A novel background field approach to the confinement-deconfinement transition

We thank the authors for their reports which have helped us improving the manuscript. We copy below our replies to both of them (the same replies can be found in the "comment" sections right below their reports). We hope that, in its present form, the paper is ready for publication.

The authors.

Reply to referee 1:

We thank the referee for his various comments. We reply point by point below and hope that the new version is now ready for publication. Our various updates of the draft (based on this and a second report) are in red for more readability.

1. Report: In this work the authors proposed a formulation to compute the effective action with a background field chosen to be consistent with the center symmetric vacuum. The formulation turned out to be useful for the description of the confinement-deconfinement transition in the Curci-Ferrari model.

I find it difficult to make positive recommendation for publication in the present form. It may contain new materials useful for readers, but the present form would cause more confusions than clarifications. When I read the introduction, I did not understand the motivation at all, and after reading through, I finally got the point. The authors said, “explicit breaking of the center symmetry by the gauge-fixing procedure”, in the introduction, and reiterated similar statements at various places, but this is very misleading. The gauge symmetry is a redundancy in the theory unlike ordinary symmetries, and the gauge-fixing procedure would not affect any gauge invariant quantities including the Polyakov loop. Therefore, for example, the Polyakov loop effective potential must have the same physical contents regardless of gauge choices such as the Polyakov gauge, the covariant gauge, the background field method, etc.

1. Response: We are surprised by the comments of the referee concerning the general motivation of our work. We state very explicitly, right in the second paragraph of the introduction, that the present considerations are relevant when approximations are involved (which is the case of all continuum approaches). Quoting ourselves: “Such an explicit breaking of the center symmetry by the gauge-fixing procedure should not alter the physical results in principle, but this is clearly an issue when approximations are involved.” 

We understand from the referee’s comment that this may not be enough so we have added explicit mentions of the role of approximations in later sections as well.

1. Report: The gauge invariant formulation of the effective action has been long known since the celebrated paper by Abbott in 1981. Generally speaking, the effective action is not necessarily gauge invariant except at the on-shell point, but it can be in the covariant background field gauge. Such transformation properties have been discussed in many literatures, e.g., in hep-th/9310195 by Korthals Altes.

2. Response: We have added references to the works of Abbott and Korthas Altes.

3. Report: Therefore, what are discussed in Secs.I, II, and III in the present paper are either misleading or already known. A “novel” approach is just to set the background to be the center symmetric minimum (that is, r = π in the SU(2) case).

4. Response: We do not quite understand the referee’s criticism here. Section I is the introduction. By definition, it presents introductory (known) material. Section II reviews generalities of (textbook) background field techniques and their application to the nonzero temperature problem and the issue of the center symmetry; In particular, we review the self-consistent background approach which is, by now, standard material, as we state explicitly in the text. But we feel that presenting this known material, emphasising its advantages and limitations for the nonzero temperature problem is useful to highlight the novelty of the present proposal, which we also describe in this section. 

It is true that the basic idea is simple,—“just” choosing a center-symmetric background—but this should not hide the fact that it is both new and, we believe, interesting and promising. Keeping explicit track of the center symmetry is of paramount importance in any approach involving approximations (which, again, is the case of all continuum approaches). To the best of our knowledge, up until our work, the only known way to do so was to use self-consistent backgrounds, which comes with some drawbacks that we review here, related to the fact that the tracker of the center symmetry is not the gauge field itself but the background, which is a gauge-fixing device. Our proposal is a different, novel way to work in a gauge-fixed setting with explicit center symmetry. It has the advantage that one directly works with the gauge field at fixed background with a genuine (Legendre transform) effective potential, from which one can directly extract vertex functions for instance (see also below our comments about the diverging susceptibility at a second order transition).

1. Report: In the first round of reading I had no idea why the authors needed to consider this, and Sec.III gave me the explanation. It is not the gauge fixing but the perturbative treatment that may look violating the center symmetry.

Indeed, the naive perturbation theory is an expansion around the empty vacuum that is not center symmetric, and this might cause some potential problems in a model that leads to perturbative confinement as in the CF model. In principle, there should not be a problem since the center symmetric vacuum must be self-consistently chosen in the confined phase, but there could be technical advantages if the calculation is initially shifted to the center symmetric point, that is understandable to some extent.

I think that the introduction and the motivation should be improved so that general readers who are not familiar with the CF model can also understand why it is needed (and of course misleading statements should be corrected). Then, the corrected manuscript could be considered for publication.

1. Response: As mentioned before, we have added some sentences to clarify the motivations of the present work which are quite general and of interest to virtually all gauge-fixed approaches to the Polyakov loop potential. As for the motivation of the CF model, we have added a sentence to address the referee’s concern.

2. Report: Now I come to questions on physics. The comparisons between the dashed curves and the solid curves in Figs.3 and 4 are the most crucial results. I thought that in the “novel” approach by the authors the confined phase at low temperature should be better described, but the results seem to show opposite. In the confined phase there is no difference, but in the deconfined phase at high temperature quantitative differences appear.

It is easy to understand why so. In the confined phase center symmetry is unbroken, and the self-consistent background approach correctly chooses the center symmetric vacuum, and no difference is seen, while in the deconfined phase the Polyakov loop value is not protected by the symmetry and quantitative differences are manifested.

1. Response: The two approaches (self-consistent background and center-symmetric background) trivially agree in the low-temperature phase for the very reason given by the referee and explicitly given in the text. We disagree though that our results “show the opposite”, namely, we are more confident in the new approach, as explained in detail in the article. The main argument is that, in an exact treatment, the electric mass must vanish at a second order transition like in the SU(2) case. This basic property may be violated when approximations are involved and this is, in fact what happens in the self-consistent background approach. That is because, in that case, the electric mass is not directly the curvature of the background field potential and its vanishing at the transition point relies on nontrivial relations between the latter and the usual Legendre transform effective potential. These relations are generally not respected when approximations are involved. 

In contrast, in the center-symmetric background approach proposed here, the relevant potential is the Legendre transform effective potential and the electric mass is directly given by its curvature and thus automatically vanishes at a continuous transition, irrespectively of the level of approximation. 

1. Report: Then, the question is which should be better, the dashed curves or the solid curves at high temperature? I would say, the dashed curve from the self-consistent background, since the "novel" approach is constructed to fit in with the confined phase, not the deconfined matter where center symmetric is lost.

Then, it is puzzling to me that the solid curves look more reasonable around Tc than the dashed curves, though, I think, the dashed curves should be better at high temperature.

In the SU(2) case the phase transition is of second order, and the electric mass should be vanishing at Tc. This is indeed so in the solid curves in Fig.3, but the dashed curves do not drop at Tc, only making a small dip. I see some explanations on page 9, but I do not think that the explanation works for SU(2). The new minimum in the self-consistent background should move continuously for a continuous transition... probably the determination of Tc is not self-consistent?

1. Response: We stress that the dashed and solid curves in Fig. 3 correspond to a gauge-variant quantity computed in different gauges (corresponding to either self-consistent or center-symmetric backgrounds). The two gauges agree in the low temperature phase but differ above the transition and there is no reason why the two curves should agree at high temperatures. Also, it is somewhat misleading to decide which gauge is better at the level of gauge-dependent quantities.

As mentioned before though, one argument in favour of the new approach is the fact that the electric mass vanishes at the continuous transition in the SU(2) case. Also, gauge-invariant quantities such as the transition temperature or the temperature dependence of the Polyakov loop are better reproduced (at one-loop order) in the novel approach. 

Finally, the referee writes that the center-symmetric background field gauge is constructed to fit in with the confined phase and seems to have in mind that it is thus not fitted to properly describe the deconfined phase. We stress that this is not so. In the self-consistent approach, the background and the average field are always identical by construction and the background thus directly characterises the (symmetric or broken) phase of the system. In the novel approach, the average and the background fields are identical in the symmetric (confined) phase but they differ in the broken (deconfined) phase. This poses no problem because what characterises the phase of the system is the average field (which can depart from the center-symmetric value) and not the background (which is nothing but a gauge-fixing device).

1. Report: Finally, I have a minor comment on Sec.IIA. The authors introduced G and G_0, but the gauge group and the boundary conditions are not related.

2. Response: It is true that any gauge transformation (irrespectively of the boundary conditions) leaves the classical action invariant. However, only those transformations that preserve the periodicity of the gauge field leave the quantum action invariant and can thus be considered as symmetries at the quantum level. This is the relevance of the group {\cal G} introduced in the text. The relevance of the subgroup {\cal G}_0 is explained in the text.

3. Report: Some representations of G may be not faithful, like the adjoint representation not faithful up to the center, but G itself is not changed by the boundary condition. The explanations given by the authors were customary until a decade ago, but many people were perplexed by conventional but rather artificial arguments. Now, the concepts have been sorted out by the language of the higher form symmetries. The center symmetry is a 1-form symmetry (or a spatial 0-form symmetry). I would not request the authors to amend descriptions in Sec.IIA; they are anyway customary explanations, though not satisfactory.

4. Response: We feel no need to introduce extra mathematical concepts. The discussion in terms of {\cal G} and {\cal G}_0 is quite simple (based on basic group theory and boundary conditions) and certainly enough for our purposes.

Reply to referee 3:

We thank the referee for the positive comments. We answer his points below and hope that the paper is now ready for publication.

1) We have removed the end of the sentence.

2a) and 2b) We do compute the Polyakov loop (3). It is true that, because the potential is here evaluated at next-to-leading order, the Polyakov loop should also be computed at next-to-leading order, adapting for instance the calculation in 1412.5672 and 1511.07690. Because r=pi+O(g^2) in the present gauge, it turns out, however, that the next-to-leading order corrections to the Polyakov loop can be entirely encapsulated in the tree-level expression for the Polyakov loop with the one-point function evaluated at next-to-leading order (which is by the way another point in favour of the new approach). There is thus no contradiction with the fact that the Polyakov loop computed here is able to actually describe lattice data.

We have clarified this by adding Eq. (25) and (26) and the text around them.

2c) We did not compute the critical exponent. Clearly, the latter is trivial at one-loop order and a meaningful calculation would require renormalisation group effects. This is beyond the scope of the present work.

2d)  It is generically the case that the perturbative results in the CF approach are better for SU(3) than for SU(2), see 2106.04256. Our feeling is that this is because the actual coupling (the Taylor coupling extracted eg from lattice calculation) is smaller for SU(3) than for SU(2).

3a) OK

3b) See above.

In the reply section below their reports, the referees can find a copy of the new arXiv version with the changes highlighted in red. For completeness, we list them here as well:

- p1: references have been added, following the recommendation of referee 1;
- p2 and p3: two sentences have been extended to insist on the fact that our discussion concerns approximations;
- p2: a few sentences have been added to motivate the Curci-Ferrari model further;
- p3: a footnote has been added to explain the relevance of the transformations that preserve the boundary conditions;
- p6 and p7: the discussion on the equivalence between the background gauge with center-symmetric background and the Landau gauge with twisted boundary conditions has been improved;
- p9: a paragraph has been added to better explain the relevance of Fig.3 and the benefits of using our new implementation of the background field method at finite temperature;
- p10: a paragraph has been added to explain why we are actually computing the expectation value of the Polyakov loop to the relevant order, in reply to referee 3.
- p11: a reference to Fig.5 has been added.