# The $h$-vectors of the edge rings of a special family of graphs

Akihiro Higashitani  Department of Pure and Applied Mathematics, Graduate
School of Information Science and Technology, Osaka University, Suita, Osaka
565-0871, Japan<EMAIL_ADDRESS>and  Nayana Shibu Deepthi
Department of Pure and Applied Mathematics, Graduate School of Information
Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
<EMAIL_ADDRESS>
###### Abstract.

The $h$-vectors of homogeneous rings are one of the most important invariants
that often reflect ring-theoretic properties. On the other hand, there are few
examples of edge rings of graphs whose $h$-vectors are explicitly computed. In
this paper, we compute the $h$-vector of a special family of graphs, by using
the technique of initial ideals and the associated simplicial complex.

###### Key words and phrases:

Edge rings, toric ideals, Gröbner basis, initial complex, $h$-vector, almost
Gorenstein

###### 2010 Mathematics Subject Classification:

Primary 13P10, Secondary 05E40

##  1\. Introduction

Throughout the paper, we assume all graphs to be connected and having no loops
and multiple edges.

Many researchers have conducted intensive studies on the edge rings and toric
ideals of graphs. Especially, Ohsugi and Hibi initiated the investigation and
have been developing the theory of edge rings. For instance, the
characterization for the edge rings to be normal was given in [10]. Note that,
almost at the same time, the same result was obtained by Simis-Vasconcelos-
Villarreal independently in [12]. For the introduction to the edge rings and
toric ideals of graphs, we refer the readers to [4, Section 5] and [15,
Section 10].

One of the most important invariants of homogeneous rings is the Hilbert
series. In fact, the Gorensteinness of homogeneous normal Cohen–Macaulay
domains is characterized by the symmetry of their $h$-vectors ([13]).
Moreover, there are many other results claiming that the $h$-vectors of
homogeneous (or semi-standard graded) normal Cohen–Macaulay rings (or domains)
have some connection with their ring-theoretic properties (see, e.g., [1, 5,
7, 16], and so on). On the other hand, the $h$-vector of an edge ring is
scarcely known. As far as we know, the $h$-vectors (or their counterparts) of
the edge rings of the following graphs have been computed:

  * •

Let $K_{m,n}$ denote the complete bipartite graph with $m+n$ vertices. Then

$h(\Bbbk[K_{m,n}];t)=\sum_{i=0}^{\min\\{m,n\\}}\binom{m-1}{i}\binom{n-1}{i}t^{i}.$

  * •

Let $K_{m}$ denote the complete graph with $m$ vertices. Then

$h(\Bbbk[K_{m}];t)=1+\frac{m(m-3)}{2}t+\sum_{i=2}^{\left\lfloor\frac{m}{2}\right\rfloor}\binom{m}{2i}t^{i}.$

  * •

The Hilbert functions of the edge rings of complete multipartite graphs were
computed in [11, Theorem 2.6].

  * •

The Hilbert series of the edge rings of bipartite graphs are described using
their interior polynomials ([8]).

(On the notations used above, see Section 2.) Regarding the results on
$K_{m,n}$ and $K_{m}$, see [14], or [15, Section 10]. Note that if the edge
rings of the graphs are normal, then the Hilbert functions (resp. $h$-vectors)
of the edge rings agree with the “Ehrhart polynomials (resp. $h^{*}$-vectors)
of the edge polytopes arising from the graphs”. The goal of this paper is to
explicitly compute the $h$-vector for some particular class of graphs in order
to suggest more examples.

Let $n\geq 2$ be an integer. We introduce a connected non-bipartite graph
$\mathcal{G}_{n}$ as shown in Figure 1. Clearly, $\mathcal{G}_{n}$ satisfies
the odd cycle condition (see Section 2).

$z_{n}$$u^{(1)}_{n}$$x_{n}$$y_{n}$$w$$u^{(2)}_{n}$$u^{(2)}_{1}$$u^{(1)}_{1}$$x_{2}$$u^{(1)}_{2}$$y_{2}$$u^{(2)}_{2}$$z_{2}$$y_{1}$$z_{1}$$x_{1}$
Figure 1. The graph $\mathcal{G}_{n}$

Our main result of this paper is the following:

######  Theorem 1.1.

The $h$-polynomial of $\Bbbk[\mathcal{G}_{n}]$ is as follows:

$h(\Bbbk[\mathcal{G}_{n}];t)=\binom{n}{0}+\left(\binom{n}{1}-1\right)t+\binom{n}{2}t^{2}+\cdots+\binom{n}{n}t^{n}=(1+t)^{n}-t.$

Moreover, $\Bbbk[\mathcal{G}_{n}]$ is almost Gorenstein but not Gorenstein if
$n\geq 3$.

The notion of almost Gorenstein homogeneous rings was introduced by Goto-
Takahashi-Taniguchi in [3], as a new class of graded rings between
Cohen–Macaulay rings and Gorenstein rings. After this work, almost Gorenstein
homogeneous rings have been studied further, e.g., in [5, 9]. On almost
Gorensteinness of edge rings, known examples of almost Gorenstein non-
Gorenstein edge rings are presumably rare. However, almost Gorenstein edge
rings arising from complete multipartite graphs were completely characterized
in [6]. According to [6, Examples 1.5-1.7], we know that $K_{2,m}$,
$K_{1,1,m}$, $K_{1,m,m}$ with $m\geq 3$ and $K_{1,1,m,m}$ with $m\geq 2$ give
almost Gorenstein but not Gorenstein edge rings, where
$K_{a_{1},\ldots,a_{r}}$ denotes the complete $r$-partite graph. Theorem 1.1
gives a new family of graphs whose edge rings are almost Gorenstein but not
Gorenstein.

A brief structure of this paper is as follows. In Section 2, we recall some
fundamental notions on homogeneous rings, and prepare some materials on the
edge rings and toric ideals of graphs for the computation of the
$h$-polynomial. In Section 3, we compute the $h$-polynomial of
$\Bbbk[\mathcal{G}_{n}]$ by using a certain initial complex of the toric ideal
of $\mathcal{G}_{n}$. In Section 4, we prove the almost Gorensteinness of
$\Bbbk[\mathcal{G}_{n}]$. On the non-Gorensteinness of
$\Bbbk[\mathcal{G}_{n}]$ for $n\geq 3$, see Remark 2.1.

##  2\. Preliminaries

First, we recall some fundamental materials like, notions on commutative
algebra, the edge rings and toric ideals of graphs.

###  2.1. Homogeneous rings

In this subsection, we collect some notions on homogeneous rings used in this
paper. We refer the readers to [2, 15] for more detailed information on
homogeneous rings.

Let $R$ be a Cohen–Macaulay homogeneous ring of dimension $d$ over a field
$\Bbbk$. Then the Hilbert series of $R$ looks as follows:

$\sum_{i\geq
0}\dim_{\Bbbk}R_{i}t^{i}=\frac{h_{0}+h_{1}t+\cdots+h_{s}t^{s}}{(1-t)^{d}},$

where $R_{i}$ denotes the homogeneous part of $R$ of degree $i$,
$\dim_{\Bbbk}$ denotes the dimension as a $\Bbbk$-vector space, and we assume
that $h_{s}\neq 0$. We call the polynomial $h_{0}+h_{1}t+\cdots+h_{s}t^{s}$
appearing in the numerator as the $h$-polynomial of $R$, denoted by $h(R;t)$,
and the sequence of the coefficients $(h_{0},h_{1},\ldots,h_{s})$ as the
$h$-vector of $R$, denoted by $h(R)$.

######  Remark 2.1.

It follows from [13] that, if $R$ is a normal Cohen–Macaulay homogeneous
domain, then $R$ is Gorenstein if and only if $h(R)$ is symmetric, i.e.,
$h_{i}=h_{s-i}$ for $i=0,1,\ldots,s$. Hence, once we get the $h$-vector of
$\Bbbk[\mathcal{G}_{n}]$, as described in Theorem 1.1, we obtain that
$\Bbbk[\mathcal{G}_{n}]$ is Gorenstein if and only if $n\leq 2$.

We denote the Cohen–Macaulay type of $R$ by $r(R)$. It is well known that
$r(R)$ coincides with the number of elements in the minimal system of
generators of the canonical module of $R$.

###  2.2. Edge rings

In this subsection, we recall the definitions of edge rings and toric ideals
of graphs, and some fundamental theorems regarding them.

First of all, let us recall the definition of edge rings. Let $G$ be a graph
on the vertex set $[d]$ with edge set $E(G)=\\{e_{1},\ldots,e_{m}\\}$. We
define a polynomial ring $R=\Bbbk[t_{1},\ldots,t_{d}]$ in $d$ variables and
another one $S=\Bbbk[x_{1},\ldots,x_{m}]$ in $m$ variables, where $\Bbbk$ is a
field. Let $\pi:S\rightarrow R$ be the ring homomorphism defined by
$\pi(x_{i})={\bf t}^{e_{i}}$ for $i=1,\ldots,m$, where ${\bf
t}^{e}:=t_{u}t_{v}$ for any edge $e=\\{u,v\\}\in E(G)$. The image
$\operatorname{Im}(\pi)$, which is a subalgebra of $R$, is called the edge
ring of $G$, and the kernel $\operatorname{Ker}(\pi)$, an ideal of $S$, is
called the toric ideal of $G$. We denote the edge ring of $G$ by $\Bbbk[G]$
and the toric ideal of $G$ by $I_{G}$. Clearly, we have the ring isomorphism
$\Bbbk[G]\cong S/I_{G}$. It is known that $\Bbbk[G]$ is $d$-dimensional if $G$
is not bipartite (cf. [10, Proposition 1.3]).

Given an edge $e=\\{i,j\\}\in E(G)$, let
$\rho(e)=\mathbf{e}_{i}+\mathbf{e}_{j}$, where
$\mathbf{e}_{1},\ldots,\mathbf{e}_{d}\in\mathbb{R}^{d}$ are the unit vectors
of $\mathbb{R}^{d}$. Let $A_{G}=\\{\rho(e):e\in E(G)\\}$. For
$\mathbb{A}\subset\mathbb{R}^{d}$, let
$\displaystyle\mathbb{A}A_{G}=\left\\{\sum_{e\in
E(G)}a_{e}\rho(e):a_{e}\in\mathbb{A}\right\\}$. In this paper, we only
consider the cases where $\mathbb{A}$ is $\mathbb{Q}_{\geq 0}$ or $\mathbb{Z}$
or $\mathbb{Z}_{\geq 0}$. We can regard $\Bbbk[G]$ as a monoid algebra of an
affine monoid $\mathbb{Z}_{\geq 0}A_{G}$. In particular, the structure of the
cone $\mathbb{Q}_{\geq 0}A_{G}$ plays a crucial role in the study of
$\Bbbk[G]$.

Next, let us describe the cone $\mathbb{Q}_{\geq 0}A_{G}$ in terms of linear
inequalities. For the description, we need to introduce some more notions on
graphs.

  * •

For a subset $W\subset V(G)$, let $G\setminus W$ be the subgraph on
$V(G)\setminus W$ with the edge set $\\{e\in E(G):e\subset V(G)\setminus
W\\}$. If $W=\\{w\\}$, then we write $G\setminus w$ instead of
$G\setminus\\{w\\}$.

  * •

For $v\in V(G)$, let $N_{G}(v)=\\{u\in V(G):\\{u,v\\}\in E(G)\\}$, and for any
subset $W\subset V(G)$, let $N_{G}(W)=\bigcup\limits_{w\in W}N_{G}(w)$.

  * •

A non-empty subset $T\subset V(G)$ is called an independent set if
$\\{v,w\\}\not\in E(G)$ for any $v,w\in T$.

  * •

We call a vertex $v$ of $G$ regular if each connected component of $G\setminus
v$ contains an odd cycle.

  * •

We say that an independent set $T$ of $V(G)$ is a fundamental set if

    * –

the bipartite graph on the vertex set $T\cup N_{G}(T)$ with the edge set
$\\{\\{v,w\\}:v\in T,w\in N_{G}(T)\\}\cap E(G)$ is connected, and

    * –

$T\cup N_{G}(T)=V(G)$, or each of the connected components of the graph
$G\setminus(T\cup N_{G}(T))$ contains an odd cycle.

It follows from [10, Theorem 1.7 (a)] that $\mathbb{Q}_{\geq 0}A_{G}$ consists
of the elements $(x_{v})_{v\in V(G)}\in\mathbb{Q}^{d}$ satisfying all the
following inequalities:

(1) $\begin{split}x_{u}&\geq 0\;\;\text{ for any regular vertex }u;\\\
\sum_{v\in N_{G}(T)}x_{v}&\geq\sum_{u\in T}x_{u}\;\;\text{ for any fundamental
set }T.\end{split}$

Next, from [4, Section 5.3], we shall recall how to describe the generators of
the toric ideal of a graph. Let $G$ be a graph. Given
$\Gamma=(e_{i_{1}},\ldots,e_{i_{2r}})$, a sequence of edges of $G$, we call
$\Gamma$ an even closed walk if $|e_{i_{j}}\cap e_{i_{j+1}}|=1$ for
$j=1,\ldots,2r$, where $i_{2r+1}=i_{1}$. Moreover, we associate a binomial
$f_{\Gamma}=f_{\Gamma}^{(+)}-f_{\Gamma}^{(-)}$, where

$f_{\Gamma}^{(+)}=\prod_{k=1}^{r}x_{i_{2k-1}}\;\text{ and
}\;f_{\Gamma}^{(-)}=\prod_{k=1}^{r}x_{i_{2k}}.$

We say that an even closed walk $\Gamma$ is primitive if there is no even
closed walk $\Gamma^{\prime}$ of $G$ with $f_{\Gamma^{\prime}}\neq f_{\Gamma}$
such that $f_{\Gamma^{\prime}}^{(+)}$ divides $f_{\Gamma}^{(+)}$ and
$f_{\Gamma^{\prime}}^{(-)}$ divides $f_{\Gamma}^{(-)}$.

######  Lemma 2.2 ([4, Lemma 5.10]).

The toric ideal $I_{G}$ is generated by the binomials $f_{\Gamma}$ for all
primitive even closed walks $\Gamma$.

Finally, we recall the condition for the normality of $\Bbbk[G]$. We say that
$G$ satisfies the odd cycle condition if for any pair of odd cycles $C$ and
$C^{\prime}$ in $G$, if $C$ and $C^{\prime}$ do not share any common vertex,
then there is a bridge between $C$ and $C^{\prime}$, i.e., there exist $v\in
V(C)$ and $v^{\prime}\in V(C^{\prime})$ such that $\\{v,v^{\prime}\\}\in
E(G)$.

We see that $\Bbbk[G]$ is normal if and only if

(2) $\displaystyle\mathbb{Z}_{\geq 0}A_{G}=\mathbb{Q}_{\geq
0}A_{G}\cap\mathbb{Z}A_{G}$

holds. (See, e.g., [2, Section 6.1].) On the normality of $\Bbbk[G]$, the
following is known:

######  Theorem 2.3 ([10, 12]).

A graph $G$ satisfies (2) if and only if $G$ satisfies the odd cycle
condition.

Remark that $\Bbbk[G]$ is a normal Cohen–Macaulay homogeneous domain if $G$
satisfies the odd cycle condition. Cohen–Macaulayness follows from Hochster’s
theorem (see [2, Theorem 6.3.5]).

###  2.3. Fundamental properties of $\Bbbk[\mathcal{G}_{n}]$

Now, let us focus on our graph $\mathcal{G}_{n}$. We identify the edges of
$\mathcal{G}_{n}$ with the variables of the polynomial ring $S$, as depicted
in Figure 1. Namely, we consider

$S=\Bbbk[x_{1},y_{1},z_{1},\ldots,x_{n},y_{n},z_{n}]$

and we regard $I_{\mathcal{G}_{n}}$ as an ideal of $S$.

For $\mathcal{G}_{n}$, we see that, from [4, Lemma 5.11], every primitive even
closed walk consists of two $3$-cycles with exactly one common vertex, and is
given by

$(x_{i},z_{i},y_{i},x_{j},z_{j},y_{j});\hskip 8.5359pt1\leq i<j\leq n.$

Hence, the toric ideal $I_{\mathcal{G}_{n}}$ is generated by the binomials:

(3) $\displaystyle x_{i}y_{i}z_{j}-z_{i}x_{j}y_{j};\hskip 8.5359pt1\leq
i<j\leq n.$

Let $<_{\text{lex}}$ be the graded lexicographic order on $S$ induced by the
ordering of the variables

(4) $\displaystyle
x_{1}<_{\text{lex}}y_{1}<_{\text{lex}}z_{1}<_{\text{lex}}\dots<_{\text{lex}}x_{n}<_{\text{lex}}y_{n}<_{\text{lex}}z_{n}.$

For the fundamental materials on initial ideals and Gröbner basis, consult,
e.g., [4, Section 1].

######  Lemma 2.4.

The binomials in (3) form a Gröbner basis of $I_{\mathcal{G}_{n}}$ with
respect to the monomial order $<_{\text{lex}}$.

###### Proof.

The result follows from the straightforward application of Buchberger’s
criterion to the set of generators (3) of $I_{\mathcal{G}_{n}}$. (For
Buchberger’s criterion, see, e.g., [4, Theorem 1.29].)

Let $f=x_{i}y_{i}z_{j}-z_{i}x_{j}y_{j}$ and
$g=x_{p}y_{p}z_{q}-z_{p}x_{q}y_{q}$ be two generators. If $i\neq p$ and $j\neq
q$, then the leading terms of $f$ and $g$ are relatively prime and thus the
$S$-polynomial $S(f,g)$ will reduce to $0$ by [4, Lemma 1.27].

Suppose $i=p$. Then

$\displaystyle S(f,g)$
$\displaystyle=\frac{\mathrm{lcm}(\operatorname{in}_{<_{\text{lex}}}(f),\operatorname{in}_{<_{\text{lex}}}(g))}{\operatorname{in}_{<_{\text{lex}}}(f)}f-\frac{\mathrm{lcm}(\operatorname{in}_{<_{\text{lex}}}(f),\operatorname{in}_{<_{\text{lex}}}(g))}{\operatorname{in}_{<_{\text{lex}}}(g)}g$
$\displaystyle=z_{q}f-z_{j}g$
$\displaystyle=z_{q}(x_{i}y_{i}z_{j}-z_{i}x_{j}y_{j})-z_{j}(x_{i}y_{i}z_{q}-z_{i}x_{q}y_{q})$
$\displaystyle=z_{i}(x_{q}y_{q}z_{j}-z_{q}x_{j}y_{j}).$

Note that, up to sign, $x_{q}y_{q}z_{j}-z_{q}x_{j}y_{j}$ is a generator of
$I_{\mathcal{G}_{n}}$ and therefore $S(f,g)$ will reduce to $0$. The $j=q$
case is similar. ∎

######  Corollary 2.5.

The initial ideal $in_{<_{\text{lex}}}\big{(}I_{\mathcal{G}_{n}}\big{)}$ of
$I_{\mathcal{G}_{n}}$ with respect to the monomial order $<_{\text{lex}}$ is
generated by the squarefree monomials

(5) $\displaystyle x_{i}y_{i}z_{j};\hskip 14.22636pt1\leq i<j\leq n.$

By this corollary, since the given monomial ideal is squarefree, we can
associate a simplicial complex whose Stanley-Reisner ideal coincides with the
initial ideal generated by (5).

##  3\. Computation of the $h$-polynomial of $\Bbbk[\mathcal{G}_{n}]$

Let $\Delta_{n}$ be the simplicial complex whose Stanley-Reisner ideal
coincides with the initial ideal of the toric ideal $I_{\mathcal{G}_{n}}$ with
respect to $<_{\text{lex}}$. (For the introduction to Stanley-Reisner theory,
consult, e.g., [2, Section 5].) Let $\mathcal{F}(\Delta_{n})$ be the set of
all facets of $\Delta_{n}$. By definition, any facet of our simplicial complex
$\Delta_{n}$ can be expressed as the maximal set that does not contain the
triplet $\\{x_{i},y_{i},z_{j}\\};\hskip 5.69046pt1\leq i<j\leq n.$ Since
$x_{n},y_{n}$ and $z_{1}$ will be contained in all the facets, with out loss
of generality, we write the facets without indicating these elements.
Therefore, any $F\in\mathcal{F}(\Delta_{n})$ can be expressed as:

$F=\bigcup\limits_{\begin{subarray}{c}i\in I\\\ 1\leq i\leq
n-1\end{subarray}}\\{x_{i}\\}\cup\bigcup\limits_{\begin{subarray}{c}j\in J\\\
1\leq j\leq
n-1\end{subarray}}\\{y_{j}\\}\cup\bigcup\limits_{\begin{subarray}{c}k\in K\\\
2\leq k\leq n\end{subarray}}\\{z_{k}\\},$

which is maximal and does not contain the triplet
$\\{x_{i},y_{i},z_{j}\\};\hskip 5.69046pt1\leq i<j\leq n.$ Let us try to get a
more concrete representation for the facets in $\mathcal{F}(\Delta_{n})$.
Consider any $F\in\mathcal{F}(\Delta_{n})$.

Case 1:

Let $i\in I\cap J$. This implies $z_{i+1},\dots,z_{n}\notin F$, since $F$ does
not contain the triplet $\\{x_{i},y_{i},z_{j}\\};\hskip 5.69046pt1\leq i<j\leq
n.$

Case 2:

Let us consider $i\in I$. If there exists some $k$ with $i<k\leq n$ such that
$z_{k}\in F$, then we have $y_{i}\notin F,$ that is, $i\in I\backslash J.$

If $z_{k}\notin F$ for all $k$ with $i<k\leq n$, then by the maximality of
$F$, we have $y_{i}\in F$ and $i\in I\cap J$.

Case 3:

Let us consider $j\in J$. If there exists some $k$ with $j<k\leq n$ such that
$z_{k}\in F$, then we have $x_{j}\notin F,$ that is, $j\in J\backslash I.$

If $z_{k}\notin F$ for all $k$ with $j<k\leq n$, then by the maximality of the
set $F$, we have $x_{j}\in F$ and $j\in I\cap J$.

According to our observations from the three situations above, any facet in
$\mathcal{F}(\Delta_{n})$ is of the form:

(6)
$\displaystyle\\{w_{1},\dots,w_{j-1}\\}\cup\\{x_{j},y_{j},\dots,x_{n-1},y_{n-1}\\}\cup\\{z_{2},\dots,z_{j}\\},$

where $w_{i}\in\\{x_{i},y_{i}\\}$ and $j=1,\dots,n.$

######  Remark 3.1 (Mac-Mullen characterization of $h$-vectors).

Here, we recall a well-known method to compute the $h$-vector of a shellable
simplicial complex. (For example, see [2, Corollary 5.1.14].)

Let $F_{1},\dots,F_{t}$ be the facets of a pure simplicial complex $\Delta$.
Let $\langle F_{1},\ldots,F_{m}\rangle$ be the unique smallest simplicial
complex which contains all $F_{i}$, $1\leq i\leq m$. The ordering of the
facets is said to be a shelling if it satisfies that $\langle
F_{i}\rangle\cap\langle F_{1},\ldots,F_{i-1}\rangle$ is generated by a non-
empty set of maximal proper faces of $F_{i}$ for all $2\leq i\leq t$. We say
that a pure simplicial complex is shellable if it has a shelling. Throughout
our further study, we may refer the subcomplex $\langle
F_{i}\rangle\cap\langle F_{1},\ldots,F_{i-1}\rangle$ as the intersection
subcomplex corresponding to the $i^{\textrm{th}}$ shelling step. Let $r_{i}$
be the number of maximal proper faces of $F_{i}$ in $\langle
F_{i}\rangle\cap\langle F_{1},\ldots,F_{i-1}\rangle$ for $2\leq i\leq t$, and
let $r_{1}=0$. Then the $h$-vector of $\Delta$,
$h(\Delta)=(h_{0},\dots,h_{d})$, is obtained by $h_{i}=|\\{j\colon
r_{j}=i\\}|.$

Now, we consider an ordering of the facets $F_{1}^{1},\ldots,F_{t_{n}}^{n}$ of
$\Delta_{n}$. From the structure of each facet as shown in (6), we have
$t_{n}=\sum\limits_{i=0}^{n-1}2^{i}$ and $|F_{i}^{n}|=2n-2$ for all $1\leq
i\leq t_{n}$. Let us consdier each facet as a $(2n-2)$-tuple of
$x_{1},y_{1},x_{2},y_{2},z_{2},\ldots,x_{n-1},y_{n-1}$, $z_{n-1},z_{n}$.
Lexicographic order $<_{L}$ in $\mathcal{F}(\Delta_{n})$ is defined by

$(a_{1},\ldots,a_{2n-2})<_{L}(b_{1},\ldots,b_{2n-2})$

if and only if either $a_{i}\neq b_{i}$ for some $i$ and
$a_{i}<_{\text{lex}}b_{i}$ with respect to (4). Now, we consider an ordering
of the facets $F_{1}^{n},\ldots,F_{t_{n}}^{n}$ of $\Delta_{n}$ such that they
are arranged in lexicographically increasing order of their corresponding
$(2n-2)$-tuple.

Let $r^{n}_{i}$ be the number of maximal proper faces of $F^{n}_{i}$ that
generates the $i^{\mathrm{th}}$ intersection subcomplex for $2\leq i\leq
t_{n}$. We define $\delta_{n}=\\{r^{n}_{2},\dots,r^{n}_{t_{n}}\\}$ with $n\geq
2$ as a multi-set.

######  Lemma 3.2.

For each $n\geq 2,$ we have

$\delta_{n+1}=\\{1,\delta_{n},2,\delta_{n}+1\\},$

where $\delta_{n}+1=\\{\alpha+1:\alpha\in\delta_{n}\\}$.

By induction on $n$, and using Lemma 3.2, we can obtain the $h$-vector of
$\Delta_{n}$. Our goal is to show that

$h(\Delta_{n})=\Bigg{(}\binom{n}{0},\binom{n}{1}-1,\binom{n}{2},\binom{n}{3},\dots,\binom{n}{n}\Bigg{)}\;\;\text{
for any}\;\;n\geq 2.$

For $n=2$, since
$\mathcal{F}(\Delta_{2})=\\{\\{x_{1},y_{1}\\},\\{x_{1},z_{2}\\},\\{y_{1},z_{2}\\}\\}$,
we obtain the $h$-vector as $(1,1,1)$ which is equal to our formula
$\Big{(}\binom{2}{0},\binom{2}{1}-1,\binom{2}{2}\Big{)}$.

By the hypothesis of induction, assume that our formula holds for an arbitrary
$n$. Therefore, the $h$-vector associated with $\Delta_{n}$ is
$\Big{(}\binom{n}{0},\binom{n}{1}-1,\binom{n}{2},\binom{n}{3},\dots,\binom{n}{n}\Big{)}$.
Let $h(\Delta_{n+1})=(h_{0}^{n+1},h_{1}^{n+1},\ldots,h_{n+1}^{n+1})$. By Lemma
3.2, we see the following:

$\displaystyle h^{n+1}_{0}=1=\binom{n+1}{0},$ $\displaystyle
h^{n+1}_{1}=1+h^{n}_{1}=1+\binom{n}{1}-1=\binom{n+1}{1}-1,$ $\displaystyle
h^{n+1}_{2}=1+h^{n}_{2}+h^{n}_{1}=1+\binom{n}{2}+\binom{n}{1}-1=\binom{n+1}{2},$
$\displaystyle
h^{n+1}_{i}=h^{n}_{i}+h^{n}_{i-1}=\binom{n}{i}+\binom{n}{i-1}=\binom{n+1}{i}\;\;\text{for}\;\;2\leq
i\leq n.$

Hence by induction, the $h$-vector associated with simplicial complex
$\Delta_{n}$ is

$\Bigg{(}\binom{n}{0},\binom{n}{1}-1,\binom{n}{2},\binom{n}{3},\dots,\binom{n}{n}\Bigg{)}\;\;\text{for
any}\;\;n\geq 2.$

Since the Hilbert series of $S/I$ coincides with that of
$S/\operatorname{in}_{<_{\text{lex}}}(I)$ in general (see, e.g., [4,
Proposition 2.6]), we conclude the above as the desired $h$-vector of
$\Bbbk[\mathcal{G}_{n}]$.

The remaining section is devoted to illustrating that our ordering of the
facets gives a shelling of $\Delta_{n}$ and establishing Lemma 3.2.

For each $n\geq 2,$ we observe that $r^{n}_{2}=1$,
$t_{n}=|\mathcal{F}(\Delta_{n})|=\sum\limits_{i=0}^{n-1}2^{i}$ and we have
$t_{n+1}=1+2t_{n}$.

As per our ordering, the facets are ordered in such a manner that the facets
consisting of $x_{1}$ comes first and after the
$\big{(}\frac{t_{n}+1}{2}\big{)}^{\textrm{th}}$ stage, the pattern of facet
ordering repeats in the exact same manner as that of
$F^{n}_{2},\dots,F^{n}_{\frac{t_{n}+1}{2}}$, and consists of same elements
except for $x_{1}$ replaced with $y_{1}$. More precisely, we have
$F_{\frac{t_{n}+1}{2}+i}=(F_{i+1}\setminus\\{x_{1}\\})\cup\\{y_{1}\\}$ for
$1\leq i\leq\frac{t_{n}-1}{2}$. Therefore, the corresponding intersection
subcomplex for each $k^{\textrm{th}}$ shelling step, $\frac{t_{n}+3}{2}\leq
k\leq t_{n}$, will definitely contain the maximal face

$F^{n}_{i+1}\setminus\\{x_{1}\\}\hskip 5.69046pt\text{for any }1\leq
i\leq\frac{t_{n}-1}{2}.$

Moreover, when we look at the first $\frac{t_{n}+1}{2}$ facets, we observe
that each of the facets differs from the preceding one by just an element.
Thus with these observations, we claim that $F_{1}^{n},\ldots,F_{t_{n}}^{n}$
is a shelling of $\Delta_{n}$.

Since the ordering pattern repeats after the
$\big{(}\frac{t_{n}+1}{2}\big{)}^{\textrm{th}}$ facet, for $1\leq
i\leq\frac{t_{n}-1}{2}$, we see that $r_{i+1}^{n}$ maximal proper faces will
always be contained in the intersection subcomplex for each $k^{\textrm{th}}$
shelling step, $\frac{t_{n}+1}{2}+i\leq k\leq t_{n}$ and the intersection
subcomplex also contains the maximal face $F_{i+1}^{n}\setminus\\{x_{1}\\}$.
Therefore, we have

$r^{n}_{\frac{t_{n}+1}{2}+i}=1+r^{n}_{i+1}\hskip 5.69046pt\text{for any }1\leq
i\leq\frac{t_{n}-1}{2}.$

Hence, we can express $\delta_{n}$ as

$\delta_{n}=\\{1,r^{n}_{3},\dots,r^{n}_{\frac{t_{n}+1}{2}},2,r^{n}_{3}+1,\dots,r^{n}_{\frac{t_{n}+1}{2}}+1\\}.$

The set $\mathcal{F}(\Delta_{n})$ consists of (6). Namely, each facet
$F^{n}_{k}$, $1\leq k\leq t_{n}$, can be denoted as:

$\bigcup\limits_{i=1}^{j-1}\\{w_{i}\\}\cup\bigcup\limits_{i=j}^{n-1}\\{x_{i},y_{i}\\}\cup\bigcup\limits_{i=2}^{j}\\{z_{i}\\},\hskip
11.38092ptj=1,\dots,n.$

For each $1\leq j\leq n$, we have $2^{j-1}$ number of facets corresponding to
it. We can show that there exists a one-to-one correspondence between the
facets $F^{n}_{k}$ and $F^{n+1}_{k+1}$ for all $n\geq 2$ and $2\leq k\leq
t_{n}$. The one-to-one correspondence is given by

$\phi\colon\Big{\\{}F_{k}^{n}\colon 2\leq k\leq
t_{n}\Big{\\}}\longrightarrow\Big{\\{}F_{k^{\prime}}^{n+1}\colon 3\leq
k^{\prime}\leq\frac{t_{n+1}+1}{2}\Big{\\}}$
$\phi(F^{n}_{k})=F^{n+1}_{k+1},\hskip 14.22636pt2\leq k\leq t_{n},$
$\bigcup\limits_{i=1}^{j-1}\\{w_{i}\\}\cup\bigcup\limits_{i=j}^{n-1}\\{x_{i},y_{i}\\}\cup\bigcup\limits_{i=2}^{j}\\{z_{i}\\}\longmapsto\\{x_{1}\\}\cup\bigcup\limits_{i=1}^{j-1}\\{w_{i+1}\\}\cup\bigcup\limits_{i=j}^{n-1}\\{x_{i+1},y_{i+1}\\}\cup\bigcup\limits_{i=2}^{j+1}\\{z_{i}\\},$

with $j=1,\dots,n.$

In order to prove this one-to-one correspondence, we show that any facet in
the set, $\Big{\\{}F_{k^{\prime}}^{n+1}\colon 3\leq
k^{\prime}\leq\frac{t_{n+1}+1}{2}\Big{\\}}$ will be of the form

$\\{x_{1}\\}\cup\bigcup\limits_{i=1}^{j-1}\\{w_{i+1}\\}\cup\bigcup\limits_{i=j}^{n-1}\\{x_{i+1},y_{i+1}\\}\cup\bigcup\limits_{i=2}^{j+1}\\{z_{i}\\},$

corresponding to the facet
$\bigcup\limits_{i=1}^{j-1}\\{w_{i}\\}\cup\bigcup\limits_{i=j}^{n-1}\\{x_{i},y_{i}\\}\cup\bigcup\limits_{i=2}^{j}\\{z_{i}\\}\;(2\leq
j\leq n)$ in the set $\Big{\\{}F_{k}^{n}\colon 2\leq k\leq t_{n}\Big{\\}}$.

According to our shelling, any facet of the simplicial complex $\Delta_{n+1}$
containing the element $\\{x_{1}\\}$ belongs to $\Big{\\{}F_{k}^{n+1}\colon
1\leq k\leq\frac{t_{n+1}+1}{2}\Big{\\}}$ with $1\leq j\leq n+1$. So, when we
consider some $F^{n+1}_{k^{\prime}}\in\Big{\\{}F_{k}^{n+1}\colon 3\leq
k\leq\frac{t_{n+1}+1}{2}\Big{\\}}$, it can be expressed as:

$F^{n+1}_{k^{\prime}}=\\{x_{1}\\}\cup\bigcup\limits_{i=2}^{j^{\prime}-1}\\{w_{i}\\}\cup\bigcup\limits_{i=j^{\prime}}^{n}\\{x_{i},y_{i}\\}\cup\bigcup\limits_{i=2}^{j^{\prime}}\\{z_{i}\\},\hskip
8.5359pt3\leq j^{\prime}\leq n+1.$

We can write it as

$\displaystyle F^{n+1}_{k^{\prime}}$
$\displaystyle=\\{x_{1}\\}\cup\bigcup\limits_{i=2}^{j^{\prime}-1}\\{w_{i}\\}\cup\bigcup\limits_{i=j^{\prime}}^{n}\\{x_{i},y_{i}\\}\cup\bigcup\limits_{i=2}^{j^{\prime}}\\{z_{i}\\}$
$\displaystyle=\\{x_{1}\\}\cup\bigcup\limits_{i=1}^{j^{\prime}-2}\\{w_{i+1}\\}\cup\bigcup\limits_{i=j^{\prime}-1}^{n-1}\\{x_{i+1},y_{i+1}\\}\cup\bigcup\limits_{i=2}^{j^{\prime}}\\{z_{i}\\}\;\;\text{for}\;\;3\leq
j^{\prime}\leq n+1.$

By expressing it in terms of $2\leq j\leq n$, we have

$F^{n+1}_{k^{\prime}}=\\{x_{1}\\}\cup\bigcup\limits_{i=1}^{j-1}\\{w_{i+1}\\}\cup\bigcup\limits_{i=j}^{n-1}\\{x_{i+1},y_{i+1}\\}\cup\bigcup\limits_{i=2}^{j+1}\\{z_{i}\\}\;\text{
for }\;2\leq j\leq n.$

Hence, we have the one-to-one correspondence. This one-to-one correspondence
guarantees that $r^{n+1}_{i+1}=r^{n}_{i}$ for $2\leq i\leq t_{n}.$

We also observe that

$t_{n+1}=1+2t_{n}\implies t_{n}=\frac{t_{n+1}-1}{2}\implies
t_{n}+1=\frac{t_{n+1}+1}{2}.$

From all the above observations, we have

$\displaystyle\delta_{n+1}$
$\displaystyle=\\{1,r^{n+1}_{3},\dots,r^{n+1}_{t_{n}+1},2,r^{n+1}_{3}+1,r^{n+1}_{t_{n}+1}+1\\}$
$\displaystyle\implies\delta_{n+1}$
$\displaystyle=\\{1,r^{n}_{2},\dots,r^{n}_{t_{n}},2,r^{n}_{2}+1,r^{n}_{t_{n}}+1\\}$
$\displaystyle\implies\delta_{n+1}$
$\displaystyle=\\{1,\delta_{n},2,\delta_{n}+1\\}.$

##  4\. On the almost Gorensteinness of $\Bbbk[\mathcal{G}_{n}]$

In this section, we are in the process of obtaining a theoretical proof to
state that $\Bbbk[\mathcal{G}_{n}]$ is almost Gorenstein for all $n\geq 2$.

We first recall the necessary and sufficient condition for a homogeneous ring
to be almost Gorenstein from [5].

######  Proposition 4.1 ([5, Corollary 2.7]).

Let $R$ be a Cohen–Macaulay homogeneous ring of dimension $d$ over a field
$\Bbbk$ and let $h(R)=(h_{0},h_{1},\ldots,h_{s})$ be its $h$-vector. Then $R$
is almost Gorenstein if and only if the following equality holds:

$\displaystyle
r(R)-1=\sum_{j=0}^{s-1}((h_{s}+\cdots+h_{s-j})-(h_{0}+\cdots+h_{j}))=:\tilde{e}(R).$

Note that $r(R)-1\leq\tilde{e}(R)$ is always satisfied, so almost
Gorensteinness is equivalent to the inequality $r(R)\geq\tilde{e}(R)+1$.

Let us assume that $R$ is a domain. Then there is an ideal $I_{R}$ which is
isomorphic to a canonical module of $R$ as an $R$-module. We know that the
number of elements of a minimal system of generators of $I_{R}$ coincides with
$r(R)$.

Let us consider $R=\Bbbk[\mathcal{G}_{n}]$. Note that $\mathcal{G}_{n}$
satisfies the odd cycle condition, so $\Bbbk[\mathcal{G}_{n}]$ is normal. By
the first part of Theorem 1.1, we can compute $\tilde{e}(R)$ as follows:

$\displaystyle\sum_{j=0}^{n-1}\left\\{\left(\binom{n}{n}+\cdots+\binom{n}{n-j}\right)-\left(\binom{n}{0}+\big{(}\binom{n}{1}-1\big{)}+\cdots+\binom{n}{j}\right)\right\\}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n-2}1=n-2.$

Hence by Proposition 4.1, it is enough to show that $r(R)\geq n-1$.
Furthermore, $r(R)$ is equal to the number of elements of a minimal system of
generators of $I_{R}$, which is the relative interior of $\mathbb{Q}_{\geq
0}A_{\mathcal{G}_{n}}\cap\mathbb{Z}A_{\mathcal{G}_{n}}$ (see [2, Theorem
6.3.5]). Let $C=\mathbb{Q}_{\geq
0}A_{\mathcal{G}_{n}}\subset\mathbb{R}^{2n+1}$. In what follows, it suffices
to show that we need at least $(n-1)$ elements as a minimal system of
generators for the relative interior of the cone $C$.

Let us denote the vertices of $\mathcal{G}_{n}$ as follows:

$\displaystyle
V(\mathcal{G}_{n})=\\{u_{i}^{(1)},u_{i}^{(2)}:i=1,\ldots,n\\}\cup\\{w\\}\;\;\text{
and we let }$ $\displaystyle
x_{i}=\\{w,u_{i}^{(1)}\\},\;y_{i}=\\{w,u_{i}^{(2)}\\}\text{ and
}z_{i}=\\{u_{i}^{(1)},u_{i}^{(2)}\\}\text{ for }i=1,\ldots,n.$

We use the following notation for each entry of $\mathbb{R}^{2n+1}$:

$\displaystyle\mathbb{R}^{2n+1}=\\{c_{1,1}\mathbf{e}_{1,1}+\cdots+c_{1,n}\mathbf{e}_{1,n}+c_{2,1}\mathbf{e}_{2,1}+\cdots+c_{2,n}\mathbf{e}_{2,n}+c^{\prime}\mathbf{e}^{\prime}:c_{1,i},c_{2,i},c^{\prime}\in\mathbb{R}\\},$

where $\mathbf{e}_{1,i},\mathbf{e}_{2,i},\mathbf{e}^{\prime}$ are the unit
vectors of $\mathbb{R}^{2n+1}$, each $\mathbf{e}_{1,i}$ (resp.
$\mathbf{e}_{2,i}$) corresponds to $u_{i}^{(1)}$ (resp. $u_{i}^{(2)}$) and
$\mathbf{e}^{\prime}$ corresponds to $w$.

For $j=1,\ldots,n-1$, let

$\alpha_{j}:=\sum_{i=1}^{n}(\mathbf{e}_{1,i}+\mathbf{e}_{2,i})+2j\mathbf{e}^{\prime}.$

In what follows, we verify that $\alpha_{j}\in
C^{\circ}\cap\mathbb{Z}^{2n+1}$, where $C^{\circ}$ denotes the relative
interior of $C$, and they should be included in a minimal system of generators
of $I_{R}$.

The first step: We check that $\alpha_{j}\in C^{\circ}$. Here, we see the
following:

  * •

Each of $u_{i}^{(1)}$s and $u_{i}^{(2)}$s is a regular vertex of
$\mathcal{G}_{n}$, while $w$ is not.

  * •

A subset $T$ of $V(\mathcal{G}_{n})$ is fundamental if and only if $T=\\{w\\}$
or $T=\\{u_{1},\ldots,u_{n}\\}$, where $u_{i}\in\\{u_{i}^{(1)},u_{i}^{(2)}\\}$
for each $i$.

Hence, it follows from (1) that
$\sum_{i=1}^{n}(c_{1,i}\mathbf{e}_{1,i}+c_{2,i}\mathbf{e}_{2,i})+c^{\prime}\mathbf{e}^{\prime}\in\mathbb{R}^{2n+1}$,
belongs to $C$ if and only if the following inequalities are satisfied:

(7) $\begin{split}&c_{1,i}\geq 0\;\;\text{and}\;\;c_{2,i}\geq 0\;\;\text{ for
any }i=1,\ldots,n,\\\ &\sum_{i=1}^{n}(c_{1,i}+c_{2,i})\geq c^{\prime},\\\
&\sum_{i\in U}c_{i,1}+\sum_{i\in[n]\setminus
U}c_{i,2}+c^{\prime}\geq\sum_{i\in[n]\setminus U}c_{i,1}+\sum_{i\in
U}c_{i,2}\;\;\text{ for any }U\subset[n].\end{split}$

It is straightforward to check that $\alpha_{j}$ satisfies these inequalities
with strict inequalities for each $j$. This implies that $\alpha_{j}\in
C^{\circ}$.

The second step: We prove that $\alpha_{j}$ cannot be written as a sum of an
element in $C^{\circ}\cap\mathbb{Z}^{2n+1}$ and an element in
$C\cap\mathbb{Z}^{2n+1}\setminus\\{{\bf 0}\\}$.

Suppose that $\alpha_{j}=\alpha^{\prime}+\beta$ for some $\alpha^{\prime}\in
C^{\circ}\cap\mathbb{Z}^{2n+1}$ and $\beta\in
C\cap\mathbb{Z}^{2n+1}\setminus\\{{\bf 0}\\}$. Let

$\alpha^{\prime}=\sum_{i=1}^{n}a_{1,i}^{\prime}\mathbf{e}_{1,i}+\sum_{i=1}^{n}a_{2,i}^{\prime}\mathbf{e}_{2,i}+a^{\prime}\mathbf{e}^{\prime}\;\text{
and
}\;\beta=\sum_{i=1}^{n}b_{1,i}\mathbf{e}_{1,i}+\sum_{i=1}^{n}b_{2,i}\mathbf{e}_{2,i}+b\mathbf{e}^{\prime}.$

Then we see that $a_{1,i}^{\prime}\geq 1$ and $a_{2,i}^{\prime}\geq 1$ for
$1\leq i\leq n$ (see (7)). Hence, $b_{1,i}\leq 0$ and $b_{2,i}\leq 0$. On the
other hand, $b_{1,i}\geq 0$ and $b_{2,i}\geq 0$ should be also satisfied, so
we obtain that $\beta=b\mathbf{e}^{\prime}$. Since $\beta\neq{\bf 0}$, by the
second inequality of (7), we have $b<0$, a contradiction to the third
inequality.

The third step: By the first and second steps, we see that
$\alpha_{1},\ldots,\alpha_{n-1}$ are required for a minimal system of
generators of $I_{R}$. This shows that $r(R)\geq n-1$, as required.

## Acknowledgements

The first named author is partially supported by JSPS Grant-in-Aid for
Scientists Research (B) 18H01134 and Scientists Research (C) 20K03513.

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# Nonleptonic decays of $\Xi_{cc}\to\Xi_{c}\pi$ with
$\Xi_{c}-\Xi_{c}^{\prime}$ mixing

Chia-Wei<EMAIL_ADDRESS>and Chao-Qiang Geng  School of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, China
University of Chinese Academy of Sciences, 100190 Beijing, China

###### Abstract

Aiming on testing the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing, we study the decays
of $\Xi_{cc}\to\Xi_{c}\pi$ with $\Xi_{cc}=(\Xi_{cc}^{++},\Xi_{cc}^{+})$,
$\Xi_{c}=(\Xi_{c}^{(\prime)+},\Xi_{c}^{(\prime)0})$ and
$\pi=(\pi^{+},\pi^{0})$. The soft-meson limit is considered along with the
pole model, and the baryon matrix elements are evaluated by the bag model with
and without removing the center-of-mass motion (CMM). We find that the four-
quark operator matrix elements are about twice larger once the unwanted CMM is
removed. We obtain that ${\cal R}={\cal
B}(\Xi_{cc}^{+}\to\Xi_{c}^{\prime+}\pi^{+})/{\cal
B}(\Xi_{cc}^{+}\to\Xi_{c}^{+}\pi^{+})=0.90\pm 0.14$ and $1.45$ with and
without removing the CMM, where the former is close to the lower bound and the
later is well consistent with ${\cal R}=1.41\pm 0.17\pm 0.10$ measured at
LHCb. In addition, we show that after including the mixing, the up-down
asymmetry of $\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)0}\pi^{+})$ flips sign.
Explicitly, we obtain that
$\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{\prime+}\pi^{0})=0.52$ and
$\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+})=0.31$ with and without the CMM
corrections, respectively, which are all negative if the mixing is absence. As
a bonus, a positive value of $\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{\prime
0}\pi^{+})$ in experiments can also serve as the evidence of the $W$-exchange
contributions.

##  I Introductions

The baryon wave functions are the precondition in evaluating the decay
quantities. It has been shown that the large $SU(3)$ flavor ($SU(3)_{F}$)
breaking effect in the singly charmed baryon semileptonic decays can be traced
back to the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing Geng:2022yxb ; SU(3)1 ;
Belle:2021crz ; ALICE:2021bli , given as

$\displaystyle|\Xi_{c}\rangle=\cos\theta_{c}|\Xi_{c}^{\overline{{\bf
3}}}\rangle+\sin\theta_{c}|\Xi_{c}^{{\bf
6}}\rangle\,,\qquad|\Xi_{c}^{\prime}\rangle=\cos\theta_{c}|\Xi_{c}^{{\bf
6}}\rangle-\sin\theta_{c}|\Xi_{c}^{\overline{{\bf 3}}}\rangle\,,$ (1)

where $\Xi_{c}^{(\prime)}=\Xi_{c}^{(\prime)+,0}$ are the physical baryons, and
$\Xi_{c}^{\overline{{\bf 3}}({\bf 6})}$ correspond to the antitriplet (sextet)
charmed baryons. At the limit of the $SU(3)_{F}$ symmetry, the physical
baryons shall have definite $SU(3)_{F}$ representations, i.e. $\theta_{c}=0$.
From the mass relations, we have found that Jenkins:1996rr

$\theta_{c}=\pm 0.137(5)\pi\,,$ (2)

with the sign unfixed. Although the mixing effect in
$\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ is the second order of $\theta_{c}$, the
effect is still sizable. Thus, it is tempting to examine the mixing effects in
the decay channels which are affected by the first order of $\theta_{c}$.

Recently, the LHCb collaboration has reported the ratio Ratio

${\cal R}(\Xi_{c}^{++}\to\Xi_{c}^{+}\pi^{+})=1.41\pm 0.17\pm 0.10\,,$ (3)

where ${\cal R}(\Xi_{cc}\to\Xi_{c}\pi)\equiv{{\cal
B}(\Xi_{cc}\to\Xi_{c}^{\prime}\pi)}/{{\cal B}(\Xi_{cc}\to\Xi_{c}\pi)}$, and
the first and second uncertainties are systematic and statistical,
respectively. It provides an ideal place to examine the mixing as it affects
both the denominator and numerator of ${\cal R}$. In the literature
Cheng:2020wmk ; Ke1 ; Gerasimov:2019jwp ; Gutsche:2018msz ; Wang:2017mqp ;
Sharma:2017txj  before the experiments, the ratio deviates largely to the
value in Eq. (3). In this work, we will show that the responsible mechanism is
precisely the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing. On the other hand, combing
several experiments, we have LHCb:2018pcs ; Belle:2019bgi ; pdg

$\frac{{\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})}{{\cal
B}(\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+})}=0.35\pm 0.20\,.$ (4)

By using ${\cal B}(\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+})>{\cal
B}(\Xi_{cc}^{++}\to\Sigma_{cc}^{++}\overline{K}^{*0}){\cal
B}(\overline{K}^{*0}\to K^{-}\pi^{+})$, ${\cal
B}(\Xi_{cc}^{++}\to\Sigma_{c}^{++}\overline{K}^{*0})=5.61\%$ Gutsche:2019iac
and ${\cal B}(\overline{K}^{*0}\to K^{-}\pi^{+})=2/3$, we obtain

${\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})>0.59\%\,,$ (5)

at $1\sigma$ confidence level. In addition, we have

${\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})=(1.33\pm 0.74)\%\,,$ (6)

by assuming that the decay of
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ contributes solely by
$\Xi_{cc}^{++}\to\Sigma_{cc}^{++}\overline{K}^{*0}$.

On the theoretical aspect, it is known that a trustworthy method in evaluating
the charm quark baryonic decays has not been given yet, since the charm quark
is neither heavy nor light enough to apply the heavy quark or $SU(4)_{F}$
symmetry. Nevertheless, it has been shown in Ref. Lcthe  that the pole model
conjunction with the current algebra and soft-meson limit can well explain the
experimental data of $\Lambda_{c}^{+}\to BP$, with $B$ and $P$ the octet
baryons and pseudoscalar mesons, respectively. As a phenomenological study,
focusing on the mixing effects, we shall follow their methodology for the
formalism. For the baryon wave functions, we will examine both the mixing
effects and the center-of-mass motion (CMM) corrections of the bag model.
Recently, it has been shown that the bag model is well consistent with the
experimental data of ${\cal B}(\Xi_{Q}\to\Lambda_{Q}\pi^{-})$ once the CMM is
removed HFC ; HFCBelle ; HFCExp ; HFCLHCb , where
$\Lambda_{Q}=(\Lambda_{c}^{+},\Lambda_{b}^{0})$ for
$\Xi_{Q}=(\Xi_{c}^{0},\Xi_{b}^{-})$.

This work is organized as follows. In Sec. II, we briefly recall the formalism
of the pole model and current algebra. In Sec. III, we give the baryon wave
functions and their matrix elements with and without the CMM. In Sec. IV, we
give the numerical results. We conclude this study in Sec. V.

##  II formalism

In general, the amplitude of ${\cal B}_{i}\to{\cal B}_{f}\pi$ is decomposed as

$i\overline{u}_{f}(A-B\gamma_{5})u_{i}\,,$ (7)

where $u_{i(f)}$ is the Dirac spinor of the initial (final) baryon, and
$A~{}(B)$ is referred to as the parity violating (conserving) amplitude. In
the pole approximation, the nonfactorizable amplitudes read as Cheng:2020wmk

$\displaystyle A^{\text{pole
}}=-\sum_{\mathcal{B}_{n}^{*}}\left[\frac{g_{\mathcal{B}_{f}\mathcal{B}_{n}^{*}\pi}b_{n^{*}i}}{M_{i}-M_{n}^{*}}+\frac{b_{fn^{*}}g_{\mathcal{B}_{n}^{*}\mathcal{B}_{i}\pi}}{M_{f}-M_{n}^{*}}\right]\,,$
(8) $\displaystyle B^{\text{pole }}=\sum_{\mathcal{B}_{n}}\left[\frac{g_{{\cal
B}_{i}\mathcal{B}_{n}\pi}a_{ni}}{M_{i}-M_{n}}+\frac{a_{fn}g_{\mathcal{B}_{n}\mathcal{B}_{i}\pi}}{M_{f}-M_{n}}\right]\,,$

where ${\cal B}_{n}^{(*)}$ are the parity even (odd) intermediate baryons,
$M_{i,f,n}^{(*)}$ correspond to the masses of ${\cal B}_{i,f,n}^{(*)}$,

$\langle{\cal B}_{2}|{\cal H}_{eff}|{\cal
B}_{1}\rangle=\overline{u}_{2}\left(a_{21}+b_{21}\gamma_{5}\right)u_{1}\,,~{}~{}~{}\langle{\cal
B}_{n}^{*}|{\cal H}_{eff}|{\cal
B}_{1}\rangle=b_{n^{*}1}\overline{u}_{n}u_{1}\,,$ (9)

${\cal B}_{1,2}\in\\{{\cal B}_{i},{\cal B}_{f},{\cal B}_{n}\\}$, and ${\cal
H}_{eff}$ represents the effective Hamiltonian. The baryon-baryon-pion
couplings of $g_{{\cal B}_{1}{\cal B}_{n}^{(*)}\pi}$ are extracted by the
Goldberg-Treiman relations

$g_{{\cal B}_{1}{\cal
B}_{2}\pi}=\frac{\sqrt{2}}{f_{\pi}}(M_{1}+M_{2})g^{A(\pi)}_{{\cal B}_{1}{\cal
B}_{2}}\,,~{}~{}~{}g_{{\cal B}_{n}^{*}{\cal
B}_{2}\pi}=\frac{\sqrt{2}}{f_{\pi}}(M_{n}^{*}-M_{2})g^{A(\pi)}_{{\cal
B}_{n}^{*}{\cal B}_{2}}\,,$ (10)

where $f_{\pi}$ is the pion decay constant, the axial vector couplings of
$g^{A(\pi)}_{{\cal B}^{\prime}{\cal B}}$ are defined by

$\langle{\cal B}^{\prime}|A^{\mu}(\pi)|{\cal
B}\rangle=\overline{u^{\prime}}\left(g^{A(\pi)}_{{\cal B}^{\prime}{\cal
B}}\gamma^{\mu}-i\overline{g}_{2}\sigma^{\mu\nu}q_{\nu}+\overline{g}_{3}q^{\mu}\right)\gamma_{5}u\,,$
(11)

$u^{(\prime)}$ is the Dirac spinor of ${\cal B}^{(\prime)}$,
$A^{\mu}(\pi^{+})=\overline{d}\gamma^{\mu}\gamma_{5}u\,,~{}A^{\mu}(\pi^{0})=\frac{1}{2}\left(\overline{u}\gamma^{\mu}\gamma_{5}u-\overline{d}\gamma^{\mu}\gamma_{5}d\right),$
and ${\cal B}^{(\prime)}\in\\{{\cal B}_{i},{\cal B}_{f},{\cal B}_{n},{\cal
B}_{n}^{*}\\}$. Note that $\overline{g}_{2,3}$ are irrelevant to this work.

To overcome the unknown baryon wave functions of ${\cal B}_{n}^{*}$, we use
the soft-meson limit and $[Q_{5}^{\pi}+Q^{\pi},{\cal H}_{eff}]=0$222 The
charge operators are defined as $Q^{\pi}=\int
d^{3}x(q^{\dagger}\sigma_{i}q)/2$ and $Q^{\pi}_{5}=\int
d^{3}x(q^{\dagger}\gamma_{5}\sigma_{i}q)/2$, where $q=(u,d)^{T}$ and
$\sigma_{i}=\sigma_{3},(\sigma_{1}\pm i\sigma_{2})/\sqrt{2}$ for
$\pi=\pi^{0},\pi^{\pm}$, respectively. The commutation relations come from
that the left-handed and right-handed currents commute. . The amplitudes of
$\Xi_{cc}\to\Xi_{c}\pi$ are summarized as Cheng:2020wmk

$\displaystyle A(\Xi_{cc}^{++}\to\Xi_{c}^{(\prime)+}\pi^{+})$ $\displaystyle=$
$\displaystyle\zeta\left(f_{\pi}^{2}a_{1}f_{1}^{(\prime)}M_{-}^{(\prime)}-c_{-}a^{(\prime)}\right)\,,$
$\displaystyle A(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)0}\pi^{+})$ $\displaystyle=$
$\displaystyle\zeta\left(f_{\pi}^{2}a_{1}f_{1}^{(\prime)}M_{-}^{(\prime)}+c_{-}a^{(\prime)}\right)\,,$
$\displaystyle A(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)+}\pi^{0})$ $\displaystyle=$
$\displaystyle\sqrt{2}\zeta c_{-}a^{(\prime)}\,,$ (12)

and

$\displaystyle
B(\Xi_{cc}^{++}\to\Xi_{c}^{(\prime)+}\pi^{+})=\zeta\left(-f_{\pi}^{2}a_{1}g_{1}^{(\prime)}M_{+}^{(\prime)}-2c_{-}a^{(\prime)}\frac{M_{cc}}{M_{-}^{(\prime)}}g^{A(\pi^{+})}_{\Xi_{cc}^{+}\Xi_{cc}^{++}}\right)\,,$
(13) $\displaystyle B(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)0}\pi^{+})=$
$\displaystyle~{}\zeta\left(-f_{\pi}^{2}a_{1}g_{1}^{(\prime)}M_{+}^{(\prime)}+c_{-}a\frac{M_{c}+M_{c}^{(\prime)}}{M_{-}}g^{A(\pi^{+})}_{\Xi_{c}^{(\prime)0}\Xi_{c}^{+}}+c_{-}a^{\prime}\frac{M^{\prime}_{c}+M_{c}^{(\prime)}}{M_{-}^{\prime}}g^{A(\pi^{+})}_{\Xi_{c}^{(\prime)0}\Xi_{c}^{\prime+}}\right)\,,$
$\displaystyle B(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)+}\pi^{0})=$
$\displaystyle~{}\sqrt{2}\zeta
c_{-}\left(-2a^{(\prime)}\frac{M_{c}}{M_{-}^{(\prime)}}g^{A(\pi^{0})}_{\Xi_{cc}^{+}\Xi_{cc}^{+}}+a\frac{M_{c}+M_{c}^{(\prime)}}{M_{-}}g^{A(\pi^{0})}_{\Xi_{c}^{(\prime)+}\Xi_{c}^{+}}+a^{\prime}\frac{M_{c}^{\prime}+M_{c}^{(\prime)}}{M_{-}^{\prime}}g^{A(\pi^{0})}_{\Xi_{c}^{(\prime)+}\Xi_{c}^{{}^{\prime}+}}\right)\,,$

where

$\zeta=\frac{G_{F}}{f_{\pi}\sqrt{2}}V_{cs}V^{*}_{ud}\,,~{}~{}~{}c_{-}=\frac{1}{2}\left(c_{1}-c_{2}\right)\,,~{}~{}~{}M_{\pm}^{(\prime)}=M_{cc}\pm
M_{c}^{(\prime)}\,,$ (14)

$M_{cc}$ and $M_{c}^{(\prime)}$ are the masses of $\Xi_{cc}$ and
$\Xi_{c}^{(\prime)}$, respectively, $G_{F}$ is the Fermi constant, $a_{1}$ is
the effective Wilson coefficient, and $V_{cs}$ and $V_{ud}$ are the Cabibbo-
Kobayashi-Maskawa matrix elements. The information of the baryon wave
functions is encapsulated in $a$, $f_{1}$ and $g_{1}$, defined by333We use the
Fierz transformation to sort $O_{-}$ defined in Ref. Cheng:2020wmk .

$\displaystyle\langle\Xi_{c}^{(\prime)+}|O|\Xi_{cc}^{+}\rangle=\langle\Xi_{c}^{(\prime)+}|2(u^{\dagger}L^{\mu}d)(s^{\dagger}L_{\mu}c)|\Xi_{cc}^{+}\rangle=\overline{u}_{c}\left(a^{(\prime)}+b^{(\prime)}\gamma_{5}\right)u_{cc}\,,$
(15)
$\displaystyle\langle\Xi_{c}^{(\prime)+}|\overline{s}\gamma^{\mu}c|\Xi_{cc}^{++}\rangle=\overline{u}_{c}\left(f_{1}^{(\prime)}(\omega^{(\prime)})\gamma^{\mu}-if^{(\prime)}_{2}(\omega^{(\prime)})\frac{\sigma^{\mu\nu}}{M_{cc}}q_{\nu}+f^{(\prime)}_{3}(\omega^{(\prime)})\frac{q^{\mu}}{M_{cc}}\right)u_{cc}\,,$
$\displaystyle\langle\Xi_{c}^{(\prime)+}|\overline{s}\gamma^{\mu}\gamma_{5}c|\Xi_{cc}^{++}\rangle=\overline{u}_{c}\left(g_{1}^{(\prime)}(\omega^{(\prime)})\gamma^{\mu}-ig^{(\prime)}_{2}(\omega^{(\prime)})\frac{\sigma^{\mu\nu}}{M_{cc}}q_{\nu}+g^{(\prime)}_{3}(\omega^{(\prime)})\frac{q^{\mu}}{M_{cc}}\right)\gamma_{5}u_{cc}\,,$
(16)

with $L^{\mu}=\gamma^{0}\gamma^{\mu}(1-\gamma_{5})$ and $u_{c(c)}$ the Dirac
spinor of $\Xi_{c(c)}$. Since $\Xi_{c}^{\overline{{\bf 3}}}$ and $\Xi_{c}^{\bf
6}$ do not have definite masses for $\theta_{c}\neq 0$, we define the
variables

$\omega^{(\prime)}=\frac{1+v^{2}}{1-v^{2}}=\frac{M_{cc}^{2}+M_{c}^{(\prime)2}-M_{\pi}^{2}}{2M_{c}^{(\prime)}M_{cc}}\,,$
(17)

with $v$ the speed of the baryons in the Briet frame. Throughout this work, we
employ the isospin symmetry, so that $\Xi_{cc}^{++}~{}(\Xi_{c}^{+})$ and
$\Xi_{cc}^{+}~{}(\Xi_{c}^{0})$ have the same masses and form factors. In
addition, we have

$g_{\Xi_{cc}^{+}\Xi_{cc}^{++}}^{A(\pi^{+})}=-\frac{1}{2}g_{\Xi_{cc}^{+}\Xi_{cc}^{+}}^{A(\pi^{0})}\,,~{}~{}~{}g_{\Xi_{c}^{\prime
0}\Xi_{c}^{(\prime)+}}^{A(\pi^{+})}=\frac{1}{2}g_{\Xi_{c}^{\prime+}\Xi_{c}^{(\prime)+}}^{A(\pi^{0})}\,,~{}~{}~{}g_{\Xi_{c}^{0}\Xi_{c}^{(\prime)+}}^{A(\pi^{+})}=\frac{1}{2}g_{\Xi_{c}^{+}\Xi_{c}^{(\prime)+}}^{A(\pi^{0})}\,.$
(18)

The above results are the general ones under the soft-meson limit, and the
unknown parts of the baryon wave functions are absorbed in the form factors
and $a^{(\prime)}$.

Plugging the mixing of Eq. (1) into Eq. (16), we arrive at

$\displaystyle f_{1}=\cos\theta_{c}f_{1}^{\overline{{\bf
3}}}(\omega)+\sin\theta_{c}f_{1}^{{\bf 6}}(\omega),~{}~{}~{}$ $\displaystyle
g_{1}=\cos\theta_{c}g_{1}^{\overline{{\bf
3}}}(\omega)+\sin\theta_{c}g_{1}^{{\bf 6}}(\omega)\,,{}$ (19) $\displaystyle
f_{1}^{\prime}=\cos\theta_{c}f_{1}^{{\bf
6}}(\omega^{\prime})-\sin\theta_{c}f_{1}^{\overline{{\bf
3}}}(\omega^{\prime}),~{}~{}~{}$ $\displaystyle
g_{1}^{\prime}=\cos\theta_{c}g_{1}^{{\bf
6}}(\omega^{\prime})-\sin\theta_{c}g_{1}^{\overline{{\bf
3}}}(\omega^{\prime})\,,$ $\displaystyle a=\cos\theta_{c}a({\overline{{\bf
3}}})+\sin\theta_{c}a({\bf 6})\,,~{}~{}~{}$ $\displaystyle
a^{\prime}=\cos\theta_{c}a({\bf 6})-\sin\theta_{c}a(\overline{{\bf
3}})\,,{}{}{}{}{}$

where $(f_{1}^{\overline{{\bf 3}}}(\omega^{(\prime)}),f_{1}^{{\bf
6}}(\omega^{(\prime)})$, $(g_{1}^{\overline{{\bf
3}}}(\omega^{(\prime)}),g_{1}^{{\bf 6}}(\omega^{(\prime)})$ and
$(a(\overline{{\bf 3}}),a({\bf 6}))$ are calculated by taking
$(\Xi_{c}^{(\prime)}=\Xi_{c}(\overline{{\bf
3}}),\Xi_{c}^{(\prime)}=\Xi_{c}({{\bf 6}}))$ in Eqs. (15) and (16). Similarly,
the axial vector couplings are modified as

$\displaystyle g_{\Xi_{c}^{\prime
0}\Xi_{c}^{\prime+}}^{A(\pi^{+})}=\cos^{2}\theta_{c}g_{{{\bf 6}}{\bf
6}}^{A}-\sin(2\theta_{c})g_{{\bf 6}\overline{{\bf 3}}}^{A}\,,$ $\displaystyle
g_{\Xi_{c}^{0}\Xi_{c}^{\prime+}}^{A(\pi^{+})}=g_{\Xi_{c}^{\prime
0}\Xi_{c}^{+}}^{A(\pi^{+})}=\cos(2\theta_{c})g_{{{\bf 6}}\overline{{\bf
3}}}^{A}+\frac{1}{2}\sin(2\theta_{c})g_{{\bf 6}{\bf 6}}^{A}\,,$ $\displaystyle
g_{\Xi_{c}^{0}\Xi_{c}^{+}}^{A(\pi^{+})}=\sin(2\theta_{c})g_{{\bf
6}\overline{{\bf 3}}}^{A}+\sin^{2}\theta_{c}g_{{{\bf 6}}{\bf 6}}^{A}\,,$ (20)

with

$\langle\Xi_{c}^{0}({\bf
R}_{2})|\overline{d}\gamma^{\mu}\gamma_{5}u|\Xi_{c}^{+}({\bf
R}_{1})\rangle=\overline{u}_{{\bf R}_{2}}\left(g^{A}_{{\bf R}_{2}{\bf
R}_{1}}\gamma^{\mu}-i\overline{g}_{2}{\sigma^{\mu\nu}}q_{\nu}+\overline{g}_{3}{q^{\mu}}\right)\gamma_{5}u_{{\bf
R}_{1}}$ (21)

and ${\bf R}_{1,2}=(\overline{{\bf 3}},{\bf 6})$. Finally, the decay widths
and up-down asymmetries are given by

$\displaystyle\Gamma=\frac{{\bf
p}_{f}}{8\pi}\frac{(M_{i}+M_{f})^{2}-M_{\pi}^{2}}{M_{i}^{2}}\left(|A|^{2}+\kappa^{2}|B|^{2}\right)\,,$
(22)
$\displaystyle\alpha=\frac{2\kappa\operatorname{Re}\left(A^{*}B\right)}{|A|^{2}+\kappa^{2}|B|^{2}}\,,$

where ${\bf p}_{f}$ is the magnitude of the pion three-momentum, and
$\kappa={\bf p}_{f}/(E_{f}+M_{f})$ with $E_{f}=\sqrt{{\bf
p}_{f}^{2}+M_{f}^{2}}$.

##  III Baryon wave functions and matrix elements

In this work, we calculate the baryon matrix elements by the bag models with
and without removing the CMM, referred to as the homogeneous bag (HB) and
static bag (SB) approaches, respectively.

The baryon wave functions concerned by this work are given as

$\displaystyle|\Xi_{cc},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}q_{a\alpha}^{\dagger}(\vec{x}_{1})c_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(ucc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi_{c}^{\overline{{\bf
3}}},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}q_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(qsc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi_{c}^{{\bf
6}},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}q_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{S_{\updownarrow}(qsc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(23)

where $q_{a\alpha}\in\\{u_{a\alpha},d_{a\alpha}\\}$, the Latin (Greek) letters
are the color (Dirac spinor) indices, and $\Psi$ describe the spatial
distributions of the quarks. In the SB, $\Psi$ read as

$\displaystyle\Psi_{A\updownarrow(q_{1}q_{2}q_{3})}^{abc(\text{SB})}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=\frac{{\cal
N}}{\sqrt{2}}\Big{(}\phi^{a}_{q_{1}\uparrow}(\vec{x}_{1})\phi^{b}_{q_{2}\downarrow}(\vec{x}_{2})-\phi^{a}_{q_{1}\downarrow}(\vec{x}_{1})\phi^{b}_{q_{2}\uparrow}(\vec{x}_{2})\Big{)}\phi^{c}_{q_{3}\updownarrow}(\vec{x}_{3})\,,$
$\displaystyle\Psi_{S\uparrow(q_{1}q_{2}q_{3})}^{abc(\text{SB})}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=\frac{{\cal
N}}{\sqrt{6}}\Big{(}2\phi^{a}_{q_{1}\uparrow}(\vec{x}_{1})\phi^{b}_{q_{2}\uparrow}(\vec{x}_{2})\phi^{c}_{q_{3}\downarrow}(\vec{x}_{3})$
$\displaystyle\quad\qquad-\phi^{a}_{q_{1}\downarrow}(\vec{x}_{1})\phi^{b}_{q_{2}\uparrow}(\vec{x}_{2})\phi^{c}_{q_{3}\uparrow}(\vec{x}_{3})-\phi^{a}_{q_{1}\uparrow}(\vec{x}_{1})\phi^{b}_{q_{2}\downarrow}(\vec{x}_{2})\phi^{c}_{q_{3}\uparrow}(\vec{x}_{3})\Big{)}\,,$
(24)

where ${\cal N}$ is the normalization constant,

$\phi_{q\updownarrow}(\vec{x})=\left(\begin{array}[]{c}\omega_{q_{+}}j_{0}({\bf
p}_{q}|\vec{x}|)\chi_{\updownarrow}\\\ i\omega_{q_{-}}j_{1}({\bf
p}_{q}|\vec{x}|)\hat{x}\cdot\vec{\sigma}\chi_{\updownarrow}\end{array}\right)~{}~{}~{}~{}\text{for}|\vec{x}|<R,$
(25)

$R$ is the bag radius, ${\bf p}_{q}$ is the magnitude of the quark three-
momentum, $\omega_{q_{\pm}}=\sqrt{E_{q}\pm M_{q}}$ with $M_{q}$ the quark mass
and $E_{q}=\sqrt{{\bf p}_{q}^{2}+M_{q}^{2}}$, $j_{0,1}$ are the spherical
Bessel functions, $\chi_{\uparrow}=(1,0)^{T}$ and
$\chi_{\downarrow}=(0,1)^{T}$.

The baryon wave functions in Eq. (III) are localized and can not be momentum
eigenstates according to the Heisenberg principle. Thus, we modify the baryon
wave functions as Liu:2022pdk

$\Psi^{\text{(HB)}}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=\int
d^{3}\vec{x}_{\Delta}\Psi^{\text{(SB)}}(\vec{x}_{1}-\vec{x}_{\Delta},\vec{x}_{2}-\vec{x}_{\Delta},\vec{x}_{3}-\vec{x}_{\Delta})\,,$
(26)

where $\Psi^{(\text{SB})}$ are the ones given in Eq. (III). With this trick,
the translational invariance of the baryons is recovered.

With the baryon wave functions, the calculations of the baryon matrix elements
are straightforward. The results of the SB approach can be found in Ref.
Cheng:2020wmk , while the form factors of the HB approach are given in Ref.
Geng:2022uyy .

Here, we sketch the method of calculating $a(\overline{{\bf 3}})$ and $a({\bf
6})$ in the HB approach. To diminish the directional dependencies in Eq. (15),
we trace over the baryon spins

$a({\bf R})=\frac{1}{2}\left(\langle\Xi_{c}^{+}({\bf
R}),\uparrow|O|\Xi_{cc}^{+},\uparrow\rangle+\langle\Xi_{c}^{+}({\bf
R}),\downarrow|O|\Xi_{cc}^{+},\downarrow\rangle\right)\,.$ (27)

with the normalization of $\overline{u}_{c(c)}u_{c(c)}=1$. By using the
anticommutation relations among the quark operators

$\left\\{q_{a\alpha}(\vec{x}),q_{b\beta}^{\dagger}\left(\vec{x}^{\prime}\right)\right\\}=\delta_{ab}\delta_{\alpha\beta}\delta^{3}\left(\vec{x}-\vec{x}^{\prime}\right)\,,$
(28)

we arrive at HFC

$\sum_{J_{z}=\updownarrow}\langle\Xi_{c}^{+}({\bf
R}),J_{z}|(u^{\dagger}L^{\mu}d)(s^{\dagger}L_{\mu}c)|\Xi_{cc}^{+},J_{z}\rangle={\cal
N}_{c}{\cal N}_{cc}\int d^{3}\vec{x}_{\Delta}{\cal
D}_{c}(\vec{x}_{\Delta}){\Upsilon}^{{\bf R}}(\vec{x}_{\Delta})\,,$ (29)

where ${\cal N}_{c(c)}$ is the normalization constant of $\Xi_{c(c)}$,

$\displaystyle{\cal D}_{c}(\vec{x}_{\Delta})=\int
d\vec{x}\phi^{\dagger}_{c}\left(\vec{x}^{+}\right)\phi_{c}\left(\vec{x}^{-}\right),$
(30) $\displaystyle\Upsilon^{{\bf R}}(\vec{x}_{\Delta})=\sum_{[\lambda]}{\cal
F}([\lambda],{\bf R})\int
d^{3}\vec{x}\phi_{u\lambda_{4}}^{\dagger}\left(\vec{x}^{+}\right)L_{\mu}\phi_{d\lambda_{2}}\left(\vec{x}^{-}\right)\phi_{s\lambda_{3}}^{\dagger}\left(\vec{x}^{+}\right)L^{\mu}\phi_{c\lambda_{1}}\left(\vec{x}^{-}\right),$

$[\lambda]=(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$,
$\vec{x}^{\pm}=\vec{x}\pm\vec{x}_{\Delta}/2$, and ${\cal F}$ are the spin-
flavor overlappings, given as

$\displaystyle\sum_{[\lambda]}{\cal F}([\lambda],\overline{\bf
3})\left(\lambda_{1}\otimes\lambda_{2}\otimes\lambda_{3}\otimes\lambda_{4}\right)=\frac{\sqrt{6}}{2}\left(\uparrow\downarrow\uparrow\downarrow-\downarrow\uparrow\uparrow\downarrow-\uparrow\downarrow\downarrow\uparrow+\downarrow\uparrow\downarrow\uparrow\right)\,,$
(31) $\displaystyle\sum_{[\lambda]}{\cal F}([\lambda],{\bf
6})\left(\lambda_{1}\otimes\lambda_{2}\otimes\lambda_{3}\otimes\lambda_{4}\right)=\frac{1}{3\sqrt{2}}\left[\left(\uparrow\downarrow+\downarrow\uparrow\right)\left(\uparrow\downarrow+\downarrow\uparrow\right)+2\uparrow\uparrow\uparrow\uparrow+2\downarrow\downarrow\downarrow\downarrow\right]\,.$

From Eq. (31), it is easy to deduce that

$\displaystyle\sum_{[\lambda]}{\cal F}([\lambda],{\bf
R})\left(\chi_{\lambda_{3}}^{\dagger}\chi_{\lambda_{1}}\right)\left(\chi_{\lambda_{4}}^{\dagger}\chi_{\lambda_{2}}\right)={\cal
C}_{\text{unflip}}^{{\bf R}}\,,$ $\displaystyle\sum_{[\lambda]}{\cal
F}([\lambda],{\bf
R})\left(\chi_{\lambda_{3}}^{\dagger}\sigma_{i}\chi_{\lambda_{1}}\right)\left(\chi_{\lambda_{4}}^{\dagger}\chi_{\lambda_{2}}\right)=0\,,$
$\displaystyle\sum_{[\lambda]}{\cal F}([\lambda],{\bf
R})\left(\chi_{\lambda_{3}}^{\dagger}\sigma_{i}\chi_{\lambda_{1}}\right)\left(\chi_{\lambda_{4}}^{\dagger}\sigma_{j}\chi_{\lambda_{2}}\right)=\delta_{ij}{\cal
C}_{\text{flip}}^{{\bf R}}\,,$ (32)

where

$\displaystyle\left({\cal C}_{\text{unflip}}^{\overline{{\bf 3}}},{\cal
C}_{\text{flip}}^{\overline{{\bf
3}}}\right)=\left(\sqrt{6},-\sqrt{6}\right)\,,~{}~{}~{}\left({\cal
C}_{\text{unflip}}^{{\bf 6}},{\cal C}_{\text{flip}}^{{\bf
6}}\right)=\left(\sqrt{2},\frac{\sqrt{2}}{3}\right)\,,$ (33)

and $\sigma_{i,j}$ are the Pauli matrices. The second and third equations of
Eq. (III) are due to that we have traced over the baryon spins so the matrix
elements can not depend on specific directions.

We decompose $\Upsilon$ in several pieces

$\Upsilon^{\bf R}(\vec{x}_{\Delta})=\int
d^{3}\vec{x}\sum_{k=1,2,3,4}\Gamma^{\bf R}_{k}(\vec{x}_{\Delta},\vec{x})$ (34)

with

$\displaystyle\Gamma_{1}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle\sum_{[\lambda]}{\cal F}([\lambda],{\bf
R})\phi_{u\lambda_{4}}^{\dagger}\left(\vec{x}^{+}\right)\phi_{d\lambda_{2}}\left(\vec{x}^{-}\right)\phi_{s\lambda_{3}}^{\dagger}\left(\vec{x}^{+}\right)\phi_{c\lambda_{1}}\left(\vec{x}^{-}\right)\,,$
$\displaystyle\Gamma_{2}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle\sum_{[\lambda]}{\cal F}([\lambda],{\bf
R})\phi_{u\lambda_{4}}^{\dagger}\left(\vec{x}^{+}\right)\gamma_{5}\phi_{d\lambda_{2}}\left(\vec{x}^{-}\right)\phi_{s\lambda_{3}}^{\dagger}\left(\vec{x}^{+}\right)\gamma_{5}\phi_{c\lambda_{1}}\left(\vec{x}^{-}\right)\,,$
$\displaystyle\Gamma_{3}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle-\sum_{[\lambda]}{\cal F}([\lambda],{\bf
R})\phi_{u\lambda_{4}}^{\dagger}\left(\vec{x}^{+}\right)V_{i}\phi_{d\lambda_{2}}\left(\vec{x}^{-}\right)\phi_{s\lambda_{3}}^{\dagger}\left(\vec{x}^{+}\right)V_{i}\phi_{c\lambda_{1}}\left(\vec{x}^{-}\right)\,,$
(35) $\displaystyle\Gamma_{4}^{\bf R}(\vec{x}_{\Delta},\vec{x})$
$\displaystyle=$ $\displaystyle-\sum_{[\lambda]}{\cal F}([\lambda],{\bf
R})\phi_{u\lambda_{4}}^{\dagger}\left(\vec{x}^{+}\right)V_{i}\gamma_{5}\phi_{d\lambda_{2}}\left(\vec{x}^{-}\right)\phi_{s\lambda_{3}}^{\dagger}\left(\vec{x}^{+}\right)V_{i}\gamma_{5}\phi_{c\lambda_{1}}\left(\vec{x}^{-}\right),$

where $V_{i}=\gamma_{0}\gamma_{i}$ with $i=1,2,3$. Plugging Eq. (III) into Eq.
(III), we obtain

$\displaystyle\Gamma_{1}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle{\cal C}_{\text{unflip}}^{{\bf
R}}\left(u_{u}^{+}u_{d}^{-}+v_{u}^{+}v_{d}^{-}\hat{x}^{+}\cdot\hat{x}^{-}\right)\left(u_{s}^{+}u_{c}^{-}+v_{s}^{+}v_{c}^{-}\hat{x}^{+}\cdot\hat{x}^{-}\right)$
$\displaystyle-{\cal C}_{\text{flip}}^{{\bf
R}}\frac{(\vec{x}_{\Delta}\times\vec{x})^{2}}{(r^{+}r^{-})^{2}}v_{u}^{+}v_{d}^{-}v_{s}^{+}v_{c}^{-}\,,$
$\displaystyle\Gamma_{2}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle-{\cal C}_{\text{flip}}^{{\bf
R}}\left(u_{u}^{+}v_{d}^{-}\hat{x}^{-}-v_{u}^{+}u_{d}^{-}\hat{x}^{+}\right)\left(u_{s}^{+}v_{c}^{-}\hat{x}^{-}-v_{s}^{+}u_{c}^{-}\hat{x}^{+}\right)\,,$
$\displaystyle\Gamma_{3}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle-\frac{{\cal C}_{\text{unflip}}^{\bf R}}{{\cal C}^{\bf
R}_{\text{flip}}}\Gamma_{2}^{\bf R}(\vec{x}_{\Delta},\vec{x})-2{\cal
C}_{\text{flip}}^{\bf
R}\left(u_{u}^{+}v_{d}^{-}\hat{x}^{-}+v_{u}^{+}u_{d}^{-}\hat{x}^{+}\right)\cdot\left(u_{s}^{+}v_{c}^{-}\hat{x}^{-}+v_{s}^{+}u_{c}^{-}\hat{x}^{+}\right)\,,$
$\displaystyle\Gamma_{4}^{\bf R}(\vec{x}_{\Delta},\vec{x})$ $\displaystyle=$
$\displaystyle-{\cal C}_{\text{flip}}^{\bf
R}\Big{[}3u_{u}^{+}u_{d}^{-}u_{s}^{+}u_{c}^{-}+v_{u}^{+}v_{d}^{-}v_{s}^{+}v_{c}^{-}\left(2+(\hat{x}^{+}\cdot\hat{x}^{-})^{2}\right)$
$\displaystyle-(u_{u}^{+}u_{d}^{-}v_{s}^{+}v_{c}^{-}+v_{u}^{+}v_{d}^{-}u_{s}^{+}u_{c}^{-})\hat{x}^{+}\cdot\hat{x}^{-}\Big{]}+{\cal
C}_{\text{unflip}}^{\bf
R}v_{u}^{+}v_{d}^{-}v_{s}^{+}v_{c}^{-}\frac{(\vec{x}_{\Delta}\times\vec{x})^{2}}{(r^{+}r^{-})^{2}}\,,$

with the abbreviation

$\phi_{q}\left(\vec{x}^{\pm}\right)=\left(\begin{array}[]{c}u^{\pm}_{q}\chi\\\
iv^{\pm}_{q}\left(\hat{x}^{\pm}\cdot\vec{\sigma}\right)\chi\end{array}\right)\,.$
(37)

Collecting Eqs. (27), (29), (30), (III) and (III), now we are able to
calculate $a({\bf R})$. Note that the formalism is reduced to the SB approach
by eliminating the $\vec{x}_{\Delta}$ integral

$a({\bf R})=\Upsilon^{\bf R}(0)\,.$ (38)

To compare with the SB approach Cheng:2020wmk , we rescale the parameters as

$a(\overline{\bf 3})=16\sqrt{6}\pi X_{2},\quad a({\bf
6})=-\frac{16\sqrt{2}\pi}{3}X_{1}\,.$ (39)

##  IV Numerical results

In crunching up the numbers, we take the bag model parameters bag

$M_{u,d}=0\,,~{}~{}~{}M_{s}=0.28~{}\text{GeV}\,,~{}~{}~{}M_{c}=1.655~{}\text{GeV}\,,~{}~{}~{}R=(5.0\pm
0.1)~{}\text{GeV}^{-1}\,.$ (40)

In the HB model, the axial vector couplings and $X_{1,2}$ are found to be

$\displaystyle\left(g^{A(\pi^{+})}_{\Xi_{cc}^{+}\Xi_{cc}^{++}}\,,~{}g^{A}_{{\bf
6}{\bf 6}}\,,~{}g^{A}_{{\bf 6}\overline{\bf
3}}\right)=(-0.259,0.522,-0.453)\,,$ (41) $\displaystyle X_{2}=(3.52\pm
0.22)10^{-4}~{}\text{GeV}^{3}\,,~{}~{}~{}X_{1}=(-2.44\pm
0.08)10^{-6}~{}\text{GeV}^{3}\,.$

while $g_{1}$ and $f_{1}$ are summarized in TABLE 1. The overlappings of
$X_{2}$ and $X_{1}$ are twice and one half larger than those of the SB
approach Cheng:2020wmk , and the same tendencies are found in the heavy-
flavor-conserving decays HFC . We emphasize that $X_{1}\propto M_{s}$ due to
the Körner-Pati-Woo theorem Korner:1970xq . As a consequence, the calculated
$X_{1}$ from the bag model shall not be fully trusted as $M_{s}$ is difficult
to be determined. Nevertheless, $X_{1}$ can be taken as zero in practice so
the final results are little affected.

Table 1:  Results of the form factors in the HB approach555 The uncertainties are smaller than the ones obtained in Ref. Geng:2022uyy  for that a smaller range of the bag radii is considered. . $f_{1}^{\overline{\bf 3}}(\omega)$ | $f_{1}^{\overline{\bf 3}}(\omega^{\prime})$ | $f_{1}^{{\bf 6}}(\omega)$ | $f_{1}^{{\bf 6}}(\omega^{\prime})$ | $g_{1}^{\overline{\bf 3}}(\omega)$ | $g_{1}^{\overline{\bf 3}}(\omega^{\prime})$ | $g_{1}^{{\bf 6}}(\omega)$ | $g_{1}^{{\bf 6}}(\omega^{\prime})$
---|---|---|---|---|---|---|---
$0.480(17)$ | $0.593(17)$ | $0.277(10)$ | $0.342(10)$ | $0.152(5)$ | $0.188(5)$ | $0.439(16)$ | $0.542(15)$

The mixing largely modifies ${\cal R}(\Xi_{c}^{++}\to\Xi_{c}^{+}\pi^{+})$, as
shown in FIG. 1. Particularly, with $\theta_{0}\equiv 0.142\pi$ we find that

$\displaystyle{\cal R}(0,\text{SB})=6.74\,,$ $\displaystyle{\cal
R}(\theta_{0},\text{SB})=5.39\,,$ $\displaystyle\quad{\cal
R}(-\theta_{0},\text{SB})=1.45\,,$ (42) $\displaystyle{\cal
R}(0,\text{HB})=0.19\pm 0.05\,,$ $\displaystyle{\cal
R}(\theta_{0},\text{HB})=0.90\pm 0.14\,,$ $\displaystyle\quad{\cal
R}(-\theta_{0},\text{HB})=0.07\,.$

Due to the large difference in $X_{1,2}$, the HB and SB approaches predict
very different ratios. However, they both require $\theta_{c}\neq 0$ to
explain the experiments. With $\theta_{c}=-\theta_{0}$, the SB approach is in
good agreement with Eq. (3), whereas with $\theta_{c}=\theta_{0}$ the HB
approach show accordance with the experimental lower bound.

Figure 1: ${\cal R}(\Xi_{c}^{++}\to\Xi_{c}^{+}\pi^{+})$ versus $\theta_{c}$.

We list out the results of the branching fractions and up-down asymmetries in
TABLE 2 along with those in the literature, where we have normalized the
branching fractions by
$(\tau(\Xi_{cc}^{++}),\tau(\Xi_{cc}^{+}))=(2.56,0.45)\times 10^{-13}s$ LifeTH
; EXPLifetime . In the literature, Ref. Gutsche:2018msz  adopts the covariant
quark model up to three-loop calculations, Ref. Sharma:2017txj  employs the
pole model but only the parity even pole is considered, Ref. Shi:2022kfa
calculates the $W$-exchange contributions by the light cone sum rule with the
heavy quark effective theory, and Refs. Gerasimov:2019jwp ; Ke1 ; Ke2
consider only the factorizable parts of the amplitudes. In the table, the
quoted values of Ref. Sharma:2017txj  are calculated by the nonrelativistic
quark model (N) and heavy quark effective theory (H) with the flavor-
independent pole, and the ones of Refs. Ke1 ; Ke2  are given by
$\theta_{c}=0.090\pm 0.013\pi$ (M) and $\theta_{c}=0$ (N) with the light-front
quark model. The results of Ref. Cheng:2020wmk  are essentially the ones of
the SB approach with $\theta_{c}=0$. Remarkably, Refs. Cheng:2020wmk  and
Gutsche:2018msz  show a good accordance, which indicates their treatments for
$\theta_{c}=0$ are reliable. However, they are inconsistent with the
experimental data of ${\cal R}(\Xi_{c}^{++}\to\Xi_{c}^{+}\pi^{+})$. We believe
that such deviations are caused by the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing. As
shown in the table, after considering the mixing, both ${\cal B}$ and ${\cal
R}$ are compatible with the current experimental data. To test our theory, we
recommend the future experiments on ${\cal
R}(\Xi_{c}^{+}\to\Xi_{c}^{0(+)}\pi^{+(0)})$, found to be

$\displaystyle{\cal
R}(\Xi_{c}^{+}\to\Xi_{c}^{0}\pi^{+})=0.25~{}(\text{SB}),~{}~{}1.17~{}(\text{HB})\,,$
$\displaystyle{\cal
R}(\Xi_{c}^{+}\to\Xi_{c}^{+}\pi^{0})=0.23~{}(\text{SB}),~{}~{}0.25~{}(\text{HB})\,.$
(43)

It is interesting to point out that the sign of
$\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+})$ is flipped by the mixing in the SB
approach. Under the factorization ansatz, the decays of
$\Xi_{cc}^{++}\to\Xi_{c}^{(\prime)+}\pi^{+}$ and
$\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+}$ behave identically, i.e. they have the
same decay widths and up-down asymmetries as shown explicitly in Refs.
Gerasimov:2019jwp ; Ke1 ; Ke2 . Therefore, the experimental measurements of
${\cal B}$ and $\alpha$ up on these decays may clarify the $W$-exchange
contributions. Especially, we recommend the future measurements on
$\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)0}\pi^{+})$ as it is essentially
negative in the factorization ansatz with $\theta_{c}=0$. It is interesting to
point out that the sign of $\alpha(\Xi_{cc}^{0}\to\Xi_{c}^{0}\pi^{+})$ is
flipped after the mixing is considered in both the SB approach and Ref. Ke2 .

Unfortunately, with the experimental value in Eq. (3), the HB and SB models
suggest opposite signs of $\theta_{c}$ as shown in Eq. (42). In
$\Xi_{c}^{0}\to\Lambda_{c}^{+}\pi^{-}$ and
$\Xi_{b}^{-}\to\Lambda_{b}^{0}\pi^{-}$, where the soft-meson limit is
trustworthy666 The ${\bf p}_{f}$ in $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ and
$\Xi_{c}^{0}\to\Lambda_{c}^{+}\pi^{-}$ are $0.96$ and $0.11$ GeV,
respectively. , it has been found that the HB approach is much more suitable
than the SB one HFC ; HFCBelle ; HFCExp ; HFCLHCb . More importantly, the HB
wave functions are self-consistent on the contrary of the SB ones. However,
the computed ${\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})$ with the HB is
much larger than Eq. (6), indicating that the branching fractions might be
overestimated. Viewing on the successes of the SB approach in the
$\Lambda_{c}^{+}$ decays Lcthe , it is likely that the CMM and finite ${\bf
p}_{f}$ corrections compensate each others. Accordingly, the sign of
$\theta_{c}$ shall be positive, suggested by the SB model. We note that the
semileptonic decays of $\Xi_{cc}\to\Xi_{c}e^{+}\nu_{e}$ are ideal places to
determine the sign of $\theta_{c}$, as they are uncontaminated by the
$W$-exchange contributions. Nonetheless, the experiments are subjected to the
difficulties imposed by the chargeless neutrinos.

Table 2:  The calculated branching fractions and up-down asymmetries (in units of %) along with the ones in the literature. All the branching fractions are normalized by $(\tau(\Xi_{cc}^{++}),\tau(\Xi_{cc}^{+}))=(2.56,0.45)\times 10^{-13}s$. For Ref. Sharma:2017txj , we quote the results of the flavor-independent pole, and the parentheses of (N) and (H) indicate the form factors are calculated by the nonrelativistic quark model and heavy quark effective theory, respectively. For Refs. Ke1 ; Ke2 , (U) and (M) are the results with and without the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing, respectively.  | HB $\theta_{c}=\theta_{0}$ | Cheng et al. Cheng:2020wmk  | Gutsche et al. Gutsche:2018msz  | Sharma & Dhir Sharma:2017txj
---|---|---|---|---
|  ${\cal B}~{}$ |  $\alpha~{}$ | ${\cal R}$ |  ${\cal B}~{}$ |  $\alpha~{}$ | ${\cal R}$ |  ${\cal B}~{}$ |  $\alpha~{}$ | ${\cal R}$ |  ${\cal B}$ (N) |  ${\cal B}$ (H) |  $\alpha$ (N) |  $\alpha$ (H) | ${\cal R}$
$\Xi_{cc}^{++}\to\Xi^{+}_{c}\pi^{+}$ | $10.3(24)$ | $-30$ | $0.90(14)$ | $0.69$ | $-41$ | $6.74$ | $0.71$ | $-57$ | $4.77$ | 6.66 | 9.30 | $-99$ | $-99$ | 0.82 (N)
$\Xi_{cc}^{++}\to\Xi^{\prime+}_{c}\pi^{+}$ | $8.91(68)$ | $-96$ | $4.65$ | $-84$ | $3.39$ | $-93$ | 5.46 | 7.51 | $-78$ | $-79$ | 0.81 (H)
$\Xi_{cc}^{+}\to\Xi^{0}_{c}\pi^{+}$ | $8.12(55)$ | $-52$ | $0.25$ | $3.84$ | $-31$ | $0.40$ |  |  |  | 0.59 | 0.95 | 55 | 34 | 0.39 (N)
$\Xi_{cc}^{+}\to\Xi^{\prime 0}_{c}\pi^{+}$ | $2.05(17)$ | $97$ | $1.55$ | $-73$ |  |  |  | 1.49 | 2.12 | 65 | 65 | 0.45 (H)
$\Xi_{cc}^{+}\to\Xi^{+}_{c}\pi^{0}$ | $8.58(104)$ | $-37$ | $0.23$ | $2.38$ | $-25$ | $0.07$ |  |  |  | 0.50 |  |  |  | $0.11$
$\Xi_{cc}^{+}\to\Xi^{\prime+}_{c}\pi^{0}$ | $1.94(24)$ | $52$ | $0.17$ | $-3$ |  |  |  | 0.054 |  |  |
| SB $\theta_{c}=-\theta_{0}$ | Shi et al. Shi:2022kfa  | Gerasimov et al. Gerasimov:2019jwp  | Ke et al. Ke1 ; Ke2
|  ${\cal B}~{}$ |  $\alpha~{}$ | ${\cal R}$ | ${\cal B}$ | ${\cal R}$ | ${\cal B}$ | ${\cal R}$ |  ${\cal B}$ (U) |  ${\cal B}$ (M) |  $\alpha~{}$ (U) |  $\alpha~{}$ (M) | ${\cal R}$
$\Xi_{cc}^{++}\to\Xi^{+}_{c}\pi^{+}$ | $2.24$ | $-93$ | $1.45$ | $6.22(194)$ | $1.42(78)$ | $7.01$ | $0.83$ | 3.48(46) | 2.14(18) | $-44(1)$ | $9(7)~{}$ | 0.56(18) (U)
$\Xi_{cc}^{++}\to\Xi^{\prime+}_{c}\pi^{+}$ | $3.25$ | $-63$ | $8.55(62)$ | $5.85$ | 1.96(24) | 3.0(1) | $-98(1)$ | $-99(1)$ | 1.41(20) (M)
$\Xi_{cc}^{+}\to\Xi^{0}_{c}\pi^{+}$ | $2.26$ | $31$ | $1.17$ |  |  | $1.23$ | $0.85$ | 0.61(8) | 0.38(3) | $-44(1)$ | $9(7)~{}$ | 0.56(18) (U)
$\Xi_{cc}^{+}\to\Xi^{\prime 0}_{c}\pi+$ | $2.64$ | $-99$ |  | $1.04$ | 0.35(4) | 0.53(2) | $-98(1)$ | $-99(1)$ | 1.41(20) (M)
$\Xi_{cc}^{+}\to\Xi^{+}_{c}\pi^{0}$ | $2.01$ | $-5$ | $0.25$ |  |  |  |  |  |  |  |  |  |  |
$\Xi_{cc}^{+}\to\Xi^{\prime+}_{c}\pi^{0}$ | $0.51$ | $-65$ |  |  |  |  |  |  |  |  |  |  |

##  V Conclusion

We have studied the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing effects in
$\Xi_{cc}\to\Xi_{c}\pi$ with the soft-meson limit. The bag model has been
employed for the baryon matrix elements with and without removing the CMM. We
have found that the CMM corrections are sizable as found in the heavy-flavor-
conserving decays. The branching fractions and up-down asymmetries have been
calculated and special attentions have been given to ${\cal R}$. In
particular, for $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ we have obtained that
$({\cal B},{\cal R})=(10.3(24)\%,0.90(14))$ and $(2.24\%,1.45)$ with and
without removing the CMM, respectively, which are consistent with the current
experimental data. To test our theory, we recommend the future experiments to
examine ${\cal R}(\Xi_{cc}^{+}\to\Xi_{c}^{0}\pi^{+})$, which have been
computed as $0.25$ and $1.17$ in the HB and SB approaches, respectively. To
probe the $W$-exchange contributions, we recommend the measurement on
$\alpha(\Xi_{cc}^{+}\to\Xi_{c}^{(\prime)+}\pi^{+})$ as they are negative in
the factorization ansatz but $0.31~{}(0.52)$ in the SB (HB) approach.

###### Acknowledgements.

We would like to thank Hai-Yang Cheng for the useful discussions. This work is
supported in part by the National Key Research and Development Program of
China under Grant No. 2020YFC2201501 and the National Natural Science
Foundation of China (NSFC) under Grant No. 12147103.

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