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Effective quark interactions from QCD

We propose a new method for an analytical, non-perturbative computation of effective quark interactions from QCD. It is based on an exact flow equation which describes the scale dependence of the effective average action for quarks in presence of gluons.


Introduction
Quantum chromodynamics as the theory of strong interactions shows very different facets at short and long distances. The high momentum behaviour is governed by asymptotic freedom [1] and the relevant degrees of freedom are quarks and gluons.
In contrast, the particles which are observed at large length scales (larger than 1 fm) are mesons and hadrons. The interactions of the pseudoscalar mesons are modeled by chiral perturbation theory [2]. The corresponding nonlinear σ-model shares the flavour symmetries of perturbative QCD. It is believed that the free phenomenological parameters of this model can ultimately be computed from the action of QCD, but this is not a simple task.
Recently, the transition from quark degrees of freedom to meson degrees of freedom has been described by a nonperturbative flow equation [3]. It is based on the concept of the effective average action Γ k [4] for which only quantum fluctuations with (covariant) momenta q 2 > k 2 are integrated out. The average action is the effective action for averages of fields. It acts like a microscope by which we can look at the theory at different length scales, with a "resolution" given by the scale k −1 . For k = 0 one recovers the usual effective action, i.e. the generating functional of the 1PI Green functions, whereas for k → ∞ Γ k equals the classical action. The dependence of Γ k on the scale k is described by an exact nonperturbative flow equation [5]. By the introduction of composite fields for the mesons the original equation formulated for quarks can be transmuted into an equivalent exact flow equation involving also the mesons [3].
In a first attempt the chiral condensate <ψψ > and the pion decay constant f π were computed along these lines [3]. In this approach the gluons have not been considered explicitly. Their effect was encoded in a phenomenologically motivated four-quark interaction, which may be thought as the result of integrating out the gluons in the defining functional integral. If one aims at a computation of the effective parameters of chiral perturbation theory from the QCD action, this shortcut has to be removed. A full computation should start at short distances with the quarkgluon description of QCD and systematically account for the quantum fluctuations of the gluon field.
Conceptually, one may integrate out the gluons at once and end with an effective quark theory. In practice, this seems almost impossible: A determination of the resulting complicated nonlocal quark interactions requires more or less a complete solution of QCD. On the other hand, the form of the flow equation at a given scale k is not sensitive to details of physics at momentum scales much below k. Only the effective propagators and vertices for modes with momenta q 2 ≈ k 2 play a role for this equation. A typical scale where the light mesons form is around 700 MeV and therefore substantially higher than the confinement scale. This situation further improves for problems involving heavier quarks as for example the charmed or beauty mesons. One may hope that very detailed features of confinement are not needed for an understanding of the mesons. An appropriate tool would therefore be a method which only integrates out the gluon fluctuations with momenta q 2 > k 2 instead of addressing the much more complicated problem of integrating out all gluon fluctuations at once. Formally, it is easy to do this at a given scale k 1 . One needs to compute the effective action for quarks and gluons Γ k 1 [ψ, A] at this scale. Solving the classical field equation for the gluon field A in dependence on ψ and inserting this classical solution into Γ k1 [ψ, A] yields exactly an effective average action Γ k 1 [ψ] involving only the quark fields. The quark-effective action Γ k 1 [ψ] may then be used as an initial value for solving a pure fermionic flow equation for k < k 1 . Obviously, the shortcoming of such an approach ist the complete omission of the effects of gluon fluctuations with q 2 < k 2 1 . Choosing a different scale k 2 for eliminating the gluons will lead to a different result. Such a sharp transition between the quarkgluon system and a description involving only quarks necessarily introduces a certain degree of arbritariness.
We propose here a more refined method which changes the classical solution for the gluon field in the course of the evolution towards lower k, thus reflecting the change in the form of Γ k [ψ, A]. The result is a smooth procedure for integrating out the gluons, where at every scale k all gluon fluctuations with q 2 > k 2 are included.
Nevertheless, we obtain a flow equation for the quark effective average action Γ k [ψ], where gluon fields do not appear explicitly. As a consequence of the inclusion of contributions from additional gluon fluctuations as the scale is lowered, correction terms appear in the flow equation for the quark interactions. In particular, we choose here a formulation where the k-dependent classical solution for A as a functional of ψ includes an effective infrared cutoff ∼ k. As a consequence, the only nonlocalities in Γ k [ψ] concern length scales shorter than k −1 . For the fermionic low momentum modes Γ k [ψ] is an effectively local action. For example, a derivative expansion is meaningful for q 2 ≪ k 2 . The expected nonlocalities arising from the complete elimination of gluon fields (for example a four-quark interaction ∼ 1 q 2 appearing already in the Born approximation) build up only step by step as k is lowered to zero. As a result of this method we will end with an exact nonperturbative flow equation for the scale dependence of Γ k [ψ]. Approximations will be needed to solve this equation but they are not limited to perturbative concepts. The correction terms reflecting the gluon fluctuations require limited knowledge about the effective gluon propagator and vertices. It is hoped that rather crude approximations for the gluonic vertices can already lead to satisfactory results.
In sect. 2 we first demonstrate our formalism for a simple model of two scalar fields. The flow of the effective average action for one of the scalar fields obtains by integrating out the other scalar field at any scale k. Subsequently this is generalized to quarks and gluons. We also give a first demonstration how this formalism describes the flow of the two-and four-point function in the effective quark theory.
The special case of heavy quarks is addressed in sect. 3. Here we argue that the evolution of the gauge field propagator is needed in this limit. In sect. 4 we compute the corresponding flow equation and discuss the scale dependence of the gluon propagator. In sect. 5 we collect all the ingredients needed for the flow equation in case of light quarks. Finally, our conclusions are contained in sect. 6.

Reduction of degrees of freedom
First we consider for simplicity two types of scalar fields, ϕ and ψ. We want to develop a formalism how to translate evolution equations for the effective average action for ϕ and ψ into corresponding equations involving only ψ. The reader may associate ϕ with the gluon fields and ψ with the quark fields. Our aim is then the construction of the effective average action for quarks out of the coupled quark-gluon system. This amounts to integrating out the gluonic degrees of freedom represented in the simplified model by ϕ. We start with the scale-dependent generating func-tional for the connected Green functions Here we denote the degrees of freedom contained in ϕ ′ (for example the Fourier modes) by ϕ ′ α and similar for ψ ′ , J and K, We have introduced an infrared cutoff quadratic in the fields and similar for ψ. This suppresses the contribution of fluctuations with small mo- are functions of q 2 as, for example, which acts like a mass term R by subtracting the infrared cutoff term k S vanishes and Γ 0 is the usual generating function for the 1PI Green functions. Using the quadratic form of ∆ k S it is straightforward to derive an exact non-perturbative evolution equation for the dependence of the effective average action on the scale k (t = ln k) [5] k + R k and the inverse propagator Γ (2) k is the second functional derivative of Γ k with respect to the fields. The matrix R k = R is block diagonal in ϕ and ψ spaces. The presence of the infrared cutoff R k inΓ (2) k guarantees infrared finiteness for the momentum integral implied by the trace even in case of massless modes. Ultraviolet finiteness is guaranteed by the exponential decay of ∂R k /∂t (2.3). A solution of the flow equation (2.6) interpolates between the classical action for k → ∞ (or k equal to some ultraviolet cutoff Λ) and the effective action for k → 0.
The generating functional for the connected Green functions for ψ obtains ¿from Correspondingly, we may introduce an effective action expressed only in terms of ψ It is easy to verify that this equation reduces in the limit R (ϕ) k = 0 to the equivalent of eq. (2.6) for fields ψ only. The corrections in the first two terms involve the explicit form of the "classical solution" ϕ k [ψ]. If one is interested in 1PI Green functions for ψ with a given number of external legs one only needs a polynomial expansion of ϕ k [ψ] up to a given order. For example, the evolution of the term ∼ ψ 4 in Γ k [ψ] needs the classical solution up to the order ψ 4 if the series ϕ k [ψ] starts with a term quadratic in ψ. Additional knowledge of the form of Γ k [φ, ψ] beyond its value forφ = 0 is only needed for the last correction term in the form of Only the ψ-dependence of the effectiveφ-propagator plays a role for the study of 1PI functions for ψ.
Let us next apply the general flow equation (2.11 explicitly to the quark-gluon system. If ψ is a Grassmann variable as appropriate for fermions the matrix R k in (2.6) becomes Also ψ * should be replaced byψ and the index summation over β should involve both ψ andψ separately. For the gauge fields we will choose here a formulation with explicit ghost variables in close analogy, but slightly different from the formulation in ref. [6]. This makes our formulation as close as possible to the language of standard perturbation theory. Details can be found in the appendix.
We start with the action including a gauge-fixing term in the background gauge and a corresponding action for the anticommuting ghost fields ξ,ξ Here S is a gauge invariant functional of the fermion fields ψ,ψ and the gauge field The background gauge fieldĀ µ appears in the gauge fixing and ghost terms We have introduced here also sources ζ for the ghost fields and note that the source K µ z couples to the gauge field fluctuation a z µ and therefore transforms homogeneously under gauge transformations as an adjoint tensor. The k-dependent version W k obtains from W by adding toŜ the infrared cutoff piece Here the fermionic cutoff reads with D µ the covariant derivative in the appropriate representation (D 2 = D µ D µ ) and r with D[Ā] an appropriate operator generalizing a covariant Laplacian in the adjoint representation which will be explained below. The matrix Z A,k accounts for an appropriate wave function renormalization. Finally, we take for the ghosts such that The k-dependent functions Z ψ,k, Z A,k and Z gh,k will be adapted to corresponding wave function renormalization constants in the kinetic terms for the fermions, gauge fields and ghosts. In principle, they can depend on the background fieldĀ. The infrared cutoff piece ∆ k S cuts off all quantum fluctuations with covariant momenta smaller than k in the functional integral defining W k . For covariant momenta larger than k the infrared cutoff is ineffective and its contribution to the propagator is exponentially suppressed.
Performing a Legendre transform and subtracting the IR cutoff piece again (c.f. k . It is derived in the appendix (A.12). We note that Γ k only involves terms with an even number of ghost fields due to the symmetrȳ In consequence, the ghost field equations have always the solutionξ = ξ = 0. We therefore can extract the propagators and vertices for the physical particles from the effective action forξ = ξ = 0: Nevertheless, the evolution equation for Γ k [ψ, A,Ā] obtains a contribution from the variation of the infrared cutoff of the ghost fields as given by Here Γ (2) k +R k in (2.28) is the matrix of second functional derivatives of Γ k +∆ k S with respect to ψ and A at fixedĀ. As compared to the more symmetric form of the flow equation (A.12) we have combined here similar pieces in the quark and ghost sector. One should remember, however, that the fermionic part (Γ (2) k + R k ) −1 is a submatrix of a larger matrix containing alsoψψ entries. For the derivation of eq. (2.28) we have exploited that the matrix of second functional derivatives of (Γ k + ∆ k S)[ψ, ξ, A,Ā] is block diagonal in the (ψ, A) and ξ components for ξ = 0. The ghost dependence of Γ k [ψ, ξ, A,Ā] appears in the evolution equation for Γ k [ψ, A,Ā] only through the term ε k which involves the second functional derivative with respect to the ghost fields Γ (gh)(2) k , which is evaluated atξ = ξ = 0 and may depend on ψ, A,Ā. As a consequence of local gauge invariance the average action Γ k must obey anomalous Slavnov-Taylor identities which are displayed in the appendix.
They constrain, in particular, the ghost dependence of Γ k . We will not pay much attention to the detailed form of Γ (gh)(2) k in the present paper and approximate it by (2.30) or a slight generalization thereof (cf. eq. (4.8)). In order to complete the formal setup of our investigation we need to specify the operator D in eq. (2.22). A good choice is is the second functional derivative of Γ k [ψ, A,Ā] with respect to A for fixedĀ and ψ = 0, evaluated at the point A =Ā. As in previous formulations [6], the effective average action Γ k [ψ, A,Ā] is gauge invariant with respect to simultaneous gauge transformations of ψ, A andĀ.
We can now apply the formalism of the last section in order to "integrate out" the gluon fields A. The classical field equation, whose solution is A k , reads where the derivative should be taken at fixedĀ. At this point A k becomes a functional of ψ andĀ. For the purpose of the present paper we only consider the special choiceĀ = 0 and omit the argumentĀ in the following. In this version R The flow equation ( in a perturbative context. As k decreases, we gradually explore the non-perturbative regime and the full quantum-effective action obtains for k = 0. Obviously, such a program is only feasible with approximations that truncate the most general form of Γ k . The truncation used in the next two sections concerns the three-gluon and four-gluon vertices which are approximated by a momentum-independent, but kdependent running gauge coupling. A similar truncation is used for the ghost contribution. In sect. 5 we truncate, in addition, the most general form of the fourand six-fermion interactions. We are interested in the evolution of the two-and four-point functions for the quarks. The respective flow equations for these quantities obtain by taking the second and fourth functional derivative of eq. (2.33) at ψ =ψ = 0. We label the different contributions on the r.h.s. of eq. (2.33) by and discuss them separately. The first term is the standard contribution of a pure fermionic theory. The remaining terms γ Aψ , γ A and γ c involve R (A) k and reflect the contributions from gluons, whereas ǫ gives the ghost contribution. The term involves only a trace over gluonic degrees of freedom and accounts for the contribution of gluon fluctuations around the ψ-dependent classical solution. The contribution of γ A to the fermionic two-and four-point functions is given by the dependence of Γ Relevant contributions to γ A therefore arise from terms in Γ k [ψ, A, 0] which are either quadratic in A and also depend on ψ or are cubic or higher-order in A.
We use a truncation where the first sort of terms is absent and no relevant contribution to γ A would be present for an abelian gauge theory. For nonabelian gauge theories we get contributions ¿from the three-and four-gluon vertices in Γ k [ψ, A].
We approximate here these vertices by the (standard) lowest order expressions which are obtained from functional derivatives of F µν F µν . More precisely, we use on the r.h.s. of the flow equation HereD µ [A] = ∂ µ − igA z µ T z represents the covariant derivative in the adjoint representation with gauge couplingg and F µν is the nonabelian field strength associated to the gauge field A µ . In this truncation the three-and four-gluon vertices are parametrized by two running parametersg(k) andZ F (k). The running renormalized gauge coupling g k is related to them by

is invariant under global gauge transformations of ψ and
A. The expression for γ A does not explicitly depend on ψ in our truncation and There is therefore no contribution from γ A to the flow equation of the fermionic two-point function. An estimate of the contribution to the four-quark interaction from γ A therefore amounts to a computation of the gluon contribution to the evolution of the term quadratic Similarly the ghost contribution ǫ is (with the approximation (2.30)) only a functional of A, containing terms quadratic in A (and higher orders). We also observe that γ A and ǫ only account for the gluon and ghost contributions to the effective gluon propagator, whereas the contribution from quark loops is implicitly contained in γ ψ . We note that the latter is not distinguished any more from any other fermionic contributions, as, for example, from an explicit four-quark interaction in describes the effect of the "classical" change in the infrared cutoff as k is lowered.
It is quadratic in the classical solution A k [ψ] and therefore gives a contribution to the fermionic four-point function. Finally the piece γ Aψ involves the explicit ψdependence of the classical solution ∼ ∂A k /∂ψ. It contributes to the running of the fermionic two-and four-point functions.

Heavy quark approximation
In the limit of infinitely large quark masses our formalism simplifies considerably. For euclidean momenta we can omit in eq. (2.33) the terms involving the inverse fermion k ) −1 since their contribution is suppressed by inverse powers of the quark masses. In the language of the last section this results in γ ψ = 0, γ Aψ = 0. The remaining computation amounts to an investigation of the pure gluon theory with static quarks. This is done most easily in the language where the gluon fields are kept explicitly and the relevant effective action is Γ k [ψ, A,Ā = 0]. If one wants to extract the effective four-quark interaction, one needs the k-dependent effective gluon propagator and the effective vertexψψA. We first describe for arbitrary quark masses the general framework how an effective four-quark interaction obtains from "gluon exchange" in the formulation where both quark and gluon degrees of freedom are kept explicitly. We then specialize to the heavy quark limit.
and parametrize the most general inverse gluon propagator by For the quark-gluon vertex we make the approximation that G ψ is a simple function not involving Dirac matri- and reads Inserting the classical solution into (3.2) and (3.3) and accounting for the term ∆ (3.8) and The curled brackets indicate contractions over not explicitly written indices (here spinor indices), i, j, k, ℓ = 1...N c are the colour indices and a, b = 1...N f the flavour indices of the quarks. By an appropriate Fierz transformation and using the identity we can split M into three terms [3] In terms of the Lorentz invariants we recognize that the quantum numbers of the fermion bilinears in M σ correspond to colour singlet, flavour non-singlet scalars in the s-channel and similarly for spinone mesons for M ρ . In analogy to ref. [3] we associate these terms with the scalar mesons of the linear σ-model and with the ρ-mesons. The bilinears in the last term M p correspond to a colour and flavour singlet spin-one boson in the t-channel.
These are the quantum numbers of the pomeron. We observe that in the heavy quark approximation where Γ (ψ) k,4 arises only from "gluon exchange", the coefficients of the quark interactions in the σ, ρ and pomeron channel (3.12) are all given by the same function F 1 .
The general quark bilinear is conveniently parametrized by the real functions The k-dependence of the functions G A , H A , G ψ , Z ψ andm a relevant for the two-and four-point functions for the quarks can now be studied using the evolution equation In the truncation where only the terms (3.2),(3.3) and (3.17) are kept, it is easy to see that the contributons to the k-dependence of G ψ , Z ψ andm a all involve quark propagators. In the heavy quark limit they can therefore be neglected for euclidean external momenta. Only the k-dependence of Γ k,2 needs to be considered. For Z ψ and G ψ we may take appropriate momentum-independent "short-distance couplings" with renormalized gauge coupling g taken at the scale k = m ψ and m ψ the heavy quark mass. We also may identify k = m ψ with the "ultraviolet cutoff" or the scale where the initial values for the flow equation are specified, i.e.
Solving the flow equation for G A (q) for k → 0 yields the effective four-quark interactions for momenta much smaller than the quark mass. For α R = 0 (see next section) the effective four-quark interaction is fully determined by We finally should mention that the heavy quark potential or the scattering amplitude for heavy quarks cannot be extracted directly from the four-point function at small momenta p 2 ≪ m 2 ψ . For these purposes the momenta appearing in M should be taken on-shell, i.e. p 2 1 = p 2 2 = p 2 3 = p 2 4 = −m 2 ψ . Their size is therefore not small as compared to m 2 ψ . For on-shell momenta the vertex function G ψ (p, q) becomes a function of q 2 . The flow of G ψ (q) does not vanish for k 2 ≪ m 2 ψ and we need to supplement the computation of the k-dependence of G A (q) by a corresponding one of G ψ (q). The heavy quark pontential can then be extracted as the

Scale dependence of the gluon propagator
In this section we compute a flow equation for the scale dependence of the effective gluon propagator in the pure Yang-Mills theory. As explained in the last section, this is the central piece which determines the heavy quark interactions. We will derive the flow equation for G A in the approximation that the vertices can be extracted from a Here we use and note that the partial derivative∂ t acts only on the explicit infrared cutoff terms R k andR k . The parts involving the effective ghost propagator P −1 gh arise from the ghost contribution ∼ −ǫ. After performing the q ′ integration the evolution equations A simplification occurs if we take for the gauge-fixing term the gauge parameter α → 0. In this limit H A diverges ∼ 1 α and b(q) approaches one. We may define a k-dependent renormalized gauge fixing parameter α R by 3 with lim k→∞ZF = 1 and lim k→∞ α R = α. The evolution equation for α R follows where we note that ∂ ∂t H A (q) has a well defined limit for q 2 → 0. We conclude that α R = 0 is a fixpoint which is infrared stable for ∂ lnZ F /∂t > 0. The approximation α R → 0 remains therefore stable in the course of the evolution. We further observe that for α R → 0 the function F 2 defined in eq. (3.8) equals F 1 /(p 1 −p 3 ) 2 and only the function G A (q) determines the effective four-quark interaction. The flow equation for G A can be written in a more compact form using We also approximate the ghost part ofΓ This yields for α R → 0 The evolution equation (4.9) is the central equation of this section. It is a partial nonlinear differential equation for G A (q 2 ; k) which can be solved numerically. We observe that the r.h.s. of the flow equation (4.9) involves not only G A (q) but also the renormalized gauge coupling g k and the gluon wave function renormalization constantZ F . The precise definition of these k-dependent constants is a somewhat subtle issue. We could defineg andZ F (and in consequence g k ) in terms of the effective three-gluon vertex ∼gZ F A 2 ∂A and four-gluon vertex ∼g 2Z F A 4 which enter the approximation for Γ (2) k used in (2.38). In this wayg andZ F are expressed in terms of third and fourth functional derivatives of Γ k [ψ = 0, A,Ā = 0] evaluated at A = 0 and projected on the appropriate index structures. One also has to choose appropriate momenta for the external legs for the effective vertices, as for example the limit where all momenta approach zero. Evolution equations for the scale dependence ofZ F ,g and g k could then be computed from appropriate functional derivatives of the flow equation (2.28). We will use here a simplification and approximateZ F by the coefficient of the q 2 term in G A . More precisely, we expand for small q 2 A (q 2 ) 2 + ... and identifyZ F with G A . There is an obvious limitation to this approximation since G A may turn negative for small k and any reasonable choice of a wave function renormalization requires positiveZ F . For the flow equation for the renormalized gauge coupling g k we rely on the fact that the first two coefficients of the β function are universal In the region of large g k we may also use nonperturbative estimates of β g 2 derived by different methods [6]. An ansatz for β g 2 combined with an estimate ofη F fixes also the evolution ofg 2 and provides all information needed for a numerical solution of the flow equation.
We concentrate first on an analytic discussion of a few prominent features of the solution of equation (4.9). The evolution equations for the mass termm 2 A and for G (1) A are easily derived by expanding the r.h.s. of eq. (4.9) in powers of q. One obtains ∂ ∂tm and the renormalized dimensionless mass term Here we have defined the integrals, with x = q ′ 2 , The evolution of the rationm 2 is characterized for small g 2 k by an approximate infrared unstable fixpoint Indeed, we can use for small g 2 k the lowest order expressions such that l 4 A,1 = l 4 1 = 1, l 6 A,2 = l 6 2 = 3 2 . We also neglect β 2 g /g 2 −η F as compared to two. For small g 2 k the general identities for the dependence of Γ k [ψ, A,Ā] on the background fieldĀ [6], or, similarly, the generalized Slavnov-Taylor identities [8], [9] imply that the k-dependent mass term is indeed described by this fixpoint (cf. Appendix). We conclude that for small g k the mass term induces only a small correction in the momentum-independent part of P A The behaviour ofm 2 A near the confinement scale where g 2 k becomes large is more complicated and best described by evaluating directly the relevant identities [6]. It is conceivable that a negative r.h.s. of (4.13) drivesm 2 A to a positive value for k → 0, but we find this scenario not very likely.
The flow equation for G (1) A can be written in the form ∂ ∂t G where we have defined the integrals and therefore findsη For a solution of (4.26) it is convenient to compareη F with where b A may depend on g k or k and reflects the deviation ¿from the one-loop βfunction for which b A = 1. As long as the k-dependence of the ratio b A /c A can be neglected, one obtainsZ In the one-loop approximation 1 g 2 (k) decreases logarithmically and reaches zero at the confinement scale Λ conf . We conclude that this is exactly the scale whereZ F would vanish. In this language the divergence of the renormalized gauge coupling is actually entirely due to the vanishing ofZ F : From (4.28) one obtains and for γ > 1 2 (cf. (4.28)) both the unrenormalized three-point vertex ∼Z Fg and four-point vertex ∼Z Fg 2 vanish at the confinement scale.
As mentioned before we should not use (4.25) for k in the vicinity of the confinement scale. We therefore propose [11] to keepZ F independent of k for k < k np where k np is defined by (4.30) 4 We have also computed ∂ ∂t H A (0) = − Nc 48π 2g 2 8l 4 A,2 + 5m 6 A,4 and find for small g 2 that The definitionZ allows to separate the issue of vanishing G A from the choice of the infrared cutoff (i.e.Z F ).
The vanishing of the term G The integral is dominated by the region x ≈ k 2 and can turn negative only if the bracket is negative in this region. As an illustration we take to be compared with and We conclude that negative m 2 A (cf. (4.18)) and positive κ tend to lower c A . A vanishing c A for large κ cannot be excluded without a more detailed investigation.

On the other side, if the integral relation
could be fulfilled for Λ conf > 0, the coefficient G A would vanish at the confinement scale Λ conf and presumably becomes negative for k < Λ conf . We emphasize that a negative value ofm 2 A or of G A for small k would independently indicate that the groundstate does not correspond to the perturbative ground state A µ = 0 [12], [13], [11].
Even before reaching the confinement scale, the gluon propagator has to be modified: Whenever the term G (1) A k 2 , the gluon propagator cannot be approximated any more by the inverse of q 2 ! As an example for a plausible form one may consider is determined by the requirement that for large q 2 one expects G A (q) = q 2 independent of k. (This holds up to neglected logarithmic corrections.) Assuming that for A q 2 is small as compared to the (q 2 ) 2 term the approximate form of the propagator would be close to the one corresponding to a confining potential.
It is obviously difficult to find an analytical answer to all these questions and it seems preferable to solve the flow equation (4.9) numerically. A numerical investigation has been performed by B. Bergerhoff and the author [14]. We show here only a few first results. In fig. 1 we plot the scale dependence of the wave function renormalizationZ F as defined bỹ with Λ QCD the two-loop confinement scale. This definition implements the idea that Z F should be taken constant for very small k (cf. eq. (4.31)) in a smooth way. One observes the decrease ofZ F as k is lowered corresponding to eq. (4.28). We have started the running in the perturbative region at k = 40 GeV with G A (q) |k=40 GeV given by the one-loop perturbative expression (containing the infrared cutoff). The latter was normalized by ∂G A /∂q 2 | q 2 =0 = 1, which explains the starting value ofZ F (40 GeV) somewhat larger than one for the definition (4.41). In order to concentrate on the deviation of G A (q) from the linear dependence on q 2 we introduce the quantity Here the mass term G A (0) is subtracted from the inverse propagator and the leading perturbative q 2 -dependence is divided out. In the classical approximation the expression in the bracket equals one, and a nonvanishing value of χ(q) is entirely due to quantum fluctuations. Since a computation of χ(q) involves a numerical derivative of a small difference, one needs a numerical solution of the flow equation for G A (q) with a relatively high precision. We observe that χ(q) plays the role of a momentum-dependent anomalous dimension. Within renormalization-group improved perturbation theory one expects for k = 0 (compare eq. (4.26)) where g 2 (µ) is the running gauge coupling at the scale µ. In figs. 2 and 3 we compare the numerical determination of χ(q) with the renormalization-group improved oneloop perturbative result (4.44), with g the two-loop running gauge coupling. Up to a scaling factor of about 10 % the two curves asymptotically coincide for large q 2 as k goes to zero. This is not a trivial result since no assumption of this type enters the flow equation (4.9). We note that a propagator ∼ q −4 for small q 2 corresponds to lim k→0 lim q 2 →0 χ(q) = 1. If a q −4 behaviour extends effectively over a certain momentum range χ(q) should develop a plateau at one in this range. As k is lowered we see in fig. 3 a sizeable increase of χ(q) for small q 2 . We also observe a tendency of an extension and flattening of the maximum somewhat below one. This tendency should stabilize as k goes to zero. We have stopped the running at k = 400 MeV since for small k 2 and q 2 the approximations leading to (4.9) become doubtful. At least part of the momentum dependence of the gluon vertices according to (A.55) should presumably be included.

Flow equations for light quarks
The formalism for integrating out the gluon degrees of freedom needs not to be restricted to the heavy quark approximation. We want to derive in this section the general flow equation for the quark two-point and four-point function corresponding to (2.33). For simplicity we mainly consider N F massless quarks -the inclusion of mass terms is straightforward -and we work in the gauge with α = 0. At some appropriate short distance scale we start with the "classical action" (cf. (3.7)) The flow equation then permits to study how Γ k changes its form as k is lowered. In particular one is interested in pole-like structures in the quark four-point function which would indicate the formation of meson bound states [15], [3].
In order to establish the flow equation we have to collect various pieces which have been discussed in the previous sections. We begin with the contribution from the gluon and ghost fluctuations γ A − ǫ + γ c which only contribute to the four-quark interaction Γ k,4 [ψ]. The direct contributions from gluon and ghost loops read The functions ∂ ∂tĜ A (q) and ∂ ∂tĤ A (q) have been computed in the last section (cf. (4.1), (4.2), or (4.9)), where we have indicated by a hat that only gluon and ghost contributons should be included here, in analogy to the heavy quark approximation. The classical solution A (0) k is given by eqs. (3.4) and (3.6), where G A and H A characterize now the gluon propagator without reference to the heavy quark approximation. We . Including also the contribution γ c which is similar in structure we obtain (In the heavy quark approximation there is no difference betweenĜ A and G A . Also G ψ is independent of k such that f 1 = ∂ ∂t F 1 (3.8).) The function f 1 involves the quark gluon vertex G ψ and the gluon propagator G A which also enters in the determination of ∂ ∂tĜ A . Up to this point the only approximations involve the three-and four-gluon vertices entering ∂ ∂tĜ A and the ghost sector. They have been discussed in sect. 4. We may in addition also truncate the quark gluon vertex and use for light quarks the ansatz The running of the renormalized gauge coupling g k is now determined by the βfunction including quark contributions andZ F may be identified with G (1) A for k > k np (cf. sect. 4). the lowest order truncation for G A (q) would be such that P A (q) is replaced by P (q) in the equation (4.9) for ∂ ∂tĜ A . The detailed discussion of the last section shows, however, that this approximation becomes in- is given by the r.h.s. of (4.9) with P A (q) = P (q) and∂ t P A (q) = ∂ ∂t P (q) −η F (P (q) − q 2 ). The first two contributions are proportional∂ t P −1 and simply reflect the change of the infrared cutoff contained in P −1 in the "classical action " (5.1). Only the last term ∼ G describes how additional quantum fluctuations are included as k is lowered -in this case the gluon and ghost contributions to the gluon propagator. If we omit the contribution ∼η F in∂ t P A (q) the funtion G(q) reads explicitly We observe that even for our simple truncations the function G(q) has a compli- and the next to leading contribution A k obtains as With the ansatz of a flavour diagonal kinetic term and mass term Γ (2) one finds the following contribution to the flow equation for the two-point function The contribution to the four-point function γ Aψ can be obtained similarly using (5.11).
For a computation of the purely fermionic contribution γ ψ we will use the following truncation for the term quartic in the fermionic fields This yields a contribution to the flow equation for the two-point function For γ (4) (ψ) we present here only the case λ ρ = λ p = λ n = 0 in the chiral limit (m a = 0). Omitting all contributions except γ (4) ψ one finds For the special case (c(q) + r (ψ) k (q)) −1 = exp − q 2 Λ 2 − exp − q 2 k 2 , η ψ = 0 this reproduces the flow equation of ref. [3]. The r.h.s. of this equation should now be supplemented by the contributions from γ A − ǫ + γ c + γ Aψ which have been discussed before. Furthermore, a better approximation should include the contributions from λ ρ etc.
Besides this, eq. (5.16) involves the explicit momentum dependence of the fermion kinetic term, i.e. the function c(q). The scale dependence of the quark propagator can be computed from γ (2) ψ and γ (2) Aψ . Combining eq. (5.15) with (5.13) and (5.12) yields the flow equation for the kinetic term The right-hand side of this equation involves the gauge coupling g k and the effective inverse gauge field propagator P A as well as the effective fermionic four-point vertices λ σ , λ ρ , λ p and λ n . It is instructive to study this equation in the "classical approximation" for the four-quark vertices λ σ , i.e.
We observe that this expression corresponds to the formal∂ t derivative of the standard one-loop correction to the fermion kinetic term in presence of an infrared cutoff in the propagator.
The fermionic wave function renormalization Z ψ can be defined by c a (p 0 ) = 1 for suitable p 0 and a. We will choose here p 0 = 0 and use c a corresponding to a light quark. Defining the anomalous dimension one obtains for m a = 0 whereṖ A = ∂P A /∂q 2 . In the perturbative limit c(q) = 1, P A (q) = P (q),∂ t P A (q) = ∂ ∂t P (q) this yields a vanishing fermionic dimension We also may extract the anomalous mass dimension by expanding eqs. (5.13) and (5.17) in linear order in m a ∂ ∂t m a (0) = ω m m a (0) (5.24) One obtains which reduces in the "classical approximation" λ σ (−q, q, 0, 0) = λ p (−q, q, 0, 0) = q 2 k λ n (−q, q, 0, 0) = 1 2 g 2 P −1 A (q) and for m a (q) = m a (0) to In lowest order perturbation theory where∂ t [(c(q) + r k (q)) 2 q 2 P (q)] −1 one recovers the standard perturbative result k (q)) 2 = 0. To summarize this section, we have explicitly derived a nonperturbative flow equation for the scale dependence of the quark propagator (5.17). We have also given a simplified non-perturbative approximation (5.20

Conclusions and discussion
In this paper we have developed a formalism for integrating out the gluon fields in order to obtain an effective action for the quarks. This is not done at once since such an approach would lead to complicated nonlocalities and a reliable direct computa- which only involves the quark fields. In the present work we have concentrated on analytical work whereas the numerical exploitation of our formulae is left to a separate investigaton [14].
Our first main result is the evolution equation (6.9) for the gluon propagator in the heavy quark limit. As described in sect. 3, its quantitative solution is connected with the heavy quark potential. The second result concerns the effective action for light quarks. The evolution equation for the two-and four-point functions can be extracted from sect. 5. The solution for the four-point function is expected to develop pole-like structures connected to mesons. This can be treated with the composite-field methods developed in ref. [3], such that one can finally make a transition to an effective theory for mesons describing the low momentum behaviour of QCD. The main difference of the present formalism as compared to ref. [3] concerns the treatment of the gluons: While ref. [3]  One may suspect that even the non-perturbative treatment of this paper breaks down for scales k of the order or below the confinement scale. Fortunately the formation of meson-bound states is expected at a scale k ϕ considerably higher than the confinement scale. For the pseudoscalar mesons a first computation indicates k ϕ ≈ 650 MeV [3]. In view of this one may hope that an understanding of the formation of mesons does not necessitate a very detailed understanding of the physics near the confinement scale. There are still several "hopes" and "expectations". A quantitative computation of the quark condensate <ψψ > and the pion decay constant f π along similar lines as in ref. [3], but using the flow equations proposed in the present paper should decide whether we are on a reasonable track for an analytical understanding of QCD.
Note added: Results for the form of the heavy quark potential have recently been obtained using flow equations [17] and are closely related to sect. 4 of the present work.
We recover the usual identities in the limit k → 0 since R = 0 implies a vanishing BRS-anomaly A BRS = 0.
The sourcesβ appear only linearly in S and it is straightforward to derive the where the connected two-point function < ac > c is related to the appropriate matrix Finally, one has the field equation for the ghost field For an abelian gauge theory the vanishing structure constants f zyw lead to im- where we have limited the matter content to a complex scalar with (T z ) ab = −δ ab .
It is clear, however, that the background field identity contains information beyond the Slavnov-Taylor identity.
Let us finally turn to solutions of the Slavnov-Taylor identity for non-abelian gauge theories. We concentrate first on a vanishing infrared cutoff R k = 0. We are Similar relations must hold for ϕ * , ψ andψ and the equation for ∂Γ ∂γ must be compatible with the solution of (A.53). We conclude that the generalized ansatz with ξ replaced byξ can only be used in the approximation where this system of equations forξ is self-consistent. Beyond this approximation the gauge invariance of the sector with ghost number zero (Γ inv ) can probably not be maintained.
In the presence of a nonvanishing infrared cutoff R k an additional piece Γ ct containing counterterms should be added. It vanishes in the limit k → 0. In the perturbative regime Γ ct contains a gluon mass term ∼ k 2 (see sect. 4). There is no reason why Γ ct should be gauge-invariant under transformations which leave the background fieldĀ fixed. We may define Γ ct by the requirement that Γ − Γ ct obeys the anomaly-free Slavnov-Taylor identity. The anomaly A