We consider the Quasilocal Quark Model of NJL type including vector and axial-vector four-fermion interaction with derivatives. The mass spectrum for the ground and first excited states is obtained. The chiral symmetry restoration sum rules in these channels are imposed as matching rules to QCD at intermediate energies and a set of constraints on parameters of QQM is performed.
hep-ph0304140.tex
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\begin{document}

\begin{center}
{\Large\bf Vector particles in Quasilocal Quark
Models}\footnote{Published in Zap. Nauch. Sem. POMI {\bf 291}
(2002), 5} \\

\vspace{4mm}

%A. A. Andrianov,
V. A. Andrianov and S. S. Afonin \\
V. A. Fock Department of Theoretical Physics, St. Petersburg State
University,
198504 St. Petersburg, Russia
\end{center}

\renewcommand{\abstractname}{Abstract}

\begin{abstract}
We consider the Quasilocal Quark Model of NJL type including vector and
axial-vector four-fermion interaction with derivatives. The mass
spectrum for the ground and first excited states is obtained.
The chiral symmetry restoration sum rules in these
channels are imposed as matching rules to QCD at intermediate energies and a
set of constraints on parameters of QQM is performed.
\end{abstract}

\section{Introduction}

%In spite of that QCD is the fundamental theory of strong interactions
%the effective quark models are still
%popular. The reason is that at low-energies one can not use in QCD
%the standard perturbation theory because of the strong
%coupling regime. At these energies the formation of light mesons
%(hadronization) occurs and corresponding physics is not described
%by the fundamental QCD degrees of freedom --- quarks and gluons.
%As a result, QCD fails in analytic calculations of hadron
%characteristics at low energies. At the same time, the different aspects of
%low-energy hadron physics can be successfully obtained by the
%QCD-inspired effective quark models where calculations are much simpler.
%An important example of such models is the models of the Nambu-Jona-Lasinio
%(NJL) type~\ci{1,2,3,4,5,6,7,8,9,10} including local
%four-fermion interactions.
%The QCD-inspired quark models with four-fermion
%interaction  are often applied for
%the effective description of low-energy QCD in the hadronization regime.

The main purpose of this paper is to describe the physics of as
ground vector (V) and axial-vector (A) meson resonances at
low energies as excited states with increasing masses at
intermediate energies in the framework of effective action of
Quasilocal Quark Models (QQM) which were introduced in~\ci{11}.
We will see that these effective QCD QQM are sufficiently general
and from the physical point of view evident to allow a good
description of wide set of mass relations for vector and
axial-vector mesons. Moreover, using properties of two-point QCD
VA correlators of quark densities at high energies (asymptotic freedom
and Operator Product Expansion (OPE)) and two-resonance ansatz for
VA meson correlators of QQM, we will obtain a number of
constraints for parameters of QQM from the so-called Chiral
Symmetry Restoration (CSR) sum rules.

It is well known that the effective QCD-inspired quark models with
four-fermion interactions of Nambu-Jona-Lasinio (NJL) type are often
applied to describe the
low-energy QCD in the hadronization regime~\ci{1,2,3,4,5,6,7,8,9,10}.
The appearance of
effective quark theory is connected with one of specific
property of QCD --- the dynamical Chiral Symmetry Breaking (CSB).
The local four-fermion interaction is involved to induce the
CSB due to strong attraction in the
scalar channel. As a consequence, the dynamical quark mass
$m_{\mbox{\scriptsize dyn}}$ is created, as well as an isospin multiplet of pions,
massless in the chiral limit, and a massive scalar meson with mass
$m_{\sigma}=2m_{\mbox{\scriptsize dyn}}$ appear. However, it is known from
experiment that there are series of meson states
with equal quantum numbers~\cite{pdg}. % and heavier masses.
%, in particular,
%$0^{-+}(\pi,\pi',\pi'',...)$; $0^{++}(\sigma
%(f_{0}),\sigma',\sigma'',...)$; $1^{--}(\rho,\rho',\rho'',...)$;
%$1^{++}(a_1,a_1',...)$.
Due to confinement, one expects an infinite number of such excited
states with increasing masses.
In such a way, the problem arises how to describe the physics of
those resonances at intermediate energies. To solve this problem one can
extend the NJL quark model
taking into account  higher-dimensional quark operators
with derivatives, i.e. quasilocal quark interactions~\ci{11}. For
sufficiently strong couplings the new
operators promote the formation of additional meson states.
Such a
quasilocal approach (see also~\ci{12,13,14,volw,volkeb}) represents a
systematic extension of the NJL-model towards the complete
effective action of QCD where many-fermion vertices with
derivatives possess the manifest chiral symmetry of
interaction, motivated by the soft momentum expansion of the
perturbative QCD effective action.
In the effective action of Quasilocal Quark Models (QQM) of NJL type
the low-energy
gluon effects are  hidden in the coupling constants.
The alternative schemes including the condensates of low-energy
gluons can be found in~\ci{6,15}.

At the same time in the large-$N_c$ approach, which is equivalent
to planar QCD~\cite{16},
%At intermediate energies the correlators of QQM can be matched
%\cite{zap} to
%the OPE of QCD correlators \cite{18}. This matching realizes the
%correspondence to QCD and improves the predictability of QQM.
%It is based on
%the large-$N_{c}$ approach which is equivalent to planar QCD
%\cite{16}. In this approximation
the correlators of color-singlet quark currents are saturated
by narrow meson resonances. In particular, the two-point
correlators of %scalar, pseudoscalar,
vector and axial-vector
quark currents are represented by the sum of related meson poles
in Euclidean space:
\be
\Pi^{C}(p^{2})=\int d^{4}x\exp(ipx)\langle \bar{q}\Gamma
q(x)\bar{q}\Gamma q(0)
\rangle=\sum_{n}\frac{Z_{n}^{C}}{p^{2}+m_{C,n}^{2}}
+D_{0}^{C}+D_{1}^{C}p^{2}, \label{cor1}
\ee
$$
C\equiv %S,P,
V,A; \qquad
%\Gamma=i,\gamma_{5},
\gamma_{\mu},\gamma_{\mu}\gamma_{5}; \qquad
D_{0},D_{1}=const.
$$
The last two terms %both in the scalar-pseudoscalar and in the
%vector-axial-vector channel
represent a perturbative contribution,
with $D_{0}$ and $D_{1}$ being contact terms required for the
regularization of infinite sums. On the other hand,
their high-energy asymptotics is
provided~\cite{18} by the perturbation theory and
OPE due to asymptotic freedom of QCD. Therefrom the
above correlators increase at large $p^{2}$,
\be
\Pi^{C}(p^{2})\mid_{p^{2}\rightarrow\infty}\sim
p^{2} \ln\frac{p^{2}}{\mu^{2}}\,.
\ee
When comparing the two expressions above one concludes that the
infinite series of resonances with the same quantum numbers
should exist in order to reproduce the perturbative asymptotics.
So at intermediate energies the correlators of QQM can be matched~\cite{zap}
to the OPE of QCD correlators. This matching realizes the
correspondence to QCD and improves the predictability of QQM.

Meantime the differences of correlators of opposite-parity currents
rapidly decrease at large momenta $p^{2}\rightarrow\infty$ %\cite{zap,25}:
%\be
%\Pi^{P}(p^{2})-\Pi^{S}(p^{2})\equiv
%\frac{\Delta_{SP}}{p^{4}}+O\left(\frac{1}{p^{6}}\right), \qquad \Delta_{SP}\simeq
%24\pi\alpha_{s}\langle\bar{q}q\rangle^{2}, \label{SP}
%\ee
%and
\cite{18,20,21}
\be
\Pi^{V}(p^{2})-\Pi^{A}(p^{2})\equiv
\frac{\Delta_{V\!\!A}}{p^{6}}-\frac{m_0^2\Delta_{V\!\!A}}{p^{8}}+
O\left(\frac{1}{p^{10}}\right),\,\, \Delta_{V\!\!A}\simeq
-16\pi\alpha_{s}\langle\bar{q}q\rangle^{2}, \label{VA}
\ee
where $m_0^2=0.8\pm0.2\,\mbox{GeV}^2$~\ci{iof}, $\alpha_s\approx0.3$
(at 1 GeV), and we have defined in the $V,A$ channels
\be
\Pi_{\mu\nu}^{V,A}(p^{2})\equiv(-\delta_{\mu\nu}p^{2}+p_{\mu}p_{\nu})
\Pi^{V,A}(p^{2}).
\ee
%and the vacuum dominance hypothesis \cite{18} in the
%large-$N_{c}$ limit is adopted.

Therefore the chiral symmetry is restored at high energies and
the
%two above
difference~\gl{VA} represents an order parameter of CSB in QCD. As it
%they
decreases rapidly at large momenta
one can perform the matching of QCD asymptotics
by means of few lowest lying resonances that gives a set of
constraints for QQM parameters from the CSR sum rules.
%They may be used
%both for obtaining some additional bounds on the model parameters
%and for calculating of some decay constants (see, some examples in
%\cite{20,23} and
%references therein).

In the present work the vector QQM model is considered with two channels
where two pairs of vector and axial-vector mesons are
generated. Respectively it is expected to reproduce the lower
part of QCD meson spectrum in the planar limit and the leading asymptotics
of chiral symmetry restoration for higher energies.
In the Sec.~2 we define the VA, $SU(2)$ QQM with two pairs of
VA-mesons and the corresponding mass spectrum for boson states is obtained.
The Sec.~3 is devoted to the $U(3)$ extension of VA QQM.
%The matching to QCD with the help of CSR sum rules is performed in the
%Sec.~4.
With the help of two-resonance ansatz for VA-mesons the correlators of QQM
are matched to the OPE of QCD correlators and a number of
constraints for parameters of QQM from CSR sum rules are performed
in the Sec.~4.

\section{VA, $SU(2)$ Quasilocal Quark Model}

%The low energy effective action of QCD in the quark
%sector has a qualitatively different structure depending on
%whether it is built at the CSB scale $\Lambda$ by means of perturbation
%theory \cite{[11],[24]} or below the CSB scale when the formation of
%light pseudoscalar meson is
%implemented manifestly \cite{[25],[23],[26]}. We will discuss only
%the first case where the QNJL model \cite{[11]}
%extends the NJL one with chiral symmetry broken due to strong
%attractive four-fermion forces in the color-singlet scalar
%channel. When dynamical quark masses
%are supposed to be sufficiently smaller than the CSB scale (cutoff) the
%minimal $n$-channel lagrangian of the QNJLM has \cite{[11],[14]}
%the following form,
The $SU(2)$ QQM Lagrangian for the two-channel vector (V) and axial-vector
(A) case in the chiral limit~$m_q = 0$ has the form~\ci{we1} (in
Euclidean space):
\be
\label{su2}
{\cal L}_{V\!\!A}=\bar{q}i\!\!\!\not{\partial\phantom{d}}\!\!\!q+
\frac{1}{4N_{f}N_{c}\Lambda^{2}}\sum_{k,l=1}^{2}
b_{kl}\left[\bar{q}\Gamma_{V,k}^iq\cdot\bar{q}\Gamma_{V,l}^iq+
\bar{q}\Gamma_{A,k}^iq\cdot\bar{q}\Gamma_{A,l}^iq\right],
\ee
$$
\Gamma_{V,k}^i\equiv i\gamma_{\mu}f_k(s)\tau^i,\qquad
\Gamma_{A,k}^i\equiv i\gamma_{\mu}\gamma_5f_k(s)\tau^i,\qquad
i=1,2,3\,,
$$
%where indices $S,P$ denote the scalar and pseudoscalar case and
%\be
%L_{SP}^{I}=\frac{1}{4N_{f}N_{c}\Lambda^{2}}\sum_{k,l=1}^{n}a_{kl}
%[\bar{q}f_{k}q\cdot\bar{q}f_{l}q+
%\bar{q}f_{k}i\gamma_{5}q\cdot\bar{q}f_{l}i\gamma_{5}q].
%\ee
where $q\equiv q_j$ ($j$ is the number of flavor $N_f$) are color
fermionic fields with $N_c$ components, $b_{kl}$ represents the
symmetric nonsingular matrix of real coupling constants, and
$\tau^i$ denote Pauli matrices. The quantities $f_k( s)$,
$s\rightarrow -\partial^2/\Lambda^2$ are the form factors
specifying the quasilocal interaction. We accept the following
sequence of action of the derivatives for the Hermitian fermion
currents:
\be
\bar q\frac{\partial^2}{\Lambda^2} q=\frac14\bar q
\biggl(\frac{\stackrel{\rightarrow}{\partial}-
\stackrel{\leftarrow}{\partial}}{\Lambda}\biggr)^2 q\, .
\ee
In addition, let us regularize the interaction vertices by
introducing the momentum cutoff
\be
\bar q q\longrightarrow\bar q\theta(\Lambda^2+\partial^2) q\,,
\label{5}
\ee
and choose the polynomial form factors  as being orthogonal on
the unit interval,
\be
\int_{0}^{1}f_{k}(s)f_{l}(s)ds=\delta_{kl}\,.
\ee
We select out here:
\be
f_{1}(s)=2-3s\,, \qquad f_{2}(s)=-\sqrt{3}s\,.
\ee
As this model interpolates the low-energy QCD action
it is supplied with a cutoff $\Lambda$ (of order of the
CSB scale) for virtual quark momenta in quark loops.

%Let us now incorporate the vector and axial-vector interaction
%with appropriate matrix of couplings $b_{kl}$,
%$$
%L_{SPVA}^{I}=\frac{1}{4N_{c}\Lambda^{2}}\sum_{k,l=1}^{2}
%\{a_{kl}[\bar{q}f_{k}q\cdot\bar{q}f_{l}q+
%\bar{q}f_{k}i\gamma_{5}q\cdot\bar{q}f_{l}i\gamma_{5}q]
%$$
%\be
%+b_{kl}[\bar{q}f_{k}i\gamma_{\mu}q\cdot\bar{q}f_{l}i\gamma_{\mu}q
%+\bar{q}f_{k}i\gamma_{\mu}\gamma_{5}q\cdot\bar{q}f_{l}i\gamma_{\mu}\gamma_{5}q]\}.
%\ee
It is convenient to pass to the auxiliary vector ($\rho_{\mu}^i$) and
axial-vector ($a_{\mu}^i$) fields,
\ba
{\cal L}_{\mbox{\scriptsize aux}}&=&\bar{q}i\!\!\!\not{\partial\phantom{d}}\!\!\!q+
\sum_{k=1}^{2}i\bar{q}\left(\Gamma_{V,k}^i\rho_{k,\mu}^i+
\Gamma_{A,k}^i a_{k,\mu}^i\right)q+ \nl
&+&N_fN_c\Lambda^{2}\sum_{k,l=1}^{2}\left(\rho_{k,\mu}^ib_{kl}^{-1}
\rho_{l,\mu}^i+a_{k,\mu}^ib_{kl}^{-1}a_{l,\mu}^i\right).
\ea

After integrating out the quark fields
$$
\biggl\langle \exp\biggl(-\int\! d^{4}x\,{\cal L}\biggr)\biggr
\rangle_{\bar q q}\equiv\exp(- S_{\mbox{\scriptsize eff}}),
$$
one comes to the bosonic effective action:
$$
S_{\mbox{\scriptsize eff}}(\rho_{\mu,k}^i, a_{\mu,k}^i)=N_fN_c\Lambda^2
\sum_{k,l=1}^{2}\{\rho_{k,\mu}^ib_{kl}^{-1}\rho_{l,\mu}^i+
a_{k,\mu}^ib_{kl}^{-1}a_{l,\mu}^i\}-N_fN_c\mbox{Tr}\ln
\left.\lefteqn{\not{\phantom{DD}}}D\right|_{\mbox{\scriptsize reg}}\,,
$$
\be
\label{eff2}
\lefteqn{\not{\phantom{DD}}}D\equiv i(\!\not{\partial}+M)+i\sum_{k=1}^{2}
\left(\Gamma_{V,k}^i\rho_{k,\mu}^i+\Gamma_{A,k}^i a_{k,\mu}^i\right),
\ee
where we have introduced the dynamic mass function
$M\equiv\sum_k\sigma_kf_k(s)$, with $\sigma_k$ being the vacuum
expectation values of scalar fields~\ci{24}.
We use the chirally invariant regularization of fermionic
determinant:
$$
\ln\det\lefteqn{\not{\phantom{DD}}}D=\mbox{Tr}^{\mbox{\scriptsize all}}
\ln\lefteqn{\not{\phantom{DD}}}D
\longrightarrow N_fN_c\mbox{Tr}\ln\left.\lefteqn{\not{\phantom{DD}}}D
\right|_{\mbox{\scriptsize reg}}\equiv\frac12N_fN_c\mbox{Tr}\ln\frac{\lefteqn{\not{\phantom{DD}}}D
\lefteqn{\not{\phantom{DD}}}D^{\dag}}{\mu^2}\,,
$$
where the constant $\mu$ is a normalization scale for quark
fields and the trace "Tr" is assumed over all degrees of freedom except
color and flavor ones. We will carry out further analysis in the mean
field approximation ($N_c\gg~1$).
Expanding the Eq.~\gl{eff2} in boson fields and retaining the quadratic
in fields part only, one obtains:
\be
S_{\mbox{\scriptsize eff}}^{(2)}(\rho_{\mu,k}^i, a_{\mu,k}^i)=\frac{1}{2}\int\frac{d^{4}p}
{(2\pi)^{4}}\sum_{k,l=1}^{2}
[\rho_{k,\mu}^iC_{kl}^{\rho,\mu\nu}\rho_{l,\nu}^i+a_{k,\mu}^i
C_{kl}^{a,\mu\nu}a_{l,\nu}^i].
\ee
%It should be noticed that when one includes the pseudoscalar (P)
%degrees of freedom the $P-A$ mixing appears. We do not consider
%this effect here since it does not influence on the mass spectrum.
The inverse propagators are defined by the corresponding second
variation of $S_{\mbox{\scriptsize eff}}(\rho_{\mu,k}^i, a_{\mu,k}^i)$:
$$
C_{kl}^{(\rho,a)\mu\nu}=2N_fN_c\Lambda^2b_{kl}^{-1}\delta_{\mu\nu}-
N_fN_c\int\frac{d^4q}{(2\pi)^4}\times
$$
\be
\label{secvar}
\mbox{tr}\left\{(i\gamma_{\mu},i\gamma_{\mu}\gamma_{5})\frac{f_{k}
\left(\left(\frac{q+p/2}{\Lambda}\right)^2\right)}{\not
q+\frac{1}{2}\!\!\not p+iM}
(i\gamma_{\nu},i\gamma_{\nu}\gamma_{5})\frac{f_{l}
\left(\left(\frac{q-p/2}{\Lambda}\right)^2\right)}{\not
q-\frac{1}{2}\!\!\not p+iM}\right\},
\ee
where the trace "tr" spans the Dirac indices only.

To obtain mass spectrum we expand~(\ref{secvar}) in a small external
momentum $p$ ($p^2/\Lambda^2\ll 1$) up to terms $\sim\! p^2$ and
calculate the corresponding loop integral using the momentum
cutoff regularization. To compensate the quadratic divergences in
this integral we parametrize the matrix of coupling constants as
follows:
\be
\label{ft}
16\pi^{2}b_{kl}^{-1}=\delta_{kl}-\frac{4}{3}
\frac{\bar\Delta_{kl}}{\Lambda^{2}}\,;
\qquad \bar\Delta_{kl}\ll\Lambda^{2}.
\ee
The general structure of~\gl{secvar} is:
\be
\label{varder}
C_{kl}^{(\rho,a)\mu\nu}=
\frac{N_fN_c}{12\pi^2}\left[\left(\hat{A}_{kl}^{(\rho,a)}p^2+
\hat{B}_{kl}^{(\rho,a)}\right)
\delta_{\mu\nu}-\hat{A}_{kl}^{(\rho,a)}p_{\mu}p_{\nu}\right]
+O\left(\frac{1}{\Lambda^2}\right),
\ee
where the two symmetric matrices - the kinetic term $\hat{A}$ and
the momentum independent part $\hat{B}$ - have been introduced:
\be
\label{aa}
\hat{A}^{(\rho,a)}\equiv\begin{pmatrix}
  4\ln\!\frac{\Lambda^{2}}{M_{0}^{2}}-\frac{15}{2} & -\frac{\sqrt3}{2} \\
  -\frac{\sqrt3}{2} & \frac{3}{2}
\end{pmatrix},
\ee
\be
\label{br}
\hat{B}^{\rho}\equiv\begin{pmatrix}
  -2\bar{\Delta}_{11} & -2\bar{\Delta}_{12} \\
  -2\bar{\Delta}_{12} & -2\bar{\Delta}_{22}
\end{pmatrix},
\ee
\be
\label{ba}
\hat{B}^{a}\equiv\begin{pmatrix}
  -2\bar{\Delta}_{11}+\sigma_{11}
   & -2\bar{\Delta}_{12}+\sigma_{12}
    \\
  -2\bar{\Delta}_{12}+\sigma_{12}
   & -2\bar{\Delta}_{22}+\sigma_{22}
\end{pmatrix},
\ee
$$
\sigma_{11}\equiv 24M_{0}^{2}\ln\!\frac{\Lambda^{2}}{M_{0}^{2}}-
\frac{477}{2}\sigma_{1}^{2}
-15\sqrt{3}\sigma_{1}\sigma_{2}+\frac{9}{2}\sigma_{2}^{2}\,,
$$
$$
\sigma_{12}\equiv -\frac{15\sqrt{3}}{2}\sigma_{1}^{2}+9\sigma_{1}\sigma_{2}
+\frac{3\sqrt{3}}{2}\sigma_{2}^{2}\,,
$$
$$
\sigma_{22}\equiv  \frac{9}{2}\sigma_{1}^{2}+3\sqrt{3}\sigma_{1}\sigma_{2}+
\frac{27}{2}\sigma_{2}^{2}\,.
$$
Here $M_0\equiv M(0)=2\sigma_1$ is the dynamic quark mass at zero
external momentum. The remaining logarithmic divergences will be
absorbed later by meson masses and renormalization of meson
fields.


%The observables should not depend on
%the cutoff $\Lambda$. The  scale invariance is achieved by
%appropriate prescription of cutoff dependence for effective
%coupling constants $b_{kl}$. Namely, we require the
%cancellation of quadratic divergences and
%parametrize the matrices of coupling constants in the vicinity
%of polycritical point as follows,
%\be
%\label{ft}
%16\pi^{2}b_{kl}^{-1}=\delta_{kl}-\frac{4}{3}
%\frac{\bar\Delta_{kl}}{\Lambda^{2}};
%\qquad \bar\Delta_{kl}\ll\Lambda^{2}.
%\ee
%The last inequalities provide the
%masses to be essentially less then the cutoff.

%For strong four-fermion coupling
%constants $a_{kl} \geq 8\pi^{2}\delta_{kl}$ in the scalar channel
%the above interaction induces the dynamical chiral symmetry
%breaking. In the language of the theory of critical phenomena it
%is equivalent to developing our model around a critical or
%scaling point (a critical surface in our case) where the
%quantum system undergoes the second-order phase transition. The
%parameters $\Delta_{kl}$ just describe the deviation from a
%critical point and determine the physical mass of scalar mesons.
%The CSB is generated by the dynamic quark mass function corresponding
%to nontrivial v.e.v. of scalar fields $\sigma_1,\sigma_2$,
%\be
%M(\tau) = \sigma_1 f_1(\tau) + \sigma_2 f_1(\tau);\qquad M_0 \equiv
%M(0) = 2 \sigma_1.
%\ee
%This CSB is characterized by the quark condensate,
%\begin{equation}
%<\bar q q > \simeq - \frac{N_c\Lambda^2}{8\pi^2}
%(\sigma_1 - \sqrt{3} \sigma_2). \label{cond2}
%\end{equation}
%We remark that this quark condensate in general is not defined solely
%by the dynamical mass $M_0$ as in the conventional (one-channel) NJL
%model but depends also from the second v.e.v. $\sigma_2$. Only if
%the dynamical mass function is strictly constant,
%$\sigma_2 = -\sqrt{3}\sigma_1 $, one reproduces the NJL relation
%\be
%<\bar q q > \simeq - \frac{N_c\Lambda^2 M_0}{4\pi^2}. \label{cond1}
%\ee

%When integrating out the quark fields  one comes
%to the bosonic effective action where the quadratic parts
%in boson field fluctuations are retained only
%to describe the meson mass spectrum:
%\be
%_{\mbox{\scriptsize eff}}^{(2)}\simeq\frac{1}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\sum_{k,l=1}^{2}
%[\rho_{k,\mu}^iC_{kl}^{\rho,\mu\nu}\rho_{l,\nu}^i+a_{k,\mu}^i
%C_{kl}^{a,\mu\nu}a_{l,\nu}^i]\,,
%\ee
%$$
%\label{varder}
%_{kl}^{(\rho,a)\mu\nu}\simeq
%\frac{N_c}{12\pi^2}\left[\left(\hat{A}_{kl}^{(\rho,a)}p^2+
%\hat{B}_{kl}^{(\rho,a)}\right)
%delta_{\mu\nu}-\hat{A}_{kl}^{(\rho,a)}p_{\mu}p_{\nu}\right],
%$$
%where the two symmetric matrices - the kinetic term $\hat{A}$ and
%the momentum independent part $\hat{B}$ - have been
%introduced. We do not consider here the $\pi-a_1$ mixing since
%this effect does not influence on the mass spectrum.

The physical mass spectrum is defined by the secular equation
\be
\label{sec}
\det(\hat{A}p^{2}+\hat{B})^{(\rho,a)}=0\,,\qquad
m_{phys}^2=-p_0^2\,.
\ee
%Several comments are in order.
%\begin{enumerate}
%\item[(i)] We neglect terms $O(\frac{1}{\Lambda^{2}})$ in all
%calculations.
%\item[(ii)] The large-log approximation
%$\ln\frac{\Lambda^{2}}{M_{0}^{2}}\gg 1$ is adopted, where $M_{0}$
%denotes the dynamical mass at zero external momentum. It is compatible with
%confinement.
%\item[(iii)] In order that the physical masses
%were insensitive to $\Lambda$,
%the effective coupling constants
%should be weakly dependent on the cutoff,
%$\Delta_{kl},\, \bar\Delta_{kl}\sim
%M_{0}^{2}\ln\frac{\Lambda^{2}}{M_{0}^{2}}$.
%\end{enumerate}
%The physical mass spectrum can be found from solutions of the
%corresponding secular equation,
%\be
%det(\hat{A}p^{2}+\hat{B})=0;\qquad m_{phys}^{2}=-p_{0}^{2}.
%\ee
As it will be seen further the consistency with CSR sum rules
imposes the following scale condition:
\be
\label{scale}
\bar\Delta_{kl}=O\left(\Lambda^2\right).
\ee
Using~\gl{aa},~\gl{br} one has for the Eq.~\gl{sec}:
$$
6\left(\ln{\!\frac{\Lambda^{2}}{M_{0}^{2}}}-2\right)p^4-
\left(8\ln{\!\frac{\Lambda^{2}}
{M_{0}^{2}}}\,\bar{\Delta}_{22}-15\bar{\Delta}_{22}+3\bar{\Delta}_{11}+
2\sqrt{3}\bar{\Delta}_{12}\right)p^2+
$$
\be
\label{sec2}
+4\,\det\hat{\bar{\Delta}}=0,
\ee
The solution of Eq.~\gl{sec2} in the large-log approximation
$\ln\!\frac{\Lambda^{2}}{M_{0}^{2}}\gg 1$ %($M_{0}$
%is the dynamic quark mass at zero external momentum)
is as follows:
\be
\label{mr1}
m_{\rho}^2=-
\frac{\det\hat{\bar{\Delta}}}{2\ln{\!\frac{\Lambda^{2}}{M_{0}^{2}}}\,
\bar{\Delta}_{22}}+c_1+\ac{} ,
\ee
\be
\label{mr2}
m_{\rho'}^2=-\frac43\bar{\Delta}_{22}+\delta+c_2+\ac{} .
\ee
To obtain the A-meson mass spectrum it is enough to make the
replacement (see \gl{br}, \gl{ba}):
$\bar{\Delta}_{kl}\rightarrow\bar{\Delta}_{kl}-1/2\sigma_{kl}$.
The result is:
\be
\label{-ma1}
m_{a_{1}}^2=-\frac{\det\hat{\bar{\Delta}}}{2\ln{\!\frac{\Lambda^{2}}
{M_{0}^{2}}}\,\bar{\Delta}_{22}}+
6M_0^2+c_1+\ac{},
\ee
\be
\label{-ma2}
m_{a_{1}'}^{2}=
-\frac43\bar{\Delta}_{22}+3\bar{\sigma}+\delta
%-\frac{d}
%{6\ln{\!\frac{\Lambda^{2}}{M_{0}^{2}}}}+
%\frac{\det\hat{\bar{\Delta}}}{2\ln{\!\frac{\Lambda^{2}}{M_{0}^{2}}}\,
%\bar{\Delta}_{22}}
+c_2+\ac{}.
\ee
The prime labels everywhere the corresponding excited meson state
and we have introduced the notations:
\be
\label{del}
\delta\equiv-\frac{6m_{\rho}^2\Lgg{}+d}{6\Lgg{}}\,,
\quad c_1 \sim c_2 = O\left(\frac{\Lambda^2}{\Lgg{2}}\right),
\ee
\be
\label{d}
d\equiv3\bar{\Delta}_{11}+2\sqrt{3}\bar{\Delta}_{12}+\bar{\Delta}_{22}\,,
\quad \bar{\sigma}\equiv\sigma_{1}^{2}+
\frac{2\sqrt{3}}{3}\sigma_{1}\sigma_{2}+3\sigma_{2}^{2}>0\,.
\ee
%The vacuum expectation values of scalar fields $\sigma_k$ were
%introduced in~\ci{24}.
As it is seen from the Eqs.~\gl{mr1}-\gl{-ma2} the scale of mass
squared for ground VA states is $O(\Lambda^2/\Lgg{})$ and for
excited ones is $O(\Lambda^2)$. Thus, the excited states turn out to be
logarithmically heavier than ground ones as it was for the scalar (S) and
pseudoscalar (P) case~\ci{24}.
This qualitative property is independent of any concrete choice of
form factors.
Combining the Eqs.~\gl{mr1}-\gl{-ma2} with the corresponding
results %for the scalar and pseudoscalar case
in~\ci{24} one obtains:
\be
\label{ma1}
m_{a_1}^{2} - m_{\rho}^{2} = 6M_{0}^{2}+\ac{}=\frac32m_{\sigma}^2+\ac{},
\ee
\be
\label{ma2}
m_{a_1'}^{2} - m_{\rho'}^{2} =
3\bar{\sigma}+\ac{}=\frac32(m_{\sigma'}^2-
m_{\pi'}^2)+\ac{}.
\ee
%The obtained mass spectrum does not depend on a concrete choice
%of form factors in the leading-log approach and
The last equalities in Eqs.~(\ref{ma1}),~(\ref{ma2}) do not depend
on model parameters. Note also, that differences of masses
squared both in the Eq.~(\ref{ma1}) and in the Eq.~\gl{ma2} are of order
$O(1)$, which indicates the chiral symmetry restoration at a scale
over 1~GeV.

%Let us comment on these spectra.
%\begin{enumerate}
%\item[(i)] The formula for $m_{\sigma}$ is a well-known Nambu relation.
%\item[(ii)] The expression for $m_{a}$ is also a known result (see, for
%example, \cite{[7]}-\cite{[10]}).
%Combining it with the Nambu
%relation we arrive to the remarkable equality:
%\be
%m_{a}^{2}-m_{\rho}^{2}\simeq\frac{3}{2}m_{\sigma}^{2},
%\ee
%which has the same form as the relation for excited meson states \gl{exc}
%for $m_\pi =0$.
%\item[(iii)] In both spectra the radial excitations are
%logarithmically heavier than the corresponding ground meson states.
%\item[(iv)] The value of mass splitting between excited states in
%both spectra is logarithmically suppressed as compared with the mass
%values. Therefore the masses of first excited meson
%states both in the scalar-pseudoscalar channel and in the
%vector-axial-vector one are approximately equal.
%\item[(v)] Another prediction is that a mass of $\sigma'$-meson is a
%little more than that of $\pi'$-meson. The situation in the
%vector-axial-vector spectrum is opposite - the
%$a'$-meson is a little heavier than $\rho'$-one.
%\item[(vi)] When taking into account the $\pi - a_1$ mixing the
%pion decay constant is determined by the physical meson masses,
%\be
%\label{fpim}
%f_{\pi}^2\simeq\frac{N_cm_{\sigma}^2m_{\rho}^2}{16\pi^2m_{a_1}^2}
%\ln\!\frac{\Lambda^{2}}{M_{0}^{2}}\, , \label{fpi}
%\ee
%wherefrom it follows that the logarithm of the cutoff
%is uniquely obtained from this relation
%in terms of physical parameters.
%\end{enumerate}

%We identify $\sigma$ with $f_{0}(400-1200)$, $\sigma'$ with
%$f_{0}(1370)$, $\pi'$ with $\pi(1300)$, $\rho$ with $\rho(770)$,
%$\rho'$ with $\rho(1450)$ and $a$ with $a_{1}(1260)$ \cite{[31]}. The
%first excitation of  the $a_{1}$ - the particle $a'$ - is not
%found yet. This second axial-vector state, perhaps, could be
%found from hadronic $\tau$ decays, although there are the strong
%phase space limitations \cite{[20]}.The experimental data give us:
%$$
%m_{\sigma}=400\div1200 \mbox{ MeV};\qquad
%m_{\sigma'}=1200\div1500 \mbox{MeV};\qquad
%m_{\pi'}=1300\pm 100 \mbox{ MeV};
%$$
%\be
%m_{\rho}=770\pm 0.8 \mbox{ MeV};\qquad
%m_{\rho'}=1465\pm 25 \mbox{ MeV};\qquad
%m_{a}=1230\pm 40 \mbox{ MeV}.
%\ee
%The prediction for mass of $\sigma$-meson is then
%\be
%m_{\sigma} \simeq 800 \mbox{ MeV}, \label{sig}
%\ee
%which is close to the averaged experimental value.
%Furthermore we have the following prediction for the mass of
%$a'$-particle,
%\be
%m_{a'}\cong 1465 \div 1850 \mbox{ MeV}.
%\ee
%The large range for a possible mass of $a'$-meson is accounted for by
%a big experimental uncertainty for the mass of $\sigma'$ and
%$\pi'$ mesons. If we accept the averaged values for them, then
%$m_{a'}-m_{\rho'}\approx 30$ MeV.

%Let us compare the paremeters of the NJL and QNJL models.
%First we notice that in both models the Nambu relation \gl{nambu}
%together with \gl{arho} leads to the large value of the
%dynamical mass $M_0 \simeq 400$ MeV for physical masses of
%$\rho$ and $a_1$ mesons. Then the physical value of the pion
%decay constant \gl{fpim} $f_\pi \simeq 90$ MeV  can be achieved
%for a cutoff $ \Lambda \simeq 1$ GeV. In turn, in the NJL it leads
%to a unacceptably high values of the quark condensate \gl{cond1},
%$<\bar q q> \simeq - (310)\mbox{MeV}^3$ whereas in the QNJL one can
%easily adjust the second v.e.v. to reproduce the condensate
%in agreement with the estimations from Chiral Perturbation Theory
%\cite{[32],[34]} (see also,\cite{iof}). In particular, the
%value $<\bar q q> \simeq - (250)\mbox{MeV}^3$ corresponds to
%$\sigma_1 \simeq -\sqrt{3} \sigma_2$ (see the discussion of this point
% in the next section).
%\bigskip

Let us comment the approximations used to
derive the meson mass spectrum: namely, the large $N_c$
and leading-log approximations. The first one is equivalent
\cite{16} to the neglect of
meson loops. The second one fits well the quarks
confinement  as
quark-antiquark threshold contributions are suppressed in two-point
functions in the leading
log approximation. The accuracy of
this approximation is controlled also  by the magnitudes of higher dimensional
operators neglected in QQM, i.e. by contributions of heavy
mass resonances not included into QQM. All these approximations
are mutually consistent. In particular, in the effective action without
gluons the quark confinement should be realized with the help of
an infinite number of quasilocal vertices with higher-order derivatives.
Then the imaginary part of quark loops can be compensated and
their momentum dependence can eventually reproduce the infinite sum of meson
resonances in
the large-$N_c$ limit. If the effective action is truncated with a finite
number of vertices and thereby deals
with only a few resonances  one has to retain only a finite number of
terms in the low-momentum expansion of quark loops in the CSB phase,
with a non-zero dynamic mass.


\section{VA, $U(3)$ extension of Quasilocal Quark Model}

In this section we build $U(3)$ version of QQM~\ci{we3}, with current quark
masses being taken into account. The corresponding extension of the
VA, $SU(2)$ QQM Lagrangian~\gl{su2} is:
\be
\label{su3}
{\cal
L}_{V\!\!A}\!=\!\bar{q}i(\hat{\not{\partial\phantom{d}}}\!\!\!+\hat{m})q\!+\!
\frac{1}{4N_{f}N_{c}\Lambda^{2}}\!\sum_{k,l=1}^{2}\!
b_{kl}^i\left[\bar{q}\Gamma_{V,k}^iq\cdot\bar{q}\Gamma_{V,l}^iq\!+\!
\bar{q}\Gamma_{A,k}^iq\cdot\bar{q}\Gamma_{A,l}^iq\right]\!,
\ee
$$
\Gamma_{V,k}^i\equiv i\gamma_{\mu}f_k(s)\lambda^i,\qquad
\Gamma_{A,k}^i\equiv i\gamma_{\mu}\gamma_5f_k(s)\lambda^i,\qquad
i=0,...,8\,,
$$
where $\lambda^i$ represent Gell-Mann matrices. The current quark mass
matrix is $\hat{m}=diag(m_u,m_d,m_s)$.
In the sequel we adopt the exact isospin symmetry $m_u=m_d$.
The symbol $\tilde{u}$ will stand everywhere for the $u$, $d$, $\bar{u}$,
$\bar{d}$ quarks. The symbol $\tilde{s}$ will denote $s$ or $\bar{s}$
quarks. The generalization of fine-tuning condition~\gl{ft} takes
the form:
\be
16\pi^{2}(b_{kl}^i)^{-1}=\delta_{kl}^i-\frac{4}{3}
\frac{\bar\Delta_{kl}^i}{\Lambda^{2}}\,;
\qquad \bar\Delta_{kl}^i\ll\Lambda^{2}.
\ee
The couplings $\bar{\Delta}_{kl}^i$ satisfy the relations
\be
\label{par2}
\bar{\Delta}_{kl}^m=\bar{\Delta}_{kl}^0\,,\quad\bar{\Delta}_{kl}^n=
\frac12\left(\bar{\Delta}_{kl}^0+\bar{\Delta}_{kl}^8\right);\quad
m=1,2,3\,,\quad n=4,5,6,7\,,
\ee
which provide the $U(3)$ Gell-Mann-Okubo relations
\be
\label{gmo}
m_{\alpha,\tilde{u}\tilde{u}}^2+m_{\alpha,\tilde{s}\tilde{s}}^2=
2m_{\alpha,\tilde{s}\tilde{u}}^2\,;\qquad
m_{\alpha,\tilde{u}\tilde{u}}=m_{\alpha,singlet}\,.
\ee
Here $\alpha\equiv V,A,V',A'$. The scheme of calculation of the
mass-spectrum for VA, $U(3)$ QQM is the same as in the Sec.~2 and
the details can be found in~\cite{we3}. We point out only some
features. First, the self-consistency of mass-spectrum imposes the
self-consistency condition~\gl{scale} for all $i$. Second, in the
effective action we did not take into account so far both the
$P-A$ mixing and different mixing terms, which are proportional to
the current quark masses, since these contributions do not
influence on the meson mass spectrum.

%The extension of mass differences~\gl{ma1},~\gl{ma2} is
The mass relations, which are independent of model parameters in
the large-log approach are:
\be
\label{u28}
m_{a_1,\tilde{u}\tilde{u}}^2-m_{\rho}^2\simeq\frac32\left(
m_{\sigma,\tilde{u}\tilde{u}}^2-m_{\pi}^2\right),\quad
m_{a_1,\tilde{s}\tilde{u}}^2-m_{K^*}^2\simeq\frac32\left(
m_{\sigma,\tilde{s}\tilde{u}}^2-m_{K}^2\right),
\ee
\be
\label{u29}
m_{a_1,\tilde{s}\tilde{s}}^2-m_{\varphi}^2\simeq\frac32\left
[m_{\sigma,\tilde{s}\tilde{s}}^2-\left(
2m_{K}^2-m_{\pi}^2\right)\right].
\ee
\be
m_{a_1',\tilde{u}\tilde{u}}^2-m_{\rho'}^2\simeq\frac32\left(
m_{\sigma',\tilde{u}\tilde{u}}^2-m_{\pi'}^2\right),\quad
m_{a_1',\tilde{s}\tilde{u}}^2-m_{{K^*}'}^2\simeq\frac32\left(
m_{\sigma',\tilde{s}\tilde{u}}^2-m_{K'}^2\right),
\ee
\be
\label{u31}
m_{a_1',\tilde{s}\tilde{s}}^2-m_{\varphi'}^2\simeq\frac32\left(
m_{\sigma',\tilde{s}\tilde{s}}^2-m_{(\eta)'}^2\right).
\ee
The $(\eta)'$ in the Eq.~\gl{u31} is the radial excitation of $\eta$ meson
and throughout the
paper the sign $\simeq$ denotes the large-log approximation.
The corresponding fits and comparisons with experiment are carried
out in~\ci{we3,we2}. The agreement with experimental data is within 10\% for
V-case and 15\% for the A-one. In such a way, the VA, $U(3)$ extension of
QQM provides the certain
 relations between meson masses of multiplets
which are independent of model parameters.

The axial anomaly was not yet considered in framework of QQM and,
thus, we did not include the $\eta'$ meson. Since the ground
P-meson masses are subject to rather $SU(3)$ Gell-Mann-Okubo
relation one has no $\eta$ meson mass in the Eq.~\gl{u29}. On the
other hand, we do not expect a strong $U_A(1)$ anomaly effect for
the case of excited $\eta'$ meson. As a result, the
relations~\gl{gmo} hold for excited P-multiplet, which is
reflected in the Eq.~\gl{u31}.


\section{Chiral symmetry restoration sum rules and constraints on parameters
of VA, $SU(2)$ QQM}

In this section we exploit the constraints based on chiral symmetry
restoration in QCD at high energies for the $SU(2)$ QQM. Expanding
the meson correlators
\gl{cor1} in powers of $p^{2}$ one arrives to the CSR sum rules:
\be
\sum_{n}Z_{n}^{V}-\sum_{n}Z_{n}^{A}=4F_{\pi}^{2}\,,
\ee
\be
\sum_{n}Z_{n}^{V}m_{V,n}^{2}-\sum_{n}Z_{n}^{A}m_{A,n}^{2}=0\,,
\ee
\be
\sum_{n}Z_{n}^{V}m_{V,n}^{4}-\sum_{n}Z_{n}^{A}m_{A,n}^{4}=\Delta_{V\!\!A}\,,
\ee
\be
\sum_{n}Z_{n}^{V}m_{V,n}^{6}-\sum_{n}Z_{n}^{A}m_{A,n}^{6}=
-m_0^2\Delta_{V\!\!A}\,.
\ee
The first two relations  are the Weinberg sum rules.
The quantity $F_{\pi}$ is the pion decay constant which is equal
in the QQM~\cite{24}:
\be
\label{f1}
F_{\pi}^2=\frac{N_fN_cM_0^2}{4\pi^2}\Lgg{}+O(1).
\ee
The residues in resonance pole contributions
in the vector and axial-vector correlators have the structure,
\be
Z_{n}^{(V,A)}=4f_{(V,A),n}^{2}m_{(V,A),n}^{2}\,,
\ee
with $f_{(V,A),n}$ being defined as electromagnetic
decay constants.

The corresponding two-point correlators for the $SU(2)$ QQM can be
calculated by variation
of external vector fields. Let us first consider the V-case. Taking
into account the external vector sources $V_{k,\mu}^i$ the Lagrangian
reads as follows:
$$
{\cal L}_{\mbox{\scriptsize aux}}^{V}=\bar{q}\left(\lefteqn{\not{\phantom{DD}}}D+
i\sum_{k=1}^2\Gamma_{V,k}^iV_{k,\mu}^i\right)q+
N_fN_{c}\Lambda^{2}\sum_{k,l=1}^{2}
\rho_{k\!,\mu}^ib_{kl}^{-1}\rho_{l\!,\mu}^i\,.
$$
After shifting the bosonic fields
$$
\rho_{k,\mu}^i\longrightarrow\rho_{k,\mu}^i-V_{k,\mu}^i\,,
$$
and integrating over fermionic degrees of freedom one comes to the
following effective action in external V-fields:
$$
S_{\mbox{\scriptsize eff}}^{V}(\rho_{k,\mu}^i,V_{k\!,\mu}^i)=-\left.N_fN_c\mbox{Tr}\ln\lefteqn
{\not{\phantom{DD}}}D\right|_{\mbox{\scriptsize reg}}+
$$
$$
+N_fN_{c}\Lambda^{2}\int d^4x\sum_{k,l=1}^{2}b_{kl}^{-1}
\left\{\rho_{k\!,\mu}^i\rho_{l\!,\mu}^i-2V_{k\!,\mu}^i\rho_{l\!,\mu}^i+
V_{k\!,\mu}^iV_{l\!,\mu}^i\right\}\,.
$$
Expanding $\left.\mbox{Tr}\ln\lefteqn{\not{\phantom{DD}}} D\right|_{\mbox{\scriptsize reg}}$
up to quadratic in fields terms, one has
$$
S_{\mbox{\scriptsize eff}}^{(2)}(\rho_{k,\mu}^i,V_{k\!,\mu}^i)=
\sum_{k\!,l=1}^{2}\left\{\frac{1}{2}\rho_{k\!,\mu}^iC_{kl}^{(\rho)\mu\nu}
\rho_{l\!,\nu}^i+N_c\Lambda^2b_{kl}^{-1}
\left[-2V_{k\!,\mu}^i\rho_{l\!,\mu}^i+
V_{k\!,\mu}^iV_{l\!,\mu}^i\right]\right\},
$$
where $C_{kl}^{(\rho)\mu\nu}$ is given by~Eq.~\gl{varder}.
Introducing the vectors
$$
\rho_{\mu}\equiv\begin{pmatrix}
  \rho_{1,\mu}^i \\
  \rho_{2,\mu}^i
\end{pmatrix}\,,\qquad V_{\mu}\equiv\begin{pmatrix}
  V_{1,\mu}^i \\
  V_{2,\mu}^i
\end{pmatrix}
$$
and taking into account~\gl{ft} (where we neglect the last term), one
integrates over $\rho_{\mu}$ with the result:
\be
\label{veff}
\frac{12\pi^2}{N_fN_c}S_{\mbox{\scriptsize eff}}(V_{\mu})=
-\frac98\left(\Lambda^4+O(\Lambda^2)\right)V_{\mu}^T\hat{H}^{\rho}_{\mu\nu}
V_{\nu}+\frac{3\Lambda^2}{4}V_{\mu}^TV_{\nu}\delta_{\mu\nu}\,,
\ee
%$$
%\hat{H}^{\rho}_{\mu\nu}\equiv\left(\hat{A}p^2+\hat{B}^{\rho}\right)^{-1}
%\left[\delta_{\mu\nu}+\hat{A}\left(\hat{B}^{\rho}\right)^{-1}\!\!
%p_{\mu}p_{\nu}\right]=
%$$
\be
\label{cr}
\hat{H}^{\rho}_{\mu\nu}\equiv\left(\hat{A}p^2+\hat{B}^{\rho}\right)^{-1}
\hat{A}\left(\hat{B}^{\rho}\right)^{-1}(-p^2\delta_{\mu\nu}+
p_{\mu}p_{\nu})+\left(\hat{B}^{\rho}\right)^{-1}\!\!\delta_{\mu\nu}\,.
\ee
The last term in the Eq.~\gl{cr} together with that of in~\gl{veff}
form a local term which will be cancelled by the same term in the
A-case.
Substituting the identity
$$
\bar{q}\gamma_{\mu}q=\frac12\left(\bar{q}f_1(s)\gamma_{\mu}q-
\sqrt{3}\,\bar{q}f_2(s)\gamma_{\mu}q\right)\,
$$
into the vector correlator
\be
\Pi_{\mu\nu}^{\rho}(p^2)=4\int\!d^{4}x\exp(ipx)\langle
\bar{q}\gamma_{\mu}q(x)\bar{q}\gamma_{\nu}q(0)\rangle,
\ee
the latter can be rewritten through the second variational
derivatives:
\be
\label{kora2}
\Pi_{\mu\nu}^{(\rho)}(p^2)=\frac{N_fN_c}{12\pi^2}
\left[\Pi_{11}^{(\rho)}+3\Pi_{22}^{(\rho)}-
2\sqrt{3}\,\Pi_{12}^{(\rho)}\right](-\delta_{\mu\nu}p^2+p_{\mu}p_{\nu})\,,
\ee
\be
\label{pr}
\hat{\Pi}^{(\rho)}\equiv\left(\hat{A}p^2+\hat{B}^{\rho}\right)^{-1}
\hat{A}\left(\hat{B}^{\rho}\right)^{-1}.
\ee
On the other hand, this correlator is parametrized as follows
(see Eq.~\gl{cor1}):
\be
\label{kora3}
\Pi_{\mu\nu}^{(\rho)}(p^2)=\left[\frac{Z_{\rho}}{p^2+m_{\rho}^2}+
\frac{Z_{\rho'}}{p^2+m_{\rho'}^2}\right]
(-\delta_{\mu\nu}p^2+p_{\mu}p_{\nu})\,.
\ee
Comparison of \gl{kora2} and \gl{kora3} allows to obtain the
corresponding residues. % $Z_k^{\rho}$\,.\,In the large-log
%approximation they are shown in the
(see Eqs.~\gl{zv1},~\gl{zv2}).
In the mean-field approximation the
vector correlator and residues can be calculated exactly which is
displayed in the Appendix.

The A-mesons must be considered together with the P-ones due to the
$P-A$ mixing. The relevant term in the effective action
$$
S_{\mbox{\scriptsize eff}}^{(2)}(\pi^i_k, a^i_{k,\mu})=\frac{1}{2}\int\frac{d^{4}p}
{(2\pi)^{4}}\sum_{k,l=1}^{2}2\pi_k^iC_{kl}^{\pi
a,\mu}a_{l,\mu}^i\,,
$$
appears by virtue of non-zero value of the corresponding second
variation of $S_{\mbox{\scriptsize eff}}(\pi^i_k, a^i_{k,\mu})$:
$$
C_{kl}^{\pi a,\mu}=-N_fN_c\int\frac{d^4q}{(2\pi)^4}
\mbox{tr}\left\{(i\gamma_{5})\frac{f_{k}\left(\left(\frac{q+p/2}{\Lambda}
\right)^2\right)}
{\!\!\not q+\frac{1}{2}\!\!\not p+iM}
(i\gamma_{\mu}\gamma_{5})\frac{f_{l}\left(\left(\frac{q+p/2}{\Lambda}
\right)^2\right)}{\!\!\not q-\frac{1}{2}\!\!\not p+iM}\right\}=\
$$
\be
=-\frac{4iN_fN_c}{(2\pi)^4}\int\frac{d^4qMf_{k}\left(\frac{q^2}{\Lambda^2}
\right)f_{l}\left(\frac{q^2}{\Lambda^2}\right)}
{\left[\left(q+\frac{1}{2}p\right)^{2}+M^{2}\right]
\left[\left(q-\frac{1}{2}p\right)^{2}+
M^{2}\right]}p_{\mu}+O\left(\frac{1}{\Lambda^2}\right).
\ee
In order to exclude the mixing terms one makes the shift of A-fields:
$$
a_{k,\mu}^i\longrightarrow a_{k,\mu}^i+D_{kl}^i\pi_l^ip_{\mu}\,,
$$
with the elements $D_{kl}^i$ being defined by the requirement of
cancellation of $P-A$ terms. This leads to changing of the kinetic
matrix~$\hat{A}^{\pi}$ due to contribution of longitudinal A-part:
\be
\label{red2}
\hat{A}^{\pi}\longrightarrow \hat{A}^{\pi}_{ren}=
\begin{pmatrix}
4\frac{m_{\rho}^2}{m_{a_1}^2}\Lgg{}+O(1) &
-\frac{\sqrt{3}}{2}+O\left(\frac{\Lgg{}}{\Lambda^2}\right)\\
-\frac{\sqrt{3}}{2}+
O\left(\frac{\Lgg{}}{\Lambda^2}\right) &
\frac32+O\left(\frac{1}{\Lambda^2}\right)
\end{pmatrix},
\ee
where the Eq.~(\ref{ma1}) and the scale~(\ref{scale}) have been
exploited. The redefinition~(\ref{red2}) does not
influence on the mass spectrum of P-mesons, but renormalizes the
pion decay constant~(\ref{f1}) is:
\be
\label{fpi2}
F_{\pi}^2 = \frac{N_fN_c M_0^2m_{\rho}^2}{4\pi^2m_{a_1}^2}
\ln\!\frac{\Lambda^2}{M_0^2}+O(1)\,.
\ee
For the model under consideration the relation~(\ref{fpi2}) fixes the
logarithm of the cutoff in terms of physical parameters.

Taking into account the point above the further calculations for the
A-case are the same as for the V-one.
As a result one finds the residues in the
meson poles. In the large-log approach one has:
\be
\label{zpi}
\tilde{Z}_{\pi}=4 F^2_{\pi} \simeq \frac{(m_{a_1}^2-m_{\rho}^2)\,\delta}
{m_{\rho}^2m_{a_1}^2m_{a_1'}^2}\,Z_1\,,\qquad
Z_1\equiv\frac{3N_fN_c\Lambda^4}{16\pi^2}\,,
\ee
\be
\label{zv1}
Z_{\rho}\simeq\frac{m_{a_1}^2}{m_{a_1}^2-m_{\rho}^2}\,4F^2_{\pi}\,,\qquad
Z_{a_1}\simeq\frac{m_{\rho}^2}{m_{a_1}^2-m_{\rho}^2}\,4F^2_{\pi}\,,
\ee
\be
\label{zv2}
Z_{\rho'}\simeq\frac{Z_1}{m_{\rho'}^2}\,,\qquad
Z_{a_1'}\simeq\frac{Z_1}{m_{a_1'}^2}\,,
\ee
where $\delta$ is given by \gl{del}.
%In the mean-field approximation the correlators and residues under
%consideration can be calculated exactly which is shown in the
%Appendix for the vector case.
%Now it is clear, that in order to adjust the scale in
%Eq.~\rf{fpi2} with the one in~\rf{zpi} we were forced to impose the
%scale~\rf{scal}.
Unlike the situation in the SP-case the residues in the VA-poles are
of the same order of magnitude:
$$
Z_{\rho}\sim Z_{a_1}\sim Z_{\rho'}\sim Z_{a_1'}=O\left(\Lambda^2\right).
$$
The statement~\gl{scale} follows from the comparison of
Eqs.~\gl{fpi2},~\gl{zpi}. The relation for~$\tilde{Z}_{\pi}$ is a
constraint on effective coupling constants of the
QQM~$\bar{\Delta}_{kl}$.

The first and the second sum rules are fulfilled
identically. The third one takes the form:
\be
\label{va3}
%m_{a_1'}^2-m_{\rho'}^2\simeq
Z_1\left(m_{a_1'}^2-m_{\rho'}^2\right)\simeq
16\pi\alpha_s\langle\bar qq\rangle^2
\qquad \mbox{or}\qquad 3Z_1\bar{\sigma}=-\Delta_{V\!\!A}\,.
\ee
Notice that the analog of Eq.~\gl{va3} in the scalar case~\ci{24}
may be cast into the form
\be
\label{sp2}
\frac{N_fN_c}{2\pi^2}\left(m_{\sigma'}^2-m_{\pi'}^2\right)\simeq
24\pi\alpha_s\langle\bar qq\rangle^2
\qquad \mbox{or}\qquad
3Z_1\bar{\sigma}=-\frac{27}{32}\Delta_{V\!\!A}\,.
\ee
The minor discrepancy between relations \gl{va3} and \gl{sp2} is
about 16\% and can be referred to the quality of two-resonance
approximation.
The fourth sum rule looks as follows~\ci{we2}:
\be
\label{vafsr}
m_{a_1'}^2\simeq m_{\rho'}^2\simeq\frac{m_0^2}{2}
\quad\mbox{or}\quad-\frac43\bar{\Delta}_{22}\cdot3\bar{\sigma}\simeq
-\frac{m_0^2\Delta_{V\!\!A}}{2Z_1}\,.
\ee
Numerical estimations~\ci{we2} show that the last sum rule fails for
QQM with the ground and first excited sets of $V\!A$ mesons.
The structure of $Z_{\rho'}$ and $Z_{a_1'}$ shows that if $m_{a_1'}\simeq
m_{\rho'}$ then $Z_{a_1'}\simeq Z_{\rho'}$ and therefore
$f_{a_1'}\simeq f_{\rho'}$. As a consequence, these residues
approximately cancel each other in sum rules and the one-channel
results for $f_{\rho}$, $f_{a_1}$ hold~\ci{we1} (see also~\ci{we4}):
\be
f_{\rho}\simeq\frac{F_{\pi}m_{a_1}}{m_{\rho}
\sqrt{m_{a_1}^{2}-m_{\rho}^{2}}}\,,
\qquad
f_{a_1}\simeq\frac{F_{\pi}m_{\rho}}{m_{a_1}
\sqrt{m_{a_1}^{2}-m_{\rho}^{2}}}\,.
\ee
After evaluating we get $f_{\rho}\approx 0.15$ and $f_{a_1}\approx
0.06$ to be compared with the experimental values
%\cite{[36]} from the
%decay $\rho^{0}\rightarrow e^{+}e^{-}$,
$f_{\rho}=0.20\pm 0.01$,
%and from the decay $a_{1}\rightarrow\pi\gamma$,
$f_{a_1}=0.10\pm 0.02$~\cite{we1}.
%We have also a
%reasonable prediction for the chiral constant $L_{10}$ \cite{[32]}
%\be
%L_{10}=\frac{1}{4}(\sum_{n}f_{A,n}^{2}-\sum_{n}f_{V,n}^{2})\simeq
%\frac{1}{4}(f_{a}^{2}-f_{\rho}^{2})\approx -5.1\cdot 10^{-3},
%\ee
%which confronts the phenomenologically accepted estimation \cite{[35]}
%$L_{10}=-(5.5\pm 0.7)\cdot 10^{-3}$.

It should be mentioned that introducing VA-meson fields influences
on some CSR constraints for SP-case obtained in~\cite{24}.
Due to the redefinition~\gl{red2}
the first SP sum rule is not fulfilled now identically but up to
terms of order $O(1/\Lambda^2)$ because of the scaling~\gl{scale}. The
second SP sum rule
%takes the form:
%\be
%\label{tsp2}
%\frac83\sigma_{22}Z_0-\frac{N_c\Lambda^4\Delta_{22}}{6\pi^2\Lgg{}}\cdot
%\frac{m_{a_1}^2-m_{\rho}^2}{m_{\rho}^2}\simeq\Delta_{SP}\,.
%\ee
%By virtue of~\gl{scale} and~\gl{ma1} the second term in
%the left-hand side of~\gl{tsp2} is of order~$O(\Lambda^2\Lgg{})$,
%while the first one is of order~$O(\Lambda^4)$, i.e. this sum rule
is not changed in the large-$\Lambda$ approximation.
The relation $Z_{\sigma}\simeq Z_{\pi}$ is not now valid.
Instead one has
$$
Z_{\sigma}\simeq\frac{m_{\rho}^2}{m_{a_1}^2}Z_{\pi}\,.
$$
The relevant physical discussions and fits for the SPVA QQM can be traced
in~\ci{we1,we2}. From the estimations~\ci{we2}, in particular,
follows that $m_{a_1'}-m_{\rho'}\approx 60$ MeV which proves a
fast restoration of chiral symmetry.

\section{Summary}

\begin{enumerate}
\item
We have shown that $SU(2)$ and $U(3)$ versions of Quasilocal Quark Model
with chirally invariant four-fermion vector and axial-vector
vertices including derivatives in fields can serve to describe the physics
of vector and axial-vector meson resonances at intermediate energies.
The corresponding
mass spectrum for the ground and first excited VA boson states was derived
in the mean field and large-log approximations. The qualitative features of
the mass spectrum obtained turned out to be the same as in the
scalar-pseudoscalar case~\cite{24}: the excited states are logarithmically
heavier than ground ones and a fast restoration of chiral symmetry over
the scale 1~GeV is predicted. Comparison with the SP, $SU(2)$ QQM permitted
to obtain two relations between boson masses independent of model
parameters:
$$
m_{a_1}^{2} - m_{\rho}^{2} \simeq \frac32m_{\sigma}^2\,,
$$
$$
m_{a_1'}^{2} - m_{\rho'}^{2}\simeq\frac32(m_{\sigma'}^2-m_{\pi'}^2).
$$
The latter relation predict the mass of the first radial excitation of
axial-vector meson in the energy range $m_{a_1}=1500\div1550$ MeV.
\item
The VA, $U(3)$ generalization of QQM allowed to derive much more relations
between masses of meson states which do not depend on any model parameters
(see Eqs.~\gl{u28}-\gl{u31}).
The agreement with experimental data is within 10\% for the V-case and
15\% for the A-one~\ci{we2}.
\item
%The chiral symmetry restoration sum rules were imposed on the vector $SU(2)$
%QQM in the Sec.~4. This method realizes the correspondence (matching) with
%QCD at intermediate energies where the OPE can be applied. As a result some
%constraints on model parameters were obtained. The residues in the VA meson
%poles were calculated in the large-log approach. For the V-case the residues
%were also calculated exactly in the Appendix.
From the expansion of QCD two-point colour-singlet current
correlators for VA-fields in inverse powers of large momentum and
the comparison with OPE one obtains the set of sum rules for the
differences of VA-correlators, which show a rapid decrease at large
momenta. Therefore, the chiral symmetry is restored at high
energies and one can perform the QCD matching by means of few
lowest lying VA-resonances that gives a set of constraints on
parameters for VA, $SU(2)$ QQM. In particular, the residues for the ground
and excited VA states turned out to be of the same order of magnitude unlike
the situation in the SP case. The inclusion of excited states did not
change the electromagnetic decay constants
of the ground VA states compared with the one-channel results:
$f_{\rho}\approx 0.15$, $f_{a_1}\approx 0.06$.
%The influence of axial-vector channel on the
%similar analysis in the SP QQM~\cite{24} was touched upon.
%It happened that
%the $P-A$ mixing renormalizes the $\pi$-meson weak decay constant and the
%residue in ground scalar pole.
\item
The main results of our work have been obtained by means of method
of effective action and on the basis of the idea of chiral
symmetry restoration at high energies and on OPE of the two-point
correlators for vector and axial-vector quark densities. All
calculations have been performed in the large-$N_c$ and
log-approximations.
\item
We conclude that the VA Quasilocal Quark Model reflects
phenomenology of low and intermediate energy meson physics and
the matching to
non-perturbative QCD based on chiral symmetry restoration at high
energies improves the predictability of the models.
%The chiral symmetry
%restoration sum rules are well saturated by two vector and two
%axial-vector resonances.
%Let us summarize the results obtained in the present work.
%\begin{enumerate}
%\item[(i)] The mass of the second axial-vector particle with $I=1$
%is predicted. It is comparable with the mass of the
%vector counterpartner: $m_{a'}=1465\div 1850$ MeV and the most
%plausible value of the mass difference is
%$m_{a'}-m_{\rho'}\approx 30$ MeV.
%\item[(ii)] The estimation on the mass of the $\sigma$-meson
%does not contradict to existing experimental data \cite{[31]}:
%$m_{\sigma} \simeq 800$ MeV.
%\item[(iii)] The couplings $f_{\rho},f_{a}$ and the chiral constant
%$L_{10}$ are evaluated from CSR Sum Rules
%as matching rules for our model to QCD at
%intermediate energies, with the result being $f_{\rho}\approx
%0.15,f_{a}\approx0.06,L_{10}\approx -5.1\cdot 10^{-3}$;\\
%\end{enumerate}

%Finally we would like to mention possible
%applications of QQM. First, such models are thought of as
%relevant for investigations of behaviour of hadron matter at high
%temperatures and nuclear densities  in the region near the
%restoration of chiral symmetry. One could
%expect that for increasing quark densities the mass-splitting is
%collapsing and therefore excitations become lighter and more
%important in hadron kinetics. Second, QQM can be used to
%describe Higgs particles in extensions of the Standard
%Model~\cite{46,45}.
\end{enumerate}


\section*{Acknowledgements}

We are grateful to A. A. Andrianov for the numerous fruitful
discussions and attention to our work.
This work is supported by Grant RFBR
01-02-17152, INTAS Call 2000 Grant (Project 587), and The Program
"Universities of Russia: Fundamental Investigations" (Grant UR.02.01.001).


\section*{Appendix}

In this Appendix we calculate in the mean field approach the two-point
vector correlator for the two-channel QQM. After introducing
external vector fields $V_{k,\mu}^i$ and integrating over $\rho_{k,\mu}^i$
one has for the quadratic in fields part of effective action:
$$
\frac{12\pi^2}{N_fN_c}S_{\mbox{\scriptsize eff}}^{(2)}(V_{\mu})=
-\frac98\Lambda^4V_{\mu}^T\hat{K}\left(\hat{A}p^2+\hat{B}^{\rho}\right)^{-1}
\hat{A}\left(\hat{B}^{\rho}\right)^{-1}\hat{K}V_{\nu}(-p^2\delta_{\mu\nu}+
p_{\mu}p_{\nu})+
$$
\be
\label{s2}
+\frac{3}{4}\Lambda^2V_{\mu}^T\left(\hat{K}-\frac32\Lambda^2\hat{K}
\left(\hat{B}^{\rho}\right)^{-1}\hat{K}\right)V_{\nu}\delta_{\mu\nu}\,,
\ee
where
%$$
%\hat{A}=\begin{pmatrix}
%  4\ln\!\frac{\Lambda^{2}}{M_{0}^{2}}-\frac{15}{2} & -\frac{\sqrt3}{2} \\
%  -\frac{\sqrt3}{2} & \frac{3}{2}
%\end{pmatrix},\qquad
%\hat{B}^{\rho}=\begin{pmatrix}
%  -2\bar{\Delta}_{11} & -2\bar{\Delta}_{12} \\
%  -2\bar{\Delta}_{21} & -2\bar{\Delta}_{22}
%\end{pmatrix},
$$
\hat{K}=\begin{pmatrix}
  1-\frac{4\bar{\Delta}_{11}}{3\Lambda^2} &
  -\frac{4\bar{\Delta}_{12}}{3\Lambda^2} \\
  -\frac{4\bar{\Delta}_{12}}{3\Lambda^2} &
  1-\frac{4\bar{\Delta}_{22}}{3\Lambda^2}
\end{pmatrix}.%\qquad
%V_{\mu}\equiv\begin{pmatrix}
%  V_{1,\mu} \\
%  V_{2,\mu}
%\end{pmatrix}.
$$

In order to obtain the correlator of local currents~\gl{cor1} one
needs to project the expression~\gl{s2} on the local current:
$$
\Pi^{V}_{\mu\nu}(p^{2})=\mbox{\rm tr}\left\{\hat{P}\frac{\delta^2S_{\mbox{\scriptsize eff}}^{(2)}}
{\delta V_{\mu}\delta V_{\nu}}\right\},\qquad
\hat P = \begin{pmatrix}
  1 & -\sqrt{3} \\
  -\sqrt{3} & 3
\end{pmatrix},
$$
where $\hat P$ is the projection operator. It is a formal way to
obtain the relation~\gl{kora2}.
%$$
%\label{proec}
%\hat P = \begin{pmatrix}
%  1 & -\sqrt{3} \\
%  -\sqrt{3} & 3
%\end{pmatrix}.
%$$
The final result is:
\ba
\label{fin}
\Pi^{V}_{\mu\nu}(p^{2})&=&\frac{Z_1\left(ap^2+b+c\right)
(-p^2\delta_{\mu\nu}+p_{\mu}p_{\nu})}{24\left(\Lgg{}-2
\right)\det\hat{\bar{\Delta}}\left(p^2+m_{\rho}^2\right)\left(p^2+
m_{\rho'}^2\right)}+\nl
&+&\frac34\Lambda^2\left(\frac{3d\Lambda^2}{4\det\hat{\bar{\Delta}}}-4\right)
\delta_{\mu\nu}\,.
\ea
In the Eq.~\gl{fin} we introduced the following notations:
$$
a\equiv -6\left(\Lgg{}-2\right)\left[d+8\varepsilon\det\hat{\bar{\Delta}}+
\varepsilon^2\det\hat{\bar{\Delta}}(\bar{\Delta}_{11}+3\bar{\Delta}_{22}-
2\sqrt{3}\bar{\Delta}_{12})\right],
$$
$$
b\equiv 8\left(\Lgg{}-2\right)\left(\bar{\Delta}_{22}+
\sqrt{3}\bar{\Delta}_{12}+\varepsilon\det\hat{\bar{\Delta}}\right)^2,
$$
$$
c\equiv \left(d+4\varepsilon\det\hat{\bar{\Delta}}\right)^2,\qquad
\varepsilon\equiv -\frac{4}{3\Lambda^2}\,,
$$
and $d$, $Z_1$ are given by~\gl{d},~\gl{zpi}.
The residues in poles of the transverse part of
correlator~\gl{fin} are:
\be
\label{ezr1}
Z_{\rho}=\frac{Z_1\left(-am_{\rho}^2+b+c\right)}{24\left(\Lgg{}-2
\right)\det\hat{\bar{\Delta}}\left(m_{\rho'}^2-m_{\rho}^2\right)}
\ee
\be
\label{ezr2}
Z_{\rho'}=\frac{Z_1\left(-am_{\rho'}^2+b+c\right)}{24\left(\Lgg{}-2
\right)\det\hat{\bar{\Delta}}\left(m_{\rho}^2-m_{\rho'}^2\right)}
\ee
One can check that the expressions \gl{ezr1}, \gl{ezr2} in the
large-log approximation turn into corresponding
quantities~\gl{zv1},~\gl{zv2}.

The exact calculation of axial-vector correlator is much more difficult
because of $P-A$ mixing.

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hep-ph0304140.extracted_bib
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(1961), 345.
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