heavy quark potential and running coupling in qcd
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Heavy quark Heavy quark potentialpotential
and running couplingand running couplingin QCDin QCD
W. SchleifenbaumW. SchleifenbaumAdvisor: H. ReinhardtAdvisor: H. Reinhardt
University of TübingenUniversity of Tübingen
EUROGRAD EUROGRAD workshopworkshop
Todtmoos 2007Todtmoos 2007
Todtmoos 2007 W. Schleifenbaum
OutlineOutline
Some basics of Yang-Mills theorySome basics of Yang-Mills theory Functional Schroedinger equationFunctional Schroedinger equation Coulomb gauge Dyson-Schwinger Coulomb gauge Dyson-Schwinger
equationsequations Quark potential & confinementQuark potential & confinement Running coupling in Landau and Running coupling in Landau and
Coulomb gaugeCoulomb gauge
Todtmoos 2007 W. Schleifenbaum
Yang-Mills theoryYang-Mills theory
Local gauge invariance of quark fields:
Lagrangian acquires gauge field through
QCD: nonabelian gauge group SU(3)
Yang-Mills Lagrangian: dynamics of gauge fields
( )( ) ( ) i xq x e q x
( )aA x
( ) ( ) L q i m q q i D m q
D gA
[ ( ), ( )] 0 , [ ( ), ( )] 0x y A x A y
41( ) ( )
4
[ , ]
a aYML d x F x F x
F A A g A A
nonabelian term
Todtmoos 2007 W. Schleifenbaum
asymptotic freedom:
running coupling:
dimensional transmutation:→ express dimensionless g in terms of
nonperturbative methods:
End of perturbative End of perturbative methodsmethods
2
22 200
1 1( ) , exp
2 ( )ln / QCDQCD
g kgk
3 50
( )( ) ( )
lng
g g g
O
lattice gauge theory continuum approach via integral equations
2( )g k
k
„The hamiltonian method forstrong interaction is dead [...]“
Todtmoos 2007 W. Schleifenbaum
Good gauge? Need unique solution
infinitesimally:
Faddeev-Popov determinant
Gauge fixingGauge fixingtasktask: separate gauge d.o.f.: separate gauge d.o.f.
QED: .... easy: QED: .... easy:
YM theory: .... hard!YM theory: .... hard!
alternative methodalternative method: fix the gauge: fix the gauge
A A11 1
U
gA A UA U U U
[ ] 0A
[ ] 0A
[ ] 0 1UA U
1[ ] [ ]ln UA G A U
UA
A
CONFIGURATION SPACE
“I am not smarter, I just think more.”
1det [ ] 0 J G A
same physics
Gribov copy
IR physics
Todtmoos 2007 W. Schleifenbaum
Coulomb gauge Coulomb gauge HamiltonianHamiltonian
Canonical quantization:
Gauß‘ law constraint:
Weyl gauge Hamiltonian:
Coulomb gauge:
0 [ ] [ ] Ui iD A A
1 2 2 22 H B
1, A G D
1 1 2 2 1 22
H J J B g J G G J
curved orbit space→ gluon confinement
heavy quark potential → quark confinement
gaugeinvariance
0 00 0
[ , ] , 0 set 0i j ij
LA i A
A
Todtmoos 2007 W. Schleifenbaum
Yang-Mills Schroedinger equation:
ansatz for vacuum wave functional:
minimizing the energy:
mixing of modes:enhanced UV modes might spoil accuracy of IR modes IR modes are enhanced as well!
Variational principleVariational principle
1
--1/2 2AωA
ψ A = J A e 1 12
, AA N
[Feuchter & Reinhardt [Feuchter & Reinhardt (2004)](2004)]
H E
„„It‘s no damn good at all!“It‘s no damn good at all!“
??*
Λ
ψ H ψ = DAJ(A)ψ (A)Hψ(A) min →
Todtmoos 2007 W. Schleifenbaum
Gap equationGap equationInitially, only one equation needs to be solved:Initially, only one equation needs to be solved:
Ghost propagator: Ghost propagator:
Ghost Dyson-Schwinger equation:Ghost Dyson-Schwinger equation:
Gap equation: (infrared expansion)Gap equation: (infrared expansion)
Cf. Landau gauge – [Alkofer & von Smekal (2001)]
1 1
1 ( ) k
0 [ , ] ?H G
F
2
1 A AG A e
gA
D
Todtmoos 2007 W. Schleifenbaum
Tree-level ghost-gluon vertexTree-level ghost-gluon vertex
0 0G A G A G A G A
Non-renormalization:
Tree-level approximation:
Check by DSE/lattice studies (Landau gauge):[W.S. et al. (2005)][Cucchieri et al. (2004)]
crucial for IR behavior!
( )ijt l
renormalizationconstant:
1 1Z
Todtmoos 2007 W. Schleifenbaum
Infrared analysisInfrared analysisPropagators in the IRPropagators in the IR
Infrared expansion of loop integralsInfrared expansion of loop integrals
1
1 12 2
1( ) , ( ) , 0
2
A Bp G p p
p p
,)(22/122 GC INABpp
)(222/122
ZC INABpp
)(
)(1,
2
12
Z
G
I
I
[Zwanziger (2004); W.S. & Leder & Reinhardt (2006)]
Two solutions :1
0.3982
Todtmoos 2007 W. Schleifenbaum
IR sector is dominated by Faddeev-Popov determinantIn a stochastic vacuum,
we have the following expectation values,
and find the same equations:
Horizon condition: [Zwanziger (1991)]
Ghost dominanceGhost dominance
0 [ ] 1YM A L
[ ] [ ] [ ]O A DA J A O A
1 1 1
1 02
1( ) 0kG k
k
Todtmoos 2007 W. Schleifenbaum
If the ghost-loop dominates the IR, it better be transverse.
In d spatial dimensions, there are two solution branches:
Infrared transversalityInfrared transversality
Only obeys transversality!
{ 3, 1/ 2}d
12
supports
14
d
( 3)/ 2( ) dG p p
Coulomb gauge: d=3
12
Todtmoos 2007 W. Schleifenbaum
Full numerical solution for Full numerical solution for =1/2=1/2
• Excellent agreement with infrared analysisExcellent agreement with infrared analysis• (in)dependence on renormalization scale(in)dependence on renormalization scale• Confinement of gluonsConfinement of gluons
[D. Epple, H. Reinhardt, W.S., PRD 75 (2007)]
Todtmoos 2007 W. Schleifenbaum
Heavy quark potentialHeavy quark potentialTwo pointlike color charges, separated by Two pointlike color charges, separated by rr
Approximation:Approximation:
(cf. ghost-gluon vertex)(cf. ghost-gluon vertex)
Solution with Solution with =1/2 gives=1/2 gives
Coulomb string tensionCoulomb string tension
2
2( )2
C ext ext
gV r G G
[D. Epple, H. Reinhardt, W.S., PRD 75 (2007)]
2 2
0 0
G G G G
G G G G
rrV Cr
C )(
C
3
3 1 22
11
2
r ip rd p
ep
Todtmoos 2007 W. Schleifenbaum
Perturbative tails & talesPerturbative tails & tales
2 22 2
( ) , ( )ln ln
G A
2 2k ke e
ffG k AA k
k k
1. Landau gauge1. Landau gauge
In the ultraviolet, QCD is asymptotically free.Free theory:
Interacting theory: (from renormalization group)
Anomalous dimensions: (scaled by
2 2
1 1( ) ,G k AA
k k
9 13, , 2 1
44 22
Todtmoos 2007 W. Schleifenbaum
running coupling:
nonperturbative UV-asymptotics:
• ghost DSE: sum rule gives correct 1/log behaviour setting gives correct and !
• ghost and gluon DSEs: sophisticated truncation of gluon DSE necessary to reproduce
nonperturbative IR-asymptotics:
• finite
•depends on renormalization prescription [WS & Leder & Reinhardt (2006)]
0 0
2
2 20
1( )
ln /g k
k
0 0
[Fischer & Alkofer (2002)]
622 2
2( ) ( ) ( , ) ( , )G A
kg k g G k AA k
ff
(0) 8.9/ CN [Lerche & von Smekal (2002)]
Todtmoos 2007 W. Schleifenbaum
2. Coulomb gauge:2. Coulomb gauge:
perturbation theory still subject to ongoing researchFree theory:
Interacting theory: (ansatz)
running coupling
solution to gap equation:
2
1( ) , ( )G k k k
k
22
2
( ) , ( ) lnln
Ge
e
fG f
2k
2kk k k
k
522 2 1
2 1( ) ( ) ( , ) ( , )G
g g Gff
kk k k
2
0 02 20
1 810, , ( ) ,2 11ln /g
k
k
[Watson & Reinhardt, arXiv:0709.0140v1]
Todtmoos 2007 W. Schleifenbaum
numerical result:
[Epple & Reinhardt & WS (2007)]
set the only scale:
→ very sensitive to accuracy of (k)
should-be result:set in ghost DSE:
3/ 112 4/ 11
1( ) ln , ,
lnG k k k k
k k
0 0
( )ZM
CN3
16)0(
Todtmoos 2007 W. Schleifenbaum
Coulomb potential: over-Coulomb potential: over-confinementconfinement
Heavy quark potential involved simple replacement
Only upper bound for Wilson loop potential (→lattice)
Lattice calculations: too large by a factor of 2-3.No order parameter for „deconfinement“.
MISSING: ’s knowledge of the quarks.
( ) ( )CV r V r
†1 2( ) ( ) q x q x
[Zwanziger (1997)]
(„No confinement without Coulomb confinement“)
Todtmoos 2007 W. Schleifenbaum
Summary and outlookSummary and outlook
minimized energy with Gaussian wave minimized energy with Gaussian wave functionalfunctional
gluon confinementgluon confinement quark confinementquark confinement computed running coupling, finite in the IRcomputed running coupling, finite in the IR need for improvement in the UVneed for improvement in the UV calculation of Coulomb string tensioncalculation of Coulomb string tension
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