∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Sardar Vallabhbhai National Institute
of Technology, Surat-395007, Gujarat, INDIA.

# Mass-Spectroscopy of [$bb][\bar{b}\bar{b}$] and [$bq][\bar{b}\bar{q}$]
tetraquark states in a diquark-antidiquark formalism

Rohit Tiwarie1,addr1  D.P. Rathaudaddr1  Ajay Kumar Raiaddr1

(Received: date / Accepted: date)

###### Abstract

In this article, we utilise the non-relativistic potential model to calculate
the mass-spectra of all bottom [$bb][\bar{b}\bar{b}$] and heavy-light bottom
[$bq][\bar{b}\bar{q}$] (q=u,d) tetraquark states in diquark-antidiquark
approximation. The four-body problem is reduced into two-body problems by
numerically solving the $Schr\ddot{o}dinger$ equation using a cornell-inspired
potential along with relativistic correction term. The splitting structure of
the tetraquark spectrum is described using spin-dependent terms (spin-spin,
spin-orbit, and tensor). We have successfully calculated and predicted the
masses of bottom mesons, diquarks and tetraquarks. The masses of S and P-wave
tetraquark states [$bb][\bar{b}\bar{b}$] and [$bq][\bar{b}\bar{q}$],
respectively, are found to be between 18.7-19.4 GeV and 10.4-11.3 GeV, in
which the masses of S-wave [$bb][\bar{b}\bar{b}$] states are less than the
2$\eta_{b}$, $\eta_{b}\Upsilon$, and 2$\Upsilon$ threshold. Additionally, we
investigated the $Z_{b}(10610)$ and $Z_{b}(10650)$ states in the current model
and found that they are 150 MeV below the $BB^{*}$ and $B^{*}B^{*}$
thresholds.

###### Keywords:

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††journal: Eur. Phys. J. A

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##  1 Introduction

In the past two decades, many exotic hadrons were experimentally found and
theoretically predicted, which increase the interest in non-conventional
hadron spectroscopy among researchers exotic1 ; exotic2 ; exotic3 ; mod2 .
Exotic hadrons are those which have quark combinations other than the
conventional baryons (qqq) and mesons ($q\bar{q}$) Gellmann ; Gyang . Some of
these exotic states have quark contents such as $qq\bar{q}\bar{q}$,
$q\bar{q}q\bar{q}$, and $qqqq\bar{q}$ etc., which are explained as multiquarks
or loosely bound molecules mod2 . In 2003, the observation of the exotic state
X(3872) aka $\chi_{c1}(3872)$ in the decay process of $B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}J/\psi$, renewed interest in exotic states, and as a
consequence, many other exotic states are still being observed experimentally
to this day belle ; cdf ; D0 . Among all the discovered non-conventional
states some of the expected to have tetraquark structure tr1 ; tr2 ; tr3 ; pdg
; silver .

All the heavy tetraquarks $QQ\bar{Q}\bar{Q}$, which often contain a charm or
bottom quarks, are of great interest to tetraquark researchers. Among the
heavy tetraquarks, the recent discovery of all-charm tetraquarks X(6900) and
the study of all-bottom tetraquarks $[bb][\bar{b}\bar{b}]$ have been important
in understanding quark confinement inside the tetraquarks cccc ; bbbb . In
2017, the CMS collaboration discovered $\Upsilon(1S)$ pair formation in pp
collisions at $\sqrt{s}=8$ TeV and an excess at 18.4 GeV in the
$\Upsilon(1S)l^{-}l^{+}$ decay channel was proposed in a subsequent
preliminary study cms ; y1s  and whereas RHIC reported similar observation at
18.2 GeV in Cu+Au collisions rhic . However, the LHCb collaboration hasn’t
found any evidence in the $\Upsilon$(1S) $\mu^{-}$ $\mu^{+}$ invariant mass
spectrum bbbb . Hence, fully bottom tetraquark searches required intensive
efforts at the experimental frontier.

While the double-heavy tetraquark sector sought the observation of di-mesonic
structure with ingredients $bb\bar{u}\bar{d}$ in 1988 bbud . In Refs.mass ;
1plus , a narrow tetraquark structure $bb\bar{u}\bar{d}$ was predicted with
spin-parity $J^{P}=1^{+}$ and mass 10389 $\pm$ 12 MeV.

The Belle collaboration reported two charged bottomonium-like resonances
$Z_{b}(10610)$ and $Z_{b}(10650)$,(hence referred to as $Z_{b}$ and
$Z^{\prime}_{b}$) in the invariant mass distributions
$e^{+}e^{-}\rightarrow\Upsilon(nS)\pi^{+}\pi^{-}$, $n=1,2,3$ and
$e^{+}e^{-}\rightarrow h_{b}(mP)\pi^{+}\pi^{-}$, $m=1,2$ zb . Both $Z_{b}$ and
$Z^{\prime}_{b}$ were predicted in the decays of vector bottomonium
$\Upsilon(10860)$ and afterwards verified in the elastic $B^{*}\bar{B}^{*}$
channels zb1 . The decays of the bottomonium-like tetraquarks (bound diquarks–
antidiquarks) into conventional bottomonium states and a pion indicate that
these resonances have a minimum quark content of four quarks zb2 ; aslam ;
aslam2 ; zb4 ; zb5 . The study of double bottom tetraquark states have been
attempted in various theoretical approaches like a quark model formalism life
, relativistic quark model nr , and quark coalescence model lhc . The study of
all bottom tetraquark states have been attempted in various approaches like
non-relativistic effective field theory nre , QCD sum rule qcdsum , the
diquark-antiquark formalism dianti , diffusion Monte Carlo approach monte , an
effective potential model QQQQ  etc. and predicted mass in the range of
18-18.8 GeV 18g .

Morover, the antiparticle of tetraquark $\bar{b}\bar{b}ud$ with the same
quantum number $J^{P}=1^{+}$, which has a mass of 10476 $\pm$ 24 $\pm$ 10 MeV
and is stable against its decay process as determined by Lattice QCD anti .
Other studies using the same methodology other1 ; other2  also support the
existence of this firmly confined tetraquark states.

Indeed, under the diquark approximation the $bb\bar{u}\bar{d}$ and
$b\bar{b}u\bar{d}$ systems are equivalent but they vary significantly when
considered on a full color basis, in which the $bb\bar{u}\bar{d}$ state is
deeply bound, while the $b\bar{b}u\bar{d}$ state is clearly unbound life ;
arxiv .

The focus of this paper is on the all-bottom tetraquak $[bb][\bar{b}\bar{b}]$,
which we shall refer to as $T_{4b}$, and the double bottom tetraquark
$[bq][\bar{b}\bar{q}]$, (q=u,d) in a diquark-antidiquark configuration. We
have utilized the non relativistic model to obtain the mass-spectra of
$T_{4b}$ and $bq\bar{b}\bar{q}$ tetraquark states, by configuring four quark
system ($QQ\bar{Q}\bar{Q}$) in two diquark [$QQ$]-antidiquark
[$\bar{Q}\bar{Q}$] system. The diquark [$QQ$] and antidiquark
[$\bar{Q}\bar{Q}$] are made up of two quarks (antiquarks) in antitriplet
(triplet) color states. After producing the mass-spectra of heavy-heavy and
heavy-light bottom mesons, the optimal set of parameters for tetra-quarks was
fitted. The $Schr\ddot{o}dinger$ equation has been solved numerically using a
Cornell-like potential model and the relativistic correction term
$\mathcal{O}(\frac{1}{m})$. Additionally, we included spin-dependent terms
(spin-spin, spin-orbit, and tensor) in order to analyse the splitting of
orbital and radial excitations.

The following is the structure of the current work: The theoretical model and
fitting procedures are presented in Sec. II, followed by an introduction. Sec.
III contains mass spectra of mesons and tetraquarks, as well as comparisons
and discussions based on our model. We concluded our work in Section IV.

##  2 Theoretical Model

We begin by introducing a non-relativistic model QQQQ  for spectroscopic
analysis of hadronic bound states comprised of heavy-heavy and heavy-light
quarks in the diquark-antidiquark model debastiani . We first describe the
four body system $QQ\bar{Q}\bar{Q}$ as two-body diquark [$QQ$] and antidiquark
[$\bar{Q}\bar{Q}$] system. The mass-spectra of $T_{4b}$ and $bq\bar{b}\bar{q}$
tetraquarks states have been calculated by solving the $Schr\ddot{o}dinger$
equation numerically using code originally developed by W. Lucha et al. lucha
which is based on the fourth-order Runge-Kutta (RK4) method. The suitable
method is to work in the center-of-mass frame while solving two-body problems
which are included in the central potential prd . To separate the angular term
and radial term of a wave function spherical harmonics can be used. In the
case of quarkonium and tetraquarks, the kinetic energy of states may be
represented in terms of the reduced mass $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$,
where $m_{1}$ and $m_{2}$ are the masses of constituents.

In the spectroscopic study, of heavy quark system the kinetic energy is
comparatively lower than the rest mass energy of the constituent quarks, hence
the reasonable approximation could be using of static potentials in a non-
relativistic model exotic2 . The spin-dependent terms are incorporated
perturbatively in the potential model. This methodology generates four free
optimal set of parameters which are fitted to mesons-spectra, later it can be
used in diquarks and tetraquarks. The Hamiltonian may be represented with an
unperturbed one-gluon exchange (OGE) potential mod2  and the relativistic
mass-correction term $V^{1}(r_{ij})$. The simple two body hamiltonian in the
center of mass frame of mesons and tetraquarks given as;

$H=\sum_{i=1}^{2}(m_{i}+\frac{p_{i}^{2}}{2m_{i}})-T_{CM}+\sum_{j>i=1}^{2}V(r_{ij})$
(1)

Here $m_{i}$ is the constituent mass and $p_{i}$ is the relative momentum of
the system, while $V(r_{ij})$ is the interaction potential. $T_{CM}$ is the
kinetic energy of the center-of-mass motion.
The time-independent radial $Schr\ddot{o}$dinger scr1 ; scr2  equation for two
body problem can be expressed as;

$\displaystyle\left[-\frac{1}{2\mu}\left(\frac{d^{2}}{d(r_{ij})^{2}}+\frac{2}{r_{ij}}\frac{d}{d(r_{ij})}-\frac{L(L+1)}{r_{ij}^{2}}\right)+V(r_{ij})\right]\times$
$\displaystyle\psi(r_{ij})=E\psi(r_{ij})\quad\quad$ (2)

where, L and E are the orbital quantum number and energy eigenvalue
respectively. By substituting $\psi(r_{ij})=r_{ij}^{-1}\phi(r_{ij})$ in Eq.(1)
modifies to;

$\left[-\frac{1}{2\mu}\left(\frac{d^{2}}{dr_{ij}^{2}}+\frac{L(L+1)}{r_{ij}^{2}}\right)+V(r_{ij})\right]\phi(r_{ij})=E\phi(r_{ij})$
(3)

The reliable and widely used potential model in the spectroscopic study of
heavy-quarkonium system is a zeroth-order $V(r_{ij})$ Cornell-like potential
cornell . The potential $V_{C+L}(r_{ij})$ represents a gluonic interaction
term between two (anti)quarks and (anti) di-quarks, referred to as the coulomb
term, as well as a linear term responsible for quark confinement.

$V_{C+L}(r_{ij})=\frac{k_{s}\alpha_{s}}{r_{ij}}+br_{ij}$ (4)

where, $\alpha_{s}$ is known as the QCD running coupling constant, $k_{s}$ is
a color factor, b is string tension. We have incorporated the relativistic
mass correction term $V^{1}(r_{ij})$ initially developed by Y. Coma et al.
koma , in the central potential. The final form of central potential is given
by;

$V(r_{ij})=V_{C+L}(r_{ij})+V^{1}(r_{ij})\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)+\mathcal{O}\left(\frac{1}{m^{2}}\right)$
(5)

The non-perturbative form of relativistic mass correction term $V^{1}(r_{ij})$
is not yet known, but leading order perturbation theory yields koma ,

$V^{1}(r_{ij})=-\frac{C_{F}C_{A}}{4}\frac{\alpha^{2}_{s}}{(r_{ij})^{2}}$ (6)

where $C_{F}=\frac{4}{3}$ and $C_{A}=3$ are the Casimir charges of the
fundamental and the adjoint representation respectively koma . The
relativistic mass correction is found to be similar to the coulombic term of
the static potential when applied to the charmonium and to be one-fourth of
the coulombic term for bottomonium. Along with the central interaction
potential $V(r_{ij})$, we have also incorporated spin-dependent interactions.
These spin-dependent terms are included perturbatively.

The non-relativistic form of kinetic energy is given by $KE_{NR}=p^{2}/2m$
while semi-relativistic form is $KE_{SR}=\sqrt{p^{2}+m^{2}}-m$ relativistic ;
dpepjc . In our previous studies ak ; vk ; dpepjc ; dpepjp ; dpijp1 ; dpfew ;
dpijp2 , we looked at the effect of relativistic correction of the kinetic
energy portion for heavy-light mesons and di-mesonic molecule systems, and
found that the contribution of higher order terms contributes less than $1\%$
to the net strength of the kinetic energy part. The proposed systems in this
paper are considerably heavier, thus we think that higher order kinetic energy
contribution should be extremely little and ignored in our calculations.
However, in Ref. relativistic , a complete numerical study of the effects of
relativistic corrections on the kinetic energy operator for $QQ\bar{q}\bar{q}$
and $QQ\bar{Q}\bar{Q}$ are presented.

###  2.1 Spin-dependent Terms

The contribution of spin-dependent potentials, i.e. a spin-spin
$V_{SS}(r_{ij})$, spin-orbit $V_{LS}(r_{ij})$, and tensor $V_{T}(r_{ij})$,
that makes significant contributions particularly for excited states, is
necessary to better understand the splitting between orbital and radial
excitations of different combinations of quantum numbers of $T_{4b}$ and
$bq\bar{b}\bar{q}$. All three spin-dependent terms are driven by the Breit-
Fermi Hamiltonian for one-gluon exchange spin1 ; spin2 , and yields;

$V_{SS}(r_{ij})=C_{SS}(r_{ij})S_{1}\cdot S_{2},$ (7)
$V_{LS}(r_{ij})=C_{LS}(r_{ij})L\cdot S,$ (8)
$V_{T}(r_{ij})=C_{T}(r_{ij})S_{12},$ (9)

The matrix element $S_{1}\cdot S_{2}$ acts on the wave function, and generates
a constant factor, but the $V_{SS}(r_{ij})$ remains a function of only
$r_{ij}$, and the expectation values of $\left\langle S_{1}\cdot
S_{2}\right\rangle$ are available through a quantum-mechanical formula
griffiths .

$\left\langle S_{1}\cdot
S_{2}\right\rangle=\left\langle\frac{1}{2}(S^{2}-S^{2}_{1}-S^{2}_{2})\right\rangle$
(10)

where, $S_{1}$ and $S_{2}$ denote the spins of constituent quarks for
quarkonium and diquarks for tetraquarks, respectively. $C_{SS}(r_{ij})$ may be
defined as follows:

$C_{SS}(r_{ij})=\frac{2}{3m^{2}}\nabla^{2}V_{V}(r_{ij})=-\frac{8k_{s}\alpha_{s}\pi}{3m^{2}}\delta^{3}(r_{ij}),$
(11)

A fair agreement may be achieved by adding spin-spin interactions in a zero-
order potentials using the $Schr\ddot{o}dinger$ equation in heavy quarkonium
spectroscopy by including the spin-spin interaction using the artefact
providing a new parameter $\sigma$ instead of the Dirac delta. So now
$V_{SS}(r_{ij})$ can be redefined as;

$V_{SS}(r_{ij})=-\frac{8\pi
k_{s}\alpha_{s}}{3m^{2}}(\frac{\sigma}{\sqrt{\pi}})^{3}\exp^{-\sigma^{2}(r_{ij})^{2}}S_{1}\cdot
S_{2},$ (12)

The expectation value of operator $\mathbf{\left\langle L\cdot
S\right\rangle}$ is mainly dependent on the total angular momentum J which is
calculated using the formula $\mathbf{J=L+S}$ ;

$\mathbf{\left\langle L\cdot
S\right\rangle=\left\langle\frac{1}{2}(J^{2}-L^{2}-S^{2})\right\rangle}$ (13)

where, L denotes the total orbital angular momentum of quarks and diquarks,
respectively, in the case of quarkonium and tetraquark. The following equation
may be used to compute $C_{LS}(r_{ij})$ ;

$C_{LS}(r_{ij})=-\frac{3k_{s}\alpha_{s}\pi}{2m^{2}}\frac{1}{(r_{ij})^{2}}-\frac{b}{2m^{2}}\frac{1}{(r_{ij})}$
(14)

The second component in the spin-orbit interaction is called Thomas
Precession, and it is proportional to the scalar term. It is thought that
confining interaction originates from the Lorentz scalar structure. In higher
excited states, the contribution of the spin-tensor becomes quite important,
which requires a little algebra and may be calculated by;

$V_{T}(r_{ij})=C_{T}(r_{ij})\left(\frac{(S_{1}\cdot(r_{ij}))(S_{2}\cdot(r_{ij}))}{(r_{ij})^{2}}-\frac{1}{3}(S_{1}\cdot
S_{2})\right)$ (15)

where;

$C_{T}(r_{ij})=-\frac{12k_{s}\alpha_{s}\pi}{4m^{2}}\frac{1}{(r_{ij})^{3}}$
(16)

The results of $(S_{1}\cdot S_{2})$ may be obtained by solving the diagonal
matrix elements for the spin-$\frac{1}{2}$ and spin-1 particles, as detailed
in the following referencesdebastiani ; prd . To solve the tensor interaction,
the simpler formulation may be used ;

$\mathbf{{S_{12}}}=12\left(\mathbf{\frac{(S_{1}\cdot(r_{ij}))(S_{2}\cdot(r_{ij}))}{(r_{ij})^{2}}-\frac{1}{3}(S_{1}\cdot
S_{2})}\right)$ (17)

and which can be redefined as ;

$\mathbf{{S_{12}}}=4\left[3\mathbf{(S_{1}\cdot\hat{(r_{ij})})(S_{2}\cdot\hat{(r_{ij})})-(S_{1}\cdot
S_{2})}\right]$ (18)

Pauli matrices and spherical harmonics with their corresponding eigenvalues
may be used to achieve the results of the $S_{12}$ term. The following
conclusions are valid for bottomonium and diquarks bethe ; thesis  ;

$\displaystyle\mathbf{\left\langle
S_{12}\right\rangle}_{\frac{1}{2}\otimes\frac{1}{2}\rightarrow S=1,l\neq
0}=-\frac{2l}{2l+3},forJ=l+1,$ (19)
$\displaystyle=-\frac{2(l+1)}{(2l-1)},forJ=l-1,and$ (20)
$\displaystyle=+2,forJ=l$ (21)

When $l=0$ and $S=0$ the $\mathbf{\left\langle S_{12}\right\rangle}$ always
vanishes, but it yields a non-zero value for excited states in mesons:
$\mathbf{\left\langle S_{12}\right\rangle}=-\frac{2}{5},+2,-4$ for $J=2,1,0$,
respectively. These value are valid only for bottomonium and diquarks that are
specifically spin-half particles, but in the case of tetraquarks when spin-1
diquarks are involved, it needs a laborious algebra, which is not discussed in
depth here, rather one can refer Refs. bethe ; thesis  for detailed
discussion. The results for tensor interaction of $T_{4b}$ will obtained by
the same formula which is used in case of bottomonium except that the
wavefunction obtained here will be of spin 1 (anti)diquark.

$\displaystyle\mathbf{S_{{d}-{\bar{d}}}}=12\left(\mathbf{\frac{(S_{d}\cdot(r_{ij}))(S_{\bar{d}}\cdot(r_{ij}))}{(r_{ij})^{2}}-\frac{1}{3}(S_{d}\cdot
S_{\bar{d}})}\right)$ (22)
$\displaystyle=\mathbf{S_{14}+S_{13}+S_{24}+S_{23}}$ (23)

where $S_{d}$ is the total spin of the diquark, $S_{\bar{d}}$ is the total
spin of the antidiquark. When the two-body problem is solved to obtain the
masses of the tetraquarks, the interaction between the two (anti) quarks
inside the (anti) diquark is identical; because (anti) diquarks are only
considered in the S-wave state, only the spin-spin interaction is relevant;
the spin-orbit and tensor are both identically zero. Because the tetraquark
radial wavefunction is obtained by treating the diquark and antidiquark as two
body problem, it is reasonable to assume that the radial-dependence of the
tensor term is the same for these four [$q\bar{q}$] interactions and can be
obtained using the radial wavefunction with Eq (14). The following functional
form for spin $\frac{1}{2}$ particles does not use any specific relation or
eigenvalues, instead relying on general angular momentum elementary theory tb
. Within this approximation, generalization of tensor operator can be consider
a sum of four tensor interaction between four quark-antiquark pair as
illustrated in thesis . A thorough discussion on tensor interaction can be
found in Ref. thesis .

###  2.2 Fitting Parameters

We have obtained the mass-spectra of mesons and tetra-quarks for three set of
fitting parameters. In the present work there are four fitting parameters (m,
$\alpha_{s}$, b, $\sigma$) for which the model mass ($m_{i}^{f}$) of the
particular tetraquark states have been calculated.
The best fitting parameters minimizes the difference between an experimental
mass ($m_{i}^{exp}$) and mass ($m_{i}^{f}$) obtained from model, which we will
call discrepancy and denoted as $\Delta=[m_{i}^{exp}-m_{i}^{f}]$. The set of
parameters pdg ; griffiths  are varied in the range of ;

0.05 $\leq$ $\alpha_{s}$ $\leq$ 0.70
0.01 Ge$V^{2}$ $\leq$ b $\leq$ 0.40 Ge$V^{2}$
0.05 GeV $\leq$ $\sigma$ $\leq$ 1.50 GeV
4.00 GeV $\leq$ $m_{b}$ $\leq$ 5.00 GeV
0.3 GeV $\leq$ $m_{q}$ $\leq$ 0.350 GeV

From the above range, the fitted parameters are tabulated in Table 1 to obtain
the mass-spectra of mesons, diquarks and tetraquarks. The mass-spectra of
bottomonium mesons ($b\bar{b}$) and bottom (anti)diquark have been obtained
from data set I. Similarly, the mass-spectra of B-mesons ($b\bar{q}$) and
heavy-light (anti)diquark obtained from data set II. The mass-spectra of
tetraquarks are computed using data set III. Indeed the mass-spectra of all
bottom tetraquarks ($T_{4b}$) are calculated using data set I as well data set
III.

Table 1: The fitting parameters for obtaining the mass spectra of $bb\bar{b}\bar{b}$ and $bq\bar{b}\bar{q}$. The quark masses $m_{b}$ = 4.783 GeV, and $m_{q}$ = 0.323 GeV have been taken from PDG pdg . Data Set | $\alpha_{s}$ |  $\sigma$ (GeV) | b (Ge$V^{2}$)
---|---|---|---
I | 0.3841 | 1.50 | 0.1708
II | 0.70 | 0.3049 | 0.0946
III | 0.3714 | 1.50 | 0.1445
Table 2: The mass-spectra of (1S-5S, 1P-3P and 1D-wave) bottomonium states [$b\bar{b}$], obtained from Data Set I. ($m_{i}^{exp}$) corresponds to recent PDG pdg . $N^{2S+1}{L}_{J}$ | $J^{PC}$ | $\langle E\rangle$ | $\langle V_{V}\rangle$ | $\langle V_{S}\rangle$ | $\langle V_{SS}\rangle$ | $\langle V_{LS}\rangle$ | $\langle V_{T}\rangle$ | $\langle V^{(1)}(r_{ij})\rangle$ | $\langle K.E.\rangle$ | $m_{i}^{f}$ | $m_{i}^{exp}$ | Meson
---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{-+}$ | -133.0 | -757.8 | 165.5 | -23.4 | 0 | 0 | -2.2 | 482.3 | 9409 |  9398.7$\pm 2.0$ | $\eta_{b}(1S)$
$1^{3}S_{1}$ | $1^{--}$ | -132.9 | -757.8 | 165.5 | 7.6 | 0 | 0 | -2.3 | 451.6 | 9440 |  9460.30$\pm 0.26$ | $\Upsilon(1S)$
$2^{1}S_{0}$ | $0^{-+}$ | 428.3 | -352.1 | 404.5 | -6.5 | 0 | 0 | -1.1 | 382.5 | 9987 | … | …
$2^{3}S_{1}$ | $1^{--}$ | 429.6 | -352.1 | 404.5 | 2.1 | 0 | 0 | -0.9 | 374.4 | 9997 |  10023.26$\pm 0.31$ | $\Upsilon(2S)$
$3^{1}S_{0}$ | $0^{-+}$ | 764.1 | -258.2 | 596.3 | -4.2 | 0 | 0 | -0.7 | 430.4 | 10325 | … | …
$3^{3}S_{1}$ | $1^{--}$ | 764.1 | -258.2 | 596.3 | 1.4 | 0 | 0 | -0.7 | 425.1 | 10331 |  10355.2$\pm 0.5$ | $\Upsilon(3S)$
$4^{1}S_{0}$ | $0^{-+}$ | 1034.1 | -213.1 | 761.0 | -3.2 | 0 | 0 | -0.5 | 489.5 | 10596 | … | …
$4^{3}S_{1}$ | $1^{--}$ | 1034.1 | -213.1 | 761.0 | 1.0 | 0 | 0 | -0.5 | 485.4 | 10601 |  10579.4$\pm 1.2$ | $\Upsilon(4S)$
$5^{1}S_{0}$ | $0^{-+}$ | 1270.4 | -185.4 | 909.1 | -2.7 | 0 | 0 | -0.4 | 549.4 | 10833 | … | …
$5^{3}S_{1}$ | $1^{--}$ | 1270.4 | -185.4 | 909.1 | 0.9 | 0 | 0 | -0.4 | 546.1 | 10837 | … | …
$1^{3}P_{0}$ | $0^{++}$ | 335.3 | -313.6 | 329.1 | 0.8 | -28.0 | -10.8 | -1.2 | 319.2 | 9863 |  9859.44$\pm 0.42$ | $\chi_{b0}(1P)$
$1^{3}P_{1}$ | $1^{++}$ | 335.3 | -313.6 | 329.1 | 0.8 | -14.0 | 5.4 | -1.2 | 319.2 | 9893 |  9892.78$\pm 0.26$ | $\chi_{b1}(1P)$
$1^{1}P_{1}$ | $1^{+-}$ | 335.3 | -313.6 | 329.1 | -2.5 | 0 | 0 | -1.2 | 322.7 | 9898 |  9899.3$\pm 0.8$ | $h_{b}(1P)$
$1^{3}P_{2}$ | $2^{++}$ | 335.3 | -313.6 | 329.1 | 0.8 | 14.0 | -1.0 | -1.2 | 319.2 | 9915 |  9912.21$\pm 0.26$ | $\chi_{b2}(1P)$
$2^{3}P_{0}$ | $0^{++}$ | 679.2 | -227.2 | 529.5 | 0.7 | -23.3 | -8.7 | -0.7 | 376.4 | 10213 |  10232.5$\pm 0.4$ | $\chi_{b0}(2P)$
$2^{3}P_{1}$ | $1^{++}$ | 679.2 | -227.2 | 529.5 | 0.7 | -11.6 | -8.7 | -0.7 | 377.1 | 10239 |  10255.46$\pm 0.22$ | $\chi_{b1}(2P)$
$2^{1}P_{1}$ | $1^{+-}$ | 679.2 | -227.2 | 529.5 | -2.3 | 0 | 0 | -0.7 | 380.2 | 10243 | - | -
$2^{3}P_{2}$ | $2^{++}$ | 679.2 | -227.2 | 529.5 | 0.7 | 11.6 | -0.8 | -0.7 | 377.1 | 10257 |  10268.65$\pm 0.22$ | $\chi_{b2}(2P)$
$3^{3}P_{0}$ | $0^{++}$ | 955.3 | -187.1 | 699.7 | 0.6 | -21.4 | -8.0 | -0.5 | 442.4 | 10492 | … | …
$3^{3}P_{1}$ | $1^{++}$ | 955.3 | -187.1 | 699.7 | 0.6 | -10.7 | 4.0 | -0.5 | 442.4 | 10515 |  10513.42$\pm 0.41$ | $\chi_{b1}(3P)$
$3^{1}P_{1}$ | $1^{+-}$ | 955.3 | -187.1 | 699.7 | -2.0 | 0 | 0 | -0.5 | 445.4 | 10519 | … | …
$3^{3}P_{2}$ | $2^{++}$ | 955.3 | -187.1 | 699.7 | 0.6 | 10.7 | -0.8 | -0.5 | 442.4 | 10532 |  10524.02$\pm 0.57$ | $\chi_{b2}(3P)$
$1^{3}D_{1}$ | $1^{--}$ | 576.4 | -213.0 | 455.2 | 0.09 | -6.2 | -1.2 | -0.9 | 333.7 | 10135 | - | -
$1^{3}D_{2}$ | $2^{--}$ | 576.4 | -213.0 | 455.2 | 0.09 | -2.0 | 1.2 | -0.9 | 333.7 | 10141 |  10163.7$\pm 1.4$ | $\Upsilon_{2}(1D)$
$1^{1}D_{2}$ | $2^{-+}$ | 576.4 | -213.0 | 455.2 | -0.2 | 0 | 0 | -0.9 | 333.7 | 10142 | - | -
$1^{3}D_{3}$ | $3^{--}$ | 576.4 | -213.0 | 455.2 | 0.09 | -4.2 | -0.3 | -0.9 | 333.7 | 10146 | - | -
Table 3: The mass-spectra of (1S-3S, 1P-3P and 1D-wave) B-mesons [$b\bar{q}$], obtained from Data Set II. ($m_{i}^{exp}$) corresponds to recent PDG pdg . $N^{2S+1}L_{J}$ | $J^{PC}$ | $\langle E\rangle$ | $\langle V_{V}\rangle$ | $\langle V_{S}\rangle$ | $\langle V_{SS}\rangle$ | $\langle V_{LS}\rangle$ | $\langle V_{T}\rangle$ | $\langle V^{(1)}(r_{ij})\rangle$ | $\langle K.E.\rangle$ | $m_{i}^{f}$ | $m_{i}^{exp}$ | Meson
---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{-+}$ | 207.3 | -427.8 | 286.0 | -9.2 | 0 | 0 | -7.1 | 358.4 | 5304 |  5279.65$\pm 0.12$ | $B^{\pm}$
$1^{3}S_{1}$ | $1^{--}$ | 207.3 | -427.8 | 286.0 | 3.0 | 0 | 0 | -6.9 | 346.6 | 5317 |  5324.70$\pm 0.21$ | $B^{*}$
$2^{1}S_{0}$ | $0^{-+}$ | 774.4 | -241.3 | 599.6 | -3.0 | 0 | 0 | -4.3 | 416.5 | 5877 | - | -
$2^{3}S_{1}$ | $1^{--}$ | 771.4 | -241.3 | 599.6 | 1.0 | 0 | 0 | -4.0 | 415.0 | 5881 | - | -
$3^{1}S_{0}$ | $0^{-+}$ | 1173.2 | -185.4 | 846.3 | -2.0 | 0 | 0 | -2.9 | 513.9 | 6277 | - | -
$3^{3}S_{1}$ | $1^{--}$ | 1173.2 | -185.4 | 846.3 | 0.7 | 0 | 0 | -2.6 | 512.3 | 6281 | - | -
$1^{3}P_{0}$ | $0^{++}$ | 626.3 | -210.5 | 489.1 | 1.1 | -28.0 | -14.1 | -4.7 | 346.4 | 5691 | - | -
$1^{3}P_{1}$ | $1^{++}$ | 623.7 | -210.5 | 489.1 | 1.1 | -14.0 | 6.9 | -4.7 | 344.2 | 5723 |  5725.9${}_{-2.7}^{+2.5}$ | $B_{1}(5721)^{+}$
$1^{1}P_{1}$ | $1^{+-}$ | 623.7 | -210.5 | 489.1 | -3.3 | 0 | 0 | -4.7 | 348.7 | 5726 |  5726.1$\pm 1.3$ | $B_{1}(5721)^{0}$
$1^{3}P_{2}$ | $2^{++}$ | 623.7 | -210.5 | 489.1 | 1.1 | 14.0 | -1.0 | -4.7 | 344.1 | 5743 |  5739.5$\pm 0.7$ | $B^{*}_{2}(5747)^{0}$
$2^{3}P_{0}$ | $0^{++}$ | 1040.0 | -160.2 | 750.4 | 0.7 | -27.3 | -12.6 | -3.4 | 449.3 | 6107 | - | -
$2^{3}P_{1}$ | $1^{++}$ | 1042.1 | -160.2 | 750.4 | 0.7 | -13.6 | -6.3 | -3.4 | 451.2 | 6141 | - | -
$2^{1}P_{1}$ | $1^{+-}$ | 1042.1 | -160.2 | 750.4 | -2.1 | 0 | 0 | -3.4 | 454.2 | 6146 | - | -
$2^{3}P_{2}$ | $2^{++}$ | 1042.1 | -160.2 | 750.4 | 0.7 | 13.6 | -1.2 | -3.4 | 451.3 | 6161 | - | -
$3^{3}P_{0}$ | $0^{++}$ | 1391.2 | -134.5 | 974.0 | 0.5 | -27.1 | -12 | -2.4 | 551.3 | 6458 | - | -
$3^{3}P_{1}$ | $1^{++}$ | 1391.2 | -134.5 | 974.0 | 0.5 | -13.5 | -6.0 | -2.4 | 551.2 | 6490 | - | -
$3^{1}P_{1}$ | $1^{+-}$ | 1391.2 | -134.5 | 974.0 | -1.6 | 0 | 0 | -2.4 | 553.2 | 6495 | - | -
$3^{3}P_{2}$ | $2^{++}$ | 1391.2 | -134.5 | 974.0 | 0.5 | 13.5 | -1.2 | -2.4 | 551.2 | 6510 | - | -
$1^{3}D_{1}$ | $1^{--}$ | 894.2 | -150.5 | 649.2 | 0.3 | -1.9 | -1.8 | -4.0 | 395.2 | 5996 | - | -
$1^{3}D_{2}$ | $2^{--}$ | 894.2 | -150.5 | 649.2 | 0.3 | -0.6 | 1.8 | -4.0 | 395.2 | 6002 | - | -
$1^{1}D_{2}$ | $2^{-+}$ | 894.2 | -150.5 | 649.2 | -1.1 | -1.9 | -1.8 | -4.0 | 397.0 | 5999 | - | -
$1^{3}D_{3}$ | $3^{--}$ | 894.2 | -150.5 | 649.2 | 0.3 | 1.2 | -0.5 | -4.0 | 395.2 | 6001 | - | -

##  3 Results and Discussion

###  3.1 Bottomonium/ B-mesons

To calculate the mass-spectra of diquarks and tetra-quarks, first, we estimate
the mass-spectra of bottomonium states [$b\bar{b}$] and B-mesons, whose
results are tabulated in Table 2 and Table 3 , respectively. The model’s
reliability and consistency have been tested by obtaining the mass-spectra of
heavy and heavy-light bottom mesons. The SU(3) color symmetry allows only
colorless quark combination $|Q\bar{Q}\rangle$ to form any color singlet state
griffiths ; thesis , as in our case [$b\bar{b}$] and [$b\bar{q}$] are mesons
and exhibits $|Q\bar{Q}\rangle:\mathbf{3\otimes\bar{3}=1\oplus 8}$
representation which leads to carry a color factor $k_{s}=-\frac{4}{3}$
debastiani . The masses of the particular [$b\bar{b}$] and [$b\bar{q}$] states
are obtained namely

$M_{(b\bar{b})}=2m_{b}+E_{b\bar{b}}+\langle V^{1}(r_{ij})\rangle$ (24)

and

$M_{(b\bar{q})}=m_{b}+m_{\bar{q}}+E_{b\bar{q}}+\langle V^{1}(r_{ij})\rangle$
(25)

The final masses obtained from the above expression constitute the
contributions from different spin-dependent terms (spin-spin, spin-orbital and
tensor) have tabulated in Table 2 and Table 3. The mass-spectra of the mesons
produced in this study are compatible with the experimental data available in
the most recent updated PDG pdg . Additionally, the current work’s findings
are consistent with those in Ref. prd , where the author computed the mass-
spectra of all-charm [$cc\bar{c}\bar{c}$], all-bottom [$bb\bar{b}\bar{b}$],
and heavy-light tetraquarks [$Qq\bar{Q}\bar{q}$] (Q = b, c and q = u, d).

There are a total of 15 bottomonium mesons ($b\bar{b}$) produced from the
model, all of which have masses fairly closed to those predicted
experimentally. In the case of bottomonium S-wave states the discrepancy
($\Delta$) is around 30 MeV. Particularly, in vector states $1^{3}S_{1}$ where
spin-spin interaction contributes repulsive strength which maximizes $\Delta$.
Similarly, there are B-mesons whose masses have also good agreement with
experimentally predicted data and have discrepancy ($\Delta$) nearly 30 MeV.
At high energy scale, discrepancy nearly 30 MeV’s between the model’s mass and
experimental data can be tolerated and the fitting parameters are assumed as
best fit.

Spin-dependent interactions are crucial in heavy quarkonium study specially in
bottomonium and charmonium because they allow for the consideration of QCD
dynamics in the heavy quark scenario, which lies between the perturbative and
non-perturbative regimes thesis . The spin-spin interaction’s involvement in
orbitally excited states is particularly intriguing. We notice that the masses
of P-wave states are more precise than those of S-wave states owing to spin-
orbit and tensor contributions (see Table 2 and Table 3). The slighter
contribution of the relativistic term, which is greater for lighter quarks,
alters the spectrum by a few MeV’s.

Table 4: The masses vector (anti)diquarks and energy eigenvalue from present work and comparison with other prior works. Units are in (MeV). Diquark | $\langle E^{0}\rangle$ | Ours | prd  | mb  | mk ; de  | mn ; zg  | Su ; RK
---|---|---|---|---|---|---|---
bb | 78.3 | 9641 | 9643 | 9845 | 9718 | 9850 | 8670$\pm$690
bq | 235.7 | 5331 | 5339 | 5465 | 5381 | 5130$\pm$110 | 5080$\pm$40

####  3.1.1 Diquarks

A (anti)diquark is a pair of (anti)quarks that interact with one another
through gluonic exchange and can form a bound state (qq) diquarks . It’s
important to remember that while constructing a diquark is a composite (qq)
system not a point-like object. The form factor, which can be represented as
the overlap integral of diquark wave functions, does indeed smear its
interaction with gluons. The Pauli principle should also be considered, which
results in the following ground state diquark limitations. To comply the Pauli
exclusion principle, which states that diquarks with the same flavour quark
should have a spin of 1, the diquark’s wavefunction must be antisymmetric
griffiths . The (qq’) diquark, which is made of quarks of various flavours,
may have spins S = 0,1 (scalar [qq’], axial vector $\\{qq^{\prime}\\}$
diquarks), while the $\\{qq\\}$ diquark, which is composed of quarks of the
same flavour, can only have spin S = 1. Because of the stronger attraction
owing to the spin–spin interaction, the scalar S diquark is frequently
referred to as a “good” diquark, while the heavier axial vector diquark is
referred to as a “bad” diquark exotic2 .

To produce the most compact diquark, we will utilise the ground state
($1^{3}S_{1}$) axial vector diquarks $\\{bb\\}$ or $\\{bq\\}$, which have no
orbital or radial excitations. Everything done for quark-quark interactions is
assumed to be identical to antiquark-antiquark interactions, with the
exception that colors are replaced by anticolors. We utilised the same
methodology as in the case of the bottom mesons to get the mass-spectra of
(anti)diquarks. According to QCD color symmetry, two quarks are combined in
the fundamental (3) representation to obtain the diquark, presented by
$\mathbf{3\otimes 3=\bar{3}\oplus 6}$. Moreover, antiquarks are combined in
the $\mathbf{\bar{3}}$ representation and can be presented as
$\mathbf{\bar{3}\otimes\bar{3}=3\oplus\bar{6}}$ debastiani ; griffiths . The
diquark-antidiquark approximation is significant because it reduces a complex
four-body problem to a simple two-body problem deutron ; deutron1 . The
hamiltonian, on the other hand, ceases replicating the meson spectra when
doing the full four-body basis treatment diquark1 . The explanation for this
is simple: the $3\otimes\bar{3}$ color coupling can be transformed into a
$1\otimes 1$ state, and also a $8\otimes 8$ state. The QCD color symmetry
produces a color factor $k_{s}=-\frac{2}{3}$ in antitriplet state and makes
the short distance part $(\frac{1}{r_{ij}})$ of the interaction attractive
griffiths .

We compared the diquark masses acquired in this work to those obtained in the
other prior investigations mentioned in Table 4. The diquark masses are
investigated in Ref. mb  using the so-called Schwinger-Dyson and Bethe-
Salpeter approach, which account for kinetic energy as well as splittings in
the spin-spin, spin-orbit, and tensor interactions. The masses of diquarks
calculated in this research are consistent with prd  and are less than those
reported in Ref. mb . Relativistic models, such as those presented in Refs. mk
; de ; mn , all predict larger diquark masses, whereas models based on QCD sum
rules, such as those presented in Refs. zg ; Su ; RK  all predict smaller
diquark masses. The discrepancies may be due to the addition of new and
updated data in this study.

Table 5:  The mass-spectra of S and P-wave $T_{4b}$ tetraquark states, obtained from Data Set I whereas $m_{th}$ corresponds to mass of two meson threshold. $N^{2S+1}{L}_{J}$ | $J^{PC}$ | $\langle E\rangle$ | $\langle V_{V}\rangle$ | $\langle V_{S}\rangle$ | $\langle V_{SS}\rangle$ | $\langle V_{LS}\rangle$ | $\langle V_{T}\rangle$ | $\langle V^{(1)}(r_{ij})\rangle$ | $\langle K.E.\rangle$ | $m_{i}^{f}$ | $m_{th}$ | Threshold
---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{++}$ | -532.8 | -1344 | 95.4 | -30.1 | 0 | 0 | -2.7 | 745.3 | 18719 | 18798 | $\eta_{b}(1S)\eta_{b}(1S)$
$1^{3}S_{1}$ | $1^{+-}$ | -532.8 | -1344 | 95.4 | -15.1 | 0 | 0 | -2.7 | 730.4 | 18734 | 18859 | $\eta_{b}(1S)\Upsilon(1S)$
$1^{5}S_{2}$ | $2^{++}$ | -532.8 | -1344 | 95.4 | 15.2 | 0 | 0 | -2.7 | 701.0 | 18764 | 18920 | $\Upsilon(1S)\Upsilon(1S)$
$2^{1}S_{0}$ | $0^{++}$ | 164.2 | -589.5 | 277.1 | -5.2 | 0 | 0 | -1.0 | 481.5 | 19441 | 19998 | $\eta_{b}(2S)\eta_{b}(2S)$
$2^{3}S_{1}$ | $1^{+-}$ | 164.2 | -589.5 | 277.1 | -2.6 | 0 | 0 | -1.0 | 478.5 | 19443 | … | …
$2^{5}S_{2}$ | $2^{++}$ | 164.2 | -589.5 | 277.1 | 2.6 | 0 | 0 | -1.0 | 473.1 | 19448 | … | …
$3^{1}S_{0}$ | $0^{++}$ | 480.1 | -427.4 | 435.3 | -3.1 | 0 | 0 | -0.6 | 475.3 | 19759 | … | …
$3^{3}S_{1}$ | $1^{+-}$ | 480.1 | -427.4 | 435.3 | -1.5 | 0 | 0 | -0.6 | 473.7 | 19760 | … | …
$3^{5}S_{2}$ | $2^{++}$ | 480.1 | -427.4 | 435.3 | 1.5 | 0 | 0 | -0.6 | 470.6 | 19764 | … | …
$1^{1}P_{1}$ | $1^{--}$ | 105.1 | -464.4 | 225.5 | -5.8 | 0 | 0 | -1.2 | 350.0 | 19381 | … | …
$1^{3}P_{0}$ | $0^{-+}$ | 105.1 | -464.4 | 225.5 | -2.9 | -25.6 | -18.1 | -1.2 | 391.0 | 19340 | 19258 | $\eta_{b}(1S)\chi_{b0}(1P)$
$1^{3}P_{1}$ | $1^{-+}$ | 105.1 | -464.4 | 225.5 | -2.9 | -12.8 | 9.0 | -1.2 | 350.7 | 19380 | 19292 | $\eta_{b}(1S)\chi_{b1}(1P)$
$1^{3}P_{2}$ | $2^{-+}$ | 105.1 | -464.4 | 225.5 | -2.9 | 12.8 | -1.8 | -1.2 | 336.0 | 19395 | 19311 | $\eta_{b}(1S)\chi_{b2}(1P)$
$1^{5}P_{1}$ | $1^{--}$ | 105.1 | -464.4 | 225.5 | 2.9 | -38.4 | -12.7 | -1.2 | 392.0 | 19338 | 19298 | $\eta_{b}(1S)h_{b}(1P)$
$1^{5}P_{2}$ | $2^{--}$ | 105.1 | -464.4 | 225.5 | 2.9 | -12.8 | 12.7 | -1.2 | 341.0 | 19390 | 19353 | $\Upsilon(1S)\chi_{b1}(1P)$
$1^{5}P_{3}$ | $3^{--}$ | 105.1 | -464.4 | 225.5 | 2.9 | 25.6 | 3.6 | -1.2 | 319.0 | 19412 | 19372 | $\Upsilon(1S)\chi_{b2}(1P)$
Table 6:  The mass-spectra of S and P-wave tetraquark states $T_{4b}$, obtained from Data Set I+III whereas $m_{th}$ corresponds to mass of two meson threshold. $N^{2S+1}{L}_{J}$ | $J^{PC}$ | $\langle E\rangle$ | $\langle V_{V}\rangle$ | $\langle V_{S}\rangle$ | $\langle V_{SS}\rangle$ | $\langle V_{LS}\rangle$ | $\langle V_{T}\rangle$ | $\langle V^{(1)}(r_{ij})\rangle$ | $\langle K.E.\rangle$ | $m_{i}^{f}$ | $m_{th}$ | Threshold
---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{++}$ | -503.4 | -1252.3 | 83.4 | -28.3 | 0 | 0 | -2.3 | 693.2 | 18749 | 18798 | $\eta_{b}(1S)\eta_{b}(1S)$
$1^{3}S_{1}$ | $1^{+-}$ | -503.4 | -1252.3 | 83.4 | -14.1 | 0 | 0 | -2.0 | 679.3 | 18764 | 18859 | $\eta_{b}(1S)\Upsilon(1S)$
$1^{5}S_{2}$ | $2^{++}$ | -503.4 | -1252.3 | 83.4 | 14.0 | 0 | 0 | -2.3 | 651.2 | 18792 | 18920 | $\Upsilon(1S)\Upsilon(1S)$
$2^{1}S_{0}$ | $0^{++}$ | 136.1 | -531.0 | 245.1 | -4.7 | 0 | 0 | -0.8 | 427.5 | 19414 | 19998 | $\eta_{b}(2S)\eta_{b}(2S)$
$2^{3}S_{1}$ | $1^{+-}$ | 136.1 | -531.0 | 245.1 | -2.3 | 0 | 0 | -0.8 | 425.6 | 19416 | … | …
$2^{5}S_{2}$ | $2^{++}$ | 136.1 | -531.0 | 245.1 | 2.3 | 0 | 0 | -0.8 | 420.2 | 19421 | … | …
$3^{1}S_{0}$ | $0^{++}$ | 422.4 | -379.4 | 387.3 | -2.8 | 0 | 0 | -0.5 | 417.5 | 19701 | … | …
$3^{3}S_{1}$ | $1^{+-}$ | 422.4 | -379.4 | 387.3 | -1.4 | 0 | 0 | -0.5 | 416.5 | 19703 | … | …
$3^{5}S_{2}$ | $2^{++}$ | 422.4 | -379.4 | 387.3 | 1.4 | 0 | 0 | -0.5 | 417.5 | 19706 | … | …
$1^{1}P_{1}$ | $1^{--}$ | 84.2 | -428.2 | 200.0 | -5.0 | 0 | 0 | -0.9 | 318.0 | 19361 | … | …
$1^{3}P_{0}$ | $0^{-+}$ | 84.2 | -428.2 | 200.0 | -2.5 | -21.5 | -15.2 | -0.9 | 316.2 | 19327 | 19258 | $\eta_{b}(1S)\chi_{b0}(1P)$
$1^{3}P_{1}$ | $1^{-+}$ | 84.2 | -428.2 | 200.0 | -2.5 | -10.4 | 7.7 | -0.9 | 316.0 | 19361 | 19292 | $\eta_{b}(1S)\chi_{b1}(1P)$
$1^{3}P_{2}$ | $2^{-+}$ | 84.2 | -428.2 | 200.0 | -2.5 | 10.5 | -1.5 | -0.9 | 316.0 | 19373 | 19311 | $\eta_{b}(1S)\chi_{b2}(1P)$
$1^{5}P_{1}$ | $1^{--}$ | 84.2 | -428.2 | 200.0 | 2.5 | -32.4 | -11.1 | -0.9 | 311.0 | 19325 | 19298 | $\eta_{b}(1S)h_{b}(1P)$
$1^{5}P_{2}$ | $2^{--}$ | 84.2 | -428.2 | 200.0 | 2.5 | -11.2 | 11.0 | -0.9 | 311.2 | 19369 | 19353 | $\Upsilon(1S)\chi_{b1}(1P)$
$1^{5}P_{3}$ | $3^{--}$ | 84.2 | -428.2 | 200.0 | 2.5 | 21.7 | -3.0 | -0.9 | 311.0 | 19388 | 19372 | $\Upsilon(1S)\chi_{b2}(1P)$
Table 7:  The mass-spectra of S and P-wave $bq\bar{b}\bar{q}$ tetraquark states, obtained from Data Set II+III, whereas $m_{th}$ corresponds to mass of two meson threshold. $N^{2S+1}{L}_{J}$ | $J^{PC}$ | $\langle E\rangle$ | $\langle V^{(0)}_{V}\rangle$ | $\langle V^{(0)}_{S}\rangle$ | $\langle V^{(1)}_{SS}\rangle$ | $\langle V^{(1)}_{LS}\rangle$ | $\langle V^{(1)}_{T}\rangle$ | $V^{(1)}(r)$ | $m_{i}^{f}$ | $m_{th}$ | Threshold
---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{++}$ | -181.1 | -764.0 | 135.0 | -50.0 | 0 | 0 | -2.0 | 10429 | 10558 | $B^{\pm}B^{\pm}$
$1^{3}S_{1}$ | $1^{+-}$ | -181.1 | -764.0 | 135.0 | -25.5 | 0 | 0 | -2.0 | 10454 | 10603 | $B^{\pm}B^{*}$
$1^{5}S_{2}$ | $2^{++}$ | -181.1 | -764.0 | 135.0 | 25.4 | 0 | 0 | -2.0 | 10505 | 10648 | $B^{*}B^{*}$
$2^{1}S_{0}$ | $0^{++}$ | 339.1 | -340.0 | 341.0 | -13.3 | 0 | 0 | -0.9 | 10987 | … | …
$2^{3}S_{1}$ | $1^{+-}$ | 339.1 | -340.0 | 341.0 | -6.6 | 0 | 0 | -0.8 | 10995 | … | …
$2^{5}S_{2}$ | $2^{++}$ | 339.1 | -340.0 | 341.0 | 6.6 | 0 | 0 | -0.9 | 11009 | … | …
$3^{1}S_{0}$ | $0^{++}$ | 637.4 | -247.2 | 507.0 | -8.4 | 0 | 0 | -0.6 | 11290 | … | …
$3^{3}S_{1}$ | $1^{+-}$ | 637.4 | -247.2 | 507.0 | -4.2 | 0 | 0 | -0.6 | 11295 | … | …
$3^{5}S_{2}$ | $2^{++}$ | 637.4 | -247.2 | 507.0 | 4.2 | 0 | 0 | -0.6 | 11303 | … | …
$1^{1}P_{1}$ | $1^{--}$ | 262.0 | -305.0 | 277.0 | -5.5 | 0 | 0 | -0.8 | 10918 | … | …
$1^{3}P_{0}$ | $0^{-+}$ | 262.0 | -305.0 | 277.0 | -2.7 | -23.0 | -17.4 | -0.8 | 10881 | 11006 | $B^{\pm}B_{1}(5721)^{+}$
$1^{3}P_{1}$ | $1^{-+}$ | 262.0 | -305.0 | 277.0 | -2.7 | -11.5 | 8.7 | -0.8 | 10918 | … | …
$1^{3}P_{2}$ | $2^{-+}$ | 262.0 | -305.0 | 277.0 | -2.7 | 11.5 | 1.7 | -1.0 | 10931 | … | …
$1^{5}P_{1}$ | $1^{--}$ | 262.0 | -305.0 | 277.0 | 2.7 | -34.4 | -12.2 | -0.8 | 10880 | … | …
$1^{5}P_{2}$ | $2^{--}$ | 262.0 | -305.0 | 277.0 | 2.7 | -11.5 | -12.2 | -1.0 | 10928 | … | …
$1^{5}P_{3}$ | $3^{--}$ | 262.0 | -305.0 | 277.0 | -5.5 | 23.0 | -3.4 | -0.8 | 10946 | … | …
Table 8: Comparison of the tetraquarks masses from the present work with others $bb\bar{b}\bar{b}$ State | $J^{PC}$ | Ours | prd  | a.v.  | ming  | kar  | guang  | deng  | qi  | xin  | ma  | Xi
---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{++}$ | 18719 | 18723 | 18754 | 19322 | 18826$\pm$25 | … | 19329 | 19255 | 19122-19344 | 18748 | 19178
$1^{3}S_{1}$ | $1^{+-}$ | 18734 | 18738 | 18808 | 19329 | ….. | 19247 | 19373 | 19251 | 19184-19354 | 18828 | 19226
$1^{5}S_{2}$ | $2^{++}$ | 18764 | 18768 | 18916 | 19341 | 18956$\pm$25 | 19249 | 19387 | 19262 | 19236-19374 | 18900 | 19236
$1^{1}P_{1}$ | $1^{--}$ | 19361 | … | … | … | … | … | … | … | … | 19281 | …
$1^{3}P_{0}$ | $0^{-+}$ | 19327 | … | … | … | … | … | … | … | … | 19288 | …
$1^{3}P_{1}$ | $1^{-+}$ | 19361 | … | … | … | … | … | … | … | … | … | …
$1^{3}P_{2}$ | $2^{-+}$ | 19373 | … | … | … | … | … | … | … | … | … | …
$1^{5}P_{1}$ | $1^{--}$ | 19325 | … | … | … | … | … | … | … | … | … | …
$1^{5}P_{2}$ | $2^{--}$ | 19369 | … | … | … | … | … | … | … | … | … | …
$1^{5}P_{3}$ | $3^{--}$ | 19388 | … | … | … | … | … | … | … | … | … | …
$bq\bar{b}\bar{q}$ State | $J^{PC}$ | Ours | aslam  | prd  | de  | mh  | farhad  | ebert  | smpatel  |  |  |
$1^{1}S_{0}$ | $0^{++}$ | 10429 | … | 10445 | 10473 | 10469 | 10462 | 10471 | 10143 |  |  |
$1^{3}S_{1}$ | $1^{+-}$ | 10454 | … | 10472 | 10494 | 10453 | … | 10492 | 10233 |  |  |
$1^{5}S_{2}$ | $2^{++}$ | 10505 | 10520 | 10523 | 10534 | …. | … | 10534 | 10413 |  |  |
$1^{1}P_{1}$ | $1^{--}$ | 10918 | 10900 | 10936 | 10807 | … | 10819 | 10807 | … |  |  |
$1^{3}P_{0}$ | $0^{-+}$ | 10881 | … | … | … | … | … | 10836 | … |  |  |
$1^{3}P_{1}$ | $1^{-+}$ | 10918 | … | … | 10939 | … | … | 10847 | … |  |  |
$1^{3}P_{2}$ | $2^{-+}$ | 10931 | … | … | … | … | … | 10854 | … |  |  |
$1^{5}P_{1}$ | $1^{--}$ | 10880 | … | … | 10944 | … | … | 10827 | … |  |  |
$1^{5}P_{2}$ | $2^{--}$ | 10928 | … | … | … | … | … | 10856 | … |  |  |
$1^{5}P_{3}$ | $3^{--}$ | 10946 | … | … | … | … | … | 10858 | … |  |  |

###  3.2 Tetraquark

Tetraquarks are color singlet states made up of a diquark and an antidiquark
in color antitriplet $\mathbf{\bar{3}}$ and triplet $\bf{3}$ configurations
respectively, that are held together by color forces de . The four-body non-
relativistic calculation is simplified to a two-body calculation using this
approximation debastiani ; prd ; de . A $T_{4b}$ and $bq\bar{b}\bar{q}$ are
color singlet states and yield a color factor $k_{s}=-\frac{4}{3}$. The
$1^{3}S_{1}$ diquark-antidiquark are combined to form color singlet tetraquark
thesis , and that can be represented as;
$|QQ|^{3}\otimes|\bar{Q}\bar{Q}|^{\bar{3}}\rangle=\mathbf{1\oplus 8}$. The
mass-spectra of all bottom ($T_{4b}$) and heavy-light bottom tetraquarks
($bq\bar{b}\bar{q}$) have been obtained with the same formulation as in the
case of mesons, namely;

$M_{(T_{4b})}=m_{bb}+m_{\bar{b}\bar{b}}+E_{[bb][\bar{b}\bar{b}]}+\langle
V^{1}(r_{ij})\rangle$ (26)

and

$M_{bq\bar{b}\bar{q}}=m_{bq}+m_{\bar{b}\bar{q}}+E_{[bq][\bar{b}\bar{q}]}+\langle
V^{1}(r_{ij})\rangle$ (27)

The masses of $T_{4b}$ and $bq\bar{b}\bar{q}$ arise mostly from the cornell
potential and the relativistic correction term $\langle V^{1}(r)\rangle$. All
spin-dependent terms have been computed for spin-1 diquarks and antiquarks
that combine to produce a color singlet tetraquark with spin $S_{T}$ = 0,1,2.
The mass-spectra of radial and orbital excitations are obtained by coupling
the total spin $S_{T}$ with the total orbital angular momentum $L_{T}$, which
results in the total angular momentum $J_{T}$. To get the tetraquark’s total
spin $S_{T}$ and quantum number $J^{PC}$, we shall utilise notations for the
spins of a diquark $S_{d}$ and an antidiquark $S_{\bar{d}}$. The interaction
of $S_{T}$ with the orbital angular momentum $L_{T}$ results in the formation
of a color singlet state $S_{T}\otimes L_{T}$.

$|T_{4Q}\rangle=|S_{d},S_{\bar{d}},S_{T},L_{T}\rangle_{J_{T}}$ (28)

To find out the quantum numbers ($J^{PC}$) of the tetra-quark states, one can
use the following formula; $P_{T}=(-1)^{L_{T}}$ and
$C_{T}=(-1)^{L_{T}+S_{T}}$.

The masses of low-lying S-wave $T_{4b}$ and $bq\bar{b}\bar{q}$ states are
anticipated to be in the range of 18-20 GeV and 10-11 GeV, respectively jw ,
in the current study, the masses are also found to be in this range. As shown
in Table 5, Table 6, and Table 7, the compactness of the 1S-states is mostly
due to the coulomb interaction. This indicates that one-gluon exchange is the
dominant mechanism behind the strong interaction between diquarks and
antidiquarks, which results in a negative energy eigenvalue E. The
contribution of the confinement term increases with the increase in radial and
orbital states.

Within the specific tetraquark mass-spectrum, the attractive strength of the
spin-spin interaction decreases as the number of radial and orbital excited
states increases. In this instance, we must bear in mind that the factors
originating from $S_{1}$ and $S_{2}$ are greater for the coupling of two
spin-1 particles than for the coupling of two spin-$\frac{1}{2}$ particles. It
is worth noting that, despite the fact that the spin-dependent terms have been
suppressed by a factor $\frac{1}{m_{qq}^{2}}$, one would anticipate them to be
less than the equivalent terms in $q\bar{q}$ mesons. The color interaction
brings diquark and antidiquark so close together that the suppression caused
by this component $\frac{1}{m_{qq}^{2}}$, is swamped by the massive
suppression at the system’s origin. On the other hand, spin-orbital and the
tensor interactions becomes more important when radial or orbital excitations
are involved. When the relativistic term $V^{1}(r_{ij})$ is incorporated in
the central potential, the mass spectrum shifts by a few MeVs.
The masses of 1S-wave all-bottom tetraquarks ($T_{4b}$) obtained from data
Sets I and I+III are 100-200 MeV below the two-meson threshold, implying that
these states may be accounted by the two meson thresholds stated in Tables 5
and 6. Initial predictions for $bb\bar{b}\bar{b}$ below the 2$\Upsilon$
threshold were made in Ref. first , and these were confirmed by subsequent
works nre ; qcdsum ; jw .

In Ref. Ciaran Hughes , the authors used a first-principles lattice non
relativistic QCD method to examine a $bb\bar{b}\bar{b}$ tetraquark bound state
and found no evidence for the mass below the lowest non-interacting
bottomonium-pair threshold in the $0^{++}$, $1^{+-}$, and $2^{++}$ channels.
Additionally, in Ref. silver , the author employs a model with only
chromomagnetic interactions and finds an unbound tetraquark and the same
conclusion was drawn within non-relativistic nonrelativistic  and semi-
relativistic quark-model studies relativistic . The existence of all-bottom
four-quark states is not ruled out by these result, but it draw attention to
the gap between the diquark approximation and a completes treatment of the
color basis. The author of Ref. a.v. , finds that $0^{++}$, $1^{+-}$, and
$2^{++}$ tetraquarks are confined by 44, 51, and 5 MeV, respectively,
utilising a phenomenologically motivated non confining potential between the
pointlike diquark and antidiquark. A tetraquark candidate has been identified
about 300 MeV below the experimental 2$\eta_{b}$ threshold in the QCD sum-
rules framework recent1 . Gang Yang et al., in Ref. Cbecchi , predicted that
the mass of the $0^{++}$ tetraquark $bb\bar{b}\bar{b}$ state would be lower
than the 2$\eta_{b}$ threshold whereas in Ref. qcdsum , a tetraquark candidate
has been found between the experimental 2$\eta_{b}$ and 2$\Upsilon$
thresholds.

Other S-wave candidates include $\eta_{b}\Upsilon$ and 2$\Upsilon$ for
$1^{+-}$ and $2^{++}$, respectively, which have a mass discrepancies $\Delta$
of less than 200 MeV with a two-meson threshold. The lowest and greatest
discrepancies for 1S-wave between the model’s mass and the two-meson threshold
are 49 MeV and 156 MeV, respectively, for data sets I and III. The discrepancy
($\Delta$) is lower in the mass-spectra obtained by data set III as compared
to data set I which shows the fitting parameters from data set III are
accurately fitted. Thus, the authors find that the $T_{4b}$ tetraquark state
may be either below or between the 2$\eta_{b}$ and 2$\Upsilon$ thresholds
(where the thresholds were established using experimental meson masses in both
instances).

Similarly, the P-wave masses match fairly well to the model’s mass and two-
meson threshold. As shown in Table 8, due to the scarcity of orbitally excited
data, this research, in conjunction with others, seems to be helpful for
studying orbitally excited states. In the P-wave states, where the
contribution of spin-orbit and tensor terms is produced, the difference
($\Delta$) between two-meson thresholds and the model’s mass falls to less
than 100 MeV. The two meson $\eta_{b}\chi_{b1}$ threshold, which contains the
quantum number ($1^{-+}$), and the $\Upsilon\chi_{b2}$ threshold, which
contains the quantum number ($3^{--}$), have the greatest and smallest
deviations from the two-meson threshold, respectively, of 88 MeV and 16 MeV.

The masses of S-wave heavy-light bottom tetraquark states $bq\bar{b}\bar{q}$,
are in good agreement with $B^{\pm}B^{\pm}$, $B^{\pm}B^{*}$, and $B^{*}B^{*}$
meson thresholds, with a difference of less than 200 MeV between the two meson
thresholds ($m_{th}$) and the model’s mass ($m^{f}_{i}$). The two most
discussed bottom resonances, $Z_{b}(10610)$ and $Z_{b}(10650)$, both with
($1^{+-}$), may be recognised as possible candidates for $bq\bar{b}\bar{q}$
states zb2 , which have a mass variation of 150 MeV from the model’s mass
($m^{f}_{i}$).

##  4 Conclusion

In the present work we have calculated the mass-spectra of all-bottom
$[bb][\bar{b}\bar{b}]$ and heavy-light bottom $[bq][\bar{b}\bar{q}]$
tetraquarks, in a non-relativistic framework that includes the cornell like
potential along with the relativistic correction term to the potential and
spin-dependent interactions. Tetraquarks have been considered to be comprised
of axial-vector diquarks and antidiquarks in a color antitriplet-triplet
($\bar{3}_{c}-3_{c}$) configuration. We first estimated the masses of bottom
mesons to fit the model’s free parameters, and then calculated the masses of
axial diquarks to get the mass spectra of corresponding tetraquarks without
violating the Pauli exclusion principle.

In this way we predict the masses of diquarks and tetra-quarks which have
atleast one bottom quark in the substructure. The doubly hidden-bottom
tetraquark states have a considerably greater energy than ordinary bottomonium
mesons and they may be differentiated experimentally from ordinary $q\bar{q}$
states. We explored the two most discussed $Z_{b}(10610)$ and $Z_{b}(10650)$
states with quantum numbers ($1^{+-}$), which are predicted as tetra-quark
aslam2  and as molecule zb2 . In the present study, the mass of the tetraquark
state $bq\bar{b}\bar{q}$ ($1^{+-}$) is about 150 MeV distant from the
experimental mass of these $Z_{b}$ states. When compared to the $BB^{*}$ and
$B^{*}B^{*}$ states, the $Z_{b}(10610)$ and $Z_{b}(10650)$ states are just a
few MeV above the two-meson threshold, indicating that these states should
have a tetraquark structure above the threshold. To get a definite conclusion
about the substructure of these $Z_{b}^{\prime}s$ states, one must investigate
their decay properties, which will be the subject of future extension work to
this study.

Due to the fact that fully bottom tetraquark states $bb\bar{b}\bar{b}$ are
heavier than heavy-light tetraquark states ($bq\bar{b}\bar{q}$), and they are
likely to be recognised below two meson thresholds, namely 2$\eta_{b}$,
2$\Upsilon$ or $\eta_{b}\Upsilon$ with masses ranging from 18.7 GeV to 19 GeV.
Our findings are in excellent accord with other non-relativistic models and
other studies cited in the literature prd ; a.v. ; ma ; Xi ; ebert .

The production of the ($QQ\bar{Q}\bar{Q}$), states is very challenging since
it necessitates the creation of two heavy quark pairs. However, new
observations of the $J/\psi\Upsilon$ psiups , pair and simultaneous
$\Upsilon\Upsilon$ cms  events have given some insight on how these doubly
hidden-charm/bottom tetra-quarks are produced. As a result, the 2$\eta_{b}$,
$\eta_{b}\Upsilon$ and 2$\Upsilon$ channels may be excellent candidates for
searching for the doubly hidden-bottom states $bb\bar{b}\bar{b}$, and LHCb,
D0, and CMS could be suitable platforms.

## Acknowledgement

Rohit Tiwari was inspired by the work of V.R. Debastiani, F.S. Navarra and Per
Lundhammar, Tommy Ohlsson on Spectroscopic study in non-relativistic model for
the tetraquark and would like to thank them for their valuable contributions
to this field.

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