  We consider theoretically the transport properties of a spinless resonant
electronic level coupled to strongly dissipative leads, in the regime of
circuit impedance near the resistance quantum. Using the Luttinger liquid
analogy, one obtains an effective Hamiltonian expressed in terms of interacting
Majorana fermions, in which all environmental degrees of freedom (leads and
electromagnetic modes) are encapsulated in a single fermionic bath. General
transport equations for this system are then derived in terms of the Majorana
T-matrix. Perturbative treatment of the Majorana interaction term yields the
appearance of a marginal, linear dependence of the conductance on temperature
when the system is tuned to its quantum critical point, in agreement with
recent experimental observations. We investigate in detail the different
crossovers involved in the problem, and analyze the role of the interaction
terms in the transport scaling functions. In particular, we show that single
barrier scaling applies when the system is slightly tuned away from its
Majorana critical point, strengthening the general picture of dynamical Coulomb
blockade.
