Oscillatory behavior of the domain wall fermions revisited

In the generic domain wall fermion formulation of chiral fermions on the lattice, the zero modes of the four-dimensional Wilson fermion operator with the negative mass parameter $-M_5$ introduce unphysical massive modes propagating in the four-dimensional spacetime. In the free fermion case, the pole mass of this kind of unphysical modes is given by $\tilde{E}=\ln(1-M_5)$, which acquires an imaginary part, $i\pi$, when $M_5>1$ and results in an oscillatory behavior of the domain wall fermion propagator in time. The existence of the unphysical modes in the presence of gauge fields is investigated in the mean field approximation, and their physical consequences are discussed. In addition, we also give a semiquantitative criterion for tuning $M_5$ in the realistic numerical study.