The incident happened on February 12 on the 4pm service between Edinburgh Waverley and Peterborough, just before the train left the station.
A man was travelling with three children and was seen by another passenger assaulting one of them, a boy aged two or three, police said.
Officers say the child was hit three times and kicked in the head and chest.
The man is described as white, of a large build and with short dark hair. He was wearing glasses, dark jeans and white Nike trainers.
He alighted the train with the three children at Alnmouth station in Northumberland.
Pc Joss Froggatt said: “I would like to speak to anyone who was on board this service who may have seen or heard the incident.
“It is unclear at this stage exactly what took place but we have had serious allegations made, so we are investigating to establish what happened.
“If you have any information, please contact me by calling 0800 40 50 40 or text 61016, quoting reference number 324 12/2/17.”