  We study four-lead junction of semi-infinite wires by using fermionic
representation in the scattering state formalism. The model of spinless
fermions with short-range Luttinger liquid type interaction is used. We find
the renormalization group (RG) equations for the conductances of the system in
the first order of fermionic interaction. In contrast to the well-known cases
of two-lead and three-lead junctions, we show that the RG equations cannot be
expressed solely in terms of conductances. The appearing ambiguity is resolved
by choosing a certain sign defined at the level of S-matrix, but hidden in
conductances. The origin of this Z_2 symmetry is traced back to the
particle-hole symmetry of our Hamiltonian. We demonstrate two distinct RG flows
from any point in the space of conductances. The scaling exponents at the fixed
points are not affected by these observations.
