Feedback in emerging extragalactic star clusters, FEAST: The relation between 3.3 $\mu$m PAH emission and Star Formation Rate traced by ionized gas in NGC 628

We present maps of ionized gas (traced by Pa$\alpha$ and Br$\alpha$) and 3.3 $\mu$m Polycyclic Aromatic Hydrocarbon (PAH) emission in the nearby spiral galaxy NGC 628, derived from new JWST/NIRCam data from the FEAST survey. With this data, we investigate and calibrate the relation between 3.3 $\mu$m PAH emission and star formation rate (SFR) in and around emerging young star clusters (eYSCs) on a scale of ${\sim}40$ pc. We find a tight (correlation coefficient $\rho$${\sim}$0.9) sub-linear (power-law exponent $\alpha$${\sim}$0.75) relation between the 3.3 $\mu$m PAH luminosity surface density and SFR traced by Br$\alpha$ for compact, cospatial (within 0.16$''$ or ${\sim}$7 pc) peaks in Pa$\alpha$, Br$\alpha$, and 3.3 $\mu$m (eYSC-I). The scatter in the relationship does not correlate well with variations in local interstellar medium (ISM) metallicity due to a radial metallicity gradient, but rather is likely due to stochastic sampling of the stellar initial mass function (IMF) and variations in the PAH heating and age of our sources. The deviation from a linear relation may be explained by PAH destruction in more intense ionizing environments, variations in age, and IMF stochasticity at intermediate to low luminosities. We test our results with various continuum subtraction techniques using combinations of NIRCam bands and find that they remain robust with only minor differences in the derived slope and intercept. An unexpected discrepancy is identified between the relations of hydrogen recombination lines (Pa$\alpha$ versus Br$\alpha$; H$\alpha$ versus Br$\alpha$).