## Periodic Reporting for period 4 - PertQCD (Automatization of perturbative QCD at very high orders.)

Reporting period: 2021-04-01 to 2021-09-30

An important goal of modern physics is to understand in a quantitative manner the fundamental laws which govern interactions at the level of elementary particles (electrons, muons, quarks, gluons, Higgs, electroweak bosons, etc).

Currently, the Large Hadron Collider probes their validity at the tiniest distances ever accessed by humans. The LHC experiments perform detailed and very precise statistical measurements of particle collisions. To figure out whether

our standard model (SM) of high energy physics is correct and to gain insights on what is not included included in the Standard Model (dark matter, heavier particles, indirect effects with implications for our understwandin of gravity),

we need to confront these data with theoretical predictions.

The aim of our project is to provide very precise theoretical predictions for the processes which take place at collider experiments.

This is a formidable task. The Standard Model of particle interactions is a quantum field theory. The mathematical objects (probability amplitudes) which give the probability of a certain event to take place at the LHC cannot be evaluated exactly by means of known mathematical methods. They can only be evaluated approximately, as a perturbative expansion around the limit of vanishing strengths of particle interactions. Currently, the first two terms (leading order and next-toleading-order) in this perturbative series are computable with automated methods for LHC processes. This yields a typical theoretical precision of approximately 15%-30%, to be contrasted with a typical (achieved or anticipated) precision of 5% or better.

In this project, we develop mathematical and computational methods for the next-to-next-to-leading (NNLO) and next-to-next-to-next-to-next-to-leading (N3LO) terms in the perturbative series. Our work, has the potential to reduce the theoretical uncertainty for most interesting LHC observables to the same level as the anticipated experimental uncertainty or better.

Currently, the Large Hadron Collider probes their validity at the tiniest distances ever accessed by humans. The LHC experiments perform detailed and very precise statistical measurements of particle collisions. To figure out whether

our standard model (SM) of high energy physics is correct and to gain insights on what is not included included in the Standard Model (dark matter, heavier particles, indirect effects with implications for our understwandin of gravity),

we need to confront these data with theoretical predictions.

The aim of our project is to provide very precise theoretical predictions for the processes which take place at collider experiments.

This is a formidable task. The Standard Model of particle interactions is a quantum field theory. The mathematical objects (probability amplitudes) which give the probability of a certain event to take place at the LHC cannot be evaluated exactly by means of known mathematical methods. They can only be evaluated approximately, as a perturbative expansion around the limit of vanishing strengths of particle interactions. Currently, the first two terms (leading order and next-toleading-order) in this perturbative series are computable with automated methods for LHC processes. This yields a typical theoretical precision of approximately 15%-30%, to be contrasted with a typical (achieved or anticipated) precision of 5% or better.

In this project, we develop mathematical and computational methods for the next-to-next-to-leading (NNLO) and next-to-next-to-next-to-next-to-leading (N3LO) terms in the perturbative series. Our work, has the potential to reduce the theoretical uncertainty for most interesting LHC observables to the same level as the anticipated experimental uncertainty or better.

We have been developing methods for the analytic computation of amplitudes at NNLO and N3LO, which exploit their anticipated physical properties.

As practical application, we achieved for the first time the evaluation of the rapidity distribution of the Higgs particle. We also computed several previously

intractable effects which now yield a very precise estimate of the number of Higgs bosons being produced at the LHC.

In separate works, we automated the evaluation of NLO corrections due to the electorweak force for generic LHC processes.

Finally, we could apply our amplitude calculation methods for a more precise simulation of newly discovered gravitational waves signal.

Analytical methods, as the ones used for the above computations, have reached maturity but are also riddled by a computation cost which grows exponentially with complexity. To address this issue, we developed appropriate numerical methods. One of them solves appropriately formulated differential equations for core "master" integrals in perturbation theory and relates full scattering amplitudes to them with linear algebra relations derived with an optimised algorithm. With this method, we are able to compute two-loop QCD amplitudes for the hadroproduction of a Higgs boson and a jet, with the exact dependence of the amplitudes on the masses of virtual top and bottom quarks which couple to the Higgs particle.

In a different approach, envisaged to have a wider applicability, we developed a formalism to cast the integrands of scattering amplitudes in a form which separates the classical configurations (corresponding to soft and collinear radiation of virtual particles) from configurations of a quantum nature (corresponding to hard scattering). The classical configurations are universal and are known analytically to very high orders in perturbation theory. They contain divergent pieces which either cancel in the combination of the radiation of real and virtual particles or are absorbed into experimentally measured factors describing the quark and gluon content of identified hadrons in the scattering. We have fully formulated the method of separating classical from quantum configurations for a large class of generic electroweak production processes at hadron colliders through the second perturbative order.

After the subtracting classical configurations in amplitudes, the reminder quantum contributions can be formally integrated over the momenta of virtual particles, as it is required physically. To achieve the latter, we developed a method based on an automated analytic integration over energies and a deformation of the contours of integration away from singularities corresponding to threshold production of intermediate states.

Configurations of radiation of both virtual and real particles need to be included together in calculable within perturbation theory ("infrared safe") physical observables. To achieve that, we have been developing the method of Local Unitarity which unifies the two types, combining them in a common integrand in which cancellations of most classical configurations (with the exception of initial state singularities) happen automatically. We have tested this framework at very high orders in perturbation theory for cross-sections or parts of them where the cancellation of divergences is complete.

''

As practical application, we achieved for the first time the evaluation of the rapidity distribution of the Higgs particle. We also computed several previously

intractable effects which now yield a very precise estimate of the number of Higgs bosons being produced at the LHC.

In separate works, we automated the evaluation of NLO corrections due to the electorweak force for generic LHC processes.

Finally, we could apply our amplitude calculation methods for a more precise simulation of newly discovered gravitational waves signal.

Analytical methods, as the ones used for the above computations, have reached maturity but are also riddled by a computation cost which grows exponentially with complexity. To address this issue, we developed appropriate numerical methods. One of them solves appropriately formulated differential equations for core "master" integrals in perturbation theory and relates full scattering amplitudes to them with linear algebra relations derived with an optimised algorithm. With this method, we are able to compute two-loop QCD amplitudes for the hadroproduction of a Higgs boson and a jet, with the exact dependence of the amplitudes on the masses of virtual top and bottom quarks which couple to the Higgs particle.

In a different approach, envisaged to have a wider applicability, we developed a formalism to cast the integrands of scattering amplitudes in a form which separates the classical configurations (corresponding to soft and collinear radiation of virtual particles) from configurations of a quantum nature (corresponding to hard scattering). The classical configurations are universal and are known analytically to very high orders in perturbation theory. They contain divergent pieces which either cancel in the combination of the radiation of real and virtual particles or are absorbed into experimentally measured factors describing the quark and gluon content of identified hadrons in the scattering. We have fully formulated the method of separating classical from quantum configurations for a large class of generic electroweak production processes at hadron colliders through the second perturbative order.

After the subtracting classical configurations in amplitudes, the reminder quantum contributions can be formally integrated over the momenta of virtual particles, as it is required physically. To achieve the latter, we developed a method based on an automated analytic integration over energies and a deformation of the contours of integration away from singularities corresponding to threshold production of intermediate states.

Configurations of radiation of both virtual and real particles need to be included together in calculable within perturbation theory ("infrared safe") physical observables. To achieve that, we have been developing the method of Local Unitarity which unifies the two types, combining them in a common integrand in which cancellations of most classical configurations (with the exception of initial state singularities) happen automatically. We have tested this framework at very high orders in perturbation theory for cross-sections or parts of them where the cancellation of divergences is complete.

''

Our computations constitute major part of the state-of-the-art in pertutbative Quantum Field Theory. The project focused on the development of new methods in perturbation theory with the aim of automation.

In the course of the development of our methods we produced computations of cross-sections and scattering amplitudes relevant to the production of the Higgs boson at the Large Hadron Collider, improving the state of the art and reducing the theoretical uncertainty for Higgs observable to an unprecedented precision.

The development of numerical methods constitutes a shift in paradigm, breaking a transition of half a century of predominantly analytic methods and calculations.

In the course of the development of our methods we produced computations of cross-sections and scattering amplitudes relevant to the production of the Higgs boson at the Large Hadron Collider, improving the state of the art and reducing the theoretical uncertainty for Higgs observable to an unprecedented precision.

The development of numerical methods constitutes a shift in paradigm, breaking a transition of half a century of predominantly analytic methods and calculations.