Logarithmic Factors in the High-Energy Behavior of Quantum Electrodynamics

Abstract

The perturbation series for electron-electron elastic scattering in quantum electrodynamics is studied in the limit of high energies. For this matrix element, in addition to the previously known terms which are proportional to $s$ (the square of the c.m. energy) and hence lead to a constant total cross section at high energies, there are found terms of the orders of magnitudes $s\ln s$, $s{\left(\ln s\right)}^2$, $s{\left(\ln s\right)}^3$, etc. For $n=1, 2, 3, \dots$, the coefficient of $s{\left(\ln s\right)}^n$ is a power series in the fine-structure constant $\alpha$, where the leading term is proportional to ${\alpha }^{2(n+1\mathrm{})\mathrm{}}$ and is due to Feynman diagrams with $n$ closed electron loops. Physically, through the optical theorem, the presence of these terms is intimately related to the production of low-energy electron-positron pairs in high-energy electron-electron scattering, but is independent of whether the spin-1 particle is a photon with zero mass or massive neutral vector meson. These leading terms of order ${\alpha }^{2(n+1\mathrm{})\mathrm{}}$ are explicitly found for all $n$, and are all imaginary, representing absorption. The procedure of summing the leading term is carried out, and the result demonstrates dramatically the importance of unitarity in the direct, or $s$, channel for high-energy processes. Generalization to two-body diffraction processes $a+b\to a^{\prime }+b^{\prime }$ is immediate.