Analytic result for the one-loop massless triangle Feynman diagram

TL;DR

This paper refines the analytic solution for the massless triangle Feynman diagram by incorporating physical momentum conservation constraints, reducing the number of hypergeometric functions needed, using an electric circuit analogy.

Contribution

It introduces a physical constraint-based approach to determine the correct analytic form of the triangle diagram, correcting previous mathematical solutions.

Findings

The correct solution involves three hypergeometric functions, not four.

The electric circuit analogy effectively identifies relevant functions.

Incorporating momentum conservation reduces the solution complexity.

Abstract

Different mathematical methods have been applied to obtain the analytic result for the massless triangle Feynman diagram yielding a sum of four linearly independent hypergeometric functions $F_4$. In this paper I work out the diagram and show that that result, though mathematically sound, is not physically correct, because it misses a fundamental physical constraint imposed by the conservation of momentum, which should reduce by one the total number of linearly independent (l.i.) functions $F_4$ in the overall solution. Taking into account that the momenta flowing along the three legs of the diagram are constrained by momentum conservation, the number of overall l.i. functions that enter the most general solution must reduce accordingly. To determine the exact structure and content of the analytic solution for the three-point function, I use the analogy that exists between Feynman diagrams and electric circuit networks, in which the electric current flowing in the network plays the role of the momentum flowing in the lines of a Feynman diagram. This analogy is employed to define exactly which three out of the four hypergeometric functions are relevant to the analytic solution for the Feynman diagram. The analogy is built based on the equivalence between electric resistance circuit networks of type "Y" and "Delta" in which flows a conserved current. The equivalence is established via the theorem of minimum energy dissipation within circuits having these structures.