#  Extremal digraphs avoiding distinct walks of length 3 with the same
endpoints

Zejun Huang, Zhenhua Lyu College of Mathmatics and Statistics, Shenzhen
University, Shenzhen, 518060, P.R. China<EMAIL_ADDRESS>School of
Science, Shenyang Aerospace University, Shenyang, 110136, P.R. China.
<EMAIL_ADDRESS>
###### Abstract

In this paper, we determine the maximum size of digraphs on $n$ vertices in
which there are no two distinct walks of length $3$ with the same initial
vertex and the same terminal vertex. The digraphs attaining this maximum size
are also characterized. Combining this with previous results, we obtain a full
solution to a problem proposed by X. Zhan in 2007 .

Key words: digraph, Turán problem, walk

2010 Mathematics Subject Classifications: 05C20, 05C35

##  1 Introduction and main results

The Turán-type extremal problem is an important topics in extremal graph
theory, which concerns the maximum number of edges in graphs containing no
given subgraphs and the extremal graphs achieving this maximum. Most of the
previous results on Turán problems concern undirected graphs and only a few
Turán problems on digraphs have been investigated; see [2, 4, 5, 8, 9, 14, 15]
and the references therein.

A simple digraph is a digraph that does not contain multiple arcs but allows
loops. A strict digraph is a loopless simple digraph. Digraphs in this paper
are simple unless otherwise stated. We follow the terminology and notations in
[1]. Directed walks, directed paths and directed cycles are abbreviate as
walks, paths and cycles, respectively. The number of vertices in a digraph is
called its order and the number of arcs its size. Given a digraph $D$ and a
set of digraphs $F$, $D$ is said to be $F$-free if $D$ contains no digraph in
$F$ as its subgraph.

Denote by $\overrightarrow{K}_{n}$ the complete digraph on $n$ vertices. A
natural Turán problem on digraphs is determining the maximum size of
$\overrightarrow{K}_{r}$-free digraphs of a given order, which has been solved
in [12]. Brown and Harary [4] determined the precise extremal size of digraphs
as well as the extremal digraphs that avoids a tournament. They also studied
digraphs avoiding a direct sum of two tournaments, or a digraph on at most 4
vertices where any two vertices are joined by at least one arc. In [9], we
determined the extremal size of digraphs avoiding an orientation of the
4-cycle as well as the extremal digraphs. By adopting dense matrices, Brown,
Erdős, and Simonovits [2, 3] presented asymptotic results on extremal digraphs
avoiding a family of digraphs. Howalla, Dabboucy and Tout [6, 7] determined
the extremal sizes of the $\overrightarrow{K}_{2}$-free digraphs avoiding $k$
paths with the same initial vertex and terminal vertex for $k=2,3$. Maurer,
Rabinovitch and Trotter [14] studied the extremal
$\overrightarrow{K}_{2}$-free transitive digraphs which contain at most one
path from $x$ to $y$ for any two distinct vertices $x,y.$

Denote by $F_{k}$ the family of digraphs consisting of two different walks of
length $k$ with the same initial vertex and the same terminal vertex. Let
$ex(n,F_{k})$ be the maximum size of $F_{k}$-free digraphs of order $n$, and
let $EX(n,F_{k})$ be the set of $F_{k}$-free digraphs of order $n$ with size
$ex(n,F_{k})$. We are interested in the following Turán -type problem.

######  Problem 1.

Given positive integers $n$ and $k$, determine $ex(n,F_{k})$ and
$EX(n,F_{k})$.

Let $D=(V,{A})$ be a digraph with vertex set
${V}=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and arc set ${A}$. Its adjacency matrix
$A_{D}=(a_{ij})$ is defined by

$a_{ij}=\left\\{\begin{array}[]{ll}1,&\textrm{if }(v_{i},v_{j})\in{A};\\\
0,&\textrm{otherwise}.\end{array}\right.$ (1.1)

Conversely, given an 0-1 matrix $A=(a_{ij})_{n\times n}$, we can define its
digraph $D(A)=({V},{A})$ on vertices $v_{1},v_{2},\ldots,v_{n}$ by (1.1),
whose adjacency matrix is $A$. Considering the entries of $A^{k}(D)$, Problem
1 is equivalent to the following matrix problem, which was proposed by X. Zhan
in 2007; see [17, p.234].

Problem 1’. Determine the maximum number of ones in a 0-1 matrix $A$ of order
$n$ such that $A^{k}$ is also a 0-1 matrix. Characterize the extremal 0-1
matrices attaining this maximum.

In 2010, Wu [16] solved Problem 1 for the case $k=2$. Huang and Zhan [11]
solved the case $k\geq n-1\geq 4$ and determined $ex(k+2,F_{k})$,
$ex(k+3,F_{k})$ when $k\geq 4$. Huang, Lyu and Qiao [10] determined
$ex(n,F_{k})$ for $4\leq k\leq n-4$, and characterized $EX(n,F_{k})$ for
$5\leq k\leq n-4$. They also characterize $EX(k+2,F_{k})$ and $EX(k+3,F_{k})$
when $k\geq 4$. Lyu [13] characterized $EX(n,F_{4})$ when $n\geq 8$. In this
paper, we solve the last remained case $k=3$ when $n\geq 16$.

To state our results, we need the following notations and definitions. For a
digraph $D=(V,A)$, we denote by $a(D)$ the size of $D$. For $i,j\in V$, if $D$
contains an arc from $i$ to $j$, we say that $j$ is a successor of $i$, and
$i$ is a predecessor of $j$, denoted $ij$ or $i\rightarrow j$. A directed walk
$xv_{1}v_{2}\cdots v_{k}y$ is called a $xy$-walk, where $x$ is its initial
vertex and $y$ is its terminal vertex. For $S,T\subset V$, we denote by $D[S]$
the subgraph of $D$ induced by $S$, $A(S,T)$ the set of arcs from $S$ to $T$,
$a(S,T)$ the cardinality of $A(S,T)$. Let

$N^{+}(u)=\\{x\in V|ux\in A\\}\quad\text{and}\quad N^{-}(u)=\\{x\in V|xu\in
A\\}$

be the out-neighbour set and in-neighbour set a vertex $u$, respectively. The
out-degree and in-degree of $u$ are $d^{+}(u)\equiv|N^{+}(u)|$ and
$d^{-}(u)\equiv|N^{-}(u)|$, respectively. Let $\Delta^{+}(D)$ denote the
maximum out-degree of $D$, which are abbreviate as $\Delta^{+}$ if there is no
confusion. Given $X\subseteq V$, $d^{+}_{X}(u)$ and $d^{-}_{X}(u)$ are the
numbers of the successors and predecessors of $u$ from $X$, respectively.

Two digraphs $D_{1}=(V_{1},A_{1})$ and $D_{2}=(V_{2},A_{2})$ are isomorphic,
written $D_{1}\cong D_{2}$, if there is a bijection $\sigma:V_{1}\rightarrow
V_{2}$ such that $(u,v)\in A_{1}$ if and only if $(\sigma(u),\sigma(v))\in
A_{2}$.

For a digraph $D=(V,A)$ with $V=\\{v_{1},v_{2},\ldots,v_{n}\\}$, a blow-up of
$D$ is obtained by replacing every vertex $v_{i}$ with a finite collection of
copies of $v_{i}$, denoted $V_{i}$, so that $xy$ is an arc for $x\in V_{i}$
and $y\in V_{j}$ if and only if $v_{i}v_{j}\in A(D)$. A blow-up of a digraph
is said to be balanced if $|V_{i}|$ and $|V_{j}|$ differ by at most one for
any pair $i\neq j$. Denote by $B(V_{1},V_{2})$ the blow-up of an arc with
vertex set $V_{1}\cup V_{2}$ such that $xy$ is an arc in $B(V_{1},V_{2})$ if
and only if $x\in V_{1},y\in V_{2}$, and $T(V_{1},V_{2},V_{3})$ the blow-up of
the transitive tournament of order 3 such that $xy$ is an arc in
$T(V_{1},V_{2},V_{3})$ if and only if $x\in V_{i},y\in V_{j}$ with $i<j$.

Given a digraph $D$, denote by $D+e$ the digraph obtained from $D$ by adding
an arc $e$, where the arc $e$ is allowed to be a loop. We define
$H(V_{1},V_{2})$ to be the digraph obtained from $B(V_{1},V_{2})$ by adding an
arc or a loop with both ends from $V_{1}$. Let

$\displaystyle T_{3,n}=$ $\displaystyle\\{$ $\displaystyle
T(V_{1},V_{2},V_{3})+e:|V_{1}|+|V_{2}|+|V_{3}|=n,|V_{i}|=\left\lfloor\frac{n}{3}\right\rfloor~{}{\rm
or}~{}\left\lceil\frac{n}{3}\right\rceil~{}{\rm for}~{}i=1,2,3,$
$\displaystyle e\text{ is an arc with both ends from }V_{2}\\},$

which is the set of digraphs obtained from a balanced blow-up of the
transitive tournament of order 3 by embedding an arc in the middle partite
set. Then each digraph in $T_{3,n}$ has one of the diagrams in Figure 1.

Figure 1: The diagrams of the digraphs in $T_{3,n}$.

Now we state our main result as follows.

######  Theorem 2.

Let $n\geq 16$ be an integer. Then

$ex(n,F_{3})=\left\lfloor\frac{n^{2}}{3}\right\rfloor+1\quad and\quad
EX(n,F_{3})=T_{3,n}.$

Notice that $T_{3,n}$ contains some loopless digraphs. By Theorem 2 we have
the following result for strict digraphs.

######  Corollary 3.

Let $D$ be an $F_{3}$-free strict digraphs on $n\geq 16$ vertices. Then

$a(D)\leq\left\lfloor\frac{n^{2}}{3}\right\rfloor+1$

with equality if and only if $D\in T_{3,n}$.

Remark. Combining Theorem 2 with previous results, we now obtain a full
solution to Problem 1 (Problem 1’).

##  2 Proofs

We will need the following lemmas to prove Theorem 2.

######  Lemma 4.

Let $D$ be a digraph on $n\geq 8$ vertices such that there are no two walks of
length 2 with the same terminal vertex. Then

$a(D)\leq\left\lfloor\frac{n^{2}}{4}\right\rfloor+1$ (2.1)

with equality if and only if $D$ is isomorphic to $H(V_{1},V_{2})$ with
$\\{|V_{1}|,|V_{2}|\\}=\\{\lfloor n/2\rfloor,\lceil n/2\rceil\\}$.

###### Proof.

Suppose $D=(V,A)$. Let $V_{1}=\\{u\in V|d^{+}(u)>0\\}$ and let
$V_{2}=V\setminus V_{1}$. Then we have

$a(V_{2},V)=0$ (2.2)

and

$d^{-}(u)\leq 1\quad{\rm for~{}all}\quad u\in V_{1}.$

Let $V^{\prime}_{1}=\\{u\in V_{1}|d^{-}(u)=0\\}$ and
$V^{\prime\prime}_{1}=\\{u\in V_{1}|d^{-}(u)=1\\}$. Then
$V_{1}=V^{\prime}_{1}\cup V^{\prime\prime}_{1}$ and $V^{\prime}_{1}\cap
V^{\prime\prime}_{1}=\emptyset$. Moreover,

$a(V_{1},V_{1})=\sum\limits_{u\in V_{1}}d^{-}(u)=|V^{\prime\prime}_{1}|.$
(2.3)

If $V^{\prime\prime}_{1}$ is empty, we have

$\displaystyle a(D)$ $\displaystyle=$ $\displaystyle
a(V_{1},V_{2})+a(V_{1},V_{1})+a(V_{2},V)$ (2.4) $\displaystyle=$
$\displaystyle a(V_{1},V_{2})+|V^{\prime\prime}_{1}|+0$ $\displaystyle\leq$
$\displaystyle|V_{1}||V_{2}|\leq\left\lfloor\frac{n^{2}}{4}\right\rfloor,$

which implies (2.1).

Now suppose $|V^{\prime\prime}_{1}|\geq 1$. Then we have

$a(V^{\prime\prime}_{1},V_{2})\leq|V_{2}|.$

Otherwise $a(V^{\prime\prime}_{1},V_{2})>|V_{2}|$ implies that there exist
$u_{1},u_{2}\in V^{\prime\prime}_{1}$ sharing a common successor in $V_{2}$,
which contradicts the given condition. Notice that

$|V^{\prime\prime}_{1}||V_{2}|+1\geq|V_{2}|+|V^{\prime\prime}_{1}|.$

We have

$\displaystyle a(D)$ $\displaystyle=$ $\displaystyle
a(V_{1},V_{2})+a(V_{1},V_{1})+a(V_{2},V)$ (2.5) $\displaystyle=$
$\displaystyle
a(V_{1}^{\prime},V_{2})+a(V^{\prime\prime}_{1},V_{2})+|V^{\prime\prime}_{1}|$
$\displaystyle\leq$
$\displaystyle|V^{\prime}_{1}||V_{2}|+|V_{2}|+|V^{\prime\prime}_{1}|$
$\displaystyle=$
$\displaystyle|V_{1}||V_{2}|-|V^{\prime\prime}_{1}||V_{2}|+|V_{2}|+|V^{\prime\prime}_{1}|$
$\displaystyle\leq$ $\displaystyle|V_{1}||V_{2}|+1$ $\displaystyle\leq$
$\displaystyle\left\lfloor\frac{n^{2}}{4}\right\rfloor+1.$

Hence, (2.1) holds.

If equality in (2.1) holds, then $|V^{\prime\prime}_{1}|\geq 1$, and all
inequalities in (2.5) are equalities. We can deduce that
$||V_{1}|-|V_{2}||\leq 1$, $a(V_{1}^{\prime},V_{2})=|V_{1}^{\prime}||V_{2}|$,
$a(V^{\prime\prime}_{1},V_{2})=|V_{2}|$ and $|V^{\prime\prime}_{1}|=1$.
Therefore, $D$ is isomorphic to $H(V_{1},V_{2})$ with
$\\{|V_{1}|,|V_{2}|\\}=\\{\lfloor n/2\rfloor,\lceil n/2\rceil\\}$. ∎

######  Corollary 5.

Let $D$ be a digraph on $n\geq 8$ vertices without two walks of length 2
sharing the same terminal vertex such that

$a(D)=\left\lfloor\frac{n^{2}}{4}\right\rfloor.$ (2.6)

Then $D$ is isomorphic to a digraph obtained from $H(V_{1},V_{2})$ with
$\\{|V_{1}|,|V_{2}|\\}=\\{\lfloor n/2\rfloor,\lceil n/2\rceil\\}$ by deleting
an arbitrary arc, or $D=H(V_{1},V_{2})$ with
$\\{|V_{1}|,|V_{2}|\\}=\left\\{n/2-1,n/2+1\right\\}$ provided $n$ is even.

###### Proof.

Define $V_{1},V_{2},V_{1}^{\prime},V_{1}^{\prime\prime}$ as in Lemma 4. If
$V_{1}^{\prime\prime}$ is empty, then we have (2.4). By (2.2) and (2.3),
equality in (2.4) holds if and only if $D$ is isomorphic to $B(V_{1},V_{2})$
with $\\{|V_{1}|,|V_{2}|\\}=\\{\lfloor n/2\rfloor,\lceil n/2\rceil\\}$.

If $|V_{1}^{\prime\prime}|\geq 1$, then we have (2.5). The condition (2.6)
leads to $|V_{1}^{\prime\prime}|=1$ and

$||V_{1}|-|V_{2}||=\left\\{\begin{array}[]{ll}1,&{\rm if}~{}n~{}{\rm
is~{}odd};\\\ 0{\rm~{}or~{}}2,&{\rm if~{}}n~{}{\rm
is~{}even}.\end{array}\right.$

Moreover, (2.2) and (2.3) implies that there is exactly one arc $e$ in
$D[V_{1}]$ and all the other arcs in $D$ have tails in $V_{1}$ and heads in
$V_{2}$. If $n$ is odd, then $||V_{1}|-|V_{2}||=1$ implies that $D$ is
isomorphic to a digraph obtained from $B(V_{1},V_{2})+e\in H(V_{1},V_{2})$ by
deleting an arc in $B(V_{1},V_{2})$. If $n$ is even, then
$|V_{1}|=|V_{2}|=n/2$ implies that $D$ is isomorphic to the digraphs obtained
from $B(V_{1},V_{2})+e$ by deleting an arc in $B(V_{1},V_{2})$, and
$||V_{1}|-|V_{2}||=2$ implies that $D$ is isomorphic to $H(V_{1},V_{2})$ with
$\\{|V_{1}|,|V_{2}|\\}=\\{n/2-1,n/2+1\\}$.

∎

Now we are ready to present the proof of Theorem 2.

Proof of Theorem 2. Let $D\in T_{3,n}$. Then $D=T(V_{1},V_{2},V_{3})+e$ with
$|V_{i}|=\lfloor{n}/{3}\rfloor$ or $\lceil{n}/{3}\rceil~{}{\rm
for}~{}i=1,2,3,$ and $e$ is an arc with both ends from $V_{2}$. It is clear
that all walks in $D$ have length $\leq 3$. Moreover, for any walk
$v_{1}v_{2}v_{3}v_{4}$ of length 3, we have $v_{2}v_{3}=e$. Therefore, there
are no two distinct walks of length 3 with the same initial vertex and the
same terminal vertex, i.e., $D$ is $F_{3}$-free. Hence,

$ex(n,F_{3})\geq a(D)=\left\lfloor\frac{n^{2}}{3}\right\rfloor+1.$ (2.7)

Now let $D=(V,A)\in EX(n,F_{3})$. Suppose $v_{0}$ is a vertex with
$d^{+}(v_{0})=\Delta^{+}$. It follows from (2.7) that

$\Delta^{+}\geq\left\lfloor\frac{n}{3}\right\rfloor+1.$ (2.8)

Let $V_{1}=N^{+}(v_{0})$ and $V_{2}=V\setminus V_{1}$. We count $a(D)$ in the
following way:

$a(D)=a(V_{1},V_{1})+a(V_{1},V_{2})+a(V_{2},V)=a(V_{1},V_{1})+\sum\limits_{u\in
V_{2}}[d^{+}(u)+d^{-}_{V_{1}}(u)].$ (2.9)

For any $u\in V_{2}$, if $d^{-}_{V_{1}}(u)\geq 2$, then $d^{+}(u)=0$ as $D$ is
$F_{3}$-free. Therefore,

$d^{+}(u)+d^{-}_{V_{1}}(u)\leq\Delta^{+}+1$ (2.10)

for all $u\in V_{2}$. Moreover, equality in (2.10) holds only if
$d^{+}(u)=\Delta^{+}$ and $d^{-}_{V_{1}}(u)=1$. Now we prove the following
claim.

Claim 1. $d^{+}(u)+d^{-}_{V_{1}}(u)\leq\Delta^{+}$ for all $u\in V_{2}$.

Proof of Claim 1. We first assert that there are at most two vertices in
$V_{2}$ satisfying the equality in (2.10). Suppose otherwise there are three
vertices $u_{1},u_{2},u_{3}\in V_{2}$ satisfying the equality in (2.10). Then
we have

$d^{-}_{V_{1}}(u_{1})=d^{-}_{V_{1}}(u_{2})=d^{-}_{V_{1}}(u_{3})=1$

and

$d^{+}(u_{1})=d^{+}(u_{2})=d^{+}(u_{3})=\Delta^{+}.$

Hence, there are $u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime}\in V_{1}$ such
that
$\\{u_{1}^{\prime}u_{1},u_{2}^{\prime}u_{2},u_{3}^{\prime}u_{3}\\}\subseteq
A$. By (2.8), there are two vertices in $\\{u_{1},u_{2},u_{3}\\}$ sharing a
common successor. Without loss of generality, we assume $u_{1}\rightarrow u$
and $u_{2}\rightarrow u$. Then $D$ contains the walks $v_{0}\rightarrow
u_{1}^{\prime}\rightarrow u_{1}\rightarrow u$ and $v_{0}\rightarrow
u_{2}^{\prime}\rightarrow u_{2}\rightarrow u$, a contradiction.

Since $D$ is $F_{3}$-free and $V_{1}$ is the out-neighbour set of $v_{0}$,
$D[V_{1}]$ does not contain two distinct walks of length 2 with a common
terminal vertex. Applying Lemma 4 on $D[V_{1}]$, we have

$a(D[V_{1}])=a(V_{1},V_{1})\leq\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor+1.$

Combining this with (2.7) and (2.9), we obtain

$\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor+\Delta^{+}(n-\Delta^{+})+3\geq
a(D)\geq\left\lfloor\frac{n^{2}}{3}\right\rfloor+1,$

which leads to

$\frac{(\Delta^{+})^{2}}{4}+\Delta^{+}(n-\Delta^{+})+3\geq\frac{n^{2}}{3}.$

It follows that

$\left\lceil\frac{2n}{3}\right\rceil-2\leq\Delta^{+}\leq\left\lfloor\frac{2n}{3}\right\rfloor+2.$
(2.11)

Suppose that there are $u_{1},u_{2}\in V_{2}$ such that

$d^{+}(u_{1})+d^{-}_{V_{1}}(u_{1})=d^{+}(u_{2})+d^{-}_{V_{1}}(u_{2})=\Delta^{+}+1.$

Then $d^{+}(u_{1})=d^{+}(u_{2})=\Delta^{+}$ and
$d^{-}_{V_{1}}(u_{1})=d^{-}_{V_{1}}(u_{2})=1$, which implies that there are
$u_{1}^{\prime},u_{2}^{\prime}\in V_{1}$ such that
$\\{u_{1}^{\prime}u_{1},u_{2}^{\prime}u_{2}\\}\subseteq A$. By (2.11), $u_{1}$
and $u_{2}$ share a common successor $u$. Therefore, $D$ contains the walks
$v_{0}\rightarrow u_{1}^{\prime}\rightarrow u_{1}\rightarrow u$ and
$v_{0}\rightarrow u_{2}^{\prime}\rightarrow u_{2}\rightarrow u$, a
contradiction.

Note that
$\Delta^{+}(n-\Delta^{+})+\left\lfloor(\Delta^{+})^{2}/4\right\rfloor$ is the
size of a complete 3-partite (undirected) graph on $n$ vertices and
$\left\lfloor n^{2}/3\right\rfloor$ is the size of a balanced 3-partite
complete (undirected) graph on $n$ vertices. We always have

$\left\lfloor
n^{2}/3\right\rfloor\geq\Delta^{+}(n-\Delta^{+})+\left\lfloor(\Delta^{+})^{2}/4\right\rfloor.$
(2.12)

Suppose there is only one vertex $u\in V_{2}$ satisfying the equality in
(2.10). Then we get

$d^{+}(u)=\Delta^{+}.$ (2.13)

Moreover, $u$ has exactly one predecessor $w$ in $V_{1}$. By (2.7), (2.9),
(2.10) and (2.12), we get

$\displaystyle a(V_{1},V_{1})$ $\displaystyle=$ $\displaystyle
a(D)-\sum\limits_{u\in V_{2}}[d^{+}(u)+d^{-}_{V_{1}}(u)]$ $\displaystyle\geq$
$\displaystyle\left\lfloor\frac{n^{2}}{3}\right\rfloor-\Delta^{+}(n-\Delta^{+})$
$\displaystyle\geq$
$\displaystyle\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor.$

Therefore, we have either

$a(V_{1},V_{1})=\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor+1$ (2.14)

or

$a(V_{1},V_{1})=\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor.$ (2.15)

If (2.14) holds, then $D[V_{1}]$ isomorphic to $H(U_{1},U_{2})$ with
$\\{|U_{1}|,|U_{2}|\\}=\\{\lfloor\Delta^{+}/2\rfloor,\lceil\Delta^{+}/2\rceil\\}$.
If (2.15) holds, applying Corollary 5 on $D[V_{1}]$, $D[V_{1}]$ is isomorphic
to a digraph obtained from $H(U_{1},U_{2})$ with
$\\{|U_{1}|,|U_{2}|\\}=\\{\lfloor\Delta^{+}/2\rfloor,\lceil\Delta^{+}/2\rceil\\}$
by deleting an arbitrary arc, or $D=H(U_{1},U_{2})$ with
$\\{|U_{1}|,|U_{2}|\\}=\left\\{\Delta^{+}/2-1,\Delta^{+}/2+1\right\\}$
provided $\Delta^{+}$ is even. In all the above cases, there is a subset of
$V_{1}$ with cardinality $\lceil\Delta^{+}/2\rceil-1$, say $V_{3}$, in which
any pair of vertices has a common successor in $V_{1}$.

Since $D$ is $F_{3}$-free, $u$ has at most one successor in $V_{3}$. Now
$n\geq 16$, $d^{+}(u)=\Delta^{+}$ and (2.11) enforce that $u$ has at least
three successors $u_{1},u_{2},u_{3}$ in $V_{2}$. On the other hand, we assert
that there is at most one vertex $v\in V_{2}$ such that

$d^{+}(v)+d^{-}_{V_{1}}(v)\leq\Delta^{+}-1.$

Otherwise we have

$\displaystyle a(D)$ $\displaystyle=$ $\displaystyle
a(V_{1},V_{1})+\sum\limits_{v\in V_{2}}[d^{+}(v)+d^{-}_{V_{1}}(v)]$
$\displaystyle\leq$
$\displaystyle\left(\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor+1\right)+(n-\Delta^{+})\Delta^{+}-1$
$\displaystyle<$ $\displaystyle\left\lfloor\frac{n^{2}}{3}\right\rfloor+1,$

a contradiction. Hence, two of $u_{1},u_{2},u_{3}$, say $u_{1}$ and $u_{2}$,
satisfy $d^{+}(u_{i})+d^{-}_{V_{1}}(u_{i})\geq\Delta^{+}-1$. It follows that
$d^{+}(u_{i})\geq\Delta^{+}-2$ for $i=1,2$. Since $u_{1}$ and $u_{2}$ have at
most one successor in $V_{3}$, they have a common successor $x$. Therefore,
$D$ contains the walks $w\rightarrow u\rightarrow u_{1}\rightarrow x$ and
$w\rightarrow u\rightarrow u_{2}\rightarrow x$, a contradiction.

Therefore, there is no vertex $u$ in $V_{2}$ such that the equality in (2.12)
holds. This completes the proof of Claim 1. ∎

By Claim 1, (2.9) and (2.12), we get

$\displaystyle a(D)$ $\displaystyle=$ $\displaystyle
a(V_{1},V_{1})+\sum\limits_{u\in V_{2}}[d^{+}(u)+d^{-}_{V_{1}}(u)]$ (2.16)
$\displaystyle\leq$
$\displaystyle\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor+1+(n-\Delta^{+})\Delta^{+}$
$\displaystyle\leq$ $\displaystyle\left\lfloor\frac{n^{2}}{3}\right\rfloor+1.$
(2.17)

Combining this with (2.7), we have

$ex(n,F_{3})=a(D)=\left\lfloor\frac{n^{2}}{3}\right\rfloor+1.$ (2.18)

Now we characterize the structure of $D$. The equality (2.18) implies that
(2.16) and (2.17) are both equalities. The equality in (2.17) leads to

$\Delta^{+}=\left\\{\begin{array}[]{ll}2n/3,&\text{if }n\equiv 0\text{ (mod
3)};\\\ \lfloor 2n/3\rfloor\text{ or }\lfloor
2n/3\rfloor+1,&\text{otherwise}.\\\ \end{array}\right.$ (2.19)

The equality in (2.16) implies

$a(V_{1},V_{1})=\left\lfloor\frac{(\Delta^{+})^{2}}{4}\right\rfloor+1$

and

$d^{+}(u)+d^{-}_{V_{1}}(u)=\Delta^{+}\quad\text{for all}\quad u\in V_{2}.$

Applying Lemma 4 on $D[V_{1}]$, $D[V_{1}]$ is isomorphic to $H(V_{3},V_{4})$
with $\\{|V_{3}|,|V_{4}|\\}=\\{\lfloor\Delta^{+}/2\rfloor,$
$\lceil\Delta^{+}/2\rceil\\}$.

Next we prove

$a(V_{1},V_{2})=0.$ (2.20)

If there is a vertex $u\in V_{2}$ such that $d^{-}_{V_{1}}(u)\geq 2$, then $u$
has no successor, which implies $d^{-}_{V_{1}}(u)=\Delta^{+}$. Therefore, all
vertices in $V_{4}$ are predecessors of $u$. Choose any $u_{1},u_{2}\in V_{3}$
and $u_{3}\in V_{4}$. We find two walks $v_{0}\rightarrow u_{1}\rightarrow
u_{3}\rightarrow u$ and $v_{0}\rightarrow u_{2}\rightarrow u_{3}\rightarrow u$
in $D$, a contradiction. Therefore, we have

$d^{-}_{V_{1}}(u)\leq 1\quad\text{and}\quad
d^{+}(u)\geq\Delta^{+}-1\quad\text{for all}\quad u\in V_{2}.$ (2.21)

Suppose $d^{-}_{V_{1}}(u)=1$ with $w$ being the predecessor of $u$ in $V_{1}$.
Then $d^{+}(u)=\Delta^{+}-1$. We assert that $u$ has at most one successor in
$V_{3}$. Otherwise if $u$ has two successors $u_{1},u_{2}$ in $V_{3}$, then
$D$ contains two walks $w\rightarrow u\rightarrow u_{1}\rightarrow u_{3}$ and
$w\rightarrow u\rightarrow u_{2}\rightarrow u_{3}$ for any $u_{3}\in V_{4}$, a
contradiction. By (2.21), $u$ has at least two successors in $V_{2}$, say
$u_{1}$ and $u_{2}$, which have a common successor $u_{3}$. Then $D$ contains
two walks $w\rightarrow u\rightarrow u_{1}\rightarrow u_{3}$ and $w\rightarrow
u\rightarrow u_{2}\rightarrow u_{3}$, a contradiction. Therefore, we have

$d^{-}_{V_{1}}(u)=0\quad\text{and}\quad d^{+}(u)=\Delta^{+}\quad\text{for
all}\quad u\in V_{2},$ (2.22)

and (2.20) holds.

Finally, we assert

$a(V_{2},V_{2})=0.$ (2.23)

Otherwise suppose $u_{1}u_{2}\in A(D[V_{2}])$. If $u_{2}$ has two successors
in $V_{2}$, say $u_{3}$ and $u_{4}$, by (2.19) and (2.22), $u_{3}$ and $u_{4}$
share a common successor $u_{5}$. Then $D$ contains two walks
$u_{1}\rightarrow u_{2}\rightarrow u_{3}\rightarrow u_{5}$ and
$u_{1}\rightarrow u_{2}\rightarrow u_{4}\rightarrow u_{5}$, a contradiction.
If $u_{2}$ has two successors $u_{3},u_{4}\in V_{3}$, then those successors
share a common successor $u_{5}$ in $V_{4}$, and $D$ also contains the above
walks, a contradiction again. It follows that

$d^{+}(u_{2})\leq|V_{2}|+2<\Delta^{+},$

which contradicts (2.22). Hence, we have (2.23).

By (2.20), (2.22) and (2.23), we see that $D[V_{2}]$ is an empty digraph and
all vertices in $V_{1}$ are successors of $u$ for all $u\in V_{2}$. Therefore,
$D=T(V_{2},V_{3},V_{4})+e\in T_{3,n}$, where both ends of the arc $e$ are from
$V_{3}$. This completes the proof. ∎

## Acknowledgement

This work was supported by Science and Technology Foundation of Shenzhen City
(No. JCYJ20190808174211224)

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# The role of the pion in the lineshape of the $X(3872)$

Angelo Esposito<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza
Università di Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy  INFN Sezione di
Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy  Davide Germani  Dipartimento
di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, I-00185 Rome,
Italy  Alfredo Glioti  Université Paris-Saclay, CNRS, CEA, Institut de
Physique Théorique, 91191, Gif-sur-Yvette, France  Antonio D. Polosa
Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2,
I-00185 Rome, Italy  INFN Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Rome,
Italy  Riccardo Rattazzi  Theoretical Particle Physics Laboratory, Institute
of Physics, EPFL, 1015 Lausanne, Switzerland  Michele Tarquini  Walter Burke
Institute for Theoretical Physics California Institute of Technology,
Pasadena, CA 91125 USA

###### Abstract

We determine the contribution of long-range pion interactions to the $X(3872)$
dynamics, assuming it is a loosely bound $D^{0}\bar{D}^{*0}$ molecule. Our
result is based on the distorted wave Born approximation in non-relativistic
quantum mechanics. Despite their long-range nature, we find that pion
interactions cannot produce a large and negative effective range. Nonetheless,
they introduce imaginary parts. In particular, they contribute to the total
decay width of the $X(3872)$ a term associated with, but not precisely
corresponding to, the $D^{*}$ width. Our approach can also be applied to the
recently discovered $T_{cc}^{+}$ state.

##  I Introduction

The $X(3872)$ was the first heavy-light four-quark resonance to be observed.
Yet, after almost two decades, its nature is still up for debate Esposito  _et
al._ (2017); Lebed  _et al._ (2017); Guo  _et al._ (2018); Brambilla  _et al._
(2020). What makes this state particularly interesting, and challenging, is
the presence of (at least) two parametric coincidences, or fine tunings.

The first tuning is given by the extreme closeness of the mass of the
$X(3872)$ to the $D^{0}\bar{D}^{*0}$ threshold. Indeed, as of today, we only
have an upper bound on the distance $B$ from threshold: $B\lesssim 120$ keV
Braaten and Hammer (2022). There are two different perspectives on this fact,
corresponding to two competing explanations for the nature of the $X(3872)$.
On the one hand, the $X(3872)$ could be a genuine tetraquark Maiani  _et al._
(2005, 2014), i.e. a compact four-quark hadron held together by gluon mediated
forces, hence present in the spectrum of QCD at distances of the order of a
fermi. In this case, given the very different color configuration of the two-
meson state and of the tetraquark, one would naïvely expect the natural scale
of $B$ to be roughly $\Lambda_{\rm QCD}$. In the face of that expectation, the
observed value of $B$ would correspond to a remarkable $1/10^{3}$ fine tuning.
However, one observes that all tetraquarks are systematically found within
$10-20$ MeV of their corresponding two-meson threshold. There could thus exist
a genuine explanation within QCD Maiani  _et al._ (2018), possibly based on
the $1/N$ expansion, for this systematic property. That would still single out
the $X(3872)$ as a $1/10^{2}$ outlier in fine tuning space.

On the other hand, the $X(3872)$ could be a very extended and loosely bound
$D^{0}\bar{D}^{*0}$ state111As $X(3872)$ is $C$-even, we would actually be
dealing with the superposition $D^{0}\bar{D}^{*0}+\bar{D}^{0}{D}^{*0}$.
Throughout our discussion the proper $C$ quantum number is always understood,
even when not properly indicated. Swanson (2004); Tornqvist (2004); Guo  _et
al._ (2018), a hadronic molecule appearing in the QCD spectrum at distances
larger than the fermi, very much like the deuteron. In this case, the
closeness to threshold would arise from a tuning of the short distance
$D^{0}\bar{D}^{*0}$ interaction, resulting in a large scattering length, $a$
(see, e.g., Braaten and Kusunoki (2004)), and in a correspondingly small
binding energy $B\sim 1/a^{2}m_{D}$. To match the observed value of $B$ one
would need $1/a\sim 13$ MeV, corresponding to a mild $1/10$ tuning with
respect to the naïvest expectation $1/a\sim\Lambda_{\rm QCD}$.

The second tuning, instead, is the fact that the mass of the $\bar{D}^{*0}$ is
almost exactly equal to the sum of the masses of the $D^{0}$ and the neutral
pion: $m_{D^{*}}-m_{D}-m_{\pi}\simeq 7$ MeV Zyla  _et al._ (2020). In
particular, this implies that, in this channel, the pion mediates long-range
interactions, and that it cannot be integrated out in an effective theory
below the QCD scale.

A promising way of discriminating between the different options for the
$X(3872)$ is offered by the study of its lineshape. In particular, the value
of the effective range, $r_{0}$, is a good discriminator Kang and Oller
(2017); Esposito  _et al._ (2022); Baru  _et al._ (2021); Kinugawa and Hyodo
(2022a, b); Song  _et al._ (2022); Sazdjian (2022); Mikhasenko (2022);
Albaladejo and Nieves (2022); Kinugawa and Hyodo (2023), as already pointed
out by Weinberg in the 60’s, when asking the same question for the deuteron
Weinberg (1965). On the one hand, for a molecular state, one expects $r_{0}$
to be controlled by the very size of the bound mesons, that is $r_{0}\sim
1\text{ fm}\sim 1/m_{\pi}$.222In our discussion, we make the rough
quantitative identification $1\text{ fm}\sim 1/m_{\pi}\sim 1/\Lambda_{\rm
QCD}$, even though $m_{\pi}$ and $\Lambda_{\rm QCD}$ are parametrically
distinguished. This expectation also abides by the modern effective field
theory perspective: unlike the scattering length $a$, the effective radius
$r_{0}$ is associated with an irrelevant operator and cannot be tuned larger
than the cut-off scale Kaplan  _et al._ (1998). On the other hand, one finds,
by explicit computation, that the presence of an interacting compact object
close to the two meson threshold produces a negative $r_{0}$ with an absolute
value larger than $1/m_{\pi}$. Again this is in agreement with the EFT
perspective: the only way to enhance an irrelevant operator is to lower the
cut-off scale, which in this case is controlled by the separation of the
tetraquark from the two meson threshold.

Now, as it was pointed out in Esposito  _et al._ (2022), a recent LHCb
analysis Aaij  _et al._ (2020a) suggests the second situation for $r_{0}$ (but
see also Baru  _et al._ (2021)). However, because of the accidental tuning we
mentioned above, the pion-mediated interaction is also characterized by a
large scale. One is thus led to wonder if that could play a role in producing
a large and negative effective range.

In this work, we address this question using a non-relativistic quantum
mechanical treatment based on the distorted wave Born approximation. An
expression for the effective range for the $X(3872)$, including the effect of
pions, already appeared in Braaten  _et al._ (2021), where a non-relativistic
effective theory for the $\bar{D}^{*0}$, the $D^{0}$ and the pion was used.
When working at leading order in the small ratio $m_{\pi}/m_{D}$, the approach
we use here considerably simplifies the problem, bypassing lengthy
calculations. When properly compared, our results are in agreement with Ref.
Braaten  _et al._ (2021).

Comparing to the most recent experimental data we find that, despite their
long-range nature, pion-mediated effects are too weak to generate the large
effective radius suggested by observations. However, our result introduces a
new structural feature: a complex effective range. Moreover, we find a pion-
induced long-range contribution to the decay width of a molecular $X(3872)$
that does not simply reduce to the decay width of the $D^{*}\to D\pi$ process.
This contribution should be added to those associated with other (short
distance) channels, like for instance $J/\psi\omega$ or $J/\psi\rho$.

We stress that our analysis can be applied with many similarities to the
recently discovered doubly-charmed $T_{cc}^{+}$ state Aaij  _et al._ (2021a,
b), which is manifestly exotic and shares several features with the $X(3872)$.
In particular, it was pointed out that the effective range of the $T_{cc}^{+}$
is negative and much larger than $1/m_{\pi}$ Mikhasenko (2022). This again
would appear to speak strongly in favor of a compact tetraquark nature, even
though this conclusion was challenged in Du  _et al._ (2022).

##  II The effective $DD^{*}$ Hamiltonian

The LHCb collaboration has recently performed a high statistics analysis of
the $B\to KX\to KJ/\psi\rho$ process, for events where the invariant mass of
the $J/\psi\rho$ pair is close to that of the $X(3872)$ Aaij  _et al._
(2020a). Plausibly assuming that the $B\to KD^{0}\bar{D}^{*0}$ and the
$D^{0}\bar{D}^{*0}\to J/\psi\rho$ vertices originate at short distance and
that they hence are approximately pointlike, the $B\to KJ/\psi\rho$ amplitude
results proportional to that for $D^{0}\bar{D}^{*0}\to D^{0}\bar{D}^{*0}$
Braaten and Kusunoki (2005), and can thus be used to extract the $X(3872)$
lineshape.

As already mentioned, since $m_{D^{*}}>m_{D}+m_{\pi}$, the $D^{*0}\to
D^{0}\pi^{0}$ decay is kinematically allowed. However, due to an accidental
fine tuning, the available phase space is small and all the particles are
endowed with a small velocity. We can therefore describe the relevant dynamics
within non-relativistic quantum mechanics, with two components of the Hilbert
space: one describing $D^{0}\bar{D}^{*0}$ (and $\bar{D}^{0}{D}^{*0}$) and the
other describing $D^{0}\bar{D}^{0}\pi^{0}$. We can represent the Hamiltonian
on this two-component Hilbert space in block diagonal form as

$\displaystyle H=\begin{pmatrix}H_{DD^{*}}&H_{I}^{\dagger}\\\
H_{I}&H_{DD\pi}\end{pmatrix}\,,$ (1)

where, in an obvious notation, $H_{DD^{*}}$ and $H_{DD\pi}$ describe evolution
within the two sectors in the absence of pion interactions, while $H_{I}$
contains the $D^{0}\bar{D}^{*0}\pi$ interaction vertex that couples the two
sectors. Forward time evolution, and the $S$-matrix, can be computed using the
retarded Green function ${\cal G}_{+}(E)=(E-H+i\epsilon)^{-1}$. As we are only
interested in the evolution in the $D^{0}\bar{D}^{*0}$ subspace we only need
the Green function reduced to the $D^{0}\bar{D}^{*0}$ block ${\cal
G}_{+}(E)|_{D^{0}\bar{D}^{*0}}$. The latter can be conveniently expressed by
integrating out the $D^{0}\bar{D}^{0}\pi^{0}$ component as ${\cal
G}_{+}(E)|_{D^{0}\bar{D}^{*0}}=(E-H_{\rm eff}(E)+i\epsilon)^{-1}$ with the
effective Hamiltonian $H_{\rm eff}(E)$ given by (e.g., Pethick and Smith,
2008),

$\displaystyle H_{\rm
eff}(E)=H_{DD^{*}}+H_{I}^{\dagger}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}\,.$ (2)

The $H_{DD^{*}}$ Hamiltonian includes the kinetic terms333As already
mentioned, we omit the $\bar{D}^{0}{D}^{*0}$ part, which has precisely the
same form. and a pointlike interaction controlled by a bare coupling,
$\lambda_{0}$,

$\displaystyle
H_{DD^{*}}=\frac{{\bm{p}}_{D^{*}}^{2}}{2m_{D^{*}}}+\frac{{\bm{p}}_{D}^{2}}{2m_{D}}-\lambda_{0}\delta^{(3)}(\bm{r})\,,$
(3)

with $\bm{r}$ the relative $D^{0}\bar{D}^{*0}$ position.444To be consistent
with power counting, at this order, one should also include higher derivative
corrections to the contact $D^{0}\bar{D}^{*0}$ interaction, as explained, for
example, in Fleming  _et al._ (2007); Jansen  _et al._ (2014). Nonetheless, we
know that such corrections give a contribution to the effective range of order
$m_{\pi}^{-1}$. Here we are interested in tracking possible contributions
substantially larger than that, and we will therefore ignore these
corrections.  The coupling $\lambda_{0}$ is assumed to bind the
$D^{0}\bar{D}^{*0}$ with a large scattering length. Since there is no
indication of any such critically large scattering length in the
$D^{0}\bar{D}^{0}\pi^{0}$ system, we model it as non-interacting:

$\displaystyle
H_{DD\pi}=-\delta+\frac{\bm{p}_{D,1}^{2}}{2m_{D}}+\frac{\bm{p}_{D,2}^{2}}{2m_{D}}+\frac{\bm{p}_{\pi}^{2}}{2m_{\pi}}\,.$
(4)

Here $\delta\equiv m_{D^{*}}-m_{D}-m_{\pi}$, and the constant term is due to
the fact that, in our notation, all energies are measured with respect to
$m_{D^{*}}+m_{D}$, and therefore the $D^{0}\bar{D}^{0}\pi^{0}$ system has a
slightly negative mass.

As far as the interaction Hamiltonian is concerned, instead, it is simpler to
write directly its matrix element between states of definite momentum and
polarization. At the lowest order in the small pion momentum, the Galilei
invariant matrix element can be written as Fleming  _et al._ (2007); Jansen
_et al._ (2014); Braaten (2015); Braaten  _et al._ (2021),

$\displaystyle\langle\bar{D}(\bm{k}_{1})D(\bm{k}_{2})\pi(\bm{q})|H_{I}|\bar{D}_{\lambda}^{*}(\bm{p}_{1})D(\bm{p}_{2})\rangle=\frac{ig}{2\sqrt{m_{\pi}}f_{\pi}m_{D^{*}}}\left(m_{D}\bm{p}_{1}-m_{D^{*}}\bm{k}_{1}\right)\\!\cdot\\!\bm{\varepsilon}_{\lambda}\,{(2\pi)}^{6}\delta^{(3)}(\bm{p}_{1}-\bm{k}_{1}-\bm{q})\delta^{(3)}(\bm{p}_{2}-\bm{k}_{2})\,,$
(5)

where $\bm{\varepsilon}_{\lambda}$ is the polarization vector of the
$\bar{D}^{*0}$. The same matrix element applies to the charge-conjugated
states. It should be stressed that the absence of time-dependent phase factors
in Eq. (5) underlies a specific choice of the non-relativistic fields. In the
present case that implies the masses appearing in the kinetic terms in Eqs.
(3) and (4), but not in the quantity $\delta$, satisfy
$m_{D^{*}}-m_{D}-m_{\pi}=0$. Henceforth, we shall everywhere set
$m_{D^{*}}=m_{D}+m_{\pi}$, except when $\delta$ is considered.

The coupling in Eq. (5) is related to the $D^{*0}\to D^{0}\pi^{0}$ decay width
by

$\displaystyle\Gamma_{*}=\frac{g^{2}\mu^{3}}{12\pi f_{\pi}^{2}}\,,$ (6)

where we defined $\mu\equiv\sqrt{2m_{\pi}\delta}\simeq 43$ MeV, and
$f_{\pi}\simeq 132$ MeV. Since no experimental value is available for the
decay of the neutral $D$-meson, $g$ can be extracted from the width of the
charged one, which gives $g^{2}\simeq 0.34$ Braaten (2015). Notice that this
result compares well with the expectation from naive dimensional analysis
Manohar and Georgi (1984); Georgi and Randall (1986), which would roughly give
$g^{2}\sim(4\pi f_{\pi})^{2}/N_{c}m_{D}^{2}\sim 0.2$.

Now, the term induced by pion exchange (the second) in the effective
Hamiltonian in Eq. (2) consists of two contributions: a
$\bar{D}^{*0}D^{0}\to\bar{D}^{*0}D^{0}$ transition and a $\bar{D}^{*0}D^{0}\to
D^{*0}\bar{D}^{0}$ one. These correspond to the two diagrams shown in Figure
1, and discussed in Jansen  _et al._ (2014); Braaten (2015); Braaten  _et al._
(2021) in a quantum field theory setup. A detailed computation of these
contributions is reported in Appendix A. Working in position space, and at
leading order in an expansion in $(m_{D^{*}}-m_{D})/m_{D}\sim m_{\pi}/m_{D}\ll
1$, the parts of the two transitions that purely contribute to the $S$-wave
read respectively

$\displaystyle\langle\bar{D}_{\lambda}^{*}(\bm{x}_{1})D(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle\simeq{}$
$\displaystyle\\!-i\frac{\Gamma_{*}}{2}\delta^{(3)}(\bm{x}_{1}-\bm{y}_{1})\delta^{(3)}(\bm{x}_{2}-\bm{y}_{2})\delta_{\lambda\lambda^{\prime}}\,,$
(7a) $\displaystyle\begin{split}\langle
D_{\lambda}^{*}(\bm{x}_{1})\bar{D}(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle\simeq{}&\\!-\\!\left[\alpha\frac{e^{i\mu
r}}{r}+\frac{g^{2}}{6f_{\pi}^{2}}\delta^{(3)}(\bm{r})\right]\\!\delta^{(3)}(\bm{x}_{1}-\bm{y}_{2})\delta^{(3)}(\bm{x}_{2}-\bm{y}_{1})\delta_{\lambda\lambda^{\prime}}\,,\end{split}$
(7b)

where $\bm{r}=\bm{x}_{1}-\bm{x}_{2}$ is the relative $D^{*}D$ position, and we
defined $\alpha\equiv g^{2}\mu^{2}/(24\pi f_{\pi}^{2})\simeq 5\times 10^{-4}$.
The quantity $\alpha$ measures the weakness of the pion-induced interaction.
For instance, Eq. (6) reads $\Gamma_{*}=2\alpha\mu$, indicating that the
$D^{*}$ is actually much narrower than what is implied by the small phase
space. The smallness of $\alpha$ simply follows from the derivative nature of
pion interactions. Finally, notice that the above leading order matrix
elements are independent of $E$: $H_{\rm eff}(E)\equiv H_{\rm eff}$.

$\bm{x}_{1},\lambda$$\bm{x}_{2}$$\bm{y}_{1},\lambda^{\prime}$$\bm{y}_{2}$$\bar{D}^{*0}$$\bar{D}^{*0}$$D^{0}$$D^{0}$$\bar{D}^{0}$$\pi^{0}$$D^{*0}$$D^{0}$$\bar{D}^{0}$$\bar{D}^{*0}$$\pi^{0}$$\bm{x}_{1},\lambda$$\bm{x}_{2}$$\bm{y}_{2}$$\bm{y}_{1},\lambda^{\prime}$

Figure 1: Diagrammatic representation of the two transition matrix elements in
Eqs. (7).

Decoupling the motion of the center of mass coordinate, the Schrödinger
equation for the relative $D^{0}\bar{D}^{*0}$ distance $\bm{r}$ then reads

$\displaystyle H_{\rm
eff}\,\psi(r)\equiv\left[-\frac{\nabla^{2}}{2\mu_{r}}-\left(\lambda_{0}+\frac{4\pi\alpha}{\mu^{2}}\right)\delta^{(3)}(\bm{r})-i\frac{\Gamma_{*}}{2}-\alpha\frac{e^{i\mu
r}}{r}\right]\psi(r)=E\,\psi(r)\,,$ (8)

with $\mu_{r}$ the reduced $D^{0}\bar{D}^{*0}$ mass. The wave function of the
$X(3872)$ corresponds to the $C=+1$ combination of the two charge conjugated
states,
$\psi=\frac{1}{\sqrt{2}}\left(\psi_{\bar{D}^{*}\\!D}+\psi_{D^{*}\\!\bar{D}}\right)$.
A detailed derivation of the results above can be found again in Appendix A.

A comment is in order. The effective potential in Eq. (8) is complex and has
an infinite range. This reflects the fact that since the $D^{*0}\to
D^{0}\pi^{0}$ decay is allowed, the intermediate pion in the scattering term
can be real, and hence propagate at arbitrarily large distances. Since the
reduced $D^{0}\bar{D}^{*0}$ subsystem does not include on-shell pions, the
potential is non-Hermitian and unitarity is not manifest. Because of that, as
we show in the next sections, there is an additional correction to the total
width of the $X(3872)$ and the effective range turns out to be complex.

##  III Effects of soft pions on the lineshape of the $X(3872)$

We now discuss the consequence of Eq. (8) on the lineshape of the $X(3872)$.
We perform our study in two steps. First, we consider the scattering problem
for positive real energies close to the (complex) $D^{0}\bar{D}^{*0}$
threshold. Secondly, we study the (unstable) bound state.

###  III.1 Scattering states: the effective range

The solution of Eq. (8), and the corresponding scattering amplitudes, can in
principle be found numerically, in complete analogy with Jackiw (1995), but
accounting for the complex terms in the Hamiltonian. However exploiting the
smallness of $\alpha$, one can study the problem analytically working in
perturbation theory.

To organize the discussion it is convenient to label the four components of
$H_{\rm eff}$ in Eq. (8) according to $H_{\rm
eff}=H_{0}-i\,\Gamma_{*}/2+V_{s}+V_{w}$, with,

$\displaystyle H_{0}=\frac{p^{2}}{2\mu_{r}}\,,\qquad
V_{s}(\bm{r})=-\left(\lambda_{0}+\frac{4\pi\alpha}{\mu^{2}}\right)\delta^{(3)}(\bm{r})\,,\qquad
V_{w}(\bm{r})=-\,{\alpha}\frac{e^{i\mu r}}{r}\,.$ (9)

As the contribution of pion exchange to $V_{s}$ is just a redefinition of the
bare coupling $\lambda_{0}$, we can absorb it in $\lambda_{0}$ and forget
about it. The genuine effects of pion exchange are then just the width
$\Gamma_{*}$ and $V_{w}$. Let us then first consider the limit in which these
contributions are neglected, and where the scattering amplitude is purely
determined by $H_{0}+V_{s}$ alone. Scattering states correspond to the $E>0$
continuum. Indicating by $k\equiv\sqrt{2\mu_{r}E}$ the relative
$D^{0}\bar{D}^{*0}$ momentum, and using the Lippmann–Schwinger formalism, the
amplitude reads

$\displaystyle
f_{s}(k)=-\frac{\mu_{r}}{2\pi}\langle\psi_{0,k}|\left[V_{s}+V_{s}\frac{1}{E-H_{0}-V_{s}+i\epsilon}V_{s}\right]|\psi_{0,k}\rangle=\frac{1}{-1/a_{s}-ik}\,,$
(10)

with $\psi_{0,k}(r)=(\sin kr)/kr$ the free $S$-wave solution and with $a_{s}$
the physical scattering length that results from the series of insertions of
$V_{s}$, after renormalization Jackiw (1995); Braaten and Kusunoki
(2004).555The Born level term in Eq. (10) gives
$a_{s}=\mu_{r}\lambda_{0}/2\pi$, but every single additional insertion of
$V_{s}$ is linearly UV divergent. Consider now the effects of $\Gamma_{*}$ and
$V_{w}$. As $V_{w}$ is sufficiently localized in space, for sufficiently small
$\alpha$, its effect on the amplitude will certainly be treatable as a
perturbation. $\Gamma_{*}$, instead, is not spatially localized: since it does
not vanish in the limit of very separated particles, its presence affects the
very definition of the asymptotic states and must be thus treated exactly.
This is however easily done. By Eq. (8), the presence of $\Gamma_{*}$ simply
implies that the asymptotic kinetic energy, $k^{2}/2\mu_{r}$, now equals
$E+i\Gamma_{*}/2$, by which the on-shell condition is now that
$E+i\Gamma_{*}/2$ is real and positive. Using
$k\equiv\sqrt{2\mu_{r}(E+i\Gamma_{*}/2)}$, Eq. (10) then simply becomes,

$\displaystyle
f_{s}(k)=-\frac{\mu_{r}}{2\pi}\langle\psi_{0,k}|\left[V_{s}+V_{s}\frac{1}{E+i\frac{\Gamma_{*}}{2}-H_{0}-V_{s}+i\epsilon}V_{s}\right]|\psi_{0,k}\rangle=\frac{1}{-1/a_{s}-ik}\,.$
(11)

In terms of the redefined $k$ the amplitude is the same as before: rather
intuitively all that matters is the kinetic energy $k^{2}/2\mu_{r}$ of the
$D^{0}\bar{D}^{*0}$ system. Notice that the amplitude analytically continued
to $E\to 0$ features a square root singularity in $\Gamma_{*}$, consistent
with the need for a full resummation of the $\Gamma_{*}$ insertions Jansen
_et al._ (2014).

Consider now $V_{w}$. At lowest order in $\alpha$, its effects are captured by
the so-called distorted wave Born approximation Weinberg (2013), resulting in
a correction to the amplitude $f_{DD^{*}}(k)$ given by (again
$k=\sqrt{2\mu_{r}(E+i\Gamma_{*}/2)}$ for the rest of this section),

$\displaystyle
f_{DD^{*}}(k)=f_{s}(k)-\frac{\mu_{r}}{2\pi}\langle\psi_{s,k}^{-}|V_{w}|\psi_{s,k}^{+}\rangle+\mathcal{O}\left(V_{w}^{2}\right)\,,$
(12)

where $|\psi_{s,k}^{\pm}\rangle$ are the asymptotic states for the Hamiltonian
$H_{0}+V_{s}-i\Gamma_{*}/2$ (e.g., Jackiw, 1995),

$\displaystyle|\psi_{s,k}^{\pm}\rangle={}$
$\displaystyle\left[1+\frac{1}{\frac{k^{2}}{2\mu_{r}}-H_{0}-V_{s}\pm
i\epsilon}\right]|\psi_{0,k}\rangle\,,$ (13a)
$\displaystyle\psi_{s,k}^{\pm}(r)={}$
$\displaystyle\frac{\sin(kr)}{kr}+\frac{1}{-1/a_{s}\mp ik}\frac{e^{\pm
ikr}}{r}=e^{\pm i\delta_{s}}\frac{\sin(kr+\delta_{s})}{kr}\,,$ (13b)

where $\delta_{s}$ is the $S$-wave scattering phase due to $V_{s}$:
$\tan\delta_{s}=-a_{s}k$.

With this at hand, the leading order correction to the scattering amplitude in
Eq. (12) is controlled by

$\displaystyle\langle\psi_{s,k}^{-}|V_{w}|\psi_{s,k}^{+}\rangle=-\,\alpha\,e^{2i\delta_{s}}\int
d^{3}r\frac{e^{i\mu r}}{r}\frac{\sin^{2}(kr+\delta_{s})}{(kr)^{2}}\,.$ (14)

The amplitude integral features a logarithmic UV divergence as a consequence
of the $\delta$-function potential that shapes the $\psi^{\pm}_{s,k}$. Indeed
turning off $V_{s}$, one would have $\delta_{s}=0$ and no UV divergence. This
UV divergence is associated with short-distance physics which we simply
parameterize by introducing a cutoff at $r=\eta$, where we expect $\eta\sim
1\text{ fm}$. At leading order as $\eta\mu\to 0$, we then get

$\displaystyle\begin{split}\langle\psi_{s,k}^{-}|V_{w}|\psi_{s,k}^{+}\rangle={}&-\frac{\pi\alpha}{k^{2}}\bigg{\\{}\left(e^{2i\delta_{s}}-1\right)^{2}\left(\gamma_{E}-i\frac{\pi}{2}+\log\eta\mu\right)+\log\left(1-\frac{2k}{\mu}\right)+e^{4i\delta_{s}}\log\left(1+\frac{2k}{\mu}\right)\bigg{\\}}+\mathcal{O}(\eta\mu)\,,\end{split}$
(15)

where $\gamma_{E}$ is the Euler–Mascheroni constant. Notice that, besides the
UV divergence controlled by $\eta$, this expression features the standard IR
divergence at $\mu\to 0$ associated with Coulombic interactions, to which our
potential reduces in this limit. Moreover, one can check that the result is
regular at $k\to 0$, corresponding to the validity of perturbation theory in
$\alpha$ even at this point.

For small relative momenta, the inverse amplitude can be expanded as follows,

$\displaystyle
f_{DD^{*}}^{-1}=-\frac{1}{a_{R}}-ik+\frac{2\alpha\mu_{r}}{\mu^{2}}\left(\frac{2}{a_{R}^{2}\mu^{2}}-1+\frac{8i}{3a_{R}\mu}\right)k^{2}+\mathcal{O}\big{(}k^{4}\big{)}\,,$
(16)

where the $\mathcal{O}\big{(}k^{2}\big{)}$ term determines the effective
range. Moreover, for consistency with perturbation theory, we truncated our
expression at order $\mathcal{O}(\alpha)$. The UV divergence has been fixed by
requiring for the $\mathcal{O}\big{(}k^{0}\big{)}$ to match the physical
scattering length, $a_{R}$. In particular, the relation between $a_{s}$ and
$a_{R}$ is,

$\displaystyle\frac{a_{s}}{a_{R}}=1-\frac{2\alpha\mu_{r}}{\mu}\bigg{[}\frac{1}{a_{R}\mu}+\gamma_{E}\mu
a_{R}+2i+\mu a_{R}\left(\log(\eta\mu)-i\frac{\pi}{2}\right)\bigg{]}\,.$ (17)

Since the $D^{0}\bar{D}^{*0}\to D^{0}\bar{D}^{*0}$ scattering cannot be
accessed experimentally, the physical scattering length must be obtained from
the distance of the $X(3872)$ from the $D^{0}\bar{D}^{*0}$ threshold. At
lowest order in $\alpha$ this is $a_{R}=1/\sqrt{2\mu_{r}B}$ (see Eq. (21)
below). Considering the current experimental bound, which for a molecular
state implies $0\text{ keV}<B\lesssim 100$ keV Aaij  _et al._ (2020b, a), one
finds the following constraints for the real and imaginary parts of the
effective range induced by pion exchange:

$\displaystyle-0.20\text{ fm}\lesssim\text{ Re}\,r_{0}\lesssim-0.16\text{
fm}\,,\qquad 0\text{ fm}\lesssim\text{ Im}\,r_{0}\lesssim 0.17\text{ fm}\,.$
(18)

As one can see, the pion is too weakly coupled to generate a large and
negative effective range in a purely molecular scenario.666Recall that by
“large” one normally means an effective range larger than its natural expected
value of roughly $1/m_{\pi}\simeq 1.4$ fm. Nonetheless, $r_{0}$ can now have
an imaginary part, even if just a small one.

###  III.2 Bound state: pole of the $X(3872)$

The unperturbed Hamiltonian $H_{0}+V_{s}$, besides a continuum of scattering
states with positive energy, features, for $a_{s}>0$, one bound state with
wave function Jackiw (1995) given by

$\displaystyle\psi_{X}(r)=\frac{1}{\sqrt{2\pi
a_{s}}}\frac{e^{-r/a_{s}}}{r}\,,$ (19)

and with energy $E_{X}\equiv-B=-\left(2a_{s}^{2}\mu_{r}\right)^{-1}$. In the
molecular hypothesis, this bound state is the very $X(3872)$. The pion-
mediated interaction now causes a shift in the energy of the molecular
$X(3872)$. The effect of the constant width is readily seen, as before, by
moving $\Gamma_{*}/2$ to the right-hand side in Eq. (8), so that
$-B=-\left(2a_{s}^{2}\mu_{r}\right)^{-1}$ now equals $E+i\Gamma_{*}/2$, that
is $E_{X}=-\left(2a_{s}^{2}\mu_{r}\right)^{-1}-i\Gamma_{*}/2$. The effect of
$V_{w}$ can instead be computed in perturbation theory and, at first order, it
gives a shift

$\displaystyle\begin{split}\Delta
E_{X}={}&\langle\psi_{X}|V_{\ell}|\psi_{X}\rangle=-\frac{\alpha}{2\pi
a_{s}}\int_{r\geq\eta}d^{3}r\frac{e^{-\frac{2r}{a_{s}}+i\mu
r}}{r^{3}}\,.\end{split}$ (20)

The real part of this expression features the same UV divergence encountered
in the previous section. Regulating the UV divergence by introducing the same
cut-off length, $\eta$, we can express the final result in terms of the
renormalized $a_{R}$ in Eq. (17) and of the other physical parameters as

$\displaystyle\begin{split}E_{X}=-\frac{1}{2\mu_{r}a_{R}^{2}}-i\frac{\Gamma_{*}}{2}-\frac{2\alpha}{a_{R}^{3}\mu^{2}}\bigg{[}1+2ia_{R}\mu-
a_{R}^{2}\mu^{2}\log\left(1+\frac{2i}{a_{R}\mu}\right)\bigg{]}\,.\end{split}$
(21)

As one can see, the term in the square bracket has an imaginary part
introducing a correction to the total width of the $X(3872)$:
$\Gamma_{X}=\Gamma_{*}+\Delta\Gamma_{X}$. This can be interpreted as the
effect of binding in $\bar{D}^{*0}\to D\pi$. Indeed $\Delta\Gamma_{X}$
vanishes in the unbound case, $a_{R}\to\infty$. Given the experimental
constraints on the distance of the $X(3872)$ from threshold, one deduces

$\displaystyle 0\text{ keV}\lesssim\Delta\Gamma_{X}\lesssim 1.9\text{ keV}\,.$
(22)

Notice that the imaginary part of the term in the square bracket in Eq. (21)
vanishes as $1/a_{R}\mu$ in the limit $a_{R}\mu\to\infty$. This further
suppresses the maximal value of $\Delta\Gamma_{X}$, given the allowed minimum
of $a_{R}\mu$ sits around $\sim 3$.

We conclude by commenting on the comparison between our results and those
found in Braaten  _et al._ (2021), where the same problem was studied in an
effective field theory. First of all, we notice that our Eq. (16) for the
effective range reproduces, at leading order in $m_{\pi}/m_{D}\ll 1$, almost
exactly the expression for the effective range in Braaten  _et al._ (2021),
exception made for the term that does not depend on the scattering length. In
Braaten  _et al._ (2021) this is written in terms of two finite constants,
which the authors call $R_{0}$ and $F_{2}$. The former is associated to the UV
divergences arising from the short distance contribution to the effective
range—i.e. to higher derivative corrections to the contact $D^{0}\bar{D}^{*0}$
interaction—while the latter is associated to an additional UV divergence
appearing when including pion effects that are subleading in the
$m_{\pi}/m_{D}\ll 1$ expansion. As explained in Section II, we neglect the
contribution coming from higher dimensional contact interaction and,
therefore, to properly compare our results with those of Braaten  _et al._
(2021), we must set $R_{0}=0$. Moreover, one can show that, at leading order
in $m_{\pi}/m_{D}\ll 1$, the $F_{2}$ constant can be calculated. When this is
done, the results for both the effective range, Eq. (16), as well as for the
energy of the $X(3872)$, Eq. (21), perfectly agree with what found in Braaten
_et al._ (2021). Finally, we stress that in Braaten  _et al._ (2021) it is
also shown that the inclusion of pion-mediated interactions induces a
breakdown of the effective range expansion. This breakdown is, however, weak,
as it is suppressed by $(m_{\pi}/m_{D})^{2}$ and, therefore, it is not present
in our results.

##  IV Conclusions

According to basic effective field theory, as synthesized in the so-called
Weinberg criterion Weinberg (1965), the effective range of the $X(3872)$, in
the hypothesis of a $D^{0}\bar{D}^{*0}$ molecular state, should be ${\cal
O}(1\text{ fm})$, while, in the hypothesis of a compact hadron, it could be
negative and with magnitude substantially larger than ${\cal O}(1\text{ fm})$.
The accidental vicinity of the $\bar{D}^{*0}$ to the $D\pi$ threshold,
however, introduces a new effective mass scale $\mu\ll 1/{\mathrm{fm}}$ in the
dynamics of the system. Even though also the strength of pion interactions
matters, the smallness of $\mu$ instills the suspicion that $\propto 1/\mu$
contributions to the effective range could affect the above clearcut picture,
also jeopardizing the validity of the results obtained in Esposito  _et al._
(2022) and Mikhasenko (2022). In this study, we showed by explicit computation
that is not the case. Our study confirms results previously obtained in the
literature Jansen  _et al._ (2014); Braaten  _et al._ (2021), but with a
technically much simpler approach based on non-relativistic scattering theory
and on the exact solution of three-dimensional $\delta$-function potentials in
quantum mechanics. One essential aspect of our computation is the occurrence
of a complex long-range potential from pion exchange, which induces shifts to
both real and imaginary parts of all the parameters describing the $X(3872)$
lineshape.

Broadly speaking, the deduction of the effective range of exotic hadrons like
the $X(3872)$ or the $T_{cc}^{+}$ from experimental data is a very relevant
matter, as it could provide a clear, model-independent determination of their
nature. While data are already available, we believe that the current
theoretical description of the lineshape of the $X(3872)$ is lacking a
systematic treatment, which includes all possible effects (contact
interactions, charged thresholds, tetraquark contribution, and so on) in a way
that can be readily interpreted from a physical viewpoint. We consider the
present study as a first step in that direction, which we plan to keep
pursuing in future work.

###### Acknowledgements.

R.R. is partially supported by the Swiss National Science Foundation under
contract 200020-213104 and through the National Center of Competence in
Research SwissMAP. A.E. and A.D.P. thank Hans Werner Hammer for an informative
discussion. A.D.P. wishes to thank Adam Szczepaniak and Kevin Ingles for
useful clarifications. We also thank the anonymous referee for providing
suggestions on how to compare our results with those of Braaten  _et al._
(2021).

##  Appendix A Effective Hamiltonian in position space

We here show in more detail how the effective Hamiltonian in Eq. (8) is
obtained. We start by computing the transition matrix elements in Eqs. (7).
Consider first eq. (7a) decomposed in momentum space

$\displaystyle\begin{split}&\langle\bar{D}_{\lambda}^{*}(\bm{x}_{1})D(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle=\int\frac{d\bm{p}_{1}}{(2\pi)^{3}}\frac{d\bm{p}_{2}}{(2\pi)^{3}}\int\frac{d\bm{k}_{1}}{(2\pi)^{3}}\frac{d\bm{k}_{2}}{(2\pi)^{3}}\\\
&\qquad\qquad\quad\times
e^{i\left(\bm{p}_{1}\cdot\bm{y}_{1}+\bm{p}_{2}\cdot\bm{y}_{2}-\bm{k}_{1}\cdot\bm{x}_{1}-\bm{k}_{2}\cdot\bm{x}_{2}\right)}\langle\bar{D}_{\lambda}^{*}(\bm{k}_{1})D(\bm{k}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{p}_{1})D(\bm{p}_{2})\rangle\,.\end{split}$
(23)

The momentum space matrix element can be evaluated using the completeness
relations for the $\bar{D}^{0}D^{0}\pi^{0}$ system,

$\displaystyle\langle\bar{D}_{\lambda}^{*}(\bm{k}_{1})D(\bm{k}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{p}_{1})D(\bm{p}_{2})\rangle=\int\frac{d\bm{q}_{1}}{(2\pi)^{3}}\frac{d\bm{q}_{2}}{(2\pi)^{3}}\frac{d\bm{q}_{3}}{(2\pi)^{3}}$
$\displaystyle\qquad\qquad\qquad\times\langle\bar{D}_{\lambda}^{*}(\bm{k}_{1})D(\bm{k}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}|\bar{D}(\bm{q}_{1})D(\bm{q}_{2})\pi(\bm{q}_{3})\rangle\langle\bar{D}(\bm{q}_{1})D(\bm{q}_{2})\pi(\bm{q}_{3})|H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{p}_{1})D(\bm{p}_{2})\rangle$
$\displaystyle\qquad\qquad\quad\simeq\left(\frac{g}{2\sqrt{m_{\pi}}f_{\pi}}\right)^{2}\\!(2\pi)^{3}\delta^{(3)}(\bm{k}_{1}-\bm{p}_{1})\delta^{(3)}(\bm{k}_{2}-\bm{p}_{2})\int
d\bm{q}\frac{\bm{q}\cdot\bm{\varepsilon}_{\lambda}\bm{q}\cdot\bm{\varepsilon}^{*}_{\lambda^{\prime}}}{E+\delta-\frac{(\bm{p}_{1}-\bm{q})^{2}}{2m_{D}}-\frac{\bm{p}_{2}^{2}}{2m_{D}}-\frac{\bm{q}^{2}}{2m_{\pi}}+i\epsilon}$
(24)
$\displaystyle\qquad\qquad\quad\simeq\frac{4\pi^{3}g^{2}}{f_{\pi}^{2}}\delta^{(3)}(\bm{k}_{1}-\bm{p}_{1})\delta^{(3)}(\bm{k}_{2}-\bm{p}_{2})\int
d\bm{q}\,\frac{\bm{q}\cdot\bm{\varepsilon}_{\lambda}\bm{q}\cdot\bm{\varepsilon}^{*}_{\lambda^{\prime}}}{\mu^{2}-q^{2}+i\epsilon}\,,$

where we expanded at lowest order in $m_{\pi}/m_{D}\ll 1$, keeping in mind the
energy $E$ will be of the order of the $D^{0}\bar{D}^{*0}$ kinetic energy,
$E\sim p^{2}/m_{D}$. Moreover, we keep $\mu=\sqrt{2m_{\pi}\delta}$ fixed,
since this is the quantity that critically controls the coordinate dependence
of the pion induce complex potential. Indeed $\mu$ acts as momentum cut off of
the effective $D^{0}\bar{D}^{*0}$ theory with the pion integrated out, and the
limit $\mu\to 0$ drastically affects the qualitative behavior of the
$D^{0}\bar{D}^{*0}$ system.

The last integral in Eq. (24) is UV divergent, and we choose to regularize it
with a cutoff $\Lambda$. Using also the orthonormality of the polarization
vectors,
$\bm{\varepsilon}_{\lambda}\cdot\bm{\varepsilon}_{\lambda^{\prime}}^{*}=\delta_{\lambda\lambda^{\prime}}$,
we obtain,

$\displaystyle\begin{split}\langle\bar{D}_{\lambda}^{*}(\bm{k}_{1})D(\bm{k}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{p}_{1})D(\bm{p}_{2})\rangle\simeq{}&-\left[\frac{16\pi^{4}g^{2}\Lambda^{3}}{9f_{\pi}^{2}}+\frac{16\pi^{4}g^{2}\Lambda\mu^{2}}{3f_{\pi}^{2}}+(2\pi)^{6}i\frac{\Gamma_{*}}{2}\right]\\\
&\times\delta^{(3)}(\bm{k}_{1}-\bm{p}_{1})\delta^{(3)}(\bm{k}_{2}-\bm{p}_{2})\delta_{\lambda\lambda^{\prime}}\,.\end{split}$
(25)

The real part is UV divergent and can be absorbed in the physical value of
$\delta$ which, with an abuse of notation, we will keep labelling by the same
symbol. By plugging the above result in Eq. (23), we obtain the transition
matrix element in position space,

$\displaystyle\langle\bar{D}_{\lambda}^{*}(\bm{x}_{1})D(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle\simeq-i\frac{\Gamma_{*}}{2}\delta^{(3)}(\bm{x}_{1}-\bm{y}_{1})\delta^{(3)}(\bm{x}_{2}-\bm{y}_{2})\delta_{\lambda\lambda^{\prime}}\,,$
(26)

where we omitted the UV divergent terms.

With similar manipulations, we can find the matrix element for the transition
$\bar{D}^{*0}D^{0}\to D^{*0}\bar{D}^{0}$ in eq. (7b) . In position space, it
reads (see also Suzuki (2005)),

$\displaystyle\langle
D_{\lambda}^{*}(\bm{x}_{1})\bar{D}(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle\simeq-\frac{g^{2}}{2f_{\pi}^{2}}\delta^{(3)}(\bm{x}_{1}-\bm{y}_{2})\delta^{(3)}(\bm{x}_{2}-\bm{y}_{1})\\!\int\\!\frac{d\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\frac{\bm{\varepsilon}_{\lambda}\cdot\bm{q}\bm{\varepsilon}_{\lambda^{\prime}}^{*}\cdot\bm{q}}{q^{2}-\mu^{2}-i\epsilon}$
$\displaystyle\qquad=\frac{g^{2}}{8\pi f_{\pi}^{2}}e^{i\mu
r}\varepsilon_{\lambda}^{i}\varepsilon_{\lambda^{\prime}}^{*j}\bigg{[}\hat{r}_{i}\hat{r}_{j}\left(\frac{3}{r^{3}}-\frac{3i\mu}{r^{2}}-\frac{\mu^{2}}{r}\right)-\delta_{ij}\left(\frac{1}{r^{3}}-\frac{i\mu}{r^{2}}\right)-\frac{4\pi}{3}\delta_{ij}\delta^{(3)}(\bm{r})\bigg{]}\delta^{(3)}(\bm{x}_{1}-\bm{y}_{2})\delta^{(3)}(\bm{x}_{2}-\bm{y}_{1})\,.$
(27)

At low enough energy, only the $S$-wave channel is relevant. To project the
above matrix element to $S$-wave, we average over the $\hat{\bm{r}}$
direction, effectively setting
$\hat{r}_{i}\hat{r}_{j}\to\frac{1}{3}\delta_{ij}$. This returns,

$\displaystyle\begin{split}\\!\langle
D_{\lambda}^{*}(\bm{x}_{1})\bar{D}(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle\simeq{}&\\!-\\!\bigg{[}\alpha\frac{e^{i\mu
r}}{r}+\frac{g^{2}}{6f_{\pi}^{2}}\delta^{(3)}(\bm{r})\bigg{]}\delta^{(3)}(\bm{x}_{1}-\bm{y}_{2})\delta^{(3)}(\bm{x}_{2}-\bm{y}_{1})\delta_{\lambda\lambda^{\prime}}\,.\end{split}$
(28)

With the transition matrix elements (26) and (28) at hand, the Schrodinger
equations for the $\bar{D}^{*0}D^{0}$ and the $D^{*0}\bar{D}^{0}$ become a
system of two coupled equations. Decoupling already the center-of-mass motion,
they read

$\displaystyle-\bigg{[}\frac{\nabla^{2}}{2\mu_{r}}+i\frac{\Gamma_{*}}{2}+\lambda_{0}\delta^{(3)}(\bm{r})\bigg{]}\psi_{\bar{D}^{*}\\!D}(r)-\bigg{[}\alpha\frac{e^{i\mu
r}}{r}+\frac{g^{2}}{6f_{\pi}^{2}}\delta^{(3)}(\bm{r})\bigg{]}\psi_{D^{*}\\!\bar{D}}(r)={}$
$\displaystyle E\,\psi_{\bar{D}^{*}\\!D}(r)\,,$ (29a)
$\displaystyle-\bigg{[}\alpha\frac{e^{i\mu
r}}{r}+\frac{g^{2}}{6f_{\pi}^{2}}\delta^{(3)}(\bm{r})\bigg{]}\psi_{\bar{D}^{*}\\!D}(r)-\bigg{[}\frac{\nabla^{2}}{2\mu_{r}}+i\frac{\Gamma_{*}}{2}+\lambda_{0}\delta^{(3)}(\bm{r})\bigg{]}\psi_{D^{*}\\!\bar{D}}(r)={}$
$\displaystyle E\,\psi_{D^{*}\\!\bar{D}}(r)\,.$ (29b)

Since the $X(3872)$ is a $C=+1$ state, its wave function as meson molecule is
$\psi=\frac{1}{\sqrt{2}}\left(\psi_{\bar{D}^{*}\\!D}+\psi_{D^{*}\\!\bar{D}}\right)$.
The corresponding Schrodinger equation is obtained by adding together the two
above and, indeed, reduces to Eq. (8).

Finally, it is interesting to show what the exchange amplitude computed above
look like away from the $m_{D^{*}}\simeq m_{D}$ limit. Recalling that, in our
notation, $m_{D^{*}}-m_{D}=m_{\pi}$, as well as Eq. (5), one can perform very
similar manipulations (albeit more tedious) and get to,

$\displaystyle\langle
D_{\lambda}^{*}(\bm{x}_{1})\bar{D}(\bm{x}_{2})|H_{I}\frac{1}{E-H_{DD\pi}+i\epsilon}H_{I}|\bar{D}_{\lambda^{\prime}}^{*}(\bm{y}_{1})D(\bm{y}_{2})\rangle=\frac{g^{2}}{4m_{\pi}f_{\pi}^{2}m_{D^{*}}^{2}}\int\frac{d\bm{p}_{1}}{(2\pi)^{3}}\frac{d\bm{p}_{2}}{(2\pi)^{3}}\frac{d\bm{k}_{1}}{(2\pi)^{3}}\frac{d\bm{k}_{2}}{(2\pi)^{3}}$
(30) $\displaystyle\qquad\qquad\quad\times
e^{i\left(\bm{p}_{1}\cdot\bm{y}_{1}+\bm{p}_{2}\cdot\bm{y}_{2}-\bm{k}_{1}\cdot\bm{x}_{1}-\bm{k}_{2}\cdot\bm{x}_{2}\right)}(2\pi)^{3}\delta^{(3)}(\bm{p}_{1}+\bm{p}_{2}-\bm{k}_{1}-\bm{k}_{2})\frac{\left(m_{D}\bm{k}_{1}-m_{D^{*}}\bm{p}_{2}\right)\cdot\bm{\varepsilon}_{\lambda}^{*}\left(m_{D}\bm{p}_{1}-m_{D^{*}}\bm{k}_{2}\right)\cdot\bm{\varepsilon}_{\lambda^{\prime}}}{E+\delta-\frac{k_{2}^{2}}{2m_{D}}-\frac{p_{2}^{2}}{2m_{D}}-\frac{(\bm{k}_{1}-\bm{p}_{2})^{2}}{2m_{\pi}}+i\epsilon}\,.$

When expanding in $m_{\pi}/m_{D}\ll 1$, one can easily check that the
corrections have the same structure as Eq. (27), with two gradients, acting
either on the potential $e^{i\mu r}/r$ or on the $\delta$-functions, with an
overall factor of $m_{\pi}/m_{D}$. Except for short distance contributions
which we are renormalizing away, both these terms scale exactly as Eq. (28)
but with an additional $m_{\pi}/m_{D}$ suppression, which makes them
negligible.

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