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 p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)
Highlights
(The title sequence is the number of independent C-even and P -even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, . . . )
New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP -even, CP -odd, C-odd, and P -odd terms beginning from order p6
While parity acts as an outer automorphism of the Lie algebra of the Euclidean spacetime symmetry group SO(d), C acts as an outer automorphism on the Lie algebra of the unbroken SU(Nf )V symmetry group of the chiral Lagrangian
Summary
We briefly review the setup of the chiral Lagrangian. Following [19, 20] (see e.g. [27] for the notation we use in the following), we consider the UV theory as the QCD Lagrangian with four external source fields — vector vμ, axial-vector aμ, scalar s, and pseudo-scalar p: LUV = LQCD + qγμ vμ + aμγ q − qs − ipγ q. Vμ and aμ are assumed to be traceless With these external fields, the global chiral symmetry (gL, gR) ∈ SU(Nf )L × SU(Nf )R satisfied by QCD can be extended into a local one. Employing the linearization recipe proposed by CCWZ [28, 29], one can find the linearly transforming building blocks under the unbroken group SU(Nf )V For the linear building blocks in eq (2.4), the chiral dimensions are respectively {uμ, Σ±, Σ± , f±μν} −→ {1, 2, 2, 2}. We summarize the transformation properties and chiral dimensions of the linear building blocks φ = {uμ, Σ±, Σ± , f±μν} in table 1
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