The color dielectric model of QCD

Prog. Part. Nucl. Phys., Vol. 29, pp. 33-85, 1992.

0146-6410/92 $15.00

Printed in Great Britain. All rights reserved.

@ 1992 Pergamon Press Ltd

The Color Dielectric Model of QCD* HANS-JURGEN PIRNERt Centerfor Theoretical Physics, Laboratoryfor Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

ABSTRACT

This paper demonstrates the emergence of valence gluons and their bound states, the glueballs from perturbative QCD. We discuss the phenomenological constraints and the theoretical method to generate effective glueball actions. We show how color dielectric confinement works naively and in the lattice model of color dielectrics. This lattice model is derived for SU(2)color by a blockspinning Monte Carlo renormalisation group procedure. We interpret the resulting long-distance action as a strongly interacting lattice string theory where the valence link gluon fields randomize in the color dielectric background which mimics the integrated out high-frequency gluon modes in the vacuum. The fluctuations of the color dielectric fields are related to color neutral glueball modes. We give the extension of this color dielectric SU(2) theory for general SU(N) with quarks and address the problems associated with combining confinement and chiral symmetry breaking. Finally we prove the efficiency of the effective theory in applications to the heavy quark system, to the baryon, to the nucleon-nucleon interaction, to baryon models and the gluon plasma transition. In all those cases the behavior of the higher energy gluons can be monitored via the color dielectric fields. An increase in the energy density from "deconfining" the higher frequency modes inside the flux tube or in thermally excited matter shows up as an increase in the value of the color dielectric field and its associated energy density. KEYWORDS QCD; color dielectric QCD; effective QCD lattice theory; glueball dynamics; soliton models; Monte Carlo renormalization group; confinement and chiral symmetry breaking; quark gluon plasma transition. I.

INTRODUCTION

1.1. Generalities

This is a review about the color dielectric-model. It consists of two parts, the first part deals with the fundamental question, "What is the color dielectric-model and how can it be related to the underlying gluondynamics * This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under contract ~DE-AC02-76ER03069, and the Bundesministerium fiir Forschung und Technologic under contract #06 HD 756. t Permanent address: Institut fiir Theoretische Physik, Heidelberg, GERMANY.

33

34

H.-J. Pirner

of Q C D " ? This has been my interest during the last four years. I hope to cover the results of this period and put them into the larger context of non-perturbative calculations of QCD in the framework of lattice theory. The second part deals with the phenomenological aspects of this model. Originally conceived as a model for quark confinement in the nucleon, the model has been applied widely over the whole field of low-energy hadron physics extending into a t t e m p t s to treat the many-soliton problem. The aim of the fundamental work was to derive the field equations of the phenomenological models. This has not been achieved. Instead a lattice string model has been derived with its own phenomenology. There are various new viewpoints, which do not find their place in the well-known "color dielectric" model. These concern the role of the color dielectric field in strong coupling lattice theory and its use as an effective dilaton field in Lagrangians with purely hadronic fields. I will try to present these more modern points of view, since I believe t h a t the conventional phenomenological models are strongly limited in their applicability. In this introduction I would like to communicate why the subject has fascinated me. There are two major aspects in this field: "How does the complexity of bound hadronic systems, e.g. nuclei, evolve out of the microscopically elementary interactions of quarks and gluons? . . . . Is it necessary to treat hadrons as composites and not as elementary fields to understand their low-energy behavior?" Let me describe these two questions in more detail. Hadron physics has three layers of complexity. The first is a genuine weakly coupled theory of quarks and gluons. It is non-linear due to the non-Abelian gauge fields and the fermion constituents. High energy short-distance reactions, like jet physics or deep inelastic scattering represent this first layer of strong interactions. The next stage of complexity arises when quarks and gluons hadronize. One can give an off-shell mass squared Q2 to the produced quarks and gluons. During the decay this virtuality Q2 decreases due to gluon radiation. This process can be reliably calculated down to Q2 > (1-2 GeV) ~. Below this scale the strong interaction confines the colored constituents into colorless hadrons. This transition and the spectroscopy of mesons and baryons represents the second layer. In bulk material one even speculates t h a t there may be a phase transition between deconfined quarks and gluons and the hadronic world as a function of temperature. For systems with baryonic charge, new structures exist. Nuclei are examples of the third form of strong interactions. They are rather weakly bound composites, where the hadronic subconstituents preserve to a large extend their structure. It has been speculated t h a t also mesons like the f0 (975 MeV) are weakly bound states of hadrons (e.g. K, ff). The length scales associated with these different layers are roughly: Asymptotic free quarks and gluons Hadrons Nuclei

< 0.1 fm ~ 1 fm ~ several fm

It seems natural to relate the structure of each higher layer to the dynamics of the layer below. Thus hadron physics becomes the physics of bound quarks and gluons. Nuclear physics is the physics of bound nucleons interacting via mesons. Our conventional thinking and research went along these lines and uncovered amazing regularities. It is only with the advent of high energy nuclear physics t h a t probes with high resolution like deep inelastic scattering pose the questions: "Is it possible to relate the third layer to the first directly?" "Can QCD achieve for nuclear physics what q u a n t u m mechanics did with molecular physics? . . . . W h a t is the most efficient hadronic theory, which lets us deal with hadronic interactions?" "Are gluons the agents of the microscopic strong interactions, involved in the interaction between color neutral hadrons whose long-range properties in the vacuum are due to gluons?" These questions are a common theme for the chapters, which will follow. The engineering of a calculational method to attack the questions may be a dry subject. May the reader keep these underlying problems in mind, if he is losing his patience. There are several reviews related to the subject. The classical reference is the book o f T . D. Lee 1 who proposed this model more t h a n ten years ago. Somewhat different aspects of color dielectric QCD are covered in the review by S. Adler. 2 Two more recent references are the work by L. Wilets s on non-topological solitons; and the review article by M. C. Birse 4 on soliton models in nuclear physics. Both concentrate on the soliton aspect of the chromo-/color-dielectric model. This review will not try to repeat their work but present complementary aspects. Please also note the article by M. K. Banerj ee in the same volume of Progress in Particle and Nuclear PA~sics. 5 Besides, there are many books on the subject of QCD, among which I can recommend as good background reading especially books by E. V. Shuryak, °, M. Creutz r and K. Bhaduri. s

35

Color Dielectric Model o f Q C D 1.2 Gluon Dynamics and Low-Energy Hadron Phenomena

Quantum chromodynamics (QCD) is the theory of colored gluons. These are eight Yang-Mills vector fields in the adjoint representation of SU(3) mediating the interactions between the three different quarks in the fundamental representation. If we define the generator of SU(3) with the 3 x 3 Gell-Mann matrices A a, then they obey the commutation relations

The gluon fields A~ = Al(z)A---~form a field tensor GAp ----Gin ~ and are elements of the Lngrangian density £ (double indices are summed)

G~p= O~A~- OpA~"Fg.f.b~ A~Ap(z) b £(,)=_l

(1.2)

G~p(Z)G:~p..(z)=_4_~.~p~,~,..(z)

= - ~ g12 t r G ^~ a e^x P

(1.3)

since

^

In Eq. (1.3) we used the scaled fields G~p = gG~p. Local gauge invariance means that the Lagrangian £(z) is invariant under local gauge transformations U E SU(3), i.e. U = e ia'(®)~'/2

.

Under local gauge transformations the transformed field tensor has the form

(1.5)

G L = U(z)C~pUt(z) , therefore the Lagrangian of Eq. (1.3) has local gauge invariance.

Let us start with this Lngrangian of Eq. (1.3) where g2 = g2(A), and A serves as n high momentum cut-off. Perturbation theory allows one to calculate the coupling constant renormalization when gauge field fluctuations with momenta A < [3[ ~ A are integrated out. There are two contributions to the integral over the gluon fluctuations which can be best visualized by imagining the gluon fields with momenta [31 < /~ as external constant magnetic field in which the fluctuations with momenta [31 > / ~ move. Partly, the fluctuations behave like spinless-chargcd excitations and give a diamagnetic response, i.e. their induced orbital moment is opposite to the external field. This response increases the energy in the Euclidean action. On the other hand, the spin of the gluon fluctuations aligns along the external field and leads to a paramagnetic response which lowers the energy. With Arc as the number of colors one gets: °, lO

Af-dia -AEpar"-

4g2(A) ~zc 12

1

[-1

4g2(A) [

4~r~ 4

~2J - - , v - -

~

~

""

,

""

aavaav "

(1.6)

(1.7)

The sum of the two gives

£ + A £ = -4g2(A----~

1-

1-

4x"

ln~

G~,vG~,~ ^6

(1.8)

Absorbing the coupling constant renormaiization into a new coupling called 1/492(.A.) one obtains the behavior of the QCD coupling constant under scale transformations 1 1 llNc A2 g'(~,-----)= g'(A-----)- -48~ - - - ~ In £-~ ;

(1.9)

36

H.-J. Pirner

For A > / k it shows that g=(h) is smaller than g=(/~). This is called asymptotic freedom. The renormali=ation group behavior of g2 is obtained, when one iterates the same procedure. From the above formula (1.8) one sees that one can do a perturbative calculation as long as g2 In ~ << 1. To understand the running coupling constant one defines the/3-function which characterizes the change of the coupling under dilatations. Set .~ = AA; g(A) ----go ~(g(A)) = d g ( g o , A ) dlnA x=l

(1.10)

llNcga

48~r2

'

One of the early speculations of Q C D was that non-linear coupled gluons form color neutral bound states. The latest latticesimulations for pure Q C D with three colors give the following bound state masses TM 12

Table 1.1 m[ll]

0++ 1.55 =t=0.12

2++ 2.18 -t- 0.16

m[12]

1.7 :t= 0.2

2.3 -4- 0.2

0-+ 2.50 q- 0.24

GeV GeV

where the scale has been set by the string tension v ~ : 420 M e V of heavy quark systems. The calculation of Ref. [11] is a calculation in SU(2)-color, at a lattice size a ~ 0.05 fm on a 32 * *4 lattice without quarks. The calculation of Ref. [12]is a SU(3)-color calculation with two degenerate light dynamical quarks of masses m ,,~ 20 M e V at a lattice size a ~ 0.1 fm on a 12 * *4 lattice. In the particle data tablezs there are many states with angular m o m e n t u m J -- 0 and positive parity. Our understanding of these states is incomplete. A partial listof these states is shown in Table 1.2. Table 1.2

fo (975 MeV) Decay Mode

Ir~r 78% Ki~ 22%

Width

33.6 + 5.6 MeV

fo (1440 MeV) a'¢c

KR

93%

7%

150 to 400 MeV

/o (1590 MeV) 7/y' dominant T/r; large 41r° large 175 ± 19 MeV

The I = 0 f0 (975 MeV) with its 2" = 1 partner ao (980 MeV) can be interpreted as hadronic bound states. The next states with large (~rTr) widths at 1440 MeV are regarded to be members of the SPo qq-nonet. There remains the /o (1590 MeV) as a gluebail candidate. One expects that a correction A M comparable to the experimental width arises if one couples dynamical fermions to the pure gluon states calculated in Table I. One should also not forget that there is a strongly attractive nonresonant ~rlr ( / = 0) amplitude which peaks around 600 MeV and has been made responsible for the attractive NN-force ("~r-particle"). In the seventies, before the discovery of QCD, the "~-resonanee" has triggered an interest in dilatation symmetry. 14 It was speculated that the low-energy meson spectrum contains two broken symmetries. One of them is chiral symmetry. It arises because quarks are massless. It manifests itself via massless pions as Goldstone particles. In addition, there is dilatation symmetry which exists if there is no mass scale. The breaking of this symmetry was related to the vacuum expectation value of the ~-field. Nowadays, we no longer believe this simple realization of dilatation symmetry breaking, since the vacuum expectation value of the ~field in the linear ~-model is related to the quark condensate. Quarks, interact via gluonic forces and become massive. Due to their nonlinear interactions gluons form glueballs. If these two phenomena are related, then the breaking of scale symmetry due to the massive glueballs may be also the intrinsic origin of chiral symmetry breaking.

37

Color Dielectric Model of QCD 1.3.

Breaking of Dilatation Symmetry and Effective Glueball Action

Global dilatations are scale transformations which transform the coordinates

' ~'v ""-4 2iv

=A-:zu=e-rzv

,

(1">0, A > I )

(1.11)

and the fields become

9(z) ~

9 ' ( z ' ) = e Z " 9 (e*z')

(1.12)

For the choice (~- > 0) the coordinate zv is dilatated with respect to z~ by the factor e *. The dimension l~ describes the transformation of the field. The canonical dimension of a massle~ scalar field is l~, = 1, since the action has no scale dependence under dilatations

S = f d ' z £(z, ~(z)) = f d'lz ' f:(z', cp'(z'))

(1.13)

Namely, setting £(~(z)) = }C0.v) ' one gets t¢ = 1. If the Lagtangian contains interaction terms g ( ~ ) -- }mayo= + g¢4 then the action changes under scale transformations as

f

d'= =

A d'=

(1.14)

The variation of the integrand A £ can be represented as a divergence of a dilatation current

8£

[

0£ + = a. La--~-~G~

8£

z~,T~ ]

(1.16)

The conserved energy momentum tensor T~ is improved in order to cancel the piece t~,{ } (82~ 2) } from the firstpart by defining (1.17) O~=T~ - -~I ( 8 . 0 " - g~,8 =) ~ ' Then the trace of O~ equals the divergence of the dilatation current on A £ A~: =

o.#~.=

Since

o~, = T,o...

(1.18)

1 2 2 = - 4 ~ + (a.~,) ~ + ~a'~"

8V

=4V-

~ =

(1.1o)

m~3

This introduction is necessary to understand the notations which are used in the discussion of the breaking of dilatational symmetry in QCD. The pure gluon theory is a theory of massless gluons, pure QCD is scale symmetric as a classical theory. The quantum fluctuations of the gluon fields, however, generate a scale, the QCD-A parameter, as can be seen from the perturbative renormalization of the classical Lagra~tge functions of Eq. (1.8). The change of S under dilations can be calculated by setting A = AA = erA and using the running coupling constant

e

1

f d~.0.V,.d,.l .=0

(1.20)

One obtains for the trace of the improved energy momentum tensor,

llN. -

^ v ~,

8~ = - 9--~2 G..G"

' + ....

~(g) (~:vG "v'4 • 2g s

(1.21)

38

H.-J. Pirner

In the above expression the leading log corrections have been summed by using the/9-function of Eq. (i.I0). A similar expression comes out when one calculates the triangle diagram with the stress tensor as one vertex and two gluons at the two other corners.15 In the treatment of Q C D a major step towards a nonperturbative phenomenology le was made by postulating a nonvanishing vacuum expectation value (1.22) This phenomenon has been called gluon condensation. The Q C D sum rules look at the correlator of two heavy quark currents, the interpolating fields of the J/~ meson. In order to fit the mass of the Y/~p such a term was necessary. In nonrelativistic language it does not correspond to a potentialIz but it plays a similar role as the linear potential in nonrelativistic quark models. In one-loop approximation an integration of the high m o m e n t u m fluctuations gives an effective Lagrangian ~

r

1 [ ii /:one-loop : - ~g2 I - ~

N

2

eg In 1 ^

A4

^

G~ ^a vG~ ^a

(1.23)

.

L Here the "external" field for G ~ v G ~ serves as a lower m o m e n t u m cut-off, which was played by A before

(cf. Eq. (1.8)). After one converts the Minkowski integral to Euclidean space one can see a hint that the Euclidean action has a minimum at a non-vanishing value of G~vG~,:

e'f ~dte" = e - f ~'"°d~"

(1.24)

with t : -i~0 and using Eq. (2.3) the Euclidean action is line FE = 2 ~ tr d/~vG~v + ~ I n It has a minimum at tr

A A

G~G~,~ =

( trd,vG~v~ 1 A4 / ~ tzG,vd~v

[

A 4 exp . - 1

96 2 1

ll-~g2 j

•

(1.2s)

(1.26)

This minimum was interpreted by Saviddi 9 as an indication for gluon condensation. There are some caveats: First, this derivation is perturbative, and second as has been pointed out by many authors (e.g., Ref. [10]), the ground state of a constant field is unstable. For an external constant magnetic field IgBzl one sees t h a t the lowest b o u n d gluon mode for Sx = +1, n = 0, has an energy •

e-:~k2z+2gBI~z+~)-2gBSz

(1.27)

which becomes imaginary, i.e. the system is unstable in time (oc e-~(s'=l, ,~=0)t). Nevertheless this one-loop formula has inspired many models. Also the color dielectric model has been influenced by the idea of a gluon condensed state. As we will see, in the lattice dielectric model gluon fields of different m o m e n t a are considered separately. The idea is t h a t the color-dielectric background formed by high m o m e n t u m gluons with momenta k > A leads to a condensate of the lattice gluons with k < A. Gluon-condensation can be related to the dilaton phenomenology. We postulate a dilaton or scalar glueball field ~ o (index G indicates it has glue in it) in such a way t h a t it has the following three properties: Its scale dimension t¢ -- 1; it obeys an effective Lagrangian, which has the same scale anomaly as QCD, i.e. with Eqs. (1.19) and (1.21)

(1.28)

Color Dielectric Model of Q C D

39

The expectation value of the (ra potential terms have to be taken at the vacuum value (~ra) = ~o of the ~a-field where the potential V(~,) has its minimum, i.e. OV/Oo'[ . . . . = 0. Further, the mass of the giueball field is m e :

( O=V('T°) / so =~

.

C1.29)

A residue of the scale invariance at the classical level is a series of low-energy theorems zs for the improved energy momentum tensor i"

f,z=~.., d=, (o IT [0;(=~)... O~,(=,)0;',(0)] [ 0) ...... t,d = (8;>,~,¢ (-4)"

(1.30)

This implies that the improved energy momentum tensor is an operator of canonical dimension le = 4. We can therefore express the trace of the improved energy momentum tensor of dimension L0 : 4 in terms of the dilaton field era(z) with canonical dimension l~ = 1. We have the proportionality of the operators 0~ = -Ber~

.

(1.31)

Integrating the linear first order differential equation resulting from (1.31) and (1.28) one gets V(era) = B~,~ (ln c,a + const.). Implementing the other constraints of Eqs. (1.28), (1.29) the model Lagrangian for the dilaton field has the form £~ : - T - V £:.---- ~ (Op.o'a) :l - B

(1.32)

q- (r~ln

with B =

m~

(1.33)

The determination of the parameters in this dilaton action at low energies is not so easy. The giueball mass m a may be deduced from the O++ glueball candidate with 1590 MeV. But one is not sure whether qq and gluon states mix. The gluon condensate (G~,~G,~) itself is phenomenologically determined from QCD sum ~

g

g

]

rules for charmonium is, 10

(0 5 - 1)GeV' This quantity is a renormalization group invariant. Assume that we sum the giuons with momenta ]k[ < A. Under a change to a higher cut-off A' > A, more g h o n modes are summed, i.e. G~,,,G~,, will increase. At the same time the coupling constant g(A') will decrease compared to g(A). So in total


For large cut-off we know the beta function ~(g) 2gs -

11 32~' + ° ( g ~ )

'

(No = 3 )

.

A low-energy action necessitates, however, an extrapolation to low cut-off momenta of ~ 500 MeV, where we have no perturbative control over the beta-function. The one-loop running coupling (/2 is obtained by integrating (1.10). Call hA = Q >> A, then

g2(Q2)__

g~

=llN~ Q= l+g04--~-aln ~-

,,~

48~ 2 Q2 l l N c l n A--~

(1.35)

40

H.-J. Pirner

since for Q large enough the second term in the denominator dominates. For A q c v ~ 100 MeV one gets for Q2 = 0.25 GeV=, g2 ~ 4.5, s value where the perturbstive ~-function is not longer trustworthy. One can try the impossible and estimate B and ~0 from the measured giuon-condensate. Then one gets with m G = 1.5 GeV and

I

~(g) G 0 ~

(-321-~12 • (0.5 - 1)GeV 4) ,~

-

(0.0174 -- 0.0348) GeV 4

the following values for B and O'o 11.48)

B ~ (22.95-

(1.36)

O'o2 = ( 0 . 0 2 8 - 0 . 0 5 5 ) G e V

~

.

(1.37)

The giuon condensed vacuum with (~I = ~o has a lower vacuum energy than the perturbative giuon vacuum. The energy difference is also sometimes called the bag-constant• We will discuss this point later, in the context of the color dielectric model. This connection may be misleading since it is not necessary is that inside hadrons we have (cry) -- 0. We call the vacuum energy £,.c (use Eq.(1.32)) e,.o = v ( ~ o = ~o) - v ( # a = o) -

£~¢

~ - (0.26 GeV

4

(1.38)

- 0.30 GeV) 4

• • 1/4 This value is larger than the original ba 8 parameter Bbsg = 145 MeV. 2° We will see what the lattice calculations give for this values in Chapter II.

1.4.

Color Dielectric Confinement

In the previous two sections the boundaries of our knowledge at high and low momenta have been outlined. QCD is a microscopic theory of gluons at high momenta and becomes an effective theory of giueballs at low momenta. In this section, the intermediate scale of physics will be discussed. For the purist such physics may not be so well-defined, since physics at the intermediate length scale will have features of both worlds; quasi-gluons, quasi-quarks and nascent glueball and other meson degrees of freedom. These quasi-particles are also called constituent particles and are nothing strange for many-body physicists, which are used to effective interactions between quasi-particles. In fact, it is the nuclear physics community which has largely followed the adventure of exploring this world during the last 15 years. Originally a constituent quark model with nonrelativistic quarks and a confining potential has been highly successful ~1, 22 Relativistic versions like the bag or color-dielectric were model built afterwards. Let us discuss how the color dielectric model originates in perturbative theory from the antiscreening effect (cf. Eq. [1.9]). The perturbative one-loop effective action (Eq. (1.8)) can be written in the form of a dielectric action with a dielectric constant e as a function of length scales L = l r / A , a n d L -- Ir/A ~> L

~,~(L) -

1 . e ( L , L ) • G~,vG~ "a "a 4g~(L)

(1.39)

The dielectric function e(L, L) < 1 gives the change of the action when giuon fluctuations with wavelength L < < L are integrated out. These high frequency fluctuations can be absorbed into some medium effect e(L, L) which sntiscreens charges• Recall that in electrodynamics the polarization charges of the molecular dipoles are aligned in the same direction as the external electric field, therefore we have a behavior in electrodynamics opposite to QCD: Electrodynamies:

~ = ~

QCD:

b = ~

= ~ + 4,=~ ;

,

~< 1 ,

P II E

;

~> 1 ;

(1.40) (1•41)

41

Color Dielectric Model o f Q C D

_ql

Fig. 1: Confinement of a color charge q in a dielectric medium with fl # 0 bordering the vacuum with ~2 = O. In the Abelian approximation to the effective dielectric QCD action it is easy to see t h a t color charges are confined in the medium with E1 # 0 bordering at a medium with ~2 = 0. For simplicity consider a half plane with z > 0 where el # 0 bordering on the region with z _~ 0 where e2 = 0. The color c h a r g e q i s situated at ~ = (a, 0, 0). By using virtual charges - q ' at ~ = ( - a , 0, 0) and q" at ~ = (a, 0, 0) one can satisfy the boundary conditions of continuous displacement field along the normal of the boundary and continuous tangential electric field. One obtains the scalar potential q

---- - -

•Irl

(~1-~)

÷

~I(~i+~2)

q

--

r2

2q

for z _> 0

(1.42)

for z < 0

(1.43)

and the force on the charge el(el + e3) 4a ~e® '

(1.44)

It is pointing in the positive z direction, i.e. away from the medium with e = 0 which represents the vacuum. Confinement looks like a natural consequence of the different dielectric constants inside and outside ofhadrons. This example of a single charge may be complicated because of infinite self-force. Let us therefore consider two charges connected by a color dielectric flux tube. 2s At the center of the flux tube the dielectric field e assumes

1-2

a value which minimizes the sum of the energy density of the color electric field ~ D /e and of the excitation energy of the vacuum W(e). W h e n W is a monotonically increasing function of ~, then this m i n i m u m will be assumed at a value e # 0 in the center of the flux tube. As one goes to the edge of the tube there is a gradual transition to • -- 0 since the constraint ~ •/} -- 41rp can be satisfied with D vanishing towards the outside of the fiux tube. The dipole moments of the high m o m e n t u m gluons compensates the electric field due to the charged sources. A n important ingredient of a serious calculation is to determine the function W . Another example is the behavior of a point charge as an isolated quark in the Abelian color dielectric approximation. From Gauss' law V]9 = 47r6(~) one finds D oc 1/r 2 ~ . In order to have an infinite energy of a free quark the energy £ = ~ f 1)2/e d3r must be diverging. This always occurs as long as in the expression for the energy £ oc ½ f R ( ~ 1 ) dr the dielectric function falls off faster than ~ for r - , o~. Then £ diverges when R --, oo and the possibility to have a free qu ark cannot occur. If the dielectric field • is produced via a scalar field ~b

42

H.-J. Pirner

with the Hamiltonian

.++= f

(,+++)"+,;,,e) d",.

then the total energy H = H~b + H+ with ~2

IIq = ~

dSr

leads to a behavior of £ cx Rs/z, of. Ref. [24]. For non-Abelian gauge theories the concept of a dielectric field presents some difficulties. Inherent in the electrodynamics approach are the linear Maxwell equations, which allow to add the effects of the applied external field and the polarized dipoles. In continuum QCD it may be argued that the nonlinear field components are forming the medium, i.e•

;)

(

i

fe~bct"~=b('~=c •

Y

•

linear part "medium" But I do not know of any calculational scheme, which constructs a theory on this suggestion. The MIT-Bag model was thought to be such a theory, since it treats non-Abelian ginons with linearised field equations. The idea behind it would be that the dielectric constant e inside the bag makes the relevant coupling g2/e < 1, such that the medium effects inside the bag can be neglected• They only play a role outside the bag, where they make up the boundary. The philosophy we will try to show in the second chapter is based on lattice theory and the renormalization group which has been highly successful for systems with zero mass particles. Instead of differentiating between linear and nonlinear terms of F~v, the method looks at gluons in different momentum groups• The nonlinear terms give Fourier components to F ~ with smaller and larger momenta than the linear terms. (Think of two gluons merging into a third gluon.) The rapidly oscillating nonlinear terms may be averaged over, when we are only interested in the long-range components of the field tensor. They form the medium. The slowly oscillating nonlinear terms remain in the effective action, which stays non-Abelian. It has been shown that Abelian dielectric or bag-like models have problems when it comes to the potential between charges in higher representations, than the fundamental representation. 2° The Abelian theory gives for a cylindrical flux tube with area F D . . . . . ] = QF or for the energy =

--

+

Bba s

V

.

/)2 contains the expectation value of the Casimir operator < (A/2)21 in the respective representation. After minimization of the energy with respect to F, one gets 2s that the energy/length or the string tension is proportional to the root of the Casimir operator

A/2

)'>

; lattice results point to a direct proportionality. 2s

Another problem of Abelian dielectric models is the possible existence of van der Waals forces. In perturbation theory we would argue that for the Abellan version of dielectric theory a color neutral system St influences a color neutral system $2 by its electric field. Take two meson flux tubes as shown in Fig. 2. Then the D fields are concentrated inside the mesons and the respective energies are £t,2 = L

Dtl2 + B

F

.

The interaction energy, however, is attractive ~2

'

f

all space

'(++,+

dr-

• .~21 dV

+f

D~ d v - ~

~2

f

D~ dV

Color Dielectric Model of QCD

43

Zr Fig. ~: Interaction of two color neutral s~stems S1 and $3 in an A belian color dielectric model. where E1,2 = D1,2/~1,2. The electric field ~;1 of the meson $1 is a dipole field which does not vanish at system S~ and produces an attractive interaction energy which goes as O(1/RS). In the above equations we have approximated e by el and e2 when we subtract the self-energies and when we calculate the E fields associated with the D field which solve the exact two meson problem. Another way s to look at the problem is to keep the full solution e for the calculation of the E fields associated with the D fields. Then there will be no E l field extending into the second meson and vice versa. A problem may arise when one subtracts the self-energies. In non-Abel/an QCD the nonlinear coupling of gauge fields exponentially suppresses the interaction at large distances compared to the range of the gluebal] field. The model we are going to show will be a lattice model with a variable coupling constant influenced by the medium. In the vacuum the remaining link fields are strongly oscillating and do not correspond to continuum gluon fields. At best, they can be considered to form a condensate. On top of this background of the gluon condensate live color neutral gluebaHs. If quark charges are present the link fields transmit the color information. The color dielectric medium leads to a random behavior of link fields and gives confinement. We will show this in more detail in Chapter II. Before we will look at the dual picture of superconductivity, namely magnetic monopole confinement; this picture gives another reason for a vsnishin 8 dielectric constant in vacuum.

44 1.5.

H.-J. Pirner Color Confinement Because of Monopole Condensation: The Dual Picture to the Meiflner Effect

For the moment let us forget dielectricsand recallthe situationin type-II superconductors. Superconductivity arises from the condensation of Cooper pairs in the superconducting material. Let us consider two magnetic monopoles in such a medium, where charge condensation has occurred. A magnetic field will arise in a flux tube connecting the two monopoles. The magnetic flux is quantized n. 27r/e where e is the electriccharge or 21r/e the magnetic charge. Inside the flux tubes the pairing of the electrons is broken by the magnetic field. The transition from the inside to the outside of the flux tube is governed by a scale, the penetration depth. The charge condensed superconducting medium expells magnetic flux and keeps it contained in a thin tube. Outside the flux tube B = p H -- 0 or the magnetic permeability /~ - 0. This phenomenon is modeled in a G i n z b u r g - L a n d a u (or Higgs) Lagrangian. The expectation value of the Higgs field I¢[ 2 plays the role of an order parameter measuring the density of Cooper pairs. 1¢12 is zero inside the flux tube and I¢12 ¢ 0 in the medium. The transition length is inversely proportional to the Higgs mass. The uniqueness of the cylindrically symmetric Higgs field ¢ -- ei'*Sf(p) gives the quantization of the magnetic flux. This picture can be translated into a model for QCD, if one replaces each field by its dual counterpart. In QCD we analyze the dual situation of two color electric charges in a medium with magnetic monopole condensation. If there is is an electric field connecting the two charges it will be confined in a flux tube if the medium has a dielectric constant e -- 0. Consequently, it should be also possible to model this phenomenon by the dual Lagrangian to the L a n d a u - G i n z b u r g theory. This has been done in fact by T. Suzuki and coUaborators. ~z The main problem is to make an Abelian projection of the non-Abelian SU(3) or SU(2) gauge theory, as proposed by 't Hooft. 2s Also measurements of the monopole density on the lattice are possible. 2°, s0 The role of the Higgs field is played by an equivalent scalar field, the glueball field in QCD which is the massive gauge composite in QCD. The monopoles themselves are also gluon field configurations, connected to the compactness of the gauge groups. II.

DERIVATION OF COLOR DIELECTRICS IN LATTICE QCD

II.1. Basics of Lattice QCD There are two major simplificationsto the problem of non-Abelian gauge theories in the lattice formulation. First the number of degrees of freedom becomes finiteby discretizingspace and time. The couplings remain local, when configuration space is discretized. In m o m e n t u m space the couplings would be nonlocai. Second, in the latticeformulation the path integral can be done, when the time coordinate is rotated to Euclidean time t : - i z o , z+ : ~ . The Euclidean Lagrangian £ E ---- (T + V) plays the same role as a Hamiltonian, but in four dimensions and by the integration over imaginary time the path integral resembles a partition function in statistical mechanics. Discretization of the continuum on a finite symmetric lattice introduces two length scales into the theory. The smallest length is the lattice constant a. The largest length is N • a, where N is the number of lattice sites in any direction. There are in general two questions associated with these two lengths. The first one is the transition to the continuum a ---* 0. In order to be meaningful as a theory of physical processes in continuous space time, the lattice theory must possess a continuum limit. Since QCD is an asymptotically free theory we know how the lattice size a or the cut-off Q = lr/a behaves as a function of the coupling constant in the limit g ---, 0. This functional dependence comes from inverting the formula of asymptotic freedom (Eq. (1.35)). Conventionally the lattice theory introduces the inverse of the square of the coupling constant as the i m p o r t a n t variable in (1.3), (1.8) ~(a) --

22v~

g2(a)

a 2 = tO2 e -~'(24~2/ltN~)

(2.1) (2.2)

One sees t h a t for g2 _.+ 0 or/3 --, oo the lattice size a goes to zero. We therefore expect to obtain continuum physics results, when/3 is large enough (/3 >/3low ). The numerical signal for being close to this region is scaling behavior, i.e. a physical mass measured in lattice units ao must vary with/3 as m2a 2 -- const, e -~24~/11N~

(2.3)

45

Color Dielectric Model o f Q C D

In reality the scaling region is limited also for large/3, since we have to pay attention to the total lattice size N • a. When/3 becomes too large (/3 >/3up) the lattice constant aup becomes too small, the total lattice size Naup can then be smaller than the physical hadron Naup ~
"

For the lattice derivation of an effectivehadronic theory the physics constraints are even tighter. The starting lattice must satisfy the constraint, i.e. /3 > ~lo~ and ~r/a > rrthad. In addition, the theory where the internal quark/gluon dynamics have been integrated out, must fit on the same lattice. Therefore, the total lattice size must be bigger than several hadron radii. This is the reason why so few attempts have been made in lattice physics to model hadronic theories, since it is a task at the edge of current abilitieseven for pure glue lattice theory. For simplicity we are going to discuss non-Abelian SU(2) chromodynamics of gluon fields (for an extensive review see Ref. [7]). That means the gluon fields have a 2 x 2 matrix representation by the Pauli matrices ~i (i = 1, 3) as

A.(z) = ~ A~ : .

(2.4)

Having discretized the three space and one time dimensions we introduce as relevant degrees of freedom the link variables fz+e,,

V. (z) = exp ig j~,

A~,d~,,= ao + i~2

(2.5)

which symbolize the integral of the gauge field between a site z of the lattice to the neighboring site z + e~, one lattice spacing away from z in the/~ = (I, 4) direction. This link variable has a 2 x 2 matrix representation Uv = a0 + i~2 given by four numbers (a0,2). It is unitary, which means that the product of a link in the positive p-direction with the same link traversed in the opposite/~-direction gives the unitary operator

V.(z)U~(z) = V.(z)V_.(z + e.) = I .

(2.6)

Unitarity restricts the four quantities (a0, 2) to the unit circle a02+22 = 1. The effect of a gauge transformation G(z) on U~(z) can be seen by expanding U~ for small gauge fields and using the standard continuum gauge transformation A . ( z ) ---* A l. : (1/i)O~,G(z)G-l(z) + G(z)A~,(z)G-l(z). We get

1 + iA~ dr. ---+ 1 + {[G(z + dr) - G ( z ) ] / d z . G - l ( z ) + iG(z)A.(z)G-l(z)} dz~ G(z + dr)(1 ÷ iA.dz.)G-l(z) .

(2.7)

Therefore a link U~,(z) transforms by applying the gauge transformations G and G -1 on the end points (z+e~,) and z of the link. Consequently, the natural choice for a gauge-invariant action is made from closed loops, the smallest of which are plaquettes with four links. Indeed this is the form of the Wegner-Wilson sl, s2 action S(U) (see Fig. 3)

S(U) :

~

1-

tr

]

(UilUl, UhjUjl) = ~ Sp(U) .

all plaquet tea

(2.8)

p

The partition function is given by the sum over all possible link configurations weighted with the "Boltsmau factor"

Z = f DUe -~s(u)

(2.9)

Here the integral can always be thought of as a multiple integration over the constrained vector variables no, at all links, i.e.

/

D U - - /line,ks

dao,zd2t6 (a~,l - 2 ~ - I )

(2.10)

46

H.-J. Pirner Time

/! I I I I I

,"

,st'e~ I |

I I I I I I

Lattice constant a

tank

space (~2)

/

/ 7*

Ix)

P

I~ space (~1} x.e 1

Fig. 3: Schematic picture of a lattice with link variablesand a basicplaquette. The QCD coupling constant g enters into the action as fl -- 4/g 2. In the strong coupling limit, i.e. for small fl, we see that the exponential allows values of S? --* 2 which are related to strong fluctuations of the links, e.g. (rl ~r u e ) =

(: 01)

For weak coupling or large fl the "Boltzmann" weight prefers identity, namely

Sp --* O, i.e.

(2.11)

trivial link variables near the

In the naive continuum limit, we expand the link matrices, the elements of the Lie group SU(2), around the identity element U~(z)= l + i A ~ ( z ) . a . (2.13) Then the action becomes the conventional Yang-Mills action. Take a plaquette with center at z~ and oriented in the (p, v) = (1, 2) plane:

flS?(U) : fl[1- ~ tr { expig A1 (z~, - ~ae,) "expig A, (z~ + lael) .exp_ig A1 (zi, + ~ae,)

(2.14)

"exp-igA, (z~- 2ael) } 3 • Expanding the gauge field in the exponential around z~ gives the field strength tensor (Eq. (1.2)).

flS?=flll-1

tr exp

Iiga=Gl=+

0(a4)] /

(2.15)

For small lattice constant a and weak coupling g the argument in the exponential is small and the exponential can be expanded. The term of order a 2 vanishes because the trace over h-matrices is zero, the leading term

Color Dielectric Model o f Q C D

47

is of order a 4. Converting the sum over vii plaquettes to a space-t/me integral, we arrive at the following expression

plsquette

The factor 1/2 comes from the symmetry under interchange of/~, v. So we can identify the lattice gauge theory with the continuum theory when/~ = 4/g 2 for SU(2). It should be noted that such a continuum limit does not exist when g becomes large, i.e. when the lattice constant a is sizeable. Strong coupling lattice theory is not a theory of strongly interacting giuons. We will call these giuoas "llnk gluons," which means that they only exist on the course-grained lattice, not as continuum fields.

II.2.

The Monte-Carlo Renormslization Group

We already encountered in Chapter I the renormalization group function, the beta-function o f Q C D (Eq. (1.10)) for pure gluonic QCD.

~(g(~))=

dg (go, A) x - - 1 dln~ -

11Ncg~ 4s~-~

(2.17)

Under a change in cut-off from A to ~A, the coupling constant increases for ~ < 1, i.e. when ln~ < 0. Starting at a scale where asymptotic freedom holds A ~ several GeV, one quickly reaches values for the coupling constant, such that a perturbative treatment of QCD is no longer justified. In principle, the theory may have new coupling between giuons. The evolution of QCD into this domain can only be calculated numerically. Even if the outset of the evolution of the theory is well-determlned perturbatively, one has very little knowledge about the non-perturbative theory. For pure glue QCD it will be a theory of 81ueballs in analogy to the discussion about the dilaton effective action in Chapter I. With the numerical Monte Carlo method we hope however not to be bound by the use of perturbative theory to handle the trace anomaly. The trace anomaly and the evolution of the energy density of the vacuum will be taken into account by the Monte Carlo Renormalization Group. The Monte Carlo Renormalization Group 88,z4,s6 is a theoretical approach which constructs an effective action directly from the QCD Lagrangian. Clearly for many experiments in nuclear physics at higher energies we need a theoretical description which at the same time contains the longrange features of confinement without giving up the elementary quark and giuon degrees of freedom. After all it is the search for these QCD degrees of freedom which motivates the experimental work. Studying the transition from asymptotically free QCD at short distances to confining hadron dynamics at long distances is by itself an exciting project. For this problem a transformation is ideal which maps a theory with a small lattice spacing a into one with a larger lattice spacing 2a. Since such a mapping or block-spin transformation can be carried out repeatedly we are led to an effective model Lagrangian for long-distance hadron physics. We define a chain of field theories at lattice sizes a0, 2a0, 4 o ¢ , . . . , 2ha0. The continuum limit a0 ---* 0 is theoretically achieved by letting n ---, oo such that lira 2ha0 ---, finite. Practically only n _~ 3 is feasible at the moment. Constructing the block variables for the effective action is a somewhat arbitrary process. Our first work 3e used a gauge fixing procedure, which does not easily vectorize. In our new work 3r we proceed in the following way. We average the variables U by taking s sum of paths from s point z on the new lattice to s point z + e~ on the new lattice which is separated by 2a0 from z on the old lattice. We block the original 164 lattice down to a 24 lattice in the following steps: 164 ---, 84 ---, 44 --* 24. The block variable depends on the physics situation. Our idea is to form a fat string of fine string bits of length a0. We want that the field variable for the fat string takes into account the energy density of the gluon fluctuations of the shorter distance scales. This way we hope to cope with the following problem. Different hsdronic objects contain a different mixture of giuon wavefunctions. If one keeps only unitary link giuons at larger distances, one has the well known strong coupling theory. One finds, however, that the ratios of various physical observables do not scale, because they are affected by the cut-off differently. We define no~-unitary block variables ~b~(z) by a superposition of old link fields ~]d(z), which are the original unitary/ink fields Us(z), when the procedure is started.

48

H.-J. Pirner

In more detail we define the block variable 6t=(z) as shown in Fig. 4:

~,(m)--N { p z

",--',

{U

t

+ v,(~)u.(, + ~,)u.(, + ~,)~v,(~,+ ~,) +

V~,(z)U~,(z+/~)Uv(z+

+ P2 ~

2/~)Uv(z + 2/~)t)

Uv(z)U.(z + v)U.(z + v + I~)e~(~ + 2/.*)t

(2.18)

v-Lt~

+ ps ~ (u~(=)u.(= + ,.)v.(= + ~,)t v.(= + ~,) v.L/~

%

+ Vt,(z)Uv(~: + tt)V,(z + tt + v)U,.(z + 2p.)t) }

o

This averaging process is iterated by replacing the fields Uu(z) by ~bu(z) in the next step. That is the reason why we added terms of the form utu = 1 in Eq. (2.18). The normalization factor is defined as

AT = Ip*l + ~ Ip=l + ],2 Ipsl

(2.19)

Pictorially the fat string contains the fluctuations of the string on the lower level.

•

(11

•

)}

Fig. J: Pictori¢l representation of the fa~ ,~ring or valence gluon opera,or *,(~).

The blocked action is defined by the integration over the original link variables U~ with the 5-function constraint (~.,. : ~.(~))

This blocked action Sblock will contain infinitelycomplicated terms in the new fields. Hopefully the series of terms in Sblo~k(~b)can be truncated to some finite set which we call S,ff(~b). In our case, truncation is done in the following way. W e drop higher powers than four in the variable ~b and neglect non-locai couplings. The reason for the first approximation is that finally we want to approximate a theory with q~'s having a small norm. Averagin 8 unit "vectors" U~(z)will always lead to a vector of norm less than unity. The locality assumption is more difficultto justify. Currently we are investigating this point in 2(2) theory, by performing tmaiyticai block spin transformations.

Color Dielectric Model of QCD

49

The effectivetheory we consider contains four terms O(1),..., 0(4) with four new coupling constants/3',/~2, A and A'

and the operators (cf. Fig. 6)

o(1) = trep 0(2) = X~,~

0(3) = x~_,~ 0(4)

=

2

(2.22)

1

,

v .l./~

where the subscript/~ varies over the positive directions, and the subscript v varies over positive and negative connections. The term ep means the product of link operators ¢ around a plaquette. In the case of unitary link fields Ikbll = IIUll = 1 the above action would reduce up to a constant to the usual Wilson plaquette action S(U), with only one term - ~18 ~ tr Up. We have introduced as a new degree of freedom the length X~(z) of P

the ¢~(z)-fields eu(z) = X#(z)U~(z) ,

(2.23)

which we call the dielectric field. In a naive continuum limit, the plaquette term of the effective action above becomes if92

trOp

f

,

(224,

P

where we have introduced a geometrically average X-field over the four link variables taken at the center of the plaquette. The X-field is a positive parity field since under reflections the ¢-fidd is odd, and the U-field is odd P [Xu(z)e i'a"(®)a] : X , ( Z ) e -i'a"(®)a (2.25) The X-field contains continuum 0 +, 2 + , . . . fields in it. From the limit a ---* 0, one sees that X~,(z) plays the role of a dielectric field, which modifies the gluon energy density to the same form as in the electrodynamics of macroscopic media. The square of the X-field is proportional to a sum of plaquettes oriented along the p-direction (see Fig. 5). We would therefore identify the glueball field with an angle average of the X2-field erG(z) = ~a Z X ~ (z) "

(2.26)

It seems easier to keep the link variable X~(z) in the effective action instead of ~rQ(z) as a site variable. The latter approach was proposed by Nielsen and Patkos ss whereas our method follows the work of Mack and collaborators, s9 There is a chance that n link variable of the form of X~(z) can make the transition to a stringlike effective action more apparent than a scalar field. The resulting string equation would again contain all the dynamics of the ground state 0 +, glueball and the higher 2 +, 4 + glueball states as well. We did, however, not yet find a method to establish such a connection. It may be necessary to block only in transverse directions to establish such a string model.

Xp:

tr

, +

'j'

:} 'k +

Fig. 5: Schematic picture of 1 tret(z)¢~,(x ) _,_ X~(X ) as a 8urn of plaquette operators.

50

H.-J. Hrner

We can study the effects of including only a finite number of terms in our effective action by simulating S,~r on a lattice of spacing 2a. For this simulation we calculate various operators, and check how well they match between the simulated color dielectric theory and the theory obtained by blocking from SU(2). By a prudent choice of the block spin parameters {Pt} (cf. Eq. (2.18)) we hope to be able to obtain matching for this lattice size, thus reducing the effects of irrelevant operators in the effective action. Once we obtain satisfactory matching on the lattice with spacing 2a, we will then be able to make further block spin transformations to lattices with spacings 4a, 8a, etc. and compare matching between the blocked theory (which originated with pure gauge SU(2)) and the simulated theory. If the matching is satisfactory then we can study further the properties of the color dielectric theory defined by our effective action.

II.3 The Use of Schwinger-Dyson Equations to Determine the Couplings of the Effective Theory There is a general understanding of how to fit a function of one variable f(z) when it is known at a finite number of points (z~). There are well-developed methods to expand the function in a set of basic functions or to make a polynomial fit etc. The problem in field theory is more complicated. After discretisation, the action is a sum over all lattice sites and/or links of products of functions at these sites and/or links, if one restricts oneself to polynomial interactions. In general, these polynomial terms can be non-local, in fact derivatives give couplings of nearest neighbors. In Eq. (2.22 / the operator 0(4) includes a coupling of the I n k field X~(z) to its perpendicular neighbor X~(z) since the u-direction is perpendicular to the/z-direction. Graphically the

different operators In Eq. (2.22) can be represented as (see Fig. 6) 0(1) = tr~bp 1

0(2) = ~ tr ~bt,,,~b.,. 1

2

(2.27)

12 ..L.

0(I)

= tr p

0(2)

:

0(3)

oc4 -

I

[/

,

,)

:

tr

2

Z/.L~

Fig. 6: Graphical ~eprese~tation o/the four operators O ( 1 ) , . . . O ( 4 ) i~ the action.

Color Dielectric Model of QCD

51

Schwinger-Dyson equations are relations between moments of field operators, which are obtained from the transformation properties of the fields. There are two relevant transformation properties of the ~,(=) fields, which lead to two types of Schwinger-Dyson equations. The first is a rescaling of the length X,(z). The second type of transformations are local gauge transformations on the ~-fields, which lead to the so-called string equations of Migdal and Makeenko. 4° As described in Ref.[36], the gauge invariance of the action and the measure leads to the equation ~1 , < ~

t t t t tr ¢1~7~e~5~4~3~2~1

--

tr ~5¢g¢7~1¢4¢3~2¢1 >

3 ( tr ~b4~bs¢2¢x)

~---

(2.28)

where the summation is over all plaquettes [] containing the links ¢i, ¢5, ~be and ~bz. The first combination of eight links on the left if arranged as a straight window, while the second combination corresponds to a crossed window, see Fig. 7.

3 4 |

•

r~

3

' g.

Tr

= 3Tr(

- Tr 1

4-

1

7

7+

5"

4

/

1

6÷

Fig. 7: Migdal-Makeenko equation in graphical form. To derive equations involving the other three couplings, we make use of the transformation properties of the integral defining the action. If we consider the expectation value obtained after blocking of some arbitrary operator, O(X, U), which is differentiable in X, we have

f [vu] f [Vx]e-".....

(

,

(2.~9)

with the measure for the variables X,

f

[vx] =

!fo

axi e(x~)

(2.30)

,

2

where the phase space function for the X'S is P(X) = Xs. The integral over Xi on the left of Eq. (2.29) can be done by parts:

< a-~ o(x,, tr)> = f[~u] j~ f01 axi P(x;) × X;=I

× P(x,)e -s ..... C~'u)O(x,,U)~,=0 - f ax,

[P(x,)e-s .....

Ix,u)]}.

(2.31)

52

H.-J. Pirner

The first term on the right-hand side of Eq. (2.31) will vanish provided t h a t the product P(x)O(x) also vanishes at X = 0 and X = 1, thus we will choose the operators O(X) to vanish at X = 1 and to be less divergent t h a n X - s at X = 0. The seven operators we started with in Ref. [37] were (Xi = X~,~):

(1-X=,.)X~,. (l-x.,.)

(1

,

1...4 ,

n----

tr¢[3 ,

¢®,. E¢[~ ,

1

-

X.,z)X.,z-~ E (X~,v+ X2-+z,v) ,

(2.32)

v J..v. "~

E

2

2

";'

+ X~,t*,Xz+#,v)

v±#

Here ¢[3 has the same meaning as ¢p before. After partial integration we obtain a Schwinger-Dyson equation for the operator O(X, U), which is an equation in the couplings of the blocked theory:

<{ b--~.,..,S~,oo.(x,v)-x-~,~},~O(x.,.,v)) =
(2.33)

Because we cannot calculate the infinity of couplings of Sbio¢k(X, U), we have to use the effective action Serf(X, V), in the Schwinger-Dyson equations. Using this procedure amounts to minimizing the "distance," in the sense of Ref. [41], between our effective action, Serf, and the optimal (but uncalculable) Sblodt. However, this does not exactly mean to minimize the differences between operator expectation values calculated for the blocked and effective theories. The effects of this procedure on the operator expectation values will be considered in the next section. From Eqs. (2.21) and (2.22), we have a serf(x,v)= cox.,-----~ -{#1 , E

trCrT/X',t~

¢®'" e~'FI

+

(2.34)

+ 4A ;. + v,'x.,

E (x.L + v±t*

The factor of four multiplying the A~ term arises because we must consider b o t h the p and the v directions when we take the derivative. Putting this expression into the Sehwinger-Dyson equation, Eq. (2.33), we obtain a system of linear equations for the four couplings of Serf. Our set of eight equations (including Eq. (2.28)) in four unknowns is overdetermined, so we use a least squares error routine to determine the best solution set for the couplings {/Y, t, 2, A, M}. We have also checked t h a t omitting one of the equations does not change our solutions to any extent. The solutions obtained in this way for/3, tt 2, A, At were not very satisfactory. 37 In the meantime B. Grossmann 42 has increased the number of operators. Using the same weight factor (1 - X®,z) he added new operators (1 - X®,~)Oi,i = 5 , 6 , 7 , 8 , 9 . o, = x~(~)

,

1

o~ = ~ (tr¢[]) Or:

=

2 1 X.(z)~ EX~

,

¢[]

¢,,,, ~

(= + v )

,

(2.35)

vJ_tt

1

2

O e = ~Z G ,

X=,~EXc]

•

He also chose a somewhat different optimization procedure. Testing out the action with more operators stabilized the results. Modifying the weight function from (1 - X) to X(1 - X) did not make a large difference.

Color Dielectric Model o f Q C D

53

We have also tried to increase the number of operators in the effective action by including terms of the form of Eq. (2.35). The main result was t h a t the terms containing other powers ofx~,~, were not sufficiently different to improve the matching. The Sehwinger-Dyson equations were not able to determine more than four eoupfings of t h a t type with good accuracy. As far as we know there is no method to find operators which are "orthogonal" to the given set. Orthogonality (see Ref. [42]) can be defined via a scalar product in the function space. The best solution in the weak-coupfing domain would be a perturbative calculation of the effective action. Because of the large number of components defining the block spin, we have not been able to perform such a calculation. Investigations of this problem for the simpler Z(2) theory are in progress.

II.4 Results for the Effective Theory The results of the blockspinning procedure depend on the choice of the weight factors pl given in Eq. (2.18). By a clever choice of the pi's one may hope t h a t the terms with large non-locality in the effective action can be reduced in importance. For pure SU(2) theory the choice (A) p~ = 1.0, p~ -- - 0 . 7 , Pa = - 0 . 1 8 has been shown 4a to be optimal. For comparison we have investigated the simplest other choice (B) pt = p2 = ps = 1.0. We did not develop a systematic method to optimize the choice of the pi's. An i m p o r t a n t criterium for the blockspin is the size of fluctuations in the effective coupling. The color dielectric phenomenon can be interpreted as a Wilson theory with a fluctuating coupling ~'X[~ -~ ~3'X4. Here X[~ is the product of four X-operators around a plaquette. The amount of fluctuations is defined by (2.36) As was shown in Ref. [44], fluctuations with ~ > (0.7 - 1.0) lead to a first-order phase transition for 2.0 < (fix 4) < 3.0 in the effective theory. At this first order phase transition, the operator expectation values change drastically. One therefore cannot expect to find a set of coupling constants, which give good matching. For the choice (A) we found t h a t tr is in this critical domain after already one blocking step, when initially 2.35 < /3 < 2.65. A typical situation is given for ~ = 2.65. For the couplings of the once blocked theory B. Grossmann 42 obtains A:

/3' : 15.0 ,

/.t2 = -11.5(5)

,

)~ = 22.3(5) ,

A' = 9.43(7) .

0.8

=2.5

0.6

fl

=

2.65

/

Ar E

0.4 v

/

0.2

0.0 -6

-7

-8

-9

-i0

-tl

-ll

-12

-13

2

Fig. 8: Hysteresis effects near the values of the couplings obtained from the Schwinger-Dyson equations. /3' -- 15.0, A = -22.3, M = 9.43 when ~ = 2.65 for blockspin A.

-14

54

H.-J. Pirner

Varying #= in small steps he could trace a hysteresis curve for the expectation value <½ tr ~bp) in the effective theory. 42 The blocked theory gives a value ~tr~bp/blo¢ k = 0.12 when the effective theory has the coupling tt 2 = -11.5. This location is indicated by a cross. In Fig. 8 the drawn curve is obtained by starting with p2 = -10.0 and then changing tt 2 in steps A/~2 = 0.03 and equilibrating the system in each twentieth MonteCarlo step. One clearly sees how the expectation value depends on the history of the system. The same phenomenon does not occur for the second choice (B) of the block spin (see Fig. 9). The resulting coupling constants for the second choice are B:

/3' = 13.04 ; 0.35

"

I

.

.

.

.

/~2 = -28.8(1) ,

I

.

.

.

.

I

.

.

.

.

I

.

.

~ = 51.0(5) ,

.

.

I

.

"

.

~' = -6.7(4) . .

.

.

I

.

.

.

.

J

.

.

.

.

I

.

.

.

.

0.30 A

0.25

¢--

0.20

"

' t?

eL-

oz i--4

0.15

x/

0.10 0.05

-28

/~,= -30

2" i q~ , -32 -34

....

-36

t .... -25

~ ....

-27.5

~ .... -30

, ....

-32.5

-35

Fig. g: No hysteresis effect near the values of the couplings for blockspin B. /3' = 13.04, tt 2 = -28.8, ~ = 51.0, ),' = -6.7 wltere/3 = 2.65.

The fluctuations are smaller, namely ¢r (/3XI-q) = 0.84 and there is not sign of metastability (cf. Fig. 0). The second blockspin (B) is therefore better. In Tables 2.1 and 2.2 we give the results of the Schwinger-Dyson equations 42 after the first and second blockspin step for blockspin (B). We rescale the couplings,/~' =/3' (X) 4, = ~ (X) 4, ~' = ~' (X) 4 and /~2 = /x2 (X)2. In addition, the rescaling factor (X), the average (X 2) and the fluctuations of ¢r(ffXo ) are indicated. The starting value/3 = 2.35(2.5) corresponds to a0 = 0.15(0.09) fm, respectively. Using the string tension tga2, Sommer 4s converted/3 to absolute lattice size a, ef. Eq. (2.3). Later, we are going to take the /3 = 2.35 run to deduce the glueball mass from the effective action. A characteristic feature of the blocking is the change of thesign of tt 2 after two blockings. Note, the smallness of (X 2) does not indicate anything. It is the decrease of/3' which signals the transition to strong coupling. The quality of the effective action can only be judged by comparing the operator expectation values obtained from the simulated effective action with the same operator expectation values of the blocked action. For the definition of the two actions, see Eqs. (2.20) and (2.21). In Table 2.3, a comparison of (Oi)block and (Oi),im (cf. gqs. (2.22) and (2.35)) is given for the choice of blockspin B and various/3-values. The distance D is defined as the rms derivation of the two sets of operators

i=l

where we use again the operators C~i = C)i/(X) d~, which appear after renormaiizing (~) to unity. The dimensions dl are given by the power of X appearing in the respective operator. After rescaling, the accuracy of the different simulations can be compared independently of the magnitude of (X). The dimensions di of an operator can be read off from Eqs. (2.22) and (2.35) as the power of X appearing in the operator.

Color Dielectric Model of QCD Table 2.1

55

Blockspin B: First Blockstep

2.35

2.5

2.65

3.0

(X) (X 2)

0.6370(I) 0.4134(1)

0.67302(8) 0.45911(9)

0.89900(5) 0.49379(7)

0.74278(8) 0.5555(1)

~(fl'Xl"])

0.6716(2)

0.8578(3)

0.8483(3)

0.8377(3)

~' ~

2.414(1) -12.8(2) 8.89(8) 2.1(1)

2.792(1) -14.9(1) 9.5(1) 1.1(2)

3.112(2) -14.1(6) 12.3(1) -1.7(2)

3.805(1) -6.2(6) 19.8(2) -11.8(2)

A'

Table 2.2

Blockspin B: Second Blockstep

fl

2.35

2.5

2.65

3.0


o.o937(1) 0.00946(2) 0.649(3) ~.1o4(8) 4.12(8) 0.58(4) -1.11(4)

0.1332(1) 0.01855(3) 0.799(2) 1.81(1) 5.2(1) 1.62(8) -1.98(7)

o.1667(2) 0.02848(6) 0.837(3) 2.3o1(7) 6.2(2) 2.9(1) -3.1(1)

o.2319(2) o.o547(1) 0.875(4) 3.21(0 8.8(4) 6.4(1) -6.6(1)

~(~'xF~) p~

Table 2.3 fl

2.35 Block

(x)

2.50 Sim.

Block

Sim.

D

Block

Sire.

Block

Sim.

o.eszo(1) o.esel(~) o.erso2(e) o.e94o(1) o.89ooo(5) o.z1924(8) o.z42zs(s) o.314o(s)

o.~499(4) o.74oe(e) 1.OlSe(2) 1.OLOS(3) 1.or81(4) 1.os11(e) 1.0335(4) 1.0~t18(6) 1.szee(~) L~5(1) o.Ts~o(z) o.sszo) (d,) 1.os~1(~) i.o~(~) i.o231(~) I.o~i(~) (d,) 1.os~s(z) 1.o~z(i) ~.o23s(4) 11.o2o~(e) (d,) Z.lle(1) 1.0BS(S) 1.os~(s) 1.o4~(i)

(d,) (d,) (d~) (d,) <¢~,> (do)

3.00

2.65

o,5z~(9) 1.o~ss(s) ~.~o~(s) ~.o~e~(z) 1.532(2) o.sez(~)

o.z11(~) 1.o~s~(e) i.o~(1) I.o~s0) 1.388(3) o.sos(s)

0.25035

0.16667

o.8943(4)

o.zsss(s)

o.z591(s)

0.880(3)

i.OlOZ(i)

I.oo32(2)

1.oo8o(2)

1.oo4(i)

1.oe2s(s) Lo~es(s) 1.2~as(e) 0.7780(6) i .o22o(a) 1.o~24(3)

1.o43~(s) 1.o131(4) 1.2s5o(1) 0.875(1) ~.olze(4) 1.OlSO(4)

1.o4o8(4)

I.o2e(~)

1.o171(4)

I.o11(2)

1.13~4(s)

1.124(5)

1 .o842(e)

1.O,116(9)

0.13866

o.3539(9)

1.oso(s)

1.0143(4)

1.010(2)

1.0148(4)

1.010(2)

1.o41~(~)

1.o23(5)

0.21022

Typically the pure x-dependent operators are very well-reproduced. The plaquette operators 01 and Os have relative errors of 14%. This is the best we could achieve. Blockspin (A) gives for the same quantities relative errors of up to 40% where 2.35
H.-J. Pirner

56 II.5 Interpretation of the Effective Theory The effective action is of the following form

i

/

{ ID

~,/.t

(2.3s)

t,.L#

The partition function Z is defined as Z = [ e-S°"(~)~¢ J

(2.39)

with

v ¢ = 1-[ d x .,. - x i , . au.(~)

(2.40)

The field variables are the I n k variables X,,~, and the unitary fields U,,~ with UtU = 1. The effective action is a lattice action, i.e. it has a cut off of A = 7c/a and is defined only on links. The coupling constants are from Table (2.2) for/3 = 2.35: /3' = 1.10(1),/~2 = 4.12(8), A = 0.58(4), A' = -1.11(4); the errors are the statistical error and systematic error from the Schwinger-Dyson equations. Part of this error can be compensated by adjusting the cut-off. For the moment let us assume that if the matching of operators is perfect, then the effective cut-off A = r/a ~ 1 GeV for a = 4a0 = 0.6 fm. I will discuss the i m p o r t a n t features of this action in general terms, so that somebody even not so familiar with the lattice formulation gets the main message and can relate it to continuum color dielectric models. The effective action contains real valued link fields Xffi,g and the unitary link matrices U~,,. The fluctuating coupling constant in front of the energy density U D formed from link matrices is on the average/3'(X[:3) ~ 1.1. The average value of the plaquette term ~ / 3 ' <61> = 0.6 < 1.0. Therefore one can construct an expansion in /3' analogous to the strong coupling expansion in vacuum. In a vacuum where X = (X) the link matrices U . ( z ) = exp

ig ~ + " Avd(~

fluctuate quite strongly. The lattice size a is too large to justify an expansion in the phase of the link fields; i.e. the gluon fields Av are no longer dynamical variables in vacuum. This may well be different when certain external fields are applied which make X >> (X). W h e n / 3 ' (X) 4 becomes large, a weak coupling expansion of the theory may be possible. The link gluons in vacuum generate a gluon condensate which is calculable s

1 tr U[](12)

~

~_a,..a,.~

We find for the gluon condensate putting a = 0.6 fm and taking (O1> / (X 4) = ½ t r ¢ [ ] / (X>4 ~ ½ tr UE] from Table 2.3 (/3 = 2.35) (Gt,~6ttv> = 0.64 GeV 4

(2.42)

The choice of blocked operators automatically avoids the corrections from perturbative physics, which play a large role at small lattice size. The condensate can be thought of in the strong coupling expansion as the sum of vacuum cluster diagrams contributing to the free energy. The first cluster is a closed plaquette (loop) of link gluons, the second consists of three-dimensional cubes , the links of which are link gluons. The gluon condensate formed out of the thick gluon link loops forms the background on which the color neutral dielectric field propagates. It is therefore best to integrate out these link fluctuations in the lowest orders of (/3'X[~). From the the strong coupling approximation z,4~ we get after a character expansion with :~j(U) as the jth character of U (cf. Eq. (4.10))

Color Dielectric Model o f Q C D

57

Neglecting all higher orders in j > 1/2 and expanding co(/3X[~) one obtains the strong coupling expression to

O n e sees t h a t the link gluons generate higher powers in the X-fields, non-loeM in the plaquette X-fields. These stabilize the x-aetion for large values of X. Note t h a t with the addition of the term (/3X[])4 a stable minimum for X exists. The color dielectric link variable must now be converted into a 0 + glueball field by averaging over all directions. In principle, the average over the space directions IL = 1, 2, 3 of X2-,~ is related to the 0 + glueball field. In the case of a symmetric lattice this averaging can be extended to include the time directions. We remark, t h a t for finite temperature or finite chemical potential the zero component X~,o plays a special role and must be considered separately from the space-like components. We choose the square of X2-,~, as an interpolating field for the glueball since it corresponds to ½ tr (¢[ t,¢®,~,) i.e. traces of closed Wilson loops on the finer lattice, which are color neutral objects and conventionally used to represent interpolating fields for glueballs.

Only recently lattice calculations tried to sum an optimal set of Wilson loops which takes care of the correct Lorentz structure and of the wavefunction of the glueball. Our choice of blockspin corresponds to one possible wavefunction. As long as we keep it fixed at each blockspinning step, the "wavefunction" of the glueball has the same configuration space structure at each length scale. It is easier to expand X=.~2 around the center of the link t h a n to use the constraint ~ra(z) =

4 ~ X~,~ 2 and ,u=l

integrate out the x-link fields. We therefore do a low m o m e n t u m approximation to the effective theory of ~ra: a

a

: ,,'G(~) + a~a,.o-~(,,,) + ~1 ay a2 . o2- ~ + . . . . The lattice action with lattice constant a = 0.6 fm looks like a continuum theory for momenta p < A =a 1 GeV. The m a x i m u m energy difference we can have is 2 GeV. Carefully considering all derivative terms and doing partial integration when necessary one arrives at the following continuum action with ~ = f d4z/a 4.

I_ 1 a4T(cra) : 1 (aO#o'.) 2 I( - fI f' + "4 2 ~12 - 16/3'4o'~)

(2.47)

a4V(o'~) = +41z'~r¢ - 6 In ~ra + (4~ + 4~') o,~ - ;3"~2.a4 p + J34"s64 G One can find the minimum of V(crG) at ~rG = ~0 and then rescale era to &a = (~a - a'o)v/f/a such t h a t the Lagrangian has a canonical form

c(~)

: ~ (a,,~) ~ + ~

~

.

(2.48)

The glueball mass is rr~B -----(2.1 GeV) 2 with the parameters obtained from the Monte Carlo renormalization group calculation in Table 2.2 for/~ = 2.35 and a = 0.6 fm. We remind the reader t h a t in previous derivations of the color dielectric model we have used 8~ the weak coupling expansion into gluonic fields after blockspinning to an effective lattice size a = 0(1) fm. This procedure is not correct. However, one can still use the low m o m e n t u m expansion for the X~,~, fields. The value of 2 GeV for the glueball is too high, which may be due to the possibility t h a t w e got too far into the strong coupling region with our effective action. We will also calculate the string tension from the effective action and compare the ratio to other lattice calculations.

58

H.-J. Pirner Ill. C O L O R

D I E L E C T R I C S F O R SU(N), N _> 3, A N D F E R M I O N S

III.1. Color Dielectric Theory in SU(3)color There are certain differences in the behavior of lattice gauge theories as a function of the gauge group. For instance the thermal properties of pure gauge SU(2) and SU(3) theory are different. The deconfinement phase transition is second order in SU(2) and first order for SU(3). The color wavefunction of the baryon in SU(2) ~b = ~ ( R B - B R ) is identical to the color wavefunction of the meson ¢ - ~ ( R / ~ + B B ) for SU(2) with the colors Red and Blue once the identification has been made that (R, B) - ( - / ~ , R). In SU(3) the baryon is really different from the meson. These examples teach that the effective theory for SU(3) may be also different from SU(2). The block variable in SU(3) ¢~,(z) is again obtained by an average over link configurations as in Eq. (2.18)

4,,(=) : ~

p,~,

(3.1)

.

i

The gluon fields are represented by a sum over 3 x 3 matrices

&(=) =

8

~ i=l

where )~i (i = 1, 8) are the Cell-Mann SU(3) matrices. The link variables

are elements of the SU(3) group. The 3 x 3 matrices ¢~(z) have a ray representation

¢.(x) -- V.(z)X~(~ ) e is'(®)

(3.2)

where Xa is a positive definite Hermitian 3 x 3 matrix, expiSa is a phase factor and Vj,(z) a unitary SU(3) matrix. This idea has been first introduced in a paper by G. Mack and K. Pinn. 4s It is a known result in mathematics 49. The index # indicates that V®,a, Xa and 0a arc associated with a link in the/~-direction. For completeness let us show how Ca can be uniquely decomposed into these three parts. The matrix Xa is determined as follows. First we calculate

¢~.

= (V.~.e'% t (V.~.e'% = X~ (V~V.) X. = ~.=

(3.3)

2 _- W D 2 W t with an unitary matrix This matrix X~ has positive eigenvalues dr, therefore we can write it as X~ W and a diagonal matrix D = d~6~j, dl > 0. Then we define Xa = V/-~a = ~ =

~(WDW')(WDW')

= WDW ' .

(3.4)

The phase 0~ can be extracted from the determinant of ~b~, namely cotg 38~ : ( Re det ~b~/Im det ¢~,) .

(3.5)

The unitary matrix V~ is given by V, =

gp~,e-i°"X'~

t = ¢~, e-iO~ W D - t W t : ¢~, e -is" ( ¢ t ¢ , ) -t/=

(3.6)

The Lorentz properties of these fields can be derived by inverting the direction of a link c t ( = ) = ¢ _ , ( = + ~) : e - , ~ . c , ) G C z ) v d c = ) .

(3.7)

Color Dielectric Model of QCD

59

or

x,(=)

: x-,,(=

+

,

=

+ +,)

.

(3.8)

Therefore, the X~, field is no~ a vector, it has even angulsr momentum, i.e. also a scalar component, whereas 0~, has vector character. The Hermitian matrix X~, has nine independent components. The reduction from 4 x N 4 link variables U~ 1 4 . . . . . with 8 degrees of freedom to 4 x (~N) link variables ~b~,.~s achieved by hawng more complex varmbhs ~ with 18 degrees of freedom on the blocked lattice. We hope, however, that after several blocking steps, the main information is carried by the color neutral part ~ ( = ) = 8t TrX~(=), which still can be identified as the color-dielectric field, and the additional color-neutral field 0~,. So far the color dielectric model has had a close analogy to the MIT Bag Model. Small X represents a non-perturbative background of a gluon condensate and large X signals a theory with weak coupling. One can speculate that for color SU(3) the bleached color neutral vector field 0r plays an important role in phenomenology. It acquires a vacuum expectation value whenever the baryon charge is non-vanishing, so it will be especially important for baryon properties and baryonic matter. As is well-known, the lattice formulation of the latter problem has until now progressed very little; perhaps this average quantity 8~ can help to bring the subject ahead. In our previous discussion of color dielectric SU(3) so we have approximated the Hermitian matrix £~(z) by a diagonal one and reduced with this method the problem to the SU(2) case where there is only one X~(z) variable. This assumption may be too hasty, since the phase space for Hermitinn matrices has the more complicated form d~ -- dPV (X~ - X~) 2 (X] - Xa2), (Xsa - X1~)2dx1~ dx• dx~ , (3.0) where dW are the degrees of freedom associated with the matrix W and X~ the diagonal matrix elements of D. The phase space suppresses the occurrence of degenerate eigenvalues. This is the phenomenon of level repulsion well known in statistical Ramiltonian matrix theory. There is current work in progress sl to treat the problem for large number of colors Nc --~ c¢. Methods of random matrix theory can be applied to get the ground state and spectrum of mean eigenvalues of ~ from the coupling constants in the SU(3) effective action. M. Rosins s= has proposed that there are terms in the effective action for SU(3) that give approximately an equal spacing. In any case, the physical picture would be one where the intrinsic state of the vacuum would be deformed in color dielectric space Xu' Naturally after considering fluctuations the symmetry would be restored. When ~ has components in 3 and 8 directions, i.e. the gauge fields can be oriented along preferred directions. This may be related to the triMity problem. The three quark colors populate an equilateral triangle in (3,8) charge space, so that only the (qqq) system and qq system can produce vanishinl5 mean A~,,sA~S fields. These are also the minimum gluon fields necessary to describe baryons and mesons, sa'°4 Until now no applications have been worked out on this aspect of color dielectric theory besides the fundamental work of Adler. The general attitude is to keep the Abelian approximation in the gluon kinetic energy and the non-Abelian SU(3) Yukawa coupling of quarks to gluons. The quark-gluon coupling contains the full S U ( 3 ) - A matrices, but local gauge invariance is given up. Generally, a scalar dielectric function is kept for SU(3) color. There are lattice calculations which calculate the matrix V~(z) in Eq. (3.2) but throw away 8 and X. The most general expression 4s, so for an effective local action in SU(3) to fourth order in ~b~ is as follows s.--*'(¢)

=

Re

+

+

))p.
+ At 13~1~ (~b~(z)¢b~.(z)~bt(f)~b,(,)) - 7 Re det ~b~(z)}

1

•

(3.10)

60

H.-J. Pirner

The first term contains the sum of the plaquettes formed with the blockspin variables ~b~. The second sum consists of quadratic and quartic local products of ~b'sas in the SU(2) case. In addition, we get for SU(3) the real part of the determinant of ~b. This term would be similar to Tr ~bt@ in SU(2). The third and fourth terms give a product of ~b'son two perpendicular adjacent links. The form of the above action preserves local gauge invariance on the blocked lattice. Under gauge transformations on the blocked sites,~b transforms as ~,,(=) ---, G(=)~,,(=)G-'(=

+ ~,) .

(3.n)

This transformation leaves the plaquette term, the trace of ~ b ~ , and det ~ invariant since det G = det G - t = 1.

In Ref. [50] we have done a weak coupling continuum limit of this effective action, and solved for a soliton made out of heavy quarks. As discussed in Chapter 2, very probably the blocked theory becomes a strong coupling theory in vacuum, so the work referred to should be taken with caution. Its calculation may illustrate some possible results in a theory with an extra bleached gluon field O~. In weak coupling approximation i.e. for small g . A~a < 1 we expand the link operators in A# and O~. Neglecting the off-diagonal elements of ~ ( z ) and integrating only over the color neutral part of X, we argue that a possible equivalent continuum Minkowski action would have the form (0~, = 0#O~ - 0~O~)

4=gauge = /~l 16X4 _{_3 2

1X%~20~vO~" 4#2X ~ - (A + At) X4 + 4 7 X s + ~ T9x ~ - 6(,X+A')x4) -

-

4

(3.12)

s~2

This action is inspired by the form of the effective lattice action and can help to get some intuition about possible physics contained in it. All gauge field kinetic terms are multiplied by the dielectricfunction X 4. The determinant term 7 det ~b produces a mass term for the 8~-field which is given by the ratio

As in conventional dielectric theory the gauge fields would be confined in regions where X # 0.

III.2. Fermions in Color Dielectric Theory It is necessary to discuss the fermion coupling to the effective blockspin @~(z) in order to understand quark confinement in the effective theory. In continuum theory gauge invariance determines the minimal coupling of quarks to the gluon fields. Under local gauge transformations the fermion field changes as s

¢'(z)

=e

°='

~(z)

.

(3.14)

The kinetic term of the quark action contains derivatives, so a local function a ( z ) would spoil gauge invariance. In order to recover gauge invariance the covariant derivative 8

0.--1

replaces the ordinary derivative £qqG = i ~ 7 ~ D ~

.

(3.16)

If one wants to translate the description of fermions to the lattice certain technical and fundamental problems arise. In order to have gauge invariant expressions quark creation and destruction operators have to be connected by a string, i.e. by link operators U~(z), when they appear at different sites. Typically

Color Dielectric Model o f Q C D

61

ff(z)U~(z)¢(z +/~)

is invariant, because the quark and antiquark are connected by a link. transformations the individual operators change as:

Under gauge

u. - . u;(=) = G ( ) ) ~ ( ~ ) C - I ( ~ + ~) ~(~) - . ~' = ~ ( ~ ) G - ' ( ~ ) ,

(3.17)

but the product of all three operators remains invariant. To form a "Lorentz" invariant one must multiply this product with 7 . and sum over/~. The expression becomes a right derivative, when we subtract the local operator (b(z)7.(z)~b(tx)

In order to have a Hermitian action one has to combine this right derivative with n left derivative

1 S1 = ~ ~

[~(~)(~.)U.(~)¢(~ + ~) - ¢(~ + ~)7~Ul¢(z)]

m,.

[Recall t h a t for the Euclidean lattice 70 :

01

and 7i =

-

(0 o,) ~

0

(3.20)

such that 7~ = 7..] Combining the

two terms, each with a factor 1/2 gives the first part of the action. The second part is local and depends on the mass of the Fermion. $2 = - ~_# ¢(z)aoM¢(z) . (3,21) $

The third term addresses to solve the fundamental problem t h a t the solution of the discretised fermion propagator contains a sin(ak~) function with k . in the first Brillouin zone - r / a < k < r/a. This sine function vanishes not only for k . ---* 0, but also at the corners of the Brillouln zone k~ao = ~r. This is the famous Fermion doubling problem. Wilson proposed as a solution to add a term with a second derivative f a0 (c9.~) ( 0 . ¢ ) d 4 z

which in analogy to the boson action generates a (cos(nob.)) term in the Fermion propagator making the mass of the unwanted fermion replica very heavy 0 ( l / a 0 ) in the limit a0 --~ 0 and k : ro/ao. This second order derivative is written as the difference of two first order differences. After shifting the summation one has 1

Ss : ~ ~ {~(z)U.(z)~(z + I~)+ ~(z + #)Ut(z)¢(z)}

(3.22)

mp.

The special feature of this action S = $1 + $2 + Ss is t h a t M # 0 even if the "real" mass m = M - 4/no goes to zero. Mack se has written the corresponding color dielectric action including terms up to second order in qb. 1 ~(z)(aoM)@(z) - ~k

S ~ mlon = - ~

• 1 + ~ ~

~

if(z)(1- %,)@(z)1 Tr ¢i (z)~b.(z)

.'.=±1,±4

(3.23)

{ i f ( z ) (1 + 7 . ) qb.(z)@(z + p) + ~ ( z +/~)(1 - 7#)¢~(z)~2(z)}

s,.=l,4

Besides a conversion of the original link variable U . ( z ) into ¢ . ( z ) by the averaging procedure, also a higher order term proportional to Ctqb~, can arise. Restrictions on the parameters M and k can be derived by calculating the continuum limit of the effective Lagrangian, see Re£ [39]. In the same same naive continuum limit we have used before in Eq. (3.12) one obtains for the Fermion effective Minkowski action 4--fermion

a /"eft

-

-----

-

(aM

4X + 4x~k)

+ [X¢(z)iTu

¢(z)¢(z)

(aO~ - 8 " - Br~---~)¢(z)] +k [(b(z)iT.¢(z)(aO",X']

(3.24,

62

H.-J. Pirner

g m

Fig. 10: $chwi~ger-Dyson equation for the quark self-erLergy in ~he Abelian color dielectric model. One can drop the last term because of current conservation (X~7~,~P is conserved). Then there is a mass-term = ( a M - 4X + 4kx 2) and a modified kinetic term left. A particular feature of the color dielectric model is a kinetic term t h a t contains the interaction with the X-field, but there are also Yukawa terms. In a first paper ss such a kinetic term was shown to simulate total quark-confinement when X --' 0 in a vacuum. Namely redefining the ~b-field as : V~ , (3.25) m

one can rewrite the kinetic and mass terms in ~ as:

In the second term we symmetrised again and used 1 This form has been taken over in various approaches to study sollton solutions for the X-field which confine quarks. W h e n in the vacuum X approaches ~ero, the effective quark mass me~ = m/X goes to infinity. The color dielectric field confines colored 81uons and quarks together. This mechanism of confinement has been criticized, when one tries to apply it to light quarks. Confinement seems only possible when a small residual quark mass is present. We believe however t h a t also zero mass quarks are confined. This has led W. Broniowski, et al. and M. K. Banerjee et al. to modify the mass term such t h a t chiral symmetry is not broken, ss It seems natural t h a t on a large distance scale there must be other relevant mesonic effective degrees of freedom. Especially the pion must play an i m p o r t a n t role at low energies since its mass is so low. Actually the philosophy in most effective field theories has been to leave out the 81uonic degrees of freedom and just model chiral symmetry correctly either with a Nambu-Jona-Lasinio model or with a Skyrme-type non-linear sigma model. The justification of this approach is the large mass gap of (1.5) GeV (the glueball mass), which freezes the gluonic vacuum in low energy physics. Let us first cover chiral symmetry breaking in the color dielectric model and then come back to this approach.

III.3. Chiral Symmetry Breaking in the Color Dielectric Model There have been various objectives to construct chixal symmetric versions of the color dielectric model. Builders of sollton models for the nucleon want to generalize the color dielectric soliton to a chiral color dielectric soliton in analogy to the work in the bag model. In this philosophy the explicit introduction of scalar and pseudoscalar meson degrees of freedom is the simplest way to solve the problem. It is more ambitious to deduce chiral symmetry breaking from the mechanism of color dielectric confinement. In the work on the chromodielectric model Wilets et al. s7 have summed one gluon exchange diagrams modified by the color dielectric medium such t h a t the effective quark gluon coupling is g2/e(X ). When the dielectric function e(X) becomes zero, the coupling is very strong and attractive between quarks and antiquarks and can produce a dynamical quark mass. Using a Schwinger-Dyson equation they sum all rainbow graphs on the quark propagator (see Fig. 10). The space-dependence of the dielectric field is ignored, in t h a t way the calculation becomes very similar to the work done ss which generated chiral symmetry breakdown from a strong gluonic Coulomb force.

Color Dielectric Model of Q C D

63

It is not clear to me how the gluon quark coupling can be very strong but the gluon-gluon coupling is negligible or can be absorbed into the dielectric function e(X). The authors of the above paper find an effective quark mass which increases to infinity on the surface of the bag lettin 8 e(X) 8 ° to zero in the vacuum. It seems to be a little tricky to combine this so!ironic solution for the nucleon with the vacuum solution of a quark condensate. Maybe similar work to the MIT bag model with a vacuum made out of so!iron bubbles which contain qq-palred states is possible. It seems, however, simpler to consider a constant ~ b from the very beginning. The quark and gluon fields would be integrated out. The so!iron is constructed from hadronic-color-neutral fields alone. This seems to me, now, the most promising approach. The color dielectric blockspin or field may help to understand which forces are important to drive the system towards chired symmetry breaking. In the 0 ++ meson sector there is possible mixing between the gluebaH and ~r-meson (qq) states. Therefore I would expect that a simple combination of the ~ model with a color dielectric x-model misses perhaps some important physics. J. Ellis e t a l . 59 have shown for the non-linear o-model that dimensional counting makes term x-dependent. A. Patkos s° has modified the mass term of the SU(3) flavor linear the kinetic -if~O~,IrO~r 1 2 sigma model and made it dependent on the gluebaH field. In Ref. [61] we have tried to explain the mechanism for chiral symmetry breaking from the underlying effective color dielectric Lagrangian, which was shown in Eqs. (3.10) and (3.12). The source for the attractive qq interaction is the bleached 81uon field 0~, which comes from the superposition ofSU(3) gluon fields (Eq. (3.2)). In SU(2) color one can make an Abe!inn projection by fixing a gauge. At long distances the Abe!inn component may carry the important physics. The importance of the Abe!inn subgroup of SU(N) has also been emphasized in the context of the dual confinement model mentioned in Section 1.4. For the color SU(3) group the bleached gluon field may carry the long-range information which is important for chiral symmetry breaking. The U(1) symmetry associated with this bleached 81non field is broken by the determinant term 7 Re det ~b~(z) which generates a mass for the 0~ field. Quarks and antiquarks can interact through the exchange of these bleached 81uons. For small X, when confinement sets in, the mass of the 0~ field becomes large: f---

ms

w ~/~

(3.27)

Hence, we are allowed to neglect the kinetic energy term and integrate out the 0~-field. This generates a quartic quark contact interaction (~7~'~)2. Neglecting the interaction of the quarks with the colored gluon fields, B~, and the color contact terms after Fierz-transformation, we arrive at the following Lagrangian (in Minkowski space):

l=0

- 2

~__o [ ( ~ ' ~ / " ! b )

q- =

(3.28) ^ '

'

where ~! are the matrices generating the Lie algebra of U(N!) (tr)~)~1 = 26~!). In practice, we shall restrict ourselves to the SU(2) × SU(2) sector, i.e. )~o = I, ~z = rl (l = 1, 2, 3). This yields a model of the Nambu-Jons-Lasinio type, where the coupling constants are dynamically generated by the X-field. We have 1 as Gx(X) = 2G2(x) : ~ ' NSfxS , (3.29) where we shall set Nc = 3, and V(X ) is the X potential, modified by reseeding the fermion field ~ = ~bv/-~ v(x)

= ~4 /~ . 2 x

2

- "Ix s + x 4 x ~) -

b A 4logx 2

(3.30)

For convenience, we prefer to work in Euclidean space. Using the standard rules, we can transform our Lagrangian into an effective Euclidean action, S(~b,~;X ). Now, with the help of a functional approach, one can change to an effective action depending on boson fields which correspond to the various channels f

64

H.-J. Pirner

the quark-quark interaction. Thus, we explicitly introduce a scalar field, s = ~ )~lst, a pseudoscalar field, ! P = ~ )VPx, a vector field, w, = ~ ),! wgy , and an axial vector field, a t, : ~ )fiat`!, according to ! ! !

f

exp {--S ( ~ , ; ; X ) }

(3.31)

-- f [dx3 [d@] [d~ td,] [apl[d~3td~lexp {-S (@,;, s, p, ~, ~'x) } , where

1

+ ~

tr! (s z + p ' ) + ~

1

tr! (w~ + a~)

(3.22)

x' + ~tr°F.vvt`v + V(x)} The final step is to integrate out the quark field. The resulting bosonized action reads S (s, p, .~, a; X) = - ~ log [i'r. (P~ + B~ + . ~ + 7,a~) + • + i ~ s ]

+fd'=[

1

~trl(s

,

1

+p')+4G--G-~tr!

X4 (w~ + a~) + ~-ig2 trcF, vF~ + V(x) ] ,

(3.33)

where Pg is the four-dimensional momentum operator satisfying the canonical commutation relation [Pg, z~] = - i ~ t ` t . . Since we are mainly interested in the problem of chiral symmetry breaking, the only relevant fields are the isoscalar component of the scalar field, namely the st-field, and the isovector pseudoscalar pion field, = ( ~ , ==, '~a). In the fonowins analysis, we shall denote these as an 0(4) vector q, = {~b=}= {~, ~}. The relevant part of the quark effective action is now given by

Sq =

- Tr log (i%,Pt., + o"+ i~.

R'rs)

= - '13' ]og'D

1 =--Tr 2

(3.34)

log:DD t .

After some straightforward manipulations, we obtain Sq = - Tr log (P= + ~b"~ + h V )

(3.35)

with ~b2 = o "= "4- ~2

,

(3.36)

V = % 0 , (# - iF. ~'rs) We now employ the heat kernel representation of the effective action, to obtain Sq = 12Tr

Joo°° d--~-~e-~(P=+4"+~'v),0'

(3.37)

which, using the coordinate-space representation of the momentum operator, Pt,, can be rewritten as 1 /d%/ sq= ~tr

d'k

f0 °° ~ e - f l [ (-i~O" +k')' +4~'+av] 1 .

(3.38)

This expression has the form of an operator acting to the right on the unit function, denoted by 1. The symbol Tr in Eq. (3.11) signifies the total trace of an operator (which includes the continuum), whereas the trace, tr, in the above equation, Eq. (3.12), acts only on Dirsc and internal symmetry indices. The methods of derivative expansion, s2 can now be applied. To second order O(02), the result for the total effective action is

sq = f d4z { W(~=) + l [g~.(~:t)6,.b+ -~' f'-g=,v,qb=d~b ,.,.:t,] Os,~=Ot,~b

(3.39)

Color Dielectric Model o f Q C D

65

with

W(¢ 2) = - 6 N 1

g,(¢.)=,N,f

f ~d4k -x T ~ log(k2 + ¢2) + ~27~

• + vCx) , ~-2

(3.40)

(2~r)4 (k 2 + ¢:~)2 ,

,

(34,)

g2(,2)=,2N, fd,~ - ~k LF( k 2k2¢2 43 (k2 :~)2) ,2 3 ] ~"') '

(3.42)

Here and subsequently, we set h = 1. From the discussion of the preceding section, it should be understood t h a t the k-integration is limited by the covariant cut-off A = lr/a. We now expand the action around the classical field (¢,) = 6~oe0. - (~b) = {5-,~-,},we obtain

=

2co -o- (eo~) -~'

The x-dependent quark mass reads

i.e. the

Introducing the fluctuating field ~ =

~ [9'(e°~) + g2(e°~)] o.5-o~5mass associated with ¢,

2N~ or

N! T

f

[ '

1-~log

dak

(2~p

(

s'o(X), is determined

1

k2+eo~ -

1+

9"I'Xs a2

=97X a .

'

.

(3.47)

by the gap equation, with

(3.44)

(3.45)

Also the x-dependent condensate and f,~ and the e-mass can be calculated, s ' In the above reference the effect of rescaling the O-field has been inadvertently omitted. Still the - l o g X2 in U(z) of Eq. (3.30) term is not su~eient to prevent t h a t for small X, e0 increases like X-s/2 and the total vacuum energy W(eo(x) ) diverges toward ( - c o ) , when X ~ 0. W e have not found a totally convincing solution to this problem. The key must be the eigenvalue spectrum of X, which is unknown as long as one has not done SU(3) calculations. M y personal speculation is that it will be distorted as such that the det ¢ term has the form 7 det ¢ ~ 7X26 • • ale

(3.46)

It may give only a second power o f x 2, thereby making ~o ~ ~ for X small with m as the constituent mass. The necessary o'0 can have a meaningful vacuum value when eo zs very high in spite o f m ~ 300 MeV. Therefore the cut-off A must be high, too. This may correct our proposal given in the above reference. 61 The qualitative conclusions obtained, however, remain correct. It is the smallness of X, when f f is fixed to the scaling region for SU(3) j3,¢~1~ ~ 6.0, which drives the chiral transition as well as the glnon condensation. In our previous work we found a threshold

Xc "~

(N!) ~

1/s

0.3 ~ 71ia

(3.47)

For X larger X~ the Goldstone phase disappears and the constituent quark mass and ( ¢ ¢ ) both go to zero. In the Wilson formulation there remain other possibilitiesto generate a finitequark mass just from interactions with the gluonic background. This would explain why quenched latticecalculations with no quark-antiquark loops stillproduce a non-vanishing (~0>. From Eq. (3.24) one sees that even when a M - 4X = 0, i.e. in the chiral limit, the interaction term proportional to k in Eq. (3.23) generates a quark mass 4x2k. Our experience with the lattice calculation seems to show that X will remain finitealso in SU(3) due to the phase-space in the x-integration (Eq. (3.9)).

66

H.-J. Pirner

III.4 Critical Evaluation of the Color Dielectric Picture This section ends the more theoretical outline of color dielectrics. The entire Chapter IV will be devoted to applications. So this seems to be a good place in the review to stop and open the discussion to questions and criticism concerning this approach. The main question can be formulated as follows. Is the introduction of the auxiliary X~(Z) field worth the work put into it? One goes through a lot of approximations truncating the complicated non-local and nonlinear action, before a simple picture with four effective couplings emerges. In addition, this picture needs simplifications to relate the auxiliary X-fieldto the glueball field. W h a t is the advantage of this theory compared to a conventional strong 4z coupling approximation to Q C D ? The traditionalstrong coupling approximation has the problem that physical observables become singular when the coupling constant approaches the continuum limit. Conventional strong coupling lattice gauge theory has only unitary link matrices on a coarse lattice as degrees of freedom. This presents certain restrictions: the ratio of the 81ueball mass in strong coupling SU(2) to the energy ~a contained in a string of length a is

~r~.GB - 4 l o g ~ . a - -iog

--4

.

This fact comes from the necessity to have a smoke ring formed by a plaquette of U-fields as interpolating field for the glueball. In the effective color dielectric theory most expressions for physical observables have similar expressions as in strong coupling gauge theory, after the integration over the unitary fields. But then the x-integration remains and it is my conjecture that the divergence of the conventional strong coupling theory is removed by this x-integration. Secondly, by constructing the glueball as an independent hadronic degree of freedom one can lower its mass. The above ratio in the color dielectric theory becomes maB

~2

as will he shown in the next section, where the string tension is calculated. The interpolating field for the glueball is the state ~ ½tr 4,~(z)~b~,(z) which costs less energy to make than the smoke ring configuration. The fat string operators ~b~(z) play the role of valence glnons. Only two valence gluons are necessary to make up a 0++-gluebali state. In my opinion the weak coupling approximations of color dielectric theory with continuum gauge fields only illustrate a possible situation, which is not calculable in a discrete lattice theory with cut-off. The lattice construction gives a condensate of link gluons for the vacuum. Even inside of flux tubes a weak coupling situation does not exist. We have not been able yet to estimate quantitatively the necessary strength of a source acting on the dielectric vacuum to change X to
=

=

'

/ (ch)o

< o

- o

,

(3.48)

2

The presence of a changed magnetic field in the three-direction and the enhanced electric field speak against an Abelian solution for the flux tube, The solution is reminiscent of the Nieisen-Olesen string ts discussed in Section 1.4. The dual of magnetic field, namely the electric field is changed between the two charges and the dual of the Cooper pair density the monopole density is expelled from the tube. The lack of monopoles may be interpreted as the reason for a smaller magnetic field between the two charges.

The calculation of the non-Abelian color dielectric flux tube will be discussed later. Qualitatively the following situation arises. The dielectric field (XS)tube is larger than (Xs),,c in vacuum, because of the increase in the

Color Dielectric M o d e l o f Q C D

67

field from the external color charges. In the Hamiltonian framework we would get an effective Hamiltonian of

the form of strong coupling Q C D 7

~IrCD~ E

2 ~ D~(~) ~--~-~/ ~ 2 + ~

t

¢

Xi

~ tr (¢ij) ¢>J

IpatiLl

+ ~V(x,('),Xy(~)) ~ i,./

(3.49)

where ¢11 = X[~ • U{=] are the dielectrically modified plaquettes in space-space directions i and j, i.e. they are related to the magnetic fields squares x- c2_x j2f,2 ' = i j and Di are the dielectric displacement fields Di = xiEi. For an external Ds field the first term in the Hamiltonian pushes Xs to larger values (Xs)tube > (Xg)vac. From the sign of A' and j3 one sees t h a t a larger (XS)tube > (X3)vac will also make (Xi)tube ~> (Xi)vac for i = 1, 2. One gains energy in this case since A' = - 1 and V = +A'X~X~sThe effective coupling ~,X~X~ = 4X~X]/g~ is therefore weaker in the tube t h a n in normal vacuum, producing a

value of (U12)tub e nearer to unity and bigger, namely (U12)tub e ~> (U1$)vac. Consequently, the magnetic energy density is diminished: 2 (B S)tub. = -~" (tr (1 -- Utm))tub. < (B~),. c

(3.50)

There have been calculations by J. Greensite so which also stress the non-Abelian nature of the flux tube. The above author explicitly criticizes the assumption t h a t the flux tube remembers the color and anticolor at the endpoints. I think lattice simulations do exclude a simple Abelian color dielectric model as advocated in the early days by Freedberg and Lee in analogy to the bag model. We discussed evidence against a bag type response, when one measures the string tension for various color charges. Lattice simulations give always t h a t the string tension is proportional to the Casimir operator of the ( A2 i )repj for j = 1, 3/2, 2, 5/2 in SU(2) e.g., whereas the Abelian bag type model would give a square root behavior. Actually j -- 1 charges can be screened by gluons. Another quantitative difference between the color dielectric model and early bag ideas is t h a t the vacuum inside the bag is only weakly modified since the glueball mass mGB ~ 2 GeV is so large. The real test of the color dielectric models lies in the applications, which include a combination of strong coupling methods with the integration over the X-fields. These have rarely been done in this form. I will try to sketch these techniques in the next chapter. Most of the work to be reviewed has been done in the framework of the continuum approximation with fermion fields. So I will try to do justice to these efforts, too.

IV.

APPLICATIONS OF C O L O R DIELECTRICS

IV.I The Heavy Quark System The heavy quark system is the best physical situation to study the pure glue dynamics of confinement. A good fit to ce-data is obtained e.g. with the Cornell potential sr which has the form A U(R) = - ~ + ~ . R

(4.1)

with A = 0.52 and ~ = 0.18GeV 2 ~ 1 G e V / f m and R is measured in GeV - t . Only at very large distances R ~ 0.6 fm would one expect to see the effects of light quarks, when the potential energy becomes equal to the threshold to produce two constituent quarks of mass m ~ 300 MeV. The string tension ~ is the quantity one wants to reproduce in the classical color dielectric model. The 1/R term can have two origins. It may be

due the exchange of a gluon

Vslu°n =

4 o~j 3 R

(4.2)

This would give us a coupling a0 ~ 0.5 (somewhat large). It may be also due ss to oscillations of the string. S. Adler 2 has pointed out t h a t the form of the color dielectric potential does influence the next to the leading order V = ~ R of the potential. It may have a 1/R or a 1 / I n R dependence.

68

H.-J. P i m e r

Let us study first the ideal situation of a long flux tube and the dynamics of the color dielectric field associated with it. To start we will simplify the problem by using the Abelian color dielectric (chromodieleetric) model in the limit where the length of the flux tube is much larger t h a n its diameter. For a calculation of the energy per unit length E / l only a transverse integration over the radial coordinate p is necessary. We assume the simplest Abelian type of color dielectric fiux tube to show the the mechanism of confinement. 2s

7=2~ ~p,(x,(p)) +U(x,~ ) + T

(4.3)

This energy has to be minimized with the Gauss law constraint:

q = 27cf pdp~(x,)E, = w~--Oa~ra, .

(4.4)

The function e(X~) plays the role of dielectric function, the other components of X are not dynamical, if one does not couple to magnetic fields. The potential function U(X,) could be obtained from integrating out the magnetic fields ¢~j in Eq. (3.49) and adding this contribution to the part from V(X,, Xj). The non-local part of V(X,, Xj) is related to the kinetic term in (4.3). The calculations in the above reference have been done with phenomenologieal parameterisations of U(X~), o',j and e(X 2) = X2(3 - 2X). The energy can be minimized by finding the optimum transition from a zone with large X, where the ~.ga/2 dominates to the vacuum where U(X) is large. The solution does not show X = 1 in the inside of the flux tube. Vacuum effects persist even inside. The string tension can be reproduced, but the radius of the flux tube is fairly high. The function X has fallen to half its value at p ~ 0.8 fm (Fig. 2 in Ref. [23]). This size of 0.8 fm looks to me too high judging on nowadays knowledge. The large radii may be due to input for the glueball mass m o s = 0.72 GeV which is too low in comparison to the more recent lattice studies. TM 12 The calculation of the string tension in the non-Abelian color dielectric lattice theory is simple, once one knows the techniques of the strong coupling approximation. One has to calculate the expectation value of a large Wilson loop in order to extract the heavy q# potential. In the limit M~ --, oo, the Euclidean Dirac equation for the quark becomes simple:

The quark cannot move, therefore we drop all space derivatives ~ and associated gauge fields A. Then

~(~, r) = e' f: A0 ~ - , , ~ ( ~ ,

0) = e' f: ~°~'~(~, r) .

(4.0)

Besides the "time" evolution with the constant mass the spinor evolves with a phase to the product of links along the time direction. Having a quark and antiquark gives two such time link products which can be connected by a string at time zero and r. In color dielectric theory the natural generalization is to take ~b-links along the Wilson plaquette. Then the potential V(R) is related to the Euclidean Wilson plaquette

( ~ . u = ~, ~x = 0) w,(R,~)=

{1

~tr 1 - I ~ 0 ( ~ , ~ , ) ~ , ( ~ , ~ ) . . . ~ , ( ~ + R ~ , , ~ ) /=1

× ,: fI ,'o + lim

W~, = e - V ( R ) ' "

:

.,:

÷

('")

e-''~''

"r--~. o o

The integration in the above expectation value over U-fields is done first, using the property t h a t

] dU U,jU~ 1 = ~6,k 1 6j . .

(4.8)

Color Dieleclric Model o f Q C D

69

In lowest order of ~e~ = ffXF3 one obtains a tilingof the Wilson plaquette operator W ~ with plaquettes ~b~ contained in the action

(4.9) []

v3-tt

=,p.

The link dependent part of the action has an expansion in powers of ~X [] in strong coupling as

where Xi are the characters of the SU(2) group ~o(U) = 1, Xt/aCU) = t r U : 2cos812, ~t(U) : ~/,(U)f~I/2(U) -~0(U) = 2cos0 + 1, etc. The higher j representations come from the coupling of spin1/2 representations. The Hemr measure of the SU(2) group in terms of the angle 0 is

f The expansion coefficientsc0(ffXO) and

fo 4,~ d0 . 2 0

/r2~r -d0- s i n ' O

bj(ffXf3)can be

obtained from the orthogonality properties of the

(4.11)

characters

f dU ~i(U)~f(U) = 6 J ' 11/"'..,xDoo., _

(4.12)

cos

co(~xD) = ~ ~ ~o

2 -- ~ X [ ] I I ( ~ X C 3 )

(4.13) "

where we used that ~ tr U[]X[] = ffXV3 cos ~ and a partial integration on sin2 ~. Similar expressions with the Bessel functions I,, give the other coefficients

bj (ffXD) =

Io (/~'XV3)

;

(4.14)

The integral in (We) involves tiling i.e. the plnquette by plsquette reduction of the R x r plsquette to a smaller size (cf. Fig. 11). The plsquettes outside of the Wilson surface are treated as before in Eqs. (2.43) and (2.44)

f V~e-S,,,C~)w(~)

(w,) =

fV~e-S..(*)

["]ew After the U-integrations, S(X) contains x-dependence which will be solved for in the mean field approximation. We differentiate between the mean field Xin : Win inside the flux tube, i.e. when s link is part of the Wilson plsquette surface mad the vacuum mean field X~¢ = Wo,t outside of the flux tube. The fluctuations of X are so short-ranged that there are only these two values, i.e. the flux tube has extension smaller than the lattice sine.

3

2

(4.16) ®,~

v.L~

[]

384

)

H.-J. Pirner

70

p.

,Xin

Xout

T m

t

#

J

<

m

~s

R

--

Fig. 11: Wilsonplaquette and tiling operation in strong coupling theory. The X-values in the Wilson surface a r e (Xin) = Win, the x-values outside of the Wilson surface (Xout) = Wout. The action St(X) in Eq. (4.15) is the same expression as S(X) in Eq. (4.16), except the summation only goes over plaquettes which are not in the Wilson surface. To determine the vacuum value of X, one takes a test X~(z) and immerses it into the mean field vacuum with X,~c = Wo,t. Self-consistency demands t h a t (X~(z)) = Wo,t

S (X,

Wo,.,.t)

= #2x2 + AX 4 -

3

In X 2 ~t4

2 ~ 8 + 2~ Ix 2 W~u, - 6 /~-~(xw~.J

3

(4.17)

4

2

/ dx eS(X'w°~dx/ / dx e-S(×'Wo"d -- Wout

J (4.18)

Using the saddle point method to do the integrals

os (x,axWo,,)~=w.o, gives Wout

:

= o

(4.19)

0.64.

The action for a link inside the plane of the Wilson plaquette contains its coupling to its inside links and to

Color Dieleclric M o d e l o f Q C D

71

~'X[~/4 in the action Sl(X), one finds for an inside llnk

outside vacuum links. Including the operator

Sl(Xin, Wout, Win) = # 1 :Xi, 1 4 - 31nx~ n + AXln

/ n '~ ~ :.,e - 4 t - ~ - x l . ~,o., -

,4

(4.20)

3--~Xin Vl/out)

The same saddle point method

OSl (Xln, Wout, Win)

= 0

(4.21)

Xln=Win

~Xin

gives Win = 0.82, a bigger value for the dielectric constant inside the flux tube. The string tension thus receives two contributions. The first contribution comes from the tiling of the Wilson plaquette with U-fields. The second contribution arises from a change of the dielectric field strength on the link inside the flux tube compared to the ones outside of the tube.

~

÷ ~,

2~ +

~'

,

s . n ~,.,./

(4.22)

F r o m Zq. (4.22) follows ~a i = 3.4 or x/~ = 0.62 G e V with a = 0.6 fm when we use the values of the effective action given in Table 2.2 for/3 = 2.35. The ratio of the glueball mass to the square root of the string tension comes out in agreement with the lattice calculations reported in the Introduction. II mG___BB = 3.4

mG__._~ = 3.7 ± 0.3

(Ref. [11]) .

The individual numbers are too large, however, which may be traced to a too small effective scale a = 4a0 = 0.6 fro. Another comment should be made here. Since the strong coupling gauge theory allows an expansion in twodimensional surfaces mapped out by the world sheet of strings, it looks reasonable at large resolution to regard the lattice color dielectric theory as a lattice string theory with additional interactions besides the area term. The plaquette term t a n assume both signs, so the stability of the theory as a function of X is not trivial, s° The additional couplings )~, A~ describe string-str ing interactions.

IV.2 Soliton Model for the Baryon

Originally the color/chromed,electric sol'ton model of the nucleon has attracted most of the attention of nuclear physicists. The review articles of Wilets s and Birse 4 give a good account of the development until 1089. I will mostly concentrate on newer papers after having outlined the principle idea. The modification of the fermion propagator in the color dielectric field leads to the hypothesis t h a t it is the kinetic term in continuum models which gives confinement by making the hopping of the quark field difficult (see Eqs. (3.25), (3.26)). In the lattice language it is the interplay of the effective mass mq = (aM - 4X + 8Xih) and the 1/X from the kinetic term which creates the scalar coupling potential. In a first paper 5s we called this mass mq, not mentioning t h a t it depends on the underlying bare quark mass M and X. The Lagrangian density contained only the X field not the 0 field, which is necessary in SU(3). Similar work has been done by A. W. Thomas et al.. ¢° £ = X~i78~b

-

m~'~

-F 1@~ (8~X):

_

Ve~'(Z) •

(4.23)

72

H.-J. Pirner

(A continuum and Minkowski notation will be used throughout this section.) After rescaling the X-field one obtains the equations of motion

(

a~+

- ¢ r ,~. V , x+~x

~b=E~b ,

o( "7') Vefr-{-

=0

,

(4.24)

In most phenomenologlcal models the vacuum value of X has been assumed X~ac = 0 and the vacuum energy V (Xvac) = 0. The effective potential governing the X-soliton is V(X ) + 3~bmq/x. It assumes a stationary value different from X = 0 when ~ b # 0. The baryon scalar density increases X. This is a mechanism similar to the mechanism in the flux tube where the non-vanishing D-field drives X to larger values. The big and small components of the wavefunctions have been calculated in Ref. [71]. There has been justified criticisms of this approach which makes a finite quark mass responsible for quark confinement. The main point is t h a t one would not be able to get confinement in the limit of mq --4 0. Also in the lattice calculations of Chu et al.,r2 the variation of different-bare quark masses does not give drastic effects on the wavefnnction. I believe t h a t one has to integrate out the colored link fields in the same way as shown for the flux tube to get a meaningful effective long-distance action. This has not been done in Ref. [61] nor in the chapter referred to chiral symmetry breaking (Chapter III.3). In the strong coupling limit/3X4 <~1, the leading term after the link integration has n form r3 which is automatically color neutral, but mixes up the four fermions interaction with the quark kinetic term. =

-a

6

(Tt~) #.r ~bi (= -I- p,)~bj(z + p)

T

X~(z)

(4.25)

It is not clear to me how to model such a term in a continuum theory for quarks. In the weak coupling limit the formulas Eq. (3.24) and (3.25) apply, but it is questionable whether the weak coupling limit of color dielectric theory is relevant for the long-distance physics. Even if Xin inside the hadron is larger t h a n the Xout outside in a vacuum, it does not propose a weak coupling expansions at a length scale a ~ 0.5 fm. The most meaningful solution is an effective meson Lagrangian without quarks, which may be cast in the form proposed by Ellis et al. s~ or in the form of a linear ~-model used by Patkos. e° Once the color glue has gone, it does not seem to be so meaningful to keep the colored quarks. Contrary to this approach one may preserve chiral symmetry and have valence quarks present implementing confinement with a term £eff -- ~ (¢r + i@75~) ~ . (4.26) X This solution has been used by many authors 5s, r4, r5 sometimes with variations in the power of X. (Also the definition of X varies.) It even has been extended to calculations of the strange particles. 7e Very recently a proposition has been made in the context of the SU(3)-color dielectric model to calculate the mean fields (~), (X) and (00) in vacuum and baryonic matter, rr I think it is i m p o r t a n t to understand the vacuum first and then turn the interest towards the nucleon. Confinement comes from the random link gauge fields and the X-field just counts the number of link fields contributing to a color-neutral process (Eq. (4.25)). Since X < 1 it limits the non-locality of the color propagation. In this respect it also confines. It would be interesting to use the color dielectric model to calculate the structure function of the nucleon. Since the model contains gluon degrees of freedom it is natural to have also some gluon m o m e n t u m fraction. A fixed soliton is not a m o m e n t u m eigenstate but can be projected on a m o m e n t u m eigenstate (see the reviews [3] and [4] for reference). The color dielectric structure function has been estimated e.g. in Ref. [78]. Our current knowledge concerning structure functions in 3 + 1 dimensions is so scarce t h a t even a toy model would be welcome to increase our understanding of the nucleon. A rather extensive calculation in 3 + 1 dimensions for the glueball with transverse color dielectric glue fields has been done on the light cone. re This calculation does not derive the effective coupling constants of the dielectric transverse lattice theory. The method uses color dielectric fields ~b~(z) as valence gluons; the Fock space for the glueball wavefunction can therefore be limited to [GG) and IGGGG) states. Before closing this chapter on the nucleon as a soliton, I would like to come back to the analysis of mean fields

for SU(3)¢olor. There we showed that the blockspin ~b~(z)

~.(.) ~.(,)e~e.c')u.(~) =

(4.2z)

Color Dieleclric Model of Q C D

73

has a ray representation which contains a positive definite Hermitian matrix ~ and a bleached color neutral vector field 0~, end the SU(3) fields U~(z). In a system with net bnryonic charge like the baryon or nuclear matter I would expect that O~(z) plays an equally important role to the trace of ~ tr ~ = X2. In a test calculation s° we used the model described in III.2 to study both mean fields X and 0o. As can be seen on Figs. 12 and 13 the X-field is correlated with the scalar density of the baryon, vanishes near the bag surface approximately as X=X~

1-

i

(4.28)

i

i

15

Ouark wave - function

I0

0.5 UG 312

~1"1Qf f

×

v a 312

0.25

0

0.5

1.0

1.5

r(fm)

Fig. 1~: Upper and lower component of the quark wave function in the b~r~/on from Ref. [80].

The &field, however, peaks near the surface. It is anti-correlated with the density and behaves roughly as

80 (1-0e

/429)

This comes about from the self-coupling terms X40~.8~v in the effective Lagrangian (Eq. (3.12)). For small quark masses the gradient of 00(v) becomes so large that the mean field equations become unstable. We regard this as an indication that pair production is possible in a vacuum and one must go over to a new vacuum taking the (qq) condensate into account. In a recent paper (Ref. [77]) the model was extended to include the chiral mass term ~ (~ + ~ s ) ~ and applied to nuclear matter. Again it shows the importance of the 0o field at large baryon density. The X-field is rather weakly varying whereas the 8o field has a much greater variation and contributes a repulsive energy to baryonic matter. It would be interesting to trace the behavior of these two fields in lattice calculations.

IV.3 Nucleon-Nucleon Interactions There has been a rather successful series of calculations of the short-range NN-interaction sl,s2 in the nonrelativistic quark model. A combination of color magnetic short-distance gluon exchange and orbital excitations

H.-J. Pirner

74 0.5

,

,

.

,

X

1.5

i O

0.25

0.5

Re\ 0

0.5

r(fm)

].0

o 15

Fig. 13: Color dielectric field 0 and bleached gluon field 8o in tl~e mean field approzimation. of the six-quark system into the [4-2] symmetric state can explain a modest repulsion between nucleons. The physics at intermediate distances is more murky and weak attractive forces have been found again with nonrelativistic quarks in a string flip picture based on U(1). ss For relativistic quarks bound in solitons calculations are rarer, since the technical effort is large and the nucleon solution is not such a safe starting point. There has been an effort s4 to calculate the N N - p o t e n t i a l on the basis of the Friedberg-Lee model as. Let me shortly resume the method. In the framework of the Hill-Wheeler equation the energy and norm kernel of the two clusters at separation distances (R, R r) are needed:

[H(R, R') - E N ( R , R')] ¢ ( R ' ) = 0 .

(4.30)

These kernels come from the Hsmiltonian including the scalar field and the six-quarks. The classical scalar field solution is converted into a coherent state wavefunction, which allows one to take overlaps between field configurations characterized by different separation distances of the two nucleons. As expected the adiabatic potential H(R, R ~) = V(R) is attractive from R = 0 to R ~ 2RN = 2 fm where 2RN is the soliton radius. Its value at R : 0 can be estimated from the six-quark solution. W i t h o u t residual gluon interaction it amounts to 350 MeV in this model. Taking into account the off-diagonal piece in the Hill-Wheeler equation we could find indications of s repulsive effect, which comes from the unwillingness of the two-nucleons to rearrange their scalar fields during the collision. The result is shown in Fig. 14. The T~ibingen group ss has experimented with a non-relativistic reduction of the relativistic color dielectric model including gluon fields but without the dielectric antiscreening. They find b o t h a short-range repulsion and an intermediate attraction. There are two more noteworthy six-quark calculations in the color dielectric framework. The first includes gluon interactions modified by the dielectric background field, sT. It divides the energy up into five parts. The mass of the object is given as M = ( H Q / - t - / H x t ) + / H M ) ÷ /H×M) -t- Eem

(4.31)

75

Color Dielectric M o d e l o f Q C D I

0

I

i

I

.

/

-100

>= -200

I

-

-300

/ /

i/yaa( ) -400

i

I

I

0

1

Fig. 1J:

1

2 x [fm]

l

i

3

4

NN-interaction in the model of Ref. [Sj]. TABLE 3.1

All masses for the nucleon (N) and dibaryon

(DB)

M

EM

E×

N

946

-110

373

654

-191

DB

1985

+25

1035

1102

-t27

93

+245

289

-206

D B - 2N

EQ

in MeV. Ecru

255

with

(4.3U) H×u = -2//u

The authors of Refs. [87,88] choose for the dielectric constant e = ~r~ = X4 in our notation. They call 2 c a = Xthelr, because they use a kinetic term } ((9~Xtheir) G l" This choice seems to be in accord with the lattice model and indeed they see t h a t phenomenology necessitates it. With the less attractive choice e = o'~ actually proposed by Nielsen and Patkos ss the authors of Ref. [89] found t h a t the model needed a low glueball mass mQB < 500 MeV. For the mass term Ref. [88] chooses a form rn/~Q which would not correspond to the weak coupling expansion. As discussed before, this term is difficult to derive in a large-distance strong coupling expansion. The calculation in the above model sr for the six-quark system with X and A~ interactions gives a slightly larger energy for the dibaryon t h a n for two isolated nucleons. In the same approach the binding energy of the H-dibaryon is estimated to range from 100 MeV to 210 MeV.

76

H.-J. P i m e r

The other alternative9° is to use the chixal mass term (b(o" + i~/5~) ~b/X and leave out the gluons. This approximation gives a slightly bound dibaryon. The mass of the three-quark system is 1300 M e V and the mass of the dibaryon is 2386 MeV. Let m e conclude this chapter with a speculative interpretation of the hadronic interactions generated by the string theory calculated on the lattice [Chapter II]. If we take X~(Z) as an interpolating field for a string bit without quarks at the end points then the positive A (~ tr ¢t¢) 2 term signals a repulsion at zero-distance between string bits. The next nearest neighbor term is given by the angle terms MX~X~ with a negative M. This interaction is attractive. W h e n one goes to larger distances then only glueball exchange is possible which fallsoff exponentially with the glueball mass. The resulting potential looks very similar to a typical molecular potential we expect for the inteaction of color neutral hadrons. Furthermore the strength of the M-term is approximately A'X4 Vattr ~ - with ~ ' = - 1 . 1 , a = 0 . 6 f m , X2 - - 0 . 4 1 - 6 0 MeV The smallness of this number is a consequence of the x4-dependence. The string tension has a In X4 dependence and is therefore larger v ~ ~ 600 MeV.

IV.4 Nuclear Matter and Quark Matter Because of the problem of the separation of the center of mass motion one may hope it is easier to treat bulk nuclear matter, t h a n the two-nucleon system. For this problem the color/chromo-dielectric model can be solved in two approximations. Either one takes plane quark waves i.e. quark matter or one tries to fabricate a lattice of locally color neutral nucleons. In the first case color neutrality may manifest itself in triple quark correlations and must play an i m p o r t a n t role at low density. In the second case the kinetic energy of nucleons is neglected and the calculation becomes unrealistic at intermediate densities. I will discuss these two methods briefly. Take the SU(3)color-model with bleached gluon field and a non-vanishing chiral symmetry violating quark mass meet as an example (see Eqs. (3.12), (3.24)). The Hamiltonian in this case is:

÷

/

(4.33) -

-

°

with v , ~ = 4 ( , ~ x ~ - ~ x ~ - ~ , x ~)

(4.34)

In Eel. [80] we choose an effective quark mass meet ~ 60 MeV which corresponds to a strange quark inside the bag with mass m.eet/x(O) ~ 140 MeV. The baryon solution has an energy of EB : 1750 MeV or a mass of 1650 MeV after cm correction. Now consider a system of 3A quarks of mass meet, in a large box of volume ft. Due to translational invariance, the classical X and 00 fields are uniform. The quark wave function solutions of the Dirac equation are plane waves. The energy of a baryon state with m o m e n t u m k is ,~ = [(m:et/X 2 + k2) 1/~ + 00]

,

(4.35)

and the total energy of the system writes

E = tl" [ f dSk

O'k ~ ~ - k),~ + [v, et(x) - 187X30o2]I

(4.36)

where the statistical factor ~1= 6 is the number of quark states (spin-color) with m o m e n t u m k, and Vdr(X) is the effective X potential given above. The quark vector density is related to the quark Fermi m o m e n t u m k! by J

,.,~.()

Color Dielectric Model of QCD 77 1 q while the baryon density is PB = iPv" Here u(k, s) is a normalized Dirnc Spinor for a state with momentum k. The source for the X-fieldis the scalar density p~ given by

dak

(4.38)

where the function F is

F(y) = T3~ ( y x / l + y ' - l n ( ~ +

1V/i--~))

(4.39)

The classical X- and 00-fields are obtained from the equation of motion

9"rx%= d~fr(X) dx

p~ ,

(4.40a)

27 yX20g : rrL.~ . 2 --~p~

.

(4.40b)

The total energy per baryon is thus

=

~

+0o + k--~-" F Vcfr(X)-~'YX oJ

,

(4.41)

with the function G given by

(4.42)

a(y) = T ~

It is very easy to show that this model automatically gives saturation even in the absence of the repulsive contribution of the 8o field. In order to illustratethis mechanism, let us consider a simplified version of the model where we ignore the vector field and keep only the X 2 term in the potential Vefr(X). W e see from the equation of motion that X goes like pl/a for not too large X. As a consequence there is a balance between the X-potential term and the quark kinetic-energy term. More precisely,the X-potential term and the related mass term behaves like p-t/3 and dominates at low density, whereas the quark kinetic energy, going like pl/S, becomes the main contribution at high density. In this simplified version, the energy and Fermi m o m e n t u m at saturation are proportional to the following quantity q = (4pmefr)I/s

(4.43)

For instance, the equation determining X is

~1/2 F(a)l/s -

kF {~ 2 '~ 1/e q \~-~j

,

(4.44)

with

1/3 x = c~ \ 4~ /

(4.45) kF

The corresponding energy per baryon is

E/A = 3 k--ff-F q (-~

+1~ - ~

,

(4.46)

the variable c~ being determined from (4.44). Numerically we find that saturation is obtained for a Fermi m o m e n t u m ki~ ~ 1.8 q. The energy per nucleon is E / A = 6.32 q and the value of X is Xqm : 0.57(rrt~fr//~) t/a.

We obtain = 1487MeV , (o)

= 2.04 , (o)

where po is the density of ordinary nuclear matter, po ~ 0.15 N f m -s.

(X)(o) = 0.23 ,

(4.47)

78

H.-J. Pirncr

The introduction of X4 corrections and the $U(3) bleached gluon field 80 does not destroy the saturation mechanism. However, the results depend on the strength of the 8-field and also on the value of the parameter q' (or in other words on the mass term) which is not yet really constrained by phenomenology. The first effect of the #-field is to bring more repulsion than the simplified model discussed above. On the other hand, it generates a Xs term which lowers the effective potential Veff(X) at large X. As a consequence "y cannot be too large in order to keep Veff(X) positive for X between zero and one. The exact calculation taking into account these effects, as well as the repulsion from the X4 term in the effective soliton potential gives the following results : 1460MeV ,

---- 1.7 ,

(X)(1) ----0.37 .

(4.48)

(1)

(0)

Notice that the energy is much smaller than the free baryon energy in this model. The corresponding saturation curve is indicated in Fig. 15 for this set of parameters (solid line), while the density dependence of the X and #0 fields are represented by dashed and dot-dashed lines, respectively.

2000

~

i

,

i

1

Quark matter set(l)

X,O E/A (MeV)

\

~

~

0.5

f

//X 1000

I 1

[ 2

l 3

P/Po Fig. 15: Solid line: equation of state for uni/orm quark matter with parameter set (1) and a quark mass of 60 Me V. The density dependence of the colordielectric fields for the same set of parameters is represented by a dashed line for the X-field and dot-dashed line for the Oo field.

W h a t is surprising about the above calculation is the self-screening of the 80-field. Due to the xS-dependence of the 8-mass the 80 field has almost no density dependence. Referring to our discussion before in the chapter on chiral symmetry breaking, I think that a possible dependence of the mass of 8~ on X 2 (Eq. (3.46)) may be more adequate for quark matter, too. In a chiral version of this Hamiltonian the authors of Ref. [77] include the energy of the negative Dirac sea. The theory then becomes unstable with respect to variations in X for vanishing baryon density

EDirac ,ea = - - ~ -

p2dp

re, eft

+ X~

(4.49)

Apparently, a Xvac : 0 would give an infinite condensation energy. This does not make sense, and a suitable renormalization procedure must be set up. 77 Very probably also after this procedure Xvac will be finite as already determined from the gluon dynamics.

Color Dielectric Model o f Q C D

79

In the soliton crystal the status of research is still similar to the literature given in the reviews, s'4 Within the Wigner-Seitz approximation a single soliton embedded in nuclear matter can be solved. The size of the Wigner-Seitz cell radius r0 is given by the nuclear density 1 P-

and equals ~ 1.2 fm at normal nuclear matter density. The energy of a quark which experiences a square well x-distribution with X = 0 outside of the free nucleon radius Rlv is E~e% Embedding the nucleon in a medium with Xmed outside gives a change in energy

Eq = Ef~,,(1 meffRNX)

(4.50)

due to the exponential tail of the quark wavefunction extending into the medium. At the same time the energy in the space between the nucleons has to be raised from Xvac = 0 to X ¢ 0 by a E = ~ lrn2~n er2 vX 2 (r0s - R ~ v ) ' Y 41r •

(4.51)

Minimizing the sum of E q and A E gives the lowering of quark energy due to deconfinement in the nuclear medium. In Ref. [55] this was the original idea of the color dielectric model with quarks which was constructed after the models of Nielsen and Patkos. The covalent binding of nucleons was thought to be related to the softening of the quark structure functions in the nucleus, the EMC effect. A follow-up on this work tried to investigate the density dependence more carefully including gluonic energies and differences in up and down quark masses. ~1 Any improvement puts this simple model at the edge of credibility. The treatment of gluonic interactions is obstructed by the separation of the nucleons into far-apart cells, and not allowing them to move and overlap. With the help of a clever choice of localized orbitals one may simulate however the nucleon-nucleon interaction more properly. °2 Without gluon interactions each energy state could have 12 quarks. In the presence of gluonic interactions it is advantageous to have cells which are color neutral with spin-l/2 like nucleons. Similar to the [6] and [4,2] symmetry combinations in the NN-problem one may categorize the band states according to their boundary conditions. This work is in progress and it will be interesting to see how the weak coupling gluon expansion can handle the problem of color neutrality. There is also some nice work ~s on finite baryon density in strong coupling approximation which may be used together with the effective color dielectric theory. Let me summarize the discussion on baryonic matter. The color dielectric model allows quarks in different nucleons to talk to each other. The importance of this effect relative to the conventional meson exchange is of very much interest. A naive estimate would limit the quark displacement from a nucleon center to a distance rh ~ 0.6 fm where the potential increase A E = i¢- rk (~ = 1 GeV/fm) equals twice the constituent mass. So at NN-separation distance l < 1.2 fin I would expect that the covalent binding via quark exchange plays an important role. For large separations only color neutral (qq) exchanges are relevant. Unfortunately, the lattice formulation of the finite density system is not at all clear. So we can get little guidance from QCD simulations.

IV.5

The Gluon Plasma Transition

The physics of heavy ion collisions and of the early universe has become an exciting subject in hadron physics. Experiments in the laboratory with large nuclei offer a chance to study hadron thermodynamics. This subject has been intensively studied in computer simulations on pure giuon theories and more recently with dynamical fermions. The relevant order parameters in the pure gluon theory is the energy of a free isolated massive quark Vq. This energy can be related to the expectation value of a product of time-like links on the lattice. At finite temperature k T all the Bose fields have to obey periodic boundary conditions. Let ~ • a~ -- h-1 be the extension of the lattice in time-like direction then (~, ~.ra.) ----~ (~, O) .

(4.52)

80

H.-J. Pirner

The order parameter for pure gluon theory is the Polyakov loop



(4.53)

(w(~)> ~< e - ~ , / ~ If (W) = 0 we get the energy of the finite quark equal to infinity, i.e. we have confinement. In the color dielectric theory it will be useful to keep the variable W(~) and X0(~) as extra degrees of freedom separately from X~(~) and U~(~). One can show that after integrating out the spatial links U~(~) there remains an effective action £ (W(~), Xg(~), X~(~) = ~ (~)) (4.54) with local coupling of the nearest neighbor Polyakov links and an interaction term between the spatial and time-like dielectric functions 94 (c > O) .

(4.55)

It is this term which at the highest temperature k T ~ 1/ae~ of the effective theory makes both Xo -2 and o'(~) larger at finite temperature. The antiscreening is decreased because of the thermal disorder of the system. In this process Xo2 becomes larger than ~. The relative increases of (r and Xo2 in the thermal system at maximum temperature are: (or)T='" ~ 1.06

(~)r=o

and

(X~)a'..= = 1.20 .

(Xo~)~:o

W e see that again we have a moderate increase of the dielectricfieldin the hot system like inside the flux tube.

V.

CONCLUSIONS

It m a y be worthwhile to put the different topics in perspective at the end of this article. Again, this is m y rather personal view. I think the relation of a color dielectric effective action to Q C D has been established for SU(2) pure glue theory. The lattice generates a dielectric theory in the strong coupling domain which has the advantage that some of the problems of normal strong coupling-theory are not appearing. Since the new dielectricfield is a good interpolating fieldfor the glueball, its mass is lower than in a naive strong coupling approach. Technically the transition from Q C D in the scaling region at very small lattice distances to the effective action can be still improved by more analytical work. In color SU(3) efforts are still to be made to explore the importance of the averaged gluon fields X2 and 0~. Especially it would be interesting to measure their correlation with the baryon density in the nucleon. Attempts to include fermions in the color dielectric picture are not totally satisfactory. The main point is to understand the condensation of quarks ( ~ ) in vacuum and at the same time the role played by X 2 in the baryon. M y impression is that (X) will be finitein the vacuum and the effective Lagrangian at large distances will be bosonic including x-terms. Then the nucleon would appear as a Skyrmion modified by the dielectric field. But one m a y at a shorter length scale arrive at a picture with valence quarks, coupled to chiral and color dielectric fields. This possibility stillhas to be demonstrated theoretically. A further open line of research concerns the role of the blocked gluon fields in the transverse dynamics of hadrons or the light-cone. It may be advantageous to use only a few transverse color dielectric modes but treat the longitudinal-time-like behavior of the valence particles constituents with a high resolution. The light-cone approach is quite promising to produce a string-like theory allowing high momentum longitudinal excitations. At different places we have emphasized the similarity of our effective action to a string picture of strong coupling QCD. The linear potential arises naturally in this approximation as the Wilson plaquette expectation obeys the area law of the minimal surface formed with lattice plaquettes. The couplings #~, A, ~ and the

Color Dielectric Model of QCD

81

resulting terms in (j3X[~)'L after the integration over the link fields may even induce string-like x-structure into the vacuum. This aspect may merit more investigations. For practitioners of the color dielectric phenomenological model the strong coupling approximation to QCD induced by the color dielectric field presents an interesting alternative to study hadronic dynamics compared to solitonic approaches. It would be interesting to check how much rotational invariance is violated in the lattice model by considering the various lattice 2++-gluebal] states. In the field of heavy ion physics the situation occurs that many strings overlap. Do they form color ropes in higher representations, or do they stay independent? This' question may be answered in the effective theory. "If you have one key, you can open a certain lock, but not all locks." Let me close with this simple comment. Hopefully this article has been a key to unlock some problems concerning gluon aspects of low-energy QCD. ACKNOWLEDGEMENTS I would like to thank John Negele and his colleagues for the kind hospitality they provided for me at the Center for Theoretical Physics, MIT. I would also like to thank Roger Gflson for his patience in typing the manuscript. REFERENCES 1. Lee, T. D. (1981) Particle Physics and Introduction to Field Theory (Harwood Academic Publishers, New York, 1981), chapters 16 and 17. 2. Adler, S. L. (1984) Rev. Mod. Phys. 56 1-40. 3. Wflets, L. (1988) Non-Topological Solitons (World Scientific, Singapore, 1988). 4. Birse, M. C. (1990) "Soliton Models for Nuclear Physics," in Progress in Particle and Nuclear Physics, 25 1, A. Feessler, ed. 5. Banerjee, M. K. "Nucleon in Nuclear Matter," University of Maryland preprint PP~91-283, and article in this volume. 6. Shuryak, E. V. (1988) The QCD Vacuum, Hadrons and the Superdense Matter (World Scientific, Singapore, 1988), chapters 1, 2 and 8. 7. Creutz, M. (1983) Quarks, Gluons and Lattices (Cambridge University Press, Cambridge, UK, 1983). 8. Bhaduri, K. (1988) Models of the Nucleon (Addison Wesley, Menlo Park, CA, 1988). 9. Savvidi, G. K. (1977) Phys. Left. BT1 133. 10. Nielsen, N. K. and and Olesen, P. (1978) Nuel. Phys. B144 376. 11. Michael, C. and Perantonis, S. (1991) Nucl. Phys. B (Proc. Suppl.) 20 177-180. 12. Biter, K. M. et el. (1991) N~cl. Phys. B (Proc. Suppl.) 20 390-393. 13. Particle Data Group, Aguilar-Benitez, M. (1990) et el., Phys. Left. B 239.

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