Multiple Stellar Populations of Globular Clusters from Homogeneous Ca--CN--CH Photometry. V. $cn^\prime_{\rm JWL}$ as a Surrogate $cn_{\rm JWL}$ Index and NGC 6723

We introduce new color indices $cn^\prime_{\rm JWL}$ (= $Ca_{\rm CTIO} - Ca_{\rm JWL}$) and $ch_{\rm JWL}$ [=$(JWL43 - b) - (b-y)$], accurate photometric measures of the CN band at $\lambda$3883 and the CH G band, respectively, in the study of the multiple populations (MPs) in globular clusters (GCs). Our photometric CN--CH relation for a large number of red-giant branch (RGB) in M5 shows that the evolutions of the CN and CH between the CN-w and CN-s populations are not continuous. Armed with our new color indices, we investigate the MPs of NGC 6723, finding the RGB populational number ratio of $n$(CN-w):$n$(CN-s) $\approx$ 35.5:64.5 ($\pm$2.8) with no radial gradient. Similar to other normal GCs with MPs, the helium abundance of the CN-s population inferred from the RGB bump magnitude is slightly enhanced by $\Delta Y$ = 0.012 $\pm$ 0.012. Our $cn_{\rm JWL}$ and $cn^\prime_{\rm JWL}$ color-magnitude diagrams clearly show the discrete double AGB populations in NGC 6723, whose bright AGB populational number ratio is in marginally agreement with that of the RGB stars within the statistical errors. Finally, our synthetic $cn_{\rm JWL}$ index is in good agreement with observations, except for the CN-w asymptotic giant branch (AGB). To mitigate the discrepancy in the CN-w AGB may require a mild nitrogen enhancement and/or a large decrement in the $^{12}$C/$^{13}$C ratio with respect to the bright RGB. ; Comment: Accepted for publication in the Astrophysical Journal; 17 Figures and 5 Tables