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Speed of sound in dense stronginteraction matter
by Jens Braun, Andreas Geißel, Benedikt Schallmo
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Submission summary
Authors (as registered SciPost users):  Jens Braun · Andreas Geißel 
Submission information  

Preprint Link:  scipost_202211_00010v2 (pdf) 
Date accepted:  20240222 
Date submitted:  20240208 12:03 
Submitted by:  Braun, Jens 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the speed of sound in stronginteraction matter in density regimes which are expected to be governed by the presence of a colorsuperconducting gap. At (very) high densities, our analysis indicates that the speed of sound approaches its asymptotic value associated with the noninteracting quark gas from below, in agreement with firstprinciples studies which do not take the presence of a colorsuperconducting gap into account. Towards lower densities, however, the presence of a gap induces an increase of the speed of sound above its asymptotic value. Importantly, even if gapinduced corrections to the pressure may appear small, we find that derivatives of the gap with respect to the chemical potential can still be sizeable and lead to a qualitative change of the density dependence of the speed of sound. Taking into account constraints on the density dependence of the speed of sound at low densities, our general considerations suggest the existence of a maximum in the speed of sound. Interestingly, we also observe that specific properties of the gap can be related to characteristic properties of the speed of sound which are indirectly constrained by observations.
Author comments upon resubmission
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(1) We agree that an at least brief discussion of the effect of the role of quark masses should be included in our manuscript. To this end, we distinguish between an explicit (constant) current quark mass $m_q$, which breaks the chiral symmetry explicitly, and inmedium selfenergy corrections to the quarks, which do not break the chiral symmetry and come with a prefactor $\sim g^2 \mu^2$ (at leading order in the coupling).
With respect to the pressure, the current quark mass $m_q$ enters the equation of state via a term $\sim \mu^2 m_q^2$ (at lowest order in $m_q$). Moreover, a term $\sim m_q^2 \Delta_0^2$ is potentially present. Since the gap enters the equation of state as $\sim \mu^2 \Delta_0^2$ at leading order, we expect that the $m_q^2$contributions to the equation of state (and also to the speed of sound) are subleading in the presence of the gap $\Delta_0$, provided that $m_q$ is small compared to the chemical potential and also small compared to the gap.
The situation is different in case of the aforementioned selfenergy corrections to the quarks. These corrections already appear in the term $\sim\mu^4$ in the equation of state. In fact, to order $g^2$, they are implicitly taken into account in the twoloop result for the pressure which appears in our definition of the coefficient $\gamma_0$ in Eq. (14).
We have added a comment on these aspects on page 8 of the revised version (from line 26 of the left column to line 9 of the right column, "In this work, we restricted our analysis ... the density range considered in this work.").
___
(2)
a) We agree with the referee that our discussion is slightly misleading at this point. The gap computed in the weakcoupling limit increases for small chemical potentials and then decreases with increasing chemical potential, as shown in Fig. 1. Only at extremely large chemical potentials, way beyond any value which would be relevant for astrophysical applications, this gap increases again with increasing chemical potential. Note that $\Delta_0/\mu$ is still decreasing, even in this regime. In any case, in Fig. 1, we do not show the gaps for such extremely large values of the chemical potential.
For our general line of arguments, it is only necessary that $\Delta_0/\mu \to 0$ for large $\mu$, which is true for both "parametrizations" of the gap considered in our present work. To clarify this point, we have slightly modified the discussion below Eq. (13), see page 4 of the revised version (line 811 counted from Eq. (13), "Assuming that ... (i.e., in the limit of infinite density n).").
b) We have added a comment on the difference in the $\mu$dependence of the gap as obtained from the weakcoupling calculation and the fRG study presented in Ref. [47], see the new footnote #4 on page 5/6. Please note that we have also included a new reference in this footnote, see Ref. [72] of the revised version of the manuscript.
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(3) We agree with the referee. We now already state in the abstract that we only consider the zerotemperature limit (see first line of the abstract of the revised version of our manuscript). Moreover, we have rephrased the paragraph around Eq. (1) to make clear that only the zerotemperature limit is considered in our present work (see page 1 of the revised version, left column, line 1218, "In the present work, ... the pressure of stronginteraction matter.").
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(4) We did not intend to introduce the density $n_min$ as a strict scale on the basis of which powercounting arguments can be made. We only intended to introduce this scale as a rough estimate for the range of validity of calculations of the speed of sound in which the presence of a gap has not been taken into account. Coming from high densities, we agree with the referee that gap effects can already be sizeable at this scale but they have not yet changed the slope of the speed of sound as a function of the density. The density $n_min$ can therefore only represent a rough estimate for the lower bound of the actual density scale above which gap effects may be safely neglected in calculations of the speed of sound.
We have adapted our discussion of this aspect on page 6 to make clear that the scale $n_{\text{min}}$ only provides a rough estimate here. To be specific, we have included a new footnote on page 6 of the revised version (see new footnote #5) and slightly modified the wording of one sentence on page 6 (right column, line 711, "Importantly, we also observe ... speed of sound at $n = n_{\text{min}}$.").
___
(5) We cannot conclusively answer this question. However, in Ref. [23], the speed of sound has already been computed in two ways using the fRG approach, namely with and without gap, and it was observed that it requires the presence of a gap to push the speed of sound above its value in the noninteracting limit when the density is decreased, see right panel of Fig. 1 in Ref. [23].
We slightly modified our discussion at the end of Sec. III and added a comment on this aspect to the manuscript, see page 8 of the revised version of our manuscript (right column, line 2133, "Interestingly, in that work, it was also observed that ... than from high densities as done in our work.").
___
(6) We expect that the presence of a color superconducting gap also leads to an increase of the speed of sound above its value in the noninteraction limit if colorflavor locking (CFL) is the dominant pairing mechanism in QCD with 2+1 quark flavors. In fact, in this case, the functional form of the expansion of the pressure given in Eq. (5) should remain unchanged (at least at the order given in this equation), i.e., the leading order gapdependent correction should be of the form $\sim \mu^2 \Delta_0^2$, where $\Delta_0$ now refers to the CFL gap. If the dependence of this gap on the chemical potential is qualitatively similar to the ones discussed for the 2SC gap in the present work, then the gap also leads to an increase of the speed of sound above its value in the noninteraction limit in the case of colorflavor locking, see also the discussion in appendix B of Ref. [48].
We added a comment on this aspect in Sec. IV ("Conclusions"), see page 9 of the revised version of our manuscript (line 45 of the left column to line 9 of the right column, "In this regard, we would like to add ... above its value in the noninteracting limit, see also the discussion in the appendix of Ref. [48]."). Moreover, we reformulated the last sentence of Sec. I of the revised version (page 2, left column, line 910, "A brief discussion ... together with our conclusions.").
___
Additional changes:
(A) There was a typo in the label of the yaxis of Fig. 1. We corrected this typo and replaced the figure accordingly.
(B) We have updated Ref. [48].
(C) We have slightly modified the wording of two sentences in the Conclusions, see page 9 (right column, line 1017, "With respect to the maximum ... astrophysical observations in Ref. [13].").
___
We hope that with these modifications the present version of our manuscript is now suitable for publication in SciPost.
List of changes
We have included a detailed list of changes in our reply letter to the referee.
Published as SciPost Phys. Core 7, 015 (2024)
Reports on this Submission
Report
The authors have carefully responded to all my questions and have modified the manuscript accordingly. I think the changes are an improvement; therefore, and for the reasons given in my first report, I recommend publication in SciPost Physics Core.
Jens Braun on 20240213 [id 4310]
Errors in usersupplied markup (flagged; corrections coming soon)
(Remark: Because of a problem in the source text (associated with a markdown command) our initially submitted reply letter is displayed in an unreadable format in the field "Author comments upon resubmission". We fixed the problem with the markdown command in the source text and make our reply letter available here for the time being.)
We are grateful to the referee for their positive report and for stating that our manuscript adds an ``interesting and carefully discussed aspect'' regarding the properties of dense stronginteracting matter. Moreover, we thank the referee for their helpful comments and suggestions regarding the presentation of our results. Below, we reply to the referee's comments and suggestions following the order in which they appear in the report:

(1) We agree that an at least brief discussion of the effect of the role of quark masses should be included in our manuscript. To this end, we distinguish between an explicit (constant) current quark mass $m_q$, which breaks the chiral symmetry explicitly, and inmedium selfenergy corrections to the quarks, which do not break the chiral symmetry and come with a prefactor $\sim g^2 \mu^2$ (at leading order in the coupling).
With respect to the pressure, the current quark mass $m_q$ enters the equation of state via a term $\sim \mu^2 m_q^2$ (at lowest order in $m_q$). Moreover, a term $\sim m_q^2 \Delta_0^2$ is potentially present. Since the gap enters the equation of state as $\sim \mu^2 \Delta_0^2$ at leading order, we expect that the $m_q^2$contributions to the equation of state (and also to the speed of sound) are subleading in the presence of the gap $\Delta_0$, provided that $m_q$ is small compared to the chemical potential and also small compared to the gap.
The situation is different in case of the aforementioned selfenergy corrections to the quarks. These corrections already appear in the term $\sim\mu^4$ in the equation of state. In fact, to order $g^2$, they are implicitly taken into account in the twoloop result for the pressure which appears in our definition of the coefficient $\gamma_0$ in Eq. (14).
We have added a comment on these aspects on page 8 of the revised version (from line 26 of the left column to line 9 of the right column, "In this work, we restricted our analysis ... the density range considered in this work.").

(2)
a) We agree with the referee that our discussion is slightly misleading at this point. The gap computed in the weakcoupling limit increases for small chemical potentials and then decreases with increasing chemical potential, as shown in Fig. 1. Only at extremely large chemical potentials, way beyond any value which would be relevant for astrophysical applications, this gap increases again with increasing chemical potential. Note that $\Delta_0/\mu$ is still decreasing, even in this regime. In any case, in Fig. 1, we do not show the gaps for such extremely large values of the chemical potential.
For our general line of arguments, it is only necessary that $\Delta_0/\mu \to 0$ for large $\mu$, which is true for both "parametrizations" of the gap considered in our present work. To clarify this point, we have slightly modified the discussion below Eq. (13), see page 4 of the revised version (line 811 counted from Eq. (13), "Assuming that ... (i.e., in the limit of infinite density n).").
b) We have added a comment on the difference in the $\mu$dependence of the gap as obtained from the weakcoupling calculation and the fRG study presented in Ref. [47], see the new footnote No. 4 on page 5/6. Please note that we have also included a new reference in this footnote, see Ref. [72] of the revised version of the manuscript.

(3) We agree with the referee. We now already state in the abstract that we only consider the zerotemperature limit (see first line of the abstract of the revised version of our manuscript). Moreover, we have rephrased the paragraph around Eq. (1) to make clear that only the zerotemperature limit is considered in our present work (see page 1 of the revised version, left column, line 1218, "In the present work, ... the pressure of stronginteraction matter.").

(4) We did not intend to introduce the density $n_\text{min}$ as a strict scale on the basis of which powercounting arguments can be made. We only intended to introduce this scale as a rough estimate for the range of validity of calculations of the speed of sound in which the presence of a gap has not been taken into account. Coming from high densities, we agree with the referee that gap effects can already be sizeable at this scale but they have not yet changed the slope of the speed of sound as a function of the density. The density $n_{\text{min}}$ can therefore only represent a rough estimate for the lower bound of the actual density scale above which gap effects may be safely neglected in calculations of the speed of sound.
We have adapted our discussion of this aspect on page 6 to make clear that the scale $n_{\text{min}}$ only provides a rough estimate here. To be specific, we have included a new footnote on page 6 of the revised version (see new footnote No. 5) and slightly modified the wording of one sentence on page 6 (right column, line 711, "Importantly, we also observe ... speed of sound at $n = n_{\text{min}}$.").

(5) We cannot conclusively answer this question. However, in Ref. [23], the speed of sound has already been computed in two ways using the fRG approach, namely with and without gap, and it was observed that it requires the presence of a gap to push the speed of sound above its value in the noninteracting limit when the density is decreased, see right panel of Fig. 1 in Ref. [23].
We slightly modified our discussion at the end of Sec. III and added a comment on this aspect to the manuscript, see page 8 of the revised version of our manuscript (right column, line 2133, "Interestingly, in that work, it was also observed that ... than from high densities as done in our work.").

(6) We expect that the presence of a color superconducting gap also leads to an increase of the speed of sound above its value in the noninteraction limit if colorflavor locking (CFL) is the dominant pairing mechanism in QCD with 2+1 quark flavors. In fact, in this case, the functional form of the expansion of the pressure given in Eq. (5) should remain unchanged (at least at the order given in this equation), i.e., the leading order gapdependent correction should be of the form $\sim \mu^2 \Delta_0^2$, where $\Delta_0$ now refers to the CFL gap. If the dependence of this gap on the chemical potential is qualitatively similar to the ones discussed for the 2SC gap in the present work, then the gap also leads to an increase of the speed of sound above its value in the noninteraction limit in the case of colorflavor locking, see also the discussion in appendix B of Ref. [48].
We added a comment on this aspect in Sec. IV ("Conclusions"), see page 9 of the revised version of our manuscript (line 45 of the left column to line 9 of the right column, "In this regard, we would like to add ... above its value in the noninteracting limit, see also the discussion in the appendix of Ref. [48]."). Moreover, we reformulated the last sentence of Sec. I of the revised version (page 2, left column, line 910, "A brief discussion ... together with our conclusions.").

Additional changes:
(A) There was a typo in the label of the yaxis of Fig. 1. We corrected this typo and replaced the figure accordingly.
(B) We have updated Ref. [48].
(C) We have slightly modified the wording of two sentences in the Conclusions, see page 9 (right column, line 1017, "With respect to the maximum ... astrophysical observations in Ref. [13].").


We hope that with these modifications the present version of our manuscript is now suitable for publication in SciPost.