$Q ¯ Q$ ( $Q \in {b, c}$ ) spectroscopy using Cornell potential

N. R. Soni nrsoni-apphy@msubaroda.ac.in B. R. Joshi brijaljoshi99@gmail.com R. P. Shah shahradhika61@gmail.com H. R. Chauhan hemangichauhan29@gmail.com J. N. Pandya jnpandya-apphy@msubaroda.ac.in Applied Physics Department, Faculty of Technology and Engineering,
The Maharaja Sayajirao University of Baroda, Vadodara 390001, Gujarat, India.

July 26, 2019

Abstract

The mass spectra and decay properties of heavy quarkonia are computed in nonrelativistic quark-antiquark Cornell potential model. We have employed the numerical solution of Schrödinger equation to obtain their mass spectra using only four parameters namely quark mass ( $m_{c}$ , $m_{b}$ ) and confinement strength ( $A_{c ¯ c}$ , $A_{b ¯ b}$ ). The spin hyperfine, spin-orbit and tensor components of the one gluon exchange interaction are computed perturbatively to determine the mass spectra of excited $S$ , $P$ , $D$ and $F$ states. Digamma, digluon and dilepton decays of these mesons are computed using the model parameters and numerical wave functions. The predicted spectroscopy and decay properties for quarkonia are found to be consistent with available experimental observations and results from other theoretical models. We also compute mass spectra and life time of the $B_{c}$ meson without additional parameters. The computed electromagnetic transition widths of heavy quarkonia and $B_{c}$ mesons are in tune with available experimental data and other theoretical approaches.

Cornell potential, decay properties, electromagnetic transition

pacs:

12.38.Bx; 12.39.Pn; 13.20.Gd; 13.40.Hq; 14.40.Pq

I Introduction

Mesonic bound states having both heavy quark and anti-quark ( $c ¯ c$ , $b ¯ b$ and $c ¯ b$ ) are among the best tools for understanding the quantum chromodynamics. Many experimental groups such as CLEO, LEP, CDF/D0 and NA50 have provided data and BABAR, Belle, CLEO-III, ATLAS, CMS and HERA-B are producing and expected to produce more precise data in upcoming experiments. Comprehensive reviews on the status of experimental heavy quarkonium physics are found in literature Eichten:2007; Godfrey:2008; Barnes:2009; Brambilla:2010; Brambilla:2014; Andronic:2016.

Within open flavor threshold, the heavy quarkonia have very rich spectroscopy with narrow and experimentally characterized states. The potential between the interacting quarks within the hadrons demands the understanding of underlying physics of strong interactions. In PDG pdg2016, large amount of experimental data is available for masses along with different decay modes. There are many theoretical groups viz. the lattice quantum chromodynamics (LQCD) Dudek:2007; Meinel:2009; Burch:2009; Liu:2012; McNeile:2012; Daldrop:2011; Kawanai:2013; Kawanai:2011; Burnier:2015; Kalinowski:2015; Burnier:2016, QCD Hilger:2014; Voloshin:2007, QCD sum rules Cho:2014; Gershtein:1995, perturbative QCD Kiyo:2013, lattice NRQCD Liu:2016; Dowdall:2013 and effective field theories Neubert:1993 that have attempted to explain the production and decays of these states. Others include phenomenological potential models such as the relativistic quark model based on quasi-potential approach Ebert:2011; Ebert:2009; Ebert:2005; Ebert:2002; Ebert:2003lepton; Ebert:2003gamma, where the relativistic quasi potential including one loop radiative corrections reproduce the mass spectrum of quarkonium states. The quasi potential has also been employed along with leading order radiative correction to heavy quark potential Gupta:1981; Gupta:1982kp; Gupta:1982; Pantaleone:1985, relativistic potential model Maung:1993; Radford:2007; Radford:2009 as well as semi relativistic potential model Gupta:1986. In nonrelativistic potential models, there exist several forms of quark anti-quark potentials in the literature. The most common among them is the coulomb repulsive plus quark confinement interaction potential. In our previous work Vinodkumar:1999; Pandya:2001; Rai:2008; Pandya:2014, we have employed the confinement scheme based on harmonic approximation along with Lorentz scalar plus vector potential. The authors of Devlani:2014; Parmar:2010; Rai:2002; Rai:2005; Rai:2006; Rai:2008prc; Patel:2008 have considered the confinement of power potential $A r^{ν}$ with $ν$ varying from 0.1 to 2.0 and the confinement strength $A$ to vary with potential index $ν$ . Confinement of the order $r^{2 / 3}$ have also been attempted FabreDeLaRipelle:1988. Linear confinement (Cornell potential) of quarks has been considered by many groups Eichten:1974; Eichten:1978; Eichten:1979; Quigg:1979; Eichten:1980; Barnes:2005; Sauli:2011; Leitao:2014; Godfrey:1985; Godfrey:2004; Godfrey:2015; Deng:2016cc; Deng:2016bb and they have provided good agreement with the experimental data for quarkonium spectroscopy along with decay properties. The Bethe-Salpeter approach was also employed for the mass spectroscopy of charmonia and bottomonia Sauli:2011; Leitao:2014; Fischer:2014. The quarkonium mass spectrum was also computed in the nonrelativistic quark model Lakhina:2006, screened potential model Deng:2016cc; Deng:2016bb and constituent quark model Segovia:2016. There are also other non-linear potential models that predict the mass spectra of the heavy quarkonia successfully Patel:2015; Bonati:2015; Gutsche:2014; Li:2009; Li:2009bb; Quigg:1977; Martin:1980; Buchmuller:1980.

In 90’s, the nonrelativistic potential models predicted not only the ground state mass of the tightly bound state of $c$ and $¯ b$ in the range of 6.2–6.3 GeV Kwong:1990; Eichten:1994 but also predicted to have very rich spectroscopy. In 1998, CDF collaboration Abe:1998 reported $B_{c}$ mesons in $p ¯ p$ collisions at $\sqrt{s}$ = 1.8 TeV and was later confirmed by D0 Abazov:2008 and LHCb Aaij:2012 collaborations. The LHCb collaboration has also made the most precise measurement of the life time of $B_{c}$ mesons Aaij:2014. The first excited state is also reported by ATLAS Collaborations Aad:2014 in $p ¯ p$ collisions with significance of $5.2 σ$ .

It is important to show that any given potential model should be able to compute mass spectra and decay properties of $B_{c}$ meson using parameters fitted for heavy quarkonia. Attempts in this direction have been made in relativistic quark model based on quasi-potential along with one loop radiative correction Ebert:2011, quasistatic and confinement QCD potential with confinement parameters along with quark masses Gupta:1996 and rainbow ladder approximation of Dyson-Schwinger and Bethe-Salpeter equations Fischer:2014.

Moreover, the mesonic states are identified with masses along with certain decay channels, therefore the test for any successful theoretical model is to reproduce the mass spectrum along with decay properties. Relativistic as well as nonrelativistic potential models have successfully predicted the spectroscopy but they are found to differ in computation of the decay properties Quigg:1977; Eichten:1978; Martin:1980; Buchmuller:1980; Gershtein:1995; Rai:2002; Rai:2005; Rai:2006; Rai:2008prc; Parmar:2010. This discrepancy motivates us to employ nonrelativistic potential of the one gluon exchange (essentially Coulomb like) plus linear confinement (Cornell potential) as this form of the potential is also supported by LQCD Bali:2000; Bali:2001; Alexandrou:2002. We solve the Schrödinger equation numerically for the potential to get the spectroscopy of the quarkonia. We first compute the mass spectra of charmonia and bottomonia states to determine quark masses and confinement strengths after fitting the spin-averaged ground state masses with experimental data of respective mesons. Using the potential parameters and numerical wave function, we compute the decay properties such as leptonic decay constants, digamma, dilepton, digluon decay width using the Van-Royen Weiskopf formula. These parameters are then used to compute the mass spectra and life-time of $B_{c}$ meson. We also compute the electromagnetic ( $E 1$ and $M 1$ ) transition widths of heavy quarkonia and $B_{c}$ mesons.

Ii Methodology

Basically, the bound state of two body system within relativistic quantum field is described in Bethe-Salpeter formalism. But the Bethe-Salpeter equation is solved only in the ladder approximations. Also Bethe-Salpeter approach in harmonic confinement is successful in low flavor sectors Isgur:1978; VijayaKumar:1997. Therefore the alternative treatment for the heavy bound state is nonrelativistic. Also due to negligible momenta of quark and anti quark compared to mass of quark-antiquark system $m_{Q, ¯ Q} ≫ Λ_{Q C D} \sim | \to p |$ , which constitutes the basis of the nonrelativistic treatment for the heavy quarkonium spectroscopy. Here for the study of heavy bound state of mesons such as $c ¯ c$ , $c ¯ b$ and $b ¯ b$ , the nonrelativistic Hamiltonian is given by

H = M + \frac{p^{2}}{2 M_{c m}} + V_{C o r n e l l} (r) + V_{S D} (r)

(1)

where

M = m_{Q} + m_{¯ Q} and M_{c m} = \frac{m_{Q} m_{¯ Q}}{m_{Q} + m_{¯ Q}}

(2)

where $m_{Q}$ and $m_{¯ Q}$ are the mass of quark and antiquark, $\to p$ is the relative momentum of the each quark and $V_{c o r n e l l} (r)$ is the quark-antiquark potential of the type coulomb plus linear confinement (Cornell potential) given by

V_{C o r n e l l} (r) = - \frac{4}{3} \frac{α_{s}}{r} + A r .

(3)

Here, $1 / r$ term is analogous to the Coulomb type interaction corresponding to the potential induced between quark and antiquark through one gluon exchange that dominates at small distances. The second term is the confinement part of the potential where the confinement strength $A$ is the model parameter. The confinement term becomes dominant at the large distances. $α_{s}$ is a strong running coupling constant and can be computed as

α_{s} (μ^{2}) = \frac{4 π}{(11 - \frac{2}{3} n_{f}) ln (μ^{2} / Λ^{2})}

(4)

where $n_{f}$ is the numbers of flavors, $μ$ is renormalization scale related to the constituent quark masses as $μ = 2 m_{Q} m_{¯ Q} / (m_{Q} + m_{¯ Q})$ and $Λ$ is a QCD scale which is taken as 0.15 GeV by fixing $α_{s}$ = 0.1185 pdg2016 at the $Z$ -boson mass.

The confinement strengths with respective quark masses are fine tuned to reproduce the experimental spin averaged ground state masses of both $c ¯ c$ and $b ¯ b$ mesons and are given in Table 1. We compute the masses of higher excited states without any additional parameters. Similar kind of work has been done by Patel:2008; Rai:2008prc; Parmar:2010 and they have considered different values of confinement strengths for different potential indices. The Cornell potential has been shown to be independently successful in computing the spectroscopy of $ψ$ and $Υ$ families. In this article, we compute the mass spectra of the $ψ$ and $Υ$ families along with $B_{c}$ meson with minimum number of parameters.

Using the parameters defined in Table 1, we compute the spin averaged masses of quarkonia. In order to compute masses of different $n^{m} L_{J}$ states according to different $J^{P C}$ values, we use the spin dependent part of one gluon exchange potential (OGEP) $V_{S D} (r)$ perturbatively. The OGEP includes spin-spin, spin-orbit and tensor terms given by Gershtein:1995; Barnes:2005; Lakhina:2006; Voloshin:2007 {dmath} V_SD (r) = V_SS (r) [S(S+1) - 32] + V_LS(r) (→L⋅→S) + V_T(r) [S(S+1)-3(S⋅^r) (S⋅^r)]

$m_{c}$	$m_{c}$	$A_{c c}$	$A_{b b}$
1.317 GeV	4.584 GeV	0.18 GeV $^{2}$	0.25 GeV $^{2}$

Table 1: Parameters for quarkonium spectroscopy

The spin-spin interaction term gives the hyper-fine splitting while spin-orbit and tensor terms gives the fine structure of the quarkonium states. The coefficients of spin dependent terms of the Eq. (II) can be written as Voloshin:2007

State	Present	PDG pdg2016	Ebert:2011	Deng:2016cc	Patel:2015	Fischer:2014	LQCD Kalinowski:2015
$1^{1} S_{0}$	2.989	2.984	2.981	2.984	2.979	2.925	2.884
$1^{3} S_{1}$	3.094	3.097	3.096	3.097	3.096	3.113	3.056
$2^{1} S_{0}$	3.602	3.639	3.635	3.637	3.600	3.684	3.535
$2^{3} S_{1}$	3.681	3.686	3.685	3.679	3.680	3.676	3.662
$3^{1} S_{0}$	4.058	4.039	3.989	4.004	4.011	–	–
$3^{3} S_{1}$	4.129	–	4.039	4.030	4.077	3.803	–
$4^{1} S_{0}$	4.448	4.421	4.401	4.264	4.397	–	–
$4^{3} S_{1}$	4.514	–	4.427	4.281	4.454	–	–
$5^{1} S_{0}$	4.799	–	4.811	4.459	–	–	–
$5^{3} S_{1}$	4.863	–	4.837	4.472	–	–	–
$6^{1} S_{0}$	5.124	–	5.155	–	–	–	–
$6^{3} S_{1}$	5.185	–	5.167	–	–	–	–
$1^{3} P_{0}$	3.428	3.415	3.413	3.415	3.488	3.323	3.412
$1^{3} P_{1}$	3.468	3.511	3.511	3.521	3.514	3.489	3.480
$1^{1} P_{1}$	3.470	3.525	3.525	3.526	3.539	3.433	3.494
$1^{3} P_{2}$	3.480	3.556	3.555	3.553	3.565	3.550	3.536
$2^{3} P_{0}$	3.897	3.918	3.870	3.848	3.947	3.833	–
$2^{3} P_{1}$	3.938	–	3.906	3.914	3.972	3.672	–
$2^{1} P_{1}$	3.943	–	3.926	3.916	3.996	3.747	–
$2^{3} P_{2}$	3.955	3.927	3.949	3.937	4.021	–	4.066
$3^{3} P_{0}$	4.296	–	4.301	4.146	–	–	–
$3^{3} P_{1}$	4.338	–	4.319	4.192	–	3.912	–
$3^{1} P_{1}$	4.344	–	4.337	4.193	–	–	–
$3^{3} P_{2}$	4.358	–	4.354	4.211	–	–	–
$4^{3} P_{0}$	4.653	–	4.698	–	–	–	–
$4^{3} P_{1}$	4.696	–	4.728	–	–	–	–
$4^{1} P_{1}$	4.704	–	4.744	–	–	–	–
$4^{3} P_{2}$	4.718	–	4.763	–	–	–	–
$5^{3} P_{0}$	4.983	–	–	–	–	–	–
$5^{3} P_{1}$	5.026	–	–	–	–	–	–
$5^{1} P_{1}$	5.034	–	–	–	–	–	–
$5^{3} P_{2}$	5.049	–	–	–	–	–	–

Table 2: Mass spectrum of

S

and

P

wave charmonia (in GeV)

V_{S S} (r) = \frac{1}{3 m_{Q} m_{¯ Q}} \nabla^{2} V_{V} (r) = \frac{16 π α_{s}}{9 m_{Q} m_{¯ Q}}

(5)

V_{L S} (r) = \frac{1}{2 m_{Q} m_{¯ Q} r} (3 \frac{d V_{V} (r)}{d r} - \frac{d V_{S} (r)}{d r})

(6)

V_{T} (r) = \frac{1}{6 m_{Q} m_{¯ Q}} (3 \frac{d V_{V}^{2} (r)}{d r^{2}} - \frac{1}{r} \frac{d V_{V} (r)}{d r})

(7)

Where $V_{V} (r)$ and $V_{S} (r)$ correspond to the vector and scalar part of the Cornell potential in Eq. (3) respectively. Using all the parameters defined above, the Schrödinger equation is numerically solved using Mathematica notebook utilizing the Runge-Kutta method Lucha:1998. The computed mass spectra of heavy quarkonia and $B_{c}$ mesons are listed in Table 2–7

State	Present	Ebert:2011	Deng:2016cc	Patel:2015	Fischer:2014
$1^{3} D_{3}$	3.755	3.813	3.808	3.798	3.869
$1^{1} D_{2}$	3.765	3.807	3.805	3.796	3.739
$1^{3} D_{2}$	3.772	3.795	3.807	3.794	3.550
$1^{3} D_{1}$	3.775	3.783	3.792	3.792	–
$2^{3} D_{3}$	4.176	4.220	4.112	4.425	3.806
$2^{1} D_{2}$	4.182	4.196	4.108	4.224	–
$2^{3} D_{2}$	4.188	4.190	4.109	4.223	–
$2^{3} D_{1}$	4.188	4.105	4.095	4.222	–
$3^{3} D_{3}$	4.549	4.574	4.340	–	–
$3^{1} D_{2}$	4.553	3.549	4.336	–	–
$3^{3} D_{2}$	4.557	4.544	4.337	–	–
$3^{3} D_{1}$	4.555	4.507	4.324	–	–
$4^{3} D_{3}$	4.890	4.920	–	–	–
$4^{1} D_{2}$	4.892	4.898	–	–	–
$4^{3} D_{2}$	4.896	4.896	–	–	–
$4^{3} D_{1}$	4.891	4.857	–	–	–
$1^{3} F_{2}$	3.990	4.041	–	–	–
$1^{3} F_{3}$	4.012	4.068	–	–	3.999
$1^{1} F_{3}$	4.017	4.071	–	–	4.037
$1^{3} F_{4}$	4.036	4.093	–	–	–
$2^{3} F_{2}$	4.378	4.361	–	–	–
$2^{3} F_{3}$	4.396	4.400	–	–	–
$2^{1} F_{3}$	4.400	4.406	–	–	–
$2^{3} F_{4}$	4.415	4.434	–	–	–
$3^{3} F_{2}$	4.730	–	–	–	–
$3^{3} F_{3}$	4.746	–	–	–	–
$3^{1} F_{3}$	4.749	–	–	–	–
$3^{3} F_{4}$	4.761	–	–	–	–

Table 3: Mass spectrum of

D

and

F

wave charmonia (in GeV)

State	Present	PDG pdg2016	Godfrey:2015	Ebert:2011	Deng:2016bb	Segovia:2016	Fischer:2014
$1^{1} S_{0}$	9.428	9.398	9.402	9.398	9.390	9.455	9.414
$1^{3} S_{1}$	9.463	9.460	9.465	9.460	9.460	9.502	9.490
$2^{1} S_{0}$	9.955	9999	9.976	9.990	9.990	9.990	9.987
$2^{3} S_{1}$	9.979	10.023	10.003	10.023	10.015	10.015	10.089
$3^{1} S_{0}$	10.338	–	10.336	10.329	10.326	10.330	–
$3^{3} S_{1}$	10.359	10.355	10.354	10.355	10.343	10.349	10.327
$4^{1} S_{0}$	10.663	–	10.523	10.573	10.584	–	–
$4^{3} S_{1}$	10.683	10.579	10.635	10.586	10.597	10.607	–
$5^{1} S_{0}$	10.956	–	10.869	10.851	10.800	–	–
$5^{3} S_{1}$	10.975	10.876	10.878	10.869	10.811	10.818	–
$6^{1} S_{0}$	11.226	–	11.097	11.061	10.997	–	–
$6^{3} S_{1}$	11.243	11.019	11.102	11.088	10.988	10.995	–
$1^{3} P_{0}$	9.806	9.859	9.847	9.859	9.864	9.855	9.815
$1^{3} P_{1}$	9.819	9.893	9.876	9.892	9.903	9.874	9.842
$1^{1} P_{1}$	9.821	9.899	9.882	9.900	9.909	9.879	9.806
$1^{3} P_{2}$	9.825	9.912	9.897	9.912	9.921	9.886	9.906
$2^{3} P_{0}$	10.205	10.232	10.226	10.233	10.220	10.221	10.254
$2^{3} P_{1}$	10.217	10.255	10.246	10.255	10.249	10.236	10.120
$2^{1} P_{1}$	10.220	10.260	10.250	10.260	10.254	10.240	10.154
$2^{3} P_{2}$	10.224	10.269	10.261	10.268	10.264	10.246	–
$3^{3} P_{0}$	10.540	–	10.552	10.521	10.490	10.500	–
$3^{3} P_{1}$	10.553	–	10.538	10.541	10.515	10.513	10.303
$3^{1} P_{1}$	10.556	–	10.541	10.544	10.519	10.516	–
$3^{3} P_{2}$	10.560	–	10.550	10.550	10.528	10.521 –
$4^{3} P_{0}$	10.840	–	10.775	10.781	–	–	–
$4^{3} P_{1}$	10.853	–	10.788	10.802	–	–	–
$4^{1} P_{1}$	10.855	–	10.790	10.804	–	–	–
$4^{3} P_{2}$	10.860	–	10.798	10.812	–	–	–
$5^{3} P_{0}$	11.115	–	11.004	–	–	–	–
$5^{3} P_{1}$	11.127	–	11.014	–	–	–	–
$5^{1} P_{1}$	11.130	–	11.016	–	–	–	–
$5^{3} P_{2}$	11.135	–	11.022	–	–	–	–

Table 4: Mass spectrum of

S

and

P

wave bottomonia (in GeV)

State	Present	PDG pdg2016	Godfrey:2015	Ebert:2011	Deng:2016bb	Segovia:2016	Fischer:2014
$1^{3} D_{3}$	10.073	–	10.115	10.166	10.157	10.127	10.232
$1^{1} D_{2}$	10.074	–	10.148	10.163	10.153	10.123	10.194
$1^{3} D_{2}$	10.075	10.163	10.147	10.161	10.153	10.122	10.145
$1^{3} D_{1}$	10.074	–	10.138	10.154	10.146	10.117	–
$2^{3} D_{3}$	10.423	–	10.455	10.449	10.436	10.422	–
$2^{1} D_{2}$	10.424	–	10.450	10.445	10.432	10.419	–
$2^{3} D_{2}$	10.424	–	10.449	10.443	10.432	10.418	–
$2^{3} D_{1}$	10.423	–	10.441	10.435	10.425	10.414	–
$3^{3} D_{3}$	10.733	–	10.711	10.717	–	–	–
$3^{1} D_{2}$	10.733	–	10.706	10.713	–	–	–
$3^{3} D_{2}$	10.733	–	10.705	10.711	–	–	–
$3^{3} D_{1}$	10.731	–	10.698	10.704	–	–	–
$4^{3} D_{3}$	11.015	–	10.939	10.963	–	–	–
$4^{1} D_{2}$	11.015	–	10.935	10.959	–	–	–
$4^{3} D_{2}$	11.016	–	10.934	10.957	–	–	–
$4^{3} D_{1}$	11.013	–	10.928	10.949	–	–	–
$1^{3} F_{2}$	10.283	–	10.350	10.343	10.338	10.315	–
$1^{3} F_{3}$	10.287	–	10.355	10.346	10.340	10.321	10.302
$1^{1} F_{3}$	10.288	–	10.355	10.347	10.339	10.322	10.319
$1^{3} F_{4}$	10.291	–	10.358	10.349	10.340	–	–
$2^{3} F_{2}$	10.604	–	10.615	10.610	–	–	–
$2^{3} F_{3}$	10.607	–	10.619	10.614	–	–	–
$2^{1} F_{3}$	10.607	–	10.619	10.647	–	–	–
$2^{3} F_{4}$	10.609	–	10.622	10.617	–	–	–
$3^{3} F_{2}$	10.894	–	10.850	–	–	–	–
$3^{3} F_{3}$	10.896	–	10.853	–	–	–	–
$3^{1} F_{3}$	10.897	–	10.853	–	–	–	–
$3^{3} F_{4}$	10.898	–	10.856	–	–	–	–

Table 5: Mass spectrum of

D

and

F

wave bottomonia (in GeV)

State	Present	Devlani:2014	Ebert:2011	Godfrey:2004	Monteiro:2016	PDG pdg2016
$1^{1} S_{0}$	6.272	6.278	6.272	6.271	6.275	6.275
$1^{3} S_{1}$	6.321	6.331	6.333	6.338	6.314	–
$2^{1} S_{0}$	6.864	6.863	6.842	6.855	6.838	6.842
$2^{3} S_{1}$	6.900	6.873	6.882	6.887	6.850	–
$3^{1} S_{0}$	7.306	7.244	7.226	7.250	–	–
$3^{3} S_{1}$	7.338	7.249	7.258	7.272	–	–
$4^{1} S_{0}$	7.684	7.564	7.585	–	–	–
$4^{3} S_{1}$	7.714	7.568	7.609	–	–	–
$5^{1} S_{0}$	8.025	7.852	7.928	–	–	–
$5^{3} S_{1}$	8.054	7.855	7.947	–	–	–
$6^{1} S_{0}$	8.340	8.120	–	–	–	–
$6^{3} S_{1}$	8.368	8.122	–	–	–	–
$1^{3} P_{0}$	6.686	6.748	6.699	6.706	6.672	–
$1^{3} P_{1}$	6.705	6.767	6.750	6.741	6.766	–
$1^{1} P_{1}$	6.706	6.769	6.743	6.750	6.828	–
$1^{3} P_{2}$	6.712	6.775	6.761	6.768	6.776	–
$2^{3} P_{0}$	7.146	7.139	7.094	7.122	6.914	–
$2^{3} P_{1}$	7.165	7.155	7.134	7.145	7.259	–
$2^{1} P_{1}$	7.168	7.156	7.094	7.150	7.322	–
$2^{3} P_{2}$	7.173	7.162	7.157	7.164	7.232	–
$3^{3} P_{0}$	7.536	7.463	7.474	–	–	–
$3^{3} P_{1}$	7.555	7.479	7.510	–	–	–
$3^{1} P_{1}$	7.559	7.479	7.500	–	–	–
$3^{3} P_{2}$	7.565	7.485	7.524	–	–	–
$4^{3} P_{0}$	7.885	–	7.817	–	–	–
$4^{3} P_{1}$	7.905	–	7.853	–	–	–
$4^{1} P_{1}$	7.908	–	7.844	–	–	–
$4^{3} P_{2}$	7.915	–	7.867	–	–	–
$5^{3} P_{0}$	8.207	–	–	–	–
$5^{3} P_{1}$	8.226	–	–	–	–
$5^{1} P_{1}$	8.230	–	–	–	–
$5^{3} P_{2}$	8.237	–	–	–	–

Table 6: Mass spectrum of

S

and

P

wave

B_{c}

meson (in GeV)

State	Present	Devlani:2014	Ebert:2011	Godfrey:2004	Monteiro:2016
$1^{3} D_{3}$	6.990	7.026	7.029	7.045	6.980
$1^{1} D_{2}$	6.994	7.035	7.026	7.041	7.009
$1^{3} D_{2}$	6.997	7.025	7.025	7.036	7.154
$1^{3} D_{1}$	6.998	7.030	7.021	7.028	7.078
$2^{3} D_{3}$	7.399	7.363	7.405	–	–
$2^{1} D_{2}$	7.401	7.370	7.400	–	–
$2^{3} D_{2}$	7.403	7.361	7,399	–	–
$2^{3} D_{1}$	7.403	7.365	7.392	–	–
$3^{3} D_{3}$	7.761	–	7.750	–	–
$3^{1} D_{2}$	7.762	–	7.743	–	–
$3^{3} D_{2}$	7.764	–	7.741	–	–
$3^{3} D_{1}$	7.762	–	7.732	–	–
$4^{3} D_{3}$	8.092	–	–	–	–
$4^{1} D_{2}$	8.093	–	–	–	–
$4^{3} D_{2}$	8.094	–	–	–	–
$4^{3} D_{1}$	8.091	–	–	–	–
$1^{3} F_{2}$	7.234	–	7.273	7.269	–
$1^{3} F_{3}$	7.242	–	7.269	7.276	–
$1^{1} F_{3}$	7.241	–	7.268	7.266	–
$1^{3} F_{4}$	7.244	–	7.277	7.271	–
$2^{3} F_{2}$	7.607	–	7.618	–	–
$2^{3} F_{3}$	7.615	–	7.616	–	–
$2^{1} F_{3}$	7.614	–	7.615	–	–
$2^{3} F_{4}$	7.617	–	7.617	–	–
$3^{3} F_{2}$	7.946	–	–	–	–
$3^{3} F_{3}$	7.954	–	–	–	–
$3^{1} F_{3}$	7.953	–	–	–	–
$3^{3} F_{4}$	7.956	–	–	–	–

Table 7: Mass spectrum of

D

and

F

wave

B_{c}

meson (in GeV)

Iii Decay properties

The mass spectra of the hadronic states are experimentally determined through detection of energy and momenta of daughter particles in various decay channels. Generally, most phenomenological approaches obtain their model parameters like quark masses and confinement/Coulomb strength by fitting with the experimental ground states. So it becomes necessary for any phenomenological model to validate their fitted parameters through proper evaluation of various decay rates in general and annihilation rates in particular. In the nonrelativistic limit, the decay properties are dependent on the wave function. In this section, we test our parameters and wave functions to determine various annihilation widths and electromagnetic transitions.

iii.1 Leptonic decay constants

The leptonic decay constants of heavy quarkonia play very important role in understanding the weak decays. The matrix elements for leptonic decay constants of pseudoscalar and vector mesons are given by

⟨ 0 | ¯ Q γ^{μ} γ_{5} Q | P_{μ} (k) ⟩ = i f_{P} k^{μ}

(8)

⟨ 0 | ¯ Q γ^{μ} Q | P_{μ} (k) ⟩ = i f_{V} M_{V} ϵ^{* μ}

(9)

where $k$ is the momentum of pseudoscalar meson, $ϵ^{* μ}$ is the polarization vector of meson. In the nonrelativistic limit, the decay constants of pseudoscalar and vector mesons are given by Van Royen-Weiskopf formula VanRoyen:1967

f_{P / V}^{2} = \frac{3 | R_{n s P / V} (0) |^{2}}{π M_{n s P / V}} {¯ C}^{2} (α_{S}) .

(10)

Here the QCD correction factor ${¯ C}^{2} (α_{S})$ Braaten:1995; Berezhnoy:1996

{¯ C}^{2} (α_{S}) = 1 - \frac{α_{s}}{π} (δ^{P, V} - \frac{m_{Q} - m_{¯ Q}}{m_{Q} + m_{¯ Q}} ln \frac{m_{Q}}{m_{¯ Q}}) .

(11)

With $δ^{P}$ = 2 and $δ^{V}$ = 8/3. Using the above relations, we compute the leptonic decay constants $f_{p}$ and $f_{v}$ for charmonia, bottomonia and $B_{c}$ mesons and are listed in Table 8 – 13.

State	$f_{p}$	$f_{p} (c o r r)$	Patel:2008	Krassnigg:2016	Lakhina:2006	LQCDBecirevic:2013	QCDSR Becirevic:2013
$1 S$	440.406	350.314	363	378	402	387(7)(2)	309 $\pm$ 39
$2 S$	350.056	278.447	275	82	240	–	–
$3 S$	313.354	249.253	239	206	193	–	–
$4 S$	290.673	231.211	217	82	–	–	–
$5 S$	274.367	218.241	202	–	–	–	–
$6 S$	261.698	208.163	197	–	–	–	–

Table 8: Pseudoscalar decay constant of charmonia (in MeV)

State	$f_{v}$	$f_{v} (c o r r)$	Patel:2008	Krassnigg:2016	Lakhina:2006	LQCDBecirevic:2013	QCDSR Becirevic:2013
$1 S$	448.096	325.876	338	411	393	418(8)(5)	401 $\pm$ 46
$2 S$	353.855	257.34	254	155	293	–	–
$3 S$	316.064	229.857	220	188	258	–	–
$4 S$	292.829	212.959	200	262	–	–	–
$5 S$	276.177	200.848	186	–	–	–	–
$6 S$	263.266	191.459	175	–	–	–	–

Table 9: Vector decay constant of charmonia (in MeV)

State	$f_{p}$	Patel:2008	Krassnigg:2016	Pandya:2001	Lakhina:2006
$1 S$	646.025	744	756	711	599
$2 S$	518.803	577	285	–	411
$3 S$	474.954	511	333	–	354
$4 S$	449.654	471	40	–	–
$5 S$	432.072	443	–	–	–
$6 S$	418.645	422	–	–	–

Table 10: Pseudoscalar decay constant of bottomonia (in MeV)

State	$f_{v}$	Patel:2008	Krassnigg:2016	Lakhina:2006	Wang:2005	LQCDColquhoun:2014
$1 S$	647.250	706	707	665	498 $\pm (20)$	649(31)
$2 S$	519.436	547	393	475	366 $\pm (27)$	481(39)
$3 S$	475.440	484	9	418	304 $\pm (27)$	–
$4 S$	450.066	446	20	388	259 $\pm (22)$	–
$5 S$	432.437	419	–	367	228 $\pm (16)$	–
$6 S$	418.977	399	–	351	–	–

Table 11: Vector decay constant of bottomonia (in MeV)

State	$f_{p}$	Patel:2008	Ebert:2002	Gershtein:1995	Eichten:1994	Monteiro:2016
$1 S$	432.955	465	503	460 $\pm (60)$	500	554.125
$2 S$	355.504	361	–	–	–
$3 S$	325.659	319	–	–	–
$4 S$	307.492	293	–	–	–
$5 S$	294.434	275	–	–	–
$6 S$	284.237	261	–	–	–

Table 12: Pseudoscalar decay constant of

B_{c}

meson (in MeV)

State	$f_{v}$	Patel:2008	Ebert:2002	Gershtein:1995	Eichten:1994
$1 S$	434.642	435	433	460 $\pm (60)$	500
$2 S$	356.435	337	–	–	–
$3 S$	326.374	297	–	–	–
$4 S$	308.094	273	–	–	–
$5 S$	294.962	256	–	–	–
$6 S$	284.709	243	–	–	–

Table 13: Vector decay constant of

B_{c}

meson (in MeV)

iii.2 Annihilation of heavy quarkonia

Digamma, digluon and dilepton annihilation decay rates of heavy quarkonia are very important in understanding the dynamics of heavy quarks within the mesons. The measurement of digamma decay widths provides the information regarding the internal structure of meson. The decay $η_{c} \to γ γ$ , $χ_{c 0, 2} \to γ γ$ was reported by CLEO-c Ecklund:2008, BABAR Lees:2010 and then BESIII Ablikim:2012 collaboration have reported high accuracy data. LQCD is found to underestimate the decay widths of $η_{c} \to γ γ$ and $χ_{c 0} \to γ γ$ when compared to experimental data Dudek:2006; ChenT:2016. Other approaches to attempt computation of annihilation rates of heavy quarkonia include NRQCD Bodwin:1994; Khan:1995; Schuler:1997; Bodwin:2006; Bodwin:2007, relativistic quark model Ebert:2003gamma; Ebert:2003lepton, effective Lagrangian Lansberg:2009; Lansberg:2006 and next-to-next-to leading order QCD correction to $χ_{c 0, 2} \to γ γ$ in the framework of nonrelativistic QCD factorization Sang:2015.

The meson decaying into digamma suggests that the spin can never be one Landau:1948; Yang:1950. Corresponding digamma decay width of a pseudoscalar meson in nonrelativistic limit is given by Van Royen-Weiskopf formula VanRoyen:1967

Γ_{n^{1} S_{0} \to γ γ} = \frac{12 α_{e}^{2} e_{Q}^{4} | n s P (0) |^{2}}{M_{η_{Q}}^{2}} [1 + \frac{α_{s}}{π} (\frac{π^{2} - 20}{3})]

(12)

Γ_{n^{3} P_{0} \to γ γ} = \frac{27 α_{e}^{2} | R_{n P}^{'} (0) |^{2} e_{Q}^{4}}{M_{χ_{Q 0}}^{4}} [1 + \frac{α_{s}}{π} (\frac{3 π^{2} - 28}{9})]

(13)

Γ_{n^{3} P_{2} \to γ γ} = \frac{36 α_{e}^{2} | R_{n P}^{'} (0) |^{2} e_{Q}^{4}}{M_{χ_{c 2}}^{4}} [1 - \frac{16}{3} \frac{α_{s}}{π}]

(14)

where the bracketed quantities are QCD next-to-leading order radiative corrections Kwong:1988; Barbieri:1981.

Digluon annihilation of quarkonia is not directly observed in detectors as digluonic state decays into various hadronic states making it a bit complex to compute digluon annihilation widths from nonrelativistic approximations derived from first principles. The digluon decay width of pseudoscalar meson along with the QCD leading order radiative correction is given by Lansberg:2009; Kwong:1988; Barbieri:1981; Mangano:1995

Γ_{n^{1} S_{0} \to g g} = \frac{α_{s}^{2} M_{η_{Q}} | R_{n s P} (0) |^{2}}{3 m_{Q}^{3}} [1 + C_{Q} (α_{s} / π)]

(15)

Γ_{n^{3} P_{0} \to g g} = \frac{3 α_{s}^{2} M_{χ_{c 0}} | R_{n P}^{'} (0) |^{2}}{m_{Q}^{5}} [1 + C_{0 Q} (α_{s} / π)]

(16)

Γ_{n^{3} P_{2} \to g g} = \frac{4}{15} \frac{3 α_{s}^{2} M_{χ_{c 2}} | R_{n P}^{'} (0) |^{2}}{m_{Q}^{5}} [1 + C_{2 Q} (α_{s} / π)]

(17)

The vector mesons have quantum numbers $1^{- -}$ and can annihilate into dilepton. The dileptonic decay of vector meson along with one loop QCD radiative correction is given by VanRoyen:1967; Kwong:1988

Γ_{n^{3} S_{1} \to ℓ^{+} ℓ^{-}} = \frac{4 α_{e}^{2} e_{Q}^{2} | n s V (0) |^{2}}{M_{V}^{2}} [1 - \frac{16 α_{s}}{3 π}]

(18)

Here, $α_{e}$ is the electromagnetic coupling constant, $α_{s}$ is the strong running coupling constant in Eq. (4) and $e_{Q}$ is the charge of heavy quark in terms of electron charge. In above relations, $| R_{n s P / V} (0) |$ corresponds to the wave function of $S$ wave at origin for pseudoscalar and vector mesons while $| R_{n P}^{'} (0) |$ is the derivative of $P$ wave wave function at origin. The annihilation rates of heavy quarkonia are listed in Table 14 - 19.

State	$Γ_{γ γ}$	$Γ_{γ γ} (c o r r)$	Li:2009	Ebert:2003gamma	Lakhina:2006	Kim:2004	PDG pdg2016
$1^{1} S_{0}$	8.582	5.618	8.5	5.5	7.18	7.14 $\pm$ 0.95	5.1 $\pm$ 0.4
$2^{1} S_{0}$	4.498	2.944	2.4	1.8	1.71	4.44 $\pm$ 0.48	2.15 $\pm$ 1.58
$3^{1} S_{0}$	3.200	2.095	0.88	–	1.21	–	–
$4^{1} S_{0}$	2.512	1.644	–	–	–	–	–
$5^{1} S_{0}$	2.074	1.358	–	–	–	–	–
$6^{1} S_{0}$	1.768	1.158	–	–	–	–	–
$1^{3} P_{0}$	11.477	11.687	2.5	2.9	3.28	–	2.34 $\pm$ 0.19
$1^{3} P_{2}$	3.107	1.412	0.31	0.50	–	–	0.53 $\pm$ 0.4
$2^{3} P_{0}$	13.235	13.477	1.7	1.9	–	–	–
$2^{3} P_{2}$	3.582	1.628	0.23	0.52	–	–	–
$3^{3} P_{0}$	14.580	14.847	1.2	–	–	–	–
$3^{3} P_{2}$	3.944	1.792	0.17	–	–	–	–
$4^{3} P_{0}$	15.746	16.034	–	–	–	–	–
$4^{3} P_{2}$	4.257	1.935	–	–	–	–	–
$5^{3} P_{0}$	16.807	17.114	–	–	–	–	–
$5^{3} P_{2}$	4.541	2.064	–	–	–	–	–

Table 14: Digamma decay width of

S

and

P

wave charmonia (in keV)

State	$Γ_{γ γ}$	Li:2009bb	Godfrey:1985	Ebert:2003gamma	Lakhina:2006	Kim:2004
$1^{1} S_{0}$	0.366	0.527	0.214	0.35	0.23	0.384 $\pm$ 0.047
$2^{1} S_{0}$	0.223	0.263	0.121	0.15	0.07	0.191 $\pm$ 0.025
$3^{1} S_{0}$	0.180	0.172	0.906	0.10	0.04	–
$4^{1} S_{0}$	0.157	0.105	0.755	–	–	–
$5^{1} S_{0}$	0.141	0.121	–	–	–	–
$6^{1} S_{0}$	0.129	0.050	–	–	–	–
$1^{3} P_{0}$	0.021	0.050	0.0208	0.038	–	–
$1^{3} P_{2}$	0.0060	0.0066	0.0051	0.008	–	–
$2^{3} P_{0}$	0.022	0.037	0.0227	0.029	–	–
$2^{3} P_{2}$	0.0058	0.0067	0.0062	0.006	–	–
$3^{3} P_{0}$	0.022	0.037	–	–	–	–
$3^{3} P_{2}$	0.0059	0.0064	–	–	–	–
$4^{3} P_{0}$	0.023	–	–	–	–	–
$4^{3} P_{2}$	0.0061	–	–	–	–	–
$5^{3} P_{0}$	0.023	–	–	–	–	–
$5^{3} P_{2}$	0.0062	–	–	–	–	–

Table 15: Digamma decay width of

S

and

P

wave bottomonia (in keV)

State	$Γ_{g g}$	$Γ_{g g} (c o r r)$	Patel:2015	Kim:2004	PDG pdg2016
$1^{1} S_{0}$	27.325	40.741	22.37	19.60	26.7 $\pm$ 3.0
$2^{1} S_{0}$	25.081	37.394	16.74	12.1	14.7 $\pm$ 0.7
$3^{1} S_{0}$	25.502	38.022	14.30	–	–
$4^{1} S_{0}$	26.365	39.310	–	–	–
$5^{1} S_{0}$	27.345	40.711	–	–	–
$6^{1} S_{0}$	28.353	42.273	–	–	–
$1^{3} P_{0}$	25.022	47.468	9.45	–	10.4 $\pm$ 0.7
$1^{3} P_{2}$	6.775	3.430	2.81	–	2.03 $\pm$ 0.12
$2^{3} P_{0}$	28.855	54.738	10.09	–	–
$2^{3} P_{2}$	7.810	3.954	7.34	–	–
$3^{3} P_{0}$	31.788	60.302	–	–	–
$3^{3} P_{2}$	8.599	4.353	–	–	–
$4^{3} P_{0}$	34.330	65.125	–	–	–
$4^{3} P_{2}$	9.281	4.699	–	–	–
$5^{3} P_{0}$	36.642	69.511	–	–	–
$5^{3} P_{2}$	9.900	5.012	–	–	–

Table 16: Digluon decay width of

S

and

P

wave charmonia (in MeV)

State	$Γ_{g g}$	Parmar:2010	Gupta:1996gg
$1^{1} S_{0}$	5.602	17.945	12.46
$2^{1} S_{0}$	4.029	–	–
$3^{1} S_{0}$	3.641	–	–
$4^{1} S_{0}$	3.473	–	–
$5^{1} S_{0}$	3.385	–	–
$6^{1} S_{0}$	3.336	–	–
$1^{3} P_{0}$	0.295	5.250	2.15
$1^{3} P_{2}$	0.079	0.822	0.22
$2^{3} P_{0}$	0.306	–	–
$2^{3} P_{2}$	0.082	–	–
$3^{3} P_{0}$	0.314	–	–
$3^{3} P_{2}$	0.084	–	–
$4^{3} P_{0}$	0.320	–	–
$4^{3} P_{2}$	0.085	–	–
$5^{3} P_{0}$	0.326	–	–
$5^{3} P_{2}$	0.087	–	–

Table 17: Digluon decay width of

S

and

P

wave bottomonia (in MeV)

State	$Γ_{ℓ^{+} ℓ^{-}}$	$Γ_{ℓ^{+} ℓ^{-}} (c o r r)$	Shah:2012	Patel:2008	Radford:2007	Ebert:2003lepton	PDG pdg2016
$1 S$	6.437	2.925	4.95	6.99	1.89	5.4	5.547 $\pm$ 0.14
$2 S$	3.374	1.533	1.69	3.38	1.04	2.4	2.359 $\pm$ 0.04
$3 S$	2.400	1.091	0.96	2.31	0.77	–	0.86 $\pm$ 0.07
$4 S$	1.884	0.856	0.65	1.78	0.65	–	0.58 $\pm$ 0.07
$5 S$	1.556	0.707	0.49	1.46	–	–	–
$6 S$	1.326	0.602	0.39	1.24	–	–	–

Table 18: Dilepton decay width of charmonia (in kev)

State	$Γ_{ℓ^{+} ℓ^{-}}$	Shah:2012	Radford:2009	Patel:2008	Ebert:2003lepton	Gonzalez:2003	PDGpdg2016
$1 S$	1.098	1.20	1.33	1.61	1.3	0.98	1.340 $\pm$ 0.018
$2 S$	0.670	0.52	0.62	0.87	0.5	0.41	0.612 $\pm$ 0.011
$3 S$	0.541	0.33	0.48	0.66	–	0.27	0.443 $\pm$ 0.008
$4 S$	0.470	0.24	0.40	0.53	–	0.20	0.272 $\pm$ 0.029
$5 S$	0.422	0.19	–	0.44	–	0.16	–
$6 S$	0.387	0.16	–	0.39	–	0.12	–

Table 19: Dilepton decay width of bottomonia (in keV)

iii.3 Electromagnetic transition widths

The electromagnetic transitions can be determined broadly in terms of electric and magnetic multipole expansions and their study can help in understanding the non-perturbative regime of QCD. We consider the leading order terms i.e. electric ( $E 1$ ) and magnetic ( $M 1$ ) dipoles with selection rules $Δ L = \pm 1$ and $Δ S = 0$ for the $E 1$ transitions while $Δ L = 0$ and $Δ S = \pm 1$ for $M 1$ transitions. We now employ the numerical wave function for computing the electromagnetic transition widths among quarkonia and $B_{c}$ meson states in order to test parameters used in present work. For $M 1$ transition, we restrict our calculations for transitions among $S$ waves only. In the nonrelativistic limit, the radiative $E 1$ and $M 1$ widths are given by Eichten:1974; Eichten:1978; Radford:2009; Brambilla:2010; Li:2010 {dmath} Γ(n^2S+1L_iJ_i →n^,2S+1L_fJ_f + γ) = 4 αe⟨eQ⟩2ω33 (2 J_f + 1) S_if^E1 —M_if^E1—^2 {dmath} Γ(n^3S_1 →n^,1S_0+ γ) = αeμ2ω33 (2 J_f + 1) S_if^M1 —M_if^M1—^2 where, mean charge content $⟨ e_{Q} ⟩$ of the $Q ¯ Q$ system, magnetic dipole moment $μ$ and photon energy $ω$ are given by

⟨ e_{Q} ⟩ = ∣ ∣ ∣ \frac{m_{¯ Q} e_{Q} - e_{¯ Q} m_{Q}}{m_{Q} + m_{¯ Q}} ∣ ∣ ∣

(19)

μ = \frac{e_{Q}}{m_{Q}} - \frac{e_{¯ Q}}{m_{¯ Q}}

(20)

and

ω = \frac{M_{i}^{2} - M_{f}^{2}}{2 M_{i}}

(21)

respectively. Also the symmetric statistical factors are given by

S_{i f}^{E 1} = m a x (L_{i}, L_{f}) {\begin{matrix} J_{i} & 1 & J_{f} L_{f} & S & L_{i} \end{matrix}}^{2}

(22)

and

S_{i f}^{M 1} = 6 (2 S_{i} + 1) (2 S_{f} + 1) {\begin{matrix} J_{i} & 1 & J_{f} S_{f} & ℓ & S_{i} \end{matrix}}^{2} {\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \frac{1}{2} & S_{f} & S_{i} \end{matrix}}^{2} .

(23)

The matrix element $| M_{i f} |$ for $E 1$ and $M 1$ transition can be written as

∣ ∣ M_{i f}^{E 1} ∣ ∣ = \frac{3}{ω} ⟨ f ∣ ∣ ∣ \frac{ω r}{2} j_{0} (\frac{ω r}{2}) - j_{1} (\frac{ω r}{2}) ∣ ∣ ∣ i ⟩

(24)

and

∣ ∣ M_{i f}^{M 1} ∣ ∣ = ⟨ f ∣ ∣ ∣ j_{0} (\frac{ω r}{2}) ∣ ∣ ∣ i ⟩

(25)

The electromagnetic transition widths are listed in Table 20 - 25 and also compared with experimental results as well as theoretical predictions.

Transition	Present	Radford:2007	Ebert:2002	Li:2009	Deng:2016cc	PDG pdg2016
$2^{3} S_{1} \to 1^{3} P_{0}$	21.863	45.0	51.7	74	22	26.3 $\pm$ 2.6
$2^{3} S_{1} \to 1^{3} P_{1}$	43.292	40.9	44.9	62	42	25.5 $\pm$ 2.8
$2^{3} S_{1} \to 1^{3} P_{2}$	62.312	26.5	30.9	43	38	25.2 $\pm$ 2.9
$2^{1} S_{0} \to 1^{1} P_{1}$	36.197	8.3	8.6	146	49	–
$3^{3} S_{1} \to 2^{3} P_{0}$	31.839	87.3	–	–	–	–
$3^{3} S_{1} \to 2^{3} P_{1}$	64.234	65.7	–	–	–	–
$3^{3} S_{1} \to 2^{3} P_{2}$	86.472	31.6	–	–	–	–
$3^{1} S_{0} \to 2^{1} P_{1}$	51.917	–	–	–	–	–
$3^{3} S_{1} \to 1^{3} P_{0}$	46.872	1.2	–	–	–	–
$3^{3} S_{1} \to 1^{3} P_{1}$	107.088	2.5	–	–	–	–
$3^{3} S_{1} \to 1^{3} P_{2}$	163.485	3.3	–	–	–	–
$3^{1} S_{0} \to 1^{1} P_{1}$	178.312	–	–	–	–	–
$1^{3} P_{0} \to 1^{3} S_{1}$	112.030	142.2	161	167	284	371 $\pm$ 34
$1^{3} P_{1} \to 1^{3} S_{1}$	146.317	287.0	333	354	306	285 $\pm$ 14
$1^{3} P_{2} \to 1^{3} S_{1}$	157.225	390.6	448	473	172	133 $\pm$ 8
$1^{1} P_{1} \to 1^{1} S_{0}$	247.971	610.0	723	764	361	257 $\pm$ 2.80
$2^{3} P_{0} \to 2^{3} S_{1}$	70.400	53.6	–	61	–	–
$2^{3} P_{1} \to 2^{3} S_{1}$	102.672	208.3	–	103	–	–
$2^{3} P_{2} \to 2^{3} S_{1}$	116.325	358.6	–	225	–	–
$2^{1} P_{1} \to 2^{1} S_{0}$	163.646	–	–	309	–	–
$2^{3} P_{0} \to 1^{3} S_{1}$	173.324	20.8	–	74	–	–
$2^{3} P_{1} \to 1^{3} S_{1}$	210.958	28.4	–	83	–	–
$2^{3} P_{2} \to 1^{3} S_{1}$	227.915	33.2	–	101	–	–
$2^{1} P_{1} \to 1^{1} S_{0}$	329.384	–	–	134	–	–
$1^{3} D_{1} \to 1^{3} P_{0}$	161.504	–	423	486	272	202 $\pm$ 42
$1^{3} D_{1} \to 1^{3} P_{1}$	93.775	–	142	150	138	81 $\pm$ 27
$1^{3} D_{1} \to 1^{3} P_{2}$	5.722	–	5.8	5.8	7.1	$\leq$ 24.8
$1^{3} D_{2} \to 1^{3} P_{1}$	165.176	317.3	297	342	285	–
$1^{3} D_{2} \to 1^{3} P_{2}$	50.317	65.7	62	70	91	–
$1^{3} D_{3} \to 1^{3} P_{2}$	175.212	62.7	252	284	350	–
$1^{1} D_{2} \to 1^{1} P_{1}$	205.93	–	335	575	362	–

Table 20:

E 1

transition width of charmonia (in keV)

Transition	Present	Radford:2007	Ebert:2002	Li:2009bb	Deng:2016bb	PDG pdg2016
$2^{3} S_{1} \to 1^{3} P_{0}$	2.377	1.15	1.65	1.67	1.09	1.22 $\pm$ 0.11
$2^{3} S_{1} \to 1^{3} P_{1}$	5.689	1.87	2.57	2.54	2.17	2.21 $\pm$ 0.19
$2^{3} S_{1} \to 1^{3} P_{2}$	8.486	1.88	2.53	2.62	2.62	2.29 $\pm$ 0.20
$2^{1} S_{0} \to 1^{1} P_{1}$	10.181	4.17	3.25	6.10	3.41	–
$3^{3} S_{1} \to 2^{3} P_{0}$	3.330	1.67	1.65	1.83	1.21	1.20 $\pm$ 0.12
$3^{3} S_{1} \to 2^{3} P_{1}$	7.936	2.74	2.65	2.96	2.61	2.56 $\pm$ 0.26
$3^{3} S_{1} \to 2^{3} P_{2}$	11.447	2.80	2.89	3.23	3.16	2.66 $\pm$ 0.27
$3^{3} S_{1} \to 1^{3} P_{0}$	0.594	0.03	0.124	0.07	0.097	0.055 $\pm$ 0.010
$3^{3} S_{1} \to 1^{3} P_{1}$	1.518	0.09	0.307	0.17	0.0005	0.018 $\pm$ 0.010
$3^{3} S_{1} \to 1^{3} P_{2}$	2.354	0.13	0.445	0.15	0.14	0.20 $\pm$ 0.03
$3^{1} S_{0} \to 1^{1} P_{1}$	3.385	0.03	0.770	1.24	0.67	–
$3^{1} S_{0} \to 2^{1} P_{1}$	13.981	–	3.07	11.0	4.25	–
$1^{3} P_{2} \to 1^{3} S_{1}$	57.530	31.2	29.5	38.2	31.8	–
$1^{3} P_{1} \to 1^{3} S_{1}$	54.927	27.3	37.1	33.6	31.9	–
$1^{3} P_{0} \to 1^{3} S_{1}$	49.530	22.1	42.7	26.6	27.5	–
$1^{1} P_{1} \to 1^{1} S_{0}$	72.094	37.9	54.4	55.8	35.8	–
$2^{3} P_{2} \to 2^{3} S_{1}$	28.848	16.8	18.8	18.8	15.5	15.1 $\pm$ 5.6
$2^{3} P_{1} \to 2^{3} S_{1}$	26.672	13.7	15.9	15.9

Q¯Q (Q∈{b,c}) spectroscopy using Cornell potential