CERN Program Library Long Writeup W5013 - CERNLIB ...

Geant 3.10 GEANT User’s Guide PHYS451
Origin : L.Urbán Submitted: 26.10.84
Revision : Revised: 16.12.93
Documentation :
1 Subroutines
Simulation of e+e- pair production by muons
CALL GPAIRM
GPAIRM generates the e + e − -pair radiated by a high energetic muon. It uses the following input and output:
input:
output:
via common block /GCTRAK/
via common block /GCKING/
GPAIRM is called automatically from the tracking routine GTMUON if, and when, the parent muon reaches its
radiation point during the tracking.
2 Method
The double differential cross-section for the process can be written [99]:
d 2 σ
dνdρ = 2
α4 3π (Zλ)2 · 1 − ν
]
[φ e +(m/M) 2 φ µ
ν
(1)
All the quantities in the expression above are defined in [PHYS450]. By computing this cross-section for
different (ν, ρ) points, it can be seen that:
1. the shape of the functions d2 σ
dνdρ
and dσ ∫
dν dρ d 2 σ
dνdρ
practically does not depend on Z
2. the dominant contribution comes from the low ν region:
ν min =(4m/E) ≤ ν ≤ 100 ∗ ν min (2)
3. in this low region (d 2 σ/dνdρ) is flat as a function of ρ
Therefore, we propose the following sampling method as a rough approximation:
1. In the low ν region the differential cross-section
∫
dσ
dν =
dρ d2 σK
dνdσ
can be approximated as:
[
dσ
dν ≈
1 − ν min
ν
] 1/2
1
v a with: a =2− ln E
10
(E in GeV) (3)
We can write:
dσ
dν
≈ f(ν)g(ν) (4)
316 PHYS451 – 1