Constraining Sub-Parsec Binary Supermassive Black Holes in Quasars with
  Multi-Epoch Spectroscopy. III. Candidates from Continued Radial Velocity
  Tests
  Quasars whose broad emission lines show temporal, bulk radial velocity (RV)
shifts have been proposed as candidate sub-parsec (sub-pc), binary supermassive
black holes (BSBHs). We identified a sample of 16 BSBH candidates based on
two-epoch spectroscopy among 52 quasars with significant RV shifts over a few
rest-frame years. The candidates showed consistent velocity shifts
independently measured from two broad lines (H$\beta$ and H$\alpha$ or Mg${\rm
\,II}$) without significant changes in the broad-line profiles. Here in the
third paper of the series, we present further third- and fourth-epoch
spectroscopy for 12 of the 16 candidates for continued RV tests, spanning
$\sim$5--15 yr in the quasars' rest frames. Cross-correlation analysis of the
broad H$\beta$ calibrated against [O${\rm\,III}]\,\lambda 5007$ suggests that 5
of the 12 quasars remain valid as BSBH candidates. They show broad H$\beta$ RV
curves that are consistent with binary orbital motion without significant
changes in the broad line profiles. Their broad H$\alpha$ (or Mg${\rm \,II}$)
lines display RV shifts that are either consistent with or smaller than those
seen in broad H$\beta$. The RV shifts can be explained by a $\sim$0.05--0.1 pc
BSBH with an orbital period of $\sim$40--130 yr, assuming a mass ratio of
0.5--2 and a circular orbit. However, the parameters are not well constrained
given the few epochs that sample only a small portion of the hypothesized
binary orbital cycle. The apparent occurrence rate of sub-pc BSBHs is
$\lesssim$13$\pm$5% among all SDSS quasars, with no significant difference in
the subsets with and without single-epoch broad line velocity offsets.
Dedicated long-term spectroscopic monitoring is still needed to further confirm
or reject these BSBH candidates.
