We study the radio properties of 706 sub-millimeter galaxies (SMGs) selected at 870$mu$m with the Atacama Large Millimeter Array from the SCUBA-2 Cosmology Legacy Survey map of the Ultra Deep Survey field. We detect 273 SMGs at $>4sigma$ in deep Karl G. Jansky Very Large Array 1.4 GHz observations, of which a subset of 45 SMGs are additionally detected in 610 MHz Giant Metre-Wave Radio Telescope imaging. We quantify the far-infrared/radio correlation through parameter $q_text{IR}$, defined as the logarithmic ratio of the far-infrared and radio luminosity, and include the radio-undetected SMGs through a stacking analysis. We determine a median $q_text{IR} = 2.20pm0.03$ for the full sample, independent of redshift, which places these $zsim2.5$ dusty star-forming galaxies $0.44pm0.04$ dex below the local correlation for both normal star-forming galaxies and local ultra-luminous infrared galaxies (ULIRGs). Both the lack of redshift-evolution and the offset from the local correlation are likely the result of the different physical conditions in high-redshift starburst galaxies, compared to local star-forming sources. We explain the offset through a combination of strong magnetic fields ($Bgtrsim0.2$mG), high interstellar medium (ISM) densities and additional radio emission generated by secondary cosmic rays. While local ULIRGs are likely to have similar magnetic field strengths, we find that their compactness, in combination with a higher ISM density compared to SMGs, naturally explains why local and high-redshift dusty star-forming galaxies follow a different far-infrared/radio correlation. Overall, our findings paint SMGs as a homogeneous population of galaxies, as illustrated by their tight and non-evolving far-infrared/radio correlation.
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