Addendum-erratum to: “Nonasymptotic critical behavior from field theory at d = 3. II. The ordered-phase case”

This note is intended to emphasize the existence of estimated Feynman integrals in three dimensions for the free energy of the O(1) scalar theory up to five loops which may be useful for other work. We also correct some misprints of the published paper. We present the corrected table I of the original paper [1] which displays the values of the Feynman integrals of the O(1) scalar theory contributing to the free energy up to five loops and their symmetry factors. The values differ from those Feynman integrals previously calculated in [2] 1 due to the necessity of introducing a “soft ” mass parameter instead of the usual renormalized (at zero-momentum) mass (see the text of the original paper [1] and [3] for details). Consequently, many of the estimates of Feynman Integrals presented in table I have been extracted from [2] by accounting for a harmless 3-d renormalization in order to get a soft-mass parameter (characterizing a minimal subtraction scheme similar to that introduced in [4]). Since the five-loop contributions to the free energy involves ϕ 3 vertices mixed to ϕ 4 vertices, Feynman integrals which are different or cannot be obtained from those previously considered in [2] have been estimated in three dimensions for the occasion 2. These are: • the four-loop integrals with Heap’s numbers 13–17 (column h in the following table I, see table IV of [5]). They involve only ϕ 3 vertices. They have been estimated and successfully compared to similar calculations extracted from [6]. • the five-loop integrals with Heap’s numbers 80–102 (see table V of [5]). They involve ϕ 3 vertices mixed with a single ϕ 4 vertex. They have been calculated exclusively for this work. • the five-loop integrals with Heap’s numbers 103–118 (see table V of [5]). They involve only ϕ 3 vertices. They have been calculated for this work and successfully compared to similar calculations extracted from [7]. 1 The unpublished Guelph report [2] may be obtained via H. Kleinert and V. Schulte-Frohlinde web site at

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