Some of these posts, including yours, border on moronic.
Cops should never enter property for any reason at all without permission, by your reasoning.
So lets change the story by just a little bit.
Instead of jumping the fence, suppose the kid had smashed down the front door and was in your house, plainly observed invasion by the cops.
By your reasoning they are supposed to wait until you invite them in?
Or do they have to wait for a warrant to be delivered from down town?
In either case, the invader could be armed and could do serious damage during the time lost.
I know you are going to take care of him yourself.
Maybe. Lots of home invasions end well for the home owner but there are quite a few that do not.
My bet is that if the cops were not in your house in milliseconds, you would be the first to complain, threaten to sue, 50 letters to the editor, face time on local TV, etc.
And so brave gunfighter that you are, you shoot the kid. Cops outside hear the gunfire, but they can not come in your castle until you invite them?
You are living in a dream world and you had better wake up and read your local laws.