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Massless vector-particle case

We have discussed above the massive vector particle case. The Lagrangian of free massless vector field has got a gauge symmetry. So we need anyway to ensure a gauge symmetry for the interaction model to support a compatibility with the free field model. In this case the Feynman gauge leads to the propagator (2) with $m=0$ and to the appearance of Faddeev-Popov ghosts [Bjorken&Drell]. However a Goldstone ghost does not appear and the longitudinal polarization becomes unphysical like temporal one. Summation over physical polarization states can be replaced by that over an extended set of polarizations like (9). See the corresponding proof in [Cheng-1984].

In the framework of amplitude technique there is no reason to include the incoming and outgoing ghost partners of massless vector particle into consideration. Longitudinal polarization becomes unphysical, but the transversal polarization vectors may be chosen of the order of unity. On the contrary, in the case of squared diagram technique the extension of polarization states is very useful and has been used in numerous calculations.

If only physical polarization states are taken into account, then for the squared diagram evaluation we must convolute free Lorentz indices, which appear after evaluation of the left and right parts of squared diagram, with the projector on physical sub-space. In the massless case this projector equals [Cheng-1984]

$$g_{\mu\nu} - \frac{k_\mu \eta_\nu + k_\nu \eta_\mu}{(k.\eta)} \;,$$

where $\eta$ is an auxiliary vector with the zero Lorentz norm. Due to the gauge invariance the sum over all diagrams does not depend on this $\eta$, but each squared diagram contribution contains it. This leads to cumbersome expressions of squared diagram contributions in comparison with the case when the unphysical polarizations are included into the sum.