Unified description of ground state mesons and baryons in a potential model

Volume 199, number 2

PHYSICS LETTERS B

17 December 1987

UNIFIED DESCRIPTION OF G R O U N D STATE M E S O N S A N D BARYONS IN A POTENTIAL MODEL A.M. B A D A L Y A N Institute o(Theoretical and Experimental Physics, 117259 Moscow, USSR Received 13 May 1987

Ground state masses are calculated using the same Cornell potential [ 5 ] for all mesons and baryons and assuming Vqq= ½Vqq. A good fit of spin-spin splittings for mesons and baryons is obtained with the QCD motivated o~( Q2 ). The resulting masses agree with the experimental ones within ~20 MeV for both mesons and baryons.

The recent d e v e l o p m e n t o f the relativistic version o f a potential m o d e l has given a n u m b e r o f new imp o r t a n t results a m o n g which we should note the linear Regge trajectories a n d a unified t r e a t m e n t o f all mesons from light to heavy ones [ I - 4 ] . The relativistic potential m o d e l ( R P M ) has given a m o r e solid f o u n d a t i o n for the o r d i n a r y nonrelativistic potential model ( N R P M ) , in particular, it became clear why the N R P M m a y in some cases be successfully applied to systems which contain light quarks. The imp o r t a n t observation was m a d e in ref. [3] that the meson spectrum o f the relativistic h a m i l t o n i a n coincides with that o f the corresponding nonrelativistic h a m i l t o n i a n if the constituent masses o f light quarks and the overall constant in a potential are redefined in a definite way. However, it is up to now unclear whether it is possible to describe all the known mesons and baryons without any m o d i f i c a t i o n of the p a r a m e t e r s of the interquark potential. O f course, it is assumed that the q u a r k - q u a r k potential in baryons and the q u a r k - a n t i q u a r k potential in mesons is connected by the colour relation V~(qsq:) = ~ Vr,,(q,(l:) ~ ½V(r;/) ,

(I)

and only pairwise interactions in three-quark systems are considered. Typically, the p a r a m e t e r s o f the interquark potential V~(q;q/) are close but not equal to the p a r a m e t e r s o f i VM(q,~I/). F o r example, in ref. [2], where the R P M was used and a good description o f the baryon spectrum was obtained, it was

necessary to decrease the string tension and to introduce an overall constant for the three-quark potential. Our m a i n purpose here is to answer this question and to d e m o n s t r a t e such a unified description o f the ground state mesons a n d baryons in the framework o f the N R P M . We use the C o u l o m b + l i n e a r interaction V ( q l q a q 3 ) = ½ Z V(r~;), i>j

V(r;j) = - K / r o + kr,/ + df ,

(2)

with the p a r a m e t e r s o f the Cornell potential [5]: tc=0.52, k = a 2, a = 2 . 3 4 GeV, r n ~ = l . 8 4 GeV, mb = 5.17 GeV. F o r the u, d, s quarks we take the following masses: m u = rod= 0.33 GeV, m s = 0 . 5 9 3 GeV. In the spirit o f ref. [3] we choose the constants df slightly d e p e n d e n t on the flavour f in the nonrelativistic case: d ( u u ) = - 0 . 9 7 5 GeV, d ( s s ) = - 0 . 9 0 5 GeV, d ( c e ) = - 0 . 8 6 5 GeV, d ( b b ) = - 0 . 7 1 5 GeV. The values of dr were o b t a i n e d by adjusting the meson centers o f gravity. F o r a m i x e d meson we take df.(q;q;) according to the relation d ( q , ( t / ) = ½[d(q,Ct,) + d ( q , q , ) ] . S p i n - s p i n splittings o f meson masses were calculated choosing the S . S interaction in a conventional form:

V,~s = ~- Tt[ eq ( Q )/m;m/]S;S:6(r,: ) ,

(3)

and considering V~. as a perturbation. Here we will 267

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PHYSICS LETTERS B

take into account only the first-order term o f the perturbative series, when the procedure of the regularization of the hyperfine interaction is not necessary. For this approach the higher order effect (as well as two-loop corrections to a~) is being absorbed into the value of the coupling constant, so that a~(Q) plays the role of some effective parameter. Then from (3) the spin-spin splitting of the S-wave meson mass is

EM.=M(3S,)-M(1So) = [8o~s(Q)/9m,rnj] IR(0) 12

(4)

Our calculation of R ( 0 ) for the Cornell potential showed that R ( 0 ) depends on #~/(#o is the reduced mass of the qd!j system), namely IR(0)]2~#/,5 with - 1.5 for light and mixed mesons and/~ ~ 1.8 for heavy quarkonia. For the coupling constant cG(QM) in (4) it is reasonable to use the Q C D motivated expression eq(Q 2) = 12n/( 3 3 - 2nf)ln( Q2 /A 2) ,

(5)

with A = A Q c D = 140 MeV. We expect that for different mesons the S . S interaction occurs at different distances, smaller than meson sizes. The best fit to experimental splittings was obtained for c~s(QM) taken at m o m e n t u m Q M = 2 # o. Such a choice of O~(QM=2pO ) is usually made in heavy quarkonia [6]. The calculated masses of vector and pseudoscalar mesons are given in table 1; their values coincide with the experimental masses with an accuracy of ~ 2 0 MeV (except for n and D-mesons where the dis-

17 December 1987

crepancy is ~ 30 MeV). Note that we obtain corect values o f E M not introducing any extra factor in (4) in contrast to ref. [7] where another value of QM = 4#o was chosen. We would like also to emphasize that the calculated spin-spin splittings of mesons depend on flayour only through as(QM) and do not depend on a choice of constants dr, but the meson centers of gravity are slightly flavour-dependent through df agreement with the analysis in ref. [ 3 ]. To obtain the spectrum and wave functions of the baryons we took the Cornell potential (2) without modification of any parameter. The hyperspherical formalism [8] was used to obtain the space wave function in the same manner as was done in refs. [ 9-11 ]. In ref. [ 10], the convergence of the hyperspherical expansion was carefully studied and it was shown that the first three harmonics usually define the masses and matrix elements with good accuracy. The masses of the ground state baryons have been calculated in ref. [ 11 ] for some power-law potentials. As was remarked by the authors of refs. [ 10,11 ], they choose, for a good fit, values oq for baryons which are much larger than cq for mesons. We have not met such a difficulty in our approach. We have also obtained the expression (7) for the matrix elements of the a-function (defining baryon spin-spin splitting) which differs from the corresponding formula in ref. [ 10]. In the hyperspherical method the Jacobi coordinates are usually used: 4=

~ - m ( ~ml m2 + m 2 ) r21'

r,,=r,-rj ,

Table 1 Masses of tSo and 3S~mesons. State

u~ ug ue ub sg se cc

M(3S,) (MeV) *

M('So) (MeV)

theory

experiment

theory

experiment

758 887 1982 5324 1022 2098 3101

770 892 2006 5325 1020 2110 3097

107 494 1832 5266 751 1970 2977

135 493 1865 5277 770 a~ 1970 2980

"~ This mass of the (ss) system is given for the mixing angle 0 = - 11 °. 268

(6)

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PHYSICS LETTERS B

Table 2 Masses of centers of gravity, M(,, for baryons.

m 3 / ~ +----m2) (m!Y~'q'm2r2 q= X/"; mM \ ml + m 2 p2 = ¢ 2 .Jl_~2,

M=ml

-k-m2 + m 3 •

17 December 1987

(6cont'd)

In (6) m is an arbitrary scale parameter which should be considered in a consistent way everywhere (rap 2 is m-independent). The matrix element 8 o- (8(to)) for a given baryon may be written in the following form:

8,)= ( 4/7~2)(/tu/m ) 3/2 JB~),s .

(7)

Here/t o is the reduced mass of the diquark (q,qj) and the factors Yo, J u will be defined below. The expression (7) for fi0 may be obtained by two different ways using either the wave function ~(~, q) with the phase volume d~dq or the wave function N(¢, q) = (mj m2m3/m2M)3/4~/({, q) with the phase volume dr2b dr31. The factor in (7) (/to/m)3/2j~ is missing in ref. [ 101. The expression (7) contains the integral

J;B=f[Z~(p)]2~3,

f[Zg(p)]2dp=l,

(8)

which is specific for a given baryon B. The value of this integral is defined by the partial wave Z~(P) with the global m o m e n t u m K = 0. The factor Yo takes into account the contribution of all other harmonics with K#0. Such a representation of 5 o (7) is convenient because the values ofy,j are close to unity. The analysis in refs. [ 1,12] has shown that in the case of baryons (q~q2q3) with equal quark masses we have Y12=~)13=))23 with an accuracy better than 1% and for baryons with unequal quark masses the y,j lie in the interval 0.75-1.25. ( m = ½GeV is chosen below.) It follows from our calculations that the values ~B~,, are almost constant for different diquarks and different baryons, therefore the main changing factor in the matrix elements 8,? is (/t,~)3/2. As a consequence, the &j are strongly dependent on the quark content of the diquark (qNj) but weakly dependent on the kind of baryon to which this diquark belongs. For example, for (udf) baryons ( f = u , s, c, b) all values of S j2 are very close and lie in the interval 4.2-4.8 GeV 3 while the matrix element 813 for A is 1.5 times smaller than for Ab. For the diquarks q,qj ( i = j ) with different flavours, e.g. uu, ss, cc, 8~2 increases by a

State

M{~ (MeV)

udu uds

1088 ~'~ 1265

udc udb usc

2420 5770 2570

ssu

1440

sss ssc

1607 2730

ccu ccs ccc

3650 3760 4777

"' M((~(exp)= ½(N+A) = 1086 MeV.

factor of 2 and 10 passing from the nucleon to f~ and f~.... correspondingly. We should like to emphasize that the matrix elements 8,~ for the baryons and [R(0)[2 for the mesons have similar/tv-dependences (for the Cornell potential they are approximately proportional to (/t/j) 3/2).

For

spin-spin

ESBW(qlq2q3)

mass

splittings

in

baryons

B = Z,>/Ess(qiqj) with

E~.s.(q,q;) =~r[as(QB)/m;m/] (S,S~) ( fi(r,;))

(9)

it is of great importance to take into account the dependence of a s(Qu) according to relation (5). For baryons A-- 140 MeV was chosen, as was for mesons. Different values of as(QB) were considered and the best fit to experiment was obtained when the values of the momentum QB were close, but not equal, to QM=2p0, namely, QB=0.833 QM = (2/to) 0.833. For the choice Q~=QM, the corresponding values of as(QR) give rise to spin-spin splittings which are systematically < 20% smaller than the experimental ones. The calculated masses of the ground state baryons are presented in tables 2 and 3. The masses of the centers of gravity Mcc are given in table 2. The absolute values of the ground state baryons have been obtained using the calculated values of Mc~ and E~s. It is seen from table 3 that our values of the baryon masses coincide with an accuracy of ~ 20 MeV with the experimental ones. For comparison we present in table 3 the baryon masses from ref. [2] where the RPM was used. This comparison shows 269

Volume 199, number 2

PHYSICS LETTERS B

Table 3 Masses of ground state baryons (in GeV) Baryon

Theory

Experiment

this work

ref. [2]

N zX A 52 I2" A, Z,. Z* A, Z. 52* E E* f~ ~ c '~ =s =* E,c =* ~tc f~,(~) n,(~) f~,~(½)

0.931 1.245 1.11 1.20 1.38 2.25 2.43 2.49 5.59 5.80 5.82 1.33 1.52 1.664 2.46 2.56 2.62 3.60 3.68 2.72 2.83 3.73

0.960 1.230 l.ll5 1.190 1.370 2.265 2.440 2.495 5.585 5.795 5,805 1.305

a~(~)

3.80

a~.~,c

4,793

1,505

1.635

0.939 1.232 1.115 1.193 1.383 2.282 2.450 5.50?

1.318 1.533 1.672 2.460

h a v i o u r of o~s(Q) (5) is taken into account. The m o m e n t u m Q is defined by the reduced mass: for mesons QM = 2/z~j and for baryons QB = (2/zij). 0.8 3 3. (iv) The obtained s p i n - s p i n splitting a n d the absolute values of the masses for mesons a n d baryons coincide with experiment with an accuracy of ~ 20 MeV. We hope that the potential constructed may be also useful for m u l t i q u a r k systems and in particular for q2c12 systems. U p to now heavy four-quark systems were usually considered [ 13 ] in the N R P M . Now we have acquired the possibility to study the low-lying states of m u l t i q u a r k systems which contain light quarks. I am very grateful to Dr. B.O. Kerbikov and Dr. M.I, Polikarpov for providing me with their new results about the baryon wave functions.

References 2.740

that the predicted masses of the baryons in the R P M and in our nonrelativistic approach are very close to each other (for E* a n d ~2 our value is more exact). It can be concluded from our analysis that: (i) The unified description of the centers of gravity for the ground states of mesons a n d baryons turns out to be possible in the N R P M . We used the Coulomb + linear potential with the f l a v o u r - i n d e p e n d e n t C o u l o m b constant and string tension but a slightly flavour-dependent constant dr. (ii) The calculated wave functions of mesons ( I R ( 0 ) 12 ) and the matrix elements 6 , for diquarks in baryons have a rather strong and similar dependence on the reduced mass of the meson or diquark. (iii) The experimental values of s p i n - s p i n splittings may be explained if the Q C D m o t i v a t e d be-

270

17 December 1987

[1] D.P. Stanley and D. Robson, Phys. Rev. Lett. 45 (1980) 235; Phys. Rev. D 21 (1980) 3180; J. Carlson, J, Kogut and V.R. Pandharipande, Phys. Rev. D 27 (1983) 233. [2] S. Godfrey and M. Isgur, Phys. Rev. D 32 (1985) 189; S. Capstick and N. Isgur, Phys. Rev. D 34 (1986) 2809. [3] J.L. Basdevant and S. Boukraa, Z. Phys. C 28 (1985) 413. [4] J.L. Basdevant and S. Boukraa, Z. Phys. C 30 (1986) 103. [5] E. Eichten et al., Phys. Rev. D 21 (1980) 203. [6] M.B. Voloshin,Yad. Fiz, 35 (1982) 1016. [7] K. Igi and S. Ono, Phys. Rev. D 32 (1985) 232. [8] Yu.A. Simonov, Yad. Fiz. 3 (1966) 461; A.M. Badalyanand Yu.A. Simonov,Yad. Fiz. 3 (1966) 765. [9] P. Hasenfratz et al., Phys. Len. B 94 (1980) 40l; A.B. Guimaraes,H.E. Coelho and R. Chanda, Phys. Rev. D 24 (1981) 1343. [10] J.M. Richard and P. Taxil, Ann. Phys. (NY) 150 (1983) 267. [ 11 ] J.M. Richard and P. Taxil, Phys. Lett. B 128 (1983) 453. [ 12 ] B.O. Kerbikov et al., preprint ITEP-86-160 (Moscow). [ 13] J.P. Ader, J.M. Richard and P. Taxil,Phys. Rev. D 25 (1982) 2370; A.M. Badalyan, B.L. Ioffe and A.V. Smilga, Nucl. Phys. B 281 (1987) 85.