Double virtual QCD corrections to tt+jet production at the LHC

In this paper collaborators and I provide the final necessary ingredients for a high-precision calculation of top-quark pair production at the Large Hadron Collider (LHC).

In the “tt + jet” process a pair of top quarks is produced alongside an additional jet of particles. This process is highly significant because approximately 50% of all top-quark events at the LHC include an associated jet. While basic predictions for this interaction have existed for years, experimental data from the LHC has become so precise that we pushed towards “Next-to-Next-to-Leading Order” (NNLO) to match that accuracy.

Calculating these processes is notoriously difficult due to “two-loop” amplitudes, which involve complex internal particle interactions. In the work we encountered two primary hurdles:

• Elliptic Structures: These are complex mathematical geometries that appear in the equations and typically block traditional methods of calculation.
• Algebraic Complexity: The sheer size of the mathematical coefficients required for these calculations often exceeds the capabilities of standard computing resources.

To solve these, we used a “hybrid approach”. We constructed a system of special functions that satisfy differential equations involving only algebraic functions, effectively bypassing the difficulties usually associated with elliptic integrals. We also utilized finite-field arithmetic to reconstruct complex analytic expressions, which helps manage the massive algebraic data.

We did not just produce a theoretical result but also developed a numerical strategy ensuring that the amplitudes are stable and fast across different physical scenarios. To make this work accessible to the broader scientific community, we released a C++ library implementing the results. By completing these “double virtual” corrections, we have enabled more accurate measurements of fundamental parameters, such as the top-quark mass. This work ensures that theoretical predictions can keep pace with the high-resolution measurements being produced by the LHC and future colliders.

Double virtual QCD corrections to tt+jet production at the LHC

Authors: Simon Badger, Matteo Becchetti, Colomba Brancaccio, Michał Czakon, Heribertus Bayu Hartanto, Rene Poncelet, Simone Zoia

Abstract: We present a leading colour computation of the double virtual contributions to top-quark pair production in association with a jet at a hadron collider at next-to-next-to-leading order in QCD. The finite remainders of the two-loop amplitudes, after subtraction of infrared and ultraviolet divergences, are extracted analytically from evaluations over finite fields by using a (potentially) overcomplete basis of special functions defined through their differential equations. We construct the colour- and spin-summed interference with the tree-level amplitudes and present a C++ library suitable for immediate use in phenomenological studies. We present new techniques for the evaluation of the special functions through direct numerical integration of differential equations which perform well across the full physical phase space.

Article on arxiv.org