The Kondo problem. I. Transformation of the model and its renormalization

A magnetic (spin-$\frac{1}{2}$) impurity in a metal shows a crossover from the asymptotically free spin at high temperatures to a singlet state at low temperatures. The susceptibility and the impurity lifetime are finite at zero temperature since the magnetic moment of the impurity is compensated by the conduction electrons. The qualitative change from infinite (free impurity spin) to a finite lifetime or susceptibility at $T=0\mathrm{}$ represents a symmetry breaking which cannot be achieved by perturbation theory. We transform the $s-d$ Hamiltonian such that the renormalization can be started with a finite relaxation time. The transformed Hamiltonian consists of a resonance level (the Toulouse limit) and a large perturbation. The resonance width acts like an infrared cutoff such that perturbation expansion at $T=0\mathrm{}$ converges term by term. We show that the transformed Hamiltonian obeys multiplicative renormalization in leading and next-leading logarithmic order and derive the scaling laws of this system.