Revisiting the SUSY μ problem and its solutions in the LHC era

The supersymmetry (SUSY)-preserving mu parameter in SUSY theories is naively expected to be of order the Planck scale while phenomenology requires it to be of order the weak scale. This is the famous SUSY mu problem. Its solution involves two steps: first forbid mu, perhaps via some symmetry, and then regenerate it of order the scale of soft SUSY-breaking terms. However, present LHC limits suggest the soft breaking scale m(soft) at lies in the multi-TeV regime while naturalness requires mu similar to m(W,Z,h) similar to 100 GeV so that a little hierarchy (LH) appears with mu << m(soft). We review 20 previously devised solutions to the SUSY mu problem and reevaluate them in light of whether they are apt to support the LH. We organize the 20 solutions as follows: (1) solutions from supergravity/superstring constructions, (2) extended minimal supersymmetric Standard Model solutions, (3) solutions from an extra local U(1)' and (4) solutions involving Peccei-Quinn symmetry and axions. Early solutions invoked a global Peccei-Quinn symmetry to forbid the mu term while relating the mu solution to solving the strong CP problem via the axion. We discuss the gravity-safety issue pertaining to global symmetries and the movement instead toward local gauge symmetries or R symmetries, either continuous or discrete. At present, discrete R symmetries of order M (Z(M)(R)) which emerge as remnants of the Lorentz symmetry of compact dimensions seem favored. Even so, a wide variety of regenerative mechanisms are possible, some of which relate to other issues such as the strong CP problem or the generation of neutrino masses. We also discuss the issue of the experimental verification or falsifiability of various solutions to the mu problem. Almost all solutions seem able to accommodate the LH.