Testing effective Polyakov loop actions
derived form a strong coupling expansion
Georg Bergner
ITP GU Frankfurt
Schladming: March 1, 2013
Intro
phases at strong coupling
tests of the effective theory
1
Introduction
2
Phase transitions from strong coupling expansion
3
Further tests of the effective theory
4
Conclusions
Conclusions
in collaboration with O. Phillipsen, J. Langelage, S. Lottini,
W. Unger, M. Neuman, M. Fromm
2/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
3/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
The final goal
critical temperature: confinement → deconfinement
critical temperature: chiral symmetry restoration
properties of the phases: (T ), p(T ), screening length, ...
4/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
QCD on the lattice
Z
Z=
Dφ e
−S[φ]
=
Z Y
dφi e −S[φ]
i
discretized continuum action
non-perturbative computations
offers different expansion schemes
(“opposite of” weak coupling perturbation theory)
gauge fields
Aµ → e igaAµ = Uµ :
matter fields ψ, φ:
1
T = L1t = aN
t
5/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
QCD on the lattice: the action
L=β
X
X
1
1 − <(TrUp ) +
ψ̄f (D[U] + mf )ψf
3
p
f
plaquette Up =
integral of group elements: Haar measure dU
integral of Grassmann fields: integrated out
Z=
Z Y
i
dUi
Y
det(D[U] + mf ) exp(−Sg [U])
f
6/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
QCD on the lattice: fermions
X
ψ̄(x)(D − m)x,y ψ(y ) =
x
+
Xh
(m + 4r )ψ̄(x)ψ(x)
x
1X
2
ψ̄(x) (γµ − r )Uµ (x)ψ(x + µ̂) + (γµ + r )Uµ† (x − µ̂)ψ(x − µ̂)
i
µ
lattice Dirac operator D: derivatives replaced by gauge
invariant difference operators
1
hopping parameter 1 + κH with κ = 2m+8r
spacial (Ui ) and temporal (U0 ) hops
Wilson-Dirac operator: additional momentum dependent mass
term ⇒ chiral symmetry breaking
fine tuning κ → κc : chiral continuum limit
7/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Confinement/Deconfinement
static quark-antiquark potential:
confinement:
linear rise at large distance r :
V (r ) = − cr + σr
deconfinement:
Yukawa type screening potential
Q t
Polyakov loop: L(x) = TrW = Tr[ N
t=0 U0 (t, x)]
L puts infinitely heavy quark in
the theory
0.5
0.45
0.4
0.35
hLi = exp(−(FQ − F0 )/T )
<|L|>
0.3
0.25
0.2
confinement FQ → ∞: hLi = 0
0.15
0.1
0.05
0
4.5
5
5.5
β
6
6.5
deconfinement: hLi =
6 0,
L around center elements
8/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Lattice QCD and finite density
Z (T , µ) = Tr(e −(H−µQ)/T )
continuum physics: extra term µψ̄γ0 ψ
on the lattice modification of D
ψ̄(x) (γ0 − r )e aµ U0 (x)ψ(x + 0̂) + (γ0 + r )e −aµ U0† (x − 0̂)ψ(x − 0̂)
γ5 D(µ)† γ5 = D(−µ∗ ) ⇒ det(D(µ)) = det(D(−µ∗ ))∗
complex determinant
all methods fail at large µ
⇒ any information about the model at finite µ is helpful,
even in unphysical limit.
⇒ need playground to test methods and find possible effects.
9/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Strong coupling expansion in lattice gauge theory
Z
Z=
[dUµ ]
Y
e
β
Tr
6
Up +Up†
p
expansion in β = 6/g 2 (opposite to weak coupling)
similar to high temperature expansion in statistical physics
simple integration rules for products of plaquette contributions
Z
Z
dU U =
†
dU U = 0;
Z
dU UU † =
1
1
3
10/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Static quark-antiquark potential in strong coupling limit
simplest example: h Wilson loop i
→
∝
RT
β
18
first contribution: Loop filled with plaquettes
confinement:
V (R) = − limT →∞ T1 loghW i = −σR
P
extension: O = n On β n
in certain region convergent series
11/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Effective action for the Polyakov loop
e
−Seff [U0 ]
Z
=
[dUi ]
Y
e
β
Tr
6
Up +Up†
p
integrating out spatial links
final result depends only on Polyakov lines L
dimensional reduction from 3 + 1D to 3D
Uµ (x, t) → U0 (x) → L(x)
no complete calculation possible
⇒ expansion of Seff e.g. in terms of interaction distance
several ways to calculate it: inverse MC, demon methods
Kästner, Wozar, Wipf, Wellegehausen], relative weights [Langfeld, Greensite], ...
Z
Z = [dL]e −Seff [L]
[Heinzl,
12/25
Intro
phases at strong coupling
tests of the effective theory
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]
13/25
Intro
phases at strong coupling
tests of the effective theory
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>
14/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Results in pure Yang-Mills
obtained with the effective action
observables from nonperturbative MC simulation of Seff
(resummation)
confinement/deconfinement
phase transition
mapping back to βc
⇒ Tc (continuum limit)
15/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Quality of the results
Next to nearest neighbors, adjoint rep. ...
... are not important for the phase transition in continuum limit.
βc relative error
effective theory
⇒ good agreement
16/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Quarks in the effective action
"
Sq = − log
#
Y
det(D − m) = −Nf Tr log(1 − κH) = Nf
f
X κl
l
l!
TrH l
⇒ truncate hopping parameter expansion
simplest contributions (no spacial hops):
single Polyakov lines (L,L∗ )
integrating out spacial links in strong coupling expansion
Q
resummation exp(−Sq ) → x det()
Y
det((1 + hW (x))(1 + h̄W ∗ (x)))2Nf
x
=
Y
2N
(1 + hL(x) + h2 L∗ (x) + h3 )(1 + h̄L∗ (x) + h̄2 L(x) + h̄3 ) f
x
[de Pietri, Feo, Seiler, Stamatescu]
17/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Quarks in the effective action
"
Sq = − log
#
Y
det(D − m) = −Nf Tr log(1 − κH) = Nf
f
X κl
l
l!
TrH l
⇒ truncate hopping parameter expansion
simplest contributions (no spacial hops):
single Polyakov lines (L,L∗ )
integrating out spacial links in strong coupling expansion
Q
resummation exp(−Sq ) → x det()
Y
det((1 + hW (x))(1 + h̄W ∗ (x)))2Nf
x
=
Y
2N
(1 + hL(x) + h2 L∗ (x) + h3 )(1 + h̄L∗ (x) + h̄2 L(x) + h̄3 ) f
x
[de Pietri, Feo, Seiler, Stamatescu]
17/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Quarks in the effective action
general form of the action
X
X
X
†
λi Ssymm.i +
hi Sasymm.i +
h̄i Sasymm.i
i
i
i
center symmetric part of the action similar to pure Yang-Mills
fermions introduce asymmetric contributions
finite µ introduces factor e ±aµ for temporal up/down hops
⇒ h 6= h̄
h(µ) = h̄(−µ) ⇒ sign problem
sign problem is mild (reweighting works in large region)
alternative algorithm: Worm algorithm
alternative algorithm: complex Langevin
18/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Quarks to the effective action
complicated pattern at higher order
quark lines building up plaquette like objects
⇒ λ(β, κ) (shift of β)
plaquette contributions to quark lines
⇒ h(κ, β)
interaction between quark lines
⇒ Li Lj interaction
19/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Results with quark matter and finite density
reproduce phase transition in heavy quark limit
phase transition at
finite densities
20/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Results with quark matter and finite density
Nuclear transition:
⇒ limited to mB ≈ 30 GeV
21/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Further test of this program
Polyakov line correlator
~ good test for effective actions
hL(~0)L† (R)i
related to static quark potential
~ = exp(−V (|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 [Greensite, Langfeld];
some nonperturbative methods might fail
22/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Test for the diagonal correlator
β = 5.4, NT = 6
β = 5.0
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
results obtained with Multilevel and Mulithit algorithm
to include strong coupling region:
need error below 10−9
deviations close to critical β; but still reasonable agreement
23/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Test for the off diagonal correlator
β = 5.4, NT = 6
20
15
15
V(R,T)/T
V(R,T)/T
β = 5.0
20
10
5
0
10
5
full SU(3)
effective model
0
0.1
0.2
0.3
0.4
0.5
0
full SU(3)
effective model
0
0.1
0.2
RT
0.3
0.4
0.5
RT
off diagonal terms remain closer to strong coupling behavior
less restoration of rotational symmetry
small improvement with next to nearest neighbor interactions
need more information about continuum limit
24/25
Intro
phases at strong coupling
tests of the effective theory
Conclusions
Conclusions and outlook
effective theories can be derived from a strong coupling series
includes explicit ordering principle,
higher orders suppressed with smaller β, larger Nt
reproduce phase transitions of pure Yang-Mills theory
quarks included in hopping parameter expansion
most important current limitation:
truncation of the κ series
⇒ higher orders included
limits Nt for the confinement/deconfinement transition
precise check: L correlator
need to understand the suppression of the interactions at
larger distances in the continuum limit
25/25