Abstract
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 Φ4 theory. As a consequence of Poincaré 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 χ invariant under translation. The four-point Carrollian amplitude is a polynomial of the two-point Carrollian amplitude whose argument is replaced with χ. 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 Φ4 theory by designing the Feynman rules for more general Carrollian amplitude.
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Acknowledgments
The work of J.L. was supported by NSFC Grant No. 12005069. The work of W.-B.Liu is supported by “the Fundamental Research Funds for the Central Universities” with No. YCJJ20242112.
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Liu, WB., Long, J. & Ye, XQ. Feynman rules and loop structure of Carrollian amplitudes. J. High Energ. Phys. 2024, 213 (2024). https://doi.org/10.1007/JHEP05(2024)213
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DOI: https://doi.org/10.1007/JHEP05(2024)213