A stringy mechanism for a small cosmological constant — multi-moduli cases

Based on the properties of probability distributions of functions of random variables, we proposed earlier a simple stringy mechanism that prefers the meta-stable vacua with a small cosmological constant [Lambda]. As an illustration of this approach, we study in this paper particularly simple but non-trivial models of the Kahler uplift in the large volume flux compactification scenario in Type IIB string theory, where all parameters introduced in the model are treated either as fixed constants motivated by physics, or as random variables with some given uniform probability distributions. We determine the value w sub(0) of the superpotential W sub(0) at the supersymmetric minima, and find that the resulting probability distribution P(w sub(0)) peaks at w sub(0) = 0; furthermore, this peaking behavior strengthens as the number of complex structure moduli increases. The resulting probability distribution P([Lambda]) for metastable vacua also peaks as [Lambda] arrow right 0, for both positive and negative [Lambda]. This peaking/divergent behavior of P([Lambda]) strengthens as the number of moduli increases. In some scenarios for [Lambda] > 0, the likely value of [Lambda] decreases exponentially as the number of moduli increases. The light cosmological moduli issue accompanying a very small [Lambda] is also mentioned.