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Explicit construction of the energy-momentum tensor in the large N limit

Carlo Pagani, Hidenori Sonoda

TL;DR

This paper constructs the energy-momentum tensor (EMT) for the O$(N)$ linear sigma model in the large $N$ limit using the exact renormalization group (ERG). The EMT is obtained as a cutoff-dependent functional that respects Ward identities for translations and rotations and evolvesaccording to a variant of the ERG equation; in the $\Lambda \to 0^+$ limit it becomes the 1PI effective action with a single EMT insertion. The authors verify the expected trace relation and show the trace vanishes at the Wilson–Fisher fixed point, confirming conformal invariance in this regime, while also elucidating the Gaussian and massive limits. They also develop the framework of cutoff-dependent composite operators to fix the EMT uniquely up to a controlled ambiguity, and provide explicit expressions for the EMT components, including $\alpha_\Lambda$ and $\beta_\Lambda(x-y)$, within the ERG formalism. Overall, the work demonstrates how the ERG can yield a concrete, symmetry-respecting EMT in a nonperturbative setting and lays groundwork for analyzing operator products and couplings to gravity in regulated theories.

Abstract

We construct the energy-momentum tensor of the O(N) linear sigma model explicitly in the large N limit using the exact renormalization group (ERG) formalism. The energy-momentum tensor is obtained as a cutoff dependent functional of N scalar field variables. Our guiding principles behind the construction are twofold: first the energy-momentum tensor must satisfy the Ward identity for translation and rotation invariance, and second the energy-momentum tensor must satisfy a variant of the exact renormalization group equation. In the limit that the momentum cutoff goes to zero, our energy-momentum tensor gives the one-particle irreducible (1PI) effective action with the insertion of a single energy-momentum tensor operator. We verify that the energy-momentum tensor constructed satisfies the expected trace formula, and that the trace vanishes at the Wilson-Fisher critical point.

Explicit construction of the energy-momentum tensor in the large N limit

TL;DR

This paper constructs the energy-momentum tensor (EMT) for the O linear sigma model in the large limit using the exact renormalization group (ERG). The EMT is obtained as a cutoff-dependent functional that respects Ward identities for translations and rotations and evolvesaccording to a variant of the ERG equation; in the limit it becomes the 1PI effective action with a single EMT insertion. The authors verify the expected trace relation and show the trace vanishes at the Wilson–Fisher fixed point, confirming conformal invariance in this regime, while also elucidating the Gaussian and massive limits. They also develop the framework of cutoff-dependent composite operators to fix the EMT uniquely up to a controlled ambiguity, and provide explicit expressions for the EMT components, including and , within the ERG formalism. Overall, the work demonstrates how the ERG can yield a concrete, symmetry-respecting EMT in a nonperturbative setting and lays groundwork for analyzing operator products and couplings to gravity in regulated theories.

Abstract

We construct the energy-momentum tensor of the O(N) linear sigma model explicitly in the large N limit using the exact renormalization group (ERG) formalism. The energy-momentum tensor is obtained as a cutoff dependent functional of N scalar field variables. Our guiding principles behind the construction are twofold: first the energy-momentum tensor must satisfy the Ward identity for translation and rotation invariance, and second the energy-momentum tensor must satisfy a variant of the exact renormalization group equation. In the limit that the momentum cutoff goes to zero, our energy-momentum tensor gives the one-particle irreducible (1PI) effective action with the insertion of a single energy-momentum tensor operator. We verify that the energy-momentum tensor constructed satisfies the expected trace formula, and that the trace vanishes at the Wilson-Fisher critical point.

Paper Structure

This paper contains 19 sections, 171 equations.

Table of Contents

  1. Introduction
  2. Review of the energy-momentum tensor and large $N$
  3. Quick review of ERG
  4. Quick review of the energy-momentum tensor and its Ward identity
  5. Quick review of the large $N$ limit
  6. Explicit construction of the energy-momentum tensor
  7. $\sigma$ term
  8. $\sigma^2$ term
  9. $\sigma^3$ term
  10. Cutoff dependent composite operators in large $N$
  11. $\sigma$ as a cutoff dependent composite operator
  12. A class of cutoff dependent composite operators
  13. An example
  14. Equation of motion operators
  15. Completion of the energy-momentum tensor
  16. ...and 4 more sections