Colour unwound - disentangling colours for azimuthal asymmetries in Drell-Yan scattering

1. A detailed analysis is given of a model calculation to test a previous published result that color entanglement effects exist in the Drell-Yan process in certain important cases. The previous result, if true, would endanger the use of standard factorization methods in these cases.
2. The analysis is sufficiently detailed to allow proper verification.
3. To support the analysis an extension of subtraction methods to the Glauber region is used. This appear to me to be new, and of importance to future work.

1. There are several places where I an unable to reproduce the authors' arguments. Some appear to be on a critical path to the paper's final result. Nevertheless, I am fairly sure that they can be repaired. See the changes list for details.
2. Indications as to to the source of the error in [1] are not adequately explicit.

This is an important paper. In [1] an argument was given that color entanglement effects break factorization in the Drell-Yan process, at least for the double-Boer-Mulders (dBM) or the double-Sivers (dS) cases. These are processes for which factorization proofs exist in the literature. The factorization proofs are intended to be applicable to all kinds of Drell-Yan process, with dBM and dS simply as special cases. So any failure in one case is a symptom that something is missing in the standard proofs, and hence that there may be a much more widespread issue. Given the importance of factorization to high-energy physics phenomenology, the subject is important and needs careful examination.

The present paper shows that the result in [1] is incorrect. A calculation is made at the lowest relevant order in a model, and the results do not give the color entanglement effect that the argument in [1] predicts.

As the authors say, the graphs are of 3-loop order, and hence to get useful results it is more or less essential to use appropriate approximations. In addition, the analysis can be made as much as possible without an integration over all components of loop momenta. Verification of the results depends on verifying and reproducing the chain of logic leading to the region analysis, the approximations, etc.

Unfortunately, I find that a number of steps in the argument need substantial clarification, at least.

Furthermore the paper appears to assert that the color entanglement effect found in [1] applies only to the dBM and dS situations, but not to the collinear case. This corresponds to the second paragraph of the "Discussion and conclusions" section of [1]: "One might wonder why the gauge connections cannot be disentangled as in the collinear case, since the color charges are entangled in both cases. ...". Unfortunately, I am completely unable to reproduce this result.

If it were true that the gauge connections were entangled in the collinear case, i.e., if (2) of [1] were correct, then I am totally unable to deduce a disentangled result for the collinear case. By explicit calculation, the results of entangled and disentangled connections differ at order $\alpha_s^2$, independently of whether the TMD or the collinear situation is considered. I am also only able to reproduce the formula with entanglement by omitting graphs with 3-gluon couplings.

My remarks apply equally to the earlier work JHEP 07(2011)065 by the same authors. They assert formula (4.4) with entanglement for the DY cross section, and then assert that if one of the hadrons is treated as collinear then the traces can be disentangled, to give (4.6). No proofs are given, and I have counterexamples to both (4.4) and the statement that (4.6) can be derived from (4.4).

Have I missed something?

1. It would be helpful if the authors could state more explicitly in the Introduction (and perhaps briefly in the Abstract) where the argument in [1] is in error. This would greatly help readers to know what issues they should focus their attention on. In the Conclusions section, it is stated that inclusion of graphs with a 3-gluon coupling is essential to the paper's result. But, on the basis of what is written I am not sure whether the failure of [1] simply lies in a failure to consider these graphs.

2. In both the Abstract and Introduction, it is stated that it is possible to assign the whole of the dBM and dS contributions to the Glauber region. However, as is explained on p. 9, the standard CSS proof shows that the Glauber contributions can also be said to cancel. There is an apparent contradiction here, and it is important that the authors point this out explicitly in the Introduction (and I think in the Abstract), and summarize how the contradiction is resolved. I get some hints about the resolution from the Conclusions, but I think it is important to bring the issue to the forefront in the Introduction, otherwise readers will be misled. I imagine that the resolution will involve two different ways of viewing the calculation.

3. In the first line of the first full paragraph on p. 3, it is stated that "a recent calculation showed that ... a color entangled contribution can arise". This needs a reference to [1] at this point. But the present paper falsifies the result of [1], at least at the lowest order, $\alpha_s^2$ at which the entanglement effect is supposed to happen. So I don't think "showed" is the correct word.

4. In addition, the experimental work referred to on p. 3 needs to have explicit references.

5. The meaning to be applied to "color-entangled" should be defined, preferably in the Introduction.

6. In the paragraph just above (1), it is stated that factorization of DY into TMDs and a hard factor was established by Bodwin and CSS. The use of the word "TMDs" indicates that TMD factorization was intended by the authors. But the referred-to papers [26-28] only claimed to prove factorization for the collinear case. The critical part of the proof concerning the Glauber region and spectator interactions does actually extend to the TMD case, but I don't know of an adequate use of it for the TMD case until [4].

7. In the definition of the BM function (5), there is a soft factor missing.

8. The notation for the Wilson lines in (7) and as used just above it needs correction, for two reasons. First, the transverse and + components of $\eta$ aren't specified. Second, the use of the arguments $a$ and $b$ as limits of a one-dimensional integral indicates that $a$ and $b$ are single numbers. But in the 3rd factor on the previous line $U$ has a $\xi$ argument is a 4-vector. Knowledgeable readers will know what is intended, but less knowledgeable readers will have trouble.

9. At the top of p. 6, it is stated that "the factorisation theorem was derived for unpolarised particles". In fact the proof, as least as given in [4], is explicitly intended to apply to all the polarised cases, and this includes the dBM and DS cases.

10. I was somewhat confused by the last paragraph on p. 6. It is perhaps worth saying more explicitly that the actual pinches of relevance only occur in the massless limit (i.e., $m/Q \to 0$); it is in this limit that the Coleman-Norton method is used. The regions of interest are neighborhoods of the pinch-singular surfaces.

11. On p. 7 a list of the relevant regions is given. Only later is it stated that Glauber-type regions are also relevant. However, before a sum over cuts, the Glauber regions are unsuppressed. I think it would be better to bring these two lists together, otherwise the categorical statement of the relevant regions is actually falsified a paragraph later. A reference for the assertions is needed, since it is quite non-trivial to justify that these are the appropriate regions and scalings, and that no other scalings need be considered.

12. It is not at all obvious to me that the soft scaling given is the appropriate one. The ultrasoft scaling is also possible. At a minimum, a reference to the discussion in [2] is needed. It might even be worth going into some detail here, since understanding the regions quantitatively is critical to the calculations. The ultrasoft region does give leading-power contributions to cut graphs for the cross section (although they do not matter, I think, for dBM and dS).

13. There is a clash in the literature between the scaling parameter $\lambda$ being treated as an external parameter, say $\Lambda_{\rm QCD}/Q$, and $\lambda$ being treated as an integration variable. The first view is predominant in the SCET literature and I think is also found in [2] and elsewhere. The second view is found in [4]. Some remarks on this might help.

14. At the end of the paragraph containing (8)-(11), it is stated that "In the inclusive DY cross section, collinear scalings with the large component pointing in arbitrary directions (corresponding to final-state jets in arbitrary directions) are also initially important, but such jets cancel after the sum over final states". While there are theorems that have some resemblance to this stated result, I am not aware of any that have this precise statement. A reference is needed. Have the authors written what they mean? In any case, for the TMD case, such configurations are actually power-suppressed.

15. At (17), the justification for the sign of the $i\epsilon$ in the denominators needs to be given or referred to, since the Glauber region is under consideration. The authors point out that the third equality in (17) fails in the Glauber region. But also the second equality is dangerous because before contours are deformed, $l^-$ is integrated through zero. To make the second equality valid, there has to be a deformation away from the pole at $l^-=0$. It is not immediately clear that this is consistent with other singularities in the $l^-$ plane. A careful discussion is needed. There is relevant material in CSS and in [4], but something needs to be said here.

16. The approximation in (17) retains the transverse component of $l$ in the $A$ factor. As already discussed in [2], this is at variance with what Collins used in [4]. The prescription in [4] appears to be important to get the Ward identity argument to work for summing over the soft attachments to a collinear factor without getting extra terms in the non-abelian case (i.e., QCD). I think some discussion is needed.

17. In the 2nd line of the 3rd paragraph of Sec. 4.1, "dBM" needs to be inserted before "cross section".

18. At the bottom of p. 11, the regions giving leading-power contributions are listed. It would be helpful to state explicitly why soft and ultrasoft gluons do not need to be considered in addition. After all the pinch surfaces are the same for both Glauber and soft momenta.

19. In Sec. 4.1, the paper introduces subtraction methods that include a treatment of the Glauber region. These appear to be new and of potentially widespread application, so they are worth mentioning in the Introduction (and possibly the Abstract). Or is there previous work along the same lines?