For our analysis of the properties of QCD at intermediate and also high densities, we employ the Wetterich equation [61] which is an RG equation for the quantum effective action Γ. Within this framework, the effective action depends on a so-called RG “time” t=ln(k/Λ) where k is the RG scale and Λ may be chosen to be the scale at which the initial condition Γk=Λ for the scale-dependent effective action Γk is fixed. In our present study, the initial condition is given by the classical (Euclidean) QCD action S for two massless quark flavors coming in three colors: S=∫d4x{14FμνaFμνa+ψ¯(i∂+g¯A-iγ0μ)ψ}.(1)Here, g¯ is the bare gauge coupling and μ is the quark chemical potential. For the values of μ considered in this work, we choose Λ≫μ to ensure that the RG flow is initialized in the perturbative high-energy regime. The gluon fields Aμa come with Lorentz (greek letters) and color (roman letters) indices and enter the definition of the field-strength tensor Fμνa=∂μAνa-∂νAμa+g¯fabcAμbAνc (a=1,…,8). Moreover, they are coupled to the quark fields ψ via the quark-gluon vertex, see Eq. (1). Note that the quark fields ψ carry color and flavor components.

The quark-gluon vertex generates a plethora of interaction channels. With respect to studies of ground-state properties, quark self-interaction channels are of particular importance as they can be directly related to the order-parameter potential of QCD. More specifically, the quark-gluon vertex induces four-quark interactions already at the one-loop level via two-gluon exchange. Schematically, this leads to corrections of the effective action of the following form: ΔΓ=∫d4x∑iλ¯i(ψ¯Oiψ)2.(2)Here, Oi determines the color, flavor, and Dirac structure of the four-quark vertex. Note that, in contrast to low-energy model studies, the four-quark couplings λ¯i are not free parameters but generated from fundamental quark-gluon interactions, λ¯i∼g¯4. Higher quark self-interactions are parametrically suppressed at high momentum scales. For example, eight quark-interactions scale as ∼g¯8. However, following the RG flow from high to low momentum scales, such higher-order interaction channels then become increasingly important. In fact, in regimes where the symmetry is broken spontaneously, eight quark interactions determine the masses of bound states of two quarks. We shall come back to this below. In particular, we shall discuss the relevance of eight-quark interactions at different scales in Sec. IV, which may also provide useful information for the construction of low-energy models at intermediate and high densities.

Still, already an analysis of the RG flow of gluon-induced four-quark interactions in the pointlike limit (“zero-momentum projection”) can provide us with an important insight into the symmetry-breaking patterns over a wide range of temperatures and quark chemical potentials, see Ref. [62] for an introduction. In fact, this has been successfully demonstrated for QCD in the vacuum limit [63], at finite temperature [64,65], and over a wide range of chemical potentials [47]. In the latter study, it has been found within a Fierz-complete two-flavor setting that the scalar-pseudoscalar channel is most dominant at low densities, in accordance with full QCD RG-flows in the vacuum limit [66,67]. At large chemical potentials, which are at the heart of the present work, the diquark channel ∼(ψ¯bτ2εabcγ5Cψ¯cT)(ψdTCγ5τ2εadeψe) is then dynamically rendered the most dominant channel, suggesting the formation of a chirally symmetric diquark condensate associated with pairing of the two-flavor color-superconductor (2SC) type [47].¹

Here, τ2 is the second Pauli matrix and, in color space, it is summed over the totally antisymmetric tensor εabc. Moreover, we have introduced C=iγ2γ0.

Although studies of the RG flow of four-quark interactions in the pointlike approximation provide a deep insight into symmetry-breaking patterns and their dependence on external control parameters, they are restricted to scales k≥kSB, where the scale kSB is associated with spontaneous symmetry breaking, such as chiral symmetry breaking or U(1)V symmetry breaking. In such a setting, symmetry breaking is indicated by a specific four-quark channel approaching criticality associated with a divergence of the corresponding coupling at the scale kSB. Below this scale, the dynamics is governed by the formation of condensates. However, an analysis of the ground-state properties of QCD in this low-energy regime k<kSB requires to go beyond the pointlike limit and to resolve the momentum dependences of the quark correlation functions. Indeed, information on bound-state and condensate formation is encoded in the momentum structure of the quark correlation functions. Such momentum dependences can be conveniently resolved by employing a Hubbard-Stratonovich transformation of at least the most dominant four-quark interaction channel. For example, as demonstrated in Ref. [28], one may perform such a transformation of gluon-induced four-quark interactions at a given scale Λ0>kSB, which then gives access to the low-energy regime. However, this introduces a dependence of the effective action on the scale Λ0 which is reflected in an uncertainty for the results for low-energy observables, see Ref. [28] for a discussion in the context of dense QCD. The dependence on this artificial scale Λ0 can be removed by employing the so-called dynamical hadronization technique [68–74], see also Ref. [75] for recent developments regarding the study of quark composites. Loosely speaking, this technique implements continuous Hubbard-Stratonovich transformations of four-quark interactions in the RG flow and thereby allows us to continuously follow the RG flow from the classical QCD action at high-momentum scales down to the deep infrared regime which is governed by the formation of bound states and condensates. We shall apply this technique in the following.

The present work should be viewed as the next step in a series of studies [28,47,76–78]. However, we do not aim at quantitative studies of thermodynamic quantities and low-energy observables. We rather aim at setting the methodological stage for subsequent new quantitative computations in this series.

Let us now be specific and construct our ansatz for the scale-dependent effective action Γk which underlies our present study of dense QCD matter. As in our previous works, see, e.g., Ref. [47], we rely on the background field approach to gauge theories [79,80] within background covariant gauges and employ the background field approximation which has been worked out in detail for applications in perturbative as well as nonperturbative settings over many years by now, see, e.g., Refs. [81–90] and, for a recent detailed fRG review on this aspect, see Ref. [91]. In this approach, the so-called background field effective action inherits gauge invariance from gauge transformations of an auxiliary background field. The equivalence of this invariance with the actual physical gauge invariance follows from the on-shell background independence of this approach and the Slavnov-Taylor identities, where the background independence is encoded in Nielsen identities. With these identities, it can then be shown that the correlation functions associated with the background field are indeed related to elements of the S-matrix [92]. In fRG studies, however, the regulator functions for fields carrying a net color charge break gauge invariance explicitly and, as a consequence, the independence of the auxiliary background field is also lost. This eventually leads to modifications of the Slavnov-Taylor and the Nielsen identities. Note that the latter also monitor the difference of correlation functions associated with the background field and those associated with the fluctuation field. In general, the construction of a manifestly gauge-invariant effective action in the spirit of the background-field approach may therefore be nontrivial within the fRG framework. In the present work, we treat the gauge sector as developed and discussed in detail in Refs. [63,65,81,82,86]. More specifically, manifest gauge invariance of the solution in these studies is maintained by identifying the full gauge field with the background field in the RG flow. Thus, in the following, we assume that the background-field two-point function can be identified with the one of the fluctuation field in the flow, which is an approximation. For a treatment of the difference of these two quantities, we refer the reader to Ref. [93]. This approximation entails that the RG flow is no longer closed [94] and only some constraints imposed by the modified Slavnov-Taylor identities are satisfied. As in previous works [63,65,81,82,86], we shall assume that corrections due to this approximation are subleading, which is at least reasonable in the (semi-)perturbative regime above the symmetry breaking scale kSB. A detailed discussion of these issues can be found in Ref. [91]. In any case, the advantage of our present approach is that it equips us with a gauge-invariant approximate solution of the effective action.

Since we would like to study the RG flow from the perturbative high-momentum regime down to the low-energy regime governed by the formation of bound states of quarks, we basically employ a combination of the classical QCD action given in Eq. (1) and an ansatz for the low-energy sector associated with complex-valued scalar diquark fields Δa describing quark composites of the form ∼(ψbTCγ5τ2εabcψc): Γk=∫d4x{ψ¯b(iγμDμbc-iμγ0)ψc+ZΔ(DμcaΔa)(DμcbΔb)*+2μZΔ(Δa(D0abΔb)*-Δa*(D0abΔb))-4μ2ZΔΔa*Δa+12λ¯csc(ψ¯bτ2iεabcγ5Cψ¯cT)(ψdTCγ5τ2iεadeψe)+12ih¯(ψbTCγ5τ2Δaεabcψc)-12ih¯(ψ¯bγ5τ2Δa*εabcCψ¯cT)+m¯2Δa*Δa+λ¯Δ(Δa*Δa)2+14ZAFμνaFμνa}+ΔΓgf+ΔΓgh.(3)Here, Dμbc=∂μδbc-ig¯AμaTbca and a, b, c are color indices. We have suppressed flavor indices for readability. Note that we do not take into account the running of the wave function renormalization of the quark fields in our present exploratory study since it depends only mildly on the RG scale, at least at small densities [66,67,69,73,74,95,96].

The diquark (Δa*)/antidiquark (Δa) fields appearing in Eq. (3) transform as an antitriplet/triplet in color space. Note that we include only these fields as effective low-energy degrees of freedom. This is motivated by the fact that the diquark channel has been found to be the most dominant interaction channel for μ≳350  MeV in a Fierz-complete study of gluon-induced four-quark interaction channels [47]. Other four-quark channels, such as the scalar-pseudoscalar interaction channel associated with pion dynamics, have been found to be clearly subdominant in this regime, provided that the U(1)A symmetry is broken explicitly. The unspecified quantities ΔΓgf and ΔΓgh in Eq. (3) are the standard background-field gauge-fixing and ghost term, respectively. In all explicit calculations, we have restricted ourselves to Feynman gauge for convenience.

A few comments are still in order at this point: In this work, we are aiming at a study of dense strong-interaction matter. To this end, we employ the diquark field as an effective degree of freedom to analyze the properties of the ground state. Since the diquark field is not a color-neutral object, the dynamical generation of a finite expectation value of this field would break the SU(3) color symmetry and therefore gauge invariance. Of course, it is known that local gauge invariance cannot be broken [97]. Moreover, the diquarks are effective degrees of freedom which do not even need to be asymptotic states in the spectrum. In any case, in (color-)superconducting systems, the physics is governed by the formation of a gap in the spectrum of fermionic excitations at the Fermi surface and the existence of such a gap is a gauge-invariant statement. The description of the formation of this gap in the fermionic excitation spectrum in terms of a diquark condensate within a fixed gauge, which effectively breaks the gauge symmetry, is only a convenient choice to get access to the low-energy dynamics [2]. In this work, we expand the effective action in the quantity Δa*Δa (summation over a is tacitly assumed), which is a gauge-invariant object. The gap in the fermionic spectrum is also constructed from this quantity. In practice, we employ a homogeneous background for the expansion and eventually evaluate the flow equations on a specific background configuration. This configuration is chosen to point into the 3-direction in color space for convenience, which may possibly lead to a residual dependence of our results for the gap on this choice. In future studies, our presently employed convenient approach to study the physics of dense QCD matter may be “outperformed” by directly computing the full momentum dependence of fermonic correlation functions in a vertex expansion and searching for signatures of a gap in these quantities, without relying on the use of diquark fields as auxiliary degrees of freedom. However, this is beyond the scope of the present work. We add that, in principle, similar issues are encountered in the description of mass generation in the electroweak sector of the Standard Model [98–102].

Of course, by construction, our ansatz for Γk does not allow for a study of the transition from a color-superconducting phase at intermediate and high densities to a phase governed by spontaneous chiral symmetry breaking at low densities. Therefore, our present work focusses on the intermediate and high density regime. Note that a quantitative analysis of the regime associated with the aforementioned transition is anyhow complicated by the fact that many four-quark interaction channels have been found to be of roughly the same strength in this regime [47]. This suggests that the ground state of QCD may exhibit a very complicated structure in this transition regime. We add that, close to the nucleonic low-density regime, the dynamics may even be governed by quarkyonic matter [103].

Let us begin our discussion of the RG flow by explaining the structure of our ansatz for the scale-dependent effective action Γk in more detail. The initial condition for Γk at the scale k=Λ≫μ is assumed to be given by the classical QCD action (1). Therefore, the values of the four-quark coupling λ¯csc, the quark-diquark coupling h¯, the bosonic wave function renormalization factor ZΔ, and the four-diquark coupling λ¯Δ should be set to zero at the initial RG scale Λ.²

We also refer to Sec. III A for a discussion of the initial conditions and the RG flow at high momentum scales.

By lowering the RG scale, starting from k=Λ, the quark-gluon vertex generates four-quark self-interactions via two-gluon exchange. With respect to this type of interaction channels, we only take into account the diquark channel as discussed above. This channel is associated with the coupling λ¯csc in Eq. (3). Once generated, this four-quark interaction channel can be removed by mapping it onto a Yukawa-type quark-diquark interaction channel associated with the coupling h¯ and a term bilinear in the diquark fields associated with the term ∼m¯2. Essentially, this corresponds to performing a Hubbard-Stratonovich transformation at a given RG scale. In the next RG step, however, the four-quark interaction λ¯csc is regenerated by the quark-gluon vertex and the quark-diquark vertex. The regenerated four-quark channel can then again be removed by mapping it onto the quark-diquark interaction channel and the term bilinear in the diquark fields. Moreover, the running of the quark-diquark coupling h¯ and the parameter m¯2 receive additional contributions from, e.g., the running of the wave function renormalization ZΔ of the diquark fields. The latter is generated itself by the quark-diquark coupling h¯. Note that, once the diquark wave function renormalization is rendered finite, the diquarks become dynamical degrees of freedom in the RG flow. It is also important to add that higher-order diquark self-interaction terms are generated via the aforementioned quark-diquark interactions. In the following, we take into account diquark self-interactions up to the four-diquark channel which is associated with the coupling λ¯Δ in Eq. (3).³

Such diquark self-interaction channels can be related to higher-order quark self-interaction channels with a nontrivial momentum structure. For example, the four-diquark channel can be related to an eight-quark interaction channel.

Employing this technique, see Appendix B for details, we find the following coupled set of flow equations for the dimensionless renormalized curvature of the effective potential εμ=(m¯2-4ZΔμ2)ZΔ-1k-2, the renormalized four-diquark coupling λΔ=λ¯ΔZΔ-2, and the renormalized quark-diquark coupling h=h¯ZΔ-12: ∂tεμ=(ηΔ-2)εμ-8h2b(2,0)(μ˜,0)+2h2εμ(1+εμ)g4b(0,4)(A)(μ˜,0,ηA),(4)∂tλΔ=2ηΔλΔ+4h4b(4,0)(μ˜,0)+4λΔh2(1+εμ)g4b(0,4)(A)(μ˜,0,ηA),(5)∂th2=ηΔh2+163g2h2b(1,2)(A)(μ˜,0,ηA)+2(1+2εμ)g4b(0,4)(A)(μ˜,0,ηA),(6)where μ˜=μ/k is the dimensionless chemical potential and g2=g¯2ZA-1 is the renormalized strong coupling. Finally, the scale-dependence of the anomalous dimension of the diquark fields is governed by ηΔ=-∂tlnZΔ=8h2d(2,0)(μ˜,0).(7)Recall that the set of couplings associated with these equations span our ansatz (3) for the scale-dependent effective action Γk. The functions b(i,j) and d(i,j) are so-called threshold functions which correspond to one-particle irreducible (1PI) Feynman diagrams with i external bosonic and j external fermionic lines, respectively. In some cases, an additional superscript (A) appears which indicates that the associated diagram contains at least one gluon line. Since we restrict ourselves to the zero-temperature limit in this work, these functions only depend on the dimensionless chemical potential μ˜ in the absence of a diquark gap. In any case, the regularization scheme dependence is also encoded in these functions. In this respect, we note that we employ a scheme which allows us to integrate out fermionic fluctuations around the Fermi surface [78], see Appendix A for its definition and brief discussion of all threshold functions entering our present study. It should be emphasized that also the anomalous dimensions ηΔ and ηA=-∂tlnZA depend on the dimensionless chemical potential μ˜. The running of the strong coupling g and its relation to the wave function renormalization ZA of the gauge fields is discussed below.

By comparing the set of flow equations (4)–(7) with our ansatz (3) for the effective action, it becomes apparent that there is no flow equation for the four-quark coupling λ¯csc. With the aid of the aforementioned dynamical hadronization technique [68–74], the contributions to this coupling are continuously transformed into contributions to the flow of the quark-diquark coupling h and the curvature εμ, such that ∂tλ¯csc=0 for any value of k. The contributions to the flow of the four-quark coupling λ¯csc therefore appear in the flow equations for the quark-diquark coupling h and the curvature εμ. In particular, these contributions are associated with the terms ∼g4 in the flow equations (4) and (6) which originally stem from two-gluon exchange box diagrams appearing in the RG flow of four-quark couplings.⁴

It is indeed possible to recover the flow equation for the four-quark coupling λ¯csc at scales above the symmetry breaking scale kSB. To be more specific, we have λcsc=h2/(2εμ) for the dimensionless four-quark coupling which relates the dimensionless four-quark coupling to the quark-diquark coupling h and the curvature εμ. From this flow equation, we deduce that the RG flow at sufficiently large scales k is governed by the two fixed points of λcsc, provided the strong coupling g2 is sufficiently small, see Sec. III C. These fixed points can be translated into fixed points for the quark-diquark coupling h and the curvature εμ, see Ref. [62] for a general discussion of this aspect.

The set of flow equations (4)–(7) describes the dynamics at high-momentum scales where the curvature εμ of the effective potential is positive. In fact, as discussed above, we shall choose initial conditions such that εμ≫1, λΔ→0, and h2→0 for k→Λ. Quark self-interactions, which are mapped onto diquark self-interactions and quark-diquark interactions in our present setting, are initially only generated by two-gluon exchange ∼g4. Following the RG flow to smaller scales k, we find that the curvature εμ decreases and eventually becomes zero at a finite scale kSB, see also our discussion in Sec. III below. At the scale kSB, spontaneous U(1)V symmetry breaking sets in.

Below the symmetry breaking scale kSB, the curvature εμ of the effective potential becomes negative and a color-superconducting ground state is formed associated with the formation of a gap in the fermionic excitation spectrum. Note that the antisymmetric flavor structure of this color-superconducting ground state corresponds to a singlet representation of the global chiral group. This implies that the formation of such a ground state does not violate the chiral symmetry. In any case, for k≤kSB, it is convenient to switch from the set of flow equations (4)–(6) to a set in which the flow equation for the curvature εμ is replaced with a flow equation for the minimum |Δ0|2=∑a|Δ0,a|2. Recall that we expand the effective action in the quantity Δa*Δa (summation over a is assumed). For convenience, we shall choose Δ0,a=Δ0δa,3 (Δ0∈R) and use κ=ZΔ|Δ0|2k-2 to parametrize the flow of the position of the minimum of the effective action. The resulting set of flow equations for scales k<kSB then reads ∂tκ=-(ηΔ+2)κ+4h2λΔb(2,0)(μ˜,h2κ),(8)∂tλΔ=2ηΔλΔ+4h4b(4,0)(μ˜,h2κ)+4λΔh2g4b(0,4)(A)(μ˜,h2κ,ηA),(9)∂th2=ηΔh2+163g2h2b(1,2)(A)(μ˜,h2κ,ηA)+2g4b(0,4)(A)(μ˜,h2κ,ηA),(10)and ηΔ=8h2d(2,0)(μ˜,h2κ).(11)In addition to the dimensionless chemical potential μ˜, the anomalous dimensions ηΔ and ηA now also depend on the so-called (diquark) gap |Δgap| which appears in the propagator of the quarks. In our conventions, we have |Δgap|=hκk. Thus, the gap |Δgap| in the quark propagator is directly related to the minimum |Δ0|2.

The gauge sector enters our flow equations (4)–(11) only via the running of the strong coupling g which is governed by the following equation: ∂tg2=ηAg2.(12)Here, ηA can be decomposed into a pure gluonic contribution ηglue and a term ηq which contains the quark contributions [63–65]: ηA=-∂tlnZA=ηglue+ηq.(13)For the purely gluonic contribution ηglue, we employ the results from previous functional RG studies [64,65,86]. There, ηglue has been computed nonperturbatively within the background field formalism which also underlies our present work. The quark contribution ηq depends on the dimensionless chemical potential μ˜ and the diquark gap: ηq=14g2d(2,0)(A)(μ˜,h2κ).(14)In the limit μ˜→0 and h2κ→0, we have d(2,0)(A)(0,0)=2/(3π2) and therefore ηq=g2/(6π2). This is nothing but the standard one-loop contribution of the quark fields to the running of the strong coupling g. We add that, in general, the running of the gauge coupling also receives corrections from quark self-interactions, such as four-quark interactions, see, e.g., Refs. [104,105]. However, within the fRG framework, it follows from an analysis of (modified) Ward-Takahashi identities that such back-reactions of the matter sector on the gauge sector are negligible, provided that the flow of the four-quark couplings is governed by the presence of fixed points [63–65]. At least above the symmetry breaking scale kSB, this is indeed the case in our present study (see also Sec. III A) which justifies that we do not take such contributions to the running of the gauge sector into account, see also Ref. [47]. For k<kSB, we shall neglect such contributions. Note that, in this regime, the situation is particularly involved anyhow because of the presence of a finite quark gap, as we shall discuss next and also in Sec. III below.

In our flow equations (4)–(14) we drop fluctuations of the diquark fields. Such fluctuation effects are associated with 1PI diagrams coming with at least one internal diquark line. Compared to the contributions that we take into account in our analysis, such contributions are subleading in an Nc-counting. Moreover, in the symmetric high-energy regime (i.e., for k>kSB), the fluctuation effects of the diquark fields are parametrically suppressed because of the large diquark mass parameter. In Sec. III, we shall see that this parameter is indeed large and only becomes small close to the symmetry breaking scale kSB. Such a suppression of fluctuation effects has already been observed and discussed in early fRG studies of chiral models in the zero-density limit [106–108].

In the regime k<kSB, which is governed by spontaneous symmetry breaking, it can no longer be argued that fluctuation effects are subleading. Whereas fluctuation effects are associated with, e.g., pion dynamics at low densities, a rigorous inclusion of fluctuations of the diquark fields at high densities requires to deal with an Anderson-Higgs-type mechanism [98–102] associated with the symmetry-breaking pattern SU(3)→SU(2) in color space (as the diquark fields carry a net color charge). As a consequence, only three of the eight gluons are massless. The remaining five gluons are effectively rendered massive by, loosely speaking, “eating up” Goldstone modes which appear in the diquark spectrum in the symmetry-broken regime, see, e.g., Ref. [6] for a review. In our present study, which mainly aims at setting the methodological stage for future more quantitative studies of dense QCD matter, we do not include this Anderson-Higgs-type mechanism but rather drop diquark fluctuations as mentioned above. A more quantitative study taking this Anderson-Higgs-type mechanism into account is deferred to future work. The general methodological groundwork for studies of this type of mechanism within the fRG framework has already been laid in studies of Abelian Higgs models [109,110] and (non-Abelian) gauged chiral Higgs-Yukawa models [111]. In any case, we shall at least estimate the effect of the appearance of the associated gap for the gluons on our present results in Sec. III below.

Let us now discuss our results for the RG flow of dense QCD matter, in particular those for the chirally symmetric (scalar) diquark condensate. To this end, we first need to specify the initial conditions of our RG flow equations at the UV scale k=Λ=10  GeV. This value of the initial scale ensures that we have Λ≫μ for all values of the quark chemical potential considered in the present work. For the dimensionless renormalized curvature εμ of the effective potential, we choose εμ=106. Thus, the diquark fields do not represent dynamical degrees of freedom at the UV scale Λ. We add that the limit εμ→∞ corresponds to the limit of a vanishing diquark wave function renormalization, ZΔ→0.

For the quark-diquark coupling h, we choose h=0.1 at k=Λ. The initial value of the four-diquark coupling λΔ is set to zero. This choice for the couplings at the scale Λ ensures that we indeed initialize the flow “in the vicinity of” the QCD action in the UV limit.⁵

We add that a finite value of the quark-diquark coupling h explicitly breaks the U(1)A symmetry. As discussed in, e.g., Ref. [47], this is required to render the four-quark coupling λcsc associated with the diquark channel ∼(ψ¯bτ2εabcγ5Cψ¯cT)(ψdTCγ5τ2εadeψe) to be most dominant at high densities [47].

From this discussion it follows that the initial value of the strong coupling is the only input parameter in our calculations. It sets the scale for all dimensionful quantities. In our present study with two massless quark flavors, we choose α=g2/(4π)=0.179±0.004 at the UV scale Λ=10  GeV which corresponds to the experimental value α=0.330±0.014 at the τ-mass scale [112].⁷

Note that the running of the strong coupling entering our calculation is compatible with the standard MS¯ running over a wide range of scales [64,65,86].

In Fig. 1, we show the RG flow of the renormalized dimensionless curvature εμ for k≥kSB (left panel), the diquark gap Δgap for k≤kSB (left panel), the (squared) quark-diquark coupling h2 (right panel), the four-diquark coupling λΔ (right panel), and the strong coupling α=g2/(4π) (right panel) over a wide range of scales for μ/ΛQCD=2. In this case, we have kSB/ΛQCD≈1.09. The gray (vertical) dashed lines in the two panels represent the point in the RG flow where k=μ. The black (vertical) dashed lines are associated with the scale k=km. The latter is an estimate for the scale at which the screening masses of the gluons exceed the scale k.⁸

For simplicity, we do not distinguish between the electric and magnetic masses.

For k>km, gluon screening effects are parametrically suppressed since mg/k<1. Note that effects associated with the quark chemical potential appearing in the quark propagator are even more suppressed, μ/k<mg/k<1. In this high-energy regime, we therefore do not expect that our results suffer significantly from the fact that we have neglected the gluon screening masses in our calculations. For these scales, the RG flow of the couplings is mainly driven by gluon exchange diagrams. Following the RG flow toward smaller scales, the strong coupling increases (see right panel of Fig. 1) and gauge fluctuations tend to drive the system toward a ground state associated with a (spontaneously) broken U(1)V symmetry. However, it should be noted that strong gauge fluctuations are in principle not required to trigger the formation of a (color-)superconducting ground state because of the presence of a Cooper instability in the system,⁹

This is different for chiral symmetry breaking which requires the gauge coupling to become sufficiently large, see, e.g., Refs. [47,62–65] for a detailed discussion.

From this line of arguments it is already clear that gluon screening effects become relevant at some point in the RG flow toward the infrared regime. To be more specific, we expect that the presence of gluon screening masses affects the dynamics for kSB<k<km. In this regime, contributions to the RG flow with at least one internal gluon line start to become parametrically suppressed since we have mg/k>1. This reduces the aforementioned “catalyzing effect” of the gluons and presumably leads to a shift of the symmetry breaking scale kSB and the gap Δgap to smaller values compared to the ones obtained in our present study. A detailed analysis of this aspect will be given elsewhere. In any case, we expect that their inclusion will not significantly alter the value of the symmetry breaking scale kSB or the dynamics for kSB<k<km, at least for sufficiently small values of the chemical potential. For example, for μ/ΛQCD=2, we have kSB/ΛQCD≈1.1 and km/ΛQCD≈2.2. Consequently, gluon screening effects are expected to be relevant only in a comparatively small regime above the symmetry breaking scale.

For increasing chemical potential, we find that the symmetry breaking scale kSB increases but only mildly, see also Fig. 2 and our discussion in Sec. III C below. In any case, a change in the hierarchy of scales sets in for increasing μ, where we eventually have kSB<km<μ. The quark dynamics is now strongly affected by the presence of the chemical potential over a wide range of scales. For μ/ΛQCD=10, for example, we have kSB/ΛQCD≈1.6. The gluon screening masses are smaller than the quark chemical potential over a wide range of scales within the regime kSB<km<μ. Nevertheless, these screening masses increase roughly linearly when μ is increased. For a given scale k, this suggests a stronger (parametric) suppression of gluonic contributions to the RG flow at large chemical potential than at small chemical potential. In other words, gluon screening effects may have a stronger impact on the RG flow over a wider range of scales when the chemical potential is increased. Correspondingly, the aforementioned “catalyzing effect” of the gauge degrees of freedom is expected to be reduced. Therefore, it is reasonable to expect that our estimates for the symmetry breaking scale kSB and the diquark gap become less reliable for large chemical potentials. In fact, after the conventional BCS-type increase of kSB for small chemical potentials, this suggests that gluon screening effects may potentially even lead to a decrease of kSB over some range of quark chemical potentials. For chemical potentials beyond those considered in this work, however, it is known that the diquark gap increases again as a function of the chemical potential [43], see also Refs. [44,45,52,54,56,116] for a discussion of the relevance of gluon screening effects.

Let us now turn to the regime k<kSB associated with spontaneous U(1)V breaking. In this regime, the situation is even more involved as it requires to deal with an Anderson-Higgs-type mechanism [98–102] associated with the breaking of the SU(3) symmetry in color space down to a SU(2) symmetry. This eventually leads to the generation of “gaps” (screening masses) for five of the eight gluons. A rigorous treatment of this mechanism is beyond the scope of the present work. We only consider two approximations in the low-energy regime k<kSB to already gain some understanding of the effect of gluon screening in the long-range limit.

In the first approximation, we simply leave the gluons ungapped for k<kSB. The corresponding results for the RG flow of the diquark gap Δgap and the various couplings are depicted by the dashed lines in Fig. 1 (and also in Fig. 2). In the second approximation associated with the solid lines for k<kSB in Fig. 1 (and also in Fig. 2), we decouple the gluon contributions from the RG flow of the matter sector, which may be viewed as adding an “infinite gap” to all gluons. In practice, we have implemented this decoupling by setting the gauge coupling to zero for k<kSB. Comparing the corresponding results with the ones for the ungapped gluons, we observe that the diquark gap is reduced by roughly a factor of two. As already discussed above, ungapped/unscreened gluons indeed act as a “catalyzer” for the formation of a (color-) superconducting ground state. This remains also true when the chemical potential is increased, see Fig. 2. For the renormalized quark-diquark coupling h and the renormalized four-diquark coupling λΔ, we observe a similar behavior. The couplings receive a significant boost in the approximation with ungapped gluons, see the right panels of Figs. 1 and 2.

Our results for the diquark gap Δgap, the symmetry breaking scale kSB, the quark-diquark coupling h, and the four-diquark coupling λΔ in the limit k→0 as a function of the quark chemical potential are summarized in Fig. 2. The (shaded) bands in Fig. 2 result from a variation of the strong coupling at the initial RG scale, see Sec. III A. Note that the variation of our results arising from a variation of the regularization scheme (associated with regulator functions) is negligible compared to the one obtained from the aforementioned variation of the initial value of the strong coupling. We refer the reader to Appendix A for the definition of the regulator functions employed in the present work and a corresponding discussion of the scheme dependence.

The observed dependence of the symmetry breaking scale kSB on the chemical potential appears consistent with the standard BCS-type scaling behavior. This is true for the case with ungapped gluons in the low-energy regime and for the case with decoupled gluon contributions as associated with (infinitely) “gapped” gluons. However, the diquark gap in the case with ungapped gluons is found to be significantly greater than the one obtained in our calculations with “gapped” gluons.

Let us now analyze the scaling behavior of the symmetry breaking scale and the diquark gap in more detail. To this end, it is convenient to reconstruct the RG flow of the four-quark coupling λcsc=h2/(2εμ) from the RG flows of the quark-diquark coupling h and the curvature εμ of the effective potential. Employing Eqs. (4) and (6), we then find ∂tλcsc=2λcsc+16λcsc2b(2,0)+163λcscg2b(1,2)(A)+g4b(0,4)(A).(15)Setting λcsc=0 (corresponding to εμ≫h2) at the initial RG scale k=Λ≫μ, the RG flow of λcsc is then dominated by the contributions ∼g4 associated with two-gluon exchange diagrams. All the other contributions to the flow of this coupling are initially subleading. Thus, we are left with ∂tλcsc=2λcsc+g4b(0,4)(A),(16)where b(0,4)(A)<0 for μ/k→0, see Appendix A.

Let us now define a scale k¯ such that the dependence of the two-gluon exchange diagrams on the chemical potential is negligible for k>k¯. In this regime, the flow equation (16) can be solved analytically. Integrating Eq. (16) from k=Λ down to k=k¯, we find λcsc(k¯)=-12b(0,4)(A)g4(k¯)+O(g6).(17)Here, we dropped terms which are subleading for Λ≫k¯.

We shall now also assume that k¯ can be chosen such that, at this scale, the gluon-induced four-quark self-interactions ∼λcsc have become strong enough to “dominate” their own RG flow. For sufficiently small values of the chemical potential μ, it may indeed be possible to choose k¯ such that the approximations underlying the derivations of Eqs. (16) and (17) are still at least reasonable. For k<k¯, the flow equation (15) of the four-quark coupling λcsc then reduces to ∂tλcsc=2λcsc+16λcsc2b(2,0),(18)where b(2,0)<0 for μ/k→0, see Appendix A.

The initial condition for the flow equation (18) at k=k¯ is given by Eq. (17). Note that b(2,0) is associated with a purely fermionic one-loop diagram with only two internal fermion lines and four external fermion lines.

From the flow equation (18) we can now obtain an estimate for the symmetry breaking scale kSB. Indeed, this scale is defined as the scale at which the curvature εμ of the effective potential becomes zero, i.e., the four-quark coupling λcsc=h2/(2εμ) diverges at this scale. Thus, we have 1/λcsc(kSB)=0. Next, we note that, for k<k¯, the flow eventually enters a regime where μ/k>1. In this regime, the loop diagram ∼λcsc2 scales as b(2,0)∼-cψ(μ2/k2) with cψ>0 being a dimensionless scheme-dependent constant.¹⁰

Note that the general dependence of this four-quark interaction on μ is scheme-independent, at least for μ/k≫1, see also Ref. [76] for a discussion.

We emphasize that our result for the dependence of the symmetry breaking scale kSB on the strong coupling differs from the one reported in, e.g., Refs. [50,117,118], see also Ref. [43]. In these seminal studies, it was found that Δgap∼kSB∼exp(-c¯′/(g2μ2)), where c¯′ is a positive constant. This g2-dependence is a consequence of the assumption λcsc∼g2. Basically, the latter can be traced back to a tree-level consideration of four-quark interactions as triggered by a one-gluon exchange. In our present work, we have taken into account loop contributions to λcsc which then alter the dependence of kSB on the strong coupling as given in Eq. (19). Starting from small chemical potentials, this change in the dependence of kSB on the strong coupling potentially induces a more rapid increase of the diquark gap Δgap when the chemical potential is increased. In any case, the scaling behavior (19) is only valid for sufficiently small values of the quark chemical potential, as discussed above. For very large chemical potentials, the diquark gap is eventually expected to increase mildly according to Δgap∼μexp(-c¯′′/g) (where c¯′′>0 is a constant) [43], such that Δgap/μ still decreases, see Ref. [6] for a detailed discussion of the diquark gap at very high densities.

We now turn to a more quantitative comparison of our present results with already existing results for the diquark gap. Of course, a direct comparison is difficult as it in principle requires to consider the diquark gap as a function of the density. Bearing this in mind, a comparison of results for the diquark gap as a function of the quark chemical potential can nevertheless be valuable to gain at least a qualitative understanding of the underlying dynamics.

To be specific, let us compare our present results for the diquark gap obtained from the computation with “gapped” gluons with those from one of the early seminal model studies in this field [48] and our recent results [28], see orange band in Fig. 2.¹²

Here, we restrict ourselves to the case with “gapped” gluons since gluonic contributions are expected to be (partially) suppressed in the low-energy regime as a consequence of the Anderson-Higgs mechanism anyhow.

In the regime associated with diquark-channel dominance, the results for the diquark gap from the aforementioned Fierz-complete calculation (see Ref. [28]) and our present study are remarkably consistent. Note that, in Ref. [28], the “transition” between the high-energy degrees of freedom and the effective low-energy degrees of freedom has been performed at a fixed scale Λ0. In principle, this scale should even carry a μ-dependence which is however at least difficult to determine a priori. In any case, the presence of this scale introduces a systematic uncertainty in the results, as indicated by the width of the orange band. We emphasize that we have removed the dependence on the scale Λ0 in our present work by implementing the dynamical hadronization technique. In the regime associated with a diquark-channel dominance, where a direct comparison of the two studies is most meaningful, we observe that the use of this technique already pays off. Indeed, the presence of the scale Λ0 in Ref. [28] also limits the range of accessible quark chemical potentials, μ≲Λ0. Since the transformation of high-energy degrees of freedom into low-energy degrees of freedom is performed continuously in our present work, the range of chemical potentials is only constrained by the requirement that the chemical potential should be sufficiently smaller than the initial RG scale Λ.

Let us now turn to a discussion of the IR values of the quark-diquark coupling h and the four-diquark coupling λΔ which often play an important role in the construction of low-energy models of dense QCD matter.

From the right panel of Fig. 2, we deduce that the quark-diquark coupling and the four-diquark coupling are smaller in the approximation with “gapped” gluons in the low-energy regime than in the approximation with ungapped gluons. However, their qualitative behavior as a function of the chemical potential is the same in the two approximations. Indeed, we observe that these two couplings decrease with increasing chemical potential in both cases. This simultaneous decrease is in accordance with our observation that the size of the gap appears to “saturate” for increasing chemical potential, as also suggested by our analytic study of the scaling behavior of the gap Δgap∼kSB, see Eq. (19). In fact, a decrease of the four-diquark coupling λΔ with increasing chemical potential tends to “pull” the position of the minimum of the effective action to larger values. This change of the position of the minimum needs to be compensated by a corresponding decrease of the quark-diquark coupling h such that the gap Δgap “saturates” for increasing chemical potential. From a phenomenological standpoint, the behavior of the quark-diquark coupling and the four-diquark coupling suggests that interactions between quarks and diquarks as well as among diquarks themselves become weaker when the density is increased, indicating that QCD is effectively described by a state of weakly coupled color-superconducting matter at (very) high densities.

Of course, the actual values of the quark-diquark coupling and the four-diquark coupling depend on the regularization scheme as specified by the regulator function in our RG flow study. However, the widths of the uncertainty bands shown in Fig. 2 are essentially determined by the variation of the strong coupling at the initial RG scale. The uncertainty arising from a variation of the regulator function is found to be much smaller, see our discussion in Appendix A for details.

From the standpoint of model building, it may be beneficial to employ the results from our RG study to constrain existing low-energy models of dense QCD matter. In the following, we shall demonstrate this aspect by considering the following quark-diquark model: SLEM=∫d4x{ψ¯a(i∂-iμγ0)ψa+12λ¯csc-1Δ¯a*Δ¯a+λ¯Δh¯4(Δ¯a*Δ¯a)2+12i(ψbTCγ5τ2Δ¯aεabcψc)-12i(ψ¯bγ5τ2Δ¯a*εabcCψ¯cT)},(20)where a, b, c are color indices and we have suppressed flavor indices for readability. The action SLEM basically represents a frequently employed low-energy model of dense QCD matter (for reviews, see Refs. [2–6]), except for the fact that we also allow for a four-diquark coupling. The inclusion of the latter is inspired by our RG study which suggests that four-diquark interactions are generated dynamically already at high scales. Therefore, such interactions should be expected to be present at scales of the order of the “hadronic” scale ΛLEM∼O(1  GeV) at which low-energy models are usually defined.

In the action SLEM defining our model, we have introduced the fields Δ¯a which are directly related to the diquark fields Δa in the ansatz (3) for the effective action underlying our fRG study. We have Δ¯a=h¯Δa. Since we shall assume that the quark-diquark coupling h¯ in our model (20) does not depend on the RG scale k, it is indeed convenient to rescale the original diquark fields in this way. In fact, Yukawa-type couplings such as the quark-diquark coupling are often treated as scale-independent quantities in low-energy model studies. In any case, the introduction of the fields Δ¯a allows us to identify the coefficient of the curvature term ∼Δ¯a*Δ¯a in Eq. (20) with the inverse of the four-quark coupling λ¯csc (up to a numerical factor), see our discussion of the relation of the curvature and the four-quark coupling in Sec. III C. Note that, by comparing the ansatz (3) for the effective action underlying our fRG study with the action SLEM of our low-energy model, we observe that the effective action (3) encompasses the action SLEM.

From a computation of the effective action ΓLEM associated with the action SLEM, we can in principle extract thermodynamic quantities which are relevant for phenomenological applications. However, this requires to fix the parameters of the model in the first place. In the following, we shall illustrate how this can be done in a mean-field study of ΓLEM. The derivation of the corresponding effective action can be found in Appendix C.

Let us start our discussion of the determination of the model parameters by considering the four-quark coupling λ¯csc in Eq. (20). Our analytic study of the four-quark coupling in Sec. III C [in particular, see the discussion of Eqs. (15)–(19)] suggests that this coupling depends only weakly on the chemical potential, provided that we consider RG scales which are sufficiently large compared to the chemical potential. This is in accordance with our numerical results where we observe that λ¯csc evaluated at scales sufficiently greater than the chemical potential shows only a very mild dependence on the chemical potential. Since we fix the model parameters at a scale ΛLEM>μ, we shall therefore assume that the parameter λ¯csc does not depend on the chemical potential. However, the value of the effective four-diquark coupling λ¯eff=λ¯Δ/h¯4 is assumed to depend on the chemical potential. The latter assumption is also in accordance with our RG results, see Fig. 3. The actual values of the model parameters λ¯csc and λ¯eff for a given value of the chemical potential are finally determined by tuning them such that we recover the value of the gap Δgap as obtained in our RG study. We emphasize again that we only consider λ¯eff to be μ-dependent. The value of the four-quark coupling λ¯csc remains constant for all values of the chemical potential considered below.

In the following we choose ΛLEM=1  GeV(≈4.8ΛQCD) which enables us to cover a reasonably large range of chemical potentials. For the four-quark coupling λ¯csc, we choose λ¯csc-1≈0.197  GeV2 (for all chemical potentials considered here). For a given value of the chemical potential, the model parameter λ¯eff is then determined by tuning it such that the value of the gap in our model study agrees with the one found in our RG study. Here, we focus on the results for the gap as obtained in the approximation with “gapped” gluons in the low-energy regime. However, we shall also comment on the case of ungapped gluons below.

In Fig. 3, we show the model parameter λ¯eff as a function of the chemical potential. There, we also present our fRG results for this quantity as obtained from an evaluation of the RG flow at the characteristic model scale k=ΛLEM. From this we deduce that the dependence of the model parameter on the chemical potential is compatible with our fRG results. Indeed, in both cases, we observe an increase with increasing chemical potential. Note that this is also the case when we evaluate the RG flow at lower scales. Finally, we add that a larger value of the four-quark coupling λ¯csc requires to choose larger values of λ¯eff to ensure that the gap Δgap as a function of the chemical potential remains unchanged.

We now use our QCD-constrained model to estimate the speed of sound of dense QCD matter. To this end, we first consider the pressure P as obtained from the effective action ΓLEM evaluated at the ground state (gs): P=-1V4ΓLEM|gs,μ+P0.(21)Here, V4 is the spacetime volume. The determination of the (vacuum) constant P0=(1/V4)ΓLEM|gs,μ=0 requires to compute the ground state in the vacuum. In QCD, the ground state is governed by spontaneous chiral symmetry breaking in the low-density regime. Since we only take into account diquarklike interaction channels (which have been found to be most dominant at high densities [28,47]), the low-density regime is not reliably accessible in our present study. However, at higher densities, derivatives of the pressure with respect to the chemical potential are accessible. A phenomenologically relevant quantity of this kind is the speed of sound cs: cs=1μ(∂P∂μ)12(∂2P∂μ∂μ)-12.(22)By solving the baryon density n, n=13∂P∂μ,(23)for the chemical potential μ, we can then compute the speed of sound as a function of the density.

In Fig. 4, we compare the speed of sound squared as a function of the density as obtained from our QCD-constrained model with results from a previous fRG study and calculations based on chiral EFT interactions at low densities. The green-shaded band associated with our model study originates from the uncertainty in the gap, see Fig. 2. Starting at high densities, we find that the speed of sound increases with decreasing density. In particular, the speed of sound is found to be greater than the one of the noninteracting quark gas in the considered density regime. Note that our present estimate for the speed of sound is in reasonable agreement with the one from Ref. [28] for n/n0≳7. This is essentially the density regime where the diquark interaction channel has been found to be most dominant in a Fierz-complete study [28,47]. For lower densities, the dynamics is governed by chiral interaction channels and therefore this regime is not accessible in our present analysis. Still, the behavior of the speed of sound at high densities observed in our present study and the one found at low(er) densities in Ref. [28] (chiral EFT and fRG) suggests the existence of a maximum in the speed of sound for n/n0≲10. Of course, a more accurate determination of the speed of sound from the full fRG flow presented in this work—rather than from our “QCD-constrained model”—is in order and will be presented elsewhere. Based on our previous studies [28,47], such a calculation then also requires the inclusion of the chiral dynamics.

We rush to add that we have also analyzed the dependence of our results on our choice for the model parameters. Indeed, we have some freedom in the model parameters since we adjust two parameters, λ¯csc and λ¯eff, to reproduce one quantity, namely the gap. Importantly, we find that the dependence on the actual choice for the parameters λ¯csc and λ¯eff is only mild and does not alter the qualitative behavior of the speed of sound as a function of the density, provided that the parameters are tuned such that the gap remains unchanged as a function of the chemical potential.

A change of the size of the gap as a function of the chemical potential affects the speed of sound. For example, the model parameters can also be adjusted such that we recover the gap obtained in the fRG calculations with ungapped gluons in the low-energy regime, which is significantly greater than the one found in the approximation with gapped gluons (see Fig. 2). This results in an increase of the speed of sound squared of up to 70% toward the lower end of the considered density range. However, the qualitative dependence of the speed of sound (squared) as a function of the density is not altered, i.e., it still increases when the density is decreased.

The robustness of our results for the speed of sound with respect to a variation of the model parameters becomes at least plausible by considering the weak-coupling limit of the effective action which is analytically accessible. In this limit of weak four-quark and four-diquark coupling, the pressure reads [2,77,122,123]: P=PSB(1+2|Δgap|2μ2+…).(24)Here, PSB=μ4/(2π2) is the pressure of the noninteracting quark gas,¹³

Gluons do not contribute to PSB in the zero-temperature limit.

Plugging now our fRG results for, e.g., the gap obtained in the approximation with gapped gluons into the expression (24) for the pressure, we can estimate the speed of sound with the aid of Eq. (22). Recall that the gap in our RG study is generated from the fundamental quark-gluon dynamics and it therefore depends on the strong coupling g. This is made explicit in Eq. (19). In any case, reassuringly, we find again that the speed of sound exceeds the value of the noninteracting quark gas and increases when the density is decreased, see Fig. 4. The width of the associated black-shaded band in Fig. 4 results from the width of the band of the gap shown in Fig. 2. Note that we have 0.3≲|Δgap|/μ≲0.6 in the considered density range.

It is also worth adding that the observed behavior of the speed of sound as a function of the density has not been observed in fRG calculations which do not take into account the formation of a gap at high densities, see Ref. [28] and Fig. 4 for an illustration.

In accordance with Ref. [28], we therefore cautiously conclude from our analysis that the appearance of a maximum in the speed of sound—which exceeds the value of the noninteracting quark gas—appears to be tightly connected to the formation of a diquark gap. Our present analysis suggests that the maximum appears in the regime n/n0≲10 for isospin-balanced QCD matter, although the determination of its exact position requires additional more advanced studies, as already indicated above. With respect to astrophysical applications, it is still worth mentioning that the analysis of constraints from neutron-star masses also strongly suggests the existence of a maximum in the speed of sound for neutron-rich matter [24,57–60]. In any case, our present findings may already provide useful information for future studies of thermodynamic quantities at supranuclear densities and also for the further development of existing models of dense QCD matter.