Musings

The Standard Model Landscape

One of the most consistently misunderstood features to emerge from the study of string compactifications is the existence of a “landscape” of metastable vacua. Surely, it is a grave defect of the theory to possess so many solutions! Obviously, the string theorist must be on the wrong track.

But, if you think about the matter, you quickly realize that the essential ingredients: coupling to gravity, and a source of violation of the null-energy condition, are rather commonplace. Arkani-Hamed, Dubovsky, Nicolis and Villadoro have a beautiful paper, in which they point out that the requisite conditions are present in the rather minimalist context of the Standard Model, coupled to gravity, with massive neutrinos. As long as there are no other light fields¹, their analysis holds for any theory containing the Standard Model, including — one hopes — string theory.

In Type II flux vacua, it is negative tension of the orientifold planes that supplies the violation of the null energy condition. In Arkani-Hamed et al’s story, it is the Casimir energy due to light fermions with periodic boundary conditions on a circle. In the simplest case, they looks at a compactification on AdS₃$\times S^1$ with positive 4D cosmological constant. At the classical level, the potential for the “radion”, the 3D scalar governing the radius of the $S^1$, is of a runaway form, and the solution, just described, is a funny way of writing dS₄.

But the Casimir effect, for fermions with periodic boundary conditions on the $S^1$, induce a 1-loop correction to the potential for the radion, which tends to make the circle shrink, rather than expand.

By happy coincidence, the neutrino masses in the real world are comparable in scale to the 4D cosmological constant, and can stabilize the radion at a finite radius of the $S^1$ (of order a few microns). The 3D effective cosmological constant is negative, with the AdS₃ radius somewhere on the order of the 4D Hubble length ($\sim 10^{25}$m).

The existence of these vacua is entirely an infrared effect. The existence of more massive degrees of freedom like, say, the electron, induce only tiny corrections of order $e^{-m_e/m_\mu}$. Indeed, the smallness of these corrections means that the scalar dual to the photon in 3D has a nearly flat potential.

For all intents and purposes, particle physics in any of these vacua is the same as in our 4D world.

There is quite a fun story to do with other “compactification” geometries.

But, I think the bottom line is that we should not be particularly alarmed at the presence of a large number of vacua. Any theory worthy of our consideration will, likely, possess a similarly large number of vacua. If we should be disturbed by anything, it is that, out of the plethora of string vacua found to date, none of them looks sufficiently like our world, rather than that there are too many that do.