Gamma Conversion into Lepton Pair

The class G4GammaConversionToMuons simulates the process of gamma conversion into muon or tau-lepton pairs. The same formulas are used in both case with the only difference in lepton mass. So, below only the case of the muon pair production will be described. Given the photon energy and \(Z\) and \(A\) of the material in which the photon converts, the probability for the conversions to take place is calculated according to a parameterized total cross section. Next, the sharing of the photon energy between the \(\mu^+\) and \(\mu^-\) is determined. Finally, the directions of the muons are generated. Details of the implementation are given below and can be also found in [BKK02].

Parameterization of the Total Cross Section

The total cross section is obtained by integration of the differential cross section Eq.(17), that is

(21)\[\sigma_{\rm tot}(E_\gamma) = \int_{x_{\rm min}}^{x_{\rm max}} \frac{d\sigma}{d x_+} \, d x_+ = 4 \, \alpha \, Z^2 \, r_c^2 \, \int_{x_{\rm min}}^{x_{\rm max}}\left(1-\frac43\,x_+x_-\right)\log(W) \, d x_+ \;.\]

\(W\) is a function of (\(x_+, E_\gamma\)) and (\(Z, A\)) of the element (see Eq.(18)). Numerical values of \(W\) are given in Table 7.

Table 7 Numerical values of \(W\) for \(x_+=0.5\) for different elements.
\(E_\gamma\) [GeV]	W for H	W for Be	W for Cu	W for Pb
1	2.11	1.594	1.3505	5.212
10	19.4	10.85	6.803	43.53
100	191.5	102.3	60.10	332.7
1000	1803	919.3	493.3	1476.1
10000	11427	4671	1824	1028.1
\(\infty\)	28087	8549	2607	1339.8

Values of the total cross section obtained by numerical integration are listed in Table 8 for four different elements. Units are in \(\mu{\rm barn}\,\), where \(1\,\mu{\rm barn} = 10^{-34}\,{\rm m}^2\,\).

Table 8 Numerical values for the total cross section
\(E_\gamma\) [GeV]	\(\sigma_{\rm tot}\), H [\(\mu{\rm barn}\,\)]	\(\sigma_{\rm tot}\), Be [\(\mu{\rm barn}\,\)]	\(\sigma_{\rm tot}\), Cu [\(\mu{\rm barn}\,\)]	\(\sigma_{\rm tot}\), Pb [\(\mu{\rm barn}\,\)]
1	0.01559	0.1515	5.047	30.22
10	0.09720	1.209	49.56	334.6
100	0.1921	2.660	121.7	886.4
1000	0.2873	4.155	197.6	1476
10000	0.3715	5.392	253.7	1880
\(\infty\)	0.4319	6.108	279.0	2042

Fig. 5 Total cross section for the Bethe-Heitler process \(\gamma \rightarrow \mu^+\mu^-\) as a function of the photon energy \(E_\gamma\) in hydrogen and lead, normalized to the asymptotic cross section \(\sigma_\infty\).

Well above threshold, the total cross section rises about linearly in \(\log(E_\gamma)\) with the slope

\[W_M = \frac{1}{4\, D_n \, \sqrt e \,m_\mu}\]

until it saturates due to screening at \(\sigma_\infty\). Fig. 5 shows the normalized cross section where

\[\sigma_\infty = \frac79 \, \sigma_0 \qquad {\rm and} \qquad \sigma_0 = 4 \, \alpha \, Z^2 \, r_c^2 \, \log(W_\infty)\;.\]

Numerical values of \(W_M\) are listed in Table 11.

Table 11 Numerical values of \(W_M\).
Element	\(W_M\) [1/GeV]
H	0.963169
Be	0.514712
Cu	0.303763
Pb	0.220771

The total cross section can be parameterized as

(22)\[\sigma_{\rm par} = \frac{28 \, \alpha \, Z^2 \, r_c^2}{9} \; \log(1 + W_M C_f E_g)\;,\]

with

\[E_g = \left(1-\frac{4 m_\mu}{E_\gamma}\right)^{t} \left(W_{\rm sat}^{s} + E_\gamma^{s} \right)^{1/s}\;.\]

and

\[W_{\rm sat} = \frac{W_\infty}{W_M} = B \, Z^{-1/3} \, \frac{4\,\sqrt e \,m_\mu^2 }{m_e}\;.\]

The threshold behavior in the cross section was found to be well approximated by \(t = 1.479 + 0.00799 D_n\) and the saturation by \(s = -0.88\). The agreement at lower energies is improved using an empirical correction factor, applied to the slope \(W_M\), of the form

\[C_f = \left[ 1 + 0.04 \log \left(1+\frac{E_c}{E_\gamma}\right)\right]\;,\]

where

\[E_c = \left[ -18.+\frac{4347.}{B \, Z^{-1/3}}\right] \;{\rm GeV}\;.\]

A comparison of the parameterized cross section with the numerical integration of the exact cross section shows that the accuracy of the parametrization is better than 2%, as seen in Fig. 6.

Fig. 6 Ratio of numerically integrated and parametrized total cross sections as a function of \(E_\gamma\) for hydrogen, beryllium, copper and lead.

Multi-differential Cross Section and Angular Variables

The angular distributions are based on the multi-differential cross section for lepton pair production in the field of the Coulomb center

(23)\[\begin{split}\frac{d\sigma}{dx_+ \, du_+ \, du_-\,d\varphi} & = \frac{4\,Z^2\alpha^3}{\pi}\,\frac{m_\mu^2}{q^4}\,u_+\,u_- \times \\ & \times \left\{ \frac{u_+^2+u_-^2}{(1+u_+^2)\,(1+u_-^2)} -2x_+x_- \left[\frac{u_+^2}{(1+u_+^2)^2}+\frac{u_-^2}{(1+u_-^2)^2}\right] -\frac{2u_+u_-(1-2x_+x_-)\,\cos\varphi}{(1+u_+^2)\,(1+u_-^2)} \right\} \,.\end{split}\]

Here

(24)\[u_\pm = \gamma_\pm \theta_\pm \quad , \qquad \gamma_\pm = \frac{E_\mu^\pm}{m_\mu} \quad,\qquad q^2=q_{\parallel}^2+q_{\perp}^2\quad,\]

where

(25)\[\begin{split}q_{\parallel}^2 &= q_{\min}^2\,(1+x_-u_+^2+x_+u_-^2)^2\,, \\ q_{\perp}^2 &= m_\mu^2\left[(u_+-u_-)^2+2\,u_+u_-(1-\cos\varphi) \right]\,.\end{split}\]

\(q^2\) is the square of the momentum \({\bf q}\) transferred to the target and \(q_{\parallel}^2\) and \(q_{\perp}^2\) are the squares of the components of the vector \({\bf q}\), which are parallel and perpendicular to the initial photon momentum, respectively. The minimum momentum transfer is \(q_{\min}=m_\mu^2/(2E_\gamma \, x_+x_-)\). The muon vectors have the components

(26)\[\begin{split}\begin{array}{rcl}\displaystyle {\bf p}_+&=&p_+\,(\;\;\;\sin\theta_+\cos(\varphi_0+\varphi/2)\,,\,\;\;\;\sin\theta_+\sin(\varphi_0+\varphi/2)\,,\,\cos\theta_+)\,,\\ {\bf p}_-&=&p_-\,( -\sin\theta_-\cos(\varphi_0-\varphi/2)\,,\, -\sin\theta_-\sin(\varphi_0-\varphi/2)\,,\,\cos\theta_-)\,, \end{array}\end{split}\]

where \(p_{\pm}=\sqrt{E_{\pm}^2-m_\mu^2}\). The initial photon direction is taken as the \(z\)-axis. The cross section of Eq.(23) does not depend on \(\varphi_0\). Because of azimuthal symmetry, \(\varphi_0\) can simply be sampled at random in the interval \((0,\,2\,\pi)\).

Eq.(23) is too complicated for efficient Monte Carlo generation. To simplify, the cross section is rewritten to be symmetric in \(u_+\), \(u_-\) using a new variable \(u\) and small parameters \(\xi,\beta\), where \(u_\pm=u \pm \xi/2\) and \(\beta = u \,\varphi\). When higher powers in small parameters are dropped, the differential cross section in terms of \(u,\xi,\beta\) becomes

(27)\[\begin{split}\frac{d\sigma}{dx_+ \, d\xi \, d\beta\, u du} &= \frac{4\,Z^2\alpha^3}{\pi} \frac{m_\mu^2}{\left(q_{\parallel}^2+m_\mu^2(\xi^2+\beta^2)\right)^2} \times \\ &\times \left\{\xi^2\left[\frac1{(1+u^2)^2}-2\,x_+x_-\,\frac{(1-u^2)^2}{(1+u^2)^4}\right]+ \frac{\beta^2(1-2x_+x_-)}{(1+u^2)^2}\right\}\,,\end{split}\]

where, in this approximation,

\[q_{\parallel}^2=q_{\min}^2\,(1+u^2)^2\,.\,\]

For Monte Carlo generation, it is convenient to replace (\(\xi,\beta\)) by the polar coordinates (\(\rho,\psi\)) with \(\xi=\rho\,\cos\psi\) and \(\beta=\rho\,\sin\psi\). Integrating Eq.(27) over \(\psi\) and using symbolically \(du^2\) where \(du^2 = 2 u \, du\) yields

(28)\[\frac{d\sigma}{dx_+\,d\rho\,du^2} =\frac{4Z^2\alpha^3}{m_\mu^2}\,\frac{\rho^3}{(q_{\parallel}^2/m_\mu^2+\rho^2)^2} \,\,\left\{\frac{1-x_+x_-}{(1+u^2)^2}-\frac{x_+x_-(1-u^2)^2}{(1+u^2)^4}\right\}.\,\]

Integration with logarithmic accuracy over \(\rho\) gives

\[\int\!\frac{\rho^3\,d\rho}{(q_{\parallel}^2/m_\mu^2+\rho^2)^2} \approx \int\limits_{q_{\parallel}/m_\mu}^1\!\frac{d\rho}{\rho} =\log\left(\frac{m_\mu}{q_{\parallel}}\right)\,.\]

Within the logarithmic accuracy, \(\log(m_\mu/q_{\parallel})\) can be replaced by \(\log(m_\mu/q_{\min})\), so that

\[\frac{d\sigma}{dx_+\,du^2}=\frac{4\,Z^2\alpha^3}{m_\mu^2}\, \left\{\frac{1-x_+x_-}{(1+u^2)^2}-\frac{x_+x_-(1-u^2)^2}{(1+u^2)^4}\right\}\, \log\left(\frac{m_\mu}{q_{\min}}\right)\,.\]

Making the substitution \(u^2 = 1/t -1\), \(du^2 = -dt \, / t^2\) gives

(29)\[\frac{d\sigma}{dx_+\,dt}=\frac{4\,Z^2\alpha^3}{m_\mu^2}\, \left[1-2\,x_+x_-+4\,x_+x_-t\,(1-t)\right]\, \log\left(\frac{m_\mu}{q_{\min}}\right) . \,\]

Atomic screening and the finite nuclear radius may be taken into account by multiplying the differential cross section determined by Eq.(27) with the factor

\[\left(F_a(q)-F_n(q)\,\right)^2\,,\]

where \(F_a\) and \(F_n\) are atomic and nuclear form factors. Please note that after integrating Eq.(28) over \(\rho\), the \(q\)-dependence is lost.

Procedure for the Generation of \(\mu ^+\mu ^-\) Pairs

Given the photon energy \(E_\gamma\) and \(Z\) and \(A\) of the material in which the \(\gamma\) converts, the probability for the conversions to take place is calculated according to the parametrized total cross section Eq.(22). The next step, determining how the photon energy is shared between the \(\mu^+\) and \(\mu^-\), is done by generating \(x_+\) according to Eq.(17). The directions of the muons are then generated via the auxilliary variables \(t,\,\rho,\,\psi\). In more detail, the final state is generated by the following five steps, in which \(R_{1,2,3,4,...}\) are random numbers with a flat distribution in the interval [0,1]. The generation proceeds as follows.

Sampling of the positive muon energy \(E_\mu^+ = x_+ \, E_\gamma\). This is done using the rejection technique. \(x_+\) is first sampled from a flat distribution within kinematic limits using

\[x_+ = x_{\rm min} + R_1 (x_{\rm max} - x_{\rm min})\]

and then brought to the shape of Eq.(17) by keeping all \(x_+\) which satisfy

\[\left(1-\frac43\,x_+x_-\right)\frac{\log(W)}{\log(W_{\rm max})} < R_2 \,.\]

Here \(W_{\rm max}= W(x_+=1/2)\) is the maximum value of \(W\), obtained for symmetric pair production at \(x_+=1/2\). About 60% of the events are kept in this step. Results of a Monte Carlo generation of \(x_+\) are illustrated in Fig. 7. The shape of the histograms agrees with the differential cross section illustrated in Fig. 4.

Fig. 7 Histogram of generated \(x_+\) distributions for beryllium at three different photon energies. The total number of entries at each energy is \(10^6\).
Generate \(t ( = \frac{1}{\gamma^2 \theta^2 + 1} )\) . The distribution in \(t\) is obtained from Eq.(29) as

\[f_1(t)\,dt=\frac{1-2\,x_+x_{-}+4\,x_+x_{-t}\,(1-t)} {1+C_1/t^2}\,dt\,,\quad 0<t\le 1\,.\]

with form factors taken into account by

\[C_1= \frac{(0.35\,A^{0.27})^2}{x_+x_-\,E_\gamma/m_\mu }\,.\]

In the interval considered, the function \(f_1(t)\) will always be bounded from above by

\[\max [f_1(t)]=\frac{1-x_+x_{-}}{1+C_1}\;.\,\]

For small \(x_+\) and large \(E_{\gamma}\), \(f_1(t)\) approaches unity, as shown in Fig. 8.

Fig. 8 The function \(f_1(t)\) at \(E_{\gamma} = 10\,{\rm GeV}\) in beryllium for different values of \(x_+\).

Fig. 9 The function \(f_1(t)\) at \(E_{\gamma} = 1\,{\rm TeV}\) in beryllium for different values of \(x_+\).

Fig. 10 Histograms of generated \(t\) distributions for \(E_{\gamma} = 10\,{\rm GeV}\) (solid line) and \(E_{\gamma} = 100\,{\rm GeV}\) (dashed line) with \(10^6\) events each.

Fig. 11 Histograms of generated \(\psi\) distributions for beryllium at four different photon energies.

The Monte Carlo generation is done using the rejection technique. About 70% of the generated numbers are kept in this step. Generated \(t\)-distributions are shown in Fig. 10.
Generate \(\psi\) by the rejection technique using \(t\) generated in the previous step for the frequency distribution

\[f_2(\psi) =\Big[1-2\,x_+x_{-}+4\,x_+x_{-}t\,(1-t)\,(1+\cos(2\psi))\Big]\;, \qquad 0\le\psi\le2\pi\,.\]

The maximum of \(f_2(\psi)\) is

\[\max [f_2(\psi)]=1-2\,x_+x_-\left[1-4\,t\,(1-t)\right]\,.\,\]

Generated distributions in \(\psi\) are shown in Fig. 11.
Generate \(\rho\). The distribution in \(\rho\) has the form

\[f_3(\rho)\,d\rho=\frac{\rho^3\,d\rho}{\rho^4+C_2}\,,\quad 0\le\rho\le \rho_{\rm max}\,,\,\]

where

\[\rho_{\rm max}^2=\frac{1.9}{A^{0.27}}\,\left(\frac{1}{t}-1\right), \,\]

and

\[C_2=\frac4{\sqrt{x_+x_-}}\left[\left(\frac{m_\mu}{2E_\gamma x_+x_-\,t} \right)^2+\left(\frac{m_e}{183 \, Z^{-1/3} \, m_\mu}\right)^2 \right]^2\,.\]

The \(\rho\) distribution is obtained by a direct transformation applied to uniform random numbers \(R_i\) according to

\[\rho=\left[C_2(\exp(\beta\,R_i)-1)\right]^{1/4}\,,\]

where

\[\beta=\log\left(\frac{C_2+\rho_{\rm max}^4}{C_2}\right)\,.\]

Generated distributions of \(\rho\) are shown in Fig. 12

Fig. 12 Histograms of generated \(\rho\) distributions for beryllium at two different photon energies. The total number of entries at each energy is \(10^6\).

Fig. 13 Histograms of generated \(\theta_+\) distributions at different photon energies.
Calculate \(\theta_+,\theta_-\) and \(\varphi\) from \(t, \rho,\psi\) with

(30)\[\gamma_{\pm} = \frac{E_{\mu}^\pm}{m_{\mu}} \qquad {\rm and} \qquad u=\sqrt{\frac1t-1}\,.\]

according to

\[\theta_+= \frac{1}{\gamma_+}\,\left(u +\frac{\rho}{2}\,\cos\psi\right)\,,\quad \theta_-= \frac{1}{\gamma_-}\,\left(u -\frac{\rho}{2}\,\cos\psi\right)\, \quad {\rm and} \quad \varphi=\frac{\rho}{u} \, \sin\psi\, . \,\]

The muon vectors can now be constructed from Eq.(26), where \(\varphi_0\) is chosen randomly between 0 and \(2\pi\). Fig. 13 shows distributions of \(\theta_+\) at different photon energies (in beryllium). The spectra peak around \(1/\gamma\) as expected.

The most probable values are \(\theta_+\sim m_\mu/E_\mu^+ = 1 / \gamma_+\). In the small angle approximation used here, the values of \(\theta_+\) and \(\theta_-\) can in principle be any positive value from 0 to \(\infty\). In the simulation, this may lead (with a very small probability, of the order of \(m_\mu/E_\gamma\)) to unphysical events in which \(\theta_+\) or \(\theta_-\) is greater than \(\pi\). To avoid this, a limiting angle \(\theta_{\rm cut}=\pi\) is introduced, and the angular sampling repeated, whenever \(\max(\theta_+,\,\theta_-)>\theta_{\rm cut}\).

Fig. 14 Angular distribution of positive (or negative) muons. The solid curve represents the results of the exact calculations. The histogram is the simulated distribution. The angular distribution for pairs created in the field of the Coulomb centre (point-like target) is shown by the dashed curve for comparison.

Fig. 15 Angular distribution in logarithmic scale. The curve corresponds to the exact calculations and the histogram is the simulated distribution.

Fig. 16 Distribution of the difference of transverse momenta of positive and negative muons (with logarithmic x-scale).

Fig. 14, Fig. 15 and Fig. 16 show distributions of the simulated angular characteristics of muon pairs in comparison with results of exact calculations. The latter were obtained by means of numerical integration of the squared matrix elements with respective nuclear and atomic form factors. All these calculations were made for iron, with \(E_\gamma=10\,{\rm GeV}\) and \(x_+=0.3\). As seen from Fig. 14, wide angle pairs (at low values of the argument in the figure) are suppressed in comparison with the Coulomb center approximation. This is due to the influence of the finite nuclear size which is comparable to the inverse mass of the muon. Typical angles of particle emission are of the order of \(1/\gamma_\pm=m_\mu/E_\mu^\pm\) (Fig. 15). Fig. 16 illustrates the influence of the momentum transferred to the target on the angular characteristics of the produced pair. In the frame of the often used model which neglects target recoil, the pair particles would be symmetric in transverse momenta, and coplanar with the initial photon.

Five-dimensional (5D) Bethe-Heitler gamma Conversion to \(\mu ^+\mu ^-\)

Since release 10.6, the G4BetheHeitler5DModel physics model for \(\gamma\)-ray conversions to e⁺e^- pairs (section Five-dimensional (5D) Bethe-Heitler gamma Conversion to e+e-) has been extended to the conversions to \(\mu^+\mu^-\) pairs, sampling the 5D Bethe-Heitler differential cross section as described in the preceding section. For decays to muon pairs, G4BetheHeitler5DModel is a low-energy complement to the former G4GammaConversionToMuons model that is using high-energy approximations, that has been verified above 10 GeV and that is described above (Gamma Conversion into Lepton Pair). G4BetheHeitler5DModel is expected to be valid down to threshold and the algorithm has been verified down to a couple of hundred keV above threshold [Ber19]. Conversions of linearly polarised or non-polarised incident photons can be generated. The energy threshold is

\[E_{\text{threshold}} = 2 (m_\mu^2 / M + m_\mu) c^2 ,\]

where M is the target mass.

For triplet conversion (\(\gamma ~ e^{-} ~ \to ~ \mu^{+}\mu^{-} ~ e^{-}\)), \(E_{\text{threshold}}\) is close to 44 GeV.
For nuclear conversion (\(\gamma ~ Z ~ \to ~ \mu^{+}\mu^{-} ~ Z\)), \(E_{\text{threshold}}\) ranges from 223 MeV for hydrogen targets down to almost \(2 m_{\mu} c^{2} \approx 211\) MeV for heavy nuclei.

Fig. 17 presents distributions of kinematic variables of interest for several incident-photon energies for \(\gamma\)-ray nuclear conversion on Argon:

\(\theta_{\mu^{+}\mu^{-}} \times E\), the pair opening angle multiplied by the incident-photon energy; the vertical line shows the most probable value of the distribution, \(3.2 m_{\mu} c^2\), obtained by Olsen in the high-energy approximation [Ols63];
\(\log_{10}{(q)}\), where q is the (nucleus) recoil momentum;
\(\phi_{\text{recoil}}\), azimutal angle of the (nucleus) recoil momentum (for a fully linearly polarized \(\gamma\)-ray beam);
\(E_{\mu^{+}} / E\), the fraction of the incident-photon energy that is carried away by the positive muon

Note that G4BetheHeitler5DModel does not take the finite size of the nucleus into account.

../../_images/muons_v4_Ar_nucl_ff_combined.png

Fig. 17 G4BetheHeitler5DModel: \(\gamma\)-ray conversions to \(\mu^+\mu^-\) pairs on Argon atoms (see text), for various incident-photon energies (bullets 400 MeV, squares 1 GeV, upper triangle 4 GeV, down triangle 10 GeV).

Bibliography

Ber19: Denis Bernard. A 5d, polarised, bethe-heitler event generator for γ to µ+µ- conversion. 2019. arXiv:1910.12501.
BKK02: H Burkhardt, S R Kelner, and R P Kokoulin. Monte carlo generator for muon pair production. Technical Report CERN-SL-2002-016-AP. CLIC-Note-511, CERN, Geneva, May 2002. URL: https://cds.cern.ch/record/558831.
KKP95: SR Kelner, RP Kokoulin, and AA Petrukhin. About cross section for high-energy muon bremsstrahlung. Technical Report, MEphI, 1995. Preprint MEPhI 024-95, Moscow, 1995, CERN SCAN-9510048.
Ols63: Haakon Olsen. Opening Angles of Electron-Positron Pairs. Phys. Rev., 131:406–415, 1963.