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LoopRefine

LoopRefine[expr]
converts the Passarino-Veltman coefficient functions in expr to analytic expressions that can be numerically evaluated.

Details and OptionsDetails and Options

  • LoopRefine converts PVA, PVB, PVC, and PVD to analytic expressions in terms of elementary functions and special functions, and are valid for real (positive or negative) external invariants and positive semi-definite internal masses.
  • With the default setting Dimensions4, LoopRefine replaces spacetime dimension in expr with 4-2ϵ (default) and retains terms through order .
  • All logarithmic UV and IR divergences are explicitly displayed as or poles. An expression generated by LoopRefine that is free of ϵ is expressly free of any logarithmic UV and IR divergences.
  • To interpret the output of LoopRefine in terms of the canonical integration measure near 4 spacetime dimensions, (1) multiply the output by , and (2) interpret poles near as follows:
  • LoopRefine 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, LoopRefine does not perform any substantial tensor algebraic manipulations.
  • With the default setting AnalyticFalse, LoopRefine checks for possible power IR divergences.
  • The precise reduction method used by LoopRefine depends on the arguments of PVA, PVB, PVC, and PVD.
  • LoopRefine automatically threads over lists.
  • The following options can be given:
  • AnalyticFalsewhether to continue to large negative values
    Dimensions4value of n for which the limit n-2ϵ is taken.
    ExplicitC0Automaticthreshold of complexity for inserting explicit analytic expressions of the scalar function
    PartAllpart of the loop integral to compute
    TargetScaleAutomaticmass variable with which ratio of 't Hooft parameter μ is formed inside logarithm
    OrganizationLTensorhow the result should be organized
  • Possible values for ExplicitC0 are:
  • All functions are always given explicitly
    Automatic functions with known simple forms are given explicitly
    NoneIR-finite functions are always left symbolic
  • Possible values for Organization are:
  • LTensorOrganizes result by tensor structures
    FunctionOrganizes result by special functions such as Log and PolyLog
    NoneLeaves result unorganized
  • Possible values for Part are:
  • AllFull integral is calculated, including the finite parts and the logarithmic UV and IR singularities.
    UVDivergentLogarithmic UV divergent part is calculated, and only terms through order are retained.
    IRDivergentLogarithmic IR divergent part is calculated, and only terms through order are retained.
    Discontinuity[s]Discontinuity across normal threshold cut in channel is calculated.
    Discontinuity[s,t]Mandelstam double spectral function in overlapping channels and is calculated.