Fully coupled functional equations for the quark sector of QCD
FFully coupled functional equations for the quark sector of QCD
Fei Gao, Joannis Papavassiliou, and Jan M. Pawlowski
1, 31
Institut f¨ur Theoretische Physik, Universit¨at Heidelberg,Philosophenweg 16, 69120 Heidelberg, Germany Department of Theoretical Physics and IFIC,University of Valencia and CSIC, E-46100 Valencia, Spain ExtreMe Matter Institute EMMI, GSI,Planckstr.1, 64291 Darmstadt, Germany
Abstract
We present a comprehensive study of the quark sector of 2 + 1 flavour QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagatorand the full quark-gluon vertex. The individual form factors of the quark-gluon vertex are expressedin a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingre-dient used as input to our equations is the Landau gauge gluon propagator with 2 + 1 dynamicalquark flavours, obtained from studies with Schwinger-Dyson equations, the functional renormal-isation group approach, and large volume lattice simulations. The appropriate renormalisationprocedure required in order to self-consistently accommodate external inputs stemming from otherfunctional approaches or the lattice is discussed in detail, and the value of the gauge coupling isaccurately determined at two vastly separated renormalisation group scales.Our analysis establishes a clear hierarchy among the vertex form factors. We identify only threedominant ones, in agreement with previous results. The components of the quark propagatorobtained from our approach are in excellent agreement with the results from Schwinger-Dysonequations, the functional renormalisation group, and lattice QCD simulation, a simple benchmarkobservable being the chiral condensate in the chiral limit, which is computed as (245 MeV) . Thepresent approach has a wide range of applications, including the self-consistent computation ofbound-state properties and finite temperature and density physics, which are briefly discussed. a r X i v : . [ h e p - ph ] F e b . INTRODUCTION In functional approaches to QCD, the task of computing quark-, gluon-, and hadroncorrelation functions is formulated in terms of closed coupled diagrammatic relations be-tween them, which must then be solved numerically. In all these approaches, such asSchwinger-Dyson equations (SDEs), functional renormalisation group (fRG), n -particle ir-reducible methods (nPI), and bound state methods (Bethe-Salpeter (BS), Faddeev- andhigher-order equations), the diagrammatic relations are built out of the propagators of thefundamental and composite QCD fields. For reviews on functional methods in QCD, see, e.g. , [1–7] (SDEs), [8–13] (fRG), and [14–16] (bound-states).Functional approaches allow for an attractively simple and versatile access to the dy-namical mechanisms that drive numerous fundamental QCD phenomena. Moreover, theirflexibility in using as external inputs correlation functions stemming from distinct non-perturbative setups ( e.g. , lattice [17–32]) is a particularly welcome feature, which increasestheir quantitative reliability and their range of applicability. However, such inputs are notalways available, prominent and important examples being QCD at finite temperature anddensity, as well as the hadron spectrum. Hence, in the past two decades, functional methodshave evolved into a self-contained quantitative approach to QCD, allowing for quantitativepredictions within a “first principle” framework, without the need to depend on externalinputs.This ongoing progress requires quantitative computations involving the full tensor struc-ture of correlation functions, and in particular that of the three- and four-point functions,that dominantly drive the dynamics of QCD. Specifically, the quark-gluon vertex is thepivotal ingredient of the matter dynamics of QCD, being intimately connected with fun-damental phenomena such as chiral symmetry breaking and quark mass generation, boundstate formation, e.g. , [33–41], and the QCD phase structure at finite temperature andchemical potential, e.g. , [42–61].To date, the quark-gluon vertices employed in most SDE studies are not based on afull solution of the corresponding dynamical equations, but are rather put together fromquark and ghost dressing functions with the aid of the Slavnov-Taylor identities (STIs);see, e.g. , [28, 62–71]. This is an operationally simple and suggestive treatment, with animpressive array of very successful applications, ranging from the properties of hadrons to the2hase structure of QCD. However, within these STI constructions, the strength associatedwith the classical tensor structure requires a phenomenological infrared enhancement, whosesize is adjusted by means of the constituent quark masses. The latter, including theirmomentum dependence, are equivalent to the physical amount of chiral symmetry breaking,and hence, in such an approach, the quantitative strength of chiral symmetry breakingis a phenomenological input rather than a prediction. To be sure, the need for such anenhancement may be attributed to the insufficient knowledge of some of the ingredientscomprising these STIs ( e.g. , quark-ghost kernel [67]). Nonetheless, in view of the results inthe present work, as well as of previous considerations within functional approaches [34, 59,60, 72–78], it seems to originate mainly from the omission of important tensor structuresthat are simply not accessible through the standard STI construction.This situation calls for a self-consistent treatment of the full quark-gluon vertex within theSDE formalism in the Landau gauge. The determination of the eight relevant form factorsfrom their dynamical equations requires the solution of the coupled system of gluon, ghostand quark propagators, the quark-gluon vertex, as well as additional vertices. The mostcomplete results in this direction have been obtained within functional methods for two-flavour QCD, see [72, 73] (quenched), and [75, 77] (unquenched). Recently, the fRG resultsof [77] have been used as input for a 2+1–flavour analysis within the SDE approach, both inthe vacuum and at finite temperature and density [59, 60]. Despite all these advances, westill lack a well-defined calculational SDE scheme, where one could unambiguously identifyand reliably compute the dominant components of this vertex, either self-consistently orwith the aid of a given input.In the present work we put forward a systematic approximation scheme for the setof functional equations governing the quark sector of QCD, by studying in detail thecoupled system of SDEs for the quark propagator (quark gap equation) and the quark-gluon vertex. Our SDE analysis reveals that the quark dynamics are dominated by three specific tensor structures of the quark-gluon vertex, in agreement with earlier considera-tions [59, 60, 72, 73, 75, 77]. It is important to stress that, apart from the dressing as-sociated with the classical tensor, the other two dominant dressings are not accessible bymeans of an STI-based construction. In fact, the numerical impact of these latter dress-ings at the level of the gap equations is crucial, furnishing directly the required amountof chiral symmetry breaking without the need to resort to artificial enhancing factors. In3ur opinion, this demonstrates conclusively that no artificial enhancement is required oncethe contributions from the appropriate tensorial structures have been properly taken intoaccount. Importantly, we also find that certain tensor structures, which in previous STItreatments seemed dominant precisely due to the use of such enhancing factors, turn outto be clearly sub-leading. Consequently, the present detailed analysis enables us to restrictour considerations to the three most relevant tensors, thus arriving at a reduced set of fullycoupled SDEs, which are solved iteratively together with the quark gap equation.A central ingredient of the system of equations considered in this work is the gluonpropagator, entering both in the gap equation and the SDE for the quark-gluon vertex. Thegluon propagator obeys its own SDE [1–5, 7], which depends on the quark propagator andfurther correlation functions, a fact that leads to a proliferation of coupled equations. Eventhough the complete treatment of such as extended system has already been implementedfor N f = 2 flavour QCD [73, 77], in the present work we prefer to maintain the focus onthe novel features of our approach rather than be sidetracked by a technically exhaustiveanalysis. To that end, we treat the gluon propagator as an external ingredient: withinour most elaborate and trustworthy approximation, we consider a renormalisation pointat large, perturbative, momenta with µ = 40 GeV, and use the SDE data for the gluonpropagator from [59, 60] as external input. These SDE data are based on the fRG two-flavour computation of [54], as are the gluon data of [57], which are also used as input, forthe purpose of estimating our systematic error. Finally, we also consider gluon data from N f = 2 + 1 lattice simulations [31, 79, 80], and a renormalisation point of µ = 4 . p < ∼ e.g. ,[77].In addition, the results obtained are particularly stable under vast changes in the valueof the renormalisation point µ . Finally, a chief advantage of this scheme is its relativeoperational simplicity and low computational cost, combined with quantitative reliabilityand systematic error control.The article is organised as follows. In Sec. II we review some general features of the SDEand fRG approaches, and introduce the notation that will be used in this work. In Sec. III4e set up the gap equation and discuss its renormalisation. Then, in Sec. IV we focus on thequark-gluon vertex, present the tensorial basis that will be employed, and derive the systemof integral equations satisfied by its form factors. In Sec. V we present a detailed discussion ofhow to implement self-consistently the renormalisation of the SDEs when an external inputis employed. In Sec. VI we discuss the procedure that fixes the values of the current quarkmasses, and introduce the light chiral condensate as our benchmark observable. In Sec. VIIwe present and discuss the central results of our analysis, with special emphasis on the quarkmass and the eight form factors of the quark-gluon vertex, evaluated at the symmetric point.Then, in Sec. VIII we confirm the stability of our results under variations of the ultraviolet(UV) cutoff, the renormalisation point, and the inputs used for the gluon propagator. InSec. IX we capitalise on the hierarchy displayed among the vertex form factors, and proposea simplified treatment that reduces the numerical cost without compromising the accuracyof the results. In Sec. X we summarise our approach and present our conclusions. Finally,we relegate in two Appendices the discussion of various technical points. II. GENERAL CONSIDERATIONS
In this section we briefly comment on certain important aspects of functional approachesthat are relevant for the ensuing analysis, and introduce the notation that will be employedin this work.
A. The action
The starting point is the classical action of QCD in covariant gauges, given by S [ φ ] = (cid:90) d x (cid:20)
14 ( F aµν ) + ¯ q (cid:0) /D + m q (cid:1) q + 12 ξ (cid:0) ∂ µ A aµ (cid:1) − ¯ c a ∂ µ D abµ c b (cid:21) , (2.1)where the ghost has a positive dispersion, typically used in fRG applications to QCD; for arecent review see [13]. The covariant derivative, D µ , and the field strength tensor, F µν , aregiven by F aµν = ∂ µ A aν − ∂ ν A aµ + g s f abc A bµ A cν , and D µ = ∂ µ − ig s A aµ t a , [ t a , t b ] = i f abc t c . (2.2)5he first two terms in (2.1) are the Yang-Mills and Dirac actions, respectively; in the latterwe have suppressed the summation over group indices in the fundamental representation,as well as Dirac and flavour indices. The remaining terms in (2.1) encode the gauge fixingand ghost sector. In (2.2), the covariant derivative in the fundamental representation reads ∂ µ − ig s A aµ T a , where T a are the corresponding generators, while that of the adjoint repre-sentation is given by ∂ µ δ ab − g s f abc A cµ . The computations in the present work are carriedout in the Landau gauge, ξ = 0. B. SDE setup and renormalisation
In contradistinction to the flow equations of the fRG approach, the SDEs depend also onderivatives of the classical QCD action in (2.1). More specifically, we need the bare action,whose parameters absorb the UV infinities of the diagrams. The mapping from bare fields, φ (0) , to renormalised finite fields, φ , is given by A (0) µ = Z / A µ , c (0) = ˜ Z / c , ¯ c (0) = ˜ Z / ¯ c , q (0) = Z / q , ¯ q (0) = Z / ¯ q , (2.3a)while for the strong coupling, masses, and gauge fixing parameters we have, correspondingly, g (0) s = Z g g , m (0) q = Z m q m q , ξ (0) = Z ξ ξ . (2.3b)Then, the bare QCD action, S bare , reads in terms of the renormalised fields and couplingparameters, S bare [ φ (0) ; g (0) s , m (0) q ] = S [ Z / A µ , ˜ Z / c, ˜ Z / ¯ c, Z / q, Z / ¯ q, Z g g s , Z m q m q ] . (2.3c)From (2.3c) we may define the renormalisation constants of the three-gluon vertex, Z , thefour-gluon vertex, Z , the ghost-gluon vertex, ˜ Z , and the quark-gluon vertex, Z f , and relatethem as Z = Z g Z / , Z = Z g Z , ˜ Z = Z g Z / ˜ Z / , Z f = Z g Z / Z . (2.3d) C. fRG setup
The central object of functional approaches to QCD is the one-particle irreducible (1PI)effective action, Γ[ φ ], where φ is a “superfield”, whose components are the fundamental6enormalised fields of QCD, including the auxiliary ghost field introduced through the gauge-fixing, φ = ( A µ , c , ¯ c , q , ¯ q ) . (2.4)While this is typically rather implicit in most SDE applications, it is commonly the startingpoint in fRG studies. Derivatives of the effective action Γ[ φ ] w.r.t. the fields are the 1PI n -point correlation functions, denoted byΓ ( n ) φ ··· φ n ( p , ..., p n ) = δ n Γ δφ ( p ) · · · φ n ( p n ) , (2.5a)where all momenta are considered as incoming. Vertices Γ ( n ) are expanded in a completetensor basis {T ( i ) φ i ··· φ in } , the standard fRG notation in QCD beingΓ ( n ) φ i ··· φ in ( p , ..., p n ) = (cid:88) i λ φ i ··· φ in ( p , ..., p n ) T ( i ) φ i ··· φ in ( p , ..., p n ) , (2.5b)with λ φ i ··· φ in denoting the scalar form factors (dressings).Note that the renormalisation factors defined in (2.3) have a natural relation to thefull dressings of the primitively divergent n -point functions in the fRG-approach, Z φ i ,k ( p ), M q,k ( p ), defined in (2.7), and λ (1) φ i ··· φ in ,k , defined in (2.5b); for a detailed account see [8, 13]. D. Running couplings
We next consider the different “avatars” of the strong running coupling α s (¯ p ) = g s (¯ p ) / π ,which can be deduced from the form factors λ (1) associated with the classical tensor struc-tures of the four fundamental QCD vertices. In particular, in the present analysis we willemploy the running couplings obtained from the ghost-gluon and quark-gluon vertices, givenby α c ¯ cA (¯ p ) = 14 π [ λ (1) c ¯ cA (¯ p )] Z A (¯ p ) Z c (¯ p ) , α q ¯ qA (¯ p ) = 14 π [ λ (1) q ¯ qA (¯ p )] Z A (¯ p ) Z q (¯ p ) , (2.6)where ¯ p is a symmetric-point configuration, and Z A , Z c and Z q are the dressings of thetwo-point functions (suppressing color),Γ (2) AA µν ( p ) = Z A ( p ) p P µν ( p ) + 1 ξ p µ p ν , Γ (2) c ¯ c ( p ) = Z c ( p ) p , Γ (2) q ¯ q ( p ) = Z q ( p ) [ i p/ + M q ( p )] , (2.7)7here we have introduced the transverse projection operator P µν ( p ) = δ µν − p µ p ν p , (2.8)usually denoted by Π ⊥ µν ( p ) in the fRG literature. Note that the above two-point functionsare the inverses of the gluon, ghost, and quark propagators, respectively.By virtue of the fundamental STIs of the theory, all QCD couplings coincide for largevalues of ¯ p , α i (¯ p ) = α s (¯ p ) , i = ( c ¯ cA , q ¯ qA, A , A ) for perturbative ¯ p = ¯ p pert . (2.9)As the momentum ¯ p gets smaller, the various α i (¯ p ) start deviating from each other, dueto differences induced by non-trivial contributions from the scattering kernels appearingin the STIs. As has been pointed out in [77], the amount of chiral symmetry breakingobtained from the gap equation appears to be particularly sensitive to the UV coincidenceof the couplings described by (2.9). The preservation of (2.9) is a indispensable feature ofany quantitatively reliable framework; in particular, special truncation schemes such as thePT-BFM [5] are tailor-made for this task. III. THE QUARK GAP EQUATION
The quark gap equation [1–7] relates the inverse quark propagator, Γ (2) q ¯ q ( p ), to its classicalcounter part, S (2) q ¯ q ( p ), to the quark and gluon propagators, and the classical and full quark-gluon vertices, see Fig. 1. Schematically it readsΓ (2) q ¯ q ( p ) = S (2) q ¯ q + Z f g s (cid:90) q G AA ( q − p ) ( − iγ ) G q ¯ q ( q ) Γ (3) q ¯ qA ( q, − p ) , (3.1)where we suppress all Lorentz and color indices, and g s stands for the gauge coupling. Thefour-dimensional momentum integration has been abbreviated by (cid:90) q := (cid:90) reg d q (2 π ) , (3.2)where the subscript “reg” indicates a suitable regularisation of the momentum integral;common choices include the dimensional regularisation or an appropriately implementedmomentum cutoff. The respective cutoff parameter ( e.g. , (cid:15) or Λ ) appears also in all renor-malisation constants, and in particular the quark-gluon vertex renormalisation, Z f , as well8 ) = ( ) + q − pp p p q p − − | {z } Σ( p ) FIG. 1: Diagrammatic representation of the quark gap equation. Gray (blue) circles denotefull propagators (vertices), black dots denote classical vertices.as the wave function renormalisation, Z , and the mass renormalisation, Z m q , of the quark.The last two factors enter into (3.1) through S (2) q ¯ q ( p ), the second derivative of the bare QCDaction, (2.3c), with respect to the renormalised quark and anti-quark fields, see (2.3a), S (2) q ¯ q ( p ) = iZ p/ + Z m q m q , (3.3)where m q denotes the bare current quark mass.The full gluon propagator, G abAA µν ( p ), in the Landau gauge, and the quark propagator, G abq ¯ q ( p ), are given by G abAA µν ( p ) = δ ab P µν ( p ) G A ( p ) , G abq ¯ q ( p ) = δ ab G q ( p ) . (3.4)In (3.4) , G A ( p ) is the scalar part of the gluon propagator, and G q ( p ) carries only the Diracstructure but not the trivial color structure. Both G A ( p and G q ( p ) can be described in termsof the scalar dressings introduced in (2.7), to wit, G A ( p ) = 1 Z A ( p ) p , G q ( p ) = 1 Z q ( p ) [ i p/ + M q ( p )] , (3.5)where M q ( p ) is the momentum-dependent mass function. Note that in the fRG-approach,for large cutoff scales, the functions Z A ( p ) and Z q ( p ) tend towards the corresponding (finite)wave function renormalisations, while M q ( p ) tends to the bare quark mass.Finally, (cid:104) Γ (3)¯ qqA (cid:105) aν ( q, − p ) denotes the quark-gluon vertex, in accordance with the generaldefinition of (2.5), with all momenta considered as incoming.The presence of the transverse projection operator P µν in (3.1) makes natural the use ofthe transversely projected version of the quark-gluon vertex. Specifically, for the purposes ofthe present work we introduce the transversely projected vertex IIΓ µ ( q, − p ), defined through P µν ( p − q ) (cid:104) Γ (3)¯ qqA (cid:105) aν ( q, − p ) = f T a c IIΓ µ ( q, − p ) , (3.6)9here f denotes the identity matrix in flavour space. Note that while (cid:104) Γ (3)¯ qqA (cid:105) aν ( q, − p )requires twelve tensors for its full decomposition, IIΓ µ ( q, − p ) is comprised by a subset of onlyeight; for more details see, e.g. , [73, 77].With the above definitions, the color contractions in (3.1) can be easily carried out, andwe arrive at the standard form of the gap equation, Z q ( p ) [ i p/ + M q ( p )] = Z ip/ + Z m q m q + Σ( p ) , (3.7)with the renormalised self-energyΣ( p ) = Z f g s C f (cid:90) q Z A ( q − p )( q − p ) γ µ Z q ( q ) [ i q / + M q ( q )] IIΓ µ ( q, − p ) , (3.8)where C f denotes the Casimir eigenvalue of the fundamental representation, with C f = 4 / SU (3).Note that Eq. (3.7) is finite due to the regularisation of the loop integral, as indicated in(3.2). As mentioned there, the cutoff-dependences of the loop integral and of Z , Z m q , and Z f , cancel against each other, giving finally rise to cutoff-independent functions Z q ( p ) and M q ( p ). As we discuss in the next section, an analogous renormalisation procedure rendersthe vertex IIΓ µ ( q, − p ) cutoff-independent.The gap equation in (3.7) can be projected on its Dirac vector and scalar parts bymultiplying it with either 1l or p/ and performing the corresponding traces. This leads us tothe standard set of coupled SDEs for Z q ( p ) and M q ( p ), Z q ( p ) p = Z p − Z f tr[ ip/ Σ( p )] , M q ( p ) = Z − q ( p )( Z m q m q + Z f tr[Σ( p )]) . (3.9)We next specify the renormalisation conditions at a given renormalisation scale µ . As iscommon to functional approaches, we employ a non-perturbative version of the momentumsubtraction (MOM) scheme, where the renormalised quantum corrections of all primitivelydivergent vertices with momenta p , ..., p n vanish at a symmetric point ¯ p = µ , when p i = ¯ p , ∀ i = 1 , ..., n . (3.10)In particular, the dressings of the two-point functions reduce to unity, Z A ( µ ) = 1 , Z c ( µ ) = 1 , Z q ( µ ) = 1 , M q ( µ ) = m q , (3.11a)10here Z c ( p ) is the dressing associated with the ghost propagator. Similarly, in the caseof the vertices, the symmetric point dressings λ (1) φ ··· φ n (¯ p ) := λ (1) φ ··· φ n ( p , ..., p n )) (cid:12)(cid:12)(cid:12) p i =¯ p of theclassical tensor structures satisfy λ (1) A ( µ ) = g s , λ (1) A ( µ ) = g s , λ (1) c ¯ cA ( µ ) = g s , λ (1) q ¯ qA ( µ ) = g s . (3.11b)Evidently, all renormalisation constants also depend on the subtraction point µ .Within the renormalisation scheme defined above, we have that Γ q ¯ q ( p = µ ) = ip/ + m q ,and the respective renormalisation factors are given by Z = 1 + tr[ ip/Z f Σ( p )] p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = µ , Z m q = 1 − tr[ Z f Σ( p )] m q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = µ . (3.12)The solution of the quark gap equation requires the knowledge of the gluon propagatorand the quark-gluon vertex, which, in turn, depend on the quark propagator and furthercorrelation functions, thus leading to an extended system of coupled integral equations,which must be solved simultaneously. Such a complete, fully back-coupled analysis, subjectto certain simplifying approximations, is indeed feasible, and has been presented withinfunctional approaches for N f = 2 flavour QCD in [73, 77]. However, the main purpose ofthe present work is the detailed analysis of the system of quark propagator and quark-gluonvertex, as well as the discussion of quantitative approximation schemes. For this reason weopt for a simpler treatment, which permits us to maintain our focus on the novel aspectsof our approach. In particular, the gluon propagator entering into both the gap equationand the vertex SDE will be treated as an external ingredient. Thus, rather than solvingits own dynamical equation, we will employ the results obtained in the unquenched latticesimulations of [31, 79, 80] and the functional analysis of [54, 57]. IV. SDE OF THE QUARK-GLUON VERTEX
In this section we set up and discuss the SDE for the IIΓ µ defined in (3.6), which enters in thequark gap equation. In the present work we consider the “one-loop dressed” approximationof this SDE, which is diagrammatically depicted in Fig. 2. This functional equation will beprojected on its different tensorial components, thus furnishing a set of dynamical equationsgoverning the respective form factors. 11 µ p − q A µ µk p − q += µ p − q + · · · B µ µk p − qqq qq p p p pk + q k + pk − q k − p FIG. 2: Diagrammatic representation of the quark-gluon SDE. Gray (blue) circles denotefull propagators (vertices), black dots denote classical vertices.The SDE for the vertex IIΓ µ is expressed asIIΓ µ ( q, − p ) = Z f g s P µν ( p − q ) ( − iγ ν ) + A µ ( q, − p ) + B µ ( q, − p ) , (4.1)with the contributions of the graphs A µ ( q, − p ) and B µ ( q, − p ) in Fig. 2 given by A µ ( q, − p ) = Z N c P µν ( p − q ) (cid:90) k Γ (0) ναβ G A ( k − q ) G A ( k − p ) IIΓ α ( k, − p ) G q ( k ) IIΓ β ( q, − k ) , B µ ( q, − p ) = − Z f N c P µν ( p − q ) × (cid:90) k G A ( k ) IIΓ α ( k + p, − p ) G q ( k + p ) ( − iγ ν ) G q ( k + q )IIΓ α ( q, − k − q ) . (4.2)In the above formulas, N c = 3 for SU (3), the vertex renormalisation constants Z and Z f were defined after (2.3c), and Γ (0) ναβ denotes the classical three-gluon vertex,Γ (0) ναβ = g s (cid:2) (2 k − p − q ) ν g αβ + (2 q − p − k ) α g νβ + (2 p − q − k ) β g αν (cid:3) , (4.3)where we have factored out the color factor f abc .The vertex IIΓ µ may be decomposed in a basis formed by eight independent tensorialstructures, denoted by T µi , which can be derived from gauge-invariant quark-gluon operators[73, 77], according to¯ q /Dq → T µ , ¯ q /D q → T µ , T µ , T µ , ¯ q /D q → T µ , T µ , T µ , ¯ q /D q → T µ . (4.4)In particular, IIΓ µ ( q, − p ) = (cid:88) i =1 λ i ( q, − p ) P µν ( q − p ) T νi ( q, − p ) , (4.5)12here the short-hand notation λ i := λ ( i ) q ¯ qA was introduced.With the aid of (4.5), and through appropriate tensor contractions, the starting SDE of(4.1) may be converted into a system of coupled integral equations for the λ i ( p, q ). Specifi-cally, one obtains λ i ( q, − p ) = Z f g s δ i + a i ( q, − p ) + b i ( q, − p ) , i = 1 , ..., a i ( q, − p ) = Z N c (cid:90) d k (2 π ) λ j ( k, − p ) λ k ( q, − k ) G A ( k − q ) G A ( k − p ) K ijk ( p, q, k ) ,b i ( q, − p ) = − Z f N c (cid:90) d k (2 π ) λ j ( k + p, − p ) λ k ( q, − k − q ) G A ( k ) (cid:101) K ijk ( p, q, k ) , (4.7)where the kernels K ijk ( p, q, k ) and (cid:101) K ijk ( p, q, k ) contain combinations of Z q , M q , and thevarious momenta; further information on their precise structure is provided in Appendix B.The renormalisation condition corresponding to (3.11) dictates that, at the symmetricpoint ¯ p = µ , we must impose Z f g s = g s − (cid:2) a ( q, − p ) + b ( q, − p ) (cid:3) p = q = µ . (4.8)This leads us to the final, explicitly renormalised coupled integral equations for the λ i ( p, q ), λ i ( p, q ) = a i ( p, q ) + b i ( p, q ) + (cid:16) g s − (cid:2) a i ( q, − p ) + b i ( q, − p ) (cid:3) p = q = µ (cid:17) δ i , (4.9)which satisfies manifestly (3.11b).As we will see in detail in Sec. VII, the numerical treatment of the system of coupledintegral equations given by Eqs.(3.9), (4.9), and (4.7), reveals a clear hierarchy among thedressings λ i . In particular, depending on their numerical impact, the λ i may be naturallyseparated into “ dominant ”, “ subleading ”, and “ negligible ”.Specifically, the three dominant components of the quark gluon vertex are λ , , , associ-ated with the tensor structures T µ ( p, q ) = − iγ µ , T µ ( p, q ) = ( p/ + q /) γ µ , T µ ( p, q ) = i p/, q /] γ µ . (4.10a)As we will see in Sec. VII , keeping only these three form factors in the coupled SDE analysis[ i.e. , the terms corresponding to i = 1 , , λ is accessible to an STI-basedderivation of the quark gluon vertex, in the spirit of the original BC construction.The three subleading components, λ , , , are associated with the basis elements T µ ( p, q ) = ( q − p ) µ , T µ ( p, q ) = i ( p/ + q /)( p − q ) µ , T µ ( p, q ) = i ( p/ − q /)( p − q ) µ . (4.10b)These three dressings may be obtained from the STI-based constructions, implemented onlyin the vacuum. Therefore, in view of the numerous applications to QCD at finite temperatureand density, SDE-based computations of these subleading tensor structures, such as the oneput forth here, are clearly preferable.Finally, the form factors associated with the tensors T µ ( p, q ) = ( p/ − q /) γ µ , T µ ( p, q ) = −
12 [ p/, q /]( p − q ) µ , (4.10c)are negligible, having no appreciable numerical impact on our benchmark observable or anyother relevant quantity (see also [59, 60]).This concludes the description of our SDE setup. V. EXTERNAL INPUT AND SELF-CONSISTENT RENORMALISATION
In this section we discuss self-consistent renormalisation schemes for the SDE with agiven external input. This issue is addressed both in general and for the given input datafor the gluon propagator used here. In addition, we detail the origin and characteristics ofthese data.In Sec. V A we elaborate on the implementation of multiplicative renormalisation in thepresent non-perturbative approach, in Sec. V B we provide an overview on the gluon propa-gator data used as input, in Sec. V C we discuss the general self-consistent determination ofthe value of the renormalised coupling α s ( µ ) at the renormalisation scale µ , and in Sec. V Dwe determine α s ( µ ) for the gluon input data specified in Sec. V C. A. Multiplicative renormalisation
The self-consistent implementation of multiplicative renormalisation at the level of thenon-perturbative SDEs constitutes a yet unresolved problem, which has been treated only14pproximately within numerical applications, see, e.g. , [7, 35, 65, 70, 81–83]. In the presentcontext, the complications stemming from this issue manifest themselves at the level of thegap equation by the presence of the factor Z f in the definition of the quark self-energy Σ( p ),and at the level of the SDE for IIΓ µ through the factors Z and Z f entering in the expressionsfor A µ and B µ , respectively.Evidently, the renormalisation constants Z , Z f display a non-trivial (“marginal”) depen-dence on the UV cutoff, which is required for rendering the diagrams finite. However, theorder-by-order cancellation known from perturbation theory does not translate straightfor-wardly to the non-perturbative setup of the SDEs. For the purposes of this analysis, weadopt the standard approximation: we assume that the cutoff-dependence encoded in Z , Z f has been successfully cancelled, and set the cutoff-independent finite parts to zero, leadingto Z , Z f → µ → ∞ , this procedure can be put forth by invokingasymptotic freedom, g s ( µ → ∞ ) →
0, and the fact that the finite parts stemming from Z , Z f → g s . This argument is further supportedwithin the setup with fRG inputs; for more details, see Appendix A. Nonetheless, theseconsiderations do not constitute a proof of the full self-consistency of this procedure, whichis the subject of ongoing work.In summary, for the numerical treatment of the system of integral equations presentedhere, we simply implement the substitutionΣ( p ) → Σ( p ) | Z f =1 , (cid:2) a i ( p, q ) , b i ( p, q ) (cid:3) → (cid:2) a i ( p, q ) , b i ( p, q ) (cid:3) Z f =1= Z . (5.1)It is evident from the discussion above that the simplifications implemented by (5.1) arebound to induce a residual cutoff- and µ -dependence to the results obtained, which arediscussed in Sec. VIII. B. Gluon propagator
The gluon propagator can be computed from its own SDE; for the most recent resultsin Yang-Mills theory, see [83–86], while for 2+1–flavour solutions of the fRG-assisted SDE,see [59, 60]. Consequently, we could extend the current system to a fully self-coupled one,the only input being the strong coupling and the current quark masses. However, in this15 p [GeV] G A ( p ) [ G e V − ] fRG-DSEfRGlatticelattice fit -1 p [GeV] G A ( p ) [ G e V − ] p [GeV] / Z A ( p ) fRG-DSEfRGlatticelattice fit -1 p [GeV] / Z A ( p ) FIG. 3: 2+1–flavour gluon propagator, G A ( p ), and dressing function 1 /Z A ( p ) = p G A ( p ).Lattice simulations: [31, 79, 80], fRG-DSE approach: [59, 60], fRG approach: [57]. Thecomputations in [57, 59, 60] are based on the 2-flavour input fRG data from [77].work we concentrate rather on the novel key ingredient, namely the computation of the fulltransversally projected quark-gluon vertex and its properties. Therefore, we simply takequantitative input data from either the lattice [31, 79, 80], the SDE [59, 60, 87, 88], or thefRG [57].For the present computation the input data have to cover the momentum regime p ∈ [0 , Λ ], where Λ is the UV cutoff of the loop integrals in the DSEs. In the presentwork we consider UV cutoffs in the range Λ = 50 − p < ∼ p > ∼ p ∈ [0 ,
40] GeV, G A ( p ) = a + bp + cp dp + ep + f p , (5.2)with the fit parameters a = 8 .
01 GeV − , b = 2 .
38 GeV − , c = 0 .
019 GeV − , d = 1 .
81 GeV − , e = 1 .
67 GeV − , f = 0 .
019 GeV − . The dressing Z − A ( p ) is obtained through multiplicationof the G A in (5.2) by p .Note also that, as can be seen in Fig. 3, the input data for G A ( p ) differ in the infrared, i.e. ,16or p < ∼ “scaling” [57, 59, 60], vs “decoupling” or “massive” [31, 79, 80]; for relateddiscussions, see, e.g. , [87, 89, 90]). Nonetheless, the quark propagator obtained using eitherof them, as well as the computed physical observables, agree within our systematic error bars.The reason for this is related to the fact that, inside the quantum loops considered here, thegluon propagator G A ( p ) is eventually multiplied by p ; as a result, the infrared differencesare largely washed out, and the relevant quantity, Z − A ( p ) = p G A ( p ), is practically identicalfor both. C. Self-consistent determination of α s ( µ ) A necessary ingredient for our analysis is the value of the dressing λ at the symmetricpoint, which, for sufficiently large values of µ is equal to the (unique) perturbative g s , orthe α s defined in (2.6) [see also (3.11b)]. If no external inputs were employed, one woulddetermine λ at µ by simply imposing that α s ( µ ) should coincide with the physical QCDcoupling, α s, phys , at a given momentum scale; for example, this can be done at the mass M Z of the Z boson, i.e. , α s ( µ ) = α s, phys ( M Z ). However, since in our study we use asexternal input the N f = 2 + 1 gluon propagator from lattice and functional methods, therenormalisation procedures adopted in those earlier computations need be incorporated intothe present SDE treatment, such that a self-consistent value for α s ( µ ) may be obtained.The self-consistent calibration of α s ( µ ) may be implemented according to two differentprocedures, (i) and (ii) , detailed below. In (i) , one compares correlation functions computedwithin the present SDE setup, whose form depends on the value of α s used, with data forthem originating from the same framework that provides the required external input. In (ii) , one invokes self-consistency conditions between the results of the current SDE approachand those derived from the STIs.We emphasize that both procedures are optimised when implemented in the perturbativeand semi-perturbative regime with, p > ∼ p pert , with p pert ≈ , (5.3)where the truncation errors are small and under control; instead, their extension to thenon-perturbative infrared regime is bound to worsen the calibration. In fact, while in theregime of (5.3) STI- as well as vertex-couplings agree at least up to two loops, they deviate17arkedly from each other as p →
0, see [77]. Moreover, the regularity assumption that isimplicit in the direct use of the STIs for the determination of transverse couplings [as in(5.4)] may fail in the infrared; for more details, see [77, 91–93].The concrete implementation of (i) and (ii) is presented in detail below. (i)
The setup in the present work only requires the data for the 2 + 1-flavour gluon propa-gator as external input from either distinct SDE approaches, the fRG, or lattice simulations.Within all these frameworks, one has access to data sets not only for this specific input butalso for additional correlation functions, as well as derived couplings, e.g. , via (2.6). Whilethe latter are not needed as explicit inputs for the SDE, they can be used as a means ofcalibrating the calculation, because they can be recomputed from their own dynamical equa-tion within the present SDE setup. In doing so, it is clear that their momentum dependencechanges as the value of α s ( µ ) is varied. Thus, the external data sets may be reproduced fora unique self-consistent choice of α s ( µ ), which calibrates our approach (for p > ∼ p pert ).If a propagator is chosen for the purpose of calibration, the ghost propagator is clearlythe best choice, as it is governed by a rather simple SDE, whose only other ingredient isthe ghost-gluon vertex, which is protected by Taylor’s non-renormalisation theorem. Forexample, if one were to use as external input the gluon propagator from the lattice, thecalibration proceeds by computing the ghost dressing function Z c ( p ) within our SDE setup,adjusting the α s ( µ ) such that the lattice data for Z c ( p ) will be best reproduced.If one of the running couplings, α i (¯ p ), is employed for the calibration, we compute itsshape within both the approach that furnishes the external input and within the SDE setup.Then, self-consistency requires that the SDE α s ( µ ) is chosen such that the difference betweenthe two α i (¯ p )’s is minimised, for ¯ p > ∼ p pert .We close with the remark that, in the present setup, all procedures mentioned above leadto α s ( µ ) that agree within the small numerical and systematic errors. We consider this animportant self-consistency check of the SDE approach put forth here. (ii) If the additional results needed for the implementation of (i) are unavailable, onecan use the STI satisfied by the quark-gluon vertex in order to fix α s ( µ ). Specifically, for¯ p > ∼ p pert one uses the relation λ (¯ p ) = g s ( µ ) L (¯ p ) , (5.4)where L is the solution of the STI for the longitudinally projected classical tensor structure,see [35, 77]. As in (i) , minimising the difference between the two sides of (5.4) singles out a18nique α s ( µ ).This concludes our general discussion of the self-consistent determination of α s ( µ ). D. Value of α s ( µ ) In this work we use two classes of data for fixing α s ( µ ), and employ both procedures, (i) and (ii) , described above; procedure (i) is used when G A ( p ) is obtained from functionalmethods, while (ii) is applied when G A ( p ) is taken from the lattice. For both classes of data,and for very different renormalisation scales [ µ = 4 . ,
40 GeV], we will produce results for thequark propagator and the pion decay constant that agree within our estimated systematicerror; this coincidence, in turn, constitutes a non-trivial check of the systematic errors. Wenext describe the determination of α s ( µ ) for both cases: Functional data sets:
Here we employ procedure (i) . The functional input for G A ( p ) isprovided by fRG [57] and SDE [59, 60] data, renormalised at µ = 40 GeV. Note that therespective SDE and fRG relations for correlation functions are expanded about their N f = 2counterparts, computed within the fRG [77]. The respective data sets also include α q ¯ qA (¯ p ),thus providing directly α s ( µ ) at µ = 40 GeV; this allows us to minimise the differencebetween input and output α q ¯ qA (¯ p ), for ¯ p > ∼ p pert . Lattice data sets:
Here we employ procedure (ii) . We use the lattice data for G A ( p )from [31, 79, 80], and, in line with our arguments, we chose the maximal lattice momentumavailable for our renormalisation scale, namely µ = 4 . L in (5.4)is computed based on the quenched computation in [67] (Fig.17, fourth panel), properlyaccounting for unquenching effects. Then, we minimise the difference between the left- andright-hand sides in (5.4).Both procedures are now applied at two rather disparate renormalisation scales µ , namely µ = 4 . µ = 40 GeV, for which we obtain the values α s (4 . . , α s (40 GeV) = 0 . , (5.5)which are fully compatible with earlier SDE and fRG considerations. Note that the value ofthe coupling at µ = M Z obtained in this setup is α s ( M Z ) = 0 .
16. The deviation from thestandard value α s, phys ( M Z ) ≈ .
12 is due to the presence of only three active flavours in ouranalysis. 19
I. CURRENT QUARK MASSES AND BENCHMARK PREDICTIONS
In this section we determine the fundamental parameters of QCD, the current-quarkmasses m q = ( m l , m s ), where we have assumed isospin symmetry with identical up anddown quark current masses: m u/d = m l . In addition, we provide results for a benchmarkobservable, namely the light chiral condensate, ∆ l = −(cid:104) ¯ l ( x ) l ( x ) (cid:105) , which allows us to evaluatethe veracity of the present approximations. In particular, we find that our ∆ l is in excellentquantitative agreement with the most recent lattice estimates reported in [94].The current quark masses m q ( µ ) at a given µ are fixed from the physical pion mass, m π ,and the ratio of strange and light current quark masses, m s ( µ ) /m l ( µ ). This procedure hasbeen used both in [57, 59, 60, 73, 77] (see also the reviews [4, 13, 15, 95]), and in latticesimulations (see, e.g. , the compilation in [94]).Note that, due to the identical (one-loop) RG-running of all m q ( µ ), the mass ratio m s ( µ ) /m l ( µ ) tends to a constant for asymptotically large µ ,lim µ →∞ m s ( µ ) m l ( µ ) = m s m l , m q ( µ ) → m q [ln( µ/ Λ QCD )] γ m , with γ m = 1233 − N f . (6.1)In the present work we use µ = 40 GeV, and compare the results to those obtained witha considerably lower µ = 4 . µ = 40 GeV, the m q ( µ ) will be determined using the values (cid:16) m q ( µ ) , µ = 40 GeV (cid:17) : m π = 138 MeV , and m s m l = 27 . (6.2) A. Pagels-Stokar formula and Gell-Mann–Oakes–Renner relation
Ideally, the pion mass, m π , and decay constant, f π , should be determined from the on-shell properties of the BS wave function of the pion. Instead, in the present work we employstandard Euclidean approximations for them, given by the Pagels-Stokar (PS) formula for f π ,and the Gell-Mann–Oakes–Renner (GMOR) relation for m π [(6.3) and (6.9), respectively].The GMOR relation is correct up to order O ( m l ) within an expansion about the chiral limit,while the PS formula is known to underestimate f π by < ∼
10% (see, e.g. , [96, 97] and thereviews [4, 15, 98]). We emphasise that this low value does not undermine the precision ofour analysis, given that f π is a derived quantity that does not feed back into the SDEs.20he PS formula reads f (PS) π = 4 N c N π (cid:90) p ¯ Z Z q ( p ) M q ( p ) (cid:2) p + M q ( p ) (cid:3) (cid:20) M l ( p ) − p M (cid:48) q ( p ) (cid:21) , (6.3)with M (cid:48) q ( p ) = ∂ p M q ( p ), and the subtracted mass function M q ( p ), M q ( p ) := M q ( p ) − m q ∂M q ( p ) ∂m q , with lim p →∞ p M q ( p ) = 0 . (6.4)Note that Eq. (6.4) applies to both m l or m s . The constant ¯ Z is linked to the quarkdressing function Z q ( p ). Specifically, the derivation of (6.4) from the pion BSE prompts theidentification ¯ Z = Z q ( p π ), where p π = − m π ; in the chiral limit, p π = 0, and we will simplyuse ¯ Z = Z l (0).The normalisation N π of the pion wave function is given by N π = 12 (cid:104) f π + (cid:112) f π + 8 N c I l (cid:105) , (6.5)with I q := (cid:90) p M q ( p ) (cid:2) p Z q ( p ) Z (cid:48)(cid:48) q ( p ) + 2 Z q ( p ) Z (cid:48) q ( p ) − p Z (cid:48)(cid:48) q ( p ) (cid:3) Z q ( p ) (cid:2) p + M q ( p ) (cid:3) , (6.6)where the abbreviations Z (cid:48) ( p ) = ∂ p Z ( p ), and Z (cid:48)(cid:48) ( p ) = ∂ p Z (cid:48) ( p ) have been used. Note that,due to the asymptotic behavior of M q ( p ) stated in (6.4), the integral is finite. The aboveexpressions for N π are approximate; a more complete treatment requires all components ofthe pion wave function, and will be considered elsewhere.It is well-known that, in the chiral limit, we must have I q,χ = 0 and N π = f π [99].However, our approximations deviate slightly from this result, furnishing a I q,χ which failsto vanish by an amount that induces a 3% discrepancy between N π and f π . We effectivelyaccount for this small error by setting I q → I q − I q,χ , (6.7)thus compensating, in a simple way, for the contributions of the omitted form factors. Thenumerical impact of this adjustment will be discussed in Sec. VI C, see (6.18).In the chiral limit we arrive at f (PS) π,χ = 84 . , (6.8)21n comparison to the FLAG estimate of f lat π,χ = 86 . f lat π = 92 . .
6) MeV forphysical quark masses.Turning to the GMOR relation, we have m π = 2 m l f π ∆ l + O ( m l ) , with ∆ l = −(cid:104) ¯ uu (cid:105) = −(cid:104) ¯ dd (cid:105) , (6.9)where m l is a µ -independent current quark mass, and ∆ l denotes the finite and µ -independentlight quark condensate; for a concise discussion of its RG properties, see [100].Since in our calculations enter the µ -dependent current quark masses, m l ( µ ), rather than m l , for the purposes of the present work we find it advantageous to capitalise on the property m l ∆ l = m l ( µ ) ∆ l ( µ ) , (6.10)and recast (6.9) in the form m π = 2 m l ( µ ) f π ∆ l ( µ ) . (6.11)Evidently, in order to extract from (6.11) the value of m l ( µ ), one requires knowledge of f π and ∆ l ( µ ). Given the aforementioned shortcomings of the PS formula, we will simply usethe physical value of f π as input in (6.9). The details of ∆ l ( µ ) will be discussed separatelybelow. B. Light chiral condensate
The benchmark observable for our computation is the µ -independent ∆ l , whose chirallimit value, ∆ l,χ , can be extracted from the UV behaviour of the corresponding constituentquark mass M l,χ ( p ) [100],lim p →∞ M l,χ ( p ) = 2 π γ m l,χ p [ln( p/ Λ QCD )] − γ m . (6.12)with γ m defined in (6.1).Of course, as mentioned above, the quantity that we will employ in (6.11) is rather ∆ l ( µ ).The latter is usually computed in lattice simulations at a given scale µ lat , which is typicallyfar lower than the one used in the present SDE approach; for more details and an overviewof the respective lattice results see [94]. For sufficiently large µ , ∆ l and ∆ l ( µ ) are related by∆ l ( µ ) = ∆ l [ln( µ/ Λ QCD )] γ m . (6.13)22 p [GeV] / Z l ( p ) physical masschiral limit p [GeV] M l ( p ) [ G e V ] physical masschiral limit FIG. 4: Quark dressing, 1 /Z q ( p ), and mass function, M q ( p ), in the chiral limit and forphysical quark masses.With m l ( µ ) defined in (6.1) and (6.13), the combination m l ( µ )∆ l ( µ ) is RG-invariant, asstated in (6.10).Combining the above relations with our SDE results for the quark propagator we arriveat our first non-trivial prediction. We use our results for M l ( p ) and 1 /Z l ( p ) in the chirallimit, shown in Fig. 4, which are necessary for our analysis; a complete discussion is providedin Sec. VII. Specifically, the value of ∆ l,χ is given by∆ l,χ = (245 MeV) , (6.14)where m s is kept fixed and m l → analyti-cally by identifying the diagrams included in each perturbative order, the current approachencodes the full two-loop running of M l ( p ), and is numerically consistent with the full three-loop running; for a more detailed discussion, see Sec. VII.We next combine (6.14) with (6.13) to obtain our prediction for ∆ l,χ ( µ ), and compare itwith the FLAG result of [94], obtained from different lattice groups. In these simulations,the renormalisation scale is µ lat = 2 GeV, and the estimate for the QCD scale is Λ latQCD =343(12) MeV in the MS scheme; instead, in the present SDE computation we have Λ QCD =293 MeV. From (6.13) and (6.14), this leads us to∆ l,χ ( µ lat ) = (270 MeV) , [∆ l,χ ( µ lat )] FLAG = (272(5) MeV) . (6.15)Taking into account the subtleties in the conversion and application of RG scales, the agree-23ent between these two values is rather impressive, providing non-trivial support for thequantitative reliability of the present approximation. We emphasise that the prediction(6.14), and hence (6.15), have been obtained without any phenomenological input on thestrength of chiral symmetry breaking.Finally, ∆ l,χ can be used for the determination of the physical ( m q (cid:54) = 0) chiral condensate∆ l through∆ l = ∆ l,χ + (∆ l − ∆ l,χ ) = ∆ l,χ + (cid:90) p (cid:32) Z Z q ( p ) M l ( p ) p + M l ( p ) − Z Z q,χ ( p ) M l,χ ( p ) p + M l,χ ( p ) (cid:33) , (6.16)where the integral is simply the difference of the loop expressions for the chiral condensates,and is finite. In (6.16) we have already set the multiplicative renormalisation factors to unity,according to the procedure described in Sec. V A. The calculated value for ∆ l is reported inthe following subsection. C. Determination of the current quark masses
With the groundwork laid in Sec. VI A and Sec. VI B, we now determine the currentquark masses. Using (6.2), (6.3), (6.11), (6.16), and a pion decay constant f π = 92 . m q at µ = 40 GeV, m q ( µ ) = 2 . , m s = 73 MeV , with m π = 138 MeV , (6.17a)and the predictions∆ l = (300 MeV) , M q (0) = 351 MeV , f (PS) π = 87 . . (6.17b)From (6.17b) we deduce that f (PS) π has an error of 5%, which is well within our conservativeestimate for the systematic error in the range of 10%. Note that this error estimate alsoextends to the GMOR relation, and hence our quark masses share this 10% uncertainty.We emphasise that this error is not inherent to our SDE-computation, but affects deducedobservables. Indeed, our benchmark result (6.15) for the chiral condensate in the chirallimit agrees with the lattice results within the statistical error of the latter (less than 1%deviation).For completeness we also report the result for f (PS) π without the correction to N π imple-mented by (6.7); we have [ f (PS) π ] I q,χ (cid:54) =0 = 86 . , (6.18)24n very good agreement with the f (PS) π in (6.17b).All results presented thus far have been obtained using the fRG data for the gluon prop-agator, renormalised at µ = 40 GeV, as input in the SDEs. In order to illustrate that ourpredictions are essentially independent of the input propagator, we also report the resultsobtained when a fit to the gluon lattice data is employed, which has the fRG perturbativebehavior built in it. Note also that the fRG and lattice data differ in the deep infrared [seeFig. 3], as commented in Sec. V B. With this particular input we arrive at m q ( µ ) = 2 . , m s = 73 MeV , with m π = 138 MeV , (6.19a)and the predictions∆ l = (301 MeV) , M q (0) = 350 MeV , f (PS) π = 88 . . (6.19b)In addition, the results for chiral condensate in the chiral limit, ∆ l,χ = (246 MeV) and∆ l,χ ( µ lat ) = (271 MeV) are in quantitative agreement with (6.14) and (6.15), obtained for µ = 40 GeV. This coincidence, and the comparison of (6.19) with (6.17) shows explicitlythat, within our estimated systematic error, the behaviour of the gluon propagator in thedeep infrared does not affect the observables considered here.This concludes the discussion of the determination of the current quark masses. Insummary, we have shown that the present SDE approach allows for a quantitatively reliablecomputation of the constituent quark mass function M q ( p ), and hence the chiral condensatein (6.15), without any phenomenological input. VII. NUMERICAL RESULTS
In this section we present and discuss the central results of our numerical analysis, fo-cusing mainly on the general features displayed by Z q , M q , and the λ i . The stability ofthese results under variations of the UV cutoff, the gluon inputs, and the RG scale will beaddressed in Sec. VIII, while the implementation of reliable simplifications will be analysedin Sec. IX.Within the setup described in Secs. IV, V and VI, the SDE system of quark propagatorand quark-gluon vertex is solved iteratively. To that end, it is convenient to use as thestarting point of the iteration input data for quark and gluon propagators and the respective25C vertex, or, when the fRG-DSE gluon is utilized [59, 60], the respective quark-gluon vertexdata. Moreover, the dressings of the three dominant tensor structures, λ , , , are fed backinto their own coupled SDEs and those of the other vertex dressings; instead, the remaining λ , , , , are only used in the quark gap equation. We have checked that this procedureaffects the key quantities only very marginally: when the sub-dominant dressings are fedback into the quark-gluon SDE, the numerical changes induced lie comfortably within theestimated error bars of our method.As discussed in Sec. VIII, the RG scale µ should be taken as large ( i.e. , as “perturbative”)as possible; therefore, we choose µ = 40 GeV. However, the lattice data for the gluonpropagator are limited by p < ∼ µ , see Sec. VI.Our main results are summarised in Fig. 5. In particular, in Fig. 5a we show all dressingsof the quark-gluon vertex, while in Fig. 5b we display the quark mass functions, M q ( p ), andthe quark wave function renormalisations 1 /Z q ( p ) (inset) for q = l, s . Note that, in orderto best expose the physical relevance of the different tensor structures and the correspond-ing dressings, we introduce dimensionless couplings, ¯ λ i (¯ p ), with i = 1 , ...,
8, see, e.g. , [73];specifically, we concentrate on the symmetric point ¯ p and define¯ λ i (¯ p ) = ¯ p n i λ i (¯ p ) Z q (¯ p ) Z / A (¯ p ) , with n = 0 , n , , = 2 , n , , = 2 , n = 3 . (7.1)In (7.1), the multiplication of λ i by ¯ p n i renders the ¯ λ i dimensionless, and the division bythe wave function renormalisations leaves us with the respective eight running couplings.For all quarks q = l, s , the products 1 / (4 π )¯ λ i (¯ p )¯ λ j (¯ p ) can be interpreted as the interac-tion strength of a one-gluon exchange between corresponding quark currents. For example, α q ¯ qA (¯ p ) = 1 / (4 π )[¯ λ (¯ p )] , see (2.6), simply measures the interaction strength or running cou-pling of a one-gluon exchange between the two quark currents ¯ q ( t ) γ µ q ( p − t ). Similarly, the26 -1 p [GeV] q u a r k - g l u o n c o u p li n g s ¯ λ ¯ λ ¯ λ ¯ λ ¯ λ ¯ λ ¯ λ ¯ λ (a) Quark-gluon couplings ¯ λ i ( p ), (7.1). Black:classical tensor structure, (2.6). Grey: chirallysymmetric non-classical tensor structures. Red:chiral-symmetry breaking tensor structures. -1 p [GeV] M q ( p ) [ G e V ] u/d quarks quarkLattice -1 p [GeV] / Z q ( p ) (b) Quark mass functions M q ( p ) and propaga-tor dressings 1 /Z q ( p ) (inset) for q = l, s , see(2.7). Lattice results from [101]: light quarkmass function M l ( p ) and dressing 1 /Z l ( p ). FIG. 5: Numerical results for the coupled system of SDEs for quark-gluon vertex (lightquark, left panel), and the light and strange quark propagators (right panel).combinations 1 / (4 π )¯ λ i (¯ p )¯ λ j (¯ p ) can be understood as the interaction strength of a one-gluonexchange between the respective tensor currents ¯ q T i q and ¯ q T j q , with the dimensionlesstensor structures T i = T i / (¯ p ) n i , with i, j = 1 , ...,
8. As in the case of the gluon dressing,one sees the prominent enhancement around ¯ p ≈ . − λ i for ¯ p → respective results of different but similar functional approaches. In our opinion, theconfirmation of this quantitative agreement, and further successful comparisons of this type,provide important information about the respective systematic error. Together with appar-ent convergence of the results in functional approaches within a systematic approximationscheme, this finally will lead to a first principle functional approach to QCD.27 p [GeV] q u a r k - g l u o n c o u p li n g s this workfRG-DSESTI scaling ¯ λ ¯ λ ¯ λ p [GeV] λ i ( p ) (a) Dominant quark gluon couplings λ , , : fullsolution SDE (this work, black), full solution[59] (fRG-DSE, red), couplings with STI, ¯ λ ,and scaling relations, ¯ λ , (STI-scaling, blue). -1 p [GeV] M l ( p ) [ G e V ] this workfRG-DSELattice -1 p [GeV] / Z l ( p ) (b) Light quark mass function M l ( p ) anddressing function 1 /Z l ( p ) (inset): full solution(this work, full black), full solution [59] (fRG-DSE, dashed red), lattice results [101] (Lattice,green). FIG. 6: Full results for dominant quark-gluon couplings ¯ λ , , (this work, black) from thepresent work in comparison to full results from [60] (fRG-DSE, red), lattice results (Lattice,green), and the approximation STI-scaling : λ , , from STI, and λ , , , from scaling relations,see discussion at the end of Sec. IX and [59, 77]. The λ i are dressings of tensor structuresin Eq. (4.10a) and measured in λ , λ [(GeV) − ] and λ [(GeV) − ].We emphasise that the present approximation includes analytically the full two-looprunning of the quark gap equation. Hence, M q ( p ) and 1 /Z q ( p ) are two-loop consistent, sincethe quark gap equation contains all tensors of the quark-gluon vertex. The SDE solution ofthe latter includes all one-loop diagrams, and hence, the numerical solution for the λ i ( p, q )encompasses the full one-loop structure analytically. Furthermore, the input gluon datacontain at least the full one-loop momentum dependence. Accordingly, all ingredients inthe quark-gap equation carry at least their full one-loop momentum dependence, and hence,the solution is analytically two-loop consistent.Of course, the vertex SDE employed (see Fig. 2) corresponds to the so-called “one-loopdressed” truncation, where vertices with no classical counterpart, such as the four-quarkvertex [60], have been omitted from the skeleton expansion. Nonetheless, the contributions28 p [GeV] q u a r k - g l u o n c o u p li n g this workone looptwo loop p [GeV] β l ¯ l A ( p ) p [GeV] q u a r k - g l u o n c o u p li n g this workone looptwo loop p [GeV] β l ¯ l A ( p ) FIG. 7: Strong coupling α l ¯ lA ( p ), see (2.6), in comparison to the one- and two-loop counter-parts, adjusted at the RG scale µ = 40 GeV (left panel) and µ = 4 . α l ¯ lA ( p ) = 1 / (4 π )¯ λ ( p ). In Fig. 7 we depict our numerical result together with theone- and two-loop α (1loop) s ( p ) and α (2loop) s ( p ), renormalised at µ = 40 GeV and µ = 4 . QCD are chosen such that also the β -functions β l ¯ lA ( p ) = p∂ p α l ¯ lA ( p ) , (7.2)match at the renormalisation scale µ .The numerical results in the present work have been obtained with µ = 40 GeV, which liesdeep in the perturbative regime. While this choice reduces the systematic error originatingfrom non-perturbative approximations to the SDEs, its successful implementation requiresa particularly accurate treatment, in order to reliably connect the wide range of momentabetween µ and the deep infrared. As can be seen in Fig. 7 (left panel), our numerical resultsfor ( α l ¯ lA , β l ¯ lA ) agree quantitatively with the respective two-loop results ( α (2loop) s , β (2loop) α s ) formomenta p > ∼ µ . We also note that the agreementwith the coupling extends even to p ≈ . p < ∼ β -function β l ¯ lA ,which measures the momentum slope, starts to deviate from the two-loop result β (2loop) α s , seeFig. 7 (inset left panel). Moreover, below p ≈ . α l ¯ lA rapidly departsfrom the perturbative two-loop coupling, signaling the onset of non-perturbative physics.Instead, while the adjustment of the one-loop coupling and its β -function at µ = 40 GeV29eads to the reasonable value of Λ QCD = 0 . p = 40 GeV and the non-perturbative infraredregime with p < ∼ − µ = 4 . α l ¯ lA agree well for momenta p > ∼ . µ = 40 GeV,the β -function reveals deviations already in the perturbative regime, see Fig. 7 (inset rightpanel). Moreover, if the one-loop coupling is renormalised at µ = 4 . p ≈ . QCD , namely Λ
QCD = 1 . µ ≥ µ min , with µ min ≈ . Z i = 1 do not apply.Finally, we note that a fully two-loop consistent analysis would require the omitted dia-grams in the quark-gluon SDE, as well as a two-loop consistent gluon input. Both tasks liewithin the technical grasp of functional approaches; for a discussion concerning the gluonpropagator, see [54, 102].In summary the agreement with the lattice as well as other functional methods is ratherimpressive, especially since no phenomenological infrared parameter is involved: the resultspresented here are obtained within a first-principle setup to QCD, the only input being thefundamental parameters of QCD. VIII. STABILITY OF THE NUMERICAL RESULTS
As we will see in detail in this section, the results obtained from our SDE analysis areparticularly stable under variations of the UV cutoff that regulates the loop integrals, thechoice of functional or lattice gluon inputs, and a vast change in the value of the RG scale.30 p [GeV] M l ( p ) [ G e V ] Λ=50[GeV]Λ=100[GeV]Λ=1000[GeV]Λ=5000[GeV] p [GeV] / Z l ( p ) (a) Light quark mass function M l ( p ) and dress-ing 1 /Z l ( p ) (inset) for different UV cutoffs. p [GeV] q u a r k - g l u o n c o u p li n g s Λ=50[GeV]Λ=100[GeV]Λ=1000[GeV]Λ=5000[GeV] (b) Quark-gluon coupling ¯ λ ( p ) of the classicaltensor structure for different UV cutoffs. FIG. 8: Numerical results for the coupled system of SDEs for different UV cutoffsΛ = 50 , , , A. Varying the UV cutoff
We have verified explicitly that our results are practically insensitive to variations mo-mentum cutoff Λ within the range Λ = 50 GeV to Λ = 5000 GeV. Note that, while λ displays a marginal (logarithmic) cutoff dependence, all remaining λ i are not subject torenormalisation. Moreover, in the Landau gauge, the logarithmic running of Z l ( p ) vanishesat one loop, and only M l ( p ) shows a one-loop logarithmic running. Accordingly, in Fig. 8we show the absence of cutoff-dependence in M l ( p ), 1 /Z l ( p ), and ¯ λ ( p ). Our results areespecially stable, and, in particular, no log Λ-dependence may be discerned.We emphasise that the detection of such a logarithmic dependence in the present systemis very difficult, due to its (Landau gauge) suppression in Z l , and the decay of M l ( p ) forlarge momenta. This leaves us with ¯ λ ( p ), whose perturbative momentum-dependence isfixed by the self-consistent determination described in Sec. V D. Note that these properties,even though they complicate the detection of residual cutoff-dependences, are a welcomefeature rather than a liability: the present setup reduces the sensitivity of the SDE systemwith respect to the subtleties of a non-perturbative numerical renormalisation.31 p [GeV] M l ( p ) [ G e V ] fRG-DSE inputfRG inputlattice fit p [GeV] / Z l ( p ) (a) Light quark mass function M l ( p ) and dress-ing 1 /Z l ( p ) (inset) for different gluon input. p [GeV] q u a r k - g l u o n c o u p li n g s ¯ λ ¯ λ ¯ λ fRG-DSE inputfRG-DSE inputLattice fit (b) Dominant quark-gluon couplings ¯ λ , , ( p )for different gluon input. FIG. 9: Numerical results for the coupled system of SDEs with µ = 40 GeV for differentgluon input data, Fig. 3 in Sec. V B: [59, 60] (fRG-DSE), [57] (fRG), [31, 79, 80] (lattice fit). B. Stability with respect to the gluon input data
We proceed with the insensitivity with respect to the gluon input data, described inSec. V B and depicted in Fig. 3. In Fig. 9 we compare the results obtained using as inputs: (a) the data from the fRG-DSE computation [59, 60] (“ fRG-DSE” input); (b) the gluonpropagator obtained with the fRG computation of [57] (“ fRG” input); and (c) the fit to thelattice data of [31, 79, 80], including an RG-consistent UV extrapolation (“ lattice fit ”). Therespective results for M l ( p ) and 1 /Z l ( p ) are shown in Fig. 9a, while those for ¯ λ , , in Fig. 9b.All results show an impressive quantitative agreement within the statistical and system-atic errors. In particular, the difference in the infrared behavior between the lattice andthe functional data used here [see Fig. 3)] does not leave any significant trace on M l ( p )and 1 /Z l ( p ), as can be seen in Fig. 9a. Accordingly, they do not influence our benchmarkprediction for the chiral condensate, given in (6.15). Moreover, the same independence isseen at the level of the ¯ λ , , ( p ), displayed in Fig. 9b. This lack of sensitivity to the infrareddetails of the input gluon propagators stems from the fact that the latter enter into four-dimensional momentum integrals, whose radial dependence, p , suppresses the deep infraredvery effectively. 32 p [GeV] M l ( p ) [ G e V ] µ =40 GeV µ =4 . GeV µ =4 . GeV rescaledrescaling factor p [GeV] / Z l ( p ) (a) M l ( p ) and Z l ( p ) using the fRG-DSE input,for µ = 40 GeV (black-solid) and µ = 4 . p [GeV] q u a r k - g l u o n c o u p li n g s ¯ λ ¯ λ ¯ λ µ =40 GeV µ =4 . GeV (b) Results for the dominant quark-gluon cou-plings ¯ λ , , , obtained with the fRG-DSE in-put, and renormalised at µ = 40 GeV and µ = 4 . FIG. 10: The µ -independence of M l ( p ) and ¯ λ , , ( p ), and multiplicative renormalisability of Z l ( p ). The RG scales differ by an order of magnitude: µ = 40 GeV and µ = 4 . C. Varying the RG scale µ Finally we test the response of our results to changes in the RG scale µ . In particular, wecompare the results obtained when all relevant quantities have been renormalised at the twovastly different scales µ = 40 GeV and µ = 4 . M l ( p ) and ¯ λ i ( p ) are formally RG-invariant quantities, and, ideally, they shouldbe µ -independent; in practice, the amount of residual µ -dependence displayed is an indicationof the veracity of the approximations employed. The results shown in Fig. 10 demonstrateclearly that the µ -dependence of these quantities lies well within the estimated error bars;in particular, the largest visible discrepancy, located at the peak of ¯ λ ( p ), is only 3.4%.On the other hand, the quantity Z l ( p ) is not RG-invariant, depending explicitly on µ , ascan be seen in the inset of Fig. 10a. However, multiplicative renormalisation, when properlyimplemented, dictates that the curves renormalised at two different values of µ , say µ and33 p [GeV] q u a r k - g l u o n c o u p li n g s ¯ λ ¯ λ ¯ λ ¯ λ (a) Quark-gluon couplings from chirally-sym-metric tensor structures T , , , . p [GeV] q u a r k - g l u o n c o u p li n g s ¯ λ ¯ λ ¯ λ ¯ λ (b) Quark-gluon couplings from of the chiral-symmetry–breaking tensor structures T , , , . FIG. 11: Quark-gluon couplings ¯ λ i ( p ), defined at the symmetric point, see (7.1). Theordering in the legends is reflecting their strength. µ , must be related by Z − l ( µ , µ ) Z − l ( p, µ ) = Z − l ( p, µ ) with µ < µ . (8.1)The operation described in (8.1) rescales the “red-dashed” curve to the “blue-dotted” onein the aforementioned inset. Note that the rescaling factor is marked on the “black-solid”curve with a blue dot; its numerical value is 0.93. Plainly, the coincidence achieved betweenoriginal and rescaled curves is excellent, indicating that multiplicative renormalisability hasbeen adequately implemented at the level of our dynamical equations. IX. RELIABLE LOW-COST APPROXIMATIONS
The numerical cost of the present work is rather modest: a full simulation with givengluon input data requires about 20 core minutes on a intel i7 chip. However, if the systemis extended by the gluon SDE in order to obtain a fully self-consistent description, thenumerical costs rises significantly. Moreover, for applications to hadron resonances, see e.g. , [15], the SDE system has to be augmented by BSE, Faddeev equations, and four-bodyequations, depending on the resonances of interest. Finally, in the study of the QCD phasestructure at finite temperature and density, a rest frame is singled out, leading to a further34 p [GeV] M l ( p ) [ G e V ] full vertex λ , only λ , only p [GeV] / Z l ( p ) (a) Quark dressings M l ( p ) and 1 /Z l ( p ) without¯ λ (dashed, red), and without ¯ λ (dotted, blue)in the vertex SDEs in comparison to the fullsolution (full, black). p [GeV] M l ( p ) [ G e V ] full vertex λ , only λ , only p [GeV] / Z l ( p ) (b) Quark dressings M l ( p ) and 1 /Z l ( p ) without¯ λ (dashed, red), and without ¯ λ (dotted, blue)in the quark gap equation in comparison to thefull solution (full, black). FIG. 12: Lack of quantitative reliability without quark-gluon couplings ¯ λ , on the righthand sides of the quark-gluon SDEs and the quark gap equation.proliferation of tensorial structures. For all the above reasons, any approach that reduces thecomputational cost without compromising the veracity of the results, is potentially usefulfor the above applications.In what follows we discuss simplified approximations of the treatment of the quark-gluonvertex, that still lead to quantitatively reliable results. To that end, we analyse the numericalimpact that the vertex dressings ¯ λ i ( p ) have on the results of the quark dressings M l ( p ) and1 /Z l ( p ). To better appreciate this discussion, we have replotted the results for the ¯ λ i ( p ),already shown in Fig. 5a: in Fig. 11 we concentrate on the ¯ λ i ( p ) in the low energy regime, i.e. ,for p < ∼ λ i ( p ) are separated in two groups, those with chiral symmetry preservingtensor structures, Fig. 11a, and those with chiral symmetry breaking ones, Fig. 11b.The main outcome of these considerations may be summarised by stating that (i) theinclusions of λ , , ( p ) is necessary and sufficient for approximating accurately the resultsof the full analysis, and (ii) λ ( p ) may be reliably obtained from STI-based constructions,while λ , ( p ) from the scaling relations put forth in [59, 60, 77].Point (i) has been established by considering the relevant SDEs approximations for the35uark-gluon vertex that include λ (which, obviously, cannot be omitted) and various subsetsof { λ i , ..., λ i n } . It is evident from Fig. 12 that the omission of either λ or λ (while keepingthe rest) leads to sizable deviations from our best results for M q ( p ) and 1 /Z q ( p ). Similarly,retaining only the special combination λ , , ( p ) reproduces very accurately our best resultsfor M q ( p ) and 1 /Z q ( p ), as shown in Fig. 12.Note also, that the hierarchy of form factors established in (i) is compatible with thatof the corresponding couplings ¯ λ i ( p ), whose relative size is shown in detail in Fig. 11. Aswe can see there, ¯ λ , , ( p ) are indeed the largest contributions; at the corresponding peaks,¯ λ > ¯ λ > | ¯ λ | . In fact, the subleading form factors are even less relevant than suggested bythe suppression of the couplings.Importantly, the above study implies that the sole use of STI-derived vertices (which, byconstruction, do not include λ , ) in either the quark-gluon SDE or the gap equation leadsto loss of quantitative precision. In particular, we have checked that the inclusion of theBC tensor structures alone in the gap equation reduces dramatically the amount of chiralsymmetry breaking, yielding M l (0) <
50 MeV.For the evaluation of point (ii) , we have computed M q ( p ) and 1 /Z q ( p ) with a dressing λ ( p ) obtained from the STI construction, while for λ , ( p ) we resort to scaling relationssuggested by the underlying gauge-invariant tensor structures [59, 60, 77]. The resultsobtained are in excellent agreement with those of the full computation, as can be seen inFig. 13.The above analysis supports the appealing possibility of implementing relatively simplebut quantitatively reliable approximations for hadron resonance computations or the phasestructure of QCD; such a setup is currently under investigation. X. SUMMARY
In this work we have considered the full set of SDEs describing the quark sector of 2+1–flavour QCD. In particular, we have coupled the gap equation of the quark propagatorwith the one-loop dressed SDE of the quark-gluon vertex, and solved the resulting systemof integral equations iteratively. The sole external ingredient used in this analysis is thegluon propagator, which has been taken from the lattice simulations of [31, 79, 80] andresults obtained from previous functional treatments [57, 59, 60]. Note, in particular, that36 p [GeV] M l ( p ) [ G e V ] full vertexvertex withSTI & scaling λ , , only (a) Quark mass function M l ( p ): full solution(full black), Vertex with STI-scaling (dashedred), λ , , only (dotted blue) p [GeV] / Z l ( p ) full vertexvertex with STI & scaling λ , , only (b) Quark dressing function 1 /Z l ( p ): full so-lution (full black), Vertex with STI-scaling(dashed red), λ , , only (dotted blue) FIG. 13: Quantitative reliability of approximations: (a) Quark-gluon couplings λ , , fromSTI, and λ , , , from scaling relations, see text and [59, 77]. (b) λ , , .the gauge coupling has been determined self-consistently, capturing correctly the analytictwo-loop running.The results of our analysis agree quantitatively with those of N f = 2 + 1 lattice simula-tions [101], and the combined (fRG and SDE) functional approach of [59, 60]. In fact, ouragreement with these latter approaches constitutes an important consistency check withinfunctional methods: SDEs and fRG represent similar but distinct non-perturbative frame-works, and the coincidence of the respective results is highly non-trivial. Moreover, the valuefor the chiral condensate, our benchmark observable, has been compared to recent latticepredictions compiled in the FLAG review [94], showing excellent agreement, see Sec. VI B.Finally, we have established that the form factors λ , , ( p ) provide the dominant numericalcontribution to the chiral infrared dynamics, as already suggested by previous studies [59, 60,77]. This allows us to devise simplified but quantitatively reliable approximations, whichmay reduce the numerical costs in the study of systems governed by a large number ofintertwined dynamical equations.In summary, the results of the current comprehensive SDE approach provide physicsresults without the need of phenomenological infrared parameters that are commonly used37xplicitly or implicitly. Moreover, we have shown how to self-consistently incorporate in ouranalysis general external inputs.In our opinion, the present comprehensive SDE approach, and in particular the combineduse of functional relations for correlation functions, is essential for a successful quantitativeinvestigation of many open physics problems in QCD, ranging from the hadron bound-stateproperties to the chiral phase structure and critical end point. Acknowledgements
We thank A.C. Aguilar and G. Eichmann for discussions. F. Gao is supported by theAlexander von Humboldt foundation. This work is supported by EMMI and the BMBFgrant 05P18VHFCA, by the Spanish Ministry of Economy and Competitiveness (MINECO)under grant FPA2017-84543-P, and the grant Prometeo/2019/087 of the Generalitat Valen-ciana. It is part of and supported by the DFG Collaborative Research Centre SFB 1225(ISOQUANT) and the DFG under Germany’s Excellence Strategy EXC - 2181/1 - 390900948(the Heidelberg Excellence Cluster STRUCTURES).
Appendix A: Mapping RG schemes
In general, for k → ∞ , all quantum fluctuations are suppressed, and the effective actionΓ k of the fRG approach tends towards the bare action of QCD. This also entails that themomentum-dependence of the dressings of correlation functions is sub-leading and dropswith powers of p /k : the dressings tend towards the renormalisation factors of the bareaction, within a momentum-cutoff renormalisation. Accordingly, the k -dependence of ef-fective action in the fRG approach translates to a dependence on the UV cutoff, Λ in thepresent SDE approach. Moreover, the dependence on the renormalisation scale µ is the same.In particular, logarithmically divergent RG factors run with log Λ /k , where k ref denotessome reference scale, typically large. This logarithmic Λ-dependence precisely cancels thatproduced by the integrated flows, where the cutoff integration runs from k = Λ → k = 0.This integrated flow agrees with the regularised SDE diagrams (again with UV cutoff Λ).Naturally, there is a very specific choice of k ref , namely k ref = Λ. Then, the logarithmic term38anishes and we can put all Z φ i ,k =Λ = 1, M q, Λ = m q , and λ A , Λ = λ c ¯ cA, Λ = λ q ¯ qA, Λ = g s , λ A , Λ = g s . This amounts to mapping the (implicit) RG scheme within the fRG to a stan-dard RG scheme in the SDEs. In practice, this has to be accompanied with setting the RGscale to µ = Λ , and finally removing the cutoff scale k .In turn, for k →
0, the fRG dressings are simply the finite dressings of the full, renor-malised theory, and no dependence on the cutoff scale is left. This property is called “RGconsistency” , see [8, 103, 104].
Appendix B: Kernels of the quark-gluon vertex SDE
The kernels K ijk ( p, q, k ) and (cid:101) K ijk ( p, q, k ) appearing in (4.7) have the general form K ijk ( p, q, k ) = (cid:88) α =1 C αijk ( p, q, k ) σ α ( k ) , (cid:101) K ijk ( p, q, k ) = (cid:88) α =1 (cid:101) C αijk ( p, q, k ) (cid:101) σ α ( p, q, k ) , (B1)with σ ( k ) = 1 Z q ( k )[ k + M q ( k )] , σ ( k ) = M q ( k ) Z q ( k )[ k + M q ( k )] , (B2)and (cid:101) σ ( p, q, k ) = 1 R ( p, q, k ) , (cid:101) σ ( p, q, k ) = M q ( q + k ) R ( p, q, k ) , (cid:101) σ ( p, q, k ) = M q ( p + k ) R ( p, q, k ) , (cid:101) σ ( p, q, k ) = M q ( q + k ) M q ( p + k ) R ( p, q, k ) , (B3)where R ( p, q, k ) := Z q ( p + k ) Z q ( q + k )[( p + k ) + M q ( p + k )][( q + k ) + M q ( q + k )] , (B4)and the closed expressions for the kinematic functions C αijk ( p, q, k ) and (cid:101) C αijk ( p, q, k ) are re-ported in the github (https://github.com/coupledSDE/FormDerive). [1] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. , 477 (1994), arXiv:hep-ph/9403224.[2] R. Alkofer and L. von Smekal, Phys. Rept. , 281 (2001), arXiv:hep-ph/0007355 [hep-ph].[3] P. Maris and C. D. Roberts, Int. J. Mod. Phys. E , 297 (2003), arXiv:nucl-th/0301049.[4] C. S. Fischer, J. Phys. G32 , R253 (2006), arXiv:hep-ph/0605173 [hep-ph].
5] D. Binosi and J. Papavassiliou, Phys. Rept. , 1 (2009), arXiv:0909.2536 [hep-ph].[6] A. Maas, Phys. Rept. , 203 (2013), arXiv:1106.3942 [hep-ph].[7] M. Q. Huber, Phys. Rept. , 1 (2020), arXiv:1808.05227 [hep-ph].[8] J. M. Pawlowski, Annals Phys. , 2831 (2007), arXiv:hep-th/0512261 [hep-th].[9] H. Gies, Lect. Notes Phys. , 287 (2012), arXiv:hep-ph/0611146 [hep-ph].[10] O. J. Rosten, Phys. Rept. , 177 (2012), arXiv:1003.1366 [hep-th].[11] J. Braun, J. Phys.
G39 , 033001 (2012), arXiv:1108.4449 [hep-ph].[12] J. M. Pawlowski, Nucl. Phys.
A931 , 113 (2014).[13] N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wsche-bor, (2020), 10.1016/j.physrep.2021.01.001, arXiv:2006.04853 [cond-mat.stat-mech].[14] I. C. Cloet and C. D. Roberts, Prog. Part. Nucl. Phys. , 1 (2014), arXiv:1310.2651 [nucl-th].[15] G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, and C. S. Fischer, Prog. Part.Nucl. Phys. , 1 (2016), arXiv:1606.09602 [hep-ph].[16] H. Sanchis-Alepuz and R. Williams, Comput. Phys. Commun. , 1 (2018),arXiv:1710.04903 [hep-ph].[17] A. Cucchieri and T. Mendes, PoS LATTICE2007 , 297 (2007), arXiv:0710.0412 [hep-lat].[18] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker, and A. Sternbeck, PoS
LAT-TICE2007 , 290 (2007), arXiv:0710.1968 [hep-lat].[19] P. O. Bowman, U. M. Heller, D. B. Leinweber, M. B. Parappilly, A. Sternbeck, L. von Smekal,A. G. Williams, and J.-b. Zhang, Phys. Rev. D , 094505 (2007), arXiv:hep-lat/0703022.[20] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker, and A. Sternbeck, Phys. Lett. B , 69 (2009), arXiv:0901.0736 [hep-lat].[21] O. Oliveira and P. J. Silva, PoS LAT2009 , 226 (2009), arXiv:0910.2897 [hep-lat].[22] J. Skullerud and A. Kizilersu, JHEP , 013 (2002), arXiv:hep-ph/0205318.[23] J. I. Skullerud, P. O. Bowman, A. Kizilersu, D. B. Leinweber, and A. G. Williams, JHEP , 047 (2003), arXiv:hep-ph/0303176.[24] A. Kizilersu, D. B. Leinweber, J.-I. Skullerud, and A. G. Williams, Eur. Phys. J. C , 871(2007), arXiv:hep-lat/0610078.[25] E. Rojas, J. P. B. C. de Melo, B. El-Bennich, O. Oliveira, and T. Frederico, JHEP , 193(2013), arXiv:1306.3022 [hep-ph].
26] O. Oliveira, A. Kızılersu, P. J. Silva, J.-I. Skullerud, A. Sternbeck, and A. G. Williams, ActaPhys. Polon. Supp. , 363 (2016), arXiv:1605.09632 [hep-lat].[27] A. Sternbeck, P.-H. Balduf, A. Kızılersu, O. Oliveira, P. J. Silva, J.-I. Skullerud, and A. G.Williams, PoS LATTICE2016 , 349 (2017), arXiv:1702.00612 [hep-lat].[28] O. Oliveira, T. Frederico, W. de Paula, and J. P. B. C. de Melo, Eur. Phys. J. C , 553(2018), arXiv:1807.00675 [hep-ph].[29] A. Athenodorou, D. Binosi, P. Boucaud, F. De Soto, J. Papavassiliou, J. Rodriguez-Quintero,and S. Zafeiropoulos, Phys. Lett. B , 444 (2016), arXiv:1607.01278 [hep-ph].[30] A. G. Duarte, O. Oliveira, and P. J. Silva, Phys. Rev. D , 074502 (2016), arXiv:1607.03831[hep-lat].[31] A. C. Aguilar, F. De Soto, M. N. Ferreira, J. Papavassiliou, J. Rodr´ıguez-Quintero, andS. Zafeiropoulos, Eur. Phys. J. C , 154 (2020), arXiv:1912.12086 [hep-ph].[32] A. C. Aguilar, F. De Soto, M. N. Ferreira, J. Papavassiliou, and J. Rodr´ıguez-Quintero,(2021), arXiv:2102.04959 [hep-ph].[33] A. Bender, W. Detmold, C. D. Roberts, and A. W. Thomas, Phys. Rev. C , 065203(2002), arXiv:nucl-th/0202082.[34] R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada, and K. Schwenzer, Annals Phys. , 106(2009), arXiv:0804.3042 [hep-ph].[35] A. C. Aguilar and J. Papavassiliou, Phys. Rev. D , 014013 (2011), arXiv:1010.5815 [hep-ph].[36] G. Eichmann, Phys. Rev. D , 014014 (2011), arXiv:1104.4505 [hep-ph].[37] D. Binosi, L. Chang, J. Papavassiliou, and C. D. Roberts, Phys. Lett. B , 183 (2015),arXiv:1412.4782 [nucl-th].[38] M. G´omez-Rocha, T. Hilger, and A. Krassnigg, Few Body Syst. , 475 (2015),arXiv:1408.1077 [hep-ph].[39] M. Gomez-Rocha, T. Hilger, and A. Krassnigg, Phys. Rev. D , 054030 (2015),arXiv:1506.03686 [hep-ph].[40] G. Eichmann, C. S. Fischer, and H. Sanchis-Alepuz, Phys. Rev. D , 094033 (2016),arXiv:1607.05748 [hep-ph].[41] D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, and C. D. Roberts, Phys. Rev. D ,096010 (2016), arXiv:1601.05441 [nucl-th].
42] J. Braun, H. Gies, and J. M. Pawlowski, Phys.Lett.
B684 , 262 (2010), arXiv:0708.2413[hep-th].[43] J. Braun, L. M. Haas, F. Marhauser, and J. M. Pawlowski, Phys. Rev. Lett. , 022002(2011), arXiv:0908.0008 [hep-ph].[44] S.-x. Qin, L. Chang, H. Chen, Y.-x. Liu, and C. D. Roberts, Phys. Rev. Lett. , 172301(2011), arXiv:1011.2876 [nucl-th].[45] C. S. Fischer, J. Luecker, and J. A. Mueller, Phys. Lett.
B702 , 438 (2011), arXiv:1104.1564[hep-ph].[46] L.-J. Luo, S. Shi, and H.-S. Zong, Mod. Phys. Lett. A , 1350105 (2013).[47] L. Fister and J. M. Pawlowski, Phys.Rev. D88 , 045010 (2013), arXiv:1301.4163 [hep-ph].[48] C. S. Fischer, L. Fister, J. Luecker, and J. M. Pawlowski, Phys. Lett.
B732 , 273 (2014),arXiv:1306.6022 [hep-ph].[49] C. S. Fischer, J. Luecker, and C. A. Welzbacher, Phys. Rev.
D90 , 034022 (2014),arXiv:1405.4762 [hep-ph].[50] N. Christiansen, M. Haas, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. Lett. , 112002(2015), arXiv:1411.7986 [hep-ph].[51] C. Shi, Y.-L. Wang, Y. Jiang, Z.-F. Cui, and H.-S. Zong, JHEP , 014 (2014),arXiv:1403.3797 [hep-ph].[52] Z.-F. Cui, F.-Y. Hou, Y.-M. Shi, Y.-L. Wang, and H.-S. Zong, Annals Phys. , 172 (2015),arXiv:1505.00310 [hep-ph].[53] G. Eichmann, C. S. Fischer, and C. A. Welzbacher, Phys. Rev. D93 , 034013 (2016),arXiv:1509.02082 [hep-ph].[54] A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev.
D97 , 054015 (2018),arXiv:1708.03482 [hep-ph].[55] R. Contant, M. Q. Huber, C. S. Fischer, C. A. Welzbacher, and R. Williams, Acta Phys.Polon. Supp. , 483 (2018), arXiv:1805.05885 [hep-ph].[56] J. Maelger, U. Reinosa, and J. Serreau, Phys. Rev. D , 014028 (2020), arXiv:1903.04184[hep-th].[57] W.-j. Fu, J. M. Pawlowski, and F. Rennecke, Phys. Rev. D , 054032 (2020),arXiv:1909.02991 [hep-ph].
58] J. Braun, M. Leonhardt, and M. Pospiech, Phys. Rev. D , 036004 (2020),arXiv:1909.06298 [hep-ph].[59] F. Gao and J. M. Pawlowski, Phys. Rev. D , 034027 (2020), arXiv:2002.07500 [hep-ph].[60] F. Gao and J. M. Pawlowski, (2020), arXiv:2010.13705 [hep-ph].[61] J. Braun, W.-j. Fu, J. M. Pawlowski, F. Rennecke, D. Rosenbl¨uh, and S. Yin, Phys. Rev. D , 056010 (2020), arXiv:2003.13112 [hep-ph].[62] J. S. Ball and T.-W. Chiu, Phys. Rev. D , 2542 (1980).[63] D. C. Curtis and M. R. Pennington, Phys. Rev. D , 4165 (1990).[64] A. I. Davydychev, P. Osland, and L. Saks, Phys. Rev. D , 014022 (2001), arXiv:hep-ph/0008171.[65] C. S. Fischer and R. Alkofer, Phys. Rev. D , 094020 (2003), arXiv:hep-ph/0301094.[66] A. C. Aguilar, D. Binosi, D. Iba˜nez, and J. Papavassiliou, Phys. Rev. D , 065027 (2014),arXiv:1405.3506 [hep-ph].[67] A. C. Aguilar, J. C. Cardona, M. N. Ferreira, and J. Papavassiliou, Phys. Rev. D , 014029(2017), arXiv:1610.06158 [hep-ph].[68] R. Bermudez, L. Albino, L. X. Guti´errez-Guerrero, M. E. Tejeda-Yeomans, and A. Bashir,Phys. Rev. D , 034041 (2017), arXiv:1702.04437 [hep-ph].[69] M. Pel´aez, U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, Phys. Rev. D , 114011(2017), arXiv:1703.10288 [hep-th].[70] A. C. Aguilar, J. C. Cardona, M. N. Ferreira, and J. Papavassiliou, Phys. Rev. D , 014002(2018), arXiv:1804.04229 [hep-ph].[71] M. Pel´aez, U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, (2020), arXiv:2010.13689[hep-ph].[72] R. Williams, Eur. Phys. J. A51 , 57 (2015), arXiv:1404.2545 [hep-ph].[73] M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev.
D91 , 054035 (2015),arXiv:1411.7978 [hep-ph].[74] A. L. Blum, R. Alkofer, M. Q. Huber, and A. Windisch, Acta Phys. Polon. Supp. , 321(2015), arXiv:1506.04275 [hep-ph].[75] R. Williams, C. S. Fischer, and W. Heupel, Phys. Rev. D93 , 034026 (2016), arXiv:1512.00455[hep-ph].
76] D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, and C. D. Roberts, Phys. Rev. D ,031501 (2017), arXiv:1609.02568 [nucl-th].[77] A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D97 , 054006 (2018),arXiv:1706.06326 [hep-ph].[78] C. Tang, F. Gao, and Y.-X. Liu, Phys. Rev. D , 056001 (2019), arXiv:1902.01679 [hep-ph].[79] P. Boucaud, F. De Soto, K. Raya, J. Rodr´ıguez-Quintero, and S. Zafeiropoulos, Phys. Rev.
D98 , 114515 (2018), arXiv:1809.05776 [hep-ph].[80] S. Zafeiropoulos, P. Boucaud, F. De Soto, J. Rodr´ıguez-Quintero, and J. Segovia, Phys. Rev.Lett. , 162002 (2019), arXiv:1902.08148 [hep-ph].[81] J. C. R. Bloch, Phys. Rev. D , 116011 (2001), arXiv:hep-ph/0106031.[82] J. C. R. Bloch, Phys. Rev. D , 034032 (2002), arXiv:hep-ph/0202073.[83] M. Q. Huber, Phys. Rev. D , 114009 (2020), arXiv:2003.13703 [hep-ph].[84] A. K. Cyrol, M. Q. Huber, and L. von Smekal, Eur. Phys. J. C , 102 (2015),arXiv:1408.5409 [hep-ph].[85] M. Q. Huber, Phys. Rev. D , 085033 (2016), arXiv:1602.02038 [hep-th].[86] M. Q. Huber, Eur. Phys. J. C , 733 (2017), arXiv:1709.05848 [hep-ph].[87] A. C. Aguilar, D. Binosi, and J. Papavassiliou, Phys. Rev. D78 , 025010 (2008),arXiv:0802.1870 [hep-ph].[88] A. C. Aguilar, D. Binosi, and J. Papavassiliou, Phys. Rev. D , 014032 (2012),arXiv:1204.3868 [hep-ph].[89] P. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene, and J. Rodriguez-Quintero,JHEP , 099 (2008), arXiv:0803.2161 [hep-ph].[90] C. S. Fischer, A. Maas, and J. M. Pawlowski, Annals Phys. , 2408 (2009), arXiv:0810.1987[hep-ph].[91] A. C. Aguilar, D. Ibanez, V. Mathieu, and J. Papavassiliou, Phys. Rev. D85 , 014018 (2012),arXiv:1110.2633 [hep-ph].[92] A. C. Aguilar, D. Binosi, C. T. Figueiredo, and J. Papavassiliou, Phys. Rev. D , 045002(2016), arXiv:1604.08456 [hep-ph].[93] A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D94 ,054005 (2016), arXiv:1605.01856 [hep-ph].
94] S. Aoki et al. (Flavour Lattice Averaging Group), Eur. Phys. J. C , 113 (2020),arXiv:1902.08191 [hep-lat].[95] C. S. Fischer, Prog. Part. Nucl. Phys. , 1 (2019), arXiv:1810.12938 [hep-ph].[96] A. Bender, G. I. Poulis, C. D. Roberts, S. M. Schmidt, and A. W. Thomas, Phys. Lett. B431 , 263 (1998), arXiv:nucl-th/9710069 [nucl-th].[97] F. Gao and Y.-x. Liu, Phys. Rev.
D97 , 056011 (2018), arXiv:1702.01420 [hep-ph].[98] A. Bashir, L. Chang, I. C. Cloet, B. El-Bennich, Y.-X. Liu, C. D. Roberts, and P. C. Tandy,Commun. Theor. Phys. , 79 (2012), arXiv:1201.3366 [nucl-th].[99] P. Maris, C. D. Roberts, and P. C. Tandy, Phys. Lett. B , 267 (1998), arXiv:nucl-th/9707003.[100] V. Miransky, Dynamical symmetry breaking in quantum field theories (1994).[101] P. O. Bowman, U. M. Heller, D. B. Leinweber, M. B. Parappilly, A. G. Williams, and J.-b.Zhang, Phys. Rev. D71 , 054507 (2005), arXiv:hep-lat/0501019 [hep-lat].[102] L. Corell, A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, SciPost Phys. , 066(2018), arXiv:1803.10092 [hep-ph].[103] J. M. Pawlowski, M. M. Scherer, R. Schmidt, and S. J. Wetzel, Annals Phys. , 165(2017), arXiv:1512.03598 [hep-th].[104] J. Braun, M. Leonhardt, and J. M. Pawlowski, SciPost Phys. , 056 (2019), arXiv:1806.04432[hep-ph]., 056 (2019), arXiv:1806.04432[hep-ph].