Nucleon QCD sum rules in the instanton medium
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Nucleon QCD Sum Rules in the Instanton Medium1 M. G. Ryskin, E. G. Drukarev*, and V. A. Sadovnikova National Research Center “Kurchatov Institute,” Konstantinov Petersburg Nuclear Physics Institute, Gatchina, Liningradskaya oblast, 188300 Russia *e-mail: drukarev.pnpi.spb.ru Received April 2, 2015
Abstract—We try to find grounds for the standard nucleon QCD sum rules, based on a more detailed description of the QCD vacuum. We calculate the polarization operator of the nucleon current in the instanton medium. The medium (QCD vacuum) is assumed to be a composition of the small-size instantons and some long-wave gluon fluctuations. We solve the corresponding QCD sum rule equations and demonstrate that there is a solution with the value of the nucleon mass close to the physical one if the fraction of the small-size instantons contribution is ws ≈ 2/3. DOI: 10.1134/S1063776115090101
1. INTRODUCTION The idea of the QCD sum rule approach is to express the characteristics of the observed hadrons in terms of vacuum expectation values of the QCD operators often referred to as condensates. This idea was suggested in [1] for the calculation of the characteristics of mesons. Later, it was used for nucleons [2]. It succeeded in describing the nucleon mass, the anomalous magnetic moment, the axial coupling constant, etc. [3]. The QCD sum rule approach is based on the dispersion relation for the function describing propagation of the system that carries the quantum numbers of a hadron. This function is usually referred to as the “polarization operator” Π(q), with q being the fourmomentum of the system. The dispersion relation (in which we do not take care of subtractions)
Im Π(k ) Π(q 2 ) = 1 dk 2 2 π k − q2
∫
2
(1)
is analyzed at large and negative values of q2. Due to the asymptotic freedom of QCD, the polarization operator can be calculated in this domain. Operator product expansion (OPE) [4] enables us to represent the polarization operator for a power series in q–2 as q2 → ‒∞. The coeffi cients of the expansion are the QCD condensates, such as the scalar quark condena |0〉, sate 〈0|q (0)q(0)|0〉, the gluon condensate〈0|GaμνG μν etc. The nonperturbative physics is contained in these condensates. A typical value of a condensate with the dimension d = n is 〈0|On|0〉 ~ (±250 MeV)n. Hence, we 1 The article is published in the original.
∑
expect the series Π(q) = 〈 0|On|0〉/(q2)n to converge n at –q2 ~ 1 GeV2. The left-hand side of Eq. (1) is calculated as an OPE series. The imaginary part on the right-hand side describes physical states with the baryon quantum number and charge equal to unity. These are the proton, described by the pole of ImΠ(k2), the cuts corresponding to systems containing a proton and pions, and so on. The right-hand side of Eq. (1) is usually approximated by the “pole + continuum” model [1, 2], in which the lowest-lying pole is written exactly, while the higher states are described by the continuum. The main aim is to obtain the value of the nucleon
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