Implications of conformal invariance in momentum space (1304.7760v4)

Abstract: We present a comprehensive analysis of the implications of conformal invariance for 3-point functions of the stress-energy tensor, conserved currents and scalar operators in general dimension and in momentum space. Our starting point is a novel and very effective decomposition of tensor correlators which reduces their computation to that of a number of scalar form factors. For example, the most general 3-point function of a conserved and traceless stress-energy tensor is determined by only five form factors. Dilatations and special conformal Ward identities then impose additional conditions on these form factors. The special conformal Ward identities become a set of first and second order differential equations, whose general solution is given in terms of integrals involving a product of three Bessel functions (`triple-K integrals'). All in all, the correlators are completely determined up to a number of constants, in agreement with well-known position space results. We develop systematic methods for explicitly evaluating the triple-K integrals. In odd dimensions they are given in terms of elementary functions while in even dimensions the results involve dilogarithms. In some cases, the triple-K integrals diverge and subtractions are necessary and we show how such subtractions are related to conformal anomalies. This paper contains two parts that can be read independently of each other. In the first part, we explain the method that leads to the solution for the correlators in terms of triple-K integrals and how to evaluate these integrals, while the second part contains a self-contained presentation of all results. Readers interested only in results may directly consult the second part of the paper.

• The paper presents a momentum space translation of conformal invariance by directly deriving conformal Ward identities for three-point functions.
• It introduces a minimal decomposition of correlators into scalar form factors and novel triple-K integrals, simplifying the treatment of divergences.
• The work offers a robust analytical framework with practical implications for cosmology and quantum field theory through effective regularization and anomaly handling.

Overview of "Implications of conformal invariance in momentum space"

The paper "Implications of conformal invariance in momentum space" by Adam Bzowski, Paul McFadden, and Kostas Skenderis tackles the constraints imposed by conformal invariance on correlation functions in field theories, specifically focusing on their momentum space representations. The work is a natural continuation of efforts to understand correlation functions within the framework of Conformal Field Theory (CFT), traditionally analyzed in position space, and extends these methodologies to momentum space.

The authors present a comprehensive treatment of three-point functions involving stress-energy tensors, conserved currents, and scalar operators, highlighting the constraints enforced by conformal invariance. By employing momentum space techniques, they address the inherent challenges and extract the conformal Ward identities directly in this domain. The analysis culminates in expressing these correlators through scalar form factors and integrals involving Bessel functions.

Key Contributions

1. Momentum Space Analysis: The paper translates the well-known position space results of correlation functions to momentum space. Although such transformations are conceptually straightforward, practical challenges arise due to singularities and renormalization requirements, particularly in even dimensions.
2. Decomposition into Form Factors: A substantial focus is on deriving a new parametrization for correlators. The authors propose a minimal decomposition using form factors, making the imposition of Ward identities more tractable and computationally efficient compared to traditional methods.
3. Triple-K Integrals: One of the core mathematical achievements of the paper is the detailed exploration of triple-K integrals, which serve as solutions to the conformal Ward identities in momentum space. These integrals represent correlations in terms of Bessel functions, providing a novel lens on the structure of CFT correlation functions.
4. Regularization and Renormalization: Addressing the divergences encountered in the triple-K integrals, the authors outline a regularization procedure that relies on analytic continuation in the spacetime and conformal dimensions. This approach ensures the theoretical consistency across various cases, especially when confounded by conformal anomalies in even dimensions.

Implications and Future Directions

The work extends conformal field theory approaches to momentum space, providing tools potentially applicable to contexts where momentum space formulations are more natural, such as in cosmology and particle physics. The novel insights into the regularization of divergences and handling of anomalies could also influence renormalization techniques in quantum field theories more broadly.

Moreover, the introduction and successful deployment of triple-K integrals might inspire further studies focusing on higher-point functions or cases involving distinct symmetry properties. The techniques might also be adapted to incorporate parity-violating or non-trivial topological terms, thereby broadening their applicability.

The results presented lay the groundwork for potential explorations into higher-order tensor decomposition strategies in more complex field theories. Future work might focus on extending these methods to other correlation functions or even other symmetries beyond the conformal case.

In essence, this paper not only provides a robust analytical framework for momentum space analysis of conformal invariance but also offers a basis for practical applications within physics, promising richer mathematical structures and insights into the behavior of correlation functions across different theoretical models.