# Low-lying mass spectra of excited doubly heavy baryons: Regge relation and
mass scaling

YongXin Song1 Duojie Jia1,2 Corresponding author<EMAIL_ADDRESS>WenXuan
Zhang1 Atsushi Hosaka3 1Institute of Theoretical Physics, College of Physics
and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
2Lanzhou Center for Theoretical Physics, Lanzhou University,
Lanzhou,730000,China  3Research Center for Nuclear Physics, Osaka University,
10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

###### Abstract

In picture of heavy-diquark-light-quark endowed with heavy-pair binding, we
explore the low-lying excited baryons $\Xi_{QQ^{\prime}}$ and
$\Omega_{QQ^{\prime}}$ containing two heavy quarks($QQ^{\prime}=cc,bb,bc$) by
applying relations of Regge trajectory and mass scaling to excitations of
light quarks and heavy diquarks, respectively. Two Regge relations, one linear
and the other nonlinear, are constructed based on semi-classical analysis of
massive QCD string so that they contain the short-distance correction due to
heavy-pair interaction when heavy diquark excited internally. Evaluating the
short-distance corrections via the binding energy of the pairs estimated by
that of the heavy-anti-heavy pairs in the observed doubly heavy mesons, we
compute effective masses of heavy diquarks in ground state, 2S- and 1P- waves
and thereby predict low-lying mass spectra of the doubly-heavy baryons up to
the 2S- and 1P- waves of the light quarks and heavy diquarks. The results
indicate that the mass level spacings due to heavy diquark excitations are
generally narrower in contrast with that due to the light quark excitations.

###### pacs:

12.39Jh,12.40.Yx,12.40.Nn, Regge trajectory, Phenomenological Models, QCD
Phenomenology, Heavy baryons

††preprint: HEP/446421

##  I Introduction

Being systems analogous to hydrogen-like atoms with notable level splitting,
by which QED interaction is tested in minute detail, the doubly heavy(DH)
baryon provides a unique opportunity to probe the fundamental theory of the
strong interaction, quantum chromodynamics(QCD) and models inspired by
itEFG:D02 ; KR:D14 . It is expected that the excited states of DH baryons can
form a set of nontrivial levels with various mass splitting and confront the
QCD interaction with the experiments in a straightforward way.

In 2017, the LHCb Collaboration at CERN discovered the doubly charmed baryon
$\Xi_{cc}^{++}$ in the $\Lambda_{c}^{+}K^{-}\pi^{{+}}\pi^{+}$ mass
spectrumAAIJ:17BC  and reconfirms it in the decay channel of
$\Xi_{c}^{+}\pi^{+}$AAIJ:18BA , with the measured mass $3621.55\pm 0.23\pm
0.30$ MeVAAIJ:20J . Before the LHCb observation, a $\Xi_{cc}^{+}$ candidate
has been observed by the SELEX Collaboration at FermilabSELEX4 ; SELEX5 ,
which gives a mass about $100$ MeV lighter than $3621.55$ MeV and has not been
confirmed by other experiments. Further, the later observation seems to
contradict with isospin symmetry of the strong interaction, by which the
$\Xi_{cc}^{+}$ ground state should be close in mass to the $\Xi_{cc}^{++}$
ground stateKR17 .

The observation of the doubly charmed baryon $\Xi_{cc}^{++}$ sets up, for the
first time, a strength scale of interaction between two heavy quarks and is of
help to understand strong interaction among heavy quarksKR:D14 ; SotoCast:D21
. For instance, the mass of the $\Xi_{cc}^{++}$ helps to “calibrate” the
binding energy between a pair of heavy quark governed by short-distance
interaction and to answer if a DH tetraquark is stable against strongly
decaying into two mesonsKR:pr2017 . For more examples, see Refs.
Eichten:pr2017 ; Luo:2017eub ; Mehen:D17 ; M3H:D21 ; M3H:D22  and the
references therein. Very recent observation of the doubly charmed
tetraquarkTccLHCb:2021  is timely to examine the assumed interquark($QQ$)
interaction in hadrons and invite conversely systematic study of the doubly
heavy baryons. Untill now, features(masses, spin-parity and decay widths) of
the $\Xi_{cc}^{++}$ and other DH baryons in their excited states remain to be
explored.

In this work, we apply Regge relations for heavy hadrons and mass scaling
among heavy hadrons(mesons and baryons) combinedly to explore excited doubly
heavy baryons $\Xi_{QQ^{\prime}}$ and $\Omega_{QQ^{\prime}}$
($QQ^{\prime}=cc,bb,bc$) with the help of the observed data of the doubly
heavy mesons, the $\Xi_{cc}^{++}$ as well as the commonly-used ground-state
masses of the DH baryons compatible with the measured data. We construct a
linear Regge relation for DH baryons, which has been applied successfully to
singly heavy hadrons in the low-lying states, and extend it to nonlinear
relation of the DH baryons with excited heavy diquark where a term of heavy-
pair binding due to the short-distance interquark interaction in pair is
incorporated. Evaluating the short-distance corrections via the binding energy
of the pairs extracted from that of the heavy-anti-heavy pairs in the observed
doubly-heavy mesons, we compute effective masses of the excited heavy diquarks
and predict low-lying mass spectra of the DH baryons, with comparison with
other calculations.

This paper is organized as follows. In Sect. II, we apply linear Regge
relation and a power law of mass scaling to estimate the spin-
independent(mean) masses of the excited DH baryons $\Xi_{QQ^{\prime}}$ and
$\Omega_{QQ^{\prime}}$ in 1p-waves in heavy-diquark-light-quark picture. In
Sect. III, we formulated spin-dependent mass splitting of the DH baryons with
the ground(1S-wave) diquark in the scheme of the $jj$ coupling. In Sect. IV,
the 1p wave masses of the DH baryons with S-wave diquark are predicted via the
relation of mass scaling. In Sect. V, we extend the Regge relation to compute
the 2S wave masses of the DH baryons with S-wave diquark. In Sect. VI, we
improve the linear Regge relation to the DH baryons with diquark excited, and
compute the mean mass shifts and the effective diquark masses due to 2S and 1P
diquark excitation. In Sect. VII, the baryons mass splittings are computed for
the DH baryons with diquark in 2S and 1P waves. We end with summary and
discussions Sect. VIII.

##  II Spin-independent mass with S-wave diquark

We use heavy-diquark-quark picture for the DH baryons $qQQ^{\prime}$($q=u,d$
and $s$) with heavy quarks $QQ^{\prime}=cc,bb$ or $bc$, in which the heavy
pair $QQ^{\prime}$ can be scalar(spin zero) diquark denoted by $[QQ^{\prime}]$
or axial-vector(spin-$1$) diquark denoted by $\\{QQ^{\prime}\\}$. The charm
diquark $cc$ and the bottom diquark $bb$ can only form axial-vector diquark,
$\\{cc\\}$ or $\\{bb\\}$, while the bottom-charm diquark $bc$ can form both,
$\\{bc\\}$ or $[bc]$. We use the notations $N_{D}L_{D}nl$ to label the quantum
numbers of the DH baryons, with the value of principal quantum number($N_{D}$)
of diquark, its orbital momentum($L_{D}$) denoted by a capital letter and the
principal quantum number($n$) for the excitations of light quark and its
orbital momentum($l$) by a lowercase letter.

To begin with, we first consider the DH baryons with $QQ^{\prime}$ in the
internal S-wave. The mass of a DH baryon $qQQ^{\prime}$($Q,Q^{\prime}=c,b$)
consists of the sum of two parts: $M=\bar{M}+\Delta M$, where $\bar{M}$ is the
spin-independent mass and $\Delta M$ is the mass splitting due to the spin-
dependent interaction. As the diquark $QQ^{\prime}$ is heavy, compared to the
light quark $q$, the heavy-light limit applies to the DH baryons
$qQQ^{\prime}$.

By analogy with Ref. JiaLH:19 , one can derive a linear Regge relation from
the QCD string model for the DH baryons with S-wave diquark(Appendix A) by
viewing the baryon to be a $q-QQ^{\prime}$ system of a massive QCD string with
diquark $QQ^{\prime}$ at one end and $q$ at the other. Denoting the mass of a
DH baryon with orbital angular momentum $l$ of light quark by $\bar{M}_{l}$,
the Regge relation takes form JiaLH:19 (Appendix A)

$(\bar{M}_{l}-M_{QQ^{\prime}})^{2}=\pi
al+\left[m_{q}+M_{QQ^{\prime}}-\frac{m_{\text{bare}QQ^{\prime}}^{2}}{M_{QQ^{\prime}}}\right]^{2},$
(1)

where $M_{QQ^{\prime}}$ is the effective mass of the S-wave heavy diquark,
$m_{q}$ the effective mass of the light quark $q$, $a$ stands for the tension
of the QCD string connecting the light quark $q$ to $QQ^{\prime}$. Here,
$m_{\text{bare}QQ^{\prime}}$ is the bare mass of $QQ^{\prime}$, given
approximately by sum of the bare masses of each quark: $m_{\text{bare
}QQ^{\prime}}$$=m_{\text{bare}Q}+m_{\text{bare}Q^{\prime}}$. Numerically, one
has

$m_{\text{bare }cc}=2.55~{}\text{GeV},m_{\text{bare
}bb}=8.36\text{GeV},m_{\text{bare }bc}=5.455\text{GeV}.$ (2)

We set the values of $m_{q}$ in Eq. (1) to be that in Table I, which was
previously evaluated in Ref. JiaLH:19  based on a combined Regge relation
matching of the available mass(experimental) data of the excited singly heavy
baryons and mesons.

Table 1: The effective masses (in GeV) of the quark masses and the string tensions $a$(in GeV2) of the $D$-$D_{s}$ and $B$-$B_{s}$ mesons in this work.  Parameters | $M_{c}$ | $M_{b}$ | $m_{n}$ | $m_{s}$ | $a({c\bar{n})}$ | $a({b\bar{n})}$ | $a({c\bar{s})}$ | $a({b\bar{s}})$
---|---|---|---|---|---|---|---|---
Input | $1.44$ | $4.48$ | $0.23$ | $0.328$ | $0.223$ | $0.275$ | $0.249$ | $0.313$

To determine $M_{QQ^{\prime}}$ in Eq. (1), we apply Eq. (1) to the ground
state($n=0=l$) to findJiaLH:19

$\displaystyle\bar{M}(1S1s)$
$\displaystyle=M_{QQ^{\prime}}+m_{q}+\frac{k_{QQ^{\prime}}^{2}}{M_{QQ^{\prime}}},$
(3) $\displaystyle k_{QQ^{\prime}}$ $\displaystyle\equiv
M_{QQ^{\prime}}v_{QQ^{\prime}}=M_{QQ^{\prime}}\left(1-\frac{m_{\text{bare}QQ^{\prime}}^{2}}{M_{QQ^{\prime}}^{2}}\right)^{1/2},$
(4)

which agrees with the mass formula predicted by heavy quark
symmetryManohar:D07 . As the experimental data are still lacking except for
the doubly charmed baryon $\Xi_{cc}^{++}$, we adopt the masses(Table II) of DH
baryons computed by Ref. EFG:D02 , which successfully predicts the ground-
state mass $M(\Xi_{cc},1/2^{+})=3620$ MeV, close to the measured data
$3621.55$ MeV for the doubly charmed baryon $\Xi_{cc}^{++}$AAIJ:17BC ;
AAIJ:20J .

Table 2: Masses(GeV) of the doubly heavy baryons in ground state. State | $J^{P}$ | Baryons | Content | Mass | $\bar{M}_{l=0}$
---|---|---|---|---|---
$1^{2}S_{1/2}$ | $\frac{1}{2}^{+}$ | $\Xi_{cc}$ | $n\\{cc\\}$ | $3.620$ | 3.6913
$1^{4}S_{1/2}$ | $\frac{3}{2}^{+}$ | $\Xi_{cc}^{\ast}$ | $n\\{cc\\}$ | $3.727$
$1^{2}S_{1/2}$ | $\frac{1}{2}^{+}$ | $\Xi_{bb}$ | $n\\{bb\\}$ | $10.202$ | 10.2253
$1^{4}S_{1/2}$ | $\frac{3}{2}^{+}$ | $\Xi_{bb}^{\ast}$ | $n\\{bb\\}$ | $10.237$
$1^{2}S_{1/2}$ | $\frac{1}{2}^{+}$ | $\Xi_{bc}$ | $n\\{bc\\}$ | $6.933$ | 6.9643
$1^{4}S_{1/2}$ | $\frac{3}{2}^{+}$ | $\Xi_{bc}^{\ast}$ | $n\\{bc\\}$ | $6.980$
$1^{2}S_{1/2}$ | $\frac{1}{2}^{+}$ | $\Xi_{bc}^{\prime}$ | $n[bc]$ | $6.963$ | 6.963

With the mass parameters $m_{q=n}$ ($n=u/d$) in Table I and $\bar{M}(1S1s)$ in
Table II, one can solve Eq. (3) to extract the heavy-diquark mass, with the
results

$\displaystyle M_{cc}$ $\displaystyle=2865.5\ \text{MeV, }M_{bb}=8916.7\
\text{MeV, }$ (5) $\displaystyle M_{[bc]}$ $\displaystyle=5891.8\ \text{MeV,
}M_{\\{bc\\}}=5892.3\ \text{MeV,}$ (6)

where the subscript $cc$($bb$) stands for the vector diquark
$\\{cc\\}$($\\{bb\\}$) for short. The mean mass of the diquark $bc$ is
$\bar{M}_{bc}=5892.3\ $MeV.

In Refs. JiaLH:19 ; DPP:Epjc21 , the singly heavy baryon families(the
$\Lambda_{c}/\Sigma_{c}$, the $\Xi_{c}/\Xi_{c}^{\prime}$, the
$\Lambda_{b}/\Sigma_{b}$ and the $\Xi_{b}/\Xi_{b}^{\prime}$) were examined in
the quark-diquark picture that the Regge slope $\pi a$ for them is the same
approximately for identical flavor constituents within one family, independent
of the light-diquark spin($=0$ or $1$). For two heavy baryons $Qqq^{\prime}$
containing the identical (light) flavor $qq^{\prime}$, the string tension $a$
is found to relies crucially on the heavy quark mass $M_{Q}$($Q=c$ or $b$),
but has little dependence on the diquark spin of $qq^{\prime}$. In the case of
the DH baryon $qQQ^{\prime}$ with given flavor $q$ one expects that the
tension $a$ depends dominantly on the flavor combinations $QQ^{\prime}$, that
is, on the diquark mass $M_{QQ^{\prime}}$ explicitly, in the heavy quark
limit.

Let us consider first the nonstrange DH baryons
$\Xi_{QQ^{\prime}}=nQQ^{\prime}$ and scale its tension from the
$D/B(=Q\bar{n})$ mesons which share similar structure consisting of one heavy
building block(diquark $QQ^{\prime}$ or $Q$) and one light constituent with
identical flavor $n$(or $\bar{n}$) and color structure ($\bar{3}_{c}\otimes
3_{c}$ or $1_{c}$). We assume, for simplicity, a power-law of the mass scaling
between the string tensions of the hadrons $D/B$ and $\Xi_{QQ^{\prime}}$:

$\frac{a_{D}}{a_{B}}=\left(\frac{M_{c}}{M_{b}}\right)^{P},$ (7)
$\frac{a_{D}}{a_{\Xi_{QQ^{\prime}}}}=\left(\frac{M_{c}}{M_{QQ^{\prime}}}\right)^{P}.$
(8)

Here, the values of the parameters ($a_{D}=a_{c\bar{n}}$,
$a_{B}=a_{b\bar{n}}$, $M_{c}$ and $M_{b}$) are previously evaluated in Ref
JiaLH:19 , as listed in Table I.

Putting the tensions and heavy-quark masses in Table I into Eq. (7) gives
$P=0.185$ and thereby predicts, by Eq. (8)

$\displaystyle a_{\Xi_{cc}}$ $\displaystyle=0.2532\text{ GeV}^{2}\text{,
}a_{\Xi_{bb}}=0.3123\text{ GeV}^{2},$ (9) $\displaystyle a_{\Xi_{bc}}$
$\displaystyle=0.2893\text{ GeV}^{2}=a_{\Xi_{bc}^{\prime}}\text{,}$ (10)

combining with Eqs. (5) and (6). Given the tensions in Eqs. (9) and Eq. (10),
and the diquark masses in Eqs. (5), (6) and the light quark mass in Table I,
one can use Eq. (1) to obtain the spin-averaged masses of the baryon system
$\Xi_{QQ^{\prime}}$ in p-wave($l=1$). The results are

$\left\\{\begin{array}[c]{l}\bar{M}\left(\Xi_{cc},1p\right)=4.0813\mathrm{\
}\text{GeV, }\bar{M}\left(\Xi_{bb},1p\right)=10.5577\mathrm{\ }\text{GeV,}\\\
\bar{M}\left(\Xi_{bc}^{\prime},1p\right)=7.3258\mathrm{\ }\text{GeV,
}\bar{M}\left(\Xi_{bc},1p\right)=7.3269\mathrm{\
}\text{GeV,}\end{array}\right\\}$ (11)

where $\Xi_{bc}^{\prime}=n[bc]$ and $\Xi_{bc}=n\\{bc\\}$ stand for the bottom-
charmed baryons with diquark spin $0$ and $1$, respectively.

The same procedure applies to the strange DH baryons, the
$\Omega_{QQ^{\prime}}=sQQ^{\prime}$ and the associated
$D_{s}$/$B_{s}$($=c\bar{s}/b\bar{s}$) mesons, for which the mass scaling,
corresponding to Eqs. (7) and (8), has the same form

$\frac{a_{D_{s}}}{a_{B_{s}}}=\left(\frac{M_{c}}{M_{b}}\right)^{P}$ (12)
$\frac{a_{D_{s}}}{a_{\Omega_{QQ^{\prime}}}}=\left(\frac{M_{c}}{M_{QQ^{\prime}}}\right)^{P}.$
(13)

Putting the tensions for the strange heavy mesons and the heavy-quark masses
in Table I to Eq. (12) gives $P=0.202$. One can use this $P$ value and Eq.
(13) and the heavy-diquark masses in Eqs. (5) and (6) to predict

$\displaystyle a_{\Omega_{cc}}$ $\displaystyle=0.2860\text{ GeV}^{2}\text{,
}a_{\Omega_{bb}}=0.3596\text{ GeV}^{2}\text{,}$ (14) $\displaystyle
a_{\Omega_{bc}}$ $\displaystyle=0.3308\
\text{GeV}^{2}=a_{\Omega_{bc}^{\ast}}\text{,}$ (15)

where $\Omega_{bc}$ and $\Omega_{bc}^{\ast}$ stand for the DH $\Omega$ baryons
with $bc$-diquark spin $0$ and $1$, respectively.

Given the tensions(including $a_{\Xi_{QQ^{\prime}}}=a_{nQQ^{\prime}}$) in Eqs.
(14) and (15), and the heavy-diquark masses in Eqs. (5) and (6) as well as the
strange quark mass $m_{s}=0.328$ GeV in Table I, one can use Eq. (1) to find
the mean(spin-averaged) masses of the baryon system $\Omega_{QQ^{\prime}}$ in
p-waves($l=1$). The results are

$\left\\{\begin{array}[c]{l}\bar{M}\left(\Omega_{cc},1p\right)=4.1895\mathrm{\
}\text{GeV, }\bar{M}\left(\Omega_{bb},1p\right)=10.6795\mathrm{\
}\text{GeV,}\\\ \bar{M}\left(\Omega_{bc}^{\prime},1p\right)=7.4430\mathrm{\
}\text{GeV, }\bar{M}\left(\Omega_{bc},1p\right)=7.4441\mathrm{\
}\text{GeV,}\end{array}\right\\}$ (16)

where $\Omega_{bc}^{\prime}=s[bc]$ stands for the strange baryons
$\Omega_{bc}$ with scalar diquark $[bc]$ and $\Omega_{bc}=s\\{bc\\}$ for the
$\Omega_{bc}$ with axial vector diquark $\\{bc\\}$.

##  III Spin-dependent mass in $jj$ coupling

In heavy diquark quark picture, two heavy quarks in a $\Xi_{QQ^{\prime}}$ or
$\Omega_{QQ^{\prime}}$ system form a S-wave color anti-
triplet$\left(\overline{\mathbf{3}}_{c}\right)$ diquark $(QQ^{\prime})$,
having spin zero($S_{QQ^{\prime}}=0$) when $QQ^{\prime}=bc$, or spin
one($S_{QQ^{\prime}}=1$) when $QQ^{\prime}=cc,bc$ and $bb$. When
$Q=Q^{\prime}$ the diquark $QQ$ must have spin one due to full antisymmetry
under exchange of two quarks. The spin $S_{QQ^{\prime}}$ can couple with the
spin $S_{q}=1/2$ of the light quark $q$($=u,d$ and $s$) to form a total spin
$S_{tot}=1\pm 1/2=1/2,3/2$ if $S_{QQ^{\prime}}=1$ or $S_{tot}=1/2$ if
$S_{QQ^{\prime}}=0$.

We assume the heavy diquark $QQ^{\prime}$ in DH baryon $qQQ^{\prime}$ to be in
internal ground (S-wave) state and consider relative p-wave excitations of the
light quark $q$ with respect to $QQ^{\prime}$ ($l=1$). Here, the relative
orbital angular momentum $l$ corresponds to the angular excitation of the
$\lambda$-mode in the quark model. In the scheme of $LS$ coupling, coupling
$S_{tot}=1/2$ to $l=1$ gives the states with the total angular momentums
$J=1/2,3/2$, while coupling $S_{tot}=3/2$ to $l=1$ leads to the states with
$J=1/2,3/2$ and $5/2$. Normally, one uses the $LS$ basis
${}^{2S_{tot}+1}P_{J}$
$=\\{^{2}P_{1/2},^{2}P_{3/2},^{4}P_{1/2},^{4}P_{3/2},^{4}P_{5/2}\\}$ to label
these multiplets in P-wave, which correspond to five allowed odd-parity states
with $J=1/2,1/2^{\prime},3/2,3/2^{\prime}$ and $5/2$.

For mass splitting $\Delta M=\langle H^{SD}\rangle$ due to spin interaction
between heavy diquark $QQ^{\prime}$ with spin $\mathbf{S}_{QQ^{\prime}}$ and
light quark $q$ with spin $\mathbf{S}_{q}$, we consider the spin-dependent
Hamiltonian $H^{SD}$ EFG:D11

$\begin{array}[c]{c}H^{SD}=a_{1}\mathbf{l}\cdot\mathbf{S}_{q}+a_{2}\mathbf{l}\cdot\mathbf{S}_{QQ^{\prime}}+bS_{12}+c\mathbf{S}_{q}\cdot\mathbf{S}_{QQ^{\prime}},\\\
S_{12}=3\mathbf{S}_{q}\cdot\hat{\mathbf{r}}\mathbf{S}_{QQ^{\prime}}\cdot\hat{\mathbf{r}}-\mathbf{S}_{q}\cdot\mathbf{S}_{QQ^{\prime}},\end{array}$
(17)

where the first two terms are spin-orbit forces, the third is a tensor force,
and the last describes hyperfine splitting. In the $LS$ basis, the two $J=1/2$
states and the two $J=3/2$ states are unmixed unless $a_{1}=a_{2}$. Otherwise
they are the respective eigenstates of a $2\times 2$ matrices
$\mathcal{M}_{J}$ representing $H^{SD}$ for $J=1/2$ and $3/2$. In the basis
$[^{2}P_{J},^{4}P_{J}]$, they can be given by the matrix KR:D2017 ; Landau
(see Appendix B also)

$\displaystyle\mathcal{M}_{J=1/2}$
$\displaystyle=\left[\begin{array}[c]{ll}\frac{1}{3}\left(a_{1}-4a_{2}\right)&\frac{\sqrt{2}}{3}\left(a_{1}-a_{2}\right)\\\
\frac{\sqrt{2}}{3}\left(a_{1}-a_{2}\right)&-\frac{5}{3}\left(a_{2}+\frac{1}{2}a_{1}\right)\end{array}\right]$
(20) $\displaystyle+b\left[\begin{array}[c]{cc}0&\frac{\sqrt{2}}{2}\\\
\frac{\sqrt{2}}{2}&-1\end{array}\right]+c\left[\begin{array}[c]{cc}-1&0\\\
0&\frac{1}{2}\end{array}\right],$ (25)

in the $J=1/2$ subspace,

$\displaystyle\mathcal{M}_{J=3/2}$
$\displaystyle=\left[\begin{array}[c]{cc}-\frac{2}{3}\left(\frac{1}{4}a_{1}-a_{2}\right)&\frac{\sqrt{5}}{3}\left(a_{1}-a_{2}\right)\\\
\frac{\sqrt{5}}{3}\left(a_{1}-a_{2}\right)&-\frac{2}{3}\left(\frac{1}{2}a_{1}+a_{2}\right)\end{array}\right]$
(28) $\displaystyle+b\left[\begin{array}[c]{cc}0&-\frac{\sqrt{5}}{10}\\\
-\frac{\sqrt{5}}{10}&\frac{4}{5}\end{array}\right]+c\left[\begin{array}[c]{cc}-1&0\\\
0&\frac{1}{2}\end{array}\right],$ (33)

in the $J=3/2$ subspace, and by

$\mathcal{M}_{J=5/2}=\frac{1}{2}a_{1}+a_{2}-\frac{b}{5}+\frac{c}{2}.$ (34)

for the $J=5/2$. One can verify that the spin-weight sum of these matrixes
over $J=1/2,3/2$ and $5/2$ is zero:

$\sum_{J}(2J+1)\mathcal{M}_{J}=0,$ (35)

as it should be for the spin interaction $H^{SD}$.

As diquark mass $M_{QQ^{\prime}}$ is much larger than the light quark mass,
all terms except for the first ($=$ $a_{1}\mathbf{l}\cdot\mathbf{S}_{q}$) in
Eq. (17) behave as $1/M_{QQ^{\prime}}$ and are then suppressed comparing with
the first($a_{1}$-) term. Due to heavy quark spin
symmetry($\mathbf{S}_{QQ^{\prime}}$ conserved), the total angular momentum of
the light quark
$\mathbf{j}=\mathbf{l}+\mathbf{S}_{q}\mathbf{=J-S}_{QQ^{\prime}}$ is conserved
and forms an set of the conserved operators {$\mathbf{J},\mathbf{j}$},
together with the total angular momentum $\mathbf{J}$ of the DH hadrons. We
use then the $jj$ coupling scheme to label the spin multiplets of the DH
baryons, denoted by the basis $|J,$ $j\rangle$(Appendix B), in which the
diquark $QQ^{\prime}$ is infinitely heavy and $\mathbf{l}\cdot\mathbf{S}_{q}$
becomes diagonal. As such, the formula for mass splitting $\Delta M$ can be
obtained by diagonalizing $\mathbf{l}\cdot\mathbf{S}_{q}$ and treating other
interactions in Eq. (17) proportional to $a_{2}$, $b$ and $c$ perturbatively.

The eigenvalues (two diagonal elements) of $\mathbf{l}\cdot\mathbf{S}_{q}$ can
be obtained to be

$\displaystyle\left\langle\mathbf{l}\cdot\mathbf{S}_{q}\right\rangle$
$\displaystyle=\frac{1}{2}\left[j(j+1)-l(l+1)-S_{q}\left(S_{q}+1\right)\right]$
$\displaystyle=-1(j=1/2)\text{ or }1/2(j=3/2).$ (36)

Computation of the diagonal mass splitting $\Delta M(J,j)=\left\langle
J,j\left|H^{SD}\right|J,j\right\rangle$ in the $jj$ coupling includes the
three steps: Firstly, one solves the eigenfunctions(the $LS$ bases)
$\left|S_{QQ^{\prime}3},S_{q3},l_{3}\right\rangle$ of the
$\mathbf{l}\cdot\mathbf{S}_{q}$ for its eigenvalues $j=1/2$ and $3/2$ in the
$LS$ coupling; Secondly, one transforms the obtained $LS$ bases into
$\left|J,j\right\rangle$ in the $jj$ coupling(Appendix B); Finally, one
evaluates the diagonal elements of the spin-interaction in Eq. (17) in the new
bases $\left|J,j\right\rangle$. The results for $\Delta M(J,j)$ are listed in
Table III.

Table 3: The matrix elements of the mass splitting operators in the p-wave heavy doubly baryons states in the $jj$ coupling. (J-j) | $\left\langle\mathbf{l}\cdot\mathbf{S}_{QQ^{\prime}}\right\rangle$ | $\left\langle\mathbf{S}_{12}\right\rangle$ | $\left\langle\mathbf{S}_{q}\cdot\mathbf{S}_{QQ^{\prime}}\right\rangle$
---|---|---|---
$(1/2,1/2)$ | $-4/3$ | $-4/3$ | $1/3$
$(1/2,3/2)$ | $-5/3$ | $1/3$ | $-5/6$
$(3/2,1/2)$ | $2/3$ | $2/3$ | $-5/6$
$(3/2,3/2)$ | $-2/3$ | $2/15$ | $-1/3$
$(5/2,3/2)$ | $1$ | $-1/5$ | $1/2$

Given Table III, one can use the lowest perturbation theory to find the mass
splitting $\Delta M(J,j)$ of the DH baryons in p-wave,

$\Delta M(1/2,1/2)=-a_{1}-\frac{4}{3}a_{2}-\frac{4}{3}b+\frac{1}{3}c,$ (37)
$\Delta
M(1/2,3/2)=\frac{1}{2}a_{1}-\frac{5}{3}a_{2}+\frac{1}{3}b-\frac{5}{6}c,$ (38)
$\Delta M(3/2,1/2)=-a_{1}+\frac{2}{3}a_{2}+\frac{2}{3}b-\frac{1}{6}c,$ (39)
$\Delta
M(3/2,3/2)=\frac{1}{2}a_{1}-\frac{2}{3}a_{2}+\frac{2}{15}b-\frac{1}{3}c,$ (40)
$\Delta M(5/2,3/2)=\frac{1}{2}a_{1}+a_{2}-\frac{1}{5}b+\frac{1}{2}c,$ (41)

which express the DH baryon mass splitting in terms of four parameters
($a_{1},a_{2},b,c$). The mass formula for the $1S1p$ states of DH baryons can
then be given by $M(J,j)=\bar{M}(1S1p)+\Delta M(J,j)$, with $\bar{M}(1S1p)$
the spin-independent masses given in Eqs. (11) and (16) in section II.

##  IV Masses of DH baryon with S-wave diquark

To evaluate spin coupling parameters ($a_{1},a_{2},b,c$) in Eq. (37) through
Eq. (41), we utilize the relations of mass scalingKR:D15 ; JiaLH:19 , which
apply successfully to the singly-heavy hadrons associated with the singly-
heavy mesons $D_{s}$ and singly-heavy baryons. For this, we list the matched
values of the spin couplings for the $D_{s}$ in Refs. KR:D15 ; JiaLH:19  and
that of the singly heavy baryons (the $\Sigma_{Q}$, the $\Xi_{Q}^{\prime}$ and
the $\Omega_{Q}$, with $Q=c,b$) in Refs. DPP:Epjc21  in Table IV.

Table 4: Spin-coupling Parameters(in MeV) of singly heavy meson $D_{s}$ and the singly heavy baryons $\Sigma_{Q}$, $\Xi_{Q}^{\prime}$ and $\Omega_{Q}$($Q=c,b$).  Hadrons | $a_{1}$ | $a_{2}$ | $b$
---|---|---|---
$D_{s}$ | $89.36$ | $40.7$ | $65.6$
$\Sigma_{c}$ | $39.96(39.34)$ | $21.75(26.82)$ | $20.70(20.05)$
$\Sigma_{b}$ | $12.99(12.65)$ | $6.42(8.62)$ | $6.45(6.45)$
$\Xi_{c}^{\prime}$ | $32.89(33.62)$ | $20.16(25.35)$ | $16.50(17.93)$
$\Xi_{b}^{\prime}$ | $9.37(10.81)$ | $6.29(8.15)$ | $5.76(5.76)$
$\Omega_{c}$ | $26.96(29.59)$ | $25.76(24.11)$ | $13.51(16.31)$
$\Omega_{b}$ | $8.98(9.51)$ | $4.11(7.75)$ | $7.61(5.24)$

Between heavy baryons $Qqq^{\prime}$ and heavy mesons(we choose the
$D_{s}=c\bar{s}$ typically, as in Refs. KR:D15 ; JiaLH:19 ), the mass scaling
relation for the spin couplings is given in Ref JiaLH:19 (see Eqs. (22-24)),
and can be expressed as

$\mathbf{l}_{qq^{\prime}}\cdot\mathbf{S}_{qq^{\prime}}:a_{1}\left[Qqq^{\prime}\right]=\frac{M_{c}}{M_{Q}}\cdot\frac{m_{s}}{m_{qq^{\prime}}}\cdot
a_{1}\left(D_{s}\right),$ (42)
$\mathbf{l}_{qq^{\prime}}\cdot\mathbf{S}_{Q}:a_{2}\left[Qqq^{\prime}\right]=\frac{M_{c}}{M_{Q}}\cdot\frac{1}{1+m_{qq^{\prime}}/M_{c}}\cdot
a_{2}\left(D_{s}\right),$ (43)
$S_{12}:b\left[Qqq^{\prime}\right]=\frac{M_{c}}{M_{Q}}\cdot\frac{1}{1+m_{qq^{\prime}}/m_{s}}\cdot
b\left(D_{s}\right),$ (44)

where $\mathbf{l}_{qq^{\prime}}$ and $\mathbf{S}_{qq^{\prime}}$ denote the
orbital angular momentum of light diquark $qq^{\prime}$ relative to the heavy
quark $Q$ and the spin of the light diquark. $m_{qq^{\prime}}$ is the mass of
the diquark $qq^{\prime}$ and $M_{Q}$ is heavy-quark mass in the singly heavy
hadrons, and the factor of the mass ratio $M_{c}/M_{Q}$ incorporates to
account for the scaling $\sim 1/M_{Q}$. In Eq. (44) for the tensor coupling
$b$, the extra recoil factor $1/\left(1+m_{qq^{\prime}}/m_{s}\right)$ is used
to take into account the correction due to the comparable heaviness between
the diquark $qq^{\prime}$ and the strange quark. Note that a similar (recoil
here) factor, $1/\left(1+m_{qq^{\prime}}/M_{c}\right)$, entering Eq. (43), has
been confirmed for the charmed and bottom baryons in P-wave and D-waveJiaLH:19
. For instance, Eq. (43) in Ref. JiaLH:19 (Eq. (60)) well reproduces the
measured masses $6146.2$ MeV and $6152.5$ MeV of the $\Lambda_{b}(6146)$ and
the $\Lambda_{b}(6152)$ observed by LHCbLamdab19 , respectively. Similar
verifications were demonstrated in Ref. DPP:Epjc21  for the excited
$\Omega_{c}$ discovered by LHCbAaij:L2017 , for which the $ss$-diquark is
comparable with the charm quark in heaviness.

Table 5: Effective masses (in GeV) of light diquarks in the singly heavy baryons shown. Data are from the Ref. JiaLH:19  and Ref. DPP:Epjc21 .  Hadrons | $\Sigma_{c}$ | $\Sigma_{b}$ | $\Xi_{c}^{\prime}$ | $\Xi_{b}^{\prime}$ | $\Omega_{c}$ | $\Omega_{b}$
---|---|---|---|---|---|---
$m_{qq}$ | $0.745$ | $0.745$ | $0.872$ | $0.872$ | $0.991$ | $0.991$

Putting the parameters of the quarks in Tables I and the light diquarks in
Table V to Eqs. (42)-(44), one can estimate the parameters $a_{1},a_{2}$ and
$b$ for the heavy baryons $\Sigma_{Q}$, $\Xi_{Q}^{{}^{\prime}}$ and
$\Omega_{c}$. We list the results in Table IV marked with parentheses adjacent
to the values(data without parentheses) of the three parameters that matched
in Ref. DPP:Epjc21  to the measured data for comparison. Evidently, the
mismatch shown is small: $\Delta a_{1}\leq 2.63$ MeV, $\Delta a_{2}\leq 5.19$
MeV and $\Delta b\leq 2.80$ MeV, and agreement between the estimates via two
scalings is notable.

In the case of the DH baryons with S-wave diquark, whose quark structure of
the $q-QQ^{\prime}$ resembles heavy meson $D_{s}$ with quark structure
$s-\bar{c}$, one expects that similar relations of mass scaling apply as well.
Replacing the heavy-quark mass $M_{Q}$ in Eqs. (42)-(44) by the diquark mass
$M_{QQ^{\prime}}$ and the light diquark mass $m_{qq^{\prime}}$ by the quark
mass $m_{q}$, one obtains

$a_{1}[QQ^{\prime}q]=\frac{M_{c}}{M_{QQ^{\prime}}}\cdot\frac{m_{s}}{m_{q}}\cdot
a_{1}\left(D_{s}\right),$ (45)
$a_{2}[QQ^{\prime}q]=\frac{M_{c}}{M_{QQ^{\prime}}}\cdot\frac{1}{1+m_{q}/M_{c}}\cdot
a_{2}\left(D_{s}\right),$ (46)
$b[QQ^{\prime}q]=\frac{M_{c}}{M_{QQ^{\prime}}}\cdot\frac{1}{1+m_{q}/m_{s}}\cdot
b\left(D_{s}\right).$ (47)

Since the hyperfine parameter $c\propto 1/(M_{QQ^{\prime}}m_{q})$ also scales
as hadron wavefunction near the origin, it should be small in p-wave, we
assume it to obey a relation of mass scaling relative to the singly charmed
baryon $\Omega_{c}=css$ as below:

$c[QQ^{\prime}q]=\left(\frac{M_{c}}{M_{QQ^{\prime}}}\right)\left(\frac{m_{ss}}{m_{q}}\right)c\left(\Omega_{c}\right).$
(48)

For the excited baryons $\Omega_{c}$, there exists five states discovered by
LHCbAaij:L2017 , with the masses $3000.4$ MeV, $3050.2$ MeV, $3065.6$ MeV,
$3095.2$ MeV and $3119.2$ MeV. Assigning these excited baryons to be in
p-wave, one can extract $c\left(\Omega_{c}\right)$ by matching the $jj$ mass
formula to the five measured masses, with the result KR:D2017 ; DPP:Epjc21

$c\left(\Omega_{c}\right)=4.07\text{ MeV.}$ (49)

The spin-averaged mass of the five excited baryons $\Omega_{c}$ is

$\bar{M}(\Omega_{c})_{1P}=\frac{1}{18}\left[(2J+1)M(\Omega_{c};J)\right]=3079.94\text{
MeV,}$ (50)

where the five observed masses $M(\Omega_{c};J)$, from the lightest to the
heaviest, are assigned to have the increasing quantum number $J$KR:D2017 .
Further, given the observed ground-state masses,
$M(\Omega_{c},1/2^{+})=2695.2\pm 1.7$ MeV and $M(\Omega_{c},3/2^{+})=2765.9\pm
2.0$ MeV ZylaPDG:D20 , one finds its spin-averaged mass in the s-wave to be

$\bar{M}(\Omega_{c})_{1s}=\frac{1}{6}\left(2M(1/2^{+})+4M(3/2^{+})\right)=2742.3\pm
1.9~{}\text{MeV.}$ (51)

Putting the values in Eq. (51) and Eq. (50) into the Regge relation for the
$\Omega_{c}=c(ss):$

$(\bar{M}-M_{c})^{2}=\pi al+\left(m_{ss}+M_{c}-1.275^{2}/M_{c}\right)^{2},$

where $M_{c}=1.44$ GeV and $m_{ss}$ is the $ss$-diquark mass, one can solve
the trajectory slope $\pi a$ and $m_{ss}$, with the resultDPP:Epjc21

$a=0.316\text{ GeV}^{2}\text{, }m_{ss}=991\text{ MeV. }$ (52)

Now, one can employ Eqs. (45)-(48) to estimate the four parameters of spin
couplings of the DH baryons using the parameters in Table I and the diquark
masses in Eqs.(5), (6)and that in Eq. (52). The results are listed
collectively in Table VI.

Table 6: Spin-coupling parameters(in MeV) in the spin interaction (17) of the DH baryons $\Xi_{QQ^{\prime}}$ and $\Omega_{QQ^{\prime}}$($Q,Q^{\prime}=c,b$). DH baryons | $a_{1}$ | $a_{2}$ | $b$ | $c$
---|---|---|---|---
$\Xi_{cc}$ | $64.05$ | $17.64$ | $19.38$ | $8.81$
$\Xi_{bb}$ | $20.58$ | $5.67$ | $6.23$ | $2.83$
$\Xi_{\\{bc\\}}$ | $31.14$ | $8.58$ | $9.42$ | $4.29$
$\Omega_{cc}$ | $44.91$ | $16.66$ | $16.48$ | $6.18$
$\Omega_{bb}$ | $14.43$ | $5.35$ | $5.30$ | $1.99$
$\Omega_{\\{bc\\}}$ | $21.84$ | $8.10$ | $8.01$ | $3.01$

As seen in Table, the parameter $a_{2}$ are smaller(three times roughly) but
comparable to $a_{1}$. The magnitudes of $b$ is as large as $a_{2}$ roughly
while $c$ is the smallest. This agrees qualitatively with the parameter
hierarchy of the spin-couplingsDPP:Epjc21 ; JiaLH:19  implied by heavy quark
symmetry.

For the bottom charmed baryons $\Xi_{bc}$ and $\Omega_{bc}$ with zero diquark
spin $S_{bc}=0$, the spin-interaction Hamiltonian is simple,

$H^{SD}=a_{1}\mathbf{l}\cdot\mathbf{S}_{q}\text{,}$ (53)

which yields, by Eq. (36), the mass splitting ($J=1/2,3/2$),

$\Delta
M(\Xi_{bc}\text{,}\Omega_{bc})=a_{1}(\Xi_{bc}\text{,}\Omega_{bc})\left[\begin{array}[c]{rr}-1&0\\\
0&1/2\end{array}\right]\text{.}$ (54)

With $a_{1}(D_{s})=89.4$ MeVKR:D15 , one can employ the relations of mass
scaling (45) relative to the $D_{s}$ to give

$\displaystyle a_{1}(\Xi_{bc})$ $\displaystyle=\left(\frac{1.440\text{
GeV}}{5.8918\text{ GeV}}\right)\left(\frac{0.328\text{ GeV}}{0.230\text{
GeV}}\right)a_{1}(D_{s})=31.14\text{ MeV,}$ $\displaystyle a_{1}(\Omega_{bc})$
$\displaystyle=\left(\frac{1.440\text{ GeV}}{5.8918\text{
GeV}}\right)a_{1}(D_{s})=21.84\text{ MeV,}$ (55)

where $M_{[bc]}=5.8918$ GeV and other mass parameters are from Table I.
Putting above parameters into Eq. (54), and adding
$\bar{M}_{L}(\Xi_{bc}$,$\Omega_{bc})$ in Eqs. (11) and (16), one obtains the
P-wave masses $\bar{M}_{L}(\Xi_{bc}$,$\Omega_{bc})+\Delta
M(\Xi_{bc}$,$\Omega_{bc})$ for the baryons $\Xi_{bc}$ and $\Omega_{bc}$(with
spin-$0$ diquark $bc$ and $J^{P}=1/2^{-}$ and $3/2^{-}$) to be

$\displaystyle M\left(\Xi_{bc}\text{,}1/2^{-}\right)$
$\displaystyle=7294.7\text{ MeV,
}M\left(\Xi_{bc}\text{,}3/2^{-}\right)=7341.4\text{ MeV,}$ (56) $\displaystyle
M\left(\Omega_{bc}\text{,}1/2^{-}\right)$ $\displaystyle=7421.2\text{ MeV,
}M\left(\Omega_{bc}\text{,}3/2^{-}\right)=7453.9\text{ MeV. }$ (57)

We list the masses of the DH baryons in their 1S1p-wave in Tables VIII to XIV,
with comparision with other predictions. They are also shown in FIGs 1-8 for
the respective states of the DH baryons. Our computation suggests that the
1S1p states of the DH baryons have increasing masses with baryon spin $J$
rising from the lowest $1/2^{-}$ to highest $5/2^{-}$.

Table 7: Mean (spin-averaged) masses(in GeV) of the excited DH baryons $\Xi_{QQ^{\prime}}$ and $\Omega_{QQ^{\prime}}$ predicted by Eq. (58).  Baryons | $\bar{M}(1D)$ | $\bar{M}(1F)$ | $\bar{M}(2S)$ | $\bar{M}(2P)$ | $\bar{M}(2D)$ | $\bar{M}(2F)$
---|---|---|---|---|---|---
$\Xi_{cc}$ | $4.373$ | $4.617$ | $4.255$ | $4.516$ | $4.742$ | $4.943$
$\Xi_{bb}$ | $10.835$ | $11.076$ | $10.722$ | $10.976$ | $11.202$ | $11.408$
$\Xi_{bc}$ | $7.615$ | $7.861$ | $7.498$ | $7.759$ | $7.989$ | $8.195$
$\Omega_{cc}$ | $4.494$ | $4.749$ | $4.370$ | $4.644$ | $4.881$ | $5.093$
$\Omega_{bb}$ | $10.975$ | $11.234$ | $10.854$ | $11.126$ | $11.369$ | $11.589$
$\Omega_{bc}$ | $7.749$ | $8.010$ | $7.625$ | $7.902$ | $8.146$ | $8.366$
Table 8:  Mass spectrum of $\Xi_{cc}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(NLn_{q}l\right)J^{P}$}\end{array}$ | Our | EFG:D02  | Lu:2017meb  | Gershtein:2000nx  | Giannuzzi:2009gh  | Yoshida:2015tia  | Shah:2017liu  | Eakins:2012jk  | Roberts:2007ni
---|---|---|---|---|---|---|---|---|---
$(1S1p)1/2^{-}$ | 3970.3 | 4053 | 3998 | 3927 | $-$ | 3947 | 3861 | 4081 | 3910
$(1S1p)1/2^{\prime-}$ | 4082.5 | 4136 | 3985 | 4052 | $-$ | 4135 | 3871 | 4073 | 4074
$(1S1p)3/2^{-}$ | 4039.9 | 4101 | 4014 | 4039 | $-$ | 3949 | 3842 | 4077 | 3921
$(1S1p)3/2^{\prime-}$ | 4100.6 | 4196 | 4025 | 4034 | $-$ | 4137 | 3851 | 4079 | 4078
$(1S1p)5/2^{-}$ | 4130.9 | 4155 | 4050 | 4047 | $-$ | 4163 | 3888 | 4089 | 4092
$(1S2s)1/2^{+}$ | 4216.3 | $-$ | 4172 | $-$ | 4183 | 4159 | 3920 | 4311 | $-$
$(1S2s)3/2^{+}$ | 4275.1 | $-$ | 4193 | $-$ | 4282 | 4131 | 3983 | 4368 | $-$
$(2S1s)1/2^{+}$ | 3926.6 | 3910 | 4004 | 3812 | $-$ | 4079 | $-$ | 4030 | 4029
$(2S1s)3/2^{+}$ | 4022.5 | 4027 | 4036 | 3944 | $-$ | 4114 | $-$ | 4078 | 4042
$(1P1s)1/2^{-}$ | 3956.0 | 3838 | 3873 | 3702 | $-$ | 4149 | $-$ | 3911 | $-$
$(1P1s)3/2^{-}$ | 4051.1 | 3959 | 3916 | 3834 | $-$ | 4159 | $-$ | 3917 | $-$
Table 9:  Mass spectrum of $\Xi_{bb}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(NLn_{q}l\right)J^{P}$}\end{array}$ | Our | EFG:D02  | Lu:2017meb  | Gershtein:2000nx  | Giannuzzi:2009gh  | Yoshida:2015tia  | Shah:2017liu  | Eakins:2012jk  | Roberts:2007ni
---|---|---|---|---|---|---|---|---|---
$(1S1p)1/2^{-}$ | 10523.1 | 10632 | 10525 | 10541 | $-$ | 10476 | 10511 | 10694 | 10493
$(1S1p)1/2^{\prime-}$ | 10559.0 | 10675 | 10504 | 10578 | $-$ | $-$ | 10514 | 10691 | $-$
$(1S1p)3/2^{-}$ | 10545.3 | 10647 | 10526 | 10567 | $-$ | 10476 | 10506 | 10691 | 10495
$(1S1p)3/2^{\prime-}$ | 10564.8 | 10694 | 10528 | 10581 | $-$ | $-$ | 10509 | 10692 | $-$
$(1S1p)5/2^{-}$ | 10574.5 | 10661 | 10547 | 10580 | $-$ | 10759 | 10518 | 10695 | 10713
$(1S2s)1/2^{+}$ | 10707.9 | 10832 | 10662 | $-$ | 10751 | 10612 | 10609 | 10940 | $-$
$(1S2s)3/2^{+}$ | 10726.8 | 10860 | 10675 | $-$ | 10770 | 10593 | 10617 | 10972 | $-$
$(2S1s)1/2^{+}$ | 10473.4 | 10441 | 10464 | 10373 | $-$ | 10571 | $-$ | 10551 | 10571
$(2S1s)3/2^{+}$ | 10506.4 | 10482 | 10480 | 10413 | $-$ | 10592 | $-$ | 10574 | 10592
$(1P1s)1/2^{-}$ | 10523.5 | 10368 | 10364 | 10310 | $-$ | 10740 | $-$ | 10470 | $-$
$(1P1s)3/2^{-}$ | 10556.4 | 10408 | 10387 | 10343 | $-$ | 10742 | $-$ | 10470 | $-$
Table 10:  Mass spectrum of $\Xi_{bc}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(NLn_{q}l\right)J^{P}$}\end{array}$ | Our | Giannuzzi:2009gh  | Shah:2017liu  | Eakins:2012jk  | Mohajery:2018qhz  | Roberts:2007ni
---|---|---|---|---|---|---
$(1S1p)1/2^{-}$ | 7273.2 | $-$ | 7156 | 7397 | 6842 | 7206
$(1S1p)1/2^{\prime-}$ | 7327.7 | $-$ | 7161 | 7390 | 6847 | 7231
$(1S1p)3/2^{-}$ | 7307.0 | $-$ | 7144 | 7392 | 6831 | 7208
$(1S1p)3/2^{\prime-}$ | 7336.6 | $-$ | 7150 | 7394 | 6837 | 7229
$(1S1p)5/2^{-}$ | 7351.3 | $-$ | 7171 | 7399 | 6856 | 7272
$(1S2s)1/2^{+}$ | 7478.6 | 7478 | 7240 | 7634 | 6919 | $-$
$(1S2s)3/2^{+}$ | 7507.2 | 7495 | 7263 | 7676 | 6939 | $-$
$(2S1s)1/2^{+}$ | 7211.8 | $-$ | $-$ | 7321 | $-$ | 7044
$(2S1s)3/2^{+}$ | 7261.0 | $-$ | $-$ | 7353 | $-$ | 7386
$(1P1s)1/2^{-}$ | 7250.3 | $-$ | $-$ | 7212 | $-$ | $-$
$(1P1s)3/2^{-}$ | 7299.2 | $-$ | $-$ | 7214 | $-$ | $-$
Figure 1: Mass spectrum of $\Xi_{cc}$ baryons (in MeV).The horizontal dashed
line shows the $\Lambda_{c}D$ threshold. Figure 2: Mass spectrum of $\Xi_{bb}$
baryons (in MeV). Figure 3: Mass spectrum of $\Xi_{bc}$ baryons (in MeV).The
horizontal dashed line shows the $\Lambda_{b}D$ threshold. Figure 4: Mass
spectrum of $\Xi_{bc}^{\prime}$ baryons (in GeV).The horizontal dashed line
shows the $\Lambda_{b}D$ threshold.

##  V DH Baryon masses in 1S and 2S waves

To extend the analysis of the orbitally excited baryons to radially excited
states, one needs a Regge relation for the radial excitations. This type of
Regge relation is proposed in Ref. JiaD:E19  previously in consideration that
the trajectory slope ratio between the radial and angular excitations is
$\pi:2$ for the heavy mesons $\bar{q}Q$ in the heavy quark limit. Extending
this ratio to the heavy-diquark-quark picture of the DH system
$q-QQ^{\prime}$, one can have a unified Regge relation for the radially and
angular excitations(view diquark $QQ^{\prime}$ as one end and light quark $q$
at the other of a QCD string). The result follows via replacing heavy quark
$Q$ there by the diquark $QQ^{\prime}$, the light antiquark $\bar{q}$ by $q$,
and $\pi al$ in Eq. (1) by $\pi a\left(l+\frac{\pi}{2}n\right)$, giving rise
toJiaD:E19

$\displaystyle\left(\bar{M}-M_{QQ^{\prime}}\right)^{2}$ $\displaystyle=\pi
a\left(l+\frac{\pi}{2}n\right)$
$\displaystyle+\left(m_{q}+M_{QQ^{\prime}}-\frac{m_{\text{bare
}QQ^{\prime}}^{2}}{M_{QQ^{\prime}}}\right)^{2}.$ (58)

Applying the parameters in Eqs. (2), (5), (6), (9), (10), (14), (15) and Table
1 to Eq. (58), one can find the mean(spin-averaged) masses of all doubly heavy
baryons ($\Xi_{cc}$, $\Xi_{bb}$, $\Xi_{bc}$, $\Omega_{cc}$, $\Omega_{bb}$,
$\Omega_{bc}$) in the radially and angularly excited states. The results are
collectively in Table VII.

For the s-waves of the DH baryons, the spin-interaction Hamiltonian is simply

$H^{SD}=c\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}\text{,}$ (59)

in which $\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}$ has the eigenvalues
$\\{-1,1/2\\}$ if $\mathbf{S}_{QQ^{\prime}}=1$($\mathbf{l}=0$). The mass
formula for the systems $q\\{QQ^{\prime}\\}$ becomes($J=1/2,3/2$),

$M(q\\{QQ^{\prime}\\})_{nS}=\bar{M}(q\\{QQ^{\prime}\\})_{nS}+c(q\\{QQ^{\prime}\\})\left[\begin{array}[c]{rr}-1&0\\\
0&1/2\end{array}\right]\text{.}$ (60)

For the s-waves of the DH system $q[QQ^{\prime}]$ with zero diquark spin
$\mathbf{S}_{QQ^{\prime}}=0$($\mathbf{l}=0$), mass of the system is simply
$\bar{M}(q[bc])_{nS}$ without splitting if there is no internal diquark
orbital excitations.

In 2s wave($n=1$), we use the mass scaling relative to the $D$ meson in 2s
wave(the $D_{0}(2550)^{0}$ with mass $2549\pm 19$ MeV and the
$D_{1}^{\ast}(2600)^{\pm 0}$ with mass $2627\pm 10$ MeVZylaPDG:D20 ) to find

$\displaystyle c\left(\Xi_{bb}(bbu)\right)$
$\displaystyle=\left(\frac{M_{c}}{M_{bb}}\right)\left[M(D(2s),1^{-}))-M\left(D(2s),0^{-}\right)\right]=12.6\text{
MeV,}$ $\displaystyle c\left(\Omega_{bb}(bbs)\right)$
$\displaystyle=\left(\frac{M_{c}}{M_{bb}}\right)\left(\frac{m_{u}}{m_{s}}\right)\left[M(D(2s),1^{-})-M\left(D(2S),0^{-}\right)\right]=8.8\text{
MeV.}$

Further application of the scaling relations between the DH baryons,

$\displaystyle c\left(\Xi_{cc}\right)_{2s}$
$\displaystyle=c\left(\Xi_{bb}\right)\left(\frac{M_{bb}}{M_{cc}}\right),c\left(\Xi_{bc}\right)_{2s}=c\left(\Xi_{bb}\right)\left(\frac{M_{bb}}{M_{bc}}\right),$
$\displaystyle c\left(\Omega_{cc}\right)_{2s}$
$\displaystyle=c\left(\Omega_{bb}\right)\left(\frac{M_{bb}}{M_{cc}}\right),c\left(\Omega_{bc}\right)_{2s}=c\left(\Omega_{bb}\right)\left(\frac{M_{bb}}{M_{bc}}\right),$
(61)

leads to the following parameter $c$ for the 2s-wave baryons $\Xi_{cc}$,
$\Xi_{bc}$, $\Omega_{cc}$ and $\Omega_{cc}$,

$\displaystyle c(\Xi_{cc})_{2s}$ $\displaystyle=39.2\text{ MeV,
}c\left(\Xi_{bc}\right)_{2S}=19.1\text{ MeV,}$ $\displaystyle
c(\Omega_{cc})_{2s}$ $\displaystyle=27.4\text{ MeV,
}c\left(\Omega_{bc}\right)_{2s}=13.3\text{ MeV.}$ (62)

With the mean masses of the 2s states in Table VII, one can employ Eq. (60) to
compute the 2s-wave(i.e., the $1S2s$-wave)masses of the DH baryons. The
results are listed in Table VIII through Table XIV, with comparison with other
calculations, and shown in FIGs 1-8, respectively.

It is of interest to apply the same strategy to the ground states to see if
one can reproduce the masses in Table II. For the 1s wave $\Xi_{cc}$ firstly,
the $c$ value can be scaled to that of the ground state $D$ mesons with mass
difference $m(D^{\ast}(2010)^{+})-m(D^{+})=140.6$ MeVZylaPDG:D20  via mass
scaling:

$c\left(\Xi_{cc}\right)_{1s}=c\left(D\right)_{1s}\left(\frac{M_{c}}{M_{cc}}\right)=(140.6\
\text{MeV})\left(\frac{1.44}{2.8655}\right)=70.7\ \text{MeV,}$

For the $\Omega_{cc}$, the relevant meson for scaling is the ground-state
$D_{s}$ mesons with mass difference $m(D_{s}^{\ast\pm})-m(D_{s}^{\pm})=143.8$
MeVZylaPDG:D20 , and the $c$ value is then

$c\left(\Omega_{cc}\right)_{1s}=c\left(D_{s}\right)_{1s}\left(\frac{M_{c}}{M_{cc}}\right)=(143.8\
\text{MeV})\left(\frac{1.44}{2.8655}\right)=72.3\ \text{MeV.}$

Using
$\bar{M}(qQQ^{\prime})_{1s}=M_{QQ^{\prime}}+m_{q}+M_{QQ^{\prime}}v_{QQ^{\prime}}^{2}$
in Eq. (3) with $v_{QQ^{\prime}}^{2}\equiv 1-$ $m_{\text{bare
}QQ^{\prime}}^{2}/M_{QQ^{\prime}}^{2}$ and putting the above $c$ values to Eq.
(60) gives

$\displaystyle(1/2,3/2)^{+}$ $\displaystyle:M(\Xi_{cc})_{1s}=3691.\allowbreak
7+70.66\left\\{-1,1/2\right\\}=\left\\{3620.8,3726.8\right\\}\ \text{MeV,}$
$\displaystyle(1/2,3/2)^{+}$
$\displaystyle:M(\Omega_{cc})_{1s}=3789.5+72.3\left\\{-1,1/2\right\\}=\left\\{3717.\allowbreak
2,3825.7\right\\}\ \text{MeV,}$

where the mean mass in 1s wave($v_{cc}^{2}=\allowbreak 0.208$)

$\bar{M}(q\\{cc\\})_{1s}=M_{cc}+m_{n,s}+M_{cc}v_{cc}^{2}=3691.\allowbreak 5\
\text{or }3789.5\ \text{MeV,}$

have been used for $q=n$ or $s$, respectively, and the two numbers in curly
braces correspond to the respective states $J^{P}=1/2^{+}$ and $3/2^{+}$.

Next, similar calculation for the 1s wave $\Xi_{bb}$ and $\Omega_{bb}$ gives

$\displaystyle c\left(\Xi_{bb}\right)_{1s}$
$\displaystyle=c\left(D\right)_{1s}\left(\frac{M_{c}}{M_{bb}}\right)=(140.6\
\text{MeV})\left(\frac{1.44}{8.9167}\right)=22.7\ \text{MeV,}$ $\displaystyle
c\left(\Omega_{bb}\right)_{1s}$
$\displaystyle=c\left(D_{s}\right)_{1s}\left(\frac{M_{c}}{M_{bb}}\right)=(143.8\
\text{MeV})\left(\frac{1.44}{8.9167}\right)=23.2\ \text{MeV,}$

and the ground state masses (with $v_{bb}^{2}=0.121$)

$\displaystyle(1/2,3/2)^{+}$ $\displaystyle:M(\Xi_{bb})_{1s}=10\allowbreak
226+22.7\left\\{-1,1/2\right\\}=\left\\{10203.3,10237.4\right\\}\ \text{MeV,}$
$\displaystyle(1/2,3/2)^{+}$
$\displaystyle:M(\Omega_{bb})_{1s}=10324+23.2\left\\{-1,1/2\right\\}=\left\\{10301.8,10335.6\right\\}\
\text{MeV,}$

where $\bar{M}(q\\{bb\\})_{1S}=M_{bb}+m_{q}+M_{bb}v_{bb}^{2}=10\allowbreak
226.0$($10\allowbreak 324.0$) MeV have been used for $q=n$(or $s$).

Finally, for the DH systems $q(bc)$, there are three ground states,
$\Xi_{bc}=$ $n\\{bc\\}$ with $J^{P}=1/2^{+}$ and $3/2^{+}$,
$\Xi_{bc}^{\prime}=$ $n[bc]$ with $J^{P}=1/2^{+}$, and three ground states
$\Omega_{bc}=$ $s\\{bc\\}$ with $J^{P}=1/2^{+}$ and $3/2^{+}$,
$\Omega_{bc}^{\prime}=$ $s[bc]$ with $J^{P}=1/2^{+}$. Notice that
$v_{bc}^{2}=0.1428(0.1429)$ for the diquark $\\{bc\\}$($[bc]$), one can
estimate the mean masses for them by Eq. (3) to be

$\displaystyle\bar{M}(n\\{bc\\})_{1s}$
$\displaystyle=M_{\\{bc\\}}+m_{n}+M_{\\{bc\\}}v_{bc}^{2}=6964.3\ \text{MeV,}$
$\displaystyle\bar{M}(s\\{bc\\})_{1s}$
$\displaystyle=M_{\\{bc\\}}+m_{s}+M_{\\{bc\\}}v_{bc}^{2}=7062.3\ \text{MeV,}$
$\displaystyle\bar{M}(n[bc])_{1s}$ $\displaystyle=6963.0\ \text{MeV,
}\bar{M}(s[bc])_{1S}=7061.0\ \text{MeV,}$

and the $c$ parameter to be

$\displaystyle c\left(\Xi_{bc}\right)_{1s}$
$\displaystyle=c\left(D\right)_{1s}\left(\frac{M_{c}}{M_{\\{bc\\}}}\right)=(140.6\
\text{MeV})\left(\frac{1.44}{5.9823}\right)=33.8\ \text{MeV,}$ $\displaystyle
c\left(\Omega_{bc}\right)_{1s}$
$\displaystyle=c\left(D_{s}\right)_{1s}\left(\frac{M_{c}}{M_{\\{bc\\}}}\right)=(143.8\
\text{MeV})\left(\frac{1.44}{5.9823}\right)=34.6\ \text{MeV.}$

The ground state masses of the baryons $\Xi_{bc}$ and $\Omega_{bc}$ are then

$\displaystyle(1/2,3/2)^{+}$
$\displaystyle:M(\Xi_{bc})_{1s}=6964.3+33.8\left\\{-1,1/2\right\\}=\left\\{6930.5,6981.2\right\\}\
\text{MeV,}$ $\displaystyle(1/2,3/2)^{+}$
$\displaystyle:M(\Omega_{bc})_{1s}=7062.3+34.6\\{-1,1/2\\}=\left\\{7027.7,7079.\allowbreak
6\right\\}\ \text{MeV,}$ $\displaystyle 1/2^{+}$
$\displaystyle:M(\Xi_{bc}^{\prime})_{1s}=6963.0\ \text{MeV,
}M(\Omega_{bc}^{\prime})_{1s}=7061.0\ \text{MeV,}$

with the two numbers in curly braces corresponding to the states
$J^{P}=1/2^{+}$ and $3/2^{+}$, respectively. Our predictions for the most
ground-state masses of the DH baryons are in consistent with that by Ref
EFG:D02 , especially, the mass splitting of the systems
$\Xi_{Q\acute{Q}^{\prime}}$ and $\Omega_{Q\acute{Q}^{\prime}}$ agree well with
the later.

Table 11:  Mass spectrum of $\Omega_{cc}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(N_{d}Ln_{q}l\right)J^{P}$}\end{array}$ | Our | EFG:D02  | Lu:2017meb  | Giannuzzi:2009gh  | Yoshida:2015tia  | Roberts:2007ni  | Shah:2016vmd  | Salehi:2019kvb  | Kiselev:2002iy
---|---|---|---|---|---|---|---|---|---
$(1S1p)1/2^{-}$ | 4102.0 | 4208 | 4087 | $-$ | 4086 | 4046 | 3989 | 3965 | 4050
$(1S1p)1/2^{\prime-}$ | 4184.1 | 4271 | 4081 | $-$ | 4199 | 4135 | 3998 | 3987 | 4145
$(1S1p)3/2^{-}$ | 4165.2 | 4252 | 4107 | $-$ | 4086 | 4052 | 3972 | 3956 | 4176
$(1S1p)3/2^{\prime-}$ | 4200.6 | 4325 | 4114 | $-$ | 4201 | 4140 | 3981 | 3978 | 4102
$(1S1p)5/2^{-}$ | 4227.1 | 4330 | 4134 | $-$ | 4220 | 4152 | 3958 | 3926 | 4134
$(1S2s)1/2^{+}$ | 4343.2 | $-$ | 4270 | 4268 | 4295 | 4180 | 4041 | 3964 | $-$
$(1S2s)3/2^{+}$ | 4384.4 | $-$ | 4288 | 4334 | 4265 | 4188 | 4096 | 3979 | $-$
$(2S1s)1/2^{+}$ | 4097.4 | 4075 | 4118 | $-$ | 4227 | $-$ | $-$ | $-$ | 3925
$(2S1s)3/2^{+}$ | 4162.0 | 4174 | 4142 | $-$ | 4263 | $-$ | $-$ | $-$ | 4064
$(1P1s)1/2^{-}$ | 4123.6 | 4002 | 3986 | $-$ | 4210 | $-$ | $-$ | $-$ | 3812
$(1P1s)3/2^{-}$ | 4190.9 | 4102 | 4020 | $-$ | 4218 | $-$ | $-$ | $-$ | 3949
Table 12:  Mass spectrum of $\Omega_{bb}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(N_{d}Ln_{q}l\right)J^{P}$}\end{array}$ | Our | EFG:D02  | Lu:2017meb  | Giannuzzi:2009gh  | Yoshida:2015tia  | Roberts:2007ni  | Shah:2016vmd  | Salehi:2019kvb  | Kiselev:2002iy
---|---|---|---|---|---|---|---|---|---
$(1S1p)1/2^{-}$ | 10651.8 | 10771 | 10605 | $-$ | 10607 | 10616 | 10646 | 10968 | 10651
$(1S1p)1/2^{\prime-}$ | 10678.2 | 10804 | 10591 | $-$ | 10796 | 10763 | 10648 | 10976 | 10700
$(1S1p)3/2^{-}$ | 10672.1 | 10785 | 10610 | $-$ | 10608 | 10619 | 10641 | 10957 | 10661
$(1S1p)3/2^{\prime-}$ | 10683.5 | 10802 | 10611 | $-$ | 10797 | 10765 | 10643 | 10963 | 10720
$(1S1p)5/2^{-}$ | 10692.3 | 10798 | 10625 | $-$ | 10808 | 10766 | 10637 | 10956 | 10670
$(1S2s)1/2^{+}$ | 10845.2 | 10970 | 10751 | 10830 | 10744 | 10693 | 10736 | 10969 | $-$
$(1S2s)3/2^{+}$ | 10858.4 | 10992 | 10763 | 10839 | 10730 | 10721 | 10617 | 10964 | $-$
$(2S1s)1/2^{+}$ | 10633.7 | 10610 | 10566 | $-$ | 10707 | $-$ | $-$ | $-$ | 10493
$(2S1s)3/2^{+}$ | 10656.8 | 10645 | 10579 | $-$ | 10723 | $-$ | $-$ | $-$ | 10540
$(1P1s)1/2^{-}$ | 10683.6 | 10532 | 10464 | $-$ | 10803 | $-$ | $-$ | $-$ | 10416
$(1P1s)3/2^{-}$ | 10706.8 | 10566 | 10482 | $-$ | 10805 | $-$ | $-$ | $-$ | 10462
Table 13:  Mass spectrum of $\Omega_{bc}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(N_{d}Ln_{q}l\right)J^{P}$}\end{array}$ | Our | Giannuzzi:2009gh  | Roberts:2007ni  | Shah:2016vmd  | Salehi:2019kvb
---|---|---|---|---|---
$(1S1p)1/2^{-}$ | 7401.8 | $-$ | 7335 | 7386 | 7476
$(1S1p)1/2^{\prime-}$ | 7441.7 | $-$ | 7346 | 7392 | 7490
$(1S1p)3/2^{-}$ | 7432.5 | $-$ | 7334 | 7373 | 7470
$(1S1p)3/2^{\prime-}$ | 7449.7 | $-$ | 7349 | 7379 | 7486
$(1S1p)5/2^{-}$ | 7463.0 | $-$ | 7362 | 7363 | 7451
$(1S2s)1/2^{+}$ | 7611.5 | 7559 | $-$ | 7480 | 7475
$(1S2s)3/2^{+}$ | 7631.7 | 7571 | $-$ | 7497 | 7485
$(2S1s)1/2^{+}$ | 7373.3 | $-$ | $-$ | $-$ | $-$
$(2S1s)3/2^{+}$ | 7407.8 | $-$ | $-$ | $-$ | $-$
$(1P1s)1/2^{-}$ | 7411.6 | $-$ | $-$ | $-$ | $-$
$(1P1s)3/2^{-}$ | 7446.1 | $-$ | $-$ | $-$ | $-$
Table 14:  Mass spectrum of $\Xi_{bc}^{\prime}/\Omega_{bc}^{\prime}$ baryons (in MeV).  $\begin{array}[c]{c}\text{State}\\\ \text{$\left(N_{d}Ln_{q}l\right)J^{P}$}\end{array}$ | Our($\Xi_{bc}^{\prime}$) | Eakins:2012jk  | Our($\Omega_{bc}^{\prime}$) | Eakins:2012jk
---|---|---|---|---
$(1S1p)1/2^{-}$ | 7294.7 | 7388 | 7421.2 | $-$
$(1S1p)3/2^{-}$ | 7341.4 | 7390 | 7453.9 | $-$
$(1S2s)1/2^{+}$ | 7497.0 | 7645 | 7624.2 | $-$
$(2S1s)1/2^{+}$ | 7243.3 | 7333 | 7396.3 | $-$
$(1P1s)1/2^{-}$ | 7281.6 | 7230 | 7434.6 | $-$
$(1P1s)1/2^{\prime-}$ | 7249.0 | 7199 | 7411.7 | $-$
$(1P1s)3/2^{-}$ | 7297.9 | 7228 | 7446.0 | $-$
$(1P1s)3/2^{\prime-}$ | 7232.7 | 7201 | 7400.3 | $-$
$(1P1s)5/2^{-}$ | 7314.2 | 7265 | 7457.4 | $-$
Figure 5: Mass spectrum of $\Omega_{cc}$ baryons (in MeV).The horizontal
dashed line shows the $\Lambda_{c}D_{s}$ threshold. Figure 6: Mass spectrum of
$\Omega_{bb}$ baryons (in MeV). Figure 7: Mass spectrum of $\Omega_{bc}$
baryons (in MeV).The horizontal dashed line shows the $\Lambda_{b}D_{s}$
threshold. Figure 8: Mass spectrum of $\Omega_{bc}^{\prime}$ baryons (in
MeV).The horizontal dashed line shows the $\Lambda_{b}D_{s}$ threshold.

##  VI Mean mass shifts due to diquark excitations

In this section, we consider excited DH baryons
$qQQ^{\prime}$($QQ^{\prime}=cc$, $bb$, $bc$) with heavy diquark $QQ^{\prime}$
in the internal 1P and 2S waves. The diquark $cc$ or $bb$ has to be in spin
singlet(asymmetry) in the internal 1P wave or spin triplet in the internal 2S
wave, while $bc$ can be either in the internal 1P or in 2S waves. As treated
before, one can decompose the excited baryon mass as $M=\bar{M}_{N,L}+(\Delta
M)_{N,L}$, where $\bar{M}_{N,L}$ is the spin-independent mass and $(\Delta
M)_{N,L}$ is mass shift(splitting) due to the spin interactions in DH baryons
$qQQ^{\prime}$.

First of all, we consider excited heavy mesons in 1P and 2S waves in QCD
string picture. For an excited heavy quarkonia $Q\bar{Q}$, a rotating-string
pictureBurnsPP:D10  in which a heavy quark at one end of the rotating string
and and an antiquark at the other with relative orbital angular momentum $L$,
infers a typical Regge relation($\bar{M}^{3/2}$ $\sim L$), akin to Eq. (1),

$\bar{M}=2M_{Q}+3\left(\frac{T^{2}L^{2}}{4M_{Q}}\right)^{1/3}.$ (63)

On the other hand, a semiclassical WKB analysis of a system of heavy quarkonia
$Q\bar{Q}$ (in the color singlet) in a linear confining potential $T|r|$ leads
to a quantization condition for its radial excitations(labeled by $N$,
Appendix C):

$\left[\bar{M}-2M_{Q}\right]^{3/2}=\frac{3\pi
T}{4\sqrt{M_{Q}}}\left(N+2c_{0}\right),$ (64)

with the constant $2c_{0}$ the quantum defect. Comparing the radial and
angular slopes (linear coefficients in $N$ and $L$, respectively, which is
$\pi:\sqrt{12}$) of the trajectory in RHS of Eq. (64) and in that of Eq. (63),
one can combine two trajectories into(Appendix C)

$\left[\bar{M}_{N,L}-2M_{Q}-B(Q\bar{Q})_{N,L}\right]^{3/2}=\frac{3\sqrt{3}T}{2\sqrt{M_{Q}}}\left(L+\frac{\pi
N}{\sqrt{12}}+2c_{0}\right),$ (65)

where a term of extra energy $B(Q\bar{Q})_{N,L}$, named heavy-pair binding
energy, enters to represent a corrections to the picture of Regge spectra due
to the effective quark masses and short-distance interquark forcesDeRuju:D75
when two heavy quarks($Q$ and $\bar{Q}$) come close each other. Such a term is
ignored in the semiclassical picture of massive QCD string as well as in the
WKB analysis (Appendix C). For the ground state of DH hadrons, a similar
binding between two heavy quarks was considered in Ref. KR:D14 ; KR:D18 .

For applications in this work, we rewrite Eq. (65) and extend it to a general
form in which $Q\bar{Q}^{\prime}$ can be $b\bar{c}$ in flavor(by heavy quark
symmetry),

$\bar{M}(Q\bar{Q}^{\prime})_{N,L}=M_{Q}+M_{Q^{\prime}}+B(Q\bar{Q}^{\prime})_{N,L}+3\frac{\left[T\left(L+\pi
N/\sqrt{12}+2c_{0}\right)\right]^{2/3}}{[2(M_{Q}+M_{Q^{\prime}})]^{1/3}}.$
(66)

Setting $N=0=L$ in Eq. (66) gives

$Tc_{0}(Q\bar{Q}^{\prime})=\frac{\sqrt{2(M_{Q}+M_{Q^{\prime}})}}{6\sqrt{3}}\left[\bar{M}(Q\bar{Q})_{1S}-M_{Q}-M_{Q^{\prime}}-B(Q\bar{Q}^{\prime})_{1S}\right]^{3/2},$
(67)

which yields numerically,

$Tc_{0}(c\bar{c})=0.0689\text{ GeV}^{2}\text{, }Tc_{0}(b\bar{b})=0.4363\text{
GeV}^{2}\text{, }Tc_{0}(b\bar{c})=0.2136\text{ GeV}^{2}\text{. }$ (68)

On the phenomenological side, one can estimate the binding energy between $Q$
and $\bar{Q}^{\prime}$ via the following relation,

$B(Q\bar{Q}^{\prime})_{N,L}=\bar{M}(Q\bar{Q}^{\prime})-\bar{M}(Q\bar{n})-\bar{M}(Q\bar{n}^{\prime})+\bar{M}(n\bar{n}^{\prime}),$
(69)

in which $\bar{M}$ represents the mean(spin-averaged) masses of the respective
mesons formed by the quark pairs $Q\bar{Q}^{\prime}$, $Q\bar{n}$ and
$n\bar{n}^{\prime}$($n=u,d$), shown in Table XV.

Table 15: Means spin-average masses (MeV) of heavy quarkonia, $B_{c}$ mesons, the $B$ mesons, $D$ mesons and the light unflavored mesons  Mesons | $c\bar{c}$ | $b\bar{b}$ | $b\bar{c}$ | $c\bar{q}$ | $b\bar{q}$ | $q\bar{q}$
---|---|---|---|---|---|---
$\bar{M}(1S)$ | 3068.65 | 9444.9 | 6320.25 | 1973.23 | 5313.40 | 619.98
$\bar{M}(2S)$ | 3627.50 | 10017.3 | 6901.20 | 2627.00 | 5917.00 | 1423.75
$\bar{M}(1P)$ | 3525.26 | 9899.7 | 6765.32 | 2435.72 | 5737.17 | 1245.79

In the case of $QQ^{\prime}=c\bar{c}$, Eq. (69) gives

$\displaystyle B(c\bar{c})_{1S}$
$\displaystyle=\bar{M}(c\bar{c})_{1S}-2\bar{M}(c\bar{n})_{1S}+\bar{M}(n\bar{n}^{\prime})_{1S},$
$\displaystyle=3068.65-2(1973.23)+619.98,$ $\displaystyle=-257.\,\allowbreak
83\text{ MeV, }$ (70)

where we have used the mean masses in the ground states,
$\bar{M}(c\bar{c})_{1S}=[3M(J/\psi)+M(\eta_{c})]/4=$ $3068.65$ MeV,
$\bar{M}(c\bar{n})_{1S}=[3M(D^{\ast})+M(D)]/4=1973.23$ MeV,
$\bar{M}(n\bar{n}^{\prime})_{1S}=[3M(\rho)+154]/4=619.98$ MeV. Here, we have
used the predicted 1S-wave mass $154.0$ MeV of the pion in Ref. EFG:D09 ,
instead of the observed pion mass as it is too light to be like mass of a
normal $n\bar{n}$ meson in that contributions from light quarks must be
concealed in Eq. (70). With the 1S wave data of the respective bottomonium,
the $B$ meson and $\bar{M}(n\bar{n}^{\prime})_{1S}=619.98$ MeV in Table XV,
one can estimate $B(b\bar{b})$ similarly via Eq. (69).

For the pair $b\bar{c}$, only the 1S and 2S wave mesons(the $B_{c}$) are
available for $\bar{M}(b\bar{c})$ experimentally, and they give, by Eq. (69),

$\displaystyle B(b\bar{c})_{1S}$
$\displaystyle=\bar{M}(b\bar{c})_{1S}-\bar{M}(b\bar{n})_{1S}-\bar{M}(\bar{c}n)_{1S}+\bar{M}(n\bar{n})_{1S},$
$\displaystyle=6320.3-5313.4-1973.2+619.98,$ $\displaystyle=-346.4\text{ MeV,
}$

for the 1S wave pair $b\bar{c}$, in which the ground-state mass
$\bar{M}(b\bar{c})_{1S}$ is estimated by the mean mass of the measured
$M(B_{c},0^{-})_{1S}=6274.5$ MeV and the predicted mass splitting $\Delta
M(B_{c})_{1S}=61$ MeV between the $1^{-}$ and $0^{-}$ states by Ref. EFG:D11 :

$\bar{M}(b\bar{c})_{1S}=\frac{1}{4}[6274.5+3\times(6274.5+\Delta
M(B_{c})_{1S})]=6320.3\text{ MeV.}$

For 2S wave $b\bar{c}$, one can use the measured 2S-wave mass
$M(B_{c},0^{-})_{2S}=6871.2$ MeV of the $B_{c}$ meson and the predicted mass
splitting $\Delta M(B_{c})_{2S}=40$ MeV by Ref. EFG:D11  to get

$\bar{M}(b\bar{c})_{2S}=\frac{1}{4}[6871.2+3\times(6871.2+\Delta
M(B_{c})_{2S})]=6901.2\text{ MeV,}$

and thereby to find

$\displaystyle B(b\bar{c})_{2S}$ $\displaystyle=6901.2-5917.0-2607.5+1423.7$
$\displaystyle=-199.6\text{ MeV,}$

where $\bar{M}(c\bar{n})_{2S}=2607.5$ MeVZylaPDG:D20 , and
$\bar{M}(b\bar{n})_{2S}=5917.0$ MeV taken from the predicted mean massJiaD:E19
of the $B$ mesons in 2S-wave. Also, $B(c\bar{c})_{2S}=-117.3$ MeV and
$B(c\bar{c})_{2S}=-392.98$ MeV. Inspired by atomic spectra in a purely
Coulombic potential, we assume the binding energy $B(Q\bar{Q}^{\prime})$ to
depend upon the reduced pair mass
$\mu_{QQ^{\prime}}=M_{Q}M_{Q^{\prime}}/(M_{Q}+M_{Q^{\prime}})$ in the
$P$-power formKR:D18 ; ZXJ:D21 ,

$B(Q\bar{Q}^{\prime})-B_{0}=k[\mu_{QQ^{\prime}}]^{P}=k\left(\frac{M_{Q}M_{Q^{\prime}}}{M_{Q}+M_{Q^{\prime}}}\right)^{P}.$
(71)

where $B_{0}$ are a constant while the parameters $k=k_{N,L}$ and $P=P_{N,L}$
depends on the radial and angular quantum numbers of the excited
$Q\bar{Q}^{\prime}$. In the ground($1S1s$) state, three values of the binding
$B(c\bar{c})=-257.83$ MeV, $B(b\bar{b})=-561.92$ MeV and $B(b\bar{c})=-346.40$
MeV in Table XVI correspond to the parameters {$B_{0}=62.\allowbreak 066$ MeV,
$P_{0,0}=0.58805$, $k_{0,0}=-388.342$}.

When $Q\bar{Q}^{\prime}$ excited to the 1P wave, one can utilizes the measured
mean masses of the charmonium and bottomonium shown in Table XV to give, by
Eq. (69),

$B(c\bar{c})_{1P}=-100.38\text{ MeV, }B(b\bar{b})_{1P}=-328.81\text{ MeV,}$

and thereby predict (for $b\bar{c}$), by Eq. (71), $P_{0,1}=$ $0.773645$,
$k_{0,1}=-0.209448$ and

$\displaystyle B(b\bar{c})_{1P}$
$\displaystyle=B_{0}+\frac{B(c\bar{c})_{1P}-B_{0}}{[(1+1.44/4.48)/2]^{P_{0,1}}},$
$\displaystyle=62.\allowbreak 066+\frac{-100.377-(62.\allowbreak
066)}{[(1+1.44/4.48)/2]^{0.773645}}$ $\displaystyle=-161.78\text{ MeV.}$ (72)

For the heavy quark-antiquark pairs $\\{c\bar{c},b\bar{b},b\bar{c}\\}$, three
values of string tension $T$ that reproduce, by Eq. (66), the mean masses in
Table XV and the binding energies in Table XVI are

$\\{T(c\bar{c}),T(b\bar{b}),T(b\bar{c})\\}=\left\\{0.21891,0.43367,0.34278\right\\}\text{
GeV}^{2}\text{.}$ (73)

Next, we consider binding energy $B(QQ^{\prime})$ of heavy quark pair
$QQ^{\prime}$ in the color antitriplet($\bar{3}_{c}$). Such a binding corrects
the energy of the QCD string connecting $Q$ and $Q^{\prime}$ in short-
distance, as indicated by heavy-quarkonia spectraZylaPDG:D20 . Note that the
$QQ^{\prime}$ interaction strength in a color triplet is half that of
$Q\bar{Q}^{\prime}$ in a color singlet by color-$SU(3)$ argument, we assume,
as in Ref. KR:D14 , that

$B(QQ^{\prime})=\frac{1}{2}B(Q\bar{Q}^{\prime}),$ (74)

with $Q\bar{Q}^{\prime}$ in color singlet($1_{c}$). It follows that for the
shift $\Delta B_{N,L}\equiv B(QQ^{\prime})_{N,L}-B(QQ^{\prime})_{0,0}$ of the
binding energy relative to the ground states($1S1s$),

$\Delta
B(QQ)_{N,L}=\frac{1}{2}[B(Q\bar{Q}^{\prime})_{N,L}-B(Q\bar{Q}^{\prime})_{0,0}].$
(75)

Applying the binding data in Table XVI to Eq. (74) and Eq. (75) gives

$\displaystyle B(cc)_{1S}$ $\displaystyle=-128.92\text{ MeV,
}B(bb)_{1S}=-280.96\text{ MeV, }B(bc)_{1S}=-173.20\text{ MeV, }$
$\displaystyle B(cc)_{2S}$ $\displaystyle=-58.65\text{ MeV,
}B(bb)_{2S}=-196.49\text{ MeV, }B(bc)_{2S}=-99.78\text{ MeV,}$ $\displaystyle
B(cc)_{1P}$ $\displaystyle=-50.19\text{ MeV, }B(bb)_{1P}=-164.41\text{ MeV,
}B(bc)_{1P}=-80.89\text{ MeV,}$ (76)

for the binding of the 1S-wave heavy pair and

$\displaystyle\vspace{-1.0cm}\Delta B(cc)_{1P}$
$\displaystyle=\frac{1}{2}(-100.38+257.83)=78.73\text{ MeV,}$
$\displaystyle\Delta B(bb)_{1P}$
$\displaystyle=\frac{1}{2}(-328.81+561.92)=116.56\text{ MeV,}$
$\displaystyle\Delta B(bc)_{1P}$
$\displaystyle=\frac{1}{2}(-161.78+346.40)=92.32\text{ MeV,}$ (77)
$\displaystyle\vspace{-1.0cm}\Delta B(cc)_{2S}$
$\displaystyle=\frac{1}{2}(-117.3+257.83)=70.\,\allowbreak 27\text{ MeV,}$
$\displaystyle\Delta B(bb)_{2S}$
$\displaystyle=\frac{1}{2}(-392.98+561.92)=84.\,\allowbreak 47\text{ MeV,}$
$\displaystyle\Delta B(bc)_{2S}$
$\displaystyle=\frac{1}{2}(-199.55+346.40)=73.\,\allowbreak 43\text{ MeV,}$
(78)

for shifts of the binding of the $1P$ and $2S$ waves of the pair, with the
results shown collectively in Table XVI.

Table 16: The binding energy$B(Q\bar{Q}^{\prime})$, $B(QQ^{\prime})$ and their shift $\Delta B$ relative to the ground state. All items are in MeV.  Binding energy | $1S$ | $2S$ | $1P$ |  $\Delta$$B(2S)$ | $\Delta B(1P)$
---|---|---|---|---|---
$B(c\bar{c})$ | -257.83 | -117.30 | -100.38 | 140.53 | 157.45
$B(b\bar{b})$ | -561.92 | -392.98 | -328.81 | 168.94 | 233.11
$B(b\bar{c})$ | -346.40 | -199.55 | -161.78 | 146.85 | 184.32
$B(cc)$ | -128.92 | -58.65 | -50.19 | 70.27 | 78.73
$B(bb)$ | -280.96 | -196.49 | -164.41 | 84.47 | 116.55
$B(bc)$ | -173.20 | -99.78 | -80.89 | 73.43 | 92.32

Finally, we consider DH baryons $qQQ^{\prime}$ with excited diquark
$QQ^{\prime}$. In the picture of QCD string, one can view the baryon as a
string system $[Q-Q]q$ consisting of a heavy subsystem of massive string
$[Q-Q]$ (with each $Q$ at one of ends) being rotating or vibrating and a light
subsystem of a light quark $q$ together with a string connected to it at the
third end. For a 1S or 1P wave diquark in DH baryon labelled by the quantum
numbers $N$ and $L$, respectively, the Regge relation similar to Eq. (66)
takes the form(Appendix D),

$\bar{M}(QQ^{\prime})_{N,L}=M_{Q}+M_{Q^{\prime}}+\Delta
B(QQ^{\prime})_{N,L}+3\frac{\left[T_{QQ^{\prime}}\left((L+\pi
N/\sqrt{12})+2c_{0}\right)\right]^{2/3}}{[2(M_{Q}+M_{Q^{\prime}})]^{1/3}}+c_{1},$
(79)

with $T_{QQ^{\prime}}$ the tension of string within the subsystem $[Q-Q]$ and
$c_{0}$ given by Eq. (67). Here, $c_{1}$ is an additive constant, defined up
to the ground state of the whole DH system.

Notice again that the interaction strength between $QQ^{\prime}$ in a color
$\bar{3}_{c}$ is half that of $Q\bar{Q}^{\prime}$ in a color singlet by a
color-$SU(3)$ argument, we shall assume

$T_{QQ^{\prime}}[\bar{3}_{c}]=\frac{1}{2}T_{Q\bar{Q}^{\prime}}[1_{c}],$ (80)

for heavy quark-quark pairs indicated. So,
$2T_{QQ^{\prime}}c_{0}=T_{Q\bar{Q}^{\prime}}c_{0}$. Accounting the excitation
energy $(\Delta\bar{M}^{(\ast)})_{N,L}=\bar{M}_{N,L}^{(\ast)}-\bar{M}_{0,0}$
of the DH baryon for the energy shift of the heavy subsystem $[Q-Q]$ relative
to its ground state, one can write the excitation energy of the DH baryon with
excited diquark($T_{Q\bar{Q}^{\prime}}=T$ given in Eq. (67)) as

$\bar{M}_{N,L}^{(\ast)}-\bar{M}_{0,0}=\Delta
B(QQ^{\prime})_{N,L}+3\frac{\left[T(L+\pi
N/\sqrt{12})/2+Tc_{0}\right]^{2/3}}{[2(M_{Q}+M_{Q^{\prime}})]^{1/3}}+C_{1},$
(81)

where $C_{1}$ is a constant related to $c_{1}$ and determined by(setting
$N=0=L$),

$C_{1}[qQQ^{\prime}]=-3\left[\frac{(Tc_{0})^{2}}{2(M_{Q}+M_{Q^{\prime}})}\right]^{1/3},$

with $\Delta B(QQ^{\prime})_{0,0}=0$. So, we obtain the baryon mass shift due
to diquark excitations,

$(\Delta\bar{M}^{(\ast)})_{N,L}=\Delta
B(QQ^{\prime})_{N,L}+3\frac{\left[T(L+\pi
N/\sqrt{12})/2+Tc_{0}\right]^{2/3}-(Tc_{0})^{2/3}}{[2(M_{Q}+M_{Q^{\prime}})]^{1/3}},$
(82)

with $Tc_{0}$ given by Eq. (67).

Using the binding data in Eqs. (77) and (78), and the values of $T$ in Eq.
(73), Eq. (82) gives rise to the mean-mass shifts of the DH baryons due to the
2S and 1P diquark excitations relative to their ground states($1S1s$),

$\displaystyle(\Delta\bar{M}^{(\ast)}(\Xi_{cc,bb,bc}))_{2S}$
$\displaystyle=\\{298.87,270.12,280.29\\}\text{ MeV,}$ (83)
$\displaystyle(\Delta\bar{M}^{(\ast)}(\Xi_{cc,bb,bc}))_{1P}$
$\displaystyle=\\{327.73,320.09,318.56\\}\text{ MeV.}$ (84)

and the same values for the $\Omega_{QQ^{\prime}}$($QQ^{\prime}=cc,bb,bc$).
This enable us to define effective masses $E_{QQ^{\prime}}$ of heavy-diquark
in its excited state via the energy shift due to diquark excitations:

$(E_{QQ^{\prime}})_{N,L}=M_{QQ^{\prime}}+(\Delta\bar{M}^{(\ast)})_{N,L},$

which gives explicitly(with $M_{cc,bb,bc}=\\{2865.5,8916.7,5892.2\\}$)

$\displaystyle(E_{cc,bb,bc})_{2S}$
$\displaystyle=\\{3.1644,9.1868,6.1726\\}\text{ GeV,}$ (85)
$\displaystyle(E_{cc,bb,bc})_{1P}$
$\displaystyle=\\{3.1932,9.2368,6.2109\\}\text{ GeV.}$ (86)

##  VII Mass splittings due to 2S and 1P diquark excitations

We consider the low-lying DH baryons with excited diquarks. For the DH baryons
$\Xi_{QQ^{\prime}}$ and $\Omega_{QQ^{\prime}}$ with excited(2S or 1P wave)
diquark $QQ^{\prime}$, the mass splitting arises from the spin interaction
between diquark with effective magnetic moments
$e_{QQ^{\prime}}(\mathbf{L+S}_{QQ^{\prime}})/(2M_{QQ^{\prime}})$ and the light
quark $q$ with spin magnetic moment of $e_{q}\mathbf{S}_{q}/m_{q}$. Here,
$e_{QQ^{\prime}}$ and $e_{q}$ stand for the charges of diquark with internal
orbital angular moment $\mathbf{L}$ and light quark with spin
$\mathbf{S}_{q}$. The spin-interaction of the baryon system is

$H^{SD}=c^{\ast}(\mathbf{L+S}_{QQ^{\prime}})\cdot\mathbf{S}_{q}\text{,}$ (87)

where the spin $\mathbf{S}_{QQ^{\prime}}$ of diquark is integral, and it has
to be unit($\mathbf{S}_{QQ^{\prime}}=1$) in its $S$ wave and to be zero
($\mathbf{S}_{QQ^{\prime}}=0$) in its $P$ wave when $QQ^{\prime}=cc$ or $bb$.
In the $bc$-diquark case, if $QQ^{\prime}=[bc]$, its spin
$\mathbf{S}_{[bc]}=0$ in its $S$ wave and $\mathbf{S}_{[bc]}=1$ in its P wave,
and if $QQ^{\prime}=\\{bc\\}$, one has $\mathbf{S}_{\\{bc\\}}=1$ in its $S$
wave and $\mathbf{S}_{\\{bc\\}}=0$ in its P wave by the symmetry of the baryon
states $q[bc]\ $and $q\\{bc\\}$. Hence, for the system $q[bc]$,
$\mathbf{J}_{d}=\mathbf{L\oplus S}_{QQ^{\prime}}$ $=\\{0\\}$ in S wave of
diquark or $\\{0,1,2\\}$ in P wave of diquark while for the system
$q\\{bc\\}$, $\mathbf{J}_{d}=\mathbf{L\oplus S}_{QQ^{\prime}}$ $=\\{1\\}$ in
the 2S and 1P wave diquark.

We consider the $2S1s$ and $1P1s$ states of the DH baryons in this section. To
find the mass splitting of the baryon multiplets, we utilize the mass scaling
relative to the ground state $D$ mesons (the $D^{\pm}$ with mass $1869.66\pm
0.05$ MeV and the $D^{\ast}(2010)^{\pm}$ with mass $2010.26\pm 0.05$
MeVZylaPDG:D20 ),

$c^{\ast}\left(\Xi_{bb}(bbq)\right)_{2S,1P}=\left(\frac{M_{c}}{(E_{bb})_{2S,1P}}\right)\left(\frac{m_{n}}{m_{q}}\right)c\left(D\right)_{1S},$
(88)

in which $c\left(D\right)_{1S}=M(D^{\ast\pm})-M\left(D^{\pm}\right)=140.6$ MeV
is one of the spin-coupling parameters(that corresponds to the contact-term)
for the $D$ mesons, as Ref. JiaLH:19 .

(1) The $2S1s$ states. For the DH system $q[bc]$, as
$\mathbf{L}=0=\mathbf{S}_{[bc]}$, the baryon mass is simply
$\bar{M}(q[bc])_{nS}$ without multiplet splitting. For the system
$q\\{QQ^{\prime}\\}$, as $\mathbf{L}=0$, the spin-interaction (87) reduces to
$H^{SD}(2S)=c^{\ast}\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}$, in which
$\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}$ has the eigenvalues
$\\{-1,1/2\\}$ for the spin of total system $J=1/2$ or $3/2$, respectively.
The mass formula for the $2S1s$ wave baryons $q\\{QQ^{\prime}\\}$ becomes(we
use $2S$ for $2S1s$ for short),

$M(q\\{QQ^{\prime}\\})_{2S}=\bar{M}_{1S1s}+(\Delta\bar{M}^{(\ast)})_{2S}+c^{\ast}(q\\{QQ^{\prime}\\})\left[\begin{array}[c]{rr}-1&0\\\
0&1/2\end{array}\right]\text{,}\begin{array}[c]{c}J=1/2,\\\ J=3/2,\end{array}$
(89)

with $\bar{M}_{1S}\equiv\bar{M}_{0,0}$ the baryon ground-state masses and
$\Delta\bar{M}^{(\ast)}$ the baryon mass shift in Eq. (82). Using Eq. (88),
one gets

$\displaystyle c^{\ast}\left(\Xi_{bb}(bbn)\right)_{2S}$
$\displaystyle=\left(\frac{M_{c}}{E_{bb}}\right)(140.6\text{ MeV})=\allowbreak
22.04\text{ MeV,}$ $\displaystyle c^{\ast}\left(\Omega_{bb}(bbs)\right)_{2S}$
$\displaystyle=\left(\frac{M_{c}}{E_{bb}}\right)\left(\frac{m_{n}}{m_{s}}\right)(140.6\text{
MeV})=\allowbreak 15.45\text{ MeV,}$ (90)

where $(E_{bb})_{2S}=9.1869$ GeV. Further application of the scaling relations
between the DH baryons

$\displaystyle c^{\ast}\left(\Xi_{cc}\right)_{2S}$
$\displaystyle=c^{\ast}\left(\Xi_{bb}\right)\left(\frac{E_{bb}}{E_{cc}}\right),c^{\ast}\left(\Xi_{bc}\right)_{2S}=c^{\ast}\left(\Xi_{bb}\right)\left(\frac{E_{bb}}{E_{bc}}\right),$
$\displaystyle c^{\ast}\left(\Omega_{cc}\right)_{2S}$
$\displaystyle=c^{\ast}\left(\Omega_{bb}\right)\left(\frac{E_{bb}}{E_{cc}}\right),c^{\ast}\left(\Omega_{bc}\right)_{2S}=c^{\ast}\left(\Omega_{bb}\right)\left(\frac{E_{bb}}{E_{bc}}\right),$
(91)

gives

$\displaystyle c^{\ast}\left(\Xi_{cc}\right)_{2S}$
$\displaystyle=63.\allowbreak 98\text{ MeV,
}c^{\ast}\left(\Xi_{bc}\right)_{2S}=32.80\text{ MeV, }$ $\displaystyle
c^{\ast}\left(\Omega_{cc}\right)_{2S}$ $\displaystyle=44.87\text{ MeV,
}c^{\ast}\left(\Omega_{bc}\right)_{2S}=23.00\text{ MeV.}$ (92)

Given the masses in Table II and the mean masses of the ground-state baryons
$\Omega_{cc}$, $\Omega_{bb}$ and $\Omega_{bc}$ in Ref. EFG:D02 ,
$\Delta\bar{M}^{(\ast)}$ in Eq. (83) and the values of $c^{\ast}$ in Eqs. (90)
and (91), one gets, by Eq. (89), the multiplet masses of the DH baryons
$\Xi_{QQ^{\prime}}$ and $\Omega_{QQ^{\prime}}$ in the $2S1s$ state, listed in
Table VIII through Table XIV, and shown in FIGs 1-8, respectively.

(2) The $1P1s$ states. For the systems $q\\{cc\\}$, $q\\{bb\\}$ and
$q\\{bc\\}$, $\mathbf{S}_{QQ^{\prime}}=0$ and the system spin
$\mathbf{J}=\mathbf{L}\oplus 1/2$ takes values $J=1/2$ and $3/2$. The spin
interaction becomes $c^{\ast}\mathbf{L}\cdot\mathbf{S}_{q}$, which equals to
$c^{\ast}diag[-1,1/2]$. So, the DH baryon mass is

$M(q\\{QQ^{\prime}\\})_{1P}=\bar{M}_{1S1s}+(\Delta\bar{M}^{(\ast)})_{1P}+c^{\ast}\left[\begin{array}[c]{cc}-1&0\\\
0&1/2\end{array}\right]\begin{array}[c]{c}J=1/2,\\\ J=3/2\text{.}\end{array}$
(93)

For the systems $q[bc]$, $\mathbf{S}_{QQ^{\prime}}=1$, and
$\mathbf{J}_{d}=\mathbf{L\oplus S}_{QQ^{\prime}}$ can take values $J_{d}=0$,
$1$ or $2$ and $J=1/2,1/2^{\prime},3/2,3/2^{\prime},5/2$. The baryon systems
can be labelled by $|J,J_{d}\rangle$ and the relation
$\mathbf{J}_{d}\cdot\mathbf{S}_{q}=[J(J+1)-J_{d}(J_{d}+1)-3/4]/2$ infers

$M(q[bc])_{1P}=\bar{M}_{1S1s}+(\Delta\bar{M}^{(\ast)})_{1P}+c^{\ast}diag\left[0,-1,\frac{1}{2},-\frac{3}{2},1\right],$
(94)

in the subspace of {$|J,J_{d}\rangle$}$=${$|1/2,0\rangle$, $|1/2,1\rangle$,
$|3/2,1\rangle$,$|3/2,2\rangle$, $|5/2,2\rangle$}.

Application of the mass scaling Eq. (88) leads to($E_{bb}=9.2368$ GeV)

$\displaystyle c^{\ast}\left(\Xi_{bb}\right)_{1P}$
$\displaystyle=\left(\frac{M_{c}}{E_{bb}}\right)(140.6\text{ MeV})=\allowbreak
21.\allowbreak 92\text{ MeV,}$ $\displaystyle
c^{\ast}\left(\Omega_{bb}\right)_{1P}$
$\displaystyle=\left(\frac{M_{c}}{E_{bb}}\right)\left(\frac{m_{n}}{m_{s}}\right)(140.6\text{
MeV})=\allowbreak 15.37\text{ MeV,}$ (95)

and

$\displaystyle c^{\ast}\left(\Xi_{cc}\right)_{1P}$
$\displaystyle=c^{\ast}\left(\Xi_{bb}\right)_{1P}\left(\frac{9.2368}{3.1933}\right)=63.40\text{
MeV,}$ $\displaystyle c^{\ast}\left(\Xi_{bc}\right)_{1P}$
$\displaystyle=c^{\ast}\left(\Xi_{bb}\right)_{1P}\left(\frac{9.2368}{6.2108}\right)=32.60\text{
MeV,}$ $\displaystyle c^{\ast}\left(\Omega_{cc}\right)_{1P}$
$\displaystyle=c^{\ast}\left(\Omega_{bb}\right)_{1P}\left(\frac{9.2368}{3.1933}\right)=44.46\text{
MeV,}$ $\displaystyle c^{\ast}\left(\Omega_{bc}\right)_{1P}$
$\displaystyle=c^{\ast}\left(\Omega_{bb}\right)_{1P}\left(\frac{9.2368}{6.2108}\right)=22.86\text{
MeV,}$ (96)

Given the masses in Table II and the mean masses of the ground-state baryons
$\Omega_{cc}$, $\Omega_{bb}$ and $\Omega_{bc}$ in Ref. EFG:D02 ,
$\Delta\bar{M}^{(\ast)}$ in Eq. (84) and the $c^{\ast}$ values in Eqs. (95)
and (96), one can apply Eq. (93) and Eq. (94) to obtain the mean mass
$\bar{M}_{1S}+(\Delta\bar{M}^{(\ast)})_{1P}$ and the multiplet masses of the
$1P1s$-wave DH baryons. The results are listed in Table XIII and XIV and shown
in FIGs. 1-8, respectively, for the $2S1s$ and $1P1s$ states of the DH
baryons.

##  VIII Summary and remarks

In this work, we apply a set of relations of mass scaling between heavy
hadrons which is based on chromodynamics similarity among these hadrons to
explore excited doubly heavy baryons $\Xi_{QQ^{\prime}}$ and
$\Omega_{QQ^{\prime}}$. The low-lying S- and P-wave masses of the excited DH
baryons are predicted in the picture of heavy-diquark-light-quark by computing
the mass parts of spin-independent and spin-dependent separately and compared
to other calculations. For the spin-independent mass of the DH baryons, two
Regge trajectories, one linear and the other nonlinear, are employed to
describe the respective excitations of the light quark and diquark internally
up to $2S$ and $1P$ waves, which are derived semiclassically from the model of
the QCD string and matched successfully to the observed heavy baryons and
mesons. In the case of diquark excitation with light quark in ground state,
the short-distance correction of heavy-pair binding are extracted from the
excitation energies of heavy diquarks phenomenologically and incorporated into
Regge relation, by which effective masses of excited diquarks are estimated
and mass scaling is constructed.

Our mass analysis suggests (see FIGs. 1-8) that the mean mass-level spacings
$\Delta\bar{M}$ of heavy diquark excitations are generally narrower than that
of light-quark excitations. This agrees qualitively with a harmonic-oscillator
approximation of the QCD string with massive ends that the level spacing of
baryons $\Delta\bar{M}\sim\omega\sim 1/m_{q}$ or $\Delta\bar{M}\sim
1/\mathrm{Mass}(QQ)$, where $\omega$ stands for oscillator frequency related
to the effective mass $m_{q}$ of the light quark or that of the diquark
$\mathrm{Mass}(QQ)$, which equals $E_{QQ}$ for excited diquark and $M_{QQ}$
for the ground state diquark. This approximation applies preferably in the
low-lying states of a baryon system with smaller size. Meanwhile, it seems
(see FIGs. 1-8) that the mass splitting of spin multiplets of the excited DH
baryons are wider for the diquark excitions than for the excitation of light
quarks, which remain to be explained in the further explorations. We hope the
oncoming experiments like LHCb and Belle to examine our predictions for the
excited DH resonances addressed.

By the way, the binding function Eq.(71), inspired by atomic spectra, mimic
the nonrelativistic spectra of a system of heavy quark and antiquark bounded
by the short-distance Coulomb-like force of single gluon exchange. In a purely
Coulombic potential $V(r)=-(4/3)\alpha_{s}/r$ the energy levels is given by
$E_{n}=-[(4/3)\alpha_{s}]^{2}\mu/(2n^{2})$ ($n=n_{r}+L+1$), linear on the
reduced mass $\mu$ of system. In a quarkonia or the heavy meson $B_{c}$, Eq.
(71) holds qualitatively as $B_{0}$ may also depend weakly on the quantum
numbers ($N,L$) and upon the effective mass of the quarksDeRuju:D75 ; KR:D18 ;
ZXJ:D21 . So, the prediction Eq.(72) for the 1P-wave binding $B(b\bar{c})$ and
ensuing 1P-wave mass calculations of the baryon $\Xi_{bc}$ and $\Omega_{bc}$
are of approximated(uncertain within $10$ MeV roughly).

Low-energy QCD interaction is known to be involved in many aspects, especially
in excited hadrons, and there exist various approaches exploring the excited
doubly heavy baryons Gershtein:2000nx ; EFG:D02 ; Lu:2017meb ;
Giannuzzi:2009gh ; Yoshida:2015tia ; Shah:2017liu ; Eakins:2012jk ;
Roberts:2007ni . The relations of Regge trajectory are established
phenomenologically and rooted in the string picture of QCD interaction. Some
remarks and discussions are in order:

(i) For the DH baryons with 1S-wave diquark, the Regge trajectory stems from
excitation of the light quark $q$ which moves relativistically and is away
from the heavy mass-center of diquark most of time so that the short-distance
interaction between the diquark $QQ^{\prime}$ and $q$ is ignorable, especially
for excited states. The string picture without short-distance correction is
appropriate in this case and the linear Regge relation Eq.(1) based on
applies.

(ii) When diquark of DH baryons excited, an improved(nonlinear) Regge relation
with a short-distance correction due to extra heavy-pair binding is required,
especially for the low-lying excitations of diquarks for which heavy-pairs are
deep in attractive Coulomb-like potential. This is supported by the energy
drop(roughly half or 1/3) of the extracted binding of the 2S and 1P wave
baryons relative to the ground state, as Table XVI and Eq. (69) indicated
phenomenologically.

(iii) While the mass predictions for excited DH baryons in this work is by no
means regorous, our estimation of the baryon spin-multiplet splitting is
general in that mass scaling used is based on the known chromodynamics
similarity between heavy baryons and heavy mesons. The parameterized spin-
dependent interactions Eq. (17) and Eq. (87) is built generally according to
Lorentz structure and tensor nature of interactionLandau ; MukNS:R93 .

ACKNOWLEDGMENTS

Y. S thanks W-N Liu for useful discussions. D. J thanks Xiang Liu and Xiong-
Fei Wang for useful discussions. This work is supported by the National
Natural Science Foundation of China under Grant No. 12165017.

##  Appendix A

For the orbitally excitations of a DH baryon $q(QQ^{\prime})$ with S-wave
diquark $QQ^{\prime}$, the classical energy and orbital angular momentum for
the rotating QCD string read

$E=\sum_{i=QQ^{\prime},q^{\prime}}\left[\frac{m_{\text{bare}i}}{\sqrt{1-\left(\omega
r_{i}\right)^{2}}}+\frac{a}{\omega}\int_{0}^{\omega
r_{i}}\frac{du}{\sqrt{1-u^{2}}}\right],$ (A1)
$l=\sum_{i=QQ^{\prime},q^{\prime}}\left[\frac{m_{i}\omega
r_{i}^{2}}{\sqrt{1-\left(\omega
r_{i}\right)^{2}}}+\frac{a}{\omega^{2}}\int_{0}^{\omega
r_{i}}\frac{u^{2}du}{\sqrt{1-u^{2}}}\right],$ (A2)

where $m_{\text{bare}QQ^{\prime}}$ and $m_{\text{bare}q}$ are the respective
bare masses of the heavy diquarks $QQ^{\prime}$ and light quark $q$, and
$\omega r_{i}=v_{i}$ are the velocity of the string end tied to the quark
$i=QQ^{\prime},q$, $a$ stands for the tension of the QCD string. We define the
effective (dynamical) masses of the heavy diquarks and the light quark in the
CM frame of the baryon by

$M_{QQ^{\prime}}=\frac{m_{\text{bare}QQ^{\prime}}}{\sqrt{1-v_{QQ^{\prime}}^{2}}},m_{q}=\frac{m_{\text{bare}q}}{\sqrt{1-v_{q}^{2}}},$
(A3)

Integrating Eq. (A1) and Eq. (A2) gives

$E=M_{QQ^{\prime}}+m_{q}+\frac{a}{\omega}\left[\arcsin\left(v_{QQ^{\prime}}\right)+\arcsin\left(v_{q}\right)\right],$
(A4)
$l=\frac{1}{\omega}\left(M_{QQ}v_{QQ^{\prime}}^{2}+m_{q}v_{q}^{2}\right)+\sum_{i=QQ^{\prime},q}\left[\arcsin\left(v_{i}\right)-v_{i}\sqrt{1-v_{i}^{2}}\right],$
(A5)

The boundary condition of string at ends linked to heavy quark gives

$\frac{a}{\omega}=\frac{(m_{\text{bare}QQ^{\prime}})v_{QQ^{\prime}}}{1-v_{QQ^{\prime}}^{2}}=\frac{M_{QQ}v_{QQ^{\prime}}}{\sqrt{1-v_{QQ^{\prime}}^{2}}}.$
(A6)

As the diquark $QQ^{\prime}$ is very heavy and moves nonrelativistically in
hadrons, $v_{QQ}$ is small in the limit of heavy
quark($m_{\text{bare}QQ^{\prime}}\rightarrow 0$). A series expansion of Eq.
(A6) upon $v_{QQ}$ gives

$\frac{a}{\omega}\simeq
M_{QQ}v_{QQ^{\prime}}+\frac{1}{2}M_{QQ}v_{QQ^{\prime}}^{3}=P_{QQ^{\prime}}+\frac{P_{QQ^{\prime}}^{3}}{2M_{QQ^{\prime}}^{2}},$
(A7)

From Eq. (A3) one has
$v_{q}=\sqrt{1-\left(m_{\text{bare}q}/m_{q}\right)^{2}}$.

Assuming $q$ to move relativistically($v_{q}\rightarrow 1$), or,
$m_{\text{bare}q}/m_{q}\ll 1$, one finds

$\arcsin\left(v_{q}\right)=\arcsin\left(\sqrt{1-\left(\frac{m_{\text{bare}q}}{m_{q}}\right)^{2}}\right)\simeq\frac{\pi}{2}-\frac{m_{\text{bare}q}}{m_{q}},$
(A8)
$\arcsin\left(v_{QQ^{\prime}}\right)=v_{QQ^{\prime}}+\frac{1}{6}v_{QQ^{\prime}}^{3}+\mathcal{O}\left(v_{QQ^{\prime}}^{5}\right),$
(A9)
$v_{QQ^{\prime}}\sqrt{1-v_{QQ^{\prime}}^{2}}=v_{QQ^{\prime}}-\frac{1}{2}v_{QQ^{\prime}}^{3}+\mathcal{O}\left(v_{QQ^{\prime}}^{5}\right),$
(A10)

Substitute the above relations into Eqs. (A4) and (A5) yields

$E\simeq
M_{QQ^{\prime}}+m_{q}+\frac{a}{\omega}\left[\frac{\pi}{2}-\frac{m_{\text{bare}q}}{m_{q}}+v_{QQ^{\prime}}+\frac{1}{6}v_{QQ^{\prime}}^{3}\right],$
(A11) $\omega l\simeq
M_{QQ^{\prime}}v_{QQ^{\prime}}^{2}+m_{q}+\frac{a}{\omega}\left[\frac{\pi}{4}-\frac{m_{\text{bare}q}}{m_{q}}\right]+\frac{\alpha}{3\omega}v_{QQ^{\prime}}^{3}.$
(A12)

Using Eq. (A7) and eliminating $\omega$, Eqs. (A11) and (A12) combines to
give, when ignoring the tiny term $m_{\text{bare}q}/m_{q}$,

$\left(E-M_{QQ^{\prime}}\right)^{2}=\pi
al+\left(m_{q}+\frac{P_{QQ^{\prime}}^{2}}{M_{QQ^{\prime}}}\right)^{2}-2m_{\text{bare}q}P_{QQ^{\prime}}.$
(A13)

where $P_{QQ^{\prime}}\equiv M_{QQ^{\prime}}v_{QQ^{\prime}}\simeq
M_{QQ^{\prime}}\left(1-m_{\text{bare}QQ^{\prime}}^{2}/M_{QQ^{\prime}}^{2}\right)^{1/2}$.
Taking the small bare-mass limit($m_{\text{bare}q}\rightarrow 0$), Eq. (A13)
leads to Eq. (1), where
$P_{QQ^{\prime}}^{2}/M_{QQ^{\prime}}=M_{QQ^{\prime}}-m_{\text{bare}QQ^{\prime}}^{2}/M_{QQ^{\prime}}$.

##  Appendix B

Given the eigenvalues $j=1/2$ and $3/2$, one solves the bases(eigenfunctions)
$\left|S_{QQ^{\prime}3},S_{q3},l_{3}\right\rangle$ of the
$\mathbf{l}\cdot\mathbf{S}_{q}$ in the $LS$ coupling. The mass formula $\Delta
M=\langle H^{SD}\rangle$ for a DH baryon $qQQ^{\prime}$ with S-wave diquark
$QQ^{\prime}$ can be obtained by diagonalizing the dominate interaction
$a_{1}\mathbf{l}\cdot\mathbf{S}_{q}$ and adding the diagonal elements of other
perturbative spin-interactions in Eq. (17). This can be done by evaluating the
matrix elements of $H^{SD}$ in the $LS$ coupling and then changing the bases
$\left|S_{QQ^{\prime}3},S_{q3},l_{3}\right\rangle$ to the new bases
$\left|J,j\right\rangle$ in the $jj$ coupling to find the mass formula $\Delta
M=\langle H^{SD}\rangle$.

For each interaction terms of Eq. (17), one can evaluate its matrix elements
by explicit construction of states with a given $J_{3}$ as linear combinations
of the baryon states $\left|S_{QQ^{\prime}3},S_{q3},l_{3}\right\rangle$ in the
$LS$ coupling where $S_{q3}+S_{QQ^{\prime}3}+l_{3}=J_{3}$. Due to the rotation
invariance of the matrix elements, it suffices to use a single $J_{3}$ for
each term. Then, one can use

$\mathbf{l}\cdot\mathbf{S}_{i}=\frac{1}{2}\left[l_{+}S_{i-}+l_{-}S_{i+}\right]+l_{3}S_{i3},$
(B1)

to find their elements by applying $\mathbf{l}\cdot\mathbf{S}_{i}$
($i=q,QQ^{\prime}$) on the the third components of angular momenta. For
projected states of baryon with given $J_{3}$, they can be expressed, in the
$LS$ coupling, as

$\displaystyle\left|{\
}^{2}P_{J=1/2},J_{3}=\frac{1}{2}\right\rangle=\frac{\sqrt{2}}{3}\left|1,-\frac{1}{2},0\right\rangle-\frac{1}{3}\left|0,\frac{1}{2},0\right\rangle$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\frac{\sqrt{2}}{3}\left|0,-\frac{1}{2},1\right\rangle+\frac{2}{3}\left|-1,\frac{1}{2},1\right\rangle,$
(B2) $\displaystyle\left|{\
}^{4}P_{J=1/2},J_{3}=\frac{1}{2}\right\rangle=\frac{1}{\sqrt{2}}\left|1,\frac{1}{2},-1\right\rangle-\frac{1}{3}\left|1,-\frac{1}{2},0\right\rangle$
$\displaystyle\quad\quad\quad-\frac{\sqrt{2}}{3}\left|0,\frac{1}{2},0\right\rangle+\frac{1}{3}\left|0,-\frac{1}{2},1\right\rangle+\frac{1}{3\sqrt{2}}\left|-1,\frac{1}{2},1\right\rangle,$
(B3) $\left|{\
}^{2}P_{J=3/2},J_{3}=\frac{3}{2}\right\rangle=\sqrt{\frac{2}{3}}\left|1,-\frac{1}{2},1\right\rangle-\sqrt{\frac{1}{3}}\left|0,\frac{1}{2},1\right\rangle,$
$\displaystyle\left|{\
}^{4}P_{J=3/2},J_{3}=\frac{3}{2}\right\rangle=\sqrt{\frac{3}{5}}\left|1,\frac{1}{2},0\right\rangle-\sqrt{\frac{2}{15}}\left|1,-\frac{1}{2},1\right\rangle$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\frac{2}{\sqrt{15}}\left|0,\frac{1}{2},1\right\rangle,$
(B4) $\left|{\ }^{4}P_{J=5/2},\quad
J_{3}=\frac{5}{2}\right\rangle=\left|1,\frac{1}{2},1\right\rangle.$

and use them to compute the matrix elements of
$\mathbf{l}\cdot\mathbf{S}_{i}$, $\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}$
and $S_{12}/2$ in the basis [${}^{2}P_{J},^{4}P_{J}$]. With some algebra, one
can find

$\left\langle\mathbf{l}\cdot\mathbf{S}_{QQ^{\prime}}\right\rangle_{J=1/2}=\left[\begin{array}[c]{cc}-\frac{4}{3}&-\frac{\sqrt{2}}{3}\\\
-\frac{\sqrt{2}}{3}&-\frac{5}{3}\end{array}\right],\left\langle\mathbf{l}\cdot\mathbf{S}_{q}\right\rangle_{J=1/2}=\left[\begin{array}[c]{cc}\frac{1}{3}&\frac{\sqrt{2}}{3}\\\
\frac{\sqrt{2}}{3}&-\frac{5}{6}\end{array}\right],$ (B5)
$\left\langle\mathbf{l}\cdot\mathbf{S}_{QQ^{\prime}}\right\rangle_{J=3/2}=\left[\begin{array}[c]{cc}\frac{2}{3}&-\frac{\sqrt{5}}{3}\\\
-\frac{\sqrt{5}}{3}&-\frac{2}{3}\end{array}\right],\left\langle\mathbf{l}\cdot\mathbf{S}_{q}\right\rangle_{J=3/2}=\left[\begin{array}[c]{cc}-\frac{1}{6}&\frac{\sqrt{5}}{3}\\\
\frac{\sqrt{5}}{3}&-\frac{1}{3}\end{array}\right],$ (B6)
$\left\langle\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}\right\rangle_{J=1/2}=\left[\begin{array}[c]{cc}-1&0\\\
0&\frac{1}{2}\end{array}\right],\quad\left\langle\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}\right\rangle_{J=3/2}=\left[\begin{array}[c]{cc}-1&0\\\
0&\frac{1}{2}\end{array}\right],$ (B7)
$\left\langle\frac{S_{12}}{2}\right\rangle_{J=1/2}=\left[\begin{array}[c]{cc}0&\frac{\sqrt{2}}{2}\\\
\frac{\sqrt{2}}{2}&-1\end{array}\right],\quad\left\langle\frac{S_{12}}{2}\right\rangle_{J=3/2}=\left[\begin{array}[c]{cc}0&-\frac{\sqrt{5}}{10}\\\
-\frac{\sqrt{5}}{10}&\frac{4}{5}\end{array}\right],$ (B8)

In the subspace of $J=5/2$, one finds

$\left\langle\mathbf{l}\cdot\mathbf{S}_{QQ^{\prime}}\right\rangle_{J=5/2}=1,\quad\left\langle\mathbf{l}\cdot\mathbf{S}_{q}\right\rangle_{J=5/2}=\frac{1}{2},$
(B9)
$\left\langle\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}\right\rangle_{J=5/2}=\frac{1}{2},\quad\left\langle\frac{S_{12}}{2}\right\rangle_{J=5/2}=-\frac{1}{5},$
(B10)

Given the above matrices, one can solve each eigenvalue $\lambda$ of
$\mathbf{l}\cdot\mathbf{S}_{q}$ and the corresponding eigenvectors for a given
$J$, and for that $J$ one can write the baryon states $\left|J,j\right\rangle$
in the $jj$ coupling. They are linear combinations of the $LS$ bases
$\left|{}^{2S+1}P_{J}\right\rangle$ with definite coefficients specified by
the respective eigenvector solved for $J$:

$\lambda=-1:\left|J=\frac{1}{2},j=\frac{1}{2}\right\rangle=\frac{1}{3}\left|1^{2}P_{1/2}\right\rangle-\frac{2\sqrt{2}}{3}\left|1^{4}P_{1/2}\right\rangle,$
(B11)
$\lambda=+\frac{1}{2}:\left|J=\frac{1}{2},j=\frac{3}{2}\right\rangle=\frac{2\sqrt{2}}{3}\left|1^{2}P_{1/2}\right\rangle+\frac{1}{3}\left|1^{4}P_{1/2}\right\rangle,$
(B12)
$\lambda=-1:\left|J=\frac{3}{2},j=\frac{1}{2}\right\rangle=\frac{2}{3}\left|1^{2}P_{3/2}\right\rangle-\frac{\sqrt{5}}{3}\left|1^{4}P_{3/2}\right\rangle,$
(B13)
$\lambda=+\frac{1}{2}:\left|J=\frac{3}{2},j=\frac{3}{2}\right\rangle=\frac{\sqrt{5}}{3}\left|1^{2}P_{3/2}\right\rangle+\frac{2}{3}\left|1^{4}P_{3/2}\right\rangle,$
(B14)
$\lambda=+\frac{1}{2}:\left|J=\frac{5}{2},j=\frac{3}{2}\right\rangle=\left|1^{4}P_{5/2}\right\rangle,$
(B15)

This gives the required baryon states in the heavy diquark limit, by which the
diagonal matrix elements of $\mathbf{l}\cdot\mathbf{S}_{QQ^{\prime}}$,
$S_{12}/2$ and $\mathbf{S}_{QQ^{\prime}}\cdot\mathbf{S}_{q}$ can be obtained.
The detailed results are collected in Table VIII.

##  Appendix C

(1)Quantization condition for heavy quarkonia $Q\bar{Q}$

Consider a heavy quarkonia $Q\bar{Q}$ (at a distance of $r$) in a linear
confining potential $T|r|$ for which the system Hamiltonian is
$H^{Q\bar{Q}}=2\sqrt{\mathbf{p}^{2}+M_{Q}^{2}}+T|r|$. Here, we have ignored
the short-distance interaction. In the frame of center-of-mass, the heavy
quark $Q$ moves equivalently in a confining potential $Tx$, with $x=r/2$, and
then the Hamiltonian for $Q$ as a half system of the quarkonia $Q\bar{Q}$,
becomes

$H^{Q}=\frac{1}{2}H^{Q\bar{Q}}=\sqrt{p_{x}^{2}+l_{q}^{2}/x^{2}+M_{Q}^{2}}+T|x|.$
(C1)

Using the semiclassical WKB analysis upon Eq. (C1) for the radial
excitations($l_{q}=0$), one has a WKB quantization conditionSonnenW:jh2014
111It is written as $\int_{x_{-}}^{x_{+}}p(x)dx=\pi N$ in the cited
literature.,

$2\int_{x_{\\_}}^{x_{+}}p_{x}(x)dx=2\pi(N+c_{0}),$ (C2)

with
$p_{x}(x)=|\mathbf{p}_{x}|=\sqrt{[\left(E(Q)-T|x|\right)^{2}-M_{Q}^{2}]}$,
$x_{+}=(E(Q)-M_{Q})/T=-x_{-}$ the classical “turning points”, $N$ the radial
quantum number, $c_{0}$ a constant. Here, $E(Q)$ is the semiclassical value of
$H^{Q}$, and the factor 2 before the integral in Eq. (C2) arises from
underlying spinor nature of a quark whose wave function returns to original
value after double cycle of journey in the position spaceJiaD:E19 . Assuming
$Q$ to be in S wave(moving radially), integration of Eq. (C2) gives

$\pi(N+c_{0})=\frac{E(Q)^{2}}{T}\left[\sqrt{1-B^{2}}+B^{2}\ln\left(\frac{1-\sqrt{1-B^{2}}}{B}\right)\right],$
(C3)

with $B\equiv M_{Q}/E(Q)$.

Above result is only for the half system. Transforming to the whole system of
quarkonia by mapping $E(Q)\rightarrow E/2$ and $N\rightarrow
N/2$SonnenW:jh2014 , Eq. (C3) gives($B\rightarrow\tilde{B}=2M_{Q}/E$)

$\displaystyle 2\pi T(N+2c_{0})$
$\displaystyle=E^{2}\left[\sqrt{1-\tilde{B}^{2}}+\tilde{B}^{2}\ln\left(\frac{1-\sqrt{1-\tilde{B}^{2}}}{\tilde{B}}\right)\right]$
$\displaystyle\simeq
E^{1/2}\left[\frac{4\sqrt{2}}{3}E^{3/2}\epsilon^{3/2}-\frac{7\sqrt{2}}{15}E^{3/2}\epsilon^{5/2}+\mathcal{O}(\epsilon^{7/2})\right]$
(C4)

where $\epsilon\equiv 1-\tilde{B}=1-2M_{Q}/E$, which is small in heavy quark
limit and $E=2M_{Q}/(1-\epsilon)\simeq 2M_{Q}$. Thus, Eq. (C4) leads to, to
the leading order of $\epsilon$,

$3\pi T(N+2c_{0})=4\sqrt{M_{Q}}(E-2M_{Q})^{3/2},$

which gives Eq. (64).

(2) Improved(nonlinear) Regge relation

Rewriting in a typical form of standard trajectory,
$\left(\bar{M}-const.\right)^{3/2}\sim$ quantum numbers, for the heavy
quarkonia system, the Regge relation Eq. (63) in Ref. BurnsPP:D10  becomes

$\left(\bar{M}-2M_{Q}\right)^{3/2}=\frac{3\sqrt{3}}{2\sqrt{M_{Q}}}TL.$ (C5)

Comparing the radial and angular slopes(linear coefficients in $N$ and $L$) in
RHS of Eq. (64) and Eq. (C5), which is $\pi:\sqrt{12}$, one can combine two
equations into one unified form:

$\left[\bar{M}_{N,L}-2M_{Q}\right]^{3/2}=\frac{3\sqrt{3}T}{2\sqrt{M_{Q}}}\left(L+\frac{\pi
N}{\sqrt{12}}+2c_{0}\right).$ (C6)

In the derivationBurnsPP:D10  of Eq. (63) and that of Eq. (64) shown in
Appendix Eq.(C1), the linear confining interaction between $Q$ and $\bar{Q}$
is assumed, with the short-distance force between them ignored. As the short-
distance force is required for low-lying quarkonia system and violates the
typical linear trajectory in Eq. (C6), one way out is to assume that the
short-distance attractive force between $Q$ and $\bar{Q}$ in the color-
singlet($1_{c}$) would provide an extra(negative) energy $B(Q\bar{Q})_{N,L}$
to the spectra of the $Q\bar{Q}$ system and to deduct $B(Q\bar{Q})_{N,L}$
(named the binding energy) from the hadron mass $\bar{M}_{N,L}$ in Eq. (C6) so
that the LHS of Eq. (C6) becomes the remaining string energy solely:
$(\bar{M}_{N,L}-2M_{Q})^{3/2}\rightarrow(\bar{M}_{N,L}-B(Q\bar{Q})_{N,L}-2M_{Q})^{3/2}$.
In doing so, the arguments by the classical string pictureBurnsPP:D10  and
that in Eq. (1) given above remain valid and the formula Eq. (C6) remains
intact formally up to a replacement
$\bar{M}_{N,L}\rightarrow\bar{M}_{N,L}-B(Q\bar{Q})_{N,L}$. This gives rise to
Eq. (65).

The binding energy $B(Q\bar{Q})_{N,L}$(in the excited states $|N,L\rangle$ of
system) depends on the quantum numbers($N,L$) of the system considered KR:D18
and thereby violates the linearity of the Regge relations Eq. (C5) and Eq.
(64). Normally, such a term is negative since when two quarks($Q$ and
$\bar{Q}$) are heavy enough to stay close each other, they both experiences an
attractive Coulomb-like force of single gluon exchange, as the observed heavy-
quarkonia spectraZylaPDG:D20  of the cascade type indicated.

##  Appendix D

In the QCD string picture, one can view a DH baryons $qQQ^{\prime}$, with
excited diquark $QQ^{\prime}$, as a string system $[Q-Q]q$, consisting of a
(heavy) subsystem of massive string $[Q-Q]$ (each $Q$ at one of ends) and a
light subsystem of a light-quark $q$ and the string connected to it.

In the semiclassical approximation, one can assume the light subsystem of $q$
and attached string to it to be in stationary state while the heavy subsystem
$[Q-Q]$ is excited to the excited state(denoted by $|N,L\rangle$, say). As
such, the excitation of the heavy subsystem $[Q-Q]$ in color
antitriplet($\bar{3}_{c}$) with string tension $T_{QQ}$ resembles the
excitation of a heavy quarkonia $Q\bar{Q}$ in color singlet($1_{c}$) with
string tension $T$ up to a color(strength) factor of string interaction which
is taken to be half commonly. Based on this similarity, one can write the
excitation energy of the heavy system, by analogy with Eq. (66),

$\bar{M}(QQ)_{N,L}-2M_{Q}=\Delta B(QQ)_{N,L}+3\frac{\left[T_{QQ}\left((L+\pi
N/\sqrt{12})+2c_{0}\right)\right]^{2/3}}{[4M_{Q}]^{1/3}}+c_{1},$ (D1)

in which the first term $\Delta B(QQ)_{N,L}=B(QQ)_{N,L}-B(QQ)_{0,0}$ in the
RSH accounts for the short-distance contribution due to heavy-quark binding
and the second term for the excited string energy of the heavy subsystem in
the $|N,L\rangle$ state, $c_{0}$ is given by Eq. (67). We add an addictive
constant $c_{1}$ since $\bar{M}(QQ)$ is defined up to the ground state of the
whole DH system containing $QQ$.

Extending Eq. (D1) to the diquark case of $QQ=bc$, one gets Eq. (79)
generally, by heavy quark symmetry.

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