Two-loop master integrals for a planar topology contributing to pp → $$ t\overline{t}j $$

Published for SISSA by Springer Received: November 17, 2022 Accepted: December 27, 2022 Published: January 26, 2023 Simon Badger,a Matteo Becchetti,a Ekta Chaubeya,b and Robin Marzuccac,d a Physics Department, Torino University and INFN Torino, Via Pietro Giuria 1, I-10125 Torino, Italy b Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589, Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France c Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark d Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland E-mail: , , , Abstract: We consider the case of a two-loop five-point pentagon-box integral configuration with one internal massive propagator that contributes to top-quark pair production in association with a jet at hadron colliders. We construct the system of differential equations for all the master integrals in a canonical form where the analytic form is reconstructed from numerical evaluations over finite fields. We find that the system can be represented as a sum of d-logarithmic forms using an alphabet of 71 letters. Using high precision boundary values obtained via the auxiliary mass flow method, a numerical solution to the master integrals is provided using generalised power series expansions. Keywords: Higher-Order Perturbative Calculations, Top Quark ArXiv ePrint: 2210.17477 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2023)156 JHEP01(2023)156 Two-loop master integrals for a planar topology contributing to pp → tt̄j Contents 1 2 Notation and definitions 3 3 Canonical form differential equations and a basis of uniform transcendental weight master integrals 3.1 Pentagon-box sector 3.2 Double-box sectors 3.3 Pentagon-bubble sector 3.4 Box-triangle sectors 3.5 Rational function reconstruction 5 9 10 11 11 12 4 Analytic structure of the differential equations 4.1 Symbol level structure 12 15 5 Numerical solution of the differential equations 5.1 Benchmark points 5.2 Numerical checks 5.3 Remark on square roots numerical evaluation 16 17 17 18 6 Conclusions 19 A UT integrals for sectors with few than five external legs 20 1 Introduction As the heaviest particle in the Standard Model (SM) of particle physics, the top quark has many important implications for the nature of the fundamental forces. The stability of the SM vacuum is highly sensitive to the value of the top mass whose precision measurement is a high priority at the Large Hadron Collider (LHC). Top quark pair production at hadron colliders is known extremely precisely both theoretically and experimentally and can be used to constrain SM parameters and parton distribution functions [1, 2]. It has been argued that top-quark pair production in association with a jet is even more sensitive to the value of the top quark mass [3–5], yet the theoretical predictions for this process are not currently at the same level of precision as the experimental measurements. Current theoretical predictions are represented by the next-to-leading order (NLO) QCD corrections [6, 7] with state-of-the-art predictions including complete decay information and interfaces with a parton shower [8–12]. Mixed QCD and EW corrections are now also available [13]. In order to match the experimental precision, see for example [14, 15], next-to-next-to-leading order –1– JHEP01(2023)156 1 Introduction (NNLO) corrections are required. Indeed, fully differential cross-section predictions at NNLO in the strong coupling would open up opportunities for the most precise determination of the top-quark mass, yet substantial computational bottlenecks remain. A lot of experience in these type of problems has been gained from the study of massless propagator five-point integrals which form a good starting point for the integrals we study in this article. The kinematic case of five massless external particles has now been fully classified into a basis numerically well-defined pentagon functions [16–21]. For the case of one off-shell external leg and four massless legs the situation is also almost complete with the planar [22–24] and the non-planar hexa-box [25] now known. This progress has allowed the calculation of several five-point two-loop scattering amplitudes [16, 19, 26–41] and led to the first NNLO theoretical predictions for 2 → 3 processes [42–46]. In this article we make a small step towards the two-loop amplitudes for pp → tt̄j by considering the computation of the master integrals associated to a five-point pentagon-box configuration with one internal massive propagator (see figure 1). This builds upon previous work considering the one-loop helicity amplitudes expanded up to O(ε2 ) in the dimensional regulator. Our methodology to determine a set of master integrals follows by the means of the differential equation method [47, 48]. In particular, we write the system of differential equations in a canonical form [49], where the dependence on the dimensional regulator factorises. The canonical form requires the identification of a uniform transcendental weight (UT) basis of master integrals and the solution to a large system of Integration-by-Parts (IBP) relations [50, 51]. For the later we employ the Laporta algorithm [52] which can be implemented within a numerical framework using finite field arithmetic [53–55]. The derivation of the differential equation system can be implemented entirely within the dataflow graphs provided by the FiniteFlow library [55] allowing us to sidestep traditional limitations due to huge intermediate expressions. The determination of a UT basis also presents a significant challenge and has a significant effect on the simplicity of the differential equation system. While considerable effort has been spent to determine automated, or semi-automated techniques for the determination of UT bases yet they are still difficult to apply to situations with a large number of kinematic scales. In this work we will describe 1 We refrain from making a stronger statement though the pattern established in pp → tt̄ would mean elliptic curves (and more complicated geometries) would only appear in closed heavy fermion loops or sub-leading colour, non-planar topologies. –2– JHEP01(2023)156 The two-loop scattering amplitudes that form part of the NNLO correction are currently unknown. In general, amplitudes with massive internal propagators represent a considerable increase in complexity compared to the massless internal propagators that have been considered so far for five particle processes. In addition to the growth in algebraic complexity that comes from the increased number of scales, the analytic complexity contained in the Feynman integrals that appear can lead to difficulties in identifying a numerically welldefined function space. In some cases, of which pp → tt̄ is one, analytic evaluation of the integrals leads to elliptic integrals that still (...truncated)