Triangle Feynman diagram in the timelike region

Abstract

In Quantum Field Theory, triangle Feynman diagram $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ is an analytic function of its variables, whose analytic structure is fully determined by the location of singularities of the propagators of particles in the loop. The form factor $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ is easily calculable in the Euclidean region of all variables, $p_i^2<0$, $i=1,2,3$. A rigorous way to obtain the form factor in the timelike region is to perform the analytic continuation from the Euclidean region using single or double dispersion representations. On the other hand, there is a simple representation of the triangle as integral over Feynman parameters. The goal of this paper is to demonstrate that all known rigorous results of dispersion representations in the regions where some of the variables $p_i^2$ are in the physical Minkowski region, are reproduced by the Feynman-parameter representation for $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ by a mere replacement $p_i^2\to p_i^2+i0$ and $m_i^2\to m_i^2-i0$, where $m_i$ are masses of particles propagating in the loop. This simple replacement takes properly into account all subtle contributions given in the context of dispersion representations by the anomalous cuts and thresholds.