In the US, the details are determined at the state level. The term "abandonment" is used very broadly, and can include a situation where a parent leaves a child without making contact for a period of time (which may result in termination of parental rights, but not a punishment). "Abandonment" as it applies in Washington state is explained here. There is what is known as a "safe haven" law, which allows a newborn (under 72 hours old) to be transferred (anonymously) to a qualified recipient (health care employee, medic, etc.), and not be liable under the criminal laws. This does not include dumping the infant in the snow.
Under RCW 9A.42.020, the parent would be guilty of criminal mistreatment in the first degree if their action "causes great bodily harm to a child or dependent person by withholding any of the basic necessities of life" (note that the law is not specific to children), and this is a class B felony. There are multiple grades of mistreatment, so if the action "creates an imminent and substantial risk of bodily injury" or "causes bodily injury or extreme emotional distress manifested by more than transient physical symptoms", this is 4th degree mistreatment, which is a misdemeanor. There are, in parallel fashion, laws against abandonment of a dependent starting at RCW 9A.42.060, punished as a class B felony down to a gross misdemeanor. The maximum penalty for a class B felony is $20,000 and 10 years in prison, and for a simple misdemeanor it is 90 days and $1,000.
In case death results, the discussion could move to the homicide statutes. Homicide by abuse is when, with
extreme indifference to human life, the person causes the death of a
child or person under sixteen years of age, a developmentally disabled
person, or a dependent adult, and the person has previously engaged in
a pattern or practice of assault or torture of said child, person
under sixteen years of age, developmentally disabled person, or
dependent person.
and this is a class A felony. If a person "recklessly causes the death of another person", this is manslaughter in the first degree (class A felony), but if it is "with criminal negligence", it is manslaughter in the second degree (a class B felony). Manslaughter charges are predicated on there not being an intent to kill. If the intent was to actually kill the child, this would be first degree homicide, where the punishment is life imprisonment. Additionally, first degree homicide can be found if "under circumstances manifesting an extreme indifference to human life, he or she engages in conduct which creates a grave risk of death to any person, and thereby causes the death of a person".
State v. Edwards is a relevant case, where a person was charged with both second degree murder and homicide by abuse, and the issue came up that "extreme indifference to human life" is not a self-evident expression. It turns out that case law in Washington interprets this, as applied to first degree murder, as meaning "indifference to human life in general", not "indifference to the life of the specific victim". After a lengthy review of principles of judicial interpretation, the court upheld the trial court's refusal to give the first-degree murder definition of indifference, that is, it is up the the jury to decide what constitutes extreme indifference, for homicides other than 1st degree murder.
