### 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 an 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.

Original language | English |
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Article number | 033B08 |

Journal | Progress of Theoretical and Experimental Physics |

Volume | 2015 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 7 2014 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

**Renormalizability of the gradient flow in the 2D O(N) non-linear sigma model.** / Makino, Hiroki; Suzuki, Hiroshi.

Research output: Contribution to journal › Article

*Progress of Theoretical and Experimental Physics*, vol. 2015, no. 3, 033B08. https://doi.org/10.1093/ptep/ptv028

}

TY - JOUR

T1 - Renormalizability of the gradient flow in the 2D O(N) non-linear sigma model

AU - Makino, Hiroki

AU - Suzuki, Hiroshi

PY - 2014/11/7

Y1 - 2014/11/7

N2 - 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 an 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.

AB - 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 an 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.

UR - http://www.scopus.com/inward/record.url?scp=84928967177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928967177&partnerID=8YFLogxK

U2 - 10.1093/ptep/ptv028

DO - 10.1093/ptep/ptv028

M3 - Article

AN - SCOPUS:84928967177

VL - 2015

JO - Progress of Theoretical and Experimental Physics

JF - Progress of Theoretical and Experimental Physics

SN - 2050-3911

IS - 3

M1 - 033B08

ER -