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On the Large Nc Expansion in Quantum Chromodynamics
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William A. Bardeen

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100 Years Werner Heisenberg
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I discuss methods based on the large Nc expansion to study nonperturbative aspects of quantum chromodynamics, the theory of the strong force. I apply these methods to the analysis of weak decay processes and the nonperturbative computation of the weak matrix elements needed for a complete evaluation of these decays in the Standard Model of elementary particle physics. • Introduction. Field theories are frequently studied through a perturbative expansion in the interaction strength or coupling
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... nstant. In many cases, a nonperturbative analysis is required to apply these theories in physical situations. The large N expansion is a nonperturbative method of analysis that makes use of particular limits for parameters unrelated to the coupling constant. For example, O(N) spin systems where N is the number of spin components can be studied by mean field methods which become exact in the large N limit. Applications of perturbative QCD, such as the cross-sections for high p_t jets, are greatly simplified using color-ordered amplitudes and a large Nc expansion where Nc is the number of colors, the quantum number associated with the QCD gauge dynamics. The large Nc expansion can also be used to study nonperturbative aspects of quantum chromodynamics with applications to the structure of chiral symmetry breaking and hadronic string theory. More recently, it has been established that there exists a duality between the large N limit of SU(N) supersymmetric Yang-Mills theory and classical supergravity in a higher dimensional space-time. In this talk, I will discuss the nature of the large Nc expansion in QCD and its application to the computation of weak decay amplitudes. • Large Nc expansion in QCD. The large Nc expansion defines a nonperturbative reordering of the QCD coupling constant expansion. Each Feynman diagram can be classified by its dependence on the strong coupling constant, α strong , and the number of colors Nc. In 1974, 't Hooft [1] argued that the structure of the theory simplified in the limit, Nc → ∞, α strong → 0 , α strong Nc fixed * . The theory is summed to all orders in the rescaled coupling constant, α strong Nc * , with corrections being formally suppressed by powers of 1/Nc. At leading order in this expansion, the gluons are constrained to be in planar Feynman diagrams. Quarks lines form boundaries of the planar surfaces formed by the gluons. In this sense, the structure of large Nc QCD is similar to an open string theory with quarks attached to the ends of the string.

doi:10.1002/9783527610853.ch7
fatcat:m37jtby6tvd6tfvfba6f5dulp4