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$
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\newcommand{\intp}{\int \frac{{\rm d}^3p}{2p^0}}
\newcommand{\intpp}{\int \frac{{\rm d}^3p'}{2{p'}^0}}
\newcommand{\intx}{\int{\rm d}^3{\rm r}}
\newcommand{\tp}{\otimes}
\newcommand{\tpp}{\tp\cdots\tp}
\newcommand{\kk}[1]{|#1\rangle}
\newcommand{\bb}[1]{\langle #1}
\newcommand{\dd}[1]{\delta_{#1}}
\newcommand{\ddd}[1]{\delta^3(#1)}
\newcommand{\vv}[1]{{\bf #1}}
\newcommand{\molp}{\Omega^{(+)}}
\newcommand{\dydxt}[2]{\frac{d#1}{d#2}}
\newcommand{\dydx}[2]{\frac{\partial#1}{\partial#2}}
$
This case is similar to a bunch of non-relativistic particles scattered by a fixed potential except that the linear momentum is
conserved.
Let the initial bunch of particles be in a state $\phi_i=U(\vv{r}_i)\phi$ corresponding to sharp momentum $\vv{p}$ and spin $\sigma$.
If there are $N$ particles in the bunch, the number of transitions per unit time will be
\begin{eqnarray*} -\dydxt{N}{t}=n_\xi=\frac{2}{\hbar}{\rm Im}\,\sum_i(\phi_i,B\phi_i) \end{eqnarray*}
where, as before, $B={\molp}^\dagger P_\xi V\molp$. By the familiar argument given above in section 11,
\begin{eqnarray*} \sum_i(\phi_i,B\phi_i)=N\frac{1}{2p^0}
\bb{\vv{p}\sigma}|B\kk{\vv{p}\sigma}
\end{eqnarray*}
Therefore
\begin{eqnarray*} n_\xi=-\dydxt{N}{t}=\frac{2N}{\hbar}\frac{1}{2p^0}
{\rm Im}\,\bb{\vv{p}\sigma}|B\kk{\vv{p}\sigma}
\end{eqnarray*}
Therefore
\begin{eqnarray*} N(t)=N(0)\exp[-t/\tau_\xi] \end{eqnarray*}
where $\tau_\xi$ is the decay constant (``lifetime").
\begin{eqnarray*} \frac{1}{\tau_\xi}&=&\frac{2}{\hbar}\frac{1}{2p^0}
{\rm Im}\,\bb{\vv{p}\sigma}|B\kk{\vv{p}\sigma}\\
&=&\frac{2\pi}{\hbar}\frac{1}{2p^0}\int d(\xi)\delta^4(P_\xi-p)
|T(\xi,\vv{p}\sigma)|^2
\end{eqnarray*}
where the T-matrix is defined by
\begin{eqnarray*} \bb{\xi}|S\kk{\vv{p}\sigma}=\bb{\xi}\kk{\vv{p}\sigma}
-2\pi i\delta^4(P_\xi-p)T(\xi,\vv{p}\sigma)
\end{eqnarray*}
