Massless Two-loop Self-energy Diagram

ai−d πd ∫ dk1 d k2 [(k1 − k)2]a1 [(k2 − k)2]a2 [(k1 − k2)]3(k 2) a4(k2 1) a5 (1.1) in d-dimensional Euclidean momentum space. It has a long and interesting history. For many years, most of the information we had about perturbative quantum field theory was coming (directly or indirectly) from this integral. All massless three-loop self-energy integrals (with integer indices) reduce to 6 master integrals [1], 5 of which are particular cases of I (1.1). Only one master integral (the non-planar one) does not reduce to I; however, using the gluing method [1], one can easily show that its value at ε = 0 is equal to the ladder integral, which reduces to I(1, 1, ε, 1, 1). At four loops [2], 15 master integrals (of 28) reduce to I, and thus can be easily expanded in ε up to high powers.