First of all ESC, should always beep initialization sound when connecting battery. It has nothing to do with arming. If your ESCs are not beeping after connecting battery and are quite, but later ESC bacon signal starts (few minutes after) this means ESC are not receiving signal from FC. This is probably your case, your flight-controller is not booting when connected via battery.
- A slowly blinking blue LED indicates that the board is booted and running the firmware; this is the normal operating mode.
- Bootloader mode. A slow fading in and out of the blue LED with the green LED on indicates that the board is in bootloader mode.
- A rapidly blinking blue LED during a bootloader update indicates an error state. An invalid bootloader image was likely detected and the update hasn’t been performed.
Just green on and no blue is probably it didn't boot at all, at least it's not a normal operation mode.
When you connect USB only but without battery, you are providing power to flight-controller via USB. However, when you power with battery only (or with both usb and battery) then power comes from only battery.
Check all your wires starting with servo header if they're all connected in right order signal,gnd,vcc. If you have short somewhere or regulator is faulty it might not provide enough power to FC, I can't think of another reason. Just please check all wires.