The rho -> gamma pi and omega -> gamma pi decays in quark-model approach and estimation of coupling for pion emission by quark

In the framework of the relativistic and gauge invariant spectral integral technique, we calculate radiative decays rho(770)-> gamma pi(140) and omega(780)-> gamma pi(140) supposing all mesons (pi, rho and omega) to be quark-antiquark states. The q anti-q wave functions found for mesons and photon lead to a reasonably good description of data ($\Gamma^{(exp)}{\rho^{\pm} \to\gamma\pi^{\pm}}=68\pm 30$keV,$\Gamma^{(exp)}{\rho^{0}\to\gamma\pi^0}=77\pm 28$keV,$\Gamma^{(exp)}{\omega\to\gamma\pi^0}=776\pm 45$ keV) that makes it possible to estimate the coupling for the bremsstrahlung emission of pion by quarks $g_{\pi }\equiv g_{\pi }(u\to d\pi )$. We have found two values for the pion bremsstrahlung coupling: $\mathrm{\mid }{g}_{\pi }\mathrm{\mid }=16.7\pm 0.3_{-2.3}^{+0.1}$ (Solution I) and $\mathrm{\mid }{g}_{\pi }\mathrm{\mid }=3.0\pm 0.3_{-2.1}^{+0.1}$ (Solution II). Within SU(6)-symmetry for nucleons, Solution I gives us for pi NN coupling the value $16.4 \le g{\pi NN}^2/(4\pi) \le 23.2$ that is in qualitative agreement with the pi N scattering data, $g_{\pi NN}^2\mathrm{/}(4\pi )\simeq 14$. For excited states, we have estimated the partial widths in Solution I as follows: $\Gamma (\rho_{2S}^\pm\to \gamma\pi)\simeq 10 - 130$keV,$\Gamma (\rho_{2S}^0\to \gamma\pi)\simeq 10 -130$keV,$\Gamma (\omega_{2S}\to \gamma\pi)\simeq 60 - 1080$ keV. The large uncertainties emphasise the necessity to carry out measurements of the meson radiative processes in the region of large masses.