The best things you can do for your tree are 1) leave it alone and let it seal off the wounds, and 2) make sure your son realizes that he shouldn't do this again. The tree will be MUCH better at taking care of itself than you will be trying to intervene.

Thanks Brandon. My son knows not to shoot arrows at any of the trees anymore. And I haven't done anything to the tree, but it doesn't look any better, either.

It will take time. The discoloration you see is not hurting the tree. The wounds need to be sealed off (by the tree itself!), and that will happen over time. I would just ignore it, if possible. The damage is actually probably fairly minimal.