Year: 2026
Type: preprint
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. We show 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 $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ through the extra replacements $p_i^2 \to p_i^2 + i0$ and $m_i^2 \to m_i^2 - i0$, where $m_i$ are masses of particles propagating in the loop. This implies that no deformation of the integration paths along the real axis of the Feynman parameters is forced by possible migrations of poles from the unphysical sheet to the physical one. The extra replacements allow us to take properly into account all subtle contributions given in the context of dispersion representations by the anomalous cuts and thresholds, and to provide the form factor on the upper boundaries of the cuts for all values of $p_i^2$.
Source: arXiv (Cornell University)
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203909821