Hi Ayat,
I’m so sorry this has happened. It is very heartbreaking to see newly hatched chicks die. Unfortunately, it is very hard to try to hand feed a finch from day 1, even for the most experienced breeders, and the chicks rarely survive. It sounds like it was too late already when you tried to feed them. Once they stop begging, they don’t live much longer. They have to be kept warm, so if the parents weren’t sitting on them, the chicks got too cold. If they are too cold they can’t digest food and they stop begging for food. For newly hatched chicks, once the yolk sac is absorbed, they need to be fed right away,and can die after only a couple of hours without being fed. If this is the first clutch for the pair, this is very common. First clutches almost always fail for different reasons, but mainly due to mistakes by the parents. I would remove the nest and rest the pair for a few months before you let them nest again. When finches are breeding, they really need an egg food for the hen when she is laying eggs and for the parents to feed the chicks. This can be cooked eggs with the shell washed, crushed and cooked with them. Or a commercial egg food made for small birds.
Take care,
Brenda