Standard Model Theory for the FCC-ee Tera-Z stage

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.