Modeling QCD

WorkshopVI

Nuclear Physics AS27 (1991) 507c-512~ North-Holland, Amsterdam

MODELING

H.-J.

507c

QCD

PIRNER

Institute

for Theoretical

A model emphasizes

Physics,

a special

Philosophenweg

aspect

19, D-6900

of the theory.

Heidelberg,

It makes a pattern

F. R. Germany’

which summarizes

in a compact fashion the feature that is most important to the model builder. QCD is a highly complex nonlinear theory with significant characteristics like confinement, chiral symmetry breaking, hadronization and nuclear binding. The colored quarks and gluons of QCD are confined into hadrons. The lowest mass bosonic excitation has almost a vanishing mass due to chiral symmetry breaking. Jets produced at short distances hadronize into collimated

beams

of hadrons

asymptotically.

Composite

hadrons

form bound

states

(e.g.

nuclei) due to their attractive residual forces. The main emphasis of this session, on QCDmodels, was on confinement, but other aspects of QCD were equally dealt with. Models preconceive intuitive understanding where we cannot yet calculate things exactly. The evolution of the QCD coupling constant in perturbation theory is known up to length scales of < 0.1 fm. The color/chromodielectric model discussed by Pirner and Wilets assumes that the vacuum of QCD d evelops medium-like properties at large distances which modify the gluonic kinetic energy as in macroscopic electrodynamics via a dielectric constant. With that notion one can understand how the effective gluonic coupling becomes big and confining for large distances. Models help to reduce experimental observations to a few parameters. A large number of hadronic masses has been efficiently described by the constituent quark model where quarks with constant masses interact via confining long range forces. This model is used in the contributions of Geiger and Oka. New models stimulate

experimental

theoretical framework. the EMC-effect.

research

to ask questions

So the quark cluster

which were not permitted

model may have speeded

in the existing

up the discovery

of

Models may modify a parameter of the theory and thereby open the possibility to make genuine controllable approximations. For QCD such parameters can be the number of space-time dimensions or the number of colors N,. In the large NC-limit QCD is equivalent to a theory of infinitely many mesons with weak coupling constants 0( $). The contribution of Mattis deals with the large N, model. A serious discussion of a model must include a clear understanding of its limitations. Ideally one should be able to check the approximations made by the model builder. For instance in the Monte Carlo Renormalization of the effective theory to the original theory

Group Pirner compares at the same length-scale.

the lattice simulation So one can see how

good the model really is. In the constituent quark model Geiger calculates the correction of the heavy quark antiquark potential due to pair creation. The additional potential is large but has the same shape as the confinement potential. In this case it can be renormalized away. A bag-model estimate of the mass of the H(AA) dibaryon has stimulated many theoretical and experimental efforts. Oka identifies an instanton induced three-body force ‘Supported

by the BMFT

03759474/91/$03.50

under contract

number

06 HD 756.

0 1991 - Elsevier Science Publishers B.V. (North-Holland)

H.-J. Pimer / Modeling QCD

508~

which does not affect

the average

stability of the H-particle question of quark models but the importance strated.

Most

models

are tailored

are used for multiquark

energy

is a theoretical of three-body to three-quark

but which unbinds

the H-particle.

The

question probably beyond the accuracy forces in the dibaryon is clearly demonand @-systems

and may fail when they

systems.

In the first contribution and Hadrons”. by a dielectric

baryon

to this parallel

session Pirner

gave a report

on “Color

Dielectrics

In color dielectrics the QCD vacuum is regarded as a medium characterised field. The attraction of this approach is that it promises to construct an

effective action for hadronic structure from the QCD lagrangian, thus relating the underlying non-abelian field theory to phenomenology. The emphasis is on the gluon aspects of hadronic physics. He presented results of lattice calculations of the coupling constants of a colour dielectric model in the pure gauge (gluon) sector and a phenomenological application of the SU(3) model to the problem of Chiral Symmetry Breaking. In the first part* he reported on work done in collaboration Signal and Wroldsen these to confinement

with

Baier,

Grossmann,

investigating the long-distance properties of the vacuum, and relating and the appearance of an effective potential for the dielectric field,

which should have a minimum near E = 0. The procedure is to start with SU(2) lattice QCD at a lattice constant a, and then to make Monte Carlo Renormalization Group transformations (RGT’s), which will generate effective theories on lattices with lattice constants 2a,4a,... It is hoped that only a small number of RGT’s are necessary before reaching the continuum limit of the effective theory. The RGT’s average out gluon effects on the short distance

scale

leaving

an effective

colour

dielectric

field which

determines

the long-range

properties of the vacuum. The RGT defines link variables, 4, on the lattice with spacing paths on the original 2a as a weighted sum over link variables U E SU(2) in appropriate lattice. scalar,

Such

a weighted

sum has a ray decomposition

x, which he identifies

with the colour

The choice of an effective action local terms up to 4th order in x: se, with the operators

= +0(l)

dielectric

in I$ is crucial.

t $0(2)

into field:

an element dz,r = xz+

Pirner

chooses

+x0(3)

$X0(4)

of SU(2)

and a

. U,,,.

a simple form containing

given by

O(1) = cWp

O(2) =

5x:,,

where the subscript /Lvaries over the positive directions, the subscript v varies over positive and negative directions, and the subscript p refers to the product of link variables around a fundamental plaquette. He tests the results of the RGT by the technique of operator matching. This consists of simulating the above effective action on a lattice with spacing 2a, using as input couplings the values determined from the RGT. He finds that the matching is good to around 10% for the color dielectric operators, however, matching of the plaquette operators is only at the 50% level. In the second part he showed results2 Zuk, dealing with the problem of chiral

of a collaboration with Mathiot, Chanfray ad symmetry breaking. Approximating the long-

H.-J. Pimer / MoaWing QCD

509c

distance gluon dynamics of SU(3)colour by colour-dielectric block-spin variables, he obtains an effective QCD theory of a scalar colour-dielectric field and a massive colour-bleached gluon field coupled to light quarks. On the quark level this theory is similar to the popular nuclear models with scalar and vector mean fields. The massive vector field produces a strong dielectric

attraction between qQ pairs, which leads to qq condensation when the colourHe calculates the qq vacuum expectation value, (&,!I), field becomes confining.

and the pion decay expansion,

gluon coupling, decreases

constant,

and integrating

fr,

cr,eff, including

inside hadronic

by evaluating

out the bosonic

the fermion

variables.

quark effects,

determinant

He finds that

is large in the vacuum,

in a derivative

the effective where (&6)

quark# 0, but

bags, where I(&,!J)[ is decreasing.

Wilets announced his intentions to improve calculations of many bag systems in the chromodielectric model. Previous work with Fai, Perry, Krein, Tang and Williams3 showed that the chromodielectric model confines quarks and can lead to chiral symmetry Therefore, these models may in principle be applied to determine the modified properties

in the medium,

high densities. Within is solved. Appropriate

and the transition

of nuclear matter

breaking. nucleonic

to the quark gluon plasma

at

the Wigner-Seitz approximation a single soliton in a nuclear medium boundary conditions give the band edges. A crucial aspect of the

model is the filling of the bands.

Without

gluonic residual interactions

each energy state can

have 12 quarks. In the presence of gluonic interactions it is advantageous to have cells which The previous calculation of Birse, Rehr are color neutral, spin l/2 objects like nucleons. and Wilets4 is proposed to be improved by including gluon residual interactions. Wilets et al. address the filling problem by invoking the independent bag - pair approximation for six-quark (two-bag) configurations. The six-quark configurations are in color-singlet states

but the three-quark

clusters

include

hidden

color states.

They

consider

two spatial

quark states and five six-quark configurations. The occupation of the two spatial states is related to the filling of the Bloch bands. The reference “cell” is spherically symmetric with a spherically symmetric d field. Th e cell contains an indefinite number of quarks, but an average of exactly three. Neighboring cells interact independent bag-pair approximation. The chromodielectric Lagrangian density is

which differs from the Friedberg-Lee coupling

term,

-g&$

Lagrangian

(which if present

with the reference

by the absence

breaks chiral invariance).

cell according

of the direct

to the

quark-sigma

Here K(U) is the dielectric

function and U(u) is a quartic self-interaction term. They treat the scalar soliton field u in the mean field approximation (MFA). Gluons are treated in the Abelian approximation, since it is assumed that the essential nonlinearity is modelled by u and a(r). The gluon fields thus satisfy Maxwell’s of the quarks are calculated regularization. equations

equations. according

They work in the Coulomb gauge. The self-energies to Fai, Perry and Wilets, employing a form factor for

OGE leads to mixing among the five six-quark

are solved in the self-consistent

configurations.

mixed-configuration

or MCMF, in analogy with mixed-configuration Hartree-Fock. Oka studies the effects of a three-body interaction induced by instantons interaction. The three-body interaction is given by

L eff =

-

~~&)&(2)&(3)[1

+

&(($I

-

+ A((&

. ~z)(AI . h) t p-m) x

‘-+‘d

. ~33]h(1)$@)$‘@)

on the dibaryon

. AZ) + perm) - &01r3 + &DIz3((ZI

n 6

The resulting

mean field approximation,

$

(h.c.)

. Z2) + perm)

H.-J. Rrner / Modeling QCD

SlOc

where Vo is an overall strength FIxa E f&l~Xb,A; are symmetric Under

the assumed

quark

of the effective interaction, and Dtss f d,.&~X:Xj and and antisymmetric color singlet operators, respectively. interaction: condensation, (&J), one can deduce the two-body

Oka and Takeuchi’ found that the effective dent attractive force and a color-magnetic is shifted observed

by the instanton hyperfine

induced

splitting,

two-body interaction consists of a spin-indepeninteraction. The ground state baryon spectrum

interaction

N-A,

for instance.

when the overall The SU(3)

strength

breaking

is fixed by the

effect is also similar

to

the conventional hyperfine interaction if one takes the ratio of u - d and u(d) - s interactions interaction, which arises among u, d and s quarks, to be - m,/m. - 0.6. The three-body does not contribute to the baryon spectroscopy. Baryon-baryon interactions have a strongly attractive direct force due to the color-spin independent potential. This direct interaction has a longer range than the exchange color-magnetic interaction and therefore is dominant at long distance. The exchange force due to the color-magnetic interaction remains and thus the NN force is repulsive at short distance. Oka et al. apply the quark cluster model to the study of two baryon systems and propose a model in which the one-gluon exchange

and the instanton

the hadron.

mediated

An NN potential

interaction

share the role in the hyperfine

with a short-range

repulsion

and a medium

splitting

of

range attraction

is then obtained by this model. In the H-particle problem they find that the three-body force unbinds the N-particle. Geiger’ studied the effect of quark pair creation on the heavy quark potential. To shed some light on which pair-creation

effects

are likely to be masked

by renormalization,

and

which ones might be expected to be observable, Geiger and Isgur have studied “unquenching” in the quark model. Specifically, they used a variation on the 3Po pair-creation model, which has been very successful at describing strong decay processes, to calculate shift of a static Qa pair due to its couplings to virtual (QQqQ) decay channels. AE,

is large

and approximately

proportional

to the distance

between

the energy The shift,

the static

sources.

The linearity of AE would suggest that the pair creation effects can be absorbed into a renormalization of the mesonic string tension. However, by considering the dependence of AE on the spin state of the static sources, they show that this masking can only be approximate: after renormalization, residual energy shifts of order 100 MeV remain. These residual shifts are largest when a particular QQ state is close to one of its virtual decay thresholds. As a specific they

illustration

have calculated

of the relevance

the shift

of such considerations

of the rho mass from its virtual

to lattice decays

QCD

studies,

to two pions

as a

function of the pion muss. The crossing of the 27r threshold produces a nonanalyticity in this shift, and thus the usual lattice procedure of calcuIating the hadron spectrum with to 2m, < mP, large current quark masses (where 2m, > m,), and then extrapolating can lead to a substanti~ underestimate of m,. Geiger andIsgur suggest that this effect is responsible, at least in part, for the so-called “N/p” problem (the nucleon-to-rho mass ratio comes out too large) of lattice &CD. There

have been

arguments

by Kiritsis,

Seki and Cohen

that

in the large

NC-limit

a

Yukawa field theory for nucleons interacting with (T and w-mesons cannot be treated in a naive loop expansion. The nucleon masses are rnN - O(N,), the meson masses /J - O(l), the meson-nucleon couplings g z NJ/‘. In Born approximation meson-baryon scattering is O(I) and nucleon-nucleon scattering O(N,). For the one-baryon loop the meson self-energy $ - g2m$ - N:, therfore the nucleon one-loop result is incompatibly with the tree result that the meson-mass p2 - 0( 1). The compositeness of the nucleon manifests itself. Mattis’ compiles

a list of conditions

for hadron

processes

to be compatible

with large

NC-&CD.

H. % Pimer / Modeling QCD

For purely

mesonic

processes,

the leading-order

511c

diagrams

in l/NC are the tree graphs.

baryon-baryon or meson-baryon processes, the prescription is a little the leading-order diagrams are those that become trees if one removes

For

more complicated: the baryon line(s).

N l/N,), the only stable non-strange Whereas all mesons are stable (P,,, ground-state positive-parity baryons that are symmetric in spinxisospin.

baryons are the In our world,

with N, = 3, these are the nucleons and the A’s. But in the large-N, world (N, odd), these states comprise an infinite tower with equal isospin and spin, I = J = f, %, f, . . . . and degenerate masses. Lhod must contain only stable states; unstable baryons will appear as resonances

in the scattering

while a meson Moreover,

of stable

is dominantly

they

restrict

particles.

Since

a baryon

a ng pair, one has, quite

the kinematic

regime

so that,

is made up of N, quarks,

naturally,

mN N N, and p N N,“.

in the center

of mass,

all internal

and external 3-momenta have magnitude 1 i 1~ N,“. This is essential: when 1 i I- N,, interactions at the baryon vertex are formally suppressed as exp( -coast x N,), and quantum hadrodynamics breaks down. Finally, the allowed meson-baryon couplings in Lhad must be consistent with the It = Jt rule and the proportionality rule. The former implies, among

other

things,

the nucleon. mesons

that

The latter

to AN, &A,

the p is primarily relates,

and so forth over the infinite

Let me close this general discussion emphasis

on existing

tensor-coupled,

models

and the w vector-coupled,

by simple proportionality

constants,

to

couplings

of

tower of I = J baryons.

by a look into the future.

may prevent

the NN

us from developing

I can imagine new efficient

that too much approximation

schemes. In this PANIC meeting Martinelli stressed that lattice gauge theory in contrast to the quark model offers the possibility of systematically improving the accuracy of the calculation. The now common soliton models regard hadrons as static blobs of quark, glue, skyrmion etc. They give us an intuitive notion of size, shape and wavefunction Only recently more attention is again given to light-cone bound state of the hadron. calculations

and to the string picture of hadrons. The light-cone approach connects to the rich phenomenology of deep inelastic scattering. The string picture has been a very successful description of hadron physics and may develop due to the progress in critical strings. I personally

see the future

of models

as efficient

links

between

experiment

and large

scale QCD-lattice calculations (see also contributions of Geiger, Thomas, Negele). Model builders must understand both languages. Experimental progress in hadron physics would come to a full stop if we rely on the progress of computing alone. On the other side QCD-simulations become pure computational problems when the physicist does not try to interfere

and propose

relevant

physical

“observable?.

I believe

that

the coming

growth

of computing power will make the QCD vacuum and hadrons transparent. Correlators of quark and/or gluon fields will give us insight into hadrons beyond their valence structure.

REFERENCES

1)

B.Grossmann, TVP-89-14.

2) J.-F.

H.-J.

H.J.Pirner,

Mathiot,

G. Chanfray

Pirner

A.I.Signal,

and H.-J.

and J. Zuk, Heidelberg

R.Baier

Pirner,

and J.Wroldsen,

Nucl.

preprint

Phys.

Birse,

J.J.

Rehr and L. Wilets,

Phys.

A500 (1989)

Preprint

HD-

605; G. Chanfray,

(1990).

B208 3) G. Fai, R.J. Perry and L. Wilets, Phys. Lett. Wilets and A.G. Williams, Phys. Lett. B212 (1988) 4) M.C.

Heidelberg

Rev.

(1988) 362.

C38 (1988)

1; G. Krein,

359.

P. Tang,

L.

512c

5)

H.-J. Pimer / Modeling QCD

M. Oka and S. Takeuchi,

University

of Pennsylvania

6) P. Geiger and N. Isgur, to be published 7) M.P. Mattis

and P.B. Arnold,

in Phys.

Los Alamos

Preprint

Rev.

Preprint

UPR-Oll’I-MT.

D.

LA-UR-90-115.