Models for CONFINEMENT

Gerard ’t HooftSpinoza InstituteUtrecht, the Netherlands

Utrecht University

Contents

First part:Absolute Quark Confinement in lattice QCDSecond part:Absolute Quark Confinement as a topologicalPhenomenonThird part: The Gluon Chain Model (Greensite, Thorn)Try to do it better: compare Renormalization Procedure(infinite) Infrared RenormalizationThe renormalized Coulomb potential (in radiation gauge)Fourth part:Gauge invariant procedureRenormalized effective actions:

an exercise in Legendre Transformations

1P B

N B e

B

What kinds of forces were holding them together?

Proton Lambda Antiproton

Pi-plus Pi-zero

The hadronic particles …

Lattice QCD (K. Wilson, London, 1974)

q

q

† †... ...

... ... ...

p q ki j r s

x

k p qi j r s

dU U U U U

C

Using the expansion21/ g

In the expansion,only terms where the energyincreases linearly with inter-quark distance survive !

21/ g

Part 1:

Part 2: Magnetic Confinement

†14, ( )A F F D D V L

In case of spontaneous "breakdown" of

(1)U I

N

S

| | F

H.B. Nielsen and P. Olesen, 1970.

Color Magnetic Super Conductivity

N S

+ _

Electric Super Conductor

Magnetic Super Conductor

G. ’t H (1974), A.M. Polyakov (1974)

The Magnetic Monopole

S. Mandelstam (1975),G. ’t H (1976)

Part 3: The gluon chain approach

Anti-quark

quark

J. Greensite and C.B. Thorn hep-ph/0112326

Ansatz for the "Wave Function": 1 2 11

( , , , ) ( )N

N ii

x x x A u

1 0 fixed (?); ,i i i Nu x x x x

Use variational principle, minimize (kin) (Coulomb)T V E=

then, improve Ansatz [ ? ]

This can be done better

The gluon chain model gives reasonable – looking "stringlike“ structures for the mesons …but confinement is not built in …The chainlike states will surely not form a complete set of states. UNITARITY ?

Describe a "modified" perturbative approach,where unitarity is guaranteed

Infinite infrared renormalization

Lowest order

Compare UV renormalization

bare( , )g A L L L L0

L

Combine this with

the higher order terms

Perturbative Confinement

gauge fix14 def

Write a aA F F g 0L L L L

14Choose: ( ) ( ') ( ')a aA F x G x x F x 0L

gauge fixPick radiation gauge: i iA A L

1 10 02 2

So,

( ) ( ) ( ) ( )i i i i i iA A G A A G A A 0L

0Anow generates a potential V between charges obeying

2 3( , ') ( ' ) ( )iG x x V x y x y

Let V be a confining potential, typically:

2 nst( ) ; | |e V x r C r xr

in space:k

2 2 2

844 ( )( )

V kk k

then

21

2

2 2

2( ) ( ) 1

2 / 2kG k k V k

k k

23 8

( ') ( ') ; | ' |r

G x x x x e r x xr

2 3( , ') ( ' ) ( )iG x x V x y x y

should be treated exactly like a renormalization counter term. Compare our procedures in the renormalization group: the coefficients (here: ) must be adjusted in such a way that the higher order correction terms, together with the contributions from , should be as insignificant as possible.

L

L

At lowest order, we should start with a Fock space ofEigen states of particles bound by the potential V . They are confined from the very beginning:

2

1 14 4

8( ) ( ) ( ) ( ')r

a a a arA F x F x F x e F x

0L

2 /14

8( ) ( ')ra arA F x e F x

L

23 8

( ') ( ') ; | ' |r

G x x x x e r x xr

Part 4: A Classically Confining Theory:

14, ( ) ( )A Z F F V J A L

; ( )i i i iD D Z E Stationary case:

212 ( )

( )D V

Z

H

212( ) min ( )

( )DD V

Z

U

U ( D ) can become any monotonically increasing function of D

Q -Q

string

;

min

minD

QD

D

DQ

D

U

U

D

DU

D

D

U

( 1)Q

Legendre Transformations:

21

2

2 2 212

d 0

dd 1/

d ;d

VDZ

VZ ZZ

U

14 ; ( ) extr -x F F f x Z x V

LWrite

dd

xVZ

( )Z

L 14

Now, in

eliminate, ( ) ( ) , A Z F F V

( )V

1

212 Z

212( ) min ( )

( )DD V

Z

U

The dual transformation

14, ( ) ( ) ( )A Z F F V L eliminate

12def

0 ;F A A F F F

: .( ) 0Z F Equations

def

1/ 4

( ) ; ;

( , ) ( )( )

G Z F G B B

B G G VZ

L

Quantum Chromodynamics is an extremely accurate theory.

At short distances, the forces become weak, so that perturbative treatment there is possible.

Calculating the QCD contributionsto high-energy scattering processes has become routine.

Interesting and important problems remain:- find a quark-gluon plasma- find more accurately converging calculation procedures ...

Utrecht University

Further References:

Nucl. Phys. B 138 (1978) 1 Nucl. Phys. B 153 (1979) 141 Nucl. Phys. B 190 (1981) 455

Acta Phys. Austriaca Suppl. XXII (1980) 531

Physics Reports 142 (1986) #6, 357 hep-th / 9903189

Erice: hep-th / 9812204 Montpellier Proceedings (2002)

The End