How do particle scattering cross sections scale with energy in colliders?

[jimgraber]

How do particle scattering cross sections scale with energy in colliders?
Particularly photons, electrons, protons, and gold or lead nucleii?
(If necessary, break this into four separate questions.)

It is stated that due to the Heisenberg uncertainty principle, it takes more energy to measure a smaller distance and the inverse wavelength energy-wavelength relationship for photons is well known.
Therefore, I would conclude that the (total scattering) cross section for photons at least would vary as the inverse square of the energy.

On the other hand, general relativity says the the longitudinal dimension shrinks with increasing velocity, but the transverse dimensions do not.
Therefore, I would conclude that the cross section of say a gold atom in RHIC or a lead atom in the LHC would basically remain constant as the energy increased.

But how about the electron, supposedly a point particle?
Due to PEP and LEP, we should have good data on this.
Looking at PDG figure 41.6,
<http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cross-section-plots.pdf>
I see lots of big spikes for resonant particles, but the underlying bacground is clearly a falling powerlaw that I think looks like it could be an inverse square law.

The proton cross sections in the next few figures, e.g. 41.7 and 41.11, show a falling elastic scattering cross section, but they also show an approximately constant total cross section.

What little I've found for RHIC supports the approximately constant cross section.

Therefore I conclude that the cross sections of photons and electrons (neglecting resonances) scale as the inverse square of the energy, but the cross sections of protons and nucleii are roughly constant with energy.

Comments or corrections?

So is it true that the photons and electrons are different from the protons and nucleii?

And why don't the protons scatter as three or six point particles (quarks and gluons) instead of one big blob?

 

[Bill_K]

According to Fig 41.6, the ratio between total e⁺e^- scattering and e⁺e^- → μ⁺μ^- scattering is roughly constant, and the latter is 4π α²/3s. Where s of course is the Mandelstam variable s, equal to the square of the center of mass energy.

From the other Figs, total pp cross section rises slowly, like ln²(s). This is explained elsewhere as due to the pomeron.

http://www.phy.bnl.gov/pp2pp/information/presentations/1999/Asilomar-99.pdf shows how complex the elastic pp cross-section is. In the high energy region it is said to fall like s^-10.

why don't the protons scatter as three or six point particles (quarks and gluons) instead of one big blob?

At high energy a pp collision is really a single parton-parton collision, with the other partons acting as spectators.

 

[jimgraber]

So why does one point particle (the electron) have a falling cross section, and three point particles (the quarks in a proton) have a rising cross section?
TIA for any reasonable explanation.

 

[Bill_K]

σ ~ ln²(s) is known as the Froissart bound, and Froissart's theorem limits σ to this growth rate or less, relying only on basic assumptions such as locality and unitarity.

Pomeron theory predates QCD, and was developed as a largely empirical explanation of the observed rising tendency in the total cross section. A pomeron is a Regge trajectory, an infinite family of particles with increasing spin, and exchange of these things is what produces the rise. The challenge is to derive this idea from something more fundamental.

The subject is complex, and I can only point you to references such as this one, which derive the behavior from perturbative QCD.

 

[mfb]

Protons do not have a fixed set of partons. If you go to smaller energy fractions x, the number of partons per energy grows beyond limit. Therefore, if you increase the beam energy, more and more partons can contribute to the inelastic cross-section. The result is that high-energic protons are similar to disks - if they collide, some partons inside will interact. This gives a cross-section with a weak energy-dependence.

Heavy ions are similar to protons. The disk approximation is even better there, as they are larger.