## Final Report Summary - 2LOOPACCURACY4LHC (2LoopAccuracy4LHC)

The ongoing experiments at the Large Hadron Collider (LHC) at CERN serve a dual purpose. One is the discovery of the Higgs boson, responsible for generating the masses of the elementary particles, and to measure its properties. With the announcement of the discovery of the particle on July 4, 2012, the LHC has already largely fulfilled one half of its purpose. The other aim of the LHC is to produce and detect new elementary particles which may be the constituents of dark matter.

The primary aim of the project has been to develop novel techniques to enable precision-level computations of processes at the LHC which are not attainable with the standard methods. The work has resulted in six articles, four of which have been published in primary journals with high impact factors (the remaining two being under current review).

The results have been presented at four high-profile international conferences, in two of which the grant recipient was a plenary speaker. The result have moreover been presented in seven invited seminars to particle physics groups at universities and research institutions.

The increasing interest in precision-level calculations of LHC processes ensures that much of the work which was carried out in the project will continue to have an impact for the foreseeable future.

Below follows a summary of the results and the impact of the six publications.

1) The papers Phys. Rev. D 93 (2016) 041701 and eprint arXiv:1612.04252 (under review) develop a new and highly efficient method for finding so-called integration-by-parts (IBP) reductions. These reductions are the technical core of calculations of LHC processes. This is so because they allow the reduction of typically millions of complicated integrals into typically only hundreds, and in turn enable equations to be set up for these remaining integrals so that they may be determined. In practice, however, generating the IBP reductions is computationally extremely expensive: for many processes of relevance to the LHC, the reductions cannot be obtained even from large-scale clusters of computers. The difficulty of this problem can be traced back to the problem of solving large systems of linear equations efficiently.

The PRD article proposes a significantly more efficient way of obtaining the IBP reductions by applying so-called unitarity cuts to break the problem of simultaneously solving all linear equations into several smaller, and much more manageable, subproblems. Along with the article, an implementation of the algorithm has been made publically available. This implementation is typically a factor of five faster than the fastest publically available IBP solver (FIRE 5 in C++ mode). The potential impact of this work is to generate IBP reductions that are needed for LHC-relevant calculations, but which are unattainable with current IBP solvers. The authors are presently working on this problem.

The eprint provides an initial step needed for the above-mentioned algorithm, namely a general algorithm for how to divide the full system of linear equations into subsystems. This article was accompanied with the publically available code Azurite.

2) The articles Phys. Rev. Lett. 114 (2015) 181602 and JHEP 07 (2015) 083 concern observables (Wilson line correlators) which measure the radiation of gluons (the mediators of the strong force) as elementary particles are produced in the LHC collisions. The articles develop a conceptually novel method for directly extracting the imaginary part of these. This is beneficial because in many diverse applications it is only the imaginary part which is of interest. The formalism developed in these papers is moreover a first step in developing a unitarity formalism for the Wilson line correlators, a formalism which has proven highly successful for amplitude calculations and become the standard technique of calculation of these over the past 15 years. The PRL article was featured on the cover.

3) The article Phys. Rev. D 92 (2015) 025015 concerns relations among the integrals involved in calculations of LHC processes, in particular relations that may be used to greatly simplify these calculations. The article shows how one such relation (known as the BDS relation), which had previously been derived through several difficult steps, can be derived almost trivally by taking a different viewpoint on the relation. The framework developed in this article, by construction completely systematic, has the potential of uncovering new relations.

4) The eprint arXiv:1701.01040 emerged as a spinoff of the developments within the framework of the proposal. It provides a Wolfram Mathematica package, published along with the article as publically available software, for evaluating multivariate residues (implementing two different algorithms, both of them completely general). Despite their use in numerous different contexts, surprisingly, such a function is not part of Mathematica’s functions. The work has generated interest from Wolfram, and the grant recipient is currently involved in an email exchange with a software engineer from Wolfram.

The above papers have involved collaboration with researchers of various nationalities: the Netherlands, Sweden, USA, Denmark, China and Italy.

The everyday interaction with the members of the host institution (whose expertise covers a wide range of research topics) has been greatly beneficial to the grant recipient.

The primary aim of the project has been to develop novel techniques to enable precision-level computations of processes at the LHC which are not attainable with the standard methods. The work has resulted in six articles, four of which have been published in primary journals with high impact factors (the remaining two being under current review).

The results have been presented at four high-profile international conferences, in two of which the grant recipient was a plenary speaker. The result have moreover been presented in seven invited seminars to particle physics groups at universities and research institutions.

The increasing interest in precision-level calculations of LHC processes ensures that much of the work which was carried out in the project will continue to have an impact for the foreseeable future.

Below follows a summary of the results and the impact of the six publications.

1) The papers Phys. Rev. D 93 (2016) 041701 and eprint arXiv:1612.04252 (under review) develop a new and highly efficient method for finding so-called integration-by-parts (IBP) reductions. These reductions are the technical core of calculations of LHC processes. This is so because they allow the reduction of typically millions of complicated integrals into typically only hundreds, and in turn enable equations to be set up for these remaining integrals so that they may be determined. In practice, however, generating the IBP reductions is computationally extremely expensive: for many processes of relevance to the LHC, the reductions cannot be obtained even from large-scale clusters of computers. The difficulty of this problem can be traced back to the problem of solving large systems of linear equations efficiently.

The PRD article proposes a significantly more efficient way of obtaining the IBP reductions by applying so-called unitarity cuts to break the problem of simultaneously solving all linear equations into several smaller, and much more manageable, subproblems. Along with the article, an implementation of the algorithm has been made publically available. This implementation is typically a factor of five faster than the fastest publically available IBP solver (FIRE 5 in C++ mode). The potential impact of this work is to generate IBP reductions that are needed for LHC-relevant calculations, but which are unattainable with current IBP solvers. The authors are presently working on this problem.

The eprint provides an initial step needed for the above-mentioned algorithm, namely a general algorithm for how to divide the full system of linear equations into subsystems. This article was accompanied with the publically available code Azurite.

2) The articles Phys. Rev. Lett. 114 (2015) 181602 and JHEP 07 (2015) 083 concern observables (Wilson line correlators) which measure the radiation of gluons (the mediators of the strong force) as elementary particles are produced in the LHC collisions. The articles develop a conceptually novel method for directly extracting the imaginary part of these. This is beneficial because in many diverse applications it is only the imaginary part which is of interest. The formalism developed in these papers is moreover a first step in developing a unitarity formalism for the Wilson line correlators, a formalism which has proven highly successful for amplitude calculations and become the standard technique of calculation of these over the past 15 years. The PRL article was featured on the cover.

3) The article Phys. Rev. D 92 (2015) 025015 concerns relations among the integrals involved in calculations of LHC processes, in particular relations that may be used to greatly simplify these calculations. The article shows how one such relation (known as the BDS relation), which had previously been derived through several difficult steps, can be derived almost trivally by taking a different viewpoint on the relation. The framework developed in this article, by construction completely systematic, has the potential of uncovering new relations.

4) The eprint arXiv:1701.01040 emerged as a spinoff of the developments within the framework of the proposal. It provides a Wolfram Mathematica package, published along with the article as publically available software, for evaluating multivariate residues (implementing two different algorithms, both of them completely general). Despite their use in numerous different contexts, surprisingly, such a function is not part of Mathematica’s functions. The work has generated interest from Wolfram, and the grant recipient is currently involved in an email exchange with a software engineer from Wolfram.

The above papers have involved collaboration with researchers of various nationalities: the Netherlands, Sweden, USA, Denmark, China and Italy.

The everyday interaction with the members of the host institution (whose expertise covers a wide range of research topics) has been greatly beneficial to the grant recipient.