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

with







where

, p

Energy and momentum of final electron in the center-of-mass frame.

c

Cosine of the scattering angle of the final electron in the center-of-mass frame.

h

Product of circular polarizations of the two initial photons.

The total crosssection is

where

where

Events are generated by the following algorithm using inverse function.

(a)

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


where and 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, the time interval and V the volume in which the macro phtons are located.

(b)

If P is too large (say, >0.1), divide the interval (and P) by an integer , and repeat the following procedure times.

(c)

Generate a random number . Reject if .

(d)

Generate another random number and solve the equation

with respect to z. Here z is defined by (). The left hand side is the integral of f from 0 to . The sign of is determined by the sign of .

(e)

Generate another random number and compute the transverse component of electron momentum by

where and are arbitrary unit vectors perpendicular to , the unit vector along the initial photon momentum in the center-of-mass frame. The latter is given by

where are the energy momentum of the photons in the original frame.
The value of should be computed from

rather than from because the latter is usually very close to unity when is much larger than the electron rest mass.

(f)

Then, the momentum of the electron in the original frame is calculated by

where . Note that must be computed from in order to avoid round off errors.

(g)

The momentum of positron is computed from the momentum conservation.