## Periodic Reporting for period 1 - InsideLoops (Colour and Kinematics of Quarks and Gluons inside Loops)

Reporting period: 2017-10-01 to 2019-09-30

The progress in physics of elementary particles has been to a large extent driven by the data obtained at the Large Hadron Collider (LHC) in Geneva, Switzerland. The data collected there describe the probability of a multitude of different outcomes for a single initial process - high-energy scattering of two protons. These protons start their life as nuclei of two hydrogen atoms. They are composed of quarks and gluons, which are bound together into protons by the strong interaction force - one of the four known fundamental forces of nature. The behavior of quarks, gluons and the corresponding strong force is known to be governed by the theory of quantum chromodynamics (QCD). Although it is a well-established theory, practical calculations in it are very complicated and require major efforts from particle theorists. This becomes an issue because the outcomes observed at the LHC are largely dominated by QCD scattering processes due to the strong interaction between the protons' constituents. Consequently, our ability to search for new fundamental physical phenomena, such as other kinds of particles and interactions, relies on our capability to make accurate predictions in QCD.

In this project the fellow has been exploring the intricate analytic structure of QCD so as to establish new methods to tackle the complexity that accompanies calculations involving the strong interaction. The theory of QCD belongs to a large class of more abstract theories, which also include the theory of the weak interaction and Maxwell's electrodynamics. Such theories are defined by supplementing the usual ``kinematic'' degrees of freedom, due to the physical spacetime, with additional ``color'' degrees of freedom, which are mathematically isolated from spacetime. Interestingly, these two, otherwise separate, kinds of degrees of freedom have recently been found to share certain algebraic properties. This color-kinematics duality of QCD and similar theories has been the main inspiration for this project.

In addition, the color-kinematics duality provides a link from QCD-like theories to gravitational theories. Gravity is the only of the four fundamental interactions, which is described not by a QCD-like theory but by Einstein's general relativity. However, we now know it to possess a double-copy structure, which is closely related to the color-kinematics duality of QCD-like theories. Using this connection, the fellow has found a novel avenue of research regarding to the computation of gravitational-wave emission from binary black-hole mergers. Such gravitational waves have only been experimentally measured in 2015 for the first time and are destined to become an invaluable source of data about the Universe. Due to the phenomenological importance of being able to compute classical observables in general relativity, the fellow has spent half of his time applying his knowledge of quantum particle theories to the classical gravitational interaction between black holes.

Following these two main avenues of research, in this project the fellow has helped establish new methods of computation in QCD and general relativity, finding new analytic results in both classes of theories and strengthening the link between them.

In this project the fellow has been exploring the intricate analytic structure of QCD so as to establish new methods to tackle the complexity that accompanies calculations involving the strong interaction. The theory of QCD belongs to a large class of more abstract theories, which also include the theory of the weak interaction and Maxwell's electrodynamics. Such theories are defined by supplementing the usual ``kinematic'' degrees of freedom, due to the physical spacetime, with additional ``color'' degrees of freedom, which are mathematically isolated from spacetime. Interestingly, these two, otherwise separate, kinds of degrees of freedom have recently been found to share certain algebraic properties. This color-kinematics duality of QCD and similar theories has been the main inspiration for this project.

In addition, the color-kinematics duality provides a link from QCD-like theories to gravitational theories. Gravity is the only of the four fundamental interactions, which is described not by a QCD-like theory but by Einstein's general relativity. However, we now know it to possess a double-copy structure, which is closely related to the color-kinematics duality of QCD-like theories. Using this connection, the fellow has found a novel avenue of research regarding to the computation of gravitational-wave emission from binary black-hole mergers. Such gravitational waves have only been experimentally measured in 2015 for the first time and are destined to become an invaluable source of data about the Universe. Due to the phenomenological importance of being able to compute classical observables in general relativity, the fellow has spent half of his time applying his knowledge of quantum particle theories to the classical gravitational interaction between black holes.

Following these two main avenues of research, in this project the fellow has helped establish new methods of computation in QCD and general relativity, finding new analytic results in both classes of theories and strengthening the link between them.

The results achieved by the fellow can be separated into those regarding QCD or the QCD-like theory, called N=2 SQCD, and those regarding gravitational interaction of massive particles or black holes.

In the single-authored paper (2018), the fellow has used a novel formalism by Arkani-Hamed, Huang and Huang (2017) to encode quarks' spin degrees of freedom. This formalism allowed him to produce new compact analytic formulae for a class of QCD scattering probability amplitudes.

In the paper with Page (2019), the fellow has considered the aforementioned color degrees of freedom of quarks and gluons. A non-standard application of ideas that are routinely used for the kinematic degrees of freedom allowed to formulate novel color decompositions for arbitrary QCD scattering amplitudes. Multiple QCD computations of Page and his other collaborators rely on the method established in this paper.

In the paper with Kaelin and Mogull (2018), the fellow has computed new probability amplitudes for scattering of quarks and gluons in a theory which can be regarded as a realistic toy model for QCD. These amplitudes were constructed in a form which exploits the duality between color and kinematics. The latest preprint with Kaelin, Mogull and Verbeek (2019) further explores the analytic structure of such amplitudes, making certain physical and mathematical properties manifest.

In the papers with Guevara and Vines (2018, 2019), the fellow has established the link between certain quantum scattering amplitudes and the classical scattering of spinning black holes. He exploited this link to provide a novel derivation of Vines' earlier result (2017), as well as to compute previously unknown contributions to the gravitational interaction potential of two spinning black holes, some of which have been confirmed since then, while the others wait for an independent cross-check.

In the paper with Johansson (2019), the fellow has presented a first detailed study of a gravitational theory that can be obtained as a double copy of QCD with massive quarks. The resulting gravitational theory contains massive spinning matter, which is the first step towards implementing realistic gravitational matter, such as spinning black holes, via the double-copy construction.

No website has been developed for the project. The results of this project were reported at several conferences, workshops and seminars, including an invited talk at Amplitudes 2019 at Trinity College Dublin.

In the single-authored paper (2018), the fellow has used a novel formalism by Arkani-Hamed, Huang and Huang (2017) to encode quarks' spin degrees of freedom. This formalism allowed him to produce new compact analytic formulae for a class of QCD scattering probability amplitudes.

In the paper with Page (2019), the fellow has considered the aforementioned color degrees of freedom of quarks and gluons. A non-standard application of ideas that are routinely used for the kinematic degrees of freedom allowed to formulate novel color decompositions for arbitrary QCD scattering amplitudes. Multiple QCD computations of Page and his other collaborators rely on the method established in this paper.

In the paper with Kaelin and Mogull (2018), the fellow has computed new probability amplitudes for scattering of quarks and gluons in a theory which can be regarded as a realistic toy model for QCD. These amplitudes were constructed in a form which exploits the duality between color and kinematics. The latest preprint with Kaelin, Mogull and Verbeek (2019) further explores the analytic structure of such amplitudes, making certain physical and mathematical properties manifest.

In the papers with Guevara and Vines (2018, 2019), the fellow has established the link between certain quantum scattering amplitudes and the classical scattering of spinning black holes. He exploited this link to provide a novel derivation of Vines' earlier result (2017), as well as to compute previously unknown contributions to the gravitational interaction potential of two spinning black holes, some of which have been confirmed since then, while the others wait for an independent cross-check.

In the paper with Johansson (2019), the fellow has presented a first detailed study of a gravitational theory that can be obtained as a double copy of QCD with massive quarks. The resulting gravitational theory contains massive spinning matter, which is the first step towards implementing realistic gravitational matter, such as spinning black holes, via the double-copy construction.

No website has been developed for the project. The results of this project were reported at several conferences, workshops and seminars, including an invited talk at Amplitudes 2019 at Trinity College Dublin.

In this project, the fellow has established new methods and achieved new analytic results in the field of elementary particle physics and its application to general relativity. The particle-physics outcomes of this project concern the smallest constituents of matter which are probed at the high-energy collider experiments such as the LHC. Such knowledge lies at the foundation of science and is for instance crucial for general science education, as evidenced by the interest expressed by the schoolchildren to whom the fellow has given an outreach lecture in Munich.

The results involving gravitational interaction help unify seemingly opposite aspects of fundamental physics: quantum theory is used to deal with classical physics by regarding such extremely large objects as black holes as if they were minuscule elementary particles. This is an example of synergy between different branches of physics, which is all the more valuable as more and more separate communities tend to appear and rarely communicate with each other, even within what from the outside may seem like a single scientific field. The impact of the work started in this project will be expressed in new links between multiple scientific sub-communities across theory and phenomenology. It is the fellow's hope that such links between the different facets of fundamental physics would mature into more consolidated and comprehensible ways of teaching physics to undergraduate students and possibly at public schools.

The results involving gravitational interaction help unify seemingly opposite aspects of fundamental physics: quantum theory is used to deal with classical physics by regarding such extremely large objects as black holes as if they were minuscule elementary particles. This is an example of synergy between different branches of physics, which is all the more valuable as more and more separate communities tend to appear and rarely communicate with each other, even within what from the outside may seem like a single scientific field. The impact of the work started in this project will be expressed in new links between multiple scientific sub-communities across theory and phenomenology. It is the fellow's hope that such links between the different facets of fundamental physics would mature into more consolidated and comprehensible ways of teaching physics to undergraduate students and possibly at public schools.