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RG Flow and Symmetry Breaking in a Weakly Coupled Model

We explore a weakly coupled $SU(N_{c})$ gauge theory, examining its fixed-point structure and the transition from infrared conformality to spontaneous symmetry breaking. Following a previous study, we couple the gauge field to $N_s$ scalars and $N_f$ fermions, take the large-$N_c$ limit with $N_s/N_c$ and $N_f/N_c$ fixed, and adjust these ratios to produce weakly coupled infrared fixed points while maintaining asymptotic freedom. We map out renormalization-group trajectories, and describe the transition to symmetry breaking as the ratio $N_s/N_c$ is increased. We examine the breaking by employing the one-loop effective potential. At tree level, a scalar-field vacuum expectation value (VEV) of a certain form is allowed, leading to masses for a subset of the scalar fields. Among the remaining massless scalars, one corresponds to a massless dilaton and the others are Nambu-Goldstone bosons absorbed by the gauge fields. The one-loop potential breaks the scale symmetry explicitly, determining the VEV and generating a dilaton mass suppressed relative to the other masses by a factor of the weak scalar coupling.