  The renormalization of conductances in a Y junction of spinless
Luttinger-liquid wires additionally coupled to acoustic longitudinal phonons is
investigated in fermionic representation. This system corresponds to geometry
of a tunneling experiment and exhibits the interplay between the Coulomb
repulsion and the attractive retarded interaction mediated by phonons. The
retardation effects related to the propagation of phonons through the junction
with arbitrary transmission and reflection amplitudes are taken into account.
The appearing logarithmic corrections to conductances of the junction are
treated in a renormalization group approach, and scaling exponents are
calculated up to infinite order in the interaction after RPA-type summation.
The fixed points and corresponding scaling exponents are considered in various
non-equilibrium regimes. We show that the boundary exponent and the bulk
anomalous dimension of fermion operator are characterized by two different
Luttinger parameters, referring to the main wire, thanks to non-local character
of phonon-mediated interaction. In the limiting case of the junction of only
two wires, the scaling exponents found by our method are in exact
correspondence with previous bosonization analysis.
