  The future 100-km circular collider FCC at CERN is planned to operate in one
of its modes as an electron-positron FCC-ee machine. We give an overview
comparing the theoretical status to the experimental demands of one of four
foreseen FCC-ee operating stages, Z-boson resonance energy physics, called the
FCC-ee Tera-Z stage for short. The FCC-ee Tera-Z will deliver the highest
integrated luminosities as well as very small systematic errors for a study of
the Standard Model (SM) with unprecedented precision. In fact, the FCC-ee
Tera-Z will allow the study of at least one more perturbative order in quantum
field theory compared to the LEP/SLC precision. The real problem is that the
present precision of theoretical calculations of the various SM observables
does not match that of the anticipated experimental measurements. The
bottle-necks to overcoming this situation are identified. In particular, the
issues of precise QED unfolding and the correct calculation of SM
pseudo-observables are critically reviewed. In an Executive Summary, we specify
which basic theoretical calculations are needed to meet the strong experimental
expectations at the FCC-ee Tera-Z. Several methods, techniques and tools needed
for higher-order multi-loop calculations are presented. By inspection of the
Z-boson partial and total decay width analyses, it is argued that at the
beginning of operation of the FCC-ee Tera-Z, the theory predictions may be
tuned to be precise enough not to limit the physics interpretation of the
measurements. This statement is based on anticipated progress in analytical and
numerical calculations of multi-loop and multi-scale Feynman integrals and on
the completion of two-loop electroweak radiative corrections to the SM
pseudo-observables this year. However, the above statement is conditional as
the theoretical issues demand a very dedicated and focused investment by the
community.
