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 $m_h\simeq 125\mathrm{GeV}$ 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, $\mu \lesssim 350\mathrm{GeV}$. 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 ${\mathbb{Z}}_{24}^R$ 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 $f_a\sim {10}^{11}\mathrm{GeV}$.

• Received 15 April 2021
• Accepted 10 July 2021

DOI:https://doi.org/10.1103/PhysRevD.104.015037

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Published by the American Physical Society