# Tical resolution is that we should just stick to

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tical resolution is that we should just stick to observables that are not sensitive to the electron side. Let me give some examples. 19.5 Initial state radiation 227
19.6 Applications of infrared divergences A process like e + e μ + μ 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 infrared safe . 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: Decay widths For situations where the cross section is important, such as Z boson production, it’s usually resonance production. In fact, the soft photons help bring the Z to the resonance peak, a process called radiative return . 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 e + e or something else that pro- duced the Z . Thus the correction to the width is a way to test for the 3 α 2 π correction. For example, suppose the Z decays to quarks and to muons. Quarks have charge 2 3 or 1 3 , 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 α and the charges of the particles in the decay. The electron side drops out completely. So we don’t need to bother with the E i 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 m γ for a log E works for final state radiation too, and there it is important. We don’t have to integrate over all photons on the μ + μ side. We can do something less inclusive (more exclusive ), and integrate only over photons up to an energy E f . If we put a cut on the photon energy, we will get logs of that energy divided by Q in the cross section. tot dE f = σ tot parenleftBigg α 2 log 2 E f 2 Q 2 + parenrightBigg (19.62) This E f is often a realistic quantity related to actual parameters of an experiment. For example, the experiment BABAR at SLAC measures the decays of B mesons to kaons and photons ( B ) . They 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,