Dilaton in scalar QFT: A no-go theorem in 4 - ε and 3 - ε dimensions

Spontaneous scale invariance breaking and the associated Goldstone boson, the dilaton, is investigated in renormalizable, unitary, interacting non-supersymmetric scalar field theories in 4 - epsilon dimensions. At leading order it is possible to construct models which give rise to spontaneous scale invariance breaking classically and indeed a mass-less dilaton can be identified. Beyond leading order, in order to have no anomalous scale symmetry breaking in QFT, the models need to be defined at a Wilson-Fisher fixed point with exact conformal symmetry. It is shown that this requirement on the couplings is incompatible with having the type of flat direction which would be necessary for an exactly massless dilaton. As a result spontaneous scale symmetry breaking and an exactly massless dilaton can not occur in renormalizable, unitary 4 - epsilon dimensional scalar QFT. The arguments apply to phi(6) theory in 3 - epsilon dimensions as well.