Instanton in the Georgi-Glashow model
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EMENTARY PARTICLES AND FIELDS Theory
Instanton in the Georgi–Glashow Model N. O. Agasian* Institute for Theoretical Experimental, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117218 Russia National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia Received August 1, 2013
Abstract—Topologically nontrivial solutions in the Georgi–Glashow model are studied. It is shown that only in quantum theory does an instanton exist as a stable field configuration. An effective action is constructed, and quantum equations of motion for the instanton are obtained on its basis. Their solutions at long and short distances are obtained. A critical size corresponding to a stable instanton in the Georgi– Glashow model is found. DOI: 10.1134/S106377881408002X
1. INTRODUCTION The instanton was the first example of a nonperturbative gluon-field fluctuation. It was discovered in 1975 by Polyakov and his coauthors [1]. An important step was made in the classical study of ’t Hooft [2], where the semiclassical tunnelling amplitude was calculated. Instantons play an important role in a number of phenomena in quantum field theory (for an overview, see the monograph of Rubakov [3] and ¨ the review article of Schafer and Shuryak [4]). At the same time, there are some problems in instanton physics in QCD. First of all, there is an infrared divergence of integrals with respect to the instanton size ρ, and this prevents evaluation of the instanton contribution to some physical quantities—for example, to the vacuum gluon condensate. Second, the area law for the Wilson loop cannot be explained within the instanton-gas model. This means that, in a semiclassical instanton–anti-instanton vacuum, there is no confinement that would be responsible for the formation of the hadron spectrum. A large number of theoretical studies were devoted to the problem of stabilizing the instanton in its scale ρ. To some extent, they reduce to attempts at stabilizing an instanton ensemble by taking into account effects of interactions between pseudoparticles [5]. The instanton-liquid model formulated phenomenologically by Shuryak[6, 7] became the most popular. Similar results for the parameters of the instanton liquid were obtained quantitatively by Diakonov and Petrov [8]. However, a further development [9] revealed that an instanton ensemble cannot be stabilized by a purely classical interaction. Thus, there is no mechanism of suppression of largesize instantons within an instanton–anti-instanton *
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ensemble alone. In addition to semiclassical instantons, however, there are other nonperturbative fields in the vacuum, which permit, among other things, solving the infrared problem for instantons. A nonperturbative QCD vacuum can be parametrized in terms of a set nonlocal gauge-invariant vacuum expectation values of the gluon-field strength [10– 13]. Within this approach, the vacuum expectation value of the Wilson loop is expressed, on the basis of the non-Abelian analog of Stokes’ theorem, in terms of the bilocal gluon-field-strength c
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