Exploring Scaling in DIS: Why Middle x-Values?

[blue2script]

Hi all!

I am just wondering why scaling in DIS is approximately realized for the middle (Bjorken-) x-values and not e.g. for the high or low x-values? Is there any depicitive reason?

Thanks for the ideas!
Blue2script

 

[vanesch]

Scaling comes about in the parton model of a hadron, where the hadron is simply seen as an almost unbound "bag" of "partons" (quarks and gluons). The bjorken -x is then nothing else (at least in the relativistic approximation of all-massless particles) of the longitudinal momentum fraction of the "hit" parton on the "whole".
As such, the interaction cross section of a particle with a hadron should factorize in a "form factor" (the probability to have a parton with fraction x of the momentum) and an "elementary cross section" which is nothing else but the interaction cross section of the incoming particle and the "free parton".
For the small x values, what happens is in fact that there are higher-order QCD diagrams in which the "original parton" presents itself as another one. That's a bit as in the case of an electron, there's a cloud of virtual e+/e- pairs around it (vacuum polarization), and at small enough scale, the interaction can be not with the original electron, but, say, with a positron of this "cloud". In the same way, an initial up quark can couple through a higher-order QCD diagram with, say, an anti-up quark with the incoming particle.
There are "evolution equations", the Altarelli-Parisi equations, which use higher-order QCD diagrams to change the quark density at a certain energy into a quark density at another energy. There have been corrections to this, which have to do with 'diffractive effects', which can be seen as scattering on "bound states" within the hadron.

I have to say that I don't really know what gives scaling violations at high x. All this is from memory from 10 years ago...

 

[cygnus2]

Well, I can give you a simple reason why there are scaling violations in the structure functions. As it happens in the naive parton model, in the infinite momentum frame the quarks are taken to be non-interacting with each other. Further, all their momentum is considered to be longitudinal. However, there is always the case where the quark can emit a gluon and acquire a momentum in the transverse direction. Moreover, the maximum momentum of the gluon that it can emit is limited by the momentum carried by the quark itself, which is $x$ times the total momentum of the proton. When you calculate this process, youy get log(x) scaling violations.

 

[blue2script]

Hi cygnus2 and vanesh! Thanks for your replies! I think I got the idea now, so let me try to summarize:

The behavior of the x-Q^2-dependence is:

1) for low x-values the density of the partons increases with increasing Q^2
2) in mid-x values the density of the partons stays constant (scaling)
3) for high-x values the density of the partons decreases with increasing Q^2

The x-value is the fraction of momentum the parton posses.

Reason:

1) If the x-value is low, the momentum of the quark is low. The cross section to emit a gluon of low moment (say, half of the quark momentum) is relatively high. That means, we will rather not find a quark with momentum fraction x but three quarks (a pair created from the gluon) with roughly the same, low momentum x/3. Then, if we probe the cloud with a high momentum transfer, we will eventually see the three quarks separately leading to a high scattering probability at low-momentum quarks-> we see a high density at low momentum.

3) If the x-value is high, the quark as a high momentum fraction. It will radiate mainly low-momentum gluons keeping itself most of the momentum. Thus, the original quark dominates the cloud and, making many experiments, we will most often see quarks at the top line of momentum whereas the low-momentum quarks are suppressed.

2) Since it lies in the middle of 1) and 3) there should be some intermediate range where the density is constant

Hope one can understand the point of my argumentation. It may be a bit confusing. If so, just tell me and I will try to clear it up.

Is this the right interpretation? Thanks again for all upcoming comments!
Blue2script