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Two-photon processes of pi^0, eta, eta", eta_c and eta_b in a coupled Schwinger-Dyson and Bethe-Salpeter approach

$pi^0, eta, eta^prime, eta_c$ and $eta_b$ are modeled as quark-antiquark bound states in a leading version of the consistently coupled Schwinger-Dyson (SD) and Bethe-Salpeter (BS) approach. We study two-photon processes of these neutral pseudoscalar mesons. In our approach, dynamical chiral symmetry breaking is obtained through the Schwinger-Dyson equation for the quark propagator which is then used in the bound-state (Bethe-Salpeter) equation in a consistent approximation. This makes possible the appearance of light pseudoscalar mesons as (pseudo)Goldstone bosons of the dynamical chiral symmetry breaking. This element is the key to the adequate description of the light sector. Secondly, the version of the coupled SD-BS approach adopted by us, had been carefully devised so that it also has the correct nonrelativistic limit when quark masses are in the opposite regime, {it i.e.}, when they become heavy. Munczek and Jain had already shown that these two elements enable the coupled SD-BS approach to yield successful meson spectroscopy over an impressive range of masses, from lightest to heaviest. We have now shown that the coupled SD-BS approach is similarly successful in the description of two--photon processes of pseudoscalar mesons over a wide range of masses. The electromagnetic interactions are incorporated through a (generalized) impulse approximation employing a dressed quark-photon vertex satisfying the Ward-Takahashi identities of QED when quarks are dynamically dressed. This is absolutely crucial in the chiral limit, where quark masses stem purely from the dynamical chiral symmetry breaking, as our approach then yields the famous axial anomaly result for $pi^0ightarrowgammagamma$ amplitude exactly and analytically, in the closed form, and independently of the details of the pion structure. Of course, it is crucial also for the off-shell extension of this process, namely for the $pi^0gamma^star ogamma$ amplitude. Although the dynamical dressing of the much heavier $c$-quarks is less important than the dressing of the light quarks, our results show that its consistent treatment is essential for the understanding of $eta_cightarrowgammagamma$ in the light of the recent data from CLEO. These data lower the partial width for $eta_cightarrowgammagamma$ below the reach of nonrelativistic calculations, while it seemingly remains too high for those Bethe-Salpeter treatments which, in contradistinction to ours, do not employ dynamically dressed quarks and accordingly dressed electromagnetic vertices, unless they fit their model parameters for that purpose. In contrast, we have not done any fitting or adjusting of the model parameters. $eta$ and $eta^prime$ are of special interest in the present context as they interpolate between the extreme regime of the lightest pseudoscalar -- the pion, and the other extreme regime of the heavy pseudoscalars $eta_c$ and $eta_b$. However, the mixing in the $eta$--$eta^prime$ system, in which the {it gluon} axial anomaly plays a special role, complicates the situation. To overcome some of the limitations due to our approximated gluon-exchange kernel, we introduce a minimal extension to the bound--state approach employed, so that the ${m U}_{ m A}{ m (1)}$ 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 model, we present the results for ${ ar f}_{eta_0}$ and ${ ar f}_{eta_8}$, the $gammagamma$ decay constants of $eta_0$ and $eta_8$, the results for $eta$ and $eta^prime$ two--photon decay widths, and for the mixing--independent $R$--ratio constructed from them. We discuss the connections thereof with the measurements to be performed on facilities such as DA$Phi$NE.

Non-perturbative QCD; Hadronic structure; Axial anomaly; Two-photon processes; pseudoscalar mesons; chiral symmetry breaking; quark models

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