5:05 PM PT -- Sources close to the investigation tell TMZ the reason the case went nowhere is because Adrian did NOT strike the boy in the forehead -- but instead the child accidentally hit his head on a carseat while Adrian was punishing him.
We're also told the form of discipline was "not impermissible." Under Texas law parents are allowed to administer reasonable punishment.
Peterson's attorney tells us an adult witness backed him up, saying Adrian "did nothing inappropriate with his son."
Adrian Peterson is under investigation for a SECOND child abuse case involving one of his other children.
This new investigation involves an incident that allegedly went down in June 2013 ... and the probe was launched after the then 4-year-old boy went home to his mother with a scar on his forehead.
The station says Adrian sent the mother a text message saying the child hit his head on a carseat -- but when the mother questioned him further ... Peterson eventually admitted hitting the child.
The mother reported the incident to Child Protective Services, but no charges were ever filed against Peterson.
Earlier today ... Peterson issued a public apology for the other case -- for which he's been indicted -- saying he was sorry for hurting the child during a visit in May.
Peterson said he'd seen a psychologist who taught him there are other ways to discipline kids.