Abstract
We comment that the recent exact six-loop and seven-loop computations of renormalization group functions for the O(N)-symmetric four dimensional phi^4 quantum field theory show hints that the associated large order behavior is dominated by instantons rather than renormalons. This is consistent with a long-standing conjecture that renormalization group functions in the minimal subtraction (MS) renormalization scheme are not sensitive to renormalons.
Highlights
INTRODUCTIONIn quantum field theory (QFT) the two main sources of divergence of perturbation theory are identified as semiclassical “instantons” (more generally, “saddles”) [1,2] or Feynman diagrammatic “renormalons” [3,4]
In quantum field theory (QFT) the two main sources of divergence of perturbation theory are identified as semiclassical “instantons” [1,2] or Feynman diagrammatic “renormalons” [3,4]
We have used the recent high perturbative order exact results of [24,25] to probe the large-order growth of the coefficients of the perturbative expansion of the beta function βðg; NÞ and of the coefficients of the epsilon expansion of the correction to scaling exponent ωðε; NÞ for OðNÞ-symmetric scalar φ4 theory in four dimensions. We suggest that these perturbative results are already showing indications of the generic form of largeorder growth in (1)
Summary
In quantum field theory (QFT) the two main sources of divergence of perturbation theory are identified as semiclassical “instantons” (more generally, “saddles”) [1,2] or Feynman diagrammatic “renormalons” [3,4]. The divergence associated with renormalons is typically related to the momentum dependence of certain classes of iterated diagrammatic structures, such as bubble chains, for example, and is closely related to the renormalization group and the operator product expansion [7–11] These divergences of perturbation theory appear as singularities in the Borel plane of the corresponding Borel transform of the perturbative expansion of the quantity that is being computed. We ask the following simple question: Do the exact results of [24,25] contain enough perturbative data to be able to see hints of large-order growth and associated nonperturbative effects in φ4 theory in four dimensions and to distinguish between instanton or renormalon effects?. The combinations of coefficients on the lhs of (10), (11), and (12) should each tend to 1 at large perturbative order k
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