# A Mixing Coupling Scheme for Spectra of Singly Heavy Baryons with Spin-1
Diquarks in P-waves

Duojie Jia1,4 111Corresponding author<EMAIL_ADDRESS>Ji-Hai Pan1 Cheng-Qun
Pang2,3,4<EMAIL_ADDRESS>1Institute of Theoretical Physics, College of
Physics and Electronic Engineering, Northwest Normal University, Lanzhou
730070, China
2College of Physics and Electronic Information Engineering, Qinghai Normal
University, Xining 810000, China
3Joint Research Center for Physics, Lanzhou University and Qinghai Normal
University, Xining 810000,China
4Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000,China

###### Abstract

A new scheme of state classification is proposed and applied to analyze masses
of the heavy baryons $\Omega_{Q}$, $\Sigma_{Q}$ and $\Xi_{Q}^{\prime}$ in
P-waves. The results confirm all excited $\Omega_{c}$ and $\Omega_{b}$ baryons
reported recently by LHCb to be bound states of a P-wave $ss$-diquark and a
respective charm or bottom quark, and thereby predict Regge trajectories for
more excited $\Omega_{c}$ and $\Omega_{b}$ baryons. We suggest one excited
$J^{P}=5/2^{-}$ $\Omega_{b}$ state to be unseen by LHCb around $6352$ MeV, and
predict P-wave masses of all spin-partners of the odd-parity baryons
$\Sigma_{c}(2800)/\Xi_{c}^{\prime}(2942)$ and
$\Sigma_{b}(6097)$/$\Xi_{b}^{\prime}(6227)$. A computation is further given in
a relativized potential quark models to explain matched values of spin
couplings of all considered baryons, by which a scaling law for these spin
couplings is discussed.

PACS number(s):12.39Jh, 12.40.Yx, 12.40.Nn

Key Words: Heavy baryon, Spectroscopy, Quantum number, Spin coupling

##  I Introduction

Scientific interest as to spectrum of strongly interacting heavy hadrons has
raised again in last two decades greatly due to the discoveries of numerous
heavy subatomic particles Tanabashi:D18 . In 2017, LHCb observed five even
narrower $\Omega_{c}$ resonances Aaij:L2017 (Table I) in the
$\Xi_{c}^{+}K^{-}$ decay channel based on $pp$ collision data. Very recently,
LHCb, again, reported the discovery of four even narrower $\Omega_{b}$
resonances around $6.3$ GeV decaying into $\Xi_{b}^{0}K^{-}$ Aaij:L2020 , with
slightly less significance. The reported masses and decay widths are listed in
Table I for both of them.

Observed patterns of masses of the excited $\Omega_{c,b}$ baryons turn out to
be perplexing and thereby rise naturally questions as to (i) why there are
four excited states of the $\Omega_{b}$ baryons while there are five for the
$\Omega_{c}$, and (ii) what their spin-parities or their inner structures are.
These questions have been addressed by many authors EFG:B08 ; EFG:D11 ;
MaltmanI:D80 ; RobertsP:A08 ; Garcilazo:G07 ; Migura:A06 ; Valcarce:A08 ;
YamaguchiH:D15 ; PerezCB:D15 ; ShahTh:C16 ; YoshidaHH:D15 ; ChenHL:D15 ;
ChenMH:D16  with recent explorations given in Refs KarlinerR:17 ;
PadmanathM:L17 ; ChenL:D17 ; WangZ:D17 ; Wang:C17 ; Agaev:U17 ; ChenHL:D17 ;
ChengC:D17 ; WangZZ:D17 ; ZhaoY:D17 ; YangP:D18 ; HuangP:D18 ; AnChen:D17 ;
Prasz:D17 ; KimPP:D17 ; ChenLM:E17 ; KarlinerR:Ax20 , where spin-parities were
assigned for the $\Omega_{c}$ states around $3.0\pm 0.1$GeV and for the
excited $\Omega_{b}$ states around $6.3\pm 0.2$GeV. Regarding the reported
$\Omega_{c}$ resonances, different interpretations were suggested, which
include the $P$-wave assignment KarlinerR:17 ; PadmanathM:L17 ; ChenL:D17 ;
WangZ:D17 ; Wang:C17 ; ShahTh:C16 , the $P$-wave and $S/D$-wave assignment
ChengC:D17 ; ZhaoY:D17 ; ChenHL:D15 , the charmed exotic systems YangP:D18 ;
HuangP:D18 ; AnChen:D17 ; Prasz:D17 ; KimPP:D17 , and entail more efforts to
answer above two questions.

For charmed baryons, spectroscopy is known to be intricate, with many excited
states expected EFG:B08 ; EFG:D11 ; MaltmanI:D80 ; RobertsP:A08 ;
Garcilazo:G07 ; PerezCB:D15 ; YoshidaHH:D15 ; ChenHL:D15 . The common practice
is to utilize the heavy quark symmetry(HQS) Manohar:D07 , which becomes exact
in the limit of heavy quark $m_{Q}\rightarrow\infty$. Assuming HQS by which
spin $\mathbf{S}_{Q}$ of heavy quark $Q$ is conserved, one can classify heavy
hadrons by their total spin $\mathbf{J}$ and the angular momentum
$\mathbf{j=J-S}_{Q}$ of light degrees of freedom(named $jj$ coupling). This is
analogous to the hydrogen-like atoms whose quantum states are well labeled by
quantum numbers of outermost electron. While this picture works well in
describing the normal charmed and bottom baryons EFG:D11 ; MaltmanI:D80 ;
RobertsP:A08  with strangeness $|S|<2$, its applicability to the
$\Omega_{c,b}$ baryons remains to be explored.

The purpose of this work is to explore quantum numbers and inner structures of
the odd-parity bound systems of a heavy $Q$-quark ($Q=c,b$) and a light spin-1
$qq$-diquark($qq=nn,ns,n=u,d$) and corresponding main feature of relative
magnitudes of the spin coupling parameters for the systems. We propose a new
scheme of state classification (named the $Jls$ mixing coupling) to perform
systematic mass analysis for the newly LHCb-reported resonances of the excited
$\Omega_{c}$ and $\Omega_{b}$, which interprets all of these resonances to be
the $P$-wave negative-parity baryons, and enables us to predict Regge
trajectories of the mean-spin masses of their excited states. We update
earlier computations for the negative-parity baryons $\Sigma_{Q}$ and
$\Xi_{Q}^{\prime}$, and explain the matched spin-couplings of these baryons in
a relativized potential quark model,

Table 1: Masses and widths of the LHCb-reported resonances $\Omega_{c}$
Aaij:L2017  and $\Omega_{b}$Aaij:L2020 , and excited baryons $\Sigma_{Q}$ and
$\Xi_{Q}^{\prime}$Tanabashi:D18 , with the proposed values of spin-parity
$J^{P}$ shown also.  $\begin{array}[]{cccc}\hline\cr\hline\cr$State$&$Mass
(MeV)$&$Width (MeV)$&$$J^{P}$ Proposed$\\\
\hline\cr\Omega_{c}(3000)^{0}&3000.4\pm 0.2\pm 0.1&4.5\pm 0.6\pm
0.3&1/2^{-}\\\ \Omega_{c}(3050)^{0}&3050.2\pm 0.1\pm 0.1&0.8\pm 0.2\pm
0.1&1/2^{-}\\\ \Omega_{c}(3066)^{0}&3065.6\pm 0.1\pm 0.3&3.5\pm 0.4\pm
0.2&3/2^{-}\\\ \Omega_{c}(3090)^{0}&3090.2\pm 0.3\pm 0.5&8.7\pm 1.0\pm
0.8&3/2^{-}\\\ \Omega_{c}(3119)^{0}&3119.1\pm 0.3\pm 0.9&1.1\pm 0.8\pm
0.4^{b}&5/2^{-}\\\ \Omega_{b}(6316)^{-}&6315.64\pm 0.31\pm 0.07^{\pm
0.5}&<2.8,&1/2^{-}\\\ \Omega_{b}(6330)^{-}&6330.30\pm 0.28\pm 0.07^{\pm
0.5}&<3.1,&1/2^{-}\\\ \Omega_{b}(6340)^{-}&6339.71\pm 0.26\pm 0.05^{\pm
0.5}&<1.5&3/2^{-}\\\ \Omega_{b}(6350)^{-}&6349.88\pm 0.35\pm 0.05^{\pm
0.5}&1.4_{-0.8}^{+0.1}\pm 1.0&3/2^{-}\\\
\Sigma_{c}(2800)^{++}&2801_{-6}^{+4}&75_{-17}^{+22}&3/2^{-}\\\
\Xi_{c}^{\prime}(2930)&2942\pm 5&36\pm 13&3/2^{-}\\\
\Sigma_{b}(6097)^{-}&6098.0\pm 1.8&29\pm 4&3/2^{-}\\\
\Xi_{b}^{\prime}(6227)^{-}&6226.9\pm 2.0&18\pm 6&1/2^{-}\\\
\hline\cr\hline\cr\end{array}$

To achieve that, we determine the mean-spin mass $\bar{M}$ and the spin
coupling strengths ($a_{1},a_{2},b,c$) as effective parameters by matching
computed and measured masses of heavy baryons considered, which implies that a
central potential plus spin-interactions describe the main features of most
established hadrons. Due to their connections with the interquark potential,
these knowledges about spin-couplings are useful and forms a basis for further
understanding QCD interaction within hadrons.

We apply our proposed scheme of the $Jls$ mixing coupling to mass analysis of
the excited $\Omega_{c}$ baryons in Sect. II, where preferred spin-parity are
assigned for them. We examine other plausible spin-parity arrangements in
Sect. III. Similar mass analysis is given for excited $\Omega_{b}$ baryons in
Sect. IV. In Sect. V, the $Jls$ mixing coupling is applied to the less strange
baryons $\Sigma_{Q}$ and $\Xi_{Q}^{\prime}$. A computation is given in the
relativized quark model to explain the matched spin-couplings in Sect. VI. We
end with conclusions and discussions in Sect. VII.

##  II Inner Structures of the $c(ss)$ Systems

In heavy quark-diquark picture, two strange quarks in a $c(ss)$ system form a
S-wave anti-color triplet($\mathbf{\bar{3}}_{c}$) diquark $(ss)$, with spin
one($S_{ss}=1$) due to the spatial symmetry under exchange of two fermions.
The diquark spin($=1$) can couple with spin $S_{c}=1/2$ of the charm quark $c$
to form a total spin $S=1\pm 1/2=1/2,3/2$. Let us consider the relative P-wave
excitations of diquark $ss$ with respect to the charm quark $c$(the relative
orbital angular momentum $L=1$). Coupling of $L=1$ with the spin $S=1/2$ gives
states with total spin $J=1/2,3/2$, while coupling with $S=3/2$ leads to
states with $J=1/2,3/2$ and $5/2$. Symbolically, they are written as, in terms
of adding of angular momentum,

$\displaystyle\left(\frac{1}{2}\right)_{S}\otimes 1_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\oplus\frac{3}{2},$
$\displaystyle\left(\frac{3}{2}\right)_{S}\otimes 1_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}^{\prime}\oplus\frac{3}{2}^{\prime}\oplus\frac{5}{2},$
(1)

In total, one has five $P$-wave states with $J=1/2,3/2,1/2^{{}^{\prime}}$,
$3/2^{\prime}$ and $5/2$ with negative parity $P=-1$ which implies $5!=120$ a
priori possible assignments for their $J^{P}$ quantum numbers.

For most heavy baryons with less strangeness, the $jj$ coupling has been
commonly used to classify their states in terms of hadron eigenstates
$|J,j\rangle$, with $J$ the eigenvalues of $\mathbf{J}$ and $j$ of
$\mathbf{j}$ respectively. In the case of doubly strange $Qss$ baryons with
the $ss$-diquark comparable with the heavy quark $Q$ in mass, the finite mass
effect of the heavy quark may become important and makes it appropriate to go
beyond the $jj$ coupling.

For this, we propose a new scheme of state classification(named the $Jls$
mixing coupling) in which the eigenfunctions (bases) of spin multiplets of the
$Q(ss)$ system($Q=c,b$) diagonalize all spin-orbit interactions between the
heavy quark $Q$ and the diquark $ss$. The spin-dependent interaction for the
$Q(ss)$ systems, in the heavy quark-diquark picture, is EFG:D11 ;
KarlinerR:D15

$H^{SD}=a_{1}\mathbf{L}\cdot\mathbf{S}_{ss}+a_{2}\mathbf{L}\cdot\mathbf{S}_{Q}+bS_{12}+c\mathbf{S}_{ss}\cdot\mathbf{S}_{Q},$
(2)
$S_{12}=3\mathbf{S}_{ss}\cdot\mathbf{\hat{r}S}_{Q}\cdot\mathbf{\hat{r}}-\mathbf{S}_{ss}\cdot\mathbf{S}_{Q},$

where the first two terms are spin-orbit interactions, the third is the tensor
energy, and the last the contact interaction between the heavy quark spin
$\mathbf{S}_{Q}$ and the diquark spin $\mathbf{S}_{ss}$. Here, $\mathbf{L}$ is
the orbital angular momentum of the system. Though four strengths $a_{1,2}$,
$b$ and $c$ of spin-couplings will be treated, in a sense, as effective
parameters in this work, they are closely related to interquark interactions
within hadrons and assumed, in potential quark modelEFG:D11 ; MaltmanI:D80 ;
RobertsP:A08 ; YoshidaHH:D15 ; ChenLM:E17 , to be computable from QCD
analogues of Breit-Fermi interaction in QED.

If $Q$ is very heavy, the first spin-orbit term
$\mathbf{L}\cdot\mathbf{S}_{ss}$ should dominate over the others in Eq.(2) if
one assumes spin-dependent interactions enter the color hyperfine interactions
through the magnetic moment $e_{i}\mathbf{S}_{i}/m_{i}$ of the quark $i=Q$ or
$ss$ with mass $m_{i}$, by analogy with the spin-relevant relativistic
correction of the heavy quarkonium Yndurain83 . As such, the heavy quark spin
$\mathbf{S}_{Q}$ decouples with the light degree of freedom and is conserved
in the heavy quark limit, making the basis eigenfunctions of
$\mathbf{L}\cdot\mathbf{S}_{ss}$(the $jj$ coupling) appropriate to classify
the baryon states. Here, the baryon spin $\mathbf{J}$ and the diquark angular
momentum $\mathbf{j=J-S}_{Q}$ are both conserved, enabling baryon states to be
labelled by the $jj$ coupling states $|J,j\rangle$. In the case of the $c(ss)$
or $b(ss)$ baryons, in which the diquark mass $m_{ss}$(near $1$ GeV) is
comparable to $M_{Q}$(about $1.5$ GeV), the terms like
$a_{2}\mathbf{L}\cdot\mathbf{S}_{Q}$ in Eq. (2) may become important. Taking
$a_{1}=a_{2}$, for instance, the spin-orbit interaction becomes proportional
to $\mathbf{L}\cdot[\mathbf{S}_{ss}+\mathbf{S}_{Q}]=\mathbf{L}\cdot\mathbf{S}$
and diagonal in the $LS$ coupling, in which $\mathbf{S}_{Q}$ first couples
$\mathbf{S}_{ss}$ to form the total quark spin
$\mathbf{S}=\mathbf{S}_{ss}+\mathbf{S}_{Q}$ and then to $\mathbf{L}$ of the
quark-diquark system to form $\mathbf{J}$, as shown in Eq. (1).

In the $Jls$ mixing coupling considered in this work, the bases
$|J,j_{LS}=j^{\prime}\rangle$ diagonalize the interaction
$a_{1}\mathbf{L}\cdot\mathbf{S}_{ss}+a_{2}\mathbf{L}\cdot\mathbf{S}_{Q}+bS_{12}$,
instead of $\mathbf{L}\cdot\mathbf{S}_{ss}$ solely, in Eq.(2). The $Jls$
scheme reduces to the $jj$ coupling when the ratio
$\epsilon=a_{2}/a_{1}$(expected to scale as $m_{ss}/M_{Q}$) tends to zero in
the heavy quark limit,

$a_{1}\mathbf{L}\cdot[\mathbf{S}_{ss}+\epsilon\mathbf{S}_{Q}]+bS_{12}\simeq
a_{1}\mathbf{L}\cdot\mathbf{S}_{ss}\text{, as }M_{Q}\rightarrow\infty\text{,}$
(3)

where $b$ is expected to be suppressed by $1/M_{Q}$ KarlinerR:D15 ; JiaLH:19 .
Finding the $Jls$ mixing eigenstates $|J,j_{LS}=j^{\prime}\rangle$ can be done
by solving the linear eigenstate equation of $2\times 2$ matrices
$\Delta\mathcal{M}_{J}$ of the mass shift interaction (2) in the spin subspace
of $J=1/2$ and $J=3/2$ KarlinerR:D15 ; JiaLH:19 (see Ref. Landau  also). In
terms of the $LS\,$bases
${}^{2S+1}P_{J}=\\{^{2}P_{1/2},^{4}P_{1/2},^{2}P_{3/2},^{4}P_{3/2},^{4}P_{5/2}\\}$,
the matrix forms of these mass shift interactions are(see appendix A of Ref.
KarlinerR:17 )

$\displaystyle\Delta\mathcal{M}_{J=1/2}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{1}{3}(a_{2}-4a_{1})&\frac{\sqrt{2}}{3}(a_{2}-a_{1})+\frac{b}{\sqrt{2}}\\\
\frac{\sqrt{2}}{3}(a_{2}-a_{1})+\frac{b}{\sqrt{2}}&-\frac{5}{3}(a_{1}+\frac{1}{2}a_{2})-b\end{array}\right]$
(9) $\displaystyle+\left[\begin{array}[]{cc}-c&0\\\
0&\frac{1}{2}c\end{array}\right],$ $\displaystyle\Delta\mathcal{M}_{J=3/2}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{2}{3}a_{1}-\frac{1}{6}a_{2}&\frac{\sqrt{5}}{3}(a_{2}-a_{1})-\frac{b}{2\sqrt{5}}\\\
\frac{\sqrt{5}}{3}(a_{2}-a_{1})-\frac{b}{2\sqrt{5}}&-\frac{1}{3}(2a_{1}+a_{2})+\frac{4b}{5}\end{array}\right]$
(15) $\displaystyle+\left[\begin{array}[]{cc}-c&0\\\
0&\frac{1}{2}c\end{array}\right],$
$\Delta\mathcal{M}_{J=5/2}=a_{1}+\frac{1}{2}a_{2}-\frac{b}{5}+\frac{c}{2}.$
(16)

Given $J^{P}$ assignments of the five $\Omega_{c}$ states(two states of
$J^{P}=1/2^{-}$, two of $J^{P}=3/2^{-}$, and one of $J^{P}=5/2^{-}$) there
should, in principle, exist one unique solution for the four parameters
$a_{1}$, $a_{2}$, $b$ and $c$. As we shall find below, there is one solution
in which all states are $P$-waves with reasonable values of parameters and the
mass pattern as reported by LHCb for the excited $\Omega_{c}$ states in Table
I, where our preferred $J^{P}$ assignments are shown.

Diagonalizing the mass shift operator
$\mathbf{L}\cdot\mathbf{S}_{ss}+\epsilon\mathbf{L}\cdot\mathbf{S}_{Q}+b_{1}S_{12}$,
with $b_{1}\equiv b/a_{1}$, one can compute the mass shifts $\Delta M$, the
eigenvalues of Eqs. (9)-(16), by treating the contact term
$c\mathbf{S}_{ss}\cdot\mathbf{S}_{Q}$ as a perturbation($c_{1}=c/a_{1}$ is
expected to be small for the relative $1P$-wave between diquark and $Q$). The
lowest order perturbation theory gives(see appendix A)

$\displaystyle\Delta M(J$ $\displaystyle=$ $\displaystyle
1/2,0^{\prime})=-\frac{a_{1}}{4}\left(6+\sqrt{\Delta_{1}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right)}+\frac{a_{2}}{a_{1}}\right)-\frac{b}{2}+c\Delta_{3}^{+}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right),$
$\displaystyle\Delta M(J$ $\displaystyle=$ $\displaystyle
1/2,1^{\prime})=-\frac{a_{1}}{4}\left(6-\sqrt{\Delta_{1}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right)}+\frac{a_{2}}{a_{1}}\right)-\frac{b}{2}+c\Delta_{3}^{-}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right),$
$\displaystyle\Delta M(J$ $\displaystyle=$ $\displaystyle
3/2,1^{\prime})=-a_{1}\left(\sqrt{\Delta_{2}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right)}+\frac{a_{2}}{4a_{1}}\right)+\frac{2b}{5}+c\Delta_{4}^{+}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right),$
$\displaystyle\Delta M(J$ $\displaystyle=$ $\displaystyle
3/2,2^{\prime})=a_{1}\left(\sqrt{\Delta_{2}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right)}-\frac{a_{2}}{4a_{1}}\right)+\frac{2b}{5}+c\Delta_{4}^{-}\left(\frac{a_{2}}{a_{1}},\frac{b}{a_{1}}\right),$
$\displaystyle\Delta M(J$ $\displaystyle=$ $\displaystyle
5/2,2^{\prime})=a_{1}+\frac{a_{2}}{2}-\frac{b}{5}+\frac{c}{2},$ (17)

where six functions $\Delta_{1,2}(\epsilon,x),\Delta_{3}^{\pm}(\epsilon,x)$
and $\Delta_{4}^{\pm}(\epsilon,x)$ are defined by

$\begin{array}[]{c}\Delta_{1}(\epsilon,x)=4+12x^{2}+4x\left(5\epsilon-2\right)-4\epsilon+9\epsilon^{2},\\\
\Delta_{2}(\epsilon,x)=1+\frac{1}{5}x^{2}-\frac{x}{5}(1+2\epsilon)-\epsilon+\frac{9}{16}\epsilon^{2}.\end{array}$
(18)
$\begin{array}[]{c}\Delta_{3}^{+}(\epsilon,x)=\frac{4-(2+6x+7\epsilon-3\sqrt{\Delta_{1}(\epsilon,x)})^{2}/(2\epsilon-2+3x)^{2}}{8+(2+6x+7\epsilon-3\sqrt{\Delta_{1}(\epsilon,x)})^{2}/(2\epsilon-2+3x)^{2}},\\\
\Delta_{3}^{-}(\epsilon,x)=\Delta_{3}^{+}\left(\sqrt{\Delta_{1}}\rightarrow-\sqrt{\Delta_{1}}\right).\end{array}$
(19)
$\begin{array}[]{c}\Delta_{4}^{+}(\epsilon,x)=\frac{10-(40-24x+5\epsilon+60\sqrt{\Delta_{2}(\epsilon,x)})^{2}/(10-10\epsilon+3x)^{2}}{20+(40-24x+5\epsilon+60\sqrt{\Delta_{2}(\epsilon,x)})^{2}/(10-10\epsilon+3x)^{2}},\\\
\Delta_{4}^{-}(\epsilon,x)=\Delta_{4}^{+}\left(\sqrt{\Delta_{2}}\rightarrow-\sqrt{\Delta_{2}}\right),\end{array}$
(20)

with $\Delta_{3,4}^{-}(\epsilon,x)$ obtained from
$\Delta_{3,4}^{+}(\epsilon,x)$ by merely replacing
$\sqrt{\Delta_{1,2}}\rightarrow-\sqrt{\Delta_{1,2}}$. This expresses the mass
shifts in terms of four parameters ($a_{1},a_{2},b,c$) nonlinearly. Expanding
Eqs. (17) to the leading order of $\epsilon,b_{1}$ and $c_{1}$, it reduces to
the linear mass shift in Ref. KarlinerR:17  in $jj$ coupling. The spin-
weighted sum of these mass shifts in Eqs. (9)-(16) vanishes: $\sum_{J}$
$Tr_{J}$($\Delta\mathcal{M}_{J})(2J+1)=0$, and the same holds for the
eigenvalues $\Delta M(J,j^{\prime})$ in Eqs. (17). Note that the sums of
eigenvalues of $\Delta\mathcal{M}_{1/2}$ and $\Delta\mathcal{M}_{3/2}$ are
equal to the traces of the respective matrices in Eq. (17). Note that an
alternative expression similar to the mass-shifts(17) is obtained recently in
Ref. Ali:jh2019 .

Adding spin-independent mass $\bar{M}$, which equals to the spin-averaged mass
of the five excited $css$ systems, the baryon mass becomes
$M(J,j_{LS}=j^{\prime})=\bar{M}+\Delta M(J,j^{\prime})$, with $\Delta
M(J,j^{\prime})$ given by Eqs. (17). Confronting $M(J,j_{LS}=j^{\prime})$ with
the observed masses in Table I leads to the values of $a_{1}$,
$a_{2}$($=\epsilon a_{1}$), $b$($=a_{1}b_{1}$) and $c$($=a_{1}c_{1}$), with
the help of the following criteria:

(i) The parameter $a_{1}$ should be positive but smaller than $119$
MeV$=M(\Omega_{c},3119)-M(\Omega_{c},3000)$.

(ii) The parameter $a_{2}$ is of same order with $a_{1}$ but no more than
$a_{1}$ roughly as $a_{2}/a_{1}=\epsilon$ scales as $m_{ss}/M_{c}$.

(iii) The parameter $b$ should be smaller than $a_{1}$ and $a_{2}$ as $b$
scales like $1/(m_{ss}M_{c})$. The parameter $c$ should be smallest, less than
$b$ as it scales as $P$-wave wave function near the origin.

We carry out mass analysis for all $5!=120$ a priori possible assignments of
$P$-wave $\Omega_{c}$ states and list the preferred assignments in Table I,
corresponding to the mean mass and the parameters KarlinerR:17

$\bar{M}=3079.94\text{MeV,}$ (21) $\\{a_{1},a_{2},b,c\\}=\\{26.96\text{,
}25.76\text{, }13.51\text{, }4.04\\}\text{(MeV),}$ (22)

with $\epsilon=0.96,b_{1}=0.50,c_{1}=0.15$. For our preferred assignment, the
inner structures of the five excited $\Omega_{c}$ baryons are

$\left\\{\begin{array}[]{r}\text{State\thinspace}{|J,j}_{LS}{\small\rangle:}\\\
{\small M(\Omega}_{c}{1P):}\\\ \text{Main
comp}.{}\end{array}\begin{array}[]{ccccc}|\frac{1}{2},0^{\prime}\rangle&|\frac{1}{2},1^{\prime}\rangle&|\frac{3}{2},1^{\prime}\rangle&|\frac{3}{2},2^{\prime}\rangle&|\frac{5}{2},2^{\prime}\rangle\\\
3000.4&3050.2&3065.6&3090.2&3119.1\\\
{}^{4}P_{1/2}\text{(}97\%\text{)}&{}^{2}P_{1/2}\text{(}97\%\text{)}&{}^{4}P_{3/2}\text{(}98\%\text{)}&{}^{2}P_{3/2}\text{(}98\%\text{)}&{}^{4}P_{5/2}\end{array}\right\\},$
(23)

where the third line gives the main component in terms of the normal $LS$
coupling (${}^{2S+1}P_{J}$, see below). One sees that the mass degeneration
within the $J=1/2$ and $J=3/2$ multiplets removed by the flip of the charm
quark spin $\mathbf{S}_{c}$. Within the same multiplet, the lower state
consists mainly of $S=3/2$ configuration
$c^{\uparrow}(s^{\uparrow}s^{\uparrow})$, while the higher consists mainly of
that with $\mathbf{S}_{c}$ flipped, $c^{\downarrow}(s^{\uparrow}s^{\uparrow})$
with $S=1/2$.

Expressing in terms of the $LS$ eigenstates $|^{2S+1}P_{J}\rangle$, with
$1/2\leq J\leq 5/2$ and $S=0,1$, the $Jls$ states in Eq. (23) become (see
Appendix B),

$\displaystyle|J$ $\displaystyle=$ $\displaystyle
1/2,j^{\prime}=0^{\prime}\rangle=-0.164|^{2}P_{1/2}\rangle+0.986|^{4}P_{1/2}\rangle,\text{
at }3000$ $\displaystyle|J$ $\displaystyle=$ $\displaystyle
1/2,j^{\prime}=1^{\prime}\rangle=0.986|^{2}P_{1/2}\rangle+0.164|^{4}P_{1/2}\rangle,\text{at
}3050$ $\displaystyle|J$ $\displaystyle=$ $\displaystyle
3/2,j^{\prime}=1^{\prime}\rangle=0.129|^{2}P_{3/2}\rangle+0.992|^{4}P_{3/2}\rangle,\text{at
}3066$ $\displaystyle|J$ $\displaystyle=$ $\displaystyle
3/2,j^{\prime}=2^{\prime}\rangle=-0.992|^{2}P_{3/2}\rangle+0.129|^{4}P_{3/2}\rangle,\text{at
}3090$ $\displaystyle|J$ $\displaystyle=$ $\displaystyle
5/2,j^{\prime}=2^{\prime}\rangle=|^{4}P_{5/2}\rangle,\text{at }3119\text{. }$
(24)

Dominately, they are
$|^{4}P_{1/2}\rangle,|^{2}P_{1/2}\rangle,|^{4}P_{3/2}\rangle,|^{2}P_{3/2}\rangle$
and $|^{4}P_{5/2}\rangle$ with the monatomically increasing mass,
respectively, as shown in Eq. (23). This is in contrast with the normal state
classification of heavy baryon systems via the $jj$ coupling(see KarlinerR:D15
, for instance),

$\displaystyle|J$ $\displaystyle=$ $\displaystyle
1/2,j=0\rangle=\sqrt{\frac{1}{3}}|1^{2}P_{1/2}\rangle+\sqrt{\frac{2}{3}}|1^{4}P_{1/2}\rangle,$
$\displaystyle|J$ $\displaystyle=$ $\displaystyle
1/2,j=1\rangle=\sqrt{\frac{2}{3}}|1^{2}P_{1/2}\rangle-\sqrt{\frac{1}{3}}|1^{4}P_{1/2}\rangle,$
$\displaystyle|J$ $\displaystyle=$ $\displaystyle
3/2,j=1\rangle=\sqrt{\frac{1}{6}}|1^{2}P_{3/2}\rangle+\sqrt{\frac{5}{6}}|1^{4}P_{3/2}\rangle,$
$\displaystyle|J$ $\displaystyle=$ $\displaystyle
3/2,j=2\rangle=\sqrt{\frac{5}{6}}|1^{2}P_{3/2}\rangle-\sqrt{\frac{1}{6}}|1^{4}P_{3/2}\rangle,$
(25)

mixing significantly between the states with $J=1/2$ and $3/2$.

It is of interest to express the mixing states (24) in terms of the $jj$
bases, giving(see Appendix B)

$\displaystyle\left|J=\frac{1}{2},j^{\prime}=0^{\prime}\right\rangle$
$\displaystyle=$ $\displaystyle
0.711\left|\frac{1}{2},j=0\right\rangle+0.703\left|\frac{1}{2},j=1\right\rangle,$
$\displaystyle\left|J=\frac{1}{2},j^{\prime}=1^{\prime}\right\rangle$
$\displaystyle=$
$\displaystyle-0.703\left|\frac{1}{2},j=0\right\rangle+0.711\left|\frac{1}{2},j=1\right\rangle,$
$\displaystyle\left|J=\frac{3}{2},j^{\prime}=1^{\prime}\right\rangle$
$\displaystyle=$ $\displaystyle
0.958\left|\frac{3}{2},j=1\right\rangle+0.286\left|\frac{3}{2},j=2\right\rangle,$
$\displaystyle\left|J=\frac{3}{2},j^{\prime}=2^{\prime}\right\rangle$
$\displaystyle=$
$\displaystyle-0.286\left|\frac{3}{2},j=1\right\rangle+0.958\left|\frac{3}{2},j=2\right\rangle,$
(26)

which mix almost equally between the $jj$ bases
$\left|\frac{1}{2},j=0,1\right\rangle$. Eq. (26) implies that the angular
momentum $\mathbf{j=J-S}_{c}$ of the diquark in our mixing states with $J=1/2$
may be observed to take values either $j=0$ or $j=1$, with almost equal
probabilities ($\allowbreak 0.51$ v.s. $0.49$), while those for the $J=3/2$
are mainly the $jj$ eigenstates $\left|3/2,j=1,2\right\rangle$ (at $0.92$),
slightly mixing the respective $\left|3/2,j=2,1\right\rangle$ (at $0.08$).
Hence, the $J=1/2$ states do not conserve $\mathbf{j=J-S}_{c}$ of the diquark
and thereby $\mathbf{S}_{c}$. This differs the $css$ states from the
nonstrange charmed baryons, all of which can be well classified by the $jj$
eigenstates EFG:D11 ; MaltmanI:D80 ; RobertsP:A08 ; KarlinerR:D15 ;
KarlinerR:17 .

One alternative solution to assignment of the five $\Omega_{c}=css$ states
involves identification of the five masses
$M(1/2,0^{\prime}),M(1/2,1^{\prime}),M(3/2,1^{\prime}),M(3/2,2^{\prime})$,
$M(5/2,2^{\prime})$ to be that at $3000$, $3066$, $3050$, $3090$ and $3119$
MeV, respectively. This gives rise to the parameters and the mean mass in Ref.
KarlinerR:17

$\displaystyle\\{a_{1},a_{2},b,c\\}$ $\displaystyle=$
$\displaystyle\\{21.40,40.75,5.67,0.44\\}~{}\text{(MeV),}$ (27)
$\displaystyle\bar{M}$ $\displaystyle=$ $\displaystyle 3078.23\text{MeV.}$
(28)

This assignment is disfavored by unreasonable relative ratio ($1:2$) between
$a_{1}$ and $a_{2}$ in the sense of the criterion (ii).

Using Eq.(17) further, we performed systematic search for all remaining
permutations among all $5!$ possibilities and find no acceptable solution for
considered permutations: the parameters obtained either have negative
(unacceptable) signs of $a_{1}$ and $a_{2}$ or unreasonable values in the
light of the criteria (i)-(iii).

Table 2: Mass and parameters for two $J^{P}$ assignments (23) and the
alternative arrangement of four LHCb-observed masses of the excited
$\Omega_{c}$ systems. All $2\ast 5=10$ possible choices of mass inputs out of
the five measured masses are considered. The predicted mass at presumably
unseen state is marked in square bracket. All parameters except the Regge
slope $a$ in MeV.
$\begin{array}[]{cccccccccccc}\hline\cr\hline\cr|\frac{1}{2},0^{\prime}\rangle&|\frac{1}{2},1^{\prime}\rangle&|\frac{3}{2},1^{\prime}\rangle&|\frac{3}{2},2^{\prime}\rangle&|\frac{5}{2},2^{\prime}\rangle&a_{1}&a_{2}&b&c&a(GeV^{2})&\bar{M}(1P)&\bar{M}(2S)\\\
\hline\cr$[2995.0]$&$3050$&$3066$&$3090$&$3119$&$27.5$&$27.0$&$15.5$&$3.6$&\lx@intercol\hfil$0.316$\hfil\lx@intercol&$3079$&$3244$\\\
$3000$&$[3049.0]$&$3066$&$3090$&$3119$&$27.2$&$25.2$&$13.7$&$4.4$&\lx@intercol\hfil$0.316$\hfil\lx@intercol&$3080$&$3244$\\\
$3000$&$3050$&$[3068.2]$&$3090$&$3119$&$26.7$&$24.8$&$15.4$&$5.0$&\lx@intercol\hfil$0.317$\hfil\lx@intercol&$3081$&$3245$\\\
$3000$&$3050$&$3066$&$[3095.4]$&$3119$&$28.2$&$23.1$&$14.4$&$2.3$&\lx@intercol\hfil$0.317$\hfil\lx@intercol&$3081$&$3246$\\\
$3000$&$3050$&$3066$&$3090$&$[3115.6]$&$26.3$&$23.7$&$14.7$&$3.2$&\lx@intercol\hfil$0.315$\hfil\lx@intercol&$3079$&$3243$\\\
$[3000.4]$&$3066$&$3050$&$3090$&$3119$&$21.4$&$40.8$&$5.7$&$0.44$&\lx@intercol\hfil$0.314$\hfil\lx@intercol&$3078$&$3242$\\\
$3000$&$[3067.4]$&$3050$&$3090$&$3119$&$20.4$&$41.9$&$6.4$&$1.2$&\lx@intercol\hfil$0.315$\hfil\lx@intercol&$3078$&$3242$\\\
$3000$&$3066$&$[3051.0]$&$3090$&$3119$&$21.4$&$40.4$&$6.1$&$0.52$&\lx@intercol\hfil$0.315$\hfil\lx@intercol&$3078$&$3242$\\\
$3000$&$3066$&$3050$&$[3090.1]$&$3119$&$21.3$&$40.8$&$5.7$&$0.59$&\lx@intercol\hfil$0.314$\hfil\lx@intercol&$3078$&$3242$\\\
$3000$&$3066$&$3050$&$3090$&$[3117.5]$&$21.4$&$39.7$&$5.7$&$-0.57$&\lx@intercol\hfil$0.314$\hfil\lx@intercol&$3078$&$3241$\\\
\hline\cr\hline\cr\end{array}$

We also made a ”recovery” test by fitting four out of the five observed
$\Omega_{c}$ masses for the $J^{P}$ assignments (23) and the alternative
solution with $M(1/2,1^{\prime})$ and $M(3/2,1^{\prime})$ interchanged,
corresponding to the parameters (27), to see if the remaining measured mass,
unused in fitting inputs, can be reproduced at the site of given spin-
states(marked by square bracket). The results for the ”recovery” test are
shown in Table II for all $2\ast 5=10$ possibilities of state arrangements for
chosen inputs and indicate that the test successes for both, whereas the
parameter values clearly favor the assignment (23) in the light of the
criterion (ii): $a_{2}$ is notably large over $a_{1}$, nearly two times, for
the alternative solution.

One can also use the mass scaling of the spin-interaction couplings, from
$D_{s}$ mesons to the $css$ baryons, to explain the values (22) for our
preferred assignment. Using the scaling relation KarlinerR:D15 ; JiaLH:19 ,
one finds

$\displaystyle a_{1}(css)$ $\displaystyle=$ $\displaystyle
a_{1}(c\bar{s})\left(\frac{m_{s}}{m_{ss}}\right)=(89.4~{}\text{MeV})\left(\frac{328}{991}\right)=29.6~{}\text{MeV,}$
(29) $\displaystyle a_{2}(css)$ $\displaystyle=$
$\displaystyle\frac{a_{2}(c\bar{s})}{1+m_{ss}/M_{c}}=\frac{40.7~{}\text{MeV}}{1+991/1440}=24.1~{}\text{MeV,}$
(30)

which are close to the values (22), but away from that in (27). Here,
$a_{1,2}(c\bar{s})$ are the parameters of the spin-orbit interactions for the
P-wave $D_{s}=c\bar{s}$ mesons, with the respective values $89.4$ MeV and
$40.7$ MeV KarlinerR:D15 . The masses($m_{s}=328$MeV and $M_{c}=1440$ MeV) of
the strange quark $s$ and the charm quark are from Regge trajectory fit of the
$D_{s}$ mesons and the charmed baryons $\Sigma_{c}/\Xi_{c}^{\prime}$ JiaLH:19
while the mass($m_{ss}=991$MeV) of the ss-diquark comes from the Regge
trajectory fit of the five excited $\Omega_{c}$ states in Table I(cf. the
discussion following (31) in Sect. III).

##  III Alternative possibilities examined

The possibility that must be checked is that not all of the $\Omega_{c}$
states in Table I are relative $P$-wave excitations between $ss$ diquark and
the charmed quark EFG:D11 ; ChenHL:D17 ; WangXZZ:D17 ; ChengC:D17 ;
AgaevAS:D17 . For instance, two $2S$ wave candidates are predicted for the
$c(ss)$ at masses $3088$ MeV ($J^{P}=1/2^{+}$) and $3123$ MeV($J^{P}=3/2^{+}$)
in Ref. EFG:D11 , not far from two higher masses($3090$ and $3119$ MeV)
reported by LHCb. This leaves possibility that some of the lower states may be
that out of the five states of $P$-waves, with other $P$-wave states unseen
somehow, probably due to locating below the $\Xi_{c}^{+}K^{-}$ threshold
($\simeq 2962$ MeV), or invisible due to the near degeneracies EFG:D11 . To
disentangle this possibility, we extend the mass analysis in Sect. II by
assuming one of the five measured masses of the $\Omega_{c}$ states in Table I
to be that of 2S state and comparing it with mass estimation of the 2S state
with the help of Regge trajectory JiaDH:Q18 ; JiaD:E19 . The results suggest
that all measured masses of the excited $\Omega_{c}$ states are too low to be
a 2S state.

We use the mass shift formula (17) to perform mass analysis for all $5!$
permutations of the $J^{P}$ arrangement with one presumed 2S state removed
from five mass inputs. Apart from the $10$ arrangements already given in Table
II, we list, in Table III, the other candidate solutions with the parameters
that are not too far away from the criteria (i)-(iii), and the thereby
predicted P-wave mass(presumably unseen) enclosed in square bracket. Though
two arrangements in the first and second data rows recover approximately the
observed masses absent in the inputs, all of these permutated arrangements are
disfavored by the criteria (i)-(iii), having unreasonable values of the
parameters listed: either $a_{2}$ or $b$ is notably large over $a_{1}$ or
negative, or $c$ is abnormally large compared with one of $a_{2}$ and $b$, or
both.

Table 3: Parameters and spin-averaged masses (MeV) for the selected
permutations of $J^{P}$ assignments of the four states out of excited
$\Omega_{c}$ levels. Only the arrangements that leads to the parameters near
to reasonable values are listed among $5\ast 4!=120$ possible permutations of
mass inputs considered, with the mass prediction at unseen level marked by
square bracket. The shown also includes the ensuing prediction for the Regge
slope $a$ (see text, in GeV2) and spin-averaged 2S-wave mass(in MeV) of the
$c(ss)$ system. $m_{ss}=991$ MeV.
$\begin{array}[]{cccccccc}\hline\cr\hline\cr|\frac{1}{2},0^{\prime}\rangle,|\frac{1}{2},1^{\prime}\rangle,|\frac{3}{2},1^{\prime}\rangle,|\frac{3}{2},2^{\prime}\rangle,|\frac{5}{2},2^{\prime}\rangle&a_{1}&a_{2}&b&c&a($
GeV$^{2})&\bar{M}(1P)&\bar{M}(2S)\\\ \hline\cr
3000,[3052.3],3066,3119,3090&23.0&-2.19&35.0&-3.43&0.313&3077.0&3240.3\\\
3000,3066,3050,3119,[3093.9]&27.8&7.86&11.0&-23.9&0.312&3076.2&3239.3\\\
3050,3066,3000,[3082.2],3119&10.7&49.9&-36.2&10.6&0.307&3070.9&3231.7\\\
3000,3090,3050,[3102.5],3119&14.6&48.9&10.1&-3.20&0.320&3083.7&3249.9\\\
3066,[3108.6],3000,3119,3090&11.2&24.8&-49.1&-38.41&0.312&3076.0&3238.9\\\
$[3008.0]$,3000,3050,3066,3119&32.3&14.2&5.77&28.9&0.302&3066.4&3225.2\\\
3050,3066,3000,[3078.7],3119&11.1&47.6&-38.8&12.7&0.306&3074.1&3230.6\\\
\hline\cr\hline\cr\end{array}$

The mass estimation of the $\Omega_{c}$ systems in $2S$ wave can be given
utilizing a Regge-like relation for the spin-averaged mass $\bar{M}$ of the
charmed baryons BChen:A15 ; JiaLH:19

$(\bar{M}-M_{c})^{2}=\pi
aL+\left[m_{d}+M_{c}\left(1-\frac{m_{barec}^{2}}{M_{c}^{2}}\right)\right]^{2},$
(31)

in which $a$ is the Regge slope, $m_{d}$ is the light diquark mass involved,
$m_{barec}=1.275$ GeV the bare mass of charm quark and $L$ the orbital angular
momentum of the systems. The charm quark mass $M_{c}=1.44$ GeV is determined
by confronting the relation (31) with the charmed baryons
$\Sigma_{c}/\Xi_{c}^{\prime}$ JiaLH:19 . In Ref. JiaD:E19 , a trajectory slope
ratio $\pi:2$ for the radially and angular excitations is suggested in
applying Eq. (31) to the heavy mesons $B/D$’s and $B_{s}/D_{s}$’s.

---
Figure 1: (Color online). Regge trajectories of the $\Omega_{c}$ baryons
relating the shifted spin-averaged mass squared to the orbital angular
momentum $L$ of the systems, with the parameters $m_{d=ss}=991$ MeV and the
Regge slope $a=0.316$GeV2 corresponding to our assignment (23). The (red)
solid circles correspond to the observed (mean) masses.

By analogy of the $c(ss)$ system with $D_{s}\,=c\bar{s}\,$ mesons, we estimate
the 2S-wave mass of the $c(ss)$ state using Eq. (31) with $a$ replaced by
$a(\pi/2)$ and $L$ by $n$, the radial quantum number of the $c(ss)$ system,

$\bar{M}(2S)=M_{c}+\sqrt{\pi
a(\pi/2)+\left(m_{d}+M_{c}\left(1-\frac{1.275^{2}}{M_{c}^{2}}\right)\right)^{2}}\text{,}$
(32)

where $m_{d}=0.991$ GeV and the Regge slope $a$ (listed in Table II and III)
are solved from (31) applying to the spin-averaged masses $\bar{M}(1S)$ and
$\bar{M}(1P)$ for a given assignment, and $M_{c}=1.44$ GeV. Here, the observed
lowest masses of the $\Omega_{c}$, $M(\Omega_{c},1/2^{+})=2695.2\pm 1.7$ MeV
and $M(\Omega_{c},3/2^{+})=2765.9\pm 2.0$ MeVTanabashi:D18 , lead to their
spin-averaged mass in S-wave

$\bar{M}(\Omega_{c},1S)=\frac{1}{6}\left(2M(1/2^{+})+4M(3/2^{+})\right)=2742.3\pm
1.9~{}\text{MeV.}$ (33)

Usage of the mean mass (21) in P-wave($L=1$) and the mass (33) in
S-wave($L=0$) in Eq. (31) gives

$\Omega_{c}:m_{d}=m_{ss}=0.991\text{ GeV, }a=0.316\text{ GeV}^{2}\text{.}$
(34)

Given trajectory parameters, one can predict Regge trajectories of the excited
baryons $\Omega_{c}$(FIG. 1) with radial quantum number $n=0$ and $1$. In
Table IV in which $n$ stands for $n+1=1,2$, we list corresponding spin-
averaged masses of the excited $\Omega_{c}$’s predicted by Eqs. (31)and (32),
which are quite helpful for further constructing whole families of the baryon
$\Omega_{c}$’s.

Table 4: Mean masses of the excited $\Omega_{Q}$’s predicted by Eqs. (31) and
(32).  $\begin{array}[]{cccccc}\hline\cr\hline\cr
State(\text{MeV})&&\bar{M}(nS)&\bar{M}(nP)&\bar{M}(nD)&\bar{M}(nF)\\\
\hline\cr\Omega_{c}($n=1$)&&2742.3&3078.3&3356.2&3598.7\\\
\Omega_{c}($n=2$)&&3241.1&3496.5&3723.6&3930.0\\\
\Omega_{b}($n=1$)&&6051.0&6342.0&6593.3&6817.7\\\
\Omega_{b}($n=2$)&&6489.2&6724.1&6936.6&7132.4\\\
\hline\cr\hline\cr\end{array}$

Depending on state arrangements having different $P$-wave mean-mass, the above
values vary slightly. The 2S-wave masses of the $c(ss)$ system estimated via
(32) are listed in Table II and III for given assignments. Evidently, the
2S-wave mass $\bar{M}(2S)\geq 3220$ MeV, and both of the higher mass states at
$3090$ MeV and $3119$ MeV are too low to be a $2S$ state candidates. The same
conclusion holds also for other low-lying masses, as shown in Table II and
III.

For other assignments searched, no solutions of the predicted unseen states,
below the $2962$ MeV threshold (of $\Xi_{c}^{+}K^{-}$) or near degeneracies
with other masses, are found up to requirements (i)-(iii) in Sect. II.

##  IV P-wave masses for $\Omega_{b}=b(ss)$ Systems

Recent observation of four $\Omega_{b}^{-}$ states Aaij:L2020  around $6.3$
GeV(Table I) makes it timely to apply mass analysis of the $\Omega_{c}=c(ss)$
states to the bottom systems $b(ss)$ which consists of a bottom quark $b$ and
a spin-1 diquark $ss$. Heaviness of the $b$ quark implies that Eqs. (17)
should certainly be applicable for the P-wave $\Omega_{b}^{-}$ states. We use
it to search the possible assignments with the following inputs:

(i) The parameter $a_{1}$ is positive and no more than the maximum gap between
the observed masses : $a_{1}<2\ast(6349.88-6315.64)$ $=68.48$ MeV. (ii) The
parameter $a_{2}$ is of same order of $a_{1}$ roughly in magnitude, but should
be smaller than $a_{1}$. (iii) The tensor parameter $b$ should be smaller than
$a_{1}$ in magnitude. (iv) The hyperfine parameter $c$ is set to zero.

Assuming one of P-wave states is unseen experimentally the search is performed
systematically via matching the observed masses of the $\Omega_{b}$ states in
Table I with that in Eq. (17) for all $4!=24$ a priori possible $J^{P}$
assignments of P-wave states and $5$ possible permutations of the unseen
$P$-wave state that is expected to exist somewhere around $6.3$ GeV.

The most preferred results are achieved by the identification of four masses
at $6315.64,6330.30$, $6339.71$ and $6349.88$ MeV as $M(1/2,0^{\prime})$,
$M(1/2,1^{\prime})$, $M(3/2,1^{\prime})$, $M(3/2,2^{\prime})$. The
corresponding parameters and the mean mass are

$a_{1}=8.98~{}\text{MeV, }a_{2}=4.11~{}\text{MeV, }b=7.61~{}\text{MeV,
}\bar{M}=6342.0~{}\text{MeV,}$ (35)

which lead, by Eqs. (17), to the following $J^{P}$ assignment,

$\left\\{\begin{array}[]{c}\text{State(}J^{P}\text{):}\\\
M(\Omega_{b},1P\text{):}\end{array}\begin{array}[]{ccccc}|1/2,0^{\prime}\rangle&|1/2,1^{\prime}\rangle&|3/2,1^{\prime}\rangle&|3/2,2^{\prime}\rangle&|5/2,2^{\prime}\rangle\\\
\lx@intercol\hfil 6315.4,\lx@intercol&\lx@intercol\hfil
6332.0\lx@intercol&\lx@intercol\hfil 6337.8\lx@intercol&\lx@intercol\hfil
6350.0\lx@intercol&\lx@intercol\hfil
<EMAIL_ADDRESS>(36)

In terms of the $LS$ basis, their inner structures are

$\displaystyle|1/2,0^{\prime}\rangle$ $\displaystyle=$
$\displaystyle-0.190|^{2}P_{1/2}\rangle+0.982|^{4}P_{1/2}\rangle,\text{at
}6316,$ $\displaystyle|1/2,1^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
0.982|^{2}P_{1/2}\rangle+0.190|^{4}P_{1/2}\rangle,\text{ at }6330,$
$\displaystyle|3/2,1^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
0.488|^{2}P_{3/2}\rangle-0.873|^{4}P_{3/2}\rangle,\text{at }6340,$
$\displaystyle|3/2,2^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
0.873|^{2}P_{3/2}\rangle+0.488|^{4}P_{3/2}\rangle,\text{at }6350,$
$\displaystyle|5/2,2^{\prime}\rangle$ $\displaystyle=$
$\displaystyle|^{4}P_{5/2}\rangle\text{ at }6351.5^{\text{Pd}}\text{. }$ (37)

which are mainly the $P$-wave states
$|^{4}P_{1/2}\rangle,|^{2}P_{1/2}\rangle$, two mixings of
$|^{4}P_{3/2}\rangle\ $and $|^{2}P_{3/2}\rangle$, and $|^{4}P_{5/2}\rangle$,
respectively.

The experimental missing of the $J=5/2$ state in Table I at
$M(5/2,2^{\prime})=6351.5$, which we predict, is most likely due to its
degeneracy with the nearby state at $M(3/2,2^{\prime})=6350$ MeV. If this is
the case, it may be hidden in the observed peak around $6350$ MeV, which,
though appears consistent with a single resonance, is actually composed of
two. This is compared to the recent assignment in Ref. KarlinerR:Ax20 , where
the mass for the unseen $J=5/2$ state ranges from $6355$ MeV to $6383$ MeV or
from $6380$ MeV to $6407$ MeV, with the favored parameters $a_{1}=10.20$ MeV,
$a_{2}+c=10.04$ MeV and $b=4.75$ MeV.

One of other plausible solutions consists of identification of the four masses
at $6315.64,6330.30$, $6339.71$ and $6349.88$ MeV as
$|1/2,0^{\prime}\rangle,|3/2,1^{\prime}\rangle$,$|3/2,2^{\prime}\rangle$ and
$|5/2,2^{\prime}\rangle$. This assignment gives parameters and spin-averaged
mass(in MeV)

$\\{a_{1},a_{2},b\\}=\\{7.39\text{, }10.32\text{, }1.93\\}\text{,
}\bar{M}=6337.4\text{,}$ (38)

where the parameter $a_{2}=10.32$ MeV is larger than $a_{1}=7.39$ MeV, not
compatible with the criterion (ii).

There are still two additional possibilities involving identification of the
four observed $\Omega_{b}$ states at higher
$j^{\prime}=1^{\prime},2^{\prime}$. The first is to identify the states at
$6330$, $6316,6340$ and $6350$ MeV with the respective states at
$M(1/2,1^{\prime})$, $M(3/2,1^{\prime})$,
$M(3/2,2^{\prime}),M(5/2,2^{\prime})$, giving the parameter set(in MeV)

$\\{a_{1},a_{2},b\\}=\\{10.86\text{, }6.70\text{,}-10.54\\}\text{,
}\bar{M}=6333.6\text{,}$ (39)

and the presumably unseen state at $M(1/2,0^{\prime})=6311.5$ MeV. This lowest
state, if exist, are most likely to be visible. Together with abnormal (by the
criteria (iii)) value $|b|\simeq 11$ MeV compared to $a_{1}=10.86$ MeV, we
disfavor this assignment.

The second is to interchange two states at $6340$ and $6350$ MeV in the
assignment above, leading to the parameter set(in MeV)

$\\{a_{1},a_{2},b\\}=\\{11.93\text{, }-8.98\text{,}-2.92\\}\text{,
}\bar{M}=6331.7\text{,}$ (40)

and the five states $|1/2,j^{\prime}=0^{\prime}\rangle$ at
$6304.7$(prediction), $|1/2,j^{\prime}=1^{\prime}\rangle$ at $6330.3$,
$|3/2,j^{\prime}=1^{\prime}\rangle$ at $6315.6$,
$|3/2,j^{\prime}=2^{\prime}\rangle$ at $6349.9$ and
$|5/2,2^{\prime}\rangle=|^{4}P_{5/2}\rangle$ at $6339.7$ MeV. Away(about $10$
MeV) from the observed masses $6315$ MeV, the presumed missing state, at
$M(1/2,0^{\prime})=6304.7$ MeV, should not be hidden in any of the four
observed peaks by LHCb. The unacceptable(negative) value of $a_{2}=-8.98$ MeV
also disfavors this assignment.

Table 5: Parameters and masses(all in MeV) for the excited $\Omega_{b}$ when
one of four observed masses was removed from the mass inputs. The removed one
(enclosed by square bracket) is reproduced by Eq. (17) approximately.
$\begin{array}[]{ccccc}\hline\cr\hline\cr|\frac{1}{2},0^{\prime}\rangle,\
|\frac{1}{2},1^{\prime}\rangle,\ |\frac{3}{2},1^{\prime}\rangle,\
|\frac{3}{2},2^{\prime}\rangle,|\frac{5}{2},2^{\prime}\rangle&a_{1}&a_{2}&b&\bar{M}(1P)\\\
\hline\cr[6314.5],6332.0,6337.8,6350.0,6351.5&9.02&4.44&7.92&6341.8\\\
6315.4,[6332.1],6337.8,6350.0,6351.5&8.91&4.27&7.53&6342.0\\\
6315.4,6332.0,[6337.7],6350.0,6351.5&8.95&4.25&7.48&6341.9\\\
6315.4,6332.0,6337.8,[6350.5],6351.5&8.99&4.01&7.12&6342.1\\\
\hline\cr\hline\cr\end{array}$ Table 6: Parameters and spin-averaged masses(in
MeV) for the selected permutations of $J^{P}$ assignments of the four excited
$\Omega_{b}$ levels, with predicted mass at unseen level(marked by square
bracket). All arrangements leading to highly unreasonable or unacceptable
parameters are not listed. The shown includes also ensuing prediction for the
Regge slope $a$ (see text) and spin-averaged $css$ masses in $2S$ wave.
$m_{ss}=991$ MeV. We denoted the five states
$|\frac{1}{2},0^{\prime}\rangle,|\frac{1}{2},1^{\prime}\rangle,|\frac{3}{2},1^{\prime}\rangle,|\frac{3}{2},2^{\prime}\rangle,|\frac{5}{2},2^{\prime}\rangle$
by $|1\rangle,|2\rangle,|3\rangle,|4\rangle,|5\rangle$ for short,
respectively.
$\begin{array}[]{cccccccccccc}\hline\cr\hline\cr|1\rangle&|2\rangle&|3\rangle&|4\rangle&|5\rangle&a_{1}&a_{2}&b&a($
GeV$^{2})&\bar{M}(1P)&\bar{M}(2S)\\\
\hline\cr$[6312.4]$&6316&6329&6340&6350&21.2&$26.9$&42.4&\lx<EMAIL_ADDRESS>6314&[6331.4]&6330&6341&6350&7.39&$10.3$&$1.93$&\lx<EMAIL_ADDRESS>6316&6330&[6321.8]&6340&6350&$7.17$&$13.3$&$-3.28$&\lx<EMAIL_ADDRESS>6316&6331&6338&[6388.1]&6350&$12.0$&$-13.9$&$24.1$&\lx<EMAIL_ADDRESS>6313&6332&6338&6351&[6352.5]&$9.4$&$4.87$&$8.37$&\lx<EMAIL_ADDRESS>$[6311.5]$&6330&6316&6340&6350&$10.9$&$6.70$&$-10.5$&\lx<EMAIL_ADDRESS>$[6304.7]$&6330&6316&6350&6340&$11.9$&$-8.98$&-2.92&\lx<EMAIL_ADDRESS>\hline\cr\hline\cr\end{array}$

In addition, we carry out ”recovery” test (Table V, all in MeV) for the
excited $\Omega_{b}$’s, in which one of four observed masses is reproduced
approximately by Eq. (17) at the site enclosed by square bracket when it was
removed from four mass inputs. The uncertainty is about $2$ at most. This
further confirms our prediction $M(5/2,2^{\prime})=6351.5$ MeV for the unseen
(excited) $\Omega_{b}$ state.

Could it be possible that some higher states of these $\Omega_{b}$ peaks are
the $2S$ excitations? Our answer to this question is negative. We employ Eq.
(17) to perform mass fitting for the four mass inputs of the $\Omega_{b}$
baryons in Table I to find the three parameters $a_{1,2}$, $b$ and
$\bar{M}(1P)$, and thereby estimate the 2S-wave mass of the $\Omega_{b}$
baryons, shown in Table VI, using Eq. (32) with $M_{c}$ replaced by
$M_{b}=4.48$ GeV (the charm bare mass $1.275$ GeV by that of the bottom quark
$4.18$ GeV). For all assignments with the parameters not far away from the
inputs (i)-(iv), we find

$\bar{M}(\Omega_{b},2S)\gtrsim 6450~{}\text{MeV,}$ (41)

which disfavors identifying newly LHCb-reported $\Omega_{b}$ to be a $2S$
excitation. During calculating, the $ss$ diquark mass, $m_{d=ss}=1.001$ GeV,
is determined from Eq. (31) applied to spin-averaged mass
$M(\Omega_{b},1S)=6061.4$ MeV of the two lowest $1S$ states, the
$\Omega_{b}(1/2^{+})$ at $6046.1\pm 1.7$ MeV and its partner
$\Omega_{b}^{\ast}$($3/2^{+}$) with mass $6046.1+$ $\Delta E(\Omega_{b},1S)$.
Here, $\Delta E(\Omega_{b},1S)$ stands for the $1S$ level-splitting between
the $\Omega_{b}^{\ast}$($3/2^{+}$) and $\Omega_{b}(1/2^{+})$, and can be
estimated from the corresponding $1S$ level-splitting of the ground states
$\Omega_{c}$ via the scaling KarlinerR:D15 ,

$\displaystyle\Delta E(\Omega_{b})$ $\displaystyle=$
$\displaystyle\left(\frac{M_{c}}{M_{b}}\right)\Delta E(\Omega_{c})$
$\displaystyle=$
$\displaystyle\left(\frac{1.44}{4.48}\right)(71~{}\text{MeV)}$
$\displaystyle\simeq$ $\displaystyle 23~{}\text{MeV.}$

which yields the spin-averaged mass in $1S$-wave:

$\displaystyle\bar{M}(\Omega_{b},1S)$ $\displaystyle=$
$\displaystyle\frac{1}{6}[6046.1\times 2+(6046.1+23)\times 4]$
$\displaystyle=$ $\displaystyle 6061.4~{}\text{MeV.}$

Hence, Eq. (41) disfavors the $2S$ state assignment for any of four observed
$\Omega_{b}$ states, as the 2S state is too high in levels to accommodate all
$\Omega_{b}$ states in Table I. Similar analysis giving Eq. (34) applies to
the P-wave bottom baryons $\Omega_{b}$ with the mean mass in Eq. (35) and
gives $a=0.318$ GeV2. This leads to the mean masses in Table IV for the
$\Omega_{b}$’s, by which one can plot Regge trajectories (FIG. 2) for them.

---
Figure 2: (Color online). Regge trajectories of the $\Omega_{b}$ baryons
relating the shifted spin-averaged mass squared to the orbital angular
momentum $L$ of the systems, with the parameters $m_{d=ss}=991$ MeV and the
Regge slope $a=0.318GeV^{2}$ corresponding to our assignment(23). The (red)
solid circles correspond to the observed (mean) masses.

##  V Excited Baryons $\Sigma_{c}/\Xi_{c}^{\prime}$ and
$\Sigma_{b}/\Xi_{b}^{\prime}$ : P-wave Masses

It is possible to apply the mixing $Jls$ coupling to computation of the P-wave
masses of the less strange heavy baryons($\Sigma_{Q}/\Xi_{Q}^{\prime}$), for
which only a single measured mass available for each one of the odd-parity
baryons, $2801$$/2942.3$ MeV for the $\Sigma_{c}(2800)/\Xi_{c}^{\prime}(2942)$
and $6098.0$$/6226.9$ MeV for the $\Sigma_{b}(6097)$/$\Xi_{b}^{\prime}(6227)$
(Table I, or see Tanabashi:D18 ). The basic idea for such a computation is to
choose our previous estimates of the parameters $a_{1,2}$, $\bar{M}$ for the
$1P$ statesJiaLH:19 , as well as the following rough estimate of the parameter
$c$(with $m_{nn}=745$ MeV, $m_{ns}=872$ MeV)

$\text{Initial: }c(cqq)\approx
c(css)\left(\frac{m_{ss}}{m_{qq}}\right)=\left\\{\begin{array}[]{rr}5.37\text{
MeV},&\Sigma_{c},\\\ 4.59\text{ MeV},&\Xi_{c}^{\prime}.\end{array}\right.$
(42)

with $c(css)=4.04$ MeV given in Eq. (22), to be three initial inputs to give a
$b$-dependent P-wave mass for each baryon using the nonlinear relation (17),
and then find optimal fit of $b$ as the initial inputs via matching
$b$-dependent mass with the measured one for each baryons(Table I). Finally,
one evolves the obtained initial inputs via the relation (17) to find stable
values of the five parameters. We list all initial parameters including
$\bar{M}(1P)$ in the Table VII, where

Table 7: Initial parameters and mean masses(all in MeV) for the less strange
heavy baryons($\Sigma_{c,b}/\Xi_{c,b}^{\prime}$).
$\begin{array}[]{cccccc}\hline\cr\hline\cr\text{Initial
input}&a_{1}&a_{2}&b&c&\bar{M}(1P)\\\
\hline\cr\Sigma_{c}(\text{MeV})&39.4&26.8&20.1&5.37&2774.1\\\
\Xi_{c}^{\prime}(\text{MeV})&33.6&25.3&17.9&4.59&2923.0\\\
\Sigma_{b}(\text{MeV})&12.7&8.61&6.45&1.73&6088.4\\\
\Xi_{b}^{\prime}(\text{MeV})&10.8&8.10&5.76&1.48&6248.2\\\
\hline\cr\hline\cr\end{array}$

the two inputs $a_{1}=39.4$ MeV/$33.6$ MeV for the P-wave baryons
$\Sigma_{c}/\Xi_{c}^{\prime}$ are scaled from that $a_{2}=26.8$ MeV/$25.3$ MeV
for the $D_{s}(1P)$ masses in Ref. JiaLH:19 , and the mean-masses
$\bar{M}=2774.1$ MeV/$2923.0$ MeV for the P-wave baryons
$\Sigma_{c}/\Xi_{c}^{\prime}$ are taken from Table III of Ref. JiaLH:19 ,
which are extrapolated from Regge-trajectory of the $\Lambda_{c}/\Xi_{c}$
spectra.

For the P-wave bottom baryons $\Sigma_{b}$/$\Xi_{b}^{\prime}$, corresponding
initial inputs (Table VII) follow from the following estimations: the initial
parameter $c$ is estimated, by scaling from the $\Omega_{c}$ to the
$\Omega_{b}$ and then to the bottom baryon $bqq$, to be about

$\text{Initial: }c(bqq)\approx
c(css)\left(\frac{M_{c}}{M_{b}}\right)\left(\frac{m_{ss}}{m_{qq}}\right)=\left\\{\begin{array}[]{rr}1.73\text{
MeV,}&\Sigma_{b},\\\ 1.48\text{ MeV,}&\Xi_{b}^{\prime}.\end{array}\right.$
(43)

with $M_{c}=1.44$ GeV and $M_{b}=4.48$ GeV[37]. The mass scaling
$a_{2}(bss)\simeq a_{2}(css)(M_{c}/M_{b})$ is used to estimate $a_{2}$ while
the HQS relation $a_{1}(bqq)=a_{1}(cqq)$ in Ref. [35] needs to be corrected,
as indicated by $a_{1}(bss)=8.98<26.96=a_{1}(css)$ and also by failure to find
reasonable parameters if setting $a_{1}(\Sigma_{b})=39.4$
MeV$=a_{1}(\Sigma_{c})$ or $a_{1}(\Xi_{b}^{\prime})=33.6$
MeV$=a_{1}(\Xi_{c}^{\prime})$. We find that choosing initial values of
$a_{1}(bqq)$ to be about one third of $a_{1}(cqq)$, as $a_{1}(bss)\simeq
a_{1}(css)/3$ indicated, yields reasonable and stable values of the parameters
during iteration of Eq. (17). In the Table VIII($\Diamond$ denotes mass
observed), we show the results of iteration starting with the initial inputs
in Table VII, with reasonable values of the parameters achieved. Note that the
obtained values of $b$ is about one half of that of $a_{1}$.

Table 8: Masses and parameters(all in MeV) refined from the initial inputs in
Table VII via iteration of the nonlinear relation (17) for the less strange
heavy baryons $\Sigma_{c}/\Xi_{c}^{\prime}$ and $\Sigma_{b}/\Xi_{b}^{\prime}$
in P-wave.  $\begin{array}[]{ccccccc}\hline\cr\hline\cr
State:&|\frac{1}{2},0^{\prime}\rangle,\ |\frac{1}{2},1^{\prime}\rangle,\
|\frac{3}{2},1^{\prime}\rangle,\
|\frac{3}{2},2^{\prime}\rangle,|\frac{5}{2},2^{\prime}\rangle&a_{1}&a_{2}&b&c&\bar{M}(1P)\\\
\hline\cr\Sigma_{c}(\text{MeV})&2668.4,2723.1,2757.3,2801.0^{\Diamond},2826.6&39.96&21.75&20.70&7.85&2776.4\\\
\Xi_{c}^{\prime}(\text{MeV})&2840.6,2881.6,2908.9,2942.3^{\Diamond},2969.5&32.89&20.16&16.50&7.17&2925.9\\\
\Sigma_{b}(\text{MeV})&6053.9,6071.8,6082.8,6098.0^{\Diamond},6104.8&12.99&6.42&6.45&1.73&6089.1\\\
\Xi_{b}^{\prime}(\text{MeV})&6226.9^{\Diamond},6235.8,6243.4,6252.3,6262.5&9.37&6.29&5.76&1.48&6249.1\\\
\hline\cr\hline\cr\end{array}$

This way of parameter relaxation via the nonlinear mass relation overcome the
difficulty in previous worksKarlinerR:D15 ; JiaLH:19  within the scheme of the
$jj$ coupling, that the masses of the baryons $\Sigma_{c}/\Xi_{c}^{\prime}$
and $\Sigma_{b}/\Xi_{b}^{\prime}$ remain undetermined due to unknown $b$: a
simple comparison of the $b$-dependent mass predictions KarlinerR:D15 ;
JiaLH:19  with experimental masses yields unreasonable values of $b$ except
for the $\Sigma_{c}$. In our computation of the P-wave masses of the
$\Sigma_{c}/\Xi_{c}^{\prime}$’s and $\Sigma_{b}/\Xi_{b}^{\prime}$’s, using of
appropriate eigenstates, the bases of the $Jls$ coupling here, is crucial to
find stable parameters which keep them within a narrow range of a few MeV
around the initial inputs in Table VII. We await upcoming measurement(mass and
spin-parity) by LHCb and Belle to verify the predictions in Table VIII.

Similar procedure can apply to the excited $\Omega_{b}$, giving nearly same
parameters and masses as Eqs. (35) and (36), only that the mass of the $J=5/2$
state evolves to $M(5/2,2^{\prime})=6352.2$ MeV, $0.7$ MeV above that in Eq.
(36), and $c$ becomes a non-vanishing but small value of $c=1.35$, as it
should be.

##  VI Spin-couplings in relativized quark model

One may ask if the parameters of spin interaction (2) can be computed based on
dynamics of quark interaction, as it is done for fine and hyperfine structures
of the hydrogen-like atoms? In this section, we employ a QCD analogues of
Breit-Fermi spin interaction(Appendix C), derived from the Lorentz-invariant
quark-antiquark scattering amplitude at the tree levelYndurain83 , to compute
the spin-couplings ($a_{1},a_{2},b,c$) and compare them with the
experimentally matched ones.

Including the spin-dependent forces(Breit-Fermi spin interaction), the quasi-
static potential between quark $1$ and antiquark $2$ is Yndurain83 ; GI:D85 ,

$\begin{array}[]{r}V^{\text{quasi-
static}}=V+S+\left(\frac{V^{\prime}-S^{\prime}}{r}\right)\mathbf{L}\cdot\left(\frac{\mathbf{S}_{1}}{2m_{d}^{2}}+\frac{\mathbf{S}_{2}}{2m_{Q}^{2}}\right)\\\
+\left(\frac{V^{\prime}}{r}\right)\mathbf{L}\cdot\left(\frac{\mathbf{S}_{1}+\mathbf{S}_{2}}{m_{1}m_{2}}\right)+\frac{1}{3m_{1}m_{2}}\left(\frac{V^{\prime}}{r}-V^{\prime\prime}\right)S_{12}\\\
+\frac{2}{3m_{1}m_{2}}\left(\mathbf{\nabla}^{2}V\right)\mathbf{S}_{1}\cdot\mathbf{S}_{2},\end{array}$
(44)

where $V$ and $S$ are the respective vector and scalar potentials, and
$V^{\prime}$, $S^{\prime}$ and $V^{\prime\prime}$ their derivatives.

One way to relativize the instant potential $V^{\text{quasi-static}}$ is to
use the replacementsGI:D85

$\displaystyle V(r)$ $\displaystyle\rightarrow$
$\displaystyle\tilde{V}(r)=\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{V}/2}V(r)\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{V}/2},$
$\displaystyle S(r)$ $\displaystyle\rightarrow$
$\displaystyle\tilde{S}(r)=\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{S}/2}S(r)\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{S}/2},$
(45)

where the relativistic factors
$m_{i}/E_{i}=m_{i}/\sqrt{m_{i}^{2}+|\mathbf{p|}^{2}}=$
$\sqrt{1-v_{i}^{2}}$($i=1,2$) rise from the length contraction $r\rightarrow
r(m/E)$ when quark $i$ moves relativistically, and these factors tend to unity
in the low-momentum ($p/m_{i}\rightarrow 0$) limit of the quarks. Here,
$\epsilon_{V,S}$ are small homogeneous factors to be determined. Applying Eqs.
(45) and (46) to a heavy quark $Q$ in color triplet($3_{c}$) and a diquark
$d=qq$($q=u,d,s$) in the anti-color triplet($\bar{3}_{c}$), four spin-
couplings become then

$\begin{array}[]{r}a_{1}=\frac{1}{2m_{d}}\left\langle\frac{\tilde{V}^{\prime}-\tilde{S}^{\prime}}{m_{d}r}+\frac{2\tilde{V}^{\prime}}{m_{Q}r}\right\rangle,\\\
a_{2}=\frac{1}{2m_{Q}}\left\langle\frac{\tilde{V}^{\prime}-\tilde{S}^{\prime}}{m_{Q}r}+\frac{2\tilde{V}^{\prime}}{m_{d}r}\right\rangle,\\\
b=\frac{1}{3m_{d}m_{Q}}\left\langle\tilde{V}^{\prime}/r-\tilde{V}^{\prime\prime}\right\rangle,c=\left\langle\frac{2\mathbf{\nabla}^{2}\tilde{V}}{3m_{d}m_{Q}}\right\rangle,\end{array}$
(46)

in which $V+S$ stands for the confining potential between $Q$ and $d$, to be
approximated by that for the heavy quarkonia $Q\bar{Q}$, and the quantum
average $\left\langle{}\right\rangle$ is made over the P-wave wavefunction
$\Psi_{Qd}$ of the $Qd$ system. For the spin interaction in Eq. (2), the
relativistic treatment (45) implies that the factor $1/m_{i}$ in Eq. (46)
becomes $1/E_{i}=(m_{i}^{2}+p^{2})^{-1/2}$, which makes sure, for example,
that the hyperfine interactions of the light quark $i$ do not blow up like
$1/m_{i}$ in the chiral limit($m_{i}\rightarrow 0$), but rather they should
have a finite limit determined by $\left\langle p^{-1}\right\rangle$, which is
in turn controlled by the radius of confinement.

To find P-wave wavefunction $\Psi_{Qd}$ of singly heavy baryons($Qd$) we
consider the relativized quark modelGI:D85  with the color-coulomb plus linear
potential $V+S$(treating the spin interaction $H^{SD}$ perturbatively)

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sqrt{m_{Q}^{2}+p^{2}}+\sqrt{m_{d}^{2}+p^{2}}+V(r)+S(r),$
$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle-k_{s}/r,S(r)=ar+C_{0}.$
(47)

where $k_{s}=4\alpha_{s}/3$ and $\alpha_{s}$ is the strong coupling.

By introducing auxiliary fields $\mu_{d}$ and $\nu$, one can solve the model
(47) using auxiliary field (AF) methodSemayBS:D2008 (Appendix C). The inputs
of $m_{d},a$, and $\alpha_{s}$ are given in Table IX, where the values of
other parameters $\xi,\zeta$ and $\epsilon_{V,S}$ for computing the spin
couplings are also listed. The solved P-wave radial wavefunction of the quark-
diquark system are given in FIG. 3.

Table 9: Inputs for the model parameters and diquark masses defined in
Hamiltonian (C1) (Appendix C) for the P-wave heavy baryons listed. The heavy
quark masses $m_{Q}=M_{Q}=1440$MeV($Q=c$) and $4480$MeV($Q=b$), and light
diquark masses $m_{d}$ are taken from Ref. JiaLH:19 . The confining potential
parameter $a$ is set by Regge trajectories of corresponding baryons, taken
from Ref. JiaLH:19  and Eq. (24) and Sect. IV.
$\begin{array}[]{cccccccc}\hline\cr\hline\cr\text{State}&m_{d}&a\text{[GeV}^{2}\text{]}&\alpha_{s}&\xi\text{(GeV}^{-1}\text{)}&\zeta\text{(GeV}^{-2}\text{)}&\epsilon_{V}&\epsilon_{S}\\\
\hline\cr\left\\{\begin{array}[]{r}\Omega_{c}\\\
\Omega_{b}\end{array}\right.&991&\begin{array}[]{r}0.316\\\
0.318\end{array}&\begin{array}[]{r}0.561\\\
0.543\end{array}&0.818&0.11&-0.10&2.65\\\
\left\\{\begin{array}[]{r}\Xi_{c}^{\prime}\\\
\Xi_{b}^{\prime}\end{array}\right.&872&\begin{array}[]{r}0.255\\\
0.307\end{array}&\begin{array}[]{r}0.590\\\
0.545\end{array}&0.820&0.12&-0.06&2.20\\\
\left\\{\begin{array}[]{r}\Sigma_{c}\\\
\Sigma_{b}\end{array}\right.&745&\begin{array}[]{r}0.212\\\
0.246\end{array}&\begin{array}[]{r}0.595\\\
0.549\end{array}&0.850&0.12&-0.05&1.80\\\ \hline\cr\hline\cr\end{array}$
---
Figure 3: (Color online). The P-wave radial wavefunction $R_{nL}(r)$ with
$n=0$ and $L=1$, numerically solved from Eq. (C8), with the inputs in Table
IX.

Upon using (46) to compute the spin couplings, we take into account non-point
nature of the diquark $d$ via adding a form factor $F(r)\equiv 1-e^{-\xi
r-\zeta r^{2}}$ EFG:D11  in the short-range potential $V$ so that

$V\rightarrow VF=-\frac{k_{s}}{r}F(r).$ (48)

Putting the improved $V$ in Eq. (48) into Eq. (46), in which the relativized
potentials become now, by Eq. (45),
$\tilde{V}=(m_{d}m_{Q}/(E_{d}^{H}E_{Q}^{H}))^{1+\epsilon_{V}}V$ and
$\tilde{S}=(m_{d}m_{Q}/(E_{d}^{N}E_{Q}^{N}))^{1+\epsilon_{S}}S$ with
$S=ar+C_{0}$ and

$E_{i}^{H,N}=[m_{i}^{2}+\langle p^{2}\rangle_{H,N}]^{1/2},$ (49)

one finds

$\begin{array}[]{r}a_{1}=\frac{1}{2m_{d}^{2}}\left\langle\left(1+\frac{2m_{d}}{m_{Q}}\right)\left(\frac{m_{d}m_{Q}}{E_{d}^{H}E_{Q}^{H}}\right)^{1+\epsilon_{V}}k_{s}\left(\frac{F}{r^{3}}-\frac{F^{\prime}}{r^{2}}\right)-\left(\frac{m_{d}m_{Q}}{E_{d}^{N}E_{Q}^{N}}\right)^{1+\epsilon_{S}}\frac{a}{r}\right\rangle,\\\
a_{2}=\frac{1}{m_{d}m_{Q}}\left\langle\left(1+\frac{m_{d}}{2m_{Q}}\right)\left(\frac{m_{d}m_{Q}}{E_{d}^{H}E_{Q}^{H}}\right)^{1+\epsilon_{V}}k_{s}\left(\frac{F}{r^{3}}-\frac{F^{\prime}}{r^{2}}\right)-\left(\frac{m_{d}m_{Q}}{E_{d}^{N}E_{Q}^{N}}\right)^{1+\epsilon_{S}}\frac{m_{d}a}{2m_{Q}r}\right\rangle,\\\
b=\frac{k_{s}}{3m_{d}m_{Q}}\left(\frac{m_{d}m_{Q}}{E_{d}^{H}E_{Q}^{H}}\right)^{1+\epsilon_{V}}\left\langle\frac{3F}{r^{3}}-\frac{3F^{\prime}}{r^{2}}+\frac{F^{\prime\prime}}{r}\right\rangle,\\\
c=\frac{2k_{s}}{3m_{d}m_{Q}}\left(\frac{m_{d}m_{Q}}{E_{d}^{H}E_{Q}^{H}}\right)^{1+\epsilon_{V}}\left[4\pi|R_{nL}^{H}(0)|^{2}-\int
drrF^{\prime\prime}|R_{nL}^{H}(r)|^{2}\right],\end{array}$ (50)

where we have used $V^{\prime}=k_{s}(F/r^{2}-F^{\prime}/r)$,
$V^{\prime\prime}=k_{s}(2F^{\prime}/r^{2}-2F/r^{3}-F^{\prime\prime}/r)$,
$S^{\prime}=a$, and the Laplace relation
$\mathbf{\nabla}^{2}(F/r)=-4\pi\delta^{3}(\mathbf{r})+F^{\prime\prime}/r$.
Here, the average is taken over $R_{0L}(r)$, and for $n_{r}=0$,

$\displaystyle E_{d}^{H}$ $\displaystyle=$
$\displaystyle\mu_{dH},E_{Q}^{H}=\left[m_{Q}^{2}+\frac{(k_{s}\mu_{H})^{2}}{(L+1)^{2}}\right]^{1/2},$
$\displaystyle E_{i}^{N}$ $\displaystyle=$
$\displaystyle\left[m_{i}^{2}+a\left(L+\frac{3}{2}\right)\sqrt{\mu/\nu}\right]^{1/2},i=Q,d,$
(51)

where the average $\langle
p^{2}\rangle_{N}=\alpha^{2}(L+3/2)=(a^{2}\mu/v)^{1/2}(L+3/2)$ in the HO
wavefunction is used again.

The AF fields ($\mu,\nu$) are solvable from the nonlinear equation (C4)
(Appendix C) given the inputs of the masses $m_{Q}=M_{Q}$ and $m_{d}$, and the
linear potential parameter $a$ extracted from Ref. JiaLH:19 , as listed in
Table IX, by which Eq. (51) gives the inverse HO length $\alpha$, and the
relativistic factors $m_{i}/E_{i}^{H,N}$. Notice that two derivatives
$F^{\prime}=(2\zeta r+\xi)e^{-\xi r-\zeta
r^{2}},F^{\prime\prime}=[2\zeta-(2\zeta r+\xi)^{2}]e^{-\xi r-\zeta r^{2}}$,
one can then compute the spin-couplings in Eqs. (50) via quantum averaging
with $R_{nL}(r)$ solved from Eq. (C8)(Appendix C). The results are shown in
Table X, which agree reasonably with the parameters ($a_{1},a_{2},b$) given in
Eq. (12) and Eq. (25) as well as that in Table VIII. One sees from Table X
that our prediction for the spin couplings $a_{2}$ is slightly overestimated
compared to that matched ones, about $4-8$MeV, and the predictions for $c$ is
anomaly larger than the matched ones, about $7-16$MeV.

Table 10: Spin coupling parameters $a_{1}$, $a_{2}$, $b$ and $c$ calculated by
Eq. (50), the AF fields ($\mu,\nu$) solved from Eq. (C4) (Appendix C) and
other related parameters for the P-wave states. All in MeV except for $a$.

$\begin{array}[]{ccccccccccc}\hline\cr\hline\cr\text{State}&\mu_{d}&\mu&\nu&\mu_{dH}&\mu_{H}&a_{B}\text{[GeV}^{-1}\text{]}&a_{1}&a_{2}&b&c\\\
\hline\cr\Omega_{c}&1291&681&909&1016&596&2.248&28.52&27.03&15.32&20.73\\\
\Omega_{b}&1379&1054&789&1034&840&1.701&10.30&10.25&5.61&9.26\\\
\Xi_{c}^{\prime}&1153&640&805&900&554&2.256&30.15&27.98&16.54&20.35\\\
\Xi_{b}^{\prime}&1273&992&787&912&757&1.882&11.42&11.06&6.14&9.82\\\
\Sigma_{c}&1017&596&728&777&504&2.283&35.46&30.96&19.10&20.44\\\
\Sigma_{b}&1117&894&703&781&665&2.130&13.93&11.15&6.61&8.93\\\
\hline\cr\hline\cr\end{array}$

One can list the resulted ratios of the spin couplings
$r_{1,2}=a_{1,2}(bqq)/a_{1,2}(cqq)$ and $r_{b}=b(bqq)/b(cqq)$ between the
bottom and charmed baryons and compare them to corresponding ratios (denoted
as $[.]_{Match}$) obtained from Eqs.(12),(25) and from Table VIII:

$\begin{array}[]{cccc}\text{ratio:}&r_{1}&r_{2}&r_{b}\\\
\Omega_{b}/\Omega_{c}:&0.361&0.379&0.366\\\
\Xi_{b}^{\prime}/\Xi_{c}^{\prime}:&0.379&0.395&0.370\\\
\Sigma_{b}/\Sigma_{c}:&0.393&0.360&0.346\end{array}\begin{array}[]{ccc}[r_{1}]_{Match}&[r_{2}]_{Match}&[r_{b}]_{Match}\\\
0.33&0.16&0.56,\\\ 0.28&0.31&0.35,\\\ 0.33&0.29&0.31,\end{array}$ (52)

One sees that the simple mass scaling law($\sim
M_{b}/M_{c}=1440/4480=\allowbreak 0.321$) holds approximately for $a_{2}$ and
$b$. In the case of $a_{1}$, the scaling law $a_{1}(bqq)=a_{1}(cqq)$
KarlinerR:D15  is broken somehow as $r_{1}\sim 1/3$ is less than unity.
Further, the matched value $r_{2}=0.16$ for the baryons $\Omega_{b}$ vs.
$\Omega_{c}$ is somehow smaller than that expected by $M_{b}/M_{c}$ and also
than our computed value $r_{2}=0.379$.

From Table XI, which shows the computed values of the relevant factors and the
averages in Eq. (50), one sees that the minus term$\sim$ $-\left\langle
m_{d}/m_{Q}\ a/r\right\rangle$ in Eq. (50) is notably suppressed and this
gives relatively larger predictions for $a_{2}$. We remark three features of
our computations:

(i) As a leading approximation, the Breit-Fermi-like interaction (44) have not
considered nonperturbative corrections that may enter at long-distance regime.
Assuming the spin interaction dominates at short distance, one expects that
the interaction (44) includes the main spin-spin and spin-orbital forces via
adding the scalar confining potential $S\sim r$ to the vector potential $V$,
and such a nonperturbative corrections to Eq. (44) is relatively small.

(ii) Static approximation of quark-diquark potential $V+S$, which is valid in
principle only for heavy quarkonia($Q\bar{Q}$), has ignored rotation of QCD
string when $L\neq 0$ LaCourseO:D89 ; KalashNef:D00 ; AgaevAS:D17 . In
orbitally-excited state, the rotation of QCD string can change the static
quark-diquark potential $V+S$ in Eq. (47), as shown in Refs. LaCourseO:D89 ;
KalashNef:D00 . For instance, this corrects $L$-dependence of the Hamiltonian,
$[m_{d}^{2}+p_{r}^{2}+L^{2}/r^{2}]^{1/2}\sim L$ into a Regge-like one,
$H\sim\sqrt{\pi aL}$ due to addition of the orbital angular moment of the
string LaCourseO:D89 .

(iii) A simple exponential form factor $F(r)$ may not be enough in the short-
range to describe the three-body quark systems of $Q(qq)$. Three-body
interaction CI:D86 ; SemayBS:D08  inside baryons may make $F(r)$ quite
nontrivial as the effective running of the color-charge of a non-point diquark
$d$ in color-antitriplet ($\bar{3}_{c}$) seen by $Q$ in color-triplet
($3_{c}$) becomes highly distance-dependent when $d$ is close to $Q$. See Ref.
CI:D86 (Eq. (18)) for other possible forms of the form factor, like
$Erf[c_{0}r]$ or its linear combinations. For simplicity, we used the
$R_{nL}^{H}(r)$ in Eq. (C11) (Appendix C) to estimate $c$ in Eq. (50) and this
may overestimate values for $c$ govern by integral of
$-F^{\prime\prime}|R_{nL}^{H}(r)|^{2}$ at short-range. Note that
$R_{nL}^{H}(0)\equiv 0$ for the P-waves.

Table 11: Values of the relevant factors (dimensionless) and the computed
averages(in GeV-3) for spin coupling $a_{1,2}$ and $b$ in Eq. (50).
$\begin{array}[]{cccccccc}\hline\cr\hline\cr\text{State}&\frac{m_{d}}{m_{Q}}&\left(\frac{m_{d}m_{Q}}{E_{d}^{H}E_{Q}^{H}}\right)^{1+\epsilon_{V}}&\left(\frac{m_{d}m_{Q}}{E_{d}^{N}E_{Q}^{N}}\right)^{1+\epsilon_{S}}&\left\langle\frac{m_{d}\
a}{2m_{Q\
}r}\right\rangle&\left\langle\frac{F}{r^{3}}-\frac{F^{\prime}}{r^{2}}\right\rangle&\left\langle\frac{a}{r}\right\rangle&\left\langle\frac{3F}{r^{3}}-\frac{3F^{\prime}}{r^{2}}+\frac{F^{\prime\prime}}{r}\right\rangle\\\
\hline\cr\Omega_{c}&0.688&0.884&0.227&0.0472&0.056&0.137&0.0994\\\
\Omega_{b}&0.220&0.866&0.276&0.0182&0.075&0.164&0.123\\\
\Xi_{c}^{\prime}&0.606&0.864&0.278&0.0309&0.049&0.102&0.090\\\
\Xi_{b}^{\prime}&0.195&0.861&0.278&0.0149&0.071&0.153&0.119\\\
\Sigma_{c}&0.518&0.833&0.313&0.0204&0.044&0.079&0.085\\\
\Sigma_{b}&0.167&0.849&0.306&0.0092&0.062&0.110&0.111\\\
\hline\cr\hline\cr\end{array}$

In addition, Eq. (50) seems to imply the scaling laws
$a_{1}\sim\mathcal{O}(m_{Q}^{0})$, $a_{2}$ $\sim 1/m_{Q}\sim b$ (see Eq. (24)
in Ref. KarlinerR:D15 )), as heavy quark symmetry expected. However, the small
variation of the averages within the baryons, manly in $1+2m_{d}/m_{Q}$ and
$k_{s}=4\alpha_{s}/3$ along with the size($\sim a_{B}$) of baryons, is
enlarged by $m_{d}^{2}$ in denominator for $a_{1}$ in Eq. (50) and this makes
$a_{1}$ changes notably between the bottom and charmed sectors. This kind of
enlargement does not happen for other spin couplings due to their overall
suppression factor $1/m_{d}m_{Q}$. Note also that the factor $1+m_{d}/2m_{Q}$
for $a_{2}$ changes slightly with heavy flavor $Q(=c,b)$, compared to the
variations of $1+2m_{d}/m_{Q}$:

$\displaystyle\left(1+\frac{m_{d}}{2m_{Q}}\right)_{c,b}$ $\displaystyle:$
$\displaystyle\\{1.34,1.11\\}_{ssQ},\\{1.30,1.09\\}_{snQ},\\{1.26,1.08\\}_{nnQ},$
$\displaystyle\left(1+\frac{2m_{d}}{m_{Q}}\right)_{c,b}$ $\displaystyle:$
$\displaystyle\\{2.37,1.44\\}_{ssQ},\\{2.21,1.39\\}_{snQ},\\{2.04,1.33\\}_{nnQ},$

As seen in Tables X and XI, the scaling law for the spin couplings breaks
moderately, especially for $a_{1}$, due to the injured heavy quark symmetry
indicated by Eq. (50).

##  VII Conclusions and Discussions

Exploring and building data set of the mean-spin mass $\bar{M}$ and the spin
coupling strengths($a_{1},a_{2},b,c$) is important as it can be a basis to
explore how quarks interact within hadrons. Although baryon is a three-body
system with subtle short-range interactions, it is useful to employ a simple
heavy quark-diquark picture for the singly heavy baryons to understand the
measured mass data of excited heavy-baryons. In this work, we have proposed a
new scheme of state classification, the $Jls$ mixing coupling, to unify
analysis of masses and inner-structures of the excited baryons $\Omega_{Q}$
and other less-strange heavy baryons with spin-1 light diquarks. Firstly, we
interpret the five $\Omega_{c}$ and four $\Omega_{b}$ excited states reported
by LHCb to be the relative $P$-wave excitations between spin-1 $ss$ diquark
and the heavy quark, the charm quark $c$ for the $\Omega_{c}$ and the bottom
quark $b$ for the $\Omega_{b}$, respectively. Further analysis via Regge
phenomenology for the low-lying excited states excludes the possibility that
some of higher states of the resonances $\Omega_{c}$ and $\Omega_{b}$ are the
$2S$ states since the $2S$ candidates are all too heavy to accommodate both of
the resonances. Secondly, we applied our formalism based on the $Jls$ mixing
coupling to update the earlier computation of all P-wave masses of the less
strange baryons $\Sigma_{c}/\Xi_{c}^{\prime}$ and
$\Sigma_{b}/\Xi_{b}^{\prime}$, for which only single measured mass available
for each of these baryons. The masses of all spin-partners of the baryons
$\Sigma_{c}(2800)/\Xi_{c}^{\prime}(2942)$ and
$\Sigma_{b}(6097)$/$\Xi_{b}^{\prime}(6227)$ are then predicted. Finally, we
demonstrated that the matched spin-couplings of the singly heavy baryons
considered can be understood in a relativized potential quark model, based on
which the scaling law for their spin coupling parameters are discussed.

With the help of the relation (17), we find that four measured masses for the
excited $\Omega_{b}$’s are enough to make a quantitative prediction
$M(5/2^{-})=6352$ MeV for one unseen state with $J^{P}=5/2^{-}$. The missing
of this $5/2^{-}$ state in LHCb experiment can be due to the degeneracy with
the nearby state $|3/2^{-},2^{\prime}\rangle$ at $6350$ MeV. The same
$J^{P}=5/2^{-}$ state is predicted as the unseen state in a recent assignment
KarlinerR:Ax20 , with two higher mass ranges predicted, about $10$ MeV away or
more.

One very recent prediction for the odd-parity $\Omega_{b}$’s is that by the LS
couplingMutuk:A20 :

$\displaystyle M(1/2^{-})=6314,6330\text{ MeV},$ $\displaystyle
M(3/2^{-})=6339,6342\text{ MeV},M(5/2^{-})=6352\text{ MeV},$ (53)

for which the explanation of near degeneracy applies only to two $J=3/2$
states and assuming one of them to be unseen is favored. In the viewpoint of
this work, with given mixing weights in Eq. (37), two LS bases
$|^{2}P_{3/2}\rangle$ and $|^{4}P_{3/2}\rangle$ are not eigenstates of the
Hamiltonian (2) for $J=3/2$ and tend to mix deeply to form the true $Jls$
eigenstates $|3/2,1^{\prime}\rangle$ and $|3/2,2^{\prime}\rangle$, with
respective masses $6337.8$ MeV and $6350.0$ MeV in Eq. (36) both matching the
measured masses($6339.7$ MeV and $6349.9$ MeV) nicely. Our mass predictions in
Eq. (36) for other states($J=3/2,5/2$) should agree with that by the LS
coupling because the bases for the LS and Jls couplings are same($J=5/2$) or
roughly same($J=3/2$), as two predictions(Eq. (53) v.s. Eq. (36)) indicated.
This favors the state at $M(5/2^{-})=6352$ MeV to be the unseen state.

Our computation via a simple heavy quark-diquark picture indicates that the
measured masses of the excited heavy-baryons considered can be understood via
a relativized potential quark model and resulted Breit-Fermi spin-orbit and
tensor forces, provided that a proper relativizing is made to these spin
interactions(forces). By the way, we confirm based on the relativized
expression of the Breit-Fermi formula (50) for the spin coupling parameters
that a mass scaling lawKarlinerR:D15  for them holds approximately for all
baryons considered. Being at the order of $\mathcal{O}(1)$, the scaling
relation $a_{1}(bqq)=a_{1}(cqq)$ for $a_{1}$ is broken most, in contrast with
other parameters at the order of $\mathcal{O}(1/m_{Q})$. Due to involved QCD
dynamics of the heavy flavor baryons(see EFG:D11 ; Manohar:D07  for instance),
our analysis for the spin-couplings is of quite approximative, with further
quantitative corrections remained to be explored.

ACKNOWLEDGMENTS

D. J thanks Xiang Liu and A. Hosaka for discussions. D. J. is supported by the
National Natural Science Foundation of China under the no. 11565023 and the
Feitian Distinguished Professor Program of Gansu (2014-2016). C.P is supported
by the National Natural Science Foundation of China under Grants No. 11965016,
and by the projects funded by Science and Technology Department of Qinghai
Province (No. 2018-ZJ-971Q and No. 2019-ZJ-A10), the Key Laboratory of IoT of
Qinghai under grant (No. 2020-ZJ-Y16).

## Appendix A

For a heavy quark-diquark system $Qd$, the bases of the mixing $Jls$ coupling
are the eigenfunctions of the mass operator
$H^{mix}=\mathbf{L}\cdot\mathbf{S}_{d}+\epsilon\mathbf{L}\cdot\mathbf{S}_{Q}+b_{1}S_{12}$,
with $b_{1}\equiv b/a_{1}$. They can be obtained by diagonalizing $H^{mix}$,
that is, diagonalizing the $2\times 2$ matrices in Eqs. (9)-(15) separately in
the subspaces of $J=1/2$, $J=3/2$. When $J=1/2$, the matrix

$H_{J=1/2}=\left[\begin{array}[]{cc}\frac{1}{3}(\epsilon-4)&\frac{\sqrt{2}}{3}(\epsilon-1)+\frac{b_{1}}{\sqrt{2}}\\\
\frac{\sqrt{2}}{3}(\epsilon-1)+\frac{b_{1}}{\sqrt{2}}&-\frac{5}{3}(1+\frac{\epsilon}{2})-b_{1}\end{array}\right]$
(A1)

can be diagonalized to give $M_{1/2}=\langle H_{J=1/2}\rangle$. Thus, adding a
perturbative correction $\sim c_{1}M_{c}$ to $M_{1/2}$, with

$M_{c}=\left[\begin{array}[]{cc}-1&0\\\ 0&\frac{1}{2}\end{array}\right],$ (A2)

leads to the mass shifts in the $J=1/2$ subspace:

$\displaystyle\Delta M(1/2,0^{\prime})/a_{1}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}-\frac{1}{2}b_{1}-\frac{1}{4}\epsilon-\frac{1}{4}\sqrt{\Delta_{1}(\epsilon,b_{1})}+c_{1}\Delta_{3}^{+},$
$\displaystyle\Delta M(1/2,1^{\prime})/a_{1}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}-\frac{1}{2}b_{1}-\frac{1}{4}\epsilon+\frac{1}{4}\sqrt{\Delta_{1}(\epsilon,b_{1})}+c_{1}\Delta_{3}^{-},$
(A3)

with the functions $\Delta_{1}(\epsilon,x),\Delta_{3}^{\pm}(\epsilon,x)$
defined in Eqs. (18) and (19).

In the $J=3/2$ subspace, a similar diagonalization of $H_{J=3/2}$ which gives
$M_{3/2}$, leads to, by adding a perturbative correction $\sim c_{1}M_{c}$,

$\displaystyle\Delta M(3/2,1^{\prime})/a_{1}$ $\displaystyle=$
$\displaystyle\frac{2}{5}b_{1}-\frac{1}{4}\epsilon-\sqrt{\Delta_{2}(\epsilon,b_{1})}+c_{1}\Delta_{4}^{+},$
$\displaystyle\Delta M(3/2,2^{\prime})/a_{1}$ $\displaystyle=$
$\displaystyle\frac{2}{5}b_{1}-\frac{1}{4}\epsilon+\sqrt{\Delta_{2}(\epsilon,b_{1})}+c_{1}\Delta_{4}^{-},$
(A4)

with the functions $\Delta_{2}(\epsilon,x)$ and $\Delta_{4}^{\pm}(\epsilon,x)$
defined in Eqs. (18) and (20). In the $J=5/2$ subspace, $\Delta M/a_{1}$ is
simply $\Delta M(5/2,2^{\prime})/a_{1}=1-b_{1}/5+\epsilon/2+c_{1}/2$. This,
combined with Eq. (A5) and (A8), proves Eq.(17). A recent Ref. Ali:jh2019
gives an alternative expression similar to the mass-shifts (17).

## Appendix B

The coefficients of Eq. (24) form normalized vectors of the eigenstates of the
operator $H^{mix}$. In the subspaces of $J=1/2$, $H^{mix}$ becomes the matrix
$H_{J=1/2}$ in Eq. (A1) with the eigenstates given in Eq. (A3), which are
$\mathbf{v}_{1}=\\{-0.208,1\\}$ and $\mathbf{v}_{2}=\left\\{4.804,1\right\\}$
using Eq. (22). Upon normalization, the eigenstates give rise to the
coefficients in the first and second lines of Eq. (24). Similar computation
leads to the coefficients in the third and fourth lines of Eq. (24).

To write the mixing bases $|J,j^{\prime}\rangle$ in terms of the $jj$ bases
$|J,j\rangle$, one writes firstly the $P$-wave $|J,j\rangle$ in terms of the
$L-S$ eigenstates $|1^{2S+1}P_{J}\rangle$, as given in Eq.(25). Its inverse is

$\displaystyle|1^{2}P_{1/2}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}|1/2,j=0\rangle+\sqrt{\frac{2}{3}}|1/2,j=1\rangle,$
$\displaystyle|1^{4}P_{1/2}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}|1/2,j=0\rangle-\sqrt{\frac{1}{3}}|1/2,j=1\rangle.$
(B1)

Putting it into the first two lines of Eq. (24) gives the first and second
lines of Eq.(26). Similarly, one can obtain the third and fourth lines of
Eq.(26) with the help of

$\displaystyle|J$ $\displaystyle=$ $\displaystyle
3/2,j=1\rangle=\sqrt{\frac{1}{6}}|1^{2}P_{3/2}\rangle+\sqrt{\frac{5}{6}}|1^{4}P_{3/2}\rangle,$
$\displaystyle|J$ $\displaystyle=$ $\displaystyle
3/2,j=2\rangle=\sqrt{\frac{5}{6}}|1^{2}P_{3/2}\rangle-\sqrt{\frac{1}{6}}|1^{4}P_{3/2}\rangle,$
(B2)

and its inverse transformation.

## Appendix C

Using auxiliary field (AF) methodSemayBS:D2008 , which re-expresses nonlinear
operator $\sqrt{O}=\min_{\lambda>0}[O/(2\lambda)+\lambda/2]$ in terms of the
AF field $\lambda$, for which the minimization is achieved when $\lambda=$
$\sqrt{O}$, one can rewrite the model (47) as

$\displaystyle H$ $\displaystyle=$ $\displaystyle
m_{Q}+\frac{p^{2}}{2m_{Q}}+\frac{m_{d}^{2}+p^{2}}{2\mu_{d}}+\frac{(ar)^{2}}{2\nu}+\frac{\mu_{d}+\nu}{2}-\frac{k_{s}}{r}+C_{0},$
(C1) $\displaystyle\mu_{d}$ $\displaystyle=$
$\displaystyle\left\langle\sqrt{m_{d}^{2}+p^{2}}\right\rangle,\nu=a\left\langle
r\right\rangle,$ (C2)

where $p=|\mathbf{p}|$ is the 3-momentum of the heavy quark $Q$ and the light
diquark $d$, and the minimization conditions (C2) for the auxiliary fields
($\mu_{d},\nu$) are assumed to hold at semi-classical level. In a sense, the
AF method linearizes the Hamiltonian $H$ in (47) in a manner similar to mean-
field method for many-body systems. Next, we solve the P-wave radial
wavefunction $R_{nL}$ of the $Qd$ system defined by Eq. (47) numerically in
two steps.

In the first step, we find two self-consistent AF fields ($\mu_{d},\nu$) via
solving Eq. (C2). For this, we ignore the (color) Coulomb term $-k_{s}/r$ to
write the Hamiltonian (C1) as that of three dimensional harmonic
oscillator(HO) with reduced mass $\mu\equiv\mu_{d}/(1+\mu_{d}/m_{Q})$, the HO
frequency $\omega=a/\sqrt{\mu\nu}$, the inverse HO length
$\alpha=\sqrt{\mu\omega}=(a^{2}\mu/v)^{1/4}$. As such, the wavefunction
$\Psi_{Qd}$ is given by that of the HO,
$\psi^{N}=R_{n_{r}L}^{N}(r)Y_{Lm}(\theta,\varphi)$ with quantum number
$N=2n_{r}+L$, where $Y_{Lm}\ $is the spherical function and

$R_{n_{r}L}^{N}(r)=\alpha^{3/2}\left[\frac{2n_{r}!}{\Gamma(n_{r}+L+3/2)}\right]^{1/2}(\alpha
r)^{L}L_{n_{r}}^{L+1/2}(\alpha^{2}r^{2})e^{-(\alpha r)^{2}/2},$ (C3)

with $L_{n_{r}}^{2L+1}$ the associated Laguerre polynomial. For the orbital
states with $n_{r}=0$ we address, one finds from Eq. (C2),

$\displaystyle\mu_{d}$ $\displaystyle\simeq$
$\displaystyle\sqrt{m_{d}^{2}+\left\langle
p^{2}\right\rangle_{N}}=[m_{d}^{2}+\alpha^{2}(L+3/2)]^{1/2},$
$\displaystyle\nu$ $\displaystyle\simeq$ $\displaystyle a\sqrt{\left\langle
r^{2}\right\rangle_{N}}=a\left[\frac{L+3/2}{\mu\omega}\right]^{1/2},$

where the HO-state averages of $p^{2}$ and $r^{2}$ are used. This leads to two
nonlinear equations

$\displaystyle\mu$ $\displaystyle\equiv$
$\displaystyle\frac{[m_{d}^{2}+a(L+3/2)(\mu/\nu)^{1/2}]^{1/2}}{1+[m_{d}^{2}+a(L+3/2)(\mu/\nu)^{1/2}]^{1/2}/m_{Q}},$
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\sqrt{a(L+3/2)}(\nu/\mu)^{1/4},$ (C4)

which can be solved by numerical iteration. The solution for ($\mu,\nu$) and
$\mu_{d}\equiv\mu/(1-\mu/m_{Q})$ are shown in Table 1(Table IX in revision).

The solution of the wavefunction $\Psi_{Qd}$ can be solved from the
Hamiltonian (C1) of the two-body $Qd$, which is

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{p^{2}}{2\mu}+\frac{1}{2}m\omega^{2}r^{2}-\frac{k_{s}}{r}+m_{Q}+\frac{m_{d}^{2}}{2\mu_{d}}+\frac{\mu_{d}+\nu}{2}+C_{0},$
(C5) $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\mu_{d}/\left[1+\frac{\mu_{d}}{m_{Q}}\right],\omega=\frac{a}{\sqrt{\mu\nu}}.$
(C6)

In the long-range where the color-Coulomb term $-k_{s}/r$ is ignorable, the
radial eigenfunctions of the Hamiltonian (C5) can be given by that of the HO
wavefunction,

$R_{n_{r}L}^{N}(r)=\alpha^{3/2}\left[\frac{2n_{r}!}{\Gamma(n_{r}+L+3/2)}\right]^{1/2}(\alpha
r)^{L}L_{n_{r}}^{L+1/2}(\alpha^{2}r^{2})e^{-(\alpha r)^{2}/2},$ (C7)

with $L_{n_{r}}^{2L+1}$ the associated Laguerre polynomial.

In the second step, we use the obtained ($\mu,\nu$) to numerically solve the
eigenstate equation for the singly heavy baryons $\Omega_{Q}$,
$\Xi_{Q}^{\prime}$ and $\Sigma_{Q}$($Q=c,b$)

$\left[-\frac{1}{2\mu
r^{2}}\frac{d}{dr}\left(r^{2}\frac{d}{dr}\right)+\frac{L(L+1)}{2\mu
r^{2}}+\frac{a^{2}}{2\nu}r^{2}-\frac{k_{s}}{r}\right]R_{nL}=ER_{nL},$ (C8)

where $R_{nL}(r)$ are the radial wavefunctions of these systems. The results
for the P-wave solutions $R_{nL}(r)$ are shown in FIG. 3.

It is useful to write the radial wavefunction $R_{n_{r}L}^{H}(r)$ in the
small-$r$ range, which is given by solution to the eigen-equation of the
Hamiltonian (C1) with the HO potential($\sim r^{2}$) ignored. It is the
solution to the hydrogen-like atoms with the effective Bohr radius
$a_{B}=1/(\mu_{H}k_{s})$ and the reduced mass
$\mu_{H}=\mu_{dH}/(1+\mu_{dH}/m_{Q})$ with
$\mu_{dH}=\left\langle\sqrt{m_{d}^{2}+p^{2}}\right\rangle_{H}$ averaged in the
hydrogen-like wavefunction $\psi^{H}=R_{n_{r}L}^{H}(r)Y_{Lm}(\theta,\varphi)$,
with

$R_{n_{r}L}^{H}(r)=\left\\{\left(\frac{2}{na_{B}}\right)^{3}\frac{n_{r}!}{2n[(n+L)!]^{3}}\right\\}^{1/2}e^{-r/(na_{B})}\left(\frac{2r}{na_{B}}\right)^{L}L_{n_{r}}^{2L+1}\left(\frac{2r}{na_{B}}\right),$
(C9)

with $n=n_{r}+L+1$. Using averaging in the hydrogen-like wavefunction
$\psi^{H}$ with $n_{r}=0$, $\langle
p^{2}\rangle_{H}=k_{s}\mu_{H}/[a_{B}(L+1)^{2}]=k_{s}^{2}\mu_{H}^{2}/(L+1)^{2}$,
one finds that the AF parameter $\mu_{dH}\simeq$ $\sqrt{m_{d}^{2}+\langle
p^{2}\rangle_{H}}$ is subjected to the self-consistent equation,

$\mu_{dH}=\left[m_{d}^{2}+\frac{1}{(L+1)^{2}}\frac{(k_{s}\mu_{dH})^{2}}{(1+\mu_{dH}/m_{Q})^{2}}\right]^{1/2}.$
(C10)

The numerical results for $\mu_{dH}\ $and thereby-resulted $\mu_{H}$ are shown
in Table V also.

In the short-range where the term $-k_{s}/r$ is dominate over the HO
potential, the radial solution of wavefunction to the Hamiltonian (C5) is that
of the hydrogen-like:

$R_{n_{r}L}^{H}(r)=\left\\{\left(\frac{2}{na_{B}}\right)^{3}\frac{n_{r}!}{2n[(n+L+1)!]^{3}}\right\\}^{1/2}e^{-r/(na_{B})}\left(\frac{2r}{na_{B}}\right)^{L}L_{n_{r}}^{2L+1}\left(\frac{2r}{na_{B}}\right),$
(C11)

with $n=n_{r}+L+1$.

## References

  * (1) P.A. Zyla et al. [Particle Data Group], Review of Particle Physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
  * (2) R. Aaij et al. (LHCb Collaboration), Observation of Five New Narrow $\Omega_{c}^{0}$ States Decaying to $\Xi K^{-}$, Phys. Rev. Lett.118, 182001 (2017).
  * (3) R. Aaij et al. (LHCb Collaboration), First Observation of Excited $\Omega_{b}^{-}$ States, Phys. Rev. Lett. 124, 082002 (2020).
  * (4) D. Ebert, R. N. Faustov, and V. O. Galkin, Masses of excited heavy baryons in the relativistic quark model, Phys. Lett. B659, 612 (2008). arXiv:0705.2957[hep-ph].
  * (5) 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 (2011) 014025, arXiv:1105.0583[hep-ph].
  * (6) K. Maltman and N. Isgur, Baryons with strangeness and charm in a quark model with chromodynamics, Phys. Rev. D 22, 1701 (1980).
  * (7) W. Roberts and M. Pervin, Heavy baryons in a quark model, Int. J. Mod. Phys. A 23, 2817 (2008); arXiv:0711.2492 [nucl-th].
  * (8) H. Garcilazo, J. Vijande, and A. Valcarce, Faddeev study of heavy baryon spectroscopy, J. Phys. G 34, 961 (2007).
  * (9) S. Migura, D. Merten, B. Metsch, and H. R. Petry, Charmed baryons in a relativistic quark model, Eur. Phys. J. A 28, 41(2006).
  * (10) A. Valcarce, H. Garcilazo, and J. Vijande, Towards an understanding of heavy baryon spectroscopy, Eur. Phys. J. A 37, 217 (2008).
  * (11) Y. Yamaguchi, S. Ohkoda, A. Hosaka, T. Hyodo, and S. Yasui, Heavy quark symmetry in multihadron systems, Phys. Rev. D 91, 034034 (2015).
  * (12) P. Pérez-Rubio, S. Collins, and G. S. Bali, Charmed baryon spectroscopy and light flavor symmetry from lattice QCD, Phys. Rev. D 92, 034504 (2015).
  * (13) Z. Shah, K. Thakkar, A. K. Rai, and P. C. Vinodkumar, Mass spectra and Regge trajectories of $\Lambda_{c}$, $\Sigma_{c}^{0}$, $\Xi_{c}^{0}$ and $\Omega_{c}^{0}$ baryons, Chin. Phys. C 40, 123102 (2016).
  * (14) T. Yoshida, E. Hiyama, A. Hosaka, M. Oka, and K. Sadato, Spectrum of heavy baryons in the quark model, Phys. Rev. D 92, 114029 (2015).
  * (15) H.-X. Chen, W. Chen, Q. Mao, A. Hosaka, X. Liu, and S. L. Zhu, $P$-wave charmed baryons from QCD sum rules, Phys. Rev. D 91, 054034 (2015).
  * (16) H.-X. Chen, Q. Mao, A. Hosaka, X. Liu, and S. L. Zhu, D-wave charmed and bottomed baryons from QCD sum rules, Phys. Rev. D 94, 114016 (2016).
  * (17) M. Karliner and J. L. Rosner, Very narrow excited $\Omega_{c}$ baryons, Phys. Rev. D 95, 114012 (2017).
  * (18) M. Padmanath and N. Mathur, Quantum numbers of recently discovered $\Omega_{c}^{0}$ baryons from lattice QCD, Phys. Rev. Lett. 119, 042001 (2017).
  * (19) B. Chen and X. Liu, New $\Omega_{c}^{0}$ baryons discovered by LHCb as the members of $1P$ and $2S$ states, Phys. Rev. D 96, 094015 (2017).
  * (20) W. Wang and R. L. Zhu, Interpretation of the newly observed $\Omega_{c}^{0}$ resonance, Phys. Rev. D 96, 014024 (2017).
  * (21) Z. G. Wang, Analysis of $\Omega_{c}(3000)$, $\Omega_{c}(3050$), $\Omega_{c}(3066$), $\Omega_{c}(3090$) and $\Omega_{c}(3119$) with QCD sum rules, Eur. Phys. J. C 77, 325 (2017).
  * (22) S. S. Agaev, K. Azizi, and H. Sundu, On the nature of the newly discovered $\Omega_{c}^{0}$ states, Eur. Phys. Lett. 118, 61001 (2017).
  * (23) H. X. Chen, Q. Mao, W. Chen, A. Hosaka, X. Liu, and S. L. Zhu, Decay properties of $P$-wave charmed baryons from light-cone QCD sum rules, Phys. Rev. D 95, 094008 (2017).
  * (24) H. Y. Cheng and C. W. Chiang, Quantum numbers of $\Omega_{c}$ states and other charmed baryons, Phys. Rev. D 95, (2017) 094018 arXiv:1704.00396 [hep-ph]
  * (25) K. L. Wang, L. Y. Xiao, X. H. Zhong, and Q. Zhao, Understanding the newly observed $\Omega_{c}$ states through their decays, Phys. Rev. D 95, 116010 (2017).
  * (26) Z. Zhao, D. D. Ye, and A. Zhang, Hadronic decay properties of newly observed $\Omega_{c}$ baryons, Phys. Rev. D 95, 114024 (2017).
  * (27) G. Yang and J. Ping, The structure of pentaquarks $\Omega_{c}^{0}$ in the chiral quark model, Phys. Rev. D 97, 034023 (2018).
  * (28) H. Huang, J. Ping, and F. Wang, Investigating the excited $\Omega_{c}^{0}$ states through $\Xi_{c}K$ and $\Xi_{c}^{\prime}K$ decay channels, Phys. Rev. D 97, 034027 (2018).
  * (29) C. S. An and H. Chen, Observed $\Omega_{c}^{0}$ resonances as pentaquark states, Phys. Rev. D 96, 034012 (2017).
  * (30) Praszałowicz, Possibility of the existence of charmed exotica, Phys. Rev. D 96, 014009 (2017); Erratum, Phys. Rev. D 96, 039902(E)(2017).
  * (31) H. C. Kim, M. V. Polyakov, M. Praszalowicz, and G. S. Yang, Strong decays of exotic and non-exotic heavy baryons in the chiral quark-soliton model, Phys. Rev. D 96, 094021 (2017).
  * (32) B. Chen, K.W. Wei, X. Liu, and T. Matsuki, Low-lying charmed and charmed-strange baryon states, Eur. Phys. J. C77, 154 (2017).
  * (33) M. Karliner and J. L. Rosner, Interpretation of excited $\Omega_{b}$ signals, arXiv:2005.12424 [hep-ph]
  * (34) A. V. Manohar and M. B. Wise, _Heavy quark physics_ , (Cambridge Univ. Press, UK, 2000).
  * (35) M. Karliner and J. L. Rosner, Prospects for observing the lowest-lying odd-parity $\Sigma_{c}$ and $\Sigma_{b}$ baryons, Phys. Rev. D92, 074026 (2015);arXiv:1506.01702 [hep-ph].
  * (36) F. J. Ynduráin, _The theory of quark and gluon interactions_ , (Springer-Verlag, NY,1983)
  * (37) Duojie Jia, Wen-Nian Liu and A. Hosaka, Regge behaviors in orbitally excited spectroscopy of charmed and bottom baryons, Phys.Rev. D101, 034016(2020); arXiv:1907.04958[hep-ph]
  * (38) L. D. Landau and E. M. Lifshitz, _Quantum Mechanics(Nonrelativistic Theory)_ , 3rd ed. (Pergamon Press, Oxford,1977).
  * (39) A. Ali, I. Ahmed, M. J. Aslam, A. Ya. Parkhomenko, A. Rehman, Mass spectrum of the hidden-charm pentaquarks in the compact diquark model, JHEP10, 256 (2019); arXiv:1907.06507[hep-ph]
  * (40) L. Y. Xiao, K. L. Wang, M. S. Liu and X. H. Zhong, Possible interpretation of the newly observed $\Omega_{b}$ states, Eur. Phys. J. C 80, 279 (2020) arXiv:2001.05110 [hep-ph]
  * (41) S. S. Agaev, K. Azizi, and H. Sundu, Interpretation of the new $\Omega_{c}^{0}$ states via their mass and width, Eur. Phys. J. C77,395 (2017)
  * (42) Duojie Jia and W-C Dong, Regge-like spectra of excited singly heavy mesons, Eur. Phys. J. Plus 134 (2019)123;arXiv:1811.04214[hep-ph]
  * (43) Duojie Jia, W-C. Dong and Atsushi Hosaka, Regge-like spectra of singly heavy hadrons, in Proc. 8th Int. Conf. Quarks and Nuclear Physics (QNP2018), JPS Conf. Proc. 26, 022021 (2019).
  * (44) B. Chen, K. W. Wei and A. Zhang, Assignments of Q and Q baryons in the heavy quark–light diquark picture. Eur. Phys. J. A 51, 82 (2015).
  * (45) H. Mutuk, A Study of Excited $\Omega_{b}$ States in Hypercentral Constituent Quark Model via Artificial Neural Network, Eur. Phys. J. A 56, 146 (2020), arXiv:2002.03695 [hep-ph].
  * (46) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * (47) C. Semay, F. Buisseret and F. Stancu, Charm and bottom baryon masses in the combined $1/N_{c}$ and $1/m_{Q}$ expansion versus the quark model, Phys. Rev. D78, 076003 (2008)
  * (48) I.C. Cloët, C. D. Roberts, Prog. Part. & Nucl. Phys. 77,1(2014)
  * (49) D.Ki Hong, Y.J Sohn, I. Zahed, Phys. Lett. B596, 191(2004)
  * (50) R. N. Cahn and J. D. Jackson, Phys. Rev. D68, 037502 (2003).
  * (51) S. Capstick’ and N. Isgur, Baryons in a relativized quark model with chromodynamics, Phys. Rev. D34, 2809 (1986).
  * (52) C. Semay, F. Buisseret and F. Stancu, Charm and bottom baryon masses in the combined $1/N_{c}$ and $1/m_{Q}$ expansion versus the quark model, Phys. Rev. D78, 076003 (2008)
  * (53) D. LaCourse and M. G. Olsson, The string potential model.1. spinless quarks, Phys. Rev. D39, 2751 (1989).
  * (54) Yu.S. Kalashnikova, A.V. Nefediev, Phys. Lett. B492, 91–97(2000)