Evaluating single-scale and/or non-planar diagrams by differential equations

TL;DR

This work extends the differential equations approach for master integrals to benchmark single-scale and non-planar Feynman diagrams by exploiting singular-limit information to fix boundary constants. By working in a basis of uniform-weight functions and solving DEs for two three-loop families—the massless form-factor with an off-shell leg and the non-planar K4 graph—the authors obtain analytic results in harmonic polylogarithms up to weight six and demonstrate consistency with expansion-by-regions and crossing symmetry. The methodology shows that the DE framework can determine boundary data without direct integrals, and it preserves the same function class for non-planar cases as in planar ones, suggesting broad applicability to complex amplitudes in gauge theories and gravity. This provides a practical route to assemble three-loop non-planar amplitudes and deepens understanding of infrared structures in high-energy theories.

Abstract

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, $p_2^2\neq 0$. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with $ε=(4-D)/2$. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small $p_2^2$ to our results at $p_2^2\neq 0$ and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called $K_4$ graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in $ε$ up to weight six.

Figures (2)

• Figure 1: A family of form-factor integrals. Here $-q^{\mu} = p_{1}^{\mu} + p_{2}^{\mu}$, $q^2 = s$ and $p_{1}^2 = 0, p_{2}^2 \neq 0$.
• Figure 2: (a) Diagram C. (b) The $K_4$ graph. All internal lines are massless and $p_{1}^2=p_{2}^2=p_{3}^2=0$. We discuss families of $K_{4}$ integrals for the cases $p_{4}^2=0$ and $p_{4}^2 \neq 0$.