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 present the first joint theory-experiment study of the electronic structure of a candidate topological semimetal with resonant inelastic X-ray scattering (RIXS). We resolve the bulk electronic states of FeSi using momentum-dependent 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 (DFT), 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 with zero adjustable parameters. While band theory predicts an excitation continuum with broad spectral features similar to the observed ones, discrepancies between theory and experiment suggest the presence of low-energy processes that DFT alone does not account for. This first study of RIXS in a topological semimetal shows that RIXS is a useful tool for revealing unanticipated behavior of bulk electronic states in this class of materials.

v2: published version