Terms Of Operator Product Expansion Research Articles

Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential frac{g}{{left|x-yright|}^2} in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.

Search for short strings ine+e−-annihilation

In this work the behavior of power corrections to Adler finction in operator product expansion (OPE) is studied, in particular the possible contribution of operator of dimension 2. The OPE terms of dimension 4 and 6 are taken into account. Various experimental data on reactions of $e^+e^-$-annihilation to pion channels ($\pi^+\pi^-$, $2\pi^+2\pi^-$, $\pi^+\pi^-2\pi^0$, $3\pi^+3\pi^-$, $2\pi^+2\pi^-2\pi^0$) are used. The high-precision fits of the experimental data are obtained and used. The method based on Borel transform of Adler function is applied. It is shown that the contribution of operator of dimension 2 is negative being compatible to zero at three standard deviations level. The strong (anti)corellation between short string and local gluon condensate is found.

O.P.E. and Power Corrections to the QCD coupling constant

Lattice data seems to show that power corrections should be convoked to describe appropriately the transition of the QCD coupling constant running from U.V. to I.R. domains. Those power corrections for the Landau-gauge MOM coupling constant in a pure Yang-Mills theory ( N f = 0) are analysed in terms of Operator Product Expansion (O.P.E.) of two- and three-point Green functions, the gluon condensate < A 2 > emerging from this study. The semi-classical picture given by instantons can be also used to look for into the nature of the power corrections and gluon condensate.

Check of QCD based on theτ-decay data analysis in the complexq2plane

The thorough analysis of the ALEPH data on hadronic $\ensuremath{\tau}$ decay is performed in the framework of QCD. The perturbative calculations are performed in three- and four-loop approximations. The terms of the operator product expansion (OPE) are accounted for up to the dimension $D=8.$ The value of the QCD coupling constant ${\ensuremath{\alpha}}_{s}{(m}_{\ensuremath{\tau}}^{2})=0.355\ifmmode\pm\else\textpm\fi{}0.025$ is found from the hadronic branching ratio ${R}_{\ensuremath{\tau}}.$ The $V+A$ and V spectral functions are analyzed using the analytical properties of the polarization operators in the whole complex ${q}^{2}$ plane. Borel sum rules in the complex ${q}^{2}$ plane along the rays, starting from the origin, are used. It is demonstrated that QCD with OPE terms is in agreement with the data for a coupling constant close to the lower error edge ${\ensuremath{\alpha}}_{s}{(m}_{\ensuremath{\tau}}^{2})=0.330.$ The restriction on the value of the gluonic condensate was found to be $〈({\ensuremath{\alpha}}_{s}/\ensuremath{\pi}{)G}^{2}〉=0.006\ifmmode\pm\else\textpm\fi{}0.012 {\mathrm{GeV}}^{2}.$ The analytical perturbative QCD is compared with the data. It is demonstrated to be in strong contradiction with experiment. The restrictions on the renormalon contribution are found. The instanton contributions to the polarization operator are analyzed in various sum rules. In the Borel transformation they appear to be small, but not in the spectral moment sum rules.