  Plasmonic nanostructures with singular geometries can exhibit a broadband
scattering response that at first glance appears to violate the lower bounds
for the radiation quality (Q) factor of small radiators, known as the Chu
limit. Here we explore this apparent contradiction, investigating the Q factor
of the resonant modes supported by two nearly touching cylinders, and analyze
how their fractional bandwidth fares in relation to the Chu limit. We first
derive lower bounds for the radiation Q factors of two-dimensional objects of
arbitrary cross-section. We then discuss the dissipation and radiation Q
factors associated with the plasmonic resonances of a cylinder dimer as a
function of its gap size. We show that the radiation Q factor is always larger
than the minimum Q and, as long as the peaks in the scattering spectrum are
well separated, their bandwidth is equal to the inverse of their Q factor. In
the limit of touching cylinders, the resonance spectra transition from discrete
to a continuum around an accumulation point, yielding a broadband response for
any finite level of material loss. Within any given frequency interval, the
response is the result of a multitude of plasmon resonances, each individually
obeying the Chu limit. Nevertheless, the connection between the Q factor and
the overall bandwidth of the scattering response is lost. Our study sheds light
onto the exotic resonant phenomena emerging when plasmonic materials are shaped
in singular geometries, and outlines their opportunities and limitations for
nanophotonics.
