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We use $(121\pm1)$ million $\Upsilon(3S)$ and $(98\pm1)$ million$\Upsilon(2S)$ mesons recorded by the BABAR detector at the PEP-II $e^+e^-$collider at SLAC to perform a study of radiative transitions involving the$\chi_{b\mathrm{J}}(1P,2P)$ states in exclusive decays with$\mu^+\mu^-\gamma\gamma$ final states. We reconstruct twelve channels in fourcascades using two complementary methods. In the first we identify both signalphoton candidates in the Electromagnetic Calorimeter (EMC), employ acalorimeter timing-based technique to reduce backgrounds, and determinebranching-ratio products and fine mass splittings. These results include thebest observational significance yet for the$\chi_{b0}(2P)\rightarrow\gamma\Upsilon(2S)$ and$\chi_{b0}(1P)\rightarrow\gamma\Upsilon(1S)$ transitions. In the second method,we identify one photon candidate in the EMC and one which has converted into an$e^+e^-$ pair due to interaction with detector material, and we measureabsolute product branching fractions. This method is particularly useful formeasuring $\Upsilon(3S)\rightarrow\gamma\chi_{b1,2}(1P)$ decays. Additionally,we provide the most up-to-date derived branching fractions, matrix elements andmass splittings for $\chi_b$ transitions in the bottomonium system. Using a newtechnique, we also measure the two lowest-order spin-dependent coefficients inthe nonrelativistic QCD Hamiltonian.

Bottomonium spectroscopy and radiative transitions involving the chi_bJ(1P,2P) states at BABAR