Wilson Line Operators Research Articles

LINE OPERATOR INDEX ON S1 × S3

We derive a general formula for an index for [Formula: see text] superconformal field theories. In the computation, 1-loop and monopole bubbling effects are taken into account. Our computation is compatible with S-dualities in the [Formula: see text] theories and the holographic principle which maps Wilson line operators in the [Formula: see text] SYMs to macroscopic objects in AdS5 × S5.

Chiral condensate in holographic QCD with baryon density

We consider the chiral condensate in the baryonic dense medium using the generalized Sakai-Sugimoto model. It is defined as the vacuum expectation value of open Wilson line that is proposed to be calculated by use of the area of world-sheet instanton. We evaluate it in confined as well as deconfined phase. In both phases, the chiral condensate has a minimum as a function of baryon density. In the deconfined phase, taking into account the chiral symmetry restoration, we classify the behavior of chiral condensate into three types. One can set the parameter of the theory such that the results, in low but sufficiently higher density, is in agreement with the expectation from QCD.

Line Operator Index on S 1 × S 3

We derive a general formula of an index for N = 2 superconformal field theories on S1 \times S3 with insertions of BPS Wilson line or 't Hooft line operator at the north pole and their anti-counterpart at the south pole of S3. One-loop and monopole bubbling effects are taken into account in the computation. As examples, we calculate the indices for N = 4 theories and N = 2 SU(2) theory with Nf = 4, and find good agreements between indices of line operators related by S-duality. The relation between Verlinde loop operators and the indices is explored. The holographic correspondence between the fundamental (anti-symmetric) Wilson line operator and the fundamental string (D5 brane) in AdS5\timesS5 is confirmed by the index comparison.

Jet physics from static charges in AdS space

Soft interactions with high-energy jets are explored in radial coordinates which exploit the approximately conformal behavior of perturbative gauge theories. In these coordinates, the jets, approximated by Wilson lines, become static charges in Euclidean AdS. The anomalous dimension of the corresponding Wilson line operator is then determined by the potential energy of the charges. To study these Wilson lines we introduce a ``conformal gauge'' which does not have kinetic mixing between radial and angular directions, and show that a number of properties of Wilson lines are reproduced through relatively simple calculations. For example, certain nonplanar graphs involving multiple Wilson lines automatically vanish. We also discuss the linear growth of the charges' imaginary potential energy with separation, and a relationship between Wilson line diagrams and Witten diagrams.

HIGH-ENERGY QCD FACTORIZATION FROM DIS TO pA COLLISIONS

The high-energy QCD factorization for Deep Inelastic Scattering and for proton-nucleus collisions using Wilson line formalism and factorization in rapidity is discussed. We show that in DIS the factorization in rapidity reduces to the k T -factorization when the 2-gluon approximation is applied, provided that the composite Wilson line operator is used in the high-energy Operator Product Expansion. We then show that the inclusive forward cross-section in proton-nucleus collisions factorizes in parton distribution functions, fragmentation functions and dipole gluon distribution function at one-loop level.

Symmetries of massless vertex operators in [formula omitted

The worldsheet sigma model of the superstring in AdS 5 × S 5 has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated with the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinite-dimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possible to define a BRST invariant open Wilson line. The central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group. In flat space, massless vertices transform covariantly only up to adding BRST-exact terms. But in AdS we show that it is possible to define vertices so that they transform exactly covariantly.

Factorization properties of soft graviton amplitudes

We apply recently developed path integral resummation methods to perturbative quantum gravity. In particular, we provide supporting evidence that eikonal graviton amplitudes factorize into hard and soft parts, and confirm a recent hypothesis that soft gravitons are modelled by vacuum expectation values of products of certain Wilson line operators, which differ for massless and massive particles. We also investigate terms which break this factorization, and find that they are subleading with respect to the eikonal amplitude. The results may help in understanding the connections between gravity and gauge theories in more detail, as well as in studying gravitational radiation beyond the eikonal approximation.

Eikonal methods applied to gravitational scattering amplitudes

We apply factorization and eikonal methods from gauge theories to scattering amplitudes in gravity. We hypothesize that these amplitudes factor into an IR-divergent soft function and an IR-finite hard function, with the former given by the expectation value of a product of gravitational Wilson line operators. Using this approach, we show that the IR-divergent part of the n-graviton scattering amplitude is given by the exponential of the one-loop IR divergence, as originally discovered by Weinberg, with no additional subleading IR-divergent contributions in dimensional regularization.

Supersymmetric open Wilson lines

In this paper we study Open Wilson Lines (OWL's) in the context of two Supersymmetric Yang Mills theories. First we consider four dimensional N=2 Supersymmetric Yang Mills Theory with hypermultiplets transforming in the fundamental representation of the gauge group, and find supersymmetric OWL's only in the superconformal versions of these theories. We then consider four dimensional N=4 SYM coupled to a three dimensional defect hypermultiplet. Here there is a semi-circular supersymmetric OWL, which is related to the ray by a conformal transformation. We perform a perturbative calculation of the operators in both theories, and discuss using localization to compute them non-perturbatively.

No generalized transverse momentum dependent factorization in the hadroproduction of high transverse momentum hadrons

It has by now been established that standard QCD factorization using transverse momentum dependent parton distribution functions fails in hadro-production of nearly back-to-back hadrons with high transverse momentum. The essential problem is that gauge invariant transverse momentum dependent parton distribution functions cannot be defined with process-independent Wilson line operators, thus implying a breakdown of universality. This has led naturally to proposals that a correct approach is to instead use a type of "generalized" transverse momentum dependent factorization in which the basic factorized structure is assumed to remain valid, but with transverse momentum dependent parton distribution functions that contain non-standard, process dependent Wilson line structures. In other words, to recover a factorization formula, it has become common to assume that it is sufficient to simply modify the Wilson lines in the parton correlation functions for each separate hadron. In this paper, we will illustrate by direct counter-example that this is not possible in a non-Abelian gauge theory. Since a proof of generalized transverse momentum dependent factorization should apply generally to any hard hadro-production process, a single counter-example suffices to show that a general proof does not exist. Therefore, to make the counter-argument clear and explicit, we illustrate with a specific calculation for a double spin asymmetry in a spectator model with a non-Abelian gauge field. The observed breakdown of generalized transverse momentum dependent factorization challenges the notion that the role of parton transverse momentum in such processes can be described using separate correlation functions for each external hadron.

SMALL-x EVOLUTION IN THE NEXT-TO-LEADING ORDER

After a brief introduction to Deep Inelastic Scattering in the Bjorken limit and in the Regge Limit we discuss the operator product expansion in terms of non local string operator and in terms of Wilson lines. We will show how the high-energy behavior of amplitudes in gauge theories can be reformulated in terms of the evolution of Wilson-line operators. In the leading order this evolution is governed by the non-linear Balitsky-Kovchegov (BK) equation. In order to see if this equation is relevant for existing or future deep inelastic scattering (DIS) accelerators (like Electron Ion Collider (EIC) or Large Hadron electron Collider (LHeC)) one needs to know the next-to-leading order (NLO) corrections. In addition, the NLO corrections define the scale of the running-coupling constant in the BK equation and therefore determine the magnitude of the leading-order cross sections. In Quantum Chromodynamics (QCD), the next-to-leading order BK equation has both conformal and non-conformal parts. The NLO kernel for the composite operators resolves in a sum of the conformal part and the running-coupling part. The QCD and [Formula: see text] kernel of the BK equation is presented.

NLO evolution of color dipoles in [formula omitted] SYM

High-energy behavior of amplitudes in a gauge theory can be reformulated in terms of the evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the conformally invariant BK equation for the evolution of color dipoles. In QCD, the next-to-leading order BK equation has both conformal and non-conformal parts, the latter providing the running of the coupling constant. To separate the conformally invariant effects from the running-coupling effects, we calculate the NLO evolution of the color dipoles in the conformal N = 4 SYM theory. We define the “composite dipole operators” with the rapidity cutoff preserving conformal invariance. The resulting Möbius-invariant kernel for these operators agrees with the forward NLO BFKL calculation of [A.V. Kotikov, L.N. Lipatov, Nucl. Phys. B 582 (2000) 19; A.V. Kotikov, L.N. Lipatov, Nucl. Phys. B 661 (2003) 19; A.V. Kotikov, L.N. Lipatov, Nucl. Phys. B 685 (2004) 405, Erratum].

Open Wilson lines and chiral condensates in thermal holographic QCD

We investigate various aspects of a proposal by Aharony and Kutasov [O. Aharony and D. Kutasov, Phys. Rev. D 78, 026005 (2008).] for the gravity dual of an open Wilson line in the Sakai-Sugimoto model or its noncompact version. In particular, we use their proposal to determine the effect of finite temperature, as well as background electric and magnetic fields, on the chiral symmetry breaking order parameter. We also generalize their prescription to more complicated worldsheets and identify the operators dual to such worldsheets.

Holographic duals of long open strings

We study the holographic map between long open strings, which stretch between D-branes separated in the bulk space-time, and operators in the dual boundary theory. We focus on a generalization of the Sakai-Sugimoto holographic model of QCD, where the simplest chiral condensate involves an operator of this type. Its expectation value is dominated by a semiclassical string world sheet, as for Wilson loops. We also discuss the deformation of the model by this operator, and, in particular, its effect on the meson spectrum. This deformation can be thought of as a generalization of a quark mass term to strong coupling. It leads to the first top-down holographic model of QCD with a non-Abelian chiral symmetry which is both spontaneously and explicitly broken, as in QCD. Other examples we study include half-supersymmetric open Wilson lines, and systems of D-branes ending on NS5-branes, which can be analyzed using world sheet methods.

Next-to-leading order evolution of color dipoles

The small-$x$ deep inelastic scattering in the saturation region is governed by the nonlinear evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the Balitsky-Kovchegov equation for the evolution of color dipoles. In the next-to-leading order the Balitsky-Kovchegov equation gets contributions from quark and gluon loops as well as from the tree gluon diagrams with quadratic and cubic nonlinearities. We calculate the gluon contribution to the small-$x$ evolution of Wilson lines (the quark part was obtained earlier).

BUBBLING GEOMETRIES FOR HALF-BPS WILSON LINES

We consider the supergravity backgrounds that correspond to supersymmetric Wilson line operators in the context of AdS/CFT correspondence. We study the gravitino and dilatino conditions of the IIB supergravity under the appropriate ansatz, and obtain some necessary conditions for a supergravity background that preserves the same symmetry as the supersymmetric Wilson lines. The supergravity solutions are characterized by continuous version of Maya diagrams. This diagram is related to the eigenvalue distribution of the Gaussian matrix model. We also consider the similar backgrounds of the 11-dimensional supergravity.

Nonperturbative Study of Non-CommutativeU(1) Gauge Theory

We study U(1) gauge theory on a 4d non-commutative torus, where two directions are non-commutative. Monte Carlo simulations are performed after mapping the regularized theory onto a U(N) lattice gauge theory in d = 2. At intermediate coupling strength, we find a phase in which open Wilson lines acquire non-zero vacuum expectation values, which implies the spontaneous breakdown of translational invariance. In this phase, various physical quantities obey clear scaling behaviors in the continuum limit with a fixed non-commutativity parameter θ, which provides evidence for a possible continuum theory. In the weak coupling symmetric phase, the dispersion relation involves a negative IR-singular term, which is responsible for the observed phase transition.

A non-perturbative study of 4d U(1) non-commutative gauge theory — the fate of one-loop instability

Recent perturbative studies show that in 4d non-commutative spaces, the trivial (classically stable) vacuum of gauge theories becomes unstable at the quantum level, unless one introduces sufficiently many fermionic degrees of freedom. This is due to a negative IR-singular term in the one-loop effective potential, which appears as a result of the UV/IR mixing. We study such a system non-perturbatively in the case of pure U(1) gauge theory in four dimensions, where two directions are non-commutative. Monte Carlo simulations are performed after mapping the regularized theory onto a U(N) lattice gauge theory in d=2. At intermediate coupling strength, we find a phase in which open Wilson lines acquire non-zero vacuum expectation values, which implies the spontaneous breakdown of translational invariance. In this phase, various physical quantities obey clear scaling behaviors in the continuum limit with a fixed non-commutativity parameter $\theta$, which provides evidence for a possible continuum theory. The extent of the dynamically generated space in the non-commutative directions becomes finite in the above limit, and its dependence on $\theta$ is evaluated explicitly. We also study the dispersion relation. In the weak coupling symmetric phase, it involves a negative IR-singular term, which is responsible for the observed phase transition. In the broken phase, it reveals the existence of the Nambu-Goldstone mode associated with the spontaneous symmetry breaking.

The functional form of open Wilson lines in gauge theories coupled to matter

The open Wilson lines are gauge-invariant operators made with a gauge transporter along an open path saturated at the end-points with matter fields. Here it is shown that numerical experiments on 3D Z2 Higgs model provide useful guidance in addressing the problem of the functional form of their vacuum expectation values. It turns out they satisfy, as long as their size does not exceed the string breaking scale, a remarkable factorization, related to finite-size scaling of the Fredenhagen-Marcu order parameter. This leads to conjecture a decay area law, like ordinary Wilson loops, with the difference that the boundary conditions of the confining string world-sheet are fixed along the path of the gauge transporter whereas are free on the straight line joining the two end-points. A consistency condition fixes uniquely the contribution of the free boundary, which turns out to be proportional to the string-breaking scale. Universal shape effects produced by quantum string fluctuations are also studied.

Graviton propagators on fuzzyG/H

We describe closed string modes by open Wilson lines in noncommutative (NC) gauge theories on compact fuzzy G/H in IIB matrix model. In this construction the world sheet cut-off is related to the spacetime cut-off since the string bit of the symmetric traced Wilson line carries the minimum momentum on G/H. We show that the two point correlation functions of graviton type Wilson lines in 4 dimensional NC gauge theories behave as 1/(momentum)^2. This result suggests that graviton is localized on D3-brane, so we can naturally interpret D3-branes as our universe. Our result is not limited to D3-brane system, and we generalize our analysis to other dimensions and even to any topology of D-brane worldvolume within fuzzy G/H.