Effective lattice theory for finite temperature Yang Mills
Georg Bergner
ITP GU Frankfurt
Lattice: July 31, 2013
Intro
PL correlators
Thermodynamics
1
Strong coupling effective action approach
2
Polyakov line correlators in the effective theory
3
Thermodynamics of SU(3) effective PL theory
4
Conclusions
Conclusions
in collaboration with O. Philipsen, J. Langelage
2/13
Intro
PL correlators
Thermodynamics
Conclusions
Introduction
Approach
effective description of finite temperature behaviour
(confined phase)
systematic derivation from the full Yang-Mills theory
Features
better access to certain regions in parameter space
tested also in heavy quark region
results for finite chemical potential possible
Further reference
talks by: J. Langelage, M. Neumann
This talk: compute/test further observables in Yang-Mills
3/13
Intro
PL correlators
Thermodynamics
Conclusions
Effective Polyakov loop action
e
−Seff [U0 ]
Z
=
[dUi ]
Y
e
β
Tr
6
Up +Up†
p
integrating out spatial links Ui
dimensional reduction from 3 + 1D to 3D
Uµ (x, t) → U0 (x) → Polyakov lines L(x)
no complete calculation possible
⇒ organization of interactions in Seff e.g. ordered by distance
several approaches: inverse MC, demon methods [Heinzl, Kästner,
Wozar, Wipf, Wellegehausen], relative weights [Langfeld, Greensite], ...
Z
Z = [dL]e −Seff [L]
4/13
Intro
PL correlators
Thermodynamics
Conclusions
Effective action from strong coupling
character expansion:
e
β
Tr
6
Up +Up†
X
=
(1 + dr ar (β)χr (Up ))
r ∈irreps.
expansion parameter u = af (resummation)
cluster expansion
+
+
−→
−→
[Polonyi, Szlachanyi]
5/13
Intro
PL correlators
Thermodynamics
Conclusions
Effective action from strong coupling and simulations
Seff = λ1 Snearest neighbors + λ2 Snext to nearest neighbors + . . .
ordering principle for the interactions
higher representations and long distances are suppressed
(u Nt ; u 2Nt ; u 2Nt +2 )
effective couplings exponentiate:
λ1 = u Nt exp(Nt P(u)) (resummation)
collect similar terms to log (resummation)
X
Snearest neighbors =
(λ1 <Li L∗j − (λ1 <Li L∗j )2 + . . .)
<ij>
=
X
log(1 + λ1 <Li L∗j )
<ij>
6/13
Intro
PL correlators
Thermodynamics
Conclusions
Simulations of the effective theory
Non-perturbative effects from MC simulation of effective theory.
as in pure SU(3) YM: 1st order phase transition, spont.
broken centre symmetry
higher representations, long distances suppressed in
continuum limit
(λ1 )c mapped back to (βc )eff → Tc
few percent difference (βc )eff to (βc )YM
7/13
Intro
PL correlators
Thermodynamics
Conclusions
Precise test of strong coupling approach
Polyakov line correlator
~ good test for effective actions
hL(~0)L† (R)i
related to free energy in presence of heavy quarks
~ = exp(−F (|R|,
~ T )/T )
hL(~0)L† (R)i
~
continuum: depends only on |R|;
lattice: dependence on the direction
(breaking of rotational symmetry)
sign for the restoration of rotational symmetry in the
continuum limit
precise check
8/13
Intro
PL correlators
Thermodynamics
Conclusions
Polyakov loop correlator
β = 5.4, Nt = 6
β = 5.0, Nt = 6
1
1
effective Model one coupling
full SU(3)
0.1
effective Model one coupling
full SU(3)
0.1
0.01
0.01
0.001
0.001
0.0001
0.0001
1e-05
1e-05
1e-06
1e-06
1e-07
1e-07
1e-08
1e-09
0
2
4
6
8
10
12
1e-08
0
2
4
6
8
10
12
YM strong coupling region: multilevel and mulithit algorithm
deviations close to (βc )YM ; but still reasonable agreement
larger deviations in off-axis correlator
next to nearest neighbor interactions: small improvement
9/13
Intro
PL correlators
Thermodynamics
Conclusions
Strong coupling off-axis correlator
Small λ1 behaviour: F (R/a, T )/T = d(R/a)Nt C (β)
d(R/a) smallest number of lattice spacings connecting points
with distance R/a on the lattice
breaking of rotational symmetry as in strong coupling YM
no UV 1/r part at small λ1
⇒ approximates strong coupling correlators
β = 5.5, Nt = 6
20
15
15
F(R,T)/T
F (R, T )/T
β = 5.0, Nt = 6
20
10
5
0
10
5
full SU(3)
effective model
small λ1 approx.
0
0.05
0.1
0.15
0.2
RT
0.25
0.3
0.35
0.4
0
full SU(3)
effective model
0
0.1
0.2
0.3
0.4
0.5
0.6
RT
10/13
Intro
PL correlators
Thermodynamics
Conclusions
Polyakov loop correlator and continuum limit
continuum behaviour at large λ1 :
effective model close to (βc )eff ↔ YM close to (βc )YM
both: restoration of rotational symmetry
YM: scaling behaviour of renormalized correlator
√
effective model: still need identify scaling region ( σ/T )
(using scale setting of YM)
T /TC = 98%
T /Tc = 98%
2
4
3
-2
FR (R, T )/T
FR (R, T )/T
Nt = 8
Nt = 10
0 Nt = 12
Nt = 14
-4
-6
-8
-10
Nt = 6
Nt = 8
Nt = 10
Nt = 12
2
1
0
-1
0
0.1
0.2
0.3
0.4
RT
0.5
0.6
0.7
-2
0.1
0.2
0.3
0.4
0.5
0.6
R/r0
11/13
Intro
PL correlators
Thermodynamics
Conclusions
Effective theory and thermodynamics
primary observable:
d p
= ∆S
dβ T 4
better agreement with YM than strong coupling expansion
useful to get small T /Tc results
10
e. model
full SU(3) YM
1
∆S
0.1
0.01
0.001
0.0001
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Nτ = 4
T /Tc
12/13
Intro
PL correlators
Thermodynamics
Conclusions
Conclusions
systematic derivation of effective PL theory:
strong coupling series
non-perturbative simulations of effective theory: reasonable
agreement with full theory in confined phase in contrast to
strong coupling results
towards continuum limit higher orders in the expansion are
important
Can we identify intermediate scaling region?
2
Tc from (λ1 )c
2 Polyakov loop correlators: must be outside perturbative region
of effective theory ⇒ close to (λ1 )c , below certain Nt
open issue: scale setting / renormalization in effective theory
2 improved strong coupling results also for thermodynamics
13/13