  We develop an Eliashberg theory for multi-scale quantum criticality,
considering ferromagnetic quantum criticality in the surface of three
dimensional topological insulators. Although an analysis based on the random
phase approximation has been performed for multi-scale quantum criticality, an
extension to an Eliashberg framework was claimed to be far from triviality in
respect that the self-energy correction beyond the random phase approximation,
which originates from scattering with $z = 3$ longitudinal fluctuations,
changes the dynamical exponent $z = 2$ in the transverse mode, explicitly
demonstrated in nematic quantum criticality. A novel ingredient of the present
study is to introduce an anomalous self-energy associated with the spin-flip
channel. Such an anomalous self-energy turns out to be essential for
self-consistency of the Eliashberg framework in the multi-scale quantum
critical point because this off diagonal self-energy cancels the normal
self-energy exactly in the low energy limit, preserving the dynamics of both $z
= 3$ longitudinal and $z = 2$ transverse modes. This multi-scale quantum
criticality is consistent with that in a perturbative analysis for the nematic
quantum critical point, where a vertex correction in the fermion bubble diagram
cancels a singular contribution due to the self-energy correction, maintaining
the $z = 2$ transverse mode. We also claim that this off diagonal self-energy
gives rise to an artificial electric field in the energy-momentum space in
addition to the Berry curvature. We discuss the role of such an anomalous
self-energy in the anomalous Hall conductivity.
