Musings

A Fearful Symmetry

Longtime readers of this blog will know that I think that discrete symmetries (particularly, those that suppress proton decay and flavour-changing neutral currents) are a serious challenge for ideas about the Landscape and, more generally, applications of the Anthropic Principle to String Theory Vacua (see, for instance, these three and posts).

If supersymmetry is broken in the visible sector below the GUT scale, then there are, generically, dimension-4 operators in the supersymmetric theory which violate baryon number. Below the supersymmetry-breaking scale, these induce the usual dimension-6 baryon number operators, suppressed, not by $M_{\text{GUT}}^{-2}$, but by $M_{\tilde q}^{-2}$, where $M_{\tilde q}$ is the squark mass. To suppress these bad operators (note that the observed proton lifetime is at least 20 orders of magnitude longer than the anthropic bound), one generally invokes a discrete symmetry. For instance, R-parity arises as a $\mathbb{Z}_2$ global symmetry, under which the quark and lepton chiral multiplets are odd, while the Higgs chiral multiplets are even.

Such discrete symmetries typically only occur on subspaces of the moduli space of very high codimension. In flux compactifications, not only must the vacuum lie on this invariant subspace, but the fluxes must be chosen to respect the symmetry as well (they appear as coupling constant in the low-energy theory). This drastically reduces the number of flux vacua.

For concreteness, let us work in the orientifold limit, with a Calabi-Yau 3-fold, $X$. The standard estimate of the number of flux vacua is roughly $L^{b_3(X)}$, where $L$ is a measure of the D3-brane charge that must be cancelled. If the desired discrete symmetry requires restricting to a subspace of the moduli space of complex dimension $n$, then the naïve estimate of the number of flux vacua respecting the symmetry is $L^{2(n+1)}$ (the extra 1 comes from $H^{(3,0)}\oplus H^{(0,3)}$). This is vastly smaller than $L^{b_3(X)}$.

There’s a very interesting recent paper which does a more careful estimate. What they find is that — in the class of examples they look at — the number of vacua respecting the discrete symmetry can be enhanced, for number-theoretic reasons, over the above naïve estimate by powers of $\log(L)$.

Where does number theory come in? Pick a point, $z$, in the complex structure moduli space and a value, $\phi$, for the axio-dilaton. View the equations $D_\alpha W=0$ as a set of (linear) equations for the fluxes. If the fluxes were real-valued, we could always solve those equations. But they’re integer-valued. For a generic point, there will be no integer-valued solutions for the allowed fluxes. For special points, the periods take values in some finite extension of $\mathbb{Q}$, and it is possible to adjust the (integer) fluxes, so as to find a solution. Consider the case where the field extension is of degree¹ $d$ and $\phi$ is of height² $H(\phi)$ in this extension. If the degree of the extension and the height, $H(\phi)$, are small enough, then one gets $$N_{\text{vacua}}\sim \left(\frac{L^{(1-d/4)}}{H(\phi)^{d/2}}\right)^{b_3}$$ as an estimate of the number of flux vacua at such a point.

At least in the simple examples they consider, summing over such points respecting the discrete symmetry produces a result enhanced by powers of $\log(L)$ over the naïve one.

They also address a related question: how many such vacua also satisfy $W_{\text{tree}}=0$ (i.e. are supersymmetric in 4D flat space, in the no-scale approximation)? Such vacua are non-generic³ as the system of equations in overdetermined. But their abundance is closely related to the statistical likelihood (if you believe in such statistical games) of low-scale supersymmetry breaking. Again, DeWolfe et al give an estimate $$N_{W=0\,\text{vacua}}\sim \left(\frac{L^{1/2}}{H(\phi)}\right)^{-d} \left(\frac{L^{(1-d/4)}}{H(\phi)^{d/2}}\right)^{b_3}$$ of the number of $W_{\text{tree}}=0$ vacua at a point where the periods lie in a degree-$d$ extension of the rationals.