Dynamical gluon mass in the instanton vacuum model

We are considering the modifications of gluon properties in Instanton Liquid Model (ILM) for the QCD vacuum. Re-scattering of a gluons on an instantons leads to the dynamical momentum-dependent gluon mass $M_g(q).$ First, we considered scalar"gluon", since there are no zero-modes problem and found its dynamical mass $M_s(q)$. At the typical phenomenological values of the average instanton size $\rho=1/3\,\,fm$ and average inter-instanton distance $R=1\,\, fm$ we got $M_s(0)=256\,\,MeV$. Further, we extended this approach to the real vector gluon with careful consideration of zero-modes and got $M^2_g(q)=2M^2_s(q).$ This modification of the gluon in the instanton media might be important for the heavy quarkonium physics at least.


I. INTRODUCTION
Without any doubt instantons represent a very important topologically nontrivial component of the QCD vacuum. The main parameters of the QCD instanton vacuum developed in the instanton liquid model (ILM) are the average instanton size ρ and inter-instanton distance R (see, for example, following reviews [1,2]). They were phenomenologically estimated as ρ = 1/3 fm, R = 1 fm and confirmed by theoretical variational calculations [1,2] and recent lattice simulations of the QCD vacuum [3][4][5][6][7]. In particular, the spontaneous breakdown of chiral symmetry is realized very well via the ILM [8]. Hence, instantons play a pivotal and significant role in describing the lightest hadrons and their interactions.
In the ILM the instanton induces strong interactions between light quarks and produce a large dynamical mass M , which was initially almost massless. Consequently, light quarks are bound and the pions as a pseudo-Goldstone boson appear as a result of spontaneous breakdown of chiral symmetry (SBCS).
On the other hand, the instantons from the QCD vacuum also interact with heavy quarks and are responsible for the generation of the heavy-heavy and heavy-light quark interactions with trace of the SBCS [9][10][11]. It is important to note that the packing parameter is given as ρ 4 /R 4 and it become very small (∼ 0.01) with the phenomenological values of ρ and R used. Then, the dynamical quark mass M is expressed as (packing parameter) 1/2 ρ −1 ∼ 365 MeV [1] while the instanton contribution to the heavy quark mass is given as ∆M ∼ (packing parameter)ρ −1 ∼ 70 MeV [12]. We see that these specific packing parameter dependencies explain the values of M and ∆M . These factors define the coupling between the light-light, heavy-light and heavy-heavy quarks induced by the instantons from the QCD vacuum.
The direct instanton effects mainly contribute to the intermediate region characterized by the instanton size (ρ 0.33 fm), as was studied in Ref. [13] in which the instanton effects are marginal but still important to be considered for a quantitative description of the heavy quarkonium spectra. One-gluon exchange is dominant at smaller distances. On the other hand, a size of the heavy quarkonium is small [14]. It means that heavy quarkonium properties might be sensitive to a modification of gluon properties in instanton media (ILM) induced by rescattering of gluons on the instantons.
Previously the dynamical gluon mass M g within ILM was estimated in [15] as M g ∼ 400 M eV with phenomenological values of ρ and R. However, this estimation was obtained by ignoring the gluon zero-modes problem and some SU (N c ) factors.
In this work, we aim at investigating the dynamical gluon mass within the ILM, extending the method developed in Ref. [16], where the formulae for the quark correlators were derive.

II. SCALAR "GLUON" PROPAGATOR
We start from the scalar massless field φ belonging to the adjoint representation as a real gluon. We have to find its propagator in the external classical gluon field in the ILM The action is defined as The scalar gluon-like propagator is given by There are no zero modes in ∆ −1 i = P 2 i and ∆ −1 = P 2 , which means the existence of the inverse operators ∆ i and ∆. Our aim is to find the propagator averaged over instanton collective coordinates∆ ≡< ∆ >= Dγ ∆. However, we start first from∆. Expanding∆ i ) carrying out further resummation, we obtain the multi-scattering series As in Ref. [16], the main contribution to the∆ can be summed up by the following equatioñ Rewriting this equation, we havē We can derive the solution of Eqs. (3) and (4) It means immediately∆ −∆ = O(N 2 ) which is negligible. We have to take a well-known results for ∆ I from Ref. [17]: F (x, y) = 1 + ρ 2 (τ x)(τ + y) x 2 y 2 = 1 + ρ 2 (xy) whereη aµν = −η aνµ is the 'tHooft symbol. We assume that the position of the instanton z = 0 and the orientation U = 1. It is clear to see from Eq. (5) that the gluon-like scalar dynamical mass operator is given by In order to average over the position z, we have to change x → x − z, y → y − z and perform integration d 4 z. Similarly, we average over the color orientation U . Introducing the orientation factor O ab = tr(U + t a U τ b ), where t a are SU (N c )-matrices, we change ∆ ab I to be O ab O a b ∆ bb I , and carry out integration dO. Here In momentum space, we find the contribution from the first term in Eq. (11) as where the form factor qρK 1 (qρ). K 1 denotes the modified Bessel function. In Fig. 1

III. REAL GLUON PROPAGATOR
The total gluon field in the ILM is A + a, where A = i A i (γ i ). The number of all collective coordinates γ i is equal to 4N c N . We have to decompose the so-called zero modes φ i µ from the total fluctuation a, which are the fluctuations along the collective coordinates γ i in the functional space. Consider first the single instanton case, based on Refs. [18,19]. All of the fluctuations will be taken with gauge fixing condition P I µ a µ = 0 imposed. Then, the quadratic part of the effective action is given as (a µ M I µν a ν ), where M I µν = P I 2 δ µν + 2iG I µν − (1 − 1/ξ)P I µ P I ν and G I µν = −i[P I µ , P I ν ]. Here ξ stands for the gauge fixing parameter. The zero modes are the solutions of the following equation In some sense the zero modes can be considered as derivatives with respect to collective coordinates of the instanton field together with the additional longitudinal term dictated by the gauge fixing condition. The projection operator to the instanton zero-modes space is defined as P I µν = i φ i µ φ i+ ν , while that to the nonzero modes space is defined as Q I µν = δ µν − P I µν . The gluon propagator S I µν is defined by the following equation The explicit solution of this equation was already derived in Ref. [17]. To generalize the formulae in Ref. [16], we introduce an artificial gluon mass m,which will be taken zero at the end of calculation. So, we define g I m,µν and take the limit of lim m→0 g I m,µν = S I µν , where We also introduce G I m,ρν , satisfying (M I µρ + m 2 δ µρ )G I m,ρν = δ µν .
As was shown in Ref. [19], we find G I m,ρν = g I m,ρν + 1 m 2 P I ρν .
It is clear to see that Now we may repeat the the same method with which we are able to obtain the averaged ILM "scalar" gluon propagator∆ given in Eqs. (3,4). First, we introduce the ILM inverse massive gluon propagator Following the way we have derived Eqs.