Singularities of Feynman Integrals

Abstract

In this paper, we study the singularities of Feynman integrals using homological techniques. We analyse the Feynman integrals by compactifying the integration domain as well as the ambient space by embedding them in higher-dimensional space. In this compactified space the singularities occur due to the meeting of compactified propagators at non-general position. The present analysis, which had been previously used only for the singularities of second-type, is used to study other kinds of singularities viz threshold, pseudo-threshold and anomalous threshold singularities. We study various one-loop and two-loop examples and obtain their singularities. We also present observations based on results obtained, that allow us to determine whether the singularities lie on the physical sheet or not for some simple cases. Thus this work at the frontier of our knowledge of Feynman integral calculus sheds insight into the analytic structure.

Figures (8)

• Figure 1: (a) Two surfaces meeting in general position (b) Two surfaces meeting in non-general position. We see that for the case of non-general position, at the point of intersection, the normal to both surfaces are parallel to each other.
• Figure 2: Bubble diagram
• Figure 3: Triangle diagram
• Figure 4: Box diagram
• Figure 5: Sunset diagram