Abstract
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.
Highlights
Gauge theories exhibit different types of infrared (IR) behavior depending on the matter-field content
Symmetry breaking is signaled by the appearance of a nonzero vacuum expectation value for the scalar field
We have studied RG flow and the onset of symmetry breaking in a weakly coupled model
Summary
Gauge theories exhibit different types of infrared (IR) behavior depending on the matter-field content. A much studied example is an SUðNcÞ gauge theory with a set of Nf massless Dirac fermions in the fundamental representation. It can be illuminating to study these issues in a context where a weak-coupling expansion is possible An interesting such model with an SUðNcÞ gauge symmetry was examined in Ref. For the ratio xs ≡ Ns=Nc below a certain critical value xs, we confirm the existence of generic RG trajectories for which the couplings run to the origin in the UV and to weak nonzero fixed points in the IR. We examine the full effective potential for the dilaton degree of freedom, in which the scale symmetry is broken explicitly It dictates the presence of a nonzero VEV and gives rise to a mass for the dilaton.
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