Infrared behavior of three-point functions in Landau gauge Yang–Mills theory

Analytic solutions for the three-gluon and ghost–gluon vertices in Landau gauge Yang–Mills theory at low momenta are presented in terms of hypergeometric series. They do not only show the expected scaling behavior but also additional kinematic divergences when only one momentum goes to zero. These singularities, which have also been proposed previously, induce a strong dependence on the kinematics in many dressing functions. The results are generalized to two and three dimensions and a range of values for the ghost propagator’s infrared exponent κ.     

Infrared divergences and a non-local gauge for superspace Yang-Mills theory

The usual superspace approach to supersymmetric gauge theories suffers from problems with infrared divergences which greatly complicate multiloop calculations. We eliminate these divergences by introducing a non-local gauge-fixing term. In the background field method this term leads to unusual quantum-background interactions. Functional methods are presented for dealing with these interactions. As an example we compute the two-loop Yang-Mills β-function using the background field method in superspace. We also show how a non-local gauge can be used in ordinary, non-supersymmetric Yang-Mills theory.     

Symmetries in a Constrained System with a Singular Higher-Order Lagrangian

A simple algorithm to construct the generator of gauge transformation for a constrained canonical system with a singular higher-order Lagrangian in field theories is developed. Based on phase-space generating functional of Green function for such a system, the generalized canonical Ward identities under the non-local transformation have been deduced. For the gauge-invariant system, based on configuration-space generating functional, the generalized Ward identities under the non-local transformation have been also derived.The conservation laws are deduced at the quantum level. The applications of the above results to the gauge invariance massive vector field and non-Abelian Chern–Simons(CS) theories with higher-order derivatives are given, a new form of gauge-ghost proper vertices, and Ward–Takahashi identity under BRS transformation and BRS charge at the quantum level are obtained. In the canonical formulation one does not need to carry out the integration over canonical momenta in phase-space path integral as usually performed.     

On unconstrained SU(2) gluodynamics with theta angle

The Hamiltonian reduction of classical SU(2) Yang–Mills field theory to the equivalent unconstrained theory of gauge invariant local dynamical variables is generalized to the case of nonvanishing -angle. It is shown that for any -angle the elimination of the pure gauge degrees of freedom leads to a corresponding unconstrained non-local theory of self-interacting second rank symmetric tensor fields, and that the obtained classical unconstrained gluodynamics with different -angles are canonically equivalent as on the original constrained level. Received: 16 November 2001 / Published online: 5 April 2002     

Renormalization and symmetry conditions in supersymmetric QED

For supersymmetric gauge theories a consistent regularization scheme that preserves supersymmetry and gauge invariance is not known. In this article we tackle this problem for supersymmetric QED within the framework of algebraic renormalization. For practical calculations, a non-invariant regularization scheme may be used together with counterterms from all power-counting renormalizable interactions. From the Slavnov–Taylor identity, expressing gauge invariance, supersymmetry and translational invariance, simple symmetry conditions are derived that are important in a twofold respect: they establish exact relations between physical quantities that are valid to all orders, and they provide a powerful tool for the practical determination of the counterterms. We perform concrete one-loop calculations in dimensional regularization, where supersymmetry is spoiled at the regularized level, and show how the counterterms necessary to restore supersymmetry can be read off easily. In addition, a specific example is given how the supersymmetry transformations in one-loop order are modified by non-local terms. Received: 23 July 1999 / Published online: 14 October 1999     

Bianchi type I cosmology in generalized Saez–Ballester theory via Noether gauge symmetry

In this paper, we investigate the generalized Saez–Ballester scalar–tensor theory of gravity via Noether gauge symmetry (NGS) in the background of Bianchi type I cosmological spacetime. We start with the Lagrangian of our model and calculate its gauge symmetries and corresponding invariant quantities. We obtain the potential function for the scalar field in the exponential form. For all the symmetries obtained, we determine the gauge functions corresponding to each gauge symmetry which include constant and dynamic gauge. We discuss cosmological implications of our model and show that it is compatible with the observational data.     

Tensor Gauge Fields in Arbitrary Representations of GL( D, $${\mathbb{R})}$$ : II. Quadratic Actions

Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in D-dimensional Minkowski space are explicitly written in a compact form by making use of Levi–Civita tensors. The field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously. Moreover, these actions allow a frame-like reformulation à la MacDowell–Mansouri, without any trace constraint in the tangent indices. Chargé de Recherches FNRS, Belgium     

String Geometry and the Noncommutative Torus

We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra ? and the noncommutative torus. We show that the tachyon algebra of ? is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d,d;ℤ) Morita equivalences between d-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang–Mills gauge theory minimally coupled to massive Dirac fermion fields. Received: 26 October 1998/ Accepted: 9 April 1999     

The perturbative proton form factor reexamined

We recalculate the proton Dirac form factor based on the perturbative QCD factorization theorem, which includes Sudakov suppression. The evolution scale of the proton wave functions and the infrared cutoffs for the Sudakov re-summation are carefully chosen such that the soft divergences from large coupling constants are diminished and perturbative QCD predictions are stabilized. We find that the King–Sachrajda model for the proton wave function leads to results which are in better agreement with experimental data than those from the Chernyak–Zhitnitsky wave function. Received: 27 November 1998 / Published online: 7 April 1999     

One-loop leading logarithms in electroweak radiative corrections

We discuss the evaluation of the collinear single-logarithmic contributions to virtual electroweak corrections at high energies. More precisely, we prove the factorization of the mass singularities originating from loop diagrams involving collinear virtual gauge bosons coupled to external legs. We discuss, in particular, processes involving external longitudinal gauge bosons, which are treated using the Goldstone-boson equivalence theorem. The proof of factorization is performed within the 't Hooft–Feynman gauge at one-loop order and applies to arbitrary electroweak processes that are not mass-suppressed at high energies. As basic ingredients we use Ward identities for Green functions with arbitrary external particles involving a gauge boson collinear to one of these. The Ward identities are derived from the BRS invariance of the spontaneously broken electroweak gauge theory. Received: 4 May 2001 / Published online: 6 July 2001     

Polynomial Invariants for Torus Knots¶and Topological Strings

We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. Received: 1 May 2000 / Accepted: 6 November 2000     

Extended Connection in Yang-Mills Theory

The three fundamental geometric components of Yang-Mills theory –gauge field, gauge fixing and ghost field– are unified in a new object: an extended connection in a properly chosen principal fiber bundle. To do this, it is necessary to generalize the notion of gauge fixing by using a gauge fixing connection instead of a section. From the equations for the extended connection’s curvature, we derive the relevant BRST transformations without imposing the usual horizontality conditions. We show that the gauge field’s standard BRST transformation is only valid in a local trivialization and we obtain the corresponding global generalization. By using the Faddeev-Popov method, we apply the generalized gauge fixing to the path integral quantization of Yang-Mills theory. We show that the proposed gauge fixing can be used even in the presence of a Gribov’s obstruction.     

Bubble Divergences from Cellular Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang–Mills theory, the Ponzano–Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called ‘bubble divergences’. A common expectation is that the degree of these divergences is given by the number of ‘bubbles’ of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian – in both cases, the divergence degree is given by the second Betti number of the 2-complex.     

An Extension of Distribution Theory and of the Paley-Wiener-Schwartz Theorem Related to Quantum Gauge Theory

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat quantum gauge field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley–Wiener–Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation. Received: 29 December 1995 / Accepted: 13 September 1996     

New Representation of the Self-Duality and Exact Solutions for Yang–Mills Equations

In this paper, we found a new representation for self-duality . In addition, exact solution class of the classical SU(2) Yang–Mills field in four-dimensional Euclidean space and two exact solution classes for SU(2) Yang–Mills when ρ is a complex analytic function are also obtained. PACS numbers: 11.15.-q Gauge field theories, 11.15.Kc Semiclassical theories in gauge fields, 12.10.-g, 12.15.-y Yang–Mills fields     

A simple and efficient numerical scheme to integrate non-local potentials

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrödinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method was proposed several years ago, with the trivially equivalent potential method, where the non-local potential is replaced by an equivalent local potential, which is state dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of ¹⁶O using the finite-range N³LO realistic interaction.     

Vortex Condensates¶for the SU(3) Chern–Simons Theory

We investigate SU(3)-periodic vortices in the self-dual Chern–Simons theory proposed by Dunne in [13, 15]. At the first admissible non-zero energy level E= 2 π, and for each (broken and unbroken) vacuum state φ₍₀₎ of the system, we find a family of periodic vortices asymptotically gauge equivalent to φ₍₀₎, as the Chern–Simons coupling parameter k→ 0. At higher energy levels, we show the existence of multiple gauge distinct periodic vortices with at least one of them asymptotically gauge equivalent to the (broken) principal embedding vacuum, when k→ 0. Received: 23 October 1999 / Accepted: 14 March 2000     

Chan–Paton soliton gauge states of the compactified open string

We study the mechanism of the enhanced gauge symmetry of the bosonic open string compactified on a torus by analyzing the zero-norm soliton (non-zero winding of the Wilson line) gauge states in the spectrum. Unlike the closed string case, we find that the soliton gauge state exists only at massive levels. These soliton gauge states correspond to the existence of enhanced massive gauge symmetries with transformation parameters containing both Einstein and Yang–Mills indices. In the T-dual picture, these symmetries exist only at some discrete values of compactified radii when N D-branes are coincident. Received: 14 May 1999 / Published online: 17 March 2000     

Superconformal Invariance and the Geography¶of Four-Manifolds

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example of gauge group SU(2) with one doublet hypermultiplet, we derive a theorem relating classical topological invariants such as the Euler character and signature to sum rules for Seiberg–Witten invariants. A short account of this paper can be found in [1]. Received: 19 December 1998 / Accepted: 7 March 1999     

On the equivalence of the Einstein–Hilbert and the Einstein–Palatini formulations of general relativity for an arbitrary connection

In the framework of the Einstein–Palatini formalism, even though the projective transformation connecting the arbitrary connection with the Levi-Civita connection has been floating in the literature for a long time and perhaps the result was implicitly known in the affine gravity community, yet as far as we know Julia and Silva were the first to realise its gauge character. We rederive this result by using the Rosenfeld–Dirac–Bergmann approach to constrained Hamiltonian systems and do a comprehensive self contained analysis establishing the equivalence of the Einstein–Palatini and the metric formulations without having to impose the gauge choice that the connection is symmetric. We also make contact with the the Einstein–Cartan theory when the matter Lagrangian has fermions.