There is water appearing in the middle of the bedroom floor under the carpet. It leaks slowly but has made the carpet soggy. It is not near a wall. There is a half bath connected to that room but the floor and under the sink is all dry. There is a hot water heater in the closet and that area is dry. We have turned off the water to those areas and it still comes up through the carpet. We are baffled.

Is the carpet on a concrete slab? Or an upper level?

My bet is on the slab. Sounds like you have a leak under the cement. You have options here, none of them cheap.
1- contact a leak detection service and have them locate and open up the slab so the leak can be repaired. **OR**
2- Consider a repipe job with plastic. Depending on the age of the house and the material are the pipes this might be your best option. Good luck, Tom