Operator Counting and Soft Blocks in Chiral Perturbation Theory

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 $\mathrm{SU}(N_f)\times \mathrm{SU}(N_f)\to \mathrm{SU}(N_f)$. 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 $\mathcal{O}(p^{10})$. 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.

31 pages, 2 figures and 6 tables. Mathematica codes added as ancillary files

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Cited by in corpus (11)

• New Selection Rules from Angular Momentum Conservation
• Complete NLO Operators in the Higgs Effective Field Theory
• Anomalous Dimensions of Effective Theories from Partial Waves
• Complete NNLO Operator Bases in Higgs Effective Field Theory
• Amplitude/Operator Basis in Chiral Perturbation Theory
• Multi-spin soft bootstrap and scalar-vector Galileon
• The ChPT: top-down and bottom-up
• Exploring the Landscape for Soft Theorems of Nonlinear Sigma Models
• Soft Phonon Theorems
• Hilbert series and higher-order Lagrangians for the $O(N)$ model
• ALPs, the on-shell way