A Monte Carlo generator for the electromagnetic production of muon pairs by annihilation of high-energy positrons with atomic electrons is described. The computer code is designed as a standard electromagnetic process for GEANT4. The formulas and algorithms are described and illustrated in detail. The code has been applied to study muon production by high energy positrons in a spoiler. The results are presented and compared to muon pair production by gamma conversion

Burkhardt, H

Kelner, S R

Kokoulin, R P

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN–AB–2003–002 (ABP)CLIC Note 554Production of muon pairs in annihilation ofhigh-energy positrons with resting electronsH.Burkhardt1, S.R. Kelner2, R.P. Kokoulin21 CERN, CH-1211 Geneva 23, Switzerland2 Moscow Engineering Physics Institute, Moscow 115409, RussiaAbstractA Monte Carlo generator for the electromagnetic production of muon pairs by annihilation of high-energy positrons with atomic electrons is described. The computer code is designed as a standardelectromagnetic process for GEANT4. The formulas and algorithms are described and illustrated indetail.The code has been applied to study muon production by high energy positrons in a spoiler. Theresults are presented and compared to muon pair production by gamma conversion.Geneva, SwitzerlandJanuary 30, 200311 IntroductionThe Monte Carlo simulation of muon production in high energy electromagnetic showers is im-portant for the design of collimation, the final focus region and detectors of linear colliders. Themain electromagnetic muon production process, the conversion of high energy photons into a pairof muons has been described recently [1]. It has meanwhile been fully implemented into Geant4 [2]and is part of the standard distribution package.Here we describe another process, the positron annihilation with atomic electrons into muons.2 Total cross sectionThe annihilation of positrons and target electrons with the creation of muon pairs in the final state(e+e− → µ+µ−) may give an appreciable contribution to the total number of muons produced inhigh-energy electromagnetic cascades. The threshold positron energy in the laboratory system forthis process with the target electron at rest isEth = 2m2µ/me −me ≈ 43.69 GeV , (1)where mµ and me are the muon and electron masses, respectively. The total cross section of theprocess on the electron is [3]σ =pi r2µ3ξ(1 +ξ2) √1− ξ , (2)where rµ = re me/mµ is the classical muon radius, ξ = Eth/E, and E is the total positron energy inthe laboratory frame. In Eq. 2, approximations are made that utilize the inequality m2e  m2µ.00.20.40.60.8150 60 70 80 100 200 300 400 500E in GeVσ in µbFigure 1: Total cross section for the process e+e− → µ+µ− as function of the positron energy E inthe laboratory system.The cross section as function of the positron energy E is shown in Fig.1. It has a maximum atE = 1.396 Eth (and not at E ≈ 1.7Eth, as written in [3], [4]). The maximum value of the crosssection is σmax = 0.5426 r2µ = 1.008 µb.23 Sampling of energies and anglesIt is convenient to simulate the muon kinematic parameters in the centre-of-mass (c.m.) system, andthen to convert into the laboratory frame.The energies of all particles are the same in the c.m. frame and equal toEcm =√12me(E + me) . (3)The muon momenta in the c.m. frame are Pcm =√E2cm −m2µ. Let us denote as x = cos θcmthe cosine of the angle between c.m. momenta of µ+ and e+. From the differential cross section(e.g., [4]) it is easy to derive that, apart from normalization, the distribution in x is described byf(x) dx = (1 + ξ + x2 (1− ξ)) dx , −1 ≤ x ≤ 1 . (4)The value of this function is contained in the interval (1 + ξ) ≤ f(x) ≤ 2. The generation of x isstraightforward using the rejection technique. Fig. 2 shows both generated and analytic distributions.00.250.50.7511.251.51.752-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1E= 500 GeV, ξ = .0874 E = 50 GeV, ξ =.874 1 + cos θcm2cos θcmEntries per binFigure 2: Generated histograms with 106 entries each and expected cos θcm distributions (dashedlines) at E = 50 and 500 GeV positron energy (in the Lab.). The asymptotic 1 + cos θ2cm distributionvalid for E →∞ is shown as dotted line.The transverse momenta of the µ+ and µ− particles are the same, both in the c.m. and the labframe, and their absolute values are equal toP⊥ = Pcm sin θcm = Pcm√1− x2 . (5)The energies and longitudinal components of the muon momenta in the laboratory system may beobtained by means of Lorentz transformation. Let us denote the velocity and Lorentz factor of thecentre-of-mass in the laboratory frame asβ =√E −meE + me, γ ≡ 1√1− β2=√E + me2me=Ecmme. (6)3We then obtain the laboratory energies and longitudinal components of momenta for positive andnegative muons as:E+ = γ (Ecm + x β Pcm) , P+‖ = γ (βEcm + x Pcm) , (7)E− = γ (Ecm − x β Pcm) , P−‖ = γ (βEcm − x Pcm) . (8)Finally, for the vectors of the muon momenta we get:P+ = (+P⊥ cos ϕ, +P⊥ sin ϕ, P+‖) , (9)P− = (−P⊥ cos ϕ,−P⊥ sin ϕ, P−‖) , (10)where ϕ is a random azimuth angle chosen between 0 and 2 pi. The z-axis is directed along themomentum of initial positron in lab frame.The maximum and minimum energies of the muons are given byEmax ≈12E(1 +√1− ξ), (11)Emin ≈12E(1−√1− ξ)=Eth2(1 +√1− ξ) . (12)The fly-out polar angles of the muons are approximatelyθ+ ≈ P⊥/P+‖, θ− ≈ P⊥/P−‖ ; (13)the maximal angle θmax ≈memµ√1− ξ is always small compared to 1.4 Implementation details, validityThe process is called AnnihiToMuPair. The source code has been added to$G4INSTALL/source/processes/electromagnetic/standardwith src/G4AnnihiToMuPair, include/G4AnnihiToMuPair.hh.The process described is assumed to be purely electromagnetic (based on virtual γ exchange).Production with Z-boson exchange and the γ−Z interference are neglected. The Z-pole correspondsto a positron energy of E = M 2Z/2me = 8136 TeV. The validity of the current implementation istherefore restricted to initial positron energies of less than about 1000 TeV.5 Cross section per atom and comparison with gamma conver-sionThe cross section of Eq. 2 was given per electron. To obtain the cross section per atom, we have tomultiply with Z, the number of electrons per atom. Muon pair production by gamma conversion inthe field of the nucleus has a cross section which scales with Z2. Gamma conversion is thereforemore important than annihilation in heavy materials like tungsten [5].Spoilers for high energy e+e− colliders will have to be constructed with low Z materials likecarbon [6], for which the cross section of both processes is rather comparable, see Table 1.4Table 1: Numerical values for the total cross section per atom for different elements. The energy ofthe incoming particle (γ in the case of pair production and e+ in the case of annihilation) is 100 GeV.Element Z σγ→µ+µ− σe+e−→µ+µ−µb µbH 1 0.1921 0.7702C 6 5.870 4.621Pb 82 886.4 63.156 Application: muon production in a spoilerThe code described here has been used to predict the muon yield from 109 positrons and electronsof 250 GeV energy, impacting perpendicular on a graphite spoiler of one radiation length thickness(18.8 cm).The computing time of the Geant4 runs was reduced (to about 2 hours on a desktop computer)by• introducing a low energy cutoff of 0.4 GeV below which all particles are locally stopped• generation of 106 (rather than 109) showers with 1000 times increased muon production crosssections.Fig. 3 shows the energy spectra of the positrons in the annihilation and the angle between the muonsproduced, as observed in the laboratory system. In case of electrons as primary particles, the annihi-lation is only possible via secondary positrons in the shower (shaded histograms in Fig. 3).Incident positrons can annihilate directly. In most cases, the positron will already have lost someenergy (mainly by bremsstrahlung), but there is also a finite probability (see the high energy peak)that the positron annihilates at nearly maximum energy.The laboratory angle between the muons in the pairs produced is limited to about 4.8 mrad(me/mµ = 4.84× 10−3, see also Section 3).01002003004005006007000 50 100 150 200 250Geant4 run corresponding to 109 e+, e- of 250 GeV on 1χ0 carbone+ incident6035 annihilations e+e- → µ+µ-795 E e+ > 245 GeVe- incident, 525 annihilationsEe+ in GeVEntries/bin01002003004005006007008000 1 2 3 4 5Angle in mradEntries/bine+ incidente- incidentFigure 3: Energy of the positron in the annihilation (left) and angle between the muons produced(right), both in the laboratory frame.5Fig. 4 shows muon energy spectra corresponding to 109 positrons or electrons impacting on thespoiler. The muon spectra for gamma conversion with initial positrons or electrons were compatiblewithin statistics (for clarity only the e+ spectrum is shown). The high energy muon spectrum isdominated by annihilation of primary positrons. Annihilation with initial electrons is only possiblevia secondary positrons. Table 2 gives actual numbers obtained in the Geant4 runs described here.110101010234γ → µ+µ-e+e- → µ+µ-109 e+ of 250 GeV on 1χ0  carbon0 50 100 150 200 250Energy in GeVParticles per 5 GeV bine+e- → µ+µ-e- incidentFigure 4: Muon energies in the laboratory system.Table 2: Number of muons produced by 109 positrons or electrons impacting on a 1 radiation lengththick graphite spoiler.Primary Particle Process muons, E > 20 GeV muons, E > 100 GeVe+, 250 GeV γ → µ+µ− 11644 1822e+, 250 GeV e+e− → µ + µ− 10872 3316e-, 250 GeV e+e− → µ + µ− 883 41CommentNote. The algorithm could also directly be applied to the simulation of τ -pairs produced in annihila-tion of positrons with target electrons. The respective threshold energy is Eth = 12.4 TeV.AcknowledgementsWe would like to thank M. Maire for his help and advice on standard electromagnetic processes inGeant4.6References[1] H. Burkhardt, S. Kelner, and R. Kokoulin, “Monte Carlo Generator for Muon Pair Production,”CERN-SL-2002-016 (AP) and CLIC Note 511, May 2002.[2] GEANT4 Collaboration, S. Agostinelli et al., “GEANT4: A simulation toolkit,”SLAC-PUB-9350, sumitted to NIM.[3] A. Akhiezer and V. Berestetskii. Quantum Electrodymanics. N.-Y., Interscience, Internat. Sci.Monographs and Texts in Physics and Astromomy, 11. Ed. by R.E. Marshak, 1965.[4] Landau and Lifshitz. Course of Theoretical Physics, vol. IV Quantum Electrodymanics,V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii.[5] L. P. Keller, “Muon background in a 1-TeV linear collider,” Contributed to 5th InternationalWorkshop on Next-Generation Linear Colliders, Stanford, CA, 13-21 Oct 1993,SLAC-PUB-6385.[6] S. Fartoukh, J. B. Jeanneret, and J. Pancin, “Heat deposition by transient beam passage inspoilers,” CERN-SL-2001-012-AP, CERN-CLIC-NOTE-477, submitted to Phys. Rev. STAB.7

Production of muon pairs in annihilation of high-energy positrons with resting electrons

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