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Deep MUSE observations have unveiled pervasive Ly$\alpha$ haloes (LAHs)surrounding high-redshift star-forming galaxies. However, the origin of theextended Ly$\alpha$ emission is still a subject of debate. We analyse theaverage spatial extent and spectral variation of the circumgalactic LAHs bystacking a sample of 155 Ly$\alpha$ emitters (LAEs) at redshift $3<z<4$ in theMUSE Extremely Deep Field (MXDF). With respect to the Ly$\alpha$ red peak ofthe target LAE, the Ly$\alpha$ line peak becomes increasingly more blueshiftedout to a projected distance of at least 60 kpc, where the velocity offset is$\approx$ 250 km/s. This signal is observed in both the mean and median stacks,and is thus a generic property of the LAE sample with typical Ly$\alpha$luminosity $\mathrm{\approx 10^{41.1} erg\,s^{-1}}$. We discuss multiplescenarios to explain the blueshift of the circumgalactic Ly$\alpha$ line. Themost plausible one is a combination of outflows and inflows. In the innerregion of the LAH, the Ly$\alpha$ photons are produced by the central starformation and then scattered within outflows. At larger radii, the infallingcool gas shapes the observed Ly$\alpha$ blueshift.

Spatially-resolved Spectroscopic Analysis of Ly$\alpha$ Haloes: Radial Evolution of the Ly$\alpha$ Line Profile out to 60 kpc