Quarks in nuclei

WORKSHOP

QUARKS

IN

II

NUCLEI

CONTRIBUTORS

J. Adam C. Chanfray A.E.L. Dieperink P. Gonzalez P.A.M. Guichon J-F. Mathiot G. Miller H.J. Pirner A.W. Schreiber A.W. Thomas E. Truhlik V. Vent0 K. Yazaki

237~

Nuclear Physics A497 (1989) 237c-246~ North-Holland, Amsterdam

QUARKS IN NUCLEI P.J. MULDERS Nations

Institute for Nuclear Physics and High-Energy section K (NIKHEF-K), P.O. Box 41882 1009 DB Amsterdam, The Netherlands

Physics,

1. INTRODUCTION In the 25 years that the quark model exists, understanding

flavor, flavor-spin

in the nonabelian

chromodynamics

(QCD) as the theory underlying

devoted

progress

to solving

gauge theories

led naturally

has been made in the

was on symmetries

and color. In the seventies the theoretical developments

and specifically

understanding

considerable

of the structure of hadrons. In the sixties the emphasis

in quantum field theories

to the emergence

the strong interactions.

QCD, of which lattice gauge theories

of models

that incorporate

confinement

and asymtotic

models. After confinement

freedom.

seem the most promising.

For the

followed route is the

a number of features of the basic theory, most notably

Well-known

examples

are quark potential

has been put in, the color magnetic

exchange interaction is responsible

of quantum

Many efforts have been

of the structure and properties of hadrons the most commonly

construction

such as

(spin-spin)

models and bag

part of the one gluon

for the hyperfine structure in the mass spectrum of baryons and

mesons. Another approach has been the construction of effective hadronic lagrangians such as in the Skyrme model. In the construction of the underlying

of effective lagrangians one attemps to preserve the symmetries

theory. For instance,

which is spontaneously

in the Skyrme model one incorporates

broken with the corresponding

chiral symmetry

appearance of the almost massless pions as

the Goldstone modes in the meson spectrum. At the basis of most of these theoretical developments instance

the measurements

were the experimental

developments,

for

of nuclear, proton and neutron form factors and the results of deep

inelastic electron scattering experiments

showing the existence of pointlike partons inside hadronic

matter. Although quarks and gluons unmis~ingly QCD is very probably degrees

of freedom

momentum excitations

the correct theory describing

for describing

transferred

the interactions

a nucleus strongly

among them, the relevant

depend on the probe and the energy and

to the nucleus. For instance in the description

of vibrational

and rotational

of the nucleus as a whole, states that are for instance excited in reactions between heavy

nuclei at the appropriate freedom,

are the underlying degrees of freedom in nuclei and

energies,

let alone their underlying

one does not even need the individual quark structure.

For higher energies

nucleon

degrees

and momenta

of

nucleons

generally provide a sufficiently accurate description of the nuclear aspects of the problem. Including moreover

the mesons,

notably the pions, and the delta as the lowest nucleon excitation

one can

obtain an accurate description of many processes involving nuclei. Nevertheless

one can pose a number of questions, some of which were addressed in the session

03X-9474j’89/$03.50 0 Elsevier Science Publishers B.V. (No~h.Holland Physics ~bIish~g Division)

‘quarks in nuclei’ at this workshop: 1. Can we understand

the existence

of nuclei with their ground state properties

dominated

by

themselves

as

nucleon degrees of freedom. 2. Could

quark effects

modifications

in low and intermediate

energy

processes

manifest

of intrinsic properties.

3. Can one learn more about the interactions between the quarks from nuclear systems. 4. How do nuclear effects enter in those cases that quarks provide the most natural description such as deep inelastic electron scattering. 5. Is it possible, examples

for instance,

of the transition

in going from low to high momentum from nucleon to quark degrees

transfer to indicate clear

of freedom as the most economic

description. 2. QCD

AND THE

STRUCTURE

A number of con~butions structure

of the hadrons.

interactions,

Few presently

doubt that QCD is the correct

with the quark and gluon theory to describe

the

although it is not yet possible to understand how the hadrons arise as the bound states

in this theory. responsible

OF HADRONS

at this workshop were directly concerned

Nevertheless

attemps

are underway

to try to really understand

how QCD is

for the structure of hadrons.

In the talk of Miller’ some prelimin~y (SCQCD) 2 were presented.

results of calculations

This is an expansion

using strong coupling

QCD

in inverse powers of the coupling constant g and

should be a valid guide to the low momentum-transfer

regime appropriate for nuclear physics where

g takes on larger values. These calculations in SCQCD are performed by treating three-dimensional space as a discrete lattice, while the time variable remains continuous. One finds a lattice hamiltonian H = HE + where

the electric

HQ

+

HM,

field contribution

(2.1) HE, the quark contribution

contribution HM are proportional to g2, 1 and gm2respectively. number of flux lines. The operator HQ is responsible

HQ and the magnetic

field

The operator HE basically counts the

for the quark kinetic energy and pair creation.

The magnetic term HM consists of plaquettes destroying and creating flux lines. One of the physical consequences

is the absence of quark exchange contributions

based on the

fact that the overlap between different flux lines is zero (see fig. 1). Only because of the action of plaquette operators and the fact that physical states are superpositions

of strings of different lengths

and shapes a nonzero exchange effect can arise. In SCQCD meson-meson caused by flux tube re~angement,

interactions

are thus not

but only the quark pair creation operator causes interactions,

much like resonance formation in TCX+ p -+ rcz

P.J. Mulders 1 Quarks in nuclei

Fig. 1: In SCQCD the overlap between diff~re~t~~

In order meson-nucleon

to discuss

nuclear

couplings

dynamics

239~

lines is zero

one only has to establish

and form factors

in reasonable

that SCQCD

agreement

leads to

with experiment.

The

preliminary results of some calculations presented by Miller indeed indicate a qualitative agreement. An attempt to study the consequences

of the QCD lagrangian

in the hadronic

regime was

described in the talk by Chanfray 3. The method, known as the color dielectric model 4, provides a method

to realize

con~g~ations

the transition

are responsible

from QCD to hadronic

for a polarizable

physics.

The short wavelength

medium with dielectric

gluon

constant E. The effective

hamiltonian,

L (Di2+ Hi’),

I;

E&f =

cw

i 2&

shows that color cannot survive in a region where E = 0 and so a (finite) domain with E f 0 will form around a color charge. Chanfray dielectric

showed

how one can extend the derivation

dielectric field x a color neutral vector field 8, appears. In mean field approximation are treated as classical remain,

of the color

theory from pure gluonic QCD in the SU(2) case to the SU(3) case. Besides the color fields the effective

where x and 6u

theory can be solved for baryons. Although problems

e.g. center of mass corrections,

the results seem promising.

The field turns out to be

surface peaked and it is expected to contribute to the repulsion in baryon-baryon

interactions.

The

model also enables one to consider the case of uniform quark nuclear matter. 3.

DEEP INELASTIC

SCATTERING

Deep inelastic scattering of leptons provides a direct picture of the quark and gluon distributions in the target. Through sum rules the connection with properties of the ground state, specifically the expectation

values of some operator is established.

In recent experiments

with

the focus has been on

the spin structure function gIN(x) 5. In his talk, Thomas 6 argued that in order to discuss the sum rules involving the spin structure function gI(x) one needs to consider models that incorporate intimately

connected

to the spin structure. For instance,

function the expectation value of the axial current enters

dx

slN(x) K

4

$0,

Qhl3rsvKQ

IN>

chiral symmetry,

as this symmetry is

in the first moment of the spin structure

P.J. Mulders / Quarks in nuclei

240~

While most of the strength is expected to come from the proton, the neutron contribution vanish. In the cloudy bag model the following nonvanishing

10 dxglN(x)

= - $Nrr

is found, where the quantities P,,

+

25

4

zpAC

-PNAn 9

considerable

result,

(3.2)

denote the probability of baryon-meson

is an interfernce term. Together with the modifications

does not

configurations

and PNA~

of the spin structure by one gluon exchange a

part (37 - 46 %) of the neutron contribution

to the Bjorken sum rule can be explained.

In a short contribution Jaffe outlined how one can calculate the structure functions for deep inelastic scattering

in the Skyrme model directly from the current communicator

in a way similar to the

derivation of the parton model from the free quark currents 7. 4. QUARKS

AND NUCLEAR

One of the fundamental

STRUCTURE

questions in the discussion of quarks in nuclei is the question how the

structure of the nucleus can be understood. shows that quark substructure

Guichon * sketched in his contribution

of nucleons

does not conflict

with the traditional

a scenario that picture of the

nucleus. His model for the nucleon is an M.I.T. bag. The bag in uniform nuclear matter is then studied in mean field approximation. scalar field (3 and an isoscalar-vector

The quarks inside the bags act as sources for an isoscalar field w, which are both treated as classical fields. A consistent

solution can be found with the correct nuclear binding energy at normal density by varying the coupling constants g,and

g, The bag radii that are considered range from 0.6 - 1.0 fm. The values

that are found for the scalar and vector potentials used in hadron field theories.

are roughly a factor three smaller than the ones

This is due to the fact that the relativistic

effect which produces

saturation is more efficient for quarks than for the nucleon. The saturation curve calculated in the model is very similar to usual nuclear matter calculations and is not much affected by corrections for Fermi motion and Pauli blocking

that have also been discussed

in the paper. The ground state

properties for nucleons at normal density such as axial charge and radius are not strongly modified in the nuclear medium. 5. QUARK

EXCHANGE

CONTRIBUTIONS

IN NUCLEI

The general starting point for the study of quark exchange

effects is a nuclear wave function

expressed in terms of quarks degrees of freedom that exhibits clustering into hadrons, i.e. colorless constituents.

The wave function is a resonating group type of wave function vab...(l,

2, 3; 4, 5, 6;...) = N-1/2 R [X(a, b,...)$,(l,

where a, b,... indicate the nucleon coordinates the CM of 1, 2, 3 constrained given by n

2, 3)&(4,

5, %.I

(5.1)

and 1, 2, 3 etc. indicate the quark coordinates

to a, etc. The antisymmetrizer,

= 1 - Z Pij, where the summation

which up to a normalization

is over all pairs of quarks, ensures

with

factor is the Pauli

241~

P.J. Mulders / Quarks in nuclei

principle at the quark level. The no~aliza~on N
of the wave function is given by

= N = ,

(5.2)

kernel N ,

b’,...; a, b ,...)

as an intermediate

= <&&,L.

I&?/ $&t,...>,

step by integrating

(5.3)

over the quark coordinates.

This kernel is

in a direct term <&&,L..]~~c&&,...> and an exchange term %(a’, b’,...; a, b ,...)

In a microscopic

calculation

= &a,, &bh . . . f N,,(a’,

b’,...; a, b ,... ).

at the quark level x(a, b,...) is usually determined

(5.4) in a variational

approach after choosing a definite form for the quark wave functions in the nucleon clusters. At the quark level the electroweak

currents are simply given by the elementary vector and axial

currents for the quarks which are pointlike fermions.

The expectation

value of a one-body

quark

operator for the nuclear state described by w,

63

J = C
can be rewritten at the nucleon level which is convenient for nuclear ai~p~cations. This gives J =c

‘XI.iNIX>+

c

N=a,b,..

where &(a’;a)

=


+ ...

(5.6)

NN pairs

<$a’/t: jil4p,

(5.7)

(5.8)

8La,

8-.i-..- _-J_ a’

a--g’

b’

b

\

-7

b

--

--

b’

Fig. 2: One-body quark operators give rise to efsecrive one and two-body nucleon operators because of antisymmetrization

at the quark level.

242~

P.J. Mulders / Quarks in nuclei

are the effective one- and two-nucleon integrating

current operators (illustrated in fig. 2), again constructed by

over the quark coordinates.

The above

illustrates

the appearance

of two-body

contributions in the calculation of matrix elements. It are these contributions approximation

which do not satisfy the separability

properties

of the impulse

such as the factorization of the nuclear form factor in a body form factor and nucleon

form factor or the factorization

of the quasi-elastic

response functions in a scaling function and the

nucleon form factor squared. Examples of this were pointed out by Dicperink in his talk 8. It is, however, not so that quark exchange contributions are there several other conventional

mechanisms

are striking quark signatures. Not only

that also lead to two-body currents, for instance

meson exchange currents, but there is also the fact that the wave functions X(a, b,...) cannot directly be identified with a conventional As was already mentioned

nuclear wave function as was pointed out by Yazaki 9 in his talk.

x could be found from a variational calculation if the hamiltonian

H(l’, 2’, 3’; 4’, 5, 6’;...; 1, 2, 3; 4, 5, 6;...) is known at the quark level. However,

H =

for any but

the lightest nuclei this is not possible so one would rather take the nuclear wave function Xc(a,b,...) from a conventional

nuclear physics calculation.

However, it is not possible to directly identify x

with xc. In order to explain this we must realize that from wherever we got our nuclear quark wave function w = R[x $I @..I, it is assumed to be the best approximation

in the variational problem

~=~,

(5.9)

where H is the hamiltonian at the quark level. Rewriting this at the nucleon level one finds 6

(5.10)

= 0,

where ti(a’,b’,...; a,b,...) = <@a~~t,~../Hk&&,...>is the hamiltonian

at the nucleon level obtained by

integrating out the quark degrees of freedom analogous as done in the normalization kernel or for the currents. This can be rewritten as (5.11)

which shows that if we are using a conventional

wave function xc, the proper comparison with a

wave function x is NV2 x N-1/2 3y112 x = l/2

=

(5.12)

xc.

Yazaki argued that the true two-body effects are given by J -CqcljNIxc> N

= C + NN

~(-)

243~

P.J. Mulders / Quarks in nuclei

=

$, - W’J)f

‘X

I{n, jN> 1X>)>

Using examples for the deuteron he illustrated the smallness of these true two-body effects among others in the form factor. Although small, the effects are not negligible. he suggests the detailed experimental

As an interesting

example

study of the 4He(e,e’n)jHe reaction.

In the talk by Dieperink 9 he stressed the deviations

from the impulse approximation

that arise

because of the two-body terms in Eq. (5.6). Although in principle drastic effects could occur 11 the effects in light nuclei turn out to be not dramatic, although they are of the same order of magnitude as meson exchange contributions.

Also in quasi-elastic

electron-nucleus

out to be small, at least for 4He, which shows a few percent

scattering the effects turn

increase

in the longitudinal

and

transverse response functions. Dieperink also pointed out that quark exchange effects give rise to a (small) effective three-body Coulomb interaction. Also in deep inelastic electron-nucleus

scattering quark exchange eFfects could be important. The

additional terms modify the quark momentum distributions in a nucleus in a way similar as observed experimentally

in the EMC measurements

In the contribution

**.

presented by Vento l3, he pointed out that there could be interesting places,

where socalled maximal Pauli effects can be expected. In ordinary (light) nuclei, the nuclear wave function

turns out to be dominated

by nucleon pairs that do not lead to dramatic Pauli effects.

Several non traditional nuclear systems such as dibaryons,

light hypemuclei

and delta-nuclei

have

been proposed as places where the quark Pauli principle is more important. Vento and collaborators have analyzed the delta-nuclei

and found sizable effects only for large momenta,

q > 4 fm-l. By

comparing

various channels, however, interesting effects at low energies might show up. They also

considered

the contributions

of quark exchange on the Coulomb energy. Energy differences

as 600 keV can occur in denser

system,

but many uncertainties

prevent

at present

as high a more

quantitative statement 14. 6.

MESON Mathiot

EXCHANGE l5 reviewed

CURRENTS

briefly the standard description

contributions

in the electromagnetic

contributions

expected

electromagnetic

current operator in nuclei and then discussed

from relativistic

corrections

in the constituent

current (MEC) successively

the

quark model, the role of

operators for six quark systems and some results from the Skyrme model.

In the standard correspond

of the meson-exchange

description

the main contribution

to two-body parts in the electromagnetic

with the continuity equation. Smaller contributions

to the meson

exchange

contributions

operator coming from the constraint associated arise from gauge invariant processes

e.g. those

involving pxy couplings and relativistic terms in the current that enter Fatthe same order as MBCs. Mathiot notes that in the constituent important contributions

quark model relativistic

corrections

in the current give

to the nucleon charge radius which have not yet been seriously investigated.

Although in bag models the quarks are treated relativistically,

many problems remain if one tries to

244~

P.J. Mulders / Quarks in nuclei

use an effective describing

With such a separation

six quark current for short distances.

the long range interactions

and quarks describing

between

hadrons

the shott range interactions

a careful

check of the continuity of the current in the transition region is necessary. In the Skyrme model the isoscalar part of the electromagnetic the Wess-Zumino

operator is proportional

term in the effective langrangian

to the baryon current which arises from

and this is connected

QCD. At present, a technical limitation for a better understanding using the Skyrme model 16 is formed by the poor knowledge

to the chiral anomaly in

of isoscalar exchange currents,

of the soliton structure for baryon

numbers larger than one. In his contribution, disintegration. equation

Truhlik

17 studied

in backward

deuteron

Calculations of MBCs in leading order in l/M satisfying the nonrelativistic

continuity

show a strong sensitivity

corrections

and short

range

meson

to the choice

phenomena.

exchange

currents

of the nucleon

For the study

form factors,

of the structure

to relativistic

of the isovector

electromagnetic

MBC Truhlik starts with a lagrangian for the nucleons and the R, p, and Al mesons

that is locally

chiral invariant

dominance.

and as such naturally

The effects of other heavy meson exchanges

are also included as well as relativistic

corrections

incorporates

the idea of vector

meson

(co, 6, rb 6) as used in the Bonn potential

to the current. From the calculations

one can

conclude that the contributions appearing due to the chiral invariance can contribute significantly and should not be omitted. Moreover, it is important to consider relativistic: effects and the con~butions of heavy mesons. 7

CONCLUSIONS It is clear that the session

mentioned

in the introduction.

understanding

of the strong interactions

theoretical considerations.

answers

to the questions

it indicates interest in a better and more fundamental

in nuclei. These questions

However, for understanding

of quantum chromodynamics 8

quarks in nuclei did not give definite Nevertheless,

cannot be answered

and modelling

the nonperturbative

only by aspects

the theoretical efforts are undispensable.

ACKNOWLEDGEMENT This work is part of the research program of the National Institute for Nuclear Physics

High-Energy the Foundation

Physics (NIKHEF, section K), Amsterdam, for Fundamental

and

made possible by financial support from

Research on Matter (FOM) and the Netherlands

Organization

Scientific Research (NWO).

9. REFERENCES 1)

G.A. Miller, ‘Strong coupling QCD and the nucleus’, contribution

2)

I. Kogut and L. Susskind, Phys. Rev. Dll (1975) 395; M. Bander, Phys. Rep. 75 (1981)

to this conference.

for

P.J. Mulders / Quarks in nuclei

24%

206; N. Isgur and J. Paton, Phys. Rev. D31 (1985) 2910. 3)

G. Chanfray, J.F. Mathiot and H.J. Pirner, ‘Color dielectric approach to the nucleon and nuclear matter’, contribution

4)

to this conference.

H.B. Nielsen and A. Patkos, Nucl. Phys. B195 (1982) 137; G. Mack, Nucl. Phys. B235 (1981) 197; H.J. Pirner, J. Wroldsen andM. Ilgenfritz, Nucl. Phys. B294 (1987) 905.

5)

EMC Collaboration,

6)

A.W. Schreiber and A.W. Thomas, ‘Corrections to the Ellis-Jaffe

J. Ashman et al., Phys. Lett. B 206 (1988) 364. sum rule’, contribution

to

this conference. 7)

R.L. Jaffe, ‘Debosonization’,

contribution

to this conference;

R.L. Jaffe, Nucl. Phys. A478

(1988) 3c.

8)

P.A.M. Guichon, ‘Quarks and the saturation of nuclear matter’, contribution conference;

9)

A.E.L. Dieperink and P.J. Mulders, ‘Quark exchange effects in electron-nucleus contribution

10)

to this

P.A.M. Guichon, Phys. Lett. 200B (1988) 235. scattering’,

to this conference.

K. Yazaki, ‘Quark exchange currents within the quark cluster model’, contribution

to this

conference.

11)

T. de Forest and P.J. Mulders, Phys. Rev. D35 (1987) 2849.

12) P. Hoodbhoy and R.L. Jaffe, Phys. Rev. D35 (1987) 113 13)

P. Gonzalez and V. Vento, ‘Quarks in light nuclei’, contribution

to this conference.

14)

F. Wang and C.W. Wong, Nucl. Phys. A432 (1985) 619; F. Wang, C.W. Wong and S. Lu, Nucl. Phys. A480 (1988) 480.

15)

J.-F. Mathiot, ‘Few-body form factors: mesonic or quark degrees of freedom’, contribution to this conference.

16) E.M. Nyman and D.O. Riska, Nucl. Phys. A468 (1987) 473. 17)

E. Truhlik and J. Adam, ‘The chiral and relativistic effects in the backward deuteron disintegration’,

contribution

to this conference.