For more than 30 years the study of scattering amplitudes in quantum field theory, gravity and string theory has sparked interest from a variety of different research fields. Motivations range from needing improved physical predictions, to searching for a new underlying mathematical formulation of a unified description of nature. Collider physics has been the main driver of developments in amplitude predictions yet this picture is now changing as astrophysical observations and gravitational wave detection are becoming more precise.
In this initiative we will study on-shell amplitudes and their application to precision predictions. The focus will be on developing new technology for analytic studies and efficient numerical evaluation able to meet the needs of modern experiments.
Analytic structure of on-shell amplitudes
Studying the analytic properties of on-shell amplitudes has led to remarkable discoveries of hidden simplicity in Quantum Field Theory. Amplitudes display a surprisingly rich mathematical structure and recent advances have led to fruitful interactions between different disciplines.
Quantum corrections from loop integrals (Feynman integrals) are of fundamental importance in any precision test of the laws of Nature. These integrals lead to a variety of interesting, yet intricate, special functions that must be understood before analytic representations can be considered. Identifying a basis of Feynman integrals, or special functions, is a key problem that prevents the improvement of theoretical predictions. The study of this problem becomes entangled with the study of differential equations that are imperative for numerical evaluation. Members of our team have even been investigating a new mathematical language for the vector space of Feynman integrals through intersection theory. We aim to develop general techniques that apply to a wide variety of amplitudes in different theories.
Amplitudes factorise in kinematic limits such as the infrared collinear limit and high-energy multi-regge limit. Through factorisation, universal building blocks are identified that can be used to build approximate amplitudes for high multiplicity and higher loop orders that capture the leading behaviour in the limit. Symmetries and universal behaviour provide strong analytic constraints on on-shell quantities. As highlighted by studies in maximally supersymmetric Yang-Mills theory, this information provides valuable input into extremely powerful bootstrap techniques (constraining a concise ansatz). Members of our initiative have played a lead role in the development of such techniques, as well as unitarity and integrand reduction methods, that apply also in physical theories.
Our aim is to develop a universal framework for the description of perturbative amplitudes through a deeper understanding of the underlying analytic structure.
Computational methods for multi-scale amplitudes
Current and future experiments require high order theoretical predictions for processes with many loops and external legs. Including mass effects is becoming increasingly important and leads to a rapid growth in complexity as more scales are included. One of the key objectives for this initiative will be to address the missing technology preventing further progress. We will focus on methods for multi-loop amplitudes at the precision frontier with multiple internal and external mass scales and build technology that will lead to a better automation of their evaluation.
As perturbative complexity grows, we reach a point where analytic expressions become overwhelming. Numerical techniques can offer a way to sidestep this issue but can be extremely computationally expensive. Members of our team have shown that semi-numerical solutions to differential equations can be a practical and efficient method to obtain phenomenological predictions when the mathematics of the function space is yet to be understood. Members of our initiative have pioneered finite field and functional reconstruction techniques that have become a key ingredient in recent state-of-the-art loop calculations. Within this approach, complex algebraic computations are replaced with fast but exact numerical evaluation over finite fields and full results are reconstructed from multiple numerical evaluations. The technique side steps large intermediate expressions in determining multi-scale amplitudes. The reduction of loop integrals into a basis of master integrals is currently one of the main bottlenecks of multi-loop and multi-scale predictions. New promising mathematical and computational methods will be developed to tackle more challenging problems of phenomenological relevance. The use of syzygy equations and techniques from algebraic and differential geometry, in combination with the aforementioned finite fields methods, has been shown to reduce the complexity of the system of identities satisfied by loop integrals by an order of magnitude, in recent results published by members of this initiative. We will continue to explore this direction to uncover deeper mathematical structures in loop integrals and use them – in combination with finite-field technology – for a more direct and efficient integral decomposition strategy.
Applications to high-energy particle physics
Turning modern amplitude methods into practical predictions requires a substantial level of expertise and effort. Using the knowledge of both the latest mathematical methods and the depth of phenomenological experience from our team members we aim to make a substantial impact on future precision collider predictions.
Progress in experimental techniques has led to increasingly precise measurements of scattering processes sensitive to the Standard Model (SM). Comprehensive tests of the SM therefore require theoretical developments on multiple fronts. Next-to-leading order (NLO) QCD corrections, which contain both one-loop amplitudes and tree-level real radiation required to cancel infrared (IR) divergences, have been almost completely automated and have become the new standard for theory prediction. Such precision is often insufficient to match experimental error and therefore an increasing range of NNLO computations are being performed as well as loop induced NLO corrections both of which require two-loop amplitudes. Mass effects and mixed QCD-EW also become mandatory if we hope to fully control perturbative uncertainties. Our team members work on a broad range of key observables measured at the LHC including the Higgs boson and its transverse momentum distributions, QCD effects in mixed photon and jet final states, Wbb and neutral current Drell-Yan processes. In all cases recent computations have been able to improve the known accuracy of phenomenological studies. Higher order corrections are also imperative for other experiments such as MUonE at CERN where a complete set of NNLO QED corrections for muon-electron scattering have recently been obtained.
In all cases our objectives are to provide flexible implementations of the theoretical tools, amplitude evaluations, IR subtraction and phase space integration, leading to a comprehensive set of differential distributions. Key processes that remain just beyond the current technology will be the main focus such as three particle final states with top quarks, Higgs and vector bosons, photons and jets.
Applications to Gravitational wave physics
The detection of gravitational waves by the LIGO and Virgo collaborations, and the anticipated huge increase in detection through LISA and the next-generation ground-based observatories requires a comparable improvement in waveform predictions from the theory community. Motivated by this fact, it has been shown that the Hamiltonian for interaction of two gravitational-wave-emitting black holes in a binary system can be studied using modern amplitude-based and on-shell techniques. Computational methods have been developed to allow analysis of physical observables like the scattering angle and the momentum impulse, which can be computed in the weak-field regime through post-Newtonian (PN) or post-Minkowskian (PM) expansions, evaluated through multi-loop scattering amplitudes within Effective Field Theory (EFT) descriptions of General Relativity.
Proposed activities and role of the various Research Units
Each unit brings a unique perspective on the goals of this amplitudes initiative which we outline below. In addition, we highlight activities in which all nodes will participate. Further, we plan to organise regular group meetings, where the progress of the various projects is discussed. Specific projects organised by topic are presented below:
Analytic structure of on-shell amplitudes
High energy behaviour [LNF, TO]
Direct evaluation of multi-leg amplitudes in general kinematics at one-loop level is highly demanding, and the extension to two-loops and beyond is ever more so. Even when such amplitudes are known, bringing them to a compact form and evaluating them efficiently are in general unsolved problems. Understanding the behaviour of amplitudes in various limits, where drastic simplifications occur, is a key strategy towards understanding them in general kinematics. The study of three-loop four-point amplitudes in the Regge limit has delivered powerful constraints on the structure of the same amplitudes in general kinematics. Studying amplitudes in the multi-Regge limit has the potential to unravel the analytic structure, as well as the colour structure, of multi-leg amplitudes. In particular, in the multi-Regge limit the state of the art is about understanding the structure of one-loop six-point amplitudes and two-loop five-point amplitudes. The LNF and TO nodes will tap their mutual expertise in general and in multi-Regge kinematics in order to evaluate, and to study the analytic structure of, two-loop five-point amplitudes.
Multi-collinear factorisation at two-loops [LNF, TO]
Progress on analytic five-point two-loop amplitudes with an off-shell leg opens a new window into the high order behaviour of QCD. Processes such as H/W/Z+4 partons contain information about triple collinear splitting (two-loop double unresolved radiations) that contributes to IR singularities at N4LO. We explore new methods for the direct extraction of these universal ingredients using direct solution of differential equations in the limit and finite field methods to avoid intermediate analytic expressions. We will also explore possible application of splitting functions for use as approximate amplitudes and for use in bootstrap methods.
Mathematical structures of scattering amplitudes [LNF, TO, PD, RM]
Analytic studies of scattering amplitudes have uncovered deep connections between Feynman integrals, Number theory and Geometry. At high loop orders with many mass scales (either internal or external), elliptic/hyper-elliptic curves or complex manifolds appear that can lead to new classes of special functions. In many cases these functions are poorly described and are still unsuitable for efficient numerical evaluation. Members of the LNF and RM nodes have unveiled the presence of such structures in the phenomenologically relevant process pp → H+jet at two-loops. Understanding the underlying geometry is a key challenge if progress is to be made. We will explore structures in differential equations and investigate the classes of functions that can appear in the next generation of precision predictions. Members of the PD node pointed to the existence of a vector space structure of Feynman integrals, by means of differential and algebraic topology. Intersection theory, Morse theory, and D-module theory will be employed to develop analytic and numerical methods for the evaluation of multi-scale Feynman integrals along with Euler-Mellin integrals and Aomoto-GKZ systems.
Computational methods for multi-scale amplitudes
Relations between Feynman integrals [BO, PD]
A key ingredient of higher-order predictions is the reduction of the Feynman integrals contributing to an amplitude to linearly independent subset of them called master integrals. The most common and successful reduction method is the Laporta algorithm that consists in generating and solving a large and sparse system of equations – also known as integration by parts (IBP) system – satisfied by the loop integrals, which includes IBP identities, Lorentz invariance identities and symmetry relations. The solution of the IBP system is a major bottleneck in high-precision phenomenological predictions. The BO and PD nodes will investigate new reduction methods that combine finite field technologies with techniques from algebraic geometry, transverse integration identities and connections with intersection theory, with the goal of unlocking new possibilities in phenomenology, especially for five-leg processes with massive internal and external states.
Differential equations [BO, PD, TO, NA, RM]
The master integrals contributing to an amplitude obey systems of differential equations (DEQs) thatcanbeobtainedbyreducingtheirderivativeswithrespecttotheinvariantsthey depend on into a linear combination of the master integrals themselves. The solution of these systems with suitable boundary conditions is currently the most powerful method for computing loop integrals. Additionally, the expansion of DEQs around special kinematic points, in alternative to the direct integration through asymptotic expansion, can be addressed within differential geometry, via restrictions of D-modules.
Members of the BO, PD and TO nodes will combine the expertise on DEQs and theory of restrictions to build new efficient analytic and semi-numerical methods for their solution and their application to cutting-edge phenomenological problems.
Evaluation of intersection numbers [BO, PD]
Intersection theory has been used to uncover the vector space structure of Feynman integrals. Linear relations, equivalent to IBPs, differential and difference equations, as well as quadratic relations can be derived by projections, using intersection numbers, which act as scalar products between integrals. Members of the BO and PD nodes plan to develop efficient algorithms for the evaluation of multivariate intersection numbers, combining number theory, differential geometry, combinatorics, and complex analysis, and to use them in the evaluation of scattering amplitudes and correlator functions.
Applications to High-Energy Particle Physics
Subtraction methods [LNF, NA]
In order for the theory predictions to match the accuracy of the experimental data at the LHC, cross sections and production rates are computed at NNLO accuracy. When fully differential NNLO computations are required, subtraction methods are used, which regularise the infrared divergences of the real radiation through a point-like subtraction. Members of the LNF and NA nodes have developed and adapted to hadron colliders a fully local subtraction scheme, which was previously known only for colourless initial states. In hadron collisions, the method has been developed so far for colourless final states. The plan is to extend the scheme to differential NNLO cross sections with any kind of final-state particles.
Scattering for five-particle processes with off-shell legs [BO, TO]
Two-loop QCD predictions for processes with five legs in which at least one is off-shell are a high priority for the next phase of LHC precision tests. Processes such as pp → W±/Z/H+2 jets require substantial effort to complete full-colour predictions involving the most complicated non-planar structures. Beyond these processes, amplitudes with more scales, such as pp → W+W-j, have interesting phenomenological applications but are not feasible with current technology. We will employ new methods for the efficient computation of these processes concentrating on finding representations suitable for fast and stable numeric evaluations. These amplitudes will then be deployed to make fully differential predictions at NNLO.
Scattering amplitudes for precision top quark physics [BO, TO]
Processes such as pp → ttj, pp → ttW±, pp → ttH are sensitive to important SM parameters such as the top mass and top Yukawa coupling. While they fall into the category of ‘two-loop five-point’ amplitudes the internal masses present a very different challenge to those with massless internal propagators. We will explore the use of semi-analytic techniques for the evaluation of missing ingredients in cases where analytic representations are not available or not feasible to use.
One of the key points of the Physics program of the LHC in the High Luminosity phase is the precise study of the properties of the Higgs boson. The Higgs sector, together with the heavy-quark sector of Particle Physics is the most promising sector where to look for New Physics beyond the Standard Model of fundamental interactions (SM). In order to reveal possible New Physics effects, however, it is mandatory to fully understand and control the SM prediction for a variety of inclusive and more exclusive observables. Members of the LNF, NA and RM nodes plan to perform a complete phenomenological analysis, including mass and renormalization-scheme effects (for top and bottom quarks) at NLO and NNLO, in particular for the Higgs total production cross section and for the pT distribution at low and high pT. The study will include QCD and QCD-EW mixed corrections, that at the needed level of precision start playing a significant role.
Scattering amplitudes and EFT operators [PD]
Effective Field Theory (EFT) can be combined with on-shell and unitarity-based methods to explore the UV properties of theories beyond the Standard Model, and in particular to calculate renormalization group coefficients such as beta functions and anomalous dimensions. Discontinuities of the form factors of the EFT composite operators can be calculated via phase-space integrals within the on-shell approach, from a product of tree-level amplitudes, even for two-loop renormalization group mixings. Equivalently, the coefficients of the UV divergences can be extracted from intersection numbers and IBPs.
These techniques will be applied to a number of new physics scenarios defined above the TeV scale (such as the Standard Model EFT, models entailing Axion-like particles, etc.) in order to analyse their impact on low-energy observables (like lepton flavour violating processes, electric dipole moments of quarks and leptons, etc.) occurring at or below the GeV scale. Such a large separation of scales demands the inclusion of running effects at two-loop order to obtain sensible predictions. While this is a tremendous task when approached with standard techniques, on-shell methods offer a simple, elegant and efficient way to reach this goal.
Applications to Gravitational Wave Physics
High-order PN-corrections [PD, NA]
Accurate modelling of waveforms for binary compact objects, accounting for spin and tidal effects, is essential to reveal insights into the internal structure of neutron stars and the nature of black holes, as well as to explore potential deviations from GR. We plan to compute 4- and 5-loop diagrams contributing to the effective potential for the compact binaries considering the spin and tidal interactions. Members at the PD node have developed automated codes based on amplitudes and Feynman integrals, which have proven to be very effective in the evaluation of the conservative Hamiltonian up to 5PN, and that we plan to extend it in order to reach the 6PN level.
High-order PM-corrections [PD, LNF]
In the gravity scattering of massive objects, such as black holes or neutron stars, the heavy-mass effective theory can be used for the efficient extraction of classical contributions from loop amplitudes, both in the conservative and in the radiative regimes. On-shell and the unitarity-based techniques and the Double-Copy relations can be used to produce compact expressions of integrands. The integral evaluation can proceed by means of IBPs and DEQs, to obtain the classical corrections. We plan to extend the application of this approach to multi-loop black-hole scattering and graviton bremsstrahlung.
