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We bootstrap the symbol of the maximal-helicity-violating four-particle form factor for the chiral part of the stress-tensor supermultiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory at two loops. When minimally normalized, this symbol involves only 34 letters and obeys the extended Steinmann relations in all partially-overlapping three-particle momentum channels. In addition, the remainder function for this form factor exhibits an antipodal self-duality: it is invariant under the combined operation of the antipodal map defined on multiple polylogarithms -- which reverses the order of the symbol letters -- and a simple kinematic map. This self-duality holds on a four-dimensional parity-preserving kinematic hypersurface. It implies the antipodal duality recently noticed between the three-particle form factor and the six-particle amplitude in this theory., Comment: 9 pages, 1 figure and 1 table; v2, minor clarifications, references added, journal version

Physical review letters 130(11), 111602 (2023). doi:10.1103/PhysRevLett.130.111602, We classify origin limits of maximally helicity violating multigluon scattering amplitudes in planar $\mathcal{N}$ = 4 super-Yang-Mills theory, where a large number of cross ratios approach zero, with the help of cluster algebras. By analyzing existing perturbative data and bootstrapping new data, we provide evidence that the amplitudes become the exponential of a quadratic polynomial in the large logarithms. With additional input from the thermodynamic Bethe ansatz at strong coupling, we conjecture exact expressions for amplitudes with up to eight gluons in all origin limits. Our expressions are governed by the tilted cusp anomalous dimension evaluated at various values of the tilt angle., Published by APS, College Park, Md.

A necessary ingredient for extending the BFKL equation to next-to-next-to-leading logarithmic (NNLL) accuracy is the one-loop central emission vertex (CEV) for two gluons which are not strongly ordered in rapidity. Here we consider the one-loop six-gluon amplitude in $\mathcal{N}=4$ super Yang-Mills (SYM) theory in a central next-to-multi-Regge kinematic (NMRK) limit, we show that its dispersive part factorises in terms of the two-gluon CEV, and we use it to extract the one-loop two-gluon CEV for any helicity configuration within this theory. This is a component of the two-gluon CEV in QCD. Although computed in the NMRK limit, both the colour structure and the kinematic dependence of the two-gluon CEV capture much of the complexity of the six-gluon amplitudes in general kinematics. In fact, the transcendental functions of the latter can be conveniently written in terms of impact factors, trajectories, single-emission CEVs and a remainder, which is a function of the conformally invariant cross ratios which characterise the six-gluon amplitudes in planar $\mathcal{N}=4$ SYM. Finally, as expected, in the MRK limit the two-gluon CEV neatly factorises in terms of two single-emission CEVs., Comment: 85 pages, 13 figures

Journal of physics / A 55(44), 443001 (2022). doi:10.1088/1751-8121/ac8380, This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments., Published by IOP Publ., Bristol

We observe that the three-gluon form factor of the chiral part of the stress-tensor multiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory is dual to the six-gluon MHV amplitude on its parity-preserving surface. Up to a simple variable substitution, the map between these two quantities is given by the antipode operation defined on polylogarithms (as part of their Hopf algebra structure), which acts at symbol level by reversing the order of letters in each term. We provide evidence for this duality through seven loops., Comment: 5+2 pages, 2 figures, 2 tables. v2: added further checks at kinematic points, some clarifications and references; version to appear in PRL

We review the bootstrap method for constructing six- and seven-particle amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory, by exploiting their analytic structure. We focus on two recently discovered properties which greatly simplify this construction at symbol and function level, respectively: the extended Steinmann relations, or equivalently cluster adjacency, and the coaction principle. We then demonstrate their power in determining the six-particle amplitude through six and seven loops in the NMHV and MHV sectors respectively, as well as the symbol of the NMHV seven-particle amplitude to four loops., Comment: 36 pages, 4 figures, 5 tables, 1 ancillary file. Contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25 September 2019, Corfu, Greece

We study the maximally-helicity-violating six-gluon scattering amplitude in planar N=4 super-Yang-Mills theory at finite coupling when all three cross ratios are small. It exhibits a double logarithmic scaling in the cross ratios, controlled by a handful of “anomalous dimensions” that are functions of the coupling constant alone. Inspired by known seven-loop results at weak coupling and the integrability-based pentagon operator product expansion, we present conjectures for the all-order resummation of these anomalous dimensions. At strong coupling, our predictions agree perfectly with the string theory analysis. Intriguingly, the simplest of these anomalous dimensions coincides with one describing the lightlike limit of the octagon, namely, the four-point function of large-charge Bogomol’nyi–Prasad–Sommerfield (BPS) operators.

We bootstrap the three-point form factor of the chiral part of the stress-tensor supermultiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, ``extended-Steinmann-like'' conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the $gg\to Hg$ and $H\to ggg$ amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as $gg\to Hg$ and $\gamma^*\to q\bar{q}g$. We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar $\mathcal{N}=4$ three-point form factor can be placed in an even smaller space of functions, with no independent $\zeta$ values at weights two and three., Comment: 46 pages, 6 figures, 9 tables. v2: conjecture in section 6 modified and minor typos corrected; version to appear in JHEP

We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders., 51 pages, 9 figures, 1 table; v2: minor typo corrections, version to appear in JHEP

We compute the symbol of the two-loop five-point scattering amplitude in N=4 supersymmetric Yang-Mills theory, including its full color dependence. This requires constructing the symbol of all two-loop five-point nonplanar massless master integrals, for which we give explicit results.

Journal of high energy physics 2019(9), 61 (2019). doi:10.1007/JHEP09(2019)061, We describe the minimal space of polylogarithmic functions that is required to express the six-particle amplitude in planar N = 4 super-Yang-Mills theory through six and seven loops, in the NMHV and MHV sectors respectively. This space respects a set of extended Steinmann relations that restrict the iterated discontinuity structure of the amplitude, as well as a cosmic Galois coaction principle that constrains the functions and the transcendental numbers that can appear in the amplitude at special kinematic points. To put the amplitude into this space, we must divide it by the BDS-like ansatz and by an additional zeta-valued constant ρ. For this normalization, we conjecture that the extended Steinmann relations and the coaction principle hold to all orders in the coupling. We describe an iterative algorithm for constructing the space of hexagon functions that respects both constraints. We highlight further simplifications that begin to occur in this space of functions at weight eight, and distill the implications of imposing the coaction principle to all orders. Finally, we explore the restricted spaces of transcendental functions and constants that appear in special kinematic configurations, which include polylogarithms involving square, cube, fourth and sixth roots of unity., Published by SISSA, [Trieste]

We compute the symbol of the two-loop five-point amplitude in $\mathcal N=8$ supergravity. We write an ansatz for the amplitude whose rational prefactors are based on not only 4-dimensional leading singularities, but also $d$-dimensional ones, as the former are insufficient. Our novel $d$-dimensional unitarity-based approach to the systematic construction of an amplitude's rational structures is likely to have broader applications, for example to analogous QCD calculations. We fix parameters in the ansatz by performing numerical integration-by-parts reduction of the known integrand. We find that the two-loop five-point $\mathcal N=8$ supergravity amplitude is uniformly transcendental. We then verify the soft and collinear limits of the amplitude. There is considerable similarity with the corresponding amplitude for $\mathcal N=4$ super-Yang-Mills theory: all the rational prefactors are double copies of the Yang-Mills ones and the transcendental functions overlap to a large degree. As a byproduct, we find new relations between color-ordered loop amplitudes in $\mathcal N=4$ super-Yang-Mills theory., Comment: 50 pages, 4 figures. v2: final journal version

We present the first fully analytic calculation of the Quantum Chromodynamics (QCD) event shape observable Energy-Energy Correlation in electron-positron annihilation at Next-To-Leading Order (NLO). This result sheds light on the analytic structure of the event shape observables beyond Leading Order (LO) and serves as a motivation to employ our methods in the investigation of other event shape observables that so far have not been calculated analytically.

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space., 70 pages, 3 figures, 4 tables; v2, minor typo corrections and clarifications

The energy-energy correlation (EEC) between two detectors in e+e− annihilation was computed analytically at leading order in QCD almost 40 years ago, and numerically at next-to-leading order (NLO) starting in the 1980s. We present the first analytical result for the EEC at NLO, which is remarkably simple, and facilitates analytical study of the perturbative structure of the EEC. We provide the expansion of the EEC in the collinear and back-to-back regions through next-to-leading power, information which should aid resummation in these regions.

The coefficient of the dimensionally regularized two-loop R^3 divergence of (nonsupersymmetric) gravity theories has recently been shown to change when non-dynamical three forms are added to the theory, or when a pseudo-scalar is replaced by the anti-symmetric two-form field to which it is dual. This phenomenon involves evanescent operators, whose matrix elements vanish in four dimensions, including the Gauss-Bonnet operator which is also connected to the trace anomaly. On the other hand, these effects appear to have no physical consequences in renormalized scattering processes. In particular, the dependence of the two-loop four-graviton scattering amplitude on the renormalization scale is simple. In this paper, we explain this result for any minimally-coupled massless gravity theory with renormalizable matter interactions by using unitarity cuts in four dimensions and never invoking evanescent operators., Comment: 21 pages, 4 figures, revtex

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal ($\bar{Q}$) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations., 43 pages, 2 figures. v2: typos corrected; version to appear in JHEP

International audience; We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar ϕ4 theory. The results are always multilinear combinations of ladder integrals, which are in turn built out of classical polylogarithms. The Steinmann relations provide a powerful constraint on such linear combinations, leading to a natural conjecture for any fishnet diagram as the determinant of a matrix of ladder integrals.

Loop-level scattering amplitudes for massless particles have singularities in regions where tree amplitudes are perfectly smooth. For example, a $2\to4$ gluon scattering process has a singularity in which each incoming gluon splits into a pair of gluons, followed by a pair of $2\to2$ collisions between the gluon pairs. This singularity mimics double parton scattering because it occurs when the transverse momentum of a pair of outgoing gluons vanishes. The singularity is logarithmic at fixed order in perturbation theory. We exploit the duality between scattering amplitudes and polygonal Wilson loops to study six-point amplitudes in this limit to high loop order in planar ${\cal N}=4$ super-Yang-Mills theory. The singular configuration corresponds to the limit in which a hexagonal Wilson loop develops a self-crossing. The singular terms are governed by an evolution equation, in which the hexagon mixes into a pair of boxes; the mixing back is suppressed in the planar (large $N_c$) limit. Because the kinematic dependence of the box Wilson loops is dictated by (dual) conformal invariance, the complete kinematic dependence of the singular terms for the self-crossing hexagon on the one nonsingular variable is determined to all loop orders. The complete logarithmic dependence on the singular variable can be obtained through nine loops, up to a couple of constants, using a correspondence with the multi-Regge limit. As a byproduct, we obtain a simple formula for the leading logs to all loop orders. We also show that, although the MHV six-gluon amplitude is singular, remarkably, the transcendental functions entering the non-MHV amplitude are finite in the same limit, at least through four loops., Comment: 64 pages, 12 figures, 1 ancillary file; v2: references added, minor typos corrected, corrected argument of NMHV finiteness in Section 2.1 and added supporting formulae in Appendix D; v3: minor typos corrected, corrected formulae (D.13)-(D.17) and the conclusions based on those equations; updated ancillary file

The analytic structure of scattering amplitudes is restricted by Steinmann relations, which enforce the vanishing of certain discontinuities of discontinuities. We show that these relations dramatically simplify the function space for the hexagon function bootstrap in planar maximally supersymmetric Yang-Mills theory. Armed with this simplification, along with the constraints of dual conformal symmetry and Regge exponentiation, we obtain the complete five-loop six-particle amplitude., Comment: 5 pages, 2 figures, 1 impressive table, and 2 ancillary files. v2: a few clarifications and references added; version to appear in PRL

In this article the authors consider why unifying the forces of nature may not be as difficult as physicists thought it would be. Topics include an overview of concepts introduced by American physicist Richard Feynman that guided physicists' understanding of particle collisions, why the authors feel Feynman's concepts have outlived their usefulness, and an in-depth examination of a novel way of analyzing particle processes developed by the authors, called the unitarity method, that bypasses the complexity of Feynmann's techniques.

We review recent NLO QCD results for W,Z + 3-jet production at hadron colliders, computed using BlackHat and SHERPA. We also include some new results for Z + 3-jet production at the LHC at 7 TeV. We report new progress towards the NLO cross section for W + 4-jet production. In particular, we show that the virtual matrix elements produced by BlackHat are numerically stable. We also show that with an improved integrator and tree-level matrix elements from BlackHat, SHERPA produces well-behaved real-emission contributions. As an illustration, we present the real-emission contributions -- including dipole-subtraction terms -- to the p_T distribution of the fourth jet, for a single subprocess with the maximum number of gluons., 6 pages, 4 figures, presented at Loops and Legs 2010

We describe BlackHat, an automated C++ program for calculating one-loop amplitudes, and the techniques used in its construction. These include the unitarity method and on-shell recursion. The other ingredients are compact analytic formulae for tree amplitudes for four-dimensional helicity states. The program computes amplitudes numerically, using analytic formulae only for the tree amplitudes, the starting point for the recursion, and the loop integrals. We make use of recently developed on-shell methods for evaluating coefficients of loop integrals, in particular a discrete Fourier projection as a means of improving numerical stability. We illustrate the good numerical stability of this approach by computing six-, seven- and eight-gluon amplitudes in QCD and comparing against known analytic results., To appear in the proceedings of 9th Workshop on Elementary Particle Theory: Loops and Legs in Quantum Field Theory, Sondershausen, Germany, 20-25 Apr 2008

Evanescent operators such as the Gauss-Bonnet term have vanishing perturbative matrix elements in exactly D=4 dimensions. Similarly, evanescent fields do not propagate in D=4; a three-form field is in this class, since it is dual to a cosmological-constant contribution. In this Letter, we show that evanescent operators and fields modify the leading ultraviolet divergence in pure gravity. To analyze the divergence, we compute the two-loop identical-helicity four-graviton amplitude and determine the coefficient of the associated (non-evanescent) R^3 counterterm studied long ago by Goroff and Sagnotti. We compare two pairs of theories that are dual in D=4: gravity coupled to nothing or to three-form matter, and gravity coupled to zero-form or to two-form matter. Duff and van Nieuwenhuizen showed that, curiously, the one-loop conformal anomaly --- the coefficient of the Gauss-Bonnet operator --- changes under p-form duality transformations. We concur, and also find that the leading R^3 divergence changes under duality transformations. Nevertheless, in both cases the physical renormalized two-loop identical-helicity four-graviton amplitude can be chosen to respect duality. In particular, its renormalization-scale dependence is unaltered., 6 pages, 1 figure, REVTeX

I provide a basic introduction to modern helicity amplitude methods, including color organization, the spinor helicity formalism, and factorization properties. I also describe the BCFW (on-shell) recursion relation at tree level, and explain how similar ideas - unitarity and on-shell methods - work at the loop level. These notes are based on lectures delivered at the 2012 CERN Summer School and at TASI 2013., Comment: 48 pages, 15 figures. Lectures presented at the 2012 European School of High-Energy Physics and at TASI 2013

We compute the interference between the resonant process $pp\ensuremath{\rightarrow}H(\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})+2\text{ }\text{jets}$ and the corresponding continuum background at leading order in QCD. For the Higgs signal, we include gluon fusion (GF) and vector boson fusion (VBF) production channels, while for the background we consider all tree-level contributions, including pure EW effects ($\mathcal{O}({\ensuremath{\alpha}}_{\mathrm{QED}}^{4})$) and QCD contributions ($\mathcal{O}({\ensuremath{\alpha}}_{\mathrm{QED}}^{2}{\ensuremath{\alpha}}_{s}^{2})$), plus the loop-induced gluon-initiated process. After convolution with the experimental mass resolution, the main effect of the interference is to shift the position of the mass peak, as in the inclusive GF case studied previously. The apparent mass shift is small in magnitude but strongly dependent on the Higgs width, potentially allowing for a measurement of, or bound on, the width itself. In the $H(\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})+2\text{ }\text{jets}$ channel, the VBF and GF contributions generate shifts of opposite signs which largely cancel, depending on the sets of cuts used, to as little as 5 MeV (toward a lower Higgs mass). The small magnitude of the shift makes this channel a good reference mass for measuring the inclusive mass shift of around 60 MeV in the Standard Model.

Electroweak vector-boson production, accompanied by multiple jets, is an important background to searches for physics beyond the Standard Model. A precise and quantitative understanding of this process is helpful in constraining deviations from known physics. We study four key ratios in $W + n$-jet production at the LHC. We compute the ratio of cross sections for $W + n$- to $W + (n-1)$-jet production as a function of the minimum jet transverse momentum. We also study the ratio differentially, as a function of the $W$-boson transverse momentum; as a function of the scalar sum of the jet transverse energy, $H_T^{\rm jets}$; and as a function of certain jet transverse momenta. We show how to use such ratios to extrapolate differential cross sections to $W+6$-jet production at next-to-leading order, and we cross-check the method against a direct calculation at leading order. We predict the differential distribution in $H_T^{\rm jets}$ for $W+6$ jets at next-to-leading order using such an extrapolation. We use the BlackHat software library together with SHERPA to perform the computations., RevTex, 60 pages, 25 tables, 19 figures

We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar $\mathcal{N} = 4$ super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a $\bar{Q}$ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N$^3$LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. We also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms., Comment: 74 pages, 12 figures, 7 tables

We describe the recently developed on-shell bootstrap for computing one-loop amplitudes in non-supersymmetric theories such as QCD. The method combines the unitarity method with loop-level on-shell recursion. The unitarity method is used to compute cut-containing parts of amplitudes, and on-shell recursion is used for the remaining rational terms., 10 pages, 9 figures, write-up of talks given by Z. Bern and D. A. Kosower at Loops & Legs 2006, Eisenach, Germany; v2: added reference

We review how (dimensionally regulated) scattering amplitudes in N=4 super-Yang-Mills theory provide a useful testing ground for perturbative QCD calculations relevant to collider physics, as well as another avenue for investigating the AdS/CFT correspondence. We describe the iterative relation for two-loop scattering amplitudes in N=4 super-Yang-Mills theory found in (C. Anastasiou et al., Phys. Rev. Lett. 91:251602 (2003),) and discuss recent progress toward extending it to three loops.

An overview of the calculation of the two-loop helicity amplitudes for scattering of two gluons into two photons is presented. These matrix elements enter into the recent improved calculation of the QCD background to Higgs boson decay into a pair of photons, which is the preferred search mode at the LHC for the case of a light Higgs boson.

We present next-to-leading-order QCD predictions for cross sections and for a comprehensive set of distributions in $\ensuremath{\gamma}\ensuremath{\gamma}+2\text{\ensuremath{-}}\mathrm{jet}$ production at the Large Hadron Collider. We consider the contributions from loop amplitudes for two photons and four gluons, but we neglect top quarks. We use BlackHat together with SHERPA to carry out the computation. We use a Frixione cone isolation for the photons. We study standard sets of cuts on the jets and the photons and also sets of cuts appropriate for studying backgrounds to Higgs-boson production via vector-boson fusion.

We introduce a generating function for the coefficients of the leading logarithmic BFKL Green’s function in transverse-momentum space, order by order in αS , in terms of single-valued harmonic polylogarithms. As an application, we exhibit fully analytic azimuthal-angle and transverse-momentum distributions for Mueller-Navelet jet cross sections at each order in αS . We also provide a generating function for the total cross section valid to any number of loops., Journal of High Energy Physics, 2014 (2), ISSN:1126-6708, ISSN:1029-8479