Light-like Wilson Loops Research Articles

Equivalence of Wilson loops inN=6super Chern-Simons matter theory andN=4SYM theory

In previous investigations, it was found that four-sided polygonal lightlike Wilson loops in $\mathcal{N}=6$ super Chern-Simons matter (ABJM) theory calculated to two-loop order have the same form as the corresponding Wilson loop in $\mathcal{N}=4$ SYM at one-loop order. Here we study lightlike polygonal Wilson loops with $n$ cusps in planar three-dimensional Chern-Simons and ABJM theory to two loops. Remarkably, the result in ABJM theory precisely agrees with the corresponding Wilson loop in $\mathcal{N}=4$ SYM at one-loop order for arbitrary $n$. In particular, anomalous conformal Ward identities allow for a so-called remainder function of conformal cross ratios for $n\ensuremath{\ge}6$, which is found to be trivial at two loops in ABJM theory in the same way as it is trivial in $\mathcal{N}=4$ SYM at one-loop order. Furthermore, the result for arbitrary $n$ obtained here, allows for a further investigation of a Wilson loop/amplitude duality in ABJM theory, for which nontrivial evidence was recently found by a calculation of four-point amplitudes that match the Wilson loop in ABJM theory.

Yangian symmetry of light-like Wilson loops

We show that a certain class of light-like Wilson loops exhibits a Yangian symmetry at one loop, or equivalently, in an Abelian theory. The Wilson loops we discuss are equivalent to one-loop MHV amplitudes in N=4 super Yang-Mills theory in a certain kinematical regime. The fact that we find a Yangian symmetry constraining their functional form can be thought of as the effect of the original conformal symmetry associated to the scattering amplitudes in the N=4 theory.

Gluon scattering amplitudes in finite temperature gauge/gravity dualities

We examine the gluon scattering amplitude in \( \mathcal{N} = 4 \) super Yang-Mills at finite temperature with nonzero R-charge densities, and in Non-Commutative gauge theory at finite temperature. The gluon scattering amplitude is defined as a light-like Wilson loop which lives at the horizon of the T-dual black holes of the backgrounds we consider. We study in detail a special amplitude, which corresponds to forward scattering of a low energy gluon off a high energy one. For this kinematic configuration in the considered backgrounds, we find the corresponding minimal surface which is directly related to the gluon scattering amplitude. We find that for increasing the chemical potential or the non-commutative parameter, the on-shell action corresponding to our Wilson loop in the T-dual space decreases. For all of our solutions the length of the short side of the Wilson loop is constrained by an upper bound which depends on the temperature, the R-charge density and the non-commutative parameter. Due to this constraint, in the limit of zeroth temperature our approach breaks down since the upper bound goes to zero, while by keeping the temperature finite and letting the chemical potential or the non-commutative parameter to approach to zero the limit is smooth.

Wilson loops in [formula omitted]-dimensional Yang–Mills theories using gravity/gauge theory correspondence

We compute the expectation values of both the time-like and the light-like Wilson loops in a strongly coupled plasma of ( p + 1 ) -dimensional Yang–Mills theories using gravity/gauge theory correspondence. From the time-like Wilson loop we obtain the velocity dependent quark–antiquark potential where the dipole is moving through the plasma with an arbitrary velocity 0 < v < 1 and also obtain expressions for the screening lengths. When the velocity v → 1 , the Wilson loop becomes light-like and we obtain the form of the jet quenching parameter in those strongly coupled plasma.

Mandelstam cuts and lightlike Wilson loops inN=4supersymmetry

We perform an analytic continuation of the two-loop remainder function for the six-point planar MHV amplitude in N=4 SUSY, found by Goncharov, Spradlin, Vergu and Volovich from the light-like Wilson loop representation. The remainder function is continued into a physical region, where all but two energy invariants are negative. It turns out to be pure imaginary in the multi-Regge kinematics, which is in an agreement with the predictions based on the Steinmann relations for the Regge poles and Mandelstam cut contributions. The leading term reproduces correctly the expression calculated by one of the authors in the BFKL approach, while the subleading term presents a result, that was not yet found with the use of the unitarity techniques. This supports the applicability of the Wilson loop approach to the planar MHV amplitudes in N=4 SUSY.

On the two-loop hexagon Wilson loop remainder function in [formula omitted] SYM

A duality relation has been proposed between the planar gluon MHV amplitudes and light-like Wilson loops in N=4 super Yang–Mills. At six-point two-loop, the results for the planar gluon MHV amplitude and for the light-like Wilson loop agree, but they both differ from the Bern–Dixon–Smirnov ansatz by a finite remainder function. Recently Del Duca, Duhr and Smirnov presented an analytical result for the two-loop hexagon Wilson loop remainder function in general kinematics. Their result is rather lengthy, and the dependence on the conformal cross ratios appears in a complicated way. Here we present an alternate, more compact representation for the two-loop hexagon Wilson loop remainder function.

Thermal transport and drag force in improved holographic QCD

We calculate the bulk viscosity, drag force and jet quenching parameter in Improved Holographic QCD. We find that the bulk viscosity rises near the phase transition but does not exceed the shear viscosity. The drag force shows the effects of asymptotic freedom both as a function of velocity and temperature. It indicates diffusion times of heavy quarks in rough agreement with data. The jet quenching parameter values computed via the light-like Wilson loop are in the lower range suggested by data.

Duality between Wilson loops and gluon amplitudes

An intriguing new duality between planar MHV gluon amplitudes and light-like Wilson loops in N=4 super Yang-Mills is investigated. We extend previous checks of the duality by performing a two-loop calculation of the rectangular and pentagonal Wilson loop. Furthermore, we derive an all-order broken conformal Ward identity for the Wilson loops and analyse its consequences. Starting from six points, the Ward identity allows for an arbitrary function of conformal invariants to appear in the expression for the Wilson loop. We compute this function at six points and two loops and discuss its implications for the corresponding gluon amplitude. It is found that the duality disagrees with a conjecture for the gluon amplitudes by Bern et al. A recent calculation by Bern et al indeed shows that the latter conjecture breaks down at six gluons and at two loops. By doing a numerical comparison with their results we find that the duality between gluon amplitudes and Wilson loops is preserved. This review is based on the author's PhD thesis and includes developments until May 2008.

Two-loop polygon Wilson loops in 𝒩 = 4 SYM

We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N=4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n=6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n=7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of "master integrals", which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the N=4 theory.

Lightlike Wilson loops from AdS/CFT

We investigate the lightlike limit of stationary spacelike string configurations on a large class of five-dimensional asymptotically AdS backgrounds. Specific examples include gravity duals which incorporate finite 't Hooft coupling, curvature-squared corrections, and chemical potentials. A universal feature of these AdS/CFT models is that the string solution with minimum action yields a lightlike Wilson loop whose leading behavior is exponentially linear, rather than quadratic, in the width of the loop. Unless there is a compelling reason for discarding the leading saddlepoint contribution to the Wilson loop, following the proposal of Liu et. al. [2] leads to zero jet-quenching parameter for all of these models.

The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude

As a test of the gluon scattering amplitude/Wilson loop duality, we evaluate the hexagonal light-like Wilson loop at two loops in N=4 super-Yang–Mills theory. We compare its finite part to the Bern–Dixon–Smirnov (BDS) conjecture for the finite part of the six-gluon amplitude. We find that the two expressions have the same behavior in the collinear limit, but they differ by a non-trivial function of the three (dual) conformally invariant variables. This implies that either the BDS conjecture or the gluon amplitude/Wilson loop duality fails for the six-gluon amplitude, starting from two loops. Our results are in qualitative agreement with the analysis of Alday and Maldacena of scattering amplitudes with infinitely many external gluons.

On T-duality and integrability for strings on AdS backgrounds

We discuss an interplay between T-duality and integrability for certain classical non-linear sigma models. In particular, we consider strings on the AdS5 ? S5 background and perform T-duality along the four isometry directions of AdS5 in the Poincar? patch. The T-dual of the AdS5 sigma model is again a sigma model on an AdS5 space. This classical T-duality relation was used in the recently uncovered connection between light-like Wilson loops and MHV gluon scattering amplitudes in the strong coupling limit of the AdS/CFT duality. We show that the explicit coordinate dependence along the T-duality directions of the associated Lax connection (flat current) can be eliminated by means of a field dependent gauge transformation. As a result, the gauge equivalent Lax connection can easily be T-dualized, i.e. written in terms of the dual set of isometric coordinates. The T-dual Lax connection can be used for the derivation of infinitely many conserved charges in the T-dual model. Our construction implies that local (Noether) charges of the original model are mapped to non-local charges of the T-dual model and vice versa.

Strong-coupling expansion of cusp anomaly and gluon amplitudes from quantum open strings in [formula omitted

An important “observable” of planar N = 4 SYM theory is the scaling function f ( λ ) that appears in the anomalous dimension of large spin twist 2 operators and also in the cusp anomaly of light-like Wilson loop. The non-trivial relation between the anomalous dimension and the Wilson loop interpretations of f ( λ ) is well-understood on the perturbative gauge theory side of the AdS/CFT duality. In the first part of this paper we present the dual string-theory counterpart of this relation, i.e., the equivalence between the closed-string and the open-string origins of f ( λ ) . We argue that the coefficient of the log S term in the energy of the closed string with large spin S in AdS 5 should be equal to the coefficient in the logarithm of expectation value of the null cusp Wilson loop, to all orders in λ − 1 / 2 expansion. The reason is that the corresponding minimal surfaces happen to be related by a conformal transformation (and an analytic continuation). As a check, we explicitly compute the leading 1-loop string sigma model correction to the cusp Wilson loop, reproducing the same subleading coefficient in f ( λ ) as found earlier in the spinning closed string case. The same function f ( λ ) appears also in the resummed form of the 4-gluon amplitude as discussed at weak coupling by Bern, Dixon and Smirnov and recently found at the leading order at strong coupling by Alday and Maldacena (AM). Here we attempt to extend the latter approach to a subleading order in λ − 1 / 2 by computing the IR singular part of the 1-loop string correction to the corresponding T-dual Wilson loop. We discuss explicitly the 1-cusp case and comment on apparent problems with the dimensional regularization proposal of AM when directly applied order by order in strong coupling (string inverse tension) expansion.

Calculating the Jet Quenching Parameter

Models of medium-induced radiative parton energy loss account for the strong suppression of high-p(T) hadron spectra in square root of (S)NN=200 GeV Au-Au collisions at BNL RHIC in terms of a single "jet quenching parameter" q. We observe that q can be given a model-independent, nonperturbative, quantum field theoretic definition in terms of the short-distance behavior of a particular lightlike Wilson loop. We then use the anti-de Sitter/conformal-field-theory correspondence to obtain a strong-coupling calculation of q in hot N=4 supersymmetric QCD, finding q(SYM)=26.69 square root of alpha(SYM)N(c)T(3) in the limit in which both N(c) and 4pialpha(SYM)N(c) are large. Thus, at strong coupling q is not proportional to the entropy density s, or to some "number density of scatterers" since, unlike the number of degrees of freedom, q does not grow like N(c)(2).

The energy of a moving quark-antiquark pair in an Script N = 4 SYM plasma

We make use of the AdS/CFT correspondence to determine the energy of an external quark-antiquark pair that moves through strongly-coupled thermal N=4 super-Yang-Mills plasma, both in the rest frame of the plasma and in the rest frame of the pair. It is found that the pair feels no drag force, has an energy that reproduces the expected 1/L (or gamma/L) behavior at small quark-antiquark separations, and becomes unbound beyond a certain screening length whose velocity-dependence we determine. We discuss the relation between the high-velocity limit of our results and the lightlike Wilson loop proposed recently as a definition of the jet-quenching parameter.

Jet quenching parameters in strongly coupled nonconformal gauge theories

Recently Liu, Rajagopal and Wiedemann (LRW) [hep-ph/0605178] proposed a first principle, nonperturbative quantum field theoretic definition of ``jet quenching parameter'' \hat{q} used in models of medium-induced radiative parton energy loss in nucleus-nucleus collisions at RHIC. Relating \hat{q} to a short-distance behavior of a certain light-like Wilson loop, they used gauge theory-string theory correspondence to evaluate \hat{q} for the strongly coupled N=4 SU(N_c) gauge theory plasma. We generalize analysis of LRW to strongly coupled non-conformal gauge theory plasma. We find that a jet quenching parameter is gauge theory specific (not universal). Furthermore, it appears it's value increases as the number of effective adjoint degrees of freedom of a gauge theory plasma increases.

Wilson loops and anomalous dimensions in cascading theories

We use lightlike Wilson loops and the AdS Conformal Field Theory correspondence to compute the anomalous dimensions of twist-2 operators in the cascading [Klebanov-Strassler (KS)] theory. The computation amounts to find a minimal surface in the uv region of the KS background, which is described by the Klebanov-Tseytlin solution. The result is similar to the one for $\mathrm{SU}(N)\mathcal{N}=4$ Super Yang-Mills theory but with N replaced by an effective, scale-dependent N. We performed also a calculation using a rotating string and find agreement. In fact, we use a double Wick rotated version of the rotating string solution which is a Euclidean world-sheet ending on a lightlike line in the boundary. It gives the same result as the rotating string for the $\mathcal{N}=4$ case, but it is more appropriate than the rotating string in the $\mathcal{N}=1$ case.

A new type of loop equations

We derive a new form of loop equations for light-like Wilson loops. In bosonic theories those loop equations close only for straight light-like Wilson lines. In the case of N=1 in ten dimensions they close for any light-like Wilson loop. Upon dimensional reduction to N=4 SYM in four dimensions, these loops become exactly the chiral loops which can be evaluated semiclassically, in the strong coupling limit, by a minimal surface in anti de-Sitter space. We show that the AdS calculation satisfies those loop equations. We also find a new fermionic loop equation derived from the gauge theory fermionic equation of motion.

Lightlike Wilson loops and gauge invariance of Yang-Mills theory in 1+1 dimensions.

A light-like Wilson loop is computed in perturbation theory up to ${\cal O} (g^4)$ for pure Yang--Mills theory in 1+1 dimensions, using Feynman and light--cone gauges to check its gauge invariance. After dimensional regularization in intermediate steps, a finite gauge invariant result is obtained, which however does not exhibit abelian exponentiation. Our result is at variance with the common belief that pure Yang--Mills theory is free in 1+1 dimensions, apart perhaps from topological effects.

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.