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On the dynamical generation of the Maxwell term and scale invariance

Gauge theories with no Maxwell term are investigated in various setups. The dynamical generation of the Maxwell term is correlated to the scale invariance properties of the system. This is discussed mainly in the cases where the gauge coupling carries dimensions. The term is generated when the theory contains a scale explicitly, when it is asymptotically free and in particular also when the scale invariance is spontaneously broken. The terms are not generated when the scale invariance is maintained. Examples studied include the large $N$ limit of the $CP^{N-1}$ model in $(2+ε)$ dimensions, a 3D gauged $ϕ^6$ vector model and its supersymmetric extension. In the latter case the generation of the Maxwell term at a fixed point is explored. The phase structure of the $d=3$ case is investigated in the presence of a Chern-Simons term as well. In the supersymmetric $ϕ^6$ model the emergence of the Maxwell term is accompanied by the dynamical generation of the Chern-Simons term and its multiplet and dynamical breaking of the parity symmetry. In some of the phases long range forces emerge which may result in logarithmic confinement. These include a dilaton exchange which plays a role also in the case when the theory has no gauge symmetry. Gauged Lagrangian realizations of the 2D coset models do not lead to emergent Maxwell terms. We discuss a case where the gauge symmetry is anomalous.