## Final Report Summary - NONPLANAR (Simplifying the non-planar sector of gauge theory.)

The basic aim of this project was to build on new insights into so-called planar physics, and especially to use these insights to understand physics which is not planar. Planarity is a property of certain scattering amplitudes; these amplitudes are basic quantities in quantum mechanics. In essence an amplitude is a complex number whose square is the probability of a certain event occuring. These are important in the context of particle colliders such as the Large Hadron Collider (LHC) at CERN. At the LHC, we want to know what is the probability that two protons colliding at high energy produce a Higgs boson, for example. To do so, we compute a particular scattering amplitude and square it.

A common method for computing these amplitudes uses Feynman diagrams. The diagrams include lines for the various particles in the process: the incoming protons and outgoing Higgs boson, for example, in the case of Higgs production. There are also other lines to draw representing the force-carring particles exchanged by the protons in the collision, or other particles such as photons which can be created in the process. A planar amplitude is simply an amplitude whose Feynman diagrams can all be drawn on a sheet of paper, with incoming and outgoing particles around the sides of the paper, and with no lines crossing one another. Diagrams which are not planar are non-planar.

The study of scattering amplitudes in the last decades has been greatly enriched by the advent of the LHC. Physicists really wanted to know the probablility of Higgs boson production with high accuracy, as well as the probability of other kinds of process (for example processes involving supersymmetric particles, and processes which might look like they involve supersymmetric particles but in fact don't involve them at all.) With this strong motivation, people worked hard at the problem of how to compute the amplitudes. The trouble is that they seemed to be really quite hard to calculate: there can be very many Feynman diagrams, and each diagram can correspond to a very lengthy mathematical expression. There can also be mathematical difficulties evaluating this expression.

Hard work on this problem was richly rewarded. We have vastly improved our ability to calculate these amplitudes, especially in the planar case. When this project started, our project team had the aim of figuring out how to determine non-planar amplitudes given that planar amplitudes have become so simple.

There are several links between planar and non-planar amplitudes, but a particular tool that we highlighted were Jacobi relations. These are linear relations satisfied by the complicated parts of amplitudes, and which relate planar and non-planar amplitudes.

The principal result of the project is quite a surprise. We found a surprising link between physics of the kind which is important at the LHC, and gravity. Gravity is an intrinsically non-planar theory: it has no planar scattering amplitudes. But somehow Jacobi relations are important in gravity. Previous work by Bern, Carrasco and Johansson showed that one can obtain amplitudes in gravity beginning from amplitudes in Yang-Mills theory (the theory of the forces which bind the proton together, a generalisation of electrodynamics) provided that the amplitudes are arranged so that all the Jacobi identities are satisfied. In this sense gravity is related to the planar sector of Yang-Mills theory.

Our discovery was that classical solutions of gravity, such as the famous Schwarzschild black hole, are related to simple classical solutions of Yang-Mills theory. Specifically the Schwarzschild black hole is related to a point charge in ordinary electrodynamics.

Much of the rest of the work we performed was to build on this discovery. The main thrust of our work was to develop this connection into a tool which may be of use in another exciting area of physics which requires tough theoretical calculations to be completed. The new area in the physics of gravitational waves, emitted for example by binary black holes.

Gravitational wave emission from bound pairs of black holes is described by Einstein's general relativity. This is a beautiful physical theory which is celebrated for its balance of simplicity and richness. But one less ideal fact about general relativity is that it is very difficult to work with using the standard method of successive approximations, known as perturbation theory. In the context of black hole binaries this is a little unfortunate, because for most of the history of these systems perturbative methods are all that we have.

In fact the present situation in gravitational wave physics is quite like the situation facing LHC physics twenty or thirty years ago. Then physicists understood the importance of computing Higgs production amplitudes to high precision. But they didn't know how to do so. It is very timely, then, that progress on scattering amplitudes might be relevant for computing gravitational waveforms with the kind of precision that is required by future observatories such as LISA or the Einstein Telescope.

The final result of the project was a detailed paper describing precisely how scattering amplitudes can be used to determine the classical radiation emitted in a certain black hole process. The process is black hole scattering: it is still easiest to use amplitudes for processes when two black holes come in, and two go out. We hope in the future to build on this to the case where the black holes orbit one another.

A common method for computing these amplitudes uses Feynman diagrams. The diagrams include lines for the various particles in the process: the incoming protons and outgoing Higgs boson, for example, in the case of Higgs production. There are also other lines to draw representing the force-carring particles exchanged by the protons in the collision, or other particles such as photons which can be created in the process. A planar amplitude is simply an amplitude whose Feynman diagrams can all be drawn on a sheet of paper, with incoming and outgoing particles around the sides of the paper, and with no lines crossing one another. Diagrams which are not planar are non-planar.

The study of scattering amplitudes in the last decades has been greatly enriched by the advent of the LHC. Physicists really wanted to know the probablility of Higgs boson production with high accuracy, as well as the probability of other kinds of process (for example processes involving supersymmetric particles, and processes which might look like they involve supersymmetric particles but in fact don't involve them at all.) With this strong motivation, people worked hard at the problem of how to compute the amplitudes. The trouble is that they seemed to be really quite hard to calculate: there can be very many Feynman diagrams, and each diagram can correspond to a very lengthy mathematical expression. There can also be mathematical difficulties evaluating this expression.

Hard work on this problem was richly rewarded. We have vastly improved our ability to calculate these amplitudes, especially in the planar case. When this project started, our project team had the aim of figuring out how to determine non-planar amplitudes given that planar amplitudes have become so simple.

There are several links between planar and non-planar amplitudes, but a particular tool that we highlighted were Jacobi relations. These are linear relations satisfied by the complicated parts of amplitudes, and which relate planar and non-planar amplitudes.

The principal result of the project is quite a surprise. We found a surprising link between physics of the kind which is important at the LHC, and gravity. Gravity is an intrinsically non-planar theory: it has no planar scattering amplitudes. But somehow Jacobi relations are important in gravity. Previous work by Bern, Carrasco and Johansson showed that one can obtain amplitudes in gravity beginning from amplitudes in Yang-Mills theory (the theory of the forces which bind the proton together, a generalisation of electrodynamics) provided that the amplitudes are arranged so that all the Jacobi identities are satisfied. In this sense gravity is related to the planar sector of Yang-Mills theory.

Our discovery was that classical solutions of gravity, such as the famous Schwarzschild black hole, are related to simple classical solutions of Yang-Mills theory. Specifically the Schwarzschild black hole is related to a point charge in ordinary electrodynamics.

Much of the rest of the work we performed was to build on this discovery. The main thrust of our work was to develop this connection into a tool which may be of use in another exciting area of physics which requires tough theoretical calculations to be completed. The new area in the physics of gravitational waves, emitted for example by binary black holes.

Gravitational wave emission from bound pairs of black holes is described by Einstein's general relativity. This is a beautiful physical theory which is celebrated for its balance of simplicity and richness. But one less ideal fact about general relativity is that it is very difficult to work with using the standard method of successive approximations, known as perturbation theory. In the context of black hole binaries this is a little unfortunate, because for most of the history of these systems perturbative methods are all that we have.

In fact the present situation in gravitational wave physics is quite like the situation facing LHC physics twenty or thirty years ago. Then physicists understood the importance of computing Higgs production amplitudes to high precision. But they didn't know how to do so. It is very timely, then, that progress on scattering amplitudes might be relevant for computing gravitational waveforms with the kind of precision that is required by future observatories such as LISA or the Einstein Telescope.

The final result of the project was a detailed paper describing precisely how scattering amplitudes can be used to determine the classical radiation emitted in a certain black hole process. The process is black hole scattering: it is still easiest to use amplitudes for processes when two black holes come in, and two go out. We hope in the future to build on this to the case where the black holes orbit one another.