The dark side of the μ: on multiple solutions to renormalisation group equations, and why the CMSSM is not necessarily being ruled out

When solving renormalisation group equations in a quantum field theory, one often specifies the boundary conditions at multiple renormalisation scales, such as the weak and grand-unified scales in a theory beyond the standard model. A point in the parameter space of such a model is usually specified by the values of couplings at these boundaries of the renormalisation group flow, but there is no theorem guaranteeing that such a point has a unique solution to the associated differential equations, and so there may exist multiple, phenomenologically distinct solutions, all corresponding to the same point in parameter space. We show that this is indeed the case in the constrained minimal supersymmetric standard model (CMSSM), and we exhibit such solutions, which cannot be obtained using out-of-the-box computer programs in the public domain. Some of the multiple solutions we exhibit have CP-even lightest Higgs mass predictions between 124 and 126 GeV. Without an exhaustive 11-dimensional MSSM parameter scan per CMSSM parameter point to capture all of the multiple solutions, CMSSM phenomenological analyses are incomplete.