Abstract
Correlators of unitary quantum field theories in Lorentzian signature obey certain analyticity and positivity properties. For interacting unitary CFTs in more than two dimensions, we show that these properties impose general constraints on families of minimal twist operators that appear in the OPEs of primary operators. In particular, we rederive and extend the convexity theorem which states that for the family of minimal twist operators with even spins appearing in the reflection-symmetric OPE of any scalar primary, twist must be a monotonically increasing convex function of the spin. Our argument is completely non-perturbative and it also applies to the OPE of nonidentical scalar primaries in unitary CFTs, constraining the twist of spinning operators appearing in the OPE. Finally, we argue that the same methods also impose constraints on the Regge behavior of certain CFT correlators.
Highlights
For interacting unitary CFTs in more than two dimensions, we show that these properties impose general constraints on families of minimal twist operators that appear in the OPEs of primary operators
We answer a version of this question for interacting unitary CFTs in d > 2 dimensions by constraining the family of minimal twist operators that appears in the OPE (1.1), independent of the rest of the theory
A concrete example of such a bound was found by Komargodski and Zhiboedov in [6] by extending the Nachtmann theorem about QCD sum rules of [7] to CFT. It was argued in [6] that for a CFT to be unitary in d > 2 dimensions, twist must be a monotonically increasing convex function of the spin for the family of minimal twist operators with even spins appearing in any reflection-symmetric OPE OO†, where O is a scalar primary
Summary
A QFT can be uniquely defined by its Euclidean correlators. Lorentzian correlators of any ordering can be obtained from Euclidean correlators by performing appropriate analytic continuations. The symbol x y represents that the point x is in the past lightcone of point y This analyticity condition is a covariant version of the standard i prescription that computes Lorentzian correlators from Euclidean correlators.. Where all xi ∈ L (or equivalently in R) and operators inside the correlator is ordered as written These analyticity and positivity conditions hold for any unitary Lorentzian QFT making them important tools that can be employed to derive very general results. These simple properties have far-reaching consequences for theories with conformal symmetry as we discuss
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