Pion and sigma meson properties in a relativistic quark model

Abstract

A variety of strong and electroweak interaction properties of a pion and a light scalar $\ensuremath{\sigma}$ meson are computed in a relativistic quark model. Under the assumption that the resulting coupling of these mesons to the constituent quarks is identical, the $\ensuremath{\sigma}$ meson mass is determined as ${M}_{\ensuremath{\sigma}}=385.4\mathrm{MeV}.$ We discuss in detail the gauging of the nonlocal meson-quark interaction and calculate the electromagnetic form factor of the pion and the form factors of the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ and $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}\ensuremath{\gamma}\ensuremath{\gamma}$ processes. We obtain explicit expressions for the relevant form factors and evaluate the leading and next-to-leading orders for large Euclidean photon virtualities. Turning to the decay properties of $\ensuremath{\sigma},$ we determine the width of the electromagnetic $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}\ensuremath{\gamma}\ensuremath{\gamma}$ transition and discuss the strong decay $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}\ensuremath{\pi}\ensuremath{\pi}.$ In a final step we compute the nonleptonic decays $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\sigma}\ensuremath{\pi}$ and $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\sigma}\ensuremath{\pi}$ relevant for the possible observation of the $\ensuremath{\sigma}$ meson. All our results are compared to available experimental data and to results of other theoretical studies.