CERN Program Library Long Writeup W5013 - CERNLIB ...

2 Method
When a muon of energy E moves in the field of an atom of charge Z, it can radiate a e + e − -pair (4 th order
QED process) with differential cross-section [99]:
⎡
∂ 2 σ
∂ν ∂ρ = 2
α4 3π (Zλ− ) 2 1 − v
( ) ⎤ 2
me
⎣φ e + φ µ ⎦ (1)
v m µ
where
and
ν = E+ + E −
E
and
ρ = E+ − E −
E + + E −
α 1/137, fine structure constant;
λ− 3.8616 × 10 −11 cm, electron Compton wavelength;
v k/E fraction of energy transferred to the pair;
T E − M kinetic energy of the muon.
E ± the energy of the e ± .
The explicit form of the terms φ e and φ µ can be found in [99]. The kinematic ranges of ν and ρ are:
4m e
E = ν min ≤ ν ≤ ν max = 1− 0.75 √ e m µ
E Z1/3
0=ρ min ≤ |ρ(ν)| ≤ ρ max (ν) =
[
1 − 6m2 µ
E 2 (1 − ν)
] √
1 − 4m e
νE
(2)
where e =2.718 ....
E c (PPCUTM in the program) is the energy cut-off; below this energy e + e − -pair are treated as continuous
energy loss, and above they are explicitly generated and v c = E c /E. The mean value of the energy lost by
the incident muon due to e + e − -pair with energy below E c is:
∫
E pair
νc
∫ ρmax(ν)
loss (Z, T, E c)=2E dν ν dρ
ν min 0
∂2 σ
∂ν ∂ρ
GeV barn/atom (3)
whereas the total cross-section for the emission of a hard e + e − -pair is:
∫ νmax
σ(Z, T, E c )=2 dν
ν c
∫ ρmax(ν)
0
dρ
∂2 σ
∂ν ∂ρ
barn/atom (4)
2.1 Parameterisation of energy loss and total cross-section
Instead of using the explicit formula (3) and (4), we have chosen to parameterise directly E pair
loss (Z, T, E c)
and σ(Z, T, E c ) as:
[
σ(Z, T, E c ) = Z[Z + ξ σ (1 + γ ln Z)] ln E ] α
max
F σ (Z, X, Y ) (5)
E pair
loss (Z, T, E c) = Z[Z + ξ l (1 + δ ln Z)]E
E
[
Ec − Ec
min
E
] β
F l (Z, X, Y ) (6)
where ξ σ,l , α, β, γ and δ are parameters obtained by a fitting procedure and E min
c
= Ev min =4m e and
E max = Ev max
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