Calculating the Decay Constant fπ

Abstract
We calculate physical quantities accompanying dynamical chiral symmetry breaking in QCD like theories. To emphasize the consistency of our approximation with chiral symmetry, we first derive an effective action for a bilocal auxiliary field standing for ψ(x)ψ(y), from which the Schwinger-Dyson equation for fermion self-energy Σ and the Bethe-Salpeter equation for the Nambu-Goldstone bound state follow. An expression for the decay constant (usually called fπ) is given exactly to the ladder approximation by solving the Bethe-Salpeter equation. Numerical calculation is done for Σ, fπ, ≪ψψ>R, with various β-functions of the gauge coupling constant. We find that fπ/≪ψψ>R, fπ/ΛQCD are rather stable against the infrared cutoff parameter, while Σ-function itself depends strongly on the infrared cutoff.