### Abstract

We discuss properties of the є-expansion in d = 4 − є dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in є in terms of the beta function coefficients. The є-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the є-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in є in terms of the coefficients of the beta function and anomalous dimensions. We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar O (1) symmetric model to O(є
^{7}).

Original language | English |
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Article number | 72 |

Journal | Journal of High Energy Physics (JHEP) |

Volume | 2020 |

Issue number | 4 |

Number of pages | 16 |

ISSN | 1126-6708 |

DOIs | |

Publication status | Published - 14. Apr 2020 |

### Keywords

- hep-th
- hep-ph
- math-ph
- math.MP
- Conformal Field Theory
- Renormalization Group