I am under investigation for a simple battery where there was no witnesses. How long will the investigation go on or if there is no probable cause?

When you say there are no witnesses, the state counts the complainant as a witness.  Even though it is one against one, they can still file a charge just based on theother person's word.  There are two parts to the investagation.  One is the cops.  They will talk to whoever they think has inforrmationa and then send it to the state attorney.  They may or may not talk to these people, such as nieghbors or friends, and will probably talk with the complainant.  Different offices take different amounts of time.  A good ballpark would be 2 to 3 weeks.  Could be shorter or much longer.  You may want to check with a criminal defense attorney in your area to see if it might be helpful to have an attorney contact the state attorney before they make a final decision.