Personally, I prefer the finish on the discontinued Ruger Mk II better than that of the Hawk, and I also like the Leonard Brownell stock design of the MkII's.
When Ruger switched over from the Mk II to the Hawk, I have read nothing about installing a new, better barrel on the Hawks.
I would suspect that the barrels remain the same, with only a cosmetic change to the finish.
I have two Ruger 77 Mk II rifles, and both are very accurate; more so than most factory rifles. So, I would doubt that Ruger would introduce new barrels to replace ones that were excellent to begin with.
In Ruger's marketing pitches describing the new Hawk, no mention was made of the Hawk getting a new, better barrel. The only things that were pitched were the new stock, new trigger, and the matte finish.