This theoretical study derives general cutting rules for quantum electrodynamics in a plane-wave background and checks their application to trident electron–positron pair production in a constant crossed field. The authors compare a direct tree-level calculation with cuts of a two-loop electron-scattering contribution, provide initial-spin-resolved direct and exchange rates, and numerically compare integrated one-step terms. The two-step terms coincide analytically; the one-step numerical results agree within the stated numerical accuracy. Recovering a correct differential distribution from an

Strong-field quantum electrodynamics — A theoretical study

One electron in. Three particles out. A new set of cutting rules connects the two.

A calculation you can look inside.

Cutting rules in strong field QED with application to trident pair production Y. V. Selivanov · A. A. Mironov · A. I. Alexeenko · A. M. Fedotov

Open the loop. Find the particles.

A quantum loop is a compact way to calculate many possibilities. A cut passes through its lines and asks which particles can appear in an outcome. Change where you cut: the same loop reveals different processes.

One loop. Three possible cuts.

Electron Photon Positron

An original explanatory schematic. Lines represent terms in a quantum calculation, not literal particle paths.

Photon emission correction

e− → e− + γ

The left cut crosses an electron and a photon. Its contribution corrects photon emission.

All cuts belong in the sum. The middle cut puts two electrons and a positron on shell: each satisfies the physical relation between energy and momentum. The side cuts contribute radiation corrections. Taking the loop’s imaginary part alone obscures terms that cancel between these processes.

The instrument illustrates the gauge-invariant two-loop polarization correction to electron forward scattering, shown in equations (4.3), (4.5) and (4.6). The central outcome is trident pair production. Direct and exchange terms contribute to the full calculation; interference from other diagrams is outside this particular illustration. The plane-wave background used here cannot create pairs spontaneously from vacuum.

Where does the momentum go?

In the two-step contribution, an electron emits a real photon, which then produces the pair. Explore how the three outgoing particles share the incoming electron’s momentum—and how the distribution changes with the quantum parameter χ.

Two-step trident distribution

Shape normalized within each view. Brightness shows relative density, not an absolute event rate. Tap the triangle to choose a momentum split.

At χ = 1, one broad maximum lies near an even sharing of momentum.

k = 0.33 · r = 0.33 · s = 0.34

The two electrons are identical. Swap their shares: the distribution stays symmetric.

This quantum parameter combines the incoming electron’s momentum with the strength of the electromagnetic background. It changes the effective conditions experienced by the electron.

Light-front momentum is energy minus momentum along the background wave’s direction. Its fractions k, r and s sum to one. For a very fast electron travelling against the wave, they approximate energy fractions.

The map is numerically recomputed from equations (7.3)–(7.14) of the paper. It shows the spin-averaged direct + exchange two-step contribution, with the common interaction-length factor set to one. Direct–exchange interference is omitted. Each χ view is divided by its own peak; changing χ compares the shape rather than the absolute rate. The moving selection halo guides the eye; the numerical distribution itself is stationary.

Two routes. One rate.

The cutting rules connect a closed quantum loop to an open particle-production process.

A direct calculation starts with the outgoing particles. A loop calculation starts with the electron scattering forward and extracts the result through cuts. For the integrated direct contribution to the trident rate, the two routes agree within numerical accuracy.

Knowing the total does not automatically tell you how momentum is distributed. Recovering that detail from loops requires additional care.

One-step contribution: loop vs. tree

Why negative? The one-step term is a signed component of the decomposition. It is not a negative probability of an event.

A real, on-shell photon connects emission and pair creation.

A virtual photon mediates a coherent contribution without an intermediate real-photon stage.

Why it matters A reliable bridge between loops and outgoing particles gives researchers a way to organize higher-order calculations in intense fields. Extending the framework to nonperturbative regimes is a future direction.

Behind the diagram.

The supplementary Mathematica calculation works through the direct trident term, showing the diagrams that connect the algebra to outgoing particles.

Cutting rules in strong field QED with application to trident pair production Y. V. Selivanov, A. A. Mironov, A. I. Alexeenko and A. M. Fedotov. Manuscript prepared for submission to JHEP. Explanatory diagrams are original; the momentum map is recomputed from the paper’s equations. The corresponding public preprint carries a CC BY-NC-ND 4.0 license.

A loop. A possible outcome.

Scroll horizontally to inspect the complete illustration.

AI-generated conceptual illustration. Original visual rendering informed by the supplementary calculation.