  We determine the fate of interacting fermions described by the Hamiltonian
$H=\textbf{p}\cdot \textbf{J}$ in three-dimensional topological semimetals with
linear band crossing, where $\textbf{p}$ is momentum and $\textbf{J}$ are the
spin-$j$ matrices for half-integer pseudospin $j\geq 3/2$. While weak
short-range interactions are irrelevant at the crossing point due to the
vanishing density of states, weak long-range Coulomb interactions lead to a
renormalization of the band structure. Using a self-consistent perturbative
renormalization group approach, we show that band crossings of the type
$\textbf{p}\cdot \textbf{J}$ are unstable for $j\leq 7/2$. Instead, through an
intriguing interplay between cubic crystal symmetry, band topology, and
interaction effects, the system is attracted to a variety of infrared fixed
points. We also unravel several other properties of higher-spin fermions for
general $j$, such as the relation between fermion self-energy and free energy,
or the vanishing of the renormalized charge. An $\text{O}(3)$ symmetric fixed
point composed of equal chirality Weyl fermions is stable for $j\leq 7/2$ and
very likely so for all $j$. We then explore the rich fixed point structure for
$j=5/2$ in detail. We find additional attractive fixed points with enhanced
$\text{O}(3)$ symmetry that host both emergent Weyl or massless Dirac fermions,
and identify a puzzling, infrared stable, anisotropic fixed point without
enhanced symmetry in close analogy to the known case of $j=3/2$.
