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 \begin{equation} \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)).\tag{33} \end{equation} 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?
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The two point function of two operators of dimensions $\Delta_I$ and $\Delta_J$ is given by \begin{equation} <\mathcal{O}_{\Delta_I} (x_1)\bar{\mathcal{O}}_{\Delta_J}(x_2)>=\frac{c\delta_{IJ}}{|x_{12}|^{2\Delta_I}} \end{equation} 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)
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$\begingroup$ arxiv.org/pdf/1301.5092.pdf In this paper the authors calculate the 3 point functions involving 2 scalar operators and a conserved current and conserved energy momentum tensor using AdS/CFT. But they have not mentioned a ward identity with the energy momentum tensor. You can look up Frandkin and Palchik's paper mentioned in the references. $\endgroup$– dbranesCommented Feb 1, 2013 at 6:07