# Amplitudes 2017 Summer School

from to (UTC)
at Appleton Tower ( Lecture Theatre 2 )
11 Crichton Street, Edinburgh, EH8 9LE, UK
 Description The study of scattering amplitudes has developped substantially over the past few years, with applications ranging from precision calculations for the LHC through new approaches to gauge theories and gravity and ending with pure mathematics. The annual meeting Amplitudes 2017 workshop will be hosted by the Higgs Centre between 10-14 July. This present summer school is held in the week preceding the workshop, and it offers an introduction to some of the central topics to be discussed at the workshop. Five lecturers will deliver sets of lectures and tutorials over the full 5 days of 3-7 July.
Go to day
• Monday, 3 July 2017
• 10:15 - 10:45 Registration 30'
• 10:50 - 11:00 Welcome 10'
• 11:00 - 11:50 1001 tales of gluons - lecture 1 50'  Speaker: Stefan Weinzierl (University of Mainz) Material:
• 12:00 - 12:50 Multiloop Scattering Amplitudes and Cluster Algebras - lecture 1 50'  Speaker: Marcus Spradlin (Brown University) Material:
• 13:00 - 14:30 Lunch Break
• 14:30 - 15:20 1001 tales of gluons - lecture 2 50'  Speaker: Stefan Weinzierl (University of Mainz) Material:
• 15:30 - 16:00 Coffee break
• 16:00 - 16:50 Multiloop Scattering Amplitudes and Cluster Algebras - lecture 2 50'  Speaker: Marcus Spradlin (Brown University) Material:
• 17:00 - 17:20 Coffee break
• 17:20 - 17:45 Azurite: finding master Integrals using algebraic geometry 25'
For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. I will introduce the package Azurite (arXiv:1612.04252) which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized- unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems Singular and Mathematica. I will also discuss some recent progresses and application of Azurite.
 Speaker: Alessandro Georgoudis (Uppsala University) Material:
• 17:50 - 18:15 Amplitudes and Correlator Integrands to Ten Loops Using Graphical Bootstraps 25'
We introduce the "triangle" and "pentagon" rules for obtaining contributing coefficients to the four-point correlation function and scattering amplitude integrands in planar, maximally supersymmetric Yang-Mills theory. Combining these graphical rules with the rung rule prove powerful enough to fully determine both functions through ten loops. We illustrate their applications, compare their relative strengths for fixing coefficients, and if time permits, survey some of the novel features of the higher-loop expressions.
 Speaker: Vuong-Viet Tran (Durham University) Material:
• 18:20 - 19:30 Welcome reception
• Tuesday, 4 July 2017
• 09:30 - 10:20 1001 tales of gluons - lecture 3 50'  Speaker: Stefan Weinzierl (University of Mainz) Material: Video
• 10:30 - 11:00 Coffee break
• 11:00 - 11:50 1001 tales of gluons - tutorial 50'  Material:
• 12:00 - 12:50 Multiloop Scattering Amplitudes and Cluster Algebras - lecture 3 50'  Speaker: Marcus Spradlin (Brown University) Material:
• 13:00 - 14:30 Lunch break
• 14:30 - 15:20 1001 tales of gluons - lecture 4 50'  Speaker: Stefan Weinzierl (University of Mainz) Material:
• 15:30 - 16:00 Coffee break
• 16:00 - 16:50 Multiloop Scattering Amplitudes and Cluster Algebras - lecture 4 50'  Speaker: Marcus Spradlin (Brown University) Material: Video
• 17:00 - 17:20 Coffee break
• 17:20 - 17:45 Maximal Cuts in Arbitrary Dimension 25'
After recalling the basics of generalized unitarity and maximal cuts, I will discuss recent work with Mads S{\o}gaard and Yang Zhang (1704.04255):
We discuss a procedure for computing maximal cuts in general D dimensions, based on the Baikov representation which simplifies the structure of the cuts. In our procedure, for a given topology, the integration splits into several regions and the number of independent regions is equal to the number of master integrals for the topology. The functions thus obtained inherit IBP and dimension shift identities from the uncut integral. Moreover, the functions, corresponding to the master integrals, obtained from the different regions, form the Wronskian matrix for the system of differential equations on the maximal cut. The discussion will be guided by an explicit example (probably the massless double box, but this remains to be decided).
 Speaker: Jorrit Bosma (ETH, Zurich) Material:
• 17:50 - 18:15 Combinatorics and Topology of Kawai–Lewellen–Tye Relations 25'
We study surprising connections between algebraic topology and string theory amplitudes which underlie Kawai–Lewellen–Tye (KLT) relations. For instance, we show that entries of the inverse KLT kernel can be understood as intersection numbers of certain cycles. In the field-theory limit, combinatorics of these intersections gives rise to the bi-adjoint scalar theory.
 Speaker: Sebastian Mizera (Perimeter Institute for Theoretical Physics) Material:
• Wednesday, 5 July 2017
• 09:30 - 10:20 Graphical Methods for Sharp Prediction: from theories entirely formal to the utterly effective - lecture 1 50'  Speaker: John Joseph Carrasco (CEA, Saclay) Material:
• 10:30 - 11:00 Coffee break
• 11:00 - 11:50 Graphical Methods for Sharp Prediction: from theories entirely formal to the utterly effective - lecture 2 50'  Speaker: John Joseph Carrasco (CEA, Saclay) Material:
• 12:00 - 12:50 Grassmannian Geometry of Scattering Amplitudes - lecture 1 50'  Speaker: Jaroslav Trnka (University of California, Davis) Material:
• 13:00 - 14:30 Lunch break
• 14:30 - 15:20 Grassmannian Geometry of Scattering Amplitudes - lecture 2 50'  Speaker: Jarosalav Trnka (University of California, Davis) Material:
• 15:30 - 16:00 Coffee break
• 16:00 - 16:50 Feynman Integrals - lecture 1 50'  Speaker: Johannes Henn (Mainz University)
• 17:00 - 17:20 Coffee break
• 17:20 - 17:45 Two-Loop Fundamental Colour-Kinematics Duality 25'
The study of loop-level scattering amplitudes in (super-)Yang-Mills theories has seen enormous progress over the last decade or so.
Unfortunately, many of these developments are limited to the planar limit which consists of only planar Feynman diagrams. Meanwhile, loop-level (super-)gravity amplitudes are difficult to calculate with ordinary Feynman diagrams. In this talk I will introduce Bern, Carrasco & Johansson’s (BCJ’s) colour-kinematics duality. The duality recycles planar tree-level amplitudes into full-colour loop- level amplitudes, while also giving loop-level gravity amplitudes by a double copy. I will also discuss the inclusion of fundamental matter multiplets, and apply these techniques to N=2 supersymmetry with extra hypermultiplets at two loops.
 Speaker: Gustav Mogull (University of Edinburgh) Material:
• 17:50 - 18:15 Unifying Relations for Scattering Amplitudes 25'
The talk is based on 1612.00868 and work to appear with Clifford Cheung and Congkao Wen.
We derive new scattering amplitudes relations which reveal a hidden unity among theories of diverse spins in general spacetime dimensions. These relations are built from a set of Lorentz invariant differential operators which transmute states of higher spin into states of lower spin. Transmutation takes amplitudes of gravity coupled to a dilaton and two-form into any amplitude in Einstein- Yang-Mills, the special Galileon, and (Dirac-)Born-field theory. Starting from Yang-Mills amplitudes, the transmutation covers any amplitude in gauged bi-adjoint scalar theory and the non-linear sigma model. As a corollary, known properties of gauge theory and gravity amplitudes like the Weinberg soft theorems, color-kinematics duality, the KLT relations, and the CHY construction are trivially inherited by the transmuted amplitudes. In the case of the non-linear sigma model, the recently found kinematic algebra becomes obvious from this construction.
 Speaker: Chia-Hsien Shen (Caltech) Material:
• Thursday, 6 July 2017
• 09:30 - 10:20 Feynman Integrals - lecture 2 50'  Speaker: Johannes Henn (Mainz University)
• 10:30 - 11:00 Coffee break
• 11:00 - 11:50 Feynman Intergrals - tutorial 50'  Speaker: Johannes Henn (Mainz University)
• 12:00 - 12:50 Graphical Methods for Sharp Prediction: from theories entirely formal to the utterly effective - lecture 3 50'  Speaker: John Joseph Carrasco (CEA, Saclay) Material:
• 13:00 - 14:30 Lunch break
• 14:30 - 15:20 Grassmannian Geometry of Scattering Amplitudes - lecture 3 50'  Speaker: Jaroslav Trnka (University of California, Davis) Material:
• 15:30 - 16:00 Coffee break
• 16:00 - 16:50 Feynman Integrals - lecture 3 50'  Speaker: Johannes Henn (Mainz University)
• 17:00 - 17:20 Coffee break
• 17:20 - 17:45 Determining the Soft Anomalous Dimension from General Constraints 25'
Bootstrapping techniques---wherein one imposes known constraints on an ansatz in the hope of uniquely determining the answer---have proven to be an increasingly powerful method for computing amplitudes in recent years. While the most significant progress in this direction has been made in planar maximally supersymmetric Yang- Mills theory, these techniques can be naturally extended to the types of quantities directly relevant to collider computations. One place these methods show particular promise is in the resummation of large logarithms, and we here report the successful bootstrap of the (correction to the all-order dipole contribution) of the massless soft anomalous dimension at three loops, up to an overall numerical factor.
 Speaker: Andrew McLeod (SLAC, Stanford University) Material:
• 17:50 - 18:15 Exploring Reggeon bound states in strongly coupled N=4 super Yang- Mills theory 25'
I will discuss scattering amplitudes in strongly coupled N=4 super Yang-Mills theory in the multi-Regge limit. In the strong coupling regime, the calculation of the scattering amplitudes reduces to the solution of a system of non-linear, coupled integral equations through the AdS/CFT-correspondence. These equations reduce to standard Bethe ansatz equations in the multi-Regge limit. In this kinematical regime, scattering amplitudes in N=4 SYM are physically described by bound states of effective particles, the so-called Reggeons. The six- point remainder function is fully described by a two-Reggeon bound state and is known at finite coupling. At eight external gluons a three-Reggeon bound state starts to appear about which much less is known. We investigate the three-Reggeon bound state in the strong coupling regime and try to make some generalisations to multiple-Reggeon bound states.
 Speaker: Theresa Abl (ETH, Zurich) Material:
• 19:15 - 22:30 Dinner ( Royal Scots Club )
arrive 19.15-19.30
• Friday, 7 July 2017
• 09:30 - 10:20 Grassmannian Geometry of Scattering Amplitudes - lecture 4 50'  Speaker: Jaroslav Trnka (University of California, Davis) Material:
• 10:30 - 11:00 Coffee break
• 11:00 - 11:50 Grassmannian Geometry of Scattering Amplitudes - tutorial 50'  Speaker: Jaroslav Trnka (University of California, Davis) Material:
• 12:00 - 12:50 Graphical Methods for Sharp Prediction: from theories entirely formal to the utterly effective - lecture 4 50'  Speaker: John Joseph Carrasco (Institute for Theoretical Physics (IPhT), CEA-Saclay and University of Paris-Saclay) Material:
• 13:00 - 14:30 Lunch break
• 14:30 - 15:20 Feynman Integrals - lecture 4 50'  Speaker: Johannes Henn (Mainz University)
• 15:30 - 16:00 Coffee break
• 16:00 - 16:50 Feynman Integrals - lecture 5 50'  Speaker: Johannes Henn (Mainz University)
• 17:00 - 17:20 Coffee break
• 17:20 - 17:45 On-shell diagrams and Ambitwistor strings in N=8 supergravity 25'
I will discuss how on-shell diagrams for N=4 Super Yang Mills can be extended to N=8 Supergravity, giving Grassmannian integral formulae for tree-level amplitudes.
I will then describe the connection between these formulae and equivalent expressions for the same amplitudes arising from ambitwistor string theory.
 Speaker: Joe Farrow (Durham University) Material:
• 17:50 - 18:15 From BCJ double copy to inflationary cosmology 25'
The BCJ double copy relates the scattering amplitudes from gauge and gravity theories. One such theory corresponds to a model first developed by Volkov and Akulov to describe Goldstone fermions (associated with spontaneous supersymmetry breaking). It was later realized that the Volkov-Akulov model is equivalent to the same constrained (nilpotent) superfields appearing in the cosmological models known as alpha-attractors. In this talk I'll explore some possible applications of the double copy procedure to inflationary cosmology.
 Speaker: Andres Luna-Godoy (University of Glasgow) Material: