标题： 卡罗尔幅度的费曼规则和环结构 显示英文标题

标题： Feynman rules and loop structure of Carrollian amplitudes

摘要： 在本文中，我们在体还原框架下推导了Carrollian振幅。 该方法中，Carrollian振幅通过傅里叶变换与散射振幅相关联。 我们提出了用于计算Carrollian振幅的费曼规则，其中傅里叶变换作为Carrollian空间中外部线的积分表示出现。 然后我们研究了无质量$\Phi^4$理论中环图级别的四点Carrollian振幅。 由于庞加莱不变性，四点Carrollian振幅可以转化为仅依赖于天球上的交叉比$z$和一个平移不变变量$\chi$的振幅。 四点Carrollian振幅是两点Carrollian振幅的多项式，其自变量被替换为$\chi$。 该多项式的系数在复数$z$平面上有分支切割。 我们还证明了重整化的Carrollian振幅满足Callan-Symanzik方程。 此外，我们通过设计更一般的Carrollian振幅的费曼规则，启用了广义的$\Phi^4$理论。 显示英文摘要

摘要： In this paper, we derive the Carrollian amplitude in the framework of bulk reduction. The Carrollian amplitude is shown to relate to the scattering amplitude by a Fourier transform in this method. We propose Feynman rules to calculate the Carrollian amplitude where the Fourier transforms emerge as the integral representation of the external lines in the Carrollian space. Then we study the four-point Carrollian amplitude at loop level in massless $\Phi^4$ theory. As a consequence of Poincar\'e invariance, the four-point Carrollian amplitude can be transformed to the amplitude that only depends on the cross ratio $z$ of the celestial sphere and a variable $\chi$ invariant under translation. The four-point Carrollian amplitude is a polynomial of the two-point Carrollian amplitude whose argument is replaced with $\chi$. The coefficients of the polynomial have branch cuts in the complex $z$ plane. We also show that the renormalized Carrollian amplitude obeys the Callan-Symanzik equation. Moreover, we initiate a generalized $\Phi^4$ theory by designing the Feynman rules for more general Carrollian amplitude.