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Integration techniques for worldline integrals

Victor M. Banda Guzman, James P. Edwards, C. Moctezuma Mata Zamora, Luis A. Rodriguez Chacon, Christian Schubert

Abstract

The worldline formalism allows one to obtain compact integral representations combining the information of large numbers of Feynman diagrams. However, their analytic calculation leads to a non-standard integration problem for which existing mathematical algorithms are of little help. Here I will summarize the state-of-the-art of worldline integration focusing on examples from QED in vacuum and in constant external fields.

Integration techniques for worldline integrals

Abstract

The worldline formalism allows one to obtain compact integral representations combining the information of large numbers of Feynman diagrams. However, their analytic calculation leads to a non-standard integration problem for which existing mathematical algorithms are of little help. Here I will summarize the state-of-the-art of worldline integration focusing on examples from QED in vacuum and in constant external fields.
Paper Structure (13 sections, 35 equations, 3 figures)

This paper contains 13 sections, 35 equations, 3 figures.

Table of Contents

  1. Worldline representation of the QED S-matrix
  2. String-inspired treatment of the worldline path integral
  3. Inclusion of a constant external field
  4. Generalizations
  5. Advantages of the worldline representation in QED
  6. The fundamental problem of worldline integration
  7. Circular integrals: the polynomial case
  8. The magic magnetic master integral
  9. One-loop four-point amplitudes
  10. Integration-by-parts algorithms
  11. Two-loop QED vacuum polarization tensors
  12. Three-loop $\beta$-function coefficients in $\phi^4$ theory
  13. Conclusions

Figures (3)

  • Figure 1: A typical multi-loop Feynman digram in QED.
  • Figure 2: Two-loop spinor QED vacuum polarisation diagrams.
  • Figure 3: Planar (P) and nonplanar (NP) diagram contributing to the three-loop $\beta$-function in $\phi^4$ theory.