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

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 $t\rightarrow\pm \infty$?