11institutetext: aApplied Physics Department, Polytechnic, The Maharaja
Sayajirao University of Baroda, Vadodara 390002, Gujarat, INDIA
bMinistry of Education, Abu Dhabi, UAE
cDepartment of Physics, Sardar Vallabhbhai National Institute of Technology,
Surat 395007, Gujarat, INDIA

# Bottomonium spectroscopy using Coulomb plus linear (Cornell) potential

Virendrasinh Khera Raghav Chaturvedib Nayneshkumar Devlani a and A. K. Raic

(Received: date / Revised version: date)

###### Abstract

The coulomb plus linear (Cornell) potential is used to investigate the mass
spectrum of bottomonium. Gaussian wave function is used in position and
momentum space to estimate values of potential and kinetic energies,
respectively. Based on our calculations, we study newly observed
${{\mathit{\Upsilon}}{(10860)}}$ as an admixture of $4^{3}S_{1}$ with
$4^{3}D_{1}$, and ${{\mathit{\Upsilon}}{(10753)}}$ as an admixture of
$6^{3}S_{1}$ with $4^{3}D_{1}$ also, we try to assign
${{\boldsymbol{\Upsilon}}{(11020)}}$ as a pure $4^{3}D_{1}$ bottomonium state.
We also study the Regge trajectories in the $(J,M^{2})$ and $(n_{r},M^{2})$
planes to help prove our association. We estimate the pseudoscalar and vector
decay constants, the radiative (Electric and Magnetic Dipole) transition
rates, and the annihilation decay width for bottomonium states.

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##  1 Introduction

Bottomonium ($b\overline{b}$ meson) was discovered as $\Upsilon$,
$\Upsilon^{\prime}$ and $\Upsilon^{\prime\prime}$ by the E288 Collaboration at
Fermilab in 1977, and were substantially studied at various $e^{+}e^{-}$
storage ringsZyla:2020zbs ; Innes:1977 . In 1982, $\chi_{bJ}$ $(2P)$ states
and in 1983, $\chi_{bJ}$ $(2P)$, $(J=1,2,3)$ states were discovered in E1
transition from $\Upsilon^{\prime\prime}$ Han:1982 ; Eigen:1982  and
$\Upsilon^{\prime}$ Klopfenstein:1983 ; Pauss:1983 , respectively. In 1984,
$\Upsilon(4S)$, $\Upsilon(5S)$ and $\Upsilon(6S)$ states were observed
Besson:1984 ; Lovelock:1985 . In 2008, after three decade of discovery of
$\Upsilon(nS)$ resonance, the BaBar Collaboration found $\eta_{b}$, the
pseudoscalar partner of the triplet state $\Upsilon(1S)$Aubert2008 . In 2012,
the Belle collaboration reported the first evidence of the $\eta_{b}(2S)$ with
mass $9999\pm 3.5_{-1.9}^{+2.8}$,$MeV/c^{2}$ using
$h_{b}(2P)\rightarrow\gamma\eta_{b}(2S)$ transition and first observation of
$h_{b}(1P)\rightarrow\gamma\eta_{b}(1S)$ and
$h_{b}(2P)\rightarrow\gamma\eta_{b}(1S)$ with mass $9402\pm 1.5\pm 1.8$
$MeV/c^{2}$ Mizuk:2012 .

In 2004, CLEO Collaboration presented the first evidence for the production of
$\Upsilon(1S)$(1D) states in the four-photon cascade, such as in a two-photon
cascade starting from the $\Upsilon(3S)\rightarrow\gamma\chi_{b}(2P_{J})$,
$\chi_{b}(2P_{J})\rightarrow\gamma\Upsilon(1D)$ and then select events with
two more subsequent photon transitions,
$\Upsilon(1D)\rightarrow\gamma\chi_{b}(1P_{J})$,
$\chi_{b}(1P_{J})\rightarrow\gamma\Upsilon(1S)$, followed by the
$\Upsilon(1S)$ annihilation into either $e^{+}e^{-}$ or
$\mu^{+}\mu^{-}$Bonvicini:2004 . In 2010, the BABAR Collaboration reported the
observation of the $J=2$ state of $\Upsilon(1^{3}D_{J})$ in the hadronic
$\pi^{+}\pi^{-}\Upsilon(1S)$ decay channel, with $\Upsilon(1S)\rightarrow
e^{+}e^{-}\,\text{or}\,\mu^{+}\mu^{-}$delAmoSanchez:2010 . In 2011, BABAR
Collaboration reported evidence for the $h_{b}(1P)$ state in the decay
$\Upsilon(3S)\rightarrow\pi{0}h_{b}(1P)$, with data sample corresponds to 28
$fb^{-1}$ of integrated luminosity at a center of mass energy of 10.355 GeV,
the mass of the $\Upsilon(3S)$ resonance and in the same year, the Belle
Collaboration reported the first observation of the spin-singlet bottomonium
states $h_{b}(1P)$ and $h_{b}(2P)$ produced via $e^{+}e^{-}\rightarrow
h_{b}(nP)\pi^{+}\pi^{-}$ transition corresponds to 121.4 $fb^{-1}$ of
integrated luminosity near the peak of the $\Upsilon(5S)$ resonance at a
center-of-mass energy $\sqrt{s}\sim 10.865$ GeVAdachi:2011 . Again in 2011,
ATLAS Collaboration observed the $\chi_{b}(nP)$ states, recorded by the ATLAS
detector during the proton-proton collisions at the LHC, run at a center-of-
mass energy $\sqrt{s}\sim 7\,TeV$ and these states were reconstructed through
their radiative decays to $\Upsilon(1S,2S)$ with
$\Upsilon\rightarrow\mu^{+}\mu^{-}$. In addition to the mass peaks
corresponding to the $\chi_{b}(1P,2P)\rightarrow\gamma\Upsilon(1S,2S)$, at a
mass of $10.530\pm 0.005(stat.)\pm 0.009(syst.)$ GeV in both the decay modes
and structure has been assigned to the $\chi_{b}(3P)$ multipletAad:2011 .

Recently in 2014, the LHCb Collaboration has determined the mass of the
$\chi_{b1}(3P)$ using its radiative decays to the $\Upsilon(1S)$ and
$\Upsilon(1S)$ mesons and the measured mass of the meson is
$m(\chi_{b1}(3P))=10515.7_{-3.9}^{+2.2}(stat.)_{-2.1}^{+1.5}(syst.)$
$MeV/c^{2}$Aaij:2014b . Many other states above open flavour have also been
observed experimentally in the Bottomonium family but their association to a
particular S,P or D state remains questionable. It is because of unknown
couplings of the many channels that become available once the threshold is
crossed. Two charged states namely ${{\boldsymbol{Z}}_{{b}}{(10610)}}$ and
${{\mathit{Z}}_{{b}}{(10650)}}$ were observed at Belle by Belle:2011aa  in
2011 in the ${{\mathit{\Upsilon}}{(5S)}}$ decays to
${{\mathit{\Upsilon}}{(nS)}}{{\mathit{\pi}}^{+}}{{\mathit{\pi}}^{-}}$ (n = 1,
2, 3) and ${{\mathit{h}}_{{b}}{(mP)}}{{\mathit{\pi}}^{+}}{{\mathit{\pi}}^{-}}$
(m = 1, 2) with $\mathit{J}{}^{P}=1{}^{+}$. Ref. Bondar:2011ev  suggested by
considering molecular nature of these states all their properties can be
explained. Recently, in 2019 one new state
${{\boldsymbol{\Upsilon}}{(10753)}}$ was observed at Belle by
Abdesselam:2019gth  in
${{\mathit{e}}^{+}}{{\mathit{e}}^{-}}\rightarrow{{\mathit{\Upsilon}}{(nS)}}{{\mathit{\pi}}^{+}}{{\mathit{\pi}}^{-}}$
(n=1,2,3) with a significance of 5.2 $\sigma$ with mass of $10753\pm 6$ MeV
and width of $36{}^{+18}_{-12}$ MeV. This new observed state can be a
candidate for ${{\mathit{\Upsilon}}{(3D)}}$ state as suggested by Chen:2019uzm
, or a compact tetraquark Ali:2009es  or hadrobottomonium Alberti:2016dru .

Lately, there has been renewed interest in the study of highest bottomonium
states ${{\Upsilon}{(10860)}}$ and ${{\mathit{\Upsilon}}{(11020)}}$ which were
first observed in 1985 at CUSB by Lovelock:1985nb . Fresh study of these two
states was conducted at BELL in 2019 by Abdesselam:2019gth , the calculated
masses found are $10885.3\pm 1.5{}^{+2.2}_{-0.9}$ and
$11000.0{}^{+4.0}_{-4.5}{}^{+1.0}_{-1.3}$ MeV, and the decay widths found are
$36.6{}^{+4.5}_{-3.9}{}^{+0.5}_{-1.1}$ and
$23.8{}^{+8.0}_{-6.8}{}^{+0.7}_{-1.8}$ MeV. Ref. Chen:2019uzm ; Molina:2020zao
suggest ${{\Upsilon}{(10860)}}$ as $5^{3}S_{1}$ bottomonium state, while
Pandya:2021  and Chaturvedi:2020  suggests it as an admixture of $5^{3}S_{1}$
and $6^{3}D_{1}$, and $5^{3}S_{1}$ and $5^{3}D_{1}$ states.
${{\mathit{\Upsilon}}{(11020)}}$ has been suggested as an admixture of
$6^{3}S_{1}$ and $5^{3}D_{1}$ by Chaturvedi:2020 ; Pandya:2021 , also it has
been well reproduced by other potential models that take into account coupled
channel effects Liu:2011yp . Further, the next generation B-factory experiment
Belle II at SuperKEKB collider with luminosity approximately 50 times greater
than Belle I will offer promising prospects for bottomonium physics.
These experimental exploration in the observation of new bottomonium states,
both conventional and unconventional has re-ignited interest and motivated
many theoreticians to carry out a comprehensive study. Comprehensive
theoretical and experimental literature can be found in Eichten:2008 ;
Brambilla:2011 ; Brambilla:2014 , also certain methods like lattice QCD
McNeile:2012qf , heavy quark effective field theory (EFT) Neubert:1993mb ,
chiral perturbation theory Pentia:2009zz , dynamical equation-based approaches
like the Schwinger–Dyson and Bethe–Salpeter equations Ricken:2000kf ;
Mitra:1990av , QCD sum rules Otsuka:2018bqq , NRQCD Mateu:2018zym , non-
relativistic EFT Brambilla:2019jfi , an effective super-symmetric approach
Nielsen:2018uyn , quark models Li:2019tbn , QCD-motivated relativistic quark
model based on the quasipotential approach Ebert:2011jc , and various
potential models have tried to compute the spectroscopic properties of
bottomonium.
Theoretical studies will allow one to single out experimental candidates and
prove to be powerful tools for understanding the quark–antiquark interaction
as expected from quantum chromodynamics (QCD). From a theoretical viewpoint,
non relativistic potential models; where quark antiquark interaction is
modeled by a potential function have been very successful to the study of
heavy quarkonium. Considering the complex structure of the QCD vacuum, it is
difficult to obtain quark–antiquark interaction potential starting from basic
principles of QCD. One of the earliest potential functions to be employed is
the Cornell potential Eichten1978 ; Eichten1980 . In the absence of closed-
form solutions for the Schrödinger equation with Cornell potential one has to
resort to various approximate methods or numerical solutionsLucha1999 . In
literature we find many studies of quarkonium with Cornell potential.
Previously Hall Hall1984  has employed a simple eigenvalue formula under some
restrictions to obtain energy eigenvalues, Vega et.al. Vega2016  have employed
variational method in the light of Supersymmetric Quantum Mechanics, Jacobs
et. al. Jacobs1986  develop a unified approach to the solution of the
Schrödinger and the spinless Salpeter equations and obtained the eigenvalues.
In the present article we employ variational method with a single Gaussian
trial wave function with one parameter; both in position space as well as
momentum space; in a Cornell potential model to calculate the mass spectrum of
the $b\bar{b}$ meson. Such a wavefunction is quite accurate in the long range
part of the potential however it has a limitation that in the short range part
the deviations can turn larger as $r\to 0$. In this regard a multi-gaussian
approach could be investigated. However, the present approach offers the
advantage that it gives an analytical expression for the wavefunction that has
a simple form. The wavefunction can then be conveniently employed for physical
application. We incorporate corrections to the kinetic energy of quarks as
well as the relativistic correction of ${\cal{O}}\left(\frac{1}{m}\right)$ to
the potential energy part of the Hamiltonian. Using our predicted masses for
the $b\bar{b}$ meson, we also plot the Regge trajectories in both the
$(M^{2}\rightarrow J)$ and $(M^{2}\rightarrow n)$ planes (where $J$ is the
spin and $n$ is the principal quantum number), as the Regge trajectories play
a important role to identify the nature of current and future experimentally
observed $b\bar{b}$ meson. We also estimate the pseudoscalar and vector decay
constants, the radiative (Electric and Magnetic Dipole) transition rates
between D, P and S wave as well as the annihilation decays for bottomonium
states.

The article is organized as follows. We present the theoretical framework for
the mass spectra in Section 2.1, the decay constants ($f_{P/V}$) in Section
2.3, the radiative (E1 and M1) transitions in Section 2.4 and annihilation
decays in Section 2.5. Results for the mass spectra, ($f_{P/V}$) decays, E1
and M1 transition width as well as annihilation decays for the $b\bar{b}$
meson, are discussed in Section 3. The Regge trajectories in the $(J,M^{2})$
and $(n_{r},M^{2})$ planes are in Section 3.5. Finally, we draw our conclusion
in Section 4.

##  2 Method

###  2.1 Cornell potential with ${\cal{O}}\left(\frac{1}{m}\right)$
corrections

Here we compute the mass spectra of bottomonium using the Coulomb plus linear
potential, the Cornell potential Eichten1978 ; Eichten1980 . Here, we
incorporate relativistic approach by adding relativistic corrections to the
kinetic energy part and ${\cal{O}}\left(\frac{1}{m}\right)$ correction to the
potential energy part kher2018 ; Koma2006 . We employ the Cornell potential
for bottomonium, as it works well for charmonium, proven by our previous work
kher2018 . Here, we employ the following Hamiltonian Gupta1995 ; Hwang1997 ;
Devlani2014 ; Kher2017b  and quark-antiquark potential Eichten1978 ; Koma2006
,

$H=\sqrt{\mathbf{p}^{2}+m_{Q}^{2}}+\sqrt{\mathbf{p}^{2}+m_{\bar{Q}}^{2}}+V(\mathbf{r}),$
(1)
$V\left(r\right)=V^{\left(0\right)}\left(r\right)+\left(\frac{1}{m_{Q}}+\frac{1}{m_{\bar{Q}}}\right)V^{\left(1\right)}\left(r\right)+{\cal
O}\left(\frac{1}{m^{2}}\right).$ (2)

The Cornell-like potential $V^{\left(0\right)}$ Eichten1978  and
$V^{\left(1\right)}\left(r\right)$ from leading order perturbation theory
areKoma2006 ,

$V^{\left(0\right)}(r)=-\frac{4\alpha_{S}\left({M^{2}}\right)}{3r}+Ar+V_{0}$
(3) $V^{\left(1\right)}\left(r\right)=-C_{F}C_{A}\alpha_{s}^{2}/4r^{2}$ (4)

where, $m_{Q}$ is the quark and $m_{\bar{Q}}$ anti-quark mass, $V_{0}$ is the
potential constant, $A$ is the potential parameter,
$\alpha_{S}\left({M^{2}}\right)$ is the strong running coupling constant and
$C_{F}=4/3$, $C_{A}=3$ are the Casimir charges Koma2006 .

To calculate the expectation values of the Hamiltonian with the Ritz
variational strategy, we use a Gaussian wave functionKher2017b ; Kher2017c ;
kher2018 , which has the form

$\displaystyle R_{nl}(\mu,r)$ $\displaystyle=$
$\displaystyle\mu^{3/2}\left(\frac{2\left(n-1\right)!}{\Gamma\left(n+l+1/2\right)}\right)^{1/2}\left(\mu
r\right)^{l}\times$ (5) $\displaystyle
e^{-\mu^{2}r^{2}/2}L_{n-1}^{l+1/2}(\mu^{2}r^{2})$

and

$\displaystyle R_{nl}(\mu,p)$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{n}}{\mu^{3/2}}\left(\frac{2\left(n-1\right)!}{\Gamma\left(n+l+1/2\right)}\right)^{1/2}\left(\frac{p}{\mu}\right)^{l}\times$
(6) $\displaystyle
e^{-{p}^{2}/2\mu^{2}}L_{n-1}^{l+1/2}\left(\frac{p^{2}}{\mu^{2}}\right)$

respectively with the Laguerre polynomial $L$ and the variational parameter
$\mu$. We estimated $\mu$ for each state, for the preferred value of $A$,
using Hwang1997 ,

$\left\langle{K.E.}\right\rangle=\frac{1}{2}\left\langle{\frac{rdV}{dr}}\right\rangle$
(7)

We expand the Hamiltonian (1), integrate the relativistic correction and
obtain the expectation values of the potential energy as well as kinetic
energy as per our previous work on charmonium and other mesonsKher2017b ;
Kher2017c ; kher2018 .

We have fitted the ground state center of the weight mass using the Eq. (8),
by fixing the potential parameters $A$, $\alpha_{s}$, $V_{0}$ and equated with
the PDG data as well as we forecast the center of weight mass for the $nJ$
state using the Eq. (9) Rai2008 ; kher2018 .

$M_{SA}=M_{P}+\frac{3}{4}(M_{V}-M_{P}),$ (8)

$M_{CW,n}=\frac{\Sigma_{J}(2J+1)M_{nJ}}{\Sigma_{J}(2J+1)}$ (9)

The spin dependent part of the potential for computing the mass difference
between different degenerate bottomonium states can be written as Barnes:2005
; Eichten:2008 ; Voloshin:2007 ; Lakhina2006 ; kher2018 .

$\displaystyle V_{SD}$ $\displaystyle=$ $\displaystyle
V_{LS}(r)\left(\vec{L}\cdot\vec{S}\right)+V_{SS}(r)\left[S\left(S+1\right)-\frac{3}{2}\right]+$
(10) $\displaystyle
V_{T}(r)\left[S\left(S+1\right)-\frac{3\left(\vec{S}\cdot\vec{r}\right)\left(\vec{S}\cdot\vec{r}\right)}{r^{2}}\right]$

where the spin-spin, spin-orbit and tensor interactions can be written in
terms of the vector and scalar parts of $V(r)$ as Voloshin:2007

$\displaystyle V_{SS}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{3m_{Q}^{2}}\nabla^{2}V_{V}=\frac{16\pi\alpha_{s}}{9m_{Q}^{2}}\delta^{3}\left(\vec{r}\right),$
(11) $\displaystyle V_{LS}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{Q}^{2}r}\left(3\frac{dV_{V}}{dr}-\frac{dV_{S}}{dr}\right),$
(12) $\displaystyle V_{T}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{6m_{Q}^{2}}\left(3\frac{d^{2}V_{V}}{dr^{2}}-\frac{1}{r}\frac{dV_{V}}{dr}\right),$
(13)

where $V_{V}(=-\frac{4\alpha_{s}}{3r})$ is the Coulomb part and $V_{S}(=Ar)$
is the confining part of Eq.(3)

In the present study, the quark masses is taken as $m_{b}=4.88$ GeV to
reproduce the ground state masses of the bottomonium states. The fitted
potential parameters are $A=0.222~{}GeV^{2}$, $\alpha_{s}=0.270$ and
$V_{0}=-0.36923~{}GeV$.

###  2.2 Mixing in bottomonium states

There are ambiguities in the structure of many of the heavy quarkonium states.
The observed states like $\Upsilon\left(4S\right)\left(10860\right)$,
$\Upsilon\left(11020\right)$ are assigned $5^{3}S_{1}$, $4^{3}D_{1}$ in the
present work, however there is also a possibility that they could be an
admixture of S-D states Badalian2010 .

Mass of a mixed state $\left(M_{nL}\right)$ is expressed in terms of two pure
states $\left(nl\,\text{ and }\,n^{\prime}l^{\prime}\right)$ asBadalian2010 ;
Shah2012

$M_{nL}=\left|\cos^{2}\theta\right|M_{nl}+\left(1-\left|\cos^{2}\theta\right|\right)M_{n^{\prime}l^{\prime}};$
(14)

here, $\theta$ is the mixing angleRadford2011 ; Shah2012 .

The wave function at origin for D-wave states is defined asBadalian2010 ;
Shah2012

$R_{D}\left(0\right)=\frac{5R_{D}^{\prime\prime}\left(0\right)}{2\sqrt{2}\omega_{b}^{2}};$
(15)

where$R_{D}^{\prime\prime}\left(0\right)$ is the second order derivative of
the wave function at origin for D state and $\omega_{b}$ is a constant having
value $5.11$ GeV. The computed states are listed in table 3.

###  2.3 Decay Constant ($f_{P/V}$)

We calculate decay constant with the QCD correction factor for bottomonium
using the Van-Royen-Weisskopf formula VanRoyen1967 ; Braaten1995 ,

$f_{P/V}^{2}=\frac{12\left|\psi_{P/V}(0)\right|^{2}}{M_{P/V}}\left(1-\frac{\alpha_{S}}{\pi}\left[2-\frac{m_{Q}-m_{\bar{q}}}{m_{Q}+m_{\bar{q}}}\ln\frac{m_{Q}}{m_{\bar{q}}}\right]\right);$
(16)

Equation(16) also gives the inequality Hwang1997a

$\sqrt{m_{v}}f_{v}\geq\sqrt{m_{p}}f_{p}$ (17)

###  2.4 Radiative Transitions

The matrix element of the EM current between the initial ($i$) and final ($f$)
quarkonium state, i.e., $\langle f\mid j_{em}^{\mu}\mid i\rangle$, affect the
radiative transition. The electric and magnetic dipole transitions are leading
order transition amplitudes Ding:2007 ; Lu2016 ; Guo:2010a ; Devlani2014 ;
kher2018 .

The E1 matrix elements are estimated by Radford2009

$\Gamma_{(E1)}\left(n^{2S+1}L_{J}\rightarrow
n^{{}^{\prime}2S^{{}^{\prime}}+1}L_{J^{{}^{\prime}}}^{{}^{\prime}}+\gamma\right)=\frac{4\alpha
e_{Q}^{2}}{3}\frac{E_{\gamma}^{3}E_{f}}{M_{i}}C_{fi}\delta_{SS^{{}^{\prime}}}\times\left|\left\langle
f\left|r\right|i\right\rangle\right|^{2}$ (18)

where photon energy $E_{\gamma}=\frac{M_{i}^{2}-M_{f}^{2}}{2M_{i}}$,
$\alpha=1/137$, the fine structure constant, $e_{Q}$ is the quark charge, and
$E_{f}$ is the energy of the final state. The angular momentum matrix element
($C_{fi}$) is

$C_{fi}=max\left(L,L^{{}^{\prime}}\right)\left(2J^{{}^{\prime}}+1\right)\left\\{\begin{array}[]{ccc}L^{{}^{\prime}}&J^{{}^{\prime}}&S\\\
J&L&1\end{array}\right\\}^{2}$ (19)

where $\left\\{:::\right\\}$ is a 6-j symbol. The matrix elements $\langle
n^{{}^{\prime}2S^{{}^{\prime}}+1}L_{J^{{}^{\prime}}}^{{}^{\prime}}\mid r\mid
n^{2S+1}L_{J}\rangle$ are evaluated using the wave-functions

$\left\langle f\left|r\right|i\right\rangle=\int
drR_{n_{i}l_{i}}\left(r\right)R_{n_{f}l_{f}}\left(R\right)$ (20)

The magnetic dipole M1 rate for transitions between $S$-wave levels is given
by Li2011 ; Bardeen2003 ; Brambilla:2011

$\Gamma_{M1}(i\rightarrow
f+\gamma)=\frac{16\alpha}{3}\mu^{2}k^{3}(2J_{f}+1)\left|\left\langle
f\right|j_{0}(kr/2)\left|i\right\rangle\right|^{2}$ (21)

where the magnetic dipole moment is

$\mu=\frac{m_{\bar{q}}e_{Q}-m_{Q}e_{\bar{q}}}{4m_{\bar{q}}m_{Q}}$

.

###  2.5 Annihilation Decays

Annihilation decays of quarkonia states are highly useful for identification
and the production of resonances as well as to recognize conventional mesons
and multi-quark structures Kwong:1987 ; Kwong:1988 .

####  2.5.1 Leptonic decays

The ${}^{3}S_{1}$ and ${}^{3}D_{1}$ states annihilate into lepton pairs
through a single virtual photon ($Q\bar{Q}\rightarrow
l^{+}l^{-},\,\,\text{where}\,\,l=e^{-},\,{\mu}^{-},\,\tau^{-}$). The leptonic
decay width of the (${}^{3}S_{1}$) and (${}^{3}D_{1}$) states of bottomonium,
including first order radiative QCD correction, is given by Segovia2016 ;
Kwong:1987 ; Bradley:1980 :

$\varGamma\left(n^{3}S_{1}\rightarrow
e^{+}e^{-}\right)=\frac{4e_{Q}^{4}\alpha^{2}\mid
R_{nS}\left(0\right)\mid^{2}}{M_{nS}^{2}}\left(1-\frac{16\alpha_{s}}{3\pi}\right)$
(22) $\varGamma\left(n^{3}D_{1}\rightarrow
e^{+}e^{-}\right)=\frac{25e_{Q}^{2}\alpha^{2}\mid
R_{nD}^{\prime\prime}\left(0\right)\mid^{2}}{2m_{Q}^{4}M_{nD}^{2}}\left(1-\frac{16\alpha_{s}}{3\pi}\right)$
(23)

where, $M_{nS}$ is the mass of the decaying $b\bar{b}$ meson state.

####  2.5.2 Decay into photons

The annihilation decay of the bottomonium states into di-photons or tri-
photons, with and/or without radiative QCD corrections are given by
Segovia2016 ; Kwong:1987 :

$\varGamma\left(n^{1}S_{0}\rightarrow\gamma\gamma\right)=\frac{3e_{Q}^{4}\alpha^{2}\mid
R_{nS}\left(0\right)\mid^{2}}{m_{Q}^{2}}\left(1-\frac{3.4\alpha_{s}}{\pi}\right)$
(24)
$\varGamma\left(n^{3}P_{0}\rightarrow\gamma\gamma\right)=\frac{27e_{Q}^{4}\alpha^{2}\mid
R_{nP}^{\prime}\left(0\right)\mid^{2}}{m_{Q}^{4}}\left(1+\frac{0.2\alpha_{s}}{\pi}\right)$
(25)
$\varGamma\left(n^{3}P_{2}\rightarrow\gamma\gamma\right)=\frac{36e_{Q}^{4}\alpha^{2}\mid
R_{nP}^{\prime}\left(0\right)\mid^{2}}{5m_{Q}^{4}}\left(1-\frac{16\alpha_{s}}{3\pi}\right)$
(26) $\varGamma\left(n^{3}S_{1}\rightarrow
3\gamma\right)=\frac{4(\pi^{2}-9)e_{Q}^{6}\alpha^{3}\mid
R_{nS}\left(0\right)\mid^{2}}{3\pi
m_{Q}^{2}}\left(1-\frac{12.6\alpha_{s}}{\pi}\right)$ (27)

####  2.5.3 Decay into gluons

The annihilation decay of the bottomonium states into di-gluons or tri-gluons
and gluons with photons and light quarks, with and/or without radiative QCD
correction, are given by Segovia2016 ; Kwong:1987 ; Kwong:1988 ; Belanger:1987
:

$\varGamma\left(n^{1}S_{0}\rightarrow gg\right)=\frac{2\alpha_{s}^{2}\mid
R_{nS}\left(0\right)\mid^{2}}{3m_{Q}^{2}}\left(1+\frac{4.4\alpha_{s}}{\pi}\right)$
(28) $\varGamma\left(n^{3}P_{0}\rightarrow gg\right)=\frac{6\alpha_{s}^{2}\mid
R_{nP}^{\prime}\left(0\right)\mid^{2}}{m_{Q}^{4}}$ (29)
$\varGamma\left(n^{3}P_{2}\rightarrow gg\right)=\frac{8\alpha_{s}^{2}\mid
R_{nP}^{\prime}\left(0\right)\mid^{2}}{5m_{Q}^{4}}$ (30)
$\varGamma\left(n^{1}D_{2}\rightarrow gg\right)=\frac{2\alpha_{s}^{2}\mid
R_{nD}^{\prime\prime}\left(0\right)\mid^{2}}{3\pi m_{Q}^{6}}$ (31)
$\varGamma\left(n^{3}S_{1}\rightarrow
3g\right)=\frac{10(\pi^{2}-9)\alpha_{s}^{3}\mid
R_{nS}\left(0\right)\mid^{2}}{81\pi
m_{Q}^{2}}\left(1-\frac{4.9\alpha_{s}}{\pi}\right)$ (32)
$\varGamma\left(n^{1}P_{1}\rightarrow 3g\right)=\frac{20\alpha_{s}^{3}\mid
R_{nP}^{\prime}\left(0\right)\mid^{2}}{9\pi m_{Q}^{4}}\ln(m_{Q}\langle
r\rangle)$ (33) $\varGamma\left(n^{3}D_{1}\rightarrow
3g\right)=\frac{760\alpha_{s}^{3}\mid
R_{nP}^{\prime\prime}\left(0\right)\mid^{2}}{81\pi m_{Q}^{6}}\ln(4m_{Q}\langle
r\rangle)$ (34) $\varGamma\left(n^{3}D_{2}\rightarrow
3g\right)=\frac{10\alpha_{s}^{3}\mid
R_{nP}^{\prime\prime}\left(0\right)\mid^{2}}{9\pi m_{Q}^{6}}\ln(4m_{Q}\langle
r\rangle)$ (35) $\varGamma\left(n^{3}D_{3}\rightarrow
3g\right)=\frac{40\alpha_{s}^{3}\mid
R_{nP}^{\prime\prime}\left(0\right)\mid^{2}}{9\pi m_{Q}^{6}}\ln(4m_{Q}\langle
r\rangle)$ (36) $\varGamma\left(n^{3}S_{1}\rightarrow\gamma
gg\right)=\frac{8(\pi^{2}-9)e_{Q}^{2}\alpha\alpha_{s}^{2}\mid
R_{nS}\left(0\right)\mid^{2}}{9\pi
m_{Q}^{2}}\left(1-\frac{7.4\alpha_{s}}{\pi}\right)$ (37)
$\varGamma\left(n^{3}P_{1}\rightarrow
q\bar{q}+g\right)=\frac{8\eta_{f}\alpha_{s}^{3}\mid
R_{nP}^{\prime}\left(0\right)\mid^{2}}{9\pi m_{Q}^{4}}\ln(m_{Q}\langle
r\rangle)$ (38)

Figure 1: Mass spectrum. Table 1: S-P-D-wave center of weight masses (in GeV).

$nL$ | This work | Expt. Zyla:2020zbs  | Others Theory $M_{CW}$
---|---|---|---
$\mu$ | $M_{CW}$ | Ref. Wang:2018  | Ref. Bhat:2017  | Ref. Deng:2017  | Ref. Lu:2016mbb  | Ref. Segovia2016  | Ref. Godfrey:2015  | Ref. Wurtz:2015  | Ref. Li:2009nr
$1S$ | 1.216 | 9.453 | 9.453 | 9.447 | 9.446 | 9.443 | 9.443 | 9.490 | 9.449 | 9.446 | 9.442
$2S$ | 0.768 | 9.995 |  | 10.010 | 10.018 | 10.009 | 10.004 | 10.009 | 9.996 | 10.015 | 10.009
$3S$ | 0.664 | 10.351 |  | 10.351 | 10.388 | 10.339 | 10.368 | 10.344 | 10.350 | 10.329 | 10.346
$4S$ | 0.613 | 10.647 |  | 10.608 | 10.702 | 10.594 |  |  | 10.632 |  | 10.607
$5S$ | 0.581 | 10.909 |  | 10.819 | 10.989 | 10.808 |  |  | 10.876 |  | 10.828
$6S$ | 0.557 | 11.148 |  | 10.998 | 11.295 | 10.995 |  |  | 11.101 |  | 11.020
$1P$ | 0.811 | 9.899 | 9.899 | 9.896 | 9.905 | 9.909 | 9.890 | 9.879 | 9.884 | 9.901 | 9.905
$2P$ | 0.678 | 10.268 | 10.260 | 10.260 | 10.276 | 10.254 | 10.263 | 10.240 | 10.252 | 10.219 | 10.258
$3P$ | 0.620 | 10.570 |  | 10.531 | 10.585 | 10.519 | 10.560 |  | 10.542 |  | 10.530
$1D$ | 0.719 | 10.149 |  | 10.149 | 10.164 | 10.154 |  | 10.123 | 10.149 | 10.164 | 10.152
$2D$ | 0.641 | 10.465 |  | 10.465 | 10.480 | 10.433 |  | 10.419 | 10.450 | 10.441 | 10.439
$3D$ | 0.598 | 10.740 |  | 10.740 | 10.767 |  |  |  | 10.706 |  | 10.677
$4D$ | 0.570 | 10.990 |  | 10.988 | 11.036 |  |  |  |  |  |

Table 2: Complete mass spectra(in GeV).

State | $J^{P}$ | This | Expt. | Others |  |
---|---|---|---|---|---|---
| work | Ref. Zyla:2020zbs  | Ref. Wang:2018  | Ref. Chen:2019uzm  | Ref. Deng:2017  | Ref. Lu:2016mbb  | Ref. Segovia2016  | Ref. Godfrey:2015  | Ref. Wurtz:2015  | Ref. Li:2009nr
$1^{1}S_{0}$ | $0^{-+}$ | 9.423 | 9.399 | 9.398 |  | 9.390 | 9.395 | 9.455 | 9.402 | 9.402 | 9.389
$1^{3}S_{1}$ | $1^{--}$ | 9.463 | 9.460 | 9.463 |  | 9.460 | 9.459 | 9.502 | 9.465 | 9.460 | 9.460
$2^{1}S_{0}$ | $0^{-+}$ | 9.983 | 9.999 | 9.989 | 9.999 | 9.990 | 9.982 | 9.990 | 9.976 | 9.998 | 9.987
$2^{3}S_{1}$ | $1^{--}$ | 10.001 | 10.023 | 10.017 | 10.023 | 10.015 | 10.011 | 10.015 | 10.003 | 10.020 | 10.016
$3^{1}S_{0}$ | $0^{-+}$ | 10.342 |  | 10.336 | 10.337 | 10.326 | 10.353 | 10.330 | 10.336 | 10.314 | 10.330
$3^{3}S_{1}$ | $1^{--}$ | 10.354 | 10.355 | 10.356 | 10.357 | 10.343 | 10.373 | 10.349 | 10.354 | 10.334 | 10.351
$4^{1}S_{0}$ | $0^{-+}$ | 10.638 |  | 10.597 | 10.627 | 10.584 |  |  | 10.623 |  | 10.595
$4^{3}S_{1}$ | $1^{--}$ | 10.650 | 10.579 | 10.612 | 10.637 | 10.597 | 10.654 | 10.607 | 10.635 |  | 10.611
$5^{1}S_{0}$ | $0^{-+}$ | 10.901 |  | 10.810 | 10.878 | 10.800 |  |  | 10.869 |  | 10.817
$5^{3}S_{1}$ | $1^{--}$ | 10.912 | 10.885 | 10.822 | 10.887 | 10.811 | 10.999 | 10.818 | 10.878 |  | 10.831
$6^{1}S_{0}$ | $0^{-+}$ | 11.140 |  | 10.991 | 11.111 | 10.988 |  |  | 11.097 |  | 11.011
$6^{3}S_{1}$ | $1^{--}$ | 11.151 | 11.000⋆ | 11.001 | 11.118 | 10.997 | 11.265 | 10.995 | 11.102 |  | 11.023
$1^{3}P_{0}$ | $0^{++}$ | 9.874 | 9.859 | 9.858 | 9.854 | 9.864 | 9.851 | 9.855 | 9.847 | 9.865 | 9.865
$1^{3}P_{1}$ | $1^{++}$ | 9.894 | 9.893 | 9.889 | 9.893 | 9.903 | 9.890 | 9.874 | 9.876 | 9.893 | 9.897
$1^{1}P_{1}$ | $1^{+-}$ | 9.899 | 9.899 | 9.894 | 9.899 | 9.909 | 9.886 | 9.879 | 9.882 | 9.900 | 9.903
$1^{3}P_{2}$ | $2^{++}$ | 9.907 | 9.912 | 9.910 | 9.911 | 9.921 | 9.899 | 9.886 | 9.897 | 9.913 | 9.918
$2^{3}P_{0}$ | $0^{++}$ | 10.248 | 10.233 | 10.235 | 10.239 | 10.220 | 10.233 | 10.221 | 10.226 | 10.194 | 10.226
$2^{3}P_{1}$ | $1^{++}$ | 10.265 | 10.255 | 10.255 | 10.259 | 10.249 | 10.257 | 10.236 | 10.246 | 10.212 | 10.251
$2^{1}P_{1}$ | $1^{+-}$ | 10.268 | 10.260 | 10.259 | 10.262 | 10.254 | 10.262 | 10.240 | 10.250 | 10.219 | 10.256
$2^{3}P_{2}$ | $2^{++}$ | 10.274 | 10.269 | 10.269 | 10.268 | 10.264 | 10.274 | 10.246 | 10.261 | 10.227 | 10.269
$3^{3}P_{0}$ | $0^{++}$ | 10.551 |  | 10.513 | 10.551 | 10.490 | 10.533 | 10.500 | 10.522 |  | 10.502
$3^{3}P_{1}$ | $1^{++}$ | 10.567 | 10.512 | 10.527 | 10.557 | 10.515 | 10.556 | 10.513 | 10.538 |  | 10.524
$3^{1}P_{1}$ | $1^{+-}$ | 10.570 |  | 10.530 | 10.556 | 10.519 | 10.560 | 10.516 | 10.541 |  | 10.529
$3^{3}P_{2}$ | $2^{++}$ | 10.576 | 10.524 | 10.539 | 10.556 | 10.528 | 10.568 | 10.521 | 10.550 |  | 10.540
$1^{3}D_{1}$ | $1^{--}$ | 10.145 |  | 10.153 | 10.136 | 10.146 | 10.136 | 10.117 | 10.138 | 10.150 | 10.145
$1^{3}D_{2}$ | $2^{--}$ | 10.149 | 10.163 | 10.162 | 10.164 | 10.153 | 10.141 | 10.122 | 10.147 | 10.161 | 10.151
$1^{1}D_{2}$ | $2^{-+}$ | 10.149 |  | 10.163 | 10.167 | 10.153 |  | 10.123 | 10.148 | 10.163 | 10.152
$1^{3}D_{3}$ | $3^{--}$ | 10.150 |  | 10.170 | 10.183 | 10.157 |  | 10.127 | 10.155 | 10.172 | 10.156
$2^{3}D_{1}$ | $1^{--}$ | 10.462 |  | 10.442 | 10.467 | 10.425 | 10.454 | 10.414 | 10.441 | 10.458 | 10.432
$2^{3}D_{2}$ | $2^{--}$ | 10.465 |  | 10.450 | 10.476 | 10.432 |  | 10.418 | 10.449 | 10.401 | 10.438
$2^{1}D_{2}$ | $2^{-+}$ | 10.465 |  | 10.450 | 10.475 | 10.432 |  | 10.419 | 10.450 | 10.447 | 10.439
$2^{3}D_{3}$ | $3^{--}$ | 10.466 |  | 10.456 | 10.478 | 10.436 |  | 10.422 | 10.455 | 10.459 | 10.442
$3^{3}D_{1}$ | $1^{--}$ | 10.736 | 10.752 | 10.675 | 10.742 |  | 10.725 | 10.653 | 10.698 |  | 10.670
$3^{3}D_{2}$ | $2^{--}$ | 10.740 |  | 10.681 | 10.744 |  |  | 10.657 | 10.705 |  | 10.676
$3^{1}D_{2}$ | $2^{-+}$ | 10.740 |  | 10.681 | 10.742 |  |  | 10.658 | 10.706 |  | 10.677
$3^{3}D_{3}$ | $3^{--}$ | 10.741 |  | 10686 | 10.740 |  |  | 10.660 | 10.711 |  | 10.680
$4^{3}D_{1}$ | $1^{--}$ | 10.985 | 11.000† | 10.871 | 10.987 |  |  | 10.853 | 10.928 |  | 10.877
$4^{3}D_{2}$ | $2^{--}$ | 10.988 |  | 10.876 | 10.986 |  |  |  | 10.934 |  | 10.882
$4^{1}D_{2}$ | $2^{-+}$ | 10.988 |  | 10.876 | 10.984 |  |  |  | 10.935 |  | 10.883
$4^{3}D_{3}$ | $3^{--}$ | 10.990 |  | 10.880 | 10.981 |  |  | 10.860 | 10.939 |  | 10.886

$\star\Upsilon\left(11020\right)$ Widely considered.
${}^{\dagger}\Upsilon\left(11020\right)$ More favoured in the present work.

Table 3: Masses of mixed states in bottomonium (in GeV). Expt. State | Mixed State | Mass of mixed states (GeV)
---|---|---
Present | ExptZyla:2020zbs
$\Upsilon\left(10860\right)$ | $4^{3}S_{1}\text{ and }4^{3}D_{1}$ | 10.886 | $10.885_{-1.6}^{+2.6}$
$\Upsilon(11020)$ | $6^{3}S_{1}\text{ and }4^{3}D_{1}$ | 11.004 | $11.000\pm 0.004$
Table 4: Pseudoscalar and vector decay constants of the bottomonium (in $\mathrm{MeV}$) Decay | State | Our Work | Expt.Ref. Zyla:2020zbs  | Ref. Pandya:2021  | Ref. Soni:2017  | Ref. Bhaghyesh:2011  | Ref. Wang:2006
---|---|---|---|---|---|---|---
$f_{V}$ | $1^{3}S_{1}$ | 640(530) | 715$\pm$5 | 653(551) | 647 | 931(645) | 498$\pm$20
| $2^{3}S_{1}$ | 383(317) | 498$\pm$8 | 563(477) | 519 | 566(439) | 366$\pm$27
| $3^{3}S_{1}$ | 338(280) | 430$\pm$4 | 507(430) | 475 | 507(393) | 304$\pm$27
| $4^{3}S_{1}$ | 319(265) | 336$\pm$18 | 466(395) | 450 | 481(356) | 259$\pm$22
| $5^{3}S_{1}$ | 308(255) |  | 434(368) | 432 | 439(341) | 228$\pm$16
| $6^{3}S_{1}$ | 301(249) |  | 406(345) | 418 |  |
$f_{P}$ | $1^{1}S_{0}$ | 639(529) |  | 654(578) | 646 | 834(694) |
| $2^{1}S_{0}$ | 382(317) |  | 564(499) | 518 | 567(472) |
| $3^{1}S_{0}$ | 338(280) |  | 508(450) | 474 | 508(422) |
| $4^{1}S_{0}$ | 319(264) |  | 467(413) | 449 | 481(401) |
| $5^{1}S_{0}$ | 308(255) |  | 435(385) | 432 |  |
| $6^{1}S_{0}$ | 301(249) |  | 406(360) | 418 |  |
Table 5: Electric dipole (E1) transitions widths of $b\overline{b}$ meson.  Meson trans. | This work | Expt. Ref.Zyla:2020zbs  | Other work ($\varGamma$ in keV)
---|---|---|---
Initial | Final | $E_{\gamma}$ | $\varGamma$ | $\varGamma$ | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref.
|  | MeV | keV | keV | Wang:2018  | Segovia2016  | Ebert2003  | Akbar:2015  | Godfrey:2015  | Chaturvedi:2020  | Brambilla:2004  | Pandya:2021
$1^{3}P_{2}$ | $1^{3}S_{1}$ | 434 | 39.99 | 34.38 | 31.4 | 39.15 | 40.2 | 37.3 | 32.8 | 39.27 | 31.6 | 15.70
$1^{3}P_{1}$ | $1^{3}S_{1}$ | 422 | 36.80 | 32.544 | 28.3 | 35.66 | 36.6 | 32.8 | 29.5 | 42.55 | 27.8 | 14.66
$1^{1}P_{1}$ | $1^{1}S_{0}$ | 465 | 24.68 | 35.77 | 34.4 | 43.66 | 52.6 | 22.9 | 35.7 | 74.63 | 41.8 | 12.52
$1^{3}P_{0}$ | $1^{3}S_{1}$ | 403 | 16.00 |  | 22.8 | 28.07 | 29.9 | 26.0 | 23.8 | 29.07 | 22.2 | 19.4
$2^{3}S_{1}$ | $1^{3}P_{2}$ | 105 | 1.78 | $2.29\pm 0.23$ | 1.86 | 2.08 | 2.46 | 2.92 | 1.88 | 2.274 | 2.04 | 0.841
$2^{3}S_{1}$ | $1^{3}P_{1}$ | 119 | 1.49 | $2.21\pm 0.22$ | 1.60 | 1.84 | 2.45 | 2.83 | 1.63 | 2.232 | 2.00 | 0.614
$2^{3}S_{1}$ | $1^{3}P_{0}$ | 124 | 0.86 | $1.22\pm 0.16$ | 0.907 | 1.09 | 1.62 | 1.85 | 0.91 | 1.30 | 1.29 | 0.302
$2^{1}S_{0}$ | $1^{1}P_{1}$ | 84 | 1.60 |  | 2.467 | 2.85 | 3.09 | 4.54 | 2.48 | 8.045 | 41.8 | 0.888
$1^{3}D_{3}$ | $1^{3}P_{2}$ | 240 | 21.68 |  | 23.9 | 24.74 | 24.6 | 31.27 | 24.3 | 5.099 | 22.6 | 6.677
$1^{3}D_{2}$ | $1^{3}P_{2}$ | 239 | 5.34 |  | 5.49 | 6.23 | 6.35 | 5.48 | 5.6 | 3.017 | 5.46 | 1.57
$1^{3}D_{2}$ | $1^{3}P_{1}$ | 251 | 20.63 |  | 18.8 | 21.95 | 23.3 | 21.03 | 19.2 | 3.940 | 20.5 | 5.242
$1^{3}D_{1}$ | $1^{3}P_{2}$ | 236 | 0.57 |  | 0.55 | 0.65 | 0.69 | 0.568 | 0.56 | 1.908 | 0.50 | 0.161
$1^{3}D_{1}$ | $1^{3}P_{1}$ | 248 | 9.91 |  | 9.51 | 12.29 | 12.7 | 10.96 | 9.7 | 2.264 | 10.7 | 2.684
$1^{3}D_{1}$ | $1^{3}P_{0}$ | 267 | 16.58 |  | 16.3 | 20.98 | 23.4 | 25.84 | 16.5 | 3.120 | 20.1 | 4.476
$2^{3}P_{2}$ | $2^{3}S_{1}$ | 272 | 20.02 | 24.645 | 14.6 | 17.50 | 16.7 | 18.99 | 14.3 | 15.66 | 14.5 |
$2^{3}P_{1}$ | $2^{3}S_{1}$ | 262 | 18.03 | 23.283 | 13.7 | 15.89 | 14.7 | 16.14 | 13.3 | 21.84 | 12.4 |
$2^{1}P_{1}$ | $2^{1}S_{0}$ | 281 | 22.09 | 40.32 | 15.0 | 17.6 | 21.4 | 12.79 | 14.1 | 23.11 | 19 |
$2^{3}P_{0}$ | $2^{3}S_{1}$ | 247 | 14.96 | 0.00012 | 11.1 | 12.8 | 6.79 | 11.88 | 10.9 | 16.84 | 9.17 |
$2^{3}P_{2}$ | $1^{3}D_{3}$ | 124 | 3.097 |  | 1.61 | 2.06 | 2.35 | 2.885 | 1.5 |  | 2.25 |
$2^{3}P_{2}$ | $1^{3}D_{2}$ | 125 | 0.569 |  | 0.339 | 0.35 | 0.449 | 1.054 | 0.3 |  | 0.434 |
$2^{3}P_{2}$ | $1^{3}D_{1}$ | 128 | 0.041 |  | 0.027 | 0.021 | 0.035 | 0.079 | 0.03 |  | 0.034 |
$2^{3}P_{1}$ | $1^{3}D_{1}$ | 119 | 0.818 |  | 0.511 | 0.41 | 0.615 | 1.46 | 0.5 |  | 0.593 |
$2^{3}P_{0}$ | $1^{3}D_{1}$ | 103 | 2.11 |  | 1.05 | 0.74 | 1.17 | 3.21 | 1.0 |  | 1.13 |
Table 6: Magnetic dipole (M1) transitions widths of $b\overline{b}$ mesons. Meson transition | This work | Other work ($\varGamma$ in keV)
---|---|---
Initial$\rightarrow$Final |  $E_{\gamma}$(MeV) |  $\varGamma$(keV) | Ref. Wang:2018  | Ref. Segovia2016  | Ref. Akbar:2015  | Ref. Godfrey:2015  | Ref. Brambilla:2004  | Ref. Ebert:2011jc
$1^{3}S_{1}\rightarrow 1^{1}S_{0}$ | 41 | $9.1\times 10^{-3}$ | $9.52\times 10^{-3}$ | $9.34\times 10^{-3}$ | $11\times 10^{-3}$ | $10\times 10^{-3}$ | $8.95\times 10^{-3}$ | $9.7\times 10^{-3}$
$2^{3}S_{1}\rightarrow 2^{1}S_{0}$ | 15 | $4.98\times 10^{-4}$ | $5.82\times 10^{-4}$ | $5.80\times 10^{-4}$ | $6.6\times 10^{-4}$ | $5.9\times 10^{-4}$ | $15.1\times 10^{-4}$ | $1.6\times 10^{-3}$
$3^{3}S_{1}\rightarrow 3^{1}S_{0}$ | 12 | $4.4\times 10^{-4}$ | $3.37\times 10^{-4}$ | $6.58\times 10^{-4}$ | 0.012 | $2.5\times 10^{-4}$ | $8.3\times 10^{-4}$ | $0.9\times 10^{-3}$
$2^{3}S_{1}\rightarrow 1^{1}S_{0}$ | 560 | 1.609 | 0.0688 | 0.0565 | 0.1729 | 0.081 | 0.00281 | $1.3\times 10^{-3}$
$2^{1}S_{0}\rightarrow 1^{3}S_{1}$ | 506 | 3.585 | 0.0706 | 0.045 | 0.00064 | 0.068 | 0.00283 | $2.4\times 10^{-3}$
Table 7: Leptonic decay widths ($\Upsilon\rightarrow\varGamma_{l^{+}l^{-}}$ in keV and $n^{1}D_{2}\rightarrow\varGamma_{l^{+}l^{-}}$ in eV ). State | This work | Expt. | Other works
---|---|---|---
$\varGamma_{l^{+}l^{-}}$ | $\varGamma_{l^{+}l^{-}}^{cf}$ | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref.
Zyla:2020zbs  | Wang:2018  | Segovia2016  | Godfrey:2015  | Bhaghyesh:2011  | Chaturvedi:2020  | Giannuzzi:2008  | Radford:2007  | Li:2009nr  | Gonzalez:2004  | Pandya:2021
$\Upsilon(1^{3}S_{1})$ | 1.074 | 0.582 | $1.340\pm 0.018$ | 1.65 | 0.71 | 1.44 | 1.809 | 1.053 | 1.237 | 1.33 | 2.31 | 1.01 | 1.224
$\Upsilon(2^{3}S_{1})$ | 0.363 | 0.197 | $0.612\pm 0.011$ | 0.821 | 0.37 | 0.73 | 0.797 | 0.562 | 0.581 | 0.61 | 0.92 | 0.35 | 0.537
$\Upsilon(3^{3}S_{1})$ | 0.274 | 0.149 | $0.443\pm 0.008$ | 0.569 | 0.27 | 0.53 | 0.618 | 0.399 | 0.270 | 0.46 | 0.64 | 0.25 | 0.402
$\Upsilon(4^{3}S_{1})$ | 0.238 | 0.129 | $0.272\pm 0.029$ | 0.431 | 0.21 | 0.39 | 0.541 | 0.282 | 0.212 | 0.35 | 0.51 | 0.22 | 0.240
$\Upsilon(5^{3}S_{1})$ | 0.216 | 0.117 | $0.31\pm 0.07$ | 0.348 | 0.18 | 0.33 | 0.481 | 0.212 |  |  | 0.42 | 0.18 |
$\Upsilon(6^{3}S_{1})$ | 0.201 | 0.109 | $0.130\pm 0.03$ | 0.286 | 0.15 | 0.27 | 0.432 | 0.166 |  |  | 0.37 | 0.15 |
$1^{3}D_{1}$ | 3.04 | 1.65 |  | 1.88 | 1.40 | 1.38 |  |  |  |  |  |  |
$2^{3}D_{1}$ | 4.47 | 2.42 |  | 2.81 | 2.50 | 1.99 |  |  |  |  |  |  |
$3^{3}D_{1}$ | 5.88 | 3.19 |  | 3.0 |  | 2.38 |  |  |  |  |  |  |
$4^{3}D_{1}$ | 7.33 | 3.97 |  | 3.0 |  | 2.18 |  |  |  |  |  |  |
Table 8: Di-photon decay widths .

State | This work | Other works ($\varGamma$ in keV) |
---|---|---|---
$\varGamma_{\gamma\gamma}$ | $\varGamma_{\gamma\gamma}^{cf}$ |  | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref.
(keV) | (keV) |  | Wang:2018  | Segovia2016  | Godfrey:2015  | Anisovich:2005  | Munz:1996  | Ebert:2003b  | Pandya:2021  | Li:2009nr  | Laverty:2009  | Chaturvedi:2020
$\eta_{b}(1^{1}S_{0})$ | 0.3364 | 0.2361 |  | 1.05 | 0.69 | 0.94 | 1.554 | 0.22 | 0.35 | 0.378 | 0.527 | 0.30 | 0.545
$\eta_{b}(2^{1}S_{0})$ | 0.1270 | 0.0896 |  | 0.489 | 0.36 | 0.41 | 1.928 | 0.110 | 0.15 | 0.263 | 0.263 | 0.14 | 0.124
$\eta_{b}(3^{1}S_{0})$ | 0.1029 | 0.0726 |  | 0.323 | 0.27 | 0.29 | 2.139 | 0.084 | 0.10 | 0.206 | 0.172 | 0.10 | 0.105
$\eta_{b}(4^{1}S_{0})$ | 0.0942 | 0.0666 |  | 0.237 |  | 0.20 |  |  |  | 0.170 | 0.105 |  | 0.068
$\eta_{b}(5^{1}S_{0})$ | 0.0900 | 0.0636 |  | 0.192 |  | 0.17 |  |  |  | 0.145 | 0.121 |  | 0.060
$\eta_{b}(6^{1}S_{0})$ | 0.0876 | 0.0619 |  | 0.152 |  | 0.41 |  |  |  | 0.123 | 0.050 |  | 0.048
$1^{3}P_{0}$ | 0.0165 | 0.0168 |  | 0.199 | 0.12 | 0.15 | 0.024 | 0.024 | 0.038 | 0.081 | 0.037 | 0.0329 | 0.068
$2^{3}P_{0}$ | 0.0169 | 0.0172 |  | 0205 | 0.14 | 0.15 | 0.023 | 0.026 | 0.029 | 0.071 | 0.037 | 0.0314 | 0.022
$3^{3}P_{0}$ | 0.0189 | 0.0192 |  | 0.180 | 0.15 | 0.13 | 0.023 |  |  | 0.062 | 0.035 |  | 0.008
$1^{3}P_{2}$ | 0.00439 | 0.00238 |  | 0.0106 | 0.00308 | 0.0093 | 0.016 | 0.0056 | 0.008 | 0.021 | 0.0066 | 0.00719 | 0.014
$2^{3}P_{2}$ | 0.00451 | 0.00245 |  | 0.0133 | 0.00384 | 0.012 | 0.015 | 0.0068 | 0.006 | 0.019 | 0.0067 | 0.00759 | 0.004
$3^{3}P_{2}$ | 0.00504 | 0.00273 |  | 0.0141 | 0.0041 | 0.013 | 0.015 |  |  | 0.016 | 0.0064 |  | 0.002

Table 9: Tri-photon decay widths (in eV). State | This work | Other works |
---|---|---|---
$\varGamma_{3\gamma}$ | $\varGamma_{3\gamma}^{cf}$ |  | Ref. Wang:2018  | Ref. Segovia2016  | Ref. Godfrey:2015  | Ref. Chaturvedi:2020
$\Upsilon(1^{3}S_{1})$ | $33.56\times 10^{-3}$ | $30.67\times 10^{-3}$ |  | $19.4\times 10^{-3}$ | $3.44\times 10^{-3}$ | $17.0\times 10^{-3}$ | $16\times 10^{-3}$
$\Upsilon(2^{3}S_{1})$ | $12.67\times 10^{-3}$ | $11.58\times 10^{-3}$ |  | $10.9\times 10^{-3}$ | $2.00\times 10^{-3}$ | $9.8\times 10^{-3}$ | $3\times 10^{-3}$
$\Upsilon(3^{3}S_{1})$ | $10.261\times 10^{-3}$ | $9.376\times 10^{-3}$ |  | $8.04\times 10^{-3}$ | $1.55\times 10^{-3}$ | $7.6\times 10^{-3}$ | $1\times 10^{-3}$
$\Upsilon(4^{3}S_{1})$ | $9.400\times 10^{-3}$ | $8.590\times 10^{-3}$ |  | $6.36\times 10^{-3}$ | $1.29\times 10^{-3}$ | $6.0\times 10^{-3}$ |
$\Upsilon(5^{3}S_{1})$ | $8.979\times 10^{-3}$ | $8.206\times 10^{-3}$ |  | $5.43\times 10^{-3}$ | $1.10\times 10^{-3}$ |  |
$\Upsilon(6^{3}S_{1})$ | $8.735\times 10^{-3}$ | $7.982\times 10^{-3}$ |  | $4.57\times 10^{-3}$ | $9.56\times 10^{-4}$ |  |
Table 10: Di-gluon decay widths. State | This work | Other works
---|---|---
$\varGamma_{gg}$ | $\varGamma_{gg}^{cf}$ | Ref. | Ref. | Ref. | Ref. | Ref. | Ref. | Ref.
(MeV) | (MeV) | Wang:2018  | Segovia2016  | Godfrey:2015  | Pandya:2021  | Laverty:2009  | Ebert2003 ; Ebert:2011jc  | Negash:2015
$\eta_{b}(1^{1}S_{0})$ | 8.219 | 11.326 | 17.9 | 20.18 | 16.6 | 5.449 | 11.49 |  | 10.865
$\eta_{b}(2^{1}S_{0})$ | 3.121 | 4.301 | 8.33 | 10.64 | 7.2 | 4.177 | 5.16 |  | 7.477
$\eta_{b}(3^{1}S_{0})$ | 2.529 | 3.485 | 5.51 | 7.94 | 4.9 | 3.449 | 3.81 |  | 6.276
$\eta_{b}(4^{1}S_{0})$ | 2.317 | 3.193 | 4.03 |  | 3.4 | 2.945 |  |  |
$\eta_{b}(5^{1}S_{0})$ | 2.214 | 3.051 | 3.26 |  | 2.9 | 2.576 |  |  |
$\eta_{b}(6^{1}S_{0})$ | 2.154 | 2.968 | 2.59 |  | 2.2 | 2.285 |  |  |
$1^{3}P_{0}$ | 0.721 | 1.34 | 3.37 | 2.00 | 2.6 | 0.906 | 0.96 | 0.653 |
$2^{3}P_{0}$ | 0.741 | 1.39 | 3.52 | 2.37 | 2.29 | 0.785 | 0.99 | 0.431 |
$3^{3}P_{0}$ | 0.828 | 1.54 | 3.10 | 2.46 | 2.2 | 0.671 |  |  |
$1^{3}P_{2}$ | 0.192 | 0.209 | 0.165 | 0.08369 | 0.147 | 0.238 | 0.33 | 0.109 |
$2^{3}P_{2}$ | 0.198 | 0.215 | 0.220 | 0.10426 | 0.207 | 0.207 | 0.35 | 0.076 |
$3^{3}P_{2}$ | 0.221 | 0.240 | 0.243 | 0.11145 | 0.227 | 0.178 |  |  |
$1^{1}D_{2}$ | 0.489 (keV) |  | 0.657(keV) | 0.37(keV) | 1.87(keV) |  |  |  |
$2^{1}D_{2}$ | 0.764 (keV) |  | 1.22(keV) | 0.67(keV) | 3.3(keV) |  |  |  |
$3^{1}D_{2}$ | 1.06 (keV) |  | 1.59(keV) |  | 4.7(keV) |  |  |  |
$4^{1}D_{2}$ | 1.38 (keV) |  | 1.86(keV) |  |  |  |  |  |
Table 11: Tri-gluon decay widths (in keV). State | This work | Other works |
---|---|---|---
$\varGamma_{3g}$ | $\varGamma_{3g}^{cf}$ | Ref. Wang:2018  | Ref. Segovia2016  | Ref. Godfrey:2015  | Ref. Pandya:2021  | Ref. Ebert:2011jc
$\Upsilon(1^{3}S_{1})$ | 114.73 | 66.42 | 50.8 | 41.63 | 47.6 | 40.0 |
$\Upsilon(2^{3}S_{1})$ | 43.33 | 25.08 | 28.4 | 24.25 | 26.3 | 26.9 |
$\Upsilon(3^{3}S_{1})$ | 35.08 | 20.31 | 21.0 | 18.76 | 19.8 | 20.6 |
$\Upsilon(4^{3}S_{1})$ | 32.14 | 18.65 | 16.7 | 15.58 | 15.1 | 16.8 |
$\Upsilon(5^{3}S_{1})$ | 30.70 | 17.78 | 14.2 | 13.33 | 13.1 | 14.1 |
$\Upsilon(6^{3}S_{1})$ | 29.87 | 17.29 | 12.0 | 11.57 | 11.0 | 11.7 |
$1^{1}P_{1}$ | 28.47 |  | 44.7 | 35.26 | 37.0 | 35.7 | 36
$2^{1}P_{1}$ | 35.10 |  | 64.6 | 52.70 | 54.0 | 34.6 | 31.5
$3^{1}P_{1}$ | 43.11 |  | 71.1 | 62.16 | 59.0 | 33.1 |
$1^{3}D_{1}$ | 9.64 |  | 10.4 | 9.97 | 8.11 | 10.6 |
$2^{3}D_{1}$ | 16.27 |  | 20.1 | 9.69 | 14.8 | 11.9 |
$3^{3}D_{1}$ | 23.67 |  | 26.0 |  | 21.2 | 11.8 |
$4^{3}D_{1}$ | 31.99 |  | 30.4 |  |  | 11.3 |
$1^{3}D_{2}$ | 0.482 |  | 0.821 | 0.62 | 0.69 |  |
$2^{3}D_{2}$ | 0.813 |  | 1.65 | 0.61 | 1.4 |  |
$3^{3}D_{2}$ | 1.18 |  | 2.27 |  | 2.0 |  |
$4^{3}D_{2}$ | 1.60 |  | 2.75 |  |  |  |
$1^{3}D_{3}$ | 1.927 |  | 2.19 | 0.22 | 2.07 | 6.0 |
$2^{3}D_{3}$ | 3.252 |  | 4.56 | 1.25 | 4.3 | 5.6 |
$3^{3}D_{3}$ | 4.732 |  | 6.65 |  | 6.6 | 5.5 |
$4^{3}D_{3}$ | 6.393 |  | 8.38 |  |  | 5.3 |
Table 12: $n^{3}S_{1}\rightarrow\gamma gg$ decay widths (in keV). State | This work | Expt. | Other works
---|---|---|---
$\varGamma_{\gamma gg}$ | $\varGamma_{\gamma gg}^{cf}$ | Ref. Zyla:2020zbs  | Ref. Wang:2018  | Ref. Segovia2016  | Ref. Godfrey:2015  | Ref. Pandya:2021
$\Upsilon(1^{3}S_{1})$ | 2.481 | 0.903 | 1.18 | 1.32 | 0.79 | 1.2 | 1.273
$\Upsilon(2^{3}S_{1})$ | 0.937 | 0.341 | 0.59 | 0.739 | 0.46 | 0.68 | 0.869
$\Upsilon(3^{3}S_{1})$ | 0.759 | 0.276 | 0.0097 | 0.547 | 0.36 | 0.52 | 0.672
$\Upsilon(4^{3}S_{1})$ | 0.695 | 0.253 |  | 0.433 | 0.30 | 0.40 | 0.548
$\Upsilon(5^{3}S_{1})$ | 0.664 | 0.242 |  | 0.370 | 0.25 |  | 0.465
$\Upsilon(6^{3}S_{1})$ | 0.646 | 0.235 |  | 0.311 | 0.22 |  | 0.389
Table 13: $n^{3}P_{1}\rightarrow q\overline{q}+g$ decay widths (in keV). State | This work | Other works
---|---|---
$\varGamma_{q\overline{q}+g}$ | Ref. Wang:2018  | Ref. Segovia2016  | Ref. Godfrey:2015  | Ref. Ebert2003  | Ref. Pandya:2021
$1^{3}P_{1}$ | 45.55 | 81.7 | 71.53 | 67 | 57 | 57.958
$2^{3}P_{1}$ | 56.16 | 117 | 106.14 | 96 | 50 | 55.396
$3^{3}P_{1}$ | 68.97 | 126 | 124.53 |  |  | 52.958

##  3 Results and Discussion

The bottomonium mass spectrum is investigated using the Cornell potential with
a Gaussian wave function and a $\mathcal{O}(1/m)$ correction to the potential
energy term and up to a ${{\cal{O}}\left({\bf p}^{10}\right)}$ expansion of
the kinetic energy term for relativistic correction of the Hamiltonian. We
have calculated Hamiltonian yield (center of weight masses) for the nS $(n\leq
6)$, $nP\;\text{and}\;nD\;(n\leq 4)$ bottomonium states and are presented in
Table 1.

###  3.1 Masses of S States

The masses of the low lying S states are well measured experimentally. The
estimated complete mass spectrum of bottomonium is presented in Table 2, with
the spectroscopic notation $n^{2S+1}L_{J}$ and is also graphically represented
in Fig.1. We have compared our results with other theoretical model
predictions. In Ref. Wang:2018  Wang et. al. employ modified Godfrey-Isgur
(GI) model, in ref. Chen:2019uzm  Chen et. al. employ the Relativistic Flux
Tube (RFT) model, in ref. Deng:2017  Deng et. al. use the Non-Relativistic
Screened Potential Model (NR-SPM), in ref. Lu:2016mbb  Lu et. al. use the
linear plus color-Coulomb (GI-type), Segovia et. al. employ the Non-
Relativistic Constituent Quark Model (NR-CQM), Godfrey et. al. in ref.
Godfrey:2015  employ Relativised Quark Model (RQM), Wurtz et. al. in ref.
Wurtz:2015  present LATTICE Field Theory calculations while Li et. al. also
employ a Screened Potential Model (SPM) as presented in Table 2. We obtained
the pseudoscalar state mass $\eta_{b}\left(1^{1}S_{0}\right)$ (9423 MeV) and
vector state mass $\Upsilon\left(1^{3}S_{1}\right)$ (9463 MeV), by adding the
spin hyperfine interaction to the fixed spin average mass for the ground
state. The estimated masses for $2^{1}S_{0}$ ($\eta_{b}$(2S)), $2^{3}S_{1}$
($\Upsilon(2S)$) and $3^{3}S_{1}$ ($\Upsilon(3S)$) states are underestimated
by 16 MeV, 22 Mev and 1 MeV respectively when compared with the experimentally
observed values.

The $\Upsilon\left(4S\right)$ is considered to be a $4^{3}S_{1}$ state with
the experimental mass $10579.4\pm 1.2$ MeV. In the present work the calculated
mass $M_{\Upsilon\left(4S\right)}=10657$ MeV is overestimated by 71 MeV as
compared to experimentally observed mass. The calculated mass is also
overestimated by other theoretical model predictions. For example the
estimated mass by RFT model in ref. Chen:2019uzm  is
$M_{\Upsilon\left(4S\right)}=10637$ MeV, the modified GI model in ref.
Wang:2018  is $M_{\Upsilon\left(4S\right)}=10612$ MeV and NR-SPM in ref.
Deng:2017  is $M_{\Upsilon\left(4S\right)}=10597$ MeV.

The state $\Upsilon\left(10860\right)$ is generally assigned to be the
$5^{3}S_{1}$ state. However other possible interpretations such as a mixing of
$\Upsilon\left(5S\right)$ with a lowest P-wave hybrid Bruschini2019  is also
considered. The experimentally measured mass is
$M_{\Upsilon\left(10860\right)}=10885.2^{+2.6}_{-1.6}$ MeV. The calculated
mass of $5^{3}S_{1}$ state in the present work is 10912 MeV which is
overestimated. The difference between our calculated and experimentally
determined mass is 27 MeV. In Table 3 we present the mass of
$\Upsilon\left(10860\right)$ state $M_{\Upsilon\left(10860\right)}=10886$ MeV
obtained as a mixture of $4^{3}S_{1}$ and $4^{3}D_{1}$ state. Thus we believe
this state could also be a mixture of S-D states. Interestingly the
relativistic flux tube (RFT) modelChen:2019uzm  and lattice QCDBicudo:2019ymo
study favour $5^{3}S_{1}$ association for ${{\Upsilon}{(10860)}}$.

The state $\Upsilon\left(11020\right)$ is widely considered to be the
$6^{3}S_{1}$ state. Its measured experimental mass is
$M_{\left(\Upsilon{11020}\right)}=11000\pm 4$ MeV. In the present work we
obtain mass of $6^{3}S_{1}$ state as $11151$ MeV which is overestimated. Thus
the present model does not favour $6^{3}S_{1}$ assignment to the
$\Upsilon\left(11020\right)$ resonance. Mass of $6^{3}S_{1}$ states in many
other theoretical model predictions also does not match with
$\Upsilon\left(11020\right)$ state. Thus $\Upsilon\left(11020\right)$ may not
be a pure 6S resonance. In ref. Badalian2010  Badalian et. al. have considered
this state to be a mixture of S-D states. In the present work mass of
$4^{3}D_{1}$ state is 10985 MeV which is slightly underestimated yet closer to
the $\Upsilon\left(11020\right)$ experimental mass. Thus if we consider mixing
between S-D states, then our model favours mixing between $6^{3}S_{1}$ and
$4^{3}D_{1}$. We have considered it to be a $4^{3}D_{1}$ to draw the Regge
trajectory in fig. 4.

The hyperfine mass splitting $\Delta
m(nS)=m\left[\Upsilon\left(nS\right)\right]-m\left[\eta_{b}\left(nS\right)\right]$
between the singlet and triplet S states, reflects the spin-dependent
interaction. Our predicted hyperfine mass splitting between 1S and 2S states
is 40 MeV and 18 MeV respectively, which are much less when compared with the
experimental and with that of other theoretical models. Our predicted
hyperfine mass splitting for higher bottomonium states $nS\;(n\geq 3)$ are in
notable difference but consistent with predictions by other theoretical
models.

####  3.1.1 Masses of P states

Until now all 1P and 2P states are confirmed by experimentZyla:2020zbs ,
calculated masses for 1P-3P states can be found in table 15. The mass
difference between the estimated value by our model and experimentally
observed values are 15 MeV for $1^{3}P_{0}$ ($\chi_{b0}(1P)$), 1 MeV for
$1^{3}P_{1}$ ($\chi_{b1}(1P)$), matches well for $1^{1}P_{1}$ ($h_{b}(1P)$),
shows 5 MeV difference for $1^{3}P_{2}$ ($\chi_{b2}(1P)$), 15 MeV for
$2^{3}P_{0}$ ($\chi_{b0}(2P)$), 10 MeV for $2^{3}P_{1}$ ($\chi_{b1}(2P)$), 8
MeV for $2^{1}P_{1}$ ($h_{b}(2P)$), 5 MeV for $2^{3}P_{2}$ ($\chi_{b2}(2P)$).

Two 3P states ${\chi_{{b1}}{(3P)}}$ and ${{\chi}_{{b2}}{(3P)}}$ have been
observed by ATLAS Aad:2011 , D0 Abazov:2012gh  and LHCb Aaij:2014caa ;
Aaij:2014b . Most models in literature overestimate the masses of
$\chi_{b1}\left(3P\right)$ and $\chi_{b2}\left(3P\right)$ states. For the
$\chi_{b1}\left(3P\right)$ ref Wang:2018  employing modified GI model predicts
10527 MeV, the RFT model in ref. Chen:2019uzm  predicts 10557 MeV, the RQM
model in ref. Godfrey:2015  predicts 10538 MeV. The present work also
overestimates the mass by 55 MeV. For the $\chi_{b2}\left(3P\right)$ ref
Wang:2018  employing modified GI model predicts 10539 MeV, the RFT model in
ref. Chen:2019uzm  predicts 10556 MeV, the RQM model in ref. Godfrey:2015
predicts 10550 MeV. Our calculated mass is overestimated by about 52 MeV
compared to the experimental massZyla:2020zbs . Also, as the 3P states are
above threshold the masses may be affected by coupled channel effect. More
theoretical and experimental efforts are desirable for the 3P bottomonium
states in future.

####  3.1.2 Masses of D states

For $1^{3}D_{2}$ ($\Upsilon(1D)$) state our calculated mass differs from the
experimental mass by 14 MeV. As mentioned earlier in introduction about the
newly observed state ${{\mathit{\Upsilon}}{(10753)}}$ in 2019 at
BelleAbdesselam:2019gth  there is still uncertainty over its structure. Based
on the present study we find the mass of $3^{3}D_{1}$ state as 10736 MeV
whereas the experimental determined mass for ${{\mathit{\Upsilon}}{(10753)}}$
is 10753 $\pm$ 6 MeV, the difference being only of 17 MeV. Hence, we assign
${{\mathit{\Upsilon}}{(10753)}}$ as $3^{3}D_{1}$ state. Same assignment for
${{\mathit{\Upsilon}}{(10753)}}$ has been made by Chen:2019uzm .
${{\mathit{\Upsilon}}{(10753)}}$ as an admixture of S and D states was ruled
out in relativistic flux tube modelChen:2019uzm  because the di-electron decay
width for admixture state can increase by 2 orders. Also, the experimental
total decay width calculated is around 36 MeV Zyla:2020zbs  which is nearby
the total decay width for $3^{3}D_{1}$ state, 54 MeV as calculated by
Wang:2018  which supports our assignment.

###  3.2 Decay Constants

The calculated vector decay constants $f_{V}$($f_{Vcor}$) and pseudoscalar
decay constants $f_{P}$($f_{Pcor}$), without and with QCD corrections, are
shown in Table 4.The vector decay constants $f_{V}$($f_{Vcor}$) for
$1^{3}S_{1}$, $2^{3}S_{1}$, $3^{3}S_{1}$ and $4^{3}S_{1}$ states are 65 Mev,
115 MeV, 92 MeV and 17 MeV lower respectively as compared to experimental
estimates. Our calculated vector decay constants are in good agreement in
comparison to Ref. Wang:2006 . Our calculated pseudoscalar decay constants
$f_{P}$($f_{Pcor}$) are lower compared to available theoretical estimates. The
$f_{Vcor}$ and $f_{Pcor}$ decay constants are underestimated, where as $f_{V}$
and $f_{P}$ decay constants are overestimated in comparison to the estimates
by other available theoretical models.

###  3.3 Radiative Transitions

The Electrical dipole (E1) transition widths
$\varGamma[2S\rightarrow(1P)\gamma]$, calculated using our predicted
bottomonium masses are shown in Table 5. Our results for the E1 transitions
width $\varGamma[2^{3}S_{1}\rightarrow(1^{3}P_{3})\gamma]$,
$\varGamma[2^{3}S_{1}\rightarrow(1^{3}P_{1})\gamma]$ and
$\varGamma[2^{3}S_{1}\rightarrow(1^{3}P_{0})\gamma]$ are 0.51 keV, 0.72 keV
and 0.36 keV lower respectively than the experimental results. Our calculated
E1 transition widths $\varGamma[2S\rightarrow(1P)\gamma]$ are comparatively in
good agreements with other theoretical estimates.

We have calculated the Electrical dipole (E1) transition widths
$\varGamma[1P\rightarrow(1S)\gamma]$, $\varGamma[1D\rightarrow(1P)\gamma]$,
$\varGamma[2P\rightarrow(2S)\gamma]$, $\varGamma[2P\rightarrow(1D)\gamma]$,
which are summarized in Table 5. Our calculated E1 transitions for
$\varGamma[1^{3}P_{2}\rightarrow(1^{3}S_{1})\gamma]$ and
$\varGamma[1^{3}P_{1}\rightarrow(1^{3}S_{1})\gamma]$ are 5.61 keV and 4.26 keV
higher respectively as compared to experimental measurements. The E1
transition width $\varGamma[1^{1}P_{1}\rightarrow(1^{1}S_{0})\gamma]$ is 11.09
keV lower when compared to experimental measurement.

Our calculated E1 transitions for
$\varGamma[2^{3}P_{2}\rightarrow(2^{3}S_{1})\gamma]$,
$\varGamma[2^{3}P_{1}\rightarrow(2^{3}S_{1})\gamma]$ and
$\varGamma[2^{1}P_{1}\rightarrow(2^{1}S_{0})\gamma]$ are 4.62 keV, 5.25 keV
and 18.23 keV lower respectively when compared to experimental measurements.
Whereas transitions for $\varGamma[2^{3}P_{0}\rightarrow(2^{3}S_{1})\gamma]$
is much higher but comparable to experimental measurement. Overall our
calculated E1 transitions for $\varGamma[1P\rightarrow(1S)\gamma]$ and
$\varGamma[2P\rightarrow(2S)\gamma]$ are comparatively in good agreement with
results of other theoretical estimates. Our calculated E1 transition widths
for $\varGamma[1D\rightarrow(1P)\gamma]$ and
$\varGamma[2P\rightarrow(1D)\gamma]$ are in close agreement with the estimates
from other theoretical models.

The M1 radiative transition
$\varGamma[\Upsilon\left(nS\right)\rightarrow\eta_{b}\left(nS\right)]$ is the
unique electromagnetic decay and until now no experimental information is
available. Our calculated M1 transition widths using bottomonium masses
predicted by our potential model for the low-lying 1S, 2S and 3S states are
shown in Table 6. Our predicted M1 transition widths are compared with the
available predictions from other theoretical models. Our calculated M1
transition width for $\varGamma[1^{3}S_{1}\rightarrow(1^{1}S_{0})\gamma]$,
$\varGamma[2^{3}S_{1}\rightarrow(2^{1}S_{0})\gamma]$ and
$\varGamma[3^{3}S_{1}\rightarrow(3^{1}S_{0})\gamma]$ are in close agreement
with the estimates by theoretical models Ref. Wang:2018 ; Segovia2016 ;
Akbar:2015 ; Godfrey:2015 , while M1 transition width for
$\varGamma[2^{3}S_{1}\rightarrow(1^{1}S_{0})\gamma]$ and
$\varGamma[2^{1}S_{0}\rightarrow(1^{3}S_{1})\gamma]$ is higher than the
estimates of other theoretical models. But the calculated M1 transition width
for $\varGamma[2^{3}S_{1}\rightarrow(1^{1}S_{0})\gamma]$ is lower than the
experimental M1 transiton width $12.5\pm 4.9$ keV Zyla:2020zbs .

We observe that radiative transitions are model dependent. In practice
radiative QCD corrections, play more critical role to the estimation of
electromagentic transitions. However estimation of radiative corrections is a
separate study in itself and lies beyond the scope of the present work.

###  3.4 Annilation Decays

We calculate the partial decay widths $\varGamma$ and $\varGamma^{cf}$ (with
QCD correction factor) of annihilation processes for $e^{+}e^{-}$, di-photon,
tri-photon, di-gluon, tri-gluon, $\gamma gg$ and $q\bar{q}+g$. The results are
compared with experimental data from the PDG Zyla:2020zbs  as well as with
other theoretical estimates.

For the bottomonium states $\Upsilon(nS)$, we calculate leptonic decay widths
$\varGamma_{l^{+}l^{-}}$ and $\varGamma_{l^{+}l^{-}}^{cf}$ in keV, tabulated
in Table 7, tri-photon decay widths $\varGamma_{3\gamma}$ and
$\varGamma_{3\gamma}^{cf}$ in eV, tabulated in Table 9, tri-gluon decay widths
$\varGamma_{3g}$ and $\varGamma_{3g}^{cf}$ in keV, tabulated in Table 11 as
well as $n^{3}S_{1}\rightarrow\gamma gg$ decay widths in keV, tabulated in
Tablet 12. Our estimated leptonic widths without and with QCD correction are
lower compared to experimentally observed widths as well as other
theoretically estimates from Ref. Godfrey:2015 ; Wang:2018 ; Li:2009nr ;
Bhaghyesh:2011 , whereas calculated leptonic widths are in good agreement with
estimates from Ref. Segovia2016 ; Chaturvedi:2020 ; Gonzalez:2004 . Our
calculated tri-photon and tri-gluon decay widths are in fine tune with
available theoretical estimates. Our calculated
$n^{3}S_{1}\rightarrow\gamma{gg}$ decay widths are higher when compared to
experimental data but are comparable to the widths calculated by other
theoretical models.

For the $\eta_{b}(nS)$ states, we calculate di-photon decay width
$\varGamma_{\gamma\gamma}$ and $\varGamma_{\gamma\gamma}^{cf}$ in keV, as well
as di-gluon decay widths $\varGamma_{gg}$ and $\varGamma_{gg}^{cf}$ in MeV and
the results are summarized in Table 8 and 10 respectively. The calculated di-
photon decay widths for 1S and 2S states agree well with the experimentally
determined decay widths when calculated excluding the first order QCD
radiative correction, while the decay width drops when the first order QCD
radiative correction is added. When comparing the calculated decay width with
the results from other theoretical approaches our results are in accordance
with the decay widths of Ref. Munz:1996 ; Ebert2003a ; Gupta:1996 ; Li:2009nr
. The decay widths of Ref. Wang:2018 ; Segovia2016 ; Godfrey:2015 ;
Anisovich:2005  are very high and not even closer to the experimental
determined results, thus this study favors our calculated decay widths. The
decay widths of all the P wave states are very low and no experimental result
is available for a bold statement.

We have calculated leptonic decay widths
$n^{3}D_{1}\rightarrow\varGamma_{l^{+}l^{-}}$, $\varGamma_{l^{+}l^{-}}^{cf}$
in eV and results are shown in Table 7. Our calculated leptonic decay widths
without QCD corrections for $n^{3}D_{1}$ are higher compared to leptonic decay
width estimated by other theoretical models. After applying QCD correction,
leptonic decay widths for $n^{3}D_{1}$ are comparatively in good agreement
with estimated by available theoretical models.

Our calculated di-photon decay widths for
$n^{3}P_{0}\rightarrow\varGamma_{\gamma\gamma}$,
$\varGamma_{\gamma\gamma}^{cf}$ and
$n^{3}P_{2}\rightarrow\varGamma_{\gamma\gamma}$,
$\varGamma_{\gamma\gamma}^{cf}$ in keV are shown Table 8. We have compared our
di-photon decay widths with the widths from available theoretical models. we
observe that our calculated di-photon decay width results for $n^{3}P_{0}$
states are lower when compared to the decay widths predicted by other
theoretical models, whereas decay widths for $n^{3}P_{2}$ states are lower in
comparison to decay widths predicted by Ref. Godfrey:2015 ; Wang:2018 ;
Chaturvedi:2020 ; Anisovich:2005 ; Segovia2016 ; Li:2009nr ; Laverty:2009 ;
Munz:1996 .

Our calculated di-gluon decay widths $n^{3}P_{0}\rightarrow\varGamma_{gg}$,
$\varGamma_{gg}^{cf}$ in MeV, $n^{3}P_{2}\rightarrow\varGamma_{gg}$,
$\varGamma_{gg}^{cf}$ in MeV and $n^{1}D_{2}\rightarrow\varGamma_{gg}$,
$\varGamma_{gg}^{cf}$ in keV are shown Table 10. For $n^{3}P_{0}$ states our
calculated di-gluon decay widths (with and without QCD correction) are lower
compared to decay widths predicted by Ref. Wang:2018 ; Segovia2016 ;
Godfrey:2015  while it is comparatively in good agreements with decay widths
predicted by Ref. Pandya:2021 ; Laverty:2009 ; Ebert2003 . Our calculated di-
gluon decay widths for the $n^{3}P_{2}$ and $n^{1}D_{2}$ states, are in good
agreements with the other theoretical estimates.

We also calculate the tri-gluon decay widths for $n^{1}P_{1}$, $n^{3}D_{1}$,
$n^{3}D_{2}$ and $n^{3}D_{3}$ states and results are tabulated in Table 11.
Tri-gluon decay widths for $n^{1}P_{1}$ state are a little bit on the lower
side. Tri-gluon decay widths for $n^{3}D_{1}$, $n^{3}D_{2}$ and $n^{3}D_{3}$
states when compared with other available theoretical estimates seems to be in
good agreement. We also calculate $n^{3}P_{1}\rightarrow q\bar{q}+g$ decay
width and results are tabulated in Table 13. Our calculated decay widths are
comparatively low with respect to Ref. Wang:2018 ; Segovia2016 ; Godfrey:2015
and in good agreements with Ref. Ebert2003 ; Pandya:2021 .

We also observe the wide variation in estimated values of annihilation decay
width, may be due to different treatments in the relativistic corrections and
the considerable uncertainties which arise from the wave function dependence
of the model.

###  3.5 Regge trajectories

We construct the Regge trajectories for the $(n,M^{2})$ and $(J,M^{2})$ planes
using bottomonium masses estimated by our potential model. The trajectories
with the same value of $J$ and differ by a quantum number corresponding to the
radial quantum number are usually called ”daughter” trajectories and their
masses are higher than those of the leading trajectory with given quantum
numbers. The linearity of Regge trajectories is interpreted as a expression of
strong forces between quarks at large distances(color confinement). The Regge
trajectories are basically nonlinear for mesons with massive quarks
Kruczenski:2004 .

The Regge trajectories in the $(J,M^{2})$ plane with $(P=(-1)^{J})$
($J^{P}=1^{-},2^{+},3^{-}$) natural and $(P=(-1)^{J-1})$
($J^{P}=0^{-},1^{+},2^{-}$) unnatural parity are plotted in Figs. 2-3. In the
diagrams, the solid triangles represent the bottomonium masses estimated by
our potential model and the hollow squares represent experimentally available
masses with the corresponding bottomonium name. The Regge trajectories for
principal quantum number $n_{r}=n-1$ in the $(n_{r},M^{2})$ plane are shown in
Figs. 4 and 5.

To calculate the $\chi^{2}$ fitted slopes ($\alpha$, $\beta$) and the
intercepts ($\alpha_{0}$, $\beta_{0}$), definitions used are Kher2017b ;
Kher2017c ; kher2018 :

$J=\alpha M^{2}+\alpha_{0}.$ (39) $n_{r}=\beta M^{2}+\beta_{0}$ (40)

The calculated slopes and intercepts are tabulated in Tables 14, 15, and 16).
The calculated masses of the bottomonium fit well into the $(n,M^{2})$ and
$(J,M^{2})$ planes. The parent Regge trajectories in both ($J,M^{2}$) and
($n_{r},M^{2}$) planes, which start from ground states, are exhibiting a
nonlinear behavior in the lower mass region, whereas the daughter
trajectories, which involve both radially and orbitally excited states, turn
out to be almost linear, equidistant and parallel. The slopes of Regge
trajectories in $(J,M^{2})$ and ($n_{r},M^{2}$) planes, increases with
increasing quark masses. The orbital momentum $\ell$ of the state is
proportional to its mass: $\ell=\alpha M^{2}(\ell)+\alpha(0)$, where the slope
$\alpha$ depends on the flavor content of the states, hence quark masses
affect the linearity of the Regge trajectories. In the framework of the hadron
string model,the radial spectrum of heavy quarkonia typically leads to strong
nonlinearities Afonin2016 .

Figure 2: Regge trajectory ($J\rightarrow M^{2}$) with natural parity.  Figure 3: Regge trajectory ($J\rightarrow M^{2}$) with unnatural parity.  Figure 4: Regge trajectory ($n_{r}\rightarrow M^{2}$) for the pseudoscalar and vector $S$ state and excited $P$ and $D$ state masses. Figure 5: Regge trajectory ($n_{r}\rightarrow M^{2}$) for the S-P-D states center of weight mass. Table 14: Slopes and intercepts of the $(J,\>M^{2})$ Regge trajectories with unnatural and natural parity.  Parity | Trajectory | $\alpha(GeV^{-2})$ | $\alpha_{0}$
---|---|---|---
Unnatural | Parent | $0.234\pm 0.023$ | $-6.587\pm 0.741$
First daughter | $0.244\pm 0.016$ | $-8.616\pm 0.634$
Second daughter | $0.235\pm 0.006$ | $-9.913\pm 0.261$
Natural | Parent | $0.145\pm 0.023$ | $-12.036\pm 2.241$
First daughter | $0.207\pm 0.020$ | $-19.754\pm 2.105$
Second daughter | $0.243\pm 0.019$ | $-25.139\pm 2.152$
Table 15: Slopes and intercepts of the $(n_{r},M^{2})$ Regge trajectories.  Meson | $J^{P}$ | $\beta(GeV^{-2})$ | $\beta_{0}$
---|---|---|---
$\eta_{b}$ | $0^{-+}$ | $0.133\pm 0.011$ | $-12.061\pm 1.133$
$\Upsilon$ | $1^{--}$ | $0.136\pm 0.010$ | $-12.367\pm 1.073$
$\chi_{b0}$ | $0^{++}$ | $0.141\pm 0.005$ | $-13.780\pm 0.543$
$\chi_{b1}$ | $1^{++}$ | $0.146\pm 0.007$ | $-14.342\pm 0.697$
$h_{b}$ | $1^{+-}$ | $0.144\pm 0.007$ | $-14.149\pm 0.784$
$\chi_{b2}$ | $2^{++}$ | $0.146\pm 0.007$ | $-14.314\pm 0.724$
$\Upsilon(^{3}D_{1})$ | $1^{--}$ | $0.162\pm 0.005$ | $-16.674\pm 0.578$
$\Upsilon(^{3}D_{2})$ | $2^{--}$ | $0.163\pm 0.006$ | $-16.780\pm 0.641$
$\Upsilon(^{1}D_{2})$ | $2^{-+}$ | $0.162\pm 0.005$ | $-16.683\pm 0.574$
$\Upsilon(^{3}D_{3})$ | $3^{--}$ | $0.162\pm 0.005$ | $-16.679\pm 0.570$
Table 16: Slopes and intercepts of $(n_{r},M^{2})$ Regge trajectory for center of weight mass.  Trajectory | $\beta(GeV^{-2})$ | $\beta_{0}$
---|---|---
S State | $0.144\pm 0.010$ | $-13.216\pm 1.049$
P State | $0.145\pm 0.007$ | $-14.25\pm 0.737$
D State | $0.162\pm 0.005$ | $-16.686\pm 0.570$

##  4 Conclusion

Our predicted masses for the $S-P-D$ states from the present study are very
close to the available experimental as well as the other theoretical
estimates. The Regge trajectories in both ($J,M^{2}$) and ($n_{r},M^{2}$)
planes are almost linear, equidistant and parallel. From Figs. 2-3, we find
that the experimental states are sitting nicely on straight lines without
deviation. The pseudoscalar $(f_{Pcor})$ and the vector $(f_{Vcor})$ decay
constants with QCD correction are close to available theoretical predictions.
The predicted radiative (E1 and M1 dipole) transition widths are in
satisfactory agreement with theoretical and experimental predictions. From
comparison of our estimated radiative transitions widths with other
theoretical estimations, we observed that due to different treatments or
parameters in the relativistic corrections in the model, the various models
have very different predictions. Although more precise experimental
measurements are required in most cases. Using the Van Royen-Weisskopf
relation, we predict annihilation Decay widths and we conclude that the
inclusion of QCD correction factors are of importance for obtaining accurate
results. The results of annihilation Decay widths in various models again show
a wide range of variations.

Acknowledgements A. K. Rai acknowledges the financial support extended by the
Department of Science of Technology, India under the SERB fast track scheme
SR/FTP /PS-152/2012.

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# Controversy of quantum congestion in two-particle quantum walks

A.D. Lobanova<EMAIL_ADDRESS>A.D. Lobanov<EMAIL_ADDRESS>Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny,
Moscow Region 141701, Russia.  A.M. Pupasov-Maksimov<EMAIL_ADDRESS>Universidade Federal de Juiz de Fora Department of Mathematics

###### Abstract

The article deals with one- and two-particle quantum walks on a graph with
Braess-like topology and analyzes the issue of network congestion in the
quantum world. Our approach to the study of congestion in quantum networks is
based on the comparison of the evolution of bosonic and fermionic many-
particle states. We consider a simple example of non-interacting particles,
where one can expect the appearance of congestion in the fermion case due to
the Pauli principle. It is shown that dependence of the transport efficiency
on the parameters of quantum graph is similar in the bosonic and fermionic
cases.

††preprint: APS/123-QED

##  I Introduction

Congested vs bare regimes of a multi-particle evolution on quantum graphs
Lobanova A.D., Pupasov-Maksimov A.M., Lobanov A.D. July 2021

Figure 1: A profile equivalent to the profile [1] in terms of geometry. Figure
2: Time dependence of the probability of transition from point 1 to point 3.
The parameters of this system are shown in Table 1. The black dashed line is
$c=0.01$, the gray line is $c=0.05$, the black line is $c=0.1$.

This paper is devoted to the study of the presence of Braess’s paradox in the
quantum world. This transport phenomenon has attracted a lot attention since
the publication of the fundamental work of Braess [1]. Braess’s paradox is
observed in transport networks of large cities, where road closures improve
the situation on the roads [2], or in electrical networks [3]; the paradox is
also observed in computer science, in electric circuits [4] and in many other
areas.

In this paper, we study one- and two-particle quantum walks on a graph which
is similar to the Braess transport network. Scanning probe electronic imaging
[5], [6] indicates the existence of an analogue of Braess’s paradox in
branched-out rectangular quantum rings [7]. However, the issue of the network
congestion in the quantum theory is far from being clear. Such quantum
phenomena as tunneling, spreading with time of quantum wave packets, state
reduction, quantum statistics etc. make it difficult to use classical
conceptions to interpret quantum transport efficiency. For instance, the
question of tunneling time through a barrier has been a puzzle in foundational
quantum mechanics for many decades [8, 9]. In [10] we have shown that two
mesoscopic samples with the same geometrical congestion (width of branches)
and Braess-like topology demonstrate completely different behavior. It is an
indication that the issue of network congestion which is crucial to the
classical Braess paradox is non-trivial in quantum theory.

Braess’s paradox appears as a result of a dynamical equilibrium in a multi-
agent system. In the quantum multi-particle systems, the quantum statistics
play an important role. If there is a limited number of available transverse
modes in the mesoscopic sample [5], [6], it is natural to assume that the
transport of fermions through such a conductor is congested due to the Pauli
principle. Here we study this assumption, isolating the possible effect of
fermion statistics in a simple model. Our approach to the congestion of a
quantum network is based on the comparison of the evolution of bosonic and
fermionic many-particle (2-partical) states. Both the bosonic and fermionic
versions of Braess’s paradox are found in the proposed system of quantum dots
(QDs) on a graph, which is quite surprising because any quantum graph should
be un-congested to multi-particle bosonic quantum walks. The results of this
paper can be used to analyze the speed of operation of computational elements
(the rate of quantum evolution) based on QDs; the algorithm offers the
opportunity to use the proposed characteristic of the time of passage of
particles along a quantum graph for non-periodic evolution.

##  II Formulation of the transport problem

Consider two-particle quantum walks on a graph. The system is a network of
quantum dots (QDs) interconnected in a certain way - four double quantum dots
(DQD) and two single quantum dots (see Fig. 3). Two non-interacting electrons
move from the initial state to the final one. We study the dynamics of
electrons in the absence of interaction between them. In experimental
realizations, the Coulomb interaction of electrons at distant dots is
suppressed by the large distance of interaction and by screening due to the
presence of metallic gates between and nearby them [11].

Quantum dots can be formed from a two-dimensional electron gas by the gate
field, and the population of electrons at these points can be controlled by
the potentials at the gates. Each position on the graph can be occupied by no
more than one electron. The position of the electron can be measured using
quantum dot contact detectors, which are located near the quantum dots in such
a way that the electron in a particular QD decreases the electric current in
the detector, increasing the potential barrier. Therefore, a lower current
detects an electron, and a higher current indicates an absence of an electron
(empty QD), respectively. A system of semiconductor QDs is mathematically
represented in the form of a graph, the vertices of which are QDs. The edges
of the graph represent possible electron tunneling transitions. Electrons
wander by tunneling through a barrier of controlled height between quantum
dots. For simplicity, we will assume that the electron spins are always up
(1/2), which can take place, for example, in a strong magnetic field [11].

Braess’s paradox appears when an additional edge is provided to the transport
network, however transportation time increases. Therefore we build two quantum
graphs where the second one has an additional edge, as in [6], [5]. Then we
compare the passage times between ”initial” and ”final” states for these
systems. An increase of the passage time for the extended graph corresponds to
an analogue of Braess’s paradox. We will consider the transport flow of two
fermions and two bosons. In the case of bosons, both quantum graphs are un-
congested, and one can expect that an additional edge can only improve
transport properties.

##  III Non-interacting fermions on a quantum graph

###  III.1 Implementation of a transport one-particle system of quantum walks
on a graph

Figure 3: Schematic representation of the systems. Coupling coefficients $l,h$
and $c$ are shown by different lines.

We start with a simple system of four single QDs (there is 1 bound state per
dot), located at the vertices of the parallelogram (Fig.1). Let the QDs be
identical, and there are nonzero integrals of the overlap of states in pairs
between all points. We define the initial Hamiltonian of the system $H_{num}$
as follows:

$H_{num}=\begin{pmatrix}0&b&0&s\\\ b&0&s&c\\\ 0&s&0&b\\\ s&c&b&0\end{pmatrix}$

Here, parameter $c$ regulates transition from QD 4 to QD 2, which is our
additional edge in terms of Braess geometry. This Hamiltonian defines the
evolution of a particle on the graph by the Schödinger equation. Eigenstates
of $4\times 4$ Hamiltonian, which can be found analytically, define time
dependence of transition probabilities: $[1\rightarrow 3]:P_{13}$,
$[1\rightarrow 2]:P_{12}$, $[1\rightarrow 4]:P_{14}$, $[1\rightarrow
1]:P_{11}$. Below we consider a particular example. Numerical values of
parameters are given in the Table 1.

Table 1: Numerical values of coupling parameters and potential energy for a system of 4 QDs.  b | s | c | $V_{0}$
---|---|---|---
0.01 | 0.01 | 0.01, 0.05, 0.1 | 0

Fig. 2 shows the time dependence of the transition probability
$[1(\text{initial})\rightarrow 3(\text{final})]$. One can observe that
transition probability has a quasi-periodic character with a clearly
detectable maximum probability of about 1. In this case, the time until
transition probability reaches its maximum value close to 1 can be considered
as a characteristic passage time.

Addition of an edge $[4\rightarrow 2]$ (parameter $c$) increases the
oscillation period between $[1\rightarrow 3]$ and thus slows down the
evolution. However, it is clear that, in contrast to Braess’s paradox, in this
case no collective effects are observed, and the considered transport system
is not congested. The change in the period of the oscillation is associated
with a shift of eigenvalues under a perturbation due to an additional
coupling. It is mostly a resonant effect originating from quantum scattering
and interference [12, 13].

In the next section, we extend the model to introduce two-particle states. In
this case, we can expect the emergence of collective effects.

###  III.2 Two-particle quantum walks on a graph

We consider four double QDs and two single QDs which form a graph shown in
Fig. 3. The potential energies of the QD
${(V_{u},V_{d},V_{1},V_{2},E_{u},E_{d})}$ and the coupling coefficients
between the QDs $(s,l,h,c)$ are schematically shown in Fig. 3.

The initial position is a double quantum dot with indices $[0,1]$, the finial
position is a double QD with indices $[8,9]$. Initial and final QDs can be
connected to external leads which provide initialization and measurement
processes by charge-qubit operations as in [14]. Note that we consider only
evolution within the system of quantum dots (manipulation, in terms of [14]).
The initial state that we consider is a pure quantum state $|\psi(0)\rangle$,
when both particles are in the initial double QD

$\displaystyle|\psi(0)\rangle_{f}=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle-|1\rangle|0\rangle)\,,$
(1)
$\displaystyle|\psi(0)\rangle_{b}=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle+|1\rangle|0\rangle)\,.$
(2)

In the absence of interactions, it is sufficient to determine one-particle
evolution and then provide proper symmetrization thus obtaining
$|\psi(t)\rangle$. In the case of fermions, only one particle can occupy a
single quantum dot due to the Pauli principle. We also consider the same one-
particle Hamiltonian to define the evolution of a pair of non-interacting
bosons.

The characteristic of transport inefficiency is the probability $P_{\perp}(t)$
that both particles are restricted to the QDs from 0 to 7,

$P_{\perp}(t)=\mathrm{Tr}\left(\hat{P}_{(0-7)}|\psi(t)\rangle\langle\psi(t)|\right)=\langle\psi(t)|\hat{P}_{(0-7)}|\psi(t)\rangle\,,$
(3)

where the projector operator to the corresponding subspace

$\hat{P}_{(0-7)}=\sum\limits_{i,j=0}^{7}|i,j\rangle_{b,f}\langle i,j|\,,$ (4)

takes into account particle statistics.

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

Figure 4: Time dependence of the probability $P_{\perp}(t)$ for the parameters
of Tables 2 and 3, the coupling coefficient for the three paths for fermions
and bosons is $c=0.3$. The black line corresponds to two paths for fermions,
the green line corresponds to three paths for fermions, the black dotted line
corresponds to two paths for bosons and the brown dotted line corresponds to
three paths for bosons.

Two numerical examples of $P_{\perp}(t)$ are given in Figs. 4a and 4b.
Parameters which correspond to Figs. 4a and 4b are shown in the Table 2 and
Table 3, respectively.

Table 2: First system parameters  l | h | s | c | $V_{u}|$ | $V_{d}|$ | $V_{1}|$ | $V_{2}|$ | $E_{u}|$ | $E_{d}$
---|---|---|---|---|---|---|---|---|---
0.1 | 0.2 | 0.25 | $\Delta$ | 0.5
Table 3: Second system parameters  l | h | s | c | $V_{u}|$ | $V_{d}|$ | $V_{1}|$ | $V_{2}|$ | $E_{u}|$ | $E_{d}$
---|---|---|---|---|---|---|---|---|---
0.04 | 0.05 | 0.25 | $\Delta$ | 0

We do not consider the following evolution, since it reduces to one particle
case and possible time delays have a purely resonant character. When
$P_{\perp}(t)$ becomes small, at least one particle reaches final QD and can
be measured there [14]. The first non-intuitive result which is seen in these
figures is that a pair of fermions will pass both quantum graphs faster
$P_{\perp,f(t)}\leq P_{\perp,b(t)}$, despite the fact that there are single-
particle channels where only one fermion can pass. However this has a simple
explanation, because the dimension of the two-particle subspace
$V_{b}=\text{Span}\\{|i,j\rangle_{b},i,j=0,7\\}$ of bosons is larger than the
dimension of the two-particle subspace of fermions,
$V_{f}=\text{Span}\\{|i,j\rangle_{f},i,j=0,7\\}$,
$\mathrm{dim}V_{b}=36>\mathrm{dim}V_{f}=28$. Note that boson and fermion
Hamiltonians coincide in our model, and taking into account that the maximum
number of mutually orthogonal states that the system can pass through per unit
of time is defined by the average energy E (above the ground state) [15], the
inequality $P_{\perp,f(t)}\leq P_{\perp,b(t)}$ follows.

Probabilities $P_{\perp}(t)$ demonstrate chaotic behavior and it is difficult
to compare which quantum graph has higher transport efficiency. An important
issue which appears when we consider quantum transport is related to the
probabilistic nature of quantum states. In a simple graph considered earlier
we saw almost periodic oscillations of probabilities. For instance, periodic
dynamics in a ring of quantum dots were observed with up to 3 pairs of quantum
dots in a ring [11]. In this case, the period of oscillations serves as a
characteristic time of quantum evolution. In a more general case of quasi-
periodic and chaotic dynamics of probabilities, there arises the problem of
characteristic time. We propose the following heuristic functional $\tau$ of
the quantum evolution

$\lambda=\frac{1}{T}\int\limits_{0}^{T}\ln(P_{\perp}(t))dt\,,\qquad\tau=\frac{\ln
2}{\lambda}\,,$ (5)

which is called half-passage time. If one considers an ensemble of identical
quantum graphs with the same two-particle initial state, then the half-passage
time corresponds to the time, when measurement in the final point will detect
at least one particle in half of the graphs from the ensemble.

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

Figure 5: Dependence of the ”half-passage” time $\tau$ on the coupling
coefficient $c$ along the third path of the systems (Tables 2, 3 ). The blue
horizontal dashed line - two paths for fermions, $c=0$. The orange dashed line
- three paths for fermions. The green horizontal dotted line - two paths for
bosons $c=0$. The red dotted line - three paths for bosons.

In Figs. 5a, 5b we plot the ”half-passage” time $\tau$ as a function of the
coupling coefficient $c$, where the initial state is $|0,1\rangle$, in the
initial double quantum dot. In Fig. 5a the half-passage time increases in both
boson and fermion cases which is similar to the increasing period of
oscillations in Fig. 2. Since bosons do not have congested channels, we can
conclude that Braess-like behavior is not related to the congestion of the
network. It is mostly a single-particle resonant effect. In the case of weakly
connected quantum graph, Fig. 5b, it is seen that there occur both Braess-like
and ”normal” behavior of transport efficiency measured by $\tau(c)$. Again,
large $c$ looks like Braess’s paradox, which has a resonant character (without
involving multi-particle effects). Note that half-passage time is longer in
the boson case, $\tau_{b}>\tau_{f}$, again due to the larger dimension of the
corresponding two-particle Hilbert space.

##  IV Conclusion

This paper considered a mathematical model of one-particle and two-particle
quantum walks on a graph. Numerical modeling has shown that, in the simplest
Braess topology, the inclusion of an additional coupling between QDs leads to
an increase in the oscillation period (evolution deceleration). Apparently,
the reason for the quantum analogue of Braess’s paradox is a change in the
spectrum of the system, in which bound states of an electron inside the system
slow down its evolution (i.e., it has a resonance capture character, see also
[12, 13]).

To study the effect of Pauli exclusion principle on network congestion, we
considered a two-particle evolution. In the case of quasi-periodic and chaotic
time dependence of transition probabilities, half-passage time (5) can be used
to measure transport efficiency. It corresponds to the time, when measurement
at the final point will detect at least one particle in half of the graphs
from the ensemble. Our numerical experiments show that pairs of bosons or
fermions demonstrate similar transport behavior in quantum graphs. Therefore,
we have come to the conclusion that the Pauli exclusion principle for a small-
sized graph does not lead to the congestion of the system. Nevertheless, a
Braess-like behavior of transport efficiency reduction is observed both for
bosons and fermions for some parameters.

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