arXiv subjects

Paul Heslop

Publications and source records attributed to Paul Heslop.

At least 37 records · Page 2Linked to original sources

arXiv2016

Wilson Loop Form Factors: A New Duality

We find a new duality for form factors of lightlike Wilson loops in planar $\mathcal N=4$ super-Yang-Mills theory. The duality maps a form factor involving an $n$-sided lightlike polygonal super-Wilson loop together with $m$ external on-shell states, to the same type of object but with the edges of the Wilson loop and the external states swapping roles. This relation can essentially be seen graphically in Lorentz harmonic chiral (LHC) superspace where it is equivalent to planar graph duality. However there are some crucial subtleties with the cancellation of spurious poles due to the gauge fixing. They are resolved by finding the correct formulation of the Wilson loop and by careful analytic continuation from Minkowski to Euclidean space. We illustrate all of these subtleties explicitly in the simplest non-trivial NMHV-like case.

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arXiv2016

Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps

We introduce two new graphical-level relations among possible contributions to the four-point correlation function and scattering amplitude in planar, maximally supersymmetric Yang-Mills theory. When combined with the rung rule, these prove powerful enough to fully determine both functions through ten loops. This then also yields the full five-point amplitude to eight loops and the parity-even part to nine loops. We derive these rules, illustrate their applications, compare their relative strengths for fixing coefficients, and survey some of the features of the previously unknown nine and ten loop expressions. Explicit formulae for amplitudes and correlators through ten loops are available at: http://goo.gl/JH0yEc.

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arXiv2016

On-Shell Diagrams for N = 8 Supergravity Amplitudes

We define recursion relations for N = 8 supergravity amplitudes using a generalization of the on-shell diagrams developed for planar N = 4 super-Yang-Mills. Although the recursion relations generically give rise to non-planar on-shell diagrams, we show that at tree-level the recursion can be chosen to yield only planar diagrams, the same diagrams occurring in the planar N = 4 theory. This implies non-trivial identities for non-planar diagrams as well as interesting relations between the N = 4 and N = 8 theories. We show that the on-shell diagrams of N = 8 supergravity obey equivalence relations analogous to those of N = 4 super-Yang-Mills, and we develop a systematic algorithm for reading off Grassmannian integral formulae directly from the on-shell diagrams. We also show that the 1-loop 4-point amplitude of N = 8 supergravity can be obtained from on-shell diagrams.

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arXiv2015

Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality

We use the soft-collinear bootstrap to construct the 8-loop integrand for the 4-point amplitude and 4-stress-tensor correlation function in planar maximally supersymmetric Yang-Mills theory. Both have a unique representation in terms of planar, conformal integrands grouped according to a hidden symmetry discovered for correlation functions. The answer we find exposes a fundamental tension between manifest locality and planarity with manifest conformality not seen at lower loops. For the first time, the integrand must include terms that are finite even on-shell and terms that are divergent even off-shell (so-called `pseudoconformal' integrals). We describe these novelties and their consequences in this letter, and we make the full correlator and amplitude available as part of this work's submission files to the arXiv.

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arXiv2015

All three-loop four-point correlators of half-BPS operators in planar N=4 SYM

We obtain the planar correlation function of four half-BPS operators of arbitrary weights, up to three loops. Our method exploits only elementary properties of the integrand of the planar correlator, such as its symmetries and singularity structure. This allows us to write down a general ansatz for the integrand. The coefficients in the ansatz are fixed by means of a powerful light-cone OPE relation between correlators with different weights. Our result is formulated in terms of a limited number of functions built from known one-, two- and three-loop conformal integrals. These results are useful for checking recent integrability predictions for the OPE structure constants.

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arXiv2015

Superconformal partial waves in Grassmannian field theories

We derive superconformal partial waves for all scalar four-point functions on a super Grassmannian space Gr(m|n,2m|2n) for all m,n. This family of four-point functions includes those of all (arbitrary weight) half BPS operators in both N=4 SYM (m=n=2) and in N=2 superconformal field theories in four dimensions (m=2,n=1) on analytic superspace. It also includes four-point functions of all (arbitrary dimension) scalar fields in non-supersymmetric conformal field theories (m=2,n=0) on Minkowski space, as well as those of a certain class of representations of the compact SU(2n) coset spaces. As an application we then specialise to N=4 SYM and use these results to perform a detailed superconformal partial wave analysis of the four- point functions of arbitrary weight half BPS operators. We discuss the non-trivial separation of protected and unprotected sectors for the <2222>, <2233> and <3333> cases in an SU(N) gauge theory at finite N. The <2233> correlator predicts a non-trivial protected twist four sector for <3333> which we can completely determine using the knowledge that there is precisely one such protected twist four operator for each spin.

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arXiv2015

On Yangian symmetry of scattering amplitudes and the dilatation operator in N=4 super Yang-Mills

It is known that the Yangian of PSU(2,2|4) is a symmetry of the tree-level S-matrix of N=4 super Yang-Mills. On the other hand, the complete one-loop dilatation operator in the same theory commutes with the level-one Yangian generators only up to certain boundary terms found by Dolan, Nappi and Witten. Using a result by Zwiebel, we show how the Yangian symmetry of the tree-level S-matrix of N=4 super Yang-Mills implies precisely the Yangian invariance, up to boundary terms, of the one-loop dilatation operator.

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arXiv2015

Bootstrapping correlation functions in N=4 SYM

We describe a new approach to computing the chiral part of correlation functions of stress-tensor supermultiplets in N=4 SYM that relies on symmetries, analytic properties and the structure of the OPE only. We demonstrate that the correlation functions are given by a linear combination of chiral N=4 superconformal invariants accompanied by coefficient functions depending on the space-time coordinates only. We present the explicit construction of these invariants and show that the six-point correlation function is fixed in the Born approximation up to four constant coefficients by its symmetries. In addition, the known asymptotic structure of the correlation function in the light-like limit fixes unambiguously these coefficients up to an overall normalization. We demonstrate that the same approach can be applied to obtain a representation for the six-point NMHV amplitude that is free from any auxiliary gauge fixing parameters, does not involve spurious poles and manifests half of the dual superconformal symmetry.

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arXiv2013

Local integrands for the five-point amplitude in planar N=4 SYM up to five loops

Integrands for colour ordered scattering amplitudes in planar N=4 SYM are dual to those of correlation functions of the energy-momentum multiplet of the theory. The construction can relate amplitudes with different numbers of legs. By graph theory methods the integrand of the four-point function of energy-momentum multiplets has been constructed up to six loops in previous work. In this article we extend this analysis to seven loops and use it to construct the full integrand of the five-point amplitude up to five loops, and in the parity even sector to six loops. All results, both parity even and parity odd, are obtained in a concise local form in dual momentum space and can be displayed efficiently through graphs. We have verified agreement with other local formulae both in terms of supertwistors and scalar momentum integrals as well as BCJ forms where those exist in the literature, i.e. up to three loops. Finally we note that the four-point correlation function can be extracted directly from the four-point amplitude and so this uncovers a direct link from four- to five-point amplitudes.

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arXiv2013

Leading singularities and off-shell conformal integrals

The three-loop four-point function of stress-tensor multiplets in N=4 super Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition to the known, ladder-type integrals. In this paper we evaluate the unknown integrals, thus obtaining the three-loop correlation function analytically. The integrals have the generic structure of rational functions multiplied by (multiple) polylogarithms. We use the idea of leading singularities to obtain the rational coefficients, the symbol - with an appropriate ansatz for its structure - as a means of characterising multiple polylogarithms, and the technique of asymptotic expansion of Feynman integrals to obtain the integrals in certain limits. The limiting behaviour uniquely fixes the symbols of the integrals, which we then lift to find the corresponding polylogarithmic functions. The final formulae are numerically confirmed. The techniques we develop can be applied more generally, and we illustrate this by analytically evaluating one of the integrals contributing to the same four-point function at four loops. This example shows a connection between the leading singularities and the entries of the symbol.

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arXiv2013

Perturbative correlation functions of null Wilson loops and local operators

We consider the correlation function of a null Wilson loop with four edges and a local operator in planar MSYM. By applying the insertion procedure, developed for correlation functions of local operators, we give an integral representation for the result at one and two loops. We compute explicitly the one loop result and show that the two loop result is finite.

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arXiv2012

Uplifting Amplitudes in Special Kinematics

We consider scattering amplitudes in planar N = 4 supersymmetric Yang-Mills theory in special kinematics where all external four-dimensional momenta are restricted to a (1+1)-dimensional subspace. The amplitudes are known to satisfy non-trivial factorisation properties arising from multi-collinear limits, which we further study here. We are able to find a general solution to these multi-collinear limits. This results in a simple formula which represents an n-point superamplitude in terms of a linear combination of functions S_m which are constrained to vanish in all appropriate multi-collinear limits. These collinear-vanishing building blocks, S_m, are dual-conformally-invariant functions which depend on the reduced m-point kinematics with 8 \leq m \leq 4l. For MHV amplitudes they can be constructed directly using, for example, the approach in Ref. [1]. This procedure provides a universal uplift of lower-point collinearly vanishing building blocks S_m to all higher-point amplitudes. It works at any loop-level l \geq 1 and for any MHV or N^kMHV amplitude. We compare this with explicit examples involving n-point MHV amplitudes at 2-loops and 10-point MHV amplitudes at 3-loops. Tree-level superamplitudes have different properties and are treated separately from loop-level amplitudes in our approach. To illustrate this we derive an expression for n-point tree-level NMHV amplitudes in special kinematics.

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arXiv2012

Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM

We study the four-point correlation function of stress-tensor supermultiplets in N=4 SYM using the method of Lagrangian insertions. We argue that, as a corollary of N=4 superconformal symmetry, the resulting all-loop integrand possesses an unexpected complete symmetry under the exchange of the four external and all the internal (integration) points. This alone allows us to predict the integrand of the three-loop correlation function up to four undetermined constants. Further, exploiting the conjectured amplitude/correlation function duality, we are able to fully determine the three-loop integrand in the planar limit. We perform an independent check of this result by verifying that it is consistent with the operator product expansion, in particular that it correctly reproduces the three-loop anomalous dimension of the Konishi operator. As a byproduct of our study, we also obtain the three-point function of two half-BPS operators and one Konishi operator at three-loop level. We use the same technique to work out a compact form for the four-loop four-point integrand and discuss the generalisation to higher loops.

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arXiv2012

Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM

We present a construction of the integrand of the correlation function of four stress-tensor multiplets in N=4 SYM at weak coupling. It does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. This symmetry holds for any gauge group, so it can be used to predict the integrand both in the planar and non-planar sectors. We demonstrate the great efficiency of graph-theoretical tools in the systematic study of the possible permutation symmetric integrands. We formulate a general ansatz for the correlation function as a linear combination of all relevant graph topologies, with arbitrary coefficients. Powerful restrictions on the coefficients come from the analysis of the logarithmic divergences of the correlation function in two singular regimes: Euclidean short-distance and Minkowski light-cone limits. We demonstrate that the planar integrand is completely fixed by the procedure up to six loops and probably beyond. In the non-planar sector, we show the absence of non-planar corrections at three loops and we reduce the freedom at four loops to just four constants. Finally, the correlation function/amplitude duality allows us to show the complete agreement of our results with the four-particle planar amplitude in N=4 SYM.

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arXiv2012

Five-loop Konishi in N=4 SYM

We present a new method for computing the Konishi anomalous dimension in N=4 SYM at weak coupling. It does not rely on the conventional Feynman diagram technique and is not restricted to the planar limit. It is based on the OPE analysis of the four-point correlation function of stress-tensor multiplets, which has been recently constructed up to six loops. The Konishi operator gives the leading contribution to the singlet SU(4) channel of this OPE. Its anomalous dimension is the coefficient of the leading single logarithmic singularity of the logarithm of the correlation function in the double short-distance limit, in which the operator positions coincide pairwise. We regularize the logarithm of the correlation function in this singular limit by a version of dimensional regularization. At any loop level, the resulting singularity is a simple pole whose residue is determined by a finite two-point integral with one loop less. This drastically simplifies the five-loop calculation of the Konishi anomalous dimension by reducing it to a set of known four-loop two-point integrals and two unknown integrals which we evaluate analytically. We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.

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arXiv2011

Wilson Loops @ 3-Loops in Special Kinematics

We obtain a compact expression for the octagon MHV amplitude / Wilson loop at 3 loops in planar N=4 SYM and in special 2d kinematics in terms of 7 unfixed coefficients. We do this by making use of the cyclic and parity symmetry of the amplitude/Wilson loop and its behaviour in the soft/collinear limits as well as in the leading term in the expansion away from this limit. We also make a natural and quite general assumption about the functional form of the result, namely that it should consist of weight 6 polylogarithms whose symbol consists of basic cross-ratios only (and not functions thereof). We also describe the uplift of this result to 10 points.

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arXiv2011

The super-correlator/super-amplitude duality: Part II

We continue the study of the duality between super-correlators and scattering super-amplitudes in planar N=4 SYM. We provide a number of further examples supporting the conjectured duality relation between these two seemingly different objects. We consider the five- and six-point one-loop NMHV and the six-point tree-level NNMHV amplitudes, obtaining them from the appropriate correlators of strength tensor multiplets in N=4 SYM. In particular, we find exact agreement between the rather non-trivial parity-odd sector of the integrand of the six-point one-loop NMHV amplitude, as obtained from the correlator or from BCFW recursion relations. Together these results lead to the conjecture that the integrands of any N^kMHV amplitude at any loop order in planar N=4 SYM can be described by the correlators of stress-tensor multiplets.

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arXiv2011

The super-correlator/super-amplitude duality: Part I

We extend the recently discovered duality between MHV amplitudes and the light-cone limit of correlation functions of a particular type of local scalar operators to generic non-MHV amplitudes in planar N=4 SYM theory. We consider the natural generalization of the bosonic correlators to super-correlators of stress-tensor multiplets and show, in a number of examples, that their light-cone limit exactly reproduces the square of the matching super-amplitudes. Our correlators are computed at Born level. If all of their points form a light-like polygon, the correlator is dual to the tree-level amplitude. If a subset of points are not on the polygon but are integrated over, they become Lagrangian insertions generating the loop corrections to the correlator. In this case the duality with amplitudes holds at the level of the integrand. We build up the superspace formalism needed to formulate the duality and present the explicit example of the n-point NMHV tree amplitude as the dual of the lowest nilpotent level in the correlator.

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