Two-pion decays of mesons and confinement singularities

We consider the two-pion decay of the $\rho$-meson, the $^3S_1q\bar q$-state of the constituent quarks -- the decay being determined by the transition $q\bar q\to \pi\pi$ contains information about confinement interactions. One can specify in this decay two types of transitions: ({\bf i}) the bremsstrahlung radiation of a pion $q\to q+\pi$ (or $\bar q\to\bar q+\pi$) with the subsequent fusion $q\bar q\to \pi$, and ({\bf ii}) the direct transition $q\bar q\to \pi\pi$. We demonstrate how in the amplitudes of the corresponding transitions the quark singularities have to disappear, {\it i.e.} what is the way the quark confinement at relatively short distances can be realized. We calculate and estimate the contributions of processes with bremsstrahlung radiation of the pion and of the direct transition $q\bar q\to \pi\pi$. The estimates demonstrate that the processes involving the direct transition $q\bar q\to \pi\pi$ are necessary, but they cannot be determined unambiguously by the decay $\rho(775)\to \pi\pi$. We conclude that for the determination of the $q\bar q\to \pi\pi$ transition more complete data on the resonance decays into the $\pi\pi$ channels are required than those available at the moment.