gerard ’t hooft spinoza institute utrecht, the netherlands utrecht university
TRANSCRIPT
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
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.
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
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