## Project Overview

Symmetry and symmetry breaking are the most fundamental concepts in modern particle theory, as symbolized by the 2008 Nobel prize in physics. In the Standard Model gauge symmetry is essential and the mass of all the particles is generated by the spontaneous symmetry breaking, a concept initiated by Nambu (Y.Nambu and G.Jona-Lasinio,1960). The CP symmetry breaking mechanism proposed is also due to nonzero quark mass and deeply tied with the origin of mass (M. Kobayashi and T. Maskawa, 1973). Thus the origin of mass and symmetry breaking is the most urgent problem of the particle physics today and as such is the main target of the LHC experiments.

In this Project we will pursue symmetries and breaking (Nonlinear Realization or nonlinear sigma model) of the symmetries in particle physics and its dynamical origin via gauge dynamics with particular emphasis on the relation to the origin of mass. In QCD the (chiral) symmetry is spontaneously broken by the gauge dynamics and appears in its nonlinear realization, giving rise to dynamical generation of the mass of quarks, or nucleons (hadrons). Before advent of QCD without knowledge of the underlying gauge dynamics, it was extremely important to clarify the general structure of the nonlinear realization (low energy effective theory) of the symmetry, which is quite independent of the details of the underlying gauge dynamics and can be directly compared with the reality. Even long after discovery of QCD, it is well known that the first principle calculation of the lattice QCD simulations is greatly improved when combined with the nonlinear realization with loop corrections (chiral perturbation).

Makawa has been persistently studying the nonlinear realization of symmetry: From chiral symmetry in the very early stage（C. Hattori, M. Kobayashi, T. Maskawa and H. Kondo; M. Kobayashi and T. Maskawa, 1970） to internal symmetries in the framework of unbroken supersymmety (M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, 1984) and supersymmetry itself at present. Without knowing QCD, they introduced U(1)A symmetry breaking term (often called 't Hooft determinant) in the nonlinear realization (M. Kobayashi and T. Maskawa, 1970) long before 't Hooft calculation in QCD. The nonlinear realization was further advanced as an extension, so-called "Hidden Local Symmetry (HLS)" theory, so as to incorporate low-lying (composite) vector particles like rho mesons as gauge bosons (M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, 1985; M. Bando, T. Kugo and K. Yamawaki, 1985 and 1988). The HLS theory was further developed into gauge theories with "extra dimensions" by identifying vector spectra (incl. higher resonances) as the Kaluza-Klein modes. It was then discovered that the HLS is a natural outcome of the holographic gauge theory derived from the string theory (T. Sakai and S. Sugimoto, 2005). Today the HLS is one of the central concepts of recent model building like the Moose Models, Little Higgs Models, Higgsless Models, Holographic Models, not to mention the old notions like Walking Technicolor, Composite Models (for Higgs, W/Z, etc.), etc.. The loop corrections (chiral perturbation) was formulated in the HLS framework and proposed a new pattern of nonlinear realization "Vector Manifestation (VM)" based on it (M. Harada and K. Yamawaki, 2001 and 2003).

The gauge dynamics responsible for the nonlinear realization in QCD was first formulated in the ladder Schwinger-Dyson equation with non-running gauge coupling in the scale-invariant form (T. Maskawa and H. Nakajima, 1974). This was later applied to the Technicolor as what we called "Scale-invariant Technicolor" with anomalous dimension of unity, later dubbed as "Walking Technicolor (WTC)" (K. Yamawaki, M. Bando and K. Matumoto, 1986), which is a conformal gauge dynamics producing a composite Higgs boson, technidilaton, associated with approximate conformal symmetry. The model has attracted world-wide attention, particularly in the context of lattice simulations and LHC phenomenology now. This is typically the dynamics with large anomalous dimension, and we also proposed "Top Quark Condensate" as another model of this class (V.A. Miransky, M. Tanabashi and K. Yamawaki, 1989) as well as "Strong ETC Technicolor" (V.A. Miransky and K. Yamawaki, 1989). It should be mentioned here that somewhat heretic dynamics called discrete light-cone quantization (DLCQ) (T. Maskawa and K. Yamawaki, 1976) is a potentially hopeful approach alternative to lattice simulations.

"Kobayashi-Maskawa Institute for the Origin of Particles and the Universe"(T. Maskawa, Director) created at Nagoya University is for cultivating new directions of research, particularly the symmetry breaking and the nonperturbative dynamics in quantum field theory. For that end a new high speed cluster computer was installed for the particle theory activity at the Institute. It is geared for the lattice simulations of gauge dynamics for the models beyond the standard model such as the WTC and the QCD matter under extreme conditions. We will also pursue the general structure of nonlinear realization for the general symmetry including the supersymmetry and HLS (including "Hidden Supergravity"). Combined with other methods developed over many decades, analytical gauge dynamics and the nonlinear realization (chiral perturbation) with HLS, the lattice simulations will give us new insights of the field theory. The present Project aims at greatly facilitating the core activity of this institute. The Project also aims at cooperation with another newly created institute ("Maskawa Toshihide Research and Education Center"; T. Maskawa, Director) at Kyoto Sangyo University.

Thus the main target of this project is to study the nonlinear realization and its gauge dynamics in relation to the origin of mass, which will be tested at LHC and the (hot/dense) QCD physics to be tested at RHIC, J-PARC and LHC.

The composite Higgs models with large anomalous dimension are now standard basis for the current activities for the Higgs physics based on the nonlinear realization and are particularly combined with the HLS, recently often renamed as "Moose", which is now widely understood as the basis for the Kaluza-Klein modes of the gauge bosons in extra dimensions and hence is a key concept of the currently fashionable model buildings on the composite Higgs models; Little Higgs, Higgsless Models, Extra Dimensions, Holography, etc.. These composite models would be tested by the LHC experiments. Furthermore, the HLS and VM have been vigorously applied to hot and dense QCD in the context of Brown-Rho scaling to be tested by heavy ion collider experiments at RHIC and J-PARC, etc.

The modern version of the WTC is often made explicit by the "Large Nf QCD", a QCD with the number of massless flavors Nf much larger than the real-life QCD, which is expected to have an infrared conformal fixed point where the theory becomes scale-invariant or conformal in such a way that the composite condensate gets resolved. Such a phase transition has an unusual feature, what we called "Conformal Phase Transition" (V.A.Miransky and K.Yamawaki, 1997). Most of our works on the WTC were done through the SD gap equation and HLS effective theory, which should be checked by a more precise first-principle method of the lattice computer simulations to make the prediction more reliable and testable at the LHC experiments. Recent developments of computer ability and algorithm enable us to attack these nonperturbative aspects of the gauge dynamics. Very recently, many groups in all over the world have rushed into the lattice simulations on the conformal phase transition.

## Grants

JSPS Grant-in Aid for Scientific Research

S (PI: Toshihide Maskawa FY 2010-2014): 2012 Interim Report

C (PI: Koichi Yamawaki FY 2011-2013)

JSPS Grant-in-Aid for JSPS Fellow

(Juni Jia: Host Scientist Koichi Yamawaki, FY2011-2013 (2 years))

Daiko Foundation

(PI: Toshihide Maskawa FY 2011-2013, 2 years)