One-loop Amplitudes in the Worldline Formalism

TL;DR

This paper reviews recent advances in the worldline formalism for calculating one-loop N-point amplitudes, highlighting its efficiency and mathematical framework involving inverse derivatives and Bernoulli numbers.

Contribution

It introduces an algorithm for computing full momentum N-point amplitudes in phi^3 theory using the worldline formalism, improving on traditional methods.

Findings

Provides master formulas for summing Feynman diagrams efficiently

Demonstrates the application to low-energy photon amplitudes in QED

Proposes a new algebraic approach involving Bernoulli numbers

Abstract

We summarize recent progress in applying the worldline formalism to the analytic calculation of one-loop N-point amplitudes. This string-inspired approach is well-adapted to avoiding some of the calculational inefficiencies of the standard Feynman diagram approach, most notably by providing master formulas that sum over diagrams differing only by the position of external legs and/or internal propagators. We illustrate the mathematical challenge involved with the low-energy limit of the N-photon amplitudes in scalar and spinor QED, and then present an algorithm that, in principle, solves this problem for the much more difficult case of the N-point amplitudes at full momentum in phi^3 theory. The method is based on the algebra of inverse derivatives in the Hilbert space of periodic functions orthogonal to the constant ones, in which the Bernoulli numbers and polynomials play a central role.