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

The G4BetheHeitler5DModel generates the five-dimensional Bethe-Heitler differential cross section, that is, with a correct target recoil momentum distribution both in magnitude and in direction (not necessarily in the pair plane). Therefore the pair itself is also kicked transversely and the photon direction does not lie in the pair plane. This model can be applied to both e⁺e^- pairs (described here) and \(\mu^+\mu^-\) pair production (section Five-dimensional (5D) Bethe-Heitler gamma Conversion to ).

The nuclear or triplet conversion of polarized or non-polarized photons on atomic or isolated-charge targets can be performed.

Total cross-section

The total cross section is inherited from the G4BetheHeitlerModel physics model. We take the nuclear, triplet share to be \(Z/(Z+1)\) , \(1/(Z+1)\) . Pure nuclear or pure triplet samples can also be generated.

Sampling of the final state

In the conversion of a high-energy photon to an \((e^+,e^-)\) pair by interaction with the field of a nucleus or of an electron of the detector, the final state contains three particles (electron, positron, “recoiling” target) and therefore is described in a five-dimensional phase space.

Bethe and Heitler (BH) have obtained an analytical expression of the differential cross section for non-polarized photons, in the first-order Born approximation and for a point-like, isolated charged-particle target [BH34]. The final-state variables are the azimuthal angles \(\phi_+\) and \(\phi_-\) and the polar angles \(\theta_+\) and \(\theta_-\) of the positron and of the electron, respectively, and the fraction of the photon energy carried away by the positron \(x_+ \equiv E_+/E\).

The differential cross section for fully polarized photons has been obtained by [BM50], put in BH notation by [May51], after which [JR76] have corrected a numerical factor. At this first order of the Born development, only the linear photon polarization takes part to the differential cross section. Therefore no polarization is transfered to the final leptons.

The generation of the probability density function for this differential cross section is made difficult by the presence of a number of divergences, in \(1/(E_+ - P_+ \cos{\theta_+})\), in \(1/(E_- - P_- \cos{\theta_-})\) and in \(1/q^4\), where \(q\) is the “recoil” momentum, that is, the momentum transfered to the target. Further more the divergences take place in functions of several of the kinematical variables in which the differential cross section is written, (\(\phi_+\), \(\phi_-\), \(\theta_+\), \(\theta_-\), \(x_+\) ), that is, in a correlated way in their space.

The correlation issue is solved in the usal way in high-energy physics: each step is performed in the appropriate Lorentz frame. The interaction between the photon and the target is performed in the center-of-mass system (CMS); an object having an invariant mass \(\sqrt{s}\) is “created”. The invariant mass of the pair is taken at random. The “decay” of that \(\sqrt{s}\) thing to the recoiling target and to the pair is generated in the CMS. The decay of the pair to an actual electron and a positron is performed in the pair Lorentz frame. The two leptons are then boosted back to the CMS. Finally every body is boosted back to the laboratory frame. The variables used to do so are defined in Table 3.

Table 3 Kinematic variables and the Lorentz frame in which they are defined.

	variable	Lorentz frame
\(\theta\)	target and pair polar angle	CMS
\(\mu\)	\(e^+e^-\) invariant mass
\(\theta_\ell\)	\(e^+\) and \(e^-\) polar angle	pair frame
\(\phi_\ell\)	\(e^+\) and \(e^-\) azimuthal angle	pair frame
\(\phi\)	target and pair azimutal angle	CMS

As the distribution of these variables still show divergences, a change of variables is used ( Table 4 ). The \(x_i, i=1\cdots 5\) are taken flat. The photon-energy-dependent bounds of the \(x_1\) segment and the \(x_i\) distributions can be found in Fig. 1 and in Fig. 3, respectively, of [Ber18].

Table 4 Relationship between the generator variables, \(x_i,i=1\cdots 5\), and the kinematic variables, and their range.

\(i\)		Jacobian	\(x_i\) range
1	\(\cos\theta = \frac{y - 1}{1 + y}\), \(y = \exp(x_1)\)	\( \frac{y} {(1+y)^2}\)	\([x_{1l},x_{1u}]\)
2	\(\mu = \mu_{min} \times (\mu_{range})^{x_2^2}\)	\(2 x_2 \log{( \mu_{range})} \mu\)	\([0,1]\)
3	\(\cos\theta_\ell = x_3\)	\(|\sin \theta_\ell|\)	\([0,\pi]\)
4	\(\phi_\ell = x_4\)	1	\([-\pi,\pi]\)
5	\(\phi = x_5\)	1	\([-\pi,\pi]\)

The algorithm is described with some detail in the documentation of the fortran demonstration model [Ber18] and its C++ implementation in [SB19]. The normalization of the phase space of the final state in this case of cascade “decays” is described in the PDG, Sect. 47.4.3 of [Gro16]. We get:

\[d \sigma = \frac{1}{(2\pi)^5} \frac{1}{32 M\sqrt{s} E} \left|{\cal M}\right|^2 | p_+^\ast | | p_r | \, d \mu \, d \Omega_+^\ast \, d \Omega_r,\]

where \(( p_+^\ast , \Omega_+^\ast )\) refers to the kinematic variables of the positron in the pair rest frame and \(( p_r, \Omega_r )\) to the kinematic variables of the target recoil in the CMS. We obtain:

\[d \sigma = H \times X \, d \mu \, d \Omega_+^\ast \, d \Omega_r, \qquad {\rm with:} \qquad H = \frac{- \alpha Z^2 r_0^2 }{ (2 \pi)^2 } \frac{ \left| p_+^\ast \right| | p_r | m^2 M}{E^3 \sqrt{s}|\vec{q}|^4} .\]

For an unpolarized photon [BH34]:

(12)\[\begin{split}X_{u} &= \left( \frac{p_+ \sin{\theta_+}}{E_+ - p_+ \cos{\theta_+}} \right)^2 (4 E_-^2 - q^2) + \left( \frac{p_- \sin{\theta_-}}{E_- - p_- \cos{\theta_-}} \right)^2 (4 E_+^2 - q^2) + \\ & \frac{2p_+ p_- \sin{\theta_+} \sin{\theta_-} \cos{(\phi_+-\phi_-)}}{(E_- - p_- \cos{\theta_-})(E_+ - p_+ \cos{\theta_+})} (4 E_+ E_- + q^2 -2 E ^2) - 2 E ^2 \frac{(p_+ \sin{\theta_+})^2 + (p_- \sin{\theta_-})^2} {(E_+ - p_+ \cos{\theta_+})(E_- - p_- \cos{\theta_-})} ,\end{split}\]

For a polarized photon [BM50, JR76, May51]:

(13)\[\begin{split}X_{f} &= 2 \left[ \left( 2 E_+ \frac{p_- \sin{\theta_-} \cos{\phi_-}}{E_- - p_- \cos{\theta_-}} + 2 E_- \frac{p_+ \sin{\theta_+} \cos{\phi_+}}{E_+ - p_+ \cos{\theta_+}} \right)^2 -q^2 \left( \frac{p_- \sin{\theta_-} \cos{\phi_-}}{E_- - p_- \cos{\theta_-}} - \frac{p_+ \sin{\theta_+} \cos{\phi_+}}{E_+ - p_+ \cos{\theta_+}} \right)^2 \right.\\ & \left. -E^2 \frac{(p_+ \sin{\theta_+})^2+(p_- \sin{\theta_-})^2+2p_+ p_- \sin{\theta_+} \sin{\theta_-} \cos{(\phi_+-\phi_-)}}{(E_- - p_- \cos{\theta_-})(E_+ - p_+ \cos{\theta_+})} \right],\end{split}\]

with \(|\vec{q}|^2 = |\vec{p_+} + \vec{p_-} - \vec{k} |^2 .\)

In case the nucleus or the electron are not isolated but are part of an atom, the screening of the target field by the (other) electrons of the atom is described by a function of \(q^2\), a coherent form factor [NFM34] for nuclear conversion, an incoherent form factor [WL39] for triplet conversion.

In contrast with the BH expressions taken at face value, for which the recoil energy is neglected and the electron energy is taken to be \(E_- = E - E_+\), here a strict energy-momentum conservation is achieved.

A number of approximations are used:

• For triplet conversions, BH neglects the \(e-\gamma\) exchange diagrams, which might be an issue at low energy (see Table 1, Fig.3 and their discussion in [Mor67]).

• Landau-Pomeranchuk-Migdal (LPM) suppression effects at very high energies are neglected.

• The finite size of the nucleus is neglected.

• Any pre-existing non-zero momentum of the target prior to the conversion, such as in the case of Compton “Doppler” broadening, is not considered.

A number of verifications of the model have been performed by comparison with analytical expressions of the distribution of one of the kinematic variables, obtained in the past from partial integrations of the BH differential cross section. In particular the distribution of the pair opening angle is found [Ber13] to take its most probable value at the high-energy asymptotic value of 1.6 MeV rad [Ols63]. The G4BetheHeitler5DModel is the only gamma-conversion physics model for which the recoil momentum distribution is found [Ber13b, GB17b] to be compatible with the analytical expression of [JLS50]. See also the extended electromagnetic example TestEm15 and its README.

The verification of the integral of the differential cross section over the full \(x_i,i=1\cdots 5\) phase space can be found in Fig. 4 of [Ber18], for atomic targets (comparison with total cross section NIST data) and for isolated charges (comparison with total cross section asymptotic expressions).

The verification of the polarization properties of the model are addressed in 5dbhpol.conv.

Bibliography

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