Sum rules for total hadronic widths of light mesons and rectilineal stitch of the masses on the complex plane

Abstract

Mass formulae for light meson multiplets derived by means of exotic commutator technique are written for complex masses and considered as complex mass sum rules (CMSR). The real parts of the (CMSR) give the well known mass formulae for real masses (Gell-Mann--Okubo, Schwinger and Ideal Mixing ones) and the imaginary parts of CMSR give appropriate sum rules for the total hadronic widths - width sum rules (WSR). Most of the observed meson nonets satisfy the Schwinger mass formula (S nonets). The CMSR predict for S nonet that the points $(m,Γ)$ form the rectilinear stitch (RS) on the complex mass plane. For low-mass nonets WSR are strongly violated due to ``kinematical'' suppression of the particle decays, but the violation decreases as the mass icreases and disappears above $\sim 1.5 GeV$. The slope $k_s$ of the RS is not predicted, but the data show that it is negative for all S nonets and its numerical values are concentrated in the vicinity of the value -0.5. If $k_s$ is known for a nonet, we can evaluate ``kinematical'' suppressions of its individual particles. The masses and the widths of the S nonet mesons submit to some rules of ordering which matter in understanding the properties of the nonet. We give the table of the S nonets indicating masses, widths, mass and width orderings. We show also mass-width diagrams for them. We suggest to recognize a few multiplets as degenerate octets. In Appendix we analyze the nonets of $1^+$ mesons.