OPE Convergence in Conformal Field Theory (1208.6449v3)

Abstract: We clarify questions related to the convergence of the OPE and conformal block decomposition in unitary Conformal Field Theories (for any number of spacetime dimensions). In particular, we explain why these expansions are convergent in a finite region. We also show that the convergence is exponentially fast, in the sense that the operators of dimension above Delta contribute to correlation functions at most exp(-a Delta). Here the constant a>0 depends on the positions of operator insertions and we compute it explicitly.

• The paper establishes that operator contributions above a threshold in unitary CFTs decay exponentially.
• It employs Laplace transforms and the Hardy-Littlewood tauberian theorem to quantify the OPE convergence rate.
• The exponential convergence of conformal blocks enables efficient truncation in bootstrap methods for CFTs.

Overview of "OPE Convergence in Conformal Field Theory"

This paper, authored by Duccio Pappadopulo, Slava Rychkov, Johnny Espin, and Riccardo Rattazzi, provides a meticulous examination of the convergence properties of the Operator Product Expansion (OPE) and the associated conformal block decomposition in unitary Conformal Field Theories (CFTs) in any number of spacetime dimensions. The authors aim to establish the conditions under which these expansions are guaranteed to converge and quantify the rate at which convergence occurs.

Convergence of OPE in CFT

The paper begins by contextualizing the relevance of CFTs in a variety of theoretical physics domains, including critical phenomena and quantum gravity. In these theories, OPE is a pivotal tool that expresses the product of two local operators as a sum of local operators. The authors restrict their initial analysis to the OPE of two identical scalar operators, a setup that simplifies the exploration of convergence.

The authors demonstrate that in unitary CFTs, the OPE converges within a finite region dictated by the spacetime positions of the operators. More specifically, they establish that the contribution of operators with dimensions exceeding a threshold $\Delta$ to the correlation functions decays exponentially as $\exp(-a\Delta)$, where the constant $a > 0$ is calculated based on the geometry of operator insertions.

Methodology and Spectral Density Analysis

A critical aspect of the authors’ methodology is the analysis of the convergence rate of the OPE series, expressed as a Laplace transform of a weighted spectral density. The density encompasses the scaling dimensions of exchanged operators and their OPE coefficients. A remarkable insight is the demonstration that the cumulative spectral density exhibits a power-law growth asymptotically, which allows the authors to conclude about the exponential convergence rate of the OPE.

The results are supported by employing the Hardy-Littlewood tauberian theorem, which bridges the asymptotic behavior of the Laplace transform with that of the spectral density. The authors show that for high-energy contributions, this spectral density's growth is significantly restricted, a finding that ties back to the physical notion that the OPE coefficients must be exponentially suppressed to maintain consistency.

Convergence of Conformal Block Decomposition

Beyond the OPE, conformal blocks are a fundamental representation in CFT for expressing correlation functions via conformal symmetry. The convergence of the conformal block decomposition is crucial for ensuring that the bootstrap methods yield accurate constraints on CFT data. The authors extend their results to show that this series also converges exponentially fast not only in the unit disk but across a larger domain in the complex plane, dictated by the conformally invariant variable $z$.

Implications and Future Directions

The findings of this paper bear significant implications for the practical application of CFTs, particularly in the context of the conformal bootstrap program, which aims to numerically solve CFTs by exploiting crossing symmetry and consistent OPE expansions. The demonstrated exponential convergence implies that practitioners can efficiently truncate the expansion series without substantial loss of accuracy, facilitating the exploration of CFTs, including those without known Lagrangian formulations.

Future research directions could involve extending these results to operators carrying spin and interacting CFTs in more than four dimensions. Additionally, understanding the crossover from power-law to exponential suppression of OPE coefficients in different settings remains an open question with substantial theoretical interest.

In summary, this paper provides a rigorous and comprehensive analysis of the convergence properties of critical expansions in CFTs. Its results secure the mathematical foundations necessary for leveraging these expansions in exploring and constraining CFT landscapes and hold promise for further advancements in theoretical and mathematical physics.