---
author:
- 'Mannque <span style="font-variant:small-caps;">Rho</span>$^{1,}$[^1]'
title: |
    Hidden Local Symmetry and the Vector Manifestation\
    of Chiral Symmetry in Hot and/or Dense Matter[^2] 
---

Introduction
============

In the current understanding of quantum chromodynamics (QCD), nearly all of the masses of low-lying hadrons (e.g., $\sim 98$% of the proton mass) come from spontaneous breaking of chiral symmetry. A hot issue raised currently by hadron/nuclear physicists is: How to un-break the broken symmetry in terrestrial laboratories or in compact stellar systems and figure out how the mass is generated to start with. I would like to address this issue in this talk using the framework Gerry Brown and I have been developing since some time, specifically, ideas based on hidden local symmetry, the “vector manifestation" of chiral symmetry and Brown-Rho scaling.

The above question belongs broadly to a class of issues connected to the origin of mass. In a glaring departure from molecules, atoms and nuclei whose masses are nearly fully, say, more than 99%, accounted for by their “elementary" constituents, the bulk of the mass of the nucleon is not given by the masses of its constituents, namely, the quarks. Thus this sets the beginning of a series of mysteries related to the question “where does the mass come from?"

The clue to the origin of the mass of quarks and leptons, the truly fundamental issue, will probably be revealed in the coming years, perhaps at LHC/CERN. What we are concerned with here is the issue – perhaps less fundamental – we can address with what we already have at our disposal and will have in the near future and deals with strongly interacting matter when it is heated to several hundred MeV in temperature or is compressed to several times the density of the nuclear matter.

The Origin of Hadron Mass
=========================

Let us take the proton as a typical hadron. The argument goes in a similar way for, say, the $\rho$ meson mass which is currently a hot topic of experimental efforts. The simplest way to proceed is to think in terms of constituent quarks whose masses are dynamically generated by the complex vacuum.

In QCD, the proton is made up of three light (“chiral") quarks, the total mass of which is tiny, a few MeV, compared with the proton mass which is $\sim$ 1000 MeV. It is believed - and the evidence is strong - that most of the proton mass comes from the “spontaneous" breaking of chiral symmetry by the vacuum, with the order parameter given by the quark condensate $\la\bar{q}q\ra$. The mass so generated must therefore be connected intimately to the quark condensate. So in the chiral limit, one should be able to write the hadron mass as a function of the condensate as m=F(|[q]{}q). As an order parameter, the condensate $\la\bar{q}q\ra$ will be assumed to go to zero when chiral symmetry is restored. It may go to zero either in a smooth way or in a discontinuous way when the system is driven by the external condition - temperature or density - to the critical point. Now even if the condensate goes to zero smoothly, the $F$ which could be a complicated function of the condensate may go to zero in a discontinuous way. At present, there is no known analytical way to settle this issue using QCD proper. So how $F$ behaves as the critical point is reached is not known. But it seems very natural to expect that F(|[q]{}q)0  [as]{} |[q]{}q)0. This is basically the basis of Brown-Rho scaling [@BR:91] and is supported by Harada-Yamawaki’s hidden local symmetry theory with the “vector manifestation" fixed point [@HY:PR]. We will call this Harada-Yamawaki theory “HLS/VM" for short.

We should however mention that this is not the only possibility. In fact, certain models can admit a non-zero $F(0)$, hence a non-zero mass, and still preserve chiral symmetry and allow a Goldstone-Wigner phase transition. For instance, one can have a linear sigma model with parity-doublet fermion fields [@parity-doublet]. In this model, the fermion mass can be non-zero in both the Goldstone and Wigner phases. The fermions can be baryons or constituent (quasi) quarks. In the former case, one can have massless mesons while having massive baryons at the critical point. In the latter case, both mesons and baryons as (weakly) bound states of quasiquarks could remain massive at the critical point with the chiral symmetry preserved. At present, neither theory nor experiment can rule out this scenario. However we find it unnatural within the HLS framework we are adopting. We shall not pursue this alternative scenario.

Hidden Local Symmetry (HLS)
===========================

The well-established perturbative QCD cannot access the highly nonperturbative and strong-coupling regime at low energy we are concerned with. The only tool available at present is effective theories that meet Weinberg’s “folk theorem" [@weinberg-folk] on effective field theories. There is some help from lattice calculations but at the present stage, not much on what happens to light-quark hadrons, e.g., nucleons, vector mesons, pions etc. when they are immersed in hot and/or dens medium.

The question we are interested in answering is: If the hadron mass vanishes at the phase transition, how can one “see" it or what is the appropriate tool for it?

In what follows, we shall mostly deal with the chiral limit. Our principal theme in addressing the above question is that [*the most important ingredient in an effective field theory that enables one to probe the regime where the effective mass of the vector meson can drop to that of the pion mass is hidden local symmetry.*]{} Our theme is anchored on the argument by Harada and Yamawaki [@HY:PR] that without hidden local symmetry, there is no consistent [*as well as*]{} simple way to allow the vector mass become as light as the pion mass. We suggest that the lack of hidden gauge symmetry is the reason why phenomenological models often used in the literature fail to observe dropping masses in hot/dense matter.

HLS as emergent gauge symmetry
------------------------------

At very low energy $E\ll \Lambda_\chi$ where $\Lambda_\chi\approx
4\pi f_\pi\sim 1$ GeV, the only relevant degrees of freedom are the pions, that is, Goldstone bosons, emerging from the spontaneous breaking of chiral symmetry $SU(N_f)_L\times SU(N_f)_R$ to $SU(N_f)_{L+R}$. The dynamics involving the pion field, $U=exp(2i\pi/F_\pi)$ [^3] is encoded in a chiral Lagrangian expanded in derivatives, the leading term of which is given by low-energy theorems. Now we make the most obvious and “trivial" observation that one can always write the $U$ field in a product form if one is willing to introduce a redundant field. Define the L/R chiral fields with the redundant field $\sigma$ \_[L,R]{}=e\^[i/F\_]{}e\^[i/F\_]{}\[chifields\] $U$ can then be rewritten as U=\_L\^\_R. In this form, we unearth a local symmetry \_[L,R]{}h(x)\_[L,R]{} with $h(x)\in SU(N_f)_V$. As it stands, we have not done anything new, so there is no new physics here. However if we elevate the local symmetry to a local gauge symmetry by introducing a gauge field – which we will denote by $V_\mu \in SU(N_f)_V$ – and endow it with a kinetic energy term, then it becomes quite a different story. First of all, this procedure allows one to go up systematically in some expansion scheme (such as chiral expansion) in energy from the low energy scale where the pionic chiral Lagrangian is applicable to a scale at which new degrees of freedom set in [@georgi-idea], thereby circumventing the breakdown of the pionic chiral theory and going beyond to the next energy scale. In our case, the scale is brought above the mass of the vector mesons $\rho$, $\omega$, $a_1$ etc. Furthermore. what is more important for our case is that thanks to local gauge symmetry, one can do a systematic chiral perturbation calculation with the vector mesons put on the same scale as the pions and access the regime where the vector mass is comparable to the pion mass.

The hidden local symmetry theory of Harada and Yamawaki[@HY:PR] – which is based on the earlier work of Bando et al. [@bandoetal] – was constructed in this way with the vector mesons $\rho$ and $\omega$ (which we label as $V_1$) as the hidden local fields together with the Goldstone pions.

This strategy of introducing a hidden gauge field can be extended to as many gauge fields as required by the energy scales one wants to deal with. There is however one important caveat here. Introducing gauge degrees of freedom in this way as a way of going up in energy scales does not necessarily lead to a unique gauge theory [@weinberg-largeNc]. Different constructions could give rise to different gauge structures at higher energies. One way that is both simple and phenomenologically appealing is the linear “moose" construction [@georgi-idea]. A version of this class of construction in which an infinite tower of gauge fields are incorporated was proposed as a “dimensionally deconstructed QCD" [@son-stephanov1]. In order to make the construction correctly represent nature up to a given scale, the theory has to be matched to a fundamental theory and “ultraviolet-completed" to go beyond.

As we will elaborate further later, if one wants to study what happens to the vector mesons in medium, this strategy is definitely needed. On a more fundamental level, one can view this as a generic phenomenon of the “emergence" of local gauge degrees of freedom. Examples are numerous, e.g., emergent gravity, emergent space-time, spin-charge separation in high-T superconductivity etc [@horowitz-polchinski].

HLS from string theory
----------------------

A novel recent development in both string theory and hadronic physics is that HLS $descends$ naturally via AdS/CFT duality from string theory to holographic QCD. This is a top-down approach to hidden local symmetry theory of QCD. One such theory which astutely implements the spontaneous breaking of chiral $SU(N_f)\times
SU(N_f)$ symmetry was constructed by Sakai and Sugimoto [@SS]. The key idea in this approach is that the strongly-coupled and hence highly non-perturbative aspect of QCD in four dimensions in the limits $\lambda\equiv g^2_{YM}N_c\rightarrow \infty$ (“’t Hooft limit") and $N_c\rightarrow \infty$ (“large $N_c$ limit") can be approximated by a readily calculable weakly-coupled gravity solution in five dimensions. Since a dimensional reduction is involved, there is a Kaluza-Klein scale $M_{KK}$ that sets the energy scale of the effective theory. What results is a pure Yang-Mills theory in five dimensions dual to strongly-coupled QCD which when reduced to four dimensions can be cast in terms of an infinite tower of local vector fields coupled gauge invariantly to the pions. Thus as in the emergent case, the same type of hidden local symmetry (with an infinite tower of vector mesons and the pions) arises from top down. Remarkably this theory describes the meson sector [@SS] as well as the baryon sector [@HRYY-PRL; @HSSY] quite well.

Perhaps the most important outcome of the development is the appearance of baryons in the theory. The hidden local symmetry theory (with the infinite tower which we will label as $V_\infty$ whenever unambiguous) is the full theory of hadrons at the scale defined by the KK mass $M_{KK}$. Since there are no explicit baryons in the theory, baryons must arise through topology, namely, solitons. In five dimensions, the soliton is an instanton but reduced to four dimensions where the baryon lives, it is a skyrmion [@skyrme]. What makes this skyrmion different from the skyrmion in the Skyrme model [@skyrme] is that the soliton involves the infinite tower of vector mesons in addition to the pions encapsulated in an instanton. The geometry in five dimensions represents the dynamics of the infinite tower of vector mesons. The bulk theory is weak-coupling and manageable in the limit $\lambda\rightarrow \infty$ and $N_c\rightarrow \infty$ and provides baryon chiral dynamics anchored on hidden gauge structure.

There is a simple prediction of the theory that can be checked against experiments. Viewed in terms of the instanton, the nucleon axial coupling $g_A$ is independent of $\lambda$ and the KK scale $M_{KK}$, so it is independent of the pion decay constant. One can also calculate in the same limit the anomalous magnetic moment of the nucleon $\mu_{an}=\mu_p -\mu_n -1$ in nuclear magneton. Both are linear in $N_c$. The ratio $R=g_A/\mu_{an}$ is therefore independent of $N_c$ in the ’t Hooft and large $N_c$ limit. The instanton calculation [@HRYY-PRL] is found to give $
 g_A\approx 0.69\frac{N_c}{3} +{\cal O}(N_c^0)$ and $\mu_{an}\approx 2.16\frac{m_N}{M_{KK}}\frac{N_C}{3}+{\cal
O}(N_c^0)$. With the ${\cal O}(1)$ term ignored for the moment, the predicted ratio for $M_{KK}\approx m_N$ as required in the meson sector [@SS] comes out to be $R\approx 0.32$. The experimental values are $g_A=1.26$ and $\mu_{an}=3.7$, so the empirical ratio $R=0.34$ gives a support to the prediction.

One can go one step further and estimate the ${\cal O}(1)$ correction with no sweat. In fact, one can argue that $1/N_c$ corrections from non-planar loop contributions come only at ${\cal
O}(1/N_c)$ and higher. The ${\cal O}(1)$ corrections are common in both $g_A$ and $\mu_{an}$ and they arise in a simple book-keeping of the spin-flavor operators involved in the hedgehog configuration that is totally independent of dynamics. The leading correction is given simply by the replacement $N_c\rightarrow N_c+2$. Thus the final prediction for $N_c=3$ is $g_A \approx 1.15$ and $\mu_{an}\approx 3.6$ in close agreement with the experimental values [^4]

Another remarkable prediction of the instanton baryon description [@HRYY-long] is that the nucleon EM form factors are completely vector-dominated just as the pionic form factor is. The photon can couple directly to the soliton which ia an extended baryon but by a suitable field re-definition, one can “eliminate" the direct photon coupling and express the form factors entirely vector-dominated with the infinite tower of vector mesons entering into the formula. This result may offer a precise meaning to what was previously considered as a “intrinsic core" attributed to a quark-gluon structure in the two-component model of the nucleon structure [@2comp-NFF; @petronzio]. It furthermore brings a basically new aspect to the solitonic structure of the baryon which is now governed by the vector mesons, not by the Goldstone (pion) field as in the skyrmion with the Skyrme model. What happens to this instanton baryon in hot/dense medium is totally unknown.

At the present stage of our understanding, the holographic approach can make predictions only in the large $\lambda$ and $N_c$ limit, restricted to the zero temperature and matter-free environment. In studying hadron properties in medium, however, it is clear that one has to be able to calculate $1/N_c$ corrections since the hadron masses are locked to the quark condensate as we argued above and in the large $N_c$ limit, the quark condensate is known to be temperature-independent [@large-N-condensate]. At present, one does not know how to compute higher-order $1/N_c$ terms in the bulk sector. Clearly further progress in this direction is needed for the approach to describe what happens in the vicinity of the chiral restoration, a matter of prime importance in the field.

Vector Manifestation
====================

HLS à la Harada and Yamawaki
----------------------------

Since one cannot yet adequately exploit the dynamics of the infinite tower in confronting nature, we will rely on the HLS theory of Harada and Yamawaki (HY) [@HY:PR] which involves only the lowest members $\rho$ (and $\omega$) of the infinite tower. Let us call it HLS$_1$ in contrast to the infinite-tower HLS theory of Sakai and Sugmoto which we will call HLS$_\infty$. The HLS$_1$ theory can be interpreted as a truncated version of HLS$_\infty$ in the following sense. One picks the scale $\Lambda_M$ as a matching scale, integrates out all the members of the tower except the lowest member $V_1$ lying below $\Lambda_M$, writes HLS Lagrangian with the $V_1$ and the pion and Wilsonian-matches at $\Lambda_M$ the HLS correlators to the QCD correlators. One can obviously include in the theory other vector mesons as for instance the axial vector meson $a_1$ etc. This gives a $bare$ Lagrangian whose parameters are given by QCD variables such as the strong coupling constant $\alpha_s$, the quark condensate $\la\bar{q}q\ra$, the gluon condensate $\la
G^2_{\mu\nu}\ra$ etc. Quantum calculations are done with the bare HLS Lagrangian so determined by renormalization group equations.

An important point to note here: [*Since the condensates in the QCD sector are background-dependent, that is, dependent on temperature, density etc, the bare Lagrangian HLS$_1$ is endowed with “intrinsic (background) dependence."*]{}

The fixed point of HLS$_1$
--------------------------

In the chiral limit and with $V_1=\rho$, there are three parameters in the HLS$_1$ Lagrangian, $g$, $F_\pi$ and $a$ where $g$ is the gauge coupling constant of HLS$_1$, $F_\pi$ is the parametric pion decay constant [^5] and $a=(F_\sigma/F_\pi)^2$ figuring in the chiral fields (\[chifields\]). Harada and Yamawaki showed that the RGEs for these parameters have the fixed point consistent with QCD [^6] g\^\*=0,  a\^\*=1.\[fixed\] This was explicitly shown to one-loop order but one can readily convince oneself that it should hold to all loop orders. This fixed point dictates how the hadronic system makes a phase transition from the chiral symmetry broken phase to the restored phase. [*The fixed point (\[fixed\]) – referred to as “vector manifestation (VM) fixed point" – is reached when the quark condensate vanishes.*]{}

In HLS$_1$ (as well as HLS$_\infty$) theory, the vector meson masses are generated by the Higgs mechanism, with the scalar $\sigma$ eaten by the vector mesons. The mass formula therefore is m\_V\^2=g\^2 F\_\^2=a g\^2 F\_\^2. It has been shown that near the critical point, either $T_c$ or $n_c$, the coupling $g$ goes to zero proportionally to $\la\bar{q}q\ra$. Therefore at the chiral restoration, the parametric mass behaves m\_V\~g\~|[q]{}q0. This is consistent with Brown-Rho scaling. The physical mass of $V_1$ also vanishes at the critical point.

The parametric pion decay constant does not exhibit any special feature as $\la\bar{q}q\ra\rightarrow 0$. However the physical pion decay constant $f_\pi=F_\pi +\delta f_\pi$ must go to zero as dictated by the low-energy theorem.

Vector dominance violation
--------------------------

An important consequence for the problem we are concerned with of the VM fixed point is that the vector dominance in the EM form factors of hadrons is drastically affected. To see this, we look at the pion EM form factor. In the HLS$_1$ theory, the photon couples to the point-like pion with a coefficient $(1-a/2)$ and through the $V_1=\rho$ meson with a coefficient $a/2$. In matter-free space, $a\approx 2$, so the form factor is vector-dominated with the direct coupling vanishing. What Harada and Yamawaki found was that the $a=2$ point sitting in free space is purely $accidental$, with $a$ lying on a RG trajectory that does not contain $a=2$. Thus if the system is slightly perturbed, $a$ quickly flows from 2 toward 1. At the VM fixed point, therefore the standard vector dominance is $strongly$ violated [^7].

“Seeing" the Dropping Mass
==========================

On-shell probes
---------------

There is a flurry of activities, both in theory and experiment, to “see" evidence for the manifestation of chiral symmetry in the behavior of masses and coupling constants in hot and/or dense medium. One would first like to observe it at a density and/or temperature readily available in laboratories before reaching the critical point and then ultimately go to the critical point. The former is looked for in precursor phenomena to the chiral phase transition. The idea is to create a hot/dense environment in which the produced particles propagate and look for signals that take a “snap-shot" of the particle propagation in the medium that is as free as possible from subsequent interactions with the medium.

Among the weakly interacting probes, the most frequently used is the electromagnetic one. There are two classes of processes that have been studied: One mediated by the $normal$ component of the chiral Lagrangian and the other by the $anomalous$ component.

The lepton pair production via virtual photon V\^\*l\^+ l\^-\[V\] where $V=\rho, \ \omega$ and $l=e, \ \mu$ is governed by the $normal$ component of the HLS$_1$ Lagrangian. Most of the past efforts to unravel chiral dynamics of hadrons in medium were directed to this process. Several heavy-ion experiments have been dedicated at CERN-SPS to expose $directly$ in-medium properties of the $\rho$ meson in hot and dense medium. The most recent and comprehensive review on the current situation regarding this class of experiments can be found in the articles by van Hees and Rapp [@rapp]. We will argue below that within the HLS/VM theory that we believe is suited to the problem, the lepton pairs in heavy-ion processes are [*neither efficient nor specific*]{} for exposing in-medium properties of the $\rho$ meson bearing on chiral symmetry, in particular, on the “dropping mass" effect.

The other class involves processes that are mediated by the $anomalous$ Wess-Zumino term in chiral Lagrangians. The process studied in this category [@metag] involves the in-medium $\omega$ meson in +A+X\^0+ X\^.\[Metag\] The coupling $\omega\pi^0\gamma$ is significant in that it arises from a chiral anomaly and is expected to behave differently in medium from that of the normal process (\[V\]) as certain anomalous processes (e.g., those involving triangle diagrams) are.

Off-shell probes
----------------

There are indirect probes that indicate how hadron masses and coupling constants behave in dense medium. A recent discussion on this matter can be found in the review by Brown et al [@BHLR06]. Most of the relevant arguments given there have been developed in a series of articles that date way back to early 1990’s [@BR:PR].

If one looks for a signal at zero temperature but at a matter density near that of the normal nuclear matter, then the mass scaling (n)=m\^\*/m m(n)/m(0) is a $parameter$ in an effective theory because what one is dealing with here is an off-shell quantity. It can however be related to the quark condensate, albeit indirectly, and hence could signal for a chiral symmetry manifestation. In order to “see" the effect in an unambiguous way, one would have to formulate the effective theory in a way consistent with chiral symmetry [*for all scales*]{} involved in the process. There are a vast variety of such formulations in the literature and a direct comparison with them is not our aim here. As an indication of the subtlety involved in the problem, let us take an exemplary case discussed by us[@BR:PR], i.e., the effective chiral Lagrangian in the mean field that can be recast in Landau Fermi-liquid theory for nuclear matter. There the scaling $\Phi (n)$ for $n\approx n_0$ (where $n_0$ is the nuclear matter density) can be related to the Landau parameter $\tilde{F}_1$ [^8] as[^9] \_1 -\_1\^3(1-1/)\[LandauF\] where $\tilde{F}_1^\pi$ is the contribution to the Landau parameter from the pion exchange which is completely fixed by chiral dynamics. One can get $\tilde{F}_1$ from the anomalous gyromagnetic ratio of the proton $ \delta g_l$ measured experimentally by using the Migdal formula that relates the two quantities [@Migdal]. Using (\[LandauF\]), one finds [@Friman-Rho] (n\_0)0.78.\[Phivalue\] In HLS$_\infty$ theory, the Landau parameter $F_1$ must be receiving contributions from all of the $V_\infty$ in the $\omega$ channel. However assuming that the lowest $\omega$ dominates, we can interpret (\[Phivalue\]) as m\_(n\_0)/m\_0.78. While this is valid only near the nuclear matter density for the “$\omega$" meson in the average sense, it is comparable – as predicted by Brown-Rho scale valid up to $\sim n_0$ – to the ratio $f_\pi (n_0)/f_\pi(0)\approx 0.80\approx \sqrt{\frac{\la\bar{q}q\ra
(n_0)}{\la\bar{q}q\ra}}$ found in deeply bound pionic atoms.

Other indirect indications for the dropping masses, such as for example the reduction of the nuclear tensor force etc., are summarized in the reviews [@BR:PR]. What may be a bit surprising is that nuclear physics has fared so successfully without ever invoking Brown-Rho scaling even though the vacuum change due to density is expected. As suggested in [@BR:PR], this may be understood in terms of a web of intricate “dualities" at low energy known variously as “quark-hadron continuity," “Cheshire Cat principle" etc., concepts that are yet to be put on a solid basis.

The import of the discussion in this subsection is that it illustrates how chiral symmetry property which is a basic property of QCD and Fermi-liquid property which is a mundane many-body effect can be compounded in $ordinary$ nuclear processes.

What the dileptons see
----------------------

Experimentalists in heavy ion physics purport to extract an in-medium “spectral function" of a vector meson quantum number, say, $\rho$ or $\omega$ as a function of invariant mass. This quantity is intended to tell us how the properties of vector mesons change when they are embedded in medium. An “ideal" snapshot for this is thought to be the dileptons mentioned above. In order to expose the effect of dense and/or hot medium, one tries to subtract all possible “trivial effects" that take place in zero-temperature, zero-density environment (e.g., “cocktail events"). Whether this can be done in a fully consistent way is not clear. The discussion given in the preceding subsection indicates that the distinction between what is trivial and what is non-trivial in the context of chiral symmetry can be highly problematic [*unless one can probe exclusively the vicinity of the VM fixed point.*]{} Let us suppose for the sake of discussion that $all$ such trivial effects can be taken out of the given experimental results and theorists are given what we shall call in-medium “ESF" (experimental spectral function). The question is: [*Can the ESF so obtained be used to verify or falsify Brown-Rho scaling or equivalently see evidence for partial or complete chiral symmetry restoration?*]{}

As stressed above, to properly address this question, it would be necessary to have at one’s disposal one complete self-consistent theoretical framework in which calculations can be done for all processes involved, avoiding mixing different models for different steps. Furthermore one should express the spectral function in terms of variables that track the chiral property of the system in terms of the order parameter, i.e., the quark condensate.

HLS$_1$ theory provides one such framework, with its local gauge invariance enabling one to access the vector fields whose mass scale can drop as low as that of the pion near the critical point.

The theoretical ingredients necessary within the framework of HY’s HLS$_1$ theory were spelled out in two unpublished articles [@BR:NA60]. They are (1) the intrinsic background (temperature, density) dependence demanded by matching to QCD [@HY:PR], (2) the violation of the vector dominance in the pion EM form factor in hot and/or dense medium that results from the vector manifestation of chiral symmetry in the HLS/VM theory [@HS:VD] and (3) many-body correlations generated by the presence of the Fermi surface, which may be considered as a quantum critical phenomenon [@BHLR06].

In order to implement all three ingredients in a consistent way, baryonic degrees of freedom are mandatory. In HLS$_\infty$ theory, they are skyrmions in an infinite tower of vector mesons if viewed in four dimensions. No such formulation exists at the moment. [^10] There is no reason why such a formulation cannot be made, so we shall assume that we have a spectral function calculated in that formulation as function of temperature/density and kinematics, and call it “TSF" (theoretical spectral function). Unfortunately, having such a TSF is not enough to directly confront the ESF. One also has to know precisely what the conditions with which the measurement of the ESF is made are. The heavy-ion experiment typically involves summing over the dilepton emissions as the system evolves in temperature and density as it expands. Many subtle and complex effects, such as for instance the “memory effects" pointed out in [@C-Greiner], will have to be all taken into account. These are formidable complications, one of the major obstructions to meaningfully confronting HLS/VM with experimental data. It is not clear that the theorists have the full control on all these.

Up to date, no one has succeeded in constructing a spectral function as a function of temperature and density that takes fully into account the three ingredients listed above. Presently available is a partial spectral function obtained by Harada and Sasaki [@HS:spectral] in which only the temperature effects in the ingredients (1) and (2) were taken into account. Lacking the crucial density dependence and also the ingredient (3), one cannot confront the Harada-Sasaki spectral function with the experimental data even if one had at one’s disposal a reliable evolution code – which one does not – and a reliable control on the experimental conditions. Even so, the analysis of Harada and Sasaki gives a clear indication as to what the dileptons might be “seeing" in hot and dense medium. First of all, mass shifts in the TSF cannot occur without the intrinsic background dependence (IBD) taken into account: Hadronic (thermal and dense) quantum (loop) corrections can broaden the spectra but do not shift the invariant mass. Next as a consequence of the IBD, a finite temperature and density take the parameter $a$ away from the vector dominance point $a=2$ which is sitting on an unstable trajectory and makes the VD violated, maximally at the VM fixed point with $a=1$. Lepton pairs will be produced more or less equally from the pions by direct coupling and through vector mesons, thereby modifying the spectral structure from that in which vector dominance is assumed, a procedure which is practiced in the field. The HLS/VM predicts unequivocally that the vector mass is shifted and the width of the vector meson gets narrower in the close vicinity of the VM fixed point. However as measured by the lepton pairs that come from both direct pions and vector mesons in the system that evolves both in temperature and density, the width will be smeared while the strength will be cut down by a factor $\sim (a/2)^2$. Since the chiral restoration effect is operative only above the “flash temperature" [@BR:PR] $T_{flash}\sim 120$ MeV (and/or the “flash density" $n_{flash}\lsim
2 n_0$), the dileptons emitted carrying the imprint of chiral restoration – at which the $\rho\pi\pi$ coupling will be near zero due to the VM – will be highly diluted, if not totally swamped, by dileptons coming from near on-shell. Unless the probe pinpoints the kinematic regime above the flash points, it is difficult to imagine how chiral properties can be clearly sorted out.

In sum, our answer to the question posed above is the $negative$ at the present stage of experimental as well as theoretical development. A corollary to this is that the null result in the analyses of [@rapp] – that mundane strong hadronic interactions can more or less explain the available dilepton data – does not necessarily invalidate the HLS/VM scenario as some people have argued. It merely indicates that whatever pertinent signals there may be are masked by the mundane processes that have nothing direct to do – though not in inconsistency – with chiral symmetry. In this regard, our conclusion is in agreement with that arrived at by Dusling, Teaney and Zahed [@zahed-na60].

The Wess-Zumino-term induced process
------------------------------------

The process (\[Metag\]) is mediated by the Wess-Zumino term that arises from the Chern-Simons action in five dimensions in AdS/QCD [@SS]. In AdS/QCD theory, this process is vector-dominated as \^0 | \^0\^\*\[omegadecay\] where $\bar{\rho}=\rho, \rho^\prime, \cdots$. It is noteworthy that the direct coupling $\omega\pi\gamma$ is totally absent. Surprisingly even in HLS$_1$ theory of Harada and Yamawaki which does not exclude this direct coupling on a fundamental ground, an analysis [@HY:PR] showed that the direct coupling in the $\omega\rightarrow \pi^0\gamma$ decay is absent. We may therefore ignore it.

For simplicity, we ignore hadronic loop corrections which do not shift masses and take into account only the IBD (intrinsic background dependent) effects. By putting asterisk for the HLS$_1$ constants with the intrinsic density dependence incorporated, we can write the amplitude for (\[omegadecay\]) as T\^\[(p,),\^0, \^\* (k,)\]\~\^ p\_k\_.\[tree\] We of course need to include hadronic loop corrections (e.g., width for the $\rho$ as well as for the $\omega$) to make a quantitative estimate. However one can note that first of all it has no $a$ dependence, hence no suppression of the type present in the dilepton vertices. Second, the near degeneracy of the $\rho$ and $\omega$ in medium could have an interesting effect, modulo of course the broadening widths associated with hadronic loops. Since anomaly-mediated processes are in general much cleaner than normal processes, it would be interesting to analyze the CBELSA/TAPS experiment taking into account the anomaly structure present in the HLS framework.

Dense Matter and Half-Skyrmions
===============================

Hadronic physics at high density relevant to the interior of compact stars is poorly understood at present. In fact we don’t know much about how hadronic matter transforms into a quark matter. Lattice QCD cannot yet access high enough density and analytic tools of QCD cannot handle the nonperturbative regime at a density which is not high enough for the color-flavor locking mechanism to dominate. Given the paucity of model-independent tools, we can be allowed to speculate on an intriguing and hitherto unexplored scenario for the transition to quark matter at high density based on the instanton/skyrmion structure in holographic QCD.

It has been observed in the study of dense baryonic matter at high density in terms of the Skyrme model put on crystals [@BYP] that a skyrmion immersed in dense medium fractionizes at a certain density $n_{meron}> n_0$ into two half-skyrmions [@halfskyrmions]. At that density, the quark condensate $\la\bar{q}q\ra$ is found to vanish but the pion decay constant remains non-zero. In terms of the condensate written as |[q]{}q=(U+U\^) what happens at $n_{meron}$ can be understood by that $\la{\Tr}(U+U^\dagger)\ra=0$ and $\chi\neq 0$. This means that chiral symmetry is restored but there is a “gap" characterized by $\chi\sim f_\pi\neq 0$. This resembles the pseudogap phenomenon in high -$T_c$ superconductivity [@pseudogap].

In the AdS/QCD picture with hidden local fields, the half-skyrmions correspond to merons, i.e., half-instantons. One can think of what’s happening as the liberation of the merons that are confined inside the instantons at normal density into two deconfined merons at $n_{meron}$, the latter becoming the relevant degrees of freedom in the “pseudogap" phase . This process in the presence of hidden gauge fields resembles what happens in the quantum deconfinement phenomenon in condensed matter physics [@QDP].[^11] If what happens at the chiral transition in dense baryonic matter is of a similar nature to what happens in the magnetic Néel-VBS transition, then there can be a variety of interesting physical phenomena associated with the topological structure of the half-skyrmions. One important immediate consequence is that there could be a strong deviation from Fermi liquid structure. Thus one could expect the “normal" matter from which the transition to a Cooper-paired state at high density takes place to be a non-Fermi liquid state. This would bring a major change to the description of the phase transition to a color-superconducting phase from the hadronic phase which has always been assumed to be in a Fermi-liquid state [@rajagopal].

Summary and Discussions
=======================

We addressed the question of how hadron masses arise and how they can be tweaked by temperature/density to disappear in the framework of hidden local symmetry at low energy. We have argued that in order to theoretically describe dropping vector-meson masses at high temperature and density, hidden local symmetry is indispensable. Furthermore hidden local symmetric fields arise naturally as emergent gauge fields bottom-up in chiral field theory as well as as low-energy degrees of freedom top-down in holographic QCD. In either way, one expects to have an infinite tower of vector fields coupled gauge invariantly to the Goldstone bosons. Baryons must emerge in this theory as skyrmions in the infinite tower of vector mesons in four dimensions encapsulated in instantons in five dimensions.

Although AdS/QCD seems to work well for chiral dynamics of low-lying hadrons in the vacuum, in describing what happens to the hadrons under extreme conditions where the vacuum change must take place for which $1/\lambda$ and $1/N_c$ corrections are crucial, we are compelled to limit ourselves to the lowest member of the tower, namely, the $\rho$ and $\omega$ (and $a_1$ if needed). This corresponds to the HLS$_1$ of Harada and Yamawaki which we interpret as a truncated HLS theory with a Wilsonian matching to QCD at a matching scale commensurate with the chiral scale. This theory has the “vector manifestation fixed point" at which hadron masses vanish (in the chiral limit) in a way equivalent to Brown-Rho scaling. We have argued that certain consequences of the VM fixed point structure must be incorporated in the theoretical framework in confronting heavy-ion experimental data and that in the absence of such analyses, the presently available experimental data can neither validate nor invalidate the scenario associated with the hidden local symmetry with the VM fixed point or Brown-Rho scaling. Finally we have suggested that the structure of baryons in terms of instantons or skyrmions embedded in an infinite tower of vector mesons could bring a drastic change to the phase transition scenario at high density.

Acknowledgments {#acknowledgments .unnumbered}
===============

Most of the ideas presented in this paper have been developed in collaboration with Gerry Brown who however should not be held responsible for the more speculative parts of the discussions. This work was supported in part by the KRF Grants KRF-2006-209-C00002.

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[^1]: e-mail address: mannque.rho@cea.fr

[^2]: Based in part on talks given at [*The 1st Asian Triangle Heavy Ion Conference*]{}, Yonsei University, Seoul, Korea, 29 June - 1 July 2006 and at [*Yukawa International Seminar on New Frontiers in QCD*]{}, Yukawa Institute, Kyoto, Japan, 3 - 8 December 2006.

[^3]: Notations: $\pi=T^a\pi^a$ and $\Tr
    (T^a T^b)=\frac 12 \delta^{ab}$.

[^4]: The instanton baryon gives an additional correction of ${\cal O}(1/(\lambda N_c))$ to $g_A$ which for the value of $\lambda N_c$ fixed in the meson sector by, say, $f_\pi$ etc, comes out to be $g_{A_{min}}\approx 0.15$, so the total $g_A\approx 1.31$ comes even closer to the experiment. The nature of approximations involved in this calculation appears to be quite similar to the quenched approximation in lattice QCD which also predicts $g_A$ close to the experimental value [@quenchedgA].

[^5]: The physical pion decay constant will be written as $f_\pi$.

[^6]: There are other fixed points in the theory but they belong to a different universality class than that of QCD.

[^7]: The violation of the standard vector dominance – phrased in terms of the lowest members $(\rho, \omega)$ of the tower – can be interpreted as the intervention of the higher members of the infinite tower. It is not clear what the “fixed point" $a=1$ in HLS/VM means in the infinite-tower structure of the vector dominance found in AdS/QCD [@HRYY-long]. It is also not clear why $a$ cannot go below 1 at some point in this model.

[^8]: $\tilde{F}$ is related to the Landau parameter $F$ by $\tilde{F}=\frac{m_N}{m_L} F$ where $m_L$ is the Landau quasiparticle mass of the nucleon.

[^9]: It should be stressed that this relation holds only in the vicinity of the nuclear saturation density, i.e., the Fermi-liquid fixed point and cannot be extrapolated to too high a density.

[^10]: Initial attempts in this direction using the Skyrme model were made by Park et al. [@BYP]. But without a proper account of the vector degrees of freedom, the result remains largely incomplete.

[^11]: Briefly, in the phase transition between the magnetically ordered Néel state and a paramagnetic valence-bond solid (VBS) state, it is believed [@QDP] that a skyrmion in (1+2) dimensions made out of the spinon field $\hat{n}$ fractionizes into two half-skyrmions (or “merons") at the phase boundary. The phase transition involves no order parameters but is characterized by new degrees of freedom, i.e., merons, specific to the critical point. Here a crucial role is played by the emergent $U(1)$ gauge degree of freedom present in the spinon field $\hat{n}$ when the latter is written in the so-called CP$^1$ parametrization $\hat{n}=z^\dagger \vec{\sigma}z$ which has a redundancy – here $U(1)$ – in a close analogy to the chiral field $U$ described above. The confinement-deconfinement process of the merons turns out to be intimately connected to the instanton in the $U(1)$ gauge field.
