Relating the CMSSM and SUGRA models with GUT scale and Super-GUT scale Supersymmetry Breaking

While the constrained minimal supersymmetric standard model (CMSSM) with universal gaugino masses, $m_{1/2}$, scalar masses, $m_0$, and A-terms, $A_0$, defined at some high energy scale (usually taken to be the GUT scale) is motivated by general features of supergravity models, it does not carry all of the constraints imposed by minimal supergravity (mSUGRA). In particular, the CMSSM does not impose a relation between the trilinear and bilinear soft supersymmetry breaking terms, $B_0 = A_0 - m_0$, nor does it impose the relation between the soft scalar masses and the gravitino mass, $m_0 = m_{3/2}$. As a consequence, $\tan \beta$ is computed given values of the other CMSSM input parameters. By considering a Giudice-Masiero (GM) extension to mSUGRA, one can introduce new parameters to the K\"ahler potential which are associated with the Higgs sector and recover many of the standard CMSSM predictions. However, depending on the value of $A_0$, one may have a gravitino or a neutralino dark matter candidate. We also consider the consequences of imposing the universality conditions above the GUT scale. This GM extension provides a natural UV completion for the CMSSM.