We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed them explicitly for3 ) and ∆(6N 2 ), which have been applied for model building in the particle physics. We also present typical flavor models by using A 4 , S 4 , and ∆(54) groups. Breaking patterns of discrete groups and decompositions of multiplets are important for applications of the non-Abelian discrete symmetry. We discuss these breaking patterns of the non-Abelian discrete group, which are a powerful tool for model buildings. We also review briefly about anomalies of non-Abelian discrete symmetries by using the path integral approach.

We investigate chiral and conformal properties of the lattice QCD with eight flavors (N f = 8) through meson spectrum using the Highly Improved Staggered Quark (HISQ) action. We also compare our results with those of N f = 12 and N f = 4 which we study on the same systematics. We find that the decay constant Fπ of the pseudoscalar meson "pion" π is non-zero, with its mass Mπ consistent with zero, both in the chiral limit extrapolation of the chiral perturbation theory (ChPT). We also measure other quantities which we find are in accord with the π data results: The ρ meson mass is consistent with non-zero in the chiral limit, and so is the chiral condensate, with its value neatly coinciding with that from the Gell-Mann-Oakes-Renner relation in the chiral limit. Thus our data for the N f = 8 QCD are consistent with the spontaneously broken chiral symmetry. Remarkably enough, while the N f = 8 data near the chiral limit are well described by the ChPT, those for the relatively large fermion bare mass m f away from the chiral limit actually exhibit a finite-size hyperscaling relation, suggesting a large anomalous dimension γm ∼ 1. This implies that there exists a remnant of the infrared conformality, and suggests that a typical technicolor ("onefamily model") as modeled by the N f = 8 QCD can be a walking technicolor theory having an approximate scale invariance with large anomalous dimension γm ∼ 1.2

We present the first observation of a flavor-singlet scalar meson as light as the pion in N f = 8 QCD on the lattice, using the Highly Improved Staggered Quark action. Such a light scalar meson can be regarded as a composite Higgs with mass 125 GeV. In accord with our previous lattice results showing that the theory exhibits walking behavior, the light scalar may be a technidilaton, a pseudo Nambu-Goldstone boson of the approximate scale symmetry in walking technicolor. PACS numbers: 11.15.Ha, 12.39.Mk, 12.60.Nz, 14.80.Tt Recently, a Higgs boson with mass around 125 GeV has been discovered at the Large Hadron Collider (LHC) [1,2]. While the current LHC data show good agreement with the Standard model Higgs boson, there exists a possibility that the Higgs boson is a composite particle in an underlying strongly coupled gauge theory. A typical example is the walking technicolor theory, featuring approximate scale invariance and a large anomalous dimension, γ m ≈ 1 [3] (see also similar works [4][5][6]). Such a theory predicts a light composite Higgs, "technidilaton" [3], emerging as a pseudo Nambu-Goldstone (NG) boson of the spontaneously broken approximate scale symmetry. It was shown [7,8] that the technidilaton is phenomenologically consistent with the current LHC data.Thus, the most urgent theoretical task to test walking technicolor theories would be to check whether or not such a light flavor-singlet scalar bound state exists from first-principle calculations with lattice gauge theory. Since the composite Higgs should be associated with the electroweak symmetry breaking, it must be predominantly a bound state of technifermions carrying electroweak charges, but not of technigluons having no electroweak charges (up to some mixings between them). Thus we look for a light flavor-singlet scalar meson in the correlator of fermionic operators on the lattice.One of the most popular candidates for walking technicolor theories is QCD with a large number of (massless) flavors (N f ) in the fundamental representation. For the past few years, we have studied the SU(3) gauge theory with N f = 4, 8, 12, and 16, in a common lattice setup [9][10][11]. (For reviews of lattice studies in search for candidates for walking technicolor theories, see [12][13][14][15].)In N f = 12 QCD we actually observed [11, 16] a flavorsinglet scalar meson (σ) lighter than the "pion" having the quantum numbers corresponding to the NG pion (π) in the broken phase. (Recently a light flavor-singlet scalar meson consistent with ours was also observed by another group [17] using a different lattice action.)We found [9] that N f = 12 QCD is consistent with a conformal theory. If it is a conformal theory, it should have no bound states ("unparticle") in the exact chiral limit, and hence a light bound state can only be formed in the presence of a fermion mass m f which explicitly (not spontaneously) breaks the scale/chiral/electroweak symmetry.Hence such a light scalar meson in N f = 12 QCD would not be a composite Higgs associated with the spontaneou...