Euclidean Coordinate-Space Perturbation Theory Calculates Three-Point Functions for Massive and Massless Fields

Understanding the interactions of fundamental particles requires increasingly complex calculations, and Christoph L. Schröder from Johannes Gutenberg-Universität Mainz, alongside Harvey B. Meyer, also at Johannes Gutenberg-Universität Mainz and CERN, are advancing the tools needed to tackle these challenges. They develop a new approach to calculating particle interactions within theoretical physics, focusing on scenarios involving particles with different masses. This work introduces a refined method for calculating key quantities called correlation functions, which describe how particles influence each other, and provides analytical solutions previously unavailable to researchers. The team’s results promise to improve the accuracy of calculations in diverse areas, including studies of extreme environments like those found in the early universe, and will aid in refining theoretical models used to interpret data from particle colliders and lattice quantum chromodynamics.

Scientists develop analytic expressions for several three-point correlation functions in theories with one massive and one massless field, working within a coordinate-space perturbation theory applicable to massive quantum field theories in general dimensional Euclidean space. This research extends earlier techniques for massless fields to incorporate propagators with mass, providing kinematics relevant to scenarios resembling quantum electrodynamics. The team systematically studied antiderivatives of products of two Bessel functions, multiplied by a power of their common argument, a crucial step for computing complex integrals, and achieved results that will prove useful in high-order perturbative calculations.

Feynman Integrals via Coordinate Space Methods

This document details a comprehensive investigation into the evaluation of Feynman integrals, focusing on coordinate space techniques as an alternative to traditional momentum space calculations. The research highlights the use of Gegenbauer polynomials to represent and manipulate integrals, enabling a systematic approach to simplification. A major application of these techniques lies in calculating the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment, a significant problem in particle physics where experimental and theoretical values currently disagree. The work explores the integration of lattice quantum chromodynamics (QCD) and quantum electrodynamics (QED) calculations to contribute to the HLbL calculation, alongside connections to high-temperature field theory and the study of quark-gluon plasma.

The document systematically details the derivation of the Gegenbauer expansion, explains the advantages of coordinate space for complex integrals, and outlines the application of these techniques to the muon g-2 calculation. Gegenbauer polynomials are presented as special polynomials used to represent and manipulate integrals in coordinate space, allowing for systematic expansion and simplification. Coordinate space integration, performed in position space rather than momentum space, offers advantages for integrals with complex topologies. Neumann’s Addition Theorem and Graf’s Addition Theorem are used to expand integrands in terms of Gegenbauer polynomials, while lattice QCD and QED provide numerical methods for calculating form factors and loop integrals. This document provides a comprehensive and technical overview of Feynman integral evaluation and the muon anomalous magnetic moment, offering a strong focus on coordinate space techniques often overlooked in standard textbooks. The clear organization, detailed derivations, and connection between analytical calculations and lattice QCD simulations make it a valuable resource for researchers in the field.

Massive Field Theory Perturbation Theory Advances

Scientists have advanced coordinate-space perturbation theory to encompass massive field theories in any number of dimensions. They developed analytical expressions for three-point correlation functions involving both massive and massless fields, building upon earlier techniques for massless diagrams. A key achievement lies in systematically studying and solving integrals involving products of Bessel functions, which arise when dealing with massive propagators, and expressing results using Gegenbauer polynomials. This allows for a more efficient calculation of certain types of diagrams compared to traditional momentum-space methods, particularly those with vertices involving numerous legs.

The team demonstrated the technique by calculating the one-loop coordinate-space propagator for a massive particle interacting with a massless one, both in a vacuum and at finite temperature. This provides a foundation for perturbative calculations in scenarios with high-degree vertices, such as those found in finite temperature or finite volume systems, and potentially in lattice quantum chromodynamics. Results show that for diagrams with three legs, the number of vertices grows twice as fast as the number of loops at high orders, suggesting a potential advantage for position-space methods. The study reveals that the number of required position-space integrals is one less than the number of vertices, offering a simplification in certain calculations. Furthermore, the developed techniques are applicable to calculations in finite volumes and within the Matsubara formalism, crucial for thermal boundary conditions, and high-order perturbative calculations of the QCD free energy. This work provides a foundation for evaluating electroweak contributions to Feynman diagrams and for treating ‘QCD blobs’ in lattice QCD calculations.

Massive Field Theories via Coordinate Space Perturbation

Scientists have advanced coordinate-space perturbation theory to encompass massive field theories in any number of dimensions. They developed analytical expressions for three-point correlation functions involving both massive and massless fields, building upon earlier techniques for massless diagrams. A key achievement lies in systematically studying and solving integrals involving products of Bessel functions, which arise when dealing with massive propagators, and expressing results using Gegenbauer polynomials. This allows for a more efficient calculation of certain types of diagrams compared to traditional momentum-space methods, particularly those with vertices involving numerous legs.

The team demonstrated the technique by calculating the one-loop coordinate-space propagator for a massive particle interacting with a massless one, both in a vacuum and at finite temperature. This provides a foundation for perturbative calculations in scenarios with high-degree vertices, such as those found in finite temperature or finite volume systems, and potentially in lattice quantum chromodynamics. While acknowledging that momentum-space methods remain advantageous when dealing with many external legs, the authors highlight the potential benefits of coordinate-space techniques for effective field theories with contact interactions involving four or more legs per vertex.