Please re-read my message. Understand that "background" means something that may not be all that obvious that it's running all the time, like Bluetooth or LTE. Are your phones jailbroken? That causes HUGE battery drain.
I'm not saying it *couldn't* be a defective battery, but I am saying that I'm skeptical of the claim "no apps running," since that's technically impossible if you include services, which of course are running all the time.
If you're sure it's really, truly not running any apps or unneeded services in the background, I'd suggest you take it in for a check. It's possible you have a defective battery, in which case it would be changed for you at no charge under warranty.