\subsection{\CTEq~  distribution functions}

There are several sets of  disribution functions   presented by CTEQ
collaboration which describe the densities of quarks and gluons in
(anti)proton. In \CompHEP~ we  have implemented the  
\CTEq\cite{strfunCTEQ} set. 
Apart from  the Feynman variable $x$  they  depends on  
the QCD-scale parameter $Q$.
  The available  range is  
\[ 
\begin{array}{lcl}
       10^{-5} & <  x  < & 1; \\ 
       1.6     & <  Q  < & 10000  \; \mbox{GeV}. 
\end{array} \]
\noindent 
For  points outside  these ranges  the  extrapolation is used and the
corresponding warning is  directed to the standard output file.

  The \CTEq~ set of functions is produced in the framework of the
next-to-leading calculations in the  standard $\overline{MS}$ scheme 
with $\alpha_s(M_Z) = 0.116$.
  See further explanations in \cite{strfunCTEQ}.
 

\subsection{\MRS~  structure functions}

 The  \MRS~ group also  presents several sets of proton structure
functions. Two of them are implemented in \CompHEP. They are
\MRS($A^\prime$) and \MRS($G$) \cite{strfunMRS}.
 
   In the $A^\prime$ case it is supposed that the sea quark and gluon
densities are parameterized at small $x$ by the $x^{-\lambda}$
function  where $\lambda$ is the same  for the gluon and for the sea
quarks. This suggestion is motivated by the QCD theory.
  The G set uses two different  $\lambda$ parameters  for the sea quarks
and for the gluon densities.  A fit of experimental data  for the G set is 
better.
     
  The next-to-leading order evaluation  and the $\overline{MS}$
factorization scheme
are  used for evaluation  of the $Q^2$ dependence of structure  function.
$\alpha_s(M_Z) = 0.113 \; [0.114]$ in the $A^\prime$ \ [$G$] case, correspondingly.

The available  range is
\[
\begin{array}{lcl}
                  10^{-5} & <  x   < & 1;  \\ 
                  5      & <  Q^2 < & 1310720  \;\mbox{GeV}^2. 
\end{array} \]
See complete explanation in \cite{strfunMRS}.



\subsection{Backscattered  photon spectrum}
    
This function describes the spectrum of photons scattered backward
from the interaction of laser light with the  high energy  electron
beam:

\[ f(x) = \left\{ 
\begin{array}{ll}
0,                                             & \mbox{for~} x>x_{max} \\
N(1-x+1/(1-x) \,(1-4x/x_0 \, (1-x/(x_0 (1-x)) \,))), & \mbox{for~} 0<x<x_{max}
\end{array} \right. \]
\noindent
where  $x_0=4.82, \; x_{max}=x_0/(1+x_0), \; N$ is a normalization factor.

The above spectrum  corresponds to the special initial condition  when 
unpolarized photons are created.
See \cite{strfunLaser} for more details. 


\subsection{Weizsaecker-Williams approximation}
\label{WW-approx}
  Weizsaecker-Williams approximation is used to describe  processes 
of electro-production in the case of  small angle of charged particle 
  scattering.
 In this case the  virtual photon emitted by the scattering particle  
appears near to the mass shell (see Fig.\ref{t^2_cancel}). It gives a 
possibility to reduce the process of
electro-production to the photo-production one with an appropriate photon 
spectrum:
$$ f(x) = (q^2\,\alpha/(2 \pi))
     (\log((1-x)/(x^2 \delta)) (1+(1-x)^2)/x -2(1-x-\delta \, x^2)/x),$$
where $\alpha$ is the  fine structure constant,
$q$ is a charge of incoming particle,
$m$ is its mass,  
$\delta = (m/Q_{max})^2.$
$Q_{max}$  sets out the  region of  photon virtuality $(P^2 >-Q_{max}^2)$ 
 which  contributes to  the process.
It is assumed  that region of large virtuality can be 
taken into account by direct calculation of electro-production.
As a rule this contribution is small enough.  

Parameters $q$, $m$, and $Q_{max}$ are defined by the user. 
See \cite{strfunWW} for the further explanations. In the case of 
  \CompHEP~  the Weizsaecker-Williams photon spectrum is 
available for charged leptons only.

\subsection{ISR and Beamstrahlung}
ISR (Initial State Radiation) is a process of photon radiation  by the
incoming electron due to its interaction with other collision particle.
 The resulting spectrum of  electron has been
calculated by Kuraev and Fadin \cite{strfunISR1}.
 In \CompHEP~ we realize the similar expression by Jadach, Skrzypek, and Ward
\cite{strfunISR2}:
\[
\begin{array}{ll}
F(x) = & \exp(\beta (3/4-Euler)) \beta (1-x)^{\beta-1}  \\ 
 & ((1+x^2)-\beta ((1+3 x^2) \ln(x)/2 + (1-x)^2)/2) /(2 \Gamma(1+\beta)),
\end{array} \] 
\noindent
where 
\[
 \begin{array}{lcl}
\alpha &=&1/137.0359895 \;\;\mbox{is the fine structure constant;} \\
\beta&=& \alpha (2 \ln(SCALE/m)-1) / pi   \\
  m    &=&0.00051099906 \;\;\;\;\;\;\mbox{is the electron mass;} \\
Euler  &=&0.5772156649 \;\;\;\;\;\;\;\;\mbox{is the Euler constant;} \\
\Gamma()  & \mbox{is} &        \mbox{the gamma function;} \\
SCALE   & \mbox{is} &          \mbox{the energy scale of reaction.}
\end{array} \] 
In the Kuraev and Fadin article  the parameter  SCALE equals to  the total 
energy of the process because  they 
considered the process of direct $e^+ e^-$ annihilation. In order to 
apply this structure function to another  processes we 
provide the user with a possibility  to define this parameter. 

Beamstrahlung is a  process of energy loss  by the
incoming electron due to its interaction with the  electron (positron)
bunch  moving in the opposite direction.    
The effective  energy spectrum of electron can be  described by 
the following function  \cite{strfunBeam}
$$
F(x)=\frac{1}{N_{cl}} [(1-E^{-N_{cl}})\,  \delta(1-x) + 
     \frac{\exp(-\eta(x))}{1-x}\,  h(\eta(x)^{1/3}, N_{cl}) ]\;\;\;,                  
$$
%
where 
\begin{eqnarray*}
\eta(x) &=& 2 /(3 \Upsilon) \, (1/x -1) \;\;\;, \\
h(z, N_{cl}) &=& \sum_{n \geq 1} \frac{z^n}{n! \Gamma(n/3)} \gamma(n+1, N_{cl})
\;\;\;,
\end{eqnarray*} 
and $\gamma$ is the incomplete  gamma function.


Function $F(x)$ depends on two parameters, $N_{cl}$ and $\Upsilon$,
which in their turn are determined by a  bunch  design:
\begin{eqnarray*}
 N_{cl}& =&  \frac{25\, \alpha^2 N}{12\, m  (\sigma_x+\sigma_y)}\;\;\;,\\
 \Upsilon &=& \frac{5\,\alpha\, N\, E}{6\, m^3 \sigma_z (\sigma_x+\sigma_y)\;,}
\end{eqnarray*}
where 
\[
\begin{array}{lcl}
   N & & \mbox{is number particles in the bunch,} \\
   \sigma_x,\sigma_y,\sigma_z & & \mbox{are sizes of bunch,} \\
   E & & \mbox{is a center-of-mass momentum.}
\end{array} \]

The Beamstrahlung spectrum  cannot be integrated by the current \CompHEP~ 
version  because it contains a $\delta$-function. 
Instead of it we provide the user with a possibility to integrate the squared 
matrix element with a convolution of Beamstrahlung and ISR  spectra.
