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 $p^2$, $p^4$, $p^6$, ...)

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

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• Amplitude/Operator Basis in Chiral Perturbation Theory
• Outer Automorphism Anomalies
• The ChPT: top-down and bottom-up
• Primary Observables for Indirect Searches at Colliders
• 23, 381, 6242, 103268, 1743183, ...: Hilbert series for CP-violating operators in SMEFT
• Hilbert series and higher-order Lagrangians for the $O(N)$ model
• Constructing Operator Basis in Supersymmetry: A Hilbert Series Approach
• Discrete symmetries and Efficient Counting of Operators
• Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory
• Reducing the O(3) model as an effective field theory
• A Large-$N$ Expansion for Minimum Bias
• BV Formalism and Partition Functions