How negative can null energy be in large N CFTs?

Smeared null energy has been shown to be bounded from below for free minimally coupled quantum field theories. This is not the case for conformally coupled free bosonic theories where states of unbounded null energy can be constructed by increasing the particle number. Little is known for interacting conformal field theories (CFTs) in dimensions larger than two. In this work we consider states that are superpositions of scalar primary operators or the stress-energy tensor itself in large N N CFTs. Within the large N N approximation we present arguments that the negative smeared null energy of such states scales at worst as the central charge of the theory, C_T CT . This provides evidence for a general bound for CFTs in d d -dimensions proportional to the central charge.