Renormalization Group Invariants and Sparticle Spectroscopy in the MSSM

We construct one-loop renormalization group (RG) invariant observables within flavor-blind minimal supersymmetric model (MSSM). The RG invariants turn out to be useful tools for sparticle spectroscopy in that they make otherwise-indirect relations among the model parameters manifest, they enable internal consistency checks of the model parameters via certain sum rules, and they directly probe the mechanism that breaks the supersymmetry. As a case study, we discuss the MSSM fine-tuning problem when the RG invariant formed by the gaugino, triscalar and Higgs bilinear soft masses vanish, and find that the sensitivity of the Z boson mass to parameters at deep ultraviolet is greatly reduced at large tan β. This method, supported by no symmetry principle at all, cannot offer a solution to the fine-tuning problem but it might be useful for constructing phenomenologically viable models. The supersymmetrization of the standard model of strong and electroweak interactions, the MSSM, provides an elegant solution to the gauge hierarchy problem in a genuinely perturbative way for all scales right up to the Planckian territory. However, supersymmetry is a blatantly broken symmetry of Nature as evidenced by the absence of superpartners mass-degenerate with known particles. Nevertheless, this breaking does not need to regenerate the hierarchy problem as long as it operates softly i.e. via dimension two and three operators in the lagrangian. The perturbative nature of the model allows one to relate measurements at the electroweak scale to physics at ultra high energies. This very hand-shaking of the infrared (IR) and ultraviolet (UV) regimes proceeds with the renormalization group (RG) flow of the lagrangian parameters. Indeed, various phenomena central to supersymmetry phenomenology i.e. gauge coupling unification[1], radiative electroweak breaking[2], and induction of flavor mixings[3] even for flavor-blind soft terms are pure renormalization effects. In fact, supression of the flavor-changing neutral currents as well as the absence of permanent electric dipole moments already imply that the soft-breaking masses cannot be all independent and arbitrarily distributed; they must be correlated by some organizing principle operating at the unification scale or above. The projection of the experimental data to ultra high energies requires solving the RGEs for all model parameters. This direct procedure, however, is disrupted by the coupled nature of the RG flows of the parameters unless the experiment is full. Besides, it would be profitable to predict certain parameters from a few measured ones with lasting consistency checks of the model as data accumulate. These disrupting aspects can be avoided by constructing RG-invariant observables at least at a given loop order. Indeed, such quantities prove highly useful not only for projecting the experimental data to high energies but also for deriving certain sum rules which enable fast consistency checks of the model. However, it should be kept in mind that, even the RG-invariant observables cannot be guaranteed to work perfectly because (i) the RG invariance holds at a given loop order and it is generically disrupted by higher loop effects, and (ii) the RGEs get modified at sparticle thresholds so that what equations must be used is not known a priori. The former is pronounced if the higher loop beta functions are large, and the latter becomes sizeable if the sparticle masses are significantly splitted. In what follows, for definiteness and simplicity, we will construct a set of RG-invariant observables for the MSSM (i) by using only the one–loop RGEs [7], and (ii) by neglecting the flavor structures of the Yukawa couplings and soft masses altogether. Our analysis is valid for energy scales above the heaviest superpartner mass, and the invariants we construct get disrupted by two-loop effects and non-trivial flavor mixings. We will briefly mention such disturbing effects. Construction of RG invariants is not new; there has already been some attempts at In general, scale and conformal invariances imply each other [4], and superconformal group involves both scale invariance and a continuous R symmetry with correlated charges [5]. Therefore, in softly broken supersymmetric theories, where the R invariance is explicitly broken, the RG invariance, if any, cannot be an all-order effect. Indeed, once the flavor mixings are switched on several one–loop RG invariants get automatically disrupted. We will therefore restrict our discussions to flavor-blind MSSM, and just note the possible connection between RG invariance and flavor violation since an approximately scale-invariant sector [4] imposes significant restrictions on the flavor structures of the soft masses [6].