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#  Proving Rho Meson be a Dynamical Gauge Boson of Hidden Local Symmetry

Koichi Yamawaki  Kobayashi-Maskawa Institute for the Origin of Particles and
the Universe (KMI), Nagoya University, Nagoya 464-8602, Japan.

###### Abstract

The rho meson has long been successfully identified with a dynamical gauge
boson of the Hidden Local Symmetry (HLS) $H_{\rm local}$ in the nonlinear
sigma model $G/H$ gauge equivalent to the model having the symmetry $G_{\rm
global}\times H_{\rm local}$, with $G=[SU(2)_{L}\times SU(2)_{R}]\simeq
O(4),H=SU(2)_{V}\simeq O(3)$, however under a hitherto unproven assumption
that its kinetic term is dynamically generated, together with an ad hoc choice
of the auxiliary field parameter “$a=2$”. We prove this assumption, thereby
solving the long-standing mystery: The rho meson kinetic term is generated
simply by the large $N$ limit of the Grassmannian model
$G/H=O(N)/[O(N-3)\times O(3)]$ gauge equivalent to $O(N)_{\rm
global}\times[O(N-3)\times O(3)]_{\rm local}$, extrapolated to $N=4$,
$O(4)_{\rm global}\times O(3)_{\rm local}$, with all the phenomenologically
successful “$a=2$ results” i.e., $\rho$-universality, KSRF relation and the
Vector Meson Dominance, realized independently of the parameter “$a$”. This in
turn establishes validity of the large $N$ dynamics at quantitative level
directly by the experiments. The relevant cutoff reads $\Lambda\simeq 4\pi
F_{\pi}$ for $N=4$, which is regarded as a matching scale of the HLS as a
“magnetic dual” to QCD. Skyrmion is stabilized by such a dynamically generated
rho meson without recourse to the underlying QCD, further signal of the
duality. The unbroken phase with massless rho meson may be realized as a novel
chiral restored hadronic phase in hot/dense QCD.

##  I Introduction

Since its proposal Bando:1984ej ; Bando:1985rf ; Bando:1984pw ;
Fujiwara:1984mp ; Bando:1987ym  (for reviews see Ref.Bando:1987br ;
Harada:2003jx ; Yamawaki:2016qux ), identifying the rho meson as a dynamical
gauge boson of the Hidden Local Symmetry (HLS) $H_{\rm local}$ has been widely
accepted in the model having the symmetry $G_{\rm global}\times H_{\rm
local}$, with $G=[SU(2)_{L}\times SU(2)_{R}]\simeq O(4)$ and
$H=SU(2)_{V}\simeq O(3)$, where its Lagrangian consists of two independent
invariants ${\cal L}_{\rm HLS}={\cal L}_{A}+a{\cal L}_{V}$, with $a$ arbitrary
parameter. This is gauge equivalent to the nonlinear sigma model, ${\cal
L}_{\rm CCWZ}$, $\grave{a}$ la Callan-Coleman-Wess-Zumino (CCWZ)
Coleman:1969sm ; Callan:1969sn  based on the manifold $G/H$: In the absence of
the kinetic term of the HLS gauge boson it is merely an auxiliary field such
that ${\cal L}_{V}=0$, and ${\cal L}_{\rm HLS}={\cal L}_{A}={\cal L}_{\rm
CCWZ}$ after gauge fixing.

Once we assume, however, that its kinetic term, ${\cal L}_{\rm kinetic}$, is
generated at quantum level by the dynamics of the nonlinear sigma model
itself, thereby put by hand to the Lagrangian, ${\cal L}_{\rm
HLS}\Rightarrow{\cal L}_{A}+a{\cal L}_{V}+{\cal L}_{\rm kinetic}$, novel
physics come out Bando:1984ej ; Bando:1985rf ; Bando:1984pw ; Fujiwara:1984mp
; Bando:1987ym ; Bando:1987br ; Harada:2003jx ; Yamawaki:2016qux : All the
successful phenomenological results, such as the universality of the rho meson
coupling ($\rho$ universality), the Kawarabayashi-Suzuki-Ryazzudin-Fayyazudin
(KSRF) relation, the Vector Meson Dominance (VMD), are derived for a
particular parameter choice $a=2$ in the resultant Lagrangian (at tree level),
in such a way that $a{\cal L}_{V}$ becomes the HLS gauge-invariant mass terms
of the $\rho$ meson which contains $\rho$ mass, $\rho$ couplings, additional
$\pi$ self-couplings, etc..

For all the phenomenological success of the HLS model of the rho meson,
however, the basic assumption of the dynamical origin of the kinetic term and
the particular parameter choice $a=2$ has never been proved within the
dynamics of HLS model itself. #1#1#1 Alternatively, we may simply assume that
the kinetic term is already generated by the underlying theory, QCD in the
case at hand, regarding the HLS model as a low energy effective theory,
including loop corrections in the sense of the derivative expansion
Harada:2003jx . In this case, however, the parameter $a$ (renormalized one) is
a completely free parameter to be adjusted to $a=2$ by hand within the HLS
model framework. It can be determined only with additional information, by
“Wilsonian matching” with the QCD parameters at UV scale $\Lambda$ as input,
$a(\Lambda^{2})\sim 1$, which then predicts $a(\mu^{2}=M_{\pi}^{2}=0)=2$ as
the infrared value (massless $\pi$ on-shell) through the (one-loop)
renormalization group for $F_{\pi}^{2}(\mu^{2}=0)$ due to $\pi$ loop alone in
the HLS model itself (see Re. Harada:2003jx  and references cited therein).

In this paper #2#2#2 Preliminary results of this paper were given as the
supplementary ones in Ref.Yamawaki:2018jvy  which is mainly addressed to a
subject on a possible dynamical gauge boson of HLS within the Higgs sector of
the Standard Model, different from the present one, that in the QCD, but the
details of the relevant calculations in the present paper may be found in
Ref.Yamawaki:2018jvy .  we resolve this long-standing mystery as simply a
consequence of the nonperturbative dynamics of the large $N$ limit of the
nonlinear sigma model based on the Grassmannian manifold
$G/H=$$O(N)/[O(N-p)\times O(p)$, with $p=3=$ fixed, which is reduced to the
relevant case $G/H=O(4)/O(3)\simeq SU(2)_{L}\times SU(2)_{R}/SU(2)_{V}$ for
the extrapolation $N\rightarrow 4$, with the rho meson being the dynamical
gauge boson of $O(3)_{\rm local}\simeq[SU(2)_{V}]_{\rm local}$.

It is in fact well known that the HLS gauge bosons in many nonlinear sigma
models, such as the $CP^{N-1}$ model with $G/H=U(N)/[U(N-1)\times
U(1)]$$\simeq SU(N)/[SU(N-1)\times U(1)]$ gauge equivalent to the model
$SU(N)_{\rm global}\times U(1)_{\rm local}$, do acquire kinetic term for
$U(1)_{\rm local}$ gauge boson at quantum level in the large $N$ limit
Eichenherr:1978qa ; Golo:1978de ; DAdda:1978vbw ; DAdda:1978dle ;
Witten:1978bc ; Arefeva:1980ms ; Haber:1980uy ; Kugo:1985jc ; Bando:1987br ;
Weinberg:1997rv ; Harada:2003jx . It was further shown (in the context
irrelevant to the rho meson physics, though) that the HLS gauge bosons
$O(p)_{\rm local}$ and $U(p)_{\rm local}$ in the Grassmannian models
$G/H=O(N)/[O(N-p)$$\times O(p)$ Brezin:1980ms  and $G/H=$$U(N)/[U(N-p)\times
U(p)]$, respectively Brezin:1980ms ; Bando:1996pg  are dynamically generated
in the large $N$ limit. However it was shown only in a specific
parameterization “covariant derivative” type which is just a particular $a=2$
choice from the onset (see Yamawaki:2018jvy  and Eq.(23)).

Here we show that not just the generation of the $O(p)_{\rm local}$ gauge
boson $\rho_{\mu}$ but also all the successful “$a=2$ results” are direct
consequences of the pure dynamics at quantum level of the large $N$ limit for
arbitrary value of $a$, thereby resolving the long-standing mystery of the rho
meson simply on the firm dynamical base. This in turn provides yet another
experimental verification of the large $N$ reliability, this time even
quantitatively, not just qualitatively #3#3#3 It is known that the large $N$
results remain qualitatively true even for the smallest value $N=2$ in the
$CP^{N-1}$ model, as checked by the equivalent $O(3)$ model exactly solvable
in 2 dimensions. Witten:1978bc .

##  II Grassmaniann $N$ Extension

Let us define the generic HLS base Bando:1985rf ; Bando:1987br  of an $N\times
N$ real matrix field
$\xi(x)=\xi({\check{\rho}}^{(p)})\cdot\xi({\check{\rho}}^{(N-p)})\cdot\xi(\pi)$,
which transforms under $G_{\rm global}\times H_{\rm local}=O(N)_{\rm
global}\times[O(N-p)\times O(p)]_{\rm local}$ as $\xi(x)\rightarrow
h(x)\cdot\xi(x)\cdot g^{-1}$ with $h(x)\in[O(N-p)\times O(p)]_{\rm
local}\,,g\in O(N)_{\rm global}$, where
$\xi(\pi)=e^{i\pi_{a}(x)X_{a}/f_{\pi}}$ is the CCWZ base for
$G/H=O(N)/[O(N-p)\times O(p)]$, with $f_{\pi}$ being the (bare/tree-level)
decay constant of the NG boson $\pi$, while
$\xi({\check{\rho}}^{(p)})=e^{i{\check{\rho}}^{(p)}(x)/f_{\rho}^{(p)}}$ and
$\xi({\check{\rho}}^{(N-p)})=e^{i{\check{\rho}}^{(N-p)}(x)/f_{\rho}^{(N-p)}}$,
with ${\check{\rho}}^{(p)}={\check{\rho}}^{(p)}_{a}S_{a}^{(p)}$ and
${\check{\rho}}^{(N-p)}={\check{\rho}}^{(N-p)}_{a}S_{a}^{(N-p)}$ being the
would-be NG bosons to be absorbed into the HLS gauge bosons $\rho^{(p)}_{\mu}$
and $\rho^{(N-p)}_{\mu}$ of $O(p)_{\rm local}$ and $O(N-p)_{\rm local}$, with
the (bare/tree-leve) decay constant $f_{\rho}^{(p)}$ and $f_{\rho}^{(N-p)}$,
respectively:

$\displaystyle\xi(x)=\xi(\pi)\,\quad\left({{\rm unitary}\,\,{\rm
gauge},\,\,{\check{\rho}}^{(N-p)}(x)=\check{\rho}}^{(p)}(x)=0\right).$ (1)

Here the generators read $X_{a}\in{\cal G}-{\cal H}$, $S_{a}^{(p)}\in{\cal
H}^{(p)}={\cal O}(p)$, $S_{a}^{(N-p)}\in{\cal H}^{{N-p}}={\cal O}(N-p)$, with
${\rm tr}(T_{a}T_{b})=2\delta_{ab}$, ${\rm tr}(S_{a}X_{b})=0$,
$T_{a}=\\{S_{a},X_{a}\\}=-T_{a}^{t}$.

To study the large $N$ limit in the Grassmannian models including the
$CP^{N-1}$ model it is customary to parameterize the HLS base as:#4#4#4
$p\times N$ degrees of freedom of $\phi_{i\beta}$ consist of $p\times(N-p)$ of
$\pi$, $p\times(p-1)/2$ of $\check{\rho}$, and $p\times(p+1)/2$ of the
constraints.

$\displaystyle\xi(x)_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{G}{N}\left(\begin{array}[]{c}\phi_{i,\beta}(x)\\\
\Phi_{k,\beta}(x)\end{array}\right),\,\,\alpha=(i,k),\,\,\beta=(j,l),\,i,j=1,\cdots,p\,\,\,;k,l=p+1,\cdots
N\,,$ (4) $\displaystyle\xi^{t}\cdot\xi$ $\displaystyle=$
$\displaystyle\frac{G}{N}\left(\phi^{t}\phi+\Phi^{t}\Phi\right)=\mbox{1}\mbox{l}\,,\quad\frac{G}{N}\equiv\
\frac{1}{{f_{\pi}}^{2}},$ $\displaystyle\xi\cdot\xi^{t}$ $\displaystyle=$
$\displaystyle\frac{G}{N}\left(\begin{array}[]{cc}\phi\phi^{t}&\phi\Phi^{t}\\\
\Phi\phi^{t}&\Phi\Phi^{t}\end{array}\right)=\left(\begin{array}[]{cc}\mbox{1}\mbox{l}_{p\times
p}&{\huge 0}\\\ {\huge
0}&\mbox{1}\mbox{l}_{(N-p)\times(N-p)}\end{array}\right)=\mbox{1}\mbox{l},$
(9)

where $G\equiv N/{f_{\pi}}^{2}$ is the (bare) coupling constant to be fixed in
the large $N$ limit (s.t. ${f_{\pi}}^{2}={\cal O}(N)$).

The covariantized Maurer-Cartan one-form reads:

$\displaystyle\hat{\alpha}_{\mu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{i}D_{\mu}\xi\cdot\xi^{t}=\frac{G}{iN}\left(\begin{array}[]{c}\partial_{\mu}\phi-i\rho^{(p)}_{\mu}\phi\\\
\partial_{\mu}\Phi-i\rho^{(N-p)}_{\mu}\Phi\end{array}\right)\cdot\left(\phi^{t}\,\,\Phi^{t}\right)={\hat{\alpha}}_{\mu,\perp}+{\hat{\alpha}}_{\mu,||},$
(12)

where ${\hat{\alpha}}_{\mu,\perp}\equiv\frac{1}{2}{\rm
tr}\left({\hat{\alpha}}_{\mu}X^{a}\right)X^{a}$,
${\hat{\alpha}}_{\mu,||}\equiv\frac{1}{2}{\rm
tr}\left({\hat{\alpha}}_{\mu}S^{a}\right)S^{a}$ are

$\displaystyle{\hat{\alpha}}_{\mu,\perp}=\alpha_{\mu,\perp}=\left(\begin{array}[]{cc}0&\frac{G}{iN}\partial_{\mu}\phi\cdot\Phi^{t}\\\
\frac{G}{iN}\partial_{\mu}\Phi\cdot\phi^{t}&0\end{array}\right),\quad{\hat{\alpha}}_{\mu,||}=\left(\begin{array}[]{cc}\frac{G}{iN}\partial_{\mu}\phi\cdot\phi^{t}-\rho^{(p)}_{\mu}&0\\\
0&\frac{G}{iN}\partial_{\mu}\Phi\cdot\Phi^{t}-\rho^{(N-p)}_{\mu}\end{array}\right),$
(17)

all transforming homogeneously as

$\displaystyle\left({\hat{\alpha}}_{\mu,\perp},{\hat{\alpha}}_{\mu,||}\right)$
$\displaystyle\rightarrow$ $\displaystyle
h(x)\cdot\left({\hat{\alpha}}_{\mu,\perp},{\hat{\alpha}}_{\mu,||}\right)\cdot
h^{-1}(x)\,,$ (18)

with $h(x)\in{\cal H}$ for $H=\left[O(N-p)\times O(p)\right]_{\rm local}$.

Thus the HLS Lagrangian consists of three independent invariants at the lowest
derivative: Yamawaki:2018jvy

$\displaystyle{\cal L}^{(N,p)}$ $\displaystyle=$ $\displaystyle{\cal
L}_{A}+a^{(p)}{\cal L}_{V}^{(p)}+a^{(N-p)}{\cal L}_{V}^{(N-p)}\,,$ (19)

where

$\displaystyle{\cal L}_{A}$ $\displaystyle=$
$\displaystyle\frac{{f_{\pi}}^{2}}{4}{\rm
tr}\left({\hat{\alpha}}^{2}_{\mu,\perp}\right)=-\frac{G}{2N}{\rm
tr}\left(\phi^{t}\partial_{\mu}\phi\cdot\Phi^{t}\partial^{\mu}\Phi\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
tr}\left(\partial_{\mu}\phi\partial^{\mu}\phi^{t}+\frac{G}{N}\left(\phi\partial_{\mu}\phi^{t}\right)^{2}\right)$
$\displaystyle({\rm unitary\,gauge})$ $\displaystyle\longrightarrow$
$\displaystyle{\cal L}_{\rm CCWZ}=\frac{{f_{\pi}}^{2}}{4}{\rm
tr}\left({\alpha}^{2}_{\mu,\perp}(\pi)\right)^{2}=\frac{1}{2}\left(\partial_{\mu}\pi_{a}\right)^{2}+\cdots\,,$
(20)

with (unitary-gauge)
${\alpha}_{\mu,\perp}\rightarrow{\alpha}_{\mu,\perp}(\pi)=\partial_{\mu}\xi(\pi)\cdot\xi^{t}(\pi)=\partial_{\mu}\xi(\pi)\cdot\xi^{\dagger}(\pi)$,
and

$\displaystyle a^{(p)}{\cal L}_{V}^{(p)}$ $\displaystyle=$
$\displaystyle\frac{a^{(p)}{f_{\pi}}^{2}}{4}{\rm
tr}\left(\left[{\hat{\alpha}}^{(p)}_{\mu,||}\right]^{2}\right)=\frac{1}{2}{\rm
tr}\left[\frac{a^{(p)}}{2}\cdot\frac{N}{G}\left(\rho^{(p)}_{\mu}-i\frac{G}{N}\phi\partial_{\mu}\phi^{t}\right)^{2}\right]$
(21) $\displaystyle=$ $\displaystyle\frac{(f_{\rho}^{(p)})^{2}}{4}{\rm
tr}\left[\rho^{(p)}_{\mu}-\frac{\partial_{\mu}{\check{\rho}}^{(p)}}{f_{\rho}^{(p)}}-\frac{i\left[\partial_{\mu}\pi,\pi\right]}{2{f_{\pi}}^{2}}+\cdots\right]^{2},$

where we should impose a bare/tree relation between the two decay constants,

$\displaystyle(f_{\rho}^{(p)})^{2}=a^{(p)}{f_{\pi}}^{2}=a^{(p)}\frac{N}{G}\,,$
(22)

to normalize the kinetic term of the would-be NG boson ${\check{\rho}}^{(p)}$
to the canonical form, and similarly for $a^{(N-p)}{\cal L}_{V}^{(N-p)}$. In
the unitary gauge, ${\check{\rho}}^{(p)}=0$, Eq.(21) reads the mass term of
$\rho_{\mu}^{(p)}$ as usual in the HLS formalism, and so does $a^{(N-p)}{\cal
L}_{V}^{(N-p)}$ the mass term of $\rho_{\mu}^{(N-p)}$.

Here we note that in contrast to $O(p)_{\rm local}$ gauge boson, the kinetic
term for the $O(N-p)_{\rm local}$ gauge boson, carrying index running
$1,\cdots,N-p$ thus subject to all the planar diagram contributions in the
large $N$ limit, is not dynamically generated #5#5#5 $O(N-p)_{\rm local}$ does
not exist for $N=4,p=3$ anyway.  and stays as an auxiliary field (i.e., ${\cal
L}^{(N-p)}_{V}=0$) as was the case in the previous calculations for $CP^{N-1}$
and Grassmannian models. #6#6#6 The $SU(N-1)_{\rm local}$ gauge boson in
$CP^{N-1}$ model with $G/H=SU(N)/[SU(N-1)\times U(1)]$, which carries the
index running through $1,\cdots,N-1$, is not dynamically generated in the
large $N$ limit, in contrast to the $U(1)_{\rm local}$ part Eichenherr:1978qa
; Golo:1978de ; DAdda:1978vbw ; DAdda:1978dle ; Witten:1978bc ; Arefeva:1980ms
; Haber:1980uy ; Kugo:1985jc ; Bando:1987br ; Weinberg:1997rv ; Harada:2003jx
. The same is true for $G/H=O(N)/[O(N-p)\times O(p)]$ and
$G/H=U(N)/[U(N-p)\times U(p)]$, with $O(N-p)_{\rm local}$ and $U(N-p)_{\rm
local}$, respectively Brezin:1980ms ; Bando:1996pg . Similarly, a popular $N$
extension $G/H=O(N)/O(N-1)$ gauge equivalent to the model $O(N)_{\rm
global}\times O(N-1)_{\rm local}$ has no dynamical gauge boson for
$O(N-1)_{\rm local}$ and is irrelevant to the rho meson.

Then without loss of generality the starting Lagrangian, Eqs.(19)-(21), is
simplified as $a^{(p)}{\cal L}^{(p)}_{V}$ $\equiv a{\cal L}_{V}$,
$a^{(N-p)}{\cal L}^{(N-p)}_{V}=0$, $\rho_{\mu}\equiv\rho^{(p)}_{\mu}$,
${\check{\rho}}={\check{\rho}}^{(p)}$,
$f_{\rho}^{2}$$\equiv(f_{\rho}^{(p)})^{2}=af_{\pi}^{2}$, etc.:

$\displaystyle{\cal L}={\cal L}_{A}+a{\cal L}_{V}=\frac{1}{2}{\rm
tr}\left[\left(\partial_{\mu}\phi\partial^{\mu}\phi^{t})+\frac{1}{2}\cdot\frac{aN}{G}\rho_{\mu}^{2}-ia\rho^{\mu}\phi\partial_{\mu}\phi^{t}\right)\right]\,+\frac{1}{2}{\rm
tr}\left[\left(1-\frac{a}{2}\right)\frac{G}{N}\left(\phi\partial_{\mu}\phi^{t}\right)^{2}-\eta\left(\phi\phi^{t}-\frac{N}{G}\mbox{1}\mbox{l}\right)\right],$
(23)

where ${\rm tr}$ and 1l should read ${\rm tr}_{p\times p}$ and
$\mbox{1}\mbox{l}_{p\times p}$, respectively, and
$(\rho_{\mu})_{ij}=\rho_{\mu}^{a}(S^{a})_{ij}$ with ${\rm
tr}(S^{a}S^{b})=2\delta^{ab}$, and the $p\times p$ matrix Lagrange multiplier
$\eta_{i,j}(x)$ is used for the constraint Eq.(9) #7#7#7 In the broken phase
this is simply equivalent to the constraint Eq.(9), while in the unbroken
phase the multiplier is only a correct description. See later discussions.  as
in the standard large $N$ arguments of $CP^{N-1}$ Eichenherr:1978qa ;
Golo:1978de ; DAdda:1978vbw ; DAdda:1978dle ; Witten:1978bc ; Arefeva:1980ms ;
Haber:1980uy ; Kugo:1985jc ; Bando:1987br ; Weinberg:1997rv ; Harada:2003jx
and other Grassmannian models Brezin:1980ms ; Bando:1996pg . For $N=4,p=3$
Eq.(23) with $O(4)_{\rm global}\times O(3)_{\rm local}$ is identical to the
standard HLS Lagrangian Bando:1984ej ; Bando:1985rf ; Bando:1984pw ;
Fujiwara:1984mp ; Bando:1987ym ; Bando:1987br ; Harada:2003jx ;
Yamawaki:2016qux  for the rho meson with $[SU(2)_{L}\times SU(2)_{R}]_{\rm
global}\times[SU(2)_{V}]_{\rm local}$. It is now clear Yamawaki:2018jvy  that
Eq.(23) coincides with that of the conventional “covariant derivative type”
Lagrangian Brezin:1980ms  for a particular choice $a=2$, with
$\phi\phi^{t}=(N/G)\mbox{1}\mbox{l}$ (see Eq.(9)).

From Eq.(23) the effective potential in the large $N$ limit for
$\langle\hbox{\boldmath$\phi$}_{i,\beta}(x)\rangle=\sqrt{N}v(\delta_{i,j},0)$
(we took $v\neq 0$ real, i.e., the unitary gauge ${\check{\rho}}(x)=0$) and
$\langle\eta_{i,j}(x)\rangle=\eta\,\delta_{i,j}$, takes the form (in $D$
dimensions):

$\displaystyle\frac{V_{\rm
eff}\left(v,\eta\right)}{Np/2}=\eta\left(v^{2}-\frac{1}{G}\right)+\int\frac{d^{D}k}{i(2\pi)^{D}}\ln\left(k^{2}-\eta\right),$
(24)

where the ($a$-dependent) 1-PI contributions are sub-leading in the large $N$
limit #8#8#8This observation is due to H. Ohki. and therefore the result is
independent of the parameter $a$, in precisely the same form as that of the
conventional “covariant derivative” parameterization of $CP^{N-1}$ and the
Grassmannian models corresponding to $a=2$ Eichenherr:1978qa ; Golo:1978de ;
DAdda:1978vbw ; DAdda:1978dle ; Witten:1978bc ; Arefeva:1980ms ; Haber:1980uy
; Kugo:1985jc ; Bando:1987br ; Weinberg:1997rv ; Harada:2003jx ; Brezin:1980ms
; Bando:1996pg , and hence yields the same gap equation:

$\displaystyle\frac{1}{Np}\frac{\partial V_{\rm eff}}{\partial v}$
$\displaystyle=$ $\displaystyle 2\eta v=0\,,$
$\displaystyle\frac{1}{Np}\frac{\partial V_{\rm eff}}{\partial\eta}$
$\displaystyle=$ $\displaystyle v^{2}-\frac{1}{G}+\frac{1}{{G}_{\rm
crit}}-v_{\eta}^{2}=0\,,$ (25)

with (for cutoff $\Lambda$) #9#9#9 The cutoff $\Lambda$ can be removed for
$2\leq D<4$ (the theory is renormalizable), introducing the renormalized
coupling at renormalization point $\mu$ as $1/{G}^{(R)}(\mu)\equiv
1/G-\int\frac{d^{D}k}{i(2\pi)^{D}}\,\frac{1}{\mu^{2}-k^{2}}\equiv\mu^{D-2}/g^{(R)}(\mu)$,
$1/G^{(R)}_{\rm
crit}\equiv\int\frac{d^{D}k}{i(2\pi)^{D}}\left(\frac{1}{-k^{2}}-\frac{1}{\mu^{2}-k^{2}}\right)$
$=\frac{\Gamma(2-D/2)}{\left(D/2-1\right)}\cdot\frac{\mu^{D-2}}{(4\pi)^{D/2}}\equiv\mu^{D-2}/g^{(R)}_{\rm
crit}$, s.t., $1/G-1/G_{\rm crit}=\mu^{D-2}\left(1/{g}^{(R)}(\mu)-1/g_{\rm
crit}^{(R)}\right)$ in the gap equation. The renormalized coupling
${g}^{(R)}(\mu)$ has an ultraviolet fixed point at $g_{\rm crit}^{(R)}$,
$0\leq g_{\rm
crit}^{(R)}=(4\pi)^{D/2}\left(D/2-1\right)/\Gamma(2-D/2)<\infty\,(2\leq D<4)$,
with the beta function
$\beta({g}^{(R)}(\mu))=\mu\partial{g}^{(R)}(\mu)/{\partial\mu}=-(D-2){g}^{(R)}(\mu)[{g}^{(R)}(\mu)-g_{\rm
crit}^{(R)}]/g_{\rm crit}^{(R)}$. While for $D=4$ the theory is not
renormalizable, $1/g_{\rm
crit}^{(R)}\sim\Gamma(2-D/2)/(4\pi)^{2}\Big{|}_{D\rightarrow
4}\sim\ln(\Lambda^{2}/\mu^{2})/(4\pi)^{2}$, with the remaining log divergence
identified in the cutoff notation.

$\displaystyle\frac{1}{{G}_{\rm
crit}}\equiv\int\frac{d^{D}k}{i(2\pi)^{D}}\,\frac{1}{-k^{2}}=\frac{1}{\left(\frac{D}{2}-1\right)\Gamma(\frac{D}{2})}\frac{\Lambda^{D-2}}{(4\pi)^{\frac{D}{2}}},$
$\displaystyle
v_{\eta}^{2}\equiv\int\frac{d^{D}k}{i(2\pi)^{D}}\left(\frac{1}{-k^{2}}-\frac{1}{\eta-k^{2}}\right)=\frac{\Gamma(2-\frac{D}{2})}{\frac{D}{2}-1}\cdot\frac{\eta^{\frac{D}{2}-1}}{(4\pi)^{\frac{D}{2}}}.$

The gap equation implies as usual the second order phase transition between
two phases, the (weak coupling) phase with the symmetry spontaneously broken
which is the same as the classical level, and the (strong coupling) phase with
that spontaneously unbroken which is a new phase at quantum level:

$\displaystyle\mbox{(i)}\quad G<G_{\rm cr}:v\neq 0\ ,\ \eta=0\,\left({\rm
broken}\,\,{\rm phase}\right)$ $\displaystyle
v^{2}=\frac{1}{G}-\frac{1}{G_{\rm crit}}>0\,\,,$ (26)
$\displaystyle\mbox{(ii)}\quad G>G_{\rm crit}:v=0\ ,\ \eta\neq 0\,\left({\rm
unbroken}\,\,{\rm phase}\right)$ $\displaystyle v_{\eta}^{2}=\frac{1}{G_{\rm
crit}}-\frac{1}{G}\,>0\,,$ (27)

with the phase transition point $v=\eta=0$. We may define a full decay
constant $F_{\pi}$ at quantum level in the large $N$ limit:

$\displaystyle F_{\pi}^{2}\equiv Nv^{2}=N\left(\frac{1}{G}-\frac{1}{G_{\rm
crit}}\right)={f_{\pi}}^{2}-\frac{N}{\left(\frac{D}{2}-1\right)\Gamma(\frac{D}{2})}\frac{\Lambda^{D-2}}{(4\pi)^{\frac{D}{2}}}\quad\rightarrow{f_{\pi}}^{2}-N\frac{\Lambda^{2}}{(4\pi)^{2}}\,\,\left(D\rightarrow
4\right)\,.$ (28)

This indicates that approaching from the broken phase to the critical point,
$F_{\pi}^{2}\rightarrow 0$, is due to the power divergence of $1/G_{\rm
crit}$(quadratic divergence for $D=4$), similarly to the “Wilsonian matching”
of the HLS model with the underlying QCD at the UV scale $\Lambda$
Harada:2003jx .

##  III Dynamical Generation of Rho Meson

Now the (amputated) two-point function of $\rho_{\mu}$ in the large $N$ limit
takes the form:

$\displaystyle\Gamma^{(\rho)}_{\mu\nu}(q)=\left(\frac{a}{2}\right)\left(\frac{N}{G}\right)g_{\mu\nu}+\left(\frac{a}{2}\right)^{2}B_{\mu\lambda}(q)\cdot
C^{\lambda}_{\nu}(q),$ $\displaystyle
C_{\mu\nu}(q)=g_{\mu\nu}+\left(\frac{a}{2}-1\right)\frac{G}{N}B_{\mu\lambda}(q)\cdot
C^{\lambda}_{\nu}(q)\,,$ (29)

where the four-$\phi$ vertex $\left(\frac{a}{2}-1\right)\frac{G}{N}$ in our
Lagrangian Eq.(23) (second line) gives rise to an infinite sum of the bubble
graph contribution $B_{\mu\nu}(q)$;

$\displaystyle\frac{1}{N}B_{\mu\nu}(q)=\frac{1}{2}\int\frac{dk^{D}}{i(2\pi)^{D}}\frac{(2k+q)_{\mu}(2k+q)_{\nu}}{\left(k^{2}-\eta\right)\left((k+q)^{2}-\eta\right)}=q^{2}f(q^{2},\eta)\cdot\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)+\left(v^{2}-\frac{1}{G}\right)\cdot
g_{\mu\nu}\,,$ (30)

with

$\displaystyle
f(q^{2},\eta)\equiv-\frac{\Gamma(2-\frac{D}{2})}{2\left(4\pi\right)^{\frac{D}{2}}\Gamma(2)}\int_{0}^{1}dx\frac{\left(1-2x\right)^{2}}{\left[x\left(1-x\right)q^{2}+\eta\right]^{2-\frac{D}{2}}}\,,$

which reads for $D\rightarrow 4$ ($\epsilon\equiv 2-D/2\rightarrow 0$ and
$1/A^{\epsilon}\simeq 1-\epsilon\ln A$):

$\displaystyle f(q^{2},0)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\cdot\frac{1}{3\left(4\pi\right)^{2}}\cdot\left[\ln\left(\frac{\Lambda^{2}}{q^{2}}\right)+\frac{8}{3}\right]\,,$
$\displaystyle f(0,\eta)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\cdot\frac{1}{3\left(4\pi\right)^{2}}\cdot\left[\ln\left(\frac{\Lambda^{2}}{\eta}\right)\right],$
(31)

where we have used the gap equation Eq.(25) and identified
$\Gamma(\epsilon)\simeq 1/\epsilon\rightarrow\ln\Lambda^{2}$.#10#10#10 The
finite part common to both phases are included in the definition of the cutoff
$\Lambda$, while the part $+8/3$ is an extra one in the broken phase
$\eta\equiv 0$, similarly to that in the $CP^{N-1}$ Weinberg:1997rv .

###  III.1 $a=2$ case

Note that for $a=2$, we have $C_{\mu\nu}=g_{\mu\nu}$ in Eq.(29), which yields
$\Gamma_{\mu\nu}^{(\rho)}(q)$:

$\displaystyle\frac{\Gamma^{(\rho)}_{\mu\nu}(q)}{N}=\left(\frac{1}{G}\right)g_{\mu\nu}+\frac{B_{\mu\lambda}(q)}{N}\cdot
g^{\lambda}_{\nu}(q)=\left(q^{2}f(q^{2},\eta)+v^{2}\right)\cdot\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)+v^{2}\cdot\frac{q_{\mu}q_{\nu}}{q^{2}}\,,$
(32)

the well-known form of one-loop dominance in the large $N$ limit in the
conventional “covariant derivative” parameterization for $CP^{N-1}$ model and
other Grassmannian models Eichenherr:1978qa ; Golo:1978de ; DAdda:1978vbw ;
DAdda:1978dle ; Witten:1978bc ; Arefeva:1980ms ; Haber:1980uy ; Kugo:1985jc ;
Bando:1987br ; Weinberg:1997rv ; Harada:2003jx ; Brezin:1980ms ; Bando:1996pg
.

For the broken phase $v\neq 0$, $\eta=0$, this is readily inverted to yield
the $\rho_{\mu}$ propagator for $a=2$:
$\langle\rho_{\mu}\rho_{\nu}\rangle(q)\equiv\langle\rho^{ij}_{\mu}\rho^{ji}_{\nu}\rangle(q)=2\langle\rho^{a}_{\mu}\rho^{a}_{\nu}\rangle(q)$:

$\displaystyle\langle\rho_{\mu}\rho_{\nu}\rangle(q)$ $\displaystyle=$
$\displaystyle-\Gamma^{(\rho)}_{\mu\nu}(q)^{-1}=\frac{1}{N}\frac{-f^{-1}(q^{2},0)}{q^{2}+f^{-1}(q^{2},0)v^{2}}\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)-\frac{1}{N}\frac{1}{v^{2}}\frac{q_{\mu}q_{\nu}}{q^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\frac{-f^{-1}(q^{2},0)}{q^{2}+f^{-1}(q^{2},0)v^{2}}\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{-f^{-1}(q^{2},0)v^{2}}\right)=2\Delta_{\mu\nu}(q),$
$\displaystyle\Delta_{\mu\nu}(q)$ $\displaystyle\equiv$
$\displaystyle\frac{g^{2}_{{}_{\rm
HLS}}(q^{2})}{q^{2}-M_{\rho}^{2}(q^{2})}\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{M_{\rho}^{2}(q^{2})}\right),$
$\displaystyle M_{\rho}^{2}(q^{2})\equiv-f^{-1}(q^{2},0)v^{2}$
$\displaystyle=$ $\displaystyle g^{2}_{{}_{\rm HLS}}(q^{2})\cdot
2F_{\pi}^{2},\quad g^{-2}_{{}_{\rm
HLS}}(q^{2})\equiv-2Nf(q^{2},0)=\frac{N}{3(4\pi)^{2}}\left[\ln\frac{\Lambda^{2}}{q^{2}}+\frac{8}{3}\right],$
(33)

which is the form of the unitary gauge (we took the unitary gauge
$\hat{\rho}(x)=0$ with $v\neq 0=$ real), with the physical pole position and
the on-shell HLS coupling given as
$q^{2}=M_{\rho}^{2}=-f^{-1}(M_{\rho}^{2},0)v^{2}=g^{2}_{{}_{\rm
HLS}}(M_{\rho}^{2})\cdot 2F_{\pi}^{2}$ and $g^{2}_{{}_{\rm HLS}}\equiv
g^{2}_{{}_{\rm HLS}}(M_{\rho}^{2})$, respectively. The relation implies the
rho meson mass is generated by the Higgs mechanism

$\displaystyle M_{\rho}^{2}=g^{2}_{{}_{\rm HLS}}\cdot F_{\rho}^{2}\,,\quad
F_{\rho}^{2}=2\cdot F_{\pi}^{2}\,,$ (34)

where $F_{\rho}$ is the decay constant of the would-be NG boson $\hat{\rho}$
(absorbed into the rho meson in the unitary gauge) at quantum level, which is
to be compared with the tree-level relation Eq.(22), with $a=a^{(p)}=2$. The
$q^{2}$ dependence of $M_{\rho}^{2}(q^{2})$ and $g^{2}_{{}_{\rm HLS}}(q^{2})$
may be regarded as the running mass and the (asymptotically non-free/infrared
free) running coupling. The resultant rho meson mass relation
$M_{\rho}^{2}=-f^{-1}(M_{\rho}^{2},0)v^{2}=g^{2}_{{}_{\rm
HLS}}(M_{\rho}^{2})\cdot 2F_{\pi}^{2}$ is independent of $N$ and can be
extrapolated into $N\rightarrow 4$ with $p=3$ for the actual rho meson.

We thus establish the dynamically generation of the rho meson as the HLS gauge
boson for $a=2$ Yamawaki:2018jvy  in exactly the same way as in the $CP^{N-1}$
model and other Grassmannian models.

In the unbroken phase, $v=0,\eta\neq 0$, on the other hand,
$\Gamma^{(\rho)}_{\mu\nu}(q)$ in Eq.(32) is transverse, implying the HLS is an
unbroken gauge symmetry. Though not invertible as it stands, it is of course
inverted by fixing the gauge as usual, to get the massless propagator
$\langle\rho_{\mu}\rho_{\nu}\rangle(q)=g^{2}_{{}_{\rm
HLS}}(q^{2},\eta)\cdot\frac{g_{\mu\nu}}{q^{2}}+$ gauge term, with
$g^{-2}_{{}_{\rm
HLS}}(q^{2},\eta)\equiv-2Nf(q^{2},\eta)\simeq-2Nf(0,\eta)\equiv
g^{-2}_{{}_{\rm HLS}}(\eta)$ which is analytic at $q^{2}=0$. $\eta$-dependence
may be regarded as the running of the coupling, asymptotically non-
free/infrared free, $g^{2}_{{}_{\rm HLS}}(\eta)\rightarrow 0\,(\eta\rightarrow
0)$, the same as that in broken phase, see Eq.(31). #11#11#11 Without gauge
symmetry ($a=0$), $\langle\alpha_{\mu,||}\,\alpha_{\nu,||}\rangle(q)$ is ill-
defined in the unbroken phase $v=0$, where the factor
$g_{\mu\lambda}+\frac{G}{N}B_{\mu\lambda}$ is pure transverse and not
invertible, in accord with the Weinberg-Witten theorem Weinberg:1980kq  on the
absence of massless spin $J\geq 1$ particles in the positive definite Hilbert
space (no gauge symmetry).  The situation is also the same as the $CP^{N-1}$
and the Grassmannian models. Note also that the massless rho meson is stable,
since it does not decay into the pions which are no longer the NG bosons and
have non-zero mass degenerate with $\check{\rho}$ (no longer the would-be NG
boson) and other degrees of freedom of $\phi_{i,\beta}$ (corresponding to the
6 constraints in the broken phase, in addition to the 3 $\pi$’s and 3
$\check{\rho}$’s for $N=4,p=3$),
$M_{\pi}^{2}=M^{2}_{\check{\rho}}=\cdots=\eta\neq 0$. Note that the phase
transition is of the second order with $v=\eta=0$ and all the spectra are
decoupled (free) massless particles:
$M_{\rho}^{2}=M^{2}_{\check{\rho}}=M_{\pi}^{2}=0$ at the phase transition
point $G=G_{\rm crit}$ (conformal).

###  III.2 Case for arbitrary value of $a$

Since we have established the dynamical generation of the rho meson for $a=2$,
the next question is wether the conclusion is dependent on the specific value
of $a=2$. Here we show that the result is independent of $a$.

For the generic case for arbitrary $a$, the large $N$ dominant diagrams are
not just the one-loop but do include an infinite sum of the bubble diagrams
coming from the extra four-vertex $\left(\frac{a}{2}-1\right)\frac{G}{N}$ as
in Eq.(29). $C_{\mu\nu}$ in Eq.(29) is solved straightforwardly though
tediously (see Ref. Yamawaki:2018jvy  for details): From Eqs.(29) and (30) we
have

$\displaystyle\frac{a}{2}C_{\mu\nu}(q)$ $\displaystyle=$
$\displaystyle\frac{a}{2}\left[g_{\mu\nu}+\left(1-\frac{a}{2}\right)\frac{G}{N}B_{\mu\nu}(q)\right]^{-1}$
$\displaystyle=$
$\displaystyle\left[1-\left(1-\frac{2}{a}\right)G\left(v^{2}+q^{2}f\right)\right]^{-1}\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)+\left[1-\left(1-\frac{2}{a}\right)Gv^{2}\right]^{-1}\frac{q_{\mu}q_{\nu}}{q^{2}}\,,$
$\displaystyle\frac{\Gamma^{(\rho)}_{\mu\nu}(q)}{N}$ $\displaystyle=$
$\displaystyle\frac{2}{G}\left(1-\frac{2}{a}\right)^{-1}\left[\frac{a}{2}C_{\mu\nu}-g_{\mu\nu}\right]\,,$
(35) $\displaystyle=$
$\displaystyle\left[\frac{f^{-1}}{q^{2}+v^{2}f^{-1}}-\left(1-\frac{2}{a}\right)G\right]^{-1}\cdot\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)+\left[\frac{1}{v^{2}}-\left(1-\frac{2}{a}\right)G\right]^{-1}\cdot\frac{q_{\mu}q_{\nu}}{q^{2}},\quad
f\equiv f(q^{2},\eta).$

This of course is reduced to Eq.(32) for $a=2$.

We finally arrive at the dynamically generated propagating HLS gauge boson for
any $a$, whose propagator in the broken phase, takes the same form of the
unitary gauge as that for $a=2$ except for the contact term (to be discussed
later): Yamawaki:2018jvy

$\displaystyle\langle\rho_{\mu}\rho_{\nu}\rangle(q)$ $\displaystyle=$
$\displaystyle-\Gamma^{(\rho)}_{\mu\nu}(q)^{-1}=\left[\frac{-f^{-1}}{q^{2}+v^{2}f^{-1}}+\left(1-\frac{2}{a}\right)G\right]\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)-\left[\frac{1}{v^{2}}-\left(1-\frac{2}{a}\right)G\right]\frac{q_{\mu}q_{\nu}}{q^{2}}$
(36) $\displaystyle=$
$\displaystyle\left(1-\frac{2}{a}\right)\,\frac{G}{N}\,g_{\mu\nu}+2\Delta_{\mu\nu}(q).$

which is reduced to Eq.(33) for $a=2$. Again the mass relation from
$\Delta_{\mu\nu}$ in the last line of Eq.(33) is independent of $N$ and thus
safely extrapolated to the realistic rho meson, $N\rightarrow 4$ with $p=3(=$
fixed).

Here the physical pole position $q^{2}=M_{\rho}^{2}=g^{2}_{{}_{\rm
HLS}}(M_{\rho}^{2})\cdot 2F_{\pi}^{2}$ and the on-shell HLS coupling
$g^{2}_{{}_{\rm HLS}}\equiv g^{2}_{{}_{\rm HLS}}(M_{\rho}^{2})$ are both
independent of $a$;

$\displaystyle M^{2}_{\rho}=g^{2}_{{}_{\rm HLS}}\cdot 2F_{\pi}^{2}\,,\quad
F_{\rho}^{2}=2\cdot F_{\pi}^{2}\,,\quad(a-{\rm independent})\,,$ (37)

which is the same as Eq.(34) but now it is an $a$-independent result, in
contrast to that of the bare quantities at tree level Eq.(22):
$f_{\rho}^{2}=a{f_{\pi}}^{2}$. This relation is a reminiscence of the KSRF II
relation, $M_{\rho}^{2}=2g^{2}_{\rho\pi\pi}F_{\pi}^{2}$, where
$g_{\rho\pi\pi}$ is the $\rho\pi\pi$ coupling. In fact in the next section we
will show $g_{\rho\pi\pi}=g_{{}_{\rm HLS}}$ (“rho-universality”) independently
of $a$, and thus derive the KSRF II relation (as well as KSRF I) independently
of $a$.

Note that the $a$-dependence is exactly cancelled in the physical part
$\Delta_{\mu\nu}(q)$ as it should be, since $a$ is actually a redundant
parameter for the auxiliary field $\rho_{\mu}$. While the $a-$ dependence
remains in the unphysical contact term $-\frac{2G}{aN}g_{\mu\nu}$ which
corresponds to the tree $\rho_{\mu}$ “propagator” with tree mass
$\frac{aN}{2G}$, it is an artifact in using the auxiliary field $\rho_{\mu}$
for the composite field
$\alpha_{\mu,||}=i\frac{G}{N}\phi\partial_{\mu}\phi^{t}$ whose two-point
function is independent of $a$ and exists even for $a=0$ (without HLS!) in the
broken phase. (They satisfy an exact relation via Ward-Takahashi identity,
$\langle\rho_{\mu}\rho_{\nu}\rangle(q)=\langle\alpha_{\mu,||}\,\alpha_{\nu,||}\rangle(q)-\frac{2G}{aN}g_{\mu\nu}$
Yamawaki:2018jvy ).

Moreover, the whole contact term is cancelled in the $\pi\pi$ scattering. The
$\pi\pi$ scattering amplitude $T_{\mu\nu}(q)$ is given as
$2T_{\mu\nu}(q)$$=-\frac{G}{N}g_{\mu\nu}$$+\langle\alpha_{\mu,||}\,\alpha_{\nu,||}\rangle(q)$,
where the first term is from the tree vertex, while the second term is only
from the loop contributions (bubble sum) dominant in the large $N$ limit,
$\langle\alpha_{\mu,||}\,\alpha_{\nu,||}\rangle(q)=$
$\left(i\frac{G}{N}\right)^{2}$$\langle\phi\partial_{\mu}\phi^{t}\,\,\phi\partial_{\nu}\phi^{t}\rangle(q)$
$=\left(i\frac{G}{N}\right)^{2}$
$\left[B_{\mu\nu}(q)+B_{\mu\lambda}(q)\cdot\left(-\frac{G}{N}\right)B^{\lambda}_{\nu}(q)+\cdots\right]$
$=\left(i\frac{G}{N}\right)^{2}B_{\mu}^{\lambda}(q)$
$\left[g_{\lambda\nu}+\frac{N}{G}\langle\alpha_{\mu,||}\,\alpha_{\nu,||}\rangle(q)\right]$
$=\left(g_{\mu\lambda}+\frac{G}{N}B_{\mu\lambda}\right)^{-1}$$\left(i\frac{G}{N}\right)^{2}B^{\lambda}_{\nu}(q)$
$=\frac{G}{N}g_{\mu\nu}+2\Delta_{\mu\nu}(q)$, with $B_{\mu\nu}(q)$ given in
Eq.(30).#12#12#12 The four-$\phi$ vertex (both for tree and loop) here is
different from Eq.(29) for the $\rho_{\mu}$ case:
$(\frac{a}{2}-1)\frac{G}{N}+(-\frac{a}{2})\frac{G}{N}=-\frac{G}{N}$ (the
additional second term is from the tree rho contribution
$(-\frac{ia}{2})(\frac{a}{2}\frac{N}{G})^{-1}(-\frac{ia}{2})$), the same as
that for $a=0$ (original nonlinear sigma model without HLS) as it should be
independent of the auxiliary field.  Then the contact term
$\frac{G}{N}g_{\mu\nu}$ is precisely cancelled in $T_{\mu\nu}$, namely the VMD
for arbitrary value of $a$. This is compared with the conventional HLS
approach where the VMD for $\pi\pi$ scattering is realized only for $a=4/3$
(not $a=2$!!) Harada:2003jx .

As seen from Eq.(33), the HLS coupling depends on the cutoff $\Lambda$ as it
should, since the nonlinear sigma model is a non-renormalizable model for
$D=4$ (see Weinberg:1997rv  for other formulation). From Eq.(33), with $N=4$
and $q^{2}=M_{\rho}^{2}\simeq(770\,{\rm MeV})^{2}$, $F_{\pi}\simeq 92\,{\rm
MeV}$, we have $\Lambda=e^{-4/3}\cdot M_{\rho}\cdot
e^{12\pi^{2}F_{\pi}^{2}/M_{\rho}^{2}}$$\simeq 1.1{\rm GeV}\simeq 4\pi
F_{\pi}$, roughly the validity scale of the chiral perturbation theory. As an
asymptotically non-free theory the kinetic term vanishes $1/g^{2}_{{}_{\rm
HLS}}(q^{2}=\mu^{2})=-2Nf(\mu^{2},0)\rightarrow 0$
($\mu^{2}\rightarrow{\tilde{\Lambda}}^{2}$) at the Landau pole
$\mu=\tilde{\Lambda}=e^{4/3}\Lambda\simeq 4.2$ GeV $\gg\Lambda\gg M_{\rho}$,
where the $\rho_{\mu}$ returns to an auxiliary field as a static composite of
$\pi$, the situation sometimes referred to as “compositeness condition”
Bardeen:1989ds  advocated in a reformulation of the top quark condensate model
Miransky:1988xi . In this viewpoint the HLS gauge bosons as bound states of
$\pi$’s develop the kinetic term as we integrate the higher frequency modes in
the large $N$ limit from $\Lambda^{2}$ down to the scale $\mu^{2}$ in the
sense of the Wilsonian renormalization group Harada:2003jx .

This also implies $g^{2}_{{}_{\rm HLS}}(q^{2}=\mu^{2})\rightarrow 0$
($\mu^{2}/{\tilde{\Lambda}}^{2}\rightarrow 0$) at approaching the phase
transition point $F_{\pi}^{2}=Nv^{2}\rightarrow 0\,(G\rightarrow G_{\rm
cr}-$). Thus the rho meson in the broken phase, with $M_{\rho}$ close enough
to the phase transition point,
$M_{\rho}/{\tilde{\Lambda}},M_{\rho}/\Lambda\rightarrow 0$, is to be
identified with a gauge boson. #13#13#13 Since the phase transition is second
order, the HLS as a gauge symmetry is crucial not only in the unbroken phase
but also in the broken phase near the phase transition point. (Just on the
phase transition point all the spectra become massless and free, i.e.,
trivially scale-symmetric.)  The result $g^{2}_{{}_{\rm HLS}}\rightarrow 0$
and $M_{\rho}^{2}\rightarrow M_{\pi}^{2}(\equiv 0)$ near the phase transition
point in the broken phase is similar to the Vector Manifestation (Ref.
Harada:2003jx  and references cited therein), both not precisely on the phase
transition point where $\rho$ and $\pi$ are just decoupled massless free
particles $M_{\rho}^{2}=M_{\pi}^{2}=0$. The latter is based on the one-loop
“Wilsonian Matching” with QCD at $\Lambda$ where the kinetic term is given
with the parameter $a=a(\Lambda^{2})\simeq 1$, which then runs down as
$a(\mu^{2})=F_{\rho}^{2}(\mu^{2})/F_{\pi}^{2}(\mu^{2})\sim
1\,(M_{\rho}^{2}<\mu^{2}<\Lambda^{2})$ (with $\rho$ loop) and further down to
$\pi$ on shell $a(0)=F_{\rho}^{2}(M_{\rho}^{2})/F_{\pi}^{2}(M_{\pi}^{2}=0)=2$
(with the $\rho$ loop decoupled for $F_{\pi}^{2}$ in $\mu^{2}<M_{\rho}^{2}$),
in contrast to the present case which is for any $a$ at all orders in the
large $N$ limit without $\rho$ loop at all.

We then have the effective action with kinetic term of rho meson $\rho_{\mu}$
and/or the composite field $\alpha_{\mu,||}$:

$\displaystyle{\cal L}_{\rm kinetic}^{{\rm eff}}=-\frac{1}{4g^{2}_{{}_{\rm
HLS}}}\cdot\frac{1}{2}{\rm tr}\rho_{\mu\nu}^{2}=-\frac{1}{4g^{2}_{{}_{\rm
HLS}}}\cdot\frac{1}{2}{\rm tr}\alpha_{\mu\nu,||}^{2},$ (38)

where
$\alpha_{\mu\nu,||}\equiv\partial_{\mu}\alpha_{\nu,||}-\partial_{\nu}\alpha_{\mu,||}-i\left[\alpha_{\mu,||},\alpha_{\nu,||}\right]$,
with $f^{2}_{\pi}=N/G\Rightarrow F^{2}_{\pi}=Nv^{2}$. For $N=4,p=3$ Eq.(38) is
precisely the Skyrme term, $\frac{1}{32e^{2}}{\rm
tr}_{{}_{SU(2)}}[L_{\mu},L_{\nu}]^{2}$, with $e^{2}=g^{2}_{{}_{\rm HLS}}$, in
the $SU(2)$ basis, where we have
$\alpha_{\mu\nu,||}=i\left[\alpha_{\mu,\perp},\alpha_{\nu,\perp}\right]$ and
$L_{\mu}\equiv\partial_{\mu}U\cdot U^{\dagger}$,
$U=\xi^{2}(\pi)=e^{i\pi^{a}\tau^{a}/F_{\pi}}$ Igarashi:1985et .

##  IV Successful “$a=2$” relations realized for any $a$

Now we derive all the phenomenologically successful relations for the rho
meson independently of $a$.

The large $N$ Green function for $\rho\pi\pi$ is given as a bubble sum, which
takes the $a-$independent form of VMD,
$\langle\rho_{\mu}(q)\phi(k)\phi(k+q)\rangle$$=\langle\alpha_{\mu,||}(q)\,\phi(k)\,\phi(q+k)\rangle$
$=2\Delta_{\mu\nu}(q)\cdot(q+2k)^{\nu}$ #14#14#14 The $\rho\pi\pi$ vertex
$\Gamma^{\rho\pi\pi,\nu}(q,k,q+k)\big{|}^{k^{2}=(k+q)^{2}=0}_{\phi-{\rm
amputated}}=\frac{a}{2}\left[g_{\mu\nu}+B_{\mu\lambda}(q)C^{\lambda}_{\nu}(q)\cdot\left(\frac{a}{2}-1\right)\frac{G}{N}\right]\cdot(q+2k)^{\nu}$
is rewritten as
$\left[g_{\mu\nu}+\Gamma^{(\rho)}_{\mu\nu}(q)\left(1-\frac{2}{a}\right)\right]\cdot(q+2k)^{\nu}$,
with
$\Gamma^{(\rho)}_{\mu\nu}(q)=-\langle\rho_{\mu}\rho_{\nu}\rangle(q)^{-1}$.
Then the Green function is $\langle\rho_{\mu}\phi\phi\rangle$
$=\langle\rho_{\mu}\rho_{\nu}\rangle(q)$$\cdot\Gamma^{\rho\pi\pi,\nu}(q,k,q+k)\big{|}^{k^{2}=(k+q)^{2}=0}_{\phi-{\rm
amputated}}=\left[\langle\rho_{\mu}\rho_{\nu}\rangle(q)-g_{\mu\nu}\cdot\left(1-\frac{2}{a}\right)\frac{G}{N}\right]\cdot(q+2k)^{\nu}=2\Delta_{\mu\nu}(q)\cdot(q+2k)^{\nu}$,
where the $a$-dependence and the contact term are all cancelled out. , where
the first equality is by the Ward-Takahashi identity #15#15#15 It follows from
the Ward-Takahashi identity $0=\int{\cal
D}\phi\frac{\delta}{\delta\rho_{\mu}(x)}$ $\left(\phi(y)\phi(z)\cdot
e^{iS[\phi,\rho_{\mu}]}\right)$ $=\int{\cal D}\phi$
$\left(\frac{aN}{2G}\right)$
$\left(\rho_{\mu}(x)-\alpha_{\mu,||}(x)\right)\cdot\phi(y)\phi(z)$ $\cdot
e^{iS[\phi,\rho_{\mu}]},$ for ${\cal L}$ given in Eq.(23). This shows
$a$-independence of $\langle\rho_{\mu}\phi\phi\rangle$ even without explicit
calculations in footnote#14, since $\langle\alpha_{\mu,||}\phi\phi\rangle$ is
obviously independent of $a$. . We may introduce “renormalized” field
$\rho_{\mu}^{(R)}\equiv$$g^{-1}_{{}_{\rm HLS}}(q^{2})\cdot\rho_{\mu}$ by
rescaling the “kinetic term” to the canonical one, i.e.,
$\Delta^{(R)}_{\mu\nu}(q)\equiv g^{-2}_{{}_{\rm
HLS}}(q^{2})\cdot\Delta_{\mu\nu}(q)$ and
$\langle\rho^{(R)}_{\mu}\phi\phi\rangle$$=\langle\alpha_{\mu,||}^{(R)}\phi\phi\rangle$$=2g_{{}_{\rm
HLS}}(q^{2})\cdot\Delta^{(R)}_{\mu\nu}(q)\cdot(q+2k)^{\nu}$, which is compared
with the definition of $g_{\rho\pi\pi}(q^{2})$,
$\langle\rho^{(R)}_{\mu}\phi\phi\rangle\equiv
2g_{\rho\pi\pi}(q^{2})\cdot\Delta^{(R)}_{\mu\nu}(q)\cdot(q+2k)^{\nu}$,
resulting in the $\rho$ universality independently of $a$:

$\displaystyle g_{\rho\pi\pi}(q^{2})=g_{{}_{\rm
HLS}}(q^{2})\quad\left(\rho\,\,{\rm universality}\right).$ (39)

It then leads to the KSRF relations (generalized for $\forall q^{2}$)
independently of $a$ #16#16#16 This is consistent with the fact that the KSRF
I relation is a low energy theorem of the HLS valid for any $a$ Bando:1984pw ;
Bando:1987br , which is proved to all order of loop expansion Harada:1993jk .
:

$\displaystyle g_{\rho}(q^{2})$ $\displaystyle=$ $\displaystyle
M_{\rho}(q^{2})F_{\rho}=2g_{\rho\pi\pi}(q^{2})F_{\pi}^{2}\,\,\left({\rm
KSRF\,I}\right),$ (40) $\displaystyle M_{\rho}^{2}(q^{2})$ $\displaystyle=$
$\displaystyle 2g_{\rho\pi\pi}^{2}(q^{2})F_{\pi}^{2}\,\,\left({\rm
KSRF\,II}\right),$ (41)

with $\langle 0|J^{{\rm em}}_{\mu}|\rho^{(R)}(q^{2})\rangle\equiv
g_{\rho}(q^{2})$$\epsilon_{\mu}(q)=M_{\rho}(q^{2})F_{\rho}\epsilon_{\mu}(q)$.

The VMD for the electromagnetic form factor $F_{{}_{{\cal B}\pi\pi}}(q^{2})$
follows also $a-$independently, similarly to the VMD in the $\pi\pi$
scattering. Here the photon field ${\cal B}_{\mu}$ is introduced by gauging
$H_{\rm global}$,
$D_{\mu}\phi\Rightarrow\partial_{\mu}\phi-i\rho_{\mu}\phi+i\phi{\cal
B}_{\mu}$. It has contributions from the ${\cal B}_{\mu}-\rho_{\mu}$ mixing
and from the “direct coupling” to $\alpha_{\mu,||}$ (with the tree contact
term cancelled by the bubble sum as in the $\pi\pi$ scattering), both coupled
to the identical VMD Green functions
$\langle\rho_{\mu}^{(R)}\phi\phi\rangle=\langle\alpha_{\mu,||}^{(R)}\phi\phi\rangle$
in a linear combination to cancel the $a$ dependence#17#17#17 $2F_{{}_{{\cal
B}\pi\pi}}(q^{2})$$\left(q+2k\right)_{\mu}=\langle J^{{\rm
em}}_{\mu}(q)\,\phi(k)\,\phi(k+q)\rangle\big{|}^{k^{2}=(k+q)^{2}=0}_{\phi-{\rm
amputated}}=-g_{\rho}(q^{2})$
$\left[\frac{a}{2}\langle\rho^{(R)}_{\mu}\,\phi\,\phi\rangle+\left(1-\frac{a}{2}\right)\langle\alpha_{\mu,||}^{(R)}\,\phi\,\phi\rangle\right]\left(q+2k\right)^{\nu}=2\cdot\left(-M_{\rho}^{2}(q^{2})\cdot\Delta^{(R)}_{\mu\nu}(q)\right)\cdot(q+2k)^{\nu}$.
:

$\displaystyle F_{{}_{{\cal
B}\pi\pi}}(q^{2})=\frac{M_{\rho}^{2}(q^{2})}{M_{\rho}^{2}(q^{2})-q^{2}},\quad
F_{{}_{{\cal B}\pi\pi}}(0)=1.$ (42)

Thus the VMD is realized, independently of $a$. #18#18#18 Although it takes
the same form as the naive VMD, $F_{{}_{{\cal
B}\pi\pi}}(q^{2})\approx\frac{M_{\rho}^{2}}{M_{\rho}^{2}-q^{2}}$ near the on-
shell, $q^{2}\approx M_{\rho}^{2}$, $M_{\rho}(q^{2})$ here has log $q^{2}$
dependence as in Eq.(33). Actually, such a $q^{2}$ dependence is necessary for
the modern version of the VMD in both the space-like and the time-like
momentum regions, see e.g., Gounaris:1968mw ; Harada:1995sj ; Benayoun:2011mm
.

##  V Conclusion and Discussions

To conclude we have proved that the rho meson is a dynamical gauge boson of
the HLS $O(3)_{\rm local}\simeq[SU(2)_{V}]_{\rm local}$ by the large $N$
dynamics of the model $G/H=O(N)/[O(N-3)\times O(3)]$, with all the successful
“$a=2$ results” being realized purely dynamically independently of $N$ for any
value of $a$, thus safely extrapolated to $N=4$, $O(4)/O(3)\simeq O(4)_{\rm
global}\times O(3)_{\rm local}\simeq[SU(2)_{L}\times SU(2)_{R}]_{\rm
global}\times[SU(2)_{V}]_{\rm local}$.

The “$a=2$ results” originally obtained for particular choice of $a=2$
Bando:1984ej ; Bando:1985rf ; Bando:1984pw ; Fujiwara:1984mp ; Bando:1987ym
are now clear to be artifacts of the combined use of the $a-$dependent tree-
level rho meson mass term and the ad hoc added kinetic term which was assumed
to be generated at quantum level without affecting the pole structure of the
dynamically generated propagator. Actually, as we demonstrated, the tree-level
parameter is no longer the true one of the pole at quantum level when the
kinetic term is generated, namely the pole position (and residue as well) of
the full propagator is shifted from the tree level one in such a way that the
$a-$ dependence is totally cancelled out. Actually, the parameter $a$ is a
redundant parameter for the auxiliary field $\rho_{\mu}$ and is irrelevant to
the physical results at quantum level as it should be for the correct
calculations. The results of the present paper revealed that it is indeed the
case in the large $N$ limit.

Further implications of the results are: Once the rho kinetic term is
generated, Eq.(38), it stabilizes the Skyrmion without ad hoc Skyrme term and
hence the nonlinear sigma model in the large $N$ limit perfectly describes via
HLS the low energy QCD for $\pi,\rho,N$ at the scale $\lesssim\Lambda\simeq
4\pi f_{\pi}$ without explicit recourse to the QCD.

The dynamically generated kinetic term, with the induced gauge coupling
$g^{2}_{\rm HLS}(q^{2})$ being asymptotically non-free/infrared free in both
broken and unbroken phases, has a cutoff $\Lambda\simeq 4\pi f_{\pi}\gg
M_{\rho}$ (and Landau pole $\tilde{\Lambda}$), so that the rho meson is
sitting near the second order phase transition point as a composite HLS gauge
boson to be matched with the underlying QCD. This implies Yamawaki:2018jvy
that the large $N$ dynamics reveals the HLS as a “magnetic gauge theory”
(infrared free in both phases) dual to the underlying QCD as the “electric
gauge theory” Harada:1999zj ; Harada:2003jx ; Komargodski:2010mc ;
Kitano:2011zk , similarly to the Seiberg duality in the SUSY QCD
Seiberg:1994bz .

If the HLS as the unbroken magnetic gauge theory is realized, say in hot/dense
QCD, we would have a new possibility for the chiral symmetry restored hadronic
phase having massless rho meson and massive $\pi,\check{\rho}$
Yamawaki:2018jvy , which is contrasted with $M_{\rho}^{2}\rightarrow
M^{2}_{\pi}(\equiv 0)$ near the phase transition point in the broken phase
(not precisely on the phase transition point) similarly to the “Vector
Manifestation” as described in the text.

It was frequently emphasized that the large $N$ results are valid even for the
small $N$ at least qualitatively as mentioned in the footnote #3. The result
of the present paper is a yet another proof of this statement, and even more,
quantitatively not just qualitatively, in perfect agreement with the
experimental facts of the rho meson.

This further implies the dynamical HLS bosons in other system described by the
large $N$ Grassmannian models. A notable case of such is the Standard Model
(SM) Higgs Lagrangian, re-parameterized Fukano:2015zua  as a scale-invariant
version of the model $G/H=O(4)/(3)\simeq O(4)_{\rm global}\times O(3)_{\rm
local}$, is precisely the same as the rho meson case, except for an extra
mode, pseudo-dilaton (SM Higgs boson) to make the model (approximately) scale-
invariant (Having no indices running through $N$, it is irrelevant to the SM
rho physics in the large $N$ limit) Yamawaki:2018jvy . This justifies the
basic assumption Matsuzaki:2016iyq  that there exists a rho meson-like vector
boson within the SM (“SM rho”) which stabilizes a skyrmion (“SM skyrmion”) as
a candidate for the dark matter existing even within the SM.

###### Acknowledgements.

We would like to thank T. Kugo who made invaluable help for the preliminary
results in Ref.Yamawaki:2018jvy . Thanks also go to H. Ohki for valuable
discussions and comments. Special thanks go to Mannque Rho for useful
questions and inviting the contribution to the special issue of the
“Symmetry”.

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