# How does the conformal Ward identity guarantee a vanishing 3-point function in this case?

I was looking through some conformal Ward identity related things when I noticed that this paper (arXiv:1212.3788) writes in their equation (33), a 3-point function between a conserved current and two scalars, complex conjugate of each other and having the same conformal dimensions. The equation reads $$\partial_\mu\langle j^\mu\mathcal{O}(y_1)\bar{\mathcal{O}}(y_2)\rangle=-iq\langle \mathcal{O}(y_1)\bar{\mathcal{O}}(y_2)\rangle(\delta(x-y_1)-\delta(x-y_2))$$ They also comment on the next page that this equation shows, Ward identity guarantees a vanishing three point function for scalar operators of unequal conformal dimensions. How to see that? Also, is this a general outcome of Ward identity that the operators inside (in such cases; may be instead of current, some other conserved tensor) the operators have to have equal conformal dimensions to have a non-vanishing correlator? A reference will be excellent. Thanks in advance!

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The two point function of two operators of dimensions $\Delta_I$ and $\Delta_J$ is given by $$<\mathcal{O}_{\Delta_I} (x_1)\bar{\mathcal{O}}_{\Delta_J}(x_2)>=\frac{c\delta_{IJ}}{|x_{12}|^{2\Delta_I}}$$ which means the right hand side of your equation vanishes if the dimensions are not equal. The derivation of this Schwinger Dyson equation associated with conserved currents can be found in Peskin and Schroeder's field theory book (Equation 9.97)