Conformal Cross-ratios Research Articles

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory

We provide a simple analytic formula for the two-loop six-point ratio function of planar N = 4 super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral \Omega^{(2)}, also plays a key role in a new representation of the remainder function R_6^{(2)} in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) \times (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) \times (parity even) part. The second non-polylogarithmic function, the loop integral \tilde{\Omega}^{(2)}, characterizes this sector. Both \Omega^{(2)} and tilde{\Omega}^{(2)} can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.

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.

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.

Y-system for scattering amplitudes

We compute super Yang–Mills planar amplitudes at strong coupling by considering minimal surfaces in AdS5 space. The surfaces end on a null polygonal contour at the boundary of AdS. We show how to compute the area of the surfaces as a function of the conformal cross ratios characterizing the polygon at the boundary. We reduce the problem to a simple set of functional equations for the cross ratios as functions of the spectral parameter. These equations have the form of thermodynamic Bethe ansatz (TBA) equations. The area is the free energy of the TBA system. We consider any number of gluons and in any kinematic configuration.

Dual conformal symmetry of 1-loop NMHV amplitudes in $ \mathcal{N} = 4 $ SYM theory

We prove that 1-loop n-point NMHV superamplitudes in N=4 SYM theory are dual conformal covariant for all numbers n of external particles (after regularization and subtraction of IR divergences). This property was previously established for n < 10 in arXiv:0808.0491. We derive an explicit representation of these superamplitudes in terms of dual conformal cross-ratios. We also show that all the 1-loop `box coefficients' obtained from maximal cuts of N^kMHV n-point functions are covariant under dual conformal transformations.

On the structure of infrared singularities of gauge-theory amplitudes

A closed formula is obtained for the infrared singularities of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. It follows from an all-order conjecture for the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory. We show that the form of this anomalous dimension is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using a diagrammatic analysis, we demonstrate that these constraints imply that to three-loop order the anomalous dimension involves only two-parton correlations, with the possible exception of a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We argue that such a function does not appear at three-loop order, and that the same is true in higher orders. Our formula predicts Casimir scaling of the cusp anomalous dimension to all orders in perturbation theory, and we explicitly check that the constraints exclude the appearance of higher Casimir invariants at four loops. Using known results for the quark and gluon form factors, we derive the three-loop coefficients of the 1/epsilon^n pole terms (with n=1,...,6) for an arbitrary n-parton scattering amplitude in massless QCD. This generalizes Catani's two-loop formula proposed in 1998.

Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes

We study the factorization of soft and collinear singularities in dimensionally- regularized fixed-angle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by light-like Wilson lines, and defining gauge-invariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilson-line velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of these cusp anomalies in any multi-leg amplitude imposes all-order constraints on the kinematic dependence of the corresponding soft anomalous dimension, relating it to the cusp anomalous dimension. For amplitudes with two or three hard partons the solution is unique: the constraints fully determine the kinematic dependence of the soft function. For amplitudes with four or more hard partons we present a minimal solution where the soft anomalous dimension is a sum over colour dipoles, multiplied by the cusp anomalous dimension. In this case additional contributions to the soft anomalous dimension at three loops or beyond are not excluded, but they are constrained to be functions of conformal cross ratios of kinematic variables.

Four-point functions of different-weight operators in the AdS/CFT correspondence

We calculate four-point correlation functions of two weight-2 and two weight-3 1/2-BPS operators in \mathcal{N}=4 SYM in the large N limit in supergravity approximation. By the AdS/CFT conjecture, these operators are dual to AdS supergravity scalar fields s_2 and s_3 with mass m^2 = -4 and m^2 = -3 respectively. This is the first non-trivial four-point function of mixed-weight operators of lowest conformal dimensions. We show that the supergravity-induced four-point function splits into a free and a part, where the quantum contribution obeys non-trivial constraints coming from the insertion procedure in the gauge theory, in particular, it depends on only one function of the conformal cross-ratios.

Correlation functions and massive Kaluza–Klein modes in the AdS/CFT correspondence

We study four-point correlation functions of 1 2 -BPS operators in N=4 SYM which are dual to massive KK modes in AdS 5 supergravity. On the field theory side, the procedure of inserting the SYM action yields partial non-renormalisation of the four-point amplitude for such operators. In particular, if the BPS operators have dimensions equal to three or four, the corresponding four-point amplitude is determined by one or two independent functions of the two conformal cross-ratios, respectively. This restriction on the amplitude does not merely follow from the superconformal Ward identities, it also encodes dynamical information related to the structure of the gauge theory Lagrangian. The dimension 3 BPS operator is the AdS/CFT dual of the first non-trivial massive Kaluza–Klein mode of the compactified type IIB supergravity, whose interactions go beyond the level of the five-dimensional gauged N=8 supergravity. We show that the corresponding effective Lagrangian has a surprisingly simple sigma-model-type form with at most two derivatives. We then compute the supergravity-induced four-point amplitude for the dimension 3 operators. Remarkably, this amplitude splits into a “free” and an “interacting” parts in exact agreement with the structure predicted by the insertion procedure. The underlying OPE fulfills the requirements of superconformal symmetry and unitarity.

On a large N degeneracy in [formula omitted] SYM and the AdS/CFT correspondence

We study the four-point correlator of 1 2 -BPS operators of weight 4 in N=4 SYM, which are dual to massive KK modes in AdS 5 supergravity. General field-theoretic arguments lead to a partially non-renormalized form of the amplitude that depends on two a priori independent functions of the conformal cross-ratios. We explicitly compute the amplitude in the large N limit at one-loop (order g 2) and in AdS 5 supergravity. Surprisingly, the one-loop result shows that the two functions determining the amplitude coincide while in the supergravity regime they are distinctly different. We discuss the possible implications of this perturbative degeneracy for the AdS/CFT correspondence.

Implications of superconformal symmetry for interacting (2,0) tensor multiplets

We study the structure of the four-point correlation function of the lowest-dimension 1/2 BPS operators (stress-tensor multiplets) in the (2,0) six-dimensional theory. We first discuss the superconformal Ward identities and the group-theoretical restrictions on the corresponding OPE. We show that the general solution of the Ward identities is expressed in terms of a single function of the two conformal cross-ratios (“prepotential”). Using the maximally extended gauged seven-dimensional supergravity, we then compute the four-point amplitude in the supergravity approximation and identify the corresponding prepotential. We analyze the leading terms in the OPE by performing a conformal partial wave expansion and show that they are in agreement with the non-renormalization theorems following from representation theory. The investigation of the (2,0) theory is carried out in close parallel with the familiar four-dimensional N=4 super-Yang–Mills theory.

Four-point functions in N=2 superconformal field theories

Four-point correlation functions of hypermultiplet bilinear composites are analysed in N=2 superconformal field theory using the superconformal Ward identities and the analyticity properties of the composite operator superfields. It is shown that the complete amplitude is determined by a single arbitrary function of the two conformal cross-ratios of the space-time variables.

Three-loop four-point correlator in N=4 SYM

We explicitly compute the complete three-loop (O( g 4)) contribution to the four-point function of chiral primary current-like operators 〈 q ̃ 2q 2 q ̃ 2q 2〉 in any finite N=2 SYM theory. The computation uses N=2 harmonic supergraphs in coordinate space. Dramatic simplifications are achieved by a double insertion of the N=2 SYM linearized action, and application of superconformal covariance arguments to the resulting nilpotent six-point amplitude. The result involves polylogarithms up to fourth order of the conformal cross ratios. It becomes particularly simple in the N=4 special case.