M

30 June 1994

N

PHYSICS LETTERS B

Physics Letters B 331 (1994) 174-178

ELSEVIER

Heavy meson spectra from relativistic B-S equations to the order 1/M Yuan-Ben Dai, Chao-Shang Huang, Hong-Ying Jin Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China

Received 9 December 1993;revised manuscript received 16 April 1994 Editor: H. Georgi

Abstract

Using the relativistic B-S equation with the kernel containing a confinement term and a gluon exchange term in a covariant generalization of the Coulomb gauge the formula for masses of heavy mesons with arbitrary spin and parity are obtained to the order 1/M. The retardation effect is taken into account. Numerical values for low-lying states of D and B systems are obtained.

Remarkable progress in the physics of hadrons containing a heavy quark has been achieved recently. It was pointed out that in the limit of the heavy quark mass M ~ ~ a spin-heavy flavor symmetry emerges in such systems [ 1 ]. Systematic expansions in 1/M can be carried out with the heavy quark effective theory [2]. Such expansions contain a series of coupling constants and form-factors which can not be determined from symmetry considerations alone. QCD sum rules have been used to calculate some of these parameters. Such expansions can also be carried out in the Bethe-Salpeter formalism [ 3 - 5 ] . In Ref. [4] it has been shown that the number of components of the general relativistic covariant B - S wave function of a heavy meson with arbitrary spin-parity is reduced from eight to two in the M ~ ~ limit. By using some model for the B - S kernel this formalism can be used to obtain various parameters in the 1/M expansions. The calculations in the lowest orders in 1 / M are made simpler by the simple form of B - S wave functions in the leading order. As a first step in calculations of the first order corrections in 1/M, in this note we have calculated the spectra of low-lying heavy meson states. We use the kernel which is the sum of a scalar confinement term and a gluon exchange term in a covariant generalization of the Coulomb gauge in the rest frame of the meson. The spectra of heavy mesons have been calculated with different relativistic wave equations and approximations in the literature [6,7]. Our calculation has the following features. (i) With the assumed kernel our method gives an exact expansion in 1/M. (ii) Our calculation is fully relativistic in the light components. The retardation effect of the transverse gluon has been taken into account. (iii) The equations used are manifestly covariant. The wave functions obtained by these equations can be used for moving mesons. In the following we shall first give a brief review of the results of Ref. [4]. After this, we shall present the derivation of 1/M corrections of B - S wave functions and the mass eigenvalues. Numerical results obtained for the spectra of low-lying heavy mesons in D and B series are given. Finally'the results obtained will be discussed. This work was supported in part by the National Science Foundationof China. 0370-2693/94/$07.00 © 1994Elsevier Science B.V. All rights reserved SSDI0370-2693 (94)00548-L

K-B. Dai et al. /Physics Letters B 331 (1994) 174-178

175

The B-S equation for a meson can be written as follows: d4q XP(P) =Sh(A1P+P) f G(P,p, q)XP(q) (2~.)4 S"i t - AzP+p)

(1)

where AI = M I / ( M 1+Mz), l~2=M2/(M1-t-M2), Ml, M2 are masses of the two quarks, P=Mv, M is the meson mass, v is the meson velocity, p is the relative momentum. Letpl = v .p, p, = p -pl/). For convenience, we shall shift the Pl variable by defining p{ = P l - [ M E / ( M l nt-M2) ]M and omit the prime in the following. In the case M~ >> Pl = O(AQcD) and MI >> M2 = m, the residue of XP(P) in ( 1 ) at the pole of the heavy quark propagator in the upper halfpj plane is of the order 1/M 2. Therefore, this propagator can be expanded as 1 +t~ Sh( )tlP +p) =

E~(1 + 0 )

2(pl+Eo+m+ie)

+

2(pl+Eo+m+ie) 2

[Pt 12( 1 + C ) 4Ml(pl +Eo + m +ie) 2

1-___._C+ /~t +(1_I 4Mi 2M1 (Pl + Eo + m + ie) I,M~]

(2)

where M - M 1 - m = Eo+E~ + O( 1/M2), E o and E1 are the first two terms in the expansion of the bound state energy. Substituting (2) into ( 1 ) we obtain the following equations:

x(p) =Xo +x~- + x / - , X°(P) = -

l+t~ f d4q plO+fit+m 2(pl+Eo+m+iE) G+(P'P'q)x°(q) (27r)4p~-W2+ie'

1-G X1- (P) = 4Ml

f

d4q

f

X~-(P) = - 2 ( p ) + E o + m + i e )

plC-t-13t + m

G + ( P ' P ' q ) x + ( q ) -(277.)4 p2 _ W2 +ie + J [ X o ] ,

d4q pl0+/3 t + m fit G ( P , P , q ) x ~ ( q ) (2,n.)4p2_W2+i e + 2M1 Xo(P) , . . . .

(3) (4) (5)

where Xo is the zeroth order wave function, X ± are first order wave functions satisfying t3X± = + X± and IPt 12 El J[Xo] = 2Ml(Pl +Eo + m + i e ) X o - Pl +Eo + m + i e X° 1+0

2

f

fit

2Ml(pl+-Eo+m+i~)

d4q pIC+13t+m G_(P,p, q)Xo(q) (27r)4p2-Wp+ie 2

1 +6 f d4q plU-l-fit +m - 4Ml(pl+Eo+m+ie) G_(P,p, q)qtXo(q) ( 2 7 r ) a p Z _ W 2 + i e l+t~ 8M,

ff

G-(P'P'q')(1-O)G-(P'q"q)x°(q) 1 +0

- 2Ml(Pl +Eo + m + i e )

f

q~O+~t+m d4q d4q ' plO+/St+m qt _-2~- -Wq . - - + l e (2"n') 4 ('2-'~-' p-~ - W-----~+ i--E

d4q pl0+/3~+m G,(P, p, q)Xo(q) (277.)4 p"-~__ W-""~;i"-E '

(6)

where Wp = ~/IP, I z + m 2, G = Go + ( 1/Ml ) G l + 0 ( 1/MI ), Go = G + + G_, and the vertex on the heavy quark line of G+ and G_ commutes and anti-commutes with 0 respectively. It has been shown in Ref. [4] that the zeroth order equation (3) has two series of solutions corresponding to degenerate doublet states of the following form:

Y.-B. Dai et al. / Physics Letters B 331 (1994) 174-178

176

xo,,,(p) = F 2 - F ~ --2- ~ 'Oo,,...,~,pt ... pF'

F~

J ~,~"~, (+,j-Eg,2j) 2j+l

1 l+t? X'oej+~(P) = r : nm...,~J+, YmPt~ ." PtJ+'( q~u-fit~b2j)

V2

(7)

2

and x~)e~(p) = W 2 j + ~ - - 7 1 1+0

X ° e J + ' ( P ) = V~

2

"r/m""~JPt~ " ' p• 5

o'1 a,2

3' 77..... ,~j+,3' Pt

tl

J 1 v'P, 2j+

q',j -fittPzj) , (g)

"..P~+l(~lj--fit~2j)

They correspond to meson doublets of the spin-parity [j ( - l), + l, (j + 1) ( l)J+, ] and [j ( - ~)', (j + 1) ( l~,] respectively. These results can be proved under the most general condition that Go contains only the Dirac matrices I and Y on the heavy quark line. This is equivalent to the condition that in the limit M~ --* w the interaction does not depend on the spin of the heavy quark. The partial wave equations for ~bu, q~zj, q4j and O2j can be found in Ref. [4]. We assume that the kernel is the sum of a static scalar confinement potential and a gluon exchange term in the Coulomb gauge in the rest frame of the meson. In order to obtain wave functions of moving mesons we generalize the kernel to the following covariant form G = G+ + G_, where (. -iG+=l®l

8"n'K ( i p _q~12+/x2) 2 -- (2'rr)3t~3(pt --qt)

f

{ (/~,-qO ®(A-qO~ - i c _ =-/-r-®,y,~-o®o+ g ~ - - q t t l :f

8"n'K

d3k ] _ 6 ® 0

(k2+].£2) 2 (27r)3] 16"nOtseff

)3(Ipt-qtl2-(p,-q,)2+ie)

167"rase ff

3 ( I p t - q t 12+ /x2) ' (9) '

(10)

p~ is taken to approach zero after solving the equation numerically. The running coupling constant o%fe is taken as a=c.(pt, qt) =

12"n27 In[Max( IPt 12, Iqt i 2 ) / a ~ c o ]

Otseff(Pt, qt) = 1

when aseff < 1, otherwise.

( 11 )

Since ½(1 + 0 ) 7 ~. ½(1 + 0 ) = ½( 1 + 0 ) v ~', the Coulomb term in G+ is reduced to 167TOtseff

3 IPt --qt I 2 1@0

(12)

in Eq. (3). The kernel G+ in (10) is independent ofp~ and q~. Defining X(Pt) = fX(p,)dp~/27r and similarly for )((qt), we find from (3) the following 3D equation: -i X°(Pt) = E o + m - W

f G+)?o d3q~t Wu--fit--m (27r) 3 2W

(13)

From Eq. (14) we know that the solution )(o (Pt) satisfies the condition - WO + fit + m

)(o(Pt)P- =)(o(Pt)

2W

m

= -~2o(p,).

From (7), (8) and (15) we obtain the relations

(14)

Y.-B. Dai et al. / Physics Letters B 331 (1994) 174-178 q~,)=-(W+m)~2j,

177

t~,j=(W-m)t~2 i •

(15)

Therefore, only one independent component remains in the zeroth order wave function. Equations satisfied by q~j and t~lj can be found in [4]. The 4D wave function Xo can be expressed in term of the 3D wave function as follows:

-i

if

(Eo + m - W

)C°(P)= E o + r n + p ~ + i E ~ + ~ z - - ~ e

)?o(P,) + Pl -- 17V+iE

d3qt W6+fit + m ) G°)(°(qt) (2rr) 3

2W

.

(16)

Substituting (17) into (4), using the adjoint equation satisfied by Xo we obtain the following formula for the first order correction to the energy levels: tr [ f AJ"a)~o(Pt)/~)?o (qt) [d4q/(2~') 4] driP~ (2~') 4]

El =

tr[fA~o(Pt)Wp~o(Pt)dnp/(2rr)4]

,

(17)

where

f,= W e IPt ]z ( 2 ~ ' ) 4 6 4 ( P - q ) 2M~

Eo + m - Wq

/I+C

-

im(-~-[(15t G - + G - 4 0

(q~+Eo+m+ie)(qj+Wq-ie)

.~ ]

Eo + m - W e (p~ + W p - i e ) ( p ~ + E o - m + i e )

(18)

In the above equation we have neglected terms contributed by the last term in (17). These terms do not vanish due to the pole in G_ but are found to be very small. We also have neglected a term proportional to ( G ) 2 c * O~seff, 2 because we have neglected in the kernel (9) terms of the same order corresponding to the crossed gluon lines and the three gluon coupling. Since the integral in the above equation converges slowly, we introduce a cut-off A in (18). With the running coupling constant as~ff the integral is not sensitive to A. We fit the experimental data with the following values of the parameters: Mb = 5 . 0 2 4 GeV,

Mc = 1.58 GeV,

AQCD=0.38GeV,

K=0.2GeV2,

m,, = 0 . 3 5 GeV,

ms =0.458 GeV,

A=I.5GeV.

(19)

The masses of the bound states in the D, Ds, B and B~ systems are shown in Table 1. The values of the parameters in (22) used by us are quite close to that used for heavy quarkonium systems in the literature. We see from the table that the results of our calculations agree well with experimental data. In particular, the mass splittings of the doublets D * - D , B *-B and D,*.-Ds, B * - B s are very close to each other, a fact which is difficult to explain with the heavy quark effective chiral lagrangian approach [ 8] and the non-relativistic quark model. This is similar to the results obtained by Liu and Zhao [7] and can be understood as follows. Since the typical momentum of the light quark in these systems is ~ 500 MeV, the relativistic effect makes the difference in Table 1 Numerical results (in MeV) for low-lyingstates of D, D,, B and Bs systemsobtained with the values of the parameters (22) Meson

D B Ds Bs

J~

0 ,/2

1?/2

0 ;-/2

1~

1~+~

2~+~

1~2

2f/2

1864(1864.5_+0.5) 5279(5278.7 -+2.1 ) 1969(1968.8_+0.7) 5372(5368.6_+7.1)

2010(2010.1+0.6) 5325(5324.6 + 2. I ) 2109(2110+2.0) 5417(6(B* -Bs)=47)

2254 5682 2369 5795

2391 5725 2507 5839

2415(2424+6) 5738 2534 5859

2461(2459.4+2.2) 5754 2584 5875

2688 6022 2823 6147

2742 6039 2879 6165

Numbers in the parenthesis are experimental data.

178

E-B. Dai et al. / Physics Letters B 331 (I 994) 174-178

masses of light quarks unimportant, the splitting is not simply proportional to 1/m. We find that the contribution of the pole in the kernel G_ to E_ is as large as ~ 100 MeV. This implies that the retardation effect is important in the heavy meson system. The B-S wave functions obtained in this article can be used to calculate various effective coupling constants and form-factors of mesons to the order 1/M. This will be done in a later work. As is shown above, for the one gluon exchange kernel in the Coulomb Gauge we reduce the integral equation to a 3D wave equation for )((p,) in the leading order of the 1/M expansion. This simplifies much the calculations in the 1/M expansion. Actually, in the Coulomb gauge this can be achieved for more complicated kernels G ( p , q ) . Consider the irreducible kernel corresponding to the exchange of arbitrary number of gluons where each gluon line connects the two quarks. One can always choose all momenta of the internal light quark lines as loop integration variables. All propagators of heavy quark lines have the form 1 / ( ~/p~+ ~)'q~+ xL+ E + iE) as M~ ~ o0, where ~/and ~' = 0 or 1 and x~ is the longitudinal part of a linear combination of integration variables. In the M~ ~ oo limit only the instantaneous Coulomb propagator contributes. Therefore the only singularity in the upper half q~ plane is from x ( q ) and the only singularity in the upper halfp~ plane is from the external light quark propagator. Therefore the same form of equation as (12) holds in this case.

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