Espectro de excitação para modelos de teorias quânticas de campo na rede: modelos puramente fermiônicos e modelos de cromodinâmica quântica / Excitation spectrum for quantum field theory models on the lattice: pure fermionic models and quantum chromodynamics models

AUTOR(ES)
DATA DE PUBLICAÇÃO

2008

RESUMO

In this thesis, we obtain, from a mathematically rigorous point of view, the low-lying energy-momentum spectrum of two $3+1$ dimensional imaginary time lattice quantum filed theory with fermion fields (we give explicit results for the case $d = 3$ and Dirac matrices): a pure fermionic model with quartic interaction in the $N$-component fermion field and a quantum chromodynamics model. For the Four-Fermion model, $\kappa$ denotes the hopping parameter, $M_0$ the fermion bare mass and $\lambda$ the interaction parameter. A polymer expansion show the existence of the model correlation functions in the thermodynamic limit, in the region where $|\frac{\kappa}|$ is small enough. The analysis of the spectrum is based on spectral representations of two- and four- point correlation functions. The analysis of such adequate correlation functions is simplified by the help of symmetries, in particular, by a {\em new} Time Reflection symmetry, which appear in the level of correlation functions. The exact determination of the spectrum is done using a detailed study of the decay rates of the correlations. Up to near the 3 particle threshold, the energy-momentum spectrum exhibits isolated dispersion curves that are identified as particles and bound states. In the one-particle subspace, the spectrum consist in just a isolated dispersion curve. The mass of the associated particle is of order $-\ln \kappa$. The two-particle spectrum shows up as solutions of a Bethe-Salpeter equation, which is solved first in a ladder approximation. The two-particle spectrum contains a two free particles band of finite width. The existence of bound states above or below the band depends on wherever the model Gaussian domination holds. A parameter $\aleph$ is given to measure the Gaussian domination. For $\aleph=0$, no bound state occurs. For $\aleph>0$, a bound state appears bellow the two-particles band. For $\aleph<0$, the bound state appears above this band. The result obtained in this ladder approximation can be extended to the full model by a rigorous control of the contributions that differ these two cases. In a second part, analog ideas are applied to analyze the spectrum of a quantum chromodynamics model. In particular, we show the existence of pentaquarks in the strong coupling regime (plaquette coupling $0 <\beta= \frac{g^2_0} \ll \kappa $). The model has a $SU(3)_c$ gauge symmetry and a $SU(2)_f$ flavor symmetry. The reveled pentaquarks are superpositions of meson-baryon bound states. Only states with an odd number of fermions and bellow the meson-baryon threshold are considered. The pentaquark are determined using a ladder approximation to the Bethe-Salpeter equation. In the dominant order in $\beta$, the bound state mass is $\approx -5 \ln\kappa$ and the binding energy is of order $\textrm(\kappa^2)$. The most strongly bounded bound state has isospin $I=\frac$. For $I=\frac$, there is no bound state. These results shows a dependence in the spins of the meson and baryon. This analysis show that a $\textrm(\kappa^2)$ quark-anti-quark exchange potential is the dominant interaction, although there is not a meson exchange interpretation.

ASSUNTO(S)

equação de bethe-salpeter estados ligados four-fermion model modelo de quatro-férmions qcd model modelo de qcd bound states espectro de excitação pentaquarks excitation spectrum pentaquark bethe-salpeter equation

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