2.2.1 Radioactive Beta-Plus-Decay
𝒖
𝒆+
𝝂𝒆
𝑾+ 𝒖
𝒖
𝒅 𝒅 𝒅
𝒑 𝒏
Figure 2.1: Feynman diagram of the β+-decay. A free proton cannot decay to a free neutron, but bound to a nucleus, the proton may "decay" to a neutron via the emission of a positron and an electron-neutrino if the daughter nucleus obtains a lower ground state energy than its parent. The fundamental transition converts an up-quark of the proton to a down-quark by the emission of aW+-boson.
Positrons can be created through electromagnetic and weak interactions. In β+ -decays, radioactive nuclei decay by emitting a positron and an electron-neutrino, because they have too few neutrons to be stable. In the table of nuclides after Emilio Sergè (plotting proton numberZ against neutron number N), theβ+-unstable iso-topes are located "above" the stable nuclei, and titled "proton rich" although many of them have a larger number of neutrons (see NNDC 2016). Energetically, a β+ -decay is only possible if the mass of the parent atom is higher than its daughter nucleus by at least 1022 keV c−2, i.e. two times the mass of a β-particle. The par-ent atom has to lose one electron to conserve the charge numberZ, and to account for the mass of the positron1. If this is not the case, the alternative and competing decay channel is the electron-capture (EC) reaction. The large family ofβ+-decays in radioactive nuclei can be described by the basic equation
A
ZXN −→Z−1AYN+1++10e+00νe, (2.7) whereAZXN is the parent isotopeX with Z protons and electrons,N neutrons, and A = Z +N nucleons. X decays to its daughter Y with one proton less, and one neutron more, and emits a positron, e+, and an electron-neutrino,νe, to account for energy and momentum conservation in this three-body decay. Energy,Q, is released in the form of kinetic energy or gamma-rays, such that
1This is counterintuitive since the atomic shells do not participate in a nuclear reaction. But because the masses of fully ionised, i.e. electron-free, nuclei are hard to measure, the atomic weights of parent and daughter are used to calculate Q-values in general.
Q=
m(AZXN)−(m(Z−1AYN+1) + 2me+ 2mν)
c2. (2.8)
Inside a β+-unstable nucleus, this may be considered as the conversion of a proton to a neutron, see Eq. (2.9). In this view, it is obvious that such a decay can only occur in a composite nucleus because the mass of the neutron is larger than that of the proton (mn−mp ≈1.293 MeV c−2, Mohr et al. (2015)), and thus energetically prohibited (Krane 1987).
p−→n+e++νe (2.9)
The reaction in Eq. (2.9) can be reduced to the emission of a W+-boson by the one of the up-quarks of the proton (valence quarks: uud), thereby converting to a down-quark to form the neutron (valence down-quarks: udd), and followed by the immediate
"decay" of theW+-boson to a positron and an electron-neutrino (see Eq. (2.10) and Fig. 2.1)
u−→d+e++νe. (2.10)
The lifetimes ofβ+-unstable nuclei range from a few nanoseconds to several billions years (Firestone et al. 1996). This reects the dierences in stability of the particular neutron-proton combination. Dierent time-scales, especially in the astrophysical context, can be traced by the measurement of positrons.
2.2.2 Decay of Leptons
Electron and positron, respectively, are the lightest lepton. Therefore, they cannot decay into smaller elementary particles, on time scales of the order of the age of the universe (mean lifetime τ(e±) > 4.6×1026 yr Olive & Particle Data Group 2014).
The second and third family of leptons, the muons (µ±) and tauons (tau leptons, τ±), have shorter lifetimes and larger masses, and decay only weakly.
Muons have a mass of mµ = 105.6583715(35) MeV c−2, and a lifetime of τµ = 2.1969811(22) µs. The dominant decay channel for muons is
µ+ −→e++νe+ ¯νµ
µ− −→e−+ ¯νe+νµ (2.11)
with an almost 100% branching ratio (Olive & Particle Data Group 2014). Muons are created in space as secondary products of cosmic-ray interactions with the inter-stellar medium (ISM) or the atmosphere of Earth (e.g. Shapiro 2012). In cosmic-ray proton-proton or proton-nuclei reactions, among many other particles, charged pions can be created (see Sec. 2.2.3), which predominantly decay into muons. In principle, muons could be created during radioactive decay, similar to theβ+-decay described
in Eq. (2.10), but with an antimuon and muon-neutrino instead. The rest mass of the muon is larger than all known decay energies and can therefore never be cre-ated during a radioactive decay (Firestone et al. 1996). If high-energy cosmic-rays interact with the atmosphere of the planet, relativistic muons can be created and detected on ground with bubble chambers, for example (Street & Stevenson 1937).
Tau leptons are even heavier (mτ = 1776.82(16) MeV c−2), and hence even shorter lived ττ = 290.3(5)×10−15 s. They have more decay channels with distinguished branching ratios. The dominant channel (≈ 65%) for tau leptons is the decay into hadrons, mainly charged mesons (kaons and pions, see Sec. 2.2.3), which on their side again decay predominantly to muons but also to electrons/positrons. Tauons have an almost equal share in their branching ratios to also decay directly either to muons (17.41(4)%) or electrons/positrons (17.83(4)%). In most of the decay channels, one of the nal products will be in the lightest charged lepton family, electrons or positrons (Olive & Particle Data Group 2014). For the discussion of the galactic positron content, the creation and decay of tauons is probably unimportant, also because they have only been measured in accelerator experiments (rst evidence by Perl et al. 1975).
2.2.3 Decay of Mesons
Mesons are bound states of dierent quark-antiquark combinations. Pions belong to the lightest meson family with three dierent combinations of up, down, anti-up, and anti-down quarks, forming either a positively, negatively, or non-charged particle (isospin triplet). In the quark model, the π+ meson is described by |π+i =
ud¯ its antiparticle is the π− and is described by charge conjugation of the π+, so that,
|π−i = |d¯ui. The neutral pion, π0, consists of a combination of up-antiup quarks, and down-antidown quarks, |π0i= √1
2 |u¯ui − dd¯
(e.g. Griths 1987).
Charged pions have a mass ofmπ± = 139.57018(35) MeV c−2, and a mean lifetime of τπ± = 2.6033(5)×10−8 s. Theπ0 meson has a mass ofmπ0 = 134.9766(6) MeV c−2, and a considerably shorter lifetime ofτπ0 = 8.52(18)×10−17 s, due to the dominant electromagnetic decay mode (Olive & Particle Data Group 2014).
Another family of mesons with strange quark constituents are the kaons. Charged kaons are|K+i=|u¯si, and|K−i=|s¯ui, with a mass ofmK± = 493.677(16) MeV c−2 and a lifetime of τK± = 1.2380(21)×10−8 s (Olive & Particle Data Group 2014).
Due to neutral kaon mixing (CP-violation, see Griths (1987) for example), there exist two distinct neutral kaon states, named KL0 ("K-long") and KS0 ("K-short"), with largely dierent lifetimes, and decay channels: |KL0i= √12 |d¯si+
sd¯
with a lifetime ofτK0
L = 5.116(21)×10−8 s, and |KS0i= √12 |d¯si − sd¯
with a lifetime of τK0
S = 8.954(4)×10−11s. Neutral kaons2have a mass ofmK0 = 497.614(24) MeV c−2 (Olive & Particle Data Group 2014).
Pions and kaons are produced as secondary particles in cosmic-ray interactions (e.g. Shapiro 2012). Highly relativistic cosmic-rays (kinetic energies > 1 GeV, see
2In the denitions of KL0
and K0S
a CP-violating asymmetry factor when mixing the states is omitted. Likewise, due to this factor,KL0 andKS0 have a mass dierence ofmK0
L−mK0
S = 3.484(6)×10−12MeV c−2. For the discussion of positrons production, this is irrelevant and only mentioned for completeness.
Sec. 5.3) react with atoms or molecules of the ISM or Earth's atmosphere, and create a large number of secondary particles, among them pions, kaons, neutrons, protons, and photons. These particles carry parts of the incident kinetic (centre of mass) energy away, creating more secondary particles, and also decay very fast. In the end, a cascade of decay particles is created, which, in case of Earth's atmosphere, can be traced back with Cherenkov telescopes, for example, to measure the direction and energy of the incident cosmic-ray (see Sec. 5.3).
Charged pions predominantly decay to muons with a branching ratio of99.98770(4)%, and only a small fraction (0.0123(4)%) is directly converted to electrons/positrons (Olive & Particle Data Group 2014). However, muons inevitably decay to elec-trons/positrons, Eq. (2.11), so that after a π+ has decayed at least one positron is created. The decay is related to theβ+-decay as the positive charge is carried away by a W+-boson to create an antimuon or positron, so that
π+ −→ µ++νµ −→e++νe+ ¯νµ+νµ
π+ −→ e++νe. (2.12)
Pi-zeros are the lightest quarkonia, i.e. bound states of quark and related antiquark, and, unlike charged pions, only decay electromagnetically. Their dominant decay channel (98.823(34)%) is the annihilation of its quark constituents to two gamma-ray photons, each of energy Eγ(π0) = 67.4883(3) MeV in the centre of mass frame.
However, it is also possible for π0 to decay into (multiple) electron-positron pairs (or Ps directly), possibly accompanied by another photon (Olive & Particle Data Group 2014).
Due to its larger mass compared to pions, kaons have more possible decay channels.
Similar to the π+-decay, K+s favour to convert to anti-muons with a probability of 63.55(11)% and only seldom decay into positrons directly (0.001581(7)%). Their hadronic decay modes also produce charged pions, which decay to muons and elec-trons/positrons, Eq. (2.12), in a second step. It is also possible for charged kaons to decay in a semileptonic way, producing neutral pions, accompanied by charged leptons. In any of the described cases, the nal result will be at least one positron (Olive & Particle Data Group 2014), e.g.
K+ −→ µ++νµ−→e++νe+ ¯νµ+νµ
K+ −→ π++π0 −→ · · · −→e++νe+ ¯νµ+νµ+ 2γ
K+ −→ π0+e++νe−→2γ+e++νe. (2.13) The energy spectrum of galactic cosmic rays reaches up to ∼ 1015 eV (Ackermann et al. 2013). The highest energies measured so far are of the order1021 eV (Watson et al. 2011), which makes it in general possible to create any kind of meson species, which may decay into positrons in the end. Forming other species involving charm, bottom, and top quarks is less likely and mesons made of such quarks will not be considered further.
2.2.4 Electron-Positron Pair Production
The time-energy uncertainty principle allows the existence of particle-antiparticle pairs for a limited time, out of the vacuum (virtual particles). If the vacuum is inuenced by the available internal energy in an arbitrary system, the emergence of real particles may be possible. In such a system, pair production may arise as another degree of freedom, if the internal energy exceeds2mc2. In high-energy environments (E >1022 keV), electron-positron pair production is hence inevitable.
2.2.4.1 Photon-Photon Interactions
If two photons with a total energy of at least the rest mass of two electrons interact with each other, it is possible to produce electron-positron pairs. This is related to the conversion and conservation of energy (equivalence of mass and energy),
E =mc2 (2.14)
which means that any form of energy can be used to create real particles - and vice versa (Einstein 1905, see also pair annihilation Sec. 2.3). The photon-photon pair-production,
γ+γ −→e++e−, (2.15)
is illustrated by the Feynman diagrams in Fig. 2.2. Theγγ-pair-creation cross sec-tion, σpair, is related to the annihilation cross section (σann, Sec. 2.3) and the cross section for the Compton eect (σcomp, Sec. 3.1.1), by considering the crossing sym-metry (i.e. the correct exchange of the Mandelstam variables or the four-momenta in the Feynman diagrams, Fig. 2.2, respectively; Greiner et al. (see 2012), and Ap-pendix B). The total pair creation cross section is given by
σpair = π 2
α2
m2e(1−β2)
(3−β4) ln
1 +β 1−β
−2β(2−β2)
(2.16) whereβ = (|p|/E)CMis the dimensionless velocity in the centre of mass (CM) frame,
|p|is the momentum of the CM frame, and E the CM energy (Olive & Particle Data Group 2014, note that in Eq. (2.16) natural units, i.e. ~ = c = 1, have been used for clarity). In the non-relativistic limit, β → 0, Eq. (2.16) can be approximated byσpairNR ≈πmα22
eβ, from which it is clear that photon-photon pair-production is only possible for β > 0. The ultra-relativistic limit, β → 1, can be expressed as σpairUR ≈ πmα22
e
1
γ2(2 ln(2γ)−1). Fig. 2.3 shows the total pair creation cross section as a function of the CM photon "Lorentz factor". The function has a maximum atβmax ≈0.701, equivalent to an incident photon energy of Eγ ≈ 716.5 keV, for a cross section of σmaxpair ≈ 1.70×10−25 cm2. The process in Eq. (2.15) is kinematically only allowed if the incident photon energies are above the pair creation threshold, Eq. (2.17).
𝒆+ 𝒆−
𝜸𝟏
𝜸𝟐
(a)
𝒆+ 𝒆−
𝜸𝟐
𝜸𝟏
(b)
Figure 2.2: Feynman diagrams of electron-positron pair production by photon-photon interactions. The incoming photon polarisation vectors are denoted by(~pi, λi), wherei= 1,2andλ= +,−, and the outgoing anti-spinors and anti-spinors for the electron and positron are denoted byu(~¯ p−, σ−)andv(~p+, σ+), respectively, whereσ±= 1,2is the spin of the particles.
Because the photons are interchangeable, the energy of one photon can be chosen as an independent variable, here E1, so that E2 becomes a function of E1,
E2 > Ethresh(E1, φ) = 2m2ec4
E1(1−cos(φ)), (2.17) where φ is the scattering angle (De Angelis et al. 2013, see also Appendix B). In the case of head-on collisions between the photons, φ = π, Eq. (2.17) reduces to E1E2 ≥ m2ec4, which would be naturally expected from the conversion of energy to mass, Eq. (2.14).
1 10 100
10−5 10−4 10−3 10−2 10−1 100
1 10 100
Centre of momentum Lorentz factor γ 10−5
10−4 10−3 10−2 10−1 100
Cross section σ(γγ→e+ e− ) [10−24 cm2 ])
Figure 2.3: Photon-photon pair production cross section as a function of centre of momentum "Lorentz factor", γ= mEcm
ec2, of either photon. Pair creation is only possible forγ >1, i.e. if the total energy of both photons together exceeds two times the rest mass of an electron. The pair creation cross section by photons reaches a maximum atβ≈0.701, corresponding toγ≈1.4, and then declines∝ln(2γ)/γ2; see text for details. Althoughσpair is of the order of the Thomson cross section, experimental proofs are dicult to achieve because of the necessity of high-energy photon beams and large photon densities.
With σpair ∼ 10−25 cm2, it is of the order of the Thomson cross section, σT =
6.6524587158(91)×10−25cm2(Mohr et al. 2015), but experimentally, photon-photon pair-production has never been observed. Pair production is only ecient at very high photon densities, but in terrestrial laboratories, such intense beams of high-energy photons are not achievable (Greiner et al. 2012). However, in compact as-trophysical sources, the optical depth of gamma-ray sources can be large enough to eciently absorb themselves (see Sec. 5.2). The CM velocity can be written as
β = |p|
E
CM
=
pECM2 −4m2e
ECM =
s
1− 2m2e
E1E2(1−cos(φ)), (2.18) so that at the maximum of σpair, at βmax ≈0.701, is reached if
E2 = 2m2e
(1−βmax2 )(1−cos(φ))E1
≈
1
(1−cos(φ))E1[TeV]
eV. (2.19)
At very high energies (Eγ & 1 TeV), the process described in Eq. (2.19) leads to the eect that the Universe becomes opaque, because such photons rapidly nd low-energy photon partners to interact with. These are much more numerous, e.g.
from star light (visible light at 500 nm wavelength equivalent to 2.5 eV photon energy) or from the cosmic microwave background (1 mm equivalent to 1 meV), so that electron-positron pairs are inevitably created, preventing high-energy photons to travel far. In particular, very-high-energy photons in the energy range between 10−2 and 102 TeV are most eciently absorbed by extragalactic background light at infrared, visible, and ultraviolet wavelengths. Between 102 and 107 TeV, cosmic microwave background photon interactions dominantly suppress the detectability, and above 107 TeV, radio background photons absorb such emission (De Angelis et al. 2013). Therefore, if such very-high-energy emission is measured, it can be considered to originate mainly in the Milky Way.
2.2.4.2 Photon Interactions with Electric Fields
For a single photon in the vacuum, it is not possible to spontaneously produce an electron-positron pair. Kinematically, energy and momentum of the photon are not conserved at the same time, so the photon needs an electromagnetically charged partner to interact with, and which is carrying away the excess momentum.
γ +AZXN −→e++e−+AZXN (2.20) Pair creation can therefore happen, e.g., in the eld of an atomic nucleus with charge −Ze, which is exhibiting an external Coulomb eld, Eq. (2.20). Fig. 2.4 shows the Feynman diagrams for pair-production in an electric eld. The reaction is related to the bremsstrahlung process when considering crossing symmetry. During bremsstrahlung, electrons (or other charged particles) scatter o protons or heavier charged particles (electric elds in general), emit real photons, and lose energy in this way (they decelerate; German "bremsen"). Similarly, a free electron cannot emit a
photon spontaneously, because energy and momentum would not be conserved in this process. In Fig. 2.4, to obtain the diagrams for bremsstrahlung, the external lines for pair production must be exchanged: The incident photon becomes the outgoing photon, and the outgoing positron becomes the incident electron (Jauch
& Rohrlich 1955; Greiner et al. 2012).
𝒊𝒁𝒆𝜸𝟎
Figure 2.4: Feynman diagrams of electron-positron pair production in the external eld of a nucleus of charge−Ze. The incoming photon polarisation vector is denoted by(~p1, λ1), whereλ= +,−, and the outgoing anti-spinors and anti-spinors for the electron and positron are denoted by¯u(~p−, σ−)and¯(~p+, σ+), respectively, where σ± = 1,2is the spin of the particles. The external Coulomb eld, Acould0 (x) = −Ze|~x|, for the position vector~xof the nucleus, takes the form−4πZeR d3q
(2π)3
exp(−i~q·~x)
|~q|2 in momentum space, where
~
q= ~p++~p−−~p1 is the momentum transferred to the nucleus (e.g. Bethe & Heitler 1934; Jauch &
Rohrlich 1955; Greiner et al. 2012).
The total cross section for pair creation of a photon with energyE in the eld of an atomic nucleus with charge−Zecannot be written out in a closed form, analytically, as stepping from the dierential cross section (Bethe & Heitler 1934) to the total one involves elliptical integrals. A compact form is given by Racah (1936), Jost et al.
(1950), and Jauch & Rohrlich (1955),
σZ,pair(E) = αZ2re2
2η2[2C2(η)−D2(η)] +
−2/27
(109 + 64η2)E2(η)−(67 + 6η2)(1−η2)F2(η) . (2.21) In Eq. (2.21), η−1 = 2mE
e ≥ 1 is the dimensionless energy of the photon, scaled by the threshold of 2me for pair creation, and re = 4π1
0
e2
mec2 = 2.8179403227(19) fm is the classical electron radius (Mohr et al. 2015). The functionsC2(η),D2(η), E2(η), and F2(η) are given in Appendix B. The pair-creation cross section in the eld of a nucleus, σZ,pair(E), is a strictly increasing function of photon energy so that with increasing photon energy, the production of electron-positron pairs is more and more probable. Near the threshold energy, i.e. in the low-energy limit, σZ,pair can be approximated by
in the high-energy limit by
σZ,pairU R (E) =αZ2r2e 28
9 ln 2E
me
− 218 27
(2.23) (Jauch & Rohrlich 1955).
Pair creation is one of the three fundamental interactions of light with matter. The other two, the Compton eect as well as the photoelectric eect will be described in Sec. 3.1.1.
Single photons do not necessarily need to interact with the electric eld of the nucleus but rather need any electric eld, so pair creation is also possible in the eld of free electron,
γ+e−−→e++e−+e−. (2.24) The cross section for creation in the eld of an electron is asymptotically identical to Eq. (2.21), omitting the charge numberZ (Haug 1975; Zdziarski 1980). The only dierence is in the low-energy limit because the threshold for pair creation in the eld of a single electron in the laboratory frame is Eγ ≥ 4mec2, and in the eld of a nucleus it is Eγ >2mec2, see Appendix B.
If in electron-electron (e-e), electron-proton (e-p), or proton-proton (p-p) collisions, the kinetic energies of interacting particles are large enough, also electron-positron pairs can be created. Kuraev & Lipatov (1974) approximated the cross section for pair creation in inelastic e-e scattering to
σee,pair ≈ α2r2e
π (1.03ρ3−6.6ρ2−11.1ρ+ 100), (2.25) where ρ= ln
p1+p2
mec
2
, and p1 and p2 are the momenta of the incident electrons, respectively. This process is only possible for highly relativistic electrons as the threshold energy in the lab frame is7mec2 (Appendix B). The cross section for pair production in e-p collisions is similar to Eq. (2.25), see Zdziarski (1980).
2.2.4.3 Photon Interactions With Magnetic Fields
Similar to the case with an (external) electric eld, it is also possible for single photons to produce electron-positron pairs in a magnetic eld. When the gyro-energy of an electron (or positron) becomes of the order of the rest mass of the particle, spontaneous pair production will be ecient (Erber 1966; Daugherty &
Harding 1983). The rate of photons that will convert to electron-positron pairs was found to depend on the parameter χ = (2m~ω
ec2)(BBsin(θ)
cr ), where ~ω is the incident photon energy, B the magnetic eld, θ the angle between the magnetic eld vector
and the photon momentum vector, and Bcr = m2ec2
~e ≈ 4.4×1013 Gauss (Toll 1952;
Erber 1966; Baier & Katkov 1968; Tsai & Erber 1974; Daugherty & Harding 1983).
Even though there is no physical particle which takes the excess photon momentum and energy, the "rigidity" of a present magnetic eld can retain energy and mo-mentum conservation. The eld, which is included in the conservation laws, does not require the momentum of the photon to be conserved orthogonal to the magnetic eld direction, Eq. (2.27). If a photon with momentum ~pγ = ω(0,sin(θ),cos(θ)) is moving with an angle θ through a homogeneous magnetic eld B~ = Bzˆ, without loss of generality, the conservation of energy and momentum reads (~=c= 1):
Energy: ω =E+k +E−j (2.26)
Momentum: pγ,z =p+,z+p−,z ⇔ωcos(θ) =p++p− (2.27) In Eq. (2.27),E+andE−are the quantised energy states of the positron and electron, respectively, given by the Landau levels of moving charged particles in a uniform magnetic eld
E+k = s
p2++m2e
1 + 2k B Bcr
, (2.28)
E−j = s
p2−+m2e
1 + 2j B Bcr
, (2.29)
with p+ and p− the momenta of the positron and electron, respectively, and k, j = 0,1,2,3, . . . the quantum levels (e.g. Johnson & Lippmann 1949). From the trans-ition of one photon to an electron-positron with discrete energy levels in a magnetic eld, the photon attenuation length α(χ) can be derived. This is a measure of how fast (at what distance) photons are being converted into pairs. In general, the num-ber of pairs n±, created during the photon path length x, can be calculated from the number of photons,nγ, by
n±=nγ(1−exp(−α(χ)x)). (2.30) The exact attenuation coecient was calculated rst by Toll (1952) and thoroughly discussed by Daugherty & Harding (1983). Because of the quantised energy levels,
n±=nγ(1−exp(−α(χ)x)). (2.30) The exact attenuation coecient was calculated rst by Toll (1952) and thoroughly discussed by Daugherty & Harding (1983). Because of the quantised energy levels,