FeynCalc manual (development version)

PolarizationSum

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.

See also

Overview, Polarization, DoPolarizationSums, Uncontract.

Examples

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]

$$-g^{\mu \nu }$$