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Revisiting the $k$-theorem with the ANEC

Nanami Nakamura, Yu Nakayama, Ung Nguyen

TL;DR

The paper tackles the monotonic decrease of charged degrees of freedom in two-dimensional quantum field theories by proving a $k$-theorem using the positivity of the averaged null energy (ANEC) and a three-point sum rule. A key advance is the careful inclusion of partial contact terms in the $TJJ$ correlator, which flips the sign of the naive sum-rule contribution and yields a correct, positive $ riangle k = k_ ext{UV}-k_ ext{IR}$. The authors validate the framework with explicit calculations for a free boson and a free Dirac fermion, both showing $ riangle k = 1$ under RG flow from UV to IR fixed points. They also discuss extensions to multiple currents, emergent symmetries, and potential higher-dimensional generalizations, highlighting the unifying role of ANEC in monotonicity theorems.

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.

Revisiting the $k$-theorem with the ANEC

TL;DR

The paper tackles the monotonic decrease of charged degrees of freedom in two-dimensional quantum field theories by proving a -theorem using the positivity of the averaged null energy (ANEC) and a three-point sum rule. A key advance is the careful inclusion of partial contact terms in the correlator, which flips the sign of the naive sum-rule contribution and yields a correct, positive . The authors validate the framework with explicit calculations for a free boson and a free Dirac fermion, both showing under RG flow from UV to IR fixed points. They also discuss extensions to multiple currents, emergent symmetries, and potential higher-dimensional generalizations, highlighting the unifying role of ANEC in monotonicity theorems.

Abstract

The fundamental theorem in renormalization group flows in two dimensions is the -theorem, which dictates that the number of degrees of freedom must decrease monotonically along the renormalization group flow. The -theorem claims that the number of charged degrees of freedom also decreases monotonically. Here, 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 -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.

Paper Structure

This paper contains 15 sections, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The review of $k$-theorem and generalization
  3. Statement and original proof
  4. Generalization to the case when the current is not conserved
  5. The free boson as an example
  6. The free fermion as an example
  7. Three-point sum rule and positivity
  8. An attempt at deriving the sum rule and a puzzle
  9. The correct sum rule and partial contact terms
  10. On the relation between the two-point and three-point sum rule
  11. A closer look at the partial contact terms
  12. The free boson
  13. The free fermion
  14. Summary and Discussion
  15. Detail of the fermion calculation

Figures (2)

  • Figure 1: $K$ for the free boson as a function $x=m|z|$
  • Figure 2: $K$ for the free fermion as function of $x = m|z|$