EFT Meets CFT: Five-Loop Renormalization of Dimension-Six Operators Enables Standard Model EFT Calculations

The precise calculation of how fundamental quantities change with energy is central to understanding both particle physics and the behaviour of matter under extreme conditions, and Johan Henriksson of Université Paris-Saclay, Stefanos R. Kousvos of the University of Pisa, and Jasper Roosmale Nepveu of National Taiwan University, along with their colleagues, have now pushed the boundaries of these calculations significantly. They compute how composite operators, which describe interactions between particles, transform at very high energies, achieving unprecedented accuracy up to five loops for many scenarios, and extending the reach of these calculations to higher dimensions. This work bridges the gap between effective field theory and conformal field theory, providing crucial insights into the behaviour of the Higgs boson within the Standard Model and enabling detailed predictions for the properties of complex systems like the Ising model and hypercubic scalar theories, ultimately paving the way for more advanced theoretical studies. The team makes all their results publicly available, offering a powerful resource for future investigations into a wide range of physical phenomena.

Multiloop Renormalization in φ4 Theories

This research investigates the interplay between effective field theory and conformal field theory through detailed calculations of multiloop renormalization constants for higher-dimensional operators in general φ4 theories. It addresses the long-standing challenge of determining the finite parts of renormalization constants, essential for precise predictions in quantum field theory. The team systematically calculated three-loop corrections to the anomalous dimensions of operators with scaling dimension Δ ≥ 4, utilising dimensional regularization and minimal subtraction schemes. These calculations, performed for a general φ4 theory, provide a comprehensive understanding of the renormalization flow and the emergence of non-trivial fixed points.

The research significantly extends previous perturbative calculations by providing complete three-loop results for a broad class of operators, including those relevant for critical phenomena and strongly correlated systems. The method involves careful consideration of loop integrals and the implementation of consistent regularization and renormalization procedures, ensuring the reliability and accuracy of the results. The team demonstrates that the finite parts of the renormalization constants exhibit a specific pattern, understood in terms of the operator product expansion and the conformal symmetry of the underlying theory. This pattern provides valuable insights into the structure of the renormalization group flow and the behaviour of the theory at different energy scales.

A key contribution is the development of a novel computational framework for efficiently calculating multiloop integrals, essential for tackling the complexity of higher-order perturbative calculations. The framework leverages modern techniques in symbolic computation and numerical integration, automating many tedious steps and reducing potential errors. The results confirm the validity of the effective field theory approach in describing the long-distance behaviour of quantum field theories and provide a precise determination of the critical exponents and scaling functions relevant for various physical systems. Furthermore, the team establishes a direct connection between the effective field theory parameters and the conformal field theory data, offering a powerful tool for analysing strongly correlated systems and exploring the non-perturbative regime of quantum field theory.

Critical Phenomena, Conformal Field Theory, and Universality

This extensive list of references focuses on critical phenomena, conformal field theory, renormalization group methods, and related topics in theoretical physics. The collection covers a broad range of research areas, including critical points in physical systems, the concept of universality, and the application of conformal field theory to calculate critical exponents. The renormalization group is a central mathematical framework for understanding how physical systems behave at different scales, with many references focusing on calculations, beta functions, and fixed points. The 1/N expansion and epsilon expansion are powerful perturbative techniques used to study systems with a large number of degrees of freedom or near a critical dimension, respectively.

Beta functions and anomalous dimensions are central quantities in renormalization group calculations, describing how coupling constants and operator dimensions change with scale. More recent references point towards connections between scattering amplitudes and effective field theory, and the geometry of the scalar sector. The references can be categorised into foundational papers, those focusing on the renormalization group and beta functions, those utilising the 1/N expansion, and those exploring conformal field theory and critical phenomena. Foundational work includes early contributions from Wegner, Polyakov, and Brezin, while Jack and Osborn have made influential contributions to background field calculations and beta functions.

Gracey is a prolific researcher in the calculation of critical exponents using the 1/N expansion, and Manashov has also made significant contributions in this area. Codello has focused on multicritical field theories, and O’Dwyer and Osborn have explored the epsilon expansion and exact renormalization group equations. Key researchers include J. A. Gracey, I.

Jack, H. Osborn, and A. N. Manashov. In conclusion, this is a highly specialised and comprehensive bibliography representing a deep dive into the theoretical tools and techniques used to understand critical phenomena and strongly correlated systems in physics.

Five Loop Anomalous Dimensions and Resummation Techniques

This work presents a comprehensive calculation of anomalous dimensions for a wide range of operators within scalar field theories, extending to five loops in most cases. The researchers developed a method to determine these dimensions, crucial for understanding how physical quantities change with energy scale, and applied it to calculate renormalization group equations for the Higgs sector of the Standard Model, up to dimension six and dimension eight, incorporating custodial symmetry. Furthermore, the team successfully determined the complete low-lying spectrum of the Ising and hypercubic scalar conformal field theories by carefully resumming perturbative expansions. The achievement lies in providing a systematic and high-order calculation of these fundamental quantities, enabling more precise predictions in particle physics and condensed matter physics.

The researchers also highlight the consistency of their approach, demonstrating how the cancellation of divergences ensures finite and physically meaningful results. While acknowledging the computational complexity of such calculations, the authors emphasize the systematic nature of their method and provide all results publicly, facilitating future research. They note that setting certain couplings to zero prematurely may introduce inconsistencies, and that the method relies on careful consideration of operator symmetries to avoid ambiguities. Future work, they suggest, could extend this approach to more complicated conformal field theories, building upon the foundation established in this study.