Name: V. Shtabovenko Date: 02/13/19-04:29:07 PM Z

Dear FeynCalc users,

I’ve found finally some time to add the support for working with Majorana
spinors.

In principle, FermionSpinSum should be now able to handle products of
amplitudes that require spinor reordering automatically. For those who
prefer more fine-grained controls, there are two new functions
FCChargeConjugateTransposed and SpinorChainTranspose.

FCChargeConjugateTransposed applies C x^T C^(-1) to x, where x is
assumed to be an object living in the Dirac space. This operation is
required when transposing spinor chains so that one can effectively
switch u und v spinors

For example

FCChargeConjugateTransposed[GA[mu, nu, rho], Explicit -> True]

FCChargeConjugateTransposed[GA[5], Explicit -> True]

SpinorChainTranspose directly implements the transposition of particular
spinor chains in the given expression. The option Select allows to
specify which types of chains should be transposed by using patterns. E.g.

SpinorChainTranspose[
  SpinorVBarD[p1, m1].GAD[mu].(GSD[p] + m).GAD[mu].SpinorUD[p2, m2]]

SpinorChainTranspose[
  SpinorUBarD[p1, m1].GAD[mu].(GSD[p] + m).GAD[mu].SpinorVD[p2, m2]]

SpinorChainTranspose[
  SpinorUBarD[p1, m1].GAD[mu].(GSD[p] + m).GAD[mu].SpinorVD[p2, m2],
  Select -> ]

As was already mentioned above, FermionSpinSum can take care of the
spinor reordering without any extra help. This behavior can be turned
off by setting the new option SpinorChainTranspose to False. This can
also be used as a cross-check, since amplitudes without Majorana spinors
should not require this additional reordering.

amp = Spinor[Momentum[p1], m, 1].Spinor[-Momentum[p2], m, 1] +
   Spinor[Momentum[p2], m, 1].Spinor[-Momentum[p1], m, 1]

FermionSpinSum[amp ComplexConjugate[amp]]

FermionSpinSum[amp ComplexConjugate[amp],
  SpinorChainTranspose -> False]

In this sense, you have the choice between doing the reordering by hand
via SpinorChainTranspose before invoking FermionSpinSum or letting
FermionSpinSum do everything automatically.

I’ve also added one tree-level MSSM example to the dev branch, which
makes use of the new capabilities.

https://github.com/FeynCalc/feyncalc/tree/master/FeynCalc/Examples/MSSM/Tree

Cheers,
Vladyslav