2 is a neutron (either
spin 0 or spin 1). We reserve the notation $\rho_{1,p}$ and $\rho_{1,n}$ to
refer to the decay function where nucleon 1 is a proton or neutron
respectively (with no conditions on nucleon 2).

Ref. Segarra:2020plg  gives the inclusive $F_{2}$ structure function for a
nucleus in terms of the decay function, by integrating the decay function over
the spectator nucleon variables (and re-defining $\vec{p}_{1}^{\perp}$ in
terms of virtuality, $v_{1}$) in order to get the light-cone density:

$\rho_{1}(\alpha_{1},v_{1})\equiv\int d^{2}\vec{p}_{1}^{\perp}\int
d^{2}\vec{p}_{2}^{\perp}\int_{0}^{A}\frac{d\alpha_{2}}{\alpha_{2}}\rho(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})\\\
\times\delta\left(v_{1}-\frac{\bar{m}p^{-}\alpha_{1}-\lvert\vec{p}_{1}^{\perp}\rvert^{2}-m_{N}^{2}}{m_{N}^{2}}\right)$
(12)

with

$p^{-}\equiv
m_{A}-\frac{m_{N}^{2}+\lvert\vec{p}_{2}^{\perp}\rvert^{2}}{\bar{m}\alpha_{2}}-\frac{m_{A-2}^{2}+\lvert\vec{p}_{\text{CM}}^{\perp}\rvert^{2}}{\bar{m}(A-\alpha_{1}-\alpha_{2})}.$
(13)

In this work, we want to predict the _neutron spectator-tagged_ structure
function, rather than the inclusive structure function. We therefore do not
integrate out the spectator variables. Instead we arrive at an expression

$F_{2}^{A,\text{tag}}(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})=\frac{1}{A\alpha_{2}}\int
d^{2}\vec{p}_{1}^{\perp}\int_{x_{B}}^{A}\frac{d\alpha_{1}}{\alpha_{1}}\\\
\left[ZF_{2}^{p}(\tilde{x},Q^{2})\rho_{pn}(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})+NF_{2}^{n}(\tilde{x},Q^{2})\rho_{nn}(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})\right]\\\
\times\left(1+v_{1}f_{\text{off}}(\tilde{x})\right),$ (14)

where $v_{1}$ takes a fixed value of
$\frac{\bar{m}p^{-}\alpha_{1}-\lvert\vec{p}_{1}^{\perp}\rvert^{2}-m_{N}^{2}}{m_{N}^{2}}$.
We further factor out the well-constrained $F_{2}^{p}$, and assume that the
ratio $F_{2}^{n}/F_{2}^{p}$ has negligible dependence on $Q^{2}$:

$F_{2}^{A,\text{tag}}(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})=\frac{1}{A\alpha_{2}}\int
d^{2}\vec{p}_{1}^{\perp}\int_{x_{B}}^{A}\frac{d\alpha_{1}}{\alpha_{1}}F_{2}^{p}(\tilde{x},Q^{2})\\\
\left[Z\rho_{pn}(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})+N\frac{F_{2}^{n}}{F_{2}^{p}}(\tilde{x})\rho_{nn}(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})\right]\\\
\times\left(1+v_{1}f_{\text{off}}(\tilde{x})\right).$ (15)

This tagged structure function can be used to define the tagged differential
cross section

$\frac{d^{6}\sigma}{dx_{B}dQ^{2}d\phi_{e}d\alpha_{2}d^{2}\vec{p}_{2}^{\perp}}=\frac{4\alpha_{EM}^{2}E^{\prime}(E-E^{\prime})}{m_{N}x_{B}EQ^{4}}\\\
\times\left[F_{1}\sin^{2}\frac{\theta_{e}}{2}+\frac{x_{B}m_{N}^{2}}{Q^{2}}F_{2}\cos^{2}\frac{\theta_{e}}{2}\right].$
(16)

We further make the approximation that $Q^{2}$ is large enough for the Callan-
Gross relation to hold, i.e., $2x_{B}F_{1}\rightarrow F_{2}$ to get our final
expression

$\frac{d^{6}\sigma}{dx_{B}dQ^{2}d\phi_{e}d\alpha_{2}d^{2}\vec{p}_{2}^{\perp}}=\frac{4\alpha_{EM}^{2}E^{\prime}(E-E^{\prime})}{m_{N}x_{B}EQ^{4}}\\\
\times\left[\frac{1}{2x_{B}}\sin^{2}\frac{\theta_{e}}{2}+\frac{x_{B}m_{N}^{2}}{Q^{2}}\cos^{2}\frac{\theta_{e}}{2}\right]F_{2}^{A,\text{tag}}(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})$
(17)

A number of inputs are needed both for the convolution model and for the GCF
description of the light-cone decay function, and we have made some specific
choices for the calculations in this work. First, for the free proton
structure function, $F_{2}^{p}$, we have employed a parameterization (Eq. 7 of
Ref. H1:1995jgh , with parameters given in Table 5) developed by the H1
Collaboration in a global fit to DIS data H1:1995jgh . Second, for the ratio
of free neutron to free proton structure functions, we have assumed the
approximate form

$\frac{F_{2}^{n}}{F_{2}^{p}}=a(1-x)^{b}+c,$ (18)

with parameter values $a=0.57,~{}b=2.2,~{}c=0.42.$, used by Segarra et al.
Segarra:2020plg , which was a simplified description of the results of the
analysis described in Ref. Segarra:2019gbp . This form is consistent with the
results of the BoNUS CLAS:2011qvj ; CLAS:2014jvt  and MARATHON
JeffersonLabHallATritium:2021usd  experiments.

Generalized Contact Formalism requires inputs for each of its factorized
components. To describe the relative motion between correlated nucleons (Eq.
10), $\rho_{\text{SRC}}$ requires universal two-body momentum distributions
$\lvert\tilde{\phi}_{j}(k)\rvert^{2}$ from a model of the nucleon-nucleon
potential. We have chosen to use the AV18 potential Wiringa:1994wb . For the
center-of-mass motion distribution of a correlated pair, given by
$\rho_{\text{CM}}$ (Eq. 9), we assume a Gaussian distribution (as in Refs.
Pybus:2020itv ; Weiss:2018tbu ; CLAS:2020mom ; CLAS:2020rue  and others) with
a width $\sigma=100$ MeV$/c$, based on the measurement presented in Ref.
LabHallA:2014wqo . Lastly, GCF requires values for the contact coefficients,
$C_{j}$. We use the values determined in Ref. Cruz-Torres:2019fum  from a
momentum-space analysis of variational Monte Carlo calculations performed with
the AV18 potential.

Figure 1: The different models used in this calculation for the universal off-
shell nucleon modification function, $f_{\text{off}}$ are shown against $x$.

In our study, we considered the impacts of different assumptions for the final
input, the off-shell nucleon modification function,
$f_{\text{off}}(\tilde{x})$. This function is not well-constrained by
available data, and therefore we consider five possibilities

  1. 1.

No modification: $f_{\text{off}}(\tilde{x})=0$. This is inconsistent with the
EMC Effect, but we use it as a useful baseline.

  2. 2.

Constant in $\tilde{x}$: $f_{\text{off}}(\tilde{x})=K$. We specifically take
the value of $K=0.96$ from the global fit analysis of Segarra:2020plg  using
the identical assumption for the free-neutron structure function.

  3. 3.

Linear in $\tilde{x}$: $f_{\text{off}}(\tilde{x})=A+B\tilde{x}$. Similarly, we
take the value of $A=0.31$ and $B=0.98$ from the global fit analysis of
Segarra:2020plg  using the identical assumption for the free-neutron structure
function.

  4. 4.

The result of Kulagin and Petti Kulagin:2004ie  (KP), who assumed a third-
order polynomial of the form

$f_{\text{off}}^{\text{KP}}(\tilde{x})=C(x_{0}-\tilde{x})(x_{1}-\tilde{x})(1+x_{0}-\tilde{x})$
(19)

with parameters $C$, $x_{0}$, and $x_{1}$. We use their values: $C=8.1$,
$x_{0}=0.448$, and $x_{1}=0.05$.

  5. 5.

The result of the CTEQ-Jefferson Lab (CJ) collaboration Accardi:2016qay ,
which used the same functional form as in Eq. 19, but with parameter values:
$C=-3.6735$, $x_{0}=0.057717$, and $x_{1}=0.36419$.

A comparison of these five possible $f_{\text{off}}$ functions is shown in
Fig. 1.

Using these inputs, we calculated the tagged structure function
$F_{2}^{A,\text{tag}}$ for Helium-4 by performing the integration in Eq. 15
over $p^{\perp}_{1}$ and $\alpha_{1}$ using Monte Carlo integration.

##  3 CLAS12 Short Range Correlations Experiment

One motivation for performing this calculation is the opportunity presented by
the CLAS12 Short Range Correlations (CLAS12-SRC) Experiment (E12-17-006A)
RGM:proposal , recently conducted at Jefferson Lab. The CLAS12-SRC Experiment
collected data from November, 2021, through February, 2022 in Experimental
Hall B, as part of Run Group M. Out-going particles were detected using the
CLAS12 Spectrometer Burkert:2020akg , a large acceptance magnetic spectrometer
designed to cover scattering angles roughly from $8^{\circ}$–$40^{\circ}$ in a
toroidal forward spectrometer, and from approximately
$35^{\circ}$–$125^{\circ}$ in a solenoidal central detector. In the
experiment, an electron beam was scattered from a variety of nuclear targets,
with the goal of identifying quasi-elastic knock-out of nucleons in SRC pairs.
Several beam energies were used, with the majority of the data being collected
with a 6 GeV beam. Since the experiment had a non-specific trigger—only
requiring the detection of a single high-energy electron—a wide range of final
states can be analyzed in the data. The targets used in the CLAS12-SRC
Experiment and the time spent on each are shown in Table 1. At the time of
this writing, data from the experiment are undergoing calibrations and
analysis.

Table 1: The targets used in CLAS12-SRC Experiment at beam energy of 6 GeV, the days spent on each target, and the number of triggers per target. Target | Days | Triggers ($\times 10^{9}$)
---|---|---
H | 2 | 1.1
D | 6 | 4.2
4He | 10 | 3.9
12C | 8 | 4.1
40Ar | 2 | 0.55
40Ca | 6 | 3.0
48Ca | 3 | 1.50
120Sn | 4 | 0.40

In addition to the forward and central spectrometers, the CLAS12-SRC
Experiment also employed the CLAS12 Backward Angle Neutron Detector (BAND)
Segarra:2020txy , covering 155∘–175∘ from the beamline. BAND is an array of
plastic scintillator bars, read out on both ends with Photomultiplier Tubes,
with a total thickness of 35 cm. BAND includes a thin layer of lead for
attenuating photons, as well as an additional thin layer of active
scintillator to serve as a charged particle veto. BAND was built for the BAND
Experiment BAND:proposal  (Jefferson Lab, E12-11-003A), aiming to measure
spectator-tagged DIS on deuterium, with the goal of detecting neutrons in the
momentum range of $\approx 200--700$ MeV$/c$. BAND was placed upstream of the
target, covering backward angles, both to avoid material obstructions and
because backward-spectator kinematics are less sensitive to final-state
interactions CLAS:2005ekq ; Strikman:2017koc .

The BAND detector allows the study of spectator neutron-tagged DIS on nuclei
using the CLAS12-SRC data. There are advantages and disadvantages with
studying tagged-DIS in nuclei as opposed to deuterium. In deuterium, there are
no other nucleons outside of the interacting nucleon-nucleon pair. In the
plane-wave limit, the momentum of spectator is exactly opposite to the initial
momentum of the struck nucleon. In larger nuclei, the residual $A-2$ system
can carry away energy and momentum. On the other hand, the magnitude of the
EMC Effect grows with nuclear size. It may be easier to study the relationship
between the EMC Effect and SRCs due to the larger effect sizes. Lastly, final
state interaction effects should grow with nuclear size, making the results
from very large nuclei harder to interpret.

In this paper, we specifically consider the EMC Effect helium-4, which has the
advantage of being relatively light. Of the nuclear targets used in the
CLAS12-SRC Experiment, it also had the longest run duration, with almost 4
billion triggers collected. The methodology of this work could easily be
applied to make predictions for the other target nuclei used in the
experiment.

##  4 Results

First, we consider the effect of virtuality-dependent modification by
calculating the ratio

$R(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})=\frac{F_{2,\text{modified}}^{A,\text{tag}}(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})}{F_{2,\text{unmodified}}^{A}(x_{B},Q^{2},\alpha_{2},\vec{p}_{2}^{\perp})},$
(20)

of modified to unmodified tagged structure functions. We have chosen to
evaluate this ratio at a fixed $Q^{2}=2$ GeV${}^{2}/c^{2}$, a typical value
for the coverage of CLAS12, given a 6 GeV electron beam. We have also chosen
to evaluate this ratio at $\lvert\vec{p}_{2}^{\perp}\rvert=0$, which is
typical for the coverage of BAND detector. For simplicity, we omit these in
our notation:

$R(x_{B},\alpha_{2})=R(x_{B},Q^{2}=2\text{~{}GeV}^{2}/c^{2},\alpha_{2},\vec{p}_{2}^{\perp}=0).$
(21)

Figures 2 and 3 show the tagged structure function ratio,
$R(x_{B},\alpha_{2})$ as functions of $\alpha_{2}$ and $x_{B}$, respectively.
We performed this calculation at three values of $x_{B}$ and three values of
$\alpha_{2}$, for the $\alpha_{2}$\- and $x_{B}$-dependence figures,
respectively. In both figures, one can see that these differing models yield
significantly varied results, and that this variance increases with
$\alpha_{2}$.

Figure 2: The results for the ratio of $F_{2}^{A}$ structure functions
modified by the specified off-shell nucleon modification function models to
the unmodified structure functions, shown against $\alpha_{2}$. Behavior to
the left of the dashed line at $\alpha_{2}=1.3$ can be dismissed due to GCF’s
limitations. Figure 3: The results for the ratio of $F_{2}^{A}$ structure
functions modified by the specified off-shell nucleon modification function
models to the unmodified structure functions, shown against $x_{B}$.

The predictions of models we considered, all of whom were tuned to existing
data, range from below 0.4 to above 1 at a $\alpha_{2}=1.6$ and $x_{B}=0.7$.
The KP model is most similar to the constant- and linear-in-$x$ models, though
it has different behavior as a function of $x_{B}$. Surprisingly, the
virtuality-dependent modification need not lead to a reduction in the
structure function. The $CJ$ model predicts that the structure function will
cause an increase from the no-modification assumption. We also note that even
at $x_{B}=0.3$, a where medium-modification effects are typically thought of
as being small, the different models predict 20–40% variation in the tagged
structure function, depending on $\alpha_{2}$.

We expect that the calculations will become less accurate with decreasing
$\alpha_{2}$. Generalized Contact Formalism relies on a separation between the
high-momentum scale of the relative momentum of nucleons within a correlated
pair, from the low-momentum scale of typical momenta of nucleons with the
nucleus. For pairs with a relative momentum comparable to the Fermi-momentum
and below, these scales cease to be separated. For $p_{2}^{\perp}=0$, this
occurs at roughly $\alpha_{2}=1.3$, which we have marked with a dashed line in
Fig. 2. We have analytically continued the calculation below $\alpha_{2}=1.3$,
but cannot say with any confidence whether the structures observed in Fig. 2
are real, or an artifact of using GCF in an improper regime.

Figure 4: The results for the double ratio of the structure function at
different $x_{B}$ values for the different off-shell nucleon modification
function models Figure 5: The results for the double ratio of the structure
function at different $\alpha_{2}$ values for the different off-shell nucleon
modification function models

In experiments, systematic effects can often be suppressed by forming ratios.
For this reason, we also consider the double ratio
$R(x_{B},\alpha_{2})/R(x_{\text{B,ref}},\alpha_{2})$ as a function of
$\alpha_{2}$ (Fig. 4) and $R(x,\alpha_{2})/R(x,\alpha_{2,\text{ref.}})$ as a
function of $x_{B}$ (Fig. 5). The reference value $x_{B,\text{ref}}=0.3$ was
chosen since structure function ratios $2F_{2}^{A}/AF_{2}^{d}$ are generally
close to unity at $x_{B}=0.3$. The reference value $\alpha_{2,\text{ref}}=1.4$
was chosen as the smallest (and hence corresponds to the least virtual
spectators) of the three values we considered.

As in the single ratios, the double ratios in Figs. 4 and 5 show a large
sensitivity to the exact form of the virtuality-dependent modification. At
$\alpha_{2}=1.6$ and $x_{B}=0.7$, the models have a spread from $\approx 0.6$
to greater than $1$ , with KP and CJ being most and least extreme,
respectively. Furthermore, the spread between predictions increases with
$x_{B}$ and $\alpha_{2}$. A measurement of
$F_{2}^{{}^{4}\text{He,tag}}(x_{B},\alpha_{2})/F_{2}^{{}^{4}\text{He,tag}}(x_{B,\text{ref}},\alpha_{2,\text{ref}})$,
normalized by a calculation of
$F_{2,\text{unmodified}}^{{}^{4}\text{He,tag}}(x_{B},\alpha_{2})/F_{2,\text{unmodified}}^{{}^{4}\text{He,tag}}(x_{B,\text{ref}},\alpha_{2,\text{ref}})$
could provide new constraints on the possible form that virtuality-dependent
modification can take. Even a measurement with 10% uncertainty would be
meaningful.

##  5 Conclusions

We have developed a formalism for calculating the spectator-tagged structure
function for scattering from a nucleon participating in a short-range
correlation, using Generalized Contact Formalism to specify the two-nucleon
light-cone density. We have performed calculations for helium-4 in kinematics
relevant for the CLAS12-SRC Experiment and find large sensitivity to the
assumed virtuality-dependent modification. Data from the CLAS12-SRC Experiment
may be able to provide constraints on the nature of this modification. The
method developed here can be naturally extended to other nuclei, for which GCF
parameters are known with reasonable uncertainty, including carbon-12 and
calcium-40.

Extending the predictive range of these calculations to lower values of
$\alpha_{2}$ will require some method for specifying
$\rho(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})$ that
does not rely on Generalized Contact Formalism. One option would be to make an
ad hoc separation between the SRC-dominated and mean-field regimes in the
nucleus, using GCF for the former, and some alternative approach for the
latter. A more sophisticated route would be to develop a two-nucleon light
cone density using ab initio techniques such as variational Monte Carlo. We
are not aware of any such calculations having been performed yet. In any case,
care must be taken to ensure that the resulting
$\rho(\alpha_{1},\vec{p}_{1}^{\perp},\alpha_{2},\vec{p}_{2}^{\perp})$
satisfies the baryon and momentum sum rules.

Spectator-tagged DIS from nuclei may offer additional insight into the
mechanisms driving the EMC Effect and the role played by short-range
correlations. The data from the CLAS12-SRC Experiment, specifically using the
BAND detector for spectator neutron tagging, offer an excellent opportunity
for such studies.

Acknowledgments

This work was supported by the US Department of Energy Office of Science,
Office of Nuclear Physics, under contract no. DE-SC0016583. The authors are
also grateful to J. R. Pybus, and T. Kutz for helpful discussions.

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