We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions (m) and codimensions (d - m), arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in d spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension d = d(c) (m). The fixed points are identified from the RG flow equations, as functions of d and m. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of (d = 3, m = 2) and (d = 2, m = 1). In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.
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