Complete CP eigen-bases of mesonic chiral Lagrangian up to p⁸-order

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 p⁸-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 $\mathcal{O}\left(p^8\right)$ \( \mathcal{O}\left({p}^8\right) \) 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.