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Breit-Wheeler Process


The differential crosssection with respect to the scattering angle of the final electron in the center-of-mass frame is given by



tex2html_wrap_inline10028, p
Energy and momentum of final electron in the center-of-mass frame.
Cosine of the scattering angle tex2html_wrap_inline10036 of the final electron in the center-of-mass frame.
Product of circular polarizations of the two initial photons.

The total crosssection is

Events are generated by the following algorithm using inverse function.

Compute tex2html_wrap_inline10028, p, a, b, G and tex2html_wrap_inline9288 for given initial parameters (reject if tex2html_wrap_inline10634, i.e., below threshold) and calculate the event probability for the given time step

where tex2html_wrap_inline10636 and tex2html_wrap_inline10638 are the weights of initial photons (number of real photons divided by that of macro photons), w the weight of the pair to be created, tex2html_wrap_inline8540 the time interval and V the volume in which the macro phtons are located.
If P is too large (say, >0.1), divide the interval tex2html_wrap_inline8540 (and P) by an integer tex2html_wrap_inline10654, and repeat the following procedure tex2html_wrap_inline10654 times.
Generate a random number tex2html_wrap_inline10658. Reject if tex2html_wrap_inline10660.
Generate another random number tex2html_wrap_inline10662 and solve the equation
with respect to z. Here z is defined by tex2html_wrap_inline10668 (tex2html_wrap_inline10670). The left hand side is the integral of f from 0 to tex2html_wrap_inline10676. The sign of tex2html_wrap_inline10678 is determined by the sign of tex2html_wrap_inline9574.
Generate another random number tex2html_wrap_inline10682 tex2html_wrap_inline10684 and compute the transverse component of electron momentum by
where tex2html_wrap_inline10376 and tex2html_wrap_inline10378 are arbitrary unit vectors perpendicular to tex2html_wrap_inline10690, the unit vector along the initial photon momentum in the center-of-mass frame. The latter is given by
where tex2html_wrap_inline10692 are the energy momentum of the photons in the original frame.
The value of tex2html_wrap_inline10694 should be computed from
rather than from tex2html_wrap_inline10696 because the latter is usually very close to unity when tex2html_wrap_inline10028 is much larger than the electron rest mass.
Then, the momentum of the electron in the original frame is calculated by
where tex2html_wrap_inline10700. Note that tex2html_wrap_inline10702 must be computed from tex2html_wrap_inline10704 in order to avoid round off errors.
The momentum of positron is computed from the momentum conservation.

next up previous contents index
Next: Virtual (almost real) photon Up: Incoherent Pair Production Previous: Incoherent Pair Production

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