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 $$ \mathfrak{su}(n) $$ 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, …)

Tidal effects for spinning particles

“…Notable achievements are the applications of the Hilbert series to the Standard Model effective field theory [45][46][47][48][49], and the extension to include gravity [50]. Non-relativistic effective field theories [51,52] and effective field theories with non-linearly realized symmetries [53] can also be constructed using Hilbert series techniques.…”

Renormalization and non-renormalization of scalar EFTs at higher orders

“…Hilbert series (more generally, the mathematical structure of polynomial rings) have recently been utilized to organize and ameliorate the difficulties surrounding the construction of EFT operator bases and to study the structure of EFTs [81][82][83][84][85][86][87] (see also the developments [33,[88][89][90][91][92][93][94][95][96][97]). The scalar EFTs we consider fall into the class where their operator bases are controlled by an underlying conformal representation theory, which can be directly used in the construction of a Hilbert series.…”

“…We emphasize that both the Hilbert series and R* techniques we develop are general, and can apply beyond the scalar EFTs we consider here. In particular, for Hilbert series, applications to spin [82][83][84][85], non-linearly realized internal symmetries [83,86], gravity [99] and non-relativistic EFTs [100,101], have all been developed. The R* method has already found application in gauge theories and the SM EFT, as already mentioned above.…”

“…Indeed, the R* method has, for instance, already been widely applied to gauge theories. Also Hilbert series technology can encompass spin [82][83][84][85], non-linearly realized internal symmetries [83,86], gravity [99] and non-relativistic EFTs [100,101], and the systematic construction of operators that involve particles of higher spin via the study of polynomial rings and conformal representation theory has recently been studied in detail [84,85], and by related methods [33,[88][89][90][91][92][93][94][95][96][97]. We expect that such on-shell techniques, and their development to off-shell objects of interest that was presented here, could also be of use in studying full loop amplitudes in EFT, i.e.…”

“…Imposing spacetime parity via the Hilbert series has been derived in appendix C of [83] (see also the application of charge conjugation symmetry to the QCD chiral Lagrangian in [86] that contains many parallels). We will summarize the results here.…”