  The hot beaming (or strahl) electrons responsible for the main electron
heat-flux in the solar wind are believed to be self-regulated by the
electromagnetic beaming instabilities, also known as the heat-flux
instabilities. Here we report the first quasi-linear theoretical approach of
the whistler unstable branch able to characterize the long-term saturation of
the instability as well as the relaxation of the electron velocity
distributions. The instability saturation is not solely determined by the drift
velocities, which undergo only a minor relaxation, but mainly from a concurrent
interaction of electrons with whistlers that induces (opposite) temperature
anisotropies of the core and beam populations and reduces the effective
anisotropy. These results might be able to (i) explain the low intensity of the
whistler heat-flux fluctuations in the solar wind (although other explanations
remain possible and need further investigation), and (ii) confirm a reduced
effectiveness of these fluctuations in the relaxation and isotropization of the
electron strahl and in the regulation of the electron heat-flux.
