I remember hearing something about where it was discerned that after a certain percent of losses armies tend to break (ie retreat). Does research on this exist? If so, at what point do armies tend to break?
There was an old, if rough rule of thumb (that I read in the Encyclopedia Britannica years ago) that an army could sustain only 30% casualties without breaking. At this point, the survivors would all feel a real fear of getting killed or wounded "next," instead of "that's what happened to the other guy."
That's all other things being equal of course. An army that saw an enemy break first, then suffered its 30th per cent casualty, would nevertheless receive a morale boost that would nullify this effect. Untrained troops tended to break at much lower casualty rates than trained troops, which was a major mark of "experience" levels. And fighting on home ground counted for a lot. At the battle of Stalingrad for instance, the "30% rule" applied only to the German army. Russian units could sustain 90%+ casualties and have the survivors fight; in one case 6 men out of 1,000 escaped a trap.
From what I have read, the level and intensity of training influences this massively. More green troops will break easily whereas veterans will tend to fight longer and harder.
Panic is another factor: the more there is, the greater the chance of an army breaking. Mercenaries when not well paid, had a tendency to break or change side -- see the Thirty Years war for numerous example of that.
Finally, a decisive key event: death of a general/leader, taking a position, or threatening the camp (with followers and your possessions) would do wonder to break the moral and route an enemy.
Green units might break, retreat or even desert without any deaths. For example, mass surrender of Germans in the west at the end of WWII, or Iraqi army at the end of the Gulf War.
While other units fight to death, specially if their culture does not allow surrender or retreat. For example Spartans on Thermopylae or Japanese on Tarawa, where survival rate was below 1%.
Hence, the answer is so vast than one should not expect to reach a number, even if that number exists, it just a mean number.