The orthodontic archwire is sometimes left slightly longer than the last bracket, and its end is bent over so that it doesn't irritate the lining of the mouth. Occasionally this end breaks off and can be swallowed. If this happens, don't worry. This little piece of wire is 4-5 mm long and will pass through the body without any problem.
If a longer piece of archwire has broken off – for example, more than 15 mm – and you suspect you've swallowed it, it's usually still not a problem, but you may want to consult a doctor to be on the safe side.
This answer is for general information. Your own case needs an examination by an orthodontist.