What about the study of born baby?

Thankyou Reshma for your question.
Neonatology is a subspecialty of pediatrics that consists of the medical care of newborn infants, especially the ill or premature newborn infant. It is a hospital-based specialty, and is usually practiced in neonatal intensive care units (NICUs). The principal patients of neonatologists are newborn infants who are ill or requiring special medical care due to prematurity, low birth weight,intrauterine growth retardation, birth defects.

Reference: Website for National Neonatology Forum : http://www.nnfi.org/

It is called Neonatology. You can read all about it in the link below.

https://en.wikipedia.org/wiki/Neonatology