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Scale vs Conformal invariance from Entanglement Entropy

For a generic conformal field theory (CFT) in four dimensions, the scale anomaly dictates that the universal part of entanglement entropy across a sphere ($\mathcal{C}_{\text{univ}}(S^{2})$) is positive. Based on this fact, we explore the consequences of assuming positive sign for $\mathcal{C}_{\text{univ}}(S^{2})$ in a four dimensional scale invariant theory (SFT). In absence of a dimension two scalar operator $\mathcal{O}_{2}$ in the spectrum of a SFT, we show that this assumption suggests that SFT is a CFT. In presence of $\mathcal{O}_{2}$, we show that this assumption can fix the coefficient of the nonlinear coupling term $\int\hspace{-.5mm} d^{4}x\sqrt{g} R\mathcal{O}_{2}$ to a conformal value.