Derivation of Intermultiplet Mass Formulas from the ChiralSU(3)⊗SU(3)Charge Algebra

Abstract

We have derived intermultiplet mass formulas by using an $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ charge algebra (involving time derivatives) and an approximation [called the $\mathrm{SU}\left(3\right)\mathrm{}$ approximation] previously proposed. The approximation assumes that the $\mathrm{SU}\left(3\right)\mathrm{}$ charge operator $V_K$ [defined, for example, in a quark model, by $V_{K^{-}}=\int d^{3}x\frac{\overline{q}\mathrm{}\left(x\right)\mathrm{}{\gamma }_4({\lambda }_4-i{\lambda }_5)q\left(x\right)\mathrm{}}{2}$] acts as an $\mathrm{SU}\left(3\right)\mathrm{}$ generator even in broken symmetry in an appropriately chosen infinite-momentum limit. We have obtained, besides the Gell-Mann-Okubo mass formulas including first-order mixing, intermultiplet mass formulas such as (i) ${m_K}^{*02\mathrm{}}-{m_{\rho }}^{02}={m_K}^{+2\mathrm{}}-{m_{\pi }}^{+2\mathrm{}}={m_K}^{**+2\mathrm{}}-{m_{A_2}}^{+2\mathrm{}}$, (ii) $m_{\rho }\simeq {m_{\omega }}^0$, (iii) $f-f^{\prime }$ mixing angle $\theta \simeq 30°$ from $2^{+}$ and $0^{-}$ mass formulas, and (iv) $\mathrm{SU}\left(6\right)\mathrm{}$-type decupletoctet baryon quadratic mass formulas. A brief discussion is presented about the $J^{\mathrm{PC}}=1^{++} \mathrm{and} 1^{+-}$ meson mass formulas.