XX Meson aka η′\eta' and Kobayashi-Maskawa-'t Hooft Six-quark Vertex -- U(1)AU(1)_A Anomaly and Generalized Nambu-Jona-Lasinio Model --

In 1970, Kobayashi and Maskawa concluded that an effective six-quark vertex with a determinantal form is necessary in the chiral effective models to account for the large mass of $X$ meson, which is now called the $\eta'$. The determinantal interaction has an $SU(3)_L \otimes SU(3)_R$ symmetry but not $U(3)_L \otimes U(3)_R$, and, hence accounts for the explicit breaking of $U(1)_A$ symmetry in quantum chromodynamics (QCD); the vertex was later derived by 't Hooft as an instanton-induced quark interaction. The vertex, which may be called the Kobayashi-Maskawa-'t Hooft (KMT) term, is widely used in quantitative analyses of hadron physics and QCD phase transitions at finite temperature and density. An account is made for the KMT term with recent extensive applications. Described are also personal experiences with Professor Maskawa and Professor Kobayashi,including an encounter with Professor Maskawa when the author first presented his work on the KMT term.Comment: A few typos are correcte

Chiral Transition and the Scalar and Vector Correlations

The properties of the scalar and vector correlations in the hot and/or dense hadronic matter close to chiral transition are discussed. Presuming that the linear realization of chiral symmetry will become appropriate at least near the critical point, we argue that the strength function in the $I=J=0$ channel will soften near the critical point, and the sigma meson will accordingly become a clearer resonance in hot and/or dense medium than in the free space. It is shown that the steep rise of the baryon-number susceptibility $\chi_B$ around the critical point seen in the lattice simulations may suggest that the interactions between quarks in the vector channel become week near the critical point; this implies that the peculiar behavior of the $\chi_B$ does not necessarily imply a proliferation of baryons and antibaryons (Skyrmions) which is though a natural scenario of chiral transition in the non-linear realization of chiral symmetry.Comment: Talk presented at International Symposium on Hadrons and Nuclei, Seoul, Feb. 20 - 22, 2001. No figures. 6 page

Uses of Envelopes for Global and Asymptotic Analysis; geometrical meaning of renormalization group equation

We give a comprehensive review of the renormalization group method for global and asymptotic analysis, putting an emphasis on the relevance to the classical theory of envelopes and the existence of invariant manifolds of the dynamics under consideration. We clarify that an essential point of the method is to convert the problem from solving differential equations to obtaining suitable initial (or boundary) conditions. We mention that the notion of envelopes is also useful for constructing global and asymptotic behavior of wave functions of quantum systems such as the ones with the quartic potential or double-well potential.Comment: Talk presented at RIMS (Kyoto) Workshop on Geometrical Methods for Asymptotic Analysis 1997.5.20 -- 5.23. LaTex, 15 page

Low Mass Scalar Mesons and Related Topics

We give a brief survey on the physical significance of the low-mass scalar mesons in QCD, and also report on recent lattice studies on the sigma and kappa mesons. The importance to explore the in-medium properties of the hadrons is mentioned.Comment: talk presented at Yukawa International Seminar(YKIS) 2006 "New Frontiers in QCD -- Exotic Hadrons and Hadronic Matter --", November 20 to December 8, 2006, at Yukawa Institute for Theoretical Physics, Kyoto University. 4 pages.tobe published in the proceedings(Supplemet of Prog. Theor. Phys.