Define scalar product of polarisation vectors

[roberpsm]

Hello,

I'm trying to set the kinematics of a 2->2 vector boson scattering at amplitude level (no polarisation sum since i am not interested in the square amplitude).
I've defined the scalar products of external momenta via ScalarProduct[p1,p1]=mV^2, ScalarProduct[p1,p2]=(s-2mV^2)/2, etc and FC works fine.
My problem is in the definition of scalar products between the polarisation vectors (others than the corresponding ones to transversality), for example ScalarProduct[p1,PolarizationVector[p2]]. Maybe I am confusing with the syntax here, I was trying different ways but no one works...

Please see the attached file for a short example (one-loop ZZ->ZZ in the SM through a bubble of Higgs boson): the products momenta.momenta are well defined in the argument of the B0 functions but the products polarisation.polarisation are still there (no products momenta.polarisation are present in this example but I am interested on them too).

Hloop-ZZtoZZ.nb.zip

I'm using Mathematica 11.3.0, FeynCalc 9.3.1 and FeynArts 3.11.

Thank you in advance!

[magnunm]

Firstly you have forgotten that the 3 and 4 polarizations are outgoing and therefore complex conjugated. The scalar product declarations should have

ComplexConjugate[PolarizationVector[p4]]

and same for p3.

Using the complex conjugated polarization vectors I still could not get your example to work however and I am not sure why...
I am able to get it to work the way I think you intended by replacing the scalar product declarations like

ScalarProduct[PolarizationVector[p1], PolarizationVector[p2]] = product12;

... etc, by explicit rules for the underlying representation of the scalar products in question:

Pair[Momentum[Polarization[p1, I, Transversality -> True], D], Momentum[Polarization[p2, I, Transversality -> True], D]] =  product12;
Pair[Momentum[Polarization[p1, I, Transversality -> True], D], Momentum[Polarization[p3, -I, Transversality -> True], D]] = product13;

... etc (note the -I for complex conjugation on p3 and p4). This solution is "brute force" and not preferable but at least it should enable you to get the results you were after until you find a proper solution. 🙂

PS: this entire block:

ScalarProduct[p1, p1] = SMP["m_Z"]^2;
ScalarProduct[p2, p2] = SMP["m_Z"]^2;
ScalarProduct[p3, p3] = SMP["m_Z"]^2;
ScalarProduct[p4, p4] = SMP["m_Z"]^2;
ScalarProduct[p1, p2] = (ss - 2 SMP["m_Z"]^2)/2;
ScalarProduct[p3, p4] = (ss - 2 SMP["m_Z"]^2)/2;
ScalarProduct[p1, p3] = -(tt - 2 SMP["m_Z"]^2)/2;
ScalarProduct[p2, p4] = -(tt - 2 SMP["m_Z"]^2)/2;
ScalarProduct[p1, p4] = -(uu - 2 SMP["m_Z"]^2)/2;
ScalarProduct[p2, p3] = -(uu - 2 SMP["m_Z"]^2)/2;

can be replaced by

SetMandelstam[ss, tt, uu, p1, p2, -p3, -p4, SMP["m_Z"], SMP["m_Z"], SMP["m_Z"], SMP["m_Z"]];

[vsht]

With

ScalarProduct[Polarization[p1, I, Transversality -> True], 
   Polarization[p2, I, Transversality -> True]] = product12;
ScalarProduct[Polarization[p1, I, Transversality -> True], 
  Polarization[p3, -I, Transversality -> True]] = product13; 
ScalarProduct[Polarization[p1, I, Transversality -> True], 
  Polarization[p4, -I, Transversality -> True]] = product14; 
ScalarProduct[Polarization[p2, I, Transversality -> True], 
  Polarization[p3, -I, Transversality -> True]] = product23;
ScalarProduct[Polarization[p2, I, Transversality -> True], 
   Polarization[p4, -I, Transversality -> True]] = product24;
ScalarProduct[Polarization[p3, -I, Transversality -> True], 
   Polarization[p4, -I, Transversality -> True]] = product34;

I get

(product12*product34*B0[ss, SMP["m_H"]^2, SMP["m_H"]^2]*SMP["e"]^4)/(128*Pi^2*SMP["cos_W"]^4*SMP["sin_W"]^4) + (product13*product24*B0[tt, SMP["m_H"]^2, SMP["m_H"]^2]*SMP["e"]^4)/(128*Pi^2*SMP["cos_W"]^4*SMP["sin_W"]^4) + 
 (product14*product23*B0[uu, SMP["m_H"]^2, SMP["m_H"]^2]*SMP["e"]^4)/(128*Pi^2*SMP["cos_W"]^4*SMP["sin_W"]^4)

Is this the expected output?

[roberpsm]

Thanks a lot for the replies, Magnus and Valdyslav. You suggestions are very useful and solve my problem!