I know that the ingredients and the manufacturing process are both important, as discussed in this question. What would be a modern approach to figuring out the ingredients?
The ingredients listed on the website are: "Carbonated Water, High Fructose Corn Syrup, Caramel Color, Phosphoric Acid, Natural Flavors, Caffeine." My ideal answer would provide information about how to get more detail about the specific ingredients (not just 'natural flavors') and their proportions.
I'd say some combination of a GCMS / LCMS. The chromatography separates the components & can often be relied upon to ppm level identification.
Then compare the MS peaks for hits against one of the standard databases e.g. NIST.
I suppose that strategy would identify most of the peaks. The left over peaks, which don't give a good match against a GCMS database, would be more challenging to identify. One would need to screen potential raw materials one suspects may have gone into the blend to see if one generates a hit.
Of course, this identifies individual chemical species. Any practical recipe on the other hand is unlikely to be based on pure species amounts but would use available ingredients which are each complex mixtures of pure species.
The GCMS / LCMS approach won't tell you what pseudo-components (e.g. lemon oil ) went in & in what proportions. So you figured out the chemical composition but not really the recipe.
Sometimes, rarely, two compounds can have very similar MS peak patterns. In those cases (or to reinforce a GCMS analysis) one could also compute the Kovats Index (KI) of each peak on the GC based on retention times alone (and the injection of a standard Hydrocarbon mix). There's a good amount of info. and databases on the KI flavor / natural product ingredients.
That would be another strategy to confirm peak identity.
On the molecular biology side of things I'm sure there are techniques using PCR etc. that could amplify and characterize traces of DNA segments that came via ingredients of plant origin. e.g. One might be able to say a certain plant oil extract was used. Although at high dilutions & after extensive processing it would be a challenge to get enough starting matter for these techniques to work.
