The paper under review is concerned with the analysis of the dynamics of a system of spins collectively coupled to a thermal environment represented by the usual Fock space model. The authors prove that if the coupling is sufficiently weak and the form factors appearing in the Hamiltonian describing the interaction satisfy certain natural analyticity conditions then the decoherence rates appearing in the convergence to the equilibrium of the density matrix of a spin system can be described explicitly without referring to the standard in quantum probability weak coupling approximation [see

*E. B. Davies*, Comm. Math. Phys. 39, 91–110 (1974;

Zbl 0294.60080)]. The proofs are based on the spectral deformation method, developed by the authors in their earlier work [see Ann. Phys. 323, No. 2, 373–412 (2008;

Zbl 1201.81083) and references therein].