The two-loop electroweak bosonic corrections to $\sin^2\theta_{\rm eff}^{\rm b}$

TL;DR

This paper calculates the two-loop bosonic electroweak corrections to the effective weak mixing angle for bottom quarks in the Standard Model, providing precise numerical predictions and new computational methods for complex Feynman integrals.

Contribution

It presents the first calculation of bosonic two-loop corrections to the $Z\bar{b}b$ vertex and introduces two approaches for evaluating complex two-loop Feynman integrals.

Findings

Relative correction of about -0.00009855 to $\sin^2\theta_{\rm eff}^{\rm b}$

Predicted $\sin^2\theta_{\rm eff}^{\rm b} = 0.232704$ for central values

Demonstrated two computational methods: sector decomposition and Mellin-Barnes techniques.

Abstract

The prediction of the effective electroweak mixing angle $\sin^2\theta_{\rm eff}^{\rm b}$ in the Standard Model at two-loop accuracy has now been completed by the first calculation of the bosonic two-loop corrections to the $Z{\bar b}b$ vertex. Numerical predictions are presented in the form of a fitting formula as function of $M_Z, M_W, M_H, m_t$ and $\Delta{\alpha}$, ${\alpha_{\rm s}}$. For central input values, we obtain a relative correction of $\Delta\kappa_{\rm b}^{(\alpha^2,\rm bos)} = -0.9855 \times 10^{-4}$, amounting to about a quarter of the fermionic corrections, and corresponding to $\sin^2\theta_{\rm eff}^{\rm b} = 0.232704$. The integration of the corresponding two-loop vertex Feynman integrals with up to three dimensionless parameters in Minkowskian kinematics has been performed with two approaches: (i) Sector decomposition, implemented in the packages FIESTA 3 and SecDec 3, and (ii) Mellin-Barnes representations, implemented in AMBRE 3/MB and the new package MBnumerics.