Operator product expansion coefficients from the functional renormalization group
Félix Rose (LPTM, Cergy-Pontoise)
In a quantum field theory, the product of two operators in the short distance limit can be expressed as a sum of operators multiplied by functions (when inserted in any correlation function).
This so-called operator product expansion (OPE) is of fundamental importance in the study of conformal field theories (CFTs) in two and higher dimensions. There, due to the the conformal symmetry, the OPE of two operators is uniquely determined by a set of numbers, the OPE (or Wilson) coefficients [1].
The determination of these coefficients has generated recent interest, with approaches from conformal bootstrap or the fuzzy sphere [3].
Here, we are interested in studying these coefficients using the non-perturbative functional renormalization group (FRG). The FRG is a versatile method, used in high-energy, condensed matter and statistical physics, particularly well-suited to determining universal quantities of critical and near-critical systems.
So far, the FRG has been able to determine with great quantitative precision c112, the leading order coefficient of the OPE of ϕ and ϕ, for the 3d O(N) models and the Ising model in dimensions 2<=d<=4 [2]. The method used, however, cannot be directly applied to other coefficients. Here, we review these results and present a different approach that can be used to determine arbitrary coefficients, which relies on finding the fixed point for the flow equation in presence of a source coupled to a composite operator, as recently proposed in Refs. [4].
[1] Di Francesco, Mathieu, and Sénéchal, Conformal Field Theory (Springer New York, 1997).
[2] Rose, Pagani, and Dupuis, Phys. Rev. D 105, 065020 (2022).
[3] Zhu, Han, Huffman, Hofmann, and He. Phys. Rev. X 13, 021009 (2023); Hu, He, and Zhu,
Phys. Rev. Lett. 131, 031601 (2023).
[4] Delamotte, De Polsi, Tissier, and Wschebor. Phys. Rev. E 109 064152 (2024); Cabrera, De Polsi, Wschebor. Phys. Rev. E 111 054126 (2025).