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2009mercury Milan v6 won't hold a charge

We replaced the battery. The dealer cannot find the problem. They asked us to leave it with them for a few days. If the car sits 36-48 hrs it won’t start. We confirmed there were no lights on or anything was left on or plugged in. We checked it by opening the door to see if the overhead light came on every 12hrs. After 48hours the car again would not start. What could be the intermittent problem?