N U C L E A R PHYSICS A

ELSEVIER Nuclear Physics A629 (1998) 160e-163e

B a r y o n r e s o n a n c e s i n a d e f o r m e d o s c i l l a t o r q u a r k m o d e l

A. Hosaka ~, H. Toki b and H. Ejiri b

~Numazu College of Technology, 3600 Ooka, Numazu, 410, Japan

bResearch Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567 Japan

We study baryon resonances in a deformed oscillator quark model. It is shown that low lying masses fit well to rotational spectra. In particular, the Roper resonance is identified with the band head of the 2hw rotational band. We also study electromagnetic transitions which are sensitive to the spatial deformation.

1. I n t r o d u c t i o n

Recent interests in baryon resonances have been stimulated by on-going and planned ex- periments at facilities such as TJNAL (CEBAF), COSY, MAMI and possibly SPring8 [1]. Measurements of various transition amplitudes are particularly useful for detailed studies of baryon structure. So far, the non-relativistic quark model has been successfully applied to a global description of baryon resonances. There a spherical confinement potentigl has been assumed, and the model shares several common aspects with, for instance, nuclear shell models. In nuclear physics, in the region far fi'om closed shells, spatially deformed states are also observed as confirmed by the characteristic pattern of rotational spectra and of E2 transitions [2].

The formation of deformed states appears to be a general property of many body sys- tems. It is then natural to expect that baryons can also deform, if they could be regarded as many-body systems of quarks and gluons. Such a deformation is related to the con- finement mechanism, and it is hoped that a detailed study of it will provide information on gluon dynamics. The purpose of the present work is to study phenomenological con- sequences of such spatial deformation in baryon resonances. This work is an extension of the previous work of Ref. [3]. To begin with we consider the effect of the deformation ex- clusively; other important effects of such as the interaction through the Nambu-Goldstone bosons [4] will be considered in future.

2. D e f o r m e d in t r ins ic s t a t e s

Let us see how deformed intrinsic states are developed in the non-relativistic quark model when oscillator parameters wx, wy and w~ are varied [5]. After removing the center of mass motion, the intrinsic energy of a baryon is given by

ENx,Nu,,% : ¢o~,(N~ q- 1) + ~u(Nu q- 1 ) q - ~ ( N ~ Jr 1), (1)

0375-9474/98/$19 © 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(97)00681-7

A. Hosaka et al./Nuclear Physics A629 (1998) 160c-163c 161c

where Ni = n) + n~ are the sum of oscillator quanta of intrinsic motions of A and p types. For a given set of (N~, Ny, N~), the minimum energy is searched. Assuming the fixed volume constraint , w=wuw ~ = w a, we find the min imum energy EI~. = 3(N~ + 1)l/3(Ny + 1)1/3(N= -t- 1)1/3to when 2 (N~+~) (Nz+l) wxwu = (N=+1)2, W=W~ = (Nu+a)2 " Some numbers and shapes are shown in Table 1. As anticipated, for excited states, deformed intrinsic states are energetical ly favored as compared to the spherical states. Whether this p roper ty changes or not when the fixed volume constraint is relaxed is an interest ing question which is related to the complicated dynamics of the QCD vacuum.

Table 1 Oscil lator parameters and energies of the deformed oscillator model.

N Ni E~i,~ w= : wy : wz shape N = 0 3 1 : 1 : 1 spherical N = 1 N~ = 1 3.780 1 : 1 : 2 prolate N = 2 N~ = 2 4.327 1 : 1 : 3 prolate

N~ = Ny = 1 4.762 2 : 2 : 1 oblate

A deformed intrinsic s ta te yields rotat ional bands of j ( j + 1)/A, where j are resonance spins and A moments of inertia. The moments of iner t ia can be computed using the cranking formula [2]. We show in Fig. 1 the result ing theoret ical masses of N = 1 (odd par i ty) and N = 2 (even par i ty) rotat ional bands as compared with experiments. We find tha t almost all four or three star resonances fit well to t h e r o t a t i o n a l spect ra for the exci tat ion energy A M < 1 GeV, or l _< 2. This is remarkable if we consider the simple

t rea tment of the presen~model . Among the resonances which do not appear in the figure, E(1385) belongs to the decouplet E(210). A(1405) and A(1520) would have a large K N component and can not be described by a simple quark model [6].

GeV

1.0

0.5

M* -M N = 2 Even Parity M* -M N = 1 Odd Parity

1=4 H~9(2220) Ho~ (2350)

1

PL~ (1720) Po (1890)

'=--5' P, ,~ ' po,(~) p,,c~) - - P~.~(1500)

D~ (22(X)) 1=3 Gi7(2190 )

Sn (1535) / ~ ( / ) D,~(1690) Su D~ I=~l O , ~ ) ) S,,O,, Sogl~670) S , ( l~) l /~(1700

q3(158....~1 ) D,~(1670) O,.~(~) S , ~ )

Theory N(28) A(28) 2("8) A(410) Theory N(28) N(48) A(28) _-~(28) ,~(48) _-(28) zl(210

Figure 1. Masses of even (N = 2) and odd (N = 1) par i ty resonances.

162c A. Hosaka et aL /Nuclear Physics A629 (1998) 160c-163c

One of interest ing consequences of the present model is tha t the first exci ted states of 1/2 +, the Roper resonance for the nucleon sector, appear as the band head of the rotat ional bands. Such resonances are observed not only in the N , but also in the A, E and A sectors. Although we should not ignore other impor tan t effects [4], it is emphasized that the deformation could be a significant driving force for pushing the 1/2 + resonances which appear too high in the spherical quark model down to the observed masses.

3. E l e c t r o m a g n e t i c t r a n s i t i o n s

Electromagnet ic transi t ions are par t icular ly sensitive and are useful to confirm the deformation [2]. The relevant ma t r ix elements are defined by

/ .

Ms, = (S I J d3 x f . . li) , (2)

where we adopt the s tandard non-relat ivist ic electromagnetic current J . The ampl i tudes are then expressed in the mult ipole basis, which can be accomplished through the multi- pole expansion of the electromagnetic field:

J

where A(ffh ) and A(jh M) are the electric and magnet ic fields of rank J and dz = h. The resonance states li) or If) are constructed by the project ion method from the

deformed intrinsic s ta te [5]. For N = 1 , 3 , . . . , odd par i ty states with l = 1 , 3 , - . . are projected out, and for N = 2 , 4 , . - . , even par i ty states with l = 0 ,2 ,4 , -~ . follow. The orbi tal angular momenta l are then coupled by the total quark spin s to yield the resonance spin j . Here we consider jus t for simplici ty only the states of s = 1/2 (28), and so, resonance spins are j = 14- 1/2.

Transit ions between rota t ional spectra may be classified into two. One is the in t raband transit ions between resonances in the same rotat ional band and the other is the in terband transi t ions for those between different bands. If deformation is large, the in t raband transi- tions are enhanced as proport ional to the intrinsic quadrupole moments Q0. Furthermore, t ransi t ion rates are determined by the Clebsh-Gordan coefficients as d ic ta ted by the Alaga rule in nuclear t ransi t ions [2]. It is ext remely interest ing to test those propert ies in future experiments. As for in terband transit ions, ampl i tudes are expected to be suppressed if intrinsic states are strongly deformed, since the overlap between sharply deformed states is suppressed.

So far, we have computed in t raband transit ions between even par i ty resonances of the N = 2 band. In Fig. 2, various E2 transi t ions at q -- 0 are shown as functions of the deformation pa ramete r d which is the ratio of the long to short axes of the prolate shape. Such a plot is useful in order to see the effect of the deformation. As d is increased, the ampl i tudes increase as well, which amounts to be, at d = 3 for N = 2, almost twice of the spherical case. The increase in the ampl i tudes is, as ant ic ipated, due to the increase in the intrinsic quadrupole moment . The effect is significant and might be observed in experiments. We also note that the spin flip processes are relat ively suppressed, since the

A. Hosaka et aL /Nuclear Physics A629 (1998) 160c-163c

E2 transitions are dominated by the ]/2 or the quadrupole moment operator [2].

163c

Various E2 transit ions at q = 0

.=e

0.2 "3 | 5/2 -> 1/'2

~ oojO, , ,o , , ,o . .o , , . , . , . , ;~®+ • , i ' i • i , i

2 4 6 8 Deformation

0.5

0.4 t 1312 - > 9 ~ ..~..:..::.'2 • . . , ~ ° ' " ; ' o , O o. 4,

0.1 q ,~'" .~. ...... Q 0 1 _la 11/2 7/2

0 0 -Jg""'" " " o . ~ , ~ . . + ~,~ I ; , . o ' " ° ' " o. . o . . ~- . . . . . . - • , , , J , , , , 0 . 0 - I ~ ~ , ~ , , , ,

2 4 6 8 2 4 6 8 Deformation Deformation

Figure 2. E2 amplitudes for intraband transitions between the N = 2 even parity band.

4. Summary and outlook

We have studied phenomenological consequences of spatial deformation in baryon res- onances. Many low lying resonances of light flavor sector have been shown to fit well to the rotational band of deformed states. Electromagnetic transitions are also computed for intraband transitions between even parity resonances, and it was found that effects of the deformation are significant. Further study of electromagnetic properties are expected in order to understand more on baryon resonances.

REFERENCES

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M. Dey, J. Dey and R.K. Bhaduri, Phys. Rev. D30 (1984) 152; M.V.N. Murthy, M. Brack, R.K. Bhaduri and B.K. Jennings, Z. Phys. C29 (1985) 385.

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