content_image/374403.jpg content_image/374404.jpg content_image/374405.jpg
where m$_{0}$ , A$_{0}$ , and λ$_{d,}$$_{0}$ are the values of m$_{φ}$ , A and λ$_{n}$ given at a scale µ$_{0}$ . Here A$_{0}$ is chosen to be real and positive (this can always be done by re-parameterizing the phase of the complex scalar field φ ), and | K$_{i}$ | < 1 are coefficients determined by the one-loop renormalization group equations. We will provide an explicit example in section in V B 1. 

## 2. Spontaneous symmetry breaking and the physical degrees of freedom 

A flat direction VEV spontaneously breaks symmetry and gives masses to the gauge bosons/gauginos similar to the Higgs mechanism [122, 123, 325, 410, 411]. A crucial point is to identify the physical degrees of freedom and their mass spectrum in presence of a non- zero flat direction VEV. Let us consider the simplest flat direction, which includes only two fields: H$_{u}$H$_{d}$ . This is also familiar from the electroweak symmetry breaking in MSSM. A clear and detailed discussion is given in [32]. 

One can always rotate the field configuration to a basis where, up to an overall phase, H 1 u = H 2 d = 0 and H 2 u = H 1 d = φ$_{0}$$_{/}$√$_{2. Here superscripts denote the weak isospin compo-}$ nents of the Higgs doublets. In this basis the complex scalar field is defined by: 

<img src='content_image/374403.jpg'>

represents a flat direction. Its VEV breaks the SU (2)$_{W}$ × U (1)$_{Y}$ down to U (1)$_{em}$ (in exactly the same fashion as in the electroweak vacuum). The W ± and Z gauge bosons then obtain masses m$_{W}$, m$_{Z}$ ∼ gφ$_{0}$ from their couplings to the Higgs fields via covariant derivatives ( g denotes a general gauge coupling). There are also: 

<img src='content_image/374404.jpg'>

Then χ$_{2}$ and χ$_{1}$$_{, R}$ ( R and I denote the real and imaginary parts of a complex scalar field respectively) acquire masses equal to m$_{W}$ and m$_{Z}$ , respectively, through the D − term part of the scalar potential. Note that 

<img src='content_image/374405.jpg'>

and χ$_{1}$$_{, I}$ are the three Goldstone bosons, which are eaten by the gauge fields via the Higgs mechanism. Therefore, out of 8 real degrees of freedom in the two Higgs doublets, there are only two light physical fields: ϕ$_{R}$, ϕ$_{I}$ . They are exactly massless when SUSY is not broken (and there is no µ term either). 

