[Paper Review] Two-Loop Master Integrals for Leading-Color $pp o t\bar{t}H$ Amplitudes with a Light-Quark Loop

F. Febres Cordero, G. Figueiredo+3 more|arXiv (Cornell University)|Dec 13, 2023

Particle physics theoretical and experimental studies94 references4 citations

TL;DR

This paper computes the two-loop master integrals for leading-color $pp \to t\bar{t}H$ amplitudes involving a closed light-quark loop, using a differential equations approach in $\epsilon$-factorized form. It presents the first analytic solution for a set of 127 two-loop seven-scale master integrals, revealing a novel nested square root structure and enabling numerical evaluation in physical phase space.

ABSTRACT

We compute the two-loop master integrals for leading-color QCD scattering amplitudes including a closed light-quark loop in $t\bar{t}H$ production at hadron colliders. Exploiting numerical evaluations in modular arithmetic, we construct a basis of master integrals satisfying a system of differential equations in $ε$-factorized form. We present the analytic form of the differential equations in terms of a minimal set of differential one-forms. We explore properties of the function space of analytic solutions to the differential equations in terms of iterative integrals which can be exploited for studying the analytic form of related scattering amplitudes. Finally, we solve the differential equations using generalized series expansions to numerically evaluate the master integrals in physical phase space. As the first computation of a set of two-loop seven-scale master integrals, our results provide valuable input for analytic studies of scattering amplitudes in processes involving massive particles and a large number of kinematic scales.

Motivation & Objective

• To compute two-loop master integrals for $pp \to t\bar{t}H$ amplitudes involving a closed light-quark loop, a critical step toward NNLO QCD corrections.
• To construct a basis of master integrals satisfying $\epsilon$-factorized differential equations for the first time in this seven-scale, massive, multi-scale process.
• To identify and analyze the function space of solutions, revealing a novel nested square root dependence in the kinematic invariants.
• To enable numerical evaluation of the master integrals in physical phase space using generalized series expansions.
• To provide a foundational input for future analytic studies of scattering amplitudes with massive particles and many kinematic scales.

Proposed method

• Employed the method of differential equations in $\epsilon$-factorized form to reduce the system of two-loop integrals to a canonical basis.
• Used numerical evaluations in modular arithmetic to construct the master integral basis and verify the $\epsilon$-factorization.
• Expressed the differential equations in terms of 152 differential one-forms, with 148 in $\textrm{d}\log$ form, simplifying the singularity structure.
• Identified a novel nested square root structure $\sqrt{\Delta_{3}^{(5)}}$ and $\sqrt{\Delta_{5}}$ in the kinematic invariants, essential for the analytic description.
• Solved the differential equations using generalized series expansions to numerically evaluate the master integrals in physical phase space.
• Provided a complete basis of pure master integrals for the $T_0$ family, including 18 one-loop integrals with explicit numerator insertions.

Experimental results

Research questions

• RQ1 What is the analytic structure of two-loop master integrals for $pp \to t\bar{t}H$ with a closed light-quark loop, particularly in the presence of multiple massive scales?
• RQ2 Can a canonical basis of master integrals be constructed for this seven-scale, leading-color two-loop amplitude using $\epsilon$-factorized differential equations?
• RQ3 What is the role of nested square roots in the kinematic dependence of the master integrals, and how does it affect the function space of solutions?
• RQ4 How can the differential equations be expressed in a compact, $\textrm{d}\log}$-formalism to reveal the singularity structure and facilitate numerical evaluation?
• RQ5 What is the numerical behavior of the master integrals in physical phase space, and can they be reliably evaluated using generalized series expansions?

Key findings

• The paper presents the first analytic computation of 127 two-loop master integrals for the $pp \to t\bar{t}H$ process with a closed light-quark loop, a key input for NNLO QCD corrections.
• A canonical basis of master integrals was constructed that satisfies $\epsilon$-factorized differential equations, with 148 out of 152 differential one-forms expressible in $\textrm{d}\log}$ form.
• The solution space features a novel nested square root dependence, specifically $\sqrt{\Delta_{3}^{(5)}}$ and $\sqrt{\Delta_{5}}$, which is essential for the analytic description of the amplitudes.
• The differential equations were solved numerically using generalized series expansions, enabling reliable evaluation of the master integrals in physical phase space.
• The master integral basis includes 18 one-loop integrals for the $T_0$ family, with explicit numerator insertions and weight-0 values provided for iterative integral solutions.
• The results are applicable not only to $t\bar{t}H$ but also to related processes such as $pp \to t\bar{t}Z$ and $e^+e^- \to t\bar{t}+2j$, extending their utility.