# Spectroscopy and decay properties of bottomonium using
instanton induced potential from QCD vacuum

Bhoomika Pandya1<EMAIL_ADDRESS>Manan Shah2<EMAIL_ADDRESS>P C
Vinodkumar1<EMAIL_ADDRESS>1Department of Physics, Sardar Patel
University,Vallabh Vidyanagar, INDIA  2P. D. Patel Institute of Applied
Sciences, CHARUSAT, Changa, INDIA

###### Abstract

Ground state and radial excitation mass spectra of bottomonium states are
computed using the Instanton Induced $q\bar{q}$ Potential obtained from
Instanton Liquid Model for QCD vacuum. The Schrodinger equation is solved for
two body problem using variational method. Hyperfine interactions are
incorporated to remove the degeneracy between spin singlet and spin triplet
states of $S$ wave masses. Spin- orbit and tensor interactions through one
gluon exchange has been added to the spin average masses of P and D wave
states. The vector decay constants and pseudoscalar decay constants are also
estimated using the Van Royan Weiskopff formula within non-relativistic limit.
The di-leptonic, di-gamma, di-gluon decays of the bottomonium states are
predicted without and with QCD corrections up to the lowest order using
numerical values of the wave function at origin. Tri-gluon decays of
respective $S$, $P$ and $D$ states are also studied. Other annihilation decay
channels of bottomonium states into $\gamma gg$ , $\gamma\gamma\gamma$ ,
$q\bar{q}+g$ are also computed. The radiative transitions of first order like
electric dipole transitions (E1) and magnetic dipole transitions (M1) are
estimated. Our estimations of the mass spectra and decay properties for
bottomonium found to be in excellent agreement with the average experimental
values reported by particle data group.

††preprint: APS/123-QED

##  I  Introduction

Quarkonia are regarded as the simple and appropriate hadronic systems to
explore the QCD aspects at the low energy regimes through its spectroscopy [2,
3, 4, 1]. Since the discovery of $\Upsilon(1S)$, $\Upsilon(2S)$ and
$\Upsilon(3S)$ at the Fermilab by E288 collaboration [5, 6]; many of its
orbital excited states such as $\chi_{b0,1,2}(1P)$ and $\chi_{b0,1,2}(2P)$
etc. are also discovered subsequently [7, 8, 9, 10]. Further, continuous
progress has been achieved in the bottomnium spectroscopy by the discovery of
$\Upsilon(4S)$, $\Upsilon(10860)$, $\Upsilon(11020)$ [12, 11]. BABAR
Collaboration discovered the spin singlet low lying $\eta_{b}(1S)$ [13] and by
the efforts of CLEO, Belle and BABAR its average mass is found to be 9399.0
$\pm$ 2.3 [14, 15, 16]. CLEO has given the successful observation of the
$\eta_{b}(2S)$ with 9974.6 $\pm$ 2.3 $\pm$ 2.1 MeV [17]but BELLE recorded the
state having a mass of 9999.0 $\pm$ 3.5 after performing 17 times the same
experiment but not for single time it founds signal nearer to 9974 MeV [18].
$\Upsilon(1^{3}D_{2})$ also well established [19, 20]and low lying $h_{b}(1P)$
and $h_{b}(2P)$ are also detected by BABAR [21]. Recently, LHCb collobaration
found the mass of $\chi_{b1}(3P)$ as $10515^{+2.2}_{-3.9}$ [22]. Two other
charged states in the bottom family, $Z_{b}(10610)$ and $Z_{b}(10650)$ are
also being reported recently [23]. Their electric charge suggest that they are
not conventional bottomonium [24, 25], so we do not include them for the
present work.

Looking to the advances in the experimental side, it is necessary to review
earlier theoretical attempts to understand the quarkonia systems. Although
there are many different approaches like Lattice QCD methods [26, 27, 28, 29],
NRQCD methods [30, 31, 32], Light front quark model [33, 36, 34, 35], various
coupled channel quark models [37, 38, 39]. Effective Lagrangian approach [40]
etc., employed to study them, still there is no consensus and discrepancy
persist in the predictions. For instance, authors [41] have used
nonrelativistic constituent quark model wherein the lowest lying bottomonium
states $\Upsilon(1S)$ and $\eta_{b}(1S)$ are about 50 MeV heigher than the PDG
[43] data but the same model give fine agreement for the higher excited state
$\Upsilon(6S)$ while the relativized quark model [42] provides very good
discription of the low-lying states but higher excited $\Upsilon(6S)$ is
overestimated by 100 MeV.

It is believed that most reliable description of the bound system is given by
Bethe-Salpeter formalism. It is a relativistic quantum field theory but
ambiguity is that the interaction kernel is not solvable from QCD and it
becomes very tough to extract hadronic information [44]. In EFTs, threshold
region is still disappointing [46]. Lattice regularized QCD beyond threshold
and full understanding of bottomonium seems difficult [47, 48, 49, 50, 51, 52,
53, 54] even though very recently they have predicted few radial excitations
also.

Currently, computational complexity of other theoretical approaches makes the
potential model approach as the most reliable which is able to meet the
expectations. Potential models provide quantitative and qualitative analysis
of the quarkonia states. Potential models account for both the shorter and
large distance behaviour of the $q$-$\bar{q}$ interaction. The most promising
non-perturbative contibution is the ”Quark Confinement” which can be
incorporated by the “Wilson loop“. Such non-perturbative effects will increase
at the larger distances [59, 2]. And such attempts are key to the lattice QCD
calculations. Apart from the Wilson loop, the contribution from Instantons
[60] is also considered for the non-perterbative effects at the larger
distance. Instantons, which are the large fluctuions of the gluon field and
corresponds to the tunneling from one minimum of the energy to the
neighbouring one. Similar fluctuations but having tunneling in the opposite
side is called ”anti-instantons”.

In the present work we aim to find the effect of the instantons on the
heaviest quarkonium system. As the form of the instanton potential contains
the term which gives the non-peerturbative effect at the larger distance as
well as coloumbian type behaviour of the shorter distance together make it
suitable for the study of the spectroscopy of bottomonium. Additionally, we
have also incorporated the spin dependent part of one gluon exchange
potential. The detailed description of the form of instanton potential for
heavy quark can be found in [61] where authors have added the spin dependent
attributes into it. This potential has a huge history [62, 63, 64] and its
central part was derived long ago [65] based on the instanton liquid model
[66, 67, 68] for the QCD Vacuum. With the use of instanton potential acting
between quark and anti-quark pair, we solved the non-relativistic schrodinger
equation by variational method. To test the predictivity of the potential we
have further calculated various decay properties like vector and pseudoscalar
decay constants, di-leptonic, di-gamma, di-gluon as well as tri-gluon decays
together with radiative transition decay widths. The computed results are
tabulated and compared with other available theoretical and experimental data
and finally we draw our own conclusions.

##  II Theoretical Framework to compute mass spectra

Evaluation of the bound state mass spectra demands the solution of the
Schrödinger equation with the potential from the instanton vacuum.
Unfortunately, the Schrödinger equation is not exactly solvable for most of
the systems. So, generally one adopt approximate methods like perturbation,
variational or WKB methods. In the present work, the variational method is
employed to calculate the ground and excited state mass spectra. For the
treatment of bottomonium on the non-relativistic footing the Hamiltonian can
be written as

$H=\frac{P^{2}}{2M}+V\left(r\right)$ (1)

Here, $P$ is the relative momenta, $M$ represents reduced mass , $V(r)$ is the
potential acting between quark and anti-quark.

Now as described earlier, the form of the potential we have used is instanton
induced potential according to the Instanton Liquid Model (ILM)[61]. The ILM
has two important parameters portraying the diluteness of the instanton liquid
[69, 66] , one is the average size of instanton $\bar{\rho}$ and other is the
average distance between instantons $\bar{R}$. Numerical values of these
parameters are different for different approches. For example, Shuryak [69]
proposed the values of these parameters as $\bar{\rho}\simeq 0.33fm$ and
$\bar{R}\simeq 1fm$. In ref [70, 72, 71] , $\bar{\rho}\simeq 0.35fm$ and
$\bar{R}\simeq 0.856fm$ were used in their $1/N_{c}$ meson loop contibution
for the light quark sector. And in this framework the range of the potential
is identified only by the size of the instanton $(\bar{\rho})$.

For $r\ll\bar{\rho}$ i.e., when the distance between quark-antiquark is
smaller than the average size of instanton, the central potential is given by
[61]

${V(r)}\simeq\frac{4{\pi}\bar{\rho}^{3}}{{\bar{R^{4}}}{N_{c}}}\left(1.345{\frac{r^{2}}{\bar{\rho}^{2}}}-0.501{\frac{r^{4}}{\bar{\rho}^{4}}}\right)$
(2)

here, $N_{c}=3$ represents the colour degrees of freedom.
For $r\gg\bar{\rho}$ i.e., the distance between quark-antiquark is higher than
the size of instanton, the central part of the potential is given by [61]

${V(r)}\simeq 2\Delta M_{Q}-{\frac{g_{np}}{r}}$ (3)

here, $\Delta M_{Q}$ is the correction to heavy-quark mass from the instanton
vacuum and is given by [61].

${\Delta
M_{Q}}={-\frac{4\pi^{4}\bar{\rho}^{3}}{3\bar{R^{4}}N_{c}}}\left(J_{0}\left(\pi\right)+\frac{1}{\pi}J_{1}\left(\pi\right)\right)$
(4)

Here, $J_{0}$ and $J_{1}$ are the Bessel functions. The coupling constant in
equation $(4)$ is defined as
$g_{np}=2\pi^{3}{\bar{\rho}^{4}}/N_{c}{\bar{R}}^{4}$ and can be considered as
nonperturbative correction to the strong coupling constant ${\alpha}_{s}(r)$.
When $r$ reaches to infinity, the potential is saturated at $2\Delta M_{Q}$
and this signifies that instanton vacuum cannot explain quark confinement
[65]. Thus for the present study, we have added a state dependent confinement
potential $V_{0}$ which has the form

${V_{0}\left(z\right)}=\left(\ln\left(z\right)+a\right)b$ (5)

Where, $z=4n+3l-1$ and the numerical values of the constants are $a$ = 0.72
MeV and $b$ = 290.4 MeV. These are our fitted model parameters which are
deduced by fitting the masses of low lying bottomonium states based on
instanton potential framework. For instanton liquid model framework, potential
is defined only for $r\ll\bar{\rho}$ and $r\gg\bar{\rho}$ and to remove the
discontinuity at $r=\bar{\rho}$ we have further added $V_{1}$ = 61.81 MeV into
eq. $(3)$. This inclusion changes the potential range from $r\ll\bar{\rho}$ to
$r\leq\bar{\rho}$ and $r\gg\bar{\rho}$ to $r\geq\bar{\rho}$.

The spin average mass of S wave $b\bar{b}$ system can be obtain as

$M_{SA}=m_{b}+m_{\bar{b}}+\langle{H}\rangle$ (6)

For the S wave further hyperfine split will give vector and pseudoscalar
components.

$V^{SS}_{Q\bar{Q}}(r)=\frac{8}{9}\frac{\alpha_{s}}{m_{Q}m_{\bar{Q}}}\vec{S_{Q}}.\vec{S}_{\bar{Q}}{4\pi}{\delta^{3}(r)}$
(7)

For the massess of $P$ and $D$ waves, we have incorporated the contribution of
the spin-orbit and tensor part of the one gluon exchange potential (OGEP) of
the form given by [80, 116, 118, 117, 81, 98]

$\displaystyle V^{LS}_{Q\bar{Q}}(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{4}\frac{N_{Q}^{2}N_{\bar{Q}}^{2}}{(E_{Q}+m_{Q})(E_{\bar{Q}}+m_{\bar{Q}})}\frac{\lambda_{Q}.\lambda_{\bar{Q}}}{2\
r}$
$\displaystyle\otimes\left[\left[\vec{r}\times(\hat{p_{Q}}-{\hat{p}}_{\bar{Q}}).(\sigma_{Q}+\sigma_{\bar{Q}})\right]\left({D^{\prime}_{0}(r)}+2{D^{\prime}_{1}(r)}\right)\right.$
$\displaystyle\left.+\left[\vec{r}\times(\hat{p_{Q}}+{\hat{p}}_{\bar{Q}}).(\sigma_{i}-\sigma_{j})\right]\left({D^{\prime}_{0}(r)}-{D^{\prime}_{1}(r)}\right)\right]$

and

$\displaystyle V^{T}_{Q\bar{Q}}(r)$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{s}}{4}\frac{N_{Q}^{2}N_{\bar{Q}}^{2}}{(E_{Q}+m_{Q})(E_{\bar{Q}}+m_{\bar{Q}})}$
(9) $\displaystyle\otimes\
\lambda_{Q}.\lambda_{\bar{Q}}\left(\left(\frac{D^{\prime\prime}_{1}(r)}{3}-\frac{D^{\prime}_{1}(r)}{3\
r}\right)S_{Q\bar{Q}}\right)$

where
$S_{Q\bar{Q}}=\left[3(\sigma_{Q}.{\hat{r}})(\sigma_{\bar{Q}}.{\hat{r}})-\sigma_{Q}.\sigma_{\bar{Q}}\right]$
with ${\hat{r}}={\hat{r}}_{Q}-{\hat{r}}_{\bar{Q}}$ as the unit vector in the
direction of $\vec{r}$.

Expression for the confined gluon propagators are taken from [80] as,

$D_{0}(r)=\left(\frac{\alpha_{1}}{r}+\alpha_{2}\right)\exp(-r^{2}c_{0}^{2}/2)$
(10)

$D_{1}(r)=\frac{\gamma}{r}\exp(-r^{2}c_{1}^{2}/2)$ (11)

For the present study we have adopted the same values of the parameters,
$\alpha_{1}=1.035$, $\alpha_{2}=0.3977$, $c_{0}=0.3418$ GeV, $c_{1}=0.4123$
GeV, $\gamma=0.8639$ as used by [81]. And the instanton potential parameters
as $\bar{\rho}\simeq 0.36fm$ and $\bar{R}\simeq 0.96fm$ are being used.

The strong running coupling constant $\alpha_{s}$ calculated as

$\alpha_{s}\left(M^{2}\right)=\frac{4\pi}{\left(11-\frac{2}{3}n_{f}\right)\left(\ln\frac{M^{2}}{\Lambda^{2}}\right)}$
(12)

where, $\Lambda_{QCD}$ is taken as 0.156 GeV which will give the PDG [43]
listed value of $\alpha_{s}$ at the $Z$ meson mass (91 GeV) range of 0.0118.
Employing $\Lambda_{QCD}$ =0.156 GeV and putting spin average mass (M) of
respective state in above equation gives $\alpha_{s}$ for corresponding state.
Here, $n_{f}$ represents the number of flavours and for bottom sector it is 4.

The harmonic oscillator trial wave-function used here is

$R_{nl}\left({r}\right)={N_{nl}}{\left(\mu{r}\right)}^{l}\exp\left({-\mu^{2}{r^{2}}/2}\right)\L_{n-1}^{l+1/2}{\left(\mu^{2}r^{2}\right)}$
(13)

with $N_{nl}$ being the normalization constant expressed as

$N_{nl}=\sqrt{\frac{2\mu^{3}\left(n-1\right)!}{\Gamma\left(n+l+1/2\right)}}$
(14)

Also, $\mu$ is the variational parameter and
${L_{n-1}^{l+1/2}}{(\mu^{2}r^{2})}$ is the associated Lagurre polynomial.

Based on the above theoretical formalism, we have computed the spectroscopic
masses of the $S$, $P$ and $D$ wave bottomonium using instanton induced
potential between the quark and anti-quark. The spin averaged masses of the
radial $(nS)$ and orbital excited states $(nP,nD)$ are computed and further
spin dependent one gluon exchange interactions are used to remove the mass
degeneracy of the bottomonium states. Results obtained here are tabulated in
the Table 1 for $S$ wave masses and in Table 2 for $P$ and $D$ wave masses. We
have compared our data set with the theoretical model predictions such as
relativistic Dirac Model [98], QCD Relativistic functional approach [113]
where authors have used two different methods rainbow-ladder truncation of
Dyson-Schwinger and Bethe-Salpeter in search of the effects of the varying
shapes of the effective running coupling on ground as well as excited states
in the channels having quantum numbers J less than or equal to 3, Constituent
quark model[41] , Relativistic quark model [42, 114], Relativistic potential
model[105] and with the recent experimental data listed by PDG[43]. Also,
Figure 1 shows the energy level diagram of the bottomonium states where we
have compared our predictions with the PDG [43]reported values.

Table 1: S-wave mass spectrum of $b\bar{b}$ bound states (in MeV). nS | State | $J^{PC}$ | Present |  PDG [43]111Particle Data Group 2018 |  [98]222Relativistic Dirac model |  [113]333Relativistic functional approach |  [41]444Constituent quark model |  [42] 555Relativistic quark model |  [107]666Non-relativistic potential model |  [114]777Relativistic quark model |  [105]888Relativistic potential model
---|---|---|---|---|---|---|---|---|---|---|---
1S | $1{{}^{3}S_{1}}$ | $1^{--}$ | 9460.75 |  9460.30 $\pm$ 0.26 | 9460.99 | 9490 | 9502 | 9465 | 9460 | 9460 | 9460
| $1{{}^{1}S_{0}}$ | $0^{-+}$ | 9412.22 |  9399.0 $\pm$ 2.3 | 9390.7 | 9414 | 9455 | 9402 | 9392 | 9398 | 9393
2S | $2{{}^{3}S_{1}}$ | $1^{--}$ | 10026.22 |  10023.26 $\pm$ 0.31 | 10024.1 | 10089 | 10015 | 10003 | 10024 | 10023 | 10023
| $2{{}^{1}S_{0}}$ | $0^{-+}$ | 9995.48 | … | 9999.3 | 9987 | 9990 | 9976 | 9991 | 9990 | 9987
3S | $3{{}^{3}S_{1}}$ | $1^{--}$ | 10364.65 |  10355.2 $\pm$ 0.5 | 10356.2 | 10327 | 10349 | 10354 | 10346 | 10355 | 10364
| $3{{}^{1}S_{0}}$ | $0^{-+}$ | 10339.00 | $\cdots$ | 10325.3 | … | 10330 | 10336 | 10323 | 10329 | 10345
4S | $4{{}^{3}S_{1}}$ | $1^{--}$ | 10594.47 |  10579.4 $\pm$ 1.2 | 10576.2 | … | 10607 | 10635 | 10575 | 10586 | 10643
| $4{{}^{1}S_{0}}$ | $0^{-+}$ | 10572.49 | $\cdots$ | 10554.4 | … | … | 10523 | 10558 | 10573 | 10364
5S | $5{{}^{3}S_{1}}$ | $1^{--}$ | 10766.14 | 10889.9 | 10758.5 | … | 10818 | 10878 | 10755 | 10869 | …
| $5{{}^{1}S_{0}}$ | $0^{-+}$ | 10746.76 | $\cdots$ | 10738.4 | … | … | 10869 | 10741 | 10851 | …
6S | $6{{}^{3}S_{1}}$ | $1^{--}$ | 11081.70 | 10992.9 | … | … | 10995 | 11243 | 10904 | 11088 | …
| $6{{}^{1}S_{0}}$ | $0^{-+}$ | 11064.47 | $\cdots$ | … | … | … | 11226 | 10892 | 11061 | …

##  III Decay properties of heavy quarkonia $Q\bar{Q}$

For any potential model, apart from the mass spectra other observables like
radiative decays and annihilation decays are important testing ground to know
the inter-quark interactions. Keeping this view, we have computed the decay
properties of $b\bar{b}$ states with no additional parameters. Further, We
have incorporated these decay properties with QCD correction whenever it is
possible.

###  III.1 Pseudoscalar and vector decay constants

Estimation of the decay constants of mesons constituting heavy quarks is
essential part of the study as it offers the information of the CKM (Cabibbo-
Kobayashi-Maskawa) matrix elements. The conventional formula within the non-
relativistic limit for the pseudoscalar and vector states of the $S$ wave
heavy quarkonia is the Van Royan-Weiskopff formula [77]. The Van Royan-
Weiskopff formula was derived under the limit that the spinors of the quark
antiquark inside the mesonic system are approximated to two component Pauli
spinors [77]. This formula relates ground state wave function at the origin to
the decay constant and is expressed as

${\textit{f}}^{2}_{P/V}\left(nS\right)=\frac{3|{R_{ns}\left(0\right)|}^{2}}{\pi
M_{ns}}{{\bar{c}^{2}}\left(\alpha s\right)}$ (15)

Where, the first order QCD corrections factor $\bar{c}^{2}\left(\alpha
s\right)$ is given by [78, 79]

${\bar{c}}\left(\alpha
s\right)=1-\frac{\alpha_{s}}{\pi}\left(\delta^{P,V}-{\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\ln\frac{m_{1}}{m_{2}}}\right)$
(16)

In the case of quarkonia, the second term inside the bracket in equation (16)
vanishes and $\delta^{V}$=8/3 in the case of vector state and $\delta^{P}$=2
for the pseudoscalar state. The calculated the pseudoscalar and vector decay
constants with and without the first order QCD corrections are presented in
Table 3 and 4 and are compared with other available model predictions.

Figure 1: $b\bar{b}$ mass spectrum  Table 2: P-wave and D-wave mass spectra of $b\bar{b}$ (in MeV) nL | State | $J^{PC}$ | Present |  PDG [43] | [98] | [113] | [41] | [42] | [107] | [114] | [105] | [108]
---|---|---|---|---|---|---|---|---|---|---|---|---
1P | $1{{}^{3}P_{2}}$ | $2^{++}$ | 9881.40 |  9912.21 $\pm$ 0.26 | 9912.3 | 9906 | 9886 | 9897 | 9908 | 9912 | 9912 | 9918
| $1{{}^{3}P_{1}}$ | $1^{++}$ | 9871.47 |  9892.78 $\pm$ 0.26 | 9901.8 | 9842 | 9874 | 9876 | 9888 | 9892 | 9891 | 9897
| $1{{}^{3}P_{0}}$ | $0^{++}$ | 9849.61 |  9859.44 $\pm$0.42 | 9889.2 | 9815 | 9855 | 9847 | 9862 | 9859 | 9861 | 9865
| $1{{}^{1}P_{1}}$ | $1^{+-}$ | 9874.56 |  9899.3$\pm$ 0.8 | 9854.1 | 9806 | 9879 | 9882 | 9896 | 9900 | 9900 | 9903
2P | $2{{}^{3}P_{2}}$ | $2^{++}$ | 10274.77 |  10268.65$\pm$ 0.22 | 10265.9 | … | 10246 | 10224 | 10268 | 10268 | 10271 | 10269
| $2{{}^{3}P_{1}}$ | $1^{++}$ | 10267.86 |  10255.46 $\pm$ 0.22 | 10258.9 | 10120 | 10236 | 10246 | 10256 | 10255 | 10255 | 10251
| $2{{}^{3}P_{0}}$ | $0^{++}$ | 10252.54 |  10232.50$\pm$ 0.40 | 10234.7 | 10254 | 10221 | 10226 | 10241 | 10233 | 10230 | 10226
| $2{{}^{1}P_{1}}$ | $1^{+-}$ | 10270.00 | … | 10264.9 | 10154 | 10240 | 10250 | 10261 | 10260 | 10262 | 10256
3P | $3{{}^{3}P_{2}}$ | $2^{++}$ | 10530.21 | … | 10516.9 | … | 10521 | 10550 | 10516 | 10550 | … | 10540
| $3{{}^{3}P_{1}}$ | $1^{++}$ | 10524.84 |  10512.1$\pm$ 2.3 | 10508.8 | 10303 | 10513 | 10538 | 10507 | 10541 | … | 10529
| $3{{}^{3}P_{0}}$ | $0^{++}$ | 10512.88 | … | 10497.6 | … | 10500 | 10552 | 10511 | 10521 | … | 10502
| $3{{}^{1}P_{1}}$ | $1^{+-}$ | 10526.50 | … | 10540.2 | … | 10516 | 10541 | 10497 | 10544 | … | 10529
4P | $4{{}^{3}P_{2}}$ | $2^{++}$ | 10717.86 | … | 10707.0 | … | 10798 | 10550 | … | 10812 | … | 10767
| $4{{}^{3}P_{1}}$ | $1^{++}$ | 10713.44 | … | 10706.5 | … | 10788 | 10538 | … | 10802 | … | 10753
| $4{{}^{3}P_{0}}$ | $0^{++}$ | 10703.56 | … | 10703.8 | … | 10781 | 10552 | … | 10781 | … | 10732
| $4{{}^{1}P_{1}}$ | $1^{+-}$ | 10714.80 | … | 10704.6 | … | 10790 | 10541 | … | 10804 | … | 10757
1D | $1{{}^{3}D_{3}}$ | $3^{--}$ | 10158.31 | $\cdots$ | 10140.4 | 10232 | 10127 | 10115 | 10177 | 10166 | 10163 | 10156
| $1{{}^{3}D_{2}}$ | $2^{--}$ | 10152.77 |  10163.7$\pm$ 1.4 | 10138.7 | 10145 | 10122 | 10147 | 10162 | 10161 | 10157 | 10151
| $1{{}^{3}D_{1}}$ | $1^{--}$ | 10144.99 | $\cdots$ | 10136.0 | … | 10117 | 10138 | 10147 | 10154 | 10149 | 10145
| $1{{}^{1}D_{2}}$ | $2^{-+}$ | 10153.80 | $\cdots$ | 10068.2 | 10194 | 10123 | 10148 | 10166 | 10163 | 10158 | 10152
2D | $2{{}^{3}D_{3}}$ | $3^{--}$ | 10459.85 | $\cdots$ | 10398.7 | … | 10422 | 10455 | 10447 | 10449 | 10456 | 10442
| $2{{}^{3}D_{2}}$ | $2^{--}$ | 10455.86 | $\cdots$ | 10397.1 | … | 10418 | 10449 | 10437 | 10443 | 10450 | 10438
| $2{{}^{3}D_{1}}$ | $1^{--}$ | 10450.23 | $\cdots$ | 10395.7 | … | 10414 | 10441 | 10428 | 10435 | 10443 | 10432
| $2{{}^{1}D_{2}}$ | $2^{-+}$ | 10456.60 | $\cdots$ | 10336.0 | … | 10419 | 10450 | 10440 | 10445 | 10452 | 10439
3D | $3{{}^{3}D_{3}}$ | $3^{--}$ | 10667.25 | … | 10620.9 | … | … | 10711 | 10652 | 10717 | … | 10680
| $3{{}^{3}D_{2}}$ | $2^{--}$ | 10664.12 | … | 10619.3 | … | … | 10705 | 10645 | 10711 | … | 10676
| $3{{}^{3}D_{1}}$ | $1^{--}$ | 10659.68 | … | 10616.8 | … | … | 10698 | 10637 | 10704 | … | 10670
| $3{{}^{1}D_{2}}$ | $2^{-+}$ | 10664.70 | … | 10564.3 | … | … | 10706 | 10646 | 10713n | … | 10677
4D | $4{{}^{3}D_{3}}$ | $3^{--}$ | 10825.12 | … | 10820.9 | … | … | 10939 | 10817 | 10963 | … | 10886
| $4{{}^{3}D_{2}}$ | $2^{--}$ | 10822.52 | … | 10819.3 | … | … | 10934 | 10811 | 10957 | … | 10882
| $4{{}^{3}D_{1}}$ | $1^{--}$ | 10818.83 | … | 10816.9 | … | … | 10928 | 10805 | 10949 | … | 10877
| $4{{}^{1}D_{2}}$ | $2^{-+}$ | 10823.00 | … | 10768.8 | … | … | 10935 | 10813 | 10959 | … | 10883
5D | $5{{}^{3}D_{3}}$ | $3^{--}$ | 10954.42 | … | 11005.2 | … | … | … | 10955 | … | … | 11069
| $5{{}^{3}D_{2}}$ | $2^{--}$ | 10951.59 | … | 11003.7 | … | … | … | 10950 | … | … | 11065
| $5{{}^{3}D_{1}}$ | $1^{--}$ | 10949.01 | … | 11001.4 | … | … | … | 10945 | … | … | 11060
| $5{{}^{1}D_{2}}$ | $2^{-+}$ | 10952.60 | … | 10956.7 | … | … | … | 10952 | … | … | 11066

###  III.2 The Leptonic Decays of bottomonium states

Descriptive study of leptonic decays of mesons not only provide underlying
dynamics of quark anti-quark annihilation [83] but also used as a probe to
study compactness of the pair of quark-antiquark [82]. In addition to that
leptonic decay rates can also help us to distinguish conventional mesons and
exotic states [84] .

The vector mesons $(1^{--})$ annihilates leptonically via single virtual
photon and leptonic partial decay width computed using Van Royen- Weisskopf
formula [77] along with the QCD correction of one loop level given as [85, 86]

$\displaystyle\Gamma(n^{3}S_{1}\rightarrow
l^{+}l^{-})=\frac{4\alpha_{e}^{2}{e_{Q}^{2}}}{M_{V}^{2}}|R_{nl}\left(0\right)|^{2}\left[1-\frac{16\alpha_{s}}{3\pi}\right]\
\ \ \ \ $ (17)

Here, $|R_{nl}(0)|^{2}$ is the square of the radial wave function at the
origin. $\alpha_{e}=1/137$ and $\alpha_{s}$ are the electromagnetic coupling
constant and strong running coupling constant respectively. $e_{Q}$ is the
charge of the heavy quark in units of electron charge. For the $D$ wave states
$n^{3}D_{1}$ leptonic decay width can be given as [87],

$\displaystyle\Gamma(n^{3}D_{1}\rightarrow
l^{+}l^{-})=\frac{25\alpha_{e}^{2}{e_{Q}^{2}}}{2m_{b}^{4}{M^{2}\left(n^{3}D_{1}\right)}}|R_{nl}^{{}^{\prime\prime}}\left(0\right)|^{2}\left[1-\frac{16\alpha_{s}}{3\pi}\right]\
\ \ \ \ $ (18)

For the $n^{3}D_{1}$ bound states annihilation into $l^{+}l^{-}$ have same
first order QCD correction factor as of $n^{3}S_{1}$ bound states [120]. The
leptonic decay widths are tabulated in Table 5 with and without QCD
corrections. We have compared our results with other available experimental as
well as theoretical data.

Table 3: The vector decay constant(in MeV) of the $S$ wave bottomonium states. State | $J^{PC}$ | $\textit{f}_{V}$ | $\textit{f}_{V}(R)$ |  PDG [43] | [95] | [98] | [97] | [96] | [99]
---|---|---|---|---|---|---|---|---|---
$1{{}^{3}S_{1}}$ | $1^{--}$ | 653.44 | 551.53 |  715 $\pm$ 5 | 649(31) | 705.4 | 706 |  498 $\pm$ (20) | 831
$2{{}^{3}S_{1}}$ | $1^{--}$ | 563.73 | 477.05 |  498 $\pm$ 8 | 481(39) | 554.9 | 547 |  366 $\pm$ (27) | 566
$3{{}^{3}S_{1}}$ | $1^{--}$ | 507.90 | 430.42 |  430 $\pm$ 4 | … | 436.8 | 484 |  304 $\pm$ (27) | 507
$4{{}^{3}S_{1}}$ | $1^{--}$ | 466.60 | 395.80 |  336 $\pm$18 | … | 332.4 | 446 |  259 $\pm$ (22) | 481
$5{{}^{3}S_{1}}$ | $1^{--}$ | 434.61 | 368.91 | … | … | 286.5 | 419 |  228 $\pm$ (16) | 458
$6{{}^{3}S_{1}}$ | $1^{--}$ | 406.39 | 345.40 | … | … |  |  |  |
Table 4: The pseudoscalar decay constant(in MeV) of the $S$ wave bottomonium states. State | $J^{PC}$ | $\textit{f}_{P}$ | $\textit{f}_{P}(R)$ | [100] | [101] |  Lattice [103] | [99] |  QCD sum rules [104] | [102]
---|---|---|---|---|---|---|---|---|---
$1{{}^{1}S_{0}}$ | $0^{-+}$ | 654.81 | 578.21 | 646.025 | 756 | 667 | 834 |  251$\pm$ 0.072 | 1016.8
$2{{}^{1}S_{0}}$ | $0^{-+}$ | 564.59 | 499.48 | 518.803 | 285 | … | 567 | … | 806.6
$3{{}^{1}S_{0}}$ | $0^{-+}$ | 508.53 | 450.35 | 474.954 | 333 | … | 508 | … | 713.4
$4{{}^{1}S_{0}}$ | $0^{-+}$ | 467.09 | 413.93 | 449.654 | 40 | … | … | … | …
$5{{}^{1}S_{0}}$ | $0^{-+}$ | 435.00 | 385.68 | 432.072 | … | … | … | … | …
$6{{}^{1}S_{0}}$ | $0^{-+}$ | 406.71 | 360.93 |  | … | … | … | … | …
Table 5: The di-leptonic decay widths (in keV) of the $S$ wave and $D$ wave bottomonium states. State | $J^{PC}$ | $\Gamma_{{\textit{l}^{+}}{\textit{l}^{-}}}$ |  $\Gamma_{{\textit{l}^{+}}{\textit{l}^{-}}}$ (R) |  PDG [43] | [105] | [41] | [107] | [98] | [80] | [109] | [108]
---|---|---|---|---|---|---|---|---|---|---|---
$1{{}^{3}S_{1}}$ | $1^{--}$ | 1.1191 | 0.7700 |  1.34 $\pm$ 0.018 | 1.33 | 0.71 | 1.20 | 1.30 | 1.809 | 1.3 | 1.60
$2{{}^{3}S_{1}}$ | $1^{--}$ | 0.7859 | 0.5442 |  0.612 $\pm$ 0.011 | 0.62 | 0.37 | 0.52 | 0.76 | 0.797 | 0.5 | 0.64
$3{{}^{3}S_{1}}$ | $1^{--}$ | 0.6171 | 0.4288 |  0.443 $\pm$ 0.008 | 0.48 | 0.27 | 0.33 | 0.45 | 0.618 | … | 0.44
$4{{}^{3}S_{1}}$ | $1^{--}$ | 0.5096 | 0.3549 |  0.272 $\pm$ 0.029 | 0.40 | 0.21 | 0.24 | 0.26 | 0.541 | … | 0.35
$5{{}^{3}S_{1}}$ | $1^{--}$ | 0.4350 | 0.3035 |  0.31 $\pm$ 0.07 | … | 0.18 | 0.19 | 0.18 | … | … | 0.29
$6{{}^{3}S_{1}}$ | $1^{--}$ | 0.3695 | 0.2586 |  | … |  |  |  | … | … |
$1{{}^{3}D_{1}}$ | $1^{--}$ | 0.0072 | 0.0050 | … | … | … | … | 0.106 | … | … | …
$2{{}^{3}D_{1}}$ | $1^{--}$ | 0.0084 | 0.0058 | … | … | … | … | 0.078 | … | … | …
$3{{}^{3}D_{1}}$ | $1^{--}$ | 0.0085 | 0.0059 | … | … | … | … | 0.051 | … | … | …
$4{{}^{3}D_{1}}$ | $1^{--}$ | 0.0084 | 0.0058 | … | … | … | … | 0.042 | … | … | …
$5{{}^{3}D_{1}}$ | $1^{--}$ | 0.0082 | 0.0057 | … | … | … | … | 0.028 | … | … | …
Table 6: The di-gamma decay widhts (in keV) of the $S$ wave and $P$ wave bottomonium states. State | $J^{PC}$ | $\Gamma_{\gamma\gamma}$ | $\Gamma_{\gamma\gamma}(R)$ | [100] | [41] | [102] | [107] | [108] | [109] | [1]
---|---|---|---|---|---|---|---|---|---|---
$1{{}^{1}S_{0}}$ | $0^{-+}$ | 0.3782 | 0.3035 | 0.387 | 0.69 | 0.738 | 0.496 | 0.527 | 0.35 | 0.214
$2{{}^{1}S_{0}}$ | $0^{-+}$ | 0.2636 | 0.2122 | 0.263 | 0.36 | 0.508 | 0.212 | 0.263 | 0.15 | 0.121
$3{{}^{1}S_{0}}$ | $0^{-+}$ | 0.2067 | 0.1668 | 0.229 | 0.27 | 0.426 | 0.135 | 0.172 | 0.10 | 0.906
$4{{}^{1}S_{0}}$ | $0^{-+}$ | 0.1705 | 0.1378 | 0.201 | … | … | … | 0.099 | 0.105 | 0.755
$5{{}^{1}S_{0}}$ | $0^{-+}$ | 0.1455 | 0.1176 | 0.193 | … | … | … | 0.078 | 0.121 | …
$6{{}^{1}S_{0}}$ | $0^{-+}$ | 0.1235 | 0.1000 | … | … | … | … |  |  | …
$1{{}^{3}P_{0}}$ | $0^{++}$ | 0.0811 | 0.1150 | 0.0196 | 0.12 | … | … | 0.050 | 0.038 | 0.0208
$2{{}^{3}P_{0}}$ | $0^{++}$ | 0.0717 | 0.1014 | 0.0195 | 0.14 | … | … | 0.037 | 0.029 | 0.0227
$3{{}^{3}P_{0}}$ | $0^{++}$ | 0.0620 | 0.0875 | 0.0194 | 0.15 | … | … | 0.037 | … | …
$4{{}^{3}P_{0}}$ | $0^{++}$ | 0.0545 | 0.0768 | 0.0192 | … | … | … | … | … | …
$5{{}^{3}P_{0}}$ | $0^{++}$ | 0.0487 | 0.0686 | … | … | … | … | … | … | …
$1{{}^{3}P_{2}}$ | $2^{++}$ | 0.0213 | 0.0147 | 0.0052 | 0.00308 | … | … | 0.0066 | 0.008 | 0.0051
$2{{}^{3}P_{2}}$ | $2^{++}$ | 0.0189 | 0.0131 | 0.0052 | 0.00384 | … | … | 0.0067 | 0.006 | 0.0062
$3{{}^{3}P_{2}}$ | $2^{++}$ | 0.0164 | 0.0114 | 0.0051 | 0.00410 | … | … | 0.0064 | … | …
$4{{}^{3}P_{2}}$ | $2^{++}$ | 0.0144 | 0.0100 | 0.0051 | … | … | … | … | … | …
$5{{}^{3}P_{2}}$ | $2^{++}$ | 0.0129 | 0.0090 | … | … | … | … | … | … | …
Table 7: The di-gluon decay widhts (in MeV) of the $S$ wave and $D$ wave bottomonium states. State | $J^{PC}$ | $\Gamma_{gg}$ | $\Gamma_{gg}(R)$ | [100] | [41] | [111] | [110] | [112] |  |
---|---|---|---|---|---|---|---|---|---|---
$1{{}^{1}S_{0}}$ | $0^{-+}$ | 5.4496 | 6.8520 | 5.448 | 20.18 | 17.945 | 11.49 | 12.46 |  |
$2{{}^{1}S_{0}}$ | $0^{-+}$ | 4.1775 | 5.2374 | 3.710 | 10.64 | … | 5.16 | … |  |
$3{{}^{1}S_{0}}$ | $0^{-+}$ | 3.4499 | 4.3182 | 3.229 | 7.94 | … | 3.80 | … |  |
$4{{}^{1}S_{0}}$ | $0^{-+}$ | 2.9454 | 3.6829 | 2.985 | … | … | … | … |  |
$5{{}^{1}S_{0}}$ | $0^{-+}$ | 2.5769 | 3.2196 | 2.832 | … | … | … | … |  |
$6{{}^{1}S_{0}}$ | $0^{-+}$ | 2.2859 | 2.8519 |  | … | … | … | … |  |
$1{{}^{3}P_{0}}$ | $0^{++}$ | 0.9056 | 1.4297 | 0.276 | 2.00 | 5.250 | 0.96 | 2.15 |  |
$2{{}^{3}P_{0}}$ | $0^{++}$ | 0.7855 | 1.2358 | 0.275 | 2.37 | … | 0.99 | … |  |
$3{{}^{3}P_{0}}$ | $0^{++}$ | 0.6713 | 1.0539 | 0.273 | 2.46 | … | … | … |  |
$4{{}^{3}P_{0}}$ | $0^{++}$ | 0.5853 | 0.9175 | 0.271 | … | … | … | … |  |
$5{{}^{3}P_{0}}$ | $0^{++}$ | 0.5192 | 0.8127 | … | … | … | … | … |  |
$1{{}^{3}P_{2}}$ | $2^{++}$ | 0.2384 | 0.2370 | 0.073 | 0.0836 | 0.822 | 0.33 | 0.22 |  |
$2{{}^{3}P_{2}}$ | $2^{++}$ | 0.2076 | 0.2064 | 0.073 | 0.1042 | … | 0.35 | … |  |
$3{{}^{3}P_{2}}$ | $2^{++}$ | 0.1778 | 0.1767 | 0.072 | 0.1114 | … | … | … |  |
$4{{}^{3}P_{2}}$ | $2^{++}$ | 0.1552 | 0.1543 | 0.072 | … | … | … | … |  |
$5{{}^{3}P_{2}}$ | $2^{++}$ | 0.1378 | 0.1370 | … | … | … | … | … |  |

###  III.3 The Di-gamma decay of bottomonium states

Adopting the case of the para positronium $(^{1}S_{0})$ state decays to two
photons to that of the ${}^{1}S_{0}$ state of quarkonium with the inclusion of
color factor, we can get the expression to compute di-gamma decay width of
pseudoscalar mesons as [77]

$\displaystyle\Gamma(n^{1}S_{0}\rightarrow\gamma\gamma)=\frac{12\alpha_{e}^{2}{e_{Q}^{4}}}{M_{p}^{2}}$
$\displaystyle|R_{nl}\left(0\right)|^{2}$ (19)
$\displaystyle\left[1-\frac{\alpha_{s}}{\pi}\left(\frac{20-\pi^{2}}{3}\right)\right]$

where bracketed quantity is the radiative correction [85].

In the case of $P$ wave states, decay width depends on the first derivative of
the radial wave function at the origin and is given by [85] .

$\displaystyle\Gamma(n^{3}P_{0}\rightarrow\gamma\gamma)=\frac{27\alpha_{e}^{2}{2^{4}}}{{M(n^{3}P_{0})}^{4}}{e_{Q}^{4}}$
$\displaystyle|R^{{}^{\prime}}_{nl}\left(0\right)|^{2}$ (20)
$\displaystyle\left[1+\frac{\alpha_{s}}{\pi}\left(\frac{\pi^{3}}{3}-\frac{28}{9}\right)\right]$

$\displaystyle\Gamma(n^{3}P_{2}\rightarrow\gamma\gamma)=\frac{36\alpha_{e}^{2}2^{4}}{5{{M(n^{3}P_{2})}^{4}}}{e_{Q}^{4}}$
$\displaystyle|R^{{}^{\prime}}_{nl}\left(0\right)|^{2}$ (21)
$\displaystyle\left[1-\left(\frac{16\alpha_{s}}{3\pi}\right)\right]$

Table 6 summarises our computed results for the di-gamma decay width of
$n^{1}S_{0}$ and $n^{3}P_{0,2}$ states along with other available data.

###  III.4 The Di-gluon decay of bottomonium states

Similar to the di-photon decay, states holding even charge conjugation can
decay into di-gluons. Expression used to compute $S$ wave spin-singlet state
decaying into two gluons together with the radiative QCD correction is given
by [85]

$\displaystyle\Gamma(n^{1}S_{0}\rightarrow
gg)=\frac{2\alpha_{s}^{2}}{3m_{Q}^{2}}|R_{nl}\left(0\right)|^{2}\left[1+\frac{4.4\alpha_{s}}{\pi}\right]$
(22)

conventionally, theoretical expression for di-gamma decay width of the
$\chi_{b_{0}}$ and $\chi_{b_{2}}$ states which depends on the first derivative
of the radial wave function at origin can be obtained from the potential
models [90] and the formula adopted here for the computation with QCD
correction factor can be written as [85, 91]

$\displaystyle\Gamma(n^{3}P_{0}\rightarrow
gg)=\frac{6\alpha_{s}^{2}2^{4}}{{{M(n^{3}P_{0})}^{4}}}|R^{{}^{\prime}}_{nl}\left(0\right)|^{2}\left[1+\frac{10.0\alpha_{s}}{\pi}\right]$
(23)

$\displaystyle\Gamma(n^{3}P_{2}\rightarrow
gg)=\frac{8\alpha_{s}^{2}2^{4}}{5{{M(n^{3}P_{2})}^{4}}}|R^{{}^{\prime}}_{nl}\left(0\right)|^{2}\left[1-\frac{0.1\alpha_{s}}{\pi}\right]$
(24)

The calculated di-gluon decay width with the other available theoretical data
are listed in Table 7.

###  III.5 The Tri-gluon decay of bottomonium states

The decay width of $S$ wave vector state annihilates into three gluons along
with QCD radiative correction is computed using the relation given by [85,
119]

$\displaystyle\Gamma(n^{3}S_{1}\rightarrow
ggg)=\frac{10\alpha_{s}^{3}2^{2}\left(\pi^{2}-9\right)}{81\pi{{M(n^{3}S_{1})}^{2}}}$
$\displaystyle|R_{nl}\left(0\right)|^{2}$ (25)
$\displaystyle\left[1-\frac{4.9\alpha_{s}}{\pi}\right]$

And in the case of $n^{1}P_{1}$ state, annihilates to the three gluons is
given by [85, 119, 41]

$\displaystyle\Gamma(n^{1}P_{1}\rightarrow
ggg)=\frac{20\alpha_{s}^{3}2^{4}}{9\pi{{M(n^{1}P_{1})}^{4}}}|R^{{}^{\prime}}_{nl}\left(0\right)|^{2}\ln\left(m_{Q}\langle
r\rangle\right)$ (26)

In the case of D-wave, $n^{3}D_{J}$ states, the three gluon decay widths are
computed using the expression given by [92]

$\displaystyle\Gamma(n^{3}D_{1}\rightarrow
ggg)=\frac{760\alpha_{s}^{3}2^{6}}{81\pi{{M(n^{3}D_{1})}^{6}}}|R^{{}^{\prime\prime}}_{nl}\left(0\right)|^{2}\ln\left(4m_{Q}\langle
r\rangle\right)$ (27)

$\displaystyle\Gamma(n^{3}D_{3}\rightarrow
ggg)=\frac{40\alpha_{s}^{3}2^{6}}{9\pi{{M(n^{3}D_{3})}^{6}}}|R^{{}^{\prime\prime}}_{nl}\left(0\right)|^{2}\ln\left(4m_{Q}\langle
r\rangle\right)$ (28)

Table 8 contains our predicted three gluon decays for $S$, $P$ and $D$ waves
where we have compared our results with other reported values.

Table 8: The Tri-gluon decay widhts (in MeV) of the $S$, $P$ and $D$ wave bottomonium states. $J^{PC}$ | State | $\Gamma_{ggg}$ | $\Gamma_{ggg}(R)$ |  PDG [43] | [41]
---|---|---|---|---|---
$1{{}^{3}S_{1}}$ | $1^{--}$ | 0.0400 | 0.0285 | 0.0441 | 0.0416
$2{{}^{3}S_{1}}$ | $1^{--}$ | 0.0269 | 0.0193 | 0.0188 | 0.0242
$3{{}^{3}S_{1}}$ | $1^{--}$ | 0.0206 | 0.0148 | 0.0072 | 0.01876
$4{{}^{3}S_{1}}$ | $1^{--}$ | 0.0168 | 0.0121 | … | …
$5{{}^{3}S_{1}}$ | $1^{--}$ | 0.0141 | 0.0102 | … | …
$6{{}^{3}S_{1}}$ | $1^{--}$ | 0.0117 | 0.0085 | … | …
$1{{}^{1}P_{1}}$ | $1^{+-}$ | 0.0357 | … | … | 0.03526
$2{{}^{1}P_{1}}$ | $1^{+-}$ | 0.0346 | … | … | 0.0527
$3{{}^{1}P_{1}}$ | $1^{+-}$ | 0.0331 | … | … | 0.0621
$4{{}^{1}P_{1}}$ | $1^{+-}$ | 0.0327 | … | … | …
$5{{}^{1}P_{1}}$ | $1^{+-}$ | 0.0309 | … | … | …
$1{{}^{3}D_{1}}$ | $1^{--}$ | 0.0106 | … | … | 0.0099
$2{{}^{3}D_{1}}$ | $1^{--}$ | 0.0119 | … | … | 0.0096
$3{{}^{3}D_{1}}$ | $1^{--}$ | 0.0118 | … | … | …
$4{{}^{3}D_{1}}$ | $1^{--}$ | 0.0113 | … | … | …
$5{{}^{3}D_{1}}$ | $1^{--}$ | 0.0108 | … | … | …
$1{{}^{3}D_{3}}$ | $3^{--}$ | 0.0060 | … | … | 0.00022
$2{{}^{3}D_{3}}$ | $3^{--}$ | 0.0056 | … | … | 0.00125
$3{{}^{3}D_{3}}$ | $3^{--}$ | 0.0055 | … | … | …
$4{{}^{3}D_{3}}$ | $3^{--}$ | 0.0053 | … | … | …
$5{{}^{3}D_{3}}$ | $3^{--}$ | 0.0051 | … | … | …
Table 9: Some other annihilation decay widhts (in keV) of the $S$ and $P$ wave bottomonium states. Transition | $\Gamma$ | $\Gamma(R)$ |  PDG [43] | [41]
---|---|---|---|---
$1{{}^{3}S_{1}}\rightarrow\gamma\gamma\gamma$ | $4.6775\times 10^{-4}$ | $1.2312\times 10^{-4}$ | $\ldots$ | $3.44\times 10^{-6}$
$2{{}^{3}S_{1}}\rightarrow\gamma\gamma\gamma$ | $3.2849\times 10^{-4}$ | $8.9895\times 10^{-5}$ | $\ldots$ | $2.00\times 10^{-6}$
$3{{}^{3}S_{1}}\rightarrow\gamma\gamma\gamma$ | $2.5794\times 10^{-4}$ | $7.2037\times 10^{-5}$ | $\ldots$ | $1.55\times 10^{-6}$
$4{{}^{3}S_{1}}\rightarrow\gamma\gamma\gamma$ | $2.1298\times 10^{-4}$ | $6.0336\times 10^{-5}$ | $\ldots$ | $1.29\times 10^{-6}$
$5{{}^{3}S_{1}}\rightarrow\gamma\gamma\gamma$ | $1.8183\times 10^{-4}$ | $5.2021\times 10^{-5}$ | $\ldots$ | $1.10\times 10^{-6}$
$6{{}^{3}S_{1}}\rightarrow\gamma\gamma\gamma$ | $1.5446\times 10^{-4}$ | $4.4933\times 10^{-5}$ | $\ldots$ | $9.56\times 10^{-7}$
$1{{}^{3}S_{1}}\rightarrow\gamma gg$ | 1.2730 | 0.7220 | 1.18 | 0.79
$2{{}^{3}S_{1}}\rightarrow\gamma gg$ | 0.8689 | 0.4982 | 0.59 | 0.46
$3{{}^{3}S_{1}}\rightarrow\gamma gg$ | 0.6718 | 0.3874 | 0.0097 | 0.36
$4{{}^{3}S_{1}}\rightarrow\gamma gg$ | 0.5485 | 0.3176 | $\ldots$ | 0.30
$5{{}^{3}S_{1}}\rightarrow\gamma gg$ | 0.4646 | 0.2698 | $\ldots$ | 0.25
$6{{}^{3}S_{1}}\rightarrow\gamma gg$ | 0.3894 | 0.2272 | $\ldots$ | 0.22
$1{{}^{3}P_{1}}\rightarrow q\bar{q}+g$ | 57.9585 | $\ldots$ | $\ldots$ | 71.53
$2{{}^{3}P_{1}}\rightarrow q\bar{q}+g$ | 55.3966 | $\ldots$ | $\ldots$ | 106.14
$3{{}^{3}P_{1}}\rightarrow q\bar{q}+g$ | 52.9585 | $\ldots$ | $\ldots$ | 124.53
$4{{}^{3}P_{1}}\rightarrow q\bar{q}+g$ | 52.4466 | $\ldots$ | $\ldots$ | $\ldots$
$5{{}^{3}P_{1}}\rightarrow q\bar{q}+g$ | 49.5181 | $\ldots$ | $\ldots$ | $\ldots$

###  III.6 Other annihilation channels of vector bottomonium states

Apart from the decays discussed above, there are other processes by which
quarkonium states can annihilate. To elaborate specifically, the decay width
of mixed strong and electromagnetic annihilation of $n^{3}S_{1}$ states into
photon and two gluons is given by [45]

$\displaystyle\Gamma(n^{3}S_{1}\rightarrow\gamma
gg)=\frac{8{\left(\pi^{2}-9\right)}\alpha{\alpha_{s}}^{2}{e_{Q}}^{2}2^{2}}{9\pi{{M(n^{3}S_{1})}^{2}}}$
$\displaystyle|R_{nl}\left(0\right)|^{2}$ (29)
$\displaystyle\left[]1-\frac{7.4\alpha_{s}}{\pi}\right]`$

For the spin triplet state of $S$ wave quarkonium the decay rate into three
photon is given by [45]

$\displaystyle\Gamma(n^{3}S_{1}\rightarrow\gamma\gamma\gamma)=\frac{16{\left(\pi^{2}-9\right)}\alpha^{3}{e_{Q}}^{6}2^{2}}{3{{M(n^{3}S_{1})}^{2}}}$
$\displaystyle|R_{nl}\left(0\right)|^{2}$ (30)
$\displaystyle\left[1-\frac{12.6\alpha_{s}}{\pi}\right]`$

Also, $n^{3}P_{1}$ state decay into light flavour meson and a single gluon.
The decay rate for such process can be written as [85]

$\displaystyle\Gamma(n^{3}P_{1}\rightarrow
q\bar{q}+g)=\frac{8\alpha_{s}^{3}n_{f}2^{4}}{9\pi{{M(n^{3}P_{1})}^{4}}}|R^{{}^{\prime}}_{nl}\left(0\right)|^{2}\ln\left(m_{Q}\langle
r\rangle\right)$ (31)

Outcomes of these decays are summarised in Table 9 along with other available
theoretical and experimental data.

###  III.7 The Electromagnetic Transition widths of bottomonium states

Bottomonium states possessed the more compactness in nature due to relatively
heavier mass of the bottom quark. In such situation especially dealing with
the radiative transition which are governed by an emission or absorption of
gamma photon, the wave length of the photon is either larger or comparable to
the size of the radiating bottomonium state. So, one expects radiative
transition in $b\bar{b}$ dominates. The leading order electromagnetic
transitions are electric dipole $(E1)$ and magnetic dipople $(M1)$
transitions.

The selection rules for electric dipole transition $(E1)$ are $\Delta l=\pm
1$, $\Delta s=0$. In contrast, for magnetic dipole transitions $(M1)$, $\Delta
l=0$, $\Delta s=\pm 1$. Within non relativistic limit, the decay width of the
$E1$ transition from the initial state $n_{i}^{(2s_{i}+1)}{l_{i}}_{J_{i}}$ to
final state $n_{f}^{(2s_{f}+1)}{l_{f}}_{J_{f}}$ can be obtained as [2]

$\Gamma(i\xrightarrow{E1}f+\gamma)=\frac{4\alpha
e_{Q}^{2}}{3}(2J^{{}^{\prime}}+1)S_{if}^{E}\omega^{3}\\\ |{\cal
E}_{if}|^{2}\times\frac{E_{f}}{M_{i}}$ (32)

where

$\omega=\frac{M_{i}^{2}-M_{f}^{2}}{2M_{i}}$ (33)

$S^{E}_{if}$ is the statistical factor and $\varepsilon_{if}$ is the overlap
integral which can be computed using initial and final state wave functions as

$\displaystyle{\cal E}_{if}=\frac{3}{\omega}\int_{0}^{\infty}$ $\displaystyle
dru_{n\ell}(r)u_{n^{{}^{\prime}}\ell^{{}^{\prime}}}(r)$ (34)
$\displaystyle\left[{\frac{\omega r}{2}}j_{0}\left(\frac{\omega
r}{2}\right)-j_{1}\left(\frac{\omega r}{2}\right)\right]$

and

$S_{if}^{E}=max(\ell,\ell^{\prime})\left\\{\begin{array}[]{ccc}J&1&J^{{}^{\prime}}\\\
\ell^{{}^{\prime}}&s&\ell\end{array}\right\\}^{2}$ (35)

The formula used for the $(M1)$ transition from initial state to final state
for the quarkonium system can be given as [93, 94]

$\displaystyle\Gamma(i\xrightarrow{M1}f+\gamma)=$
$\displaystyle\frac{4\alpha{e_{Q}}^{2}}{3{m_{Q}}^{2}}(2J^{{}^{\prime}}+1)S_{if}^{M}$
(36) $\displaystyle\ \omega^{3}|{\langle f|{j_{0}}\left(\frac{\omega
r}{2}\right)|i\rangle|}^{2}\times\frac{E_{f}}{M_{i}}$

The statistical factor for the $(M1)$ transition can be given as

$\displaystyle S_{if}^{M}=6\left(2s+1\right)$
$\displaystyle\left(2s{{}^{\prime}}+1\right)$ (41)
$\displaystyle\left\\{\begin{array}[]{ccc}J&1&J^{{}^{\prime}}\\\
s^{{}^{\prime}}&l&s\end{array}\right\\}^{2}\left\\{\begin{array}[]{ccc}1&\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&s^{{}^{\prime}}&s\end{array}\right\\}^{2}$

It is to be noted that the term $E_{f}/M_{i}$ is acting as the relativistic
correction factor to the radiative transition width where $E_{f}$ is the
energy of the final state and $M_{i}$ being the mass of the initial state.
Possible $E1$ and $M1$ transitions including relativistic correction factor
are tabulated in Table X and XI along with the available theoretical and
experimental data for comparison.

Table 10: The E1 transition decay widhts (in keV) of bottomonium states. Initial | Final | $\Gamma_{E1}$ | $\Gamma_{E1}\left(R\right)$ |  PDG [43] | [100] | [115] | [41] | [42]
---|---|---|---|---|---|---|---|---
$1{{}^{3}P_{2}}$ | $1{{}^{3}S_{1}}$ | 15.7013 | 15.0471 | … | 57.530 | 31.8 | 39.15 | 32.8
$1{{}^{3}P_{1}}$ | $1{{}^{3}S_{1}}$ | 14.6574 | 14.0602 | … | 54.927 | 31.9 | 35.66 | 29.5
$1{{}^{3}P_{0}}$ | $1{{}^{3}S_{1}}$ | 12.5171 | 12.0327 | … | 49.530 | 27.5 | 28.07 | 23.8
$1{{}^{1}P_{1}}$ | $1{{}^{1}S_{0}}$ | 19.4593 | 18.5864 | … | 72.094 | 35.8 | 43.66 | 35.7
$1{{}^{3}D_{1}}$ | $1{{}^{3}P_{0}}$ | 4.4757 | 4.3472 | … | 9.670 | 19.8 | 20.98 | 16.5
$1{{}^{3}D_{1}}$ | $1{{}^{3}P_{1}}$ | 2.6836 | 2.6123 | … | 6.313 | 13.3 | 12.29 | 9.7
$1{{}^{3}D_{1}}$ | $1{{}^{3}P_{2}}$ | 0.1606 | 0.1564 | … | 0.394 | 1.02 | 0.65 | 0.56
$1{{}^{3}D_{2}}$ | $1{{}^{3}P_{1}}$ | 5.2421 | 5.0988 | … | 11.489 | 21.8 | 21.95 | 19.2
$1{{}^{3}D_{2}}$ | $1{{}^{3}P_{2}}$ | 1.5736 | 1.5321 | … | 3.583 | 7.23 | 6.23 | 5.6
$1{{}^{3}D_{3}}$ | $1{{}^{3}P_{2}}$ | 6.6766 | 6.4976 | … | 14.013 | 32.1 | 24.74 | 24.3
$1{{}^{1}D_{2}}$ | $1{{}^{1}P_{1}}$ | 6.8415 | 6.6567 | … | 14.821 | 30.3 | 17.23 | 24.9
$2{{}^{3}S_{1}}$ | $1{{}^{3}P_{0}}$ | 0.3021 | 0.2968 |  1.22 $\pm$ 0.11 | 2.377 | 1.09 | 1.09 | 0.91
$2{{}^{3}S_{1}}$ | $1{{}^{3}P_{1}}$ | 0.6140 | 0.6045 |  2.21 $\pm$ 0.19 | 5.689 | 2.17 | 1.84 | 1.63
$2{{}^{3}S_{1}}$ | $1{{}^{3}P_{2}}$ | 0.8412 | 0.8291 |  2.29 $\pm$ 0.20 | 8.486 | 2.62 | 2.08 | 1.88
$2{{}^{1}S_{0}}$ | $1{{}^{1}P_{1}}$ | 0.8877 | 0.8770 | … | 10.181 | 3.41 | 2.85 | 2.48
$2{{}^{3}P_{2}}$ | $2{{}^{3}S_{1}}$ | 4.5037 | 4.3961 |  15.1 $\pm$ 5.6 | 28.848 | 15.3 | 17.50 | 14.3
$2{{}^{3}P_{1}}$ | $2{{}^{3}S_{1}}$ | 4.1508 | 4.0542 |  19.4 $\pm$ 5.0 | 26.672 | 15.3 | 15.89 | 13.3
$2{{}^{3}P_{0}}$ | $2{{}^{3}S_{1}}$ | 3.4322 | 3.3572 | … | 23.162 | 14.4 | 12.80 | 10.9
$2{{}^{1}P_{1}}$ | $2{{}^{1}S_{0}}$ | 5.9958 | 5.8377 | … | 35.578 | 16.2 | 17.60 | 14.1
$2{{}^{3}P_{2}}$ | $1{{}^{3}S_{1}}$ | 6.6263 | 6.1221 |  9.8$\pm$ 2.3 | 29.635 | 12.5 | 11.38 | 8.4
$2{{}^{3}P_{1}}$ | $1{{}^{3}S_{1}}$ | 6.4644 | 5.9762 |  8.9 $\pm$ 2.2 | 28.552 | 10.8 | 9.13 | 5.5
$2{{}^{3}P_{0}}$ | $1{{}^{3}S_{1}}$ | 6.1146 | 5.6606 | … | 26.769 | 5.54 | 5.44 | 2.5
$2{{}^{1}P_{1}}$ | $1{{}^{1}S_{0}}$ | 7.4768 | 6.8844 | … | 34.815 | 16.1 | 14.90 | 13.0
$3{{}^{3}S_{1}}$ | $2{{}^{3}P_{0}}$ | 0.2151 | 0.2128 |  1.20 $\pm$ 0.12 | 3.330 | 1.21 | 1.21 | 1.03
$3{{}^{3}S_{1}}$ | $2{{}^{3}P_{1}}$ | 0.4175 | 0.4137 |  2.56 $\pm$ 0.26 | 7.936 | 2.61 | 2.13 | 1.91
$3{{}^{3}S_{1}}$ | $2{{}^{3}P_{2}}$ | 0.5585 | 0.5536 |  2.66 $\pm$ 0.27 | 11.447 | 3.16 | 2.56 | 2.30
$3{{}^{1}S_{0}}$ | $2{{}^{1}P_{1}}$ | 0.4586 | 0.4555 | … | 13.981 | 4.25 | 2.60 | 2.96
$3{{}^{3}S_{1}}$ | $1{{}^{3}P_{0}}$ | 0.6434 | 0.6122 |  0.055 $\pm$ 0.010 | 0.594 | 0.097 | 0.15 | 0.01
$3{{}^{3}S_{1}}$ | $1{{}^{3}P_{1}}$ | 1.7078 | 1.6277 |  0.018 $\pm$ 0.010 | 1.518 | 0.0005 | 0.16 | 0.05
$3{{}^{3}S_{1}}$ | $1{{}^{3}P_{2}}$ | 2.6850 | 2.5627 |  0.20 $\pm$ 0.03 | 2.354 | 0.14 | .083 | 0.45
$3{{}^{1}S_{0}}$ | $1{{}^{1}P_{1}}$ | 4.3163 | 4.1267 | … | 3.385 | 0.67 | 0.0084 | 1.30
Table 11: The M1 transition decay widhts (in eV) of bottomonium states. Initial | Final | $\Gamma_{M1}$ | $\Gamma_{M1}\left(R\right)$ |  PDG [43] | [100] | [115] | [41] | [42]
---|---|---|---|---|---|---|---|---
$1{{}^{3}S_{1}}$ | $1{{}^{1}S_{0}}$ | 3.7986 | 3.7827 | … | 37.668 | 10 | 9.34 | 10
$1{{}^{3}P_{2}}$ | $1{{}^{1}P_{1}}$ | 0.0197 | 0.0197 | … | … | 0.095 | 0.089 | 0.096
$1{{}^{1}P_{1}}$ | $1{{}^{3}P_{1}}$ | 0.0018 | 0.0018 | … | … | 0.0094 | 0.0115 | 0.012
$1{{}^{1}P_{1}}$ | $1{{}^{3}P_{0}}$ | 0.3191 | 0.3183 | … | … | 0.90 | 0.86 | 0.89
$2{{}^{3}S_{1}}$ | $2{{}^{1}S_{0}}$ | 1.7856 | 1.7801 | … | 5.619 | 0.59 | 0.58 | 0.59
$3{{}^{3}S_{1}}$ | $3{{}^{1}S_{0}}$ | 1.0341 | 1.0315 | … | 2.849 | 3.9 | 0.66 | 0.25

##  IV Results and Discussion

The spectroscopic masses of S, P and D waves of the bottomonia are computed
based on the instanton induced potential in the non-relativistic frame work.
The present results are compared with available experimental as well as with
other model predictions in Table 1 for S wave masses and in Table 2 for P wave
and D wave masses. Our results are found to be in very good agreement with the
experimental values of the respective states. Our estimation for the
$1^{3}S_{1}$ is 9460.75 MeV which is in excellent agreement with the PDG
listed mass of $1^{3}S_{1}(9460.30\pm 0.26)$. For the case of spin singlet
pseudoscalar state $1^{1}S_{0}$ our finding is 9412.22 MeV which is roughly 13
MeV higher than PDG listed mass $(9399.0\pm 2.3$ MeV). The mass splitting of
1S state $(1^{3}S_{1}-1^{1}S_{0})$ is found to be 48 MeV which is relatively
consistent with the PDG listed mass split for 1S bottomonium (61 MeV). We
observe that the S wave mass predictions upto the $4S$ states are very close
to the experimental values (PDG average). The deviations observed in the case
of 5S and 6S states are somewhat higher with reference to the respective
experimental values. Similar agreements are also seen in the case of P wave
states and the D waves with respect to the existing experimental values.

As far as the identification of the excited bottomonion states are concerned,
the masses, decay constant, di gamma, di leptonic decay width are primarily
important. Comparing the results of the vector decay constants by our
formalism with the experimental result of PDG [43] we found that our results
are fairly in good agreement. For the orbitally excited states our results
with the radiative corrections are in very good agreement with PDG [43]
reported values. For instance, the vector decay constant of 3S state we have
predictrd is $430.42$ MeV and value of vector decay constant reported by PDG
[43] is also $430\pm 4$ MeV. This indicates that radiative corrections plays
an important role in decay mechanism. We found that radiatively corrected
results are also compatible with LQCD [95] predictions.

In the case of pseudoscalar decay constant, due to the unavailability of PDG
data we have compared our data set with the results of other theoretical model
[100], Lattice QCD [103] and QCD sum rules [104]. We found our results in good
agreement with the Cornell potential predictions [100]. Also in the case of
Lattice QCD, the pseudosalar decay constant for $1^{1}S_{0}$ state is (667
MeV) which is in close agreement with our prediction (654.81 MeV). One can see
from the Table 4 that predictions from QCD sum rule [104] is roughly three
times lower than all other estimations for $1^{1}S_{0}$ state. Also, the
results in the case of [102] (potential model study) are considerably higher
than other estimations of the pseudoscalar decay constant. It is important to
note that both the vector and the pseudosalar decay constants relie upon the
numerical value of the square of the wave function at the origin, so the
choice of the potential and parameters may affect on the predictions.

We have also computed the Di-lepton, Di-gamma and Di-gluon decay widths and
results are summarized in Table 5, 6 and 7. Our results are compared with the
respective values reported in PDG [43]. In the case of di-leptonic decay of
the $n^{3}S_{1}$ it is observed that the radiative corrections are important
for higher $(n>2)$ radial excited states. The results predicted by [105]
(Relativistic potential model) are much closer to our predictions with
radiative corrections. Besides that our predicted results with radiative
correction are also in a close agreement with [108] (Screened potential
model). Our findings are also comparable with other predictions. We have been
able to compute the $n^{3}D_{1}$ annhilation into electron positron pair but
we don’t find more estimations inliterature for the detailed comparison.

The di-gamma decay widths are consistent with the outcomes of Cornell
potential model [100] for pseudoscalar $n^{3}S_{0}$ states. It is found that
the predictions reported by [102] are almost two times higher than all the
other reported values. For the $n^{3}P_{0}$ states annihilation into
$\gamma\gamma$, our results are comparable with other predictions. Our
predictions are found to be higher than other results in the case of
$n^{3}P_{2}$ states.

For $n^{3}S_{0}$ decays into $gg$ our estimated results without radiative
correction are consistent with [100] and considerably lower than that from the
[41, 110, 111, 112]. For $n^{3}P_{0}$ states, our computed results agree well
with the [110] without QCD correction while they are in accordance with
results of [41, 112] with radiative correction. For these states estimations
of [100] are comparable with our predictions. For the $n^{3}P_{2}$ our
predictions are in line with that of [112] and [110] . For the S wave vector
state decaying into $ggg$ our results are in agreement with the decay width
listed by PDG [43]. Also they are comparable with the [41] especially for
$1^{3}S_{1}$ state where our prediction is 0.040 MeV without radiative
correction and their prediction is 0.041 MeV. For the $n^{1}P_{1}$,
$n^{3}D_{1}$ and $1^{3}D_{3}$ we do not include the radiative correction
because such correction will be very much lower. The results are consistent
[41]. Intrestingly, for $1^{1}P_{1}$ state our predicted tri-gluon decay width
is 0.035 MeV which is the same as reported by [41].

Table 9 summaries some of the other annihilation decay of the $n^{3}S_{1}$
into $\gamma\gamma\gamma$ and $\gamma gg$. As far as the $\gamma\gamma\gamma$
concerns, we find that present results are considerably higher than [41]. For
$\gamma gg$ it is again consistent with the PDG [43] and [41]. The radiatively
corrected results are allmost matching with the [41]. Results for
$n^{3}P_{1}\rightarrow q{\bar{q}}+g$ are few keV lower than [41] but still
comparable. For all these decays one has to wait for the experimental
confirmation.In general, We can conclude that our instanton potential
predictions and those from the constituent quark model predictions [41] are in
a good agreement for the $ggg$ as well as $\gamma gg$ decays.

We present some of the allowed electric dipole transitions (E1) and magnetic
dipole transitions (M1) of the first order in Tables 10 and 11 respectively.
Our predicted transition widths are compared with the available other
predictions and experimental values wherever it is available. Looking to the
results, we find that for $1P\rightarrow 1S$ results are much lower than that
reported in [100]. And similarly we find that our results deviates from [100]
for every transitions except from $1D\rightarrow 1P$, $2P\rightarrow 1S$ where
the predictions do agree. For other transitions like $2P\rightarrow 1S$ our
predicted outcomes are comparable with the data sets of [115], [41] and [42].
Particularly, for $2^{3}P_{0}$ into $1^{3}S_{1}$ our transition width is (5.66
keV) in excellent agreement with the [115] (5.54 keV) and [41] (5.44 keV). On
the experimental side, transition width from $1P\rightarrow 1S$,
$1D\rightarrow 1P$ is not yet listed in PDG. For the magnetic dipole
transitions our predictions are more or less comparable with other data set.
The discrepancies in the theoretical predictions may vary from one model to
the other due to the choice of potential which plays an important role in the
predictions of mass which in turn creates differences in phase space which
affects the transition widths. Also sometimes it is a choice of different wave
function which effects the predictins of the transition widths. Also,
instanton vacuum ptential plays a vital role in obtaining mass spectroscopy
and other relevent properties of bottomonium. Finally, we hope that our
predicted results using instanton effects on heavy quarks will be helpful in
search of the new quarkonium physics experimentally as well as theoretically.

##  V Acknowledgement

We acknowledge the partial support from DST-SERB, India through the major
research project: (SERB/F/8749/2015-16).

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