We calculate the complete $O({\alpha }^2)$ initial state radiation corrections to $e^{+}{e}^{-}$ annihilation into a neutral vector boson in a direct analytic computation without any approximation. The corrections are represented in terms of iterated incomplete (elliptic) integrals over alphabets of square-root valued letters. Performing the limit $s\gg m_e^2$, we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in $m_e^2/s$. In this way, we also confirm the validity of the factorization of massive partons in the Drell-Yan process to $O({\alpha }^2)$. We add non–logarithmic terms at $O({\alpha }^2)$ which have not been considered in previous calculations. The final results in the limit $s\gg m_e^2$ can be given in terms of Nielsen integrals.

我们计算完整 $Ø（{\alpha }^2）$ 初始状态辐射校正 ${Ë}^{+}{Ë}^{-}$在没有任何近似的情况下，通过直接解析计算将into灭变成中性矢量玻色子。校正值是通过对平方根值字母的字母进行迭代的不完整（椭圆）积分表示的。执行极限$s\gg {米}_{Ë}^2$，我们发现与Ref。的早期结果存在差异。[1]，并确认参考文献中获得的结果。[2]使用了大规模算子矩阵元素的有效方法，该方法适用于除功率校正以外的所有方法${米}_{Ë}^2/s$。这样，我们还确认了Drell-Yan过程中大规模partons分解的有效性。$Ø（{\alpha }^2）$。我们在以下位置添加非对数项$Ø（{\alpha }^2）$以前的计算中没有考虑过。最终结果在极限$s\gg {米}_{Ë}^2$ 可以根据尼尔森积分给出。