Operator Counting and Soft Blocks in Chiral Perturbation Theory

TL;DR

This paper systematically enumerates independent operators in chiral perturbation theory up to order p^{10} using an on-shell method, introducing soft blocks that generate tree amplitudes and relate to low energy constants.

Contribution

It presents a novel on-shell approach to count and classify operators in ChPT, extending the enumeration up to N^4LO and clarifying relations among operators through soft blocks.

Findings

Agreement with existing results up to NNNLO.

Predictions made for N^4LO operator structures.

Soft blocks serve as seeds for tree amplitude generation.

Abstract

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 ${\rm SU}(N_f)\times {\rm SU}(N_f)\to {\rm SU}(N_f)$. In the limit of massless $N_f$ 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 ${\cal O}(p^{10})$. Given the massless on-shell condition and total momentum conservation, soft blocks are homogeneous polynomials of kinematic invariants exhibiting the 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 NNNLO, and make a prediction at N$^4$LO.