Colour dielectrics — a way to understand quark bags

Nucleiul'bysicsA446(1985)431c436c North-Holland,Anmterdam

431c

COLOUR DIELECTRICS - A WAY TO UNDERSTAND QUARK BAGS

Hans J. PIRNER TheoreticalPhysics Division, CERN, 1211 Geneva 23, Switzerland

We discuss the colour dielectric model which introduces as new effective variable the colour dielectric field for quark confinement. The model may be constructed from the QCD action and leads to the very successful bag model of hadrons. It allows to calculate the change of nucleon properties in nuclei which the EMC effect indicates.

1. CALCULATING THE BAG FROM LATTICE GAUGE THEORY Everybody

is

familiar

with

dielectrics

from

its

first course

in

electrodynamics. A macroscopic amount of matter has of the order of 1O23 charges in it, therefore it is impractical to solve the Maxwell's equations from the charge and current distributionsof all these charges. It is better to introduce a dielectric constant characterizingthe medium. In QCD not only the quarks have colour charges but also the gluons carry colour. So even vacuum will have a colour dielectric field. From the macroscopicHamiltonian

it is easily seen that colour displacement fields+Diand magnetic fieldsf, cannot subsist in a medium where E = 0, because the energy associatedwith them would be infinite. The vacuum therefore has E = 0. The dynamics of confinement is reduced to find the value of the dielectric field in every point in space around the colour charges. The region with E = 0 forms a bag around the E = 1 region inside of the hadron, where colour is present and not influenced by large scale vacuum fluctuations. The colour dielectric model132 proposes a local order parameter x(x) which is related to the dielectric constant E as E = x4 and can be calculated by averages over the gluon charges in a small volume L4:

0375-9474/85/%[email protected]. (North-HollandPhysicsPublishingDivision)

Qn the lattice with a lattice constanr:a, one starts with a11 Wilson loops which fit into a box I4 = (28)' and paas through xC. From the mcnrents

gioea

A dxp one i @ reconsCructsthe effective aeCion of the x-field given by a Wnetic term and a by

the -tntegral over all link variablea Urf = exp If,“j

fourth order polynomial potential U(x)

Note after the first averaging sCep $ 0, so one has Co xesrart the procedure by calculating the mome*Co

for

a larger

cell

of size f2L)4 * f4af4‘ * 0.2 GeV. The QCU whether confinement Pavours surface

Hopefully, this procedure converges fast when 1fnl-t h sffective potential U(x) datermines) e.g.,

dominated (SLAC) or volume (MIT) bag solutions. In the effective Lagrangian also Che quarks can be included wing local gauge invariance.I do not know how Co J.nCegrateout the quark fields+ When the scale size becomes vary large, probably choral fields %(x) and S(X) should be considered. The success of quenched QGD calculationsattributesa less important r&e to Chess fields. The effectfve Lagranglan of the cofour-dielectricmodel is:

and mq as current mass. The effecCive potentialIi(x)- 0 at x J 0 snd can be expanded as

H.J. Pirner / Colour dielectrics

3 awaX* 1

U(x) = The mass

mGB

correlation Before in

x2

x=o

O+ bound

state in pure gluon theory is given by the

length of the x field in the effective Lagrangian.

discussing

relation

experimental

to the others

dielectric

model

bag models

0; x2-

=f+,

of lightest

433c

consequences

mentioned

a

dynamical

field

soliton model.

to

describe

let me place it

Of course,

the colour

like the MIT and cloudy

is closest to the other bag models3,

and the non-topological

introduces

of this model,

in this meeting.

In contrast

confinement

to the first, it

and

can

be

used

in

scattering calculations. It is very similar to the Friedberg-Lee soliton 4 model , the main difference consists in the coupling of the soliton field. In the colour current

dielectric

quark

Friedberg-Lee negative colour

mass

model the effective quark mass mq/x is identical

inside

model

the

the bag

quark

mass

(x=1) is

and

large

infinite outside

outside

of the bag -1

inside. This last feature does not go well with asymptotic

dielectric

theory,

one can calculate

the string

tension,

to the

(x=0). In the GeV and

freedom. In once U(x) is

given. One also obtains the profile of the string, which in the strong coupling models

is

infinitely

indication

thin.

for collective

dynamical

confinement

In

spectroscopy,

bag motion5

the

Roper

resonance

gives

an

. Fake six-quark states disappear when

is taken into account for NN-scattering6.

2. TESTING THE BAG INSIDE AND OUTSIDE OF THE NUCLEUS There hadrons

has been

to

relation

description calculates

deep

sea

separates

outside change effects depths inflated

The

important

inelastic

helps

to avoid

quark

colour-dielectric

colour

a lot of fine work' with regard to low energy properties of

in bag models.

scattering

feature of the quark bag model is its 8 . Hopefully a translational invariant

the infinities'

distributions.

The

model are essentially

the inside I = 1 region

of the static bag model, when one

valence

quark

distributions

given by the confinement

from x = 0 outside

in

the

radius, which

for a free nucleon.

The

dielectric

field in nuclear matter, however, changes to x * 0.02 # 0 N of the bag and lets the quarks tunnel outside of the nucleon bags. The

of the x-vacuum

costs

energy,

the quark

tunnelling

wins

energy,

together make the nucleon bound in the nucleus. The effective UN

of a nucleon

nucleon

in the nucleus

can be related. We

results of the following Table.

and the rms confinement

find, using

the formulas

both

potential

radius of the

of Ref.

2, the

H.J. Pirner / colour dielectrfes

434C

He3

A

I

Be9

Cl*

I

Pe56

AU197

J [fml

0.655

0.658

0.695

0.746

0.805

XN

0.0073

0.0076

0.0116

0.016

0.020

‘N

lMevl -15.5

z [MeV]

-16.1

-24.6

-33.9

-42.4

-17

-25

-36" (Ni=)

-44*

(Li6) pth (x-0.62)

1.05

1.05

1.02

pexp

1.09

1.06

1.05

1

(Pb*O*)

1

0.97

1

0.95

10 The shift g of the quasi-elastic peak in electron scattering should be compared

to the theoretical potential depth WN. A change of the quark 11 distributions in the nucleus is directly shown in the EXC and SLAC results .

At x = 0.62, the ratios p - F2(x,Q2,A)/F2(~,Q2,Fe) xIo 62 are given in the last

I

.

two rows of the Table. They are calculated with a scaling violation parameter b = -0.25 and
is correct. Nucleons with their centres closer than deform

clusters and thereby the confinement size also increases. If we calculate for all the nuclei the modified confinementsize as a function of dc and then fit to the SLAC data, we find d

C

= 1.3 fm as critical centre-centredistance. At

this dc the average probabilitiesfor a n-quark cluster in Fes6 are pB = 45X, p6 = 201, 4 = 11% and p12+p15+... = 24%. Certainly a calculation based on six-quark clusters alone can be quite misleading. In fact we interpret these high cluster probabilities as an indication that with Increasing effective radius of the nucleon, the quarks percolate through the nucleus and the nucleus becomes colour-conductingas proposed in Ref. 12. Experiments at large Q* make other models than the quark model an academic exercise.

The

Skyrme model would have to include an infinity of mesons to

obtain scaling. At low energy, the bag models are less predictive. I expect, however, a qualitative description of the transition from low to high density nuclear

matter.

The leakage of colour given by xN = 0.02 in the colour

L. Wilets/ Quark-solitonmodel of nucleon structure

dielectric interacting

model

makes

the

(O(g2/x4>>.

meaningful

calculation 14 investigated .

It

quarks is

only

in

the

their

inter-nucleon

small

of their interaction

density

43%

medium

which

energy. This problem

may

strongly allow

a

is currently

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