Radiative corrections of $O(\a)$ for pion beta decay in the light-front quark model

If the CKM matrix element $V_{ud}$, that can be derived from superallowed nuclear decays, neutron decay and pion beta decay, is used for a precision test of the unitarity of the CKM matrix, the combination of the present world data seems to indicate a small violation of the unitarity condition for the first row. While an accurate calculation of the radiative corrections of $O(\a)$ (RC) is crucial in order to determine the value of $V_{ud}$ as precisely as possible, the theoretical analysis has been limited in the past by the rather crude estimate of the effect of the hadronic structure. Only the contribution due to the axial current depends on the hadronic environment. We develop a strategy to deal with the influence of the hadronic structure on the decay properties of the simplest hadron, the pion, and calculate the contribution of the axial vector current to the RC, using a light-front model for the pion. Its $q\bar{q}$ bound state structure is well described by two parameters, constituent quark mass and confinement scale, that have been fixed by a comparison with the data. We take into consideration three different groups of two-loop diagrams, and derive their light-front representations. We discuss the associated zero-mode problem and show that the respective light-front amplitudes are free of spurious contributions. There is only a small model dependent uncertainty of the final result for the RC for pion beta decay.