There appear to be a few issues. First, $PATH is not valid R code. Second, you are as far as I can tell, just simply putting to unquoted strings next to each other. You can do something like this:
Also, I am not sure if this is system dependent, but my PATH values are separated by semi-colons and not colons. I only used a colon in my sep = argument because it looked like you were trying to use a colon in your example code. You should verify what is used in your PATH by inspecting your old_path (or Sys.getenv("PATH")) variable once it is defined to determine which you need. You can put whatever it is as the sep= argument in the paste function