Solving Gauge Anomaly Equations in the Standard Model using the Method of Chords

TL;DR

This paper extends a geometric method called the Method of Chords to solve anomaly cancellation equations for the Standard Model gauge group, including non-Abelian groups, on curved backgrounds, by reducing the problem to a Diophantine equation.

Contribution

It generalizes the Method of Chords from Abelian to non-Abelian gauge groups and applies it to the Standard Model on curved backgrounds, deriving a cubic Diophantine equation.

Findings

The anomaly equations reduce to a homogeneous cubic Diophantine equation.

The method provides a systematic way to find solutions for charges in the Standard Model.

Extension of the geometric solution technique to non-Abelian gauge theories.

Abstract

In a recent paper, Allanach et al introduced a geometric method to solve the anomaly cancellation equations for a $U(1)$ gauge theory with an arbitrary number of charges - the Method of Chords known in Diophantine analysis. We extend their result to non-Abelian gauge groups, and show that this method can be used to find the general solution to the anomaly cancellation equations for a theory with the Standard Model gauge group on a curved background. Given $K$ charges in $(\textbf{2}, \textbf{3})$, $L$ charges in $(\textbf{2}, \textbf{1})$, $M$ charges in $(\textbf{1}, \textbf{3})$, and $N$ charges in $(\textbf{1}, \textbf{1})$ representations of $SU(2)\times SU(3)$, the equations reduce to a homogeneous cubic Diophantine equation in $K+L+M+N-4$ variables.