Quark--antiquark states and their radiative transitions in terms of the spectral integral equation. {\Huge I.} Bottomonia

In the framework of the spectral integral equation, we consider the $b\bar b$ states and their radiative transitions. We reconstruct the $b\bar b$ interaction on the basis of data for the levels of the bottomonium states with $J^{PC}=0^{-+}$, $1^{--}$, $0^{++}$, $1^{++}$, $2^{++}$ as well as the data for the radiative transitions $\Upsilon(3S) \to\gamma\chi_{bJ}(2P) $ and $\Upsilon(2S) \to\gamma \chi_{bJ}(1P) $ with $J=0,1,2$. We calculate bottomonium levels with the radial quantum numbers $n\le 6$, their wave functions and corresponding radiative transitions. The ratios $Br[\chi_{bJ}(2P)\to\gamma\Upsilon(2S)]/Br[\chi_{bJ}(2P)\to\gamma\Upsilon(1S)]$ for $J=0,1,2$ are found in the agreement with data. We determine the $b\bar b$ component of the photon wave function using the data for the $e^+e^-$ annihilation, $e^+e^- \to\Upsilon(9460)$, $\Upsilon(10023)$, $\Upsilon(10036)$, $\Upsilon(10580)$, $ \Upsilon(10865)$, $\Upsilon(11019)$, and predict partial widths of the two-photon decays $\eta_{b0}\to\ggam$, $\chi_{b0}\to\ggam$, $\chi_{b2}\to\ggam$ for the radial excitation states below $B\bar B$ threshold ($n\le 3$).