# Using FeynArts/FormCalc for single diagram evaluation

Hi folks,

I was wondering if someone knows enough about the FormCalc package to help me out: I am using FeynArts to generate Feynman-diagrams for certain tree-level processes and FormCalc to calculate the squared amplitude of these processes (http://en.misho-web.com/phys/feynlecture.html gives some neat examples on how this is done). Now this works fine if one considers all tree-level-diagrams, i.e. the complete process at tree level, but fails when only evaluating one of several diagrams (e.g. just calculating the squared 4-vertex-diagram of the gluon+gluon -> gluon+gluon process, instead of all squared diagrams and interference terms).
My first naive idea was that FormCalc might use identities which are correct only for physical processes (i.e. all diagrams to a certain order considered) and not on the single diagram level; but I don't know if such identities exist and furthermore I didn't find anything in the code (which is to advanced for me anyway).
If someone has an idea any hint would be appreciated.

Edit: Further testing showed that the problem seems to be related just to vectorbosons in initial or final states. Because of this I was playing aroung with changing the polarizationsum (in the PolarizationSum.frm) to -d_([mu],[nu]). However even in this configuration a single squared diagram including vectorbosons does not give the result one calculates via Feyman-rules per hand using this simplified polarizationsum. So the problem isn't just the introduction of gauge-dofs in single diagrams which cancel for the whole process.

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## Answers and Replies

Hepth
Gold Member
What do you mean by fails? Can you attach a test notebook? I use feyncalc and feynarts but not formcalc much. I have the code though to look at it. It DOES have massive vector spin sums built in right?

I'll take a look though.

Hi,

I attached a notebook with my calculation. So what I wanted to do there is calculate only the 4-vertex squared, using the sum over all gaugefield-polarizations equal to -gαβ instead of -gαβ+(some function involving a vector η perpendicular to physical polarizations). So I used the option "GaugeTerms->False" in the PolarizationSum call. But this does not give me the result I (and others) calculated when calculating the squared diagram with this reduced 'completeness relation' by hand; it's off by a factor of 6, and that's what I mean by "fails".
I tested that implementing the reduced 'completeness relation' in the form code in the file PolarizationSum.frm gives (for this calculation) the same result as using the option "GaugeTerms->False"; so it should give the result calculated by hand.

Now I am confused and curious as to what does FormCalc actually do for a single diagram involving initial or final gauge fields; the endresult for the whole process (without excluding internal lines) is exactly what you find in the literature. In addition, there seems to be no problem when calculating single diagrams without gaugefields in initial or final states, so this has to be related to the treatment of gaugefields.

On a sidenote: Curiously, using "GaugeTerms->False" does in general not reproduce the result with reduced 'completeness relation' in the PolarizationSum.frm file; for example in the s-channel diagram squared of the gg->gg process.

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Hepth
Gold Member
Hmm, doing it real fast by hand in feyncalc I get if I just sum over polarizations and indices, the 7776 g^4 you mention, BUT we're not SUMMING over all polarizations. When you have 2 incoming vector bosons you have to AVERAGE over initial polarizations, SUM over final, right? So there should be a factor of 1/3? for each of the incoming, correct? Or it might be more complicated than just merely (1/3)^2. I don't know if there is some argument to only taking the longitudinal/transverse part of each incoming polarization or whatnot.

Code:
AMP = (-I g^2) (SUNF[a, b, e].SUNF[c, d,
e].(MT[\[Alpha], \[Gamma]].MT[\[Beta], \[Delta]] -
MT[\[Alpha], \[Delta]].MT[\[Beta], \[Gamma]]) +
SUNF[a, c, e].SUNF[b, d,
e].(MT[\[Alpha], \[Beta]].MT[\[Gamma], \[Delta]] -
MT[\[Alpha], \[Delta]].MT[\[Gamma], \[Beta]]) +
SUNF[a, d, e].SUNF[b, c,
e].(MT[\[Alpha], \[Beta]].MT[\[Delta], \[Gamma]] -
MT[\[Alpha], \[Gamma]].MT[\[Delta], \[Beta]])) // Calc;
AMPCC = Calc[
Calc[AMP] /. {a -> aa, b -> bb, c -> cc, d -> dd,
e -> ee, \[Alpha] -> \[Alpha]2, \[Beta] -> \[Beta]2, \[Delta] -> \
\[Delta]2, \[Gamma] -> \[Gamma]2}];
ASQ = SUNSimplify[(1/3)^2 Calc[
AMP.AMPCC.(MT[\[Alpha], \[Alpha]2].MT[\[Beta], \[Beta]2].MT[\
\[Delta], \[Delta]2].MT[\[Gamma], \[Gamma]2].SUNDelta[a, aa].SUNDelta[
b, bb].SUNDelta[c, cc].SUNDelta[d, dd])],
Explicit -> True] /. {CA -> 3, CF -> 4/3}
gives -864 g^4

Hi,

Thanks for looking into this. But I don't think it's an normalization/averaging problem.
Of course you're right: if you want to compute the averaged amplitude you have to sum over final and average over initial states (for vector bosons this means, dividing by a factor of (Nc^2-1)*2 for every initial boson, i.e. the number of gluons times the number of physical polarizations). But I didn't do that and FormCalc doesn't do this either for PolarizationSum and Bosons. Taking your calculation (and correcting the sign error taking the complex conjugate of I) without the averaging gives exactly 7776 g^4, the result FormCalc should give, but doesn't. I am fairly sure that this is the right result because taking all other diagrams for this process I calculated the invariant amplitude you'd find in the literature.