The body armour that stops rifle bullets use ceramic plates to do it. Some of them can stop a 7.62mm NATO round at point blank range. An arrow would not be able to cut through such plates in the same way it might do against a soft kevlar vest.
But anyway bodyarmour will not be the main reason for why bows are totally outclassed in a modern battlefield as rifles are superior in every way, rate of fire, accuracy, range and amount of ammo. Bows existed in a time where melee was important but today firepower is everything. For that little melee that can still be done today a bayonent, riflebutt or unarmed martial arts works fine.
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