Vacuum seagull: Evaluating a three-loop Feynman diagram with three mass scales

We study a three-loop five-propagator Feynman integral, which we call the vacuum seagull, with arbitrary masses and spacetime dimension using the symmetries of Feynman integrals method. It is our first example with potential numerators. We determine the associated group GGL(3) which happens to be five dimensional and the associated set of five differential equations. G is determined by a geometric approach which we term "current freedom." We find the generic G-orbit to be co-dimension 0 and hence the method is maximally effective, and the diagram reduces to a line integral over simpler diagrams. For a reduced parameter space with three mass scales, we are able to present explicit results in terms of special functions for general dimension along with the ϵ expansion around d=4. This might be the first such example.