- Open Access
Exact 3D Conformal Blocks from Fractional Calculus
Phys. Rev. Lett. 135, 211603 – Published 19 November, 2025
DOI: https://doi.org/10.1103/2jmh-qdhj
Abstract
We uncover a striking connection between conformal blocks and fractional calculus. By employing a modified form of half derivatives, we derive explicitly the exact form of the three-dimensional conformal block, expressed as the product of two hypergeometric functions. This result provides a rigorous proof of Hogervorst’s formula, conjectured nearly a decade ago. Furthermore, we demonstrate its implications for the conformal bootstrap, potentially leading to new analytical techniques and numerical tools that deepen our understanding of conformal field theory.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (26)
- S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Phys. Rev. D 86, 025022 (2012).
- D. Poland and D. Simmons-Duffin, Nat. Phys. 12, 535 (2016).
- D. Simmons-Duffin, in New Frontiers in Fields and Strings: TASI 2015 Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics (World Scientific, Singapore, 2017), pp. 1–74.
- D. Poland, S. Rychkov, and A. Vichi, Rev. Mod. Phys. 91, 015002 (2019).
- S. Rychkov and N. Su, Rev. Mod. Phys. 96, 045004 (2024).
- M. Isachenkov and V. Schomerus, Phys. Rev. Lett. 117, 071602 (2016).
- F. A. Dolan and H. Osborn, Nucl. Phys. B678, 491 (2004).
- M. Hogervorst, J. High Energy Phys. 09 (2016) 017.
- S. Pal, J. Qiao, and S. Rychkov, Commun. Math. Phys. 402, 2169 (2023).
- S. Caron-Huot, Y. Gobeil, and Z. Zahraee, J. High Energy Phys. 02 (2023) 190.
- D. Simmons-Duffin, J. High Energy Phys. 03 (2017) 086.
- D. Simmons-Duffin, J. High Energy Phys. 04 (2014) 146.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley & Sons, New York, 1993).
- I. N. Sneddon, in Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974 (Springer, New York, 2006) pp. 37–79.
- F. Dolan and H. Osborn, arXiv:1108.6194.
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/2jmh-qdhj for additional derivations.
- M. Hogervorst and S. Rychkov, Phys. Rev. D 87, 106004 (2013).
- M. Hogervorst and B. C. van Rees, J. High Energy Phys. (2017) 193.
- S. Caron-Huot, J. High Energy Phys. 09 (2017) 078.
- D. Simmons-Duffin, D. Stanford, and E. Witten, J. High Energy Phys. 11 (2018) 140.
- C. Behan, Phys. Rev. D 97, 094020 (2018).
- M. Hogervorst, arXiv:1703.08159.
- D. Karateev, P. Kravchuk, and D. Simmons-Duffin, J. High Energy Phys. 01 (2018) 130.
- J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin, J. High Energy Phys. 03 (2019) 052.
- H.-Y. Chen and H. Kyono, J. High Energy Phys. 10 (2019) 149.
- The data underlying the figures and the implementation of the T-transform are available on Zenodo at 10.5281/zenodo.17517603 together with a Mathematica notebook that reproduces the benchmarks.
