We present a set of formulas to extract the longitudinal proton structure function F_L(x, Q²) and the reduced cross section σ_r(x, Q²) by using Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations and Altarelli-Martinelli equation at the leading order (LO) and the next-to-leading order (NLO) approximations in perturbative quantum chromodynamics (QCD). These formulas are obtained by the Laplace transform method at the virtualities higher than \({Q_{0}^{2}}\). We show that, the obtained equations for F_L and σ_r depend only on the parton distribution functions (PDF’s) at the initial scale \({Q_{0}^{2}}\). We obtain the corresponding numerical results in a range of the virtuality \({Q_{0}^{2}}\leq Q^{2} \leq 800\) GeV² and Bjorken scale 10^− 4 ≤ x ≤ 1 and compare them with the results achieved by MSTW2008 (Martin et al. Eur. Phys. J 63(2), 189–285, 2009), WT (White and Thorne Phy. Rev. D 75(3), 034005, 2007) and Dipole (b-Sat) model (H. Kowalski, L. Motyka, G. Watt, Phys. Rev. D 74(7), 074016, 2006) predictions and also with data released by the Hadron Electron Ring Accelerator (HERA). Our numerical results show an acceptable agreement with the deep inelastic scattering (DIS) experimental data throughout over a wide range of the Bjorken scale x and the virtuality Q², and then can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects.

我们提出了一组公式，通过使用 Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) 演化方程来提取纵向质子结构函数F _L ( x , Q ² ) 和缩减截面σ _r ( x , Q ² )和 Altarelli-Martinelli 方程在微扰量子色动力学 (QCD) 中处于领先阶 (LO) 和次领先阶 (NLO) 近似。这些公式是在高于\({Q_{0}^{2}}\)的虚拟性下通过拉普拉斯变换方法获得的。我们表明，获得的F _L和σ _r方程仅依赖于初始尺度\({Q_{0}^{2}}\)的部分分布函数 (PDF )。我们在一定范围内得到相应的数值结果\({Q_{0}^{2}}\leq Q^{2} \leq 800\) G e V ²和 Bjorken scale 10 ^{− 4} ≤ x ≤ 1并将它们与 MSTW2008 (Martin et al. Eur. Phys. J 63 (2), 189–285, 2009)、WT (White and Thorne Phy. Rev. D 75 (3), 034005, 2007) 取得的结果进行比较和偶极子 (b-Sat) 模型 (H. Kowalski, L. Motyka, G. Watt, Phys. Rev. D 74(7), 074016, 2006) 预测以及强子电子环加速器 (HERA) 发布的数据。我们的数值结果与深非弹性散射 (DIS) 实验数据在 Bjorken 尺度x和虚拟性Q ²的广泛范围内具有可接受的一致性，然后可以应用于大型强子对撞机和未来圆形对撞机项目的分析.