Renormalizability of the gradient flow in the 2D $O(N)$ non-linear sigma model

Abstract

It is known that the gauge field and its composite operators evolved by the Yang--Mills gradient flow are ultraviolet (UV) finite without any multiplicative wave function renormalization. In this paper, we prove that the gradient flow in the 2D $O(N)$ non-linear sigma model possesses a similar property: The flowed $N$-vector field and its composite operators are UV finite without multiplicative wave function renormalization. Our proof in all orders of perturbation theory uses a $(2+1)$-dimensional field theoretical representation of the gradient flow, which possesses local gauge invariance without gauge field. As application of the UV finiteness of the gradient flow, we construct the energy--momentum tensor in the lattice formulation of the $O(N)$ non-linear sigma model that automatically restores the correct normalization and the conservation law in the continuum limit.

32 pages, 15 figures, the tittle has been changed, the final version to appear in PTEP

References in corpus (4)

• Perturbative analysis of the gradient flow in non-abelian gauge theories
• High-precision scale setting in lattice QCD
• The Yang-Mills gradient flow in finite volume
• Gradient Flow of O(N) nonlinear sigma model at large N

Cited by in corpus (11)

• Locally smeared operator product expansions in scalar field theory
• Gradient Flow of O(N) nonlinear sigma model at large N
• Gradient flow exact renormalization group
• Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy
• Asymptotically Free Theory with Scale Invariant Thermodynamics
• Fixed Point Structure of Gradient Flow Exact Renormalization Group for Scalar Field Theories
• Encoding field theories into gravities
• Gradient Flow: Perturbative and Non-Perturbative Renormalization
• Functional Renormalization Group Analysis of $O(3)$ Nonlinear Sigma Model and Non-Abelian Bosonization Duality
• Perturbative analysis of the Wess-Zumino flow
• Topology of the O(3) non-linear sigma model under the gradient flow