Ginsparg-Wilson operators and a no-go theorem

If one uses a general class of Ginsparg-Wilson operators, it is known that CP symmetry is spoiled in chiral gauge theory for a finite lattice spacing and the Majorana fermion is not defined in the presence of chiral symmetric Yukawa couplings. We summarize these properties in the form of a theorem for the general GinspargWilson relation. The Ginsparg-Wilson relation[1] provides a convenient framework for the analyses of chiral symmetry in lattice theory[2]. It has been recently pointed out by Hasenfratz[3] that the overlap operator[4] has a conflict with CP symmetry in chiral gauge theory for any finite lattice spacing a. We pointed out that the lattice chiral symmetry of the Ginsparg-Wilson operator has a certain conflict with the definition of the Majorana fermion in the presence of Yukawa couplings[5]. (The breaking of CP symmetry in a different context was also mentioned in[5]). These analyses are either based on the simple form of the Ginsparg-Wilson relation and its explicit solution[3], or on the generalized forms of Ginsparg-Wilson relation but still on their explicit solutions[5]. It may be useful to formulate these properties in a more abstract and general setting to understand the general features of these complications. In this paper we provide such an analysis. We deal with a hermitian lattice operator defined by H = aγ5D = H † = aDγ5 (1) where D stands for the lattice Dirac operator and a is the lattice spacing. We analyze the general Dirac operator defined by the algebraic relation γ5H +Hγ5 = 2H f(H) (2)