Show that the invariant amplitude, (4.41), for "spinless"...

Show that the invariant amplitude, (4.41), for "spinless" electron-positron scattering can be written as
$$
\text { IR }_{\mathrm{e}^{-} \mathrm{e}^{+}}(s, t, u)=e^{2}\left(\frac{s-u}{t}+\frac{t-u}{s}\right) .
$$

Key Concepts

Feynman Diagrams and Exchange Channels

The derivation of the invariant amplitude involves considering the Feynman diagrams that contribute to the scattering process, typically including different exchange channels such as the s-channel and t-channel. These diagrams provide a visual and calculational tool for applying the Feynman rules, which eventually lead to the amplitude's expression in terms of the Mandelstam variables.

Spinless Approximation

In the context of the problem, the spinless approximation is taken for electron-positron scattering to eliminate complications arising from spin degrees of freedom. This simplification allows one to focus solely on the contribution of the exchange channels in the scattering process, resulting in an amplitude expression that depends only on the Mandelstam variables and the coupling constant (here denoted by e).

Invariant Amplitude

This is a central quantity in scattering theory that represents the probability amplitude for a given process to occur. It is constructed to be Lorentz invariant, meaning that its value does not depend on the choice of reference frame, and it encapsulates all the dynamical information about the interaction described by a quantum field theory.

Mandelstam Variables

These are a set of three kinematic invariants (commonly denoted s, t, and u) that are used to describe the energy and momentum transfer in a scattering process. They provide a convenient and frame-independent way to express the invariant amplitude and to analyze the kinematics of the process, especially when applying crossing symmetry and other analytical tools.

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