Repeat the above calculation for an incident virtual photon...

Repeat the above calculation for an incident virtual photon of mass $k^2 \equiv-Q^2$. Continue to use (6.111). Show that for $\gamma^* \mathrm{e}^{-} \rightarrow$ $\gamma^{-}$(where $\gamma^*$ denotes a virtual photon), $$ \overline{|\mathscr{R}|^2}=2 e^4\left(-\frac{u}{s}-\frac{s}{u}+\frac{2 Q^2 t}{s u}\right) . $$ We shall make use of this result in Chapter 10 .

Repeat the above calculation for an incident virtual photon of mass $k^2 \equiv-Q^2$. Continue to use (6.111). Show that for $\gamma^* \mathrm{e}^{-} \rightarrow$ $\gamma^{-}$(where $\gamma^*$ denotes a virtual photon),
$$
\overline{|\mathscr{R}|^2}=2 e^4\left(-\frac{u}{s}-\frac{s}{u}+\frac{2 Q^2 t}{s u}\right) .
$$

We shall make use of this result in Chapter 10 .

Key Concepts

Feynman Diagram Calculations

Feynman diagrams provide a pictorial representation of the interactions involved in quantum field theory processes. The calculation of scattering amplitudes involves applying Feynman rules to these diagrams, including vertex factors, propagators, and external state wavefunctions. These calculations allow for the systematic evaluation of cross sections and other observable quantities, and are particularly important in processes involving virtual particles such as the off?shell photon in the given example.

Polarization Sums and Averaging

In scattering processes, particularly in quantum electrodynamics, the calculation of observable quantities often requires summing over the polarization states of the particles and averaging over initial spin configurations. This procedure, which involves carefully handling the polarization vectors of both real and virtual photons, ensures that the theoretical predictions are directly comparable to experimental results, where the initial states are typically unpolarized.

Virtual Photon

A virtual photon is an off-shell mediator of the electromagnetic interaction whose four?momentum squared does not vanish, i.e., k² ? 0. In calculations it is treated similarly to real photons but with modifications that account for its virtuality, such as propagator factors and altered phase space. Its role is essential in quantum electrodynamics, where it facilitates forces between charged particles, even though it cannot be observed as an isolated entity.

Mandelstam Variables

Mandelstam variables (s, t, and u) are Lorentz-invariant quantities that characterize the energy and momentum transfer in scattering processes. They simplify the analysis of scattering amplitudes by providing a concise description of the kinematics independent of the reference frame. These invariants are used extensively in the evaluation of Feynman diagrams and are crucial in relating different channels of a scattering event.

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