You can buy diamond paste, also referred to as “diamond compound,” from a number of online sources, as well as jewelry or dental supply retailers. The abrasiveness of the paste is measured by microns, symbolized by the Greek lowercase mu (µ).

If using a polishing cloth doesn’t seem to work, try a more high-powered rotary tool with a felt tip attached.

If the scratch is extremely deep -- deeper than a hairline scratch -- crystal replacement by a professional may be the best option.