Meson loop corrections to the NJL model

We present a symmetry preserving approach to1/N (N being the number of colors) corrections in the Nambu-Jona-Lasinio model. This is achieved through an explicit 1/N expansion where N is considered as a free parameter. Verification and use of the pertinent Ward identity to the next to leading order in 1/N is made in obtaining the Goldberger-Treiman relation. Although the present results are not new, the approach may be useful for deriving higher (than one loop) corrections to the NJL model in a systematic way.

Authorship

Francisco Peña

Universidade Federal de Minas Gerais, Instituto de Ciências Exatas , Departamento de Física, Belo Horizonte, Minas Gerais, BrazilUniversidade Federal de Minas GeraisBrazilBelo Horizonte, Minas Gerais, BrazilUniversidade Federal de Minas Gerais, Instituto de Ciências Exatas , Departamento de Física, Belo Horizonte, Minas Gerais, Brazil

M. Carolina Nemes

Alex H. Blin

Universidade de Coimbra, Departamento de Física , Coimbra, PortugalUniversidade de CoimbraPortugalCoimbra, PortugalUniversidade de Coimbra, Departamento de Física , Coimbra, Portugal

Brigitte Hiller

Universidade de Coimbra, Departamento de Física , Coimbra, PortugalUniversidade de CoimbraPortugalCoimbra, PortugalUniversidade de Coimbra, Departamento de Física , Coimbra, Portugal

SCIMAGO INSTITUTIONS RANKINGS

Universidade de Coimbra, Departamento de Física , Coimbra, PortugalUniversidade de CoimbraPortugalCoimbra, PortugalUniversidade de Coimbra, Departamento de Física , Coimbra, Portugal

Figures

Figures (4)

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Figure 1.
The effective action with the explicit dependence on N of the several one-loop quark contributions. The dashed lines represent the meson fields. To the N^1/2 term contributes only the scalar meson; the propagator term, of order N⁰, stands for the pion and the sigma mesons; the three point function proportional to N^1/2 involves either three scalars or one scalar coupled to two pions; the four point function of order N^-1 stands for either all pions or all scalars or two pions and two scalars.

Thumbnail

Figure 2.
The next to leading order in N to the gap equation. The internal meson line stands for a pion or a sigma.

Thumbnail

Figure 3.
The quark propagator to the next to leading order in the 1/N expansion. The thin lines denote fermion propagators in the dominant order, the dashed internal lines stand for pion and scalar insertions. The external line of the tadpole is a scalar.

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Figure 4.
The meson propagator up to next to leading order in the 1/N expansion. In the present work we consider only the pion propagator. The internal line stands in the upper two bubbles for a pion and a scalar and in the lower graph for a pion and a scalar insertion. The same types of diagrams emerge for the calculation of the weak pion decay constant, if one interprets the simple dashed line as well as one of the external lines in the bubble diagrams to be the axial vector current and the other the pseudoscalar meson.

Figure 1. The effective action with the explicit dependence on N of the several one-loop quark contributions. The dashed lines represent the meson fields. To the N^1/2 term contributes only the scalar meson; the propagator term, of order N⁰, stands for the pion and the sigma mesons; the three point function proportional to N^1/2 involves either three scalars or one scalar coupled to two pions; the four point function of order N^-1 stands for either all pions or all scalars or two pions and two scalars.

Figure 2. The next to leading order in N to the gap equation. The internal meson line stands for a pion or a sigma.

Figure 3. The quark propagator to the next to leading order in the 1/N expansion. The thin lines denote fermion propagators in the dominant order, the dashed internal lines stand for pion and scalar insertions. The external line of the tadpole is a scalar.

Figure 4. The meson propagator up to next to leading order in the 1/N expansion. In the present work we consider only the pion propagator. The internal line stands in the upper two bubbles for a pion and a scalar and in the lower graph for a pion and a scalar insertion. The same types of diagrams emerge for the calculation of the weak pion decay constant, if one interprets the simple dashed line as well as one of the external lines in the bubble diagrams to be the axial vector current and the other the pseudoscalar meson.

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Meson loop corrections to the NJL model

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