Which of 不谢 and 不用谢 is correct?
which is the correct word to say "you are welcome"?

Both 不谢 and 不用谢 are correct.

MDBG

不谢


  don't mention it / not at all


不用谢


  You're welcome / Don't mention it


Hell, I have had an old lady tell me: 不用谢谢 - but I wouldn't recommend using this one. 

In the north you might hear 甭(béng)谢, 甭 being a contraction of 不 and 用.

You can also use 不客气


  you're welcome / don't mention it / impolite / rude / blunt