Landau Singularities and Higher-Order Roots

TL;DR

This paper explores the apparent contradiction between Landau's classification of Feynman diagram singularities and the existence of higher-order polynomial roots in some integrals, proposing that such singularities occur only in higher co-dimension kinematic limits.

Contribution

The authors analyze specific examples of higher-order roots in Feynman integrals and demonstrate that these singularities arise only in higher co-dimension limits, extending Landau's framework.

Findings

Higher-order polynomial roots in Feynman integrals occur in higher co-dimension limits.

Such singularities are not covered by Landau's original classification.

Concrete examples involve cube-roots in four dimensions and degree six polynomial roots in two dimensions.

Abstract

Landau's work on the singularities of Feynman diagrams suggests that they can only be of three types: either poles, logarithmic divergences, or the roots of quadratic polynomials. On the other hand, many Feynman integrals exist whose singularities involve arbitrarily higher-order polynomial roots. We investigate this apparent paradox using concrete examples involving cube-roots in four dimensions and roots of a degree six polynomial in two dimensions, and suggest that these higher-order singularities can only be approached via kinematic limits of higher co-dimension than one, thus evading Landau's argument.