content_image/345926.jpg content_image/345928.jpg content_image/345927.jpg
where a$_{ig}$ and b$_{ij}$ are constants, m$_{g}$ is the gaugino mass, h$_{a}$ the Yukawa coupling, A is the A -term, and t = ln M$_{X}$/q . The full RG equations have been listed in [31, 32, 34]. 

However, let us consider some of the salient features of the MSSM one-loop RG equations. The one-loop RG equations for the three gaugino mass parameters are determined by: 

<img src='content_image/345926.jpg'>

where i = 1 , 2 , 3 correspond to U (1) , SU (2) , SU (3). An interesting property is that the three ratios m$_{i}$/g 2 i are RG scale independent. Therefore at the GUT scale, where the gauge couplings unify at M$_{GUT}$ ∼ 2 × 10 16 GeV, it is assumed that gauginos masses also unify with a value m$_{1}$$_{/}$$_{2}$ . Then at any scale: 

where g$_{GUT}$ is the unified gauge coupling at the GUT scale. The RG evolution due to Yukawa interactions are small except for top. The ones relevant to flat directions, involving the Higgs doublet H$_{u}$ which couples to the top quark, the right-handed stop ˜ u$_{3}$ , the left- handed doublet of third generation squarks ˜ Q$_{3}$ and the A − parameter A$_{t}$ associated with the top Yukawa interaction. The RG equations read [31, 32, 34] 

<img src='content_image/345928.jpg'>

<img src='content_image/345927.jpg'>

Here q denotes the logarithmic scale; this could be an external energy or momentum scale, but in the case at hand the relevant scale is set by the VEV(s) of the fields themselves. h$_{t}$ is the top Yukawa coupling, while g$_{i}$ and m$_{i}$ are respectively the gauge couplings and soft breaking gaugino masses of U (1)$_{Y}$ × SU (2) × SU (3) . If h$_{t}$ is the only large Yukawa coupling (i.e. as long as tan β is not very large), the beta functions for (mass) 2 of squarks of the first and second generations and sleptons only receive significant contributions from gauge/gaugino loops. A review of these effects can be found in [34, 403]. 

