  Unitary conformal field theories (CFTs) are believed to have positive
(non-negative) energy correlators. Energy correlators are universal observables
in higher-dimensional CFTs built out of integrated Wightman functions of the
stress-energy tensor. We analyze energy correlators in parity invariant
four-dimensional CFTs. The goal is to use the positivity of energy correlators
to further constrain unitary CFTs. It is known that the positivity of the
simplest one-point energy correlator implies that 1/3 <= a/c <= 31/18 where a
and c are the Weyl anomaly coefficients. We use the positivity of higher point
energy correlators to show that CFTs with extremal values of a/c have trivial
scattering observables. More precisely, for a/c=1/3 and a/c=31/18 all energy
correlators are fixed to be the ones of the free boson and the free vector
theory correspondingly. Similarly, we show that the positivity and finiteness
of energy correlators together imply that the three-point function of the
stress tensor in a CFT cannot be proportional to the one in the theory of free
boson, free fermion or free vector field.
