More varieties of 4-d gauge theories: product representations

TL;DR

This paper explores the structure and rarity of anomaly-free, chiral gauge theory representations, especially product representations, using arithmetic geometry and number theory, with implications for phenomenology.

Contribution

It generalizes the classification of anomaly-free representations to product cases and analyzes their geometric and number-theoretic properties, revealing the scarcity of chiral representations.

Findings

Projective varieties of product representations are rational for all m and n.

Chiral representations are asymptotically rarer than non-chiral ones, with specific growth rates.

Examples include an asymptotically-free gauge theory with algebra su(7).

Abstract

Recently, we used methods of arithmetic geometry to study the anomaly-free irreducible representations of an arbitrary gauge Lie algebra. Here we generalize to the case of products of irreducible representations, where it is again possible to give a complete description. A key result is that the projective variety corresponding to $m$-fold product representations of the Lie algebra $\mathfrak{su}_n$ is a rational variety for every $m$ and $n$. We study the simplest case of $\mathfrak{su}_3$ (corresponding to the strong interaction) in detail. We also describe the implications of a number-theoretic conjecture of Manin (and related theorems) for the number of chiral representations of bounded size $B$ (measured roughly by the Dynkin labels) compared to non-chiral ones, giving a precise meaning to the sense in which the former (which are those most relevant for phenomenology) are rare compared to the latter. As examples, we show that, for both irreducible representations of $\mathfrak{su}_5$ and once-reducible product representations of $\mathfrak{su}_3$ that are non-anomalous, the number of chiral representations is asymptotically between $B (\log B)^5$ and $B^\frac{4}{3}$, while the number of non-chiral representations is asymptotically $B^2$. Despite this rarity of chiral, anomaly-free, product representations, we show that there are examples relevant for phenomenology, including one that gives an asymptotically-free gauge theory with Lie algebra $\mathfrak{su}_7$.