2, 12, 117, 1959, 45171, 1170086, ...: A Hilbert series for the QCD chiral Lagrangian

TL;DR

This paper uses Hilbert series techniques to systematically enumerate operators in the mesonic QCD chiral Lagrangian, including anomalous and symmetry-odd terms, extending to very high chiral orders.

Contribution

It extends Hilbert series methods to include external fields and charge conjugation, providing new enumeration results up to order p^{16} in the chiral expansion.

Findings

Enumerates anomalous operators at order p^8

Classifies CP, C, and P odd terms from order p^6 onward

Extends Hilbert series to high orders in chiral expansion

Abstract

We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $\frak{su}(n)$ Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order $p^8$, as well as enumeration of $CP$-even, $CP$-odd, $C$-odd, and $P$-odd terms beginning from order $p^6$. The method is extendable to very high orders, and we present results up to order $p^{16}$. (The title sequence is the number of independent $C$-even $P$-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions $p^2$, $p^4$, $p^6$, ...)