Hamilton approch to Yang-Mills Theory in Coulomb Gauge

H. Reinhardt

Tübingen

Collaborators:D. Campagnari, D. Epple, C. Feuchter, M. Leder, M.Quandt, W. Schleifenbaum, P. Watson

Plan of the talk

• Hamilton approach to continuum Yang-Mills theory in Coulomb gauge

• Variational solution of the YM Schrödinger equation: Dyson- Schwinger equations

• Numerical Results• Infrared analysis of the DSE• Topological susceptibility• `t Hooft loop• Conclusions

Classical Yang-Mills theory

24

41 ))((2 xFxdLg

AAAAxF ,)(

Lagrange function:

field strength tensor

Canonical Quantization of Yang-Mills theory

)()(/)( momenta xExALx ai

ai

ai

)( scoordinatecartesian xAa

0)( :gauge Weyl 0 xAa0)(0 xa

)(/)( :onquantizati xAix ak

ak

))()(( 22321 xBxxdH

Gauß law: mD

)()( :)x U(invariance gauge residual AAU

Coulomb gauge

mD Gauß law:

|| 1m( D ) , ( A )

resolution of Gauß´ law

)()(*)(| AAAJDAcurved space

Faddeev-Popov )()( DDetAJ

A 0, A A

|| , / i A

YM Hamiltonian in Coulomb gauge

)( 2||||1121 BJJJJH

-arises from Gauß´law =neccessary to maintain gauge invariance -provides the confining potential

Coulomb term11

C 2

1 1 2 112

m

H J J

J ( D ) ( )( D ) J

color density: A

Christ and Lee

aim: solving the Yang-Mills Schrödinger eq.

for the vacuum by the variational principle

with suitable ansätze for

H DAJ(A) (A)H (A) min

metric of the space of gauge orbits: )( DDetJ

Dyson-Schwinger equations

Importance of the Faddeev-Popov determinant

ˆDet( D )

defines the metric in the space of gauge orbitsand hence reflects the gauge invariance

12

1A exp A A

Det D

2*

12*

12

21 |

)(r , )(

drdrr

rJr

rQM: particle in a L=0-state

vacuum wave functional

x x´ determined from

H min

variational kernel

DSE (gap equation)

ghost propagator 1 dG ( D )

ghost form factor dAbelian case d=1

gluon propagator 11

2AA

gluon DSE (gap equation) 2 2 2k k k

k k ...

2

12

ln Det D

A A

curvature

gluon self-energy

Dyson-Schwinger Equations

ghost DSE

d( ),....

Regularization and renormalization:momentum subtraction scheme renormalization constants:

ultrviolet and infrared asymtotic behaviour of the solutions to the Schwinger Dyson equations is independent of the renormalization constants except for )(d

In D=2+1 is the only value for which the coupled Schwinger-Dyson equation have a self-consistent solution

)( criticaldd

critical

1

d( ) d :

d (k 0) 0

horizon condition Zwanziger

Numerical results (D=3+1)

k : d k 1/ ln k k k

k 0 : d k 1/ k k 1/ k

k 0

k k finite (renormalization) const.

ghost form factor gluon energy and curvature

Coulomb potential

4k 0

V(k) 1/ k

Coulomb form factor f

Schwinger-Dyson eq.

rigorous result to 1-loop: Rf p const

external static color sources

electric field

ghost propagator

1 DE

The color electric flux tube

missing: back reaction of the vacuum to the external sources

The dielectric „constant“ of the Yang-Mills vacuum

)/()( 1 dD

EDDdE , , Maxwell´s displecement

The Yang-Mills vacuum is a perfect color dia-electric

dielectric „constant“

)(/1)( kdk

k

)(k

comparison with lattice d=3

lattice: L. Moyarts, dissertation

D=3+1

Infrared behaviour of lattice GF: not yet conclusive too small lattices

related work: A.P. Szczepaniak, E. S. Swanson, Phys. Rev. 65 (2002) 025012 A.P. Szczepaniak, Phys. Rev. 69(2004) 074031

different ansatz for the wave functional did not include the curvature of the space of gauge orbits i.e. the Faddeev- Popov determinant

present work: C. Feuchter & H. R. hep-th/0402106, PRD70(2004) hep-th/0408237, PRD71(2005)

W. Schleifenbaum, M. Leder, H.R. PRD73(2006) D. Epple, H. R., W. Schleifenbaum, in prepration

full inclusion of the curvature

measure for the curvature

2

12

ln Det D

A A

Importance of the curvature

Szczepaniak & SwansonPhys. Rev. D65 (2002)

• the = 0 solution does not produce a linear confinement potential

kSzczepania &Swanson 0

rkpresent wo ansatz) (radial 21

AADDetA 21exp)(

Infrared limit = independent of

Robustness of the infrared limit

0/ 0/ dHdHto 2-loop order:

oft independen is AA

Infrared analysis of the DSE

A exp( S A / 2)

generating functional

vacuum wave functional:

d=4 Landau gauge functional integral

d=3 Coulomb gauge canonical quantization S A ...?

2

21g

S A F ghost dominance in the infrared

S A 0

A 1

strong coupling

Z j exp jA

DADet( D )exp( S A jA)

Analytic solution of DSE in the infraredLG: Lerche, v. Smekal Zwanziger, Alkofer, Fischer,…CG: Schleifenbaum, Leder, H.R.

gluon propagator2

AD(p)

p 2

BG(p)

p ghost propagator

basic assumption:Gribov´s confinment scenario at work

horizon condition:2 -1G(p) d(p) / p , d (p=0)=0 0

2 d 4 sum rule:

ghost DSE (bare ghost-gluon vertex)

Landau gauge d=4 2 0 infrared divergent ghost form factor 0

infrared finite gluon form factor <0

2 1 Coulomb gauge d=3

solution of gluon DSE

1.18, ( 1.0)

0.796(0.85), 1.0(0.99)

Coulomb gauge d=2 2 2 0.4

running coupling

sum rule for the infrared exponents from ghost DSE

2p 0 const

2

22 5Rg16p p G p D p

3 4

Fischer, Zwanzigerinterpolating gauges

Topological susceptibility

4d x FF(x)FF(0)

Witten-Veniciano formula:

,2

2F

2m

N F

Topological susceptibility in Hamilton approach

3 3U(x) : x R S U SU(2)

33winding number : n U (S ) Z

spatial gauge transformation:

i n UUA e A vacuum :

explicit realization: CSi S AA e A UA A

UCS CSS A S A n U Chern-Simon action:

2

2

0

1 d E( ), E( ) H

V d

topological susceptibility:

vanishes to all orders in g

2

2

2132

n n

n B 0V 0 B (x) 0

E

Identify our variational wave functional with the restriction of the gauge invarinant to Coulomb gauge A A 0

exact cancelation of the Abelian part of BB

very preliminary result (D. Campagnari -Diploma thesis) (very crude parametrization of the ghost and gluon GFs) :

4180MeV

Input: 2-loop formula for the running coupling

large variety of wave functionals produce the same DSE more sensitive observables than energy

Coulomb potential = upper bound for true static quark potential (Zwanziger)confining Coulomb potential (=nessary but) not suffient for confinement

Wilson loop 1N

C

W A C tr P exp A

order parameter of YMT

temporal Wilson loopexp( A(C)) confined phase

W[A](C)exp( P(C)) deconfine phase

difficult to calculate in continuum theory due to path ordering

spatialT 0 : W C

disorder parameter of YMT

spatial ´t Hooft looop

exp( P(C)) confined phaseV[A](C)

exp( A(C)) deconfine phase

continuum representation: H.R: Phys.Lett.B557(2003)

3V(C) exp i d x (C)(x) (x)A a ai i(x) / i A (x)

1 2L(C ,C )1 2W (C ) (C ) ZA

V(C)-center vortex generator

center vortex field CA

´t Hooft loop

1 2L(C ,C )1 2 2 1V(C )W(C ) Z W(C )V(C )defining eq.

1 2

Z (non trivial) center element

L(C ,C ) Gauß´ linking number

1C

2C

´t Hooft loop electric flux

C

E

Wilson loop magnetic flux

C

B

3V(C) exp i d x (C)(x) (x)A a ai i(x) / i A (x)

V C A A (C)A

didxexp iap x x a , p QM:

*V(C) DA (A) (A (C))A

wave functionals in Coulomb gauge satisfy Gauß´law and hence should be regarded as the gauge invariant wave functional restricted to transverse gauge fields.

´t Hooft loop in Coulomb gauge

*V(C) DA Det( D ) (A ) (A (C))A

infrared properties of K(p) determine the large R-behaviour of S(R)

p 0

K p 0 p p c(finite)

Det( D ) exp A A

representation (correct to 2 loop) H. R. & C.F. PRD71

2

12

ln Det D

A A

V(C) exp( S(C)) 0

S(C) dpK(p)h(C,p)

h(C;p)-geometry of the loop C

2

12

(p)K(p) (p) 1

(p)

properties of the YM vacuum

planar circular loop C with radius R

from gap equation k 0

k k finite (renormalization) const : c

renormalization condition:

c=0 produces wave functional which in the infrared approaches thestrong coupling limit A 1

V C exp perimeter(C)

V C exp perimeter(C) log(perimeter(C))

c 0

V C exp area(C)

neglect curvature 0

Summary and Conclusion

• Variational solution of the YM Schrödinger equation in Coulomb gauge

• Quark and gluon confinement• IR-finite running coupling• Curvature in gauge orbit space (Fadeev –Popov

determinant) is crucial for the confinement properties

• Topological susceptibility• ´t Hooft loop: perimeter law for a wave functional

which in the infrared shows strict ghost dominance

Thanks to the organizers