Anomalous Gauge Theories Research Articles

Parisi–Wu quantization, CP violation and beyond the standard model

The Parisi–Wu scheme of quantization opens up the possibility of using anomalous fermionic gauge theories. An analysis of ultra-violet divergences reveals that the structure of counter terms is different from what is expected in conventional quantization schemes. In this paper, it is argued that there exists a possible mechanism of CP violation that requires at least three generations of quarks, a result well known from a phenomenological analysis of mass mixing of quarks. A few observations on possible ways of going beyond the Standard Model are included.

Holography for 2D N=(0,4) quantum field theory

We study the correspondence between ${\mathrm{AdS}}_{3}$ massive IIA supergravity vacua and two-dimensional $\mathcal{N}=(0,4)$ quiver quantum field theories. After categorizing all kinds of gravity solutions, we demystify the ones that seem to reflect anomalous gauge theories. In particular, we prove that there are bound states of D-branes on the boundary of the space that provide the dual quiver theory with exactly the correct amount of flavor symmetry in order to cancel its gauge anomalies. Then we propose that the structure of the field theory should be complemented with additional bifundamental matter, which we argue may only be $\mathcal{N}=(4,4)$ hypermultiplets. Finally, we construct a Bogomol'nyi-Prasad-Sommerfield (BPS) string configuration and use the old and new supersymmetric matter to build its dual ultraviolet operator. During this holographic synthesis, we uncover some interesting features of the quiver superpotential and associate the proposed operator with the same classical mass of its dual BPS string.

Implications of the weak gravity conjecture in anomalous quiver gauge theories

We argue a smallness of gauge couplings in abelian quiver gauge theories, taking the anomaly cancellation condition into account. In theories of our interest there exist chiral fermions leading to chiral gauge anomalies, and an anomaly-free gauge coupling tends to be small, and hence can give a non-trivial condition of the weak gravity conjecture. As concrete examples, we consider $U(1)^{k}$ gauge theories with a discrete symmetry associated with cyclic permutations between the gauge groups, and identify anomaly-free $U(1)$ gauge symmetries and the corresponding gauge couplings. Owing to this discrete symmetry, we can systematically study the models and we find that the models would be examples of the weak coupling conjecture. It is conjectured that a certain class of chiral gauge theories with too many $U(1)$ symmetries may be in the swampland. We also numerically study constraints on the couplings from the scalar weak gravity conjecture in a concrete model. These constraints may have a phenomenological implication to model building of a chiral hidden sector as well as the visible sector.

Supersymmetric nonlinear sigma models as anomalous gauge theories

We revisit supersymmetric nonlinear sigma models on the target manifold $CP^{N-1}$ and $SO(N)/SO(N-2)\times U(1)$ in four dimensions. These models are formulated as gauged linear models, but it is indicated that the Wess-Zumino term should be added to the linear model since the hidden local symmetry is anomalous. Applying a procedure used for quantization of anomalous gauge theories to the nonlinear models, we determine the form of the Wess-Zumino term, by which a global symmetry in the linear model becomes smaller in the action than the conventional one. Moreover, we analyze the resulting linear model in the $1/N$ leading order. Consequently, we find that the model has a critical coupling constant similar to bosonic models. In the weak coupling regime, the $U(1)$ local symmetry is broken but supersymmetry is never broken. In contrast to the bosonic case, it is impossible to find stable vacua in the strong coupling regime as far as in the $1/N$ leading order. These results are straightforwardly generalized to the case of the hermitian symmetric space.

Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids

Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called "Symmetry Enriched Topological phases" (SET). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group $G_g$), implementing a global symmetry (denoted by $G_s$) is anomalous although symmetry charge carried by topological point-like excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in 3D system alone. Such anomalous state construction generalizes the "2D surface topological order" to 3D. A concrete example with $G_g=\mathbb{Z}_2\times\mathbb{Z}_4$ and $G_s=\mathbb{Z}_2$ is calculated.

Heat kernel approach to the relations between covariant and consistent currents in chiral gauge theories

We apply the heat kernel method to relations between covariant and consistent currents in anomalous chiral gauge theories. Banerjee et al. have shown that the relation between these currents is expressed by a "functional curl" of the covariant current. Using the heat kernel method, we evaluate the functional curl explicitly in arbitrary even dimensions. We also apply the heat kernel method to evaluate Osabe and Suzuki's results of the difference between covariant and consistent currents in two and four dimensions. Applying the arguments of Banerjee et al. to gravitational anomalies, we investigate the relationship between the covariant and consistent energy-momentum tensors. The relation is found to be expressed by a functional curl of the covariant energy-momentum tensor.

Universal Superspace Unitary Operator and Nilpotent (Anti-)Dual-BRST Symmetries: Superfield Formalism

We exploit the key concepts of the augmented version of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism to derive the superspace (SUSP)dualunitary operator and its Hermitian conjugate and demonstrate their utility in the derivation of the nilpotent and absolutely anticommuting (anti-)dual-BRST symmetry transformations for a set of interesting models of theAbelian1-form gauge theories. These models are the one (0+1)-dimensional (1D) rigid rotor and modified versions of the two (1+1)-dimensional (2D) Proca as well as anomalous gauge theories and 2D model of a self-dual bosonic field theory. We show theuniversalityof the SUSPdualunitary operator and its Hermitian conjugate in the cases ofallthe Abelian models under consideration. These SUSP dual unitary operators, besides maintaining the explicit group structure, provide the alternatives to the dual horizontality condition (DHC) and dual gauge invariant restrictions (DGIRs) of the superfield formalism. The derivations of thedualunitary operators and corresponding (anti-)dual-BRST symmetries are completelynovelresults in our present investigation.

Universal Superspace Unitary Operator for Some Interesting Abelian Models: Superfield Approach

Within the framework of augmented version of superfield formalism, we derive the superspace unitary operator and show its usefulness in the derivation of Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a set of interesting models for the Abelian 1-form gauge theories. These models are (i) a one (0+1)-dimensional (1D) toy model of a rigid rotor, (ii) the two (1+1)-dimensional (2D) modified versions of the Proca and anomalous Abelian 1-form gauge theories, and (iii) the 2D self-dual bosonic gauge field theory. We provide, in some sense, the alternatives to the horizontality condition (HC) and the gauge invariant restrictions (GIRs) in the language of the above superspace (SUSP) unitary operator. One of the key observations of our present endeavor is the result that the SUSP unitary operator and its Hermitian conjugate are found to be thesameforallthe Abelian models under consideration (including the 4DinteractingAbelian 1-form gauge theories with Dirac and complex scalar fields which have been discussed earlier). Thus, we establish theuniversalityof the SUSP operator for the above Abelian theories.

Ward identities and gauge independence in general chiral gauge theories

Using the Batalin-Vilkovisky formalism, we study the Ward identities and the equations of gauge dependence in potentially anomalous general gauge theories, renormalizable or not. A crucial new term, absent in manifestly nonanomalous theories, is responsible for interesting effects. We prove that gauge invariance always implies gauge independence, which in turn ensures perturbative unitarity. Precisely, we consider potentially anomalous theories that are actually free of gauge anomalies thanks to the Adler-Bardeen theorem. We show that when we make a canonical transformation on the tree-level action, it is always possible to re-renormalize the divergences and re-fine-tune the finite local counterterms, so that the renormalized $\Gamma $ functional of the transformed theory is also free of gauge anomalies, and is related to the renormalized $\Gamma $ functional of the starting theory by a canonical transformation. An unexpected consequence of our results is that the beta functions of the couplings may depend on the gauge-fixing parameters, although the physical quantities remain gauge independent. We discuss nontrivial checks of high-order calculations based on gauge independence and determine how powerful they are.

Anomalous U(1) models in four and five dimensions and their anomaly poles

We analyze the role played by anomaly poles in an anomalous gauge theory by discussing their signature in the corresponding off-shell effective action. The origin of these contributions, in the most general kinematical case, is elucidated by performing a complete analysis of the anomaly vertex at perturbative level. We use two independent (but equivalent) representations: the Rosenberg representation and the longitudinal/transverse (L/T) parameterization, used in recent studies of g−2 of the muon and in the proof of non-renormalization theorems of the anomaly vertex. The poles extracted from the L/T parameterization do not couple in the infrared for generic anomalous vertices, as in Rosenberg, but we show that they are responsible for the violations of unitarity in the UV region, using a class of pole-dominated amplitudes. We conclude that consistent formulations of anomalous models require necessarily the cancellation of these polar contributions. Establishing the UV significance of these terms provides a natural bridge between the anomalous effective action and its completion by a nonlocal theory. Some additional difficulties with unitarity of the mechanism of inflow in extra dimensional models with an anomalous theory on the brane, due to the presence of anomaly poles, are also pointed out.

On free 4D Abelian 2-form and anomalous 2D Abelian 1-form gauge theories

We demonstrate a few striking similarities and some glaring differences between (i) the free four- (3+1)-dimensional (4D) Abelian 2-form gauge theory, and (ii) the anomalous two- (1+1)-dimensional (2D) Abelian 1-form gauge theory, within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism. We demonstrate that the Lagrangian densities of the above two theories transform in a similar fashion under a set of symmetry transformations even though they are endowed with a drastically different variety of constraint structures. With the help of our understanding of the 4D Abelian 2-form gauge theory, we prove that the gauge-invariant version of the anomalous 2D Abelian 1-form gauge theory is a new field-theoretic model for the Hodge theory where all the de Rham cohomological operators of differential geometry find their physical realizations in the language of proper symmetry transformations. The corresponding conserved charges obey an algebra that is reminiscent of the algebra of the cohomological operators. We briefly comment on the consistency of the 2D anomalous 1-form gauge theory in the language of restrictions on the harmonic state of the (anti-) BRST and (anti-) co-BRST invariant version of the above 2D theory.

Four-dimensional lattice chiral gauge theories with anomalous fermion content

In continuum field theory, it has been discussed that chiral gauge theories with Weyl fermions in anomalous gauge representations (anomalous gauge theories) can consistently be quantized, provided that some of gauge bosons are permitted to acquire mass. Such theories in four dimensions are inevitablly non-renormalizable and must be regarded as a low-energy effective theory with a finite ultraviolet (UV) cutoff. In this paper, we present a lattice framework which enables one to study such theories in a non-perturbative level. By introducing bare mass terms of gauge bosons that impose ``smoothness'' on the link field, we explicitly construct a consistent fermion integration measure in a lattice formulation based on the Ginsparg-Wilson (GW) relation. This framework may be used to determine in a non-perturbative level an upper bound on the UV cutoff in low-energy effective theories with anomalous fermion content. By further introducing the St\"uckelberg or Wess-Zumino (WZ) scalar field, this framework provides also a lattice definition of a non-linear sigma model with the Wess-Zumino-Witten (WZW) term.

Anomalous gauge theories revisited

A possible formulation of chiral gauge theories with an anomalous fermion content is re-examined in light of the lattice framework based on the Ginsparg-Wilson relation. It is shown that the fermion sector of a wide class of anomalous non-abelian theories cannot consistently be formulated within this lattice framework. In particular, in 4 dimension, {\it all} anomalous non-abelian theories are included in this class. Anomalous abelian chiral gauge theories cannot be formulated with compact $\U(1)$ link variables, while a non-compact formulation is possible at least for the vacuum sector in the space of lattice gauge fields. Our conclusion is not applied to effective low-energy theories with an anomalous fermion content which are obtained from an underlying anomaly-free theory by sending the mass of some of fermions to infinity. For theories with an anomalous fermion content in which the anomaly is cancelled by the Green-Schwarz mechanism, a possibility of a consistent lattice formulation is not clear.

Topological mass generation and duality transformations in diverse dimensions

In this seminar we shall discuss the issue of duality transformation in the context of f topological mass generation in diverse dimensions. Particular emphasis will be given to the mass generation mechanism as the result of interference between self and anti self-dual components, as disclosed by the soldering formalism. Since this is a gauge embedding procedure derived from an old algorithm of second-class constraint conversion used by the author to approach anomalous gauge theories, a quick review of the subject will be presented. The problem of classification of the electromagnetic duality groups that is closely related will also be briefly discussed. Particular emphasis will be given to a new approach to duality based on the soldering embedding to tackle to problem of mass generation by topological mechanisms in D=3 and D=4 dimensions including the couplings to dynamical matter and nonlinear cases.

QED from six-dimensional vortex and gauge anomalies

Starting from an anomaly-free Abelian Higgs model coupled to gravity in a 6-dimensional space-time we construct an effective four-dimensional theory of charged fermions interacting with U(1) Abelian gauge field and gravity, both localised near the core of a Nielsen-Olesen vortex configuration. We show that an anomaly free theory in 6-dimensions can give rise to an anomalous theory in D=4, which suggests a possibility of consistent regularisation of abelian anomalous chiral gauge theories in four dimensions. We also show that the spectrum of charged bulk fermions has a mass gap.

Structural Aspects of the Fermion-Boson Mapping in Two-Dimensional Gauge and Anomalous Gauge Theories with Massive Fermions

Using a synthesis of the functional integral and operator approaches we discuss the fermion-boson mapping and the role played by the Bose field algebra in the Hilbert space of two-dimensional gauge and anomalous gauge field theories with massive fermions. In QED2 with quartic self-interaction among massive fermions, the use of an auxiliary vector field introduces a redundant Bose field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. In anomalous chiral QED2 with massive fermions the effect of the chiral anomaly leads to the appearance in the mass operator of a spurious Bose field combination. This phase factor carries no fermion selection rule and the expected absence of Θ-vacuum in the anomalous model is displayed from the operator solution. Even in the anomalous model with massive Fermi fields, the introduction of the Wess–Zumino field replicates the theory, changing neither its algebraic content nor its physical content.

Noncompact chiral U(1) gauge theories on the lattice

A new, adiabatic phase choice is adopted for the overlap in the case of an infinite volume, noncompact abelian chiral gauge theory. This gauge choice obeys the same symmetries as the Brillouin-Wigner (BW) phase choice, and, in addition, produces a Wess-Zumino functional that is linear in the gauge variables on the lattice. As a result, there are no gauge violations on the trivial orbit in all theories, consistent and covariant anomalies are simply related and Berry's curvature now appears as a Schwinger term. The adiabatic phase choice can be further improved to produce a perfect phase choice, with a lattice Wess-Zumino functional that is just as simple as the one in continuum. When perturbative anomalies cancel, gauge invariance in the fermionic sector is fully restored. The lattice effective action describing an anomalous abelian gauge theory has an explicit form, close to one analyzed in the past in a perturbative continuum framework.

Refining the anomaly consistency condition

In the extended antifield formalism, a quantum BRST differential for anomalous gauge theories is constructed. Local BRST cohomological classes are characterized, in addition to the form degree and the ghost number, by the length of their descents and of their lifts, and this both in the standard and the extended antifield formalism. It is shown that during the BRST invariant renormalization of a local BRST cohomological class, the anomaly that can appear is constrained to be a local BRST cohomological class with a shorter descent and a longer lift than the given class. As an application of both results, a simple approach to the Adler-Bardeen theorem for the non-Abelian gauge anomaly is proposed. It applies independently of the gauge fixing, of power counting restrictions, and does not rely on the use of the Callan-Symanzik equation.

Anomalous gauge theories with antisymmetric tensor fields

We derive a dual theory of the four-dimensional anomalous U(1) gauge theory with a Wess--Zumino (WZ) term and with a St\"uckelberg type mass term by means of a duality transformation at each of the classical and quantum levels. It is shown that in the dual anomalous U(1) gauge theory, the $BF$ term with a rank-two antisymmetric tensor field plays the roles of the WZ term as well as the mass term of the U(1) gauge field. Similar anomalous U(1) gauge theory with $BF$ term is considered in six-dimensions by introducing a rank-four antisymmetric tensor field. In addition to this theory, we propose a six-dimensional anomalous U(1) gauge theory including an extended $BF$ term with a rank-two antisymmetric tensor field, discussing a difference between the two theories. We also consider a four-dimensional anomalous SU(2)$\times$U(1) gauge theory with $BF$ term and recognize a crucial role of the $BF$ term in cancelling the non-abelian chiral anomaly.

Partition functions of chiral gauge theories on the two dimensional torus and their duality properties

Two different families of abelian chiral gauge theories on the torus are investigated: the aim is to test the consistency of two-dimensional anomalous gauge theories in the presence of global degrees of freedom for the gauge field. An explicit computation of the partition functions shows that unitarity is recovered in particular regions of parameter space and that the effective dynamics is described in terms of fermionic interacting models. For the first family, this connection with fermionic models uncovers an exact duality which is conjectured to hold in the nonabelian case as well.