Using numerical stochastic perturbation theory, we determine the first 35 infinite volume coefficients of the perturbative expansion in powers of the strong coupling constant α of the plaquette in SU(3) gluodynamics. These coefficients are obtained in lattice regularization with the standard Wilson gauge action. The on-set of the dominance of the dimension four renormalon associated to the gluon condensate is clearly observed.We determine the normalization of the corresponding singularity in the Borel plane and convert this into the MS scheme. We also comment on the impact of the renormalon on non-perturbative determinations of the gluon condensate.

We determine the non-perturbative gluon condensate of four-dimensional SU(3) gauge theory in a model-independent way. This is achieved by carefully subtracting high order perturbation theory results from non-perturbative lattice QCD determinations of the average plaquette. No indications of dimension two condensates are found. The value of the gluon condensate turns out to be of a similar size as the intrinsic ambiguity inherent to its definition. We also determine the binding energy of a B meson in the heavy quark mass limit.PACS numbers: 12.38. Gc,12.38.Bx,11.55.Hx,12.38.Cy,11.15.Bt The operator product expansion (OPE) [1] is a fundamental tool for theoretical analyses in quantum field theories. Its validity is only proven rigorously within perturbation theory, to arbitrary finite orders [2]. The use of the OPE in a non-perturbative framework was initiated by the ITEP group [3] (see also the discussion in Ref. [4]), which postulated that the OPE of a correlator could be approximated by the following series:where the expectation values of local operators O d are suppressed by inverse powers of a large external momentum Q ≫ Λ QCD , according to their dimensionality d.The Wilson coefficients C d (α) encode the physics at momentum scales larger than Q. These are well approximated by perturbative expansions in the strong coupling parameter α. The large-distance physics is described by the matrix elements O d that usually have to be determined non-perturbatively. Almost all QCD predictions of relevance to particle physics phenomenology are based on factorizations that are generalizations of the above generic OPE.For correlators where O 0 = 1, the first term of the OPE expansion is a perturbative series in α. In pure gluodynamics the first non-trivial gauge invariant local operator has dimension four. Its expectation value is the so-called non-perturbative gluon condensate

We determine the infinite volume coefficients of the perturbative expansions of the self-energies of static sources in the fundamental and adjoint representations in SU(3) gluodynamics to order α 20 in the strong coupling parameter α. We use numerical stochastic perturbation theory, where we employ a new second order integrator and twisted boundary conditions. The expansions are obtained in lattice regularization with the Wilson action and two different discretizations of the covariant time derivative within the Polyakov loop. Overall, we obtain four different perturbative series. For all of them the high order coefficients display the factorial growth predicted by the conjectured renormalon picture, based on the operator product expansion. This enables us to determine the normalization constants of the leading infrared renormalons of heavy quark and heavy gluino pole masses and to translate these into the modified minimal subtraction scheme (MS). We also estimate the four-loop β-function coefficient of the lattice scheme.

We compute the static self-energy of SU(3) gauge theory in four spacetime dimensions to order α 20 in the strong coupling constant α. We employ lattice regularization to enable a numerical simulation within the framework of stochastic perturbation theory. We find perfect agreement with the factorial growth of high order coefficients predicted by the conjectured renormalon picture based on the operator product expansion.PACS numbers: 11.15. Bt,12.38.Cy,12.38.Bx,11.10.Jj,12.39.Hg Little is known about properties of quantum field theories from first principles. This is particularly so for asymptotically free gauge theories such as quantum gluodynamics. One of the most salient features of this theory is the confinement of charged objects. Yet this property has not been proven, and the best evidence comes from the linearly rising static potential at large distances obtained in lattice simulations. Another expected property is the asymptotic nature of perturbative weak coupling expansions. In four dimensional non-Abelian gauge theories one particular pattern of asymptotic divergence should be determined by the structure of the operator product expansion (OPE). It is usually named renormalon [1] or, more specifically, infrared renormalon. Its existence has also not been proven but only tested assuming the dominance of β 0 -terms, which amounts to an effective Abelianization of the theory, or in the two dimensional O(N ) model [2], where it is suppressed by powers of 1/N . Moreover, the possible non-existence or irrelevance of renormalons in Quantum Chromodynamics has been suggested in several papers, see, e.g. [3,4] and references therein. This has motivated dedicated high order perturbative expansions of the plaquette, e.g. [5][6][7][8], in lattice regularization, with conflicting conclusions. Powers as high as α 20 were achieved in the most recent simulation [9]. However, the expected asymptotic behaviour was not seen. If confirmed, this non-observation would cast doubt on the well-accepted lore of the OPE and renormalon physics (see [10] for a comprehensive review), and would significantly affect the phenomenological analysis of data from high energy physics experiments on the decay of heavy hadrons, heavy quark masses, the running coupling parameter, parton distributions, etc.. Therefore, this issue should be clarified unambiguously.In this letter we present compelling numerical evidence that the expected renormalons indeed exist not only in models but in real gluodynamics. We also argue why previous analyses based on the plaquette have failed to detect them. The vital and new ingredients of our study are as follows. (a) We consider a perturbative series whose leading renormalon is dictated by a dimension d = 1 operator, rather than by the d = 4 plaquette. (b) Using a higher order integrator and employing twisted boundary conditions, among other improvements, we are able to obtain results of unprecedented precision on an extensive set of spacetime volumes. (c) We carefully extrapolate to the infinite volume limi...

Using Numerical Stochastic Perturbation Theory (NSPT), we calculate the static self-energy of SU(3) gauge theory up to order α 20. Simulations on a large set of different lattice volumes allow for a careful treatment of finite size effects. The resulting infinite volume perturbative series of the static self-energy is in remarkable agreement with the predicted asymptotic behaviour of high order expansions, namely with a factorial growth of perturbative coefficients known as renormalon.