Amplitude/Operator Basis in Chiral Perturbation Theory

We establish a systematic construction of the on-shell amplitude/operator basis for Chiral Perturbation Theory (ChPT) in $D=4$ spacetime dimensions and with an arbitrary number of flavors $N_f$. For kinematic factors, we employ spinor-helicity variables to construct the soft blocks, which are local amplitudes satisfying the Adler's zero condition, as well as to take into account the reduction in the kinematic basis due to the Gram determinant, which arises at $O(p^{10})$ when the number of multiplicity $N$ in an amplitude becomes large: $N>D$. For flavor factors, we include group-theoretic relations at small $N_f$, $N_f\le N$, which decreases the flavor basis. The result is obtained by adapting the Young tensor method of constructing the operator basis for generic effective field theories to the case of non-linearly realized symmetries. Working in the massless quark limit, we present purely mesonic operators for both even- and odd-parity at $O(p^6)$ and $O(p^8)$ for $N=6$ and arbitrary $N_f$, and establish a direct correspondence between the amplitude basis and the operator basis. Furthermore, the redundancy due to the Gram determinant is studied at $O(p^{10})$ for $N=6, 8$ and 10.