  In nodal-line semimetals, Coulomb interactions and short-range correlated
disorder are both marginal perturbations to the clean non-interacting
Hamiltonian. We analyze their interplay using a weak-coupling renormalization
group approach. In the clean case, the Coulomb interaction has been found to be
marginally irrelevant, leading to Fermi liquid behavior. We extend the analysis
to incorporate the effects of disorder. The nodal line structure gives rise to
kinematical constraints similar to that for a two-dimensional Fermi surface,
which plays a crucial role in the one-loop renor- malization of the disorder
couplings. For a two-fold degenerate nodal loop (Weyl loop), we show that
disorder flows to strong coupling along a unique fixed trajectory in the space
of symmetry inequiv- alent disorder couplings. Along this fixed trajectory, all
symmetry inequivalent disorder strengths become equal. For a four-fold
degenerate nodal loop (Dirac loop), disorder also flows to strong coupling,
however the strengths of symmetry inequivalent disorder couplings remain
different. We show that feedback from disorder reverses the sign of the beta
function for the Coulomb interaction, causing the Coulomb interaction to flow
to strong coupling as well. However, the Coulomb interac- tion flows to strong
coupling asymptotically more slowly than disorder. Extrapolating our results to
strong coupling, we conjecture that at low energies nodal line semimetals
should be described by a noninteracting nonlinear sigma model. We discuss the
relation of our results with possible many-body localization at zero
temperatures in such materials.
