## Final Report Summary - CRUNCHLOOPS (Towards systematization of NNLO theoretical predictions for<br/>advanced phenomenology at the LHC)

The physics program of the Large Hadron Collider (LHC) at CERN will help us answer some of the most fundamental unresolved questions in physics. The LHC, in the first few years of its operation, has so far discovered previously unobserved hadrons (like the xp (3P) bottomonium state) and a massive 125 GeV particle which is apparently confirmed to be the long-sought Higgs boson. LHC has also recorded the first observations of the very rare decay of the Bs meson into two muons and has created a quark-gluon plasma. The Nobel prize in physics 2013 was awarded jointly to Francois Englert and Peter Higgs for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed at the LHC through the discovery of the predicted fundamental particle by the ATLAS and CMS experiments.

At the LHC, any verification of present knowledge or solid new proof of new physics will only come after copious efforts of comparing experimental data against accurate theoretical predictions. For most of the processes at the LHC, perturbative calculations in quantum field theory at the next-to-leading order (NLO) are sufficient whereas for some processes next-to-next-to-leading order (NNLO) is needed, a highly non trivial theoretical task. Furthermore, apart from the so called fixed-order calculations which are based on taking into account only the first few terms of the perturbative expansion, resummation programmes will have to be considered in cases where large logarithms of some scale appear to all orders such that they endanger the validity of the perturbative expansion itself.

One of the most important resummation programmes in high energy Quantum Chromodynamics (QCD) is the Balitsky-Fadin-Kuraev-Lipatov (BFKL) formalism. The BFKL dynamics, not only takes care of resumming large logarithms in the centre-of-mass energy that may spoil the convergence of the perturbative expansion at high energies but is also in the core of addressing fundamental issues on factorization between soft and hard physics. Moreover, BFKL can offer a theoretical bridge connecting QCD and N = 4 Supersymmetric Yang-Mills (N = 4 SYM) quantum field theory, sometimes considered as the harmonic oscillator of the 21st century since it seems to provide, in the planar limit, the first example of a solvable non trivial quantum field theory in four dimensions. The discovery of an unexpected simplicity in the structure of scattering amplitudes and the connection with string theory provided by the anti de Sitter/conformal field theory (AdS/CFT) duality support this reasoning.

In this project, we proposed to apply a novel technique, the Loop-Tree Duality method between loop Feynman integrals and phase-space integrals, to NLO (and subsequently to NNLO) QCD calculations for LHC phenomenology. Secondly, we proposed to create a public library with the results of all the two-loop Feynman integrals necessary for some of the most important LHC processes that are needed to NNLO. For a generic 2 -> 2 NNLO QCD process at the LHC one can reduce the necessary Feynman integrals needed to be computed to a smaller basis of so-called Master Integrals (MI). The exact basis is naturally process-dependent, nevertheless, many of the processes share common Feynman diagram topologies which reduces the total number of the MI needed to be in the Library. A third fieldwork emerged in the course of the project, this one in connection to the BFKL formalism whereas the first two are connected to fixed order calculations. It was directed to two different approaches for studying high energy scattering amplitudes. One was toward the development of a Monte Carlo code for studying physical observables at the LHC and the other was toward more formal grounds like the applicability of Lipatov's effective action to the description of high energy processes in QCD.

There was very good progress in all subfields of this project.

Firstly, it was shown that the Loop-Tree Duality method exhibits attractive theoretical aspects and nice properties which are manifested by a direct physical interpretation of the singular behavior of the loop integrand. After applying the Loop-Tree Duality method, integrand singularities cancel among dual integrals. The remaining singularities, excluding UV divergences, can be interpreted in terms of causality and are restricted to a finite region of the loop three-momentum space, which is of the size of the external particle momenta. As a result, a local mapping at the integrand level is possible between one-loop and tree-level matrix elements to cancel soft and collinear divergences. One can anticipate that a similar analysis at higher orders of the loop-tree duality relation is expected to provide equally interesting results.

In principle, any cancelation of divergencies would not be of great use unless one can perform the remaining three-dimensional integration in a numerically stable way. This was proven to be achievable for a wide range of tried cases of one-loop Feynman integrals with up to six external legs. An implementation of the method currently exists in Mathematica and most importantly in a fast C++ integration routine.

With regard to the MI public library, in the last couple of years, several groups published results presenting analytic expressions for the two-loop MI of several 2 -> 2 processes. This fact means that the virtual radiative corrections can now be addressed in a fully analytic way for most of the cases.

Lastly, a number of achievements were obtained from the work on the BFKL formalism. An iterative method for solving the BFKL equation numerically was proposed several years ago. The iterative method was implemented anew in a C++ Monte Carlo code that calculates the gluon Green's function for different BFKL kernels: forward LO and NLO kernels for QCD and N = 4 SYM in the fundamental representation and both forward and non-forward LO and NLO kernels for N = 4 SYM in the adjoint representation. This led to a number of studies with emphasis put on optimizing the code to be ready for phenomenological studies at the LHC.

The applicability of Lipatov's effective action to the description of high energy processes in QCD was also explored. This effective action is not obtained through a reduction of the number of degrees of freedom, but rather adds a new degree of freedom through the Reggeized gluon. Several non-trivial checks of the validity of this prescription, including one-loop and two-loop computations, have been performed finding agreement with previous results in the literature. The most complicated case would be that of the two-loop gluon Regge trajectory.

At the LHC, any verification of present knowledge or solid new proof of new physics will only come after copious efforts of comparing experimental data against accurate theoretical predictions. For most of the processes at the LHC, perturbative calculations in quantum field theory at the next-to-leading order (NLO) are sufficient whereas for some processes next-to-next-to-leading order (NNLO) is needed, a highly non trivial theoretical task. Furthermore, apart from the so called fixed-order calculations which are based on taking into account only the first few terms of the perturbative expansion, resummation programmes will have to be considered in cases where large logarithms of some scale appear to all orders such that they endanger the validity of the perturbative expansion itself.

One of the most important resummation programmes in high energy Quantum Chromodynamics (QCD) is the Balitsky-Fadin-Kuraev-Lipatov (BFKL) formalism. The BFKL dynamics, not only takes care of resumming large logarithms in the centre-of-mass energy that may spoil the convergence of the perturbative expansion at high energies but is also in the core of addressing fundamental issues on factorization between soft and hard physics. Moreover, BFKL can offer a theoretical bridge connecting QCD and N = 4 Supersymmetric Yang-Mills (N = 4 SYM) quantum field theory, sometimes considered as the harmonic oscillator of the 21st century since it seems to provide, in the planar limit, the first example of a solvable non trivial quantum field theory in four dimensions. The discovery of an unexpected simplicity in the structure of scattering amplitudes and the connection with string theory provided by the anti de Sitter/conformal field theory (AdS/CFT) duality support this reasoning.

In this project, we proposed to apply a novel technique, the Loop-Tree Duality method between loop Feynman integrals and phase-space integrals, to NLO (and subsequently to NNLO) QCD calculations for LHC phenomenology. Secondly, we proposed to create a public library with the results of all the two-loop Feynman integrals necessary for some of the most important LHC processes that are needed to NNLO. For a generic 2 -> 2 NNLO QCD process at the LHC one can reduce the necessary Feynman integrals needed to be computed to a smaller basis of so-called Master Integrals (MI). The exact basis is naturally process-dependent, nevertheless, many of the processes share common Feynman diagram topologies which reduces the total number of the MI needed to be in the Library. A third fieldwork emerged in the course of the project, this one in connection to the BFKL formalism whereas the first two are connected to fixed order calculations. It was directed to two different approaches for studying high energy scattering amplitudes. One was toward the development of a Monte Carlo code for studying physical observables at the LHC and the other was toward more formal grounds like the applicability of Lipatov's effective action to the description of high energy processes in QCD.

There was very good progress in all subfields of this project.

Firstly, it was shown that the Loop-Tree Duality method exhibits attractive theoretical aspects and nice properties which are manifested by a direct physical interpretation of the singular behavior of the loop integrand. After applying the Loop-Tree Duality method, integrand singularities cancel among dual integrals. The remaining singularities, excluding UV divergences, can be interpreted in terms of causality and are restricted to a finite region of the loop three-momentum space, which is of the size of the external particle momenta. As a result, a local mapping at the integrand level is possible between one-loop and tree-level matrix elements to cancel soft and collinear divergences. One can anticipate that a similar analysis at higher orders of the loop-tree duality relation is expected to provide equally interesting results.

In principle, any cancelation of divergencies would not be of great use unless one can perform the remaining three-dimensional integration in a numerically stable way. This was proven to be achievable for a wide range of tried cases of one-loop Feynman integrals with up to six external legs. An implementation of the method currently exists in Mathematica and most importantly in a fast C++ integration routine.

With regard to the MI public library, in the last couple of years, several groups published results presenting analytic expressions for the two-loop MI of several 2 -> 2 processes. This fact means that the virtual radiative corrections can now be addressed in a fully analytic way for most of the cases.

Lastly, a number of achievements were obtained from the work on the BFKL formalism. An iterative method for solving the BFKL equation numerically was proposed several years ago. The iterative method was implemented anew in a C++ Monte Carlo code that calculates the gluon Green's function for different BFKL kernels: forward LO and NLO kernels for QCD and N = 4 SYM in the fundamental representation and both forward and non-forward LO and NLO kernels for N = 4 SYM in the adjoint representation. This led to a number of studies with emphasis put on optimizing the code to be ready for phenomenological studies at the LHC.

The applicability of Lipatov's effective action to the description of high energy processes in QCD was also explored. This effective action is not obtained through a reduction of the number of degrees of freedom, but rather adds a new degree of freedom through the Reggeized gluon. Several non-trivial checks of the validity of this prescription, including one-loop and two-loop computations, have been performed finding agreement with previous results in the literature. The most complicated case would be that of the two-loop gluon Regge trajectory.