An instanton picture O.P.E. (A2) condensate?

arXiv:hep-ph/0209031v1 3 Sep 2002 1 An Instanton Picture O.P.E. DA2ECondensate? Ph. Boucauda, F. De Sotob, A. Doninic, J.P. Leroya, A. Le Yaouanca, J. Michelia, H. Moutarded, O. P`enea, J. Rodr´ıguez-Quinteroe aLaboratoire de Physique Th´eorique, Universit´e de Paris XI, Batiment 210, 91405 Orsay-Cedex, France. bDpto. de F´ısica At´omica Molecular y Nuclear, Universidad de Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. cI.N.F.N., Roma I and Dip. Fisica, Universit´a di Roma ”La Sapienza”, P. le A. Moro 2, 00185 Rome, Italy dCentre de Physique Th`eorique Ecole Polytechnique, 91128 Palaiseau Cedex, France eDpto. de F.A., E.P.S. La R´abida, Universidad de Huelva, 21819 Palos de la Fra., Spain. Gluon two- and three-point Green Functions computed in Landau gauge from the lattice show the existence of power corrections to the purely perturbative expressions, that can be explained through an Operator Product Expansion as the influence of a non gauge invariant mass dimension two condensate. The relationship of this condensate with topological properties of QCD, namely instantons, will be studied, giving a first estimate of the contribution of instantons to this condensate based in the direct lattice measure, after a cooling process, of the instanton liquid properties. 1. QCD coupling constant, O.P.E. and A2condensate. Lattice calculations of the QCD coupling constant and gluon propagator based in the Green Functions Method [1], suggest the necessity to add power corrections to the purely perturbative expressions to correctly describe their running [3]. An Operator Product Expansion (O.P.E.) analysis of the Green functions in Landau gauge1re- lates this power corrections to the existence of a non-perturbative A2condensate [4], through expressions: G(2) O.P.E.(p2) = G(2) P ert.(p2) + cA2R,µ p2, αO.P.E s(p2) = αP ert. s(p2) + c′A2R,µ p2,(1) 1In the lattice we will work in the minimum A2Landau gauge, ∂µAµ= 0, so all gauge dependent quantities, will be expressed in this particular gauge. where perturbative expressions are developed at three loops, and the functions cand c′include the Wilson coefficient of the expansion, and the anomalous dimension of the condensate at leading logarithm. By performing a combined fit of lattice results to expressions in (1), in two different MOM schemes, a value of ΛMS is extracted, in fairly good agreement with the one obtained by the ALPHA collaboration [2], by a completely different method. A value of the condensate comes out from the analysis. The physical meaning of this condensate is still an open question, and a lot of work is being devoted to its study during last years, for example, in relation to confinement [5]. The aim in this work will be to study the possible semiclassical contribution to this condensate coming from instantons, and whether they might explain the presence of power corrections in Green Functions. 2 2. The role of instantons. Instantons have been extensively studied as a possible description of the QCD vacuum (See [7] for a general overview), and so as a major source of QCD properties at low energies. In relation with the aim of this work, an ensemble of noninteracting instantons (I) and antiinstantons (I) in Landau gauge would give a contribution to the A2condensate; A2inst ≈N VZd4xAa µ(x)Aa µ(x) = 12π2ρ2n, (2) where Aa µ(x) is the standard ’t Hooft Polyakov instanton gauge field [6], ρthe average radius, and n=NI+NI Vthe density. If we accept the phenomenological values assigned to nand ρby the Instanton Liquid Model [7] (n∼0.5fm−4and ρ∼1/3fm), the instantonic contribution will be A2Inst. ∼0.5GeV2. We will perform, however, our own analysis, thus testing the latter approach. 2.1. Cooling. In principle, a direct measure of A2in the lattice should be possible, but the presence of the UV divergent part is hardly separable from the soft, instantonic one. The other possibility is to perform a cooling procedure, that will allow us to compute the number and size of instantons, giving an indirect measure of the A2through (2). We will use the traditional cooling method [8], even if it introduces a number of known biases, as I−I annihilation, and a modification of instanton sizes and lattice spacing. The approach proposed here is to compute instanton properties for different number of cooling sweeps, and extrapolate back to the thermalised situation in order to recover their physical meaning2. 2.2. Shape Recognition. Instantons will be localised in cooled lattices via a geometrical method (Described in [10].) 2The use of improved cooling methods, as the one developed in [9], could improve this approach, as radii evolution is minimised, but I −I annihilation is unavoidable, so the extrapolation will be anyway necessary. that accepts a topological charge lump as an instanton when the ratio of the integral over a given fraction of the topological charge at the maximum, α, and its theoretical counterpart, ǫ, is ∼1, for a range of values of α: ǫ=Rx/ |Qρ(x)| |Qρ(0)|≥αd4xQρ(x) 1−3α1/2+ 2α3/4(3) Once the lump has been identified as an instanton, the radius will be computed from the size of the cluster where the integral has been developed. 2.3. A naive model of annihilation. With the method outlined above, we compute the density and size of instantons in a lattice, for different numbers of cooling sweeps, nc, obtaining values with a strong dependence on nc(See figure), that avoids to obtain any physical information at fixed nc. As a first approach to the understanding of this evolution, we will make a simple model, where instantons annihilate with antiinstantons (Being so ∆N=NI−NIa constant) proportionally to their packing ratio, and to the number of antiinstantons, so that the equation for the evolution of N=NI+NIis: ∂N ∂nc =−λ 2Vρ4(nc)(N(nc)2−∆N2).(4) If we assume ρ(nc) = cte, the solution of Eq. (4) will give N(nc)∼N(0) 1+λnc, the expression used in [10], as a first order approach, but our cooling procedure modifies instanton’s size (See figure), in a way thah we phenomenologically parametrise as: ρ(nc) = ρ(0)(1 + aln(1 + nc)) .(5) We will include (5) in equation (4), with ρ(0) the extrapolated radius at the thermalised situation and aa constant to determine. After performing a combined fit of our lattice results to the expressions (5) and the one coming from the integration of (4), we can fix the initial values of the density, n(0) and the radius ρ(0), 3 and the two constants that govern the evolution, λand a. 0 50 100 150 200 nc 0 0.5 1 1.5 A2 (GeV2) 1 10 100 nc 0.3 0.4 0.5 0.6 0.7 ρ (fm) Figure 1. Results of the combined fit for the instanton density and radius as a function of the number of cooling sweeps for a 244lattice at β=6.0. 3. Results and conclussion. The result of the combined fit gives a value of the instantonic contribution to A2 Ins∼ 0.4GeV2, however the result of the extrapolation is highly dependent on the value of ρ, which due to the logarithmic behaviour is hardly reliable. We therefore prefer the value at the maximum, 1.12(11) GeV2, as a crude estimation of A2 Ins. This semiclassical evaluation of A2, which does not run with the scale, is difficult to relate to that appearing in the O.P.E. expansion, which does depend on the renormalisation scheme and scale. The typical scale of instantons is ρ−1∼0.7GeV. Unluckly it is not possible to run the A2 O.P.E.to such a low energy, where pertubative QCD is not valid. The lowest reacheble energy scale is 2.6GeV [4,10]; A2 O.P.E.(2.6GeV)= 1.4(3)(3)GeV2,(6) the first error coming from the OPE determination of the condensate renormalised at 10 GeV, and the second from higher orders in the running. 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