Direct instantons in QCD nucleon sum rules.

We study the role of direct (i.e. small-scale) instantons in QCD correlation functions for the nucleon. They generate sizeable, nonperturbative corrections to the conventional operator product expansion, which improve the quality of both QCD nucleon sum rules and cure the long-standing stability problem, in particular, of the chirally odd sum-rule. 12.38.Lg, 14.20.Dh Typeset using REVTEX 1 QCD sum rules, introduced by Shifman, Vainshtein and Zakharov (SVZ) [1], provide a systematic, nonperturbative framework for the calculation of hadron properties. They have been intensely studied and applied over the last decade and produced the most exhaustive model-independent analysis of hadron properties to date. The sum-rule approach is based on the comparison of two “dual” descriptions for correlation functions of hadronic currents, in terms of quarks and gluons on the left-hand side (LHS) and in terms of hadrons on the right-hand side (RHS). The RHS uses a simple parametrization of the spectral function in terms of the hadron parameters, such as mass, overlap with the source current and continuum threshold. The QCD calculation on the LHS employs a non-perturbative operator-product expansion (OPE). Long-distance bulk properties of the physical vacuum are efficiently parametrized in terms of the vacuum expectation values of composite quark-gluon operators (“condensates”), which are independent of the hadron considered. The short-distance physics is contained in the perturbatively calculated Wilson coefficients. The inverse renormalization scale of the operators, μ, serves as the dividing line separating long and short distances. Both sides of the sum rules are then Borel-transformed, and the hadron parameters are determined by fitting the two sides in the fiducial region, i.e. in the range of Borel mass values in which both descriptions of the correlator are expected to be adequate. The quality of this fit is the only intrinsic criterion for the accuracy and reliability of the sum rules. If it is met sufficiently well, the resulting hadron parameters will be approximately independent of the Borel mass in the fiducial region. The nucleon sum rules, however, do not show such a stability plateau, despite many improvements over the last decade [2]. It seems that some relevant physics in the fiducial region (around 1 GeV) is missing in the OPE. In this letter we suggest that small-size instantons [3], termed “direct” by SVZ, provide the dominant part of this physics. Instantons [4] are classical solutions of the euclidean Yang-Mills equation. Due to the infrared complexities of QCD, their quantum properties and vacuum distribution cannot yet be derived from first principles. A consistent picture of their importance and bulk 2 features has been established, however, by extensive phenomenological [5,6], analytical [7] and numerical studies (in the instanton liquid model [6] and on the lattice, e.g. in ref. [8]). They indicate, in particular, that the average instanton size ρc in the vacuum is considerably smaller than the average separation R between instantons [6]: ρc ≃ 1 3 fm, R ≃ 1 fm. (1) Most of the contributions of these rather small instantons to the correlation function are ignored in the conventional OPE: As the scale of these fields is smaller than the inverse renormalization scale (taken around μ ≃ 0.5GeV), they would contribute to the Wilson coefficients, but do not show up in the perturbative evaluation. The aim of our paper is to calculate the leading instanton contributions to the nucleon correlator, using an instanton size-distribution [6] in accordance with the bulk features (1), and to study their effects in the nucleon sum rules. We do not use the detailed assumptions of the above-mentioned model calculations. The instanton contributions are mediated mainly by the quark zero-modes [9] in the instanton background field. Due to their particular chiral and color properties, the magnitude of instanton effects is channel dependent. Even if they can e.g. be safely neglected in the vector and axial-vector channel, they play a dominant role in the pseudoscalar sum rules [10] (and more generally in the spin-0 channel), where the conventional OPE was known to fail [1]. Our expectation of sizeable instanton effects in the nucleon sum rules is mainly based on parallels with the pseudoscalar sector. The interpolating fields (see below) in the nucleon correlator contain spin-0 diquarks, which receive zero-mode contributions of the same order as the pseudoscalars. Furthermore, the magnitude of the nucleon correlator at distances around 1 2 fm is much larger than the perturbative contribution [11], which is reminiscent of the strongly attractive correlations due to instantons in the pseudoscalar channel. Recently, Dorokhov and Kochelev [12] made a first attempt to calculate instanton contributions to the nucelon sum rule. However, as we will outline in the course of this paper, 3 we do not agree with their results and many of their conclusions. The QCD nucleon sum-rules are based on the correlation function