search vs go after

search is a noun but go after is not a noun.

Word	Noun	Adjective	Verb	Adverb
search	Yes	No	Yes	No
go after	No	No	Yes	No

As verbs, go after is a hyponym of search; that is, go after is a word with a more specific, narrower meaning than search:

search (noun)	go after (noun)
the activity of looking thoroughly in order to find something or someone
boarding and inspecting a ship on the high seas
an investigation seeking answers
the examination of alternative hypotheses
an operation that determines whether one or more of a set of items has a specified property

search (verb)	go after (verb)
inquire into	go in search of or hunt for
try to locate or discover, or try to establish the existence of	go after with the intent to catch
subject to a search
search or seek