HCN $J$=4-3, HNC $J$=1-0, $\mathrm{H^{13}CN}$ $J$=1-0, and
  $\mathrm{HC_3N}$ $J$=10-9 Maps of Galactic Center Region II.: Physical
  Properties of Dense Gas Clumps and Probability of Star Formation
  We report a statistical analysis exploring the origin of the overall low star
formation efficiency (SFE) of the Galactic central molecular zone (CMZ) and the
SFE diversity among the CMZ clouds using a wide-field HCN $J$=4-3 map, whose
optically thin critical density ($\sim10^7\,\mathrm{cm}^{-3}$) is the highest
among the tracers ever used in CMZ surveys. Logistic regression is performed to
empirically formulate star formation probability of 195 HCN clumps, 13 of which
contain star formation signatures. The explanatory parameters in the best-fit
model are reduced into the virial parameter $\alpha_{\mathrm{vir}}$ without
significant contribution from other parameters, whereas the performance of the
model without $\alpha_{\mathrm{vir}}$ is no better than that using randomly
generated data. The threshold $\alpha_{\mathrm{vir}}$ is 6, which translates
into a volume density ($n_{\mathrm{H_2}}$) of $10^{4.6}\,\mathrm{cm}^{-3}$ with
the $n_{\mathrm{H_2}}$-$\alpha_{\mathrm{vir}}$ correlation. The scarcity of the
low-$\alpha_{\mathrm{vir}}$ clumps, whose fraction to all HCN clumps is 0.1,
can be considered as one of the immediate causes of the suppressed SFE. No
correlation between the clump size or mass and star formation probability is
found, implying that HCN $J$=4-3 does not immediately trace the mass of
star-forming gas above a threshold density. Meanwhile, star-forming and
non-star-forming clouds are degenerate in the physical parameters of the CS
$\mathit{J}$=1-0 clouds, highlighting the efficacy of the HCN $\mathit{J}$=4-3
line to probe star-forming regions in the CMZ. The time scale of the
high-$\alpha_{\mathrm{vir}}$ to low-$\alpha_{\mathrm{vir}}$ transition is
$\lesssim2$ Myr, which is consistent with the tidal compression and X1/X2 orbit
transition models but possibly does not fit the cloud-cloud collision picture.
