There is nothing in the Constitution about “professional” police and that is a potential problem.
Back in the day, citizens kept the peace in their own neighborhoods. If someone violated the common law, your neighbors (the posse) would come to your aid and apprehend the hooligan. If the trouble involved a lot of bad guys, the citizens formed a militia to deal with it.
If there was a question about whether someone deserved to be charged with a crime, the grand jury would decide that question, then a jury of your peers would decide guilt or innocence.
However, today’s police are rapidly crossing the line from peacekeepers to a standing army, and that is a very serious issue. The Framers hated the idea of the government owning an army without an enemy, because history showed that such an army would always be used against the citizenry. Always.
We have a problem but no easy answers.
What day was it in America that had no law, except whatever the locals made up?