Classical Electrodynamics · Problem 15.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-annihilation-soft-photon

Problem

A situation closely related to that of Problem 15.4b is the emission of radiation caused by the disappearance of charges and magnetic moments in the annihilation of electrons and positrons to form hadrons in high-energy storage ring experiments. If the differential cross section for the process

e^+e^-\to

hadrons is

d\sigma_0

, without the emission of photons, calculate the cross section for the same process accompanied by a soft photon (

\hbar\omega\to 0

) in the energy interval

d(\hbar\omega)

around

\hbar\omega

. Compare your results with the quantum-mechanical expressions:

\frac{d^2\sigma}{d\Omega\, d(\hbar\omega)} = \frac{\alpha}{\pi^2}\frac{d\sigma_0(s')}{\hbar\omega}\sqrt{\frac{s'^2-4m^2 s'}{s^2-4m^2 s}}\cdot\frac{\left[\beta^2\sin^2\theta + \frac{\hbar^2\omega^2}{s'+2m^2}(1-\beta^4\cos^4\theta)\right]}{(1-\beta^2\cos^2\theta)^2}

\frac{d\sigma}{d(\hbar\omega)} = \frac{4\alpha}{\pi}\frac{d\sigma_0(s')}{\hbar\omega}\sqrt{\frac{s'^2-4m^2 s'}{s^2-4m^2 s}}\left\{\frac{1}{2}\left[\frac{1+\beta^2}{2\beta}\ln\left(\frac{1+\beta}{1-\beta}\right) - 1\right] + \frac{\hbar^2\omega^2}{s'+2m^2}\left[\frac{1}{\beta}\ln\left(\frac{1+\beta}{1-\beta}\right) - 1\right]\right\}

where

s = (p_1+p_2)^2

s' = (p_1+p_2-k)^2

\beta =

electron

v/c

in c.m. frame. Neglect the emission of radiation by any of the hadrons, all assumed to be much heavier than the electrons. The factors proportional to

\omega^2

in the numerators of these expressions can be attributed to the disappearance of the magnetic moments. If you have not included such contributions in your semiclassical calculation, you may consider doing so.

Related theory

Soft Photon Theorem

Browse the problem library