  The one-loop free energy of the four-dimensional compact QED, which is known
to be equivalent to the vector Sine-Gordon model, is calculated in the strong
coupling regime. In the case, when the norm of the strength tensor of the
saddle-point value of the corresponding Sine-Gordon model is much larger than
the typical inverse area of a loop in the gas of the monopole rings, the
obtained free energy decays exponentially versus this norm. In the opposite
case, when the dominant configuration of the Sine-Gordon model is identically
zero, the resulting free energy decays with the growth of loops as an exponent
of the inverse square of their typical area.

High Energy Physics - Theory