A brain counts as food and I don't think zombies can go vegetarian.
Here are some more words to prove my point:
reason justification yes logic duh see proof proof proof reason justification yes logic duh see proof proof proof reason justification yes logic duh see proof proof proof yes
The zombie virus lives inside of the brain and a virus' soul purpose is to spread itself. If the zombie eats the brain and then it will be destroyed which means the victim won't become a zombie because the virus will die without a host. So that victim won't infect other humans and there will only be one zombie alive.
The Zombie is already dead.It feels hunger, but no longer needs nutrients to survive.So it makes no sense that it would die once again.
Besides, most zombies would die out naturally this way, and we won't have any cool zombie shooting games and movies.This really sucks, So they cannot die.