Isospin QCD as a Laboratory for Dense QCD

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The study of strongly interacting material is important for high-energy nuclear physics and asrtophysics. Because of the complexity of QCD at  the region of hadron physics, models are useful tools to explore the properties of QCD matter. This manuscript extend their previous work with two-flavor to three-flavor case to study the impact of strangeness. The results are reasonable. The manuscript is well prepared, I recommend the publication of the manuscript. To address the following issues is helpful to improve the reasonableness of the work.

1. In table 2, the masses of several mesons are listed, which is consistent with experimental data well, except the meson a0(980). What is the possible reason?

2. The manuscript said that the EOS is dominated by mesons at low density but taken over by quarks at high density. How to match the two cases?

Author Response

Comment 1: In table 2, the masses of several mesons are listed, which is consistent with experimental data well, except the meson a0(980). What is the possible reason? Response 1: For the pseudo-scalar octet, the chiral symmetry constrains the spectra. The eta’ meson is reproduced by tuning the strength of the UA(1) symmetry breaking. For scalar mesons we need more dynamical information beyond symmetry and the results depend on the dynamical details. In general it is difficult to reproduce the spectra of the scalar mesons. Indeed the scalar meson nonet is often regarded as exotic hadrons for which simple valence quark picture does not work. So within our simple model we do not expect good agreement in scalar meson spectra. We have added one sentence just above the Table 1. Comment 2: The manuscript said that the EOS is dominated by mesons at low density but taken over by quarks at high density. How to match the two cases? Response 2: In the quark-meson model we do not have confinement in strict sense and quark contributions already exist near the onset of pion condensates (although they are highly suppressed in the hadronic region). The EOS is given by sum of hadronic and quark contributions. In dilute regime the quark contributions are very small and the EOS is dominated by the hadronic contributions. At high density the hadronic contributions are saturated but the quark contributions keep growing. There is an intermediate regime where these two contributions are comparable.

Reviewer 2 Report

Comments and Suggestions for Authors

universe-3085193

Title: Isospin QCD as a laboratory for dense QCD

Authors: Toru Kojo, Daiki Suenaga, Ryuji Chiba

 

The paper is devoted to modelling the cold baryon free QCD matter at finite isospin density. The study is performed within the phenomenological quark-meson model and focuses on the continuous picture of the quark-hadron transition. Inclusion of strangeness is a new element of the work. The authors examine the bulk properties, compare them to the available lattice QCD data and discuss the mesonic mass spectrum. The topic is of scientific importance and is addressed with relevant methods and degree of accuracy. In general, the paper deserves publication in the Universe journal. At the same time, the manuscript includes statements and expressions which are not transparent. In addition, there are several puzzling elements requiring clarification. Thus, I recommend the paper for publication after addressing the criticism explained below.

 

1. In the abstract the authors make a strong statement about describing the quark-hadron continuity on the microscopic level. A phenomenological model explicitly introducing composite mesons as elementary degrees of freedom hardly can address this question microscopically. The mentioned statement should be relaxed.

2. The momentum integral in the gap equation (2) includes the soft-gluon generated interaction kernel and the BCS type ratio of the gap to the particle energy shifted by the chemical potential. Why the single particle Fermi distribution is absent?

3. Eq. (4) is obtained for large mu and assumes a finite pairing gap. How this relates to pQCD reporting the gap to asymptotically vanish at large mu?

4. Why the current quark mass is neglected in the Dirac part of the Lagrangian (6)?

5. Section 2 assumes the same N_c scaling of the quark-meson and three-meson couplings leading to g/K=const. The values from Table 1 contradict this. Why?

6. The expression between Eqs. (17) and (18) assumes the same mass for all the pion states. How was it obtained? Why is it different from Eqs. (32) and (33)? Do these later expressions correspond to the BEC phase so the pion masses always coincide with twice the isospin chemical potential? What is the vacuum pion mass?

7. Eq. (47) is just the zero point term. Where are the excitation parts of the quark thermodynamic potential?

8. I recommend to change the subscript index “g” in Section 6.1 to anythin else since that notation was previously used for gluons.

9. The authors discuss the overlap density of pions but completely ignore the density where the double quark decay of these mesons onsets. I recommend considering the second quantity.

10. The lattice data from Ref. [21] are available for large mu and contrary to the considered model assume c_s^2->1/3-0 and Delta->0+0. This should be mentioned and the corresponding discussion related to conformality should be properly updated.

11. The single particle distribution (96) is not just the Fermi one. How was it obtained. The corresponding explanation should be added.

Author Response

Comment 1: In the abstract the authors make a strong statement about describing the quark-hadron continuity on the microscopic level. A phenomenological model explicitly introducing composite mesons as elementary degrees of freedom hardly can address this question microscopically. The mentioned statement should be relaxed.

Response 1: We understand the referee’s concern but our writing just says “to study the quark-hadron continuity,… we use a quark meson model….”. This statement is not quite strong statement; we do think that the studies of the quark-meson model clarify “some aspects” of the continuity/crossover. Of course the present study cannot give the final answer since our model does not manifestly contain the dynamics of confinement.

If we take sigma and pi as external fields in a generating functional and then integrate quarks up to some energy scale, the resulting Lagrangian is like a quark-meson model at some renormalization scale. In this view, having mesons explicitly in Lagrangian does not necessarily mean that mesons are elementary. Also coupling quarks to mesons open the decay channels, and one can describe the dissociation of mesons to some extent.

Comment 2: The momentum integral in the gap equation (2) includes the soft-gluon generated interaction kernel and the BCS type ratio of the gap to the particle energy shifted by the chemical potential. Why the single particle Fermi distribution is absent?

Response 2: In the BCS gap equation the Fermi distribution does not manifestly show up, but there are such effects in Eq.(2); if one takes Delta -> 0  limit in Eq(2), sqrt[ (E-mu)^2 ] becomes + (E-mu) for E>mu, and - (E-mu) for E<mu. The necessity of this classification generates the step functions (Fermi distributions). 

Comment 3: Eq. (4) is obtained for large mu and assumes a finite pairing gap. How this relates to pQCD reporting the gap to asymptotically vanish at large mu?

cThe weak coupling estimate is based on a propagator with the form of ~ alpha_s(mu)/p^2 that assumes the typical momentum transfer of p ~ mu. But this form is too crude. For a small momentum transfer, one should use ~alpha_s (p)/p^2 where alpha_s(p) can be much bigger than alpha_s (mu). If one neglects such running in momentum “p” and use supposedly typical scale “mu”, the coupling approaches 0 at asymptotically high density and hence the gap would disappear. We are arguing that, even at large mu, there remains a soft momentum transfer process for which the effective coupling alpha_s(p) remains much stronger than alpha_s (mu). Unless soft processes are cutoff by the color screening, there is no reason why the gap vanishes. The question to be clarified is how fast the color screening is developed, and this is still an open issue.

Comment 4: Why the current quark mass is neglected in the Dirac part of the Lagrangian (6)?

Response 4: Thanks for pointing out this. Actually we also did analyses including current quark mass explicitly. Including the current quark masses, the condensate and effective mass is no longer proportional and we need additional renormalization procedures for newly generated terms. This sort of treatment is conceptually more accurate but it increases the volume of the material considerably. On the other hand it does not improve our descriptions much, so we decided to show the simpler version in which the explicit breaking is summarized into the hadronic Lagrangian.

Comment 5: Section 2 assumes the same N_c scaling of the quark-meson and three-meson couplings leading to g/K=const. The values from Table 1 contradict this. Why?

Response 5: The table 1 does not display how the couplings change with respect to Nc. Since we are not looking at the Nc scaling of g and K here, g/K needs not be an Nc-independent constant.

Comment 6: The expression between Eqs. (17) and (18) assumes the same mass for all the pion states. How was it obtained? Why is it different from Eqs. (32) and (33)? Do these later expressions correspond to the BEC phase so the pion masses always coincide with twice the isospin chemical potential? What is the vacuum pion mass?

Response 6: Thanks for pointing out this. The notation was certainly confusing. Eqs.(17) and (18) use the “vacuum” pion mass. In Eqs.(32) and (33) mean the pion mass in medium. At the onset of the BEC, the medium pion mass simply coincides with the vacuum mass, so mu_I = m_pi^{vac}/2 holds. 

To avoid reader’s confusion, we attache “vac” for the vacuum pion mass.

Comment 7: Eq. (47) is just the zero point term. Where are the excitation parts of the quark thermodynamic potential?

Response 7:  In the BCS, the chemical potential is hidden into “xi” and the thermodynamics is already expressed. Meanwhile the finite temperature is not displayed as we are working on the zero temperature case only. 

Comment 8:  I recommend to change the subscript index “g” in Section 6.1 to anything else since that notation was previously used for gluons.

Response 8:  For gluons we use “g_s”, not “g”, and the notations are distinguishable.

Comment 9:  The authors discuss the overlap density of pions but completely ignore the density where the double quark decay of these mesons onsets. I recommend considering the second quantity.

Response 9:  Thanks for this recommendation. Actually we plan to study the meson correlators with quark loops in the present model. After such studies we can begin to discuss the dissociation of mesons.

Comment 10:  The lattice data from Ref. [21] are available for large mu and contrary to the considered model assume c_s^2->1/3-0 and Delta->0+0. This should be mentioned and the corresponding discussion related to conformality should be properly updated.

Response 10:  We went over Ref. [21] but could not find the statement on the assumption which the referee quoted. Could you please tell us where the assumption is mentioned?

Comment 11:  The single particle distribution (96) is not just the Fermi one. How was it obtained. The corresponding explanation should be added.

Response 11:  The quark propagator in the presence of the pion condensate takes the Nambu-Gor’kov form. Using this propagator one can compute the occupation probability. This is the standard treatment.

Reviewer 3 Report

Comments and Suggestions for Authors

Report of the Referee

Manuscript Ref.: Universe-3085193 

Title: "Isospin QCD as a laboratory for dense QCD"

==========================================

The authors presented a comprehensive study on QCD theory with the isospin chemical potential applied to the neutron star (NS) physics.  The equation of state (EoS) interpolates hadronic and quark matter degrees of freedom at microscopic level. In particular, a three-flavors analysis is carried out in order to investigate the effect of the strangeness. This sophisticated model is able to give predictions for the sound speed peak and negative trace anomaly and at the same time is consistent with the lattice results. Therefore, the demonstrations presented in the paper are of high interest in the research area of EoSs for compact objects. The main uncertainties in the calculations and the limitations on the methodology are carefully discussed.

 

The paper is very well presented and the references are adequate. The topic is hot and of interest for the researchers studying neutron stars physics and compact astrophysical objects. The subject of the manuscript is a scientific breakthrough and the quality of the paper corresponds to the level of the journal “Universe”. The manuscript is well-written, presenting ideas and methods clearly and analyzing the results thoroughly. The work is in suitable form for publication.

 

I do not have suggestions for substantial changes, but I have minor points which I would like to ask the authors to address. Recently, the CMS-LHC collaboration reported the most precise measurement to date of the sound speed at the quark–gluon plasma which offers new insights into this extremely hot state of matter [Rept.Prog.Phys. 87 (2024) 7, 077801]. The measurement is done for very central lead-lead collisions at the effective temperature of 220 MeV and is consistent with the predictions from lattice QCD. How effective is the corrent model if extrapolated to such very high temperatures ? Along these lines, in Refs. [Astrophys.J. 950 (2023) 2, 107,Phys.Rev.Lett. 128 (2022) 20, 202701] the QCD input to NS EoS  interpolates between low and high densities which leads to stable and causal EoS (see also Ref. Nature Phys. 16 (2020) 9, 907-910).  This sort of connection between low-density information from nuclear theory and high-density constraints from perturbative QCD is also employed in Ref. [Phys.Rev.C 107 (2023) 5, L052801]. How the present study is compared to those works? I think the underlying methodology is different and the model considers a distinct role played by the strangeness effects. I suggest some comments on these issues could be introduced in the discussion section.  

 

 

For the reasons presented above, the manuscript is suitable in my view for the Universe journal.

 

 

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I found some  wording issues in the text (not exhaustive):

 

Page 3, Line 104: "these process"--> "this process" or "these processes".

Page 16, Line 406: "in Table. 1 and 2"--> "in Tables 1 and 2" or "in Table 1 and Table 2".

Page 19, Line 453: "tell which degrees of freedom is relevant"--> "tell which of degrees of freedom is relevant" or "tell which degrees of freedom are relevant"?

Page 23, Line 579: "How the transition between these two regimes occur..." --> "How the transition between these two regimes occurs..."?

 

Author Response

Comment 1: I do not have suggestions for substantial changes, but I have minor points which I would like to ask the authors to address. 

Recently, the CMS-LHC collaboration reported the most precise measurement to date of the sound speed at the quark–gluon plasma which offers new insights into this extremely hot state of matter [Rept.Prog.Phys. 87 (2024) 7, 077801]. The measurement is done for very central lead-lead collisions at the effective temperature of 220 MeV and is consistent with the predictions from lattice QCD. How effective is the current model if extrapolated to such very high temperatures? 

Response 1:  To describe the finite temperature behaviors, at least we should improve our model, e.g., by coupling the present model to the Polyakov loop which suppresses colored thermal excitations. Thermal gluons should be also included. Although the Polyakov loop extended quark-meson models contain some conceptual issues, if only qualitative trends, such as a dip of the sound speed near the critical temperature, are concerned, the models can give reasonable descriptions capturing key aspects. 

Comment 2: Along these lines, in Refs. [Astrophys.J. 950 (2023) 2, 107, Phys.Rev.Lett. 128 (2022) 20, 202701] the QCD input to NS EoS  interpolates between low and high densities which leads to stable and causal EoS (see also Ref. Nature Phys. 16 (2020) 9, 907-910).  This sort of connection between low-density information from nuclear theory and high-density constraints from perturbative QCD is also employed in Ref. [Phys.Rev.C 107 (2023) 5, L052801]. How the present study is compared to those works? 

Response 2: The present construction of EOS manifestly describes the domain intermediate between low and high density limits. The previous studies mentioned by the referee interpolate only EOS, not microscopic physics. This limitation becomes problematic when we extend the framework to finite temperature; one needs the thermodynamic stability and causality not only for one-dimensional thermodynamic variable (e.g., chemical potential mu) but also in any directions in two-dimensional thermodynamic variables (mu and T). The latter interpolation is highly nontrivial and a physics informed strategy should be taken.

I think the underlying methodology is different and the model considers a distinct role played by the strangeness effects. I suggest some comments on these issues could be introduced in the discussion section.  

Response 2: We thank the referee for this constructive suggestion. We add sentences 573-576 to explain in which aspects our model studies are distinct from studies based on inference.

Finally we would like to thank the referee for pointing out typos. We have corrected all typos pointed out by the referee and also corrected typos found by ourselves.