In a unitary CFT with a lowest-twist scalar primary, the scalar contributes to the lightcone OPE only at subleading order, and no positive averaged operator analogous to the ANEC exists for a scalar integral.
The averaged null energy condition for general quantum field theories in two dimensions
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abstract
It is shown that the averaged null energy condition is fulfilled for a dense, translationally invariant set of vector states in any local quantum field theory in two-dimensional Minkowski spacetime whenever the theory has a mass gap and possesses an energy-momentum tensor. The latter is assumed to be a Wightman field which is local relative to the observables, generates locally the translations, is divergence-free, and energetically bounded. Thus the averaged null energy condition can be deduced from completely generic, standard assumptions for general quantum field theory in two-dimensional flat spacetime.
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Light cone OPE in a CFT with lowest twist scalar primary
In a unitary CFT with a lowest-twist scalar primary, the scalar contributes to the lightcone OPE only at subleading order, and no positive averaged operator analogous to the ANEC exists for a scalar integral.