Heavy quark potential and running coupling in QCD

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


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


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 [...]“


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


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


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 →


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


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


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


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


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


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)]


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


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


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)]


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]


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(


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“)


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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Heavy quark potential and running coupling in QCD

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