Good reference, and Alain is quite correct. In the US Army, the branch-of-service insignia and associated colors of piping worn by a soldier are determined by the branch which trains that soldier in his specialty, not by the type of unit to which that soldier is assigned.
For example, my first unit was an independent company which was under an otherwise-inactive Infantry regiment. My original specialty however was radio operator, an MOS which was formally trained by the US Army Signal School but could be found in every type of unit the US Army had. While my parent unit was Co. G (Abn.), 143rd Infantry Regiment, I wore the branch-of-service insignia of the Signal Corps -- not the Infantry -- even though I never once held an assignment to a Signal unit. I did not have authorization to wear the blue cord and crossed musket insignia of the Infantry until I had been formally awarded that MOS at a later point.