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$O(α_s)$ corrections to the polar angle dependence of the longitudinal spin-spin correlation asymmetry in $e^+e^-\to q\bar q$

We provide analytical results for the $O(α_s)$ corrections to the polar angle dependence of the longitudinal spin-spin correlation asymmetry in $e^+e^-\to q\bar q$. For top quark pair production the $O(α_s)$ corrections to the longitudinal spin-spin asymmetry are strongly polar angle dependent and can amount up to $4\%$ in the $q^2$-range from above $t\bar t$ threshold up to $\sqrt{q^2}=1000$ GeV. The $O(α_s)$ radiative corrections to the correlation asymmetry are below $1\%$ in the forward direction where the cross section is largest. In the $e^+e^-\to b\bar b$ case the $O(α_s)$ corrections reduce the asymmetry value from its $m_b=0$ value of $-100\%$ to approximately $-96\%$ for $q^2$-values around the $Z$ peak and are practically independent of the value of the polar angle theta. This reduction can be traced to finite anomalous contributions from residual mass effects which survive the $m_b\to 0$ limit. We discuss the role of the anomalous contributions and the pattern of how they contribute to spin-flip and non-flip terms.