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Generating Function of Loop Reduction by Baikov Representation

Chang Hu, Wen-Di Li, Xiang Li

TL;DR

This work develops a generating-function framework within the Baikov representation to compute reduction coefficients for multi-loop Feynman integrals. By expressing integrals in Baikov variables and employing residue calculus, the authors derive closed-form generating functions that encode reduction coefficients for top and sub-topologies, with explicit one-loop and higher-loop examples including tadpole, bubble, sunset, and vacuum diagrams. The method yields results in terms of hypergeometric functions and, after careful analytic treatment, reduces to purely rational coefficients for the relevant masters, demonstrating practical simplifications and structural insights. The approach shows promise for extending to non-standard propagators and more complex topologies, offering a unifying view of integral reduction and analytic structure that complements existing IBP and unitarity techniques.

Abstract

In this work, we study the computation of reduction coefficients for multi loop Feynman integrals using generating functions constructed within the Baikov representation. Compared with traditional Feynman rules, the Baikov formalism offers a more structured and transparent framework, especially well suited for analyzing the reduction problem. We emphasize that, in a variety of nontrivial cases including several one loop and selected multi loop examples the generating functions can be explicitly computed in closed form, often involving hypergeometric or elementary functions. These analytic expressions signifi cantly simplify the determination of reduction coefficients and enhance their interpretability. The results demonstrate the practicality and potential of this approach, suggesting that the use of generating functions within the Baikov representation can serve as a powerful and flexible tool in modern Feynman integral reduction, even though its full scope for generic multi-loop topologies remains to be explored.

Generating Function of Loop Reduction by Baikov Representation

TL;DR

This work develops a generating-function framework within the Baikov representation to compute reduction coefficients for multi-loop Feynman integrals. By expressing integrals in Baikov variables and employing residue calculus, the authors derive closed-form generating functions that encode reduction coefficients for top and sub-topologies, with explicit one-loop and higher-loop examples including tadpole, bubble, sunset, and vacuum diagrams. The method yields results in terms of hypergeometric functions and, after careful analytic treatment, reduces to purely rational coefficients for the relevant masters, demonstrating practical simplifications and structural insights. The approach shows promise for extending to non-standard propagators and more complex topologies, offering a unifying view of integral reduction and analytic structure that complements existing IBP and unitarity techniques.

Abstract

In this work, we study the computation of reduction coefficients for multi loop Feynman integrals using generating functions constructed within the Baikov representation. Compared with traditional Feynman rules, the Baikov formalism offers a more structured and transparent framework, especially well suited for analyzing the reduction problem. We emphasize that, in a variety of nontrivial cases including several one loop and selected multi loop examples the generating functions can be explicitly computed in closed form, often involving hypergeometric or elementary functions. These analytic expressions signifi cantly simplify the determination of reduction coefficients and enhance their interpretability. The results demonstrate the practicality and potential of this approach, suggesting that the use of generating functions within the Baikov representation can serve as a powerful and flexible tool in modern Feynman integral reduction, even though its full scope for generic multi-loop topologies remains to be explored.

Paper Structure

This paper contains 25 sections, 120 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Review
  3. Baikov representation
  4. Reduction Coefficient
  5. Examples
  6. Generating Function in Baikov Representation
  7. Methodological Framework
  8. One loop examples
  9. Tadpole
  10. Massive bubble
  11. One loop diagram for the heavy quark potential
  12. Higher loop examples
  13. Two loop and three loop vacuum diagram
  14. The massless sunset-type diagram with a vertical propagator
  15. The sunset-type diagram with four propagators
  16. ...and 10 more sections

Figures (5)

  • Figure 1: One-loop diagram for the heavy quark potential. The wavy line denotes a propagator for the static source
  • Figure 2: (a) Two loop vacuum diagram, (b) Three loop vacuum diagram
  • Figure 3: The massless sunset-type diagram with a vertical propagator
  • Figure 4: The sunset-type diagram with four propagators
  • Figure 5: (a) Feynman diagrams corresponding to case A and case B. Wavy lines denote propagators for the static source, (b) Three loop One external momentum diagram.