Ultrasoft Fermionic Modes at High Temperature

Abstract: A possible collective fermionic excitation in the ultrasoft energy-momentum region is examined in Yukawa model with scalar coupling and quantum electrodynamics (QED) with g being coupling constant at extremely high temperature T where the fermion mass is negligible. We analytically sum up the ladder diagrams for the vertex correction in the leading order in QED, which is not necessary in the Yukawa model, and find that the fermion pole exists at \omega = \pm p/3-i\zeta with ultrasoft momentum p both for the Yukawa model and QED; \zeta is the sum of the damping rates of fermion and boson with hard momenta. We also obtain the expression of the residue of the pole, which is as small as of order g^2. We show that the fermion propagator and the vertex function satisfy the Ward-Takahashi identity in QED. Thus we establish the existence of an ultrasoft fermionic mode at extremely high temperature, which was originally called phonino and was suggested in the context of supersymmetry and its breaking at finite T. We discuss the possible origin of such an ultrasoft fermionic mode without recourse to supersymmetry. The case of QCD is briefly mentioned.