Nnlo QCD Predictions for Triboson Production at TeV Enable Precision Tests of Electroweak Couplings at the LHC

The precise measurement of rare particle interactions increasingly tests the Standard Model of particle physics, and the production of multiple bosons provides a crucial window into the behaviour of fundamental forces. Paolo Garbarino, Massimiliano Grazzini, Stefan Kallweit, and Chiara Savoini, from the Universities of Zürich and Technical University of Munich, have now calculated the most accurate theoretical predictions to date for the production of a boson alongside two others at the Large Hadron Collider. This achievement involves a complex calculation incorporating next-to-next-to-leading-order quantum corrections, significantly refining previous estimates and reducing theoretical uncertainties to just a few percent. The team’s results demonstrate substantial corrections to earlier predictions, enhancing their accuracy and paving the way for more precise tests of the Standard Model at the LHC.

Researchers employ increasingly complex calculations, extending beyond the simplest approximations to incorporate higher-order corrections. These corrections are vital for reducing theoretical uncertainties and achieving the desired level of precision.

Scientists utilize sophisticated techniques to remove infinities through a process called renormalization, and employ subtraction methods to handle the complex integrals that arise in higher-order calculations. Calculating these corrections requires evaluating intricate loop integrals, often using specialized software packages. Modern techniques, such as the calculation of pentagon functions, are also employed to further refine the accuracy of the predictions. Accurate modeling of the internal structure of protons, described by parton distribution functions, is also crucial for these calculations. The study pioneers a complex calculation involving quartic gauge-boson couplings, fundamental interactions between bosons, and delivers predictions essential for the precision era of collider physics. Researchers employed the qT-subtraction formalism, extending it to handle the production of massive tribosons.

This approach addresses infinite quantities that arise in the calculations and was implemented within the Matrix framework, a powerful tool for multi-particle calculations. To manage the complexity of the NNLO calculation, scientists introduced a technical cut-off and carefully extrapolated the results. A sophisticated rescue system, utilizing quadruple-precision arithmetic, was developed to maintain numerical stability. The team implemented recently available two-loop amplitudes into a dedicated C++ library, building upon existing amplitude libraries and incorporating a robust numerical rescue system. Researchers achieved a conservative numerical error of a few parts per thousand for the predicted NNLO fiducial cross section, demonstrating the reliability and stability of the results. The results demonstrate substantial NNLO corrections, enhancing next-to-leading-order predictions by approximately one percent, with perturbative uncertainties estimated to be around the one percent level. To achieve this level of precision, the team employed the qT-subtraction formalism, introducing a technical cut-off to manage calculations and then extrapolating to obtain predictions independent of this cut-off.

The extrapolation procedure, applied bin-by-bin to kinematic distributions, ensures results are free from unwanted corrections. Measurements confirm the numerical stability of the calculations, allowing for a conservative assignment of a few parts per thousand error to the predicted fiducial NNLO cross section. The team developed a fully automated computation within the Matrix framework, leveraging existing tools for efficient amplitude evaluation. Numerical results for proton-proton collisions at a center-of-mass energy of 13 TeV, using standard fiducial cuts, show the impact of these advanced calculations. The team implemented a sophisticated numerical rescue system, utilizing quadruple-precision arithmetic and symmetry properties of the amplitude, to maintain stability throughout the complex calculations. Scientists achieved this by calculating quantum chromodynamic (QCD) radiative corrections, significantly improving the precision of theoretical predictions for this process. Results demonstrate that these NNLO corrections enhance existing predictions by approximately 23%, while also reducing the uncertainty to around 5%. Researchers observed a notable impact of the corrections on kinematic distributions, particularly in regions where the radiation zero effect is diminished by the inclusion of higher-order terms. The authors acknowledge that their calculation is exact except for an approximation made in the finite part of the two-loop contribution, which introduces a residual uncertainty of less than 1%.