Post on 26-Jun-2020
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
3 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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)
4 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
5 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
6 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
7 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
8 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
9 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
10 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 = 0T = 1.5Tc
10 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 = 0T = 1.5TcT = 3.0Tc
10 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
Anisotropic gluon propagator up to T = 6Tc
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
11 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
12 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
13 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
13 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 )
14 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
15 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
16 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
17 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 copies100 copies
17 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 copies100 copies500 copies
17 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 copies100 copies500 copies
5000 copies
17 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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 ]
18 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
19 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
20 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
T = 1.5Tc,8× 323, β = 2.25T = 1.5Tc,16× 323, β = 2.49
20 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
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
20 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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
21 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
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〉]
22 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen
Thank you.Contact:
Faculty of ScienceInstitute for Theoretical PhysicsAuf der Morgenstelle 1472076 Tübingen
hannes.vogt@uni-tuebingen.de
23 | Hannes Vogt - Coulomb Gauge: From Zero to Finite Temperature c© 2013 University of Tuebingen