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On the complex structure of YangMills theory
by Jan Horak, Jan M. Pawlowski, Nicolas Wink
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Submission summary
Authors (as registered SciPost users):  Jan Horak · Jan M. Pawlowski · Nicolas Wink 
Submission information  

Preprint Link:  scipost_202209_00032v2 (pdf) 
Date submitted:  20230712 11:58 
Submitted by:  Horak, Jan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider the coupled set of spectral DysonSchwinger equations in YangMills theory for ghost and gluon propagators. Within this setup, we perform a systematic analytic evaluation of the constraints on generalised spectral representations in YangMills 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.
List of changes
 comment about allowed violations of the spectral representation in abstract and intro
 paragraph explaining how the spectral DSE is suitable for studying general complex structure beyond the spectral rep at the beginning of sec IV
references about gauge dependence of complex structure suggested by second referee in the first paragraph of sec IV A
 added paragraph and revised first part of sec V to clarify the role of the gluon mass parameter in the DSE, in particular in distinction to the CF model
Current status:
Reports on this Submission
Report
I think the authors managed to make some improvements. However, the requested revision was not aimed at specific sentences and details, but at the complete structure of the article. I will discuss the points as adressed in my first report and the reply of the authors:
1) This has been discussed again in the second report of ref 1 and in the reply of Dr Horak. Some comments have been added on the CF model, and other comments have been deleted because they caused confusion. I think the current way of presenting is still confusing, or even more so. First of all, the added sentences
"Note that this procedure has to be carefully distinguished from massive extensions of YangMills theory such as the CurciFerrari model. There, the ghost and gluon gap equations are evaluated at a given loop order with fixed ghost and (massive) gluon propagator input. "
What does 'there' refer to? The authors distinguish between a 'procedure' on the one hand, and a 'model' on the other hand. The CurciFerrari model has been, and can be used, in many different ways, as it is a model, and not a method. The model is not fixed to a given loop order or input.
The authors state that using the term YM model is standard practice, and they seem to be right about this. For me, a question however remains: if they would have started not from a YM model, but a CF model, would they have had a different outcome? I think that eventually, their concept of the YM model is too wide (as the YM model should include e.g. BRST invariance) and their concept of the CF model is too narrow (it is not restricted to one methodology). I do not see why one cannot call this the CF model. As was said in v1:
In the past decade many studies have also exploited massive extensions of YangMills, formulated in terms of the CurciFerrari (CF) model with mass terms for ghosts and gluons,or by simply adding a mass term for the gluon after the gaugefixing. Note that in the numerical computations in the present work we follow the latter approach.
These sentences were removed in v2, and no direct connection to the current work was made in the paragraph anymore. This does not clear things up, but makes them more vague. Since the statements in the first sentences are identical (the CF model IS adding a mass term after the gaugefixing), I think the above statement will do once this is cleared up.
2) This point was dealt with adequately by the authors with the insertion of a new paragraph.
3) I think the authors are right that they have some distinct results and this work deserves to be published, especially as they say, as a "starting point for future analysis". Therefore I think that if point (1) is cleared up, this can contribute in a more clear way to the consisting efforts of understanding complex structures. It will also do justice to sentences with predictions on cancellations, if they are given in the context of a BRSTbreaking, and most likely nonunitary, model.
Requested changes
Requested changes: more clarity in the text on point (1).
Report
Second Report on the manuscript
scipost_202209_00032v2 (resubmitted modified version)
“On the complex structure of YangMills theory”
by Jan Horak, Jan M. Pawlowski and Nicolas Wink
In my opinion, the authors did not make the main changes which were suggested by the referees.
Moreover, their answers to the main criticisms are not satisfactory.
Thus, despite the relevance of the subject, I cannot suggest publication in the present form.
The new version of the manuscript is still misleading: it should be said clearly, in the abstract and in the introduction, that the theory they are studying is not YM theory, but an effective model, the oneloop CurciFerrari model with self consistent propagators, as admitted at the end of Sec.IV in the old version.
In the revised version, they cut that sentence at the end of Sec. IV, because it was "misinterpreted by the referee".
But they say in Sec. V: "Strictly speaking, by the presence of an additional parameter the described approach only constitutes a model of YangMills theory, additionally breaking gauge invariance."
That model is CurciFerrari model!
The claimed difference between their model and the CF model seems to be just the approximation which is used, rather than the Lagrangian: the new added sentence in Sec. V says: "Note that this procedure has to be carefully distinguished from massive extensions of YangMills theory such as the CurciFerrari model. There, the ghost and gluon gap equations are evaluated at a given loop order with fixed ghost and (massive) gluon propagator input. In contradistinction, in functional approaches, the equations are solved selfconsistently, with full propagators inside the loops, while aiming at eliminating the introduced gluon mass parameter."
Thus, they are studyng the same CF model by a selfconsistent approach to a truncated set of oneloop SDE, while in the usual perturbative approach the loops contain the bare propagator. Actually, Eqs.(27),(29) are the SDE of the CF model. With a finite mass parameter!
A model is defined by its Lagrangian, is not defined by the approximation method. Here, the difference has to do with the approximate method and with the (failed) attempt to find a selfconsistent solution for the CF model. The failure could be expected, since the CF model predicts complexconjugated poles which prevent from any selfconsistent solution with bare vertices, because of the interesting discussion on complex singularities in Appendix C. By the way, as also required by Ref.2, Appendix C is an important section of the paper and should have been inserted in the main text.
The "gradual lowering" of the mass parameter makes sense, but as a sort of optimization of the CF model in order to reproduce YM theory. However, as discussed in 2107.05352, the mass parameter must be sent to zero, otherwise the studied model is not YM theory but CF. In fact, in their answer, the authors claim that "it is possible to dynamically generate a gluon mass gap in the coupled system of ghost and gluon propagator DSEs with bare vertices, as shown in arXiv:2107.05352, Sec. IV Scenario A."
But that does not seem to be correct: Scenario A is discussed in Sec.III where only a scaling solution is found when the mass parameter is sent to zero. Scenario B, where a finite mass is added, requires the existence of a pole in the gluonghost vertex and would not be consistent with a bare vertex. Thus, no dynamical mass is found with bare vertices, unless it is added by hand as a model for YM theory.
In summary, if a model is used instead of YM theory, it shoud be said from the beginning, in the abstract and in the introduction.
If the authors insist that YM theory is studied instead of CF, they should explain what the difference is: if one would study the CF model by the very same selfconsistent solution of SDEs, the result would be the same (the SDEs are the same). The authors did not answer this simple question.
About the consistency of the approximation, it was observed by the referees that the approximations are not consistent with the CF model. Then, the lack of self consistency is a consequence of the chosen model.
As observed by Ref.2, it is "not clear why this research was conducted with oneloop results already obstructing the extraction of a KL spectral density function."
In their answer, the authors say that there is no compatibility problem with their spectral functional approach which "covers a large class of functions for the propagators".
The point is that the chosen model, CF model with bare vertices, is known to give rise to complex poles. Then, there are two kind of problems: a) the numerical method is not suited for dealing with complex poles, and a real pole is introduced instead, for simulating the complex poles; b) as shown in Appendix C, no selfconsistent solution can exist with complex poles. We can only conclude that the spectral method is not the best method for investigating the existence of complex poles in the CF model. Moreover, the need of an added part in the spectral function is the sign of complex anomalies. In that sense, it could be regarded as a proof of existence for the complex poles. If there were no complex poles, then there would be no need of a real pole structure for simulating their presence (the added part was real only because their numerical method did not allow the insertion of a complex polepart).
Author: Jan Horak on 20230810 [id 3890]
(in reply to Report 1 on 20230731)The referee still insists on the fact that our work treats the CF model, selfconsistently using DSEs, since our setup features a mass parameter in the gluon DSE. Since 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 selfenergy, which is then sent to zero. Since we are renormalizing the gluon DSE at a nonvanishing 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 decouplingtype solutions with nonvanishing 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 finetuning 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 oneloop analysis in the CF model, where complex poles are found. In our opinion, the current analysis is a first but important step towards a selfconsistent solution of the ghostgluon system. In YangMills theory this certainly necessitates feeding back the ghost and gluon propagators or rather their spectral functions into the loop. Note that this is highly nonlinear due to the occurrence of both propagators in both gap equations. This minimally selfconsistent solution is first studied here in the present work and is qualitatively different from the oneloop 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 setup, 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 wellsuited 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 wellsuited 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 setup a clear path towards a fully selfconsistent 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 YMtheory, 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 selfconsistently. 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 YangMills 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.