Bethe-Salpeter $q\overline{q}$ dynamics under harmonic confinement

A Bethe-Salpeter (BS) treatment of $q\overline{q}$ proposed recently for equal-mass quarks in the instantaneous approximation is extended to unequal masses. An explicit mass formula is obtained in the dimensionless form $F\left(M\right)=N+\mathrm{const}$, where $F\left(M\right)\mathrm{}$, a nonlinear function of $M$, represents a central supermultiplet ${\mathrm{}\left(\mathrm{mass}\right)\mathrm{}}^2$ for each $N$ excitation, formally corrected for spin and flavor variations among members. These universality features of $F\left(M\right)\mathrm{}$ are borne out by the data to a significant extent up to the charmed sector, in terms of a single reduced spring constant $\tilde{\omega }$ (=0.15 GeV) and the quark masses ($m_q$). A fully covariant and transparent construction is found for the Bethe-Salpeter wave functions for $P$, $V$, and $A_1$ mesons consistent with their equations of motion and satisfying the BS normalization without the need for a conventional decomposition into (±±) components. Tests of these wave functions are offered through the predictions of (a) $P$, $V\to l\overline{l}$ and $\tau \to (\pi , \rho , A_1){\nu }_{\tau }$ decays, (b) electromagnetic radii of $P$ mesons, and (c) weak form factor in $K_{l3\mathrm{}}$ decay, all in excellent agreement with experiment, without extra parameters.