# The Graph Minor Relation Satisfies the Twin Alternative Conjecture

Jorge Bruno

###### Abstract.

In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the
equivalence class of a tree under the embeddability relation is, up to
isomorphism, either trivial or infinite. In 2022 LaFlamme, et al. provided a
rigorous exposition of a conter-example to TAC developed by Tetano in his 2008
PhD thesis. Also in 2022, the present author provided a positive answer to TAC
for the topological minor relation. Along with embeddability and the
topological minor, the graph minor relation completes the triad of the most
widely studied graph relations. In this paper we provide a positive answer to
TAC for the the graph minor.

##  1\. Introduction

Consider the following local operations on a graph [5]:

  1. (1)

Edge removal ($\backslash e$):

  2. (2)

Edge contraction ($/e$):

  3. (3)

Vertex removal ($\backslash v$):

  4. (4)

Deg-2 vertex dissolution ($/v$):

For
$\mathcal{L}\subseteq\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\backslash
e,/e,\backslash v,/v}\\}$ then $H\leq_{\mathcal{L}}G$ if $H$ is obtained from
$G$ by a sequence of operations from $\mathcal{L}$. The table below summarises
the most commonly studied graph relations arising from the above local graph
operations.111Since all trees are connected, the induced and regular versions
of the above are equivalent. Therefore, given that our results concern only
trees, we make no distinction between the two versions of any of the above
graph relations.

Relation | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\backslash e}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/e}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\backslash v}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/v}$
---|---|---|---|---
subgraph/embeddable ($\leq$) | $\bullet$ |  | $\bullet$ |
induced subgraph/strongly embeddable |  |  | $\bullet$ |
topological minor ($\leq^{\sharp}$) | $\bullet$ |  | $\bullet$ | $\bullet$
induced topological minor |  |  | $\bullet$ | $\bullet$
graph minor ($\leq^{*}$) | $\bullet$ | $\bullet$ | $\bullet$ |
induced graph minor |  | $\bullet$ | $\bullet$ |

Two trees $T,S$ are mutually embeddable, $T\equiv S$, (resp. topologically
equivalent, $T\equiv^{\sharp}S$, and mutual minors, $T\equiv^{*}S$) if $T\leq
S\leq T$ (resp. $T\leq^{\sharp}S\leq^{\sharp}T$ and $T\leq^{*}S\leq^{*}T$).
The symbol $\cong$ is reserved for the isomorphism relation. Since any deg-2
vertex dissolution can be interpreted as an edge dissolution we get that
$T\cong S\Rightarrow T\leq S\Rightarrow T\leq^{\sharp}S\Rightarrow
T\leq^{*}S$. The equivalence class of a tree $T$ with respect to
$\equiv,\equiv^{\sharp}$ and $\equiv^{*}$ is denoted by $[T]$, $[T]_{\sharp}$
and $[T]_{*}$, respectively.

The authors of [3] proved that $|[T]|=1$ or $\infty$ for any rayless tree and
conjectured the same must be true of any tree: the Tree Alternative Conjecture
(TAC) stated that the number of isomorphism classes of trees mutually
embeddable with a given tree $T$ is either 1 or infinite. TAC is certainly
true for finite trees (since mutually embeddable finite trees are necessarily
isomorphic) and has been confirmed for a number of nontrivial classes of
infinite trees - [3], [10] and [12]. In particular, in [12] it is shown that
TAC holds for all rooted trees, where a rooted tree $(T,r)$ is one with a
distinguished vertex $r\in V(T)$. In 2022 a counterexample to TAC was found by
LaFlamme et al. [CITE] where for each $n\in\mathbb{N}$ an unrooted and locally
finite tree $T_{n}$ is constructed with $|[T_{n}]|=n$.

One way to naturally generalise this line of enquiry is to extend TAC to the
topological and graph minor relations. Given
$\hat{R}\in\\{\equiv,\equiv^{\sharp},\equiv^{*}\\}$ we define

TAC($\hat{R}$). For any tree $T$,

$|[T]_{\hat{R}}|\not\in\mathbb{N}\smallsetminus\\{1\\}.$

RootedTAC($\hat{R}$). For any tree $T$ and any $r\in V(T)$,

$|[(T,r)]_{\hat{R}}|\not\in\mathbb{N}\smallsetminus\\{1\\}.$

As stated earlier, TAC($\equiv$) is false and RootedTAC($\equiv$) is true.
RootedTAC($\equiv^{\sharp}$) and TAC($\equiv^{\sharp})$ are true [1] and in
this manuscript we prove the following result.

Main Theorem. TAC($\equiv^{*}$) and RootedTAC($\equiv^{*}$) hold.

Our approach to proving the Main Theorem is to deal with large and small trees
separately: a small tree is one where every ray is eventually bare and a ray
$R=v_{1}\ldots$ is eventually bare provided $\exists M\in\mathbb{N}$ where,
for all $n\geq M$, $\deg(v_{n})=2$. Trees that are not small are said to be
large. Since all rayless trees are small, the following result holds.

######  Corollary 1.1.

For any $\hat{R}\in\\{\equiv,\equiv^{\sharp},\equiv^{*}\\}$,
RootedTAC($\hat{R})$ holds. Further, TAC($\hat{R})$ also holds for the
collection of all small trees.

##  2\. Background

An equivalent interpretation for the graph minor relation employs the notion
of components: $T\leq^{*}S$ precisely when there exists a function
$\mu:V(T)\to C(S)$, where $C(S)$ is the collection of components of $S$, so
that

  1. (1)

$\mu(v)\cap\mu(w)=\emptyset$ whenever $v\not=w$, and

  2. (2)

$\exists u_{v}u_{w}\in E(S)$ with $u_{v}\in\mu(v)$ and $u_{w}\in\mu(w)$
whenever $vw\in E(T)$.

We call the function $\mu$ a model of $T$ in $S$. Composing models is done as
follows: given models $\mu_{1}:V(T)\to C(S_{1})$ and $\mu_{2}:V(S_{1})\to
C(S_{2})$ we compose $\mu_{1}$ with $\mu_{2}$ as
$\mu_{2}\circ^{*}\mu_{1}:V(T)\to C(S_{2})$ where
$(\mu_{2}\circ^{*}\mu_{1})(v)$ is the component of $S_{2}$ composed of all
components $\mu_{2}(w)$, $w\in\mu_{1}(v)$, and the edges joining them.

A rooted tree $(T,r)$ is one with a distinguished vertex $r\in V(T)$. All
rooted trees naturally give rise to a tree order on its set of vertices:
$v\leq_{T}w$ precisely when the unique path from $r$ to $w$ contains $v$. When
considering rooted trees the functional definitions for the graph relation
considered in this paper must preserve the tree order; a tree $(T,r)$ is a
rooted graph minor of another tree $(S,s)$ precisely when there exists a model
$\mu$ of $T$ in $S$ with $v\leq_{T}w\Leftrightarrow\min_{S}\\{t\in
V(\mu(v))\\}\leq_{S}\min_{S}\\{t\in V(\mu(w))\\}$. Compositions of rooted
models are defined in the obvious way. The equivalence class of a tree $T$
with respect to the graph minor relation is denoted by $[T]_{*}$ and if the
tree is rooted then we write $[(T,r)]_{*}$.

We use some basic set-theoretic notation and terminology which can be found in
[7] or [8]. Recall that $\aleph_{0}$ is the cardinality of the natural numbers
and we use the set-theoretic notation $\omega$ to denote the the set of
natural numbers, $\mathbb{N}$, with $0$ included. The continuum $\mathfrak{c}$
or $2^{\aleph_{0}}$ denotes the cardinality of the set of real numbers. A
cardinal $\lambda$ is cofinal in $\kappa$ if for every $\alpha\in\kappa$ there
exist a $\beta\in\lambda$ with $\beta\geq\alpha$. The cofinality of a cardinal
$\kappa$, cf$(\kappa)=\min\\{\alpha\mid\lambda\text{ is cofinal in
}\kappa\\}$. A cardinal $\kappa$ is regular if cf($\kappa$)$=\kappa$ and
singular, otherwise. Given a singular cardinal $\kappa$ and a cardinal
$\lambda<\kappa$ the interval $[\lambda,\kappa]$ contains infinitely many
cardinals smaller than $\kappa$.

###  2.1. Large Trees

Our approach to establishing the Main Theorem is to deal with large and small
trees separately. In [1] it is shown that for any large tree $T$ and any $r\in
V(T)$ both $|[T]_{\sharp}|$ and $|[(T,r)]_{\sharp}|\geq 2^{\aleph_{0}}$. It
follows that the Main Theorem holds for all large trees.

######  Theorem 2.1.

TAC($\equiv^{*}$) and RootedTAC($\equiv^{*}$) hold for all large trees.

##  3\. Small Trees

We turn our attention to proving the Main Theorem for small trees: in Section
3.1 we prove establish it for rooted trees and then extend it to all trees in
Section 3.2. First, we focus on some basic preliminary facts. Given a tree
$T$, $F(T)=\\{v\in V(T)\mid\deg(v)>2\\}$ and $inf(T)=\\{v\in
V(T)\mid\deg(v)=\infty\\}$. If $T$ is small and locally finite then clearly
$F(T)$ must be finite.

######  Lemma 3.1.

Let $T,S$ be trees and $\mu:V(T)\to C(S)$ be a model of $T$ in $S$. It follows
that

  1. (1)

if $S$ is small then so is $T$,

  2. (2)

if $v\in F(T)$ then $F(S)\cap\mu(v)\not=\emptyset$, and

  3. (3)

if $v\in inf(T)$ then $inf(S)\cap\mu(v)\not=\emptyset$ or there exists a ray
in $S$ that is not eventually bare and contained in $\mu(v)$.

###### Proof.

(1) is obvious were (2) false then $\mu(v)$ would be a path and would be
connected to at most two other connected components of $S$.

For (3) take any $v\in inf(T)$ and let $u_{i}$, $i\in\mathbb{N}$, be any
infinite collection of distinct neighbours of $v$. Because $\mu:V(T)\to C(S)$
is a model of $T$ in $S$ it follows that there must exist an infinite sequence
of distinct edges $w_{1}v_{1}\in E(S)$ with $v_{i}\in\mu(v)$ and
$w_{i}\in\mu(u_{i})$. Therefore, if $\mu(v)\cap inf(S)=\emptyset$ then the
vertices $v_{i}$ span a ray $R$ in $S$ that is not eventually bare. ∎

Given a collection $A\subseteq V(T)$ we denote $\overline{A}$ as the closure
of $A$ in $T$: $\overline{A}$ is $A$ along with all paths in $T$ joining any
pair of vertices in $A$. The following result is key in establishing the Main
Theorem for small trees. Briefly, it states that all four equivalence
relations (i.e., $\cong,\equiv,\equiv^{\sharp},\equiv^{*}$) coincide on the
collection of small locally finite trees.

######  Theorem 3.1.

Let $T,S$ be locally finite and small. It follows that $T\cong
S\Leftrightarrow T\equiv^{*}S$.

###### Proof.

The authors of [1] prove that $T\cong S\Leftrightarrow T\equiv^{\sharp}S$ (see
Theorem 5). In turn, we focus on showing that $T\equiv^{*}S$ implies
$T\equiv^{\sharp}S$. Suppose that $\mu:V(T)\to C(S)$ witnesses $T\leq^{*}S$
and recall that by Lemma 3.1 for each $v\in F(T)$ it must be that
$V(\mu(v))\cap F(S)\not=\emptyset$. Hence, $F(T)\leq F(S)$ and since
$T\geq^{*}S$ we also get that $F(S)\leq F(T)$. It follows that $V(\mu(v))\cap
F(S)=\\{w\\}$ for any $v\in F(T)$ since $|F(S)|=|F(T)|$. It then follows that
any edge contraction employed to turn $T$ into $S$ must be performed on an
edge $vw\in E(T)$ with $\deg(v),\deg(w)\leq 2$. Any such edge contraction is a
deg-2 vertex dissolution. Hence, $T\leq^{\sharp}S$ and, by symmetry,
$T\geq^{\sharp}S$. ∎

######  Remark 3.2.

The converse of the above is obviously not true. That said, Theorem 2.1
implies that any such collection must be composed of small trees only.

###  3.1. RootedTAC($\equiv^{*}$)

For a rooted tree $(T,r)$ we let $succ(v)$ denote the collection of immediate
successors of a vertex $v$ and $(T_{v},v)$ denote the full subtree of $(T,r)$
generated by all $w\geq_{T}v$.

######  Lemma 3.2.

For any rooted tree $(T,r)$ and any $v\in V(T)$ we have
$|[(T_{v},v)]_{*}|\leq|[(T,r)]_{*}|$ .

###### Proof.

It suffices to prove the claim for any vertex in $succ(r)$. Hence, fix a $v\in
succ(r)$ and take any $(S,s)\in[(T_{v},v)]_{*}$. Construct the tree
$(T^{\prime},r)$ as follows: for any $w\in succ(v)$

  * •

replace $(T_{w},w)$ with $(S,s)$ if $(T_{w},w)\equiv^{*}(S,s)$, and

  * •

leave $(T_{w},w)$ as it is, otherwise.

By design, $(T^{\prime},r)\equiv^{*}(T,r)$ and $(T^{\prime},r)\not\cong(T,r)$.
∎

In view of the above and given a small tree $(T,r)$ it follows that the only
way for $\aleph_{0}>|[(T,r)]_{*}|>1$ is if $|[(T_{v},v)]_{*}|=1$ for all $v\in
succ(r)$. Indeed, since increasing sequences of vertices in a small tree end
on a terminal vertex or span an eventually bare ray it follows that any
sequence $\left(|[(T_{v_{n}},{v_{n}})]_{*}|\right)_{n\in\mathbb{N}}$ with
$v_{n}\in succ(v_{n-1})$ must have a tail of 1’s. In light of this
observation, we finish this section by proving that any tree $(T,r)$ with
$|[(T_{v},v)]_{*}|=1$ for any $v\in succ(r)$ can not have a finite non-trivial
equivalence class.

######  Theorem 3.3.

For any small tree $(T,r)$ with $|[(T_{v},v)]_{*}|=1$, $\forall v\in succ(r)$,
it follows that $|[(T,r)]_{*}|=1$ or $\infty$.

###### Proof.

Let $(T,r)$ and $(S,s)$ be two non-isomorphic trees with
$(S,s)\in[(T,r)]_{*}$, and models $\mu_{T}:V(T)\to C(S)$ and $\mu_{S}:V(S)\to
C(T)$. Further assume that $|[(T_{v},v)]_{*}|=1$ for any $v\in succ(r)$. We
seek to prove that $|[(T,r)]_{*}|\geq\aleph_{0}$.

Put $\mathcal{I}=\\{(X_{\alpha},\alpha)\mid\alpha\in\kappa\\}$ as the set of
all isomorphism types of all rooted trees $(T_{v},v)$ and $(S_{w},w)$ with
$v\in succ(r)$ and $w\in succ(s)$. Define $f_{T}:\\{(T_{v},v)\mid v\in
succ(r)\\}\to\kappa$ so that $f_{T}(T_{v},v)=\alpha$ provided
$(T_{v},v)\cong(X_{\alpha},\alpha)$ (resp. $f_{S}:\\{(S_{w},w)\mid w\in
succ(s)\\}\to\kappa$ so that $f_{S}(S_{w},w)=\alpha$ provided
$(S_{w},w)\cong(Y_{\alpha},\alpha)$). Since $(T,r)\not\cong(S,s)$ there must
exist a $\beta\in\kappa$ for which, without loss of generality,
$|f_{T}^{-1}(\beta)|<|f_{S}^{-1}(\beta)|$. Put $\eta=|f_{T}^{-1}(\beta)|$ and
$\zeta=|f_{S}^{-1}(\beta)|$, and enumerate the elements of $f_{T}^{-1}(\beta)$
and $f_{S}^{-1}(\beta)$ as $(T_{i},i)$ and $(S_{j},j)$ with $i\in\eta$ and
$j\in\zeta$. We are then left with two following complementary distinct
scenarios.

CASE 1: $\forall j\in\zeta$, $\exists i_{j}\in\eta$ with
$\mu_{S}(S_{i},i)\subseteq(T_{i_{j}},{i_{j}})$.

CASE 2: $\exists j\in\zeta$ with $\mu_{S}(S_{j},j)\subseteq(T_{v},v)\not\in
f_{T}^{-1}(\beta)$.

Before moving further with this proof, let us establish some more notation:
put $(T^{\prime},r)$ as $(T,r)$ but with all full subtrees $(T_{i},i)$,
$i\in\eta$, removed - this also includes the roots $i$ of each $(T_{i},i)$ and
the edge joining $i$ with $r$. In what follows we fix a $(T_{k},k)\in
f_{T}^{-1}(\beta)$ and for each cardinal $\lambda$ denote
$(T^{\prime}_{\lambda},r)$ as $(T^{\prime},r)$ with $\lambda$ copies of
$(T_{k},k)$ attached to $r$ \- we label each such copy
$(T^{\prime}_{\lambda_{l}},{\lambda_{l}})$, $l\in\lambda$. Clearly,
$(T^{\prime}_{\eta},r)\cong(T,r)$ and
$(T^{\prime}_{\nu},r)\not\cong(T^{\prime}_{\xi},r)$ for any pair of cardinals
$\nu\not=\xi$.

CASE 1: Observe that $\zeta=\bigcup_{i\in\eta}|A_{i}|$, where

$A_{i}=\\{(S_{j},j)\in
f_{S}^{-1}(\beta)\mid\mu_{S}(S_{j},j)\subseteq(T_{i},i)\\}.$

Since $\eta<\zeta$ we must consider separately the cases of whether or not
$\zeta=|A_{\gamma}|$ for some $\gamma\in\eta$. If $\zeta=|A_{\gamma}|$ for
some $\gamma\in\eta$ then for each pair of cardinals $\nu,\xi\leq\zeta$ we
have that $(T^{\prime}_{\xi},r)\equiv^{*}(T^{\prime}_{\nu},r)\equiv^{*}(T,r)$.
Indeed, the former is true since $\zeta=|A_{\gamma}|$ implies that $(T_{k},k)$
admits $\zeta$ disjoint models of itself. Hence, if $\zeta\geq\aleph_{0}$ then
we are done. Otherwise, a simple inductive argument shows that $(T_{k},k)$
admits $\aleph_{0}$ copies of itself. Thus,
$(T^{\prime}_{n},r)\equiv^{*}(T,r)$ for all $n\in\mathbb{N}$ and
$|[(T,r)]_{*}|\geq\zeta\geq\aleph_{0}$. Finally, if $\zeta\not=|A_{i}|$ for
any $i\in\eta$ then $\zeta$ must be a singular cardinal and, in turn, there
exists at least $\aleph_{0}$ many cardinals $\alpha\in[\eta,\zeta]$. Clearly,
for each cardinal $\alpha\in[\eta,\zeta]$,
$(T^{\prime}_{\alpha},r)\equiv^{*}(T,r)$. This completes the proof Theorem for
Case 1.

CASE 2:  Fix a $\gamma\in\zeta$ for which
$\mu(S_{\gamma},\gamma)\subseteq(T_{v},v)\not\in f_{T}^{-1}(\beta)$. Next, for
each $n\in\omega$, put $w_{n}\in succ(s)$ so that
$(\mu_{T}\circ^{*}\mu_{S})^{n}[(S_{\gamma},\gamma)]\subseteq(S_{w_{n}},w_{n})$
and $v_{n}\in succ(r)$ with
$\mu_{S}\circ^{*}(\mu_{T}\circ^{*}\mu_{S})^{n}[(S_{\gamma},\gamma)]\subseteq(T_{v_{n}},v_{n})$.
Clearly, $(S_{w_{0}},w_{0})=(S_{\gamma},\gamma)$ and
$(T_{v_{0}},v_{0})=(T_{v},v)$, and

$(S_{w_{0}},w_{0})\leq^{*}(T_{v_{0}},v_{0})\leq^{*}(S_{w_{1}},w_{1})\leq^{*}(T_{v_{1}},v_{1})\leq^{*}\ldots$

Observe that $w_{n}\not=w_{0}$ for any $n\in\omega$ since then
$(S_{\gamma},\gamma)\cong(T_{v},v)$ from the assumption that
$|[(T_{u},u)]_{*}|=1$ for any $u\in succ(r)$. The same logic dictates
$(S_{w_{n}},w_{n})\not\in f_{S}^{-1}(\beta)$ and $(T_{v_{n}},v_{n})\not\in
f_{T}^{-1}(\beta)$ for any $n>1$. Thus, we get two sequences of non-repeating
elements: $(w_{n})_{n\in\omega}$ and $(v_{n})_{n\in\omega}$.

If $\eta$ is finite then $(T^{\prime}_{\lambda},\lambda)\geq^{*}(T,r)$ for
each finite $\lambda\geq\eta$ and given any $\lambda\geq\eta$ let
$\mu_{\lambda}:V(T^{\prime}_{\lambda})\to C(T)$ with

  * •

$\mu_{\lambda}[(T^{\prime}_{\lambda_{l}},{\lambda_{l}})]=\mu_{S}\circ^{*}(\mu_{T}\circ^{*}\mu_{S})^{l}[(S_{\gamma},\gamma)]$
for each $l\leq\lambda$,

  * •

$\mu_{\lambda}[(T_{v_{n}},{v_{n}})]=\mu_{S}\circ^{*}(\mu_{T}\circ^{*}\mu_{S})^{n+\lambda+1}[(S_{\gamma},\gamma)]$
for each $n\in\omega$, and

  * •

the identity everywhere else.

Since $(S_{\gamma},\gamma)\cong(T_{k},k)$, $\mu_{\lambda}$ is then a model of
$(T^{\prime}_{\lambda},\lambda)$ in $(T,r)$ for each $\lambda\geq\eta$.

If $\eta$ is infinite and
$|\\{i\in\zeta\mid\mu_{S}[(S_{i},i)]\subseteq(T_{v},v)\mid(T_{v},v)\in
f^{-1}_{T}(\beta)\\}|=\zeta$ then we are in a scenario just like the one
treated in Case 1. Otherwise,
$|\\{i\in\zeta\mid\mu_{S}[(S_{i},i)]\subseteq(T_{v},v)\mid(T_{v},v)\not\in
f^{-1}_{T}(\beta)\\}|=\zeta$ and it is simple to see why
$(T^{\prime}_{\lambda},\lambda)\equiv^{*}(T,r)$ for all
$\lambda\leq\zeta>\eta\geq\aleph_{0}$. This then finishes the proof for Case 2
and the Theorem itself. ∎

###  3.2. Unrooted

We are now ready to establish Theorem 1 for all unrooted trees. We do this by
proving Theorem 3.4 as an adaptation of a fixed-point theorem of Polat and
Sabidussi from [11]. To that end we make use of Lemma 1.1 from [11].

######  Lemma 3.3.

[Lemma 1.1, [11]] Let $G$ be a rayless graph,
$(A_{\alpha})_{\alpha\in\text{Ord}}$ a decreasing sequence of subsets of
$V(G)$ so that

  1. (1)

$A_{\alpha}=\bigcup_{\beta<\alpha}A_{\beta}$ and

  2. (2)

each $A_{\alpha}$ induces a connected subgraph of $G$.

By $\kappa$ denote the smallest ordinal for which the sequence
$(A_{\alpha})_{\alpha\in\text{Ord}}$ becomes constant. If
$A_{\kappa}=\emptyset$, then $\kappa$ is a successor ordinal.

Let $\mu\in GM(T)$ \- the collection of all models of $T$ in $T$ \- and let
$v\in inf(T)$. By Lemma 3.1, since $T$ is small then $\mu(v)$ has a vertex of
infinite degree. It follows that
$\mu\upharpoonright\overline{inf(T)}:\overline{inf(T)}\to
C\left(\overline{inf(T)}\right)$ is a model of $\overline{inf(T)}$ in
$\overline{inf(T)}$. It is then relatively straight forward to apply the above
lemma to obtain the following fixed-element result.

######  Theorem 3.4.

Given a small tree $T$ there exists a vertex or edge of $T$ fixed by all
$\mu\in\text{GE}(T)$.

###### Proof.

Let $T$ be small and infinite, and set $T_{1}=\overline{\text{inf}(T)}$:
$T_{1}$ is connected, rayless, properly contained in $T$ \- since $T_{1}$ does
not contain any end-points of $T$ \- and for any $\mu\in\text{GE}(T)$ we have
$\mu\upharpoonright T_{1}$ is a model of $T_{1}$ in $T_{1}$. For all
$\alpha\in\text{Ord}$ construct $T_{\alpha}$ as

$T_{\alpha}=\begin{cases}\overline{\text{inf}(T_{\beta})}&\text{ if
$\alpha=\beta+1$ and}\\\ \bigcap_{\beta<\alpha}T_{\beta},&\text{
otherwise.}\end{cases}$

Let $\kappa$ denote the smallest ordinal for which $(T_{\alpha})$ is constant.
This happens when $T_{\kappa}=\emptyset$ which means, by Lemma 3.3, that
$\kappa=\lambda+1$ for some $\lambda\in\text{Ord}$, and $T_{\lambda}$ is
locally finite and rayless. By design, for any $\mu\in GM(T)$,
$\mu\upharpoonright T_{\lambda}$ is a model of $T_{\lambda}$ in $T_{\lambda}$
and, by Theorem 3.1, it is equivalent to an isomorphism of $T_{\lambda}$. It
follows by [6] that some vertex or edge of $T_{\lambda}$ is fixed by all such
$\mu$. ∎

We should remark that given a model $\mu$ of $T$ in $T$, a fixed-vertex $v\in
V(T)$ of $\mu$ is not a “true” fixed-element of $\mu$ since $\mu(v)=\\{v\\}$.
The same holds for the fixed edge. This is merely a technicality and not
relevant to our objectives. Having established Theorem 3.4, the arguments
employed for Theorem 1 in [3] apply directly to the following theorem and,
thus, we omit the proof.

######  Theorem 3.5.

For any small tree $T$, TAC($\equiv^{*}$) holds. In turn, the Main Theorem
holds for all small unrooted trees.

##  4\. Conclusion

One way to naturally generalise this line of enquiry is to extend
TAC($\hat{R}$) is by asking about the size of an equivalence class of a tree
up to a stronger relation. Given
$\hat{R}\in\\{\equiv,\equiv^{\sharp},\equiv^{*}\\}$ and
$\hat{S}\in\\{\cong,\equiv,\equiv^{\sharp}\\}$ we write
$[T]_{\hat{R}}/\hat{S}$ for the equivalence class of $T$ with respect to
$\hat{R}$ up to $\hat{S}$.

TAC($\hat{R},\hat{S}$). For any tree $T$ and any $r\in V(T)$,

$|[T]_{\hat{R}}/\hat{S}|\not\in\mathbb{N}\smallsetminus\\{1\\}.$

RootedTAC($\hat{R},\hat{S}$). For any tree $T$ and any $r\in V(T)$,

$|[(T,r)]_{\hat{R}}/\hat{S}|\not\in\mathbb{N}\smallsetminus\\{1\\}.$

Combining all of the results developed thus far we obtain that
TAC($\equiv,\cong$) is false, RootedTAC($\hat{R},\cong$) is true for any
$\hat{R}\in\\{\equiv,\equiv^{\sharp},\equiv^{*}\\}$, and TAC($\hat{R},\cong$)
is true for any $\hat{R}\in\\{\equiv^{\sharp},\equiv^{*}\\}$. We pose the
following conjectures.

Conjecture 1. TAC($\hat{R},\hat{S}$) holds for any
$\hat{R}\in\\{\equiv^{\sharp},\equiv^{*}\\}$ and
$\hat{S}\in\\{\cong,\equiv,\equiv^{\sharp}\\}$.

Conjecture 2. For any $\hat{R}\in\\{\equiv,\equiv^{\sharp},\equiv^{*}\\}$ and
$\hat{S}\in\\{\cong,\equiv,\equiv^{\sharp}\\}$

  1. (1)

RootedTAC($\hat{R},\hat{S}$) holds.

  2. (2)

Restricted to small trees, TAC($\hat{R},\hat{S}$) also holds.

We deal with these and other affine topics in [2].

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# Ground state heavy tetraquark production in heavy quark fragmentation

S. Mohammad Moosavi Nejada,b<EMAIL_ADDRESS>Nahid Amiria (a)Faculty of
Physics, Yazd University, P.O. Box 89195-741, Yazd, Iran  (b)School of
Particles and Accelerators, Institute for Research in Fundamental Sciences
(IPM), P.O.Box 19395-5531, Tehran, Iran

(August 27, 2024)

###### Abstract

During recent years, the study of exotic hadrons including tetraquarks and
pentaquarks has attracted a lot of interests and more studies are in progress
experimentally and theoretically. It is well-known that at sufficiently large
transverse momentum the dominant production mechanism for standard heavy
hadrons (mesons/baryons) is actually the fragmentation so that the same
mechanism is also proposed for the production of heavy exotic hadrons. This
work is the first attempt to study the direct fragmentation of a heavy quark
into ground state heavy tetraquarks in leading order of perturbative QCD. In
this regard, we will present an analytical expression for the fragmentation
production of neutral hidden flavor tetraquarks $(Qq\bar{Q}\bar{q})$ which
includes most of the kinematical and dynamical properties of the process.

##  I Introduction

Since the early days of the quark model it has been assumed the possibility of
the existence of exotic multiquark hadrons with the content of the valence
quarks/antiquarks different from those in standard mesons and baryons. In this
context, the exotic hadrons such as tetraquarks (composed of two quarks and
two antiquarks) and pentaquarks (including four quarks and one antiquark) had
been anticipated by Murray Gell-Mann and George Zweig in their 1964 papers
GellMann:1964nj . The only necessary condition for these new multiquark states
is to be color singlets. However, the absence of convincing experimental
evidences for such exotic structures made their investigation of marginal
interest for several decades. Nevertheless, the situation dramatically changed
in the last two decades when the first explicit experimental evidence of the
existence of these exotic hadrons became available, for recent reviews see
Refs. Liu ; Brambilla ; Yang  and references therein. Within the past decade
several exotic hadrons, the majority pertaining to the class of heavy
tetraquarks with at least two heavy quarks, were clearly confirmed by the
BELLE, BESIII and LHCb experimental collaborations. The first experimental
evidence for a heavy tetraquark was the $X(3872)$ state Belle:2003nnu
observed in 2003 by the Belle collaboration in exclusive $B^{\pm}\to
K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays. The decay of this state as well as the
state $Z_{c}(3900)^{\pm}$ into the $J/\psi$ meson (observed in BESIII)
confirmed that these exotic particles are formed of two quarks and two
antiquarks. Very recently, a narrow structure around 6.9 GeV, named as
$X(6900)$, is observed by the CERN LHCb in the process $pp\to J/\psi J/\psi X$
Aaij . This is the first candidate of fully-heavy tetraquarks observed in
experiment which is proposed to be either the first radial (2S) excitation or
the second orbital (1D) excitation of the hidden-charm tetraquark
$cc\bar{c}\bar{c}$.
Nowadays, convincing candidates for both the exotic tetraquark
($qq\bar{q}\bar{q}$) and pentaquark ($qqqq\bar{q}$) states are found. In the
literature, there is no definite consensus about the composition of these
states and different interpretations for the tetraquark candidates are
proposed, for example: molecules composed from two mesons loosely bound by the
meson exchange, compact tetraquarks composed from a diquark and antidiquark
bound by strong forces, hadroquarkonia composed of a heavy quarkonium embedded
in a light meson, kinematic cusps, etc. The distinction between various models
is a very complicated experimental task. Among exotic hadrons (tetraquarks,
pentaquarks, hexaquarks, etc), heavy tetraquarks are of particular interest
since the presence of a heavy quark increases the binding energy of the bound
system and, in conclusion, the possibility that such tetraquarks will have
masses below the thresholds for decays to mesons with open heavy flavor. In
this case the strong decays, which proceed through the quark and antiquark
rearrangements, are kinematically forbidden and the corresponding tetraquarks
can decay only weakly or electromagnetically and thus they should have a tiny
decay width.

When the strong interactions are concerned, the investigation of production
mechanism of heavy standard/exotic hadrons is a powerful tool to understand
the dynamics of strong interactions. Basically, two different mechanisms are
assumed for the production of heavy hadrons: recombination and fragmentation
Martynenko:1995hg . In the second mechanism, the fragmentation refers to the
process of a parton which carries large transverse momentum and subsequently
forms a jet containing the expected hadron Braaten:1993rw . At sufficiently
large transverse momentum of heavy hadron production, the direct leading order
production scheme (recombination mechanism) is normally suppressed while the
fragmentation mechanism becomes dominant, though it is formally of higher
order in the strong coupling constant $\alpha_{s}$ Braaten:1993rw ;
Kramer:2001hh . In Refs. MoosaviNejad:2020vqc ; MoosaviNejad:2016qdx ;
MoosaviNejad:2020nsl ; Nejad:2013vsa ; MoosaviNejad:2014ntq ;
Salajegheh:2019ach ; MoosaviNejad:2016scq ; Salajegheh:2019nea , we calculated
the perturbative QCD fragmentation function (FF) for a heavy quark to fragment
into the $S$-wave heavy mesons and baryons at leading-order (LO) and next-to-
leading order (NLO). In Ref. Cheung:2004ji , it has been pointed out that the
dominant production mechanism for pentaquarks consisting of a heavy quark is
heavy quark fragmentation, similar to heavy-light mesons and baryons
consisting of a heavy quark. So, this mechanism is also adopted in our work
for the production of heavy tetraquark states and we, for the first time,
study the production mechanism of heavy tetraquarks in the fragmentation of a
heavy quark at lowest order of perturbative QCD. In our calculation,
tetraquarks are assumed as four-quark $Qq\bar{Q}\bar{q}$ states which are
tightly bound by the color forces. Our analytical results are presented for
the fragmentation function of heavy quark $Q$ into neutral hidden flavor
tetraquarks $Qq\bar{Q}\bar{q}$ with $Q=c,b$ and $q=u,d,s$. Having these FFs it
would be possible to evaluate the production rate of these heavy states in
$e^{+}e^{-}$ annihilation and hadron colliders. In fact, the specific
importance of FFs is for their model independent predictions of the cross
sections at colliders. However, due to a large number of decay modes the
results from $e^{+}e^{-}$ annihilation seem to be small but sizable rates are
expected in energetic hadron colliders where a large number of them is
produced. To assess the prospects for the experimental study of heavy
tetraquarks in present and future colliders, it is important to know the
accurate predictions for their production rates and, in conclusion, their FFs.

This paper is organized as follows. In Sec. II, we present the theoretical
formalism for the heavy tetraquark FF in the Suzuki model and present our
numerical results, and our discussions and conclusions are given in the last
section.

##  II Heavy tetraquark fragmentation

A motivation for studying the production of multiquark structures is for
better understanding of the dynamics of strong interactions and the
confinement mechanism. It is well known that the dominant mechanism to produce
standard heavy hadrons at sufficiently large transverse momentum is
fragmentation so this mechanism is valid for the production of heavy
tetraquarks too. Fragmentation refers to the production of a parton with a
large transverse momentum which subsequently decays to form a jet containing
an expected hadron. It is hence important to obtain the corresponding
fragmentation function in order to properly estimate the production rate of a
specific hadronic state. According to the QCD factorization theorem, the cross
section for the production of hadron $H$ in the typical scattering process
$A+B\rightarrow H+X$, can be expressed as

$\displaystyle
d\sigma=\sum_{a,b,c}\int_{0}^{1}dx_{a}\int_{0}^{1}dx_{b}\int_{0}^{1}dzf_{a/A}(x_{a},Q)f_{b/B}(x_{b},Q)d\hat{\sigma}(a+b\rightarrow
c+X)D_{c}^{H}(z,Q),$ (1)

where $\it{a}$ and $\it{b}$ are incident partons in the colliding initial
hadrons $A$ and $B$, respectively, $f_{a/A}$ and $f_{b/B}$ are the
nonperturbative parton distribution function (PDFs) at the scale $Q^{2}$ of
the partonic subprocess $a+b\rightarrow c+X$, $\it{c}$ is the fragmenting
parton and $X$ stands for any unobserved jet. Here, $D_{c}^{H}(z,Q)$ is the FF
at the scale $Q$ which can be obtained by evolving from the initial FF
$D_{c}^{H}(z,\mu_{0})$ using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) renormalization group equations dglap1 . In electron-positron
annihilation, one does not need to deal with the nonperturbative PDFs as in
hadron collisions. This process has in general less contributions by
background processes compared to hadron collisions and the uncertainties
introduced by parton density functions are not appeared. The process of
inclusive heavy hadron production in electron-positron annihilation, i.e.,
$e^{+}e^{-}\to(\gamma,Z)\to H+X$, has the cross section as Delpasand:2020vlb ;
Soleymaninia:2019yqz ; Soleymaninia:2017xhc

$\displaystyle\frac{d\sigma}{dx}(x,s)=\sum_{a}\int_{x}^{1}\frac{dy}{y}\frac{d\sigma}{dy}(y,\mu_{R},\mu_{F})D_{a}^{H}(\frac{x}{y},\mu_{F}),$
(2)

where, $a$ stands for the partons $a=g,u,\bar{u},\cdots,b,\bar{b}$ and
$\mu_{F}$ and $\mu_{R}$ are the factorization and renormalization scales,
respectively. Generally, one can choose two different values for these scales,
but a common choice consists of setting $\mu_{R}=\mu_{F}=\sqrt{s}$. Here,
$x=2E_{H}/\sqrt{s}$ measures the energy of hadron in units of the beam energy
in the center-of-mass frame. At lowest-order of perturbative QCD, the cross
section of relevant partonic subprocesses is given by Baier:1979sp

$\displaystyle\frac{d\sigma}{dy}(e^{+}e^{-}\to
q\bar{q})=N_{c}\sigma_{0}(V_{q_{i}}^{2}+A_{q_{i}}^{2})\delta(1-y).$ (3)

Here, $N_{c}=3$ is the number of quark colors, $V_{q_{i}}$ and $A_{q_{i}}$ are
the effective vector and axial-vector couplings of quark $q_{i}$ to the photon
and $Z$ boson. For small energies ($\sqrt{s}\ll M_{Z}$), for the summation of
squared effective electroweak charges we have
$V_{q_{i}}^{2}+A_{q_{i}}^{2}=e_{q_{i}}^{2}$ where $e_{q_{i}}$ is the electric
charge of the quark $q_{i}$ Kneesch:2007ey . In Eq. (3),
$\sigma_{0}=4\pi\alpha^{2}/(3s)$ is the leading order total cross section of
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ for the massless muons, in which $\alpha$ is the
Sommerfeld fine-structure constant. Therefore, for the cross section of
$e^{+}e^{-}\to H+X$ at LO, on has

$\displaystyle\frac{d\sigma^{LO}}{dx}(x,s)=N_{c}\sigma_{0}\sum_{q}e_{q}^{2}[D_{q}^{H}(x,\mu)+D_{\bar{q}}^{H}(x,\mu)]=\frac{8\pi\alpha^{2}}{s}\sum_{q}e_{q}^{2}D_{q}^{H}(x,\mu),$
(4)

where we assumed $D_{q}^{H}=D_{\bar{q}}^{H}$ for $q=u,d,s,c,b$.
Conventionally, the fragmentation mechanism is described by the function
$D_{i}^{H}(z,\mu)$ which refers to the probability for a parton $i$ at the
fragmentation scale $\mu$ to fragment into a hadron $H$ carrying away a
fraction $z$ of its momentum. The FFs are related to the low-energy part of
the hadroproduction processes and forms the nonperturbative aspect of QCD but,
fortunately, it was found that these functions for heavy hadron productions
are analytically calculable by virtue of perturbative QCD (pQCD) with limited
phenomenological parameters. Historically, the first theoretical effort to
describe the procedure of heavy hadron production was made by Bjorken
Bjorken:1977md  by using a naive quark-parton model and in the following the
pQCD scheme was applied by Suzuki Suzuki:1977km , Amiri and Ji Amiri:1986zv .
Among them, an elaborate model of fragmentation which does contain spin
information has been proposed by Suzuki Suzuki:1985up . This model includes
most of the kinematical and dynamical properties of the splitting process and
gives a detailed insight about the fragmentation process. In this approach,
Suzuki calculates the fragmentation function of a heavy quark using the
convenient Feynman diagrams for the parton level of the process and also by
considering the wave function of heavy bound state which contains the
nonperturbative dynamic of hadroproduction process. In fact, the Suzuki model
mixes a perturbative picture with nonperturbative dynamics of fragmentation
and not only predicts the $z$-dependence of the FFs, but also their dependence
on the transverse momentum of the hadron relative to the jet.

Figure 1: The lowest order Feynman diagram contributing to the fragmentation
of a heavy quark Q into a heavy tetraquark $X(Qq\bar{Q}\bar{q})$. The four
momenta are also labeled.

Here, using the Suzuki model we focus on the heavy tetraquark FF for which the
only Feynman diagram at leading order in $\alpha_{s}$ is shown in Fig. 1. Any
other possible Feynman diagram is related to higher orders in $\alpha_{s}$.
Considering this diagram we first set the relevant four-momenta as

$\displaystyle p_{\mu}=[p_{0},\boldsymbol{p}_{T},p_{L}]\quad,$
$\displaystyle\quad s_{\mu}=[s_{0},\boldsymbol{0},s_{L}]$ $\displaystyle
s_{\mu}^{\prime}=[s_{0}^{\prime},\boldsymbol{p}_{T},s_{L}^{\prime}]\quad,$
$\displaystyle\quad t_{\mu}=[t_{0},\boldsymbol{0},t_{L}]$ (5) $\displaystyle
t_{\mu}^{\prime}=[t_{0}^{\prime},\boldsymbol{0},t_{L}^{\prime}]\quad,$
$\displaystyle\quad r_{\mu}=[r_{0},\boldsymbol{0},r_{L}]$

where we assumed that the produced tetraquark $X$ moves along the
$\hat{z}$-axes (fragmentation axes). The four-momentum of tetraquark is also
written as $\bar{P}_{\mu}=[\bar{P}_{0},\boldsymbol{0},\bar{P}_{L}]$ so that
$\bar{P}_{L}=s_{L}+r_{L}+t_{L}+t^{\prime}_{L}$. In the Suzuki model the FF for
the production of unpolarized ground state tetraquark is defined as
Suzuki:1985up

$\displaystyle D_{Q}^{X}(z,\mu_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{1+2s_{Q}}\sum_{spin}\sum_{color}\int|T_{X}|^{2}\delta^{3}(\boldsymbol{\bar{P}}+\boldsymbol{s^{\prime}}-\boldsymbol{p})d^{3}\boldsymbol{\bar{P}}d^{3}\boldsymbol{s^{\prime}},$
(6)

where $\mu_{0}$ is the fragmentation scale, $s_{Q}$ refers to the spin of the
fragmenting quark and the summation is going over the spins and colors. Four
momenta are as labeled in Fig. 1. In computing Eq. (6), following Ref.
Suzuki:1985up , we adopt the infinite momentum frame where the fragmentation
variable in the usual light-cone form, i.e.
$z=(\bar{P}_{0}+\bar{P}_{L})/(p_{0}+p_{L})$, is reduced to a more popular one
as $z=\bar{P}_{0}/p_{0}=E_{X}/E_{Q}$. From experimental point of view, this
definition is more suitable since refers to the energy fraction of the
fragmenting heavy quark which is taken away by the produced tetraquark. This
scaling variable ranges as $0\leq z\leq 1$.
In Eq. (6), $T_{X}$ is the probability amplitude of the tetraquark production
which, at the large momentum transfer, is expressed in terms of the hard
scattering amplitude $T_{H}$ and the process-independent distribution
amplitude $\Phi_{B}$ as Brodsky:1985cr ; Amiri:1985mm

$\displaystyle
T_{X}(\bar{P},s^{\prime})=\int[dx]T_{H}(\bar{P},s^{\prime},x_{i})\Phi_{B}(x_{i},Q^{2}),$
(7)

where $[dx]=\Pi_{i=1}^{4}dx_{i}\delta(1-\sum_{i=1}^{4}x_{i})$. Here, $x_{i}$’s
stand for the tetraquark energy fractions carried by its constituent quarks
which are defined as

$\displaystyle x_{1}=\frac{s_{0}}{\bar{P}_{0}}\quad,\quad
x_{2}=\frac{r_{0}}{\bar{P}_{0}}\quad,\quad
x_{3}=\frac{t_{0}^{\prime}}{\bar{P}_{0}}\quad,\quad
x_{4}=\frac{t_{0}}{\bar{P}_{0}}\cdot$ (8)

The energy conservation law requires that $\sum_{i=1}^{4}x_{i}=1$.

The scheme applied to describe the probability amplitude $T_{X}$ as the
convolution in Eq. (7) is convenient to absorb the soft behavior of the bound
state (tetraquark in our example) into the hard scattering amplitude $T_{H}$
Brodsky:1985cr . The short-distance part of the amplitude, i.e. $T_{H}$, can
be computed perturbatively from quark-gluon subprocesses at leading-order or
higher-order QCD approximations. The long-distance distribution amplitude
$\Phi_{B}$ which contains the bound state nonperturbative dynamic of the
outgoing tetraquark, is the probability amplitude for a $Qq\bar{Q}\bar{q}$-set
to evolve into a particular bound state. The underlying link between hadronic
phenomena in QCD at long- and short-distances is the hadronic wave function.
In fact, the nonperturbative aspect of the hadroproduction processes is
contained in the bound state of the hadron which is described by the wave
function $\Phi_{B}$. Following Ref. Suzuki:1985up  and according to the
Lepage-Brodsky’s approach Lepage:1980fj  we neglect the relative motion of the
constituent quarks inside the tetraquark therefore we assume, for simplicity,
that the constituent quarks are emitted collinearly with each other and move
along the $\hat{z}$-axes. In fact, in the Suzuki model a multiquark state is
replaced by collinear constituents with neglecting their Fermi motion and the
nonperturbative aspect of the hadroproduction is included in the wave function
of the heavy tetraquark state. By this simplicity, the relative motion of the
constituent quarks is effectively nonrelativistic and this allows one to
estimate the nonrelativistic wave function of heavy tetraquark as a delta
function form, see also Nejad:2013vsa . Therefore, the distribution amplitude
for a S-wave heavy tetraquark with neglecting the Fermi motion reads
GomshiNobary:1994eq

$\displaystyle\Phi_{B}\approx f_{B}\delta(x_{i}-\frac{m_{i}}{m_{X}}).$ (9)

where $m_{X}$ is the tetraquark mass and $f_{B}$ refers to the decay constant
of hadron. In Ref. MoosaviNejad:2018ukp , we studied the effect of meson wave
function on its FF by considering a typical mesonic wave function which is
different of the delta function and is the nonrelativistic limit of the
solution of Bethe-Salpeter equation with the QCD kernel Brodsky:1985cr .
There, we showed that the Fermi motion effect can be neglected with a
reasonable approximation.
With the delta function approximation for the tetraquark wave function (9), we
are assuming that the contribution of each constituent from the tetraquark
energy is proportional to its mass, i.e. $x_{i}=m_{i}/m_{X}$ where
$m_{X}=2(m_{q}+m_{Q})$. Considering the definitions of fragmentation variable
($z=\bar{P}_{0}/p_{0}$) and the fractions $x_{i}$’s presented in Eq. (8) one
also may write the parton energies in terms of the initial heavy quark energy
($p_{0}$) as: $s_{0}=x_{1}zp_{0}$, $r_{0}=x_{2}zp_{0}$,
$t_{0}^{\prime}=x_{3}zp_{0}$, $t_{0}=x_{4}zp_{0}$ and
$s_{0}^{\prime}=(1-z)p_{0}$ where $x_{1}=x_{4}=m_{Q}/m_{X}$ and
$x_{2}=x_{3}=m_{q}/m_{X}$.

In Eq. (7), the amplitude $T_{H}$ is, in essence, the partonic cross section
to produce a set of heavy quarks with certain quantum numbers that in the QCD
perturbation theory considering the Feynman rules is written as

$\displaystyle
T_{H}=\frac{16\pi^{2}\alpha_{s}(2m_{Q})\alpha_{s}(2m_{q})m_{X}m_{Q}^{2}C_{F}}{2\sqrt{2\bar{P}_{0}s_{0}^{\prime}p_{0}}}\frac{\Gamma}{D_{0}},$
(10)

where, $D_{0}=\bar{P}_{0}+s_{0}^{\prime}-p_{0}$ is the energy denominator,
$\alpha_{s}(\mu)$ is the strong coupling constant at the renormalization scale
$\mu$ and $C_{F}$ is the color factor which is calculated using color line
counting rule.
In Eq. (10), $\Gamma$ represents an appropriate combination of the propagators
and the spinorial parts of the amplitude. Considering Fig. 1 for the
transition $Q\to X_{Qq\bar{Q}\bar{q}}$ the transition amplitude at leading
order is written as

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
G\bar{u}(t)\gamma^{\mu}u(p)\bigg{[}\bar{u}(t^{\prime})\gamma_{\mu}(\displaystyle{\not}q_{2}+m_{q})\gamma_{\nu}v(r)\bigg{]}v(s)\gamma^{\nu}\bar{u}(s^{\prime})$
(11)

where $G$ is related to the denominator of propagators as

$\displaystyle
G=\frac{1}{q_{1}^{2}(q_{2}^{2}-m_{q}^{2})q_{3}^{2}}=\frac{1}{8(m_{Q}^{2}-p\cdot
t)^{2}(m_{Q}^{2}+s\cdot s^{\prime})^{2}(m_{Q}^{2}+r\cdot s+r\cdot
s^{\prime}+s\cdot s^{\prime})^{2}}.$ (12)

In squaring the amplitude $\Gamma$, one needs the following dot products of
the relevant four-momenta:

$\displaystyle r\cdot t^{\prime}=m_{q}^{2}\quad,\quad s\cdot t=m_{Q}^{2}$
$\displaystyle r\cdot s=t\cdot t^{\prime}=s\cdot t^{\prime}=m_{q}m_{Q},$
$\displaystyle p\cdot s=p\cdot
t=\frac{m_{X}m_{Q}}{2z}+\frac{zm_{Q}}{2m_{X}}(m_{Q}^{2}+p_{T}^{2}),$
$\displaystyle p\cdot
s^{\prime}=\frac{1}{2}(m_{Q}^{2}+p_{T}^{2})(1-z+\frac{1}{1-z})-p_{T}^{2},$
(13) $\displaystyle r\cdot s^{\prime}=s^{\prime}\cdot
t^{\prime}=\frac{zm_{q}}{2m_{X}(1-z)}(m_{Q}^{2}+p_{T}^{2})+\frac{m_{X}m_{q}(1-z)}{2z},$
$\displaystyle s\cdot s^{\prime}=t\cdot
s^{\prime}=\frac{m_{Q}m_{X}(1-z)}{2z}+\frac{zm_{Q}}{2m_{X}(1-z)}(m_{Q}^{2}+p_{T}^{2}),$
$\displaystyle p\cdot t^{\prime}=p\cdot
r=\frac{m_{q}m_{X}}{2z}+\frac{zm_{q}}{2m_{X}}(m_{Q}^{2}+p_{T}^{2}).$

To obtain the FF for an unpolarized S-wave heavy tetraquark
$X_{Qq\bar{Q}\bar{q}}$, considering Eqs. (6-10) one has

$\displaystyle
D_{c}^{X}(z,\mu_{0})=(4\pi^{2}f_{B}\alpha_{s}(2m_{q})\alpha_{s}(2m_{Q})m_{X}m_{Q}^{2}C_{F})^{2}\int\frac{d^{3}\boldsymbol{\bar{P}}d^{3}\boldsymbol{s^{\prime}}\sum_{spin}|\Gamma|^{2}\delta^{3}(\boldsymbol{\bar{P}}+\boldsymbol{s^{\prime}}-\boldsymbol{p})}{\bar{P}_{0}p_{0}s_{0}^{\prime}(\bar{P}_{0}+s_{0}^{\prime}-p_{0})^{2}}.$
(14)

To perform the phase space integrations we start with the following integral

$\displaystyle\int\frac{d^{3}\boldsymbol{\bar{P}}\delta^{3}(\boldsymbol{\bar{P}}+\boldsymbol{s^{\prime}}-\boldsymbol{p})}{\bar{P}_{0}p_{0}(\bar{P}_{0}+s_{0}^{\prime}-p_{0})^{2}}=\int\frac{\bar{P}_{0}d^{3}\boldsymbol{\bar{P}}\delta^{3}(\boldsymbol{\bar{P}}+\boldsymbol{s^{\prime}}-\boldsymbol{p})}{p_{0}(\bar{P}_{0}^{2}-(p_{0}-s_{0}^{\prime})^{2})^{2}}=\frac{zp_{0}}{p_{0}(m_{X}^{2}-(p-s^{\prime})^{2})^{2}}=\frac{z}{(m_{X}^{2}-2m_{Q}^{2}+2p\cdot
s^{\prime})^{2}}.$ (15)

According to the Suzuki approach, in Eq. (14) instead of integrating over the
transverse momentum of outgoing heavy quark Q, we simply replace the
integration variable by its average value, i.e.

$\displaystyle\int\frac{d^{3}\boldsymbol{s^{\prime}}}{s_{0}^{\prime}}g(z,s^{\prime}_{T})=\int
g(z,s^{\prime}_{T})\frac{ds^{\prime}_{L}d^{2}s^{\prime}_{T}}{s_{0}^{\prime}}=m_{Q}^{2}g(z,\left\langle
p_{T}^{2}\right\rangle),$ (16)

where the average value $\langle p_{T}^{2}\rangle$ is a free parameter which
can be specified phenomenologically.

Putting all in Eq. (14), the FF of heavy tetraquark $X$ reads

$\displaystyle D_{Q}^{X}(z,\mu_{0})$ $\displaystyle=$ $\displaystyle
N\frac{z\times\Sigma_{spin}\Gamma\bar{\Gamma}}{(m_{X}^{2}-2m_{Q}^{2}+2p\cdot
s^{\prime})^{2}}=N\frac{z\times\Sigma_{spin}\Gamma\bar{\Gamma}}{[m_{X}^{2}-(m_{Q}^{2}+\left\langle
p_{T}^{2}\right\rangle)(1+z-\frac{1}{1-z})]^{2}},$ (17)

where
$N=(16\pi^{2}f_{B}\alpha_{s}(2m_{q})\alpha_{s}(2m_{Q})m_{q}m_{Q}^{4}C_{F})^{2}$
but it is related to the normalization condition: $\int dzD(z,\mu_{0})=1$
Suzuki:1985up ; Amiri:1986zv . Using the traditional trace technique the
squared amplitude ($\sum|\Gamma|^{2}$) reads

$\displaystyle\Sigma_{spin}\Gamma\bar{\Gamma}=\frac{G^{2}}{z^{3}(1-z)}\bigg{(}z^{6}\left\langle
p_{T}^{2}\right\rangle^{3}+\alpha z^{4}\left\langle
p_{T}^{2}\right\rangle^{2}+\beta z^{2}\left\langle
p_{T}^{2}\right\rangle-\gamma\bigg{)},$ (18)

where

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
m_{Q}^{2}(11z^{2}-18z+12)+6m_{q}^{2}(2z^{2}-z+2)+4m_{q}m_{Q}(3z^{2}-4z+6),$
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle
m_{Q}^{4}(19z^{4}-84z^{3}+168z^{2}-144z+48)+4m_{Q}^{2}m_{q}^{2}(6z^{4}-37z^{3}+102z^{2}-112z+72)$
$\displaystyle-8m_{q}^{4}(z^{3}-6z^{2}+6z-6)-16m_{Q}m_{q}^{3}(3z^{3}-12z^{2}+14z-12)+8m_{q}m_{Q}^{3}(3z^{4}-22z^{3}+54z^{2}-52z+24),$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-64m_{Q}m_{q}^{5}(z^{3}+3z^{2}-10z+6)+8m_{q}^{4}m_{Q}^{2}(z^{5}-6z^{4}+42z^{3}-158z^{2}+236z-120)+16m_{q}^{3}m_{Q}^{3}(3z^{5}-$
$\displaystyle
24z^{4}+98z^{3}-196z^{2}+192z-80)-2m_{q}^{2}m_{Q}^{4}(6z^{6}-71z^{5}+422z^{4}-1232z^{3}+1872z^{2}-1424z+480)-$
$\displaystyle
4m_{q}m_{Q}^{5}(3z^{6}-40z^{5}+182z^{4}-424z^{3}+544z^{2}-352z+96)-32m_{q}^{6}(z^{3}-3z+2)-m_{Q}^{6}(z^{2}-2z+4)(3z^{2}-8z+4)^{2}.$

In the Suzuki model, the fragmentation function depends on both the
fragmentation variable $z$ and the parameter $\left\langle
p_{T}^{2}\right\rangle$ (the transverse momentum of produced hadron relative
to the parent jet). In Ref. GomshiNobary:1994eq , it is shown that the choice
of $\left\langle p_{T}^{2}\right\rangle=1$ GeV2 is an extreme value for this
quantity and any higher value will produce the peak even at lower-z regions.
In this work we adopt this value for the transverse momentum of heavy
tetraquark.
The $z$-dependence of FFs is not yet calculable at each scale, however once
they are computed at the initial scale $\mu_{0}$ their $\mu$ evolution can be
specified by the DGLAP evolution equations dglap1 . Here, we set the initial
scale of fragmentation to $\mu_{0}=m_{X}+m_{Q}$ which is the minimum value of
the invariant mass of the fragmenting parton. Then, the FF presented in Eq.
(17) should be regarded as a model for the heavy quark fragmentation $Q$ into
the heavy tetraquark $X(Qq\bar{Q}\bar{q})$ at the initial scale $\mu_{0}$. To
present our numerical analysis, we adopt the following input parameters
Patrignani:2016xqp :

$\displaystyle m_{c}=1.67\leavevmode\nobreak\ \textrm{GeV},\quad
m_{b}=4.78\leavevmode\nobreak\ \textrm{GeV},\quad
f_{B}=0.25\leavevmode\nobreak\
\textrm{GeV},\quad\alpha_{s}(2m_{c})=0.26,\quad\alpha_{s}(2m_{b})=0.18.$ (20)

Note that, in Ref. Patrignani:2016xqp  two values are reported for the quark
masses. The above masses show the pole masses, i.e. the renormalized quark
masses in the on-shell renormalization scheme, which correspond to the values
$m_{c}=1.27\pm 0.02$ GeV and $m_{b}=4.18\pm 0.03$ GeV for the $\overline{MS}$
masses (the renormalized quark mass in the modified minimal subtraction scheme
tHooft:1973mfk  which is intimately related to the use of dimensional
regularization tHooft:1972tcz ). The relation between the pole quark mass and
the $\overline{MS}$ mass is known to three loops in QCD perturbation theory,
details are given in Ref. Melnikov:2000qh . In our analysis, to show the
effect of variation of heavy quark masses on their corresponding FFs we will
consider the charm and bottom quark masses as $1.27\leq m_{c}\leq 1.67$ GeV
and $4.18\leq m_{b}\leq 4.78$ GeV.

Figure 2: Fragmentation of $c$-quark into heavy tetraquark
$X_{cu\bar{c}\bar{u}}$ at lowest-order of perturbative QCD. The uncertainty
band due to the variation of charm mass is also shown. Figure 3: The $b\to
X_{bs\bar{b}\bar{s}}$ FF at leading order perturbative QCD along with the
uncertainty band of bottom mass.

In Fig. 2, our phenomenological prediction for the FFs of charm into the
neutral hidden-charm state $X(cu\bar{c}\bar{u})$ is shown at the starting
scale $\mu_{0}=m_{c}+m_{X}$. The uncertainty band due to the variation of
charm quark mass is also plotted. Assuming $m_{u}\approx m_{d}$ this result is
also valid for the transitions $c\to
X(cd\bar{c}\bar{d}),X(cu\bar{c}\bar{d}),X(cd\bar{c}\bar{u})$.
In Fig. 3, the b-quark FF into the neutral hidden-bottom tetraquark including
strangeness flavor, i.e. $b\to X(bs\bar{b}\bar{s})$, is shown considering the
mass uncertainty band. Since the b-quark is heavier than the c-quark, as we
expected, the peak of b-quark FF shifted toward higher values of z.

Besides the heavy tetraquark FFs, their first moment is also of
phenomenological interest and subject to experimental determination. It
corresponds to the average fraction of energy that the tetraquark receives
from the fragmenting heavy quark and is defined as Kneesch:2007ey

$\displaystyle\left\langle
z\right\rangle_{Q}(\mu)=\frac{1}{B_{Q}(\mu)}\int_{0}^{1}zD_{Q}^{X}(z,\mu)dz,$
(21)

where, $B_{Q}(\mu)$ stands for the branching fraction which is defined as
$B_{Q}(\mu)=\int_{0}^{1}D_{Q}^{X}(z,\mu)dz$, where $Q=c,b$. For the tetraquark
$X(cu\bar{c}\bar{u})$ the branching fraction and the average energy fraction
are $B_{c}=2\times 10^{-3}$ and $\left\langle
z\right\rangle_{c}(\mu_{0})=0.635$, respectively, and the corresponding ones
for the tetraquark $X(bu\bar{b}\bar{u})$ read $B_{b}=1.1\times 10^{-4}$ and
$\left\langle z\right\rangle_{b}=0.629$.
Considering the inclusive differential cross section given in Eq. (4), one can
compute the inclusive cross section for the production of heavy tetraquark at
LO as

$\displaystyle\sigma^{LO}(e^{+}e^{-}\to
X_{Qq\bar{Q}\bar{q}}+jets)=\frac{8\pi\alpha^{2}}{s}\sum_{q}e_{q}^{2}\int_{0}^{1}dzD_{q}^{X}(z,\mu),$
(22)

By ignoring the contributions which are appeared at higher orders of
perturbation, one has

$\displaystyle\sigma^{LO}(e^{+}e^{-}\to
X_{Qq\bar{Q}\bar{q}}+jets)\approx\frac{8\pi\alpha^{2}}{s}e_{Q}^{2}B_{Q}(s).$
(23)

For example, the contributions of $c\to X_{bu\bar{b}\bar{u}}$, $b\to
X_{cu\bar{c}\bar{u}}$ or $g\to X_{bu\bar{b}\bar{u}}/X_{cu\bar{c}\bar{u}}$ are
related to very complicated Feynman diagrams which are in higher orders in
$\alpha_{s}$, because they do not occur directly. Then, their contributions
are ignorable in comparison with the one occurs directly, i.e., $b\to
X_{bu\bar{b}\bar{u}}$ and $c\to X_{cu\bar{c}\bar{u}}$ FFs (17). Finally,
having the branching fraction $B_{Q}(\mu)$ for the transition $B\to X$ at the
center-of-mass energy $\mu=s$ one can compute the corresponding cross section
at electron-positron annihilation.

##  III Conclusion

Besides the standard mesons and baryons there exist hadrons made of more than
three quarks/antiquarks or valence gluons. These new types of hadrons consist
of exotic states including glueballs, hybrids, tetraquarks, pentaquarks,
hexaquarks, etc. In last two decades a large number of these exotic states,
especially tetraquarks and pentaquarks, have been observed in the particle
factories. However, some of them are now well-established there are also some
doubts on existence of some other members. Therefore, exact determinations of
the nature, structure and quantum numbers of these exotic states need more
experimental efforts. Specifically, there are many exotic states which are
proposed in theory but their existence need to be confirmed by the
experiments.

It should be noted that, most of heavy multiquark particles discovered in the
experiment have hidden-charm or hidden-bottom quark structure, i.e., they
contain $c\bar{c}$ or $b\bar{b}$ in their inner structures. The study of their
production mechanism and using it as a probe to the structure of hadrons are
among the most active research fields in particle and nuclear physics. In
fact, the challenge is to understand the nonperturbative transition from high
energy $e^{+}e^{-}$, photon-hadron, and hadron-hadron collisions to physical
exotic states.
It is well-known that the dominant mechanism to produce heavy hadrons at high
transverse momentum is fragmentation; the production of a high energy parton
followed by its fragmentation into the heavy bound states. In this work, using
the Suzuki model we, for the first time, studied the fragmentation function of
a heavy quark into an unpolarized S-wave heavy tetraquark at leading order
perturbative QCD. To be specific, we analytically computed the FF of bottom
and charm quark into the neutral hidden flavor tetraquarks
($X|Qq\bar{Q}\bar{q}>$) at the initial scale $\mu_{0}=m_{X}+m_{Q}$. Using the
extracted FFs one can compute the production rates of heavy tetraquarks in
hadron colliders.
To have more accurate FFs, one can think of other effects such as the Fermi
motion of constituents, the tetraquark mass effect, NLO radiative corrections,
etc. Although, including the effects such as the Fermi motion of constituents
leads to very complicated integrals which should be solved numerically
MoosaviNejad:2018ukp .

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