We present an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their individual segments, reducing the number of diagrams needed to be calculated. The important step lies in the observation that different linear path topologies can be related to each other using their color structure. This framework allows one to easily switch results between different Wilson line topologies, which is helpful when testing different structures against each other.
Realistic heterotic string models require the presence of the Wilson lines. We study the effect of continuous Wilson lines on the heterotic Yukawa couplings and find that the presence of continuous Wilson lines affects the magnitude of the twisted Yukawa couplings resulting in their stronger hierarchy.
We point out a simple realization of gauge symmetry enhancement by Wilson lines in QFT with twisted compactification in extra dimensions. We illustrate this in the field contents taken from heterotic supergravity in arbitrary dimensions toroidally compactified.
We study the relationship between the family number of chiral fermions and the Wilson line phases, based on the orbifold family unification. We find that flavor numbers are independent of the Wilson line phases relating extra-dimensional components of gauge bosons, as far as the standard model gauge symmetry is respected. This feature originates from a hidden quantum-mechanical supersymmetry.
We compute the triple pomeron vertex from the Wilson line formalism, including both planar and non-planar contributions, and get perfect agreement with the result obtained in the Extended Generalized Logarithmic Approximation based on reggeon calculus.
We re-derive the R symmetries for the Z6-II orbifold with non-trivial Wilson lines and find expressions for the R charges which differ from those in the literature.
We study torus/orbifold models with magnetic flux and Wilson line background. The number of zero-modes and their profiles depend on those backgrounds. That has interesting implications from the viewpoint of particle phenomenology.
Wilson lines are key objects in many QCD calculations. They are parallel transporters of the gauge field that can be used to render non-local operator products gauge invariant, which is especially useful for calculations concerning validation of factorization schemes and in calculations for constructing or modelling parton density functions. We develop an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their Wilson segments, reducing the number of diagrams needed to be calculated. We show how different linear path topologies can be related using thei…
We investigate four-dimensional N=1 Type IIB orientifolds with continuous Wilson lines, and their T-dual realizations as orientifolds with moving branes. When continuous Wilson lines become discrete the gauge symmetry is enhanced and the T-dual orientifold corresponds to branes sitting at the orbifold fixed points. There is a field theoretic analog describing these phenomena as D- and F-flat deformations of the T-dual model, where the branes sit at the origin (original model without Wilson lines) as well as a deformation of the T-dual model where sets of branes sit at the fixed points (the mod…
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 st…
We present general formulae to compute Wilson line correlators with the Color Glass Condensate described by the McLerran-Venugopalan model. We explicitly construct a complete and non-orthogonal set of color-singlet bases and write matrix elements down, so that the exponential of the matrix leads to the Wilson line correlators. We further develop a systematic perturbative expansion of dipole Wilson line correlators in terms of $1/N_c$ where $N_c$ is the color number. As a phenomenological application we calculate the flow harmonics $v_n\{m\}$ in the dipole model and discuss the $N_c$ scaling.
We study the stabilization of an extra-dimensional radius in the presence of a Wilson line phase of an extra $U(1)$ gauge symmetry on a five-dimensional space-time, using the effective potential relating both the radion and the Wilson line phase at the one-loop level. We find that the radion can be stabilized by the introduction of a small number of fermions.
In the Randall-Sundrum compactification of AdS$_5$ with detuned brane tensions, supersymmetry can be spontaneously broken by a non-trivial Wilson line for the graviphoton. The supersymmetry breaking vanishes in the tuned limit. This effect is equivalent to supersymmetry breaking by Scherk-Schwarz boundary conditions.
We provide a recursive diagrammatic prescription for the exponentiation of gauge theory amplitudes involving products of Wilson lines and loops. This construction generalizes the concept of webs, originally developed for eikonal form factors and cross sections with two eikonal lines, to general soft functions in QCD and related gauge theories. Our coordinate space arguments apply to arbitrary paths for the lines.
Wilson lines, being comparators that render non-local operator products gauge invariant, are extensively used in QCD calculations, especially in small-$x$ calculations, calculations concerning validation of factorisation schemes and in calculations for constructing or modelling parton density functions. We develop an algorithm to express piecewise path ordered exponentials as path ordered integrals over the separate segments, and apply it on linear segments, reducing the number of diagrams needed to be calculated. We show how different linear path topologies can be related using their colour s…
We study the flavor structure of 4D effective theories, which are derived from extra dimensional theories with magnetic fluxes and non-Abelian Wilson lines. We study zero-mode wavefunctions and compute Yukawa couplings as well as four-point couplings. In our models, we also discuss non-Abelian discrete flavor symmetries such as $D_4$, $Δ(27)$ and $Δ(54)$.
We investigate the appearance of straight infinite Wilson lines lying on the self-dual plane in the context of the Self Dual sector of the Yang Mills theory and in a connection to the Lagrangian implementing the MHV vertices (MHV Lagrangian) according to the Cachazo-Svrcek-Witten method. It was already recognized in the past by two of the authors, that such Wilson line functional provides the field transformation of positive helicity fields between the Yang-Mills theory on the light-cone and the MHV Lagrangian. Here we discuss in detail the connection to the Self Dual sector and we provide a n…
We consider six-dimensional supergravity compactified on orbifolds with Wilson lines and bulk flux. Torus Wilson lines are decomposed into Wilson lines around the orbifold fixed points, and twisted boundary conditions of matter fields are related to fractional localized flux. Both, orbifold singularities and flux lead to chiral fermions in four dimensions. We show that in addition to the standard bulk and fixed point anomalies the Green-Schwarz term also cancels the four-dimensional anomaly induced by the flux background. The two axions contained in the antisymmetric tensor field both contribu…
It is shown that the anomaly inflow mechanism can be implemented using Wilson line in odd dimensional gauge theories. An action of Wess-Zumino-Witten (WZW) type can be constructed using Wilson line. The action is understood in the odd dimensional bulk space-time rather than in the even dimensional boundary. This action is not gauge invariant. It gives anomalous gauge variations of the consistent form on boundary space-times. So it can be used to cancel the quantum anomalies localized on boundary space-times. This offers a new way to cancel the gauge anomaly and construct anomaly-free gauge the…
We consider five dimensional theories compactified on the orbifold S^1/Z_2 and prove that spontaneous local supersymmetry breaking by Wilson lines and by the Scherk-Schwarz mechanism are equivalent. Wilson breaking is triggered by the SU(2)_R symmetry which is gauged in off-shell N=2 supergravity by auxiliary fields. The super-Higgs mechanism disposes of the would-be Goldstinos which are absorbed by the gravitinos to become massive. The breaking survives in the flat limit, where we decouple all gravitational interactions, and the theory becomes softly broken global supersymmetry.
We consider the description of form factors of local and Wilson line operators (reggeon amplitudes) in N=4 SYM within the framework of four dimensional ambitwistor string theory. We present the explicit expressions for string composite operators corresponding to stress-tensor operator supermultiplet and Wilson line operator insertion. It is shown, that corresponding tree-level string correlation functions correctly reproduce previously obtained Grassmannian integral representations. As by product we derive four dimensional tree-level scattering equations representations for the mentioned form …
We address the implementation of non-Abelian Wilson lines in D=4 N=1 Type IIB orientifold constructions. We present an explicit three-family example with the gauge group (U(4)xU(2)xSU(2)xSU(2))^2x(U(6)xSp(4))^2 and give the particle spectrum and the trilinear superpotential. Emphasizing the new subtleties associated with the introduction of non-Abelian Wilson lines, we show that the Abelian gauge anomalies are cancelled by the Green-Schwarz-type mechanism, and calculate the Fayet-Iliopoulos terms and gauge coupling corrections. The analysis thus sets a stage for further investigations of the p…
A way to break supersymmetry in perturbative superstring theory is the string version of the Scherk-Schwarz mechanism. There, the fermions and bosons have mass splitting due to different compactification boundary conditions. We consider the implementation of this mechanism in abelian orbifold compactifications with Wilson line backgrounds. For $Z_N$ and $Z_N\times Z_M$ orbifolds, we give the possible $U(1)$ R-symmetries which determine the mass splitting, and thus, the supersymmetry breaking at the perturbative level. The phenomenlogical viability of this mechanism implies some dimension(s) to…
We discuss the divergence structure of Wilson line operators with partially overlapping segments on the basis of the cyclic Wilson loop as an explicit example. The generalized exponentiation theorem is used to show the exponentiation and factorization of power divergences for certain linear combinations of associated loop functions.
We investigate various aspects of a proposal by Aharony and Kutasov arXiv:0803.3547 [hep-th] for the gravity dual of an open Wilson line in the Sakai-Sugimoto model or its non-compact 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.
We study the structure of soft gluon corrections to multi-leg scattering amplitudes in a non-Abelian gauge theory by analysing the corresponding product of semi-infinite Wilson lines. We prove that diagrams exponentiate such that the colour factors in the exponent are fully connected. This completes the generalisation of the non-Abelian exponentiation theorem, previously proven in the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary representations of the colour group. Our proof is based on the replica trick in conjunction with a new formalism where multiple emissions f…
We examine the high energy (Regge) limit of gravitational scattering using a Wilson line approach previously used in the context of non-Abelian gauge theories. Our aim is to clarify the nature of the Reggeization of the graviton and the interplay between this Reggeization and the so-called eikonal phase which determines the spectrum of gravitational bound states. Furthermore, we discuss finite corrections to this picture. Our results are of relevance to various supergravity theories, and also help to clarify the relationship between gauge and gravity theories.
We develop techniques to construct general discrete Wilson lines in four-dimensional N=1 Type IIB orientifolds, their T-dual realization corresponds to branes positioned at the orbifold fixed points. The explicit order two and three Wilson lines along with their tadpole consistency conditions are given for D=4 N=1 Z_6 Type IIB orientifold. The systematic search for all models with general order three Wilson lines leads to a small class of inequivalent models. There are only two inequivalent classes of a potentially phenomenologically interesting model that has a possible SU(3)_{color} x SU(2)_…
Unlike the Wilson line in QED the Wilson line in QCD contains path ordering. In this paper we get rid of the path ordering in the light-like Wilson line in QCD by simplifying all the infinite number of non-commuting terms in the SU(3) pure gauge. We prove that the light-like Wilson line in QCD naturally emerges when path integral formulation of QCD is used to prove factorization of soft and collinear divergences at all order in coupling constant in QCD processes at high energy colliders.
We develop a new classical action that in addition to $\mathrm{MHV}$ vertices contains also $\mathrm{N^kMHV}$ vertices, where $1\leq k \leq n-4$ with $n$ the number of external legs. The lowest order vertex is the four-point MHV vertex -- there is no three point vertex and thus the amplitude calculation involves fewer vertices than in the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action. The action is obtained by a canonical transformation of the Yang-Mills action in the light-cone gauge, where the field transformations are based on the Wilson line functiona…
Boundary operators are gauge invariant operators whose form factors correspond to boundary contributions of BCFW shifts. In gauge theory, the boundary operators contain infinite series, which are constrained by gauge symmetry. We compute the boundary operators of all possible BCFW shifts in Yang-Mills theory and QCD, and show that the infinite series can be elegantly organized into Wilson lines, which are natural building blocks for non-local gauge invariant operators. We comment on their connection to jet functions and gauge invariant off-shell amplitudes. We also verify our results by studyi…
The dynamics of Wilson lines integrated along a warped extra dimension has been unknown. We study a five dimensional SU(N) pure gauge theory with Randall-Sundrum warped compactification on S^1/Z_2. We clarify the notion of large gauge transformations that are non-periodic on the covering space for this setup. We obtain Kaluza-Klein expansions of gauge and ghost fields for the most general twists and background gauge field configurations, which break the gauge symmetry at classical level in general. We calculate the one-loop effective potential and find that the symmetry corresponding to the su…
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.
We propose a novel inflationary scenario in string theory in which the inflaton field is a 'Wilson line' degree of freedom in the worldvolume of a probe Dp-brane, in a warped flux compactification. Kinetic terms for Wilson line fields on the world volume of a D-brane take a nonstandard Dirac-Born-Infeld (DBI) form. Thus, we work in the framework of DBI inflation. This extends the original slow roll Wilson line inflationary scenario, where only the quadratic piece was considered. Warped DBI Wilson line inflation offers an attractive alternative to ordinary (position field) DBI inflation, inasmu…
By using path integral formulation of QCD and QED we prove that the factorization theorem is valid for light-like Wilson line but is not valid for non-light-like Wilson line. This conclusion is shown to be consistent with Ward identity and Grammer-Yennie approximation. Hence we find that the factorization theorem is violated in all the previous studies which used the non-light-like Wilson line in the definition of the (transverse momentum dependent) parton distribution function and fragmentation function at high energy colliders.
Effective theories for the thermal Wilson line are constructed in an SU(N) gauge theory at nonzero temperature. I propose that the order of the deconfining phase transition for Z(N) Wilson lines is governed by the behavior of SU(N) Wilson lines. In a mean field theory, the free energy in the deconfined phase is controlled by the condensate for Z(N) Wilson lines. Numerical simulations on the lattice, and the mean field theory for Z(3) Wilson lines, suggest that about any finite temperature transition in QCD, the dominant correlation length increases by a large, uniform factor, of order five.
Orbifold compactifications with continuous Wilson lines have very interesting characteristics and as a consequence they are candidates to obtain realistic models. We perform an analysis of the soft supersymmetry-breaking terms arising in this type of compactifications. We also compare these results with those of orbifolds without including continuous Wilson lines. Their phenomenological properties turn out to be similar.
We study hard $1\to 2$ final-state parton splittings in the medium, and put special emphasis on calculating the Wilson line correlators that appear in these calculations. As partons go through the medium their color continuously rotates, an effect that is encapsulated in a Wilson line along their trajectory. When calculating observables, one typically has to calculate traces of two or more medium-averaged Wilson lines. These are usually dealt with in the literature by invoking the large-$N_c$ limit, but exact calculations have been lacking in many cases. In our work, we show how correlators of…
To study the deconfining phase transition at nonzero temperature, I outline the perturbative construction of an effective theory for straight, thermal Wilson lines. Certain large, time dependent gauge transformations play a central role. They imply the existence of interfaces, which can be used to determine the form of the effective theory as a gauged, nonlinear sigma model of adjoint matrices. Especially near the transition, the Wilson line may undergo a Higgs effect. As an adjoint field, this can generate eigenvalue repulsion in the effective theory.
At high energies the particles move very fast so their trajectories can be approximated by straight lines collinear to their velocities. The proper degrees of freedom for the fast gluons moving along the straight lines are the Wilson-line operators -- infinite gauge factors ordered along the straight line. I review the study of the high-energy scattering in terms of Wilson-line degrees of freedom.
The Makeenko-Migdal loop equation is non-linear and first order in the area derivative, but we show that for simple loops in QCD$_2$ it is possible to reformulate this equation as a linear equation with second order derivatives. This equation is a bound state Schrödinger equation with a three dimensional Coulomb potential. Thus, loop dynamics leads to a surprising new picture of confinement, where this phenomenon is due to a (bound state) localization in loop space, with the Wilson loops decaying exponentially outside a characteristic radius.
We study a novel non-Abelian matrix configuration of probe D-branes in AdS5. This configuration gives rise to a new D-brane phenomenon related to the known "Myers effect" in the context of holography. It is dual to a deformation of the field theory by a Wilson line threaded fermion bilinear. We study the two point function of these short Wilson lines from both the non-Abelian DBI action and a classical string world sheet calculation and identify the region where they agree. We also study a related configuration where the non-Abelian nature of the embedding functions is enhanced by a background…
The effects of the soft gluon emission in hard scattering processes at the phase boundary are resummed in the soft-collinear effective theory (SCET). In SCET, the soft gluon emission is decoupled from the energetic collinear part, and is obtained by the vacuum expectation value of the soft Wilson-line operator. The form of the soft Wilson lines is universal in deep inelastic scattering, in the Drell-Yan process, in the jet production from e+e- collisions, and in the gamma* gamma* -> pi0 process, but its analytic structure is slightly different in each process. The anomalous dimensions of the s…
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 rela…
It is well known that high-energy scattering of a meson from some hadronic target can be described by the interaction of that target with a color dipole formed by two Wilson lines corresponding to fast quark-antiquark pair. Moreover, the energy dependence of the scattering amplitude is governed by the evolution equation of this color dipole with respect to rapidity. Similarly, the energy dependence of scattering of a baryon can be described in terms of evolution of a three-Wilson-lines operator with respect to the rapidity of the Wilson lines. We calculate the evolution of the 3-quark Wilson l…
The contribution of quarks to the effective potential for the phase of the Wilson line is computed at nonzero temperature and quark chemical potential to one and two loop order. At zero temperature, regardless of the value of the quark chemical potential, the effective potential for the phase of the Wilson line vanishes. At nonzero temperature, for special values of the phase the free energy of quarks equals that of bosons; at nonzero chemical potential, such quarks can ``Bose condense'', albeit with negative density.
We present an explicit realization of the Chen et al. approach to the proton spin decomposition in terms of Wilson lines, generalizing the light-front gauge-invariant extensions discussed recently by Hatta. Particular attention is drawn to the residual gauge freedom by further separating the pure-gauge term into contour and residual terms. We show that the kinetic orbital angular momentum operator can be expressed in terms of the Wigner operator only when the momentum variable is integrated over. Finally, we confirm from twist-2 arguments that the advanced, retarded and antisymmetric light-fro…
We write down explicit expressions for the $x$-evolution (equivalent to energy or rapidity evolution) of $2 n$ ($n = 1, 2, ...$) Wilson lines using the JIMWLK equation and the Color Glass Condensate formalism. We investigate the equation in the weak gluon field limit (linear regime) by expanding the Wilson lines in powers of the gluon field and show that it reduces to the BJKP equation describing the evolution of a state of $2 n$ Reggeized gluons with energy. We also make available for download a {\it Mathematica} program which provides this expression for any value of $n$.
We study the dynamics of Wilson lines near the light cone in QCD. Lattice simulations of the near light cone Hamiltonian in SU(2) show that the correlation mass at strong coupling vanishes at intermediate coupling, which signals a continuum transition on the light cone. We point out the possible relevance of this mass for high energy scattering.
We estimate the effect of threshold corrections coming from Wilson lines to gauge coupling unification in the weakly coupled heterotic string with orbifold compactification. By expressing the corrections in terms of an effective field theory calculation we are able to estimate the minimal threshold corrections in a realistic model without constructing the full string theory. Using this we show that the effect of the gauge boson Kaluza Klein excitations is systematically to reduce the string prediction for the unification scale. In the case of discrete Wilson lines the effect of the Wilson line…
We discuss the relation between the infrared singularities of on-shell partonic form factors and parton distribution functions (PDFs) near the elastic limit, through their factorisation in terms of Wilson-line correlators. Ultimately we identify the difference between the anomalous dimensions controlling single poles of these two quantities to all loops in terms of the closed parallelogram Wilson loop. To arrive at this result we first use the common hard-collinear behaviour of the two to derive a relation between their respective soft singularities, and then show that the latter is manifested…
We calculate the time independent four-point function in high temperature (T) QCD and obtain the leading momentum dependent terms. Furthermore, we relate these derivative interactions to derivative terms in a recently proposed finite T effective action based on the SU(3) Wilson Line and its trace, the Polyakov Loop. By this procedure we thus obtain a perturbative matching at finite T between QCD and the effective model. In particular, we calculate the leading perturbative QCD-correction to the kinetic term for the Polyakov Loop.
In a gauge theory at nonzero temperature the eigenvalues of the Wilson line form a set of gauge invariant observables. By constructing the corresponding partition function for the phases of these eigenvalues, we prove that the trivial vacuum, where the phases vanish, is a minimum of the free energy.
We analyze the off-shell scattering amplitudes in the framework of the light-front perturbation theory. It is shown that the previously derived recursion relation between tree level off-shell amplitudes in this formalism actually resums whole classes of graphs into a Wilson line. More precisely, we establish a correspondence between the light-front methods for the computation of the off-shell amplitudes and the approach which makes use of the matrix elements of straight infinite Wilson lines, which are manifestly gauge invariant objects. Furthermore, since it is needed to explicitly verify the…
It is shown that, in QCD, the same universal function $Γ_{cusp}(\vartheta, α_\s)$ determines the infrared behaviour of the on-shell quark form factor, the velocity-dependent anomalous dimension in the heavy quark effective field theory (HQET) and the renormalization properties of the vacuum averaged Wilson lines with a cusp. It is demonstrated that a combined use of the methods developed in the relevant different branches of quantum field theory essentially facilitates the all-order study of the asymptotic and analytic properties of this function.
We perform simulations of an effective theory of SU(2) Wilson lines in three dimensions. We include a non-perturbative "fuzzy-bag" contribution which is added to the one-loop perturbative potential for the Wilson line. We confirm that, at moderately weak coupling, this leads to eigenvalue repulsion in a finite region above the deconfining phase transition which shrinks in the extreme weak-coupling limit. A non-trivial Z(N) symmetric vacuum arises in the confined phase.
We argue that soft-collinear effective theory, as is currently formulated, lacks an ingredient which is essential for obtaining a gauge invariant definitions of certain non-perturbative matrix elements relevant for semi-inclusive high-energy processes. Those processes and the companion cross-sections are to be analyzed and measured at the Large Hadron Collider. The missing feature is a gauge link constructed only from the transverse component of the gluon field denoted by the "$T$-Wilson line"'. It is neither soft nor collinear. As such it is a new feature of the theory.
I start with an elementary observation about the pressure in the deconfined phase of a SU(3) gauge theory without quarks. This suggests a ``fuzzy'' bag model for the analogous pressure in QCD, with dynamical quarks. I then sketch how the deconfined phase might be described using an effective theory of Wilson lines. To leading order in weak coupling, the effective electric field appears in a form familiar from the lattice theory of Banks and Ukawa.
This is the second paper on hadronic wave functions in the light front formulation. We first consider only confinement effects, using the classical Nambu-Gotto string with massive end points in the light cone gauge. We derive the light-front Hamiltonian and show how to solve it, using an expansion in suitable basis functions. We next discuss the correlators of Wilson lines on the light front, leading to an effective Hamiltonian that includes spin effects. At the end, we consider a separate problem of instanton-induced interactions, at the origin of the pion as a massless Goldstone mode on the …
We compute the effective potential for the phase of the Wilson line at high temperature in the standard model to one loop order. Besides the trivial vacua, there are metastable states in the direction of $U(1)$ hypercharge. Assuming that the universe starts out in such a metastable state at the Planck scale, it easily persists to the time of the electroweak phase transition, which then proceeds by an unusual mechanism. All remnants of the metastable state evaporate about the time of the $QCD$ phase transition.
