  The fate of relativistic pair beams produced in the intergalactic medium by
very high energy emission from blazars remains controversial in the literature.
The possible role of resonance beam plasma instability has been studied both
analytically and numerically but no consensus has been reached. In this paper,
we thoroughly analyze the development of this type of instability. This
analysis takes into account that a highly relativistic beam loses energy only
due to interactions with the plasma waves propagating within the opening angle
of the beam (we call them parallel waves), whereas excitation of oblique waves
results merely in an angular spreading of the beam, which reduces the
instability growth rate. For parallel waves, the growth rate is a few times
larger than for oblique ones, so they grow faster than oblique waves and drain
energy from the beam before it expands. However, the specific property of
extragalactic beams is that they are extraordinarily narrow; the opening angle
is only $\Delta\theta\sim 10^{-6}-10^{-5}$. In this case, the width of the
resonance for parallel waves, $\propto\Delta\theta^2$, is too small for them to
grow in realistic conditions. We perform both analytical estimates and
numerical simulations in the quasilinear regime. These show that for
extragalactic beams, the growth of the waves is incapable of taking a
significant portion of the beam's energy. This type of instability could at
best lead to an expansion of the beam by some factor but the beam's energy
remains nearly intact.
