**Research projects**

The following research projects will be undertaken by the SAGEX Early Stage Researchers:

**Project 1: Form factors and Higgs amplitudes from N = 4 super Yang-Mills to QCD**

**Institute:** Queen Mary University of London

**Objectives**: Scattering amplitudes in QCD involving a Higgs boson and several gluons have fascinating connections to much simpler quantities, namely form factors of protected operators in N = 4 SYM. Project 1 will explore this connection in two ways: firstly, by studying form factors with four particles (corresponding to Higgs + four-gluon processes); and then by performing an analysis of the corrections due to the finiteness of the top quark mass, which can be described in the language of effective field theory, and correspond to form factors of higher-dimensional operators in N = 4 SYM.

**First supervisor**: GabrieleTravaglini.

**Second supervisor**: Chris White.

**Mentor**: Charalampos Anastasiou.

**Expected results:** Calculation of four-point form factor of half-BPS operators and three-point form factors of unprotected dimension six operators at two loops in N = 4 SYM; related calculations of Higgs amplitudes, including finite top-mass effects, in QCD and a comparison with N = 4 SYM.

**Planned Secondments**: Three months to Wolfram Research; short term visits to other network members (ETH Zurich, University of Copenhagen). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 2: From amplitudes to the dilatation operator of N = 4 super Yang-Mills

**Institute: **Queen Mary University of London

**
Objectives**: Compute the two-loop dilatation operator in N = 4 SYM in specific sectors such as the scalar SO(6) sector from two-point correlation functions using unitarity. Study the intriguing relation between the Yangian symmetry of the dilatation operator and that of the superamplitudes, with the goal of finding constraints which may eventually determine the complete two-loop dilatation operator of the theory, whose expression is presently not known.

**First supervisor**: Andreas Brandhuber.

**Second supervisor**: Chris White.

**Mentor**: Matthias Staudacher.

**Expected results:** Computation of the two-loop dilatation operator in subsectors such as SU(2|3) and SO(6); computation of the complete dilatation operator and understanding of Yangian symmetry at two loops.

**Planned Secondments:** Three months to Maplesoft; short term visits to other network members (Humboldt University, Oxford University). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 3: Uncovering the kinematic algebra behind colour-kinematic duality

**Institute: **Humboldt University Berlin

**
Objectives**: The colour-kinematic duality is a highly nontrivial property relating gauge and gravity amplitudes. Gravitational amplitudes can then be miraculously found by simply replacing colour factors in Yang-Mills amplitudes with kinematic ones. However, the algebraic underpinnings of this hidden kinematic algebra are largely unknown. Curiously the subsector of self-dual Yang-Mills was identified with area-preserving dif-feomorphisms known from the lightcone quantisation of the relativistic membrane. The goal is to find how this algebra extends outside the self-dual sector.

**First supervisor**: Jan Plefka.

**Second supervisor**: Johannes Broedel.

**Mentor**: Tristan McLoughlin.

**Milestones and expected results**: Construction of numerators for six and seven gluon amplitudes and extraction of generators; use lessons learned to elucidate the infinite-dimensional symmetry algebra underlying the S-matrix.

**Planned Secondments**: Three months to Wolfram Research; short term visits to other network members (University of Copenhagen, Trinity College Dublin). Further secondment at Danske Bank, DreamQuark, Maersk Tankers or Milde Marketing.

###### Project 4: Integrability for amplitudes and correlators

**Institute: **Humboldt University Berlin

**
Objectives**: Correlation functions in N = 4 SYM in a light-like limit yield amplitudes. Recently, integrable system descriptions for planar amplitudes and correlators have been proposed; the two approaches are related but significantly different. A first aim is to understand how the correlator integrability can be mapped to the construction of amplitudes in the limiting procedure. This is to be studied first on specific examples at lower loops, then extending then to higher loops. The goal is to obtain non-trivial kinematics directly from integrability, starting with four-point functions of stress-tensor multiplets.

**First supervisor**: Matthias Staudacher.

**Second supervisor**: Johannes Broedel.

**Mentor**: Lionel Mason.

**Milestones and expected results**: Test and extend the integrable systems description of planar amplitudes and correlation functions in N = 4 SYM in examples with the aim of extracting kinematics directly from integrability; study in detail the octagon operator for four-point functions.

**Planned Secondments**: Three months to Maplesoft; short term visits to other network members (Durham University, Oxford University). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 5: 4D ambitwistor theory for N = 8 supergravity

**Institute: **DESY

**
Objectives**: The goal is to set up a string-theoretic worldsheet description of four-dimensional N = 8 super-gravity and use it in order to study loop-level scattering amplitudes. Recently, an ambitwistor model has been described whose physical states and tree-level scattering amplitudes coincide with the ones of the supergravity. It is intended to set up a systematic loop expansion for this model (starting from one loop) and to study the resulting amplitudes by comparing special limits with available results from N = 8 supergravity.

**First supervisor**: Volker Schomerus.

**Second supervisor**: Georgios Papathanasiou.

**Mentor**: Arthur Lipstein.

**Expected results:** Computation of supergravity amplitudes using ambitwistor strings at tree and then one-loop level; an extension to higher loops and a comparison with known results.

**Planned Secondments**: Three months to Maplesoft; short term visits to other network members (Durham University, University of Copenhagen). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 6: Differential equations for phase-space integrals and Cutkosky rules

**Institute: **DESY

**
Objectives**: Computation of phase-space integrals for 2 -> n scattering processes of massless and massive particles, as needed for high-precision predictions at the LHC. Established techniques for higher-loop integrals include the method of differential equations and systematic ways to integrate the Laurent expansion in the parameter of dimensional regularisation in terms of e.g. hyperlogarithms over a given alphabet of words. Much less systematic work has been performed for phase-space integrals for high multiplicities of the final state, and this project aims to close this gap.

**First supervisor**: Johannes Bluemlein.

**Second supervisor:** Sven Moch.

**Mentor**: Carsten Schneider.

**Expected results**: Calculations of phase-space integrals for 2 -> 2 for massless and massive particles; a systematic extension of this, developing efficient methods for 2 -> n (n= 2, 3, 4).

**Planned Secondments:** Three months to RISC Software GmbH; short term visits to network partner University of Hamburg. Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 7: Amplitudes and correlation functions as generalised polytopes

**Institute: **Durham University**
Objectives**: The Amplituhedron gives a description of planar amplitudes in N = 4 SYM as a purely geometrical object, generalising the volume of a polytope in an extended twistor space. Correlation functions of gauge-invariant operators in N = 4 SYM are intimately related to planar amplitudes, giving them in multiple lightlike limits. Furthermore loop-level integrands are equivalent to higher-point tree-level correlators. The project will investigate and generalise this geometric structure both for correlators and amplitudes.

**First supervisor**: Paul Heslop.

**Second supervisor:** Arthur Lipstein.

**Mentor**: John Joseph Carrasco.

**Expected results**: A generalised polytope interpretation for correlators and amplitudes at tree level for four- and five-point examples; development of a systematic understanding for higher loops and more legs.

**Planned Secondments:** Three months to Wolfram Research; short term visits to other network members (ETH Zurich, University of Copenhagen). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 8: Perturbative simplicity in lower dimensions

**Institute: **Durham University

**
Objectives**: The exact S-matrices for a variety of integrable quantum field theories in two dimensions have been known for many years. These theories can often also be studied perturbatively using standard Feynman diagrams, where the integrability manifests itself in a priori surprising cancellations and simplifications, whose underlying mechanism is still ill-understood. This project will look at this phenomenon for affine Toda field theories, where hints of deeper structure already exist in, for example, the relationship between on-shell diagrams for singularities in amplitudes to planar projections of certain higher-dimensional polytopes. Our study will be firstly performed at tree level and then extended to loops.

**First supervisor:** Patrick Dorey.

**Second supervisor**: Paul Heslop.

**Mentor**: Tristan McLoughlin.

**Expected results**: Link integrability to miraculous Feynman diagram cancellations in tractable two-dimensional models, first at tree and then one-loop level; generalise to higher loops and identify underlying structures.

**Planned Secondments:** Three months to Maplesoft; short term visits to other network members (University of Copenhagen, Trinity College Dublin). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 9: Local loop-level recursion for nonplanar theories

**Institute: **CEA Saclay**
Objectives:** It is known through generalised unitarity methods that tree-level data encodes all necessary information for all-loop quantisation. Promoting this to analytic loop-level recursion would engender all-loop order insight through analysis of tree-level data, as well as providing a natural non-planar generalisation of the amplituhedron. For this to be useful for phenomenological theories, results must be amiable to integration and lining up with potential integral basis. This means achieving local representations. Here the power of the colour-kinematics to relate non-planar and planar information in a local graph basis has tremendous promise. It is likely sufficient to require colour-kinematics only up to edges privileged by recursion.

**First supervisor**: John Joseph Carrasco.

**Second supervisor:** Pierre Vanhove.

**Mentor**: Zvi Bern.

**Expected results**: Establish new multi-loop-level recursion relations, starting with finite-colour theories at four-point one-loop; this will subsequently be extended to higher loops and legs with the goal of recursing up to three-loops.

**Planned Secondments**: Three months to Wolfram Research; short term visits to other network members (Queen Mary University of London, UCLA). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 10: Two-loop QCD amplitudes for next-to-next-to-leading order calculations at the LHC

**Institute: **CEA Saclay

**
Objectives**: Future studies at the LHC will require precision calculations at the next-to-next-to-leading order (NNLO) in perturbative QCD, for processes with external quarks, gluons, electroweak vector bosons, photons, and Higgs bosons. The project will implement selected unitarity-based approaches for two-loop amplitudes into the existing BlackHat library. It will include the development of necessary two-loop integral libraries. The code will then be applied to NNLO phenomenology of selected processes.

**First supervisor:** David Kosower.

**Second supervisor:** John Joseph Carrasco.

**Mentor**: Lance Dixon.

**Expected results**: Development of a two-loop integral software library. Warmup: Reproduce the numerous known N = 4 SYM two-loop examples. Expected results: Two-loop numerical unitarity implementation; NNLO QCD phenomenology.

**Planned Secondments: **Three months to RISC Software GmbH; short term visits to other network members (DESY, SLAC) Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 11: Scattering equations, kinematic algebra and tree and loop amplitudes

**Institute: **University of Copenhagen

**
Objectives**: We will investigate the CHY formalism, realising explicit links to other insightful representations, e.g. the string-based Bern-Kosower rules, the Grassmannian and Amplituhedron formalism by Arkani-Hamed et al, and ambitwistor strings. We will also investigate how scattering equations can be best employed in practical computations, e.g. using the newly developed concepts of Q-cuts and integration rules for scattering equations. Connections between KLT and BCJ/monodromy relations will be studied with the goal of finding a kinematic algebra for amplitudes valid at tree level, first, and then loop level.

**First supervisor:** Emil Bjerrum-Bohr.

**Second supervisor:** Jacob Bourjaily.

**Mentor**: Michael Green.

**Expected results:** Initially relationships of scattering equations (CHY) to KLT, Grassmannian and Amplituhedron formalisms will be established at tree level; then this will be promoted to loop level and practical tools to evaluate scattering equations at loop level will be investigated.

**Planned Secondments**: Three months to Maplesoft; short term visits to other network members (DESY, Queen Mary University of London). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 12: Applications of amplitude results in effective field theory

**Institute: **University of Copenhagen

**
Objectives**: Several powerful amplitude techniques will be applied to Standard Model extensions with effective operator couplings, and we will systematically analyse corrections from higher-derivative couplings. As a first step we will compute tree amplitudes via on-shell recursion. Next, unitarity will be used to generate loop results. We will then explore how the standard unitarity-based programme can incorporate explicit cut-off scales in the (effective) theory. For Run 2 of the LHC, this will lead to efficient tests for the existence of higher-derivative corrections to the Standard Model, critical for identifying any such new physics.

**First supervisor**: Poul Damgaard.

**Second supervisor**: Emil Bjerrum-Bohr.

**Mentor**: Andreas Brandhuber.

**Expected results**: As a warm-up, computation of new higher-derivative corrections to Standard Model amplitudes which are relevant for the LHC at tree level; extension to the one- and possibly two-loop level and investigate the applicability of unitarity in the presence of cut-off scales.

**Planned Secondments**: Three months to Wolfram Research; short term visits to other partners (CEA, Queen Mary University of London). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 13: Soft limits and symmetries in perturbative gauge theory and gravity

**Institute: **Trinity College Dublin

**
Objectives**: How can the symmetries of gauge and gravitational theories can be used to constrain the form of amplitudes and form factors? Spontaneously broken symmetries are related to universal limits of amplitudes where one or more of the particles becomes soft. Our aim is to have a transparent formulation of the connection between symmetries and soft limits in a broad context of quantum field theories. A specific goal will be to understand to what extent such soft limits can be used to determine complete amplitudes in N = 4 SYM and N = 8 supergravity, first at tree and then loop level.

**First supervisor**: Tristan McLoughlin.

**Second supervisor**: Ruth Britto.

**Mentor**: Jan Plefka.

**Expected results**: First, find a relation between double-soft limits and asymptotic symmetry algebra for gauge bosons and gravitons for tree amplitudes; next extend these results to loop level and develop a generalised “inverse-soft” construction of amplitudes.

**Planned Secondments:** Three months to Maplesoft; short term visits to other partners (Humboldt University, UCLA). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

###### Project 14: Perturbative amplitude computations and integrability

**Institute: **Trinity College Dublin

**
Objectives:** The functions appearing in Yang-Mills theory are highly constrained, most notably with a high degree of supersymmetry. The aim is to describe the space of allowed functions, in view of physical constraints and integrability, and to characterise the functions in a way that leads to efficient computations. We will study properties of individual Feynman integrals and the nature of cancellations that take place in their sum.

**First supervisor:** Ruth Britto.

**Second supervisor:** Tristan McLoughlin.

**Mentor:** Patrick Dorey.

**Expected results:** First, analyse several known amplitudes in order to identify useful variables and uncover hidden structures. Then, formulate underlying principles and relations on which to base future computations.

**Planned Secondments:** Three months to Wolfram Research; short term visits to other partners (Durham University, Humboldt University). Further secondment at Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.

**Project 15: Computer algebra for special functions**

**Institute: **RISC/RISC Software GmbH

**
Objectives**: Many problems in SAGEX can be formulated as huge sums of complicated integrals or as strongly-coupled systems of difference/differential equations. Our goal is to discover better representations of these and extract the desired physical information. There are two key subtasks: 1. To generalise the existing summation/integration algorithms and recurrence/differential equation solvers to handle not only indefinite nested sums and integrals, but also elliptic functions. 2. To extend our symbolic toolbox for special functions to compute asymptotic expansions needed to handle functions in current and future calculations.

**First supervisor:** Carsten Schneider.

**Second supervisor:** Peter Paule.

**Mentor**: Johannes Bluemlein.

**Expected results**: First, write new freely available Mathematica packages for summation and integration and apply these to SAGEX problems. Second, explore new classes of special (e.g. elliptic) functions, and extend the Mathematica packages to cover these cases.

**Planned Secondments**: 2 months at DESY to complement computer algebra training with relevant physics; 3 months at Wolfram Research. Short-term visits to University of Hamburg, Danske Bank, DreamQuark, Maersk Tankers, or Milde Marketing.