Renormalization equation at arbitrary impedance and transmission

Derive the renormalization equation governing the energy dependence of effective transmission coefficients for two-terminal quantum contacts with arbitrary transmission probabilities at a general dimensionless environmental impedance.

Background

For weak environmental coupling, the paper presents a renormalization-group equation for the transmission coefficient of each channel in a two-terminal quantum contact, namely, dT_i/dξ = -2zT_i(1−T_i), where z is the dimensionless environmental impedance. This result applies in the small-impedance regime. The authors state that the corresponding equation at general impedance is unavailable when the transmissions are arbitrary, leaving the full renormalization problem unresolved beyond the weak-coupling description.