  A massive real scalar dark matter particle $S$ can couple to Standard Model
leptons or quarks through a vector-like fermionic mediator $\psi$, a scenario
known as the Vector-like portal. Due to helicity suppression of the
annihilation cross section into a pair of SM fermions, it has been shown in
previous works that radiative corrections, either at one-loop or through
radiation of gauge bosons, may play a significant role both in determining the
relic abundance and for indirect detection. All previous works considered the
limit of massless final state quarks or leptons. In this work, we focus on a
technical issue, which is to reliably determine the annihilation cross sections
taking into account finite fermion masses. Following previous works in the
framework of simplified supersymmetric dark matter scenarios, and building on
an analogy with Higgs decay into fermions, we address the issue of infrared and
collinear divergences that plagues the cross section by adopting an effective
operator description, which captures most of the relevant physics and give
explicit expressions for the annihilation cross sections. We then develop
several approximations for the differential and total cross sections, which
simplify greatly their expressions, and which can then be used in various
phenomenological studies of similar models. Finally, we describe our method to
compute the final gamma-ray spectrum, including hadronisation of the heavy
fermions, and provide some illustrative spectra for specific dark matter
candidates.
