  The Future Circular Collider (FCC) at CERN, a proposed 100-km circular
facility with several colliders in succession, culminates with a 100 TeV
proton-proton collider. It offers a vast new domain of exploration in particle
physics, with orders of magnitude advances in terms of Precision, Sensitivity
and Energy. The implementation plan foresees, as a first step, an Electroweak
Factory electron-positron collider. This high luminosity facility, operating
between 90 and 365 GeV centre-of-mass energy, will study the heavy particles of
the Standard Model, Z, W, Higgs, and top with unprecedented accuracy. The
Electroweak Factory $e^+e^-$ collider constitutes a real challenge to the
theory and to precision calculations, triggering the need for the development
of new mathematical methods and software tools. A first workshop in 2018 had
focused on the first FCC-ee stage, the Tera-Z, and confronted the theoretical
status of precision Standard Model calculations on the Z-boson resonance to the
experimental demands. The second workshop in January 2019, which is reported
here, extended the scope to the next stages, with the production of W-bosons
(FCC-ee-W), the Higgs boson (FCC-ee-H) and top quarks (FCC-ee-tt). In
particular, the theoretical precision in the determination of the crucial input
parameters, alpha_QED, alpha_QCD, M_W, m_t at the level of FCC-ee requirements
is thoroughly discussed. The requirements on Standard Model theory calculations
were spelled out, so as to meet the demanding accuracy of the FCC-ee
experimental potential. The discussion of innovative methods and tools for
multi-loop calculations was deepened. Furthermore, phenomenological analyses
beyond the Standard Model were discussed, in particular the effective theory
approaches. The reports of 2018 and 2019 serve as white papers of the workshop
results and subsequent developments.
