Regularization Of Chiral Gauge Theories Research Articles

Chiral Gauge Theory at the Boundary between Topological Phases.

I demonstrate how chiral fermions with an exact gauge symmetry can appear on the d-dimensional boundary of a finite volume (d+1)-dimensional manifold, without any light mirror partners. The condition for the d-dimensional boundary theory to be local is that gauge anomalies cancel and that the volume be large. This can likely be achieved on a lattice and provides a new paradigm for the lattice regularization of chiral gauge theories.

Symmetric Mass Generation

The most well-known mechanism for fermions to acquire a mass is the Nambu–Goldstone–Anderson–Higgs mechanism, i.e., after a spontaneous symmetry breaking, a bosonic field that couples to the fermion mass term condenses, which grants a mass gap for the fermionic excitation. In the last few years, it was gradually understood that there is a new mechanism of mass generation for fermions without involving any symmetry breaking within an anomaly-free symmetry group, also applicable to chiral fermions with anomaly-free chiral symmetries. This new mechanism is generally referred to as the symmetric mass generation (SMG). It is realized that the SMG has deep connections with interacting topological insulator/superconductors, symmetry-protected topological states, perturbative local and non-perturbative global anomaly cancellations, and deconfined quantum criticality. It has strong implications for the lattice regularization of chiral gauge theories. This article defines the SMG, summarizes the current numerical results, introduces an unifying theoretical framework (including the parton-Higgs and the s-confinement mechanisms, as well as the symmetry-extension construction), and presents an overview of various features and applications of SMG.

Six-dimensional regularization of chiral gauge theories

We propose a regularization of four dimensional chiral gauge theories using six-dimensional Dirac fermions. In our formulation, we consider two different mass terms having domain-wall profiles in the fifth and the sixth directions, respectively. A Weyl fermion appears as a localized mode at the junction of two different domain-walls. One domain-wall naturally exhibits the Stora-Zumino chain of the anomaly descent equations, starting from the axial U(1) anomaly in six-dimensions to the gauge anomaly in four-dimensions. Another domain-wall implies a similar inflow of the global anomalies. The anomaly free condition is equivalent to requiring that the axial U(1) anomaly and the parity anomaly are canceled among the six-dimensional Dirac fermions. Since our formulation is based on a massive vector-like fermion determinant, a non-perturbative regularization will be possible on a lattice. Putting the gauge field at the four-dimensional junction and extending it to the bulk using the Yang-Mills gradient flow, as recently proposed by Grabowska and Kaplan, we define the four-dimensional path integral of the target chiral gauge theory.

Lattice regularization of chiral gauge theories to all orders of perturbation theory

In the framework of perturbation theory, it is possible to put chiral gauge theories on the lattice without violating the gauge symmetry or other fundamental principles, provided the fermion representation of the gauge group is anomaly-free. The basic elements of this construction (which starts from the Ginsparg-Wilson relation) are briefly recalled and the exact cancellation of the gauge anomaly, at any fixed value of the lattice spacing and for any compact gauge group, is then proved rigorously through a recursive procedure.

Geometrical aspects of chiral anomalies in the overlap

The set of one dimensional lowest energy eigenspaces used to construct the overlap induces a two form on gauge orbit space which is the locally exact curl of Berry's connection. If anomalies do not cancel, examples of two dimensional closed sub-manifolds of orbit space are produced over which the integral of the above two form does not vanish. Based on these observations, a natural definition of covariant currents is obtained, a simple way to calculate chiral anomalies on the lattice is found, and indications for how to construct an ideal regularization of chiral gauge theories are seen to emerge.

A Wilson-Majorana regularization for lattice chiral gauge theories

We discuss the regularization of chiral gauge theories on the lattice introducing only physical degrees of freedom. This is obtained by writing the Wilson term in a Majorana form, at the expense of the U(1) symmetry related to fermion number conservation. The idea of restoring chiral invariance in the continuum by introducing a properly chosen set of counterterms to be added to the tree level action is checked against one-loop perturbative calculations.

Regularization of chiral gauge theories

The regularization of chiral gauge theories is reviewed from the “overlap” point of riew. This is a brief and biased review containing no references.

Kaplan-Narayanan-Neuberger lattice fermions pass a perturbative test.

We test perturbatively a recent scheme for implementing chiral fermions on the lattice, proposed by Kaplan and modified by Narayanan and Neuberger, using as our testing ground the chiral Schwinger model. The scheme is found to reproduce the desired form of the effective action, whose real part is gauge invariant and whose imaginary part gives the correct anomaly in the continuum limit, once technical problems relating to the necessary infinite extent of the extra dimension are properly addressed. The indications from this study are that the Kaplan--Narayanan--Neuberger (KNN) scheme has a good chance at being a correct lattice regularization of chiral gauge theories.

Regularization of chiral gauge theories

In this paper we present an explicit two-loop calculation which shows that, as a consequence of the breaking of chiral gauge symmetry induced by the regularization, non-gauge invariant counterterms are present in the ghost sector of a chiral gauge theory. This occurrence prevents one from inverting the Faddeev-Popov procedure and removing the gauge fixing in numerical simulations.

A note on “anomalous” chiral gauge theories

We show that the lattice regularization of chiral gauge theories proposed by Kaplan, when applied to a (2 + 1)-dimensional domain wall, produces a (1 + 1)-dimensional theory at low energy even if the gauge anomaly produced by chiral fermions does not cancel. But the corresponding statement is not true in higher dimensions.

Chiral lattice theories without redundant variables

We discuss a lattice regularization of chiral gauge theories which does not make use of unphysical auxiliary degrees of freedom. Chiral fermions are described by Weyl spinors and the fermion doubling problem is solved by a Wilson term which explicitely breaks fermion number. The resulting anomalous baryon number non-conservation, in the continuum limit, is discussed.

A lattice regularization of chiral gauge theories

A new lattice regularization procedure, invariant under global chiral transformations of the fermionic fields, is proposed and applied to a perturbative study in two and four dimensions. A general discussion on the consistency of a perturbative analysis and the constraint it implies on the regularization is presented.

The dimensional regularization of chiral gauge theories and generalized Slavnov-Taylor identities

We shall consider the problem of the spurious anomalies which appear in the Slavnov-Taylor identities when the 't Hooft-Veltman prescription is adopted for the γ 5 matrix in dimensional regularization. We shall formulate a correct renormalization scheme which forces the spurious anomalies to be absent and makes the 't Hooft-Veltman prescription for γ 5 completely consistent and manageable. Then our renormalization scheme is applied to some interesting examples where the spurious anomalies have been recently found, and it is shown to work well.