  The standard theory of dynamical Coulomb blockade [$P(E)$ theory] in
ultra-small tunnel junctions has been formulated on the basis of phase-phase
correlations by several authors. It was recently extended by several
experimental and theoretical works to account for novel features such as
electromagnetic environment-based renormalization effects. Despite this
progress, aspects of the theory remain elusive especially in the case of linear
arrays. Here, we apply path integral formalism to re-derive the Cooper-pair
current and the BCS quasi-particle current in single small Josephson junctions
and extend it to include long Josephson junction arrays as effective single
junctions. We consider renormalization effects of applied oscillating voltages
due to the impedance environment of a single junction as well as its
implication to the array. As is the case in the single junction, we find that
the amplitude of applied oscillating electromagnetic fields is renormalized by
the same complex-valued weight $\Xi(\omega) = |\Xi(\omega)|\exp i\eta(\omega)$
that rescales the environmental impedance in the $P(E)$ function. This weight
acts as a linear response function for applied oscillating electromagnetic
fields driving the quantum circuit, leading to a mass gap in the thermal
spectrum of the electromagnetic field. The mass gap can be modeled as a pair of
exotic `particle' excitation with quantum statistics determined by the argument
$\eta(\omega)$. In the case of the array, this pair corresponds to a bosonic
charge soliton/anti-soliton pair injected into the array by the electromagnetic
field. Possible application of these results is in dynamical Coulomb blockade
experiments where long arrays are used as electromagnetic power detectors.
