How is the parton model justified by QCD?

[petergreat]

In the parton model for deep inelastic scattering, the momenta of the partons are assumed to be longitudinal and on-shell. What's peculiar (at least rhetorically) is that even the "virtual" sea quarks are assumed to be on-shell. How does QCD justify these assumptions? Are there systematic corrections coming from non-vanishing transverse momentum and off-shell partons? Or are such corrections vanishingly small in some appropriate limit?

 

[tom.stoer]

You will find all these QCD effects in the deep inelastic scattering results:
- there is a so-called scaling limit of DIS recovered by tree-level QCD which corresponds to free = non-interacting partons;
- there is a systematic way to calculate higher loops which results in scaling viaolation (confirmed experimentally);
- and there are the so-called nucleon structure functions F_n(x,Q²) which contain the non-perturbative QCD effects including sea quarks;

I think you should try to understand these structure functions plus certain sum rules which can be derived by integrating over the scaling variable x.

 

[petergreat]

Yes, loop effects which violate Bjorken scaling can be calculated, but the *initial* incoming parton is still taken to be light-like, with no virtuality and no transverse momentum. Why do we have $$F(x,Q^2)$$ instead of $$F(x,P_T,t,Q^2)$$, where $$P_T$$ and $$t$$ are transverse momentum and virtuality, and $$Q^2$$ is the energy scale that is probed?

 

[petergreat]

In short, how does one prove rigorously, starting from the QCD Lagrangian, that in DIS only light-like parton momentum need to be considered?
Also, how does one prove that all non-perturbative effects can be absorbed into the parton distribution functions? I think such a proof must be non-perturbative in nature. Anything involving Feynman diagrams doesn't count.

 

[tom.stoer]

I don't think that light-like is a crucial assumption; m=0 is just an approximation which does not affect the relevant physics.

I never thought about your question in detail; for me the structure of the matrix element simply followed from Lorentz covariannce which dictates that there are some trivial kinematical factors plus non-trivial functions involving F(x,Q²). The Lorentz structure of the matrix element is valid w/o ever referring to perturbation theory.

 

[Vanadium 50]

In short, how does one prove rigorously, starting from the QCD Lagrangian, that in DIS only light-like parton momentum need to be considered?
Also, how does one prove that all non-perturbative effects can be absorbed into the parton distribution functions? I think such a proof must be non-perturbative in nature. Anything involving Feynman diagrams doesn't count.

You can't, because it is not true. The parton model is an approximation. A very good approximation, but it is not exact.

 

[petergreat]

I don't think that light-like is a crucial assumption.

The proton is not made up of free quarks. Why should they be on-shell (i.e. light-like)? It doesn't bother me as much when you take up and down quarks to be on-shell. But why are the sea quarks also taken to be on-shell? Aren't they virtual particles by definition?

You can't, because it is not true. The parton model is an approximation. A very good approximation, but it is not exact.

I guess it may be true in some appropriate limit. For example, I suspect when the collision energy tends to the infinity, the fractional error tends to zero. (Of course, I mean the QCD-corrected parton model where scaling is violated.)

 

[Vanadium 50]

I'm not sure there is anything deeper than "it's an approximation that works where it works". For example, in the case you describe, I know that it will not work for the process pp goes to 4 jets. At high collision energy, double parton scattering dominates.

 

[petergreat]

I've read some more about this. QCD factorization theorems guarantee that for certain processes such as DIS, Drell-Yan etc. the parton model cross section is asymptotically correct as $\Lambda_{QCD}/Q \to 0$, i.e. as center-of-mass collision energy tends to infinity. But double parton scattering that Vanadium mentioned are for processes for which no factorization theorem has been established, so nothing rigorous can be said of such cases.

 

[humanino]

In DIS, the transverse momentum is integrated over and part of the definition for inclusive structure functions. As it was said previously, those also contribute into scaling violations.

But please note that there are transverse momentum dependent PDFs (TMDs), I am not sure what the exact status of factorization theorems is at the moment because the theory is still under development. Even if some (no?) case(s) have reach consensus (there are several scales in the problem), it is believed it can be done consistently at least.

You can measure TMDs in semi inclusive DIS, data is being analyzed and taking more on schedule already.

TMDs have already been related to other new observable, generalized parton distributions (GPDs), which all stem from the most general two-body correlation function, namely Wigner function. Although no process is known to measure Wigner function of partons in hadrons, GPDs factorization theorems are well established. GPDs can be measured in exclusive processes, again data is being analyzed and more is on schedule.

There has also been attempts to link GPDs as a first approximation for double parton scattering (4 jets or others).

 

[kaksmet]

The reason the normal (collinear) PDFs does not depend on transverse momenta is because the transverse momenta has been integrated over. If one does not perform the integrals one get so called unintegrated/transverse momentum dependent distributions.

For transverse momentum dependent distributions (TMDs) factorization has only recently been proven for the Drell-Yan process ( see Collins, Foundations of perturbative QCD)

Factorization is generally very difficult to prove, and especially when not integrating over the transverse momenta. There are also attempts to find factorization breaking effects.

One last comment is that there are also attempts to generalize the factorization for Drell-Yan into a factorization for double Drell-Yan

 

[humanino]

see Collins, Foundations of perturbative QCD

You are right that this is the most authoritative reference on the question, and as a matter of fact he provided a modification for the very definition of TMDs in order to overcome the difficulties associated with their factorization. I am only saying that the fine details have not been fully polished, at least if they have, it was in the last few months.

So from his book, just before he begins 13.7.1 "Definition of TMDs" :
"It is an urgent problem to completely fill in the details of the proofs"

and he then refers to the exercises at the end, where he repeats several times "Publish the result if you are the first to solve this problem."

For a briefer and free version, see also New definition of TMD parton densities (where reference number 1 is the book in question)