Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

Abstract

Subdivergences constitute a major obstacle to the evaluation of Feynman integrals and an expression in terms of finite quantities can be a considerable advantage for both analytic and numeric calculations. We report on our implementation of the suggestion by F. Brown and D. Kreimer, who proposed to use a modified BPHZ scheme where all counterterms are single-scale integrals. Paired with parametric integration via hyperlogarithms, this method is particularly well suited for the computation of renormalization group functions and easily automated. As an application of this approach we compute the 6-loop beta function and anomalous dimensions of the φ 4 model. Loops and Legs in Quantum Field Theory

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

“…the two critical exponents η( ) = 2γ φ (g * ( )) and ν( ) = 2 + γ m 2 (g * ( )) −1 at the Wilson-Fisher fixed point to O( 6 ). We find complete agreement with the results reported in the Mathematica file accompanying [11]. The same holds true for the additional critical exponents α, β, γ and δ related to η and ν through the scaling relations γ = ν(2 − η), (4 − )ν = 2 − α, βδ = β + γ and α + 2β + γ = 2.…”

“…. , 5 we find complete agreement with the reported results in the Mathematica file accompanying [11]. This is a check of our formal manipulations.…”

“…, b i−1 only and does not receive corrections from higher orders. So for the scalar O(N) symmetric model where the beta function is known to six loops [11] we can calculate the first six coefficients g i , i = 0, . .…”

“…It does not receive corrections from higher orders and is exact to this order. Again for the scalar O(N) symmetric model using the six loop beta function, six loop field anomalous dimension γ φ and six loop mass anomalous dimension γ m 2 computed in [11] we can compute the correction to scaling ω( ) = β (g * ( ), ) and…”

“…The three and four loop calculation [2,3] as well as the five loop calculation [4][5][6][7][8][9] long stood as the state-of-the-art of higher loop calculations in d = 4 − dimensions. But recently the six loop calculations initiated in [10,11] were brought to the end in [12] for general N. In addition an impressive seven loop calculation in the special case of N = 1 has appeared [13]. These now constitute the highest loop order calculations of the renormalization group functions which include the beta function, the field anomalous dimension and the mass anomalous dimension.…”