Very constrained minimal supersymmetric standard models

We consider very constrained versions of the minimal supersymmetric extension of the standard model (VCMSSMs) which, in addition to constraining the scalar masses $m_0$ and gaugino masses $m_{1/2}$ to be universal at some input scale, impose relations between the trilinear and bilinear soft supersymmetry-breaking parameters $A_0$ and $B_0$. These relations may be linear, as in simple minimal supergravity models, or nonlinear, as in the Giudice-Masiero mechanism for generating the Higgs-mixing $\mu$ term. We discuss the application of the electroweak vacuum conditions in VCMSSMs, which may be used to make a prediction for $\tan \beta$ as a function of $m_0$ and $m_{1/2}$ that is usually unique. We baseline the discussion of the parameter spaces allowed in VCMSSMs by updating the parameter space allowed in the CMSSM for fixed values of $\tan \beta$ with no relation between $A_0$ and $B_0$ assumed a priori, displaying contours of $B_0$ for a fixed input value of $A_0$, incorporating the latest CDF/D0 measurement of $m_t$ and the latest BNL measurement of $g_{\mu }-2$. We emphasize that phenomenological studies of the CMSSM are frequently not applicable to specific VCMSSMs, notably those based on minimal supergravity, which require $m_0=m_{3/2}$ as well as $A_0=B_0+m_0$. We then display $(m_{1/2},m_0)$ planes for selected VCMSSMs, treating in a unified way the parameter regions where either a neutralino or the gravitino is the lightest supersymmetric particle. In particular, we examine in detail the allowed parameter space for the Giudice-Masiero model.