Scale versus conformal invariance from entanglement entropy

For a generic conformal field theory (CFT) in four dimensions, the scale anomaly dictates that on conformally flat backgrounds, the universal part of entanglement entropy across a sphere [C-univ(S-2)] is positive. Based on this fact, we explore the consequences of assuming a positive sign for C-univ(S-2) on such backgrounds in a four-dimensional scale-invariant theory (SFT). In the absence of a dimension two scalar operator O-2 in the spectrum of a SFT, we show that this assumption suggests that SFT is a CFT. In the presence of O-2, we show that this assumption can fix the coefficient of the nonlinear coupling term integral d(4)x root gRO(2) to a conformal value.