  The pair-condensed unpolarized spin-$1/2$ Fermi gases have a collective
excitation branch in their pair-breaking continuum (V.A. Andrianov, V.N. Popov,
1976). We study it at zero temperature, with the eigenenergy equation deduced
from the linearized time-dependent BCS theory and extended analytically to the
lower half complex plane through its branch cut, calculating both the
dispersion relation and the spectral weights (quasiparticle residues) of the
branch. In the case of BCS superconductors, so called because the effect of the
ion lattice is replaced by a short-range electron-electron interaction, we also
include the Coulomb interaction and we restrict ourselves to the weak coupling
limit $\Delta/\mu\to 0^+$ ($\Delta$ is the order parameter, $\mu $ the chemical
potential) and to wavenumbers $q=O(1/\xi)$ where $\xi$ is the size of a pair;
when the complex energy $z_q$ is expressed in units of $\Delta$ and $q$ in
units of $1/\xi$, the branch follows a universal law insensitive to the Coulomb
interaction. In the case of cold atoms in the BEC-BCS crossover, only a contact
interaction remains, but the coupling strength $\Delta/\mu$ can take arbitrary
values, and we study the branch at any wave number. At weak coupling, we
predict three scales, that already mentioned $q\approx 1/\xi$, that
$q\approx(\Delta/\mu)^{-1/3}/\xi$ where the real part of the dispersion
relation has a minimum and that $q\approx (\mu/\Delta)/\xi\approx k_{\rm F}$
($k_{\rm F}$ is the Fermi wave number) where the branch reaches the edge of its
existence domain. Near the point where the chemical potential vanishes on the
BCS side, $\mu/\Delta\to 0^+$, where $\xi\approx k_{\rm F}$, we find two scales
$q\approx(\mu/\Delta)^{1/2}/\xi$ and $q\approx 1/\xi$. In all cases, the branch
has a limit $2\Delta$ and a quadratic start at $q=0$. These results were
obtained for $\mu>0$, where the eigenenergy equation admits at least two branch
points $\epsilon_a(q)$ and $\epsilon_b(q)$ on the positive real axis, and for
an analytic continuation through the interval $[\epsilon_a(q),\epsilon_b(q)] $.
We find new continuum branches by performing the analytic continuation through
$[\epsilon_b(q),+\infty[$ or even, for $q$ low enough, where there is a third
real positive branch point $\epsilon_c(q)$, through
$[\epsilon_b(q),\epsilon_c(q)]$ and $[\epsilon_c(q),+\infty[$. On the BEC side
$\mu<0$ not previously studied, where there is only one real positive branch
point $ \epsilon_a(q)$, we also find new collective excitation branches under
the branch cut $[\epsilon_a (q),+\infty[$. For $\mu>0$, some of these new
branches have a low-wavenumber exotic hypoacoustic $z_q\approx q^{3/2}$ or
hyperacoustic $z_q\approx q^{4/5}$ behavior. For $\mu<0$, we find a
hyperacoustic branch and a nonhypoacoustic branch, with a limit $2\Delta$ and a
purely real quadratic start at $q=0$ for $\Delta/|\mu|<0.222$.
