In present work we study the production of ground and excited charmonium states in $e^{-}{e}^{+}\to \gamma {\chi }_{cJ}(nP)(J=0,1)$, and $e^{-}{e}^{+}\to \gamma {\eta }_c(nS)$, through leading order (LO) diagrams, which proceed through exchange of a virtual photon that couples to γ and ${\eta }_c/{\chi }_{cJ}$ through the triangular quark loop diagram, in the framework of $4\times 4$ Bethe-Salpeter equation (BSE), at center of mass energies $\sqrt{s}=10.6$ GeV (Belle energy), and 4.6 GeV (BESIII energy). The amplitude simplifies to a general form required by Lorentz-covariance, in terms of its form factors. The cross sections for these processes with leading order diagrams alone at $\sqrt{s}=10.6$ GeV provide a sizable contribution, which might be mainly due to the BSE being a fully relativistic approach that incorporates the relativistic effect of quark spins and can also describe internal motion of constituent quarks within the hadron in a relativistically covariant manner. Our results are compared with recent Belle data at 10.6 GeV, and BESIII data at 4.6 GeV, as well as other models. Plots of cross sections versus the center-of mass energy, $\sqrt{s}$ reveal mild fluctuations in their behaviour for all the three processes in the low energy ($\sqrt{s}=4\text{–}6$ GeV) region, and are analyzed in terms of the form factors.

在目前的工作中，我们研究了基态和激发态的产生$e^{-}{e}^{+}\to \gamma {\chi }_{CJ}（n磷）（J=0,1）$， 和$e^{-}{e}^{+}\to \gamma {\eta }_C（nS）$，通过主序 (LO) 图，该图通过交换耦合到γ和的虚拟光子来进行${\eta }_C/{\chi }_{CJ}$通过三角夸克环图，在框架中$4\times 4$Bethe-Salpeter 方程 (BSE)，位于质心能量$\sqrt{s}=10.6$GeV（Belle 能量）和 4.6 GeV（BESIII 能量）。就其形状因子而言，幅度简化为洛伦兹协方差所需的一般形式。这些过程的横截面以及主要顺序图仅位于$\sqrt{s}=10.6$GeV 提供了相当大的贡献，这可能主要是因为 BSE 是一种完全相对论的方法，它结合了夸克自旋的相对论效应，并且还可以以相对论协变的方式描述强子内组成夸克的内部运动。我们的结果与最近的 10.6 GeV 的 Belle 数据、4.6 GeV 的 BESIII 数据以及其他模型进行了比较。横截面与质心能量的关系图，$\sqrt{s}$揭示了低能中所有三个过程的行为的轻微波动（$\sqrt{s}=4\text{–}6$GeV）区域，并根据形状因素进行了分析。