https://doi.org/10.1140/epjc/s10052-025-14852-0
Regular Article - Theoretical Physics
General formulas for loop-induced decays of 
and their applications
1
Institute of Fundamental and Applied Sciences, Duy Tan University, 70000, Ho Chi Minh City, Vietnam
2
Faculty of Natural Sciences, Duy Tan University, 50000, Da Nang, Vietnam
3
VNUHCM-University of Science, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam
a
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Received:
3
April
2025
Accepted:
21
September
2025
Published online:
10
October
2025
Abstract
Within the framework of the Standard Model Higgs extensions, including the Two-Higgs-Doublet Model with vector-like fermions and the Triplet-Higgs Model, we derive general one-loop contributions to the rare decay process 
The analytical expressions are formulated with Passarino–Veltman scalar functions, which represent the scalar coefficients of one-loop Lorentz-covariant tensor integrals. These functions are written in accordance with the input conventions of the LoopTools and Collier packages, facilitating the efficient numerical generation of decay rates and their distributions using these computational tools. As part of our phenomenological study, we examine the branching ratios of this decay channel within the viable parameter space of the considered models. Our results indicate that the branching ratio can reach 
in the Two-Higgs-Doublet Model and 
in the Triplet-Higgs Model at specific points within the allowed parameter regions. Furthermore, the inclusion of vector-like fermions in the loop leads to a significant modification of the partial decay rates, with an observed variation of approximately 
Additionally, we explore the dependence of the branching ratios on key model parameters, including the charged Higgs mass, mixing angles, and the soft-breaking scale, providing deeper insights into the phenomenological implications of these Higgs-extended frameworks.
© The Author(s) 2025
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Funded by SCOAP3.
