Transition form factors γ^*to γ f₂(1270) and γ^*to γ a₂(1320) in the e^+e^- collisions

The predictions for transition form factors $\gamma^*\to \gamma f_2(1270)$ and $\gamma^*\to \gamma a_2(1320)$ and corresponding $e^+e^-\to \gamma^*\to f_2\gamma$ and $e^+e^-\to \gamma^*\to a_2\gamma$ cross sections are obtained for the energy region up to $2$ GeV. These predictions are coordinated with the recent Belle data on the $\gamma^*(Q^2)\gamma\to f_2$ transition. It is shown that the QCD asymptotics of the amplitudes of the reactions $e^+e^-\to \gamma^*\to f_2\gamma$ and $e^+e^-\to \gamma^*\to a_2\gamma$ can be reached only by taking into account a compensation of contributions of $\rho(770)$, $\omega(782)$ with contributions of their radial excitations. The relation $\sigma(e^+e^-\to \gamma^*\to f_2\gamma)/\sigma(e^+e^-\to \gamma^*\to a_2\gamma)\approx 25/9$, obtained with the help of the $q\bar q$ model of $a_2$ and $f_2$ and QCD, is used to determine the $\sigma(e^+e^-\to \gamma^*\to a_2\gamma)$ at high energies. Recent BABAR measurement of the $e^+e^-\to f_2\gamma\to\pi^+\pi^-\gamma$ cross section at $10.58$ GeV gives hope for detailed investigation of the $\gamma^*(s)\to f_2\gamma$ and $\gamma^*(s)\to a_2\gamma$ transition form factors at high energy region.