Vacuum polarization of scalar fields near Reissner-Nordstr\"{o}m black   holes and the resonance behavior in field-mass dependence
  We study vacuum polarization of quantized massive scalar fields $\phi$ in equilibrium at black-hole temperature in Reissner-Nordstr\"{o}m background. By means of the Euclidean space Green's function we analytically derive the renormalized expression $<\phi^{2}>_{H}$ at the event horizon with the area $4\pi r_{+}^{2}$. It is confirmed that the polarization amplitude $<\phi^{2}>_{H}$ is free from any divergence due to the infinite red-shift effect. Our main purpose is to clarify the dependence of $<\phi^{2}>_{H}$ on field mass $m$ in relation to the excitation mechanism. It is shown for small-mass fields with $mr_{+}\ll1$ how the excitation of $<\phi^{2}>_{H}$ caused by finite black-hole temperature is suppressed as $m$ increases, and it is verified for very massive fields with $mr_{+}\gg1$ that $<\phi^{2}>_{H}$ decreases in proportion to $m^{-2}$ with the amplitude equal to the DeWitt-Schwinger approximation. In particular, we find a resonance behavior with a peak amplitude at $mr_{+}\simeq 0.38$ in the field-mass dependence of vacuum polarization around nearly extreme (low-temperature) black holes. The difference between Scwarzschild and nearly extreme black holes is discussed in terms of the mass spectrum of quantum fields dominant near the event horizon. 
Akira Tomimatsu and Hiroko Koyama