First part

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15 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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Gerard ’t Hooft

Spinoza Institute

Utrecht, the Netherlands

Utrecht University

Contents

First part:

Absolute Quark Confinement in lattice QCD

Second part:

Absolute Quark Confinement as a topological

Phenomenon

Third part:


The Gluon Chain Model (Greensite, Thorn)

Try to do it better: compare Renormalization Procedure

(infinite) Infrared Renormalization

The renormalized Coulomb potential (in radiation gauge)

Fourth part:

Gauge invariant procedure

Renormalized effective actions:


an exercise in Legendre Transformations

1
P 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 k
i j r s
x
k p q
i j r s
dU U U U U
C

 

 

Using the expansion

2
1/
g
In the expansion,

only terms where the energy

increases linearly with inter
-

quark distance survive !

2
1/
g
Part 1:


Part 2:

Magnetic Confinement




1
4
,( )
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 1
1
(,,,) ( )
N
N i
i
x x x A u
 


  

1 0
fixed (?)
;,
i i i N
u 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 L
0

L
Combine this with

the higher order
terms

Perturbative Confinement



gauge fix
1
4
def
Write
a a
A F F
 
     
g 0
L L L L


1
4
Choose:
( ) (') (')
a a
A F x G x x F x
 
   
0
L


gauge fix
Pick radiation gauge:
i i
A A

  
L


1 1
0 0
2 2
So,
( ) ( ) ( ) ( )
i i i i i i
A A G A A G A A
 

         
0
L
0
A
now generates a potential
V

between charges


obeying

2 3
(,') (') ( )
i
G x x V x y x y

   
Let
V

be a confining potential, typically:

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



   
in space:

k
2 2 2
8
4
4 ( )
( )
V k
k k



  
then



2
1
2
2 2
2
( ) ( ) 1
2/2
k
G k k V k
k k

  

    
 
2
3
8
(') (');|'|
r
G x x x x e r x x
r






     
2 3
(,') (') ( )
i
G 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 of

Eigen states of particles bound by the potential
V .

They are confined from the very beginning:



2
1 1
4 4
8
( ) ( ) ( ) (')
r
a a a a
r
A F x F x F x e F x


   



   
0
L


2/
1
4
8
( ) (')
r
a a
r
A F x e F x

 




  
L
2
3
8
(') (');|'|
r
G x x x x e r x x
r






     
Part 4:

A Classically Confining Theory:



1
4
,( ) ( )
A Z F F V J A
   
  
   
L
;( )
i i i i
D D Z E
 
  
Stationary case:

2
1
2
( )
( )
D
V
Z


 
H
2
1
2
( ) min ( )
( )
D
D V
Z



 
 
 
 
U
U
(
D
)

can become any monotonically increasing

function of
D

Q

-
Q








string
;
min
min
D
Q
D
D
D
Q
D



 
 

 
 
U
U
D



D
U


D
D
U

( 1)
Q

Legendre

Transformations:



2
1
2
2 2 2
1
2
d 0
d
d 1/
d
;
d
V
D
Z
V
Z Z
Z

 
 
 
U


1
4

;( ) extr -
x
F F f x Z x V
 


  
L
Write
d
d
x
V
Z

( )
Z



 
   
  
L
1
4
Now, in
eliminate
,( ) ( ),
A Z FF V
( )
V

1

2
1
2
Z


2
1
2
( ) min ( )
( )
D
D V
Z



 
 
 
 
U
The dual transformation



1
4
,( ) ( ) ( )
A Z F F V
 
   
  
L
eliminate
1
2
def
0
;
F A A
F
F F
       
 




 
    


:
.
( ) 0
Z F
 

 
Equations
def
1/4
( );;
(,) ( )
( )
G Z F G B B
B G G V
Z
      
 

 


   
  
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 contributions

to 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