Restore the mass m of the electron and show that at high energy, s →∞, the integrated cross sect

step 1 The energy of the electron is given by EE equals HF minus HF prime plus MEC squared and the mome

Restore the mass m of the electron and show that at high...

Restore the mass $m$ of the electron and show that at high energy, $s \rightarrow \infty$, the integrated cross section for Compton scattering is $$ \sigma=\frac{1}{64 \pi^2 s} \int \sqrt{|\Omega|^2} d \Omega \rightarrow \frac{2 \pi \alpha^2}{s} \log \left(\frac{s}{m^2}\right) . $$ Note that at high energy the dominant contribution comes from $9 \pi_2$, via a glancing collision in which the $u$-channel electron is almost on mass shell.

Restore the mass $m$ of the electron and show that at high energy, $s \rightarrow \infty$, the integrated cross section for Compton scattering is
$$
\sigma=\frac{1}{64 \pi^2 s} \int \sqrt{|\Omega|^2} d \Omega \rightarrow \frac{2 \pi \alpha^2}{s} \log \left(\frac{s}{m^2}\right) .
$$

Note that at high energy the dominant contribution comes from $9 \pi_2$, via a glancing collision in which the $u$-channel electron is almost on mass shell.

Key Concepts

Compton Scattering

Compton scattering is the process in which a photon interacts with a charged particle, typically an electron, resulting in a change in the photon's energy and momentum. It is a fundamental interaction in quantum electrodynamics (QED), and its analysis involves computing scattering amplitudes and cross sections using Feynman diagram techniques.

Mandelstam Variables

Mandelstam variables, such as s, t, and u, are used to parameterize the energy and momentum transfer in scattering processes. The variable s represents the square of the total energy in the center-of-momentum frame and is crucial in high-energy approximations for simplifying the calculation of cross sections.

High Energy Limit Approximations

The high energy limit refers to the regime where the square of the total energy (s) is much larger than the square of the particle masses, allowing for significant simplifications in theoretical calculations. In this limit, contributions to scattering cross sections often manifest as logarithmic dependencies arising from the interplay between large energies and small masses.

On-Shell and Nearly On-Shell Propagation

In scattering theory, particles are considered on-shell when they satisfy their mass-shell condition. In high-energy collisions, intermediate particles may be nearly on-shell, which can enhance certain contributions to the scattering amplitude. This near-on-shell behavior, especially in channels like the u-channel for an electron, can lead to pronounced logarithmic terms in the integrated cross section.

Regularization via Particle Mass

Restoring the mass of the electron in theoretical calculations serves as an important regularization mechanism. By maintaining a finite mass, the calculations avoid divergences that would otherwise occur in the high-energy limit, and the resulting logarithmic factors, such as log(s/m^2), encapsulate the transition between high-energy behavior and the effects of the finite mass.

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