Wilson Loop Expectation Value Research Articles

THE STRING CALCULATION OF WILSON LOOPS IN TWO-DIMENSIONAL YANG-MILLS THEORY

We demonstrate that the large N expansion of Wilson loop expectation values in SO (N) and Sp (N) Yang-Mills theory on orientable and nonorientable surfaces has a natural description as a weighted sum over covers of the given surface. The sum takes the form of the perturbative expansion of an open string theory. The derivation makes contact with the classification of branched covers by Gabai and Kazez. Comparison with the analogous results for the chiral sectors of QCD 2 is instructive for both cases.

String calculation of QCD Wilson loops on arbitrary surfaces.

Compact string expressions are found for non-intersecting Wilson loops in SU(N) Yang-Mills theory on any surface (orientable or nonorientable) as a weighted sum over covers of the surface. All terms from the coupled chiral sectors of the 1/N expansion of the Wilson loop expectation values are included.

Large N 2D Yang-Mills Theory and Topological String Theory

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A=0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's ``\Omega^{-1} points.'' We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.

Resummation of large infrared corrections using Wilson loops

We discuss a method to resum systematically large soft gluon corrections to hard hadronic cross sections such as the structure function F( x, Q 2) for x → 1. We show that all singularities as x → 1 are factorized into a Wilson loop expectation value which is a function of the ratio of the infrared scale (1− x) Q 2 and the short distance cutoff μ 2 introduced by the factorization procedure. Thus we obtain the x → 1 behavior by studying the appropriate renormalization group equation which controls the μ-dependence of Wilson loops. Next-to-leading corrections are calculated. The method can be generalized to other distributions.

Partonic distributions for large x and renormalization of Wilson loop

We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone. Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost. Nevertheless we establish the renormalization group equation for the light-like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions. By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.

Wilson loop and contour shapes.

By computing the Wilson loop expectation value {ital W}({ital C}) in the two-dimensional Schwinger model on {ital R}{sup 1}{times}{ital S}{sup 1}, we show that nonleading terms depend on the shape of the contours both on {ital R}{sup 1}{times}{ital R}{sup 1} and {ital R}{sup 1}{times}{ital S}{sup 1}. We also find that the rhombic contour and triangular contour lead to the same static potentials both on {ital R}{sup 1}{times}{ital R}{sup 1} and {ital R}{sup 1}{times}{ital S}{sup 1}. The binding energy is also affected by the choice of contour shapes on the {ital R}{sup 1}{times}{ital S}{sup 1} model. This indicates that the model on {ital R}{sup 1}{times}{ital S}{sup 1} deserves more study.

A simple statistical system simulating dynamical ferminns in the strong coupling limit

It is shown, that in the strong coupling limit a system of dynamical fermions interacting via U(1)-gauge fields is equivalent to a system of vectors, which live on the odd sites and point on the even sites of a cubic lattice. Preliminary results for the chiral condensate and Wilson loop expectation values at β = 0 are presented. The latter appear as coefficients in a loop expansion of the fermionic determinant, which can be used to go beyond the quenched approximation.

The flux tube and the flux tube breaking amplitude in the harmonic approximation

In the flux tube model the amplitude for meson hadronic decay A to B+C is given by an overlap integral containing a factor gamma bca which represents the amplitude that the initial flux tube a should break at a specific point, forming the two flux tubes b and c of the final two mesons. The authors estimate the factor gamma bca in a harmonic approximation for the decays of both ordinary quark model mesons (a=flux tube ground state) and the lowest vibrational hybrids (a=first excited state of flux tube). They also discuss the relationship of the flux tube model to the leading term in the evaluation of the Wilson loop expectation value.

Comparison of an exact fermion Monte Carlo algorithm with the pseudofermion method using staggered fermions.

We present first results obtained from a simulation of lattice OCD with staggered fermions by a Monte Carlo algorithm that incorporates the effect of the fermion determinant exactly. The chiral condensate $\overline{\ensuremath{\chi}}\ensuremath{\chi}$ and various Wilson-loop expectation values are measured on a ${4}^{4}$ space-time lattice and the results are compared with the same quantities measured on the same-size lattice with the pseudofermion method. A quantitatively good agreement is found.

Dielectric lattice gauge theory

Dielectric lattice gauge theory models are introduced. They involve variables Φ(b) ∈ G that are attached to the links b = ( x + e μ , x) of the lattice and takee their values in the linear space G which consists of real linear combinations of matrices in the gauge group G. The polar decomposition Φ( b) = U( b) σ μ ( x) specifies an ordinary lattice gauge field U( b) and a kind of dielectric field ε ijασ iσ j ∗δ ij . A gauge invariant positive semidefinite kinetic term for the Φ-field is found, and it is shown how to incorporate Wilson fermions in a way which preserves Osterwalder-Schrader positivity. Theories with G = SU(2) and without matter fields are studied in some detail. It is proved that confinement holds, in the sense that Wilson-loop expectation values show an area law decay, if the euclidian action has certain qualitative features which imply that Φ=0 (i.e. dielectric field ) is the unique maximum of the action.

Monte Carlo calculation with unquenched Wilson fermions

A Monte Carlo updating procedure taking into account the virtual quark loops is described. It is based on high-order hopping parameter expansion of the quark determinant for Wilson fermions. In a first test run Wilson loop expectation values are measured on a 6 4 lattice at β = 5.70 using 16th order hopping parameter expansion for the quark determinant.

String tension in SU(3) gauge theory on a 16 4 lattice

Wilson loop expectation values are measured with improved statistics at β=5.40 and 5.70 by the Monte Carlo method. In this coupling constant region the (dimensionless) string tension drops more rapidly than required by the one-loop renormalization formula.

SU(3) gluon condensate from lattice mc data

The SU(3) gluon condensate is determined from a fit to the Creutz ratios of Wilson loop expectation values taken from new Monte Carlo data. The result agrees very well with the phenomenological value.

Glueball States in Reduced Large-NHamiltonians

It is shown that the reduced large-N/sub c/ Hamiltonian can reproduce not only the Wilson-loop expectation values but also the glueball spectrum of the infinite-volume lattice gauge theory. With the addition of fermions, the resulting quantum matrix model can potentially be used to calculate a large set of strong-interaction features.

Wilson loop dependence on the contour shape

By computing exactly the Wilson loop expectation value for two-dimensional quantum electrodynamics, we show that the nonleading terms depend drastically on the shape of the contour.

String tension in SU(3) lattice gauge theory

Wilson loop expectation values have been determined in SU(3) lattice gauge theory without fermions using Monte Carlo methods and considering lattices of up to 10 4 sites. A heat bath technique has been developed in order to enhance the statistical independence of successive lattice configurations.

Phase diagram of SO(2) lattice gauge theory in four dimensions

Monte-Carlo studies of SO(2) lattice gauge theory in four dimensions are presented. The results include Wilson loop expectation values for a 64 lattice, the string tension, plaquette-plaquette and loop-plaquette correlations. The possibility of measuring the glueball mass is also discussed.

Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory

A rigorous lower bound is obtained for the Wilson loop expectation value $〈A[C]〉$ in the four-dimensional U(1) lattice gauge theory with an action of the Villain form. The bound, which holds for ${g}^{2}l0.168$, has the form of a Coulomb interaction and guarantees that electric charges are not confined. Using the strong-coupling results of Osterwalder and Seiler, one concludes that this model has at least one phase transition. Using the work of Elitzur, Pearson, and Shigemitsu, one also concludes that the four-dimensional Villain $Z(N)$ lattice gauge theory has at least three phases if $Ng37$. Finally, it is also shown that the infinite-volume limit of $〈A[C]〉$ exists.

The area law as a consequence of the loop wave equation

An argument is presented showing that the Wilson loop expectation value decays according to the area law, if it is assumed to satisfy the free loop wave equation.