Octet baryons in the independent-quark-model approach based on the dirac equation with a power-law potential

Volume 120B, number 4,5,6 PHYSICS LETTERS 13 January 1983

OCTET BARYONS IN THE INDEPENDENT-QUARK-MODEL APPROACH

BASED ON THE DIRAC EQUATION WITH A POWER-LAW POTENTIAL

N. BARIK’ and M. DAS

Department of Physics, Utkal University, Vatzi Vihar, Bhubanewar-751004, Orissa, India

Received 2 August 1982

Several properties of octet baryons such as (i) the magnetic moment, (ii) (GA/G,), for neutron @decay and (iii) the

charge radius of the proton have been calculated in a simple independentquark model under the assumption that the indi-

vidual constituent quarks are confined, in first approximation, by a relativistic power-law potential V,(r) = (1 + p) (awlru

+ VO) with a, v > 0. In view of the simplicity of the model, the results obtained are quite encouraging.

The phenomenology of constituent quark dynam- ics in hadrons had been reasonably successful through the “bag-model” approach [ 11, which implies that the observed properties of hadrons are not too much inconsistent with the picture of constituent quarks moving relatively independently within the hadron. In these models, the fmite region of space called the “bag” containing the relativistic quarks, is a futed spherical cavity inside which the effective quark interactions are relatively weak to be treated perturbatively. The confinement of relativistic quarks is achieved here by providing a futed finite boundary in terms of the bag radius or equivalently the vacuum pressure. If(i) the idea of the independent constituent quarks in hadrons and (ii) the mechanism of confinement of these quarks to hadronic dimensions are the two basic

ingredients of these models leading to reasonable success, then one can make a simpler alternative approach based on independent-quark Dirac equation with some average quark-interaction potential of suitable Lorentz structure. Such a scheme has been followed by many authors in the recent past [2,3]. For example, a scalar potential in linear form was used by Critchfield [2] to confine the relativistic individual quarks in nucleons, where as the authors of ref. [3] have used various potential forms (linear, harmonic or Coulomb plus linear) with a Lorentz structure of equally mixed

’ Present address: Department of Theoretical Physics, The

Schuster Laboratory, The University, Manchester Ml3 9PL, UK.

scalar and vector parts. In such schemes confinement of individual constituent quarks in hadrons had been achieved through some average potential with suitable Lorentz structure without any finite boundary restric- tions of the “bag models”. Here the confining potential replaces the effects of external pressure on the bag. Implications of such a scheme in the context of quark confinement and relativistic consistency has been studied [4] by one of the present authors in reference with the heavy meson spectra. It has been found there [4] that a power-law potential with a Lorentz structure in the form of an equal admixture of scalar and vector parts, not only can guarantee

relativistic quark confmement but also can generate CT and bb bound-state masses in reasonable agreement with experiment. The power-law potential form has beenutilized recently by various authors with remark-

able success in the non-relativistic potential model studies of mesons [5] and s2- baryon [6]. The successful applications of this purely phenomenological potential in the above mentioned works makes it tempt- ing to extrapolate the investigation of ref. [4] to the low-lying baryon states in the spin l/2+ baryon octet. Therefore in the present work, we intend to under- stand the static electromagnetic properties of the octet baryons in the frame work of the independent- quark model based on the Dirac equation with a confining power-law potential.

Here we assume that the constituent quarks of the baryons move independently in an average flavour- independent central potential of the form

0 03 1-9 163/83/0000-0000/$03 .OO 0 1983 North-Holland 403


Vq(r)=( l + 3 ) V ( r ) = ( l +3)(ao+lrv+ Vo), (1)

where a and v > 0 and r is the radial distance from the baryon centre of mass. It is further assumed that the independent quark of rest mass mq obey the Dirac equation so that the four-component quark wave function qZq(r) satisfies the equation (with h = c = 1)

[0t. P + 3mq + Vq(r)] q~q(r) = Eq@q(r) . (2)

Putting

*q(r) = t q~8(r) 1'

one finds

[Eq - mq - 2 V(r)] ~kA(r) - ~-P~kB(r ) = 0 , (3)

. . P ~ A ( r ) -- (Eq + mq) ~kB(r ) = 0 . (4)

Following the usual prescription [7] for the separa- tion o f radial and angular parts, one can write ~k A (r) =f ( r ) y1.'~, and t~B(r ) = ig(r)Ylia.l., where ~ . stands for the ~o~rmalized spin-angula'/t~unction (e'ig~nfunc- tion o f J 2, Jz, L2 and S 2) formed by the following combination o f Pauli spinors with spherical harmon- ics,

Y¢31+l/2,l = + [(/+13 + 1/2)/(2l + 1)l 1/2 ~3 -1/2 X1,

+ [(/-T-j3 + 1/2)/(21 + 1)] 1/2 Y~13+1/2 X4 • (5)

Then separating the angular parts, the radial equations for g(r) and f ( r ) will be

g(r) = (Eq + mq) -1 (df(r) /dr - [(k - 1)/r]f(r)) (6)

and

r-2(d/dr) (r 2 dr/dr) + ((Eq + mq) [Eq - mq - 2 V(r)]

- k (k - 1)/r2)f(r) = 0 . (7)

Here k is the eigenvalue of (0- L + 1) which takes values k = -+(] + 1/2) for which l A = j ~- 1/2 and l B = j +- 1/2. For both values of k, eq. (7) reduces identi- cally to

r-Z(d/dr) [r 2 df(r)/dr] +((Eq + mq) [Eq - m q - 2 V(r)]

--/A(/A + 1)/r2}f(r) = O. (8)

It has been noticed earlier [8] that in the above form

of eq. (8), spin-orbi t interaction is absent due to ex- act cancellation of such terms coming from the vector and the scalar parts of the potential in eq. (1) taken in equal proportion. This aspect of the model need not be treated as a serious drawback since the contri- bution of spin-orbi t interaction term to the baryon mass splittings is already known [9] to be negligible.

We now assume further that all the three constituent quarks (q = u, d, s) of the octet baryons are in their ground states with JP = 1/2 +. This is quite like the assumption made in bag-model calculations [10]. Then for the ground state with J = 1/2 = J3, k = 1 and l h = 0, eq. (8), on put t ingf(r ) = Uq(r)/r and V(r) = av+lr v + VO, reduces to

d2Uq(r)/dr 2 + [(Eq +mq)

× ( E q - m q - 2 V o - 2 a v + l r ° ) ] U q ( r ) = O . (9)

Now we define a dimensionless variable p = r/r 0 with the scale factor r 0 suitably chosen as,

r0 = (22~qaV+l)-1/(o+2), (10)

where kq = Eq + mq. Then eq. (9) can be transformed to a convenient form:

d2Uq(r)/dp 2 + (eq - pO)Uq(p) = 0 , (11)

where

eq = (Eq - mq -- 2VO)(2av+l/Xq]2) -2/(v+2) • (12)

The solution of the eigenvalue equation (11) by any standard numerical method or by WKB approximation method would give eq and the normalized function Uq(p) for a particular choice of o > 0 and independent of any other parameter such as a, V 0 and mq. As for example the WKB solution gives [11 ]

e wKB = [3x/~TP(3/2 + 1/v)/ZF(1 + 1/v)] 2v/(v+2) (13)

and

u : K B ( p ) = [const./(eq -- 0v) 1/4 ] 1 / o ~q

) < c o s ( f d p ' ( e q - p ' ° ) - l / 2 - r r / 4 ) . (14) 0

Once eq is known, relation (12) can be inverted to obtain the individual quark binding energy Eq. For doing so, we substitute in eq. (12)

Eq - mq - 2V 0 =aXq (15)

404


and

2mq + 2Vo = abq , (16)

which lead to a root equation for x9 as

x;ti2)/u(xq + bs) =22/u e$+2J1v. (17)

Getting xq from (17) by any standard numerical method we can obtain from (15) the quark binding

energy Eq as

Eq=2Votmqtaxq. (18)

The corresponding ground-state wave function \kq(r) takes the form

\kq(f) =N ( cpq Wxf

q (~*mq)~,wX~ ’ 1 (19)

where p,(r) = C [U,(r)/r] e(f3, cp) is the normalized radial-angular part with a normalization constant C.

Nq stands for the overall normalization of @q(r), which can be easily obtained as

Ni= [l+Ai’(E,-mq - 2vo -2&W,)] -1 ,(20)

The angular brackets appearing here and elsewhere in this paper mean the expectation values with respect

to Vq (r). With the assumption that SU(3) is broken in the

quark rest masses as mu = md # m,, we can present

now some consequences of the model in terms of de- rived expressions for some of the measurable quantities of the octet baryons which are obtained simply by adding the contributions of each individual constituent quark. Following the lines of ref. [3], we obtain expressions for (i) the ratio (GA/Gv), for the /3- decay of neutron, (ii) the mean square charge radius (r2), of the proton, and (iii) the mass MB and magnetic moments & of octet baryons in the following manner. We find

‘(GA/G& = ; (4N; - 1) , (21)

(A, = (2N&)

X [(E, - Vo) (r2), - u”+~W+~)~ + 3/2&l , (22)

MB = CEq (23) 4

and

,-$ = $@rl~qu;ier), (24)

where ]Bt ) is the regular W(6) state corresponding to any of the octet baryons and pq is the bound constituent quark magnetic moment obtainable here as

[31

pq = (2MpNi/\)~q n.m. (25)

It is interesting to note that the bound quark magnetic moment pq Q-N~/X, instead of being a 1/2mq

normally assumed. The explicit expressions of PR for all the baryons in the octet obtained in the usual man-

ner using SU(6) states are listed in table 2 below. The angular brackets (P) , (r2jU and (P+2)u appearing in the relations (20)-(23) can be evaluated numerically or

by the WKB approximation method through the gener-

al expression

(P>, = r: q((p”N, ,

where

(26)

((pa>), = ydp P”$@) I

jdp U;(P). (27) 0 0

In the WKB method, neglecting the oscillation of

U,(p) as given in (14) one obtains [l l]

((p”)), = P]dp ,P(eq - P’)-~‘~, i PC

Jdp(e, - P’)-~”

0 0

= c$V3((crt1)/u, 1/2)/B(llu, l/2) . (28)

The outcome of the model through eqs. (18)-(28) depends very much on our choice of the potential parameters u, a and V. and the quark mass parameters

“lU = md and m,. Before making a choice of the potential parameters the following discussions will be in order. We hope that the phenomenological average potential used in this model is a substitute for the long-range part of the quark two-body interactions, since at baryonic dimensions, one can expect only the long-range part of the interaction to be dominant. For

the short-range one-gluon exchange interaction it is well known that the two-body quark-quark potential Vqq in a baryon is half the quark-antiquark potential Vqc in a meson. However, any definite relationship of

such type corresponding to the long-range part of the interaction is not yet known in clear terms. It is only

405


quite natural to expect that the quarks belonging to mesons and baryons may be acted by different long- range potential. Dosch and Miller [ 121 have shown that in a lattice approximation without including vacuum polarization, the three-body potential for bary-

ons may be written approximately as 0.54 times the sum of the two-body qi potential. However, one does not really know what happens in the continuum limit. Therefore, it may not be totally unreasonable if we take the average central potential for quarks in a baryon to be about the same as that in a meson obtained in ref. [4]. The results so obtained may provide justifications a posteriori for such an assumption.

Therefore following ref. [4], we set the potential parameters as

(u, a, Vo) = (0.1, 1 S562 GeV, -1.89 GeV) . (29)

Then using appropriate expressions based on the WKB approach described earlier for computing (GA/

G,), , (r2), , vp, Mp and cl,, , we get a good estimate of the quark rest mass parameters m, = md and m,. With a typical choice of m, and m, as

(m,, ms) = (185.60 MeV, 408.5 MeV) (30)

and the potential parameters as given in (29), we find

it possible to get an overall agreement with the exper-

imental values of (GA/Gv)n, pp, Mp and pi\. The results of this calculation based on the WKB approach

are presented in table 1 in comparison with the corresponding experimental values. The charge radius of the proton is found to be (r2$,/2 = 1.0063 fm, as compared to its experimental value of 0.88 f 0.03 fm. The neutron charge radius is obviously zero here in contradiction with its experimental value (r2+!,/2 = -0.12 fm. This is of course the case with most of the models of such a kind including the bag model. The values of the ratio (GA/G,), obtained here are compar-

able to typical bag-model calculations. We further no- tice that /+, with its calculated value 2.7809 nm, is quite close to the experimental value, unlike in the bag models where it comes out somewhat too small. Now we compute the magnetic moments of all the baryons in the octet by using the constituent quark magnetic moments according to relations (24) and (25). The constituent quark moments vu, yd and ps as given in table 1 are obtained through relation (25).

The predictions for the octet baryon magnetic moments are listed separately in table 2 in comparison

Table 1

Outcome of the model with parameters and solutions along with the results for several measurable quantities in (i) WKB and (ii)

numeriacl method of calculations.

Model inputs

and outcome

Potential

parameters b,a, vo,) (0.1, 1.5562GeV, -1.89 GeV)

quark rest m,=md 185.6 146.95

masses (MeV) % 408.5 387.3

quark binding

energies (MeV)

bound quark

magnetic moments

(n.m.)

observable

quantities

WU 1.8539 1.91547

pd -0.92697 -0.9517

fiS -0.61397 -0.61397

(GA/%), 1.0856 1.0086 1.254

(r* $I* (fm) 1.0063 1.056 0.88kO.03

pp (n.m.) 2.7809 2.8732 -2.793

c1~ (n.m.) -0.61397 -0.61397 -0.6138-0.0047

MD (MeV) 938.28 938.31 938.28tO.003

Calculations

(i) WKB (ii) numerical

Exp.

1.229 1.2364

312.76 312.77

438.27 443.48

406

1 4 - yu + ys -3Crd +3Ps - 3-1’2!.‘u + 3-‘12pd


Table 2 Baryon magnetic moments in nuclear magnetons in independent quark model as calculated by (i) WKB approximation method and

(ii) numerical method.

WKB Numerical

results results

Exp. Ref.

2.7809 2.8732 2.793

-1.8539 -1.91547 -1.913

-0.61397 -0.61397 -0.6283*0.0047 [131

2.6766 2.7586 2.33 to.13 iI41

-1.0313 -1.072 -1.41 to.25 1151 -0.89 kO.14 1161

-1.4366

-0.5096

-1.6056

-1.457 -1.250 ~0.014 1151

-0.4994 -0.69 fO.04 [151 -1.6588 -1.82 i-O.18 [I71

-0.25

wih the corresponding experimental values. We find

the agreement fairly good. So far we have based our phenomenology on the

WKB calculations which are straightforward and simpler to work with. However, we now repeat our calculations by adopting standard numerical methods. We find that the same potential parameters as in (29) with slightly different quark masses mu = 146.95 MeV and m, = 387.3 MeV yield the quantities of our interest not drastically very different from the corresponding WKB values. These results are also presented in table 1 and 2. In this analysis we have not attempt- ed any detail fit for which we do not claim the parameters to be strictly unique. However, we have checked that by choosing the potential parameters around half of its present value, the quantities like charge radius of the proton in particular, come out much too worse. We further point out that we have not emphasized very much on the mass spectrum of the octet baryons which is highly degenerate here. The degeneracy can be removed with the considera- tion of possible gluonic corrections as is done in bag model calculations of Degrand et al. [ 11. Considera- tions of such corrections with a detail phenomenol-

ogical fit to the mass spectrum and other electromagnetic properties of the hadrons will be taken up in a separate communication. Nevertheless the overall predictions of the model in the simplest form, for the baryonic magnetic moments, (CA/G,), for the P-decay of neutron and charge radius of the proton,

in reasonably good agreement with corresponding ex-

perimental values can be interpreted as a support in favour of our earlier contention that the average po-

tential in such models for quarks belonging to baryons or meson should be about the same. Similar con- clusion was also reached by Ferreira, Helayel and Zagury in ref. [2].

We are thankful to Professor B.B. Deo for his constant inspirations. We also thank the Computer Centre, Utkal University for its timely cooperation in the computational work.

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Volume 120B, number 4,s ,6 PHYSICS LETTERS 13 January 1983

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Octet baryons in the independent-quark-model approach based on the dirac equation with a power-law potential