Confinement Mechanism in the Field Correlator Method

Abstract

Confinement in QCD results from special properties of vacuum fluctuations of gluon fields. There are two numerically different scales, characterizing nonperturbative QCD vacuum dynamics: “small” one, corresponding to gluon condensate, critical temperature etc, which is about 0.1–0.3 GeV, and a “large” one, given by inverse confining string width, glueball and gluelump masses, and so forth, which is about 1.5–2.5 GeV. We discuss the origin of this hierarchy in a picture where confinement is ensured by quadratic colorelectric field correlators of the special type. These correlators, on the other hand, can be calculated via gluelump Green's function, whose dynamics is defined by the correlators themselves. In this way one obtains a self-consistent scheme, where string tension can be expressed in terms of Λ_QCD.

1. Introduction

Confinement of color is the most important property of quantum chromodynamics (QCD), ensuring stability of matter in the Universe. Attempts to understand physical mechanism of confinement are incessant since advent of constituent quark model and QCD; for reviews, see, for example, [1–4]. Among many suggestions, one can distinguish three major approaches as follows.

(i)Confinement is due to classical field lumps like instantons or dyons. (ii)Confinement can be understood as a kind of Abelian-like phenomenon, for example, according to seminal suggestion by Hooft [5] and Mandelstam [6] of dual Meissner scenario. (iii)Confinement results from properties of quantum stochastic ensemble of nonperturbative (np) fields filling QCD vacuum.

The first approach has a body of successful phenomenology; see review in [7]. On the other hand, as is well known, instanton gas model lacks confinement. As for the second approach, it has gained much popularity in the lattice community, and studies of various projected objects such as Abelian monopoles and center vortices are going on (see, e.g., [8, 9]). It is worth mentioning that despite tremendous recent progress in lattice calculations, the problem of identifying correct gauge-invariant picture of the QCD vacuum is extremely difficult on the lattice as well as in the continuum. For example, the results of [10, 11]—lowering deconfinement critical temperature with rising magnitude of external Abelian chromomagnetic (not chromoelectric!) field—are not easily incorporated to the conventional dual superconductor picture with space-time independent monopole condensate and presumably suggest the formation of nontrivial inhomogeneous condensates. It seems to be correct to conclude that the exact pattern of the Abelian confinement scenario is not yet fixed.

The development of the third approach in a systematic way started in 1987 [12–15] (see, e.g., [16] as example of earlier investigations concerning stochasticity of QCD vacuum). The nontrivial structure of np vacuum can be described by a set of nonlocal gauge-invariant field strength correlators (FCs). The discussion presented below is in the framework of this scenario.

QCD sum rules [17–19] were suggested as an independent approach to npQCD dynamics, not addressing directly confinement mechanism. The key role is played by gluon condensate 𝐺₂, which is defined as np average of the following type: 𝐺₂=𝛼_𝑠𝜋⟨𝐹^𝑎_𝜇𝜈(0)𝐹^𝑎_𝜇𝜈(0)⟩. (1.1) In the sum rule framework, it is a universal quantity characterizing QCD vacuum as it is, while it enters power expansion of current-current correlators with channel-dependent coefficients. This is the cornerstone of QCD sum rules ideology.

It is worth stressing from the very beginning that 𝐺₂ and other condensates are usually considered as finite physical quantities. Naively in perturbation theory, one would get 𝐺₂∼𝑎⁻⁴, where 𝑎 is a space-time ultraviolet cutoff (e.g., lattice spacing). It is always assumed that this “hard’’ contribution is somehow subtracted from (1.1) and the remaining finite quantity results from “soft’’ np fields, in some analogy with Casimir effect where modification of the vacuum by boundaries of typical size 𝐿 yields nonzero shift of energy-momentum tensor by the amount of order 𝐿⁻⁴. In QCD, the role of 𝐿 is played by dynamical scale Λ⁻¹_QCD.

The idea of np gluon condensate has proved to be very fruitful. However the relation of 𝐺₂ to confinement is rather tricky. Let us stress that since there can be no local gauge-invariant order parameter for confinement-deconfinement transition, 𝐺₂ is not an order parameter, and, in particular, neither perturbative nor np contributions to this quantity vanish in either phase. (Let us repeat that following [17, 18], we define 𝐺₂ in (1.1) as purely np object.) On the other hand, one can show (see details in [3]) that the scale of deconfinement temperature is set by the condensate, or, more precisely, by its electric part: 𝑇_𝑐∼𝐺₂^1/4. Let us remind the original estimate for the condensate [17, 18] given by 𝐺₂=0.012GeV⁴, with large uncertainties. One clearly sees some tension between this “small’’ scale and a “large’’ mass scale in QCD given by, for example, the mass of the lightest 0⁺⁺ glueball, which is about 1.5 GeV.

The existence of two numerically different np scales in strong interaction physics is a common feature one encounters in many models. As an early example, let us mention seminal discussion [20] concerning the difference between confinement scale and chiral symmetry breaking scale. Another example is given by instanton model [21] where the mean instanton size (taken to be about 0.3 Fm) is assumed to be smaller than mean instanton radius (about 1 Fm) and this fact is nothing but the “diluteness’’ of the instanton gas in this model. Dual superconductor picture [8, 9] provides more recent example of this sort: for the QCD vacuum understood as the so-called type-II superconductor, the mass of diagonal field—the “photon’’ (inverse London penetration depth)—is parametrically larger than the effective Higgs mass. Correspondingly the nonabelian electric field penetrates the vacuum forming Abrikosov fluxes (in dual analogy with the magnetic fields in conventional metal superconductors).

Of immediate interest are recent lattice studies of relevant nonperturbative scales [22]. The authors project each lattice link variable to specific zone in the momentum space bound by some infrared and ultraviolet cuts—the momentum ring—and study dependence of various nonperturbative quantities of physical interest on the location and width of this ring. This analysis demonstrates numerically that the relevant scales are indeed different for different quantities. It is far from obvious, however, what gauge-invariant statements based on these results can be made (since the approach is gauge dependent).

The connections between these models and the approach discussed here were studied in many papers (see reviews in [12–15] and references therein). For example, direct analogy between 𝜆 and effective instanton size was analyzed in [23]. The relation between dual superconductor picture and field correlator method was studied in details in [24, 25]. Notice that in both cases the main advantage of field correlator method is its manifest gauge invariance.

The phenomenon of this hierarchy of scales is of prime importance for the stochastic scenario. It was suggested in [12–15]; see also, [26–28] that qualitative physical explanation is the following: QCD vacuum is filled by short-distance correlated nonperturbative gluon fields. In other words, besides gluon condensate 𝐺₂, there is another important dimensionfull parameter characterizing np dynamics of vacuum fields: correlation length 𝜆 (also denoted as 𝑇_𝑔 in some papers). The gauge-invariant dimensionless product (𝐺₂𝜆⁴)^1/2 is about a few percents (see below). This leads to numerous consequences, some of which are discussed in the rest of this paper.

An immediate question to ask is about the origin of this smallness. To answer it is one of the goals of the present paper. The key observation is the following: correlation length 𝜆 is nothing but inverse mass of certain gluelump. Gluelumps (for details, we refer the reader to [29, 30]) are rather unusual objects from perturbative field theory point of view. In some sense they are analogous to heavy-light mesons. The latter can be approximated as a bound state of fundamental color charge (light quark) in the static field of fundamental color anticharge (heavy antiquark). In the limit of heavy mass 𝑀 going to infinity, the difference between the meson mass and 𝑀 stays constant and the object as a whole is of course color singlet. Analogously, gluelump is a singlet bound state of adjoint color charge “valent gluons’’ in the static field of “infinitely heavy’’ adjoint color source. Roughly speaking, gluelump is what replaces gluon propagator in gauge-invariant formulation of the theory. Of course there are no elementary heavy particles with adjoint charge in QCD, and in this sense, gluelump cannot be “detected’’ as a real particle. Nevertheless, gluelumps happen to be very useful objects in discussions of the most fundamental properties of confining QCD vacuum (see more on that below).

Let us illustrate the appearance of numerical hierarchy of the kind we are discussing on a simple phenomenological example. Consider leading the following 𝜌-meson Regge trajectory: 𝐽=𝛼^′_𝐽𝑀²(𝐽)+0.48,𝛼^′_𝐽=0.89GeV⁻². (1.2) This corresponds to the product 𝜎𝛼^′_𝐽=12𝜋=0.16, (1.3) where 𝜎=0.18GeV² is the confining string tension. One can say that the mass squared is a factor “2𝜋’’ larger than string tension. On the other hand, in stochastic picture one has, parametrically, 𝜎∝𝜆²𝐺₂. Here the smallness of string tension is controlled by two powers of 𝜆. Thus to get self-consistent picture with the desired hierarchy, roughly speaking, one is to make manifest this “1/2𝜋’’ factor in 𝜆². This is the essence of the phenomenon discussed in the present paper.

Let us come back to the stochastic picture. Its crucial feature found in [31–33] is that the lowest, quadratic, nonlocal FC, ⟨𝐹_𝜇𝜈(𝑥)𝐹_𝜆𝜎(𝑦)⟩, describes all np dynamics with very good accuracy. It was also shown that a simple exponential form of quadratic correlators found on the lattice [34–38] allows to calculate all properties of lowest mesons, glueballs, hybrids, and baryons, including Regge trajectories, lepton widths, etc.; see reviews in [2, 27, 28] and references therein. Since the correlation length 𝜆 entering these exponents is small, potential relativistic quantum-mechanical picture is applicable and all QCD spectrum is defined mostly by string tension 𝜎 (which is an integral characteristic of the nonlocal correlator; see below) and not by its exact profile. This is discussed in details in the next section.

However to establish the confinement mechanism unambiguously, one should be able to calculate vacuum field distributions, that is, field correlators, self-consistently. In the long run, it means that one is to demonstrate that it is essential property of QCD vacuum fields ensemble to be characterized by correlators, which support confinement for temperatures 𝑇 below some critical value 𝑇_𝑐 and deconfinement at 𝑇>𝑇_𝑐.

Attempts to achieve this goal have been undertaken in [39, 40]; however, the resulting chain of equations is too complicated to use in practice.

Another step in this direction is done in [41], where FCs are calculated via gluelump Green's functions and self-consistency of this procedure was demonstrated for the first time. These results were further studied and confirmed in [42].

The main aim of this paper is to present this set of equations as a self-consistent mechanism of confinement and to clarify qualitative details of the FC-gluelump connection. In particular we demonstrate how the equivalence of color-magnetic and colorelectric FCs for 𝑇=0 ensures the Gromes relation [43, 44]. We also show, that self-consistency condition for FC as gluelump Green's function allows to connect the mass scale to 𝛼_𝑠, and in this way, to express Λ_QCD via string tension 𝜎.

Lattice computations play important role as a source of independent knowledge about vacuum field distributions. Recently a consistency check of this picture was done on the lattice [45] by measurement of spin-dependent potentials. The resulting FCs are calculated and compared with gluelump predictions in [46] demonstrating good agreement with analytic results; in particular small vacuum correlation length 𝜆∼0.1 Fm is shown to correspond to large gluelump mass 𝑀₀≈2 GeV.

The paper is organized as follows. In the next section, we remind the basic expressions for Green's functions of 𝑞𝑞 and 𝑔𝑔 systems in terms of Wegner-Wilson loops, and qualitative picture of FCs dynamics is discussed. Section 3 is devoted to the expressions of spin-dependent potentials in terms of FC. We also shortly discuss the lattice computations of FC. In Section 4 FC are expressed in terms of gluelump Green's functions, while in Section 5 the self-consistency of resulting relations is studied. Our conclusions are presented in Section 6.

2. Wegner-Wilson Loop, Field Correlators, and Green's Functions

Since our aim is to study confinement for quarks (both light and heavy) and also for massless gluons, we start with the most general Green's functions for these objects in a proper physical background using Fock-Feynman-Schwinger representation. The formalism is built in such a way that gauge invariance is manifest at all steps. A reader familiar with this technique can skip this section and go directly to Section 3.

First, we consider the case of mesons made of quarks while gluon bound states are discussed below. For quarks, one forms the initial and final state operators Ψ_in,out(𝑥,𝑦)=𝜓^†(𝑥)Γ_in,outΦ(𝑥,𝑦)𝜓(𝑦), (2.1) where 𝜓^†, 𝜓 are quark operators, Γ is a product of Dirac matrices, that is, 1, 𝛾_𝜇,𝛾₅,(𝛾_𝜇𝛾₅),…, and Φ(𝑥,𝑦)=𝑃exp(𝑖𝑔∫^𝑥_𝑦𝐴_𝜇𝑑𝑧_𝜇) is parallel transporter (also known as Schwinger line or phase factor). The meson Green's function can be written (in quenched case) as𝐺_𝑞𝑞(𝑥,𝑦∣𝑥^′,𝑦^′)=⟨Ψ⁺_out(𝑥^′,𝑦^′)Ψ_in(𝑥,𝑦)⟩_𝐴,𝑞=⟨Γ^†Φ(𝑥^′,𝑦^′)𝑆_𝑞(𝑦^′,𝑦)ΓΦ(𝑦,𝑥)𝑆_𝑞(𝑥,𝑥^′)⟩_𝐴, (2.2) and the quark Green's function 𝑆_𝑞 is given in Euclidean space time (see [47, 48] and references therein) as 𝑆_𝑞(𝑥,𝑦)=(𝑚+𝐷)⁻¹=(𝑚−𝐷)^∞₀𝑑𝑠(𝐷𝑧)_𝑥𝑦exp(−𝐾)𝑃_𝐴exp(𝑖𝑔^𝑥_𝑦𝐴_𝜇𝑑𝑧_𝜇)𝑃_𝜎(𝑥,𝑦,𝑠), (2.3) where 𝐾 is kinetic energy term as 𝐾=𝑚²1𝑠+4^𝑠₀𝑑𝜏(𝑑𝑧_𝜇(𝜏))𝑑𝜏², (2.4) where 𝑚 is the pole mass of quark, and quark trajectories 𝑧_𝜇(𝜏) with the end points 𝑥 and 𝑦 are being integrated over in (𝐷𝑧)_𝑥𝑦.

The factor 𝑃_𝜎(𝑥,𝑦,𝑠) in (2.3) is generated by the quark spin (color-magnetic moment) and is equal to 𝑃_𝜎(𝑥,𝑦,𝑠)=𝑃_𝐹exp[𝑔^𝑠₀𝜎_𝜇𝜈𝐹_𝜇𝜈(𝑧(𝜏))𝑑𝜏], (2.5) where 𝜎_𝜇𝜈=(1/4𝑖)(𝛾_𝜇𝛾_𝜈−𝛾_𝜈𝛾_𝜇), and 𝑃_𝐹(𝑃_𝐴) in (2.5) and (2.3) are, respectively, ordering operators of matrices 𝐹_𝜇𝜈(𝐴_𝜇) along the path 𝑧_𝜇(𝜏).

With (2.2) and (2.3), one easily gets 𝐺_𝑞𝑞(𝑥,𝑦;𝐴)=^∞₀𝑑𝑠^∞₀𝑑𝑠^′(𝐷𝑧)_𝑥𝑦(𝐷𝑧^′)_𝑥𝑦𝑒^{−𝐾−𝐾′}Tr(Γ(𝑚−𝐷)𝑊_𝜎(𝑥,𝑦)Γ^†(𝑚^′−𝐷^′)), (2.6) where Tr means trace operation both in Dirac and color indices, while 𝑊_𝜎(𝑥,𝑦)=𝑃_𝐴exp(𝑖𝑔_{𝐶(𝑥,𝑦)}𝐴_𝜇𝑑𝑧_𝜇)𝑃_𝜎(𝑥,𝑦,𝑠)𝑃^′_𝜎(𝑥,𝑦,𝑠^′). (2.7) In (2.7) the closed contour 𝐶(𝑥,𝑦) is formed by the trajectories of quark 𝑧_𝜇(𝜏) and antiquark 𝑧^′_𝜈(𝜏^′), and the ordering 𝑃_𝐴 and 𝑃_𝐹 in 𝑃_𝜎, 𝑃^′_𝜎 is universal, that is, 𝑊_𝜎(𝑥,𝑦) is the Wegner-Wilson loop (W-loop) for spinor particle.

The factors (𝑚−𝐷) and (𝑚^′−𝐷^′) in (2.6) need a special treatment when being averaged over fields. As shown in Appendix 1 of [49], one can use the simple replacement 𝑚−𝐷→𝑚−𝑖𝑝,𝑝_𝜇=12(𝑑𝑧_𝜇)𝑑𝜏_𝜏=𝑠. (2.8)

The representation (2.6) is exact in the limit 𝑁_𝑐→∞, when internal quark loops can be neglected, and is a functional of gluonic fields 𝐴_𝜇, 𝐹_𝜇𝜈, which contains both perturbative and np contributions, not specified at this level.

The next step is averaging over gluon fields, which yields the physical Green's function 𝐺_𝑞𝑞: 𝐺_𝑞𝑞(𝑥,𝑦)=⟨𝐺_𝑞𝑞(𝑥,𝑦;𝐴)⟩_𝐴. (2.9) The averaging is done with the usual Euclidean weight exp(−Action/ℏ), containing all necessary gauge-fixing and ghost terms.

To proceed, it is convenient to use nonabelian Stokes theorem (see [2] for references and discussion) for the first factor on the r.h.s. in (2.7) (corresponding to the W-loop for scalar particle) and to rewrite it as an integral over the surface 𝑆 spanned by the contour 𝐶=𝐶(𝑥,𝑦): ⟨𝑊(𝐶)⟩=⟨Tr𝑃exp(𝑖𝑔_𝐶𝐴_𝜇(𝑧)𝑑𝑧_𝜇)⟩=⟨Tr𝒫exp(𝑖𝑔_𝑆𝐹(𝑢)𝑑𝑠(𝑢))⟩=Tr𝒫_𝑥exp^∞_𝑛=2(𝑖𝑔)^𝑛_𝑆⟨⟨𝐹(𝑢₁)⋯𝐹(𝑢_𝑛)⟩⟩𝑑𝑠₁⋯𝑑𝑠_𝑛=exp^∞_𝑛=2Δ^(𝑛)[𝑆]. (2.10) Here 𝐹(𝑢_𝑖)𝑑𝑠_𝑖=Φ(𝑥₀,𝑢_𝑖)𝐹^𝑎_𝜇𝜈(𝑢_𝑖)𝑡^𝑎Φ(𝑢_𝑖,𝑥₀)𝑑𝑠_𝜇𝜈(𝑢_𝑖), and 𝑢_𝑖, 𝑥₀ are the points on the surface 𝑆 bound by the contour 𝐶=𝜕𝑆. The double brackets ⟨⟨⋯⟩⟩ stay for irreducible correlators proportional to the unit matrix in the color space (and therefore, only spacial ordering 𝒫_𝑥 enters (2.10)). Since (2.10) is gauge-invariant, one can make use of generalized contour gauge [50–52], which is defined by the condition Φ(𝑥₀,𝑢_𝑖)≡1. Notice that throughout the paper we normalize trace over color indices as Tr𝟏=1 in any given representation.

Since (2.10) is an identity, the r.h.s. does not depend on the choice of the surface, which is integrated over in 𝑑𝑠_𝜇𝜈(𝑢). On the other hand, it is clear that each irreducible 𝑛-point correlator ⟨⟨𝐹⋯𝐹⟩⟩ integrated over 𝑆 (these are the functions Δ^(𝑛)[𝑆] in (2.10)) depends, in general case, on the choice one has made for 𝑆. Therefore it is natural to ask the following question: is there any hierarchy of Δ^(𝑛)[𝑆] as functions of 𝑛 for a given surface 𝑆?

To get the physical idea behind this question, let us take the limit of small contour 𝐶. In this case one has for the np part of the W-loop in fundamental representation the following: −log⟨𝑊(𝐶)⟩_𝑛𝑝∼𝐺₂⋅𝑆², (2.11) where 𝑆 is the minimal area inside the loop 𝐶. The np short-distance dynamics is governed by the vacuum gluon condensate 𝐺₂; see (1.1). Higher condensates and other possible vacuum averages of local operators, in line with Wilson expansion, have been introduced and phenomenologically estimated in [19]; for a review see [53].

Suppose now that the size of W-loop is not small, that is, it is larger than some typical dynamical scale 𝜆=𝒪(1GeV) to be specified below. Let us now ask the following question: if one still wants to expand the loop formally in terms of local condensates, what would the structure of such expansion be? It is easy to see from (2.10) that there are two subseries in this expansion. The first one is just an expansion in 𝑛, that is, in the number of field strength operators. It is just ⟨𝑊(𝐶)⟩=1+Δ⁽²⁾[𝑆]+Δ⁽³⁾1[𝑆]+2(Δ⁽²⁾[𝑆])²+⋯. (2.12) The second series is an expansion of each Δ^(𝑛)[𝑆] in terms of local condensates, that is, expansion in powers of derivatives. Taking for simplicity the lowest 𝑛=2 term, this expansion reads Δ⁽²⁾[𝑆]=_𝑆𝑑𝑠_𝑥𝑆𝑑𝑠_𝑦=⟨⟨𝐹(𝑥)𝐹(𝑦)⟩⟩_𝑆𝑑𝑠_𝑥𝑆𝑑𝑠_𝑦[⟨⟨𝐹(𝑥)𝐹(𝑥)⟩⟩+(𝑥−𝑦)_𝜇⟨⟨𝐹(𝑥)𝐷_𝜇𝐹(𝑥)⟩⟩+(𝑥−𝑦)_𝜇(𝑥−𝑦)_𝜈⟨⟨𝐹(𝑥)𝐷_𝜈𝐷_𝜇𝐹(𝑥)⟩⟩/2+⋯]. (2.13)

It is worth mentioning (but usually skipped in QCD sum rules analysis) that the latter series contain ambiguity related to the choice of contours in parallel transporters. The expression (2.13) is written for the simplest choice of straight contour connecting the points 𝑥 and 𝑦 (and not 𝑥 and 𝑥₀ or 𝑦 and 𝑥₀). This is the choice we adopt in what follows. In general case each condensate in (2.13) is multiplied by some contour-dependent function of 𝑥, 𝑦, and 𝑥₀. It is legitimate to ask whether this approximation, that is, replacement of ⟨TrΦ(𝑥₀,𝑥)𝐹_𝜇𝜈(𝑥)Φ(𝑥,𝑥₀)Φ(𝑥₀,𝑦)𝐹_𝜌𝜎(𝑦)Φ(𝑦,𝑥₀)⟩ (2.14) by ⟨Tr𝐹_𝜇𝜈(𝑥)Φ(𝑥,𝑦)𝐹_𝜌𝜎(𝑦)Φ(𝑦,𝑥)⟩ (2.15) affects our final results. Since the only difference between the above two expressions is profiles of the transporters, this is the question about contour dependence of FCs. We will not discuss this question in details in the present paper and refer an interested reader to [12–15]. Here we only mention that the result (weak dependence on transporters' choice) is very natural in Abelian dominance picture since for Abelian fields the transporters exactly cancel.

Let us stress that both expansions (2.12) and (2.13) are formal since the loop is assumed to be large. Moreover, from dimensional point of view, the terms of these two expansions mix with each other. For example, one has nonzero v.e.v. of two operators of dimension eight: Δ⁽²⁾[𝑆]→⟨𝐹_𝜇𝜈𝐷⁴𝐹_𝜇𝜈⟩,Δ⁽³⁾[𝑆]→⟨𝐹_𝜇𝜎𝐹_𝜈𝜎𝐷²𝐹_𝜇𝜈⟩, (2.16) with no a priori reason (one could have such reasons if there is some special kinematics in the problem, but this is not the case.) to drop any of them. Moreover, even if one assumes that all these condensates of high orders can be self-consistently defined analogously to 𝐺₂, it remains to be seen whether the whole series converges or not.

Here the physical picture behind FC method comes into play. It is assumed (and confirmed a posteriori in several independent ways) that if the surface 𝑆 is chosen as the minimal one, the dominant contribution to (2.10) results from v.e.v.s of the operators 𝐹𝐷^𝑘𝐹, and this subseries can be summed up as (2.13). In this language, nonlocal two-point FC can be understood as a representation for the sum of infinite sequence of local terms (2.13).

Physically the dominance of ⟨𝐹𝐷^𝑛𝐹⟩ terms (and hence, of two-point FC) corresponds to the fact that the correlation length is small yielding a small expansion parameter 𝜉=𝐹𝜆²≪1 (𝐹 is average modulus of np vacuum fields), or, in other words, that typical inverse correlation length 𝜆⁻¹ characterizing ensemble of QCD vacuum fields is parametrically larger than fields themselves. The dimensionless parameter 𝜉 in terms of condensate is given by (𝐺₂𝜆⁴)^1/2, and it is indeed small: of the order of 0.05 according to lattice estimates. One can say that often repeated statement “QCD vacuum is filled by strong and strongly fluctuating chromoelectric and chromomagnetic fields’’ is only partly correct, namely, in reality the fields are not so strong as those of 𝜆⁻¹. Technically one can say that we sum up the leading subseries in (2.10) in the same spirit as one does in covariant perturbation theory [54] for weak but strongly varying potentials.

Thus, for understanding of confining properties of QCD vacuum dynamics, not only scale of np fields given by 𝐺₂ but also another quantity—the vacuum correlation length 𝜆— which defines the nonlocality of gluonic excitations is important. It is worth repeating here that physically 𝜆 encodes information about properties of the series (2.13) and has the same theoretical status as that of the corresponding np condensates in the following sense: knowledge of all terms in (2.13) would allow to reconstruct 𝜆. Of course, it is impossible in practice and one has to use other methods to study large distance asymptotics of FC. On more phenomenologic level, it was discussed in [55–59], while rigorous definition was given in the framework of the FC method (often referred to as Stochastic Vacuum Model.) (see [12–15, 26, 60] for reviews).

It is important to find also the world-line representation for gluon Green's function, which is done in the framework of background perturbation theory (see [61–65] for details). In this way one finds (where 𝑎, 𝑏 are adjoint color indices and dash sign denotes adjoint representation) that 𝐺^𝑎𝑏_𝜇𝜈(𝑥,𝑦)={^∞₀𝑑𝑠(𝐷𝑧)_𝑥𝑦𝑒^−𝐾𝑃_𝑎exp(𝑖𝑔^𝑥_𝑦𝐴_𝜇𝑑𝑧_𝜇)𝑃_Σ(𝑥,𝑦,𝑠)}^𝑎𝑏_𝜇𝜈, (2.17) where 𝑃_Σ(𝑥,𝑦,𝑠)=𝑃_𝐹exp(2𝑖𝑔^𝑠₀𝐹_𝜆𝜎(𝑧(𝜏))𝑑𝜏). (2.18) All the above reasoning about asymptotic expansions of the corresponding W-loops is valid here as well.

Thus the central role in the discussed method is played by quadratic (Gaussian) FC of the form 𝐷_{𝜇𝜈,𝜆𝜎}≡𝑔²⟨Tr𝐹_𝜇𝜈(𝑥)Φ(𝑥,𝑦)𝐹_𝜆𝜎(𝑦)Φ(𝑦,𝑥)⟩, (2.19) where 𝐹_𝜇𝜈 is the field strength and Φ(𝑥,𝑦) is the parallel transporter. Correlation lengths 𝜆_𝑖 for different channels are defined in terms of asymptotics of (2.19) at large distances: exp(−|𝑥−𝑦|/𝜆_𝑖). The physical role of 𝜆_𝑖 is very important since it distinguishes two regimes: one expects validity of potential-type approach describing the structure of hadrons of spatial size 𝑅 and at temporal scale 𝑇_𝑞 for 𝜆_𝑖≪𝑅,𝑇_𝑞, while in the opposite case, when 𝜆_𝑖≫𝑅,𝑇_𝑞 the description in terms of spatially homogeneous condensates can be applied.

At zero temperature, the 𝑂(4) invariance of Euclidean space-time holds and FC (2.19) is represented through two scalar functions 𝐷(𝑧),𝐷₁(𝑧) (where 𝑧≡𝑥−𝑦) as follows(all treatment in this paper as well as averaging over vacuum fields is done in the Euclidean space. Notice that only after all Green's functions are computed, analytic continuation to Minkowskii space-time can be accomplished): 𝐷_{𝜇𝜈,𝜆𝜎}(𝑧)=(𝛿_𝜇𝜆𝛿_𝜈𝜎−𝛿_𝜇𝜎𝛿_𝜈𝜆1)𝐷(𝑧)+2[𝜕𝜕𝑧_𝜇(𝑧_𝜆𝛿_𝜈𝜎−𝑧_𝜎𝛿_𝜈𝜆𝜕)+𝜕𝑧_𝜈(𝑧_𝜎𝛿_𝜇𝜆−𝑧_𝜆𝛿_𝜇𝜎)]𝐷₁(𝑧). (2.20)

One has to distinguish from the very beginning perturbative and np parts of the correlators 𝐷(𝑧), 𝐷₁(𝑧). Beginning with the former, one easily finds at tree level that 𝐷^𝑝,0(𝑧)=0,𝐷₁^𝑝,0(𝑧)=𝐶₂(𝑓)4𝛼_𝑠𝜋1𝑧⁴, (2.21) where 𝐶₂(𝑓) is fundamental Casimir 𝐶₂(𝑓)=(𝑁²_𝑐−1)/2𝑁_𝑐. At higher orders, situation becomes more complicated. Namely, one has at 𝑛-loop order 𝐷^𝑝,𝑛(𝑧)=𝐷₁^𝑝,0(𝑧)⋅𝐺^(𝑛)(𝑧),𝐷₁^𝑝,𝑛(𝑧)=𝐷₁^𝑝,0(𝑧)⋅𝐺₁^(𝑛)(𝑧), (2.22) where the gauge-invariant functions 𝐺^(𝑛)(𝑧), 𝐺₁^(𝑛)(𝑧) have the following general structure: 𝐺^(𝑛)(𝑧),𝐺₁^(𝑛)(𝑧)∼𝛼^𝑛_𝑠[𝑐_𝑛(log𝜇𝑧)^𝑛+⋯], (2.23) where 𝑐_𝑛 is numerical coefficient and 𝜇 is renormalization scale, and we have omitted subleading logarithms and constant terms in the r.h.s.; explicit expressions for the case 𝑛=1 can be found in [66, 67].

Naively one could take perturbative functions 𝐷^𝑝,𝑛(𝑧), 𝐷₁^𝑝,𝑛(𝑧) written above and use them for computations of W-loop or static potentials in Gaussian approximation. This, however, would be incorrect. The reason is that at any given order in perturbation expansion over 𝛼_𝑠 one should take into account perturbative terms of the given orders coming from all FCs, and not only from the Gaussian one. For example, at one loop level, the function 𝐷^𝑝,1(𝑧) is nonzero which naively would correspond to area law at perturbative level. Certainly this cannot be the case since self-consistent renormalization program for W-loops is known not to admit any terms of this sort [68, 69]. Technically in FC language, the correct result is restored by cancelation of contributions proportional to 𝐷^𝑝,1(𝑧) to all observables by the terms coming from triple FC (see details in [70]).

In view of this general property of cluster expansion, it is more natural just to take relation 𝐷^𝑝,𝑛(𝑧)≡0 as valid at arbitrary 𝑛, having in mind that proper number of terms from (2.12) has to be included to get this result. Thus we assume that the following decomposition takes place: 𝐷(𝑧)=𝐷^𝑛𝑝(𝑧),𝐷₁(𝑧)=𝐷^𝑝₁(𝑧)+𝐷₁^𝑛𝑝(𝑧), (2.24) and 𝐷(𝑧) has a smooth limit when 𝑧→0. As for 𝐷₁(𝑧), its general asymptotic at 𝑧→0 reads as 𝐷₁𝑐(𝑧)=𝑧⁴+𝑎₂𝑧²+𝑂(𝑧⁰), (2.25) where 𝑐 and 𝑎₂ weakly (logarithmically) depend on 𝑧. Notice that by 𝐷₁(0), we always understand np “condensate’’ part in what follows: 𝐺₂=6𝑁_𝑐𝜋²(𝐷^𝑛𝑝(0)+𝐷₁^𝑛𝑝(0)). (2.26)

The term 1/𝑧² deserves special consideration. One may argue that at large distances the simple additivity of perturbative and np contributions to 𝑉_𝑄𝑄 potential is violated [71]. It is interesting whether this additivity holds at small, distances and, in particular, mixed condensate 𝑎₂ was suggested in [72] and argued to be phenomenologically desirable.

As we demonstrate below, our approach is self-consistent with 𝑎₂=0, that is, when perturbative-np additivity holds at small distances and there is no dimension-two condensate. However, at intermediate distances, a complicated interrelation occurs, which does not exclude (2.25) being relevant in this regime.

We proceed with general analysis of (2.19). As proved in [12–15], 𝐷(𝑧), 𝐷₁(𝑧) do not depend on relative orientation of plane (𝜇𝜈), plane (𝜆𝜎), and vector 𝑧_𝛼. This orientation can be of the following 3 types: (a) planes (𝜇𝜈) and (𝜆𝜎) are perpendicular to the vector 𝑧_𝛼; (b) planes (𝜇𝜈) and (𝜆𝜎) are parallel or intersecting along one direction and 𝑧_𝛼 lies in one of planes; (c) planes (𝜇𝜈) and (𝜆𝜎) are perpendicular and 𝑧_𝛼 lies necessarily in one of them.

It is easy to understand that this classification is 𝑂(4) invariant, and therefore, one can assign indices 𝑎,𝑏,𝑐 to 𝐷_{𝜇𝜈,𝜆𝜎}, resulting in general case in three different functions. Physically speaking, case (a) refers to, for example, color magnetic fields 𝐹_𝑖𝑘, 𝐹_𝑙𝑚 with 𝑧_𝛼 in the 4th direction, and the corresponding FCs will be denoted as 𝐷_⟂(𝑧). Case (b) refers to, for example, the color electric fields 𝐸_𝑖(𝑥), 𝐸_𝑘(𝑦), connected by the same temporal links along 4th axis, and the corresponding FCs are denoted as 𝐷_‖(𝑧). Finally, in case (c) only 𝐷₁ part survives in (2.20) and it will be given by the subscript 𝐸𝐻, 𝐷₁^𝐸𝐻.

In case of zero temperature, when 𝑂(4) invariance holds, these three functions 𝐷_⟂(𝑧), 𝐷_‖(𝑧), and 𝐷₁^𝐸𝐻(𝑧) are easily expressed via 𝐷(𝑧), 𝐷₁(𝑧)∶

𝐷_⟂(𝑧)=𝐷(𝑧)+𝐷₁𝐷(𝑧),_‖(𝑧)=𝐷(𝑧)+𝐷₁(𝑧)+𝑧²𝜕𝐷₁(𝑧)𝜕𝑧²,𝐷₁^𝐸𝐻(𝑧)≡𝐷₁(𝑧) (2.27) (note that just 𝐷_⟂, 𝐷_‖ were measured on the lattice in [34–38]). For nonzero temperature, the correlators of color-electric and color-magnetic fields can be different and in general there are five FCs: 𝐷^𝐸(𝑧), 𝐷^𝐸₁(𝑧), 𝐷^𝐻(𝑧), 𝐷^𝐻₁(𝑧), and 𝐷₁^𝐸𝐻(𝑧). As a result, one obtains the full set of five independent quadratic FCs which can be defined as follows: 𝑔²⟨Tr𝐻_𝑖(𝑥)Φ𝐻_𝑗(𝑦)Φ⁺⟩=𝛿_𝑖𝑗(𝐷^𝐻(𝑧)+𝐷^𝐻₁(𝑧)+𝐳²𝜕𝐷^𝐻₁𝜕𝑧²)−𝑧_𝑖𝑧_𝑗𝜕𝐷^𝐻₁𝜕𝑧²,𝑔²⟨Tr𝐸_𝑖(𝑥)Φ𝐸_𝑗(𝑦)Φ⁺⟩=𝛿_𝑖𝑗(𝐷^𝐸(𝑧)+𝐷^𝐸₁(𝑧)+𝑧²₄𝜕𝐷^𝐸₁𝜕𝑧²)+𝑧_𝑖𝑧_𝑗𝜕𝐷^𝐸₁𝜕𝑧²,𝑔²⟨Tr𝐻_𝑖(𝑥)Φ𝐸_𝑗(𝑦)Φ⁺⟩=𝑒_𝑖𝑗𝑘𝑧_𝑘𝑧₄𝜕𝐷₁^𝐸𝐻𝜕𝑧². (2.28) It is interesting that the structure (2.28) survives for nonzero temperature, where 𝑂(4) invariance is violated: it tells that (2.28) give the most general form of the FCs. Note that at zero temperature, 𝐷^𝐻=𝐷^𝐸, 𝐷^𝐻₁=𝐷^𝐸₁ at coinciding point; the 𝑂(4) invariance requires that colorelectric and colormagnetic condensates coincide: 𝑔²⟨Tr𝐻_𝑖(𝑥)𝐻_𝑖(𝑥)⟩=𝑔²⟨Tr𝐸_𝑗(𝑥)𝐸_𝑗(𝑥)⟩; hence, 𝐷^𝐻(0)+𝐷^𝐻₁(0)=𝐷^𝐸(0)+𝐷^𝐸₁(0). (2.29) Here we assume that the np part of all FCs is finite.

At 𝑧≠0 for zero temperature, FCs can be expressed through 𝐷_⟂(𝑧), 𝐷_‖(𝑧) which do not depend on whether 𝐸_𝑖 or 𝐻_𝑘 enters in them, since, for example, 𝐷^𝐸_‖ can be transformed into 𝐷^𝐻_‖ by an action of 𝑂(4) group elements (the same for 𝐷^𝐸_⟂, 𝐷^𝐻_⟂). From this, one can deduce that both 𝐷₁ and 𝐷 do not depend on subscripts 𝐸, 𝐻 for zero temperature and 𝐷₁^𝐸𝐻 coincides with 𝐷₁.

Below we will illustrate this coincidence by concrete calculations of 𝐷^𝐸, 𝐷^𝐻, and so forth, through the gluelump Green's functions and show that the corresponding correlation lengths satisfy the relations (with obvious notations) 𝜆^𝐸=𝜆^𝐻≡𝜆;𝜆^𝐸₁=𝜆^𝐻₁=𝜆₁^𝐸𝐻≡𝜆₁. (2.30) As shown below, all correlation lengths 𝜆_𝑗 appear to be just inverse masses of the corresponding gluelumps 𝜆_𝑗=1/𝑀_𝑗.

Having analytic expressions for FCs, one might ask that how to check them versus experimental and lattice data. On experimental side in hadron spectroscopy, one measures masses and transition matrix elements, which are defined by dynamical equation, and the latter can be used of potential type due to smallness of 𝜆_𝑗. In static potential, only integrals over distance enter and the spin-independent static potential can be written as [12–15, 73, 74] 𝑉_𝑄𝑄(𝑟)=2^𝑟₀(𝑟−𝜆)𝑑𝜆^∞₀𝑑𝜈𝐷^𝐸(𝜆²+𝜈²)+^𝑟₀𝜆𝑑𝜆^∞₀𝐷^𝐸₁(𝜆²+𝜈²). (2.31)

At this point one should define how perturbative and np contributions combine in 𝐷^𝐸, 𝐷^𝐸₁, and this analysis was done in [70]. Making use of (2.21) together with definition of the string tension [12–15], one yeilds 𝜎^𝐸=12𝑑²𝑧𝐷^𝐸(𝑧). (2.32) One obtains from (2.31) the standard form of the static potential for 𝑁=3 at distances 𝑟≫𝜆^𝐸as 𝑉_𝑄𝑄(𝑟)=𝜎^𝐸𝑟−4𝛼_𝑠(𝑟)𝜆3𝑟+𝒪(^𝐸𝑟,𝛼²_𝑠). (2.33) As argued below, 𝜆^𝐸≈0.1 Fm, so that the form (2.33) is applicable in the whole range of distances 𝑟>0.1 Fm provided that asymptotic freedom is taken into account in 𝛼_𝑠(𝑟) in (2.33).

At the smallest values of 𝑟, 𝑟≲𝜆^𝐸, one would obtain a softening of confining term, 𝜎𝑟→𝑐𝑟² [12–15], which is not seen in accurate lattice data at 𝑟≳0.2 Fm [75, 76], imposing a stringent limit on the value of 𝜆^𝐸; 𝜆^𝐸≲0.1 Fm; see discussion in [76].

3. Spin-Dependent Potentials and FC

We now turn to the spin-dependent interactions to demonstrate that FC can be extracted from them [73, 74, 77–79]. 𝑉^′₁(𝑟),𝑉^′₂(𝑟),𝑉₃(𝑟),𝑉₄(𝑟), plus static term 𝑉^′_𝑄𝑄(𝑟) contain in integral form all five FCs: 𝐷^𝐸,𝐷^𝐸₁,𝐷^𝐻,𝐷^𝐻₁,𝐷₁^𝐻𝐸, and one can extract the properties of the latter from the spin-dependent potentials. This procedure is used recently for comparison of analytic predictions [46] with the lattice data [45].

To express spin-dependent potentials in terms of FC, one can start with the W-loop (2.7), entering as a kernel in the 𝑞𝑞 Green's function, where quarks 𝑞, 𝑞 can be light or heavy. Since both kernels 𝑃_𝜎, 𝑃^′_𝜎 contain the matrix 𝜎_𝜇𝜈𝐹_𝜇𝜈, the terms of spin-spin and spin-orbit types appear. As before, we keep Gaussian approximation, and in this case spin-dependent potentials can be computed not only for heavy but also for light quarks. They have the following Eichten-Feinberg form [80]: 𝑉^(diag)_𝑆𝐷𝜎(𝑅)=(¹𝐿₁4𝜇²₁−𝜎₂𝐿₂4𝜇²₂1)(𝑅𝑑𝜀+𝑑𝑅2𝑑𝑉₁(𝑅))+𝜎𝑅𝑑𝑅₂𝐿₁−𝜎₁𝐿₂2𝜇₁𝜇₂1𝑅𝑑𝑉₂+𝜎𝑑𝑅¹𝜎₂𝑉₄(𝑅)12𝜇₁𝜇₂+(3𝜎₁𝑅𝜎₂𝑅−𝜎₁𝜎₂𝑅²)𝑉₃12𝜇₁𝜇₂𝑅². (3.1)

Spin-spin interaction appears in 𝑊_𝜎 in (2.7) which can be written as 𝑔exp{−²2^𝑠1₀𝑑𝜏₁^𝑠2₀𝑑𝜏₂⟨𝝈⁽¹⁾𝐁𝝈⁽¹⁾𝐄𝝈⁽¹⁾𝐄𝝈⁽¹⁾𝐁_𝑧1𝝈⁽²⁾𝐁𝝈⁽²⁾𝐄𝝈⁽²⁾𝐄𝝈⁽²⁾𝐁_𝑧2⟩}, (3.2) where 𝑧_1,2=𝑧(𝜏_1,2), and spin-orbit interaction arises in (2.19) from the products ⟨𝜎_𝜇𝜈𝐹_𝜇𝜈𝑑𝑠_𝜌𝜆𝐹_𝜌𝜆⟩.

It is clear that the resulting interaction will be of matrix form (2×2)×(2×2) (not accounting for Pauli matrices). If one keeps only diagonal terms in 𝜎_𝜇𝜈𝐹_𝜇𝜈 (as the leading terms for large 𝜇_𝑖≈𝑀) then one can write for the spin-dependent potentials the representation of the Eichten-Feinberg form (3.1).

At this point one should note that the term with 𝑑𝜀/𝑑𝑅 in (3.1) was obtained from the diagonal part of the matrix (𝑚−𝐷)𝜎_𝜇𝜈𝐹_𝜇𝜈, namely, as product 𝑖𝜎_𝑘𝐷_𝑘⋅𝜎_𝑖𝐸_𝑖; see [73, 74, 77, 78] for details of derivation, while all other potentials 𝑉_𝑖, 𝑖=1,2,3,4 are proportional to FCs ⟨𝐻_𝑖Φ𝐻_𝑘Φ⟩. One can relate FCs of colorelectric and colormagnetic fields to 𝐷^𝐸, 𝐷^𝐸₁, 𝐷^𝐻, 𝐷^𝐻₁ defined by (2.28) with the following result: 1𝑅𝑑𝑉₁𝑑𝑅=−^∞_−∞𝑑𝜈^𝑅₀𝑑𝜆𝑅𝜆(1−𝑅)𝐷^𝐻1(𝜆,𝜈),𝑅𝑑𝑉₂=𝑑𝑅^∞_−∞𝑑𝜈^𝑅₀𝜆𝑑𝜆𝑅²[𝐷^𝐻(𝜆,𝜈)+𝐷^𝐻₁(𝜆,𝜈)+𝜆²𝜕𝐷^𝐻₁𝜕𝜆²𝑉],₃=−^∞_−∞𝑑𝜈𝑅²𝜕𝐷^𝐻₁(𝑅,𝜈)𝜕𝑅²,𝑉₄=^∞_−∞𝑑𝜈(3𝐷^𝐻(𝑅,𝜈)+3𝐷^𝐻₁(𝑅,𝜈)+2𝑅²𝜕𝐷^𝐻₁𝜕𝑅²1),𝑅𝑑𝜀(𝑅)=𝑑𝑅^∞_−∞𝑑𝜈^𝑅₀𝑑𝜆𝑅[𝐷^𝐸(𝜆,𝜈)+𝐷^𝐸₁(𝜆,𝜈)+(𝜆²+𝜈²)𝜕𝐷^𝐸₁𝜕𝜈²]. (3.3)

One can check that Gromes relation [43, 44] acquires the form [79]

𝑉^′₁(𝑅)+𝜀^′(𝑅)−𝑉^′₂(𝑅)=^∞_−∞𝑑𝜈[^𝑅₀𝑑𝜆(𝐷^𝐸(𝜆,𝜈)−𝐷^𝐻1(𝜆,𝜈))+2𝑅(𝐷^𝐸₁(𝑅)−𝐷^𝐻₁(𝑅))]. (3.4)

For 𝑇=0, when 𝐷^𝐸=𝐷^𝐻, 𝐷^𝐸₁=𝐷^𝐻₁, the Gromes relations are satisfied identically; however, for 𝑇>0 electric and magnetic correlators are certainly different and Gromes relation is violated, as one could tell beforehand; since for 𝑇>0, the Euclidean 𝑂(4) invariance is violated.

4. Gluelumps and FC

In this section we establish a connection of FC 𝐷(𝑧) and 𝐷₁(𝑧) with the gluelump Green's functions.

To proceed, one can use the background field formalism [61–65, 81–83], where the notions of valence gluon field 𝑎_𝜇 and background field 𝐵_𝜇 are introduced, so that total gluonic field 𝐴_𝜇 is written as 𝐴_𝜇=𝑎_𝜇+𝐵_𝜇. (4.1) The main idea we are going to adopt here is suggested in the second paper of [39, 40]. To be self-contained, let us explain it here in simple form. First, we single out some color index 𝑎 and fix it at a given number. Then in color components we have, by definition, that 𝐴^𝑐_𝜇=𝛿^𝑎𝑐𝑎^𝑎_𝜇+(1−𝛿^𝑎𝑐)𝐵^𝑐_𝜇, (4.2) and there is summation neither over 𝑎 in the first term nor over 𝑐 in the second. As a result, the integration measure factorizes _𝑐𝒟𝐴^𝑐_𝜇=𝒟𝑎^𝑎_{𝜇𝑐≠𝑎}𝒟𝐵^𝑐_𝜇, (4.3) so one first averages over the fields 𝐵^𝑐_𝜇, and after that one integrates over the fields 𝑎^𝑎_𝜇. The essential point is that the integration over 𝒟𝐵^𝑐_𝜇 provides colorless adjoint string attached to the gluon 𝑎^𝑎_𝜇, which keeps the color index “𝑎’’ unchanged in the course of propagation of gluon 𝑎 in background made of 𝐵. This is another way of saying that gluon field of each particular color enters QCD Lagrangian quadratically. Despite the fact that such separation of the measure is a kind of trick, the formation of adjoint string is the basic physical mechanism behind this background technic, and it is related to the properties of gluon ensemble: even for 𝑁_𝑐=3, one has one color degree of freedom 𝑎^𝑎_𝜇 and 7 fields 𝐵^𝑐_𝜇; also confining string is a colorless object, and therefore, the singled-out adjoint index “𝑎’’ can be preserved during interaction process of the valence gluon 𝑎^𝑎_𝜇 with the background (𝐵^𝑐_𝜇). These remarks to be used in what follows make explicit the notions of the valence gluon and background field.

We assume the background Feynman gauge 𝐷_𝜇𝑎_𝜇=0 and assign field transformations as follows: 𝑎_𝜇→𝑈⁺𝑎_𝜇𝑈,𝐵_𝜇→𝑈⁺(𝐵_𝜇+𝑖𝑔𝜕_𝜇)𝑈. (4.4) As a result, the parallel transporter Φ(𝑥,𝑦)=𝑃exp𝑖𝑔∫^𝑥_𝑦𝐵_𝜇(𝑧)𝑑𝑧 keeps its transformation property, and every insertion of 𝑎_𝜇 between Φ transforms gauge covariantly. In what follows we will assume that Φ is made entirely of the field 𝐵_𝜇. Note that in the original cluster expansion of the W-loop 𝑊(𝐵+𝑎) in FC [81–83] 𝑊(𝐵+𝑎)=⟨Tr𝑃exp𝑖𝑔_𝐶(𝐵_𝜇+𝑎_𝜇)𝑑𝑧_𝜇⟩, (4.5) it does not matter whether (𝐵_𝜇+𝑎_𝜇) or 𝐵_𝜇 enters Φ^′s since those factors cancel in the sum. Thus we assume from the beginning that Φ^′s are not renormalized, since such renormalization is unphysical anyway according to what have been said above.

As for renormalization having physical meaning, it is known [68, 69] to reduce to the charge renormalization and renormalization of perimeter divergences on the contour 𝐶. For static quarks, the latter reduces to the mass renormalization.

In background field formalism, one has an important simplification: the combination 𝑔𝐵_𝜇 is renorminvariant [61–65, 68, 69], and so is any expression made of 𝐵_𝜇 only. This point is important for comparison of FC with those of the lattice correlators in [45, 46]. Renormalization constants for field strengths 𝑍_𝑏 used there account for finite size of plaquettes and they are similar to the lattice tadpole terms.

Now we turn to the analytic calculation of FC in terms of gluelump Green's function. To this end, we insert (4.1) into (2.19) and have for 𝐹_𝜇𝜈(𝑥) the following: 𝐹_𝜇𝜈(𝑥)=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇−𝑖𝑔[𝐴_𝜇,𝐴_𝜈]=𝐷_𝜇𝑎_𝜈−𝐷_𝜈𝑎_𝜇−𝑖𝑔[𝑎_𝜇,𝑎_𝜈]+𝐹^(𝐵)_𝜇𝜈. (4.6) Here, the term 𝐹^(𝐵)_𝜇𝜈 contains only the field 𝐵^𝑏_𝜇. It is clear that when one averages over field 𝑎^𝑎_𝜇 and sums finally over all color indices 𝑎, one actually exploits all the fields with color indices from 𝐹^(𝐵)_𝜇𝜈, so that the term 𝐹^(𝐵)_𝜇𝜈 can be omitted, if summing over all indices 𝑎 is presumed to be done at the end of calculation.

As a result, 𝐷_{𝜇𝜈,𝜆𝜎} can be written as 𝐷_{𝜇𝜈,𝜆𝜎}(𝑥,𝑦)=𝐷⁽⁰⁾_{𝜇𝜈,𝜆𝜎}+𝐷⁽¹⁾_{𝜇𝜈,𝜆𝜎}+𝐷⁽²⁾_{𝜇𝜈,𝜆𝜎}, (4.7) where the superscripts 0, 1, 2 refer to the power of 𝑔 coming from the term 𝑖𝑔[𝑎_𝜇,𝑎_𝜈].

We can address now an important question about the relation between FCs and gluelump Green's functions. We begin with 1-gluon gluelump, whose Green's function reads 𝐺^(1𝑔)_𝜇𝜈(𝑥,𝑦)=⟨Tr_𝑎𝑎_𝜇(𝑥)Φ(𝑥,𝑦)𝑎_𝜈(𝑦)⟩. (4.8) According to (4.4), 𝐺^(1𝑔)_𝜇𝜈(𝑥,𝑦) is a gauge-invariant function.

As shown in [41], the first term in (4.7) is connected to the functions 𝐷^𝐸₁, 𝐷^𝐻₁ and it can be written as follows: 𝐷⁽⁰⁾_{𝜇𝜈,𝜆𝜎}𝑔(𝑥,𝑦)=²2𝑁²_𝑐{𝜕𝜕𝑥_𝜇𝜕𝜕𝑦_𝜆𝐺^(1𝑔)_𝜈𝜎(𝑥,𝑦)+perm}+Δ⁽⁰⁾_{𝜇𝜈,𝜆𝜎}, (4.9) where Δ⁽⁰⁾_{𝜇𝜈,𝜆𝜎} contains contribution of higher FCs, which we systematically discard.

On the other hand, one can find 𝐺^(1𝑔)_𝜇𝜈(𝑥,𝑦) from the expression [47, 48] written as 𝐺^(1𝑔𝑙)_𝜇𝜈(𝑥,𝑦)=Tr_𝑎^∞₀𝑑𝑠(𝐷𝑧)_𝑥𝑦exp(−𝐾)⟨𝑊^𝐹_𝜇𝜈(𝐶_𝑥𝑦)⟩, (4.10) where the spin-dependent W-loop is 𝑊^𝐹_𝜇𝜈(𝐶_𝑥𝑦)=𝑃𝑃_𝐹{exp(𝑖𝑔𝐵_𝜆𝑑𝑧_𝜆)exp𝐹}_𝜇𝜈, (4.11) and the gluon spin factor is exp𝐹≡exp(2𝑖𝑔∫^𝑠₀𝑑𝜏𝐹_𝐵(𝑧(𝜏))) with 𝐹_𝐵 made of the background field 𝐵_𝜇 only.

Analogous expression can be constructed for Green's function of 2-gluon gluelump. It is given by the following expression : 𝐺^(2𝑔𝑙)_{𝜇𝜈,𝜆𝜎}(𝑥,𝑦)=⟨Tr_𝑎(𝑓^𝑎𝑏𝑐𝑓^𝑑𝑒𝑓𝑎^𝑎_𝜇(𝑥)𝑎^𝑏_𝜈(𝑥)𝑇^𝑐Φ(𝑥,𝑦)𝑇^𝑓×𝑎^𝑑_𝜆(𝑦)𝑎^𝑒_𝜎(𝑦))⟩≡𝑁²_𝑐(𝑁²_𝑐−1)(𝛿_𝜇𝜆𝛿_𝜈𝜎−𝛿_𝜇𝜎𝛿_𝜈𝜆)𝐺^(2𝑔𝑙)(𝑧). (4.12) At small, distances 𝐺^(2𝑔𝑙)(𝑥,𝑦) is dominated by the perturbative expansion terms 𝐺^(2𝑔𝑙)1(𝑧)∼𝑧⁴, (4.13) however, as we already discussed in details, all these perturbative terms are canceled by those from higher FCs (triple, quartic, etc.); therefore, expansion in fact starts with np terms of dimension four.

To identify the np contribution to 𝐺^(2𝑔𝑙)(𝑧), we rewrite it as follows: 𝐺^(2𝑔𝑙)(𝑧)=^∞₀𝑑𝑠₁^∞₀𝑑𝑠₂(𝐷𝑧₁)_0𝑥(𝐷𝑧₂)_0𝑥Tr𝑊_Σ(𝐶₁,𝐶₂), (4.14) where the two-gluon gluelump W-loop 𝑊_Σ(𝐶₁,𝐶₂) is depicted in Figure 6 and can be written as (in the Gaussian approximation) Tr𝑊_Σ(𝐶₁,𝐶₂1)=exp{−2_𝑠𝑑𝜋_𝜇𝜈(𝑢)𝑑𝜋_𝜆𝜎(𝑣)⟨𝐹_𝜇𝜈(𝑢)Φ𝐹_𝜆𝜎(𝑣)⟩}, (4.15) and the total surface 𝑆 consists of 3 pieces, as shown in Figure 4, as follows: 𝐹_𝜇𝜈𝑑𝜋_𝜇𝜈(𝑢)=𝐹_𝜇𝜈𝑑𝑠_𝜇𝜈(𝑢)−2𝑖𝑔𝑑𝜏(𝐹(𝑢)), (4.16) where (𝐹) has Lorentz tensor and adjoint color indices 𝐹^𝑎𝑏_𝑖𝑗 and lives on gluon trajectories, that is, on the boundaries of 𝑆_𝑖. In full analogy with (4.9), we have that 𝑔𝐷(𝑧)=⁴(𝑁²_𝑐−1)2𝐺^(2𝑔𝑙)(𝑧). (4.17)

The crucial point is that Green's functions (4.9), (4.14) can be calculated in terms of the same FCs 𝐷(𝑧), 𝐷₁(𝑧). Indeed one has that 𝐺^(𝑤)(𝑧)=⟨𝑓|exp(−𝐻_𝑤|𝑧|)|𝑖⟩, (4.18) where the index 𝑤 stays for 1-gluon or 2-gluon gluelump Hamiltonians. The latter are expressed via the same FCs 𝐷(𝑧), 𝐷₁(𝑧) (see [84–86] and references therein): 𝐻_𝑤=𝐻₀[𝜇]+Δ𝐻_𝐿[𝜇,𝜈]+Δ𝐻_Coul[𝐷₁]+Δ𝐻_string[𝐷,𝜈], (4.19) where the last term 𝐻_string[𝐷,𝜈] depends on 𝐷(𝑧) via the adjoint string tension 𝜎_adj=94^∞₀𝑑²𝑧𝐷(𝑧), (4.20) and Hamiltonian depends on einbein fields 𝜇 and 𝜈. The term corresponding to perimeter Coulomb-like interaction Δ𝐻_Coul[𝐷₁] depends on 𝐷₁(𝑧) (compare with (2.31), (3.1)).

The self-consistent regimes correspond to different asymptotics of the solutions to these equations. In Coulomb phase of a gauge theory, both 𝐻_1𝑔, 𝐻_2𝑔 exhibit no mass gap, that is, large-𝑧 asymptotic of (4.18) is power like. The function 𝐷(𝑧) vanishes in this phase and W-loop obeys perimeter law. In the confinement phase realized in Yang-Mills theory at low temperatures, a typical large-𝑧 pattern is given by 𝐷(𝑧),𝐷₁(𝑧)∼exp(−|𝑧|𝜆_𝑖), (4.21) that is, there is a mass gap for both 𝐻_1𝑔, 𝐻_2𝑔.

Confining solutions are characterized by W-loops obeying area law. In other words, Hamiltonians expressed in terms of interaction kernels depending on 𝐷(𝑧), 𝐷₁(𝑧) exhibit mass gap if these kernels are confining. On the other hand, the same mass gap plays a role of inverse correlation length of the vacuum. This should be compared with well-known mean-field technique. In our case the role of mean-field is played by quadratic FC, which develops nontrivial Gaussian term 𝐷(𝑧). Notice that the exponential form of its large distance asymptotics exp(−|𝑧|/𝜆) (and not, let's say, exp(−𝑧²/𝜆²)) is dictated by spectral expansion of the corresponding Green's function at large distances.

Full solution of the above equations is a formidable task not addressed by us here. Instead, as a necessary prerequisite, we check below different asymptotic regimes and demonstrate self-consistency of the whole picture.

We begin with small distance region. The np part of contribution to 𝐷₁(𝑧) at small distances comes from two possible sources: the area law term (first exponent and the exp𝐹 term in (4.11)). Both contributions are depicted in Figures 1 and 2, respectively. We will disregard the term exp𝐹 in this case, since for the one-gluon gluelump it does not produce hyperfine interaction and only gives rise to the np shift of the gluon mass, which anyhow is eliminated by the renormalization (see the appendix for more details). This is in contrast to the two-gluon gluelump Green's function generating 𝐷^𝐸, 𝐷^𝐻, where the hyperfine interaction between two gluons is dominating at small distances.

Figure 1 

Figure 2 

As a result, (4.8) without the exp𝐹 term yields 𝐷₁(𝑧)=4𝐶₂(𝑓)𝛼_𝑠𝜋1𝑧⁴+𝑔²𝐺12². (4.22) It is remarkable that the sign of the np correction is positive.

At large distances, one can use the gluelump Hamiltonian for one-gluon gluelump from [30] to derive the asymptotics [41] 𝐷₁(𝑧)=2𝐶₂(𝑓)𝛼_𝑠𝑀₀⁽¹⁾𝜎_adj𝑒|𝑧|^{−𝑀0(1)|𝑧|},|𝑧|𝑀₀⁽¹⁾≫1, (4.23) where 𝑀₀⁽¹⁾=(1.2÷1.4) GeV for 𝜎_𝑓=0.18 GeV2 [29, 30].

We now turn to the FC 𝐷(𝑧) as was studied in [41]. The relation (4.17) connects 𝐷(𝑧) to the two-gluon gluelump Green's function, studied in [30] at large distances. Here we need its small-𝑧 behavior and we will write it in the form 𝐺^(2𝑔𝑙)(𝑧)=𝐺_𝑝^(2𝑔𝑙)(𝑧)+𝐺^(2𝑔𝑙)_𝑛𝑝(𝑧), (4.24) where 𝐺_𝑝^(2𝑔𝑙)(𝑧) contains purely perturbative contributions which are subtracted by higher-order FCs, while 𝐺^(2𝑔𝑙)_𝑛𝑝(𝑧) contains np and possible perturbative-np interference terms. We are interested in the contribution of the FC ⟨𝐹𝐹⟩ to 𝐺^(2𝑔𝑙)(𝑧), when 𝑧 tends to zero.

One can envisage three types of contributions as follows.

(a) The first one is due to product of surface elements 𝑑𝑠_𝜇𝜈𝑑𝑠_𝜆𝜎, which gives for small surface the factor exp(−𝑔²⟨𝐹𝐹⟩𝑆²_𝑖/24𝑁_𝑐), similar to the situation discussed for 𝐷₁. This is depicted in Figure 4.

(b) The second one is contribution of the type 𝑑𝜏₁𝑑𝜏₂⟨𝐹𝐹⟩, where 𝑑𝜏 and 𝑑𝜏₂ belong to different gluon trajectories, 1 and 2, as depicted in in Figure 5. This is the hyperfine gluon-gluon interaction, taken into account in [30] in the course of the gluelump mass calculations (regime of large 𝑧); however, in that case mostly the perturbative part of ⟨𝐹𝐹⟩ contributes (due to 𝐷₁). Here we keep in ⟨𝐹𝐹⟩ only the 𝑁𝑃 part and consider the case of small 𝑧.

(c) Again the third is the term 𝑑𝜏^𝑖𝑑𝜏_𝑗, but now 𝑖=𝑗. This is actually a part of the gluon selfenergy correction, which should be renormalized to zero, when all (divergent) perturbative contributions are added. As we argued in Appendix of [41], we disregard this contribution here, as well as in the case of 𝐷₁(𝑧) (one-gluon gluelump case). Now we treat both contributions (a) and (b).

(a)In line with the treatment of 𝐷₁, (4.22), one can write the term representing two-gluon gluelump as two nearby one-gluon gluelumps which yield for 𝐷(𝑧) the following: Δ𝐷_𝑎𝑔(𝑧)=−⁴𝑁_𝑐𝐺₂4𝜋². (4.25)(b)In this case one should consider the diagram given in Figure 5. which yields the answer (for details see the appendix) 𝐺_𝑏^(2𝑔𝑙)(𝑧)=4𝑁²_𝑐𝑁²_𝑐𝑑−1⁴𝑤𝑑⁴𝑤^′𝐷(𝑤−𝑤^′)(4𝜋²)⁴(𝑤−𝑧)²𝑤²(𝑤^′−𝑧)²𝑤^′2, (4.26) which contributes to 𝐷(𝑧) as Δ𝐷_𝑏(𝑧)=2𝑁²_𝑐𝑔⁴ℎ(𝑧), (4.27) where at small 𝑧, ℎ(𝑧)≈(𝐷(𝜆₀)/64𝜋⁴)log²(𝜆₀√𝑒/𝑧)and 𝜆₀ is of the order of correlation length 𝜆.

At large distances, one uses the two-gluon gluelump Hamiltonian as in [29] and finds the corresponding spectrum and wave functions; see [29] and Appendix 5 of [41] for details. As a result, one obtains in this approximation 𝑔𝐷(𝑧)=⁴(𝑁²_𝑐−1)20.1𝜎²_𝑓𝑒^{−𝑀0(2)|𝑧|},𝑀₀⁽²⁾|𝑧|≫1, (4.28) where 𝑀₀⁽²⁾ is the lowest two-gluon gluelump mass found in [30] to be about 𝑀₀⁽²⁾=(2.5÷2.6) GeV.

We will discuss the resulting properties of 𝐷(𝑥) and 𝐷₁(𝑥) in the next section.

5. Discussion of Consistency

We start with the check consistency for 𝐷(𝑧). As is shown above, 𝐷(𝑧) has the following behavior at small 𝑧∶𝐷(𝑧)≈−4𝑁_𝑐𝛼²_𝑠(𝜇(𝑧))𝐺₂+𝑁²_𝑐𝛼²_𝑠(𝜇(𝑧))2𝜋²𝐷(𝜆₀)log²(𝜆₀√𝑒𝑧). (5.1) Since 𝛼_𝑠(𝜇(𝑧))∼2𝜋/𝛽₀log(Λ𝑧)⁻¹, the first term is subleading at 𝑧→0 and the last term on the r.h.s. tends to a constant 𝑁𝐷(0)=^2𝑐2𝜋²𝐷(𝜆₀)(2𝜋𝛽₀)². (5.2) From (5.2) one can infer that 𝐷(0)≈0.15𝐷(𝜆₀) for 𝑁_𝑐=3, where 𝜆₀≳𝜆^𝐸. So 𝐷(𝑧) is an increasing function of 𝑧 at small 𝑧, 𝑧≲𝜆₀, and for 𝑧≫𝜆, one observes exponential falloff. The qualitative picture illustrating this solution for 𝐷(𝑧) is shown in Figure 5.

This pattern may solve qualitatively the contradiction between the values of 𝐷(0) estimated from the string tension 𝐷_𝜎(0)≃𝜎/𝜋𝜆²≈0.35 GeV4 and the value obtained in naive way from the gluon condensate 𝐷_𝐺2(0)=(𝜋²/18)𝐺₂≈(0.007÷0.012) GeV4. One can see that 𝐷_𝜎(0)≈(30÷54)𝐷_𝐺2(0). This seems to be a reasonable explanation of the mismatch discussed in Section 1.

As shown in [41], the large distance exponential behavior is self-consistent, since (assuming that it persists for all 𝑧, while small-𝑧 region contributes very little) from the equality 𝜎=𝜋𝜆²𝐷_𝜎(0), comparing with (4.28), one obtains 0.1⋅8𝜋²𝛼²_𝑠(𝑁²_𝑐−1)𝜎²_𝑓=𝜎_𝑓𝜋𝜆², (5.3) where in 𝛼_𝑠(𝜇) the scale 𝜇 corresponds roughly to the gluelump average momentum (inverse radius) 𝜇₀≈1 GeV. Thus (5.3) yields 𝛼_𝑠(𝜇₀)≈0.4 which is in reasonable agreement with 𝛼_𝑠 from other systems [87].

We end this section with discussion of three points as follows.

(1) 𝐷(𝑧) and 𝐷₁(𝑧) have been obtained here in the leading approximation, when gluelumps of minimal number of gluons contribute 2 for 𝐷(𝑧) and 1 for 𝐷₁(𝑧). In the higher orders of 𝑂(𝛼_𝑠), one has an expansion of the type 𝐷(𝑧)=𝐷^(2𝑔𝑙)(𝑧)+𝑐₁𝛼³_𝑠𝐷^(1𝑔𝑙)(𝑧)+𝑐₂𝛼³_𝑠𝐷^(3𝑔𝑙)𝐷(𝑧)+⋯,₁(𝑧)=𝐷^(1𝑔𝑙)(𝑧)+𝑐^′₁𝛼³_𝑠𝐷^(2𝑔𝑙)(𝑧)+⋯. (5.4) Hence the asymptotic behavior for 𝐷(𝑧) will contain exponent of 𝑀₀⁽¹⁾|𝑧| too, but with a small preexponent coefficient.

(2) The behavior of 𝐷(𝑧), 𝐷₁^(𝑛𝑝)(𝑧) at small 𝑧 is defined by NP terms of dimension four, which are condensate 𝐺₂ as in (2.11) and the similar term from the expansion of exp𝐹, namely, ⟨exp𝐹⟩=1+4𝑔^2𝑠₀𝑑𝜏^𝑠′₀𝑑𝜏^′⟨𝐹(𝑢(𝜏))𝐹(𝑢(𝜏^′))⟩+⋯, (5.5) therefore, one does not encounter mixed terms like 𝑂(𝑚²/𝑧²); however, if one assumes that expansion of ⟨𝐹(0)𝐹(𝑧)⟩ starts with terms of this sort as it was suggested in [72], then one will have a self-consistent condition for the coefficient in front of this term.

(3) To study the difference between 𝐷^𝐸₁, 𝐷^𝐻₁ at 𝑇≤𝑇_𝑐, one should look at (4.9) and compare the situation, when 𝜇=𝜆=4, 𝜈=𝜎=𝑖 and take 𝑧=𝑥−𝑦 along the 4th axis (for 𝐷^𝐸₁), while for 𝐷^𝐻₁ one takes 𝜇=𝜆=𝑖, 𝜈=𝜎=𝑘, and the same 𝑧. One can see that in both cases one ends up with the one-gluon gluelump Green's function 𝐺_𝜇𝜈 (4.10) which in the lowest (Gaussian) approximation is the same 𝐺_𝜇𝜈=𝛿_𝜇𝜈𝑓(𝑧), and 𝑓(𝑧) is the standard lowest mass (and lowest angular momentum) Green's function.

Hence 𝐷^𝐸₁=𝐷^𝐻₁ in this approximation, and for 𝑇=0, this is an exact relation, as discussed above. Therefore,

𝜆₁=𝜆^𝐸₁=𝜆^𝐻₁=1𝑀₀⁽¹⁾,𝑀₀⁽¹⁾≈1.2÷1.4GeV,𝜆₁≅0.2÷0.15Fm. (5.6) The value 𝑀₀⁽¹⁾ in (5.6) is taken from the calculations in [30]. The same is true for 𝐷^𝐸, 𝐷^𝐻, as it is seen from (4.12), where 𝐺^(2𝑔𝑙) corresponds to two-gluon subsystem angular momentum 𝐿=0 independently of 𝜇𝜈, 𝜆𝜎.

Hence one obtains 𝜆≡𝜆^𝐸=𝜆^𝐻=1𝑀₀⁽²⁾,𝜆≅0.08Fm,𝑀₀⁽²⁾≈2.5GeV, (5.7) where the value 𝑀₀⁽²⁾≈2.5GeV is taken from [30].

6. Conclusions

We have derived, following the method of [61–65], the expressions for FC 𝐷(𝑧), 𝐷₁(𝑧) in terms of gluelump Green's functions. This is done in Gaussian approximation. The latter are calculated using Hamiltonian where np dynamics is given by 𝐷^𝑛𝑝(𝑧), 𝐷₁^𝑛𝑝(𝑧). In this way one obtains self-coupled equations for these functions, which allow two types of solutions: (1) 𝐷^𝑛𝑝(𝑧)=0, 𝐷₁^𝑛𝑝(𝑧)=0, that is, no np effects at all; (2) 𝐷^𝑛𝑝(𝑧), 𝐷₁^𝑛𝑝(𝑧) are nonzero and defined by the only scale, which should be given in QCD, for example, string tension 𝜎 or Λ_QCD. All other quantities are defined in terms of these basic ones. We have checked consistency of self-coupled equations at large and small distances and found that to the order 𝒪(𝛼_𝑠) no mixed perturbative-np terms appear. The function 𝐷₁(𝑧) can be represented as a sum of perturbative and np terms, while 𝐷(𝑧) contains only np contributions.

We have found a possible way to explain the discrepancy between the average values of field strength taken from 𝐺₂ and from 𝜎 by showing that 𝐷(𝑧) has a local minimum at 𝑧=0 and grows at 𝑧∼𝜆. Small value of 𝜆 and large value of the gluelump mass 𝑀_𝑔𝑙=1/𝜆≈2.5GeV explain the lattice data for 𝜆≈0.1Fm. Thus the present paper argues that relevant degrees of freedom ensuring confinement are gluelumps, described self-consistently in the language of FCs.

Appendix

Nonperturbative Contributions to 𝐷₁(𝑧) and 𝐷(𝑧) at Small 𝑧

We start with 𝐷₁(𝑧) expressed via gluelump Green's function in (4.8), (4.10) 𝐺_𝜇𝜈(𝑧)=𝛿_𝜇𝜈𝑓(𝑧).

The leading NP behavior at small 𝑧, proportional to ⟨𝐹(𝑥)𝐹(𝑦)⟩, is obtained from the amplitudes, shown in Figures 1 and 2. For the amplitude of Figure 1, one can use the vacuum average of the W-loop (2.11) in the limit of small contours and neglect the factor exp𝐹 in (4.11). One obtains the adjoint loop ⟨𝑊(𝐶_𝑥𝑦𝑔)⟩=exp(−²⟨(𝐹^𝑎_𝜇𝜈(0)²)⟩24𝑁_𝑐𝛾𝑆²), (A.1) where 𝛾=𝐶₂(adj)/𝐶₂(𝑓)=2𝑁²_𝑐/(𝑁²_𝑐−1) (original derivation in [41] referred to the fundamental loop). Proceeding as in Appendix 2 of [41], one obtains the NP correction Δ𝐷₁ as Δ𝐷_1𝑎=𝐶₂𝑔(adj)²12𝑁_𝑐𝐺₂=𝑔²𝐺12², (A.2) where we have introduced the standard definition [17, 18] 𝐺₂≡(𝛼_𝑠/𝜋)⟨𝐹^𝑎_𝜇𝜈(0)𝐹^𝑎_𝜇𝜈(0)⟩. One can check that Δ𝐷₁=𝒪(𝑁⁰_𝑐).

We turn now to the amplitude of Figure 2. The corresponding 𝑓(𝑥) for 𝑥→0 can be written as follows: Δ𝑓_1𝑏(𝑥)=𝑑⁴𝑢𝑑⁴𝑣𝐺(0,𝑢)𝐺(𝑢,𝑣)𝐺(𝑣,𝑥)4𝜋²𝑁²_𝑐𝐺₂, (A.3) where the gluon Green's function 𝐺(𝑥,𝑦) for small |𝑥−𝑦| can be replaced by the perturbative part: 𝐺(𝑥,𝑦)→𝐺₀(𝑥,𝑦)=1/4𝜋²(𝑥−𝑦)².

Taking into account that Δ𝐷₁(𝑥)=−(2𝑔²/𝑁²_𝑐)(𝑑/𝑑𝑥²)Δ𝑓(𝑥), one obtains for 𝑥→0 the following: Δ𝐷_1𝑏(𝑥)=2𝑔²𝐺₂𝐼(𝑥), (A.4) where we have defined 𝑑𝐼(𝑥)=−𝑑𝑥²𝑑⁴𝑢𝑑⁴𝑣(4𝜋²)²𝑢²(𝑢−𝑣)²(𝑣−𝑥)². (A.5)

The integral 𝐼(0) diverges at small 𝑣 and large 𝑢,𝑣. The latter divergence is removed since at large arguments 𝐺(𝑥,𝑦) is damped by confining force, keeping the propagating gluon nearby the 4th axis 𝑥₄=𝑦₄=0 where the static gluonic source resides. One can estimate 𝐼(𝑥→0)∼𝒪((ln(𝜆²/𝑥²))^𝛼). Since 𝐺₂ is connected to 𝐷, 𝐷₁, then 𝐺₂=3𝑁_𝑐𝜋²(𝐷^𝐸(0)+𝐷^𝐸₁(0)+𝐷^𝐻(0)+𝐷^𝐻₁(0)), (A.6) and one can see in (A.4) that coefficients of 𝐷^𝐸₁ on both sides of (A.4) have the same order of magnitude and the same sign, suggesting a self-consistency on this preliminary level. However, as we argued before and in [41], this contribution is actually gluon mass renormalization, which is zero. This is especially clear when ⟨𝐹𝐹⟩ in Figure 2 is replaced by its perturbative part.

We turn now to the most important case of the confining FC 𝐷^𝐸(𝑥). The corresponding two-gluon gluelump amplitude is depicted in Figure 3, and the 𝑛𝑝 contributions to 𝐷^𝐸(𝑥) are given in Figures 4 and 5.

Figure 3 

Figure 4 

Figure 5 

Figure 6 

We start with the amplitude of Figure 4, which can be represented as a doubled diagram of Figure 1, and therefore, the contribution to the two-gluon gluelump Green's function 𝐺^(2𝑔𝑙)(𝑥) will be 𝐺^(2𝑔𝑙)1(𝑥)=(4𝜋²𝑥²)²−𝛾𝐺₂24𝑁_𝑐(4𝜋²). (A.7) Now using relation (2.11), one obtains for the np contribution of Figure 4Δ𝐷_𝑎𝑔(𝑥)=⁴(𝑁²_𝑐−1)2Δ𝐺^(2𝑔𝑙)𝑔(𝑥)=−⁴𝑁_𝑐4𝜋²𝐺₂. (A.8) Note that this contribution is finite at 𝑥→0, however, with the negative sign. We now turn to the np contribution of Figure 5, which can be obtained from (4.12) inserting there the product of operators 𝐹_𝑓𝑔(𝑤)𝑑⁴𝑤𝐹_{𝑓′𝑔′}(𝑤^′)𝑑⁴𝑤^′, where 𝑓𝑔(𝑓^′𝑔^′) are adjoint color indices, 𝐹_𝑓𝑔=𝐹^𝑎𝑇^𝑎_𝑓𝑔=𝐹^𝑎(−𝑖)𝑓_𝑎𝑓𝑔. Using the formulas 𝑓_𝑎𝑏𝑐𝑓_{𝑎′𝑏𝑐}=𝑁_𝑐𝛿_𝑎𝑎′,𝑓_𝑎𝑏𝑐𝑓_𝑎𝑑𝑒𝑓_𝑏𝑑𝑓𝑓_𝑐𝑒𝑔=𝑁²_𝑐2𝛿_𝑓𝑔, (A.9) one arrives at the expression Δ𝐷_𝑏(𝑥)=2𝑁²_𝑐𝑔⁴𝑑ℎ(𝑥),ℎ(𝑥)=⁴𝑤𝑑⁴𝑤^′𝐷(𝑤−𝑤^′)𝐺(𝑤)𝐺(𝑤^′)𝐺(𝑤−𝑥)𝐺(𝑤^′−𝑥), (A.10) where 𝐺(𝑦) is the gluon Green's function in the two-gluon gluelump; the gluon is connected at large distances by two strings to another gluon and to the static gluon source; see Figure 5. At small 𝑥 (we take it along axis 4 for convenience), the integral in ℎ(𝑥) grows when 𝐺(𝑦) is close to the 4th axis and becomes the free gluon 𝐺(𝑦)→𝐺₀(𝑦)=1/4𝜋²𝑦². As a result, one obtains ℎ(𝑥)→ℎ₀𝑑(𝑥)=⁴𝑤𝑑⁴𝑤^′(4𝜋²)⁴𝐷(𝑤−𝑤^′)𝑤²𝑤^′2(𝑤−𝑥)²(𝑤^′−𝑥)², (A.11) and one should have in mind that the integral for 𝑥→0 diverges both at small and at large 𝑤, 𝑤^′. The small 𝑤, 𝑤^′ region will give the terms 𝑂(ln(𝜆²₀/𝑥²)), and at large 𝑤, 𝑤^′, the integral is protected by the falloff of 𝐺(𝑦) at large 𝑦 due to confinement; therefore, we must imply in the integrals in (A.11) the upper limits for 𝑤, 𝑤^′ at some 𝜆₀≳1/𝜎_adj. Then introducing for 𝑤(𝐰,𝑤₄) polar coordinates |𝐰|=𝜌sin𝜃, 𝑤₄=𝜌cos𝜃 (and the same for 𝑤^′), one arrives at the integral ^𝜋₀sin²𝜃𝑑𝜃𝜌²−2𝑥𝜌cos𝜃+𝑥²=𝜋2𝜌²(𝜌≥𝑥), (A.12) or (𝜋/2𝑥²)(𝑥≥𝜌), and finally one has the estimate of contribution to (A.10) from the region of small 𝑤,𝑤^′(𝑤|𝑤|,|𝑤^′|≲𝜆₀) as ℎ(𝑥)≈𝐷(𝜆₀)4(4𝜋²)²log²(𝜆₀√𝑒𝑥). (A.13) And finally for 𝐷(𝑥), one obtains (omitting the perturbative term in (A.7)) 𝑔𝐷(𝑥)=−⁴𝑁_𝑐4𝜋²𝐺₂+𝑁²_𝑐2𝜋²𝛼²_𝑠𝐷(𝜆₀)log²(𝜆₀√𝑒𝑥). (A.14)

Note that the last term on the r.h.s. dominates at small 𝑥 and compensates the decrease of the 𝛼²_𝑠 term since 𝛼²_𝑠(𝑥)ln²(𝜆₀√𝑒𝑥)≅(4𝜋ln(𝜆₀√𝑒/𝑥)𝛽₀ln(1/𝑥²Λ²))→(4𝜋𝛽₀)²(𝑥→0). (A.15) Hence,

𝐷(𝑥<𝜆₀𝑁)≈^2𝑐2𝜋²(4𝜋𝛽₀)²𝐷(𝜆₀)<𝐷(𝜆₀). (A.16)

Acknowledgments

This work is supported by the grant for support of scientific schools NS-4961.2008.2. The work of the second author is also supported by INTAS-CERN fellowship 06-1000014-6576.