So before you say it is the CPU overheating it is at 40 degrees while playing black ops 3 while recording. I just wanted help to narrow down the possible issues thanks.
Is it a generic brand of PSU? THat could be the issue of why it just shuts down, PSUs that are generic are most likely really cheaply made, hence why they fail so easily under load.
Main culprits for sudden shutdown without any error messages are weak PSU or system overheating (could be some other component overheating, the CPU isn't the only component that needs to stay cool as you seem to think).