π- andK-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 ⌫ H and the dressed-quark propagator in the chiral limit. In addition to a term proportional to ␥ 5 , ⌫ H necessarily contains qualitatively and quantitatively important terms proportional to ␥ 5 ␥• P and ␥ 5 ␥•kk•P, where P is the total momentum of the bound state. The axial-vector vertex contains a bound state pole described by ⌫ 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-quarkquark 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. ͓S0556-2813͑97͒04112-5͔

Hadron properties and Dyson–Schwinger equations

“…They relate even to radial excitations and/or hybrids [43,44,45], and heavy-light [46,47] and heavy-heavy mesons [39]. The results have been illustrated using a renormalisation-group-improved rainbow-ladder truncation [38,48], which also provided a prediction of the electromagnetic pion form factor [49]. In addition, algebraic parametrisations of the dressed-quark propagators and meson bound-state amplitudes obtained from such studies continue to be useful.…”

“…Notwithstanding these truths, M(p 2 ) is a more fundamental tracer for DCSB. The condensate is only a small part of the information that the mass function contains, amongst other pieces are pointwise consequences of Goldstone's theorem [32], definitions and an explanation of the constituent-quark mass [35,38,39], and a related σ-term [40]. 8 Harking back to Sect.…”

QCD modeling of hadron physics

“…Furthermore, these two truncations are mutually consistent in the sense that the combination produces vector and axial-vector vertices satisfying their respective Ward identities. In the axial case, this ensures that in the chiral limit the ground state pseudoscalar mesons are the massless Goldstone bosons associated with chiral symmetry breaking [5,6]. In the vector case, this ensures, in combination with impulse approximation, electromagnetic current conservation [7].…”

“…with qq 0 the renormalization-point-independent vacuum quark condensate [6]. It is a longstanding prediction of DSE studies in QCD that both the mass function M (p 2 ) and the wavefunction renormalisation function Z(p 2 ) receive strong momentum-dependent corrections at infrared momenta, see e.g.…”

Consequences of fully dressing quark–gluon vertex function with two-point gluon lines

“…(1) that, in principle, one must first solve the DSE for the gluon 2-point function as well as the DSE for the dressed quark-gluon vertex. In the renormalization group improved rainbow approximation [6], one replaces the combination of those two objects by bare vertex times an effective gluon 2-point function. The latter has a phenomenological infrared part that joins smoothly to the 1-loop renormalization group result for the ultraviolet running coupling.…”