We present a covariant formulation and model calculations of the leading-twist time-reversal even transverse momentum-dependent quark distribution functions (TMDs) for a spin-one target. Emphasis is placed on a description of these 3-dimensional distribution functions which is independent of any constraints on the spin quantization axis. We apply our covariant spin description to all nine leading-twist time-reversal even ρ meson TMDs in the framework provided by the Nambu-Jona-Lasinio model, incorporating important aspects of quark confinement via the infrared cutoff in the proper-time regularization scheme. In particular, the behavior of the 3-dimensional TMDs in a tensor polarized spin-one hadron are illustrated. Sum rules and positivity constraints are discussed in detail. Our results do not exhibit the familiar Gaussian behavior in the transverse momentum, and other results of interest include the finding that the tensor polarized TMDs-associated with spin-one hadronsare very sensitive to quark orbital angular momentum, and that the TMDs associated with the quark operator γ + γ T γ 5 would vanish were it not for dynamical chiral symmetry breaking. In addition, we find that 44% of the ρ meson's spin is carried by the orbital angular momentum of the quarks, and that the magnitude of the tensor polarized quark distribution function is about 30% of the unpolarized quark distribution. A qualitative comparison between our results for the tensor structure of a quark-antiquark bound state is made to existing experimental and theoretical results for the two-nucleon (deuteron) bound state.

In order to describe the hadronization of polarized quarks, we discuss an extension of the quark-jet model to transverse momentum dependent fragmentation functions. The description is based on a product ansatz, where each factor in the product represents one of the transverse momentum dependent splitting functions, which can be calculated by using effective quark theories. The resulting integral equations and sum rules are discussed in detail for the case of inclusive pion production.In particular, we demonstrate that the 3-dimensional momentum sum rules are satisfied naturally in this transverse momentum dependent quark-jet model. Our results are well suited for numerical calculations in effective quark theories, and can be implemented in Monte-Carlo simulations of polarized quark hadronization processes. PACS numbers: 13.60.Hb, 13.60.Le, 12.39.Ki 3 Like Γ µ and σ µ , the quantity s µ is not a Lorentz 4-vector, but Einstein's summation convention still applies[50].

This study investigated modulation of the long period stacking order (LPSO) structure in aged Mg 97 Zn 1 Y 2 alloys using conventional transmission electron microscopy (TEM) and aberration-corrected high-angle annular dark field-scanning transmission electron microscopy (HAADF-STEM). The irregular stacking sequence of a fragment of 24R-type LPSO acts as a catalyst for the transformation from 18R-to 14H-type LPSO. The elementary step of the transformation from 18R-to 24R-type takes place by the ledge-pair movement on different (0001) Mg planes with Shockley partial dislocations. Each ledge has a transition region in front of it. The transition regions are HCP-type stacking sequence with lower Zn and Y concentrations than those of the FCC-type enrichment layer. The solute elements migrate easily in the region, where solute elements produce a kind of diffusion field. Therefore, structural modulation occurs by a mechanism resembling diffusionaldisplacive transformation. Local strain analysis using HAADF-STEM images has elucidated that lattice spacing of (0001) Mg in the FCC-type enrichment layer is shorter than that in the HCP-type transition region. These structural and compositional irregularities are an elementary step in the transformation of LPSO in Mg 97 Zn 1 Y 2 alloys. A diffusionaldisplacive type transformation mechanism in LPSO has been proposed.

The structure of hadrons is described well by the Nambu-Jona-Lasinio (NJL) model, which is a chiral effective quark theory of QCD. In this work we explore the electromagnetic structure of the pion and kaon using the three-flavor NJL model in the proper-time regularization scheme, including effects of the pion cloud at the quark level. In the calculation there is only one free parameter, which we take as the dressed light quark (u and d) mass. In the regime where the dressed light quark mass is approximately 0.25 GeV, we find that the calculated values of the kaon decay constant, current quark masses, and quark condensates are consistent with experiment and QCD based analyses. We also investigate the dressed light quark mass dependence of the pion and kaon electromagnetic form factors, where comparison with empirical data and QCD predictions also favors a dressed light quark mass near 0.25 GeV.