On-shell effective field theory: A systematic tool to compute power corrections to the hard thermal loops

Abstract

We show that effective field theory techniques can be efficiently used to compute power corrections to the hard thermal loops (HTL) in a high temperature T expansion. To this aim, we use the recently proposed on-shell effective field theory (OSEFT), which describes the quantum fluctuations around on-shell degrees of freedom. We provide the OSEFT Lagrangian up to third order in the energy expansion for QED, and use it for the computation of power corrections to the retarded photon polarization tensor for soft external momenta. Here soft denotes a scale of order eT , where e is the gauge coupling constant. We develop the necessary techniques to perform these computations, and study the contributions to the polarization tensor proportional to e 2 T 2 , e 2 T and e 2 T 0 . The first one describes the HTL contribution, the second one vanishes, while the third one provides corrections of order e 2 to the soft photon propagation. We check that the results agree with the direct calculation from QED, up to local pieces, as expected in an effective field theory.

The HTL Lagrangian at NLO: The photon case

“…The first term in Eq. (1) is of order O(e 2 Q 2 ) where Q is the external momentum, and was obtained in [12,13]. The second term is of order O(e 4 T 2 ) and is obtained for the first time in this work.…”

“…Both Π L and Π T contain three different contributions: the LO HTL result, the power corrections to the 1-loop HTL result (calculated in ref. [12]), and the 2-loop result calculated in this paper. For convenience we gather these results in Eq.…”

“…The contribution from this diagram is identical to the one from the self graph that is shown, and we will take it into account by including a factor of two. 1 Note that our conventions differ from those used in [12] and [13]: more specifically, our definitions for Π µν and for Π T both differ by a minus sign with the ones used there. We also take this opportunity to point out some misprints in these works: the definition of the Debye mass after formula (45) in [12] should read m D = e 2 T 2 /3 and Eq.…”

“…This was one of the motivations to apply the On-Shell Effective Field Theory (OSEFT) [11] to this problem in ref. [12], where the leading power corrections to the photon self-energy were calculated. The complete leading power corrections for massless QED, namely including the fermion sector, were worked out in ref.…”

Two-loop thermal spectral functions with general kinematics

“…If one relaxes the HTL assumption that the external momentum K 2 is much smaller than T 2 , the complete self-energies must be evaluated numerically [22]. Effective field theory methods have also been developed recently to compute associated power corrections [23]. The imaginary parts involve phase space integrals for 1 ↔ 2 'decays' which are needed for our master diagrams II and III as well as certain terms arising in V and VI.…”