Fridel sum rules for one- and two-channel Kondo models and unitarity paradox via bosonization-refermionization approach

We calculate the single-particle Green's functions and scattering amplitudes of the one-channel and channel-anisotropic two-channel Kondo models at the Toulouse and Emery-Kivelson lines, respectively, where exact solutions via the bosonization-refermionization approach are admitted. We demonstrate that in this approach the Friedel sum rules -- the relations between the trapped spin and ``flavor'' moments and the scattering phase shifts in the Fermi-liquid regime -- arise naturally and elucidate on their subtleties. We also recover the ``unitarity paradox'' [1,2] -- the vanishing of the single-particle scattering amplitude at the channel-symmetric point of the two-channel Kondo model -- stemming from non-Fermi-liquid behavior. We discuss the implications of these results for the development of composite pairing in heavy fermion systems.\\[4pt] [1] A. W. W. Ludwig and I. Affleck, Phys. Rev. Lett. 67, 3160 (1991).\\[0pt] [2] J. M. Maldacena and A. W. W. Ludwig, Nucl. Phys. B. 506, 565 (1997).