  The properties of stable Luttinger liquid phases in models with a
non-conserved number of particles are investigated. We study the Luttinger
liquid phases in one-dimensional models of hard-core boson and spinless fermion
chains where particles can be created and annihilated three by three on
adjacent sites. We provide an intuitive and systematic method based on flow
equations approach, which accounts for additional terms in the correlations
generated by the $\mathbb{Z}_3$-symmetric interactions. We find that despite
the emergence of U(1) symmetry under renormalization, the observables are still
affected by its breaking in the bare Hamiltonian. In particular, the standard
bosonization mapping becomes insufficient to capture the full behavior of
correlation functions.
