Single particle transition rates

When discussing the cluster decomposition principle in Vol 1 of his QFT textbooks, Weinberg defines $S_{q’q}^C$ (connected part of the $S$-matrix) for single-particle transitions to just be $S_{q’q}$, which then says is equal to $delta(q’-q)$. Here $q$ labels quantum numbers, so for example $delta(q’-q)=delta^3(mathbf{p}’-mathbf{p})delta_{sigma’sigma}$ where $mathbf{p}$ is momentum and $sigma$ is spin along $z$ direction (in the particle’s rest frame).

Why is this true? $S_{q’q}=delta(q’-q)$ is clearly the case for a non-interacting field theory, but I can’t see how it holds in interacting theories. In particular, we have the expression
begin{align}
S_{betaalpha} = delta(beta-alpha) -2pi i delta(E_beta-E_alpha) T_{betaalpha}^+
end{align}
where to first order in perturbation theory for an interaction $V$, $T_{betaalpha}^+=<Phi_beta|VPhi_{alpha}>$. I don’t see why this contribution is zero when $V$ is nonzero.

Edit: He does say later that "we are here assuming that single-particle states are stable, so that there are no transitions between single-particle states and any others, such as the vacuum". Would this amount to imposing $T_{q’q}^+$ by hand? And why does this make sense?