Show that the invariant amplitude, (4.41), for "spinless" electron-positron scattering can be written as
$$
\pi_{\mathrm{e}^{-} \mathrm{e}^{+}}(s, t, u)=e^2\left(\frac{s-u}{t}+\frac{t-u}{s}\right) .
$$
Comment on the symmetry of 9 under $s \leftrightarrow t$.
Let us look back at the amplitude for "spinless" electron-electron scattering. The amplitude (4.40) is derived taking, of course, the process to be $A B \rightarrow C D$,
that is, the $s$ channel process. In terms of invariant variables, (4.40) becomes
$$
\pi_{\mathrm{e}^{-} \mathrm{e}^{-}}=e^2\left(\frac{u-s}{t}+\frac{t-s}{u}\right) .
$$
The resulting cross section is sketched in Fig. 4.8, and the origin of the forward and backward peaks is identified; $-t$ and $-u$ are the squares of the three-momentum transferred in Figs. 4.4a and 4.4c, respectively, that is, of the momentum carried by the virtual photon. When the photon has a very small momentum squared $\left(-q^2\right)$, that is, almost on its mass shell, then by the uncertainty principle the range of the interaction is very large. Interactions with small deflections therefore occur with large cross sections.