Distribution of supersymmetry μ parameter and Peccei-Quinn scale fa from the landscape

A scan of soft supersymmetry (SUSY) breaking parameters within the string theory landscape with the minimal supersymmetric standard model assumed as the low energy effective field theory-using a power-law draw to large soft terms coupled with an anthropic selection of a derived weak scale to be within a factor of 4 of our measured value-predicts a peak probability of m(h) similar or equal to 125 GeV with sparticle masses typically beyond the reach of LHC Run 2. Such multiverse simulations usually assume a fixed value of the SUSY conserving superpotential mu parameter to be within the assumed anthropic range, mu less than or similar to 350 GeV. However, depending on the assumed solution to the SUSY mu problem, the expected mu term distribution can actually be derived. In this paper, we examine two solutions to the SUSY mu problem. The first is the gravity-safe-Peccei-Quinn model based on an assumed Z(24)(R) discrete R-symmetry which allows a gravity-safe accidental, approximate Peccei-Quinn global symmetry to emerge which also solves the strong CP problem. The second case is the Giudice-Masiero solution wherein the mu term effectively acts as a soft term and has a linear draw to large values. For the first case, we also present the expected landscape distribution for the Peccei-Quinn scale f(a); in this case, weak scale anthropics limits its range to the cosmological sweet zone of around f(a) similar to 10(11) GeV.