IL NUOVO CIMENTO VOL. 98 A, N. 1 Luglio 1987

Calculat ion of the Masses of the Ground-State Baryons as a

Test of the Hypothesis that the Potential is a Combination of a

Coulomb and a Linear Potential .

C. S. KALMAN and B. TRAN

Elementary Particle Physics Group, Concordia University 1455 de Maisonneuve Blvd., West, Montreal, P. Q., Canada H3G 1M8

R. L. HALL

Department of Mathematics and Elementary Particle Physics Group Concordia University - 1455 de Maisonneuve Blvd., West, Montreal, P. Q., Canada H3G 1M8

(ricevuto il 20 Marzo 1987)

Summary. - - The hypothesis that the interquark potential in the baryon is the sum of a Coulomb and a linear potential is evaluated in terms of the model of Isgur and Karl as modified by Kalman, Hall and Misra. Six parameters are used to fit the eight ground-state baryon masses. The closeness of the predicted values to the experimental values verifies the hypothesis.

PACS. 14.20. - Baryons and baryon resonances (including antiparticles). PACS. 12.40.Qq. - Potential models. PACS. 12.70. - Hadron mass formulae.

Within the f ramework of quantum chromodynamics, the short-range potential be tween two quarks or a quark and an ant iquark is clearly Coulombic. F o r large distances lattice gauge theory (1) and str ing models (2) lead one to expect a linear confining potential.

(') K.G. WILSON: Phys. Rev. D, 10, 2445 (1974); Phys. Rep. C, 23, 331 (1976); J. KOGUT and L. SUSSKIND: Phys. Rev. D, 11, 395 (1975). (2) K. JOHNSON and C. THORN: Phys. Rev. D, 13, 1934 (1976).

125

126 c . s . KALMAN, B. TRAN and R. L. HALL

Such a combination provides a good fit to meson mass spectra (3). Gromes and Stamatescu (4) studied this possibility in baryons using a harmonic-oscillator basis by writing

where ot is the reduced mass, and U(r) is treated perturbatively. They note that if V(r) is purely linear, then the approximation is quite good. For a sum of linear and Coulomb potentials, the method is still appropriate provided that the potential is essentially determined by the linear part.

On phenomenological grounds and sometimes on the basis of pure speculation, a number of other potentials have been suggested (s). In this paper, we will evaluate the hypothesis that the potential is the sum of a Coulomb and a linear potential, by using eq. (1) in the model of Isgur and Karl(*) as modified by Kalman, Hall and Misra(7).

Consider then a Hamiltonian

3

(2) H = ~ mi + Ho + Hhyr, i - I

where m~ are the quark masses,

9

where as is the quark-gluon fine structure constant and r~ is the separation between a pair of quarks and S, is the spin of the i-th quark,

(4) Ho = ~ p2/2mi + ~ V(r~j) - Pi 2 m~ , i , < j

(5) 11 ] v(r,~) = k ~ + u( r , j ) .

(3) E. EICHTEN, K. GfYrrRFIED, T. KINOSHITA, K. D. LANE and T.-M. YAN: Phys. Rev. D, 17, 3090 (1978). (4) D. GROMES and I. O. STAMATESCU: Nucl. Phys. B, 112, 213 (1976). (5) A full review of the harmonic-oscillator potential as a candidate for the confining potential is given by A. N. MITRA: Proc. Indian Natl. Sci. Acad., Part A, 47, 167 (1981). The logarithmic potential is considered as the confining potential by D. GROMES and I. O. STAMATESCU: Z. Phys. C, 3, 43 (1979); B. K.BHADURI and Z. RACZ: Phys. Rev. D, 22, 2261 (1980). The Martin potential, V(r) = A + B~, is investigated by R. K. BHADUm, L. E. COHLER and Y. NOGOMI: Nuovo Cimento, 65, 376 (1981); S. ONO and F. SCHOBERL: Phys. Lea. B, 118, 419 (1982); J. M. RICHARD and P. TAXIL: Ann. Phys. (N. Y.), 150, 267 (1983); Phys. Lea. B, 128, 453 (1983). (8) N. ISGUR and G. KARL: Phys. Rev. D, 20, 1191 (1979). C) C. S. KALMAN, R. L. HALL and S. K. MISRA: Phys. Rev. D, 21, 1908 (1980).

CALCULATION OF THE MASSES OF THE GROUND-STATE BARYONS ETC. 127

No spin-orbit coupling is included. Isgur and Karl (8) obtain a good fit to the baryon spectra without specifying the form of U(r~j) and without including any spin-orbit coupling. They argue that there is a strong cancellation between the two-body spin-orbit effects related to the short-range and long-range potentials. It is unclear whether three-body spin-orbit forces should be included. For a discussion of this last point, see the recent review article by Hey and Kelly (').

Following Isgur and Karl(S), we write

(6) an ~ = 12 f U(xla) x 2~ exp [ - x~ d x , n = 1, 2, 3, e = (~mJ2) i , 0

where m is the mass of the u-quark. In terms of the integrals {a~}, the contributions in first-order perturbation of the Coulomb, linear, and simple- harmonic-oscillator terms contained in the anharmonic potential

(7) U(rii) = - a~/ri~ + arii +

are given by the equation

(8)

where

(9) 6M = 10 - 11 i ]

- 15 27 - . 4 - 8

Note that the result of this calculation is independent of ~,. Using the values of the parameters obtained by Isgur and Karlff), we find that

(10) f - (~,as/rii) = - 1287MeV ,

(~br,~)= - 784MeV and

( ~ar~j) = 1421 MeV,

( ~ � 8 9 .

Thus in eq. (5) the harmonic-oscillator term ~'.�89 and the harmonic-oscillator contributions contained within the anharmonic potential U(r) do not cancel.

(8) hl. ISGUR and G. KARL: Phys. Rev. D, 19, 2653 (1979). (9) A. J. G. HEY and R. L. KELLY: Phys. Rep., 96, 71 (1983).

128 c . s . KALMAN, B. TRAN and S. L. HALL

A major triumph C ~ of the model of Isgur and Karl is the correct prediction in sign and magnitude of A5-~(1830) and Zs-~(1765) relative to the ground state. However, if the parameters obtained in their fit to the positive-parity baryons are applied to the problem of the negative-parity baryons, the mass difference between A 5-~ and ~5~ will be reduced from 50 to 15 MeV; this is no longer consistent with experiment. Kalman and Hall (11) noted that the resolution of this difficulty is the method of calculation of the nonharmonic part of the potential. Isgur and Karl (s) obtain the value of the contribution of this term in the SU(3) limit (ms = m~). Kalman, Hall and Misra(~) instead generalize eq. (6)1 writing

(11) 0

(12) ~(t) = (oJm. t/2) ~ 1

a,(t) V~= = 12 f U[x/a(t)] x ~" exp [ - x z] dx,

(13) m~ = m, ,

t = 411 + 3(m~/nv, tJ -1 1

3ml m3 m ~ - 2m1+ m3 ,

n = 1, 21 3,

where in the various strangeness (S) sectors

I m l = m " l S = 0 1 - 1 , m3=m, , S = 0 1 - 2 1 (14)

m z = m , 1 S = - 2 , 3 , m3=m81 S = - 1 1 - 3 .

Kalman and Hall (1,) show that, in such a consistent model, the mass difference between the A 5-~ and Z 5-~ is restored to a value in agreement with experiment. A corresponding full fit to the ground-state P-wave baryons was made by Kalman (~) using the values of the parameters found in this fit. It follows that

f --(~_~asrij}----- l l l2MeV1 (~'.arii)=513MeV1

(15) ( ~ b r } ) = - 2 4 7 M e V and (~ �89

We observe that the harmonic-osciUator term ~ �89 k ~ still does not cancel with the harmonic-oscillator contributions contained within the anharmonic potential U(r).

In these calculations by Kalman (~), the parameters were obtained by fitting the masses of the ground-state baryons. The masses of the negative-parity baryons were then predicted (1~).

(1o) N. ISGUR and G. KARL: Phys. Lett. B, 72, 109 (1977); 74, 353 (1978); Phys. Rev. D, 18, 4187 (1978). (") C. S. KALMAN and R. L. HALL: Phys'. Rev. D, 25, 217 (1982). (~) C. S. KALMAN: Phys. Rev. D, 26, 2326 (1982). (,3) Some slight adjustments of the parameters were made by comparison with the experimental values of ~b-~, As-~, j~-~ and ~ 7§

CALCULATION OF THE MASSES OF THE GROUND-STATE BARYONS ETC. 129

As noted by Isgur and Karl (e), in order to fit the ground state, a mixing between the ground state and the first excited positive-parity baryons (caused by the hyperfine interaction) must be included. In that work, the compoSition and masses 9f the excited positive-parity baryons obtained by Isgur and Karl (s) were used. In the present work, the elements of the mixing matrix cor- responding to the first excited positive-parity baryons are recalculated based on eq. (11). Kalman and MukerjiC0 in their fit to the + and ~" spectrum note that in addition to mixing caused by the hyperfine interaction, the anharmonic potential U also has an off-diagonal contribution. Similar contributions to the mixing matrix are also included here. If we now force an exact cancellation of the harmonic-oscillator terms, that is to say, we explicitly take the anharmonic U term to have the form given by eq. (7), then the a,(t), n = 1, 2, 3, defined by eq. (11) can be approximately evaluated in terms of a~, co and one free parameter ,a~, (identical to a of eq. (7)). This is done following Kalman, Hall and Misra C), by constructing quadratic approximations about a, = a,(1), n = 1, 2, 3; explicitly

( 1 6 ) a l = D - 3 ~ / 2 - E ,

(17) a~ = 2D - 15co/4 - E ,

(18) a8 = 6D - 105~/8 - 2 E ,

where D = 6 V~ a/(m~ o,7:) t, and E = 4as[m~ co/2=] t. Since the harmonic-oscillator potential has been cancelled out of eqs. (7) and

(10), ~ only is used in a fit by means of Gaussian wave functions. Clearly a bet ter fit should be obtained if one value of co(oil) is used for the ground state and a second value of ~(~) is used for the excited states. There are thus six parameters to be fitted: m~, x = mJm~, 0~1, ~ and as. The best fit yields m~ = 904 MeV, x = 0.740, o~1 = 1200 MeV, o~ = 3000 MeV, a = 0.481 GeV, a, = 0.388. The results of the fit to the eight ground-state masses are shown in table I. It is gratifying to note that the best fit yields a value of as identical with that obtained by the Cornell theory group (15) in their fit to heavy quarkonium. Richard and Taxil (5) had noted as a problem for the nonrelativistic quark models (NRQM) the discrepancy in the value of as in earlier NRQM models with the values deduced from heavy-meson spectra. Perhaps this is because our fit unlike earlier ones includes the effects of mixing with the first excited positive-parity baryons.

A concern about the calculation arises from our use of first-order perturbation. This approach allows us to discuss the potential -as/r i~+ brij within the framework of the Isgur-Karl model. If our first-order calculation can

(,4) C. S. K h L ~ N and N. MUKERJI: P h y s . Rev . D, 27, 2114 (1983). (15) E. EICItTEN, K. GOTTRFIED, T. KINOSHITA, K. D. LANE and T.-M. YAN: P h y s . Rev . D, 17, 3090 (1978); 21, 313 (1980).

9 - II Nuovo Cimento A.

130 C. S. KALMAN, B. TITAN and R. L. HALL

TABLE I. - Calculation of the masses of the ground-state baryons.

Particle Mass (MeV) % deviation from experiment

Experiment Calculation

939 960 2.2 h 1236 1235 0.1 A 1116 1114 0.2 vl/2 1193 1162 2.6 Z ~ 1385 1385 - - 2 l~z 1318 1288 2.3 --~ 1533 1559 1.7 [2 1672 1690 1.1

subsequent ly be shown to be essential ly correct , then our conclusions may be summar ized as follows: the closeness of our predicted values to the exper imenta l values means tha t the in te rquark potent ial may be closely approx imated by the Coulomb plus l inear potential.

***

We are gra teful to the Na tura l Sciences and Engineer ing Research Council of Canada for part ial financial suppor t (Grants No. A0358 and A3438).

�9 R I A S S U N T O (*)

Si valuta l'ipotesi secondo cui il potenziale tra i quark nel barione ~ la somma di un potenziale lineare e di Coulomb, in termini del modello di Isgur e Karl modificato da Kalman, Hall e Misra. Si usano sei parametri per approssimare le otto masse dei barioni hello stato fondamentale. La vicinanza dei valori previsti a quelli sperimentali verifica tale ipotesi.

(*) Traduzione a cura della Redazione.

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