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Incoming and outgoing ghosts

A solution of the above problem looks as a chess sacrifice. The main idea is to include incoming and outgoing unphysical states into the consideration. In the t'Hooft-Feynman gauge the masses of ghost partners equal the mass of parent vector particle. Let us consider ghost states as unphysical polarizations alike the temporal one (3). Note that the Faddeev-Popov ghost states and the temporal state have a negative norm, whereas the Goldstone state has a positive norm [Bjorken&Drell]. So the unphysical polarizations can give a positive as well as a negative contribution to the polarization sum.

The main statement is that a contribution of all unphysical polarizations to the sum of squared matrix element over polarizations equals zero [Baulieu-1985]. As a result

$$\sum_{i \in S_{phys}} A_i A_i^{*} = \sum_{i \in S_{all}} \sigma(i) A_i A_i^{*}\;,$$ (9)

where $i$ is a multi-index for polarization states; $A_i$ is an amplitude of the process; $S_{phys}$ is a set of physical polarization states; $S_{all}$ is a full set of polarizations including unphysical ones; $\sigma(i)=\pm 1$ depending on a signature of the Hilbert space norm of the polarization state $i$.

A drawback due to enlarging a set of polarization states is clear: we have much more matrix element terms for evaluation and subsequent summation. To see an advantage of this trick let us sum the temporal (3) and longitudinal (8) polarization contributions. Note that both of them have components of order of $k$, but for calculation of the corresponding sum we can alter the basis of polarization states

$$e^0_{\mu} e^0_{\nu} - e^3_{\mu} e^3_{\nu} = e'^0_{\mu} e'^0_{\nu} - e'^3_{\mu} e'^3_{\nu}\;,$$ (10)

and in such a way to have new basis vectors of order of unity in spite of possible large value of the momentum $k$

$$\begin{eqnarray*} e'^0=(1,0,0,0)\;;\\ e'^3=(0,0,0,1)\;. \end{eqnarray*}$$

In other words, the inclusion of the temporal polarization replaces the normalization condition (4) by

$$e^0_{\mu} e^0_{\nu}- e^1_{\mu} e^1_{\nu} - e^2_{\mu} e^2_{\nu} - e^3_{\mu} e^3_{\nu} = g_{\mu\nu}\;.$$ (11)

which does not lead to the fatal requirement on a value of polarization vectors. In the case of unpolarized calculation by the squared diagram technique we substitute $g_{\mu\nu}$ for the polarization sum according to (11). But now we have to add incoming and outgoing ghost states to the polarization sum in order to compensate the contribution of temporal polarization state.