From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD
Jens Braun, Leonard Fister, Jan M. Pawlowski, Fabian Rennecke
FFrom Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD
Jens Braun,
1, 2
Leonard Fister, Jan M. Pawlowski,
4, 2 and Fabian Rennecke
4, 2 Institut f¨ur Kernphysik (Theoriezentrum), Technische Universit¨at Darmstadt,Schloßgartenstraße 2, D-64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany Institut de Physique Th´eorique, CEA Saclay, F-91191 Gif-sur-Yvette, France Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
We present an analysis of the dynamics of two-flavour QCD in the vacuum. Special attention ispaid to the transition from the high energy quark-gluon regime to the low energy regime governedby hadron dynamics. This is done within an functional renormalisation group approach to QCDamended by dynamical hadronisation techniques. The latter allow us to describe conveniently thetransition from the perturbative high-energy regime to the nonperturbative low-energy limit withoutsuffering from a fine-tuning of model parameters. In the present work, we apply these techniques totwo-flavour QCD with physical quark masses and show how the dynamics of the dominant low-energydegrees of freedom emerge from the underlying quark-gluon dynamics.
PACS numbers: 12.38.Aw, 11.30.Rd , 12.38.Gc
I. INTRODUCTION
For an accurate first-principles description of the dy-namics of QCD, a reliable inclusion of hadronic statesis of great importance. This holds in particular for anapproach aiming at the hadron spectrum or the phasestructure of QCD at finite density. In the present work ontwo-flavour QCD we develop a theoretical framework fortaking into account the fluctuation dynamics of quarks,gluon and hadrons. This approach is based on previousfunctional renormalisation group studies [1, 2] and a re-lated quantitative study in the quenched limit [3]. Thepresent work and [3] are first works within a collaboration(fQCD) aiming at a quantitative functional renormalisa-tion group framework for QCD [4]. This framework allowsto dynamically include hadronic states as they emergefrom the microscopic quark and gluon degrees of freedom.We use the functional renormalisation group (FRG)approach for QCD, for reviews see [5–14], and [15–21]for reviews on related work. In order to describe thetransition from quarks and gluons to hadrons, we extendthe dynamical hadronisation technique (or rebosonisa-tion), introduced in Refs. [7, 22–24]. For the first time,this technique is applied here to dynamical two-flavourQCD with physical quark masses. It is shown how thedominant hadronic low-energy degrees of freedom andtheir dynamics emerge from the underlying quark-gluondynamics. The hadronisation technique, as further de-veloped in the present work, already applied in Ref. [3]in a quantitative study of quenched QCD. In the latterwork, a large number of interaction channels has beentaken into account, aiming at full quantitative precision.Here, we exploit the results from [3] as well as resultson the scale-dependent glue sector of Yang-Mills theoryfrom Refs. [18, 25, 26]. This enables us to concentrate onthe RG flows of the most relevant couplings from a morephenomenological point of view, paying special attentionto unquenching effects.In summary, the aim of this work is threefold: Firstly, we aim at a detailed understanding of the fluctuationphysics in the transition region between the high energyquark-gluon regime to the low energy hadronic regime.Secondly, we want to initiate the quest for the minimal setof composite operators that have to be taken into accountfor reaching (semi-)quantitative precision, while keepingthe study analytic. This deepens the understanding ofthe fluctuation physics by only taking into account therelevant operators. Moreover, it is also of great interestfor low energy effective models. Thirdly, we discuss fullunquenching effects in terms of the matter back-couplingto the glue sector that is important for QCD regimes withdominant quark fluctations such as QCD at high densitiesor many flavours.The paper is organised as follows: In Sect. II we intro-duce the ansatz for the quantum effective action whichwe are considering in the present work. The generalframework of dynamical hadronisation is then discussedin detail in Sect. III, where we also give a discussionof the RG flow in the gauge sector and the role of thequark-gluon vertex. Our results for two-flavour QCD arethen presented in Sect. IV. While our analysis suggeststhat the use of dynamical hadronisation techniques onlyyields mild quantitative corrections in low-energy modelstudies, its use is indispensable from both a qualitativeand a quantitative point of view for a unified descriptionof the dynamics of QCD on all scales. Our conclusions aregiven in Sect. V. Some technical details as well as a briefdiscussion about the effect of dynamical hadronisation onlow-energy models are discussed in the appendices.
II. THE EFFECTIVE ACTION
Our aim is to describe two-flavour QCD in d = 4 Eu-clidean dimensions at vanishing temperature and densityin a vertex expansion. The starting point is the micro-scopic gauge fixed QCD action. Thus, we include thequark-gluon, three- and four-gluon vertices as well as the a r X i v : . [ h e p - ph ] J u l ghost-gluon vertex and the corresponding momentum-dependent propagators. Four-quark interactions are dy-namically generated at lower scales and we therefore takethe scalar-pseudoscalar channel into account in our trun-cation. This is by far the dominant four-quark channel,as it exhibits quark condensation, see [3].On even lower energy scales, bound state degrees offreedom appear and eventually become dynamical. Toproperly take this into account, we introduce a scale-dependent effective potential V k which includes arbitraryorders of mesonic self-interactions. Since dynamics in thissector is dominated by the lightest mesons, we restrictour analysis to pions and the sigma-meson and theircorresponding momentum-dependent propagators. Wetherefore assume a strong axial anomaly, i.e. U (1) A ismaximally broken. As a consequence, the meson sector inthe chiral limit exhibits an O (4) flavor symmetry. Notethat this is also reflected in the four-quark interaction:the scalar-pseudoscalar channel ∼ λ q,k is invariant under SU (2) V × SU (2) A but violates U (1) A symmetry, see (1).Explicit chiral symmetry breaking is included via a sourceterm − cσ . It is directly related to a finite current quarkmass and, as a consequence, non-zero pion masses. Thisimplies that we have a chiral crossover transition ratherthan a second order phase transition. The meson sector iscoupled to the quark sector by a field-dependent Yukawacoupling h k ( φ ). That way, arbitrarily high orders ofquark-antiquark multi-meson correlators are included [27].We elaborate on the physics picture in Sect. IV.The key mechanism to consistently describe the dy-namical generation of bound state degrees of freedom inthis work is dynamical hadronisation, and is discussed inSect. III A. In summary, this yields the following scale-dependent effective action,Γ k = (cid:90) x (cid:26) F aµν F aµν + Z c,k ¯ c a ∂ µ D abµ c b + 12 ξ ( ∂ µ A aµ ) + Z q,k ¯ q ( γ µ D µ ) q − λ q,k (cid:104) (¯ q T q ) − (¯ qγ (cid:126)T q ) (cid:105) + h k ( φ ) (cid:104) ¯ q ( iγ (cid:126)T (cid:126)π + T σ ) q (cid:105) + 12 Z φ,k ( ∂ µ φ ) + V k ( ρ ) − cσ (cid:27) + ∆Γ glue , (1)with the O (4) meson field φ = ( σ, (cid:126)π ) and ρ = φ / D µ = ∂ µ − iZ / A,k g k A aµ t a is the Dirac operator, with the strongcoupling g k = (cid:112) πα s,k and the gluonic wave-functionrenormalisation Z A,k . With this definition the covariantderivative D µ is renormalisation group invariant. The lastterm in the first line, ∆Γ glue , stands for the non-trivialghost-gluon, three-gluon and four-gluon vertex corrections,for further details see Sect. III C and in particular Eq. (70).The full momentum dependence of the pure gauge sectoris taken into account in the gluon and ghost dressingfunctions Z A,k and Z c,k . This is crucial for the correctIR behaviour of the gauge sector. Due to asymptotic freedom the effective action at theinitial cutoff scale Λ relates to the classical (gauge-fixed)QCD action,Γ k → Λ (cid:39) (cid:90) x (cid:26) F aµν F aµν + ¯ q (cid:0) γ µ D µ + m UV q (cid:1) q + ¯ c a ∂ µ D abµ c b + 12 ξ ( ∂ µ A aµ ) (cid:27) . (2)The quark mass m UV q at the UV scale Λ is directly relatedto the coupling c in Eq. (1). The other couplings appear-ing in our ansatz (1) for the effective action are generateddynamically in the RG flow.In this work, we use Hermitian gamma matrices so that { γ µ , γ ν } = 2 δ µν . (3)The commutator for the SU ( N c ) generators reads [ t a , t b ] = if abc t c and, hence, the trace is positive, Tr t a t b = δ ab . (cid:126)T are the SU ( N f ) generators and T = √ N f N f × N f . Forthe field strength tensor we use the relation F µν = iZ / A,k g k [ D µ , D ν ] (4)= Z / A,k t a (cid:16) ∂ µ A aν − ∂ ν A aµ + Z / A,k g k f abc A bµ A cν (cid:17) . For more details on the gauge part of our truncation seeSect. III C. All masses, wave-function renormalisationsand couplings are scale-dependent. The scalar potentialand the Yukawa coupling are expanded about a scale-independent point κ , ∂ t κ = 0. As shown in [27] thisyields a rapid convergence of the expansion V k ( ρ ) = N V (cid:88) n =1 v n,k n ! ( ρ − κ ) n ,h k ( ρ ) = N h (cid:88) n =0 h n,k n ! ( ρ − κ ) n . (5)Note that the quark and meson mass functions (two-point functions at vanishing momentum) depend on themeson fields. The masses are given by the mass functionsevaluated at the physical minimum ρ ,k = σ / V k ( ρ ) − cσ , m q,k = 12 h k ( ρ ,k ) ρ ,k ,m π,k = V (cid:48) ( ρ ,k ) ,m σ,k = V (cid:48) ( ρ ,k ) + 2 ρ ,k V (cid:48)(cid:48) ( ρ ,k ) , (6)where m q,k is the constituent quark mass. The currentquark mass m UV q is related to the symmetry breakingsource c via the mass function at the ultraviolet scale, m UV q = h Λ v , Λ c , (7)while c does not occur explicitly in the flow equation as itis the coefficient of a one-point function. This entails thatthe flows of the effective action in the chiral limit andthat in QCD with non-vanishing current quark massesagree, see also [27]. The difference solely relates to thesolution of the equation of motion for the σ -field, δ Γ k =0 δσ (cid:12)(cid:12)(cid:12)(cid:12) σ = σ EoM = 0 . (8)If expanding the flow in powers of the mesonic fields asdone in the present work, the expansion point has tobe close to σ EoM , such that it is within the radius ofconvergence of the expansion.
III. QUANTUM FLUCTUATIONS
Quantum fluctuations are computed with the func-tional renormalisation group. For QCD related reviewsand corresponding low-energy models, we refer the readerto Refs. [5–14]. A consistent description of the dynam-ical transition from quark-gluon degrees of freedom tohadronic degrees of freedom is achieved by the dynamicalhadronisation technique. Loosely speaking, it is a way ofstoring four-quark interaction channels, which are reso-nant at the chiral phase transition, in mesonic degrees offreedom and therefore allows for a unified description ofthe different degrees of freedom governing the dynamicsat different momentum scales.
A. Functional RG & dynamical hadronisation
The starting point of the functional renormalisationgroup is the scale-dependent effective action Γ Λ at aUV-cutoff scale Λ. In the case of QCD, Λ is a large,perturbative energy scale and correspondingly Γ Λ is themicroscopic QCD action with the strong coupling constantand the current quark masses as the only free parametersof the theory. From there, quantum fluctuations are suc-cessively included by integrating out momentum shellsdown to the RG scale k . This yields the scale-dependenteffective action Γ k , which includes all fluctuations frommomentum modes with momenta larger than k . By lower-ing k we resolve the macroscopic properties of the systemand eventually arrive at the full quantum effective actionΓ = Γ k =0 . The RG evolution of the scale-dependent effec-tive action is given by the Wetterich equation [28], which in the case of QCD with Φ = ( A, q, ¯ q, c, ¯ c, φ ) reads ∂ t Γ k [Φ] =12 Tr (cid:0) G AA,k [Φ] · ∂ t R Ak (cid:1) − Tr (cid:0) G c ¯ c,k [Φ] · ∂ t R ck (cid:1) − Tr (cid:0) G q ¯ q,k [Φ] · ∂ t R qk (cid:1) + 12 Tr (cid:0) G φφ,k [Φ] · ∂ t R φk (cid:1) . (9)Here, the regulator functions R Φ i k ( p ) can be understoodas momentum-dependent masses that introduce the sup-pression of infrared modes of the respective field Φ i , andare detailed in App. C. The derivative ∂ t is the totalderivative with respect to the RG scale t = ln( k/ Λ) withsome reference scale Λ. The traces sum over discrete andcontinuous indices of the fields, including momenta andspecies of fields. The first line on the right hand side of(9) is the flow in the pure glue sector, the second line cre-ates the matter fluctuations. G k [Φ] denote the scale andfield-dependent full propagators of the respective fields,e.g. for the quarks G q ¯ q,k [Φ] = (cid:18) δ Γ k [Φ] δq ( − p ) δ ¯ q ( p ) + R qk (cid:19) − . (10)In the following, we will not encounter mixed two-pointfunctions. Hence, it is sufficient to define these expres-sion for the combinations quark–anti-quark, meson-meson,gluon-gluon (both transverse) and ghost–anti-ghost. Forthe rest of the manuscript, we drop the redundant secondfield-index for the two-point functions and the propaga-tors. In a slight abuse of notation we define the scalarparts of the two-point functions of the quark, meson,gluon and ghost asΓ (2) q,k ( p ) ≡ δ Γ k [Φ] δq ( − p ) δ ¯ q ( p ) , Γ (2) φ,k ( p ) ≡ δ Γ k [Φ] δφ ( − p ) δφ ( p ) , Γ (2) A,k ( p ) ≡ δ Γ k [Φ] δA ( − p ) δA ( p ) , Γ (2) c,k ( p ) ≡ δ Γ k [Φ] δc ( − p ) δ ¯ c ( p ) . (11)With this we define the corresponding wave-function renor-malisations and (scalar parts of the) propagators Z Φ i ,k ( p ) = ∆Γ (2)Φ i ,k ( p ) / ∆ S (2)Φ i ( p ) (cid:12)(cid:12)(cid:12) scalar part ,G Φ i ,k ( p ) = (cid:16) Z Φ i ,k ( p )∆ S (2)Φ i + R Φ i k ( p ) (cid:17) − (cid:12)(cid:12)(cid:12) scalar part , (12)with Φ i = q, φ, A or c . The scalar part is the coefficientof the tensor structure of the expressions above. In (12)we have ∆Γ (2)Φ i ,k ( p ) = Γ (2)Φ i ,k ( p ) − Γ (2)Φ i ,k (0) for all fieldsexcept for the gluon, where ∆Γ (2) A,k ( p ) = Γ (2) A,k ( p ). Thesame holds true for ∆ S (2)Φ i . At k = 0 and the fields setto their vacuum expectation value, G Φ i ,k =0 ( p ) is the fullpropagator. The above definitions are exemplified withthe full gluon propagator, G abA,k ( p ) = 1 Z A,k ( p ) p + R Ak Π ⊥ δ ab , (13)with the transversal projection operator Π ⊥ , see (C2). Forour calculations, we use four-dimensional Litim regulators R k , [29], for details see App. C.In the infrared regime of QCD, the dynamical degrees offreedom are hadrons, while quarks and gluons are confinedinside hadrons. This entails that a formulation in termsof local composite fields with hadronic quantum numbersis more efficient in this regime. Note that these compositefields are directly related to hadronic observables at theirpoles.Let us illustrate this at the relevant example of thescalar-pseudoscalar mesonic multiplet at a given cutoffscale k . At a fixed large cutoff scale, where the mesonicpotential V k ( ρ ) is assumed to be Gaußian, we can resortto the conventional Hubbard-Stratonovich bosonisation:the local part of the scalar–pseudo-scalar channel of thefour-quark interaction with coupling λ q,k , see the secondline in (1), can be rewritten as a quark-meson term, seethe third line in (1), on the equations of motion for φ ,that is φ EoM . This leads to λ q,k = h k v ,k , φ j, EoM = h k v ,k ¯ q τ j q , (14)where v is the curvature mass of the mesonic field and τ = ( γ (cid:126)T , iT ), j ∈ { , , , } . Note that (14) is onlyvalid for Z φ ≡ V k ( ρ ) = v ρ .Moreover, mis-counting of degrees of freedom may occurfrom an inconsistent distribution of the original four-fermiinteraction strength to the Yukawa coupling and the four-fermi coupling. The dynamical hadronisation techniqueused in the present work, and explained below, resolvesthese potential problems.One advantage of the bosonised formulation concernsthe direct access to spontaneous chiral symmetry breakingvia the order parameter potential V k ( ρ ): spontaneoussymmetry breaking is signaled by v = 0 at the symmetrybreaking scale k χ which relates to a resonant four-quarkinteraction. It also facilitates the access to the symmetry-broken infrared regime.Let us now assume that we have performed the abovecomplete bosonisation at some momentum scale k (cid:29) k χ .There, the above conditions for the bosonisation in (14)are valid. Hence, we can remove the four-fermi term com-pletely in favour of the mesonic Yukawa sector. However,four-quark interactions are dynamically re-generated fromthe RG flow via quark-gluon and quark-meson interac-tions, see Fig. 1.Indeed, these dynamically generated contributions dom-inate due to the increase of the strong coupling α s,k fora large momentum regime, leading to a quasi-fixed pointrunning of the Yukawa coupling, see Refs. [3, 22, 23] andSec. IV. Thus, even though λ q,k was exactly replaced DYNAMIC AL HADRONIZATION • translate 4-quark interaction into Yukawa coupling at scale ! • but: 4-fermi coupling immediately re-generated during RG-flow ⇤ ⇡ ⇤ + ⇤ dk • (some) double counting of 4-fermi interaction at lower scales ! • missing 4-fermi interaction at high scales • no continuous change of d.o.f. continuous translation of UV to IR degrees of freedom Need unified description in terms of one scale dep. effective action! dynamical hadronization (re-bosonization)
Fabian Rennecke, ITP Heidelberg & EMMI ERG, 22.09.2014 conventional bosonization:
Figure 1: Re-generation of four-quark interactions from theRG-flow. + k dk ! Figure 2: sasa scalar-pseudoscalar mesonic multiplet at a given cuto↵scale k . At a fixed large cuto↵ scale, where the mesonicpotential V k ( ⇢ ) is assumed to be Gaußian, we can resortto the conventional Hubbard-Stratonovich bosonisation:the local part of the scalar–pseudo-scalar channel of thefour-quark interaction with coupling q,k , see the secondline in (1), can be rewritten as a quark-meson term, seethe third line in (1), on the equations of motion for ,that is EoM . This leads to q,k = h k v ,k , j, EoM = h k v ,k ¯ q ⌧ j q , (13)where v is the curvature mass of the mesonic field and ⌧ = ( ~T , iT ), j , , , } . Note that (13) is onlyvalid for Z ⌘ V k ( ⇢ ) = v ⇢ .Moreover, mis-counting of degrees of freedom may occurfrom an inconsistent distribution of the original four-fermiinteraction strength to the Yukawa coupling and the four-fermi coupling. The dynamical hadronisation techniqueused in the present work, and explained below, resolvesthese potential problems.One advantage of the bosonised formulation concernsthe direct access to spontaneous chiral symmetry breakingvia the order parameter potential V k ( ⇢ ): spontaneoussymmetry breaking is signaled by v = 0 at the symmetrybreaking scale k which relates to a resonant four-quarkinteraction. It also facilitates the access to the symmetry-broken infrared regime.Let us now assume that we have performed the abovecomplete bosonisation at some momentum scale k k .There, the above conditions for the bosonisation in (13)are valid. Hence, we can remove the four-fermi term com-pletely in favour of the mesonic Yukawa sector. However,four-quark interactions are dynamically re-generated fromthe RG flow via quark-gluon and quark-meson interac-tions, see Fig. 1.Indeed, these dynamically generated contributions dom-inate due to the increase of the strong coupling ↵ s,k fora large momentum regime, leading to a quasi-fixed point running of the Yukawa coupling, see Refs. [3, 22, 23] andalso our discussions below. Thus, even though q,k wasexactly replaced by m ,k and h k at a scale k k , thereis still a non-vanishing RG-flow of q,k at lower scales.Note, however, that we have explicitly checked that thisis only a minor quantitative e↵ect as long as one considerslow-energy e↵ective models, see App. A.In summary, it is not possible to capture the full dy-namics of the system in the quark-gluon phase with theconventional Hubbard-Stratonovich bosonisation. As aconsequence, within conventional bosonisation, the scalewhere composite fields take over the dynamics from fun-damental quarks and gluons is not an emergent scalegenerated by the dynamics of QCD, but is fixed by handby the scale where the Hubbard-Stratonovich transforma-tion is performed.In the present approach we employ dynamical hadro-nisation instead of the conventional bosonisation. It isa formal tool that allows for a unified description of dy-namically changing degrees of freedom and consequentlyis not plagued by the shortcomings of conventional boson-isation discussed above. It has been introduced in [22]and was further developed in [7, 23, 24]. The construc-tion works for general potentials V k ( ⇢ ) (more preciselygeneral k [ ]), and implements the idea of bosonisingmulti-fermion interactions at every scale k rather just atthe initial scale. Consequently, the resulting fields of thisbosonisation procedure, i.e. the mesons, become scale-dependent and can be viewed as hybrid fields: while theyact as conventional mesons at low energies, they encodepure quark dynamics at large energy scales.Here we follow the dynamical hadronisation set-upput forward in [7] and outline the derivation of the flowequation in the presence of scale-dependent meson fields.The starting point is the functional integral represen-tation of the scale-dependent e↵ective action k withscale-dependent meson fields. To this end, we define thedynamical superfield ˆ k = ( ˆ ', ˆ k ), where the microscopicfields are combined in ˆ ' = ( ˆ A µ , ˆ q, ˆ¯ q, ˆ c, ˆ¯ c ) and the scale-dependent meson fields, in our case pions and the sigmameson, are represented by the O (4) field ˆ k = (ˆ ~⇡ k , ˆ k ).The path integral representation of k reads e k [ k ] = Z D ˆ ' exp n S [ ˆ ' ] S k [ ˆ k ] (14)+ ( k + S k ) k ( ˆ k k ) + S k [ k ] , where we defined the expectation value of the fields k = h ˆ k i and used J = ( k + S k ) k and S k [ k ] = 12 k R k k . (15)To arrive at the evolution equation for k [ k ], we take thescale derivative @ t = k ddk of Eq. (14). The RG evolutionof the scale-dependent composite meson fields is of the Figure 1: Re-generation of four-quark interactions from theRG-flow. by m φ,k and h k at a scale k (cid:29) k χ , there is still a non-vanishing RG-flow of λ q,k at lower scales. Note, however,that we have explicitly checked that this is only a minorquantitative effect as long as one considers low-energyeffective models, see App. A.In summary, it is not possible to capture the full dy-namics of the system in the quark-gluon regime withthe conventional Hubbard-Stratonovich bosonisation. Asa consequence, with the conventional bosonisation, thescale where composite fields take over the dynamics fromfundamental quarks and gluons is not an emergent scalegenerated by the dynamics of QCD, but is fixed by handby the scale where the Hubbard-Stratonovich transforma-tion is performed.In the present approach we employ dynamical hadro-nisation instead of the conventional bosonisation. It isa formal tool that allows for a unified description of dy-namically changing degrees of freedom and consequentlyis not plagued by the shortcomings of conventional boson-isation discussed above. It has been introduced in [22]and was further developed in [7, 23, 24]. The construc-tion works for general potentials V k ( ρ ) (more preciselygeneral Γ k [Φ]), and implements the idea of bosonisingmulti-fermion interactions at every scale k rather just atthe initial scale. Consequently, the resulting fields of thisbosonisation procedure, i.e. the mesons, become scale-dependent and can be viewed as hybrid fields: while theyact as conventional mesons at low energies, they encodepure quark dynamics at large energy scales.Here we follow the dynamical hadronisation set-upput forward in [7] and outline the derivation of the flowequation in the presence of scale-dependent meson fields.The starting point is the functional integral represen-tation of the scale-dependent effective action Γ k withscale-dependent meson fields. To this end, we define thedynamical superfield ˆΦ k = ( ˆ ϕ, ˆ φ k ), where the microscopicfields are combined in ˆ ϕ = ( ˆ A µ , ˆ q, ˆ¯ q, ˆ c, ˆ¯ c ) and the scale-dependent meson fields, in our case pions and the sigmameson, are represented by the O (4) field ˆ φ k = (ˆ (cid:126)π k , ˆ σ k ).The path integral representation of Γ k reads e − Γ k [Φ k ] = (cid:90) D ˆ ϕ exp (cid:110) − S [ ˆ ϕ ] − ∆ S k [ ˆΦ k ] (15)+ δ (Γ k + ∆ S k ) δ Φ k ( ˆΦ k − Φ k ) + ∆ S k [Φ k ] (cid:27) , where we defined the expectation value of the fields Φ k = (cid:104) ˆΦ k (cid:105) and used J = δ (Γ k + ∆ S k ) δ Φ k and ∆ S k [Φ k ] = 12 Φ k R k Φ k . (16)Note that the functional integral in (15) contains onlythe fundamental fields ˆ ϕ of QCD. Composite operatorssuch as the (scale dependent) mesons are introduced viacorresponding source terms in the Schwinger functional,see [7].To arrive at the evolution equation for Γ k [Φ k ], we takethe scale derivative ∂ t = k ddk of Eq. (15). The RG evolu-tion of the scale-dependent composite meson fields is ofthe form ∂ t ˆ φ k = ˙ A k ¯ q τ q + ˙ B k ˆ φ k . (17)The first part of this equations reflects the bound statenature of the mesons. The second part corresponds to ageneral rescaling of the fields. The coefficients ˙ A k and ˙ B k ,which we call hadronisation functions, are specified below.Note that the right hand side of (17) only involves thequark mean fields q = (cid:104) ˆ q (cid:105) , ¯ q = (cid:104) ˆ¯ q (cid:105) . An explicit solution tothis equation is given byˆ φ k = C k e B k ¯ q τ q , (18)with ˙ A k = ˙ C k e B k . This reflects the quark-antiquarknature of the meson. Eq. (17) leads to the followingidentity for the flow of the hadronisation field (cid:104) ∂ t ˆ φ k (cid:105) = ˙ A k ¯ q τ q + ˙ B k φ k , (19)and furthermore (cid:104) ∂ t ˆ φ k (cid:105) = ∂ t φ k . Taking (17) into account,the scale derivative of (15) gives a modified version of theflow equation (9). While the gauge and quark parts ofthe equation remain unchanged, the mesonic part nowreads: ∂ t (cid:12)(cid:12) φ Γ k [Φ k ] = 12 Tr (cid:104) G φφ,k [Φ] · (cid:16) ∂ t R φk + 2 R φk ˙ B k (cid:17)(cid:105) − Tr (cid:20) δ Γ k [Φ] δφ i (cid:16) ˙ A k ¯ q τ i q + ˙ B k φ i (cid:17)(cid:21) . (20)The first line of (20) corresponds to the mesonic part of theflow equation (9) with a shift in the scale derivative of theregulator owing to the part of ∂ t φ k which is proportionalto φ k itself. Note that (20) remains valid for the moregeneral flow of the super-field [7] ∂ t ˆΦ i,k = ˙ A ij,k · F j,k [Φ k ] + ˙ B ij,k [Φ k ] ˆΦ j,k , (21)where F [Φ k ] is any functional of the mean super-fieldΦ k . We emphasise that the one-loop nature of the flowequation is not spoiled as long as ∂ t ˆΦ i,k is at most linearin the quantum field ˆΦ i,k . It can, in fact, be an arbitraryfunction of the mean fields Φ i,k without altering the prop-erties of the flow equation. The meson regulator has the form (see App. C) R φk ( p ) = Z φ,k p r B ( p /k ) , (22)and its corresponding scale derivative can convenientlybe written as ∂ t R φk ( p ) = (cid:0) ∂ t (cid:12)(cid:12) Z − η φ,k (cid:1) R φk ( p ) , (23)with the anomalous dimension of the scale-dependentmesons, η φ,k = − ∂ t Z φ,k Z φ,k . (24)This choice of the regulator functions implies that theflow equations of RG-invariant quantities only containthe anomalous dimension which stems from the scalederivative of the regulator whereas the wave-functionrenormalisations drop out completely. With this, we canrewrite (20) into: ∂ t (cid:12)(cid:12) φ Γ k [Φ k ] = 12 Tr (cid:104) G φφ,k [Φ] · (cid:16) ∂ t (cid:12)(cid:12) Z − ( η φ,k − B k ) (cid:17) R φk (cid:105) − Tr (cid:20) δ Γ k [Φ] δφ i (cid:16) ˙ A k ¯ q τ i q + ˙ B k φ i (cid:17)(cid:21) . (25)It is now obvious that the first line of the modified flowequation above gives the original flow equations withoutscale-dependent fields, but with a shifted meson anoma-lous dimension: η φ,k → η φ,k − B k . (26)The other coefficient, ˙ B k , in (17) is at our disposal, andwe may use it to improve our truncation.The second line of (20) induces additional contributionsin particular to the flows of the four-quark and the Yukawacoupling, owing to the particular ansatz we made for ∂ t φ k .This allows us to specify the hadronisation procedure: wechoose the coefficient ˙ A k such that the flow of the four-quark interaction λ q,k vanishes within our truncation, ∂ t λ q,k = 0. This way, all information about the multi-quark correlations are stored in the flow of the Yukawacoupling. Thus, h k encodes the multi-quark correlationsin the quark-gluon regime and the meson–constituent-quark correlations in the hadronic regime, including adynamical transition between these different regimes. B. Hadronised flow equations
In the following we specify the hadronisation procedureand give the resulting modified flow equations of thescale-dependent parameters of the truncation (1). Thesemodifications are given by explicitly evaluating the secondline of (20). Note that the explicit form of the modifiedflow equations depends on the details of our projectionprocedures, see also App. B.In the following, we rescale all fields with their respec-tive wave-function renormalisation, ¯Φ = (cid:112) Z Φ ,k Φ andintroduce the RG-invariant parameters¯ g k = g k Z q,k Z / A,k , ¯ λ q,k = λ q,k Z q,k , ¯ c k = cZ φ,k , (27)¯ λ n,k = λ n,k Z nφ,k , ¯ h n,k = h n,k Z q,k Z (2 n +1) / φ,k , ¯ κ k = Z φ,k κ . Note that the parameters defined in (27) do scale withthe infrared cutoff scale k , but are invariant under gen-eral RG-transformations (reparameterisations) of QCD.For example, ¯ g k is nothing but the running strong cou-pling. The RG-invariant dimensionless masses are definedaccordingly as¯ m q,k = m q,k k Z q,k and ¯ m π/σ,k = m π/σ,k k Z / φ,k . (28)Note that we rescale mesonic parameters with the wave-function renormalisation Z φ,k of the scale-dependentmesons φ k . The constant source c as well as the expan-sion point κ have only canonical running after rescaling,given only by the running of Z φ,k , see Eq. (B3). Conse-quently, we also rescale the hadronisation functions and,in addition, define them to be dimensionless:˙¯ A k = k Z / φ,k Z − q,k ˙ A k , ˙¯ B k = ˙ B k . (29)With this, we proceed now to the modified flow equationsof these RG-invariant quantities.For the flow of the four-quark interaction ¯ λ q,k we find: ∂ t (cid:12)(cid:12) φ ¯ λ q,k = 2 η q,k ¯ λ q,k + ∂ t ¯ λ q,k (cid:12)(cid:12) η φ,k → ˜ η φ,k − B k + (cid:18) ¯ h k (¯ ρ ) + 2¯ ρ ¯ h (cid:48) k (¯ ρ ) 4 N f N c − N f N c + 1 (cid:19) ˙¯ A k . (30)Here, ∂ t ¯ λ q,k denotes the flow without dynamical hadroni-sation which is specified in App. B. As already discussedabove, this contribution is subject to a shift in the mesonanomalous dimension, indicated by η φ,k → η φ,k − B k .Following the discussion in the previous section, wechoose ˙¯ A k such that the flow of ¯ λ q,k vanishes. This isachieved by the following choice:˙¯ A k = − (cid:18) ¯ h k (¯ ρ ) + 2¯ ρ ¯ h (cid:48) k (¯ ρ ) 4 N f N c − N f N c + 1 (cid:19) − × ∂ t ¯ λ q,k (cid:12)(cid:12) η φ,k → η φ,k − B k . (31)Together with the initial condition ¯ λ q, Λ = 0, this yields ∂ t (cid:12)(cid:12) φ ¯ λ q,k = 0 . (32)The flow of the Yukawa coupling assumes the following form: ∂ t (cid:12)(cid:12) φ ¯ h k = (cid:18) η q,k + 12 η φ,k (cid:19) ¯ h k + ∂ t ¯ h k (cid:12)(cid:12) η φ,k → ˜ η φ,k − B k − k (cid:0) p + ¯ V (cid:48) k (¯ ρ ) (cid:1) ˙¯ A k − (cid:0) ¯ h k + 2¯ ρ ¯ h (cid:48) k (cid:1) ˙¯ B k , (33)where ¯ h k = ¯ h k (¯ ρ ) is implied and ∂ t ¯ h k is specified inApp. B. From Eq. (31), it is now clear that the flowof the quark interaction and, therefore, all informationabout the multi-quark correlations within our truncationis incorporated into the flow of the hadronised Yukawacoupling.It is left to specify the hadronisation function ˙¯ B k , whichalso enters (33). We see from Eq. (18) that it correspondsto a phase factor of the hadronisation field. It can beused to improve the current approximation by absorbinga part of the momentum-dependence of the mesonic wave-function renormalisation and the Yukawa coupling. Thiswill be discussed elsewhere. Here, we use˙ B k ≡ , (34)for the sake of simplicity. We see that our hadronisationprocedure enforces a vanishing four-quark interaction.The effect of four-quark correlations is then stored inthe Yukawa coupling, which now serves a dual purpose:while it captures the current-quark self-interactions in thequark-gluon regime, it describes the meson–constituent-quark in the hadronic regime. C. Gauge sector
In this section, we discuss the gauge sector of the trun-cation given in (1). Most importantly, this permits todistinguish the quark-gluon coupling from pure gluody-namics. This directly signals the transition from the per-turbative quark-gluon regime at large momenta, where allcouplings scale canonically, to the hadronic regime wherenon-perturbative effects are dominant.The couplings induced from three-point functions playa dominant role in the description of interactions. Hence,we solve the flow equations for all three-point functionsin QCD, the quark-gluon, three-gluon and ghost-gluonvertices. In addition, the effects from the four-gluonvertex are important [18, 25, 26]. Thus, we employ anansatz which has proven to be accurate in previous studies[25, 26]. For the computation presented here, we takethe gluon and ghost propagators from pure gauge theoryas input [18, 25, 26] and augment them by unquenchingeffects. In the perturbative domain this procedure isaccurate, as the error is order α s,k . At scales below theconfinement transition the gluon is gapped and thereforedecouples from the dynamics.Perturbation theory gives a direct relation between thenumber of gluon legs m attached to the vertex Γ ( n ) andthe order in the strong coupling, Γ ( n ) ∼ (4 πα s,k ) m/ . Nev- @ GeV D ê H Z A , k H p L p + R k A H p LL k = = = = p [GeV] / Z Y M A , k ( p ) p + R A k ( p ) Figure 2: The regulated gluon propagator from pure Yang-Mills theory as a function of the momentum for various k . Weuse this as an external input for our QCD computations. ertheless, the RG running is different, although purely in-duced by the external legs attached. Their wave-functionrenormalisations cancel exactly those from the propaga-tors, see (38) below. As a result of this truncation, theflow equations for couplings depend on the anomalousdimensions only.In this analysis we restrict ourselves to classical tensorstructures of the gauge action S [Φ]. Omitting colourand Lorentz indices for clarity, we parametrise the quark-gluon, three- and four-gluon and the ghost-gluon verticesas Γ (¯ qAq ) k = Z A,k Z q,k g ¯ qAq,k S (3)¯ qAq , Γ ( A ) k = Z A,k g A ,k S (3) A , Γ ( A ) k = Z A,k g A ,k S (4) A , Γ (¯ cAc ) k = Z A,k Z c,k g ¯ cAc,k S (3)¯ cAc . (35)The classical tensor structures S ( n )Φ ... Φ n are obtained from(2) by S ( n )Φ ... Φ n = δ n Γ Λ δ Φ . . . δ Φ n (cid:12)(cid:12)(cid:12)(cid:12) g k =1 , (36)where we have omitted indices for clarity. In this work,we use as input the gluon/ghost two-point functionsΓ (2) , YM A/c,k ( p ) computed in [18, 25, 26] for pure Yang-Millstheory, Γ (2) , YM A,k = Z YM A,k ( p ) p Π ⊥ , Γ (2) , YM c,k = Z YM c,k ( p ) p , (37)where the identity matrix in adjoint color space is im-plied. In Figs. 2 and 3 we show this input as a functionof the momentum p for various k . Note that we showthe regulated gluon propagator and ghost dressing func- @ GeV D p ê H Z c , k H p L p + R kc H p LL p [GeV] p / Z Y M c , k ( p ) p + R c k ( p ) Figure 3: The regulated ghost dressing function from pureYang-Mills theory as a function of the momentum for various k . We use this as an external input for our QCD computations.The labelling is the same as in Fig. 2. tion with optimized regulators R A/ck = ( Z YM A/c,k ( k ) k − Z YM A/c,k ( p ) p ) θ ( k − p ).We want to emphasise that a particular strength of theapproach presented here is that it is independent of thespecific form of the input in the sense that Yang-Millspropagators from any given method can be used. We haveexplicitly checked that our results are not altered if we usee.g. lattice input. In this case, the input dressing functionsare of the form Z YM A/c ( p ) = Z YM A/c,k =0 ( p ) and the RG-scaledependence can be introduced by the identification p = k .In order to make full use of the non-trivial input we usehere, we expand the flow equation for the gluon propagatorin QCD about that in Yang-Mills theory. We use thefreedom in defining the cutoff function R Ak to simplify theanalysis. This is done by choosing the same prefactor Z A,k for the gluon regulator as for the vertex parametrisationsin (35), see Eq. (C1). Note that the gluon propagatorenters in loop integrals with momenta p (cid:46) k . If weestimate the full gluon propagator (13) with the simpleexpression (with the tensor structure omitted for clarity) G A,k ( p ) ≈ Z A,k p + R Ak = 1 Z A,k p (1 + r B ( p /k )) , (38)i.e. we only consider the fully p -dependent Z A,k ( p ) eval-uated at p = k , the system of flow equations consideredis tremendously simplified. The error of such a simpleestimate relates to p (cid:18) Z A,k ( p ) p + R Ak − Z A,k p + R Ak (cid:19) n (39)= p n (cid:32) Z A,k − Z A,k ( p ) (cid:0) Z A,k ( p ) p + R Ak (cid:1) (cid:0) Z A,k p + R Ak (cid:1) (cid:33) n , where the factor p stems from the momentum integration ∼ dp p . The expression in (39) occurs with powers n ≥ p (cid:46) k . For small momenta it tends towards zero whileits value for maximal momenta p ≈ k is proportional tothe difference Z A,k − Z A,k ( k ). Consequently, we choose Z A,k = Z A,k ( k ) . (40)We have checked that the difference between full flowsand approximated flows is less than 5% for all k .Within approximations (35) and (38), the gluon propaga-tor enters flow equations only via the anomalous dimen-sion η A,k with η A,k = − ∂ t Z A,k Z A,k . (41)As a consequence of (40), η A,k has two contributions fromthe full dressing function Z A,k ( p ), ∂ t Z A,k = ∂ t Z A,k ( p ) (cid:12)(cid:12) p = k + k ∂Z A,k ( p ) ∂p (cid:12)(cid:12)(cid:12)(cid:12) p = k . (42)The first term stems from the genuine k -dependence ofthe dressing function, while the second term results fromits momentum dependence. As it is the case for any flowof a coupling in a gapped theory (away from potentialfixed points), the first term of (42) vanishes in the limit k →
0, lim k → ∂ t Z A,k ( p ) (cid:12)(cid:12) p = k = 0 . (43)The second term of (42) carries the information about themomentum dependence of the dressing function and inparticular of the (bare) mass gap m gap at small momenta.The gluon propagator exhibits a gap at small momentumscales and hence the dressing function of the full quantumtheory, Z A,k =0 ( p ), is of the formlim p → Z A,k =0 ( p ) ∝ m p . (44)This implies for the second term in (42)lim k → k∂ p ln ( Z A,k ( p )) (cid:12)(cid:12)(cid:12) p = k = − . (45)Thus, the second term of (42) generates a non-vanishinggluon anomalous dimension η A,k , as defined in (41) for k → k -dependence and the momentum dependence of the gluondressing function is both highly non-trivial and indispens-able in any satisfactory truncation, even on a qualitativelevel. The RG-scale dependence alone does not sufficeto capture the non-perturbative physics of YM theory orQCD in the gauge sector, as it misses the confining prop-erties of the theory. Being of primary importance, the gluon mass gap emerges from the non-trivial momentumdependence of the propagator. We remark that this is incontrast to the chiral properties of the matter sector ofQCD, where approximations based on solely k -dependentparameters at least qualitatively capture all the relevantphysics.It is crucial that Z A,k does not appear explicitly, andhence flows do only depend on η A,k , the vertex couplings g , masses and further couplings. Note that this is onlypartially due to the approximation in (38). It mainlyrelates to the parameterisations (35) of the vertices whichstores most of the non-trivial information in the associatedvertex couplings α i = g i π , with i = ¯ cAc , A , A , ¯ qAq . (46)This freedom directly relates to the reparametrisationinvariance of the theory and hence to RG invariance. Theabove discussion in particular applies to the anomalousdimension itself: first, we note that the glue part η glue ,k of the anomalous dimension η A,k only depends on thevertex couplings: η glue ,k = η glue ,k ( α ¯ cAc , α A , α A ) . (47)In the semi-perturbative regime these couplings agree dueto the (RG-)modified Slavnov–Taylor identities [7, 30–32],which themselves do not restrict the couplings in the non-perturbative transition regime, see Ref. [3]. In turn, inthe non-perturbative regime the couplings differ alreadydue to their different scalings with the gluonic dressing Z A,k . For small cutoff scales k →
0, this dressing divergesproportional to the QCD mass gap,lim k → Z A,k ∝ ¯ m = m k . (48)This is a slight abuse of notation since ¯ m in (48) is notrenormalised as the other dimensionless mass ratios ¯ m .Here it simply relates to the wave-function renormalisation Z A,k defined in (40). Hence, it is not RG-invariant andshould not be confused with the physical mass gap ofQCD. It is related with the latter upon an appropriaterenormalisation.As a consequence, while we expect α ¯ cAc ≈ α ¯ qAq downto small scales, the purely gluonic couplings should besuppressed to compensate the higher powers of diverging Z A,k present in the vertex dressing in (35). This alsoentails that we may parameterise the right hand sidewith powers of 1 /α i . For i = ¯ cAc, ¯ qAq , for example,we expect 1 /α i . In accordance with this observation, weparameterise the difference of the various vertex couplingsin η glue with the gap parameter ¯ m gap defined in (48) andconclude for the gluon anomalous dimension of QCD η A,k = η glue ,k ( α s , ¯ m gap ) + ∆ η A,k ( α ¯ qAq , ¯ m q ) , (49)where α s stands for either α ¯ cAc or α A . We shall check - - a s , k h H a s , k L h + h - h C Figure 4: The UV and IR branches of η YM A , η + and η − , as afunction of the strong coupling. that our results do not depend on this choice which justi-fies the identification of the couplings in (49). Note thatthis does not entail that the couplings agree but thatthey differ only in the regime where the glue fluctuationsdecouple. Moreover, in the present approximation α A isnot computed separately but identified with α A .A simple reduction of (49) is given by η A,k = η YM A,k + ∆ η A,k ( α ¯ qAq , ¯ m q ) . (50)This amounts to a gluon propagator, where the vacuumpolarisation is simply added to the Yang-Mills propagator.This approximation has been used in an earlier work,[1, 2, 10], and subsequently in related Dyson-Schwingerworks, see e.g. [33–36].The term ∆ η A,k is the quark contribution to the gluonanomalous dimension, and is computed with S [ ]. Omitting colourand Lorentz-indices for clarity, we parametrise the quark-gluon, three- and four-gluon and the ghost-gluon verticesas (¯ qAq ) k = Z A,k Z q,k g ¯ qAq,k S (3)¯ qAq , ( AAA ) k = Z A,k g AAA,k S (3) AAA , ( AAAA ) k = Z A,k g AAAA,k S (4) AAAA , (¯ cAc ) k = Z A,k Z c,k g ¯ cAc,k S (3)¯ cAc , (34)with the tensor structures S ( n ) ... n obtained by takingderivatives of the classical action S with respect to thefields entering the vertex before setting the field expec-tation values to their vacuum expectation value and thebare coupling to unity.In this work, we take the two-point functions com-puted in [28, 29], (2) , YM A/c,k ( p ) for the gluon/ghost, as input,whose Z YM A/c,k we define similar to (12). The correspondinganomalous dimensions are given by ⌘ YM A/c,k = @ t Z YM A/c,k Z YM A/c,k . (35)In order to make full use of this non-trivial input weexpand the flow equation for the gluon propagator in QCDabout that in Yang-Mills theory. We use the freedomin defining the cuto↵ function R Ak , see Appendix C, tosimplify the analysis. This is done by choosing the sameprefactor Z A,k for the gluon regulator as for the vertexparameterisations in (34). Note that the gluon propagatorenters in loop integrals with momenta p . k . If weestimate the full gluon propagator (13) with the simpleexpression G A,k ( p ) ⇡ Z A,k p + R Ak = 1 Z A,k p (1 + r B ( p /k )) , (36)i.e. the p -dependence of Z A,k ( p ) is neglected but evaluatedat p = k , the system of flow equations considered is greatlysimplified. The error of such a simple estimate relates to p ✓ Z A,k ( p ) p + R Ak Z A,k p + R Ak ◆ n = p n ⇥ Z A,k Z A,k ( p ) ⇤ Z A,k ( p ) p + R Ak Z A,k p + R Ak ! n (37)The expression in (37) occurs with powers n p . k .For small momenta it tends towards zero while its valuefor maximal momenta p ⇡ k is proportional to the ⌘ A,k = Z A,k N c @ @p ⇧ ? ( p ) ·
48 4. Setting the stage
Vacuum polarisation of the gluon
The vacuum polarisation of the gluon has already been calculated in Ref. [89] in a one-loopRG improved approximation and is given by ⌘ A q = N f M
43 14 ⇡ ↵ s
11 + e ⇡i +
1+ ¯ M ¯ µ ¯ T
11 + e ⇡i +
1+ ¯ M +¯ µ ¯ T . (4.33)The equation we derive here has been studied simultaneously in the same truncation by F.Rennecke, see [160]. Here we give the full results within our truncation and at finite chemicalpotential and temperature and also include wave function renormalisations parallel Z andperpendicular Z to the heat bath, renormalising the zero and the vector component of themomentum.Figure 4.5: The vacuum polarisation of the gluon through the quark.We implement the 3 d regulator given by Eqn. (4.16). To determine the vacuum polarisa-tion of the gluon, i.e. ⌘ A q , we must project onto the lhs of the flow of (2) AA @ t (2) AA = ⇣ ˙ Z A n + ˙ Z A ~p ⌘ ⇧ , dµ ab + 1 ⇠ ⇧ , dµ ab p , (4.34)where the n are the bosonic Matsubara frequencies and we want to project onto the trans-verse component relative to the heat bath (as we are in Landau gauge there is only thestandard transverse part of the propagator but there is a transverse and a longitudinal com-ponent with respect to the heat bath) and there we want the flow of the wave functionrenormalisation proportional to the vector component of the momentum. So we have to per-form two derivatives with respect to the momentum p at vanishing momentum. Dividing bythe negative of the wave function renormalisation we are left with the desired contributionto the anomalous dimension, i.e. the vacuum polarisation of the gluon by the quarks. Therhs is simply given by the same manipulations we have just performed on the lhs and whichwe then apply to the diagram given in Fig. 4.5.So we have to derive the rhs of ⌘ A q = N c
1) 1 Z A ( @ p ✓ ⇧ , dµ ab ⇥ ⇤◆ p =0 ) . (4.35)and actually all we have to do is to calculate the quantity in the curly brackets. The trace ! p =0 di↵erence Z A,k Z A,k ( k ). Consequently, we choose Z A,k = Z A,k ( k ) . (38)We have checked that the di↵erence between full flowsand approximated flows is less than 5%.Within approximation (36) and (34) the gluon propagatoronly enters via the anomalous dimension ⌘ A,k with ⌘ A,k = @ t Z A,k Z A,k . (39)Most importantly, Z A,k does not appears explicitly. Thisalso applies to the anomalous dimension itself which isproportional to ↵ s as the only parameter. Note thatthe couplings ↵ s, ¯ cAc , ↵ s,AAA , ↵ s,A occur. For now, weneglect the di↵erence of the di↵erent vertex couplings andconclude that ⌘ A,k = ↵ s,k ↵ YM s,k ⌘ YM A,k + ⌘ A,k , (40)where ⌘ A,k is the quark contribution to the gluon anoma-lous dimension. It is defined asHere, p is the modulus of the external momentum and⇧ ? is the transversal projection operator defined in (C2).Note that the dots represent the full vertices and the linesthe full propagators. The crossed circle represents theregulator insertion. For N f = 2 and N c = 3 we find ⌘ A,k = 124 ⇡ g qAq,k (1 + ¯ m q,k ) ⇥ ⇥ ⌘ q,k + 8 ¯ m q,k (1 ⌘ q,k ) ¯ m q,k ⇤ . (41)Note that the Yang-Mills anomalous dimension alsocontains a resummation term and its full dependenceon ↵ s is of the type ↵ s / (1 + c ↵ s ). In (40) we have notconsidered the change in c ↵ s . Also, we have checked thatthe results in the matter sector do not change if takingeither ↵ s, ¯ cAc , ↵ s,AAA = ↵ s,A in (40) in the current work.The same local approximation can be applied to theghost, leading to ⌘ c,k = ↵ s,k ↵ YM s,k ⌘ YM c,k , (42)where ↵ s,k = ↵ s, ¯ cAc,k . This modification is used in theequation for the ghost-gluon vertex.Finally, this allows us to determine the ghost and gluon (51)Here, p is the modulus of the external momentum andΠ ⊥ is the transversal projection operator defined in (C2).Note that the dots represent full vertices and the linesstand for full propagators. The crossed circle representsthe regulator insertion. For N f = 2 and N c = 3 we find∆ η A,k = 124 π g qAq,k (1 + ¯ m q,k ) − × (cid:2) − η q,k + 4 ¯ m q,k − (1 − η q,k ) ¯ m q,k (cid:3) . (52)The approximation (51) works well as long as the quarkcontribution has only a mild momentum dependence.This is the case due to the gapping of the quarks viaspontaneous chiral symmetry breaking, and has beenchecked explicitly. A necessary check for the validity ofthis equation is that it reduces to the perturbative resultin the corresponding limit, i.e. η q,k , ¯ m q,k →
0. Indeed, - - - a s , k h Y M H k L h YM + h YM - Figure 5: The UV and IR branches of η YM A,k ( k ), which isdefined in (59). (52) reduces to one-loop perturbation theory in this case,∆ η A,k = g qAq,k / (6 π ).This leaves us with the task of determining η glue ,k ( α s , ¯ m ), the pure glue contribution to η A,k . Theloop expression for η glue only consists of Yang-Mills dia-grams. As it depends solely on the value of the coupling α s we arrive at η glue ( α s , ¯ m QCDgap ) = η YM A ( α s , ¯ m QCDgap ) , (53)i.e. the pure gauge part of the gluon anomalous dimensionof QCD is identical to the gluon anomalous dimensionof pure Yang-Mills theory however driven by the QCDcouplings. η YM A can be determined in Yang-Mills theoryor in quenched QCD as a function of α s and ¯ m gap .For using (53), a trackable form of η YM A as well as ¯ m QCDgap is required. To this end, we first note that η ( α s,k ) is amulti-valued function in both Yang-Mills theory/quenchedQCD and QCD, see Fig. 4. The two branches meet at k = k peak (peak of the coupling) with ∂ t α s,k | k = k peak = 0 . (54)We have a UV branch η + ( α s , ¯ m gap ) for k > k peak andan IR branch η − ( α s , ¯ m gap ) for k < k peak . In Fig. 4 weshow η YM A as a function of the coupling. Interestingly, η + ( α s,k ) is well-described by a quadratic fit in α s upto couplings close to α s,k peak . In turn, η − ( α s,k ) is well-described as a function of the cutoff scale as indicatedby (48). In the deep IR the gluon dressing function isdetermined by the bare gap, Z A,k → ∝ m /k , see alsothe discussion around (45). Hence we havelim k → η A,k = 2 . (55)This is seen in Fig. 4. We also see in this figure thatthe whole IR branch η − is almost constant. This impliesthat the mass gap which suppresses α s,k develops quicklyaround k ≈ k peak and remains roughly constant for the restof the flow for k (cid:46) k peak . This allows us to parametrise0the IR-branch in terms of the RG-scale, η − = 2 − c − k , with c − = 2 − η YM A ( α peak ) k , (56)where the mass gap ¯ m relates to η YM A ( α peak ). Note thatthe quality of these simple fits entails that the transitionfrom the semi-perturbative regime to the non-perturbativeIR regime happens quite rapidly and asymptotic fits inboth areas work very well. In summary we arrive at thefinal representation of η YM A with η YM A,k ( α s,k ) = η + ( α s,k ) θ ( α s,k − α s, peak )+ η − ( k ) θ ( α s, peak − α s,k ) . (57)Inserting (57) on the right hand side of (53) gives usa closed equation for η A,k in (49). Its integration alsoprovides us with the QCD mass gap.The same analysis as for η A, k can be applied to theghost anomalous dimension η c, k leading to a similar rep-resentation with the only difference that η c,k =0 = 0. Itturns out that an even simpler global linear fit givesquantitatively reliable results for matter correlations, η c,k ( α s,k ) = α s,k α η YM c,k ( α ) , (58)where α s,k = α ¯ cAc,k , see Fig. 4. This modification is usedin the equation for the ghost-gluon vertex. Note thatthis overestimates ghost-gluon correlations in the deepinfrared where the glue-sector has decoupled from thematter sector. Hence this is of no relevance for the physicsof chiral symmetry breaking discussed in the present work.We are now in a position to finally determine the ghostand gluon propagators at vanishing cutoff scale in dynam-ical QCD. Again, we could use the α, ¯ m gap representationfor extracting the full dressing function Z A,k ( p ) on the ba-sis of the results. To that end, the momentum-dependentflows as functions of α, ¯ m gap are required, η YM A,k ( p ) = − ∂ t Z YM A,k ( p ) Z YM A,k ( p ) , ∂ t ∆ η A,k ( p ) , (59)where ∆ η A,k ( p ) stands for the momentum-dependent flowof the vacuum polarisation. The first term in (59) againis well approximated in terms of a low order polynomialin α s . This is expected because is relates directly to thestandard anomalous dimension of the gluon. In Fig. 5 itis shown for momentum p = k as a function of α s,k . Thedefinition of η YM A,k ( p ) implies that only the first term in(42) contributes here. Thus, for vanishing k (44) holdsand hence lim k → η YM A,k ( k ) = 0 as observed in Fig. 5.An already very good estimate for the dressing functionis Z A,k =0 ( p ) (cid:39) Z A,k = p ( p ) = Z A,k = p , (60)as the flow of the propagators decay rapidly for momenta @ GeV D ê Z A ê Z A , k = p ê Z A ,0 H p L Figure 6: Comparison of the momentum dependent gluondressing function Z A, ( p ) and Z A,k = p . larger than the cutoff scale, p (cid:38) k . Moreover, the mo-mentum derivative of the dressing is only large in theUV-IR transition regime. In Fig. 6, the inverse dress-ing 1 /Z A, ( p ) and its approximation 1 /Z A,p are shown.Clearly, there are only minor deviations in the UV-IRtransition regime. The same argument holds true to aneven better degree for the quark contribution, and wehave checked the smoothness of the flow ∆Γ
A,k ( p ). Thisleads to a very simple, but quantitative estimate for thefull dressing function with Z glue A/c,k =0 ( p ) (cid:39) Z YM A/c,k =0 ( k α ) Z YM A/c,k α Z glue A/c,k = p , (61)with Z glue A/c,k = exp (cid:26) − (cid:90) p Λ dkk η glue A/c,k (cid:27) , (62)where Z A/c, Λ = 1, and k α = k ( α s,k ) is the YM-cutoffvalue that belongs to a given coupling α s .In summary we conclude that, based on Fig. 6, an al-ready quantitative approximation to the fully unquenchedpropagator is done if putting the ratio in (61) to unity.This leads to Z A/c ( p ) (cid:39) exp (cid:26) − (cid:90) p Λ dkk η A/c,k (cid:27) , (63)with η A/c,k defined in (49). In the non-perturbativeregime diagrams involving an internal gluon are sup-pressed with the generated gluon mass. Hence, albeitthe approximation by itself may get less quantitative inthe infrared, the error propagation in the computation issmall.In summary this leaves us with relatively simple ana-lytic flow equations for the fully back-coupled unquench-ing effects of glue and ghost propagators. A full erroranalysis of the analytic approximations here will be pub-1lished elsewhere, and is very important for the reliableapplication of the present procedure to finite temperatureand density.In the following, we will outline the definition andderivation of the gluonic vertices we use. First of all, weonly take into account the classical tensor structure ofthe vertices. Moreover, throughout this work, we definethe running coupling at vanishing external momentum.Together with our choice for the regulators, this has theadvantage that the flow equations are analytical equa-tions. In particular, loop-momentum integrations canbe performed analytically. This approximation is semi-qunatitative as long as the dressing of the classical tensorstructures do not show a significant momentum depen-dence, and the other tensor structures are suppressed.This approximation is motivated by results on purelygluonic vertices, see Refs. [25, 37–44], which show non-trivial momentum-dependencies only in momentum regionwhere the gluon sector already starts to decouple fromthe system. In turn, the tensor structures and momentumdependences of the quark-gluon vertex are important, seethe DSE studies [45–47] and the recent fully quantitativeFRG study [3]. To take this effectively into account, weintroduce an infrared-strength function for the strongcouplings, which is discussed at the end of this sectionand in App. D.To extract the flow of the quark-gluon coupling g ¯ qAq ,we use the following projection procedure, ∂ t g ¯ qAq = 18 N f ( N c − × lim p → Tr (cid:18) γ µ t a ∂ t Γ k δqδA aµ δ ¯ q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Φ=Φ , (64)which leads to the equation ∂ t g ¯ qAq,k =12 ( η A,k + 2 η q,k ) g ¯ qAq,k − v ( d ) g ¯ qAq,k ¯ h k (cid:110) N ( m )2 , ( ¯ m q,k , ¯ m σ,k ; η q,k , η φ,k )+ ( N f − N ( m )2 , ( ¯ m q,k , ¯ m π,k ; η q,k , η φ,k ) (cid:111) + g qAq,k v ( d ) N c N ( g )2 , ( ¯ m q,k ; η q,k , η A,k )+ g qAq,k g A ,k v ( d ) N c N ( g )1 , ( ¯ m q,k ; η q,k , η A,k ) . (65)The threshold functions appearing on the right-hand sidecan be found in the App. C. For the quark-gluon ver-tex, no ghost diagrams are present. Furthermore, themesonic contributions dominate in the infrared. Thesecontributions have the same sign as the gluonic ones andtherefore lead to an effective infrared enhancement ofthe quark-gluon vertex. The three-gluon vertex g A ,k is @ GeV D a s a q Aq a AAA a c Ac a - Loop
Figure 7: The running of the different strong couplings incomparison to the 1-loop running. Since perturbation theorybreaks down at the scale where the strong couplings start todeviate from each other, we show the 1-loop running onlydown to 1 GeV. defined via ∂ t g A ,k = i N c ( N c −
1) lim p → ∂ ∂p (66)Tr (cid:18) δ µν p σ f abc ∂ t Γ k δA ( p ) aµ δA ( − p ) bν δA cσ (0) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Φ=Φ . Note that in the limit of vanishing external momentumthe flow is independent of the kinematic configurationin the projection procedure. Thus, we find for the flowequation for N c = 3 and N f = 2 ∂ t g A ,k = 32 η A,k g A ,k − π g qAq,k (cid:16) − η q,k (cid:17) (1 + 2 ¯ m q,k )(1 + 2 ¯ m q,k ) + 364 π g A ,k (11 − η A )+ 164 π g cAc,k (cid:16) − η C,k (cid:17) , (67)with the ghost anomalous dimension η C,k = − ( ∂ t Z C,k ( k )) /Z C,k ( k ). The second line in (67)corresponds to the quark-triangle diagram and thethird and fourth line are the gluon- and ghost-trianglediagrams, respectively. Note that the third line alsoincludes the contribution from the diagram containingthe four-gluon vertex, which we approximate as explainedbelow.Within our approximation, the ghost-gluon vertex g ¯ cAc,k has only canonical running since the diagramsthat contribute to the flow of g ¯ cAc,k are proportional tothe external momentum. Thus, at vanishing external2momentum they vanish and we are left with: ∂ t g ¯ cAc,k = (cid:18) η A,k + η C,k (cid:19) g ¯ cAc,k . (68)Lastly, we comment on our approximation for the four-gluon vertex g A ,k . For the sake of simplicity, we restricthere to a semi-perturbative ansatz for this vertex, whichensures that g A ,k has the correct perturbative running.To this end, we set g A ,k = g A ,k . (69)This approximation is valid for k (cid:38) . glue in (1):∆Γ glue = (cid:90) x (cid:26) (cid:18) F (cid:12)(cid:12)(cid:12) g k = g A ,k − F (cid:12)(cid:12)(cid:12) g k (cid:19) + ¯ c a ∂ µ (cid:18) D abµ (cid:12)(cid:12)(cid:12) g k = g ¯ cAc,k − D abµ (cid:12)(cid:12)(cid:12) g k (cid:19) c b + Z q,k ¯ q γ µ (cid:18) D µ (cid:12)(cid:12)(cid:12) g k = g ¯ qAq,k − D µ (cid:12)(cid:12)(cid:12) g k (cid:19) q (cid:27) , (70)where we used the abbreviation F = F aµν F aµν . We seethat ∆Γ glue corrects for distinctive coupling strengths forinteraction terms. While perturbation theory ensures thatall couplings agree in the UV, non-perturbative effectslead to differing behaviour in the mid-momentum and IRregime.The result for the different running couplings discussedhere is shown in Fig. 7. While they all agree with eachother and follow the perturbative running at scales k (cid:38) g ¯ cAc,k vanish withinour approximation, they give finite contributions at non-vanishing momenta. This was studied in more detail inthe case of quenched QCD [3]. Indeed, it turned out thatboth, momentum dependencies and the inclusion of non-classical vertices, lead to large quantitative effects. It wasshown there that within an extended truncation the ap-proach put forward in the present work leads to excellentquantitative agreement with lattice QCD studies. @ GeV D h k h L= = h L= = h L= = h L= = h L= = h L= = Figure 8: Yukawa coupling as a function of the RG scale forvarious initial scales Λ and initial conditions h Λ . We take the findings in [3] as a guideline for a phe-nomenological modification of the gauge couplings. Ef-fectively this provides additional infrared strength to thegauge couplings in the non-perturbative regime with k (cid:46) IV. RESULTS
First we summarize the system of flow equations usedin the present work. The effective potential ¯ V k (¯ ρ ) andthe Yukawa coupling ¯ h k (¯ ρ ) are expanded about a fixedbare field as shown in (5). These expansions are alreadyfully converged for N V = 5 and N h = 3, for a detaileddiscussion see [27]. The flow equations for the effectivepotential and its expansion coefficients are given by (B1)and (B2). For the Yukawa coupling they are given by (B6)and (B7) in the case of scale-independent meson fields.The latter are modified by dynamical hadronisation whichresults in (33) for the final flow of the Yukawa coupling.The flows of the renormalised expansion point ¯ κ k andthe explicit symmetry breaking ¯ c k are purely canonicaland given by (B3). In order to accurately capture thephysics in the IR, we choose the expansion point suchthat it matches the minimum of the renormalised effectivepotential at k = 0, ¯ κ k =0 = ¯ ρ ,k =0 , cf. [27] for details.Owing to dynamical hadronisation, the flow of the fourquark interaction ¯ λ q,k for scale-independent fields entersthrough the flow of the Yukawa coupling and is given by(B8). Following our construction discussed in Sect. III B,the flow of ¯ λ q,k vanishes in the presence of the scale-dependent mesons. The RG flows of the quark-gluon, thethree-gluon and the ghost-gluon couplings are given by(65), (67) and (68). Owing to our construction of thevertices, see (35), non-trivial momentum dependencies of3 @ GeV D ê Z A , k = p QCDYM H input L QCD H reduced L Figure 9: Comparison between the quenched and theunquenched running gluon propagators 1 /Z YM A,k ( k ) and1 /Z A,k ( k ) as defined in Eq. (61). We also show the curve forQCD (reduced) where the gluon propagator is a direct sum ofYang-Mills propagator and vacuum polarisation, see Eq. (50). the propagators enter solely through the correspondinganomalous dimensions η Φ ,k . For the mesons and quarksthey are given by (B11) and (B13). The parametrisationof the gluon and ghost anomalous dimensions is discussedin section III C. The gluon anomalous dimension η A,k isdefined by (49) and contains the pure gauge part and thevacuum polarization. The vacuum polarization is givenby (52). The pure gauge part is constructed from the fullgluon anomalous dimension of pure Yang-Mills theory,which we use as an input. It is computed from (53) and(57) with α s,k = α ¯ cAc,k . The ghost anomalous dimensionof QCD is computed from (58), where we also augmentthe input from pure Yang-Mills theory by correcting forthe differences between the strong couplings of YM andQCD, which, in turn, are computed here. Together withthe fact that we evaluate all flows at vanishing externalmomentum, this leads to a set of ordinary differentialequations in the RG scale k which can easily be solved.The starting point of the present analysis is the micro-scopic action of QCD. We therefore initiate the RG flow atlarge scales, deep in the perturbative regime. The initialvalues for the strong couplings are fixed by the value ofthe strong coupling obtained from 1-loop perturbationtheory. Since the different strong couplings we use here(see Eq. (46)) need to be identical in the perturbativeregime, they consequently have the same initial value α s . It is shown in Fig. 7 that indeed the different strongcouplings agree to a high degree of accuracy with the 1-loop running of the strong coupling for scales k > α s implicitly determines the absolute physical scale. Herewe choose α s, Λ = 0 . ≈
20 GeV. Aquantitative determination requires the determination ofthe RG-condition in relation to standard ones such as the @ GeV D ê Z q , k H unquenched L ê Z q , k H quenched L Blank k m q , k H unquenched L k m q , k H quenched L Figure 10: Dressing function (red) and mass (blue) of thequark as function of the RG scale at vanishing momentum.We compare our present model (solid) to the quenched model(dashed) with the parameters fixed to match those of [3].
MS-scheme as well as the extraction of α s,k =0 ( p = Λ), us-ing Λ as the renormalisation point. This goes beyond thescope of the present paper and we shall restrict ourselvesto observables that are ratios of scales, our absolute scalesare determined in terms of Λ = 20 GeV. The other micro-scopic parameter of QCD, the current quark mass, is inour case fixed by fixing the symmetry breaking parameter c . We choose ¯ c Λ = 3 . which yields a infrared pionmass of M π, = 137 MeV; M k = k ¯ m k is the renormalizeddimensionful mass.Note that the masses defined in Eq. (6), and hence inparticular M π, , are curvature masses, i.e. the Euclideantwo-point functions evaluated at vanishing momentum.However, it is the pole masses, defined via the poles of thepropagators, that are measured in the experiments. More-over, curvature and pole masses do not necessarily agree.In the present work, this difference is potentially of im-portance for the accurate determination of the pion mass.Now we use that curvature and pole masses are close forweakly momentum dependent wave function renormalisa-tions, for a detailed discussion see [48]. There it also hasbeen shown that the pion wave function renormalisation isindeed weakly momentum dependent, and pion curvatureand pole mass deviate by less than 1%. It has been alsoshown in [48] that the large deviation of pion pole andcurvature masses seen in previous works, [49], originatesin the local potential approximation (LPA). Moreover, ascale-dependent, but momentum-independent, wave func-tion renormalisation already removes the discrepanciesseen in LPA, and the results agree well within the 1% level.In summary, curvature and pole mass of the pion agreeon the 1% level. The inclusion of momentum-independentrunning wave function renormalisations, as in the presentwork, guarantees quantitative reliability for this issue.Since mesons are not present in the perturbative regime,we only have to make sure that this sector is decoupled4 @ GeV D m a ss @ G e V D quark masspion masssigma massmass > k - - Figure 11: The renormalized quark, pion and sigma masses asa function of the RG scale. The inset figure shows the massesfor a larger range of scales. The shaded gray area indicateswhich fields contribute dynamically: masses within the grayarea exceed the cutoff scale and the corresponding fields aretherefore decoupled from the dynamics. On the other hand,fields with masses within the white area are dynamical. at the initial scale. We therefore choose M π, Λ = M σ, Λ =10 Λ . Our results are independent of the choice of theinitial masses and the Yukawa coupling as long as theinitial four-fermi coupling related to it is far smaller than α s . This is demonstrated for the Yukawa coupling inFig. 8, where we see that, with initial values that differby many orders of magnitude, we always get the samesolution in the IR. Loosely speaking, the memory of theinitial conditions is lost in the RG flow towards the IRregime due to the presence to a pseudo fixed-point onintermediate scales, see also Ref. [23].In the present work we have studied the unquenchingeffects due to the full back-coupling of the matter dy-namics to the glue sector. In an earlier work,[2, 10], wedirectly identified η glue ,k = η YM A,k at the same cutoff scale k , see Eq. (50). This simply adds the vacuum polarisa-tion to the Yang-Mills propagator without feedback. Itis well-adapted for taking into account qualitatively evenrelatively large matter contributions to the gluonic flow:the main effect of the matter back-coupling is the modifi-cation of scales, most importantly Λ QCD , which is alreadycaptured well in (one-loop) perturbation theory, if the ini-tial scale is not chosen too large. This approximation hasalso been subsequently used in related Dyson-Schwingerworks, see e.g. [33–36], extending the analysis also to finitedensity. Here, we improve these approximations by takinginto account the back-reaction of matter fluctuations onthe pure gauge sector. Furthermore, the gluon vacuumpolarization was based on a one-loop improved approxi-mation in previous FRG studies. Here, we compute thefull vacuum polarization self-consistently.In Fig. 9 we show the quenched and unquenched gluonpropagators. The quenched gluon propagator is a FRGinput from [18, 26]. We clearly see that the screeningeffects of dynamical quarks decrease the strength of the @ GeV D ê H + m k L quarkpionsigma Figure 12: Dimensionless RG-invariant propagators as func-tions of the RG scale. gluon propagator. Fig. 9 also shows the partially un-quenched results (denoted by “QCD (reduced)” in Fig. 9)for the propagator. Here, partially unquenched refersto an approximation, where the gluon propagator is adirect sum of Yang-Mills propagator and vacuum polari-sation, see Eq. (50). It shows deviations from the fullyunquenched computation. This is seemingly surprising asit is well-tested that partial unquenching works well evenat finite temperature, see e.g. [2, 10, 33–36]. However,we first notice that the importance of quark flucutationsis decreased at finite temperature due to the Matsub-ara gapping of the quarks relative to the gluons. Thisimproves the reliability of the partial unquenching re-sults. Moreover, in these works the infrared strength isphenomenologically adjusted with the constituent quarkmass in the vacuum. This effectively accounts for thedifference between unquenching and partial unquenching.Note that this finding rather supports the stability andpredictive power of functional approaches.On the other hand this also entails that the full un-quenching potentially is relevant in situations where thevacuum balance between pure glue fluctuations and quarkfluctuations is changed due to an enhancement of thequark fluctuations. Prominent cases are QCD with alarge number of flavours, and in particular QCD at fi-nite density. Indeed, (49) even shows the self-amplifyingeffect at large quark flucutations: The sign of the correc-tion by ∆ η A,k is such that when it grows large, the ratio α s, QCD /α s, YM decreases as does η glue and the importanceof the matter fluctuations is further increased. A moredetailed study of this dynamics in the above mentionedsituations is deferred to a subsequent publication.Using the same parameters as in Ref. [3], we com-pare the quenched and unquenched quark propagators inFig. 10. As for the gluon propagator, Fig. 9, we see largeunquenching effects. Unquenching results in smaller quarkmasses (blue lines) and larger wave function renormaliza-tions Z q,k , and, therefore, enhanced quark fluctuations,as expected. Furthermore, we see that the generation of5 - - @ GeV D Z f , k Figure 13: Wave-function renormalization of the mesons. constituent quark masses takes place at smaller scales inthe unquenched case. This can again be traced back toscreening effects: The effects of gauge fluctuations aresuppressed in the presence of dynamical quarks and leadto weaker gauge couplings. Since the strength of the gaugecouplings triggers chiral symmetry breaking, criticality ofthe four-quark interactions is reached later in the flow forweaker gauge couplings. Hence, chiral symmetry breakingtakes place at smaller scales in the presence of dynamicalquarks.The results for the different running gauge couplings α ¯ qAq , α ¯ cAc and α A discussed in Sec. III C are shownin Fig. 7. At scales k (cid:38) Z A,k .Owing to our construction for the vertices and the gluonpropagator, (35) and (38), all non-trivial informationsabout the gauge sector are encoded in the gauge couplings.In particular, they genuinely involve powers of Z / A,k thatcorrespond the number of external gluon legs attached tothem. Hence, the more external gluonic legs the couplinghas, the more its strength is suppressed by the emerginggluon mass gap. This explains why the three-gluon vertex α A is much weaker in the non-perturbative regime than α ¯ qAq and α ¯ cAc : it is suppressed by Z / A,k , while the quark-gluon and ghost gluon couplings are only suppressed by Z / A,k . The gluon dressing function as we defined it herediverges for k →
0, and, thus, all gauge couplings becomezero in this limit.The fact that α ¯ cAc is weaker than α ¯ qAq can be at-tributed to the neglected momentum dependencies in thissector. Since all diagrams that drive the flow of the ghost-gluon vertex are proportional to the external momentum,they vanish for our approximation and α ¯ cAc only runscanonically, see (68). If these momentum-dependencies @ GeV D m k @ G e V D m p , k m s , k Figure 14: The masses m π/σ,k = (cid:113) Γ (2) σ/π (0) = kZ / φ,k ¯ m π/σ,k ofthe mesons. were taken into account, the ghost-gluon vertex wouldeven be stronger than the quark-gluon vertex, at least inthe quenched case [3].The present approach allows an easy access to the rela-tive importance of quantum fluctuations of the respectivefields: we find that for the renormalised, dimensionlessmass being larger than one,¯ m = m Z Φ k ≥ , (71)all threshold functions that depend on the propagator ofthe respective field mode are suppressed with powers of1 / ¯ m . This entails that the dynamics of the system is notsensitive to fluctuations of this field. In turn, for ¯ m ≤ m = 1is not a strict boundary for the relevance of the dynamics.In Figs. 11 and 12 we show ¯ m for the matter fields.In the shaded area the condition (71) applies, and therespective matter fields do not contribute to the dynamics.This already leads to the important observation that theresonant mesonic fluctuations are only important for thedynamics in a small momentum regime with momenta p (cid:46)
800 MeV, see also Fig. 12. While the σ - and quark-modes decouple rather quickly at about 300 - 400 MeV,the (cid:126)π as a pseudo-Goldstone mode decouples at its massscale of about 140 MeV.In turn, in the ultraviolet regime, the mesonic modesdecouple very rapidly, see Fig. 12 for the size of thepropagator measured in units of the cutoff. At about 800MeV this ratio is already 0.1 and above this scale themesonic modes are not important, and QCD quickly iswell-described by quark-gluon dynamics without resonantinteractions. This observation is complementary to thefact that the initial condition of the Yukawa coupling doesnot play a role for the physics at vanishing coupling, seeFig. 8. For all initial cutoff scales Λ (cid:38) h at k = 0.We add that the Yukawa coupling relates to the ratiobetween constituent quark mass and the vacuum expec-tation value of the field ¯ σ ,¯ h = ¯ m q ¯ σ . (72)Note that it cannot be tuned and is a predicition of thetheory. On the other hand, in low-energy model studies,the (renormalised) quantities ¯ m q and ¯ σ correspondingto physical observables are related to model parameters,and have to be tuned such that ¯ m q and ¯ σ assume theirphysical values.The decoupling of meson degrees of freedom is alsoreflected in the behaviour of the meson wave-functionrenormalisation Z φ,k shown in Fig. 13. Starting at scales k >
500 MeV, Z φ,k decreases very rapidly towards the UV.There, it is about seven orders of magnitude smaller thanin the hadronic regime, where it is O (1). Furthermore,the masses m π/σ,k = Γ (2) σ/π ( p = 0) = Z φ,k M π/σ,k becomescale-independent for k >
800 MeV as shown in Fig. 14.This implies that the meson sector becomes trivial be-yond this scale. We see that the drastic decrease of themeson wave-function renormalisation triggers the largerenormalised meson masses M π/σ,k = m π/σ,k /Z φ,k shownin Fig. 11, which are responsible for the suppression of thedynamics of the meson sector at scales k >
800 MeV. Inturn, this implies that if we start with decoupled mesonsin the UV as in the present case, i.e. initial meson massesmuch larger than the cutoff, the running of Z φ,k drivesthe meson masses to their small values in the IR. Withoutthis peculiar behaviour of the meson wave function renor-malisation, the meson masses would never become smallerthan the cutoff scale and hence meson dynamics could notbe generated dynamically. The fact that our results areindependent of the exact value of the initial renormalisedmeson mass M φ, Λ (cid:38) Λ implies that the running of Z φ,k depends on the initial value M φ, Λ . Indeed, if we choosean initial meson mass that is one order of magnitudesmaller (larger), Z φ,k falls off two orders of magnitudeless (more). This is a direct consequence of the definitionof the renormalised mass, c.f. (28) with M k = k ¯ m k , andthe observation that the running of the meson masses isexclusively driven by Z φ,k in the UV, cf. Fig. 14. Notethat this behaviour of Z φ,k has consequences also for lowenergy models in the local potential approximation, sincefor scales larger than about 800 MeV, the effect of runningwave-function renormalisations can not be neglected.Finally, we discuss further consequences of our findingsfor low energy effective models. To that end we note thatthe gluon modes decouple at momenta below 500 − V. CONCLUSIONS & OUTLOOK
In the present work, we have set up a non-perturbativeFRG approach to QCD, concentrating on the effects of afull unquenching of the glue sector. We also provided adetailed study of the fluctuation physics in the transitionregion from the quark-gluon regime to the hadronic regime.This includes a discussion of the relative importance ofthe fluctuations of quark, meson and glue fluctuations. Adetailed discussion is found in the previous section.Here we simply summarise the main results. Firstly,we have shown that the full back-coupling of the matterfluctuations in the glue sector also plays a quantitativerole in the vacuum. In the present two-flavour case, itaccounts for about 10-15% of fluctuation strength in thestrongly correlated regime at about 1 GeV. This hintsstrongly at the importance of these effects in particular atfinite density, where the importance of quark fluctuationsis further increased and the effect is amplified.Secondly, the still qualitative nature of the present ap-proximation necessitates the adjustment of the infraredcoupling strength, fixed with the constituent quark mass.However, the inclusion of dynamical hadronisation whichre-enforces the four-fermion running, this phenomeno-logical tuning is much reduced. In future work we planto utilise the findings of the quantitative study [3] inquenched QCD for improving our current approximationtowards quantitative precision, while still keeping its rela-tive simplicity.Finally, we have also discussed how low energy effectivemodels emerge dynamically within the present set-up dueto the decoupling of the glue sector: the present resultsand their extensions can be used to systematically improvethe reliability of low energy effective models by simplycomputing the effective Lagrangian of these models attheir physical UV cutoff scale of about 500 - 700 MeV.Moreover, the temperature- and density-dependence of7the model parameters at this UV scale can be computedwithin the present set up.Future work aims at a fully quantitative unquenchedstudy by also utilising the results of [3], as well as studyingthe dynamics at finite temperature and density.
Acknowledgments — We are greatful to Lisa M. Haasfor many discussions and collaboration in an early stageof the project. We thank Tina Herbst, Mario Mitterand Nils Strodthoff for discussions and collaboration onrelated projects. J.B. acknowledges support by HIC forFAIR within the LOEWE program of the State of Hesse.Moreover, this work is supported by the Helmholtz Al-liance HA216/EMMI and by ERC-AdG-290623. L.F. issupported by the European Research Council under theAdvanced Investigator Grant ERC-AD-267258.
Appendix A: Dynamical hadronisation and lowenergy effective models
In low energy models of QCD, such as (Polyakov-loopenhanced) Nambu–Jona-Lasinio models or quark-mesonmodels, gluons are considered to be integrated out andone is left with effective four-quark interactions, eitherexplicitly or in a bosonised formulation. The latter isparticularly convenient as the phase with spontaneousbroken chiral symmetry is easily accessible. There, theformulation of the effective theory is usually based on theconventional Hubbard-Stratonovich bosonization ratherthan dynamical hadronisation. Following our argumentsgiven in Sect. III A, the question arises whether dynamicalhadronisation leads to quantitative and/or qualitativecorrections in the context of low energy effective model.Since the matter part of our truncation (1) is that ofa quark-meson model, we will consider here the specialcase of the quark-meson model defined by switching off allgluon contributions in (1). To see the effect of dynamicalhadronisation, we look at the ratios of IR observablesobtained with and without dynamical hadronisation. Tothis end, we choose Λ LE = 1 GeV as a typical UV-cutoffscale and use the same set of initial conditions in bothcases. For results see Tab. I. f π / ˜ f π M q / ˜ M q M π / ˜ M π M σ / ˜ M σ . Table I: Effect of dynamical hadronisation on a quark-mesonmodel: The quantities with/without a tilde are the resultsobtain from a solution of the flow equations of the quark-mesonmodel with/without dynamical hadronisation techniques.
We see that the effect of dynamical hadronisation onphysical observables of a low-energy quark-meson model(without gluons) is negligible, since it only gives correc-tions of less than 1%. This does not change if we varythe UV-cutoff within the range of typical values for thistype of models, i.e Λ LE ∈ [0 . , .
5] GeV. Furthermore, it implies in particular that the mis-counting problem dis-cussed in Sect. III A is less severe in low energy models.This observation can be understood by looking at theflow of the four-quark interaction λ q,k , see Eq. (B8). Incase of the quark-meson model, only the meson box dia-grams ∼ h k contribute to the flow, see also Fig. 1, whilethe gluon box diagrams are neglected. In the chirallysymmetric regime, the mesons are decoupled and thecorresponding contributions to the flow are therefore sup-pressed. Furthermore, in the hadronic regime, the quarksacquire a large constituent mass and, in addition, thepions become light. Therefore, the contribution from dy-namical hadronisation to the flow of the Yukawa coupling(33), ∼ ¯ m π,k ∂ t ¯ λ q,k , is suppressed by these two effects inbroken regime. Thus, following our present results, inparticular Fig. 11, the only regime where dynamical hadro-nisation can play a role in a low-energy model is in thevicinity of chiral symmetry breaking scale. However, sincethis region is small compared to range of scales consideredeven in low-energy models, only very small correctionsrelated to the re-generation of four-quark interactions areaccumulated from the RG flow.Note, however, that we checked this statement only invacuum and it might not be true in medium, especially atlarge chemical potential where quark fluctuations are en-hanced. This can potentially lead to larger, non-negligiblecorrections from dynamical hadronisation. We also em-phasise that we used the same initial conditions for ourcomparison of the RG flow of the quark-meson model withand without dynamical hadronisation techniques. How-ever, usually the parameters of low-energy models arefixed in the vacuum, independent of the model truncation.Once the parameters are fixed, these models are thenused to compute, e.g., the phase diagram of QCD at finitetemperature and chemical potential. In this case, it maystill very well be that the use of dynamical hadronisationtechniques yield significant corrections. Appendix B: Flow equations of the couplings
In this appendix, we briefly discuss the derivation of theflow equations of the couplings before dynamical hadroni-sation techniques are applied.We expand the effective potential and the Yukawa cou-pling about a fixed expansion point κ , see (5). The advan-tage of such an expansion is that it is numerically stable,inexpensive and it converges rapidly [27]. This allows usto take the full field-dependent effective potential V k ( ρ )and Yukawa coupling h k ( ρ ) into account in the presentanalysis.The flow equation of the effective potential includingthe symmetry breaking source, V k ( ρ ) − cσ , is obtainedby evaluating (9) for constant meson fields, φ ( x ) → φ and vanishing gluon, quark and ghost fields. In this case,the effective action reduces to Γ k = Ω − ( V k ( ρ ) − cσ ),where Ω is the space-time volume. The flow of the effective8potential ¯ V k (¯ ρ ) = V k ( ρ ) is then given by: ∂ t | ρ ¯ V (¯ ρ ) =2 k v ( d ) (cid:110)(cid:2) ( N f − l B ( ¯ m π,k ; η φ,k )+ l B ( ¯ m σ,k ; η φ,k ) (cid:3) − N f N c l F ( ¯ m q,k ; η q,k ) (cid:111) , (B1)where v ( d ) = (2 d +1 π d/ Γ( d/ − and the treshold func-tions l B and l F are given in Eq. (C4). The flows of thecouplings in (5) can be derived from the above equationvia: ∂ n ¯ ρ ∂ t | ρ ¯ V (¯ ρ ) (cid:12)(cid:12)(cid:12) ¯ ρ =¯ κ k =( ∂ t − nη φ,k )¯ λ n,k − ¯ λ n +1 ,k ( ∂ t + η φ,k )¯ κ k . (B2)Rescaling the expansion point and the symmetry breakingsource in order to formulate RG invariant flows introducesa canonical running for these parameters: ∂ t ¯ κ k = − η φ ¯ κ k ,∂ t ¯ c = 12 η φ ¯ c . (B3)The renormalised minimum of the effective potential¯ ρ ,k = ¯ σ ,k /
2, which determines the pion decay constantat vanishing IR-cutoff, ¯ σ ,k =0 = f π , and serves as an or-der parameter for the chiral phase transition, is obtainedfrom: ∂ ¯ ρ (cid:2) ¯ V k (¯ ρ ) − ¯ c k ¯ σ (cid:3)(cid:12)(cid:12)(cid:12) ¯ ρ ,k = 0 . (B4)All physical observables such as f π and the masses are de-fined at vanishing cutoff-scale k = 0 and at the minimumof the effective potential ¯ ρ = ¯ ρ ,k =0 .We define the field-dependent Yukawa coupling via therelation m q,k ( ρ ) = σh k ( ρ ) at vanishing external momen-tum and constant meson fields, leading to the followingprojection: ∂ t h k ( ρ ) = − σ i N c N f lim p → Tr (cid:18) δ ∂ t Γ k δq ( − p ) δ ¯ q ( p ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ρ ( x )= ρ . (B5) The resulting flow is given by: ∂ t | ¯ ρ ¯ h (¯ ρ ) = (cid:18) η q,k + 12 η φ,k (cid:19) ¯ h k (¯ ρ ) − v ( d )¯ h k (¯ ρ ) (cid:104) ( N f − L ( F B )1 , ( ¯ M q,k , ¯ m π,k ; η q,k , η φ,k ) − L ( F B )1 , ( ¯ m q,k , ¯ m σ,k ; η q,k , η φ,k ) (cid:105) + 8 v ( d )¯ h k (¯ ρ ) ¯ h (cid:48) k (¯ ρ ) ¯ ρ (cid:2) ¯ h k (¯ ρ ) + 2¯ ρ ¯ h (cid:48) k (¯ ρ ) (cid:3) × L ( F B )1 , ( ¯ m q,k , ¯ m σ,k ; η q,k , η φ,k ) − v ( d ) k (cid:2)(cid:0) h (cid:48) k (¯ ρ ) + 2¯ ρ ¯ h (cid:48)(cid:48) k (¯ ρ ) (cid:1) l B ( ¯ m σ,k ; η φ,k )+ 3¯ h (cid:48) k (¯ ρ ) l B ( ¯ m π,k ; η φ,k ) (cid:3) − ξ ) C ( N c ) v ( d ) g qAq,k ¯ h k (¯ ρ ) × L ( F B )1 , ( ¯ m q,k , η q,k , η A,k ) , (B6) ξ is the gauge fixing parameter, which we set to zerosince we use Landau gauge in this work. The function L ( F B )1 , is given in Eq. (C5). The flows of the renormalisedcouplings in (5) are: ∂ n ¯ ρ ∂ t | ρ ¯ h (¯ ρ ) (cid:12)(cid:12)(cid:12) ¯ ρ =¯ κ k =( ∂ t − nη φ,k )¯ h n,k − ¯ h n +1 ,k ( ∂ t + η φ,k )¯ κ k . (B7)It was shown in Ref. [27], already a φ expansion of theeffective potential, corresponding to N V = 2 in (5) givesquantitatively precise results for small temperatures anddensities. On the other hand, a leading order expansion ofthe Yukawa coupling, i.e. N h = 0, is not sufficient since theexpansion is not yet converged. Here, we choose N h = 3 toensure that we take the effect of the full field-dependentYukawa coupling into account. Note that we have tochoose N V ≥ N h for numerical stability and thereforechoose N V = 5.For the flow of the four-quark coupling we choose theprojections in [12]. This yields ∂ t ¯ λ q,k = − g qAq,k (cid:18) N c − N c (cid:19) v ( d ) L ( F B )1 , ( ¯ m q,k ; η q,k , η A,k )+ ¯ h k (¯ κ k ) (cid:18) N c + 1 (cid:19) v ( d ) × L ( F B )1 , , ( ¯ m q,k , ¯ m π,k , ¯ m σ,k ; η q,k , η φ,k ) . (B8)The treshold functions L ( F B )1 , and L ( F B )1 , , are shown inEq. (C5). In Eq. (B8), we anticipate full dynamical hadro-nisation for the four fermi interaction. This leads to a van-ishing four-quark coupling ¯ λ q,k = 0 on the right-hand side:the self-coupling diagram proportional to ¯ λ q,k is dropped.Furthermore, we neglect contributions from higher orderquark-meson vertices related to field-derivatives of ¯ h k (¯ ρ ),since they are subleading.9The anomalous dimensions are related to the flow ofthe wave-function renormalisations, η = − ∂ t Z/Z . The Z ’s on the other hand encode the non-trivial momentumdependence of the propagators. Here, as already discussedabove, we approximate the full momentum, scale and fielddependence of the anomalous dimensions by only scale-dependent ones in the leading order expansion in the fieldsin analogy to (5): Z φ,k ( p , ρ ) = Z φ,k ( κ ) and Z q,k ( p , ρ ) = Z q,k ( κ ) . (B9)For the meson anomalous dimension, we therefore use thefollowing projection: η φ,k = − Z φ,k lim p → ∂ ∂ | p | Tr (cid:18) δ ∂ t Γ k δπ i ( − p ) δπ i ( p ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ρ = κ , (B10)where the choice of i = 1 , , O (3) symmetry of the pions. This yields η φ,k =8 v ( d ) k − ¯ κ k ¯ U (cid:48)(cid:48) k (¯ κ k ) M , ( ¯ m π,k , ¯ m σ,k )+ 2 N c N f v ( d ) ¯ h k (¯ κ k ) (cid:2) M ( ¯ m q,k ; η q,k )+ 12 k − ¯ κ k ¯ h k (¯ κ k ) M ( ¯ m q,k ; η q,k ) (cid:21) . (B11)The functions M , and M / are defined in Eq. (C6).Note that it is crucial that the functional derivatives in(B10) are with respect to the pions, since sigma-derivativeswould contaminate the flow with contributions propor-tional to σZ (cid:48) φ,k ( ρ ).For the anomalous dimension of quarks, we use theprojection η q,k = − N f N c Z q,k (B12) × lim p → ∂ ∂ | p | Tr (cid:18) γ µ p µ δ ∂ t Γ k δq ( − p ) δ ¯ q ( p ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ρ = κ , which yields η q =2 v ( d ) C ( N c ) g qAq (cid:2) (3 − ξ ) M , ( ¯ m q,k , η A,k ) − − ξ ) ˜ M , ( ¯ m q,k , η q,k , η A,k ) (cid:3) + 12 v ( d )[ (cid:0) ¯ h k (¯ κ k ) + 2¯ κ k ¯ h (cid:48) k (¯ κ k ) (cid:1) ×M , ( ¯ m q,k , ¯ m σ,k ; η φ,k )+ ( N f − h k (¯ κ k ) M , ( ¯ m q,k , ¯ m π,k ; η φ,k ) (cid:3) . (B13) The corresponding threshold functions can be in Eq. (C6).Some of the flow equations in this work were derivedwith the aid of an extension of DoFun [51] which utilizesForm [52] and FormLink [53]. It was developed and firstused by the authors of [3]. Appendix C: Threshold functions
Here, we collect the threshold functions which enter theflow equations and encode the regulator and momentumdependence of the flows. Note that it is here, where thesubstitution η φ,k → η φ,k − B k has to be made accordingto (26).Throughout this work, we use 4 d regulator functionsof the form: R φk ( p ) = Z φ,k p r B ( p /k ) ,R qk ( p ) = Z q,k γ µ p µ r F ( p /k ) ,R A ,µνk ( p ) = Z A,k p r B ( p /k ) Π ⊥ µν ( p ) , (C1)with the transverse projectorΠ ⊥ µν ( p ) = δ µν − p µ p ν p . (C2)Note that in the approximation at hand the ghost reg-ulator does not enter. The optimised regulator shapefunctions r B/F ( x ) are given by [29]: r B ( x ) = (cid:18) x − (cid:19) Θ(1 − x ) ,r F ( x ) = (cid:18) √ x − (cid:19) Θ(1 − x ) . (C3)The threshold functions for the effective potential are l Bn ( ¯ m B ; η B ) = 2( δ n, + n ) d (cid:18) − η B d + 2 (cid:19) (1 + ¯ m B ) − ( n +1) ,l Fn ( ¯ m F ; η F ) = 2( δ n, + n ) d (cid:18) − η F d + 1 (cid:19) (1 + ¯ m F ) − ( n +1) , (C4)and that for the Yukawa coupling and the four-quarkcoupling are0 L ( F B )1 , ( ¯ m F , ¯ m B ; η F , η B ) = 2 d (1 + ¯ m F ) − (1 + ¯ m B ) − (cid:40)(cid:18) − η F d + 1 (cid:19) (1 + ¯ m F ) − + (cid:18) − η B d + 2 (cid:19) (1 + ¯ m B ) − (cid:41) ,L ( F B )1 , ( ¯ m F ; η F , η B ) = 2 d (1 + ¯ m F ) − (cid:40) (cid:18) − η B d + 2 (cid:19) − (cid:18) − η F d + 1 (cid:19) + 2(1 + ¯ m F ) − (cid:18) − η F d + 1 (cid:19)(cid:41) ,L ( F B )1 , , ( ¯ m F , ¯ m B , ¯ m B ; η F , η B ) = 2 d (1 + ¯ m F ) − (1 + ¯ m B ) − (1 + ¯ m B ) − (cid:40)(cid:2) (1 + ¯ m B ) − + (1 + ¯ m B ) − (cid:3) (C5) × (cid:18) − η B d + 2 (cid:19) + (cid:2) m F ) − − (cid:3) (cid:18) − η F d + 1 (cid:19)(cid:41) . For the anomalous dimensions, we have M ( ¯ m F ; η F ) = (cid:0) m F (cid:1) − , M , ( ¯ m B , ¯ m B ; η B ) = (1 + ¯ m B ) − (1 + ¯ m B ) − M , ( ¯ m F , ¯ m B ; η F , η B ) = (cid:18) − η B d + 1 (cid:19) (1 + ¯ m F ) − (1 + ¯ m B ) − (C6) M ( ¯ m F ; η F ) = (cid:0) m F (cid:1) − + 1 − η F d − (cid:0) m F (cid:1) − − (cid:18)
14 + 1 − η F d − (cid:19) (cid:0) m F (cid:1) − ˜ M , ( ¯ m F , η F , η B ) = 2 d − (cid:0) m F (cid:1) − (cid:40) (cid:18) η F d − (cid:19) + (cid:18) − η B d + 1 (cid:19) + (cid:18) − η F d (cid:19) (cid:0) m F (cid:1) − (cid:41) . Finally, for the flow of z ¯ qAq we use N ( m )2 , ( ¯ m F , ¯ m B ; η F , η B ) = 1 d (cid:18) − η F d + 1 (cid:19) (1 + ¯ m B ) − (cid:110) m F (1 + ¯ m F ) − + (1 + ¯ m F ) − (cid:111) + 1 d (cid:18) − η B d + 2 (cid:19) (1 + ¯ m B ) − (cid:110) ¯ m F (1 + ¯ m F ) − + (1 + ¯ m F ) − (cid:111) , N ( g )2 , ( ¯ m F ; η F , η A ) = 1 d (cid:18) − η F d + 1 (cid:19) ¯ m F (1 + ¯ m F ) − + 12 d (cid:18) − η A d + 2 (cid:19) ¯ m F (1 + ¯ m F ) − , N ( g )1 , ( ¯ m F ; η F , η A ) = 1 d + 1 (cid:18) − η F d + 2 (cid:19) (cid:110) m F (1 + ¯ m F ) − − (1 + ¯ m F ) − (cid:111) + 4 d + 1 (cid:18) − η A d + 3 (cid:19) (1 + ¯ m F ) − . (C7) Appendix D: Infrared parameter
In our study, we introduced an “infrared-strength” func-tion ς a,b ( k ) which we define as ς a,b ( k ) = 1 + a ( k/b ) δ e ( k/b ) δ − , (D1) with b > δ >
1. Note that the specific form of ς a,b ( k )is irrelevant for our result as long as it has the propertiesspecified below. It defines a smooth step function centered1around b with interpolates smoothly between ς a,b ( k (cid:29) b ) = 1 and ς a,b ( k (cid:28) b ) = 1 + a . (D2)Thus, for b = O (1 GeV), ς a,b ( k ) gives an IR-enhancement,while it leaves the perturbative regime unaffected. Wethen modify the gauge couplings as g s,k −→ ς a,b ( k ) g s,k , (D3)where g s,k = g ¯ qAq,k , g A ,k , g ¯ cAc,k . We choose the sameparameters a and b for every gauge coupling. Accordingly,the flow equations of the gauge couplings then are ∂ t g s,k −→ g s,k ∂ t ς a,b ( k ) + ς a,b ( k ) ∂ t g s,k . (D4) We have found that our results do not depend stronglyon the precise value of b as long as it is O (1 GeV). To bespecific, we choose b = 1 . δ = 3 in the following.The parameter a is adjusted such that we get physicalconstituent quark masses in the infrared. Here, a =0 .
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