  The energy spectrum of topological semimetals contains protected degeneracies
in reciprocal space that correspond to Weyl, Dirac, or multifold fermionic
states. To exploit the unconventional properties of these states, one has to
access the electronic structure of the three-dimensional bulk. In this work, we
resolve the bulk electronic states of candidate topological semimetal FeSi
using momentum-dependent resonant inelastic X-ray scattering (RIXS) at the Fe
$L_3$ edge. We observe a broad excitation continuum devoid of sharp features,
consistent with particle-hole scattering in an underlying electronic band
structure. Using density functional theory, we calculate the electronic
structure of FeSi and derive a band theory formulation of RIXS in the fast
collision approximation to model the scattering process. We find that band
theory qualitatively captures the number and position of the main spectral
features, as well as the overall momentum dependence of the RIXS intensity. Our
work paves the way for targeted studies of band touchings in topological
semimetals with RIXS.
