As charge carrier of the macroscopic superconductivity, the Cooper pair is a
composite particle of two paired electrons, which has both center-of-mass and
inner-pair degrees of freedom. In most cases, these two different degrees of
freedom can be well described by the macroscopic Ginzburg-Landau theory and the
microscopic Bardeen-Cooper-Schrieffer (BCS) theory, respectively. Near the
superconducting phase transition where the Cooper pair is fragile and unstable
because of the small binding energy, there are non-trivial couplings between
these two different degrees of freedom due to such as finite energy and/or
momentum transfer. The non-trivial couplings make the original derivation of
the Ginzburg-Landau theory from the BCS theory fail in principle as where these
two different degrees of freedom should not be decoupled. In this manuscript,
we will present a renormalization formalism for an extended Ginzburg-Landau
action for the superconducting phase transition where there is finite energy
transfer between the center-of-mass and the inner-pair degrees of freedom of
Cooper pairs. This formalism will provide a theoretical tool to study the
unusual effects of the inner-pair time-retarded physics on the superconducting
phase transition.Comment: 10 pages, 4 figure