QM Research Seminar: Analytic Reconstruction and Algebro-Geometric Structures in QCD: Two-Loop $pp \rightarrow Vjj$ in the leading-colour approximation

Loop amplitudes are central to precision collider phenomenology, yet their computation remains a major challenge due to the complexity of intermediate steps. A powerful modern strategy is to bypass this complexity through numerical evaluations over finite fields, followed by the reconstruction of the amplitudes’ analytic form from a set of numerical samples using number-theoretic and algebro-geometric techniques.

In this talk, I will present the recent computation of the two-loop leading-colour amplitudes for the production of a heavy electroweak vector boson, $V=\\\{W^\pm,Z,\gamma^*\\\}$, in association with two light jets at hadron colliders ($pp\rightarrow Vjj$) [arXiv:2503.10595], including leptonic decays of the electroweak boson ($V\rightarrow \ell\bar{\ell}$). Compared to the previous state-of-the-art result [arXiv:2110.07541], the new approach achieves a three-order-of-magnitude reduction in size (1.4 GB to 1.9 MB), while also lowering the number of required numerical samples from one million to fifty thousand.

I will outline the core ingredients of this computation: a basis-change algorithm in the vector space of rational functions based on correlations among multivariate residues; numerical sampling in number fields with non-Archimedean metrics (finite fields and $p$-adic numbers); the use of redundant spinor-helicity variables organised via polynomial quotient rings; and the role of primary decompositions in identifying allowed multivariate partial fraction decompositions. These methods not only yield efficient and stable results ready for phenomenological applications, but also expose structural features that we expect to generalise to more complex multi-loop amplitudes.