Quantum synchrotron radiation in the case of a field with finite extension

Tobias N Wistisen

doi:10.1103/physrevd.92.045045

Abstract

The semiclassical operator method of Baier and Katkov allows one to obtain the spectrum of synchrotron radiation in a way similar to the classical derivation but which is fully valid also in the quantum case of very strong electromagnetic fields. In the usual calculation the extension of the field is taken to be infinite. In this paper we apply a numerical routine based on the semiclassical operator method to the case of a constant field but with a finite extension. For large extensions of the field one obtains the usual result of quantum synchrotron radiation, while in the limit of small extension of the field one obtains a spectrum resembling that of bremsstrahlung. We derive a formula for the radiation spectrum in this limit. In the transition toward shorter field extensions one finds that the power-spectrum increases for soft photons and slightly diminishes for harder photons. It is found that in the classical case the total power emitted decreases as the field extension decreases while in the quantum case the total power emitted is first increased and then decreases. Such an effect could be important for future ${e}^{+}{e}^{\ensuremath{-}}$ colliders such as the ILC or CLIC where the dominant energy and luminosity loss is due to synchrotron radiation by an ${e}^{\ensuremath{-}}/{e}^{+}$ in the field of the opposing bunch, often termed ``beamstrahlung.'' In this paper we also discuss how these effects, in the quantum case could be measured in an experiment using thin aligned single crystals and high energy electrons available at e.g. the CERN SPS North Area, and in the classical case could already be relevant at existing accelerators with conventional magnets providing the electromagnetic field.

Highlights

In the field of strong field QED, processes whose description defy the usual application of the perturbative expansion in orders of α are of major interest
In [10] it is discussed how this method can be rewritten in a form similar to the classical formula of radiation emission as seen in e.g. [11] which allows one to find the radiation emission spectrum in any electromagnetic field by numerical methods
As seen in e.g. [1,12] a more traditional approach to the problem consists of finding the exact wave function of the Dirac equation where the constant crossed field is included exactly [13], and treating the radiation emission to first order in perturbation theory

Summary

INTRODUCTION

In the field of strong field QED, processes whose description defy the usual application of the perturbative expansion in orders of α are of major interest. In [10] it is discussed how this method can be rewritten in a form similar to the classical formula of radiation emission as seen in e.g. [1,12] a more traditional approach to the problem consists of finding the exact wave function of the Dirac equation where the constant crossed field is included exactly [13], and treating the radiation emission to first order in perturbation theory. The semiclassical operator method is powerful here, in the sense that finding this wave function is unnecessary—the classical motion in the field is sufficient.

THEORETICAL FORMALISM

SYNCHROTRON RADIATION

OF NUMERICAL RESULTS

THE SPECTRUM IN THE LIMIT OF A SHORT EXTENSION OF THE FIELD

DISCUSSION