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Quark-antiquark potential with retardation and radiative contributions and the heavy quarko


HUB-EP-99/58

Quark-antiquark potential with retardation and radiative contributions and the heavy quarkonium mass spectra
D. Ebert, R. N. Faustov? and V. O. Galkin?
Institut f¨r Physik, Humboldt–Universit¨t zu Berlin, Invalidenstr.110, D-10115 Berlin, Germany u a

arXiv:hep-ph/9911283v1 9 Nov 1999

Abstract
The charmonium and bottomonium mass spectra are calculated with the systematic account of all relativistic corrections of order v 2 /c2 and the oneloop radiative corrections. Special attention is paid to the contribution of the retardation e?ects to the spin-independent part of the quark-antiquark potential, and a general approach to accounting for retardation e?ects in the long-range (con?ning) part of the potential is presented. A good ?t to available experimental data on the mass spectra is obtained. I. INTRODUCTION

The investigation of the meson properties in the framework of constituent quark models is an important problem of the elementary particle physics. At present a large amount of experimental data on the masses of ground and excited states of heavy and light mesons has been accumulated [1]. By comparing theoretical predictions with experimental data, one can obtain a valuable information on the form of the quark-antiquark interaction potential. Such information is of great practical interest since at present it is not possible to obtain the q q potential in the whole range of distances from the basic principles of QCD. As it is ? well known, the growing of the strong coupling constant with distance makes perturbation theory inapplicable at large distances (in the infrared region). In this region it is necessary to account for nonperturbative e?ects connected with the complicated structure of the QCD vacuum. All this leads to a theoretical uncertainty in the q q potential at large and interme? diate distances. It is just in this region of large and intermediate distances that most of the basic meson characteristics are formed. This makes it possible to investigate the low-energy region of strong interaction by studying the mass spectra and decays of mesons. Some recent investigations [2–4] have shown that there could be also a linear (in radius) correction to the perturbative Coulomb potential at small distances (in contradiction with

leave of absence from Russian Academy of Sciences, Scienti?c Council for Cybernetics, Vavilov Street 40, Moscow 117333, Russia.

? On

1

OPE predictions). The estimates of the slope yield that it could be of the same order of magnitude as the slope of the long-range con?ning linear potential. It means then that the widely used Cornell potential (the sum of the Coulomb and linear con?ning terms) is really a correct one in the static limit both at large and at small distances. The relativistic properties of the quark-antiquark interaction potential play an important role in analysing di?erent static and dynamical characteristics of heavy mesons. The Lorentz-structure of the con?ning quark-antiquark interaction is of particular interest. In the literature there is no consent on this item. For a long time the scalar con?ning kernel has been considered to be the most appropriate one [5]. The main argument in favour of this choice is based on the nature of the heavy quark spin-orbit potential. The scalar potential gives a vanishing long-range magnetic contribution, which is in agreement with the ?ux tube picture of quark con?nement of [6], and allows to get the ?ne structure for heavy quarkonia in accord with experimental data. However, the calculations of electroweak decay rates of heavy mesons with a scalar con?ning potential alone yield results which are in worse agreement with data than for a vector potential [7,8]. The radiative M1-transitions in quarkonia such as e. g. J/ψ → ηc γ are the most sensitive to the Lorentz-structure of the con?ning potential. The relativistic corrections for these decays arising from vector and scalar potentials have di?erent signs [7,8]. In particular, as it has been shown in ref. [8], agreement with experiments for these decays can be achieved only for a mixture of vector and scalar potentials. In this context, it is worth remarking, that the recent study of the q q ? interaction in the Wilson loop approach [9] indicates that it cannot be considered as simply a scalar. Moreover, the found structure of spin-independent relativistic corrections is not compatible with a scalar potential. A similar conclusion has been obtained in ref. [10] on the basis of a Foldy-Wouthuysen reduction of the full Coulomb gauge Hamiltonian of QCD. There, the Lorentz-structure of the con?nement has been found to be of vector nature. The scalar character of spin splittings in heavy quarkonia in this approach is dynamically generated through the interaction with collective gluonic degrees of freedom. Thus we see that while the spin-dependent structure of (q q ) interaction is well established now, the spin-inde? pendent part is still controversial in the literature. The uncertainty in the Lorentz-structure of the con?ning interaction complicates the account for retardation corrections since the relativistic reconstruction of the static con?ning potential is not unique. In our previous paper [11] we gave some possible prescription of such reconstruction which, in particular, provides the ful?lment of the Barchielli, Brambilla, Prosperi (BBP) relations [12] following from the Lorentz invariance of the Wilson loop. Here we generalize this prescription and discuss its connection with the known quark potentials and the implications for the heavy quarkonium mass spectra. The other important point is the inclusion of radiative corrections in the perturbative part of the quark potential. There have been considerable progress in recent years and now the perturbative QCD corrections to the static potential are known up to two loops [13,14] though for the velocity dependent and spin-dependent potentials only one-loop corrections are calculated [15–17]. The paper is organized as follows. In Sec. II we describe our relativistic quark model. The approach to accounting for retardation e?ects in the q q potential in the general case is ? presented in Sec. III. The resulting heavy quark potential containing both spin-independent and spin-dependent parts with the account of one-loop radiative corrections is given in 2

Sec. IV. We use this potential for the calculations of the heavy quarkonium mass spectra in Sec. V. Section VI contains our conclusions and discussion of the results.
II. RELATIVISTIC QUARK MODEL

In the quasipotential approach a meson is described by the wave function of the bound quark-antiquark state, which satis?es the quasipotential equation [18] of the Schr¨dinger o type [19] p2 b2 (M) ? ΨM (p) = 2?R 2?R where the relativistic reduced mass is ?R = and Ea , Eb are given by Ea = M 2 ? m2 + m2 b a , 2M Eb = M 2 ? m2 + m2 a b . 2M (3) Ea Eb M 4 ? (m2 ? m2 )2 a b = , 3 Ea + Eb 4M (2) d3 q V (p, q; M)ΨM (q), (2π)3 (1)

Here M = Ea + Eb is the meson mass, ma,b are the masses of light and heavy quarks, and p is their relative momentum. In the centre of mass system the relative momentum squared on mass shell reads b2 (M) = [M 2 ? (ma + mb )2 ][M 2 ? (ma ? mb )2 ] . 4M 2 (4)

The kernel V (p, q; M) in Eq. (1) is the quasipotential operator of the quark-antiquark interaction. It is constructed with the help of the o?-mass-shell scattering amplitude, projected onto the positive energy states. Constructing the quasipotential of the quark-antiquark interaction we have assumed that the e?ective interaction is the sum of the usual one-gluon exchange term with the mixture of long-range vector and scalar linear con?ning potentials, where the vector con?ning potential contains the Pauli interaction. The quasipotential is then de?ned by [20] V (p, q; M) = ua (p)?b (?p) ? u 4 ? ν αs D?ν (k)γa γb 3 (5)

+VV (k)Γ? Γb;? + VS (k) ua (q)ub (?q), a

where αS is the QCD coupling constant, D?ν is the gluon propagator in the Coulomb gauge D 00 (k) = ? 4π , k2 D ij (k) = ? 4π ij k i k j δ ? 2 k2 k , D 0i = D i0 = 0, (6)

and k = p ? q; γ? and u(p) are the Dirac matrices and spinors 3

uλ (p) = with ?(p) = √

?(p) + m 2?(p)

1
σp ?(p)+m

χλ ,

(7)

p2 + m2 . The e?ective long-range vector vertex is given by Γ? (k) = γ? + iκ σ?ν k ν , 2m (8)

where κ is the Pauli interaction constant characterizing the anomalous chromomagnetic moment of quarks. Vector and scalar con?ning potentials in the nonrelativistic limit reduce to VV (r) = (1 ? ε)Ar + B, VS (r) = εAr, reproducing Vconf (r) = VS (r) + VV (r) = Ar + B, (10)

(9)

where ε is the mixing coe?cient. The expression for the quasipotential for the heavy quarkonia, expanded in v 2 /c2 without retardation corrections to the con?ning potential, can be found in Ref. [20]. The structure of the spin-dependent interaction is in agreement with the parameterization of Eichten and Feinberg [21]. All the parameters of our model like quark masses, parameters of the linear con?ning potential A and B, mixing coe?cient ε and anomalous chromomagnetic quark moment κ are ?xed from the analysis of heavy quarkonium masses (see below Sec. V) and radiative decays. The quark masses mb = 4.88 GeV, mc = 1.55 GeV and the parameters of the linear potential A = 0.18 GeV2 and B = ?0.16 GeV have usual values of quark models. The value of the mixing coe?cient of vector and scalar con?ning potentials ε = ?1 has been determined from the consideration of the heavy quark expansion for the semileptonic B → D decays [22] and charmonium radiative decays [8]. Finally, the universal Pauli interaction constant κ = ?1 has been ?xed from the analysis of the ?ne splitting of heavy quarkonia 3 PJ - states [20]. Note that the long-range magnetic contribution to the potential in our model is proportional to (1 + κ) and thus vanishes for the chosen value of κ = ?1. In the present paper we will include into consideration the retardation corrections as well as one-loop radiative corrections.
III. GENERAL APPROACH TO ACCOUNTING FOR RETARDATION ? EFFECTS IN THE QQ POTENTIAL

For the one-gluon exchange part of the q q potential it is quite easy to isolate the retar? dation contribution. Indeed due to the vector current conservation (gauge invariance) we have the well-known relation on the mass shell 1 ? ua (p)?b (?p)γa γb? ua (q)ub (?q) ? u k2 4

= ??a (p)?b (?p) u u
2 k 2 = k0 ? k2 ;

0 0 1 (γ a · k)(γ b · k) γa γb + 2 γa · γb ? ua (q)ub (?q), 2 k k k2 k0 = ?a (p) ? ?a (q) = ?b (q) ? ?b (p); k = p ? q.

(11)

The left-hand side and the right-hand side of this relation are easily recognized to be in the Feynman gauge and the Coulomb gauge, respectively. Now, if the nonrelativistic expansion in p2 /m2 is applicable, we can immediately extract the retardation contribution. Namely 2 we expand the left-hand side of eq. (11) in k0 /k2 :
2 1 ? ? 1 ? k0 = 2 k0 ? k2 k2 k4

and get with needed accuracy [23] ? ua (p)?b (?p) ? u
0 0 γa γb k2 1+ 0 k2 k2

?

γa · γb ua (q)ub (?q). k2

(12)

In the right-hand side of eq. (11) one should use the identity following from the Dirac equation ua (p)?b (?p)(γ a · k)(γ b · k)ua (q)ub (?q) ? u 0 0 = ua (p)?b (?p)γa γb ua (q)ub (?q)(?a (p) ? ?a (q))(?b (q) ? ?b (p)). ? u
2 After de?ning k0 as a symmetrized product [23,24] 2 k0 = (?a (p) ? ?a (q))(?b (q) ? ?b (p))

(13)

2 and dropping k0 in the denominator we obtain the expression which is identical to eq. (12). In this way we obtain the well-known Breit Hamiltonian (the same as in QED [23]) if we further expand eq. (13) in p2 /m2

(p2 ? q2 )2 2 k0 ? ? . = 4ma mb

(14)

This treatment allows also for the correct Dirac limit in which the retardation contribution vanishes when one of the particles becomes in?nitely heavy [25]. For the con?ning part of the q q potential the retardation contribution is much more ? inde?nite. It is a consequence of our poor knowledge of the con?ning potential especially in what concerns its relativistic properties: the Lorentz structure (scalar, vector, etc.) and the 2 dependence on the covariant variables such as k 2 = k0 ? k2 . Nevertheless we can perform some general considerations and then apply them to a particular case of the linearly rising potential. To this end we note that for any nonrelativistic potential V (?k2 ) the simplest 2 relativistic generalization is to replace it by V (k0 ? k2 ). In the case of the Lorentz-vector con?ning potential we can use the same approach as before even with more general vertices containing the Pauli terms, since the mass-shell vector currents are conserved here as well. It is possible to introduce alongside with the “diagonal gauge” the so-called “instantaneous gauge” [26] which is the generalization of the Coulomb gauge. The relation analogous to eq. (11) now looks like (up to the terms of order of p2 /m2 ) 5

2 VV (k0 ? k2 )?a (p)?b (?p)Γ? Γb? ua (q)ub (?q) = ua (p)?b (?p) VV (?k2 )Γ0 Γ0 u u ? u a a b

? VV (?k2 )Γa · Γb + VV′ (?k2 )(Γa · k)(Γb · k) where

ua (q)ub (?q),

(15)

2 2 ′ VV (k0 ? k2 ) ? VV (?k2 ) + k0 VV (?k2 ) =

and as in the case of the one-gluon exchange above we put (p2 ? q2 )2 2 k0 = (?a (p) ? ?a (q))(?b (q) ? ?b (p)) ? ? = 4ma mb (16)

again with the correct Dirac limit. 2 For the case of the Lorentz-scalar potential we can make the same expansion in k0 , which yields
2 ′ 2 VS (k0 ? k2 ) ? VS (?k2 ) + k0 VS (?k2 ). =

(17)

2 But in this case we have no reasons to ?x k0 in the only way (13). The other possibility is to take a half sum instead of a symmetrized product, namely to set (see e. g. [24,25]) 2 k0 =

1 1 1 1 + 2 . (?a (p) ? ?a (q))2 + (?b (q) ? ?b (p))2 ? (p2 ? q)2 = 2 2 8 ma mb

(18)

The Dirac limit is not ful?lled by this choice, but this cannot serve as a decisive argument. Thus the most general expression for the energy transfer squared, which incorporates both possibilities (16) and (18) has the form
2 k0 = λ(?a (p) ? ?a (q))(?b (q) ? ?b (p)) + (1 ? λ)

1 (?a (p) ? ?a (q))2 + (?b (q) ? ?b (p))2 , (19) 2

where λ is the mixing parameter. After making expansion in p2 /m2 we obtain (p2 ? q2 )2 1 1 1 2 k0 ? ?λ + 2 + (1 ? λ) (p2 ? q)2 = 2 4ma mb 8 ma mb 2λ 1 1 1 (1 ? λ) (k · p)2 + 2(k · p)(k · q) + (k · q)2 . + 2 ? = 2 8 ma mb ma mb
2 Thus as expected k0 ? O(p2/m2 ) ? 1. Then the Fourier transform of the potential 2 2 V (k0 ? k2 ) ? V (?k2 ) + k0 V ′ (?k)2 = 2 with k0 given by eq. (20) can be represented as follows [25]

(20)

1 1 2λ 1 d3 k 2 + 2 ? V (k0 ? k2 )eik·r = V (r) + (1 ? λ) 3 2 (2π) 4 ma mb ma mb 1 × V (r)p2 + V ′ (r) (p · r)2 , r W 6

(21)

where {. . .}W denotes the Weyl ordering of operators and V (r) = d3 k V (?k2 )eik·r . (2π)3 (22)

In the case of the one-gluon exchange potential we had λ = 1, VC (?k2 ) = ? 4 4παs ; 3 k2 VC (r) = ? 4 αs . 3 r (23)

As for the con?ning potential we assume it to be a mixture of scalar and vector parts. In the nonrelativistic limit we adopt the linearly rising potential V0 (r) = Ar; V0 (?k2 ) = ? 8πA , (k2 )2 (24)

which we split into scalar and vector parts by introducing the mixing parameter ε. The possible constant term in V0 has been discussed in [11]. V0 = VS + VV ; VS = εV0 ; VV = (1 ? ε)V0 . (25)

Hence the retardation contribution (21) from scalar and vector potentials has the form 1 1 1 (1 ? λS,V ) + 2 4 m2 mb a ? 2λS,V ma mb 1 ′ VS,V (r)p2 + VS,V (r) (p · r)2 r ,
W

(26)

where we use the general Ansatz (19), (20) for both the scalar and vector potentials for the sake of completeness. The other spin-independent corrections in our model had been calculated earlier [20,11] 1 1 1 1 + 2 ?VV (r) + VV (r)p2 (1 + 2κ) 2 8 ma mb ma mb ? 1 2 1 1 + 2 2 ma mb VS (r)p2 . (27)

W

W

Adding to the above expression the retardation contributions (26) and the nonrelativistic parts (23) and (25) we obtain the complete spin-independent q q potential: ? VSI (r) = VC (r) + V0 (r) + VVD (r) + where the velocity-dependent part
C V S VVD (r) = VVD (r) + VVD (r) + VVD (r), (p · r)2 4 αs 2 (p · r)2 1 1 C VC (r) p2 + ? p + = VVD (r) = 2ma mb r2 2ma mb 3 r r2 W 1 1 1 2λV 1 V VVD (r) = + 2 ? VV (r)p2 + (1 ? λV ) 2 W ma mb 4 ma mb ma mb 2 (1 ? λV ) 1 1 (p · r) = (1 ? ε) + 2 × VV (r)p2 + VV′ (r) 2 r 4 ma mb W

1 8

1 1 + 2 ?[VC (r) + (1 + 2κ)VV ], 2 ma mb

(28)

(29) ,
W

7

× Ar p2 +
S VVD (r) =

(p · r)2 r2

+
W W

1 2

1 1 + 2 2 ma mb
2

VV (r)p2 (p VV′ (r)

× VV (r)p + ε =? 4

· r)2 r

λV λV (p · r)2 (1 ? ε) Ar 1 ? p2 ? ma mb 2 2 r2 1 1 2λS 1 + (1 ? λS ) + 2 ? 2 4 ma mb ma mb
W

,
W

1 (p · r)2 1 + 2 Ar (1 + λS )p2 + (λS ? 1) m2 mb r2 a 2 ελS (p · r) ? Ar p2 + . 2ma mb r2 W

W

Making the natural decomposition VVD (r) = 1 (p · r)2 p2 Vbc (r) + Vc (r) ma mb r2 +
W

1 1 + 2 2 ma mb

p2 Vde (r) ?

(p · r)2 Ve (r) r2

W

(30) we obtain from eqs. (29) Vbc (r) = ? λS λV 2αs ?ε Ar, + (1 ? ε) 1 ? 3r 2 2 λV λS 2αs Ar, ? (1 ? ε) +ε Vc (r) = ? 3r 2 2 1 Vde (r) = [(1 ? ε) (1 ? λV ) ? ε(1 + λS )] Ar, 4 1 Ve (r) = ? [(1 ? ε) (1 ? λV ) + ε(1 ? λS )] Ar. 4 ε Vde + Ve = ? Ar. 2

(31)

The following simple relations hold: Vbc ? Vc = (1 ? ε)Ar; (32)

The exact BBP relations [12] (see also [27]) in our notations look like 1 1 Vde ? Vbc + (VC + V0 ) = 0, 2 4 r d(VC + V0 ) 1 =0 Ve + Vc + 2 4 dr

(33)

1 1 (in the original version Vbc ≡ ?Vb ? 3 Vc and Vde ≡ Vd + 3 Ve ). The functions (31) identically satisfy the BBP relations (33) independently of values of the parameters ε, λV , λS but only with the account of retardation corrections. In our model [20,11] we have ε = ?1 and λV = 1, if we assume further that λS = 1 [11] then we get

Vbc (r) = ?

2αs 3 + Ar; 3r 2

Vc (r) = ? 8

2αs 1 ? Ar; 3r 2

1 Vde (r) = Ar; 2

Ve (r) = 0.

(34)

Our expressions (28) and (29) for purely vector (ε = 0) and purely scalar (ε = 1) interactions and for κ = 0, λS = λV = 1 coincide with those of Ref. [25]. In the minimal area low (MAL) and ?ux tube models [28] 2αs + 3r 1 Vde (r) = ? Ar; 6 Vbc (r) = ? 1 2αs 1 Ar; Vc (r) = ? ? Ar; 6 3r 6 1 Ve = ? Ar. 6

(35)

To obtain these expressions one should set in relations (31), (32) 2 ε= ; 3 λV + 2λS = 1. (36)

Thus one gets a family of values for λV and λS . The most natural choice reads as λV = 1, λS = 0, (37)

which resembles the Gromes proposal [24]: the symmetrized product for the vector potential and the half sum for the scalar potential. But still the Dirac limit is not ful?lled in this case. Expression (28) for VSI contains also the term with the Laplacian: 1 8 1 1 + 2 ?[VC (r) + (1 + 2κ)VV (r)]. m2 mb a (38)

In the MAL and some other models these terms look like [28] 1 8 1 1 + 2 ?[VC (r) + V0 (r) + Va (r)] 2 ma mb (39)

and usually it is adopted that ?Va (r) = 0. Lattice simulations [29] suggest that b ?VaL (r) = c ? , r b ? 0.8GeV2 . = (41) (40)

In our model expression (38) can be recast as follows 1 1 1 ? + 2 ?[VC (r) + V0 (r) + Va (r)], 2 8 ma mb ? Va (r) = (1 + 2κ)(1 ? ε)V0 (r) ? V0 (r) and for the adopted values ε = ?1, κ = ?1 A ? ?Va (r) = ?3?(Ar) = ?6 , r 6A ? 1.1GeV2 , = (43)

(42)

which is close to the lattice result (41) but di?ers from the suggestion (40). 9

IV. HEAVY QUARK-ANTIQUARK POTENTIAL WITH THE ACCOUNT OF RETARDATION EFFECTS AND ONE LOOP RADIATIVE CORRECTIONS

At present the static quark-antiquark potential in QCD is known to two loops [13,14]. However the velocity dependent and spin-dependent parts are known only to the one-loop order [15,16]. Thus we limit our analysis to one-loop radiative corrections. The resulting heavy quark-antiquark potential can be presented in the form of a sum of spin-independent and spin-dependent parts. For the spin-independent part using the relations (28), (29) with λV = 1 and including one-loop radiative corrections in MS renormalization scheme [15,16] we get VSI (r) = ?
2 4 β0 αs (?2 ) ln(?r) ? 4 αV (?2 ) + Ar + B ? 3 r 3 2π r 2 1 1 1 4 αV (?2 ) 4 β0 αs (?2 ) ln(?r) ? + + 2 ? ? ? + (1 ? ε)(1 + 2κ)Ar 8 m2 mb 3 r 3 2π r a 1 (p · r)2 4 αV ? + ? p2 + 2ma mb 3 r r2 W 2 2 (p · r)2 ln(?r) 1 4 β0 αs (? ) 2 ln(?r) p + ? ? 3 2π r r2 r r W 1 (p · r)2 1?ε ε 1 + + 2 Ar p2 ? ? 2ma mb 4 m2 mb r2 a W ελS 1 1 1 1 (p · r)2 ? + + Ar p2 + 2 2 m2 m2 ma mb r2 a b W 1 1 1 1 + + Bp2 , + 4 m2 m2 ma mb a b

(44)

where αV (?2 ) = αs (?2 ) 1 + ? 31 10 ? nf , 3 9 2 β0 = 11 ? nf . 3 a1 = Here nf is a number of ?avours and ? is a renormalization scale. For the dependence of the QCD coupling constant αs (?2 ) on the renormalization point ?2 we use the leading order result αs (?2 ) = 4π . β0 ln(?2 /Λ2 ) (46) a1 γE β0 + 4 2 αs (?2 ) , π (45)

Comparing this expression for VSI with the decomposition (30) we ?nd Vbc (r) = ?
2 ? 1 ? ε ελS 2 αV (?2 ) 2 β0 αs (?2 ) ln(?r) Ar + B, ? + ? 3 r 3 2π r 2 2 2 2 αV (?2 ) 2 β0 αs (?2 ) ln(?r) 1 1 ? ε ελS ? Vc (r) = ? ? Ar, ? ? + 3 r 3 2π r r 2 2

10

ε Vde (r) = ? (1 + λS )Ar + B, 4 ε Ve (r) = ? (1 ? λS )Ar. 4

(47)

It is easy to check that the BBP relations are exactly satis?ed. The spin-dependent part of the quark-antiquark potential for equal quark masses (ma = mb = m) with the inclusion of radiative corrections [15,17] can be presented in our model [20] as follows: VSD = a L · S + b 3 (Sa · r)(Sb · r) ? (Sa · Sb ) + c Sa · Sb , r2 αs (?2 ) 1 1 β0 ? β0 1 4αs (?2 ) 1+ nf ? + γE ?2 + ln a= 2 3 2m r π 18 36 2 2 m β0 A A + ? 2 ln(mr) ? + 4(1 + κ)(1 ? ε) 2 r r 2 2 αs (? ) 1 25 β0 ? β0 1 4αs (? ) 1+ nf + + γE ?3 + ln b= 3m2 r3 π 6 12 2 2 m β0 A + ? 3 ln(mr) + (1 + κ)2 (1 ? ε) 2 r αs (?2 ) 23 5 3 4 8παs (?2 ) 1+ ? nf ? ln 2 δ 3 (r) c= 2 3m 3 π 12 18 4 αs (?2 ) β0 2 ln(?/m) 1 1 ln(mr) + γE 1 + ? ? + ?2 nf ? π 8π r π 12 16 r A , +(1 + κ)2 (1 ? ε) r (48)

(49)

(50)

(51)

where L is the orbital momentum and Sa,b , S = Sa + Sb are the spin momenta. The correct description of the ?ne structure of the heavy quarkonium mass spectrum requires the vanishing of the vector con?nement contribution. This can be achieved by setting 1 + κ = 0, i.e. the total long-range quark chromomagnetic moment equals to zero, which is in accord with the ?ux tube [6] and minimal area [30,28] models. One can see from Eq. (48) that for the spin-dependent part of the potential this conjecture is equivalent to the assumption about the scalar structure of con?nement interaction [5].
V. HEAVY QUARKONIUM MASS SPECTRA

Now we can calculate the mass spectra of heavy quarkonia with the account of all relativistic corrections (including retardation e?ects) of order v 2 /c2 and one-loop radiative corrections. For this purpose we substitute the quasipotential which is a sum of the spinindependent (44) and spin-dependent (48) parts into the quasipotential equation (1). Then we multiply the resulting expression from the left by the quasipotential wave function of a bound state and integrate with respect to the relative momentum. Taking into account the accuracy of the calculations, we can use for the resulting matrix elements the wave functions

11

of Eq. (1) with the static potential

1

VNR (r) = ?

4 αV (?2 ) ? + Ar + B. 3 r

(52)

As a result we obtain the mass formula (ma = mb = m) b2 (M) =W + a L·S + b 2?R where W = VSI + p2 , 2?R 3 (Sa · r)(Sb · r) ? (Sa · Sb ) r2 + c Sa · Sb , (53)

1 L · S = (J(J + 1) ? L(L + 1) ? S(S + 1)), 2 3 6( L · S )2 + 3 L · S ? 2S(S + 1)L(L + 1) (Sa · r)(Sb · r) ? (Sa · Sb ) = ? , r2 2(2L ? 1)(2L + 3) 3 1 S(S + 1) ? , S = Sa + Sb , Sa · Sb = 2 2 and a , b , c are the appropriate averages over radial wave functions of Eqs. (49)-(51). We use the usual notations for heavy quarkonia classi?cation: n2S+1 LJ , where n is a radial quantum number, L is the angular momentum, S = 0, 1 is the total spin, and J = L ? S, L, L + S is the total angular momentum (J = L + S). The ?rst term on the right-hand side of the mass formula (53) contains all spinindependent contributions, the second term describes the spin-orbit interaction, the third term is responsible for the tensor interaction, while the last term gives the spin-spin interaction. To proceed further we need to discuss the parameters of our model. There is the following set of parameters: the quark masses (mb and mc ), the QCD constant Λ and renormalization ? point ? (see Eqs. (46), (44), (48)) in the short-range part of the QQ potential, the slope A and intercept B of the linear con?ning potential (10), the mixing coe?cient ε (9), the longrange anomalous chromomagnetic moment κ of the quark (8), and the mixing parameter λS in the retardation correction for the scalar con?ning potential (26). As it was already discussed in Sec. II, we can ?x the values of the parameters ε = ?1 and κ = ?1 from the consideration of radiative decays [8] and comparison of the heavy quark expansion in our model [22,33] with the predictions of the heavy quark e?ective theory. We ?x the slope of the linear con?ning potential A = 0.18 GeV2 which is a rather adopted value. In order to reduce the number of independent parameters we assume that the renormalization scale ? in the strong coupling constant αs (?2 ) is equal to the quark mass. 2 We also varied the quark

static potential includes also some radiative corrections [16]. The remaining radiative correction term with logarithm in (44), also not vanishing in the static limit, is treated perturbatively. numerical analysis showed that this is a good approximation, since the variation of ? does not increase considerably the quality of the mass spectrum ?t.
2 Our

1 This

12

masses in a reasonable range for the constituent quark masses. The numerical analysis and comparison with experimental data lead to the following values of our model parameters: mc = 1.55 GeV, mb = 4.88 GeV, Λ = 0.178 GeV, A = 0.18 GeV2 , B = ?0.16 GeV, ? = mQ (Q = c, b), ε = ?1, κ = ?1, λS = 0. The quark masses mc,b have usual values for constituent quark models and coincide with those chosen in our previous analysis [20] (see Sec. II). The above value of the retardation parameter λS for the scalar con?ning potential coincides with the minimal area low and ?ux tube models [28], with lattice results [29] and Gromes suggestion [24]. The found value for the QCD parameter Λ gives the following values for the strong coupling constants αs (m2 ) ≈ 0.32 and αs (m2 ) ≈ 0.22. c b The results of our numerical calculations of the mass spectra of charmonium and bottomonium are presented in Tables I and II. We see that the calculated masses agree with experimental values within few MeV and this di?erence is compatible with the estimates of the higher order corrections in v 2 /c2 and αs . The model reproduces correctly both the positions of the centres of gravity of the levels and their ?ne and hyper?ne splitting. Note that the good agreement of the calculated mass spectra with experimental data is achieved by systematic accounting for all relativistic corrections (including retardation corrections) of order v 2 /c2 , both spin-dependent and spin-independent ones, while in most of potential models only the spin-dependent corrections are included. The calculated mass spectra of charmonium and bottomonium are close to the results of our previous calculation [20] where retardation e?ects in the con?ning potential and radiative corrections to the one-gluon exchange potential were not taken into account. Both calculations give close values for the experimentally measured states as well as for the yet unobserved ones. The inclusion of radiative corrections allowed to get better results for the ?ne splittings of quarkonium states. Thus we can conclude from this comparison that the inclusion of retardation e?ects and spin-independent one-loop radiative corrections resulted only in the slight shift (≈ 10%)in the value of the QCD parameter Λ and an approximately two-fold decrease of the constant B. 3 Such changes of parameters almost do not in?uence the wave functions. As a result the decay matrix elements involving heavy quarkonium states remain mostly unchanged. 4 We plot the reduced radial wave functions u(r) = rR(r) for charmonium and bottomonium in Figs. 1 and 2.

that in Ref. [20] we included this constant both in vector and scalar parts, while the present analysis indicates that the better ?t can be obtained if the constant B is included only in the vector part (9). changes in decay matrix elements are of the same order of magnitude as the contributions of the higher order relativistic and radiative corrections.
4 The

3 Note

13

VI. CONCLUSIONS

In this paper we have considered the heavy quarkonium spectroscopy in the framework of the relativistic quark model. Both relativistic corrections of order v 2 /c2 and one-loop radiative corrections to the short-range potential have been included into the calculation. Special attention has been devoted to the role and the structure of retardation corrections to the con?ning interaction. Our general analysis of the retardation e?ects has shown that we have a good theoretical motivation to ?x the form of retardation contributions to the vector potential in the form (15) which corresponds to the parameter λV = 1 in the generalized expression (26). On the contrary, the structure of the retardation contribution to the scalar potential is less restricted from general analysis. This means that it is not possible to ?x the value of λS in (26) on general grounds. Our numerical analysis has shown that the value of λS = 0 is preferable. Thus for the energy transfer squared we have the symmetrized product (16) for the vector potential and a half sum (18) for the scalar potential, in agreement with lattice calculations [29] and minimal area law and ?ux tube models [28]. The found structure of the spin-independent interaction (44) with the account of retardation contributions satis?es the BBP [12] relations (33), which follow from the Lorentz invariance of the Wilson loop. In our calculations we have used the heavy quark-antiquark interaction potential with the complete account of all relativistic corrections of order v 2 /c2 and one-loop radiative corrections both for the spin-independent and spin-dependent parts. The inclusion of these corrections allowed to ?t correctly the position of the centres of gravity of the heavy quarkonium levels as well as their ?ne and hyper?ne splittings. Moreover, the account for radiative corrections results in a better description of level splittings. The values of the main parameters of our quark model such as the slope of the con?ning linear potential A = 0.18 GeV2 , the mixing coe?cient ε = ?1 of scalar and vector con?ning potentials and the long-range anomalous chromomagnetic quark moment κ = ?1 used in the present analysis are kept the same as they were ?xed from the previous consideration of radiative decays [8] and the heavy quark expansion [22,33]. The value of ε = ?1 implies that the con?ning quark-antiquark potential in heavy mesons has predominantly a Lorentz-vector structure, while the scalar potential is anticon?ning and helps to reproduce the initial nonrelativistic potential. On the other hand, the value of κ = ?1 supports the conjecture that the long-range con?ning forces are dominated by chromoelectric interaction and that the chromomagnetic interaction vanishes, which is in accord with the dual superconductivity picture [35] and ?ux tube model [6]. The presented results for the charmonium and bottomonium mass spectra agree well with the available experimental data. It is of great interest to consider the predictions for the masses of the 1 S0 and D levels of bottomonium, which have not yet been observed experimentally. The di?culty of their experimental observation is that these states (except 3 D1 ) cannot be produced in e+ e? collisions, since their quantum numbers are not the same as the quantum numbers of the photon. Therefore, in search for these states one must investigate decay processes of vector (3 S1 ) levels. We discussed the possibility of observation of these states in radiative decays in Ref. [8]. Note that the small value predicted for the hyper?ne splitting M(Υ) ? M(ηb ) ? 60 MeV leads to di?culties in observation of the ηb = state. 14

Recently it was argued [34] that the account of relativistic kinematics substantially modi?es the description of the charmonium ?ne structure and, in particular, leads to considerably larger values of the 23 PJ splittings than in the nonrelativistic limit. Both our previous calculation [20] and the present one con?rm this observation. Our prediction for the charmonium ? 23 P0 mass lies close to the prediction of Ref. [34] and slightly lower than the D D ? threshold. ? and close to D D ? thresholds makes threshold ? However, the fact that this state is above D D e?ects very important and can considerably in?uence the quark model prediction.
ACKNOWLEDGMENTS

We thank A.M. Badalyan, G. Bali, N. Brambilla, M.I. Polikarpov and V.I. Savrin for useful discussions of the results. Two of us (R.N.F. and V.O.G.) are grateful to the particle theory group of Humboldt University for the kind hospitality. The work of R.N.F. and V.O.G. was supported in part by the Deutsche Forschungsgemeinschaft under contract Eb 139/1-3.

15

REFERENCES
[1] [2] [3] [4] [5] C. Caso et al., Particle Data Group, Eur. Phys. J. C 3, 1 (1998). Yu.A. Simonov, hep-ph/9902233 (1999). G.S. Bali, Phys. Lett. B 460, 170 (1999). F.V. Gubarev, M.I. Polikarpov and V.I. Zakharov, hep-ph/9908293 (1999). H.J. Schnitzer, Phys. Rev. Lett. 35, 1540 (1975); W. Lucha, F.F. Sch¨berl and D. o Gromes, Phys. Rep. 200, 127 (1991); V.D. Mur, V.S. Popov, Yu.A. Simonov and V.P. Yurov, Zh. Eksp. Teor. Fiz. 78, 1 (1994) [J. Exp. Theor. Phys. 78, 1 (1994)]; A.Yu. Dubin, A.B. Kaidalov and Yu.A. Simonov, Phys. Lett. B 323, 41 (1994); Yu.A. Simonov, Phys. Usp. 39, 313 (1996). W. Buchm¨ ller, Phys. Lett. B 112, 479 (1982). u R. McClary and N. Byers, Phys. Rev. D 28, 1692 (1983). V.O. Galkin and R.N. Faustov, Yad. Fiz. 44, 1575 (1986) [Sov. J. Nucl. Phys. 44, 1023 (1986)]; V.O. Galkin, A.Yu. Mishurov and R.N. Faustov, Yad. Fiz. 51, 1101 (1990) [Sov. J. Nucl. Phys. 51, 705 (1990)]. N. Brambilla and A. Vairo, Phys. Lett. B 407, 167 (1997). A.P. Szczepaniak and E.S. Swanson, Phys. Rev. D 55, 3987 (1997). D. Ebert, R.N. Faustov and V.O. Galkin, Eur. Phys. J. C 7, 539 (1999). A. Barchielli, N. Brambilla and G.M. Prosperi, Nuovo Cim. A 103, 59 (1990). M. Peter, Phys. Rev. Lett. 78, 602 (1997); Nucl. Phys. B 501, 471 (1997). Y. Schr¨der, Phys. Lett. B 447, 321 (1999); hep-ph/9909520 (1999). o S. Gupta and S.F. Radford, Phys. Rev. D 24, 2309 (1981); ibid. 25, 3430 (1982); S. Gupta, S.F. Radford and W.W. Repko, Phys. Rev. D 26, 3305 (1982). S. Titard and F.J. Yndurain, Phys. Rev. D 49, 6007 (1994); ibid. 51, 6348 (1995). J. Pantaleone, S.-H.H. Tye and Y.J. Ng, Phys. Rev D 33, 777 (1986). A.A. Logunov and A.N. Tavkhelidze, Nuovo Cimento 29, 380 (1963). A.P. Martynenko and R.N. Faustov, Teor. Mat. Fiz. 64, 179 (1985). V.O. Galkin, A.Yu. Mishurov and R.N. Faustov, Yad. Fiz. 55, 2175 (1992) [Sov. J. Nucl. Phys. 55, 1207 (1992)]. E. Eichten and F. Feinberg, Phys. Rev. D 23, 2724 (1981). R.N. Faustov and V.O. Galkin, Z. Phys. C 66, 119 (1995). A.I. Akhiezer and V.B. Berestetskii, Quantum Electrodynamics (Interscience Publishers, New York, 1965). D. Gromes, Nucl. Phys. B 131, 80 (1977). M.G. Olson and K.J. Miller, Phys. Rev. D 28, 674 (1983). W. Celmaster and F.S. Henyey, Phys. Rev. D 17, 3268 (1978). Yu-Qi Chen and Yu-Ping Kuang, Z. Phys. C 67, 627 (1995). N. Brambilla and A. Vairo, Phys. Rev. D 55, 3974 (1997). G.S. Bali, A. Wachter and K. Schilling, Phys. Rev. D 56, 2566 (1997). N. Brambilla, P. Consoli and G.M. Prosperi, Phys. Rev. D 50, 578 (1994). Y.F. Gu and S.F. Tuan, hep-ph/9910423 (1999). K.W. Edwards et al. (CLEO Collaboration), Phys. Rev. D 59, 032003 (1999). D. Ebert, R.N. Faustov and V.O. Galkin, Phys. Lett. B 454, 365 (1998); CERN-TH/99175, hep-ph/9906415, Phys. Rev. D to be published. 16

[6] [7] [8]

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

[34] A.M. Badalyan, V.L. Morgunov and B.L.G. Bakker, hep-ph/9906247 (1999); A.M. Badalyan and V.L. Morgunov, hep-ph/9901430 (1999). [35] M. Baker, J.S. Ball, N. Brambilla, G.M. Prosperi and F. Zachariasen, Phys. Rev. D 54, 2829 (1996).

17

TABLES
TABLE I. Charmonium mass spectrum. State (n(2S+1) L 11 S 13 S
0 1 J)

Particle ηc J/Ψ χc0 χc1 χc2
′ ηc Ψ′

Theory 2.979 3.096 3.424 3.510 3.556 3.583 3.686 3.798 3.813 3.815

Experiment [1] 2.9798 3.09688 3.4173 3.51053 3.55617 3.594 3.686 3.7699?

Experiment [31] 2.9758

13 P0 13 P1 13 P2 21 S0 23 S1 13 D1 13 D2 13 D3 23 P0 23 P1 23 P2 31 S0 33 S1 23 D1 23 D2 23 D3
?

3.4141

χ′ c0 χ′ c1 χ′ c2
′′ ηc Ψ′′

3.854 3.929 3.972 3.991 4.088 4.194 4.215 4.223

4.040? 4.159?

Mixture of S and D states

18

TABLE II. Bottomonium mass spectrum. State (n(2S+1) LJ ) 11 S0 13 S1 13 P0 13 P1 13 P2 21 S0 23 S1 13 D1 13 D2 13 D3 23 P0 23 P1 23 P2 31 S0 33 S1 23 D1 23 D2 23 D3 33 P0 33 P1 33 P2 41 S0 43 S1 χ′′ b0 χ′′ b1 χ′′ b2
′′′ ηb Υ′′′

Particle ηb Υ χb0 χb1 χb2
′ ηb Υ′

Theory 9.400 9.460 9.864 9.892 9.912 9.990 10.020 10.151 10.157 10.160

Experiment [1] 9.46037 9.8598 9.8919 9.9132

Experiment [32]

9.8630 9.8945 9.9125

10.023

χ′ b0 χ′ b1 χ′ b2
′′ ηb Υ′′

10.232 10.253 10.267 10.328 10.355 10.441 10.446 10.450 10.498 10.516 10.529 10.578 10.604

10.232 10.2552 10.2685

10.3553

10.580

19

FIGURES

0.6 0.4 u(r)/GeV1/2 0.2 0 -0.2 -0.4 -0.6 0 2 4 6 r . GeV 8 10 12

FIG. 1. The reduced radial wave functions for charmonium. The solid line is for 1S, bold line for 2S, long-dashed line for 1P , dashed-dotted line for 2P , and dotted line for 1D states.

0.75 0.5 u(r)/GeV1/2 0.25 0 -0.25 -0.5 -0.75 0 2 4 6 8

r . GeV

FIG. 2. The same as in Fig. 1 for bottomonium and long-short-dashed line for 3S state.

20


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