Introducing my own group coefficients.

[daniilkrichevskiyDM]

Dear all,
I would be grateful if someone could help me with the following issue.

In FeynArts, I have created my model file for the Hidden Local Symmetry theory which describes pions and vector mesons. Rules for vertices go with some group coefficient, e.g. the interaction between 2 pions and vector meson has the following rule

C[ S[1, { iPi } ], S[1, { jPi } ], V[1, { A } ] ] == cLO* { { c3*fAik[A, iPi, jPi ] } }

fAik is just a placeholder for the structure coefficient (it is important for me to use my own group objects because I have many of them and, moreover I work with Sp, not SU theory). Say, I calculate 2 pion scattering which goes through an intermediate vector meson. Then the diagram must be summed over the the indices of the vector meson ohn the internal line. This is precisely what I get (see the screenshot. ) The sum should be performed over the dummy index Vec5.

The problem arises when I want to square the amplitude. When I take the conjugate, the dummy index is not renamed, and consequently when I square the amplitude, I get an answer where the group coefficient which should have different dummy indices have same instead (and consequently, I can not perform the sum correctly).

Thank you!
Daniil

[vsht]

Hi,

it's not quite straightforward to do, because introducing custom matrices with special properties is not something FeynCalc really supports out of the box.

You can declare your fAik as a tensor, but to have the dummy index renaming working it's easier to give it some index heads already familiar to FeynCalc, say LorentzIndex. Then the following should work

DeclareFCTensor[fAik]
ex = fAik[LorentzIndex[Vec5], LorentzIndex[i], LorentzIndex[l]] fAik[
   LorentzIndex[Vec5], LorentzIndex[j], LorentzIndex[k]]
ex ComplexConjugate[ex, Conjugate -> {fAik}]

But you will still need to write your own code for handling all contractions between those tensors or doing the sums.

Cheers,
Vlad

[daniilkrichevskiyDM]

Hi Vlad,

Thank you very much, this helped!

Best,
Daniil