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 $\gamma^{*} \mathrm{e}^{-} \rightarrow$ $\gamma \mathrm{e}^{-}$(where $\gamma^{*}$ denotes a virtual photon), $$ \overline{|\mathscr{T}|^{2}}=2 e^{4}\left(-\frac{u}{s}-\frac{s}{u}+\frac{2 Q^{2} t}{s u}\right) $$ 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 $\gamma^{*} \mathrm{e}^{-} \rightarrow$ $\gamma \mathrm{e}^{-}$(where $\gamma^{*}$ denotes a virtual photon),
$$
\overline{|\mathscr{T}|^{2}}=2 e^{4}\left(-\frac{u}{s}-\frac{s}{u}+\frac{2 Q^{2} t}{s u}\right)
$$
We shall make use of this result in Chapter 10 .

Key Concepts

Virtual Photon

A virtual photon is an off-shell mediator of the electromagnetic interaction that does not satisfy the standard energy?momentum relation for real photons. In calculations, its squared four-momentum is not zero but given by k² = -Q², where Q² characterizes the degree of its off-shellness. This concept is central in perturbative quantum electrodynamics (QED) when describing processes involving intermediate states that are not directly observable.

Mandelstam Variables

Mandelstam variables (s, t, and u) are Lorentz-invariant quantities used to describe the kinematics of scattering processes. They concisely capture the energy and momentum transfer between particles in different channels. Their invariance under Lorentz transformations makes them indispensable for expressing scattering amplitudes and cross sections in theoretical calculations.

Spin-Averaged Squared Amplitude

The spin-averaged squared amplitude is the modulus squared of the scattering amplitude summed (or averaged) over initial and final spin states. It provides a gauge-invariant measure of the probability for a scattering process to occur. In QED calculations, averaging over spin states simplifies the comparison with experimental results and ensures that unobserved spin degrees of freedom do not affect the predicted cross sections.

Perturbative QED Calculations

Perturbative QED is the framework used to compute scattering amplitudes using Feynman diagrams and rules. It involves expanding the interaction in a power series over the coupling constant. This method allows for systematic calculation of processes such as photon-electron scattering, including cases where the photon is virtual, by considering contributions from different channels and employing established techniques like loop integrals and renormalization.

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