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On the complex structure of Yang-Mills theory

by Jan Horak, Jan M. Pawlowski, Nicolas Wink

Submission summary

Authors (as registered SciPost users): Jan Horak · Jan M. Pawlowski · Nicolas Wink
Submission information
Preprint Link: scipost_202209_00032v3  (pdf)
Date submitted: 2023-09-21 16:12
Submitted by: Horak, Jan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • High-Energy Physics - Phenomenology
Approach: Theoretical


We consider the coupled set of spectral Dyson-Schwinger equations in Yang-Mills theory for ghost and gluon propagators. Within this set-up, we perform a systematic analytic evaluation of the constraints on generalised spectral representations in Yang-Mills theory that are most relevant for informed spectral reconstructions. Furthermore, we provide numerical results for the coupled set of ghost and gluon spectral functions for a range of potential mass gaps of the gluon, while allowing for small violations of the spectral representation. The analyses are accompanied by thorough discussion of the limitations and extensions of the present work.

Current status:
Awaiting resubmission

Author comments upon resubmission

We acknowledge the point 1) addressed by referee 1 concerning the confusion regarding the distinction between the CF model and functional approaches to YM theory. We attempt to clarify the matter by adding further explanation in the respective paragraph quoted by referee 1.

List of changes

- Further explanation to clarify distinction between the common loop expansion in the CF model and functional approaches to YM theory, including the interpretation of the gluon mass parameter and BRST invariance in section V, just above section V.A

Reports on this Submission

Anonymous Report 2 on 2023-11-10 (Invited Report)


I plainly agree with the report of referee 1. I cannot recommend publication because the points adressed in the first two reports by both referees were not sufficiently adressed. There are valuable calculations in the article, but the authors fail to put them in the right context.

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Anonymous Report 1 on 2023-10-6 (Invited Report)


I agree entirely with the last report of the other referee (Report 2 on 2023-9-12), but I do not see any real improvement in the last version.
In my opinion, in the new (third) version, the authors did not manage to clarify point (1) of that report. That point was also one of my main criticisms and it was the minimal effort which was expected for publication. Thus, I cannot recommend publication in the present form.

Actually, the authors still refuse to answer a very simple question which was raised by both referees:
"if they would have started not from a YM model, but a CF model, would they have had a different outcome?"
I do not think so.

In the new comments, above Sec. V-A, the authors admit that the difference between CF and YM would be the "interpretation" of the mass parameter: the mass is a free parameter in CF, while it is tuned in YM by a procedure
which aims "at eliminating the introduced gluon mass parameter and restoring BRST invariance".

However, in the numerical calculations, the equations are solved "for a family of input gluon mass parameters". It does not seem that the mass is fixed by restoring BRST. Even in the figures, it seems that different values of the mass parameter are discussed, as a free parameter. Thus, according to that "interpretation" the model which is studied is CF.

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Author:  Nicolas Wink  on 2023-11-22  [id 4141]

(in reply to Report 1 on 2023-10-06)

The referee still insists on the fact that our work treats the CF model, self-consistently using DSEs, since our setup features a mass parameter in the gluon DSE. However, this mass parameter is always present in numerical DSE approaches to the gluon propagator, the referee is claiming nothing different but that every numerical DSE study on the gluon propagator to date, which does not exhibit a scaling solution or a massless pole in the longitudinal sector, is treating the CF model. Us calling the theory under investigation YM theory represents our opinion, which is standard in the DSE community, as e.g. the references arXiv:0802.1870, 0810.1987, 1005.4598, 1501.07150, 2003.13703, 2007.11505 and 2107.05352 demonstrate.

In short, this criticism solely concerns how we label our setup, and while we appreciate the standpoint of the referee, we think that our labelling is a commonly used one, and, in our opinion, it is more to the point, keeping in mind the clearly stated aims of the work. More importantly, we describe our setup in the manuscript very thoroughly and transparently. Our presentation is neither misleading nor unclear, and the interested readers can form their own opinion of how our setup should be called.

One of the authors of the submitted manuscript is also an author of the quoted work 2107.05352, and we believe that there is a misunderstanding on the side of the referee with respect to the quoted statements of this work. The mass parameter present in Scenario A is not comparable to the one in the current work. In 2107.05352, the value of the mass parameter is taken to be identical to the renormalization point of the gluon self-energy, which is then sent to zero. Since we are renormalizing the gluon DSE at a non-vanishing scale independent of the mass parameter, the vanishing mass in 2107.05352 corresponds to a finite mass in our case. In fact, as stated in our reply to the previous report, the vanishing mass limit of 2107.05352 is the limit that we attempt to achieve by gradually lowering our mass parameter. It is well known that the scaling solution, which by the referee is regarded as the only case in which YM theory is studied, appears as the closure of decoupling-type solutions with non-vanishing mass parameters. The only way to numerical obtain these types of solutions is by tuning exactly the mentioned mass parameter, that is, by solving the corresponding quadratic fine-tuning problem. In other words, there is simply no way to arrive at what the referee considers YM theory without tuning through what he considers to be the CF model. This has also been discussed very explicitly already in 1605.01856 by one of the authors, where the renormalization group nature of this tuning is apparent.

The last paragraph of the recent report deserves some comments.

To begin with, the referee refers to statements in the report before which were specific to a one-loop analysis in the CF model, where complex poles are found. In our opinion, the current analysis is a first but important step towards a self-consistent solution of the ghost-gluon system. In Yang-Mills theory this certainly necessitates feeding back the ghost and gluon propagators or rather their spectral functions into the loop. Note that this is highly non-linear due to the occurrence of both propagators in both gap equations. This minimally self-consistent solution is first studied here in the present work and is qualitatively different from the one-loop analysis in the CF model. Moreover, while one hence may expect that these complex conjugate poles persist at least for large explicit mass in the present set-up, this is shown in the present work. For smaller mass parameters we do not think that the one loop analysis in the CF model is relevant as the system is tuned towards a strongly correlated regime.

Importantly, we show in the present work, that augmenting the gluon spectral function with complex conjugate poles does not lead to a consistent solution of the system. We cannot see how this can be regarded as a proof of the existence of cc poles at all. Our work includes a lengthy analysis of the consequences of these findings and none of the referees so far addresses this analysis.

Finally, the referee concludes that the spectral approach is not well-suited to address complex poles in the propagator. We beg to differ; this extension is readily done and indeed we did it in the present work including explicit numerical results. There is no technical obstacle at all, and we indeed believe that the present versatile spectral setup is specifically well-suited for a situation with complex singularities with its clear separation of the spectral part and the complex singularities. The obstruction in the present work came from the fact, that cc poles are not consistent in the present approximation even for larger explicit mass parameters. We discussed the limitations of this approximation at length and set-up a clear path towards a fully self-consistent analysis. Again, we believe that this analysis allows the reader to come to their own conclusion about the obstructions and their origin.

As we already mentioned repeatedly, the idea of the numerical study was to reach the confining region of YM-theory, marked by a scaling solution, corresponding to the case of vanishing mass in 2107.05352 and a finite value of the mass parameter in our setup. On the trajectory to this solution, parametrized by the value of the mass parameter, complex poles appear in the, as we call it ourselves, unphysical massive regime of the theory. Due the proof in Appendix C, these poles hinder tuning the system towards the scaling solution while still solving the system self-consistently. We hope that this explanation clarifies the matter.

In summary, we believe that the present work constitutes a major step forward towards the spectral resolution of Yang-Mills theory (with or without complex singularities) and we hope that with these further explanations the referees can judge the present work in view of its clearly described aims and not by their aims they would have if working on this subject.

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