Concentration is for the duration of the spell
#Concentration is for the duration of the spell# TheThe 1 minute duration of the spell is for both holding the spell until you deliver a hit as well as for maintaining the Condition effect of the smite (in this case, Frightened condition.)
This is supported by Jeremy Crawford via Sage Advice
Some smite spells, such as wrathful smite, do have an ongoing effect that requires concentration