The highest precision for theoretical predictions at the high LHC energy requires the calculation of parton distribution function (PDFs) and the structure functions which include heavy quarks that include perturbative QCD and QED corrections. In this paper, the first, we review calculation of PDFs with QED corrections that we obtain them from the QCD⊗QED DGLAP evolution equations in Mellin space in NLO QCD and NLO QED approximations. We give a detailed explanation of the approach that we follow to determine the longitudinal structure functions up to $\mathcal{O}({\alpha }_s^2)$ in QCD, $\mathcal{O}(\alpha {\alpha }_s)$ and $\mathcal{O}({\alpha }^2)$ in QED approximations. We show how our method gives an exact representation for the total longitudinal structure functions, $F_L(x,Q^2)$, including the heavy quark structure functions in terms of the PDFs based on the Mellin transform techniques, valid to all orders in QED and QCD. In this work, we investigate the effects of the $\mathcal{O}(\alpha {\alpha }_s)$ and $\mathcal{O}({\alpha }^2)$ corrections to PDFs on the longitudinal structure functions. We compare our results with APFEL and experimental data in the different values of $Q^2$. The results are shown that the QED corrections have a small but non-negligible impact on the structure functions.

高 LHC 能量理论预测的最高精度需要计算部分子分布函数 (PDF) 和结构函数，其中包括包含微扰 QCD 和 QED 校正的重夸克。在本文中，首先，我们回顾了具有 QED 校正的 PDF 计算，我们从 NLO QCD 和 NLO QED 近似中的梅林空间中的 QCD⊗QED DGLAP 演化方程获得它们。我们详细解释了我们所遵循的确定纵向结构函数的方法$\mathcal{哦}({\alpha }_{秒}^2)$ 在 QCD 中， $\mathcal{哦}(\alpha {\alpha }_{秒})$ 和 $\mathcal{哦}({\alpha }^2)$在 QED 近似值中。我们展示了我们的方法如何准确表示总纵向结构函数，$F_{升}(X,{问}^2)$，包括基于梅林变换技术的 PDF 中的重夸克结构函数，对 QED 和 QCD 中的所有阶有效。在这项工作中，我们调查了$\mathcal{哦}(\alpha {\alpha }_{秒})$ 和 $\mathcal{哦}({\alpha }^2)$对纵向结构函数的 PDF 进行更正。我们将我们的结果与 APFEL 和不同值的实验数据进行比较${问}^2$. 结果表明，QED 修正对结构函数的影响很小但不可忽略。