Pion and sigma meson properties in a relativistic quark model

A variety of strong and electroweak interaction properties of a pion and a light scalar $\sigma$ 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 $\sigma$ meson mass is determined as $M_{\sigma }=385.4\mathrm{}\mathrm{MeV}.$ 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 ${\pi }^0\to \gamma \gamma$ and $\overrightarrow{\sigma }\gamma \gamma$ 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 $\sigma ,$ we determine the width of the electromagnetic $\overrightarrow{\sigma }\gamma \gamma$ transition and discuss the strong decay $\overrightarrow{\sigma }\pi \pi .$ In a final step we compute the nonleptonic decays $\overrightarrow{D}\sigma \pi$ and $\overrightarrow{B}\sigma \pi$ relevant for the possible observation of the $\sigma$ meson. All our results are compared to available experimental data and to results of other theoretical studies.