Applications of infrared divergences
A process like
is not well-defined in perturbation theory because of the infrared divergence.
We found that we need to sum over additional photon emissions to get a finite quantity. Quantities which
are insensitive to the infrared regulator are called
. There is a theorem due to Kinoshita-Lee
and Nauenberg (KLN theorem) which says that any physically observable quantity is infrared safe.
Here are a few examples of the importance of infrared safety in practice:
For situations where the cross section is important, such as
boson production, it’s usually resonance
production. In fact, the soft photons help bring the
to the resonance peak, a process called
. In this case, it’s just the decay width you measure, so you only need the final state loops. The
decay width is calculable, finite, and does not depend on whether it was
or something else that pro-
. Thus the correction to the width is a way to test for the
For example, suppose the
decays to quarks and to muons. Quarks have charge
, so the
photon loop correction will be smaller for the width to quarks than the width to muons. Thus the ratio of
partial widths directly measures
and the charges of the particles in the decay. The electron side drops
out completely. So we don’t need to bother with the
cutoff or the vertex correction loop.
Protons in QCD
For QCD, where you don’t have free quarks, the wave-packets have a natural size – the proton. The
proton is a bundle of quarks and gluons, where the gluons are mostly soft, and can be thought of to a first
approximation like the photons we have been using in this discussion.
These virtual and soft gluons
strongly influence the distribution of energies of quarks inside the proton, in a calculable way. This is the
physical basis of
parton distribution functions
. So there’s a lot of physics in initial state radiation, it just
doesn’t have much to do with total cross sections in QED.
Final state energy cuts.
Swapping a log
for a log
works for final state radiation too, and there it
important. We don’t have
to integrate over all photons on the
side. We can do something less
integrate only over photons up to an energy
. If we put a cut on the photon energy, we will get logs of
that energy divided by
in the cross section.
is often a realistic quantity related to actual parameters of an experiment. For example, the
experiment BABAR at SLAC measures the decays of
mesons to kaons and photons (
are only sensitive to photons harder than 1.6
GeV. Softer photons are below the threshold sensitivity of
their detector. Thus this log is a very important quantitative feature of the cross section they measure,