The one-word answer is "absolutism", but that doesn't help, does it?
Louis XIV is a bit outside my area of expertise, but as I understand it there are several factors at work, including:
Before Louis XIV came to power, Cardinal Richelieu oversaw the razing of castles and fortresses around France, as well as the elimination of city walls, particularly in protestant areas. This made building and protecting a force more difficult and dangerous.
The move to a standing army also made raising an opposing force more dangerous. Significantly, Louis XIV moved authority over French forces out of the hands of provincial and fortress governors to marshals and generals responsible directly to the king, so he could count on his army to keep nobles in line.
Court life at Versailles not only distracted nobles from conspiring against the king, but the great expense of it literally indebted many of them to the monarchy; without royal subsidies many nobles could not have continued to maintain their lavish lifestyle at court. This represented an inversion of sorts, as previously the cost of raising an army had necessitated the monarchy's use of private armies, whereas under Louis XIV the cost of court life (and the time spent there) made raising private armies difficult.
Court life also presented nobles with alternatives to armed conflict, primarily in the form of hunting. The significance of hunting in court life must not be understated, and there is a great deal of evidence to suggest it provided a substitute for warfare, even in artistic representations of noblemen.
None of these factors alone (or others) truly prevented nobles from having a private army, but they made it more difficult or dangerous to accomplish or otherwise occupied the time of France's rich and powerful in other ways. Indeed, there were instances of privately organized and financed armies put to limited use, and town militias and other organizations continued to exist, but these were primarily in service of the king. "In other words," writes Roy McCullough, "the Crown maintained a monopoly on the right to use coercive force, but had little compunction about farming this right out to a variety of coercive institutions when necessary."^1