Analytic computing methods for precision calculations in quantum field theory

An overview is presented on the current status of main mathematical computation methods for the multiloop corrections to single-scale observables in quantum field theory and the associated mathematical number and function spaces and algebras. At present, massless single-scale quantities can be calculated analytically in QCD to 4-loop order and single mass and double mass quantities to 3-loop order, while zero-scale quantities have been calculated to 5-loop order. The precision requirements of the planned measurements, particularly at the FCC-ee, form important challenges to theory, and will need important extensions of the presently known methods