Observing the Minkowskian dynamics of the pion on the null-plane

A dynamical model is applied to the study of the pion valence light-front wave function, obtained from the actual solution of the Bethe-Salpeter equation in Minkowski space, resorting to the Nakanishi integral representation. The kernel is simplified to a ladder approximation containing constituent quarks, an effective massive gluon exchange, and the scale of the extended quark-gluon interaction vertex. These three input parameters carry the infrared scale ΛQCD and are fine-tuned to reproduce the pion weak decay constant, within a range suggested by lattice calculations. Besides fπ, we present and discuss other interesting quantities on the null-plane, like: (i) the valence probability, (ii) the dynamical functions depending upon the longitudinal or the transverse components of the light-front (LF) momentum, represented by LF-momentum distributions and distribution amplitudes, and (iii) the probability densities both in the LF-momentum space and the 3D space given by the Cartesian product of the covariant Ioffe-time and transverse coordinates, in order to perform an analysis of the dynamical features in a complementary way. The proposed analysis of the Minkowskian dynamics inside the pion, though carried out at the initial stage, qualifies the Nakanishi integral representation as an appealing effective tool, with still unexplored potentialities to be exploited for addressing correlations between dynamics and observable properties.