Triangle Feynman diagram in the timelike region
Abstract
In Quantum Field Theory, triangle Feynman diagram F(p12,p22,p32|m1,m2,m3) 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(p12,p22,p32|m1,m2,m3) is easily calculable in the Euclidean region of all variables, pi2<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 pi2 are in the physical Minkowski region, are reproduced by the Feynman-parameter representation for F(p12,p22,p32|m1,m2,m3) through the extra replacements pi2 → pi2 + i0 and mi2 → mi2 − i0, where mi 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 pi2.
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BibTeX
@misc{ivanov2026triangle,
title = {{Triangle Feynman diagram in the timelike region}},
author = {Mikhail A. Ivanov and Dmitri Melikhov and Silvano Simula},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.18894},
eprint = {2608.18894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Triangle Feynman diagram in the timelike region
AU - Mikhail A. Ivanov
AU - Dmitri Melikhov
AU - Silvano Simula
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.18894
UR - https://arxiv.org/abs/2608.18894
AB - 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 → p_i^2 + i0 and m_i^2 → 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.
ER -