Employing a renormalization group analysis that allows for an unbiased treatment of competing physical ingredients, we systematically trace how the interplay between Cooper pairing and disorder scatterings governs the emergence or suppression of Cooper instability in the low-energy regime of fractional Dirac semimetals.In the clean limit, we find that the emergence of Cooper instability requires surpassing a finite interaction threshold $|λ_c|$, and depends sensitively on both the fractional exponent $α$ and the transfer momentum $\mathbf{Q}=(Q,ϕ)$. Specifically, bigger values of $α$ enhance the tendency toward BCS instability. For $α\in(0.001,0.61)$, the $(Q,ϕ)$ parameter space separates into two distinct regions: Zone-\uppercase\expandafter{\romannumeral1}, where Cooper instability is suppressed, and Zone-\uppercase\expandafter{\romannumeral2}, where it is allowed. In the presence of disorders, we demonstrate that they can either promote or suppress Cooper instability. Disorder of type $Δ_1$ or $Δ_2$ enhances superconductivity by reducing the critical interaction threshold $|λ_c|$ and expanding the superconducting phase space (Zone-\uppercase\expandafter{\romannumeral2}). In sharp contrast, either $Δ_0$ or $Δ_3$ suppresses Cooper pairing by increasing $|λ_c|$ and shrinking the available phase space (Zone-\uppercase\expandafter{\romannumeral1}). Although Cooper instability can be enhanced when promotive disorders ($Δ_1$, $Δ_2$) coexist with a single suppressive disorder ($Δ_0$ or $Δ_3$), the suppressive influence of $Δ_{0,3}$ generally dominates the promotive effects of $Δ_{1,2}$ in the presence of all sorts of disorders. These results would be helpful for further studies of fractional Dirac semimetals and alike materials.
