Coulomb Gauge on the lattice · 29.07.2013, Hannes Vogt, Giuseppe Burgio, Markus Quandt, Hugo...

Faculty of ScienceInstitute for Theoretical Physics

Coulomb Gauge on the latticeFrom Zero to Finite Temperature

29.07.2013, Hannes Vogt, Giuseppe Burgio, Markus Quandt, Hugo Reinhardt

Content

• Gluon propagator- in SU(3) and SU(4) at T = 0- at T > 0 on anisotropic lattices

• Ghost propagator at T > 0

• Coulomb potential- Effect of Gribov copies- Results at T > 0

• Summary and Outlook

2 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen

The gluon propagator

D(|p| ,p0) =1

9V〈Aa

i (|p| ,p0)Aai (− |p| ,−p0)〉

• Static propagator shows scaling violations- p0 dependence in the non-instantaneous propagator after

residual gauge fixing• Scaling violations resolved

- in lattice Hamiltonian limit ξ →∞ (anisotropic lattices)- by taking the static propagator at fixed p0


The gluon propagator

• Non-instantaneous bare propagator factorizes

D(|p| ,p0) = fβ(|p|)gβ(z), z =p0

|p|

• Identify gβ(z) from fit of

gβ(z) =D(|p| ,p0)

D(|p| ,0), g(0) ≡ 1

• Integrate out the z dependence

D(|p|) =1Nt

∑p0

|p| D(|p| ,p0)

gβ(z)


The gluon propagator at T=0

• In SU(2) on isotropic lattices: Gribov form

D(|p|) =1√

|p|2 + M4

|p|2

• For SU(3) and SU(4) modify the mid-momentum region:

D(|p|) =1√

|p|2 +γM2 + M4

|p|2

• On anisotropic lattices:

D(|p|) =1√

|p|2 +γM2+αM3

|p| + M4

|p|2


The gluon propagator at T=0

• In SU(2) on isotropic lattices: Gribov formD(|p|)|p|

=1√

|p|4 + M4

• For SU(3) and SU(4) modify the mid-momentum region:

D(|p|)|p|

=1√

|p|4 +γM2 |p|2 + M4

• On anisotropic lattices:

D(|p|)|p|

=1√

|p|4 +γM2 |p|2+αM3 |p| + M4


SU(3) gluon propagator, V = 244

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8

D(|p|)/|p|[1/G

eV2 ]

|p| [GeV ]

Gribov formula, M = 1.02 GeVmodified Gribov formula, M = 0.80 GeV

β = 5.7β = 5.9β = 6.0


SU(4) gluon propagator, V = 244

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8

D(|p|)/|p|[1/G

eV2 ]

|p| [GeV ]

Gribov formula, M = 0.95 GeVmodified Gribov formula, M = 0.78 GeV

β = 10.550β = 10.700β = 11.085


Finite Temperature & anisotropic lattices

• Temperature given by T = 1Ntat

• IR analysis on isotropic lattices limited to low temperature(T < 1.5Tc)• Solution: anisotropic lattices ξ = as/at

- up to T = 6Tc at ξ = 4 possible

• Setup:- SU(2)- anisotropy ξ = as/at = 4- lattice volume V = Nt × 323

- 100 configurations- gauge fixing: Simulated annealing and Overrelaxation

with varying number of copies


Anisotropic gluon propagator up to T = 3Tc

5 sets of configurations for each T (β ∈ [2.25,2.64])

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

D(|p|)/|p|[1/G

eV2 ]

|p| [GeV ]

T = 0




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

D(|p|)/|p|[1/G

eV2 ]

|p| [GeV ]

T = 0T = 1.5Tc




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

D(|p|)/|p|[1/G

eV2 ]

|p| [GeV ]

T = 0T = 1.5TcT = 3.0Tc



3 or 4 sets of configurations for each T (β ∈ [2.49,2.64])

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14

D(|p|)/|p|[1/G

eV2 ]

|p| [GeV ]

T = 0T = 1.5TcT = 3.0TcT = 4.0TcT = 6.0Tc


The ghost propagator

G(p) =1

3Ns

⟨∑x ,y

eip(x−y) [M−1]aa(x ,y)

⟩

• ghost dressing function d(|p|) = |p|2 G(|p|)• continuum results: IR power-law

d(|p|) ∼ 1|p|κ

• asymptotic freedom: UV power-law with log corrections:

d(|p|) ∼ 1

logγ |p|m

• continuum sum rules violated


Ghost dressing function at fixed β = 2.5 and ξ0

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7

d(|p|)

|p| [GeV ]

192× 483,T = 0128× 323,T = 0

16× 323,T = 1.5Tc


Ghost dressing function at fixed β = 2.5 and ξ0

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7

d(|p|)

|p| [GeV ]

192× 483,T = 0128× 323,T = 0

16× 323,T = 1.5Tc8× 323,T = 3.0Tc6× 323,T = 4.0Tc4× 323,T = 6.0Tc


Ghost dressing function at T = 1.5Tc

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 2 4 6 8 10 12 14

d(|p|)

|p| [GeV ]

β = 2.40,Nt = 12β = 2.49,Nt = 16β = 2.54,Nt = 20β = 2.59,Nt = 24β = 2.64,Nt = 32

γ = 0.63,m = 0.22GeVκ = 0.47

UV: equally well fitted with fixed γ = 0.5 (m = 0.44GeV )


Ghost dressing function at T = 3.0Tc

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 2 4 6 8 10 12 14

d(|p|)

|p| [GeV ]

β = 2.49,Nt = 8β = 2.54,Nt = 10β = 2.59,Nt = 12β = 2.64,Nt = 16

γ = 0.42,m = 0.83GeVκ = 0.46


Coulomb Potential

Vc(p) =1

3Ns

⟨∑x ,y

eip(x−y) [M−1(−∆)M−1]aa(x ,y)

⟩

Vc(|p|) ∼ |p|4 at |p| → 0 ⇐⇒ Vc(r ) ∼ σcr at r →∞

• Problems:- strong Gribov copy effects- Conjugate Gradient inversion is costly- extrapolation to the string tension based on few datapoints

lim|p|→0

Vc(|p|) = 8πσc


Gribov problem (T = 0, β = 2.2)

10

15

20

25

30

35

40

45

50

55

0 0.5 1 1.5 2 2.5 3

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

5 copies



10

15

20

25

30

35

40

45

50

55

0 0.5 1 1.5 2 2.5 3

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

5 copies100 copies



10

15

20

25

30

35

40

45

50

55

0 0.5 1 1.5 2 2.5 3

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

5 copies100 copies500 copies



10

15

20

25

30

35

40

45

50

55

0 0.5 1 1.5 2 2.5 3

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

5 copies100 copies500 copies

5000 copies


Gribov problemraw data, configurations with highest contribution separated

0

50

100

150

200

250

300

0 0.5 1 1.5 2 2.5 3

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]


Remarks on gauge fixing and the Coulomb potential

Compare the functional value F g[U] to Vc(p = (1,1,1))at fixed timeslice in 10,000 gauge copies

• extreme values of Vc not correlated to F g[U]

• in average Vc gets smaller at increasing F g[U]

• the highest value of F g[U] is found in more than 10 copies• exceptional configurations are found in ≈ 1% of the copies

Want to try yourself? http://www.culgt.com


Coulomb potential at T ≥ 0

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6 7

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

T = 0,128× 323, β = 2.25T = 0,128× 323, β = 2.49



0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6 7

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

T = 0,128× 323, β = 2.25T = 0,128× 323, β = 2.49

T = 1.5Tc,8× 323, β = 2.25T = 1.5Tc,16× 323, β = 2.49



0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6 7

|p|4

Vc(|p|)[G

eV2 ]

|p| [GeV ]

T = 0,128× 323, β = 2.25T = 0,128× 323, β = 2.49

T = 1.5Tc,8× 323, β = 2.25T = 1.5Tc,16× 323, β = 2.49T = 3.0Tc,4× 323, β = 2.25T = 3.0Tc,8× 323, β = 2.49


Summary

• The static gluon propagator obtained by integrating out the p0dependence is multiplicatively renormalizable• At finite temperature:

- No quantity is sensitive to deconfinement up to 1.5Tc- Differences set in at larger T- The gluon propagator is further IR-suppressed at higher

temperature• Extrapolation of σc from Vc has large uncertainties:

- Large Gribov copy effect- Large outliers need much higher statistics- UV scaling is defective: go to weak coupling limit


Outlook

• Control the exceptional configuration in ghost and Vc inversion→ deflation techniques• T-dependence of Vc calculated from the temporal gluon

propagator 〈A0A0〉 or partial Polyakov lines [log〈U0U†0〉]


Thank you.Contact:

Faculty of ScienceInstitute for Theoretical PhysicsAuf der Morgenstelle 1472076 Tübingen

[email protected]


Coulomb Gauge on the lattice · 29.07.2013, Hannes Vogt, Giuseppe Burgio, Markus Quandt, Hugo...

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