mega$ and the only solution for this condition is that the three-point energy correlator is identically zero. Analogously, we can proceed to higher point functions and conclude that all of them are zero at non-coincident points This is exactly what we get in the theory of free boson. Thus, we conclude that all energy correlators in a CFT with the three-point function being purely bosonic are identical to the ones in the theory of free boson.

In the case of free theories all energy correlators with more than two detectors are zero because there is no particle production in the theory. In an abstract CFT the lack of particle production could be defined in this way. The two-point energy correlator is given by the delta function in free theories because the state created by the stress tensor is a state of two freely propagating particles. In an abstract CFT the fact that the two-point energy correlator is given by the delta function is, hence, the analog of the S-matrix triviality.

Now we switch to the maximum value of ${a \over c}$ and assume that the three-point function of the stress tensor is given by the purely vector structure $n_v \neq 0$, $n_b = 0$, $n_f = 0$. The one-point energy correlator takes the form And we repeat exactly the same type of analysis as described above for the case of boson. Namely we choose $q = (q^0 , \vec 0)$ and $\hat n = (1,0,0,1)$. The condition that the one-point function  is zero becomes $\eps_{1 2}=0$, $\eps_{11} = \eps_{22}$.

The zero flux constraint   allows us to fix five out of seven functions We are left with two functional degrees of freedom given by $f_4$ and $f_6$. In addition to this there is another solution localized at $\xi = 1$. We can relax $\eps_{1 2}=0$, $\eps_{11} = \eps_{22}$ and impose the positivity of the energy correlator. The solution exists only when detectors are at the opposite points of the celestial sphere and the energy correlators are distributions and not functions.

The final result is that the energy correlators take the form Again the CFT wit