We show that coupling the Standard Model to a Lorentz symmetry violating sector may co-exist with viable phenomenology, provided that the interaction between the two is mediated by higher-dimensional operators. In particular, if the new sector acquires anisotropic scaling behavior above a "Hořava-Lifshitz" energy scale Λ HL and couples to the Standard Model through interactions suppressed by M pl , the transmission of the Lorentz violation into the Standard Model is protected by the ratio Λ 2 HL /M 2 pl . A wide scale separation, Λ HL ≪ M pl , can then make Lorentz-violating terms in the Standard Model sector within experimental bounds without fine-tuning. We first illustrate our point with a toy example of Lifshitz-type neutral fermion coupled to photon via the magnetic moment operator, and then implement similar proposal for the Hořava-Lifshitz gravity coupled to conventional Lorentz-symmetric matter fields. We find that most radiatively induced Lorentz violation can be controlled by a large scale separation, but the existence of instantaneously propagating non-Lifshitz modes in gravity can cause a certain class of diagrams to remain quadratically divergent above Λ HL . Such problematic quadratic divergence, however, can be removed by extending the action with terms of higher Lifshitz dimension, resulting in a completely consistent setup that can cope with the stringent tests of Lorentz invariance.

The unique ghost-free mass and nonlinear potential terms for general relativity are presented in a diffeomorphism and local Lorentz invariant vierbein formalism. This construction requires an additional two-index Stückelberg field, beyond the four scalar fields used in the metric formulation, and unveils a new local SL(4) symmetry group of the mass and potential terms, not shared by the Einstein-Hilbert term. The new field is auxiliary but transforms as a vector under two different Lorentz groups, one of them the group of local Lorentz transformations, the other an additional global group. This formulation enables a geometric interpretation of the mass and potential terms for gravity in terms of certain volume forms. Furthermore, we find that the decoupling limit is much simpler to extract in this approach; in particular, we are able to derive expressions for the interactions of the vector modes. We also note that it is possible to extend the theory by promoting the two-index auxiliary field into a Nambu-Goldstone boson nonlinearly realizing a certain spacetime symmetry, and show how it is "eaten up" by the antisymmetric part of the vierbein.

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.

We show, using exact lattice chirality, that partition functions of lattice gauge theories with vectorlike fermion representations can be split into "light" and "mirror" parts, such that the "light" and "mirror" representations are chiral. The splitting of the full partition function into "light" and "mirror" is well defined only if the two sectors are separately anomaly free. We show that only then is the generating functional, and hence the spectrum, of the mirror theory a smooth function of the gauge field background. This explains how ideas to use additional non-gauge, high-scale mirror-sector dynamics to decouple the mirror fermions without breaking the gauge symmetry-for example, in symmetric phases at strong mirror Yukawa couplingare forced to respect the anomaly-free condition when combined with the exact lattice chiral symmetry. Our results are also useful in explaining a paradox posed by a recent numerical study of the mirror-fermion spectrum in a toy would-be-anomalous two-dimensional theory. In passing, we prove some general properties of the partition functions of arbitrary chiral theories on the lattice that should be of interest for further studies in this field.

We perform a Monte-Carlo study of the lattice two-dimensional gauged XY-model. Our results confirm the strong-coupling expansion arguments that for sufficiently small values of the spin-spin coupling the "gauge symmetry breaking" terms decouple and the long-distance physics is that of the unbroken pure gauge theory. We find no evidence for the existence, conjectured earlier, of massless states near a critical value of the spin-spin coupling. We comment on recent remarks in the literature on the use of gauged XY-models in proposed constructions of chiral lattice gauge theories. I. MOTIVATION AND SUMMARYThe gauged two-dimensional lattice XY-model has been studied for quite some time. Its simplicity allows for analytic studies via lattice dualities and strong-coupling expansions [1] as well as for numerical analysis, in either a Hamiltonian [2] or Euclidean (see, e.g., [3]) formulation. The model exhibits many phenomena found in more complicated higher-dimensional theories, notably the presence of both confining and Higgs phases [1,2,4,5].The motivation for this short study stems from our interest in using the gauged XY-model, as well as its nonabelian higher-dimensional analogues [4,6,7], in lattice constructions of chiral gauge theories. The lattice formulation of chiral gauge theories is an outstanding problem with no practical solution yet, despite much recent progress; for reviews, see [8]. While not our topic here, we note that the recent constructions of [9,10] have some attractive features (for an earlier proposal of similar flavor, see [11]). Most importantly, they may offer a way around the difficult and unsolved problem of the explicit nonperturbative construction of the fermion measure for general chiral lattice gauge theories. Ref. [9] aims to achieve this by combining older ideas [12] to use non-gauge strong dynamics to decouple the mirror fermions in a vectorlike theory with the recently discovered exact lattice chirality of the Neuberger-Dirac operator.A concrete two-dimensional realization of the proposal [9] has the gauged XY-model as an essential part and was studied in various limits in [10]. Recently, [13] considered the perturbative spectrum of the models of [9] at nonzero gauge coupling. It was claimed there that the gauge boson is always massive and, therefore, the construction of [9, 10] was argued to be irrelevant to the study of unbroken chiral gauge theories, on account of this fact alone.This brings us to the main topic of this paper. The arguments of [13] do not invoke either the fermions or the strong mirror dynamics (admittedly, complicated and not yet completely understood), but concern only the spectrum of the gauged XY-model, arguing that it only * Electronic address: poppitz@physics.utoronto.ca † Electronic address: ywshang@physics.utoronto.ca has a phase with a massive gauge boson. This conclusion contradicts strong-coupling expansion arguments for the decoupling of the "gauge-breaking" terms for small values of the spin-spin coupling. As these arguments have been made many time...