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Gluon condensates and effective gluon mass
by Jan Horak, Friederike Ihssen, Joannis Papavassiliou, Jan M. Pawlowski, Axel Weber and Christof Wetterich
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Submission summary
Authors (as registered SciPost users):  Jan Horak · Jan M. Pawlowski 
Submission information  

Preprint Link:  scipost_202202_00009v1 (pdf) 
Date submitted:  20220206 20:11 
Submitted by:  Horak, Jan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Phenomenological 
Abstract
Lattice simulations along with studies in continuum QCD indicate that nonperturbative quantum fluctuations lead to an infrared regularisation of the gluon propagator in covariant gauges in the form of an effective masslike behaviour. In the present work we propose an analytic understanding of this phenomenon in terms of gluon condensation through a dynamical version of the Higgs mechanism, leading to the emergence of color condensates. Within the functional renormalisation group approach we compute the effective potential of covariantly constant field strengths, whose nontrivial minimum is related to the color condensates. In the physical case of an SU(3) gauge group this is an octet condensate. The value of the gluon mass obtained through this procedure compares very well to lattice results and the mass gap arising from alternative dynamical scenarios.
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Reports on this Submission
Anonymous Report 2 on 2022328 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202202_00009v1, delivered 20220328, doi: 10.21468/SciPost.Report.4794
Report
Here we present a referee report for the paper ``Gluon condensates and effective gluon mass'' by Jan Horak, Friederike Ihssen, Joannis Papavassiliou, Jan M. Pawlowski, Axel Weber, and Christof Wetterich.
In this present manuscript, the authors compute the effective potential for a constant field strength $F_{\mu\nu}$, whose nontrivial minimum is associated with the formation of a color condensate. In the sequence, they establish a connection between the formation of the color gluon condensate to the emergence of an effective gluon mass in the YangMills Green's functions, performing an average over color directions. As a result, the authors found that the condensate value $\langle F \rangle^2$ is in good agreement with previous phenomenological estimates. In addition, the determination of the scale of the gluon mass agrees rather well with previous the lattice and SchwingerDyson results.
As stressed throughout the manuscript, the present analysis may be faced as a starting point for a systematic exploration of the connection between gluon condensates and the gluon mass gap. The two crucial simplified working hypotheses were: {\it(i)} compute the gaugeinvariant effective potential for constant field strength $F_{\mu\nu}$ and {\it(ii)} the coloraveraging procedure, which leads to the appearance of averaging factor $f_{av}(N_c)$ in the gluon mass expression.
In my opinion, the general idea and the manuscript results are quite interesting. For these reasons, the article merits publication in SciPost. However, before that, the authors should clarify the following point
One of the authors' main results is the relation between the gluon mass and the condensate $\langle F \rangle^2 $ given by Eq.~(34). Back in 88, using operator product expansion, Lavelle & Schaden in Ref. [41] found a similar color factor connecting the gluon selfenergy and the condensate $\langle G \rangle^2 $. Since the color averaging procedure is admittedly the largest source of systematic error in the present work, it would be nice if the authors included a paragraph saying that a similar color structure was found previously in [41] and discussed if there is any reason for such different methods lead to precisely the same color factor.
Anonymous Report 1 on 2022328 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202202_00009v1, delivered 20220328, doi: 10.21468/SciPost.Report.4789
Report
The authors investigate a scenario for the origin of the infrared behavior of the gluon propagator in covariant gauges in the form of an effective masslike behavior as observed in many nonperturbative studies both on the lattice as well as in the continuum. They propose a Higgslike mechanism generated dynamically by the theory where color condensates play the role of a Higgs field. Their study provides evidence for this scenario on the basis of a nonperturbative estimate of the effective action using the functional RG. Remarkably, the quantitative result for their estimate of the gluonmass parameter obtained from their continuum method compares favorably to lattice results and other approaches.
This paper successfully combines two key ideas: the phenomenological description of lowenergy properties of QCD using color condensates on the one hand, and the possibility to compute phenomenologically relevant quantities based on nonperturbative propagators and the functional RG in background formalism on the other. In this way, the so far existing gap of a conceptual understanding for the gluon mass behavior observed in Landaugauge propagators has been closed; or at least, a promising way for closing this gap has been identified.
The paper is very well written, technical details are comprehensively explained in the Appendices and the manuscript has been thoroughly prepared. I can fully recommend the paper for publication in SciPost.
I have one question which the authors may wish to address in the final version:
In the par. below (70), the authors claim that the result of eq.(70) should be considered as an upper bound, since it is "composed by the condensates of both $F^2$ and $F\tilde{F}$".
This statement seems confusing: I understand that the selfdual background field does not allow a clean distinction between operators built from $F^2$ and those built from $F\tilde{F}$. Therefore, an effective action (density) of the form W($F^2$,$F\tilde{F}$) is (mis)identified as $W$($F^2$,$F\tilde{F}$=$F^2$) $= W$($F^2$). This makes clear that the desired effective potential for $F^2$ may be contaminated by terms arising from $F\tilde{F}$ operators. However, I do not see that this necessarily leads to an overestimation of the condensate. Corrections could equally go into the opposite direction.
Also, I am not certain whether the authors indeed wanted to make the statement that they expect the occurrence of an $F\tilde{F}$ condensate. Wouldn't such a condensate be an unwanted source of CP violation? (Of course, a local topological charge density is expected to be nonzero, but I would assume that it averages to zero globally in order to preserve CP.)
Requested changes
I also found a couple of typos which the authors may wish to correct:
 Eq.(19): "$...+ a_\mu$" > "$...+ a_\mu^a$"
 line below Eq.(19): "... as the fields strength." > "... as the field strength."
 Par. below Eq.(27): "At this state ..." > "At this stage ..."
 1st par., right col. on p.5: "... color blind strength..." > "... color blind value ..." ?
 below (39c): "... function of the Laplacian $D$ ..." > "... function of the covariant derivative D ..."
 1st sentence, right col on p.12: "Now we us ..." > "Now we use ..."
 Ref. [107] is incomplete.
The authors thank the referee for his report and careful reading of our manuscript.
The referee adresses that the possible contribution of $F \tilde F$terms to our condensate value eq. (70) might as well be negative. Furthermore, such a contribution represents a source of CP violation, as the referee points out.
We softened the statement below eq. (70) about the direction $F \tilde F$contribution to our result and added a comment about the CPviolating nature of such contributions. The typos listed under requested changes have been fixed.
Author: Jan Horak on 20220406 [id 2362]
(in reply to Report 2 on 20220328)The authors thank the referee for his report.
The referee adresses that color averaging factor in eq. (34) of the manuscript is identical to that one found by Lavelle & Schaden in [41]. In the expression for the gaugeinvariant effective potential, eq. (18) of [41], the factor $(N_c^2  1)/N_c$ indeed appears, but is multiplying the quark condensate $\langle \bar \psi \psi \rangle$, which we do not consider. The color prefactor in front of the gluon condensate $\langle G^2 \rangle$ (where G is the field strength) is $N_c$ however, which is not identical to our result. In fact, [41] finds the same linear large $N_c$scaling of the condensate term in the effective potential as we do particular, which we found to be an important consistency check for our color averaging procedure, cf. Sec. IIIc.
The inverse of our color factor $(N_c^2  1)/N_c$ also appears in front of the $\langle G^2 \rangle$ condensate in the gluon selfenergy in eq. (11) of [41]. This we consider to be coincidental.