Properties of the є-expansion, Lagrange inversion and associahedra and the O (1) model

Thomas A. Ryttov*

*Corresponding author for this work

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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 languageEnglish
Article number72
JournalJournal of High Energy Physics (JHEP)
Volume2020
Issue number4
Number of pages16
ISSN1126-6708
DOIs
Publication statusPublished - 14. Apr 2020

Keywords

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

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