Shell model of the nucleon

Nuclear Physics A507 (1990) 245c-250c North-Holland

245c

S H E L L M O D E L OF T H E N U C L E O N M BOLSTERLI Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U S A J A PARMENTOLA West Virginia University, Morgantown, West Virginia 26506, U S A A shell model of the nucleon IS described an which the quark-gluon core of the nucleon and the virtual pxons around the core separately occupy single-particle states T h e main characteristics of a n emerging shell model of the nucleon can b e seen In the cloudy-bag model t ( C B M ) a n d the Los Alamos Sohton model ~ (LASM), two models of the nucleon in which t h e virtual plons t h a t s u r r o u n d the nucleon's quark-gluon core are put into a single t-shell (all t h e virtual plons have isospin 1) like the single j-shell t h a t is popular in nuclear shell-model calculations

In these models the emission a n d absorption of the pions

by the nucleon core are governed by a Harmltonian of the Yukawa type with a form factor for t h e core-pion interaction t h a t is determined by ideas a b o u t chiral current conservation T h e C B M resembles the shell model of the a t o m , in t h a t t h e nucleon core is t r e a t e d as static or fixed

T h e LASM IS more hke the nuclear shell model, in t h a t the core and the pIons

are t r e a t e d democratically, with b o t h being allowed to move under t h e dynamical action of the H a m l l t o n i a n

Since the properties of the C B M have been discussed extensively in the

literature, in the rest of this p a p e r we c o n c e n t r a t e on the LASM T h e r e are other nucleon models of the shell-model type t h a t assume t h a t the nucleon consists of c o n s t i t u e n t quarks a n d meson fields, r a t h e r t h a n t r e a t i n g the core as a unit T h e simplest of these is the M I T bag model, In which the quarks occupy single-particle states within a confining cavity with a residual quark-quark interaction t h a t is t r e a t e d as a perturbation

Subsequent models have a t t e m p t e d to derive t h e features of the M I T

bag model from a q u a n t u m field theory of interacting quarks a n d scalar meson field

For

example, in t h e model of Goldflam and Wilets, 3 the quarks occupy single-particle states, and the scalar meson field is t r e a t e d in the mean-field approximation, which can be regarded as an a p p r o x i m a t i o n in which the q u a n t a of the scalar field all occupy t h e same s s t a t e In this t r e a t m e n t , the scalar field simulates the bag or confinement region T h e r e are extensions * of these models t h a t restore chiral s y m m e t r y by adding an interaction of t h e quarks with a pion field, t h e n t h e r e are, in effect, three sets of single-particle states, one each for t h e quarks, scalar mesons, a n d pi mesons

In contrast, t h e LASM assumes t h a t the internal s t a t e of the

quark-gluon core is the same for all motIonal states of the core, In much the same way as the internal s t a t e of t h e nucleon IS assumed to b e always t h e same In t h e various states of the nuclear shell model

Furthermore, the LASM t r e a t s t h e core as a unit whose internal

0375-9474 / 90 / $3 50 © Elsevier Science Publishers B V (North-Holland)

246c

M Bolsterh, J A Parmentola / Shell model of the nucleon

s t r u c t u r e appears t h r o u g h form factors t h a t descmbe its interaction with the electromagnetm field, these form factors are taken from the M I T bag model, but can easily be modified to incorporate new reformation a b o u t the internal quark-gluon s t r u c t u r e of the core As in the nuclear shell model, in the LASM a localized s t a t e of the system is used to develop a p p r o x i m a t e wave functions

T h e result is a state, of t h e type t h a t is often called

a nontopologIcal sohton, m which the motion of t h e core a n d the surrounding plon field are determined in a self-consistent way T h e total energy of the nucleon in the LASM is the sum of a positive kinetic energy of the core a n d a negative interaction energy of the core wlth the plon field t h a t it generates by the emission a n d absorption of virtual pions T h e p a r a m e t e r s t h a t describe the core motion are chosen so as to rmmrmze the expectation value of the H a m l l t o m a n m the nucleon ground state, the minimization deterrmnes b o t h the a p p r o x i m a t e core-motion wave function and the form of the pion field It is not unreasonable to regard the Hartree-Fock s t a t e of a nuclear H a m d t o n l a n as a n o t h e r example of a nontopological sohton state In first approximation, each c o m p o n e n t of the sohton state of the LASM H a m l l t o n l a n contams the core in one of the four states in Its s~/~ shell, while virtual q u a n t a of the plon field occupy various of the nine states in a p m n p shell (the n u m b e r of substates is t h r e e for lsospln by three for angular m o m e n t u m for a total of nine) As in other shell models, the isospIn and angular m o m e n t u m of the occupied core s t a t e and the occupied plon p states must be vector coupled to give t h e t o t a l spin a n d isospm of the nucleon s t a t e

T h e locahzed s t a t e in which

the respective particles are restricted to just these two shells, one for t h e plons and one for the core, has been used to c o m p u t e values for the proton and n e u t r o n electric form factors and the

profon elecfric form facfor 0.8-

R=-.35

0.6-

0.4R=.65 0.2R=.55 0

I

0

10

I

I

I

20

30

40

Q2 in fm -2

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M Bolsterh, J A Parraentola I Shell model of the nucleon

Psa and Ptl plon-nucleon phase shifts, with the results shown m the figures The descnptaon of the nucleon electric form factors m the LASM is analogous to the shell-model description of the electrac form factors of nuclei In first approxlm&tlon the quark-gluon core is treated as a point Ferrmon m its lowest self-consistently determined s-state

For computing electromagnetic

properties, the extended nature of the core is included by folding the intrinsic charge density of the core with the probabdaty density of the point core There is an additional contribution to the charge density from tile plon field around the dynamic core The results agree quahtatlvely with the experamental data for the proton electrac form factor and the paon-nucleon P33 phase shaft when the core radius has a value of about 0 5 fm Core motaon has the effect of making the nucleon core appear larger than ats mtrmsac saze, that as, ats charge density is spread out by ats motaon Therefore, sarmlar agreement wath the data as obtained m the LASM wath an mtrmsac radms that is smaller than the core radaus m the CBM There as poor agreement wath the experamental Pit phase shaft The mare feature of the shell model, whach as common to both the nucleon and nuclear shell models, is the davlsaon of the single-particle Halbert space into two subspaces of whach one contains the smgle-partacle modes that are amportant for the ground and low-lying states of the system whde the other contains the rest of the smgle-partacle Halbert space The first

,3

5 3 phase shift in radians .

....... .

o.OO...~oO'°°°°°°'°'°°

1 0 -- 1

-

11 p h a s e shift in radians R = 0 . 3 5 frn R = 0 . 4 5 frn

-

R = 0 . 5 5 fin R = 0 . 6 5 fro!

-2 1

2

3

4

pion total energy in pion m a s s e s

5

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M Bolsterh, J A Parmentola / Shell model of the nucleon

and second subspaces can be called the spaces of the strongly and weakly mteractxng modes, respectively, although they are strongly or weakly interacting only m the low-lying states of the system

As the excitation energy increases, the separation of the modes into strongly

and weakly interacting loses its value

In general, the shell model attempts to treat the

strongly interacting modes as exactly as possible The weakly interacting modes are treated perturbatlvely m an expansion in the number of weakly interacting quanta

A shell model is

useful If the set of strongly interacting modes is not too large for modern computers to deal with A major difference from the atomlc and nuclear shell models, where the number of electrons or nucleons is conserved, is that m the nucleon shell model the number of virtual plons within the nucleon is not conserved, and the state vector has components with different numbers of plon field quanta

The interaction Hamlltoman of the Yukawa form connects basis states m

which the number of plon quanta changes by one

A related interesting feature of the shell

model of the nucleon that is not present m the shell model of the nucleus is the posslblhty of large-scale occupation af a partacular paon shell

Sance the paons are Bosons, there is no

restnctaon on the number of paons that can occupy a single t-shell

In the case of strong

coupling, it as necessary to use varaous coherent-state technaques m order to generate paomc states that adequately represent the paon parts of the ground state Compensating somewhat the large occupation of the plon states as the fact that only one of the core states as occupied in each component of the nucleon wave function There are varaous posslballtaes for generating excited states of the nucleon m this sort of shell model Farst there are the anternal excltataons of the core that are the subject of models, like the MIT bag model, that use an assumed quark-quark anteractlon to compute dascrete elgenvalues of a Hamlltoman that as assumed to describe the internal states of the core This first type of excatatlon has not yet been incorporated into the LASM Then there are states m whach the paon field around the core as excited, sometimes these paomc excitations are dxvlded into rotataonal and vlbrataonal modes of excxtatlon

Excltataons of this second type

are contained in the LASM Hamlltoman (and an the CBM and Skyrme model Harmltomans) and thear low-lying excltataon energaes have been computed

Finally, there are excitations

that are umque to the LASM, namely, excltat~ons of the core motaon, that is, excatataon of the core from its presumed ground 0s state into other orthogonal states

Although the

LASM Harmltoman contains the posslbahty of such core-motion excatat~ons, the corresponding excitation energies have not yet been computed In terms of shells, the descnptaon of the s wave paon-nucleon phase shafts may reqmre the consaderataon of two more strongly coupled modes, an s shell for the paons and the lowest p shell of the core Which of the two possible shells for the core, P~/2 or P3/2, is more important ~emams to be determined

Eventually, the locahzed approxamataon to the ground state wave

functaon of the nucleon will contain components wath the core m varaous single-core states accompamed by paon quanta dlstrabuted over a set of strongly coupled smgle-plon states

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M Bolsterh, JA Parmentola / Shell model of the nucleon

What can we hope to learn from the shell model of the nucleon7 In the nuclear shell model, the axm has been to obtain a set of parameters that specify the effective two-nucleon mteractmn m nuclei With this set of parameters, the shell model gives a prescription for calculating energy levels and transition matrix elements for nuclear states

Slrmlarly, the

nucleon shell model can be expected to produce a descnptmn of the pmn-core interaction This parametrlzatxon of the plon-core anteractmn will be useful for descrabxng the nucleon and its excited states, it wall also gxve reformation about the internal structure of the nucleon core that will be valuable as a means of characterizing nucleon state vectors and understanding them at an mtmtave level ACKNOWLEDGMENT This work was performed under the auspaces of the U S Department of Energy REFERENCES 1 A Chodos and C B Thorn, Phys Rev D 12 (1982)59, M V Barnhlll, W K Cheng, and A Halprm, Phys Rev D 20 (1979) 727, G E Brown and M Rho, Phys Lett 82b (1979) 177, G A Miller, A W Thomas, and S Th~berge, Phys Lett 91B (1980) 192, S Th6berge, G A Miller, and A W Thomas, Can J Plays 60 (1982)59 M Bolsterh and J A Parmentola, Phys Rev D 34 (1986) 2112, M Bolsterh and J A Parmentola, m Chiral Sohtons edxted by K -F Lm (World Scientific 1987) 507, M Bolsterh and J A Parmentola, m ToplcaJ Conference on Nuclear Chromodynarmcs edited by J Qm and D Sxvers (World Scmntlfic 1988) 63, M Bolsterh and J A Parmentola, Phys Rev D 39 (1989) 4744 R Goldflam and L Wflets, Phys Rev D 25 (1982) 1951, L Wdets, in Chlral Sohtons edited by K -F Lm (World Scmntafic 1987) 362 M C Blrse and M K Banerjee, Phys Lett 136B (1984)284, M K Banerjee, W Bromowskl, and T D Cohen, m Chlral Solltons e&ted by K - F Lm (World Scientific 1987) 255, K Goeke, M Harvey, U - J Wlese, F Grummer, and J N Urbano, Z Plays A 326 (1987) 339, Th Melssner, F Grummer, K Goeke, and M Harvey, Phys Rev D 39 (1989) 1903