Physics Monkey said:
Thanks! This argument I understand, but how do we know that the potentials don't have compact flat directions like the mexican hat? And how much evidence do we have that the moduli potentials don't also depend on continuous parameters?
There are two sources of flat directions. First, a scalar field may have no potential at all, so it is not fixed. Second, we can have a compact flat direction when the potential depends only on the complex modulus |\phi_i| so that the phase does not appear in the potential. In a supersymmetric theory, we have superpotentials. These are holomorphic, so as long as a scalar field enters into the superpotential, so does its phase. As long as the F-flatness conditions can be solved, the phases will be fixed when we compute the roots of the superpotential. There won't be any flat directions.
I think the only caveat to the above argument is if a field only enters into the superpotential linearly, so that there is no mass term. In this case we cannot guarantee that the F-flatness conditions fix the value of the field.
So the challenge is simply to generate a superpotential that contains all moduli. For IIB complex structure moduli, this is rather simple. The presence of 3-form flux generates a term
W_G \sim \int_X G^{(3)}\wedge \Omega(z_i) ,
where the (3,0)-form \Omega(z_i) depends on all complex structure moduli. I think it's generic that mass terms are generated from this formula, since \Omega depends quadratically on the covariantly constant spinor, so should be at least quadratic in the z_i.
It is a bit more difficult to compute the superpotential for Kaehler moduli, since it is nonperturbative, but there are solid constructions such as
http://arxiv.org/abs/arXiv:1003.1982 that stabilize all Kaehler moduli.
Since susy is highly non-generic from the point of view of field theory, I personally find it unconvincing to invoke low scale SUSYsusy. Even supposing susy were required at very high scales for consistency or something (quite a claim already),
Well low scale SUSY is not something created by string theorists, but by phenomenologists that want to solve the hierarchy problem. Of course, SUSY makes computations much easier, but there is a strong motivation in the absence of direct evidence.
it seems to me that the vacua with susy breaking at a high scale will vastly outnumber the vacua with low scale susy breaking. I freely admit that I have no clean framework for making this statement, only the rough intuition that susy is highly non-generic, requiring the tuning of many relevant operators to zero.
I'm not that big on promoting the landscape, but from that perspective, the relative paucity of vacua with low-scale SUSY would be encouraging, if in fact low-scale SUSY is found in nature. It's hard to make other suggestions, since in the absence of a selection mechanism, we don't really know whether SUSY is preferred or not.
But if we accept that we'll generically be left with some strongly interacting non-susy gauge theory at high scales, well then I would imagine that computation of the masses will be next to impossible. Of course, if low energy susy is found then the story seems quite different as you say.
Yes, it's clear that SUSY, at the moment, is crucial to computations. This problem would likely face any theory that spit out an effective field theory a few orders of magnitude below the Planck scale.
. Low scale susy is much more natural in fluxless G2 compactifications of M-theory, where one can stabilize all moduli non-perturbatively and the large hierarchy of scales can be easily generated. The reason for the superpotential being purely non-perturbative is the PQ-type shift symmetry, inherited from the gauge symmetry of the 11-D supergravity 3-form, which all the complexified moduli possess. This symmetry automatically forbids any perturbative contributions to the superpotential but can be broken by non-perturbative effects, i.e. gaugino condensation or the membrane instantons. So, the scale of susy breaking is given by
. Guys, just look at the data.
