Study of Heavy Quarkonium with Energy Dependent Potential

doi:10.4236/jmp.2012.310189

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Journal of Modern Physics, 2012, 3, 1530-1536

http://dx.doi.org/10.4236/jmp.2012.310189 Published Online October 2012 (http://www.SciRP.org/journal/jmp)

Study of Heavy Quarkonium with Energy

Dependent Potential

Pramila Gupta, Indira Mehrotra

Nuclear and Particle Physics Group, Department of Physics, University of Allahabad, Allahabad, India

Email: [email protected]

Received August 10, 2012; revised September 9, 2012; accepted September 17, 2012

ABSTRACT

Heavy quark systems (cc and bb ) have been studied in the nonrelativistic framework using energy dependent inter-

quark potential of the form harmonic oscillator with a small linear term as energy dependent as perturbation plus a in-

verse square potential. This potential admits exact analytical solution of the Schrodinger equation. Mass spectra, lep-

tonic decay width, root mean square radii



r, the expectation value of the radius





rand 1r have been

estimated for different quantum mechanical states for cc and bb systems. It is observed that energy dependent term

in the potential leads to saturation of the mass spectra and degree of saturation is governed by the magnitude of pertur-

bation. The calculated values of leptonic decay widths for 1s state are in very good agreement with the experimental

data both for cc and bb systems.

Keywords: Heavy Quark; Mass Spectrum; Energy Dependent Potential

1. Introduction

Energy spectrum of heavy quarkonium are a rich source

of information on the nature of the interquark force at

distances <0.1 fm and >1.0 fm. Many features such as

mass spectra and decay properties of heavy quarkonia

could be described by applying the ordinary nonrelativis-

tic Schrödinger wave equation to the two body quark-

antiquark system. It is well known that the non-relativis-

tic approach is justified when large quark masses are

involved and level spacing between the energy levels is

less than the constituent masses.

A variety of forms for the interquark potential have

been used in heavy quarkonia mass spectroscopy. These

can be broadly classified as 1) QCD motivated potential

[1-5] and 2) purely phenomenological potential [6-9]. A

comprehensive list of potential models is described in the

work of Lichtenberg [10]. All the potentials have almost

similar behavior in the range of 0.1 fm r



0.1 fm,

the characteristic interval of cc and bb systems but

differ from each other outside this range. At present it is

not possible to obtain exact interquark potential in the

entire range of distances from the first principal of

Quantum Chromodynamics. Moreover major shortcom-

ing of the existing potentials is that they have not been

able to account for the observed saturation pattern in the

experimental spectra.

A different category of potentials, which are energy

dependent, have been known in physics for a long time.

They occur in relativistic quantum mechanics at various

places like with Pauli Schrodinger equation [11], in the

Hamiltonian formulation of relativistic many body prob-

lem in covariant formulation with constraints in nonlin-

ear Hamiltonian evolution equation, and also in soliton

propagation. In the non-relativistic physics energy de-

pendent potentials offer the possibility of studying non-

linear effects in the framework of Schrodinger equation.

Lombard [12] have for the first time used an energy

dependent potential to study the bound state properties of

cc and bb systems. Initially they had used one di-

mensional harmonic oscillator potential with a linear

energy dependent term as perturbation to study the effect

of energy dependence on the energy eigenvalues. Later

they solved the problem for three dimensions with dif-

ferent power law potentials (harmonic, linear and Cou-

lomb) with energy dependence. Their main conclusion

has been that energy dependence saturates the mass

spectra. It is well known that any realistic interquark po-

tential has two components: asymptotic and confinement.

However Lombard’s potential has only one component

either confinement (harmonic and linear) or asymptotic

(Coulomb). In view of this in the present work we have

used a more realistic interquark potential of the form

harmonic plus inverse square with small linear energy

P. GUPTA, I. MEHROTRA 1531

dependence on the confining harmonic oscillator poten-

tial. The choice of linear dependence is motivated by the

fact that it leads to a coherent theory [13]. The combina-

tion of harmonic oscillator and inverse square potential

was first of all adopted by Joshi and Mitra [14] in a the-

ory based on the Schrodinger equation for studying the

heavy meson spectroscopy. Later it has been used by Iyer

et al. [15] and Ryes et al. [16] in the study of hadron

spectroscopy. In all these studies a better fit was obtained

for the potential with a term proportional to 1/r2 because

it has singularity for r  0 that improves the behavior in

this region. Also one great advantage is that our potential

admits exact analytical solution for the radial part of the

Schrodinger equation. This is a great advantage in view

of the high nonlinearity of the differential equation to be

solved. Using this potential we have calculated the mass

spectrum, the root mean square radii, average radii, lep-

tonic decay width and 1r for cc and bb systems.

The latter two properties are sensitive to the asymptotic

part of the potential. The aim is to study the effect of

energy dependence on the low as well as high excitation

states of the system.

2. Details of Calculation

In the present work, energy spectrum of heavy quark-

onium systems (cc and bb ) have been studied in the

framework of non-relativistic Schrödinger equation using

interquark potential as spin independent harmonic oscil-

lator with a small linear term energy dependent plus in-

verse square potential given by

 

nl nl

VrEm rEr





, (1)

where ω, g and γ are constants.

The three dimensional Schrodinger equation in the

center-of-mass system is



 

,,, ,,,

,,,

2nlnlmnl nlm

VrEr Er





 





 





,

(2)

where the reduced mass µ in terms of quark mass

and antiquark mass

m is



.

In natural units is considered.

1c

The wave function is written as

 

,, ,

nlm lm



 



Now putting









, 2

tmr





, 2E









and (1ll )2g





 , Equation (2) reduces to







tuttu t













 



 (3)





ut s are the solution of the radial equation, which are

bounded at infinity and are zero at the origin. As t tends

to , the bounded solution behaves like



exp 2t



and

since t = 0 is a singularity of Equation (3) we seek for a

solution in the form,

 

exp 2l

uttRt









 (4)

in which, on account of boundary condition,



has to be

positive. Substituting Equation (4) into Equation (3) and

taking







11218

4lg





  (5)

This leads to the equation

  

244

tR ttRtR t









 l



 



 (6)

Apart from the constant factor the nonsingular solution

of Equation (6) is the Confluent Hypergeometric series

[17]



,2 ;

44 2

Rt Ft









 







. (7)





ut increases without bound as t   unless the series

F reduces to a polynomial. This occurs only if

1;0,1,2

44 rr





 , (8)

which implies that the energy eigenvalue

22 222

116

Ea a



 



  (9)

where



4221 8an lg



 .

,nl is the classified eigen energy by principal quan-

tum number and angular quantum number ℓ ≤ (n − 1) and

the quark mass is connected to the physical mass as

qq() 1q s



.

The parameters



, g and



are obtained from fit to

the experimental energies of triplet states of 2s and 3s

and center of gravity of 1p with respect to 1S in Equa-

tion (9). In the case of energy independent potential (



0) the two parameters



and g are obtained from fitting

to the theoretically estimated values of 2s and 1p to the

corresponding experimental data. These in turn are used

to predict eigenvalues of higher excited levels from the

energy eigenvalue Equation (9). All the experimental

data for cc and bb are taken from recent compilation

of Particle Data Group 2008 [18]. In the literature, the

charm quark mass is chosen between 1.2 < mc <1.8 GeV

P. GUPTA, I. MEHROTRA

1532

whereas that of the bottom quark is between 4.5 < mb <

5.4 GeV. In the present work we have chosen the mass to

be 1.5 GeV for the charm quark and 5.0 GeV for the

bottom quark, which is almost at the mid values of above

ranges. While solving non linear energy eigenvalue Equ-

ation (9) only negative roots of



are accepted because

negative values of



can compress the spectrum which

is experimentally observed. Once the parameters of the

potential are fixed it is substituted in the reduced radial

Schrodinger equation given by

of s wave at the origin. Mns is the mass of bound triplet

(vector) state,



is the electromagnetic fine structure

constant and eq the charge of quark in units of the elec-

tron charge











,,,

nlnlnl nl

ur rE Eur





 





2,Vll



 (10)

This equation is solved numerically in MATHE-

MATICA 8.0 by software program obtained by Lucha et

al. [19] for each quantum state separately. The exact nu-

merical solution of wave functions so obtained are used

to calculate the leptonic decay width, r, 1r and

Vector state with spin one, negative parity (3S1) of

quark antiquark pair can annihilate into lepton pair

through single virtual photon. Leptonic decay width of

3S1 states of cc and bb quarkonia is the physical

quantity which is very sensitive to the form of potential.

Leptonic decay widths are calculated according to Van

Royen-Weisskoph formula [20]. This formula is true for

energy dependent potential also.



 





22 2

Γ01 3π

nS eeMqq





 



 





(11)

where





is the bound state radial wave function



23 ,13eee e

In the present work we have taken strong coupling

constants as





20.37

am  and .



20.26

am 

Expectation values of r, 1/r and bound state root mean

square radii (rms) are obtained from



urr







, (12)

where x stands for r, 1/r and r2.

3. Results and Discussion

The potential parameters obtained from fit to experimen-

tal energy levels are given for cc and bb systems in

Tables 1 and 2 respectively. These exhibit flavor de-

pendence. Parameters of set A correspond to energy in-

dependent potential and are compared with the parame-

ters obtained by Ryes et al. [16] for harmonic oscillator

plus inverse square potential. Parameters of set B are for

energy dependent potential used in present work. These

are compared with the values obtained by Lombard et al.

[12] for an energy dependent harmonic oscillator poten-

tial. The quark masses used are slightly different in each

work.

The corresponding potentials as a function of distance

are plotted in Figures 1 and 2. The energy dependent per-

turbation leads to slight variation of confinement poten-

tial for different states. When the energy of the state in-

creases with radial excitation, the classically allowed

region for harmonic oscillator potential is greater.

Table 1. Spectroscopic parameters obtained from fit to experimental data for cc system.

Parameters Present work (A) Ref [16] Present work (B) Ref [12]

mq (GeV) 1.50 3.812 1.50 1.207

γ (GeV−1) 0 0 −0.117 −0.433



(GeV) 0.174 0.285 0.203 0.55

g (GeV−1) −0.073 −0.0655 −0.155 0

Table 2. Spectroscopic parameters obtained from fit to experimental data for bb system.

Parameters Present work (A) Ref [16] Present work (B) Ref [12]

mq (GeV) 5.00 7.093 5.00 4.401

γ (GeV−1) 0 0 −0.102 −0.455



(GeV) 0.176 0.2090 0.187 0.530

g (GeV−1) −0.0425 −0.0352 −0.044 0

P. GUPTA, I. MEHROTRA 1533

-1

-0.5

0.5

1.5

r (GeV

-1

)

V(r) (GeV)



= 0

Figure 1. Quark interquark potential curves as a function

of r for energy dependent (1s and 4s states) and energy in-

dependent case (γ = 0) for cc .

-1

-0.5

0.5

1.5

r (GeV

-1

)

V(r) (GeV)



= 0

Figure 2. Quark interquark potential curves as a function

of r for energy dependent (1s and 4s states) and energy in-

dependent case (γ = 0) for bb .

The energy spectra of charmonium and bottonium

system with respect to E1s are shown in Figures 3 and 4

respectively. We compare our plotted mass spectra results

with and without energy dependence with experimental

data and also with the work of Lombard et al. [12].

Position of different ns states with respect to 1s is

shown in Table 3. It is observed that the ,1,

in-

creases slowly with principal quantum number. This in-

crease is less rapid for energy dependent potential,

showing saturation effect as compared to energy inde-

pendent case (

–

ns s



= 0). The eigenvalues E1ℓ are displayed

up to ℓ = 20 for different values of



in Figure 5 for

the cc system. The interesting feature of the result is

that



determines the level of saturation. On decreasing



, maximum value of eigenvalue E1ℓ decreases and

reaches an upper limit. In contrast for energy independ-

ent case E1ℓ increases regularly towards with increasing ℓ.

Similar plot up to ℓ = 30 is shown for the bb system in

400

600

800

1000

1200

1400

Mass (MeV)

1p 1p 1p

2s 2s 2s 2s

1d 1d 1d

2p 2p 2p 2p

3s 3s 3s

(a) (b) (c) (d)

Figure 3. Mass spectrum of charmonium system with re-

spect to ground state (E1s) with (a) experimental data (b) γ =

0 (c) γ ≠ 0 (d) Lombard’s work. The levels with asterisk are

used as input data in the parameter fitting.

400

600

800

1000

1200

1400

Ma ss (M eV)

(a) (b) (c) (d)

2s 2s 2s

1p 1p

1d 1d

2p 2p 2p

3s 3s 3s

4s 4s

Figure 4. Mass spectrum of bottonium system with respect

to ground state (E1s) with (a) experimental data (b) γ = 0 (c)

γ ≠ 0 (c) Lombard’s work. The levels with asterisk are used

as input data in the parameters fitting.

Table 3. Spacing between the various radial excitations for ℓ

= 0.

cc bb

γ = 0 γ ≠ 0 γ = 0 γ ≠ 0

E4S-E1S (GeV)1.25 1.20 1.26 1.21

E5S-E1S (GeV)1.65 1.61 1.61 1.57

E6S-E1S (GeV)1.98 1.96 1.97 1.95

Figure 6. The saturation in bb is reached more slowly

compared to cc system. We have studied the effect of

varying quark mass on the value of the parameters for

cc and bb systems. It turns out that all the parameters

vary with quark mass. The variation of g with mc and mb

is given in Table 4 keeping



and ω constant.

Leptonic decay width of vector meson for 1s state and

its ratio with those of other states are listed and also

compared with experimental data in Table 5. Leptonic

P. GUPTA, I. MEHROTRA

1534

05 10 15 20

E1l

 = 0

 = -.117

 = -.4

 = -.8

γ = 0

γ = -0.117

γ = -0.4

γ = -0.8

Figure 5. Behavior of spectrum for different values of γ in

cc system.

05 10 15 20 25 30

 = 0

 = -.102

 = -.25

 = -.6

γ = 0

γ = -0.102

γ = -0.25

γ = -0.6

Figure 6. Behavior of spectrum for different values of γ in

system.

Table 4. Values of parameter g for different masses of cc

and bb system.

cc bb

Mass (GeV) g (GeV−1) Mass (GeV) g (GeV−1)

1.2 −0.194 4.4 −0.045

1.5 −0.155 5.0 −0.040

1.7 −0.130 5.5 −0.036

decay width for 1s state obtained in the present work are

very close to experimental data and almost double of the

value of obtained by Lombard. This shows the impor-

tance of using asymptotic term in the potential.

Results for r, 1r and 2

r computed for

different quantum states of cc and bb systems are

listed in Tables 6 and 7. Also given for comparison are

the results of the calculation of Boroum et al. [21] for

r and 1r with global potential and of Chen Hong

et al [22] for root mean square radius with QCD based

potential. Root mean square radii for different 1s state

have slightly smaller values as compared with Lombard

potential reported as 0.49 (for cc ) and 0.26 (for bb ).

Our estimated values fall well in the range of the results

of other calculation.

It is observed that bb system has smaller radii than

the cc system. Root mean square radii of various states

of cc and bb fall within the interval 0.1 to 1 fm be-

cause all the potentials are similar in this range. It means

that average size (radius) of the cc system is greater

Table 5. Ratio of Leptonic decay width of different states with that of 1s state.

cc bb

Calculated value Experimental value [18] Calculated value Experimental value [18]

 (1s) (kev) 5.15 (2.65) 5.55 ± 0.14 1.05 (0.47) 1.32 ± 0.018

 

2s 1s

ee ee

 

 0.61 0.45 ± 0.08 0.68 0.46 ± 0.03

 

3s 1s

ee ee



 0.50 0.16 ± 0.04 0.63 0.33 ± 0.03

 

4s 1s

ee ee

 

 0.43 0.11 ± 0.04 0.57 0.23 ± 0.02

Table 6. r, 1r and 2

r for cc .

r (GeV−1) 1r (GeV) 2

r (fm)

Present work [21] Present work [21] Present work [22]

1s 2.790 2.618 0.507 0.491 0.456 0.401

2s 4.612 4.761 0.396 0.325 0.898 0.801

3s 5.9 0.343 1.252 1.242

1p 4.266 3.751 0.316 0.307 0.698 0.639

2p 5.588 0.239 1.113 1.101

P. GUPTA, I. MEHROTRA 1535

Table 7. r, 1r and 2

r for bb .

r (GeV−1) 1r (GeV) 2

r (fm)

Present work [21] Present work [21] Present work [22]

1s 1.574 1.823 0.8706 0.685 0.206 0.196

2s 2.523 3.100 0.6946 0.486 0.510 0.490

3s 3.207 0.6701 0.790 0.781

1p 2.306 2.446 0.4927 0.467 0.415 0.395

2p 3.017 0.44212 0.715 0.693

than the bb system i.e. heavy quarkonium have smaller

radii.

4. Summary and Conclusion

Heavy quarkonia system (and bb) have been studied in

the framework of non-relativistic Schrodinger equation

with energy dependent potential. Such potentials consti-

tute a way to include nonlinear effects in the Schrodinger

equation. The interquark potential used in the present

work is of harmonic oscillator plus inverse square form

with a small energy dependent term in harmonic oscilla-

tor part. This potential is more realistic compared to the

energy dependent harmonic oscillator potential used in

the work of Lombard which does not have the asymptotic

term. The energy dependence in the potential results in

the compression of the spectrum. As the quantum num-

bers increase the energy eigenvalues increase an upper

bound. Though the choice of potential parameters is not

unique, the above form represents a class of potentials

which can give a satisfactory explanation of the satura-

tion pattern in the spectra as well as the correct order of

magnitude of Leptonic decay widths both for cc and

bb systems. Our results for the Leptonic decay width of

1s state (1s) are much closer to the experimental values

compared to those obtained by Lombard (values given

within bracket in Table 5) for both cc and bb sys-

tems. This shows the importance of the asymptotic term

in the interquark potential in determining the wavefunc-

tion at the origin.

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