I've seen countless videos where people survive the initial accident, and lie there cut in half, but still alive. You can survive without your lower half, but you must have medical attention fairly fast to control the bleeding. Your vital organs are above your waist, but the change in blood pressure is what will probably kill you--your body's circulatory system is what gets affected right away.
I saw a video last year online, which showed a man cut in half by a train. What appeared to be police, were walking around this man as he lied there still alive. They spoke to him, and got his wallet out of his back pocket on his lower half which was a few feet away. They didn't render any First Aid, because I don't think they knew what to do with him. It's like they were waiting for him to die. It was sad, but what can you do? He was in pain, but I have to believe it was more mental shock.