A nuclear Fermi liquid exhibits competing pairing instabilities in different spin, isospin, and orbital channels. In a Fermi-surface renormalization group (RG) treatment, the channel that develops a pole first is determined not only by its tree-level attraction but also by its one-loop RG coefficient. We illustrate this mechanism in a minimal $S/P$ -wave model. A spherical Fermi surface establishes the reference competition between the lowest even- and odd-parity interactions. Axial deformation changes the relevant Fermi-surface integrals and lifts the degeneracy between longitudinal and transverse $P$ -wave components. In isospin-asymmetric matter, neutron--proton Fermi-momentum splitting restricts the simultaneous low-energy contribution of the two species and can terminate the $np$ running at finite threshold scales. Our calculations are intended as controlled one-loop RG illustrations rather than as quantitative nuclear-matter calculations. We show how the Fermi-surface geometry and composition can change the ordering of competing pairing instabilities.A nuclear Fermi liquid exhibits competing pairing instabilities in different spin, isospin, and orbital channels. In a Fermi-surface renormalization group (RG) treatment, the channel that develops a pole first is determined not only by its tree-level attraction but also by its one-loop RG coefficient. We illustrate this mechanism in a minimal $S/P$-wave model. A spherical Fermi surface establishes the reference competition between the lowest even- and odd-parity interactions. Axial deformation changes the relevant Fermi-surface integrals and lifts the degeneracy between longitudinal and transverse $P$-wave components. In isospin-asymmetric matter, neutron--proton Fermi-momentum splitting restricts the simultaneous low-energy contribution of the two species and can terminate the $np$ running at finite threshold scales. Our calculations are intended as controlled one-loop RG illustrations rather than as quantitative nuclear-matter calculations. We show how the Fermi-surface geometry and composition can change the ordering of competing pairing instabilities.

nucl-thCross-listedarXiv:2608.15759Aug 18, 2026

S/P

-wave model. A spherical Fermi surface establishes the reference competition between the lowest even- and odd-parity interactions. Axial deformation changes the relevant Fermi-surface integrals and lifts the degeneracy between longitudinal and transverse

P

-wave components. In isospin-asymmetric matter, neutron--proton Fermi-momentum splitting restricts the simultaneous low-energy contribution of the two species and can terminate the

np

running at finite threshold scales. Our calculations are intended as controlled one-loop RG illustrations rather than as quantitative nuclear-matter calculations. We show how the Fermi-surface geometry and composition can change the ordering of competing pairing instabilities.