# The canonical trace of
Cohen-Macaulay algebras of codimension 2

Antonino Ficarra  Dedicated with my deepest gratitude to the memory of my
maestro and friend,
Professor Jürgen Herzog (1941-2024)  Antonino Ficarra, Departamento de
Matemática, Escola de Ciências e Tecnologia, Centro de Investigação,
Matemática e Aplicações, Instituto de Investigação e Formação Avançada,
Universidade de Évora, Rua Romão Ramalho, 59, P–7000–671 Évora, Portugal
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the present paper, we investigate a conjecture of Jürgen Herzog. Let $S$ be
a local regular ring with residue field $K$ or a positively graded
$K$-algebra, $I\subset S$ be a perfect ideal of grade two, and let $R=S/I$
with canonical module $\omega_{R}$. Herzog conjectured that the canonical
trace $\operatorname{tr}(\omega_{R})$ is obtained by specialization from the
generic case of maximal minors. We prove this conjecture in several cases, and
present a criterion that guarantees that the canonical trace specializes under
some additional assumptions. As the final conclusion of all of our results, we
classify the nearly Gorenstein monomial ideals of height two.

###### Key words and phrases:

Nearly Gorenstein rings, canonical trace, Hilbert-Burch theorem

###### 2020 Mathematics Subject Classification:

Primary 13P10; Secondary 05E40

##  1\. Introduction

Let $(R,{\mathfrak{m}},K)$ be either a local ring or a positively graded
$K$-algebra. Let ${\mathfrak{m}}$ be the (graded) maximal ideal of $R$. Assume
that $R$ is Cohen-Macaulay and admits a canonical module $\omega_{R}$. The
canonical trace of $R$ is then defined as

$\operatorname{tr}(\omega_{R})\ =\
\sum_{\varphi\in\operatorname{Hom}_{R}(\omega_{R},R)}\varphi(\omega_{R}).$

It turns out that $R$ is a Gorenstein ring if and only if
$\operatorname{tr}(\omega_{R})=R$. Thus, the canonical trace measures, in a
certain sense, how far is $R$ from being Gorenstein.

The ring $R$ is called nearly Gorenstein if $\operatorname{tr}(\omega_{R})$
contains ${\mathfrak{m}}$. Nearly Gorenstein rings were introduced Herzog,
Hibi and Stamate in [19], although they were implicitly considered in the work
of Huneke and Vraciu [21]. Many people are interested to develop their theory,
classify nearly Gorenstein rings, and compare the nearly Gorenstein property
with other properties of $R$. See [8, 11, 12, 14, 16, 20, 22, 23].

Let ${\bf x}=x_{1},\dots,x_{n}$ be a regular sequence on $R$, and denote by
“$\overline{\phantom{l.}}$” the reduction modulo ${\bf x}$. It is well-known
that the canonical module specializes, that is
$\overline{\omega_{R}}=\omega_{\overline{R}}$. One would guess that the
canonical trace specializes as well:
$\operatorname{tr}(\omega_{\overline{R}})=\operatorname{tr}(\omega_{R})\overline{R}$?
It is easy to see that
$\operatorname{tr}(\omega_{R})\overline{R}\subseteq\operatorname{tr}(\omega_{\overline{R}})$.
However, in [12, Example 3.3] we discovered a counterexample to this
expectation. Let $R=S/I$ where $S=K[x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}]$,
$I=(x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{1}x_{2},x_{1}x_{3},x_{2}y_{3})$ and
let ${\bf x}=x_{1}-y_{1}$. Then
$\operatorname{tr}(\omega_{\overline{R}})\neq\operatorname{tr}(\omega_{R})\overline{R}$.
This example is remarkable, because $I$ is the edge ideal of a Cohen-Macaulay
very well-covered graph, see [6, 7].

Now, let $S$ be a Noetherian local ring or a positively graded $K$-algebra,
and let $I\subset S$ be an ideal, which we assume to be homogeneous if $S$ is
graded. We furthermore assume that $S$ is regular. We will explain this
assumption in a moment.

We set $R=S/I$, and as before we assume that $R$ is Cohen-Macaulay with
canonical module $\omega_{R}$. The simplest Cohen-Macaulay algebras of this
form are those such that $I$ is a perfect ideal of grade two. Recall that $I$
is perfect if $\textup{grade}\,R=\operatorname{pd}R$. These algebras are
characterized by the celebrated Hilbert-Burch theorem [5].

######  Theorem 1.1.

([4, Theorem 1.4.17], [9, Theorem 20.15]) With the notation introduced, assume
that $I$ has a free resolution

${\mathbb{F}}\ \ :\ \ 0\rightarrow
S^{m-1}\xrightarrow{\varphi}S^{m}\rightarrow I\rightarrow 0.$

Then, there exists a $S$-regular element $a$ such that $I=aI_{m-1}(\varphi)$.
If $I$ is projective, then $I=(a)$, otherwise $\operatorname{pd}I=1$ and $I$
is perfect of grade two.

Conversely, if $a$ is a non-zero divisor on $S$,
$\operatorname{grade}I_{m-1}(\varphi)\geq 2$ and the map $S^{m}\rightarrow I$
sends the $i$th basis element to $(-1)^{i}a$ times the minor obtained from
$\varphi$ by leaving out the $i$th row, then ${\mathbb{F}}$ is the minimal
free resolution of $I=aI_{m-1}(\varphi)$.

Fixing bases of $S^{m-1}$ and $S^{m}$, we may represent $\varphi$ by a matrix
$A$. Any such $A$ is called an Hilbert-Burch matrix of $I$. Furthermore, we
may compute $I_{j}(\varphi)$ as $I_{j}(A)$ for all $j$ and this ideal is
independent from the particular Hilbert-Burch matrix $A$. Tensoring
${\mathbb{F}}$ with $R=S/I$, we obtain the exact sequence

$0\rightarrow C\xrightarrow{\alpha}R^{m-1}\xrightarrow{\psi}R^{m}\rightarrow
I/I^{2}\rightarrow 0,$

where $C=\operatorname{Ker}\psi$ is the kernel of
$\psi=\varphi\otimes\textup{id}_{R}$, and $\alpha$ is the inclusion map.

Since, as we assumed before, $S$ is regular, we can use the so-called
Vasconcelos trick [19, Corollary 3.2] to obtain that
$\operatorname{tr}(\omega_{R})=I_{1}(\alpha)$.

Let now $S=K[x_{1},\dots,x_{n}]$ be a graded polynomial ring over $K$. Let
$\mu(I)$ be the minimal number of generators of $I$. The Hilbert-Burch Theorem
1.1 implies that any graded perfect ideal $I$ of grade two, with $\mu(I)=m$,
arises as a specialization of the ideal of maximal minors $I_{m-1}(X)$ of a
generic matrix $X=(x_{ij})$ of indeterminates. Indeed, if $A=(f_{ij})$ is an
Hilbert-Burch matrix of $I$, and $R^{\prime}=S[X]/I_{m-1}(X)$, then $R\cong
R^{\prime}/({\bf g})$ where ${\bf g}=x_{11}-f_{11},\ldots,x_{m,m-1}-f_{m,m-1}$
is a regular sequence on $R^{\prime}$. By [12, Theorem 1.1], we have
$\operatorname{tr}(\omega_{K[X]/I_{m-1}(X)})=I_{m-2}(X)/I_{m-1}(X)$. In this
setting, we prove in Lemma 2.3 that

$I_{m-2}(\psi)=I_{m-2}(A)R\ \subseteq\
\operatorname{tr}(\omega_{R})=I_{1}(\alpha).$

These facts, and several experimental evidences, lead to the following
conjecture which Jürgen Herzog posed to the author of this paper in private
conversations.

######  Conjecture 1.2.

(Herzog) Let $S$ be either a regular local ring with residue class field $K$
or a regular positively graded $K$-algebra. Let $I$ be a perfect ideal of
grade two and let $A$ be an Hilbert-Burch matrix of $I$. Set $R=S/I$. Then,

$\operatorname{tr}(\omega_{R})=I_{\mu(I)-2}(A)R.$

In this paper we address this conjecture, and we settle it in important cases.
Along the way we characterize perfect monomial ideals of grade two which are
generically Gorenstein. More generally, we show that Conjecture 1.2 holds if
we assume in addition that $I$ is generically Gorenstein. Furthermore, we
prove that Conjecture 1.2 holds for any Cohen-Macaulay monomial ideal $I$ of
$S=K[x,y]$, and again we classify the nearly Gorenstein algebras $S/I$ in this
class. Finally, we classify the nearly Gorenstein monomial ideals of height
two of $S=K[x_{1},\dots,x_{n}]$.

Before breaking out the content of the paper, let us stress the relevance of
the Hilbert-Burch Theorem 1.1 across the fifty-five years of Jürgen Herzog
research. Herzog first paper ever [17], titled Generators and relations of
abelian semigroups and semigroup rings, contains among other fundamental and
basic results on semigroup rings, an explicit description of the minimal free
resolution of the semigroup ring of any numerical semigroup with three
generators. As Herzog remarked many times in his lifetime, this is just a
special case of the Hilbert-Burch theorem, which, however, he did not know at
the time as a student.

The Hilbert-Burch theorem gained again Herzog interest in the last few months
of his life, as a motivation for Conjecture 1.2, but also in the study of the
so-called Fitting ideals $\textup{Fitt}_{j}(I)$ of an ideal $I$. This led to a
joint paper [10] with Eisenbud, Herzog, Moradi and the author of this paper.
Such article, the third to last paper of Herzog, deals with the problem of
characterizing those ideals $I$ of a Noetherian ring such that
$I=\textup{Fitt}_{j}(I)$ for some $j$. In the case $j=1$, a certain converse
of the Hilbert-Burch Theorem 1.1 was proved.

The outlines of the paper are as follows. In Section 2 we present a wide class
of perfect ideals of grade two for which Conjecture 1.2 holds true. Let
$S=K[x_{1},\dots,x_{n}]$ be a positively graded polynomial ring, and let
$I\subset S$ be a perfect graded ideal of grade two. It is proved in Theorem
2.1 that, if in addition $I$ is generically Gorenstein, then the canonical
trace is the expected one, namely it is generated by the submaximal minors
$I_{\mu(I)-2}(X)$, modulo $I$, of any Hilbert-Burch matrix $X$ of $I$. This
result was already proved in [12, Corollary 3.5] by a different method. Our
new proof is based on Lemma 2.2 and some results of Avramov and Herzog [3].

On the other hand, the inclusion
$I_{\mu(I)-2}(X)R\subseteq\operatorname{tr}(\omega_{R})$ holds true always as
we show in Lemma 2.3. Next, we characterize the perfect monomial ideals of
height two which are generically Gorenstein. It turns out that these ideals
are parameterized by finite simple graphs $G$ with $t$ edges and integer
sequences ${\bf a}$ and ${\bf b}$ of length $t$ such that a certain graph
$G({\bf a,b})$, constructed from such data, is cochordal. This is the content
of Theorem 2.4. The implication (b) $\Rightarrow$ (c) of the proof of this
theorem is in part implicitly contained in the proof of [1, Proposition 2.2].

In Section 3 we consider, among monomial ideals, the simplest class of perfect
monomial ideals of grade two, which are not necessarily generically
Gorenstein. Namely, the Cohen-Macaulay monomial ideals in $S=K[x,y]$. For this
class of ideals, we obtain in Lemma 3.3 a fairly explicit Hilbert-Burch
matrix. As a consequence, we are able to show in Theorem 3.1 that Conjecture
1.2 holds for this class of ideals. Hence, in Proposition 3.5 we successfully
classify the nearly Gorenstein ideals within this class.

As the final conclusion of all of our results, in Theorem 4.1 we classify the
nearly Gorenstein monomial ideals of height two.

##  2\. Perfect generically Gorenstein ideals of height two

Let $S=K[x_{1},\ldots,x_{n}]$ be a graded polynomial ring over a field $K$
with $\deg x_{i}=d_{i}$ for $1\leq i\leq n$. Since $S$ is Cohen-Macaulay, by
[4, Corollary 2.1.4] for any ideal $I\subset S$ we have
$\operatorname{grade}I=\textup{height}(I)$. Thus $I$ is perfect if and only if
$I$ is Cohen-Macaulay.

Recall that $I\subset S$ is called generically Gorenstein if $S_{P}/I_{P}$ is
Gorenstein for any minimal prime $P\in\operatorname{Ass}(I)$. Whereas, $I$ is
called generically a complete intersection if $S_{P}/I_{P}$ is a complete
intersection for any minimal prime $P\in\operatorname{Ass}(I)$. Recall that
given an ideal $J$ of a Noetherian local (or graded) ring $T$ we have
$\mu(J)\geq\operatorname{height}(J)$, and if equality holds $J$ is called a
complete intersection. If $T$ is Cohen-Macaulay, then $J$ is a complete
intersection if and only if $J$ is generated by a regular sequence.

The goal of this section is to prove

######  Theorem 2.1.

Let $I\subset S$ be a graded perfect ideal of height two with $\mu(I)=m$. Let
$X$ be an Hilbert-Burch matrix of $I$. Set $R=S/I$, and assume that $R$ is
generically Gorenstein. Then

$\operatorname{tr}(\omega_{R})=I_{m-2}(X)/I.$

This result was already shown in [12, Corollary 3.5] with a different method.
Here, we provide a different and somewhat easier argument.

We shall need the following

######  Lemma 2.2.

Let $I\subset S$ be a Cohen-Macaulay ideal of height two. Then, the following
conditions are equivalent.

  1. (a)

$I$ is generically Gorenstein.

  2. (b)

$I$ is generically a complete intersection.

###### Proof.

That (b) implies (a) is clear. Conversely, assume (a) and let
$P\in\operatorname{Ass}(I)$ be a minimal associated prime. Then $I_{P}$ is
again a Cohen-Macaulay ideal of height two, and by assumption $S_{P}/I_{P}$ is
Gorenstein. Thus $\mu(I_{P})\geq 2$ and the Cohen-Macaulay type of $I_{P}$ is
one. By the Hilbert-Burch Theorem 1.1 we deduce that $I_{P}$ is generated by
two elements. Hence $I_{P}$ is a complete intersection.

Let $I\subset S$ be a perfect graded ideal of height two, generically a
complete intersection (equivalently, generically Gorenstein), and minimally
generated by $m$ homogeneous polynomials ${\mathbf{f}}=f_{1},\ldots,f_{m}$
with $\deg f_{i}=a_{i}$ for $1\leq i\leq m$. Furthermore, let $X$ be an
Hilbert-Burch matrix of $I$ with homogeneous entries. Set $R=S/I$, and
consider the graded minimal free $S$-resolution

$0\rightarrow\bigoplus_{j=1}^{m-1}S(-b_{j})\xrightarrow{X}\bigoplus_{i=1}^{m}S(-a_{i})\rightarrow
I\rightarrow 0$

of $I$, and the induced exact sequence

$0\rightarrow
C\rightarrow\bigoplus_{j=1}^{m-1}R(-b_{j})\xrightarrow{\psi}\bigoplus_{i=1}^{m}R(-a_{i})\rightarrow
I/I^{2}\rightarrow 0,$

where $C=\operatorname{Ker}\psi$ and $I/I^{2}$ is the so-called conormal
module of $I$.

###### Proof of Theorem 2.1.

Let $X^{\prime}$ be the $(m-1)\times m$-matrix with entries the indeterminates
$x_{ij}$, for $1\leq i\leq m$ and $1\leq j\leq m-1$, and let
$S^{\prime}=S[X^{\prime}]$ and $R^{\prime}=S^{\prime}/I^{\prime}$, where
$I^{\prime}=I_{m-1}(X^{\prime})$. We set $\deg x_{ij}=b_{j}-a_{i}$ for all $i$
and $j$. Then $I^{\prime}$ is a graded ideal in the non-standard graded
polynomial ring $S^{\prime}$ and $R$ is a specialization of $R^{\prime}$. In
other words, the sequence ${\bf g}=x_{11}-f_{11},\dots,x_{m,m-1}-f_{m,m-1}$ is
a homogeneous regular sequence on $S^{\prime}$ and $R^{\prime}$ with
$S^{\prime}/({\bf g})=S$ and $R^{\prime}/({\bf g})=R$.

We have the exact sequence

$0\rightarrow\bigoplus_{j=1}^{m-1}S^{\prime}(-b_{j})\xrightarrow{X^{\prime}}\bigoplus_{i=1}^{m}S^{\prime}(-a_{i})\rightarrow
I^{\prime}\rightarrow 0$

and the induced exact sequence

$\displaystyle 0\rightarrow
C^{\prime}\rightarrow\bigoplus_{j=1}^{m-1}R^{\prime}(-b_{j})\xrightarrow{\psi^{\prime}}\bigoplus_{i=1}^{m}R^{\prime}(-a_{i})\rightarrow
I^{\prime}/I^{\prime 2}\rightarrow 0,$ (1)

where $C^{\prime}=\operatorname{Ker}\psi^{\prime}$.

We denote by the overline “$\overline{\phantom{ll}}$” the reduction modulo
$({\bf g})$. Then $\overline{R^{\prime}}=R$, and the exact sequence (1)
induces the exact sequences

$\displaystyle\overline{C^{\prime}}\rightarrow\bigoplus_{j=1}^{m-1}R(-b_{j})\rightarrow\overline{\operatorname{Im}\psi^{\prime}}\rightarrow
0$ (2)

and

$\displaystyle\overline{\operatorname{Im}\psi^{\prime}}\rightarrow\bigoplus_{i=1}^{m}R(-a_{i})\rightarrow\overline{I^{\prime}/I^{\prime
2}}\rightarrow 0.$ (3)

Now we are going to show that:

  1. (i)

$\overline{I^{\prime}/I^{\prime 2}}=I/I^{2}$;

  2. (ii)

$\overline{C^{\prime}}\rightarrow\bigoplus_{j=1}^{m-1}R(-b_{j})$ is injective;

  3. (iii)

$\overline{\operatorname{Im}\psi^{\prime}}\rightarrow\bigoplus_{i=1}^{m}R(-a_{i})$
is injective.

From (i), (ii) and (iii) it follows that

$0\rightarrow\overline{C^{\prime}}\rightarrow\bigoplus_{j=1}^{m-1}R(-b_{j})\xrightarrow{\psi}\bigoplus_{i=1}^{m}R(-a_{i})\rightarrow
I/I^{2}\rightarrow 0,$

is exact, and this implies that $C=\overline{C^{\prime}}$. This then yields
the desired conclusion.

Proof of (i): We have the exact sequence

$\displaystyle\overline{I^{\prime
2}}\rightarrow\overline{I^{\prime}}\rightarrow\overline{I^{\prime}/I^{\prime
2}}\rightarrow 0.$ (4)

Since ${\bf g}$ is a regular sequence on $R^{\prime}$ and $S^{\prime}$, the
exact sequence $0\rightarrow I^{\prime}\rightarrow S^{\prime}\rightarrow
R^{\prime}\rightarrow 0$ induces the exact sequence

$0\rightarrow\overline{I^{\prime}}\rightarrow S\rightarrow R\rightarrow 0.$

Since the image of the map $\overline{I^{\prime}}\rightarrow S$ is equal to
$I$ we see that $\overline{I^{\prime}}$ is identified with $I$ via the
injective map $\overline{I^{\prime}}\rightarrow S$. Under this identification,
the image of $\overline{I^{\prime 2}}\rightarrow\overline{I^{\prime}}$ in the
exact sequence (4) is identified with $I^{2}$. Thus, (4) shows that
$\overline{I^{\prime}/I^{\prime 2}}=I/I^{2}$.

Proof of (ii): Here and in the proof of (iii) we use the following facts: Let
$T$ be a polynomial ring over a field $K$, and let $J\subset T$ be a graded
perfect ideal of height two, minimally generated by the homogeneous elements
${\bf h}=h_{1},\ldots,h_{m}$. Let $H_{i}(J;T)$ be the Koszul homology
$H_{i}({\bf h};T)$. This notation is justified because Koszul homologies do
not depend on the minimal set of generators of $J$ [4, Proposition 1.6.21].
Then,

  1. (a)

[3, Theorem (2.1)] The Koszul homology modules $H_{i}(J;T)$ are maximal Cohen-
Macaulay $T/J$-modules.

  2. (b)

[3, Theorem (3.2)] Let $\deg h_{i}=a_{i}$ for $1\leq i\leq m$. Then the
natural sequence

$0\rightarrow H_{1}(J;T)\rightarrow\bigoplus_{i=1}^{m}(T/J)(-a_{i})\rightarrow
J/J^{2}\rightarrow 0$

is exact, if $J$ is generically a complete intersection.

We may apply this to $I^{\prime}\subset S^{\prime}$. It follows from (1) that
$\operatorname{Im}\psi^{\prime}=H_{1}(I^{\prime};S^{\prime})$, since
$R^{\prime}=S^{\prime}/I^{\prime}$ is a domain by [4, Theorem 7.3.1(c)].
Therefore,

$0\rightarrow
C^{\prime}\rightarrow\bigoplus_{j=1}^{m-1}R^{\prime}(-b_{j})\rightarrow\operatorname{Im}\psi^{\prime}\rightarrow
0$

is an exact sequence of maximal Cohen-Macaulay modules. Hence this sequence
remains exact after reduction modulo the regular sequence ${\bf g}$. This
proves (ii).

Proof of (iii): The statement (iii) is equivalent to saying that the natural
map
$\overline{\operatorname{Im}\psi^{\prime}}\rightarrow\operatorname{Im}\psi$ is
an isomorphism. Therefore by (b) it amounts to show the the natural map
$\overline{H_{1}(I^{\prime};S^{\prime})}\rightarrow H_{1}(I;S)$ is an
isomorphism.

The desired isomorphism follows by induction on the length of the sequence
${\bf g}$ from the following claim: let $J\subset T$ be as above a perfect
ideal of height two, which is not necessarily a generically complete
intersection. Let $g\in T$ be a homogeneous polynomial of degree $d$ which is
regular on $T/J$. Then the natural map

$\overline{H_{1}(J;T)}\rightarrow H_{1}(J;\overline{T})$

is an isomorphism. (Note that $\overline{T/J}=\overline{T}/\overline{J}$ and
$H_{1}(J;\overline{T})=H_{1}(\overline{J};\overline{T})$).

Observing that for any partial sequence of ${\bf g}$, the residue class ring
modulo this partial sequence is again a polynomial ring, we can proceed with
the induction by using the statement of the claim.

For the proof of the claim we consider the short exact sequence

$0\rightarrow T(-d)\xrightarrow{g}T\rightarrow\overline{T}\rightarrow 0,$

which induces the long exact sequence of Koszul homology

$\displaystyle\cdots\rightarrow
H_{1}(J;T(-d))\xrightarrow{g}H_{1}(J;T)\rightarrow
H_{1}(J;\overline{T})\rightarrow H_{0}(J;T(-d))\xrightarrow{g}H_{0}(J;T).$

Note that $H_{0}(J;T)\cong T/J$. Since we assume that $g$ is regular on $T/J$,
the map $H_{0}(J;T(-d))\xrightarrow{g}H_{0}(J;T)$ is injective. This implies
that

$H_{1}(J;T(-d))\xrightarrow{g}H_{1}(J;T)\rightarrow
H_{1}(J;\overline{T})\rightarrow 0$

is exact. Since the kernel of the second map is the submodule of $H_{1}(J;T)$
generated by $g$, we see that the natural map
$\overline{H_{1}(J;T)}\rightarrow H_{1}(J;\overline{T})$ is an isomorphism.

On the other hand, the inclusion
$I_{m-2}(X)R\subseteq\operatorname{tr}(\omega_{R})$ is always true, without
the generically Gorenstein assumption on $R$. For $m\geq 1$, we set
$[m]=\\{1,2,\dots,m\\}$.

######  Lemma 2.3.

Let $I\subset S$ be a graded perfect ideal of height two with $\mu(I)=m$ and
let $a\in S$ be a non-zero element. Set $J=aI$ and $R=S/J$. The minimal free
resolution of $J$ induces an exact sequence

$0\rightarrow C\xrightarrow{\alpha}R^{m-1}\xrightarrow{\psi}R^{m}\rightarrow
J/J^{2}\rightarrow 0,$

where $C=\operatorname{Ker}\psi$ and $\alpha$ is the inclusion map. Then

$aI_{m-2}(\psi)\subseteq I_{1}(\alpha).$

###### Proof.

Let ${\bf f}_{1},\dots,{\bf f}_{m-1}$ and ${\bf e}_{1},\dots,{\bf e}_{m}$ be
two bases for $R^{m-1}$ and $R^{m}$. By the Hilbert-Burch Theorem 1.1, $\psi$
is represented by the image in $R$ of an homogeneous $m\times(m-1)$ Hilbert-
Burch matrix $X=(a_{ij})$ of $I$ with respect to the given bases. For an
element $f\in S$, we denote again by $f$ the image in $R$.

For two subsets $A\subset[m]$, $B\subset[m-1]$ of the same size, we denote by
$[A|B]$ the minor of $X$ whose rows are indexed by $A$ and whose columns are
indexed by $B$.

For any $A\subset[m]$ with $|A|=m-2$ we set

$c_{A}\ =\ \sum_{j=1}^{m-1}(-1)^{j+1}a[A|[m-1]\setminus\\{j\\}]{\bf f}_{j}.$

Note that

$\displaystyle\psi(c_{A})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m-1}(-1)^{j+1}a[A|[m-1]\setminus\\{j\\}]\sum_{i=1}^{m}a_{ij}{\bf
e}_{i}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}(\sum_{j=1}^{m-1}a(-1)^{j+1}a_{ij}[A|[m-1]\setminus\\{j\\}]){\bf
e}_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{m}\pm a[A_{i}|[m-1]]{\bf
e}_{i},$

where $A_{i}$ is obtained from $A$ by adding the $i$th column of $X$ to $A$.
If $i\in A$, then $[A_{i}|[m-1]]=0$, and if $i\notin A$, then
$a[A_{i}|[m-1]]\in aI_{m-1}(X)=aI=J$ and hence is equal to $0$ in $R$.
Therefore, $c_{A}\in C$ for all $A\subset[m]$ with $|A|=m-2$. This shows that
$aI_{m-2}(\psi)\subseteq I_{1}(\alpha)$, and concludes the proof.

Next, we provide a structure theorem for perfect monomial ideals of height two
which are generically Gorenstein. To state this result, we need some
preparation.

Hereafter, $S$ denotes the standard graded polynomial ring. That is $\deg
x_{i}=1$ for all $i$. Let $u={\bf x^{a}}=x^{a_{1}}\cdots x_{n}^{a_{n}}\in S$
be a monomial, where ${\bf a}=(a_{1},\dots,a_{n})\in{\mathbb{Z}}_{\geq
0}^{n}$. The polarization of $u$ is the monomial

$u^{\wp}=\prod_{i=1}^{n}(\prod_{j=1}^{a_{i}}x_{i,j})=\prod_{\begin{subarray}{c}1\leq
i\leq n\\\ a_{i}>0\end{subarray}}x_{i,1}x_{i,2}\cdots x_{i,a_{i}}.$

Whereas, the polarization of a monomial ideal $I\subset S$, with minimal
monomial generating set $G(I)$, is defined as the monomial ideal $I^{\wp}$
with minimal generating set $G(I^{\wp})=\\{u^{\wp}:u\in G(I)\\}$, in the
polynomial ring $S^{\wp}$ in the variables $x_{i,j}$. Given monomial ideals
$I_{1},\dots,I_{t}\subset S$, we have $(I_{1}\cap\dots\cap
I_{t})^{\wp}=I_{1}^{\wp}\cap\dots\cap I_{t}^{\wp}$.

Let $G$ be a finite simple graph with vertex set $\\{x_{1},\dots,x_{n}\\}$
with $t$ edges, and let ${\bf a}:a_{1},\dots,a_{t}$ and ${\bf
b}:b_{1},\dots,b_{t}$ be two sequences of positive integers. Suppose that
$E(G)=\\{\\{x_{i_{\ell}},x_{j_{\ell}}\\}:1\leq\ell\leq t\\}$. We associate to
$G$, ${\bf a}$ and ${\bf b}$ the height-unmixed height two monomial ideal of
$S=K[x_{1},\dots,x_{n}]$,

$I_{G,{\bf a},{\bf b}}\ =\
\bigcap_{\ell=1}^{t}(x_{i_{\ell}}^{a_{\ell}},x_{j_{\ell}}^{b_{\ell}}).$ (5)

We define the intersection graph $G({\bf a,b})$ of $I_{G,{\bf a},{\bf b}}$ as
follows:

The vertex set of $G({\bf a,b})$ is

$V(G({\bf a,b}))\ =\ \bigcup_{i=1}^{n}\big{\\{}x_{i,j}\ :\ 1\leq
j\leq\max_{\ell,k}(\\{a_{\ell}:a_{\ell}=i\\}\cup\\{b_{k}:b_{k}=i\\})\big{\\}},$

and the edge set of $G({\bf a,b})$ is

$E(G({\bf a,b}))\ =\
\bigcup_{\ell=1}^{t}(\bigcup_{p=1}^{a_{\ell}}\bigcup_{q=1}^{b_{\ell}}\\{x_{i_{\ell},p},x_{j_{\ell},q}\\}).$

Finally, we recall that a graph $G$ is called chordal if it has no induced
cycles of length bigger than three. Whereas, $G$ is called cochordal if the
complementary graph $G^{c}$ is chordal. Here, the complementary graph $G^{c}$
of $G$ is the graph with the same vertex set of $G$ and whose edges are the
non edges of $G$.

For unexplained facts of the theory of Alexander duality on monomial ideals we
refer the reader to [18, Subsections 1.5.3 and 8.1.2].

######  Theorem 2.4.

Let $I\subset S=K[x_{1},\dots,x_{n}]$ be a monomial ideal. Then, the following
conditions are equivalent.

  1. (a)

$I$ is a perfect monomial ideal of height two which is generically a complete
intersection.

  2. (b)

$I$ is a perfect monomial ideal of height two which is generically Gorenstein.

  3. (c)

$I=I_{G,{\bf a,b}}$ for some graph $G$ with $t$ edges, and some integer
sequences ${\bf a}$, ${\bf b}$ of length $t$ such that $G({\bf a,b})$ is a
cochordal graph.

###### Proof.

(a) $\Rightarrow$ (b) This implication is trivial.

(b) $\Rightarrow$ (c) Since $I$ is perfect, $S/I$ is Cohen-Macaulay, and thus
$I$ is height-unmixed of height two. We are going to prove that $I=I_{\bf
G,{\bf a,b}}$ for some graph $G$ and some integer sequences ${\bf a,b}$ such
that $G({\bf a,b})$ is cochordal. Let $I=Q_{1}\cap\dots\cap Q_{t}$ be the
standard primary decomposition of $I$, cfr [18, Subsection 1.3.2, page 12].
Then $Q_{\ell}=(x_{i_{\ell}}^{a_{\ell}},x_{j_{\ell}}^{b_{\ell}})$ for all
$1\leq\ell\leq t$. Let $P_{\ell}=\sqrt{Q_{\ell}}=(x_{i_{\ell}},x_{j_{\ell}})$
for all $\ell$. Hence $\textup{Ass}(I)=\\{P_{1},\dots,P_{t}\\}$. Applying
localization and [2, Proposition 4.9] we obtain

$I_{P_{k}}=(\bigcap_{\ell=1}^{t}Q_{\ell})_{P_{k}}=\bigcap_{\ell=1}^{t}(Q_{\ell})_{P_{k}}=\bigcap_{\ell\
:\ P_{\ell}=P_{k}}(Q_{\ell})_{P_{k}}.$ (6)

We may replace ordinary localization with the so-called monomial localization.
Indeed, recall that for a monomial ideal $L\subset S$, the monomial
localization of $L$ with respect to $P=(x_{i_{1}},\dots,x_{i_{r}})$ is the
monomial ideal $L(P)$ in the polynomial ring
$S(P)=K[x_{i_{1}},\dots,x_{i_{r}}]$ obtained by applying the substitutions
$x_{i}\mapsto 1$ for $x_{i}\notin P$. Since we have the equality
$L(P)S_{P}=LS_{P}$, we can indeed replace ordinary localization with monomial
localization. Thus, equation (6) becomes

$I(P_{k})=\bigcap_{\ell\ :\ P_{\ell}=P_{k}}Q_{\ell}.$ (7)

Since $I$ is generically Gorenstein, $I(P_{k})$ must be a Gorenstein ideal of
$K[x_{i_{k}},x_{j_{k}}]$. Let $p$ be the projective dimension of $I(P_{k})$.
The Cohen-Macaulay type of $I(P_{k})$ must be $\beta_{p}(I(P_{k}))=1$. Notice
that $I(P_{k})$ has at least two generators. It follows from Lemma 3.3, in the
next section, that $p=1$ and $\mu(I(P_{k}))=\beta_{1}(I(P_{k}))+1$ is exactly
two. To simplify the notation, put $x=x_{i_{\ell}}$ and $y=x_{j_{\ell}}$. Thus
$I(P_{k})=(x^{c_{1}}y^{d_{1}},x^{c_{2}}y^{d_{2}})$ with $c_{1}>c_{2}\geq 0$
and $0\leq d_{1}<d_{2}$. Since $I(P_{k})$ is Cohen-Macaulay and not a
principal ideal, Lemma 3.4 implies that $c_{2}=d_{1}=0$. Thus
$I(P_{k})=(x^{c_{1}},y^{d_{2}})$ is $(x,y)$-primary and since (7) is again a
standard primary decomposition of $I(P_{k})$, it follows that there is only
one $\ell$ such that $P_{\ell}=P_{k}$, namely $\ell=k$.

Summarizing our argument, we have shown that all components
$Q_{1},\dots,Q_{t}$ have a different radical. Thus
$\\{i_{p},j_{p}\\}\neq\\{i_{q},j_{q}\\}$ for all $1\leq p<q\leq t$. This fact
shows that $I=I_{G,{\bf a,b}}$ for some finite simple graph $G$ on vertex set
$\\{x_{1},\dots,x_{n}\\}$ with $t$ edges and with ${\bf a}:a_{1},\dots,a_{t}$
and ${\bf b}:b_{1},\dots,b_{t}$.

Since polarization commutes with intersections, we have

$I^{\wp}=I_{G,{\bf
a,b}}^{\wp}=\bigcap_{\ell=1}^{t}(\prod_{p=1}^{a_{\ell}}x_{i_{\ell},p},\prod_{q=1}^{b_{\ell}}x_{j_{\ell},q})=\bigcap_{\ell=1}^{t}[\bigcap_{p=1}^{a_{\ell}}\bigcap_{q=1}^{b_{\ell}}(x_{i_{\ell},p},x_{j_{\ell},q})]=I_{G({\bf
a,b}),{\bf 1,1}},$

where ${\bf 1}:1,1,\dots,1$ is a sequence of length
$\sum_{\ell=1}^{t}a_{\ell}b_{\ell}$.

It follows from [18, Corollary 1.6.3] that $S/I$ is a Cohen-Macaulay ring if
and only if $S^{\wp}/I^{\wp}$ is such. Notice that the Alexander dual
$(I^{\wp})^{\vee}$ of $I^{\wp}$ is the edge ideal of the graph $G({\bf a,b})$.
It follows from the Eagon-Reiner criterion [18, Theorem 8.1.9] that
$S^{\wp}/I^{\wp}$ is Cohen-Macaulay if and only if $(I^{\wp})^{\vee}$ has a
linear resolution. By Fröberg theorem [18, Theorem 9.2.3] this is equivalent
to $G({\bf a,b})$ being a cochordal graph. Assertion (c) follows.

(c) $\Rightarrow$ (a) Suppose $I=I_{G,{\bf a,b}}$ and $G({\bf a,b})$ is
cochordal. It follows by the very definition and structure of $I_{G,{\bf
a,b}}$, that this ideal has height two and that $S/I$ is generically a
complete intersection. As before, we have $I^{\wp}=I_{G({\bf a,b}),{\bf 1,1}}$
and by Fröberg theorem [18, Theorem 9.2.3] $(I^{\wp})^{\vee}$ has linear
resolution because it is the edge ideal of a cochordal graph. By [18,
Corollary 1.6.3] and the Eagon-Reiner criterion [18, Theorem 8.1.9] we
conclude that $I$ is perfect.

##  3\. The canonical trace of Cohen-Macaulay algebras $K[x,y]/I$

Let $S=K[x,y]$. The set of monomial ideals $I\subset S$ is in bijection with
the set of all pairs $({\bf a},{\bf b})$ of integer sequences satisfying the
conditions

${\bf a}:a_{1}>a_{2}>\cdots>a_{m}\geq
0\,\,\,\,\,\,\,\,\textup{and}\,\,\,\,\,\,\,\,{\bf b}:0\leq
b_{1}<b_{2}<\cdots<b_{m}.$ (8)

Indeed, if $I$ is a monomial ideal and $G(I)$ is its minimal monomial
generating set, then
$G(I)=\\{x^{a_{1}}y^{b_{1}},x^{a_{2}}y^{b_{2}},\dots,x^{a_{m}}y^{b_{m}}\\}$
for two sequences ${\bf a}$ and ${\bf b}$ as above. Conversely, if ${\bf a}$
and ${\bf b}$ are two sequences as above, then
$x^{a_{1}}y^{b_{1}},x^{a_{2}}y^{b_{2}},\dots,x^{a_{m}}y^{b_{m}}$ is the
minimal generating set of a monomial ideal $I\subset S$.

Hereafter, we set $I_{\bf
a,b}=(x^{a_{1}}y^{b_{1}},x^{a_{2}}y^{b_{2}},\dots,x^{a_{m}}y^{b_{m}})$, where
${\bf a}$ and ${\bf b}$ are as in (8).

The aim of this section is to prove:

######  Theorem 3.1.

Let $I\subset S=K[x,y]$ be a monomial ideal such that $R=S/I$ is Cohen-
Macaulay. Let $X$ be an Hilbert-Burch matrix of $I$. Then

$\operatorname{tr}(\omega_{R})=I_{\mu(I)-2}(X)/I.$

Theorem 3.1 follows from the following slightly more general statement.

######  Theorem 3.2.

Let $I=I_{\bf a,b}\subset S=K[x,y]$ be a monomial ideal with $\mu(I)=m\geq 2$.
Let ${\mathbb{F}}:0\rightarrow S^{m-1}\xrightarrow{\varphi}S^{m}\rightarrow
I\rightarrow 0$ be the minimal free resolution of $I$. Set $R=S/I$. We have
the exact sequence

$0\rightarrow C\xrightarrow{\alpha}R^{m-1}\xrightarrow{\psi}R^{m}\rightarrow
I/I^{2}\rightarrow 0,$

where $C=\operatorname{Ker}\psi$, $\alpha$ is the inclusion map and
$\psi=\varphi\otimes\textup{id}_{R}$. In particular,

$I_{1}(\alpha)\ =\ x^{a_{m}}y^{b_{1}}I_{m-2}(\psi).$

For the proof of this result we need the following lemma.

######  Lemma 3.3.

Let $m\geq 2$. The minimal free resolution ${\mathbb{F}}$ of $I_{\bf a,b}$ is

${\mathbb{F}}\ \ :\ \ 0\rightarrow\bigoplus_{i=1}^{m-1}S{\bf
f}_{i}\xrightarrow{\varphi}\bigoplus_{i=1}^{m}S{\bf
e}_{i}\xrightarrow{\varepsilon}I_{\bf a,b}\rightarrow 0,$

where ${\bf f}_{i}$ has multidegree $x^{a_{i}}y^{b_{i+1}}$, for $1\leq i\leq
m-1$, ${\bf e}_{i}$ has multidegree $x^{a_{i}}y^{b_{i}}$ and $\varepsilon({\bf
e}_{i})=(-1)^{i}x^{a_{i}}y^{b_{i}}$, for $1\leq i\leq m$, and $\varphi$ is
represented with respect to the given bases by the following Hilbert-Burch
matrix:

$X\ =\ \left(\begin{matrix}\phantom{-}y^{b_{2}-b_{1}}&&&\\\
-x^{a_{1}-a_{2}}&\phantom{-}y^{b_{3}-b_{2}}&&\\\ &-x^{a_{2}-a_{3}}&\ddots&\\\
&&\ddots&\phantom{-}y^{b_{m}-b_{m-1}}\\\
&&&-x^{a_{m-1}-a_{m}}\end{matrix}\right).$ (9)

###### Proof.

Let ${\bf c}:c_{1}>\dots>c_{m-1}>c_{m}=0$ and ${\bf
d}:0=d_{1}<d_{2}<\dots<d_{m}$, where $c_{i}=a_{i}-a_{m}$ and
$d_{i}=b_{i}-b_{1}$ for $1\leq i\leq m$. Since
$x^{a_{1}-a_{m}},y^{b_{m}-b_{1}}\in I_{\bf c,d}$ and these elements form a
regular sequence, we have $\operatorname{depth}I_{\bf c,d}=2$.

It is easily seen that
$x^{a_{m}}y^{b_{1}}\cdot[[m]\setminus\\{i\\}\,|\,[m-1]]=(-1)^{m-i}x^{a_{i}}y^{b_{i}}$
for all $1\leq i\leq m$. Therefore $x^{a_{m}}y^{b_{1}}I_{m-2}(X)=I_{\bf a,b}$.
Since $I_{m-2}(X)=I_{\bf c,d}$ and $\operatorname{depth}I_{m-2}(X)=2$, Theorem
1.1 implies that ${\mathbb{F}}$ is indeed the minimal free resolution of
$I_{\bf a,b}$.

Given integers $k\leq h$, we denote by $[k,h]$ the interval
$\\{k,k+1,\dots,h\\}$.

Let $\psi=\varphi\otimes\textup{id}_{R}$. We denote the bases $\\{{\bf
f}_{i}\otimes 1\\}_{1\leq i\leq m-1}$ and $\\{{\bf e}_{i}\otimes 1\\}_{1\leq
i\leq m}$ again by $\\{{\bf f}_{i}\\}_{1\leq i\leq m-1}$ and $\\{{\bf
e}_{i}\\}_{1\leq i\leq m}$, respectively. A matrix representing $\psi$ with
respect to these bases is the $m\times(m-1)$ matrix whose entries are the
images in $R=S/I$ of the entries of the matrix $X$ given in (9). We denote
this matrix again by $X$, and we denote the residue class of any element $f\in
S$ in $R$ again by $f$.

###### Proof of Theorem 3.2.

Let ${\bf w}=\sum_{i=1}^{m-1}w_{i}{\bf f}_{i}\in\operatorname{Ker}\psi$.
Proceeding by induction on $m\geq 2$, we will show that $w_{i}\in
x^{a_{m}}y^{b_{1}}I_{m-2}(\psi)$ for all $i$, with the base case $m=2$ being
trivial. This implies that $I_{1}(\alpha)\subseteq
x^{a_{m}}y^{b_{1}}I_{m-2}(\psi)$. Since the opposite inclusion holds by Lemma
2.3, the desired equality follows.

By Lemma 3.3, $I_{j}(\psi)=I_{j}(X)$ for all $j$, where $X$ is given in (9).

The resolution ${\mathbb{F}}$ of $I$ is multigraded. Therefore, we may assume
that each $w_{i}$ is a monomial of $R$. Let $p=\min\\{i:w_{i}\neq 0\\}$ and
$q=\max\\{i:w_{i}\neq 0\\}$. Then

$\displaystyle\psi({\bf w})\ $ $\displaystyle=\ (y^{b_{p+1}-b_{p}}w_{p})\,{\bf
e}_{p}$ $\displaystyle+\
\sum_{i=p+1}^{q}(y^{b_{i+1}-b_{i}}w_{i}-x^{a_{i-1}-a_{i}}w_{i-1})\,{\bf
e}_{i}$ $\displaystyle-\ (x^{a_{q}-a_{q+1}}w_{q})\,{\bf e}_{q+1}\ =\ 0.$

All the coefficients in this expression must be zero in $R=S/I$. Since $I$ is
a monomial ideal, by [18, Corollary 1.1.3] we have

$y^{b_{p+1}-b_{p}}w_{p},\,\,x^{a_{q}-a_{q+1}}w_{q}\in I$ (10)

and

$\textup{either}\,\,\,\,\,\,\,\,y^{b_{i+1}-b_{i}}w_{i},\,\,x^{a_{i-1}-a_{i}}w_{i-1}\in
I\,\,\,\,\,\,\,\,\textup{or}\,\,\,\,\,\,\,\,y^{b_{i+1}-b_{i}}w_{i}=x^{a_{i-1}-a_{i}}w_{i-1},$

for $p<i\leq q$.

Notice that if $y^{b_{i+1}-b_{i}}w_{i}$, $x^{a_{i-1}-a_{i}}w_{i-1}\in I$ for
some $p<i\leq q$, then both the elements ${\bf
w}^{\prime}=\sum_{j=p}^{i-1}w_{j}{\bf f}_{j}$ and ${\bf
w}^{\prime\prime}=\sum_{j=i}^{q}w_{j}{\bf f}_{j}$ are again in
$\operatorname{Ker}\psi$. Hence, we may furthermore assume that

$y^{b_{i+1}-b_{i}}w_{i}=x^{a_{i-1}-a_{i}}w_{i-1}\,\,\,\,\,\textup{for
all}\,\,\,\,p<i\leq q.$ (11)

Let $J=(I:y^{b_{p+1}-b_{p}})$ and $L=(I:x^{a_{q}-a_{q+1}})$. By [18,
Proposition 1.2.2] we have
$J=(x^{a_{j}}y^{\max\\{b_{j}+b_{p}-b_{p+1},0\\}}:1\leq j\leq m)$ and
$L=(x^{\max\\{a_{j}-a_{q}+a_{q+1},0\\}}y^{b_{j}}:1\leq j\leq m)$. By equation
(10), $w_{p}\in J$ and $w_{q}\in L$.

Hence, $x^{a_{h}}y^{\max\\{b_{h}+b_{p}-b_{p+1},0\\}}$ divides $w_{p}$ for some
$1\leq h\leq m$. Using equation (11) for $i=p+1$, we see that
$x^{a_{h}+a_{p}-a_{p+1}}$ divides
$x^{a_{p}-a_{p+1}}w_{p}=y^{b_{p+2}-b_{p+1}}w_{p+1}$ and so divides $w_{p+1}$.
Using again (11) for $i=p+2$, we see that $x^{a_{h}+a_{p}-a_{p+2}}$ divides
$w_{p+2}$. Iterating this reasoning, we obtain that

$x^{a_{h}+a_{p}-a_{i}}\,\,\,\,\textup{divides}\,\,\,\,w_{i}\,\,\,\,\,\textup{for
all}\,\,\,\,p\leq i\leq q.$ (12)

Similarly, $x^{\max\\{a_{k}-a_{q}+a_{q+1},0\\}}y^{b_{k}}$ divides $w_{q}$ for
some $1\leq k\leq m$. Using (11) for $i=q$ we see that
$y^{b_{k}+b_{q+1}-b_{q}}$ divides
$y^{b_{q+1}-b_{q}}w_{q}=x^{a_{q-1}-a_{q}}w_{q-1}$. Hence
$y^{b_{k}+b_{q+1}-b_{q}}$ divides $w_{q-1}$. Using again (11) for $i=q-1$, we
see that $y^{b_{k}+b_{q+1}-b_{q-1}}$ divides $w_{q-2}$. Iterating this
procedure, we see that

$y^{b_{k}+b_{q+1}-b_{i+1}}\,\,\,\,\textup{divides}\,\,\,\,w_{i}\,\,\,\,\,\textup{for
all}\,\,\,\,p\leq i\leq q.$ (13)

We may assume that the following two conditions are satisfied.

  1. (i)

$k<h$, and

  2. (ii)

$(\min\\{p,k\\},\max\\{q+1,h\\})\neq(1,m)$.

Indeed, suppose (i) does not hold. Then $k\geq h$. Since $p\leq i$, then
$a_{p}-a_{i}\geq 0$. Thus, by equation (12), $x^{a_{h}}$ divides $w_{i}$ for
$p\leq i\leq q$. Similarly, since $i\leq q$, then $b_{q+1}-b_{i+1}\geq 0$, and
from equation (13) we obtain that $y^{b_{k}}$ divides $w_{i}$ for $p\leq i\leq
q$. Hence $x^{a_{h}}y^{b_{k}}$ divides $w_{i}$ for all $i$. Since $h\leq k$,
then $a_{k}\leq a_{h}$ and so $x^{a_{k}}y^{b_{k}}$ divides $w_{i}$ for all
$i$. Notice that

$x^{a_{k}}y^{b_{k}}\ =\ x^{a_{m}}y^{b_{1}}\cdot
x^{a_{k}-a_{m}}y^{b_{k}-b_{1}}\ =\
\pm\,x^{a_{m}}y^{b_{1}}\cdot[[m]\setminus\\{k\\}\,|\,[m-1]].$

Hence $w_{i}\in x^{a_{m}}y^{b_{1}}I_{m-1}(X)\subset
x^{a_{m}}y^{b_{1}}I_{m-2}(X)$ for all $i$.

Next, suppose (ii) does not hold. Then, one of the following possibilities
occurs: (a) $(p,q)=(1,m-1)$, (b) $(p,h)=(1,m)$, (c) $(k,q)=(1,m-1)$, (d)
$(k,h)=(1,m)$.

Assume that (a) holds. Since $a_{h}\geq a_{m}$ and $b_{k}\geq b_{1}$, the
equations (12) and (13) imply that
$x^{a_{1}-a_{i}+a_{m}}y^{b_{m}-b_{i+1}+b_{1}}$ divides $w_{i}$ for all $i$.
Since

$x^{a_{m}}y^{b_{1}}\cdot x^{a_{1}-a_{i}}y^{b_{m}-b_{i+1}}\ =\
\pm\,x^{a_{m}}y^{b_{1}}\cdot[[m]\setminus\\{1,m\\}\,|\,[m-1]\setminus\\{i\\}]$

for all $i$, we conclude that $w_{i}\in x^{a_{m}}y^{b_{1}}I_{m-2}(X)$ for all
$i$. The cases (b), (c) and (d) can be treated similarly.

The conditions (i) and (ii) imply that $[\min\\{p,k\\},\max\\{q+1,h\\}]$ is a
proper subset of $[m]$. Say it is contained in either $[r,m]$ for some $r>1$
or $[1,s]$ for some $s<m$.

Assume that $[\min\\{p,k\\},\max\\{q+1,h\\}]\subseteq[r,m]$. The other case is
analogous. Then $p,k\geq r$. Let
$I_{1}=(x^{a_{r}}y^{b_{r}},x^{a_{r+1}}y^{b_{r+1}},\dots,x^{a_{m}}y^{b_{m}})$
and set $R_{1}=S/I_{1}$. Let ${\mathbb{F}}_{1}:0\rightarrow
S^{m-r}\xrightarrow{\varphi_{1}}S^{m-r+1}\rightarrow I_{1}\rightarrow 0$ be
the minimal free resolution of $I_{1}$ described in Lemma 3.3, where $S^{m-r}$
has basis ${\bf f}_{1,1},\dots,{\bf f}_{1,m-r}$, $S^{m-r+1}$ has basis ${\bf
e}_{1,1},\dots,{\bf e}_{1,m-r+1}$ and $\varphi_{1}$ is described by the
Hilbert-Burch matrix $X_{1}$ given in (9). Notice that $X_{1}$ is the
submatrix of $X$ obtained by deleting the first $r-1$ rows and columns. Let
$C_{1}$ be the kernel of $\psi_{1}=\varphi_{1}\otimes\textup{id}_{R_{1}}$.
Since $h>k\geq r$, equations (10) and (11) imply that the element ${\bf
w}_{1}=\sum_{j=p}^{q}w_{j}{\bf f}_{1,j}\in C_{1}$. Since
$\mu(I_{1})=m-r<\mu(I)$, by induction on $m$ we obtain that each $w_{i}$ is
divided by a monomial $v_{i}=x^{a_{m}}y^{b_{r}}M_{i}\in
x^{a_{m}}y^{b_{r}}I_{m-(r-1)-2}(X_{1})$ for all $p\leq i\leq q$, where $M_{i}$
is a $(m-(r-1)-2)$-minor of $X_{1}$. Notice that

$X\ =\ \left(\begin{array}[]{ccc|ccccc}\phantom{-}y^{b_{2}-b_{1}}&&&&&&&\\\
-x^{a_{1}-a_{2}}&\ddots&&&&&&\\\
&\ddots&\\!\\!\\!\phantom{-}y^{b_{r}-b_{r-1}}&&&&&\\\
\hline\cr&&\\!\\!\\!-x^{a_{r-1}-a_{r}}&&&&&\\\ &&&&&&&\\\ &&&&&\,\,{\Huge
X_{1}}&&\\\\[0.7pt] &&&&&&&\end{array}\right).$ (14)

For all $i$, let $N_{i}$ be the $(m-2)$-minor of $X$ whose rows are the first
$r-1$ rows of $X$ together with those of $M_{i}$, and whose columns are the
first $r-1$ columns of $X$ together with those of $M_{i}$. From equation (14),
we see that $y^{b_{r}-b_{1}}M_{i}=N_{i}$ for all $i$. Hence $w_{i}\in
x^{a_{m}}y^{b_{1}}I_{m-2}(X)$ for all $i$, as desired.

######  Lemma 3.4.

The ring $K[x,y]/I_{\bf a,b}$ is Cohen-Macaulay if and only if $m=1$ or $m>1$
and $a_{m}=b_{1}=0$.

###### Proof.

Set $S=K[x,y]$ and $I=I_{\bf a,b}$. If $m=1$, then $I$ is a principal ideal
and thus $S/I$ is Cohen-Macaulay. Let $m>1$. Then [13, Proposition 5.1(c)]
implies that ${\mathfrak{m}}=(x,y)\in\operatorname{Ass}(I)$. Since $S/I$ is
Cohen-Macaulay, $I$ is height-unmixed, and so
$(x),(y)\notin\operatorname{Ass}(I)$. Hence [13, Proposition 5.1(a)-(b)]
implies that $a_{m}=b_{1}=0$. Conversely, if $a_{m}=b_{1}=0$, then
$x^{a_{1}},y^{b_{m}}\in I$ and $\dim S/I=0$. Thus $\operatorname{depth}S/I=0$
as well, and $S/I$ is Cohen-Macaulay.

###### Proof of Theorem 3.1.

Let $I=I_{\bf a,b}$ and $\mu(I)=m$. If $m=1$, then $R$ is Gorenstein and
$\operatorname{tr}(\omega_{R})=R=I_{-1}(X)/I$. Let $m>1$. Then $a_{m}=b_{1}=0$
by Lemma 3.4, and the assertion follows from Theorem 3.2 and [19, Corollary
3.2].

We have the following classification.

######  Proposition 3.5.

Let $I\subset S=K[x,y]$ be a monomial ideal such that $R=S/I$ is Cohen-
Macaulay. Then $R$ is nearly Gorenstein if and only if $\mu(I)=1$, $\mu(I)=2$
or $\mu(I)=3$ and $I=(x^{a},x^{b}y^{c},y^{d})$ for some integers $a>b\geq 0$,
$0\leq c<d$ satisfying the conditions: (i) $a-b=1$ or $b=1$, (ii) $d-c=1$ or
$c=1$, and (iii) $b+c\geq 1$.

###### Proof.

If $\mu(I)=1$ or $\mu(I)=2$, then $R$ is a complete intersection, and thus a
(nearly) Gorenstein ring. If $\mu(I)>3$, then
$\operatorname{tr}(\omega_{R})\subseteq{\mathfrak{m}}^{2}$, where
${\mathfrak{m}}=(x,y)R$. Finally assume $\mu(I)=3$. Then, by Lemma 3.4,
$I=(x^{a_{1}},x^{a_{2}}y^{b_{2}},y^{b_{3}})$ where $a_{1}>a_{2}\geq 0$, $0\leq
b_{2}<b_{3}$ and $a_{2}+b_{2}\geq 1$. By equation (9), an Hilbert-Burch matrix
of $I$ is

$\left(\begin{matrix}\phantom{-}y^{b_{2}}&0\\\
-x^{a_{1}-a_{2}}&\phantom{-}y^{b_{3}-b_{2}}\\\
0&-x^{a_{2}}\end{matrix}\right).$

Hence
$\operatorname{tr}(\omega_{R})=(x^{a_{1}-a_{2}},x^{a_{2}},y^{b_{3}-b_{2}},y^{b_{2}})/I$,
and the assertion follows.

##  4\. Nearly Gorenstein monomial ideals of height two

We conclude the paper with the classification of the nearly Gorenstein
monomial ideals of height two. For a monomial $u\in S$, we set
$\operatorname{supp}(u)=\\{x_{i}:x_{i}\ \textup{divides}\ u\\}$.

######  Theorem 4.1.

Up to a relabeling of the variables, the only nearly Gorenstein monomial
ideals in $S=K[x_{1},\dots,x_{n}]$ of height two are:

  1. (a)

$(u,v)$ for two monomials $u,v\in S$ with
$\operatorname{supp}(u)\cap\operatorname{supp}(v)=\emptyset$.

  2. (b)

$(x_{1}^{a},x_{1}^{b}x_{2}^{c},x_{2}^{d})$ with $n=2$ and $a>b\geq 0$, $0\leq
c<d$ satisfying the conditions: (i) $a-b=1$ or $b=1$, (ii) $d-c=1$ or $c=1$,
and (iii) $b+c\geq 1$.

  3. (c)

$(x_{1}^{a}x_{2}^{b},x_{1}x_{3},x_{2}x_{3})$ with $a\geq 1$, $b\geq 0$,
$a+b\geq 2$ and $n=3$.

  4. (d)

$(x_{1}x_{2}^{b},x_{2}^{b+1},x_{1}x_{3})$ with $b\geq 1$ and $n=3$.

  5. (e)

$(x_{1}x_{3},x_{1}x_{4},x_{2}x_{4})$ and $n=4$.

###### Proof.

Let $I\subset S$ be a nearly Gorenstein monomial ideal of height two and set
$R=S/I$. By [19, Proposition 2.3(a)], $I$ is Gorenstein on the punctured
spectrum, and so is generically Gorenstein. Theorem 2.1 implies that
$\operatorname{tr}(\omega_{R})=I_{m-2}(X)R$ where $m=\mu(I)$ and $X$ is an
Hilbert-Burch matrix of $I$. Since each non-zero entry of $X$ has positive
degree, $\operatorname{tr}(\omega_{R})\subseteq{\mathfrak{m}}^{m-2}$, with
${\mathfrak{m}}=(x_{1},\dots,x_{n})R$. Thus, if $I$ is nearly Gorenstein, then
$m\leq 3$. We cannot have $m=1$, otherwise $I$ has height one.

If $m=2$, then $I=(u,v)$. If
$x_{i}\in\operatorname{supp}(u)\cap\operatorname{supp}(v)$, then
$\operatorname{height}(I)=1$ against the assumption. Thus
$\operatorname{supp}(u)\cap\operatorname{supp}(v)=\emptyset$ and case (a)
follows. Conversely, if (a) holds, then $I=(u,v)$ is a complete intersection,
and thus is (nearly) Gorenstein.

Assume now $m=3$, then $\operatorname{tr}(\omega_{R})\subseteq{\mathfrak{m}}$
and so we must have $\operatorname{tr}(\omega_{R})={\mathfrak{m}}$. Since
$m=3$, then $n\geq 2$. If $n=2$, then Proposition 3.5 implies that case (b)
holds.

Hence, we may assume $n\geq 3$. Let $G(I)=\\{u_{1},u_{2},u_{3}\\}$. Then, we
have the following matrix arising from the so-called Taylor resolution (see
[18, Section 7.1]):

$T=\left(\begin{matrix}\phantom{-}\frac{\operatorname{lcm}(u_{1},u_{2})}{u_{1}}&\phantom{-}\frac{\operatorname{lcm}(u_{1},u_{3})}{u_{1}}&0\\\
-\frac{\operatorname{lcm}(u_{1},u_{2})}{u_{2}}&0&\phantom{-}\frac{\operatorname{lcm}(u_{2},u_{3})}{u_{2}}\\\
0&-\frac{\operatorname{lcm}(u_{1},u_{3})}{u_{3}}&-\frac{\operatorname{lcm}(u_{2},u_{3})}{u_{3}}\end{matrix}\right).$

It follows from [18, Lemma 9.2.4] that an Hilbert-Burch matrix of $I$ is an
appropriate $3\times 2$ submatrix of $T$. Let $X$ be such an Hilbert-Burch
matrix. Then $X$ has only four non-zero entries. Since we must have
$\operatorname{tr}(\omega_{R})=(x_{1},\dots,x_{n})R$, but we also have
$\operatorname{tr}(\omega_{R})=I_{1}(X)R$, we deduce that $n$ is at most four.

Let $n=3$. Up to relabeling and exchange of rows, we have

$X=\left(\begin{matrix}\phantom{-}x_{1}&0\\\ -x_{2}&\phantom{-}w\\\
0&-x_{3}\end{matrix}\right)\,\,\,\,\,\,\textup{or}\,\,\,\,\,\,X=\left(\begin{matrix}\phantom{-}x_{1}&0\\\
-x_{2}&\phantom{-}x_{3}\\\ 0&-w\end{matrix}\right),$ (15)

where $w$ is a suitable monomial.

In the first case $I=I_{2}(X)=(x_{1}w,x_{1}x_{3},x_{2}x_{3})$. Note that
$x_{3}$ cannot divide $w$, otherwise we would have $m<3$. Thus
$w=x_{1}^{a}x_{2}^{b}$ for some $a,b\geq 0$ with $a+b\geq 1$. Case (c)
follows. Conversely, if (c) holds and
$I=(x_{1}^{a}x_{2}^{b},x_{1}x_{3},x_{2}x_{3})$ for some $a\geq 1$, $b\geq 0$
and $n=3$, then $I$ is generically Gorenstein, putting
$w=x_{1}^{a-1}x_{2}^{b}$, the first matrix $X$ in (15) is an Hilbert-Burch
matrix of $I$ and so $I$ is nearly Gorenstein.

In the second case $I=I_{2}(X)=(x_{1}x_{3},x_{1}w,x_{2}w)$. Notice that
$x_{1}$ and $x_{3}$ cannot divide $w$, otherwise $I$ would have height one.
Thus $w=x_{2}^{b}$ with $b\geq 1$ and case (d) follows. Conversely, if (d)
holds and $I=(x_{1}x_{2}^{b},x_{2}^{b+1},x_{1}x_{3})$ with $b\geq 1$ and
$n=3$, then $I$ is generically Gorenstein, putting $w=x_{2}^{b}$ the second
matrix $X$ given in (15) is an Hilbert-Burch matrix of $I$ with
$I_{1}(X)={\mathfrak{m}}$ and $I$ is nearly Gorenstein.

Finally, assume $n=4$. Then, up to relabeling and exchange of rows,

$X=\left(\begin{matrix}\phantom{-}x_{1}&0\\\ -x_{2}&\phantom{-}x_{3}\\\
0&-x_{4}\end{matrix}\right).$ (16)

Thus $I=I_{2}(X)=(x_{1}x_{3},x_{1}x_{4},x_{2}x_{4})$ and case (e) follows.
Conversely, if (e) holds, then the $I=(x_{1}x_{3},x_{1}x_{4},x_{2}x_{4})$ is
generically Gorenstein, the matrix $X$ given in (16) is an Hilbert-Burch
matrix of $I$ and thus $I$ is nearly Gorenstein.

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# Theoretical study on the contributions of $\omega$ meson to the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ and $J/\psi\pi^{+}\pi^{-}\pi^{0}$ decays

Hao-Nan Wang<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of
Nuclear Science, Institute of Quantum Matter, South China Normal University,
Guangzhou 510006, China  Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  Qian Wang<EMAIL_ADDRESS>Guangdong
Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China  Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China  Theoretical Physics
Center for Science Facilities, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  Ju-Jun Xie<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  School of Nuclear Sciences and Technology, University of Chinese
Academy of Sciences, Beijing 101408, China  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

With the newly measurements of $X(3872)\to J/\psi\pi^{+}\pi^{-}$ from LHCb
Collaboration, in addition to the dominant contribution from the $\rho^{0}$
meson, we perform a theoretical study on the contribution of $\omega$ meson in
the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ and $X(3872)\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ processes. It is found that the recent
experimental measurements on the $\pi^{+}\pi^{-}$ invariant mass distributions
can be well reproduced, and the ratio of the couplings between $X(3872)\to
J/\psi\rho^{0}$ and $X(3872)\to J/\psi\omega$ is also evaluated. Within the
parameters extracted from the $\pi^{+}\pi^{-}$ invariant mass distributions of
the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ process, the branching fractions of the
$X(3872)\to J/\psi\omega$ channel relative to that of the $X(3872)\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ channel and the $\pi^{+}\pi^{-}\pi^{0}$ invariant
mass distributions of the $X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decay are
calculated, which gives a hint for the further high-statistic experiment.

##  I Introduction

In 2003, the $\chi_{c1}(3872)$ (also known as $X(3872)$), as an exotic
candidate, was discovered by the Belle experiment in the
$J/\psi\pi^{+}\pi^{-}$ channel Belle:2003nnu . An updated analysis was done in
2011 Belle:2011vlx . Ten years after its observation, its quantum number has
been well determined to be $I^{G}(J^{PC})=0^{+}(1^{++})$ LHCb:2013kgk .
Nowadays the $X(3872)$ is well established. The ”OUR AVERAGE” value in the
2020 version and 2021 updated of the Review of Particle Physics (RPP)
ParticleDataGroup:2020ssz  for the mass of $X(3872)$ is $3871.65\pm 0.06$ MeV,
which is near the $D^{0}\bar{D}^{*0}$ mass threshold. As its mass is much
lower than the one of $\chi_{c1}(2P)$ predicted by the $c\bar{c}$ quark model
Godfrey:1985xj , it cannot be accepted by normal charmonium picture.
Furthermore, its width is extremely narrow compared with other hadrons that
have similar energy. In 2020, its Breit-Wigner width was measured and it is
$1.39\pm 0.24\pm 0.10$ MeV LHCb:2020xds  or $0.96^{+0.19}_{-0.18}\pm 0.21$ MeV
LHCb:2020fvo , depending on the assumed lineshape.

Although the $X(3872)$ have been intensively studied in various pictures, for
instance the $D\bar{D}^{*}+c.c.$ molecular picture, the compact tetraquark
picture, the normal charmonium with a mixture of the $D\bar{D}^{*}+c.c.$
molecule and so on. The nature of the $X(3872)$ state is still puzzling (more
details can be seen in these review articles Lebed:2016hpi ; Chen:2016qju ;
Esposito:2016noz ; Olsen:2017bmm ; Guo:2017jvc ; Ali:2017jda ;
Brambilla:2019esw ; Liu:2019zoy ; Guo:2019twa ; Dong:2021bvy ; Chen:2022asf ).
Because its mass is very close to the $D^{0}\bar{D}^{*0}$ mass threshold, it
could have a large $D^{0}\bar{D}^{*0}$ molecular component Guo:2019qcn ;
Zhang:2020mpi , which leads a large isospin breaking effect Gamermann:2009fv ;
Li:2012cs ; Meng:2021kmi ; Tornqvist:2004qy ; Takeuchi:2011uh ;
Terasaki:2009in ; Takeuchi:2014rsa . This isospin breaking effect has been
found by Belle Collaboration Belle:2011vlx  in the ratio of three- and two-
pion branching fractions

$\displaystyle
R^{\mathcal{B}_{3\pi}/\mathcal{B}_{2\pi}}=\frac{\mathcal{B}[X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-}\pi^{0}]}{\mathcal{B}[X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-}]}=0.8\pm 0.3.$ (1)

With $\mathcal{B}_{2\pi}=(3.8\pm 1.2)\%$ quoted in RPP
ParticleDataGroup:2020ssz , one can easily obtain $\mathcal{B}_{3\pi}=(3.0\pm
1.5)\%$. If we assumed that the two-pion final state is dominated by the
$\rho^{0}$ meson, while the three-pion is dominated by the $\omega$ meson, it
is found that $X(3872)$ has large isospin violation effects in these two above
decays, since they have the same rate within one sigma uncertainty.

The above two- and three-pion transition modes are our focus due to the recent
new measurements from LHCb collaboration LHCb:2022bly . As the $X(3872)$
decays to $J/\psi\pi^{+}\pi^{-}$ with $\pi^{+}\pi^{-}$ from $\rho^{0}$ meson
Belle:2003nnu ; CDF:2005cfq , the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ is isospin
breaking process due to the isospin singlet property to the $X(3872)$. The
isospin conserving decay $X(3872)\to J\psi\omega\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ is important to understand the internal structure
of $X(3872)$, though it has small phase space. The $X(3872)\to J\psi\omega$
decay has been observed with a significance of more than $5\sigma$ by the
BESIII collaboration BESIII:2019qvy . Previously, the Belle and BABAR
collaborations also found evidence for the $X(3872)\to J/\psi\omega$ decay
Belle:2005lfc ; BaBar:2010wfc , but their measurements are with large
uncertainties. Recently, sizeable $\omega$ contribution to $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay are observed by the LHCb collaboration
LHCb:2022bly . These new measurements can be used to study the effects of the
isospin-violating $\rho^{0}$ and $\omega$ mixing. Indeed, in Ref.
Hanhart:2011tn  this effect was taken into account in the analysis of the
$J/\psi\pi^{+}\pi^{-}$ and $J/\psi\pi^{+}\pi^{-}\pi^{0}$ invariant mass
distributions in the decays $X(3872)\to J/\psi\rho^{0}$ and $X(3872)\to
J/\psi\omega$, where they focused on the determination of the quantum numbers
of $X(3872)$. Besides, based on the $D\bar{D}^{*}$ molecular picture of
$X(3872)$, the isospin breaking effects of the $X(3872)\to J/\psi\rho^{0}$ and
$X(3872)\to J/\psi\omega$ decays were investigated in Refs. Gamermann:2009fv ;
Li:2012cs ; Meng:2021kmi ; Suzuki:2005ha ; Liu:2006df ; Gamermann:2009uq ;
Coito:2010if ; Albaladejo:2015dsa ; Wu:2021udi .

In this work, with the new measurements of the LHCb collaboration LHCb:2022bly
, and following the work of Ref. Hanhart:2011tn , we study the invariant mass
distributions of the $\pi^{+}\pi^{-}$ and $\pi^{+}\pi^{-}\pi^{0}$ final states
in the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ and $X(3872)\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ decays, respectively, where we will focus on the
role played by the $\omega$ meson to the $X(3872)\to J/\psi\pi^{+}\pi^{-}$
decay and the ratio of the effective couplings of $X(3872)$ to
$J/\psi\rho^{0}$ and $J/\psi\omega$.

The paper is organized as follows. In Sec. II, we present the theoretical
formalism of the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ and $X(3872)\to
J/\psi\pi^{+}\pi^{-}\pi^{0}$ decays, and in Sec. III, we show our numerical
results and discussions, followed by a short summary in Sec. IV.

##  II formalism

The effective Lagrangian method is an useful tool in describing the various
processes around the resonance region. The model used in the present work can
give a reasonable description of the experimental data for the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay, and our calculation offers some important clues
for the mechanisms of the decays of $X(3872)\to J/\psi\pi^{+}\pi^{-}$ and
$J/\psi\pi^{+}\pi^{-}\pi^{0}$. In this section, we introduce the theoretical
formalism and ingredients to study the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ and
$X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decays by using the effective
Lagrangian method.

###  II.1 Feynman diagrams and effective interaction Lagrangian densities

Following previous analyses of Ref. Hanhart:2011tn , we assume that the
$X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay is mediated by the $\rho^{0}$ and
$\omega$ meson, while the $X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decay is
mediated by the $\omega$ meson. The corresponding basic tree-level diagrams
are shown in Fig. 1. For the $\omega\to\pi^{+}\pi^{-}\pi^{0}$ decay, we take
the $\rho$ meson as an intermediate state. The $\omega$ meson firstly couples
to $\pi\rho$ and then the $\rho$ meson decays into $\pi\pi$ in the final
state. 111In the calculation, we consider only the process of
$\omega\to\pi^{0}\rho^{0}\to\pi^{0}\pi^{+}\pi^{-}$, and finally multiply by an
isospin factor three to the total decay. This treatment will not change the
three pion lineshape. On the other hand, we also consider the contribution of
$\omega$ meson to the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay with $\omega$
decaying into $\pi^{+}\pi^{-}$. Note that, in Ref. Hanhart:2011tn , the
$\omega\to\rho^{0}$ mixing was taken into account for the contribution of
$\omega$ meson to the $\pi^{+}\pi^{-}$ production, where the transition
amplitude is described by a real parameter. Here, we consider an effective
coupling for the $\omega\to\pi^{+}\pi^{-}$ decay.

Figure 1: Feynman diagrams for the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay
with $\rho^{0}$ and $\omega$ contribution (a) and $X(3872)\to
J/\psi\pi^{0}\pi^{+}\pi^{-}$ decay with $\omega$ contribution (b).

To obtain the decay amplitudes of the processes shown in Fig. 1, we need the
effective interactions for these interaction vertexes, which can be described
by the effective Lagrangian densities as used in Refs. Lucio-Martinez:2000now
; Xie:2008ts ; Wang:2020duv ; Janssen:1994uf :

$\displaystyle\mathcal{L}_{XJ/\psi V}$ $\displaystyle=$ $\displaystyle
g_{X}\varepsilon^{\mu\nu\rho\sigma}\partial_{\mu}X_{\nu}J/\psi_{\rho}V_{\sigma},$
(2) $\displaystyle\mathcal{L}_{\omega\rho\pi}$ $\displaystyle=$ $\displaystyle
g_{\omega\rho\pi}\varepsilon^{\alpha\beta\gamma\tau}\partial_{\alpha}\omega_{\gamma}\partial_{\beta}\rho_{\tau}\phi_{\pi},$
(3) $\displaystyle\mathcal{L}_{V\pi\pi}$ $\displaystyle=$ $\displaystyle
g_{V\pi\pi}V_{\mu}\partial^{\mu}(\phi_{\pi^{+}}-\phi_{\pi^{-}}),$ (4)

where $X$ and $V$ represent $X(3872)$ and $\rho^{0}/\omega$ meson,
respectively. While $g_{X}$, $g_{\omega\rho\pi}$, $g_{\omega\pi\pi}$, and
$g_{\rho\pi\pi}$ are the coupling constants of the corresponding vertexes. In
this work, we will use $g_{X_{1}}$ and $g_{X_{2}}$ for the effective coupling
constants of $X(3872)$ to $J/\psi\rho^{0}$ and $J/\psi\omega$, respectively.
Furthermore, we will take $g_{X}$, $g_{\omega\rho\pi}$, $g_{\rho\pi\pi}$, and
$g_{\omega\pi\pi}$ as real and their values will be discussed below.

###  II.2 Invariant decay amplitudes

With the effective interaction Lagrangian densities given above, the invariant
decay amplitudes for these diagrams shown in Fig. 1 can be written as

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$ $\displaystyle{\cal
M}_{\rho}+{\cal M}_{\omega}$ (5) $\displaystyle=$
$\displaystyle-i\left(\frac{g_{X_{1}}g_{\rho\pi\pi}}{D_{\rho}(q^{2})}F_{\rho}(q^{2})+\frac{g_{X_{2}}g_{\omega\pi\pi}}{D_{\omega}(q^{2})}F_{\omega}(q^{2})\right)\times$
$\displaystyle\varepsilon^{\mu\nu\rho\sigma}p_{\mu}\epsilon_{\nu}(p)\epsilon_{\rho}^{\ast}(p_{4})(p_{1}-p_{2})_{\sigma},$
$\displaystyle\mathcal{M}_{b}$ $\displaystyle=$
$\displaystyle\frac{g_{X_{2}}g_{\omega\rho\pi}g_{\rho\pi\pi}}{D_{\omega}(q^{2}_{1})D_{\rho}(q^{2}_{2})}F_{\omega}(q_{1}^{2})F_{\rho}(q_{2}^{2})\varepsilon^{\mu\nu\rho\sigma}p_{\mu}\epsilon_{\nu}(p)\epsilon_{\rho}^{\ast}(p_{4})$
(6)
$\displaystyle\times\left(-g_{\sigma\gamma}+\frac{q_{1\sigma}q_{1\gamma}}{m_{\omega}^{2}}\right)\varepsilon^{\alpha\beta\gamma\tau}q_{1\alpha}q_{2\beta}(p_{2}-p_{1})_{\tau},$

where $p$, $p_{1}$, $p_{2}$, $p_{3}$, $p_{4}$ are the four-momenta of
$X(3872)$, $\pi^{-}$, $\pi^{+}$, $\pi^{0}$, $J/\psi$, while $q_{1}$ and
$q_{2}$ represent the four-momenta for the intermediate $\omega$ and
$\rho^{0}$ mesons. $D_{\rho}(q^{2})$ and $D_{\omega}(q^{2}_{1})$ are the
denominators of the propagators for the $\rho^{0}$ and $\omega$ meson, which
are

$\displaystyle D_{\rho}(q^{2})$ $\displaystyle=$ $\displaystyle
q^{2}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}(q^{2}),$ (7) $\displaystyle
D_{\omega}(q^{2}_{1})$ $\displaystyle=$ $\displaystyle
q^{2}_{1}-m^{2}_{\omega}+im_{\omega}\Gamma_{\omega}.$ (8)

Since the major decay channel of $\omega$ meson is the $\pi^{+}\pi^{-}\pi^{0}$
and its width is narrow, we take $m_{\omega}=782.65$ MeV and
$\Gamma_{\omega}=8.49$ MeV as quoted in the Review of Particle Physics
ParticleDataGroup:2020ssz . For the width of $\rho$ meson, since it is large
and the predominant decay mode is $\pi\pi$, we take that $\Gamma_{\rho}$ is
energy dependent, which is given by Hanhart:2010wh ; Zhang:2017eui

$\displaystyle\Gamma_{\rho}(q^{2})=\Gamma_{0}\frac{|\textbf{p}_{\pi}|^{3}}{|\textbf{p}_{\pi}^{0}|^{3}}\frac{m_{\rho}^{2}}{q^{2}},$
(9)

where

$\displaystyle|\textbf{p}_{\pi}|=\frac{\sqrt{q^{2}-4m_{\pi}^{2}}}{2},~{}~{}~{}|\textbf{p}^{0}_{\pi}|=\frac{\sqrt{m_{\rho}^{2}-4m_{\pi}^{2}}}{2}.$
(10)

In evaluating the decay amplitudes of $X(3872)\to J/\psi\pi^{+}\pi^{-}$, we
include the form factors for $\rho^{0}$ and $\omega$ mesons since they are not
point like particles Liu:1995st . In this work we adopt here the common scheme
used in many previous works Xie:2008ts ; Xie:2014kja ; Xie:2014zga :

$\displaystyle
F_{\rho/\omega}(q^{2})=\frac{\Lambda_{\rho/\omega}^{4}}{\Lambda_{\rho/\omega}^{4}+(q^{2}-m_{\rho/\omega}^{2})^{2}},$
(11)

where we assume that $\Lambda_{\rho}=\Lambda_{\omega}=\Lambda$ and they are
determined to fit with the recent LHCb measurements. Note that a Blatt-
Weisskopf barrier factor was used for the $P$-wave decay of a vector to
$\pi\pi$ in Ref. LHCb:2022bly .

Besides, the coupling constants, $g_{\rho\pi\pi}$, $g_{\omega\pi\pi}$, and
$g_{\omega\rho\pi}$, are determined from the experimentally observed partial
decay widths of $\rho\to\pi\pi$, $\omega\to\pi\pi$, and
$\omega\to\rho\pi\to\pi\pi\pi$, respectively. With the effective interaction
Lagrangians shown in Eq. (4), these partial decay widths $\rho\to\pi\pi$ and
$\omega\to\pi\pi$ can be easily calculated. The coupling constants are related
to the partial decay widths as

$\displaystyle\Gamma_{\rho\to\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{\rho\pi\pi}}{6\pi}\frac{p_{\rho}^{3}}{m^{2}_{\rho}},$
(12) $\displaystyle\Gamma_{\omega\to\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{\omega\pi\pi}}{6\pi}\frac{p^{3}_{\omega}}{m^{2}_{\omega}},$
(13)

where $p_{\rho}$ and $p_{\omega}$ are the three momenta of the $\pi$ meson in
the $\rho$ or $\omega$ rest frame, respectively. With
$\Gamma_{\rho\to\pi\pi}=149.1$ MeV and $\Gamma_{\omega\to\pi\pi}=0.133$ MeV as
quoted in the Ref. ParticleDataGroup:2020ssz , we obtain $g_{\rho\pi\pi}=5.97$
and $g_{\omega\pi\pi}=0.18$, respectively. Note that from the partial decay
width, one can only obtain the absolute value of the coupling constant, but
not the phase. In this work, we assume that $g_{\rho\pi\pi}$ and
$g_{\omega\pi\pi}$ are real and positive. In fact, these values obtained here
were used in Refs. Janssen:1994uf ; Janssen:1996kx ; Meissner:1987ge  for
other processes.

In Ref. Chen:2017jcw , it was found that that $\omega$-$\rho^{0}$ mixing plays
the major role in the evaluating the partial decay width of
$\omega\to\pi^{+}\pi^{-}$, and its contribution is two orders of magnitude
larger than that from the direct $\omega\pi\pi$ coupling. However, here we
obtained the coupling constant $g_{\omega\pi\pi}$ with the experimental
results of the $\omega\to\pi^{+}\pi^{-}$ decay. In other words, we have taken
the effective $\omega\pi\pi$ coupling as a constant, and determined with the
experimental partial decay width of $\omega\to\pi\pi$, rather than the mixing
between $\rho^{0}-\omega$ with the explicit $\rho^{0}$ propagator Chen:2017jcw
.

In addition, the value of $g_{\omega\rho\pi}$ is determined with the partial
decay width of $\omega\to\pi^{+}\pi^{-}\pi^{0}$, which reads

$\displaystyle\text{d}\Gamma_{\omega\to\pi^{+}\pi^{-}\pi^{0}}=\frac{1}{(2\pi)^{3}}\frac{1}{32m_{\omega}^{3}}|\mathcal{M}|^{2}\text{d}M_{\pi^{+}\pi^{-}}^{2}\text{d}M_{\pi^{-}\pi^{0}}^{2},$
(14)

where

$\displaystyle\mathcal{M}=\frac{-ig_{\omega\rho\pi}g_{\rho\pi\pi}}{D_{\rho}(q^{2}_{\rho})}\varepsilon_{\mu\nu\rho\sigma}q^{\mu}_{\omega}q_{\rho}\epsilon^{\nu}(q_{\omega})(p_{\pi^{+}}-p_{\pi^{-}})^{\sigma},$
(15)

where $q_{\omega}$, $q_{\rho}$, $p_{\pi^{+}}$, and $p_{\pi^{-}}$ stand for the
four-momenta of $\omega$, $\pi^{+}$, and $\pi^{-}$, respectively. With
$\Gamma_{\omega\to\pi^{+}\pi^{-}\pi^{0}}=7.74$ MeV, we obtain
$g_{\omega\rho\pi}=0.046$ ${\rm MeV}^{-1}$ for the case of $\Gamma_{\rho}$
energy dependent and $g_{\omega\rho\pi}=0.05$ ${\rm MeV}^{-1}$ for the case of
$\Gamma_{\rho}$ as a constant. One see that the affect of the $\Gamma_{\rho}$
energy dependent is rather small and can be neglected.

###  II.3 Invariant mass distributions

With the formalism and ingredients given above, the calculations of the
invariant mass distribution for the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ and
$X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decays are straightforward
ParticleDataGroup:2020ssz . The invariant $\pi^{+}\pi^{-}$ mass distribution
of the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay is given by

$\displaystyle\frac{\text{d}\Gamma_{X{3872}\to
J/\psi\pi^{+}\pi^{-}}}{\text{d}M_{\pi^{+}\pi^{-}}}=\frac{1}{24(2\pi)^{4}M_{X}^{2}}\times$
$\displaystyle\int\Sigma|\mathcal{M}_{2\pi}|^{2}|\mathbb{p}_{1}^{\ast}||\mathbb{p}_{4}|\text{d}\cos{\theta_{1}}\text{d}\phi_{1},$
(16)

where $\mathbb{p}_{1}^{\ast}$ and ($\theta_{1}$, $\phi_{1}$) are the three-
momentum and decay angle of the outing $\pi^{+}$ (or $\pi^{-}$) in the center-
of-mass (c.m.) frame of the final $\pi^{+}\pi^{-}$ system, $\mathbb{p}_{4}$ is
the three-momentum of the final $J/\psi$ meson in the rest frame of $X(3872)$,
and $M_{\pi^{+}\pi^{-}}$ is the invariant mass of the final $\pi^{+}\pi^{-}$
system.

For the invariant $\pi^{+}\pi^{-}\pi^{0}$ mass distributions of the
$X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decay, it is given by,

$\displaystyle\frac{\text{d}\Gamma_{X(3872)\to
J/\psi\pi^{+}\pi^{-}\pi^{0}}}{\text{d}M_{\pi^{+}\pi^{-}\pi^{0}}}=\frac{1}{16(2\pi)^{7}M_{X}^{2}}\int\Sigma|\mathcal{M}_{3\pi}|^{2}\times$
$\displaystyle|\mathbb{p}_{1}^{\ast}||\mathbb{p}_{3}^{\prime}||\mathbb{p}_{4}|\text{d}M_{\pi^{+}\pi^{-}}\text{d}\cos{\theta_{1}}\text{d}\phi_{1}\text{d}\cos{\theta_{2}}\text{d}\phi_{2},$
(17)

with $M_{\pi^{+}\pi^{-}\pi^{0}}$ the invariant mass of $\pi^{+}\pi^{-}\pi^{0}$
system. The definitions of these variables in the phase space integration are
given in Appendix A.

Besides, in Eqs. (16) and (17), we take

$\displaystyle{\cal M}_{2\pi}$ $\displaystyle=$ $\displaystyle{\cal
M}_{\rho}+e^{i\varphi}{\cal M}_{\omega},$ (18) $\displaystyle{\cal M}_{3\pi}$
$\displaystyle=$ $\displaystyle{\cal M}_{b}.$ (19)

Note that we have included a free parameter $\varphi$ which stands for the
relative phase between $\omega$ and $\rho^{0}$ terms for the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay. On the other hand, more details for the
integration of the multi-body phase space can be found in Refs. Xie:2015zga ;
Xie:2018gbi ; Jing:2020tth .

##  III Numerical results and discussions

In this section, we present the numerical results for the invariant mass
distribution of $\pi^{+}\pi^{-}$ of the $X(3872)\to J/\psi\pi^{+}\pi^{-}$
decay. To compare the theoretical invariant mass distributions with the
experimental measurements, we introduce an extra global normalization factor
$C$, which will be fitted to the experimental data. In the calculation, the
masses, widths and spin-parities of the involved particles are listed in Table
1.

Table 1: Masses, widths and spin-parities of the involved particles in this work. Particle | Mass (MeV) | Width (MeV) | Spin-parity ($J^{P}$)
---|---|---|---
$X(3872)$ | 3871.69 | 1.19$\pm$0.21 | $1^{+}$
$J/\psi$ | 3096.9 | — | $1^{-}$
$\rho^{0}$ | 775.26 | 149.1$\pm$0.8 | $1^{-}$
$\omega$ | 782.66 | 8.68$\pm$0.13 | $1^{-}$
$\pi^{+}/\pi^{-}$ | 139.57 | — | $0^{-}$
$\pi^{0}$ | 134.97 | — | $0^{-}$

We perform four parameters ($R_{X}\equiv g_{X_{1}}/g_{X_{2}}$,
$Cg_{X_{2}}^{2}$, $\Lambda$, and $\varphi$) $\chi^{2}$ fits to the
experimental data on the $\pi^{+}\pi^{-}$ invariant mass distributions. We
will study two types of fit: one takes the total width of $\rho$ energy
dependent, while the other one takes the total width of $\rho$ as a constant.
The fitted parameters and the corresponding $\chi^{2}/\mathrm{d.o.f.}$ are
shown in Table 2. We have checked that the results of the two fits are very
similar, this indicates that the affects of the energy dependent of the $\rho$
total width $\Gamma_{\rho}$ is very small and can be neglected. It is worth to
mention that the obtained ratio $R_{X}$ is very similar with these values
$0.29\pm 0.04$ and $0.26^{+0.08}_{-0.05}$ obtained in Ref. LHCb:2022bly  and
Ref. Hanhart:2011tn , respectively.

Although the two parameters $R_{X}$ and $Cg_{X_{2}}^{2}$ can be obtained from
the fit directly, the physical couplings $g_{X_{1}}$ and $g_{X_{2}}$ can only
be extracted with the further inputs. With the value $R_{X}=0.25\pm 0.01$, the
coupling $g_{X_{2}}=0.33\pm 0.06$ can be extracted from the the branching
ratio $\mathcal{B}(X(3872)\to J/\psi\pi^{+}\pi^{-})=(3.8\pm 1.2)\%$
ParticleDataGroup:2020ssz . Consequently, the coupling $g_{X_{1}}=0.08\pm
0.02$ and the normalization factor $C=(7.48\pm 2.78)\times 10^{-5}$ can be
obtained. The above values have been listed in Table 2 as well as those for
the constant $\Gamma_{\rho}$ case.

Table 2: The fitted and determined parameters in this work. The second and third columns are the fitted results of the $\Gamma_{\rho}(q^{2})$ energy-dependent fit and fixed $\Gamma_{0}$ fit, respectively. Number |  1 |  2
---|---|---
$\rho$ meson width |  $\Gamma_{\rho}(q^{2})$ energy dependent |  $\Gamma_{0}$ constant
$R_{X}$ | $0.25\pm 0.01$ | $0.30\pm 0.01$
$Cg_{X_{2}}^{2}~{}(\times 10^{4})$ | $8.31\pm 0.45$ | $7.10\pm 0.48$
$\Lambda$ (MeV) | $612\pm 18$ | $598\pm 16$
$\varphi~{}(\degree)$ | $128.9\pm 8.0$ | $134.5\pm 7.7$
$\chi^{2}/\mathrm{d.o.f.}$ | $0.6$ | $0.6$
$g_{X_{1}}$ | $0.08\pm 0.02$ | $0.09\pm 0.02$
$g_{X_{2}}$ | $0.33\pm 0.06$ | $0.31\pm 0.06$
$C~{}(\times 10^{5})$ | $7.48\pm 2.78$ | $7.48\pm 2.81$

With the central values of Table 2 for the case of $\Gamma_{\rho}$ energy
dependent, the $\pi^{+}\pi^{-}$ invariant mass distribution is shown by the
red curve in Fig. 2. Note that the results for the case of $\Gamma_{\rho}$ as
a constant are very similar with the ones obtained for the case of
$\Gamma_{\rho}$ energy dependent. In Fig. 2, the red-solid curve stands for
the total contributions from the $\rho^{0}$ and $\omega$ mesons, the blue-
dash-dotted and green-dash-dotted curves correspond to the contribution from
only the $\rho^{0}$ and $\omega$, respectively, while the black-dash-dotted
stands for their interference. The band accounts for the corresponding $68\%$
confidence-level interval deduced from the distributions of the fitted
parameters shown in Table 2. One can see that the total numerical results can
explain the experimental data quite well. Furthermore, the contribution of
$\rho$ meson is predominant in the whole energy region consider for the
$M_{\pi^{+}\pi^{-}}$, while the contribution of $\omega$ meson is crucial to
the $\pi^{+}\pi^{-}$ invariant mass distributions at high energy of
$M_{\pi^{+}\pi^{-}}$.

Figure 2: Invariant mass distribution of $\pi^{+}\pi^{-}$ for the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay, compared with the experimental data taken from
Ref. LHCb:2022bly . The blue-dash-dotted, green-dash-dotted curves and black-
dash-dotted are the contributions from the $\rho^{0}$, $\omega$, and their
interference, respectively. The red solid one is their total contribution. The
band accounts for the corresponding $68\%$ confidence-level interval deduced
from the distributions of the fitted parameters shown in Table 2

With the fitted couplings $g_{X_{1}}$ and $g_{X_{2}}$, one can easily obtain
the $\omega$ contribution, without the $\rho^{0}$-$\omega$ interference terms,
to the total $X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay as

$\displaystyle\frac{\Gamma_{X(3872)\to J/\psi\omega\to
J/\psi\pi^{+}\pi^{-}}}{\Gamma_{X(3872)\to J/\psi\pi^{+}\pi^{-}}}=(4.7\pm
2.4)\%$ (20)

for the case of $\Gamma_{\rho}$ energy dependent and

$\displaystyle\frac{\Gamma_{X(3872)\to J/\psi\omega\to
J/\psi\pi^{+}\pi^{-}}}{\Gamma_{X(3872)\to J/\psi\pi^{+}\pi^{-}}}=(4.0\pm
2.1)\%$ (21)

for the case of $\Gamma_{\rho}$ as a constant. These values agree with the
value $(1.9\pm 0.4\pm 0.3)\%$, obtained by the LHCb analysis in Ref.
LHCb:2022bly  within one standard deviation.

It is customary to apply the so-called narrow width approximation in the case
where a particle decays into two particles and one of them with narrow width
subsequently decays into other two particles in the final state Cheng:2020iwk
; Cheng:2020mna . Since the width of $\omega$ meson is so narrow, we can
extract the branching fraction of the quasi-two-body decay $X(3872)\to
J/\psi\omega$

$\displaystyle{\cal B}(X(3872)\to J/\psi\omega)$ (22) $\displaystyle=$
$\displaystyle\frac{{\cal B}(X(3872)\to J/\psi\omega\to
J/\psi\pi^{+}\pi^{-})}{{\cal B}(\omega\to\pi^{+}\pi^{-})}$ $\displaystyle=$
$\displaystyle(11.5\pm 7.5)\%,$

within the narrow width approximation. Furthermore, we extract the branching
ratio fraction between the $J/\psi\omega$ and $J/\psi\pi^{+}\pi^{-}$ modes as
defined in Ref. BESIII:2019qvy .

$\displaystyle\mathcal{R}\equiv\frac{{\cal B}(X(3872)\to J/\psi\omega)}{{\cal
B}(X(3872)\to J/\psi\pi^{+}\pi^{-})}=3.0\pm 2.2$ (23)

for the $\Gamma_{\rho}$ energy-dependent case, and

$\displaystyle{\cal B}(X(3872)\to J/\psi\omega)=(9.9\pm 6.3)\%,$ (24)
$\displaystyle\mathcal{R}=2.6\pm 1.9$ (25)

for the constant $\Gamma_{\rho}$ case. The two values of $\mathcal{R}$ are in
agreement with the experimental measurements $1.6^{+0.4}_{-0.3}\pm 0.2$ by
BESIII collaboration BESIII:2019qvy  within uncertainties.

Next we turn to the $X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decay. With the
values of $g_{X_{1}}$ and $g_{X_{2}}$, we obtain:

$\displaystyle{\cal B}[X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}]=(1.1\pm
0.5)\%,$ (26)

for the $\Gamma_{\rho}$ energy-dependent case and

$\displaystyle{\cal B}[X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}]=(1.4\pm
0.6)\%,$ (27)

for the constant $\Gamma_{\rho}$ case. Those values are in agreement with both
the experimental measurements ParticleDataGroup:2020ssz  and the theoretical
calculations in Ref. Aceti:2012cb .

Figure 3: Invariant mass distribution of $\pi^{+}\pi^{-}\pi^{0}$ for the
$X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decay, compared with the experimental
data taken from Ref. BaBar:2010wfc .

Finally, with these model parameters determined by fitting to the
$\pi^{+}\pi^{-}$ invariant mass distributions for the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay, we calculate the $\pi^{+}\pi^{-}\pi^{0}$
invariant mass distributions for the $X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$
decay. The numerical results are shown as the red curve in Fig. 3. The error
band of the theoretical calculations are obtained from the uncertainty of the
parameter $g_{X_{2}}$ which stems from the uncertainties of both the two pion
invariant mass distribution of the $J/\psi\pi^{+}\pi^{-}$ decay mode, i.e. the
fitted parameter $R_{X}$, and the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ branching
ratio. To compare the theoretical invariant mass distributions with the
experimental measurements, we have to introduce again an extra global
normalization factor $C_{3\pi}$. In Fig. 3 the red-solid curve has been
adjusted to the strength of the inverse-second data point of BABAR
BaBar:2010wfc  by taking $C_{3\pi}=6.07\times 10^{4}$ for the $\Gamma_{\rho}$
energy-dependent case. For the constant $\Gamma_{\rho}$ case, the value of
$C_{3\pi}$ is $5.77\times 10^{4}$, and the line shape of the
$\pi^{+}\pi^{-}\pi^{0}$ invariant mass distributions is almost the same. As
shown in Fig. 3, most of the experimental data locate in the theoretical one
sigma region. This can be tested by future precise measurements for the
$X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ channel. In addition, further more
precise measurements of the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ channel can also
help to reduce the uncertainty of the couplings $g_{X_{1}}$ and $g_{X_{2}}$.
The $g_{X_{2}}$ can also constrain the data in $J/\psi\pi^{+}\pi^{-}\pi^{0}$
channel.

##  IV Summary

In summary, we have performed a theoretical calculation for the processes of
$X(3872)\to J/\psi\pi^{+}\pi^{-}$ and $J/\psi\pi^{+}\pi^{-}\pi^{0}$. For the
$X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay, in addition to the dominant
contribution from the $\rho^{0}$ meson, the contribution of the intermediate
$\omega$ meson with an effective $\omega\pi\pi$ coupling is also considered in
our framework. It is found that the recent LHCb experimental measurements on
the $\pi^{+}\pi^{-}$ invariant mass distributions LHCb:2022bly  can be well
reproduced. Meanwhile, the ratio of the couplings between $X(3872)\to
J/\psi\rho^{0}$ and $X(3872)\to J/\psi\omega$ is determined, which is
consistent with the previous analysis in Refs. Hanhart:2011tn ; LHCb:2022bly .

Furthermore, with the model parameters determined from the $\pi^{+}\pi^{-}$
invariant mass distribution of the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ decay,
the $X(3872)\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ branching fraction and the
corresponding $\pi^{+}\pi^{-}\pi^{0}$ invariant mass distributions are
extracted, which are also in agreement with the available experimental data
with large errors. This kind of results could be tested by the future precise
measurements.

## Acknowledgments

We would like to thank Prof. Gang Li for useful discussions. This work is
partly supported by the National Natural Science Foundation of China under
Grant Nos. 12075288, 11735003, 11961141012, 12035007, the Youth Innovation
Promotion Association CAS, Guangdong Provincial funding with Grant No.
2019QN01X172, Science and Technology Program of Guangzhou No. 2019050001. Q.W.
is also supported by the NSFC and the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the funds provided to the Sino-German
Collaborative Research Center TRR110 “Symmetries and the Emergence of
Structure in QCD” (NSFC Grant No. 12070131001, DFG Project-ID 196253076-TRR
110).

##  Appendix A Four-body phase space

In this appendix, we provide the definitions of those variables in the phase
space integration of Eq. (17), which are explicitly shown in Fig. 4. The
$\mathbb{p}_{1}^{\ast}$ and ($\theta_{1}$, $\phi_{1}$) are the three-momentum
and decay angles of the outing $\pi^{-}$ in the $\pi^{+}\pi^{-}$ center-of-
mass (c.m.) frame. The $\mathbb{p}^{\prime}_{3}$ and ($\theta_{2}$,
$\phi_{2}$) are the three-momentum and decay angles of the outing $\pi^{0}$ in
the $\pi^{+}\pi^{-}\pi^{0}$ c.m. frame. The $\mathbb{p}_{4}$ is the three-
momentum of the final $J/\psi$ meson in the $X(3872)$ rest frame.

Figure 4: The definitions of these variables in the phase space integration of
the $X(3872)\to J/\psi\omega\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ decay.

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