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Unveiling Regions in multi-scale Feynman Integrals using Singularities and Power Geometry

B. Ananthanarayan, Abhishek Pal, S. Ramanan, Ratan Sarkar

TL;DR

ASPIRE presents an algebro-geometric framework to identify all regions in multi-scale Feynman integrals by solving the Landau equations in alpha-parameter space and applying Power Geometry. The method centers on analyzing the Newton polytope of the combined Symanzik polynomial $G = U + F$ and uses Gröbner-basis transformations to map neighborhoods of singular points to canonical coordinates, yielding region exponents that correspond to hard, collinear, Glauber, and related configurations. It is implemented in Mathematica and validated on representative one-loop and two-loop diagrams, reproducing known regions and revealing additional scaleful structures in certain cases. The approach aligns with prior MoR methods (e.g., ASY) while providing a systematic algebro-geometric route to uncovered regions and setting the stage for extensions to non-planar topologies and subleading contributions.

Abstract

We introduce a novel approach for solving the problem of identifying regions in the framework of Method of Regions by considering singularities and the associated Landau equations given a multi-scale Feynman diagram. These equations are then analyzed by an expansion in a small threshold parameter via the Power Geometry technique. This effectively leads to the analysis of Newton Polytopes which are evaluated using a Mathematica based convex hull program. Furthermore, the elements of the Gröbner Basis of the Landau Equations give a family of transformations, which when applied, reveal regions like potential and Glauber. Several one-loop and two-loop examples are studied and benchmarked using our algorithm which we call ASPIRE.

Unveiling Regions in multi-scale Feynman Integrals using Singularities and Power Geometry

TL;DR

ASPIRE presents an algebro-geometric framework to identify all regions in multi-scale Feynman integrals by solving the Landau equations in alpha-parameter space and applying Power Geometry. The method centers on analyzing the Newton polytope of the combined Symanzik polynomial and uses Gröbner-basis transformations to map neighborhoods of singular points to canonical coordinates, yielding region exponents that correspond to hard, collinear, Glauber, and related configurations. It is implemented in Mathematica and validated on representative one-loop and two-loop diagrams, reproducing known regions and revealing additional scaleful structures in certain cases. The approach aligns with prior MoR methods (e.g., ASY) while providing a systematic algebro-geometric route to uncovered regions and setting the stage for extensions to non-planar topologies and subleading contributions.

Abstract

We introduce a novel approach for solving the problem of identifying regions in the framework of Method of Regions by considering singularities and the associated Landau equations given a multi-scale Feynman diagram. These equations are then analyzed by an expansion in a small threshold parameter via the Power Geometry technique. This effectively leads to the analysis of Newton Polytopes which are evaluated using a Mathematica based convex hull program. Furthermore, the elements of the Gröbner Basis of the Landau Equations give a family of transformations, which when applied, reveal regions like potential and Glauber. Several one-loop and two-loop examples are studied and benchmarked using our algorithm which we call ASPIRE.

Paper Structure

This paper contains 27 sections, 97 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formalism
  3. Method of Regions
  4. ASY Implementation
  5. Determination of the regions using the analytic structure of the propagator
  6. Singularities and Threshold processes
  7. Regions from Feynman graph in the Alpha Parametric form
  8. Newton Polytope and Power Geometry
  9. Algorithm: ASPIRE
  10. Demonstration of the algorithm and unveiling the associated regions
  11. One-loop examples
  12. Example 1: Two-point one loop diagram
  13. Example 2: A five point one loop diagram
  14. A two-loop fish diagram
  15. One loop Scalar Triangle Diagrams in Sudakov Limits
  16. ...and 12 more sections

Figures (12)

  • Figure 1: Self energy diagram for a scalar field with mass $m$
  • Figure 2: Convex Hull of the Projected set of vector exponents
  • Figure 3: Types of singularities: P1 is a simple pole, P2 and P3 are Pinched Singularities and P4 is an End Point Singularity. While the contour between the points $A$ and $B$ can be deformed so as to avoid the simple pole $P1$, the same is not true for the pinch and end-point singularities.
  • Figure 4: Newton polytope of support
  • Figure 5: A two point one loop diagram
  • ...and 7 more figures