Minkowski space Bethe-Salpeter equation within Nakanishi representation

• Autor(a) principal: Gutiérrez Gómez, Cristian Leonardo [UNESP]
• Data de Publicação: 2016
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://hdl.handle.net/11449/144735
• topic: Bethe-Salpeter equation; Nakanishi integral representation; Light-front dynamics; Equação de Bethe-Salpeter; Representação integral de Nakanishi; Dinâmica na frente de luz

The work presented in this thesis was dedicated in exploring bound-state solutions of the Bethe-Salpeter equation directly in the Minkowski space. For that, we consider a method that combines the Nakanishi integral representation for the Bethe-Salpeter amplitude, developed by Noboru Nakanishi in the sixties, together with the projection of the Bethe-Salpeter amplitude onto the null-plane, also known as the light-front projection. This approach, besides of allowing to compute the binding energies, which are accessible from the usual Euclidean calculation, enables to obtain the Bethe-Salpeter amplitude in the Minkowski space and the light-front wave function. The feasibility of such an approach is further verified by comparing the Bethe-Salpeter amplitude obtained directly in the Euclidean space with the corresponding amplitude obtained by solving the Bethe-Salpeter equation, using the Nakanishi integral representation, once the Wick rotation is performed to this latter. The success of the approach when applied to study the bound state problem of two-scalar particles exchanging another scalar particle in the ground state, as well as the corresponding study at the zero-energy limit, has encouraged us to extend this method to another interesting problems. In particular, we start by extending the method to study problems in (2+1) dimensions due to the increasing interest in the condensed-matter physics, like the study of Dirac electrons in graphene. In this initial examination we restrict to the scalar model, which enables us to access to the main difficulties that we will face when studying the fermion-fermion bound state problem. Hence, this calculation can be considered as the first step towards the implementation of the method to real fermionic problems. The previous calculations have been performed by considering the ladder approximation for the irreducible interacting kernel for s-wave states. Therefore, one of the extensions that is explored in this thesis is the effect of introducing the next contribution in the interacting kernel, known as the scalar-scalar cross-ladder contribution. The effects in the eigenvalues and the light-front wave functions are analyzed in detail, by considering the computed results. A particular interesting subject, extensively studied in this thesis, is concerned to the spectrum of the Bethe-Salpeter equation for the scalar-scalar bound-state problem. The spectrum of excited states obtained with the Nakanishi integral representation approach is compared with that obtained in the Euclidean calculation. Besides, the ratio energies excited/ground of the relativistic spectrum is reduced to the non-relativistic one by choosing small binding energies and the mass of the exchanged boson approaching to zero. The valence light-front wave function and the impact-parameter space valence wave function are displayed, revealing the main features of excited states known from the non-relativistic framework. In the analysis of the spectrum, we also studied the transverse-momentum amplitudes for the ground and first-excited state, which can be equivalently obtained in the Minkowski or Euclidean spaces. Finally, we focus on the study of electromagnetic elastic form factors within the Bethe-Salpeter approach. Aware that the correct calculation of form factors should be performed in the Minkowski space, the calculation of the elastic form factor is carried out with the standard impulse approximation and in addition the effect of the next contribution to the form factor is studied.