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 $\phi$ in equilibrium at the black-hole temperature in a Reissner-Nordström 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 redshift effect. Our main purpose is to clarify the dependence of $〈{\phi }^2{〉}_H$ on the field mass m in relation to the excitation mechanism. It is shown for small-mass fields with ${\mathrm{mr}}_{+}\ll 1$ how the excitation of $〈{\phi }^2{〉}_H$ caused by a finite black-hole temperature is suppressed as m increases, and it is verified for very massive fields with ${\mathrm{mr}}_{+}\gg 1$ that $〈{\phi }^2{〉}_H$ decreases in proportion to $m^{-2}$ with an amplitude equal to the DeWitt-Schwinger approximation. In particular, we find a resonance behavior with a peak amplitude at ${\mathrm{mr}}_{+}\simeq 0.38$ 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.