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

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.

arXiv commentv2,29 pages, plain latex, 12 figures, 3 tables, mathematica programs given as ancillary files,corresponds to version to appear in European Physical Journal C; compared to v1, sign of U polynomial corrected everywhere and in the new notebooks. The sign did not alter any of the results. Extensive discussions added, and clarifications inserted. Reference added

References in corpus (15)

• Asymptotic expansion of Feynman integrals near threshold
• FIESTA 4: optimized Feynman integral calculations with GPU support
• Analytic Regularization in Soft-Collinear Effective Theory
• Expansion by regions: revealing potential and Glauber regions automatically
• Geometric approach to asymptotic expansion of Feynman integrals
• Critical points and number of master integrals
• Foundation and generalization of the expansion by regions
• The method of regions and next-to-soft corrections in Drell-Yan production
• Two-loop electroweak next-to-leading logarithmic corrections to massless fermionic processes
• The two-loop vector form factor in the Sudakov limit
• The expansion by regions in pi K scattering
• Analytic representation of $F_K/F_{\pi }$ in two loop chiral perturbation theory
• NUMERICAL APPROACH TO TWO-LOOP THREE POINT FUNCTIONS WITH MASSES
• Analytic representations of $m_K$, $F_K$, $m_{\eta }$ and $F_{\eta }$ in two loop $SU(3)$ chiral perturbation theory
• Asymptotic Expansions of Feynman Integrals on the Mass Shell in Momenta and Masses

Cited by in corpus (19)

• Collider Physics at the Precision Frontier
• Landau Discriminants
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• On the status of expansion by regions
• Principal Landau Determinants
• The on-shell expansion: from Landau equations to the Newton polytope
• High-Energy Expansion of Two-Loop Massive Four-Point Diagrams
• Projected transverse momentum resummation in top-antitop pair production at LHC
• Identifying regions in wide-angle scattering via graph-theoretical approaches
• Asymptotic expansions through the loop-tree duality
• Dissecting polytopes: Landau singularities and asymptotic expansions in $2\to 2$ scattering
• Asymptotic analysis of Feynman diagrams and their maximal cuts
• Subleading Effects in Soft-Gluon Emission at One-Loop in Massless QCD
• Expansion by regions meets angular integrals
• Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model
• Expansion by regions with pySecDec
• Region analysis of $H\to \gamma \gamma$ via a bottom quark loop
• Singularities of Feynman Integrals
• Unregulated Divergences of Feynman Integrals