resummation: screened perturbation theory versus weak coupling · versus weak coupling jens o....

IntroductionEffective field theory

Screened perturbation theoryConclusions and outlook

Resummation: screened perturbation theoryversus weak coupling

Jens O. Andersen1

1Department of PhysicsNorwegian University of Science and Technology

Talk given at R+R workshopApril 12, 2010

1Collaborators: Eric Braaten, Mike Strickland, Emmanuel Petitgirard, Lars Leganger, Lars Kyllingstad.

Jens O. Andersen Resummation: screened perturbation theory vs weak coupling



Outline

1 Introduction

2 Effective field theory

3 Screened perturbation theory

4 Conclusions and outlook




Introduction

Phase diagram of QCD 2

250 500 750 1000 1250 1500 1750 2000Baryon chemical potential @MeVD

25

50

75

100

125

150

175

200

Tem

pera

ture@M

eVD

Quark-gluon plasma

Early Universe

Hadron phase 2SC

NQCFL

2Taken from Shovkovy’s homepage




Introduction

Weak-coupling expansion of the pressure through orderα

5/2s

3.

3Zhai and Kastening, PRD 52, 7232 (1995): Braaten and Nieto PRD 53, 3421(1996), Boyd et al (lattice)




Introduction

Pressure:

Can be calculated to order α5/2 either using resummedperturbation theory or effective field theory

Poor convergence properties

Breaks down at order α3 due to infrared divergences(magnetic mass problem)Three scales in the problem> T , gT , and g2T

Can use effective field theory methods to calculate effectivetheory for scale g2T

Put it on the lattice

g6 contribution from scale T can be calculated fromperturbation theory.




Dimensional reduction

Fields (anti)periodic in Euclidean time β = 1/T

Free propagators

∆ =1

p20 + p2

,

Nonzero Matsubara modes have a mass of order T anddecouple.

Construct an effective field theory in three dimensions forstatic mode

Effective field theory valid for soft scale gT




Recipe

Recipe:

Write down the most general three-dimensional Lagrangianconsistent with the symmetries

Determine the coefficients by matching

Power counting

Use the effective Lagrangian in actual calculations

Leff =12

(∇φ)2 +12

m2φ2 +g2

324φ4 + ...

Also include coefficient of unit operaror f !Coefficients of Leff are power series in g2




Coefficient of the unit operator f found by calculatingvacuum diagrams the full theory with bare propagators.




Fhard = −π2T 4

90

{1− 5

4α

+154

[L +

13γE +

3145

+43ζ ′(−1)

ζ(−1)− 2

3ζ ′(−3)

ζ(−3)

]α2

+1516

[π2

ε− 12L2 −

(1084

45+ 8γE − 8π2 + 32

ζ ′(−1)

ζ(−1)

−16ζ ′(−3)

ζ(−3)

)L− 134

9− 25

3γ2

E −127ζ(3)

+3115γE −

π2

2+ 4γEπ

2 − 2069

ζ ′(−1)

ζ(−1)− 16

3γ1 + 8γE

ζ ′(−3)

ζ(−3)

+43γEζ ′(−1)

ζ(−1)− 8

(ζ ′(−1)

ζ(−1)

)2

− 203ζ ′′(−1)

ζ(−1)

−23

C′ball + 2Ca

triangle + π2Cbtriangle

]α3 +O(ε)

}.




g23 = g2T

[1− 3g2

(4π)2 (L + γE )− 3g2

(4π)2

(L2 + 2γEL +

π2

8− 2γ1

)ε

].




m̃2 =1

24g2T 2

{1 +

g2

(4π)2

[L + 2− γE + 2

ζ ′(−1)

ζ(−1)

]− 6g4

(4π)4

[52

L2 +1918

L +2851864

− 9548γ2

E (1)

+3γEL− 119144

γE −1

144ζ(3)− 7γ1

+ζ ′(−1)

ζ(−1)

(2L +

11372

+1712γE

)− 1

4ζ ′′(−1)

ζ(−1)+

2532π2

−2γE log(2π) + 2 log2(2π)− 124

C′ball +

14

CI

]+O(ε)

}, (2)

∆m2 =g4T 2

24(4π)2ε

[1− 6g2

(4π)2 (L + γE )− 6g2

(4π)2

(L2 + 2γEL +

π2

8− 2γ1

)ε

]=

g43

24(4π)2ε. (3)




Calculations in the effective theory

Leff0 =

12

(∇φ)2 +12

m2φ2

Leffint =

g23

24φ4




F (s) = −m3T12π

+g2

3m2T8(4π)2 +

g43mT

96(4π)3 [8Lm + 9− 8 log 2]

+g6

3T768(4π)4 [−4 + 16 log 2− 16Lm − 42ζ(3)

+π2(1 + 2 log 2 + 4Lm)]

−g8

3T288m(4π)5

[L2

m +14

Lm − 2Lm log 2− 1564− 3

8π2

+98π2 log 2 +

234

log 2 + 6 log2 2

−6 log 3− 8116ζ(3) + 5Li2(1

4) + 9C4j

],




Hard contribution to the pressure




Soft contribution to the pressure




Total pressure to order g7




Can seletively resum higher orders by not expanding theparameters m2 and g2

3




Screened perturbation theory

Screened perturbation theory 4

Weak-coupling expansion is an expansion about an idealgas of massless particles

L =12

(∂µφ)2 +g2

24φ4

Perhaps better to expand about an ideal gas of massiveparticles

L0 =12

(∂µφ)2 +12

m2φ2

Lint = −12

m21φ

2 +g2

24φ4

Variational calculations4

Karsch, Patkos, and Petreczky, PLB 401 69 (1997).





Recipe

Treat m2 O(1) term

Treat 12 m2

1φ2 and g2φ4 as perturbations on equal footing.

Calculate physical quantities and set m1 = m at the end

Feynman rules for SPT

Mass prescription





Feynman graphs through four loops





Prescriptions for mass parameter m

Debye mass

p2 + m2 + Σ(0,p) = 0 .

Tadpole mass

m2t = g2 ∂F

∂m2

∣∣∣∣m1=m

= g2〈φ2〉

Variational mass

∂F∂m2 = 0




Prescriptions for mass parameter m cont’d

All gap equations are the same at one loop:

m2 =12

g2∑p0

∫d3p

(2π)31

P2 + m2

=12α(µ∗)

[J1(βm)T 2 −

(2 log

µ

µ∗+ 1)

m2]

J1(βm) = 8β2∫ ∞

0

dp p2

(p2 + m2)1/2

1eβ(p2+m2)1/2 − 1

J1(0) =4π2

3

Gap equations differ at two loop and higher




Prescriptions for mass parameter m cont’d

Debye mass not well defined in nonabelian gauge theoriesbeyond leading order.

Tadpole mass not well defined in nonabelian gaugetheories since 〈φ2〉 → 〈AµAµ〉

Variational mass well defined for gauge theories

Screening mass and tadpole mass do not have solutions forall values of g




Miscellaneous

Two-loop SPT-result for the pressure, the gap equation andthe entropy are the same as those for the two-loop 2PIeffective action (but different renormalization!) 5

TS2 =1

(4π)2

[2J0T 4 + J1m2T 2

]J0(βm) =

163β4∫ ∞

0

dp p4

(p2 + m2)1/21

eβ(p2+m2)1/2 − 1

J0(0) =16π4

45

Entropy of ideal gas of massive particles.

5Blaizot, Iancu, and Rebhan PRD 63, 065003 (2001), JOA, Braaten and Strickland PRD 63, 105008 (2001)




Mass expansions

At four loops, we cannot calculate sum-integrals(semi)-analyticallyCarry out an m/T expansion, where m is of order gTExpansions converges reasonably fastOne-loop example:

∑p0

∫d3p

(2π)3 log(p20 + p2 + m2) =

∫d3p

(2π)3 log(p2 + m2)

+′∑

p0

∫d3p

(2π)3

[m2

P2 −12

m4

P4 + ...

]More involved for multiloop diagrams




Numerical results

m/T expansions of two, three, and four-loopapproximations. Weak-coupling expansion for comparison




Numerical results

m/T expansions of two, three, and four-loopapproximations to order g7. Weak-coupling expansion forcomparison




Conclusions and outlook

We have calculated the free energy to order g7 atweak-coupling

Can use evolution equation for parameters in Leff to sumup g2n+3 logn(g) etc

Complicated calculations of loop diagrams in fourdimensions necessary to get hard g6 contribution in QCD

Screened perturbation theory shows remarkable stability

We are working on three-loop hard-thermal-loopperturbation theory for QED and QCD


resummation: screened perturbation theory versus weak coupling · versus weak coupling jens o....

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