The Landau equations describe the singularities of the second Symanzik polynomial, which in turn appears as the denominator polynomial of the corresponding Feynman integral. There is therefore a strong connection between the singularities of the second Symanzik polynomial and singularities of the Feynman integral itself.
Before turning to the investigation of the Landau equations, we addressed a more fundamental question: how can one describe the region in the kinematic parameter space in which a given Feynman integral converges? It was known that if the second Symanzik polynomial is strictly copositive, that is, positive on the positive real orthant, then the associated Feynman integral converges. This region of parameter space is referred to as the Euclidean region in the physics literature. In a previous paper, we provided an effective method to characterize the Euclidean region under the assumption that the kinematic parameters are sufficiently generic, thereby excluding physically relevant situations in which some particles may be massless.
During the first month of the present project, we began to study the problem of copositivity itself. Using a classical representation theorem from real algebraic geometry, combined with techniques from toric geometry, we derived a new effective method for detecting copositivity. While this approach can be applied to determine the Euclidean region even in the case of massless particles, it is more general: it applies not only to Symanzik polynomials but to arbitrary polynomials.
After that paper was uploaded to arxiv and submitted for publication, I continued to study certificates of nonnegativity and extended them using tools from toric and tropical geometry. In parallel, I implemented an initial version of a Julia package that computes the irreducible components of the Landau discriminant that intersect the positive real orthant, that is, those giving rise to positive solutions of the Landau equations. At present, neither of these projects is in a final form, both require further investigation and research.
