Vacuum polarization of scalar fields near Reissner-Nordström black holes and the resonance behavior in field-mass dependence

We study vacuum polarization of quantized massive scalar fields $\phi$ inequilibrium at black-hole temperature in Reissner-Nordstr\"{o}m background. Bymeans of the Euclidean space Green's function we analytically derive therenormalized expression $_{H}$ at the event horizon with the area$4\pi r_{+}^{2}$. It is confirmed that the polarization amplitude$_{H}$ is free from any divergence due to the infinite red-shifteffect. Our main purpose is to clarify the dependence of $_{H}$ onfield mass $m$ in relation to the excitation mechanism. It is shown forsmall-mass fields with $mr_{+}\ll1$ how the excitation of $_{H}$caused by finite black-hole temperature is suppressed as $m$ increases, and itis verified for very massive fields with $mr_{+}\gg1$ that $_{H}$decreases in proportion to $m^{-2}$ with the amplitude equal to theDeWitt-Schwinger approximation. In particular, we find a resonance behaviorwith a peak amplitude at $mr_{+}\simeq 0.38$ in the field-mass dependence ofvacuum polarization around nearly extreme (low-temperature) black holes. Thedifference between Scwarzschild and nearly extreme black holes is discussed interms of the mass spectrum of quantum fields dominant near the event horizon