Null-plane Bethe-salpeter dynamics: Mass spectra, decay constants of pseudoscalar mesons, and the pion form factor

A new relativistic definition of the reduced mass (${\mu }_{12}$) of a $q\overline{q}$ pair, so as to be in conformity with the standard Wightman-Garding definition of its relative four-momenta $q_{\mu }$, is introduced into the kernel of an ongoing Bethe-Salpeter (BS) program on a two-tier basis. The new definition of ${\mu }_{12}$ involving the hadron mass $M$ is found to produce a natural Regge asymptotic behavior ($M^2\sim N$) in the hadron mass spectra, while retaining the property of an asymptotically linear ($\sim r$) confinement in the three-dimensional structure of the BS kernel. The relativistic structure of ${\mu }_{12}$ is responsible for a significant improvement in the fits to the ground-state masses of $q\overline{q}$ and $Q\overline{q}$ mesons as compared to its nonrelativistic definition $\frac{m_1{m}_2}{(m_1+m_2)}$. The leptonic decay constants $f_p$ and the charge radii thus calculated are also in excellent agreement with data ($\pi ,k$) where available, while $f_p$ predictions for $Q\overline{q}$ mesons have good overlap with recent lattice predictions. Further, the scaling property ($\sim k_{\mu }^{-2\mathrm{}}$) of the hadron's electromagnetic form factor at large $k^2$ is a consequence of the "on-shell" form of its null-plane wave function. All these results (which are indicated in the barest outline) are preceded by a perspective summary of the theoretical premises and practical working of the BS equation with a four-fermion interaction kernel as a necessary background on a two-tier basis.