A 5D, polarised, Bethe-Heitler event generator for γ→ μ + μ − conversion
AA 5D, polarised, Bethe-Heitler event generator for γ → µ + µ − conversion D. Bernard,LLR, Ecole Polytechnique, CNRS/IN2P3, 91128 Palaiseau, FranceOctober 29, 2019
Abstract
I describe a five-dimensional, polarised, Bethe-Heitler event generator of γ -ray conversions to µ + µ − , based on a generator for conversion to e + e − developed in the past. Verifications are per-formed from close-to-threshold to high energies. keywords :gamma rays, pair conversion, muon pairs, gamma factory, event generator, Bethe-Heitler, Geant4 In the past I wrote an event generator of γ -ray conversions to an e + e − pair [1] that we used forthe simulation of the HARPO experiment [2, 3]. The generator samples the exact, that is, five-dimensional, Bethe-Heitler differential cross section. Later I developed a new version of that code, thefortran model [4] of an algorithm appropriate to event generation in Geant4. The Geant4 toolkit isused for the detailed simulation of scientific experiments that involve the interaction of “elementary”particles with matter, in particular with a detector [5–7]. The generator has been made available asthe G4BetheHeitler5DModel physics model of Geant4 since release 10.5 [8, 9], see also SubSubSect.6.5.4 of [10].A Gamma Factory project is being proposed at CERN in the photon energy range 1 < E <
400 MeV [11], with photon intensities orders of magnitude above that presently available. Such afacility could be a powerful source of muons by gamma-ray conversions to µ + µ − , and therefore also apowerful source of neutrinos. A precise simulation of photon conversion to muons pairs was thereforeintensely desirable [12].The γ → µ + µ − physics model presently available in Geant4, G4GammaConversionToMuons isbased on the algorithm described in [13] and in Sect. 6.7 of [10] and in references therein. Thecalculation makes use of high-energy approximations and the model has been verified above incidentphoton energies of 10 GeV.In this note I characterise a modified version of the fortran model of conversions to e + e − docu-mented in [4] for conversions to µ + µ − and that is valid down to threshold, either for • conversions – on a nucleus (nuclear conversion) γ Z → µ + µ − Z ; – on an electron (triplet conversion) γ e − → µ + µ − e − . • on a target in an atom, or on an isolated target (“QED”)1 a r X i v : . [ phy s i c s . d a t a - a n ] O c t for linearly polarised or non-polarised incident photons.The differential cross section for conversions of non-polarised photons was obtained by Bethe &Heitler [14], and that to linearly polarised photons by [15–17] . The final state is five-dimensional,even when the recoiling target cannot be observed. It, the final state, can be defined by the polarangles θ + and θ − , and the azimuthal angles φ + and φ − , of the negative lepton ( − ) and of the positivelepton (+), respectively, and the fraction x + of the energy of the incident photon carried away by thethe positive lepton, x + ≡ E + /E (Table 1 of [4]). q is the momentum “transferred” to the target. Incase conversion takes place in the field of an isolated nucleus or electron, the Bethe-Heitler expressionis used, while when the nucleus or the electron is part of an atom, the screening of the target field bythe other electrons of the atom is described by a form factor, a function of q [18] (nuclear) or [19](triplet).Please note that in contrast with Bethe-Heitler [14], I do not assume that the energy carried awayby the recoiling target be negligible (compare the differential element of eq. (20) of [14] to that of eqs.(1)-(3) of [4]).The differential cross section diverges at small q and, for high-energy leptons, in the forwarddirection ( θ ≈ /q divergence is a particularly daunting nuisance as the expression for q involves several of the variables that describe the final state. The inconvenience for event generation isovercome as is usual in particle physics, by performing each step of the conversion (gamma-target → recoiling target-pair; pair → two leptons) in its own centre-of-mass frame (Sect. 3 of [4] and referencestherein).The problem is then reduced to using the Von Neumann acceptance-rejection method (Sect. 40.3of [24]) from a mock-up probability density function (pdf) that is the simple product of five independentpdfs, of five judiciously chosen variables x i , i = 1 · · · The bounds of x i , i = 2 · · · x , though, the bounds must be provided and they depend on the energy ofthe incident photon, see Fig. 1 of [4] for conversions to e + e − . The x bounds for conversions to µ + µ − are similar and are shown in Fig. 1.For better readability, energy-variation plots are presented as a function of the available energy E (cid:48) = E − E threshold , (1)where the energy threshold is E threshold = 2( m µ /M + m µ ) , (2)where M is the target mass. • For triplet conversion to µ + µ − , E threshold is close to 43.9 GeV. • For nuclear conversion to µ + µ − , E threshold ranges from 223.2 MeV for hydrogen targets down toalmost 2 m µ ≈ . x bounds are clearly not quite optimal for triplet conversion and might be tightened a bitfurther separately. The CPU gain for routine Geant4 use would be small though, due to the 1 /Z suppression of triplet with respect to nuclear and due to the much higher threshold energy. The polarised differential cross section first obtained by [15] was put in Bethe-Heitler form by [16], after which amisprint was corrected by [17]. nucl E’ x , m i n x , m ax -100-80-60-40-20020 MeV GeV TeV PeV triplet E’ x , m i n x , m ax Figure 1: Variation with the available energy E (cid:48) of the maximum and of the minimum values forvariable x , for photon conversions to µ + µ − . Left: nuclear conversion; Right: triplet conversion.“raw” isolated charged target (star), and the following atoms: helium (bullet), neon (square), argon(upper triangle), xenon (down triangle) and radon (plusses). The lines denote the bounds that areused by the generator. P = 1 and P = 0 samples are plotted for each photon-energy value. The acceptance-rejection method needs the knowledge of the maximum value of the differential crosssection, so as to determine a value of the constant C (see Sect. 5 of [4] or Sect. 40.3 of [24]). Thevariation of the maximum is shown in Fig. 2. As the maxima for all target masses were obtained fromthe same set of photon energies, and as the threshold varies with target mass for nuclear conversion,the available energy E (cid:48) also varies with target mass. In the nuclear conversion plot, for example, for E = 217 .
40 MeV photon conversion on helium, E threshold is ≈ .
26 MeV, that is, E (cid:48) ≈
140 keV. Theparametrisation of the maximum is obtained with the same expression as for e + e − (eq. (9) of [4])with a different set of parameters.In practice, for safety, C is enlarged by a factor of 1.5 with respect to what can be seen on Fig. 2. Figs. 1 – 3 are based on samples of 10 simulated events, and Figs. 4 – 6 on samples of 10 simulatedevents. “Raw” nuclear samples, that is, simulations of conversions of an isolated target (“QED”) havebeen produced assuming an argon nucleus mass. 3 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 rawHeNeAr XeRn nucl -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 rawHeNe Ar XeRn triplet MeV GeV TeV PeV
E’ 1
MeV GeV TeV PeV E’ Figure 2: Top: Variation of the maximum value of the pdf (arbitrary units) with the available energy E (cid:48) . Left: nuclear conversion; Right: triplet conversion. “raw” isolated charged target (star), and thefollowing atoms: helium (bullet), neon (square), argon (upper triangle), xenon (down triangle) andradon (plusses). The curves denote the fit to the present data of the expression of eq. (9) of [4]).Bottom: residues of the fit, with the 1.5 safety factor marked with a dashed line. P = 1 and P = 0samples are plotted for each photon-energy value. The measurement of the polarisation angle ϕ and of the linear polarisation fraction P of a gamma-raybeam can be performed by the analysis of the distribution of the event azimuthal angle ϕ d N d ϕ ∝ (1 + A P cos[2( ϕ − ϕ )]) , (3)Figure 3 shows the polarisation asymmetry, A , obtained from the ( P = 1) samples, definingthe event azimuthal angle as the bisector of the electron and of the positron azimuthal angles, ϕ ≡ ( φ + + φ − ) / E’ A MeV GeV TeV PeVHeNeArXeRnrawnucl E’ A MeV GeV TeV PeVHeNeArXeRnrawtriplet π Figure 3: Polarisation asymmetry calculated on simulated event samples as a function of E (cid:48) . Left:nuclear conversion. Right: triplet conversion. “raw” isolated charged target (star), and the followingatoms: helium (bullet), neon (square), argon (upper triangle), xenon (down triangle) and radon(plusses). The horizontal lines denote the low- and high-energy approximations of π/ / A ≈ (cid:18) (cid:19) log (cid:18) Em µ c (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) log (cid:18) Em µ c (cid:19) − (cid:18) (cid:19) . (4)At low energy the obtained values agree with the asymptotic value of π/ The distributions of the fraction x + ≡ E + /E of the energy of the incident photon that is taken awayby the positive lepton is shown in Fig. 4. They compare nicely with the published representations ofthe singly-differential cross section obtained by integration on all other variables (Fig. 5 of [14]). The distribution of the (log of the) recoil momentum is shown in Fig. 5. Figure 6 shows the distributions of the pair opening angle, θ + − , normalised to 1 /E for conversionson argon. At high energies, they peak at the most probable value, ˆ θ + − , computed by Olsen in the5 +
100 GeVtriplet1 TeV10 TeV100 TeV x + Figure 4: Distributions of the fraction of the energy of the incident photon that is taken away by thepositive lepton, x + ≡ E + /E , for various values of E (conversions on argon). -5 -2.5 0 2.5 5
100 TeV10 TeV1 TeV100 GeV10 GeV1 GeV log q(MeV/c)nucl -5 -2.5 0 2.5 5
100 TeV10 TeV1 TeV100 GeVlog q(MeV/c)triplet Figure 5: (log of the) recoil momentum spectra for gamma conversions on argon atoms.high-energy approximation [23], that is indicated by a vertical line:ˆ θ + − = 3 . mc E , (5)where m is the mass of the pair lepton. Here m = m µ andˆ θ + − ≈ . E . (6)6
100 TeV10 TeV1 TeV100 GeV10 GeV1 GeV Θ +- * E (rad MeV)nucl
100 TeV10 TeV1 TeV100 GeV Θ +- * E (rad MeV)triplet Figure 6: Distributions of the product of the pair opening angle and of the photon energy, θ + − × E ,for nuclear conversions on argon for 1 GeV (bullet), 10 GeV (full square), 100 GeV (upper triangle),1 TeV (down triangle), 10 TeV (circles) and 100 TeV (empty squares). The vertical value shows themost probable value of ≈ . rad · MeV computed by Olsen in the high-energy approximation [23](see also eqs. (5)-(6)).
I have adapted the sampler of the Bethe-Heitler differential cross section that I had written for gammaconversions to e + e − pairs [4] to the generation of gamma conversions to µ + µ − , by simply tuning theparametrisations of the bounds for variable x (Sect. 2) and of the maximum differential cross section(Sect. 3). Verifications of various distributions obtained by the algorithm have been performedsuccessfully from threshold to PeV energies (Sect. 4).The Geant4 physics model corresponding to the present algorithm is about to be made availableto users in release 10.6 [25]. I’d like to express my gratitude to Vladimir Ivantchenko who brought the need for a correct low-energy γ → µ + µ − event generator to my attention and to Igor Semeniouk and Mihaly Novak for stimulatingdiscussions. References [1] D. Bernard, “Polarimetry of cosmic gamma-ray sources above e + e − pair creation threshold,” Nucl. Instrum.Meth. A (2013) 765.[2] P. Gros et al. , “Performance measurement of HARPO: A time projection chamber as a gamma-ray telescopeand polarimeter,” Astropart. Phys. (2018) 10.
3] D. Bernard [HARPO Collaboration], “HARPO, a gas TPC active target for high-performance γ -ray as-tronomy; demonstration of the polarimetry of MeV γ -rays converting to e + e − pair,” Nucl. Instrum. Meth.A (2019) 405.[4] D. Bernard, “A 5D, polarised, Bethe-Heitler event generator for γ → e + e − conversion,” Nucl. Instrum.Meth. A (2018) 85.[5] S. Agostinelli et al. [GEANT4 Collaboration], “GEANT4: A Simulation toolkit,” Nucl. Instrum. Meth. A (2003) 250.[6] J. Allison et al. , “Recent Developments in Geant4,” Nucl. Instrum. Meth. A (2016) 186.[7] J. Apostolakis et al. , “Progress in Geant4 Electromagnetic Physics Modelling and Validation,” J. Phys.Conf. Ser. (2015) 072021.[8] I. Semeniouk and D. Bernard, “C++ implementation of Bethe-Heitler, 5D, polarized, γ → e + e − pair con-version event generator,” Nucl. Instrum. Meth. A (2019) 290.[9] V. Ivanchenko et al., “Progress of Geant4 electromagnetic physics developments and applications”, EPJWeb of Conferences , 02046 (2019).[10] Geant4 Physics Reference Manual, Release 10.5, March 5 2019 (Geant4 User Documentation).[11] M. Krasny et al. , “The CERN Gamma Factory Initiative: An Ultra-High Intensity Gamma Source,”9th International Particle Accelerator Conference, Vancouver, Canada, 29 Apr - 4 May 2018,doi:10.18429/JACoW-IPAC2018-WEYGBD3.[12] V. Ivantchenko,. “Muon pair production Monte Carlo”, Gamma Factory meeting, CERN, 25-28 March2019 (indico).[13] H. Burkhardt, S. R. Kelner and R. P. Kokoulin, “Monte Carlo generator for muon pair production,”CERN-SL-2002-016-AP, CLIC-NOTE-511.[14] H. Bethe and W. Heitler, “On the Stopping of Fast Particles and on the Creation of Positive Electrons”,Proceedings of the Royal Society of London A, (1934) 83.[15] T. H. Berlin and L. Madansky, “On the Detection of gamma-Ray Polarization by Pair Production”, Phys.Rev. (1950) 623.[16] M. M. May, “On the Polarization of High Energy Bremsstrahlung and of High Energy Pairs”, Phys. Rev. (1951) 265.[17] Jauch and Rohrlich, The theory of photons and electrons , (Springer Verlag, 1976).[18] N.F. Mott, H.S.W. Massey, “The Theory of Atomic Collisions”, University Press, Oxford, 1934.[19] J.A. Wheeler and W.E. Lamb, “Influence of atomic electrons on radiation and pair production”, Phys.Rev. (1939) 858 (errata in (1956) 1836).[20] P. Gros et al. [HARPO Collaboration], “ γ -ray telescopes using conversions to e + e − pairs: event generators,angular resolution and polarimetry,” Astropart. Phys. (2017) 60.[21] P. Gros et al. [HARPO Collaboration], “ γ -ray polarimetry with conversions to e + e − pairs: polarizationasymmetry and the way to measure it,” Astropart. Phys. (2017) 30.[22] V. F. Boldyshev and Y. P. Peresunko, “Electron-positron pair photoproduction on electrons and analysisof photon beam polarization,” Yad. Fiz. (1971) 1027.[23] H. Olsen, “Opening Angles of Electron-Positron Pairs,” Phys. Rev. (1963) 406.[24] M. Tanabashi et al. [Particle Data Group], “Review of Particle Physics,” Phys. Rev. D (2018) 030001,on pdglive.[25] I. Hˇrivn´acˇov´a for the Geant4 EM working group, “ Geant4 electromagnetic physics progress ”, the 24thInternational Conference on Computing in High-Energy and Nuclear Physics (CHEP 2019), 4 - 8 Nov.2019, Adelaide, Australia.(2018) 030001,on pdglive.[25] I. Hˇrivn´acˇov´a for the Geant4 EM working group, “ Geant4 electromagnetic physics progress ”, the 24thInternational Conference on Computing in High-Energy and Nuclear Physics (CHEP 2019), 4 - 8 Nov.2019, Adelaide, Australia.