From what you have described is the crime of assault and battery. The seriousness of the crime depends upon the degree of injury the parties sustained from a physical standpoint.Whether or not law enforcement decides to press charges based upon the conduct that you have described is up to the invidual officer taking the report to decide and for the district attorney's office to decide if they want to actually file charges. The ultimate result depends upon how hard one of the parties involved in the incident wishes to push the matter as to a criminal proceeding.