Cut a piece of PVC pipe, equal in size to the drain pipe's diameter, 12 to 18 inches longer than the sidewalk's width, using a circular saw. Cut one of the PVC pipe's ends at a 45-degree angle, with the circular saw.
Dig a 12-inch-deep and 2-inch-wide trench perpendicular to the sidewalk, with a duckbill shovel, in the location you wish to run the drain underneath. Dig the trench right up to the sidewalk's edge. Continue the trench on the opposite side of the sidewalk so you know when you have passed all the way under.
Place a zero-degree spray nozzle on a pressure-washer wand. Lay the wand in the trench a few inches from the sidewalk's edge. Turn on the pressure washer and allow the water to loosen and blast away the dirt.
Turn off the pressure washer and remove the wand from the hole. Lay the PVC pipe in the hole with the pointed end pointing toward the sidewalk. Hit the flat end with a hammer to push the pipe about 4 inches into the loosened soil.
Pull the pipe out and remove the soil plug inside. Lay the pressure washer wand in the trench again and aim it into the hole left by the PVC pipe. Turn the pressure washer on and loosen the soil. Turn the pressure washer off and remove the wand.
Insert the PVC pipe into the hole, pointed end first. Hammer on the flat end to push the pipe 4 inches deeper into the hole. Pull the pipe out and remove the soil plug. Continue to alternate the pressure-washer wand and PVC pipe until you have tunneled underneath the sidewalk.