# Scattering Amplitudes: from Geometry to Experiment (SAGEX)

The Centre for Research in String Theory has been invited to coordinate the preparation of the grant agreement of the Marie Skłodowska-Curie Innovative Training Network of the European Commission: * SAGEX "Scattering Amplitudes: from Geometry to Experiment". *The network will support

**15 Early-Stage Researcher (ESR) PhD positions to begin in 2018/19. Project descriptions can be found below.**

**Applications will be accepted from April 2018. **The positions will all last three years, and will allow all students to participate in an exciting programme comprising schools, workshops, and secondments at academic as well as private-sector partners (see the overview of the project below). The positions will be hosted at the following beneficiaries:

- QMUL (2 PhD positions, ESR 1 and ESR 2)
- Humboldt University Berlin (2 PhD positions, ESR 3 and ESR 4)
- Deutsches Elektronen-Synchrotron (DESY) (2 PhD positions, ESR 5 and ESR 6)
- Durham University (2 PhD positions, ESR 7 and ESR 8)
- Niels Bohr Institute (2 PhD positions, ESR 9 and ESR 10)
- CEA Saclay (2 PhD positions, ESR 11 and ESR 12)
- Trinity College Dublin (2 PhD positions, ESR 13 and ESR 14)
- RISC/RISC GmbH (1 PhD position, ESR 15)

The SAGEX consortium also includes academic partner organisations:

- University of Oxford
- ETH Zurich
- University of California, Los Angeles
- SLAC National Accelerator Laboratory (Stanford University)
- University of Hamburg;

plus private sector partners

- Wolfram Research
- Maplesoft
- Danske Bank
- DreamQuark
- Mærsk Tankers
- Milde Marketing Science Communication
- Ekaterina Eremenko Films.

**Overview of the project and of the training offered to all Early Stage Researchers of SAGEX**

*Scattering amplitudes, describing the observations of high-energy collider experiments, provide a window into the fundamental structures predicted by relativistic quantum theories. By identifying and exploiting seemingly disparate **concepts from abstract geometry, symbolic big data, and phenomenological calculations, the SAGEX network will train the next generation of researchers in the new tools, approaches and insights that will make possible previously intractable analyses directly relevant to current and near-future particle physics experiments. Assembling an unprecedented team of top scholars in mathematics, theoretical physics, and symbolic computation with major industry partners including Wolfram Research, Maplesoft, and RISC GmbH, we intend to leverage our successes, expertise, and world-class scientific challenges to provide a unique training opportunity for Early Stage Researchers (ESRs) in Europe.*

*Training of the network's ESRs will comprise an integrated curriculum of local and intensive network courses, schools, and engagement with active cutting-edge research. They will be seconded to at least one academic and one private sector **partner of the project, building bridges between academic and industrial communities. SAGEX is set up to integrate several existing and highly recognised annual conferences with a series of new schools, workshops and industry partnerships. Through developing invaluable analytic, computational and soft skills, the ESRs emerging from this action will be eminently employable with the potential to become the next generation of European leaders in academia, industry, and the public sector. Finally, the training we offer, and the research carried out, will be made open and available, boosting not only Europe's continued leadership in the field of high-energy theoretical physics, but strategically allowing non-partner institutions with less domain experience to train scholars in relevant, in-demand skills.*

**Scientists in charge of the projects**

- QMUL: Andreas Brandhuber and Gabriele Travaglini
- Humboldt University Berlin: Jan Plefka and Matthias Staudacher
- Deutsches Elektronen-Synchrotron: Johannes Blümlein and Volker Schomerus
- Durham University: Patrick Dorey and Paul Heslop
- Niels Bohr Institute: Emil Bjerrum-Bohr and Poul Henrik Damgaard
- CEA Saclay: John Joseph Carrasco and David Kosower
- Trinity College Dublin: Ruth Britto and Tristan McLoughlin
- RISC/RISC GmbH: Carsten Schneider

**Interested applicants are strongly encouraged to contact as soon as possible the scientists in charge of the projects.**

## How to apply and eligibility requirements

The application process will open in April 2018. Relevant information on how to apply will appear on this site. Positions at different institutions will require separate applications. *Eligibility requirements:* There are strict *eligibility requirements* within Marie Skłodowska-Curie Innovative Training Networks. At the time of appointment, applicants must not have resided or carried out their main activity (work, studies) in the country for more than 12 months in the 3 years immediately before their appointment; AND shall also be in the first four years of their research careers at the time of appointment and have not been awarded a doctoral degree.

## Contact

Please email directly the scientist(s) in charge of the project(s) you are interested in.

**List of available projects:**

**ESR 1 (QMUL): Form factors and Higgs amplitudes from N = 4 super Yang-Mills to QCD ****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.

**ESR 2 (QMUL): From amplitudes to the dilatation operator of N = 4 super Yang-Mills****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.

**ESR 3 (Humboldt): Uncovering the kinematic algebra behind colour-kinematic duality****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

**ESR 4 (Humboldt): Integrability for amplitudes and correlators****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.

**ESR 5 (DESY): 4D ambitwistor theory for N = 8 supergravity****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.

**ESR 6 (DESY): Differential equations for phase-space integrals and Cutkosky rules****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.

**ESR 7 (Durham): Amplitudes and correlation functions as generalised polytopes****Objectives**: The Amplituhedron gives a description of planar amplitudes in N = 4 SYM as a purely geo-metrical 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.

**ESR 8 (Durham): Perturbative simplicity in lower dimensions****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 Feyn-man 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.

**ESR 9 (CEA): Local loop-level recursion for nonplanar theories**

Objectives: We know through generalised unitarity methods that tree-level data encodes all necessary in-formation 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.

**ESR 10 (CEA): Two-loop QCD amplitudes for next-to-next-to-leading order calculations at the LHC****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: Repro-duce 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.

**ESR 11 (NBI): Scattering equations, kinematic algebra and tree and loop amplitudes****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.

**ESR 12 (NBI): Applications of amplitude results in effective field theory****Objectives**: Several powerful amplitude techniques will be applied to Standard Model extensions with effec-tive 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.

**ESR 13 (TCD): Soft limits and symmetries in perturbative gauge theory and gravity****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.

**ESR 14 (TCD): Perturbative amplitude computations from integrability****Objectives**: Integrability of planar N = 4 SYM is a powerful tool for computing certain correlation functions and amplitudes at any coupling, however its conjectured validity remains unproven and deeply intriguing. Our goal is to learn how integrability emerges, at all loops, by analysing explicit perturbative computations of amplitudes and correlation functions in SYM. The necessary control over computations will be achieved by introducing deformations such as twists, q-deformations and Wilson lines that suppress or enhance different pieces in the sum over Feynman diagrams, while preserving the property of integrability.**First supervisor**: Britto. **Second supervisor**: McLoughlin. **Mentor**: Dorey.**Milestones and expected results:** First, identify a "simplifying" deformation in N = 4 SYM where higher-loop integrability can be derived from first principles. Second, understand the non-deformed case, and apply the insights gained to discover new algebraic relations between loop integrals.**Planned Secondments**: Three months to Wolfram; short term visits to other partners (UDUR, HU). Further secondment at Danske Bank, DreamQuark, Mærsk, or Milde Marketing.

**ESR 15 (RISC/RISC GmbH): Computer algebra for special functions****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.