Towards holographic color superconductivity in QCD

Dear Editor and Anonymous Referee,

We would like to thank you for the professional handling of our manuscript and for recommending it for publication in your journal. Below, we address the referee’s comments and summarize the revisions made to our manuscript.

Best regards, Christian Ecker, on behalf of all authors

Referee: 1) Around eq. (3) the authors assume they can ignore the tachyon field dual to the gauge-invariant quark bilinear, because the latter has a vanishing vev in the deconfined phase. Shouldn't this be a consequence of the analysis rather than an assumption? It is not clear if this is the case or not.

Reply: The vanishing of the vev in the deconfined phase is indeed not an assumption. A branch of black hole solutions which has a nonzero vev does exist in this model, but it has lower free energy than the solution with zero vev. In early studies, [76,77], it was found that the phase with nonzero vev only appears in a limited part of the phase diagram. In the current article we are using the version of the model fitted carefully to lattice data in [83], and in this version the black hole solution with zero vev is always energetically preferred to the solution with nonzero vev. This was checked in Ref. [83] both at T=0 and at mu=0, but the plots of the article do not show the unstable branch because they focus on the physically relevant phases. To clarify this, we modified the first paragraph in section 2.2.

Referee: 2) The combination on the LHS of eq. (11) seems to be obtained by summing eqs. (2) and (7), and neglecting the term proportional to F_MN in (7). However, it is not clear at this stage that this corresponds to the weakly-coupled limit where \phi goes to -\infty, as the explicit form of w(\phi) is given later in eq. (14). Is there a simple argument for this?

Reply: The standard behavior of the gauge field near the boundary in five dimensions, consistent with UV dimension of the baryon number operator is schematically $\Phi \sim \mu_q + n_q r^2$ as $r \to 0$. This is obtained in our setup if $w(\phi)$ tends to a constant at the boundary (which is indeed our choice later in Eq. (17)). The effect of the charge term near the boundary is through the combination $e^{-4A}(\Phi')^2$ which is $\sim r^6$ and therefore highly suppressed with respect to the logarithmic flow of Eqs. (12) and (13). To clarify this, we added footnote 2 as a further explanation.

Referee: 3) The authors make some choices about the potentials that would require some clarifications. These are: - a choice of a potential set in eqs. (15-17) among a one-parameter family that is allowed by constraints by lattice data - a choice of asymptotic behavior of Vf in eq. (16) that departs from the string-frame flat result of eq. (14) - a choice of a free parameter (namely, the AdS radius, or W0) - a choice of a potential set in (27) It is not clear why the authors made these choices, and if and how they affect (at least qualitatively) their final results. It would be worthy to comment about this in section 2.3.

Reply: We agree that our choices for the potentials deserve a more explicit justification. We address these points below and have added comments in Secs. 2.3 and 2.4. and in Appendix A.1. -Choice of the “7a” potential set (V-QCD family) and the values of parameters: Given that the current article is our first work on this topic, we chose to work only with a single set of potentials, corresponding to a specific value of $W_0$ as it is stated just after equation (18). Within the class of potentials which agree well with constraints from QCD phenomenology and lattice results, we do not expect strong dependence on results for the paired instabilities, because earlier work for the equation of state (e.g. [83,90]) and modulated instabilities [101] show only mild dependence on the parameters. In order to check this roughly, we have computed BF bounds for different V-QCD potential sets, corresponding to various values of $W_0$, and add them in the Appendix A.1. After choosing the value of $W_0$, the other fit parameters in the DBI action are determined by lattice data to a good accuracy. The value of the AdS radius can be scaled out by using symmetry and does not affect the solutions. The results shows that the threshold value of the BF bound changes mildly for different potential sets, implying the general features are similar. Moreover, since the current study does not take into account backreaction, carrying out a detailed analysis of the model dependence at this stage does not make sense to us. Therefore we have chosen a single intermediate set, which reproduces the lattice thermodynamics at small baryon chemical potential, and yields an EOS that lies in the phenomenologically preferred band at zero temperature (in particular, it is compatible with all the known astrophysical constraints for neutron stars). -Asymptotic behavior of $V_f$ vs. string-frame flatness: Among the set of potential functions considered, the asymptotics of the dilaton dependence in $V_f$ is least strongly constrained, because observables are insensitive to this particular choice. This happens because the geometries in the deconfined phase do not probe asymptotically large values of $\phi$ whereas in the confined phase chiral symmetry is broken and the IR behavior of the flavor potential is dominated by the tachyon dependence instead of the dilaton. Here we have followed earlier literature, and chose the asymptotics with an integer value power law of $e^\phi$. This choice was made in early literature, before the link to string frame flatness was understood, and we have not changed it because the value does not seem to play any major role for the solutions. As pointed out in footnote 4 both the string-frame value $v_f = 7/3$ and our choice $v_f = 2$ lie within the interval allowed by phenomenological constraints as we point out in footnote 4. The precise choice is expected to have negligible effect on the results, since the difference is small and value of $\phi$ are bounded from above in the black hole geometries. We have modified footnote 4. - The choice of the potentials for the charged scalar: We make here the simplest choice that reproduces the expected UV behavior with fixed boundary dimension and the expected IR asymptotics (which are flat in string frame). One could introduce more complicated functions, but given the qualitative nature of our study we have decided to restrict ourselves to this choice. Note that typically the potential functions (e.g., in the DBI action, Eqs. (15)-(17)), which lead to reasonable phenomenology are relatively structureless monotonic functions. Therefore there is no obvious motivation to introduce additional terms in the potentials of the charged scalar, as there is also no data to which to fit the potentials. Note that the Ansatz in Eq. (27) is similar to Eqs. (15)-(17) apart from the terms $\proto V_{1}$, $V_2$, $W_1$, and $W_2$, which are fixed by the perturbative RG flow, and dropped in the case of charged scalar, where we do not have a perturbative calculation to match to.

Referee: 4) After eq. (23) the authors discuss that if the scalar is embedded in a flavor bifundamental, then no non-zero vev would preserve the full U(N) vectorial symmetry. The added scalar is only charged under the vector U(1), so it seems that they are describing a deconfined phase where baryon number is spontaneously broken, rather than a paired phase. It is then not clear that the phase they discuss should be phenomenologically relevant for QCD. Moreover, as they correctly mention after eq. (27), the scalar cannot be a naive diquark operator, as this is not gauge invariant. It is then not clear why they do not interpret instead the scalar as a higher-order gauge-invariant quark operator (this would change the UV scaling dimension from 3 to a higher one, though).

Reply:--- This is an important point but unfortunately it is challenging to improve. Let us however stress that we definitely want to construct something that breaks the baryon number: apart from the baryon number being (most likely, at least in CFL) broken in the paired phase, we also want to eventually achieve a model in the bulk (as we explain in the conclusions) where the flux supporting the AdS$_2$ geometry is absorbed by the condensate, leading to a geometry with zero entropy at zero temperature. But it is not obvious to us how to make this closer to what is expected for a specific paired phases such as the CFL phase. Perhaps a fruitful direction would be to follow Ref. [74] (also cited in the conclusions) which came out while we were finishing the current article. This reference suggests another approach which is more closely motivated by the top-down construction in Ref. [64]. As for the higher-dimensional operator (such as the $\Lambda\Lambda$ in the case of CFL), we think that the condensation of a diquark operator should be the leading effect, and what we are constructing in the article could be phenomenologically closer to mimicking such a condensation. Moreover, the condensation is driven by the AdS$_2$ region in the geometry. Because this region is at strong coupling, the potentials controlling the behavior of the fields can be changed essentially independently in this region from the boundary behavior, which determines the dimension of the dual operator. Therefore, we expect that simply changing the boundary dimension of the dual field would not change our results much.

Referee: 5) Are there any lattice/phenomenological/holographic results confirming the fact that a spatially modulated phase is favored with respect to a paired quark phase, as the results of section 3 suggest?

Reply:--- We are not aware of lattice or earlier holographic studies that would compare an inhomogeneous phase to a paired phase. But while we were preparing the response, a new article arXiv:2512.20510 appeared, which discusses inhomogeneous phases using functional renormalization group methods. This article seems to confirm the earlier indications (e.g., arXiv:2406.11312) of a widespread region with inhomogeneities in the phase diagram. However, they do not compare to paired phases, but it seems to us that the critical temperatures they report are higher than usually found for paired phases in model computations. Moreover, one should keep in mind that the nature of the modulated phase which we discuss in our manuscript (modulated current on the field theory side) appears to be different from the modulated phases considered in the QCD literature, where typically the chiral condensate is modulated. We did also find a relatively recent model computation, arXiv:2012.07520, which explicitly compares these two kinds of phases in a NJL model. However, their results depend on the model parameters. Since the main focus of this manuscript is the paired phase rather than inhomogeneous phases, we chose not to discuss these studies further but did add the references in the introduction.

Referee: 6) Why does the probe analysis of section 4 suggest the transition to be second order? This is mentioned in footnote 3, but in principle it would require to verify the continuity of the (derivative of the) free energy at the transition. What is the argument of the authors in favor of this?

Reply:--- In the probe limit analysis, we have modeled the effect of backreaction through including a quartic term in the potential of the fermion field. This is seen to lead to a second order transition. This and the fact that second order transitions are typical in similar constructions in the literature (even though first order is also possible) makes us think the the transition is likely to be of second order, but we do not have a strong argument. We think that the claim in footnote 3 (footnote 4 in the new version) was too strong. We reworded, and moved the comments about the order of the transition from the footnote to the end of section 5.1.

Referee: 7) At the end of section 5.1 the authors claim that the backreaction of the scalar field is negligible, based on the fact that the transition is second order. However, they also claim that the transition should be second order, if the backreaction is neglected. Since the backreaction has not been computed, it is not clear what is an assumption and what has been verified.

Reply:--- We thank the referee for pointing this out, and have reworded the confusing comments in the manuscript at the end of section 5.1.

Referee: 8) As a consequence of the stability analysis, the authors observe that the modulated phase is favored with respect to the paired phase. It is not clear then how the phase diagram in Figure 1 should be interpreted, given that the modulated phase is favored with respect to the homogeneous paired phase. In particular, do the authors expect this condensed quark matter to exist in their model? How does this compare with real QCD expectations?

Reply:--- Figure 1 (figure 2 in the revised version) simply represents the current status, as we have not yet computed the inhomogeneous ground state. We do not know yet if the condensed matter state will exist in the final phase diagram. There are regions in the phase diagram which only show the inhomogeneous instability, which suggests that there is an inhomogeneous ground state without the paired condensate. But both kind of condensates could occur simultaneously, and our observation that the inhomogeneous instabilities grow faster does not rule out the possibility of a ground state of purely paired condensate. As for real QCD, it is hard to make useful comments because most results in the region of low temperature and relatively high density, which is our focus here, are coming from models and may not be quantitatively reliable. See however the discussion of point 5) above, which indicates that our results may be in rough agreement with functional renormalization group analysis. We revised Fig. 2 to additionally display the previously computed onset of modulated instability in the non-Abelian sector of the model.

Dear Editor and Anonymous Referee,

We would like to thank you for the professional handling of our manuscript and for recommending it for publication in your journal. Below, we address the referee’s comments and summarize the revisions made to our manuscript.

Best regards, Christian Ecker, on behalf of all authors

Referee: 1) In Section 3.2 the authors claim that spatially modulated instabilities, which appear at high μB/T in the Witten-Sakai-Sugimoto model and at lower μB/T in V-QCD, are model independent and persist across a wide class of bottom up holographic models fitted with lattice data. Do these ones also include the Einstein-Maxwell-dilaton models described, for example in Rougemont et al. "Hot QCD phase diagram from holographic Einstein–Maxwell–Dilaton models" Prog.Part.Nucl.Phys. 135 (2024) 104093? Are these instabilitities a feature of holographic models? Are they also expected to appear in QCD?

Reply: Indeed, the generic emergence of modulated instabilities in both V-QCD and Einstein–Maxwell–Dilaton (EMD) bottom-up models tuned to QCD has been demonstrated in [arXiv:2405.02399] and [arXiv:2405.02392]. Such instabilities are expected to be a common feature of holographic models, including those presented in “Hot QCD phase diagram from holographic Einstein–Maxwell–Dilaton models” (Prog. Part. Nucl. Phys. 135 (2024) 104093), which share key characteristics with the aforementioned EMD frameworks. That is, while the extent of the instability was not checked for precisely the fits discussed in the review by Rougemont et al., it was checked for similar fits by other authors in [arXiv:2405.02392], and the dependence on the details of the fit was observed to be so small that it is clear that the main findings also apply to the models of this review.

However, we stress that the class of models to which our comment applies is limited to (chirally symmetric) holographic models precisely fitted to lattice data for the equation of state and baryon number susceptibilities of QCD. In particular, while there are indications for the existence of inhomogeneous phases in QCD, there are several factors which make a direct quantitative connection of the holographic studies to real (N=3) QCD challenging. These include i) effects of chiral symmetry breaking through quark masses ii) effects missed due to the inherently large-N limit of holographic studies iii) similarly, effects missed due to the inherently infinite coupling.

Referee: 2) In Section 5.1, the authors describe the phase diagram from Fig. 1 that shows a first order transition line. Does the corresponding equation of state (EoS) agree with lattice QCD data from zero to small μB/T? Reference [90] does not show a comparison between lattice QCD and the resulting EoS from the effective model either. Also, does the inclusion of the quark-pairing mechanism affect the agreement between V-QCD and lattice QCD?

Reply: For simplicity, our manuscript focuses on the parameter set 7a introduced in [arXiv:1809.07770], which represents a canonical intermediate choice. This set not only lies well within the constrained band of model-agnostic parametrizations at zero temperature (see, e.g., Fig. 2 in [arXiv:2402.11013])—being sufficiently soft to satisfy the tidal deformability constraints inferred from the GW170817 event—but is also stiff enough to prevent a prompt collapse and to allow for an extended, potentially second-long-lived post-merger remnant, as suggested by the EM counterpart observed in this event. Indeed, the fit to lattice data corresponding to this specific parameter set is not shown in our manuscript but can be found in Fig. 1 (Yang–Mills sector only) and Fig. 2 (full model) of [arXiv:1809.07770]. In Section 2.3, we have added a new plot demonstrating the consistency of the thermodynamic properties of the model obtained from potential choice 7a with the 2+1 flavor lattice data from the Wuppertal–Budapest and HotQCD collaborations. Since we constrain the model using lattice QCD data at small baryon chemical potential and high temperature—where quark pairing is not realized—pairing is not expected to affect the parameter fit. Furthermore, we note that the present study does not include the backreaction of the condensate on the gluon and flavor sectors. Consequently, at this level of approximation, the paired phase cannot influence the fit to lattice data.

Requested changes: 1) Figures that showcase comparisons between the V-QCD EoS from this novel framework and lattice QCD data are highly recommended to be included in the manuscript, similarly to what has been done in Ref. [119]

Reply: We agree with the referee that including a direct comparison of the lattice data with the V-QCD model used will improve the manuscript and make it more self-contained. We have therefore added a new Figure 1 in Section 2.3, which now shows a comparison of the interaction measure, pressure, and baryon number susceptibility at zero chemical potential as functions of temperature, as obtained from the V-QCD 7a model fit used throughout the manuscript, together with the corresponding lattice data from the Wuppertal–Budapest and HotQCD collaborations.