c-theorem and improvement in non-compact conformal field theories

Energy-momentum tensor in general conformal field theories have improvement ambiguity and it can affect the argument in deriving $c$-theorem. While the derivation of Zamolodchikov's c-theorem is still formally valid with the improved energy-momentum tensor, the behavior of the c-function proposed by Zamolodchikov becomes different. When the theory suffers an IR divergence due to the non-compactness, the $c$-function can be unbounded and may even show non-monotonicity. We find that one of the three-point function sum rule proposed by Hartman and Mathys motivated by the averaged null energy condition, however, is agnostic about the improvement and gives the (effective) Virasoro central charge, when the IR theory is gapped.