# Heavy quark dominance in orbital excitation of singly and doubly heavy
baryons

Zhen-Yu Li1<EMAIL_ADDRESS>Guo-Liang Yu2<EMAIL_ADDRESS>Zhi-Gang
Wang2<EMAIL_ADDRESS>Jian-Zhong Gu3<EMAIL_ADDRESS>1 School of
Physics and Electronic Science, Guizhou Education University, Guiyang 550018,
China
2 Department of Mathematics and Physics, North China Electric Power
University, Baoding 071003, China
3 China Institute of Atomic Energy, Beijing 102413,China

###### Abstract

A mechanism of the heavy quark dominance in the orbital excitation is proposed
in this paper which is testified to be reasonable for singly and doubly heavy
baryons. In the relativistic quark model, an analysis of the Hamiltonian
figures out the mechanism that the excitation mode with lower energy levels is
always associated with the heavy quark(s), and the splitting of the energy
levels is suppressed by the heavy quark(s). So, the heavy quarks dominate the
orbital excitation of singly and doubly heavy baryons. Furthermore, a physical
understanding of this mechanism is given in a semi-classical way. Accordingly,
the predicted mass spectra of singly and doubly heavy baryons confirm the
rationality of this mechanism. In addition, an interesting consequence of this
mechanism is that a heavy-light meson is more likely to be produced in the
strong decay of the high-orbital excited states, which is supported by
experiments. This mechanism is rooted in the breakdown of the mass symmetry.
Therefore, it may be also valid for other multi-quark systems, such as the
tetraquarks $Qqqq$ and $QQqq$, or the pentaquarks $Qqqqq$ and $QQqqq$.

Key words: Heavy quark dominance, Orbital excitation, Heavy baryon,
Relativistic quark model, Mass symmetry breaking.

###### pacs:

13.25.Ft; 14.40.Lb

##  I Introduction

Quantum Chromodynamics (QCD), the theory of quarks and gluons, has been
developed for 50 yearsf01 . An important prediction of QCD is the existence of
heavy baryons. The study of heavy baryon spectroscopy has always been one of
the hot topics. It contributes to a deeper understanding of the properties of
heavy quarks, such as the heavy quark symmetry, chiral dynamics, feature of
strong interactions, and relevant models inspired by QCD. Until now, there
have been various methods to analyze baryon spectra (see f01 ; fp01  and Refs.
therein). Among them, quark potential models are currently an indispensable
tool for guiding our complete understanding of the multitude of observed
baryons and their propertiesf02 .

Copley, Isgur and Karl first studied the charmed baryons in a quark potential
model with hyperfine interactionsf03 . While the first prediction of the full
spectrum of baryons including heavy-quark baryons was presented by Capstick
and Isgur in 1986, where the relativistic quark model was developedf04 ; f05 .
Twenty years later, Roberts and Pervin analyzed the heavy baryons
systematically and exhaustively, using a non-relativistic quark modelf06 . All
of the above studies tried to give a unified theoretical description of all
baryons including heavy-quark ($Q$) baryons, by solving the three-quark system
explicitly. These works continue to serve as guidelines for experimental and
theoretical researches. However, their calculations predict more heavy-quark
baryon excited states than those presented in the spectra extracted from data.
It is actually a general problem with baryon spectroscopy, which first
appeared in the light-quark ($q$) baryons and was called the ‘missing
resonance’ problemf01 ; fp01 ; f02 .

A possible solution to this problem is to partially freeze the dynamical
degrees of freedom in a three-quark system. For example, if a baryon consists
of a quark and a diquark, the reduction of the number of internal degrees of
freedom would lead to a more sparsely populated spectrumf02 . Based on this
idea, Ebert, Faustov and Galkin calculated the spectra of the singly heavy
baryons in the heavy quark-light diquark picture in the framework of the QCD-
motivated relativistic quark model, where the excitations between the two
light quarks are frozenf07 ; f07p . It was found that the available
experimental data can be well described and the model predicts significantly
fewer states than the genuine three-body picturef07p . Later, some theoretical
works were carried out to investigate the singly heavy baryons with similar
methodsf08 ; f08p ; f08p2 ; f08p3 ; f08p4 . These methods have also been
applied to the spectroscopy of doubly heavy baryons, where the light quark-
heavy diquark structure is assumedf09 ; f09p1 ; f09p2 ; f09p3 ; f09p4 .
However, the reliability of the extension of this method has yet to be tested.
‘It is telling that this simple diagnostic is difficult to apply since so
little is known of the excited baryon spectrum.’f01

For a three-quark system, quark models feature the orbital excitations in two
relative coordinates. As shown in Fig.1, they are often taken to be the Jacobi
coordinates $\bm{\rho}$ and $\bm{\lambda}$, which is consistent with the
flavor sectors of the light quark pair ($q_{1}q_{2}$) in a singly heavy baryon
or the heavy quark pair ($Q_{1}Q_{2}$) in a doubly heavy baryon (see Sect.II
A). In 2016, Yoshida et al. dealt rigorously with the three-quark systems, and
analyzed the low-lying excitations of singly and doubly heavy baryons,
respectivelyf10 . They concluded that the excitation of the $\lambda$-mode
becomes dominant in low-lying states of singly heavy baryons. While the
$\rho$-mode becomes dominant in low-lying states of doubly heavy baryons. In
fact, these features have been mentioned earlierf03 ; f06 , and in particular,
they were expressed systematically in Ref.f06 . Later, Chen et al. noticed
that almost all observed single heavy baryons could be explained as the
$\lambda$-mode excited states (in the diquark picture) in theoryf10p .

Figure 1: Jacobi coordinates for the three-quark systems. The heavy quarks are
represented by bigger balls and the light quarks by smaller balls. They are
numbered for ease of use in calculations.

Inspired by the above theoretical achievements, in 2022, the spectra of singly
heavy baryons were studied systematically, in which all orbital excitations
were assumed in the $\lambda$-modef11 ; f11p  and the $\rho$-mode was used
alone to study the excitations of doubly heavy baryonsf12 ; f12p ; f12p2  in
the framework of the relativistic quark modelf04 ; f05 . The calculated
results are satisfactory, especially, the predicted spectra of singly heavy
baryons can reproduce almost all experimental data nicely. Then, the
rationality of this approach and the physical nature behind it need to be
carefully considered.

As shown in Fig.1, the determination of the Jacobi coordinates $\bm{\lambda}$
in a singly heavy baryon and $\bm{\rho}$ in a doubly heavy baryon is
associated with the heavy quark(s). The same is true of the corresponding
orbital excitation modes. Hence, it implies that the orbital excitation may be
dominated by the heavy quarks. In this paper, we will comprehensively analyze
the heavy quark dominance (HQD) mechanism in orbital excitations of singly and
doubly heavy baryons, and discuss the reasonability of the HQD approximation.

The remainder of this paper is organized as follows. In Sect.II, the HQD
mechanism is analyzed with the Hamiltonian of the relativistic quark model and
a semi-classical approach. In Sect.III, the HQD approximation is tested in
singly heavy baryons. Sect.IV discusses the HQD approximation in doubly heavy
baryons. The signature of the HQD mechanism in the strong decay is analyzed in
Sect.V. And Sect.VI is reserved for the conclusions.

##  II Analysis of the HQD approximation with the Hamiltonian

###  II.1 Hamiltonian and the wave function

In our previous studies on the spectroscopy of singly and doubly heavy
baryons, we adopted the relativistic quark model f04 ; f05 . In the
calculation, the Gaussian expansion method (GEM) and the infinitesimally-
shifted Gaussian (ISG) basis functions have been used as well f12p3 . For the
convenience of discussion, a brief introduction of the Hamiltonian and the
wave function is presented here. In the relativistic quark model, the
Hamiltonian for a three-quark system reads,

$\displaystyle H$ $\displaystyle=H_{0}+H^{conf}+H^{so}+H^{hyp}$ (1)
$\displaystyle=\sum_{i=1}^{3}\sqrt{p_{i}^{2}+m_{i}^{2}}+\sum_{i<j}(H^{conf}_{ij}+H^{so}_{ij}+H^{hyp}_{ij}),$

where the interaction terms $H^{conf}_{ij}$, $H^{so}_{ij}$ and $H^{hyp}_{ij}$
are the confinement, spin-orbit and hyperfine interactions, respectively. The
confinement term $H^{conf}_{ij}$ includes the one-gluon exchange potentials
$G_{ij}$ and linear confining potentials $S_{ij}$, in which the smearing
transformations are performed. And the interactions have been modified with
the momentum-dependent factors.

With the requirement of the flavor $SU(3)_{F}$ subgroups for the light quark
pair ($q_{1}q_{2}$), the singly heavy baryons belong to either a sextet
$(\mathbf{6}_{F})$ of the flavor symmetric states, or an antitriplet
$(\bar{\mathbf{3}}_{F})$ of the flavor antisymmetric states. Thus,
$\Lambda_{Q}$ and $\Xi_{Q}$ belong to the $\bar{\mathbf{3}}_{F}$ sector;
$\Sigma_{Q}$, $\Xi_{Q}^{{}^{\prime}}$ and $\Omega_{Q}$ belong to the
$\mathbf{6}_{F}$ sector. Here $Q$ denotes charm ($c$) quark or bottom ($b$)
quark. Analogously, in the doubly heavy baryons, $\Xi_{bc}$ and $\Omega_{bc}$
belong to the $\bar{\mathbf{3}}_{F}$ sector, $\Xi_{bb(cc)}$,
$\Xi_{bc}^{{}^{\prime}}$, $\Omega_{bb(cc)}$ and $\Omega_{bc}^{{}^{\prime}}$
belong to the $\mathbf{6}_{F}$ sector, with the requirement of the heavy quark
pair ($Q_{1}Q_{2}$) as shown in Fig.1.

Based on the above discussion and the heavy quark effective theory (HQET) f06
; f12p4 ; f12p5 , the spin and orbital wave function of the state is written
as

$\displaystyle|l_{\rho}\ l_{\lambda}\ L\ s\ j\ J\ M_{J}\rangle$
$\displaystyle=\\{[(|l_{\rho}\ m_{\rho}\rangle|l_{\lambda}\
m_{\lambda}\rangle)_{L}\times(|s_{1}\ m_{s_{1}}\rangle|s_{2}\
m_{s_{2}}\rangle)_{s}]_{j}$ (2) $\displaystyle\times|s_{3}\
m_{s_{3}}\rangle\\}_{JM_{J}}.$

$l_{\rho}$($l_{\lambda}$), $L$ and $s$ are the quantum numbers of the relative
orbital angular momentum $\textbf{\emph{l}}_{\rho}$
($\textbf{\emph{l}}_{\lambda}$), total orbital angular momentum L, and total
spin of the quark pair $\mathbf{s}$, respectively. $j$ denotes the quantum
number of the coupled angular momentum of L and s, so that the total angular
momentum $J=j\pm\frac{1}{2}$. And $M_{J}$ is the 3rd component of J. Then, the
baryon state is simply labeled with $nL(J^{P})_{j}$, in which $n$ is the
quantum number of the radial excitation.

The orbital part of the above wave function $(|l_{\rho}\
m_{\rho}\rangle|l_{\lambda}\ m_{\lambda}\rangle)_{L}$ is expanded in terms of
the Gaussian basis functions. Then, the matrix element of each Hamiltonian
term can be obtained on these bases with the Jacobi coordinates $\bm{\rho}$
and $\bm{\lambda}$ as shown in Fig.1 (for details, see f04 ; f11 ; f11p ).
Thus, the eigenvalues of the Hamiltonian for the orbital excited states can be
calculated. Particularly, the contribution of each term to the eigenvalues can
be analyzed, which will be done in the next subsection.

###  II.2 HQD mechanism in the orbital excitation

For the $L$-wave excitation with
L=$\textbf{\emph{l}}_{\rho}$+$\textbf{\emph{l}}_{\lambda}$, there are an
infinite number of orbital excitation modes. Taking $L=1$ as an example, the
excitation modes $(l_{\rho},l_{\lambda})_{L}$ are $(1,0)_{1}$, $(0,1)_{1}$,
$(1,1)_{1}$, $(2,1)_{1}$, $(1,2)_{1}$, $(2,2)_{1}$, and so on. Here
$(L,0)_{L}$ is the so-called $\rho$-mode, and $(0,L)_{L}$ the so-called
$\lambda$-mode.

Now we analyze the excitation energies of the $\lambda$-mode and the
$\rho$-mode with $L=1$. Tables 1 and 2 in the appendix list the details of
each term contribution to the eigenvalues of the Hamiltonian. For comparison,
Types I, II and III in each table are presented, which correspond to the
light-quark baryon, singly heavy baryon and doubly heavy baryon, respectively.
$H$ in the last column is the total energy (the mass of the state). $\Delta$
in the brackets is the excitation energy of each term with respect to the
ground state.

As shown in Tables 1 and 2, the energy levels of each
$(l_{\rho},l_{\lambda})nL$ mode are dominated by the sum of $H_{0}$, $\sum
G_{ij}$ and $\sum S_{ij}$. While $H^{hyp}$ and $H^{so}$ are responsible for
the splitting of the levels. This feature can be seen more clearly in Figs.2
and 3. For the convenience of the following discussion, we define

$\displaystyle H_{mode}=H_{0}+\sum G_{ij}+\sum S_{ij},$ (3)

whose eigenvalues stand for the energy levels of the excitation modes
$(l_{\rho},l_{\lambda})_{L}$. Then, we use $\langle\Delta H_{mode}\rangle$ to
represent the excitation energies (relative energy levels) of the modes with
respect to the pure $S$-wave mode $(0,0)_{0}$.

Compared with Type I, as shown in Fig.2, $m_{3}$ in Type II is replaced by a
heavy quark mass ($m_{b}=4977$ MeV), which leads to two results in Type II:
(1) The excitation energy of the $(0,1)1P$ mode is suppressed; (2) The
corresponding energy splitting is also reduced. As a whole, the energy level
of the $\lambda$-mode and the energy splitting are both suppressed due to the
mass of the heavy quark. The same is true in Type III, nevertheless, here the
excitation energy of the $\rho$-mode is suppressed which is caused by the
heavy quark pair. The similar phenomenon can be seen for the $\mathbf{6}_{F}$
sector as shown in Fig.3. The conclusion in this work is the same as that of
Refs.f06 ; f10 . We can uniformly summarize that the lower orbital excitation
energy is caused by the heavy quark(s) in singly and doubly heavy baryons.
This is the HQD mechanism.

Figure 2: Relative energy levels $\langle\Delta H_{mode}\rangle$ and the
energy splittings of the $\rho$-mode and the $\lambda$-mode for the
$\bar{\mathbf{3}}_{F}$ sector with Types I, II and III. The energy values are
in MeV. Figure 3: Same as Fig.2, but for the $\textbf{6}_{F}$ sector.

This mechanism can be understood in a semi-classical way. The orbital
excitation generates the rotational kinetic energy $E_{r.k.}$, which increases
the kinetic energies of quarks and the radial distance between quarks. The
former contributes directly to $H_{{}_{0}}$. While the latter raises the
values of $\sum G_{ij}$ and $\sum S_{ij}$ due to the one-gluon exchange
potential $G(r)=-\frac{4\alpha_{s}(r)}{3r}$ and linear confinement potential
$S(r)=b^{\prime}r+c^{\prime}$. Here $b^{\prime}$ and $c^{\prime}$ are
constants. $\alpha_{s}(r)$ is the running coupling constant and has been
parameterized f05 . Therefore, the rotational kinetic energy $E_{r.k.}$
determines $\langle\Delta H_{mode}\rangle$ finally. While the rotational
kinetic energy in a given state $\langle E_{r.k.}\rangle$ depends on the
moment of inertia of the state $\langle I\rangle$ through $\langle
E_{r.k.}\rangle=\frac{\langle L^{2}\rangle}{2\langle I\rangle}$. So, a larger
$\langle I\rangle$ will cause a smaller $\langle E_{r.k.}\rangle$, which will
eventually lead to a lower $\langle\Delta H_{mode}\rangle$.

For a two-body system rotating in a plane, the following relationships exist,
$m_{A}r_{A}=m_{B}r_{B}$ and $r_{A}+r_{B}=r$. Here $m_{A(B)}$ denotes the mass
of $A(B)$ particle. $r_{A(B)}$ is the corresponding distance to the center of
mass. Then, the moment of inertia of the system
$I=m_{A}r_{A}^{2}+m_{B}r_{B}^{2}=\frac{m_{A}m_{B}}{m_{A}+m_{B}}r^{2}$. For the
$P$-wave orbital excitation of singly heavy baryons, the $\lambda$-mode means
($l_{\rho},l_{\lambda}$)=(0,1). If we take the baryon as a two-body system
with $m_{A}=m_{Q}\gg m_{q}$ and $m_{B}=m_{q_{1}}+m_{q_{2}}\sim 2m_{q}$,
$\langle I_{\lambda}\rangle\sim 2m_{q}\langle r_{\lambda}^{2}\rangle$ is
obtained based on the above discussion. In the $\rho$-mode with
($l_{\rho},l_{\lambda}$)=(1,0), the rotation occurs in the two light quarks.
And $\langle I_{\rho}\rangle\sim\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$
as $m_{A}\sim m_{B}\sim m_{q}$. In this way, for both of the
$(\bar{\mathbf{3}}_{F})$ and $(\mathbf{6}_{F})$ sectors, we can obtain the
moment of inertia $\langle I\rangle$ for Types I, II and III. By using the
calculated values of $\langle r_{\rho}^{2}\rangle^{1/2}$ and $\langle
r_{\lambda}^{2}\rangle^{1/2}$, we can estimate the value of the rotational
kinetic energy $\langle E_{r.k.}\rangle$ and compare it with the excitation
energy $\langle\Delta H_{mode}\rangle$.

The used variables and comparison results are listed in Table 3. It is shown
that for Type II (the singly heavy baryons), the estimated values of $\langle
E_{r.k.}\rangle$ in the $\lambda$-mode are lower than those in the
$\rho$-mode, being the bigger moment of inertia $\langle I\rangle$ in the
$\lambda$-mode due to the influence of the heavy quark. By comparison, it can
be seen that the estimated $\langle E_{r.k.}\rangle$ is near $\langle\Delta
H_{mode}\rangle$. For Type III (the doubly heavy baryons), we can conclude
that the heavy quarks dominate the lower $\langle E_{r.k.}\rangle$ in the
$\rho$-mode, which qualitatively agrees with $\langle\Delta H_{mode}\rangle$
as shown in Table 3.

The above discussion confirms our understanding that the rotational kinetic
energies $E_{r.k.}$ of the excitation modes involving heavy quark(s) are
significantly lower, which determines the relative energy levels of the
excitation modes. Therefore, the HQD mechanism has been understood in the
semi-classical way. On the other hand, the suppression of the energy levels
splitting for heavy baryons can be explained in the heavy quark limit f06 ;
f13p ; f13p2 . We may conclude that the excitation mode with lower energy
levels is always associated with the heavy quark(s), and the energy level
splitting is also suppressed by the heavy quark(s). So, the heavy quarks
dominate the orbital excitation of singly and doubly heavy baryons.

##  III The test of HQD approximation in singly heavy baryons

###  III.1 The orbital excitation of singly heavy baryons

For further investigating the HQD approximation in singly heavy baryons, the
evolution of the excitation energies with $m_{3}$ for different modes is
plotted in the cases of the $P$-, $D$\- and $F$-wave states. At the same time,
the relative energies of the $(1,1)1S(\frac{1}{2}^{+})$ and
$(2,2)1S(\frac{1}{2}^{+})$ states with respect to the pure $S$-wave state
$(0,0)1S(\frac{1}{2}^{+})$ are also given for comparison, as shown in Fig.4.

In the $\bar{\mathbf{3}}_{F}$ sector, the energies of the
$(1,1)1S(\frac{1}{2}^{+})$ and $(2,2)1S(\frac{1}{2}^{+})$ states are obviously
higher than that of the pure $S$-wave state. For the $P$-, $D$\- and $F$-wave
states, the excitation energies of most modes are quickly suppressed as
$m_{3}$ increases, except for the $\rho$-mode. And the $\lambda$-mode is
indeed the one with the lowest excitation energy.

It is reasonable to believe that the excitation mode with the lowest energy is
most stable and has the greatest probability of being observed experimentally.
Then, all excitation modes with higher energies can be approximately ignored.
Thus, the ‘true’ ground state comes from the pure $S$-wave state, and the
‘real’ orbital excited state originates from the $\lambda$-mode, which is
dominated by the heavy quark(s). This is the HQD approximation as mentioned
before in this paper.

The situation of the $\mathbf{6}_{F}$ sector is similar to that of the
$\bar{\mathbf{3}}_{F}$ sector that the $\lambda$-mode is dominant. But the
phenomenon is not so obvious when $m_{3}\sim$ 1.6 GeV for the $P$-wave states,
where the excitation energy of the $(0,1)1P(\frac{1}{2}^{+})_{1}$ state is
only slightly lower than that of the $(1,0)1P(\frac{1}{2}^{+})_{1}$ state. It
implies that $c$ quark ($m_{c}=1.628$GeV) is not heavy enough and hardly
treated as a heavy quark, which was also pointed out in some theoretical works
f13 . Even so, the HQD approximation is still applicable to the
$\mathbf{6}_{F}$ sector.

Figure 4: The evolution of excitation energies with $m_{3}$ for different
modes of singly heavy baryons. Here only the modes with $J=L\pm\frac{1}{2}$
are selected. $m_{3}$ changes from 0.2 GeV to 5.0 GeV. The excitation energies
are measured from the pure $S$-wave states $1S(\frac{1}{2}^{+})_{0}$ for the
$\bar{\mathbf{3}}_{F}$ and $1S(\frac{1}{2}^{+})_{1}$ for the $\mathbf{6}_{F}$,
respectively. And $m_{1}=m_{2}=0.22$ GeV is used.

###  III.2 Predicted excited states with HQD approximation

With the HQD approximation, the excited spectra of singly heavy baryons were
systemically investigated with a unified set of parameters f11 ; f11p . The
results show that the predicted masses reproduce the experimental data well.
As shown in Table 4, for the well established baryons in experiment f14 ; f14p
, the calculated masses are very close to the data, apart from
$\Lambda_{c}(2940)^{+}$ f11 ; f15 . The maximum difference of the mass values
between theory and experiment is generally no more than 20 MeV.

For the observed baryons whose $J^{P}$ values are not yet confirmed f14 ; f16
; f16p1 ; f16p2 ; f16p3 ; f16p4 , the calculated results are also helpful for
their rational assignments. As shown in Table 5, these baryons have been
divided into several groups. In particular, some members in the same family
were grouped together and labeled with $1P(\mathbf{6}_{F})$. With the help of
the mass spectra, one can see them more clearly. As shown in Fig.5, for
$\Sigma_{c(b)}$, $\Xi_{c(b)}^{{}^{\prime}}$ and $\Omega_{c(b)}$, there are
five excited states for the $1P$-wave, and their energies are almost
degenerate. The five states are $1P(\frac{3}{2}^{-})_{2}$,
$1P(\frac{1}{2}^{-})_{1}$, $1P(\frac{1}{2}^{-})_{0}$,
$1P(\frac{3}{2}^{-})_{1}$ and $1P(\frac{5}{2}^{-})_{2}$, and increase in
energy. Some baryons of these $1P$ states have been observed as shown in Table
5 and Fig.5. However, because their energies are so close to each other, the
identification of these baryons becomes very difficult. So they can only be
labeled with $1P(\mathbf{6}_{F})$, as shown in Table 5.

It is worth noting that the predicted spectra were obtained in 2022. After
that, four baryons $\Omega_{c}(3185)^{0}$, $\Omega_{c}(3327)^{0}$ f16p3 ,
$\Xi_{b}(6095)^{0}$ and $\Xi_{b}(6087)^{0}$ f16p4  were observed by the LHCb
collaboration. As expected, $\Omega_{c}(3185)^{0}$, $\Xi_{b}(6095)^{0}$ and
$\Xi_{b}(6087)^{0}$ can be directly assigned to suitable states in Table 5.
And $\Omega_{c}(3327)^{0}$ f16p3 , for the same reason mentioned above, should
be labeled with $1D(\mathbf{6}_{F})$, where a total of six quasi-degenerate
states are overlapped together (for details, see f11 ). In general, almost all
baryons can be in good agreement with the predicted states. The exception is
that $\Lambda_{c}(2940)^{+}$ f11 ; f15 , $\Xi_{c}(3123)^{+}$ f11p ; f16p2  and
$\Omega_{c}(3120)^{+}$ f14 ; f11  can not be assigned reasonably into the
spectra of Fig.5, which needs further research and confirmation in theory and
experiment .

Fig.5 shows the predicted spectra agree well with the experimental data. And
the ‘missing resonance’ problem does not seem to exist here. It makes one more
confident that the HQD mechanism is reasonable and the HQD approximation is
feasible. In addition, Fig.5 shows an obvious shell structure of the spectrum.
It implies that $L$ might be approximated as a good quantum number. If this is
true, the $\gamma$ transitions would occur between the neighbouring orbits.
This feature has been partially observed in experiment. For example,
$\Xi_{c}(2815)^{0}\rightarrow\Xi_{c}^{0}\gamma$ is the $\gamma$ transition
between the $1P(\frac{3}{2}^{-})$ and $1S(\frac{1}{2}^{+})$ states f14 .
Actually, the statement of ‘$L$-wave’ has been adopted in many works such as
Refs. f17a0 ; f17 ; f17p0 ; f17p1 ; f17p2 ; f17p3 ; f17p4 ; f17p5 ; f17p6 ;
f17p7 ; f17p8 ; f17p9 ; f17p10 ; f17p11 . In this work, it is further
identified.

Figure 5: Calculated spectra of singly heavy baryons and the relevant
experimental data. ‘++’,‘+’, ‘0’ and ‘-’ in the brackets indicate the charge
states of baryons. The solid black circles denote the baryons in Table 1, and
the open circles are the ones in Table 2. The three bigger red circles
represent $\Lambda_{c}(2940)^{+}$, $\Xi_{c}(3123)^{+}$, and
$\Omega_{c}(3120)^{+}$, respectively, which can not be assigned very well.
More excited states can be found in Refs. f11 ; f11p .

##  IV HQD mechanism in doubly heavy baryons

The doubly heavy baryons $\Xi_{cc}$($\Omega_{cc}$), $\Xi_{bb}$($\Omega_{bb}$)
and $\Xi^{\prime}_{bc}$($\Omega^{\prime}_{bc}$) belong to the $\mathbf{6}_{F}$
sector, and $\Xi_{bc}$($\Omega_{bc}$) belongs to the $\bar{\mathbf{3}}_{F}$
sector, as mentioned in Sect.II A. To examine the HQD mechanism in the
$\mathbf{6}_{F}$ sector, we set $m_{2}=m_{1}$ and change $m_{1}$ from 0.3 GeV
to 5.1 GeV. For the $\bar{\mathbf{3}}_{F}$ sector, $m_{2}$ is set to be
$\frac{m_{c}}{m_{b}}\times m_{1}$, which stands for $\Xi_{bc}$ (or
$\Omega_{bc}$) baryon when $m_{1}$ approaches to 5.0 GeV (see the caption in
Fig.6).

As shown in Fig.6, the orbital excitation of doubly heavy baryons presents the
same feature as that of singly heavy baryons. In this case, nevertheless, the
mode with the lowest energy is the $\rho$-mode and the lowest orbital
excitation energy is still dominated by the heavy quarks. Hence, the HQD
mechanism remains valid and the HQD approximation is feasible for doubly heavy
baryons.

Based on the HQD approximation, the spectra of doubly heavy baryons were
investigated in the $\rho$-mode f12 ; f12p ; f12p2 . The predicted ground
state mass of $\Xi_{cc}$ is 3640 MeV, which is very close to the experimental
data 3621.6$\pm$0.4 MeV f14 . The spectrum also presents an obvious shell
structure as shown in Fig.7. It is expected that more doubly heavy baryons
will be observed and can be used to further test the HQD approximation.

In addition, some works have noticed the excitation of the heavy quark pair is
important and can not be ignoredf09p1 ; f09p3 . In those works, the spectra of
the doubly heavy baryons therefore contain the excitation of the heavy quark
pair. And the heavy quark pair was not treated as a point-like diquark, which
was used in some other worksf18p ; f18p2 ; f18p3 . Unlike both of them, the
HQD approximation freezes the orbital excitation of the $\lambda$-mode and
only considers the $\rho$-mode, which predicts fewer excited states than the
formerf09p1 ; f09p3 , and lowers the excited energy levels than the latterf18p
; f18p2 ; f18p3 .

Figure 6: Evolution of excitation energies with $m_{1}$ for different modes of
doubly heavy baryons, where $m_{1}$ changes from 0.3 GeV to 5.1 GeV. The value
of $m_{3}$ is equal to 0.22 GeV. $m_{2}$ is set as $m_{2}=m_{1}$ for the
$\mathbf{6}_{F}$ sector. For the $\bar{\mathbf{3}}_{F}$ sector, $m_{2}$ is set
as $m_{2}=\frac{m_{c}}{m_{b}}\times m_{1}$, which corresponds to $\Xi_{bc}$ or
$\Omega_{bc}$ when $m_{1}=4.977$ GeV. Figure 7: Same as Fig.5, but for doubly
heavy baryons. The solid black circle denotes the observed baryon
$\Xi_{cc}^{++}$. More excited states can be found in Refs. f12 ; f12p ; f12p2
.

##  V Signature of HQD mechanism in strong decay

If the HQD mechanism is reasonable, an interesting phenomenon would occur in
the strong decay. The calculation shows that for a singly heavy baryon, the
root-mean-square value of $r_{\lambda}$ increases with the orbital excitation
f11 ; f11p . It means the heavy quark is further away from the light quark
pair, as the orbital quantum number increases. For a high-orbital excited
state, when the strong decay occurs, the heavy quark is therefore likely to
capture only one light quark. So, the heavy-light meson is more likely to be
produced. From the Particle Data Group (PDG) f14 , one can see the $D^{0}p$
decay mode occurs in $\Lambda_{c}(2860)^{+}$ and $\Lambda_{c}(2880)^{+}$,
which are identified as the $D$-wave states in theory f07p ; f11 ; f19 .

Similarly, the $\Lambda D^{+}$ decay modes are also observed experimentally
for $\Xi_{c}(3055)^{+}$ and $\Xi_{c}(3080)$. So, they are assigned to the
$1D(\frac{3}{2}^{+},\frac{5}{2}^{+})$ doublet states of the
$\bar{\mathbf{3}}_{F}$ sector in Table 5. So far, the two-body strong decay
mode containing a heavy-light meson has been only observed in the high-orbital
excited states. The same phenomenon is expected to occur in doubly heavy
baryons, which might be helpful in searching for more doubly heavy baryons in
experiments.

##  VI Conclusions

The singly and doubly heavy baryons are special, due to the large mass values
of charm and bottom quarks. Since these heavy quarks are much more massive
than the light quarks, they break the mass symmetry. So, the $SU(3)_{F}$
flavor symmetry valid for light-quark baryons does not work here. On the other
hand, the heavy quark limit is not fully applicable here either, because the
masses of the heavy quarks are much less than infinity. Inspired by the
progress of the heavy baryon spectroscopy in recent decades, we speculate a
mechanism of the heavy quark dominance (HQD) in the orbital excitation of
singly and doubly heavy baryons.

By analyzing the Hamiltonian of the relativistic quark model for a three-quark
system, we tried to understand the HQD mechanism. It is shown that the
excitation mode with lower energy levels is always associated with the heavy
quark(s), and the splitting of the energy levels is suppressed by the heavy
quark(s) as well. In other words, the heavy quarks dominate the orbital
excitation of singly and doubly heavy baryons. In a semi-classical way, we
show that the rotational kinetic energy generated by the orbital excitation is
positively related to the excitation energy of the mode. And the smaller
rotational kinetic energy is determined by the heavy quark(s). So, the heavy
quarks eventually dominate the lower orbital excitation for singly and doubly
heavy baryons. Thus, we get a physical understanding of this mechanism.

The HQD mechanism is then tested by comparing various orbital excitation modes
for singly heavy baryons. It is shown that the excitation mode dominated by
the heavy quark(s) has lower energy levels, therefore, it is most stable and
has the greatest probability of being observed experimentally, which justifies
the HQD approximation. With the HQD approximation, the predicted excited
spectra of singly heavy baryons can reproduce the available experimental data
nicely. Additionally, the shell structure of the mass spectra implies $L$ can
be regarded as a good quantum number approximately, which is supported by the
$\gamma$ transition in experiment. Thus, the HQD mechanism is proved to be
reasonable in singly heavy baryons. For doubly heavy baryons, their orbital
excitations are also dominated by the heavy quarks. The corresponding mass
spectra with the HQD approximation are expected to be tested in future
experiments. The HQD mechanism also predicts an interesting phenomenon that
the heavy-light meson is more likely to be produced in the strong decay of the
high-orbital excited states for singly and doubly heavy baryons. This kind of
strong decay mode may be the signature of the HQD mechanism as well.

The HQD mechanism is rooted in the breakdown of the mass symmetry in a multi-
quark system. Therefore, it should be valid not only for singly and doubly
baryons, but also for the multi-quark systems containing one or two heavy
quarks, such as the tetraquarks $Qqqq$ and $QQqq$, or the pentaquarks $Qqqqq$
and $QQqqq$.

## Acknowledgements

We thank the reviewers of the series of work on the heavy baryon spectroscopy,
for their helpful comments, questions and suggestions, which prompted our
consideration of this work. This research was supported by the National
Natural Science Foundation of China (Grant Nos. 11675265, 12175068), the
Continuous Basic Scientific Research Project (Grant No. WDJC-2019-13), and the
Leading Innovation Project (Grant No. LC 192209000701).

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## Appendix

Table 1: Contribution of each term to the eigenvalues of $H$ for the states of the $(0,0)_{0}$, $(0,1)_{1}$ and $(1,0)_{1}$ modes of the $\bar{\mathbf{3}}_{F}$ sector (all quantities are in MeV). $(l_{\rho},l_{\lambda})nL(J^{P})_{j}$ | $H_{{}_{0}}~{}(\Delta)$ | $G_{12}~{}(\Delta)$ | $G_{13(23)}~{}(\Delta)$ | $\sum G_{ij}~{}(\Delta)$ | $S_{12}~{}(\Delta)$ | $S_{13(23)}~{}(\Delta)$ | $\sum S_{ij}~{}(\Delta)$ | $H_{mode}~{}(\Delta)$ | $H^{hyp}~{}(\Delta)$ | $H^{so}~{}(\Delta)$ | $H~{}(\Delta)$
---|---|---|---|---|---|---|---|---|---|---|---
Type I ($m_{1}=m_{2}=m_{3}=220$ MeV) |
$(0,0)1S(\frac{1}{2}^{+})_{0}$ | 737 (0) | -45 (0) | -48 (0) | -141 (0) | 245 (0) | 152 (0) | 549 (0) | 1145 (0) | -168 (0) | 0 (0) | 978 (0)
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 762 (25) | -45 (0) | -31 (17) | -107 (34) | 299 (54) | 305 (153) | 909 (360) | 1564 (419) | -168 (0) | -53 (-53) | 1364 (386)
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 762 (25) | -45 (0) | -31 (17) | -107 (34) | 299 (54) | 305 (153) | 909 (360) | 1564 (419) | -141 (27) | 24 (24) | 1435 (457)
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | 770 (33) | -31 (14) | -38 (10) | -108 (33) | 499 (254) | 205 (53) | 909 (360) | 1571 (426) | -148 (20) | -52 (-52) | 1395 (417)
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | 770 (33) | -31 (14) | -38 (10) | -108 (33) | 499 (254) | 205 (53) | 909 (360) | 1571 (426) | 58 (226) | -47 (-47) | 1607 (629)
Type II ($m_{1}=m_{2}=220$ MeV and $m_{3}=4977$ MeV) |
$(0,0)1S(\frac{1}{2}^{+})_{0}$ | 5473 (0) | -45 (0) | -48 (0) | -141 (0) | 239 (0) | 112 (0) | 463 (0) | 5795 (0) | -173 (0) | 0 (0) | 5622 (0)
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 5482 (9) | -45 (0) | -28 (20) | -102 (39) | 290 (51) | 201 (89) | 692 (229) | 6072 (277) | -167 (6) | -7 (-7) | 5898 (276)
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 5482 (9) | -45 (0) | -28 (20) | -102 (39) | 290 (51) | 201 (89) | 692 (229) | 6072 (277) | -162 (11) | 4 (4) | 5913 (291)
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | 5505 (32) | -31 (14) | -36 (12) | -103 (38) | 491 (252) | 157 (45) | 806 (343) | 6208 (413) | -14 (159) | -50 (-50) | 6170 (548)
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | 5505 (32) | -31 (14) | -36 (12) | -103 (38) | 491 (252) | 157 (45) | 806 (343) | 6208 (413) | 10 (183) | -49 (-49) | 6194 (572)
Type III ($m_{1}=4977$ MeV, $m_{2}=1628$ MeV and $m_{3}=220$ MeV) |
$(0,0)1S(\frac{1}{2}^{+})_{0}$ | 6853 (0) | -88 (0) | -55 (0) | -199 (0) | 66 (0) | 124 (0) | 313 (0) | 6967 (0) | -12 (0) | 0 (0) | 6955 (0)
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 6870 (17) | -88 (0) | -27 (28) | -143 (56) | 74 (8) | 261 (137) | 596 (283) | 7323 (356) | -13 (-1) | -26 (-26) | 7304 (349)
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 6870 (17) | -88 (0) | -27 (28) | -143 (56) | 74 (8) | 261 (137) | 596 (283) | 7323 (356) | -9 (3) | 13 (13) | 7317 (362)
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | 6856 (3) | -33 (55) | -43 (12) | -120 (79) | 192 (126) | 156 (32) | 504 (191) | 7240 (273) | -16 (-4) | -7 (-7) | 7220 (265)
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | 6856 (3) | -33 (55) | -43 (12) | -120 (79) | 192 (126) | 156 (32) | 504 (191) | 7240 (273) | 8 (20) | -7 (-7) | 7243 (288)
Table 2: Contribution of each term to the eigenvalues of $H$ for the states of the $(0,0)_{0}$, $(0,1)_{1}$ and $(1,0)_{1}$ modes of the $\textbf{6}_{F}$ sector (all quantities are in MeV). $(l_{\rho},l_{\lambda})nL(J^{P})_{j}$ | $H_{0}~{}(\Delta)$ | $G_{12}~{}(\Delta)$ | $G_{13(23)}~{}(\Delta)$ | $\sum G_{ij}~{}(\Delta)$ | $S_{12}~{}(\Delta)$ | $S_{13(23)}~{}(\Delta)$ | $\sum S_{ij}~{}(\Delta)$ | $H_{mode}~{}(\Delta)$ | $H^{hyp}~{}(\Delta)$ | $H^{so}~{}(\Delta)$ | $H~{}(\Delta)$
---|---|---|---|---|---|---|---|---|---|---|---
Type I ($m_{1}=m_{2}=m_{3}=220$ MeV) |
$(0,0)1S(\frac{1}{2}^{+})_{1}$ | 737 (0) | -45 (0) | -48 (0) | -141 (0) | 245 (0) | 152 (0) | 549 (0) | 1145 (0) | -131 (0) | 0 (0) | 1015 (0)
$(0,0)1S(\frac{3}{2}^{+})_{1}$ | 737 (0) | -45 (0) | -48 (0) | -141 (0) | 245 (0) | 152 (0) | 549 (0) | 1145 (0) | 110 (241) | 0 (0) | 1256 (241)
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 762 (25) | -45 (0) | -31 (17) | -107 (34) | 299 (54) | 305 (153) | 909 (360) | 1564 (419) | 21 (152) | -25 (-25) | 1571 (556)
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 762 (25) | -45 (0) | -31 (17) | -107 (34) | 299 (54) | 305 (153) | 909 (360) | 1564 (419) | 45 (176) | 12 (12) | 1615 (600)
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | 770 (33) | -31 (14) | -38 (10) | -108 (33) | 499 (254) | 205 (53) | 909 (360) | 1571 (426) | -167 (-36) |  $\sim$0 (0) | 1404 (389)
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | 770 (33) | -31 (14) | -38 (10) | -108 (33) | 499 (254) | 205 (53) | 909 (360) | 1571 (426) | 45 (176) |  $\sim$0 (0) | 1615 (600)
Type II ($m_{1}=m_{2}=220$ MeV and $m_{3}=4977$ MeV) |
$(0,0)1S(\frac{1}{2}^{+})_{1}$ | 5473 (0) | -45 (0) | -48 (0) | -141 (0) | 239 (0) | 112 (0) | 463 (0) | 5795 (0) | 24 (0) | 0 (0) | 5820 (0)
$(0,0)1S(\frac{3}{2}^{+})_{1}$ | 5473 (0) | -45 (0) | -48 (0) | -141 (0) | 239 (0) | 112 (0) | 463 (0) | 5795 (0) | 54 (30) | 0 (0) | 5849 (29)
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 5482 (9) | -45 (0) | -28 (20) | -102 (39) | 290 (51) | 201 (89) | 692 (229) | 6072 (277) | 38 (14) | -3 (-3) | 6107 (287)
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 5482 (9) | -45 (0) | -28 (20) | -102 (39) | 290 (51) | 201 (89) | 692 (229) | 6072 (277) | 43 (19) | 1 (1) | 6116 (296)
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | 5505 (32) | -33 (12) | -36 (12) | -103 (38) | 491 (252) | 157 (45) | 805 (342) | 6207 (412) | -31 (-55) |  $\sim$0 (0) | 6178 (358)
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | 5505 (32) | -33 (12) | -36 (12) | -103 (38) | 491 (252) | 157 (45) | 805 (342) | 6207 (412) | -5 (-29) |  $\sim$0 (0) | 6204 (384)
Type III ($m_{1}=m_{2}=4977$ MeV and $m_{3}=220$ MeV) |
$(0,0)1S(\frac{1}{2}^{+})_{1}$ | 10199 (0) | -137 (0) | -56 (0) | -250 (0) | 18 (0) | 118 (0) | 254 (0) | 10203 (0) | -10 (0) | 0 (0) | 10192 (0)
$(0,0)1S(\frac{3}{2}^{+})_{1}$ | 10199 (0) | -137 (0) | -56 (0) | -250 (0) | 18 (0) | 118 (0) | 254 (0) | 10203 (0) | 9 (19) | 0 (0) | 10211 (19)
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 10216 (17) | -137 (0) | -27 (29) | -191 (59) | 21 (3) | 259 (141) | 539 (285) | 10564 (361) |  $\sim$0 (10) | -2 (-2) | 10563 (371)
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 10216 (17) | -137 (0) | -27 (29) | -191 (59) | 21 (3) | 259 (141) | 539 (285) | 10564 (361) | 3 (13) | 1 (1) | 10569 (377)
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | 10201 (2) | -53 (84) | -50 (6) | -152 (98) | 115 (97) | 138 (20) | 391 (137) | 10440 (237) | -12 (-2) |  $\sim$0 (0) | 10428 (236)
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | 10201 (2) | -53 (84) | -50 (6) | -152 (98) | 115 (97) | 138 (20) | 391 (137) | 10440 (237) | 5 (15) |  $\sim$0 (0) | 10445 (253)
Table 3: Calculated $\langle r_{\rho}^{2}\rangle^{1/2}$ and $\langle r_{\lambda}^{2}\rangle^{1/2}$ (in fm) for the states of the $(0,0)_{0}$, $(0,1)_{1}$ and $(1,0)_{1}$ modes. And comparison of the rotational kinetic energy $\langle E_{r.k.}\rangle$ and the excitation energy $\langle\Delta H_{mode}\rangle$ (in MeV) of the above modes. The data in boldface are used in calculations ($m_{Q}=4977$ MeV and $m_{q}=220$ MeV).  the $\bar{\mathbf{3}}_{F}$ sector | the $\textbf{6}_{F}$ sector
---|---
$(l_{\rho},l_{\lambda})nL(J^{P})_{j}$ | $\langle r_{\rho}^{2}\rangle^{1/2}$ | $\langle r_{\lambda}^{2}\rangle^{1/2}$ | $\langle I\rangle$ | $\langle E_{r.k.}\rangle$ | $\langle\Delta H_{mode}\rangle$ | $(l_{\rho},l_{\lambda})nL(J^{P})_{j}$ | $\langle r_{\rho}^{2}\rangle^{1/2}$ | $\langle r_{\lambda}^{2}\rangle^{1/2}$ | $\langle I\rangle$ | $\langle E_{r.k.}\rangle$ | $\langle\Delta H_{mode}\rangle$
Type I ($m_{1}=m_{2}=m_{3}=220$ MeV) | Type I ($m_{1}=m_{2}=m_{3}=220$ MeV)
$(0,0)1S(\frac{1}{2}^{+})_{0}$ | 0.534 | 0.520 | - | - | - | $(0,0)1S(\frac{1}{2}^{+})_{1}$ | 0.609 | 0.471 | - | - | -
- | - | - | - | - | - | $(0,0)1S(\frac{3}{2}^{+})_{1}$ | 0.672 | 0.595 | - | - | -
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 0.566 | $\mathbf{0.717}$ | $\frac{2}{3}m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 516$ | 419 | $(0,1)1P(\frac{1}{2}^{-})_{1}$ | 0.701 | $\mathbf{0.770}$ | $\frac{2}{3}m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 447$ | 419
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 0.587 | $\mathbf{0.798}$ | $\frac{2}{3}m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 414$ | 419 | $(0,1)1P(\frac{3}{2}^{-})_{1}$ | 0.718 | $\mathbf{0.823}$ | $\frac{2}{3}m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 390$ | 419
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | $\mathbf{0.827}$ | 0.500 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 516$ | 426 | $(1,0)1P(\frac{1}{2}^{-})_{1}$ | $\mathbf{0.842}$ | 0.503 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 498$ | 426
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | $\mathbf{0.900}$ | 0.628 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 436$ | 426 | $(1,0)1P(\frac{3}{2}^{-})_{1}$ | $\mathbf{0.920}$ | 0.633 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 417$ | 426
Type II ($m_{1}=m_{2}=220$ MeV, $m_{3}=4977$ MeV) | Type II ($m_{1}=m_{2}=220$ MeV, $m_{3}=4977$ MeV)
$(0,0)1S(\frac{1}{2}^{+})_{0}$ | 0.519 | 0.407 | - | - | - | $(0,0)1S(\frac{1}{2}^{+})_{1}$ | 0.633 | 0.430 | - | - | -
- | - | - | - | - | - | $(0,0)1S(\frac{3}{2}^{+})_{1}$ | 0.644 | 0.451 | - | - | -
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 0.534 | $\mathbf{0.577}$ | $2m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 265$ | 277 | $(0,1)1P(\frac{1}{2}^{-})_{1}$ | 0.666 | $\mathbf{0.621}$ | $2m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 229$ | 277
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 0.539 | $\mathbf{0.590}$ | $2m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 253$ | 277 | $(0,1)1P(\frac{3}{2}^{-})_{1}$ | 0.670 | $\mathbf{0.630}$ | $2m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 222$ | 277
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | $\mathbf{0.852}$ | 0.440 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 486$ | 413 | $(1,0)1P(\frac{1}{2}^{-})_{1}$ | $\mathbf{0.869}$ | 0.444 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 467$ | 412
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | $\mathbf{0.865}$ | 0.462 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 472$ | 413 | $(1,0)1P(\frac{3}{2}^{-})_{1}$ | $\mathbf{0.883}$ | 0.466 | $\frac{1}{2}m_{q}\langle r_{\rho}^{2}\rangle$ | $\sim 452$ | 412
Type III ($m_{1}=4977$ MeV, $m_{2}=1628$ MeV, $m_{3}=220$ MeV) | Type III ($m_{1}=m_{2}=4977$ MeV, $m_{3}=220$ MeV)
$(0,0)1S(\frac{1}{2}^{+})_{0}$ | 0.370 | 0.479 | - | - | - | $(0,0)1S(\frac{1}{2}^{+})_{1}$ | 0.297 | 0.470 | - | - | -
- | - | - | - | - | - | $(0,0)1S(\frac{3}{2}^{+})_{1}$ | 0.299 | 0.483 | - | - | -
$(0,1)1P(\frac{1}{2}^{-})_{1}$ | 0.391 | $\mathbf{0.654}$ | $m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 412$ | 356 | $(0,1)1P(\frac{1}{2}^{-})_{1}$ | 0.313 | $\mathbf{0.666}$ | $m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 398$ | 361
$(0,1)1P(\frac{3}{2}^{-})_{1}$ | 0.395 | $\mathbf{0.696}$ | $m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 364$ | 356 | $(0,1)1P(\frac{3}{2}^{-})_{1}$ | 0.314 | $\mathbf{0.688}$ | $m_{q}\langle r_{\lambda}^{2}\rangle$ | $\sim 373$ | 361
$(1,0)1P(\frac{1}{2}^{-})_{1}$ | $\mathbf{0.562}$ | 0.503 | $\frac{1}{4}m_{Q}\langle r_{\rho}^{2}\rangle$ | $\sim 99$ | 273 | $(1,0)1P(\frac{1}{2}^{-})_{1}$ | $\mathbf{0.456}$ | 0.496 | $\frac{1}{2}m_{Q}\langle r_{\rho}^{2}\rangle$ | $\sim 75$ | 237
$(1,0)1P(\frac{3}{2}^{-})_{1}$ | $\mathbf{0.566}$ | 0.522 | $\frac{1}{4}m_{Q}\langle r_{\rho}^{2}\rangle$ | $\sim 97$ | 273 | $(1,0)1P(\frac{3}{2}^{-})_{1}$ | $\mathbf{0.459}$ | 0.508 | $\frac{1}{2}m_{Q}\langle r_{\rho}^{2}\rangle$ | $\sim 74$ | 237
Table 4: Contrast of the calculated results f11 ; f11p  and data for the well established singly heavy baryons f14 . Baryon | $M_{exp.}$ | $J^{P}$ | $M_{cal.}$ | $nL(J^{P})$ | Baryon | $M_{exp.}$ | $J^{P}$ | $M_{cal.}$ | $nL(J^{P})$
---|---|---|---|---|---|---|---|---|---
$\Lambda_{c}^{+}$ | 2286.46$\pm$0.14 | $\frac{1}{2}^{+}$ | 2288 | $1S(\frac{1}{2}^{+})$ | $\Xi_{c}(2970)^{+}$ | 2964.3$\pm$1.5 | $\frac{1}{2}^{+}$ | 2949 | $2S(\frac{1}{2}^{+})(\bar{\mathbf{3}}_{F})$
$\Lambda_{c}(2765)^{+}$ f14p  | 2766.6${}^{+3.6}_{-7.1}$ | $\frac{1}{2}^{+}$ | 2764 | $2S(\frac{1}{2}^{+})$ | $\Xi_{c}(2970)^{0}$ | 2967.1$\pm$1.7 | $\frac{1}{2}^{+}$ | 2949 | $2S(\frac{1}{2}^{+})(\bar{\mathbf{3}}_{F})$
$\Lambda_{c}(2595)^{+}$ | 2592.25$\pm$0.28 | $\frac{1}{2}^{-}$ | 2596 | $1P(\frac{1}{2}^{-})$ | $\Omega_{c}^{0}$ | 2695.2$\pm$1.7 | $\frac{1}{2}^{+}$ | 2699 | $1S(\frac{1}{2}^{+})$
$\Lambda_{c}(2625)^{+}$ | 2628.11$\pm$0.19 | $\frac{3}{2}^{-}$ | 2631 | $1P(\frac{3}{2}^{-})$ | $\Omega_{c}(2770)^{0}$ | 2765.9$\pm$0.28 | $\frac{3}{2}^{+}$ | 2762 | $1S(\frac{3}{2}^{+})$
$\Lambda_{c}(2860)^{+}$ | 2856.1${}^{+2.3}_{-6.0}$ | $\frac{3}{2}^{+}$ | 2875 | $1D(\frac{3}{2}^{+})$ | $\Lambda_{b}^{0}$ | 5619.60$\pm$0.17 | $\frac{1}{2}^{+}$ | 5622 | $1S(\frac{1}{2}^{+})$
$\Lambda_{c}(2880)^{+}$ | 2881.63$\pm$0.24 | $\frac{5}{2}^{+}$ | 2891 | $1D(\frac{5}{2}^{+})$ | $\Lambda_{b}(5912)^{0}$ | 5912.19$\pm$0.17 | $\frac{1}{2}^{-}$ | 5898 | $1P(\frac{1}{2}^{-})$
$\Sigma_{c}(2455)^{++}$ | 2453.97$\pm$0.14 | $\frac{1}{2}^{+}$ | 2457 | $1S(\frac{1}{2}^{+})$ | $\Lambda_{b}(5920)^{0}$ | 5920.09$\pm$0.17 | $\frac{3}{2}^{-}$ | 5913 | $1P(\frac{3}{2}^{-})$
$\Sigma_{c}(2455)^{+}$ | 2452.65${}^{+0.22}_{-0.16}$ | $\frac{1}{2}^{+}$ | 2457 | $1S(\frac{1}{2}^{+})$ | $\Lambda_{b}(6070)^{+}$ | 6072.3$\pm$2.9 | $\frac{1}{2}^{+}$ | 6041 | $2S(\frac{1}{2}^{+})$
$\Sigma_{c}(2455)^{0}$ | 2453.75$\pm$0.14 | $\frac{1}{2}^{+}$ | 2457 | $1S(\frac{1}{2}^{+})$ | $\Lambda_{b}(6146)^{0}$ | 6146.2$\pm$0.4 | $\frac{3}{2}^{+}$ | 6137 | $1D(\frac{3}{2}^{+})$
$\Sigma_{c}(2520)^{++}$ | 2518.41${}^{+0.22}_{-0.18}$ | $\frac{3}{2}^{+}$ | 2532 | $1S(\frac{3}{2}^{+})$ | $\Lambda_{b}(6152)^{0}$ | 6152.5$\pm$0.4 | $\frac{5}{2}^{+}$ | 6145 | $1D(\frac{5}{2}^{+})$
$\Sigma_{c}(2520)^{+}$ | 2517.4${}^{+0.7}_{-0.5}$ | $\frac{3}{2}^{+}$ | 2532 | $1S(\frac{3}{2}^{+})$ | $\Sigma_{b}^{+}$ | 5810.56$\pm$0.25 | $\frac{1}{2}^{+}$ | 5820 | $1S(\frac{1}{2}^{+})$
$\Sigma_{c}(2520)^{0}$ | 2518.48$\pm$0.20 | $\frac{3}{2}^{+}$ | 2532 | $1S(\frac{3}{2}^{+})$ | $\Sigma_{b}^{-}$ | 5815.64$\pm$0.27 | $\frac{1}{2}^{+}$ | 5820 | $1S(\frac{1}{2}^{+})$
$\Xi_{c}^{+}$ | 2467.71$\pm$0.23 | $\frac{1}{2}^{+}$ | 2479 | $1S(\frac{1}{2}^{+})(\bar{\mathbf{3}}_{F})$ | $\Sigma_{b}^{*+}$ | 5830.32$\pm$0.27 | $\frac{3}{2}^{+}$ | 5849 | $1S(\frac{3}{2}^{+})$
$\Xi_{c}^{0}$ | 2470.44$\pm$0.28 | $\frac{1}{2}^{+}$ | 2479 | $1S(\frac{1}{2}^{+})(\bar{\mathbf{3}}_{F})$ | $\Sigma_{b}^{*-}$ | 5834.74$\pm$0.30 | $\frac{3}{2}^{+}$ | 5849 | $1S(\frac{3}{2}^{+})$
$\Xi_{c}^{{}^{\prime}+}$ | 2578.2$\pm$0.5 | $\frac{1}{2}^{+}$ | 2590 | $1S(\frac{1}{2}^{+})(\mathbf{6}_{F})$ | $\Xi_{b}^{-}$ | 5797.0$\pm$0.6 | $\frac{1}{2}^{+}$ | 5806 | $1S(\frac{1}{2}^{+})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}^{{}^{\prime}0}$ | 2578.7$\pm$0.5 | $\frac{1}{2}^{+}$ | 2590 | $1S(\frac{1}{2}^{+})(\mathbf{6}_{F})$ | $\Xi_{b}^{0}$ | 5791.9$\pm$0.5 | $\frac{1}{2}^{+}$ | 5806 | $1S(\frac{1}{2}^{+})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}(2645)^{+}$ | 2645.1$\pm$0.3 | $\frac{3}{2}^{+}$ | 2658 | $1S(\frac{3}{2}^{+})(\mathbf{6}_{F})$ | $\Xi_{b}(5935)^{-}$ | 5935.02$\pm$0.05 | $\frac{1}{2}^{+}$ | 5943 | $1S(\frac{1}{2}^{+})(\mathbf{6}_{F})$
$\Xi_{c}(2645)^{0}$ | 2646.16$\pm$0.25 | $\frac{3}{2}^{+}$ | 2658 | $1S(\frac{3}{2}^{+})(\mathbf{6}_{F})$ | $\Xi_{b}(5945)^{0}$ | 5952.3$\pm$0.6 | $\frac{3}{2}^{+}$ | 5971 | $1S(\frac{3}{2}^{+})(\mathbf{6}_{F})$
$\Xi_{c}(2790)^{+}$ | 2791.9$\pm$0.5 | $\frac{1}{2}^{-}$ | 2789 | $1P(\frac{1}{2}^{-})(\bar{\mathbf{3}}_{F})$ | $\Xi_{b}(5955)^{-}$ | 5955.33$\pm$0.13 | $\frac{3}{2}^{+}$ | 5971 | $1S(\frac{3}{2}^{+})(\mathbf{6}_{F})$
$\Xi_{c}(2790)^{0}$ | 2793.9$\pm$0.5 | $\frac{1}{2}^{-}$ | 2789 | $1P(\frac{1}{2}^{-})(\bar{\mathbf{3}}_{F})$ | $\Xi_{b}(6100)^{-}$ | 6100.3$\pm$0.6 | $\frac{3}{2}^{-}$ | 6097 | $1P(\frac{3}{2}^{-})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}(2815)^{+}$ | 2816.51$\pm$0.25 | $\frac{3}{2}^{-}$ | 2819 | $1P(\frac{3}{2}^{-})(\bar{\mathbf{3}}_{F})$ | $\Omega_{b}^{-}$ | 6045.2$\pm$1.2 | $\frac{1}{2}^{+}$ | 6043 | $1S(\frac{1}{2}^{+})$
$\Xi_{c}(2815)^{0}$ | 2819.79$\pm$0.30 | $\frac{3}{2}^{-}$ | 2819 | $1P(\frac{3}{2}^{-})(\bar{\mathbf{3}}_{F})$ | $\Lambda_{c}(2940)^{+}$ | 2939.6${}^{+1.3}_{-1.5}$ | $\frac{3}{2}^{-}$ | (?) | (?)
Table 5: The observed baryons with unconfirmed $J^{P}$ and the corresponding calculated results f11 ; f11p . Baryon | $M_{exp.}$ | $M_{cal.}$ | $nL(J^{P})$ | Baryon | $M_{exp.}$ | $M_{cal.}$ | $nL(J^{P})$
---|---|---|---|---|---|---|---
$\Sigma_{c}(2800)^{++}$ f14  | 2801${}^{+4}_{-6}$ |  | $1P(\mathbf{6}_{F})$ |  $\Sigma_{b}(6097)^{+}$ f14  | 6095.8$\pm$1.7 |  | $1P(\mathbf{6}_{F})$
$\Sigma_{c}(2800)^{+}$ f14  | 2792${}^{+14}_{-5}$ |  | $1P(\mathbf{6}_{F})$ |  $\Sigma_{b}(6097)^{-}$ f14  | 6098.0$\pm$1.8 |  | $1P(\mathbf{6}_{F})$
$\Sigma_{c}(2800)^{0}$ f14  | 2806${}^{+5}_{-7}$ |  | $1P(\mathbf{6}_{F})$ |  $\Xi_{b}(6227)^{-}$ f14  | 6227.9$\pm$0.9 |  | $1P(\mathbf{6}_{F})$
|  |  |  |  $\Xi_{b}(6227)^{0}$ f14  | 6226.8$\pm$1.6 |  | $1P(\mathbf{6}_{F})$
$\Xi_{c}(3055)^{+}$ f14  | 3055.9$\pm$0.4 | 3063 | $1D(\frac{3}{2}^{+})(\bar{\mathbf{3}}_{F})$ |  $\Xi_{b}(6327)^{0}$ f14  | 6327.28$\pm$0.35 | 6320 | $1D(\frac{3}{2}^{+})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}(3080)^{+}$ f14  | 3077.2$\pm$0.4 | 3076 | $1D(\frac{5}{2}^{+})(\bar{\mathbf{3}}_{F})$ |  $\Xi_{b}(6333)^{0}$ f14  | 6332.69$\pm$0.28 | 6327 | $1D(\frac{5}{2}^{+})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}(3080)^{0}$ f14  | 3079.9$\pm$1.4 | 3076 | $1D(\frac{5}{2}^{+})(\bar{\mathbf{3}}_{F})$ |  |  |  |
$\Omega_{c}(3000)$ f14  | 3000.41$\pm$0.22 |  | $1P(\mathbf{6}_{F})$ |  $\Omega_{b}(6316)^{-}$ f14  | 6315.6$\pm$0.6 |  | $1P(\mathbf{6}_{F})$
$\Omega_{c}(3050)^{+}$ f14  | 3050.19$\pm$0.13 |  | $1P(\mathbf{6}_{F})$ |  $\Omega_{b}(6330)^{-}$ f14  | 6330.3$\pm$0.6 |  | $1P(\mathbf{6}_{F})$
$\Omega_{c}(3065)^{0}$ f14  | 3065.54$\pm$0.26 |  | $1P(\mathbf{6}_{F})$ |  $\Omega_{b}(6340)^{-}$ f14  | 6339.7$\pm$0.6 |  | $1P(\mathbf{6}_{F})$
$\Omega_{c}(3090)^{0}$ f14  | 3090.1$\pm$0.5 |  | $1P(\mathbf{6}_{F})$ |  $\Omega_{b}(6350)^{-}$ f14  | 6349.8$\pm$0.6 |  | $1P(\mathbf{6}_{F})$
$\Xi_{c}(2923)^{0}$ f16  | 2923.04$\pm$0.59 |  | $1P(\mathbf{6}_{F})$ |  $\Omega_{c}(3185)^{0}$ f16p3  | 3185.1$\pm$1.7 | 3197 | $2S(\frac{3}{2}^{+})(\mathbf{6}_{F})$
$\Xi_{c}(2939)^{0}$ f16  | 2938.55$\pm$0.52 |  | $1P(\mathbf{6}_{F})$ |  $\Omega_{c}(3327)^{0}$ f16p3  | 3327.1$\pm$1.2 |  | $1D(\mathbf{6}_{F})$
$\Xi_{c}(2964)^{0}$ f16  | 2964.88$\pm$0.54 |  | $1P(\mathbf{6}_{F})$ |  $\Xi_{b}(6095)^{0}$ f16p4  | 6095.36$\pm$0.68 | 6097 | $1P(\frac{3}{2}^{-})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}(2930)^{+}$ f16p1  | 2942.3$\pm$5.9 |  | $1P(\mathbf{6}_{F})$ |  $\Xi_{b}(6087)^{0}$ f16p4  | 6087.24$\pm$0.76 | 6084 | $1P(\frac{1}{2}^{-})(\bar{\mathbf{3}}_{F})$
$\Xi_{c}(3123)^{+}$ f16p2  | 3122.9$\pm$1.3 | (?) | (?) |  $\Omega_{c}(3120)^{+}$ f14  | 3119.1$\pm$1.0 | (?) | (?)