# Talk:Parton distribution functions (definition)

Referee A:

This contribution by John Collins gives a clear and concise account of the definition for parton densities, both for its physical motivation and its technical implementation.

It is only in the section on the IR point-of-view where I think that the article would gain from clarification or reformulation of several points.

In the second paragraph (starting with "Fundamentally this is a mistaken view") the author states that in a model theory where all fields are massive, the bare parton densities have no collinear or soft divergences. This is true, but I think the relevant point in this context is that the on-shell partonic cross section (rather than the parton distribution) has no such divergences. In fact, the "IR point-of-view" draws on the relation between UV divergences of parton densities and collinear divergences of partonic cross sections when all partons are massless. I think that this relation, explained in the article, is useful and important, both conceptually and for practical calculations. (Of course, the reasons for its simplicity need to be understood in order to prevent inappropriate generalizations.) In this light, it may be useful to discuss the case of massive partons after the discussion of eqs. (29) to (34), rather than before.

In addition, it may be helpful to state that the correspondence between singularities that is obtained with massless partons and dimensional regularization is a particularly simple (perhaps deceivingly simple) example of a more general correspondence: in any other scheme, the UV subtractions in parton densities and the subtractions in the partonic cross sections must properly match each other, so as to achieve a proper factorization of dynamics at high and low scales and to avoid double counting. (The general need for subtractions is mentioned earlier in the article, but only briefly, and an unprepared reader may well miss this point in the present context.)

I think that, as they stand, the formulations in the last paragraph of the section (starting with "This result is physically misleading") are problematic. Especially the statement "But these collinear divergences are not present in the actual theory" could be misunderstood. What is probably meant is something like "actual QCD where quarks and gluons are confined" as opposed to "perturbation theory for massless quarks and gluons", but this should be spelled out more explicitly. The same holds for the notion of "true divergences".

Apart from this criticism, I only have a few comments.

1. In the opening phrase, one may wish to change "scattering with hadron beams" into "scattering with hadrons beams or targets". (Our experimental colleagues may especially appreciate this point.)

2. After eq. (2) it may be useful to point out that other conventions for the kernels P exist (regarding their overall factor and their sign).

3. In eq. (5) the symbol $\hat{F}(Q)$ needs a subscript $1$, as far as I see.

4. At the end of the first paragraph of the section "The IR point-of-view", the notion of "mass divergences" appears for the first time and is not explained. It may be simpler to repeat the term "collinear divergences", or to point out the equivalence of the two terms.

I conclude with a suggestion. Unless this aspect is to be covered in other articles of this collection, one could easily make the present article more general by removing the "MS-bar" in its title. In my view, this would only require an explicit remark that there are other schemes for UV renormalization, and maybe a brief explanation of the DIS scheme. I leave it, however, to the author and editor to decide whether to follow up on this suggestion.

Referee B:

The article aims to explain the definition of parton distribution functions in perturbative QCD. It is detailed in many technical aspects, but requires the reader to have in-depth knowledge on many aspects of modern quantum field theories.

Since Scholarpedia is developing quickly, it may not be needed to expand further upon the following topics, which should however be hyperlinked at the appropriate place into this article as soon as they become available: QCD, renormalization, dimensional regularization, Wilson line, Feynman rules. Without previous knowledge on these topics, the article is very hard to understand.

Concerning the structuring of the article, the section on Feynman rules in the context of parton distribution functions should be placed before the discussion on UV divergences and renormalization, since it is required for the understanding of the formulae in this paragraph.

The sections on 'polarized parton densities', 'the IR point-of-view' and 'dealing with heavy quarks' are only superficially worked out. They contain many previously undefined technical terms, and are definitely only accessible to experts working in the field. They should be taken out altogether, or completely rewritten to be accessible to a larger audience.

The critisism of the 'IR point-of-view' in this article reflects the point-of-view of the author, and the arguments which are put forward are not convincing. As a matter of fact, parton distributions are most commonly used in an effective theory approximation of QCD: namely massless QCD with N_F quark flavours. In the perturbative treatment of this theory, infrared divergences are present (and arguments about a confinement radius shielding these divergences are outside perturbation theory). The correspondence between IR and UV divergences in different effective theories has been subject of much development in recent years (largely triggered by Smirnov, 2002), leading to a multiplicative formulation of the corresponding factorization and renormalization properties. Statements about this formulation to be 'mistaken' and 'misleading' clearly reflect only the view of the author, and are misplaced of a scholarly text.

Response to Referee A

I agree with all the comments of referee A, and I have at last found the time to make appropriate changes.

For the particular issue of the title, I agree that it would be better to remove "MS-bar". It presumably needs action by the editor to do this (Dave Soper).

Response to Referee B

It is correct that to understand this article, a reader needs a good knowledge of quantum field theory. But that is the minimum needed to understand the technical definition of a parton density. When I look over other articles on theoretical high energy physics and on quantum and statistical field theory, I see that many also require a good knowledge of QFT.

Even so, links are needed to concepts on which the article depends, and I have added the main ones.

Concerning the sections on polarized parton densities and heavy quarks, it seems to be that some such sections are needed, to get a suitable amount of completeness. The polarized case forms a natural generalization of the simplest parton densities, so not to mention them is wrong. Heavy quarks need to be mentioned, because this is the one area in which the parton densities commonly used in phenomenology deviate from the MS-bar definitions. (The commonly used fits for parton densities are those labelled CTEQ, MRST, NNPDF and HERAPDF. All, as far as I know, use the MS-bar definition for their primary results, with the exception of the treatment of heavy quarks.) However, a detailed treatment of these areas would excessively extend the article, so these sections are primarily pointers to appropriate places in the literature.

I have revised the article to put these two sections as subsections of a section "Further developments", to show their status. I also took the opportunity to add a third subsection on "Transverse momentum dependent parton densities", which again is merely an identification of the concept together with pointers to where further information can be found.

As to the section on the IR point of view, I stand by what I wrote. I have modified the beginning of the section to quote a much better example of the problem I was complaining about.

Referee B states that "parton distributions are most commonly used in an effective theory approximation of QCD: namely massless QCD with N_F quark flavours." For that being the most common approach, I am not at all sure this is correct. In any case, there is a fundamental flaw in the statement. A massless approximation to QCD (or any other QFT) is only appropriate when masses can be neglected. This can be true in a calculation of a graph when all the lines have virtualities much bigger than the masses. But a parton density, by construction and its use in factorization, is to contain dependence on low scales. Thus it is a clearly incorrect approximation to neglect masses in parton densities.

This can be trivially seen from an example mentioned in the section in question. Consider a QFT all of whose fields have nonzero mass. Compute parton densities in the theory (e.g., with low-order Feynman graphs). These have no mass divergences, but only UV divergences which are cancelled by UV renormalization. These are correct parton densities to use in a factorization property for cross sections like that for DIS. But if the masses are replaced by zero in the calculation of the parton densities, incorrect and unphysical collinear divergences are obtained; these divergences are not present in the actual theory, which by definition has nonzero masses. Note that the presence of nonzero masses does not in any sense violate a correct proof of factorization.