Revisiting the $k$-theorem with the ANEC

TL;DR

This paper provides a rigorous proof of the $k$-theorem in two-dimensional quantum field theories using the averaged null energy operator, extending the approach used for the $c$-theorem.

Contribution

The authors adapt the sum rule approach based on the ANE operator to establish the monotonic decrease of the current central charge, including the treatment of partial contact terms.

Findings

The proof confirms the monotonic decrease of the current central charge $k$ along RG flows.

Inclusion of partial contact terms is essential for a consistent sum rule.

The approach generalizes the method used for the $c$-theorem to the $k$-theorem.

Abstract

The fundamental theorem in renormalization group flows in two dimensions is the $c$-theorem, which dictates that the number of degrees of freedom must decrease monotonically along the renormalization group flow. The $k$-theorem claims that the number of charged degrees of freedom also decreases monotonically. Here, $k$ is the current central charge defined by the two-point function of the current. A recent derivation of the c-theorem by Hartman and Mathys, which uses the three-point function sum rule and the positivity of the averaged null energy (ANE) operator, motivates us to seek a similar proof of the k-theorem. In the case of the $k$-theorem, the partial contact terms need to be taken into consideration. While ignoring the partial contact terms yields contradictory results, our careful analysis incorporating them leads to the correct sum rule and a complete proof based on the positivity of the ANE operator.