go after vs search

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

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

As verbs, search is a hypernym of go after; that is, search is a word with a broader meaning than go after:

Other hypernyms of go after include look for, seek.

go after (noun)	search (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

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