Major root cuts or removal, since the sidewalk is close enough to the tree to be impacted by those roots, would potentially cause serious damage to the tree. We can't say with certainty what the risk is to its long-term survival, though, so suggest you consult with a certified arborist or licensed tree expert who can assess the situation in person. Perhaps there is a way to raise the new sidewalk slightly so that roots are not injured.

Miri