Chiral Lattice Gauge Theories via Mirror–fermion Decoupling: A Mission (Im)possible?

Abstract

This is a review of the status and outstanding issues in attempts to construct chiral lattice gauge theories by decoupling the mirror fermions from a vectorlike theory. In the first half, we explain why studying nonperturbative chiral gauge dynamics may be of interest, enumerate the problems that a lattice formulation of chiral gauge theories must overcome, and briefly review our current knowledge. We then discuss the motivation and idea of mirror-fermion decoupling and illustrate the desired features of the decoupling dynamics by a simple solvable toy model. The role of exact chiral symmetries and matching of 't Hooft anomalies on the lattice is also explained. The second, more technical, half of the article is devoted to a discussion of the known and unknown features of mirror-decoupling dynamics formulated with Ginsparg-Wilson fermions. We end by pointing out possible directions for future studies.

Novel phases in strongly coupled four-fermion theories

“…It has been proposed that such quartic interactions can be used in the context of domain wall fermion theories to provide a path to achieve chiral lattice gauge theories [22][23][24]. If indeed the current model avoids symmetry-breaking phases it may be possible to revisit the original Eichten-Preskill proposal for the construction of chiral lattice gauge theories using strong four-fermion terms in the bulk to lift fermion doubler modes [18,25]. However, it is not clear to the authors how such constructions can work in detail; the model described here uses reduced staggered rather than Wilson or naive fermions which negates a simple transcription of the four-fermion interaction appearing in this model to those earlier constructions.…”

On the decoupling of mirror fermions

“…These theories also tend to have a pronounced "sign problem" (really, a complex phase problem). Aspects of these problems have been reviewed in [8,25,26] (there are proposed solutions to some of the problems mentioned above-for instance, the recent approach of [27], which involves non-perturbative gauge fixing).…”

“…In the case of Abelian chiral gauge theories, the phase can be determined for arbitrary gauge backgrounds and a way of implementing it in practice has been suggested in [28,29]. 2 Fur- 1 We only give an illustrative list of particle-physics models using chiral gauge dynamics; for a more complete list of references, see the Introduction in the recent review article [8].…”

“…The approach studied in this paper would formulate such models without having to solve Lüscher's consistency conditions. We give a brief review of the main ideas in the next Section and recommend [8] for an extensive review.…”

“…Using the lattice strong coupling expansion, it was shown long ago in [33] that such interactions can render all participating fermions massive. A toy model (that has all the features of [33], but is simpler to study) of decoupling of fermions with strong multi-fermion interactions, showing how all states become gapped in the strong-coupling limit is constructed in Section 3.2 in [8].…”