Reorganization of Perturbation Theory for Thermal QCD

Eric Braaten
The Ohio State University

{Problem: Weak-coupling expansion appears to diverge!

The series appears to converge only for such extremely small values of
that T is approaching the grand unification scale.

Arnold and Zhai, Phys. Rev. D50, 7603 (1994) hep-ph/9408276;
Phys. Rev. D51, 1906 (1995) [hep-ph/9410360].
Kastening and Zhai, Phys. Rev. D52, 7232 (1995) [hep-ph/9507380].
Braaten and Nieto, Phys. Rev. Lett. 76, 1417 (1995) [hep-ph/9508406];
Phys. Rev. D53, 3421 (1995) [hep-ph/9510408].

Solution: Reorganization of perturbation theory?

Constraints

should give a sequence of approximations with successively smaller errors
should reduce to the weak-coupling expansion in the limit
should match on to lattice gauge theory results (equation of state, etc.)
for intermediate values of

Possible approaches

1. Pade approximation
2. Screened perturbation theory
   and its generalization to gauge theories, HTL perturbation theory
3. Luttinger-Ward or -derivable approach
   and HTL-inspired approximations to it

Hatsuda, Phys. Rev. D56, 8111 (1997) [hep-ph/9708257];
Kastening, Phys. Rev. D56, 8107 (1997) [hep-ph/9708219].

Karsch, Patkos, and Petreczky, Phys. Lett. B401, 69 (1997) [hep-ph/9702376].

Andersen, Braaten, and Strickland, Phys. Rev. Lett. 83, 2139 (1999)
[hep-ph/9902327]; [hep-ph/9905337]; [hep-ph/9908323].

Vanderheyden and Baym, J. Stat. Phys. 93, 843 (1998) [hep-ph/9803300];
Peshier, Kampfer, Pavlenko, and Soff, Europhys. Lett. 43, 381 (1998)
[hep-ph/9801344].

Blaizot, Iancu, and Rebhan, Phys. Rev. Lett. 83, 2906 (1999)
[hep-ph/9906340]; [hep-ph/9910309].
Peshier, [hep-ph/9910451].

Advantages and Disadvantages

HTL Perturbation Theory

(Andersen, Braaten, and Strickland)

exact gauge invariance:
approximate thermodynamic consistency: higher order in
running of via renormalization group
tractable at order (calculation in progress)
can be generalized to order , but magnetic mass problem at order

-derivable approach

(Vanderheyden and Baym)

approximate gauge invariance: higher order in
exact thermodynamic consistency:
incorrect running of
intractable (requires solving integral equation for gluon propagator)

HTL approximation to -derivable entropy

(Blaizot, Iancu, and Rebhan)

gauge invariant by neglecting gauge-dependent terms
approximate thermodynamic consistency
running of via simple substitution
tractable
no obvious generalization to higher order in