Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

TL;DR

The paper develops and applies an algorithm to analytically solve univariate differential equation systems that factorize at first order, representing solutions as iterated integrals over a finite alphabet with associated constants. The method uncouples the system, solves homogeneous first-order factors via d'Alembertian integrals, and constructs inhomogeneous solutions by variation of constants, producing ε-expansions that remain within the space of iterated hyperexponential integrals. It is demonstrated on the color-planar and complete light-quark non-singlet three-loop heavy-quark vector form factors, yielding analytic results in terms of harmonic and cyclotomic harmonic polylogarithms up to weight 6, with extensive numerical validation and ancillary material. The approach is basis-free and broadly applicable to first-order-factorizing systems, offering a practical tool for high-precision QCD calculations and related perturbative quantum field theory problems, supported by a numerical implementation for cyclotomic HPLs and accompanying software. The work highlights the interplay between differential equations, iterated integrals, and modern special functions in producing compact, checkable predictions for collider phenomenology.

Abstract

We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the coefficient matrix of the differential equations. These systems appear in a large variety of higher order calculations in perturbative Quantum Field Theories. We apply this method to calculate the master integrals of the three--loop massive form factors for different currents, as an illustration, and present the results for the vector form factors in detail. Here the solution space emerging is given by the cyclotomic harmonic polylogarithms and their associated special constants. No special basis representation of the master integrals is needed. The algorithm can be applied as well to more general cases factorizing at first order, which are based on more general alphabets, iterated integrals and associated constants.

Figures (6)

• Figure 1: The color--planar topologies
• Figure 2: The $n_l$ topologies
• Figure 3: The $O(\varepsilon^0)$ contribution to the vector three-loop form factors $F_{V,1}^{(3)}$ (left) and $F_{V,2}^{(3)}$ (right) as a function of $x \in [0,1]$. Dash-dotted line: leading color contribution of the non--singlet form factor; Full line: sum of the complete non--singlet $n_l$-contributions for $n_l =5$ and the color--planar non--singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.
• Figure 4: The $O(\varepsilon^0)$ contribution to the vector three-loop form factors ${\sf Re}[F]_{V,1}^{(3)}$ (left) and ${\sf Im}[F]_{V,1}^{(3)}$ (right) as a function of $x \in [-1,0]$. Full red line: expansion around $x=0$; Full green line: expansion around $x = -1$; Dotted line: $n_l$-contributions. Dash-dotted line: leading color contribution of the non--singlet form factor; Full black line: sum of the complete non--singlet $n_l$-contributions for $n_l =5$ and the color--planar non--singlet form factor.
• Figure 5: The $O(\varepsilon^0)$ contribution to the vector three-loop form factors ${\sf Re}[F]_{V,2}^{(3)}$ (left) and ${\sf Im}[F]_{V,2}^{(3)}$ (right) as a function of $x \in [-1,0]$. Full red line: expansion around $x=0$; Full green line: expansion around $x = -1$; Dotted line: $n_l$-contributions. Dash-dotted line: leading color contribution of the non--singlet form factor; Full black line: sum of the complete non--singlet $n_l$-contributions for $n_l =5$ and the color--planar non--singlet form factor.