### Abstract

The three-loop β-function for the two-dimensional non-linear σ-model with a Wess- Zumino-Witten term is computed by using the covariant background field method. The β-function is proved to satisfy an integrability condition. The explicitly constructed action can be interpreted as the (α′)^{2} correction to the tree string effective action with the antisymmetric tensor field coupling included. The three-loop β-function is explicitly dependent, in general, on the two arbitrary constants, which can be fixed by a choice of the dimensional regularisation prescription for the Levi-Civita symbol, ε{lunate}^{μν}. When restricted to the purely metric case, the β-function satisfies the integrability condition, which leads to the curvature cubed correction to the bosonic string effective action previously computed by Metsaev and Tseytlin from string amplitude considerations. When restricted to the Wess-Zumino-Witten model defined on a group manifold, the perturbative β-function we find up to three loops is in agreement with the non-perturbative result of the conformal field theory for the derivative of the β-function at a fixed point. A complete classification of the three-loop divergent integrals with the results of the one- and two-loop subdivergence subtraction procedure are given.

Original language | English |
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Pages (from-to) | 447-498 |

Number of pages | 52 |

Journal | Nuclear Physics, Section B |

Volume | 332 |

Issue number | 2 |

DOIs | |

Publication status | Published - 5 Mar 1990 |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics, Section B*,

*332*(2), 447-498. https://doi.org/10.1016/0550-3213(90)90105-M