Replying to Getting a piece of 2" pipe under a sidewalk

I'm going to fix some gutters for my mom this spring or as soon as the snow melts and the ground thaws. To make them right I am going to run a gutter straight to the ground, and put a piece of PVC pipe under her sidewalk and drain it into the yard. I would like to use a piece of 3" pipe but if I must I will use a piece of 2" and bell it up to 3" for the gutter to drain into. Now the problem I'm running into is how to get the pipe under the sidewalk. I can dig as much as needed on either side but would rather not demo out any concrete. The sidewalk is about 30" wide and I'm assuming it's a 4" thick slab. I would like to do minimal damage to the yard when I do this as well. I have access to numerous roto hammers, a mini electric jack hammer, and pretty much any digging tool made. I also could possibly get a mole and bore it out but that seems like over kill and I would need to rent a large air compressor to use the mole. Anyone have any suggestions to make this easier or do I need to dig holes on either side of the concrete and start hand tunnelling? Thanks.