You should NOT be bottlenecked that much by your CPU with that card.
a) run the game in Windowed mode so you can also monitor the GPU usage in CCC (if you ALT-TAB so the game is in the background it often stops rendering the game)
b) run Task Manager and monitor the CPU processes (TM->Performance->CPU usage history).
It's CTRL-ALT-DEL, then select Task Manager. Make sure to view ALL threads, not just the average CPU.
If NONE of your CPU threads are hitting 100% then you aren't CPU bottlenecked.http://www.tomshardware.com/reviews/radeon-hd-7970-benchmark-tahiti-gcn,3104-9.html
*It seems very possible to drop to 45FPS if you have maxed out the settings. My advice is to adjust Anti-Aliasing and other settings so that you achieve 60FPS with VSYNC enabled most of the time. A small drop in quality is worth it to avoid screen tearing (VSYNC disabled) or major stutter (VSYNC enabled but dropping below 60FPS).