Is the chiral U(1) theory trivial?
  The chiral U(1) theory differs from the corresponding vector theory by an imaginary contribution to the effective action which amounts to a phase factor in the partition function. The vector theory, i.e. QED, is known to be trivial in the continuum limit. It is argued that the presence of the phase factor will not alter this result and the chiral theory is non-interacting as well. 
V. Bornyakov, A. Hoferichter and G. Schierholz