Arborvitae browning has many causes - need for water (not this year), root damage, winter sun scald, damage to the branch. You note the branch does not appear damaged. It could be the multiple years of drought caused it. With this being the only portion appearing brown it is suggested to leave for a season to see if it recovers. If it stays brown you can prune it out. If the brown area becomes more widespread I would recommend you call a local tree care company and request an arborist evaluate.