Complete CP-eigen Bases of Meson-Baryon Chiral Lagrangian up to $p^5$-order

Chiral perturbation theory describes the low energy dynamics of mesons and baryons in terms of the nonlinear Goldstone boson and fermion degrees of freedom. Through the Young tensor technique, we construct the on-shell operator bases for the meson-baryon system up to $p^5$-order, using the chiral dimension power counting and heavy baryon expansion. For the Lorentz structure, additional treatments on off-shell external sources and operators with higher derivatives are necessarily considered, while for the internal structure, the invariant tensor basis is converted into the trace basis equivalently, and Cayley-Hamilton relations are utilized to classify different CP eigen-operators. Finally we present the complete operator set of $C$+$P$+, $C$+$P$-, $C$-$P$+, and $C$-$P$- eigen-operators at the $p^5$-order, and obtain the operator counting from the Hilbert series.