SciPost Submission Page
Symmetry restoration and the gluon mass in the Landau gauge
by Urko Reinosa, Julien Serreau, Rodrigo Carmo Terin, Matthieu Tissier
Submission summary
As Contributors:  Julien Serreau 
Preprint link:  scipost_202010_00005v1 
Date submitted:  20201007 22:18 
Submitted by:  Serreau, Julien 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the generation of a gluon screening mass in YangMills theory in the Landau gauge. We propose a gaugefixing procedure where the Gribov ambiguity is overcome by summing over all Gribov copies with some weight function. This can be formulated in terms of a local field theory involving constrained, nonlinear sigma model fields. We show that a phenomenon of radiative symmetry restoration occurs in this theory, similar to what happens in the standard nonlinear sigma model in two dimensions. This results in a nonzero gluon screening mass, as seen in lattice simulations.
Current status:
Editorincharge assigned
Author comments upon resubmission
We thank the referee for the careful reading of our manuscript and for his/her constructive remarks. We answer his/her points and questions below and detail the changes in the manuscript (which appear in red in the revised version).
Referee: It is not clear to me how eq.(4) is defined for configurations that live on the socalled Gribov horizon. Such configurations represent zero modes of the operator F and, therefore, the denominator seems to be illdefined. The authors could provide a comment on that.
We have added a discussion about this point (footnote 1).
R: The parameter ζ introduces a mass term for the FaddeevPopov ghosts and the authors claim that it should not be taken as a problem. However, in the standard formulation of the Landau gauge, mass terms for the ghosts are forbidden by the Ward identities. In the standard (Refined) GribovZwanziger construction this remains true. Can the authors comment about that and explain how to reconcile such things? Or am I missing something?
We are not completely sure what the referee means here. The point is that the Ward identities he/she refers to are those of the standard formulation of the Landau gauge, i.e., if we understand correctly, those of the FaddeevPopov gaugefixing. But symmetry identities are specific to each particular gaugefixing and may not apply to others. For instance, a gluon mass term would also be forbidden by the standard SlavnovTaylor identities. The construction we propose here is a different formulation of the Landau gauge, which comes with its own symmetry identities. In particular, the shift symmetry cbar > cbar + constant, which forbids a mass term for the ghosts in either the FP or the GW constructions, is simply not a symmetry of the gaugefixing proposed here.
R: The gluonmass parameter is associated with the parameter β which is akin to a gauge parameter. This is very confusing to me. Are the authors claiming that the mass parameter that is generated by averaging over Gribov copies is gauge dependent and therefore can be taken to any value? Does it mean that β cannot enter correlation functions of gaugeinvariant correlators?
As explained below Eq. (3), gaugeinvariant observables are clearly independent of beta (and zeta) by construction. In a systematic expansion scheme (like perturbation theory), we expect this property to be respected order by order.
As for the gluon mass discussed here, it is not a gauge invariant quantity. We see no reason why it should be gauge independent. This, however, does not imply that it can take any value. For instance, as the phase diagram in Fig 2 shows, the value of the generated mass (only in the “massive” phase) is bounded.
Thinking about this point, we have realized that there may be an ambiguity in what one understands by “mass generation”. We have added some comments to clarify these points.
R: Can the authors connect the mass parameter that they obtain to the socalled Gribov parameter which is generated in the standard elimination of infinitesimal Gribov copies in the Landau gauge by the restriction of the path integral to the Gribov region?
No, we do not know how to do such a connection in a precise sense. We only have a qualitative intuition that our procedure favors configurations similar to those which are selected by the GZ restriction, i.e. near the first Gribov horizon. Indeed, the first horizon is enhanced by the denominator in (4) whereas higher Gribov regions are supressed by the exponential weight. It is certainly very hard to make a precise connection between the present proposal and the GZ procedure beyond this qualitative level, simply because these are in fact different gaugefixing procedures. Relating the dimensionful parameters of these two to one another is certainly nontrivial. We have added a discussion on this point.
R: Below eq.(4), the authors say that F represents the FP operator in the Landau gauge and write F[A,U]=F[A^U] Can they explain what do they mean by this equality?
We simply mean that the function F[A,U] is, in fact, a function of A^U only and not of A and U separately. The revised version makes this clear.
R: It seems that this averaging method does not (strongly) rely on the choice of the Landau gauge and also on the type" of copies, i.e., if they are infinitesimal or large". This would be a strong advantage with respect to the restriction to the Gribov region, which has a strong dependence of those aspects. Can the authors make comments about the extension to other gauges and if this is compatible with BRST invariance?
It is true that our approach does not rely on the type of copies, as mentioned by the referee. However, the path integral formulation of our procedure requires that the Gribov copies one averages over correspond to extrema of the functional f[A,U] in the weight (4). In other words, the chosen gaugefixing must have a formulation as an extremization problem. Only a few gauges are known to have this property. The most wellknown one is the Landau gauge, which extremizes the functional (2). The only other example known to us is the class of CurciFerrariDelbourgoJarvis gauges, which have been studied in terms of a similar averaging procedure as the one proposed here in Serreau et al. Phys. Rev. D 89 (2014) 125019. Unfortunately, the case of linear gauges does not fulfil this requirement. We have added a discussion on this point.
As for the question of the BRST symmetry, the standard BRST symmetry is softly broken by the averaging procedure (non flat weight over the Gribov copies). We have not identified a modified BRST symmetry in the present case.
R: My general comment is that the authors focus on the averaging procedure to eliminate Gribov copies, but mostly do not make any reference to what has been done using the restriction to the Gribov region" method. It would be beneficial for their work to connect their results with the other perspective in some way.
We agree with the referee. We have added some comments in this direction.
Referee: It is not clear to me how eq.(4) is defined for configurations that live on the socalled Gribov horizon. Such configurations represent zero modes of the operator F and, therefore, the denominator seems to be illdefined. The authors could provide a comment on that.
We have added a discussion about this point (footnote 1).
R: The parameter ζ introduces a mass term for the FaddeevPopov ghosts and the authors claim that it should not be taken as a problem. However, in the standard formulation of the Landau gauge, mass terms for the ghosts are forbidden by the Ward identities. In the standard (Refined) GribovZwanziger construction this remains true. Can the authors comment about that and explain how to reconcile such things? Or am I missing something?
We are not completely sure what the referee means here. The point is that the Ward identities he/she refers to are those of the standard formulation of the Landau gauge, i.e., if we understand correctly, those of the FaddeevPopov gaugefixing. But symmetry identities are specific to each particular gaugefixing and may not apply to others. For instance, a gluon mass term would also be forbidden by the standard SlavnovTaylor identities. The construction we propose here is a different formulation of the Landau gauge, which comes with its own symmetry identities. In particular, the shift symmetry cbar > cbar + constant, which forbids a mass term for the ghosts in either the FP or the GW constructions, is simply not a symmetry of the gaugefixing proposed here.
R: The gluonmass parameter is associated with the parameter β which is akin to a gauge parameter. This is very confusing to me. Are the authors claiming that the mass parameter that is generated by averaging over Gribov copies is gauge dependent and therefore can be taken to any value? Does it mean that β cannot enter correlation functions of gaugeinvariant correlators?
As explained below Eq. (3), gaugeinvariant observables are clearly independent of beta (and zeta) by construction. In a systematic expansion scheme (like perturbation theory), we expect this property to be respected order by order.
As for the gluon mass discussed here, it is not a gauge invariant quantity. We see no reason why it should be gauge independent. This, however, does not imply that it can take any value. For instance, as the phase diagram in Fig 2 shows, the value of the generated mass (only in the “massive” phase) is bounded.
Thinking about this point, we have realized that there may be an ambiguity in what one understands by “mass generation”. We have added some comments to clarify these points.
R: Can the authors connect the mass parameter that they obtain to the socalled Gribov parameter which is generated in the standard elimination of infinitesimal Gribov copies in the Landau gauge by the restriction of the path integral to the Gribov region?
No, we do not know how to do such a connection in a precise sense. We only have a qualitative intuition that our procedure favors configurations similar to those which are selected by the GZ restriction, i.e. near the first Gribov horizon. Indeed, the first horizon is enhanced by the denominator in (4) whereas higher Gribov regions are supressed by the exponential weight. It is certainly very hard to make a precise connection between the present proposal and the GZ procedure beyond this qualitative level, simply because these are in fact different gaugefixing procedures. Relating the dimensionful parameters of these two to one another is certainly nontrivial. We have added a discussion on this point.
R: Below eq.(4), the authors say that F represents the FP operator in the Landau gauge and write F[A,U]=F[A^U] Can they explain what do they mean by this equality?
We simply mean that the function F[A,U] is, in fact, a function of A^U only and not of A and U separately. The revised version makes this clear.
R: It seems that this averaging method does not (strongly) rely on the choice of the Landau gauge and also on the type" of copies, i.e., if they are infinitesimal or large". This would be a strong advantage with respect to the restriction to the Gribov region, which has a strong dependence of those aspects. Can the authors make comments about the extension to other gauges and if this is compatible with BRST invariance?
It is true that our approach does not rely on the type of copies, as mentioned by the referee. However, the path integral formulation of our procedure requires that the Gribov copies one averages over correspond to extrema of the functional f[A,U] in the weight (4). In other words, the chosen gaugefixing must have a formulation as an extremization problem. Only a few gauges are known to have this property. The most wellknown one is the Landau gauge, which extremizes the functional (2). The only other example known to us is the class of CurciFerrariDelbourgoJarvis gauges, which have been studied in terms of a similar averaging procedure as the one proposed here in Serreau et al. Phys. Rev. D 89 (2014) 125019. Unfortunately, the case of linear gauges does not fulfil this requirement. We have added a discussion on this point.
As for the question of the BRST symmetry, the standard BRST symmetry is softly broken by the averaging procedure (non flat weight over the Gribov copies). We have not identified a modified BRST symmetry in the present case.
R: My general comment is that the authors focus on the averaging procedure to eliminate Gribov copies, but mostly do not make any reference to what has been done using the restriction to the Gribov region" method. It would be beneficial for their work to connect their results with the other perspective in some way.
We agree with the referee. We have added some comments in this direction.
Submission & Refereeing History
Submission 2004.12413v1 on 28 April 2020