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## Coherent Pair Creation

When a high energy photon goes through a strong transverse field, it can create a real electron-positron pair. This process is known as `coherent pair creation' and is characterized by the parameter

where is the energy of the initial photon. ( and are defined in eq.(84).) The probability of the process is exponentially small () when is small.

Let us denote the energy and polarization vector of initial photon and final positron/ electron by , (), , and . The transition rate is obtained by the following replacement in the formula (92):

Ignoring the terms related to the polarization correlation between the final electron and positron, we obtain

where is defined as

is the modified Bessel function and is defined in eq.(87). Their arguments are defined in eq.(99). The transition rate summed over the final polarization is

Figure 9: Function for three values of . The solid, dotdash and dashed curves are for , 1, -1, respectively. The curves for are normalized such that , and those for are drawn with the same scale as the corresponding curves.

The function is plotted in Fig.9 as a function of .

1ex

The total rate for given is approximately

where

In order to make the spectrum function flatter, CAIN introduces the variable y (-1<y<1) instead of (0<x<1):

The spectrum function with respect to y then becomes

where y/2 is used because the range of y is 2. The constant

is chosen so that for any . is plotted in Fig.10.

Figure 10: Function as a function of y for three values of =0.3, 2, 40 and for . The parameter is indicated by the line mode and is by crosses (no cross for ).

Now, the event generation goes as

(1)
Compute and reject if (the rate is too small).
(2)
Generate a uniform random number 0<p<1 and compute .
(3)
Reject if .
(4)
Generate another uniform random numbers 0<q<1 and -1<y<1 and compute . (Instead of q, one can also use .)
(5)
Reject if .
(6)
Accept and compute x from eq.(104) and , .
(7)
Compute the final polarization of from

The formula for is obtained by replacing x by 1-x.
(8)
In the case of rejection, the polarization of the photon should be changed according to eq.(12):

The error of the formula (103) does not cause any inaccuracy of event generation (but causes inefficiency).

Next: Incoherent Pair Production Up: Physics and Numerical Methods Previous: Enhancement factor of the

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998