The dynamical retardation effect for the mass spectrum of bound qq̄ systems

NUCLEAR PHYSICS A ELSEVIER

Nuclear

Physics A 587 (1995) 758-768

The dynamical retardation effect for the mass spectrum of bound qS systems T. Kopaleishvili, A. Rusetsky High Energy Physics Institute,

Tbilisi State Uniuersity, 9, University st. 380 086 7’bilisi, Georgia Received

7 October

1994

Abstract The Logunov-Tavkhelidze quasipotential approach is used in order to evaluate the first-order corrections to the bound-qq-system masses due to the dynamical retardation effect. The confining qq kernel is taken to be a 4-dimensional oscillator in the momentum space, being the covariantized version of the static 3-dimensional oscillator potential. The parameters arising in the kernel are determined via the comparison of the quasipotential to the static 3-dimensional kernel in the nonrelativistic limit p/m ---)0. It turned out that the first-order perturbative corrections due to the retardation are of a relatively small size and do not exceed 30% in the light-quark sector. The ambiguities arising in the transition from the 3-dimensional to the 4-dimensional description of the qq system under study are discussed.

1. Introduction The Bethe-Salpeter (BS) equation provides the natural basis for the unified relativistic treatment of bound qq systems in the framework of the constituent quark model both in the light-quark (u, d, s) and heavy-quark (c, b) sectors. However, due to the lack of the probability interpretation of the 4-dimensional (4D) BS amplitude as well as due to the serious mathematical diseases which are inherent in the BS approach to the bound-state problem, various 3-dimensional (3D) reduction schemes of the original BS equation [l-3] are often used. Within this approach the retardation effect in the kernel of the BS equation is, in principle, fully taken into account. However, in practice, the 3D reduction of the BS equation is, as a rule, carried out with the use of the instantaneous (static) approximation for the kernel of the BS equation. If only this approximation is used in the BS equation, then this equation reduces to the Salpeter equation [4] for the 3D wave function which has the probability interpretation. The Salpeter equation 0375-9474/95/$09.50 0 1995 El sevier Science SSDI 0375-9474(94)00723-3

B.V. All rights reserved

T. Kopaleishvili, A. Ruse&

/Nuclear

Physics A 587 (1995) 758-768

759

without further approximations was used for the description of the bound qq systems in Refs. [5-lo], whereas some additional approximations were made in Refs. [11,12]. The instantaneous approximation for the kernel of the BS equation was used also in Refs. [13,14], where the problem was considered in the framework of null-plane dynamics, as well as in Ref. [15], where the effective three-dimensional free two-fermion propagator was chosen in the form guaranteeing the correct one-body limit when the second particle mass tends to infinity. As a next step within the approach considered above, there arose the problem of taking into account the retardation effect in the kernel of the BS equation (the dynamical retardation effect). To our knowledge, the first attempt in this direction was made in Refs. [16,17], namely, in our previous papers [161 we constructed the first-order quasipotential for the qq interaction using the Logunov-Tavkhelidze method [1,3] and estimated the retardation effect for the masses of bound qq systems of light quarks, using the oscillator-type confinement qq interaction potential. It turned out that for some mesons the dynamical retardation effect is large enough, rendering the treatment of this effect as perturbation doubtful. An analogous result was obtained in Ref. [17], where the corrections coming from the retardation were calculated for the bound q?j system of heavy quarks for the linear-confinement-potential case. Here it is necessary to note that the procedure used in Refs. [16,17] to estimate the retardation effect for meson masses is not unique. In fact, from QCD one cannot directly obtain the explicit expression for the 4D kernel of the BS equation, corresponding to the confining part of the qq interaction. For this reason in Refs. [16,17] the following approach was used: starting from the 3D instantaneous (static) kernel of the BS equation corresponding to the given form of the confining potential (square or linear), 4D analogues of these kernels were constructed merely by covariantization of these 3D expressions, keeping in mind that this procedure is correct for the one-gluon exchange part of the q?j potential. However, there arises a new problem. Namely, the 4D confining kernel constructed with the use of the method described above needs infrared regularization which introduces additional dimensional constants into the expression of this kernel, and some additional physical constraints are needed in order to determine these constants. In the present work we shall dwell upon this problem in detail.

2. The first-order retardation effect in the quasipotential approach In order to estimate the retardation effect for the masses of bound qo systems, below we use the basic relations of the Logunov-Tavkhelidze approach formulated in Ref. [l] for two spinless particles and generalized in Ref. [3] for the case of two fermions. The quasipotential in this approach is defined by the expression ~=G+JZGG-~,

(1)

where K is the full 4D BS kernel and G and G, are, respectively, the full and free

T. Kopaleishvili, A. Rusetsky / Nuclear Physics A 587 (1995) 758-768

760

2-fermion Green functions. The procedure follows:

“-” for any operator

A is defined as

(2) Here P and p, P’ are, respectively, the total and relative 4-momenta. The quantities c and ~$a are related to the corresponding quantities G and G,, in the following way (in c.m.f.1 G‘o(M,, P? P’) = (S,(M,;

P, P’) = (2”)36(3’(P

= -i[ M, + io - h,(P) G = co

+

WG

- h*( -P)]

= Go + G,KG

-P’)(m43~

PI

-‘r;r;))rPr;K

+ &( 1 - r;$ll)

(3)

= G + &( 1 - &,“n), (4)

where

and mj and M, are, respectively, the quark and bound qq system masses. The equal-time BS wave function

where +(P”, p) is the 4D BS wave function, obeys the following quasipotential equation: p&-h(P)

-M-P)]@(P)

=~$+,,..,,,;

PY P’)@(P’)>

(7)

where I&&&;

P, P’)

= -i&@(ME3;

P, P’> = (PI!?

I P?].

(8)

The first-order quasipotential is obtained from (1) if the full Green function G is replaced by the free Green function Go. As a result we have I’$,(&;

P, P’)

= -~~~~~(PI~O’~G~~~‘IP’).

(9)

It is necessary to note that if the kernel of the BS equation is taken in the instantaneous approximation, i.e. when K(P? P, P’) -K,,(P;

P, P’>,

(10)

T. Kopaleishvili, A. Ruse&y/Nuclear

Physics A 587 (1995) 758-768

761

then it can be easily checked that the solution of the Salpeter equation with the bound-state mass Mi,

(11) where v,,(PY P’) =Z%4(-M,,(r,,

P’),

(12)

obeys the corresponding quasipotential equation in the static limit. On the other hand, for the first-order quasipotential in the static limit we have v$&(Mn;

P, P')* v,,(P, P')qP')Y;Y;.

(13)

and, consequently, we do not arrive at the Salpeter equation in the first-order approximation. However, if in the Salpeter wave function cp= pp(++)+ p(--), where cp(f ) = A$* )Ay )p it were possible to neglect the negative-energy component cp(--‘, then the use’of the potential (13) would yield the same equation as the use of the potential (12). Thus, in this case the full quasipotential equation and the first-order quasipotential equation reduce to the same equation for the wave function cp(++) in the static limit. Therefore, if in the calculations of the mass spectrum of the bound qq system the negative-energy component of the wave function can be neglected, then for the estimation of the dynamical retardation effect, which is determined by the difference AK=K(P;

P, p’)

-K,,(p,

P’),

(14)

the first-order quasipotential equation can be used. In our previous paper [lo] we have investigated the effect of the account of the negative-energy component of the wave function in the mass spectrum of the bound qq system within the approach based on the Salpeter equation. Taking the spin structure of the confining part of the potential to be zy!rt + (1 - x)Z X Z (0 GX G l), it was demonstrated that when the solutions of the equation exist (X z 0.2-0.3), then the contribution of the negative-energy component of the wave function to the calculated meson masses is negligible. Therefore the use of the first-order quasipotential equation for the estimation of the dynamical retardation effect in the bound qa system masses seems to be justified. To construct the first-order quasipotential (9) for the kernel K(P; p, p’> we used the expression corresponding to the confining part of the qq interaction which, in general, cannot be built directly from QCD. We start from the 3D oscillator-type confining potential

which is an approximate version of the potential used in Ref. 1101for light quarks (the constant A, from Ref. [lo] is very small, otherwise r2 in Eq. (15) should be substituted by r*/ 1 +m,m,A,r*) and as@*) is the QCD running coupling

162

T. Kopaleishvili, A. Rusetsky /Nuclear Physics A 587 (1995) 758-768

constant. In order to find a 4D generalization of this potential (in the momentum space) we can choose either of two possibilities: l Taking into account the relation r2= ;mO[ -$(?)I,

(16)

which in the momentum space reads as (21r)~A,6’~‘( 4) = lims $

l

(17)

we can generalize the last expression merely by the replacement =q 2 . Writing down the potential (1.5) in the momentum space, I/,(q) = +s(~:2)(~1*12~~Ag

-q*

+ v,)(2~)3~‘3’(q)>

+ 4”’ - q2

(18)

we can consider the covariant 4D version of this expression, V,(Ma, p, p’) = -i&(M,; = $4

P, p’)

42,q

x25TA,s(p”-p’o),

i/J

,,w;a;

-

V(J(2Tf)j6’3’(

p -p’) (19)

where A, is an additional free parameter having the dimension of mass, and the renormalized parameter to does not necessarily coincide with V,. The values of the parameters A, and V, should be detnermined with the use of appropriate physical constraints. As to the operator 0, in Eq. (191, it determines the spin structure of the confining part of the potential. In the present paper we use the second way for deriving the covariant version of the oscillator-type confining potential. Note that we attempt to treat the operator (14) with HP; p, p’> defined by Eq. (19) as a perturbation in the calculation of the size of the dynamical retardation effect, i.e. we solve Eq. (7) with the first-order quasipotential (9) by the perturbative method, calculating the mass M, in the first approximation, whereas the wave function 4(p) is taken in the zeroth approximation. Keeping in mind the result obtained in Ref. [lo], namely that the negative-energy component c&-) of the full wave function cp= c#++) + &-) can be neglected, we can consider the equation Cm, = m2 = ml (&-2w)$++)(p)

=/-$

A~‘A$+)V~~~,(MB; p, p’)A’:‘A$+‘c$++‘( p’) (20)

instead of Eq. (7). Below we consider the retardation effect which stems purely from the confining qq interaction and neglect the one-gluon exchange part of the potential. Note that the meson masses within the constituent quark model are

T. Kopaleishvili, A. Ruse&

predominantly we obtain

determined

/ NuclearPhysics A 587 (1995) 758-768

763

by the confining part of the qq interaction. In this case

4+‘( PI WY -P) q&w (M,;

P’>4+)(P’)4+)(-P’)

P,

$,(m;,)~(,+)(p)A(,+)(

= (2rr)36(3)(~ +)

-p)A$+)( -p)Gc

i 2 X

(MB-2w+iO)w

‘p-t

2 + m2/w2

2m2/w2 - (M,-2~+i0)~

5

w p

li’:‘( p’)d2+)( -p’)

+ (MB-2W+iO)w

Next we fix the values of the constants A, and v,. For this purpose it is natural to demand that in the nonrelativistic limit p/m --f 0 the quasipotential is reduced to the potential (18). Then we have 2A,(M,) V&V,)

= -(MB--2m), = VO+ y

2

3

i

i

m(Mn-2m)

+ (M,-2m)2

1*

(22)

Further, taking into account the result obtained in Ref. [lo], according to which for the confining potential with the spin structure dC =xyFyt + (1 -x)Z X Z the mass spectrum reveals the weak dependence on the value of x, provided that the solution of Salpeter equation exists (X k 0.2-0.3), we fix the value of x at 0.5. We look for the solution of Eq. (20) in the following form [lo]:

Using the partial-wave expansion

/q”‘(P) = c

(nlL!W&)R~+,!(p),

n = $,

s = 0,l

(24)

LSJM,

for the radial wave functions R’(L+si(p),the following equation can be obtained (the mixing between the states with L = J - 1 and L = J -t1 is not taken into account since its contribution to the calculated meson masses can be shown to be negligible): (MB - 2W)ZQ?(P)

= I& ,

( ME%,P)ZG?(P),

(25)

T. Kopaleishvili, A. Ruse&y /Nuclear

764

Physics A 587 (1995) 758-768

where Q,l,(&~

P)

= - 4’Ys(4m*) ~~~~~~~~j~(l+~)(~A’iv,j 2J(J+

mofj 3m4 + i -2w”+V-J) 4 I

1) 1 w-m ---w2 w+m

2J+l

2 + m2/w2 + 3m4/w4 + 2(1+ m2/w2)pd/dp

I

m2

w4’dp

-

+

1

m2/w2)

(M, - 2w + io)2

+

=

- 2m)

m2/w2 (26)

+ (M, - 2m)* II

and A; = d2/dp2 + (2/p) d/dp - L(L + l>/p2. The partial-wave decomposition of the Salpeter equation for the radial wave functions Rp$(p): (Mj - 2w)W&P)

3( 1 + m2/w2) 2m(M,

2w( M, - 2w + i0)

_ (m*/w*)(l

d

equation

leads to a similar

Q?(P)Rgj(P),

(27)

where \

I$$(

p) = - &(4m*)

[ 12( 1 + I141. w2)(Y+:+vO)

3m4 -3+(L-J)

25(5+1)

1 w-m

2J+l

w* w+m

m2

d

(28)

3. The dynamical input and the calculation of the first-order perturbative corrections The first-order perturbative corrections to the meson masses due cal retardation effect can be obtained directly from Eqs. (25)-(28). the meson mass in the form M, = Mj + SM$) and replacing M, expression for the quasipotential (26), we immediately arrive at equality:

to the dynamiWriting down by Mi in the the following

SM;“imp2 dp( RI+,:)’ zz imp2 dp@i!(p)[t,,,,(M::,

P)-(M:-~~)]W(P).

(29)

T. Kopaleishuili,A. Rusetsky/ Nuclear PhysicsA 587 (I 995) 758-768

765

Here we have used the self-adjointness of the operators Qeff,JA4& p) and $,,,,04& p) as well as the bound-state equation (27). The expression for Kr,,.A@, P) cant ains the energy denominators which become singular in the integration region if Mi > 2m. In order to ensure that 6Mf) calculated from Eq. (29) is real, in analogy with Refs. 116,171,we have neglected the imaginary part of &&M:, P), cho osing the principal-value prescription for the singular energy denominators appearing in Eq. (26). In the calculations of the energy shift 6M#’ we follow the conventions and parametrization of Ref. [lo]. For the QCD coupling constant as(Q2) the familiar expression is used, 4Q2)

(30)

= $5, f

where IZ~is the number of colours (n, = 3 in the light-quark sector) and A is the QCD scale parameter. We take the value A = 120 MeV, as in Ref. [lo]. The u, d, s quark masses are fixed at the following values: m, = md = 280 MeV, m, = 400 MeV. The values of the parameters wO, V, are o0 = 710 MeV, V, = 525 MeV. The unknown radial wave functions R (G] from (291, which are the solutions of Eq. (27), are expanded in the basis of the nonrelativistic oscillator wave functions RnL(p) which are solutions of Eq. (27) in the nonrelativistic limit [lo], RG(

P) = iY WJR*L( n=O

PI*

(31)

Substituting Eq. (31) into Eq. (27) and truncating the infinite sum at some fixed value N,,, we obtain a system of homogeneous linear algebraic equations which allows for the determination of the eigenvalues M$“’ as well as the eigenvectors c;;J$$Z,i = 1, . . . ) Iv,,. It is easy to check that due to the self-adjointness of the

Table 1 First-order

retardation

uii, 1+2-+ 1++ I-1-o++ 2++ ss I-1-z++ 1++

in the light-quark

sector

I@ (MeV)

SM$‘) (MeV)

PC7701 ~(1600) a,(9801 a,(1320)

960 1283 1563 1283 989 1607 1283 1335

- 206 15 279 15 - 181 -88 15 116

0.22 0.011 0.18 0.011 0.18 0.055 0.011 0.087

$(1020) 60680) f,(1270) f,(1420)

1227 1864 1570 1540

- 357 - 181 2 -72

0.29 0.097 0.0015 0.047

Mesons

o-+

corrections

da rr(l40) S,(1235) ~,(1670) A,(12601

766

T Kopaleishuili, A. Ruse&/Nuclear

Physics A 587 (1995) 758-768

operator &,, all eigenvalues are real. Increasing then N,,,,, we ensure that the lowest eigenvalues remain stable, whereas the absolute value of the n-th component of the eigenvector CAt)y), corresponding to the ith energy level, rapidly decreases with the increase of I n - i I. We interpret this as the existence of stable solutions to the initial Salpeter equation (27), given by Eq. (31) where the summation is carried out from n = 0 to n = N,,, provided N,, exceeds the saturation point. In practical calculations the choice N,,, = 5 already provides an extremely high accuracy for the lowest energy levels in all partial-wave channels, so the use of the above-described numerical algorithm for the problem under consideration seems to be appropriate. Having obtained the solutions to the Salpeter equation (271, the calculation of the first-order retardation corrections with the use of Eq. (29) is then a straightforward task. The results of such calculations in the light-quark sector are presented in Table 1.

4. Results and discussion As mentioned above, in the present paper an attempt is made at an evaluation of the first-order perturbative corrections to the meson masses due to the dynamical retardation effect. As one can easily observe from Table 1, the retardation corrections do not exceed 30% for any of the mesons considered here. Moreover, the corrections in n-excited states are much less than in the corresponding ground state. Further, it can be seen that the retardation corrections tend to increase the level splitting between L-exited states both for the r/A, and p/J, trajectories, e.g. for the masses of ‘L, mesons for L = 0,1,2 we obtain the values 960, 1283, 1563 MeV in the static approximation, whereas the retardation corrections change these values to 754, 1297, 1842 MeV. Note that we have neglected the one-gluon exchange part of the potential as well as set the value of the parameter A, from Ref. [lo] without altering the other parameters of the model. Consequently, we do not expect an accurate description of the observed meson masses, having focused our attention on the evaluation of the relative size of the retardation effect. A few remarks, however, should be made concerning the approach used in the present paper to the evaluation of the retardation corrections. As already mentioned, the choice of the 4D kernel corresponding to the 3D confining qq interaction potential is by no means unique and different prescriptions for the construction of the kernel may, in general, lead to different predictions. In particular, constructing the 4D confining kernel, one has to fix the value of an additional parameter (with the dimension of mass) appearing due to the infrared singular behaviour of the confining interaction. Note that this parameter can emerge either from the beginning as A,, in Eq. (19) or after the regularization of infrared-divergent kernels as a regulator mass [17,18]. Instead of introducing a new parameter, one can impose a constraint directly on the bound-state equation [19]. Moreover, the parameters which already enter the 3D confining potential, such as the constant term V,,, need the renormalization in 4 dimensions [19]. The variety of

i? Kopaleishvili, A. Ruse&y /Nuclear

Physics A 587 (1995) 758-768

767

constraints which are used in order to fix the values of the parameters entering the 4D interaction kernel renders the identification of the retardation corrections less transparent, e.g. the comparatively small size of retardation corrections obtained in the present paper as compared with the results given in Refs. [16,17] is partially caused by the appropriate choice of the constraint for determining the values of A, and v, (Eq. (22)). Another remark concerns the applicability of the perturbative method to the evaluation of retardation corrections in the presence of the confining interaction. Our initial 4D model (19) is, in fact, equivalent to the FKR model [20] which suffers from the existence of the unnorrnalizable timelike excitations (see also Ref. [21]). The three-dimensional reduction scheme for BS equation used in the Logunov-Tavkhelidze approach must, in principle, suppress such excitations. However, if these excitations are not totally excluded, this can manifest itself in the divergence of the perturbative series for energy levels, though the first-order corrections are relatively small. Further, note that the use of the pe;turbative expansion is complicated due to the dependence of the quasipotential V&T& (26) on the eigenvalue M, we are looking for. The presence of singular energy denominators in the quasipotential can also be a source of divergence of perturbative expansion. Indeed, a more careful investigation of subsequent perturbative corrections is needed in order to draw definite conclusions about the size of the retardation effect in bound q?j systems. This aspect of the problem is now under study.

Acknowledgments The authors express their deep gratitude to A. Chanturia for his assistance in preparing the manuscript. One of the authors (A.R.) acknowledges support under grant UR 94-6.7-2042

References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]

A.A. Logunov and A.N. Tavkhelidze, Nuovo Cimento 29 (1963) 380. R. Blankenbeckler and R. Sugar, Phys. Rev. 142 (1966) 1051. A.A. Khelashvili, Dubna preprint JINR P2-4327 (1969) [in Russian]. E.E. Salpeter, Phys. Rev. 87 (19.52) 328. C. Long, Phys. Rev. D 30 (1984) 1970. A. Archvadze, M. Chachkhunashvili and T. Kopaleishvili, Kiev preprint ITP-85-131E (1985). M. Chachkhunashvili and T. Kopaleishvili, Few-Body Systems 6 (1989) 1. A. Archvadze, M. Chachkhunashvili and T. Kopaleishvili, Sov. J. Nucl. Phys. 54 (1991) 1109. A. Archvadze, M. Chachkhunashvili and T. Kopaleishvili, Few-Body Systems 14 (1993) 53. A. Archvadze, M. Chachkhunashvili, T. Kopaleishvili and A. Rusetsky, Nucl. Phys. A 581 (1995) 460. 1111A.N. Mitra, Z. Phys. C 8 (1981) 25. [12] A. Gara, B. Durand and L. Durand, Phys. Rev. D 42 (1990) 1651. [13] N.N. Singh, Y.K. Mathur and A.N. Mitra, Few-Body Systems 1 (1986) 47.

768

T. Kopaleishvili, A. Ruse&y /Nuclear Physics A 587 (1995) 758-768

[14] K.K. Gupta, A.N. Mitra and N.N. Singh, Phys. Rev. D 42 (1990) 1604. [15] P.C. Tiemeijer and J.A. Tjon, Phys. Rev. C 48 (1993) 896. [16] T. Kopaleishvili and A. Rusetsky, Russian J. Nucl. Phys. 56 (7) (1993) 168. [17] N. Brambilla and G.M. Prosperi, Phys. Rev. D 46 (1992) 1096; D 48 (1993) 2360. [18] N. Lucha, F. Schdberl and D. Gromes, Phys. Reports 200 (1991) 127. [19] F. Gross and J. Milana, Phys. Rev. D 43 (1991) 2401. 1201 R.P. Feynman, L. Kisslinger and F. Ravndal, Phys. Rev. D 3 (1971) 127. [21] S.N. Biswas, S.R. Choudhury, K. Dutta and A. Goyal, Phys. Rev. D 26 (1982) 1983