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

Preprint Link:  scipost_202209_00032v1 (pdf) 
Date submitted:  20220916 11:20 
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, which gives us access to the ghost and gluon spectral functions. The setup is used for a systematic analytic evaluation of the constraints on generalised spectral representations in YangMills theory that are most relevant for informed spectral reconstructions. We also provide numerical results for the coupled set of spectral functions for a large range of potential mass gaps of the gluon, and discuss the limitations and extensions of the present work.
Current status:
Reports on this Submission
Strengths
1)Interesting approach used in a new context.
2) Clear figures and accompanying text.
3) Many interesting leads for further research into the spectral properties of YM theories .
Weaknesses
1) The structure of the paper is does not bring about it in a clear way what was actually done.
2) The CF model which was used should be more explicitly treated.
3) Not clear why this research was conducted with oneloop results already obstructing the extraction of a KL spectral density function.
Report
First of all, I plainly agree with the comments of the first referee. I will therefore try to keep my own comments complementary instead of redundant. Overall, my feeling while reading this paper is that while much information was given and the techniques were interesting, arguments were not always conclusive and information that is substantial to the paper was not presented as its main body. I would therefore only recommend publication after a thorough revision of the structure of this work:
1 From the abstract and first pages, one gets a completely different idea about the content of the paper than from the actual paper. As ref1 mentioned, the numerical analysis is employed for a CurciFerrari model (or, if you like, a mess term added by hand after gaugefixing YM). One would therefore expect an introduction into this very model and its features, including references to the vast amount of work that has been done around this model, see [2106.04526 [hepth]] and references therein. In this introduction one should also go deeper into the disadvantages of this model, such as a breaking of the BRST invariance, and how this might affect the outcome of their calculations.
2 Related to (1), it has to be clarified how the BPHZ renormalisation, which is mentioned in the introduction, is related to the mass term that is added by hand. How does the ‘occurence’ of a mass term relate to the inserted mass, and when the mass is inserted, what is the role of the BPHZ renormalization mentioned in app C? This should be more explicit.
3 The techniques presented in this work are interesting, but they have already been presented elsewhere by the same authors. In my opinion the authors should start by arguing why they expect their treatment of the CurciFerrari model to add any new knowledge about the structure of YangMills theories. This is nontrivial because the complex poles in the oneloop corrections of the gluon have already been treated in ref [40] and also in comparison with the unitary Higgs model in [1905.10422 [hepth]]. See also [1004.1607 [hepph]], where comments on the spectral density function of the CF model were first given.
4 The imbalance already mentioned by ref1 between the main body of the paper and the appendices is also because valuable information that should be in the main body has been put in appendices. Appendix C, for example, contains the main calculation and I would therefore expect it to be in the main part of the paper.
Requested changes
The above mentioned structure changes should be implemented to make clear what is the actual achievement of this paper.
Some further smaller changes:
1 As is scarcely mentioned in the paper, the analysis is done in Landau gauge, while other gauges are not mentioned, nor the fact that in other gauges one could get different results for the spectral function, see e.g. [2008.07813 [hepth]]. Changing the parameter \xi in eq. could give completely different results. It is not a problem to work in the Landau gauge, but one should not do general statements without always mentioning this gauge, such as for example at the end of section II.
2Interpunction is not always correct with sentences like: In the present, work.. and a grammar check would also improve the text. Bird eyes view = bird’s eye view, with help= with the help, respective= the respective, originates in= originates from, satisfied on the level= satisfied at the level.
3The symbol \xi is used twice in different contexts, as a gauge parameter and as the temporal screening length.
Strengths
1 The spectral functional approach is used for the first time in the context of YM theory.
2 Many details and discussions which will be useful for further work on the subject.
3 The analysis gives a further indirect proof of existence of complex conjugated poles in the gluon propagator
Weaknesses
1 The complex structure of the manuscript, with many important details which are discussed in appendices.
2 The real conclusions are not clearly discussed in the abstract and in the introduction: they seem to be concealed in the intricate structure of the manuscript.
3 There are misleading statements and extrapolations to YangMills theory, while an effective model is studied instead.
4 Because of the complex poles, the effective model has no self consistent solutions and the spectral approach does not work properly.
Report
Report on the manuscript
Scipost_202209_00032v1
“On the complex structure of YangMills theory”
by Jan Horak, Jan M. Pawlowski and Nicolas Wink
The manuscript presents a large body of work and discusses a first attempt to use the "spectral functional approach", as developed by the same authors in Ref.[1], in the special context of YM theory.
The paper is sound and contains many interesting details.
However, the main results and conclusions, which can only be drawn by a deeper reading, seem to be hidden in the very complex structure of the paper, with so many important aspects developed in the appendices.
Actually, by a careful reading, the main result of the manuscript seems to be a "negative" one, but nonetheless an important result which would be worth publishing if the main conclusions were well underlined in the abstract and in the introduction (which are quite vague in the present version).
As the author might confirm, the main conclusion is that a direct spectral approach, based on KällenLehman representation, is not suited for the study of a oneloop selfconsistent CurciFerrari effective model for YM theory, since the oneloop model predicts complex conjugate poles, in contrast with the standard KL representation.
That is important as it is a confirmation that the standard KällenLehman spectral function is not enough and a complex structure should be added to the gluon spectral representation (as discussed by several authors, see for instance Sec.II B of Ref.[40] and references therein).
The failure of the approach, which is not able to deal with complex poles in its present formulation, is a clear evidence of the existence of the poles (the results are not consistent if a polepart is not added by hand).
On the other hand, the failure could be a consequence of the effective model and an improved approximation might work for YM theory.
Thus, the authors should specify more clearly,
in the abstract, introduction and conclusions, that:
1) the theory that 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)
2) the model is not compatible with their spectral approach which is based on KL spectral representation, confirming the existence of complex poles (as expected by a oneloop approximation, see Ref.[40]).
3) the approximations they are using (selfconsistent and oneloop) are not consistent with the CF model. Then the lack of self consistency is a consequence of the model.
Even if the result might appear negative, the work is important for further investigations. Then I suggest that the paper should be published, provided that the above main conclusions are well underlined and that the authors cut any misleading statement about the existence of complex poles in YM theory.
In fact, in the present version, the reader is led to believe that the existence of complex conjugate poles is unlikely in YM theory.
For instance, in Sec.IV, there are misleading statements, like "In YangMills theory with bare vertices, a pair of complex poles in the gluon propagator... cannot be part of a analytically consistent solution of YangMills theory...." (Sec.IV) or "a selfconsistent solution with complex
conjugate poles and no further branch cuts does not exist" (Sec.VI).
In the first place, there is nothing like "YangMills theory with bare vertices". YM theory has not bare vertices. The oneloop theory with bare vertices and a mass parameter is not Yang Mills theory.
But the reader is led to believe that the statements might be also valid for YangMills theory, without any proof.
About the above conclusions, 13, some comments are in order:
1) Since a mass parameter and a mass counterterm are added by hand to the Lagrangian, the model which is studied is CurciFerrari model. That should be stated clearly. The Lagrangian is not BRST invariant and I would expect that the longitudinal parts of the gluon propagator and self energy are not zero. Thus the solution would not be self consistent. The authors should comment about it.
In YM theory, a mass would be generated by the Schwinger mechanism which relies on the special pole structure of the vertex. With bare vertices, there is no way to recover a dynamical mass, unless it is added by hand. However, the pole structure must be related to the mass generation mechanism and a trivial added mass does not know anything about the vertex structure. Thus, any reference to YM theory should be avoided since the effective model which is studied basically is a oneloop truncation of the CurciFerrari model.
It should be avoided the repetitive mention of "YM theory with bare vertices" and replace it by oneloop Curci Ferrari model.
Moreover, if there were complex poles in the exact gluon propagator of YM theory, then the complex poles, with their mass scale, would arise by the same mechanism which generate the mass. Thus, the same mechanism would cancel all spurious divergences (without adding mass counterterms) and, as discussed in the manuscript, a very complex cancellation would arise from the vertex structure in order to generate self consistent solutions of the full set of DSEs.
In that case, one would expect that the delicate cancellation would be spoiled if the vertices were replaced by the bare ones.
Thus, the proliferating of complex analytic structures is not a valid argument against the existence of complex poles in the exact YM theory.
2) The method is based on the assumption, in eqs.(32),(33), that a spectral function exists and KL representation holds. However, it was shown in Ref.[40] that a oneloop calculation gives complex poles and the KL representation does not work. Actually, we read in Sec.VA that "in all cases the spectral difference is fit quite well by a single pair of complex conjugate poles, suggesting that the violation is mainly due to a single pair of these poles".
That is a strong evidence for the existence of complex poles, confirming previous studies.
The authors should give more relevance to that evidence.
However, the existence of complex poles invalidates the spectral method which was based on KL spectral representation. Thus, we conclude that the spectral method, in its present version, cannot be used for the gluon.
The attempt to introduce a fit by a real pole cannot be conclusive. Moreover I expect that the fit does not work when the polepart is the larger contribution, as it is the case in oneloop calculations[40] (see other comments below).
3) The oneloop and selfconsistent approximations are not compatible for the SDEs of the Curci Ferrari model. Thus the shortcomings might come from the attempt to use the two approximations together. We know that oneloop CF model gives complex poles for the gluon propagator [40]. Then, no self consistent solution can exist because of the discussion in Sec.IVB on the propagation of nonanalyticities. Actually, a self consistent solution of the full set of SDEs might exist, but we do not know, because the equations are infinite. The oneloop truncation, with bare vertices, is an approximation. It can be improved going to two loop or higher orders. The attempt for a selfconsistent solution is equivalent to the sum of an infinite number of graphs, to all orders. But it is an arbitrary class of graphs, since the vertices are not replaced by selfconsistent vertices. As said before, there is no self consistent solution of the oneloop equations with bare vertices. Then, I would trust more a two or three loop expansion with bare propagators inside the loops. In that case, no propagation of nonanalyticities would occur.
Other minor points:
a) The structure of the paper is very complex, with many references to appendices, which does not help. The authors should avoid unnecessary references to very short appendices (like F and G).
For instance, the screening length could be defined on page 10 when it is shown in figure 6.
b) In the text below fig.8, the mass gap is negative (3.69,0.31) and units are not specified. Are they internal units? And by the way, how are the internal units defined? The mass scale is negative in figure 6, but a positive value of 3 GeV^2 is found in the caption. Which one is correct? The scale in fig.8 is positive but a misprint in the caption does not help (<  2.7). Some better description of units would be helpful.
c) There are two contributions to the gluon propagator, a part arising from the ordinary spectral function, the other from the added real pole in eq.(37). What is the ratio between the two parts?
And how does it change going towards the boundary of the mass range? In other oneloop calculations, the ordinary spectral contribution is very small compared to the pole part which is large and is a good approximation for the whole propagator. I would expect that the self consistent solution disappears when the realpole part becomes large (as it cannot give a good approximation for the whole propagator).
d) On page 12, the statement: “In short, none of our solutions is in the confining region, for more details see... In consequence, a statement about the complex structure of YangMills in the confining phase within the chosen approximation cannot be made.”
It is an important statement, but can only be found on pag.12. It should be emphasized from the beginning.
e) A spectral dimensional renormalisation is mentioned in the manuscript. It would respect all internal symmetries of the theory (with no quadratic divergence). Why are the authors using a momentum cutoff scheme, which requires a mass counterterm? Has that to do with the choice of CurciFerrari model where the original BRST is broken anyway?
Requested changes
The authors should specify more clearly,
in the abstract, introduction and conclusions, that:
1) the theory that 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)
2) the model is not compatible with their spectral approach which is based on KL spectral representation, confirming the existence of complex poles (as expected by a oneloop approximation, see Ref.[40]).
3) the approximations they are using (selfconsistent and oneloop) are not consistent with the CF model. Then the lack of self consistency is a consequence of the model.
Moreover
4) It should be avoided the repetitive mention of "YM theory with bare vertices" and replace it by oneloop Curci Ferrari model.
5) the authors should cut any misleading statement about the existence
(or non existence) of complex poles in YM theory.
Reply referee report 1
We thank the reviewers for their suggestions and believe that they have substantially improved the paper. Below, we reply to their comments as numbered in the report.
1) The idea of our approach is to approach the strongly correlated, confining region of YM theory by gradually lowering the mass parameter. This parameter cannot be left unattended in functional approaches, see arXiv:2107.05352 and arXiv:1605.01856 for thorough discussions in the context of DSEs and the fRG, respectively. The approach has been applied successfully to calculate scaling, decoupling and CF type solutions in arXiv:1605.01856, see Fig. 9. This was not stated clearly in the previous version of the manuscript. We therefore added a paragraph in the beginning of Sec. V which thoroughly explains our approach, including a careful distinction to the CF model.
Additionally, since the remark relating our numerical procedure to massive extensions of YM at the end of Sec. IV appeared to have been misinterpreted by the referee, we deleted it and deferred the detailed explanation of our procedure to the beginning of Sec. V. Additionally, we added further references to the CF model (1004.1607 and 1105.2475).
In fact, 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. This motivates to study the system in the timelike domain using the spectral functional approach.
2) Indeed, our numerical results are not consistent with a KL representation for the gluon propagator. This does not spoil compatibility with the spectral functional approach, however, as we state in Sec. IV, above eq. (34):
‘The spectral formulation employed in Section III enables us to study the general complex structure of ghost and gluon DSE, as it covers a large class of functions for the propagators and is by no means restricted to propagators satisfying the KL representation. In particular, a gluon propagator with a pair of complex conjugate poles is realised by collapsing the (gluonic) spectral integrals at complex spectral values corresponding to complex conjugate pole positions, multiplied by the respective residues. ’
In order to make the versatility of the spectral functional approach more apparent, a respective paragraph was added at the beginning of Sec. IV. Furthermore, we modified parts of the abstract and introduction, explicitly stating that our numerical results in Sec. V allow for violations of the spectral representation.
We emphasize that our results do not allow for the interpretation as confirmation of complex poles in YM however, see also point e) in the referee report. We discuss possible avenues to investigate the consistency of solutions with complex singularities in the massive extensions of YM theory, such as the CF model, in point 3).
3) In our system of coupled DSEs, removing the inconsistencies with the CF model the referee mentions amounts to including nontrivial vertices. As we discuss in Sec. IV B, we consider it to be unlikely for nontrivial vertices to render solutions with complex poles in the gluon propagator consistent, since this
‘either requires (A) a respective ghostgluon vertex that counteracts the loss of the spectral representation of the ghost, or (B) combinations of diagrams and vertices in the gluon gap equation prohibit the backpropagation of the additional branch cuts of the ghost.‘
The reasons for qualifying above scenarios as unlikely is given further in the text (page 8 & 9), and can be summarized as follows:
(A) The simplest possibility is a cancelation of the effects of the complex poles of the gluon propagator in the ghost selfenergy in order to keep the spectral representation of the ghost intact. This involves the scattering kernel of the ghostgluon vertex. Such a cancellation has not been observed in the literature yet though. A cancellation of complex singularities via the WTI part of the electron photonvertex, as for example known from the electron gap equation in QED, does not carry over here: The STI part of the ghostgluon vertex involves the ghost dressing function and not that of the gluon, which would be needed for countering the complex singularities.
(B) Another way out is to prevent the backpropagation of the additional branch cuts in the ghost propagator into the gluon propagator eventually causing the observed inconsistency. This possibility comes at the cost of accepting for a violation of the KL representation of the ghost propagator, which, in turn, is a wellaccepted fact in the literature. Going even further, to our knowledge, there is no work in the literature which found such a violation yet. This led to our decision to not further consider this possibility.
Still, a statement about whether the observed inconsistency is due to the inconsistency of our setup could be made by a corresponding perturbative study with ‘two or three loop expansion with bare propagators’, as mentioned by the referee. In fact, we propose such a study towards the end of Sec. IV B,
‘Accordingly, the backpropagation of the ghost propagator’s nonanalyticities into the gluon DSE at least requires a perturbative threeloop computation. While certainly being challenging, this may be within the technical range of perturbative computations in the CF model, and is very desirable.’
Performing such a study could shed light on sources of inconsistencies of potential solutions to YM containing complex poles in the gluon propagator, but have not been performed yet to our knowledge.
We reply to the further minor points addressed by the referee below.
a) Owing to the complexity of the subject under discussion, we try to increase readability of our manuscript by outsourcing technical details to the appendices. Appendix F is indeed very short, and we integrated it into the main text for the resubmission
b) Correct descriptions of the units have been added now.
c) The ratio of pole to spectral contribution is plotted in Fig. 8, encoded in the spectral violation defined by eq. (38). It measures the deviation of the full Euclidean propagator from the purely spectral one, hence describing the weight carried by the pure pole contribution.
d) We added emphasis on the mentioned fact in the abstract and conclusion.
e) Implementing spectral dimensional renormalisation requires analytic solution of the spectral integrals using the known UV asymptotics of the propagators in the loops. The spectral integrands (see, e.g., eq. (A9) including the definitions below) are too complex to find such analytic solutions, however.
Author: Jan Horak on 20230712 [id 3802]
(in reply to Report 2 on 20221219)Reply referee report 2
We thank the reviewers for their suggestions and believe that they have substantially improved the paper. Below, we reply to their comments as numbered in the report.
1) The idea of our approach is to approach the strongly correlated, confining region of YM theory by gradually lowering the mass parameter. This parameter cannot be left unattended in functional approaches, see arXiv:2107.05352 and arXiv:1605.01856 for thorough discussions in the context of DSEs and the fRG, respectively. The approach has been applied successfully to calculate scaling, decoupling and CF type solutions in arXiv:1605.01856, see Fig. 9. This was not stated clearly in the previous version of the manuscript. We therefore added a paragraph in the beginning of Sec. V which thoroughly explains our approach, including a careful distinction to the CF model.
Additionally, since the remark relating our numerical procedure to massive extensions of YM at the end of Sec. IV appeared to have been misinterpreted by the referee, we deleted it and deferred the detailed explanation of our procedure to the beginning of Sec. V.
The mentioned reference (2106.04526) is already cited in the manuscript, we moved the citation to a better fitting spot, however. Additionally, we added further references to the CF model (1004.1607 and 1105.2475).
2) BPHZ renormalisation entails the presence of a mass counterterm to cure quadratic divergencies in the gluon DSE, as for example in the tadpole diagram. This introduces the mass parameter into the system. We explain our renormalisation procedure in detail in Sec. III C, explicitly stating that
‘In turn, in spectral BPHZ renormalisation quadratic divergences are present, which is a wellknown property of the BPHZ scheme in gauge theories and originates in it being a momentum cutoff scheme. For a detailed discussion see [14,76,77,88] where also the direct link to Wilsonian cutoffs in the fRG approach and the ensuing modified SlavnovTaylor identities (STIs) is discussed. In short, momentum cutoff schemes such as BPHZtype schemes necessitate a gluon mass counterterm, which is adjusted such that the STIs are satisfied. Accordingly, the occurrence of mass counterterms in YangMills theory in a BPHZtype scheme is a property of the scheme and restores gauge consistency and does not (necessarily) signal its breaking
In the present spectral BPHZ setup, the spectral divergences are cured by introducing counterterm, including a gluon mass counterterm, through the renormalisation constants in (21) and taking epsilon to 0 before computing the spectral integrals. Then, gauge invariance is restored by adjusting the finite part of this counterterm such that the STIs are satisfied on the level of the renormalised correlation function. For discussions about the treatment of quadratic divergences in functional approaches to YangMills theory, see e.g. [46, 47, 55, 79].’
The parameter choices for the gluon mass parameter used in numerical computations are explicitly stated in Sec. V. We also hope that the added paragraph mentioned above in 1) helps clarifying this matter.
3) Our manuscript contains two distinct results: In Sec. IV, we present an analytic study of YangMills theory using the spectral DSEs, assuming a gluon propagator with complex poles. This study shows that in the present truncation (bare vertices), a gluon propagator with complex poles cannot be part of any consistent solution to the system under consideration. We further argue in Sec. IV, why we consider it unlikely that this is any different in full YangMills theory (see also 3) in our reply to referee I), and state explicitly in the conclusions that
‘A central aspect of our analytic study of the complex structure of YangMills theory in Section IV is, that the existence of complex conjugate poles in the gluon propagator leads to a violation of the spectral representation for the ghost, at least for the case of bare vertices. For this not to carry over to full YM theory, an intricate cancellation of the complex poles in the gluon propagator by the full ghostgluon vertex is required. In our opinion, this is unlikely to occur in YangMills theory or QCD’.
The revealed mechanism of propagation of nonanalyticities in coupled functional equations have, to our knowledge, not yet been described for the case of YangMills theory. In our opinion, they add to the debate about the complex structure of the gluon propagator by constituting a strong argument against the presence of complex poles. This is mentioned prominently in our conclusion, cf. Sec. VI.
The second main result is presented in Sec. V, which are the numerical solutions to the considered system of DSEs. We allow for small violations of the spectral representations by mapping them to simple poles on the real axis, thereby evading the scenario described in Sec. IV. These results represent an attempt to solve the spectral DSEs in the confining region of YangMills theory by the procedure describe above in 1). As stated at the beginning of Sec. V, we were not able to access that reason with the described techniques, however, as explicitly stated in the conclusion:
‘Solving the system for more QCDlike regions is hindered by increasing violation of the spectral representation, which is accounted for approximatively.’
Still, we chose to document the results, as a starting point for future analysis. In particular, the described technique of mapping the complex to real poles in the gluon propagator can be helpful for moving towards the confining region, since complex poles represent a generic feature in initial conditions for the considered system.
4) Owing to the complexity of the subject under discussion, we try to increase readability of our manuscript by outsourcing technical details to the appendices. For this reason, we chose to consistently move the details of the calculations for Sec. III, IV and V to appendices. Appendix F however is indeed very short, and we integrated it into the main text for the resubmission.
We reply to the further minor points addressed by the referee below.
a) We explicitly added reference to Landau gauge for the general remarks at the end of Sec. II. In addition, we added a sentence about the gaugedependence of possible results and reference to 1905.10422 and 2008.07813 in Sec. IV A.
b) We performed another spell and grammar check and eliminated the mentioned mistakes.
c) Since for both, \xi is the common symbol, we added a remark to avoid confusion at the top of page 12, where the screening mass/length is mentioned first.