Electron-positron production in kinematic conditions of PrimEx
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1 Electron-positron production in kinematic conditions of PrimEx Alexandr Korchin Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine 1
2 We consider photoproduction of e + e pairs on a nucleus with atomic weight A and atomic number Z: γ + A e + + e + A at photon energies ω a few GeV s and very small momentum transfer Q relevant for PrimEx. Cross section of this process consists of the contributions listed below in order of importance. Bethe-Heitler mechanism of pair production on the nucleus (coherent process) with screening effects due to atomic electrons and Coulomb distortion. Pair production on atomic electrons with excitation of all atomic states. It contains correlation effects due to presence of other electrons and nucleus. Quantum Electrodynamical (QED) radiative corrections (of order α/π with respect to dominant contributions): (i) virtual-photon loops and (ii) realphoton process γ + A e + + e + A + γ, where the final photon has the energy ω δω (energy resolution in experiment). 2
3 Nuclear incoherent contribution quasi-elastic, or quasi-free process on the proton γ+p e + +e +p. Nuclear coherent contribution, or virtual Compton Scattering (CS) two-step mechanism γ +A γ + A e + + e + A. Below these contributions are reviewed in some detail. Pair production on nucleus The exclusive cross section on the nucleus has the form d 4 σ A dθ dθ + dφ = Z2 α 3 2πω 3 Q 4 F A( Q 2 ) f at ( Q 2 ) 2 T 2, where α = 1/ is the fine-structure constant, θ +, θ are the lepton polar angles, φ is the azimuthal angle between the plane spanned by the momenta k, p + and the plane spanned by k, p, k = (ω, k) is the photon 4-momentum, p + = (ɛ +, p + ) and p = (ɛ, p ) is the positron and electron 4-momentum respectively and 3
4 m e is the electron mass. Q = k p + p is 3-momentum transfered to the nucleus. In no-recoil approximation Q 0 = ω ɛ + ɛ = 0. T 2 is a kinematic factor T 2 = p + p sin θ + sin θ [ p2 + ξ+ 2 (4ɛ 2 Q 2 ) sin 2 θ + p2 ξ 2 (4ɛ 2 + Q 2 ) sin 2 θ + 2ω2 ξ + ξ (p 2 + sin 2 θ + +p 2 sin 2 θ ) + 2p +p ξ + ξ (2ɛ ɛ 2 Q 2 ) sin θ + sin θ cos φ] where ξ ± = ɛ ± p ± cos θ ± and p ± = (ɛ 2 ± m 2 e) 1/2. x + ɛ + /ω (x ɛ /ω) is fraction of energy carried by positron (electron), and x + + x = 1. f at ( Q 2 ) is the atomic form factor describing charge distribution of electrons ρ at (r). Screening of the nucleus charge by atomic electrons becomes important if Q < a 1 at, where a at 137Z 1/3 /m e cm is the atomic size. Since the main contribution to cross section 4
5 comes from the region close to the minimal possible value Q min m e ( m eω 2ɛ + ɛ ) = m e 2x + x ( m e ω ), then at high energies, where ω >> m e and x +, x are not too close to zero, Q min is always less than a 1 at 10 kev, and the screening is very important. In calculation the Thomas-Fermi-Moliere model [Moliere] is used f at ( Q 3 2 a iq 2 ) = 1, Q 2 + b 2 i /b2 0 i=1 with parameters b 0, b i and a i. Further, F A ( Q 2 ) is the nuclear charge form factor (Fourier transform of ρ A (r)) which behaves like F A ( Q 2 ) Q 2 r 2 A where r 2 A is the m.s.r. of the nucleus. In the conditions of PrimEx this form factor can always be replaced by unity since Q < a 1 at << r 1 nucl. 5
6 Energy distribution of the positrons has the form [Bethe and Heitler]: dσ A d 4 σ A = dθ dθ + dφ dθ + dθ dφ = Z 2 α 3 m 2 eω 3[(ɛ2 + + ɛ 2 )(φ ɛ +ɛ (φ log Z 4f)]. log Z 4f) f f((αz) 2 ) is the Coulomb distortion function [Bethe and Maximon] f((αz) 2 ) = (αz) 2 n=1 1 n[n 2 + (αz) 2 ]. For the 12 C nucleus f Functions φ 1,2 depend on the parameter γ = 100Z 1/3 m e ωx + x and account for the screening effect. If γ >> 1 then screening is unimportant, while for γ 0 screening is essential. 6
7 In the conditions of PrimEx screening leads to reduction of cross sections by a factor depending on x +. The functions φ 1,2 are calculated in the Thomas- Fermi-Moliere model. Note that the recoil of the nucleus can be neglected as the energy of the final nucleus is of the order 1 kev in the conditions of PrimEx. Pair production on atomic electrons The corresponding cross section has the form d 4 σ e dθ dθ + dφ = Z α3 2πω 3 Q 4H( Q 2 ) T 2, where H( Q 2 ) is a correlation factor related to all atomic excitations. One can also say that it accounts for screening effects due to the presence of other electrons and the nucleus. It can be chosen as [Tsai]: H( Q 2 ) = a 4 Q 4 (1 + a 2 Q 2 ) 2 7
8 with a = 1194Z 2/3 /(e 1/2 m e ) which follows from the value of radiation logarithm L rad in (see below). The energy distribution of positrons has the form dσ e = Z α3 m 2 eω 3[(ɛ2 ++ɛ 2 )(ψ log Z)+2 3 ɛ +ɛ (ψ log Z)]. The functions ψ 1,2 [Wheeler and Lamb] depend on the parameter ε = 100Z 2/3 m e ωx + x. Values φ 1 (0) and ψ 1 (0) are directly related to the unit radiation length of material X 0 = A Z 2 (L rad f) + ZL, rad where in the Thomas-Fermi-Moliere model L rad = 1 4 φ 1(0) 1 3 log(z) = log( Z 1/3 ), L rad = 1 4 ψ 1(0) 2 3 log(z) = log(1194 Z 2/3 ). For example, for carbon one obtains X 0 = g/cm 2. 8
9 Cross section can be further corrected for recoil of the target electron where dσ e dσ e (1 δ rec ), δ rec = ( m e ω ) 4L3 /3 3L L L/9 218/27 with L log(2ω/m e ). This is a small effect of order at high photon energies. Radiative corrections QED radiative corrections are included in the method of Weizsacker-Williams following the work of Mork. There are 16 Feynman diagrams which describe virtualphoton loops (vertex modification, lepton self-energy insertions, box diagrams and vacuum polarization), and real-photon emission. The latter contribution depends on the energy resolution. Specifically, if the energy difference 9
10 ω ɛ + ɛ is determined with accuracy δω then the emitted photon may have the energy ω ω max = δω. The energy resolution in pair production experiments δω/ω varies from 1.70% to 1.85%. The radiative corrections modify the lowest-order cross section dσ A,e dσ A,e (1 + δ RC ), and for the correction factor δ RC we obtain δ RC = G 1 + G 2 log( δω ω ) + G 3(log(R)) 1. Function G 2 originates from real-photon contribution, G 1 from real- and virtual-photon contributions and G 3 from vacuum-polarization diagrams. The quantity R is defined as m e R = [( ) 2 + ( Z1/3 2ωx + x 183 )2 ] 1/2, and R 2x + x ω/m e if screening is unimportant, while R 183Z 1/3 for complete screening relevant for PrimEx. 10
11 Radiative correction due to vacuum polarization is small and therefore δ RC virtually depends only on x +. Typical radiation correction is shown on Fig. 1. For the other photon energies the corresponding δ RC practically coincides with the curve in Fig. 1. Nuclear incoherent contribution The photon interacts with the nucleon inside the nucleus as with a free nucleon. The cross section of this quasi-free (quasi-elastic) incoherent process in the Fermigas model can approximately be written as follows d 4 σ N dθ dθ + dφ = Z α3 2πω 3 Q 4 F p( Q 2 ) 2 I F ( Q 2 ) T 2. The factor I F ( Q 2 ) takes into account Pauli blocking for the final proton I F ( Q 2 ) = 3 Q Q (1 2 4p F 12p 2 ), if Q 2p F, F = 1, if Q > 2p F. 11
12 Radiative correction δ RC [%] 1.5 e + e - production on 12 C 1.0 dσ / dε + = dσ 0 / dε + (1 + δ RC ) ω = GeV ε + / ω Figure 1: Radiative correction δ RC as a function of the fraction of energy carried by positron. 12
13 Here p F is the Fermi momentum for the nucleus A, in particular, for 12 C nucleus, one finds from electron scattering 12 C(e, e ) experiments [Whitney et al.] p F = 221 MeV. The contribution from the neutrons is neglected, as well as magnetic moments of the nucleons. The latter contribution is proportional to a very small factor Q 2 /(4MN 2 ). F p ( Q 2 ) is the proton charge form factor (for example, in the dipole parameterization) F p ( Q 2 ) = (1 + Q GeV 2) 2. This form factor at very small values of Q is practically equal to unity. The energy distribution is obtained by integration of this cross section over the angles of the leptons dσ N = Z 4α3 m 2 eω 3 β F 2 { (ɛ2 + + ɛ 2 )B 13
14 +4ɛ + ɛ [ β F + β F (1 + β F )B]}, where β F = 3m e 2p F (1 m eω 2ɛ + ɛ ) 2, β F B = log( ). 1 + β F At very small Q, of the order m e or less, the incoherent cross section is suppressed due to the Pauli blocking effect. Rough estimate gives β F , β F 2 log(β F) 10 2, therefore it is at most Z of the dominant Bethe- Heitler cross section on the nucleus. Nuclear coherent contribution The nuclear coherent contribution is the most complicated and least known mechanism of pair production. The corresponding amplitude of the virtual Compton scattering (CS) on the nucleus, M V CS, describes the process 14
15 γa γ A e + e A. We need this amplitude for almost forward scattering in the energy region a few GeV s, i.e. above the region of excitation of baryon resonances. At these enrgies this is diffractive process which can be viewed as the t channel exchanges by Regge trajectories associated with the so-called Pomeron and a few scalar and tensor mesons. At very high energies and small momentum transfer we can express the virtual CS amplitude on the nucleus in terms of the spin-averaged Compton scattering amplitude on the free proton (neglecting the neutron contributions): M V CS ZM (p) V CS. The magnitude of M V CS 2 is extremely small, therefore it is sufficient to keep only the interference between the dominant Bethe-Heitler amplitude on the nucleus and virtual CS amplitude 2Re(M A M VCS ). The virtual CS amplitude on the proton, M (p) V CS, can be related to forward scattering Compton amplitude with real photons f 1 (ω). The energy dependence of this amplitude is known 15
16 from high-energy CS experiments [Armstrong et al.]. In particular, for the real part we obtain parameterization Re f 1 (ω) = α M p [ ( ω ω 0 ) 1/2 ], where ω 0 =1 GeV and M p = MeV is the proton mass. Here we consider only kinematics for pair production at very small (but not equal) angles and the region δ + δ 1, δ ± = ɛ ±θ ± m e, δ + δ < m e ɛ ±, φ π < m e ɛ ±, which gives the dominant logarithmic contribution to the energy distribution in the conditions of PrimEx. In these conditions we find the ratio of the interference cross section and the nucleus Bethe-Heitler cross section R (x x + ) m 2 e ωm p [ ( ω ω 0 ) 1/2 ]. 16
17 At photon energy 5 GeV, for example, one obtains R (x x + ). For the energy distribution we get the estimate dσ inter 10 7 (x x + ) dσ A. This minor contribution can be completely neglected. Note that the interference results in asymmetry between the positron and electron yields. The cross section is not anymore symmetrical in variables x +, x and depends on which lepton (e + or e ) has greater energy, i.e. σ(x +, x ) x >x + > σ(x +, x ) x+ >x. This is an example of the charge asymmetry caused by the BH virtual CS interference in general case [Bjorken, Drell and Frautschi] 2Re(M A M VCS ) M A 2 = σ(e+, e ) σ(e, e + ) σ(e +, e ) + σ(e, e + ). 17
18 mechanism contribution in % nuclear Bethe-Heitler atomic electrons nuclear incoherent (quasielastic) nuclear coherent (virtual CS) 10 5 total Table 1: Various contributions to cross section at photon energy 4.91 GeV. x + = 0.4 and x = 0.6 In order to minimize the virtual CS contribution the symmetric conditions for the leptons ɛ + = ɛ, θ + = θ and φ = π are the most appropriate. In this case the interference term 2Re(M BH M VCS ) vanishes identically and one is left with virtual CS amplitude squared M V CS 2, which can be safely omitted. The cross section (energy distribution) including all 18
19 above contributions is thus dσ = dσ A + dσ e + dσ N. Finally, the total cross section σ tot = is presented in Table 2. ω me m e dσ 19
20 photon energy in GeV σ tot in mb Table 2: Total cross section of pair production on carbon at various photon energies 20
21 90 80 ω = 4.91 GeV δω / ω = 1.76 % dσ/dε + [mb/gev] ε + / ω Figure 2: Energy distribution of positrons in e + e production on carbon. The photon energy is 4.91 GeV. 21
22 dσ/dε + [mb/gev] ω = 5.46 GeV δω / ω = 1.77 % ε + / ω Figure 3: The same as in Fig. 2 but for the photon energy 5.46 GeV. 22
arxiv:nucl-th/ v1 3 Jul 1996
MKPH-T-96-15 arxiv:nucl-th/9607004v1 3 Jul 1996 POLARIZATION PHENOMENA IN SMALL-ANGLE PHOTOPRODUCTION OF e + e PAIRS AND THE GDH SUM RULE A.I. L VOV a Lebedev Physical Institute, Russian Academy of Sciences,
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