  In a Bose superfluid, the coupling between transverse (phase) and
longitudinal fluctuations leads to a divergence of the longitudinal correlation
function, which is responsible for the occurrence of infrared divergences in
the perturbation theory and the breakdown of the Bogoliubov approximation. We
report a non-perturbative renormalization-group (NPRG) calculation of the
one-particle Green function of an interacting boson system at zero temperature.
We find two regimes separated by a characteristic momentum scale $k_G$
("Ginzburg" scale). While the Bogoliubov approximation is valid at large
momenta and energies, $|\p|,|\w|/c\gg k_G$ (with $c$ the velocity of the
Bogoliubov sound mode), in the infrared (hydrodynamic) regime $|\p|,|\w|/c\ll
k_G$ the normal and anomalous self-energies exhibit singularities reflecting
the divergence of the longitudinal correlation function. In particular, we find
that the anomalous self-energy agrees with the Bogoliubov result
$\Sigan(\p,\w)\simeq\const$ at high-energies and behaves as $\Sigan(\p,\w)\sim
(c^2\p^2-\w^2)^{(d-3)/2}$ in the infrared regime (with $d$ the space
dimension), in agreement with the Nepomnyashchii identity $\Sigan(0,0)=0$ and
the predictions of Popov's hydrodynamic theory. We argue that the hydrodynamic
limit of the one-particle Green function is fully determined by the knowledge
of the exponent $3-d$ characterizing the divergence of the longitudinal
susceptibility and the Ward identities associated to gauge and Galilean
invariances. The infrared singularity of $\Sigan(\p,\w)$ leads to a continuum
of excitations (coexisting with the sound mode) which shows up in the
one-particle spectral function.
