Baryon spectrum in a potential quark model

IL NUOVO CIMENTO VOL. 102 A, N. 3 Settembre 1989

Baryon Spectrum in a Potential Quark Model.

C. S. KALMAN and B. TRAN

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

(ricevuto il 30 Settembre 1988)

Summary. -- A quark model suggested by quantum chromodynamics is used to calculate the entire spectrum of baryons containing u, d, s, c, b quarks. A short-range Coulomb potential is used together with a long-range linear potential modified by a harmonic-oscillator potential. This combination leads to a good description of the masses of both positive- and negative- parity baryons based on the five quark masses and four additional parameters related to the chosen potential.

PACS 24.90 - Other topics in nuclear reactions and scattering: general.

1 . - Introduction.

In this paper a quark model of baryons is presented. Only eight parameters are used to describe both the positive- and negative-parity baryons containing u, d, s, c and b quarks.

The model is developed within the framework introduced by De Rujula, Georgi and Glashow(1). At small distances, QCD suggests a Coulomb-type potential -as~r, where a~ is the quark-gluon fine-structure constant. At large distances one expects a confining potential. The exact form of the confining potentials is unknown. Lattice gauge theory (2) and string models (3) lead one to expect a linear confining potential. Such a combination of a linear confining potential plus Coulombic-type short-distance potential plus one-gluon-exchange

(1) A. DE RUJULA, H. GEORGI and S. L. GLASHOW: Phys. Rev. D, 12, 147 (1975). (2) K.G. WILSON: Phy8. Rev. D, 10, 2445 (1974); Phys Rev. C, 23, 331 (1976); J. KOGUT and L. SUSSKIND: Phys. Rev. D, 11, 395 (1975). (3) K. JOHNSON and C. THORN: Phys. D, 13, 1934 (1976).

835

836 c . s . KALMAN and B. TRAN

forces provides provide a good fit to meson mass spectral (4). For baryons the situation is much less clear. One might assume the potential between two quarks in baryons has the form

(1.1) V(r~j) = - CFas/rij + arij,

where rij is the separation between quark i and j, C~ is the color factor of baryons taken to be 2/3 as suggested by QCD and a is a constant describing the string strength. Another possibility is that suggested by a branched string emanating from some central point in between the three quarks. It is probably not necessary to give special consideration to such models since it has been shown in an appropriate treatment of lattice gauge theory (5) that a sum of linear two-body forces may have the same effect as such a three-body force. A number of the other potentials have been suggested; the harmonic-oscillator potential by Mitra (6), the Martin potential V(r) = A + B r ~ by Bhaduri, Cohler and Nogami (7) and by Ono and Schoberl(8), the logarithmic potential by Gromes and Stamatescu (9) and also by Bhaduri and Racz (10). Gromes and Stamatescu (9.1~) made a general study of baryons using a harmonic-oscillator basis by writing

(1.2) 1 2 2] V'(rij) = -~1 ~ 2 r 2 " + V(rij) - ~ f~o~ rijJ = Vo(r~j) + U(rij)

where t~ is the reduced mass and V(rij) is the potential. U(r~j) is treated perturbatively. Such a treatment takes advantage of the fact that the harmonic- oscillator potential is the only potential which can be solved exactly for any number of particles (12). Barbour and Ponting (13) reexamined the problem using a stochastic variational method. These papers describe nonstrange baryons with some success. Isgur and Karl(14) used an unspecified U(rij) in a highly successfully calculation of the excited positive-parity baryons. In this model different parameter sets have to be applied to analyze the positive- and negative- parity baryons. Kalman (1~) in a similar approach was able to describe a broad

(4) E. D, 17, (5) H. (5) A. (7) R. (8) S. (9) D. (18) B. (11) D.

EICHTEN, K. GOTTFRIED, T. KINOSHITA, K. D. LANE and T. M. YAN: Phys. Rev. 3090 (1978). D. DOSCH and V. F. MULLER: Nucl. Phys. B, 116, 470 (1976). N. MITRA: Proc. Indian Nat. Sci. Acad., Part A, 47, 167 (1981). K. BHADURI, a. E. COHLER and Y. NOGAMI: Nuovo Cimento A, 65, 376 (1981). Oso and F. SCHOBERL: Phys. Lett. B, 118, 419 (1982). GROMES and I. O. STAMATESCU: Z. Phys. C, 3, 43 (1979). K. BHADURI and V. RACZ: Phys. Rev. D, 22, 1161 (1980). GROMES and I. O. STAMATESCU: Nucl. Phys. B, 112, 213 (1976).

(12) R. L. HALL and B. SCHWESINGER: J. Math. Phys. (N.Y.), 20, 2481 (1979). (13) j. M. BARBOUR and D. K. PONTING: Z. Phys. C, 4, 119 (1980). (14) N. ISGUR and G. KARL: Phys. Rev. D, 19, 2653 (1979). (15) C. S. KALMAN: Nuovo Cimento A, 94, 219 (1986).

B A R Y O N S P E C T R U M IN A P O T E N T I A L Q U A R K MODEL 837

spectrum of baryons containing c, b quarks in addition to u, d and s quarks with a single set of parameters. In this paper the combination of Coulomb and linear potential given by eq. (1.1) is modified by a harmonic-oscillator potential. This combination behaves in a manner similar to a combination of a Coulomb and a logarithmic potential at short and medium range. As illustration, fig. 1 shows the

cD

"4" ~'-2

- 3

f

J - - I I I I I I I I I

0 2 4 6 8 10 r' (GeV -~)

Fig. 1. - The graphs of the potentials: 1) V(r)=- 0.667/r+ 0.08r-0.004r 2 and 2) V(r) = - 0.667/r + 0.191nr.

potential used in this paper for the situation where the coefficient of the harmonic-oscillator term corresponds to the ground state of the ~ oscillator (this coefficient varies with states which will be explained in sect. 3) and a combination of a Coulomb and logarithmic potential which is the form of the potential used by Gromes and Stamatescu (9). It should be noted that as pointed out by Gromes and Stamatescu (9) the logarithmic potential only describes the interaction at medium range and the potential may have a different form at long range. Furthermore, in reality, the energy of the states corresponds to the short- and medium-range behavior of the chosen potential.

In this work a single set of parameters is used for all baryon states. The mixing between the ground-state and the first-excited positive-parity baryons as well as the mixing between the first- and the second-excited negative-parity baryons are taken into account. The importance of this interband mixing was shown by Isgur and Karl (1~).

Until now no true relativistic potential model exists. The approach taken by

(16) N. ISGUR and G. KARL: Phys. Rev. D, 20, 1101 (1979).

8 3 8 c . s . K A L M A N and B. T R A N

many model builders is to adopt relativistic kinematics(~,18). This is an interesting half-way house, but there are no objective criteria to show that such models are more realistic. A recent study of the unequal-mass quarkonium spectra by Kalman and D'Souza (19) using only a few undetermined parameters and a nonrelativistic approach yields results which are extremely close to experimental data and which are similar to those obtained by Godfrey and Isgur (20) in a model based on relativistic kinematics.

2. - The harmonic-oscil lator model.

Because the present model employs the harmonic-oscillator wave functions basis and also is an extension of the harmonic-oscillator potential model, it is useful to go over this model. The model employs a Hamiltonian of the form

(2.1) H = ~ mi + H0, i

where m~ are the quark masses and

2 1 kr~j (2.2) H0 = ~ 2 ~ i + ~

In terms of Jacobi relative coordinates

2 E m i i

1 (2.3a) p---- - - ( r l - - r2),

(2.3b) ~t= 1 (rl+r2-2r~),

eq. (2.2) decouples into a description of two independent harmonic oscillators with the same spring constant k

(2.4) P~

H o - 2m:

_ - - + p2 + 3k(~2+~2) 2m---~ 2

(17) j. FRANKLIN: Phys. Rev. D, 21, 241 (1980); D. B. LICHTENBERG: Quark cluster potential model with relativistic kinematics, Indiana University publication IUHET 88 (1983). Invited talk presented at the Indiana University workshop on manifestation of hadron structure in Nuclear Physics, McCormick's Creek State Park, October 19-21, 1983. (is) S. CAPSTICK and N. ISGUR: Phys. Rev. D, 34, 2809 (1986). (19) C. S. KALMAN and I. D'SouzA: Nuovo Cimento A, 96, 286 (1986). (20) S. GODFREY and N. ISGUR: Phys. Rev. D, 32, 189 (1985).

BARYON SPECTRUM IN A POTENTIAL QUARK MODEL 839

This particular decomposition is valid provided that at least two of the quark masses are equal. The masses of u and d quarks are taken to be the same since the difference between them is just of the order of the hadron electromagnetic interaction. Baryons containing three different quark masses can still be included but require a different decomposition than that provided by the Jacobi coordinates introduced in eq. (2.3). In this paper the convention will be used that in all cases the two quarks comprising the p oscillator are the ones with the same masses. Hence the reduced masses of m r and m~, of the p and ~ oscillators, respectively, have the form

(2.5a) m~ = ml,

3 ml m3 (2.5b) m), - ,

2ml + m8

where for the various hadrons.

(2.6a) m l = m u ; X, 5, A, Ac, Ab, Z, Zc, Zb,

(2.6b) = ms ; ~, ~,

(2.6c) = m e ; E ~ , ~ ,

(2.6d) = m b ; Eb,~b,

(2.6e) m3= mu; uV, A, --, E,c, Eb,

(2.6f) =m~; A, Z, ~2,

(2.6g) = m~ ; A~, Z~, ~ ,

(2.6h) = m b ; Ab, ~'b, ~b .

It will be convenient to use the notation

(2.7)

and

(2.8a)

(2.8b)

xl --" m~/ms , x2 = m~/mc , x~ = mu/mb ,

0) 2 = 3 k / m ~ , ~4 = 3 k m u ,

~o~ = 3 k / m j , ~4 = 3 k m ¢ , j = p,)~.


The eigenfunction of the Hamiltonian for the ground s ta te is the wave function

(2.9) ~ o = ~booo ¢ooo

and the eigenfunctions of the Hamiltonian for N = 1 band can be chosen as

(2.10a) 'F~I ----" ~000 ¢111,

(2.10b) '° ~J'I1 = ~000 ¢111 °

F o r baryons with equal quark masses, the eigenfunctions for N = 2, 3 can be

chosen as

for N = 2 :

(2.11a) 1 F6 = - - - : (~ooo ¢~oo + ~2oo ¢ooo),

(2.11 b) T~o = 1

(~110 ¢110 -- ~111¢11-1 -- ~11-I¢111),

(2.11c) T~o = 1 (#2oo ¢ooo - - ~000 ¢200) ,

(2.11d) TIA1 = 1 (~111¢110 -- ~110 ¢111) ,

1 (2.11e) T ~ - = - (~ooo ¢ ~ + ~2= ¢ooo),

(2.11f) ~1~22 ---- ~111 ¢111,

(2.1 lg) ~F~ = 1

and for N = 3:

(/~222 ¢oo0 -/~000 ¢222).

(2.12a) ,F , _ 1 vr3¢222~111), - ~ (¢ooo ¢~ -

(2.12b) 1 IFfl ---- ~ (~333 ¢000 -- V3~111 ¢222),

(2.12c) T': 1 (-V~333 ~000 _]_ ~)111 ¢222:) 33 ~--" "~


(2.12d) ~22 -- 1 (~f3 ~000 ¢333 nt- ~222 ¢111), 33 --

(2.12e) ~F~2 = - - 1

(2.12f) ~ = 1 (~221 ¢111 -- V2~222 ¢110) ,

1 (2.12g) W~, - x/ -d -- - - (5 ~200 ¢111 -- V ~ ~000 ¢311 -- 2 v r3 ~222 ¢11-1 "4-

+ v +221¢11o- v +=o ¢11,),

(2.12h) W¢l- 1 -- - - (-- 5 ~111 ¢200 + V ~ 3 1 1 ¢000 "~ 2 V 3 ~11-1 ¢222 --

-- V r~ ~110 ¢221 "~ V ~ ~111 ¢220),

(2.12i) ~'~1 ~-~ - - 1 ( 5 +200 ¢111 -- ~ V 3 +000 ¢311 "4- 2 V 3 +222 ¢11.1 -- 2

-- V/6 ¢~221 ¢110 "~- V ~ +220 ¢111) ,

(2.12j) W~, - 1 ( 5 + m ¢ 2 o o _ ~ V~+311~OOO.4_2V~11.1~222__ 2

-- ~/-6 ~110 ¢221 -~- ~/r2 ~111 ¢220) ,

(2.12k) F~ = 1 (V3~b2ooCm + V5~b~¢3n),

(2.12/) ~ i ~ - - 1 ( V ~ ~111 ¢200 _{_ V ~ ~311 ¢000) ,

For states containing quarks with two distinct masses, the eigenstates with N = 2, 3 are quite distinct from those shown in eqs. (2.11) and (2.12) since the degeneracy between the p and ~ normal modes has been broken (see eq. (2.8b)). The eigenfunctions used in this paper for the corresponding baryons have the form:

for N = 2:

(2.13a) ~ - ~ o o - ~ooo ¢2oo,

842

(2.13b)

(2.13c)

(2.13d)

(2.13e)

(2.13f)

(2.13g)

and for N = 3:

(2.14a)

(2.14b)

(2.14c)

(2.14d)

(2.14e)

(2.14f)

(2.14g)

C. S. K A L M A N and B. T R A N

~ ) = ~ 3 (~,o ¢11o -/~111 ¢1~-~ - ~1-~ ¢~1),

~r~ = ~2oo ¢ooo,

}r~ = ~ (/~m Cno- ~o 6~),

T~ =/~22~ ¢ooo,

~J"22 ---- ~111 ¢111,

~f i = ~ooo ¢=~

F~° =/~333 ¢ooo,

3 3 -- ~111 ~222,

~ J " ),),~, - . t . A 33 -- ~000 ~383

(~221 ¢111 -- V2~222 ¢110) ,

g•_ 1 22- - ( / ~ ¢ ~ - V ~ ~ o ¢ ~ ) ,

F~i ° =/~311 ¢ooo,

(2.14h)

(2.14i)

(2.14j)

(2.14k)

T ~ _ 1 ( V ~ ¢ ~ - ~ - V ~ 2 2 ~ ¢ ~ o + ~ o C m ) ,

~ _ 1 (~/6~11-1¢222- v r 3 @110 ¢221 -~-~111¢220) 11

~j" ),),), _ _ ~ - / ~ ¢~,

~ 1 2 ' = ~200 ¢111,

(2.14/) ~ n - ~bln ¢200,


in which ~b~m, Cnt~ are the eigenfunctions of the ~ and Z oscilators, respectively. The eigenvalues corresponding to @nlm Cn'l'm' are given by

3 (2.15) Eo = (n + -~)o~ + (n' + 3 ) ~ .

3 . - T h e m o d e l .

The model employs a Hamiltonian of the form

P~ ~ + 2 V(r~j) - (3.1) H = ~ mi + 2mi] i<j

where

2 2 m~ i

- - H h y p ,

(3.2) V(r~j) = - 2~/3r~5 + ar~5

and

( 3 . 3 a ) Hhyp = ~ /~h3yp , i<j

where

(3.3b)

Following Gromes and Stamatescu(9,19 the Hamiltonian is rewritten as

(3.4) H = H° + ~ IV(r~i)- 1 bkr~jl + Hhyp= H° + ~ U(ro) + 2 ~

where H0 is given by eq. (2.2). In this model the potential is modified by attaching a coefficient b to the harmonic-oscillator term (see eq. (1.2)) We now turn off the hyperfine interaction. The variational method is employed to determine % and a~. In the case of three equal quark masses we can make use of the resulting symmetry and it follows that

(3.5)

54 - I l N u o v o C imen to A .


TABLE I. - Spectrum of the nucleon.

State Calculation Experimental mass Status

4DM 7+/2 2.006 1.950 + 2.050 * *

2Ds 5+/2 1.781 1.670 + 1.690 * * * * 2DM 5+/2 1.996 4DM 5+/2 2.042

2Ds 3+/2 1.780 4SM 3+/2 1.915 2DM 3+/2 1.995 4DM 3+/2 2.018 2PA 3+/2 2.094

2Ss 1+/2 0.958 2 S , s 1 +/2 1.458 2SM 1+/2 1.805 2DM 1+/2 1.973 2PA 1+/2 2.096

4F M 9-/2 2.258

2F M 7-/2 2.123 4F M 7-/2 2.271 2F s 7-/2 2.291 2FA 7-/2 2.359 4D M 7-/2 2.373

4P M 5-/2 1.648 4p~ 5-/2' 2.105 2F M 5-/2 2.124 4FM 5-/2 2.253 2rs 5-/2 2.290 tp~ 5-/2 2.297 2D M 5-/2 2.357 2FA 5-/2 2.366 2DM 5-/2 2.392

2P M 3-/2 1.470 4pM 3-/2 1.680 2ps 3-/2 1.841 2p~ 3-/2' 1.973 4p~ 3-/2' 2.158 4F M 3-/2 2.245 2p~ 3-/2 2.271 4p~ 3-/2 2.314 2D M 3-/2 2.343 4D M 3-/2 2.392 2PA 3-/2 2.414

1.690+1.800 * * * *

0.939 * * * * 1.400+1.480 * * * * 1 . 6 8 0 + 1 . ~ 0 * * *

2 . 1 3 0 + 2 . 2 ~ * * * *

1.660 + 1.690 * * * *

1.510 + 1.530 * * * * 1.670 + 1.730 * * * *

BARYON SPECTRUM IN A POTENTIAL QUARK MODEL

TABLE I (continued).

845


2P M 1-/2 1.469 4P M 1-/2 1.614 2Ps 1-/2 1.842 2p~ 1-/2' 1.968 4p~ 1-/2' 2.058 4p~ 1-/2 2.274 2p~ 1-/2 2.276 4D M 1 - / 2 2 . 3 9 3

2PA 1-/2 2.417

1.520 + 1.560 * * * * 1.620 + 1.680 * * * *

****: Good, clear and unmistakable. *** : Good but in need of clarification or not absolutely certain. ** : Not established; needs confirmation. * : Evidence weak; likely to disappear. The ratings here are taken from the Particle Data Group.

this considerably simplifies the calculation. In the case of two distinct quark masses we still have the symmetry between quark 1 and 2. Hence

(3.6)

The changing of the variables

(3.7)

where

t(3/8)1/2(%/~) 2 1/V2

4 (3.8) t = 1 + 3 ( ~ / ~ ) 2

is used to calculate (~]U13]/~}. These are also applied for the calculation of the hyperfine interactions later.

The variational process is applied separately to the p and ~ oscillators and to different orbital angular-momentum sectors. In the first case by using eq. (3.5) we have to deal only with the ~ oscillator. In the second case the calculation of (~l U131{~ } involves both p and ~ variables and the variational method is applied as follows.

To determine the % a basis of {~lmCZlm}, n = 2i + l, i = 0, 1 {Oi} is used. We diagonalize the two-by-two matrix {(~]H]~j}}, i , j = 0 , 1 and minimize the

846

TABLE II. - The spectrum of A.



tDs 7+/2 1.943 1 .910- 1.960 * * * *

4Ds 5+/2 1.956 1 .890- 1.920 * * * * 2DM 5+/2 2.016

ass 3+/2 1.215 1 .230- 1.240 * * * * 4 t Ss 3+/2 1.788 1.500 + 1.900 * * * 4Ds 3+/2 1.950 1 .860- 2.160 * * * ~DM 3+/2 2.018

2SM 1+/2 1.919 1.850 + 1.950 aDs 1+/2 1.937

aF s 9-/2 2.297

2F M 7-/2 2.262 4Fs 7-/2 2.315

4P s 5-/2 2.020 ~FM 5-/2 2.262 4Fs 5-/2 2.307 2D M 5 - / 2 2.366

2PM 3-/2 1.653 4Ps 3-/2 2.020 ~P~ 3-/2' 2.116 4F s 3-/2 1.284 2p~ 3-/2 2.304 2DM 3-/2 2.367

1.890-1.960 * * *

1.630-1.740 * * * *

2P M 1-/2 1.655 1.600 - 1.650 * * * * tP s 1-/2 2.020 1.850 + 2.000" * * * 2p~ 1-/2' 2.117 ~P~ 1-/2 2.295

****: Good, clear and unmistakable. *** : Good but in need of clarification or not absolutely certain. The ratings here are taken from the Particle Data Group.

e igenva lues . I n this process the coefficient b is also i n se r t ed to the ha rmonic -

osci l la tor t e r m in H0 (see eq. (2.2)). The re are two reasons for doing this . F i r s t

t he m a t r i x e l e m e n t m a y be b r o k e n down and we could no t have the m i n i m u m

va lue of the e igenva lues w h e n b is large. Secondly, we find t ha t by doing th is the


TABLE I I I . - The spectmem of h.

847


4D~ 7+/2 2.218 2.020 + 2.120 *

eD~, 5+/2 1.930 1.815 + 1.825 * * * * ~D~,~ 5+/2 2.089 2.090+2.140 * * * * 2D~ 5+/2 2.186 4D~ 5+/2 2.242 4p~ 5+/2 2.278

~D~ 3+/2 1.929 2D~ 3+/2 2.087 4S~) 3+/2 2.145 eD~ 3+/2 2.189 4D~ 3+/2 2.213 4p~ 3+/2 2.266 2P~ 3+/2 2.281

2S 1+/2 1.134 2S~ 1+/2 1.622 2S~, 1+/2 1.788 2S~ 1+/2 2.076 4D~, 1+/2 2.186 2p~ 1+/2 2.282 4p~, 1+/2 2.303

4F~,~ 9-/2 2.111 tF~ ~ 9-/2 2.569

2 F ~ 9-/2 2.095 I F ~ 7-/2 2.127 2F)~ 7-/2 2.193 2 F ~ 7-/2 2.414 4 F ~ 7-/2 2.594 4 D ~ 7-/2 2.621 2 F ~ 7-/2 2.830

4p 5 -/2 1.654 4 I P~ ~ 5-/2 1.972 ~ F ~ 5-/2 2.094 4F~ 5-/2 2.119 4 p ~ 5-/2 2.129 2F~ ~ 5-/2 2.193 2 F ~ 5-/2 2.413 2D 5 /2 2.521 4 F ~ 5-/2 2.577 4D;),)~ 5-/2 2. 595 2D~ ~ 5-/2 2.624 2 F ~ 5-/2 2.823 4p 5-/2 2.891

1.850 + 1.910 * * * *

1.116 * * * * 1.560 + 1.700 * * * 1.750 + 1.850 * * *

2.090+2.110 * * * *

1.810 + 1.830 * * * *


TABLE III (continued).


2p~. 3-/2 1.547 2p~ 3-/2 1.612 4p~ 3-/2 1.743 2 p ~ 3-/2 1.865 4p~), 3-/2 1.976 ~P~ 3-/2 2.053 4F~p~ 3-/2 2.103 4p~p~ 3-/2 2.166 2 p ~ 3-/2 2.198 2p~ 3-/2 2.412 2D~ 3-/2 2.521 4F ~ 3-/2 2.552 4D~ 3-/2 2.599 2D~ 3-/2 2.634 2p~ 3-/2 2.728 4 p~ 3-/2 2.889 2 p ~ 3-/2 2.997

2p~ 1-/2 1.525 4p~, 1-/2 1.604 2pp 1-/2 1.658 2p~z~, 1-/2 1.859 4p~z~, 1-/2 1.929 2p~ 1-/2 2.038 4p~ 1-/2 2.087 2 p ~ 1-/2 2.209 2p~ 1-/2 2.413 4Dp~ 1-/2 2.589 2p~ 1-/2 2.723 4 p ~ 1-/2 2.883 2 p~ 1-/2 2.995

1.520 * * * *

1.685 + 1.695 * * * *

1.400 - 1.410 * * * *

1.660- 1.680 * * * * 1.720 - 1.850 * * *

****: Good, clear and unmistakable. *** : Good but in need of clarification or not absolutely certain. * : Evidence weak; likely to disappear. The ratings here are taken from the Particle Data Group.

mixing by the U-term is ex t remely small and so as far as the potential is concerned, we achieve almost the same effect as taking into account the mixing by the potential in a large basis. That is the eigenfunctions of the potential used are ve ry well approximated by the harmonic-oscillator wave functions. ~ is de termined similarly by using the basis { ~ t ~ ¢ ~ } , n = 2i + l, i = 0, 1. Here we use these wave functions instead of eqs. (2.9)-(2.14) for the reason of simplicity and because we are in teres ted in the two lowest radially excited s tates in each


TABLE IV. - The spectrum of E.

849


4D~ 7+/2 1.939 2.025 + 2.040 * * * * ~D~,~ 7+/2 2.266

~D~ 5+/2 1.905 1.900 + 12.935 4D~ 5+/2 1.962 2.010+ 2.110 ~D~ 5+/2 2.210 ~D~, 5+/2 2.267 ~D~ 5+/2 2.281

4S 3+/2 1.395 4S~ 3+/2 1.887 ~D~ 3+/2 1.905 4D~ 3+/2 1.945 4S~ 3+/2 2.013 ~D~ 3+/2 2.211 4D~ 3+/2 2.259 2D~ 3+/2 2.269 ~P~ 3+/2 2.281

2S 1+/2 1.279 2S~ 1+/2 1.710 4D~ 1+/2 1.920 2S~ 1+/2 1.971 ~S~ 1+/2 2.140 4 D ~ 1+/2 2.251 2p~ 1+/2 2.278

4F~ ~ 9-/2 2.298 4F~, 9-/2 2,756

2F~ 7-/2 2.109 2F~, 7-/2 2.249 4F~ 7-/2 2.305 2F~ 7-/2 2.482 4D~ 7-/2 2.528 a F ~ 7-/2 2.770 2F~,~, 7-/2 2.869

4p~ 5-/2 1.879 2F~ 5-/2 2.109 tp~,~. 5-/2 2.235 2F~. 5-/2 2.251 4Fp;~ 5-/2 2.302 2Fo~, 5-/2 2.482

1.385 1.800 + 1,925

2.070 + 2,140

1.193 1.630 + 1.690

1.830 + 1.985

1.770 + 1.780

850

TABLE IV (continued).



2D~, 5-/2 2.517 4D~ 5-/2 2.527 2D~ 5-/2 2.619 4 p ~ 5-/2 2.712 4 p ~ 5-/2 2.747 4F~,~ 5-/2 2.772 2F~, 5-/2 2.869

2p~ 3-/2 1.631 2p~ 3-/2 1.813 4p~ 3-/2 1.901 ~P~ 3-/2 1.926 2p~ 3-/2 2.095 2 p ~ 3-/2 2.223 4 p ~ 3-/2 2.262 4Fp~ 3-/2 2.296 2D~ 3-/2 2.513 4D~ 3-/2 2.534 2D~ 3-/2 2.618 2 p ~ 3-/2 2.648 4 p ~ 3-/2 2.720 2 p ~ 3-/2 2.726 4F~,~ 3-/2 2.750 4 p ~ 3-/2 2.774 2 p ~ 3-/2 3.004

2pp 1-/2 1.629 2p~ 1-/2 1.804 4p~ 1-/2 1.872 2 p ~ 1-/2 1.926 2p~ 1-/2 2.093 4 p ~ 1-/2 2.216 2p~>,~ 1-/2 2.228 4D~ 1-/2 2.520 2p~, 1-/2 2.645 ap~?~ 1-/2 2.701 2 p ~ 1-/2 2.718 4 p ~ 1-/2 2.755 2 p ~ 1-/2 3.008

1.665 - 1.685

1.900 + 1.950

1.730 - 1.800

* * * *

* * *

****: Good, clear and unmistakable. *** : Good but in need of clarification or not absolutely certain. ** : Not established; needs confirmation. * : Evidence weak; likely to disappear. The ratings here are taken from the Particle Data Group.

BARYON S P E C T R U M IN A P O T E N T I A L QUARK MODEL

T A B L E V . - The spectrum of A c.

851


tD~ 7+/2 3.340

2D~ 5+/2 3.139 2D~, 5+/2 3.174 ~D~ 5+/2 3.327 4D~ 5+/2 3.368 4p~ 5+/2 3.414

2D~ 3+/2 3.139 eD~ 3+/2 3.174 4S~ 3+/2 3.261 2D~ 3+/2 3.326 4D~ 3+/2 3.338 4p~ 3+/2 3.409 2p~ 3+/2 3.416

2S 1+/2 2.282 2S~ 1+/2 2.759 eS~ 1+/2 2.991 2S~ 1+/2 3.228 4D~ 1+/2 3.315 2p~ 1+/2 3.412 4p~ 1+/2 3.439

tF~ 9-/2 3.345 4F~ 9-/2 3.659

2F~ 7-/2 3.315 2F~ 7-/2 3.342 tF~ 7-/2 3.356 eF~ 7-/2 3.518 4F~ 7-/2 3.682 4D~ 7-/2 3.717 2F~ 7-/2 3.935

4p~ 5-/2 2.864 t p ~ 5-/2 3.154 2F~ 5-/2 3.315 ap~ 5-/2 3.338 2F~ 5-/2 3.345 4F~ 5-/2 3.352 2F~ 5-/2 3.517 2D~ 5-/2 3.659 4F~ 5-/2 3.669 4D~ 5-/2 3.689 2D~ 5-/2 3.720 2F~ 5-/2 3.921 4p~ 5-/2 3.959

2.282

852

TABLE V (continued).

c. s. KALMAN and B. TRAN


2p~ 3-/2 2.669 2p~ 3-/2 2.843 4p~ 3-/2 2.924 2p~), 3-/2 3.086 4p~ 3-/2 3.179 2p~ 3-/2 3.199 4F~ 3-/2 3.339 2p~ 3-/2 3.348 4p~ 3-/2 3.395 2p~,~ 3-/2 3.528 2p~ 3-/2 3.642 4F~ ~,~ 3-/2 3.662 2D~ 3-/2 3.694 4D~ 3-/2 3.732 2D~ 3-/2 3.877 2p~.~ 3-/2 3.949 4p~ 3-/2 4.061

2p~ 1-/2 2.653 4p~ 1-/2 2.811 2p~ 1-/2 2.887 2p;~,~ 1-/2 3.061 4p~.~ 1-/2 3.144 2p~ 1-/2 3.192 4p~ 1-/2 3.268 2pp~ 1-/2 3.398 2p~ 1-/2 3.531 4D~ 1-/2 3.683 2p~ 1-/2 3.876 4pp~,~ 1-/2 3.940 2p~ 1-/2 4.058

****: Good, clear and unmistakable. The ratings here are taken from the Particle Data Group.

sector. In the model by Gromes and Stamatescu(9) a s and ~ are determined by minimizing each state separately.

For the case where b > l . 0 as in our case, the potential will become unconfining at some point. At the first this seems to be an unphysical potential. However, as mentioned earlier the energy of the states corresponds to the short- and medium-range behavior of the potential and the potential used is valid only in these ranges. We take the same view as Gromes and Stamatescu (9) that the potential may be different than that used in this paper, at long range. Examining


TABLE VI. - The spectrum of Z¢.

853


4D~ 7+/2 3.149 4D~ 7+/2 3.364

2D~ 5+/2 3.137 4D~ 5+/2 3.165 2D~ 5+/2 3.343 ~D~,~ 5+/2 3.364 tD~ 5+/2 3.370

as 3+/2 2.556 tS~ 3+/2 3.002 2D~ 3+/2 3,136 tD~ 3+/2 3.156 4S~ 3+/2 3,216 2D~ 3+/2 3.343 2D~ 3+/2 3.362 4D~ 3+/2 3.368 2p~ 3+/2 3.406

2S 1+/2 2.515 2S~ 1+/2 2.931 4D~ 1+/2 3.138 2S~ 1+/2 3.197 2S~ 1+/2 3.265 4D~ 1+/2 3.359 2p~ 1+/2 3,406

4F~ 9-/2 3.429 4F~ 9-/2 3.878

2F~p 7-/2 3.345 2F~ 7-/2 3.399 t F ~ 7-/2 3.435 2F~ 7-/2 3.579 4D~ 7-/2 3.667 4F~ 7-/2 3.886 2F~ 7-/2 3.986

4p~ 5-/2 3.002 4p~ 5-/2 3.331 2F~ 5-/2 3.345 2F~ 5-/2 3.399 4Fp~ 5-/2 3.433

2.426 + 2.460

854 C.S. KALMAN and B. TRAN

TABLE VI (continued).


eF~ 5-/2 3.579 eD~ 5-/2 3.660 4D~ 5-/2 3.670 ~D~ 5-/2 3.714 tP~ 5-/2 3.836 4F~ 5-/2 3.878 4p~ 1-/2 3.909 2F~ 5-/2 3.987

2p~ 3-/2 2.845 2p~ 3-/2 2.969 4p~ 3-/2 3.008 2p~ 3-/2 3.106 2p~ 3-/2 3.274 4p~x 3-/2 3.340 2p~ 3-/2 3.375 4F~ 3-/2 3.427 2D~ 3-/2 3,656 4D~ 3-/2 3.673 2D~ 3-/2 3.714 2p~ 3-/2 3.819 4p~ 3-/2 3.840 2p~ 3-/2 3.864 4F~ 3-/2 3.876 tp~.~ 3-/2 3.922 ~P~ 3-/2 4.074

2pp 1-/2 2.844 2p~ 1-/2 2.966 4p), 1-/2 3.001 2P~ 1-/2 3.106 2p~,~ 1-/2 3.273 4p~ 1-/2 3.326 2p~ 1-/2 3.375 4D~ 1-/2 3.669 2p~ 1-/2 3.816 4p~ 1-/2 3.835 2ppp~ 1-/2 3.858 4p~ 1-/2 3.904 2p~ 1-/2 4.075

**: Not established; needs confirmation. The ratings here are taken from the Particle Data Group.


TABLE VII. - The spectrum of AD.

855


4D~ 7+/2 6.501

~D~ 5+/2 6.274 eD~ 5+/2 6.318 2D~ 5+/2 6.495 4D~ 5+/2 6.529 4p~. 5+/2 6.579

2D~ 3+/2 6.274 2D~;. 3+/2 6.318 4S~ 3+/2 6.417 2D~ 3+/2 6.492 4D~ 3+/2 6.501 2p~, 3+/2 6.577 4p~ 3+/2 6.581

2S 1+/2 5.425 eS~ 1+/2 5.885 2S~ 1+/2 6.132 2S~ 1+/2 6.403 4D~ 1+/2 6.479 2p~ 1+/2 6.574 4p~ 1+/2 6.604

4F~ 9-/2 6.464 4F~ 9-/2 6.809

2F~)~ 7-/2 6.458 2F~ 7 /2 6.466 aF~ 7-/2 6.472 2Fo~ 7-/2 6.664 4F~ 7-/2 6.829 4D~ 7-/2 6.868 2F~ 7-/2 7.101

4pp 5-/2 5.983 4p~ 5-/2 6.254 2F~ 5-/2 6.457 ~F~ 5-/2 6.462 4p~ 5-/2 6.468 4F~ 5-/2 6.470 2F~ 5-/2 6.662 4F~ 5-/2 6.814 2D~ 5-/2 6.831


TABLE VII (continued).


4D~ 5-/2 6.843 ~D~ 5-/2 6.873 ~P~ 5-/2 7.079 ~F~ 5-/2 7.113

2p~ 3-/2 5.828 2p~ 3-/2 5.973 4p~ 3-/2 6.058 2p~ 3-/2 6.202 4p~ 3-/2 6.257 ~P~ 3-/2 6.338 4F~ 3-/2 6.460 2p~p~ 3-/2 6.477 ap~ 3-/2 6.519 2p~ 3-/2 6.692 4F~ 3-/2 6.793 ~D~ 3-/2 6.828 4D~ 3-/2 6.849 ~D~ 3-/2 6.884 ~P~ 3-/2 7.053 4p~ 3-/2 7.094 ~P~ 3-/2 7.225

2p~ 1-/2 5.817 4p~ 1-/2 5.929 ~Pp 1-/2 6.012 2p~ 1-/2 6.172 tp;~ 1-/2 6.259 ~P~ 1-/2 6.330 4p~ 1-/2 6.397 2p~p~ 1-/2 6.529 2p~ 1-/2 6.694 4D~ 1-/2 6.836 2p~ 1-/2 7.052 4p~ 1-/2 7.085 2p~ 1-/2 7.223

fig. 1 it is clear that in the regions of interest to this paper, the potential used in this paper is confining. For excited states, the potential will be more confining than in the ground state due to the lower values of a~ and a~.

Equations (2.9)-(2.14) are now used to construct the baryon wave functions. The construction of the complete baryon wave functions and the treatment of the hyperfine interaction is given in appendices A and B, respectively. The U-term


TABLE VIII. - The spectrum of Eb.

857


4D~ 7+/2 6.278 4D~,~, 7+/2 6.512

2D~ 5+/2 6.274 tD~ 5+/2 6.292 2D~ 5+/2 6.506 2D~, 5+/2 6,512 4D~)~ 5+/2 6.515

4S 3+/2 5.690 4S~ 3+/2 6.117 2D~ 3+/2 6.270 4D~ 3+/2 6.288 48~ 3+/2 6.350 2D~ 3+/2 6.506 2D~ 3+/2 6.511 4D~,~, 3+/2 6.514 ~P~ 3+/2 6.571

eS 1+/2 5.675 2S)~ 1+/2 6.088 4D~ 1+/2 6.270 2S~ 1+/2 6.342 2S~ 1+/2 6,426 4D~ 1+/2 6.509 2p~ 1+/2 6.571

4F~;~ 9-/2 6.577 4F~)~ 9-/2 7.033

2F~ 7-/2 6,465 2F~ 7-/2 6.556 4F~ 7-/2 6,582 2F~),~ 7-/2 6.715 4D~ 7-/2 6.838 4F~ 7-/2 7.039 2F~ 7-/2 7.152

4p~ 5-/2 6.155 2F~ 5-/2 6.464 4p~ ~), 5-/2 6.476 eF~ 5-/2 6.556

858

TABLE VIII (continued).


aF~ 5-/2 6.581 2F~ 5-/2 6.715 2D~ 5-/2 6.830 aD~ 5-/2 6.842 2D~ 5-/2 6.868 4p~ 5-/2 7.007 4F~ 5-/2 7.034 4p~ 5-/2 7.082 ~F~ 5-/2 7.153

2pp 3-/2 5.970 2p~ 3-/2 6.126 ap~ 3-/2 6.156 2p~ 3-/2 6.214 2p~ 3-/2 6.419 4p~ 3-/2 6.479 2p~ 3-/2 6.503 4F~ 3-/2 6.575 2D~ 3-/2 6.827 4D~ 3-/2 6.841 2Dp~ 3-/2 6.868 ~P~p~ 3-/2 6.982 4ppp~ 3-/2 7.006 4F~ 3-/2 7.030 2p~ 3-/2 7.049 2p~ ~ 3-/2 7.097 ~P~ 3-/2 7.242

2p~ 1-/2 5.970 2p~ 1-/2 6.124 4p~ 1-/2 6.157 2p~ 1-/2 6.214 2p~ 1-/2 6.419 4p~ 1-/2 6.475 2p~ 1-/2 6.503 4D~ 1-/2 6.842 2p~ 1-/2 6.981 4pp~ 1-/2 7.008 2p~ 1-/2 7.039 4p~ 1-/2 7.081 2p~ 1-/2 7.242



TABLE IX. - The spectrum of 5,.

859


4D~ 7+/2 1.925 4D~ 7+/2 2.449

eD~ 5+/2 1.852 4D~ 5+/2 1.943 ~D~ 5+/2 2.283 2D~ 5+/2 2.450 4 D ~ 5+/2 2.463

tS 3+/2 1.446 2D~ 3+/2 1.851 4S~ 3+/2 1.898 4D~ 3+/2 1.932 4S). ;~ 3+/2 2.103 2D~ 3+/2 2.285 2p,,~, 3+/2 2.321 ~D~ 3+/2 2.446 ~D~ 3+/2 2.459

2S 1+/2 1.349 2S:~ 1+/2 1.722 4D~ 1+/2 1.905 ~S~ 1+/2 1.932 ~S~ 1+/2 2.233 ~P~ 1+/2 2.321 4D~,~ 1+/2 2.441

4F~ 9-/2 2.447 tF~,~ 9-/2 2.797

2F~ 7-/2 2.024 2F~)~ 7 /2 2.379 4Fop~ 7-/2 2.454 4D~ 7-/2 2.503 2F~. 7-/2 2.755 ~F~ 7-/2 2.802 4F),~,~ 7-/2 2.807

4p~ 5-/2 2.005 ~F~ 5-/2 2.024 eF~ 5-/2 2.378 4p~ 5-/2 2.389 4F~ 5 /2 2.450 4D~ 5-/2 2.486

1.530 1.820 (3/2?)

1.317

55 - I1 Nuovo Cimento A.

860

TABLE IX (continued).

c. s. KALMAN and B. TRAN


~D~ 5-/2 2.500 t p ~ 5-/2 2.540 t p ~ 5-/2 2.565 ~D~ 5-/2 2.752 ~F~ 5-/2 2.755 eF~,) 5-/2 2.801 4F:~:~ ~ 5-/2 2.804

2p~ 3-/2 1.526 2p~ 3-/2 1.910 2p~ 3-/2 2.030 tp~ 3-/2 2.039 2 p ~ 3-/2 2.095 2 p , 3-/2 2.366 4 p ~ 3-/2 2.425 4F~ 3-/2 2.443 2 p ~ 3-/2 2.460 2D~ 3-/2 2.489 4D~ 3-/2 2.507 4 p ~ 3-/2 2.551 2 p ~ 3-/2 2.559 4 p ~ 3-/2 2.575 2D~,~ 3-/2 2.752 4F~)~ 3-/2 2.793 e P ~ 3-/2 2.978

2p~ 1-/2 1.526 2p~ 1-/2 1.898 4p~, 1-/2 1,989 2p~ 1-/2 2.031 2p~, 1-/2 2.095 4p~;. 1-/2 2.343 ~P~ 1-/2 2.384 ~ P ~ 1-/2 2.460 4D~ 1-/2 2.483 4 p ~ 1-/2 2.530 4p~p~ 1-/2 2.549 2p~ 1-/2 2.567 2 p ~ 1-/2 2.981

****: Good, clear and unmistakable. *** : Good but in need of clarification or not absolutely certain. The ratings here are taken from the Particle Data Group.


TABLE X. - The spectrum of ~¢.

861


4D,, 7+/2 3.864 4 D ~ 7+/2 4.552

2D~ 5+/2 3,814 4D, 5+/2 3.873 2D~x 5+/2 4.417 2D~ 5+/2 4.553 4Dxx 5+/2 4.559

4S 3+/2 3.309 2D~ 3+/2 3.811 4S~ 3+/2 3.818 4D~ 3+/2 3.867 4S~ 3+/2 4.321 ~D~ 3+/2 4,417 2p~ 3+/2 4,442 2D~ 3+/2 4.551 4D),~, 3+/2 4.557

2S 1+/2 3.292 2S~ 1+/2 3.731 4D~ 1+/2 3.858 2S~ 1+/2 4,122 2S,:~ 1+/2 4.377 2p~ 1+/2 4,442 4D~ 1+/2 4,548

tF 9-/2 4.568 4F~ 9 /2 4,841

2F~ 7-/2 3.897 2F 7-/2 4.544 4F 7 /2 4.573 4D 7-/2 4.589 2F~ ~ 7 /2 4.817 4F)~) 7-/2 4.844 2F~.~ 7 /2 4.857

2F~ 5-/2 3.897 4p; 5 /2 4.130 4 p P~,:~ 5 /2 4,439 2F~,~), 5-/2 4.543

862

TABLE X (continued).


4F~ 5-/2 4.570 4p~ 5-/2 4.573 4D~ 5-/2 4.580 ~D~ 5-/2 4.587 t p ~ 5-/2 4.606 ~F~ 5-/2 4.816 ~D~ 5-/2 4.837 4 F ~ 5-/2 4.842 2F),~ 5-/2 4.857

2p~ 3-/2 3.102 2p~ 3-/2 3.979 2p~ 3-/2 4.050 tp~ 3-/2 4.148 2p~ 3-/2 4.211 2p~ 3-/2 4.435 4p~ 3-/2 4.453 4F~ 3-/2 4.564 2p~ 3-/2 4.572 4p~p:~ 3-/2 4.577 2D~ 3-/2 4.582 2p~ 3-/2 4.587 4D~ 3-/2 4.591 4p~,~ 3-/2 4.619 2pp~ 3-/2 4.837 4F~ 3-/2 4.840 2p~ 3-/2 4.954

2p~ 1-/2 3.102 ep~ 1-/2 3.978 2p~ 1-/2 4.048 tp~ 1-/2 4.118 ~P~ 1-/2 4.211 4p~ 1-/2 4.425 2p~ 1-/2 4.438 4p~ 1-/2 4.564 2ppp~ 1-/2 4.569 4D~ 1-/2 4.578 2p~ 1-/2 4.585 4p~ 1-/2 4.604 2p~ 1-/2 4.955



TABLE XI. - The spectrum of ~b.

863


4D~ 7+/2 9.890 ~D~.), 7+/2 10.619

2D~ 5+/2 9.856 4D~ 5+/2 9.895 ~D~ 5+/2 10.499 ~D~ 5+/2 10.619 4D)~), 5+/2 10.622

4S 3+/2 9.137 aS~ 3+/2 9.796 2D~ 3+/2 9.856 4D~ 3+/2 9.892 2D~, 3+/2 10.499 2p~ 3+/2 10.505 4S~ 3+/2 10.603 2D~ 3+/2 10.619 4D~ 3+/2 10.621

2S 1+/2 9.133 2S~ 1+/2 9.745 tD~ 1+/2 9.887 2S~ 1+/2 10.455 2S~ 1+/2 10.481 2p~ 1+/2 10.505 4D),~, 1+/2 10.617

tF~ 9-/2 10.625 4F~ 9-/2 10.928

2F~ 7-/2 9.886 2F~ 7-/2 10.621 4F~, 7-/2 10.628 tD~ 7-/2 10.639 2F~ 7-/2 10.908 4F~ 7-/2 10.929 2F~ 7-/2 10.931

2F~ 5-/2 9.886 tp~ 5-/2 10.139 4p~, 5-/2 10.420 tpp~ 5-/2 10.551 2F~ 5-/2 10.621

864

TABLE XI (continued).


4Fo~ 5-/2 10.627 4D~ 5-/2 10.633 ~D~ 5-/2 10.637 4p~,~ 5-/2 10.704 ~F~ 5-/2 10.908 ~D~ 5-/2 10.920 4F~.~ 5-/2 10.929 ~F~ 5-/2 10.931

2p~), 3-/2 8.854 2p~ 3-/2 10.003 2p~ 3-/2 10.127 4p~ 3-/2 10.149 2p~ 3-/2 10.269 4p~p~ 3-/2 10.427 2p~ 3-/2 10.493 4p~, 3-/2 10.554 2p~ 3-/2 10.566 4F~ 3-/2 10.623 ~D~ 3-/2 10.634 4D~ 3-/2 10.640 2p~,~ 3-/2 10.675 4p~), 3-/2 10.713 2D~,~ 3-/2 10.920 4F~ 3-/2 10.927 2p~. 3-/2 10.996

2p~ 1-/2 8.854 2p~ 1-/2 10.003 2p~ 1-/2 10.125 4p~ 1-/2 10.133 2p~ 1-/2 10.269 4p~.~ 1-/2 10.414 2p~, 1-/2 10.492 4p~ 1-/2 10.550 2p~ 1-/2 10.566 4D~ 1-/2 10.632 2p~ 1-/2 10.672 4 p ~ 1-/2 10.701 2p~,~ 1-/2 10.996



TABLE XII. - The spec t rum of ~.

865


4/9 s 7+/2 2.263

4/9 s 5+/2 2.268 219 M 5+/2 2.344

tSs 3+/2 1.620 4 S ' s 3 +/2 2.020 4D s 3+/2 2.264 2D M 3+/2 2.344

2S M 1+/2 2.219 4D s 1+/2 2.259

4F s 9-/2 2.575

2F M 7-/2 2.538 4Fs 7-/2 2.582

4P s 5-/2 2.250 2F M 5-/2 2.538 4Fs 5-/2 2.579 2D M 5 /2 2.663

2P M 3-/2 2.022 4P s 3-/2 2.250 2p~ 3-/2' 2.356 4F s 3-/2 2.569 2p~ 3-/2 2.576 2D M 3 /2 2.663

2P M 1-/2 2.022 4P s 1-/2 2.250 2p~ 1 /2' 2.356 2p~ 1-/2 2.573

1.672 * * * *

****: Good, clear and unmistakable. The ratings here are taken from the Particle Data Group.

(eq. (3.4)) and the hyper f ine in te rac t ion are t r e a t ed as per tu rba t ion . The

mixings by these t e r m s within each band and be tween dif ferent bands are t aken

into account .

The calculation of the b a r y o n masses depends on the qua rk masses , ~s, a, b

and an overal l cons tan t C. F i t t i ng the S = 0 and S = - 1 s t r angenes s sec tor

resu l t s in m~ = md = 0.23 GeV, xl -- 0.38, ~s = 1.0, a = 0.08 GeV 2, b = 1.68 and

C = - 0 . 6 GeV. In the papers by Gromes and S t ama te scu (1~) and B a r b o u r and

866

TABLE XIII. - The spectrum of g2 c.



4Ds 7+/2 5.103

4Ds 5+/2 5.105 2D M 5+/2 5.219

4Ss 3+/2 4.364 4S~ 3+/2 4.722 4Ds 3+/2 5.104 2DM 3+/2 5.219

2SM 1+/2 5.026 4Ds 1+/2 5.101

4F s 9-/2 5.446

~FM 7-/2 5.376 4Fs 7-/2 5.449

4Ps 5-/2 4.870 2FM 5-/2 5.376 4F s 5-/2 5.448 20 M 5-/2 5.586

2P M 3-/2 4.848 4P s 3-/2 4.870 2p~ 3-/2' 5.106 4F s 3-/2 5.445 2p~ 3-/2 5.450 2D M 3-/2 5.586

2P M 1 -/2 4.848 4P s 1-/2 4.870 2p~ 1-/2 5.106 2p~ 1-/2 5.450

Ponting(TM) they have, in the corresponding par ts , m ~ = 0 . 3 G e V , ~ = 0 . 5 , a = 0.05 GeV 2, C -- - 0.32 GeV and mu = 1.12 GeV, as = 1.01, a = 0.042 GeV 2,

C = - 0.67 GeV, respect ively. The calculation of the ~ and ~ sectors is based on this result . The calculation of

ba ryons containing c and b quarks requires in addition x2 and x3 which are

obtained by fit t ing the ground s ta tes of Ac and Ab, respect ively, yielding

x2 = 0.1237 and x3 = 0.0455. The calculated baryon masses are given in table I-

X I V toge the r with the available exper imenta l data.


TABLE XIV. - The spectrum of [ib.

867


419 s 7+/2 13.541

4D s 5+/2 13.542 20 M 5+/2 13.830

4S s 3+/2 12.359 4 i Ss 3+/2 12.856 4D s 3+/2 13.541 2t9 M 3+/2 14.830

2S M 1+/2 13.485 41) s 1+/2 13.540

4Fs 9-/2 14.050

2F M 7-/2 13.895 4F s 7-/2 14.051

4P s 5- /2 12.888

2F M 5- /2 13.895

4F s 5-/2 14.050 2DM 5-/2 14.419

4P s 3-/2 12.889 2P M 3-/2 13.236 2p~ 3-/2' 13.512 4Fs 3-/2 14.049 2p~ 3-/2 14.179 2D M 3- /2 14.419

4P s 1-/2 12.888 2P M 1-/2 13.236 2p~ 1-/2' 13.512 2p~ 1-/2 14.179

4. - Discussion of the results and comparison with other works.

This section is devoted to the discussion of the resul t and the comparison with o ther works. We see tha t there are many unobserved states. The answer to this

problem m a y be obtained from Koniuk and Isgur(21) and For sy th and

(21) R. KONIUK and N. ISGUR: Phys. Rev. Lett., 44, 545 (1980); Phys. Rev. D, 21, 1869 (1980).

868 C . S . KALMAN and B. TRAN

Cutkovsky(2~) in which it has been shown that a large number of states essentially decouple from the partial wave analysis. Therefore care should be taken in comparison with experimental data in the case of nonunique states because we could not be certain which one corresponds to an experimental state. Nevertheless we see that in most of the cases, the experimental states seem to correspond to the lowest calculational states which would seem to be reasonable for the comparison.

Until now accommodating the negative-parity baryons into the same framework with the positive-parity baryons seems to be a problem to model using a nonrelativistic approach. Isgur and Karl('4~16,~) and Kalman(~5~u) successfully described the negative-parity and positive-parity baryons but with a high price of using a different parameter sets. Forsyth (25) obtained good results but a new parameter has to be introduced for the (56, 1-) multiplet. In term of a universal parameter set Barbour and Ponting('~) had some success with the positive-parity section but the negative-parity section is not as good.

So far the only successful description of both positive- and negative-parity baryons is the work by Capstick (~8) using the relativized quark model. In this model relativistic kinematics are used and some other terms are put in by hand to describe the momentum dependence of the interaction.

The model used by Barbour and Ponting('3) is similar to the model here. Compared to this work, the result obtained here is a considerable improvement. Their result shows a rather high value for the (56', 0 ÷) N -- 2 state and rather low values for the first-excited negative-parity states. This problem is considerably improved in the result here. Of course it can be said that the reason is that there is one more parameter in this model but it should be remembered that the work here covers much larger ground than that of Barbour and Pointing and also that this problem seems to persist even in the relativized model. Similarly, the model here is able to accommodate the (56, 1-) N = 3 states which have to be lowered by introducing a new parameter S~6 -- 200 MeV in Forsyth (5~). The high values of these states are also found in the relativized model (,8). Finally the point should be stressed that the model here uses less parameters than used by Forsyth and by Capstick in their model. Not counting the masses of the c and b quarks, the model here uses six parameters as compared to thirteen in the relativized quark model. Forsyth's model only considers the nonstrange section and employs nine parameters compared to five employed to calculate the nonstrange sector in this model.

(zs) C. P. FORSYTH and R. E. CUTKOVSKY: Z. Phys. C, 18, 219 (1983). (es) N. ISGUR and G. KARL: Phys. Rev. D, 18, 4187 (1978). (24) C. S. KALMAN: Phys. Rev. D, 26, 2326 (1982).


5. - Conclusion.

Transitions between states should be examined in order to have a clearer understanding of the spectrum. This is expected to be done in the near future. It should be kept in mind that this model employs only basic parameters (besides an overall constant). Of course, it is not expected that the parameters reflect the true values but rather the ,,effective, ones. In addition to the constituent quark masses, the values of as and a should be regarded as the effective values which include any relativistic effects, the variation of ~ with energy, etc. Even though good results are obtained by adding a harmonic-oscillator potential, its role has yet to be clarified. It is unclear whether this potential really corresponds to what happens inside baryons or simply reflects the closeness with a logarithmic potential discussed earlier. Despite this, since the predicted spectra are in good agreement with the experimental data it seems clear that the baryon spectrum can be presented very well within a nonrelativistic scheme.

We are grateful to the Natural sciences and Engineering research Council of Canada for partial financial support (Grant No. A0358 and Fellowship No. 800).

A P P E N D I X A

This appendix is devoted to the construction of the complete baryon wave functions. 0nly the highest states of total angular momentum and the highest m- states in each J-state are given. The lower states can be obtained by the usual combination of orbital angular momentum and spin. Following Isgur and Karl (~4) the construction of baryons containing quarks with equal masses follows the SU(6) type and the one with two distinct quark masses follows the uds type.

The spin wave functions used have the form

(A.1) z~=-~---1 ( t $ I' - $ t 1'),

1 (A.2) z , ~ = - (I' $ $ - $ 1' $) ,

(A.3) x~ = 1

(A.4) z ~ = 1

( 2 I ' 1 ' $ - t $ 1 ' - $ 1' 1'),

(1' $ $ + $1' $ - 2 5 $1'),

(A.5) X~/2 = I' I' 1', etc.

870 C. S. KALMAN and B. TRAN

And the flavor wave functions,

(A.6) ¢~ =

(A.7) ¢~n =

(A.8) ~ - Cp-

(A.9) ¢~ =

(A.10) ¢] =

(A. 11) ¢~ =

1 (udu - duu) ,

1 ( udd - dud ) ,

1

1

(2uud - u d u - duu) ,

(udd + dud - 2 ddu) ,

u u u , etc.,

1 (ud - du) s ,

(A.12) Cz= 1 ( u d + d u ) s .

The flavor wave functions of the other particles are similar. Each state in SU(6) type is presented in the form A~,'J e in which A is the

name of the particle and s, l, s', J, P represent spin, orbital angular momentum, spatial symmetry, total angular momentum and parity, respectively. In the uds type spatial symmetry is replaced by the excitation mode.

The wave functions of the nucleons are given by

N = 0:

1+ - ! (z~ ¢~ + z~ ¢9 ~ o . (A.13) N2S~ 2 V ~

N = I :

(A.14) N 2 P M - - _ _ 3- _ 1 ¢~ ¢~ 2 2 [(X~ - Z~ ¢~) T~'~ + (Z'~ + Z~+ ¢9 ~r~l],

(A.15) 5- 1 [(¢; T~l + ¢~ ~1). N4PM y = z~12

N = 2 :

(A.16) 7 + 1 N4DM ~- = Z~e ~ (¢; ~F~ + ¢~ ~F~),

(A. 17) 5 + _ 1 (z'~ ¢~ + z~+ ¢~') ~ ,

N2D 2


(A. 18) N2DM - - - - 5 + _ 1 [(Z% ¢~ - Z'~ ¢~) W~z + (Z'+ ¢) + Z~+ ¢~) ~r~u] 2 2

(A.19) 3 + 1 ¢~ ~ N4SM -~- =)~/~ ~ (¢~ ~°~o+ ~oo),

3 + _ 1 ( z ~ ¢ ~ _ z ~ ¢ . 9 F , ~ , (A.20) NzP'4 2 V ~

(A.21) NeS~ 1+ - 1 (z~ ¢: + if+ ¢~) ~F~'o,

(A.22) N2SM 1+2 - 21 [(Z.~_ ¢~ _ z~ ¢~') ~ o + (z~ ¢~ + Z~ ¢:) ~F~o]

N = 3 :

(A.23) N2p~3- _ 1 (X'~¢~+~¢~)~°[~, 2

(A.24) N2P'i - - _ _ 3- _ 1 [(Z~ ¢': - z~+ ¢;) W~ + (z~- ¢~ + Z~ ¢':) }F'~I] 2 2

(A.25) N2p~ . . . . 3- _ 1 [(z# ¢~ - z# ¢~) }F#~ + (Z~- ¢~ + Z~ ¢~) ~F;'~] 2 2

(A.26) N2PA 3- _ 1 (z~ - ¢~_ x~ + ¢~) }FIA1, 2

(A.27) N4pM ~ = z~/2 V ~

(A.28) 5- 1 ¢~ N4pM ' ~ Z~/2 (¢': ~1'1 "~- ~)~'

(A.29) N2DM 5 - _ 1 2 2 [(Z~ ¢" - Z~ ¢~) }F~2 + (Z~ ¢~ + Z~ ¢~) F22],

(A.30) 7- 1 ¢~ ~ N4DM -~- = Z~/2 - ~ (¢~ }g~2 + ~22),

(A.31) N e F s 7 - _ 1 (Z~_¢~+Z~¢~.)T~, 2


(A.32) N 2 F M - - _ _ 7- _ 1 T~ + (z~¢~+ ~.o~}Fo~ 2 2 [(z'~ ¢ ' : - z~ ¢0 ~ z + ~ ~sJ,

(A.33) N4FM 9- 1 . ~ ~ , -~- = zi/2 -~ (¢~ F~3 + ~ 3 ~ ,

(A.34) N2FA 7- _ _~1 (Z~ ¢~ - Z~+ ¢~) T~3. 2 v~

For the zl we have

N = O :

(A.35)

N = 1:

(A.36) A 2P M 3- 1

2 v~

N = 2 :

(A.37)

(A.38)

(A.39)

5 + _ 1 A 4DM 2

v~ - - ( z ~ ~22 + z~+ 22) ~ ,

A4S, 3 + - 8 ~ } F s ' s - - ~ ~ ~ ~3/2 00

(A.40) A 2S M 1 + 1

2 v~ (z ~+ ~oo + z~+ ~o) ¢~ .

N = 3 :

(A.41) A 2p~ 3- 1

2 v~

(A.42) A 2p~ 3 -~ 1

2 v~ (Z+ ~11 +Z+ ~11)¢ s,

(A.43) A4P~ ~ Z312 11,


(A.44) zl 4P M 5- 1

2 - - + X + W22) ¢ ,

(A.45) A 2FM 7- 1 2

- - -1- Z + T 3 3 ) ~

o - (A.46) /t4F~-~ - -~ ~ Y~ ~ Z3/2 33-

Z

For the A we have

N = 0:

1 ÷ (A.47) A 2S ~ = ¢~,Z~ ~roo.

N = I :

- - 8 9 (A.48) A4p: -~- = ¢~X3/2 F n ,

(A.49) A4P) 3~ 2 = ¢~,Z~+ ~F~,

Q -

(A.50) A4P~ 2 = ¢~'z~+ ~ "

N = 2:

7 + s (A.51) A 4D.:~ -~- = ¢~Z312 T ~ ,

(A.52)

(A.53) 5 + A2D~ ~ = ¢~Z~ ~ ,

(A.54) +

(A.55) 5 + . 2 ,r~p)t A2D~ ~ = ~ Z + ~ ,


3 ÷

3 + ~ 9), (A.57) A 2p~), 2 - __ ¢), Z + ~Fll,

(A.58) A2S:.~ - ~ = ¢~Z+ ~ o ,

1 + (A.59) A~S~ ~ = ¢~Z~ T~ ~,

1 + (A.60) A ~S~ -~- = ¢~ Z + }F~o,

N = 3 :

3- (A.61) A 2 P ~ "-~ = ¢~Z~ ~ ~11

3 - ~ i/f ~)~)~ (A.62) A 2 p ~ ~" = ¢~)~+ ~11 ,

(A.63) 2 , 3- . ~ ~ , A Po~ --~ = ~ Z + h~ll '

(A.64) A~P~ 3-- 2 = ¢~Z~ ~ 1 ~,

(A.65) 2 , _~ ' A P ~ = ¢~Z~ ~ i ~ ,

(A.66) A2P~p ~ = ¢~Z~+ ~ ,

(A.67) 4 , _~_ = - ~ }F~' A P ~ ~Z3/2 11 ,

_ ~ _ _ _ . s ~p,~)~ (A.68) A 4 p ~ ~Z3/2 11 ,

(A.69) A 4p~ 5~ 2 -- ¢~Z~/2 }[f~P ,

BARYON SPECTRUM IN

(A. 70) A 2D~., 5-- 2

(A.71) A2D~,j, 5-- 2

(A. 72) A 4D:~,>, 7-- 2

7- (A.73) AZF.,~

7- (A. 74) d ~F,.,,

(A.75)

(A.76)

(A.77)

(A.78)

and for S

N = 0 :

A P O T E N T I A L QUARK MODEL

z 3 3

33 ,

A2F~ 7-- 2 = ¢~z~ ~ ' ,

A 2F~ ~ = ¢~z~ ~° ,

- - ~3/2 33 2 = ¢~

A 4 F _ 9- =~. s ~:~

+

(A.79) Z2S - ~ = Czz~ ~'oo.

N = I :

5- . s ~ , (A.80) S 4 P z ~ = ~zZ~/2 11,

(A.81) S2P~ 3-- 2 = CzZ~; ~ 1 ,

(A.82) S2P~ 3-- 2 = CsZ~+ ¥ ~ .

56 - l l Nuovo Cimento A.

875


(A.83)

(A.84)

(A.85)

(A.86)

(A.87)

(A.88)

(A.89)

(A.90)

(A.91)

(A.92)

(A.93)

N = 2 :

7 + . s F,~ Z 4 D ~ - ~ = 9 ~ X ~ 22,

7 + . = T~ 214D~ -~- = ~,~Zl/2 22,

5 + "U2Op~ - ~ = $s X ~ + ~ ,

5 + 22D~ -~- = ¢~X~+ ~ ,

5 ÷ 22Dp~ -~- = CzX~ T~,

3 + . = T , ~ 24S~ -~- =9~X3~ 00,

24S~ - ~ = ¢~Xl~ ~ ,

3 +

1 +

1+ = ~Foo,

1 + z~S~ y = ¢=z~- ~ .

N = 3:

(A.94) 22P~.~ 3~ 2 - " ~ '~"~'~

(A.96) 22P'. , -~- " ~ " ~ ' ~ZX+ T l l

BARYON SPECTRUM IN

3 - ), ~ (A.97) 22P~, - ~ = CzZ+ ~'~,

(A.98) .~ 2p~a 3- ), ~' y = ¢=z+ ~'~7,

(A.99) 2 2 P ~ 3-- 2 = Czz~+ ~ ,

(A.100) Z4P~, = ~Z~/z 'n ,

(A.101) 24P.~,~ 5-- 2 = CzZ~/2 ~'~}',

(A. 102) --,v4P~ -~- = CzZ~/2 ~w~,~, ~11

-~ - ~)')' (A.103) 2 2 D ~ = CzX~ ~22 ,

(A.104) Z 2 D ~ = CzZ~+ ~F2e ,

(A.105) 2 4 D ~ 7-- 2 = ¢z~/2 ~F~z ~" ,

7- (A. 106) 22F .~ - ~ = Czz~+ ~'~,~ ~, *33

(A.107) 2ZF~, - ~ = CzZ~+ 33 ,

(A.108) ~F~:~, ~ =¢~x÷ ~,

(A.109) 2~Fo~ -~ = Czz~, ~g,

(A.110) , ~ 4 F ~ ~zZ~/2 33 ,

9- ~,~ (A.111) Z4F.~oo, ~ = CzZ~/z aa •

A POTENTIAL QUARK MODEL

The wave functions of the other particles are constructed similarly.

877

878

APPENDIX B


In this appendix the hyperfine interaction is reviewed. This interaction has the form

(B.la) Hhyp=~,H[yp,

2 ~ Si" Sj ~(rij) + - Si" Sj . (B.lb) HhiJ-- 3mimy r~L (3Si" rij~

The first term is called the contact term and the second is the tensor term. This interaction is treated as a perturbation. As already mentioned for the baryons containing quarks with equal masses we have

= ~ T-liJ _ ,~ K/'12 (B.2) Hhyp .d~ z.[ hyp - - o , . l hyp , i<j

and for the baryons containing two distinct masses

(B.3) Hhyp = ~ H~yp = H~2yp + 2H~3yp. i<j

The calculation of the 12-component of the contact term is straightforward. The 23-component is also easy calculated by replacing

(B.4) ~a(r23)=(2/3)3/2 ~3() _ 1V3 P)"

The tensor term is calculated by using(~)

(B.5) lj2j.~ r-~ij|

where

transformation (3.7)-(3.8) is used to calculate the 23-component.

s )

3 j~ J~/ is the 63" symbol and <ll II> is the reduced matrix. The 32 3sJ

(25) M. WEISSBLUTH: Atoms and Molecules (Academic Press, New York, N.Y., 1978).


• R I A S S U N T O (*)

Si usa un modello di quark della cromodinamiea quantistiea per caleolare l'intero spettro dei barioni che contengono i quark u, d, s, c, b. Si usa un potenziale di Coulomb a corto raggio insieme ad un potenziale lineare a lungo raggio modificato mediante un potenziale di oscillatore armonico. Questa combinazione porta ad una descrizione buona (?) delle masse dei barioni con parit~t sia negativa che positiva basata sulle masse di cinque quark e quattro parametri aggiuntivi correlati con il potenziale scelto.

(*) Traduzione a cura della Redazione.

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Pe3mMe (*). - - MoIle~h KBapKoB, npe;ano~o~enHan B paMKaX KBaHTOBO~ XpOMO,/IHHaMHKH, vICllO/Ib3yeTcg /IYi~ ablqrlcaean~ cneKrpa 6apnonoB, coaepxamnx u, d, s, c, b KBapKm I/IcrlOJlb3yeTcfl KOpOTKOJIeI~ICTBytOI/~H~ Ky:IOHOBCKI, ffI rlOTeHILHaYl BMeCTe C ~IJIHHHO~IeI~ICTByIOIIIHM JIHHeI~IHbIM IIOTeHIIHaJIOM, MOaHqbHHHpOBaHHblM C I1OMOIRblO

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BM6paHHblM HOTeHHFIaJIOM.

(*) Flepe6eOeuo peOam4ue&

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Baryon spectrum in a potential quark model