How do I trace a damaged security system wire?
First, cut power to the alarm system. 

Get your hands on a multimeter with a resistance setting. Run a long extension cord (unplugged) from the broken sensor to the panel. Put a probe on each end of the cord (ground pin is easy to remember; it only matters that you're both on the same conductor). 

If you're lucky, you and an assistant will probe wires until you get a no-resistance reading. That's the intact wire of the pair. The rest writes itself.

If you're not lucky, you'll have cut both conductors and will have to run around identifying all your wires, in order to find the severed one by elimination.

Or, for around 40-80 dollars, you can get a tone generator and find your break in 5 minutes. The upside to this plan is that you won't have to run new wire -- you'll just locate and fix the existing. I own this one, though there are more/less expensive options: Fluke toner