# Electroweak pseudo-observables and Z-boson form factors at two-loop accuracy

## TL;DR

This paper provides precise Standard Model predictions for electroweak pseudo-observables related to the Z-boson, incorporating two-loop radiative corrections and QCD effects, enhancing the accuracy of electroweak precision tests.

## Contribution

It presents the first complete two-loop electroweak corrections to Z-boson pseudo-observables, with simple parameterization formulas and an analysis of theoretical uncertainties.

## Key findings

- Complete two-loop electroweak corrections calculated
- Parameterization formulas provided for observables
- Estimated size of missing higher-order corrections

## Abstract

We present Standard Model predictions for the complete set of phenomenologically relevant electroweak precision pseudo-observables related to the Z-boson: the leptonic and bottom-quark effective weak mixing angles $\sin^2\theta_{\rm eff}^\ell$, $\sin^2\theta_{\rm eff}^b$, the Z-boson partial decay widths $\Gamma_f$, where $f$ indicates any charged lepton, neutrino and quark flavor (except for the top quark), as well as the total Z decay width $\Gamma_Z$, the branching ratios $R_\ell$, $R_c$, $R_b$, and the hadronic cross section $\sigma_{\rm had}^0$. The input parameters are the masses $M_Z$, $M_H$ and $m_t$, and the couplings $\alpha_s$, $\alpha$. The scheme dependence due to the choice of $M_W$ or its alternative $G_\mu$ as a last input parameter is also discussed. Recent substantial technical progress in the calculation of Minkowskian massive higher-order Feynman integrals allows the calculation of the complete electroweak two-loop radiative corrections to all the observables mentioned. QCD contributions are included appropriately. Results are provided in terms of simple and convenient parameterization formulae whose coefficients have been determined from the full numerical multi-loop calculation. The size of the missing electroweak three-loop or QCD higher-order corrections is estimated. We briefly comment on the prospects for their calculation. Finally, direct predictions for the $Z{\bar f}f$ vector and axial-vector form-factors are given, including a discussion of separate order-by-order contributions.