Group Meeting at Mainz 14th June 2012
"The cross section for the ISR process is composed by two parts:the radiator function and the cross section for the non-ISR process.
The radiator function says that photons with small energies are preferred. However, the cross section of the non-ISR process is maximum for
hadronic invariant masses at the threshold. Because the photon energy is completely related to the invariant mass of the hadronic system,
low invariant masses mean high photon energies. In the end, the effect of the radiator function is compensated by the total cross section
of the non-ISR process (e+e- toNNbar) and the ISR cross section still has a maximum for low invariant masses (high photon energies).
If you want to see the cross section, you can find it in page 4 of my talk in Orsay which I attach here
One can make this argument more quantitative as follows
x = 2 E_gamma / sqrt(s) is the fraction of the incident electron energy carried by the photon it radiates. The radiator function goes as 1/x.
The e+e- -> pbar p cross section goes as 1/(1-x)^5 because q^2 goes as (1-x) and the form factors go as 1/q^4. Therefore, the ISR cross section
goes as 1/[x(1-x)^5], which is strongly peaked at x=1 (as well as at x=0). The infinities here come from neglecting the electron mass (i.e. they
go away in real life). Therefore, E_gamma peaks at sqrt(s)/2 (the full energy of one of the electrons from which it is radiated).
E_gamma also peaks at zero energy. These photons are radiative corrections to the e+e- -> pbar p reaction at q^2=s.
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