Sorcha - polishing is indeed originally a grinding process (and I think this also applies for the english word). Every such process causes scratches, but if these are smaller than about one micron they become invisible because the wavelength of visible light is about that value. Polishing means grinding by using an agent (?) with grains of smaller size in order to grind visible scratches away. As far as I know the grains in industrial polishing pastes are still larger than those in tooth paste. So if you don't add larger grains (e.g. from a dirty cloth) the CD won't be damaged. BTW medical cotton wool (hoping that's correct) is also suitable to be used instead of a cloth.

Tooth paste works also well cleaning jewelry. I have some silver piece which is gold-plated at one spot and that spot always develops a very thin violet layer of some oxide (I would be interested in what kind of gold that is). Tooth paste removes that layer at once without damaging the glossy surface.

So I would in any case try tooth paste first, because with the appropriate care I could see no risk at all. Only if the scratches are too deep it becomes complicated. They can still be ground away but one must use a coarser grinding agent first, which will make the surface opaque and then polish these scratches away. But the more you grind the thinner becomes the lacquer and that might have an optical effect by defocusing the laser beam.

I would be very careful with things that stick to the CD. If it doesn't work, how am I to remove it?

Good luck.

Joerg