Society finch dying??? Please help?

I would put a heating pad under the outside bottom of the birds cage and cover cage halfway with a blanket and put in a dim quiet room. You are sure it is a male and not a female as some females will do this when they start to lay eggs. His feathers will not grow back until the next molt and may not at all if he is not healthy. I would call the vet and see what they say. May want to have him put to sleep if he is suffering. Is the bird eating or drinking? Can try feeding him. There is a product called rescue remedy sold at health food stores and in wegmans natural food dept. You can mix with a little drop of water and give him that that does help them with stress and pain. Best advice though is warm and as comfy as possible and to the vet if can.