The forward-backward asymmetry for massive bottom quarks at the $Z$ peak at next-to-next-to-leading order QCD

We compute the order ${\alpha }_s^2$ QCD corrections to the $b$-quark forward-backward asymmetry in $e^{+}{e}^{-}\to b\bar{b}$ collisions at the $Z$ boson resonance, taking the non-zero mass of the $b$ quark into account. We determine these corrections with respect to both the $b$-quark axis and the thrust axis definition of the asymmetry. We compute also the distributions of these axes with respect to the electron beam. If one neglects the flavor singlet contributions to the $b$-quark asymmetry, as was done in previous computations for massless $b$ quarks, then the second-order QCD corrections for $m_b\mathrm{\ne }0$ are smaller in magnitude than the corresponding corrections for $m_b=0$. Including the singlet contributions slightly increases the magnitude of the corrections. The massive ${\alpha }_s^2$ corrections to the $b$-quark forward-backward asymmetry slightly diminish the well-known tension between the bare $b$-quark asymmetry and the standard model fit from $2.9\sigma$ to $2.6\sigma$.