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Potential regime for heavy quarks dynamics and Lorentz nature of confinement

Propagation of the heavy quark in the field of a static antiquark source is studied in the framework of effective Dirac equation. The model of QCD vacuum is described by bilocal gluonic correlators. In the heavy quark limit the effective interaction is reduced to the potential one with 5/6 Lorentz scalar and 1/6 Lorentz vector linear confinement, while spin-orbit term is in agreement with Eichten-Feinberg-Gromes results. New spin-independent corrections to the leading confining regime are identified, which arise due to the nonlocality of the interaction in time direction and quark Zitterbewegung.

The most consistent derivation of the spin-dependent potentials was performed in the framework of the Wilson loop approach [1,2], where the potentials were expressed in terms of expectation values of gluonic field insertions into a Wilson loop corresponding to the propagation of qq pair, and fundamental relations between static and spin-dependent potentials were established [2].
One of the challenging problems is the long range part of the spin-orbit force which is sensitive to the Lorentz nature of the static confining interaction ε(r). If it is a local Lorentz scalar potential, one has while for the time component of the Lorentz vector it is Phenomenological analysis of heavy quarkonia spectra clearly prefers possibility (2), but it was shown within Vacuum Background Correlators Method [3][4][5][6] and in the framework of the Coulomb gauge QCD Hamiltonian [7] that relation (2) is respected at large distances without ad hoc assumption of scalar confinement. It was demonstrated in [3][4][5][6] that nonrelativistic reduction of the long range interaction is not the whole story, and additional contributions to the spin-orbit force exist, which are due to the nonlocality of the QCD generated interaction.
The aim of the present paper is to study systematically the nonlocality corrections to the effective long range interaction for heavy quark. We have found a new constraint on the parameters of the interaction, which appears to be crucial for the selfconsistent potential-type dynamics of heavy quarks. We adopt the approach suggested recently in [8] and, in a more simple version, in [9], that allows, as a byproduct, to establish explicitly the Lorentz nature of confinement. Our derivation is restricted to the case of a heavy quark propagating in the field of an infinitely heavy antiquark colour source, and is only the first step towards defining the full dynamics of heavy quarkonia.
The starting point of approach [8] is the Green function S qQ for the qQ system, written in the Euclidean space as where SQ(x, y) is the propagator of the static antiquark placed at the origin. To consider the one-body limit it is convenient to choose the modified Fock-Schwinger gauge [10] in which SQ(x, y) is simply Integration over gluonic field A µ in (4) can be performed with the result where L ef f (ψ, ψ + ) is the effective quark Lagrangian: Using gauge invariance of the vacuum one has in the limit N C → ∞, yielding the Schwinger-Dyson equation with the self-energy part M(x, z) given by where S(x, y) = 1 N C < ψ β (x)ψ + β (y) > is the colour trace of the quark Green function. As in gauge (5) Green function (6) of the static source is unity in the colour space, quantity S(x, y) completely defines propagation of the colourless qQ object.

Gauge condition (5) can be rewritten as
so the average K µν can be expressed in terms of field strength correlator < F a µν (x)F b λρ (y) >, for which we use the parametrization [3,4]: where the second term ∆ (1) is a full derivative and does not contribute to the confinement.
As we are interested only in long range force, we consider only the term proportional to D(x − y) in (15), which, in contrast to ∆ (1) , contributes to the area law with the string tension Function D(u 4 , | u|) is actually a function of u 2 4 + u 2 due to Lorentz invariance, but in our apparently non-invariant treatment we keep dependences on | u| and |u 4 | separately as in (16).
Finally, for average K µν (x, y) one has (τ = x 4 − y 4 ): The system of equations (11), (12) may be rewritten in terms of wave functions as This equation is essentially nonlinear, since the eigenfunctions ψ n enter the spectral representation for S(x, z). In the heavy quark limit we solve it perturbatively, substituting the free Green function S 0 (x, z) into the self-energy part. The resulting linear equation was considered in [8,9].
As correlators (15) are defined in the Euclidean space, it is convenient to formulate the eigenvalues problem for equation (18) in the Euclidean space too and perform the Wick rotation to the Minkowski one afterwards arriving at the Dirac-type equation with operatorM given bŷ where ε = ε( k) = k 2 + m 2 and γ-matrices are Euclidean ones OperatorM is nonlocal both in space and time, and our strategy is to find the leading local limit, to establish the first order nonlocal corrections and to check if they are small. To this end we approximately rewrite the expression in the curly brackets in (20) in the form introducing the quarks binding energy ε 0 = E − m.
The leading local confining interaction is obtained after omitting the terms proportional to ε 0 , γ k and k 2 in (21), and yieldŝ It is relevant now to comment on the Lorentz nature of confinement and to clarify some related confusions. The underlying interaction is bilinear in vector verteces, as it is clearly seen from the effective Lagrangian (8): everywhere including resulting expression (22) the interaction contains γ µ . . . γ ν product. As it will be shown below, for heavy quarks this structure actually reduces to γ 0 . . . γ 0 product, and we agree at this point with the statement made in [7]: the interaction is time-like vector one. However, this almost trivial observation does not straightforwardly help to answer another question, which is usually put in connection with the problem of the Lorentz structure: the effective local confining interactionM 0 ( x, z) can be proportional either to unity or to γ 0 , and it is added either to the mass term (scalar confinement) or to the energy term (vector confinement) in the effective Dirac equation (19). The answer to this question depends on the behaviour of the function K µν .
The correlator D(u 4 , | u|) should decrease in all directions in the Euclidean space so that the string tension (16) were finite, and the Dirac structure of the confining potential depends on the correlation length T g which governs this decrease (r = | x|): with large distance (r ≫ T g ) behaviour for mT g ≫ 1, and for mT g ≪ 1. Both regimes were established in [8,9]. In what follows we demonstrate that only the regime mT g ≫ 1 is selfconsistent. Now we consider various corrections to the leading interaction (23). The terms proportional to ( γ k) in (21) are calculated by means of integration by parts and givê Problems start with the terms proportional to k 2 in (21). To bring these terms into local form one should integrate by parts twice: and only momentum dependent terms can be expressed in terms of integrals of the function D(τ, λ): Moreover, the momentum independent contribution appears to diverge in the T g → 0 limit. Indeed, and for large r (r ≫ T g ) one has the following asymptotics for expression (29): where α + , α − and β are some coeffitients of order unity depending on the explicit form of function D(τ, λ). It is clear from expression (29) and asymptotics (30) that for the case mT g ≫ 1 V k 2 is indeed a corection to the leading regime (23), (24). For mT g ≪ 1 this correction is larger than the leading interaction, and we conclude in such a way that the embarrasing regime (25) is not potential, and, moreover, for such light quarks one should turn to the full system of equations (11) and (12).
The term proportional to the binding energy is simply and does not cause many additional problems, it is small comparing to the confinement term if ε 0 T g ≪ 1, as it was already found in [9]. The effective Schroedinger equation is obtained from Dirac equation (19) by the standard Foldy-Wounthuysen (FW) reduction, and displays a lot of pleasant features for mT g ≫ 1. For the upper Dirac component of the quark wave function the resulting Hamiltonian is where ε E (r) is the standard confining interaction and Expressions (33) and (34) were obtained from equation (19) in [9], and coincide with the ones given by the Vacuum Background Correlators Method [3][4][5][6]. As it was shown in [5], form (34) for the spin-orbit force is equivalent to the form obtained by Gromes [2] in terms of Wilson loop expectations.
As for r ≫ T g ε E = σr and V LS = − σ l 4m 2 r , these expressions mimic scalar confinement (2) at large distances. Nevertheless, confining interaction (24) is not a scalar one, at large distances it is 5/6 scalar and 1/6 vector, and is due to the electric correlator K 44 only as one should expect for the nonrelativistic particle.
There are three sources for the spin-orbit force (34). One piece comes from the FW reduction of confinement (23), it is purely magnetic and gives this requirement is indeed new? In the N C → ∞ limit correlators (15) are given by the pure Yang-Mills theory, with single nonperturbative mass scale. On the other hand, model (15) contains two dimensional parameters, correlation length T g and D(0), which is proportional to the gluonic condensate and is related to the string tension σ via equation (16). So it is not surprising that two phenomenological quantities, √ σ and T −1 g , should be of the same order of magnitude, and indeed they are. The commonly accepted value of the string tension gives √ σ ≈ 0.4GeV , while lattice measurements [12] give for the correlation length value T −1 g ∼ 1GeV , so the requirement mT g ≫ 1 does not bring drastic changes into our understanding of heavy quark dynamics. Much more interesting are new corrections (35) and (36), which are potentially large and even at large distances depend on the explicit form of vacuum correlation function D.
We acknowledge useful discussions with Yu.A.Simonov. This work is supported by grants 96-02-19184a and 97-02-16404 of Russian Fundamental Research Foundation and by INTAS 94-2851 and 93-0079ext.