  Three-dimensional line-nodal superconductors exhibit nontrivial topology,
which is protected by the time-reversal symmetry. Here we investigate four
types of short-range interaction between the gapless line-nodal fermionic
quasiparticles by carrying renormalization group analysis. We find that such
interactions can induce the dynamical breaking of time-reversal symmetry, which
alters the topology and might lead to six possible distinct superconducting
states, distinguished by the group representations. After computing the
susceptibilities for all the possible phase-transition instabilities, we
establish that the superconducting pairing characterized by $id_{xz}$-wave gap
symmetry is the leading instability in noncentrosymmetric superconductors.
Appropriate extension of this approach is promising to pick out the most
favorable superconducting pairing during similar topology-changing transition
in the polar phase of $^3$He.
