Complete $CP$ Eigen-bases of Mesonic Chiral Lagrangian up to $p^8$-order

TL;DR

This paper constructs complete mesonic chiral Lagrangian operator bases up to order $p^8$, using Young tensor techniques, and classifies them by $CP$ eigenstates for $SU(2)$ and $SU(3)$.

Contribution

It provides a systematic construction of $CP$-classified mesonic operators up to $p^8$ order using Young tensor methods, matching Hilbert series results.

Findings

Constructed 567 $C$+$P$+ operators for $SU(2)$ at $p^8$

Classified operators into $CP$ eigenstates for $SU(2)$ and $SU(3)$

Confirmed operator counts match Hilbert series calculations

Abstract

Chiral perturbation theory systematically describes the low energy dynamics of meson and baryons using nonlinear Nambu-Goldstone fields. Using the Young tensor technique, we construct the pure mesonic effective operators up to $p^8$-order, one-to-one corresponding to contact amplitudes with the on-shell Adler zero condition. The off-shell external sources, non-vanishing under equation-of-motion conditions, are also added to the operator bases. We also show the invariant tensor bases using the Young tableau is equivalent to the trace bases with Cayley-Hamilton relations. Separated into different $CP$ eigenstates, at $\mathcal{O}(p^8)$ we obtain the operator lists of the 567 $C$+$P$+ operators, 483 $C$+$P$- operators, 376 $C$-$P$+ operators, and 408 $C$-$P$- operators for $SU(2)$ case, while there are 1959 $C$+$P$+ operators, 1809 $C$+$P$- operators, 1520 $C$-$P$+ operators, and 1594 $C$-$P$- operators for $SU(3)$ case, consistent with results using the Hilbert series.