Warped Passages - Lisa Randall [199]
Although I have focused on a two-brane scenario with a specific warp factor, the features we have considered are likely to be more general than this particular example. With extra dimensions there is good reason to expect disparate masses. The particle physics intuition that masses should be more or less the same is violated, and a wide range of masses is expected. Particles located in different locations would naturally have different masses. Their shadows change as you move around. In our four-dimensional world, the result would be a range of sizes and masses, and that is what we observe.
Further Developments
When our paper explaining the hierarchy in terms of warped geometry appeared in 1999, most of our colleagues didn’t recognize that it was a genuinely new theory, very different from the large dimensions idea. Joe Lykken said to me, “Reaction built slowly. Eventually everyone understood that this paper [and another that Chapter 22 will describe] were big and new and generic and opened a whole new arena of ideas, but not at first.”
For months after our paper came out, I was asked to give talks about my work on “large extra dimensions.” I kept having to object that the beauty of our theory is precisely that the dimensions are not large! In fact, Mark B. Wise, a (very aptly named) Caltech particle theorist, laughed at the title I was assigned for a plenary talk in the closing session of the Lepton-Photon Conference of 2001, the major particle conference where experimenters present important results. The organizers had given my talk a title that referred to all research on extra dimensions except my own!
Mark and his then student, Walter Goldberger, were two of the first to understand the merits of the warped scenario. But they also recognized that Raman and I had left a potential gap in our result that needed to be filled. We had assumed that brane dynamics would naturally lead to branes that are a modest distance apart. However, we had not explicitly said how the distance between the two branes is established. This wasn’t just a detail; our theory’s role as a solution to the hierarchy problem depended on being able to readily stabilize the two branes a small but finite distance apart. It was possible that the inverse exponential function of the distance (which we wanted to be extremely tiny), rather than the distance itself, would turn out naturally to be a modest number. If so, the predicted hierarchy between the weak scale mass and the Planck scale mass would be a modest number, and not the (much smaller) inverse exponential of that number—and our solution wouldn’t work.
Goldberger and Wise did the important research that closed this potentially treacherous loophole in the theory Raman and I had presented. They demonstrated that the distance between the two branes is a modest number, and the inverse of the exponential of that distance is extremely tiny, exactly as was required for our solution to work.
Their idea was elegant, and turned out to be of more general validity than anyone realized at the time. As it happens, any stabilization model is very similar to theirs. Goldberger and Wise suggested that in addition to the graviton, there was a massive particle that lived in the five-dimensional bulk. They assigned properties to this particle that made it act like a spring. In general, a spring has a favored length; any larger or smaller length would carry energy that would make the spring move. Goldberger and Wise had introduced a particle (and associated field) for which the equilibrium configuration for the field and the branes would involve a modest brane separation—again, just what our solution to the hierarchy required.
Their solution relied on two competing effects, one that favors widely separated branes and another that favors nearby branes. The result is a stable compromise position. The combination of the two counteracting