ηandη′in a coupled Schwinger-Dyson and Bethe-Salpeter approach

Abstract

Extending our earlier treatments of π 0 , η c and η b , we study the η-η ′ system and its γγ decays using a model which is a leading version of the consistently coupled Schwinger-Dyson (SD) and Bethe-Salpeter (BS) approach. The electromagnetic interactions are incorporated through a (generalized) impulse approximation consistent with this bound-state approach, so that the Ward-Takahashi identities of QED are preserved when quarks are dynamically dressed. To overcome some of the limitations due to the ladder approximation, we introduce a minimal extension to the bound-state approach employed, so that the U A (1) problem is avoided. Pointing out which of our predictions hold in the coupled SD-BS approach in general, and which are the consequences of the specific, chosen model, we present the results for the axial-current decay constants of η 8 , η 0 , and of their physical combinations η and η ′ , the results for the γγ-decay constants of η 0 and η 8 , for the two-photon decay widths of η and η ′ , and for the mixing-independent R-ratio constructed from them.

ηandη′mesons and dimension 2 gluon condensate⟨A2⟩

“…The description [25,26,42] of η and η ′ is especially noteworthy, as it is successful in spite of the limitations of the DS approach in the ladder approximation. For this description, the crucial issues are the meson mixing and construction of physical meson states.…”

“…Typically, as in Refs. [21,25,33,39,40,41] for example, it combines the modeled nonperturbative component (which is strong enough to obtain required DχSB) and the known perturbative QCD contribution. Our gluon condensate-induced α eff (Q 2 ) (15) also behaves nonperturbatively at low Q 2 and perturbatively at very high Q 2 , as seen both in its earlier derivation [15] and the alternative derivation in Sec.…”

“…For the DS approach, they are formulated in Refs. [25,26], where solving of appropriate BS equations (17) yields the eigenvalues of the squared masses, M 2 uū , M 2 dd , M 2 ss , and M 2 us , of the respective quark-antiquark bound states |uū , |dd , |ss , and |us . The last one is simply the kaon, and M us is its mass M K .…”

“…Nevertheless, for η and η ′ this is not enough because of the influence of the gluon anomaly and thus we explain how its effects are included in the manner of Refs. [25,26]. The implementation of the anomaly and the SU(3) flavor symmetry breaking, as well as their interplay, are presented independently of any concrete dynamics in subsection 3 A, and in detail, because of some renewed interest in the flavor dependence of the mixing in the η-η ′ complex (e.g., Ref.…”

Dynamical chiral symmetry breaking and a critical mass

“…The truncation has been used widely; e.g., Refs. [17,28,29], and references thereto, and an efficacious implementation preserves the one-loop ultraviolet behaviour of perturbative QCD. However, a model assumption is required for the behaviour of the kernel in the infrared; viz., on the interval Q 2 < ∼ 1 GeV 2 , which corresponds to length-scales…”