We present improved photometric measurements for the host galaxies of 206 spectroscopically confirmed type Ia supernovae discovered by the Dark Energy Survey Supernova Program (DES-SN) and used in the first DES-SN cosmological analysis. Fitting spectral energy distributions to the $griz$ photometric measurements of the DES-SN host galaxies, we derive stellar masses and star-formation rates. For the DES-SN sample, when considering a 5D ($z$, $x_1$, $c$, $alpha$, $beta$) bias correction, we find evidence of a Hubble residual `mass step, where SNe Ia in high mass galaxies ($>10^{10} textrm{M}_{odot}$) are intrinsically more luminous (after correction) than their low mass counterparts by $gamma=0.040pm0.019$mag. This value is larger by $0.031$mag than the value found in the first DES-SN cosmological analysis. This difference is due to a combination of updated photometric measurements and improved star formation histories and is not from host-galaxy misidentification. When using a 1D (redshift-only) bias correction the inferred mass step is larger, with $gamma=0.066pm0.020$mag. The 1D-5D $gamma$ difference for DES-SN is $0.026pm0.009$mag. We show that this difference is due to a strong correlation between host galaxy stellar mass and the $x_1$ component of the 5D distance-bias correction. To better understand this effect, we include an intrinsic correlation between light-curve width and stellar mass in simulated SN Ia samples. We show that a 5D fit recovers $gamma$ with $-9$mmag bias compared to a $+2$mmag bias for a 1D fit. This difference can explain part of the discrepancy seen in the data. Improvements in modeling correlations between galaxy properties and SN is necessary to determine the implications for $gamma$ and ensure unbiased precision estimates of the dark energy equation-of-state as we enter the era of LSST.
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