A piece of glass sliced through my cat's tail. What should I do?

One of my cats knocked a painting off the wall and a piece of broken glass sliced about halfway through the tail of my other cat (he was lounging on the floor beneath the painting when it happened). Nothing was severed, but the cut seems to be pretty deep. It’s towards the tip of his tail. It isn’t bleeding and he seems to be okay, though the end of his tail droops when he walks around. I washed his tail with some warm soapy water and rinsed it, as a cat care website suggested. What should I do now? Do I need to take him to the vet? I love my cat but I can’t afford several hundred dollars in vet bills if it is not an emergency.