Well, dying will show up as lost tissue. Sometimes when branches die they go white first (as the tissue dies off), then they will turn green (often from excess phosphate/algae as XeniaMania points out).
If you are new to sps, sometimes it's hard to recognize when the tissue is gone. If the tissue is still there, you should be able to see the polyps, even if they are retracted. If you see no evidence of tissue and/or polyps (compare to the part of the coral that appears healthy, top left), those branches are dead. Often, this tissue loss will migrate through the whole colony as STN (slow tissue necrosis, looks likes what you may have), or RTN (rapid tissue necrosis - the whole colony dies in a matter of hours, couple days at most).
If the branches at the lower right are void of tissue/polyps, keep a close eye on the colony. If you see the death spreading, the best option is to frag off some of the healthy branches to try to save the colony.
If the branches at the lower right are turning green but still exhibit polyp extension and the tissue is apparent, the colony is adjusting to your tank/environment/lighting. But still watch this area closely as it could be indicative of inadequate lighting on this part of the colony.
If you a new to sps, I'd really encourage you to find a source for tank raised frags, hopefully from someone local. I know that requires greater patience for them to grow out, but your success ratio will probably be much higher than purchasing small colonies like this at the LFS. Cheaper, hardier, and less chance that you'll become discouraged.
Good luck! Your setup reads nice, and you seem to have all the necessary high-end hardware. Pay close attention to maintaing stable alk/ca, practice good husbandry, and you should be on your way.