Gauge Freedom in Chiral Gauge Theory with Vacuum Overlap

In the vacuum overlap formalism[1] of a generic chiral gauge theory, gauge symmetry is explicitly broken by the complex phase of fermion determinant. In order to restore the gauge invariance, gauge average—the integration along gauge orbit— is invoked. Then, what is required for the dynamical nature of the gauge freedom at β = ∞ (pure gauge limit) is that the global gauge symmetry is not broken spontaneously and the bosonic fields of the gauge freedom have large mass compared to a typical mass scale of the theory so as to decouple from physical spectrum[2,1]. However, through the analysis of the waveguide model[3,4], it has been claimed that this required disordered nature of the gauge freedom causes the vector-like spectrum of fermion[5]. In this argument, the fermion correlation functions at the waveguide boundaries were examined. One may think of the counter parts of these correlation functions in the overlap formalism by putting creation and annihilation operators in the overlap of vacua with the same signature of mass. Let us refer this kind of correlation function as boundary correlation function and the correlation function in the original definition as overlap correlation function. We should note that the boundary correlation functions are no more the observables in the sense defined in the overlap formalism; they cannot be expressed by the overlap of two vacua with their phases fixed by the Wigner-Brillouin