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Critical points and number of master integrals

Roman N. Lee, Andrei A. Pomeransky

TL;DR

The paper tackles counting master integrals in multiloop Feynman diagrams by linking the MI count to the critical points of Symanzik-related polynomials. It proposes a geometric approach where the number of MIs equals the sum of Milnor numbers of proper critical points of the combined polynomial G=F+U in the parametric representation (or P0 in Baikov), with Lefschetz thimbles providing a basis of integration contours. The authors develop an algorithm and a Mathematica package, Mint, to automate MI counting and demonstrate its effectiveness on a large 4-loop on-shell g-2 family, including handling rare non-isolated critical point cases. This framework offers a rigorous, geometry-based alternative to brute-force IBP reductions and suggests potential broader applications in algebraic geometry and integral reductions.

Abstract

We consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals.

Critical points and number of master integrals

TL;DR

The paper tackles counting master integrals in multiloop Feynman diagrams by linking the MI count to the critical points of Symanzik-related polynomials. It proposes a geometric approach where the number of MIs equals the sum of Milnor numbers of proper critical points of the combined polynomial G=F+U in the parametric representation (or P0 in Baikov), with Lefschetz thimbles providing a basis of integration contours. The authors develop an algorithm and a Mathematica package, Mint, to automate MI counting and demonstrate its effectiveness on a large 4-loop on-shell g-2 family, including handling rare non-isolated critical point cases. This framework offers a rigorous, geometry-based alternative to brute-force IBP reductions and suggests potential broader applications in algebraic geometry and integral reductions.

Abstract

We consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals.

Paper Structure

This paper contains 6 sections, 31 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Parametric and Baikov representation
  3. Number of master integrals, basis of $M$-cycles and critical points
  4. Algebraic treatment
  5. Example: 4-loop onshell $g-2$ integrals
  6. Conclusion

Figures (8)

  • Figure 1: Contour basis in the cut plane. Out of 5 contours $\Gamma_{1},\ldots\Gamma_{5}$ only 4 are independent, e.g. $\Gamma_{5}=-\Gamma_{1}-\Gamma_{2}-\Gamma_{3}-\Gamma_{4}$
  • Figure 2: Saddle-point contours $\Gamma_{+}\left(z^{(i)}\right)$ and $\Gamma_{-}\left(z^{(i)}\right)$ in the cut plane (respectively, solid and dashed curves with arrows).
  • Figure 3: Finding reduction rules for $J\left(n_{1},n_{2},n_{3}\right)$ with LiteRed.
  • Figure 4: Example of using Mint.
  • Figure 5: Finding the master integrals with $\mathbf{FindMIs}$.
  • ...and 3 more figures