Iterative structure of finite loop integrals

TL;DR

The paper develops a four-dimensional, uniform-weight differential-equation framework for finite Feynman integrals, revealing a block-triangular, weight-graded structure that enables iterative, Chen-iterated-integral solutions. Applied to planar light-by-light scattering up to three loops, it yields explicit UT bases, canonical differential equations, and Mandelstam representations, with efficient numerical and analytic forms. The approach unifies one-, two-, and three-loop analysis within a single formalism, providing systematic, scalable tools for high-precision multi-loop computations in four dimensions and offering broad potential for QCD and N=4 SYM applications.

Abstract

In this paper we develop further and refine the method of differential equations for computing Feynman integrals. In particular, we show that an additional iterative structure emerges for finite loop integrals. As a concrete non-trivial example we study planar master integrals of light-by-light scattering to three loops, and derive analytic results for all values of the Mandelstam variables $s$ and $t$ and the mass $m$. We start with a recent proposal for defining a basis of loop integrals having uniform transcendental weight properties and use this approach to compute all planar two-loop master integrals in dimensional regularization. We then show how this approach can be further simplified when computing finite loop integrals. This allows us to discuss precisely the subset of integrals that are relevant to the problem. We find that this leads to a block triangular structure of the differential equations, where the blocks correspond to integrals of different weight. We explain how this block triangular form is found in an algorithmic way. Another advantage of working in four dimensions is that integrals of different loop orders are interconnected and can be seamlessly discussed within the same formalism. We use this method to compute all finite master integrals needed up to three loops. Finally, we remark that all integrals have simple Mandelstam representations.

Figures (5)

• Figure 1: Families of massive one- and two-loop integrals for light-by-light scattering. Possible irreducible numerator factors at two loops are not shown.
• Figure 2: Families of massive three-loop integrals for light-by-light scattering. Possible irreducible numerator factors are not shown.
• Figure 3: Hierarchy of one-loop functions. The integrals are classified according to their (transcendental) weight, shown in the leftmost column. Each arrow corresponds to one non-zero element of the derivative matrix $A$, cf. eq. (\ref{['Amatrix1loop']}). The fact that arrows only link integrals in adjacent rows is the statement that the matrix is block triangular. Solid and dashed lines denote massive and massless propagators, respectively.
• Figure 4: Hierarchy of master integrals up to two loops. The integrals are classified according to their (transcendental) weight, shown in the leftmost column. Each arrow corresponds to one non-zero element of the derivative matrix $A$, cf. eq. (\ref{['Amat2loop']}). The fact that arrows only link integrals in adjacent rows is the statement that the matrix is block triangular. The result for an integral can immediately be written down by summing over all paths leading up from the tadpole integral $g_{1}=1$. Each path gives a contribution to an iterated integral, with the integration kernels being specified by the 'letters' written next to the corresponding arrows. Solid and dashed lines denote massive and massless propagators, respectively. Note that the pictures are intended to give an idea of how the integrals look like, but omit details such as e.g. numerator factors.
• Figure 5: Master integrals the integral family, organized by the number of propagators. Each integral stands for a set of integrals sharing the same propagators, with possible numerator factors not shown in the pictures.