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 -g^{\mu \nu}+\frac{k^{\mu} k^{\nu}}{k^2}, i.e. the sum over the 3 physical polarizations of a massive on-shell vector boson with m = k^2.PolarizationSum[mu, nu] or PolarizationSum[mu, nu, k, 0] gives -g^{\mu \nu }. This corresponds to the summation over all 4 polarizations of a massless vector boson, 2 of which are unphysical if the particle is on-shell.PolarizationSum[mu, nu, k, n] yields -g^{\mu \nu}+\frac{k^{\mu }n^{\nu}+k^{\nu }n^{\mu }}{k \cdot n} - \frac{n^2 k^{\mu}k^{\nu}}{(k \cdot n)^2} which is the so-called axial-gauge polarization sum that picks up only the two physical polarizations of a massless vector boson. Here n is an auxiliary vector that must satisfy n \cdot k \neq 0. The physical results will not depend on n, yet in practice it is often convenient to identify n with one of the 4-vectors already present in the calculation. For example, in a final state with multiple gluons denoted by their momenta k_i, the vector n for the i-th gluon could be a k_j with j \neq i. 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 D-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.
PolarizationSum[\[Mu], \[Nu]]-\bar{g}^{\mu \nu }
PolarizationSum[\[Mu], \[Nu], k]\frac{\overline{k}^{\mu } \overline{k}^{\nu }}{\overline{k}^2}-\bar{g}^{\mu \nu }
PolarizationSum[\[Mu], \[Nu], k, Dimension -> D]\frac{k^{\mu } k^{\nu }}{k^2}-g^{\mu \nu }
FCClearScalarProducts[]; SP[k] = 0;
PolarizationSum[\[Mu], \[Nu], k, n]-\frac{\overline{n}^2 \overline{k}^{\mu } \overline{k}^{\nu }}{(\overline{k}\cdot \overline{n})^2}-\bar{g}^{\mu \nu }+\frac{\overline{k}^{\nu } \overline{n}^{\mu }}{\overline{k}\cdot \overline{n}}+\frac{\overline{k}^{\mu } \overline{n}^{\nu }}{\overline{k}\cdot \overline{n}}
FCClearScalarProducts[]
PolarizationSum[\[Mu], \[Nu], k, 0, Dimension -> D]
-g^{\mu \nu }
FCClearScalarProducts[]
PolarizationSum[\[Mu], \[Nu], k, 0, Dimension -> D, VirtualBoson -> True]-g^{\mu \nu }