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IL NUOVO CIMENTO VOL. 110 A, N. 8 Agosto 1997 NOTE BREVI

The semirelativistic kinetic term of the heavy quarkonia

in the oscillator-representation method

C. ITOH(1), T. MINAMIKAWA(2), K. MIURA(3) and T. WATANABE(4)

(1_{) Department of Physics, Meiji-Gakuin University - Totsuka, Yokohama 244, Japan}

(2_{) Department of Physics, Tokyo University of Mercantile Marine - Etchujima, Tokyo 135, Japan} (3_{) Department of Physics, Nippon Ika University - Kawasaki 211, Japan}

(4_{) Department of Physics, Asia University - Musasino, Tokyo 180, Japan}

(ricevuto il 10 Ottobre 1997; approvato il 3 Novembre 1997)

Summary. — On the basis of the oscillator-representation method, we discuss that the effects of the semirelativistic kinetic term of the charmonium 1 P , 1 D , 2 S are considerably large compared to those of the bottomonium.

PACS 12.39.Jh – Nonrelativistic quark model. PACS 12.39.Pn – Potential models.

Nonrelativistic quark models [1] are successful for the description of the masses of the bound states of heavy quarks. However, some analyses [2] show that the introduction of

k

m2

1 p2 instead of p2_{/( 2 m) as the kinetic term produces better}

results in the hadron mass spectroscopy. Many authors treat the charm quark as heavy. However, is this really a good approximation?

In this paper we estimate the semirelativistic kinetic term of quark-antiquark bound states in the linear potential [3, 4] on the basis of the oscillator-represen-tation (OR) method [5]. The Hamiltonian of a quark-antiquark system in the confining linear potential V(r2_{)(4kr1d) is given by 2}

k

m2_{1 p}2_{1 V(r}2). We apply the OR method to the Hamiltonian, and obtain

H 4e01 1 2 m (: p 2 : 1u2_{: r}2 : ) 1: 2

k

m2 1 p22 p 2 2 m 1 V(r 2 ) 2 u 2_r2 2 m : , (1)

where : : denotes the normal order product. The second term of the Hamiltonian (1) describes the harmonic oscillator and is taken as the unperturbed Hamiltonian H0. The

last term of the Hamiltonian (1) is the perturbed Hamiltonian H1, and is written as

follows: H14



d3_k ( 2 p)3[KA(k 2_{) e}2u/4 k2_{: e} 2(ik Q p): 1VA(k2) e21 /4 uk 2 : e2(ik Q r): ] , (2)

C.ITOH,T.MINAMIKAWA,K.MIURAandT.WATANABE 892 where

.

/

´

K A(k2 ) 4



d3 q 2

k

m2_{1 q}2exp [2ikQq] , V A(k2 ) 4



d3_sV(s2 ) exp [2ikQs] , (3) e2(x) 4exp [x]212x2 1 2x 2_. (4)

e0is the ground-state energy and is given by

e04

1

G( 3 /2 )[K01 V0]b 41, (5)

and m and u are parameters determined by OR conditions G

g

5

2

h

u 12mK1Nb 414 0 , G

g

5

2

h

u 12mV1Nb 414 0 , (6)

which mean that H1does not include :p2: and :r2: terms. Knand Vnare given by

Kn4

g

b2 ¯ ¯b

h

n



0 Q du u1 /2_b3 /2_e2bu₂

_k

_m2 1 uu , (7) Vn4

g

b2 ¯ ¯b

h

n



0 Q du u1 /2b3 /2e2bu_V

g

u u

h

. (8)

Introducing the creation and annihilation operators aj† and aj ( j 41, 2, 3) [5], we can

write H0as H04 v(a†a) , v 4 u m , (a † a) 4

!

i a† i ai. (9)

The 1 S , 2 S , 1 P , 1 D bound states of H0are given by

.

/

´

N1Sb4N0b, N2Sb4621/2(a†_a† )N0b , N1Pb4221/2(a† 16a2†)N0b, a3†N0b , N1Db4821/2(a† 16a2†)2N0b, 221/2a3†(a1†6a2†)N0b, 1221/2(3a3†a3†2(a†a†))N0b, (10)

respectively, where N0b is the ground state.

Finally we obtain the bound-state energy E(A) of the state A up to the second order in the perturbation as follows:

.

/

´

E( 1 S) 4e01 e2( 1 S); e1( 1 S) 40 , e2( 1 S) 42 2 pv n 42

!

Q Bn

N

1 n!I 1 n

N

2 , (11)

THE SEMIRELATIVISTIC KINETIC TERM OF THE HEAVY QUARKONIA ETC. 893

.

`

/

`

´

E( 2 S) 4e01 2 v 1 e1( 2 S) 1e2( 2 S); e1( 2 S) 4 4 3 kpI 1 2 , e2( 2 S) 42 128 45 pv

N

I 2 2 2 1 4I 1 3

N

2 2 1 3 pv n 42

!

Q ( 2 n 12)(2n13) BnQ Q

N

1 n!

g

I 1 n 2 4 2 n 13I 2 n 111 4 ( 2 n 12)(2n13)I 1 n 12

h

N

2 , (12)

.

/

´

E( 1 P) 4e01 v 1 e2( 1 P); e1( 1 P) 40 , e2( 1 P) 42 16 45 pvNI 2 2 N22 2 3 pv n 42

!

Q ( 2 n 13)Bn

N

1 n!(I 1 n 2 2 2 n 13I 2 n 11)

N

2 , (13)

.

`

/

`

´

E( 1 D) 4e01 2 v 1 e1( 1 D) 1e2( 1 D); e1( 1 D) 4 8 15 kpI 1 2 , e2( 1 D) 42 448 225 pv

N

I 2 2 2 1 7I 1 3

N

2 2 2 15 pv n 42

!

Q ( 2 n 13)(2n15) BnQ Q

N

1 n!

g

I 1 n 2 4 2 n 13 I 2 n 111 4 ( 2 n 13)(2n15)I 1 n 12

h

N

2 , (14)

where ei(A) denotes the i-th–order perturbed energy of the state A, Bnstands for the

beta function B(n , 3 /2 ), and I1

n 4 [Kn1 (21 )nVn]b 41, In24 [Kn2 (21 )nVn]b 41.

(15)

The bound-state energies in the case of the kinetic term being nonrelativistic are obtained by replacing

k

m2_{1 p}2_{by m 1p}2/( 2 m) in the above formulae, which reduce to Kn4 0 for n D 1 in eq. (15).

Table I shows the first- and second-order perturbed energies, representing that the calculation to second order in H1is a good approximation, since e2 has the sufficiently

small value.

The values of the parameter k of the linear potential and the charm and bottom quark masses are determined by fitting to the experimental data of the cc– and bb– masses [6]. We obtain mb4 5.280 GeV, mc4 1.903 GeV, k 4 0.2969 GeV2 and d 42

1.5976 GeV, and ub4 0.7177 GeV2, vb4 0.2636 GeV, uc4 0.3836 GeV2, vc4

0.3606 GeV for the case of the semirelativistic kinetic terms, and ub4 0.7032 GeV2,

vb4 0.26646 GeV, uc4 0.3561 GeV2, vc4 0.3743 GeV for the case of the nonrelativistic

kinetic terms. Our results show that the total energy differences between

TABLEI. – The first- and second-order perturbed energies in the linear potential in units of MeV.

1S 2S 1P 1D 1S 2S 1P 1D semirelativistic 2e1( bb – ) 0 70 0 28 2e1( cc–) 0 107 0 43 2e2( bb – ) 2 12 4 11 2e2( cc–) 3 13 5 11 nonrelativistic 2e1( bb – ) 0 67 0 27 2e1( cc – ) 0 94 0 37 2e2( bb – ) 1 13 4 12 2e2( cc – ) 2 18 6 17

C.ITOH,T.MINAMIKAWA,K.MIURAandT.WATANABE 894

Fig. 1. – The relation of the energy difference (DE) between the semi- and non-relativistic energies and the quark mass (m) in the linear potential.

nonrelativistic and semirelativistic cases in the (1S, 2S, 1P, 1D) states of the charmonium (mc4 1.9 GeV) are (17, 52, 29, 43) MeV, respectively, and those of the

bottomonium (mb4 5.3 GeV) are (3, 10, 5, 8) MeV, respectively. Thus the nonrelativistic

approximation is good for these states of the bottomonium. However, for the 2 S state of the charmonium the difference is 52 MeV, which is so large. We see that the semi-relativistic effect cannot be neglected for the charmonium even in the 2 S state.

In order to understand this mass dependence more clearly, we estimate the energy differences DE of the bound states between in the case of the semirelativistic kinetic term and in the case of the nonrelativistic kinetic term as a function of the quark mass m. Figure 1 shows the result. We choose the value of the parameter k as k 4 0.2 GeV2_{, which is near those of more realistic models [4]. We can see that for the}

bottom quark mass region m B5 GeV, the energy differences are smaller than 7 MeV for 1 S , 2 S , 1 P , 1 D states, so that the semirelativistic effects are negligible for these states. We can also see that for the small mass region below B2 GeV, the energy differences are considerably large and sensitive to the quark mass for 2 S , 1 P , 1 D states, so that the semirelativistic effects are important for these states.

R E F E R E N C E S

[1] EICHTENE. J. and QUIGGC., Phys. Rev. D, 49 (1994) 5845; ITOHC., MINAMIKAWAT., MIURAK. and WATANABET., Nuovo Cimento A, 109 (1996) 569.

[2] ITOHC., MINAMIKAWAT., MIURAK. and WATANABET., Prog. Theor. Phys., 61 (1979) 548; Phys.

Rev. D, 40 (1989) 3660; GUPTAS. N. and JOHNSONJ. M., Phys. Rev. D, 51 (1995) 168; 53 (1996) 312.

[3] EICHTENE., GOTTFRIEDK., KINOSHITAT., KOGUTJ., LANEK. D. and YAN T.-M., Phys. Rev.

Lett., 34 (1975) 369.

[4] For summary of the potential models, for example, LUCHAW., SCHO¨BERLF. F. and GROMES D., Phys. Rep., 200 (1991) 127.

[5] DINEYHAMM., EFIMOVG. V., ANBOLDG. and NEDELKOS. N., Oscillator Representation in

Quantum Physics (Springer-Verlag) 1995.