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LoopRefineSeries

LoopRefineSeries[f,{s,s0,n}]
generates a Taylor series expansion of f containing Passarino-Veltman functions about the point s=s0 to order (s-s0)n.

LoopRefineSeries[f,{s,s0,ns},{t,t0,nt},...]
successively finds Taylor series expansions with respect to s, then t, etc.

Details and OptionsDetails and Options

  • LoopRefineSeries generates an expansion effectively by computing the necessary derivatives of the Passarino-Veltman functions PVA, PVB, PVC, and PVD, and converting them using an internal version of LoopRefine.
  • LoopRefineSeries is unable to construct a series expansion near (Landau) singular points, e.g. near normal thresholds or near zero internal mass.
  • Set option AnalyticTrue to continue ϵ to large (and negative) values, rendering all Landau singularities sufficiently differentiable to permit a series expansion. »
  • LoopRefineSeries replaces spacetime dimension with 4-2ϵ and retains terms through order .
  • All logarithmic ultraviolet divergences and mass singularities are explicitly displayed as or poles. To interpret the output of LoopRefineSeries in terms of the canonical integration measure , (1) multiply the result by , and (2) interpret poles near as follows:
  • output:implied expression:
    1/epsilon-gamma_(E)+ln(4pi)
    1/(epsilon^2)+1/epsilon(-gamma_(E)+ln(4pi))+(gamma_(E)^2)/2-gamma_(E)ln(4pi)+1/2ln^2(4pi)
  • Besides reorganizing the result, LoopRefineSeries does not perform any substantial tensor algebraic manipulations.
  • LoopRefineSeries yields expressions valid for real (positive or negative) external invariants and positive semi-definite internal masses.
  • LoopRefineSeries has the same options as LoopRefine.
  • LoopRefineSeries generates a SeriesData object.