Lattice Chiral Gauge Theories Research Articles

Noncommutative chiral gauge theories on the lattice with manifest star-gauge invariance

We show that noncommutative U(r) gauge theories with a chiral fermion in the adjoint representation can be constructed on the lattice with manifest star-gauge invariance in arbitrary even dimensions. Chiral fermions are implemented using a Dirac operator which satisfies the Ginsparg-Wilson relation. A gauge-invariant integration measure for the fermion fields can be given explicitly, which simplifies the construction as compared with lattice chiral gauge theories in ordinary (commutative) space-time. Our construction includes the cases where continuum calculations yield a gauge anomaly. This reveals a certain regularization dependence, which is reminiscent of parity anomaly in commutative space-time with odd dimensions. We speculate that the gauge anomaly obtained in the continuum calculations in the present cases can be cancelled by an appropriate counterterm.

Chiral gauge theory on the lattice with domain wall fermions

We investigate a $U(1)$ lattice chiral gauge theory with domain wall fermions and compact gauge fixing. In the reduced model limit, our perturbative and numerical investigations show that there exist no extra mirror chiral modes. The longitudinal gauge degrees of freedom have no effect on the free domain wall fermion spectrum consisting of opposite chiral modes at the domain wall and at the anti-domain wall which have an exponentially damped overlap.

Lattice chiral gauge theories

I review the substantial progress which has been made recently with the non-perturbative construction of chiral gauge theories on the lattice. In particular, I discuss three different approaches: a gauge invariant method using fermions which satisfy the Ginsparg-Wilson relation, and two gauge non-invariant methods, one using different cutoffs for the fermions and the gauge fields, and one using gauge fixing. Open problems within all three approaches are addressed.

Anomaly cancellation condition in lattice gauge theory

We study the gauge anomaly A defined on a 4-dimensional infinite lattice while keeping the lattice spacing finite. We assume that (I) A depends smoothly and locally on the gauge potential, (II) A reproduces the gauge anomaly in the continuum theory in the classical continuum limit, and (III) U (1) gauge anomalies have a topological property. It is then shown that the gauge anomaly A can always be removed by local counterterms to all orders in powers of the gauge potential, leaving possible breakings proportional to the anomaly in the continuum theory. This follows from an analysis of nontrivial local solutions to the Wess–Zumino consistency condition in lattice gauge theory. Our result is applicable to the lattice chiral gauge theory based on the Ginsparg–Wilson Dirac operator, when the gauge field is sufficiently weak ∥ U ( n , μ )−1∥< ϵ ′ , where U ( n , μ ) is the link variable and ϵ ′ a certain small positive constant.

Phase diagram and spectrum of gauge-fixed Abelian lattice gauge theory

We consider a lattice discretization of a covariantly gauge-fixed abelian gauge theory. The gauge fixing is part of the action defining the theory, and we study the phase diagram in detail. As there is no BRST symmetry on the lattice, counterterms are needed, and we construct those explicitly. We show that the proper adjustment of these counterterms drives the theory to a new type of phase transition, at which we recover a continuum theory of (free) photons. We present both numerical and (one-loop) perturbative results, and show that they are in good agreement near this phase transition. Since perturbation theory plays an important role, it is important to choose a discretization of the gauge-fixing action such that lattice perturbation theory is valid. Indeed, we find numerical evidence that lattice actions not satisfying this requirement do not lead to the desired continuum limit. While we do not consider fermions here, we argue that our results, in combination with previous work, provide very strong evidence that this new phase transition can be used to define abelian lattice chiral gauge theories.

Chiral gauge theories on the lattice with exact gauge invariance

A recently proposed formulation of chiral lattice gauge theories is reviewed, in which the locality and gauge invariance of the theory can be preserved if the fermion representation of the gauge group is anomaly-free.

The restoration of gauge symmetry in the overlap formalism

The overlap formalism is one of the most promising approaches to the lattice chiral gauge theory. The gauge symmetry violation on the lattice resides minimally in the phase of the fermion determinant. The crucial issue is whether the gauge symmetry is restored in the continuum limit for the anomaly-free fermion content. We propose some specific criteria for the gauge symmetry restoration and investigate them numerically for the 2D chiral U(1) gauge theory.

More on lattice BRST invariance

In the gauge-fixing approach to (chiral) lattice gauge theories, the action in the U(1) case implicitly contains a free ghost term, in accordance with the continuum abelian theory. On the lattice there is no BRST symmetry and, without fermions, the partition function is strictly positive. Recently, Neuberger pointed out in hep-lat/9801029 that a different choice of the ghost term would lead to a BRST-invariant lattice model, which is ill-defined nonperturbatively. We show that such a lattice model is inconsistent already in perturbation theory, and clearly different from the gauge-fixing approach.

Remark on lattice BRST invariance

A recently claimed resolution to the lattice Gribov problem in the context of chiral lattice gauge theories is examined. Unfortunately, I find that the old problem remains.

Domain-wall model in an asymptotic-free dynamics

We investigate a possibility that the rough gauge problem, which have appeared to be a main reason for failures of lattice chiral gauge theories, is cured by an asymptotic-free dynamics. Taking the domain-wall model in 2(+1) dimensions with SU(2) gauge group, we carry out the quenched simulation of gauge fields in the extra dimension. By studying fermion spectra in several volumes, we show that the chiral zero modes exist on the wall without having the spontaneous symmetry breaking thanks to the asymptotic-free dynamics. This result suggests that the rough gauge problem is solved in some class of lattice chiral fermions as well as in 4 dimensions if an asymptotic-free dynamics is incorporated.

Chiral fermions on the lattice through gauge fixing: Perturbation theory

We study the gauge-fixing approach to the construction of lattice chiral gauge theories in one-loop weak-coupling perturbation theory. We show how the infrared properties of the gauge degrees of freedom determine the nature of the continuous phase transition at which we take the continuum limit. The fermion self-energy and the vacuum polarization are calculated, and confirm that, in the Abelian case, this approach can be used to put chiral gauge theories on the lattice in four dimensions. We comment on the generalization to the non-Abelian case.

Gauge-fixing approach to lattice chiral gauge theories

We report on recent progress with the definition of lattice chiral gauge theories, using a lattice action that includes a discretized Lorentz gauge-fixing term. This gauge-fixing term has a unique global minimum, and allows us to use perturbation theory in order to study the influence of the gauge degrees of freedom on the fermions. For the abelian case, we find, both in perturbation theory and numerically, that the fermions remain chiral, and that there are no doublers.

A Wilson-Yukawa Model with undoubled chiral fermions in 2D

We consider the fermion mass spectrum in the strong coupling vortex phase (VXS) of a lattice fermion-scalar model with a global U(1) L × U(1) R , in two dimensions, in the context of a recently proposed two-cutoff lattice formulation. The fermion doublers are made massive by a strong Wilson-Yukawa coupling, but in contrast with the standard formulation of these type of models, in which the light fermion spectrum was found to be vector-like, we find massless fermions with chiral quantum numbers at finite lattice spacing. When the global symmetry is gauged, this model is expected to give rise to a lattice chiral gauge theory.

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.

Further discussions on a possible lattice chiral gauge theory

In a possible $SU_L(2)$ lattice chiral gauge theory with a large multifermion coupling, we try to further clarify the threshold phenomenon: the possibility that the right-handed three-fermion state turns into the virtual states of its constituents (free chiral fermions) in the low-energy limit. Provided this phenomenon occurs, we discuss the chiral gauge coupling, Ward identities and the gauge anomaly within the gauge-invariant prescription of the perturbative chiral gauge theory.

Lattice chiral gauge theories with finely-grained fermions

The importance of lattice gauge field interpolation for our recent non-perturbative formulation of chiral gauge theory is emphasized. We illustrate how the requisite properties are satisfied by our recent four-dimensional non-abelian interpolation scheme, by going through the simpler case of U(1) gauge fields in two dimensions.

Domain-wall fermions with U(1) dynamical gauge fields in (4 + 1)-dimensions

We carry out a numerical simulation of a domain-wall model in (4 + 1) dimensions, in the presence of a quenched U(1) dynamical gauge field only in an extra dimension, corresponding to the weak coupling limit of a (4-dimensional) physical gauge coupling. Our numerical data suggest that the zero mode seems absent in the symmetric phase, so that it is difficult to construct a lattice chiral gauge theory in the continuum limit.

The emergence of a heavy quark family on a lattice

Within the framework of the “Rome approach” for a lattice chiral gauge theory, the four-quark interaction with flavour symmetry is included. We analyse spontaneous symmetry breaking and compute composite modes and their contributions to the ground state energy. As a result, it is shown that the emergence of a heavy quark family is the energetically favoured solution.

Domain-wall fermions with U(1) dynamical gauge fields

We have carried out a numerical simulation of a domain-wall model in $(2+1)$-dimensions, in the presence of a dynamical gauge field only in an extra dimension, corresponding to the weak coupling limit of a ( 2-dimensional ) physical gauge coupling. Using a quenched approximation we have investigated this model at $\beta_{s} ( = 1 / g^{2}_{s} ) =$ 0.5 ( ``symmetric'' phase), 1.0, and 5.0 (``broken'' phase), where $g_s$ is the gauge coupling constant of the extra dimension. We have found that there exists a critical value of a domain-wall mass $m_{0}^{c}$ which separates a region with a fermionic zero mode on the domain-wall from the one without it, in both symmetric and broken phases. This result suggests that the domain-wall method may work for the construction of lattice chiral gauge theories.

Progress in lattice chiral gauge theories

Some key features of continuum chiral fermions are shown to be satisfied by the overlap.