The QCD Vacuum as a Disordered Chromomagnetic Condensate

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The Authors extended a variational approach to the vacuum ground-state wavefunctional in the presence of an Abelian chromomagnetic field from SU(2) to SU(3). They identify three different one-loop instabilities which they try to overcome by a variational ansatz. They find that the variational ansatz is energetically not preferred over the perturbative ground-state. Nevertheless the variational approach motivated a new picture of the QCD vacuum, consisting of independent chromomagnetic domains. The authors identified a topological phase transition separating the perturbative ground state from the disordered chromomagnetic condensate. The author could successfully show that this new picture is indeed in agreement with some phenomenological properties of the QCD vacuum as the gluon condensate. Their picture also explains the color Meissner effect. The profile of the flux tube as measured in lattice QCD calculations was well described by their variational approach. 

The manuscript is well written and pedagogically structured in several subsections. Sufficient references to the literature are given to follow the calculations and line of thoughts. It is quite remarkable how well the quantitative description of the flux tube profile deduced from the variational ansatz works. I fully support publication, however,  before publication I would like the authors to comment on a few points:

 

1.  In Eq. (3.13) it is assumed that b_u(p₁,p₂) factorizes. To what extent is this assumption justified?

 

2. Is there an operator that could be calculated in lattice QCD and that would serve as an order-disorder operator for the independent chromomagnetic domains?

 

3. It is not clear to me what the differences are between the fits (6.69) and (6.70). Both fits seem to refer to the same ansatz (6.68) and the same data. 

 

4. Is there a typo in equation (6.72)? GeV^2 → GeV

 

5. In line 1158 (after eq. 7.1) it should likely read “Bjorken-Drell convention’’. 

•  

 

Comments on the Quality of English Language

There are some minor problems with typos and the english, e.g.

• Line 364 “the” →  “there” 

• Line 624: “have showed” → “have shown” 

• Line 916: “was showed” → “was shown”

• Caption Fig. 8 “e” → “and”

• Line 1161: “neglects” → “neglecting” 

• Line 1272: “is showed” → “was shown

Author Response

Reply to the Referee 1

      

 

 I thank the Referee for the useful comments aimed at improving my paper.

 

 

  1)  The variational parameters are found by solving

      the integral-differential equations obtained by minimization

      of the unstable-mode vacuum energy functional. It turns out

      that the integral-differential equations depend on the second

      derivative with respect to x_3 and on the combination p_3^2-gH

      (or gH/2) since the unstable modes lie on the lowest Landau levels.

      As a consequence the variational parameter \rho(p_2,p_3) factorizes

      into a function of p_2 times a function of p_3 (called \rho(p_3)

      in the paper) and likewise the induced background fields

      are given by the product of the functions f(x_3) with u(x_1,x_2).

 

  2) The Referee  is addressing a very good question. However, it is known that for a

    order-disorder  topological phase transition without symmetry breaking  there is no

    local order parameter. As a consequence, the eventual order parameter for these kinds

    of  phase transitions must be  non-local and gauge-invariant. Unfortunately, presently

    I do not have an answer to the Referee's question, but I am well aware that this is

    a very important point that needs careful considerations.

 

 

  3)    Eq. (6.69) reports the fits performed by the author of Ref.[109] to

        the full data set. Since in Fig. 7 I reported only a representative

        set of data, to check the consistency of the displayed data,

        I performed the same fits as in Ref.[109] to the data displayed in the figure.

        Since I obtained values of the fitting parameters in good agreement

        with Ref.[109], I concluded that the representative set of data were

        good enough for the purpose of the paper.

 

  4)    The Referee is correct, there is a misprint. I replaced Gev^2 with GeV

        in Eq.(6.72)

 

  5)    On pag. 42, after Eq. (7.1), I replaced " Bjorken-Drell convection"

         with  “Bjorken-Drell convention"

   

  6)    I have performed the following changes:

 

        line 364 “the” replaced with “there”

        line 624: “have showed” replaced with “have shown”

        line 916: “was showed” replaced with “was shown”

 

        Caption Fig. 8 “e” replaced with “and”

 

        line 1161: “neglects” replaced with “neglecting”

 

        line 1272: “is showed” replaced with “was shown"

Reviewer 2 Report

Comments and Suggestions for Authors

Referee report for

Manuscript No: universe-2854799

Authors: Paolo Cea

Title: The QCD vacuum as a disordered chromomagnetic condensate

 

In the present manuscript the author aims at obtaining theoretical insights into the QCD vacuum state via studying primarily the pure gauge sector of the theory in an external Abelian chromomagnetic field. The study is interesting and discusses many different aspects of the QCD vacuum state and the confinement mechanism.

The author motivates it as an attempt towards understanding confinement from first principles. However, in my opinion the present study rather constitutes a very much phenomenally driven account invoking numerous ad hoc assumptions. Correspondingly, I would classify it as a microscopic physics inspired model calculation.

Let me also note that I consider the present manuscript as quite difficult to read, mainly because the covered rich material is neither self-contained nor presented in a particularly logical and easily accessible way.

Nevertheless, I tend to *recommend publication of the author's manuscript in Universe once the comments given and questions raised below are satisfactory addressed* by the author.

 

(1) I believe that the manuscript world severely benefit from a clear introductory statement detailing why at all one can expect relevant insights into the QCD vacuum and the confinement mechanism from the study of a constant Abelian chromomagnetic field that, in constant to a standard magnetic field, cannot be physically realized.

 

(2) I would like to ask the author to provide details why the study of a strictly constant external field is to be considered as sufficient

for the present purpose. What justifies neglecting the inevitable inhomogeneities in the transition between different field domains from the outset.

 

(3) In line 96 the author emphasizes that the calculation is non-perturbative even at one-loop. I propose to specify here in which quantity this calculation is to be considered as non-perturbative.

 

(4) Please define the meaning of the three horizontal lines in the argument of (2.18).

 

(5) Does the factorization assumption in-between (3.12) and (3.13)come with any restrictions?

 

(6) I suggest changing the upper label * on x_3 in (3.30)-(3.32) because the same notation is used for complex conjugation in this work. The same comment applies to quantities in (5.51) as well as (6.11).

 

(7) I have a question concerning the expressions (3.38) and (7.21): does the presence of the UV cutoff \Lambda not simply indicate the need of a renormalization, and the terms quadratic in it can be considered as defining the (would be) physical Maxwell-like term "-H^2/4" (such as done in QED, e.g., when renormalizing the Heisenberg-Euler Lagrangian)?

 

(8) In line 374 the author introduces the notation \rho_\beta(p_2,p_3). However, (3.35)depends only on p_3. Why does he explicitly refer to p_2 here?

 

(9) Is "admits" really the correct word in line 426? I suppose "implies" would be more appropriate here.

 

(10) I would appreciate some details on how (5.35) comes about. Perhaps a reference could be provided here.

 

(11) In lines 667 et seq. the author shows that there are no long range color correlations for separations larger than L_D. Admittedly, I got a bit confused here: what about the original calculation of the author before enforcing a partitioning of the system in domains of extent L_D? Does this not also correspond to one of the viable configurations and would precisely enable such long range correlations? The argument is that this ordered one extremely unlikely to ne realized?

 

(12) What is the role of (6.39)? It is provided to indirectly state that  \sqrt{\beta/6}g = 1 and thus defines the lattice coupling?

 

(13) In lines 1063-1084 the author notes that "theoretical predictions are in reasonable agreement with lattice data ..." What theoretical predictions does the author have in mind here? Probably a single reference or several references can be provided here.

 

(14) In the context of (6.71) probably the analogy with (1.1) could be highlighted and commented on.

 

(15) What is the logic behind considering 3 spinor components in (7. 25) and why is the 3rd component assumed to vanish here?

 

(16) Finally, the author emphasizes at various instances of the present work how good his analytical results are in line with the numerical results of lattice simulations. However, his study is only accounting for constant-field one-loop effects. What does this imply concerning both derivative and higher-loop order corrections? Is this in line with expectations and standard folklore on the topic? I believe that some comments on that would be both in order and highly appreciated in the present context.

Comments on the Quality of English Language

Apart from several typos and a number of grammatical mistakes, for instance in lines 39, 225, 228, 302, 310, 325, 333, 336, 344, 364 and also in the remainder of the manuscript, the text is fine.

Author Response

   

         Reply to the Referee 2

 

 

 1), 2), 3), 7), 16)

          The aim of the paper was to stimulate the high-energy

          comunity to address the problem of confinement from first principles.

          Firstly, I never said in the paper that I have solved the confinement

          problem. Indeed, the Abstract begin with "An attempt is made ..."

          and in section Summary and concluding remarks I said that the results

          of the paper "has been useful leading to a proof of concept that

          confinement can be understood. ... Nevertheless, with with present paper

          we hope to stimulate further studies to reach a complete quantitative

          understanding of confinement in quantum chromodynamics."

          I don't have problems to stress this also in the Introduction.

          So that, following the Referee's suggestion, I added at the end

          of Section 1 the following statement:

 

          " It is worthwhile to stress that we are not claiming that we have

          solved completely the confinement problem in QCD. However, we feel

          that the results of the present paper could offer a promising path

          towards the complete understanding of confinement in quantum

          chromodynamics."

 

          As discussed in the Introduction, the main motivation to consider

          background chromomagnetic fields was the old observation that a

          ferro-chromomagnetic  state could lower the vacuum energy. The crucial

          aspect of the chromomagnetic fields is the presence of chromomagnetic

          instabilities. In fact these instabilities led to the picture of

          the QCD vacuum as a disordered chromomagnetic condensate. I have also

          performed the relevant calculations in the case of an Abelian chromomagnetic

          field directed along the direction 8 in color space and

          considered the Nielsen-Ninomiya theta_3 vacuum. In both cases one

          is led to the disordered chromomagnetic condensate vacuum. So that,

          I believe that even non-uniform chromomagnetic background fields

          that allow the instabilities would led to the same conclusion.

          It is evident that  a magnetic background field that is coupled

          directly only to quarks cannot generate any instabilities.

          The stabilization of the one-loop tachyonic modes is a truly

          non-perturbative calculation as highlighted by the non-analytic

          form of the induced background fields and the condensation energy

          (gH)^2/g^2. Concerning the high-order contributions, I would like to

          stress that the induced background fields are solitons arising from

          the variational field equations and therefore are non subject to

          radiative corrections. Finally, I would remark that the vacuum

          energy density contains ultraviolet-divergent terms that the renormalization

          of the coupling constant, quark masses and quantum fields does not

          suffice to ensure that the vacuum energy remains finite. In general,

          one needs a further additive and moltiplicative renormalization.

          Only in the one-loop approximation where the vacuum energy density

          coincides with the effective potential one obtains that the usual

          renormalization procedure eliminates the divergencies in the

          vacuum energy. In the approximation adopted in the paper the

          ultraviolet divergency is eliminated by a further additive renoemalization.

 

                 

   4)     Problem fixed.

 

  5), 8) 

        The variational parameters are found by solving the integral-differential

        equations obtained by minimization  of the unstable-mode vacuum energy

        functional. Since  the integral-differential equations depend on the second

        derivative with respect to x_3 and on the combination p_3^2-gH

        (or gH/2) due to the fact that  the unstable modes lie on the lowest Landau  

        levels. Therefore, the variational parameter \rho(p_2,p_3) factorizes

        into a function of p_2 times a function of p_3 that I called \rho(p_3)

        in the paper to avoid a proliferation of symbols. It follows that the induced

        background fields  are given by the product of the functions f(x_3) with

        u(x_1,x_2).         

 

  6)    I replaced x_3^* with  \overline{x}_3 from Eq. (3.30) up to Eq. (5.6).

 

  9)    On line 426 I replaced "admits" with "implies".

 

  10)   I imposed the conservative constraint that L_D/\pi should be greater than

        1/\sqrt(gH/2) such that there are enough unstable modes in a given

        chromomagnetic domain.

 

  11)   The Referee is correct, indeed that arguments developed in Section 5

        led to the conclusion that the ordered vacuum wavefunctional is extremely

        unlikely to be realized.

 

  12)  The overall factor \sqrt{\beta/6} affects only the normalization

       of the flux-tube chromoelectric field. Likewise the smearing

       procedure introduces an unknown moltiplicative renormalization constant.

       So that, as discussed in Section 6 the overall normalization of the

       Chromoelectric fields must be fixed by comparing the calculated string tension

       to the reference value

      

  13)   I am referring to unpublished calculations I performed by assuming

        vector-meson dominance together with the duality between quark-antiquark

         bound states and asymptotically free quarks. Basically, this approach

         is a generalisation of the hadron resonance gas hypothesis. I assumed

         that the background magnetic field is coupled mainly to vector mesons.

         After that, summing over the tower of vector meson (with zero width)

         one gets the contribution to the vacuum energy density due to the magnetic

         background field. One needs to assume that the light quarks have a

         small effective mass (around 20 MeV) so that \Sigma can be

         evaluated by mass derivative. Unfortunately, the resulting calculations

         were too long, so that I didn't include these into the paper to avoid

         a further increase  of the paper size.

   

  14)  After Eq. (6.72) I added the following sentence:

       " It is interesting to note that Eq. (6.71) agrees with Eq. (1.1).

         However, before addressing to  any conclusions it should be desirable

         to investigate the behaviour of the critical temperature versus

         applied chromomagnetic fields in QCD with dynamical quarks at the

         Physical point."

 

  15)  I was interested in the zero-energy solutions of the Dirac equation.

       It is easy to see that a non-zero 3rd color component in the quark

       Dirac spinor would give rise to a positive contribution to the energy.

 

 

        Finally, I have  implemented the following changes:

 

        line 39 “has been” replaced with “have been”

        line 225 “As concern” replaced with “As concerns”

        line 228 “at variance of” replaced with “at variance with”

        line 310 “that that” replaced with “that”

        line 325 “this last equations” replaced with “these last equations"

        line 333 “these” replaced with “this”

        line 336 “in such a way to” replaced with “in such a way as to”

        line 344 “energies” replaced with “energy”

        line 364 “energetically favoured” replaced with “favoured energetically"

 

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

I am very happy with the author's response. I feel that all my comments and questions have been addressed appropriately and satisfactorily. In turn, I am glad to recommend the publication of the present manuscript in the journal "Universe".

One final remark: I never intended to imply that I, in any way, feel that the author claims to have proved confinement in the present contribution. In fact, I only felt that the notion of a "theoretical/mathematical" approach (vs. the study of lattice simulations) may be a bit missleading and potentially even distract readers from going through the author's nice (I would say:) phenomenological considerations presented in the present work.