Exploring correlations in the stochastic Yang-Mills vacuum

The correlation lengths of nonperturbative-nonconfining and confining stochastic background Yang-Mills fields are obtained by means of a direct analytic path-integral evaluation of the Green functions of the so-called one- and two-gluon gluelumps. Numerically, these lengths turn out to be in a good agreement with those known from the earlier, Hamiltonian, treatment of such Green functions. It is also demonstrated that the correlation function of nonperturbative-nonconfining fields decreases with the deviation of the path in this correlation function from the straight-line one.

heavy adjoint source [14]. In Yang-Mills theory, they define the correlation lengths of the field-strengths' two-point function in the same way as, in full QCD, physically existing heavylight mesons define the correlation length of a nonlocal quark condensate ψ (x)Φ xx ′ ψ(x ′ ) [11,15] (here Φ xx ′ is a phase factor along the straight-line path, which is provided by the heavy-quark propagator). Unlike the fundamental case, the adjoint case allows for two different types of heavy-light objects -those with a single gluon, called one-gluon gluelumps, and those with two gluons, called two-gluon gluelumps (cf. Fig. 1 below). The first case is similar to the above-mentioned nonlocal quark condensate, whereas the second case is conceptually different, as it corresponds to two gluons connected together with the heavy source by three fundamental strings. Specifically, it has been shown in Ref. [12] that the Green functions of one-and two-gluon gluelumps define respectively the correlation lengths of nonperturbative-nonconfining and confining stochastic Yang-Mills fields. Furthermore, these Green functions have been explored in Ref. [12] by using respectively one-and twobody relativistic Hamiltonians with linear potentials.
In the next Section, we accomplish the full calculation of quantum-mechanical path integrals representing those Green functions, which has been started in Ref. [13]. This analysis will in particular allow us to find the above-mentioned two vacuum correlation lengths, and compare them with those obtained in Ref. [12] within the Hamiltonian approach. Such a calculation turns out to be possible by virtue of an effective parametrization of minimal areas swept out by the strings in the gluelumps. This parametrization, suggested in Ref. [16], has been successfully used there to account for confinement of virtual gluons in the polarization operator. In Section III, we summarize the main results of the paper. In Appendix A, we calculate the Green function of a one-gluon gluelump for paths deviating from the straight-line one, and show that the corresponding correlation function of nonperturbative-nonconfining background fields decreases with the deformation of the path.

II. ANALYTIC CALCULATION OF THE VACUUM CORRELATION LENGTHS
Stochastic vacuum model suggests the following parametrization for the two-point correlation function of gluonic field strengths [1]: T a is a phase factor along some path interconnecting the points x ′ and x. For the rest of the present Section, this path is chosen along the straight line. Furthermore, T a 's are the SU(N c ) generators in the fundamental representation, the average . . . is taken with respect to the Yang-Mills action 1 The functions D 1 (x) and D(x) parametrize respectively the nonperturbative-nonconfining and confining self-interactions of stochastic background fields. In this Section, we calculate D 1 (x) and D(x) analytically, using their relation to the infra-red Green functions of one-and twogluon gluelumps [12]: where C f = (N 2 c −1)/(2N c ) is the quadratic Casimir operator of the fundamental representation. Equations (2) hold at large distances, |x| ≥ O(σ , where σ f is the string tension in the fundamental representation [12]. In this infra-red regime, perturbative contributions are negligible, and the Wilson-loop averages appearing in the Green functions G(x) and S(x) exhibit the minimal-area law (cf. below). In Ref. [12], the leading infra-red asymptotes of these Green functions have been obtained in terms of the lowest eigenvalues of respectively one-and two-body Hamiltonians with the linear potential. Below, the full infra-red Green functions G(x) and S(x) will be found by means of a direct calculation of the corresponding path integrals with the minimal-area ansätze.
We start with the Green function of a one-gluon gluelump, where the minimal surface of area S min is swept out by the adjoint string of tension σ, which connects the gluon to the adjointly charged source. While the effects of non-staticity of the source are considered in Appendix A, here we assume the source to be static, i.e. evolving entirely along the x 4 -axis. This means that only the x 4 -coordinate of the point x in Eq. (3) is nonvanishing, i.e.
In what follows, we continue with the notations G(x) and S(x), expressing these functions in terms of L ≡ |x|. We calculate the path integral in Eq. (3) by approximating S min through the Cauchy-Schwarz inequality [16]: It reduces the path integral to that of a harmonic oscillator with a variable frequency. Indeed, with A > 0, and changing the proper-time variable as s → µ = L 2s , we have where Dz denotes the integration over trajectories, which start and end up at z = 0.
The kinetic term µż 2 µ 2 means, of course, that the auxiliary parameter µ can be viewed as an effective gluon mass. Furthermore, the approximate equality "≃" in Eq. (5) is understood in the sense of approaching the upper limit for S min in Eq. (4). We see then that the resulting path integral over z 4 (τ ) is that of a free particle, whereas the integral over z(τ ) is that of a harmonic oscillator. It reads where ω ≡ σ 2µL/λ is the frequency of the oscillator. Changing further the integration variable from λ to ξ ≡ σL 3/2 / √ 2µλ, we can perform the µ-integration: In the infra-red limit l ≫ 1 of interest, this integral can be evaluated analytically as follows: . (8) Here, the replacement of the lower limit of integration in the second integral by 0 was legitimate, since the saddle-point value ξ s.p. = 2l/3, in the limit l ≫ 1 at issue, is larger than 1 (and therefore the contribution of the integration region 0 < ξ < 1 to the whole integral is exponentially small). Thus, the leading result in the large-|x| limit reads By means of Eq. (2), it yields the following function D 1 (x): This expression can be compared with the one from Ref. [12], where the value M 0 ≃ 1.5 GeV was obtained from the Schrödinger equation with the linear potential. To this end, we evaluate the adjoint string tension via the so-called Casimirscaling hypothesis [17]. This hypothesis, supported both by lattice simulations [18] and analytic studies [3,19], suggests proportionality of the string tension, in a given represen- turns out to be very close to the quoted value of M 0 . Note that, in principle, a gluon is confined up to arbitrarily large distances only at N c ≫ 1, whereas at N c ∼ 1 it can be screened by other gluons. In the large-N c limit, σ → 2σ f , and the exponents in Eqs. (10) and (11) coincide numerically extremely well, since in that case √ 6σ ≃ 1.5 GeV.
We proceed now to the Green function of the two-gluon gluelump: Trying out the Cauchy-Schwarz inequality, Eq. (4), for each of the three distances separately would involve integrations over three auxiliary parameters. In order to reduce this number to just one, it is more useful to apply the Cauchy-Schwarz inequality in the form Now, so long as the common square root for the three distances is assembled, we can again apply the Cauchy-Schwarz inequality in the form of Eq. (4) with only one auxiliary integration: Similar to the one-gluon gluelump, integrations over z 4 (τ ) andz 4 (τ ) in this formula yield We observe now that, if the terms z 2 andz 2 in Eq. (13) were absent, the path integral would be that of two (mutually interacting but otherwise free) particles, which was calculated in Ref. [16]. Here, however, we have to deal with two (also mutually interacting) harmonic oscillators. Nevertheless, such a path integral Dz Dz can still be calculated. This turns out to be possible upon the diagonalization of the action by virtue of the known fact that two positively definite quadratic forms (that are, the kinetic and the potential energies) can be diagonalized simultaneously. Passing to the new integration trajectories u(τ ) and v(τ ) according to the formulae we obtain the diagonalization conditions The solution to these equations is straightforward: where we have chosen for concreteness the "+" sign in front of the last square root. Then the kinetic-and the potential-energy terms read The Jacobian stemming from the change of integration trajectories in Eq. (14) is, of course, also µ-andμ-dependent, namely Dz Dz = |1 − αβ| Du Dv = 1 + α 2 μ µ Du Dv.
Thus, we have reduced the path integral Dz Dz to the product of two well-known path integrals for non-interacting harmonic oscillators. Introducing finally the dimensionless vari- we can write down the result in the following form: where α ≡ α(ν,ν) and β ≡ β(ν,ν) are given by Eq. (15). The remaining ordinary integrations in this formula have been performed numerically. In Fig. 2, we plot the so-calculated agrees remarkably well with its value of 2.56 GeV found in Ref. [12] within the Hamiltonian approach.

III. CONCLUDING REMARKS
In this paper, we have analytically confirmed the statement that nonperturbativenonconfining and confining self-interactions of stochastic background Yang-Mills fields can have different correlation lengths. Namely, we have obtained these lengths from the Green functions of one-and two-gluon gluelumps, by means of an analytic treatment of the quantum-mechanical path integrals for one and two gluons (inter)connected by strings with the static adjoint source. The resulting inverse correlation lengths are given by Eqs. (12) and (16). They turn out to be in a remarkably good agreement with those found in Ref. [12] by means of a different, Hamiltonian, approach. We have also demonstrated that, in ac-cordance with the lattice results [5], the two-point correlation function of nonperturbativenonconfining background fields decreases with the deformation of the path interconnecting these two points. On the technical side, for the calculation of the Green function of a twogluon gluelump, a novel method has been developed of an analytic evaluation of the path integral with the common minimal surface formed by two gluons together with the adjointly charged source.
In the forthcoming studies, we plan to incorporate dynamical quarks in this theoretical framework. Their appearance can lead to the breaking of not only fundamental strings, but also of the adjoint ones -through the creation ofq-g-q hybrids. An effective reduction, due to the string-breaking, of string world sheets in the spatial directions can lead to their enhancement in the temporal direction, that is, to the increase of the vacuum correlation lengths. In this way, vacuum correlation lengths in full QCD can exceed those obtained here in pure Yang-Mills theory by a factor of 3, as suggested by the lattice simulations [4] and the phenomenology of hadronic collisions [20].
concrete form will be specified below. Accordingly, the minimal-area ansatz (4) gets modified, and takes the form The emerging path integral Dz turns out to be similar to that of Eq. (6), and reads (cf. Ref. [21]) The corresponding path-dependent Green function G w (x) generalizes Green function G(x), and can be written as [cf. Eq. (7)] The function introduced here, vanishes in the limiting case w = 0. Performing again the µ-integration exactly, and using Eq. (2) with G(x) replaced by G w (x), we obtain the path-dependent correlation function where l is defined in Eq. (7). To calculate this integral, we parametrize the path w(τ ) in the form which provides smooth approximations to step-like paths used in Ref. [5]. Namely, we consider two types of smooth paths: Each of these paths lies in the (1,4)-plane, deviating from the x 4 -axis to the maximum distance L/2, which is still compatible with L. In general, for larger deviations, one can expect cancellations of contributions stemming from the mutually backtracking pieces of the path.
Consider first the path w (1) (τ ). The corresponding function (A2) reads The ξ-integration in Eq. (A3) can be performed in the same way as in Eq.  One can then prove numerically that the absolute value of the second term on the left-hand side of Eq. (A4) exceeds the third term by at least one order of magnitude. Therefore, the third term can be disregarded compared to the second one, for the reason of smallness of f ′ (ξ). Accordingly, approximating f (ξ) by some constant f , we obtain the saddle-point value ξ s.p. = 2l √ 1 + f /3, which is much larger than 1 in the limit √ l ≫ 1 of interest.
And indeed, for ξ ≫ 1, one can see a very weak variation of the function f (ξ), which is illustrated by Fig. 3, where this function is plotted in the range ξ ∈ [5,30]. We therefore approximate f (ξ) by its limiting value at ξ ≫ 1, i.e. set f ≃ 1.23.
Then, given that ξ s.p. lies deeply inside the region (1, ∞), we can replace the lower limit of integration by 0 [cf. Eq. (8)], and obtain Comparing this expression with Eq. also exhibits a rapid levelling-off, similar to its counterpart given by Eq. (A5) [cf. Fig. 3].
The difference is that the limiting value of the function f (ξ) at ξ ≫ 1 appears in this case larger, namely f ≃ 4.86, instead of 1.23. Accordingly, the suppression factor in Eq. (A6) becomes 4 √ 1 + f ≃ 1.56, instead of 1.22. Such an increase of the factor 4 √ 1 + f is quite fast, in spite of its slow, fourth-root, functional dependence. Thus, our analysis suggests a substantial suppression of contributions to the two-point correlation function, which stem