Operator counting and soft blocks in chiral perturbation theory

Chiral perturbation theory (ChPT) is a low-energy effective field theory of QCD and also a nonlinear sigma model based on the symmetry breaking pattern SU(Nf)×SU(Nf)→SU(Nf). In the limit of massless Nf quarks, we enumerate the independent operators without external sources in ChPT using an on-shell method, by counting and presenting the soft blocks at each order in the derivative expansion, up to O(p10). Given the massless on-shell condition and total momentum conservation, soft blocks are homogeneous polynomials of kinematic invariants exhibiting Adler's zero when any external momentum becomes soft and vanishing. In addition, soft blocks are seeds for recursively generating all tree amplitudes of Nambu-Goldstone bosons without recourse to ChPT, and in one-to-one correspondence with the "low-energy constants"which are the Wilson coefficients. Relations among operators, such as those arising from equations of motion, integration by parts, Hermiticity, and symmetry structure, manifest themselves in the soft blocks in simple ways. We find agreements with the existing results up to next-to-next-to-next-to-leading order (NNNLO) and make a prediction at N4LO.