Bare mass to physical mass in the limit of vanishing interaction as $t\rightarrow \pm\infty$

Question

In the Quantum Field Theory by Itzykson and Zuber (page 202), they assume that the coupling terms in the Equation of motion (of an interacting theory) vanishes smoothly as $t\rightarrow \pm \infty$. If I understand correct, this implies that at asymptotic times there is no interaction and asymptotic states are free states. Now the question is, if the interaction vanishes, the bare mass $m_0$ change to renormalized or physical mass. But one always assumes the asymptotic free states to have physical mass $m$.

But if Itsykson and Zuber is correct, how will $m_0$ be driven to $m$ without interaction at $t\rightarrow\pm \infty$?

Answer 1

The theory doesn't tend to become free, only the on-shell particles become non-interacting with each other because of the large distance between them. Say, in QED there always exist radiative corrections to the electron propagator, no matter how far it is from other particles.