标题： 天弦积分形式及其展开 显示英文标题

标题： Celestial String Integrands & their Expansions

摘要： 我们将一环四点型 $\mathrm{I}$ 开放超弦胶子振幅转化为包括可（非）定向平面和非平面部分的球面关联函数。这需要对散射弦的能量进行Mellin变换，并且对开放弦世界面模空间进行积分。完成前者后，我们得到具有剩余世界面积分的天文学弦积分因子 $\Psi\left(\beta\right)$，其中 $\beta$ 与所考虑的共形主算符的共形标度维度相关。采用另一种方法，即首先对开放超弦振幅进行 $\alpha'$-展开，然后进行Mellin变换，我们得到了一个完全积分表达式，捕获了 $\beta$-平面上的极点结构。同样的分析在树级上也进行了，得到了类似的结果。 我们通过为$\beta$的特定值求解$\Psi\left(\beta\right)$，始终重现了$\alpha'$-展开假设的结果。 在所有方法中，我们都发现对$\alpha'$的依赖性减少为一个简单的总体因子$\left(\alpha'\right)^{\beta-3}$（在圈数和树级$\left(\alpha'\right)^{\beta}$处），与先前的文献一致。 显示英文摘要

摘要： We transform the one-loop four-point type $\mathrm{I}$ open superstring gluon amplitude to correlation functions on the celestial sphere including both the (non-)orientable planar and non-planar sector. This requires a Mellin transform with respect to the energies of the scattered strings, as well as to integrate over the open-string worldsheet moduli space. After accomplishing the former we obtain celestial string integrands with remaining worldsheet integrals $\Psi\left(\beta\right)$, where $\beta$ is related to the conformal scaling dimensions of the conformal primary operators under consideration. Employing an alternative approach of performing an $\alpha'$-expansion of the open superstring amplitude first and Mellin transforming afterwards, we obtain a fully integrated expression, capturing the pole structure in the $\beta$-plane. The same analysis is performed at tree-level yielding similar results. We conclude by solving $\Psi\left(\beta\right)$ for specific values of $\beta$, consistently reproducing the results of the $\alpha'$-expansion ansatz. In all approaches we find that the dependence on $\alpha'$ reduces to that of a simple overall factor of $\left(\alpha'\right)^{\beta-3}$ at loop and $\left(\alpha'\right)^{\beta}$ at tree level, consistent with previous literature.