Its easy to see why violence on the field would occur: athletes are in the heat of the moment, when their hormones are pumping at full strength and they're incredibly focused. In light of this, it makes little sense to punish someone on the field severly, especially considering the people punishing them are the ones profiting from them being there in the first place.
Violence in the field of sports should be treated the same as violence off of the sporting arena. An affront to ones civil liberties is prevalent regardless of where the offense took place. Violence on the field is the same as violence off the field and commands the same treatment.