approximately twice that of the DM. In such scenarios, one has to consider DM annihilation
processes at energies where the tree-level propagator of the mediator becomes singular.
A standard way to avoid such singularities is to employ a Breit-Wigner (BW) ansatz for
the propagator with a constant decay width [17]. Then, the amplitude gets regularized
by the non-zero width of the mediator and can thus get drastically enhanced by many
orders of magnitude, when compared to other non-resonant contributions. In Quantum
Field Theory (QFT), the BW form of the propagator usually arises from a Dyson series
summation of self-energy graphs of the mediator. In the so-called on-mass-shell (OS)
scheme of renormalization [18], the dispersive parts of the self-energies renormalize the
masses, whilst their absorptive part is related to the decay width of the mediator.
From the phenomenological point of view, the channel of resonant DM annihila-
tion turns out to be an attractive option, as it leads to suppressed DM-nucleon cross-
sections thereby avoiding the tight constraints emanating from the null results of direct
DM searches. Therefore, it should not be too surprising that this resonant region in
question may become the only viable region in the parameter space of a given model with
DM mass in the GeV range that survives after all direct detection limits on the DM-nucleon
cross-section were imposed [19, 20].
As was first noted in [14], the BW approximation with a constant decay width in the
propagator of the mediator can become very inaccurate close to the DM production thresh-
old, especially when the respective DM channel contributes significantly to the decay width
of the mediator. To partially remedy this problem, one is compelled to use an effective
BW propagator with momentum-dependent or running width for the exchanged particle
in the s-channel. In this article, we will go beyond the previously used non-relativistic
approach [14]. In general, an s-dependent width results from the imaginary (absorptive)
part of the self-energy of the mediator. This quantity is only gauge-independent when
evaluated at the pole of the propagator, but becomes gauge-variant in the off-shell region.
This was a well-known problem in QFT and pertains to the question whether a consistent
gauge-independent definition of off-shell Green’s function for unstable particles exists in
spontaneously broken gauge theories [21]. To deal with this issue, a number of recipes
and methods have been put forward by several authors, such as the Laurent series expan-
sion [22, 23], the complex mass scheme [24, 25], the fermion loop scheme [26], and the
effective theory approach [27].
An elegant and equally consistent gauge-independent framework to address the afore-
mentioned problem is the so-called Pinch Technique (PT) [28–32]. The PT preserves basic
properties of QFT, such as analyticity, unitarity and the gauge invariance of the classical
action. The PT resummation approach to unstable particles [30, 31, 33] was extensively
studied in the literature originally within the context of the SM [34–38] and more recently
in two Higgs-doublet models [39–42].
In this paper we discuss the problems that arise in the relativistic treatment of resonant
DM annihilation processes in spontaneously broken gauge theories and show how these can
be avoided in a resummation approach implemented by the PT. As an archetypal model,
we consider a gauged U(1)
X
complex-scalar extension of the SM that includes a massive
stable gauge boson X as a candidate particle for a Vector DM (VDM). We explicitly demon-
– 2 –
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