Periodic Reporting for period 1 - QCDchallenge (QCD challenges in precision physics at the LHC)
Okres sprawozdawczy: 2023-10-01 do 2025-09-30
The next decade of experiments at CERN's Large Hadron Collider (LHC) will put highly stringent tests on the Standard Model of particle physics. This will require to compare the experimental measurements of the very high precision, which will approach the percent level, with the theoretical predictions based on the Standard Model. It is mandatory for theory to match the same level of precision of the measurements, in order to detect the slightest deviation from the Standard Model and exploit the full potential of the experiments.
On the theory side, the calculation of the radiative corrections in Quantum Chromodynamics (QCD) at the required accuracy is pushing the boundaries of the state-of-the-art techniques.
The project QCDchallenge tackled four-loop corrections to the scale evolution of the parton densities of the proton, which enters the interpretation of nearly every measurement at the LHC, and provided new insight into the high-energy limit of QCD scattering amplitudes.
On the theory side, the calculation of the radiative corrections in Quantum Chromodynamics (QCD) at the required accuracy is pushing the boundaries of the state-of-the-art techniques.
The project QCDchallenge tackled four-loop corrections to the scale evolution of the parton densities of the proton, which enters the interpretation of nearly every measurement at the LHC, and provided new insight into the high-energy limit of QCD scattering amplitudes.
Despite the early termination of the project, all the main scientific objectives have been achieved.
1) My collaborators and I computed moments of the splitting functions that determine the scale evolution of the parton densities. We extended a methodology based on the renormalisation of composite operators, which we had previously developed in the context of Yang-Mills theory, to the case of QCD. We first reproduced a set of moments of the known three-loop splitting functions. Finally, we computed the first ten moments of all the splitting functions at four loops, which corresponds to the Next-to-Next-to-Next-to-Leading Order (N3LO) in QCD perturbation theory.
2) We constructed an approximate four-loop scale evolution of the parton densities by using the ten moments of the splitting functions, which we had computed, and information on their kinematic limits. We estimated that the uncertainty of the approximation is of the order of 1% across the kinematic regions that are relevant for the description of the proton collisions at the LHC. We released the approximate splitting functions in a publicly available fortran code.
3) Using the same procedure adopted to construct the approximate splitting functions, we reconstructed the exact analytic result for one gauge invariant contribution to the quark-to-gluon splitting function at four loops.
4) We developed a technique to reconstruct the anomalous dimensions of classes of unphysical operators that are required in the procedure of composite operator renormalisation at the basis of the moment calculation. We obtain the one-loop anomalous dimensions, reproducing the results in the literature and providing new results for one class of operators.
In parallel I have been working on the high-energy limit of QCD amplitudes. Together with my collaborators
5) We formulated the high-energy factorisation through the Next-to-Next-to Leading Logarithms (NNLL), for the QCD amplitudes with five external partons. The factorisation formula involves two contributions, known as Regge pole and Regge cut. We developed a framework to disentangle the Regge cut from the Regge pole and we computed Regge cuts up to two loops.
6) We computed the two-loop QCD amplitudes with five external partons, with full information of the colour structure, in the high energy limit. We verified the high-energy factorisation formula, by removing the two-loop Regge cuts and isolating the Regge pole contribution. We verified the universality of the Regge pole, by comparing the results for three different processes, namely gluon-gluon to 3 gluon, quark-gluon to quark-gluon-gluon and quark-quark to quark-quark-gluon scattering. We determined the two-loop Lipatov vertex, which characterises the contribution of the Regge pole to the five-leg amplitudes and is a key ingredient for high-energy factorisation at NNLL.
7) We studied the high-energy factorisation, at the level of the NNLL, of the amplitudes for the production of one Higgs boson in association with one jet. We established that the Regge cuts must be absent through three-loop order. We verified that in the high energy limit the two-loop amplitudes, which are known in the literature, are indeed described purely by a Regge pole, involving a universal impact factor for the Higgs boson, which we determined at the two-loop level. We verified that the Regge-pole ansatz correctly reproduces the infrared singularities of the amplitudes at three loops.
1) My collaborators and I computed moments of the splitting functions that determine the scale evolution of the parton densities. We extended a methodology based on the renormalisation of composite operators, which we had previously developed in the context of Yang-Mills theory, to the case of QCD. We first reproduced a set of moments of the known three-loop splitting functions. Finally, we computed the first ten moments of all the splitting functions at four loops, which corresponds to the Next-to-Next-to-Next-to-Leading Order (N3LO) in QCD perturbation theory.
2) We constructed an approximate four-loop scale evolution of the parton densities by using the ten moments of the splitting functions, which we had computed, and information on their kinematic limits. We estimated that the uncertainty of the approximation is of the order of 1% across the kinematic regions that are relevant for the description of the proton collisions at the LHC. We released the approximate splitting functions in a publicly available fortran code.
3) Using the same procedure adopted to construct the approximate splitting functions, we reconstructed the exact analytic result for one gauge invariant contribution to the quark-to-gluon splitting function at four loops.
4) We developed a technique to reconstruct the anomalous dimensions of classes of unphysical operators that are required in the procedure of composite operator renormalisation at the basis of the moment calculation. We obtain the one-loop anomalous dimensions, reproducing the results in the literature and providing new results for one class of operators.
In parallel I have been working on the high-energy limit of QCD amplitudes. Together with my collaborators
5) We formulated the high-energy factorisation through the Next-to-Next-to Leading Logarithms (NNLL), for the QCD amplitudes with five external partons. The factorisation formula involves two contributions, known as Regge pole and Regge cut. We developed a framework to disentangle the Regge cut from the Regge pole and we computed Regge cuts up to two loops.
6) We computed the two-loop QCD amplitudes with five external partons, with full information of the colour structure, in the high energy limit. We verified the high-energy factorisation formula, by removing the two-loop Regge cuts and isolating the Regge pole contribution. We verified the universality of the Regge pole, by comparing the results for three different processes, namely gluon-gluon to 3 gluon, quark-gluon to quark-gluon-gluon and quark-quark to quark-quark-gluon scattering. We determined the two-loop Lipatov vertex, which characterises the contribution of the Regge pole to the five-leg amplitudes and is a key ingredient for high-energy factorisation at NNLL.
7) We studied the high-energy factorisation, at the level of the NNLL, of the amplitudes for the production of one Higgs boson in association with one jet. We established that the Regge cuts must be absent through three-loop order. We verified that in the high energy limit the two-loop amplitudes, which are known in the literature, are indeed described purely by a Regge pole, involving a universal impact factor for the Higgs boson, which we determined at the two-loop level. We verified that the Regge-pole ansatz correctly reproduces the infrared singularities of the amplitudes at three loops.
This project delivered the state-or-the-art results for the evolution of the parton densities. Indeed, the project provided highly precise approximations of the N3LO splitting functions, which are ready for precision phenomenology at the LHC. For instance, these can be employed in new fits of the parton densities which include N3LO corrections, as required for LHC phenomenology at percent-level precision. In addition, the results for fixed moments of the splitting functions provide analytic insight into the exact N3LO splitting functions, which are unknown to date.
The project made significant steps towards establishing high-energy factorisation to the next level of logarithmic accuracy. It shed light on the interplay between two different analytic structures, dubbed Regge cut and Regge pole. Finally, it determined two new building blocks, the two-loop Lipatov vertex and the two-loop Higgs impact factor, which govern the high-energy factorisation of multiparton scattering amplitudes in QCD and of Higgs production in association with jets, respectively.
The project made significant steps towards establishing high-energy factorisation to the next level of logarithmic accuracy. It shed light on the interplay between two different analytic structures, dubbed Regge cut and Regge pole. Finally, it determined two new building blocks, the two-loop Lipatov vertex and the two-loop Higgs impact factor, which govern the high-energy factorisation of multiparton scattering amplitudes in QCD and of Higgs production in association with jets, respectively.