I enter without Piper into Diamon City, then her little sister shouts "PIPER" and runs off. I have never to be seen again. Does anybody know what happen?

I've seen this before.

For some reason, Nat's response to Piper entering Diamond City was triggered. So her AI package is taking her off to find Piper in order to talk to her. And Piper is presumably in whichever settlement you dismissed her to.

She can't die, so you can either 1. reload and take Piper with you to activate the conversation properly or 2. Just wait for Nat to come back. Which she probably will eventually.