Chiral Perturbation in the Hidden Local Symmetry and Vector Manifestation of Chiral Symmetry ∗

In this talk I summarize our recent works on the chiral phase transition in the large flavor QCD studied by the hidden local symmetry (HLS). Bare parameters in the HLS are determined by matching the HLS with the underlying QCD at the matching scale through the Wilsonian matching. This leads to the vector manifestation of the Wigner realization of the chiral symmetry in which the symmetry is restored by the massless degenerate pion (and its flavor partners) and rho meson (and its flavor partners) as the chiral partner. Chiral phase transition in QCD is discussed in various contexts such as the large flavor QCD and the hot and/or dense QCD, etc. Recently, in Refs. 1, 2 and 3, we studied the chiral phase transition in the large flavor QCD using the hidden local symmetry (HLS) model [4], which is an effective field theory of QCD including vector and pseudoscalar mesons. The chiral symmetry restoration in the large Nf QCD (< 11 2 Nc) was implied by the fact that the coupling at the infrared fixed point becomes very small [5]. Such a restoration was indeed observed by various methods like lattice simulation [6], ladder Schwinger-Dyson equation [7, 8], dispersion relation [9], instanton calculus [10], etc. In Ref. 1 we pointed that the chiral restoration takes place for large Nf also in the HLS by its own dynamics. Inclusion of the quadratic divergences in the renormalization group equations (RGE’s) was essential to obtain the phase transition. Here I should emphasize that thanks to the gauge symmetry in the HLS it is possible to perform a systematic loop expansion including the vector mesons in addition to the pseudoscalar mesons [11, 12, 13, 14, 2] in a way to extend the chiral perturbation theory [15, 16]. There the loop expansion corresponds to the derivative expansion, so that the one-loop calculation of the RGE is reliable in the low-energy region. The quadratic divergence in the RGE’s yields the quadratic running of (square of) the decay constant F 2 π (μ), where μ is the renormalization point. What is shown in Ref. 1 is that the order parameter Fπ(0) can become zero for larger Nf even when Fπ(Λ) 6= 0, where Fπ(Λ) is not the order parameter but just a parameter of the bare Lagrangian defined at the cutoff Λ where the matching with QCD is made. In Ref. 2 we proposed a novel way of matching the HLS with the underlying QCD in the sense of a Wilsonian RGE, namely, including quadratic divergences in the HLS (“Wilsonian matching”). The basic tool of the Wilsonian matching is the Operator Product Expansion (OPE) of QCD for the axialvector and vector current correlators, which are equated with those from the HLS at the matching scale Λ. This determines without much ambiguity the bare parameters of the HLS defined at the scale Λ in terms of the QCD parameters. It is shown that the physical quantities for the π and ρ system are calculated by the Wilsonian RGE’s from the bare parameters in remarkable agreement with experiment. In Ref. 3 we applied the Wilsonian matching for the large Nf QCD, and proposed “Vector Manifestation” (VM) of the chiral symmetry as a novel manifestation of the Wigner realization in which the vector meson denoted by ρ (ρ meson and its flavor partner) becomes massless at the chiral phase transition point. Accordingly, the (longitudinal) ρ becomes the chiral partner of the Nambu-Goldstone (NG) boson denoted by π (pion and its flavor partners). ∗Talk presented at 8th International Symposium on Particle Strings and Cosmology (PASCOS 2001), Chapel Hill, North Carolina, 10-15 Apr 2001. This talk is based on the works done in collaboration with Prof. Yamawaki [1, 2, 3].