Correpondence between the one-loop three-point vertex and the Y- and $\Delta$- electric resistor networks

TL;DR

This paper analyzes the three-point Feynman diagram using electric circuit analogies to identify the physically relevant hypergeometric functions, ensuring the solution respects momentum conservation in higher-loop calculations.

Contribution

It introduces a novel approach linking Feynman diagrams to electric resistor networks to determine the correct analytic structure of the three-point function.

Findings

Identifies the physically relevant hypergeometric functions for the diagram.

Shows the importance of momentum conservation in reducing independent functions.

Establishes a correspondence between Feynman diagrams and resistor networks.

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 of two variables $F_4$. These are defined for especific regions of convergence for the ratios of the squares of momentum variables. In this paper I work out the diagram and show that that result, though mathematically sound, is not physically acceptable when it is embedded in higher loops - meaning further momentum integrations - 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 that can be embedded in higher loops, 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.