Iterative structure of finite loop integrals

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.