Gauge anomaly cancellations in SU(2)× U(1) electroweak theory on the lattice

Abstract

We consider the cohomological classification of the 4+2-dimensional topological field, which is proposed by Lüscher, for SU(2) L × U(1) Y electroweak theory. The dependence on the admissible abelian gauge field of U(1) Y is determined through topological argument, with SU(2) L gauge field fixed as background. We then show the exact cancellation of the local gauge anomaly of the mixed type SU(2) L 2 × U(1) Y at finite lattice spacing, as well as U(1) Y 3 , using the pseudo reality of SU(2) L and the anomaly cancellation conditions in the electroweak theory given in terms of the hyper-charges of U(1)1 The topological aspect of the non-abelian anomaly for Weyl fermions defined based on the overlap formalism / the Ginsparg-Wilson relation has been examined by D.H. Adams in close relation to the argument of L. Alvarez-Gaumé and P. Ginsparg in the continuum theory [13,14]. The global SU(2) anomaly has been examined by H. Neuberger and O. Bär and I. Campos [15,16] in detail. A lattice implementation of the η-invariant and its relation to the effective action for chiral Ginsparg-Wilson fermions has been examined by T. Aoyama and Y.K. in [17]. Non-compact formulation of abelian chiral gauge theories has been considered recently by Neuberger [18].

Lattice 2001: Reflections

“…(16,17,18) are the first, almost trivial steps in a highly complex construction which led to a breakthrough concerning chiral gauge theories [68]. For anomaly free complex representations for U (1), or SU (2) × U (1) gauge groups the theory is constructed non-perturbatively [69,70], for a general compact group it is constructed to all orders of PT [71]. At present, chiral gauge theories obtained this way have a certain ambiguity which I would like to discuss briefly.…”

“…Going over to a different basis, the measure changes by a gauge field dependent phase. The difficult part of defining a chiral gauge theory is to fix this phase (or the basis) so that the final theory is local and gauge invariant [68][69][70][71]. The absolute value | O F | is, however, independent of the basis chosen.…”

Analytic progress on exact lattice chiral symmetry

“…The issues in this approach are the local cohomology problem and the proof of the global integrability condition. Fortunately, the cohomology problem can be solved for the U(1) part by the trick to treat the SU(2) gauge field as a background [40,13]. As to the global integrability condition, it is proved for "gauge loops" in the space of the U(1) gauge fields with arbitrary SU(2) gauge field in the background.…”

CP breaking in lattice chiral gauge theory

“…Discovery of gauge covariant local lattice Dirac operators [1,2], which satisfy the GinspargWilson relation [3], paved a way to a manifestly local and gauge invariant lattice formulation of anomaly-free chiral gauge theories [4]- [8]. (See also related early work in ref.…”