#  $\lambda$\- and $\rho$-trajectories for the hidden bottom/charm tetraquarks
$(Qq)(\bar{Q}\bar{q}^{\prime})$

Jia-Qi Xie<EMAIL_ADDRESS>School of Physics and Information Engineering,
Shanxi Normal University, Taiyuan 030031, China  He Song<EMAIL_ADDRESS>School of Physics and Information Engineering, Shanxi Normal University,
Taiyuan 030031, China  Xia Feng<EMAIL_ADDRESS>School of Physics and
Information Engineering, Shanxi Normal University, Taiyuan 030031, China
Jiao-Kai Chen<EMAIL_ADDRESS><EMAIL_ADDRESS>(corresponding
author) School of Physics and Information Engineering, Shanxi Normal
University, Taiyuan 030031, China

###### Abstract

We propose the Regge trajectory relations for the heavy tetraquarks
$(Qq)(\bar{Q}\bar{q}^{\prime})$ $(Q=b,\,c;\,q,\,q^{\prime}=u,\,d,\,s)$ with
hidden bottom and charm. By employing the new relations, both the
$\lambda$-trajectories and the $\rho$-trajectories for the tetraquarks
$(Qq)(\bar{Q}\bar{q}^{\prime})$ can be discussed. The masses of the
$\lambda$-mode excited states and the $\rho$-mode excited states are
estimated, and they agree with other theoretical predictions. We show that the
behaviors of the $\rho$-trajectories are different from those of the
$\lambda$-trajectories. The $\rho$-trajectories behave as
$M{\sim}x_{\rho}^{1/2}$ $(x_{\rho}=n_{r},\,l)$ while the
$\lambda$-trajectories behave as $M{\sim}x_{\lambda}^{2/3}$
$(x_{\lambda}=N_{r},\,L)$. Moreover, the Regge trajectory behaviors for other
types of tetraquarks are investigated based on the spinless Salpeter equation.
We show that both the $\lambda$-trajectories and the $\rho$-trajectories are
concave downwards in the $(M^{2},\,x)$ plane. The Regge trajectories for the
tetraquarks containing the light diquark and/or the light antidiquark also are
concave in the $(M^{2},\,x)$ plane when the masses of the light constituents
are included and the confining potential is linear.

$\lambda$-trajectory, $\rho$-trajectory, tetraquark, mass

##  I Introduction

As a type of exotic hadrons, tetraquarks will enhance the study of hadrons and
will provide new probes for the understanding of QCD Jaffe:1976ig ;
Jaffe:1976ih ; pdg:2024 ; Gross:2022hyw ; Bicudo:2022cqi ; Liu:2019zoy ;
Jaffe:2004ph ; Esposito:2016noz . The heavy tetraquarks
$(Qq)(\bar{Q}\bar{q}^{\prime})$ $(Q=b,\,c;\,q,q^{\prime}=u,\,d,\,s)$ with
hidden bottom and charm have been studied through various methods. These
approaches include the relativistic quark model based on the quasipotential
approach in QCD Ebert:2008se ; Faustov:2021hjs ; Ebert:2008kb , the non-
relativistic potential model Tiwari:2022azj ; Lundhammar:2020xvw , the
spinless Salpeter equation Ferretti:2020ewe ; Giannuzzi:2023vrx , the
relativistic four-body Faddeev-Yakubovsky equation Hoffer:2024alv , the
dynamical diquark model Giron:2020qpb , the homogeneous Lippmann-Schwinger
integral equation Hadizadeh:2015cvx , the effective Hamiltonian in the adopted
chromomagnetic interaction model Wu:2018xdi , the chiral quark model
Liu:2022vyy , the sum rules Chen:2017dpy ; Wang:2019mxn , the relativistic
four-quark equations in the framework of coupled-channel formalism
Gerasyuta:2008cb , the constituent quark model Maiani:2004vq , among others.

The Regge trajectory111A Regge trajectory of hadrons is assumed to be written
as $M=m_{R}+\beta_{x}(x+c_{0})^{\nu}$ $(x=l,\,n_{r})$, where $M$ is mass of
the bound state, $l$ is the the orbital angular momentum, and $n_{r}$ is the
radial quantum number. $m_{R}$ and $\beta_{x}$ are parameters. For simplicity,
the plots in the $(M,\,x)$ plane Chen:2022flh ; Chen:2023cws , in the
$(M,\,(x-c_{0})^{\nu})$ plane Burns:2010qq , in the $(M^{2},\,x)$ plane
Chen:2018nnr ; Chen:2021kfw , in the $((M-m_{R})^{2},\,x)$ plane Chen:2023djq
; Chen:2023web  or in the $((M-m_{R})^{1/{\nu}},\,x)$ plane [see Fig. 6 in
this work] are all called the Chew-Frautschi plots. The Regge trajectories can
be plotted in these different planes.  is one of the effective approaches
widely used in the study of hadron spectra Regge:1959mz ; Chew:1961ev ;
Chew:1962eu ; Nambu:1978bd ; Collins:1971ff ; Inopin:1999nf ; Irving:1977ea .
Ref. Selem:2006nd  gives a compact and predictive unified picture of mesons,
baryons, and tetraquarks with the loaded flux-tube model. Ref. Nielsen:2018uyn
presents relations between the Regge trajectories of mesons, baryons, and
tetraquarks based on the Holographic light front QCD. In Ref.
Sonnenschein:2018fph , the masses of mesons, baryons, glueballs and
tetraquarks are predicted by the Regge trajectories obtained from the
Holography Inspired Stringy Hadron model. In Ref. MartinContreras:2020cyg ,
mesons, hybrid mesons and multiquark states are discussed by the nonlinear
Regge trajectories in the model AdS/QCD with dilaton. In Refs. Chen:2023web ;
Chen:2023djq , we discuss the universal descriptions of the heavy-heavy and
heavy-light systems by using the Regge trajectories. In preceding works, the
Regge trajectories for the tetraquarks are the $\lambda$-trajectories. To our
knowledge, no studies address the $\rho$-trajectories for tetraquarks although
there are many studies on the $\rho$-excited states of tetraquarks. In the
diquark picture, the hidden bottom tetraquarks are composed of one heavy-light
diquark $(Qq)$ and one heavy-light antidiquark $(\bar{Q}\bar{q}^{\prime})$. By
applying the Regge trajectory relation for the heavy-light diquarks
Chen:2023cws  and the Regge trajectory formula for the heavy-heavy systems
Chen:2023djq , we present the Regge trajectory relations for the hidden
bottom/charm tetraquarks. The masses of the $\lambda$-mode excited and
$\rho$-mode excited tetraquarks with hidden bottom and charm are estimated.
Both the $\lambda$-trajectories and $\rho$-trajectories are investigated.

The paper is organized as follows: In Sec. II, the Regge trajectories for the
hidden bottom/charm tetraquarks are discussed. The conclusions are presented
in Sec. III. The discussions on the Regge trajectory behaviors for various
tetraquarks are in appendix.

##  II Regge trajectories for the tetraquarks with hidden bottom and charm

In this section, with the help of the diquark Regge trajectories Chen:2023cws
; Feng:2023txx ; Chen:2023ngj , we investigate the Regge trajectory behaviors
for various tetraquarks. The Regge trajectory relations are presented, which
can be employed to discuss both the $\lambda$-trajectories and the
$\rho$-trajectories.

###  II.1 Preliminary

In the diquark picture, tetraquarks consist of one diquark and one
antidiquark, see Fig. 1. $\rho_{1}$ ($\rho_{2}$) separates the quarks
(antiquarks) in the diquark (antidiquark), and $\lambda$ separates the diquark
and the antidiquark. There exist three excited modes: the $\rho_{1}-$mode
involves the radial and orbital excitation in the diquark, the $\rho_{2}-$mode
involves the radial and orbital excitation in the antidiquark, and the
$\lambda-$mode involves the radial or orbital excitation between the diquark
and antidiquark. Consequently, there exist three series of Regge trajectories:
two series of $\rho$-Regge trajectories and one series of $\lambda$-Regge
trajectories.

In $SU_{c}(3)$, there is attraction between quark pairs $(qq^{\prime})$ in the
color antitriplet channel and this is just twice weaker than in the color
singlet $q\bar{q}^{\prime}$ in the one-gluon exchange approximation. Only the
$\bar{3}_{c}$ representation of $SU_{c}(3)$ is considered in the present work;
the $6_{c}$ representation is not considered. The tetraquarks considered are
composed of a diquark and an antidiquark in color $\bar{3}$ and $3$
configuration.

Figure 1: Schematic diagram of the tetraquarks in the diquark-antidiquark
picture.

In the diquark picture, the state of tetraquark is denoted as

$\left((q_{1}q^{\prime}_{1})^{{\bar{3}_{c}}}_{n_{1}^{2s_{1}+1}l_{1j_{1}}}(\bar{q}_{2}\bar{q}^{\prime}_{2})^{{{3}_{c}}}_{n_{2}^{2s_{2}+1}l_{2j_{2}}}\right)^{1_{c}}_{N^{2j+1}L_{J}},$
(1)

where $\bar{3}_{c}$ denotes the color antitriplet state of diquark, and
$1_{c}$ represents the color singlet state of tetraquark. [The superscript
$1_{c}$ is often omitted because the tetraquarks are colorless.] The diquark
$(q_{1}q^{\prime}_{1})$ is $\\{q_{1}q^{\prime}_{1}\\}$ or
$[q_{1}q^{\prime}_{1}]$. The antidiquark $(\bar{q}_{2}\bar{q}^{\prime}_{2})$
is $\\{\bar{q}_{2}\bar{q}^{\prime}_{2}\\}$ or
$[\bar{q}_{2}\bar{q}^{\prime}_{2}]$. $\\{qq^{\prime}\\}$ and $[qq^{\prime}]$
indicate the permutation symmetric and antisymmetric flavor wave functions,
respectively. $N=N_{r}+1$, where $N_{r}=0,\,1,\,\cdots$.
$n_{1,2}=n_{r_{1,2}}+1$, where $n_{r_{1,2}}=0,\,1,\,\cdots$. $N_{r}$,
$n_{r_{1}}$ and $n_{r_{2}}$ are the radial quantum numbers of the tetraquark,
diquark 1, antidiquark 2, respectively. The completely antisymmetric states
for the diquarks in $\bar{3}_{c}$ are listed in Table 8.
$\vec{J}=\vec{L}+\vec{j}$, $\vec{j}=\vec{j}_{1}+\vec{j}_{2}$,
$\vec{j}_{1}=\vec{s}_{1}+\vec{l}_{1}$, $\vec{j}_{2}=\vec{s}_{2}+\vec{l}_{2}$.
$\vec{J}$, $\vec{j}_{1}$ and $\vec{j}_{2}$ are the spins of tetraquark,
diquark 1, antidiquark 2, respectively. $\vec{j}$ is the summed spin of
diquarks and antidiquarks in the tetraquark. $L$, $l_{1}$ and $l_{2}$ are the
orbital quantum numbers of tetraquark, diquark 1 and antidiquark 2,
respectively. $\vec{s}_{1}$ ($\vec{s}_{2}$) is the summed spin of quarks
(antiquarks) in the diquark (antidiquark).

###  II.2 Regge trajectory relations for the hidden bottom/charm tetraquarks

The tetraquarks with hidden bottom and charm consists of one heavy-light
diquark $(Qq)$ and one heavy-light antidiquark $(\bar{Q}\bar{q}^{\prime})$.
Both $(Qq)$ and $(\bar{Q}\bar{q}^{\prime})$ are heavy because the bottom/charm
quark are heavy. Therefore, the tetraquarks $(Qq)(\bar{Q}\bar{q}^{\prime})$
are the heavy-heavy systems in case of the $\lambda$-mode excitations.
Meanwhile, the Regge trajectories for $(Qq)(\bar{Q}\bar{q}^{\prime})$ exhibit
the properties of the heavy-light systems in case of the $\rho$-mode
excitations. According to Eqs. (C.3) and (C.3), we list the relations for the
tetraquarks with hidden bottom and charm

$\displaystyle M$
$\displaystyle=m_{R{\lambda}}+\beta_{x_{\lambda}}(x_{\lambda}+c_{0x_{\lambda}})^{2/3}\;(x_{\lambda}=L,\,N_{r}),$
$\displaystyle M_{\rho_{1}}$
$\displaystyle=m_{R\rho_{1}}+\beta_{x_{\rho_{1}}}\sqrt{x_{\rho_{1}}+c_{0x_{\rho_{1}}}}\;(x_{\rho_{1}}=l_{1},\,n_{r_{1}}),$
$\displaystyle M_{\rho_{2}}$
$\displaystyle=m_{R\rho_{2}}+\beta_{x_{\rho_{2}}}\sqrt{x_{\rho_{2}}+c_{0x_{\rho_{2}}}}\;(x_{\rho_{2}}=l_{2},\,n_{r_{2}}),$
(2)

where

$\displaystyle m_{R{\lambda}}$ $\displaystyle=M_{\rho_{1}}+M_{\rho_{2}}+C,$
$\displaystyle m_{R\rho_{1}}$ $\displaystyle=m_{Q}+m_{q}+C/2,$ $\displaystyle
m_{R\rho_{2}}$ $\displaystyle=m_{Q}+m_{q^{\prime}}+C/2,$
$\displaystyle\beta_{L}$
$\displaystyle=\frac{3}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fL},$
$\displaystyle\beta_{N_{r}}$
$\displaystyle=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fN_{r}},$
$\displaystyle\mu_{\lambda}$
$\displaystyle=\frac{M_{\rho_{1}}M_{\rho_{2}}}{M_{\rho_{1}}+M_{\rho_{2}}},$
$\displaystyle\beta_{l_{1}}$
$\displaystyle=\sqrt{2\sigma}c_{fl_{1}},\quad\beta_{n_{r_{1}}}=\sqrt{\pi\sigma}c_{fn_{r_{1}}},$
$\displaystyle\beta_{l_{2}}$
$\displaystyle=\sqrt{2\sigma}c_{fl_{2}},\quad\beta_{n_{r_{2}}}=\sqrt{\pi\sigma}c_{fn_{r_{2}}}.$
(3)

$M$, $M_{\rho_{1}}$, $M_{\rho_{2}}$, $m_{Q}$, and $m_{q}$ are the tetraquark
mass, the diquark mass, the antidiquark mass, the heavy quark mass, and the
light quark mass, respectively. $\sigma$ is the string tension. $C$ is a
fundamental parameter. $c_{fx}$ are theoretically equal to one but are fitted
in practice. $c_{0x}$ vary with Regge trajectories.

###  II.3 Parameters

The quark masses, the string tension $\sigma$ and the parameter $C$ are from
Ref. Faustov:2021hjs . The parameters for the heavy-light diquarks $(bu)$,
$(bs)$, $(cu)$, and $(cs)$ are from Ref. Chen:2023cws  and listed in Table 1.
With these parameters determined, the $\rho-$modes and the diquark masses can
be discussed, see Eqs. (II.2) and (II.2). To discuss the masses of the hidden
bottom/charm tetraquarks and the $\lambda-$modes excited states, the
parameters $c_{fL}$, $c_{fN_{r}}$, $c_{0{L}}$ and $c_{0N_{r}}$ should be
determined, see Eqs. (42) and (43). More discussions are in section D. Once
all parameters are determined, the Regge trajectories can be discussed and can
be used to estimate the masses of the tetraquarks with hidden bottom and
charm.

Table 1: The values of parameters Chen:2023cws ; Faustov:2021hjs . |  $m_{u,d}=0.33\;{{\rm GeV}}$, $m_{s}=0.50\;{{\rm GeV}}$, $m_{b}=4.88\;{{\rm GeV}}$,
---|---
|  $m_{c}=1.55\;{{\rm GeV}}$, $\sigma=0.18\;{{\rm GeV}^{2}}$, $C=-0.3\;{{\rm
GeV}}$,
$(bu)$ |  $c_{fn_{r_{1}}}=c_{fn_{r_{2}}}=0.988$, $c_{fl_{1}}=c_{fl_{2}}=0.965$,
|  $c_{0n_{r_{1}}}(1^{1}s_{0})=c_{0n_{r_{2}}}(1^{1}s_{0})=0.125$,
|  $c_{0n_{l_{1}}}(1^{1}s_{0})=c_{0n_{l_{2}}}(1^{1}s_{0})=0.18$,
|  $c_{0n_{r_{1}}}(1^{3}s_{1})=c_{0n_{r_{2}}}(1^{3}s_{1})=0.155$,
|  $c_{0n_{l_{1}}}(1^{3}s_{1})=c_{0n_{l_{2}}}(1^{3}s_{1})=0.22$,
$(bs)$ |  $c_{fn_{r_{1}}}=c_{fn_{r_{2}}}=0.953$, $c_{fl_{1}}=c_{fl_{2}}=0.919$,
|  $c_{0n_{r_{1}}}(1^{1}s_{0})=c_{0n_{r_{2}}}(1^{1}s_{0})=0.08$,
|  $c_{0n_{l_{1}}}(1^{1}s_{0})=c_{0n_{l_{2}}}(1^{1}s_{0})=0.115$.
|  $c_{0n_{r_{1}}}(1^{3}s_{1})=c_{0n_{r_{2}}}(1^{3}s_{1})=0.11$,
|  $c_{0n_{l_{1}}}(1^{3}s_{1})=c_{0n_{l_{2}}}(1^{3}s_{1})=0.16$,
$(cu)$ |  $c_{fn_{r_{1}}}=c_{fn_{r_{2}}}=1.000$, $c_{fl_{1}}=c_{fl_{2}}=1.038$,
|  $c_{0n_{r_{1}}}(1^{1}s_{0})=c_{0n_{r_{2}}}(1^{1}s_{0})=0.065$,
|  $c_{0n_{l_{1}}}(1^{1}s_{0})=c_{0n_{l_{2}}}(1^{1}s_{0})=0.095$,
|  $c_{0n_{r_{1}}}(1^{3}s_{1})=c_{0n_{r_{2}}}(1^{3}s_{1})=0.17$,
|  $c_{0n_{l_{1}}}(1^{3}s_{1})=c_{0n_{l_{2}}}(1^{3}s_{1})=0.19$,
$(cs)$ |  $c_{fn_{r_{1}}}=c_{fn_{r_{2}}}=1.016$, $c_{fl_{1}}=c_{fl_{2}}=1.015$,
|  $c_{0n_{r_{1}}}(1^{1}s_{0})=c_{0n_{r_{2}}}(1^{1}s_{0})=0.03$,
|  $c_{0n_{l_{1}}}(1^{1}s_{0})=c_{0n_{l_{2}}}(1^{1}s_{0})=0.055$,
|  $c_{0n_{r_{1}}}(1^{3}s_{1})=c_{0n_{r_{2}}}(1^{3}s_{1})=0.095$,
|  $c_{0n_{l_{1}}}(1^{3}s_{1})=c_{0n_{l_{2}}}(1^{3}s_{1})=0.135$,

###  II.4 $\lambda-$ and $\rho-$trajectories for the hidden bottom tetraquarks

Table 2: The masses of the $\lambda$-mode excited states of the hidden bottom tetraquarks (in ${{\rm GeV}}$). The notation in Eq. (1) is rewritten as $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$. $n=u,\,d$. Eqs. (II.2), (42) and (43) are used. $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$ | $(bn)(\bar{b}\bar{n})$ | $(bn)(\bar{b}\bar{s})$ | $(bs)(\bar{b}\bar{s})$
---|---|---|---
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 10.46 | 10.57 | 10.68
$|1^{1}s_{0},1^{1}s_{0},2^{1}S_{0}\rangle$ | 10.91 | 11.02 | 11.12
$|1^{1}s_{0},1^{1}s_{0},3^{1}S_{0}\rangle$ | 11.21 | 11.32 | 11.43
$|1^{1}s_{0},1^{1}s_{0},4^{1}S_{0}\rangle$ | 11.47 | 11.57 | 11.68
$|1^{1}s_{0},1^{1}s_{0},5^{1}S_{0}\rangle$ | 11.70 | 11.80 | 11.91
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 10.46 | 10.57 | 10.68
$|1^{1}s_{0},1^{1}s_{0},1^{1}P_{1}\rangle$ | 10.78 | 10.89 | 11.00
$|1^{1}s_{0},1^{1}s_{0},1^{1}D_{2}\rangle$ | 11.01 | 11.12 | 11.22
$|1^{1}s_{0},1^{1}s_{0},1^{1}F_{3}\rangle$ | 11.20 | 11.31 | 11.41
$|1^{1}s_{0},1^{1}s_{0},1^{1}G_{4}\rangle$ | 11.37 | 11.48 | 11.59
$|1^{1}s_{0},1^{1}s_{0},1^{1}H_{5}\rangle$ | 11.53 | 11.64 | 11.74

Figure 2: The $\lambda-$trajectories for the hidden bottom tetraquarks. Circles represent the predicted data and the blue lines are the $\lambda-$trajectories, see Eq. (II.2). Data are listed in Table 2. Table 3: Comparison of theoretical predictions for the masses of the hidden bottom tetraquarks (in GeV). The charge conjugation $C$ is defined only for $q=q^{\prime}$. $J^{PC}$ | $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$ | Tetraquark | Our | FGM Faustov:2021hjs  | FS Ferretti:2020ewe  | TR Tiwari:2022azj  | HK Hadizadeh:2015cvx
---|---|---|---|---|---|---|---
$0^{++}$ | $|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | $(bn)(\bar{b}\bar{n})$ | 10.46 | 10.471 |  |  | 10.410
|  | $(bn)(\bar{b}\bar{s})$ | 10.57 | 10.572 | 10.407 | 10.522 |
|  | $(bs)(\bar{b}\bar{s})$ | 10.68 | 10.662 |  | 10.711 | 10.613
| $|1^{1}s_{0},1^{1}s_{0},2^{1}S_{0}\rangle$ | $(bn)(\bar{b}\bar{n})$ | 10.91 | 10.917 |  |  | 10.899
|  | $(bn)(\bar{b}\bar{s})$ | 11.02 | 11.018 | 10.909 | 11.085 |
|  | $(bs)(\bar{b}\bar{s})$ | 11.12 | 11.111 |  | 11.244 | 11.100
$1^{--}$ | $|1^{1}s_{0},1^{1}s_{0},1^{1}P_{1}\rangle$ | $(bn)(\bar{b}\bar{n})$ | 10.78 | 10.807 |  |  | 10.806
|  | $(bn)(\bar{b}\bar{s})$ | 10.89 | 10.907 | 10.790 | 11.024 |
|  | $(bs)(\bar{b}\bar{s})$ | 11.00 | 11.002 |  | 11.184 | 11.009

The tetraquarks with hidden bottom include the neutral states
$(bu)(\bar{b}\bar{u})$ and $(bd)(\bar{b}\bar{d})$, the charged states
$(bu)(\bar{b}\bar{d})$ and $(bd)(\bar{b}\bar{u})$, the states with open
strangeness $(bs)(\bar{b}\bar{n})$ and $(bn)(\bar{b}\bar{s})$ ($n=u,\,d$), and
the states with hidden strangeness $(bs)(\bar{b}\bar{s})$. We assume
$m_{u}=m_{d}$, therefore, the masses for diquarks $(bd)$ and $(bu)$ are equal
and it holds for the tetraquarks as well.

When calculating the $\lambda$-mode excitations, the scalar diquark and the
scalar antidiquark are used. The radially and orbitally excited states of the
$\lambda-$mode are calculated by using Eqs. (II.2), (42) and (43) and
parameters in Table 1. The calculated results are listed in Table 2. More
results can be easily obtained in a similar manner. The $\lambda-$trajectories
for the hidden bottom tetraquarks are shown in Fig. 2.

Table 4: The masses of the $\rho$-mode excited states of the hidden bottom tetraquarks (in ${{\rm GeV}}$). The notation in Eq. (1) is rewritten as $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$. $(M^{\prime},\,M^{\prime\prime})$ is for $\left(|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle,\;|n_{2}^{2s_{2}+1}l_{2j_{2}},n_{1}^{2s_{1}+1}l_{1j_{1}},N^{2j+1}L_{J}\rangle\right)$. For $(bn)(\bar{b}\bar{n})$ and $(bs)(\bar{b}\bar{s})$, $M^{\prime}=M^{\prime\prime}$. $n=u,\,d$. Eqs. (II.2), (42) and (43) are used. $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$ | $(bn)(\bar{b}\bar{n})$ | $(bn)(\bar{b}\bar{s})$ | $(bs)(\bar{b}\bar{s})$
---|---|---|---
| $M^{\prime}=M^{\prime\prime}$ | $(M^{\prime},\,M^{\prime\prime})$ | $M^{\prime}=M^{\prime\prime}$
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 10.46 | (10.57, 10.57) | 10.68
$|2^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 10.98 | (11.08, 11.10) | 11.21
$|3^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 11.27 | (11.37, 11.38) | 11.49
$|4^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 11.49 | (11.60, 11.61) | 11.71
$|5^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 11.69 | (11.79, 11.79) | 11.90
$|1^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 10.49 | (10.60, 10.60) | 10.71
$|2^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 10.99 | (11.09, 11.11) | 11.22
$|3^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 11.27 | (11.38, 11.39) | 11.50
$|4^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 11.50 | (11.61, 11.61) | 11.72
$|5^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 11.69 | (11.80, 11.80) | 11.91
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 10.44 | (10.55, 10.55) | 10.66
$|1^{1}p_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 10.82 | (10.93, 10.94) | 11.05
$|1^{1}d_{2},1^{1}s_{0},1^{5}S_{2}\rangle$ | 11.04 | (11.15, 11.16) | 11.26
$|1^{1}f_{3},1^{1}s_{0},1^{7}S_{3}\rangle$ | 11.22 | (11.32, 11.33) | 11.43
$|1^{1}g_{4},1^{1}s_{0},1^{9}S_{4}\rangle$ | 11.37 | (11.47, 11.47) | 11.58
$|1^{1}h_{5},1^{1}s_{0},1^{11}S_{5}\rangle$ | 11.50 | (11.60, 11.59) | 11.70
$|1^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 10.47 | (10.58, 10.59) | 10.69
$|1^{3}p_{2},1^{1}s_{0},1^{5}S_{2}\rangle$ | 10.83 | (10.94, 10.95) | 10.06
$|1^{3}d_{3},1^{1}s_{0},1^{7}S_{3}\rangle$ | 11.05 | (11.16, 11.17) | 11.27
$|1^{3}f_{4},1^{1}s_{0},1^{9}S_{4}\rangle$ | 11.22 | (11.33, 11.33) | 11.44
$|1^{3}g_{5},1^{1}s_{0},1^{11}S_{5}\rangle$ | 11.37 | (11.48, 11.47) | 11.58
$|1^{3}h_{6},1^{1}s_{0},1^{13}S_{6}\rangle$ | 11.50 | (11.61, 11.60) | 11.71

Figure 3: The $\rho-$trajectories for the hidden bottom tetraquarks. Circles
indicate the predicted data and the blue lines are the $\rho-$trajectories,
see Eq. (II.2). Data are listed in Table 4.

As discussed in Section D, the values of $c_{fx_{\lambda}}$ and
$c_{0x_{\lambda}}$ vary with the masses of diquark and antidiquark,
$M_{\rho_{1}}$ and $M_{\rho_{2}}$, see Eqs. (II.2), (42) and (43). This means
that $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$ vary with different states of
diquark or antidiquark as we discuss the $\rho-$trajectories, even though
$\lambda-$mode remains ground state. Using Eqs. (II.2), (42) and (43) and
parameters in Table 1, the masses of the $\rho-$mode excited states are
calculated and listed in Table 4.

The mass of the state
$|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$
for $(bq)(\bar{b}\bar{q^{\prime}})$ is equal to that of the state
$|n_{2}^{2s_{2}+1}l_{2j_{2}},n_{1}^{2s_{1}+1}l_{1j_{1}},N^{2j+1}L_{J}\rangle$
for $(bq^{\prime})(\bar{b}\bar{q})$. For the $\rho-$mode excited states, the
mass of the state
$|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$
with definite parity and charge parity (for $q=q^{\prime}$)
$1/\sqrt{2}\big{[}(bq)(\bar{b}\bar{q^{\prime}})\pm(bq^{\prime})(\bar{b}\bar{q})\big{]}$
is equal to that of state

$\displaystyle\frac{1}{\sqrt{2}}\Big{(}|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$
$\displaystyle\pm|n_{2}^{2s_{2}+1}l_{2j_{2}},n_{1}^{2s_{1}+1}l_{1j_{1}},N^{2j+1}L_{J}\rangle\Big{)}$
(4)

for $(bq)(\bar{b}\bar{q^{\prime}})$ if $n_{1}^{2s_{1}+1}l_{1j_{1}}$ is not
identical to $n_{2}^{2s_{2}+1}l_{2j_{2}}$. The corresponding masses should be
$M=(M^{\prime}+M^{\prime\prime})/2$, where $M^{\prime}$ and $M^{\prime\prime}$
are listed in Table 4. Therefore, the corresponding tetraquarks will be
degenerate in mass. The $\rho-$trajectories are shown in Fig. 3.

The obtained results for the $\lambda-$mode excited states are in agreement
with other theoretical predictions, see Table 3. To our knowledge, the highly
radial and orbital excited states of the $\lambda-$mode states,
$|1^{1}s_{0},1^{1}s_{0},4^{1}S_{0}\rangle$,
$|1^{1}s_{0},1^{1}s_{0},5^{1}S_{0}\rangle$,
$|1^{1}s_{0},1^{1}s_{0},1^{1}F_{3}\rangle$,
$|1^{1}s_{0},1^{1}s_{0},1^{1}G_{4}\rangle$,
$|1^{1}s_{0},1^{1}s_{0},1^{1}H_{5}\rangle$, and the $\rho-$trajectories of the
hidden bottom/charm tetraquarks are discussed for the first time.

###  II.5 $\lambda-$ and $\rho-$trajectories for the hidden charm tetraquarks

Table 5: The masses of the $\lambda$-mode excited states of the hidden charm tetraquarks (in ${{\rm GeV}}$). The notation in Eq. (1) is rewritten as $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$. $n=u,\,d$. Eqs. (II.2), (42) and (43) are used. $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$ | $(cn)(\bar{c}\bar{n})$ | $(cn)(\bar{c}\bar{s})$ | $(cs)(\bar{c}\bar{s})$
---|---|---|---
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 3.83 | 3.94 | 4.05
$|1^{1}s_{0},1^{1}s_{0},2^{1}S_{0}\rangle$ | 4.39 | 4.50 | 4.60
$|1^{1}s_{0},1^{1}s_{0},3^{1}S_{0}\rangle$ | 4.80 | 4.90 | 5.00
$|1^{1}s_{0},1^{1}s_{0},4^{1}S_{0}\rangle$ | 5.15 | 5.25 | 5.34
$|1^{1}s_{0},1^{1}s_{0},5^{1}S_{0}\rangle$ | 5.46 | 5.56 | 5.65
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 3.84 | 3.95 | 4.05
$|1^{1}s_{0},1^{1}s_{0},1^{1}P_{1}\rangle$ | 4.23 | 4.33 | 4.44
$|1^{1}s_{0},1^{1}s_{0},1^{1}D_{2}\rangle$ | 4.52 | 4.63 | 4.73
$|1^{1}s_{0},1^{1}s_{0},1^{1}F_{3}\rangle$ | 4.78 | 4.88 | 4.98
$|1^{1}s_{0},1^{1}s_{0},1^{1}G_{4}\rangle$ | 5.01 | 5.11 | 5.21
$|1^{1}s_{0},1^{1}s_{0},1^{1}H_{5}\rangle$ | 5.23 | 5.32 | 5.42

Figure 4: The $\lambda-$trajectories for the hidden charm tetraquarks. Circles represent the predicted data and the blue lines are the $\lambda-$trajectories, see Eq. (II.2). Data are listed in Table 2. Table 6: Comparison of theoretical predictions for the masses of the hidden charm tetraquarks (in GeV). $C$ is defined only for $q=q^{\prime}$. $J^{PC}$ | $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$ | Tetraquark | Our | FGM Faustov:2021hjs  | FS Ferretti:2020ewe  | TR Tiwari:2022azj  | HK Hadizadeh:2015cvx
---|---|---|---|---|---|---|---
$0^{++}$ | $|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | $(cn)(\bar{c}\bar{n})$ | 3.83 | 3.812 |  |  | 3.739
|  | $(cn)(\bar{c}\bar{s})$ | 3.94 | 3.922 | 3.852 | 3.955 |
|  | $(cs)(\bar{c}\bar{s})$ | 4.05 | 4.051 |  | 4.045 | 3.946
| $|1^{1}s_{0},1^{1}s_{0},2^{1}S_{0}\rangle$ | $(cn)(\bar{c}\bar{n})$ | 4.39 | 4.375 |  |  | 4.357
|  | $(cn)(\bar{c}\bar{s})$ | 4.50 | 4.481 | 4.383 | 4.500 |
|  | $(cs)(\bar{c}\bar{s})$ | 4.60 | 4.604 |  | 4.620 | 4.558
$1^{--}$ | $|1^{1}s_{0},1^{1}s_{0},1^{1}P_{1}\rangle$ | $(cn)(\bar{c}\bar{n})$ | 4.23 | 4.244 |  |  | 4.231
|  | $(cn)(\bar{c}\bar{s})$ | 4.33 | 4.350 | 4.234 | 4.413 |
|  | $(cs)(\bar{c}\bar{s})$ | 4.44 | 4.466 |  | 4.556 | 4.464

The discussions on the hidden charm tetraquarks are the same as those on the
hidden bottom tetraquarks. We assume $m_{u}=m_{d}$, therefore, the masses for
diquarks $(cd)$ and $(cu)$ are equal and it holds for the tetraquarks as well.

When considering the $\lambda$-mode excitations, the scalar diquark and the
scalar antidiquark are used. Employing Eqs. (II.2), (42), (43), and parameters
in Table 1, the masses of the $\lambda$-mode excited states are estimated, see
Table 5. The obtained results for the $\lambda-$mode excited states are in
agreement with other theoretical predictions, see Table 6. The
$\lambda-$trajectories are shown in Fig. 4.

When considering the $\rho-$mode excited states, the $\lambda$-mode state is
the ground state, $L=0$ and $N_{r}=0$. The masses of the $\rho-$mode excited
states are calculated, see Table 7.

Table 7: The masses of the $\rho$-mode excited states of the hidden charm tetraquarks (in ${{\rm GeV}}$). The notation in Eq. (1) is rewritten as $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$. $(M^{\prime},\,M^{\prime\prime})$ is for $\big{(}|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle,$ $|n_{2}^{2s_{2}+1}l_{2j_{2}},n_{1}^{2s_{1}+1}l_{1j_{1}},N^{2j+1}L_{J}\rangle\big{)}$. For $(cn)(\bar{c}\bar{n})$ and $(cs)(\bar{c}\bar{s})$, $M^{\prime}=M^{\prime\prime}$. $n=u,\,d$. Eqs. (II.2), (42) and (43) are used. $|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$ | $(cn)(\bar{c}\bar{n})$ | $(cn)(\bar{c}\bar{s})$ | $(cs)(\bar{c}\bar{s})$
---|---|---|---
| $M^{\prime}=M^{\prime\prime}$ | $(M^{\prime},\,M^{\prime\prime})$ | $M^{\prime}=M^{\prime\prime}$
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 3.83 | (3.94, 3.94) | 4.05
$|2^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 4.40 | (4.50, 4.56) | 4.67
$|3^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 4.70 | (4.80, 4.87) | 4.97
$|4^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 4.93 | (5.03, 5.11) | 5.21
$|5^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 5.12 | (5.23, 5.31) | 5.41
$|1^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 3.95 | (4.05, 4.04) | 4.15
$|2^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 4.44 | (4.54, 4.59) | 4.69
$|3^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 4.72 | (4.83, 4.89) | 4.99
$|4^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 4.95 | (5.05, 5.12) | 5.23
$|5^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 5.14 | (5.25, 5.32) | 5.42
$|1^{1}s_{0},1^{1}s_{0},1^{1}S_{0}\rangle$ | 3.83 | (3.94, 3.95) | 4.06
$|1^{1}p_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 4.28 | (4.38, 4.42) | 4.52
$|1^{1}d_{2},1^{1}s_{0},1^{5}S_{2}\rangle$ | 4.52 | (4.63, 4.66) | 4.76
$|1^{1}f_{3},1^{1}s_{0},1^{7}S_{3}\rangle$ | 4.71 | (4.82, 4.85) | 4.95
$|1^{1}g_{4},1^{1}s_{0},1^{9}S_{4}\rangle$ | 4.87 | (4.98, 5.00) | 5.11
$|1^{1}h_{5},1^{1}s_{0},1^{11}S_{5}\rangle$ | 5.01 | (5.12, 5.14) | 5.25
$|1^{3}s_{1},1^{1}s_{0},1^{3}S_{1}\rangle$ | 3.91 | (4.02, 4.03) | 4.13
$|1^{3}p_{2},1^{1}s_{0},1^{5}S_{2}\rangle$ | 4.30 | (4.41, 4.44) | 4.55
$|1^{3}d_{3},1^{1}s_{0},1^{7}S_{3}\rangle$ | 4.54 | (4.65, 4.68) | 4.78
$|1^{3}f_{4},1^{1}s_{0},1^{9}S_{4}\rangle$ | 4.73 | (4.83, 4.86) | 4.96
$|1^{3}g_{5},1^{1}s_{0},1^{11}S_{5}\rangle$ | 4.89 | (4.99, 5.02) | 5.12
$|1^{3}h_{6},1^{1}s_{0},1^{13}S_{6}\rangle$ | 5.03 | (5.13, 5.16) | 5.26

Figure 5: The $\rho-$trajectories for the hidden charm tetraquarks. Circles
indicate the predicted data and the blue lines are the $\rho-$trajectories,
see Eq. (II.2). Data are listed in Table 4.

The mass of the state
$|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$
for $(cq)(\bar{c}\bar{q^{\prime}})$ is equal to that of the state
$|n_{2}^{2s_{2}+1}l_{2j_{2}},n_{1}^{2s_{1}+1}l_{1j_{1}},N^{2j+1}L_{J}\rangle$
for $(cq^{\prime})(\bar{c}\bar{q})$. For the $\rho-$mode excited states, the
mass of the state
$|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$
with definite parity and charge parity (for $q=q^{\prime}$)
$1/\sqrt{2}\big{[}(cq)(\bar{c}\bar{q^{\prime}})\pm(cq^{\prime})(\bar{c}\bar{q})\big{]}$
is equal to that of state

$\displaystyle\frac{1}{\sqrt{2}}\Big{(}|n_{1}^{2s_{1}+1}l_{1j_{1}},n_{2}^{2s_{2}+1}l_{2j_{2}},N^{2j+1}L_{J}\rangle$
$\displaystyle\pm|n_{2}^{2s_{2}+1}l_{2j_{2}},n_{1}^{2s_{1}+1}l_{1j_{1}},N^{2j+1}L_{J}\rangle\Big{)}$

for $(cq)(\bar{c}\bar{q^{\prime}})$ if $n_{1}^{2s_{1}+1}l_{1j_{1}}$ is not
identical to $n_{2}^{2s_{2}+1}l_{2j_{2}}$. The corresponding masses should be
$M=(M^{\prime}+M^{\prime\prime})/2$, where $M^{\prime}$ and $M^{\prime\prime}$
are listed in Table 7. Therefore, the corresponding tetraquarks will be
degenerate in mass. The $\rho-$trajectories are shown in Fig. 5.

###  II.6 Discussions

By employing the Regge trajectory formulas in (II.2), the masses for the more
highly excited states can be calculated easily. We argue that the calculation
of the masses of highly excited state is instructive although the highly
excited states will be unstable. This is because various models or approaches
to calculate the masses of tetraquarks often use different parameter values.
These models, despite using different parameter values, often give agreeable
predictions for the ground state and the lower excited states. The theoretical
predictions for the highly excited states are expected to show differences,
which will be instructive.

The Regge trajectories take different form and behave differently in various
energy regions Chen:2022flh ; Chen:2021kfw . The hidden charm tetraquark case
is similar to the hidden bottom tetraquark case. Both the diquark $(Qq)$ and
antidiquark $(\bar{Q}\bar{q}^{\prime})$ are the heavy-light systems;
therefore, the $\rho$-trajectories of the hidden bottom/charm tetraquarks
behave as $M{\sim}x_{\rho}^{1/2}$, see Eq. (II.2). For the $\lambda$-mode, the
tetraquark $(Qq)(\bar{Q}\bar{q}^{\prime})$ is the heavy-heavy system; hence,
the $\lambda$-trajectories of the hidden bottom/charm tetraquarks behave as
$M{\sim}x_{\lambda}^{2/3}$.

According to Eq. (II.2), there are three series of Regge trajectories: two
series of $\rho$-trajectories and one series of $\lambda$-trajectories.
Correspondingly, there are three series of masses. When $q=q^{\prime}$, there
is degeneracy in the $\rho$-trajectories, corresponding to the degeneracy in
masses of the $\rho$-excited states.

For the hidden bottom/charm tetraquarks, both the $\lambda$-trajectories and
the $\rho$-trajectories are concave downwards in the $(M^{2},\,x)$ planes if
the confining potential is linear. In the subsection C.4, we show that for all
tetraquarks, not only the $\lambda$-trajectories but also the
$\rho$-trajectories are concave downwards in the $(M^{2},\,x)$ planes when the
masses of the light constituents are considered and the confining potential is
linear.

##  III Conclusions

In this work, we obtain the Regge trajectory formulas for the hidden
bottom/charm tetraquarks $(Qq)(\bar{Q}\bar{q}^{\prime})$ by using the Regge
trajectory formula for the heavy-heavy systems and the newly proposed Regge
trajectory relation for the heavy-light diquarks. We apply the obtained Regge
trajectory formulas to investigate the tetraquarks with hidden bottom and
charm. The masses of the $\lambda$-mode excited and $\rho$-mode excited
tetraquarks are estimated. The calculated results are in agreement with other
theoretical predictions. Both the $\lambda$-trajectories and
$\rho$-trajectories are discussed.

For the hidden bottom/charm tetraquarks, there are three series of the Regge
trajectories: two series of $\rho$-trajectories and one series of
$\lambda$-trajectories. They exhibit different behaviors. The
$\rho$-trajectories behave as $M{\sim}x_{\rho}^{1/2}$ because the $\rho$-mode
excitations are in the heavy-light diquark $(Qq)$ and antidiquark
$(\bar{Q}\bar{q}^{\prime})$. In contrast, the $\lambda$-trajectories behave as
$M{\sim}x_{\lambda}^{2/3}$ because the $\lambda$-mode excitations occur
between the diquark and the antidiquark, with the the diquark $(Qq)$ and
antidiquark $(\bar{Q}\bar{q}^{\prime})$ forming a heavy-heavy system
$(Qq)(\bar{Q}\bar{q}^{\prime})$.

Moreover, the behaviors of the Regge trajectories for various tetraquarks are
investigated based on the spinless Salpeter equation. It is shown that both
the $\lambda$-trajectories and the $\rho$-trajectories for all tetraquarks are
concave downwards in the $(M^{2},\,x)$ plane when the masses of the light
constituents are considered and the confining potential is linear.

##  Appendix A States of diquarks

Table 8: The completely antisymmetric states for the diquarks in $\bar{3}_{c}$ Feng:2023txx . $j$ is the spin of the diquark $(qq^{\prime})$, $s$ denotes the total spin of two quarks, $l$ represents the orbital angular momentum. $n=n_{r}+1$, $n_{r}$ is the radial quantum number, $n_{r}=0,1,2,\cdots$.  Spin of diquark | Parity | Wave state | Configuration
---|---|---|---
( $j$ ) | $(j^{P})$ | $(n^{2s+1}l_{j})$ |
j=0 | $0^{+}$ | $n^{1}s_{0}$ | $[qq^{\prime}]^{{\bar{3}_{c}}}_{n^{1}s_{0}}$
| $0^{-}$ | $n^{3}p_{0}$ | $[qq^{\prime}]^{{\bar{3}_{c}}}_{n^{3}p_{0}}$
j=1 | $1^{+}$ |  $n^{3}s_{1}$, $n^{3}d_{1}$ |  $\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n^{3}s_{1}}$, $\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n^{3}d_{1}}$
| $1^{-}$ |  $n^{1}p_{1}$, $n^{3}p_{1}$ |  $\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n^{1}p_{1}}$, $[qq^{\prime}]^{{\bar{3}_{c}}}_{n^{3}p_{1}}$
j=2 | $2^{+}$ |  $n^{1}d_{2}$, $n^{3}d_{2}$ |  $[qq^{\prime}]^{{\bar{3}_{c}}}_{n^{1}d_{2}}$, $\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n^{3}d_{2}}$
| $2^{-}$ |  $n^{3}p_{2}$, $n^{3}f_{2}$ |  $[qq^{\prime}]^{{\bar{3}_{c}}}_{n^{3}p_{2}}$, $[qq^{\prime}]^{{\bar{3}_{c}}}_{n^{3}f_{2}}$
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$

In this section, we list the completely antisymmetric states for the diquarks
in $\bar{3}_{c}$ Feng:2023txx .

##  Appendix B States of tetraquarks

The states of tetraquark in the diquark picture are listed in Table 9.

Table 9: The states of tetraquark in the diquark picture. The diquark notation is in 8. Here, $q$ and $q^{\prime}$ represent both the light quarks and the heavy quarks. The superscript $1_{c}$ is omitted. $J^{P}$ | $(l_{1},l_{2},L)$ | Configuration
---|---|---
$0^{+}$ | $(0,0,0)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{1}s_{0}}\right)_{N^{1}S_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{1}S_{0}}$,
| $(1,1,0)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{0}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{0}}\right)_{N^{1}S_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{1}S_{0}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{1}S_{0}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{2}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{1}S_{0}}$,
|  |  $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{1}S_{0}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{1}S_{0}}$
| $(0,1,1)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}P_{0}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}P_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{0}}\right)_{N^{3}P_{0}}$,
|  |  $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}P_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}P_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{3}P_{0}}$
| $(1,0,1)$ |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{3}P_{0}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{3}P_{0}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{0}}$,
|  |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{0}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{0}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{2}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{0}}$
| $\cdots$ | $\cdots$
$0^{-}$ | $(0,0,1)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{0}}$, $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{3}P_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{0}}$,
| $(1,0,0)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{0}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{1}s_{0}}\right)_{N^{1}S_{0}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{1}S_{0}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{1}S_{0}}$,
| $(0,1,0)$ |  $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{1}s_{0}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{3}p_{0}}\right)_{N^{1}S_{0}}$, $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{1}p_{1}}\right)_{N^{1}S_{0}}$, $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{3}p_{1}}\right)_{N^{1}S_{0}}$,
| $\cdots$ | $\cdots$
$1^{+}$ | $(0,0,0)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{1}s_{0}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}S_{1}}$,
| $(1,1,0)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{0}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}S_{1}}$,
|  |  $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{0}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{3}S_{1}}$,
|  |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{0}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{3}S_{1}}$,
|  |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{2}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{2}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{2}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{3}S_{1}}$,
| $(0,1,1)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}P_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{0}}\right)_{N^{3}P_{1}}$,
|  |  $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{3}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{3}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{3}P_{1}}$
|  |  $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{1}p_{1}}\right)_{N^{5}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{1}}\right)_{N^{5}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{3}p_{2}}\right)_{N^{5}P_{1}}$
| $(1,0,1)$ |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{3}P_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{3}P_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{1}}$,
|  |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{2}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{1}}$
|  |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{5}P_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{5}P_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{3}p_{2}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{3}s_{1}}\right)_{N^{5}P_{1}}$
| $\cdots$ | $\cdots$
$1^{-}$ | $(0,0,1)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{1}s_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{1}}$, $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{3}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}P_{1}}$,
|  |  $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{1}s_{0}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{1}s_{0}}\right)_{N^{1}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{1}P_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{5}P_{1}}$,
| $(1,0,0)$ |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{{\bar{3}_{c}}}_{n_{1}^{1}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{1}s_{0}}\right)_{N^{3}S_{1}}$,
|  |  $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{1}}[\bar{q}\bar{q}^{\prime}]^{3_{c}}_{n_{2}^{1}s_{0}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{{\bar{3}_{c}}}_{n_{1}^{3}p_{2}}\\{\bar{q}\bar{q}^{\prime}\\}^{3_{c}}_{n_{2}^{3}s_{1}}\right)_{N^{3}S_{1}}$
| $(0,1,0)$ |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{3}p_{0}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{1}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{1}s_{0}}\\{\bar{q}\bar{q}^{\prime}\\}^{{\bar{3}_{c}}}_{n_{2}^{1}p_{1}}\right)_{N^{3}S_{1}}$,
|  |  $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{3}p_{1}}\right)_{N^{3}S_{1}}$, $\left([qq^{\prime}]^{3_{c}}_{n_{1}^{1}s_{0}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{3}p_{1}}\right)_{N^{3}S_{1}}$, $\left(\\{qq^{\prime}\\}^{3_{c}}_{n_{1}^{3}s_{1}}[\bar{q}\bar{q}^{\prime}]^{{\bar{3}_{c}}}_{n_{2}^{3}p_{2}}\right)_{N^{3}S_{1}}$
| $\cdots$ | $\cdots$

##  Appendix C Regge trajectory behaviors for various tetraquarks

In this section, the Regge trajectory behaviors for various tetraquarks are
investigated.

###  C.1 Spinless Salpeter equation

The spinless Salpeter equation Godfrey:1985xj ; Ferretti:2019zyh ;
Bedolla:2019zwg ; Durand:1981my ; Durand:1983bg ; Lichtenberg:1982jp ;
Jacobs:1986gv  reads as

$\displaystyle M\Psi_{d,t}({{\bf
r}})=\left(\omega_{1}+\omega_{2}\right)\Psi_{d,t}({{\bf
r}})+V_{d,t}\Psi_{d,t}({{\bf r}}),$ (5)

where $M$ is the bound state mass (diquark or tetraquark). $\Psi_{d,t}({{\bf
r}})$ are the diquark wave function and the tetraquark wave function,
respectively. $V_{d,t}$ denotes the diquark potential and the tetraquark
potential, respectively (see Eq. (7)). $\omega_{1}$ is the relativistic energy
of constituent $1$ (quark or diquark), and $\omega_{2}$ is of constituent $2$
(quark or antidiquark),

$\omega_{i}=\sqrt{m_{i}^{2}+{\bf p}^{2}}=\sqrt{m_{i}^{2}-\Delta}\;\;(i=1,2).$
(6)

$m_{1}$ and $m_{2}$ are the effective masses of constituent $1$ and $2$,
respectively.

When using diquark in multiquark systems, the interactions between quark and
quark, diquark and quark, diquark and diquark are needed. In Refs.
Faustov:2021hjs , these interactions are constructed with the help of the off-
mass-shell scattering amplitude, which is projected onto the positive energy
states. The interactions also can be established by expanding the interactions
of the quark-antiquark system to the quark-quark system, and then to the
diquark-antidiquark systems or the diquark-quark systems Lundhammar:2020xvw .
Furthermore, the effect of the finite size of diquark is treated differently.
In Refs. Faustov:2021hjs , the size of diquark is taken into account through
corresponding form factors. At times, diquark is taken as being pointlike
Ferretti:2019zyh ; Lundhammar:2020xvw , and we use this approximation in the
present work.

Following Refs. Ferretti:2019zyh ; Bedolla:2019zwg ; Ferretti:2011zz ;
Eichten:1974af , we employ the potential

$\displaystyle V_{d,t}$
$\displaystyle=-\frac{3}{4}\left[V_{c}+{\sigma}r+C\right]\left({\bf{F}_{i}}\cdot{\bf{F}_{j}}\right)_{d,t},$
(7)

where $V_{c}\propto{1/r}$ is a color Coulomb potential or a Coulomb-like
potential due to one-gluon-exchange. $\sigma$ is the string tension. $C$ is a
fundamental parameter Gromes:1981cb ; Lucha:1991vn . The part in the bracket
is the Cornell potential Eichten:1974af . ${\bf{F}_{i}}\cdot{\bf{F}_{j}}$ is
the color-Casimir,

$\langle{(\bf{F}_{i}}\cdot{\bf{F}_{j}})_{d}\rangle=-\frac{2}{3},\quad\langle{(\bf{F}_{i}}\cdot{\bf{F}_{j}})_{t}\rangle=-\frac{4}{3}.$
(8)

###  C.2 Regge trajectory relations for various systems

For the heavy-heavy systems, $m_{1},m_{2}{\gg}{{|\bf p|}}$, Eq. (5) reduces to

$\displaystyle M\Psi_{d,t}({{\bf r}})$ $\displaystyle=$
$\displaystyle\left[(m_{1}+m_{2})+\frac{{{\bf
p}}^{2}}{2\mu}\right]\Psi_{d,t}({{\bf r}})$ (9)
$\displaystyle+V_{d,t}\Psi_{d,t}({{\bf r}}),$

where

$\mu=m_{1}m_{2}/(m_{1}+m_{2}).$ (10)

By employing the Bohr-Sommerfeld quantization approach Brau:2000st  and using
Eqs. (7) and (9), we obtain the parameterized relation Chen:2022flh ;
Chen:2021kfw

$M=m_{R}+\beta_{x}(x+c_{0x})^{2/3}\,\,(x=l,\,n_{r},\,L,\,N_{r}),$ (11)

with

$\beta_{x}=c_{fx}c_{x}c_{c},\quad m_{R}=m_{1}+m_{2}+C^{\prime},$ (12)

where

$C^{\prime}=\left\\{\begin{array}[]{cc}C/2,&\text{diquarks},\\\
C,&\text{tetraquarks}.\end{array}\right.$ (13)
$\sigma^{\prime}=\left\\{\begin{array}[]{cc}\sigma/2,&\text{diquarks},\\\
\sigma,&\text{tetraquarks}.\end{array}\right.$ (14)

$c_{x}$ and $c_{c}$ are

$c_{c}=\left(\frac{\sigma^{\prime 2}}{\mu}\right)^{1/3},\quad
c_{l,L}=\frac{3}{2},\quad c_{n_{r},N_{r}}=\frac{\left(3\pi\right)^{2/3}}{2}.$
(15)

$c_{fx}$ are equal theoretically to one and are fitted in practice. In Eq.
(11), $m_{1}$, $m_{2}$, $c_{x}$ and $\sigma$ are universal for the heavy-heavy
systems. $c_{0x}$ vary with different Regge trajectories.

For the heavy-light systems ($m_{1}\to\infty$ and $m_{2}\to 0$), Eq. (5)
simplifies to

$\displaystyle M\Psi_{d,t}({{\bf r}})=\left[m_{1}+{{|\bf
p|}}+V_{d,t}\right]\Psi_{d,t}({{\bf r}}).$ (16)

By employing the Bohr-Sommerfeld quantization approach Brau:2000st  and using
Eq. (16), the parameterized formula can be written as Chen:2022flh ;
Chen:2021kfw

$M=m_{R}+\beta_{x}\sqrt{x+c_{0x}}\;(x=l,\,n_{r},\,L,\,N_{r}).$ (17)

$\beta_{x}$ is in Eq. (12), and with

$c_{c}=\sqrt{\sigma^{\prime}},\quad c_{l,L}=2,\quad
c_{n_{r},N_{r}}=\sqrt{2\pi}.$ (18)

For the heavy-light systems, the common choice of $m_{R}$ is Selem:2006nd ;
Chen:2021kfw ; Jakhad:2023mni ; Chen:2014nyo ; Chen:2017fcs ; Veseli:1996gy ;
Jia:2018vwl

$m_{R}=m_{1}.$ (19)

The usual Regge trajectory, Eq. (17) with (19), is obtained in the limit
$m_{1}\to\infty$ and $m_{2}\to 0$. There are different ways of including the
light constituent’s mass Selem:2006nd ; Nielsen:2018uyn ; Sonnenschein:2018fph
; MartinContreras:2020cyg ; Chen:2023cws ; Chen:2022flh ; Chen:2023ngj ;
Chen:2014nyo ; Afonin:2014nya ; Sergeenko:1994ck . Two modified formulas are
proposed in Ref. Chen:2023cws , which can universally describe both the heavy-
light mesons and the heavy-light diquarks. One is Eq. (17) with $m_{R}$ in
(12), where $m_{2}$ is the light constituent’s mass. Another reads

$M=m_{R}+\sqrt{\beta_{x}^{2}(x+c_{0x})+\kappa_{x}m^{3/2}_{2}(x+c_{0x})^{1/4}}$
(20)

if $m_{2}{\ll}M$, where

$m_{R}=m_{1}+C^{\prime},\quad\kappa_{x}=\frac{4}{3}\sqrt{{\pi}\beta_{x}},$
(21)

where $\beta_{x}$ is in (12). Eq. (17) with (12) is an extension of
Afonin:2014nya

$M=m_{1}+m_{2}+\sqrt{a(n_{r}+{\alpha}l+b)}$ (22)

and the formula Chen:2022flh

$(M-m_{1}-m_{2}-C)^{2}=\alpha_{x}(x+c_{0})^{\gamma}$ (23)

while (20) with (21) is based on the results in Selem:2006nd ;
Sonnenschein:2018fph . As $m_{2}=0$, these two modified formulas, Eq. (17)
with (12) and Eq. (20) with (21), become identical. As $m_{2}=0$ and $C$ is
neglected, these two modified formulas reduce to the usual Regge trajectory
formula for the heavy-light systems, i.e., (17) with (19). Although they give
different behavior of $m_{2}$, Eq. (17) with (12) and Eq. (20) with (21)
produce consistent results for $l,\,n_{r}<10$ and have the same behavior
$M{\sim}x^{1/2}$ Chen:2023cws .

For the light systems ($m_{1},\,m_{2}\to 0$), Eq. (5) simplifies to

$\displaystyle M\Psi_{d,t}({{\bf r}})=\left[2{{|\bf
p|}}+V_{d,t}\right]\Psi_{d,t}({{\bf r}}).$ (24)

By employing the Bohr-Sommerfeld quantization approach Brau:2000st , the
parameterized formula can be written as Chen:2022flh ; Chen:2021kfw

$M=\beta_{x}\sqrt{x+c_{0x}}\;(x=l,\,n_{r},\,L,\,N_{r}).$ (25)

For the light systems, the parameters read as

$c_{c}=\sqrt{\sigma^{\prime}},\;c_{l,L}=2\sqrt{2},\;c_{n_{r},N_{r}}=2\sqrt{\pi}.$
(26)

According to the results in Refs. Selem:2006nd ; Sonnenschein:2018fph ,
similar to Eq. (20) Chen:2023cws , we suggest

$M=C^{\prime}+\sqrt{\beta_{x}^{2}(x+c_{0x})+\kappa_{x}\left(m_{1}^{3/2}+m_{2}^{3/2}\right)(x+c_{0x})^{1/4}}.$
(27)

Based on Eq. (22) Afonin:2014nya  and (23), we obtain a modified Regge
trajectory relation for the light-light systems in Ref. Chen:2023ngj

$M=m_{R}+\beta_{x}\sqrt{x+c_{0x}},$ (28)

where

$c_{c}=\sqrt{\sigma^{\prime}},\;c_{l,L}=2\sqrt{2},\;c_{n_{r},N_{r}}=2\sqrt{\pi}.$
(29)

$m_{R}$ is in Eq. (12) with $m_{1,2}$ being the masses of the light
constituents.

Table 10: The coefficients for the heavy-heavy systems (HHS), the heavy-light systems (HLS), and the light-light systems (LLS). | HHS | HLS | LLS
---|---|---|---
$\nu$ | $2/3$ | $1/2$ | $1/2$
$c_{c}$ | $\left({\sigma^{\prime 2}}/{\mu}\right)^{1/3}$ | $\sqrt{\sigma^{\prime}}$ | $\sqrt{\sigma^{\prime}}$
$c_{l,\,L}$ | $3/2$ | $2$ | $2\sqrt{2}$
$c_{n_{r},\,N_{r}}$ | ${\left(3\pi\right)^{2/3}}/{2}$ | $\sqrt{2\pi}$ | $2\sqrt{\pi}$

When Eq. (17) with (12) is employed to discuss the heavy-light systems, and
Eq. (28) with (12) is used to discuss the light systems, summarising Eqs.
(11), (17) with (12) and (28) with (12), we have a general form of the Regge
trajectories Chen:2022flh ; plan

$\displaystyle M=$ $\displaystyle
m_{R}+\beta_{x}(x+c_{0x})^{\nu}\,\,(x=l,\,n_{r},\,L,\,N_{r}),$ $\displaystyle
m_{R}=$ $\displaystyle
m_{1}+m_{2}+C^{\prime},\quad\beta_{x}=c_{fx}c_{x}c_{c},$ (30)

where ${\nu}$, $c_{x}$ and $c_{c}$ are listed in Table 10. $c_{fx}$ are
theoretically equal to one and are fitted in practice. $c_{0x}$ vary with
different Regge trajectories. Eq. (C.2) can be employed to discuss various
systems including the heavy-heavy systems, the heavy-light systems, and the
light-light systems: diquarks, mesons, baryons, and tetraquarks Chen:2023djq ;
Chen:2023ngj ; Chen:2023web .

It should be noticed that the general form (C.2) is provisional. Because there
are different methods to include the masses of the light constituents. To
distinguish the better one, more theoretical and experimental data are needed.
In addition to this, the parameter values are universal for both the heavy-
heavy systems and the heavy-light systems Chen:2023cws ; Feng:2023txx , while
the parameter values should change for the light systems Chen:2023ngj  to
obtain agreeable results.

Table 11: Behaviors of three series of Regge trajectory for different tetraquarks in the diquark picture. $Q=c,\,b$ and $q=u,\,d,\,s$. $M$ is the mass of the tetraquark. $x_{\lambda}=L,\,N_{r}$ are the angular momentum quantum number and the radial quantum number for the $\lambda$ mode. $x_{\rho_{1}}=l_{1},\,n_{r_{1}}$ are for the $\rho_{1}$ mode and $x_{\rho_{2}}=l_{2},\,n_{r_{2}}$ are for the $\rho_{2}$ mode. ${\,{\boldsymbol{?}}}$ denotes the indefinite result.  |  | Traj. behavior $M{\sim}x^{\nu}$ | Traj. behavior $M^{2}{\sim}x^{\nu^{\prime}}$
---|---|---|---
|  |  $\rho_{1}$-mode |  $\rho_{2}$-mode |  $\lambda$-mode |  $\rho_{1}$-mode |  $\rho_{2}$-mode |  $\lambda$-mode
1 | $(q_{1}q^{\prime}_{1})(\bar{q}_{2}\bar{q}^{\prime}_{2})$ |  $x_{\rho_{1}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{1}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\rho_{2}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{2}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$ |  $x_{\rho_{1}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{1}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\rho_{2}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{2}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\lambda}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$
2 | $(q_{1}q^{\prime}_{1})(\bar{Q}_{2}\bar{q}_{2})$ |  $x_{\rho_{1}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{1}}^{3/4}{\,{\boldsymbol{?}}}$ | $x_{\rho_{2}}^{1/2}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$ |  $x_{\rho_{1}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{1}}^{3/4}{\,{\boldsymbol{?}}}$ | $x_{\rho_{2}}^{1/2}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$
3 | $(q_{1}Q_{1})(\bar{q}_{2}\bar{q}^{\prime}_{2})$ | $x_{\rho_{1}}^{1/2}$ |  $x_{\rho_{2}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{2}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$ | $x_{\rho_{1}}^{1/2}$ |  $x_{\rho_{2}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{2}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$
4 | $(q_{1}q^{\prime}_{1})(\bar{Q}_{2}\bar{Q}^{\prime}_{2})$ |  $x_{\rho_{1}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{1}}^{3/4}{\,{\boldsymbol{?}}}$ | $x_{\rho_{2}}^{2/3}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$ |  $x_{\rho_{1}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{1}}^{3/4}{\,{\boldsymbol{?}}}$ | $x_{\rho_{2}}^{2/3}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$
5 | $(Q_{1}Q^{\prime}_{1})(\bar{q}_{2}\bar{q}^{\prime}_{2})$ | $x_{\rho_{1}}^{2/3}$ |  $x_{\rho_{2}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{2}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$ | $x_{\rho_{1}}^{2/3}$ |  $x_{\rho_{2}}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\rho_{2}}^{3/4}{\,{\boldsymbol{?}}}$ |  $x_{\lambda}^{1/2}{\,{\boldsymbol{?}}}$, $x_{\lambda}^{2/3}{\,{\boldsymbol{?}}}$
6 | $(q_{1}Q_{1})(\bar{Q}_{2}\bar{q}_{2})$ | $x_{\rho_{1}}^{1/2}$ | $x_{\rho_{2}}^{1/2}$ | $x_{\lambda}^{2/3}$ | $x_{\rho_{1}}^{1/2}$ | $x_{\rho_{2}}^{1/2}$ | $x_{\lambda}^{2/3}$
7 | $(q_{1}Q_{1})(\bar{Q}_{2}\bar{Q}^{\prime}_{2})$ | $x_{\rho_{1}}^{1/2}$ | $x_{\rho_{2}}^{2/3}$ | $x_{\lambda}^{2/3}$ | $x_{\rho_{1}}^{1/2}$ | $x_{\rho_{2}}^{2/3}$ | $x_{\lambda}^{2/3}$
8 | $(Q_{1}Q^{\prime}_{1})(\bar{Q}_{2}\bar{q}_{2})$ | $x_{\rho_{1}}^{2/3}$ | $x_{\rho_{2}}^{1/2}$ | $x_{\lambda}^{2/3}$ | $x_{\rho_{1}}^{2/3}$ | $x_{\rho_{2}}^{1/2}$ | $x_{\lambda}^{2/3}$
9 | $(Q_{1}Q^{\prime}_{1})(\bar{Q}_{2}\bar{Q}^{\prime}_{2})$ | $x_{\rho_{1}}^{2/3}$ | $x_{\rho_{2}}^{2/3}$ | $x_{\lambda}^{2/3}$ | $x_{\rho_{1}}^{2/3}$ | $x_{\rho_{2}}^{2/3}$ | $x_{\lambda}^{2/3}$

###  C.3 Behaviors of the Regge trajectories for various tetraquarks

The Regge trajectories behave differently in different energy regions
Chen:2022flh ; Chen:2021kfw . The Regge trajectory behaviors for baryons are
discussed in Ref. plan . In this subsection, we discuss the Regge trajectory
behaviors for various tetraquarks.

Taken as an example, tetraquarks $(cc)(\bar{q}\bar{q}^{\prime})$ are
discussed, where $q$ and $q^{\prime}$ are the light quarks, $u$, $d$, or $s$.
The diquark $(cc)$ is the doubly heavy diquark, therefore, the diquark Regge
trajectories are clear, see Eq. (11),

$M_{\rho_{1}}{\sim}x^{2/3}_{\rho_{1}}\,\;(x_{\rho_{1}}=l_{1},\,n_{r_{1}}),$
(31)

where $M_{\rho_{1}}$ is mass of the diquark $(cc)$. The antidiquark
$(\bar{q}\bar{q}^{\prime})$ is the light diquark. Although they have
indefinite forms [see Eqs. (27) and (28) with (12)], the light-diquark Regge
trajectories have the same behavior,

$M_{\rho_{2}}{\sim}x^{1/2}_{\rho_{2}}\,\;(x_{\rho_{2}}=l_{2},\,n_{r_{2}}),$
(32)

where $M_{\rho_{2}}$ is mass of the antidiquark $(\bar{q}\bar{q}^{\prime})$.
Generally speaking, the diquark Regge trajectories are not the same as the
$\rho$-trajectories of tetraquarks.

When discussing the $\lambda$-mode, the Regge trajectory relation for the
tetraquarks $(cc)(\bar{q}\bar{q}^{\prime})$ is not clear. We will be
confronted with two problems. One is when the light antidiquark can be
regarded as being heavy, which will affect not only the behaviors of the
${\lambda}$-trajectories but also the behaviors of the $\rho$-trajectories.
Another is how to introduce the mass of the light constituent if the
antidiquark is light, which will affect the behavior of the
$\rho$-trajectories. If the light antidiquark $(\bar{q}\bar{q}^{\prime})$ is
regarded as being light, the tetraquarks $(cc)(\bar{q}\bar{q}^{\prime})$ are
the heavy-light systems, therefore, $\lambda$-trajectories behave as

$M{\sim}x^{1/2}_{\lambda}\;(x_{\lambda}=L,\,N_{r})$ (33)

no matter whether Eq. (17) with (12) or Eq. (20) with (21) is chosen as the
$\lambda$-trajectory formula. When Eq. (20) is chosen as the
$\lambda$-trajectory relation, it will give
$M{\sim}M_{\rho_{1}},\,M_{\rho_{2}}^{3/2}$ while Eq. (17) with (12) gives
$M{\sim}M_{\rho_{1}},\,M_{\rho_{2}}$ due to different ways to include the
light constituent’s mass $M_{\rho_{2}}$ [corresponding to $m_{2}$ in Eq. (17)
with (12) and Eq. (20) with (21)]. They give two different behaviors of the
$\rho$-trajectories. Using Eqs. (31) and (32), Eq. (17) with (12) gives
$M{\sim}x_{\rho_{1}}^{2/3},\,x_{\rho_{2}}^{3/4}$ while Eq. (20) with (21)
gives $M{\sim}x_{\rho_{1}}^{2/3},\,x_{\rho_{2}}^{1/2}$. [When discussing the
$\rho$-trajectories, highly excited antidiquarks are involved. The first
radially and orbitally excited $(ud)$ are about $1.3$ GeV. The masses of the
$1^{1}p_{1}$ states of $(us)$ and $(ss)$ are much heavier, $1.46$ GeV and
$1.65$ GeV Chen:2023ngj , respectively. They approximate or even are larger
than the mass of the charm quark. Therefore, the light antidiquark can be
regarded as being heavy.] If the light antidiquark is regarded as being heavy,
$\lambda$-trajectories behave as

$M{\sim}x_{\lambda}^{2/3}$ (34)

according to Eq. (11) with (12). Using Eqs. (11), (12), (31), and (32), we
have
$M{\sim}M_{\rho_{1}},\,M_{\rho_{2}}{\sim}x_{\rho_{1}}^{2/3},\,x_{\rho_{2}}^{1/2}$.

Other types of tetraquarks can be discussed by similar way. The Regge
trajectory behaviors of different types of tetraquarks are listed in Table 11.

When the diquark (antidiquark) is doubly heavy or heavy-light, it is clear how
to introduce its large mass, see Eqs. (11) with (12) and Eq. (17) with (12) or
Eq. (20) with (21). After choosing the Regge trajectory formulas for the
heavy-light diquarks, for example, (17) with (12), the Regge trajectory
formulas can be written explicitly, and then the behaviors of the
$\lambda$-trajectory and the $\rho$-trajectory are definite. Using Eq. (C.2),
the Regge trajectory relations for the tetraquarks
$(Q_{1}q_{1})(\bar{Q}_{2}\bar{q}_{2})$ read

$\displaystyle M$
$\displaystyle=m_{R{\lambda}}+\beta_{x_{\lambda}}(x_{\lambda}+c_{0x_{\lambda}})^{2/3}\;(x_{\lambda}=L,\,N_{r}),$
$\displaystyle M_{\rho_{1}}$
$\displaystyle=m_{R\rho_{1}}+\beta_{x_{\rho_{1}}}\sqrt{x_{\rho_{1}}+c_{0x_{\rho_{1}}}}\;(x_{\rho_{1}}=l_{1},\,n_{r_{1}}),$
$\displaystyle M_{\rho_{2}}$
$\displaystyle=m_{R\rho_{2}}+\beta_{x_{\rho_{2}}}\sqrt{x_{\rho_{2}}+c_{0x_{\rho_{2}}}}\;(x_{\rho_{2}}=l_{2},\,n_{r_{2}}),$
(35)

where

$\displaystyle m_{R{\lambda}}$ $\displaystyle=M_{\rho_{1}}+M_{\rho_{2}}+C,$
$\displaystyle m_{R\rho_{1}}$ $\displaystyle=m_{Q_{1}}+m_{q_{1}}+C/2,$
$\displaystyle m_{R\rho_{2}}$ $\displaystyle=m_{Q_{2}}+m_{q_{2}}+C/2,$
$\displaystyle\beta_{L}$
$\displaystyle=\frac{3}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fL},$
$\displaystyle\beta_{N_{r}}$
$\displaystyle=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fN_{r}},$
$\displaystyle\mu_{\lambda}$
$\displaystyle=\frac{M_{\rho_{1}}M_{\rho_{2}}}{M_{\rho_{1}}+M_{\rho_{2}}},$
$\displaystyle\beta_{l_{1}}$
$\displaystyle=\sqrt{2\sigma}c_{fl_{1}},\quad\beta_{n_{r_{1}}}=\sqrt{\pi\sigma}c_{fn_{r_{1}}},$
$\displaystyle\beta_{l_{2}}$
$\displaystyle=\sqrt{2\sigma}c_{fl_{2}},\quad\beta_{n_{r_{2}}}=\sqrt{\pi\sigma}c_{fn_{r_{2}}}.$
(36)

The Regge trajectory relations for the tetraquarks
$(Q_{1}Q^{\prime}_{1})(\bar{Q}_{2}\bar{q}_{2})$ read

$\displaystyle M$
$\displaystyle=m_{R{\lambda}}+\beta_{x_{\lambda}}(x_{\lambda}+c_{0x_{\lambda}})^{2/3}\;(x_{\lambda}=L,\,N_{r}),$
$\displaystyle M_{\rho_{1}}$
$\displaystyle=m_{R\rho_{1}}+\beta_{x_{\rho_{1}}}(x_{\rho_{1}}+c_{0x_{\rho_{1}}})^{2/3}\;(x_{\rho_{1}}=l_{1},\,n_{r_{1}}),$
$\displaystyle M_{\rho_{2}}$
$\displaystyle=m_{R\rho_{2}}+\beta_{x_{\rho_{2}}}\sqrt{x_{\rho_{2}}+c_{0x_{\rho_{2}}}}\;(x_{\rho_{2}}=l_{2},\,n_{r_{2}}),$
(37)

where

$\displaystyle m_{R{\lambda}}$ $\displaystyle=M_{\rho_{1}}+M_{\rho_{2}}+C,$
$\displaystyle m_{R\rho_{1}}$
$\displaystyle=m_{Q_{1}}+m_{Q^{\prime}_{1}}+C/2,$ $\displaystyle
m_{R\rho_{2}}$ $\displaystyle=m_{Q_{2}}+m_{q_{2}}+C/2,$
$\displaystyle\beta_{L}$
$\displaystyle=\frac{3}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fL},\;\beta_{N_{r}}=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fN_{r}},$
$\displaystyle\mu_{\lambda}$
$\displaystyle=\frac{M_{\rho_{1}}M_{\rho_{2}}}{M_{\rho_{1}}+M_{\rho_{2}}},\;\mu_{\rho_{1}}=\frac{m_{Q_{1}}m_{Q^{\prime}_{1}}}{m_{Q_{1}}+m_{Q^{\prime}_{1}}},$
$\displaystyle\beta_{l_{1}}$
$\displaystyle=\frac{3}{2}\left(\frac{\sigma^{2}}{4\mu_{\rho_{1}}}\right)^{1/3}c_{fl_{1}},\;\beta_{l_{2}}=\sqrt{2\sigma}c_{fl_{2}},$
$\displaystyle\beta_{n_{r_{1}}}$
$\displaystyle=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{4\mu_{\rho_{1}}}\right)^{1/3}c_{fn_{r_{1}}},\quad\beta_{n_{r_{2}}}=\sqrt{\pi\sigma}c_{fn_{r_{2}}}.$
(38)

The tetraquark Regge trajectory relations for
$(Q_{1}q_{1})(\bar{Q}_{2}\bar{Q}^{\prime}_{2})$ can be obtained easily by the
similar way as obtaining the Regge trajectory formulas for
$(Q_{1}Q^{\prime}_{1})(\bar{Q}_{2}\bar{q}_{2})$. The Regge trajectory
relations for the tetraquarks
$(Q_{1}Q^{\prime}_{1})(\bar{Q}_{2}\bar{Q}^{\prime}_{2})$ read

$\displaystyle M$
$\displaystyle=m_{R{\lambda}}+\beta_{x_{\lambda}}(x_{\lambda}+c_{0x_{\lambda}})^{2/3}\;(x_{\lambda}=L,\,N_{r}),$
$\displaystyle M_{\rho_{1}}$
$\displaystyle=m_{R\rho_{1}}+\beta_{x_{\rho_{1}}}(x_{\rho_{1}}+c_{0x_{\rho_{1}}})^{2/3}\;(x_{\rho_{1}}=l_{1},\,n_{r_{1}}),$
$\displaystyle M_{\rho_{2}}$
$\displaystyle=m_{R\rho_{2}}+\beta_{x_{\rho_{2}}}(x_{\rho_{1}}+c_{0x_{\rho_{1}}})^{2/3}\;(x_{\rho_{2}}=l_{2},\,n_{r_{2}}),$
(39)

where

$\displaystyle m_{R{\lambda}}$ $\displaystyle=M_{\rho_{1}}+M_{\rho_{2}}+C,$
$\displaystyle m_{R\rho_{1}}$
$\displaystyle=m_{Q_{1}}+m_{Q^{\prime}_{1}}+C/2,$ $\displaystyle
m_{R\rho_{2}}$ $\displaystyle=m_{Q_{2}}+m_{Q^{\prime}_{2}}+C/2,$
$\displaystyle\beta_{L}$
$\displaystyle=\frac{3}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fL},\;\beta_{N_{r}}=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{\mu_{\lambda}}\right)^{1/3}c_{fN_{r}},$
$\displaystyle\mu_{\lambda}$
$\displaystyle=\frac{M_{\rho_{1}}M_{\rho_{2}}}{M_{\rho_{1}}+M_{\rho_{2}}},\;\mu_{\rho_{1}}=\frac{m_{Q_{1}}m_{Q^{\prime}_{1}}}{m_{Q_{1}}+m_{Q^{\prime}_{1}}},$
$\displaystyle\mu_{\rho_{2}}$
$\displaystyle=\frac{m_{Q_{2}}m_{Q^{\prime}_{2}}}{m_{Q_{2}}+m_{Q^{\prime}_{2}}},\;\beta_{l_{1}}=\frac{3}{2}\left(\frac{\sigma^{2}}{4\mu_{\rho_{1}}}\right)^{1/3}c_{fl_{1}},\;$
$\displaystyle\beta_{n_{r_{1}}}$
$\displaystyle=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{4\mu_{\rho_{1}}}\right)^{1/3}c_{fn_{r_{1}}},$
$\displaystyle\beta_{l_{2}}$
$\displaystyle=\frac{3}{2}\left(\frac{\sigma^{2}}{4\mu_{\rho_{2}}}\right)^{1/3}c_{fl_{2}},$
$\displaystyle\beta_{n_{r_{2}}}$
$\displaystyle=\frac{(3\pi)^{2/3}}{2}\left(\frac{\sigma^{2}}{4\mu_{\rho_{2}}}\right)^{1/3}c_{fn_{r_{2}}}.$
(40)

Due to ambiguity, the tetraquark Regge trajectory formulas for the tetraquarks
containing the light diquark or the light antidiquark are not given.

###  C.4 Discussions

The form of the Regge trajectories depends not only on energy regions but also
on the confining potential Chen:2022flh ; Chen:2021kfw . In this work, the
employed confining potential is linear. Moreover, various mixings are not
considered when discussing the behaviors of the tetraquark Regge trajectories,
see Table 11. When the mixings are considered, the behaviors of the tetraquark
Regge trajectories will become complex.

In potential models, the curvature of the Regge trajectories is related to the
dynamic equation and the confining potential Chen:2018bbr . In Ref.
Chen:2023web , we suggest that the Regge trajectories for the diquarks,
mesons, baryons and tetraquarks are concave downwards in the $(M^{2},\,x)$
planes. In fact, the baryon trajectories and the tetraquark Regge trajectories
discussed in Refs. Chen:2023web ; Chen:2023djq  actually are the
$\lambda$-trajectories. We can see from Table 11 that both the
$\lambda$-trajectories and the $\rho$-trajectories for various tetraquarks are
all concave downwards in the $(M^{2},\,x)$ plane. For the light systems, the
trajectories approximate linear as the masses are neglected and become concave
when the masses are considered.

If the diquark quark contents have the same flavors as the antidiquark
contents, there are degeneracy in masses. The $\rho_{1}$-mode excited states
have the same masses as the $\rho_{2}$-mode excited states.

##  Appendix D Determination of $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$ for
the $\lambda$-modes

In this section, we determine the values of $c_{fx_{\lambda}}$ and
$c_{0x_{\lambda}}$ for the $\lambda$-modes of the tetraquarks with hidden
bottom and charm. Apply Eq. (C.2) to fit the Regge trajectories for the doubly
heavy mesons, the $\lambda$-modes of heavy-heavy baryons composed of one heavy
quark and one doubly heavy (or one heavy-light) diquark and the
$\lambda$-modes of heavy-heavy tetraquarks composed of one doubly heavy (or
heavy-light) diquark and one doubly heavy (or heavy-light) antidiquark. The
quality of a fit is measured by the quantity $\chi^{2}$ defined by

$\chi^{2}=\frac{1}{N-1}\sum^{N}_{i=1}\left(\frac{M_{fi}-M_{ei}}{M_{ei}}\right)^{2},$
(41)

where $N$ is the number of points on the trajectory, $M_{fi}$ is fitted value
and $M_{ei}$ is the experimental value or the theoretical value of the $i$-th
particle mass. The parameters are determined by minimizing $\chi^{2}$.

Table 12: The spin averaged masses of the radially excited states of mesons and baryons, and the masses of tetraquarks (in ${{\rm GeV}}$). | $1S$ | $2S$ | $3S$ | $4S$ | $5S$ | $6S$
---|---|---|---|---|---|---
$c\bar{c}$ | $3.0687$ | $3.6740$ | $4.0273$ | $4.4115$ | $4.8305$ | $5.1640$
$c\bar{b}$ | 6.3184 | 6.8793 | 7.2500 | 7.6030 | 7.9430 |
$b\bar{b}$ | 9.4450 | 10.0173 | 10.3486 | 10.5778 | 10.8645 | 11.0812
$\Omega_{ccb}$ | 7.9940 | 8.4097 |  |  |  |
$\Omega_{cbb}$ | 11.2107 | 11.6967 |  |  |  |
$[cs][\bar{c}\bar{s}]$ | 4.051 | 4.604 |  |  |  |
$[bs][\bar{b}\bar{s}]$ | 10.662 | 11.111 |  |  |  |
Table 13: The spin averaged masses of the orbitally excited states of mesons and baryons, and the masses of tetraquarks (in ${{\rm GeV}}$). | $1S$ | $1P$ | $1D$ | $1F$ | $1G$ | $1H$
---|---|---|---|---|---|---
$c\bar{c}$ | 3.0687 | 3.5253 | 3.8186 | 4.0720 | 4.3435 | 4.5931
$c\bar{b}$ | 6.3184 | 6.7486 | 7.0261 | 7.2720 | 7.4879 |
$b\bar{b}$ | 9.4450 | 9.8997 | 10.1629 | 10.3467 | 10.5127 | 10.6711
$\Omega_{ccb}$ | 7.9940 | 8.2643 | 8.4741 |  |  |
$\Omega_{cbb}$ | 11.2107 | 11.5261 | 11.8053 |  |  |
$[cs][\bar{c}\bar{s}]$ | 4.051 | 4.466 | 4.728 |  |  |
$[bs][\bar{b}\bar{s}]$ | 10.662 | 11.002 | 11.216 |  |  |

The masses used are listed in Tables 12 and 13. For the mesons and baryons,
the listed data are the spin averaged masses. If the corresponding states are
experimentally determined, the PDG data from Ref. pdg:2024  are used. For the
undetermined states, the theoretical data from Ref. Ebert:2011jc ;
Faustov:2021qqf  are used. (In Ref. Ebert:2011jc , the mass of the
$\Omega_{cbb}$ $(7/2)^{+}$(1D) state is not provided, so we use the value
$11.807$ ${{\rm GeV}}$.) The theoretical data for the tetraquarks are from
Faustov:2021hjs ; Faustov:2022mvs . Beside the bound states’ masses, some
parameters are listed in Table 1. The masses of the axial vector diquarks are
$m_{\\{cc\\}}=3.14$ ${{\rm GeV}}$, $m_{\\{bb\\}}=9.63$ ${{\rm GeV}}$
Feng:2023txx . The masses of the scalar diquarks $[cs]$ and $[bs]$ are
calculated by using the parameters in Table 1 and Eq. (C.2) or (C.3)
Chen:2023cws .

The fitted parameters are listed in Table 14. The fitted Regge trajectories
are plotted in the $((M-m_{R})^{3/2},\;x)$ plane, see Fig. 6. The results are
consistent with those in Ref. Chen:2023djq .

The fitted $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$ vary across different
Regge trajectories for different states and for different systems, see Fig. 7
and Table 14. In case of the baryons and tetraquarks, these parameters are
sensitive to diquark masses. For example, $c_{fN_{r}}=1.218$ when $[bc]=6.38$
${{\rm GeV}}$ Feng:2023txx  while $c_{fN_{r}}=1.027$ when $[bc]=6.519$ ${{\rm
GeV}}$, as determined by fitting the radial Regge trajectory for the
tetraquark $[bc][\bar{b}\bar{c}]$ Faustov:2022mvs .

According to Eq. (C.3), $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$ are needed
to determine a Regge trajectory as $m_{R_{\lambda}}$ can be calculated by
using Eq. (C.3) and parameters in Table 1. Two or more states on the Regge
trajectory are needed to obtain $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$. We
choose $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$ by fitting these parameters
from other systems. By fitting the data in Table 14, we have

$\displaystyle c_{fL}=$ $\displaystyle
1.116+0.013\mu_{\lambda},\;c_{0L}=0.540-0.141\mu_{\lambda},$ (42)
$\displaystyle c_{fN_{r}}=$ $\displaystyle
1.008+0.008\mu_{\lambda},\;c_{0N_{r}}=0.334-0.087\mu_{\lambda},$ (43)

where $\mu_{\lambda}$ is the reduced masses, see Eq. (10) or (C.3). Eqs. (42)
and (43) are obtained when $\mu_{\lambda}<3.83$ ${{\rm GeV}}$. And the
formulas for the $\mu_{\lambda}>3.83$ ${{\rm GeV}}$ are not considered here.
As more experimental data or more theoretical data become available, the
fitted formulas will be refined.

Table 14: The fitted $c_{fx_{\lambda}}$ and $c_{0x_{\lambda}}$. | $c\bar{c}$ | $c\bar{b}$ | $b\bar{b}$ | $\Omega_{ccb}$ | $\Omega_{cbb}$ | $[cs][\bar{c}\bar{s}]$ | $[bs][\bar{b}\bar{s}]$
---|---|---|---|---|---|---|---
$(c_{fN_{r}},\,c_{0N_{r}})$ | (0.994, 0.191) | (1.028, 0.135) | (1.046, 0.0) | (0.994, 0.333) | (1.037, 0.347) | (1.025, 0.255) | (1.022, 0.087)
$(c_{fL},\,c_{0L})$ | (1.120, 0.320) | (1.154, 0.231) | (1.187, 0.0) | (1.047, 0.559) | (1.137, 0.537) | (1.151, 0.399) | (1.164, 0.133)

Figure 6: The radial and orbital Regge trajectories (the black lines) for
mesons, baryons and tetraquarks. The dots are the spin averaged masses of
mesons and baryons and the masses of tetraquarks. The used data are listed in
Tables 12, 13 and 14.

Figure 7: The fitted $c_{fx_{\lambda}}$ (fig. 7) and $c_{0x_{\lambda}}$ (fig.
7), ($x_{\lambda}=L,\,N_{r}$). The data are listed in Table 14.
$\mu_{\lambda}$ is the reduced masses, see Eq. (10) or (C.3). The black filled
circles (mesons), the red filled squares (tetraquarks), and the green filled
triangles (baryons) are the fitted $c_{fL}$ or $c_{0L}$. The black empty
circles (mesons), the red empty squares (tetraquarks), and the green empty
triangles (baryons) are the fitted $c_{fN_{r}}$ or $c_{0N_{r}}$. The black
lines ($c_{fL}$ and $c_{0L}$) and the dashed lines ($c_{fN_{r}}$ and
$c_{0N_{r}}$) are the linear fit, see Eqs. (42) and (43).

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