Calculation of the anomalous $γπ^* \to ππ$ form factor

Abstract

The form factor for the anomalous process $γπ^* \to ππ$, $F^{3π}(s,t,u)$, is calculated as a phenomenological application of the QCD Dyson-Schwinger equations. The chiral-limit value dictated by the electromagnetic, anomalous chiral Ward identity, $F^{3π}(0,0,0)= e N_c/(12π^2 f_π^3)$, is reproduced, {\it independent} of the details of the modelling of the gluon and quark 2-point Schwinger functions. Using a parametrisation of the dressed $u$-$d$ quark 2-point Schwinger function that provides a good description of pion observables, such as $π$-$π$ scattering-lengths and the electromagnetic pion form factor, $F^{3π}(s,t,u)$ is calculated on a kinematic range that proposed experiments plan to explore. Our result confirms the general trend of other calculations; i.e., a monotonic increase with $s$ at fixed $t$ and $u$, but is uniformly larger and exhibits a more rapid rise with $s$.