Power-suppressed Thermal Effects from Heavy Particles
  In quantum field theory, heavy particles with mass $M$ much greater than the temperature $T$ give not only effects suppressed by the Boltzmann factor $e^{-M/T}$, but also effects suppressed by powers of $T/M$. We show that power-suppressed terms in equilibrium observables arise from effective interactions among light particles due to virtual heavy particles. We study a model introduced by Matsumoto and Yoshimura in which study a model introduced by Matsumoto and Yoshimura in which heavy bosons interact only through a term that allows pair annihilation into light particles. We construct an effective Lagrangian for the light field by integrating out the heavy field, and use it to calculate the leading power-suppressed terms in the energy density. The thermal average of the Hamiltonian density for the heavy field includes a term proportional to $T^6/M^2$, but we show that this term can be eliminated by a field redefinition and therefore cannot have any physical significance. 
Eric Braaten and Yu Jia