# Non commutative Multidimensional Stronger Central Sets Theorem and it’s
polynomial extension

Dibyendu De  and  Sujan Pal<EMAIL_ADDRESS>Department of
Mathematics, University of Kalyani, Kalyani, Nadia-741235, West Bengal, India.
<EMAIL_ADDRESS>Department of Mathematics, University of Kalyani,
Kalyani, Nadia-741235, West Bengal, India

###### Abstract.

Hindman’s theorem and van der Waerden’s theorem are two classical Ramsey
theoretic results, the first one deals with finite configurations and the
second one deals with infinite configurations. The Central Sets Theorem due to
Furstenberg is a strong simultaneous extension of both theorems, which also
applies to general commutative semigroups. Beiglboeck provided a common
extension of the Central Sets Theorem and Milliken-Taylor Theorem in
commutative case. Furstenberg’s original Central Sets Theorem was proved in
[F] for $\left(\mathbb{N},+\right)$ for finitely many sequences at a time.
Bergelson and Hindman provided a non commutative version of this Theorem [BH].
The first author of this article jointly with Hindman and Strauss provided a
non-commutative version of Central Sets Theorem using arbitrary many sequences
at a time [DHS]. In this work we will provide a non-commutative extension of
Beiglboeck’s Theorem. We also provide polynomial generalization of
Beiglboeck’s theorem.

###### Key words and phrases:

Central sets, Central Sets Theorem, Algebra of Stone-Čech compactification of
discrete semigroup.

##  1\. Introduction

Van der Waerden’s Theorem [VDW] states that for any partition of the positive
integers $\mathbb{N}$ one of the cells of the partition contains arbitrarily
long arithmetic progressions. Hindman’s Theorem [H, Theorem 3.1] states that
for any partition of the positive integers $\mathbb{N}$ one of the cells
contains all possible finite sums of some sequence. Central Sets Theorem
provided a joint extension of this two theorems. To state the Central Sets
Theorem let us formulate some notations. By $P_{f}(X)$ we denote the set of
all finite nonempty subsets of $X$. For a sequence
$\left(x_{n}\right)_{n=1}^{\infty}$ in $\mathbb{N}$ we set

$\text{FS}\left(x_{n}\right)=\left\\{\sum_{t\in\alpha}x_{t}:\alpha\in\mathcal{P}_{f}\left(\mathbb{N}\right),\alpha\neq\emptyset\right\\}.$

A set $A$ is called an IP-set if and only if there exists a sequence
$\left(x_{n}\right)_{n=1}^{\infty}$ in $\mathbb{N}$ such that
$\text{FS}\left(x_{n}\right)\subset A$. (This definitions makes perfect sense
in any semigroup $(S,\cdot)$ and we use it in this context. FS is an
abbriviation of finite sums and will be replaced by FP if we use
multiplicative notation for the semigroup operation).

To state about Central Sets Theorem, let us introduce some algebraic
preliminaries on Stone-Čech compactification.

Let $\left(S,\cdot\right)$ be a discrete semigroup and $\beta S$ be the Stone-
Čech compactification of the discrete semigroup $S$ and $\cdot$ on $\beta S$
(which we represent by the same symbol on $S$) is the extension of $\cdot$ on
$S$. The points of $\beta S$ are ultrafilters and principal ultrafilters are
identified by the points of $S$. The extension is unique extension for which
$\left(\beta S,\cdot\right)$ is compact, right topological semigroup with $S$
contained in its topological center. That is, for all $p\in\beta S$ the
function $\rho_{p}:\beta S\to\beta S$ is continuous, where $\rho_{p}(q)=q\cdot
p$ and for all $x\in S$, the function $\lambda_{x}:\beta S\to\beta S$ is
continuous, where $\lambda_{x}(q)=x\cdot q$. For $p,q\in\beta S$, $p\cdot
q=\left\\{A\subseteq S:\left\\{x\in S:x^{-1}A\in q\right\\}\in p\right\\}$,
where $x^{-1}A=\left\\{y\in S:x\cdot y\in A\right\\}$.

There is a famous theorem due to Ellis [HS, Corollary 2.39] that if $S$ is a
compact right topological semigroup then the set of idempotents
$E\left(S\right)\neq\emptyset$.

A non-empty subset $I$ of a semigroup $T$ is called a left ideal of $S$ if
$TI\subset I$, a right ideal if $IT\subset I$, and a two sided ideal (or
simply an ideal) if it is both a left and right ideal. A minimal left ideal is
the left ideal that does not contain any proper left ideal. Similarly, we can
define minimal right ideal.

Any compact Hausdorff right topological semigroup $T$ has the smallest two
sided ideal,

$\begin{array}[]{ccc}K(T)&=&\bigcup\left\\{L:L\text{ is a minimal left ideal
of }T\right\\}\\\ &=&\bigcup\left\\{R:R\text{ is a minimal right ideal of
}T\right\\}\end{array}.$

Given a minimal left ideal $L$ and a minimal right ideal $R$, $L\cap R$ is a
group, and in particular contains an idempotent. If $p$ and $q$ are
idempotents in $T$ we write $p\leq q$ if and only if $pq=qp=p$. An idempotent
is minimal with respect to this relation if and only if it is a member of the
smallest ideal $K(T)$ of $T$. See [HS] for an elementary introduction to the
algebra of $\beta S$ and for any unfamiliar details.

######  Definition 1.1.

Let $S$ be a discrete semigroup and let $C$ be a subset of $S$. Then $C$ is
central if there is an idempotent $p$ in $K\left(\beta S\right)$ such that
$C\in p$.

The notion of central sets originally introduced by Furstenberg [F] using
topological dynamics and latter by Bergelson and Hindman proved this notation
to be equivalent with algebraic definition using algebra of $\beta\mathbb{N}$.

######  Theorem 1.2.

(Original Central Sets Theorem) Let $l\in\mathbb{N}$ and for each
$i\in\left\\{1,2,...,l\right\\}$ , let $\left(y_{i,n}\right)_{n=1}^{\infty}$be
a sequence in $\mathbb{Z}$. Let $C$ be a central subset of $\mathbb{N}.$ Then
there exist sequences $\left(a_{n}\right)_{n=1}^{\infty}$ in $\mathbb{N}$ and
$\left(H_{n}\right)_{n=1}^{\infty}$ in
$\mathcal{P}_{f}\left(\mathbb{N}\right)$ such that

$\left(1\right)$ for all $n$, $\max H_{n}<\min H_{n+1}$ and

$\left(2\right)$ for all $F\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ and all
$i\in\left\\{1,2,...,l\right\\},$

$\sum_{n\in F}\left(a_{n}+\sum_{t\in H_{n}}y_{i,t}\right)\in C.$

K. Milliken and A. Taylor ([M],[T]) found a quite natural common extension of
the Theorems of Hindman and Ramsey: For a sequence $(x_{n})_{n=1}^{\infty}$ in
$\mathbb{N}$ and $k\geq 1$ put

$\left[\text{FS}\left(x_{n}\right)\right]{}_{<}^{k}:=\left\\{\left\\{\sum_{t\in\alpha_{1}}x_{t},\ldots,\sum_{t\in\alpha_{k}}x_{t}\right\\}:\alpha_{1}<\ldots<\alpha_{k}\in\mathcal{P}_{f}\left(\mathbb{N}\right)\right\\}$

$\alpha<\beta$ for $\alpha,\beta\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ if
and only if $\max\alpha<\min\beta$.

Let $[S]^{k}$ denote the k-element subsets of a set $S$. The Milliken-Taylor
theorem says that for any finite partition
$[\mathbb{N}]^{k}=\cup_{i=1}^{r}A_{i}$, there exist some
$i\in\left\\{1,2,\ldots,r\right\\}$ and a sequence $(x_{n})_{n=1}^{\infty}$ in
$\mathbb{N}$ such that
$\left[\text{FS}\left(x_{n}\right)\right]{}_{<}^{k}\subseteq A_{i}$.

Define $\Phi=\left\\{f\in\mathbb{N}^{\mathbb{N}}:\text{ for each
}n\in\mathbb{N},f\left(n\right)\leq n\right\\}$.

As a combine extension of Milliken-Taylor Theorem and Central Sets Theorem,
Beiglboeck established the following result.

######  Theorem 1.3.

[B, Theorem 1.4] Let $(S,\cdot)$ be a commutative semigroup and assume that
there exists a non principal minimal idempotent in $\beta S$. For each
$l\in\mathbb{N}$, let $\langle y_{l,n}\rangle_{n=0}^{\infty}$ be a sequence in
$S$. Let $k,r\geq 1$ and let $[S]^{k}=\cup_{i=1}^{r}A_{i}$. There exist
$i\in\left\\{1,2,\ldots,r\right\\}$, a sequence
$\left(x_{n}\right)_{n=1}^{\infty}$ in $S$ and a sequence
$\alpha_{0}<\alpha_{1}<\ldots$ in $\mathcal{P}_{f}(\mathbb{N})$ such that for
each $g\in\Phi$,

$\left[\text{FP}\left(\left\langle
x_{n}\prod_{t\in\alpha_{n}}y_{g(n),t}\right\rangle_{n=0}^{\infty}\right)\right]_{<}^{k}\subseteq
A_{i}.$

There are several extensions of Central Sets Theorem [BH], [BJM], [DHS],
[HMS], [GBP] in the literature. In [HMS] authors established a version of
Central Sets Theorem for commutative semigroups considering countable
infinitely many sequenses at a time and in [BH] authors established the non
commutative version of Central Sets Theorem. We state the Theorem from [HS].

For our purppose let us introduce the following notations. In a noncommutative
semigroup, by $\prod_{t\in F}x_{t}$ we mean the product taken in increasing
order of indices. In the following $\mathcal{T}$ is the collection of all
sequences in $S$.

######  Definition 1.4.

Let $m\in\mathbb{N}$. Then we denote

$\mathscr{i}\mathscr{i}\mathcal{\mathscr{I}}_{m}=\left\\{\left(H_{1},H_{2},\ldots,H_{m}\right)\in\mathcal{P}_{f}\left(\mathbb{N}\right)^{m}:H_{t}<H_{t+1},t\in\left\\{1,2,\ldots,m-1\right\\},m>1\right\\},\text{
and}$
$\mathscr{J}_{m}=\left\\{\left(t\left(1\right),t\left(2\right),\ldots,t\left(m\right)\right)\in\mathbb{N}^{m}:t\left(1\right)<t\left(2\right)<\ldots<t\left(m\right)\right\\}.$

Let $\left(S,\cdot\right)$ be a semigroup.

(a) Given $n\in\mathbb{N}$, $a\in S^{m+1}$, $t\in\mathscr{J}_{m}$ and
$f\in\mathcal{T}$

$x\left(m,a,t,f\right)=\left(\prod_{j=1}^{m}a\left(j\right)\cdot
f\left(t\left(j\right)\right)\right)\cdot a\left(m+1\right).$

(b) $A\subset S$ is a $J$-set if and only if for each
$F\in\mathcal{P}_{f}\left(\mathcal{T}\right)$ there exist $m\in\mathbb{N}$,
$a\in S^{m+1}$, and $t\in\mathscr{J}_{m}$ such that for each $f\in F$ ,
$x\left(m,a,t,f\right)\in A$.

(c) $J\left(S\right)=\left\\{p\in\beta S:\text{ for all }A\in p\text{, then
$A$ is a J-set}\right\\}$.

(d) $A\subset S$ is a C-set if and only if there exist

$m:\mathcal{P}_{f}\left(\mathcal{T}\right)\to\mathbb{N},\qquad\alpha\in\times_{F\in\mathcal{P}_{f}\left(\mathcal{T}\right)}S^{m\left(F\right)+1},\text{
and
}\tau\in\times_{F\in\mathcal{P}_{f}\left(\mathcal{T}\right)}\mathscr{I}_{m\left(F\right)}$

such that

(i) If $F\varsubsetneq G$ in $\mathcal{P}_{f}\left(\mathcal{T}\right)$ then
$\tau\left(F\right)\left(m\left(F\right)\right)<\tau\left(G\right)\left(1\right)$
and

(ii) For any $n\in\mathbb{N}$, if $G_{1}\varsubsetneq
G_{2}\varsubsetneq\ldots\varsubsetneq G_{n}$ in
$\mathcal{P}_{f}\left(\mathcal{T}\right)$ and for each
$i\in\left\\{1,2,\ldots,n\right\\}$, $f_{i}\in G_{i}$ then

$\prod_{i=1}^{n}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right)\in
A.$

######  Theorem 1.5.

[HS, Theorem 14.15] Let $S$ be a semigroup, let $A$ be a central subset of
$S$, and for each $l\in\mathbb{N}$, let $\langle
y_{l,n}\rangle_{n=1}^{\infty}$ be a sequence in $S$. Given $l,m\in\mathbb{N}$
and $a\in S^{m+1}$, and let $H\in\mathscr{I}_{m}$ let

$w(a,H,l)=\left(\left(\prod_{i=1}^{m}a\left(i\right)\cdot\prod_{t\in
H(t)}y_{l,t}\right)\right)\cdot a(m+1).$

There exist sequences $\left\langle m(n)\right\rangle_{n=1}^{\infty}$,
$\left\langle a_{n}\right\rangle_{n=1}^{\infty}$ and $\left\langle
H_{n}\right\rangle_{n=1}^{\infty}$ such that

  1. (1)

for each $n\in\mathbb{N}$, $m(n)\in\mathbb{N}$, $a_{n}\in S^{m(n)+1},$and
$H_{n}\in\mathscr{I}_{m\left(n\right)}$ and $\max H_{n,m\left(n\right)}<\min
H_{n+1,1},$and

  2. (2)

for each $f\in\Phi$, $FP\left(\left\langle
w(a_{n},H_{n},f\left(n\right)\right\rangle_{n=1}^{\infty}\right)\subseteq A$.

In [BJM] author established the polynomial version of the original Central
Sets Theorem for commutative semigroups. In [DHS] authors gave a version of
Central Sets Theorem to stronger form using arbitrary many sequences at a time
known as new and Stronger Central Sets Theorem.

######  Theorem 1.6.

[DHS, Theorem 2.2](Strong Central Sets Theorem for commutative semigroup) Let
$\left(S,+\right)$ be a commutative semigroup and let $C$ be a central subset
of $S$. Then there exist functions
$\alpha:\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)\to S\text{ and
}H:\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)\to\mathcal{P}_{f}\left(\mathbb{N}\right)$
such that

$\left(1\right)$ If $F,G\in\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)$ and
$F\subsetneq G$ then $\max H\left(F\right)<\min H\left(G\right)$ and

$\left(2\right)$ If
$m\in\mathbb{N},G_{1},G_{2},....,G_{m}\in\mathcal{P}_{f}\left(S^{\mathbb{N}}\right)$;
$G_{1}\subsetneq G_{2}\subsetneq....\subsetneq G_{m}$; and for each

$i\in\left\\{1,2,....,m\right\\},\left(y_{i,n}\right)_{n=1}^{\infty}\in
G_{i},$ then

$\sum_{i=1}^{m}\left(\alpha\left(G_{i}\right)+\sum_{t\in
H\left(G_{i}\right)}y_{i,t}\right)\in C.$

There are two extension of the above Theorem. In one direction in the same
article authors established a non-commutative version of the Theorem.

######  Theorem 1.7.

[DHS, Corollary 3.10](Strong Central Sets Theorem for non-commutative
semigroup) Let $\left(S,\cdot\right)$ be a semigroup and let $C$ be a central
subset of $S$. Then there exist functions
$m:\mathcal{P}_{f}\left(\mathcal{T}\right)\to\mathbb{N},\qquad\alpha\in\times_{F\in\mathcal{P}_{f}\left(\mathcal{T}\right)}S^{m\left(F\right)+1},$and
$\tau\in\times_{F\in\mathcal{P}_{f}\left(\mathcal{T}\right)}\mathscr{I}_{m\left(F\right)}$
such that

  1. (1)

If $F\varsubsetneq G$ in $\mathcal{P}_{f}\left(\mathcal{T}\right)$ then
$\tau\left(F\right)\left(m\left(F\right)\right)<\tau\left(G\right)\left(1\right)$
and

  2. (2)

For any $n\in\mathbb{N}$, if $G_{1}\varsubsetneq
G_{2}\varsubsetneq\ldots\varsubsetneq G_{n}$ in
$\mathcal{P}_{f}\left(\mathcal{T}\right)$ and for each
$i\in\left\\{1,2,\ldots,n\right\\}$, $\left\langle
y_{i,t}\right\rangle_{t=1}^{\infty}\in G_{i}$ then

$\prod_{i=1}^{n}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),\left\langle
y_{i,t}\right\rangle_{t=1}^{\infty}\right)\in A.$

In the second section of the article we shall establish multidimensional
version of this Theorem. In the other direction in [GBP] authors established
polynomial version of the above Theorem.

######  Theorem 1.8.

(Polynomial Strong Central Sets Theorem) Let $A\subseteq\mathbb{N}$ be a
central set and $T\subseteq\mathcal{P}_{f}\left(\mathbb{P}\right)$,
$\mathbb{P}$ is the set of polynomials from $\mathbb{N}$ to $\mathbb{N}$
vanishes as $0$. Then there exist functions
$\alpha:\mathcal{P}_{f}\left(\mathbb{N}^{\mathbb{N}}\right)\to S\text{ and
}H:\mathcal{P}_{f}\left(\mathbb{N}^{\mathbb{N}}\right)\to\mathcal{P}_{f}\left(\mathbb{N}\right)$
such that

$\left(1\right)$ If
$F,G\in\mathcal{P}_{f}\left(\mathbb{N}^{\mathbb{N}}\right)$ and $F\subsetneq
G$ then $\max H\left(F\right)<\min H\left(G\right)$ and

$\left(2\right)$ If
$m\in\mathbb{N},G_{1},G_{2},....,G_{m}\in\mathcal{P}_{f}\left(\mathbb{N}^{\mathbb{N}}\right)$;
$G_{1}\subsetneq G_{2}\subsetneq....\subsetneq G_{m}$; and

for each $i\in\left\\{1,2,....,m\right\\},f_{i}\in G_{i},$ for all $P\in T$,

$\sum_{i=1}^{m}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{m}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\in A.$

In our work we shall extend 1.3 in the context of noncommutative semigroup. In
the third section of the article we shall establish multidimensional version
of Theorem 1.8.

##  2\. Noncommutative multidimensional central sets theorem

Let us recall the following technical lemma from [B].

######  Lemma 2.1.

[B, Lemma 3.1] Let $S$ be a set, let $e\in\beta S\setminus S$, let $k,r\geq 1$
and let $\left[S\right]{}^{k}=\cup_{i=1}^{r}A_{i}$. For each
$i\in\left\\{1,2,\dots,r\right\\}$, each $t\in\left\\{1,2,\dots,k\right\\}$
and each $E\in\left[S\right]{}^{t-1}$, define $B_{t}\left(E,i\right)$ by
downward induction on $t$:

  1. (1)

For $E\in\left[S\right]{}^{k-1}$, $B_{k}\left(E,i\right):=\left\\{y\in
S\setminus E:E\cup\left\\{y\right\\}\in A_{i}\right\\}$.

  2. (2)

For $1\leq t<k$ and $E\in\left[S\right]{}^{t-1}$,

$B_{t}\left(E,i\right):=\left\\{y\in S\setminus
E:B_{t+1}\left(E\cup\left\\{y\right\\},i\right)\in e\right\\}.$

Then there exists some $i\in\left\\{1,2,\dots,r\right\\}$ such that
$B_{1}\left(\emptyset,i\right)\in e.$

We need to introduce a few notations before introducing a key lemma. Let $S$
be a set. A set of ordered tuples $T$ of $S$ is said to be a tree if all of
its initial segments belong to $T$. More formally let by $S^{<\omega}$ we mean
collection of all functions from $\left\\{0,\ldots,n-1\right\\}$ to $S$, where
$n\in\mathbb{N}$. Now a non empty set $T\subseteq S^{<\omega}$ is a tree in
$S$ if $f\in T$ iff for any $A=\\{0,1,\ldots,m\\}\subseteq domf$,
$f\mid_{A}\in T$.

A function $f\in S^{\left\\{0,\ldots,n-1\right\\}}$ can be identified by
$\left\langle
f\left(0\right),f\left(1\right),\ldots,f\left(n-1\right)\right\rangle$. If
$s\in S$ then $f^{\frown}s:=\left\langle
f\left(0\right),f\left(1\right),\ldots,f\left(n-1\right),s\right\rangle$. For
$f\in S^{<\omega}$ we put $T\left(f\right)=\left\\{s\in S:f^{\frown}s\in
T\right\\}$.

The following is an important lemma for our purpose.

######  Lemma 2.2.

Let $\left(S,\cdot\right)$ be a semigroup such that there exists an idempotent
$e\in\beta S\setminus S$, let $k,r\geq 1$ and let
$\left[S\right]{}^{k}=\bigcup_{i=1}^{r}A_{i}$. Then there exist
$i\in\left\\{1,2,\dots,r\right\\}$ and a tree $T\subseteq S^{<\omega}$ such
that for all $f\in T$, and $\alpha_{1}<\alpha_{2}<\ldots<\alpha_{k}\subseteq
domf,\>\alpha_{j}\in\mathcal{P}_{f}\left(\omega\right),\>j\in\left\\{1,2,\ldots,k\right\\}$
one has:

  1. (1)

$T\left(f\right)\in e$.

  2. (2)

$\left\\{\prod_{t\in\alpha_{1}}f\left(t\right),\prod_{t\in\alpha_{2}}f\left(t\right),\ldots,\prod_{t\in\alpha_{k}}f\left(t\right)\right\\}\in
A_{i}$.

###### Proof.

[B, Lemma 3.2]. ∎

The following theorem is non-commutative extension of Beiglboeck theorem. In
fact, we will be using the same techniques for induction.

######  Theorem 2.3.

Let $\left(S,\cdot\right)$ be a commutative semigroup and assume that there
exists a non principal minimal idempotent in $\beta S$. Let $k,r\geq 1$ be
integers and let $\left[S\right]{}^{k}=\cup_{i=1}^{r}A_{i}$. There exist
$i\in\left\\{1,2,\dots,r\right\\}$ and functions

$m:\mathcal{P}_{f}\left(\mathcal{T}\right)\to\mathbb{N},\qquad\alpha\in\times_{F\in\mathcal{P}_{f}\left(\mathcal{T}\right)}S^{m\left(F\right)+1},\text{
and
}\tau\in\times_{F\in\mathcal{P}_{f}\left(\mathcal{T}\right)}\mathscr{I}_{m\left(F\right)}$

such that

  1. (1)

If $F\varsubsetneq G$ in $\mathcal{T}$, then
$\tau\left(F\right)\left(m\left(F\right)\right)<\tau\left(G\right)\left(1\right)$
and

  2. (2)

For any sequnece $G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq G_{n}$ in
$\mathcal{P}_{f}\left(\mathcal{T}\right),$ $f_{i}\in G_{i}$ and
$\alpha_{1}<\alpha_{2}<\dots<\alpha_{k}\leq n$ in
$\mathcal{P}_{f}\left(\mathbb{N}\right)$, we have

$\left(\prod_{i\in\alpha_{1}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right),\ldots,\prod_{i\in\alpha_{k}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right)\right)\in
A_{i}.$

###### Proof.

Fix a minimal idempotent $e\in\beta S\setminus S$. Let
$i\in\left\\{1,2,\dots,r\right\\}$ and $T\subseteq S^{<\omega}$ be a tree as
provided by lemma 2.2. We shall inductively construct sequences
$G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq G_{n}$ in
$\mathcal{P}_{f}\left(\mathcal{T}\right)$ such that for any
$\alpha_{1}<\alpha_{2}<\dots<\alpha_{k}\leq n$ ,

$\left\langle\prod_{i\in\alpha_{1}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right),\ldots,\prod_{i\in\alpha_{k-1}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right)\right\rangle\in
T$

By lemma 2.2 we have

$G_{n_{k}}=T\left(\left\langle\prod_{i\in\alpha_{1}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right),\ldots,\prod_{i\in\alpha_{k-1}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right)\right\rangle\right)\in
e.$

Then $G_{n_{k}}$ is a central set in $S$. Then we have $f_{i}\in G_{i}$,
$\prod_{i\in\alpha_{k}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right)\in
G_{n_{k}}$, where $\alpha_{k-1}<\alpha_{k}\leq n.$ Then

$\left\langle\prod_{i\in\alpha_{1}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right),\ldots,\prod_{i\in\alpha_{k}}x\left(m\left(G_{i}\right),\alpha\left(G_{i}\right),\tau\left(G_{i}\right),f_{i}\right)\right\rangle\in
T.$

using lemma 2.2 we get our desire result. ∎

##  3\. Multidimensional polynomial central sets theorem

In [GBP] authors proved a polynomial generalization of Stronger Central Sets
Theorem followed by polynomial generalization of Central Sets Theorem in
[BJM]. In [BJM] authors proved the polynomial version of Central sets Theorem
for countable abelian group taking finiterly many sequenses at a time. But in
[GBP] authors deal with semigroup $\left(\mathbb{N},+\right)$ taking all
sequences at a time. Almost verbatim we get the polynomial generalization of
Stronger Central Sets Theorem for countable commutative group. First let us
recall the definition of polynomial for countable commutative group.

######  Definition 3.1.

A map $f:G\to H$ between countable commutative groups we say that $f$ is a
polynomial map of degree 0 if it is constant. We say that $f$ is a polynomial
map of degree $d$, $d\in\mathbb{N}$, if it is not a polynomial map of degree
$d-1$ and for every $h\in H$, the map $x\to f\left(x+h\right)-f\left(x\right)$
is a polynomial of degree $\leq d-1$. Finally we denote by
$\mathbb{P}\left(G,H\right)$ the set of all polynomial maps $f:G\to H$ with
$f\left(0\right)=0$. Note that homomorphisms are elements of
$\mathbb{P}\left(G,H\right)$ having degree 1.

######  Theorem 3.2.

(Polynomial CST) Let $\left(S,\cdot\right)$ be a countable commutative group
and $F\in\mathcal{P}_{f}\left(\left(S^{j}\right){}^{\mathbb{N}}\right)$, let
$T\in\mathcal{P}_{f}\left(\mathbb{P}\right)$, where $\mathbb{P}$ is set of all
polynomials from $S^{j}$ to $S$ vanishes at zero and let $A$ be a central
subset of $S$. Then there exist sequences $\left\langle
b_{n}\right\rangle_{n=1}^{\infty}$ in $S$ and $\left\langle
H_{n}\right\rangle_{n=1}^{\infty}$ in $\mathcal{P}_{f}\left(\mathbb{N}\right)$
such that

  1. (1)

For each $n\in\mathbb{N}$, $\max H_{n}<\min H_{n+1}$ and

  2. (2)

For each $f\in F$, each $P\in T$ and each
$K\in\mathcal{P}_{f}\left(\mathbb{N}\right)$

$\sum_{n\in K}b_{n}+P\left(\sum_{n\in K}\sum_{t\in
H_{n}}f\left(t\right)\right)\in A.$

In [GBP] authors established a stronger version of the above thorem in
$\left(\mathbb{N},+\right)$ which we extend for countable commutative group.
First we need the following lemma.

######  Lemma 3.3.

Let $j,m\in\mathbb{N}$, let $S$ be a countable commutative group and let
$\mathbb{P}$ be a set of all polynomials from $S^{j}$ to $S$ vanishes at
$\boldsymbol{0}$. Let $A\subset S$ be a central set in $S$ and
$F\in\mathcal{P}_{f}\left(\left(S^{j}\right){}^{\mathbb{N}}\right)$. Then for
every $T\in\mathcal{P}_{f}\left(\mathbb{P}\right)$ there exist $a\in S$ and
$H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ such that $a+P\left(\sum_{t\in
H}f\left(t\right)\right)\in A$, for all $f\in F$ and $P\in T$.

###### Proof.

[BJM, Corollary 2.12]. ∎

The lemma has a following stronger version.

######  Lemma 3.4.

Let $j,m\in\mathbb{N}$, let $S$ be a countable commutative group and let
$\mathbb{P}$ be a set of all polynomials from $S^{j}$ to $S$ vanishes at
$\boldsymbol{0}$. Let $A\subset S$ be a central set in $S$ and
$F\in\mathcal{P}_{f}\left(\left(S^{j}\right){}^{\mathbb{N}}\right)$. Then for
every $T\in\mathcal{P}_{f}\left(\mathbb{P}\right)$ there exist $a\in S$ and
$H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ with $\min H>m$ such that
$a+P\left(\sum_{t\in H}f\left(t\right)\right)\in A$, for all $f\in F$ and
$P\in T$.

###### Proof.

Let $j,m\in\mathbb{N}$, $T\in\mathcal{P}_{f}\left(\mathbb{P}\right)$ and
$F\in\mathcal{P}_{f}\left(\left(S^{j}\right){}^{\mathbb{N}}\right)$, for each
$f\in F$ define $g_{f}\in\left(S^{j}\right){}^{\mathbb{N}}$ by
$g_{f}\left(t\right)=f\left(t+m\right)$, $t\in\mathbb{N}$. For this
$K=\left\\{g_{f}:f\in
F\right\\}\in\mathcal{P}_{f}\left(\left(S^{j}\right){}^{\mathbb{N}}\right)$,

there exist $a\in S$ and $L\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ such
that $a+P\left(\sum_{t\in L}g_{f}\left(t\right)\right)\in A$, for all $f\in F$
and $P\in T$.

Therefore $a+P\left(\sum_{t\in L}f\left(t+m\right)\right)\in A$, for all $f\in
F$ and $P\in T$.

i,e, $a+P\left(\sum_{t\in H}f\left(t\right)\right)\in A$, for all $f\in F$ and
$P\in T$. Where $H=L+m>m$. ∎

In the following $\mathcal{T}_{j}$ is the collection of all sequences in
$S^{j}.$

######  Theorem 3.5.

Let $\left(S,\cdot\right)$ be a countable commutative semigroup, let
$T\in\mathcal{P}_{f}\left(\mathbb{P}\right)$, where $\mathbb{P}$ be a set of
all polynomials from $S^{j}$ to $S$ vanishes at $\boldsymbol{0}$. Let
$A\subset S$ be a central set. Then there exist functions
$\alpha:\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)\to S$,
$H:\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)\to\mathcal{P}_{f}\left(\mathbb{N}\right)$,
Such that

  1. (1)

If $F\subsetneq G$ then $H\left(F\right)<H\left(G\right)$

  2. (2)

For any $n\in\mathbb{N}$, $G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq
G_{n}$ in $\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)$, we have for each
$i\in\left\\{1,2,....,n\right\\},f_{i}\in G_{i},$ and for all $P\in T$,

$\sum_{i=1}^{n}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{n}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\in A.$

###### Proof.

Choose a minimal idempotent $p\in\beta S$ with $A\in p$. For
$F\in\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)$, we shall use induction on
cardinality of $F$ , define $\alpha\left(F\right)\in S$ and
$H\left(F\right)\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ for witnessing
$\left(\ref{1},\ref{2}\right)$.

If $F=\left\\{f\right\\}$, as $p$ is minimal idempotent, the set
$A^{*}=\left\\{x\in A:-x+A\in p\right\\}$ belongs to $p$ [HS, Corollary 4.14].
Hence $A^{*}$ is a central set. So by [BJM, Corollary 2.12], there exist $a\in
S$ and $H\in\mathcal{P}_{f}\left(\mathbb{N}\right)$ such that

$\forall P\in T,\ a+P\left(\sum_{t\in H}f\left(t\right)\right)\in A^{*}.$

By setting $\alpha\left(\left\\{f\right\\}\right)=a$ and
$H\left(\left\\{f\right\\}\right)=H$, conditions $\left(\ref{1}\right)$ and
$\left(\ref{2}\right)$ are satisfied.

Now assume that $\mid F\mid>1$, $\alpha\left(G\right)$ and $H\left(G\right)$
have been defined for all proper subsets $G$ of $F$. Let
$K=\bigcup\left\\{H\left(G\right):\emptyset\neq G\subsetneq
F\right\\}\in\mathcal{P}_{f}\left(\mathbb{N}\right)$, $m=\max K$ and

Let

$R=\left\\{\begin{array}[]{c}\sum_{i=1}^{n}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\mid n\in\mathbb{N}\\\ \emptyset\neq
G_{1}\subsetneq G_{2}\subsetneq\cdots\subsetneq G_{n}\subsetneq F,\\\ f_{i}\in
G_{i},\forall i=1,2,..,n.\end{array}\right\\}$

$M=\left\\{\begin{array}[]{c}\sum_{i=1}^{n}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{n}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\mid n\in\mathbb{N}\\\
\emptyset\neq G_{1}\subsetneq G_{2}\subsetneq\cdots\subsetneq G_{n}\subsetneq
F,\\\ f_{i}\in G_{i},\forall i=1,2,..,n,\,P\in T.\end{array}\right\\}$

Then $R$ is a fininte subset of $S^{j}$, $M$ is a finite subset of $S$ and by
induction hypothesis , $M\subseteq A^{*}$.

Let

$B=A^{*}\cap\left(\bigcap_{x\in M}\left(-x+A^{*}\right)\right)\in p.$

For $P\in T$ and $d\in R$, let us define the polynomial
$Q_{p,d}\in\mathbb{P}\left(S^{j},S\right)$ by

$Q_{p,d}\left(y\right)=P\left(y+d\right)-P\left(d\right).$

Degree of $Q_{p,d}$ is one degree lesser than $P$.

Let $D=T\cup\left\\{Q_{p,d}\mid P\in T\text{ and }d\in R\right\\}$.

From lemma 3.4, there exist $\gamma\in\mathcal{P}_{f}\left(\mathbb{N}\right)$
with $\min\left(\gamma\right)>m$ and $a\in S$ such that

$\forall Q\in D,f\in F\ a+Q\left(\sum_{t\in\gamma}f\left(t\right)\right)\in
B.$

We set $\alpha\left(F\right)=a$ and $H\left(F\right)=\gamma$. Now we varify
conditions (1) and (2).

Since $\min\left(\gamma\right)>m$, (1) is satisfies.

to varify (2), let $n\in\mathbb{N}$ and
$G_{1},G_{2},...,G_{n}\in\mathcal{P}_{f}\left(\mathcal{T}_{0}\right),\,G_{1}\subsetneq
G_{2}\subsetneq.....\subsetneq G_{n}=F$ and $f_{i}\in G_{i},i=1,2,...,n$.

For $n=1$, then $G_{1}=G_{n}=F$, $\alpha\left(G_{n}\right)+P\left(\sum_{t\in
H\left(G_{n}\right)}f_{n}\left(t\right)\right)=a+P\left(\sum_{t\in\gamma}f\left(t\right)\right)\in
B\subseteq A^{*}$ for all $P\in T\subseteq D$, $f\in F$.

If $n>1$, then

$\sum_{i=1}^{n}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{n}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)$

$=\alpha\left(G_{n}\right)+\sum_{i=1}^{n-1}\alpha\left(G_{i}\right)+P\left(\sum_{t\in
H\left(G_{n}\right)}f_{i}\left(t\right)+\sum_{i=1}^{n-1}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)$

$=a+\sum_{i=1}^{n-1}\alpha\left(G_{i}\right)+P\left(\sum_{t\in\gamma}f_{n}\left(t\right)+\sum_{i=1}^{n-1}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)$

Since $G_{n}=F$ and $\alpha\left(F\right)=a,H\left(F\right)=\gamma$.

$=a+\sum_{i=1}^{n-1}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{n-1}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)+$
$P\left(\sum_{t\in\gamma}f_{n}\left(t\right)+\sum_{i=1}^{n-1}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)-P\left(\sum_{i=1}^{n-1}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)$
$=a+y+Q_{p,d}\left(\sum_{t\in\gamma}f_{n}\left(t\right)\right)$

where
$y=\sum_{i=1}^{n-1}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{n-1}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\in M$,

$d=\sum_{i=1}^{n-1}\sum_{t\in H\left(G_{i}\right)}f_{i}\left(t\right)\in R$
and $P\in T$ so $Q_{p,d}\in D$.

So we have

$a+Q_{p,d}\left(\sum_{t\in\gamma}f_{n}\left(t\right)\right)\in
B\subseteq-y+A^{*}$

Therefore

$\sum_{i=1}^{n}\alpha\left(G_{i}\right)+P\left(\sum_{i=1}^{n}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\in A^{*}.$

This completes the induction argument, hence the proof. ∎

Now we shall extend Beiglboeck Multidimentional Central Sets Theorem for
countable commutative group and polynomial version of Stronger Central Sets
Theorem.

######  Theorem 3.6.

Let $\left(S,\cdot\right)$ be a countable commutative group and assume that
there exists a non principal minimal idempotent in $\beta S$. Let
$k,r\left(\geq 1\right)\in\mathbb{N}$ and let $[S]^{k}=\cup_{i=1}^{r}A_{i}$.
There exist $j\in\left\\{1,2,\dots,r\right\\}$ , and functions
$\alpha:\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)\to S$,
$H:\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)\to\mathcal{P}_{f}\left(\mathbb{N}\right)$
such that

  1. (1)

If $F\subsetneq G$ then $H\left(F\right)<H\left(G\right)$

  2. (2)

For $G_{1}\subsetneq G_{2}\subsetneq\ldots\subsetneq G_{n_{k}}$ in
$\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)$, for every
$\beta_{1}<\beta_{2}<\ldots<\beta_{k}\leq n_{k}$ in
$\mathcal{P}_{f}\left(\mathbb{N}\right)$ and for every
$F\in\mathcal{P}_{f}\left(\mathbb{P}\left(S^{j},S\right)\right)$, $f_{i}\in
G_{i}$ we have

$\left\langle\begin{array}[]{c}\sum_{i\in\beta_{1}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{1}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right),\ldots,\\\
\sum_{i\in\beta_{k}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{k}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\end{array}\right\rangle\in
A_{j},$

for all $P\in F$.

###### Proof.

As the theorem 2.2 we fix a minimal idempotent $e\in\beta S\setminus S$. Let
$i\in\left\\{1,2,\dots,r\right\\}$ and $T\subseteq S^{<\omega}$ be a tree as
provided by lemma 2.2. For the sequence $G_{1}\subsetneq
G_{2}\subsetneq\ldots\subsetneq G_{n_{k}}$ in
$\mathcal{P}_{f}\left(\mathcal{T}_{j}\right)$ such that for
$\beta_{1}<\beta_{2}<\ldots<\beta_{k-1}\leq n_{k-1}$ , and fixed
$F\in\mathcal{P}_{f}\left(\mathbb{P}\left(S^{j},S\right)\right)$ . We shall
use induction such that

$\left\langle\begin{array}[]{c}\sum_{i\in\beta_{1}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{1}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right),\ldots,\\\
\sum_{i\in\beta_{k-1}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{k-1}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\end{array}\right\rangle\in T.$

$f_{i}\in G_{i}$, for all $P\in F$.

By lemma 2.2, we have

$C=T\left(\left\langle\begin{array}[]{c}\sum_{i\in\beta_{1}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{1}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right),\ldots,\\\
\sum_{i\in\beta_{k-1}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{k-1}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\end{array}\right\rangle\right)\in
e.$

Then $C$ is a central set in $S$. Then for
$F\in\mathcal{P}_{f}\left(\mathbb{P}\left(S^{j},S\right)\right)$ and
$G_{n_{k-1}+1}\subsetneq G_{n_{k-1}+2}\subsetneq\ldots\subsetneq G_{n_{k}}$,
$f_{i}\in G_{i}$,
$\sum_{i\in\beta_{k}}\left(\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{k}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\right)\in C\text{ for all }P\in
F$, where
$\beta_{k-1}<\beta_{k}\subseteq\left\\{n_{k-1}+1,\ldots,n_{k}\right\\}$.

So we have,

$\left\langle\begin{array}[]{c}\sum_{i\in\beta_{1}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{1}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right),\ldots,\\\
\sum_{i\in\beta_{k}}\alpha\left(G_{i}\right)+P\left(\sum_{i\in\beta_{k}}\sum_{t\in
H\left(G_{i}\right)}f_{i}\left(t\right)\right)\end{array}\right\rangle\in T,$

as we wanted to show. ∎

Acknowledgment: The first author is supported by NBHM(RP) fellowship
R&DII/9678.

The second author acknowledges the Grant CSIR-UGC NET fellowship with file No.
09/106(0199)/2019-EMR-I of CSIR-UGC NET.

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# Detecting nuclear mass distribution in isobar collisions via charmonium

Jiaxing Zhao<EMAIL_ADDRESS>SUBATECH, Université de Nantes, IMT
Atlantique, IN2P3/CNRS, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France
Physics Department, Tsinghua University, Beijing 100084, China  Shuzhe Shi
<EMAIL_ADDRESS>Center for Nuclear Theory, Department of Physics and
Astronomy, Stony Brook University, Stony Brook, New York 11794–3800, USA

###### Abstract

The collective properties of final state hadrons produced in the high
statistics ${}_{44}^{96}$Ru+${}_{44}^{96}$Ru and
${}_{40}^{96}$Zr+${}_{40}^{96}$Zr collisions at
$\sqrt{s_{\mathrm{NN}}}=200~{}\mathrm{GeV}$ are found to be significantly
different. Such differences were argued to be precise probes of the difference
in nucleon distribution in the isobar nuclei. We investigate the $J/\psi$
production in the isobar collision via a relativistic transport approach. By
comparing the isobar systems according to equal centrality bin and equal
multiplicity bin, we find that the yield ratio of $J/\psi$ is sensitive to the
differences in both the number of binary collisions and the medium evolution.
Besides, the elliptic flow $v_{2}$ of $J/\psi$ is qualitatively different from
the light hadrons, and the ratio between Ru+Ru and Zr+Zr collisions is
sensitive to the medium evolution. The charmonium production provides an
independent probe to study the nucleon distribution in the isobar system.

_Introduction._ — At the mean-field level, nuclear properties like spin,
parity of ground state, magic number, and $\beta$ decays can be well described
by the nuclear shell model Caurier  _et al._ (2005). In the shell model, we
know most of the nuclei in their ground state are deformed and non-uniform,
except the Doubly-Magic Nucleus. Under the polarization effect of valence
nucleons, nuclei with an unfilled shell or sub-shell will be deformed and
generates collective motion induced by the interaction between valence
nucleons and shell structure Rowe (2010). The deformation depends closely on
the number of protons and neutrons in the nucleus and has been studied in low-
energy nuclear physics for many years, e.g., see reviews Heyde and Wood
(2011); Wood  _et al._ (1992). In relativistic heavy-ion collisions, the
nuclei are fully destroyed at the beginning of the collisions, but the
deformed information will still show its fingerprint in the final observables
Rosenhauer  _et al._ (1986); Shuryak (2000). The deformed nuclear-nuclear
collisions will lead to an anisotropic overlap region. The spatial anisotropy
of the collision overlap region in the initial state will transform into a
momentum anisotropy of the produced hadrons in the final state Ollitrault
(1992); Voloshin and Zhang (1996); Qiu and Heinz (2011); Filip  _et al._
(2009); Alver and Roland (2010); Carzon  _et al._ (2020). Therefore,
relativistic heavy-ion collisions may supply a new way to probe nuclear
deformation.

Recently, a high statistics heavy-ion collision, which collides
${}_{44}^{96}$Ru+${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr+${}_{40}^{96}$Zr with
beam energy $\sqrt{s_{\text{NN}}}=200~{}\text{GeV}$, is performed by the STAR
Collaboration at the Relativistic Heavy Ion Collider (RHIC) Abdallah  _et al._
(2022)111For the rest of the paper, we will refer to the ${}_{44}^{96}$Ru and
${}_{40}^{96}$Zr nucleus as Ru and Zr, separately.. Such a contrast experiment
originally proposed to search for the chiral magnetic effect (CME) Kharzeev
(2006); Kharzeev  _et al._ (2008); Fukushima  _et al._ (2008), under the
presumption that the same baryon number would lead to the same non-CME
background while the different electric number would induce a sizable
difference in the CME signal. Unexpectedly, the bulk properties, namely the
charge multiplicity and elliptic and triangular flows, are found to be
different in the isobar systems Abdallah  _et al._ (2022). While such a
subtlety prevents one to make a conclusive statement on the existence of CME
in heavy-ion collisions and calls for more efforts in better quantification of
the background Feng  _et al._ (2022); Kharzeev  _et al._ (2022), it brings a
new opportunity to study the nucleon distribution in relativistic heavy-ion
collisions Giacalone (2020a, b); Jia and Zhang (2021); Jia (2022); Zhang and
Jia (2022); Jia  _et al._ (2022); Giacalone  _et al._ (2021); Li  _et al._
(2020); Xu  _et al._ (2021a, 2022, b); Zhao  _et al._ (2022a); Bally  _et al._
(2022); Nijs and van der Schee (2021). In such experiments, a new phase of
matter, called Quark Gluon Plasma (QGP), is created, and the final state
particles can be well explained by a hydrodynamic description of the system
evolution, (see e.g. Refs. Gale  _et al._ (2013); Schenke  _et al._ (2011);
Shen  _et al._ (2016)). The initial state of the evolution is the consequence
of multiple nucleon-nucleon collisions, and it provides an unique opportunity
of measuring multi-nucleon correlation within a nucleus, regardless of the
fact that the mapping between initial condition and final observables is
complicated.

Figure 1: 3D shapes of Ru and Zr nucleus.

In Ru and Zr nuclei, the spatial distribution of nucleons are usually
described by a deformed Woods–Saxon distribution Fricke  _et al._ (1995),

$\displaystyle\rho(r,\theta,\phi)$ $\displaystyle=$
$\displaystyle{\rho_{0}\over 1+e^{[r-R(\theta,\phi)]/a_{0}}},$ $\displaystyle
R(\theta,\phi)$ $\displaystyle=$ $\displaystyle
R_{0}(1+\beta_{2}Y_{2,0}+\beta_{3}Y_{3,0}+...),$ (1)

where the parameters $R_{0}$ and $a_{0}$ are respectively called radius and
diffusiveness. The maximum nuclear density $\rho_{0}$ is given by the
normalization condition that $\int\rho(r,\theta,\phi)\mathrm{d}^{3}\bm{r}=A$.
$Y_{\ell,m}$’s are spherical harmonics. $\beta_{2}$ and $\beta_{3}$ control
the quadrupole and octupole deformations, respectively. Both lower-energy
nuclear experiments and high-energy heavy-ion collisions suggest that the Ru
nucleus has a sizable quadrupole deformation while the Zr nucleus has an
octupole deformation Mach  _et al._ (1990); Hofer  _et al._ (1993); Zhang and
Jia (2022); Xu  _et al._ (2021a). Particularly, it is found in Zhang and Jia
(2022) and Nijs and van der Schee (2021), which respectively describes the
medium evolution by a particle transport model and hydrodynamics, that the
high-precision measurements in the isobar collisions can be described by the
parameter set $R_{0}^{\mathrm{Ru}}=5.09~{}\textrm{fm}$,
$a_{0}^{\mathrm{Ru}}=0.46~{}\textrm{fm}$, $\beta_{2}^{\mathrm{Ru}}=0.162$,
$\beta_{3}^{\mathrm{Ru}}=0.0$, $R_{0}^{\mathrm{Zr}}=5.02~{}\textrm{fm}$,
$a_{0}^{\mathrm{Zr}}=0.52~{}\textrm{fm}$, $\beta_{2}^{\mathrm{Zr}}=0.06$, and
$\beta_{3}^{\mathrm{Zr}}=0.2$. The corresponding three-dimensional shapes are
shown in Fig. 1.

Charmonium states like $J/\psi$, which are boundstates of a charm quark and
its anti-quark, are independent probes of the initial condition. Owing to the
heavy mass, charm quarks and charmonium states are dominantly produced in the
initial hard scattering in heavy-ion collisions. After the initial production,
they interact with the QGP and their properties get modified. The spectrum and
flow information of charmonium states in heavy-ion collisions reflect both the
QGP properties and initial condition Matsui and Satz (1986); Grandchamp and
Rapp (2002); Zhao and Rapp (2010); Du and Rapp (2015); He  _et al._ (2022);
Yan  _et al._ (2006); Liu  _et al._ (2009); Zhou  _et al._ (2014); Chen
(2019); Zhao  _et al._ (2022b); Chen  _et al._ (2020); Blaizot  _et al._
(2016); Katz and Gossiaux (2016). In heavy-ion collisions, the initially
produced charmonium bound states might dissociate into scattering states due
to the static color-screening effect and dynamic dissociation Matsui and Satz
(1986); Satz (2006); Peskin (1979); Bhanot and Peskin (1979); Brambilla  _et
al._ (2013). Some of them can survive to the end of the QGP evolution and be
detected. Another source of final state charmonium particle comes from the
recombination of uncorrelated charm and anti-charm quarks in the QGP
Grandchamp and Rapp (2002); Zhao and Rapp (2010); Du and Rapp (2015); He  _et
al._ (2022); Yan  _et al._ (2006); Liu  _et al._ (2009); Zhou  _et al._
(2014); Chen (2019). We note that initial production rates of both charm
quarks and charmonium states are proportional to the number of binary nucleon-
nucleon collisions, denoted as $N_{\mathrm{coll}}$. In the hypothetical
situation that the bulk background being the same in the isobar system, the
survived charmonium production rate is proportional to $N_{\mathrm{coll}}$,
whereas the recombination production rate is proportional to the square of
charm quark number, hence $\propto N_{\mathrm{coll}}^{2}$. Compared to light
flavor observables, charmonium states are sensitive to different aspects of
the initial state and bulk evolution, and henceforth serve as an independent
probe of the QGP. In this work, we study the differences between the
properties of the $J/\psi$ particles produced in Ru+Ru and those in Zr+Zr
collisions. We aim to provide an independent probe for the deformations in Ru
and Zr nuclei.

Figure 2: (Left) The centrality dependence of the maximum temperature of the
medium (upper), nuclear modification factor $R_{AA}$ of $J/\psi$ in Ru+Ru
(middle) and Zr+Zr (lower) collisions at
$\sqrt{s_{\textrm{NN}}}=200~{}\textrm{GeV}$. Vertical dashed lines indicate
the position of the cusp, whereas the horizontal dashed line label the
dissociation temperature of $J/\psi$. (Right) Same as Left but for equal-
multiplicity bins.

_Charmonium transport._ — We start by describing the charmonium dissociation
and recombination in the QGP. Since charmonium states are heavy and color
neutral, they are unlikely to be thermalized with the medium. Their phase
space distribution, $f_{\psi}({\bm{p}},{\bm{x}},\tau)$ with
$\psi\in\\{J/\psi,\chi_{c},\psi^{\prime}\\}$, is governed by a transport
equation which includes both initial production and regeneration Yan  _et al._
(2006); Liu  _et al._ (2009); Zhou  _et al._ (2014); Chen (2019),

$\displaystyle\left[\cosh(y-\eta)\partial_{\tau}+{\sinh(y-\eta)\over\tau}\partial_{\eta}+{\bm{v}}_{T}\cdot\nabla_{T}\right]f_{\psi}$
(2) $\displaystyle=$ $\displaystyle-\alpha\,f_{\psi}+\beta,$

where $y=(1/2)\ln[(E+p_{z})/(E-p_{z})]$ is the momentum rapidity,
${\bm{v}}_{T}={\bm{p}}_{T}/E_{T}$ is the transverse velocity,
$E_{T}=\sqrt{m_{\psi}^{2}+{\bm{p}}_{T}^{2}}$ is the transverse energy, and
$\nabla_{T}\equiv(\partial_{x},\partial_{y})$ is the transverse gradient. The
second(third) term on the left hand side arises from the free streaming of
$\psi$ which leads to the leakage effect in the longitudinal(transverse)
direction. The anomalous suppression and regeneration mechanisms in the QGP
medium are respectively reflected in the loss term $\alpha$ and gain term
$\beta$. Charmonia in hot QGP medium suffer Debye screening Matsui and Satz
(1986). With increasing temperature, the interaction between a pair of heavy
quarks is more screened, while the averaged size of a charmonium state
increases. When the averaged size $\langle r\rangle$ diverges, the charmonium
is dissociated. We solve the two-body Schrödinger equation with the finite-
temperature potential and define the dissociation temperature $T_{d}$ as the
temperature when $\langle r\rangle\to\infty$. We find
$T_{d}=(2.3,1.2,1.1)~{}T_{c}$ for $J/\psi$, $\chi_{c}$, and $\psi^{\prime}$,
respectively Zhao and Zhuang (2022). When the local temperature is higher than
the dissociation temperatures $T_{d}$, the charmonium state disappears, and
the regeneration only happens when the temperature is lower than $T_{d}$. In
addition to the Debye screening, charmonium states suffer dynamical
dissociation in the QGP, such as gluon dissociation process, $g+\psi\to
c+\bar{c}$ Peskin (1979); Bhanot and Peskin (1979). The charmonium gluo-
dissociation cross-section $\sigma_{g\psi}^{c\bar{c}}$ can be derived, via the
operator-production-expansion (OPE) method, in vacuum Peskin (1979); Bhanot
and Peskin (1979) and extended to the finite temperature Liu  _et al._ (2009);
Zhou  _et al._ (2014); Chen (2019). Here we adopt the setup of Liu  _et al._
(2009). Taking only the gluo-dissociation as the loss term and its inverse
process, $c+\bar{c}\to g+\psi$, as the gain term in this study, $\alpha$ and
$\beta$ can be explicitly expressed as Yan  _et al._ (2006)

$\displaystyle\alpha({\bm{p}},{\bm{x}},\tau)$ $\displaystyle=$
$\displaystyle{1\over
2E_{T}}\int{d^{3}{\bm{p}}_{g}\over(2\pi)^{3}2E_{g}}W_{g\psi}^{c\bar{c}}(T,s)f_{g}({\bm{p}}_{g},{\bm{x}},\tau)$
$\displaystyle\times\Theta(T({\bm{x}},\tau)-T_{c}),$
$\displaystyle\beta({\bm{p}},{\bm{x}},\tau)$ $\displaystyle=$
$\displaystyle{1\over
2E_{T}}\int{d^{3}{\bm{p}}_{g}\over(2\pi)^{3}2E_{g}}{d^{3}{\bm{p}}_{c}\over(2\pi)^{3}2E_{c}}{d^{3}{\bm{p}}_{\bar{c}}\over(2\pi)^{3}2E_{\bar{c}}}$
(3) $\displaystyle\times
W_{c\bar{c}}^{g\psi}(T,s)f_{c}({\bm{p}}_{c},{\bm{x}},\tau)f_{\bar{c}}({\bm{p}}_{\bar{c}},{\bm{x}},\tau)$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p+p_{g}-p_{c}-p_{\bar{c}})$
$\displaystyle\times\Theta(T({\bm{x}},\tau)-T_{c}),$

where $E_{g}$, $E_{c}$, and $E_{\bar{c}}$ are the gluon, charm quark and anti-
charm quark energies, respectively, and ${\bm{p}}_{g}$, ${\bm{p}}_{c}$ and
${\bm{p}}_{\bar{c}}$ are their momenta. The Mandelstam variable $s$ is the
$g\psi$ interaction energy. $W_{g\psi}^{c\bar{c}}$ is the dissociation
probability while $W_{c\bar{c}}^{g\psi}$ the regeneration probability. Their
related due to the detailed balance between the two processes, and the
explicit expressions are given in Polleri  _et al._ (2004). $\Theta$ is the
Heaviside step function to guarantee the calculation in the QGP phase above
the critical temperature $T_{c}$. The latter is taken to be
$T_{c}=160~{}\text{MeV}$.

The gluon distribution $f_{g}$ follows the Bose-Einstein distribution, whereas
charm quarks with low transverse momenta are found to be thermalized according
to experimental measurement of open-charm meson Adamczyk  _et al._ (2017),
despite of their large mass. As a first-order approximation, we take a thermal
distribution for charm/anti-charm momentum,
$f_{c}({\bm{p}}_{c},{\bm{x}},\tau)={N_{c}\rho_{c}({\bm{x}},\tau)\over
e^{p_{c}^{\mu}u_{\mu}/T}+1}$, where $N_{c}$ is the normalization factor to
ensure $\int{N_{c}\over
e^{p_{c}^{\mu}u_{\mu}/T}+1}{d^{3}{\bm{p}}_{c}\over(2\pi)^{3}2E_{c}}=1$.
Neglecting the density change due to transverse expansion, the density in
coordinate space is governed by the nuclear geometry of the colliding system
Liu  _et al._ (2009),

$\displaystyle\rho_{c}({\bm{x}},\tau)$ $\displaystyle=$
$\displaystyle{\rho_{\mathrm{coll}}({\bm{x}})}{\cosh\eta\over\tau}{d\sigma_{pp}^{c\bar{c}}\over
dy},$ (4)

where $d\sigma^{c\bar{c}}_{pp}/dy$ is charm quark production cross section per
unit rapidity in p+p collisions. We take
$d\sigma^{c\bar{c}}_{pp}/dy=0.162~{}\mathrm{mb}$, which corresponds to
$\sqrt{s_{\mathrm{NN}}}=200~{}\text{GeV}$ Zhao and Zhuang (2022). The binary
collision density $\rho_{\mathrm{coll}}({\bm{x}})$ is given by the initial
condition of the bulk background. In this work, we adopt the boost invariant,
event-by-event fluctuating initial conditions generated by the Monte Carlo
Glauber model (MCGlauber) Miller  _et al._ (2007). We generate millions of
initial profiles and bin them into different centrality or multiplicity
classes. Events within the same class are then averaged after being overlapped
according to the center of mass and aligned according to the second-order
participant plane. With such event-averaged initial conditions, we solve the
medium evolution taking the data validated music hydrodynamic simulation
package Gale  _et al._ (2013); Schenke  _et al._ (2011, 2010); McDonald  _et
al._ (2017), with taking the s95p Equation of State Huovinen and Petreczky
(2010), a constant shear viscosity $\eta/s$ = 0.08, and vanishing bulk
viscosity Policastro  _et al._ (2001); Kovtun  _et al._ (2005); Bernhard  _et
al._ (2016). The hydrodynamic evolution provides the space-time profile of
temperature $(T)$ and flow velocity $(u^{\mu})$ of the bulk background in the
calculation of charmonium transport and gluon and charm quark distribution.
The centrality and multiplicity dependence of the maximum temperature of the
medium produced in Ru+Ru and Zr+Zr collisions are shown in the upper panels of
Fig. 2. One can find them to be close to identical in central collisions but
visibly different in peripheral collisions.

Figure 3: The centrality (left) and multiplicity (right) dependence of the
yield ratio ($N_{\text{AA}}^{\text{RuRu}}/N_{\text{AA}}^{\text{ZrZr}}$) and
double ratio ($R_{\text{AA}}^{\text{RuRu}}/R_{\text{AA}}^{\text{ZrZr}}$) of
prompt $J/\psi$ produced in Ru+Ru and Zr+Zr collisions at
$\sqrt{s_{\textrm{NN}}}=200~{}\textrm{GeV}$. Dotted lines represent the
regeneration contribution. While thin and thick solids lines are the initial
and total results, respectively. The filled dots are the ratio of number of
collisions $N_{\text{coll}}$ between Ru+Ru and Zr+Zr, and open dots are its
square.

Besides the hot nuclear matter effect which affects charmonium motion through
the above discussed anomalous suppression and regeneration in the QGP, there
is also the cold nuclear matter effect which changes the initial condition of
the transport equation (2). The cold nuclear matter effect includes mainly the
nuclear absorption Gerschel and Hufner (1988), Cronin effect Cronin  _et al._
(1975), and nuclear shadowing Mueller and Qiu (1986). Considering a finite
formation time of charmonia and small nuclear size of Ru and Zr in high energy
collisions, the nuclear absorption is neglected in this study. The initial
charmonium distribution in heavy-ion collisions is constructed by a
superposition of the charmonium distribution in p+p collisions with
considering the cold nuclear matter effects,

$\displaystyle\begin{split}f_{\psi}({\bm{p}},{\bm{x}},\tau_{0})=\;&{(2\pi)^{3}\over
E\tau_{0}}\rho_{\text{coll}}({\bm{x}})f^{\mathrm{Cronin}}_{\psi}({\bm{p}},{\bm{x}})\\\
&\times\mathcal{R}_{g}^{P}({\bm{p}},{\bm{x}})\mathcal{R}_{g}^{T}({\bm{p}},{\bm{x}}),\end{split}$
(5)

where the differential cross section Zha  _et al._ (2016); Zhao and Zhuang
(2022),

$\displaystyle f^{\mathrm{Cronin}}_{\psi}={a\over 2\pi\langle
p_{T}^{2}\rangle_{A}}\left(1+b^{2}{p_{T}^{2}\over\langle
p_{T}^{2}\rangle_{A}}\right)^{-n}{d\sigma^{\psi}_{pp}\over dy},$ (6)

encodes the Cronin effect that enhance the transverse momentum compared to p+p
collisions

$\displaystyle\langle p_{T}^{2}\rangle_{A}\approx\langle
p_{T}^{2}\rangle_{p}+a_{gN}\,\rho_{0}^{-1}\sqrt{\rho_{\text{coll}}({\bm{x}})/\sigma_{\textrm{NN}}^{inel}}.$
(7)

The parameters $a=2b^{2}(n-1)$,
$b=\Gamma(\frac{3}{2})\Gamma(n-\frac{3}{2})/\Gamma(n-1)$, $n=3.93$, and
$\langle p_{T}^{2}\rangle_{p}=3.05~{}\textrm{GeV}^{2}$ are fitted to match the
experimental data for p+p collisions Zha  _et al._ (2016); Zhao and Zhuang
(2022); Adam  _et al._ (2018). The Cronin parameter, which describes the
averaged charmonium transverse momentum square obtained from the gluon
scattering with a unit of length of nucleons, is set as
$a_{gN}=0.1~{}\textrm{GeV}^{2}/\textrm{fm}$ Zhao and Rapp (2010); Zhao and
Zhuang (2022) in order to fit experimental data in p-A and low-energy A-A
collisions Topilskaya  _et al._ (2003); Adare  _et al._ (2008). The mean
trajectory length of the two gluons in the two nuclei before the $c\bar{c}$
formation has been estimated as
$\rho_{0}^{-1}\sqrt{\rho_{\text{coll}}({\bm{x}})/\sigma_{\textrm{NN}}^{inel}}$,
where $\rho_{0}$ is the maximum nuclear density in Eq. (1). The direct
production cross sections
$d\sigma^{\\{J/\psi,\chi_{c},\psi^{\prime}\\}}_{pp}/dy=\\{0.6,0.3,0.1\\}\times
d\sigma^{J/\psi,\mathrm{prompt}}_{pp}/dy$ Andronic  _et al._ (2016), where
$d\sigma^{J/\psi,\mathrm{prompt}}_{pp}/dy=716.7~{}\mathrm{nb}$ is prompt
production cross section Adam  _et al._ (2018).

$\mathcal{R}_{g}^{P}$ and $\mathcal{R}_{g}^{T}$ in Eq. (5) are spatial-
dependent shadowing factors for gluon in projectile and target. Assuming that
the inhomogeneous shadowing is proportional to the parton path length through
the nucleus, the spatial-dependent shadowing factor can be estimated as,

$\mathcal{R}_{g}^{P,T}({\bm{p}},{\bm{x}})\approx
1+{A[R_{g}(x_{g}^{P,T},Q^{2})-1]\sqrt{\rho_{\text{coll}}({\bm{x}})}\over
N_{\text{coll}}^{\text{cen}}/\sqrt{\sigma_{\textrm{NN}}^{inel}}},$ (8)

where the shadowing modification factor $R_{g}$ can be simulated by the EPS09
package Helenius  _et al._ (2012).
$x_{g}^{P,T}=(\sqrt{m_{\psi}^{2}+p_{T}^{2}}/\sqrt{s_{\textrm{NN}}})e^{\pm y}$
is gluon longitudinal momentum fraction with $y$ being the charmonium
rapidity, and the $+$($-$) sign is chose for projectile(target). The momentum
transfer squared can be taken as $Q^{2}=m_{\psi}^{2}+p_{T}^{2}$. For both Ru
and Zr, the nuclear mass number is $A=96$. $N_{\text{coll}}^{\text{cen}}$ is
the number of binary collision for the most central collisions.

_Results._ — Experimental measurement of prompt $J/\psi$ mesons include feed-
down contributions from the excited charmonium states, i.e., $\chi_{c}$ and
$\psi^{\prime}$. With initial condition given in Eq. (5), we solve the
transport equations (2) for $J/\psi$, $\psi^{\prime}$ and $\chi_{c}$ on top of
event-averaged hydrodynamic background corresponding to different collision
systems and different centrality or multiplicity bins. Then we decay
$\chi_{c}$ and $\psi^{\prime}$ into $J/\psi$, with branch ratios respectively
being $22\%$ and $61\%$ Zyla  _et al._ (2020), and obtain the properties of
prompt $J/\psi$ in the final state. It shall be noted that the non-prompt
contribution from $B$-decays can be neglected at $200~{}\textrm{GeV}$ Ru+Ru
and Zr+Zr collisions, due to the small production cross-section of the latter
at low beam energy Andronic  _et al._ (2016). Therefore, inclusive production
rate of $J/\psi$ is dominated by the prompt one.

We first study the nuclear modification factor of $J/\psi$ in both Ru+Ru and
Zr+Zr collisions, which is defined as the ratio of production rate in AA to
that in p+p collisions, scaled by the inverse of number of binary collisions,

$\displaystyle R_{\text{AA}}=\frac{N_{\text{AA}}}{N_{pp}\,N_{\text{coll}}}\,.$
(9)

In Fig. 2, we present both the centrality and multiplicity dependence of
prompt $J/\psi$ $R_{AA}$ in Ru+Ru and Zr+Zr collisions at
$\sqrt{s_{\textrm{NN}}}=200~{}\textrm{GeV}$, with contributions from initial
production and regeneration separated. We observe a clear cusp located the
$5-10\%$ centrality bin or the $\sim 230$ multiplicity bin. For collisions
more central than the cusp, the highest temperature is higher than the
dissociation temperature of $J/\psi$, $T_{d}=2.3~{}T_{c}$, which is indicated
by the horizontal dashed lines. $J/\psi$ mesons initially produced at the
$T>T_{d}$ region suffer a strong suppression. Meanwhile, one can find the
regeneration contribution plans an important role in final total yield,
especially at central collisions even in small systems like Ru+Ru and Zr+Zr
collisions.

Then we move on to compare the $J/\psi$ production in the isobar systems by
taking the ratio between observables in Ru+Ru and those in Zr+Zr collisions.
In the upper panels of Fig. 3, we show the ratio of total yield as well as
those for initial and regeneration productions. In the hypothetical case of
identical background in the isobar systems, one would expect the ratio of
initial production (thin solid lines) to be the same as that of
$N_{\text{coll}}$ (filled square), whereas the ratio of regeneration
production (thin dash lines) to be identical to that of $N_{\text{coll}}^{2}$
(open square)222It shall be worth noting that the difference due to cold
nuclear effect is negligible. We observe that the former is below the
$N_{\text{coll}}$ ratio, because higher temperature and larger volume in Ru+Ru
collisions. Meanwhile, the ratio of regeneration production agree well with
the $N_{\text{coll}}^{2}$ ratio at central collisions, and goes above such a
expectation for peripheral collisions owning to the longer evolution time in
Ru+Ru collisions, and therefore more $J/\psi$ particles are produced via
regeneration. As for the double ratio, by definition, it equals to the yield
ratio times the inverse ratio of the $N_{\text{coll}}$. So, the double ratio
is smaller than 1 and decreases monotonously at peripheral collisions. In
comparing the left and right panels, one can clearly see that all ratios are
closer to unity when we compare the isobar system within the same multiplicity
bin than the comparison for the same centrality bin.

Figure 4: (Upper panel) The centrality dependence of the elliptic flow $v_{2}$
of light hadrons $\pi$ and $J/\psi$ produced in Ru+Ru and Zr+Zr collisions at
$\sqrt{s_{\textrm{NN}}}=200~{}\textrm{GeV}$. (Lower panel) The ratio of
$v_{2}$ in Ru+Ru and Zr+Zr collisions for $J/\psi$ and $\pi^{\pm}$.

The nucleon distribution in the isobar nuclei influences the anisotropy of the
medium created in heavy-ion collisions. The latter can be measured by the
momentum anisotropy of final state hadrons, which is defined as the Fourier
coefficient of their azimuthal distribution and referred to as $v_{n}$
Ollitrault (1992). The second flow coefficient, called elliptic flow $v_{2}$,
is mostly a collective flow response to the ellipsoidal shape of the overlap
region in non-central collisions. The difference in the light flavor $v_{2}$
has been found to be sensitive to the initial anisotropy contributed by the
deformation Zhang and Jia (2022); Jia (2022), and it would be interesting to
measure $v_{2}^{J/\psi}$ which probes the medium anisotropy in a different
way. Results are shown in Fig. 4. The $v_{2}$ of $J/\psi$ is influenced by two
origins. One is regeneration in which the anisotropy is similar to that of
$\pi$ in both the centrality trend and order of magnitude. The other is the
initial production which is isotropic initially and pick up some small amount
of anisotropy via path-dependent dissociation in the anisotropic medium
background. In central collisions, the anisotropy of the background is small,
whereas in peripheral collisions the temperature is small and the dissociation
effect is weak. Both these factors make $v_{2}$ of initially produced $J/\psi$
flat and small for all centrality ranges. As is shown in Fig. 2, going from
the most central to peripheral collisions, the regeneration fraction decreases
and henceforth the overall $v_{2}$ of $J/\psi$ decreases, which is opposite to
the trend of light hadrons. A small tilde (enhancement) in $v_{2}$ for the
most central collisions is also observed, because the existence of $T>T_{d}$
region induces extra anisotropy for initially produced $J/\psi$. With the Ru-
to-Zr ratio shown in the lower panel, we expect $\sim 5-10\%$ difference in
the elliptic flow of $J/\psi$. It shall be worth noting that such a ratio is
greater than that of $v_{2,\pi}$. With details shown in the Appendix, the
difference of $v_{2,J/\psi}$ in the isobar system is contributed dominantly by
the fact that there is a greater portion of $J/\psi$ that is created via
regeneration production in Ru+Ru collisions than in Zr+Zr. Thus, a precise
measurement of $v_{2,J/\psi}$ difference serves as an accurate probe of the
regeneration fraction.

_Summary._ — Via a relativistic transport approach, we investigate the yield
and elliptic flow of $J/\psi$ in isobar collisions. The evolution of the hot
medium produced in Ru+Ru and Zr+Zr collisions are described by relativistic
viscous hydrodynamics with event averaged initial conditions. Aiming to detect
the difference in nucleon distributions between Ru and Zr, we compare the
observables in the isobar system according to the same centrality bin and the
same multiplicity bin. Due to the charm produced via initial binary
collisions, the yield ratio can clearly show the difference between Ru and Zr.
The elliptic flow $v_{2}$ of $J/\psi$ in isobar collisions is qualitatively
different from the light hadrons, and the ratio between Ru+Ru and Zr+Zr
collisions is around $5-10\%$ in non-central collisions. These heavy flavor
particles serve as an independent probe to characterize the number of binary
collisions in the initial state and therefore the nucleon distribution in the
isobar nuclei.

The regeneration production of charmonium is proportional to the square of the
production cross-section of charm quark, with the latter being proportional to
the number of binary collision and also increasing with collision energy. At
LHC energy charmonium states are produced dominantly via regeneration and we
expect the difference in nucleon distribution to be better reflected there.

In addition, the deformation of the nucleus is better reflected in ultra-
central collisions, in which one may select events with preference on the so
called body-body collisions. In such events, the deformation axes of the
projectile and target are parallel to each other, and both of them are
perpendicular to the beam direction. Therefore, the deformation of the nucleon
distribution is translated into the eccentricity of the initial condition, and
henceforth the final state anisotropy. Comparison of both light and heavy
flavor observables will be reported in our future publication.

Acknowledgement: We thank Haojie Xu and Pengfei Zhuang for helpful discussion.
J.Z. is supported by the NSFC Grant Nos. 12175165 and 12047535, and the
European Union’s Horizon 2020 research and innovation program under grant
agreement No 824093 (STRONG-2020). S.S. acknowledges support by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, Grants
Nos. DE-FG88ER41450 and DE-SC0012704.

##  Appendix A Source of difference in elliptic flow

Figure 5: (Left) Sources of $v_{2,J/\psi}$ difference. (Right) Fraction of
regeneration production in the final state $J/\psi$ in isobar collisions.

In this appendix, we discuss the source of difference in $v_{2,J/\psi}$
between Ru+Ru and Zr+Zr collisions. We separately compare the $v_{2}$ of
initial production and regeneration productions. Denoting $f_{\mathrm{reg}}$
as the fraction of regeneration production in all the final state $J/\psi$
particles, one can perform the separation as
$v_{2,\mathrm{tot}}=(1-f_{\mathrm{reg}})v_{2,\mathrm{ini}}+f_{\mathrm{reg}}v_{2,\mathrm{reg}}$.
Then, one can show that

$\displaystyle\begin{split}&\Delta v_{2}^{\mathrm{tot}}\\\
=\;&(1-f_{\mathrm{reg}}^{\mathrm{Ru}})v_{2,\mathrm{ini}}^{\mathrm{Ru}}+f_{\mathrm{reg}}^{\mathrm{Ru}}v_{2,\mathrm{reg}}^{\mathrm{Ru}}-(1-f_{\mathrm{reg}}^{\mathrm{Zr}})v_{2,\mathrm{ini}}^{\mathrm{Zr}}-f_{\mathrm{reg}}^{\mathrm{Zr}}v_{2,\mathrm{reg}}^{\mathrm{Zr}}\\\
=\;&\Big{(}1-\frac{f_{\mathrm{reg}}^{\mathrm{Ru}}+f_{\mathrm{reg}}^{\mathrm{Zr}}}{2}\Big{)}(v_{2,\mathrm{ini}}^{\mathrm{Ru}}-v_{2,\mathrm{ini}}^{\mathrm{Zr}})-(f_{\mathrm{reg}}^{\mathrm{Ru}}-f_{\mathrm{reg}}^{\mathrm{Zr}})\frac{v_{2,\mathrm{ini}}^{\mathrm{Ru}}+v_{2,\mathrm{ini}}^{\mathrm{Zr}}}{2}+\frac{f_{\mathrm{reg}}^{\mathrm{Ru}}+f_{\mathrm{reg}}^{\mathrm{Zr}}}{2}(v_{2,\mathrm{reg}}^{\mathrm{Ru}}-v_{2,\mathrm{reg}}^{\mathrm{Zr}})+(f_{\mathrm{reg}}^{\mathrm{Ru}}-f_{\mathrm{reg}}^{\mathrm{Zr}})\frac{v_{2,\mathrm{reg}}^{\mathrm{Ru}}+v_{2,\mathrm{reg}}^{\mathrm{Zr}}}{2}\\\
=\;&\Big{(}1-\frac{f_{\mathrm{reg}}^{\mathrm{Ru}}+f_{\mathrm{reg}}^{\mathrm{Zr}}}{2}\Big{)}(v_{2,\mathrm{ini}}^{\mathrm{Ru}}-v_{2,\mathrm{ini}}^{\mathrm{Zr}})+\frac{f_{\mathrm{reg}}^{\mathrm{Ru}}+f_{\mathrm{reg}}^{\mathrm{Zr}}}{2}(v_{2,\mathrm{reg}}^{\mathrm{Ru}}-v_{2,\mathrm{reg}}^{\mathrm{Zr}})+(f_{\mathrm{reg}}^{\mathrm{Ru}}-f_{\mathrm{reg}}^{\mathrm{Zr}})\Big{(}\frac{v_{2,\mathrm{reg}}^{\mathrm{Ru}}+v_{2,\mathrm{reg}}^{\mathrm{Zr}}}{2}-\frac{v_{2,\mathrm{ini}}^{\mathrm{Ru}}+v_{2,\mathrm{ini}}^{\mathrm{Zr}}}{2}\Big{)}\\\
\equiv\;&(1-\bar{f}_{\mathrm{reg}})\Delta
v_{2,\mathrm{ini}}+\bar{f}_{\mathrm{reg}}\Delta
v_{2,\mathrm{reg}}+(\bar{v}_{2,\mathrm{reg}}-\bar{v}_{2,\mathrm{ini}})\Delta
f_{\mathrm{reg}}\,.\end{split}$ (10)

In the left panel of Fig. 5, we plot these three sources separately, together
with the full value of $\Delta v_{2}^{\mathrm{tot}}$. It turns out that the
last term, which is caused by the fact that there is a greater portion of
$J/\psi$ that is created via regeneration production in Ru+Ru collisions than
in Zr+Zr. As noted in the main text that ${v}_{2,\mathrm{reg}}$ is much
greater than ${v}_{2,\mathrm{ini}}$, the difference in $f_{\mathrm{reg}}$
makes the greatest contribution to the difference in elliptic flow.

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