Why is the mission impossible? Decoupling the mirror Ginsparg–Wilson fermions in the lattice models for two-dimensional Abelian chiral gauge theories

摘要

In the mirror fermion approach with Ginsparg-Wilson fermions, it has beenargued that the mirror fermions do not decouple: in the 345 model with Dirac-and Majorana-Yukawa couplings to XY-spin field, the two-point vertex functionof the (external) gauge field in the mirror sector shows a singular non-localbehavior in the PMS phase. We re-examine why the attempt seems a "Mission:Impossible" in the 345 model. We point out that the effective operators tobreak the fermion number symmetries ('t Hooft operators plus others) in themirror sector do not have sufficiently strong couplings even in the limit oflarge Majorana-Yukawa couplings. We observe also that the type of Majorana massterm considered there is singular in the large limit due to the nature of thechiral projection of the Ginsparg-Wilson fermions, but a slight modificationwithout such singularity is allowed by virtue of the very nature. We thenconsider a simpler four-flavor axial gauge model, the 1$^4$(-1)$^4$ model, inwhich the U(1)$_A$ gauge and Spin(6)(SU(4)) global symmetries prohibit thebilinear terms, but allow the quartic terms to break all the other continuousmirror-fermion symmetries. In the strong-coupling limit of the quarticoperators, the model is well-behaved and simplified. Through Monte-Carlosimulations in the weak gauge coupling limit, we show a numerical evidence thatthe two-point vertex function of the gauge field in the mirror sector shows aregular local behavior, and we still argue that all you need is killing thecontinuous mirror-fermion symmetries with would-be gauge anomalies non-matched.Finally, by gauging a U(1) subgroup of the U(1)$_A$$imes$ Spin(6)(SU(4)) ofthe previous model, we formulate the $2 1 (-1)^3$ chiral gauge model and arguethat the induced fermion measure term satisfies the required locality propertyand provides a solution to the reconstruction theorem.