Direct (E ,E−) Pair Creation by Muon

Home , Pair production

− Direct (e+, e ) pair creation by muon 1

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+ Direct (e , e−) pair creation by muon

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' $ Direct (e+, e−) pair creation by muon

Creation of a (e+, e−) pair by virtual photon in the Coulomb ﬁeld of the nucleus (for momentum conservation).

− µ + nucleus µ + e+ + e + nucleus −→ e- e+

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It is one of the most important processes of muon interaction. At TeV muon energies, pair creation cross section exceeds those of other muon interaction processes in a wide region of energy transfers : 100 MeV 0.1 E ≤ ≤ µ Average energy loss for pair production increases linearly with muon energy, and in TeV region this process contributes over 50 % to the total energy loss rate.

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'energy transfers $

Main contribution to the total cross section is given by transferred energies: 5MeV 0.01 E ≤ ≤ µ The contribution to average muon energy loss is determined mostly by region: − 10 3 E 0.1 E µ ≤ ≤ µ Thus, to adequatly describe the number of pairs produced, average energy loss and stochastic energy loss distribution one need to reproduce with a suﬃcient accuracy the diﬀerential cross section behaviour in a wide range of energy transfers:

5MeV 0.1 E ≤ ≤ µ

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l a ) ot

2 t 100 e

m F c

1 − g

e 10 M (

Fe ion x d

/ s E 1 r m l

d i e c a r u p b n e e e F F F

0.1 1 10 102 103 104 105 Muon energy (GeV)

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diﬀerential cross section

The differential cross section is given by Kokoulin et al. [Koko71]. It includes : screening of the field of the nucleus • correction for finite nuclear size • contribution from the atomic electrons [Keln97] • ... • See [Koko71] for a complete discussion.

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'diﬀerential cross section $

The diﬀerential cross section per atom can be written as : dσ 4 α2 r2 1 v = e − [Z(Z + ζ)] F (Z, E, ) (1) d 3π with ρmax m 2 F (Z, E, ) = Φ (v, ρ) + e Φ (v, ρ) dρ (2) e m µ Z0 " µ # where

− = + + = total energy of the created pair; v = /E − ρ = (+ )/ = asymmetry coeﬃcient; −

The functions Φe, Φµ, ζ can be found in [Koko00].

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limits

2 min = 4mec 3√e = E m c2 Z1/3 max − 4 µ ρmin = 0 6(m c2)2 ρ = 1 µ 1 min max − E(E ) − − r

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' $ Energetic pairs and truncated energy loss rate

One may wish to take into account separately the high-energy pairs

emitted above a given threshold cut (miss detection, explicit simulation ...). Those pairs must be excluded from the mean energy loss count.

dE cut dσ = n d − dx at d <cut Zmin nat is the number of atoms per volume. Then, the truncated total cross-section for emitting ’hard’ pairs is: max dσ σ(E, ) = d cut ≤ ≤ max d Zcut

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The muon deﬂection angle is of the order of: mc2 θ = E

Above 1000 TeV the LPM suppression mechanism may have an ∼ eﬀect.

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' $ number of interactions per cm in Iron. (cut 100 MeV)

Tables for MUON - in Iron

BREM X-sec (1/cm) PAIR X-sec (1/cm) MUNU X-sec (1/cm)

muon kinetic energy (GeV)

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' $ 1 TeV muon in 10 meter of Fe (ﬁeld 5 tesla). direct pair creation only

WORL RUN NR 1 28/ 9/ 0 EVENT NR 13 5

100 cm

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' $ 10 meter of Fe : muons 100 GeV, 1 TeV, 5 TeV. left : brems only right : brems + direct pair creation

RUN NR 1 RUN NR 1 WORL 28/ 9/ 0 WORL 28/ 9/ 0 EVENT NR 123456 EVENT NR 123456

100 cm 100 cm

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'energy deposited by high energy muons in a bloc of Iron $

Muon in 2x2x2 m Iron 30 E peak (GeV)

Geant4

15 Geant3

2 3 4 10 10 10 Emu10 (GeV)

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'energy spectrum of 1 TeV muons after 3 m of iron $

1 TeV muon in 3x3x3 m iron 120

100 G4

TRAMU

0 9400 9500 9600 9700 9800 9900 10000 Edep(MeV) 2 3m iron. muon 1TeV/c x 10

Geant4&Tutorial October 25, 2006% Direct (e+, e−) pair creation by muon 16-1 References

[Lohm85] W.Lohmann,R.Kopp,R.Voss, Cern Report 85-03 (1985)

[Koko69] R.P.Kokoulin and A.A.Petrukhin, Proc. 11th Intern. Conf. on Cosmic Rays, Budapest, 1969 [Acta Phys. Acad. Sci. Hung.,29, Suppl.4,p.277, 1970].

[Koko71] R.P.Kokoulin and A.A.Petrukhin, Proc. 12th Int. Conf. on Cosmic Rays, Hobart, 1971, vol.6, p.2436. [Keln97] S.R.Kelner, Phys. Atomic Nuclei, 61 (1998) 448.

[Koko00] S.R.Kelner, R.P.Kokoulin, A.Rybin. Geant4 Physics Reference Manual, Cern (2000)

[pdg] D.E. Groom et al. Particle Data Group . Rev. of Particle Proper- ties. Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/

Geant4 Tutorial October 25, 2006