I am not an expert but have some limitied experience with alarm systems, so follow my advice at your own risk
if you have simply chopped the wire then both ends that are now flapping around in that hole should have matching colours.
There is a likelyhood that the wires themselves are carrying 12v dc, so be careful not to touch them together.
method 1 - proper way to do it (imo)
with everything turned off & the battery backups removed, tape a new wire to the one u have cut, leave it on the roll & then goto the alarm panel & try pulling it back, it should run the new cable for u if there are no obstructions. once u get it there, goto the other end, estimate how much you need from the hole to the pir sensor, add a meter or two extra & then cut the roll & tape that end to the other cut piece & pull through from the sensor, when you remove the wires from the sensor, write them down & replace exactly as you found them, repeat this at the alarm panel.
method 2 - usually quick repair, if done properly is usually adequate, good if you cannot easily run new cable
turn off the alarm system & remove any fuse from the spur, unplug it, and remove the battery backup either from the alarm panel or bell box or both.
carefully peel back the outer sheath, usually white, this will reveal several cores, all different colours. do this on both sides of the break.
personally i would solder each core together not forgetting to tin the wires properly first. once u have soldered them, then use heat shrink sleeving and a hot air gun / lighter to shrink the sleeving.
if you cannot do it like this then i would recommend a waterproof junction box and taking both ends & twisting them together, then put the pair into the junction block so both screws clamp the twisted pair.
once you have done this check again you have the correct colours matched up. just give it another check before you turn the power on. dont reconnect the batteries again immediately, ensure it all works properly first before reconnecting them last.
the alarm probably sounded because each pir, alarm panel, keypad & bell box have tamper switches, which are normally closed. if u cut the cable then the alarm is telling you someone has opened the housing or tampered with it. Its actually a good sign because alot of the alarms we repair / upgrade have not had this circuit connected.
lastly when you test it, remember if any of the housings are open or something moves in front of a pir sensor, window / door open it will trigger the alarm, i know this sounds daft but the number of times i've watched an apprentice scratch his head because the alarm keeps sounding while he's got the alarm panel open with the protected front door wide open & a dog wandering around in front of the pir
hope this helps & good luck