# Umehara algebra and complex submanifolds of indefinite complex space forms
111Keywords: complex submanifold, holomorphic isometric embedding, indefinite
complex space form, Nash algebraic; 2010 Subject classes: 32H02, 32Q40, 53B35.

Xu Zhang222Corresponding author. , Donghai Ji

###### Abstract

The Umehara algebra is studied with motivation on the problem of the non-
existence of common complex submanifolds. In this paper, we prove some new
results in Umehara algebra and obtain some applications. In particular, if a
complex manifolds admits a holomorphic polynomial isometric immersion to one
indefinite complex space form, then it cannot admits a holomorphic isometric
immersion to another indefinite complex space form of different type. Other
consequences include the non-existence of the common complex submanifolds for
indefinite complex projective space or hyperbolic space and a complex manifold
with a distinguished metric, such as homogeneous domains, the Hartogs
triangle, the minimal ball, the symmetrized polydisc, etc, equipped with their
intrinsic Bergman metrics, which generalizes more or less all existing
results.

##  1 Introduction

The study of holomorphic isometric embedding between complex manifolds is a
classical problem in complex differential geometry. Starting with Bochner’s
paper [1], such questions have been studied extensively by many authors (e.g.
[2, 7, 8, 11, 16, 17, 18, 22, 25].) In his PhD. Thesis [2], E. Calabi obtained
the existence, uniqueness and global extension of a local holomorphic isometry
from a complex Kähler manifold into a complex space form. In particular,
Calabi proved that any complex space form cannot be locally isometrically
embedded into another complex space form with a different curvature sign with
respect to their canonical Kähler metrics. The key idea in Calabi’s work is
his diastasis function, which is a Hermitian symmetric, potential function of
the Kähler metric that does not have the pluriharmonic terms. Using the
diastasis function, Calabi was able to reduce the metric tensor equation to
functional identities involving diastasis functions.

Using the diastasis function, Umehara studied an interesting question whether
two complex space forms can share a common complex submanifold with induced
metrics and he proved that two complex space forms with different curvature
signs cannot share a common Kähler submanifold [22]. Umehara later defined the
so called Umehara algebra and generalized Calabi’s existence and uniqueness
results for holomorphic isometric embeddings from a complex manifold with an
indefinite Kähler metric into an indefinite complex space form [23].

Two Kähler manifolds $M_{1},M_{2}$ share a common complex submanifold if a
complex submanifold of $M_{1}$ endowed with the induced metric is
biholomorphically isometric to a complex submanifold of $M_{2}$ endowed with
the induced metric as well. Di Scala and Loi showed in [8] that Hermitian
symmetric spaces of compact type and of non-compact type do not share common
complex submanifolds. In addition, the fact that Euclidean spaces and
Hermitian symmetric spaces of compact types do not share common complex
submanifolds follows from Umehara’s result [22] and the classical Nakagawa-
Takagi embedding of Hermitian symmetric spaces of compact type into complex
projective spaces. Finally, it was shown by Huang and Yuan in [12] that
Euclidean spaces and Hermitian symmetric spaces of non-compact types do not
share common complex submanifolds. In [4], Cheng, Di Scala and Yuan
generalized the problem to indefinite complex space forms. In particular, they
proved that the indefinite Euclidean space cannot share a common complex
submanifold with an indefinite complex projective space or an indefinite
complex hyperbolic space. However, whether or not an indefinite complex
projective space and an indefinite complex hyperbolic space can share a common
complex submanifold is an interesting open problem [4, 24]. Such problem of
existence/non-existence of common complex submanifold for various different
complex manifolds remains an active research problem lately and is studied
extensively by different authors (cf. [5, 6, 14, 19, 21, 26]).

The powerful method to attack such problem is to use Umehara algebra. As
developed in [23], if one can prove that certain function does not belong to
the Umehara algebra or its quotient field, the negative answer to the problem
thus follows. Umehara algebras are intensively studied in [4, 14] and
interesting common complex submanifolds problem are discussed there as their
consequences.

In this paper, we further consider Umehara algebra and one key feature is that
it may involve non-Nash algebraic functions. The main theorem in section 2
describes that certain real analytic functions are not contained in the
Umehara algebra or its field of fractions (cf. Theorem 2.1). On one hand, such
Umehara algebra leads to a partial answer to the problem of the common complex
submanifolds for indefinite complex projective and hyperbolic space. More
precisely, we prove that, if a Kähler manifold admits a holomorphic polynomial
isometric immersion into one indefinite complex space form, then it cannot
admits a holomorphic isometric immersion into another indefinite complex space
form of different type (cf. Corollary 3.2). On the other hand, it has deep
consequences in the problem of the common complex submanifolds for indefinite
complex space forms and a complex manifold with a distinguished metric. In
particular, we are able to provide the sufficient condition for a Kähler
manifold that does not share common complex submanifolds with an indefinite
complex space form (cf. Corollary 3.3). The examples include bounded
homogeneous domains, the Hartogs triangle, the minimal ball, the symmetrized
polydisc, etc, equipped with their intrinsic Bergman metrics, which
generalizes more or less all existing results. The method in this paper is
developed from the ideas of Huang and Yuan in [11, 12] for Nash algebraic
functions and we now further generalize it to certain non-Nash algebraic
functions.

The paper is organized as follows: in section 2, we state the main theorem in
the context of the Umehara algebra and give the proof; in section 3, the
applications of the holomorphic isometric embeddings are provided. In
particular, we show that if a Kähler manifold admits a holomorphic polynomial
isometric immersion into one indefinite complex space form then it cannot
admits a holomorphic isometric immersion into another indefinite complex space
form of different type. Moreover, we give the sufficient condition for a
Kähler manifold that does not share common complex submanifolds with an
indefinite complex space form.

Acknowledgement: We thank the referees for very helpful suggestions and
comments.

##  2 The Umehara algebra

###  2.1 The statement of the main theorem

Umehara introduces in [23] an associate algebra on a complex manifold $M$ and
uses the algebra to study the holomorphic isometric embedding between complex
manifolds. Since the interest here is the local existence of a complex
submanifold at some point $p$ in $M$, we modify Umehara’s definition as
follows. Use $\mathcal{O}_{p}$ to denote the local ring of germs of
holomorphic functions at $p$ and let
$\tilde{\mathcal{O}}_{p}=\\{\chi\in\mathcal{O}_{p}|\chi(p)=0\\}$.

Define

$\Lambda_{p}:=\left\\{f|f=\sum_{j=1}^{\kappa}a_{j}|\chi_{j}|^{2},\chi_{j}\in\mathcal{O}_{p},a_{j}\in\mathbb{R}\right\\},$

and let $K_{p}$ be the field of fractions of $\Lambda_{p}$. It is shown in
[23] (cf. Theorem 3.2 in [23]) that every $f\in\Lambda_{p}$ can be written as
$f=h+\overline{h}+\sum_{j=1}^{\kappa}a_{j}|\chi_{j}|^{2}$ for
$h\in\mathcal{O}_{p},a_{j}\in\mathbb{R}$ and linearly independent
$\chi_{1},\cdots,\chi_{\kappa}\in\tilde{\mathcal{O}}_{p}$. Moreover, define

$\tilde{\Lambda}_{p}:=\left\\{f=a_{0}+\sum_{j=1}^{\kappa}a_{j}|\chi_{j}|^{2}\in\Lambda_{p}|\chi_{j}\in\tilde{\mathcal{O}}_{p},a_{j}\in\mathbb{R}\right\\}.$

Let
$\prod_{j=1}^{t}\Lambda_{p}^{\mu_{j}}=\left\\{\prod_{j=1}^{t}h_{j}^{\mu_{j}}|h_{1},\cdots,h_{t}\in\Lambda_{p}\right\\}$
for $\mu_{1},\cdots,\mu_{t}\in\mathbb{R}$. In particular,
$\Lambda_{p}^{1}\cdot\Lambda_{p}^{-1}=K_{p}$ is the field of fractions of
$\Lambda_{p}$. Note that the germs of real numbers, denoted by
$\mathbb{R}_{p}$, belong to $\tilde{\Lambda}_{p}$.

The main theorem of the paper is as follows.

######  Theorem 2.1.

Let $p$ be a fixed point on a complex manifold $M$. Let
$\chi_{1},\cdots,\chi_{m}\in\mathcal{O}_{p}$ be non-constant germs of
holomorphic functions at $p$ and denote $\chi=(\chi_{1},\cdots,\chi_{m})$. The
following statements hold:

  * (i)

If $\chi_{1},\cdots,\chi_{m}\in\tilde{\mathcal{O}}_{p}$ are germs of
holomorphic polynomials, $a_{1},\cdots,a_{m}$ are non-zero real numbers and
$\alpha<0$, then

$\left(1+\sum_{i=1}^{m}a_{i}|\chi_{i}|^{2}\right)^{\alpha}\not\in\Lambda_{p}\setminus\mathbb{R}_{p}.$

  * (ii)

Let $\chi_{1},\cdots,\chi_{m}\in\tilde{\mathcal{O}}_{p}$ and
$H(z,\overline{w})=R\left(\prod_{j=1}^{\kappa_{1}}H_{j}^{\mu_{j}}(z,\overline{w})\right)\prod_{j=\kappa_{1}+1}^{\kappa_{1}+\kappa_{2}}H_{j}^{\mu_{j}}(z,\overline{w})$,
where
$H_{1}(z,\overline{w}),\cdots,H_{\kappa_{1}+\kappa_{2}}(z,\overline{w})$ are
Hermitian symmetric, Nash algebraic functions and do not have non-constant
pluriharmonic terms, $R(\cdot)$ is a rational function with real coefficients
and $\mu_{1},\dots,\mu_{\kappa_{1}+\kappa_{2}}\in\mathbb{R}$. Then

$\exp\left(H(\chi,\overline{\chi})\right)\not\in\Lambda_{p}^{\mu}\setminus\mathbb{R}_{p},$
(1)

and

$\log\left(H(\chi,\overline{\chi})\right)\not\in\tilde{\Lambda}_{p}^{\mu}\setminus\mathbb{R}_{p}.$
(2)

  * (iii)

Let $H(z,\overline{w})$ be a Hermitian symmetric, Nash algebraic function.
Then

$\exp\left(H(\chi,\overline{\chi})\right)\not\in\tilde{\Lambda}_{p}^{\mu}\setminus\mathbb{R}_{p},$
(3)

and

$\log
H(\chi,\overline{\chi})\not\in\tilde{\Lambda}^{\mu}_{p}\setminus\mathbb{R}_{p}.$
(4)

If, in addition, $H(z,\overline{w})$ does not have non-constant pluriharmonic
terms and $\chi_{1},\cdots,\chi_{m}\in\tilde{\mathcal{O}}_{p}$, then

$\exp\left(H(\chi,\overline{\chi})\right)\not\in\prod_{j=1}^{\kappa}\Lambda_{p}^{\nu_{j}}\setminus\mathbb{R}_{p},$
(5)

and

$\log
H(\chi,\overline{\chi})\not\in\prod_{j=1}^{\kappa}\tilde{\Lambda}_{p}^{\nu_{j}}\setminus\mathbb{R}_{p}.$
(6)

###  2.2 Proof of the main theorem

We first prove Part (i) and Part (ii), (iii) will be proved via a different
method.

For Part (i), we argue by contradiction and suppose
$\left(1+\sum_{i=1}^{m}a_{i}|\chi_{i}|^{2}\right)^{\alpha}\in\Lambda_{p}\setminus\mathbb{R}_{p}$.
Namely, there exists an open neighborhood $U\subset M$ of $p$,
$h\in\mathcal{O}_{p}$ and linearly independent
$g_{1},\cdots,g_{n}\in\tilde{\mathcal{O}}_{p}$ such that

$\left(1+\sum_{i=1}^{m}a_{i}|\chi_{i}|^{2}\right)^{\alpha}=h+\overline{h}+\sum_{j=1}^{r}|g_{j}|^{2}-\sum_{j=r+1}^{n}|g_{j}|^{2}$

on $U$. We choose a holomorphic coordinate $\\{z\\}$ on $U$ with $z(p)=0$. By
the standard argument to get rid of $\partial\overline{\partial}$ (cf. [7,
17]), since there is no non-constant pluriharmonic term on the left hand side,
it follows that $h+\overline{h}=1$.

First of all, we will show that each $g_{j}$ can be written as a polynomial of
$\chi_{1},\cdots,\chi_{m}$. By polarization, the identity above is equivalent
to

$\left(1+\sum_{i=1}^{m}a_{i}\chi_{i}(z)\overline{\chi}_{i}(w)\right)^{\alpha}=1+\sum_{i=1}^{r}g_{i}(z)\overline{g}_{i}(w)-\sum_{j=r+1}^{n}g_{j}(z)\overline{g}_{j}(w),$
(7)

where $(z,w)\in U\times\hbox{conj}({U})$,
$\hbox{conj}({U})=\\{z\in{\mathbb{C}}|\overline{z}\in U\\}$, and
$\overline{\chi}_{i}(w)=\overline{\chi_{i}(\overline{w})}$.
Taking $l$-th derivative of the equation (7) in $w$ for $l=1,2,\cdots$, and
then evaluating at $w=0$, we have the following matrix equation:

$P=A\cdot Q,$

where

$A=\begin{bmatrix}\cdots\frac{\partial\overline{g}_{i}}{\partial
w}(0)\cdots-\frac{\partial\overline{g}_{j}}{\partial w}(0)\cdots\\\ \vdots\\\
\cdots\frac{\partial^{l}\overline{g}_{i}}{\partial
w^{l}}(0)\cdots-\frac{\partial^{l}\overline{g}_{j}}{\partial
w^{l}}(0)\cdots\\\ \vdots\\\ \end{bmatrix}_{\infty\times
n,}~{}Q=\begin{bmatrix}\vdots\\\ g_{i}(z)\\\ \vdots\\\ g_{j}(z)\\\
\vdots\end{bmatrix}_{n\times 1,}\text{and}~{}P=\begin{bmatrix}\vdots\\\
p_{j}\\\ \vdots\end{bmatrix}_{\infty\times 1,}$

with each $p_{k}$ being polynominal function in
$\chi_{1}(z),\cdots,\chi_{m}(z)$.

Now we show rank$(A)=n$. Suppose rank$(A)=d<n$. Without loss of generality, we
assume that the first $d$ columns are linearly independent in the coefficient
matrix $A$, denoted by $L_{1},L_{2},\cdots,L_{d}$. Then, for any $d^{\prime}$
with $d<d^{\prime}$, the $d^{\prime}$-th columns is linear combination of
$L_{1},L_{2},\cdots,L_{d}$, i.e.

$L_{d^{\prime}}=\sum_{i=1}^{d}C_{i}L_{i}.$

In other words, $\overline{g}_{d^{\prime}}$ can be written as linear
combination of $\\{\overline{g}_{1},\cdots,\overline{g}_{d}\\}$ by the Taylor
expansion, meaning that $\\{g_{1},\cdots,g_{n}\\}$ is not linear independent.
This is a contradiction.

Since rank$(A)=n$, there exist $k=k_{1},\cdots,k_{n}$ such that $k_{1}$-row to
$k_{n}$-row in matrix $A$ are linearly independent and all other rows can be
written as linear combinations of $k_{1}$-row up to $k_{n}$-row. Re-organize
the matrices $A$ and $P$ by deleting rows other than $k_{1}$-row to
$k_{n}$-row, and denote the corresponding matrices by $A_{n},P_{n}$
respectively. We obtain the matrix equation

$P_{n}=A_{n}\cdot Q.$

Since $A_{n}$ is nondegenerate, $Q=A_{n}^{-1}P_{n}$. In other words, each
$g_{i},g_{j}$ can be written as the linear combination of these $p_{j}$. As a
consequence, they are still polynomials functions in
$\chi_{1}(z),\cdots,\chi_{m}(z)$ and thus are polynomials as well since
$\chi_{1},\cdots,\chi_{m}$ are polynomials.

Therefore, we reach a contradiction to (7) as the right hand side is a
polynomial while the left hand side is not. This completes the proof of Part
(i).

If $H(\chi,\overline{\chi})$ is constant, then Part (ii), (iii) are trivial.
Otherwise, we also argue by contradiction. Suppose the conclusion of Part (ii)
and (iii) is false. Namely, there exist an open neighborhood $U$ of $p$,
$a_{j}\in\mathbb{R}$ and
$g_{1},\cdots,g_{k},h_{1},\cdots,h_{l}\in\mathcal{O}_{p}$ or
$\tilde{\mathcal{O}}_{p}$ accordingly, such that

$\exp\left(H(\chi,\overline{\chi})\right)=\prod_{j=1}^{\kappa}\left(a_{j}+\sum_{\alpha=1}^{k_{j}}|g_{\alpha}|^{2}-\sum_{\beta=1}^{l_{j}}|h_{\beta}|^{2}\right)^{\nu_{j}}$
(8)

and

$\log\left(H(\chi,\overline{\chi})\right)=\prod_{j=1}^{\kappa}\left(a_{j}+\sum_{\alpha=1}^{k_{j}}|g_{\alpha}|^{2}-\sum_{\beta=1}^{l_{j}}|h_{\beta}|^{2}\right)^{\nu_{j}}$
(9)

on $U$. We again choose a holomorphic coordinate $\\{z\\}$ on $U$ with
$z(p)=0$. We will need the following algebraicity result as in [14] (cf. Lemma
2.3 in [26]).

######  Lemma 2.2.

Writing
$S=\\{\phi_{1},\cdots,\phi_{k+l+m}\\}=\\{\chi_{1},\cdots,\chi_{m},g_{1},\cdots,g_{k},h_{1},\cdots,h_{l}\\}$,
then there exists a maximal algebraic independent subset
$\\{\phi_{1},\cdots,\phi_{d}\\}\subset S$ over the field $\cal{R}$ of rational
functions on $U$, and Nash algebraic functions
$\hat{\phi}_{j}(t,X_{1},\cdots,X_{d})$ defined in a neighborhood $\hat{U}$ of
$\\{(s,\phi_{1}(s),\cdots,\phi_{d}(s))|s\in U\\}$, such that
$\phi_{j}(t)=\hat{\phi}_{j}(t,\phi_{1}(t),\cdots,\phi_{d}(t))$ for all $1\leq
j\leq k+l+m$ after shrinking $U$ toward the origin if needed.

By polarization as in (7), equations (8), (9) are equivalent to

$\begin{split}\exp\left(H(\chi(z),\overline{\chi}(w))\right)&=\prod_{j=1}^{\kappa}\left(a_{j}+\sum_{\alpha=1}^{k_{j}}g_{\alpha}(z)\overline{g}_{\alpha}(w)-\sum_{\beta=1}^{l_{j}}h_{\beta}(z)\overline{h}_{\beta}(w)\right)^{\nu_{j}}\\\
\end{split}$ (10)

and

$\begin{split}\log\left(H(\chi(z),\overline{\chi}(w))\right)&=\prod_{j=1}^{\kappa}\left(a_{j}+\sum_{\alpha=1}^{k_{j}}g_{\alpha}(z)\overline{g}_{\alpha}(w)-\sum_{\beta=1}^{l_{j}}h_{\beta}(z)\overline{h}_{\beta}(w)\right)^{\nu_{j}}\\\
\end{split}$ (11)

for $(z,w)\in U\times\hbox{conj}({U})$.

Denote $X=(X_{1},\cdots,X_{d})$. For
$(s,X,t)\in\hat{U}\times\hbox{conj}({U})$, now define

$\begin{split}\Psi_{1}(s,X,t)=H(\hat{\chi}(s,X),\overline{\chi}(t))-\sum_{j=1}^{\kappa}\nu_{j}\log\left(a_{j}+\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(t)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(t)\right),\end{split}$
(12)

$\Psi_{2}(s,X,t)=\log
H(\hat{\chi}(s,X),\overline{\chi}(t))-\prod_{j=1}^{\kappa}\left(a_{j}+\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(t)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(t)\right)^{\nu_{j}}.$
(13)

######  Claim 2.3.

Under the corresponding assumptions,
$\Psi_{j}^{(l)}(s,X):=\frac{\partial^{l}}{\partial t^{l}}\Psi_{j}(s,X,0)$ is
Nash algebraic for any $j=1,2$ and $l\in\mathbb{N}$.

For $j=1$, since, under the assumption of Theorem 2.1.(ii),
$H_{1}(z,0),\cdots,H_{\kappa_{1}+\kappa_{2}}(z,0)$ are all constants,
$\frac{\partial^{l}}{\partial t^{l}}H(\hat{\chi}(s,X),\overline{\chi}(0))$ is
Nash algebraic. Under the assumption of Theorem 2.1.(iii),
$\frac{\partial^{l}}{\partial t^{l}}H(\hat{\chi}(s,X),\overline{\chi}(0))$ is
obviously Nash algebraic. Thus $\Psi_{1}^{(l)}(s,X)$ is Nash algebraic. For
$j=2$, under the assumption of Theorem 2.1.(ii), $\frac{\partial^{l}}{\partial
t^{l}}\log H(\hat{\chi}(s,X),\overline{\chi}(0))$ is Nash algebraic by the
same argument. Under the assumption of Theorem 2.1.(iii),
$\frac{\partial^{l}}{\partial t^{l}}\log
H(\hat{\chi}(s,X),\overline{\chi}(0))$ is also obviously Nash algebraic. The
other term is also Nash algebraic by noting
$a_{j}+\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(0)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(0)=a_{j}$.
Thus $\Psi_{2}^{(l)}(s,X)$ is Nash algebraic.

######  Lemma 2.4.

For any $j=1,2$ and $l>0$,

$\Psi_{j}^{(l)}(s,X)\equiv 0.$

###### Proof.

Suppose that $\Psi_{j}^{(l)}(s,X,0)$ is not constant. There exists a
holomorphic polynomial
$P(s,X,y)=A_{\hat{d}}(s,X)y^{\hat{d}}+\cdots+A_{0}(s,X)$ of degree $\hat{d}$
in $y$, with $A_{0}(s,X)\not\equiv 0$ such that
$P(s,X,\Psi_{j}^{(l)}(s,X,0))\equiv 0$. As
$\Psi_{j}(s,\phi_{1}(s),\cdots,\phi_{d}(s),t)\equiv 0$ for any
$(s,t)\in{U}\times\hbox{conj}({U})$, it follows that
$\Psi_{j}^{(l)}(s,\phi_{1}(s),\cdots,\phi_{d}(s))\equiv 0$ and therefore
$A_{0}(s,\phi_{1}(s),\cdots,\phi_{d}(s))\equiv 0$. This means that
$\\{\phi_{1}(s),\cdots,\phi_{d}(s)$} are algebraic dependent over $\cal{R}$.
This is a contradiction and it follows that $\Psi_{j}^{(l)}(s,X)$ is a
constant. Therefore,
$\Psi_{j}^{(l)}(s,X)=\Psi_{j}^{(l)}(s,\phi_{1}(s),\cdots,\phi_{d}(s))\equiv
0.$ ∎

By the Taylor expansion, it follows that
$\Psi_{j}(s,X,t)\equiv\Psi_{j}(s,X,0)$ for any $t$ near 0. We now obtain:

$\Psi_{1}(s,X,t)=H(\hat{\chi}(s,X),\overline{\chi}(0))-\sum_{j=1}^{\kappa}\nu_{j}\log\left(\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(0)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(0)\right),$

equivalent to

$\exp\left\\{H(\hat{\chi}(s,X),\overline{\chi}(t))-H(\hat{\chi}(s,X),\overline{\chi}(0))\right\\}=\prod_{j=1}^{t}\left(\frac{a_{j}+\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(t)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(t)}{a_{j}+\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(0)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(0)}\right)^{\nu_{j}};$
(14) $\Psi_{2}(s,X,t)=\log
H(\hat{\chi}(s,X),\overline{\chi}(0))-\prod_{j=1}^{\kappa}a_{j}^{\nu_{j}},$

equivalent to

$\exp\left\\{\prod_{j=1}^{\kappa}a_{j}^{\nu_{j}}\right\\}\frac{H(\hat{\chi}(s,X),\overline{\chi}(t))}{H(\hat{\chi}(s,X),\overline{\chi}(0))}=\exp\left(\prod_{j=1}^{\kappa}\left(a_{j}+\sum_{\alpha=1}^{k_{j}}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(t)-\sum_{\beta=1}^{l_{j}}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(t)\right)^{\nu_{j}}\right).$
(15)

In order to achieve a contradiction, we need the following lemmas.

######  Lemma 2.5.

  * •

There exists some $t_{1}\in\hbox{conj}({U})$ such that
$H(\hat{\chi}(s,X),\overline{\chi}(t_{1}))$ is not constant in
$(s,X)\in\hat{U}$.

  * •

There exists some $t_{2}\in\hbox{conj}({U})$ such that
$\sum_{\alpha}\hat{g}_{\alpha}(s,X)\overline{g}_{\alpha}(t_{2})-\sum_{\beta}\hat{h}_{\beta}(s,X)\overline{h}_{\beta}(t_{2})$
is not constant in $(s,X)\in\hat{U}$.

###### Proof.

Suppose not. We may choose
$t_{1}=t_{2}=\overline{s},X=(\phi_{1}(s),\cdots,\phi_{d}(s))$ such that

$H(\hat{\chi}(s,\phi_{1}(s),\cdots,\phi_{d}(s)),\overline{\chi}(\overline{s}))=H(\chi(s),\overline{\chi(s)})=\text{constant},$

and

$\begin{split}~{}&\sum_{\alpha}\hat{g}_{\alpha}(s,\phi_{1}(s),\cdots,\phi_{d}(s))\overline{g}_{\alpha}(\overline{s})-\sum_{\beta}\hat{h}_{\beta}(s,\phi_{1}(s),\cdots,\phi_{d}(s))\overline{h}_{\beta}(\overline{s})\\\
&=\sum_{\alpha}|g_{\alpha}(s)|^{2}-\sum_{\beta}|h_{\beta}(s)|^{2}=\text{constant}.\end{split}$

Both identities contradict to our initial assumptions. ∎

######  Lemma 2.6.

Let $V\subset{\mathbb{C}}^{k}$ be a connected open set. Let
$H(\xi),H_{1}(\xi),\cdots,H_{\kappa_{1}+\kappa_{2}}(\xi)$ be holomorphic Nash
algebraic functions on $V$,
$\mu,\mu_{1},\cdots,\mu_{\kappa_{1}+\kappa_{2}}\in\mathbb{R}\setminus\\{0\\}$
and $R(\cdot)$ be a holomorphic rational function. Assume that

$\exp\left\\{R\left(\prod_{j=1}^{\kappa_{1}}H_{j}^{\mu_{j}}(\xi)\right)\prod_{j=\kappa_{1}+1}^{\kappa_{1}+\kappa_{2}}H_{j}^{\mu_{j}}(\xi)\right\\}=H^{\mu}(\xi),$
(16)

or

$R\left(\prod_{j=1}^{\kappa_{1}}H_{j}^{\mu_{j}}(\xi)\right)\prod_{j=\kappa_{1}+1}^{\kappa_{1}+\kappa_{2}}H_{j}^{\mu_{j}}(\xi)=\exp\left\\{H^{\mu}(\xi)\right\\},$
(17)

for $\xi\in V$. Then $H(\xi)$ is constant.

Lemma 2.6 is a further generalization of Lemma 2.1 in [26] and can be proved
by the similar argument. The idea is very simple: when the right hand sides
approaches the infinity, the left hand side grow exponentially fast while the
right hand side at most polynomially in (16) and vice versa in (17). We will
not give the detail here.

Now let us see how to reach the contradiction. We first consider cases
(1),(3),(5). Under the assumption of Theorem 2.1.(i) or the second case of
(ii), since $H(z,\overline{w})$ does not contain the non-constant
pluriharmonic terms, $H(\hat{\chi}(s,X),\overline{\chi}(0))$ is constant.
Since the left hand of (14) is non-constant, we reach the contradiction by
(16),(17) in Lemma 2.6, respectively. Under the assumption of the first case
of (ii) in Theorem 2.1, since $\kappa=1$ and the denominator on the right hand
side is constant, the right hand side is non-constant in (14). This is again a
contradiction to (16). Now we consider cases (2),(4),(6). Under the assumption
of Theorem 2.1.(i) or the second case of (ii), since $H(z,\overline{w})$ does
not contain the non-constant pluriharmonic terms,
$H(\hat{\chi}(s,X),\overline{\chi}(0))$ is constant. Since the left hand of
(15) is non-constant, we reach the contradiction by (16),(17) in Lemma 2.6,
respectively. Under the assumption of the first case of (ii) in Theorem 2.1,
since $\kappa=1$, the right hand side is non-constant in (15) and we reach a
contradiction to (17). This finishes the proof of the theorem. ∎

##  3 Applications to the holomorphic isometric embedding

Theorem 2.1 has wide applications in the study of non-existence of the common
complex submanifolds, that may generalize various results obtained in
[12][4][5][26].

###  3.1 Indefinite complex space forms

The indefinite Kähler metric $\omega_{\mathbb{C}^{N}_{\kappa}}$ on the
indefinite complex Euclidean space
$\mathbb{C}^{N}_{\kappa}=\mathbb{C}^{N}=\\{(z_{1},\cdots,z_{N})|z_{j}\in\mathbb{C},j=1,\cdots,N\\}(0\leq\kappa\leq
N)$ is given by

$\omega_{{\mathbb{C}}_{\kappa}^{N}}=4\sqrt{-1}\partial\overline{\partial}\left(\sum_{i=\kappa+1}^{N}|z_{i}|^{2}-\sum_{j=1}^{\kappa}|z_{j}|^{2}\right).$

The indefinite complex projective space
${\mathbb{C}}{\mathbb{P}}_{\kappa}^{N}(b)(0\leq\kappa\leq N)$ of positive
constant holomorphic sectional curvature $b>0$ is the open complex submanifold
$\\{(\xi_{0},\cdots,\xi_{N})\in\mathbb{C}^{N+1}|\sum_{i=0}^{N-\kappa}|\xi_{i}|^{2}-\sum_{j=0}^{\kappa-1}|\xi_{N-j}|^{2}>0\\}/{\mathbb{C}}^{*}$
of ${\mathbb{C}}{\mathbb{P}}^{N}$. The indefinite Kähler metric of
${\mathbb{C}}{\mathbb{P}}_{\kappa}^{N}(b)$ is given by

$\omega_{{\mathbb{C}}{\mathbb{P}}_{\kappa}^{N}(b)}=\frac{4\sqrt{-1}}{b}\partial\overline{\partial}\log\left(\sum_{i=0}^{N-\kappa}|\xi_{i}|^{2}-\sum_{j=0}^{\kappa-1}|\xi_{N-j}|^{2}\right).$

The indefinite complex hyperbolic space
${\mathbb{C}}{\mathbb{H}}_{\kappa}^{N}(b)(0\leq\kappa\leq N)$ of negative
constant holomorphic sectional curvature $b<0$ is obtained from
${\mathbb{C}}{\mathbb{P}}^{N}_{N-\kappa}(-b)$ with indefinite Kähler metric

$\omega_{{\mathbb{C}}{\mathbb{H}}_{\kappa}^{N}(b)}=-\frac{4\sqrt{-1}}{(-b)}\partial\overline{\partial}\log\left(\sum_{i=0}^{\kappa}|\xi_{i}|^{2}-\sum_{j=0}^{N-\kappa-1}|\xi_{N-j}|^{2}\right).$

Without loss of generality, assuming $\xi_{0}\not=0$, then under inhomogeneous
coordinates $(z_{1},\cdots,z_{N})=(\xi_{1}/\xi_{0},\cdots,\xi_{N}/\xi_{0})$,
the metrics are given by

$\omega_{{\mathbb{C}}{\mathbb{P}}_{\kappa}^{N}(b)}=\frac{4\sqrt{-1}}{b}\partial\overline{\partial}\log\left(1+\sum_{i=1}^{N-\kappa}|z_{i}|^{2}-\sum_{j=0}^{\kappa-1}|z_{N-j}|^{2}\right)$

and

$\omega_{{\mathbb{C}}{\mathbb{H}}_{\kappa}^{N}(b)}=-\frac{4\sqrt{-1}}{(-b)}\partial\overline{\partial}\log\left(1+\sum_{i=1}^{\kappa}|z_{i}|^{2}-\sum_{j=0}^{N-\kappa-1}|z_{N-j}|^{2}\right).$

In particular, when $\kappa=0$,
${\mathbb{C}}^{N,0},{\mathbb{C}}{\mathbb{P}}_{\kappa}^{N}(b),{\mathbb{C}}{\mathbb{H}}_{\kappa}^{N}(b)$
are just the standard complex Euclidean, projective, hyperbolic space,
respectively.

Furthermore, suppose that $D$ is a complex manifold such that there exist
holomorphic maps
$F=(F_{1},\cdots,F_{N}):D\rightarrow{\mathbb{C}}{\mathbb{P}}^{N}_{\kappa}$ and
$L=(L_{1},\cdots,L_{N^{\prime}}):D\rightarrow{\mathbb{C}}{\mathbb{H}}_{\kappa^{\prime}}^{N^{\prime}}(b)$)
with
$F^{*}\omega_{{\mathbb{C}}{\mathbb{P}}^{N}_{\kappa}}=L^{*}\omega_{{\mathbb{C}}{\mathbb{H}}_{\kappa^{\prime}}^{N^{\prime}}(b)}$.
Fixing $x\in D$, without loss of generality, we assume $F(x)=0,L(x)=0$ by
composing the automorphisms on
${\mathbb{C}}{\mathbb{P}}_{\kappa^{\prime}}^{N^{\prime}}(b)$ and
${\mathbb{C}}{\mathbb{H}}_{\kappa^{\prime}}^{N^{\prime}}(b)$. By the standard
argument to get rid of $\partial\overline{\partial}$ (cf. [7]), since there is
no non-constant pluriharmonic terms on both sides of the identity, we have

$1+\sum_{i=1}^{N^{\prime}-\kappa^{\prime}}|L_{i}(z)|^{2}-\sum_{j=0}^{\kappa^{\prime}-1}|L_{N^{\prime}-j}(z)|^{2}=\left(\sum_{i=1}^{N-\kappa}|F_{i}(z)|^{2}-\sum_{j=1}^{\kappa}|F_{N-\kappa+j}(z)|^{2}\right)^{b}.$

The following question was raised in [4, 24]:

######  Question 3.1.

Let $f_{1},\cdots,f_{m}$ be non-constant germs of linearly independent
holomorphic functions at $0\in\mathbb{C}$ such that
$f_{1}(0)=\cdots=f_{m}(0)=0$ and $a_{1},\cdots,a_{m}$ be non-zero real
numbers, do there exist germs of holomorphic functions $g_{1},\cdots,g_{n}$
and non-zero real numbers $b_{1},\cdots,b_{n}$ and $\alpha<0$ such that

$1+\sum_{i=1}^{m}a_{i}|f_{i}|^{2}=\left(1-\sum_{j=1}^{n}b_{j}|g_{j}|^{2}\right)^{\alpha}?$

This question was motivated by a classical theorem of Umehara, who showed that
when all $a_{i},b_{j}$ are positive, the answer is ”no” [23]. In other words,
the complex projective space and the complex hyperbolic space does not share a
common complex submanifold. As a consequence of Theorem 2.1, we show that
under an additional assumption, the question for the indefinite complex
projective space and the indefinite complex hyperbolic space also has a
negative answer. However, the general question is still open.

######  Corollary 3.2.

Let $f_{1},\cdots,f_{m}\in\mathcal{O}_{p}$ be non-constant germs of
holomorphic polynomials at $p$ such that $f_{1}(p)=\cdots=f_{m}(p)=0$. Assume
$\alpha<0$. For non-zero real numbers $a_{1},\cdots,a_{m}$, then there do not
exist $g_{1},\cdots,g_{n}$ and nonzero real numbers $b_{1},\cdots,b_{n}$, such
that

$\left(1+\sum_{i=1}^{m}a_{i}|f_{i}|^{2}\right)^{\alpha}=\sum_{j=1}^{n}b_{j}|g_{j}|^{2}$

near $p$.

###  3.2 Kähler manifolds with distinguished Kähler potentials

Consider two real analytic complex manifolds $(M_{1},\omega_{M_{1}})$ and
$(M_{2},\omega_{M_{2}})$, with
$(M_{2},\omega_{M_{2}})=({\mathbb{C}}^{N}_{\kappa},\omega_{{\mathbb{C}}^{N}_{\kappa}})$,
$({\mathbb{C}}{\mathbb{P}}^{N}_{\kappa}(b),\omega_{{\mathbb{C}}{\mathbb{P}}^{N}_{\kappa}(b)})$
or
$({\mathbb{C}}{\mathbb{H}}^{N}_{\kappa}(b),\omega_{{\mathbb{C}}{\mathbb{H}}^{N}_{\kappa}(b)})$.
Here $\omega_{M_{1}}$ is possibly indefinite or degenerate and even can be an
arbitrary closed real analytic $(1,1)$-form. Suppose that $D$ is a complex
manifold such that there exist holomorphic maps
$F=(F_{1},\cdots,F_{N}):D\rightarrow M_{1}$ and
$L=(L_{1},\cdots,L_{N^{\prime}}):D\rightarrow M_{2}$ with
$F^{*}\omega_{M_{1}}=L^{*}\omega_{M_{2}}$. Fixing $p\in D$ and choosing
holomorphic coordinate $\\{z\\}$ near $p$ with $z(p)=0$, without loss of
generality, we assume $L(p)=0$ by composing the automorphisms on $M_{2}$. Let
$\\{\xi\\}$ be the holomorphic coordinate with $\xi(F(p))=0$ and by
$\partial\bar{\partial}$-lemma, $(M_{1},\omega_{M_{1}})$ locally admits a
potential function $\varphi$ with
$\omega_{M_{1}}=4\sqrt{-1}\partial\overline{\partial}\varphi(\xi,\overline{\xi})$.
Calabi defined the diastasis function
${\mathcal{D}}(\xi,\overline{\eta})=\varphi(\xi,\overline{\xi})+\varphi(\eta,\overline{\eta})-\varphi(\xi,\overline{\eta})-\varphi(\eta,\overline{\xi})$,
which is a Hermitian symmetric function without non-constant pluriharmonic
terms. Moreover,
$\omega_{M_{1}}(\xi)=4\sqrt{-1}\partial_{\xi}\overline{\partial}_{\xi}{\mathcal{D}}(\xi,\overline{\eta})$
(cf. [2]).

By the standard argument to get rid of $\partial\overline{\partial}$ (cf.
[7]), since there is no non-constant pluriharmonic terms on both sides of the
identity, we have:

$\log\left(1+\sum_{i=1}^{N-\kappa}|L_{i}(z)|^{2}-\sum_{j=0}^{\kappa-1}|L_{N^{\prime}-j}(z)|^{2}\right)^{\mu}={\mathcal{D}}(F(z),\overline{F(z)})$

when $M_{2}={\mathbb{C}}{\mathbb{P}}^{N}_{\kappa}(b)$ or
${\mathbb{C}}{\mathbb{H}}^{N}_{\kappa}(b)$, and

$\sum_{i=\kappa+1}^{N}|L_{i}(z)|^{2}-\sum_{j=1}^{\kappa}|L_{j}(z)|^{2}={\mathcal{D}}(F(z),\overline{F(z)})$

when $M_{2}={\mathbb{C}}^{N}_{\kappa}$. Note that since $L_{j}(0)=0$ for all
$j$,

$\left(1+\sum_{i=1}^{N-\kappa}|L_{i}(z)|^{2}-\sum_{j=0}^{\kappa-1}|L_{N-j}(z)|^{2}\right)^{\mu}\in\tilde{\Lambda}_{p}^{\mu},~{}~{}~{}~{}\sum_{i=\kappa+1}^{N}|L_{i}(z)|^{2}-\sum_{j=1}^{\kappa}|L_{j}(z)|^{2}\in\tilde{\Lambda}_{p}.$

In general, if the polarization $\varphi(\xi,\overline{\eta})$ of the
potential function $\varphi$ is Nash algebraic, the diastasis function
${\mathcal{D}}(\xi,\overline{\eta})$ is Nash algebraic as well. Similarly, if
$\exp\\{\varphi(\xi,\overline{\eta})\\}$ is Nash algebraic, then
$\exp\\{{\mathcal{D}}(\xi,\overline{\eta})\\}=\frac{\exp\\{\varphi(\xi,\overline{\xi})\\}\exp\\{\varphi(\eta,\overline{\eta})\\}}{\exp\\{\varphi(\xi,\overline{\eta})\\}\exp\\{\varphi(\eta,\overline{\xi})\\}}$
is also Nash algebraic. This fits into the framework of Theorem 2.1.(iii). In
fact, this even applies to the product space of
${\mathbb{C}}{\mathbb{P}}^{N}_{\kappa}(b)$ and
${\mathbb{C}}{\mathbb{H}}^{N}_{\kappa}(b)$. As a consequence, we have:

######  Corollary 3.3.

Assume $(M_{1},\omega_{M_{1}})$ locally admits a potential function
$\varphi(\xi,\overline{\eta})$ with
$\omega_{M_{1}}=4\sqrt{-1}\partial\overline{\partial}\varphi(\xi,\overline{\xi})$.

  1. 1.

If $\varphi(\xi,\overline{\eta})$ is Nash algebraic, then
$(M_{1},\omega_{M_{1}})$ and $(M_{2},\omega_{M_{2}})$ do not share common
complex submanifolds for
$(M_{2},\omega_{M_{2}})=\left(\prod_{j=1}^{m}N_{j},\oplus_{j=1}^{m}c_{j}\omega_{N_{j}}\right)$
with $N_{j}={\mathbb{C}}{\mathbb{P}}^{N_{j}}_{\kappa_{j}}(b_{j})$ or
${\mathbb{C}}{\mathbb{H}}^{N_{j}}_{\kappa_{j}}(b_{j})$.

  2. 2.

If $\exp\\{\varphi(\xi,\overline{\eta})\\}$ is Nash algebraic, then
$(M_{1},\omega_{M_{1}})$ and $(M_{2},\omega_{M_{2}})$ do not share common
complex submanifolds for
$(M_{2},\omega_{M_{2}})=\left(\prod_{j=1}^{m}N_{j},\oplus_{j=1}^{m}c_{j}\omega_{N_{j}}\right)$
with $N_{j}={\mathbb{C}}^{N_{j}}_{\kappa_{j}}(b_{j})$.

######  Corollary 3.4.

The indefinite complex Euclidean spaces and the bounded domain with Nash
algebraic Bergman kernel function equipped with the Bergman metric do not have
common complex submanifolds.

Note that the following bounded domains have Nash algebraic Bergman kernel
function: (a) bounded homogeneous domains (cf. Proposition 1 in [5]), (b) the
Hartogs triangle and its generalizations [9], (c) the minimal ball [20], (d)
the symmetrized polydisc [10, 3] and (e) certain Hartogs domains over bounded
homogeneous domains (for instance, Cartan-Hartogs domains) [13]. Therefore,
they do not share common complex submanifolds with the indefinite complex
Euclidean spaces.

######  Example 3.5.

Let $\tilde{\varphi}(\xi,\eta)$ be any Nash algebraic, irrational function in
$\mathbb{C}^{2}$ such that $\tilde{\varphi}(\xi,\bar{\eta})$ is Hermitian
symmetric. Let $D$ be a planar domain such that
$\tilde{\varphi}(\xi,\bar{\xi})$ is $C^{2}$-smooth up to the boundary of $D$.
Then there exists $C>0$, such that
$\varphi(\xi,\bar{\xi})=C|\xi|^{2}+\tilde{\varphi}(\xi,\bar{\xi})$ is a
strictly plurisubharmonic function in $D$. $\varphi(\xi,\bar{\xi})$ is thus an
example of Nash algebraic and irrational potential function and
$\sqrt{-1}\partial\overline{\partial}\varphi(\xi,\overline{\xi})$ provides a
Kähler metric on $D$. This example satisfies the assumption in Part 1 of
Corollary 3.3. One can construct many other examples of Kähler metrics on
domains in $\mathbb{C}^{n}$ with Nash algebraic and irrational potential
functions in a similar manner.

######  Remark 3.6.

It follows from Theorem 3.1 in [15] that, there exist rational functions
$h_{1},\cdots,h_{m}$ and positive numbers $\mu_{1},\cdots,\mu_{m}$ such that
$\log\left(\prod_{j=1}^{m}h_{j}^{\mu_{j}}(z)\right)$ is the potential function
of the homogeneous Kähler metric on the bounded homogeneous domain. By
polarization, Calabi’s diastasis function fits into the form of $\log H(z,w)$
in Theorem 2.1.(ii). As a consequence of Theorem 2.1.(ii)(2), a bounded
homogeneous domain equipped with a homogeneous Kähler metric and an indefinite
complex Euclidean space do not share common complex submanifolds. This
recovers Theorem 1.1.(ii) in [15].

Statement: Data sharing not applicable to this article as no datasets were
generated or analysed during the current study.

## References

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Harbin, Heilongjiang, 150080, China.


# Strong decays of exotic and nonexotic heavy baryons
in the chiral quark-soliton model

Hyun-Chul Kim  Department of Physics, Inha University, Incheon 22212, Republic
of Korea<EMAIL_ADDRESS>School of Physics, Korea Institute for Advanced
Study (KIAS), Seoul 02455, Republic of Korea  Maxim V. Polyakov  Institut für
Theoretische Physik II, Ruhr-Universität Bochum, D–44780 Bochum, Germany
<EMAIL_ADDRESS>Petersburg Nuclear Physics Institute,
Gatchina, St. Petersburg 188 300, Russia  Michał Praszałowicz  M. Smoluchowski
Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków,
Poland<EMAIL_ADDRESS>Ghil-Seok Yang  Department of Physics,
Soongsil University, Seoul 06978, Republic of Korea<EMAIL_ADDRESS>
###### Abstract

In the large $N_{c}$ limit both heavy and light baryons are described by the
universal mean field, which allows us to relate the properties of heavy
baryons to light ones. With the only input from the decays of light octet
baryons (due to the universality of the chiral mean field), excellent
description of strong decays of both charm and bottom sextets is obtained. The
parameter-free prediction for the widths of exotic antidecapentaplet
($\overline{\bm{15}}$) baryons is also made. The exotic heavy baryons should
be anomalously narrow despite of large phase space available. In particular,
the widths of $\Omega_{c}(3050)$ and $\Omega_{c}(3119)$, interpreted as
members of $\overline{\bm{15}}$-plet, are very small: 0.48 MeV and 1.12 MeV
respectivly. This result is in very good agreement with the measurements of
the LHCb Collaboration and provides natural and parameter-free explanation of
the LHCb observation that $\Omega_{c}(3050)$ and $\Omega_{c}(3119)$ have
anomalously small widths among five recently observed states.

Heavy baryons, mean field approach, Mass splittings of SU(3) baryons, Chiral
soliton model, Flavor symmetry breaking, Exotic pentaquarks.

###### pacs:

12.39.Hg, 14.20.Lq, 14.20.Mr, 11.30.Qc

††preprint: INHA-NTG-05/2017

##  I Introduction

In a recent paper Kim:2017jpx  we have proposed to interpret two narrow
$\Omega_{c}$ resonances reported recently by the LHCb Collaboration
Aaij:2017nav  as exotic pentaquark states belonging to the SU(3)
representation $\overline{\bf 15}$ (antidecapentaplet) Diakonov:2010tf . To
this end we have used the chiral quark soliton model ($\chi$QSM)
Diakonov:1987ty  (for a review see Ref. Christov:1995vm  and references
therein) modified in the spirit of heavy quark symmetry Isgur:1989vq ;
Isgur:1991wq ; Georgi:1990um  to accommodate one heavy quark Diakonov:2010tf ;
Yang:2016qdz . The $\chi$QSM is based on an old argument by Witten
Witten:1979kh , which says that in the limit of a large number of colors
($N_{c}\rightarrow\infty$), $N_{c}$ relativistic valence quarks generate
chiral mean fields represented by a distortion of a Dirac sea that in turn
interact with the valence quarks themselves. In this way a self-consistent
configuration called a soliton is formed. The mean fields exhibit so called
hedgehog symmetry, which means that neither quark spin ($\bm{S}_{q}$) nor
quark isospin ($\bm{T_{q}}$) are ”good” quantum numbers. Instead a grand spin
$\bm{K}=\bm{S}_{q}+\bm{T}_{q}$ is a good quantum number.

In order to project out spin and isospin quantum numbers one has to rotate the
soliton, both in flavor and configuration spaces. These rotations are then
subsequently quantized semiclassically and the collective Hamiltonian is
computed. The model predicts rotational baryon spectra that satisfy the
following selection rules:

  * •

allowed SU(3) representations must contain states with hypercharge
$Y^{\prime}=N_{c}/3$,

  * •

the isospin $\bm{T}^{\prime}$ of the states with $Y^{\prime}=N_{c}/3$ couples
with the soliton spin $\bm{J}$ to a singlet: $\bm{T}^{\prime}+\bm{J}=0$.

In the case of light positive parity baryons the lowest allowed
representations are $\mathbf{8}$ of spin 1/2, $\mathbf{10}$ of spin 3/2, and
also exotic pentaquark representation $\overline{\mathbf{10}}$ of spin 1/2
with the lightest state corresponding to the putative $\Theta^{+}(1540)$.
Chiral models in general predict that pentaquarks are light
Praszalowicz:2003ik ; Diakonov:1997mm  and – in some specific models – narrow
Diakonov:1997mm .

Figure 1: Lowest lying SU(3) flavor representations allowed by the constraint
$Y^{\prime}=2/3$. The first exotic representation, $\overline{\mathbf{15}}$
contains the putative pentaquark states $\Omega_{c}$ with $\Omega_{c}^{0}$
marked in red.

In order to construct a heavy baryon in the $\chi$QSM we have to strip off one
light quark from the valence level and quantize the soliton with a new
constraint $Y^{\prime}=(N_{c}-1)/3$, which modifies the above selection rules
in the following way:

  * •

allowed SU(3) representations must contain states with hypercharge
$Y^{\prime}=(N_{c}-1)/3$,

  * •

the isospin $\bm{T}^{\prime}$ of the states with $Y^{\prime}=(N_{c}-1)/3$
couples with the soliton spin $\bm{J}$ to a singlet:
$\bm{T}^{\prime}+\bm{J}=0$.

This $N_{c}-1$ light quark configuration is then coupled to a heavy quark $Q$
to form a color singlet. The lowest allowed SU(3) representations are shown in
Fig. 1. They correspond to the soliton in representation in
$\overline{\mathbf{3}}$ of spin 0 and to ${\mathbf{6}}$ of spin 1. Therefore
the baryons constructed from such a soliton and a heavy quark form an SU(3)
antitriplet of spin 1/2 and two sextets of spin 1/2 and 3/2 that are subject
to a hyperfine splitting. The next allowed representation of the rotational
excitations corresponds to the exotic $\overline{\mathbf{15}}$ of spin 0 or
spin 1 Kim:2017jpx . The spin 1 soliton has lower mass and when it couples
with a heavy quark it forms spin 1/2 or 3/2 exotic multiplets that should be
hyperfine split similarly to the ground state sextets by $\sim 70$ MeV. In
Ref. Kim:2017jpx  we have proposed to interpret two LHCb states:
$\Omega^{0}_{c}(3050)$ and $\Omega^{0}_{c}(3119)$ as $1/2^{+}$ and $3/2^{+}$
pentaquarks belonging to the SU(3) $\overline{\mathbf{15}}$. As can be seen
from Fig. 1 they belong to the isospin triplets, and therefore should have
charged partners of the same mass, which allows, in principle, for rather
straightforward experimental verification of this interpretation.

In the present paper we calculate strong decay widths of nonexotic and exotic
heavy quark baryons (both charm and bottom) in an approach proposed many years
ago by Adkins, Nappi and Witten Adkins:1983ya  and expanded in Ref.
Diakonov:1997mm , which is based on the Goldberger-Treiman relation where
strong decay constants are expressed in terms of the axial-vector current
couplings. We show that by fixing these axial-vector current couplings from
the hyperon decays in the light sector, we can predict strong decay widths of
heavy baryons with only one free parameter related to the modification of
these couplings due to the fact that the soliton is constructed from $N_{c}-1$
rather than $N_{c}$ light quarks. We test our approach against experimentally
known charm and bottom baryon decay widths, and then show that the decay
widths of $\Omega^{0}_{c}(3050)$ and $\Omega^{0}_{c}(3119)$ are small and
compatible with the LHCb measurements. Overall agreement of the predicted
decay widths with experiment is more than satisfactory and exceeds the
expected model accuracy, which is believed to be at the level of 10% – 20%.

The rotational states described above correspond to positive parity. Negative
parity states generated by the soliton configurations with one light quark
excited to the valence level from the Dirac sea have been discussed in Ref.
Kim:2017jpx . It has been shown that the remaining three LHCb $\Omega^{0}_{c}$
states can be accommodated in such an approach. The formalism used in the
present paper has to be rather strongly modified to describe negative parity
state decay widths, and we plan to address this issue in a separate
publication.

Other authors have also considered a possibility that at least some of the
LHCb $\Omega_{c}$ states may be interpreted as pentaquarks Yang:2017rpg ;
Huang:2017dwn ; An:2017lwg ; Anisovich:2017aqa  in different variants of the
quark (or quark-diquark) models. However, the states considered in these
papers have negative parity and are isospin singlets. A simpler interpretation
that they are $p$-wave or radial and $p$-wave excitations of the $ss$-diquark
$c$-quark system has been put forward in Refs. Karliner:2017kfm ; Wang:2017hej
; Wang:2017vnc ; Cheng:2017ove ; Chen:2017gnu ; Ali:2017wsf . Both radial and
$p$-wave excitations have been also studied in the framework of the QCD sum
rules Agaev:2017jyt ; Chen:2017sci ; Wang:2017zjw ; Aliev:2017led ;
Azizi:2017bgs , in a phenomenological approach Zhao:2017fov  and on the
lattice Padmanath:2017lng . A nonperturbative holographic QCD has been applied
to investigate heavy, regular, and exotic baryons in Refs. Liu:2017xzo . The
variety of quantum number assignments proposed in the above references has
been possible because all the above approaches (including the one in the
present paper) suffer from systematic uncertainties that exceed tiny
experimental errors of the heavy baryon masses. An ongoing analysis of spin
and parity of the LHCb $\Omega_{c}$ states is therefore of utmost importance
to discriminate different theoretical models.

The paper is organized as follows. In Sec. II we describe briefly the
formalism for description of the baryon strong decays in the $\chi$QSM. In the
following Sec. III we test the mean field picture of heavy baryons against the
known strong decays of ground-state baryon sextets. In Sec. IV the parameter-
free prediction for decays of exotic $\overline{\bm{15}}$-plet baryons is
made. Particular attention is paid to strong decays of $\Omega_{c}(3050)$ and
$\Omega_{c}(3119)$ which we interpret as members of the exotic
antidecapentaplet. Conclusions are presented in Sec. V.

##  II Heavy baryons and their strong decays in the chiral quark-soliton model

In our picture the $\overline{\bm{3}}$, sextet and exotic $\overline{\bm{15}}$
baryons shown in Fig. 1 are rotational excitations of the meson mean field,
which is essentially the same as for light baryons. The corresponding wave
function of the light sector is given in terms of the Wigner rotational $D(A)$
matrices

$\displaystyle\Psi_{(B\,;\,-Y^{\prime}\,S\,S_{3})}^{(\mathcal{R\,};\,B)}(A)$
$\displaystyle=\langle A|\mathcal{R},\,B,\,(-Y^{\prime},S,S_{3})\rangle$
$\displaystyle=\sqrt{\text{dim}(\mathcal{R})}\,(-)^{S_{3}-Y^{\prime}/2}D_{(Y,\,T,\,T_{3})(Y^{\prime},\,S,\,-S_{3})}^{(\mathcal{R})\ast}(A)$
(1)

where $\mathcal{R}$ denotes the SU(3) representation of the light sector,
$B=(Y,T,T_{3})$, stands for the SU(3) quantum numbers of a baryon in question,
and the second index of the $D$ function, $(Y^{\prime},S,-S_{3})$, corresponds
to the soliton spin. For the heavy baryons $Y^{\prime}=(N_{c}-1)/3$. $A(t)$
denotes relative configuration space – SU(3) group space rotation matrix.

The total wave function of a heavy baryon of spin $J$ is constructed by
coupling (1) to a heavy quark ket $\left|1/2,s_{3}\right\rangle$, with a
pertinent SU(2) Clebsch-Gordan coefficient:

$|\mathcal{R},\,B,\,J,J_{3}\rangle={\displaystyle\sum\limits_{S_{3},s_{3}}}\left(\begin{array}[c]{cc}S&1/2\\\
S_{3}&s_{3}\end{array}\right|\left.\begin{array}[c]{c}J\\\
J_{3}\end{array}\right)\,\left|1/2,s_{3}\right\rangle\,|\mathcal{R},\,B,\,(-Y^{\prime},S,S_{3})\rangle$
(2)

The masses of various multiplets are obtained by sandwiching the collective
Hamiltonian between the wave function (2); see for details Refs. Kim:2017jpx ;
Yang:2016qdz . To calculate the decays of the heavy baryons one has to
sandwich the corresponding decay operator between the wave functions (2).
Following Ref. Diakonov:1997mm  we use in this paper the decay operator
describing the emission of a $p$-wave pseudoscalar meson $\varphi$, as in the
case of regular baryons, with possible rescaling of the coefficients $G_{i}$
(see below),

$\mathcal{O}_{\varphi}=\frac{3}{M_{1}+M_{2}}\sum_{i=1,2,3}\left[G_{0}D_{\varphi\,i}^{(8)}-G_{1}\,d_{ibc}D_{\varphi\,b}^{(8)}\hat{S}_{c}-G_{2}\frac{1}{\sqrt{3}}D_{\varphi\,8}^{(8)}\hat{S}_{i}\right]p_{i}.$
(3)

We are considering decays $B_{1}\rightarrow B_{2}+\varphi$, where $M_{1,2}$
denote masses of the initial and final baryons respectively and $p_{i}$ is the
c.m. momentum of the outgoing meson of mass $m$:

$\left|\vec{p}\,\right|=p=\frac{1}{2M_{1}}\sqrt{(M_{1}^{2}-(M_{2}+m)^{2})(M_{1}^{2}-(M_{2}-m)^{2})}$
(4)

The decay width is related to the matrix element of $\mathcal{O}_{\varphi}$
squared, summed over the final and averaged over the initial spin and isospin
denoted as $\overline{\left[\ldots\right]^{2}}$; see the appendix of Ref.
Diakonov:1997mm  for details of the corresponding calculations,

$\Gamma_{B_{1}\rightarrow B_{2}+\varphi}=\frac{1}{2\pi}\overline{\left\langle
B_{2}\left|\mathcal{O}_{\varphi}\right|B_{1}\right\rangle^{2}}\,\frac{M_{2}}{M_{1}}p.$
(5)

Here factor $M_{2}/M_{1}$ is the same as in heavy baryon chiral perturbation
theory (HBChPT); see e.g., Ref. Cheng:2006dk . This factor arises because in
the heavy quark effective theory and in HBChPT the velocities of $B_{1}$ and
$B_{2}$ are the same up to corrections of order $O(1/m_{Q})$, which we
neglect.

The pseudoscalar meson-baryon couplings can be related to the transition
($B_{1}\to B_{2}$) axial-vector constants with the help of the Goldberger-
Treiman relation, see Ref. Yan:1992gz  for the derivation in the case of heavy
baryons. Using the Goldberger-Treiman relation we obtain for the couplings
$G_{0,1,2}$

$\left\\{G_{0},G_{1},G_{2}\right\\}=\frac{M_{1}+M_{2}}{2F_{\varphi}}\frac{1}{3}\left\\{-a_{1},a_{2},a_{3}\right\\},$
(6)

where constants111 For the reader’s convenience we give the relations of the
constants $a_{1,2,3}$ to nucleon axial charges in the chiral limit:
$g_{A}=\frac{7}{30}\left(-a_{1}+\frac{1}{2}a_{2}+\frac{1}{14}a_{3}\right)$,
$g_{A}^{(0)}=\frac{1}{2}a_{3}$,
$g_{A}^{(8)}=\frac{1}{10\sqrt{3}}\left(-a_{1}+\frac{1}{2}a_{2}+\frac{1}{2}a_{3}\right)$
$a_{1,2,3}$ enter the definition of the axial-vector current
Praszalowicz:1998jm  and have been extracted from the semileptonic decays of
the baryon octet in Ref. Yang:2015era ,

$a_{1}=-3.509\pm 0.011,\;a_{2}=3.437\pm 0.028,\;a_{3}=0.604\pm 0.030.$ (7)

For the decay constants $F_{\varphi}$ we have chosen the convention in which
$F_{\pi}=93$ MeV and $F_{K}=1.2\,F_{\pi}=112$ MeV.

The final formula for the decay width in terms of axial-vector constants
$a_{1,2,3}$ reads as follows:

$\displaystyle\Gamma_{B_{1}\rightarrow B_{2}+\varphi}$
$\displaystyle=\frac{1}{72\pi}\frac{p^{3}}{F_{\varphi}^{2}}\frac{M_{2}}{M_{1}}G_{\mathcal{R}_{1}\rightarrow\mathcal{R}_{2}}^{2}\times$
(8) $\displaystyle\times
3\frac{\dim\mathcal{R}_{2}}{\dim\mathcal{R}_{1}}\left[\begin{array}[c]{cc}\bm{8}&\mathcal{R}_{2}\\\
01&Y^{\prime}S_{2}\end{array}\right|\left.\begin{array}[c]{c}\mathcal{R}_{1}\\\
Y^{\prime}S_{1}\end{array}\right]^{2}\left[\begin{array}[c]{cc}\bm{8}&\mathcal{R}_{2}\\\
Y_{\varphi}T_{\varphi}&Y_{2}T_{2}\end{array}\right|\left.\begin{array}[c]{c}\mathcal{R}_{1}\\\
Y_{1}T_{1}\end{array}\right]^{2}$ (17)

Here $\mathcal{R}_{1,2}$ are the SU(3) representations of the initial and
final baryons, $[...|..]$ are the SU(3) isoscalar factors. The decay constants
$G_{\mathcal{R}_{1}\rightarrow\mathcal{R}_{2}}$ are calculated from the matrix
elements of (3) as

$\displaystyle\overline{\bm{15}}_{1}$
$\displaystyle\rightarrow\overline{\bm{3}}_{0}\qquad
G_{\overline{\bm{3}}}=-a_{1}-\frac{1}{2}a_{2}=0.44,$
$\displaystyle\overline{\bm{15}}_{1}$
$\displaystyle\rightarrow\bm{6}_{1}\qquad
G_{\bm{6}}=-a_{1}-\frac{1}{2}a_{2}-a_{3}=-0.16,$ $\displaystyle\bm{6}_{1}$
$\displaystyle\rightarrow\overline{\bm{3}}_{0}\qquad
H_{{\overline{\bm{3}}}}=-a_{1}+\frac{1}{2}a_{2}=3.88,$ (18)

where numerical values have been calculated with the help of Eq. (7). With
these definitions of the couplings the formulas for the decay widths averaged
over the initial isospin and summed over the final isospin read as follows:

$\displaystyle\Gamma_{\Sigma(\bm{6}_{1})\rightarrow\Lambda(\overline{\bm{3}}_{0})+\pi}$
$\displaystyle=\frac{1}{72\pi}\frac{p^{3}}{F_{\pi}^{2}}\frac{M_{\Lambda(\overline{\bm{3}}_{0})}}{M_{\Sigma(\bm{6}_{1})}}H_{{\overline{\bm{3}}}}^{2}\frac{3}{8},$
$\displaystyle\Gamma_{\Xi(\bm{6}_{1})\rightarrow\Xi(\overline{\bm{3}}_{0})+\pi}$
$\displaystyle=\frac{1}{72\pi}\frac{p^{3}}{F_{\pi}^{2}}\frac{M_{\Xi(\overline{\bm{3}}_{0})}}{M_{\Xi(\bm{6}_{1})}}H_{{\overline{\bm{3}}}}^{2}\frac{9}{32},$
$\displaystyle\Gamma_{\Omega(\overline{\bm{15}}_{1})\rightarrow\Xi(\overline{\bm{3}}_{0})+K}$
$\displaystyle=\frac{1}{72\pi}\frac{p^{3}}{F_{K}^{2}}\frac{M_{\Xi(\overline{\bm{3}}_{0})}}{M_{\Omega(\overline{\bm{15}}_{1})}}G_{{\overline{\bm{3}}}}^{2}\frac{3}{10},$
$\displaystyle\Gamma_{\Omega(\overline{\bm{15}}_{1})\rightarrow\Omega(\bm{6}_{1})+\pi}$
$\displaystyle=\frac{1}{72\pi}\frac{p^{3}}{F_{\pi}^{2}}\frac{M_{\Omega(\bm{6}_{1})}}{M_{\Omega(\overline{\bm{15}}_{1})}}G_{\bm{6}}^{2}\frac{4}{15},$
$\displaystyle\Gamma_{\Omega(\overline{\bm{15}}_{1})\rightarrow\Xi(\bm{6}_{1})+K}$
$\displaystyle=\frac{1}{72\pi}\frac{p^{3}}{F_{K}^{2}}\frac{M_{\Xi(\bm{6}_{1})}}{M_{\Omega(\overline{\bm{15}}_{1})}}G_{\bm{6}}^{2}\frac{2}{15}.$
(19)

When we need a decay width for a specific isospin combination, the widths of
Eqs. (19) have to be multiplied by a pertinent SU(2) Clebsch-Gordan
coefficient.

Note that in the $\chi$QSM couplings (7) are expressed in terms of the inertia
parameters Praszalowicz:1998jm ,

$a_{1}=A_{0}-\frac{B_{1}}{I_{1}},\;a_{2}=2\frac{A_{2}}{I_{2}},\;a_{3}=2\frac{A_{1}}{I_{1}}.$
(20)

Since all inertia parameters scale as $N_{c}$, we see that formally $a_{1}$
contains both leading and the first subleading term in $N_{c}$, whereas
$a_{2,3}$ scale as $N_{c}^{0}$. This can be the best seen in the
nonrelativistic (NR) limit for small soliton size, where Praszalowicz:1995vi

$A_{0}\rightarrow-N_{c},\;\frac{B_{1}}{I_{1}}\rightarrow
2,\;,\;\frac{A_{2}}{I_{2}}\rightarrow 2,\;\frac{A_{1}}{I_{1}}\rightarrow 1$
(21)

or

$a_{1}\rightarrow-(N_{c}+2),\;a_{2}\rightarrow 4,\;a_{3}\rightarrow 2$ (22)

Remember that in this limit the axial-vector coupling constant

$g_{A}=\frac{7}{30}\left(-a_{1}+\frac{1}{2}a_{2}+\frac{1}{14}a_{3}\right)\rightarrow\frac{5}{3},$
(23)

which is equal to the naive quark model result for $g_{A}$, whereas for the
phenomenological values (7) $g_{A}=1.23.$

In the case of heavy baryons all inertia parameters should be rescaled by
approximately222Strictly speaking rescaling by a factor $(N_{c}-1)/N_{c}$
should work well only for quantities dominated by valence levels. As the
contribution of the sea quarks to some quantities can be sizeable one may
expect 10%–20% variations of that rescaling factor. $(N_{c}-1)/N_{c}$, which
does not change the scaling of their ratios; however it does change the value
of $a_{1}$(strictly speaking the $A_{0}$ part of $a_{1}$). Therefore for heavy
baryons we should use

$A_{0}\rightarrow\tilde{A}_{0}=\frac{N_{c}-1}{N_{c}}A_{0}.$ (24)

Unfortunately from the fits to the experimental data we do not know separately
the values of $A_{0}$ and $B_{1}/I_{1}$. We know only these values in the NR
limit (21). Making the rather bold assumption that NR relation $B_{1}=2A_{1}$
holds also for realistic soliton sizes, we approximate $B_{1}/I_{1}\sim
a_{3}$, which gives $A_{0}=a_{1}+a_{3}$. Therefore, following Eq.(24), we make
the following replacement:

$a_{1}\rightarrow\tilde{a}_{1}=\left[\frac{N_{c}-1}{N_{c}}\left(a_{1}+a_{3}\right)-a_{3}\right]\sigma.$
(25)

Here $\sigma$ is a correction factor that takes into account possible
deviations from the assumption that $B_{1}=2A_{1}$ and possible deviation of
the rescaling factor from $(N_{c}-1)/N_{c}$. The parameter $\sigma$
characterizes the modification of the mean field when one goes from $N_{c}$
quarks in light baryons to $N_{c}-1$ quarks in heavy baryons. In the ideal
case $\sigma$ should be close to unity; however in practice, as we shall see
in the following, a $15\%$ correction is required to get satisfactory
description of the decay widths. Such small modification of the mean field is
fully compatible with the expected size of $1/N_{c}$ corrections.

It is interesting to calculate the decay constants (18) in the nonrelativistic
quark model limit (22) with $a_{1}\rightarrow-(N_{c}+1)$. One can see that in
this limit $G_{\bm{6}}=0$. This means that the decay channels of the putative
heavy pentaquarks to the sextets should be strongly suppressed. This situation
is very similar to the suppression of $\Theta^{+}(1530)$ decay width
Diakonov:1997mm .

One should remember that in the large $N_{c}$ limit flavor representations of
the light sector should be generalized Karl:1985qy ; Dulinski:1987er ;
Piesciuk:2007xq , and in the present case the standard generalization takes
the following form QlargeNc :

$\displaystyle"\bm{\overline{3}}"$ $\displaystyle=(0,q+1),$
$\displaystyle"\bm{6}"$ $\displaystyle=(2,q),$
$\displaystyle"\overline{\bm{15}}"$ $\displaystyle=(1,q+2)$ (26)

with $q=(N_{c}-3)/2$. With this generalization the pertinent SU(3) Clebsch-
Gordan coefficients acquire $N_{c}$ dependence. For example

$G_{\overline{\bm{3}}}=-a_{1}-\frac{N_{c}-1}{4}a_{2},$ (27)

which means that $G_{\overline{\bm{3}}}$ is suppressed in the large $N_{c}$ NR
limit (22), since the leading $N_{c}$ terms cancel out. This cancellation is
similar as in the case of high-dimensional exotic representations of light
baryons Piesciuk:2007xq . We see that large $N_{c}$ counting together with the
arguments based on the nonrelativistic limit explain the numerical hierarchy
of the decay couplings (18). Full $N_{c}$ dependence of the mass splittings
and decay widths will be discussed elsewhere QlargeNc .

##  III Decay widths of sextet heavy baryons

A number of decay widths are measured for heavy baryons. In Ref. Cheng:2006dk
decays of charm sextet $\Sigma_{c}$, both of spin 1/2 and 3/2, have been
fitted with the help of the formula analogous to (19) in terms of a single
coupling $g_{2}$. The updated phenomenological value is presently $g_{2}=0.56$
with a 5% error. Coupling $g_{2}$ can be expressed in terms of
$H_{{\overline{\bm{3}}}}$:

$\left|g_{2}\right|=\frac{1}{4\sqrt{3}}H_{{\overline{\bm{3}}}}=\frac{1}{4\sqrt{3}}\left(-\tilde{a}_{1}+\frac{1}{2}a_{2}\right).$
(28)

To fit the phenomenological value of 0.56 we need a correction factor
$\sigma=0.85$. Other than that there is no extra freedom and all decay widths
are genuine predictions of the present model.

Two comments are in order here. Decay widths of different $T_{3}$ states show
small isospin violations, which are mainly due to a small phase volume and
hence due to the sensitivity to the mass difference of $\pi^{\pm}-\pi^{0}$.
Decay couplings are calculated in the isospin symmetric limit. Secondly,
different decays of particles within the same initial and final SU(3)
multiplets ($\mathcal{R}_{1}\rightarrow\mathcal{R}_{2}$) are related by the
SU(3) symmetry, which is more general than the present model. $\chi$QSM
relates couplings of different SU(3) multiplets, like
$H_{{\overline{\bm{3}}}}$, $G_{\overline{\bm{3}}}$ or $G_{\bm{6}}$, which are
expressed as different combinations of $a_{1,2,3}$ (7).

Below, in Tables 1 and 2, we list our results and the experimental values both
for the charm and bottom baryons. Note that in the case of
$\Xi_{Q}^{q}(\bm{6}_{1},3/2)$ (where $q$ denotes the pertinent charge) decays
to $\Xi_{Q}(\overline{\bm{3}}_{0},1/2)+\pi$ the width is summed with the CG
weighting
$\frac{1}{3}(\Xi_{Q}^{q}(\overline{\bm{3}}_{0},1/2)+\pi^{0})+\frac{2}{3}(\Xi_{Q}^{q\mp
1}(\overline{\bm{3}}_{0},1/2)+\pi^{\pm})$. When no charges are specified the
width is averaged over initial and summed over final isospin.

# | Decay | $\begin{array}[c]{c}\text{This}\\\ \text{work}\end{array}$ | Exp.
---|---|---|---
$1$ | $\Sigma_{c}^{++}(\bm{6}_{1},1/2)\rightarrow\Lambda_{c}^{+}(\overline{\bm{3}}_{0},1/2)+\pi^{+}$ | $1.93$ | $1.89_{-0.18}^{+0.09}$
$2$ | $\Sigma_{c}^{+}(\bm{6}_{1},1/2)\rightarrow\Lambda_{c}^{+}(\overline{\bm{3}}_{0},1/2)+\pi^{0}$ | $2.24$ | $<4.6$
$3$ | $\Sigma_{c}^{0}(\bm{6}_{1},1/2)\rightarrow\Lambda_{c}^{+}(\overline{\bm{3}}_{0},1/2)+\pi^{-}$ | $1.90$ | $1.83_{-0.19}^{+0.11}$
$4$ | $\Sigma_{c}^{++}(\bm{6}_{1},3/2)\rightarrow\Lambda_{c}^{+}(\overline{\bm{3}}_{0},1/2)+\pi^{+}$ | $14.47$ | $14.78_{-0.19}^{+0.30}$
$5$ | $\Sigma_{c}^{+}(\bm{6}_{1},3/2)\rightarrow\Lambda_{c}^{+}(\overline{\bm{3}}_{0},1/2)+\pi^{0}$ | $15.02$ | $<17$
$6$ | $\Sigma_{c}^{0}(\bm{6}_{1},3/2)\rightarrow\Lambda_{c}^{+}(\overline{\bm{3}}_{0},1/2)+\pi^{-}$ | $14.49$ | $15.3_{-0.5}^{+0.4}$
$7$ | $\Xi_{c}^{+}(\bm{6}_{1},3/2)\rightarrow\Xi_{c}(\overline{\bm{3}}_{0},1/2)+\pi$ | $2.35$ | $2.14\pm 0.19$
$8$ | $\Xi_{c}^{0}(\bm{6}_{1},3/2)\rightarrow\Xi_{c}(\overline{\bm{3}}_{0},1/2)+\pi$ | $2.53$ | $2.35\pm 0.22$
Table 1: Charm sextet baryons decay widths in MeV. Experimental data are taken from Particle Data Group Patrignani:2016xqp . # | Decay | $\begin{array}[c]{c}\text{This}\\\ \text{work}\end{array}$ | Exp.
---|---|---|---
$1$ | $\Sigma_{b}^{+}(\bm{6}_{1},1/2)\rightarrow\Lambda_{b}^{0}(\overline{\bm{3}}_{0},1/2)+\pi^{+}$ | $6.12$ | $9.7_{-3.0}^{+4.0}$
$2$ | $\Sigma_{b}^{-}(\bm{6}_{1},1/2)\rightarrow\Lambda_{b}^{0}(\overline{\bm{3}}_{0},1/2)+\pi^{-}$ | $6.12$ | $4.9_{-2.4}^{+3.3}$
$3$ | $\Xi_{b}^{{}^{\prime}}(\bm{6}_{1},1/2)\rightarrow\Xi_{c}(\overline{\bm{3}}_{0},1/2)+\pi$ | $0.07$ | $<0.08$
$4$ | $\Sigma_{b}^{+}(\bm{6}_{1},3/2)\rightarrow\Lambda_{b}^{0}(\overline{\bm{3}}_{0},1/2)+\pi^{+}$ | $10.96$ | $11.5\pm 2.8$
$5$ | $\Sigma_{b}^{-}(\bm{6}_{1},3/2)\rightarrow\Lambda_{c}^{0}(\overline{\bm{3}}_{0},1/2)+\pi^{-}$ | $11.77$ | $7.5\pm 2.3$
$6$ | $\Xi_{b}^{0}(\bm{6}_{1},3/2)\rightarrow\Xi_{b}(\overline{\bm{3}}_{0},1/2)+\pi$ | $0.80$ | $0.90\pm 0.18$
$7$ | $\Xi_{b}^{-}(\bm{6}_{1},3/2)\rightarrow\Xi_{b}(\overline{\bm{3}}_{0},1/2)+\pi$ | $1.28$ | $1.65\pm 0.33$
Table 2: Bottom sextet baryons decay widths in MeV. Experimental data are
taken from Particle Data Group Patrignani:2016xqp .

We see remarkably good agreement of the $\chi$QSM results with the
experimental widths for both charm and bottom baryons. To better illustrate
this we have plotted in Figs. 2 and 3 the results collected in Tables 1 and 2.
We stress that for the calculation of the heavy baryon widths essentially we
did not need any new parameter – everything was fixed in the light baryon
sector. The scaling factor $\sigma$ introduced in Eq. (25) is not really a new
parameter; it rather characterizes the $\sim$15 % modification of the
universal meson mean field due to $1/N_{c}$ corrections.

##  IV Decay widths of exotic antidecapentaplet baryons

With all decay constants fixed (7) from the decays of light baryons we can now
predict the widths of the putative pentaquark $\Omega_{c}(3050)$ and
$\Omega_{c}(3119)$ states that are collected in Tables 3 and 4. Note that the
exotic $\Omega_{c}$’s from $\overline{\bm{15}}$ have the isospin one and hence
the decay mode to $\Omega_{c}(\bm{6})+\pi$ is allowed. However, the
corresponding decay constant $G_{6}=-\tilde{a}_{1}-\frac{1}{2}a_{2}-a_{3}$ is
zero in the nonrelativistic small soliton size limit, analogously as for the
corresponding coupling of the light pentaquark $\Theta^{+}$. Therefore, we
expect that the $\Omega_{c}(\bm{6})+\pi$ decay mode should be strongly
suppressed.

# | Decay | $\begin{array}[c]{c}\text{This}\\\ \text{work}\end{array}$ | Exp.
---|---|---|---
| $\Omega_{c}(\overline{\bm{15}}_{1},1/2)\rightarrow\Xi_{c}(\overline{\bm{3}}_{0},1/2)+K$ | 0.339 | $-$
| $\Omega_{c}(\overline{\bm{15}}_{1},1/2)\rightarrow\Omega_{c}(\bm{6}_{1},1/2)+\pi$ | 0.097 | $-$
| $\Omega_{c}(\overline{\bm{15}}_{1},1/2)\rightarrow\Omega_{c}(\bm{6}_{1},3/2)+\pi$ | 0.045 | $-$
9 | Total | 0.48 | $0.8\pm 0.2\pm 0.1$
Table 3: $\Omega_{c}(\overline{\bm{15}}_{1},1/2)$ partial and total decay widths in MeV. Experimental value is from the LHCb measurement Aaij:2017nav . # | Decay | $\begin{array}[c]{c}\text{This}\\\ \text{work}\end{array}$ | Exp.
---|---|---|---
| $\Omega_{c}(\overline{\bm{15}}_{1},3/2)\rightarrow\Xi_{c}(\overline{\bm{3}}_{0},1/2)+K$ | 0.848 | $-$
| $\Omega_{c}(\overline{\bm{15}}_{1},3/2)\rightarrow\Xi_{c}(\bm{6}_{1},1/2)+K$ | 0.009 | $-$
| $\Omega_{c}(\overline{\bm{15}}_{1},3/2)\rightarrow\Omega_{c}(\bm{6}_{1},1/2)+\pi$ | 0.169 | $-$
| $\Omega_{c}(\overline{\bm{15}}_{1},3/2)\rightarrow\Omega_{c}(\bm{6}_{1},3/2)+\pi$ | 0.096 | $-$
10 | Total | 1.12 | $1.1\pm 0.8\pm 0.4$
Table 4: $\Omega_{c}(\overline{\bm{15}}_{1},3/2)$ partial and total decay
widths in MeV. Experimental value is from the LHCb measurement Aaij:2017nav

Note that the kaon momentum in the decay of $\Omega_{c}(3050)$, $p_{K}=275$
MeV, is quite close to the pion momentum in the decay of $\Delta$,
$p_{\pi}=228$ MeV, yet the $\Delta$ decay width is two orders of magnitude
larger than the one of $\Omega_{c}(3050)$. From Tables 3 and 4 we see that,
despite the large phase volume available, the predicted decay widths are very
small and are in agreement with the small ($\sim 1$ MeV) decay widths observed
by the LHCb collaboration (see also Fig. 2). Note that $\Omega_{c}(3050)$ and
$\Omega_{c}(3119)$ are the narrowest states among five LHCb $\Omega_{c}$’s and
our mean field picture gives a natural, parameter-free explanation of this
observation. In the large $N_{c}$ nonrelativistic limit discussed at the end
of Sec. II the decay constant to $\bm{6}$ is strongly suppressed, whereas the
decay constant to $\overline{\bm{3}}$ is suppressed in the leading order of
$N_{c}$.

Figure 2: Decay widths of the charm baryons. Numbers on the horizontal axis
label the decay modes as listed in Tables 1, 3 and 4. Red full circles
correspond to our theoretical predictions. Dark green triangles correspond to
the experimental data Patrignani:2016xqp . Data for decays 4 – 7 of
$\Sigma_{c}(\bm{6}_{1},3/2)$ (down-triangles) have been divided by a factor of
5 to fit within the plot area. Widths of two LHCb Aaij:2017nav  $\Omega_{c}$
states that we interpret as pentaquarks are plotted as black full squares.
Figure 3: Decay widths of the bottom baryons. Numbers on the horizontal axis
label the decay modes as listed in Table 2. Red full circles correspond to our
theoretical predictions. Dark green triangles correspond to the experimental
data Patrignani:2016xqp . Data for decays 6 and 7 of $\Xi_{b}(\bm{6}_{1},3/2)$
(down-triangles) have been multiplied by a factor of 5 to be better visible on
the plot.

The results in this section also imply that other members of the exotic
antidecapentaplet are expected to be anomalously narrow. All their partial
decay widths can be easily computed in our model with the help of general
formula (8). As an illustration we quote here the result for the decay widths
of other explicitly exotic members of the antidecapentaplet, $\Xi_{c}^{3/2-}$
and $\Xi_{c}^{3/2++}$, which have the minimal quark content ($cdds\bar{u}$)
and ($cuus\bar{d}$). The masses of these states are predicted in Ref.
Kim:2017jpx  to be 2931 and 3000 MeV for the $J^{P}=1/2^{+}$ and
$J^{P}=3/2^{+}$ multiplets, respectively. The predictions for the partial
widths333Note that the decay $\Xi_{c}^{3/2}\to\Lambda_{c}+K$ is forbidden by
the isospin symmetry. of exotic $\Xi_{c}^{3/2}$ are given in Table 5. We see
that, indeed, the widths are anomalously small. Interesting is that the width
of the isospin-1/2 $\Xi_{c}$ from $\overline{\bm{15}}$-plet is even smaller
($<1$ MeV), with the dominant decay mode: $\Lambda_{c}+K$.

Decay | $J=1/2$ | $J=3/2$
---|---|---
$\Xi^{3/2}_{c}(\overline{\bm{15}}_{1},J)\rightarrow\Xi_{c}(\overline{\bm{3}}_{0},1/2)+\pi$ | 1.67 | 2.49
$\Xi^{3/2}_{c}(\overline{\bm{15}}_{1},J)\rightarrow\Xi_{c}(\bm{6}_{1},1/2)+\pi$ | 0.045 | 0.079
$\Xi^{3/2}_{c}(\overline{\bm{15}}_{1},J)\rightarrow\Xi_{c}(\bm{6}_{1},3/2)+\pi$ | 0.022 | 0.046
$\Xi^{3/2}_{c}(\overline{\bm{15}}_{1},J)\rightarrow\Sigma_{c}(\bm{6}_{1},1/2)+K$ | $-$ | 0.019
Total | 1.74 | 2.64
Table 5: Predictions in MeV for the partial and total decay widths of
explicitly exotic $\Xi^{3/2}_{c}(\overline{\bm{15}}_{1},J)$.

##  V Conclusions

In the large $N_{c}$ limit both heavy and light baryons are described by the
universal mean field. This allows us to relate the properties of heavy baryons
to the light ones. The goal of the present paper was twofold: first, to test
the mean-filed picture of baryons against the data on strong decays of sextets
of charm and bottom baryons. Second, to make predictions for the decay widths
of exotic antidecapentaplet ($\overline{\bm{15}}$) baryons.

With the only input from the decays of light octet baryons (due to the
universality of the chiral mean field) we have obtained excellent description
of strong decays of both charm and bottom sextets. The agreement is
illustrated in Fig. 2 for charm baryon decays and in Fig. 3 for bottom decays.
We have also shown that going from $N_{c}$ light quarks in light baryons to
$N_{c}-1$ light quarks in heavy baryons the mean field is modified by about
15%. That moderate modification is an agreement with expected size of
$1/N_{c}$ corrections.

Given the excellent agreement of our calculation with the measured widths of
ground-state heavy baryons we made parameter-free predictions for the decays
of exotic $\overline{\bm{15}}$-plet baryons. We have shown that the widths of
$\overline{\bm{15}}$ baryons must be anomalously small, due to essentially the
same mechanism as in the case of narrow anti-decuplet light baryons. In
particular, for $\Omega_{c}(3050)$ and $\Omega_{c}(3119)$, which we
interpreted in Ref. Kim:2017jpx  as belonging to the antidecapentaplet, we
obtained widths of 0.48 MeV and 1.12 MeV correspondingly. Experimentally,
these two (among five) states have the smallest widths Aaij:2017nav  which are
in agreement with our parameter-free calculations, see Tables 3 and 4 and Fig.
2. The parametrical suppression of the pertinent decay constants has been
discussed at the end of Sec. II. We have shown that theoretical arguments
based on large $N_{c}$ and NR limits explain the numerical hierarchy of the
decay couplings (18).

For the complete description of the LHCb $\Omega_{c}$ states, estimates of
strong decays for negative parity baryons are needed. In the $\chi$QSM
negative parity baryons correspond to the configuration with one quark excited
from the Dirac see to the empty valence level (see Ref. Kim:2017jpx ).
Therefore one expects that the decay operator will depend on a new parameter
related to this transition, similarly to the case of mass splittings for these
states. Furthermore, negative parity particles will have $s$-wave and/or
$d$-wave decays, which are not possible if the parity is positive. Finally,
for heavier states, decays to light baryons and heavy mesons are possible.
Such decays require new theoretical treatment within the framework of the
$\chi$QSM since the Goldberger-Treiman relation is not directly applicable in
this case. All these issues require further study therefore we have not
attempted to address them in the present paper.

The results of the present study reinforce our conclusions from Ref.
Kim:2017jpx  that the two narrowest $\Omega^{0}_{c}$ states reported recently
by the LHCb collaboration in Ref. Aaij:2017nav  correspond to the exotic SU(3)
multiplet, namely the antidecapentaplet ($\overline{\bm{15}}$). As seen from
Fig. 1 these states belong to the isospin triplet, rather than the singlet.
Therefore this quantum number assignment can be relatively easily verified
experimentally.

###### Acknowledgements.

H.-Ch. K. is grateful to J.Y. Kim for discussion. The present work was
supported by Basic Science Research Program through the National Research
Foundation of Korea funded by the Ministry of Education, Science and
Technology (Grant No. NRF-2015R1A2A2A04007048 (H.-Ch. K.) and Grant No.
NRF-2016R1C1B1012429 (Gh.-S. Y.)). The work of M.V.P. is supported by Sino-
German Collaborative Research Centre 110. M.V.P. is also grateful to the
Department of Theoretical Physics of Irkutsk State University for hospitality
and support. The work of M.P. has been performed in the framework of COST
Action CA15213 THOR.

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