Extremal fixed points and Diophantine equations

The coupling constants of fixed points in the $\epsilon$ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting's theorem, Siegel's theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples.