Pion Mass and the PCAC Relation in the Overlap Fermion Formalism (II): Gauged Gross-Neveu Model on a Lattice

Abstract

In the previous paper hep-lat/9905001, we studied the overlap fermion formalism by using solvable model with spontaneous chiral symmetry breaking, the gauged GrossNeveu model in 2 dimensions. There the interaction terms are local and invariant under the ordinary chiral transformation but they break Lüsher’s extended chiral symmetry on a lattice which is respected by the kinetic term of fermion. Then we showed that the flavour-nonsinglet pion mass is proportional to the bare fermion mass and the PCAC relation is satisfied in the continuum limit. This means that the bare mass is only the parameter which measures deviation from the exact chiral symmetry in the continuum limit. In this paper we modify the interaction terms of the model to be invariant under the extended chiral transformation. We show that the pion mass is proportional to the bare fermion mass and the PCAC relation is satisfied at finite lattice spacing. e-mail address: ikuo@hep1.c.u-tokyo.ac.jp e-mail address: nagao@hep1.c.u-tokyo.ac.jp 1 Lattice fermion formulation is one of the most important problem in the lattice field theory. Recently a promising formalism was proposed, the overlap fermion formalism[1], and after that there appeared a lot of works on that. Importance of the Ginsparg and Wilson (GW) relation[2] was stressed there. In a previous paper[3], we studied the overlap fermion by using a solvable model, the gauged Gross-Neveu model in 2 dimensions. There the kinetic term of fermion is given by the overlap formula. The kinetic term of the overlap formula is not invariant under the ordinary chiral transformation but there exists an extended chiral symmetry in the overlap fermion[4]. The four-Fermi interaction term in Ref.[3] is local one which is invariant under the ordinary chiral transformation but breaks the extended chiral symmetry. We showed that the pion mass is proportional to the bare femion mass in the continuum limit, though neither the ordinary nor the extended chiral symmetry is an exact symmetry of the system even if the bare fermion mass MB = 0. This result means that the overlap fermion is better than the Wilson fermion in which a fine tunning between the Wilson parameter and the bare fermion mass is necessary. On the other hand a drawback of the overlap fermion is its nonlocality or more precisely “quasi-locality” as discussed in Ref.[5]. Anyway the above result is rather encouraging for formulation of lattice field theory with a symmetry. For example, it is difficult to formulate supersymmetry (SUSY) as an exact symmetry on a lattice but the above result suggets the possibility that SUSY is restored in the continuum limit if SUSY is formulated properly on a lattice as an approximate symmetry[6]. In this paper we shall revisit the same model but modify the interaction terms in order to respect the extended chiral symmetry. As a result, only the fermion bare mass breaks the extended chiral symmetry (and the measure of the path-integral of fermions), whereas the interaction terms become nonlocal. The model is given by the Similar modification is discussed in Ref.[7].

Cite this paper

@inproceedings{Ichinoseand1999PionMA, title={Pion Mass and the PCAC Relation in the Overlap Fermion Formalism (II): Gauged Gross-Neveu Model on a Lattice}, author={Ikuo Ichinoseand and Keiichi Nagao}, year={1999} }