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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 [Kuraev&Fadin]. In CompHEP  we realize the similar expression by Jadach, Skrzypek, and Ward [Jadach-1991]:

$$\begin{array}{ll} F(x) = & \exp(\beta (3/4-Euler)) \beta (1-... ... x^2) \ln(x)/2 + (1-x)^2)/2) /(2 \Gamma(1+\beta)), \end{array}$$

where

$$\begin{array}{lcl} \alpha &=&1/137.0359895 \;\;\mbox{is the... ...& \mbox{is} & \mbox{the energy scale of reaction.} \end{array}$$

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 [Chen-1992]

$$F(x)=\frac{1}{N_{cl}} [(1-E^{-N_{cl}})\, \delta(1-x) + \frac{\exp(-\eta(x))}{1-x}\, h(\eta(x)^{1/3}, N_{cl}) ]\;\;\;,$$

where

$$\begin{eqnarray*} \eta(x) &=& 2 /(3 \Upsilon) \, (1/x -1) \;\;\;, \\ h(z, N_{cl... ...n \geq 1} \frac{z^n}{n! \Gamma(n/3)} \gamma(n+1, N_{cl}) \;\;\;, \end{eqnarray*}$$

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:

$$\begin{eqnarray*} N_{cl}& =& \frac{25\, \alpha^2 N}{12\, m (\sigma_x+\sigma_y)}... ...frac{5\,\alpha\, N\, E}{6\, m^3 \sigma_z (\sigma_x+\sigma_y)\;,} \end{eqnarray*}$$

where

$$\begin{array}{lcl} N & & \mbox{is number particles in the b... ...,} \\ E & & \mbox{is a center-of-mass momentum.} \end{array}$$

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.