Abstract
When matter fields are included in chiral perturbation theory, the nonvanishing mass in the chiral limit introduces a new energy scale so that the loop diagrams including such matter field propagators spoil the usual power counting. However, the power counting breaking terms can be absorbed into counterterms in the chiral Lagrangian. In this paper, we systematically derive these terms to leading one-loop order (next-to-next-to leading order in the chiral expansion) at once by calculating the generating functional using the path integral. They are then absorbed by counterterms in the next-to-leading order Lagrangian. The method can be extended to calculating power counting breaking terms for other matter fields.
Highlights
When matter fields are included in chiral perturbation theory, the nonvanishing mass in the chiral limit introduces a new energy scale so that the loop diagrams including such matter field propagators spoil the usual power counting
A very important step was made by Ellis and Tang [14, 15]. They noted that the soft-momentum part of a loop diagram is infrared singular and the power counting breaking (PCB) terms, coming from the hard-momentum modes only, are a local polynomial in small momenta and Goldstone boson masses and can be absorbed into the low-energy constants (LECs) of the most general chiral Lagrangian
We will study the one-loop generating functional of correlation functions with up to four external particles for the chiral Lagrangian with spinless matter fields in the fundamental representation of SU(N ). Such a theory can be applied to study kaonpion scattering by treating kaons as matter fields [7], and it is expected to have a better convergence of the chiral expansion than that of the normal SU(3) ChPT which treats kaons as Goldstone bosons as well
Summary
We denote the matter fields and Goldstone boson fields as P and φ, respectively. To introduce the effective Lagrangian, one has to specify the power counting rules. PCB terms show up when matter field propagators enter the loop integrals which are calculated using dimensional regularization with the MS scheme This is due to the existence of the new energy scale mP. To construct the effective Lagrangians respecting the chiral symmetry, we collect the Goldstone bosons in a N × N unitary matrix U(x), U(x) = u2(x) = exp iφ , F0. It is convenient for building up effective Lagrangians respecting the symmetry constraints to construct the matter fields so that they transform under. Based on the power counting for the Feynman graphs, the calculation of relevant physical observables up to O p3 requires the effective chiral Lagrangian.
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