# Energy dependency of the total cross section for different species

Comparing the plots for the total (inelastic) cross sections as a function of the centre of mass energy for $pp$ and $e^+e^-$ collisions:

one notes that the trend at high energy is opposite: the $pp$ cross section increases while the $e^+e^-$ decreases. Is there a (simple) explanation for this?

• Note that the electron cross-section, away from resonances, seems to be proportional to $1/\sqrt E$ (if center-of-mass energy $E$ is what's on that unlabeled horizontal axis). Also note that the e-e plot would fit only in the left half of the hadron plot, so maybe stuff starts happening again at higher energy. I don't know, though.
– rob
Jan 29, 2016 at 0:01
• @rob Yes, that's obviously energy [GeV], as easily inferred from the masses of the resonances. Regarding the stuff happening at higher energy: one expects resonances at the Higgs and at the Top masses, but there are no hints that the general trend should be altered. Jan 29, 2016 at 7:51
• There's probably a simple interpretation for the $1/\sqrt E$ cross section in e-e. Thermal (milli-eV) neutrons have the same shape, and the usual explanation is that the cross section is proportional to inverse of the speed, or to the "dwell time" near a nucleus (but of course that particular argument doesn't hold for relativistic electrons). If you can predict the $1/\sqrt E$ cross section for electrons you should find yourself making an assumption that is broken for protons.
– rob
Jan 29, 2016 at 19:13
• Can someone give me a peer-reviewed reference of where the $e^+e^-$ cross section graph comes from? Mar 5 at 13:58

Roughly, the reason why the proton-proton cross section grows with Mandelstam $$s$$ is because the parton density functions (PDF's) of a proton, in particular it's gluon PDF, grows faster than $$s$$, thus outcompeting the natural fall off rate of $$\frac{1}{s}$$ of a two particle cross section. The growth of the electron PDF's on the other hand do not outcompete the $$\frac{1}{s}$$ falloff rate.
If you don't know what a parton density function is: The parton density function of a hadron is a measure of how many particles it contains. You might complain that a proton only has three particles, two up quarks and a down, but this description is only accurate at low energies and is mainly useful for hadron spectroscopy (classifying the hadrons). When calculating a cross section in QCD, the framework is such that you presume that there is some likelihood of finding any particle inside of a proton, for example $$P_{\gamma}(E_{\gamma})$$ may denote the likelihood of finding a photon of energy $$E$$ inside of the proton. Without going into too many details, the intuition is obvious, the more particles inside of a hadron, the greater the cross section.
• Interesting, maybe it would be useful to elaborate a bit more on how/why PDFs depend on $s$? Jan 4, 2021 at 20:55