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

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

N. R. Soni    B. R. Joshi    R. P. Shah    H. R. Chauhan    J. N. Pandya 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 (, ) and confinement strength (, ). The spin hyperfine, spin-orbit and tensor components of the one gluon exchange interaction are computed perturbatively to determine the mass spectra of excited , , and 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 meson without additional parameters. The computed electromagnetic transition widths of heavy quarkonia and 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 (, and ) 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 with varying from 0.1 to 2.0 and the confinement strength to vary with potential index . Confinement of the order 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 and 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 mesons in collisions at = 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 mesons Aaij:2014. The first excited state is also reported by ATLAS Collaborations Aad:2014 in collisions with significance of .

It is important to show that any given potential model should be able to compute mass spectra and decay properties of 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 meson. We also compute the electromagnetic ( and ) transition widths of heavy quarkonia and 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 , 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 , and , the nonrelativistic Hamiltonian is given by

 H=M+p22Mcm+VCornell(r)+VSD(r) (1)

where

 M=mQ+m¯Q    and    Mcm=mQm¯QmQ+m¯Q (2)

where and are the mass of quark and antiquark, is the relative momentum of the each quark and is the quark-antiquark potential of the type coulomb plus linear confinement (Cornell potential) given by

 VCornell(r)=−43αsr+Ar. (3)

Here, 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 is the model parameter. The confinement term becomes dominant at the large distances. is a strong running coupling constant and can be computed as

 αs(μ2)=4π(11−23nf)ln(μ2/Λ2) (4)

where is the numbers of flavors, is renormalization scale related to the constituent quark masses as and is a QCD scale which is taken as 0.15 GeV by fixing = 0.1185 pdg2016 at the -boson mass.

The confinement strengths with respective quark masses are fine tuned to reproduce the experimental spin averaged ground state masses of both and 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 and families. In this article, we compute the mass spectra of the and families along with 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 states according to different values, we use the spin dependent part of one gluon exchange potential (OGEP) 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)]

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

 VSS(r)=13mQm¯Q∇2VV(r)=16παs9mQm¯Q (5)
 VLS(r)=12mQm¯Qr(3dVV(r)dr−dVS(r)dr) (6)
 VT(r)=16mQm¯Q(3dV2V(r)dr2−1rdVV(r)dr) (7)

Where and 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 mesons are listed in Table 27

## 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|¯Qγμγ5Q|Pμ(k)⟩=ifPkμ (8)
 ⟨0|¯QγμQ|Pμ(k)⟩=ifVMVϵ∗μ (9)

where is the momentum of pseudoscalar meson, 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

 f2P/V=3|RnsP/V(0)|2πMnsP/V¯C2(αS). (10)

Here the QCD correction factor Braaten:1995; Berezhnoy:1996

 ¯C2(αS)=1−αsπ(δP,V−mQ−m¯QmQ+m¯QlnmQm¯Q). (11)

With = 2 and = 8/3. Using the above relations, we compute the leptonic decay constants and for charmonia, bottomonia and mesons and are listed in Table 813.

### 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 , 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 and 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 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

 Γn1S0→γγ=12α2ee4Q|nsP(0)|2M2ηQ[1+αsπ(π2−203)] (12)
 Γn3P0→γγ=27α2e|R′nP(0)|2e4QM4χQ0[1+αsπ(3π2−289)] (13)
 Γn3P2→γγ=36α2e|R′nP(0)|2e4QM4χc2[1−163αsπ] (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

 Γn1S0→gg=α2sMηQ|RnsP(0)|23m3Q[1+CQ(αs/π)] (15)
 Γn3P0→gg=3α2sMχc0|R′nP(0)|2m5Q[1+C0Q(αs/π)] (16)
 Γn3P2→gg=4153α2sMχc2|R′nP(0)|2m5Q[1+C2Q(αs/π)] (17)

The vector mesons have quantum numbers 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

 Γn3S1→ℓ+ℓ−=4α2ee2Q|nsV(0)|2M2V[1−16αs3π] (18)

Here, is the electromagnetic coupling constant, is the strong running coupling constant in Eq. (4) and is the charge of heavy quark in terms of electron charge. In above relations, corresponds to the wave function of wave at origin for pseudoscalar and vector mesons while is the derivative of wave wave function at origin. The annihilation rates of heavy quarkonia are listed in Table 14 - 19.

### 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 () and magnetic () dipoles with selection rules and for the transitions while and for transitions. We now employ the numerical wave function for computing the electromagnetic transition widths among quarkonia and meson states in order to test parameters used in present work. For transition, we restrict our calculations for transitions among waves only. In the nonrelativistic limit, the radiative and 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⟨eQ2ω33 (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 of the system, magnetic dipole moment and photon energy are given by

 ⟨eQ⟩=∣∣∣m¯QeQ−e¯QmQmQ+m¯Q∣∣∣ (19)
 μ=eQmQ−e¯Qm¯Q (20)

and

 ω=M2i−M2f2Mi (21)

respectively. Also the symmetric statistical factors are given by

 SE1if=max(Li,Lf){Ji1JfLfSLi}2 (22)

and

 SM1if=6(2Si+1)(2Sf+1){Ji1JfSfℓSi}2{1121212SfSi}2. (23)

The matrix element for and transition can be written as

 ∣∣ME1if∣∣=3ω⟨f∣∣∣ωr2j0(ωr2)−j1(ωr2)∣∣∣i⟩ (24)

and

 ∣∣MM1if∣∣=⟨f∣∣∣j0(ωr2)∣∣∣i⟩ (25)

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