Open AccessArticle

Proving Rho Meson Is 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

Symmetry 2023, 15(12), 2209; https://doi.org/10.3390/sym15122209

Submission received: 16 October 2023 / Revised: 4 December 2023 / Accepted: 13 December 2023 / Published: 18 December 2023

(This article belongs to the Special Issue Symmetries and Ultra Dense Matter of Compact Stars II: Emergence of Symmetries in Strong Nuclear Correlations)

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Abstract

The rho meson has long been successfully identified with a dynamical gauge boson of Hidden Local Symmetry (HLS)

H_{local}

in the non-linear sigma model

G / H

gauge equivalent to the model having the symmetry

G_{global} \times H_{local}

, with

G = [S U {(2)}_{L} \times S U {(2)}_{R}] ≃ O (4), H = S U {(2)}_{V} ≃ 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)}_{global} \times {[O (N - 3) \times O (3)]}_{local}

, extrapolated to

N = 4

O {(4)}_{global} \times O {(3)}_{local}

, with all the phenomenologically successful “

a = 2

results”, i.e.,

ρ

-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 the quantitative level directly by the experiments. The relevant cutoff reads

Λ ≃ 4 π F_{π}

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, a further signal of the duality. The unbroken phase with a massless rho meson may be realized as a novel chiral-restored hadronic phase in the hot/dense QCD.

Keywords:

hidden local symmetry; rho meson; large N limit; dynamical generation; grassmannian manifold; kawarabayashi–suzuki–ryazuddin–fayyazuddin relation; vector meson dominance; rho meson universality; skyrmion; second order phase transition

1. Introduction

Since its proposal [1,2] (for reviews see [3,4,5]), identifying the rho meson as a dynamical gauge boson of Hidden Local Symmetry (HLS)

H_{local}

has been widely accepted in the model as having the symmetry

G_{global} \times H_{local}

, with

G = [S U {(2)}_{L} \times S U {(2)}_{R}] ≃ O (4)

and

H = S U {(2)}_{V} ≃ O (3)

, where its Lagrangian consists of two independent invariants

L_{HLS} = L_{A} + a L_{V}

, with a arbitrary parameter. This gauge is equivalent to the non-linear sigma model,

L_{CCWZ}

\overset{‘}{a}

la Callan–Coleman–Wess–Zumino (CCWZ) [6,7] 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

L_{V} = 0

, and

L_{HLS} = L_{A} = L_{CCWZ}

after gauge fixing.

Once we assume, however, that its kinetic term,

L_{kinetic}

, is generated at the quantum level by the dynamics of the non-linear sigma model itself, thereby put by hand to the Lagrangian,

L_{HLS} \Rightarrow L_{A} + a L_{V} + L_{kinetic}

, novel physics come out [1,2,3,4,5]: All the successful phenomenological results, such as the universality of the rho meson coupling (

ρ

universality), the Kawarabayashi–Suzuki–Ryazuddin–Fayyazuddin (KSRF) relation, and 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 L_{V}

becomes the HLS gauge-invariant mass terms of the

ρ

meson, which contains

ρ

mass,

ρ

couplings, additional

π

self-couplings, etc. For application of the HLS to the nuclear physics, see [8].

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 the HLS model itself. 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 [4]. In this case, however, the parameter a (the renormalized one) is a completely free parameter to be adjusted to

a = 2

by hand within the HLS model framework. It can only be determined with additional information, by “Wilsonian matching” with the QCD parameters at UV scale

Λ

as the input,

a (Λ^{2}) \sim 1

, which then predicts

a (μ^{2} = M_{π}^{2} = 0) = 2

as the infrared value (massless

π

on-shell) through the (one-loop) renormalization group for

F_{π}^{2} (μ^{2} = 0)

due to the

π

loop alone in the HLS model itself (see [4] and references cited therein).

In this paper, we resolve this long-standing mystery as simply a consequence of the non-perturbative dynamics of the large N limit of the non-linear 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) ≃ S U {(2)}_{L} \times S U {(2)}_{R} / S U {(2)}_{V}

for the extrapolation

N \to 4

, with the rho meson being the dynamical gauge boson of

O {(3)}_{local} ≃ {[S U {(2)}_{V}]}_{local}

. Preliminary results of this paper were given as the supplementary ones in [9], 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 of the QCD, but the details of the relevant calculations in the present paper may be found in [9].

It is in fact well known that the HLS gauge bosons in many non-linear sigma models, such as the

C P^{N - 1}

model with

G / H = U (N) / [U (N - 1) \times U (1)]

≃ S U (N) / [S U (N - 1) \times U (1)]

gauge equivalent to the model

S U {(N)}_{global} \times U {(1)}_{local}

, do acquire a kinetic term for the

U {(1)}_{local}

gauge boson at the quantum level in the large N limit [3,4,10,11,12,13,14,15,16,17,18]. It was further shown (in a context irrelevant to rho meson physics, however) that the HLS gauge bosons

O {(p)}_{local}

and

U {(p)}_{local}

in the Grassmannian models

G / H = O (N) / [O (N - p)

\times O (p)

[19] and

G / H =

U (N) / [U (N - p) \times U (p)]

, respectively [19,20], 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 [9] and Equation (10)).

Here, we show not only the generation of the

O {(p)}_{local}

gauge boson

ρ_{μ}

but also that all the successful “

a = 2

results” are direct consequences of the pure dynamics at the quantum level of the large N limit for the 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. (It is known that the large N results remain qualitatively true even for the smallest value

N = 2

in the

C P^{N - 1}

model, as checked by the equivalent

O (3)

model exactly solvable in two dimensions [14]).

2. Grassmaniann N Extension

Let us define the generic HLS base [2,3] of an

N \times N

real matrix field

ξ (x) = ξ ({\overset{ˇ}{ρ}}^{(p)}) \cdot ξ ({\overset{ˇ}{ρ}}^{(N - p)}) \cdot ξ (π)

, which transforms under

G_{global} \times H_{local} = O {(N)}_{global} \times {[O (N - p) \times O (p)]}_{local}

ξ (x) \to h (x) \cdot ξ (x) \cdot g^{- 1}

with

h (x) \in {[O (N - p) \times O (p)]}_{local}, g \in O {(N)}_{global}

, where

ξ (π) = e^{i π_{a} (x) X_{a} / f_{π}}

is the CCWZ base for

G / H = O (N) / [O (N - p) \times O (p)]

, with

f_{π}

being the (bare/tree-level) decay constant of the NG boson

π

, while

ξ ({\overset{ˇ}{ρ}}^{(p)}) = e^{i {\overset{ˇ}{ρ}}^{(p)} (x) / f_{ρ}^{(p)}}

and

ξ ({\overset{ˇ}{ρ}}^{(N - p)}) = e^{i {\overset{ˇ}{ρ}}^{(N - p)} (x) / f_{ρ}^{(N - p)}}

, with

{\overset{ˇ}{ρ}}^{(p)} = {\overset{ˇ}{ρ}}_{a}^{(p)} S_{a}^{(p)}

and

{\overset{ˇ}{ρ}}^{(N - p)} = {\overset{ˇ}{ρ}}_{a}^{(N - p)} S_{a}^{(N - p)}

being the would-be NG bosons to be absorbed into the HLS gauge bosons

ρ_{μ}^{(p)}

and

ρ_{μ}^{(N - p)}

O {(p)}_{local}

and

O {(N - p)}_{local}

, with the (bare/tree-leve) decay constant

f_{ρ}^{(p)}

and

f_{ρ}^{(N - p)}

, respectively:

\begin{matrix} ξ (x) = ξ (π) ({unitary gauge, {\overset{ˇ}{ρ}}^{(N - p)} (x) = \overset{ˇ}{ρ}}^{(p)} (x) = 0) . \end{matrix}

(1)

Here, the generators read

X_{a} \in G - H

S_{a}^{(p)} \in H^{(p)} = O (p)

S_{a}^{(N - p)} \in H^{N - p} = O (N - p)

, with

tr (T_{a} T_{b}) = 2 δ_{a b}

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

C P^{N - 1}

model, it is customary to parameterize the HLS base as (

p \times N

degrees of freedom of

ϕ_{i β}

consist of

p \times (N - p)

π

p \times (p - 1) / 2

\overset{ˇ}{ρ}

, and

p \times (p + 1) / 2

of the constraints):

\begin{matrix} ξ {(x)}_{α β} & = & \frac{G}{N} (\begin{matrix} ϕ_{i, β} (x) \\ Φ_{k, β} (x) \end{matrix}), α = (i, k), β = (j, l), i, j = 1, \dots, p; k, l = p + 1, \dots N, \\ ξ^{t} \cdot ξ & = & \frac{G}{N} (ϕ^{t} ϕ + Φ^{t} Φ) = 𝟙, \frac{G}{N} \equiv \frac{1}{{f_{π}}^{2}}, \end{matrix}

(2)

\begin{matrix} ξ \cdot ξ^{t} & = & \frac{G}{N} (\begin{matrix} ϕ ϕ^{t} & ϕ Φ^{t} \\ Φ ϕ^{t} & Φ Φ^{t} \end{matrix}) = (\begin{matrix} 𝟙_{p \times p} & 0 \\ 0 & 𝟙_{(N - p) \times (N - p)} \end{matrix}) = 𝟙, \end{matrix}

(3)

where

G \equiv N / {f_{π}}^{2}

is the (bare) coupling constant to be fixed in the large N limit (s.t.

{f_{π}}^{2} = O (N)

The covariantized Maurer–Cartan one-form reads:

\begin{matrix} {\hat{α}}_{μ} & \equiv & \frac{1}{i} D_{μ} ξ \cdot ξ^{t} = \frac{G}{i N} (\begin{matrix} \partial_{μ} ϕ - i ρ_{μ}^{(p)} ϕ \\ \partial_{μ} Φ - i ρ_{μ}^{(N - p)} Φ \end{matrix}) \cdot (ϕ^{t} Φ^{t}) = {\hat{α}}_{μ, ⊥} + {\hat{α}}_{μ, | |}, \end{matrix}

(4)

where

{\hat{α}}_{μ, ⊥} \equiv \frac{1}{2} tr ({\hat{α}}_{μ} X^{a}) X^{a}

{\hat{α}}_{μ, | |} \equiv \frac{1}{2} tr ({\hat{α}}_{μ} S^{a}) S^{a}

are

\begin{matrix} {\hat{α}}_{μ, ⊥} & = & α_{μ, ⊥} = (\begin{matrix} 0 & \frac{G}{i N} \partial_{μ} ϕ \cdot Φ^{t} \\ \frac{G}{i N} \partial_{μ} Φ \cdot ϕ^{t} & 0 \end{matrix}), \\ {\hat{α}}_{μ, | |} & = & (\begin{matrix} \frac{G}{i N} \partial_{μ} ϕ \cdot ϕ^{t} - ρ_{μ}^{(p)} & 0 \\ 0 & \frac{G}{i N} \partial_{μ} Φ \cdot Φ^{t} - ρ_{μ}^{(N - p)} \end{matrix}), \end{matrix}

all transforming homogeneously as

\begin{matrix} ({\hat{α}}_{μ, ⊥}, {\hat{α}}_{μ, | |}) & \to & h (x) \cdot ({\hat{α}}_{μ, ⊥}, {\hat{α}}_{μ, | |}) \cdot h^{- 1} (x), \end{matrix}

(5)

with

h (x) \in H

for

H = {[O (N - p) \times O (p)]}_{local}

Thus, the HLS Lagrangian consists of three independent invariants at the lowest derivative [9]:

\begin{matrix} L^{(N, p)} & = & L_{A} + a^{(p)} L_{V}^{(p)} + a^{(N - p)} L_{V}^{(N - p)}, \end{matrix}

(6)

where

\begin{matrix} L_{A} & = & \frac{{f_{π}}^{2}}{4} tr ({\hat{α}}_{μ, ⊥}^{2}) = - \frac{G}{2 N} tr (ϕ^{t} \partial_{μ} ϕ \cdot Φ^{t} \partial^{μ} Φ) \\ = & \frac{1}{2} tr (\partial_{μ} ϕ \partial^{μ} ϕ^{t} + \frac{G}{N} {(ϕ \partial_{μ} ϕ^{t})}^{2}) \\ (unitary gauge) & ⟶ & L_{CCWZ} = \frac{{f_{π}}^{2}}{4} tr {(α_{μ, ⊥}^{2} (π))}^{2} = \frac{1}{2} {(\partial_{μ} π_{a})}^{2} + \dots, \end{matrix}

(7)

with (unitary-gauge)

α_{μ, ⊥} \to α_{μ, ⊥} (π) = \partial_{μ} ξ (π) \cdot ξ^{t} (π) = \partial_{μ} ξ (π) \cdot ξ^{†} (π)

, and

\begin{matrix} a^{(p)} L_{V}^{(p)} & = & \frac{a^{(p)} {f_{π}}^{2}}{4} tr ({[{\hat{α}}_{μ, | |}^{(p)}]}^{2}) = \frac{1}{2} tr [\frac{a^{(p)}}{2} \cdot \frac{N}{G} {(ρ_{μ}^{(p)} - i \frac{G}{N} ϕ \partial_{μ} ϕ^{t})}^{2}] \\ = & \frac{{(f_{ρ}^{(p)})}^{2}}{4} tr {[ρ_{μ}^{(p)} - \frac{\partial_{μ} {\overset{ˇ}{ρ}}^{(p)}}{f_{ρ}^{(p)}} - \frac{i [\partial_{μ} π, π]}{2 {f_{π}}^{2}} + \dots]}^{2}, \end{matrix}

(8)

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

\begin{matrix} {(f_{ρ}^{(p)})}^{2} = a^{(p)} {f_{π}}^{2} = a^{(p)} \frac{N}{G}, \end{matrix}

(9)

to normalize the kinetic term of the would-be NG boson

{\overset{ˇ}{ρ}}^{(p)}

to the canonical form, and similarly for

a^{(N - p)} L_{V}^{(N - p)}

. In the unitary gauge,

{\overset{ˇ}{ρ}}^{(p)} = 0

, Equation (8) reads the mass term of

ρ_{μ}^{(p)}

as usual in the HLS formalism, and so does

a^{(N - p)} L_{V}^{(N - p)}

, the mass term of

ρ_{μ}^{(N - p)}

Here, we note that in contrast to the

O {(p)}_{local}

gauge boson, the kinetic term for the

O {(N - p)}_{local}

gauge boson, carrying index running

1, \dots, N - p

, thus subject to all the planar diagram contributions in the large N limit, is not dynamically generated and (

O {(N - p)}_{local}

does not exist for

N = 4, p = 3

anyway), and stays as an auxiliary field (i.e.,

L_{V}^{(N - p)} = 0

) as was the case in the previous calculations for

C P^{N - 1}

and the Grassmannian models. The

S U {(N - 1)}_{local}

gauge boson in the

C P^{N - 1}

model with

G / H = S U (N) / [S U (N - 1) \times U (1)]

, which carries the index running through

1, \dots, N - 1

, is not dynamically generated in the large N limit, in contrast to the

U {(1)}_{local}

part [3,4,10,11,12,13,14,15,16,17,18]. 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)}_{local}

and

U {(N - p)}_{local}

, respectively [19,20]. Similarly, a popular N extension

G / H = O (N) / O (N - 1)

gauge equivalent to the model

O {(N)}_{global} \times O {(N - 1)}_{local}

has no dynamical gauge boson for

O {(N - 1)}_{local}

and is irrelevant to the rho meson.

Then, without loss of generality, the starting Lagrangian Equations (6)–(8), is simplified as

a^{(p)} L_{V}^{(p)}

\equiv a L_{V}

a^{(N - p)} L_{V}^{(N - p)} = 0

ρ_{μ} \equiv ρ_{μ}^{(p)}

\overset{ˇ}{ρ} = {\overset{ˇ}{ρ}}^{(p)}

f_{ρ}^{2}

\equiv {(f_{ρ}^{(p)})}^{2} = a f_{π}^{2}

, etc.:

\begin{matrix} L = L_{A} + a L_{V} & = & \frac{1}{2} tr [(\partial_{μ} ϕ \partial^{μ} ϕ^{t}) + \frac{1}{2} \cdot \frac{a N}{G} ρ_{μ}^{2} - i a ρ^{μ} ϕ \partial_{μ} ϕ^{t})] \\ + & \frac{1}{2} tr [(1 - \frac{a}{2}) \frac{G}{N} {(ϕ \partial_{μ} ϕ^{t})}^{2} - η (ϕ ϕ^{t} - \frac{N}{G} 𝟙)], \end{matrix}

(10)

where

tr

and

𝟙

should read

{tr}_{p \times p}

and

𝟙_{p \times p}

, respectively, and

{(ρ_{μ})}_{i j} = ρ_{μ}^{a} {(S^{a})}_{i j}

with

tr (S^{a} S^{b}) = 2 δ^{a b}

, and the

p \times p

matrix Lagrange multiplier

η_{i, j} (x)

is used for the constraint Equation (3) as in the standard large N arguments of

C P^{N - 1}

[3,4,10,11,12,13,14,15,16,17,18] and other Grassmannian models [19,20]. (In the broken phase this is simply equivalent to the constraint Equation (3), while in the unbroken phase the multiplier is only a correct description. See later discussions).

For

N = 4, p = 3

Equation (10) with

O {(4)}_{global} \times O {(3)}_{local}

is identical to the standard HLS Lagrangian [1,2,3,4,5] for the rho meson with

{[S U {(2)}_{L} \times S U {(2)}_{R}]}_{global} \times {[S U {(2)}_{V}]}_{local}

. It is now clear [9] that Equation (10) coincides with that of the conventional “covariant derivative type” Lagrangian [19] for a particular choice

a = 2

, with

ϕ ϕ^{t} = (N / G) 𝟙

(see Equation (3)).

From Equation (10), the effective potential in the large N limit for

〈 ϕ_{i, β} (x) 〉 = \sqrt{N} v (δ_{i, j}, 0)

(we took

v \neq 0

real, i.e., the unitary gauge

\overset{ˇ}{ρ} (x) = 0

), and

〈 η_{i, j} (x) 〉 = η δ_{i, j}

, takes the form (in D dimensions):

\begin{matrix} \frac{V_{eff} (v, η)}{N p / 2} = η (v^{2} - \frac{1}{G}) + \int \frac{d^{D} k}{i {(2 π)}^{D}} ln (k^{2} - η), \end{matrix}

(11)

where the (a-dependent) 1-PI contributions are sub-leading in the large N limit (this 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

C P^{N - 1}

and the Grassmannian models corresponding to

a = 2

[3,4,10,11,12,13,14,15,16,17,18,19,20], and hence yields the same gap equation:

\begin{matrix} \frac{1}{N p} \frac{\partial V_{eff}}{\partial v} & = & 2 η v = 0, \\ \frac{1}{N p} \frac{\partial V_{eff}}{\partial η} & = & v^{2} - \frac{1}{G} + \frac{1}{G_{crit}} - v_{η}^{2} = 0, \end{matrix}

(12)

with (for cutoff

Λ

)

\begin{matrix} \frac{1}{G_{crit}} \equiv \int \frac{d^{D} k}{i {(2 π)}^{D}} \frac{1}{- k^{2}} = \frac{1}{(\frac{D}{2} - 1) Γ (\frac{D}{2})} \frac{Λ^{D - 2}}{{(4 π)}^{\frac{D}{2}}}, \\ v_{η}^{2} \equiv \int \frac{d^{D} k}{i {(2 π)}^{D}} (\frac{1}{- k^{2}} - \frac{1}{η - k^{2}}) = \frac{Γ (2 - \frac{D}{2})}{\frac{D}{2} - 1} \cdot \frac{η^{\frac{D}{2} - 1}}{{(4 π)}^{\frac{D}{2}}} . \end{matrix}

The cutoff

Λ

can be removed for

2 \leq D < 4

(the theory is renormalizable), introducing the renormalized coupling at renormalization point

μ

1 / G^{(R)} (μ) \equiv 1 / G - \int \frac{d^{D} k}{i {(2 π)}^{D}} \frac{1}{μ^{2} - k^{2}} \equiv μ^{D - 2} / g^{(R)} (μ)

1 / G_{crit}^{(R)} \equiv \int \frac{d^{D} k}{i {(2 π)}^{D}} (\frac{1}{- k^{2}} - \frac{1}{μ^{2} - k^{2}})

= \frac{Γ (2 - D / 2)}{(D / 2 - 1)} \cdot \frac{μ^{D - 2}}{{(4 π)}^{D / 2}} \equiv μ^{D - 2} / g_{crit}^{(R)}

, s.t.,

1 / G - 1 / G_{crit} = μ^{D - 2} (1 / g^{(R)} (μ) - 1 / g_{crit}^{(R)})

in the gap equation. The renormalized coupling

g^{(R)} (μ)

has an ultraviolet fixed point at

g_{crit}^{(R)}

0 \leq g_{crit}^{(R)} = {(4 π)}^{D / 2} (D / 2 - 1) / Γ (2 - D / 2) < \infty (2 \leq D < 4)

, with the beta function

β (g^{(R)} (μ)) = μ \partial g^{(R)} (μ) / \partial μ = - (D - 2) g^{(R)} (μ) [g^{(R)} (μ) - g_{crit}^{(R)}] / g_{crit}^{(R)}

. While for

D = 4

the theory is not renormalizable,

1 / g_{crit}^{(R)} \sim Γ (2 - D / 2) / {(4 π)}^{2} |_{D \to 4} \sim ln (Λ^{2} / μ^{2}) / {(4 π)}^{2}

, with the remaining log divergence identified in the cutoff notation.

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:

\begin{matrix} (i) & G < G_{cr} : v \neq 0, η = 0 (broken phase) \\ v^{2} & = \frac{1}{G} - \frac{1}{G_{crit}} > 0, \\ (i i) & G > G_{crit} : v = 0, η \neq 0 (unbroken phase) \end{matrix}

(13)

\begin{matrix} v_{η}^{2} = \frac{1}{G_{crit}} - \frac{1}{G} > 0, \end{matrix}

(14)

with the phase transition point

v = η = 0

. We may define a full decay constant

F_{π}

at quantum level in the large N limit:

\begin{matrix} F_{π}^{2} \equiv N v^{2} = N (\frac{1}{G} - \frac{1}{G_{crit}}) & = & {f_{π}}^{2} - \frac{N}{(\frac{D}{2} - 1) Γ (\frac{D}{2})} \frac{Λ^{D - 2}}{{(4 π)}^{\frac{D}{2}}} \\ ⟶ & {f_{π}}^{2} - N \frac{Λ^{2}}{{(4 π)}^{2}} (D \to 4) . \end{matrix}

(15)

Thisindicates that approaching from the broken phase to the critical point,

F_{π}^{2} \to 0

, is due to the power divergence of

1 / G_{crit}

(quadratic divergence for

D = 4

), similarly to the “Wilsonian matching” of the HLS model with the underlying QCD at the UV scale

Λ

[4].

3. Dynamical Generation of Rho Meson

Now the (amputated) two-point function of

ρ_{μ}

in the large N limit takes the form:

\begin{matrix} Γ_{μ ν}^{(ρ)} (q) = (\frac{a}{2}) (\frac{N}{G}) g_{μ ν} + {(\frac{a}{2})}^{2} B_{μ λ} (q) \cdot C_{ν}^{λ} (q), \\ C_{μ ν} (q) = g_{μ ν} + (\frac{a}{2} - 1) \frac{G}{N} B_{μ λ} (q) \cdot C_{ν}^{λ} (q), \end{matrix}

(16)

where the four-

ϕ

vertex

(\frac{a}{2} - 1) \frac{G}{N}

in our Lagrangian Equation (10) (second line) gives rise to an infinite sum of the bubble graph contribution

B_{μ ν} (q)

;

\begin{matrix} \frac{1}{N} B_{μ ν} (q) & = & \frac{1}{2} \int \frac{d k^{D}}{i {(2 π)}^{D}} \frac{{(2 k + q)}_{μ} {(2 k + q)}_{ν}}{(k^{2} - η) ({(k + q)}^{2} - η)} \\ = & q^{2} f (q^{2}, η) \cdot (g_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) + (v^{2} - \frac{1}{G}) \cdot g_{μ ν}, \end{matrix}

(17)

with

\begin{matrix} f (q^{2}, η) \equiv - \frac{Γ (2 - \frac{D}{2})}{2 {(4 π)}^{\frac{D}{2}} Γ (2)} \int_{0}^{1} d x \frac{{(1 - 2 x)}^{2}}{{[x (1 - x) q^{2} + η]}^{2 - \frac{D}{2}}}, \end{matrix}

which reads for

D \to 4

(

ϵ \equiv 2 - D / 2 \to 0

and

1 / A^{ϵ} ≃ 1 - ϵ ln A

\begin{matrix} f (q^{2}, 0) & = & - \frac{1}{2} \cdot \frac{1}{3 {(4 π)}^{2}} \cdot [ln (\frac{Λ^{2}}{q^{2}}) + \frac{8}{3}], \\ f (0, η) & = & - \frac{1}{2} \cdot \frac{1}{3 {(4 π)}^{2}} \cdot [ln (\frac{Λ^{2}}{η})], \end{matrix}

(18)

where we have used the gap equation Equation (12) and identified

Γ (ϵ) ≃ 1 / ϵ \to ln Λ^{2}

. The finite part common to both phases is included in the definition of the cutoff

Λ

, while the part

+ 8 / 3

is an extra one in the broken phase

η \equiv 0

, similarly to that in the

C P^{N - 1}

[18].

3.1. $a = 2$ Case

Note that for

a = 2

, we have

C_{μ ν} = g_{μ ν}

in Equation (16), which yields

Γ_{μ ν}^{(ρ)} (q)

\begin{matrix} \frac{Γ_{μ ν}^{(ρ)} (q)}{N} = (\frac{1}{G}) g_{μ ν} + \frac{B_{μ λ} (q)}{N} \cdot g_{ν}^{λ} (q) = (q^{2} f (q^{2}, η) + v^{2}) \cdot (g_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) + v^{2} \cdot \frac{q_{μ} q_{ν}}{q^{2}}, \end{matrix}

(19)

the well-known form of one-loop dominance in the large N limit in the conventional “covariant derivative” parameterization for

C P^{N - 1}

model and other Grassmannian models [3,4,10,11,12,13,14,15,16,17,18,19,20].

For the broken phase

v \neq 0

η = 0

, this is readily inverted to yield the

ρ_{μ}

propagator for

a = 2

〈 ρ_{μ} ρ_{ν} 〉 (q) \equiv 〈 ρ_{μ}^{i j} ρ_{ν}^{j i} 〉 (q) = 2 〈 ρ_{μ}^{a} ρ_{ν}^{a} 〉 (q)

\begin{matrix} 〈 ρ_{μ} ρ_{ν} 〉 (q) & = & - Γ_{μ ν}^{(ρ)} {(q)}^{- 1} = \frac{1}{N} \frac{- f^{- 1} (q^{2}, 0)}{q^{2} + f^{- 1} (q^{2}, 0) v^{2}} (g_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) - \frac{1}{N} \frac{1}{v^{2}} \frac{q_{μ} q_{ν}}{q^{2}} \\ = & \frac{1}{N} \frac{- f^{- 1} (q^{2}, 0)}{q^{2} + f^{- 1} (q^{2}, 0) v^{2}} (g_{μ ν} - \frac{q_{μ} q_{ν}}{- f^{- 1} (q^{2}, 0) v^{2}}) = 2 Δ_{μ ν} (q), \\ Δ_{μ ν} (q) & \equiv & \frac{g_{_{HLS}}^{2} (q^{2})}{q^{2} - M_{ρ}^{2} (q^{2})} (g_{μ ν} - \frac{q_{μ} q_{ν}}{M_{ρ}^{2} (q^{2})}), \\ M_{ρ}^{2} (q^{2}) \equiv - f^{- 1} (q^{2}, 0) v^{2} = g_{_{HLS}}^{2} (q^{2}) \cdot 2 F_{π}^{2}, \\ g_{_{HLS}}^{- 2} (q^{2}) \equiv - 2 N f (q^{2}, 0) = \frac{N}{3 {(4 π)}^{2}} [ln \frac{Λ^{2}}{q^{2}} + \frac{8}{3}], \end{matrix}

(20)

which is the form of the unitary gauge (we took the unitary gauge

\hat{ρ} (x) = 0

with

v \neq 0 =

real), with the physical pole position and the on-shell HLS coupling given as

q^{2} = M_{ρ}^{2} = - f^{- 1} (M_{ρ}^{2}, 0) v^{2} = g_{_{HLS}}^{2} (M_{ρ}^{2}) \cdot 2 F_{π}^{2}

and

g_{_{HLS}}^{2} \equiv g_{_{HLS}}^{2} (M_{ρ}^{2})

, respectively. The relation implies the rho meson mass is generated by the Higgs mechanism

\begin{matrix} M_{ρ}^{2} = g_{_{HLS}}^{2} \cdot F_{ρ}^{2}, F_{ρ}^{2} = 2 \cdot F_{π}^{2}, \end{matrix}

(21)

where

F_{ρ}

is the decay constant of the would-be NG boson

\hat{ρ}

(absorbed into the rho meson in the unitary gauge) at the quantum level, which is to be compared with the tree-level relation Equation (9), with

a = a^{(p)} = 2

. Equation (20) suggests even more general results than the on-shell relation Equation (21):

\begin{matrix} M_{ρ}^{2} (q^{2}) = g_{_{HLS}}^{2} (q^{2}) \cdot F_{ρ}^{2}, F_{ρ}^{2} = 2 \cdot F_{π}^{2} . \end{matrix}

(22)

We shall come back to this point later when discussing VMD.

The

q^{2}

dependence of

M_{ρ}^{2} (q^{2})

and

g_{_{HLS}}^{2} (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_{ρ}^{2} = - f^{- 1} (M_{ρ}^{2}, 0) v^{2} = g_{_{HLS}}^{2} (M_{ρ}^{2}) \cdot 2 F_{π}^{2}

is independent of N and can be extrapolated into

N \to 4

with

p = 3

for the actual rho meson.

We thus establish the dynamic generation of the rho meson as the HLS gauge boson for

a = 2

[9] in exactly the same way as in the

C P^{N - 1}

model and other Grassmannian models.

In the unbroken phase,

v = 0, η \neq 0

, on the other hand,

Γ_{μ ν}^{(ρ)} (q)

in Equation (19) is transverse, implying the HLS is an unbroken gauge symmetry. Although not invertible as it stands, it is of course inverted by fixing the gauge as usual, to get the massless propagator

〈 ρ_{μ} ρ_{ν} 〉 (q) = g_{_{HLS}}^{2} (q^{2}, η) \cdot \frac{g_{μ ν}}{q^{2}} +

gauge term, with

g_{_{HLS}}^{- 2} (q^{2}, η) \equiv - 2 N f (q^{2}, η) ≃ - 2 N f (0, η) \equiv g_{_{HLS}}^{- 2} (η)

which is analytic at

q^{2} = 0

η

-dependence may be regarded as the running of the coupling, asymptotically non-free/infrared free,

g_{_{HLS}}^{2} (η) \to 0 (η \to 0)

, the same as that in broken phase, see Equation (18). Without gauge symmetry (

a = 0

〈 α_{μ, | |} α_{ν, | |} 〉 (q)

is ill defined in the unbroken phase

v = 0

, where the factor

g_{μ λ} + \frac{G}{N} B_{μ λ}

is pure transverse and not invertible, in accordance with the Weinberg–Witten theorem [21] in 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

C P^{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

\overset{ˇ}{ρ}

(no longer the would-be NG boson) and other degrees of freedom of

ϕ_{i, β}

(corresponding to the six constraints in the broken phase, in addition to the 3

π

’s and 3

\overset{ˇ}{ρ}

’s for

N = 4, p = 3

M_{π}^{2} = M_{\overset{ˇ}{ρ}}^{2} = \dots = η \neq 0

. Note that the phase transition is of the second order with

v = η = 0

and all the spectra are decoupled (free) massless particles:

M_{ρ}^{2} = M_{\overset{ˇ}{ρ}}^{2} = M_{π}^{2} = 0

at the phase transition point

G = G_{crit}

(conformal).

3.2. 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

(\frac{a}{2} - 1) \frac{G}{N}

as in Equation (16).

C_{μ ν}

in Equation (16) is solved straightforwardly though tediously (see [9] for details): From Equations (16) and (17) we have

\begin{matrix} \frac{a}{2} C_{μ ν} (q) & = & \frac{a}{2} {[g_{μ ν} + (1 - \frac{a}{2}) \frac{G}{N} B_{μ ν} (q)]}^{- 1} \\ = & {[1 - (1 - \frac{2}{a}) G (v^{2} + q^{2} f)]}^{- 1} (g_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) \\ + {[1 - (1 - \frac{2}{a}) G v^{2}]}^{- 1} \frac{q_{μ} q_{ν}}{q^{2}}, \\ \frac{Γ_{μ ν}^{(ρ)} (q)}{N} & = & \frac{2}{G} {(1 - \frac{2}{a})}^{- 1} [\frac{a}{2} C_{μ ν} - g_{μ ν}], \\ = & {[\frac{f^{- 1}}{q^{2} + v^{2} f^{- 1}} - (1 - \frac{2}{a}) G]}^{- 1} \cdot (g_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) \\ + {[\frac{1}{v^{2}} - (1 - \frac{2}{a}) G]}^{- 1} \cdot \frac{q_{μ} q_{ν}}{q^{2}}, f \equiv f (q^{2}, η) . \end{matrix}

(23)

This of course is reduced to Equation (19) for

a = 2

We finally arrive at the dynamically generated propagating HLS gauge boson for any a, whose propagator in the broken phase and takes the same form of the unitary gauge as that for

a = 2

except for the contact term (to be discussed later): [9]

\begin{matrix} 〈 ρ_{μ} ρ_{ν} 〉 (q) = - Γ_{μ ν}^{(ρ)} {(q)}^{- 1} & = & [\frac{- f^{- 1}}{q^{2} + v^{2} f^{- 1}} + (1 - \frac{2}{a}) G] (g_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) \\ - [\frac{1}{v^{2}} - (1 - \frac{2}{a}) G] \frac{q_{μ} q_{ν}}{q^{2}} \\ = & (1 - \frac{2}{a}) \frac{G}{N} g_{μ ν} + 2 Δ_{μ ν} (q) . \end{matrix}

(24)

which is reduced to Equation (20) for

a = 2

. Again, the mass relation from

Δ_{μ ν}

in the last line of Equation (20) is independent of N and thus safely extrapolated to the realistic rho meson,

N \to 4

with

p = 3 (=

fixed).

Here, the physical pole position

q^{2} = M_{ρ}^{2} = g_{_{HLS}}^{2} (M_{ρ}^{2}) \cdot 2 F_{π}^{2}

and the on-shell HLS coupling

g_{_{HLS}}^{2} \equiv g_{_{HLS}}^{2} (M_{ρ}^{2})

are both independent of a;

\begin{matrix} M_{ρ}^{2} = g_{_{HLS}}^{2} \cdot 2 F_{π}^{2}, F_{ρ}^{2} = 2 \cdot F_{π}^{2}, (a - independent), \end{matrix}

(25)

which is the same as Equation (21) but now it is an a-independent result, in contrast to that of the bare quantities at tree level Equation (9):

f_{ρ}^{2} = a {f_{π}}^{2}

. This relation is a reminiscence of the KSRF II relation,

M_{ρ}^{2} = 2 g_{ρ π π}^{2} F_{π}^{2}

, where

g_{ρ π π}

is the

ρ π π

coupling. In fact, in the next section we will show

g_{ρ π π} = g_{_{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

Δ_{μ ν} (q)

as it should be, since a is actually a redundant parameter for the auxiliary field

ρ_{μ}

. While the

a -

dependence remains in the unphysical contact term

- \frac{2 G}{a N} g_{μ ν}

which corresponds to the tree

ρ_{μ}

“propagator” with tree mass

\frac{a N}{2 G}

, it is an artifact in using the auxiliary field

ρ_{μ}

for the composite field

α_{μ, | |} = i \frac{G}{N} ϕ \partial_{μ} ϕ^{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 [9]:

\begin{matrix} 〈 ρ_{μ} ρ_{ν} 〉 (q) = 〈 α_{μ, | |} α_{ν, | |} 〉 (q) - \frac{2 G}{a N} g_{μ ν} . \end{matrix}

(26)

This implies that both are identical at a limit

a \to \infty

, which is relevant to the physics of the off-shell rho meson, like the skyrmion physics.

Moreover, the whole contact term is cancelled in the

π π

scattering (VMD). The

π π

scattering amplitude

T_{μ ν} (q)

is given as

\begin{matrix} 2 T_{μ ν} (q) = - \frac{G}{N} g_{μ ν} + 〈 α_{μ, | |} α_{ν, | |} 〉 (q) = 2 Δ_{μ ν} (q), (VMD) . \end{matrix}

(27)

where the

g_{μ ν}

term is from the tree vertex, while

〈 α_{μ, | |} α_{ν, | |} 〉 (q)

is only from the loop contributions (bubble sum) dominant in the large N limit, giving the result consistent with the Ward–Takahashi identity, Equation (26), with Equation (24):

\begin{matrix} 〈 α_{μ, | |} α_{ν, | |} 〉 (q) & = & {(i \frac{G}{N})}^{2} 〈 ϕ \partial_{μ} ϕ^{t} ϕ \partial_{ν} ϕ^{t} 〉 (q) \\ = & {(i \frac{G}{N})}^{2} [B_{μ ν} (q) + B_{μ λ} (q) \cdot (- \frac{G}{N}) B_{ν}^{λ} (q) + \dots] \\ = & {(i \frac{G}{N})}^{2} B_{μ}^{λ} (q) [g_{λ ν} + \frac{N}{G} 〈 α_{μ, | |} α_{ν, | |} 〉 (q)] = {(g_{μ λ} + \frac{G}{N} B_{μ λ})}^{- 1} {(i \frac{G}{N})}^{2} B_{ν}^{λ} (q) \\ = & \frac{G}{N} g_{μ ν} + 2 Δ_{μ ν} (q), \end{matrix}

(28)

with

B_{μ ν} (q)

given in Equation (17). The four-

ϕ

vertex (both for tree and loop) here is different from Equation (16) for the

ρ_{μ}

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{i a}{2}) {(\frac{a}{2} \frac{N}{G})}^{- 1} (- \frac{i a}{2})

), the same as that for

a = 0

(original non-linear sigma model without HLS) as it should be independent of the auxiliary field. Then, the contact term

\frac{G}{N} g_{μ ν}

is precisely canceled in

T_{μ ν}

, namely the VMD for arbitrary value of a, now with the generalized relation Equation (22) not just the on-shell one Equation (21). This is compared with the conventional HLS approach where the VMD for

π π

scattering is realized only for

a = 4 / 3

(not

a = 2

!!) [4].

We then have the effective action with kinetic term of rho meson

ρ_{μ}

, which becomes that of the composite field

α_{μ, | |}

a \to \infty

as implied in Equation (26):

\begin{matrix} L_{kinetic}^{eff} & = & - \frac{1}{4 g_{_{HLS}}^{2}} \cdot \frac{1}{2} tr ρ_{μ ν}^{2}, \end{matrix}

(29)

\begin{matrix} ⟶ & - \frac{1}{4 g_{_{HLS}}^{2}} \cdot \frac{1}{2} tr α_{μ ν, | |}^{2} (a \to \infty), \end{matrix}

(30)

where

α_{μ ν, | |} \equiv \partial_{μ} α_{ν, | |} - \partial_{ν} α_{μ, | |} - i [α_{μ, | |}, α_{ν, | |}]

, with

f_{π}^{2} = N / G \Rightarrow F_{π}^{2} = N v^{2}

. For

N = 4

p = 3 .

Equation (30) is precisely the Skyrme term,

\frac{1}{32 e_{Skyrme}^{2}} {tr}_{_{S U (2)}} {[L_{μ}, L_{ν}]}^{2}

, with

e_{Skyrme}^{2} = g_{_{HLS}}^{2}

, in the

S U (2)

basis, where we have

α_{μ ν, | |} = i [α_{μ, ⊥}, α_{ν, ⊥}]

and

L_{μ} \equiv \partial_{μ} U \cdot U^{†}

U = ξ^{2} (π) = e^{i π^{a} τ^{a} / F_{π}}

[22].

In the conventional HLS arguments with

a = 2

and assumed rho meson kinetic term, the kinetic term can still stabilize the skyrmion but only for the low-energy region

q^{2} ≪ M_{ρ}^{2} = O (a)

, where the rho meson mass term effectively becomes the constraint

ρ_{μ} = α_{ν, | |}

[22]. In the large N limit here, on the other hand, with the induced rho meson kinetic term we can freely take the value a at any

q^{2}

region including

q^{2} > M_{N}^{2}

to describe the nucleon with mass

m_{N}

identified with the skyrmion, without affecting all the

a —

independent successful relations of the rho meson to be shown in the next section. Thus, the system

(π, ρ, N)

can be well described by the non-linear sigma model at large N without ad hoc rho meson kinetic term, with the free parameter choice of a just for the detailed description of the nucleon.

As seen from Equation (20), the HLS coupling depends on the cutoff

Λ

as it should, since the non-linear sigma model is a non-renormalizable model for

D = 4

(see [18] for other formulation). From Equation (20), with

N = 4

and

q^{2} = M_{ρ}^{2} ≃ {(770 MeV)}^{2}

F_{π} ≃ 92 MeV

, we have

Λ = e^{- 4 / 3} \cdot M_{ρ} \cdot e^{12 π^{2} F_{π}^{2} / M_{ρ}^{2}}

≃ 1.1 GeV ≃ 4 π F_{π}

, roughly the validity scale of the chiral perturbation theory. As an asymptotically non-free theory the kinetic term vanishes

1 / g_{_{HLS}}^{2} (q^{2} = μ^{2}) = - 2 N f (μ^{2}, 0) \to 0

(

μ^{2} \to {\tilde{Λ}}^{2}

) at the Landau pole

μ = \tilde{Λ} = e^{4 / 3} Λ ≃ 4.2

GeV

≫ Λ ≫ M_{ρ}

, where the

ρ_{μ}

returns to an auxiliary field as a static composite of

π

, the situation sometimes referred to as “compositeness condition” [23] advocated in a reformulation of the top quark condensate model [24]. In this viewpoint, the HLS gauge bosons as bound states of

π

’s develop the kinetic term as we integrate the higher frequency modes in the large N limit from

Λ^{2}

down to the scale

μ^{2}

in the sense of the Wilsonian renormalization group [4].

This also implies

g_{_{HLS}}^{2} (q^{2} = μ^{2}) \to 0

(

μ^{2} / {\tilde{Λ}}^{2} \to 0

) at approaching the phase transition point

F_{π}^{2} = N v^{2} \to 0 (G \to G_{cr} -

). Thus, the rho meson in the broken phase, with

M_{ρ}

close enough to the phase transition point,

M_{ρ} / \tilde{Λ}, M_{ρ} / Λ \to 0

, is to be identified with a gauge boson. 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_{_{HLS}}^{2} \to 0

and

M_{ρ}^{2} \to M_{π}^{2} (\equiv 0)

near the phase transition point in the broken phase is similar to the Vector Manifestation (see [4] and references cited therein), both not precisely on the phase transition point where

ρ

and

π

are just decoupled massless free particles

M_{ρ}^{2} = M_{π}^{2} = 0

. The latter is based on the one-loop “Wilsonian Matching” with QCD at

Λ

where the kinetic term is given with the parameter

a = a (Λ^{2}) ≃ 1

, which then runs down as

a (μ^{2}) = F_{ρ}^{2} (μ^{2}) / F_{π}^{2} (μ^{2}) \sim 1 (M_{ρ}^{2} < μ^{2} < Λ^{2})

(with

ρ

loop) and further down to

π

on shell

a (0) = F_{ρ}^{2} (M_{ρ}^{2}) / F_{π}^{2} (M_{π}^{2} = 0) = 2

(with the

ρ

loop decoupled for

F_{π}^{2}

μ^{2} < M_{ρ}^{2}

), in contrast to the present case which is for any a at all orders in the large N limit without

ρ

loop at all.

4. 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

ρ π π

is given as a bubble sum, which takes the

a —

independent form of VMD, as shown in Appendix D of Ref. [9]:

\begin{matrix} 〈 ρ_{μ} (q) ϕ (k) ϕ (k + q) 〉 & = & 〈 ρ_{μ} ρ_{ν} 〉 (q) \cdot Γ^{ρ π π, ν} (q, k, q + k) |_{ϕ - amputated}^{k^{2} = {(k + q)}^{2} = 0} \\ = & [〈 ρ_{μ} ρ_{ν} 〉 (q) - g_{μ ν} \cdot (1 - \frac{2}{a}) \frac{G}{N}] \cdot {(q + 2 k)}^{ν} \\ = & 2 Δ_{μ ν} (q) \cdot {(q + 2 k)}^{ν}, \end{matrix}

(31)

where the

ρ π π

vertex is given as

\begin{matrix} Γ^{ρ π π, ν} (q, k, q + k) |_{ϕ - amputated}^{k^{2} = {(k + q)}^{2} = 0} & = & \frac{a}{2} [g_{μ ν} + B_{μ λ} (q) C_{ν}^{λ} (q) \cdot (\frac{a}{2} - 1) \frac{G}{N}] \cdot {(q + 2 k)}^{ν} \\ = & [g_{μ ν} + Γ_{μ ν}^{(ρ)} (q) (1 - \frac{2}{a}) \frac{G}{N}] \cdot {(q + 2 k)}^{ν}, \end{matrix}

(32)

with

Γ_{μ ν}^{(ρ)} (q) = - 〈 ρ_{μ} ρ_{ν} 〉 {(q)}^{- 1}

given in Equation (16). Then the a-dependence and the contact term are all cancelled out. It is also shown by the Ward–Takahashi identity

0 = \int D ϕ \frac{δ}{δ ρ_{μ} (x)}

(ϕ (y) ϕ (z) \cdot e^{i S [ϕ, ρ_{μ}]})

= \int D ϕ

(\frac{a N}{2 G})

(ρ_{μ} (x) - α_{μ, | |} (x)) \cdot ϕ (y) ϕ (z)

\cdot e^{i S [ϕ, ρ_{μ}]},

such that

\begin{matrix} 〈 ρ_{μ} (q) ϕ (k) ϕ (k + q) 〉 = 〈 α_{μ, | |} (q) ϕ (k) ϕ (q + k) 〉 \end{matrix}

(33)

for

L

given in Equation (10) even without explicit calculations, since

〈 α_{μ, | |} ϕ ϕ 〉

is obviously independent of a.

We may introduce “renormalized” field

ρ_{μ}^{(R)} \equiv

g_{_{HLS}}^{- 1} (q^{2}) \cdot ρ_{μ}

by rescaling the “kinetic term” to the canonical one, i.e.,

Δ_{μ ν}^{(R)} (q) \equiv g_{_{HLS}}^{- 2} (q^{2}) \cdot Δ_{μ ν} (q)

and

〈 ρ_{μ}^{(R)} ϕ ϕ 〉

= 〈 α_{μ, | |}^{(R)} ϕ ϕ 〉

= 2 g_{_{HLS}} (q^{2}) \cdot Δ_{μ ν}^{(R)} (q) \cdot {(q + 2 k)}^{ν}

, which is compared with the definition of

g_{ρ π π} (q^{2})

〈 ρ_{μ}^{(R)} ϕ ϕ 〉 \equiv 2 g_{ρ π π} (q^{2}) \cdot Δ_{μ ν}^{(R)} (q) \cdot {(q + 2 k)}^{ν}

, resulting in the

ρ

universality independently of a:

\begin{matrix} g_{ρ π π} (q^{2}) = g_{_{HLS}} (q^{2}) (ρ universality) . \end{matrix}

(34)

It then leads to the KSRF relations (generalized for

\forall q^{2}

as in Equation (22)) independently of a:

\begin{matrix} g_{ρ} (q^{2}) & = & M_{ρ} (q^{2}) F_{ρ} = 2 g_{ρ π π} (q^{2}) F_{π}^{2} (KSRF I), \end{matrix}

(35)

\begin{matrix} M_{ρ}^{2} (q^{2}) & = & 2 g_{ρ π π}^{2} (q^{2}) F_{π}^{2} (KSRF II), \end{matrix}

(36)

with

〈 0 | J_{μ}^{em} | ρ^{(R)} (q^{2}) 〉 \equiv g_{ρ} (q^{2})

ϵ_{μ} (q) = M_{ρ} (q^{2}) F_{ρ} ϵ_{μ} (q)

, where

M_{ρ}^{2} (q^{2})

and

g_{_{HLS}} (q^{2})

Δ_{μ ν} (q)

are given in Equation (20). This is consistent with the fact that the KSRF I relation is a low-energy theorem of the HLS valid for any a [3], which is proved to all order of loop expansion [25].

The VMD for the electromagnetic form factor

F_{_{B π π}} (q^{2})

also follows

a —

independently, similarly to the VMD in the

π π

scattering shown in Equation (27). Here, the photon field

B_{μ}

is introduced by gauging

H_{global}

D_{μ} ϕ \Rightarrow \partial_{μ} ϕ - i ρ_{μ} ϕ + i ϕ B_{μ}

in Equation (10) with

α_{μ, | |} = i \frac{G}{N} ϕ \partial_{μ} ϕ^{t}

. It has contributions from the

B_{μ} - ρ_{μ}

mixing and from the “direct coupling” to

α_{μ, | |}

(with the tree contact term canceled by the bubble sum as in the

π π

scattering), both coupled to the identical VMD Green functions

〈 ρ_{μ}^{(R)} ϕ ϕ 〉 = 〈 α_{μ, | |}^{(R)} ϕ ϕ 〉

, Equation (33), in a linear combination to cancel the a dependence:

\begin{matrix} 2 F_{_{B π π}} (q^{2}) {(q + 2 k)}_{μ} & = & 〈 J_{μ}^{em} (q) ϕ (k) ϕ (k + q) 〉 |_{ϕ - amputated}^{k^{2} = {(k + q)}^{2} = 0} \\ = & - g_{ρ} (q^{2}) [\frac{a}{2} 〈 ρ_{μ}^{(R)} ϕ ϕ 〉 + (1 - \frac{a}{2}) 〈 α_{μ, | |}^{(R)} ϕ ϕ 〉] {(q + 2 k)}^{ν} \\ = & 2 \cdot (- M_{ρ}^{2} (q^{2}) \cdot Δ_{μ ν}^{(R)} (q)) \cdot {(q + 2 k)}^{ν} . \end{matrix}

(37)

Namely,

\begin{matrix} F_{_{B π π}} (q^{2}) = \frac{M_{ρ}^{2} (q^{2})}{M_{ρ}^{2} (q^{2}) - q^{2}}, F_{_{B π π}} (0) = 1 . \end{matrix}

(38)

Thus the VMD is realized independently of a.

Although it takes the same form as the naive VMD,

F_{_{B π π}} (q^{2}) \approx \frac{M_{ρ}^{2}}{M_{ρ}^{2} - q^{2}}

near the on-shell,

q^{2} \approx M_{ρ}^{2}

M_{ρ} (q^{2})

here has log

q^{2}

dependence as in Equation (20). 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., [26,27,28].

We now have the effective action at arbitrary value of a for the rho meson after rescaling the

ρ_{μ}

field

ρ_{μ} \to g_{_{HLS}} ρ_{μ}

with

g_{_{HLS}} = g_{ρ π π}

and

B_{μ} \to e B_{μ}

\begin{matrix} L_{ρ}^{eff} = \frac{1}{2} \cdot 2 F_{π}^{2} [g_{_{HLS}}^{2} ({(ρ_{μ}^{+})}^{2} + {(ρ_{μ}^{-})}^{2}) + (ρ_{μ}^{0}, B_{μ}) (\begin{matrix} g_{_{HLS}}^{2} & e g_{_{HLS}} \\ e g_{_{HLS}} & e^{2} \end{matrix}) (\begin{matrix} {ρ^{0}}^{μ} \\ B^{μ} \end{matrix})] - \frac{1}{2} tr ρ_{μ ν}^{2} . \end{matrix}

(39)

After diagonalization this gives for arbitrary a:

\begin{matrix} M_{ρ^{0}}^{2} = (g_{_{HLS}}^{2} + e^{2}) \cdot 2 F_{π}^{2} = M_{ρ^{\pm}}^{2} + 2 e^{2} F_{π}^{2}, \end{matrix}

(40)

in perfect agreement with the experiment, the same result as that of the conventional HLS prediction with assumed kinetic term of rho meson for

a = 2

[4].

However, in contrast to the above a-independent results of the rho meson, mostly on-shell and some off-shell results, the quantities related to the off-shell rho meson in principle do depend on a as the rho meson propagator in Equation (24) has the a-dependent contact term. A typicality of such is the skyrmion physics to be stabilized by the Skyrme term as we mentioned in the last section.

5. Conclusions and Discussions

To conclude, we have proved that the rho meson is a dynamical gauge boson of the HLS

O {(3)}_{local} ≃ {[S U {(2)}_{V}]}_{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) ≃ O {(4)}_{global} \times O {(3)}_{local} ≃ {[S U {(2)}_{L} \times S U {(2)}_{R}]}_{global} \times {[S U {(2)}_{V}]}_{local}

The “

a = 2

results” originally obtained for particular choice of

a = 2

[1,2,3,4] 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 the 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 canceled out. Actually, the parameter a is a redundant parameter for the auxiliary field

ρ_{μ}

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 as follows: Once the rho kinetic term is generated, Equation (29), it stabilizes the Skyrmion without ad hoc Skyrme term, with a free parameter a, and hence the non-linear sigma model in the large N limit perfectly describes via HLS the low energy QCD for

π, ρ, N

at the scale

≲ Λ ≃ 4 π f_{π}

without explicit recourse to the QCD.

The dynamically generated kinetic term, with the induced gauge coupling

g_{HLS}^{2} (q^{2})

being asymptotically non-free/infrared free in both broken and unbroken phases, has a cutoff

Λ ≃ 4 π f_{π} ≫ M_{ρ}

(and Landau pole

\tilde{Λ}

), 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 [9] 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” [4,29,30,31], similarly to the Seiberg duality in the SUSY QCD [32].

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 a massless rho meson and massive

π, \overset{ˇ}{ρ}

[9], which is contrasted with

M_{ρ}^{2} \to M_{π}^{2} (\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 [14]. 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 [5] as a scale-invariant version of the model

G / H = O (4) / (3) ≃ O {(4)}_{global} \times O {(3)}_{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) [9]. This justifies the basic assumption [33] 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.

Funding

This research received no external funding.

Data Availability Statement

Detailed information relevant to this paper is given in Ref. [9].

Acknowledgments

We would like to thank T. Kugo who made invaluable help for the preliminary results in [9]. 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”.

Conflicts of Interest

The author declares no conflict of interest.

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Yamawaki, K. Proving Rho Meson Is a Dynamical Gauge Boson of Hidden Local Symmetry. Symmetry 2023, 15, 2209. https://doi.org/10.3390/sym15122209

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Proving Rho Meson Is a Dynamical Gauge Boson of Hidden Local Symmetry

Abstract

1. Introduction

2. Grassmaniann N Extension

3. Dynamical Generation of Rho Meson

3.1. $a = 2$ Case

3.2. Case for Arbitrary Value of a

4. Successful “a = 2” Relations Realized for Any a

5. Conclusions and Discussions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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Proving Rho Meson Is a Dynamical Gauge Boson of Hidden Local Symmetry

Abstract

1. Introduction

2. Grassmaniann N Extension

3. Dynamical Generation of Rho Meson

3.1. a = 2 Case

3.2. Case for Arbitrary Value of a

4. Successful “a = 2” Relations Realized for Any a

5. Conclusions and Discussions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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3.1. $a = 2$ Case