Abstract
We consider conformal perturbation theory for n-point functions on the sphere in general 2D CFTs to first order in coupling constant. We regulate perturbation integrals using canonical hard disk excisions of size ϵ around the fixed operator insertions, and identify the full set of counter terms which are sufficient to regulate all such integrated n-point functions. We further explore the integrated 4-point function which computes changes to the structure constants of the theory. Using an sl(2) map, the three fixed locations of operators are mapped to 0, 1, and ∞. We show that approximating the mapped excised regions to leading order in ϵ does not lead to the same perturbative shift to the structure constant as the exact in ϵ region. We explicitly compute the correction back to the exact in ϵ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in ϵ region to obtain agreement with the exact results for structure constants in this theory.
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Acknowledgments
We thank the Mainz Institute for Theoretical Physics for hospitality where part of this work was done. We are grateful to Luis Apolo, Scott Collier, and Christoph Keller for helpful discussions. We wish to thank the participants of the workshop “Exact Results and Holographic Correspondences” at MITP for many interesting conversations. BAB is thankful for funding support from Hofstra Univeristy including startup funds and faculty research and development grants, and for support from the Scholars program at KITP, which is supported in part by grants NSF PHY-1748958 and PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP), where some of this work was completed. IGZ is supported by the Cluster of Excellence Precision Physics, Fundamental Interactions, and Structure of Matter (PRISMA+, EXC 2118/1) within the German Excellence Strategy (Project-ID 390831469).
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Burrington, B.A., Zadeh, I.G. Conformal perturbation theory for n-point functions: structure constant deformation. J. High Energ. Phys. 2024, 78 (2024). https://doi.org/10.1007/JHEP06(2024)078
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DOI: https://doi.org/10.1007/JHEP06(2024)078