Abstract
In this paper, we study the implications of conformal invariance in momentum space for correlation functions in quantum mechanics. We find that three point functions of arbitrary operators can be written in terms of the 2F1 hypergeometric function. We then show that generic four-point functions can be expressed in terms of Appell’s generalized hypergeometric function F2 with one undetermined parameter that plays the role of the conformal cross ratio in momentum space. We also construct momentum space conformal partial waves, which we compare with the Appell F2 representation. We test our expressions against free theory and DFF model correlators, finding an exact agreement. We then analyze five, six, and all higher point functions. We find, quite remarkably, that n-point functions can be expressed in terms of the Lauricella generalized hypergeometric function, EA, with n − 3 undetermined parameters, which is in one-to-one correspondence with the number of conformal cross ratios. This analysis provides the first instance of a closed form for generic momentum space conformal correlators in contrast to the situation in higher dimensions. Further, we show that the existence of multiple solutions to the momentum space Ward identities can be attributed to the Fourier transforms of the various possible time orderings. Finally, we extend our analysis to theories with \( \mathcal{N} \) = 1, 2 supersymmetry, where we find that the constraints due to the superconformal ward identities are identical to identities involving hypergeometric functions.
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Acknowledgments
We would like to thank Fernando Alday, M. Ali, Sachin Jain, Sunil Mukhi, Kostas Skenderis and Farman Ullah for valuable discussions. D K.S would also like to thank the organizers of the 18th Kavli Asian Winter School (YITP-W-23-13) for providing a stimulating environment and Precision Wires India Ltd for String Theory and Quantum Gravity research at IISER Pune for making the trip possible. S.Y acknowledges support from INSPIRE fellowship from the Department of Science and Technology, Government of India.
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K.S., D., Mazumdar, D. & Yadav, S. n-point functions in conformal quantum mechanics: a momentum space odyssey. J. High Energ. Phys. 2024, 85 (2024). https://doi.org/10.1007/JHEP08(2024)085
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DOI: https://doi.org/10.1007/JHEP08(2024)085