Distribution of the Number of Generations in Flux Compactifications

Flux compactification of string theory generates an ensemble with a large number of vacua called the landscape. By using the statistics of various properties of low-energy effective theories in the string landscape, one can therefore hope to provide a scientific foundation to the notion of naturalness. This article discusses how to answer such questions of practical interest by using flux compactification of F-theory. It is found that the distribution is approximately in a factorized form given by the distribution of the choice of 7-brane gauge group, that of the number of generations $N_{\mathrm{g}\mathrm{e}\mathrm{n}}$ and that of effective coupling constants. The distribution of $N_{\mathrm{g}\mathrm{e}\mathrm{n}}$ is approximately Gaussian for the range $\mathrm{\mid }{N}_{\mathrm{g}\mathrm{e}\mathrm{n}}\mathrm{\mid }\lesssim 10$. The statistical cost of higher-rank gauge groups is also discussed.