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.
Highlights
Chiral perturbation theory (ChPT) describes the lowenergy dynamics of QCD [1], in particular interactions of mesons and baryons below the energy scale Λ ∼ Oð1 GeVÞ, and is an integral component of our understanding of many nuclear processes
The success of ChPT lies in the key observation that, in most cases, symmetry alone is sufficient to capture the long wavelength dynamics of a physical system; short wavelength fluctuations are encoded in an infinite number of “coupling constants,” or Wilson coefficients in the language of renormalization group evolution, which are given as inputs in the effective field theory (EFT)
In this work we would like to take up a relatively modest task: counting the number of independent parity-even operators in ChPT, in the massless quark limit, and with the external sources turned off, up to Oðp10Þ [next-to-nextto-next-to-next-to-leading order (N4LO)]; and we provide an “on-shell” basis for the operators
Summary
Chiral perturbation theory (ChPT) describes the lowenergy dynamics of QCD [1], in particular interactions of mesons and baryons below the energy scale Λ ∼ Oð1 GeVÞ, and is an integral component of our understanding of many nuclear processes. At a fixed order n in the momentum expansion, there are an infinite number of operators carrying n derivatives and an arbitrary even power of pion fields ðπ=fπÞ2k This does not imply there are an infinite number of LECs at a fixed derivative order, because operators carrying different powers of π=fπ are often related to one another by the broken symmetry group in a nonlinear way. At Oðp4Þ there are four invariant operators in general and, four LECs. At higher orders in the momentum expansion, it is customary to rescale each derivative ∂μ → ∂μ=Λ and each pion field πa → πa=fπ, the structure of ChPT reads. As well as an expository description of the Mathematica notebook, are relegated to the Appendixes
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