<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi></mml:math>- and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi></mml:math>-meson Bethe-Salpeter amplitudes

Abstract

Independent of assumptions about the form of the quark-quark scattering kernel $K$, we derive the explicit relation between the flavor-nonsinglet pseudoscalar-meson Bethe-Salpeter amplitude ${\ensuremath{\Gamma}}_{H}$ and the dressed-quark propagator in the chiral limit. In addition to a term proportional to ${\ensuremath{\gamma}}_{5}$, ${\ensuremath{\Gamma}}_{H}$ necessarily contains qualitatively and quantitatively important terms proportional to ${\ensuremath{\gamma}}_{5}\ensuremath{\gamma}\ensuremath{\cdot}P$ and ${\ensuremath{\gamma}}_{5}\ensuremath{\gamma}\ensuremath{\cdot}\mathrm{kk}\ensuremath{\cdot}P$, where $P$ is the total momentum of the bound state. The axial-vector vertex contains a bound state pole described by ${\ensuremath{\Gamma}}_{H},$ whose residue is the leptonic decay constant for the bound state. The pseudoscalar vertex also contains such a bound state pole and, in the chiral limit, the residue of this pole is related to the vacuum quark condensate. The axial-vector Ward-Takahashi identity relates these pole residues, with the Gell-Mann--Oakes--Renner relation a corollary of this identity. The dominant ultraviolet asymptotic behavior of the scalar functions in the meson Bethe-Salpeter amplitude is fully determined by the behavior of the chiral limit quark mass function, and is characteristic of the QCD renormalization group. The rainbow-ladder Ansatz for $K$, with a simple model for the dressed-quark-quark interaction, is used to illustrate and elucidate these general results. The model preserves the one-loop renormalization group structure of QCD. The numerical studies also provide a means of exploring procedures for solving the Bethe-Salpeter equation without a three-dimensional reduction.