  The semimetal-superconductor quantum phase transition on the two-dimensional
(2D) surface of a 3D topological insulator is conjectured to exhibit an
emergent $\mathcal{N}=2$ supersymmetry, based on a renormalization group (RG)
analysis at one-loop order in the $\epsilon$ 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 $\epsilon^3$,
obtaining the more accurate value $\nu\approx 0.985$ 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.
