Reducing the fine-tuning of gauge-mediated SUSY breaking

Despite their appealing features, models with gauge-mediated supersymmetry breaking (GMSB) typically present a high degree of fine-tuning, due to the initial absence of the top trilinear scalar couplings, At=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t=0$$\end{document}. In this paper, we carefully evaluate such a tuning, showing that is worse than per mil in the minimal model. Then, we examine some existing proposals to generate At≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t\ne 0$$\end{document} term in this context. We find that, although the stops can be made lighter, usually the tuning does not improve (it may be even worse), with some exceptions, which involve the generation of At\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t$$\end{document} at one loop or tree level. We examine both possibilities and propose a conceptually simplified version of the latter; which is arguably the optimum GMSB setup (with minimal matter content), concerning the fine-tuning issue. The resulting fine-tuning is better than one per mil, still severe but similar to other minimal supersymmetric standard model constructions. We also explore the so-called “little At2/m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t^2/m^2$$\end{document} problem”, i.e. the fact that a large At\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t$$\end{document}-term is normally accompanied by a similar or larger sfermion mass, which typically implies an increase in the fine-tuning. Finally, we find the version of GMSB for which this ratio is optimized, which, nevertheless, does not minimize the fine-tuning.