# Heavy baryon spectrum with chiral multiplets of scalar and vector diquarks

Yonghee Kim<EMAIL_ADDRESS>Department of Physics, Kyushu
University, Fukuoka 819-0395, Japan  Yan-Rui Liu<EMAIL_ADDRESS>School of
Physics, Shandong University, Jinan 250100, China  Makoto Oka
<EMAIL_ADDRESS>Advanced Science Research Center, Japan Atomic Energy
Agency (JAEA), Tokai 319-1195, Japan  Nishina Center for Accelerator-Based
Science, RIKEN, Wako 351-0198, Japan  Kei Suzuki
<EMAIL_ADDRESS>Advanced Science Research Center, Japan
Atomic Energy Agency (JAEA), Tokai 319-1195, Japan

###### Abstract

Chiral effective theory of scalar and vector diquarks is formulated according
to the linear sigma model. The main application is to describe the ground and
excited states of singly heavy baryons with a charm or bottom quark. Applying
the potential quark model between the diquark and the heavy quark ($Q=c,b$),
we construct a heavy-quark–diquark model. The spectra of the positive- and
negative-parity states of $\Lambda_{Q}$, $\Sigma_{Q}$, $\Xi^{(^{\prime})}_{Q}$
and $\Omega_{Q}$ are obtained. The masses and interaction parameters of the
effective theory are fixed partly from the lattice QCD data and also from
fitting low-lying heavy baryon masses. We find that the negative parity
excited states of $\Xi_{Q}$ (flavor $\bar{\bf 3}$) are different from those of
$\Lambda_{Q}$, because of the inverse hierarchy of the pseudoscalar diquark.On
the other hand, $\Sigma_{Q},\Xi^{\prime}_{Q}$ and $\Omega_{Q}$ (flavor ${\bf
6}$) baryons have similar spectra. We compare our results of the heavy-
quark–diquark model with experimental data as well as the quark model.

##  I Introduction

Recent observation of multiquark exotic hadron stimulates discussion on
various different structures of hadrons. Multiquark correlations are often
manifested as clusters of quarks. The simplest cluster is the diquark made of
two quarks Gell-Mann (1964); Ida and Kobayashi (1966); Lichtenberg and Tassie
(1967); Lichtenberg (1967); Souza and Lichtenberg (1967); Lichtenberg  _et
al._ (1968); Carroll  _et al._ (1968); Lichtenberg (1969); Anselmino  _et al._
(1993); Jaffe (2005). It was suggested that the diquark plays important roles
in the baryon spectra and decays. It is also a main component in the color
superconducting phase of dense matter. For the properties of diquarks, such as
the mass and size, lattice QCD simulations have been used. Hess  _et al._
(1998); Orginos (2006); Alexandrou  _et al._ (2006); Babich  _et al._ (2007);
DeGrand  _et al._ (2008); Green  _et al._ (2010); Bi  _et al._ (2016);
Watanabe and Ishii (2021).

In studying the properties of the diquark, singly heavy baryons ($Qqq$) are
useful. They consist of one heavy (charm or bottom) quark ($Q=c,b$) and two
light (up, down or strange) quarks ($q=u,d,s$), which can be represented by a
diquark ($qq$) characterized by $SU(3)$ Lichtenberg (1975); Lichtenberg  _et
al._ (1982, 1983); Fleck  _et al._ (1988); Ebert  _et al._ (1996, 2008); Kim
_et al._ (2011); Ebert  _et al._ (2011); Chen  _et al._ (2015); Jido and
Sakashita (2016); Kumakawa and Jido (2017); Harada  _et al._ (2020); Kim  _et
al._ (2020); Dmitrašinović and Chen (2020); Kawakami  _et al._ (2020); Suenaga
and Hosaka (2021). Here the $P$-wave excited states of a singly heavy baryon
are classified into two kinds of orbital excitations, the $\lambda$-mode and
the $\rho$-mode. The $\lambda$-mode is an excitation of the coordinate between
the heavy quark and the diquark. On the other hand, the $\rho$-mode is an
excitation of the diquark cluster Copley  _et al._ (1979); Yoshida  _et al._
(2015).

The most popular light diquarks for hadron spectroscopy is the scalar diquark
with the spin-parity $J^{P}=0^{+}$, color $\bar{\bf 3}$ and flavor $\bar{\bf
3}$. It appears most frequently in hadrons such as the ground states of
$\Lambda_{Q}$ and $\Xi_{Q}$. In Ref. Harada  _et al._ (2020), the authors have
applied the chiral effective theory based on $SU(3)_{R}\times SU(3)_{L}$
chiral symmetry for this diquark. There the scalar and pseudoscalar diquarks
are chiral partners to each other, i.e., belonging to the same representation
of chiral symmetry. Their mass difference comes from spontaneous chiral
symmetry breaking, and they become degenerate when chiral symmetry is
recovered. This theory also shows that $U_{A}(1)$ anomaly ’t Hooft (1976,
1986) leads to inverse hierarchy of diquarks masses, that is, the non-strange
pseudoscalar diquark is heavier than that containing one strange quark,
$M(ud,0^{-})>M(ds/su,0^{-})$. As a consequence, the diquark cluster model of
charmed baryons predicts that the $\rho$-mode excited states are also
reversely ordered as
$M_{\rho}(\Lambda_{c},1/2^{-})>M_{\rho}(\Xi_{c},1/2^{-})$.

In the previous work, we calculated the spectrum of $\Lambda_{Q}$ and
$\Xi_{Q}$ baryons as two body systems of a spin $0$ diquark and a heavy quark
Kim  _et al._ (2020). In this paper, we extend our approach to include spin-1
($1^{+}$ and $1^{-}$) diquarks and consider the flavor sextet baryons,
$\Sigma_{Q}$, $\Xi^{\prime}_{Q}$ and $\Omega_{Q}$. We present a chiral
effective theory of the scalar/pseudoscalar ($0^{\pm}$) and vector/axial-
vector ($1^{\pm}$) diquarks with $SU(3)_{R}\times SU(3)_{L}$ symmetry. We also
construct the heavy-quark–diquark model for the flavor sextet baryons,
$\Sigma_{Q}$, $\Xi^{\prime}_{Q}$ and $\Omega_{Q}$. The parameters of the
effective Lagrangian and the potential of the heavy-quark–diquark model are
determined so that they reproduce the observed baryon masses. We find that the
spectra of $\Lambda_{Q}$ and $\Xi_{Q}$ contain the $\lambda$-mode of the
scalar diquark, $\rho$-mode from the pseudoscalar diquark and also from the
vector diquark. We compare the results with experimental data and also the
quark model.

This paper is organized as follows. In Sec. II, we introduce diquarks and
formulate chiral effective theory of diquarks. In Sec. III, we construct the
heavy-quark–diquark potential model and determine parameters of the potential
between the heavy quark and the diquark. In Sec. IV, we show numerical results
of the singly heavy baryon spectrum. At last, Sec. V is devoted to our
conclusion and outlook.

##  II Chiral effective theory of diquark

In this section, we introduce a chiral effective theory of diquarks and mass
formulas of diquarks given by spontaneous chiral symmetry breaking, both for
the scalar and pseudoscalar diquarks Harada  _et al._ (2020); Kim  _et al._
(2020), and for the vector and axial-vector diquarks.

###  II.1 Scalar and pseudoscalar diquarks

We here consider diquarks belonging to the color $\bf\overline{3}$
representation given in Table 1. They are the diquarks appearing in the
baryon, color-singlet three-quark system.

Table 1: Local diquark operators in color $\bf\overline{3}$. $C=i\gamma^{0}\gamma^{2}$ is the charge conjugation operator. The subscripts $S$ and $A$ indicate the flavor symmetric ($S$, 6) and antisymmetric ($A$, $\bar{\bf 3}$) operators, respectively.  Diquark | Operator | $J^{P}$ | Color | Flavor
---|---|---|---|---
scalar | $(q^{T}C\gamma^{5}q)^{\overline{3}}_{A}$ | $0^{+}$ | $\overline{3}$ | $\overline{3}$
pseudoscalar | $(q^{T}Cq)^{\overline{3}}_{A}$ | $0^{-}$ | $\overline{3}$ | $\overline{3}$
vector | $(q^{T}C\gamma^{\mu}\gamma^{5}q)^{\overline{3}}_{A}$ | $1^{-}$ | $\overline{3}$ | $\overline{3}$
axial-vector | $(q^{T}C\gamma^{\mu}q)^{\overline{3}}_{S}$ | $1^{+}$ | $\overline{3}$ | 6

To consider the diquark from the viewpoints of chiral symmetry, we use the
chiral projection operators
$P_{R,L}\equiv(1\pm\gamma^{5})/2,~{}\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$.
We split the quark operator $q^{a}_{i}$ into the right quark as
$q^{a}_{R,i}=P_{R}q^{a}_{i}$ and the left quark as
$q^{a}_{L,i}=P_{L}q^{a}_{i}$, where $a$ and $i$ are color and flavor indices
of the quark, respectively. Under the chiral $SU(3)_{R}\times SU(3)_{L}$
transform given by

$\displaystyle\begin{split}&q^{a}_{R,i}\rightarrow(U_{R})_{ij}q^{a}_{R,j},~{}~{}U_{R}\in
SU(3)_{R},~{}~{}~{}~{}~{}(3,1)\\\
&q^{a}_{L,i}\rightarrow(U_{L})_{ij}q^{a}_{L,j},~{}~{}U_{L}\in
SU(3)_{L},~{}~{}~{}~{}~{}(1,3)\end{split}$ (1)

the diquarks of spin 0 and 1 transform as $(\bar{3},1)$, $(1,\bar{3})$ and
$(3,3)$ representations, respectively, as shown in Table 2.

Table 2: Diquark operators in the chiral basis. Chiral operator | Spin | Color | Chiral
---|---|---|---
$d^{a}_{R,i}=\epsilon_{abc}\epsilon_{ijk}(q^{bT}_{R,j}Cq^{c}_{R,k})$ | $0$ | $\overline{3}$ | $(\overline{3},1)$
$d^{a}_{L,i}=\epsilon_{abc}\epsilon_{ijk}(q^{bT}_{L,j}Cq^{c}_{L,k})$ | $0$ | $\overline{3}$ | $(1,\overline{3})$
$d^{a,\mu}_{ij}=\epsilon_{abc}(q^{bT}_{L,i}C\gamma^{\mu}q^{c}_{R,j})$ | $1$ | $\overline{3}$ | $(3,3)$

Chiral effective theory for the scalar ($0^{+}$) and pseudoscalar ($0^{-}$)
diquarks with color $\bf\overline{3}$ and flavor $\bf\overline{3}$ is given
according to Ref. Harada  _et al._ (2020), by

$\displaystyle\mathcal{L}=\mathcal{L}_{S}+\frac{1}{4}{\rm
Tr}[\partial^{\mu}\Sigma^{\dagger}\partial_{\mu}\Sigma]-V(\Sigma),$ (2)
$\displaystyle\begin{split}\mathcal{L}_{S}&=\mathcal{D}_{\mu}d_{R,i}(\mathcal{D}^{\mu}d_{R,i})^{\dagger}+\mathcal{D}_{\mu}d_{L,i}(\mathcal{D}^{\mu}d_{L,i})^{\dagger}\\\
&-m_{S0}^{2}(d_{R,i}d_{R,i}^{\dagger}+d_{L,i}d_{L,i}^{\dagger})\\\
&-\frac{m_{S1}^{2}}{f_{\pi}}(d_{R,i}\Sigma_{ij}^{\dagger}d_{L,j}^{\dagger}+d_{L,i}\Sigma_{ij}d_{R,j}^{\dagger})\\\
&-\frac{m_{S2}^{2}}{2f_{\pi}^{2}}\epsilon_{ijk}\epsilon_{lmn}(d_{R,k}\Sigma_{li}\Sigma_{mj}d_{L,n}^{\dagger}+d_{L,k}\Sigma^{\dagger}_{li}\Sigma^{\dagger}_{m,j}d_{R,n}^{\dagger}).\\\
\end{split}$ (3)

Here $\Sigma$ represents meson field of chiral $(\bar{3},3)$ representation
Gell-Mann and Levy (1960); Lévy (1967). This contains nonet scalar $\sigma$
and pseudoscalar $\pi$ mesons, whose chiral transform is given by

$\Sigma_{ij}=\sigma_{ij}+i\pi_{ij}\rightarrow
U_{L,ik}\Sigma_{km}U^{{\dagger}}_{R,mj},~{}~{}~{}(\overline{3},3)$ (4)

In Eq.(2), $V(\Sigma)$ is the potential for the meson field $\Sigma$ and is
supposed to induce spontaneous chiral symmetry breaking (SCSB), represented by
the vacuum expectation values of the scalar meson $\sigma$ as

$\langle\Sigma_{ij}\rangle=\langle\sigma_{ij}\rangle=f_{\pi}\delta_{ij}~{},~{}~{}~{}\langle\pi_{ij}\rangle=0,$
(5)

where $f_{\pi}\simeq 92$ MeV is the pion decay constant. Then $\pi$ is
regarded as the massless Nambu-Goldstone bosons. Also the parity
transformation is given as
$\mathcal{P}~{}:~{}\Sigma\rightarrow\Sigma^{{\dagger}}$.

Since the diquark is not a color-singlet state, we introduce the color-gauge-
covariant derivative,
$\mathcal{D}^{\mu}=\partial^{\mu}+igT^{\alpha}G^{\alpha,\mu}$, with the gluon
field $G^{\mu}$ and the color $SU(3)$ generator for the color $\bf\bar{3}$
representation $T^{\alpha}$. All the color indices are contracted and not
explicitly written.

For each term in Eq. (3), the parameters $m_{S0}^{2},m_{S1}^{2}$, and
$m_{S2}^{2}$ give the masses of scalar and pseudoscalar diquarks. When the
chiral symmetry is restored, the scalar and pseudoscalar diquarks are
degenerate with the chiral invariant mass $m_{S0}$. In the ordinary vacuum,
the mass splitting is induced by SCSB and is expressed by $m_{S1}$ and
$m_{S2}$. Besides, the $m_{S1}^{2}$ term brings about the $U_{A}(1)$ symmetry
breaking ’t Hooft (1976, 1986).

When the flavor $SU(3)$ symmetry is broken, the masses of scalar and
pseudoscalar diquarks, $M(0^{+})$ and $M(0^{-})$, satisfy the relation given
by Harada  _et al._ (2020)

$\displaystyle[M_{qs}(0^{+})]^{2}-[M_{qq}(0^{+})]^{2}=[M_{qq}(0^{-})]^{2}-[M_{qs}($
$\displaystyle 0^{-})]^{2}$ (6) $\displaystyle>0.$

The index $q$ stands for $u$ or $d$ quark and $s$ for the $s$ quark, which
means the constituent quark of diquarks. Here we can find the inverse mass
hierarchy for the pseudoscalar diquarks, where the non-strange diquark is
heavier than the singly strange diquark, $M_{qq}(0^{-})>M_{qs}(0^{-})$.

###  II.2 Chiral effective Lagrangian including vector and axial-vector
diquarks

Next we introduce the vector ($1^{-}$) and axial-vector ($1^{+}$) diquarks,
which are both color $\bf\overline{3}$ but have different flavor with each
other. The chiral diquarks are given in the third line of Table 2,

$d^{a,\mu}_{ij}=\epsilon^{abc}(q^{bT}_{L,i}C\gamma^{\mu}q^{c}_{R,j})=\epsilon^{abc}(q^{bT}_{R,j}C\gamma^{\mu}q^{c}_{L,i}).$
(7)

This contains both right and left quark operators, belonging to chiral $(3,3)$
representation. According to Eq. (1), the chiral transformation is given as

$\begin{split}&d^{a,\mu}_{ij}\rightarrow
U_{L,ik}~{}d^{a,\mu}_{km}~{}U^{T}_{R,mj},~{}~{}~{}(3,3)\\\
&d^{a,\mu{\dagger}}_{ij}\rightarrow
U^{T{\dagger}}_{R,ik}~{}d^{a,\mu{\dagger}}_{km}~{}U^{{\dagger}}_{L,mj},~{}~{}~{}(\overline{3},\overline{3})\\\
\end{split}$ (8)

As the spatial inversion interchanges the left and right diquarks, the vector
diquarks transform as

$\mathcal{P}~{}:~{}d^{a,\mu}_{ij}\rightarrow-d^{a}_{\mu,ji},$ (9)

and then obtain the Lorentz vector and axial-vector diquarks, as

$\displaystyle
V^{a,\mu}_{ij}=\frac{1}{\sqrt{2}}(d^{a,\mu}_{ij}-d^{a,\mu}_{ji})=\epsilon_{abc}\frac{1}{\sqrt{2}}(q_{i}^{bT}C\gamma^{\mu}\gamma^{5}q_{j}^{c}),$
(10) $\displaystyle
A^{a,\mu}_{ij}=\frac{1}{\sqrt{2}}(d^{a,\mu}_{ij}+d^{a,\mu}_{ji})=\epsilon_{abc}\frac{1}{\sqrt{2}}(q_{i}^{bT}C\gamma^{\mu}q_{j}^{c}).$
(11)

Thus we see that vector and axial-vector diquarks are chiral partners,
belonging to $(3,3)$ representation of chiral symmetry, satisfying
$V^{a,\mu}_{ij}=-V^{a,\mu}_{ji}$, $A^{a,\mu}_{ij}=A^{a,\mu}_{ji}$. Note that
the vector diquark belongs to flavor antisymmetric $\bf\overline{3}$
representation, while the axial-vector diquark to symmetric $\bf 6$
representation.

We now construct the full effective Lagrangian with both the scalar and vector
diquarks as

$\displaystyle\mathcal{L}=\mathcal{L}_{S}+\mathcal{L}_{V}+\frac{1}{4}{\rm
Tr}[\partial^{\mu}\Sigma^{\dagger}\partial_{\mu}\Sigma]-V(\Sigma)+\mathcal{L}_{V-S},$
(12)

in which the effective Lagrangian for vector and axial-vector diquarks is
expressed as

$\displaystyle\begin{split}\mathcal{L}_{V}&=\frac{1}{2}{\rm
Tr}[F^{\mu\nu}F^{{\dagger}}_{\mu\nu}]+m_{V0}^{2}{\rm
Tr}[d^{\mu}d^{{\dagger}}_{\mu}]\\\ &+\frac{m_{V1}^{2}}{f_{\pi}^{2}}{\rm
Tr}[\Sigma^{{\dagger}}d^{\mu}\Sigma^{T}d^{{\dagger}T}_{\mu}]\\\
&+\frac{m_{V2}^{2}}{f_{\pi}^{2}}[{\rm
Tr}\\{\Sigma^{T}\Sigma^{{\dagger}T}d_{\mu}^{{\dagger}}d^{\mu}\\}+{\rm
Tr}\\{\Sigma\Sigma^{{\dagger}}d^{\mu}d^{{\dagger}}_{\mu}\\}].\\\ \end{split}$
(13)

For the first term of Eq. (13),
$F^{\mu\nu}=\mathcal{D}^{\mu}d^{\nu}-\mathcal{D}^{\nu}d^{\mu}$ is the strength
of chiral vector diquark fields. Similarly to the scalar and pseudoscalar
diquarks, masses of vector and axial-vector diquarks are given by three
parameters $m_{V0}^{2},m_{V1}^{2}$ and $m_{V2}^{2}$. $m_{V0}$ is the chiral
invariant mass of vector and axial-vector diquarks, and their mass splitting
is given by the $m_{V1}^{2}$ and $m_{V2}^{2}$ terms.

We neglect terms that includes ${\rm Tr}[\Sigma\Sigma^{{\dagger}}]$, or more
than two $\Sigma$’s. All the color indices are contracted and the self-
interaction terms of gluons are omitted as same as Eq. (3).

The last term of the effective Lagrangian $\mathcal{L}_{V-S}$ describes the
coupling of scalar/vector diquarks and the meson. We neglect this term since
it does not contribute to the mass formulas of diquarks.

###  II.3 Masses of vector and axial-vector diquarks

Chiral symmetry is exact if the light quark masses are zero. In this limit,
the effective quark masses of quarks, of oder 300–500 MeV, are given in the
linear sigma model by the condensates of the scalar meson $\sigma$, as $m_{\rm
eff}=g_{s}\langle\sigma\rangle$, where $g_{s}\simeq 3$ is the quark-meson
coupling constant. In reality, chiral symmetry is explicitly broken by the
non-zero quark masses. In order to include this effect, we superpose the bare
quark mass and the scalar condensate as

$\displaystyle\mathcal{M}_{\rm eff}=\mathcal{M}+g_{s}\langle\Sigma\rangle,$
(14)

where $\mathcal{M}={\rm diag}(m_{u},m_{d},m_{s})$ is the current quark mass
matrix. The vacuum expectation value $\langle\Sigma\rangle$ may also contain
the symmetry breaking with
$\langle\sigma_{11}\rangle=\langle\sigma_{22}\rangle=f_{\pi}\simeq 92$ MeV for
the $u,d$ quarks and $\langle\sigma_{33}\rangle=f_{s}=2f_{K}-f_{\pi}\simeq
128$ MeV for the $s$ quark. By neglecting $u$ and $d$ quark masses, we obtain
the effective quark mass matrix as

$\displaystyle\mathcal{M}_{\rm eff}\simeq g_{s}f_{\pi}{\rm
diag}(1,1,1+\epsilon)\equiv g_{s}f_{\pi}X,$ (15)
$\displaystyle\epsilon=\frac{f_{s}}{f_{\pi}}\left(1+\frac{m_{s}-g_{s}f_{\pi}}{g_{s}f_{s}}\right)\simeq
2/3.$ (16)

We now introduce the explicit chiral symmetry breaking to the diquark
effective Lagrangian by the replacement Harada  _et al._ (2020)

$\displaystyle\Sigma\rightarrow\tilde{\Sigma}\equiv\Sigma+\mathcal{M}/g_{s},$
(17)

with its vacuum expectation value,

$\displaystyle\langle\tilde{\Sigma}\rangle=\mathcal{M}_{\rm
eff}/g_{s}=f_{\pi}X.$ (18)

By applying Eq. (17) to Eq. (13) and using the notations of Eqs. (10) and
(11), we can read the mass terms for the vector and axial-vector diquarks as

$\displaystyle\mathcal{L}_{Vmass}$ $\displaystyle=\frac{1}{2}m_{V0}^{2}{\rm
Tr}[A^{\mu}A^{{\dagger}}_{\mu}]+\frac{1}{2}m_{V0}^{2}{\rm
Tr}[V^{\mu}V^{{\dagger}}_{\mu}]$ (19) $\displaystyle+\frac{1}{2}m_{V1}^{2}{\rm
Tr}[XA^{\mu}XA^{{\dagger}}_{\mu}]-\frac{1}{2}m_{V1}^{2}{\rm
Tr}[XV^{\mu}XV^{{\dagger}}_{\mu}]$ $\displaystyle+m_{V2}^{2}{\rm
Tr}[X^{2}A^{\mu}A^{{\dagger}}_{\mu}]+m_{V2}^{2}{\rm
Tr}[X^{2}V^{\mu}V^{{\dagger}}_{\mu}].$

Then the masses of axial-vector and vector diquarks, $M(1^{+})$ and
$M(1^{-})$, are given to the leading order in $\epsilon$ by

$\displaystyle\left[M_{qq}(1^{+})\right]^{2}=m_{V0}^{2}+m_{V1}^{2}+2m_{V2}^{2},$
(20)
$\displaystyle\left[M_{qs}(1^{+})\right]^{2}=m_{V0}^{2}+m_{V1}^{2}+2m_{V2}^{2}$
$\displaystyle+\epsilon(m_{V1}^{2}+2m_{V2}^{2}),$ (21)
$\displaystyle\left[M_{ss}(1^{+})\right]^{2}=m_{V0}^{2}+m_{V1}^{2}+2m_{V2}^{2}$
$\displaystyle+2\epsilon(m_{V1}^{2}+2m_{V2}^{2}),$ (22)
$\displaystyle\left[M_{qq}(1^{-})\right]^{2}=m_{V0}^{2}-m_{V1}^{2}+2m_{V2}^{2},$
(23)
$\displaystyle\left[M_{qs}(1^{-})\right]^{2}=m_{V0}^{2}-m_{V1}^{2}+2m_{V2}^{2}$
$\displaystyle+\epsilon(-m_{V1}^{2}+2m_{V2}^{2}).$ (24)

From Eqs. (20)–(22), we find the generalized Gell-Mann–Okubo mass formula for
the axial-vector diquarks, Gell-Mann (1962); Okubo (1962)

$\displaystyle[M_{ss}(1^{+})]^{2}-[M_{qs}(1^{+})]^{2}$
$\displaystyle=[M_{qs}(1^{+})]^{2}-[M_{qq}(1^{+})]^{2}$ (25)
$\displaystyle=\epsilon(m_{V1}^{2}+2m_{V2}^{2}).$

This equation indicates the inequality,

$\displaystyle M_{ss}(1^{+})+M_{qq}(1^{+})<2M_{qs}(1^{+})$ (26)

One can check this relation from the masses of the singly heavy baryons. From
the spin-averaged masses, $M(\Sigma_{c},1/2^{+},3/2^{+})=2496.6$ MeV,
$M(\Xi^{\prime}_{c},1/2^{+},3/2^{+})=2623.5$ MeV, and
$M(\Omega_{c},1/2^{+},3/2^{+})=2742.3$ MeV Zyla  _et al._ (2020), we find the
average of the $\Sigma_{c}$ and $\Omega_{c}$ masses is smaller than the
$\Xi^{\prime}_{c}$ mass as is expected from the inequality (26).

On the other hand, from Eqs. (23) and (24), the square mass difference between
these vector diquarks is given by

$\displaystyle[M_{qs}(1^{-})]^{2}-[M_{qq}(1^{-})]^{2}=\epsilon(-m_{V1}^{2}+2m_{V2}^{2}).$
(27)

Comparing with Eq. (25), the mass difference by the number of constituent
strange quarks is different between the vector and axial-vector diquarks,
which is determined by the parameter $m_{V1}^{2}$.

##  III Heavy baryon spectrum

In this section, we construct a nonrelativistic potential model for singly
heavy baryons, composed of a heavy quark and a diquark (heavy-quark–diquark
model). We determine the unknown model parameters such as the diquark masses
and the coefficients of heavy-quark–diquark potential from experimental data
of heavy baryons. We refer to a general analysis of the heavy baryon spectrum
to Ref. Chen  _et al._ (2017a).

###  III.1 Heavy-quark–diquark model

We apply a nonrelativistic potential model to the system consisting of a heavy
quark and a diquark. In Table 3, we summarize the baryon states that we
consider in the present calculation. We consider the S (scalar), P
(pseudoscalar), V (vector) and A (axial-vector) diquarks. Excited states are
labeled by the number of nodes in the radial wave function, $n$, and the
relative angular momentum, $L$.

In the quark model, the $P$-wave excited states of baryons are often
classified into the $\lambda$ and $\rho$ modes. This is based on the quark
model notation that denotes the relative coordinate between the heavy quark
and the diquark as $\lambda$, while the internal coordinate of the diquark is
called $\rho$. In the heavy-quark–diquark model, the $\lambda$ modes are given
by the excitations between the heavy quark and the diquark. On the contrary,
the $\rho$ coordinate is not explicitly treated dynamically, and the $\rho$
mode excitations are realized by the excited diquarks, such as P and V
diquarks. We further consider the nodal excitations in the relative S wave
states. Possible spin-parity $J^{P}$ combinations of the baryon are listed in
the last column of Table 3. All these states will appear in the final
calculation.

Table 3: Quantum numbers of the ground and excited states of the singly heavy baryons in the heavy-quark–diquark model.  | Diquark ($J^{P}$) |  $n$ |  $L$ |  $J^{P}$
---|---|---|---|---
|  | 0 | 0 |  $1/2^{+}$
| S ($0^{+}$) | 0 | 1 |  $1/2^{-},3/2^{-}$
$\Lambda_{c/b}$ |  | 0 | 2 |  $3/2^{+},5/2^{+}$
$\Xi_{c/b}$ |  | 1 | 0 |  $1/2^{+}$
| P ($0^{-}$) | 0 | 0 |  $1/2^{-}$
| V ($1^{-}$) | 0 | 0 |  $1/2^{-},3/2^{-}$
$\Xi^{\prime}_{c/b}$ |  | 0 | 0 |  $1/2^{+},3/2^{+}$
$\Sigma_{c/b}$ | A ($1^{+}$) | 0 | 1 |  $(1/2^{-})^{2},(3/2^{-})^{2},5/2^{-}$
$\Omega_{c/b}$ |  | 0 | 2 |  $1/2^{+},(3/2^{+})^{2},(5/2^{+})^{2},7/2^{+}$
|  | 1 | 0 |  $1/2^{+},3/2^{+}$

Following the previous study Kim  _et al._ (2020), we employ the
nonrelativistic Hamiltonian for the heavy-quark–diquark model given by

$H=\frac{{\bm{p}}^{2}}{2\mu}+M_{Q}+M_{d}+V_{0}+V_{S},$ (28)

and the relative momentum and the reduced mass are defined by

$\displaystyle{\bm{p}}=\frac{M_{Q}{\bm{p}}_{d}-M_{d}{\bm{p}}_{Q}}{M_{Q}+M_{d}},$
(29) $\displaystyle\mu=\frac{M_{Q}M_{d}}{M_{Q}+M_{d}}.$ (30)

where the indices $Q$ and $d$ denote the heavy quark and the diquark,
respectively. Also $\bm{p}_{Q/d}$ and $M_{Q/d}$ are the momentum and mass,
respectively. We solve the Schr$\ddot{\rm o}$dinger equation for this
Hamiltonian, applying the Gaussian expansion method Kamimura (1988); Hiyama
_et al._ (2003).

The two-body potential is given in terms of the relative coordinate
$r=|{\bm{r}}_{d}-{\bm{r}}_{Q}|$ and consists of the spin-independent central
force, $V_{0}(r)$ and the spin-dependent term, $V_{S}(r)$. The spin-
independent potential $V_{0}(r)$ consists of the Coulomb term with the
coefficient $\alpha$, the linear term with $\lambda$ and a constant shift $C$,

$V_{0}(r)=-\frac{\alpha}{r}+\lambda r+C.$ (31)

We choose three different choices of the parameters taken from the quark
models by Yoshida et al. (Y) Yoshida  _et al._ (2015), Silvestre-Brac (S)
Silvestre-Brac (1996), and Barnes et al. (B) Barnes  _et al._ (2005).

$V_{S}$ is the sum of the spin-spin potential $V_{ss}(r)$, the spin-orbit
potential $V_{so}(r)$, and the tensor potential $V_{ten}(r)$, taken from Refs.
Yoshida  _et al._ (2015); Silvestre-Brac (1996); Barnes  _et al._ (2005). The
spin-spin potential for the B-potential is taken as a Gaussian function,

$\displaystyle
V_{ss}(r)=({\bm{s}}_{d}\cdot{\bm{s}}_{Q})\frac{\kappa_{Q}}{M_{d}M_{Q}}\left(\frac{\Lambda}{\sqrt{\pi}}\right)^{3}\exp{(-\Lambda^{2}r^{2})}.$
(32)

On the other hand, the Y- and S-potentials have the Yukawa form as

$\displaystyle
V_{ss}(r)=({\bm{s}}_{d}\cdot{\bm{s}}_{Q})\frac{\kappa_{Q}}{M_{d}M_{Q}}\frac{\Lambda^{2}}{r}\exp{(-\Lambda
r)}.$ (33)

Here $\Lambda$ is a cutoff parameter and $({\bm{s}}_{d}\cdot{\bm{s}}_{Q})$ is
the inner product of the diquark spin ${\bm{s}}_{d}$ and the heavy-quark spin
${\bm{s}}_{Q}$. The coefficient $\kappa_{Q}$ will be determined for the charm
and bottom sectors separately in the heavy-quark–diquark model.

The spin–orbit potential $V_{so}(r)$ consists of the symmetric ($+$) and
antisymmetric ($-$) terms as

$\displaystyle V_{so}(r)$ $\displaystyle=V_{so}^{+}(r)+V_{so}^{-}(r)$ (34)
$\displaystyle V_{so}^{+}(r)$ $\displaystyle=\eta\frac{(1-e^{-\Lambda
r})^{2}}{r^{3}}$ (35)
$\displaystyle\times\left(\frac{1}{M_{d}^{2}}+\frac{1}{M_{Q}^{2}}+\frac{4}{M_{d}M_{Q}}\right)({\bm{L}}\cdot{\bm{S}}^{+}),$
$\displaystyle V_{so}^{-}(r)$ $\displaystyle=\eta\frac{(1-e^{-\Lambda
r})^{2}}{r^{3}}\left(\frac{1}{M_{d}^{2}}-\frac{1}{M_{Q}^{2}}\right)({\bm{L}}\cdot{\bm{S}}^{-}),$
(36)

with ${\bm{S}}^{\pm}\equiv{\bm{s}}_{d}\pm{\bm{s}}_{Q}$. The tensor potential
$V_{ten}(r)$ is given by

$\displaystyle V_{ten}(r)$ $\displaystyle=2\eta\frac{(1-e^{-\Lambda
r})^{2}}{M_{d}M_{Q}r^{3}}$ (37)
$\displaystyle\times\left[\frac{3({\bm{s}}_{d}\cdot{\bm{r}})({\bm{s}}_{Q}\cdot{\bm{r}})}{r^{2}}-{\bm{s}}_{d}\cdot{\bm{s}}_{Q}\right]$

We note that the spin–orbit and tensor potentials were originally included
only in the Y-potential but we apply the same form also for the S- and
B-potentials. In Eqs. (35)–(37), $\eta$ is the common coupling strength that
is determined later for the heavy-quark–diquark model.

###  III.2 Diquark masses and potential parameters

Choices of the heavy-quark–diquark model parameters are summarized in Table 4.
Among them, the heavy-quark masses, $M_{Q}$, the strength of the Coulomb
potential, $\alpha$, that of the linear confinement potential, $\lambda$, and
the cutoff parameter, $\Lambda$, are taken from the quark models, Ref. Yoshida
_et al._ (2015); Silvestre-Brac (1996); Barnes  _et al._ (2005). The constant
shifts $C_{Q}$ in $V_{0}(r)$ are determined by fitting the masses of the
ground $\Lambda_{Q}$ baryon as in Ref. Kim  _et al._ (2020).

Table 4: Parameters of the heavy-quark–diquark model. The values with asterisk (*) are taken from the potential models by Yoshida et al. (Y-pot.) Yoshida  _et al._ (2015), by Silvestre-Brac (S-pot.) Silvestre-Brac (1996), and by Barnes et al. (B-pot.) Barnes  _et al._ (2005). $\mu$ included in the parameter of the Y-pot. is the reduced mass, Eq. (30). | Potential models
---|---
Parameters |  Y-pot. Yoshida  _et al._ (2015) |  S-pot. Silvestre-Brac (1996) |  B-pot. Barnes  _et al._ (2005)
$\alpha$ | 60/$\mu$* |  $0.5069$* |  $0.7281$*
$\lambda({\rm GeV}^{2})$ |  $0.165$* |  $0.1653$* |  $0.1425$*
$C_{c}(\rm GeV)$ | $-0.831$ | $-0.707$ | $-0.191$
$C_{b}(\rm GeV)$ | $-0.819$ | $-0.696$ | $\ldots$
$\Lambda(\rm GeV)$ |  $0.691$* |  $0.434$* |  $1.0946$*
$\kappa_{c}$ | $0.8586$ | $1.1570$ | $2.7258$
$\kappa_{b}$ | $0.6635$ | $0.8145$ | $\ldots$
$\eta$ | $0.2494$ | $0.2494$ | $0.2494$
$M_{c}(\rm GeV)$ |  $1.750$* |  $1.836$* |  $1.4794$*
$M_{b}(\rm GeV)$ |  $5.112$* |  $5.227$* | $\ldots$

We omit the bottom part of the B-potential parameters, because the original
B-potential was not applied to the bottom baryons.

In Table 5, the diquark masses and the parameters of the effective Lagrangian
are summarized. The values for the scalar and pseudoscalar diquarks are taken
from Ref. Kim  _et al._ (2020). In particular, the mass of the $ud$ scalar
diquark, $M_{qq}(0^{+})$, is the input value from lattice QCD simulations Bi
_et al._ (2016). The vector diquark masses, $M_{qq}(1^{+}),M_{qs}(1^{+})$, and
$M_{qq}(1^{-})$, are determined from some spin-averaged charmed baryon masses
as inputs:

$\displaystyle M(\Sigma_{c},1/2^{+},3/2^{+})=2496.60\hbox{ MeV},$
$\displaystyle M(\Xi^{\prime}_{c},1/2^{+},3/2^{+})=2623.52\hbox{ MeV},$
$\displaystyle M_{\rho}(\Lambda_{c},1/2^{-},3/2^{-})=2927.65\hbox{ MeV},$

where $M_{\rho}(\Lambda_{c},1/2^{-},3/2^{-})$ denotes the spin averaged mass
of the $\rho$-mode excited states of $\Lambda_{c}$. We assign the
$\Lambda_{c}$ at $M(\Lambda_{c},3/2^{-})=2939.60$ MeVZyla  _et al._ (2020) as
a $\rho$-mode excited state, and assume that the splitting between $1/2^{-}$
and $3/2^{-}$ is the same as the lower ($\lambda$-mode) $\Lambda_{c}$ baryons,
i.e., $M(\Lambda_{c};3/2^{-})-M(\Lambda_{c},1/2^{-})=35.86$ MeV. The other
diquark masses are given by the mass formulas (20)–(24) with $\epsilon=2/3$.

Table 5: Diquark masses and parameters of the chiral effective Lagrangian determined with each potential model. The values for scalar and pseudoscalar diquarks are taken from Ref. Kim  _et al._ (2020).  | Potential model
---|---
|  Y-pot. Yoshida  _et al._ (2015) |  S-pot. Silvestre-Brac (1996) |  B-pot. Barnes  _et al._ (2005)
Masses of S, P diquarks ($\rm MeV$) Kim  _et al._ (2020)
$M_{qq}(0^{+})~{}$ | 725 | 725 | 725
$M_{qs}(0^{+})~{}$ | 942 | 977 | 983
$M_{qq}(0^{-})~{}$ | 1406 | 1484 | 1496
$M_{qs}(0^{-})~{}$ | 1271 | 1331 | 1341
Masses of V, A diquarks ($\rm MeV$)
$M_{qq}(1^{+})~{}$ | 973 | 1013 | 1019
$M_{qs}(1^{+})~{}$ | 1116 | 1170 | 1179
$M_{ss}(1^{+})~{}$ | 1242 | 1309 | 1320
$M_{qq}(1^{-})~{}$ | 1447 | 1527 | 1540
$M_{qs}(1^{-})~{}$ | 1776 | 1883 | 1901
Parameters in $\mathcal{L}_{S}$ ($\rm MeV^{2}$) Kim  _et al._ (2020)
$~{}~{}m_{S0}^{2}~{}$ | $(1119)^{2}$ | $(1168)^{2}$ | $(1176)^{2}$
$~{}~{}m_{S1}^{2}~{}$ | $(690)^{2}$ | $(746)^{2}$ | $(754)^{2}$
$~{}~{}m_{S2}^{2}~{}$ | $-(258)^{2}$ | $-(298)^{2}$ | $-(303)^{2}$
Parameters in $\mathcal{L}_{V}$ ($\rm MeV^{2}$)
$~{}~{}m_{V0}^{2}~{}$ | $(708)^{2}$ | $(714)^{2}$ | $(714)^{2}$
$~{}~{}m_{V1}^{2}~{}$ | $-(757)^{2}$ | $-(808)^{2}$ | $-(816)^{2}$
$~{}~{}m_{V2}^{2}~{}$ | $(714)^{2}$ | $(765)^{2}$ | $(773)^{2}$

The masses of the vector and axial-vector diquarks in the Y-potential model
are smaller than those with the other two potentials. This behavior comes from
the difference of the Coulomb interaction. Since the reduced mass of diquark
and charm quark is about 600 MeV, the strength of the Coulomb potential,
$\alpha$, in the Y-potential model is about 0.1 and is much smaller than that
in the other models. The small value of $\alpha$ makes the binding energy
smaller and the required diquark masses become smaller. Note that for each
potential model, the diquark masses satisfy the inequality Eq. (26).

###  III.3 Strengths of the spin-dependent potentials

The coefficients of the spin-spin potential, $\kappa_{Q}$, and the spin-orbit
and tensor forces, $\eta$, are determined by known masses of the low-lying
heavy baryons and summarized in Table 4. We take the strengths of the spin-
spin potential, $\kappa_{c}$ and $\kappa_{b}$, independently. They are
determined so as to reproduce the mass splittings,
$M(\Sigma_{c},1/2^{+})=2453.54$ MeV and $M(\Sigma_{c},3/2^{+})=2518.13$ MeV,
and $M(\Sigma_{b},1/2^{+})=5813.10$ MeV and $M(\Sigma_{b},3/2^{+})=5832.53$
MeV, respectively.

The parameter $\eta$ for $V_{so}$ and $V_{ten}$ is determined from the
splitting of the $\lambda$-mode excitations of $\Lambda_{c}$. Mass difference
of $M(\Lambda_{c},1/2^{-})=2592.25$ MeV and $M(\Lambda_{c},3/2^{-})=2628.11$
MeV comes from the spin-orbit interaction

$V_{so}(r)=\eta~{}\frac{(1-e^{-\Lambda
r})^{2}}{r^{3}}\left(\frac{2}{M_{Q}^{2}}+\frac{4}{M_{d}M_{Q}}\right)~{}({\bm{L}}\cdot{\bm{s}}_{Q}),$
(38)

which is reduced from Eqs. (35) and (36) for the S diquark, $s_{d}=0$. In
order to reduce the number of the independent parameters, we assume that
$\eta$ is independent of the heavy-quark mass. Furthermore, for all the three
potentials, we use the same value $\eta=0.2494$ obtained with the Y-potential.

###  III.4 Diquarks towards chiral restoration

At high temperature and/or high baryon density, the QCD matter is expected to
make a transition to a phase in which chiral symmetry is restored. In terms of
the order parameter, the quark condensate $\langle\bar{q}q\rangle$ may vanish
through the transition. It is of great interest to study how the diquarks are
affected by the change of the chiral order parameter. For such a study, the
effective theory based on the linear sigma model is most suitable though it
may be too simplified. We here take the simplest approach for the chiral
symmetry restoration by changing the chiral condensate $\langle\Sigma\rangle$
towards zero in due course.

Figure 1: Dependence of the diquark masses on the chiral symmetry breaking
parameter $x$.

In the present chiral effective theory, the diquark masses of the V and A
diquarks will become degenerate to $m_{V0}$ when the chiral condensate
$\langle\Sigma\rangle$ becomes zero. In Table 5, one sees that $m_{V0}$ is
about 700 MeV and is about 400 MeV lower than the mass of S and P diquarks in
the chiral limit, $m_{S0}$. In contrast, the diquark masses in the ordinary
vacuum is aligned so that the A diquark is heavier than the S diquark. Let us
vary $\langle\Sigma\rangle$ by multiplying a factor $x$ and change $x$ from
the vacuum, $x=1$, to the chiral symmetry restored phase, $x=0$.
Correspondingly, the mass formulas for the non-strange S and A diquarks are
given by

$\displaystyle
M(0^{+})=\sqrt{m^{2}_{S0}-(x+\epsilon)m^{2}_{S1}-x^{2}m^{2}_{S2}},$ (39)
$\displaystyle M(1^{+})=\sqrt{m^{2}_{V0}+x^{2}(m^{2}_{V1}+2m^{2}_{V2})}.$ (40)

Table 6: Masses of the ground-state baryons with the S and A diquark. The values with asterisk (*) denote the input values from PDG Zyla  _et al._ (2020). |  | Baryon masses in the ground state (MeV)
---|---|---
Baryon ($J^{P}$) | Diquark ($J^{P}$) | Potential model |
|  |  Y-pot. Yoshida  _et al._ (2015) |  S-pot. Silvestre-Brac (1996) |  B-pot. Barnes  _et al._ (2005) | Experiment Zyla  _et al._ (2020)
$\Lambda_{c}$ ($1/2^{+}$) | S ($0^{+}$) | 2286* | 2286* | 2286* | 2286.46
$\Sigma_{c}$ ($1/2^{+}$) | A ($1^{+}$) | 2452 | 2453 | 2441 | 2453.54
$\Sigma_{c}$ ($3/2^{+}$) | A ($1^{+}$) | 2516 | 2517 | 2521 | 2518.13
$\Xi_{c}$ ($1/2^{+}$) | S ($0^{+}$) | 2469* | 2469* | 2469* | 2469.42
$\Xi_{c}^{\prime}$ ($1/2^{+}$) | A ($1^{+}$) | 2583 | 2583 | 2571 | 2578.80
$\Xi_{c}^{\prime}$ ($3/2^{+}$) | A ($1^{+}$) | 2642 | 2643 | 2647 | 2645.88
$\Omega_{c}$ ($1/2^{+}$) | A ($1^{+}$) | 2700 | 2710 | 2689 | 2695.20
$\Omega_{c}$ ($3/2^{+}$) | A ($1^{+}$) | 2755 | 2758 | 2762 | 2765.90
$\Lambda_{b}$ ($1/2^{+}$) | S ($0^{+}$) | 5620* | 5620* | $\ldots$ | 5619.60
$\Sigma_{b}$ ($1/2^{+}$) | A ($1^{+}$) | 5810 | 5797 | $\ldots$ | 5813.10
$\Sigma_{b}$ ($3/2^{+}$) | A ($1^{+}$) | 5829 | 5816 | $\ldots$ | 5832.53
$\Xi_{b}$ ($1/2^{+}$) | S ($0^{+}$) | 5796 | 5785 | $\ldots$ | 5794.45
$\Xi_{b}^{\prime}$ ($1/2^{+}$) | A ($1^{+}$) | 5934 | 5914 | $\ldots$ | 5935.02
$\Xi_{b}^{\prime}$ ($3/2^{+}$) | A ($1^{+}$) | 5952 | 5933 | $\ldots$ | 5953.82
$\Omega_{b}$ ($1/2^{+}$) | A ($1^{+}$) | 6047 | 6022 | $\ldots$ | 6046.10
$\Omega_{b}$ ($3/2^{+}$) | A ($1^{+}$) | 6064 | 6040 | $\ldots$ | $\ldots$
Table 7: Masses of the $\rho$-mode excited-state baryons with the P and V diquark.  |  | Baryon masses in the $\rho$-mode state (MeV)
---|---|---
Baryon ($J^{P}$) | Diquark ($J^{P}$) | Potential model |
|  |  Y-pot. Yoshida  _et al._ (2015) |  S-pot. Silvestre-Brac (1996) |  B-pot. Barnes  _et al._ (2005) | Experiment Zyla  _et al._ (2020)
$\Lambda_{c}$ ($1/2^{-}$) | P ($0^{-}$) | 2890 | 2890 | 2890 | $\ldots$
$\Lambda_{c}$ ($1/2^{-}$) | V ($1^{-}$) | 2894 | 2892 | 2881 | $\ldots$
$\Lambda_{c}$ ($3/2^{-}$) | V ($1^{-}$) | 2943 | 2944 | 2949 | (2939.60)
$\Xi_{c}$ ($1/2^{-}$) | P ($0^{-}$) | 2765 | 2758 | 2758 | (2793.25)
$\Xi_{c}$ ($1/2^{-}$) | V ($1^{-}$) | 3209 | 3215 | 3205 | $\ldots$
$\Xi_{c}$ ($3/2^{-}$) | V ($1^{-}$) | 3252 | 3261 | 3267 | $\ldots$
$\Lambda_{b}$ ($1/2^{-}$) | P ($0^{-}$) | 6207 | 6174 | $\ldots$ | $\ldots$
$\Lambda_{b}$ ($1/2^{-}$) | V ($1^{-}$) | 6233 | 6197 | $\ldots$ | $\ldots$
$\Lambda_{b}$ ($3/2^{-}$) | V ($1^{-}$) | 6249 | 6214 | $\ldots$ | $\ldots$
$\Xi_{b}$ ($1/2^{-}$) | P ($0^{-}$) | 6084 | 6051 | $\ldots$ | $\ldots$
$\Xi_{b}$ ($1/2^{-}$) | V ($1^{-}$) | 6540 | 6497 | $\ldots$ | $\ldots$
$\Xi_{b}$ ($3/2^{-}$) | V ($1^{-}$) | 6554 | 6513 | $\ldots$ | $\ldots$
Table 8: Masses of the $\lambda$-mode excited-state baryons with the S and A diquark. |  | Baryon masses in the $\lambda$-mode state (MeV)
---|---|---
Baryon ($J^{P}$) | Diquark ($J^{P}$) | Potential model |
|  |  Y-pot. Yoshida  _et al._ (2015) |  S-pot. Silvestre-Brac (1996) |  B-pot. Barnes  _et al._ (2005) | Experiment Zyla  _et al._ (2020)
$\Lambda_{c}$ ($1/2^{-}$) | S ($0^{+}$) | 2589 | 2676 | 2700 | (2592.25)
$\Lambda_{c}$ ($3/2^{-}$) | S ($0^{+}$) | 2625 | 2716 | 2749 | (2628.11)
$\Sigma_{c}$ ($1/2^{-}$) | A ($1^{+}$) | 2717 | 2807 | 2830 | $\ldots$
$\Sigma_{c}$ ($1/2^{-}$) | A ($1^{+}$) | 2751 | 2847 | 2897 | $\ldots$
$\Sigma_{c}$ ($3/2^{-}$) | A ($1^{+}$) | 2781 | 2881 | 2919 | $\ldots$
$\Sigma_{c}$ ($3/2^{-}$) | A ($1^{+}$) | 2811 | 2916 | 2964 | $\ldots$
$\Sigma_{c}$ ($5/2^{-}$) | A ($1^{+}$) | 2844 | 2953 | 2994 | $\ldots$
$\Xi_{c}$ ($1/2^{-}$) | S ($0^{+}$) | 2754 | 2852 | 2887 | (2793.25)
$\Xi_{c}$ ($3/2^{-}$) | S ($0^{+}$) | 2787 | 2890 | 2933 | (2818.50)
$\Xi_{c}^{\prime}$ ($1/2^{-}$) | A ($1^{+}$) | 2845 | 2942 | 2971 | $\ldots$
$\Xi_{c}^{\prime}$ ($1/2^{-}$) | A ($1^{+}$) | 2876 | 2979 | 3033 | $\ldots$
$\Xi_{c}^{\prime}$ ($3/2^{-}$) | A ($1^{+}$) | 2902 | 3010 | 3052 | $\ldots$
$\Xi_{c}^{\prime}$ ($3/2^{-}$) | A ($1^{+}$) | 2926 | 3037 | 3090 | $\ldots$
$\Xi_{c}^{\prime}$ ($5/2^{-}$) | A ($1^{+}$) | 2958 | 3073 | 3120 | $\ldots$
$\Omega_{c}$ ($1/2^{-}$) | A ($1^{+}$) | 2960 | 3064 | 3098 | $\ldots$
$\Omega_{c}$ ($1/2^{-}$) | A ($1^{+}$) | 2989 | 3100 | 3156 | $\ldots$
$\Omega_{c}$ ($3/2^{-}$) | A ($1^{+}$) | 3013 | 3127 | 3173 | $\ldots$
$\Omega_{c}$ ($3/2^{-}$) | A ($1^{+}$) | 3033 | 3150 | 3206 | $\ldots$
$\Omega_{c}$ ($5/2^{-}$) | A ($1^{+}$) | 3063 | 3185 | 3235 | $\ldots$
$\Lambda_{b}$ ($1/2^{-}$) | S ($0^{+}$) | 5914 | 6018 | $\ldots$ | (5912.20)
$\Lambda_{b}$ ($3/2^{-}$) | S ($0^{+}$) | 5927 | 6033 | $\ldots$ | (5919.92)
$\Sigma_{b}$ ($1/2^{-}$) | A ($1^{+}$) | 6043 | 6143 | $\ldots$ | $\ldots$
$\Sigma_{b}$ ($1/2^{-}$) | A ($1^{+}$) | 6065 | 6171 | $\ldots$ | $\ldots$
$\Sigma_{b}$ ($3/2^{-}$) | A ($1^{+}$) | 6079 | 6187 | $\ldots$ | $\ldots$
$\Sigma_{b}$ ($3/2^{-}$) | A ($1^{+}$) | 6117 | 6234 | $\ldots$ | $\ldots$
$\Sigma_{b}$ ($5/2^{-}$) | A ($1^{+}$) | 6129 | 6247 | $\ldots$ | $\ldots$
$\Xi_{b}$ ($1/2^{-}$) | S ($0^{+}$) | 6069 | 6178 | $\ldots$ | $\ldots$
$\Xi_{b}$ ($3/2^{-}$) | S ($0^{+}$) | 6080 | 6192 | $\ldots$ | $\ldots$
$\Xi_{b}^{\prime}$ ($1/2^{-}$) | A ($1^{+}$) | 6164 | 6269 | $\ldots$ | $\ldots$
$\Xi_{b}^{\prime}$ ($1/2^{-}$) | A ($1^{+}$) | 6183 | 6293 | $\ldots$ | $\ldots$
$\Xi_{b}^{\prime}$ ($3/2^{-}$) | A ($1^{+}$) | 6195 | 6307 | $\ldots$ | $\ldots$
$\Xi_{b}^{\prime}$ ($3/2^{-}$) | A ($1^{+}$) | 6227 | 6346 | $\ldots$ | $\ldots$
$\Xi_{b}^{\prime}$ ($5/2^{-}$) | A ($1^{+}$) | 6238 | 6359 | $\ldots$ | $\ldots$
$\Omega_{b}$ ($1/2^{-}$) | A ($1^{+}$) | 6273 | 6383 | $\ldots$ | $\ldots$
$\Omega_{b}$ ($1/2^{-}$) | A ($1^{+}$) | 6290 | 6404 | $\ldots$ | $\ldots$
$\Omega_{b}$ ($3/2^{-}$) | A ($1^{+}$) | 6301 | 6418 | $\ldots$ | $\ldots$
$\Omega_{b}$ ($3/2^{-}$) | A ($1^{+}$) | 6329 | 6452 | $\ldots$ | $\ldots$
$\Omega_{b}$ ($5/2^{-}$) | A ($1^{+}$) | 6339 | 6464 | $\ldots$ | $\ldots$

From the numerical values in all the potential models in Table 5, we find that
the $m_{S1}$ term is dominant in the S diquark mass, while the $m_{V2}$ term
is dominant in the A diquark. Fig. 1 illustrates the masses of the $0^{\pm}$
and $1^{\pm}$ diquarks as functions of $x$ for the Y-potential. One clearly
sees that the mass of the S diquark will increase when we change $x$ from 1
towards 0, for the Y-potential, starting from 725 MeV at $x=1$ reaching 967
MeV at $x=0$. On the other hand, the mass of the A diquark will decrease from
973 MeV ($x=1$) to 708 MeV ($x=0$). Thus there is a crossing in between.
Namely, the inversion of the S and A diquarks will occur when the chiral
symmetry is restored at finite temperature/baryon density. In our Y-potential
parameter set, the crossing occurs at around $x=0.6$.

It is also noted that at the chiral restoration, the masses of the non-strange
S and P diquarks are not degenerate due to the contribution of the strange
quark mass induced by the $U_{A}(1)$ anomaly. Here, for simplicity, we assume
that the $U_{A}(1)$ anomaly stays intact under the chiral symmetry
restoration.

This behavior of the diquark masses towards restoration of chiral symmetry may
change the baryon spectrum significantly at finite temperature and/or baryon
density. It may also affect the appearance of color superconductivity at dense
matter, where the A diquarks may be lighter than the S diquarks.

##  IV Numerical results

###  IV.1 Spectrum of singly charmed baryons

Figure 2: The energy spectra of singly charmed baryons from Tables 6, 7, and
8, given by the Y-potential. The colors of lines show the types of constituent
diquarks as S (cyan), P (purple), V (magenta), and A (green). For the negative
parity states, they are classified into the $\rho$-mode and the $\lambda$-mode
with the symbols ${\bm{\rho}}$ and ${\bm{\lambda}}$, respectively. Figure 3:
Contribution of the spin-dependent terms of the Y-potential to the energy
splitting in the $\lambda$-mode states of $\Sigma_{c}$. The black dotted lines
indicate the splitting and/or mixing of the states by adding the potential.
Figure 4: The energy spectra of singly bottom baryons from Tables 6, 7, and 8,
given by the Y-potential. The colors of lines show the types of constituent
diquarks as S (cyan), P (purple), V (magenta), and A (green). For the
negative-parity states, they are classified into the $\rho$-mode and the
$\lambda$-mode with the symbols ${\bm{\rho}}$ and ${\bm{\lambda}}$,
respectively.

Numerical results of masses of singly charmed baryons are summarized in the
upper parts of Tables 6, 7, and 8, where we show the ground states,
$\rho$-mode states, and $\lambda$-mode states, respectively.

First we explain which values are used to determine the diquark masses and the
potential parameters. The values with asterisk in Table 6 are the experimental
data taken from PDG Zyla  _et al._ (2020). Also, in Table 7, the mass of
$\Lambda_{c}(1/2^{-})$ with the P diquark, 2890 MeV, is taken from Ref.
Yoshida  _et al._ (2015). They are used to determine the parameter $C_{c}$ and
the masses of the S and P diquarks Kim  _et al._ (2020). In addition, for
determining $\kappa_{c}$ and $\eta$, we used the observed mass differences
between the $\Sigma_{c}(1/2^{+})$ and $\Sigma_{c}(3/2^{+})$ and between the
$\lambda$-modes of the $\Lambda_{c}(1/2^{-})$ and $\Lambda_{c}(3/2^{-})$.

The energy spectra of singly charmed baryons from the Y-potential are shown in
Fig. 2, where the difference in color indicates the type of the constituent
diquarks inside baryons: S (cyan), P (purple), V (magenta), and A (green).
Note that the results of $\Lambda_{c}$ and $\Xi_{c}$ baryons including the S
or P diquark were already reported in the previous work Kim  _et al._ (2020).
However, in Ref. Kim  _et al._ (2020), the spin-orbit potential was neglected
for simplicity, so that the $J^{P}=1/2^{-}$ and $3/2^{-}$ states in the
$\lambda$-modes were degenerate.

$\Lambda_{c}$ and $\Xi_{c}$ can include three types of constituent diquarks
with flavor $\bar{\bm{3}}$. First we focus on the ground states and the
$\lambda$-mode states with the S diquark and the $\rho$-mode states with the P
diquark, which is explained in the previous work Kim  _et al._ (2020). In each
state, the masses of singly charmed baryons including S diquark become heavier
with increasing the number of constituent strange quarks. On the other hand,
for the $\rho$-modes of baryons with the P diquark, the non-strange baryon is
heavier than that of including one strange quark as
$M_{\rho}(\Lambda_{c},1/2^{-})>M_{\rho}(\Xi_{c},1/2^{-})$. This is caused by
the inverse-mass hierarchy of P diquarks, Eq. (6). Another diquark appearing
in the spectrum of $\Lambda_{c}$ and $\Xi_{c}$ is the V diquark, which makes
the heaviest $\rho$-mode states. The masses of $\Xi_{c}$ are about 300 MeV
heavier than those of $\Lambda_{c}$. This coincides with the mass difference
between the strange and non-strange V diquarks is
$M_{qs}(1^{-})-M_{qq}(1^{-})=329$ MeV, as shown in Table 5.

Next we discuss the spectra of $\Sigma_{c}$, $\Xi^{\prime}_{c}$, and
$\Omega_{c}$. These baryons include the A diquark, which gives both the ground
states and $\lambda$-mode excited states. In our model, the masses of these
baryons become heavier as the total angular momentum $J$ is larger. For
instance, in the ground states, the masses for $J^{P}=3/2^{+}$ are about 50
MeV larger than those of $1/2^{+}$. Also the masses of $\lambda$-modes with
$J^{P}=1/2^{-}$, $3/2^{-}$, and $5/2^{-}$ are in order.

The mass differences between $\Sigma_{c}$ and $\Xi^{\prime}_{c}$ with the same
$J^{P}$ are about 130 MeV, and those between $\Xi^{\prime}_{c}$ and
$\Omega_{c}$ are about 120 MeV. This behavior is regarded as a generalization
of the conventional Gell-Mann–Okubo mass formula Gell-Mann (1962); Okubo
(1962). Values of these mass differences are almost from those of A diquarks,
$M_{qs}(1^{+})-M_{qq}(1^{+})=143$ MeV and $M_{ss}(1^{+})-M_{qs}(1^{+})=126$
MeV, which are smaller than that of V diquarks
($M_{qs}(1^{-})-M_{qq}(1^{-})=329$ MeV). This is due to the negative value of
the parameter $m_{V1}^{2}<0$, where the square mass difference between the A
diquarks in Eq. (25) becomes smaller than that between the V diquarks in Eq.
(27).

Comparing with each potential, there is no much difference in the ground state
and the $\rho$-modes according to Tables 6 and 7. However, in the
$\lambda$-mode state in Table 8, the masses given by the S- and B-potentials
are approximately 100 MeV heavier than those of the Y-potential. In other
words, the excitation energy from the ground state to the $\lambda$-mode is
approximately 200–300 MeV for the Y-potential and 300–400 MeV for the S- and
B-potentials. This difference is caused by the coefficients of the Coulomb
potential, $\alpha$, whose value is approximately $0.1$ for the Y-potential
and is smaller than the other two potentials. As a result, the wave function
obtained from the Y-potential becomes broader, and its difference between the
ground state and $\lambda$-mode becomes smaller. Therefore, the excitation
energy decreases as $\alpha$ becomes smaller, as explained in the previous
work Kim  _et al._ (2020). According to the Table 8, about the masses of the
$\lambda$-mode of $\Lambda_{c}(1/2^{-})$ and $\Lambda_{c}(3/2^{-})$, the
numerical results by the Y-potential are mostly closest to the experimental
data from PDG Zyla  _et al._ (2020) than the other two potentials. From now,
we mainly use the Y-potential for the two-body calculation.

Lastly, we discuss the splitting of energy spectrum by the spin dependent
potentials. In Fig. 3, we sketch the splitting behaviors in the
$\lambda$-modes of $\Sigma_{c}$ by adding the spin-spin ($V_{ss}$), spin-orbit
($V_{so}$), and tensor ($V_{ten}$) potentials. At first, two $J^{P}=1/2^{-}$,
two $J^{P}=3/2^{-}$, and one $J^{P}=5/2^{-}$ states are degenerate, and the
spin-spin potential splits these five degenerate states into two parts. The
lower states with $J^{P}=1/2^{-}$ and $3/2^{-}$ are degenerate, and the higher
states with $J^{P}=1/2^{-},3/2^{-}$, and $5/2^{-}$ also do. Here, the mass
difference between the lower and higher parts is about 20 MeV. After this, the
symmetric term of spin-orbit potential $V_{so}^{+}$ splits these five states
completely, and the antisymmetric term $V_{so}^{-}$ mixes the states with the
same $J^{P}$. We can clearly see that the splitting by the spin-orbit
potential changes the ordering of total angular momentum $J$ into
$1/2^{-}<1/2^{-}<3/2^{-}<3/2^{-}<5/2^{-}$, and the splitting range is
approximately 120 MeV. Finally, the contribution from the tensor potential is
very small.

###  IV.2 Spectrum of singly bottom baryons

Similarly to singly charmed baryons, the numerical results for singly bottom
baryons are summarized in the lower parts of Tables 6, 7, and 8. Here only the
mass of $\Lambda_{b}(1/2^{+})$ including the S diquark is the input value for
determining the parameter $C_{b}$ Kim  _et al._ (2020). The ground states of
$\Sigma_{b}(1/2^{+})$ and $\Sigma_{b}(3/2^{+})$ are used to determine the
parameter $\kappa_{b}$.

The energy spectra of singly bottom baryons from the Y-potential are shown in
Fig. 4. We can see that the spectrum of singly bottom baryons are similar to
those of singly charmed baryon. In particular, we can find the inverse-mass
hierarchy of the $\rho$-mode excitations as
$M_{\rho}(\Lambda_{b},1/2^{-})>M_{\rho}(\Xi_{b},1/2^{-})$, which is caused by
the inverse hierarchy of the pseudoscalar diquarks in Eq. (6).

By comparing Figs. 2 and 4, one sees the heavy-quark mass dependence of masses
of singly heavy baryons. The spin-spin potential $V_{ss}$, Eqs. (32) and (33),
and the tensor potential $V_{ten}$, Eq. (37), are inversely proportional to
both the diquark mass and heavy-quark mass ($V_{ss,ten}\propto 1/M_{d}M_{Q}$).
Also, as in Eqs. (35) and (36), some terms of the spin-orbit potential
$V_{so}$ are inversely proportional to the heavy-quark mass $M_{Q}$. In the
heavy-quark limit ($M_{Q}\rightarrow\infty$), these terms become zero, and the
surviving term in the spin-orbit potential is

$\displaystyle V_{so}(r)=\eta\frac{(1-e^{\Lambda
r})^{2}}{r^{3}}\frac{2}{M_{d}^{2}}({\bm{L}}\cdot{\bm{s}}_{d}),$ (41)

which is independent of the heavy-quark spin ${\bm{s}}_{Q}$. This limit
induces characteristic structures of hadron mass spectra: (i) the heavy-quark
spin (HQS) doublet with $J=j\pm 1/2$, which is a pair of two states
characterized by the angular momentum $j$ of the light components and the
heavy-quark spin $1/2$, and (ii) the HQS singlet, where the angular momentum
of light components is zero ($j=0$).

For example, the ground states of $\Sigma_{Q}$, $\Xi^{\prime}_{Q}$, and
$\Omega_{Q}$ with the A diquark consist of two states with $J=1/2$ and $3/2$
using the diquark spin $j=1$. The mass difference between $J^{P}=1/2^{+}$ and
$3/2^{+}$ is approximately 60 MeV for the charmed baryons, while that for the
bottom baryons is about 20 MeV. Such a smaller mass difference for the bottom
baryons is an example of the HQS doublet structure. In our spectra, similar
HQS doublet structures can be also found in the $\lambda$ modes of
$\Lambda_{Q}$ and $\Xi_{Q}$ with the S diquark (using the orbital angular
momentum $j=1$) and the $\rho$ modes of $\Lambda_{Q}$ and $\Xi_{Q}$ with the V
diquark (using the diquark spin $j=1$). Furthermore, the spectra of the
$\lambda$ modes of $\Sigma_{Q}$, $\Xi^{\prime}_{Q}$, and $\Omega_{Q}$ consist
of two HQS doublets ($J=1/2$ and $3/2$ using $j=1$, and $J=3/2$ and $5/2$
using $j=2$) and one HQS singlet ($J=1/2$ using $j=0$ due to the cancellation
of the diquark spin and orbital angular momentum).

###  IV.3 Spectrum of positive-parity excited states

Table 9: Masses of radially-excited-state baryons with the number of node $n=1$ and the $D$-wave excited-state baryons with the orbital angular momentum of the $\lambda$-coordinate $L=2$. Y-potential is used for this calculation.  Charmed baryon ($J^{P}$) |  $n$ |  $L$ | Mass (MeV) | Experiment Zyla  _et al._ (2020) | Bottom Baryon ($J^{P}$) |  $n$ |  $L$ | Mass (MeV) | Experiment Zyla  _et al._ (2020)
---|---|---|---|---|---|---|---|---|---
$\Lambda_{c}(1/2^{+})$ | 1 | $0$ | 2825 | (2765) | $\Lambda_{b}(1/2^{+})$ | 1 | $0$ | 6121 | …
$\Lambda_{c}(3/2^{+})$ | 0 | $2$ | 2869 | 2856.10 | $\Lambda_{b}(3/2^{+})$ | 0 | $2$ | 6172 | 6146.20
$\Lambda_{c}(5/2^{+})$ | 0 | $2$ | 2897 | 2881.63 | $\Lambda_{b}(5/2^{+})$ | 0 | $2$ | 6178 | 6152.50
$\Sigma_{c}(1/2^{+})$ | 1 | $0$ | 2974 | … | $\Sigma_{b}(1/2^{+})$ | 1 | $0$ | 6274 | …
$\Sigma_{c}(1/2^{+})$ | 0 | $2$ | 3016 | … | $\Sigma_{b}(1/2^{+})$ | 0 | $2$ | 6304 | …
$\Sigma_{c}(3/2^{+})$ | 1 | $0$ | 3013 | … | $\Sigma_{b}(3/2^{+})$ | 1 | $0$ | 6286 | …
$\Sigma_{c}(3/2^{+})$ | 0 | $2$ | 3043 | … | $\Sigma_{b}(3/2^{+})$ | 0 | $2$ | 6316 | …
$\Sigma_{c}(3/2^{+})$ | 0 | $2$ | 3053 | … | $\Sigma_{b}(3/2^{+})$ | 0 | $2$ | 6330 | …
$\Sigma_{c}(5/2^{+})$ | 0 | $2$ | 3076 | … | $\Sigma_{b}(5/2^{+})$ | 0 | $2$ | 6341 | …
$\Sigma_{c}(5/2^{+})$ | 0 | $2$ | 3094 | … | $\Sigma_{b}(5/2^{+})$ | 0 | $2$ | 6365 | …
$\Sigma_{c}(7/2^{+})$ | 0 | $2$ | 3115 | … | $\Sigma_{b}(7/2^{+})$ | 0 | $2$ | 6373 | …
$\Xi_{c}(1/2^{+})$ | 1 | $0$ | 2976 | … | $\Xi_{b}(1/2^{+})$ | 1 | $0$ | 6260 | …
$\Xi_{c}(3/2^{+})$ | 0 | $2$ | 3017 | … | $\Xi_{b}(3/2^{+})$ | 0 | $2$ | 6307 | …
$\Xi_{c}(5/2^{+})$ | 0 | $2$ | 3043 | … | $\Xi_{b}(5/2^{+})$ | 0 | $2$ | 6313 | …
$\Xi^{\prime}_{c}(1/2^{+})$ | 1 | $0$ | 3088 | … | $\Xi^{\prime}_{b}(1/2^{+})$ | 1 | $0$ | 6381 | …
$\Xi^{\prime}_{c}(1/2^{+})$ | 0 | $2$ | 3131 | … | $\Xi^{\prime}_{b}(1/2^{+})$ | 0 | $2$ | 6411 | …
$\Xi^{\prime}_{c}(3/2^{+})$ | 1 | $0$ | 3124 | … | $\Xi^{\prime}_{b}(3/2^{+})$ | 1 | $0$ | 6392 | …
$\Xi^{\prime}_{c}(3/2^{+})$ | 0 | $2$ | 3155 | … | $\Xi^{\prime}_{b}(3/2^{+})$ | 0 | $2$ | 6423 | …
$\Xi^{\prime}_{c}(3/2^{+})$ | 0 | $2$ | 3164 | … | $\Xi^{\prime}_{b}(3/2^{+})$ | 0 | $2$ | 6434 | …
$\Xi^{\prime}_{c}(5/2^{+})$ | 0 | $2$ | 3186 | … | $\Xi^{\prime}_{b}(5/2^{+})$ | 0 | $2$ | 6444 | …
$\Xi^{\prime}_{c}(5/2^{+})$ | 0 | $2$ | 3200 | … | $\Xi^{\prime}_{b}(5/2^{+})$ | 0 | $2$ | 6465 | …
$\Xi^{\prime}_{c}(7/2^{+})$ | 0 | $2$ | 3221 | … | $\Xi^{\prime}_{b}(7/2^{+})$ | 0 | $2$ | 6472 | …
$\Omega_{c}(1/2^{+})$ | 1 | $0$ | 3194 | … | $\Omega_{b}(1/2^{+})$ | 1 | $0$ | 6480 | …
$\Omega_{c}(1/2^{+})$ | 0 | $2$ | 3236 | … | $\Omega_{b}(1/2^{+})$ | 0 | $2$ | 6511 | …
$\Omega_{c}(3/2^{+})$ | 1 | $0$ | 3227 | … | $\Omega_{b}(3/2^{+})$ | 1 | $0$ | 6491 | …
$\Omega_{c}(3/2^{+})$ | 0 | $2$ | 3259 | … | $\Omega_{b}(3/2^{+})$ | 0 | $2$ | 6522 | …
$\Omega_{c}(3/2^{+})$ | 0 | $2$ | 3268 | … | $\Omega_{b}(3/2^{+})$ | 0 | $2$ | 6532 | …
$\Omega_{c}(5/2^{+})$ | 0 | $2$ | 3288 | … | $\Omega_{b}(5/2^{+})$ | 0 | $2$ | 6541 | …
$\Omega_{c}(5/2^{+})$ | 0 | $2$ | 3300 | … | $\Omega_{b}(5/2^{+})$ | 0 | $2$ | 6559 | …
$\Omega_{c}(7/2^{+})$ | 0 | $2$ | 3320 | … | $\Omega_{b}(7/2^{+})$ | 0 | $2$ | 6567 | …
Figure 5: The energy spectra of $\Lambda_{c}$ and $\Sigma_{c}$ from Tables 6,
7, 8, and 9, given by the Y-potential. The colors of lines show the parities
(Blue: positive parity. Red: negative parity.). For the blue spectra, they are
classified by $n=0$ or $1$ and $L=0(S)$ or $2(D)$, expressed as the symbol
$nL=0S,1S$, and $0D$. Figure 6: The energy spectra of $\Lambda_{b}$ and
$\Sigma_{b}$ from Tables 6, 7, 8, and 9, given by the Y-potential.

In the previous subsections, we have discussed the (positive-parity) ground
states and negative-parity excited states. Here, we study two types of
positive-parity excited states. The first is the radially excited states,
where the number of nodes is $n=1$ (i.e., the principal quantum number is
$2$), and the orbital angular momentum is zero ($L=0$). The second is the
$D$-wave excited states, where the orbital angular momentum in the $\lambda$
coordinate is $L=2$. These states have positive parity and are composed of the
S or A diquarks. The numerical results from the Y-potential are summarized in
Table 9, where the left and right parts show the singly charmed and bottom
baryons, respectively. Among them, we show the energy spectra of
$\Lambda_{c/b}$ and $\Sigma_{c/b}$ in Figs. 5 and 6, respectively.

From these figures, we find that the masses of the radially excited states are
lower than those of the corresponding $D$-wave excited states. For example,
$\Lambda_{c}(1/2^{+},1S)$ is lighter than $\Lambda_{c}(3/2^{+},0D)$ and
$\Lambda_{c}(5/2^{+},0D)$. However, the mass difference between these states
are not so much large. Also, for the $1S$ and $0D$ states of $\Lambda_{c/b}$
and $\Sigma_{c/b}$, as the total angular momentum becomes larger, the mass
becomes heavier. For the singly bottom baryons, we can clearly see the HQS
doublet structures of the positive-parity excited states. As a result of this
structure, the range of the energy spectrum of bottom baryons tends to shrink.

We compare the masses of the positive-parity excited states and those of
negative-parity $\rho$\- and $\lambda$-modes. For $\Lambda_{c/b}$, we can see
that their masses are close to each other, and the $\rho$-modes are slightly
heavier than the positive-parity excited states. On the other hand, the
$\lambda$-modes are lying in between the ground state and the positive-parity
excited states. This tendency for the $\lambda$-modes is also seen in the
spectrum of $\Sigma_{c/b}$.

###  IV.4 Comparison with experimental data

Figure 7: Energy spectra of singly charmed baryons from our model with the
Y-potential (blue or red dotted lines), and from the experimental data in PDG
Zyla  _et al._ (2020) (black solid lines). Here we put the spectra of
$\Xi_{c}$ and $\Xi^{\prime}_{c}$ together. Figure 8: Energy spectra of singly
bottom baryons from our model with the Y-potential (blue or red dotted lines),
and from the experimental data in PDG Zyla  _et al._ (2020) (black solid
lines). Here we put the spectra of $\Xi_{b}$ and $\Xi^{\prime}_{b}$ together.
Note that $\Lambda_{b}(1/2^{+})$ with the mass of 6072 MeV and $\Omega_{b}$
excited states with the asterisk (*) are the experimental values from Refs.
Aaij  _et al._ (2020a, b).

Here, we compare the numerical results and experimental data. In the last
column of Tables 6, 7, 8, and 9, we show the masses of the baryons whose spin
and parity are specified by PDG Zyla  _et al._ (2020). In Figs. 7 and 8, we
show the energy spectra of singly heavy baryons, where both our results and
experimental data are shown.

The masses of the ground states of $\Lambda_{c/b}(1/2^{+})$ and
$\Xi_{c}(1/2^{+})$ are the input values. Also the mass differences between the
ground states of $\Sigma_{c/b}(1/2^{+})$ and $\Sigma_{c/b}(3/2^{+})$ and
between the $\lambda$-mode states of $\Lambda_{c}(1/2^{-})$ and
$\Lambda_{c}(3/2^{-})$ are the input values, where the obtained mass values
are similar to the observed data. Therefore the spectra of
$\Xi^{(^{\prime})}_{b}$ and $\Omega_{c/b}$ are the pure outputs from our
model.

It is obvious that, in the ground states, all the mass values from our model
are near the experimental values. On the other hand, for the $\lambda$-mode
states of $\Omega_{c/b}$, the masses from our model seem to be lower than the
observed values. This is because we have not adjusted the potential parameters
of the Coulomb interaction term $\alpha$ and the linear confinement term
$\lambda$ for our heavy-quark–diquark model.

In Table 9, $\Lambda_{c/b}(3/2^{+})$ and $\Lambda_{c/b}(5/2^{+})$ are
experimentally observed. We expect that these states are the $D$-wave excited
states ($L=2$) from the viewpoints of our diquark model. Although the masses
from our model are about 20 MeV larger than the experimental values, the mass
splitting between the $3/2^{+}$ and $5/2^{+}$ states is almost same as the
experimental value.

Next we comment on the $n=1$ excited states with $1/2^{+}$. The mass of the
first excited state of $\Lambda_{c}(1/2^{+})$ is known to be 2765 MeV. Its
isospin is determined by the Belle collaboration Tanida  _et al._ (2019),
while $J^{P}$ is not determined yet but may be assumed as $1/2^{+}$. In
addition, recently, the LHCb Collaboration have observed the first excited
state of $\Lambda_{b}(1/2^{+})$ whose mass is 6072 MeV Aaij  _et al._ (2020a).
These states may be analogous to the Roper resonance in the nucleon spectrum.
Our results, $\Lambda_{c}(1/2^{+})=2825$ MeV and $\Lambda_{b}(1/2^{+})=6121$
MeV, are approximately 50 MeV heavier than the experimental masses. Thus, so
far, we cannot provide an interpretation based on the heavy-quark–diquark
picture.

For the other excited states $\Sigma_{c/b}$, $\Xi_{c/b}$, and $\Omega_{c/b}$,
$J^{P}$ is still unknown, except for $\Xi_{c}(1/2^{-})$ and
$\Xi_{c}(3/2^{-})$. It may be difficult to determine the unknown total angular
momenta from our model, but at least we can expect the signs of parity. For
instance, $\Sigma_{c}(2800)$, $\Sigma_{b}(6097)$, the five states of
$\Omega_{c}$, and the four observed states of $\Omega_{b}$ may be
$\lambda$-mode states with negative parity. On the other hand, it is not
conclusive to identify the quantum numbers of the five higher states of
$\Xi_{c}^{(^{\prime})}$ and $\Xi_{b}(6227)$. Note that, recently,
$\Xi_{c}^{(^{\prime})}(2930)$ is reinterpreted as $\Xi_{c}(2923)$ and
$\Xi_{c}(2939)$ Aaij  _et al._ (2020c), and the spin and parity of
$\Xi_{c}(2970)$ are determined as $1/2^{+}$ Moon  _et al._ (2020). Also, the
new state $\Xi_{b}(6100)$ is recently observed Sirunyan  _et al._ (2021).

###  IV.5 Comparison with three-quark model

Figure 9: Energy spectra of $\Lambda_{c}$ and $\Sigma_{c}$ from our model with
Y-potential (blue or red dotted lines), and from the three-quark model in Ref.
Yoshida  _et al._ (2015) (black solid lines). Figure 10: Energy spectra of
$\Lambda_{b}$ and $\Sigma_{b}$ from our model with Y-potential (blue or red
dotted lines), and from the three-quark model in Ref. Yoshida  _et al._ (2015)
(black solid lines).

Here, we compare the numerical results from our heavy-quark–diquark model and
those from a constituent quark model containing two light quarks and one heavy
quark, which we call the three-quark model. In Ref. Yoshida  _et al._ (2015),
$\Lambda_{c/b}$, $\Sigma_{c/b}$, and $\Omega_{c/b}$ baryons were analyzed,
where the Schr$\ddot{\rm o}$dinger equation with three quarks was solved by
the Gaussian expansion method Kamimura (1988); Hiyama  _et al._ (2003).

In Figs. 9 and 10, we show the comparison of the energy spectra of
$\Lambda_{c/b}$ and $\Sigma_{c/b}$. We can see that the masses from the three-
quark model tend to be heavier than those from our model. Also, in the excited
states, the width of energy splitting from three-quark model is narrower than
our model. For example, the value of splitting between the
$\Lambda_{c}(1/2^{-})$ and $\Lambda_{c}(3/2^{-})$ is 36 MeV for our model,
while Ref. Yoshida  _et al._ (2015) obtained 2 MeV. This difference between
two models is caused by the choice of potential parameters. In Ref. Yoshida
_et al._ (2015), most of parameters were determined so as to reproduce the
masses of hyperons (baryons containing three light quarks with at least one
strange quark), while some parameters in our model are determined from the
singly charmed or bottom baryons. As a result, the coefficient of the spin-
orbit potential is $\eta=0.026$ in Ref. Yoshida  _et al._ (2015), while
$\eta=0.2494$ in our model. Since the strength of the spin-orbit potential is
related to the mass splitting, thus the mass splitting by the spin-orbit
potential in Ref. Yoshida  _et al._ (2015) is much smaller than our results.

Furthermore, due to the difference of the spin-orbit potential parameter
$\eta$, the ordering of total angular momenta $J$ is different from that in
our model. For example, in Fig. 3, the ordering of the negative-parity
$\Sigma_{c}$ from our model changes from
$1/2^{-}=3/2^{-}<1/2^{-}=3/2^{-}=5/2^{-}$ to
$1/2^{-}<1/2^{-}<3/2^{-}<3/2^{-}<5/2^{-}$ adding the spin-orbit potential
$V_{so}$. On the other hand, the ordering from Ref. Yoshida  _et al._ (2015)
is $1/2^{-}<3/2^{-}<1/2^{-}<3/2^{-}<5/2^{-}$ as shown in Fig. 9, since the
strength of spin-orbit potential is weak.

In conclusion, comparing with the experimental data in PDG Zyla  _et al._
(2020), the results from our heavy-quark–diquark model are closer than those
from the three-quark model in Ref. Yoshida  _et al._ (2015).

##  V Conclusion

In this paper, chiral properties and dynamics of the color-$\bar{\bf 3}$,
spin-0 and -1 diquarks are studied. The scalar (S, $0^{+}$) diquark and
pseudoscalar (P, $0^{-}$) diquarks are paired into a chiral
$(\bar{3},1)+(1,\bar{3})$ representation, while the axial-vector (A, $1^{+}$)
and vector (V, $1^{-}$) diquarks are assigned to a chiral $(3,3)$
representation. The S, P, and V diquarks form anti-triplet ($\bar{\bf 3}$)
representationss in flavor $SU(3)$, while the A diquark belongs to a $SU(3)$
sextet (6).

We have then constructed a chiral effective theory of the diquarks in the form
of the linear-sigma model. The masses of the S and P diquarks are given in
term of three mass parameters, $m_{S0}^{2}$, $m_{S1}^{2}$ and $m_{S2}^{2}$.
The first one $m_{S0}^{2}$ is chiral invariant, while the latter two generate
the diquark masses associated with a symmetry breaking condensate of the meson
field $\Sigma$. Furthermore, the second one $m_{S1}^{2}$ represents the
contribution of the axial $U(1)$ anomaly. The mass splitting of the S and P
diquarks are induced by the symmetry breaking terms. In turn, if the chiral
symmetry is fully restored, the S and P diquarks become degenerate, having the
mass $m_{S0}$.

For the A and V diquarks, we also have three mass parameters, $m_{V0}^{2}$,
$m_{V1}^{2}$ and $m_{V2}^{2}$, where the first one is chirally invariant and
the other two are from the chiral symmetry breaking terms. Similarly to the S
and P diquarks, the mass difference between the A and V diquarks is induced by
the $m_{V1}^{2}$ and $m_{V2}^{2}$ parameters.

We have obtained mass formulas for the S, P, A and V diquarks by further
taking into account explicit flavor symmetry breaking due to a heavy mass of
strangeness. We find a few interesting relations among the diquark masses. One
is the inverse hierarchy of the P diquarks, for which the diquark with a
strange quark becomes lighter than the non-strange one,
$M_{qq}(0^{-})>M_{qs}(0^{-})$. The other is a generalized Gell-Mann–Okubo mass
formula for the A diquarks,
$[M_{ss}(1^{+})]^{2}-[M_{qs}(1^{+})]^{2}=[M_{qs}(1^{+})]^{2}-[M_{qq}(1^{+})]^{2}$,
where the squared diquark masses are equally spaced with the increased number
of strange quarks.

In order to quantify the masses of the diquarks in a more realistic manner, we
study the spectrum of singly heavy baryons using the heavy-quark–diquark
model. The heavy-quark–diquark model describes singly heavy baryons, such as
$\Lambda_{c/b}$ and $\Sigma_{c/b}$, as two-body systems of a heavy quark and a
diquark. The potential between the two is given by a linear confinement plus
Coulomb potential with additional spin-dependent forces.

The parameters of the chiral effective Lagrangian and the heavy-quark–diquark
potential are fixed so as to reproduce the low-lying masses of the singly
heavy baryons. We have calculated the negative- and positive-parity excited
states of the singly heavy baryons as well as the ground states. The obtained
spectra are compared with the current experimental data and also the standard
three-quark descriptions in the constituent quark model.

The specific features of the diquark spectrum are also realized in the baryon
spectrum. The inverse mass hierarchy of the P diquarks causes the inverse mass
hierarchy of singly heavy baryons in the $\rho$-mode excited state Kim  _et
al._ (2020). The generalized Gell-Mann–Okubo mass formula is reflected in the
spectrum of $\Sigma_{Q},\Xi^{\prime}_{Q}$ and $\Omega_{Q}$ baryons.

The energy spectra from our heavy-quark–diquark model are mostly consistent
with the experimental data in the ground state. However, our results are
lighter than the observed masses in the excited states. In order to correctly
reproduce these experimental data, we need to readjust the parameters of the
central potentials. Also we have compared our results with the spectra from
the three-quark model in Ref. Yoshida  _et al._ (2015). Here the energy
splittings from our model are larger than those from the constituent three-
quark model, and this difference is caused by the different strength of the
spin-orbit potential. As a consequence, our model reproduced the results
closer to the observed spectra.

In this paper, we have introduced the well-known $S$-wave diquarks, S
($0^{+}$) and A ($1^{+}$), and their chiral partners. By applying them to the
singly heavy baryons, we find all the ground-state baryons are consistent with
the three-body quark models, while the $P$-wave excited states are not fully
recovered. A missing piece is the flavor $\bf 6$ vector diquarks. They should
appear in the negative-parity baryons with flavor $\bf 6$, i.e., $\Sigma_{Q}$,
$\Xi^{\prime}_{Q}$ and $\Omega_{Q}$ excitations. It is necessary to introduce
a set of tensor diquarks with color $\bar{\bf 3}$ and flavor $\bf 6$ Shuryak
and Zahed (2004); Hong  _et al._ (2004); Shuryak (2005).

Several future directions of the use of the diquark effective theory are quite
interesting. So far, we considered only color-$\bar{\bf 3}$ diquarks, while
the other choice, the color-6 diquark is possible. It may not appear in the
singly heavy baryon systems as it cannot make color singlet state with a heavy
quark. It, however, may appear in multi-quark systems, such as tetraquarks,
$\bar{Q}\bar{Q}qq$ and pentaquarks, $\bar{Q}qqqq$ Chen  _et al._ (2016);
Hosaka  _et al._ (2016); Chen  _et al._ (2017b); Olsen  _et al._ (2018); Ali
_et al._ (2018); Liu  _et al._ (2019).

It is also extremely interesting and important to reveal how chiral symmetry
works on the diquarks at finite temperature and baryon/quark density including
the color superconducting phase Alford  _et al._ (1998); Rapp  _et al._
(1998). In fact, we have found that in the course of chiral symmetry
restoration, the masses of the S and A diquarks may be inverted and the A
diquarks may become dominant at the vicinity of chiral phase transition.

## Acknowledgments

This work was supported in part by JSPS KAKENHI Grants No. JP20K03959 (M.O.),
No. JP19H05159 (M.O.), No. JP20K14476 (K.S.), No. JP17K14277 (K.S.), No.
JP21J20048 (Y.K.), and also by NNSFC (No. 11775132) (Y.R.L.).

## References

  * Gell-Mann (1964) M. Gell-Mann, “A Schematic Model of Baryons and Mesons,” Phys. Lett. 8, 214–215 (1964).
  * Ida and Kobayashi (1966) Masakuni Ida and Reido Kobayashi, “Baryon resonances in a quark model,” Prog. Theor. Phys. 36, 846 (1966).
  * Lichtenberg and Tassie (1967) D. B. Lichtenberg and L. J. Tassie, “Baryon Mass Splitting in a Boson-Fermion Model,” Phys. Rev. 155, 1601–1606 (1967).
  * Lichtenberg (1967) D. B. Lichtenberg, “Electromagnetic mass splittings of baryons in a boson-fermion model,” Nuovo Cimento 49, 435 (1967).
  * Souza and Lichtenberg (1967) P. D. De Souza and D. B. Lichtenberg, “Electromagnetic Properties of Hadrons in a Triplet-Sextet Model,” Phys. Rev. 161, 1513–1522 (1967).
  * Lichtenberg  _et al._ (1968) D. B. Lichtenberg, L. J. Tassie, and P. J. Keleman, “Quark-Diquark Model of Baryons and $SU(6)$,” Phys. Rev. 167, 1535–1542 (1968).
  * Carroll  _et al._ (1968) J. Carroll, D. B. Lichtenberg, and J. Franklin, “Electromagnetic properties of baryons in a quark-diquark model with broken $SU(6)$,” Phys. Rev. 174, 1681–1688 (1968).
  * Lichtenberg (1969) D. B. Lichtenberg, “Baryon supermultiplets of $SU(6)\times O(3)$ in a quark-diquark model,” Phys. Rev. 178, 2197–2200 (1969).
  * Anselmino  _et al._ (1993) Mauro Anselmino, Enrico Predazzi, Svante Ekelin, Sverker Fredriksson, and D. B. Lichtenberg, “Diquarks,” Rev. Mod. Phys. 65, 1199–1233 (1993).
  * Jaffe (2005) R.L. Jaffe, “Exotica,” Physics Reports 409, 1–45 (2005).
  * Hess  _et al._ (1998) M. Hess, F. Karsch, E. Laermann, and I. Wetzorke, “Diquark masses from lattice QCD,” Phys. Rev. D 58, 111502 (1998), arXiv:hep-lat/9804023 .
  * Orginos (2006) Konstantinos Orginos, “Diquark properties from lattice QCD,” PoS LAT2005, 054 (2006), arXiv:hep-lat/0510082 .
  * Alexandrou  _et al._ (2006) C. Alexandrou, Ph. de Forcrand, and B. Lucini, “Evidence for diquarks in lattice QCD,” Phys. Rev. Lett. 97, 222002 (2006), arXiv:hep-lat/0609004 .
  * Babich  _et al._ (2007) Ronald Babich, Nicolas Garron, Christian Hoelbling, Joseph Howard, Laurent Lellouch, and Claudio Rebbi, “Diquark correlations in baryons on the lattice with overlap quarks,” Phys. Rev. D 76, 074021 (2007), arXiv:hep-lat/0701023 .
  * DeGrand  _et al._ (2008) Thomas DeGrand, Zhaofeng Liu, and Stefan Schaefer, “Diquark effects in light baryon correlators from lattice QCD,” Phys. Rev. D 77, 034505 (2008), arXiv:0712.0254 [hep-ph] .
  * Green  _et al._ (2010) Jeremy Green, John Negele, Michael Engelhardt, and Patrick Varilly, “Spatial diquark correlations in a hadron,” PoS LATTICE2010, 140 (2010), arXiv:1012.2353 [hep-lat] .
  * Bi  _et al._ (2016) Yujiang Bi, Hao Cai, Ying Chen, Ming Gong, Zhaofeng Liu, Hao-Xue Qiao, and Yi-Bo Yang, “Diquark mass differences from unquenched lattice QCD,” Chin. Phys. C40, 073106 (2016), arXiv:1510.07354 [hep-ph] .
  * Watanabe and Ishii (2021) Kai Watanabe and Noriyoshi Ishii, “Building diquark model from Lattice QCD,” (2021) arXiv:2105.07969 [hep-lat] .
  * Lichtenberg (1975) D. B. Lichtenberg, “Charmed Baryons in a Quark-Diquark Model,” Nuovo Cim. A 28, 563 (1975).
  * Lichtenberg  _et al._ (1982) D. B. Lichtenberg, W. Namgung, E. Predazzi, and J. G. Wills, “Baryon Masses in a Relativistic Quark–Diquark Model,” Phys. Rev. Lett. 48, 1653 (1982).
  * Lichtenberg  _et al._ (1983) D. B. Lichtenberg, W. Namgung, J. G. Wills, and E. Predazzi, “Light and Heavy Hadron Masses in a Relativistic Quark Potential Model With Diquark Clustering,” Z. Phys. C 19, 19 (1983).
  * Fleck  _et al._ (1988) S. Fleck, B. Silvestre-Brac, and J. M. Richard, “Search for Diquark Clustering in Baryons,” Phys. Rev. D 38, 1519–1529 (1988).
  * Ebert  _et al._ (1996) Dietmar Ebert, Thorsten Feldmann, Christiane Kettner, and Hugo Reinhardt, “A Diquark model for baryons containing one heavy quark,” Z. Phys. C 71, 329–336 (1996), arXiv:hep-ph/9506298 .
  * Ebert  _et al._ (2008) D. Ebert, R. N. Faustov, and V. O. Galkin, “Masses of excited heavy baryons in the relativistic quark–diquark picture,” Phys. Lett. B 659, 612–620 (2008), arXiv:0705.2957 [hep-ph] .
  * Kim  _et al._ (2011) Kyungil Kim, Daisuke Jido, and Su Houng Lee, “Diquarks: A QCD sum rule perspective,” Phys. Rev. C 84, 025204 (2011), arXiv:1103.0826 [nucl-th] .
  * Ebert  _et al._ (2011) D. Ebert, R. N. Faustov, and V. O. Galkin, “Spectroscopy and Regge trajectories of heavy baryons in the relativistic quark–diquark picture,” Phys. Rev. D 84, 014025 (2011), arXiv:1105.0583 [hep-ph] .
  * Chen  _et al._ (2015) Bing Chen, Ke-Wei Wei, and Ailin Zhang, “Investigation of $\Lambda_{Q}$ and $\Xi_{Q}$ baryons in the heavy quark–light diquark picture,” Eur. Phys. J. A51, 82 (2015), arXiv:1406.6561 [hep-ph] .
  * Jido and Sakashita (2016) Daisuke Jido and Minori Sakashita, “Quark confinement potential examined by excitation energy of the $\Lambda_{c}$ and $\Lambda_{b}$ baryons in a quark–diquark model,” PTEP 2016, 083D02 (2016), arXiv:1605.07339 [nucl-th] .
  * Kumakawa and Jido (2017) Kento Kumakawa and Daisuke Jido, “Excitation energy spectra of the $\Lambda_{c}$ and $\Lambda_{b}$ baryons in a finite-size diquark model,” PTEP 2017, 123D01 (2017), arXiv:1708.02012 [nucl-th] .
  * Harada  _et al._ (2020) Masayasu Harada, Yan-Rui Liu, Makoto Oka, and Kei Suzuki, “Chiral effective theory of diquarks and the $U_{A}(1)$ anomaly,” Phys. Rev. D 101, 054038 (2020), arXiv:1912.09659 [hep-ph] .
  * Kim  _et al._ (2020) Yonghee Kim, Emiko Hiyama, Makoto Oka, and Kei Suzuki, “Spectrum of singly heavy baryons from a chiral effective theory of diquarks,” Phys. Rev. D 102, 014004 (2020), arXiv:2003.03525 [hep-ph] .
  * Dmitrašinović and Chen (2020) V. Dmitrašinović and Hua-Xing Chen, “Chiral $SU_{L}(3)\times SU_{R}(3)$ symmetry of baryons with one charmed quark,” Phys. Rev. D 101, 114016 (2020).
  * Kawakami  _et al._ (2020) Yohei Kawakami, Masayasu Harada, Makoto Oka, and Kei Suzuki, “Suppression of decay widths in singly heavy baryons induced by the $U_{A}(1)$ anomaly,” Phys. Rev. D 102, 114004 (2020), arXiv:2009.06243 [hep-ph] .
  * Suenaga and Hosaka (2021) Daiki Suenaga and Atsushi Hosaka, “Novel pentaquark picture for singly heavy baryons from chiral symmetry,” (2021), arXiv:2101.09764 [hep-ph] .
  * Copley  _et al._ (1979) L. A. Copley, Nathan Isgur, and Gabriel Karl, “Charmed Baryons in a Quark Model with Hyperfine Interactions,” Phys. Rev. D 20, 768 (1979), [Erratum: Phys.Rev.D 23, 817 (1981)].
  * Yoshida  _et al._ (2015) Tetsuya Yoshida, Emiko Hiyama, Atsushi Hosaka, Makoto Oka, and Katsunori Sadato, “Spectrum of heavy baryons in the quark model,” Phys. Rev. D92, 114029 (2015), arXiv:1510.01067 [hep-ph] .
  * ’t Hooft (1976) Gerard ’t Hooft, “Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle,” Phys. Rev. D 14, 3432–3450 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * ’t Hooft (1986) Gerard ’t Hooft, “How Instantons Solve the $U(1)$ Problem,” Phys. Rept. 142, 357–387 (1986).
  * Gell-Mann and Levy (1960) M Gell-Mann and M. Levy, “The axial vector current in beta decay,” Nuovo Cim. 16, 705 (1960).
  * Lévy (1967) M Lévy, “Currents and symmetry breaking,” Il Nuovo Cimento A (1965-1970) 52, 23–49 (1967).
  * Gell-Mann (1962) M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
  * Okubo (1962) Susumu Okubo, “Note on unitary symmetry in strong interactions,” Prog. Theor. Phys. 27, 949–966 (1962).
  * Zyla  _et al._ (2020) P.A. Zyla _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * Chen  _et al._ (2017a) Hua-Xing Chen, Wei Chen, Xiang Liu, Yan-Rui Liu, and Shi-Lin Zhu, “A review of the open charm and open bottom systems,” Rept. Prog. Phys. 80, 076201 (2017a), arXiv:1609.08928 [hep-ph] .
  * Kamimura (1988) M. Kamimura, “Nonadiabatic coupled-rearrangement-channel approach to muonic molecules,” Phys. Rev. A38, 621 (1988).
  * Hiyama  _et al._ (2003) E. Hiyama, Y. Kino, and M. Kamimura, “Gaussian expansion method for few-body systems,” Prog. Part. Nucl. Phys. 51, 223 (2003).
  * Silvestre-Brac (1996) B. Silvestre-Brac, “Spectrum and static properties of heavy baryons,” Few Body Syst. 20, 1 (1996).
  * Barnes  _et al._ (2005) T. Barnes, S. Godfrey, and E. S. Swanson, “Higher charmonia,” Phys. Rev. D72, 054026 (2005), arXiv:hep-ph/0505002 [hep-ph] .
  * Aaij  _et al._ (2020a) Roel Aaij _et al._ (LHCb), “Observation of a new baryon state in the ${\Lambda}_{\mathrm{b}}^{0}{\pi}^{+}{\pi}^{-}$ mass spectrum,” JHEP 06, 136 (2020a), arXiv:2002.05112 [hep-ex] .
  * Aaij  _et al._ (2020b) Roel Aaij _et al._ (LHCb), “First observation of excited $\Omega_{b}^{-}$ states,” Phys. Rev. Lett. 124, 082002 (2020b), arXiv:2001.00851 [hep-ex] .
  * Tanida  _et al._ (2019) K. Tanida _et al._ (Belle), “Experimental determination of the isospin of $\Lambda_{c}(2765)^{+}/\Sigma_{c}(2765)^{+}$,” in  _18th International Conference on Hadron Spectroscopy and Structure_ (2019) arXiv:1908.06235 [hep-ex] .
  * Aaij  _et al._ (2020c) Roel Aaij _et al._ (LHCb), “Observation of New $\Xi_{c}^{0}$ Baryons Decaying to $\Lambda_{c}^{+}K^{-}$,” Phys. Rev. Lett. 124, 222001 (2020c), arXiv:2003.13649 [hep-ex] .
  * Moon  _et al._ (2020) T. J. Moon _et al._ (Belle), “First Determination of the Spin and Parity of a Charmed-Strange Baryon, $\Xi_{c}(2970)^{+}$,” (2020), arXiv:2007.14700 [hep-ex] .
  * Sirunyan  _et al._ (2021) Albert M Sirunyan _et al._ (CMS), “Observation of a new excited beauty strange baryon decaying to $\Xi^{-}_{b}\pi^{+}\pi^{-}$,” (2021), arXiv:2102.04524 [hep-ex] .
  * Shuryak and Zahed (2004) Edward Shuryak and Ismail Zahed, “A Schematic model for pentaquarks using diquarks,” Phys. Lett. B 589, 21–27 (2004), arXiv:hep-ph/0310270 .
  * Hong  _et al._ (2004) Deog Ki Hong, Young Jin Sohn, and Ismail Zahed, “A Diquark chiral effective theory and exotic baryons,” Phys. Lett. B 596, 191–199 (2004), arXiv:hep-ph/0403205 .
  * Shuryak (2005) Edward V. Shuryak, “Toward dynamical understanding of the diquarks, pentaquarks and dibaryons,” J. Phys. Conf. Ser. 9, 213–217 (2005), arXiv:hep-ph/0505011 .
  * Chen  _et al._ (2016) Hua-Xing Chen, Wei Chen, Xiang Liu, and Shi-Lin Zhu, “The hidden-charm pentaquark and tetraquark states,” Phys. Rept. 639, 1–121 (2016), arXiv:1601.02092 [hep-ph] .
  * Hosaka  _et al._ (2016) Atsushi Hosaka, Toru Iijima, Kenkichi Miyabayashi, Yoshihide Sakai, and Shigehiro Yasui, “Exotic hadrons with heavy flavors: X, Y, Z, and related states,” PTEP 2016, 062C01 (2016), arXiv:1603.09229 [hep-ph] .
  * Chen  _et al._ (2017b) Hua-Xing Chen, Wei Chen, Jun He, Xiang Liu, and Shi-Lin Zhu, “The hidden-charm multiquark states,” PoS INPC2016, 374 (2017b), arXiv:1701.01551 [hep-ph] .
  * Olsen  _et al._ (2018) Stephen Lars Olsen, Tomasz Skwarnicki, and Daria Zieminska, “Nonstandard heavy mesons and baryons: Experimental evidence,” Rev. Mod. Phys. 90, 015003 (2018), arXiv:1708.04012 [hep-ph] .
  * Ali  _et al._ (2018) Ahmed Ali, Luciano Maiani, Anatoly V. Borisov, Ishtiaq Ahmed, M. Jamil Aslam, Alexander Ya. Parkhomenko, Antonio D. Polosa, and Abdur Rehman, “New Look at Hidden Charm Tetra and Pentaquark States,” Acta Phys. Polon. B 49, 1315–1324 (2018).
  * Liu  _et al._ (2019) Yan-Rui Liu, Hua-Xing Chen, Wei Chen, Xiang Liu, and Shi-Lin Zhu, “Pentaquark and Tetraquark states,” Prog. Part. Nucl. Phys. 107, 237–320 (2019), arXiv:1903.11976 [hep-ph] .
  * Alford  _et al._ (1998) Mark G. Alford, Krishna Rajagopal, and Frank Wilczek, “QCD at finite baryon density: Nucleon droplets and color superconductivity,” Phys. Lett. B 422, 247–256 (1998), arXiv:hep-ph/9711395 .
  * Rapp  _et al._ (1998) R. Rapp, Thomas Schäfer, Edward V. Shuryak, and M. Velkovsky, “Diquark Bose condensates in high density matter and instantons,” Phys. Rev. Lett. 81, 53–56 (1998), arXiv:hep-ph/9711396 .