Pp total cross section and asymtotic freedom

[gtoeroe]

Hi!

On page 12 of http://pdg.lbl.gov/2009/reviews/rpp2009-rev-cross-section-plots.pdf one can see the total cross section increasing beyond a certain threshold. At higher energy protons (and their quarks) have a shorter wave length and so they can become closer an closer and the strong force should become smaller and smaller. Then the cross section should decrease. Where is my mistake?

gtoeroe

 

[mfb]

Protons have a substructure and a finite size. The roughly flat cross-section over a large energy range is just something like the geometric size of the protons.

The cross-section for events like "half the collision energy ends up as transverse energy" goes down, but you get more and more collisions with small momentum exchange relative to the collision energy - and the PDFs rise quickly if you go to smaller x.

The small rise is not well understood, and there are multiple ideas, but that is a small effect.

 

[gtoeroe]

Well, the increase within the curve is small, but compared to what one would expect it is not small. If one interprets "asymptotic freedom" as "vanishing interaction at high energy" then a decrease like in the coulomb range should be expected. So we have something here what QCD can't explain?

 

[mfb]

What you would expect from the model "the proton has a finite size which determines the cross-section" is a flat line.
What you would expect from the model "we have PDFs as function of x and Q^2, and integrate the cross-sections starting from some minimal momentum exchange" is not far away from a flat line if you take care of multi-parton interactions (otherwise it is rising with energy).

It is hard to get the exact shape with QCD without any experimental input, but it is easy to get a reasonable approximation, and the remaining difference is probably just our lack of knowledge about nonperturbative QCD.
Relevant concept: Pomerons

 

[gtoeroe]

What you would expect from the model "the proton has a finite size which determines the cross-section" is a flat line.
What you would expect from the model "we have PDFs as function of x and Q^2, and integrate the cross-sections starting from to some minimal momentum exchange" is not far away from a flat line if you take care of multi-parton interactions (otherwise it is rising with energy).

It is hard to get the exact shape with QCD without any experimental input, but it is easy to get a reasonable approximation, and the remaining difference is probably just our lack of knowledge about nonperturbative QCD.
Relevant concept: Pomerons

There is no doubt that the calculation of a strongly interacting process is a very difficult assignment. However, even without doing such calculations one can apply fundamental laws of physics and derive conclusion pertaining to hadronic structure. Below are a few points that are relevant to the proton-proton scattering data depicted on p. 11 of http://pdg.lbl.gov/2012/reviews/rpp2012-rev-cross-section-plots.pdf (hereafter, these data are called “the p-p data”):

1. The data of electron-proton scattering prove that for high energy the process is described as a collision of the electron with a single quark. This property holds even for the energy which was available nearly 50 years ago. The term Deep Inelastic Scattering describes this process. See e.g. https://en.wikipedia.org/wiki/Deep_inelastic_scattering.
2. The electron-proton high energy scattering data show that the total cross-section decreases with increasing energy (see e.g. the Rosenbluth formula) and that for a heavy collision, the portion of elastic cross-section decreases even faster. Therefore, the data of electron-proton scattering is completely different from that of the p-p data, where, at very high energy, both the elastic and the total cross-sections increase.
3. Fundamental wave properties indicate that as the projectile’s energy increases and its associated wave-length decreases, the effective scattering region decreases. This is the reason for the decrease of the electromagnetic electron-proton cross-section. Therefore, unless new dynamical factors arise or the interaction near the origin grows much faster than 1/r, the total cross-section should decrease with an increasing scattering energy.) This is a general scattering property of every quantum particle. On top of that, one should also note that QCD’s “asymptotic freedom” means that the quark-quark interaction practically vanishes at small distance. It follows from QCD that the proton-proton scattering cross-section should decrease much faster than the electron-proton cross-section. This QCD result is completely inconsistent with the p-p data.

What you would expect from the model "the proton has a finite size which determines the cross-section" is a flat line.
What you would expect from the model "we have PDFs as function of x and Q^2, and integrate the cross-sections starting from to some minimal momentum exchange" is not far away from a flat line if you take care of multi-parton interactions (otherwise it is rising with energy).

It is hard to get the exact shape with QCD without any experimental input, but it is easy to get a reasonable approximation, and the remaining difference is probably just our lack of knowledge about nonperturbative QCD.
Relevant concept: Pomerons

 

[mfb]

The electron-proton high energy scattering data show that the total cross-section decreases with increasing energy (see e.g. the Rosenbluth formula)

This is a formula for elastic scattering, and with electron/proton collisions instead of proton/proton collisions so you would not expect the same result. Also, where did you integrate it over all phase space?

Fundamental wave properties indicate that as the projectile’s energy increases and its associated wave-length decreases, the effective scattering region decreases. This is the reason for the decrease of the electromagnetic electron-proton cross-section.

This argument is faulty as it ignores the proton PDFs and the minimal momentum exchange to be considered a collision. At high energies you have more particles in the proton that can contribute to a collision.

On top of that, one should also note that QCD’s “asymptotic freedom” means that the quark-quark interaction practically vanishes at small distance.

The coupling goes down for higher energies, but that just means the cross-section for jets with 1/4 the proton energy (random example) goes down. So what, that's not the total cross-section.