Analytic Computing Methods for Precision Calculations in Quantum Field Theory

TL;DR

The paper surveys analytic computing methods for precision quantum field theory calculations, detailing how multi-loop, single-scale observables are approached through a spectrum of techniques and function spaces. It connects the decoupling properties of differential equations to the solvability of master integrals, and catalogs analytic strategies (PSLQ, difference equations, MB representations, hyperlogarithms, DEs, and symbolic summation) that generate and simplify results expressed in harmonic, cyclotomic, and binomial-sqrt alphabets. It also discusses the structure-function factorization approach for universal logarithms and the emergence of advanced function spaces, including elliptic and modular structures, necessary for higher-loop and multi-scale problems. The article sketches a practical roadmap for achieving FCC-era precision, highlighting current capabilities, bottlenecks, and the need for sustained, interdisciplinary collaboration in theory, mathematics, and computer algebra.

Abstract

An overview is presented on the current status of main mathematical computation methods for the multi-loop corrections to single scale observables in quantum field theory and the associated mathematical number and function spaces and algebras. At present massless single scale quantities can be calculated analytically in QCD to 4-loop order and single mass and double mass quantities to 3-loop order, while zero scale quantities have been calculated to 5-loop order. The precision requirements of the planned measurements, particularly at the FCC-ee, form important challenges to theory, and will need important extensions of the presently known methods.