We report a super-linear correlation for the star formation law based on new CO($J$=1-0) data from the CARMA and NOBEYAMA Nearby-galaxies (CANON) CO survey. The sample includes 10 nearby spiral galaxies, in which structures at sub-kpc scales are spatially resolved. Combined with the star formation rate surface density traced by H$alpha$ and 24 $mu$m images, CO($J$=1-0) data provide a super-linear slope of $N$ = 1.3. The slope becomes even steeper ($N$ = 1.8) when the diffuse stellar and dust background emission is subtracted from the H$alpha$ and 24 $mu$m images. In contrast to the recent results with CO($J$=2-1) that found a constant star formation efficiency (SFE) in many spiral galaxies, these results suggest that the SFE is not independent of environment, but increases with molecular gas surface density. We suggest that the excitation of CO($J$=2-1) is likely enhanced in the regions with higher star formation and does not linearly trace the molecular gas mass. In addition, the diffuse emission contaminates the SFE measurement most in regions where star formation rate is law. These two effects can flatten the power law correlation and produce the apparent linear slope. The super linear slope from the CO($J$=1-0) analysis indicates that star formation is enhanced by non-linear processes in regions of high gas density, e.g., gravitational collapse and cloud-cloud collisions.
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