PolarizationSum[mu, nu, ... ]
represents the sum over a polarization vector and its complex conjugate with two free indices. Depending on its arguments the function returns different polarization sums for massive or massless vector bosons.
PolarizationSum[nu, nu, k]
returns , i.e. the sum over the 3 physical polarizations of a massive on-shell vector boson with .PolarizationSum[mu, nu]
or PolarizationSum[mu, nu, k, 0]
gives . This corresponds to the summation over all polarizations of a massless vector boson, of which are unphysical if the particle is on-shell.PolarizationSum[mu, nu, k, n]
yields which is the so-called axial-gauge polarization sum that picks up only the two physical polarizations of a massless vector boson. Here is an auxiliary vector that must satisfy . The physical results will not depend on , yet in practice it is often convenient to identify with one of the 4-vectors already present in the calculation. For example, in a final state with multiple gluons denoted by their momenta , the vector for the -th gluon could be a with . Notice that when using this polarization sum in a QCD calculation, one doesn’t have to consider diagrams with ghosts in the final states.To obtain a -dimensional polarization sum use the option Dimension
.
If you need to calculate a polarization sum depending on a 4-momentum that is not on-shell, use the option VirtualBoson
.
Overview, Polarization, DoPolarizationSums, Uncontract.
[\[Mu], \[Nu]] PolarizationSum
[\[Mu], \[Nu], k] PolarizationSum
[\[Mu], \[Nu], k, Dimension -> D] PolarizationSum
[]; SP[k] = 0;
FCClearScalarProducts
[\[Mu], \[Nu], k, n] PolarizationSum
[]
FCClearScalarProducts
[\[Mu], \[Nu], k, 0, Dimension -> D] PolarizationSum
[]
FCClearScalarProducts
[\[Mu], \[Nu], k, 0, Dimension -> D, VirtualBoson -> True] PolarizationSum