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We study a 3-loop 5-propagator Feynman Integral, which we call the vacuumseagull, with arbitrary masses and spacetime dimension using the Symmetries ofFeynman Integrals method. It is our first example with potential numerators. Wedetermine the associated group $G \subset GL(3)$ which happens to be 5dimensional and the associated set of 5 differential equations. $G$ isdetermined by a geometric approach which we term "current freedom". We find thegeneric $G$-orbit to be co-dimension 0 and hence the method is maximallyeffective, and the diagram reduces to a line integral over simpler diagrams.For a reduced parameter space with 3 mass scales we are able to presentexplicit results in terms of special functions. This might be the first suchexample.

The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales