3-loop tadpoles with substructure from 12 elliptic curves
October 23, 2023
January 22, 2024
The generic 2-loop kite integral has 5 internal masses. Its
completion by a sixth propagator gives a 3-loop tadpole
whose substructure involves 12 elliptic curves. I show how
to compute all such kites and their tadpoles, with 200 digit
precision achieved in seconds, thanks to the procedure of
the arithmetic geometric mean for complete elliptic
integrals of the third kind. The number theory of 3-loop
tadpoles poses challenges for packages such as HyperInt by
Erik Panzer and MZIteratedIntegral by Kam Cheong Au. In
particular, I obtain three surprising empirical reductions
to classical polylogarithms of totally massive tadpoles.
These have been checked at 600-digit precision.
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