A phase cell approach to Yang-Mills theory

In this paper the basic local stability result is obtained, in a form valid in both small field and large field regions. To achieve this, some modifications are made in both the action and the renormalization group transformation. Though there is some sacrifice of elegance in these modifications, the establishment of this local stability estimate yields the most basic ingredient of the phase cell cluster expansion, good estimates for all the actions.Incidental to the estimates of this paper we establish some results on lattice geometry, interesting in their own right. A bound on the minimum area of a loop of lengthl, ind dimensions, is obtained asl ²/8(1–1/d). This, a best possible bound, was obtained for us by A. Blass. We also construct a radial maximal tree for the lattice ind dimensions. We hope to stimulate someone to find a better construction of radial trees.This work was supported in part by the National Science Foundation under Grant No. PHY 85-02074     

A phase cell approach to Yang-Mills theory

For the abelian Yang-Mills theory, a one-to-one correspondence is established between continuum gauge potentials and compatible lattice configurations on an infinite sequence of finer and finer lattices. The compatibility is given by a block spin transformation determining the configuration on a lattice in terms of the configuration on any finer lattice. Thus the configuration on any single lattice is not an approximation to the continuum field, but rather a subset of the variables describing the field.It is proven that the Wilson actions on the lattices monotonically increase to the continuum action as one passes to finer and finer lattices. Configurations that minimize the continuum action, subject to having the variables fixed on some lattice, are studied.This work was supported in part by the National Science Foundation under Grant No. PHY-85-02074     

A phase cell approach to Yang-Mills theory V. Analysis of a chunk

In the present formalism the Yang-Mills field is constructed as a non-linear sum of excitations, small field excitations, the modes, and large field excitations, the chunks. The chunk excitations, herein studied, are each described by a finite number of group element variables. The continuum field associated to the excitation in general has point gauge singularities (arising from the non-trivial ₃(G)). We find estimates for plaquette assignments, edge assignments, and the smoothness of edge assignments, at all scales. The central conceptual motor in our constructions and estimates is a split up of the field at each length scale, locally, into a pure gauge field, and a deviation field. An example is presented establishing the general inevitability of gauge singularities, as a consequence of fall off requirements on the continuum field of an excitation.This work was supported in part by the National Science Foundation under Grant No. PHY-87-01329     

String-like phase in Yang-Mills theory

A dynamical mechanism is discussed leading to the phase where the string of the color flux tube is formed in Yang-Mills theory. The pairing force between massless gauge fields in the color single channel forms a tachyon or a Cooper state and leads to the instability of the normal vacuum. A stable vacuum is required to remain massless after the condensation of pairs. Then the stability of the new vacuum leads to the condition on the 't Hooft index α which is equivalent to the condition of the existence of the string solutions.     

The high-temperature phase of Landau-gauge Yang-Mills theory

The properties of the high-temperature phase of Yang-Mills theory in Landau gauge are investigated by extending an earlier study on the infinite-temperature limit to finite temperatures. To this end the Dyson-Schwinger equations for the propagators of the gluon and the Faddeev-Popov ghost are solved analytically in the infrared and numerically at non-vanishing momenta. Gluons, polarized transversely with respect to the heat bath are found to comply with the Gribov-Zwanziger and the Kugo-Ojima scenario, while longitudinally polarized gluons are screened. Therefore the high-temperature phase is strongly interacting. It is furthermore conjectured that Yang-Mills theory undergoes a first-order phase transition. Indications are found that at high temperatures the thermodynamic properties are nearly those of an ideal gas, although long-range interactions prevail.Received: 5 April 2005, Revised: 26 April 2005, Published online: 8 June 2005PACS: 11.10.Wx, 11.15.-q, 12.38.-t, 12.38.Aw, 12.38.Lg, 12.38.Mh, 14.70.Dj     

World-line approach to the Bern-Kosower formalism in two-loop Yang-Mills theory

Based on the world-line formalism with a sewing method, we derive the Yang-Mills effective action in a form useful to generate the Bern-Kosower-type master formulae for gluon scattering amplitudes at the two-loop level. It is shown that four-gluon (φ⁴ type sewing) contributions can be encapsulated in the action with three-gluon (φ³ type) vertices only, the total action thus becoming a simple expression. We then derive a general formula for a two-loop Euler-Heisenberg type action in a pseudo-abelian su(2) also studied.     

A string-like self-dual solution of Yang-Mills theory

We attempt to obtain Nielsen-Olesen strings in pure Yang-Mills theory without Higgs scalars. This is inspired by recent work interpreting the Prasad-Sommerfield monopole as a static self-dual Yang-Mills solution in which A₄^a plays the role of the Higgs field. In similar fashion, we make A₃^aA₄^a serve as the two required isovector fields in an ansatz independent of x₃ and x₄. The condition of self-duality results in a single Painlevé equation of the third kind (or equivalently, a radial sinh-Gordon equation in 2 + 0 dimensions), the solution of which determines A_μ^a. We make use of the extensive analysis of the former equation carried out by Wu, McCoy and collaborators in the context of the scaling limit of the two-dimensional Ising model. Their simplest solution yields a flux value of ?2π/e just as in the Nielsen-Olesen model and flux is quantized in multiples of this unit. The string tension (action per unit time per unit distance) diverges as r^?2 (In r)^?2 as r → 0 for the same Wu-McCoy solution.     

Classical solutions to topologically massive Yang-Mills theory

Classical solutions to (2 + 1)-dimensional Yang-Mills theory in the presence of the Chern-Simons invariant are considered. The SO(3)-invariant solutions to the Euclidean field equations are complex, whereas the equations in Minkowski space-time possess real SO(2, 1)-invariant solutions. The field equations for time independent axially symmetric vector potentials are derived and some solutions are obtained. The behavior of general Euclidean spacetime solutions is discussed. It is also shown that, because of the gauge dependence of the Chern-Simons invariant, the reduced field equations cannot be uniquely obtained from the reduced action. Applications of the results to the infrared structure of finite temperature QCD are discussed; in particular, it is argued that the Chern-Simons invariant cannot be consistently incorporated as a gauge-invariant magnetic mass term in a three-dimensional effective long distance theory at high temperatures.     

Yang-Mills theory and uniformization

We define a notion of a stable system of Hodge bundles. A stable system of Hodge bundles has a Hermitian-Yang-Mills metric and, if certain Chern classes vanish, this gives a complex variation of Hodge structure. We use these ideas to obtain a criterion for a variety to be uniformized by a bounded symmetric domain.     

A non-perturbative contribution to the vacuum energy in supersymmetric Yang-Mills theory

It is shown that the quantum fluctuations around an interacting instanton-anti-instanton field configuration induce negative vacuum energy in supersymmetric Yang-Mills theory.     

A unified approach to phase diagrams in field theory and statistical mechanics

We construct the phase diagram of any system which admits a low-temperature polymer or cluster expansion. Such an expansion turns the system into a hard-core interacting contour model with small, but not necessarily positive, activities. The method uses some of Zahradnik's ideas [Z1], but applies equally well to systems with complex interactions. We give two applications. First, to low-temperatureP()₂ models with complex couplings; and second, to a computation of asymptotics of partition functions in periodic volumes. If the index of a supersymmetric field theory is known, the second application would help determine the number of phases in infinite volume.Alfred P. Solan Research Fellow. Supported in part by the National Science Foundation under Grants PHY87-064220, DMS 88-58073, and PHY/DMS 86-45122     

A lagrangian formulation of the Wess approach to supersymmetric Yang-Mills theories

We obtain a superspace lagrangian density which reproduces the Wess formulation of supersymmetric gauge theories. The present approach, which is a first-order formalism, gives rise to a lagrangian density polynomial in the Yang-Mills and auxiliary superfields and appears to be stable under quantization and renormalization.     

Gravity as Yang-Mills gauge theory

We propose a Yang-Mills field theory of gravity based on a unitary phase-gauge-invariance of the lagrangian where the gauge transformations are those of the SU(2) × U(1) symmetry of the 2-spinors. In the classical limit this microscopic theory results in Einstein's metrical theory of gravity, where we restrict ourselves in a first step to its linearized version.     

Yang-Mills gauge theory and gravitation

We point out that Yang's and Einstein's gravitational equations can be obtained from a geometric approach of Yang-Mills gauge theory in a sourceless case, under a decomposition of the Poincaré algebra. Otherwise, Einstein's equations cannot be derived from a Yang-Mills gauge equation when sources are inserted in the rotational sector of that algebra. A gauge Lagrangian structure is also discussed.     

Geometric regularization of Yang-Mills theory

A continuum regularization of Yang-Mills theory with a possible non-perturbative interpretation is introduced in the euclidean formalism. The regularization preserves Lorentz covariance and gauge invariance and involves not only a regularization of the action but also a regularization of the functional measure in terms of a nuclear-riemannian structure of the space of gauge orbits.