The extended Lyα haloes (LAHs) have been found to be prevalent around high-redshift star-forming galaxies. However, the origin of the LAHs is still a subject of debate. The spatially resolved analysis of Lyα profiles provides an important diagnostic. We analyse the average spatial extent and spectral variation of the circumgalactic LAHs by stacking a sample of 155 Lyα emitters (LAEs) at redshifts of 3 < z < 4 in the MUSE Extremely Deep Field. Our analysis reveals that, with respect to the Lyα line of the target LAE, the peak of the Lyα line at large distances becomes increasingly more blueshifted up to a projected distance of 60 kpc (≈3× virial radius), with a velocity offset of ≈250 km/s. This trend is evident in both the mean and median stacks, suggesting that it is a general property of our LAE sample, which typically has a Lyα luminosity of ≈1041.1 erg s−1. However, due to the absence of systemic redshift data, it remains unclear whether the Lyα line peak at large projected distances is less redshifted compared to the inner regions or truly blueshifted with respect to the systemic velocity. We explore various scenarios to explain the large-scale kinematics of the Lyα line.
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