Is it possible to get a "perfect ring" scratched xbox 360 disk working again?

An odd noise emanated from my xbox, tonight, and I figured it was due to the fact it was standing up (something I had never done before), so I decided to put it on it’s side. An epic sadness came over my face as I turned my 360 and realized I didn’t hit the power button first, a grinding noise was heard, and then in a frenzy I opened my cd drive and saw the damage. A perfect circle of a scratch on my before flawless CoD4 disk. Thinking it was cosmetic, I closed the drive again and again, yet it wouldn’t read the disk.

I looked online for solutions and tried the two most plausible, toothpaste (because of the grit) and windex (as it is a glass cleaner) and neither worked.

Any methods that get deep scratches out? Please, I love this damn game.