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.

arXiv comment32 pages, 15 figures, the tittle has been changed, the final version to appear in PTEP

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Cited by in corpus (23)

• Locally smeared operator product expansions in scalar field theory
• Gradient Flow of O(N) nonlinear sigma model at large N
• Proof of the renormalizability of the gradient flow
• Derivation of a Gradient Flow from ERG
• Determination of Reference Scales for Wilson Gauge Action from Yang--Mills Gradient Flow
• Confinement-Deconfinement Crossover in the Lattice $\mathbb{C}{P}^{N-1}$ Model
• Background field method in the gradient flow
• Gradient flow exact renormalization group
• 4D $\mathcal{N}=1$ SYM supercurrent in terms of the gradient flow
• Geometries from field theories
• Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy
• The Yang-Mills gradient flow and lattice effective action
• Asymptotically Free Theory with Scale Invariant Thermodynamics
• Fixed Point Structure of Gradient Flow Exact Renormalization Group for Scalar Field Theories
• Supersymmetric gradient flow in the Wess-Zumino model
• Topological Susceptibility of the 2d O(3) Model under Gradient Flow
• Encoding field theories into gravities
• Gradient Flow: Perturbative and Non-Perturbative Renormalization
• Axial $U(1)$ anomaly in a gravitational field via the gradient flow
• Perturbative analysis of the Wess-Zumino flow
• Functional Renormalization Group Analysis of $O(3)$ Nonlinear Sigma Model and Non-Abelian Bosonization Duality
• Topology of the O(3) non-linear sigma model under the gradient flow
• Flow equation for the large $N$ scalar model and induced geometries