# Conventions for Levi Civita contraction

Hi again,

i am considering the process e+e - > t t~ / A and comparing the squared matrix element. FORM, FeynCalc and a calculation by hand give the same result and for a given phase space point i obtain

0.00418983, whereas MadGraph gives 0.00516665. I am able to reproduce the value from MadGraph if i change the signs in the terms generated by the gamma5 contributions coming from the vertices. So i get a relation of the form

eps_(\mu \sigma \nu \rho)*eps^(\mu \sigma \alpha \beta) = 2*g^(\alpha \rho)*g^(\beta \nu) - 2*g^(\alpha \nu)*g^(\beta \rho) with eps^(0123) = +1

which leads my result, but changing the overall sign on the right hand side gives the result from MadGraph. Therefore i am wondering what kind of conventions MadGraph is using which cause the difference.

Another question regarding the process e+ e- > t t ~ g / Z: the squared matrix elements are here the same, but i get a difference in the value for the cross section (my result 0.06272 pb, Madgraph 0.054 pb). I checked that i use the same values (especially no running for alphaS, no widths at all, same cuts (there is only one relevant cut for the gluon energy which i set to 20 GeV), cms energy 1 TeV). Do you have any idea what might cause this deviation?

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2019-08-17
2019-08-17
 Olivier Mattelaer (olivier-mattelaer) said on 2019-08-14: #1

Our definition is based on this relation:
4i epsilon() = Tr(GGGG Gamma5)

rewritten in the UFO convention:
4j* aloha_obj.Epsilon('mu','nu','rho', 'sigma') =
gamma('mu','i',1)*gamma('nu',1,2)*gamma('rho',2,3)*gamma('sigma',3,4)*gamma5(4,'i')

Note that we have
eps = Epsilon('mu','nu','rho', 'sigma')
self.assertEqual(eps.get_rep([0,1,2,3]), -1)

> Another question regarding the process e+ e- > t t ~ g / Z: the squared matrix elements are here the same, but i get a difference in the value for the cross section (my result 0.06272 pb, Madgraph 0.054 pb). I checked that i use the same values (especially no running for alphaS, no widths at all, same cuts (there is only one relevant cut for the gluon energy which i set to 20 GeV), cms energy 1 TeV). Do you have any idea what might cause this deviation?

My first guess here: energy is not a "good" variable to cut in general since it depend of the frame. MG5aMC will cut in the CMS frame. Do you do the same?

Cheers,

Olivier

> On 14 Aug 2019, at 19:37, Marcel <email address hidden> wrote:
>
> New question #682897 on MadGraph5_aMC@NLO:
>
> Hi again,
>
> i am considering the process e+e - > t t~ / A and comparing the squared matrix element. FORM, FeynCalc and a calculation by hand give the same result and for a given phase space point i obtain
>
> 0.00418983, whereas MadGraph gives 0.00516665. I am able to reproduce the value from MadGraph if i change the signs in the terms generated by the gamma5 contributions coming from the vertices. So i get a relation of the form
>
> eps_(\mu \sigma \nu \rho)*eps^(\mu \sigma \alpha \beta) = 2*g^(\alpha \rho)*g^(\beta \nu) - 2*g^(\alpha \nu)*g^(\beta \rho) with eps^(0123) = +1
>
> which leads my result, but changing the overall sign on the right hand side gives the result from MadGraph. Therefore i am wondering what kind of conventions MadGraph is using which cause the difference.
>
> Another question regarding the process e+ e- > t t ~ g / Z: the squared matrix elements are here the same, but i get a difference in the value for the cross section (my result 0.06272 pb, Madgraph 0.054 pb). I checked that i use the same values (especially no running for alphaS, no widths at all, same cuts (there is only one relevant cut for the gluon energy which i set to 20 GeV), cms energy 1 TeV). Do you have any idea what might cause this deviation?
>
> --

 Marcel (realmald) said on 2019-08-14: #2

The first relation is the same as in FeynCalc. The sign of eps^{0123} should not matter at the end of the calculation since the signs will ultimately cancel out. I am still not sure why the deviation appears.

Regarding the second question: I am working in the cms of the two inc. particles and cut the gluon energy in order to prevent singularities.

 Marcel (realmald) said on 2019-08-15: #3

Here is the matrix element squared which i calculated by hand and with FORM:

(1/(9 cw^4 (Mz^2 -
qs.qs)^2))gw^4 (4 mt^2 sw^2 (-3 + 16 sw^2 - 40 sw^4 + 32 sw^6) p1.p2 + (9 - 60 sw^2 + 148 sw^4 - 160 sw^6 + 128 sw^8) p1.p4 p2.p3 + 4 sw^4 (13 - 40 sw^2 + 32 sw^4) p1.p3 p2.p4)

With this matrix element i obtain the result above for a given phase space point.
I have also compared the result from MadGraph with RECOLA, both results fit perfectly. Another Levi-Civita contraction might lead to a different sign in the gamma5 induced terms which would effectively cause in the fact that i need to replace the momenta p3 <-> p4 OR p1 <-> p2 which would give the result from MadGraph and RECOLA.

 Marcel (realmald) said on 2019-08-17: #4

How does one obtain the couplings for this process? Consider for example the easier process Z -> e+ e-. MadGraph gives back the constants

GC_51 = (0.000000e+00,2.880442e-01)
GC_59 = (0.000000e+00,8.230988e-02)

but how do you get these values?

 Olivier Mattelaer (olivier-mattelaer) said on 2019-08-17: #5

You can type display couplings to see the full definition list

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________________________________
Sent: Saturday, August 17, 2019 4:17:25 PM
To: Olivier Mattelaer <email address hidden>
Subject: Re: [Question #682897]: Conventions for Levi Civita contraction

How does one obtain the couplings for this process? Consider for example
the easier process Z -> e+ e-. MadGraph gives back the constants

GC_51 = (0.000000e+00,2.880442e-01)
GC_59 = (0.000000e+00,8.230988e-02)

but how do you get these values?

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