  The ionizing photon production efficiency, $\xi_{ion}$, is a critical
parameter that provides a number of physical constraints to the nature of the
early Universe, including the contribution of galaxies to the timely completion
of the reionization of the Universe. Here we use KECK/MOSFIRE and ZFOURGE
multi-band photometric data to explore the $\xi_{ion}$ of a population of
galaxies at $z\sim2$ with $log_{10}(M_*/M_\odot)\sim9.0-11.5$. Our 130 \Halpha\
detections show a median $log_{10}(\xi_{ion}[Hz/erg])$ of $24.8\pm0.5$ when
dust corrected using a Calzetti et al. (2000) dust prescription. Our values are
typical of mass/magnitude selected $\xi_{ion}$ values observed in the $z\sim2$
Universe. Using BPASSv2.2.1 and Starburst99 stellar population models with
simple parametric star-formation-histories (SFH), we find that even with models
that account for effects of stellar evolution with binaries/stellar rotation,
model galaxies at $log_{10}(\xi_{ion}[Hz/erg])\lesssim25.0$ have low H$\alpha$
equivalent widths (EW) and redder colors compared to our $z\sim2$ observed
sample. We find that introducing star-bursts to the SFHs resolve the tension
with the models, however, due to the rapid time evolution of $\xi_{ion}$,
H$\alpha$ EWs, and rest-frame optical colors, our Monte Carlo simulations of
star-bursts show that random distribution of star-bursts in evolutionary time
of galaxies are unlikely to explain the observed distribution. Thus, either our
observed sample is specially selected based on their past SFH or stellar models
require additional mechanisms to reproduce the observed high UV luminosity of
galaxies for a given production rate of hydrogen ionizing photons.
