Wilson Loop Average Research Articles

Infrared Yang-Mills wave functional due to percolating center vortices

Inspired by the center-vortex dominance in the infrared sector of $SU(N)$ Yang-Mills theory observed on the lattice, we propose a vacuum wave functional localized on an ensemble of correlated center vortices endowed with stiffness and magnetic monopoles that change the orientation of the vortex flux. In the electric-field representation, this wave functional becomes an effective partition function for N complex scalar fields. The inclusion of both oriented and nonoriented vortices as well as so-called N-vortex matchings leads to an effective potential that has only a center symmetry left. In the center-vortex condensed phase, this symmetry is spontaneously broken. In this case, the Wilson loop average can be approximated by a solitonic saddle point localized around the minimal surface. The asymptotic string tension thus obtained displays Casimir scaling.

How to extract the dominant part of the Wilson loop average in higher representations

In previous works, we have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement. This framework improves the Abelian projection in the gauge-independent manner. For quarks in the fundamental representation, we have demonstrated some numerical evidences for the restricted field dominance in the string tension, which means that the string tension extracted from the restricted part of the Wilson loop reproduces the string tension extracted from the original Wilson loop. However, it is known that the restricted field dominance is not observed for the Wilson loop in higher representations if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation. In this paper, therefore, we focus on confinement of quarks in higher representations. By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable gauge-invariant operators constructed from the restricted field to reproduce the correct behavior of the original Wilson loop averages for higher representations. Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators. We find that the proposed operators well reproduce the behavior of the original Wilson loop average, namely, the linear part of the static potential with the correct value of the string tension, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.

Lattice study of area law for double-winding Wilson loops

We study the double-winding Wilson loops in the SU(N) Yang-Mills theory on the lattice. We discuss how the area law falloff of the double-winding Wilson loop average is modified by changing the enclosing contours C1 and C2 for various values of the number of color N. By using the strong coupling expansion, we evaluate the double-winding Wilson loop average in the lattice SU(N) Yang-Mills theory. Moreover, we compute the double-winding Wilson loop average by lattice Monte Carlo simulations for SU(2) and SU(3). We further discuss the results from the viewpoint of the Non-Abelian Stokes theorem in the higher representations.

Magnetic monopole versus vortex as gauge-invariant topological objects for quark confinement

First, we give a gauge-independent definition of chromomagnetic monopoles in [Formula: see text] Yang–Mills theory which is derived through a non-Abelian Stokes theorem for the Wilson loop operator. Then we discuss how such magnetic monopoles can give a nontrivial contribution to the Wilson loop operator for understanding the area law of the Wilson loop average. Next, we discuss how the magnetic monopole condensation picture are compatible with the vortex condensation picture as another promising scenario for quark confinement. We analyze the profile function of the magnetic flux tube as the non-Abelian vortex solution of [Formula: see text] gauge-Higgs model, which is to be compared with numerical simulations of the [Formula: see text] Yang–Mills theory on a lattice. This analysis gives an estimate of the string tension based on the vortex condensation picture, and possible interactions between two non-Abelian vortices.

Double-winding Wilson loops in the SU(N) Yang-Mills theory

We consider double-winding, triple-winding and multiple-winding Wilson loops in the $SU(N)$ Yang-Mills gauge theory. We examine how the area law falloff of the vacuum expectation value of a multiple-winding Wilson loop depends on the number of color $N$. In sharp contrast to the difference-of-areas law recently found for a double-winding $SU(2)$ Wilson loop average, we show irrespective of the spacetime dimensionality that a double-winding $SU(3)$ Wilson loop follows a novel area law which is neither difference-of-areas nor sum-of-areas law for the area law falloff and that the difference-of-areas law is excluded and the sum-of-areas law is allowed for $SU(N)$ ($N \ge 4$), provided that the string tension obeys the Casimir scaling for the higher representations. Moreover, we extend these results to arbitrary multi-winding Wilson loops. Finally, we argue that the area law follows a novel law, which is neither sum-of-areas nor difference-of-areas law when $N\ge 3$. In fact, such a behavior is exactly derived in the $SU(N)$ Yang-Mills theory in the two-dimensional spacetime.

Differential expansion and rectangular HOMFLY for the figure eight knot

Differential expansion (DE) for a Wilson loop average in representation R is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of 3d Chern–Simons theory. Especially simple is the relation between the DE for the trefoil 31 and for the figure eight knot 41. Since arbitrary colored HOMFLY for the trefoil are known from the Rosso–Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfill this program for all rectangular representation R=[rs], i.e. make a plausible conjecture for the rectangularly colored HOMFLY of the figure eight knot, which generalizes the old result for totally symmetric and antisymmetric representations.

Low-energy effective Yang-Mills theory for quark and gluon confinement

We derive a gauge-invariant low-energy effective model of the Yang-Mills theory. We find that the effective gluon propagator belongs to the Gribov-Stingl type and agrees with it when a mass term which breaks nilpotency of the BRST symmetry is included. We show that the effective model with gluon propagator of the Gribov-Stingl type exhibits both quark and gluon confinement: the Wilson loop average has the area law and the Schwinger function violates reflection positivity. However, we argue that both quark and gluon confinement can be obtained even in the absence of such a mass term.

Gauge-invariant gluon mass, infrared Abelian dominance, and stability of magnetic vacuum

We give an argument for deriving analytically the infrared “Abelian” dominance in a gauge invariant way for the Wilson loop average in SU(2) Yang– Mills theory. In other words, we propose a possible mechanism for realizing the dynamical Abelian projection in the SU(2) gauge-invariant manner without breaking color symmetry. This supports validity of the dual superconductivity picture for quark confinement. We also discuss the stability of the vacuum with magnetic condensation as a by-product of this result.

Improved heavy quark potential at finite temperature from anti-de Sitter supergravity

We improve the heavy quark potential, extracted from the Wilson loop average in the Ads/CFT approach by taking the quantum fluctuation of the only radial coordinate of Ads 5 which is transverse to the world-sheet of the classical Nambu-Goto string in the static gauge, and obtain the universal Lüscher-Symanzik-Weisz/Lüscher term.

Exactly soluble QCD and confinement of quarks

An exactly soluble non-perturbative model of the pure gauge QCD is derived as a weak coupling limit of the lattice theory in plaquette formulation [B. Rusakov, Phys. Lett. B 398 (1997) 331 ]. The model represents QCD as a theory of the weakly interacting field strength fluxes. The area law behavior of the Wilson loop average is a direct result of this representation: the total flux through macroscopic loop is the additive (due to the weakness of the interaction) function of the elementary fluxes. The compactness of the gauge group is shown to be the factor which prevents the elementary fluxes contributions from cancellation. There is no area law in the non-compact theory.

Curvature Expansion for the Gluodynamics String Including Perturbative Gluonic Contributions

Perturbation theory, which represents a Wilson loop in the SU(2) gluodynamics as an integral over all the orientations in color space, in the nonperturbative QCD vacuum and the non-Abelian Stokes theorem is applied to a derivation of the correction to the string effective action in the lowest order in the coupling constant g. This correction is due to the interaction of perturbative gluons with the string worldsheet and affects only the coupling constant of the rigidity term, while its contribution to the string tension of the Nambu–Goto term vanishes. The obtained correction to the rigidity coupling constant multiplicatively depends on the color "spin" of the representation of the Wilson loop under consideration and a certain path integral, which includes the background Wilson loop average.

From the Feynman-Schwinger representation to the nonperturbative relativistic bound state interaction

We write the four-point Green function in QCD in the Feynman-Schwinger representation and show that all the dynamical information is contained in the Wilson loop average. We work out the QED case in order to obtain the usual Bethe-Salpeter kernel. Finally we discuss the QCD case in the nonperturbative regime giving some insight into the nature of the interaction kernel.

Schwinger-Dyson Equation for Supersymmetric Yang-Mills Theory: Manifestly Supersymmetric Form

We study our Schwinger-Dyson equation as well as the large $N_{c}$ loop equation for supersymmetric Yang-Mills theory in four dimensions by the N=1 superspace Wilson-loop variable. We are successful in deriving a new manifestly supersymmetric form in which a loop splitting and joining are represented by a manifestly supersymmetric as well as supergauge invariant operation in superspace. This is found to be a natural extension from the abelian case. We solve the equation to leading order in perturbation theory or equivalently in the linearized approximation, obtaining a desirable nontrivial answer. The super Wilson-loop variable can be represented as the system of one-dimensional fermion along the loop coupled minimally to the original theory. One-loop renormalization of the one-point Wilson-loop average is explicitly carried out, exploiting this property. The picture of string dynamics obtained is briefly discussed.

Schwinger-Dyson and large Nc loop equation for supersymmetric Yang-Mills theory

We derive an infinite sequence of Schwinger-Dyson equations for N = 1 supersymmetric Yang-Mills theory. The fundamental and the only variable employed is the Wilson-loop geometrically represented in N = 1 superspace: it organizes an infinite number of supersymmetrizing insertions into the ordinary Wilson-loop as a single entity. In the large N c limit, our equation becomes a closed loop equation for the one-point function of the Wilson-loop average.

The SU ( N) Wilson loop average in two dimensions

We solve explicitly a closed, linear loop equation for the SU(2) Wilson loop average on a two-dimensional plane and generalize the solution to the case of the SU( N) Wilson loop average with an arbitrary closed contour. Furthermore, the flat space solution is generalized to any two-dimensional manifold for the SU(2) Wilson loop average and to any two-dimensional manifold of genus O for the SU( N) Wilson loop average. The SU( N) Wilson loop average folows an area law W( C) = Σ r P' r exp[ - Σ i j ri 2 S i ], where j ri 2 is the quadratic Casimir operator for the window with area S i . Only certain combinations of the Casimir operators are allowed in the sum over i. We give a physical interpretation of the constants P r 1 in the case of a non self-intersecting composed path C and of the constraints determining in which combinations the Casimir operators occur.

Spin, statistics, and geometry of random walks: T. Jaroszewicz and P. S. Kurzepa. Department of Physics, University of California Los Angeles, Los Angeles, California 90024

The purpose of this paper is to review the recent progress in understanding quark confinement. The emphasis of this review is placed on how to obtain a manifestly gauge-independent picture for quark confinement supporting the dual superconductivity in the Yang–Mills theory, which should be compared with the Abelian projection proposed by ’t Hooft. The basic tools are novel reformulations of the Yang–Mills theory based on change of variables extending the decomposition of the SU(N) Yang–Mills field due to Cho, Duan–Ge and Faddeev–Niemi, together with the combined use of extended versions of the Diakonov–Petrov version of the non-Abelian Stokes theorem for the SU(N) Wilson loop operator.Moreover, we give the lattice gauge theoretical versions of the reformulation of the Yang–Mills theory which enables us to perform the numerical simulations on the lattice. In fact, we present some numerical evidences for supporting the dual superconductivity for quark confinement. The numerical simulations include the derivation of the linear potential for static interquark potential, i.e., non-vanishing string tension, in which the “Abelian” dominance and magnetic monopole dominance are established, confirmation of the dual Meissner effect by measuring the chromoelectric flux tube between quark–antiquark pair, the induced magnetic-monopole current, and the type of dual superconductivity, etc. In addition, we give a direct connection between the topological configuration of the Yang–Mills field such as instantons/merons and the magnetic monopole.We show especially that magnetic monopoles in the Yang–Mills theory can be constructed in a manifestly gauge-invariant way starting from the gauge-invariant Wilson loop operator and thereby the contribution from the magnetic monopoles can be extracted from the Wilson loop in a gauge-invariant way through the non-Abelian Stokes theorem for the Wilson loop operator, which is a prerequisite for exhibiting magnetic monopole dominance for quark confinement. The Wilson loop average is calculated according to the new reformulation written in terms of new field variables obtained from the original Yang–Mills field based on change of variables. The Maximally Abelian gauge in the original Yang–Mills theory is also reproduced by taking a specific gauge fixing in the reformulated Yang–Mills theory. This observation justifies the preceding results obtained in the maximal Abelian gauge at least for gauge-invariant quantities for SU(2) gauge group, which eliminates the criticism of gauge artifact raised for the Abelian projection. The claim has been confirmed based on the numerical simulations.However, for SU(N) (N≥3), such a gauge-invariant reformulation is not unique, although the extension along the line proposed by Cho, Faddeev and Niemi is possible. In fact, we have found that there are a number of possible options of the reformulations, which are discriminated by the maximal stability group H̃ of G, while there is a unique option of H̃=U(1) for G=SU(2). The maximal stability group depends on the representation of the gauge group, to that the quark source belongs. For the fundamental quark for SU(3), the maximal stability group is U(2), which is different from the maximal torus group U(1)×U(1) suggested from the Abelian projection. Therefore, the chromomagnetic monopole inherent in the Wilson loop operator responsible for confinement of quarks in the fundamental representation for SU(3) is the non-Abelian magnetic monopole, which is distinct from the Abelian magnetic monopole for the SU(2) case. Therefore, we claim that the mechanism for quark confinement for SU(N) (N≥3) is the non-Abelian dual superconductivity caused by condensation of non-Abelian magnetic monopoles. We give some theoretical considerations and numerical results supporting this picture. Finally, we discuss some issues to be investigated in future studies.

Action-variational approach to SU(3) pure gauge theory

A variational method with respect to a trial action is applied to the SU(3) lattice gauge theory. From the evaluation of the Wilson-loop average, the SU(3) string tension is extracted.

The area law of the Wilson loop and vacuum field correlators

The Wilson loop average in the background vacuum fields is expressed via the cumulant expansion in terms of irreducible correlators of the gluon field. Assuming a fast fall-off of the latter at large distance one obtains the area law for large contours.

On the random-surface representation of U(∞) lattice gauge theory

A recently proposed expansion of the Wilson loop average in U(∞) lattice gauge theory as a sum over noninteracting planar surfaces is substantially simplified. The local factors in the surface amplitude no longer depend on the inverse coupling β. The amount of surfaces involved is highly reduced.

Loop-space quantum formulation of free electromagnetism

A procedure for direct quantization of free electromagnetism in the loop-space is proposed. Explicit solutions for the loop-dependent vacuum and the Wilson loop-average are given. It is shown that elementary lines of magnetic field appear as extremals in the vacuum state as a result of the regularization procedure.