Hidden Local Symmetry at Loop -- A New Perspective of Composite Gauge Boson and Chiral Phase Transition

TL;DR

This paper develops a Wilsonian, loop-level effective field theory for QCD-like theories using Hidden Local Symmetry (HLS), treating vector mesons as dynamical gauge bosons and including quadratic divergences to capture chiral phase transitions. It demonstrates that Wilsonian matching to QCD fixes the bare HLS parameters and predicts the Vector Manifestation (VM) at chiral restoration, where m_rho and F_pi vanish and rho becomes the chiral partner of the massless pions in the large $N_f$ limit. The work shows that Vector Dominance is not fundamental but emergent and may be violated near the VM point, and it connects vector-meson dynamics to conventional CHPT through vector-meson saturation of low-energy constants. It also emphasizes the physical meaning of quadratic divergences in EFTs, illustrating their role in phase transitions across models such as the NJL, CP^{N-1}, and the Standard Model, and highlighting implications for hot/dense QCD and beyond-Standard-Model constructions.

Abstract

We develop an effective field theory of QCD and QCD-like theories, based on the hidden local symmetry (HLS) model. We formulate the chiral perturbation theory with loops of rho as well as pi and further include quadratic divergences which are crucial to the chiral phase transition. Detailed calculations of the one-loop renormalization-group equation of the parameters of the HLS model are given, with the bare parameters (defined at cutoff Lambda) determined by the matching (``Wilsonian matching'') with the underlying QCD at Lambda through the operator-product expansion of current correlators. The Wilsonian matching predicts low energy phenomenology for the three-flavored QCD in remarkable agreement with the experiments. When the chiral symmetry restoration takes place in the underlying QCD, the HLS model uniquely leads to the Vector Manifestation (VM) as a new pattern of Wigner realization of chiral symmetry, with the rho becoming degenerate with the massless pi as the chiral partner. The VM is in fact realized in the large Nf QCD when Nf to Nf^{crit}-0, with Nf^{crit} = 5 Nc/3 being in rough agreement with the lattice simulation for Nc=3. The VM can be realized also in hot and/or dense QCD.

Figures (43)

• Figure 1: Schematic view of $P$-wave $\pi\pi$ scattering amplitude.
• Figure 2: Electromagnetic form factor in the HLS: (a) the direct $\gamma\pi\pi$ interaction from ${\cal L}_{\rm A}$ term in the Lagrangian (\ref{['leading Lagrangian 0']}); (b) the direct $\gamma\pi\pi$ interaction from $a {\cal L}_{\rm V}$ term; (c) the $\gamma\pi\pi$ interaction mediated by $\rho$ exchange.
• Figure 3: Diagrams contributing to the $\pi\pi$ scattering in the HLS: (a) contribution from the contact 4$\pi$-interaction from ${\cal L}_{\rm A}$ term in the Lagrangian (\ref{['leading Lagrangian 0']}); (b) contribution from the contact 4$\pi$-interaction from $a {\cal L}_{\rm V}$ term; (c) contribution from the $\rho$-exchange. The diagram (c) implicitly includes three diagrams: $s$-channel, $t$-channel and $u$-channel $\rho$-exchange diagrams.
• Figure 4: Effective $\pi^0\gamma^\ast\gamma^\ast$ vertex: (a) direct $\pi^0\gamma\gamma$ interaction $\propto (1-c_4)$; (b) and (c) through $\pi^0 \omega\gamma$ and $\pi^0\rho^0\gamma$ interactions $\propto (c_4-c_3)$; (d) through $\omega\rho^0\pi^0$ interaction $\propto c_3$.
• Figure 5: Effective $\omega\pi^0\gamma^\ast$ vertex: (a) direct $\omega\pi^0\gamma$ interaction $\propto (c_3-c_4)$; (b) through $\omega\rho^0\pi^0$ interaction $\propto c_3$.