YM-Thermodynamics 2024, 30: One-Loop Polarization Tensor of the Massless Mode

The first approximation of an external photon on the naive mass shell can be used to check the polarization tensor Π. Doing so will not be entirely selfconsistent, but will serve as a "quick and dirty" sanity check. For selfconsistency, one would have to set the mass shell to the screening function G, though this, and the zero mass shell approach yield functionally identical answers at desired temperatures.

First, begin with the transverse part of the massless mode for the SU(2) case. Due to the unbroken U(1) invariance, Π is 4D-transverse for any p² of the photonic TLM mode. Using the Euclidean signature, and the resulting decomposition, identifies two functions which determine the TLM mode's propagation. This can be Wick-rotated to real-time, which places p along the z-axis.

This is consistent with previous approaches for F = G = 0. The longitudinal structure describes the quantum propagation and reduces to the free limit. At the zero mass shell the 00-component of Π can be calculated and confirmed to be finite for the temporal limit. Resummation of the 00-component generating the mass shell for p² = F, this structure breaks down. Instead, the function G needs to adhere to the dispersion law for the transversely propagating TLM mode.

The full function G can be calculated from the loop diagrams.

By this equation, the energy of propagating TLM modes are reduced compared to the free case, and propagation of TLM modes from the interaction with TLH modes is very small. At high temperatures, there is a power-like suppression of |G/T²|. The effect of G on the shifting photon propagation away from the mass shell is to be read as the constraint for momentum transfer in the 4-vertex. All other 1-loop diagrams may be ignored, so no further constraints are necessary. The longitudinal plasma modes can be identified in a similar fashion.