You can run a piece of PVC under the sidewalk with a good water hose and a hammer...
First dig a trench back into the yard a little bit longer than the width of the sidewalk.
Get a piece of 4" Schedule 80 PVC slightly longer than the width of the sidewalk.
place one end of the PVC under the sidewalk where you need to pass through and tap it in place with the hammer.
Now put the garden hose in the PVC and turn the water on. As the water sprays out the end of the pipe it will erode the soil and you can tap the pipe it will move ahead a bit. Continue running water and tapping the pipe until it has passed completely under the sidewalk.