I did a similar thing a while ago with satellite dish coax. Dish was out by road and not on the house.
Dig a hole on either side of the sidewalk. One side will be more like a trench. doesn't have to be very deep.
Get a piece of galvanized pipe the length to match the sidewalk, or pretty close. Thread one end and put a cap on it. Get someone to align it in the trench (holding it). The capped end goes forward so the dirt doesn't plug the pipe. You drive it through the ground under the sidewalk with a large mallet. Works best when the ground is wet. It drove straight through for us. Then unscrew the capped end and you can thread the cord through.
Was surprised how well it worked.
(If you don't have the ability to thread and cap the leading end probably some well placed duct tape will do the same)
This would be the easiest way to go under. You might also try sticking a high with an inline nozzle inside an uncapped pipe to essentially hydroblast your way through if it's real tough going. Going to make a very large mess though. A pressure washer with a 0 degree tip would do the same thing.