Wilson Line Operators Research Articles

Correlators in higher-spin AdS3 holography from Wilson lines with loop corrections

We study the correlators of the 2d W_N minimal model in the semiclassical regime with large central charge from bulk viewpoint by utilizing open Wilson lines in sl(N) Chern-Simons gauge theory. We extend previous works for the tree level of bulk theory to incorporate loop corrections in this paper. We offer a way to regularize divergences associated with loop diagrams such that three point functions with two scalars and a higher spin current agree with the values fixed by the boundary W_N symmetry. With the prescription, we reproduce the conformal weight of the operator corresponding to a bulk scalar up to the two loop order for explicit examples with N=2,3.

Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields

We first prove that, in Vasiliev’s theory, the zero-form charges studied in Sezgin E and Sundell P 2011 (arXiv:1103.2360 [hep-th]) and Colombo N and Sundell P 20 (arXiv:1208.3880 [hep-th]) are twisted open Wilson lines in the noncommutative Z space. This is shown by mapping Vasiliev’s higher-spin model on noncommutative Yang–Mills theory. We then prove that, prior to Bose-symmetrising, the cyclically-symmetric higher-spin invariants given by the leading order of these n-point zero-form charges are equal to corresponding cyclically-invariant building blocks of n-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On the higher spin gravity side, our computation reproduces the results of Didenko V and Skvortsov E 2013 J. High Energy Phys. JHEP04(2013)158 using an alternative method amenable to the computation of subleading corrections obtained by perturbation theory in normal order. On the free CFT side, our proof involves the explicit computation of the separate cyclic building blocks of the correlation functions of n conserved currents in arbitrary dimension using polarization vectors, which is an original result. It is shown to agree, for , with the results obtained in Gelfond O A and Vasiliev M A 2013 Nucl. Phys. B 876 871–917 in various dimensions and where polarization spinors were used.

Grassmannian integral for general gauge invariant off-shell amplitudes in mathcal{N}=4 SYM

In this paper we consider tree-level gauge invariant off-shell amplitudes (Wilson line form factors) in mathcal{N}=4 SYM with arbitrary number of off-shell gluons or equivalently Wilson line operator insertions. We make a conjecture for the Grassmannian integral representation for such objects and verify our conjecture on several examples. It is remarkable that in our formulation one can consider situation when on-shell particles are not present at all, i.e. we have Grassmannian integral representation for purely off-shell object. In addition we show that off-shell amplitude with arbitrary number of off-shell gluons could be also obtained using quantum inverse scattering method for auxiliary mathfrak{g}mathfrak{l}left(4Big|4right) super spin chain.

An index for ray operators in 5d En SCFTs

We construct an index for BPS operators supported on a ray in five dimensional superconformal field theories with exceptional global symmetries. We compute the En representations (for n = 2, . . . , 7) of operators of low spin, thus verifying that while the expression for the index is only SO(2n − 2) × U(1) invariant, the index itself exhibits the full En symmetry (at least up to the order we expanded). The ray operators we studied in 5d can be viewed as generalizations of operators constructed in a Yang-Mills theory with fundamental matter by attaching an open Wilson line to a quark. For n ≤ 7, in contrast to local operators, they carry nontrivial charge under the ℤ9 − n ⊂ En center of the global symmetry. The representations that appear in the ray operator index are therefore different, for n ≤ 7, from those appearing in the previously computed superconformal index. For 3 ≤ n ≤ 7, we find that the leading term in the index is a character of a minuscule representation of En. We also discuss the case n = 8, which presents a unique technical challenge, and remains an open problem.

Improved quasi parton distribution through Wilson line renormalization

Recent developments showed that hadron light-cone parton distributions could be directly extracted from spacelike correlators, known as quasi parton distributions, in the large hadron momentum limit. Unlike the normal light-cone parton distribution, a quasi parton distribution contains ultraviolet (UV) power divergence associated with the Wilson line self energy. We show that to all orders in the coupling expansion, the power divergence can be removed by a “mass” counterterm in the auxiliary z-field formalism, in the same way as the renormalization of power divergence for an open Wilson line. After adding this counterterm, the quasi quark distribution is improved such that it contains at most logarithmic divergences. Based on a simple version of discretized gauge action, we present the one-loop matching kernel between the improved non-singlet quasi quark distribution with a lattice regulator and the corresponding quark distribution in dimensional regularization.

Energy loss at NLO in a high-temperature Quark-Gluon Plasma

We present an extension of the Arnold-Moore-Yaffe kinetic equations for jet energy loss to NLO in the strong coupling constant. A novel aspect of the NLO analysis is a consistent description of wider-angle bremsstrahlung (semi-collinear emissions), which smoothly interpolates between 2↔2 scattering and collinear bremsstrahlung. We describe how many of the ingredients of the NLO transport equations (such as the drag coefficient) can be expressed in terms of Wilson line operators and can be computed using a Euclidean formalism or sum rules, both motivated by the analytic properties of amplitudes at light-like separations. We conclude with an outlook on the computation of the shear viscosity at NLO.

Collinearly improved JIMWLK evolution in Langevin form

The high-energy evolution of Wilson line operators, which at leading order is described by the Balitsky-JIMWLK equations, receives large radiative corrections enhanced by single and double collinear logarithms at next-to-leading order and beyond. We propose a method for resumming such logarithmic corrections to all orders, at the level of the Langevin formulation of the JIMWLK equation. The ensuing, collinearly-improved Langevin equation features generalized Wilson line operators, which depend not only upon rapidity (the logarithm of the longitudinal momentum), but also upon the transverse size of the color neutral projectile to which the Wilson lines belong. This additional scale dependence is built up during the evolution, via the condition that the successive emissions of soft gluons be ordered in time. The presence of this transverse scale in the Langevin equation furthermore allows for the resummation of the one-loop running coupling corrections.

Off-shell amplitudes as boundary integrals of analytically continued Wilson line slope

One of the methods to calculate tree-level multi-gluon scattering amplitudes is to use the Berends-Giele recursion relation involving off-shell currents or off-shell amplitudes, if working in the light cone gauge. As shown in recent works using the light-front perturbation theory, solutions to these recursions naturally collapse into gauge invariant and gauge-dependent components, at least for some helicity configurations. In this work, we show that such structure is helicity independent and emerges from analytic properties of matrix elements of Wilson line operators, where the slope of the straight gauge path is shifted in a certain complex direction. This is similar to the procedure leading to the Britto-Cachazo-Feng-Witten (BCFW) recursion, however we apply a complex shift to the Wilson line slope instead of the external momenta. While in the original BCFW procedure the boundary integrals over the complex shift vanish for certain deformations, here they are non-zero and are equal to the off-shell amplitudes. The main result can thus be summarized as follows: we derive a decomposition of a helicity-fixed off-shell current into gauge invariant component given by a matrix element of a straight Wilson line plus a reminder given by a sum of products of gauge invariant and gauge dependent quantities. We give several examples realizing this relation, including the five-point next-to-MHV helicity configuration.

Jet-medium interactions at NLO in a weakly-coupled quark-gluon plasma

We present an extension to next-to-leading order in the strong coupling constant $g$ of the AMY effective kinetic approach to the energy loss of high momentum particles in the quark-gluon plasma. At leading order, the transport of jet-like particles is determined by elastic scattering with the thermal constituents, and by inelastic collinear splittings induced by the medium. We reorganize this description into collinear splittings, high-momentum-transfer scatterings, drag and diffusion, and particle conversions (momentum-preserving identity-changing processes). We show that this reorganized description remains valid to NLO in $g$, and compute the appropriate modifications of the drag, diffusion, particle conversion, and inelastic splitting coefficients. In addition, a new kinematic regime opens at NLO for wider-angle collinear bremsstrahlung. These semi-collinear emissions smoothly interpolate between the leading order high-momentum-transfer scatterings and collinear splittings. To organize the calculation, we introduce a set of Wilson line operators on the light-cone which determine the diffusion and identity changing coefficients, and we show how to evaluate these operators at NLO.

Diffeomorphism-invariant observables and their nonlocal algebra

Gauge-invariant observables for quantum gravity are described, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics. These can be thought of as operators that create a particle, together with its inseparable gravitational field, and reduce to usual field operators of quantum field theory in the weak-gravity limit; they include both Wilson-line operators, and those creating a Coulombic field configuration. We also describe operators creating the field of a particle in motion; as in the electromagnetic case, these are expected to help address infrared problems. An important characteristic of the quantum theory of gravity is the algebra of its observables. We show that the commutators of the simple observables of this paper are nonlocal, with nonlocality becoming significant in strong field regions, as predicted previously on general grounds.

Cusped Wilson lines in symmetric representations

We study the cusped Wilson line operators and Bremsstrahlung functions associated to particles transforming in the rank-$k$ symmetric representation of the gauge group $U(N)$ for ${\cal N} = 4$ super Yang-Mills. We find the holographic D3-brane description for Wilson loops with internal cusps in two different limits: small cusp angle and $k\sqrt{\lambda}\gg N$. This allows for a non-trivial check of a conjectured relation between the Bremsstrahlung function and the expectation value of the 1/2 BPS circular loop in the case of a representation other than the fundamental. Moreover, we observe that in the limit of $k\gg N$, the cusped Wilson line expectation value is simply given by the exponential of the 1-loop diagram. Using group theory arguments, this eikonal exponentiation is conjectured to take place for all Wilson loop operators in symmetric representations with large $k$, independently of the contour on which they are supported.

OPE of the stress tensors and surface operators

Abstract We demonstrate that the divergent terms in the OPE of a stress tensor and a line (co-dimension two) operator of general shape in three dimensional spacetime cannot be constructed only from local geometric data depending only on the shape of the line. We verify this holographically for Wilson line operators or equivalently the twist operator corresponding to computing the entanglement entropy using the Ryu-Takayanagi formula. We discuss possible implications of this result.

Solution of the NLO BFKL Equation and γ*γ* Cross-Section at NLO

The analytic NLO cross section of the γ*γ* scattering process at high-energy is calculated. The result is obtained factorizing the scattering amplitude in rapidity space using the high-energy Operator Product Expansion in terms of composite Wilson line operators. The next-to-leading logarithmic resummation is included employing the solution of the NLO BFKL equation which obtained from the linearization of the evolution in rapidity space of the composite Wilson line operators.

Higher order topological invariants from the Chern-Simons action

It is well known that for a field theory with Chern-Simons action, expectation values of Wilson line operators are topological invariants. The standard result is expressed in terms of the Gaussian linkings of closed curves defining the operators. We show how a judicious choice of Wilson lines leads to higher order topological linkings.

The γ * γ * total cross section in next-to-leading order BFKL and LEP2 data

We study the total cross section for the collision of two highly-virtual photons at large energies, taking into account the BFKL resummation of energy logarithms with full next-to-leading accuracy. A necessary ingredient of the calculation, the next-to-leading order impact factor for the photon to photon transition, has been calculated by Balitsky and Chirilli using an approach based on the operator expansion in Wilson lines. We extracted the result for the photon impact factor in the original BFKL calculation scheme comparing the expression for the photon-photon total cross section obtained in BFKL with the one recently derived by Chirilli and Kovchegov in the Wilson-line operator expansion scheme. We perform a detailed numerical analysis, combining different, but equivalent in next-to-leading accuracy, representations of the cross section with various optimization methods of the perturbative series. We compare our results with previous determinations in the literature and with the LEP2 experimental data. We find that the account of Balitsky and Chirilli expression for the photon impact factor reduces the BFKL contribution to the cross section to very small values, making it impossible to describe LEP2 data as the sum of BFKL and leading-order QED quark box contributions.

Rapidity evolution of Wilson lines at the next-to-leading order: Balitsky–JIMWLK equation at NLO

Abstract Scattering amplitudes of proton–Nucleus or Nucleus–Nucleus collisions at high energy are described by matrix elements of Wilson line operators — infinite gauge factors ordered along the straight line trajectories of the fast moving particles. The energy dependence of such amplitudes is described by the evolution equation of Wilson lines with respect to the rapidity parameter — the Balitsky–JIMWLK evolution equation. Most of the current phenomenology of high-energy and high-density QCD is based on the leading-order evolution equation with only running coupling corrections. I will present the derivation of the Balitsky–JIMWLK evolution equation at the next-to-leading order.

Wilson lines and gauge invariant off-shell amplitudes

We study matrix elements of Fourier-transformed straight infinite Wilson lines as a way to calculate gauge invariant tree-level amplitudes with off-shell gluons. The off-shell gluons are assigned “polarization vectors” which (in the Feynman gauge) are transverse to their off-shell momenta and define the direction of the corresponding Wilson line operators. The infinite Wilson lines are first regularized to prove the correctness of the method. We have implemented the method in a computer FORM program that can calculate gluonic matrix elements of Wilson line operators automatically. In addition we formulate the Feynman rules that are convenient in certain applications, e.g. proving the Ward identities. Using both the program and the Feynman rules we calculate a few examples, in particular the matrix elements corresponding to gauge invariant g * g * g * g and g * g * g * g * g processes. An immediate application of the approach is in the high energy scattering, as in a special kinematic setup our results reduce to the form directly related to Lipatov’s vertices. Thus the results we present can be directly transformed into Lipatov’s vertices, in particular into RRRP and RRRRP vertices with arbitrary “orientation” of reggeized gluons. Since the formulation itself is not restricted to high-energy scattering, we also apply the method to a decomposition of an ordinary on-shell amplitude into a set of gauge invariant objects.

Balitsky-JIMWLK evolution equation at NLO

Wilson line operators are infinite gauge factors ordered along the straight lines of the fast moving particles. Scattering amplitudes of proton-Nucleus or Nucleus-Nucleus collisions at high-energy are written in terms of matrix elements of these operators and the energy dependence of such amplitudes is obtained by the evolution equation with respect to the rapidity parameter: the Balitsky-JIMWLK evolution equation. A brief description of the derivation of the Balitsky-JIMWLK evolution equation at leading order and next- to-leading order will be presented.

Webs and posets

The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.

Rapidity evolution of Wilson lines at the next-to-leading order

At high energies particles move very fast so the proper degrees of freedom for the fast gluons moving along the straight lines are Wilson-line operators - infinite gauge factors ordered along the line. In the framework of operator expansion in Wilson lines the energy dependence of the amplitudes is determined by the rapidity evolution of Wilson lines. We present the next-to-leading order hierarchy of the evolution equations for Wilson-line operators.