Final multiplicity of a QED cascade in generalized Heitler model

TL;DR

This paper derives an exact formula for the final multiplicity of leptons in a generalized QED cascade model, revealing independence from free paths and asymptotic proportionality to seed energy, with implications for high-energy cascade characteristics.

Contribution

It provides an exact solution for the final lepton count in a generalized Heitler model, extending previous models and deriving approximate formulas for cascade properties.

Findings

Final lepton number is independent of photon and lepton free paths.

Final lepton number asymptotically proportional to seed particle energy.

Original Heitler model is recovered as a special case.

Abstract

We consider a generalized Heitler model for QED cascade. An exact formula for the final number of leptons is obtained by solving the kinetic equations. We demonstrate that in such a model the final number of leptons does not depend on photon and lepton free paths. We derive approximate formulas for the main characteristics of cascades at high energy, including the final number of leptons and the cascade depth. We show that in general the final number of leptons is asymptotically proportional to the energy of seed particle. It is also demonstrated how the original Heitler model is reproduced as a special case.