  The relationship between galaxy star formation rates (SFR) and stellar masses
($M_\ast$) is re-examined using a mass-selected sample of $\sim$62,000
star-forming galaxies at $z \le 1.3$ in the COSMOS 2-deg$^2$ field. Using new
far-infrared photometry from $Herschel$-PACS and SPIRE and $Spitzer$-MIPS 24
$\mu$m, along with derived infrared luminosities from the NRK method based on
galaxies' locations in the restframe color-color diagram $(NUV - r)$ vs. $(r -
K)$, we are able to more accurately determine total SFRs for our complete
sample. At all redshifts, the relationship between median $SFR$ and $M_\ast$
follows a power-law at low stellar masses, and flattens to nearly constant SFR
at high stellar masses. We describe a new parameterization that provides the
best fit to the main sequence and characterizes the low mass power-law slope,
turnover mass, and overall scaling. The turnover in the main sequence occurs at
a characteristic mass of about $M_{0} \sim 10^{10} M_{\odot}$ at all redshifts.
The low mass power-law slope ranges from 0.9-1.3 and the overall scaling rises
in SFR as a function of $(1+z)^{4.12 \pm 0.10}$. A broken power-law fit below
and above the turnover mass gives relationships of $SFR \propto M_{*}^{0.88 \pm
0.06}$ below the turnover mass and $SFR \propto M_{*}^{0.27 \pm 0.04}$ above
the turnover mass. Galaxies more massive than $M_\ast \gtrsim 10^{10}\ M_{\rm
\odot}$ have on average, a much lower specific star formation rate (sSFR) than
would be expected by simply extrapolating the traditional linear fit to the
main sequence found for less massive galaxies.
