Fierz-complete NJL model study III: Emergence from quark-gluon dynamics

Our understanding of the dynamics and the phase structure of dense strong-interaction matter is to a large extent still built on the analysis of low-energy models, such as those of the Nambu–JonaLasinio-type. In this work, we analyze the emergence of the latter class of models at intermediate and low energy scales from fundamental quark-gluon interactions. To this end, we study the renormalization group flow of a Fierz-complete set of four-quark interactions and monitor their strength at finite temperature and quark chemical potential. At small quark chemical potential, we find that the scalar-pseudoscalar interaction channel is dynamically rendered most dominant by the gauge degrees of freedom, indicating the formation of a chiral condensate. Moreover, the inclusion of quark-gluon interactions leaves a significant imprint on the dynamics as measured by the curvature of the finite-temperature phase boundary which we find to be in accordance with lattice QCD results. At large quark chemical potential, we then observe that the dominance pattern of the fourquark couplings is changed by the underlying quark-gluon dynamics, without any fine-tuning of the four-quark couplings. In this regime, the scalar-pseudoscalar interaction channel becomes subleading and the dominance pattern suggests the formation of a chirally symmetric diquark condensate. In particular, our study confirms the importance of explicit UA(1) breaking for the formation of this type of condensate at high densities.

Our understanding of the dynamics and the phase structure of dense strong-interaction matter is to a large extent still built on the analysis of low-energy models, such as those of the Nambu-Jona-Lasinio-type. In this work, we analyze the emergence of the latter class of models at intermediate and low energy scales from fundamental quark-gluon interactions. To this end, we study the renormalization group flow of a Fierz-complete set of four-quark interactions and monitor their strength at finite temperature and quark chemical potential. At small quark chemical potential, we find that the scalar-pseudoscalar interaction channel is dynamically rendered most dominant by the gauge degrees of freedom, indicating the formation of a chiral condensate. Moreover, the inclusion of quark-gluon interactions leaves a significant imprint on the dynamics as measured by the curvature of the finite-temperature phase boundary which we find to be in accordance with lattice QCD results. At large quark chemical potential, we then observe that the dominance pattern of the fourquark couplings is changed by the underlying quark-gluon dynamics, without any fine-tuning of the four-quark couplings. In this regime, the scalar-pseudoscalar interaction channel becomes subleading and the dominance pattern suggests the formation of a chirally symmetric diquark condensate. In particular, our study confirms the importance of explicit UA(1) breaking for the formation of this type of condensate at high densities.

I. INTRODUCTION
Low-energy models of the theory of the strong interaction (Quantumchromodynamics, QCD) are still considered very valuable for a variety of reasons. In particular in the high-density regime, which is at least difficult to access with lattice Monte Carlo techniques, the Nambu-Jona-Lasinio (NJL) model [1,2] and its various variations and relatives (see, e.g., Refs. [3][4][5][6][7][8] for reviews), such as quark-meson (QM) models, allow us to gain some insight into the plethora of symmetry breaking patterns that may potentially be realized in this regime, see Refs. [9][10][11][12] for reviews. From a phenomenological point of view, this regime attracts significant interest from the astrophysics side. For example, studies of neutron stars require a knowledge of the equation of state of strong-interaction matter as input. However, the latter are currently still plagued by (significant) uncertainties at least at high density, see, e.g., Ref. [13] for a recent review. In order to constrain the equation of state further [14], we eventually need a better understanding of the symmetry breaking patterns of QCD guided by first-principle approaches.
In two preceding works [15,16], we have studied the relevance of Fierz completeness of four-quark selfinteractions in NJL-type models at finite temperature and quark chemical potential. Our beyond mean-field renormalization-group (RG) analysis of the "hierarchy" of the various interaction channels in terms of their relative strengths allowed us to gain insight into the symmetry breaking patterns and the structure of the ground state. Particularly at high density, we found the aspect of Fierz completeness to be of great importance, leading to an increased phase transition temperature as compared to conventional NJL model studies. At least naively, this observation might have crucial implications for the prop-erties of cold dense quark matter at low temperatures as an increase of the critical temperature may point to an increase of the size of the gap in the low-energy spectrum.
The four-quark couplings appearing in the ansatz of NJL-type models are usually considered as fundamental parameters. In fact, owing to the non-renormalizability of NJL-type models in four space-time dimensions, both on the perturbative as well as on the non-perturbative level (see, e.g., Refs. [17,18]), the ultraviolet (UV) cutoff scale Λ becomes a parameter of the model, too. Against this background, the regularization scheme becomes also part of the definition of the model. The initial values of the four-quark couplings are then chosen such that a given set of low-energy observables is reproduced in the vacuum limit. In Ref. [16], guided by the findings of RG studies of QCD [19][20][21] and in order to relate to conventional NJL model studies, we have set all fourquark couplings to zero initially, except for the scalarpseudoscalar coupling. As the only remaining parameter, we adjusted this coupling to fix low-energy observables in the vacuum limit. However, this scale fixing procedure underlying many low-energy model studies can be potentially problematic. The distinct role of the scalarpseudoscalar interaction channel at the initial UV cutoff scale can be indeed questioned since a specific four-quark interaction channel is reducible by means of Fierz transformations. Yet adopting more complex initial conditions by also taking into account four-quark couplings other than the scalar-pseudoscalar interaction channel may suffer from the fact that the parameters cannot be uniquely determined by a given set of low-energy observables. Indeed, the values of the low-energy observables may in general be reproduced by various different parameter sets or certain parameters might be even left undetermined at all. The remaining ambiguity has been found to affect the phase structure significantly, see, e.g., Refs. [9,[22][23][24]. Moreover, boundary conditions which are defined in the vacuum limit are possibly inappropriate for computations at finite external control parameters, such as temperature T and/or quark chemical potential µ. Considering NJL-type models to be rooted in QCD, the RG evolution of gluon-induced four-quark interactions in fact suggests a dependence of these model parameters on external control parameters [25]. In particular at finite quark chemical potential, as also discussed in detail in Refs. [15,16] (see, e.g., Refs. [9,11,12,26] for reviews), effective degrees of freedom associated with four-quark interaction channels other than the scalar-pseudoscalar interaction channel are expected to become important or even dominant. In such a situation, a choice for the initial conditions with a specifically tuned scalar-pseudoscalar coupling is therefore unfortunate as it may potentially bias the outcome in terms of symmetry breaking patterns along the finite-temperature phase boundary.
Thus far, we have not yet discussed the role of the UV cutoff scale. In the context of NJL-type models, we have to deal with the existence of a finite UV extent, i.e., the cutoff scale Λ is limited by a validity bound which in turn limits the model's range of applicability in terms of external parameters. The origin of this bound is actually twofold: First, NJL-type models eventually become unstable in the UV and develop a Landau pole at a certain scale. Second, these models have a phenomenological UV extent beyond which the description of the physics in terms of the models' effective fields becomes invalid and the knowledge of the underlying fundamental dynamics, i.e., quark-gluon dynamics, is ultimately required. As a consequence, a choice of the UV cutoff scale within the a priori unknown validity bound either limits the applicable range of external parameters or, for external parameters outside of this range, implies that the initial effective action is already a complicated object itself, see Ref. [27] for a detailed discussion of these issues. Considering NJL-type models to be embedded in QCD, a possibility to resolve this problem is the determination of the model parameters by employing RG studies of the fundamental theory, see, e.g., Refs. [25,28,29]. In the past 20 years, tremendous progress has been made within the functional RG framework in the development of a "topdown" approach to QCD, see, e.g., Refs. [20,21,. The only input in such an approach is given by the fundamental parameters of QCD, i.e., the current quark masses and the value of the strong coupling in the perturbative high-momentum regime. These functional RG studies are basically free of additional model parameters. In this context, even very good quantitative agreement of results from lattice QCD and functional RG studies has been demonstrated at zero and finite temperature for QCD with different flavor numbers, see, e.g., Refs. [21,35,37,38,50].
Whereas the aforementioned RG studies aiming at quantitative precision constitute essential advances towards predictive first-principle investigations of the QCD phase diagram, eventually even at high densities, we aim to take another important step toward such a top-down first-principle approach to analyze the phase structure of QCD at high densities with functional methods. With our analysis of the Fierz -complete NJL model with two quark flavors in Ref. [16], we have gained valuable insight into the quark dynamics. In the present work, we now incorporate gluodynamics by extending our Fierz -complete ansatz with dynamical gauge degrees of freedom, following earlier functional RG studies [36,37,39,45]. In full QCD, the values of the four-quark couplings are no longer fundamental parameters since these self-interactions are fluctuation-induced by the dynamics of the gauge fields. Taking this aspect into account, the aforementioned issue associated with the determination of model parameters -such as ambiguities related to the possibility to Fierz -transform given initial conditions and the potential existence of more than one parameter set reproducing equally well a given set of low-energy observables, or the dependence of the initial conditions on external control parameters -can in principle be resolved. More specifically, including gauge dynamics and thus resolving the fundamental microscopic degrees of freedom allows the initialization of the RG flow at a large scale Λ associated with the perturbative regime, which effectively corresponds to starting in the vacuum as we have T /Λ 1 and µ/Λ 1. In this way, the finite UV extent as implied by the validity bound of NJL-type models is surmounted and the limit on the range of applicability in terms of external parameters is lifted.
Working in the chiral limit, the strong coupling g s is the only parameter which is set at a the initial UV scale Λ. By integrating out fluctuations from this scale Λ on, the quark-gluon vertex gives rise to 1PI box diagrams with two-gluon exchange which dynamically generate the four-quark interaction channels. Depending on the strength of the strong coupling and the external parameters, the quark sector may then be driven to criticality, signaling the onset of spontaneous symmetry breaking, e.g., chiral symmetry breaking or diquark condensation of a specific type. Following the approach developed in our two preceding works [15,16], we shall consider the RG flow of the four-quark couplings in the pointlike limit to study the QCD phase structure at finite temperature and quark chemical potential. In particular, we analyze the "hierarchy" of the four-quark couplings in terms of their strength which shall prove very valuable in order to gain insight into the symmetry properties of the QCD ground state in the low-energy limit.
This work is organized as follows: In Sec. II, we discuss general aspects of the formalism and concepts underlying our study. We begin with a discussion of the relation of the quark-gluon vertex and four-quark interactions in Subsec. II A. In Subsec. II B, we then briefly review the relation of the RG flow of four-fermion couplings to the onset of phase transitions, including a discussion of the general structure of the RG flow equations for the fourquark couplings. The scale fixing procedure underlying our present work is discussed in Subsec. II C. The QCD phase structure and symmetry breaking patterns at finite temperature and density are then analyzed in Sec. III. There, we also compare the results for the phase boundary to the one obtained from our previous Fierz -complete NJL model study [16]. Moreover, we discuss the effect of explicit U A (1) symmetry breaking and comment on the curvature of the finite-temperature phase boundary at small chemical potential resulting from various different approaches. Our conclusions can be found in Sec. IV.

A. Quark-gluon vertex and four-quark interactions
In the present work, we employ the functional RG approach [51] to study the RG flow of QCD starting from the Euclidean QCD action (see Refs. [17,52,53] for reviews): whereḡ s is the bare gauge coupling and µ is the quark chemical potential. The gluon fields A a µ enter the definition of the field-strength tensor F a µν in the usual way. We emphasize that we exclusively consider the case of quarks coming in N c = 3 colors and N f = 2 flavors.
In the RG flow, the quark-gluon vertex generates quark self-interactions already at the one-loop level via twogluon exchange. This gives rise to contributions to the quantum effective action, e.g., of the following form: where the elements L j form a ten-component Fierzcomplete basis B of pointlike four-quark interactions to be specified below. The various terms are associated with corresponding bare couplingsλ i and vertex renormalization factors Z j . By construction, the couplings are not parameters of our calculation but solely generated by quark-gluon dynamics. This is an important difference to, e.g., NJL-type model studies where such couplings represent input parameters. In the following, we focus on the RG flow of pointlike projected four-quark correlation functions Γ (4) which eventually corresponds to a calculation of the effective action at leading order of the derivative expansion. To be specific, we define the four-quark couplings associated with the vertex of the form (2) as follows Here, α, β, γ, δ denote collective indices for color, flavor, and Dirac structures determined by a specific element L j of our Fierz -complete basis. We add that this zero-momentum projection does not represent a Silver-Blaze-symmetric point [15,54,55]. However, it matches the standard definition of four-quark couplings in conventional low-energy models (see Refs. [3,4,9,10,56] for reviews) and BCS-type models (see Refs. [11,12,26,57] for reviews).
Let us now specify the elements L j of our Fierzcomplete basis B of pointlike four-quark interactions. Since Poincaré invariance is explicitly broken in our calculations at finite temperature and quark chemical potential, we are only left with rotational invariance among the spatial components of the various possible channels. Moreover, a finite quark chemical potential also explicitly breaks the charge conjugation symmetry. Therefore, with respect to the fundamental symmetries associated with charge conjugation, time reversal, and parity, only invariance under parity transformations and time reversal transformations remain intact. Assuming finally invariance of the channels under SU (N c ) ⊗ SU L (2) ⊗ SU R (2) ⊗ U V (1), we end up with a Fierz -complete basis composed of ten elements [16]. Guided by QCD low-energy phenomenology, we choose four of the ten channels such that they are invariant under SU (N c ) ⊗ SU L (2) ⊗ SU R (2) ⊗ U V (1) transformations but break the U A (1) symmetry explicitly: where, e.g., ψ γ 5 τ i ψ 2 ≡ ψ γ 5 τ i ψ ψ γ 5 τ i ψ and the T a 's denote the generators of SU (N c ). Moreover, we introduced charge conjugated fields ψ C = Cψ T and ψ C = ψ T C with C = iγ 2 γ 0 being related to the charge conjugation operator. The remaining six channels can then be chosen to be even invariant under Of course, this basis is not unique. In principle, we can combine elements of the basis to perform a basis transformation. However, as indicated above, our present choice is motivated by the structure of the four-quark channels conventionally employed in QCD low-energy models. In fact, the scalar-pseudoscalar channel associated with pion dynamics and chiral symmetry breaking is given by the channel L (σ-π) . The channel associated with the element L (S+P )− can be rewritten as (up to a numerical constant) where the determinant is taken in flavor space. This channel is associated with the presence of topologically non-trivial gauge configurations violating the U A (1) symmetry. Indeed, such configurations can be recast into a four-quark interaction channel of the form (14) in the case of two-flavor QCD [58][59][60][61][62][63][64]. The channel (7) may be viewed as a version of the channel L (S+P )− with a non-trivial color structure. We add that, for the phenomenologically more relevant three-flavor case, the explicit breaking of the U A (1) symmetry has been suggested already before the two-flavor case in order to explain the mass splitting of the η and η in the mesonic mass spectrum [65,66]. Finally, we have included a channel which allows us to "measure" the status of the formation of a diquark condensate of the type which carries a net baryon and net color charge. 1 Note transformations and the corresponding condensate leaves the chiral symmetry intact. However, the formation of such a diquark condensate comes along with the breakdown of the U V (1) symmetry, as expected for a BCS-type condensate. 2 With respect to our discussion of the effect of U A (1) symmetry breaking, we add that we can use our Fierzcomplete set of pointlike four-quark interactions to monitor the strength of U A (1) symmetry breaking. Indeed, requiring that the effective action Γ is invariant under U A (1) transformations, we find the following two sum rules which are satisfied simultaneously if the U A (1) symmetry is intact: see Ref. [16] for a more detailed discussion. We shall come back to the issue of U A (1) symmetry breaking in Subsec. III B.
To summarize, from here on, we shall consider the RG flow of pointlike four-quark interactions as generated by 1 Here, f ≡ the quark-gluon vertex. The latter implies that the initial conditions of the four-quark interactions are set to zero at the initial RG scale Λ, i.e., they do not represent parameters of our study as it is the case for QCD low-energy models. We emphasize again that our set of four-quark interactions is Fierz -complete, i.e., any other pointlike four-quark interaction invariant under the symmetries specified above is reducible by means of Fierz transformations.
Since we consider the RG flow of the four-quark interactions at leading order of the derivative expansion (i.e., in the pointlike limit as indicated above), the quark wave-function renormalization factors do not receive contributions directly from the four-quark interactions, see Ref. [17] for a detailed discussion. Contributions to the quark wave-function renormalizations resulting from the coupling of the quarks to the gluons have been found to be small at this order in earlier studies [32,35,39], at least for RG scales relevant for the present work. Therefore, we do not take into account the running of the quark wave-function renormalization factors and defer it to future work. Finally, we note that quark self-interactions of higher order are also generated dynamically in the RG flow. However, at leading order of the derivative expansion, they do not contribute to the RG flow of the fourquark self-interactions and are thus not included in our present study. In any case, since the RG flow equations of the four-quark couplings depend on the strong coupling, we also need to take into account the running of the strong coupling which we shall discuss below.

B. Phase transitions and four-quark interactions
The four-quark couplings depend on the chemical potential, the temperature, and the RG scale k. Although the scale dependence implies that part of the information on the momentum dependence is still taken into account in our RG analysis in an effective manner [67], the pointlike approximation underlying our present work ignores relevant information of four-quark correlation functions. To be more specific, bound-state information is encoded in the momentum structure of the quark correlation functions. Therefore, our present approximation only allows us to study the symmetric high-energy regime [17] whereas the low-energy regime is only accessible in the absence of (spontaneous) symmetry breaking which is associated with bound-state formation, as it is the case at, e.g., high temperature. For our purposes, this is nevertheless sufficient as it enables us to study the approach towards the symmetry-broken low-energy regime. Indeed, symmetry breaking is ultimately triggered by a specific four-quark channel approaching criticality as indicated by a divergence of the corresponding coupling. Such a seeming Landau-pole-type behavior of four-quark couplings can be traced back to the formation of condensates since the pointlike four-quark couplings can be shown to be proportional to the inverse mass parame-ters of a generalized Ginzburg-Landau effective potential for the order parameters in a (partially) bosonized formulation [17,68,69]. Thus, if the size of all four-quark couplings is found to be bounded in the RG flow, the system remains in the symmetric regime [17,[35][36][37]67]. The observation of a divergent four-quark coupling for a given temperature and quark chemical potential may therefore be considered as an indicator that the orderparameter potential develops a non-zero ground-state expectation value in the direction associated with a specific four-quark channel. The critical temperature T cr (µ) at a given value of the quark chemical potential above which no spontaneous symmetry breaking occurs is then defined as the smallest temperature for which the four-quark couplings still remain finite in the infrared (IR) limit associated with k → 0 [15-17, 36, 37]. However, we emphasize that our present approach is only able to detect phase transitions of second order as the definition of the critical temperature is associated with a change from positive to negative curvature of the order parameter potential at the origin. In case of a first-order phase transition, a non-trivial minimum of the potential is formed but the curvature at the origin remains positive. Consequently, our criterion for the detection of a phase transition does not allow to detect first-order transitions. Still, it allows us to detect the line of metastability [15], see for the first NJL model analysis of this aspect [70]. In any case, the nontrivial assumption entering our analysis of the QCD phase structure in the present work is that it is possible to relate the dominance pattern of the four-quark couplings to the symmetry-breaking pattern in terms of condensates, see Refs. [15,16,67,71] for a detailed discussion of this aspect.
It is reasonable to expect that the symmetry-breaking patterns associated with the various four-quark channels change when we vary, e.g., the quark chemical potential. For example, the diquark channel may become more relevant than the scalar-pseudoscalar channel at high density. The most dominant channel can be identified by requiring that the modulus of the coupling of this channel is greater than the ones of all the other four-quark couplings. For such an analysis to be meaningful, it is therefore ultimately required to consider a Fierz -complete set of four-quark couplings.
For the derivation of the RG flow equations of the fourquark couplings, we employ the Wetterich equation [51] which is an RG equation for the quantum effective action Γ. Within this framework, the effective action Γ depends on the RG scale k which is related to the so-called RG "time" t = ln(k/Λ). Note that the scale k defines an infrared (IR) cutoff scale and Λ may be chosen to be the scale at which we fix the initial conditions of the RG flow of the couplings. The general structure of the RG flow equations for the dimensionless renormalized couplings λ i = Z i k 2λ i is given by

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A n e 6 p f X S b d R 9 6 7 q j Y d m r X V X 1 F G G M z i H S / D g G l p w D 2 3 o A I E E n u E V 3 p z U e X H e n Y / l a M k p d k 7 h D 5 z P H 5 l N k g 8 = < / l a t e x i t > g s < l a t e x i t s h a 1 _ b a s e 6 4 = " v g 2 R O S i n X c K M N S f X P j t B l F y T Q R g = " > A A A B 8 3 i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s z U g i 4 L I r i s Y B / Q G U o m z b S h S W Z I M k I Z + h t u X C j i 1 p 9 x 5 9 + Y a W e h r Q c C h 3 P u 5 Z 6 c M O F M G 9 f 9 d k o b m 1 v b O + X d y t 7 + w e F R 9 f i k q + N U E d o h M Y 9 V P 8 S a c i Z p x z D D a T 9 R F I u Q 0 1 4 4 v c 3 9 3 h N V m s X y 0 c w S G g g 8 l i x i B B s r + e O h L 7 C Z K J H p + b B a c + v u A m i d e A W p Q Y H 2 s P r l j 2 K S C i o N 4 V j r g e c m J s i w M o x w O q / 4 q a Y J J l M 8 p g N L J R Z U B 9 k i 8 x x d W G W E o l j Z J w 1 a q L 8 3 M i y 0 n o n Q T u Y J 9 a q X i / 9 5 g 9 R E N 0 H G Z J I a K s n y U J R y Z G K U F 4 B G T F F i + M w S T B S z W R G Z Y I W J s T V V b A n e 6 p f X S b d R 9 6 7 q j Y d m r X V X 1 F G G M z i H S / D g G l p w D 2 3 o A I E E n u E V 3 p z U e X H e n Y / l a M k p d k 7 h D 5 z P H 5 l N k g 8 = < / l a t e x i t > +µ < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 v q k s 3 w S K 4 c c j 0 Y a e 7 g g g u K 9 g H t E P J p J k 2 N J k Z k o x Q h v 6 C G x e K u P W H 3 P k 3 Z t o K K n r g w u G c e 7 n 3 H j / m T G m E P q z c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 6 q g o k Y S 2 S c Q j 2 f O x o p y F t K 2 Z 5 r Q X S 4 q F z 2 n X n 1 5 l f v e e S s W i 8 2 v q k s 3 w S K 4 c c j 0 Y a e 7 g g g u K 9 g H t E P J p J k 2 N J k Z k o x Q h v 6 C G x e K u P W H 3 P k 3 Z t o K K n r g w u G c e 7 n 3 H j / m T G m E P q z c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 6 q g o k Y S 2 S c Q j 2 f O x o p y F t K 2 Z 5 r Q X S 4 q F z 2 n X n 1 5 l f v e e S s W i 8

/ E S s H A E p q 4 X b B h H k A = " > A A A B 6 3 i c d V D L S g M x F M 3 U V 6
2 v q k s 3 w S K 4 c c j 0 Y a e 7 g g g u K 9 g H t E P J p J k 2 N J k Z k o x Q h v 6 C G x e K u P W H 3 P k 3 Z t o K K n r g w u G c e 7 n 3 H j / m T G m E P q z c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 6 q g o k Y S 2 S c Q j 2 f O x o p y F t K 2 Z 5 r Q X S 4 q F z 2 n X n 1 5 l f v e e S s W i 8 o n Q T u Y J 9 a q X i / 9 5 g 9 R E N 0 H G Z J I a K s n y U J R y Z G K U F 4 B G T F F i + M w S T B S z W R G Z Y I W J s T V V b A n e 6 p f X S b d R 9 6 7 q j Y d m r X V X 1 F G G M z i H S / D g G l p w D 2 3 o A I E E n u E V 3 p z U e X H e n Y / l a M k p d k 7 h D 5 z P H 5 l N k g 8 = < / l a t e x i t > g s < l a t e x i t s h a 1 _ b a s e 6 4 = " v g 2 R O S i n X c K M N S f X P j t B l F y T Q R g = " > A A A B 8 3 i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s z U g i 4 L I r i s Y B / Q G U o m z b S h S W Z I M k I Z + h t u X C j i 1 p 9 x 5 9 + Y a W e h r Q c C h 3 P u 5 Z 6 c M O F M G 9 f 9 d k o b m 1 v b O + X d y t 7 + w e F R 9 f i k q + N U E d o h M Y 9 V P 8 S a c i Z p x z D D a T 9 R F I u Q 0 1 4 4 v c 3 9 3 h N V m s X y 0 c w S G g g 8 l i x i B B s r + e O h L 7 C Z K J H p + b B a c + v u A m i d e A W p Q Y H 2 s P r l j 2 K S C i o N 4 V j r g e c m J s i w M o x w O q / 4 q a Y J J l M 8 p g N L J R Z U B 9 k i 8 x x d W G W E o l j Z J w 1 a q L 8 3 M i y 0 n o n Q T u Y J 9 a q X i / 9 5 g 9 R E N 0 H G Z J I a K s n y U J R y Z G K U F 4 B G T F F i + M w S T B S z W R G Z Y I W J s T V V b A n e 6 p f X S b d R 9 6 7 q j Y d m r X V X 1 F G G M z i H S / D g G l p w D 2 3 o A I E E n u E V 3 p z U e X H e n Y / l a M k p d k 7 h D 5 z P H 5 l N k g 8 = < / l a t e x i t > with τ = T /k andμ τ = µ/(2πT ). The temperature-and chemical-potential-dependent coefficients A  Fig. 1. Terms proportional to λ j g 2 s are generated by the triangle diagrams (b) depicted in Fig. 1. Finally, terms proportional to g 4 s are associated with the box diagrams (c) shown in Fig. 1. We add that we have dropped an explicit dependence of these loop diagrams on the anomalous dimensions of the quark and gluon fields as these additional contributions have been found to be subleading in the symmetric regime [32,[35][36][37]72].
For the derivation of the set of RG flow equations (18), we have made extensive use of existing software packages [73,74]. Because of the size of the resulting system of equations for the Fierz -complete set of four-quark couplings, we dealt with the flow equations only numerically and therefore refrain from listing these equations explicitly here. An explicit representation of the flow equations for the purely fermionic part as parameterized by the matrices A (i) mn can be found in our preceding work [16], including a discussion of the regularization scheme also underlying our present work. For the same regularization scheme, an explicit representation of a Fierz -complete set of flow equations for the four-quark interactions in the vacuum limit in case of an SU (N c ) ⊗ U L (2) ⊗ U R (2) symmetry can be found in Ref. [35], where the contributions proportional to λ j g 2 s and g 4 s have been taken into account as well.
In our present approximation, the RG flow of the gauge sector enters the flow equations of the four-quark couplings only via the running of the strong coupling. In our numerical analysis in Sec. III, we employ the running coupling computed non-perturbatively in Refs. [36,37] where the same regularization scheme has been used as in the present work. Let us be more explicit at this point. In Refs. [36,37], the running of the gauge coupling has been computed at zero and finite temperature, but not at finite quark chemical potential. An extension of these non-perturbative studies to finite quark chemical potential is beyond the scope of the present work and therefore deferred to future work. To estimate the effect of a quark chemical potential on the running of the strong coupling and thereby on the flow of the four-quark couplings, we employ the running strong coupling from Refs. [36,37] as obtained for (pure) Yang-Mills (YM) theory (N f = 0) and for QCD with two massless quark flavors. The reason behind is that, at the one-loop level, the quark contribution to the β function of the strong coupling at finite quark chemical potential and zero temperature has been found to be identical to the one in the vacuum limit for RG scales k ≥ µ and to vanish identically for k < µ, see, e.g., Refs. [41,50,75]. This implies that, at least at the one-loop level at zero temperature, the RG flow of the strong coupling is identical to the one in the vacuum limit for k ≥ µ but identical to the one in YM theory for k < µ. Loosely speaking, the YM coupling and the QCD coupling in the vacuum limit may therefore be viewed as two extremes of the zero-temperature QCD coupling at finite quark chemical potential. The stepwise change of the β function of the strong coupling at k = µ is then smeared out at finite temperature.
In addition to corrections to the strong coupling originating from the presence of a finite quark chemical potential, one may be worried about possible corrections to the strong coupling arising from the presence of quark self-interactions. However, provided the flow of the four-quark couplings is governed by the presence of fixed points [72], as it is the case in the symmetric regime [35][36][37], it follows from the analysis of (modified) Ward -Takahashi identities that the back-reaction of the four-quark couplings on the strong coupling is negligible. For our present analysis, the use of the strong coupling from Refs. [36,37] in our set of flow equations for the four-quark couplings is therefore not only a convenient but also a consistent approximation.
We close this discussion by adding that the mechanism underlying the dynamical generation of the fourquark couplings and the role of the running gauge coupling for spontaneous symmetry breaking can be already understood in simple terms by analyzing analytically the fixed-point structure of the RG flow equations for the four-quark couplings. Here, we refrain from repeating this general line of arguments and only refer the reader to the discussions given in Refs. [35][36][37][76][77][78], see Ref. [17] for an introduction.
C. Scale-fixing procedure As already mentioned above, the initial values of the four-quark interaction channels are not considered as fundamental parameters in our present approach and are set to zero at the UV scale Λ. This corresponds to a U A (1)symmetric scenario according to the sum rules (16) and (17). The case with explicit U A (1)-symmetry breaking is discussed separately in Subsec. III B. In any case, the four-quark couplings must be generated dynamically by quark-gluon interactions which removes a potential bias in the parameter choice as present in low-energy model studies. In fact, within our present study, we rather predict the values of the four-quark couplings from the underlying quark-gluon dynamics. This is a feature that has already been used in Ref. [14] to constrain the equation of state of symmetric nuclear matter.
The only free parameter in our study is the value of the running gauge coupling g s at the initial RG scale Λ. This value is adjusted at the UV scale Λ = 10 GeV to obtain a critical temperature T cr (µ = 0) ≡ T 0 = 132 MeV at zero quark chemical potential, being the value of the chiral phase transition temperature at µ = 0 found in very recent lattice QCD studies [79]. 3 In Sec. III, we shall use the scale T 0 as a reference scale, i.e., we "measure" all physical observables in units of this scale. The chosen value for Λ ensures that T /Λ 1 and µ/Λ 1 for the range of external parameters considered in the present work and therefore allows us to avoid cutoff artefacts (and to reduce regularization-scheme dependences) [27].
In order to obtain the critical temperature T cr (µ = 0) = 132 MeV at µ = 0, we tune the running gauge coupling at the initial scale Λ. For example, for our study with a gauge coupling with two quark flavors, this amounts to choosing α s (Λ = 10 GeV) = g 2 s /(4π) = 0.2137. Evolved to the Z-boson mass scale M Z = 91.19 GeV, the value of the gauge coupling fixed in this way is then about 6% greater than the experimental results [80]. The necessity of a larger initial value for the running gauge coupling in order to trigger criticality in the quark sector has been observed before in studies employing approximations of the present type, see, e.g., Ref. [45]. Only most advanced RG truncations with a very accurate treatment of momentum structures do not require such an "enhancement", see, e.g., Refs. [20,21]. The latter is of particular importance for a quantitative description of low-energy observables within the symmetry-broken regime. For our present work aiming at a study of the onset of spontaneous symmetry breaking and an analysis of symmetry breaking patterns, we expect that this plays a secondary role.

A. Symmetry breaking patterns
Let us now study the phase diagram in the plane spanned by the temperature and the quark chemical po- 3 The lattice QCD study presented in Ref. [79] considering two degenerate massless quarks and a physical strange quark mass finds the chiral phase transition Tcr = 132 +3 −6 MeV at zero quark chemical potential.  Figure 2. Phase boundary associated with the spontaneous breakdown of at least one of the fundamental symmetries of QCD as accessible by our Fierz -complete ansatz (red and blue lines). The red line has been obtained by employing a strong coupling with two massless quark flavors. To illustrate the effect of quark contributions to the strong coupling, we also show the phase boundary as obtained when the strong coupling from pure Yang-Mills theory is employed, see main text for details. The gray boxes labeled "1" and "2" specify two exemplarily points for which the RG flows of the fourquark couplings are shown in Fig. 3. tential for the U A (1)-symmetric case, i.e., with all fourquark couplings set to zero at the initial RG scale Λ. Recall that we have defined the critical temperature T cr (µ) at a given value of the quark chemical potential to be the smallest temperature for which all four-quark couplings still remain finite in the IR limit k → 0 [15-17, 36, 37]. Keep also in mind that a divergence in the flow of one four-quark coupling at a critical scale k cr (T, µ) entails corresponding divergences in all other channels as well. However, as discussed in detail in Refs. [15,16], the four-quark couplings in general develop distinct relative strengths and it is possible to identify a dominant four-quark channel, i.e., the modulus of the associated coupling is (significantly) greater than the absolute values of all the other four-quark couplings, see our general discussion above.
As a first non-trivial result, we observe a dominance of the scalar-pseudoscalar coupling in the vacuum limit (T = µ = 0) with the couplings diverging at a symmetry breaking scale k cr (µ = 0)/T 0 ≈ 2.62. Close to the symmetry breaking scale, we find that the modulus of the scalar-pseudoscalar coupling is at least two times greater than the modulus of all other couplings, suggesting that the QCD ground state is governed by chiral symmetry breaking in the vacuum limit. At first glance, this is very similar to the situation observed in NJL-and quark-meson-type studies. However, the crucial difference is given by the initial conditions for the four-quark couplings. In the present case, we can exclude that the dominance of the scalar-pseudoscalar channel may have been triggered by the choice for the initial conditions of the four-quark couplings since they are set to zero at the initial scale and are therefore only generated dynamically. Thus, the dominance of the scalar-pseudoscalar coupling indicating chiral symmetry breaking is solely generated by quark-gluon dynamics. This is in line with earlier RG flow studies of QCD in the vacuum limit [19,20] and it has indeed been observed that this dominance entails the formation of a corresponding chiral condensate [20,21] governing the low-energy dynamics.
Increasing the temperature at vanishing quark chemical potential, the dominance of the scalar-pseudoscalar interaction channel persists even up to high temperatures beyond the critical temperature. The red line in Fig. 2 depicts the critical temperature as a function of the quark chemical potential which has been obtained with the running gauge coupling as computed in Refs. [36,37] for N f = 2, here denoted by α QCD . Following the phase boundary from small to large chemical potential, we first observe that the dominance of the scalar-pseudoscalar interaction channel persists up to µ/T 0 ≈ 1.7 as indicated by the red solid line. To illustrate the relative strengths of the various four-quark couplings in this regime, we show the scale dependence of the (dimensionful) renormalized couplings at µ/T 0 = 1.0 for T /T 0 ≈ 0. 95 T cr (µ)/T 0 (i.e., right above the phase transition) in the left panel of Fig. 3. This point is indicated in the phase diagram by the little gray box labeled "1". For illustrational purposes, the various couplings are normalized by the value of the scalar-pseudoscalar couplingλ (σ-π) at k = 0. We readily observe that the dynamics is clearly dominated by the scalar-pseudoscalar coupling. In fact, its modulus is at least two times greater than the modulus of all other couplings. According to our line of argument, this dominance indicates that in this regime the phase boundary continues to be governed by chiral symmetry breaking.
Following the phase transition line, we then encounter that the exclusive dominance of the scalar-pseudoscalar channel goes away and a small region from approximately µ/T 0 ≈ 1.7 to µ/T 0 = µ χ /T 0 ≈ 2.0 opens up (as depicted by the red dotted line in Fig. 2). In this region, we observe that the scalar-pseudoscalar channel, the CSC channel, as well as the (S + P ) adj − -, (V + A) adjand (V −A) adj ⊥ -channel are most dominant, meaning that these channels are significantly greater compared to the remaining interaction channels. Such a region of "mixed" dominances might potentially indicate a metastable or mixed phase [71]. It is noteworthy that four out of this set of five dominant channels are adjoint interaction channels which may point to a non-trivial color-structure of the ground state in this regime. However, the appearance of this regime may also very well be a consequence of the U A (1)-preserving initial conditions as the resulting U A (1)-symmetric RG flow affects the development of dominances. Indeed, the "entanglement" of several equally strong four-quark couplings is lifted by taking into account U A (1)-violating effects, see Subsec. III B.
For µ/T 0 2.0, depicted by the red dashed line in Fig. 2, we eventually observe a clear dominance of the  CSC channel indicating the emergence of a diquark condensate δ a . This dominance is again illustrated by the scale dependence of the couplings in this region shown in the right panel of Fig. 3. There, we show the RG flow for µ/T 0 = 4.0 and T /T 0 ≈ 0.74 T cr (µ)/T 0 (i.e., right above the phase transition). In the phase diagram, the corresponding point is depicted by the little gray box labeled "2" in Fig. 3. Note that, in the right panel of Fig. 3, the couplings are now normalized by the modulus of the dominant CSC coupling, |λ csc | at k = 0. We observe that the modulus of any other four-quark coupling is at most less than half the value of the CSC coupling, except for the (S + P ) adj − coupling. The latter assumes the same value in the IR as the CSC coupling, only with opposite sign. The reason for this behavior is that the boundary conditions with all four-quark couplings initially set to zero at the UV scale Λ leaves the axial U A (1) symmetry intact, as already mentioned above. Since the RG flow preserves the symmetries of the initial effective average action, the sum rules (16) and (17) are exactly fulfilled at all scales k, with the first sum rule implyinḡ λ csc = −λ (S+P ) adj − . In fact, the sum rules show that two of the 10 four-quark couplings of our Fierz -complete basis are not independent in the U A (1)-symmetric case. We shall discuss the effect of U A (1)-violating initial discussions in Subsec. III B.
In Fig. 4, we show the dominance pattern of the four-quark couplings along the finite-temperature phase boundary presented in Fig. 2 (as obtained with the twoflavor coupling α QCD ). The values of the dimensionful renormalized couplings are shown for k → 0 as functions of the quark chemical potential for temperatures right above the critical temperature T cr (µ), where we have normalized the couplings by the scalar-pseudoscalar couplingλ (σ-π) for µ = 0 at k = 0 for convenience. As already mentioned above, we first observe a clear dominance of the scalar-pseudoscalar interaction channel for µ/T 0 1.7, followed by a change of the dominance pattern to a dominance of the CSC coupling for µ/T 0 2. In the region of CSC dominance, the intact U A (1) symmetry is again encoded by the fact that the values of the CSC and the (S + P ) adj − -coupling are identical, up to their signs. The dominance of the CSC coupling for µ/T 0 2 is as clearly visible as the dominance of the scalar-pseudoscalar coupling for µ/T 0 1.7. In fact, the ratio of the modulus of the second largest coupling and the largest coupling is even smaller in this high-density regime. Note that, in Fig. 4, the values of the four-quark couplings are extracted at a temperature T slightly above the critical temperature T cr (µ) where (T − T cr (µ))/T 0 ≈ 0.004. By this, we ensure that the RG flow is fully located in the symmetric regime and the flow can be followed down to k = 0. However, owing to this small distance to the phase transition line, the aforementioned region with "mixed" dominances is not fully resolved in this figure.
The distinct dominance and evident "hierarchy" among the four-quark self-interactions in the low-and high-density regime is nicely illustrated in Fig. 4. In view of this result, it is tempting to speculate whether such a  Figure 4. (Dimensionful) renormalized four-quark couplings at k = 0 as functions of the quark chemical potential for temperatures (T − Tcr(µ))/T0 ≈ 0.004 (i.e., slightly above the respective critical temperature Tcr), illustrating the "hierarchy" of the four-quark couplings in terms of their relative strength along the phase boundary (i.e., along the red line in Fig. 2). For convenience, the values of the four-quark couplings are normalized by the value of the scalar-pseudoscalar couplinḡ λ (σ-π) for µ = 0 at k = 0. change in the "hierarchy" points to the existence of a nearby tricritical point in the phase diagram. Yet, this aspect is speculative as the presently employed approximations do not allow a definite answer to this question. However, we emphasize that the change in the "hierarchy" from a dominance of the scalar-pseudoscalar coupling to a dominance of the CSC coupling at µ χ /T 0 ≈ 2.0 is a non-trivial outcome of our study as it is completely determined by the dynamics of the system itself. The four-quark couplings are initially set to zero at the UV scale Λ and are dynamically generated by quark-gluon dynamics in the RG flow. Thus, the dynamics is not affected by any kind of fine-tuning of the boundary conditions of the four-quark couplings which would potentially favor particular channels. At this point, let us once more bring to attention that the dominance of a four-quark coupling only indicates the onset of the formation of an associated condensate. It does neither guarantee the actual formation (e.g., IR fluctuations could restore the associated symmetries) nor does it strictly exclude the possible formation of other condensates associated with subdominant couplings. Our analysis based on the dominance pattern of the fourquark couplings must therefore be taken with some care, see also our discussion in our preceding works [15,16]. Still, in the context of condensed-matter physics, the appearance of a clear dominance of a given channel has been found to be a precursor of the formation of a corresponding condensate [71].
In Fig. 2, we also included results for the finitetemperature phase boundary from a computation where we have used a running gauge coupling as obtained in YM theory (depicted by the blue line labeled α YM ). The phase boundary as well as the dominances agree almost perfectly with the results from the computation using the running coupling α QCD . To be specific, we observe a dominance of the scalar-pseudoscalar coupling at small quark chemical potential, a regime characterized by a "mixed" dominance pattern between 1.7 µ/T 0 2.0, and finally a clear dominance of the CSC coupling at large quark chemical potential. This is noteworthy since the YM coupling α YM grows more rapidly than the QCD coupling α QCD when the scale k is lowered. However, the effect of this difference in the scale dependence is also to a certain extent compensated by our scale fixing procedure. The initial value of the YM gauge coupling has to be chosen smaller than for the α QCD coupling in order to obtain the same critical temperature T 0 ≡ T cr (µ = 0) = 132 MeV at zero quark chemical potential. Nevertheless, this observation may be viewed as an indication that the "hierarchy" of the various couplings in terms of their relative strength is predominantly determined by the dynamics within the quark sector whereas the gauge sector as associated with the details of the running coupling is mostly required to bring the quark sector close to criticality in the first place. Once the gauge sector have rendered the four-quark couplings sufficiently large, they become relevant operators in the RG flow and the quarks start to develop their "own effective dynamics". Then, the details of the gauge sector play a subleading role, at least in the present approximation. Loosely speaking, one may therefore state that we are basically left with an NJL-type model once the fourquark couplings have been rendered sufficiently large by the underlying quark-gluon dynamics.

B. Role of UA(1) symmetry
The initial conditions of the RG flow chosen so far leave the axial U A (1) symmetry intact. All couplings of the four-quark self-interactions are set to zero at the UV scale Λ and are solely generated dynamically by quarkgluon interactions in the RG flow. In, e.g., Refs. [36,37], U A (1)-violating channels have been omitted based on the assumption that they become relevant only in the lowenergy regime governed by spontaneous symmetry breaking. Our Fierz -complete basis B composed of the 10 fourquark interaction channels is effectively reduced to eight interaction channels in case of the U A (1) symmetry being intact [16]. Recall that the sum rules (16) and (17) imply that two of the couplings associated with the four U A (1)-violating interaction channels of our basis B are not independent. These sum rules are exactly fulfilled at all scales in the symmetric phase and for all k > k cr in the phase governed by spontaneous symmetry breaking. In the latter case, the U A (1) symmetry may potentially still be broken spontaneously below the symmetry breaking scale k cr . However, this cannot be resolved within our present approximation.
In our preceding study of the phase structure of the NJL model and the role of Fierz completeness [16], we have observed that U A (1) symmetry breaking affects the dominances of the four-quark couplings in terms of their relative strength along the finite-temperature phase boundary. In particular, we have found explicit U A (1) symmetry breaking to be important for the formation of the conventional CSC ground state at intermediate and large values of the chemical potential. In this work, the four-quark couplings are now dynamically generated by quark-gluon dynamics. Following the critical temperature T cr (µ) as a function of the quark chemical potential, we observe two regions characterized by different distinct "hierarchies" of the four-quark couplings which are remarkably robust against a variation of the running gauge coupling, see Fig. 2 for the U A (1)-symmetric case. The scalar-pseudoscalar interaction channel dominates the dynamics at small quark chemical potential, signaling the formation of the chiral condensate, whereas at higher quark chemical potential the dominance of the conventional CSC coupling suggests the formation of a diquark condensate. The latter is observed in spite of the intact U A (1) symmetry in our considerations thus far. Only for 1.7 µ/T 0 2.0, we observe a regime which is characterized by several equally strong four-quark interaction channels, see our discussion in the previous subsection. Moreover, the dominance of the CSC coupling is always accompanied by an equally strong (S + P ) adj − -channel as a direct consequence of the intact U A (1) symmetry: The sum rule (16) ties the modulus of the CSC coupling to the modulus of the (S + P ) adj − coupling. In order to probe the role of explicit U A (1)-symmetry breaking in our present study, we now analyze the RG flow for U A (1)-violating boundary conditions for the fourquark couplings. The strength of U A (1) breaking is effectively controlled by the initial value of the (S + P ) − coupling since the associated four-quark interaction channel is directly related to the so-called 't Hooft determinant (14), see Refs. [3,[58][59][60][61][62][63]. In the following, we therefore vary only the initial condition of this coupling but still set the initial values of the other four-quark couplings to zero at the UV scale Λ. For a given initial value of the (S + P ) − coupling, we then adjust the UV value of the gauge coupling g s (Λ) such that the value of the critical temperature at zero chemical potential remains unchanged, T cr (µ = 0) = 132 MeV. This ensures comparability between our results for different initial conditions for the (S + P ) − coupling.
As also done in Refs. [15,16], we begin by analyzing the fate of the U A (1) symmetry at finite temperature and quark chemical potential with the aid of the sum rules (16) and (17). To this end, we first normalize the two sum rules as follows: where The quantities S  of the two cases. Interestingly, we observe the exact same qualitative behavior as found in Ref. [16] for the NJL model. At small quark chemical potential close to the critical temperature T cr (µ), U A (1) breaking is driven by the scalar-pseudoscalar channel associated with pion dynamics and becomes stronger toward the IR as indicated by increasing values of R 1 and R 2 , with R 2 R 1 . At large chemical potential, we find that the strength of U A (1)-symmetry breaking becomes also stronger as the phase boundary is approached from above, but now driven by the dynamics of diquark degrees of freedom as associated with the CSC channel. As a consequence, R 1 and R 2 are of the same order of magnitude since both depend on the CSC coupling, see Eqs. (16) and (17). In either case, for increasing temperature, U A (1) breaking as measured by the sum rules remains more and more on its initial level as determined by the U A (1)-violating boundary conditions in the UV regime since quark fluctuations become more and more thermally suppressed.
Let us now compare the phase diagram as obtained with the U A (1)-symmetric initial conditions employed in the previous subsection, i.e., with all four-quark cou-plings initially set to zero, to the phase diagrams resulting from U A (1)-violating initial conditions. The strength of the explicit U A (1) breaking at the initial UV scale Λ is controlled by the value of the (dimensionless) renormalized coupling of the (S + P ) − channel ("'t Hooft channel") which we choose to assume the values λ (UV) (S+P )− = 0.01, 0.1, 1.0 . In the following, we shall only present results from computations using the running gauge coupling for two massless quark flavors. As in the U A (1)symmetric case, the dependence on our specific choice for the coupling is found to be very mild anyhow. In Fig. 6, the various phase diagrams as obtained with these three different choices for the boundary conditions are shown. It is remarkable how little the critical phase temperature as a function of the quark chemical potential is affected by the strength of the initial explicit breaking of the U A (1) symmetry, although the strength in terms of the initial value of the (S + P ) − coupling is varied over three orders of magnitude. 4 Across the entire range of chemical potentials shown in this figure, the variation of the critical temperature for any given value of the chemical potential is less than 2%. 5 In all cases, we observe a dominance of the scalar-pseudoscalar coupling at small quark chemical potential, depicted by the solid lines in Fig. 6. The regime of "mixed" dominances for chemical potentials between 1.7 µ/T 0 2.0 appearing in case of U A (1)-symmetric initial conditions vanishes for all considered U A (1)-violating initial conditions. Choosing 0.01 ≤ λ (UV) (S+P )− ≤ 1.0 for the initial value of the coupling associated with the U A (1)-violating (S + P ) − channel, we indeed find that the dominance changes directly from the scalar-pseudoscalar channel to the CSC channel within the region 1.8 µ χ /T 0 2.0, as indicated by the dashed lines in Fig. 6. This is similar to what has been found in a Fierz -complete NJL-model study [16].
The "hierarchy" of the various four-quark couplings in terms of their relative strength along the phase boundary is shown in Fig. 7, exemplarily for the initial coupling λ (UV) (S+P )− = 1.0. This figure shows again the values of the (dimensionful) renormalized couplings for k → 0 as functions of the quark chemical potential for temperatures (T − T cr (µ))/T 0 ≈ 0.002 (i.e., slightly above the critical temperature T cr (µ) for a given quark chemical potential µ). The values are normalized by the scalarpseudoscalar couplingλ (σ-π) for k → 0 at zero chemical potential. We again observe a clear dominance of the scalar-pseudoscalar interaction channel for µ/T  Figure 7. "Hierarchy" of the four-quark couplings in terms of their relative strength along the phase boundary (blue line in Fig. 6). To obtain this figure, we have evaluated the renormalized four-quark couplings for k → 0 as functions of the quark chemical potential for temperatures (T − Tcr(µ))/T0 ≈ 0.002 (i.e., slightly above the critical temperature Tcr(µ) for a given quark chemical potential). The values of the four-quark couplings are shown for UA(1)-violating initial conditions with λ (UV) (S+P ) − = 1.0 for the 't Hooft coupling. To normalize the couplings, we have used the scalar-pseudoscalar coupling for k → 0 and zero quark chemical potential. and a clear dominance of the CSC interaction channel for larger values of the chemical potential. Compared to the case with U A (1)-symmetric boundary conditions (see Fig. 4), these dominances appear even more pronounced in the present case. In particular, the CSC channel is not accompanied anymore by an equally strong (S + P ) adj − coupling. The latter now assumes considerably smaller values for µ/T 0 2.0 in comparison to the computation with intact U A (1) symmetry, whereas the CSC coupling assumes even slightly larger values. From this, we conclude that the breaking of the U A (1) symmetry plays an important role in "shaping the hierarchy" of the four-quark interaction channels and thus in the formation of associated condensates as indicated by their dominances. This observation confirms the importance of explicit U A (1) breaking for the formation of the conventional CSC ground state at large chemical potential as already discussed in Ref. [16] for NJL-type models. In this respect, we also refer again to early seminal works on color superconductivity, see, e.g., Refs. [81][82][83][84][85][86][87][88][89][90]. Noteworthy, we find that the change in the "hierarchy" from a dominance of the scalar-pseudoscalar coupling to a dominance of the CSC coupling at µ χ /T 0 ≈ 2.0 is remarkably insensitive to the initial strength of explicit U A (1) symmetry breaking as controlled by the initial coupling λ (UV) (S+P )− associated with the "'t Hooft channel". We emphasize that this change in the "hierarchy" of fourquark couplings is a non-trivial outcome completely determined by the dynamics of the system itself.
Let us finally compare our results with those from a Fierz -complete NJL model study [16]. In Fig. 6, we also show the finite-temperature phase boundary resulting from a Fierz -complete NJL model study (black line). The corresponding flow equations have been presented in Ref. [16]. In this computation, the initial scalarpseudoscalar coupling at the UV scale Λ/T 0 ≈ 75.76 has been tuned such that we obtain T cr (µ = 0) = 132 MeV for the critical temperature at µ = 0. The remaining initial four-quark couplings are set to zero. Interestingly, we find that the finite-temperature phase boundary agrees well with the one obtained from our present study with a two-flavor running gauge coupling α QCD , at least for small quark chemical potential. For µ/T 0 0.5, however, the two phase boundaries start to deviate from each other significantly. Indeed, at the largest quark chemical potential shown in Fig. 6, µ/T 0 = 4.4, the critical temperature resulting from the NJL model computation is T cr /T 0 ≈ 0.366. In contrast to that, at the same quark chemical potential, the computation including dynamical gauge fields yields a critical temperature that exceeds the one from the NJL model study by a factor of two: T cr (µ = 4.4 T 0 )/T 0 ≈ 0.731. This observation may have further phenomenological consequences. For example, in standard BCS theory, the critical temperature can be directly related to the size of the diquark gap δ a at zero temperature, i.e., T cr ∼ |δ| [26,57,91]. Thus, the observed increase of the critical temperature in our results from the computation including dynamical gauge degrees of freedom may hint to a larger diquark gap at T = 0.
Recall that we use different initial conditions for the four-quark couplings in our NJL model study and our study with dynamical gauge fields. This is required since different mechanisms are at play which drive the quark sector to criticality. To be specific, in our NJL model study, it is required to choose a sufficiently large initial scalar-pseudoscalar coupling to ensure that the RG flow diverges at a finite symmetry breaking scale k cr for sufficiently low temperatures, signaling the onset of spontaneous symmetry breaking. Apart from the value of the coupling associated with the 't Hooft channel, the fourquark couplings in our QCD study are solely dynamically generated and the quark sector is driven to criticality by the gauge coupling becoming sufficiently large, see Refs. [17,[35][36][37] for a detailed discussion of the latter mechanism. Of course, this difference also affects the results at large quark chemical potential, although the gauge coupling plays a less prominent role in this regime as the dynamics is largely controlled by the appearance of a BCS-type instability. One may argue that the initial conditions chosen in case of the NJL model study actually favor the scalar-pseudoscalar coupling and do not sufficiently support the dynamics associated with the formation of a diquark condensate or other channels which may become relevant at large chemical potential. Indeed, the boundary conditions enforce that the dynamics are initially driven by the scalar-pseudoscalar self-interaction, at least over a wide range of RG scales. Still, at large chemical potential, the CSC channel is found to dominate the dynamics even in case of the NJL model. As dis-Setting curvature κ mean field (NJL, one channel) [9,92] [93][94][95][96] 0.034 . . . 0.070 Table I. Curvature κ of the two-flavor finite-temperature phase boundary as obtained from different studies, see main text for details. Note that the curvature range for the meanfield studies reflects the difference between the chiral limit and the case of physical pion masses. The fRG results in this work have been obtained in the chiral limit.
cussed in detail in Refs. [16], this can be understood from an analysis of the fixed-point structure. Our study taking into account gauge degrees of freedom comes without the requirement of an initial tuning of a specific four-quark coupling. It is therefore intriguing that the "hierarchy" of the various interaction channels in both computations changes at approximately the same quark chemical potential from the scalar-pseudoscalar coupling to the CSC coupling, see Fig. 6. This observation indicates that the "hierarchy" of the interaction channels in terms of their strength is determined to a large extent by the interplay of the various four-quark couplings themselves, see also our discussion in the previous subsection. As a closing remark, we would like to add that the comparison of the different phase boundaries shown in Figs. 2 and 6 have to be taken with some care. Although all computations yield approximately the same critical scale k cr /T 0 ≈ 2.6 (symmetry-breaking scale) in the vacuum limit, which sets the scale for low-energy observables O ∼ k cr (see, e.g., Ref. [76]), our present approximation does not allow to check whether the different studies indeed lead to the same values of low-energy observables in the IR limit. This potential issue complicates a comparison of our results for the phase boundary, as well as our subsequent comparison of the curvature of the phase boundary.

C. Curvature of the phase boundary
Finally, we briefly comment on the curvature κ of the finite-temperature phase boundary at small chemical potential: The results for the curvature as obtained from various different studies are summarized in Tab. I. Compared to our Fierz -complete NJL model study, we find the curvature to be significantly decreased in our study including dynamical gauge degrees of freedom. In fact, the curvature is reduced by approximately 40%.
The curvature of a standard one-channel NJL-model study in the mean-field approximation is even more than four times greater than in our present study with gauge degrees of freedom. This holds true for all settings with gauge degrees of freedom considered in this work, including computations using a Yang-Mills coupling, a strong coupling for the two-flavor case, as well as computations taking into account explicit U A (1) symmetry breaking (labelled "fRG (QCD)" in Tab. I). Note that the results from our Fierz -complete fRG calculations including gauge degrees of freedom listed in Tab. I have been obtained with the strong coupling for the two-flavor case. For the case with explicit U A (1) symmetry breaking, we have chosen λ (UV) (S+P )− = 1.0 for the initial condition of the coupling associated with the 't Hooft channel. A summary of results for the curvature from lattice QCD calculations with two flavors can also be found in Tab. I. We observe that the results from our Fierz -complete studies taking into account gauge degrees of freedom are well in accordance with those from lattice QCD studies. We add that low-energy model studies indicate that the inclusion of IR fluctuation effects tend to further lower the value of the curvature [92,[97][98][99]. For the 2 + 1-flavor case, more recent lattice results for the curvature are also available, indicating κ ∼ 0.034 . . . 0.21 [100]. The curvature found in a very recent 2 + 1-flavor fRG-QCD study with physical masses is also in accord with the latter results [50]. In any case, a direct comparison of our present results to those from 2 + 1-flavor studies is only possible to a limited extent, if at all. Still, in general, it is reasonable to expect from our present study that the issue of Fierzincompleteness also affects the results for the curvature in the 2 + 1-flavor case, as it does in the 2-flavor case (see Tab. I). Indeed, it has also been observed in a Fierzincomplete two-channel study of a 2 + 1-flavor NJL-type model that four-quark interaction channels other than the scalar-pseudoscalar channel can significantly impact the value of the curvature [101].

IV. CONCLUSIONS
In this work, we have analyzed the RG flow of fourquark interactions in the pointlike limit in a Fierzcomplete fashion starting from the classical QCD action in the UV limit. Working in the chiral limit, the only parameter of our study in the U A (1)-symmetric limit is given by the strong coupling g s which we fixed at a large initial UV scale in the perturbative regime.
With this setting at hand, we found that the inclusion of gluodynamics leads to an increase of the critical temperatures at large quark chemical potential in comparison to the results from a corresponding Fierz -complete NJL model study. Assuming that the critical temperature can be related to the size of the zero-temperature gap in a color superconducting phase of quark matter, our results therefore suggest that the diquark gap is likely to be greater than the one found in corresponding NJL model studies, at least within the range of chemical potentials considered in our present work.
Toward the IR limit, the treatment of the four-quark interactions in the pointlike limit does of course not allow us to access the phase governed by spontaneous symmetry breaking. The introduction of mesonic auxiliary fields by means of a Hubbard -Stratonovich transformation or applying the more advanced technique of dynamical hadronization, see Refs. [32,52,69,102] and, e.g., Refs. [20,21,39,45,50] for their application to QCD, would enable us to study the dynamics even within regimes governed by spontaneous symmetry breaking. In order to gain nevertheless some insight into the structure of the ground state emerging in case of spontaneous symmetry breaking, we followed the approach of our NJL model studies in Refs. [15,16] and analyzed the "hierarchy" of the four-quark couplings.
Our RG analysis of the U A (1)-symmetric case revealed a clear dominance of the scalar-pseudoscalar interaction channel associated with chiral symmetry breaking at small quark chemical potential. Very importantly, this dominance is not triggered by a specific choice for the initial conditions of the four-quark couplings since all four-quark couplings are set to zero at the initial RG scale, i.e., they are solely gluon-induced in the RG flow. For µ/T 0 2.0, we then observe a change in the "hierarchy". In this regime, the CSC channel associated with the formation of the most conventional color superconducting condensate in two-flavor QCD now dominates the quark dynamics. We emphasize that the dominance pattern as a function of the quark chemical potential as well as the actual position µ χ , where the dominance pattern changes from a scalar-pseudoscalar dominance to a CSC dominance, are found to be a remarkably robust feature. The little influence of the considered different running gauge couplings may be viewed as an indication that the dominance pattern is largely determined within the quark sector. The gauge sector as associated with the details of the running coupling is mostly required to bring the quarks close to criticality. Once the four-quark couplings have been rendered sufficiently large by the underlying quark-gluon dynamics, the quarks develop their "own dynamics" and the details of the gauge sector start to play a subleading role, at least in the present approximation. At this point, loosely speaking, we are then basically left with an NJL-type model. Note again that the dynamics in our present study is not "contaminated" by any kind of fine-tuning of the initial conditions for the four-quark couplings which would in general favor particular channels. However, the analysis based on the "hierarchy" of the four-quark couplings must nevertheless be taken with some care. In fact, it is clear that a dominance of a specific four-quark coupling does neither guarantee the formation of an associated condensate in the IR nor does it exclude the formation of other condensates. Still, it should also be mentioned that corresponding studies in the context of condensed-matter theory show that the appearance of a clear dominance of a specific four-fermion coupling is indeed a precursor for the formation of the associated condensate [71].
In order to probe the effect of explicit U A (1)-symmetry breaking on the phase boundary and the dominance pattern of the four-quark couplings, we implemented U A (1)violating initial conditions in form of finite values for the coupling associated with the so-called 't Hooft channel. Even at large chemical potential, the considered strengths of explicit U A (1) symmetry breaking at the initial UV scale showed surprisingly little effect on the shape of the phase boundary. The same is true for the actual position of the point µ χ , where the dominance pattern of the four-quark couplings changes qualitatively. Compared to the U A (1)-symmetric RG flow, however, U A (1)violating initial conditions affect the "hierarchy" of the four-quark couplings along the phase boundary which is clearly visible in, e.g., the amplification of the dominance of the scalar-pseudoscalar interaction channel at small chemical potential as well as of the dominance of the CSC channel at large chemical potential. From this, we conclude that explicit U A (1) symmetry breaking plays indeed an important role in the formation of the condensates, in particular with respect to the formation of the conventional CSC ground state at large chemical potential [81][82][83][84][85][86][87][88][89][90]. Future extensions of our present work should of course include a direct computation of the 't Hooftcoupling within our RG flow by following earlier works in this direction [63].
Of course, our present study can be further improved in various directions. Still, our analysis already provides an important insight into the dynamics underlying the QCD phase structure at finite temperature and quark chemical potential and consolidates the findings obtained from our preceding Fierz -complete NJL-type model study [16]. Moreover, the inclusion of gauge degrees of freedom combined with the Fierz -complete set of four-quark interactions enables us to identify the relevant effective lowenergy degrees of freedom and to determine, or at least constrain, from first principles the couplings of a suitably constructed truncation for the low-energy sector, in particular at large quark chemical potential. In the future, this may prove very valuable, e.g., to study the thermodynamics of quark matter at high density. Indeed, first steps into this direction have already been taken very recently [14].