Multi-Loop Techniques for Massless Feynman Diagram Calculations
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ulti-Loop Techniques for Massless Feynman Diagram Calculations1 A. V. Kotikova, * and S. Teberb aBogoliubov bSorbonne
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980 Russia Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, Paris, F-75005 France *e-mail: [email protected]
Abstract—We review several multi-loop techniques for analytical massless Feynman diagram calculations in relativistic quantum field theories: integration by parts, the method of uniqueness, functional equations and the Gegenbauer polynomial technique. A brief, historically oriented, overview of some of the results obtained over the decades for the massless 2-loop propagator-type diagram is given. Concrete examples of up to 5-loop diagram calculations are also provided. DOI: 10.1134/S1063779619010039
CONTENTS 1. INTRODUCTION 2. BASICS OF FEYNMAN DIAGRAMS 2.1. Notations 2.2. Massless Vacuum Diagrams (and Mellin–Barnes Transformation) 2.3. Massless One-Loop Propagator-Type Diagram 3. MASSLESS TWO-LOOP PROPAGATOR-TYPE DIAGRAM 3.1. Basic Definition 3.2. Symmetries 3.3. Brief Historical Overview of Some Results 4. METHODS OF CALCULATIONS 4.1. Parametric Integration 4.2. Integration by Parts 4.2.1. Presentation of the method 4.2.2. Example of an important application 4.3. The Method of Uniqueness 4.4. Transformation of Indices 4.4.1. Splitting a line into loop 4.4.2. Adding a new propagator and a new loop 4.4.3. Conformal transform of the inversion 4.5. Fourier Transform and Duality 4.6. Functional Equations
4.6.1. Solution of the functional equations 4.6.2. Expansion of I(1 + a5ε) 4.7. Final Results for J(D, p, 1 + a1ε, 1 + a2ε, 1 + a3ε, 1 + a4ε, 1 + a5ε) 4.8. Three-Loop Master Integrals 5. MORE EXAMPLES OF CALCULATIONS 5.1. Three and Four Loop Calculations in the Φ4-Model 5.2. Five Loop Calculations in the Φ4-Model 5.2.1. Diagram “a” 5.2.2. Diagram “d” 6. CONCLUSIONS 7. APPENDIX A. GEGENBAUER POLYNOMIAL TECHNIQUE 7.1. Presentation of the Method 7.2. One-Loop Integral 7.3. One-Loop Integral with Traceless Products 7.4. One-Loop Integral with Heaviside Functions 7.5. Application to I(1 + a) 8. REFERENCES
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1. INTRODUCTION In relativistic quantum field theories, the first perturbative techniques using covariance and gauge invariance were independently developed in the 1940s by Tomonaga [1], Schwinger [2] and Feynman [3] and
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unified by Dyson [4]. This led to the discovery of the concept of renormalization as an attempt to give a meaning to divergent integrals appearing in the perturbative series. Specific ideas of the field theoretic renormalization group (RG) were then developed starting from the 50s [5–11]. Early success came from their application to quantum electrodynamics (QED) with unprecedented agreement between high precision experiments and high precision theoretical computations of measurable quantitie
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