The semimetal-superconductor quantum phase transition on the two-dimensional (2D) surface of a 3D topological insulator is conjectured to exhibit an emergent N=2 supersymmetry, based on a renormalization group (RG) analysis at one-loop order in the ϵ expansion. We provide additional support for this conjecture by performing a three-loop RG analysis and showing that the supersymmetric fixed point found at this order survives the extrapolation to 2D. We compute critical exponents to order ϵ^3,

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... aining the more accurate value ν≈ 0.985 for the correlation length exponent and confirming that the fermion and boson anomalous dimensions remain unchanged beyond one loop, as expected from non-renormalization theorems in supersymmetric theories. We further couple the system to a dynamical U(1) gauge field, and argue that the transition becomes fluctuation-induced first order. We discuss implications of this result for quantum phase transitions between certain symmetry-preserving correlated surface states of 3D topological insulators.