Massive renormalization schemeand perturbation theory at finite temperature
QFT Workshop

YITP, Kyoto

September 1st, 2015

Based on work done with N. Wschebor, arXiv 1409.4795(Phys.Lett. B741 (2015) 310-315)

Jean-Paul Blaizot, IPhT- Saclay

What is the origin of the strongly coupledcharacter of the quark-gluon plasma?
A puzzling situation- The coupling constant is not small, but not huges0.3 ÷ 0.4- Strict perturbation does not work, but successful resummations exist- Our present understanding of early stages of HI collisions relies on weakcoupling concepts

Clue?
-

«Strong coupling» behavior may appear at weak coupling, whenmany degrees of freedom contribute coherently (e.g. collective
phenomena, BCS, CGC, etc) 

-The quark-gluon plasma is a multiscale system

Strict perturbation theory breaks downBut
- this has (almost) nothing to do with QCD

QCD at finite temperature
Perturbation theory is ill behaved

Similar difficulty in the caseof scalar field theory at finite temperature

Scalar field theory with quartic couplingScalar field theory with quartic coupling

Still, asymptotic freedom works !

Pressure for SU(3) YM theory at (very) high temperature(from G. Endrodi et al, arXiv: 0710.4197)

A continuing effort•

Calculate higher orders….

⇣⌘
Pressure in scalar theory is known up to order O g ln g
8J.O. Andersen, L. Kyllingstad and L.E. Leganger, arXiv:0903.4596•reorganize perturbation theory,resum, 2PI, NPRG, HTL pert. th.
etc)

DIMENSIONAL REDUCTION
Integration over the hard modes(gT ≤ Λ E ≤ T)

Di = ∂i − igE AiIn leading ordergE ≈ g T

Integration over the soft modes

Non perturbative contributionmE ≈ gTλE ≈ g 4 T(g 2T ≤ Λ M ≤ gT)

State of the art in high order perturbative calculations( from M. Laine, Y Schroeder, hep-ph/0603048)

( from J. O. Andersen, Nan Su, M. Strickland, arXiv: 1005.1603

Why is pertubation theory so bad at finite
temperature while it is fine at zerotemperature (for comparable values of thecoupling constant) ?

uctuations when  . m, are manifest order by order in perturbation theory.
We shall now present
such a scheme.
Expansion
parameter and thermal
We consider the theory of afluctutations
scalar field ' with the action
ZZn1o22mg
BS['] =d⌧ dd x @µ '@µ ' +'2 + B '4 ,224!
0(4)here mB and gB denote the bare mass and coupling constant, respectively.
he upper bound of the integration over the imaginary time ⌧ is= 1/T ,here T is the temperature. In line with the previous discussion, we introduce
specific renormalization scheme with the following, temperature dependent,
normalization conditions:
m21=(2)(p = 0, ! = 0, T ),d (2)= dp2 (p2 = µ2 , ! = 0, T ),Suggests a breakdown of perturbation theory when2.g T

1 In QCD, the long wavelength “magnetic” fluctuations have a mass of order g 2 T , which ist large enough to prevent the breakdown of perturbation theory.

Weakly AND strongly coupled …
Degrees of freedom with different wavelengths aredifferently coupled.
Expansion parameter

Dynamical scalesNon perturbative renormalization group

Courtesy, A. Ipp

Non perturbative renormalization groupat finite temperature( J.-P B, A. Ipp, N. Wschebor, 2010 )g3

P ! P01.00

1.10

6g log gg4

1.05

0.98

g5

0.96

LPA "exp#
LPA "Litim#

0.94

P # P0

1.02

1.00
0.95
0.90

2g

0.85

0.92

2 PIBMW

0.90
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.80
0.7

0.0m!T
JPB, A. Ipp, N. Wschebor, arXiv:1007.0991

FigurePressurea function of thJPB,
A. Ipp, 3:R. (Color
Mendez online)
Galain, N.Left:
Wschebor,
arXiv: as
hep-ph/0610004

The coupling grows with decreasing k

But !
• Dimensional reduction at high temperature• Dynamical generation of a thermal massm ⇠ gT

Massive, decoupling, scheme

Renormalization conditions (at finite temperature)Issues• Temperature dependent counterterms• Fixing the parameters (m, g)One-loop running in massive scheme
1.00.82

g =1T=0m= 10 22⇡T

g2

0.60.4m= 10 32⇡T

0.2

0.0

-14-12-10-8-6-4-20tt = ln(µ/⇤)

122g = 1210g2

864

T=020-14-12-10-8-6t-4-20

Leading order calculation

The renormalization condition implies

Relate thermal mass to zero temperature mass

Note: unusual calculation !

Self-consistent equation for the thermal mass22
m0 = mg2
IT (m)
2

1

2

massless scheme order gmassive thermal scheme
2PI
3
massless scheme order g

m /T

2

0,8

2

0,60,40,200

510152g

20

25

1,1
massive thermal scheme
2PI2
massless scheme order g

1,05massless scheme order g

3

P/P0

10,95

0,90,850,80510152g

2025

1,1

1,05

P/P0

10,95

2

massless scheme order gmassive thermal scheme
2PI
3massless scheme order g

0,90,85

00,10,20,322m /T

0,40,5

Summary• An appropriate choice of renormalization scheme

can greatly improve perturbation theory at finite
temperature• The proposed massive scheme leads to a well
behaved perturbative expansion• The idea of expanding around a massive theory is
not new (screened perturbation theory, optimized
perturbation theory, etc), but the present
implementation is conceptually and technicallysimpler.

• Higher order corrections are being calculated.