Bethe–Heitler Process in QED

• The Bethe–Heitler process is a QED phenomenon where high-energy photons convert into lepton pairs via interactions with an external charged target, with recoil ensuring momentum conservation.
• It features complex kinematics, including singular behavior at low momentum transfer and requires precise treatment of screening effects, finite-mass corrections, and polarization asymmetries.
• This process is pivotal in various applications, from fixed-target experiments and numerical simulations to modeling astrophysical phenomena such as cosmic-ray energy loss and blazar emissions.

The Bethe–Heitler process denotes a class of QED reactions in which pair creation or real-photon emission occurs in the electromagnetic field of an external charged target. In its canonical form, it is the conversion of a high-energy photon into a charged lepton–antilepton pair in the field of a nucleus or an electron, $\gamma+Z\to \ell^+\ell^-+Z$ or $\gamma+e^-\to \ell^+\ell^-+e^-$; in modern usage, the same label also covers closely related external-field channels such as $l+N\to l'+N'+\gamma$, the embedded fixed-target process $e^-Z\to e^-Ze^+e^-$, and the photohadronic reaction $p+\gamma\to p+e^+e^-$ (Bernard, 2019).

1. Canonical process and basic kinematics

In vacuum, the decay $\gamma\to \ell^+\ell^-$ is forbidden by four-momentum conservation: a single massless particle cannot decay into two massive ones while conserving energy and momentum. Bethe–Heitler pair creation becomes possible because an external charged particle supplies the recoil momentum $q$ needed to satisfy four-momentum conservation. The two standard channels are nuclear conversion,

$\gamma + Z \to \ell^+ + \ell^- + Z,$

and triplet conversion on an electron,

$\gamma + e^- \to \ell^+ + \ell^- + e^- .$

The same tree-level QED structure applies to $\ell=e$ and $\gamma+e^-\to \ell^+\ell^-+e^-$0, but the lepton mass changes the threshold, the angular scales, and the energy dependence of the cross section (Bernard, 2019).

When the recoil is unobserved, the final state is still five-dimensional. A convenient set of variables is the energy fraction $\gamma+e^-\to \ell^+\ell^-+e^-$1 together with the polar and azimuthal angles $\gamma+e^-\to \ell^+\ell^-+e^-$2 and $\gamma+e^-\to \ell^+\ell^-+e^-$3 of the two leptons. In exact treatments the recoil momentum is then fixed by four-momentum conservation, and the process remains singularity-prone because the dominant region is small momentum transfer, $\gamma+e^-\to \ell^+\ell^-+e^-$4, together with increasingly forward lepton emission at high energy (Bernard, 2019).

Muon-pair production illustrates the same structure on a different mass scale. For nuclear conversion, the threshold photon energy is

$\gamma+e^-\to \ell^+\ell^-+e^-$5

which approaches $\gamma+e^-\to \ell^+\ell^-+e^-$6 for heavy nuclei, whereas triplet conversion on electrons has a threshold near $\gamma+e^-\to \ell^+\ell^-+e^-$7. This large separation of scales is one reason why $\gamma+e^-\to \ell^+\ell^-+e^-$8 requires dedicated exact generators rather than a trivial re-use of electron-pair approximations (Bernard, 2019).

2. Variant usages and amplitude structures

Historically, the Bethe–Heitler name is attached not only to $\gamma+e^-\to \ell^+\ell^-+e^-$9, but also to real-photon emission in external fields. In lepton–nucleon scattering the Bethe–Heitler reaction is

$l+N\to l'+N'+\gamma$0

with the emitted real photon radiated from the lepton line and the nucleon probed through its electromagnetic form factors $l+N\to l'+N'+\gamma$1, $l+N\to l'+N'+\gamma$2, or equivalently $l+N\to l'+N'+\gamma$3, $l+N\to l'+N'+\gamma$4. In this channel the Bethe–Heitler amplitude is the dominant reference process in many DVCS and VCS analyses because it shares the same external state as deeply virtual Compton scattering while remaining a pure QED process on the lepton side (Barbaro et al., 2013).

In fixed-target electron scattering, the same physics appears in the embedded reaction

$l+N\to l'+N'+\gamma$5

treated in the external-field approximation with an infinitely heavy nucleus. Two amplitude topologies are distinguished. In timelike Bethe–Heitler, the beam electron radiates a timelike photon $l+N\to l'+N'+\gamma$6 that converts into the pair. In spacelike Bethe–Heitler, the nucleus-emitted spacelike photon couples directly to the produced pair while the beam electron exchanges a spacelike photon $l+N\to l'+N'+\gamma$7. For $l+N\to l'+N'+\gamma$8 final pairs the amplitude must also be antisymmetrized under interchange of the scattered beam electron and the produced electron, reflecting Fermi–Dirac statistics (Pustyntsev et al., 18 May 2026).

A further extension occurs in hadronic astrophysics, where Bethe–Heitler refers to photopair production by protons,

$l+N\to l'+N'+\gamma$9

Here the proton survives the interaction and loses only a small fraction of its energy per collision, but the process can still become radiatively important because it operates below the photopion threshold and can inject large numbers of secondary pairs into a source or along a propagation path (Petropoulou et al., 2014).

This suggests that “Bethe–Heitler” now functions less as a single exclusive reaction label than as a family name for closely related external-field and embedded pair-conversion or bremsstrahlung amplitudes.

3. Differential cross sections, screening, recoil, and polarization

For photon conversion in an external field, the fully differential five-dimensional Bethe–Heitler cross section has the generic structure

$e^-Z\to e^-Ze^+e^-$0

with $e^-Z\to e^-Ze^+e^-$1. The $e^-Z\to e^-Ze^+e^-$2 enhancement at small momentum transfer is the defining kinematic singularity of the process; for atomic targets it is regularized by screening through a form factor $e^-Z\to e^-Ze^+e^-$3, while exact-recoil formulations retain the full relativistic kinematics rather than neglecting recoil energy as in older approximations (Bernard, 2019).

In the fixed-target reaction $e^-Z\to e^-Ze^+e^-$4, the heavy-target limit yields a fully differential lab-frame expression

$e^-Z\to e^-Ze^+e^-$5

or, equivalently, a phase-space form in terms of $e^-Z\to e^-Ze^+e^-$6. In this formulation the nuclear field is written as

$e^-Z\to e^-Ze^+e^-$7

making the familiar $e^-Z\to e^-Ze^+e^-$8 scaling explicit at amplitude-squared level and embedding finite-size nuclear effects in $e^-Z\to e^-Ze^+e^-$9 (Pustyntsev et al., 18 May 2026).

Polarization enters already at tree level. For linearly polarized photons, the event-plane azimuth is modulated as

$p+\gamma\to p+e^+e^-$0

with $p+\gamma\to p+e^+e^-$1. For nuclear conversion to $p+\gamma\to p+e^+e^-$2, the polarization asymmetry tends to $p+\gamma\to p+e^+e^-$3 near threshold for heavy targets, while at high energy it approaches the Boldyshev–Peresunko asymptotic behavior and tends toward a value close to $p+\gamma\to p+e^+e^-$4. Triplet conversion behaves differently: its asymmetry tends to zero at threshold (Bernard, 2019).

4. Finite-mass effects and radiative corrections

Finite lepton mass matters in a more subtle way than simple phase-space replacement $p+\gamma\to p+e^+e^-$5. In the Bethe–Heitler reaction $p+\gamma\to p+e^+e^-$6, the full massive-lepton tensor modifies the cross section through genuine mass-dependent structures in the propagators and tensor coefficients. These effects become especially important near propagator quasi-singularities, where one of the lepton propagator denominators approaches zero and the mass regularizes what would otherwise be a plane-wave singularity. After the erratum discussed in the source material, the dramatic enhancement previously claimed at $p+\gamma\to p+e^+e^-$7 does not survive, but the need for full mass dependence near quasi-singular kinematics remains (Barbaro et al., 2013).

For $p+\gamma\to p+e^+e^-$8, soft-photon QED corrections already show a strong mass hierarchy. In the kinematics of the planned MAMI measurement, the soft-photon calculation gives corrections of the percent level for muons and of order $p+\gamma\to p+e^+e^-$9 for electrons. The central analytic result is a multiplicative correction factor $\gamma\to \ell^+\ell^-$0 in which the infrared divergences of virtual and real soft radiation cancel exactly, while the remaining mass-dependent logarithms differ strongly between electron and muon channels because $\gamma\to \ell^+\ell^-$1 and $\gamma\to \ell^+\ell^-$2 are numerically very different (Heller et al., 2018).

A full one-loop treatment of the same reaction sharpens this picture. The complete QED calculation includes lepton self-energies, half-off-shell vertex corrections, vacuum polarization, and leptonic box diagrams with full lepton-mass dependence. It also shows that two-photon-exchange processes with both photons attached to the proton line vanish after averaging over di-lepton angles, and that radiation off the proton is relatively small. In the proposed lepton-universality observable, radiative corrections shift the cross-section ratio by about $\gamma\to \ell^+\ell^-$3, whereas a $\gamma\to \ell^+\ell^-$4 difference between the effective electric form factors $\gamma\to \ell^+\ell^-$5 and $\gamma\to \ell^+\ell^-$6 produces an effect of about $\gamma\to \ell^+\ell^-$7. The soft-photon approximation overestimates the correction, so the full one-loop result is required for a test at the $\gamma\to \ell^+\ell^-$8 level (Heller et al., 2019).

5. Strong-field, nonlinear, and angular-momentum generalizations

The Bethe–Heitler process in combined strong laser and atomic fields can be treated quasiclassically, exact in both $\gamma\to \ell^+\ell^-$9 and the laser background, through the Green’s function of the Dirac equation in a localized potential plus an arbitrary plane wave. In that regime the total cross section becomes a function of the invariant parameter

$q$0

rather than the usual laser intensity parameter alone. The strong laser field can substantially modify the cross section at already available photon energies and laser parameters; for $q$1 the paper reports a suppression of about $q$2, interpreted as an analog of the Landau–Pomeranchuk–Migdal effect for Bethe–Heitler pair production in a coherent external field (Piazza et al., 2012).

In the explicitly nonlinear Bethe–Heitler process in a bichromatic laser field, the reaction is driven by multiphoton absorption from the laser background rather than by a single real photon. The symmetry properties of the emitted $q$3 angular distributions are then governed by the vector potential characterizing the laser field rather than by the corresponding electric field. This is a distinctly strong-field statement: in the Volkov-state formulation the decisive phase structure is built from $q$4, and the resulting angular asymmetries follow the symmetry of the vector potential even when the electric-field symmetry differs (Krajewska et al., 2012).

A separate refinement concerns angular momentum. If the final electron and positron are described by twisted states with definite total-angular-momentum projection, the spin angular momentum of an incident circularly polarized plane-wave photon is converted not only into the leptons’ spin angular momentum but also into their orbital angular momentum. The computed averages show that the orbital contribution surpasses the spin contribution, while their orientations coincide, and that both depend on photon energy and on the opening angle of the emitted leptons. This places Bethe–Heitler pair creation among the QED processes that explicitly realize spin-to-orbital angular-momentum transfer (Lei et al., 2023).

6. Exact generators, transport models, and laboratory exploitation

Modern numerical work increasingly treats Bethe–Heitler as a fully differential event-generation problem rather than a total-cross-section lookup. Bernard’s exact five-dimensional polarized generator for $q$5 samples the full differential Bethe–Heitler cross section using auxiliary variables, exact recoil kinematics, screening form factors, and a von Neumann acceptance–rejection algorithm. It is validated from threshold to $q$6, reproduces the expected opening-angle scaling $q$7, and was motivated in part by the CERN Gamma Factory, by Geant4 detector simulations, and by gamma-ray polarimetry (Bernard, 2019).

In particle-in-cell simulations of dense targets, Bethe–Heitler pair creation appears as the secondary stage following bremsstrahlung. A screened implementation based on a Thomas–Fermi–Debye potential extends the standard high-energy Bethe–Heitler cross section to arbitrarily ionized plasmas and integrates it into a weighted pairwise Monte Carlo scheme. In solid-density copper, increasing the target temperature from $q$8 to $q$9 can raise the total Bethe–Heitler cross section by about $\gamma + Z \to \ell^+ + \ell^- + Z,$0 at $\gamma + Z \to \ell^+ + \ell^- + Z,$1 and by up to $\gamma + Z \to \ell^+ + \ell^- + Z,$2 at $\gamma + Z \to \ell^+ + \ell^- + Z,$3, because the Debye component lengthens the effective screening scale. In foil-transport simulations this channel then competes with Coulomb trident production and grows more strongly with target thickness (Martinez et al., 2019).

At MAGIX@MESA, Bethe–Heitler pair production is both an irreducible QED background and the structural template for visible new-physics production. The reaction $\gamma + Z \to \ell^+ + \ell^- + Z,$4 is evaluated from full timelike and spacelike Bethe–Heitler amplitudes in the external-field approximation, with realistic tantalum form factors and asymmetric spectrometer acceptances. Replacing the timelike photon by a narrow scalar, pseudoscalar, vector, or axial-vector mediator turns the same Bethe–Heitler topology into a resonance search channel. In this setup MAGIX is projected to probe vector or axial-vector couplings down to $\gamma + Z \to \ell^+ + \ell^- + Z,$5 in the tens-of-MeV mass range (Pustyntsev et al., 18 May 2026).

7. Astrophysical Bethe–Heitler pair production

In high-energy astrophysics, Bethe–Heitler usually denotes the proton–photon channel

$\gamma + Z \to \ell^+ + \ell^- + Z,$6

which acts as a continuous proton energy-loss mechanism and a source of secondary pairs. In leptohadronic blazar models, this process injects pairs whose synchrotron emission fills the gap between the low-energy synchrotron hump and the photopion-induced gamma-ray hump. For typical BL Lac parameters, the resulting “$\gamma + Z \to \ell^+ + \ell^- + Z,$7 bump” appears in the $\gamma + Z \to \ell^+ + \ell^- + Z,$8–$\gamma + Z \to \ell^+ + \ell^- + Z,$9 band and accompanies $\gamma + e^- \to \ell^+ + \ell^- + e^- .$0–$\gamma + e^- \to \ell^+ + \ell^- + e^- .$1 neutrino emission, making three-hump SEDs a diagnostic of hadronic activity (Petropoulou et al., 2014).

A similar mechanism has been proposed for ultraluminous X-ray sources. In the transparent funnel of a super-Eddington accretion flow, primary relativistic electrons cool too quickly to dominate the nonthermal population, but relativistic protons interacting with disk photons can produce Bethe–Heitler pairs that outnumber the primaries. In the ULX models summarized in the source material, those secondaries radiate through synchrotron and inverse Compton processes in the $\gamma + e^- \to \ell^+ + \ell^- + e^- .$2–$\gamma + e^- \to \ell^+ + \ell^- + e^- .$3 range with luminosities from $\gamma + e^- \to \ell^+ + \ell^- + e^- .$4 up to $\gamma + e^- \to \ell^+ + \ell^- + e^- .$5, potentially detectable from Galactic ULXs by future MeV missions (Romero et al., 3 Sep 2025).

In reconnection-driven flares from M87*, disk photons can likewise act as Bethe–Heitler targets for relativistic protons accelerated in magnetospheric current sheets. The resulting pairs emit synchrotron photons above $\gamma + e^- \to \ell^+ + \ell^- + e^- .$6; these are then attenuated by the same disk photon field, launching an electromagnetic cascade whose softer synchrotron photons enhance both photopion production and $\gamma + e^- \to \ell^+ + \ell^- + e^- .$7 opacity down to tens of GeV. In that regime Bethe–Heitler is not a small correction but a dynamical driver of the flare spectrum and of the neutrino yield (Petropoulou et al., 26 Feb 2026).

For ultrahigh-energy cosmic-ray propagation, a common misconception is to identify the Bethe–Heitler interaction length with the proton horizon. The relevant quantity is instead the mean energy-loss distance, because each $\gamma + e^- \to \ell^+ + \ell^- + e^- .$8 collision has small inelasticity. The cited cosmological calculation shows that the Bethe–Heitler mean free path on the CMB can be short while the energy-loss distance remains of order gigaparsecs; multiple scatterings therefore blur arrival directions long before they catastrophically drain proton energy (Ruffini et al., 2015).