  We investigate the effects of local vibrational excitations in the
nonsymmetrized current noise $S(\omega)$ of a nanojunction. For this purpose,
we analyze a simple model - the Holstein model - in which the junction is
described by a single electronic level that is coupled to two metallic leads
and to a single vibrational mode. Using the Keldysh Green's function technique,
we calculate the nonsymmetrized current noise to the leading order in the
charge-vibration interaction. For the noise associated to the latter, we
identify distinct terms corresponding to the mean-field noise and the vertex
correction. The mean-field result can be further divided into an elastic
correction to the noise and in an inelastic correction, the second one being
related to energy exchange with the vibration. To illustrate the general
behavior of the noise induced by the charge-vibration interaction, we consider
two limit cases. In the first case, we assume a strong coupling of the dot to
the leads with an energy-independent transmission whereas in the second case we
assume a weak tunneling coupling between the dot and the leads such that the
transport occurs through a sharp resonant level. We find that the noise
associated to the vibration-charge interaction shows a complex pattern as a
function of the frequency $\omega$ and of the transmission function or of the
dot's energy level. Several transitions from enhancement to suppression of the
noise occurs in different regions, which are determined, in particular, by the
vibrational frequency. Remarkably, in the regime of an energy-independent
transmission, the zero order elastic noise vanishes at perfect transmission and
at positive frequency whereas the noise related to the charge-vibration
interaction remains finite enabling the analysis of the pure
vibrational-induced current noise.
