hamilton approch to yang-mills theory in coulomb gauge
DESCRIPTION
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. related work: D. Zwanziger A.P. Szczepaniak, E. S. Swanson, …. Plan of the talk. - PowerPoint PPT PresentationTRANSCRIPT
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
related work:
D. Zwanziger
A.P. Szczepaniak, E. S. Swanson, …
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• ghost-gluon and 3-gluon vertex• 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
Attempts to solve the Schrödinger equation with
gauge invariant wave functionals
K. Johnson,… gauge invariant variables
Karabali, Kim, Nair strong coupling expansion of the (D=2+1) YM wave functional
Greensite gradient expansion
projection techniquesKogan, Kovner,...Heineman, Martin, VautherinSchröder, H.R.
more efficient way: resolve Gauß´law explicitly by fixing the gauge
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
Importance of the Faddeev-Popov determinant
ˆDet( D )
defines the metric in the space of gauge orbitsand hence reflects the gauge invariance
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
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
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
comparison with lattice d=3
lattice: L. Moyarts, dissertation
D=3+1
Infrared behaviour of lattice GF: not yet conclusivetoo small lattices, see talk by A. Maas
previous 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
ghost gluon vertex
D=4 Landau gauge(Taylor): non-renormalization in all orders in g becomes bare for vanishing incoming ghost momentumd=3 Coulomb gauge: similar behaviour renormalization can be ignored
3- gluon vertex
a b c abc abci 1 j 2 k 3 ijk 1 2 3 ijk 1 2 3A (p ),A (p ),A (p ) : (p ,p ,p ) f (p ,p ,p )
Asumption: color structure of the bare vertex single form factor
2 d / 2 2 3 / 21 2 3
2 1.775 2 1.77
p p p : p F p (p ) (p)
d 3 , 0.85 F p (p ) (numer. F p (p ) )
ijk 1 2 3 1 ik 1 2 3jp ,p ,p i p F p ,p ,p perm.
3-gluon vertex in Coulomb gauge
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
´t Hooft loop
´t HooftMünsterTomboulisSamuelBhattacharya et alDel Debbio, Di Giacomo, B. LuciniChernodub et al.Korthals-Altes, Kovner,..de Focrand, D´Elia, Pepe, v. Smekal,….Quandt, H.R., Engelhardt….Recent review: Greensite
disorder parameter of YMT
spatial ´t Hooft loop
exp( P(C)) confined phase
V[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
3V(C) exp i d x (C)(x) (x)A a ai i(x) / i A (x)
gauge dependent but produces gauge invariant results when acting on gauge invariant states
2 3i; x d x x( )
iA
aaexp T Z
C
C
E
a 2 ai iV(C ) exp i d (x( ))
´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 • Infrared analysis and numerical solution of the resulting DSE• Quark and gluon confinement• Curvature in gauge orbit space (Fadeev –Popov determinant) is
crucial for the confinement properties• Ghost-gluon vertex = IR-finite• 3-gluon vertex= IR-divergent• ´t Hooft loop: perimeter law for a wave functional which in the
infrared shows strict ghost dominance• Current projects:
– QCD string– Topological susceptibility
Thanks to the organizers