  Since the proposal of monopole Cooper pairing in [Phys. Rev. Lett. 120,
067003 (2018)], considerable research efforts have been dedicated to the study
of Cooper pairing order parameters constrained (or obstructed) by the
nontrivial normal-state band topology at Fermi surfaces in 3D systems. In the
current work, we generalize the topologically obstructed pairings between Chern
states (like the monopole Cooper pairing) by proposing Euler obstructed Cooper
pairing in 3D systems. The Euler obstructed Cooper pairing widely exists
between two Fermi surfaces with nontrivial band topology characterized by
nonzero Euler numbers; such Fermi surfaces can exist in 3D PT-protected
spinless-Dirac/nodal-line semimetals with negligible spin-orbit coupling, where
PT is the space-time inversion symmetry. An Euler obstructed pairing channel
must have pairing nodes on the pairing-relevant Fermi surfaces, and the total
winding number of the pairing nodes is determined by the sum or difference of
the Euler numbers on the Fermi surfaces. In particular, we find that when the
normal state is time-reversal invariant and the pairing is weak, a
sufficiently-dominant Euler obstructed pairing channel with zero total momentum
leads to nodal superconductivity. If the Fermi surface splitting is small, the
resultant nodal superconductor hosts hinge Majorana zero modes. The possible
dominance of the Euler obstructed pairing channel near the superconducting
transition and the robustness of the hinge Majorana zero modes against disorder
are explicitly demonstrated using effective or tight-binding models. Our work
presents the first class of higher-order nodal superconductivity originating
from the topologically obstructed Cooper pairing.
