Euclidean Wilson Loop Research Articles

Nonperturbative potential for the study of quarkonia in QGP

A thermal potential can be defined to facilitate understanding the behavior of quarkonia in quark-gluon plasma. A nonperturbative evaluation of this potential from lattice QCD is difficult, as it involves real-time corelation function, and has often involved the use of Bayesian analysis, with its associated systematics. In this work we show that using the properties of the static quarkonia thermal correlation functions, one can directly extract a thermal potential for quarkonia from Euclidean Wilson loop data. This leads to a controlled extraction, and allows us to judge the suitability of various model potentials. We also discuss the phenomenology of quarkonia in the gluonic plasma.

Holographic calculations of Euclidean Wilson loop correlator in Euclidean anti-de Sitter space

The correlation functions of two or more Euclidean Wilson loops of various shapes in Euclidean anti-de Sitter space are computed by considering the minimal area surfaces connecting the loops. The surfaces are parametrized by Riemann theta functions associated with genus three hyperelliptic Riemann surfaces. In the case of two loops, the distance $L$ by which they are separated can be adjusted by continuously varying a specific branch point of the auxiliary Riemann surface. When $L$ is much larger than the characteristic size of the loops, then the loops are approximately regarded as local operators and their correlator as the correlator of two local operators. Similarly, when a loop is very small compared to the size of another loop, the small loop is considered as a local operator corresponding to a light supergravity mode.

Wilson loops and minimal surfaces beyond the wavy approximation

We study Euclidean Wilson loops at strong coupling using the AdS/CFT correspondence, where the problem is mapped to finding the area of minimal surfaces in Hyperbolic space. We use a formalism introduced recently by Kruczenski to perturbatively compute the area corresponding to boundary contours which are deformations of the circle. Our perturbative expansion is carried to high orders compared with the wavy approximation and yields new analytic results. The regularized area is invariant under a one parameter family of continuous deformations of the boundary contour which are not related to the global symmetry of the problem. We show that this symmetry of the Wilson loops breaks at weak coupling at an a priori unexpected order in the perturbative expansion. We also study the corresponding Lax operator and algebraic curve for these solutions.

Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(lambda,s) depending on a complex parameter lambda known as the spectral parameter. This family has remarkable properties. As a function of lambda, X(lambda,s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(lambda,s) has an essential singularity at the origin lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.

Minimal Area Surfaces, Euclidean Wilson loops and Riemann Theta Functions

The AdS/CFT correspondence relates Wilson loops in supersymmetric N = 4 gauge theory to minimal area surfaces in anti-de Sitter space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3 space. Using known mathematical results for such minimal area surfaces we describe an infinite parameter family of analytic solutions for closed Wilson loops. Further, we also consider examples of surfaces with two boundaries dual to correlators of two Wilson loops.

Notes on Euclidean Wilson loops and Riemann theta functions

The AdS/CFT correspondence relates Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory to minimal area surfaces in ${\mathrm{AdS}}_{5}$ space. In this paper we consider the case of Euclidean flat Wilson loops, which are related to minimal area surfaces in Euclidean ${\mathrm{AdS}}_{3}$ space. Using known mathematical results for such minimal area surfaces, we describe an infinite parameter family of analytic solutions for closed Wilson loops. The solutions are given in terms of Riemann theta functions, and the validity of the equations of motion is proven based on the trisecant identity. The world sheet has the topology of a disk, and the renormalized area is written as a finite, one-dimensional contour integral over the world-sheet boundary. An example is discussed in detail with plots of the corresponding surfaces. Further, for each Wilson loops we explicitly construct a one parameter family of deformations that preserve the area. The parameter is the so-called spectral parameter.

General computations without fixing the gauge

Within the framework of a manifestly gauge invariant exact renormalization group for $SU(N)$ Yang-Mills, we derive a simple expression for the expectation value of an arbitrary gauge invariant operator. We illustrate the use of this formula by computing the $\mathrm{O}({g}^{2})$ correction to the rectangular, Euclidean Wilson loop with sides $T\ensuremath{\gg}L$. The standard result is trivially obtained, directly in the continuum, for the first time without fixing the gauge. We comment on possible future applications of the formalism.

MASSIVE AXIAL GAUGE IN THE EXACT RENORMALIZATION GROUP APPROACH

The Exact Renormalization Group (ERG) approach to massive gauge theories in the axial gauge is studied and the smoothness of the massless limit is analysed for a formally gauge invariant quantity such as the Euclidean Wilson loop.

SU( N ) Quantum Yang–Mills theory in two dimensions: A complete solution

A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang–Mills continuum theories with gauge groups SU(N) and U(1) and space-time topologies R1×R1 and R1×S1. (For the U(1) theory, we also consider the S1×S1 topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder–Schrader axioms for gauge theories is proposed. These axioms are satisfied in the present model.

EUCLIDEAN WILSON LOOP AND AXIAL GAUGE

A critical analysis is given in order to set up a well-defined perturbative expansion for Yang-Mills Euclidean theories within the axial gauge choice, using Cauchy principal value prescription to handle the spurious singularities. It is shown that, following a mathematically meaningful and unambiguous procedure, the exponential behavior of the Euclidean Wilson loop does not occur at variance with the covariant and planar gauge choices. A comparison with previous similar approaches and results is worked out in order to clearly understand the reasons for the breakdown of gauge invariance in the present case. In particular we can conclude that the so far proposed alternative prescriptions, which have been claimed to restore gauge invariance, should be carefully reexamined, at least for the Euclidean formulation.