  We develop a field-theoretic approach to massless Dirac fermions in a
supercritical Coulomb potential. By introducing an Aharonov--Bohm solenoid at
the potential center, the critical Coulomb charge can be made arbitrarily small
for one partial-wave sector, where a perturbative renormalization group
analysis becomes possible. We show that a scattering amplitude for reflection
of particle at the potential center exhibits the renormalization group limit
cycle, i.e., log-periodic revolutions as a function of the scattering energy,
revealing the emergence of discrete scale invariance. This outcome is further
incorporated in computing the induced charge and current densities, which turn
out to have power-law tails with coefficients log-periodic with respect to the
distance from the potential center. Our findings are consistent with the
previous prediction obtained by directly solving the Dirac equation and can in
principle be realized by graphene experiments with charged impurities.
