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

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 $\mathfrak{su}(n)$ 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 $p^2$, $p^4$, $p^6$, ...)

Comments: 18 pages + appendices, 3 figures, 10 tables, 1 auxiliary file; v2: additional appendix on p^4 chiral Lagrangian, additional references. Matches version published in JHEP