Differential cross-section for pair production

The Bethe-Heitler differential cross-section with the Coulomb correction for a photon of energy E to produce a -pair with one of the particles having energy $\epsilon E$ ($\epsilon$ is the fraction of the photon energy carried by one particle of the pair) is given as in []: $\{d\sigma (Z,E,\epsilon )d\; \epsilon \}=\; \{r$₀²αZ [ Z +ξ(Z) ]E²}[ε²+ ( 1 -ε) ²][ Φ₁(δ) - {F (Z)2}]+ {23}ε(1-ε) [Φ₂(δ) - {F (Z)2}]

where $\Phi$_i(δ) are the screening functions depending on the screening variable $\delta$

$f$c(Z)	$=$	$a(1/(1+a)+0.20206-0.0369a+0.0083*a2-0.002a3)$
$a$	$=$	$(Z)2$
$\alpha$	$=$	$1/137$

The kinematical range for the variable $\epsilon$ is $\{mE\}\le \epsilon \le 1\; -\; \{mE\}$

The cross-section is symmetric with respect to the interchange of $\epsilon$ with $1-\epsilon$ , so we can restrict $\epsilon$ to lie in the range $\epsilon$₀= m/E≤ε≤1/2

After some algebraic manipulations we can decompose the cross-section as (note: the normalisation is not important):

$\{d\sigma d\epsilon \}$

$=$

$\sum$i=12αifi(ε) gi(ε)

where

$\alpha$1	$=$	$\{(0.5\; -\; \epsilon$0)23}F10	$\alpha$2	$=$	$\{12\}F$20
$f$1(ε)	$=$	$\{\; 3(0.5\; -\; \epsilon$0)3}(ε-0.5)2	$f$2(ε)	$=$	$\{\; 10.5-\epsilon$0}
$g$1(ε)	$=$	$F$1/ F10	$g$2(ε)	$=$	$F$2/ F20

and

$F$1	$=$	$F$1(δ) = 3Φ1(δ)-Φ2(δ)-F(Z)	$F$10	$=$	$F$1(δmin)
$F$2	$=$	$F$2(δ) = {32}Φ1(δ)+ {12}Φ2(δ)-F(Z)	$F$20	$=$	$F$2(δmin)
$\delta$min	$=$	$4\{\; 136mZ\{13\}3E\}$

$\delta$_min is the minimal value of the screening variable $\delta$ . It can be seen that the functions $f$_i(ε) are normalised and that the functions $g$_i(ε) are ``valid" rejection functions.

Therefore, if $r$_i are uniformly distributed random numbers ($0\le r$_i≤1 ), we can sample the $\epsilon$ (x in the program) in the following way:

1. select i to be 1 or 2 according to the following ratio:

   $BPAR=\; \{\; \alpha$₁α₁+α₂}

   If $r$₀< BPAR then i=1, otherwise if $r$₀≥BPAR

   i=2;

2. sample $\epsilon$ from $f$₁(ε) . This can be performed by the following expressions:

   

3. calculate the rejection function $g$_i(ε) . If $r$₂≤g_i(ε) , accept $\epsilon$ , otherwise return to step 1.

It should be mentioned that we need a step just after sampling $\epsilon$

in the step 2, because the cross-section formula becomes negative at large $\delta$

and this imposes an upper limit for $\delta$ ;

$\delta$_max= exp[{42.24-F(Z)8.368}]- 0.952

If we get a $\delta$ value using the sampled $\epsilon$ such that $\delta >\; \delta$_max , we have to start again from the step 1. After the successful sampling of $\epsilon$ , GPAIRG generates the polar angles of the electron with respect to an axis defined along the direction of the parent photon. The electron and the positron are assumed to have a symmetric angular distribution. The energy-angle distribution is given by Tsai [,] as following:

$\{d\; \sigma dpd\; \Omega \}$	$=$	$\{2\; \alpha 2e2\pi k\; m4\}[\; \{2x(1-x)(1+l)2\}-\{12lx(1-x)(1+l)4\}](Z2+Z)\; .$
		$+\; .\; [\; \{2x2-2x+1(1+l)2\}+\; \{4lx(1-x)(1+l)4\}][\; X-2Z2f((\alpha Z)2)]$

where k is the photon energy, p and E are the momentum and the energy of the electron of the $e+e-$ -pair, $x=E/k$ and $l\; =\; E2\theta 2/m2$ . This distribution is quite complicated to sample and, furthermore, considered as function of the variable $u\; =\; E\; \theta /m$ , it shows a very weak dependence on Z, E, k, $x\; =\; E/k$ . Thus, the distribution can be approximated by a function $f(u)\; =\; C\; (\; u\; e-au+\; d\; u\; e-3au)$

where

$C$	$=$	$\{9a29\; +\; d\}$
$a$	$=$	$0.625$
$d$	$=$	$27.0$

The sampling of the function $f(u)$ can be done in the following way ($r$_i are uniformly distributed random numbers in [0,1]):

1. choose between $u\; e-au$ and $d\; u\; e-3au$ , with relative probability given by $9/(9+d)$ and $d/(9+d)$ respectively; if $r$₁< 9/(9+d) then b=a, else b=3a;
2. sample $u\; e-bu$ , $u=-(lnr$₂+ lnr₃)/b .

The azimuthal angle, $\phi$ , is generated isotropically.

This information together with the momentum conservation is used to calculate the momentum vectors of both decay products and to transform them to the GEANT coordinate system. The choice of which particle in the pair is the electron/positron is made randomly.