Two-Parton Contribution to the Heavy-Quark Forward-Backward Asymmetry in NNLO QCD

Forward-backward asymmetries, $A_{FB}^Q$, are important observables for the determination of the neutral-current couplings of heavy quarks in inclusive heavy quark production, $e^{+}{e}^{-}\to {\gamma }^{\ast },Z^{\ast }\to Q+X$. In view of the measurement perspectives on $A_{FB}^Q$ at a future linear collider, precise predictions of $A_{FB}^Q$ are required for massive quarks. We compute the contribution of the $Q\bar{Q}$ final state to $A_{FB}^Q$ to order ${\text{as}}^2$ in the QCD coupling. We provide general formulae, and we show that this contribution to $A_{FB}^Q$ is infrared-finite. We evaluate these two-parton contributions for $b$ and $c$ quarks on and near the $Z$ resonance, and for $t$ quarks above threshold. Moreover, near the $t\bar{t}$ threshold we obtain, by expanding in the heavy-quark velocity $\beta$, an expression for $A_{FB}^{t \bar t}$toorder$\as^2$andNNLLin$\beta$. This quantity is equal, to this order in $\beta$, to the complete forward-backward asymmetry $A_{FB}^t$.