· FeynCalc

Name: Vladyslav Shtabovenko Date: 10/13/14-05:38:52 PM Z

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Previous message: L.X. Xu: “e+e- annihilation’s M squared(calculate with FeynArts and Feyncalc) differ by an overall minus sigh from Peskin\&Schroeder”
In reply to: L.X. Xu: “e+e- annihilation’s M squared(calculate with FeynArts and Feyncalc) differ by an overall minus sigh from Peskin\&Schroeder”

Hi,

first of all it would be nice to know which FeynCalc and Mathematica
version are you using?

I evaluated your code with the latest development snapshot
and obtained

-((2 e^4 me^4)/Pair[Momentum[k1], Momentum[p1]]^2) + (4 e^4 me^2)/
  Pair[Momentum[k1], Momentum[p1]] - (2 e^4 me^4)/
  Pair[Momentum[k2], Momentum[p1]]^2 + (4 e^4 me^2)/
  Pair[Momentum[k2], Momentum[p1]] - (4 e^4 me^4)/(
  Pair[Momentum[k1], Momentum[p1]] Pair[Momentum[k2],
    Momentum[p1]]) + (2 e^4 Pair[Momentum[k1], Momentum[p1]])/
  Pair[Momentum[k2], Momentum[p1]] + (
  2 e^4 Pair[Momentum[k2], Momentum[p1]])/
  Pair[Momentum[k1], Momentum[p1]]

which is precisely the negative of Eq. 5.105 in Peskin. The correct
matrix element squared is given by -1 times Eq. 5.105, as stated in the
sentence under this equation: “The overall minus sign is the result of
the crossing relation (5.68) and should be removed”. This way in my case
FeynCalc produces the correct result with your code.

Now coming to the last part of your question.

Personally, I prefer to evaluate FeynArts ampltiudes without
CreateFCAmp, but this is just a matter of taste. Furthermore, I
usually set $FAVerbose to 0, since otherwise FeynArts displays lots
of additional information that I’m not interested in.

For the evaluation of this amplitude it is rather convenient to
introduce Mandelstam variables (SetMandelstam). Note that here FeynCalc
assumes all momenta to be ingoing, which is why we need to insert the
photon momenta with the minus sign.

Since ComplexConjugate doesn’t rename the dummy indices (I’m thinking to
improve on that), you can use the new function (development snapshot
only) FCRenameDummyIndices that does exactly that. Then, instead of
replacing the polarization sums by the metric by hand, it is more
convenient to handle this by a replacement rule:

{Pair[LorentzIndex[x1_],Momentum[Polarization[y_,I]]]Pair[LorentzIndex[x2_],Momentum[Polarization[y_,-I]]]
:>-MT[x1,x2]}

This makes everything much more automatic ;)
Finally, since you want to have the result in terms of momenta and not
Mandelstam variables, it is necessary to reintroduce them. First you
need to clear the downvalues of ScalarProduct via ClearScalarProducts
and then specify the values of the on-shell momenta. After that one can
eliminate s in favor of t, u and ME and replace t and u by (p1-k1)^2 and
(p1-k2)^2.

Cheers,
Vladyslav

On 12/10/14 19:45, L.X.Xu wrote:
> hi,
> I am using feynarts and feyncalc to calculate the process: e+e-annihilation into a pair of photon. When I am doing the polarization sum of final state photon, I just replace the polarization vector by metric tensor,here is the mathematica code for this process:
>
> Quit[];
>
> $LoadPhi = True;
> $LoadFeynArts = True;
>
> $Configuration = “QED”;
> $Lagrangians = {“QED”[1], “QED”[2]};
>
> «HighEnergyPhysics`Feyncalc`
>
> SetOptions[FourVector, FeynCalcInternal -> False];
>
>
> tops = CreateTopologies[0, 2 -> 2];
> Paint[tops, AutoEdit -> False, ColumnsXRows -> {4, 1}];
>
> inserttops =
> InsertFields[tops, {F[2, {1}], -F[2, {1}]} -> {V[1], V[1]},
> InsertionLevel -> {Classes}, LastSelections -> {F[2, {1}]}];
> Paint[inserttops, AutoEdit -> False, ColumnsXRows -> {3, 1}];
>
> M20 = CreateFCAmp[inserttops] /. {ME -> me, EL -> e} // Total
> M21 = ComplexConjugate[M20] /. {\[Mu]1 -> m1, \[Mu]2 -> m2}
> M22 = M20*M21 // Expand
>
>
> M23 = M22 /.
> Pair[LorentzIndex[m1, D], Momentum[Polarization[p3, I], D]] Pair[
> LorentzIndex[m2, D], Momentum[Polarization[p4, I], D]] Pair[
> LorentzIndex[\[Mu]1, D], Momentum[Polarization[p3, -I], D]] Pair[
> LorentzIndex[\[Mu]2, D], Momentum[Polarization[p4, -I], D]] ->
> Pair[LorentzIndex[m1, D], LorentzIndex[\[Mu]1, D]] Pair[
> LorentzIndex[m2, D], LorentzIndex[\[Mu]2, D]]
>
> M24 = 1/4*FermionSpinSum[M23] // Contract
> M25 = M24 /. DiracTrace -> TR // Simplify
>
> M26 = M25 /. {Pair[Momentum[p2], Momentum[p2]] -> me^2,
> Pair[Momentum[p3], Momentum[p3]] -> 0,
> Pair[Momentum[p4], Momentum[p4]] -> 0,
> PropagatorDenominator[Momentum[p2, D] + Momentum[p3, D], me] ->
> 1/(2 Pair[Momentum[p2], Momentum[p3]]),
> PropagatorDenominator[Momentum[p2, D] + Momentum[p4, D], me] ->
> 1/(2 Pair[Momentum[p2], Momentum[p4]])}
>
> M27 = M26 /. {Pair[Momentum[p2], Momentum[p3]] ->
> Pair[Momentum[p1], Momentum[p4]],
> Pair[Momentum[p2], Momentum[p4]] ->
> Pair[Momentum[p1], Momentum[p3]],
> Pair[Momentum[p3], Momentum[p4]] ->
> Pair[Momentum[p1], Momentum[p2]] + me^2}
>
> M28 = M27 /.
> Pair[Momentum[p1],
> Momentum[p2]] -> -Pair[Momentum[p1], Momentum[p4]] -
> Pair[Momentum[p1], Momentum[p3]] - me^2 // Expand
>
> M29 = M28 /. {Pair[Momentum[p1],
> Momentum[p3]] -> -Pair[Momentum[p1], Momentum[k1]],
> Pair[Momentum[p1],
> Momentum[p4]] -> -Pair[Momentum[p1], Momentum[k2]]}
>
>
> I am wondering why the final result differ by an overall minus sign from Peskin and if there are any better way to perform the whole process???
>
> Thanks for Help!!!!!!!!!
>

text/x-objcsrc attachment: PairAnnih.m