Musings

The idea is to take $N$ real scalar fields, $\phi_i$, with a decoupled potential

which is the sum of general quartic potentials,

Here, $M_*$ is an ultraviolet cutoff, above which supersymmetry (or similar) kicks in. The dimensionless couplings, $a_{i j}$, in these potentials are assumed to be chosen randomly from some probability distribution, with the obvious provisos that

• $a_{i 4}\gt 0$, so that the potential is bounded from below.
• $a_{i 2}\lt 0$, sufficient to ensure that there are two minima.
• $|a_{i j}|\lt 1$ (maybe we should say $\ll$), so that we can trust the effective field theory below $M_*$.

Generically, each such potential will have a pair of nondegenerate minima, located at $\phi_{i\pm}$,

It is easy to arrange that the tunnelling rate is sufficiently suppressed so that the higher of the two vacua is stable on cosmological timescales. Thus we obtain a theory with $2^N$ vacua, labelled by $\{\eta\}$, with randomly-chosen vacuum energies, $V_{\{\eta\}}= \sum_{i=1}^N V_{i,\eta_i}$.

Note that we’ve assumed no cross-couplings between the $\phi_i$. Even if they aren’t coupled to anything else, the $\phi_i$ couple to gravity, and 1-loop diagrams will induce cross-couplings of the form $$\frac{1}{16 \pi^2} \sum_{i,j} \frac{V_i(\phi_i) V_j(\phi_j)}{M_{\text{pl}}^4}$$ However, we need to take the cutoff, $M_*$, sufficiently low so that the corrections to Newton’s constant are under control, $$M_{\text{pl}}^2\gg \frac{N}{16\pi^2} M_*^2$$ and, with that proviso, the cross-coupling terms are parametrically small compared to $V$.

Similarly, when we couple the $\phi_i$ to Standard Model fields, we assume that the couplings are a sum of independent (randomly-chosen) couplings, $c_i(\phi_i/M_*)$, to the $\phi_i$, with no cross-couplings. Again, Standard Model loops induce cross-couplings which are parametrically small compared to $\sum_i c_i(\phi_i/M_*)$.

Why are we emphasizing the absence of cross-terms? Because we want to apply the Central Limit Theorem to study the distribution of values for $c=c_0 +\sum_i c_i$. In the absence of cross-couplings, the large-$N$ limit gives us a Gaussian distribution with mean

and variance

where $c_{av} = \frac{1}{2}(c(\phi_+) + c(\phi_-))$, $c_{dif} = \frac{1}{2}(c(\phi_+) - c(\phi_-))$ and $\overline{c_{av}}$, $\overline{c_{dif}}$ indicate the ensemble average of these parameters over the probability distribution for the couplings of the $\phi_i$.

If, as is typically the case, $\overline{c_{av}}\sim \overline{c_{dif}}$ in order of magnitude, then

(or even much smaller if $c_0$ dominates over $N \overline{c_{av}}$). For large $N$, one doesn’t scan a large range of coupling-constant space.

On the other hand, for some couplings, one can sometimes arrange for $|\overline{c_{av}}|\ll \overline{c_{dif}}$. For instance, consider a supersymmetric theory with superpotentials

(The absence of a quadratic term can be assured by a discrete $\mathbb{Z}_4$ R-symmetry.) Such a theory has $W_{av}=0$. So the range of values for the cosmological constant effectively scanned by this model is

For sufficiently large $N$, one can find vacua which cancel SUSY-breaking contribution to the cosmological constant, $\Lambda_0$ to an accuracy $2^{-N} M_*^4$.

They devote the latter half of the paper to discussing various models in which “most” coupling are effectively frozen, but a few (the cosmological constant, the Higgs mass, the $\mu$-parameter, …) can be fine-tuned.

I have a few quibbles with some of the details. Most revolve around taking $N\sim 10^2$-$10^3$, as is typically considered in string constructions, and as they do here.

1. While we argued that the cross-couplings are parametrically small, they are not numerically much smaller than $1/\sqrt{N}$ for $N$ in this range. So we’re not really quite able to justify using the Central Limit Theorem
2. It’s not really quite true that $\Lambda_{\text{QCD}}$ (and other parameters related to dimensionless couplings by dimensional transmutation) are effectively frozen. $$\frac{\delta\Lambda_{\text{QCD}}}{\Lambda_{\text{QCD}}} = \log(\Lambda_{\text{QCD}}/M_{\text{GUT}}) \frac{\delta\alpha_{\text{GUT}}}{\alpha_{\text{GUT}}}$$ For this range of $N$, $\alpha_{\text{GUT}}$ may vary only by a few percent, but the variation in $\Lambda_{\text{QCD}}$ is substantial. This has a big impact when one tries to argue for the tuning of the cosmological constant and the electroweak scale (respectively, from Weinberg’s argument about structure formation and from the atomic principle)

I could list some more, but I don’t want to spoil the fun of reading a really thought-provoking paper. I also encourage you to read Nima’s comment on Luboš’s blog.

Update (1/20/2005)

I led a very lively discussion, in our brown bag seminar today, on this subject. In it, Willy Fischler pointed out that there’s a radiative correction, much larger than the one mentioned above, which leads to cross-couplings in the scalar potential. One-loop diagrams (with scalars running around the loop) lead to a correction to $1/8\pi G_N$ proportional to $\sum_i V_i''(\phi_i)$. When you do the Weyl rescaling so that the Einstein-Hilbert term has its canonical coefficient, this introduces a correction to the scalar potential of the form

which is not parametrically small (down by a factor of $1/N$) compared to $\sum_i V_i(\phi_i)$. This really does seem to create a cloud over the radiative stability of the decoupled form of the scalar potential, and hence on the applicability of the Central Limit Theorem.