# Characterizing finite groups whose enhanced power graphs have universal
vertices

David G. Costanzo  School of Mathematical and Statistical Sciences, O-110
Martin Hall, Box 340975, Clemson University, Clemson, SC 29634, USA
<EMAIL_ADDRESS>,  Mark L. Lewis  Department of Mathematical
Sciences, 1300 Lefton Esplanade, Kent State University, Kent, OH 44242, USA
<EMAIL_ADDRESS>,  Stefano Schmidt  ,  Eyob Tsegaye  Department of
Mathematics, Princeton University, Princeton, NJ 08540<EMAIL_ADDRESS>and
Gabe Udell  Department of Mathematics, Malott Hall, 212 Garden Ave, Cornell
University, Ithaca, NY 14853<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking
$G\setminus\\{1\\}$ as the vertex set of $\Delta(G)$ and by drawing an edge
between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$
be the set consisting of the universal vertices of $\Delta(G)$ along the
identity element. For a solvable group $G$, we present a necessary and
sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection
between $\Delta(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes
and the diameter of $\Delta(G)$ is $2$.

###### Key words and phrases:

enhanced power graph, universal vertices, diameter

###### 2010 Mathematics Subject Classification:

Primary 20D25; Secondary 05C25

##  1\. Introduction

In this paper, all groups are finite. Let $G$ be a finite group. The enhanced
power graph of $G$ is the graph whose vertex set is $G\setminus\\{1\\}$ and
there is an edge between $x,y\in G\setminus\\{1\\}$ if $\langle x,y\rangle$ is
cyclic. The enhanced power graph of $G$ will be denoted by $\Delta(G)$. This
graph was first studied in 2008 under the name “cyclic graph” in [9]. That
paper was motivated by the paper [2] where they essentially considered the
complement graph: in particular, they considered the graph, which they called
the noncyclic graph, where the vertices of this graph is the set $G$ and there
is an edge between $x$ and $y$ whenever $\langle x,y\rangle$ is noncyclic.

It appears that the name enhanced power graph was introduced in the paper [1].
Apparently, the authors of that paper were unaware that the graph had been
previously studied under the name cyclic graph in [9] and the followup paper
[10]. We recommend the reader consult the excellent expository paper [7] for
an account of graphs and groups. Since that paper has been published, there
has been an explosion in the number of papers published on the enhanced power
graph, and we are not going to try to reference all of those papers. Also,
there has been an expository paper that focuses just on enhanced power graphs
of groups and it is worth consulting this paper [12].

In graph theory, a universal vertex is a vertex that is adjacent to all the
other vertices in the graph. Said differently, if a graph has $n$ vertices,
then a universal vertex is a vertex of degree $n-1$. We note that we have seen
a number of different names used for such vertices in the literature. In
particular, we have seen these vertices called complete vertices, cone
vertices, and dominating vertices. We note that dominating vertex seems to be
favored among the literature for enhanced power graph papers and in fact was
the term we use in [8]. However, universal vertex seems to be the term favored
in graph theory, and so, we have decided to use that term here.

We define $K(G)$ to be the set of all universal vertices in the enhanced power
graph of $G$, along with the identity of $G$. We discuss the background for
$K(G)$ in Section 2. In this paper, we will characterize the groups where
$K(G)>1$ and then we will see that when $G$ is a $\\{p,q\\}$-group, we will be
able to determine when $K(G)>1$ by looking at $\Delta(G)$. We note that we are
not able to say much when $p$ is odd or when $p=2$ and a Sylow $2$-subgroup is
cyclic. We see that when $p=2$ and a Sylow $2$-subgroup is not cyclic, the
structure of $G$ is rather limited. Our first main theorem characterizes the
situation where $K(G)>1$. We will define the notation used in this theorem in
Section 2.

######  Theorem A.

Let $G$ be a solvable group, let $p$ be a prime, and let $P$ be a Sylow
$p$-subgroup of $G$. Then $p$ is a prime divisor of $|K(G)|$ if and only if
one of following situations occurs:

  1. (1)

$P$ is cyclic and $(Z(G)\cap P)\neq 1$.

  2. (2)

The prime $p=2$ and one of the following situations occurs.

    1. (a)

$P$ is generalized quaternion, $G$ has a normal $2$-complement $Q$, and
$Z(P)\leq C_{P}(Q)$.

    2. (b)

$Z(P)\leq C_{P}(O_{2^{\prime}}(G))$ and $G/O_{2^{\prime}}(G)$ is isomorphic to
either $\textnormal{SL}_{2}(3)$ or $\widetilde{\textnormal{GL}}_{2}(3)$. If in
addition $3$ divides $|K(G)|$, then
$G/O_{2^{\prime}}(G)\cong\textnormal{SL}_{2}(3)$.

If conclusion (2) occurs, then $|K(G)|_{2}=2$.

We now turn to the case where $G$ is a $\\{p,q\\}$-group for distinct primes
$p$ and $q$. We prove the following theorem:

######  Theorem B.

Let $p$ and $q$ be distinct primes, and let $G$ be a $\\{p,q\\}$-group. Then
$\textnormal{diam}(\Delta(G))=2$ if and only if $|K(G)|>1$.

We would like to conclude this introduction with an open question. In
particular, we wonder how much Theorem B can be extended beyond groups that
are divisible by two primes. While it seems too much to hope that for a
general group $G$ or even a general solvable group $G$ that the enhanced power
graph of $G$ has diameter $2$ if and only if $K(G)>1$. Recall that $K(G)\leq
Z(G)$. We wonder if it might be true that $\textnormal{diam}(\Delta(G))=2$ if
and only if $|Z(G)|>1$. The meager evidence that we have been able to gather
suggests that this should be true at least when $|G|$ is divisible by three
primes.

This research was conducted during a summer REU in 2020 at Kent State
University with the funding of NSF Grant DMS-1653002. We thank the NSF and
Professor Soprunova for their support.

##  2\. Solvable groups with $K(G)$ nontrivial

Let $G$ be a group, and define

$K(G)=\\{x\in G\mid\langle x,g\rangle\>\textnormal{is cyclic for all}\>g\in
G\\}.$

Hence, $K(G)$ is the set of universal vertices for the enhanced power graph of
$G$ along with the identity element of $G$. This set seems to have been
originally studied independent of the cyclic graph in [15]. In Theorem 1 of
[15], the authors prove that $K(G)$ is a subgroup of $G$, and they note that
it is contained in the center of $G$. In Theorem 2 of that paper, they prove
that $K(G)$ is the intersection of the maximal cyclic subgroups of $G$, and in
Theorem 4 of that paper, they prove that $K(G/K(G))=1$. The intersection of
the maximal cyclic subgroups of $G$ is also considered in the dissertation
[17], and he also proves that $K(G)\leq Z(G)$ and $K(G/K(G))=1$ (see
Definition 5.1.7 and Lemma 5.1.8 in that dissertation).

The set $K(G)$ was considered for several special cases in [3]. They consider
the cases of a group $G\times Z_{n}$ where ${\rm gcd}(|G|,n)=1$ (Theorem 3.1
of [3]), when $G$ is abelian (Theorem 3.2 of [3]), and when $G$ is a non-
abelian $p$-group for some prime $p$ (Theorem 3.3 of [3]). In addition, they
showed that if $G$ is a nonabelian simple group, then $K(G)=1$ (see Theorem
3.4 of [3]). The set $K(G)$ is considered for nilpotent groups in Proposition
4.5 of [13].

We need a reformulation of a result in our previous paper (Theorem 1.4 of
[8]). We note that a result similar to Theorem 1.4 of [8] has been proved by
Cameron in Theorem 9.1 (b) of [7] and by Mahmoudifar and Babai in the Main
Theorem of [14]. We note that [14] was submitted in 2018, two years before [8]
was submitted, even though it appeared after it. Also, Cameron’s paper was
submitted in April 2021 which was before [8] appeared in print (although [8]
did appear electronically in December 2020.) In addition, we remark that Lemma
2.7 of [4] proves a version of this next lemma. For completeness and to keep
this paper somewhat self contained, we provide a proof of the reformulation
for the reader’s convenience.

######  Lemma 2.1.

Let $G$ be a group and let $p$ be a prime. Then $G$ has unique subgroup of
order $p$ and that subgroup is central in $G$ if and only if $p$ divides
$|K(G)|$.

###### Proof.

Assume that $G$ has a unique subgroup of order $p$, say $Z$, and $Z\leq Z(G)$.
Of course, $Z$ is cyclic, and so we can write $Z=\langle z\rangle$ for some
$z\in G$. If $g\in G$ is a $p^{\prime}$-element, then $\langle z,g\rangle$ is
cyclic since $g$ and $z$ commute and have coprime orders. If $g\in G$ has
order divisible by $p$, then $\langle g\rangle$ has a subgroup of order $p$,
and so $Z\leq\langle g\rangle$ by the uniqueness of $Z$. Hence, $\langle
g,z\rangle=\langle g\rangle$ is cyclic. We conclude that $z\in K(G)$.

Conversely, assume that $p$ divides $|K(G)|$. As we mentioned previously, the
set $K(G)$ is actually a subgroup of $G$, and so, under the hypothesis that
$p$ divides $|K(G)|$, we know that $K(G)$ has a subgroup $Z$ of order $p$. As
$K(G)\leq Z(G)$, the subgroup $Z$ is central. Let $X$ be a subgroup of $G$
such that $|X|=p$. Write $X=\langle x\rangle$ and $Z=\langle z\rangle$. As
$z\in K(G)$, we know that $\langle x,z\rangle$ is cyclic. Hence, $\langle
x,z\rangle$ has exactly one subgroup of order $p$. Since $X$ and $Z$ are
subgroups of $\langle x,z\rangle$ such that $|X|=|Z|=p$, we deduce that $X=Z$.
Since $X$ was an arbitrary subgroup of $G$ of order $p$, we conclude that $Z$
is the only subgroup of $G$ with order $p$. ∎

We close the discussion on the background of universal vertices by mentioning
that [5] used Theorem 1.8 of [8] in Corollary 4.2 of [5] to explicitly
determine $K(G)$ when $G$ is a nilpotent group.

We now work to characterize solvable groups $G$ with $K(G)>1$. In the
following lemma, we shall write $\widetilde{\textnormal{GL}}_{2}(3)$ for the
group that is isoclinic to $\textnormal{GL}_{2}(3)$ of order $48$ that has a
generalized quaternion group of order $16$ as a Sylow $2$-subgroup. We note
that the solvability hypotheses on this lemma is necessary since, for example,
$\textnormal{SL}_{2}(5)$ would be a nonsolvable counterexample.

In Lemma 2.3, we will use some structural properties of the automorphism group
of a generalized quaternion group of order at least 16. More specifically, we
need the fact that if $Q$ is a generalized quaternion group of order at least
16, then $\textnormal{Aut}(Q)$ is a $2$-group. This fact is mentioned briefly
on page 321 in [6], but we include the elementary proof here for completeness.
Of course, we shall use a few basic properties of a generalized quaternion
group; these properties can be found in Theorem 4.1 and Theorem 4.2 in [16].

######  Lemma 2.2.

If $Q$ is a generalized quaternion group of order $2^{n}$, where $2^{n}\geq
16$, then $A=\textnormal{Aut}(Q)$ is a $2$-group.

###### Proof.

We know that $Q$ has a cyclic maximal subgroup $M$ such that $|Q:M|=2$.
Clearly $Q^{\prime}\leq M$, and so $M\leq C_{Q}(Q^{\prime})$. As $Q$ has
nilpotence class $n-1\geq 3$, we know that $Q^{\prime}$ is noncentral, and so
$C_{Q}(Q^{\prime})=M$ by the maximality of $M$. The centralizer of a
characteristic subgroup is characteristic, and so $M$ is a characteristic
subgroup of $Q$.

For each $a\in A$, let $\hat{a}$ be the automorphism of $M$ obtained by
restricting $a$ to $M$. (This restriction makes sense as $M$ is characteristic
in $Q$.) The map $a\mapsto\hat{a}$ is homomorphism from $A$ into
$\textnormal{Aut}(M)$ with kernel $C_{A}(M)$. Hence, $A/C_{A}(M)$ embeds into
$\textnormal{Aut}(M)$ by the $N/C$-Theorem. The automorphism group of a cyclic
$2$-group is a $2$-group, and so $A/C_{A}(M)$ is a $2$-group.

Let $p$ be a prime divisor of $|C_{A}(M)|$, and let $a\in C_{A}(M)$ with
$o(a)=p$. As $a\in C_{A}(M)$, we know that $M\leq C_{Q}(a)$. If $C_{Q}(a)=Q$,
then $a=1$, contrary to our choice of $a$. Hence, $M=C_{Q}(a)$. If
$\varOmega=Q\setminus M$, then $a$ permutes the elements of $\varOmega$. In
fact, $a$ has no fixed points on $\varOmega$, as $M=C_{Q}(a)$. So, $a$
partitions $\varOmega$ into orbits of length $p$. As
$|\varOmega|=2^{n}-2^{n-1}=2^{n-1}$, the prime $p$ must divide $2^{n-1}$,
which forces $p=2$. We deduce that $C_{A}(M)$ is also a $2$-group. Indeed, $A$
is a $2$-group. ∎

We now prove a critical lemma concerning solvable groups that have a
generalized quaternion Sylow $2$-subgroup.

######  Lemma 2.3.

If $G$ is a solvable group and a Sylow $2$-subgroup of $G$ is generalized
quaternion, then one of the following situations occurs:

  1. (1)

$G$ has a normal $2$-complement

  2. (2)

$G/O_{2^{\prime}}(G)\cong\textnormal{SL}_{2}(3)$

  3. (3)

$G/O_{2^{\prime}}(G)\cong\widetilde{\textnormal{GL}}_{2}(3)$.

###### Proof.

We start by making a reduction. Let $U=O_{2^{\prime}}(G)$. Since $G$ is
solvable, the factor group $G/U$ is solvable. If $T$ is a Sylow $2$-subgroup
of $G$, then $TU/U\cong T$ is a Sylow $2$-subgroup of $G/U$; thus, a Sylow
$2$-subgroup of $G/U$ is generalized quaternion. In particular, $G/U$
satisfies the same hypotheses as $G$. As $O_{2^{\prime}}(G/U)$ is trivial, we
may assume without loss of generality that $U=1$. Note that under the
assumption that $U$ is trivial, (1) occurs if and only if $G$ is a $2$-group.

Set $Q=O_{2}(G)$. Since $Q$ is a subgroup of a generalized quaternion group,
$Q$ has a unique subgroup of order $2$ and is therefore either cyclic or
generalized quaternion. If $Q$ is cyclic, then $\textnormal{Aut}(Q)$ is a
$2$-group. If $Q$ is generalized quaternion of order at least $16$, then
$\textnormal{Aut}(Q)$ is a $2$-group. By the $N/C$-Theorem, $G/C_{G}(Q)$
embeds in $\textnormal{Aut}(Q)$. So, if $Q$ is either cyclic or generalized
quaternion of order at least $16$, then $G/C_{G}(Q)$ is a $2$-group. The group
$G$ is solvable and $U=1$; thus, $C_{G}(Q)\leq Q$ by the Hall-Higman Lemma
1.2.3 (see [11], Theorem 3.21). Hence, $G$ is a $2$-group and we have
conclusion (1).

Assume now that $Q$ is quaternion of order $8$. As mentioned in the previous
paragraph, the Hall-Higman Lemma 1.2.3 applies and tells us that $C_{G}(Q)\leq
Q$. In particular, $C_{G}(Q)=Z(Q)$ and so $|C_{G}(Q)|=2$. We note that
$Q/C_{G}(Q)$ is isomorphic to the Klein $4$-group. If $G=Q$, then we are done.
Assume that $Q<G$. The section $G/C_{G}(Q)$ embeds in $\textnormal{Aut}(Q)$ by
the $N/C$-Theorem. As $Q$ is quaternion of order $8$,
$\textnormal{Aut}(Q)\cong S_{4}$. The only subgroups of $S_{4}$ that properly
contain a subgroup isomorphic to the Klein $4$-group are $A_{4}$ and $S_{4}$.
Thus, $G/C_{G}(Q)\cong A_{4}$ or $G/C_{G}(Q)\cong S_{4}$. If $G/C_{G}(Q)\cong
A_{4}$, then $G\cong\textnormal{SL}_{2}(3)$. If $G/C_{G}(Q)\cong S_{4}$, then
since the only central extension of $S_{4}$ that has a generalized quaternion
Sylow $2$-subgroup is $\widetilde{\textnormal{GL}}_{2}(3)$, we conclude that
$G\cong\widetilde{\textnormal{GL}}_{2}(3)$. ∎

We now are ready to prove Theorem A.

###### Proof of Theorem A.

Suppose that $p$ divides $|K(G)|$. If $p$ is odd, then, using Lemma 2.1, we
immediately deduce that we have conclusion (1). So assume that $p=2$. A Sylow
$2$-subgroup of $G$ is either cyclic or generalized quaternion. If a Sylow
$2$-subgroup of $G$ is cyclic, then we again obtain conclusion (1). Assume
that a Sylow $2$-subgroup of $G$ is generalized quaternion. The subgroup
$K(G)$ is cyclic, and so if $|K(G)|_{2}>2$, then $K(G)$ has an element of
order $4$, say $u$. Now $U=\langle u\rangle$ is a normal $2$-subgroup of $G$
and so $U\leq P$. Since $U\leq K(G)\leq Z(G)$, we have that $U\leq Z(P)$. But
$|Z(P)|=2$, and so we have a contradiction. We conclude that $|K(G)|_{2}=2$.
In particular, $K(G)\cap P=Z(P)$. If $G$ has a normal $2$-complement $Q$, then
we have (2a) as $Z(P)\leq C_{P}(Q)$. Assume that $G$ does not have a normal
$2$-complement. Using Lemma 2.3, we have that $G/O_{2^{\prime}}(G)$ is
isomorphic to either $\textnormal{SL}_{2}(3)$ or
$\widetilde{\textnormal{GL}}_{2}(3)$. If $3$ divides $|K(G)|$, then $G$ has a
cyclic Sylow $3$-subgroup and $G$ is $3$-nilpotent by Lemma 2.1 and Corollary
5.30 in [11]. Because $\widetilde{\textnormal{GL}}_{2}(3)$ does not have a
normal Sylow $2$-subgroup, $G/O_{3}(G)\cong\textnormal{SL}_{2}(3)$.

If (1) occurs, then $p$ divides $|K(G)|$ by Lemma 2.1. Assume (2). Let
$U=O_{2^{\prime}}(G)$ and let $Z=Z(P)$. Because $G/U\cong
X\in\\{\textnormal{SL}_{2}(3),\widetilde{\textnormal{GL}}_{2}(3)\\}$, we
deduce that $ZU/U=Z(G/U)$, and so $ZU/U$ is normal in $G/U$. By the
Correspondence Theorem ([11], X.21), $ZU$ is normal in $G$. As $Z$ is a Sylow
$2$-subgroup of $ZU$, we have that $G=ZUN_{G}(Z)=UN_{G}(Z)$ by the Frattini
Argument (see Lemma 1.13 in [11]). By our hypothesis, $Z\leq C_{P}(U)$, and so
$U\leq C_{G}(Z)$. Hence, $G=N_{G}(Z)$ and $Z$ is normal in $G$. Since $Z$ is a
normal subgroup of $G$ such that $|Z|=2$, we have that $Z\leq Z(G)$. If $T$ is
a subgroup of $G$ such that $|T|=2$, then $\langle Z,T\rangle=ZT$ is a
$2$-subgroup of $G$. Because the Sylow 2-subgroups of $G$ are generalized
quaternion, we deduce that $T=Z$. We therefore appeal to Lemma 2.1 and
conclude that $2$ divides $|K(G)|$. ∎

##  3\. $\\{p,q\\}$-groups

Let $p$ and $q$ be distinct prime numbers, and let $G$ be a $\\{p,q\\}$-group.
In this section, we present a necessary and sufficient condition for
$\textnormal{diam}(\Delta(G))=2$. It turns out that
$\textnormal{diam}(\Delta(G))=2$ precisely when a universal vertex exists. We
use the notation $\sim$ to denote an edge between $x$ and $y$ in $\Delta(G)$,
and $G^{\\#}=G\setminus\\{1\\}$.

We start with an easy lemma concerning elements of prime order in a
$\\{p,q\\}$-group $G$ with $\textnormal{diam}(\Delta(G))=2$.

######  Lemma 3.1.

Let $p$ and $q$ be distinct primes. If $G$ is a $\\{p,q\\}$-group such that
$\textnormal{diam}(\Delta(G))=2$, then $x\sim y$ for every element $x$ of
order $p$ and for every element $y$ of order $q$.

###### Proof.

Let $x\in G^{\\#}$ be an arbitrary element of order $p$ and let $y\in G^{\\#}$
be an arbitrary element of order $q$. If $x\sim y$, then the result holds.
Since $\textnormal{diam}(\Delta(G))=2$, there exists some $z\in G^{\\#}$ such
that $x\sim z\sim y$. Of course, at least one of primes $p,q$ divides $o(z)$.
Without loss of generality, we may assume that $p$ divides $o(z)$. Since
$\langle x,z\rangle$ is cyclic, $\langle x\rangle=\langle z^{n}\rangle$ for
some integer $n$. Now, $x,y\in\langle z,y\rangle$. As $\langle z,y\rangle$ is
cyclic, we conclude that $x\sim y$. ∎

We now prove our main result. Notice that sufficiency is essentially a group-
theoretic formulation of the existence of a universal vertex in the
corresponding enhanced power graph. Hence, this result yields Theorem B.

######  Theorem 3.2.

Let $p$ and $q$ be distinct primes, and let $G$ be a $\\{p,q\\}$-group. Then,
$\textnormal{diam}(\Delta(G))=2$ if and only if $G$ has a unique subgroup of
order $p$ or a unique subgroup of order $q$ and that subgroup is central in
$G$.

###### Proof.

Assume that the group $G$ has a unique subgroup $Z$ of prime order and that
$Z\leq Z(G)$. Write $Z=\langle z\rangle$ and, without loss of generality, we
may assume that $|Z|=p$. Now, let $g\in G^{\\#}$. If $p$ does not divide
$o(g)$, then $g\sim z$ since commuting elements with coprime orders generate a
cyclic subgroup. If $p$ does divides $o(g)$, then, since $Z$ is the unique
subgroup of order $p$, we have that $\langle g^{n}\rangle=Z$ for some integer
$n$. So, again, $g\sim z$. Hence, $z$ is a universal vertex in $\Delta(G)$,
which yields the bound $\textnormal{diam}(\Delta(G))\leq 2$. As $G$ is non-
cyclic, we conclude that $\textnormal{diam}(\Delta(G))=2$.

Next, assume that $\textnormal{diam}(\Delta(G))=2$. We start by establishing
the existence of a unique subgroup of prime order. If $G$ has a unique
subgroup of order $p$, then we are done. So, suppose that $P_{1}=\langle
x_{1}\rangle$ and $P_{2}=\langle x_{2}\rangle$ are distinct subgroups of order
$p$. Note that $d(x_{1},x_{2})=2$. Write $x_{1}\sim y\sim x_{2}$ for $y\in
G^{\\#}$. Replacing $y$ if necessary, we may assume that $o(y)=q$. Let $w$ be
an element of order $q$. Consider the products $x_{1}y$ and $x_{2}w$. The
elements $x_{1}$ and $y$ have coprime orders. The elements $x_{2}$ and $w$
have coprime orders. These elements are commuting pairs of elements by Lemma
3.1. So, $\langle x_{1},y\rangle=\langle x_{1}y\rangle$ and $\langle
x_{2},w\rangle=\langle x_{2}w\rangle$. Now, if $x_{1}y\sim x_{2}w$, then
$\langle x_{1}\rangle=\langle x_{2}\rangle$, a contradiction. Hence,
$d(x_{1}y,x_{2}w)=2$. Write $x_{1}y\sim z\sim x_{2}w$ for $z\in G^{\\#}$.
Observe that this situation forces $z$ to be a $q$-element, and so we may
assume that $o(z)=q$. Now, $\langle y\rangle=\langle z\rangle=\langle
w\rangle$. In particular, $\langle y\rangle$ is the unique subgroup of $G$ of
order $q$.

In the previous paragraph, we established that $G$ has a unique subgroup of
prime order. Without loss of generality, we proceed by assuming that $G$ has a
unique subgroup of order $p$, say $P=\langle x\rangle$. If $P\leq Z(G)$, then
we are done. Assume that $P\not\leq Z(G)$, and so there exists some $g\in G$
such that $P\not\leq C_{G}(g)$. If $p$ divides $o(g)$, then $P=\langle
x\rangle\leq\langle g\rangle$ since $P$ is the unique subgroup of order $p$.
But then, $P\leq C_{G}(g)$, a contradiction. Hence, $g$ must be a $q$-element.
As we proved in Lemma 3.1, an element of order $p$ is adjacent to an element
of order $q$ in $\Delta(G)$. Hence, $o(g)=q^{l}$, where $l\geq 2$. For some
positive integer $n$, the subgroup $Q=\langle g^{n}\rangle$ has order $q$. We
claim that $Q$ is the unique subgroup of $G$ with order $q$ and that $Q\leq
Z(G)$. Let $Q_{0}=\langle y\rangle$ be a subgroup of order $q$. Note that
$\langle x,y\rangle=\langle xy\rangle$ and, by our hypothesis, $d(xy,g)\leq
2$. If $xy\sim g$, then $P=\langle x\rangle\leq C_{G}(g)$, a contradiction.
Hence, $d(xy,g)=2$. Write $xy\sim h\sim g$ for $h\in G^{\\#}$. If $p$ divides
$o(h)$, then $\langle x\rangle=\langle h^{t}\rangle\leq C_{G}(g)$ for some
integer $t$. But then we have arrived at another contradiction. The element
$h$ is therefore a $q$-element, and so $Q_{0}=\langle y\rangle=\langle
h^{s}\rangle=\langle g^{n}\rangle=Q$ for some positive integer $s$. Since
$Q_{0}$ was an arbitrary subgroup of $G$ of order $q$, we conclude that $Q$ is
the unique subgroup of $G$ with order $q$, as wanted. Finally, observe that
the section $G/C_{G}(P)$ embeds in $\textnormal{Aut}(P)$ and the section
$G/C_{G}(Q)$ embeds in $\textnormal{Aut}(Q)$. By arithmetical considerations,
one of these sections is trivial. Since $P$ is non-central, we must have that
$G=C_{G}(Q)$, and so $Q\leq Z(G)$. ∎

We now prove Theorem B.

###### Proof of Theorem B.

Assume that $\textnormal{diam}(\Delta(G))=2$. By Theorem 3.2, we know that $G$
either has a unique subgroup of order $p$ or a unique subgroup of order $q$
and that subgroup is central. By Lemma 2.1, we know that either $p$ or $q$
divides $|K(G)|$, and so $K(G)>1$.

Conversely, suppose $K(G)>1$. Then either $p$ or $q$ divides $|K(G)|$. In
light of Lemma 2.1, we see that $G$ has either a unique subgroup of order $p$
or and a unique subgroup of order $q$ that is central. Applying Theorem 3.2,
we see that $\Delta(G)$ has diameter $2$, as desired. ∎

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# Hadronization effects on transverse momentum dependent jet fragmentation
function in small systems

Xiang-Pan Duan(段香盼)  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
$200433$, China  Shanghai Research Center for Theoretical Nuclear Physics,
NSFC and Fudan University, Shanghai $200438$, China  Wenbin Zhao(赵文彬)
<EMAIL_ADDRESS>Department of Physics and Astronomy, Wayne State
University, Detroit, Michigan 48201, USA  Guo-Liang Ma(马国亮<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai $200433$, China  Shanghai Research
Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai
$200438$, China

###### Abstract

The transverse momentum $j_{\rm T}$-dependent jet fragmentation functions have
been investigated in proton+proton (p $+$ p) and proton+lead (p $+$ Pb)
collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ with a multiphase transport
model containing both the simple quark coalescence mechanism and a new hybrid
hadronization mechanism with coalescence and fragmentation processes.
Hadronized by the new hadronization mechanism, the AMPT model achieves a
quantitative description of the $j_{\rm T}$-dependent jet fragmentation
functions measured by ALICE. Besides, no obvious jet-medium interaction and
cold nuclear matter effects on the $j_{\rm T}$-dependent jet fragmentation
functions in p $+$ Pb collisions were observed. We found the $j_{\rm
T}$-dependent jet fragmentation functions are dominated by the quark
coalescence contribution for the new hadronization mechanism, which can be
decomposed into narrow and wide parts. The root mean square value of the wide
part depends on the jet radius $R$ and jet transverse momentum
$p_{\rm{T,jet}}$, which is sensitive to different hadronization mechanisms and
their components. Therefore, the $j_{\rm T}$-dependent jet fragmentation
functions are proposed as a sensitive probe to study the non-perturbative
hadronization effect of jets in small colliding systems.

##  I Introduction

High-energy heavy-ion collisions produce a hot and dense quantum
chromodynamics (QCD) matter, called the quark-gluon plasma (QGP) Gyulassy and
McLerran (2005), which has been observed at the Relativistic Heavy Ion
Collider (RHIC) Arsene et al. (2005); Back et al. (2005); Adams et al. (2005);
Adcox et al. (2005); Shen and Yan (2020) and the Large Hadron Collider (LHC)
Aad et al. (2010); Aamodt et al. (2011); Chatrchyan et al. (2012a); Song et
al. (2017). Jets, collimated sprays of particles produced in early-stage hard
QCD scattering processes of heavy-ion collisions Sterman and Weinberg (1977);
Feynman et al. (1978); Wang (1997), loss energy when traversing the hot and
dense QGP matter Bjorken (1982); Baier et al. (2000); Qin et al. (2009). This
phenomenon is referred to as jet quenching Gyulassy and Plumer (1990); Wang
and Gyulassy (1992); Qin and Wang (2015); Cao and Wang (2021), which has been
widely used as a hard probe to detect the properties of QGP matter. Not only
is substantial evidence of jet quenching observed in experimental measurements
Adams et al. (2003); Aamodt et al. (2012); Chatrchyan et al. (2014a); Aad et
al. (2015), but also many theoretical calculations has simulated this
phenomenon in high-energy heavy-ion collisions collisions Baier et al. (1997);
Wang (2004); Ma (2013a, 2014a); Burke et al. (2014); Milhano and Zapp (2016);
Cao et al. (2018); Gao et al. (2018); Chang et al. (2020); Luo et al. (2022);
Chen et al. (2020); Wan et al. (2019); Shi et al. (2019); Zhang and Cheng
(2017); Xu et al. (2015).

As one of jet properties, the jet fragmentation function (JFF) describes the
momentum distribution of hadrons inside a jet Workman and Others (2022); Aad
et al. (2011). The JFFs can be characterized by the longitudinal momentum
fraction $z_{\rm h}$ and the transverse momentum $j_{\rm T}$ carried by the
hadrons inside the jet cone. Both longitudinal and transverse momentum JFFs
reflect the jet internal structure affected by both the evolution of the QCD
parton shower and its hadronization process. The JFFs are only determined
phenomenologically due to the non-perturbative nature in principle. It has
been argued that the $z_{\rm h}$-dependence is sensitive to the standard
collinear fragmentation functions, while the $j_{\rm T}$-dependence probes the
transverse momentum-dependent fragmentation functions Kang et al. (2017,
2019). For jet quenching, the experimental measurements in high-energy heavy-
ion collisions have shown that JFFs are strongly modified in the presence of
the QGP, and thus can be used as a sensitive probe for the QCD medium
Chatrchyan et al. (2012b, 2014b); Aad et al. (2014); Aaboud et al. (2019).
Since many physical effects can affect JFFs, JFFs provide an important tool to
study the rich physics associated with jet Hwa and Yang (2004); Majumder et
al. (2006); Salgado and Wiedemann (2004); Ma (2013b).

The transverse momentum of the hadron perpendicular to the jet axis, $j_{\rm
T}$, gives the transverse structure of the distribution of jet constitute
particles. The $j_{\rm T}$-dependent JFFs have been recently measured by the
two-particle correlation using the leading-track momentum vector as the axis
Angelis et al. (1980); Adler et al. (2006a, b); Acharya et al. (2019) or by
jet reconstruction method with the jet axis determined by jet finding
algorithm Aaltonen et al. (2009); Aad et al. (2011); Aaij et al. (2019);
Acharya et al. (2021). The measured $j_{\rm T}$-dependent JFFs have been
fitted with a two-component function including a Gaussian function for the
narrow part and an inverse Gamma function for the wide part Acharya et al.
(2019, 2021), which have been compared with Herwig and the PYTHIA 8 and
indicates that it is necessary to properly include the $j_{\rm T}$
distribution into the models. In addition, the $j_{\rm T}$-dependent jet
fragmentation functions in p $+$ Pb collisions have been also experimentally
compared to those in p $+$ p collisions, which shows no significant cold
nuclear matter effect in p $+$ Pb collisions. It provides an important
baseline for measuring the jet quenching effect on JFFs in relativistic heavy-
ion collisions.

The QCD hadronization process is not yet fully understood due to its non-
perturbative nature. Fragmentation and quark coalescence are two main well-
known phenomenological mechanisms for hadronization, both of which can give
good descriptions to experimental measurements of final hadrons, although they
work for different kinematics in a complementary way Greco et al. (2003);
Fries et al. (2003); Hwa and Yang (2004). There have been some efforts to
unify the two hadronization mechanisms Hwa and Yang (2004); Majumder et al.
(2006); Han et al. (2016). It has been demonstrated by a “Hydro-Coal-Frag”
model that the interplay between hydrodynamic freeze-out at low-$p_{\rm T}$,
parton coalescence at intermediate $p_{\rm T}$ and string fragmentation at
high $p_{\rm T}$ can simultaneously explain the nuclear modification $R_{\rm
AA}$, elliptic flow $v_{2}$ of charged and identified hadrons as well as their
flavor dependence in the full range of $p_{\rm T}$ for Pb $+$ Pb collisions at
the LHC Zhao et al. (2022). The same model also provided a good description of
measured $p_{\rm T}$ spectra and differential elliptic flow $v_{2}$($p_{\rm
T}$) of identified hadrons over the $p_{\rm T}$ range up to 6 GeV for p $+$ Pb
collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ Zhao et al. (2020). It
indicates that the hybrid hadronization works for both large and small
colliding systems. In Ref. Ma (2013b), we have studied the jet-medium
interaction effect on jet fragmentation function in high-energy heavy-ion
collisions based on a multiphase transport (AMPT) model, and show existence of
distinct competition between fragmentation and coalescence for jet
hadronization in different $z_{\rm h}$ ranges for different centrality bins.
In the present work, we calculate the jet transverse momentum $j_{\rm
T}$-dependent fragmentation functions using jet reconstruction method in p $+$
p and p $+$ Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ with the
AMPT model with a simple quark coalescence hadronization mechanism Lin et al.
(2005) and a new hybrid hadronization mechanism containing both dynamical
quark coalescence and string fragmentation schemes Zhao et al. (2020, 2022).

The paper is organized as follows. In Sec. II, we introduce the framework of
the AMPT model with a jet-triggering technique and two hadronization
mechanisms, and the method of jet reconstruction and jet observables. Section.
III presents the results on transverse momentum $j_{\rm T}$-dependent jet
fragmentation functions and discusses the relevant physics on jet
hadronization. A summary is given in Sec. IV.

##  II Methodology

###  II.1 The AMPT model with old and new hadronization mechanisms

In this work, we calculate transverse momentum dependent jet fragmentation
functions using a string-melting version of AMPT model with two different
hadronization mechanisms. The string melting version of the AMPT model
contains four main stages of relativistic heavy-ion collisions Lin et al.
(2005); Lin and Zheng (2021): the initial condition, parton cascade,
hadronization, and hadronic rescatterings.

(1) Initial condition. The heavy ion jet interaction generator (HIJING) model
Wang and Gyulassy (1991); Gyulassy and Wang (1994) simulates the initial
condition for p $+$ p, p $+$ A, and A $+$ A collsions. The transverse density
profile of the colliding nuclues is taken to as a Woods-Saxon distribution.
The multiple scatterings among incoming nucleons produce the spatial and
momentum distributions of minijet partons and soft excited strings. The
nuclear shadowing effect is included via an impact-parameter-dependent but
$Q^{2}$(and flavor)-independent parameterization. A jet triggering technique
is used to produce dijet event in the HIJING model, which includes several
hard QCD processes with the initial and final state radiations:
$g+g\rightarrow g+g$, $g+g\rightarrow q+\bar{q}$, $q+g\rightarrow q+g$,
$q+\bar{q}\rightarrow g+g$, $q_{1}+q_{2}\rightarrow q_{1}+q_{2}$, and
$q_{1}+\bar{q_{1}}\rightarrow q_{2}+\bar{q_{2}}$ Sjostrand (1994). In the
string melting mechanism, the primordial hadrons that would be produced from
the excited Lund strings, minijet and jet partons in the HIJING model are
melted into primordial quarks and antiquarks according to the flavor and spin
structures of their valence quarks. The quarks are generated by string melting
after a formation time:

$t_{\rm f}=E_{\rm H}/m^{2}_{\rm T,H},$ (1)

where $E_{\rm H}$ and $m_{\rm T,H}$ represent the energy and transverse mass
of the parent hadron. The initial positions of quarks from melted strings are
calculated from those of their parent hadrons using straight-line
trajectories. Under the initial conditions, a partonic plasma with a dijet is
produced.

(2) Parton cascade. The ZPC model describes parton interactions with two-body
elastic scatterings Zhang (1998). The parton cross section, $\sigma$, is
calculated by the leading-order pQCD for gluon-gluon interaction,

$\frac{d\sigma}{dt}=\frac{9\pi\alpha_{\rm
s}^{2}}{2}\left(1+\frac{\mu^{2}}{s}\right)\frac{1}{\left(t-\mu^{2}\right)^{2}},$
(2)

where $\alpha_{\rm s}$ is the strong coupling constant, $\mu$ the Debye
screening mass, and $s$ and $t$ are the usual Mandelstam variables. The parton
collisions can not only drive the expansion of the QGP, but also lead to the
interactions between jet parton shower and the QGP medium until the freeze out
of the whole partonic system.

(3) Hadronization. In this study, hadronization can be performed by either an
old quark coalescence model or a new hybrid hadronization model, respectively.
(i) The old quark coalescence model is the original mechanism in the AMPT
model Lin et al. (2005), which simply combines two or three nearest partons
into hadrons including mesons and baryons, regardless of the relative momentum
among the coalescing partons. The three-momentum is conserved during
coalescence. The hadron species are determined by the flavor and invariant
mass of combined partons. The simple quark coalescence model includes the
formation of all mesons and baryons listed in the HIJING program. (ii) The new
hybrid hadronization model contains the quark coalescence and string
fragmentation. In the quark coalescence process, the momentum distributions of
mesons and baryons are defined as Han et al. (2016),

$\begin{split}\frac{dN_{\rm M}}{d^{3}\mathbf{P}_{\rm M}}=&g_{\rm M}\int
d^{3}\mathbf{x}_{1}d^{3}\mathbf{p}_{1}d^{3}\mathbf{x}_{2}d^{3}\mathbf{p}_{2}f_{\rm
q}\left(\mathbf{x}_{1},\mathbf{\rm p}_{1}\right)f_{\bar{\rm
q}}\left(\mathbf{x}_{2},\mathbf{p}_{2}\right)\\\ &\times W_{\rm
M}(\mathbf{y},\mathbf{k})\delta^{(3)}\left(\mathbf{P}_{\rm
M}-\mathbf{p}_{1}-\mathbf{p}_{2}\right),\\\ \frac{dN_{\rm
B}}{d^{3}\mathbf{P}_{\rm B}}=&g_{\rm B}\int
d^{3}\mathbf{x}_{1}d^{3}\mathbf{p}_{1}d^{3}\mathbf{x}_{2}d^{3}\mathbf{p}_{2}d^{3}\mathbf{x}_{3}d^{3}\mathbf{p}_{3}f_{\rm
q_{1}}\left(\mathbf{x}_{1},\mathbf{p}_{1}\right)\\\ &\times f_{\rm
q_{2}}\left(\mathbf{x}_{2},\mathbf{p}_{2}\right)f_{\rm
q_{3}}\left(\mathbf{x}_{3},\mathbf{p}_{3}\right)W_{\rm
B}\left(\mathbf{y}_{1},\mathbf{k}_{1};\mathbf{y}_{2},\mathbf{k}_{2}\right)\\\
&\times\delta^{(3)}\left(\mathbf{P}_{\rm
B}-\mathbf{p}_{1}-\mathbf{p}_{2}-\mathbf{p}_{3}\right),\end{split}$ (3)

where $g_{\rm M,B}$ is the statistical factor for forming a meson or baryon of
certain spin, $f{\rm{}_{q,\bar{q}}{(\rm\bf x,p)}}$ are the quark and antiquark
phase-space distribution functions and $W_{\rm M,B}$ is the smeared Wigner
function of mesons or baryons as a function of the relative coordinates and
momenta (${\bf y}$ and ${\bf k}$). The Wigner function of a meson in the
$n$-th excited state is given by $W_{\rm
M,n}(\bf{y},\bf{k})=\frac{\emph{v}^{\rm n}}{\rm n!}e^{-\emph{v}}$ with
$\emph{v}=\frac{1}{2}\left(\frac{{\bf y}^{2}}{\sigma\rm_{M}^{2}}+{\bf
k}^{2}\sigma\rm_{M}^{2}\right)$, where ${\bf y}$ and ${\bf k}$ are the
relative coordinates and momenta between the two constituent quark and
antiquark in the meson. The Wigner function of a baryon in the $n_{1}$-th and
$n_{2}$-th excited states is given by $W_{\rm
B,n_{1},n_{2}}(\mathbf{y}_{1},\mathbf{k}_{1};\mathbf{y}_{2},\mathbf{k}_{2})=\frac{\emph{v}_{1}^{n_{1}}}{n_{1}!}e^{-\emph{v}_{1}}\cdot\frac{\emph{v}_{2}^{n_{2}}}{n_{2}!}e^{-\emph{v}_{2}}$,
where $\emph{v}_{\rm i}=\frac{1}{2}\left(\frac{{\bf y}_{\rm
i}^{2}}{\sigma_{\rm Bi}^{2}}+{\bf k}_{\rm i}^{2}\sigma_{\rm Bi}^{2}\right)$
with ${\bf y}_{\rm i}$ and ${\bf k}_{\rm i}$ being the relative coordinates
and momenta among the three constituent quarks in the baryon. Following Zhao
et al. (2020, 2022), the excited meson states up to $n=10$ and excited baryon
states up to $n_{1}+n_{2}=10$ are included. The width parameters $\sigma_{\rm
M}$, $\sigma_{\rm B1}$ and $\sigma_{\rm B2}$ are determined by the radii of
formed hadrons Beringer et al. (2012). The remnant partons, which can not find
the coalescence partners, are merged into the strings in the fragmentation
part. Using the “hadron standalone mode” of PYTHIA 8 Sjostrand et al. (2008),
these strings will fragment into hadrons. The new hadronization mechanism has
been shown to reproduce the $z_{\rm h}$ and $j_{\rm T}$ distributions of the
PYTHIA jets that have been hadronized by the default Lund string fragmentation
Han et al. (2016). Recently, the new hybrid hadronization mechanism connected
to the coupled linear Boltzmann transport-hydrodynamic model and hadronic
afterburner well described the experimental data of $R_{\rm AA}$, $v_{2}$ and
its hadron flavor dependence from low to high $p_{\rm T}$ in Pb $+$ Pb at
$\sqrt{s_{NN}}$=5.02 TeV Zhao et al. (2022).

(4) Hadronic rescatterings. A relativistic transport (ART) model Li and Ko
(1995) simulates resonance decays and hadronic reactions including both
elastic and inelastic scatterings for baryon-baryon, baryon-meson, and meson-
meson interactions. The mesons include $\pi$, $\rho$, $\omega$, $\eta$, $K$,
$K^{*}$, and $\phi$ for all possible charges. The baryons and antibaryons
include $N$, $\Delta$, $N^{*}(1440)$, $N^{*}(1535)$, $\Lambda$, $\Sigma$,
$\Xi$, and $\Omega$.

In this study, we use the string-melting version of AMPT model with old and
new hadronization mechanisms to simulate p $+$ p and p $+$ Pb collisions at
$\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$. We denote the results from old and new
hadronization mechanisms as “AMPT” and “New Hadronization”, respectively. The
parton cross section is set to $3~{}\rm mb$, which can effectively both
reproduce the collective flow Ma and Bzdak (2014); Bzdak and Ma (2014); Bozek
et al. (2015); He et al. (2016) and simulate between jet and partonic matter
interactions in p $+$ Pb collisions, since the AMPT model with such a parton
cross section has successfully described many experimental observables of
reconstructed jets, such as $\gamma$-jet imbalance Ma (2013c), dijet asymmetry
Ma (2013a), jet fragmentation function Ma (2013b, 2014b), jet shape Ma
(2014a), jet anisotropies Nie and Ma (2014), jet transport coefficient Zhou et
al. (2020), and the redistribution of lost energy Gao et al. (2018); Luo et
al. (2022). In addition, the cross section of $0~{}\rm mb$ represents the
reference without jet-medium interactions. We will focus on the difference of
$j_{\rm T}$-dependent JFFs between “AMPT” and “New Hadronization”. Such
difference reflects the effect of hadronization process, as they share the
same partonic final state and hadronic rescatterings for small colliding
systems.

Figure 1: Top panels: The $j_{\rm T}$-dependent jet fragmentation functions of
charged particles for four $p_{\rm{T,jet}}$ ranges in p $+$ p collisions at
$\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ from the AMPT model with old and new
hadronization mechanisms, compared to the ALICE data Acharya et al. (2021).
Middle panels: Same as the top panels but for p $+$ Pb collisions at
$\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$. Bottom panels: The ratios of the $j_{\rm
T}$-dependent jet fragmentation function in p $+$ Pb collisions (0 or 3 mb) to
that in p $+$ p collisions (0 mb) for the AMPT model with the old and new
hadronization mechanisms.  Figure 2: Upper panel: The $j_{\rm T}$-dependent
jet fragmentation functions of charged particles for four $p_{\rm{T,jet}}$
ranges in p $+$ Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ from the
AMPT model with old (dashed curve) and new hadronization mechanisms (solid
curve), compared to the ALICE data Acharya et al. (2021). Bottom panel: The
ratios of the $j_{\rm T}$-dependent jet fragmentation function for the high
$p_{\rm{T,jet}}$ range to that for 40 $<p_{\rm{T,jet}}<$ $60~{}{\rm GeV}$/$c$.
Figure 3: Upper panel: The $j_{\rm T}$-dependent jet fragmentation functions
of charged particles for four $p_{\rm{T,jet}}$ ranges in p $+$ Pb collisions
at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ from the coalescence and fragmentation
parts in the new hadronization mechanism. Bottom panel: The contributions of
the coalescence and fragmentation parts to the total $j_{\rm T}$-dependent jet
fragmentation function. Figure 4: The $j_{\rm T}$-dependent jet fragmentation
function of charged particles for $60<p_{\rm{T,jet}}<80~{}{{\rm GeV}/c}$ in p
$+$ Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ from the AMPT model
with the new hadronization mechanism, which is fitted by Eq. (6) including
both narrow and wide parts. Figure 5: The RMS values for the narrow and wide
parts for different hadronization mechanisms and their components in p $+$ Pb
collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$, compared to the ALICE data
Acharya et al. (2021).

###  II.2 The analysis method on $j_{\rm T}$-dependent JFFs

The transverse momentum of hadrons within a jet, $j_{\rm T}$, is defined as
the perpendicular component of the momentum of constituent particle with
respect to the reconstructed jet axis,

$j_{\rm T}=\frac{\left|\vec{p}_{\text{jet }}\times\vec{p}_{\text{track
}}\right|}{\left|\vec{p}_{\text{jet }}\right|},$ (4)

where $\vec{p}_{\text{jet}}$ is the momentum of the jet and
$\vec{p}_{\text{track}}$ is the momentum of the tracks. The $j_{\rm
T}$-dependent jet fragmentation functions of charged particles are calculated
according to,

$D(j_{\rm T})=\frac{1}{N_{\text{jets }}}\frac{1}{j_{\rm{T,ch}}}\frac{{\rm
d}N_{\rm{ch}}}{{\rm d}j_{\rm{T},\rm{ch}}},$ (5)

where $N_{\text{jets}}$ is the number of the reconstructed jets and
$N_{\text{ch}}$ is the number of the charged particles inside the
reconstructed jet cone. In the ALICE experiment, the $j_{\rm T}$-dependent jet
fragmentation function has been fitted by the following two-component function
including a Gaussian function and an inverse $\Gamma$ function,

$\frac{1}{N_{\text{jets }}}\frac{{\rm d}N}{j_{\rm{T},\rm{ch}}{\rm
d}j_{\rm{T},\rm{ch}}}=\frac{B_{2}}{B_{1}\sqrt{2\pi}}e^{-\frac{j_{\rm{T}}^{2}}{2B_{1}^{2}}}+\frac{B_{3}B_{5}^{B_{4}}}{\Gamma\left(B_{4}\right)}\frac{e^{-\frac{B_{5}}{j_{\rm{T}}}}}{j_{\rm{T}}^{B_{4}+1}},$
(6)

where $B_{\rm i}$ (i=1-5) are the free parameters and $\rm\Gamma$ is the gamma
function. The root-mean-square (RMS) value of the $j_{\rm T}$-dependent jet
fragmentation function for the narrow component can be obtained from the
Gaussian function Acharya et al. (2021),

$\sqrt{\left\langle j_{\rm{T}}^{2}\right\rangle}=\sqrt{2}B_{1},$ (7)

and the RMS value for the wide component can be extracted from the inverse
gamma function,

$\sqrt{\left\langle
j_{\rm{T}}^{2}\right\rangle}=\frac{B_{5}}{\sqrt{\left(B_{4}-2\right)\left(B_{4}-3\right)}}.$
(8)

Note that in the Ref. Acharya et al. (2019), the ALICE experiment found that
the final state radiation of jet shower leads to the production of wide
component, and defines the remainder other than the final state radiation in
the JFF as ‘hadronization’, which is different from our definition of
hadronization, by which we mean that hadronization is the process of
transition from partons to hadrons.

In this work, the $j_{\rm T}$-dependent jet fragmentation function inside the
jet is studied by using the anti-$k_{\rm T}$ algorithm Cacciari et al. (2008),
which is implemented using a FastJet framework Cacciari et al. (2012). All
particles, containing charged and neutral particles, are considered for the
jet reconstruction with a fixed resolution parameter, i.e. jet cone size $R$.
However, only charged particles within the jet cones are used to calculate the
distributions of jet fragmentation observables. The background for the
reconstructed jet is estimated by finding a second cone, which is rotated 90
degrees in $\phi$ from the observed jet cone. The second cone includes all
charged particles that can be found within this cone with the same $R$ size as
same as the original jet cone, which is applied to calculate the background
distribution subtracted from jet fragmentation $j_{\rm T}$-dependent jet
fragmentation function. We use the same kinematic cuts as the ALICE experiment
Acharya et al. (2021), where the transverse momenta of particles need to
exceed $0.15~{}{\rm GeV}/c$ and the jet pseudorapidity satisfy
$\left|\eta_{\rm{jet}}\right|<0.25$.

##  III Results and discussion

###  III.1 $j_{\rm T}$-dependent JFFs

Fig 1 shows the transverse momentum $j_{\rm T}$-dependent jet fragmentation
functions of charged particles inside jets in p $+$ p collisions (top panels)
and p $+$ Pb collisions (middle panels) at $\sqrt{s_{\rm NN}}=5.02~{}{\rm
TeV}$ for four $p_{\rm{T,jet}}$ intervals (between 40 and $150~{}{\rm GeV}/c$)
from the AMPT model with old and new hadronization mechanisms, compared to the
ALICE data Acharya et al. (2021). In general, more low $j_{\rm T}$ particles
are produced by the new hadronization mechanism than that by the old simple
coalescence model. Remarkably, the results from the new hadronization
mechanism are in good agreement with the experimental data for low and
intermediate $j_{\rm T}$ regions, although it slightly overestimates the
experimental data at the high $j_{\rm T}$ region 111This could be due to the
absence of gluons in the AMPT output partons.. In addition, the bottom panels
present the ratios of the JFF in p $+$ Pb to that in p $+$ p collisions for
different parton interaction cross sections in ZPC part. Two sets of parton
interaction cross sections, $3~{}\rm mb$ and $0~{}\rm mb$, are utilized to
mimic with and without jet-medium interactions in p $+$ Pb collisions. We find
that the $j_{\rm T}$-dependent JFFs in p $+$ Pb collisions, with different
cross sections, are consistent with that in p $+$ p collisions, almost
regardless of parton interaction cross section. It indicates that there is no
obvious modification effect from jet-medium interactions or the cold nuclear
matter effect on the $j_{\rm T}$-dependent JFFs in p $+$ Pb collisions.

Upper panel in Fig 2 shows the $j_{\rm T}$-dependent jet fragmentation
functions of charged particles inside jets for four $p_{\rm{T,jet}}$ intervals
in p $+$ Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ from the AMPT
model with old and new hadronization mechanisms, compared to the ALICE data.
The ratios of the $j_{\rm T}$-dependent JFF for high $p_{\rm{T,jet}}$
intervals with respect to that for the lowest $p_{\rm{T,jet}}$ interval (40
$<p_{\rm{T,jet}}<$ $60~{}{\rm GeV}$/$c$) are presented in the bottom panel in
Fig 2. These ratios increase slowly with $j_{\rm T}$ in the low and
intermediate $j_{\rm T}$ regions ($j_{\rm T}<1.0$ GeV/c). At $j_{\rm T}>1.0$
GeV/c, the ratios increases rapidly. These features are caused by the fact
that the high-$p_{\rm{T,jet}}$ jet includes more particles inside jet cone,
especially more large-$j_{\rm T}$ particles, than the low-$p_{\rm{T,jet}}$
jets, which is consistent with the ALICE measurements Acharya et al. (2021).
The ratio from new hadronization mechanism are higher than that from the old
hadronization mechanism, especially for high-$p_{\rm T,jet}$ jets, suggesting
that such ratio could be used as a sensitive probe to study the effect of
different hadronization mechanisms on the JFFs.

To further investigate the individual contributions from quark coalescence and
string fragmentation to the total JFFs in the new hybrid hadronization
mechanism, we compare the $j_{\rm T}$-dependent JFFs of charged particles from
quark coalescence, string fragmentation and the total for four
$p_{\rm{T,jet}}$ ranges in p $+$ Pb collisions at $\sqrt{s_{\rm
NN}}=5.02~{}{\rm TeV}$ in upper panel of Fig 3. We find that most hadrons
inside a jet are produced by coalescence process. To learn about the exact
contribution percentages from two hadronization mechanisms, the ratios of the
quark coalescence-contributed JFF to the total JFF and the fragmentation-
contributed JFF to the total JFF are shown as functions of $j_{\rm T}$ in the
bottom panel of Fig 3. We find that $j_{\rm T}$-dependent JFFs are dominantly
contributed by the quark coalescence part. The contribution of the string
fragmentation part increases with $j_{\rm T}$ and reaches up to 25% at $j_{\rm
T}\sim$ 3 GeV/$c$.

The $j_{\rm T}$-dependent jet fragmentation functions have been fitted with a
two-component function in Eq. (6) including a Gaussian function (narrow part
or low-$j_{\rm T}$ part) and an inverse gamma function (wide part or
high-$j_{\rm T}$ part). Figure 4 shows our fitting to the $j_{\rm
T}$-dependent JFF for $60<p_{\rm{T,jet}}<80~{}{{\rm GeV}/c}$ in p $+$ Pb
collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$ from the AMPT model with
the new hadronization mechanism. We can see that the two-component function
can well describe our results. This is understandable because the final state
radiation is also turned on during our initialization of the jet in HIJING
part. Based on the fitting function, we calculate the root-mean-square (RMS)
values for narrow and wide parts based on Eqs. (7) and (8). In Fig 5, the RMS
values from the different hadronization processes and their components are
shown as functions of $p_{\rm{T,jet}}$, compared to the ALICE measurements. We
find that all RMS values of the narrow part are consistent with the ALICE
data, independent of $p_{\rm{T,jet}}$. It indicates that the RMS value of
narrow part is not sensitive to hadronization mechanism. However, for the wide
part, the AMPT model with the old hadronization mechanism fails and
overestimates the ALICE data. On the other hand, the new hadronization
mechanism describes well the measured RMS values of wide part which increases
with $p_{\rm{T,jet}}$. Note that the experimental data are also described well
by the coalescence part in the new hadronization mechanism, which indicates
that the $j_{\rm T}$-dependent JFFs are dominantly contributed by coalescence
process. The above comparisons reveal that the RMS value of wide part is more
sensitive to different hadronization mechanisms than the narrow part.

###  III.2 $R$ size dependence of $j_{\rm T}$-dependent JFFs

Figure 6: Upper panel: The $j_{\rm T}$-dependent jet fragmentation functions
of charged particles with three jet radii $R$ = 0.3, 0.4 and 0.5 for
$60<p_{\rm{T,jet}}<80~{}{{\rm GeV}/c}$ in p $+$ Pb collisions at $\sqrt{s_{\rm
NN}}=5.02~{}{\rm TeV}$ from the AMPT model with old (dashed curve) and new
hadronization (solid curve) mechanisms. Bottom panel: The ratios of the
$j_{\rm T}$-dependent jet fragmentation function for a large jet radius $R$ to
that for the smallest jet radius $R$ = 0.3. Figure 7: The RMS values for the
narrow and wide parts for jets with three jet radii $R$ from the AMPT model
with old and new hadronization mechanisms in p $+$ Pb collisions at
$\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$.

Varying the size of the jet cone can incorporate the particles with different
$j_{\rm T}$ into the jet cone. Three jet cone size radii, $R=$ 0.3, 0.4, and
0.5, are used in our analysis to study the jet cone size dependence of the
$j_{\rm T}$-dependent JFFs. The $j_{\rm T}$-dependent JFFs for
$60<p_{\rm{T,jet}}<80~{}{{\rm GeV}/c}$ in p $+$ Pb collisions at $\sqrt{s_{\rm
NN}}=5.02~{}{\rm TeV}$ from the AMPT model with old and new hadronization
mechanisms are shown in the upper panel of Fig 6. The bottom panel of Fig 6
shows the ratios of the $j_{\rm T}$-dependent JFF with respect to that in the
lowest jet radius $R=0.3$. We find that the ratio increases slowly with
$j_{\rm T}$ in the low and intermediate $j_{\rm T}$ ($j_{\rm T}<$ 1.1 GeV/c)
regions, while the ratio increases rapidly with $j_{\rm T}$ in the high
$j_{\rm T}$ region, which indicates that the jets with a large jet cone size
carry more associated particles, especially for the high-$j_{\rm T}$
particles, compared to the jet with the small jet cone size. On the other
hand, we find that the ratio for new hadronization mechanism is larger than
that for old hadronization mechanism. The difference is more obvious for a
larger jet cone size, which suggests that jets with a larger jet cone size are
more sensitive to different hadronization mechanisms.

The corresponding RMS values for different jet cone sizes $R$ as functions of
$p_{\rm{T,jet}}$ are shown in Fig 7. Note that we do not compare our results
with experimental data, since the experimental data for $R=$ 0.3 and 0.5 are
not available. There is no obvious difference among different jet cone sizes
for the RMS value of narrow part, also independent of hadronization mechanism.
On the other hand, the RMS value of wide part increases with the jet radius
$R$ and jet transverse momentum $p_{\rm{T,jet}}$. However, the AMPT model with
the old hadronization mechanism gives larger RMS values of wide part than the
new hadronization mechanism. Therefore, we propose that the RMS value of wide
part could also be a sensitive probe to study different hadronization
mechanisms.

##  IV Summary

The $j_{\rm T}$-dependent jet fragmentation functions have been investigated
in p $+$ p and p $+$ Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm TeV}$
with a multiphase transport (AMPT) model using old simple quark coalescence
mechanism and a new hadronization mechanism which includes both dynamical
quark coalescence and string fragmentation schemes. The $j_{\rm T}$-dependent
JFFs from the new hadronization mechanism are in better agreement with the
ALICE experimental data than the old simple quark coalescence mechanism.
Within the new hadronization mechanism framework, the $j_{\rm T}$-dependent
JFFs are dominated by the coalescence, while the contribution of the
fragmentation increases with $j_{\rm T}$. The ratios of the $j_{\rm
T}$-dependent JFFs in p $+$ Pb collisions to those in p $+$ p collisions are
consistent with unity, which indicate there is no significant effects of jet
quenching or cold nuclear matter in p $+$ Pb collisions. On the other hand, we
find that the $j_{\rm T}$-dependent JFFs become wider with $p_{\rm T,jet}$,
which are sensitive to different hadronization mechanisms. The RMS values of
the JFFs for narrow and wide parts have been extracted by a two-component
function fitting for different hadronization mechanisms. Unlike the
insensitivity for the RMS value of narrow part, the RMS value of wide part
increases with the increasing of $p_{\rm{T,jet}}$ and jet cone size $R$. Since
the RMS value of wide part is very sensitive to different hadronization
mechanisms, we propose that the wide part of JFFs is a good probe to
investigate the non-perturbative QCD hadronization effect on jets in small
colliding systems. Our current work focuses only on small colliding systems,
and future studies of $j_{\rm T}$-dependent JFFs in high-energy A $+$ A
collisions are expected to explore the effect of hot QCD medium modifications
on jet hadronization.

###### Acknowledgements.

X.-P.D. and G.-L.M. are supported by the National Natural Science Foundation
of China under Grants No.12147101, No. 11890714, No. 11835002, No.
11961131011, No. 11421505, the Strategic Priority Research Program of Chinese
Academy of Sciences under Grant No. XDB34030000, and the Guangdong Major
Project of Basic and Applied Basic Research under Grant No. 2020B0301030008.
W. Z. is supported by the National Science Founda-tion (NSF) under grant
numbers ACI-2004571 within the framework of the XSCAPE project of the JETSCAPE
collaboration.

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