Loop-by-loop differential equations for dual (elliptic) Feynman integrals

A bstract We present a loop-by-loop method for computing the differential equations of Feynman integrals using recently developed dual form formalism. give explicit prescriptions fibration multi-loop forms. Then, we test our formalism on simple, but non-trivial, example: two-loop three-mass elliptic sunrise family integrals. obtain an ? -form equation within correct function space in sequence relatively simple algebraic steps. In particular, none these steps relies analysis q -series. discuss interesting properties satisfied by basis as well its relation to known integrands. The underlying K3-geometry three-loop four-mass integral is also discussed. Finally, speculate how construct “good” at three-loop.