MUSEQuBES: Calibrating the redshifts of Ly {\alpha} emitters using stacked circumgalactic medium absorption profiles

Lyman$-\alpha$ (Ly$\alpha$) emission lines are typically found to be redshifted with respect to the systemic redshifts of galaxies, likely due to resonant scattering of Ly$\alpha$ photons. Here we measure the average velocity offset for a sample of 96 $z\approx3.3$ Ly$\alpha$ emitters (LAEs) with a median Ly$\alpha$ flux (luminosity) of $\approx 10^{-17}~\rm erg~cm^{-2}~s^{-1}$ ($\approx10^{42}~\rm erg~s^{-1}$) and a median star formation rate (SFR) of $\approx1.3 \rm M_{\odot} yr^{-1}$ (not corrected for possible dust extinction), detected by the Multi-Unit Spectroscopic Explorer as part of our MUSEQuBES circumgalactic medium (CGM) survey. By postulating that the stacked CGM absorption profiles of these LAEs, probed by 8 background quasars, must be centered on the systemic redshift, we measure an average velocity offset, V$_{\rm offset} = 171 \pm 8$ $\rm km s^{-1}$, between the Ly$\alpha$ emission peak and the systemic redshift. The observed V$_{\rm offset}$ is lower by factors of $\approx1.4$ and $\approx2.6$ compared to the velocity offsets measured for narrow-band selected LAEs and Lyman break galaxies, respectively, which probe galaxies with higher masses and SFRs. Consistent with earlier studies based on direct measurements for individual objects, we find that the V$_{\rm offset}$ is correlated with the full width at half-maximum of the red peak of the Ly$\alpha$ line, and anti-correlated with the rest-frame equivalent width. Moreover, we find that $V_{\rm offset}$ is correlated with SFR with a sub-linear scaling relation, V$_{\rm offset}\propto \rm SFR^{0.16\pm0.03}$. Adopting the mass scaling for main sequence galaxies, such a relation suggests that V$_{\rm offset}$ scales with the circular velocity of the dark matter halos hosting the LAEs.