In order to trace the instantaneous star formation rate at high redshift, and hence help understanding the relation between the different emission mechanisms related to star formation, we combine the recent 4 Ms Chandra X-ray data and the deep VLA radio data in the Extended Chandra Deep Field South region. We find 268 sources detected both in the X-ray and radio band. The availability of redshifts for $sim 95$ of the sources in our sample allows us to derive reliable luminosity estimates and the intrinsic properties from X-ray analysis for the majority of the objects. With the aim of selecting sources powered by star formation in both bands, we adopt classification criteria based on X-ray and radio data, exploiting the X-ray spectral features and time variability, taking advantage of observations scattered across more than ten years. We identify 43 objects consistent with being powered by star formation. We also add another 111 and 70 star forming candidates detected only in the radio or X-ray band, respectively. We find a clear linear correlation between radio and X-ray luminosity in star forming galaxies over three orders of magnitude and up to $z sim 1.5$. We also measure a significant scatter of the order of 0.4 dex, higher than that observed at low redshift, implying an intrinsic scatter component. The correlation is consistent with that measured locally, and no evolution with redshift is observed. Using a locally calibrated relation between the SFR and the radio luminosity, we investigate the L_X(2-10keV)-SFR relation at high redshift. The comparison of the star formation rate measured in our sample with some theoretical models for the Milky Way and M31, two typical spiral galaxies, indicates that, with current data, we can trace typical spirals only at z<0.2, and strong starburst galaxies with star-formation rates as high as $sim 100 M_odot yr^{-1}$, up to $zsim 1.5$.
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