Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

N O N - S T R A N G E BARYON M A S S E S W I T H C O L O R H Y P E R F I N E AND P I O N E X C H A N G E I N T E R A C T I O N S "A"

H.J. WEBER J. W. Beams Laboratory and Department of Physics, University of Virginia, Charlottesville, VA 22901, USA

and

H.T. WILLIAMS Department of Physics, Washington and Lee University, Lexington, VA 24450, USA

Received 2 September 1987

The masses of strangeness zero baryons, with quarks in the S, P, and D shells of a harmonic oscillator potential, are calculated in the nonrelativistic constituent quark model. While the color hyperfine interaction is known to be successful in describing the mass splittings, inclusion of pion exchanges markedly improve the mass predictions of nucleon resonances, particularly those of negative parity. Fitting to all well established non-strange baryon masses below ~ 2 GeV/c 2 significantly improves previous results.

Within the constituent quark model [ 1 ], mass spectroscopy is based upon use o f a nonrelativistic average harmonic-oscillator confinement potential to bind three quarks into the S, P, and D shells. Degen- eracy splitting is provided by the color hyperfine in- teraction (chf) ,

Vchf= 3m---~as (4n~rl .tTEt~(r)q_S12/2r3 ) , (1)

where

S12 = 3trl • P o'2 • r tlrl • o'2, ~=r / r .

This is a spin-dependent but color neutral effective two-body potential, suggested by the color magnetic part of the one-gluon exchange between quarks at short distances [2 ].

Use o f the constituent quark effective mass, mq ~ mN / 3, implicitly includes nonperturbative QCD effects o f various ranges. Instantons are known to break chiral symmetry spontaneously and yield a bi- linear quark-quark interaction o f scalar and pseu- doscalar character [ 3 ]. In an instanton fluid model, Shuryak [ 4 ] obtains mq ~ 200 MeV/c 2 and a constit-

Work supported in part by the US National Science Foundation.

uent quark size of about 1/3 fm. This model consid- ers only relatively short-range ( < ~ 2 / 5 fm) con- tributions. At long range (low frequencies), the quark condensate contributes to the pion mass and decay constant, while the gluon condensate is often related to the confinement potential [4]. Nonetheless there is presently no adequate QCD treatment of pion dy- namics, as is done to lowest order in chiral bag models and to all orders in skyrmion models. This, despite the fact that constituent quarks and pion dynamics are widely recognized as the appropriate degrees of freedom at low energy and long range [ 5 ].

Towards the goal of an improved QCD treatment o f pion dynamics, we estimate here the effects of the one-pion-exchange potential (OPEP) ,

V~ ( r) = ( G m J 2 m q )2( lh "T2 / 12~z)

X {~rl "oz [ e x p ( - m ~ r ) / r - (4rr/m 2 )~(r) ]

+ S l 2 ( l + 3 / m ~ r + 3 / m Z r Z ) e x p ( - - m ~ r ) / r } , (2)

on the low mass spectrum of strangeness-zero bar- yons in the context of the constituent quark model. This potential is generated by a p ion-quark vertex of the form iG~r.q/2mq, where the coupling constant G is related to the pion-nucleon coupling constant by

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Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

g~NN = 5/3 m N / m q G ,

2 /4n=14.5 (3) grcNN

We assume that the constituent quark mass in- cludes the pionic quark self-energy among other short- range contributions. Therefore, when the OPEP be- tween constituent quarks is diagonalized, both time- ordered pion exchange diagrams and the pionic quark self-energy diagrams (to all orders ) are accounted for. These are the major ingredients of the Chew-Low contribution to the baryon masses.

The basis of three-quark states in a harmonic oscil- lator well with size parameter a ~ mq can be charac- terized by SU(6) supermultiplets: (56, 0 +) for the ground states; (70, 1 - ) for the 1 hco excitations of negative parity; and (56', 0+), (70, 0+), (56, 2+), (70, 2+), and (20, 1 +) for the 2 ho9 excitations of positive parity. In the pure harmonic oscillator model, all the 2 ho9 supermultiplets are degenerate in mass, however, a non-harmonic central potential will break the degeneracy into a simple pattern [ 1 ]:

E(56 ' , 0+) = E o - A E ,

E(70, 0 +) = E o - 0 . 5 AE,

E(56, 2+) = E o - 0 . 4 AE,

E(70, 2 +) = E o - 0 . 2 AE,

E(20, 1 + ) =Eo •

The mass of the (70, 1 - ) multiplet, E_, is indepen- dent of the above masses. Without attempting to characterize the deviation from purely harmonic of the average potential seen by the quarks, one can use the three mass parameters - E_, Eo, and AE - to characterize its effects.

We thus diagonalize a hamiltonian which contains an anharmonic central well (that produces the super- multiplet splitting discussed above), the residual color hyperfine interaction ofeq. ( 1 ), and the OPEP ofeq. (2). The zero-range portion of the pion exchange po- tential has evoked some debate [ 6 ] and there are ar- guments both for and against its inclusion in this context. Hence, calculations are made here with and without the term containing the Dirac delta function in eq. (2), as well as with the delta function replaced by a gaussian of unit area and full width at half max- imum of 480 MeV (representing a constituent quark

radius of 1/3 fm). The five parameters, i.e; or: har- monic oscillator well size, Ors: color hyperfine strength, E_: negative-parity ( 1 hog) supermultiplet mass, and Eo, zXE: positive-parity (2 hog) mass parameters, are varied to produce the best fit of calculated baryon masses when compared to the results presented by the Particle Data Group [ 7 ]. This variation is done sep- arately for the cases

(a) no residual quark-quark pion exchange potential,

(b) quark-quark OPEP with no contact term, (c) quark-quark OPEP including contact term, (d) quark-quark OPEP with "smeared" contact

term. The calculation yields the masses of 19 isospin-1/2,

strangeness-zero baryons; and 11 isospin-3/2, strangeness-zero baryons. Not all of these correspond to experimentally confirmed particles; others, while listed in the data tables of ref. [ 7 ], have large exper- imental uncertainties. For the purposes of the pres- ent calculation, the quality of a fit is determined by comparison of theory and experiment for 14 baryons which result from this model and which are listed in the data tables with the designation " . . * *: good, clear, and unmistakable." The root mean square deviation

l~ \i/2 R M S = (Mexp -Mthy )2 /14 )

is calculated for each set of the five variational pa- rameters, and is minimized relative to the five. The minimization procedure is further constrained by discarding any fit in which the nucleon mass N 1/2+(939) and that of the A 3/2+(1232) are not within 5 MeV of the experimental masses.

The results of this minimization procedure are pre- sented in tables 1 and 2. Table 1 shows the masses which result in the best fit. The experimental masses in parentheses are those which are found in the data tables, but with a certainty less than the • * * * list- ings. The absence of an entry in the "Expt." column corresponds to a particle predicted by the model which does not correspond to a listing in the baryon tables of ref. [ 2 ]. The five columns following the ex- perimental results correspond to the cases ( a ) - ( d ) above, and

(e) the results generated by the present calculation

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Table 1 Strangeness-zero baryon masses in MeV.

PHYSICS LETTERS B 28 April1988

Expt. Chf Chf+ pion Chf+ pion Chf+ pion Isgur, [ MeV ] only no contact contact smeared Karl

N 1/2- 1535 1504 1525 1510 1530 1489 1650 1650 1669 1725 1721 1656

N 3 /2 - 1520 1542 1526 1525 1540 1532 (1700) 1730 1653 1591 1627 1748

N 5 /2- 1675 1662 1662 1649 1671 1670 N 1/2 + 939 940 940 938 936 891

1440 1374 1450 1448 1465 1448 (1710) 1691 1744 1728 1753 1729

1878 1945 1951 1962 1902 (2100) 2037 1996 2081 2062 2058

N 3/2 + 1720 1694 1713 1739 1731 1715 1819 1872 1818 1853 1869 1915 1938 1914 1933 1954 1944 1961 1986 1989 1977 2033 1995 2060 2051 2057

N 5/2 + 1680 1694 1717 1744 1736 1715 1920 1939 1920 1939 1956

(2000) 1986 1960 1955 1964 2026 N 7/2 + 1924 1937 1968 1972 1954 D 1/2- 1620 1676 1660 1636 1662 1685 D 3 /2 - 1700 1676 1660 1636 1662 1685 D 1/2 + 1910 1821 1848 1834 1859 1876

1863 1911 1915 1922 1924 D 3/2 + 1232 1237 1233 1237 1235 1230

(1600) 1685 1761 1640 1701 1799 1920 1886 1939 1887 1908 1948

1949 1966 1962 1974 1984 D 5/2 + 1905 1889 1924 1896 1915 1944

(2000) 1943 1958 1959 1968 1978 D 7/2 + 1950 1966 1978 1960 1971 2002

RMS dev.[MeV] 36.7 27.2 39.5 33.5 33.2

w h e n t he p a r a m e t e r s o f I sgur a n d K a r l ' s c a l c u l a t i o n

o f D s ta te b a r y o n s a re i n s e r t e d [ l ].

T a b l e 2 shows t he va lues o f the f ive v a r i a t i o n a l pa-

r a m e t e r s w h i c h give t he lowes t v a l u e o f t he R M S de-

Table 2 Model parameters for fits of table 1.

v i a t i o n . T h e u n c e r t a i n t y in t h e va lues o f E _ a n d Eo

is + 10 M e V ; in t he v a l u e s o f ot a n d E is + 5 M e V ;

a n d in t he v a l u e o f ors is + 0.1. In t he cases o f t h e

v a l u e o f ot for t he " s m e a r e d " c o n t a c t t e r m , a n d val -

Chf Chf+ pion Chf+ pion Chf+ pion Isgur, only no contact contact smeared Karl

a [MeV] 290 289 300 296 287 as 2.2 2.2 0.71 1.24 2.6 E_ [MeV] 1610 1590 1630 1630 1610 Eo[MeV] 2010 2000 2040 2030 2020 AE[MeV] 508 420 420 405 420

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Volume 205, number 2,3 PHYSICS LETTERS B

Table 3 Strangeness-zero, negative-parity baryon masses in MeV.

28 April 1988

Expt. Chf Chf+ pion Chf+ pion Chf+ pion Isgur, [ MeV ] only no contact contact smeared Karl

N 1/2- 1535 1520 1530 1529 1523 1488 1650 1644 1656 1658 1655 1656

N 3/2- 1520 1552 1530 1529 1524 1532 (1700) 1712 1672 1666 1663 1748

N 5/2- 1675 1655 1663 1662 1659 1670 D 1/2- 1620 1666 1665 1663 1660 1685 D 3/2- 1700 1666 1665 1663 1660 1685

RMS dev.[MeV] 28.7 24.4 24.4 24.6 33.8

ues of as for the contact and the " smea red" contact cases, more precis ion was necessary to satisfy the constraints on the nucleon and delta masses, and the l isted parameters are accurate to _+ 1 in the last digit. Tables 3 and 4 give the results when the fit is done using only six negat ive-par i ty states which are ra ted "good, clear, and unmis takab le . "

The results indicate that the masses are best pre- d ic ted in the model in which pion exchange is in- cluded, but the contact (del ta funct ion) te rm is excluded. Inclusion o f the contact te rm produces a worse fit than the model with no p ion exchange, and inclusion o f a finite width te rm in place o f the del ta funct ion produces results between those o f the no- contact term case and the delta-function contact term. Decreasing the size o f the " smear ing" of the contact te rm causes the results to approach those o f the pure contact term; increasing the size improves the fit, but never to the accuracy of the no-contact term results. Wi th the inclusion o f the contact term, the nucleon and delta masses become very sensit ive funct ions o f the parameters as and a requir ing more precise def- in i t ion o f these parameters than the other cases. The

results of Isgur and Karl presented in the last column show a relat ively good RMS deviat ion, but a t tent ion

is called to the value (891 MeV) o f the nucleon mass p roduced using their parameters . Thei r parameters were chosen so as to produce a best fit for the D state baryons so no restriction was put on the nucleon mass in that calculation. A different set o f parameters was found to produce the best fit o f the ground state bar- yons in their model [ 8 ].

There exists the possibil i ty of double counting when both the OPEP and effective color hyperf ine poten- tials are used in their entirety. When the OPEP con- tact te rm is omit ted, there can be no double counting between the color hyperf ine and the scalar OPEP Yukawa potential . The corresponding tensor poten- t ials may produce redundancy, however. To investi- gate this possibil i ty, a calculat ion o f the fourteen " , . . . " baryon masses was made with both the OPEP contact term and tensor potential switched off. The resulting best-fit RMS devia t ion was 38.6 MeV, only marginal ly bet ter than the complete OPEP and considerably worse than the best results of OPEP without contact te rm (table 1 ).

Table 4 Model parameters for fits of table 3.

Chf Chf+ pion Chf+ pion Chf+ pion Isgur, only no contact contact smeared Karl

a [MeV] 260 265 245 250 260 as 2.6 3.1 2.9 2.9 3.5 E_ [MeV] 1610 1595 1620 1610 1610

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References

[1 ] N. Isgur and G. Karl, Phys. Rev. D 21 (1980) 3175, and references therein.

[2] A. De Rtijula, H. Georgi and S. Glashow, Phys. Rev. D 12 (1975) 147.

[ 3 ] G. 't Hooft, Phys. Rev. D 24 (1976) 3432. [4] E.V. Shuryak, Nucl. Phys. B 214 ( 1983 ) 239; Phys. Rep. 115

(1984) 151, and references therein.

[5] P.D. Simic, Phys. Rev. D 34 (1986) 1903, and references therein.

[6] M. Weyrauch and H.J. Weber, Phys. Lett. B 171 (1986) 13. [7] Particle Data Group, M. Aguilar-Benitez et al., Review of

particle physics, Phys. Lett. B 170 (1986) 1. [ 8 ] N. Isgur and G. Karl, Phys. Rev. D 20 ( 1979 ) 1191.

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