[Paper Review] Analytical Calculation of Two-Loop Feynman Diagrams

Roberto Bonciani|ArXiv.org|Oct 14, 2004

Algebraic and Geometric Analysis3 citations

TL;DR

This paper reviews the Laporta algorithm for reducing two-loop Feynman diagrams to master integrals and the differential equations method for their analytical evaluation. It emphasizes the use of harmonic polylogarithms (HPLs) and their generalizations to express results in terms of transcendental functions, enabling precise analytical control over physical observables at NNLO precision in high-energy physics calculations.

ABSTRACT

We review the Laporta algorithm for the reduction of scalar integrals to the master integrals and the differential equations technique for their evaluation. We discuss the use of the basis of harmonic polylogarithms for the analytical expression of the results and some generalization of this basis to wider sets of transcendental functions.

Motivation & Objective

• To provide a comprehensive review of the Laporta algorithm for reducing two-loop scalar integrals to master integrals (MIs) in multi-scale quantum field theory calculations.
• To present the differential equations technique as a method for analytically evaluating master integrals arising from two-loop diagrams.
• To investigate the optimal basis of special functions—specifically harmonic polylogarithms (HPLs)—for expressing analytical results with manifest analytical structure.
• To extend the HPL framework to include generalized functions capable of handling multiple thresholds and complex kinematic structures, such as those involving square roots or multiple scales.
• To support numerical evaluation accuracy and asymptotic behavior analysis through closed algebraic and transformation properties of the function basis.

Proposed method

• Apply the Laporta algorithm to reduce all scalar integrals of a given topology to a minimal set of master integrals (MIs) using integration-by-parts (IBP), Lorentz invariance, and symmetry identities.
• Construct a system of coupled first-order differential equations in the kinematic variable $x = s/a$ for the MIs, derived from the parametric representation of the integrals.
• Solve the differential equations using series expansions around singular points (e.g., $x=0$ and $x=1$) with boundary conditions determined from known limits.
• Express the solutions in terms of harmonic polylogarithms (HPLs), defined via iterated integrals over a basis of rational functions $g(a;x)$ with $a \in \{-1, 0, 1\}$.
• Generalize the HPL basis to include additional functions such as $g(\pm4;x)$, $g(c,x)$, $g(\pm r,x)$, and $g(\pm1/4;x)$ to handle new thresholds at $x=4$, complex poles, or square roots.
• Ensure the generalized HPLs maintain closure under integration and shuffle algebra, preserving analytical control and enabling efficient numerical evaluation.

Experimental results

Research questions

• RQ1 How can two-loop scalar integrals with multiple scales be systematically reduced to a minimal set of master integrals?
• RQ2 What is the most effective method for analytically solving the differential equations that govern the master integrals in multi-scale processes?
• RQ3 Why are harmonic polylogarithms (HPLs) an optimal basis for expressing analytical results in two-loop calculations?
• RQ4 How can the HPL basis be extended to handle physical configurations involving additional thresholds, such as $x=4$ or complex poles?
• RQ5 What algebraic and transformation properties of generalized HPLs ensure their suitability for numerical evaluation and asymptotic analysis?

Key findings

• The Laporta algorithm enables a complete and automated reduction of two-loop scalar integrals to a finite set of master integrals using IBP and symmetry identities.
• The differential equations method allows for the analytical evaluation of master integrals by solving a system of ODEs in the kinematic variable $x$, with solutions expressed in terms of harmonic polylogarithms.
• Harmonic polylogarithms (HPLs) form a shuffle algebra and are closed under argument transformations, ensuring analytical control and efficient numerical evaluation.
• The standard HPL basis with $a \in \{-1, 0, 1\}$ is sufficient for diagrams with thresholds only at $x=0$ and $x=1$, but must be extended for more complex kinematics.
• Generalized HPLs incorporating functions like $g(\pm4;x)$, $g(c,x)$, $g(\pm r,x)$, and $g(\pm1/4;x)$ allow for the analytical treatment of diagrams with thresholds at $x=4$, complex poles, or square roots.
• The extended HPL basis preserves the key properties of the original HPLs—closure under integration and shuffle algebra—enabling robust analytical and numerical analysis of two-loop amplitudes.