{"title":"Schwinger–Dyson Equation for Quarks in a QCD Inspired Model","authors":"V. I. Shilin, V. N. Pervushin","doi":"10.1134/S1547477123060341","DOIUrl":null,"url":null,"abstract":"<p>We discuss formulation of QCD in Minkowski–spacetime and effect of an operator product expansion by means of normal ordering of fields in the QCD Lagrangian. The formulation of QCD in the Minkowski–spacetime allows us to solve a constraint equation and decompose the gauge field propagator in the sum of an instantaneous part, which forms a bound state, and a retarded part, which contains the relativistic corrections. In Quantum Field Theory, for a Lagrangian with unordered operator fields, one can make normal ordering by means of the operator product expansion, then the gluon condensate appear. This gives us a natural way of obtaining a dimensional parameter in QCD, which is missing in the QCD Lagrangian. We derive a Schwinger–Dyson equation for a quark, which is studied both numerically and analytically. The critical value of the strong coupling constant <span>\\({{\\alpha }_{s}} = {4 \\mathord{\\left/ {\\vphantom {4 \\pi }} \\right. \\kern-0em} \\pi }\\)</span>, above which a nontrivial solution appears and a spontaneous chiral symmetry breaking occurs, is found. For the sake of simplicity, the considered model describes only one flavor massless quark, but the methods can be used in more general case. The Fourier-sine transform of a function with log-power asymptotic was performed.</p>","PeriodicalId":730,"journal":{"name":"Physics of Particles and Nuclei Letters","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physics of Particles and Nuclei Letters","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1134/S1547477123060341","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"PHYSICS, PARTICLES & FIELDS","Score":null,"Total":0}
引用次数: 0
Abstract
We discuss formulation of QCD in Minkowski–spacetime and effect of an operator product expansion by means of normal ordering of fields in the QCD Lagrangian. The formulation of QCD in the Minkowski–spacetime allows us to solve a constraint equation and decompose the gauge field propagator in the sum of an instantaneous part, which forms a bound state, and a retarded part, which contains the relativistic corrections. In Quantum Field Theory, for a Lagrangian with unordered operator fields, one can make normal ordering by means of the operator product expansion, then the gluon condensate appear. This gives us a natural way of obtaining a dimensional parameter in QCD, which is missing in the QCD Lagrangian. We derive a Schwinger–Dyson equation for a quark, which is studied both numerically and analytically. The critical value of the strong coupling constant \({{\alpha }_{s}} = {4 \mathord{\left/ {\vphantom {4 \pi }} \right. \kern-0em} \pi }\), above which a nontrivial solution appears and a spontaneous chiral symmetry breaking occurs, is found. For the sake of simplicity, the considered model describes only one flavor massless quark, but the methods can be used in more general case. The Fourier-sine transform of a function with log-power asymptotic was performed.
期刊介绍:
The journal Physics of Particles and Nuclei Letters, brief name Particles and Nuclei Letters, publishes the articles with results of the original theoretical, experimental, scientific-technical, methodological and applied research. Subject matter of articles covers: theoretical physics, elementary particle physics, relativistic nuclear physics, nuclear physics and related problems in other branches of physics, neutron physics, condensed matter physics, physics and engineering at low temperatures, physics and engineering of accelerators, physical experimental instruments and methods, physical computation experiments, applied research in these branches of physics and radiology, ecology and nuclear medicine.