
\subsection{The problem}

Every time  when we are  trying to create a model containing a massive 
vector particle we meet a problem caused by  
bad asymptotics of its propagator

\begin{equation}
   \frac{i}{  (2\pi)^4}\,  \frac{g_{\mu\nu} -k_\mu k_\nu/m^2}{m^2-k^2}\;.
\label{mass_vect_phys_prop}
\end{equation}

The $( g_{\mu\nu} - k_\mu k_\nu/m^2  )$ factor gives a projection on
 physical degrees of freedom in the polarization space. This term appears
because a 4-component vector
field is used to describe a particle with 3 degrees of freedom.  The $k_\mu
k_\nu/m^2$ term  leads to a  fast  growth of  amplitudes
at high energies, what breaks unitarity and  is not compatible with 
the renormalizability of  theory. 
 

The problem mentioned above is  solved 
in the framework of  gauge field theories where the  gauge symmetry is 
responsible for  mutual cancellation of rapidly growing  contributions
of separate diagrams \cite{BD}.  
 Let  our model of particle interaction be based on a gauge
theory.  Then on a step of 
numerical evaluation  we  expect a  cancellation  of
contributions which come from various Feynman diagrams. Consequently,
finite precision numerical calculations may lead to wrong results.
   Accompanying problem
is a  cumbersome  expression for  each diagram as a result 
of appearance of mutually canceling  terms. 

At the same time  there is a freedom in formulation of  Feynman rules
for gauge theories caused by an  ambiguity of  gauge fixing terms \cite{BD}. 
 These terms   modify the quadratic 
part of the Lagrangian and consequently may improve the vector 
particle propagator.  	Indeed, in the case of t'Hooft-Feynman  gauge the 
propagator  of vector particle takes the form
\begin{equation}
   \frac{i}{ (2\pi)^4}  \frac{g_{\mu\nu}}{m^2 - k^2}\;.
\label{hf_ propagator}
\end{equation}
that  provides a formulation of the theory
where the problem of vector particle propagator is solved explicitly.

A price for this solution  is an appearance  of three 
additional unphysical particles in the model. They are 
a couple of Faddeev-Popov ghosts and one Goldstone ghost.
All of them have  scalar type propagators with the same mass $m$.
Opposite to (\ref{mass_vect_phys_prop}) the propagator (\ref{hf_ propagator}) 
does not vanish when it projected onto  the temporal polarization state
\begin{equation}
 e^0=k/m\;\;,
\label{temporal}
\end{equation}
that also leads to the  appearance of  additional unphysical state.
The  main principles of gauge invariance guarantee \cite{BD} that
an  expression for the amplitude should  be  the same for any  gauge if 
only physical incoming and outgoing states are considered.


Generally the t'Hooft-Feynman  gauge solves the problem of 
cancellations.  But while  calculating  
processes with incoming or outgoing massive vector particles, 
we meet a similar problem. Indeed, we need to multiply 
the diagram contributions  by polarization vectors. The polarization 
vectors $(e^1,e^2,e^3)$ constitute an orthonormal basis 
in the sub-space orthogonal to a momentum $k$.  Due to the relation
\begin{equation}
  e^1_{\mu} e^1_{\nu} + e^2_{\mu} e^2_{\nu} + e^3_{\mu} e^3_{\nu} =
 k_\mu k_\nu/m^2  -g_{\mu\nu}
\label{polar_sum} 
\end{equation}      
at  least one of the polarization vectors  has large (of order of $k$)
components for any 
choice of  polarization  basis.  Let  vector  $k$ have the 
components
\begin{equation}
   k=(\sqrt {m^2 +  p^2}, 0,0,p)\;.
\end{equation}
Then the polarization vectors can  be chosen as
\begin{eqnarray}
e^1=(0,1,0,0)\;;\\
e^2=(0,0,1,0)\;;\\
\label{longitudinal}
e^3=(p/m,0,0,\sqrt {1 +  p^2/m^2)}\;.
\end{eqnarray}

The first two vectors correspond to transversal polarizations and the third
one corresponds to  a longitudinal one. We see that the
longitudinal vector has large ( of order of  $k$) components.
It may imply a fast  increase of cross-sections of processes with the
longitudinal
polarizations at high energies or an appearance of cancellations between various 
diagrams. Indeed the second case is realized and,
hence, we have got a problem with diagram  cancellations.   
When evaluating  squared diagrams  we  get 
 cumbersome  expressions again as a result of  substitution of the projector
(\ref{polar_sum}).



\subsection{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 (\ref{temporal}).  
Note that the Faddeev-Popov ghost states and the  temporal
state have a negative norm, whereas the Goldstone state has a
positive norm \cite{BD}. 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 \cite{LB}. As a result
\begin{equation}
\sum_{i \in S_{phys}} A_i A_i^{*} = \sum_{i \in S_{all}} \sigma(i)  A_i
A_i^{*}\;,
\label{phys.eq.full}
\end{equation}
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 (\ref{temporal})  and  
longitudinal (\ref{longitudinal}) 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
\begin{equation}
  e^0_{\mu} e^0_{\nu} - e^3_{\mu} e^3_{\nu} =   e'^0_{\mu} e'^0_{\nu} - e'^3_{\mu}
e'^3_{\nu}\;, 
\end{equation}
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 (\ref{polar_sum})  
by
\begin{equation}
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}\;.
\label{polar_sum_0}
\end{equation}
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 (\ref{polar_sum_0}).  
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. 

%It seams that nobody uses this summation  in the framework of amplitude
%calculation technique because of long series of  needed summation over 
%polarizations.
%For the by-hand calculation  in the framework of  squared diagram technique 
%it looks hard enough for the same reason, namely one should  consider a
%lot  of channels with ghost particles. However for the automatic calculation
%package like  \CompHEP~  this problem is not so serious.


\subsection{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 (\ref{hf_ propagator}) with  $m=0$  and  
to the appearance of  Faddeev-Popov ghosts \cite{BD}. 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 (\ref{phys.eq.full}). See 
the corresponding proof in \cite{CL}.

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

 
\subsection{Summation of ghost diagrams  in \CompHEP}

\CompHEP~ uses the  squared diagram technique with summation over
polarizations. Basically  one squared diagram corresponds to the
\begin{equation}
\sum_{i \in S_{all}} A_i^k A_i^{*l}
\end{equation}  
contribution in a squared matrix element, where $A^k$ and $A^l$ 
are the amplitudes related to some Feynman diagrams.

If we follow an idea of the previous section and take into account 
the ghost incoming particles,  a number of squared diagrams 
increases significantly. For example, in  the simplest case of 
$e^-,\gamma \rightarrow  n_e, W^-$  process one squared diagram of
Fig.\ref{ghostDiagrams}(a) is accompanied by three ghost diagrams with 
the similar topology.  

Let us note that all diagrams of Fig.\ref{ghostDiagrams} have the same 
denominators. The  numerators of these diagrams are 
 polynomials of scalar products of momenta. The powers of these polynomials 
are the same for all diagrams, so one might expect that the  symbolic 
sum of all these diagrams has approximately the same size as one term. 

Following  this note \CompHEP~ calculates the  symbolic sum of all sets of 
diagrams which become identical after replacing the ghost particles by their
parents. 

\subsection{Gauge symmetry and cancellations} 
                      \label{cancellations} 

Cancellation of diagram contributions 
is an essential point  both  for  symbolic and  numerical processing,
because a relatively small variation  of one diagram contribution may lead  to 
a significant error. Such variation can be caused either by finite
precision of floating operations or by correction of Feynman rules, for
instance, by including particle widths into consideration, or  by 
removal of  some diagram subset. 
We would like to stress  again these obstacles  
to warn the user.
  
 There are two well known  examples of gauge cancellations.
The first one is the ultraviolet  cancellation of terms originating from
the propagators of massive vector particles.
This problem could  be resolved  by the calculation of squared matrix element
in the t'Hooft - Feynman gauge. 

The second example is the cancelation  of double pole $(t^{-2})$ 
terms  of  t-channel photon 
propagator. There is a wide class of processes where the  incoming 
electron goes out in the forward direction emitting a virtual  photon
like in Fig.\ref{t^2_cancel}. The corresponding 
diagrams have got  the $1/t$ pole, where $t$ is the squared momentum of the virtual
photon.  For the above kinematics   the photon appears very close to its
mass shell ($ t \approx 0$), hence
this configuration gives a large contribution to the cross section. 

For the squared  matrix element  we expect the $1/t^2$ pole, 
but it appears to be reduced 
up to $1/t$ pole  \cite{strfunWW} in the zero-electron-mass limit. 
This fact is caused by electro-magnetic U(1) gauge  invariance. 
If diagrams of Fig.\ref{t^2_cancel} type contribute to your process, we
 strongly recommend to 
 to set the {\it 'Gauge invariance'} switch 
{\it ON} (see Section \ref{Breit-Wigner}) to prevent the  gauge  symmetry
breaking by width terms.
Another way to solve this problem is the usage of  the Weizsaecker-Williams
approximation (see Section \ref{WW-approx}).
