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Investigation of meson loop effects in the Nambu-Jona-Lasinio model

Technische Universität
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  • Mathematics
  • Physics


The influence of mesonic fluctuations on quantities in the Nambu--Jona-Lasinio model are examined. To that end different approximation schemes are introduced which guarantee the consistency with several relations following from chiral symmetry. In particular this refers to the Goldstone theorem, which states the existence of massless bosons, in our case pions, if the symmetry of the Lagrangian is spontaneously broken. The Gell-Mann--Oakes--Renner relation describes the behavior of the pion mass for small current quark masses, which explicitly break chiral symmetry. Three different schemes are presented. These are the inclusion of the ring sum in a so-called ``Phi-derivable-method'', an expansion in powers of 1/Nc and an expansion up to one-meson loop in the effective action formalism. Since the two latter schemes are used for explicit calculations, it is explicitely proved that the Goldstone theorem as well as the Gell-Mann--Oakes--Renner relation hold within those schemes. The influence of meson-loop effects on the quark condensate, the pion mass, the pion decay constant and properties of rho- and sigma-meson are investigated. First we focus on the determination of a consistent set of parameters. In the 1/Nc-expansion scheme it is possible to find a set of parameters which allows to simultaneously describe the quantities in the pion sector and those related to the rho-meson, whereas this turns out to be not possible within the expansion of the effective action. Results for the sigma-meson are also discussed. Here, similarly to the rho-meson, mesonic intermediate states are essential. Besides, the relation of our model to hadronic models is discussed. In the last part of this thesis the behavior of the quark condensate at nonzero temperature is studied. In the low-temperature region agreement with the model independent chiral perturbation theory result in lowest order can be obtained. The perturbative 1/Nc-expansion scheme does not allow for an examination of the chiral phase transition at higher temperatures, whereas this is possible within the meson-loop approximation scheme. A first order phase transition is found.

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