**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:** QMUL

**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. This project 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**: Travaglini. **Second supervisor**: White. **Mentor**: Anastasiou.

**Milestones and expected results:** First calculate 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; next perform related calculations of Higgs amplitudes, including finite top-mass effects, in QCD and compare with N = 4 SYM.

**Planned Secondments**: Three months to Wolfram; short term visits to other partners (ETH, NBI). Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

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

**Institute:**QMUL**
Objectives**: This project has a twofold goal. Firstly, we will 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. Next we will 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**: Brandhuber.

**Second supervisor**: White.

**Mentor**: Staudacher.

**Milestones and expected results**: Initially, compute the two-loop dilatation operator in subsectors such as SU(2|3) and SO(6); then determine the complete dilatation operator and understand Yangian symmetry at two loops.

**Planned Secondments:** Three months to Maplesoft; short term visits to other partners (HU, OU). Further secondment at Danske Bank, DreamQuark, Mærsk, 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**: Plefka.

**Second supervisor**: Broedel.

**Mentor**: McLoughlin.

**Milestones and expected results**: Begin with construction of numerators for six and seven gluon amplitudes and then extract generators; next use lessons learned to elucidate the infinite-dimensional symmetry algebra underlying the S-matrix.

**Planned Secondments**: Three months to Wolfram; short term visits to other partners (NBI, TCD). Further secondment at Danske Bank, DreamQuark, Mærsk or Milde Marketin

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

**Institute: **Humboldt University Berlin**
Objectives**: Correlation functions in N = 4 SYM in a lightlike 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 see 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, extending then to higher loops. The plan is to obtain non-trivial kinematics directly from integrability, starting with four-point functions of stress-tensor multiplets.

**First supervisor**: Staudacher.

**Second supervisor**: Broedel.

**Mentor**: Mason.

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

**Planned Secondments**: Three months to Maplesoft; short term visits to other partners (UDUR, OU). Further secondment at Danske Bank, DreamQuark, Mærsk, 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**: Schomerus.

**Second supervisor**: Teschner.

**Mentor**: Lipstein.

Milestones and expected results: Start out by computing supergravity amplitudes using ambitwistor strings at tree and then one-loop level; at a later stage extend this to higher loops and compare with known results.

**Planned Secondments**: Three months to Maplesoft; short term visits to other partners (UDUR, NBI). Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

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

**Institute: **DESY**
Objectives**: We will compute 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 at closing this gap.

**First supervisor**: Bluemlein.

**Second superviso**r: Moch.

**Mentor**: Schneider.

**Milestones and expected results**: First calculate phase-space integrals for 2 -> 2 for massless and massive particles; then systematically extend this and develop efficient methods for 2 -> n (n= 2, 3, 4).

**Planned Secondments:** Three months to RISC; short term visits to UH. Further secondment at Danske Bank, DreamQuark, Mærsk, 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**: Heslop.

**Second supervisor:**Lipstein.

**Mentor**: Carrasco.

**Milestones and expected results**: Find a generalised polytope interpretation for correlators and amplitudes at tree level for four- and five-point examples; using these initial results develop a systematic understanding for higher loops and more legs.

**Planned Secondments:** Three months to Wolfram; short term visits to other partners (ETH, NBI). Further secondment at Danske Bank, DreamQuark, Mærsk, 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:**Dorey.

**Second supervisor**: Heslop.

**Mentor**: McLoughlin.

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

**Planned Secondments:** Three months to Maplesoft; short term visits to other partners (NBI, TCD). Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

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

**Institute: **CEA Saclay**
Objectives:** We know 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**: Carrasco.

**Second supervisor:**Vanhove.

**Mentor**: Bern.

**Milestones and 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; short term visits to other partners (QMUL, UCLA). Fur-ther secondment at Danske Bank, DreamQuark, Mærsk, 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:**Kosower.

**Second supervisor:**Carrasco.

**Mentor**: Dixon.

**Milestones and 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; short term visits to other partners (DESY, SLAC) Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

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

**Institute: **Niels Bohr 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:**Bjerrum-Bohr.

**Second supervisor:**Bourjaily.

**Mentor**: Green.

Milestones and 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 partners (DESY, QMUL). Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

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

**Institute: **Niels Bohr 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**: Damgaard.

**Second supervisor**: Bjerrum-Bohr.

**Mentor**: Brandhuber.

**Milestones and expected results**: As a warm-up compute new higher-derivative corrections to Standard Model amplitudes which are relevant for the LHC at tree level; next extend this 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; short term visits to other partners (CEA, QMUL). Further secondment at Danske Bank, DreamQuark, Mærsk, 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**: McLoughlin.

**Second supervisor**: Britto.

**Mentor**: Plefka.

**Milestones and 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 (HU, UCLA). Further secondment at Danske Bank, DreamQuark, Mærsk, 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:**Britto.

**Second supervisor:**McLoughlin.

**Mentor:**Dorey.

Milestones and 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; short term visits to other partners (UDUR, HU). Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

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

**Institute: **RISC 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:**Schneider.

**Second superviso**r: Paule.

**Mentor**: Bluemlein.

**Milestones and 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. Short-term visits to UH, Danske Bank, DreamQuark, Mærsk, or Milde Marketing.