## Periodic Reporting for period 1 - MathAm (Mathematical Structures in Scattering Amplitudes)

**Reporting period:**2015-09-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

One of the main ingredients that allow physicists to derive predictions from quantum field theory are the so-called scattering amplitudes, a set of mathematical quantities that encode the probabilities of how quantum particles interact. While the basic definition of a scattering amplitude dates back to the early days of quantum field theory, explicit computations of the so-called quantum loop corrections are still one of the major bottlenecks in theoretical particle physics. Since scattering amplitudes play such a central role in quantum physics, understanding their properties and mathematical structures is crucial in order to improve our knowledge of the fundamental laws of nature. Moreover, scattering amplitudes are not only interesting objects to study in their own right, but they are the main theoretical tool to make predictions for collider experiments like the LHC.

Despite their importance and ubiquity in modern theoretical physics, the explicit computation of scattering amplitudes is often still a bottleneck. One of the main issues when computing scattering amplitudes is the necessity to evaluate certain classes of integrals, known as Feynman integrals, which allow one to compute quantum corrections to a scattering process due to the exchange of virtual quanta. These integrals often evaluate to special functions that are poorly understood even from the mathematical side. In recent years it was realised that there are deep and far-reaching connections between scattering amplitudes and certain areas of modern mathematics, like algebraic geometry and number theory.

The goal of the project MathAm is to apply cutting-edge techniques from modern mathematics to the computation of multi-loop multi-leg scattering amplitudes in quantum field theory. Guided by advances modern mathematics in the last decade, the goal of the project is to derive a radically new viewpoint on scattering amplitudes in quantum field theory from which the properties and the simplicity of amplitudes will be manifest at every step of the computation. The results obtained by MathAm will not only shed light on the mathematical underpinnings of quantum field theory and scattering amplitudes in general, but they will also enable MathAm to make predictions for collider experiments like the LHC at an unprecedented level of accuracy.

Despite their importance and ubiquity in modern theoretical physics, the explicit computation of scattering amplitudes is often still a bottleneck. One of the main issues when computing scattering amplitudes is the necessity to evaluate certain classes of integrals, known as Feynman integrals, which allow one to compute quantum corrections to a scattering process due to the exchange of virtual quanta. These integrals often evaluate to special functions that are poorly understood even from the mathematical side. In recent years it was realised that there are deep and far-reaching connections between scattering amplitudes and certain areas of modern mathematics, like algebraic geometry and number theory.

The goal of the project MathAm is to apply cutting-edge techniques from modern mathematics to the computation of multi-loop multi-leg scattering amplitudes in quantum field theory. Guided by advances modern mathematics in the last decade, the goal of the project is to derive a radically new viewpoint on scattering amplitudes in quantum field theory from which the properties and the simplicity of amplitudes will be manifest at every step of the computation. The results obtained by MathAm will not only shed light on the mathematical underpinnings of quantum field theory and scattering amplitudes in general, but they will also enable MathAm to make predictions for collider experiments like the LHC at an unprecedented level of accuracy.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"In the following we describe in detail the work performed on each working package during the first 18 months of the project.

After the discovery of the Higgs boson by the LHC in 2012, the study of the properties of the Higgs boson is one of the most pressing questions in modern particle physics and one of the main objectives of the experiments carried out at the LHC at CERN. Since no new physics has been discovered by the LHC, we have entered a new era of precision physics where precise experimental measurements are confronted to state-of-the art theoretical predictions. Precise theoretical predictions require the computation of quantum corrections to scattering processes, and it has been known for a long time that the quantum corrections which affect the production of a Higgs boson at the LHC are large, leading ultimately to a sizeable uncertainty on theoretical predictions for this process. Towards the end of 2014, the experimental uncertainty on the production probability of a Higgs boson was challenging the theoretical prediction, calling for an updated and improved theoretical calculation. This calculation required the computation of quantum correction of the third order, a daunting task which had never been performed in the past for an LHC observable. We have successfully completed this challenging computation, which required the computation of complicated multi-loop integrals that were considered out of reach by conventional techniques. All of these integrals could be computed analytically by exploiting novel cutting-edge mathematical tools whose use in physics was pioneered by the members of MathAm, like the Hopf algebra of multiple polylogarithms and novel algorithms for symbolic integrations. As a result, we were able to obtain the most precise theoretical predictions for the production of a Higgs boson at the LHC, including all third order quantum corrections due to the strong interaction (JHEP 1605 (2016) 058). The results obtained by the MathAm project are by now the official reference for the LHC experiments for the production probability of a Higgs boson. Furthermore, they clearly demonstrate the power of MathAm: pushing the boundaries in our understanding of multi-loop scattering amplitudes relevant to modern particle physics by applying cutting-edge techniques from modern mathematics.

Precision QCD predictions usually require the computation of quantum corrections of at least second order in the strong interaction. This requires the evaluation of two-loop scattering amplitudes. The latter are divergent, and the divergences cancel once the amplitude is combined with the result of computation of the amplitudes with fewer loops, but with additional radiation in the final state. Delicate procedures have been developed over the last 20 years to perform this combination. One such procedure had been developed by Del Duca, Somogyi and Trocsanyi 15 years ago, at least in the context of electron-positron colliders. In order to be validated and to be used in practise, however, the procedure required the computation of a large collection of very complicated integrals that were thought beyond reach by current technology. Using and extending the cutting-edge mathematical tools developed in the context of Higgs physics, we have been able to perform all the relevant integrals. This has not only shown that the procedure by Del Duca, Somogyi and Trocsanyi is consistent (in the sense that all divergences cancel as they should), but we have been able to apply their method to the computation of the second order corrections to the production of three jets in electron-positron collisions. While this computation had been performed by another group already a decade ago, we have shown that the method by Del Duca, Somogyi and Trocsanyi has a better numerical convergence than previous methods. In addition, we have been able to present the first computation of some physical observables to second order that could not be computed by other techniques (Phys.Rev.Lett. 117 (2016) no.15, 152004; Phys.Rev. D94 (2016) no.7, 074019).

An important aspect of MathAm is that it applies cutting-edge ideas from modern mathematics to the computation of scattering amplitudes. In particular, we aim at understanding at a deeper mathematical level the structure of these fundamental quantities. A particularly clean environment to study the mathematical properties of scattering amplitudes is the so-called N=4 Super Young-Mills (SYM) theory. While N=4 SYM itself has no direct phenomenological impact, it shares still many of properties of more realistic theories like QCD. Hence, it is a toy model to study the mathematical structure of more realistic theories. In recent years, a lot of connections between N=4 SYM and the geometry of Grassmannians, cluster algebras and polytopes have been conjectured. If these conjectures are true, they will have far reaching consequences for our understanding of scattering amplitudes in general. Unfortunately, the connections are not always concrete, and it is not always clear how to use them in concrete applications. We have shown that in a certain kinematic limit, known as the "multi-Regge limit", all the geometric concepts can be made explicit. In particular, we have shown that the geometry underlying the multi-Regge limit is a special incarnation of a Grassmannian, known as the moduli space of genus zero curves with marked points. The function theory on that space is well studied in the mathematical literature, and we have shown how we can use these results from mathematics to obtain new results for scattering amplitudes in this limit for many loops and many legs (JHEP 1608 (2016) 152)."

After the discovery of the Higgs boson by the LHC in 2012, the study of the properties of the Higgs boson is one of the most pressing questions in modern particle physics and one of the main objectives of the experiments carried out at the LHC at CERN. Since no new physics has been discovered by the LHC, we have entered a new era of precision physics where precise experimental measurements are confronted to state-of-the art theoretical predictions. Precise theoretical predictions require the computation of quantum corrections to scattering processes, and it has been known for a long time that the quantum corrections which affect the production of a Higgs boson at the LHC are large, leading ultimately to a sizeable uncertainty on theoretical predictions for this process. Towards the end of 2014, the experimental uncertainty on the production probability of a Higgs boson was challenging the theoretical prediction, calling for an updated and improved theoretical calculation. This calculation required the computation of quantum correction of the third order, a daunting task which had never been performed in the past for an LHC observable. We have successfully completed this challenging computation, which required the computation of complicated multi-loop integrals that were considered out of reach by conventional techniques. All of these integrals could be computed analytically by exploiting novel cutting-edge mathematical tools whose use in physics was pioneered by the members of MathAm, like the Hopf algebra of multiple polylogarithms and novel algorithms for symbolic integrations. As a result, we were able to obtain the most precise theoretical predictions for the production of a Higgs boson at the LHC, including all third order quantum corrections due to the strong interaction (JHEP 1605 (2016) 058). The results obtained by the MathAm project are by now the official reference for the LHC experiments for the production probability of a Higgs boson. Furthermore, they clearly demonstrate the power of MathAm: pushing the boundaries in our understanding of multi-loop scattering amplitudes relevant to modern particle physics by applying cutting-edge techniques from modern mathematics.

Precision QCD predictions usually require the computation of quantum corrections of at least second order in the strong interaction. This requires the evaluation of two-loop scattering amplitudes. The latter are divergent, and the divergences cancel once the amplitude is combined with the result of computation of the amplitudes with fewer loops, but with additional radiation in the final state. Delicate procedures have been developed over the last 20 years to perform this combination. One such procedure had been developed by Del Duca, Somogyi and Trocsanyi 15 years ago, at least in the context of electron-positron colliders. In order to be validated and to be used in practise, however, the procedure required the computation of a large collection of very complicated integrals that were thought beyond reach by current technology. Using and extending the cutting-edge mathematical tools developed in the context of Higgs physics, we have been able to perform all the relevant integrals. This has not only shown that the procedure by Del Duca, Somogyi and Trocsanyi is consistent (in the sense that all divergences cancel as they should), but we have been able to apply their method to the computation of the second order corrections to the production of three jets in electron-positron collisions. While this computation had been performed by another group already a decade ago, we have shown that the method by Del Duca, Somogyi and Trocsanyi has a better numerical convergence than previous methods. In addition, we have been able to present the first computation of some physical observables to second order that could not be computed by other techniques (Phys.Rev.Lett. 117 (2016) no.15, 152004; Phys.Rev. D94 (2016) no.7, 074019).

An important aspect of MathAm is that it applies cutting-edge ideas from modern mathematics to the computation of scattering amplitudes. In particular, we aim at understanding at a deeper mathematical level the structure of these fundamental quantities. A particularly clean environment to study the mathematical properties of scattering amplitudes is the so-called N=4 Super Young-Mills (SYM) theory. While N=4 SYM itself has no direct phenomenological impact, it shares still many of properties of more realistic theories like QCD. Hence, it is a toy model to study the mathematical structure of more realistic theories. In recent years, a lot of connections between N=4 SYM and the geometry of Grassmannians, cluster algebras and polytopes have been conjectured. If these conjectures are true, they will have far reaching consequences for our understanding of scattering amplitudes in general. Unfortunately, the connections are not always concrete, and it is not always clear how to use them in concrete applications. We have shown that in a certain kinematic limit, known as the "multi-Regge limit", all the geometric concepts can be made explicit. In particular, we have shown that the geometry underlying the multi-Regge limit is a special incarnation of a Grassmannian, known as the moduli space of genus zero curves with marked points. The function theory on that space is well studied in the mathematical literature, and we have shown how we can use these results from mathematics to obtain new results for scattering amplitudes in this limit for many loops and many legs (JHEP 1608 (2016) 152)."

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The computation of the Higgs cross section through third order is not only the most precise determination of the Higgs cross section to date, but it is at the same time the first computation ever performed at this order. As a consequence, the results of MathAm go beyond the current state of the art and have broken new ground in precision computations for hadron colliders. Moreover, this result will have a lasting impact on the physics program carried out at the Large Hadron Collider (LHC) at CERN. Indeed, the results obtained by the MathAm group are by now the official reference for the LHC experiments for the production probability of a Higgs boson. They will therefore play a crucial role in the study of the properties of the Higgs boson, and hence in improving our understanding of one of the fundamental laws of nature.