Breakdown of the Chiral Luttinger Liquid in One-Dimension
  We have developed a fermionic boot-strap method to solve a class of chiral one dimensional fermion model which cannot be tackled by bosonization. Using this scheme, we show that Luttinger liquid behavior in a gas of four interacting chiral Majorana fermions is highly sensitive to the velocity degeneracy. Upon changing the velocity of a one chiral fermion, a sharp bound (or anti-bound) state splits off from the original Luttinger liquid continuum, cutting off the X-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency. 
A. F. Ho and P. Coleman