Dyson-Schwinger Equations in QCD

Craig D. Rob erts

Physics Division, 203, Argonne National Lab oratory,

Argonne, Illinois 60439-4843, USA

Abstract. These three lectures describ e the nonp erturbative derivation of Dyson-

Schwinger equations in quantum eld theory, their renormalisati on and, using the

exp edient of a simple one-parameter mo del, their application to the calculation of

hadronic observables and the prop erties of QCD at nite temp erature.

1 The Dyson-Schwinger Equations

The Dyson-Schwinger equations [DSEs] provide a nonp erturbative, Poincar e in-

variant approach to solving quantum eld theory, in which the fundamental

elements are the Schwinger functions. The Schwinger functions are moments of

the measure. For example, in a Euclidean quantum eld theory describing a self-

interacting scalar eld, x, sp eci ed by a measure d[], which will involve the

classical Euclidean action for the theory and, p erhaps, gauge xing terms, etc.,

then

hx :::x i d[] x :::x ; 1

1 n 1 n

where d[] represents a functional integral, is an n-p ointSchwinger function.

In a manner analogous to that when a probability measure is involved, a quantum

eld theory is completely sp eci ed if all of the moments of its measure are known.

The fo cus of lattice eld theory is a numerical estimation of these Schwinger

functions. The DSEs provide a continuum framework for their calculation.

The DSEs are an in nite tower of coupled equations, with the equation for

a given n-p oint function involving at least one m>n-p oint function. A tractable

problem is only obtained if one truncates the system. Truncations that preserve

the global symmetries of a theory; for example, chiral symmetry in QCD, are

relatively simple to e ect [Bender, et al. 1996]. It is more dicult to preserve

lo cal gauge symmetries, although much progress in this direction has b een made

in Ab elian gauge theories Bashir and Pennington, 1994.

One systematic means of truncating the system is a weak coupling expansion.

In this way one readily nds that the DSEs contain p erturbation theory in the

sense that, for a given theory, the weak coupling expansion of the equations

generates all of the diagrams obtainable in p erturbation theory. In this way,

at the very least, the DSEs can b e used as a generating-to ol for p erturbation

An intro duction to the functional integral formulation of quantum eld theory can

b e found in Itzykson and Zub er 1980 and Rivers 1987.

Dyson-Schwinger Equations in QCD 213

Σ D = γ

S Γ

Fig. 1. A diagrammatic representation of the DSE for a fermion self-energy, p

2 2

i p[Ap 1] + B p : S p=1=[i p + p] is the connected fermion 2-p oint

function; D k is the connected gauge b oson 2-p oint function; and p; k is the

1-particle irreducible fermion{gauge-b oson 3-p oint function. Both D k and p; k

satisfy their own DSEs and this illustrates the coupling of the equation for a given

n-p oint function to those involving m>n-p oint functions.

theory. This can also b e used as a constraint on alternative truncation schemes;

i.e., they must b e such as to preserve the feature that p erturbative results are

recovered in the weak coupling limit.

There are many familiar examples of DSEs. For example, the gap equation

that describ es Co op er pairing in ordinary sup erconductivity is simply a trun-

cated DSE for a 2-p oint electron Schwinger function; Bethe-Salp eter equations

[BSEs], which describ e relativistic two-b o dy b ound states, are DSEs for 4-p oint

functions; and covariantFadde'ev equations, which describ e relativistic three-

b o dy b ound states, are DSEs for 6-p oint functions.

The statement that a theory is solved if all of its Schwinger functions are

known can b e lo osely re-expressed as the statement that all observable S -matrix

amplitudes can b e expressed in terms of the Schwinger functions of the elemen-

tary elds in the theory. This entails that one can connect observables to the

fundamental parameters of the theory via these Schwinger functions.

The 2-p oint functions in a given theory contain imp ortant information. For

example: in a gauge theory, the form and analytic prop erties of the gauge-b oson

2-p oint function can provide information ab out whether, due to interactions,

the gauge b oson acquires a gauge invariant mass that screens the interaction

Schwinger-mass generation or a strong enhancement at small momenta that

can b e a signal of con nement. Either of these prop erties will in uence the

propagation characteristics of other mo des in the theory and hence physical

observables. In all gauge theories the fermions act as a source of the gauge eld

and their propagation characteristics are strongly a ected by their interaction

with their self-generated gauge eld. This is describ ed by the DSE for the fermion

2-p oint function, illustrated in Fig. 1, in which the gauge-b oson 2-p oint function

app ears as a driving term.

For example, in QCD, whether the gauge-b oson 2-p oint function is nite

or strongly enhanced at small momenta determines whether chiral symmetry is

214 Craig D. Rob erts

dynamically broken and/or whether quarks are con ned.

Euclidean Metric. Herein I employ a Euclidean metric formulation of eld

theory; i.e., a non-negative metric for real vectors:

a b a b a b 2

i i

i=1

where is the Kronecker-delta. In this case, Q is a spacelikevector if Q > 0.

My Dirac matrices satisfy

[ ] = ; f ; g =2

and . One realisation of this algebra is provided by

5 4 1 2 3

E 0 E j

; i ;j=1;2;3 4

4 i

where are the usual contravariant Dirac matrices in Minkowski space.

I adopt the p oint of view that the Euclidean formulation is primary; i.e.,

that a eld theory should b e de ned in Euclidean space, which is the p ersp ective

employed in constructive eld theory and, usually as a pragmatic arti ce, in the

lattice formulation and numerical simulation of eld theories. The Schwinger

functions can then b e calculated and the question of the existence of the Wight-

man functions, and hence the Minkowski space propagators, addressed subse-

quently. [A fuller discussion of these p oints can b e found in Sec. 2.3 of Rob erts

and Williams 1994.]

This is imp ortant b ecause the analytic structure of a nonp erturbatively

dressed Schwinger function is not necessarily the same as that of its p erturba-

tive seed. Given this, one cannot know a priori the singularities in the integrand

of the integral equation. Hence the true consequences of rotating the momen-

tum space integration contour, as one do es in a Wick rotation, are unclear;

i.e., the correct form of the \Wick rotated" equation mayinvolve contributions

from p oles, branch cuts, etc., that cannot b e anticipated based on the p ertur-

bative form of the Schwinger functions involved. To elucidate this, the following

Euclidean , Minkowski Transcription Rules are valid at each order in p erturba-

tion theory:

Momentum Space Con guration Space

Z Z Z Z

M E

4 M 4 E 4 M 4 E

d k ! i d k 1. d x !i dx 1.

E E

2. =k !i k 2. =@ ! i @

E E

3. k q !k q

3. A= !i A

E E E E

4. k x !k x , 4. A B !A B

by which I mean that the correct Minkowski space integral for a given diagram

in p erturbation theory is obtained by applying these transcription rules to the

Euclidean integral. However, for skeleton diagrams; i.e., those in which each line

and vertex represents a fully dressed n-p oint function, this cannot b e guaranteed.

Dyson-Schwinger Equations in QCD 215

1.1 Quantum Electro dynamics in d-Dimensions, QED

I will use QED to illustrate the nonp erturbative derivation and form of the

DSEs. The generating functional or partition function for QED is

E E E E

Z [ ; ;J ]= 5

Z Z

E E E E d fE fE fE fE E E

d ; ;A exp ; d x + + A J

E E E E fE

where there is an implicit normalisation Z [ =0; =0;J =0]=1; ,

fE E

, J are auxiliary source elds;

E E E E

d ; ;A 6

Y Y Y

fE fE E E E E

D x D x DA x exp S [ ; ;A ]

indicates a functional integration and sp eci es the measure; and the action is

1 1

E E E E E E E E d E

F F @ + A 7 S [ ; ;A ]= d x

4 2

f f

fE E E E E fE

+ @ + m + ie A

0 0

f =1

E E E E E

with F = @ A @ A the eld strength tensor and f lab elling the fermion

\ avour".

This generating functional contains all the information ab out the eld theory;

the b ound state sp ectrum, reaction rates, etc. - one must only enquire of it in

the appropriate manner.

The nonp erturbative derivation of the DSEs follows from the simple obser-

vation that the integral of a total derivative is zero, assuming appropriate and

sensible b oundary conditions; e.g., that the elds vanish on the b oundary of

the compacti cation of Euclidean space used in rigorously de ning Eq. 5. For

example [the sup erscript E is implicit hereafter],

d f f f f

d x + +A J

[ ]

e : 8 ;;A J x 0= d

A x

This leads to the following functional di erential equation for the generating

functional

0= ; ; J x Z [ ; ; J ] ; 9

A x J

216 Craig D. Rob erts

where the functional di erential op erator is obtained from

;;A = 10

A x

@ @ + 1 @ @ A x+ e xi x

by making the obvious replacements of the arguments.

The generating functional for connected Schwinger functions is de ned via

Z [ ; ; J ] exp G [ ; ; J ] : 11

n-p ointSchwinger functions obtained as functional derivatives of G [ ; ; J ] are

connected in the sense that, diagrammatically, each of the n-p oints is connected

byanunbroken line to each of the others. Equation 9 yields

1 G

J x= @ @ + 1 : 12 @ @

J x

The generating functional for 1-particle irreducible Schwinger functions is

intro duced via the Legendre transformation

h i

d f f f

G [ ; ; J ] [;;A ]+ d x + +A J 13

G G G

A x ; x ; x ;

where

J x x

J x= x= ; x= ; :

and hence

A x x

; ;A ] are 1- The n-p oint functions obtained as functional derivatives of [

particle irreducible in the sense that, diagrammatically, the n-p oints remain

connected if one internal line is cut.

All of the quantities needed in the nonp erturbative derivation of DSEs have

now b een intro duced. To illustrate this I will present the derivation of the DSE

for the photon vacuum p olarisation, k .

DSE for the Photon Vacuum Polarisation. Consider the connected, 2-p oint

fermion Schwinger function

f f

x x; y ; [A ] h S yi = 14

r rs s

f f

y x

s r

=0=

and observe that

2 2

15 d z

f h

x z

y z

h f

z x

t r

d fg d

d z = x y ;

s t

Dyson-Schwinger Equations in QCD 217

f 2 g

from which one identi es the inverse of S x; y ; [A ] as = x y.

rs s t

Setting the external fermion sources to zero; i.e., =0= , one has

j =0 =j , b ecause the measure is even under ! and

=0= =0=

!. This corresp onds to the statement that the vacuum exp ectation value

of the fermion elds is zero. In this case Eq. 9 b ecomes

= @ @ + 1 @ @ A x; J 16

A x; J

e tr i S x; x; [A x; J ] :

Acting now with =A y and setting J x = 0, whichentails A x=0,

yields

d 1

@ @ x y x; y 17 D x; y = @ @ + 1

where

D x; y 18

A x A y

A =0= =

is the inverse of the connected 2-p oint photon Schwinger function and the photon

vacuum p olarisation is the 1-particle irreducible 2-p oint function

x; y e tr i S x; x; [A ] 19

A y

d d f f f

= e d z d z tr S x; z y; z ;z S z ;x 20

1 2 1 1 2 2

where I have de ned

; 21 ie y; z ;z

1 2

A y

z z

2 1

J =0= =

which is the 1-particle irreducible 3-p ointSchwinger function describing the

fermion-photon vertex.

Equation 20 is the one sought. It describ es the mo di cation of the photon

propagator by fermion lo op insertions. While its p erturbative form and content

are well know, the derivation presented here is nonp erturbative. At no p ointwas

it necessary to employ a notion of weak-coupling in order to justify a step. This

derivation serves as an archetypal example. With Eq. 20 relating the 1-particle

irreducible photon 2-p oint function to the connected fermion 2-p oint function

and the 1-particle irreducible fermion-photon 3-p oint function, one has another

illustration of a p oint made in the intro duction; i.e., the DSE for a given n-p oint

function involves at least one m>n-p oint function.

218 Craig D. Rob erts

S Π iΓ i γ a) =

S b) = + D D

D 0 0 Π D

Fig. 2. A diagrammatical representation of the DSEs for the photon vacuum p olarisa-

tion and connected 2-p ointSchwinger function [Euclidean propagator]; Eqs. 18 and

20.

The DSE for the 1-particle irreducible fermion 2-p oint function [fermion self-

energy] is illustrated in Fig. 1. Apart from itself, via the connected fermion 2-

p oint function, this DSE involves the connected photon 2-p oint function and the

1-particle irreducible fermion-photon 3-p oint function, . One observes that

the DSEs for the vacuum p olarisation and the fermion self-energy would form a

closed coupled pair if were known.

satis es its own DSE, illustrated in Fig. 3. The driving term for this equa-

tion is the Bethe-Salp eter kernel for fermion-antifermion scattering, K , which

involves a countable in nityofskeleton diagrams and cannot b e written in a

closed form. This is the p oint where it b ecomes clear that the tower of DSEs

must b e truncated in order to formulate a tractable problem.

There are twoways to pro ceed. One can develop a systematic truncation

pro cedure for K , one which preserves whatever symmetries are deemed to b e

imp ortant, and simultaneously solve the system of coupled equations for ,

and . This is a computationally intensive path but less so than numerical sim-

ulations of the lattice formulation of QED and it is feasible with contemp orary

computing resources. Hitherto this approach has not b een explored. A simpler

alternativeistodevelop an Ansatz for ,or K, one that preserves as many

of the symmetries of QED as p ossible, and explore variations of this Ansatz to

discover those features and results that are robust. This technique has b een used

extensively and ecaciously to the p oint where it is now p ossible to obtain gauge

invariant results in studies of dynamical chiral symmetry breaking D SB in

strong-coupling QED [Bashir and Pennington 1994]. This illustration of the

results that are p ossible suggests that the e ort required to follow the rst path

maynow b e justi able. An indication of how one might pro ceed systematically

to construct the Bethe-Salp eter kernel, K ,isgiven by Bender et al. 1996.

Renormalisation. Nonp erturbative renormalisation of the DSEs is straight-

forward. One intro duces the renormalised elds and coupling via the usual de -

Dyson-Schwinger Equations in QCD 219

i S iΓ i γ iΓ = + K

i S

= + K + K K +

= + M

Fig. 3. DSE for the 1-particle irreducible fermion-photon 3-p oint function.

nitions

p p

f f

1 f

f 0

Z ;A Z A ;e e ; 22

2 3

0 0

Z Z

where the subscript \0" denotes \bare", to obtain the renormalised action for

QED

1 Z

S [;;A ]= d x Z F F + @ A 23

4 2

h i

f f f

f f f f

+ Z @ + m + iZ e A :

2 0 1

f =1

Gauge invariance of the renormalised action entails the Ward identity Z =

Z and the result that the longitudinal piece of the photon propagator is not

mo di ed byinteractions:

k k =0; 24

and hence that the gauge parameter is multiplicatively renormalised by the con-

stant Z ; i.e., = Z .

3 0 3

The renormalised DSEs are obtained by applying the pro cedure outlined

ab ove to the generating functional constructed using the renormalised action.

For example, the renormalised photon vacuum p olarisation is

2 2

k = k k k k 25

where

2 0 2 0

k = k 0 26

with the regularised vacuum p olarisation satisfying

0 2 0 2

k k k k k 27

i h

d q

1 1

f f f f

k q; k S q k tr S q + e = Z

2 2

2 f

220 Craig D. Rob erts

Π a) 1 = + 2

1 1 + 6 + + 2

b) = + D

D 0 D0 Π D

Fig. 4. DSE for the gluon vacuum p olarisatio n and connected gluon 2-p oint function:

solid line - quark; spring - gluon; dotted-line - ghost.

where each term on the right-hand-side is renormalised and the DSEs are solved

with the on-shell renormalisation b oundary conditions

f 0 2

Z 2 A p

2 f 2

p =m

f f

f f 0 2

m Z m +B p 28

2 f 2

2 0

p =m

0 2

Z 1 p ;

p =0

f 0 f 0

where A and B are the regularised vector and scalar fermion self-energies,

which arise in renormalising Fig. 1.

1.2 Quantum Chromo dynamics

The DSEs for QCD are obtained using the metho ds describ ed ab ove but starting

with a generating functional whose measure is provided by the QCD action. The

form of the equation for the 1-particle irreducible quark 2-p oint function is very

close to that of the analogous equation for the electron:

a a 4

d `

S `D p ` `; p ; 29 p=g

2 2 2

where here D is the connected gluon 2-p oint function, is the 1-particle

irreducible quark-gluon 3-p oint function and f =2g are the generators of

a=1

SU 3-colour.

The QCD analogue of Eq. 20 is, however, very di erent. It is illustrated in

Fig. 4 in a general covariant gauge. The complications arise b ecause QCD is a

non-Ab elian gauge theory and hence admits gluon self-couplings. The rst dia-

gram in Fig. 4 is the direct analogue of the electron-lo op insertion in the photon

vacuum p olarisation. The remaining terms are particular to QCD, indicating the

Dyson-Schwinger Equations in QCD 221

e ects of: 1-particle irreducible gluon 3- and 4-p oint functions; the 1-particle irre-

ducible ghost-gluon 3-p oint function; and the connected ghost 2-p oint function.

These additional contributions provide for asymptotic freedom and con nement,

a fact discussed in more detail in Sec. 2.

Renormalised Quark Dyson-Schwinger Equation. Just as in QED , the

renormalised DSEs in QCD can b e obtained in a straightforward manner from

the generating functional constructed from the renormalised action. Of course,

there are more elds and hence more renormalisation constants and more iden-

tities b etween them; the so-called Slavnov-Taylor identities, which are the non-

Ab elian analogue of the Ward identities in QED .

As an example, since I will use it in the following, the renormalised DSE for

the connected quark 2-p oint function is

4 1 2

d q S p = Z i p + m +Z g D p q S q q; p 30

2 0 1

where: m is the Lagrangian quark bare-mass; is the regularisation cuto ;

4 4 4

d q d q=2 ; all functions in the integrand are renormalised; Z is the renor-

malisation constant of the quark-gluon-vertex; and Z is the renormalisation con-

stant of the quark wave-function. The renormalisation constants are functions

of and the renormalisation p oint, . They satisfy the Slavnov-Taylor identity

2 2 2 2 YM 2 2 YM 2 2 YM

Z ; =Z ; =Z ; =Z ; , where Z is the gluon

2 1

3 1 3

wave-function renormalisation constant and Z is the 3-gluon-vertex renor-

malisation constant. [Recall that in QED, which do es not have gauge b oson

2 2 2 2

self-interactions, one has the Ward identity: Z ; =Z ; .]

1 2

A qualitatively new feature is the fact that, b ecause of asymptotic freedom,

the renormalisation b oundary conditions must b e applied at large spacelike mo-

mentum, where p erturbation theory can b e de ned. The quark 2-p oint function

2 2

is renormalised such that, for p = large and spacelike,

f 1 2

S p = i p + m ; 31

2 2

p =

where m is the renormalised current-quark mass, which states that the

quarks are free at large spacelike momentum. The failure of p erturbation theory

at small-p entails that one cannot employ an on-shell renormalisation prescrip-

tion in p erturbation theory. In addition, the connected 2-p oint functions in a

con ning theory do not necessarily have a Lehmann representation and hence,

2 2

In simple terms, a scalar function F p has a Lehmann, or sp ectral, representation

if there is a non-negative sp ectral density s such that F p = ds : For a

p +s

free scalar particle one obviously has s=sm . The generalisati on to fermions

is relatively straightforward. Discussions of the role and imp ortance of the Lehmann

representation in quantum eld theory can b e found in Itzykson and Zub er 1980, Chap. 5, for example.

222 Craig D. Rob erts

nonp erturbatively, there may not even b e a mass-shell for the elementary exci-

tations; i.e., quarks and gluons.

The renormalised self energy is de ned via

1 2

S p i p + p i p [1+ p] + m + p 32

V R S

or alternatively

1 2 2

S p= i pAp + Bp : 33

I de ne the regularised self energy via

1 2 2 0

S p=Z ; [i p + m ] + p; 34

2 0

where

0 0 0

p=i p p+ p; 35

V S

4 2

d q = Z g D p q S q q; p : 36

The renormalisation condition of Eq. 31 entails

2 2 0 2 2

Z ; 2 A ; 37

2 2 2 2 0 2 2

m Z ; m +B ; 38

R 2 0

and hence

2 2 0 2 2 0 2 2

Ap ; =1+Ap ; A ; ; 39

2 2 2 0 2 2 0 2 2

B p ; =m +B p ; B ; : 40

2 Hadron Observables

Hadronic physics is characterised bytwo imp ortant facts: con nement, the ab-

sence of coloured asymptotic states; and D SB, the dynamical enhancementof

the current-quark mass without the exp edient of an auxiliary scalar eld. Both

of these e ects can b e understo o d via the nonp erturbative dressing of quark and

gluon 2-p ointSchwinger functions, which is therefore fundamentally imp ortant

in QCD. The DSEs are an ideal means of understanding and elucidating these e ects.

Dyson-Schwinger Equations in QCD 223

2.1 Chiral Symmetry Breaking

In covariant gauges the bare 2-p oint quark Schwinger function is

S p= ; 41

i p + m

where m is the bare current-quark mass, which is the source of explicit chiral

symmetry breaking in the QCD action. The presence of this term is measured

by the quark condensate

d p

hqq i tr [S p] ; 42

where indicates that the integral is necessarily divergent, b eing the exp ec-

tation value of a bilo cal op erator at zero relative separation, and must b e regu-

larised.

The scalar part of the quark self-energy, B p , is mo di ed by the interaction

of the quark with its own gluon eld, as describ ed by the DSE in Fig. 1. In

p erturbation theory one nds [e.g., Pascual and Tarrach 1984, pp. 67-70]

3 p

B p =m 1 ln + ::: ; 43

4 m

i.e., the bare term and each p erturbative correction is prop ortional to the current-

quark mass. Hence, in a p erturbative analysis, hqq i / m , whichvanishes in

the p erturbativechiral limit: m ! 0.

Dynamical chiral symmetry breaking is the statement that hqq i 6= 0 in the

chiral limit and, as observed ab ove, this is impossible in p erturbation theory;

i.e., it is an intrinsically nonp erturbative e ect.

The relation b etween hqq i and S pentails that Eq. 29 is an ideal to ol for

studying this e ect. The kernel of this integral equation involves the connected

gluon 2-p oint function, which in Landau gauge has the form

k k 1 1

: 44 D k =

2 2 2

k k 1+ k

Extensive study of Eq. 29 [Rob erts and Williams 1994, Sec. 6.1] has shown

2 2 2 2

that, with Gk g =[1 + k ], one automatically has D SB if Gk =0

4 and Gk is a monotonically decreasing function of its argument. In order

to set a scale, this minimal value of the coupling required to induce D SB may

b e compared with G 0 0:1, the value in QED, which is 100 times weaker.

QED

At this p oint one automatically asks:

1. Is the value of G0 in QCD this large and, if so, what is the mechanism

that drives the enhancement?

224 Craig D. Rob erts

2. Given that one has D SB in the manner describ ed here, how do es that

yield massless pseudoscalar mo des without ne-tuning; i.e., howdoesthe

dynamical generation of a fermion mass ensure the presence of massless

Goldstone mo des in the sp ectrum?

2 2

3. Is the b ehaviour of Gk in the vicinityof k = 0 connected with quark

con nement?

Presently the most reliable information ab out the form of the connected

gluon 2-p oint function is provided by studies of the gluon DSE illustrated in

2 2 2

Fig. 4. They indicate that for k 1 2 GeV , Gk is small and calculable

2 2

in p erturbation theory but for k < 1 GeV it is essentially nonp erturbative,

with a singularityat k = 0 such that the connected gluon 2-p oint function

do es not have a Lehmann representation and is b est describ ed by a distribution

in the neighb ourho o d of this p oint [Rob erts 1996]. This strong enhancement

in the intermediate and infrared regions is driven by the gluon vacuum p olar-

isation contribution asso ciated with the 3-gluon vertex, the second diagram in

Fig. 4 [Brown and Pennington 1989], and necessarily entails D SB without

ne-tuning. These results answer question 1.

Meson b ound states in QCD are describ ed by an homogeneous BSE. In gauge

theories the BSE for pseudoscalar b ound states is intimately related to the DSE

for the connected quark 2-p oint function as a result of the chiral Ward identity

f f

1 1

5 f 1 1

P Pi + i S k P ; 45 k ; P = S k+

5 5

2 2

where =2 are the generators of SU 3- avour, which relates the 1-

f =1

particle irreducible, avour-o ctet, pseudovector-quark 3-p oint function to the

connected quark 2-p oint function. It is this relation b etween these equations

that ensures the presence of pseudoscalar Goldstone mo des as a consequence of

D SB [Rob erts 1996], which answers question 2.

2.2 Quark Con nement

The answer to question 3 is \Yes" but this cannot b e established in p ertur-

bation theory where, at arbitrary nite order, one obtains a connected quark

2-p oint function that has a Lehmann representation, which provides for quark

pro duction thresholds in colour-singlet ! singlet S -matrix amplitudes.

Con nement is related to the analytic prop erties of the connected n-p oint

functions and is sensitive to the true infrared prop erties of the gauge theory.

In studying con nementIhave found it useful to analyse the prop erties of the

connected n-p oint functions in con guration space. To understand why, consider

the scalar part of the quark 2-p oint function in con guration space

d p

i~px~+p

^ ~x; e p ; 46

S S

Dyson-Schwinger Equations in QCD 225

2 2

where S p i p p + p , and de ne

V S

M ln j j 47

where

d x ^ ~x; : 48

For a massive free fermion p= , which has an obvious Lehmann

2 2

p +M

representation. In this case M =M; i.e., M isolates the particle mass,

which is nite and hence one has a real asymptotic fermion state.

As an alternative, consider the mo del de ned by Eq. 29 with [Burden et al.

1992]

k k

2 4 2 4 1

g D k 4 k and i k; k @ S k : 49

In this case Eq. 29 b ecomes

z = 2[m z z] 50

V S

z z =2[ zz +1+m z1] 51

V S

2 2

with m m and S p i z z + z; p = z:

V S

This di erential equation has the solution

p p

m J 4m z

p p

z =Ce + dK m zJ ze ; 52

S 1 1

2m z z

which completely determines z via Eq. 50.

In the chiral limit,m ! 0, z = Ce ,M= ; i.e., in contrast to

the free particle case, the mass function is -dep endent and diverges as !1,

which means that the particle is con ned. The result is qualitatively unchanged

form 6= 0 and is tied to the fact that in this mo del z and z are entire

V S

functions with an essential singularity on the b oundary of the complex-z plane;

such functions do not have a Lehmann representation.

It is true in general that M is a sensitive prob e of the analytic structure

of the quark 2-p oint function and its b ehaviour clearly signals whether or not

S p has a Lehmann representation. The absence of a Lehmann representation

for the connected 2-p oint functions of the elementary excitations in QCD is a

sucient condition for con nement since it ensures the absence of quark and

gluon pro duction thresholds in colour-singlet!singlet S -matrix amplitudes. In

this mo del a gluon 2-p oint function that do es not have a Lehmann representation

and which is describ ed by a distribution in the neighb ourho o d of k = 0, yields a

quark 2-p oint function that also do es not have a Lehmann representation. Gluon

2-p oint functions that do not have a Lehmann representation but which are not

enhanced in the infrared yield a quark 2-p oint function that has a Lehmann

representation, a nite mass and hence are not con ned [Hawes, et al. 1994].

This illustrates a sense in which one gives an armative answer to question 3.

226 Craig D. Rob erts

2.3 Hadron Observables: A Mo del Study

This is an ideal p oint to illustrate the application of DSEs in the study of hadron

observables. The renormalised connected gluon 2-p oint function in Eq. 30 is

primarily resp onsible for D SB, con nement and the quantitative features of

hadron observables. In a general covariant gauge it has the form

k k k k

2 2 2

g D k = k +g 53

2 4

k k

where is the gauge parameter and

k : 54

2 2

[1 + k ] k

The fact that the longitudinal -dep endent piece of this 2-p oint function is not

mo di ed byinteractions is a fundamental feature of QCD and is a consequence

of the Slavnov-Taylor identities [Pascual and Tarrach 1984, pp. 42-45].

2 2

In QED, k is prop ortional to the running coupling constant k

QED

[Itzykson and Zub er 1980, Chap. 13]. In QCD, b ecause of the presence of ghost

elds, this is not true. However, in constructing a simple DSE mo del of QCD

gh gh

here, I will employ the Abelian approximation, in which one sets Z = Z ; i.e.,

1 3

one enforces equalitybetween the renormalisation constants for the ghost-gluon-

YM YM

vertex and ghost wave-function. This entails Z = Z and hence Z = Z

1 2

3 1

in Eq. 30; and leads to [Rob erts and Williams 1994, Chap. 5]

k =4 ; 55

2 2

which motivates a mo del choice for k whose form at large spacelike-k is

given by the running coupling constant.

Icho ose [Frank and Rob erts 1996]

2 3

1e 12

6 7

4 2 2 2 2

k+ k = 4 d 4 m ;d= ; 56

4 5

k 33 2N

which has one free parameter, m . This gluon 2-p oint function do es not have

a Lehmann representation and hence asymptotic gluon states are precluded in

the mo del; i.e., gluons are con ned. It preserves the large-k b ehaviour of the

QCD running coupling constant up to ln k corrections and the rst term in

the sum, which dominates in the infrared, is a regularised infrared singularity

that mo dels the b ehaviour found by Brown and Pennington 1989.

Dyson-Schwinger Equations in QCD 227

Schwinger Function in Con guration Space. Since Eq. 56 is a mo del for

the gluon 2-p oint function, the ratio of the co ecients of the two terms in the

sum is arbitrary.To see why the particular value used is chosen, consider the

Fourier transform

2 2

4 ik x 2 2 2 x m

d k e k =d m + : 57 x e

>From this it is clear that, for m x 1; i.e., at short spacetime distances in

the ultraviolet,

2 4

x = +Ox 58

and hence the ratio of co ecients I havechosen ensures that the long-range

e ects asso ciated with the infrared term are completely cancelled in the ultra-

violet. The length x 1=m can b e said to mark the b oundary b etween the

t t

infrared and ultraviolet domains in the mo del.

In addition to the renormalised gluon 2-p oint function, Eq. 30 involves the

renormalised 1-particle irreducible quark-gluon 3-p oint function. In this illustra-

tive study I employ the rainbow truncation, which means using

rainbow

p; q = p; q : 59

In my opinion, this often used truncation can only b e quantitatively reliable

in Landau gauge, = 0,ifever. I observe this b ecause in studies of D SB in

strong-coupling quenched-QED, the critical coupling obtained using the rain-

bow truncation is only 12 larger than that obtained using the most sophisti-

cated 1-particle irreducible fermion-photon 3-p oint function presently available

=1:05 c.f. 0:93 [Bashir and Pennington 1994]. This is the b est one can

do. The critical coupling and other quantities that should b e gauge-parameter

indep endent are very sensitiveto in rainbow truncation. In QED or QCD,

working in Landau gauge minimises the quantitative e ects of this spurious

gauge-parameter dep endence.

UV Analysis. My DSE mo del of QCD is fully sp eci ed using the Ab elian

approximationZ = Z and Eqs. 56 and 59 in Eq. 30. Making use of

1 2

2 2 2

the fact that, at large-k , k 1=k , one can analyse the DSE to nd that

2 2

Alarge p 1 and B large p is obtained from

1 1 By

dy y B x= x y + y x ; 60

4 x y y +By

where x = p , x is the step function and the angular integral has b een evalu-

2 2

ated explicitly, which is straightforward when k =1=k .

dB x dB x

>From Eq. 40 it is obvious that = and hence, assuming B x !

dx dx

0as x!1, Eq. 60 is equivalentto

d d

x B x + B x=0; 61

dx dx 4

228 Craig D. Rob erts

with =4Z d>1 for Z 1, sub ject to the ultraviolet b oundary condition

2 2

[xB x] = Z ; m ; 62

2 0

2 2

which is equivalentto B =m . The conversion of the integral equation

into a di erential equation, valid in the ultraviolet, is an often used technique. It

provides a means of verifying the form of anynumerically determined solution

of the integral equation and allows for an analysis of the ultraviolet b ehaviour

of the mo del.

, Eq. 61 can b e written Intro ducing the variable z =ln

d d

B z +2 B z +B z =0; 63

dz dz

which is the equation for a damp ed harmonic oscillator. Such an oscillator system

exhibits critical b ehaviour at = 1: no oscillations taking place for 1; i.e.,

any displacement from the mean p osition relaxes back to this p osition without

passing through it. In the present context, this b ehaviour is actually asso ciated

with D SB.

1+ with the ultravi- The solution of Eq. 63 is B z =e cos z

olet b oundary condition sp ecifying cos = m but with the overall scale, ,

only b eing determined by the solution of the complete integral equation. One

observes from this that, when the ultraviolet b ehaviour of the gauge-b oson 2-

p oint function is 1=k , nite solutions for the fermion 2-p oint function will have

2 2 2 2

B p ; that oscillates ab out zero, with the rst zero at some p > . This

is not the b ehaviour one exp ects in QCD and may b e a defect of my simple

mo del. It is tied to the fact that I have neglected ln k corrections to the 2- and

3-p oint functions in this mo del; i.e., I have neglected the QCD anomalous dimen-

sions. However, as long as one cho oses suciently large then the calculated

quantities will not b e sensitive to this feature.

The renormalisation condition of Eq. 40 requires that Z ; m +

2 0

0 2 2 2

Bp ; b e indep endentof. This means that Z ; m is merely a

2 0

subtraction constant that one must vary to ensure this outcome - it is not

aphysical quantity. One nds that a xed value of m entails a given

value of Z ; m , whichvaries with , and that for any m 2 [0; 1:

2 0 R

lim Z ; m = 0. I de ne the chiral limit in this mo del as m =

2 0 R

Bound States. In quantum eld theory 2-b o dy b ound states are describ ed

by the homogeneous Bethe-Salp eter equation, which is derived under the as-

sumption that the 1-particle irreducible 4-p oint function, in a channel with the

quantum numb ers of the b ound state under consideration, has a simple P -p ole

contribution, where P is the total momentum variable [Itzykson and Zub er

Dyson-Schwinger Equations in QCD 229

1980, Chap. 10]. I will not go into detail here but simply write the homoge-

neous BSE for the pseudoscalar quark-antiquark pion b ound state in ladder

truncation:

1 1

4 2 0

d qg D k q S q + k ; P = Z P q ; P S q P 64

2 2

where k is the relative quark-antiquark momentum, P is the total momentum

and, since I treat u and d quarks as having equal mass, the general form of the

regularised Bethe-Salp eter amplitude is

0 0 0 0

k ; P = [iF k ; P + PF k; P+ kk PF k;P] : 65

1 2 3

The equation is solved sub ject to the renormalisation b oundary condition

0 0

F k ; P =F k;P F k;Pj ; 66

2 2

1 1

k =

which is consistent with the renormalisation of Eq. 30. This pro cedure ensures

the preservation of the axial-vector Ward identity, which underlies Goldstone's

theorem.

In this illustrative study I assume that F 0 F , whichintro duces errors

2 3

of 10, and then Eq. 64 yields the following equation for F

0 4 + +

F d qp q q q + F q;P; 67 k ; P = 4Z

+ 1 2

V V S S

P and = q . This equation can b e solved by employing where q q

a Tschebyshev expansion:

k P

i 2 2 i

68 F k ; P F k ;P U cos ; cos

2 2

k P

i=0

i 1

where fU g are the orthogonal Tschebyshev functions of the second kind,

i=1

i j

dx 1 x U x U x= : In practice, for equal mass which satisfy

0 2 2

quarks, it is a go o d approximation to retain only F k ; P and this certainly

suces for a p edagogical example.

A detailed discussion of the straightforward realisation of Goldstone's theo-

rem in the context of the DSEs can b e found in Rob erts 1996. Here one simply

2 0 2 2

observes that, for P = 0, Eq. 67 is identical to the equation for B k ; .

Hence, in the chiral limit, m = 0, the existence of D SB, as manifest in

a dynamically generated scalar contribution to the quark self energy, necessar-

ily entails the existence of a massless P = 0 pseudoscalar excitation in the

sp ectrum of the mo del whose Bethe-Salp eter amplitude is

2 0 2 2 2 2

k ; P =0=i F k ; P = i B k ; ; 69

5 5 m =0

230 Craig D. Rob erts

where N is the canonical normalisation constant, which is discussed in Itzyk-

son and Zub er 1980, Chap. 10. Conventionally the amplitude is normalised

such that the asso ciated p ole contribution to the 1-particle irreducible 4-p oint

function has unit residue. For P = 0, with the truncations used here,

2 2

N = dxxB x 70

m =0

h i h i

2 2

00 0 2 0 0 00 0 2

; 2[ + x ] x x

V S V S

V V V S V S S

0 2

where, in this equation, , etc., denote di erentiation with resp ect to x = p .

Ihave indicated that for zero current-quark mass the pion BSE yields a

massless pion. In reality the u and d quarks have small but nonzero current-

masses. Using Eqs. 30, 67 and 69, one can derive the following expression

[Frank and Rob erts 1996]

2 2

m N = hm qq i = 71

B x N

m =0 c

m 6=0

m =0

dx x B x x B x x ;

m 6=0 m =0

R R

S S

2 B x

m 6=0

0 R

which provides an accurate estimate of the mass obtained in solving Eq. 67 for

realistic u and d current-quark masses.

At the same level of approximation, the electroweak pion decay constant,

+ +

whichcharacterises the decay ! , is given by[Frank and Rob erts

1996]

h i

0 0 0

dxxF x + f = : 72

V S S V

1 V S

The di erence b etween f and N is an artifact of neglecting F and F in

2 3

Eq. 65. As illustrated in Table 1, this is a 10 e ect in the present case.

One can derive formulae for other hadronic observables, such as the scat-

tering lengths that describ e ! scattering [Rob erts et al. 1994]. The

quantities presented here, however, illustrate the general features: each observ-

able is given byintegrals involving the b ound state Bethe-Salp eter amplitudes

and the dressed quark 2-p oint function. In most cases the gluon 2-p oint function

do es not app ear explicitly but it is always present implicitly in the momentum

dep endence of these other n-p oint functions.

Phenomenology. One can now determine whether the simple, illustrative

mo del develop ed in the last few pages can provide a quantitatively go o d de-

scription of hadronic observables. There are two parameters: the mass-scale, m ,

whichcharacterises the b oundary b etween the infrared and ultraviolet regimes

in the mo del; and the current-quark mass, m . Cho osing a renormalisation

p oint deep in the ultraviolet, =48fm 9:47 GeV, and using the formu-

lae presented ab ove, and in Rob erts et al. 1994, a - t to the exp erimental

Dyson-Schwinger Equations in QCD 231

Table 1. Comparison of calculated observables with exp eriment; a are the scat-

tering lengths and r is the electromagnetic pion charge radius. The argumentin

square-brackets indicates whether B or the Bethe-Salp eter amplitude obtained in

m =0

solving Eq. 67 was used to calculate a given quantity. In the tting, B was used

m =0

to approximate this amplitude, which led to m =0:69 GeV and m =1:1 MeV.

t R

This value of m corresp onds to x =0:29 fm.

t t

Calculated Exp eriment

m [B ] 138.7 MeV 138.3 0.5 MeV

MF 0

m [F ] 137.2

BSE

m 139.5

f [F ] 92.4 MeV 92.4 0.3 MeV

f [B ] 92.3

N [F ] 102

r [F ]N [F ] 0.24 dimensionless 0.31 0.004

0 0

a [F ] 0.16 0.21 0.02

a [F ] -0.041 -0.040 0.003

[F ] a 0.028 0.038 0.003

a [F ] 0.0022 0.0017 0.0003

a [F ] 0.0013

quantities listed in Table 1 using Eq. 69 yields the calculated results in the

table. The agreement, with such a simple mo del and with such little e ort, is

excellent. The small discrepancies b etween the calculated and observed values of

the scattering lengths and charge radius arise b ecause this calculation neglects

nal-state interactions [Alkofer, et al. 1995]. Exploring the b ehaviour of

Eq. 71 on m 2 [0; 0:02] GeV, one nds

2 2 3 2 2 3

m N = 20:45 m +2:6 m + 150 m ; 73

R R

and therefore identi es hqq i = 0:45 GeV . The linear term contributes 96

of the total at the tted value of m .

The quark mass function, M p=Bp=Ap, obtained using the tted value

of m is illustrated in Fig. 5 [Maris and Rob erts 1997]. Dynamical chiral sym-

metry breaking is manifest in the fact that the chiral limit solution is nonzero.

One observes a signi cant infrared enhancementinMp for each current-quark

mass value, which is due to the infrared enhancement in the gluon 2-p oint func-

tion. This is a qualitatively signi cant result that is observable in hadronic pro-

cesses. Its strength for a given quark can b e quanti ed in the ratio M =m ,

where M is the Euclidean constituent-quark mass, de ned as the solution of f

232 Craig D. Rob erts

3.0 chiral limit u−quark s−quark c−quark b−quark 2.0 p=M(p) M(p)

1.0

0.0 0.0 2.0 4.0 6.0 8.0 10.0

p (GeV)

Fig. 5. Quark mass function obtained with m =0:69 GeV in Eq. 56 for a varietyof

values of m , which corresp ond to u ! b-quark values. The solution of p = M p

de nes M , the Euclidean constituent-quark mass.

p = M p, and m is the current-quark mass:

f f

f : u=d s c b t

: 74

M =m : 400 20 5 2:5 ! 1

The dynamical enhancement of the mass is extremely imp ortant for the light

quarks and, although it diminishes with increasing current-quark mass, it re-

mains signi canteven for the b-quark.

3 Finite Temp erature

The equilibrium thermo dynamics of a theory of a self interacting scalar eld at

temp erature T is describ ed by the generating functional

Z Z Z

Z [j; T =1= ] d [] exp d d xj~x; ~x; 75

R R

E 3 E

with d []= D~x; exp S [] ; S []= d d x L x; ;

x~; 2[0; ]

L x; is the Euclidean Lagrangian density for the self-interacting b oson eld

, which satis es ~x; =0=~x; = . The justi cation of this result and

its generalisation to gauge theories is discussed by Kapusta 1989 and Rothe 1992, for example.

Dyson-Schwinger Equations in QCD 233

Qualitatively, one sees that at nite-T the O 4 symmetry of Euclidean

space b ecomes an O 3; 1 symmetry.Further, the fact that one dimension is

b ounded by1=T means that as T !1 a d-dimensional theory is reduced to a

d 1-dimensional one; i.e., one has a dimensional reduction and, for example,

lim QED = QED .

T !1

4 3

The nite-T b ehaviour of QCD is of interest b ecause of the p ossibility that,

with increasing T , QCD may undergo a transition to a phase in which quarks and

gluons are weakly interacting, even at small-q ; i.e., a quark-gluon plasma may

form. The natural scale in QCD is 200 MeV and any such transition

QCD

would require T , whichis10 ro om-T . Such temp eratures are on the

QCD

astrophysical and cosmological scale. Standard cosmology assumes a large matter

and radiation density at high temp erature. The manner in which this system

co ols in uences presentday observables and scenarios for grand uni cation, with

the nite-T b ehaviour of a given theory providing a means for its falsi cation.

These sp eculations have led to the construction of a Relativistic Heavy Ion

Collider RHIC at Bro okhaven National Lab oratory. Due to b e completed in

197

1999, it will use counter-circulating, colliding 100 A GeV Au b eams to generate

a total centre-of-mass energy of 40 TeV, in an e ort to pro duce an equilibrated

quark-gluon plasma.

The study of phase transitions requires nonp erturbative metho ds and Karsch

1995 discusses the present status of lattice simulations of nite-T QCD. The

DSEs provide a complimentary means of studying this problem; a computation-

ally less intensive approach that is easily extended to nite density and to the

study of a broad range of scattering observables - two regimes that are currently

inaccessible to lattice simulations. Finite density presents a problem in principle

for lattice simulations b ecause the Euclidean action is complex, which prevents

the use of naive probabilistic metho ds in evaluating the functional integral.

The nite-T DSE formalism can b e develop ed via a straightforward applica-

tion of the metho ds discussed in Sec. 1 to the generating functional of the nite-T

theory. I will not explore this explicitly herein but instead app eal to analogy.

At nite-T the free fermion 2-p oint function is S p=1=i ~p~+i ! + m,

4 n

where ! =2n+1T is the fermion Matsubara frequency, arising b ecause

the b oundary condition for the fermion eld is ~x; =0=~x; = ,

which ensures Fermi-Dirac statistics. The dressed fermion 2-p oint function can

b e written

S p; ! = 76

i ~pA~ p; ! + i ! C p; ! + Bp; !

n 4 n n n

= i ~ p~ p; ! i ! p; ! + p; ! : 77

A n 4 n C n B n

The renormalised 2-p oint function has the form

0 1 A

i ~p~+Z i ! + m +p; ! 78 S p; ! =Z

2 4 n 0 n n

where the regularised self energy is

0 0 0 0

p; ! =i ~p~ p; ! + i ! p; ! + p; ! 79

n n 4 n n n

A C B

234 Craig D. Rob erts

and one observes that there are two renormalisation constants. The quark DSE

b ecomes a system of three coupled nonlinear integral equations:

4 1

0 2

p; ! = g D p q; ! ! tr [P S q; ! q; ! ; p; ! ] 80

k k l F l l k

3 4

l;q

A 2

where F = A; B ; C ; P Z =p~i ~p~,P Z ,P Z =! i , and

A B 1 C 1 k 4

R R

P 3

d q

. The renormalisation conditions are T

l=1

l;q

S p; ! = i ~ p~ + i ! + m : 81

0 4 0 R

2 2

p +! =

At nite-T , in Landau gauge, the connected gluon 2-p oint function has the

form

2 L T

g D p; =P p; p; +P p p; ; =2n T 82

n n F n G n n

0 ; and=or =4;

p p

i j

P p 83

; ; =1;2;3

T L

with P p+P p; p = p p = p p ; ; =1;:::;4. A \Debye-

mass" for the gluon app ears as a T -dep endent contribution to . do es not

F G

receive a simple, analogous contribution and hence one also observes the O 3; 1

symmetry in the gluon sector.

3.1 Finite-T QCD: A Mo del Study

To illustrate DSE metho ds and the prop erties of eld theories at nite-T ,I

discuss a simple nite-T extension of the mo del develop ed in Sec. 2.3 [Bender et

al, 1996], with the connected gluon 2-p oint function de ned via: p;

F n

D p; ; m and p; Dp; ; 0;

n D G n n

2 2 2 2

[p + +m =4m

]

n t

2 1e

2 2 3

D p; ; m 4 d m ; 84 p+

n 0 n

2 2 2

T p + + m

2 2 2

where d =12=33 2N and the \Debye-mass" is m =cT ,c=4 dc, c =

N =3+N =6, whichvanishes at T = 0. Employing the Ab elian approximation:

c f

A A

Z = Z , Z = Z ; the rainbow truncation, Eq. 59; and requiring that m

1 2 t

1 2

and m retain the values xed in the T = 0 studies, means that this is a

parameter-free extension of the mo del to nite-T .

To study a phase transition it is necessary to identify an order parameter;

i.e., the exp ectation value of some op erator that characterises the transition.

Identifying the order parameter is often the most dicult task. If X is the

op erator then hX i 0 in the disordered phase but hX i6 0 in the ordered

I note that, as T ! 0, the mo del and results of Sec. 2.3 are recovered exactly.

Dyson-Schwinger Equations in QCD 235

phase. An example is the magnetisation in a ferromagnet. Phase transitions can

b e divided into two classes: 1st order, where the order parameter b ecomes zero

discontinuously, as in solid-liquid melting transitions; and 2nd order, where the

order parameter falls continuously to zero, suchasinspontaneous magnetisation.

Second order transitions are characterised by critical exp onents, which can b e

studied using the renormalisation group.

Zero temp erature QCD is characterised byD SB and con nement, and it

is the transition to a phase in which one has a restoration of chiral symmetry

and/or decon nement that is of interest. The order parameter for chiral sym-

metry breaking is the quark condensate and, b ecause of Eq. 42, an equivalent

order parameter is B ~p =0;! . An order parameter for decon nementis

more dicult to identify.

Ihave discussed con nement in Sec. 2.2, indicating that the analytic struc-

ture of a dressed n-p oint function can b e quite di erent to that of its bare

counterpart. This b eing the case, in nonp erturbative studies it is not p ossible,

in general, to p erform the Matsubara sum in Eq. 80 analytically - it must b e

evaluated numerically.Further, the p ossible nonexistence of a Lehmann repre-

sentation complicates or precludes a real-time formulation of the nite-T theory,

whichintro duces a barrier to the study of the non-equilibrium thermo dynamics

of the theory.

A Continuum Order Parameter for Decon nement. The discussion of

Sec. 2.2 intro duces an obvious means for studying decon nement via the con g-

uration space Schwinger function and the mass-function of Eq. 47. Consider

1 2

x; =0 T dp p sinpx p; ! 85

B B n

0 0

n=1

x : 86

n=0

2 2

x +M ! M

and x=Me , from For a free fermion, ~p; ! =

2 2 2 B n

+p +M !

whichitisobvious that the n = 0 mo de dominates the sum in Eq. 86. In

2 2 2

this case, the mass function M x; T T + M and ln x =

one observes immediately that: 1 the mass function isolates the free-particle

p ole; and 2 nite-T e ects only b ecome imp ortant for T , where, in an

interacting theory, M is most naturally identi ed with the Euclidean constituent-

quark mass intro duced in conjunction with Fig. 5. In the 2- avour DSE mo del

for QCD under consideration here, M 450 MeV and hence one can exp ect

u=d

that nite-T e ects only b ecome imp ortantat T 150 MeV.

2 2 2

p + +M

Consider the mo del 2-p oint function: D p; = , which has

n 2 2 2 2 4

p + +M +b

complex conjugate p oles displaced from the negative real-p axis by a distance

b , therefore no Lehmann representation and hence can b e interpreted as the

connected 2-p oint function for a con ned excitation. In this case one obtains

236 Craig D. Rob erts

0.8 ∆(x) 0.6

0.4

x α 1/b 0.2

0.0

−0.2 0.0 1.0 2.0

x (fm)

Fig. 6. x calculated at T = 5 MeV in my exemplary 2- avour DSE mo del of

QCD. The presence of a zero signals quark con nement in the mo del at T 0.

0 Mx

x= e cos [bx].Itisimmediately obvious that the exp onential mo dulat-

ing factor isolates the real part of the p ole; and the p erio d of the oscillations,

the imaginary part. This provides a continuum test for con nement: in a given

x and the presence of oscillations is a clear mo del/theory, one calculates

signal of the absence of a Lehmann representation and hence con nementofthe

asso ciated excitation. This approach has b een used very successfully by Maris

1995.

x, calculated from the solution of Eqs. 78{81 at T = 5 MeV in

my exemplary DSE mo del of 2- avour QCD, is illustrated in Fig. 6. The zero

is a clear signal for quark con nement in this mo del at T 0. A continuum

order parameter for decon nementisnowobvious. One de nes = , where

r is the p osition of the rst zero in x. measures the distance of the

p ole in the 2-p oint function from the real axis. If at some temp erature, T ,

T = 0 then the p oles have migrated to the real axis, one has recovered a free

particle 2-p oint function and the quark is decon ned. is the continuum order

parameter for decon nement, valid for b oth light and heavy quarks, and T is

the temp erature at which thermal uctuations overwhelm the con nement mass

scale.

With m and m xed at T = 0, it is straightforward to solve Eqs. 78{

t R

81 for arbitrary T and use the solutions to study the nite-T b ehaviour of the

mo del and its predictions for the T -dep endence of physical observables via the

Dyson-Schwinger Equations in QCD 237

χ (GeV)

0.8

0.4

ν (GeV)

0.0 0.0 50.0 100.0 150.0

T (MeV)

Fig. 7. The T -dep endence of the chiral symmetry, , and con nement, , order pa-

rameters. The parameters of the tted curves are given in Eq. 87.

generalisations of Eqs. 70{72 [Bender, et al. 1996].

The b ehaviour of the chiral symmetry and con nement order parameters is

presented in Fig. 7. The mo del predicts coincident, second order chiral symmetry

restoration and decon nement transitions. Fitting the b ehaviour to curves of

the form a1 T=T on T 2 [120; 150] MeV, yields the critical exp onents and

temp erature in Eq. 87. The critical exp onents are equal, within what I consider

to b e a reasonable estimate of the errors involved, 10.

a =1:1 GeV ; =0:33 ;T = 150 MeV

a =0:16 GeV ; =0:30 ;T = 150 MeV :

These results are in agreement with recent lattice simulations [Karsch 1995],

which yield T 150 MeV and =0:32 0:09. Ra jagopal 1995 has argued

that QCD with 2 massless avours is in the same universality class as the N =

4 Heisenb erg magnet, whichwould yield 0:37. In analysing the ts, the

di erence b etween this and 0:33 in Eq. 87 is not statistically signi cant,

however, the transitions cannot b e describ ed by a mean- eld critical exp onent,

=0:5.

Employing the DSE solutions in the nite-T generalisations of Eqs. 70{72,

this mo del predicts that, as exp ected from the arguments following Eq. 86, f

and m are approximately indep endentofT for T 0:7 T . Another illustration

238 Craig D. Rob erts

of the slow resp onse of observables to increases in T is provided by the observa-

tion that, for T =0:9T , there is only a 20 suppression of the width .

c !`

However, as one reaches T there is a dramatic e ect: the pion p ole contribution

to the 1-particle irreducible, 4-p oint quark-antiquark Schwinger function is elim-

inated; i.e., the pion disapp ears from the sp ectrum. The nature of any residual

quark-antiquark correlations ab ove T , which this result shows are to o weak to

supp ort b ound states, is the sub ject of ongoing study.

4 Closing Remarks

These lectures are limited in scop e, fo cusing on the formal derivation of the DSEs

and an heuristic application of the techniques using a simple mo del. They do not

describ e the extensive bodyofwork that explores the phenomenological applica-

tion of these metho ds to exclusive scattering pro cesses nor the essential progress

made in developing and understanding the truncation pro cedures that provide

the foundation of the recent phenomenological successes. For those interested

in the more complex phenomenological applications, the articles byPenning-

ton 1996, Pichowsky and Lee 1996 and Tandy 1997 provide a springb oard

for pursuing them; while those interested in more formal asp ects can trace the

discussion from the articles of Bashir and Pennington 1994 and Hawes et al.

1996.

Acknowledgements. This work was supp orted by the US DepartmentofEn-

ergy, Nuclear Physics Division, under contract numb er W-31-109-ENG-38.

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