First off.. pigeons unfortunately lose legs in captivity and the wild, all the time,for many reasons, and they recover fully on own, never even losing their socal status, ability to breed or care soley even for Young. I've seen this professionally and personally (I've never had happen to mine, but had clients neighbors and friends happen a lot). Just pluck the feathers around area carefully at base fast making sure not to cause further damage or hopefully more discomfort (birds deal with pain great fortunately), giving it a drop daily of some luquil poultry vitamin ect supplement, and use cheap triple antibiotic ointment gotten at any store (DO NOT USE *PLUS PAIN* TYPES, ONLY PLAIN ANTIBIOTIC OINTMENT.. because some plus pain type meds can be toxic). Feed it chicken layer pellets instead of seed grain legume ect mixes, and you'll see majot improvement in about a week.