Is the magnitude of the Peccei–Quinn scale set by the landscape?

The value of the Higgs boson mass plus the lack of signal at LHC13 has led to a naturalness crisis for supersymmetric models. In contrast, rather general considerations of the string theory landscape imply a mild statistical draw towards large soft SUSY breaking terms tempered by the requirement of proper electroweak symmetry breaking where SUSY contributions to the weak scale are not too far from $$m_{weak}\sim 100$$mweak∼100 GeV. Such a picture leads to the prediction that $$m_h\simeq 125$$mh≃125 GeV while most sparticles are beyond current LHC reach. Here we explore the possibility that the magnitude of the Peccei–Quinn (PQ) scale $$f_a$$fa is also set by string landscape considerations within the framework of a compelling SUSY axion model. First, we examine the case where the PQ symmetry arises as an accidental approximate global symmetry from a more fundamental gravity-safe $$\mathbb {Z}_{24}^R$$Z24R symmetry and where the SUSY $$\mu $$μ parameter arises from a Kim-Nilles operator. The pull towards large soft terms then also pulls the PQ scale as large as possible. Unless this is tempered by rather severe (unknown) cosmological or anthropic bounds on the density of dark matter, then we would expect a far greater abundance of dark matter than is observed. This conclusion cannot be negated by adopting a tiny axion misalignment angle $$\theta _i$$θi because WIMPs are also overproduced at large $$f_a$$fa. Hence, we conclude that setting the PQ scale via anthropics is highly unlikely. Instead, requiring soft SUSY breaking terms of order the gravity-mediation scale $$m_{3/2}\sim 10$$m3/2∼10–100 TeV places the mixed axion–neutralino dark matter abundance into the intermediate scale sweet zone where $$f_a\sim 10^{11}$$fa∼1011–$$10^{12}$$1012 GeV. We compare our analysis to the more general case of a generic SUSY DFSZ axion model with uniform selection on $$\theta _i$$θi but leading to the measured dark matter abundance: this approach leads to a preference for $$f_a\sim 10^{12}$$fa∼1012 GeV.