High-energy behaviour of Fermi theory

TL;DR

The paper addresses the high-energy fate of the massless four-fermion amplitude in Fermi theory by deriving recurrence relations for leading UV divergences through the Bogolyubov-Parasyuk R-operation and translating them into renormalization group equations that sum leading logarithms to all orders. Numerical solutions in the asymptotic regime show a stark operator-dependent dichotomy: the unit operator exhibits a Landau pole and potential unitarity violation, whereas the V-A operator is asymptotically free and unitarity is restored by radiative corrections, with cross-sections scaling as $d\sigma/d\Omega \sim 1/(s\log^2(s/\mu^2))$ in both cases. The results, which resemble the high-energy behavior of theories with intermediate gauge bosons, demonstrate a concrete framework for predicting UV behavior in non-renormalizable theories via leading-log summation. This work thus provides a method to reconcile non-renormalizable Fermi theory with unitary high-energy behavior and highlights the crucial role of operator structure in determining asymptotics.

Abstract

We consider the 4-fermion scattering amplitude in massless Fermi theory. Based on the Bogolyubov-Parasyuk theorem, which guarantees locality of the counter terms, we derive the recurrence relations for ultraviolet divergences of diagrams that establish a connection between successive orders of perturbation theory. We check their validity up to three loops comparing them with explicit calculation made earlier. Then we construct the corresponding RG equation that sums up the leading logarithmic contributions in all orders of perturbation theory. Numerical analysis of these equations in the asymptotic regime $s\sim t\sim u \sim E^2 \to \infty$ is performed for two cases: the unit and the V-A operator in the fermion current. We found out that for the unit operator the high energy behaviour of the theory in the leading order is characterized by the presence of the Landau pole, while for the V-A operator the theory is asymptotically free. Therefore, in the latter case, radiative corrections restores unitarity, which is violated at the tree level. We compare the obtained behaviour of the amplitude with one in the theory with the intermediate gauge bosons and found an overlap between them.

Figures (7)

• Figure 1: Action of the $R'$-operation on an n-loop diagram
• Figure 2: Recurrence relation for a four-point diagram
• Figure 3: Feynman diagram for four-fermion scattering. All momenta are incoming here.
• Figure 4: The dependence of the amplitude $A(y)$ on energy in different approximations of perturbation theory.
• Figure 5: Comparative analysis of the function $A(y)$: NA is the numerical solution and PT is a finite number of terms of perturbation theory