Evaluation of the D0 function

$LoadPhi = True ; $LoadFeynArts = True ;

Needs["HighEnergyPhysics`FeynCalc`"] ;

FeynCalc 4.2.0
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FeynArts 3.1

by Hagen Eck, Sepp Kueblbeck, and Thomas Hahn

last revised 12 Feb 03

patched for use with FeynCalc by Frederik Orellana

PHI comes with the function VeltmanD0. The implementation of VeltmanD0 is only a first attempt at implementing the formulae from 't Hooft and Veltman directly in Mathematica. IT SEEMS NOT TO WORK, so it is recomended to use LoopTools for numerical evaluations of D0.

This is the definition of i π^2D _ 0 of Passarino and Veltman (equation 6.1):

FeynAmpDenominator[PropagatorDenominator[Momentum[q1, SpaceDimensions], ParticleMass[1]], PropagatorDenominator[Momentum[q1, SpaceDimensions] - Momentum[p1, SpaceDimensions], ParticleMass[2]], PropagatorDenominator[Momentum[q1, SpaceDimensions] - Momentum[p1, SpaceDimensions] - Momentum[p2, SpaceDimensions], ParticleMass[3]], PropagatorDenominator[Momentum[q1, SpaceDimensions] - Momentum[p1, SpaceDimensions] - Momentum[p2, SpaceDimensions] - Momentum[p3, SpaceDimensions], ParticleMass[4]]]

1/(q _ 1^2 - (m _ 1^(ó  ))^2) . ((q _ 1 - p _ 1)^2 - (m _ 2^(ó  ))^2) . ((-p _ 1 - p _ 2 + q _ 1)^2 - (m _ 3^(ó  ))^2) . ((-p _ 1 - p _ 2 - p _ 3 + q _ 1)^2 - (m _ 4^(ó  ))^2)

FeynCalc reduces the above to D0 with exactly the same arguments as FF (LoopTools) expects:

oon = OneLoop[q1, %]

i π^2 D _ 0 ( p _ 1^2 , p _ 2^2 , p _ 3^2 , p _ 1^2 + 2 p _ 1 · p _ 2 + 2 p _ 1 · p _ 3 + p _ 2^2 + 2 p _ 2 · p _ 3 + p _ 3^2 , p _ 1^2 + 2 p _ 1 · p _ 2 + p _ 2^2 , p _ 2^2 + 2 p _ 2 · p _ 3 + p _ 3^2 , (m _ 1^(ó  ))^2 , (m _ 2^(ó  ))^2 , (m _ 3^(ó  ))^2 , (m _ 4^(ó  ))^2 )

Evaluation with FF (LoopTools):

looptools = Install["LoopTools"] ;

D0[600, 40, 50, 60, 70, 80, 40, 40, 50, 60]

-0.00005734960395441325` + 0.0001258450679504652` i

Uninstall[looptools] ;

The smartest evaluation is using the three-point funcion:

VeltmanD0[600, 40, 50, 60, 70, 80, 40, 40, 50, 60, D0Evaluation -> "C0", SmallEpsilon -> 0.01] // N

-0.00022865195660159838` + 1.12823669711954`*^-20 i

Converted by Mathematica  (July 10, 2003)