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Electroweak Interactions in a Chiral Effective Lagrangian for Nuclei
[chapter]

Brian D., Xilin Zhang

2012
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Advances in Quantum Field Theory
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Covariant meson-baryon effective field theories of the nuclear many-body problem (often called quantum hadrodynamics or QHD) have been known for many years to provide a realistic description of the bulk properties of nuclear matter and heavy nuclei. 4 www.intechopen.com 2 Will-be-set-by-IN-TECH ago, a QHD effective field theory (EFT) was proposed (Furnstahl et al., 1997) that includes all of the relevant symmetries of the underlying QCD. In particular, the spontaneously broken SU(2) L ⊗ SU(2) R
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... n SU(2) L ⊗ SU(2) R chiral symmetry is realized nonlinearly. The motivation for this EFT and illustrations of some calculated results are discussed in example. This QHD EFT has also been applied to a discussion of the isovector axial-vector current in nuclei (Ananyan et al., 2002) . This QHD EFT has three desirable features: (1) It uses the same degrees of freedom to describe the currents and the strong-interaction dynamics; (2) It respects the same internal symmetries, both discrete and continuous, as the underlying QCD; and (3) Its parameters can be calibrated using strong-interaction phenomena, like π N scattering and the properties of finite nuclei (as opposed to EW interactions with nuclei). In this work, we focus on the introduction of EW interactions in the QHD EFT, with the Delta (1232) resonance (Δ) included as manifest degrees of freedom. To realize the symmetries of QCD in QHD EFT, including both chiral symmetry SU(2) L ⊗ SU(2) R and discrete symmetries, we apply the background-field technique (Gasser & Leutwyler, 1984; Serot, 2007) . Based on the EW synthesis in the Standard Model, a proper substitution of background fields in terms of EW gauge bosons in the lagrangian, as constrained by the EW interactions of quarks (Donoghue et al., 1992) , leads to EW interactions of hadrons at low energy. This lagrangian has a linear realization of the SU(2) V isospin symmetry and a nonlinear realization of the spontaneously broken SU(2) L ⊗ SU(2) R (modulo SU(2) V ) chiral symmetry (when the pion mass is zero). It was shown in Ref. (Furnstahl et al., 1997) that by using Georgi's naive dimensional analysis (NDA) (Georgi, 1993) and the assumption of naturalness (namely, that all appropriately defined, dimensionless couplings are of order unity), it is possible to truncate the lagrangian at terms involving only a few powers of the meson fields and their derivatives, at least for systems at normal nuclear densities (Müller & Serot, 1996) . It was also shown that a mean-field approximation to the lagrangian could be interpreted in terms of density functional theory (Kohnso that calibrating the parameters to observed bulk and single-particle nuclear properties (approximately) incorporates many-body effects that go beyond Dirac-Hartree theory. Explicit calculations of closed-shell nuclei provided such a calibration and verified the naturalness assumption. This approach therefore embodies the three desirable features needed for a description of electroweak interactions in the nuclear many-body problem. Moreover, the technical issues involving spin-3/2 degrees of freedom in relativistic quantum field theory are also discussed here (Krebs et al., 2010; Pascalutsa, 2008) . Following the construction of the lagrangian, we apply it to calculate certain matrix elements to illustrate the consequences of chiral symmetries in this theory, including the conservation of vector current (CVC) and the partial conservation of axial-vector current (PCAC). To explore the discrete symmetries, we talk about the manifestation of G parity in these current matrix elements. This chapter is organized as follows: After a short introduction, we discuss chiral symmetry and discrete symmetries in QCD in the framework of background fields. The EW interactions of quarks are also presented, and this indicates the relation between the EW bosons and background fields. Then we consider the nonlinear realization of chiral symmetry and other symmetries in QHD EFT, as well as the EW interactions. Following that, we outline the lagrangian with the Δ included. Subtleties concerning the number of degrees of freedom 76 Advances in Quantum Field Theory

doi:10.5772/38331
fatcat:iowqujxoebhnhdgvylvvhgcglu