Landau Singularities and Higher-Order Roots

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.

49 pages; 2 figures, ancillary file includes details of examples discussed. Content matches version to be published in Physical Review D, including an additional example with more complete analysis and more extensive explanation of the mathematical context