  We study theoretically the quantum critical phenomenon of the phase
transition between the trivial insulator and the topological insulator in (3+1)
dimensions, which is described by a Dirac fermion coupled to the
electromagnetic field. The renormalization group (RG) equations for the running
coupling constant \alpha, the speed of light c and electron v are derived. The
almost exact analytic solutions to these RG equations are obtained to reveal
that (i) c and v approach to the common value with combination c^2v being
almost unrenormalized, (ii) the RG flow of \alpha is the same as that of usual
QED with c^3 being replaced by c^2v, and (iii) there are two crossover
momentum/energy scales separating three regions of different scaling behaviors.
The dielectric and magnetic susceptibilities, angle-resolved photoemission
spectroscopy (ARPES), and the behavior of the gap are discussed from this
viewpoint.
