I am looking for the Longbow itself being the decisive weapon, not the Longbow men, as was the case in Agincourt. This could mean some significant disadvantages to the enemy, including things like: the Longbows inflicted a high number of casualties or the Longbows being a strong force multiplier that it helped the English army win.
I didn't include Agincourt because the Longbow was not the decisive factor IMO; it was the mud and the battlefield as a whole.