  An electron or electron-positron beam streaming through a plasma is
notoriously prone to micro-instabilities. For a dilute ultrarelativistic
infinite beam, the dominant instability is a mixed mode between longitudinal
two-stream and transverse filamentation modes, with a phase velocity oblique to
the beam velocity. A spatiotemporal theory describing the linear growth of this
oblique mixed instability is proposed, which predicts that spatiotemporal
effects generally prevail for finite-length beams, leading to a significantly
slower instability evolution than in the usually assumed purely temporal
regime. These results are accurately supported by particle-in-cell (PIC)
simulations. Furthermore, we show that the self-focusing dynamics caused by the
plasma wakefields driven by finite-width beams can compete with the oblique
instability. Analyzed through PIC simulations, the interplay of these two
processes in realistic systems bears important implications for upcoming
accelerator experiments on ultrarelativistic beam-plasma interactions.
