## Periodic Reporting for period 1 - Soft Gluons (Soft Gluon Physics and Multi-Loop Calculations)

Reporting period: 2015-10-01 to 2017-09-30

The fundamental interactions in Nature are studied nowadays by means of large accelerators, like the Large Hadron collider (LHC), at CERN. Two proton beams are accelerated close to the velocity of light and collide at selected points, producing a plethora of particles. The type and rate at which particles are produced is dictated by the fundamental interactions: the study of these collisions gives us access to the dynamics of particle scattering and thus to the structure of the fundamental interactions.

Our current knowledge of fundamental interactions is based on the Standard Model of Particle Physics, which describes well all physical phenomena from the scale of everyday life down to the Fermi scale (~ 10^(-15) meters). Experiments at the LHC allows us to explore for the first time physics at shorter distance scales, and new phenomena are expected to appear. However, any such phenomenon will occur in the background of Standard Model interactions, and therefore will show up as small deviations from the theoretical predictions obtained using the Standard Model.

In order to identify these deviations we need precise theoretical predictions. There has been a worldwide effort among theoretical physicists over the past few years, devoted to producing accurate predictions for scattering processes at the LHC. A large part of this effort has been aimed at calculating scattering amplitudes, the basic object describing a scattering process. These functions are given in terms of complicated integrals, and there exist no general methods to perform these calculations. For this reason, much effort has been directed to studying the analytic structure of scattering amplitudes.

This is the context in which the present action has been conceived. The aim of this project has been to study scattering amplitudes in specific kinematic limits of interest, namely, particle scattering near threshold, and in the high-energy limit. The objectives were to understand the factorisation structure of amplitudes in these limits and identify iterative structures which would allow one to calculate them as iterated integrals.

The project has been successful, and the outcome of this research will be useful in the first place to the community of theoretical physicists. The results obtained will help them to further characterise analytic properties of scattering amplitudes, and to produce precise calculations, which in turn can be contrasted with experimental data from the LHC. This work will be useful for the society as a whole, as it contributes towards the common effort of advancing our knowledge of quantum field theory, and thus our knowledge of fundamental interactions in Nature.

Our current knowledge of fundamental interactions is based on the Standard Model of Particle Physics, which describes well all physical phenomena from the scale of everyday life down to the Fermi scale (~ 10^(-15) meters). Experiments at the LHC allows us to explore for the first time physics at shorter distance scales, and new phenomena are expected to appear. However, any such phenomenon will occur in the background of Standard Model interactions, and therefore will show up as small deviations from the theoretical predictions obtained using the Standard Model.

In order to identify these deviations we need precise theoretical predictions. There has been a worldwide effort among theoretical physicists over the past few years, devoted to producing accurate predictions for scattering processes at the LHC. A large part of this effort has been aimed at calculating scattering amplitudes, the basic object describing a scattering process. These functions are given in terms of complicated integrals, and there exist no general methods to perform these calculations. For this reason, much effort has been directed to studying the analytic structure of scattering amplitudes.

This is the context in which the present action has been conceived. The aim of this project has been to study scattering amplitudes in specific kinematic limits of interest, namely, particle scattering near threshold, and in the high-energy limit. The objectives were to understand the factorisation structure of amplitudes in these limits and identify iterative structures which would allow one to calculate them as iterated integrals.

The project has been successful, and the outcome of this research will be useful in the first place to the community of theoretical physicists. The results obtained will help them to further characterise analytic properties of scattering amplitudes, and to produce precise calculations, which in turn can be contrasted with experimental data from the LHC. This work will be useful for the society as a whole, as it contributes towards the common effort of advancing our knowledge of quantum field theory, and thus our knowledge of fundamental interactions in Nature.

Leonardo Vernazza has focused on developing new calculation techniques for particle scattering near threshold and in the high-energy limit. Both these correspond to particular kinematic limits of scattering processes, and require resummation of high-order corrections in perturbation theory. In each case the simplification associated with the kinematic limit provides means to compute such high-order corrections, which are inaccessible with present techniques for general kinematics.

An important example is the case of logarithms associated with partonic threshold. Large logarithms arise as an effect of dynamical enhancement of soft radiation in conjunction with small phase space. Techniques for resumming threshold logarithms at leading power have been known for a while. However, in order to match the precision goals set by the LHC experiments, it is necessary to resum an additional tower of logarithms occurring at the next-to-leading power (NLP). Leonardo Vernazza studied NLP logarithms with collaborators at the university of Amsterdam, Queen Mary (London) and Turin, establishing a factorisation theorem describing how these logarithms arise in the case of the Drell-Yan. They have exploited this factorisation theorem to show that scattering cross sections near threshold exhibit a universal structure at next-to-leading order (NLO) in the strong coupling, and at NLP in the threshold expansion. This result is very interesting, because it allows them to provide approximate predictions at NLO for processes such as double Higgs production, which are otherwise difficult to calculate.

Leonardo Vernazza has been working on this subject during the entire timeframe of the fellowship. Results have been published into two papers, listed as no. 1 and 3 in section “Publications”.

Leonardo Vernazza also exploited the possibility given by the Higgs Centre to organise a Workshop on this subject: https://higgs.ph.ed.ac.uk/workshops/threshold-logarithms-beyond-leading-power. The workshop provided a great occasion to disseminate his results and discuss the topic with world-leading experts in the field.

During the timeframe of the fellowship Leonardo Vernazza studied also the the high-energy limit of scattering processes, in collaboration with the supervisor, Einan Gardi, and Simon Caron Huot (McGill University, Montreal) as well as with Joscha Reichel, a PhD student in Edinburgh. In this limit the structure of the amplitude simplifies, and it is possible to calculate it to higher orders in perturbation theory. With this motivation in mind we developed a framework to calculate scattering amplitudes in the high-energy limit, in the context of the shockwave formalism. This work allowed us to establish an explicit connection between concepts from Regge theory and perturbative QCD in the high-energy limit. We have derived the factorisation structure of amplitudes in the presence of Regge cuts, and provided a way to calculate the latter as iterated integrals in perturbation theory, up to next-to-next-to leading logarithmic accuracy. Furthermore, we achieved a breakthrough concerning the calculation of two-parton scattering at next-to-leading logarithmic accuracy. At this logarithmic accuracy we have been able to calculate the infrared divergent part of the amplitude to all orders in the strong coupling, and thus extract the soft anomalous dimension, a function responsible for generating all the infrared singularities in an amplitude, to all orders in the strong coupling.

These results have been published into two papers, listed as no. 2 and 4 in section “Publications”.

Furthermore, we exploited the possibility given by the Higgs Centre to organise a workshop on this subject: http://higgs.ph.ed.ac.uk/workshops/iterated-integrals-and-regge-limit. The workshop has provided a great occasion to disseminate our results and discuss the subject with world-leading experts in the field.

An important example is the case of logarithms associated with partonic threshold. Large logarithms arise as an effect of dynamical enhancement of soft radiation in conjunction with small phase space. Techniques for resumming threshold logarithms at leading power have been known for a while. However, in order to match the precision goals set by the LHC experiments, it is necessary to resum an additional tower of logarithms occurring at the next-to-leading power (NLP). Leonardo Vernazza studied NLP logarithms with collaborators at the university of Amsterdam, Queen Mary (London) and Turin, establishing a factorisation theorem describing how these logarithms arise in the case of the Drell-Yan. They have exploited this factorisation theorem to show that scattering cross sections near threshold exhibit a universal structure at next-to-leading order (NLO) in the strong coupling, and at NLP in the threshold expansion. This result is very interesting, because it allows them to provide approximate predictions at NLO for processes such as double Higgs production, which are otherwise difficult to calculate.

Leonardo Vernazza has been working on this subject during the entire timeframe of the fellowship. Results have been published into two papers, listed as no. 1 and 3 in section “Publications”.

Leonardo Vernazza also exploited the possibility given by the Higgs Centre to organise a Workshop on this subject: https://higgs.ph.ed.ac.uk/workshops/threshold-logarithms-beyond-leading-power. The workshop provided a great occasion to disseminate his results and discuss the topic with world-leading experts in the field.

During the timeframe of the fellowship Leonardo Vernazza studied also the the high-energy limit of scattering processes, in collaboration with the supervisor, Einan Gardi, and Simon Caron Huot (McGill University, Montreal) as well as with Joscha Reichel, a PhD student in Edinburgh. In this limit the structure of the amplitude simplifies, and it is possible to calculate it to higher orders in perturbation theory. With this motivation in mind we developed a framework to calculate scattering amplitudes in the high-energy limit, in the context of the shockwave formalism. This work allowed us to establish an explicit connection between concepts from Regge theory and perturbative QCD in the high-energy limit. We have derived the factorisation structure of amplitudes in the presence of Regge cuts, and provided a way to calculate the latter as iterated integrals in perturbation theory, up to next-to-next-to leading logarithmic accuracy. Furthermore, we achieved a breakthrough concerning the calculation of two-parton scattering at next-to-leading logarithmic accuracy. At this logarithmic accuracy we have been able to calculate the infrared divergent part of the amplitude to all orders in the strong coupling, and thus extract the soft anomalous dimension, a function responsible for generating all the infrared singularities in an amplitude, to all orders in the strong coupling.

These results have been published into two papers, listed as no. 2 and 4 in section “Publications”.

Furthermore, we exploited the possibility given by the Higgs Centre to organise a workshop on this subject: http://higgs.ph.ed.ac.uk/workshops/iterated-integrals-and-regge-limit. The workshop has provided a great occasion to disseminate our results and discuss the subject with world-leading experts in the field.

The results obtained during the course of the fellowship open up the way to several lines of research. The reason is that the work done so far constitute the foundation of new calculation methods, both in case of particle scattering near threshold and in the high-energy limit. This methods can now be exploited and provide the basis for several applications, to be investigated in the following years.

Concluding, the work done within this action has been very successful. It has allowed Leonardo Vernazza to develop an extensive and original research program, which he will be able to exploit in the next years. The outcome will be useful for the whole community of particle physicist, as it will allow researcher to understand fundamental properties of scattering amplitudes, and obtain precise predictions for the physics program at the LHC.

Concluding, the work done within this action has been very successful. It has allowed Leonardo Vernazza to develop an extensive and original research program, which he will be able to exploit in the next years. The outcome will be useful for the whole community of particle physicist, as it will allow researcher to understand fundamental properties of scattering amplitudes, and obtain precise predictions for the physics program at the LHC.