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Extragalactic studies have demonstrated there is a moderately tight($\approx$0.3 dex) relationship between galaxy stellar mass ($M_{\star}$) andstar formation rate (SFR) that holds for star-forming galaxies at $M_{\star}\sim 3 \times 10^8$-10$^{11}~M_{\odot}$, i.e., the "star formation mainsequence." However, it has yet to be determined whether such a relationshipextends to even lower mass galaxies, particularly at intermediate or higherredshifts. We present new results using observations for 714 narrowbandH$\alpha$-selected galaxies with stellar masses between $10^6$ and$10^{10}~M_{\odot}$ (average of $10^{8.2}~M_{\odot}$) at $z \approx$ 0.07-0.5.These galaxies have sensitive UV to near-infrared photometric measurements andoptical spectroscopy. The latter allows us to correct our H$\alpha$ SFRs fordust attenuation using Balmer decrements. Our study reveals: (1) for low-SFRgalaxies, our H$\alpha$ SFRs systematically underpredict compared to FUVmeasurements, consistent with other studies; (2) at a given stellar mass($\approx $10$ ^{8}~M_{\odot}$), log(specific SFR) evolves as $ A \log(1+z) $with $ A = 5.26 \pm 0.75 $, and on average, specific SFR increases withdecreasing stellar mass; (3) the SFR-$M_{\star}$ relation holds for galaxiesdown to $\sim$10$^6~M_{\odot}$ ($\sim$1.5 dex below previous studies), and overlookback times of up to 5 Gyr, follows a redshift-dependent relation of$\log{({\rm SFR})} \propto \alpha \log(M_{\star}/M_{\odot}) + \beta z$ with$\alpha = 0.60 \pm 0.01$ and $\beta = 1.86 \pm 0.07$; and (4) the observeddispersion in the SFR-$M_{\star}$ relation at low stellar masses is$\approx$0.3 dex. Accounting for survey selection effects using simulatedgalaxies, we estimate the true dispersion is $\approx$0.5 dex.

monthly notices of the royal astronomical society

The Metal Abundances across Cosmic Time ($\mathcal{MACT}$) Survey. III  -- The relationship between stellar mass and star formation rate in extremely  low-mass galaxies