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Register a new userThe scalar gluonium correlator: large-beta_0 and beyond
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Matthias Jamin
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The investigation of the scalar gluonium correlator is interesting because it carries the quantum numbers of the vacuum and the relevant hadronic current is related to the anomalous trace of the QCD energy-momentum tensor in the chiral limit. After reviewing the purely perturbative corrections known up to next-next-to-leading order, the behaviour of the correlator is studied to all orders by means of the large-beta_0 approximation. Similar to the QCD Adler function, the large-order behaviour is governed by the leading ultraviolet renormalon pole. The structure of infrared renormalon poles, being related to the operator product expansion are also discussed, as well as a low-energy theorem for the correlator that provides a relation to the renormalisation group invariant gluon condensate, and the vacuum matrix element of the trace of the QCD energy-momentum tensor.
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We present a formulation to give renormalon-free predictions consistently with fixed order perturbative results. The formulation has a similarity to Lees method in that the renormalon-free part consists of two parts: one is given by a series expansion which does not contain renormalons and the other is the real part of the Borel integral of a singular Borel transform. The main novel aspect is to evaluate the latter object using a dispersion relation technique, which was possible only in the large-$beta_0$ approximation. Here, we introduce an ambiguity function, which is defined by inverse Mellin transform of the singular Borel transform. With the ambiguity function, we can rewrite the Borel integral in an alternative formula, which allows us to obtain the real part using analytic techniques similarly to the case of the large-$beta_0$ approximation. We also present detailed studies of renormalization group properties of the formulation. As an example, we apply our formulation to the fixed-order result of the static QCD potential, whose closest renormalon is already visible.
We propose a clear definition of the gluon condensate within the large-$beta_0$ approximation as an attempt toward a systematic argument on the gluon condensate. We define the gluon condensate such that it is free from a renormalon uncertainty, consistent with the renormalization scale independence of each term of the operator product expansion (OPE), and an identical object irrespective of observables. The renormalon uncertainty of $mathcal{O}(Lambda^4)$, which renders the gluon condensate ambiguous, is separated from a perturbative calculation by using a recently suggested analytic formulation. The renormalon uncertainty is absorbed into the gluon condensate in the OPE, which makes the gluon condensate free from the renormalon uncertainty. As a result, we can define the OPE in a renormalon-free way. Based on this renormalon-free OPE formula, we discuss numerical extraction of the gluon condensate using the lattice data of the energy density operator defined by the Yang--Mills gradient flow.
We revisit, improve and confirm our previous results [1-3] from the scalar digluonium sum rules within the standard SVZ-expansion at N2LO {it without instantons} and {it beyond the minimal duality ansatz} : one resonance $oplus$ QCD continuum parametrization of the spectral function. We select different unsubtracted sum rules (USR) moments of degree $leq$ 4 for extracting the two lowest gluonia masses and couplings. We obtain in units of GeV: $(M_{G},f_G)=[1.04(12),0.53(17)]$ and $[1.52(12),0.57(16)]$. We attempt to predict the masses of their first radial excitations to be $M_{sigma} simeq 1.28(9)$ GeV and $M_{G_2}simeq 2.32(18)$ GeV. Using a combination of the USR with the subtracted sum rule (SSR), we estimate the conformal charge (subtraction constant $psi_G(0)$ of the scalar gluonium two-point correlator at zero momentum) which agrees completely with the Low Energy Theorem (LET) estimate. Combined with some low-energy vertex sum rules (LEV-SR), we confront our predictions for the widths with the observed $I=0$ scalar mesons spectra. We confirm that the $sigma$ and $f_0(980)$ meson can emerge from a maximal (destructive) ($bar uu+bar dd$) meson - $(sigma_B$) gluonium mixing [10]. The $f_0(1.37)$ and $f_0(1.5)$ indicate that they are (almost) pure gluonia states (copious decay into $4pi$) through $sigmasigma$, decays into $etaeta$ and $etaeta$ from the vertex $U(1)_A$ anomaly with a ratio $div$ to the square of the pseudoscalar mixing angle sin$^2theta_P$.
We study how topological defects manifest themselves in the equal-time two-point field correlator. We consider a scalar field with Z_2 symmetry in 1, 2 and 3 spatial dimensions, allowing for kinks, domain lines and domain walls, respectively. Using numerical lattice simulations, we find that in any number of dimensions, the correlator in momentum space is to a very good approximation the product of two factors, one describing the spatial distribution of the defects and the other describing the defect shape. When the defects are produced by the Kibble mechanism, the former has a universal form as a function of k/n, which we determine numerically. This signature makes it possible to determine the kink density from the field correlator without having to resort to the Gaussian approximation. This is essential when studying field dynamics with methods relying only on correlators (Schwinger-Dyson, 2PI).
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