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

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.