Self-consistent meson mass spectrum

A dual-topological-unitarization (or dual-fragmentation) approach to the calculation of hadron masses is presented, in which the effect of planar "sea"-quark loops is taken into account from the beginning. Using techniques based on analyticity and generalized ladder-graph dynamics, we first derive the approximate "generic" Regge-trajectory formula $\alpha \mathrm{}\left(t\right)=max(S_1+S_2, S_3+S_4)-\frac{1}{2}+2\mathrm{}{\hat{\alpha }}^{\prime }[s_a+\frac{1}{2}(t-\Sigma {m_i}^2\left)\right]$ for any given hadronic process 1+2→3+4, where $S_i$ and $m_i$ are the spins and masses of $i=1, 2, 3, 4, \mathrm{and} \sqrt{s_a}$ is the effective mass of the lowest nonvanishing contribution ($a$) exchanged in the crossed channel. By requiring a minimization of secondary (background, etc.) contributions to $a$, and demanding simultaneous consistency for entire sets of such processes, we are then able to calculate the masses of all the lowest pseudoscalar and vector $q\overline{q}$ states with $q=u, d, s$, and the Regge trajectories on which they lie. By making certain additional assumptions we are also able to do this with $q=u, d, c$ and $q=u, d, b$. Our only arbitrary parameters are $m_{\rho }$, $m_{K^{*}}$, $m_{\psi }$, and $m_{\Upsilon }$, one of which merely serves to fix the energy scale. In contrast to many other approaches, a small $\frac{{m_{\pi }}^2}{{m_{\rho }}^2}$ ratio arises quite naturally in the present scheme.