  Among the transparent conducting oxides, the perovskite barium stannate is
most promising for various electronic applications due to its outstanding
carrier mobility achieved at room temperature. However, most of its important
characteristics, such as band gaps, effective masses, and absorption edge,
remain controversial. Here, we provide a fully consistent picture by combining
state-of-the-art {\it ab initio} methodology with forefront electron
energy-loss spectroscopy and optical absorption measurements. Valence electron
energy-loss spectra, featuring signals originating from band gap transitions,
are acquired on defect-free sample regions of a BaSnO$_{3}$ single crystal.
These high-energy-resolution measurements are able to capture also very weak
excitations below the optical gap, attributed to indirect transitions. By
temperature-dependent optical absorption measurements, we assess band-gap
renormalization effects induced by electron-phonon coupling. Overall, we find
for the effective electronic mass, the direct and the indirect gap, the optical
gap, as well as the absorption onsets and spectra, excellent agreement between
both experimental techniques and the theoretical many-body results, supporting
also the picture of a phonon-mediated mechanism where indirect transitions are
activated by phonon-induced symmetry lowering. This work demonstrates a
fruitful connection between different high-level theoretical and experimental
methods for exploring the characteristics of advanced materials.
