# Decays of Roper-like singly heavy baryons in a chiral model

Daiki Suenaga<EMAIL_ADDRESS>Research Center for Nuclear Physics,
Osaka University, Ibaraki, 567-0048, Japan  Atsushi Hosaka
<EMAIL_ADDRESS>Research Center for Nuclear Physics, Osaka
University, Ibaraki, 567-0048, Japan  Advanced Science Research Center, Japan
Atomic Energy Agency (JAEA), Tokai 319-1195, Japan

###### Abstract

We present a chiral model describing decays of the Roper-like
$\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$ as well as those of the heavy quark
spin-doublet $\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$, by focusing on chiral representations
of diquarks inside the heavy baryons. Based on our chiral model together with
the heavy quark spin symmetry, we derive a relation satisfied by axial charges
of the heavy baryons that controls magnitude of their one pion (kaon) decays.
Besides, it is shown that the large decay widths of the Roper-like
$\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$, which cannot be implemented by
conventional non-relativistic quark models, are explained by reasonable values
of the axial charges. In addition, we apply the chiral model to bottom baryons
and predict decay widths of the undiscovered Roper-like $\Xi_{b}$. We expect
that our present investigation leads to elucidation of dynamical properties of
the excited heavy baryons from the viewpoint of chiral symmetry.

##  I Introduction

With the help of recent development of hadron experiments at, e.g., KEK, LHC
and SLAC, properties of singly heavy baryons have been unveiled. Since singly
heavy baryons include only one heavy quark ($c$ or $b$ quark) whose mass is
larger than the typical energy scale $\Lambda_{\rm QCD}$ of Quantum
Chromodynamics (QCD), those baryons are useful to understand properties of the
remaining diquark inside the baryons which is colorful Manohar  _et al._
(2000).

Among singly heavy baryons, the Roper-like baryons such as $\Lambda_{c}(2765)$
and $\Xi_{c}(2970)$ for the charm sector are particularly of interest. Those
baryons are heavy quark spin-singlet (HQS-singlet) whose spin and parity are
$J^{P}=1/2^{+}$ Moon  _et al._ (2021),111Although the spin and parity of
$\Lambda_{c}(2765)$ have not been established, they seem to be $J^{P}=1/2^{+}$
Arifi  _et al._ (2020). which are the same as the corresponding ground-state
$\Lambda_{c}$ and $\Xi_{c}$. Their masses are larger than the ground-state
ones by about $500$ MeV Zyla  _et al._ (2020). This mass difference is similar
to the Roper resonance $N(1440)$ in the nucleon sector Roper (1964), and such
a universality of the mass difference seems to persist to the bottom sector
Sirunyan  _et al._ (2020); Aaij  _et al._ (2020). In addition, it is known
that the large decay widths of $\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$ whose
values are tens of MeV are not easily explained by the conventional non-
relativistic quark model Copley  _et al._ (1979); Yoshida  _et al._ (2015)
giving the widths of only a few MeV Nagahiro  _et al._ (2017).222In Ref. Arifi
_et al._ (2021), we showed that a quark model including relativistic
corrections can explain the large decay widths of Roper-like baryons. In
addition, the ${}^{3}P_{0}$ model can reproduce the decay widths as well as
the masses of them satisfactorily Chen  _et al._ (2017).

In this paper, we study decay properties of the heavy baryons, especially
focusing on the large decay widths of the Roper-like baryons, by means of a
chiral model together with the heavy quark spin symmetry Kawakami and Harada
(2018, 2019); Harada  _et al._ (2020); Dmitrašinović and Chen (2020); Kawakami
_et al._ (2020); Suenaga and Hosaka (2021). The chiral model describes the
heavy baryons based on chiral symmetry, which is one of the fundamental
symmetries in QCD, exhibited by the diquark inside the baryons. Hence, the
model is useful to examine interaction properties among the heavy baryons and
light pseudo-scalar mesons, regarded as the Nambu-Goldstone (NG) bosons
associated with the chiral symmetry breaking. In other words, our present
investigation leads to the better understanding of such dynamical natures of
the heavy baryons from the viewpoint of chiral symmetry Kawakami and Harada
(2018, 2019); Kawakami  _et al._ (2020).

In Ref. Suenaga and Hosaka (2021), based on a chiral model we argued that the
Roper-like $\Lambda_{c}(2765)$, $\Xi_{c}(2970)$ and the ground-state
$\Lambda_{c}$, $\Xi_{c}$ are superpositions of a three-quark state $Qqq$ and a
pentaquark states $Qqq\bar{q}q$, and the former was shown to be mostly
$Qqq\bar{q}q$ while the latter $Qqq$. In this paper, we generalize the
discussions in Ref. Suenaga and Hosaka (2021) by focusing on chiral
representations of the diquark. Besides, we extend the model by incorporating
heavy quark spin-doublet (HQS-doublet) baryons such as
$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$ carrying $J^{P}=1/2^{+},3/2^{+}$. Based
on such a model, we examine decays of the Roper-like $\Lambda_{c}(2765)$,
$\Xi_{c}(2970)$ as well as those of the HQS-doublet
$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$.

With respect to the heavy quark spin symmetry, investigation of the bottom
baryons are more useful. For the bottom baryons, only the Roper-like
$\Lambda_{b}$ was discovered and the Roper-like $\Xi_{b}$ is still missing
Zyla  _et al._ (2020). Thus, in this paper we apply our chiral model to the
bottom baryons as well, and present predictions of decay widths of the
undiscovered Roper-like $\Xi_{b}$.

Our present paper is organized as follow. In Sec. II we construct our chiral
Lagrangian to investigate decays of the heavy baryons, indicating important
points presented in Ref. Suenaga and Hosaka (2021). Based on the Lagrangian,
in Sec. III we show analytic expressions of the decay widths. In Sec. IV
numerical results of the decay widths for the charmed baryons are shown.
Besides, in Sec. V we apply the model to bottom baryons. In Sec. VI, we
comment on the important relation derived in our chiral model that axial
charges of the heavy baryons satisfy, and discuss decays of the parity
partners (or the so-called chiral partners) Harada  _et al._ (2020); Kawakami
_et al._ (2020). And in Sec. VII we summarize the present study.

##  II Model

The Roper-like baryons $\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$ decay into the
HQS-doublet baryons $\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$ by emitting a pion (and a kaon). In
this section we construct a Lagrangian describing those decays based on chiral
symmetry and heavy quark spin symmetry.

###  II.1 HQS-singlet baryons

In this subsection, we introduce HQS-singlet baryon fields describing both the
Roper-like $\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$, and the ground-state
$\Lambda_{c}$ and $\Xi_{c}$. For this purpose, to begin with, we focus on the
diquarks inside the baryons.

The diquarks inside the HQS-singlet baryons are Lorentz scalar. Thus, possible
interpolating fields for the diquarks including only two quarks are Liu and
Oka (2012); Harada  _et al._ (2020)

$\displaystyle(d_{R})^{a}_{i}$ $\displaystyle\sim$
$\displaystyle\epsilon_{ijk}\epsilon^{abc}(q_{R}^{T})^{b}_{j}C(q_{R})^{c}_{k}\
,$ $\displaystyle(d_{L})^{a}_{i}$ $\displaystyle\sim$
$\displaystyle\epsilon_{ijk}\epsilon^{abc}(q_{L}^{T})^{b}_{j}C(q_{L})^{c}_{k}\
,$ (1)

where $q_{R}=\frac{1+\gamma_{5}}{2}q$ and $q_{L}=\frac{1-\gamma_{5}}{2}q$ are
right-handed and left-handed quark fields, respectively. The symbol $C$ is the
charge-conjugation operator $C=i\gamma^{2}\gamma^{0}$, and the superscript
$a,b,\cdots$ and subscript $i,j,\cdots$ represent color and flavor indices,
respectively. Hence, the chiral representations of the diquarks in Eq. (1) are

$\displaystyle d_{R}$ $\displaystyle\sim$
$\displaystyle({\bm{1}},\bar{\bm{3}})\ ,\ \ d_{L}\sim(\bar{\bm{3}},{\bm{1}})\
.$ (2)

In order to describe the Roper-like baryons $\Lambda_{c}(2765)$ and
$\Xi_{c}(2970)$ in addition to the ground-state ones $\Lambda_{c}$ and
$\Xi_{c}$ collectively, we need to introduce the other diquarks
$d_{R}^{\prime}$ and $d_{L}^{\prime}$, whose Lorentz structures, and
accordingly chiral representations, are identical to those of $d_{R}$ and
$d_{L}$. Here, we assign the chiral representations of $d^{\prime}_{R}$ and
$d_{L}^{\prime}$ as

$\displaystyle d^{\prime}_{R}$ $\displaystyle\sim$
$\displaystyle(\bar{\bm{3}},{\bm{1}})\ ,\ \
d^{\prime}_{L}\sim({\bm{1}},\bar{\bm{3}})\ ,$ (3)

where the representations are flipped from those of $d_{R}$ and $d_{L}$ in Eq.
(2), following Ref. Suenaga and Hosaka (2021).333In this reference, we
introduced the so-called mirror diquarks
$\displaystyle(d^{\prime}_{R})_{i}^{a}$ $\displaystyle\sim$
$\displaystyle\epsilon_{jkl}\epsilon^{abc}(q_{R}^{T})^{b}_{k}C(q_{R})^{c}_{l}[(\bar{q}_{L})^{d}_{i}(q_{R})^{d}_{j}]\
,$ $\displaystyle(d^{\prime}_{L})_{i}^{a}$ $\displaystyle\sim$
$\displaystyle\epsilon_{jkl}\epsilon^{abc}(q_{L}^{T})^{b}_{k}C(q_{L})^{c}_{l}[(\bar{q}_{R})^{d}_{i}(q_{L})^{d}_{j}]\
,$ (4) which are regarded as tetraquark states for $d_{R}^{\prime}$ and
$d_{L}^{\prime}$. The mirror diquarks give the flipped chiral representations
as in Eq. (3), because of the $\bar{q}_{L}q_{R}$ ($\bar{q}_{R}q_{L}$)
component in Eq. (4).

Interpolating fields of HQS-singlet baryons composed of $d_{R}$, $d_{L}$,
$d_{R}^{\prime}$ and $d_{L}^{\prime}$ are given by attaching a heavy quark $Q$
to the diquarks as

$\displaystyle(B_{R})_{i}\sim Q^{a}(d_{R})_{i}^{a}\ \ $ , $\displaystyle\ \
(B_{L})_{i}\sim Q^{a}(d_{L})_{i}^{a}\ ,$
$\displaystyle(B_{R}^{\prime})_{i}\sim Q^{a}(d_{R}^{\prime})_{i}^{a}\ \ $ ,
$\displaystyle\ \ (B_{L}^{\prime})_{i}\sim Q^{a}(d_{L}^{\prime})_{i}^{a}\ .$
(5)

From the baryon fields in Eq. (5), we can construct a chiral invariant
Lagrangian with Eqs. (2) and (3). Here, for later use we define anti-symmetric
matrices of the baryons in terms of the flavor space as Kawakami and Harada
(2019)

$\displaystyle({\cal B}_{R})_{ij}\equiv\epsilon_{ijk}(B_{R})_{k}\ \ $ ,
$\displaystyle\ \ ({\cal B}_{L})_{ij}\equiv\epsilon_{ijk}(B_{L})_{k}\ ,$
$\displaystyle({\cal
B}_{R}^{\prime})_{ij}\equiv\epsilon_{ijk}(B_{R}^{\prime})_{k}\ \ $ ,
$\displaystyle\ \ ({\cal
B}_{L}^{\prime})_{ij}\equiv\epsilon_{ijk}(B_{L}^{\prime})_{k}\ .$ (6)

The anti-symmetric structures in Eq. (6) manifestly reflect the anti-symmetric
chiral representations in Eqs. (2) and (3). Hence, the $SU(3)_{L}\times
SU(3)_{R}$ chiral transformation laws of the baryons defined in Eq. (6) become
simply

$\displaystyle{\cal B}_{R}\overset{\rm ch.}{\to}g_{R}{\cal B}_{R}g_{R}^{T}\ \
$ , $\displaystyle\ \ {\cal B}_{L}\overset{\rm ch.}{\to}g_{L}{\cal
B}_{L}g_{L}^{T}\ ,$ $\displaystyle{\cal B}_{R}^{\prime}\overset{\rm
ch.}{\to}g_{L}{\cal B}_{R}^{\prime}g_{L}^{T}\ \ $ , $\displaystyle\ \ {\cal
B}_{L}^{\prime}\overset{\rm ch.}{\to}g_{R}{\cal B}_{L}^{\prime}g_{R}^{T}\ ,$
(7)

without changing the representations in Eqs. (2) and (3), where $g_{R(L)}\in
SU(3)_{R(L)}$. Besides, the parity transformation laws read

$\displaystyle{\cal B}_{R}\overset{\rm P}{\to}-\gamma^{0}{\cal B}_{L}\ \ $ ,
$\displaystyle\ \ {\cal B}_{L}\overset{\rm P}{\to}-\gamma^{0}{\cal B}_{R}\ ,$
$\displaystyle{\cal B}_{R}^{\prime}\overset{\rm P}{\to}-\gamma^{0}{\cal
B}_{L}^{\prime}\ \ $ , $\displaystyle\ \ {\cal B}_{L}^{\prime}\overset{\rm
P}{\to}-\gamma^{0}{\cal B}_{R}^{\prime}\ .$ (8)

The basis of ${\cal B}^{(\prime)}_{R(L)}$ is useful to construct a chiral
invariant Lagrangian, since the chiral representations are manifestly
exhibited. They are, however, not parity eigenstates as seen from Eq. (8). The
parity eigenstates are given by

$\displaystyle{\cal B}^{(\prime)}_{+}=\frac{1}{\sqrt{2}}({\cal
B}^{(\prime)}_{R}-{\cal B}^{(\prime)}_{L})\ ,\ \ {\cal
B}^{(\prime)}_{-}=\frac{1}{\sqrt{2}}({\cal B}^{(\prime)}_{R}+{\cal
B}^{(\prime)}_{L})\ ,$ (9)

where the subscript “$+$” or “$-$” corresponds to the parity eigenvalue. Thus,
defining parity eigenstates $B_{\pm}^{(\prime)}$ from $B_{R(L)}^{(\prime)}$ in
Eq. (5) in a similar manner to Eq. (9), the matrix ${\cal B}^{(\prime)}_{\pm}$
is explicitly expressed as

$\displaystyle{\cal B}_{\pm}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&B_{\pm,i=3}&-B_{\pm,i=2}\\\
-B_{\pm,i=3}&0&B_{\pm,i=1}\\\ B_{\pm,i=2}&-B_{\pm,i=1}&0\\\
\end{array}\right)\ ,$ (13) $\displaystyle{\cal B}^{\prime}_{\pm}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&B^{\prime}_{\pm,i=3}&-B^{\prime}_{\pm,i=2}\\\
-B^{\prime}_{\pm,i=3}&0&B^{\prime}_{\pm,i=1}\\\
B^{\prime}_{\pm,i=2}&-B^{\prime}_{\pm,i=1}&0\\\ \end{array}\right)\ .$ (17)

In Ref. Suenaga and Hosaka (2021), based on the spontaneous breakdown of
chiral symmetry, it was shown that in general the mass eigenstates are given
by superpositions of ${B}_{\pm}$ and ${B}^{\prime}_{\pm}$. That is, the mass
eigenstates read

$\displaystyle\left(\begin{array}[]{c}B_{\pm,i}^{L}\\\ B_{\pm,i}^{H}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{B_{\pm}^{i}}&\sin\theta_{B_{\pm}^{i}}\\\
-\sin\theta_{B_{\pm}^{i}}&\cos\theta_{B_{\pm}^{i}}\\\
\end{array}\right)\left(\begin{array}[]{c}B_{\pm,i}\\\ B_{\pm,i}^{\prime}\\\
\end{array}\right)\ ,$ (24)

where the superscripts $L$ and $H$ refer to lower and higher masses. In later
analysis, we will only focus on the positive-parity $B_{+}^{L}$ and
$B_{+}^{H}$, where $B_{+}^{L}$ corresponds to the ground-state $\Lambda_{c}$
and $\Xi_{c}$ while $B_{+}^{H}$ the Roper-like $\Lambda_{c}(2765)$ and
$\Xi_{c}(2970)$. Explicitly, $B_{+}^{L}$ and $B_{+}^{H}$ correspond to the
experimentally observed baryons as

$\displaystyle B^{L}_{+,i=1}=\Xi_{c}^{0}\ \ ,\ \
B^{H}_{+,i=1}=\Xi_{c}(2970)^{0}\ ,$ $\displaystyle B^{L}_{+,i=2},=\Xi_{c}^{+}\
\ ,\ \ B^{H}_{+,i=2}=\Xi_{c}(2970)^{+}\ ,$ $\displaystyle
B^{L}_{+,i=3}=\Lambda_{c}^{+}\ \ ,\ \ B^{H}_{+,i=3}=\Lambda_{c}(2765)^{+}\ .$
(25)

###  II.2 HQS-doublet baryons

In this subsection, we introduce HQS-doublet baryon fields describing
$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$. The diquark inside the baryons must
carry spin one since those baryons belong to the spin-doublet with the total
spin $J=1/2,3/2$. The simplest diquark field constructed by two quarks is
Lorentz vector, which is uniquely given by Liu and Oka (2012); Kawakami and
Harada (2019); Harada  _et al._ (2020)

$\displaystyle
d_{(i,j)}^{a,\mu}\sim\epsilon^{abc}(q_{L}^{T})^{b}_{j}C\gamma^{\mu}(q_{R})^{c}_{i}\
.$ (26)

Here, it should be noted that the Lorentz vector diquarks including two
$q_{L}$’s ($q_{R}$’s) do not exist due to the Pauli principle.

Similarly to Eq. (5), interpolating fields of HQS-doublet baryons $S^{\mu}$
are given by attaching a heavy quark $Q$ to the diquark in Eq. (26), i.e.,

$\displaystyle S^{\mu}_{ij}\sim Q^{a}d_{(i,j)}^{a,\mu}\ .$ (27)

Hence, the $SU(3)_{L}\times SU(3)_{R}$ chiral and parity transformation laws
of the baryon $S^{\mu}$ read

$\displaystyle S^{\mu}\overset{\rm ch.}{\to}g_{R}S^{\mu}g_{L}^{T}\ \ \ ,\ \ \
S^{\mu}\overset{\rm P}{\to}-\gamma^{0}S^{T}_{\mu}\ ,$ (28)

respectively, from the diquark (26). When we separate $S^{\mu}$ into two
matrices which are flavor symmetric and anti-symmetric, i.e., the flavor
sextet $S^{\mu}_{6}$ and anti-triplet $S_{\bar{3}}^{\mu}$ parts as

$\displaystyle S^{\mu}=S_{6}^{\mu}+S_{\bar{3}}^{\mu}\ ,$ (29)

the parity transformation in Eq. (28) yields

$\displaystyle S_{6}^{\mu}\overset{\rm P}{\to}-\gamma^{0}S_{6\mu}\ \ \ ,\ \ \
S_{\bar{3}}^{\mu}\overset{\rm P}{\to}\gamma^{0}S_{\bar{3}\mu}\ .$ (30)

In other words, $S^{\mu}_{6}$ contains positive-parity baryons while
$S^{\mu}_{\bar{3}}$ negative-party ones.444Our definition of the diquark (26)
reads $d^{a\mu}\overset{\rm P}{\to}-(d^{a}_{\mu})^{T}$ under the proper parity
transformation having an extra minus sign. Thus, we can assign the HQS-doublet
baryons as

$\displaystyle
S_{6}^{\mu}=\left(\begin{array}[]{ccc}\Sigma_{c}^{I_{z}=1}&\frac{1}{\sqrt{2}}\Sigma_{c}^{I=0}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime
I=1/2}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{I=0}&\Sigma_{c}^{I=-1}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime
I=-1/2}\\\ \frac{1}{\sqrt{2}}\Xi_{c}^{\prime
I=1/2}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime I=-1/2}&\Omega_{c}\\\
\end{array}\right)^{\mu}\ ,$ (34) (35)

and

$\displaystyle
S_{\bar{3}}^{\mu}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&\Lambda_{c1}&\Xi_{c1}^{I=1/2}\\\
-\Lambda_{c1}&0&\Xi_{c1}^{I=-1/2}\\\ -\Xi_{c1}^{I=1/2}&-\Xi_{c1}^{I=-1/2}&0\\\
\end{array}\right)^{\mu}\ .$ (39) (40)

For later use, following the useful field definition proposed in Ref. Falk
(1992), we decompose $S_{6(\bar{3})}^{\mu}$ into $J=3/2$ part
$S_{6(\bar{3})}^{*\mu}$ and $J=1/2$ part $S_{6(\bar{3})}$ as

$\displaystyle
S_{6(\bar{3})}^{\mu}=S_{6(\bar{3})}^{*\mu}+\frac{1}{\sqrt{3}}(\gamma^{\mu}+v^{\mu})\gamma_{5}S_{6(\bar{3})}\
,$ (41)

with $v^{\mu}$ being a velocity of the baryon.

In later analysis, we will leave only the flavor sextet part $S^{\mu}_{6}$
where $\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$ are contained. In the decomposition in
Eq. (41), $\Sigma_{c}(2455)$ and $\Xi_{c}^{\prime}$ belong to $S_{6}$ while
$\Sigma_{c}(2520)$ and $\Xi_{c}(2645)$ belong to $S_{6}^{*\mu}$, and their
assignment in the flavor space is understood from Eq. (35).

###  II.3 Interaction Lagrangian

Here, we construct a Lagrangian describing interactions among the HQS-singlet
baryons ${\cal B}^{(\prime)}_{R(L)}$ and the HQS-doublet ones $S^{\mu}$.

From the transformation laws in Eqs. (7), (8) and (28) [or Eq. (30)], an
interaction Lagrangian which is invariant under the $SU(3)_{L}\times
SU(3)_{R}$ chiral, $SU(2)_{h}$ heavy quark spin, and parity transformations
reads

$\displaystyle{\cal L}_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}g_{A}}{f_{0}}{\rm Tr}\Big{[}\bar{\cal
B}_{R}\partial_{\mu}\Sigma^{\dagger}S^{T\mu}+\bar{\cal
B}_{L}\partial_{\mu}\Sigma S^{\mu}+{\rm h.c.}\Big{]}$ $\displaystyle+$
$\displaystyle\frac{\sqrt{2}g_{A}^{\prime}}{f_{0}}{\rm Tr}\Big{[}\bar{\cal
B}_{L}^{\prime}\partial_{\mu}\Sigma^{\dagger}S^{T\mu}+\bar{\cal
B}_{R}^{\prime}\partial_{\mu}\Sigma S^{\mu}+{\rm h.c,}\Big{]}\ .$ (42)

In this Lagrangian,

$\displaystyle\Sigma=S+iP$ (43)

is a light meson matrix containing a scalar nonet $S$ and a pseudo-scalar
nonet $P$, and we have left interaction terms having one
$\partial_{\mu}\Sigma$ or $\partial_{\mu}\Sigma^{\dagger}$. Besides, $f_{P}$
is an averaged pseudo-scalar decay constant:
$f_{0}=\frac{f_{\pi}+f_{K}}{2}=101\,{\rm MeV}$. In this way $g_{A}$ and
$g_{A}^{\prime}$ become dimensionless. Making use of Eq. (9), the Lagrangian
(42) turns into

$\displaystyle{\cal L}_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{2g_{A}}{f_{0}}{\rm Tr}\Big{[}-i\bar{\cal
B}_{+}\partial_{\mu}PS_{6}^{\mu}-\bar{\cal
B}_{+}\partial_{\mu}SS_{\bar{3}}^{\mu}+i\bar{\cal
B}_{-}\partial_{\mu}PS_{\bar{3}}^{\mu}$ $\displaystyle+$
$\displaystyle\bar{\cal B}_{-}\partial_{\mu}SS_{6}^{\mu}+{\rm
h.c.}\Big{]}+\frac{2g_{A}^{\prime}}{f_{0}}{\rm Tr}\Big{[}-i\bar{\cal
B}_{+}^{\prime}\partial_{\mu}PS_{6}^{\mu}$ $\displaystyle-$
$\displaystyle\bar{\cal
B}_{+}^{\prime}\partial_{\mu}SS_{\bar{3}}^{\mu}+i\bar{\cal
B}_{-}^{\prime}\partial_{\mu}PS_{\bar{3}}^{\mu}+\bar{\cal
B}_{-}^{\prime}\partial_{\mu}SS_{6}^{\mu}+{\rm h.c.}\Big{]}\ ,$

In this Lagrangian, since $S_{6}^{\mu}$ and $S_{\bar{3}}^{\mu}$ carry the
positive- and negative-parity HQS-doublets, respectively, the parity
invariance of the Lagrangian derives a selection rule for their couplings; for
instance, the coupling between $S_{6}^{\mu}$ and ${\cal B}_{+}^{(\prime)}$ is
mediated by a pseudo-scalar meson $P$, while that between $S_{\bar{3}}^{\mu}$
and ${\cal B}_{+}^{(\prime)}$ is by a scalar meson $S$. Moreover, chiral
symmetry demands that the couplings for parity partners Harada  _et al._
(2020); Kawakami  _et al._ (2020) of the heavy baryons should be commonly
controlled by $g_{A}$ and $g_{A}^{\prime}$. In fact, the couplings for ${\rm
Tr}[\bar{S}_{6}^{\mu}\partial_{\mu}P{\cal B}_{+}]$ and ${\rm
Tr}[\bar{S}_{\bar{3}}^{\mu}\partial_{\mu}P{\cal B}_{-}]$ are identical.
Discussions related to this structure will be given in Sec. VI.2.

In what follows, we focus on only the positive-parity heavy baryons in our
chiral model (LABEL:LIntParity). As mentioned in Sec. II.1 and Sec. II.2,
those baryons are provided by ${\cal B}_{+}^{(\prime)}$ and $S_{6}^{\mu}$.
Hence, we can neglect ${\cal B}_{-}^{(\prime)}$ and $S_{\bar{3}}$ in Eq.
(LABEL:LIntParity), i.e., the Lagrangian is reduced to

$\displaystyle{\cal L}_{\rm int}$ $\displaystyle=$ $\displaystyle
i\frac{2g_{A}}{f_{0}}{\rm Tr}\Big{[}\bar{S}_{6}^{\mu}\partial_{\mu}P{\cal
B}_{+}-\bar{\cal B}_{+}\partial_{\mu}PS_{6}^{\mu}\Big{]}$ (45)
$\displaystyle+$ $\displaystyle i\frac{2g_{A}^{\prime}}{f_{0}}{\rm
Tr}\Big{[}\bar{S}_{6}^{\mu}\partial_{\mu}P{\cal B}^{\prime}_{+}-\bar{\cal
B}^{\prime}_{+}\partial_{\mu}PS_{6}^{\mu}\Big{]}\ ,$

where the pseudo-scalar nonet regarded as the NG bosons associated with the
chiral symmetry breaking is explicitly given by

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}\frac{\eta_{N}+\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\eta_{N}-\pi^{0}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K}^{0}&\eta_{S}\\\ \end{array}\right)\ .$ (49)

Before closing this section, we mention axial charges of the heavy baryons.
Couplings among the heavy baryons and an axial gauge field ${\cal A}_{\mu}$
can be read off by replacing the derivative $\partial_{\mu}\Sigma$ to the
following covariant derivative, in the Lagrangian (42):

$\displaystyle D_{\mu}\Sigma=\partial_{\mu}\Sigma+\frac{i}{2}\\{{\cal
A}_{\mu},\Sigma\\}\ .$ (50)

That is, in terms of $S_{6}^{\mu}$ and $B_{+}^{(\prime)}$ we can read the
couplings as

$\displaystyle{\cal L}_{\rm axial}$ $\displaystyle=$ $\displaystyle ig_{A}{\rm
Tr}\Big{[}\bar{S}_{6}^{\mu}{\cal A}_{\mu}{\cal B}_{+}-\bar{\cal B}_{+}{\cal
A}_{\mu}S_{6}^{\mu}\Big{]}$ (51) $\displaystyle+$ $\displaystyle
ig_{A}^{\prime}{\rm Tr}\Big{[}\bar{S}_{6}^{\mu}{\cal A}_{\mu}{\cal
B}^{\prime}_{+}-\bar{\cal B}^{\prime}_{+}{\cal A}_{\mu}S_{6}^{\mu}\Big{]}\ .$

In obtaining Eq. (51), we have assumed that the vacuum expectation value (VEV)
of $\Sigma$ is diagonal in the flavor space as
$\langle\Sigma\rangle=\frac{f_{0}}{2}{\bm{1}}$. That is, we assume the
$SU(3)_{L+R}$ symmetry is satisfied in our present investigation, and indeed
this assumption seems to be reasonable as will be seen later. Equation (51)
implies that $g_{A}$ and $g_{A}^{\prime}$ are understood to be the axial
charges defined in this way.

##  III Analytic evaluation of decay widths

Our chiral model (45) describes the ground-state $\Lambda_{c}$ and $\Xi_{c}$,
the Roper-like $\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$, and the HQS-doublet
$\\{\Sigma_{c}(2450),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$, together with the NG bosons in a
simple form. Therefore, in Sec. IV we will study decay widths of both the
Roper-like and the HQS-doublet baryons, in order to fully examine dynamical
properties of the baryons interacting with the NG bosons from chiral symmetry.
Before presenting the numerical results, in this section we clarify our
procedure by showing analytic expressions of decay widths of
$\Lambda_{c}(2765)\to\Sigma_{c}(2455)\pi$,
$\Lambda_{c}(2765)\to\Sigma_{c}(2520)\pi$, and
$\Sigma_{c}(2455)\to\Lambda_{c}\pi$, $\Sigma_{c}(2520)\to\Lambda_{c}\pi$ in
detail.

First, we show how to compute decays of the Roper-like $\Lambda_{c}(2765)$.
From the Lagrangian (45), couplings describing interactions of
$\Lambda_{c}(2765)$-$\Sigma_{c}(2455)$-$\pi$ and
$\Lambda_{c}(2765)$-$\Sigma_{c}(2520)$-$\pi$ are read off as

$\displaystyle{\cal L}_{\rm int}$ $\displaystyle\to$
$\displaystyle\frac{i\sqrt{2}}{f_{0}}g_{A,3}^{+H}\Big{[}\big{(}\bar{\Sigma}_{Q}^{I=1}\big{)}^{\mu}\partial_{\mu}\pi^{+}{B}_{+,3}^{H}$
$\displaystyle-$
$\displaystyle\big{(}\bar{\Sigma}_{Q}^{I=0}\big{)}^{\mu}\partial_{\mu}\pi^{0}{B}_{+,3}^{H}-\big{(}\bar{\Sigma}_{Q}^{I=-1}\big{)}^{\mu}\partial_{\mu}\pi^{-}{B}_{+,3}^{H}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{i\sqrt{2}}{f_{0}}g_{A,3}^{+L}\Big{[}\bar{B}_{+,3}^{L}\partial_{\mu}\pi^{-}(\Sigma_{Q}^{I=1})^{\mu}$
$\displaystyle-$
$\displaystyle\bar{B}_{+,3}^{L}\partial_{\mu}\pi^{0}(\Sigma_{Q}^{I=0})^{\mu}-\bar{B}_{+,3}^{L}\partial_{\mu}\pi^{+}(\Sigma_{Q}^{I=-1})^{\mu}\Big{]}\
,$

where we have used Eq. (24) and have defined

$\displaystyle g_{A,3}^{+H}$ $\displaystyle\equiv$ $\displaystyle
g_{A}\sin\theta_{B_{+}^{3}}-g_{A}^{\prime}\cos\theta_{B_{+}^{3}}\ ,$
$\displaystyle g_{A,3}^{+L}$ $\displaystyle\equiv$ $\displaystyle
g_{A}\cos\theta_{B_{+}^{3}}+g_{A}^{\prime}\sin\theta_{B_{+}^{3}}\ .$ (53)

It should be noted that, in our notation, the Roper-like $\Lambda_{c}(2765)$
and the ground-state $\Lambda_{c}$ are denoted by $B_{+,3}^{H}$ and
$B_{+,3}^{L}$, respectively, from Eq. (25), while the HQS-doublet
$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ is provided by
$\big{(}{\Sigma}_{Q}^{I=\pm 1,0}\big{)}^{\mu}$. Hence, $g_{A,3}^{+H}$ and
$g_{A,3}^{+L}$ in Eq. (53) correspond to the axial charges related to
$\Lambda_{c}(2765)$-$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ sector and
$\Lambda_{c}$-$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ sector, respectively.

From Eq. (LABEL:ModelReduced) together with the spin decomposition in Eq.
(41), transition amplitudes of Fig. 1 are evaluated as

$\displaystyle\left|{\cal M}_{1(a)}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{2(g_{A,3}^{+H})^{2}}{3f_{0}^{2}}\left|\bar{U}_{f}(P_{f})\left(\gamma^{\mu}-\frac{P_{f}^{\mu}}{M_{f}}\right)\gamma_{5}p^{\pi}_{\mu}U_{i}(P_{i})\right|^{2}$
$\displaystyle\left|{\cal M}_{1(b)}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{2(g_{A,3}^{+H})^{2}}{f_{0}^{2}}\left|\bar{U}_{f}^{\mu}(P_{f})p^{\pi}_{\mu}U_{i}(P_{i})\right|^{2}\
,$ (54)

where ${\cal M}_{1(a)}$ and ${\cal M}_{1(b)}$ correspond to Fig. 1 (a) and to
Fig. 1 (b), respectively. In Eq. (54), $P^{\mu}_{i}$ and $P^{\mu}_{f}$ are
momenta carried by the heavy baryons of the initial and final states,
respectively, and $p^{\mu}_{\pi}$ (or $p_{\mu}^{\pi}$) is that by the emitted
pion as indicated in the figure. The masses of initial- and final-state heavy
baryons are denoted by $M_{i}$ and $M_{f}$, and the velocity $v^{\mu}$ has
been replaced by $P^{\mu}_{i}/M_{i}$ or $P^{\mu}_{f}/M_{f}$. Besides,
$U_{i}(P_{i})$ [$U_{f}(P_{f})$] is the Dirac spinors for the spin-$1/2$ heavy
baryon, while $U_{f}^{\mu}(P_{f})$ is the Rarita-Schwinger spinor for the
spin-$3/2$ heavy baryon. That is, those spinors satisfy the following
relations Falk (1992):

$\displaystyle
U_{i}(P_{i})\bar{U}_{i}(P_{i})={\ooalign{\hfil/\hfil\crcr$P$}}_{i}+M_{i}\ ,$
(57) $\displaystyle
U_{f}(P_{f})\bar{U}_{f}(P_{f})={\ooalign{\hfil/\hfil\crcr$P$}}_{f}+M_{f}\ ,$
(60)

and

$\displaystyle U_{f}^{\mu}(P_{f})\bar{U}_{f}^{\nu}(P_{f})$ $\displaystyle=$
$\displaystyle-({\ooalign{\hfil/\hfil\crcr$P$}}_{f}+M_{f})\Bigg{(}g^{\mu\nu}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}$
$\displaystyle-\frac{2}{3M_{f}^{2}}P_{f}^{\mu}P_{f}^{\nu}+\frac{P_{f}^{\mu}\gamma^{\nu}-\gamma^{\mu}P_{f}^{\nu}}{3M_{f}}\Bigg{)}\
.$

Figure 1: Feynman diagrams for (a) $\Lambda_{c}(2765)\to\Sigma_{c}(2455)\pi$
and for (b) $\Lambda_{c}(2765)\to\Sigma_{c}(2520)\pi$.

Thus, from Eq. (54) the decay widths are evaluated as

$\displaystyle\Gamma[\Lambda_{c}(2765)\to\Sigma_{c}(2455)\pi]=\frac{(g_{A,3}^{+H})^{2}}{6\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi)\ ,$
$\displaystyle\Gamma[\Lambda_{c}(2765)\to\Sigma_{c}(2520)\pi]=\frac{(g_{A,3}^{+H})^{2}}{3\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi)\ ,$ (65)

for all isospin channels, where we have defined

$\displaystyle F_{1}(M_{i},E_{f};\phi)$ $\displaystyle\equiv$
$\displaystyle\frac{(E_{f}+M_{f})|{\bm{p}}_{\phi}|}{M_{i}}$ (66)
$\displaystyle\times$
$\displaystyle\Bigg{[}\frac{(E_{f}E_{\phi}+|{\bm{p}}_{\phi}|^{2})^{2}}{M_{f}^{2}}-m_{\phi}^{2}\Bigg{]}\
.$

In this equation, $E_{f}=\sqrt{|{\bm{P}}_{f}|^{2}+M_{f}^{2}}$ and
$E_{\phi}=\sqrt{|{\bm{p}}_{\phi}|^{2}+m_{\phi}^{2}}$ are dispersion relations
of the final-state heavy baryon and of $\phi$, respectively, with $\phi$
representing the emitted NG boson ($m_{\phi}$ is its mass), in the rest frame
of the decaying heavy baryon. The momentum conservation leads to
${\bm{p}}_{\phi}=-{\bm{P}}_{f}$ and

$\displaystyle|{\bm{p}}_{\phi}|=\frac{\sqrt{\big{[}M_{i}^{2}-(M_{f}+m_{\phi})^{2}\big{]}\big{[}M_{i}^{2}-(M_{f}-m_{\phi})^{2}\big{]}}}{2M_{i}}\
.$

We note that a spin average of the initial-state heavy baryon is taken in
obtaining the formulae in Eq. (65). Besides, the coefficients of all isospin
channels are identical since $\Lambda_{c}(2765)$ carries isospin zero which is
trivial; the square of Clebsch-Gordan coefficients for all the channels are
identical. The formulae show that the analytic form of the decay width of
$\Lambda_{c}(2765)\to\Sigma_{c}(2520)\pi$ is exactly twice that of
$\Lambda_{c}(2765)\to\Sigma_{c}(2455)\pi$, which can be understood by the
difference of spin degeneracies of the final-state heavy baryon. Such a
symmetric form has been provided by the $SU(2)_{h}$ heavy quark spin symmetry
in our present model.

Figure 2: Feynman diagrams for (a) $\Sigma_{c}(2455)\to\Lambda_{c}\pi$ and for
(b) $\Sigma_{c}(2520)\to\Lambda_{c}\pi$.

Next, from Eq. (LABEL:ModelReduced) the amplitudes for decays of the HQS-
doublet $\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ are obtained as

$\displaystyle\left|{\cal M}_{2(a)}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{2(g_{A,3}^{+L})^{2}}{3f_{0}^{2}}\left|\bar{U}_{f}(P_{f})p^{\pi}_{\mu}\left(\gamma^{\mu}+\frac{P_{i}^{\mu}}{M_{i}}\right)\gamma_{5}U_{i}(P_{i})\right|^{2}\
,$ $\displaystyle\left|{\cal M}_{2(b)}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{2(g_{A,3}^{+L})^{2}}{f_{0}^{2}}\left|\bar{U}_{f}(P_{f})p^{\pi}_{\mu}U_{i}^{\mu}(P_{i})\right|^{2}\
,$ (68)

where ${\cal M}_{2(a)}$ and ${\cal M}_{2(b)}$ correspond to Fig. 2 (a) and
Fig. 2 (b), respectively. The notation should be understood similarly to Eq.
(54). Thus, from Eq. (68) the analytic forms of decay widths of
$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ read

$\displaystyle\Gamma[\Sigma_{c}(2455)\to\Lambda_{c}\pi]=\frac{(g_{A,3}^{+L})^{2}}{6\pi
f_{0}^{2}}F_{2}(M_{i},E_{f};\pi^{-})\ ,$
$\displaystyle\Gamma[\Sigma_{c}(2520)\to\Lambda_{c}\pi]=\frac{(g_{A,3}^{+L})^{2}}{6\pi
f_{0}^{2}}F_{2}(M_{i},E_{f};\pi^{-})\ ,$ (69)

for all isospin channels, with

$\displaystyle
F_{2}(M_{i},E_{f};\phi)=\frac{(E_{f}+M_{f})|{\bm{p}}_{\phi}|^{3}}{M_{i}}\ .$
(70)

Equation (69) shows that the analytic formulae of all the channels are
identical because $\Lambda_{c}$ carries trivial isospin as in Eq. (65). We
note that, unlike Eq. (65), the factor for decays of $\Sigma_{c}(2455)$ and
$\Sigma_{c}(2520)$ coincide due to the $SU(2)_{h}$ heavy quark spin symmetry,
since their spin averages have been taken Cho (1994).

The analytic formulae of decay widths of the remaining $\Xi_{c}$ baryons are
obtained in a similar way, by assuming the isospin symmetry. The resultant
formulae are summarized in Appendix. A

##  IV Results

In this section, we investigate decay widths of the Roper-like
$\Lambda_{c}(2765)$, $\Xi_{c}(2970)$ as well as those of the HQS-doublet
$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and $\Xi_{c}(2645)$ based on the
analytic expressions given in Sec. III and in Appendix. A. We note that
$\Xi_{c}^{\prime}$ does not decay by the strong interactions since no
thresholds open.

Baryon | $J^{P}$ | Mass | Total width
---|---|---|---
$\Lambda_{c}^{+}$ | $1/2^{+}$ | $2286.46$ | No strong decay
$\Xi_{c}^{+}$ | $1/2^{+}$ | $2467.95$ | No strong decay
$\Xi_{c}^{0}$ | $1/2^{+}$ | $2470.99$ | No strong decay
$\Sigma_{c}(2455)^{++}$ | $1/2^{+}$ | $2453.97$ | $1.89$
$\Sigma_{c}(2455)^{+}$ | $1/2^{+}$ | $2452.9$ | $<4.6$
$\Sigma_{c}(2455)^{0}$ | $1/2^{+}$ | $2453.75$ | $1.83$
$\Sigma_{c}(2520)^{++}$ | $3/2^{+}$ | $2518.41$ | $14.78$
$\Sigma_{c}(2520)^{+}$ | $3/2^{+}$ | $2517.5$ | $<17$
$\Sigma_{c}(2520)^{0}$ | $3/2^{+}$ | $2518.48$ | $15.3$
$\Xi_{c}^{\prime+}$ | $1/2^{+}$ | $2578.2$ | No strong decay
$\Xi_{c}^{\prime 0}$ | $1/2^{+}$ | $2578.7$ | No strong decay
$\Xi_{c}(2645)^{+}$ | $3/2^{+}$ | $2645.1$ | $2.14$
$\Xi_{c}(2645)^{0}$ | $3/2^{+}$ | $2646.16$ | $2.35$
$\Lambda_{c}(2765)^{+}$ | $1/2^{+}$ | $2766.6$ | 50
$\Xi_{c}(2970)^{+}$ | $1/2^{+}$ | $2967.1$ | $20.9$
$\Xi_{c}(2970)^{0}$ | $1/2^{+}$ | $2965.9$ | $?$
Table 1: Experimental values of masses and decay widths of the heavy baryons
in units of MeV, indicated in the Particle Data Group (PDG) Zyla  _et al._
(2020). In this table only the central values are shown.

Before showing the main results, we present properties of the axial charges.
Axial charges for negative-parity HQS-singlet baryons $g_{A,i}^{-H}$ and
$g_{A,i}^{-L}$ are defined in a similar way to $g_{A,i}^{+H}$ and
$g_{A,i}^{+L}$ from the Lagrangian (LABEL:LIntParity) with the mixing formula
(24):

$\displaystyle g_{A,i}^{-H}$ $\displaystyle\equiv$ $\displaystyle
g_{A}\sin\theta_{B_{-}^{i}}-g_{A}^{\prime}\cos\theta_{B_{-}^{i}}\ ,$
$\displaystyle g_{A,i}^{-L}$ $\displaystyle\equiv$ $\displaystyle
g_{A}\cos\theta_{B_{-}^{i}}+g_{A}^{\prime}\sin\theta_{B_{-}^{i}}\ .$ (71)

Then, from Eq. (53) [and Eq. (86)] we can confirm that the following relation
holds:

$\displaystyle(g_{A,i}^{+H})^{2}+(g_{A,i}^{+L})^{2}=(g_{A,i}^{-H})^{2}+(g_{A,i}^{-L})^{2}=R^{2}\
,$ (72)

where $R\equiv\sqrt{g_{A}^{2}+g_{A}^{\prime 2}}$ represents the magnitude of
axial charges, which is independent of the flavor index $i$ due to the
$SU(3)_{L+R}$ flavor symmetry. Discussions on Eq. (72) will be given in Sec.
VI.

Focussing on the positive-parity parts, the relation (72) implies that once
$g_{A,i}^{+L}$ is determined, the other $g_{A,i}^{+H}$ can be fixed by the
“radius $R$” solely regardless of the mixing angles $\theta_{B_{+}^{i}}$.
Hence, first we fix the parameter $g_{A,i}^{+L}$ by fitting experimentally
observed decays of $\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\Xi_{c}(2645)$ listed in Table 1. And next, with the help of Eq. (72) we
examine decay widths of the Roper-like $\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$
by varying $R$.

Owing to the $SU(2)_{h}$ heavy quark spin symmetry, the axial charges
contributing to decays of the HQS-doublet baryons are common among the spin
partners. Based on this fact, using the experimental values in Table 1, we
evaluate the axial charges $g_{A,i}^{+L}$ by the following averages:

$\displaystyle g_{A,3}^{+L}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}g_{A,3}^{+L}\Big{|}_{\Sigma_{c}(2455)^{++}}+g_{A,3}^{+L}\Big{|}_{\Sigma_{c}(2455)^{0}}$
(73)
$\displaystyle+g_{A,3}^{+L}\Big{|}_{\Sigma_{c}(2520)^{++}}+g_{A,3}^{+L}\Big{|}_{\Sigma_{c}(2520)^{0}}\Big{)}$
$\displaystyle=$ $\displaystyle 0.512\ ,$

and

$\displaystyle g_{A,1}^{+L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{(}g_{A,1}^{+L}\Big{|}_{\Xi_{c}(2645)^{+}}+g_{A,1}^{+L}\Big{|}_{\Xi_{c}(2645)^{0}}\Bigg{)}$
(74) $\displaystyle=$ $\displaystyle 0.511\ ,$

where $g_{A,i}^{+L}>0$ has been assumed.555In the non-relativistic quark
model, those axial charges are expected to be about $1$. Hence, in order to
reproduce the axial charges of about $0.5$ as in Eq. (73) [Eq. (74)] in the
quark model, we need to include a phenomenological suppression parameter by
hand Yan  _et al._ (1992). We also note that such a suppression is
systematically realized by including relativistic corrections to the non-
relativistic quark model Arifi  _et al._ (2021).  In Eq. (73), we have not
included the widths of $\Sigma_{c}(2455)^{+}$ and $\Sigma_{c}(2520)^{+}$ to
estimate $g_{A,3}^{+L}$ since only the upper limit is known as indicated in
Table 1. Also, in Eq. (74) only decays of $\Xi_{c}(2645)$ are taken into
account since $\Xi_{c}^{\prime}$ does not decay, and the isospin symmetry has
been taken: $g_{A,1}^{+L}=g_{A,2}^{+L}$. Interestingly, Eqs (73) and (74) show
that the $SU(3)_{L+R}$ flavor symmetry is realized as a good approximation for
the axial charges $g_{A,3}^{+L}$ and $g_{A,1}^{+L}$. Therefore, in what
follows, we assume the $SU(3)_{L+R}$ flavor symmetry for the axial charges to
satisfy $g_{A,3}^{+L}=g_{A,1}^{+L}\equiv g_{A}^{+L}$, where

$\displaystyle
g_{A}^{+L}=\frac{1}{2}\Big{(}g_{A,3}^{+L}+g_{A,1}^{+L}\Big{)}=0.512\ .$ (75)

It should be noted that such an assumption leads to
$\theta_{B_{+}^{1}}=\theta_{B_{+}^{2}}=\theta_{B_{+}^{3}}\equiv\theta_{B_{+}}$
for the mixing angle defined in Eq. (24).

Figure 3: $R=\sqrt{g_{A}^{2}+g_{A}^{\prime 2}}$ dependence of the decay widths of $\Lambda_{c}(2765)$ (red) and $\Xi_{c}(2970)^{+}$ (blue). The dashed lines are the corresponding experimental values (the central values) indicated in PDG. Decaying baryon | Channel | Width | PDG value
---|---|---|---
$\Lambda_{c}(2765)^{+}$ | $\Sigma_{c}(2455)\pi$ |  $12.7$ \- $41.6$ |
| $\Sigma_{c}(2520)\pi$ |  $10.2$ \- $33.4$ |
| Sum |  $23.0$ \- $75.0$ | ($50$)
$\Xi_{c}(2970)^{+}$ | $\Sigma_{c}(2455)\bar{K}$ |  $0.704$ \- $2.30$ |
| $\Xi_{c}^{\prime}\pi$ |  $6.83$ \- $22.3$ |
| $\Xi_{c}(2645)$ |  $7.03$ \- $22.9$ |
| Sum |  $14.6$ \- $47.6$ | ($20.9^{+2.4}_{-3.5}$)
$\Xi_{c}(2970)^{0}$ | $\Sigma_{c}(2455)\bar{K}$ |  $0.577$ \- $1.88$ |
| $\Xi_{c}^{\prime}\pi$ |  $6.78$ \- $22.1$ |
| $\Xi_{c}(2645)$ |  $6.97$ \- $22.7$ |
| Sum |  $14.3$ \- $46.7$ |
Table 2: Partial and total decay widths of $\Lambda_{c}(2765)$ and
$\Xi_{c}(2970)$ for $R=0.53$ \- $0.57$ in units of MeV.

In Fig. 3, the resultant $R=\sqrt{g_{A}^{2}+g_{A}^{\prime 2}}$ dependence of
decay widths of $\Lambda_{c}(2765)^{+}$ (red) and $\Xi_{c}(2970)^{+}$ (blue)
is plotted. In this figure, the red and blue dashed lines are the experimental
values of $\Lambda_{c}(2765)^{+}$ and $\Xi_{c}(2970)^{+}$ indicated in PDG
Zyla  _et al._ (2020), respectively. Figure 3 shows that the decay widths of
$\Lambda_{c}(2765)^{+}$ and $\Xi_{c}(2970)^{+}$ whose magnitudes are tens of
MeV, which cannot be explained by the conventional non-relativistic quark
model Nagahiro  _et al._ (2017), are reasonably obtained for $R\sim 0.53$ \-
$0.57$. In the figure, we have not shown the result for $\Xi_{c}(2970)^{0}$
since PDG does not indicate the decay width explicitly. For convenience, in
Table 2, we show partial and total decay widths of $\Lambda_{c}(2765)$ and
$\Xi_{c}(2970)$ for $R=0.53$ \- $0.57$. From the table, one can see that the
decay widths are very sensitive to the value of $R$, i.e., the magnitude of
axial charges.

When we choose the value of $R$ to be $R=0.55$, the resultant axial charge
$g_{A}^{+H}$ reads $g_{A}^{+H}=0.201$ (with an assumption of $g_{A}^{+H}>0$).
To summarize, the magnitude of axial charges related to
$\Lambda_{c}(2765)$-$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ sector and
$\Lambda_{c}$-$\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ sector are
$g_{A}^{+H}\approx 0.201$ and $g_{A}^{+L}=0.512$, respectively.

Baryon | $J^{P}$ | Mass | Total width
---|---|---|---
$\Lambda_{b}^{0}$ | $1/2^{+}$ | $5619.60$ | No strong decay
$\Xi_{b}^{-}$ | $1/2^{+}$ | $5797.0$ | No strong decay
$\Xi_{b}^{0}$ | $1/2^{+}$ | $5791.9$ | No strong decay
$\Sigma_{b}^{+}$ | $1/2^{+}$ | $5810.56$ | $5.0$
$\Sigma_{b}^{0}$ | $1/2^{+}$ | ? | ?
$\Sigma_{b}^{-}$ | $1/2^{+}$ | $5815.64$ | $5.3$
$\Sigma_{b}^{*+}$ | $3/2^{+}$ | $5830.32$ | $9.4$
$\Sigma_{b}^{*0}$ | $3/2^{+}$ | ? | ?
$\Sigma_{b}^{*-}$ | $3/2^{+}$ | $5834.74$ | $10.4$
$\Xi_{b}^{\prime 0}$ | $1/2^{+}$ | ? | ?
$\Xi_{b}^{\prime}(5935)^{-}$ | $1/2^{+}$ | $5935.02$ | $<0.08$
$\Xi_{b}(5945)^{0}$ | $3/2^{+}$ | $5952.3$ | $0.90$
$\Xi_{b}(5955)^{-}$ | $3/2^{+}$ | $5955.33$ | $1.65$
$\Lambda_{b}(6070)^{0}$ | $1/2^{+}$ | $6072.3$ | 72
Table 3: Experimental values of masses and decay widths of the bottom baryons
in units of MeV, indicated in PDG Zyla  _et al._ (2020). In this table only
the central values are shown. We note that the spin and parity of
$\Lambda_{b}(6070)$ are assumed to be $J^{P}=1/2^{+}$ as suggests by the
recent experimental data Sirunyan  _et al._ (2020); Aaij  _et al._ (2020).

##  V Bottom baryons

In this section, we investigate bottom baryons in a similar way to Sec. IV.
The experimentally observed masses and widths of the relevant bottom baryons
are listed in Table 3.

As done in Sec. IV, we determine $g_{A,i}^{+L}$ for bottom baryons by fitting
decay widths of the HQS-doublet $\Sigma_{b}$’s and $\Xi_{b}$’s. Using the
available experimental values in Table 3, we can get the following averaged
values:

$\displaystyle g_{A,3}^{+L}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}g_{A,3}^{+L}\Big{|}_{\Sigma_{b}^{+}}+g_{A,3}^{+L}\Big{|}_{\Sigma_{b}^{-}}+g_{A,3}^{+L}\Big{|}_{\Sigma_{b}^{*+}}+g_{A,3}^{+L}\Big{|}_{\Sigma_{b}^{*-}}\Big{)}$
(76) $\displaystyle=$ $\displaystyle 0.488\ ,$

and

$\displaystyle g_{A,1}^{+L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}g_{A,1}^{+L}\Big{|}_{\Xi_{b}(5945)^{0}}+g_{A,1}^{+L}\Big{|}_{\Xi_{b}(5955)^{-}}\Big{)}$
(77) $\displaystyle=$ $\displaystyle 0.577\ ,$

together with the isospin symmetry, for the bottom baryons. Similarly to the
charmed baryons, we have assumed $g_{A,i}^{+L}>0$. Equations (76) and (77)
imply that the violation of $SU(3)_{L+R}$ flavor symmetry for bottom baryons
is slightly larger than that for charmed baryons, but still the deviation is
small. Hence, again we assume the $SU(3)_{L+R}$ flavor symmetry as
$g_{A,3}^{+L}=g_{A,1}^{+L}\equiv g_{A}^{+L}$, with $g_{A}^{+L}$ being
estimated as

$\displaystyle
g_{A}^{+L}=\frac{1}{2}\Big{(}g_{A,3}^{+L}+g_{A,1}^{+L}\Big{)}=0.532\ .$ (78)

In this estimate the deviation reads

$\displaystyle\delta
g_{A}^{+L}\equiv\frac{g_{A}^{+L}-g_{A,3}^{+L}}{g_{A}^{+L}}=\frac{g_{A,1}^{+L}-g_{A}^{+L}}{g_{A}^{+L}}=8.37\,\%\
,$ (79)

which is still less than ten percent.

A candidate for the Roper-like $\Lambda_{b}$ is $\Lambda_{b}(6070)$ as
indicated in Table 3, while the Roper-like $\Xi_{b}$ has not been
experimentally observed. Thus, here we fix the value of $R$ for bottom baryons
by fitting the width of $\Lambda_{b}(6070)$ from
$\Lambda_{b}(6070)\to\Sigma_{b}\pi$ and
$\Lambda_{b}(6070)\to\Sigma_{b}^{*}\pi$ channels. The resultant value is

$\displaystyle R=0.612\ .$ (80)

We note that the unknown masses of $\Sigma_{b}^{0}$ and $\Sigma_{b}^{*0}$ have
been estimated by averaging masses of available isospin partners:
$M_{\Sigma_{b}^{0}}=\frac{1}{2}(M_{\Sigma_{b}^{+}}+M_{\Sigma_{b}^{-}})=5813.1$
MeV, and
$M_{\Sigma_{b}^{*0}}=\frac{1}{2}(M_{\Sigma_{b}^{*+}}+M_{\Sigma_{b}^{*-}})=5832.53$
MeV, in obtaining Eq. (80). $R$ for the charmed baryons has been considered to
be $R\sim 0.53$ \- $0.57$, thus Eq. (80) shows that the heavy quark flavor
symmetry for the axial charges seems to be satisfied well. The value of
$g_{A}^{+H}$ for bottom baryons are estimated as $g_{A}^{+H}=0.303$ with
$g_{A}^{+H}>0$ assumed.

One theoretical candidate for the undiscovered Roper-like $\Xi_{b}$ is the
$\Xi_{b}(6255)$ predicted in Ref. Chen  _et al._ (2018). Therefore, we employ
the mass of $\Xi_{b}(6255)$ as that of the Roper-like $\Xi_{b}$, and examine
its partial and total widths together with $\Lambda_{b}(6070)$ ones. The
results are summarized in Table 4. In this table we have used $R=0.612$, and
assumed that the unknown mass of $\Xi_{b}^{\prime 0}$ is identical to that of
$\Xi_{b}^{\prime}(5935)^{-}$ from isospin symmetry.

Decaying baryon | Channel | Width | (PDG value)
---|---|---|---
$\Lambda_{b}(6070)^{0}$ | $\Sigma_{b}\pi$ | $29.6$ |
| $\Sigma_{b}^{*}\pi$ | $42.4$ |
| Sum | $72$ | ($72\pm 11\pm 2$)
$\Xi_{b}(6255)^{0}$ | $\Xi_{b}^{\prime}\pi$ | $16.9$ |
| $\Xi_{b}^{\prime*}\pi$ | $26.7$ |
| Sum | $43.5$ |
$\Xi_{b}(6255)^{-}$ | $\Xi_{b}^{\prime}\pi$ | $16.8$ |
| $\Xi_{b}^{\prime*}\pi$ | $27.0$ |
| Sum | $43.8$ |
Table 4: Partial and total decay widths of $\Lambda_{b}(6070)$ and
$\Xi_{b}(6255)$ for $R=0.612$ in units of MeV. In this table
$\Xi_{b}^{\prime}$ and $\Xi_{b}^{\prime*}$ collectively denote
$\\{\Xi_{b}^{\prime 0},\Xi_{b}^{\prime}(5935)^{-}\\}$ and
$\\{\Xi_{b}(5945)^{0},\Xi_{b}(5955)^{-}\\}$, respectively. Figure 4:  Mass
dependence of the decay width of Roper-like $\Xi_{b}$. The vertical black line
represents the mass of $\Xi_{b}(6255)$ predicted in Ref. Chen  _et al._
(2018).  Figure 5: $R=\sqrt{g_{A}^{2}+g_{A}^{\prime 2}}$ dependence of the
decay widths of $\Lambda_{b}^{0}(6070)$ (red) and the Roper-like $\Xi_{b}^{-}$
(blue). The red dashed line is the experimental values (the central value)
indicated in PDG, and the vertical black line corresponds to $R=0.612$.

In order to examine other possibilities of the Roper-like $\Xi_{b}$, we show
the mass dependence of its decay width in Fig. 4 for $R=0.612$. In this figure
the vertical black line represents the mass of $\Xi_{b}(6255)$ predicted in
Ref. Chen  _et al._ (2018), corresponding to the values indicated in Table 4.
We have shown only the width of charged $\Xi_{b}^{-}$, since the results for
$\Xi_{b}^{0}$ is almost identical to the $\Xi_{b}^{-}$ one. Figure 4 indicates
that the decay width of Roper-like $\Xi_{b}$ is expected to be tens of MeV,
when we assume its mass is larger than the mass of ground-state $\Xi_{b}$ by
about $500$ MeV ($\sim 6.3$ GeV) as naively anticipated. Depicted in Fig. 5 is
$R$ dependence of the decay width of Roper-like $\Lambda_{b}(6070)$ (red) and
theoretically predicted $\Xi_{b}(6255)$ (blue). The red dashed horizontal line
indicates the PDG value for the decay width of $\Lambda_{b}(6070)$, and the
vertical black line corresponds to $R=0.612$ in Eq. (80). Figures 4 and 5
provide future experiments with useful information on the decay width of the
undiscovered Roper-like $\Xi_{b}$.

##  VI Discussions

###  VI.1 Discussions on the relation (72)

In this subsection we discuss the relation satisfied by axial charges of the
heavy baryons presented in Eq. (72). This relation is one of the unique
features of our present model, controlling couplings among the heavy baryons
and the light mesons. Here, we summarize its properties:

  * •

The relation holds regardless of the mixing angles for the HQS-singlet baryons
in Eq. (24). In other words, the relation is derived only by a fact that the
ground-state and the Roper-like heavy baryons are described by mixings of the
components (5), and information on the detailed mixing properties is not
essential.

  * •

The relation does not include the flavor index $i$, so that it links axial
charges for the different flavor sectors with the identical magnitude, as long
as $SU(3)_{L+R}$ flavor symmetry works well as explicitly shown in the present
investigation.

  * •

Moreover, the relation includes the axial charges associated with the
negative-parity HQS-singlet baryons in a similar way to those with the
positive-parity ones, sharing the identical magnitude $R$. This is a
significant consequence of employing the chiral model, and is expected to play
an important role in pursuing the parity partners Harada  _et al._ (2020);
Kawakami  _et al._ (2020) as will be further discussed in Sec. VI.2.

In the remainder of this subsection, we discuss higher-order corrections to
the relation (72). In Sec. II.3 we have constructed the interaction Lagrangian
for the heavy baryons and a NG boson with one $\partial_{\mu}\Sigma$ or
$\partial_{\mu}\Sigma^{\dagger}$. As for one NG boson emission decays of the
heavy baryons, momentum carried by the NG boson is of ${\cal O}(\Lambda_{\rm
QCD})$ which is small compared to heavy baryon masses. For this reason, our
truncation that only one $\partial_{\mu}\Sigma$ or
$\partial_{\mu}\Sigma^{\dagger}$ is left seems to work well based on the
spirit of heavy quark symmetry.

Apart from the above corrections, it may be possible to include higher-oder
corrections with respect to non-derivative terms of $\Sigma$,
$\Sigma^{\dagger}$, $\Sigma^{T}$ and $\Sigma^{*}$ to Eq. (42). For instance,

$\displaystyle{\cal L}^{\rm higher}_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{4\sqrt{2}\tilde{g}_{A}}{f_{0}^{3}}$ (81)
$\displaystyle\times$ $\displaystyle{\rm Tr}\Big{[}\bar{\cal
B}_{R}(\partial_{\mu}\Sigma^{\dagger})\Sigma\Sigma^{\dagger}S^{T\mu}+\bar{\cal
B}_{L}(\partial_{\mu}\Sigma)\Sigma^{\dagger}\Sigma S^{\mu}+{\rm h.c.}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{4\sqrt{2}\tilde{g}_{A}^{\prime}}{f_{0}^{3}}$
$\displaystyle\times$ $\displaystyle{\rm Tr}\Big{[}\bar{\cal
B}_{L}^{\prime}(\partial_{\mu}\Sigma^{\dagger})\Sigma\Sigma^{\dagger}S^{T\mu}+\bar{\cal
B}_{R}^{\prime}(\partial_{\mu}\Sigma)\Sigma^{\dagger}\Sigma S^{\mu}+{\rm
h.c.}\Big{]}$ $\displaystyle+$ $\displaystyle\cdots\ ,$

is possible as one of the simplest next-to-leading oder contributions. We note
that the even number of $\Sigma$, $\Sigma^{\dagger}$, $\Sigma^{T}$ and
$\Sigma^{*}$ with one derivatives are forbidden due to chiral symmetry. Under
the spontaneous breakdown of chiral symmetry the $\Sigma$ is replaced by its
VEV: $\langle\Sigma\rangle=\frac{f_{0}}{2}{\bm{1}}$ when its fluctuations are
neglected, and thus the Lagrangian (81) turns into

$\displaystyle{\cal L}^{\rm higher}_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\tilde{g}_{A}}{f_{0}}{\rm Tr}\Big{[}\bar{\cal
B}_{R}\partial_{\mu}\Sigma^{\dagger}S^{T\mu}+\bar{\cal
B}_{L}\partial_{\mu}\Sigma S^{\mu}+{\rm h.c.}\Big{]}$ (82) $\displaystyle+$
$\displaystyle\frac{\sqrt{2}\tilde{g}_{A}^{\prime}}{f_{0}}{\rm
Tr}\Big{[}\bar{\cal
B}_{L}^{\prime}\partial_{\mu}\Sigma^{\dagger}S^{T\mu}+\bar{\cal
B}_{R}^{\prime}\partial_{\mu}\Sigma S^{\mu}+{\rm h.c.}\Big{]}$
$\displaystyle+$ $\displaystyle\cdots\ ,$

by abbreviating multi NG boson interactions. Apparently Eq. (82) exhibits the
same structure as in Eq. (42). The structure has arisen since the VEV of
$\Sigma$ is of the form $\frac{f_{0}}{2}{\bm{1}}$ proportional to the unit
matrix, and such a coincidence is also true for any higher-order
contributions. Therefore, argument for $g_{A}$ and $g_{A}^{\prime}$ below Eq.
(42) can be applied here in a similar way; the couplings incorporating higher-
order contributions defined by

$\displaystyle G_{A}\equiv g_{A}+\tilde{g}_{A}+\cdots\ ,\ \
G^{\prime}_{A}\equiv g^{\prime}_{A}+\tilde{g}^{\prime}_{A}+\cdots\ ,$ (83)

coincide with the axial charges as derived in Eq. (51), and the relation for
the axial charges shown in Eq. (72) is found to hold at any higher order as

$\displaystyle(G_{A,i}^{+H})^{2}+(G_{A,i}^{+L})^{2}=(G_{A,i}^{-H})^{2}+(G_{A,i}^{-L})^{2}={\cal
R}^{2}\ ,$ (84)

with ${\cal R}=\sqrt{G_{A}^{2}+G_{A}^{\prime 2}}$ and

$\displaystyle G_{A,i}^{\pm H}$ $\displaystyle\equiv$ $\displaystyle
G_{A}\sin\theta_{B_{\pm}^{i}}-G_{A}^{\prime}\cos\theta_{B_{\pm}^{i}}\ ,$
$\displaystyle G_{A,i}^{\pm L}$ $\displaystyle\equiv$ $\displaystyle
G_{A}\cos\theta_{B_{\pm}^{i}}+G_{A}^{\prime}\sin\theta_{B_{\pm}^{i}}\ .$ (85)

The above demonstration implies that the relation (84) is derived simply as
consequences of the inclusion of two HQS-singlet baryons, belonging to
(anti-)fundamental representations of $SU(3)_{L}\times SU(3)_{R}$ as in Eqs.
(2) and (3), which are necessary to describe both the Roper-like as well as
the ground-state singly heavy baryons.

###  VI.2 Decays of the parity partners

In addition to the positive-parity heavy baryons examined in this article,
pursuing the undiscovered negative-parity ones, whose existences are naturally
predicted as parity partners Harada  _et al._ (2020); Kawakami  _et al._
(2020), are of great importance toward elucidation of chiral symmetric
properties of the heavy baryons. In this subsection we comment on predictions
in association with such partners.

The axial charges associated with the negative-parity HQS-singlet baryons
satisfy the relation (72) [or more generally Eq. (84)] sharing the identical
magnitude $R$ to those with the positive-parity ones. Those structures are one
of the notable features provided by the parity partners in a chiral model with
the linear representation. Hence, our present model can predict decay widths
where $S_{\bar{3}}^{\mu}$ is involved without introducing additional
parameters.

The $S_{\bar{3}}^{\mu}$ denotes the HQS-doublet baryons with
$J^{P}=1/2^{-},3/2^{-}$ belonging to the flavor anti-symmetric representation.
For charmed baryons, $S_{\bar{3}}^{\mu}$ contains
$\\{\Lambda_{c}(2595),\Lambda_{c}(2625)\\}$ and
$\\{\Xi_{c}(2790),\Xi_{c}(2815)\\}$ indicated in PDG. Thus, the interaction
Lagrangian (LABEL:LIntParity) can predict the decay widths of one NG boson
emission processes such as $\Xi_{c}(1/2^{-})\to\Xi_{c}(2790)\pi$, etc., in
principle when the threshold opened.666Here, $\Xi_{c}(1/2^{-})$ is the
undiscovered parity partner to the ground-state $\Xi_{c}$. The negative-parity
$\Lambda_{c}(1/2^{-})$ is also defined in the same way. Detailed estimation of
such partial decay widths requires understandings of the masses of
$\Xi_{c}(1/2^{-})$, in addition to our finding that the magnitude of axial
charges is $R\approx 0.55$ in Sec. IV.

Moreover, for the decay of $\Lambda_{c}(1/2^{-})$, as argued in Ref. Kawakami
_et al._ (2020) the $S$-wave decay of $\Lambda_{c}(1/2^{-})\to\Lambda_{c}\eta$
would not be neglected if the threshold is opened, and this may be true for
decays of $\Xi_{c}(1/2^{-})$. These decays cannot be described by our present
model (LABEL:LIntParity), implying that extension of our present model is
inevitable. The main decay modes are largely dependent on the masses of
$\Lambda_{c}(1/2^{-})$ and $\Xi_{c}(1/2^{-})$, and we leave those examinations
on the undiscovered parity partners to future study.

##  VII Summary

In this paper, we have investigated decay properties of the Roper-like singly
heavy baryons $\Lambda_{c}(2765)$ and $\Xi_{c}(2970)$, as well as those of the
HQS-doublet ones $\\{\Sigma_{c}(2455),\Sigma_{c}(2520)\\}$ and
$\\{\Xi_{c}^{\prime},\Xi_{c}(2645)\\}$ within a chiral model. Based on the
model, we have derived a relation satisfied by axial charges of the heavy
baryons that controls magnitude of their one pion (kaon) decays. Employing
reasonable values of the axial charges, our chiral model has successfully
explained large decay widths of the Roper-like $\Lambda_{c}(2765)$ and
$\Xi_{c}(2970)$. In addition to the investigation of charmed baryons, we have
applied our model to bottom baryons to estimate a decay width of the
undiscovered Roper-like $\Xi_{b}$, predicting tens of MeV.

We expect that our present investigation leads to elucidation of dynamical
properties of the excited heavy baryons from the viewpoint of chiral symmetry.
In particular, the relation that axial charges of the heavy baryons satisfy
[Eq. (72)] is one of the unique features of making use of a chiral model
together with the $SU(2)_{h}$ heavy quark spin symmetry, which cannot be
derived by quark models. Besides, the estimated value of the axial charges is
expected to be a benchmark for the investigation of the heavy baryons in the
future. In addition, we emphasize that pursuing the undiscovered parity
partners of the ground-state $\Lambda_{c}$ and $\Xi_{c}$ is indispensable for
further unveiling chiral structures of the singly heavy baryons.

## acknowledgement

D.S. thanks D. Jido for useful comments on the mirror diquarks. A.H. was
supported in part by Grants-in Aid for Scientific Research on Innovative Areas
(No. 18H05407).

Figure 6: Feynman diagrams for (a) $\Xi_{c}(2970)\to\Sigma_{c}(2455)\bar{K}$,
(b) $\Xi_{c}(2970)\to\Xi_{c}^{\prime}\pi$, and (c)
$\Xi_{c}(2970)\to\Xi_{c}(2645)\pi$. Figure 7: A Feynman diagram for
$\Xi_{c}(2645)\to\Xi_{c}\pi$.

##  Appendix A Analytic expressions of decay widths of $\Xi_{c}(2970)$ and
$\Xi_{c}(2645)$

In this appendix we summarize analytic expressions of decay widths of the
Roper-like $\Xi_{c}(2970)$, and those of the HQS-doublet $\Xi_{c}(2645)$. The
relevant diagrams are illustrated in Figs. 6. We note that $\Xi_{c}^{\prime}$
does not decay by the strong interactions since no thresholds open.

In the following calculations, we will assume isospin symmetry for the mixing
angles: $\theta_{B_{+}^{1}}=\theta_{B_{+}^{2}}$, and hence we will focus on
only $i=1$. As in Eqs. (LABEL:ModelReduced) and (53), the decay widths are
controlled by axial charges defined by

$\displaystyle g_{A,1}^{+H}$ $\displaystyle\equiv$ $\displaystyle
g_{A}\sin\theta_{B_{+}^{1}}-g_{A}^{\prime}\cos\theta_{B_{+}^{1}}\ ,$
$\displaystyle g_{A,1}^{+L}$ $\displaystyle\equiv$ $\displaystyle
g_{A}\cos\theta_{B_{+}^{1}}+g_{A}^{\prime}\sin\theta_{B_{+}^{1}}\ .$ (86)

Here, it should be noted that $\Xi_{c}$ baryons carry isospin-$1/2$, so that
coefficients for decay widths in different isospin channels can vary. For this
reason we present all the decay channels explicitly. Following the procedure
explained in Sec. III, the decay widths for the Roper-like $\Xi_{c}(2970)$ are
evaluated as

$\displaystyle\Gamma[\Xi_{c}(2970)^{0}\to\Sigma_{c}(2455)^{+}{K}^{-}]=\frac{(g_{A,1}^{+H})^{2}}{12\pi
f_{0}^{2}}$ $\displaystyle\hskip 147.95424pt\times F_{1}(M_{i},E_{f};{K}^{-})\
,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{0}\to\Sigma_{c}(2455)^{0}\bar{K}^{0}]=\frac{(g_{A,1}^{+H})^{2}}{6\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\bar{K}^{0})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{+}\to\Sigma_{c}(2455)^{++}{K}^{-}]=\frac{(g_{A,1}^{+H})^{2}}{6\pi
f_{0}^{2}}$ $\displaystyle\hskip 147.95424pt\times F_{1}(M_{i},E_{f};{K}^{-})\
,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{+}\to\Sigma_{c}(2455)^{+}\bar{K}^{0}]=\frac{(g_{A,1}^{+H})^{2}}{12\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\bar{K}^{0})\ ,$

for Fig. 6 (a),

$\displaystyle\Gamma[\Xi_{c}(2970)^{0}\to\Xi_{c}^{\prime+}\pi^{-}]=\frac{(g_{A,1}^{+H})^{2}}{12\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{-})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{0}\to\Xi_{c}^{\prime
0}\pi^{0}]=\frac{(g_{A,1}^{+H})^{2}}{24\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{0})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{+}\to\Xi_{c}^{\prime
0}\pi^{+}]=\frac{(g_{A,1}^{+H})^{2}}{12\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{+})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{+}\to\Xi_{c}^{\prime+}\pi^{0}]=\frac{(g_{A,1}^{+H})^{2}}{24\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{0})\,,$ (88)

for Fig. 6 (b), and

$\displaystyle\Gamma[\Xi_{c}(2970)^{0}\to\Xi_{c}(2645)^{+}\pi^{-}]=\frac{(g_{A,1}^{+H})^{2}}{6\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{-})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{0}\to\Xi_{c}(2645)^{0}\pi^{0}]=\frac{(g_{A,1}^{+H})^{2}}{12\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{0})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{+}\to\Xi_{c}(2645)^{0}\pi^{+}]=\frac{(g_{A,1}^{+H})^{2}}{6\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{+})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2970)^{+}\to\Xi_{c}(2645)^{+}\pi^{0}]=\frac{(g_{A,1}^{+H})^{2}}{12\pi
f_{0}^{2}}F_{1}(M_{i},E_{f};\pi^{0})\ ,$

for Fig. 6 (c). In those equations, the notation is understood similarly to
that used in Eq. (65). From the formulae we can find that, for instance, the
decay width of $\Xi_{c}(2970)^{0}\to\Sigma_{c}(2455)^{0}\bar{K}^{0}$ is twice
that of $\Xi_{c}(2970)^{0}\to\Sigma_{c}(2455)^{+}{K}^{-}$. Such a difference
is understood by the Clebsch-Gordan coefficients with respect to the isospin.
In addition, similarly to the formulae for $\Lambda_{c}(2765)$ demonstrated in
Sec. III, the $SU(2)_{h}$ heavy quark spin symmetry is manifestly seen.

Similarly, the analytic formulae for decay widths of $\Xi_{c}(2645)$
corresponding to the diagram in Fig. 7 are

$\displaystyle\Gamma[\Xi_{c}(2645)^{0}\to\Xi^{+}_{c}\pi^{-}]=\frac{(g_{A,1}^{+L})^{2}}{12\pi
f_{0}^{2}}F_{2}(M_{i},E_{f};\pi^{-})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2645)^{0}\to\Xi^{0}_{c}\pi^{0}]=\frac{(g_{A,1}^{+L})^{2}}{24\pi
f_{0}^{2}}F_{2}(M_{i},E_{f};\pi^{0})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2645)^{+}\to\Xi^{0}_{c}\pi^{+}]=\frac{(g_{A,1}^{+L})^{2}}{12\pi
f_{0}^{2}}F_{2}(M_{i},E_{f};\pi^{+})\ ,$
$\displaystyle\Gamma[\Xi_{c}(2645)^{+}\to\Xi^{+}_{c}\pi^{0}]=\frac{(g_{A,1}^{+L})^{2}}{24\pi
f_{0}^{2}}F_{2}(M_{i},E_{f};\pi^{0})\,,$ (90)

where the notation is the same as in Eq. (69). Again differences by a factor
of two appear due to the Clebsch-Gordan coefficients with respect to the
isospin.

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