Scalar theory pressure
beyond the local potential approximation
Andreas Ipp
Vienna University of Technology
Jean-Paul Blaizot, AI, Nicolás Wschebor,
Nucl.Phys.A849:165,2011

YITP Kyoto, Sept 1, 2011

QCD phase diagram

2

QCD pressure
Perturbative expansion of QCD pressure converges badly
P/P0

Perturbation theory:

1.4
1.2

g4

g2: Shuryak; Chin (1978)
g3: Kapusta (1979)
g4 ln g: Toimela (1983)
g4 : Arnold, Zhai (1994)
g5 : Zhai, Kastening (1995),Braaten, Nieto (1996)
6g ln g: Kajantie, Laine,Rummukainen, Schröder(2002)6g (partly): Di Renzo, Laine,Miccio,Schröder, Torrero (2006)

g3

1.5
0.8

g2
g5

0.6
1.5

22.5

3

3.5

4

4.5

5

T/TC
Lattice data: G. Boyd et al. (1996); M. Okamoto et al. (1999).
3

QCD pressureSelf–consistent 2PI resummation works for
S/S01

Φ-derivable approximation

NLABlaizot, Iancu, Rebhan,
PRD63 (2001)

0.9
0.8

T 2.5 T C

HTLtested at large NfBlaizot, AI, Rebhan, Reinosa,
PRD72 (2005)

0.7
0.6

1.5

22.5

3

3.5

Lattice data: G. Boyd et al. (1996).

44.5

5T/TC

2PI: 2 Particle Irreducible diagrams
HTL: Hard Thermal Loop

4

Scalar theory
Poor convergence appears also in other systems

QEDScalar φ4 theory

Andersen, Strickland, Su, PRD80, 085015 (2009).

R. Parwani and H. Singh,PRD51, 4518 (1995),

Also: Large N φ4 theory
Drummond, Horgan, Landshoff, Rebhan, Nucl. Phys. B524, 579 (1998).

5Outline
Introduction
Thermodynamics at finite temperature
Non-perturbative renormalization group
BMW approximation (Blaizot–Mendez-Galain–Wschebor)
BMW at finite temperature
Role of the regulator
Numerical results
Summary
6Perturbation theory at finite T
12 1
Scalar O(N) theory L= [∂  x] − m20  2  x− g 2 [ 2  x]222
3
1/T
F.T.
dq
3T ∑∫...
3action S =∫0 d ∫ d x L x
(2 π)
ωwith 1/T =0npressure

screening mass

Matsubara frequency  n =2  T n

scattering amplitude

7

Screening mass
Leading order from perturbation theory:3dq
1222m D =12 g T ∑ ∫+δm(2 π)3 ω 2n+q 2ωn2 ∞6g= 2 ∫ dq q n(q)π 0=g 2 T 21n(q)= q/Te −1Resum IR divergent diagrams to obtain g3 contribution:∑d3q11m =12 g T ∑ ∫−322222(2π)ω+q+mω+qω
nn2
D2n2()∞

6g−m2= 2 T ∫ dq 22πq+m0
3
3=− g 2 T m=− g 3 T 2ππ(for ω n=0)8

Running of the coupling
scattering amplitude2∂g
9 426 g =
= 2 g O  g ∂ 2

T =0g2T 0g29 4 g ln2 '2−9 3g2⇔119'=log
g 2 g  ' 2 2  2implies Landau pole22 Landau = exp
29 g

coupling diverges at Landau pole
coupling is bound and vanishes at small scales
9Self-consistent 2PI resummation
Baym (1962); Luttinger, Ward (1960); Cornwall, Jackiw, Tomboulis (1974)

Pressure expressed through dressed propagators:
4d k−1−P =∫n(ω)(ℑ
log
G−ℑ ΠG)+T Φ[G ]/V4(2 π)Φ [G ] =+ ...

+with G−1 =G−10 +Π

2PI: 2 Particle Irreducible diagrams

Gap equation:∂ Φ[G ]
Π=2=
∂G++ ...
10

Flow equation
effective action

1d3q∂t   []= T ∑ ∫∂ R QG  Q ; 3 t 2 n
2 −1propagator G

2Q ; ≡  Q ,−Q ; R Q

=Tetradis, Wetterich 1993

regulator

four-momentum vectors:

Q=n , qQ=∣Q∣
Matsubara frequency  n=2  n T

momentum derivative: ∂ t ≡ ∂
scalar field: ≡1 2
211

Flow equation
effective potential

1d3q∂t V κ (ρ)= T ∑ ∫∂ R (Q)G κ (Q ; ρ)
3 t κ
2 n(2 π)
−12
GQ;
≡
propagator  Q ,−Q ; R Q

=Infinite tower of coupled
differential equations:approximation necessary3dq
22∂t   P ,−P ; =T ∑ ∫∂RQG Q ; 3 t 2 n{×  3 P ,Q ,−P−Q ; G  QP ;  3
 −P , PQ ,−Q ;

}

1 4
−   P ,−P , Q ,−Q ; 212Local potential approximation
1d 3q∂t V  = T ∑ ∫∂ R QG  Q ; 3 t 2 n
2 =−1
2GQ;≡propagator  Q ,−Q ; R Q

2∂V2222  Q ,−Q ; =Q m  with m ≡∂ 2Applying BMW method at the bottom of the hierarchy,
one recovers the local potential approximation.
13

BMW method
effective potential−11d 3q
2
∂t V  = T ∑ ∫∂ R Q [   Q ,−Q ; R Q ]3 t 2 n2 3d q2∂ t  P ,−P ; =T ∑ ∫∂RQG Q ; 3 t 
2 n2{ 33×  P ,0 ,−P−0 ; G  QP ;  −P , P0 ,−0 ;

}1−  4 P ,−P , 0 ,−0 ; 2

BMW method:“set loop momentum Q=0 in the n-point
functions on the r.h.s of the flow equation”
Blaizot–Mendez-Galain–Wschebor, 2006

close equations with:22

∂    P , 3  P ,−P , 0 ; =,∂ 4 P ,−P ,0 ,0 ; =2∂    P , ∂214

Technicalities
The second flow equation contains also information about the potential2 P=0 ,−P=0 ; =∂ 2 V  ∂2In order to respect this relation, it is better to use the setV   , 2
2   P ;= 2P;−

P=0;
−P

It is convenient to define the Z-factor in the regulator.

Z =∣∂  2  p 0=0, p , ∂p2=1
2P =0, =0∂    p0 =0, p , ∂p2∣2P =0,=015

3D vs. 4D regulator
LPA: 3D regulator possible.

R  n ,q= 2 −q 2  2−q 2 
Litim regulator

BMW: Euclidean method.
4D regulator more appropriateR κ (Q)=Z κ κ 2 r (Q / κ)α q2analytical treatment of
Matsubara sums possible

12n  1T∑
=222 n −i n  
But: does not respect
Euclidean invariance;
frequency is not regulated

2or r (q)=α e−β q −γ q

4r (q)= qe −1
with β=1/2, γ=1/24
2

Exponential regulator

Need to sum overMatsubara frequencies.

16Applying the flow equation
classical theory
Effective action links classicaland quantum theory

=

Coupling is obtained from zerotemperature flow at momentum

=2 Tg 2 T 
T =0T 0quantum theory

=017

Integration domains
Tκ∂t V (ρ)−∂ t VT =0
κ

(ρ)

=nmax

Λ d3q
1
1 Λ d4q
TT =0= T ∑ ∫∂R(Q)G(Q;ρ)−∂R(Q)G(Q ; ρ)∫κκ
3 t κ4 t κ
2 n=−n2(2 π)
(2 π)
max

Euclidean invariance imposesfour-dimensional sphere
Summation over Matsubara
frequencies implies a fourdimensional cylinder
Mismatch can give rise tospurious pressure contributions
18

Flow of couping
weak coupling

1g = V ' ' ∣=082
strong coupling

Region of flow shrinks but noqualitative difference
19Flow of Z-factor

 2

Z=

∂    p0 =0, p , ∂p2∣

small anomalous dimension at d=42

P =0, =020

Flow of mass and pressurem 2 =V ' ∣=0P =−V  0 

numerically challenging
21

Mass and pressure at finite T

2PI and the FRG methods LPA and BMWagree wellmuch better behaved than perturbation theory

2PI: 2 particle irreducible
LPA: local potential approximation
BMW: Blaizot, Mendez-Galain, Wschebor

22
LPA (Litim): Blaizot, AI, Mendez-Galain, Wschebor, NPA (2007)
BMW: Blaizot, AI,Wschebor, NPA (2011)Pressure as function of mass

Comparison of renormalization-scale
independent quantities

2PI: 2 particle irreducible
LPA: local potential approximation
BMW: Blaizot, Mendez-Galain, Wschebor
SPT: Screened perturbation theory
23

SPT: Andersen and Kyllingstad, Phys. Rev. D78, 076008 (2008)

Dependence on temperature
LPA2q2−β q 2−γ q 4r (q)=α er (q)=α q /(e −1)classical theory

=Flow ofcouplingg 2 T T 0T =0=0quantum theory24Regulator dependence
LPA

2q2r (q)=α q /(e −1)−β q 2−γ q 4r (q)=α e

Even though regulators seem similar, the pressure
is sensitive to the choice of the regulator
25

Flow of the potential
LPA

2q2r (q)=α q /(e −1)−β q 2−γ q 4r (q)=α eDifference of the flow ofthe potential at finite
and zero temperatureR Q=Z   2 r Q /26

Oscillations due to regulator
Oscillations can be understood in complex q0 plane−11d3q(2)∂t V κ (ρ)= T ∑ ∫∂ R (Q) ( Γ κ (Q ,−Q ; ρ)+ Rκ (Q) )3 t κ
2 n(2 π)Pole structure q2r q= qe −1
2

Difficulty of finding suitable regulator for realtime quantities?
Analytic continuation from imaginary to real axis?
Heavy Rotation [AKB48]

27

OutlookCalculation of real-timequantities at finite
temperature like particle
width are not readily
accessible within LPA.
Need approximation
scheme beyond LPA.
But: analytic continuation
may turn out to be tricky.

Particle width: imaginary
part of 2-point function

Perturbative calculation of
damping rate
Parvani (1992), Wang and Heinz (1996)28

Summary
Non-perturbative renormalization group
Approximation necessary
BMW at finite temperature
4D regulator more appropriate
Results for pressure and mass similar to
local potential approximation
OutlookCalculate real time quantities
29