\documentclass[final,preprint,aps,showpacs,tightenlines,fleqn]{revtex4}
\usepackage{bm}
\usepackage{dcolumn}

\begin{document}
\preprint{PNU-NTG-02/2002} 
\title {A test of the instanton vacuum 
with low-energy theorems of the axial anomaly}
\author{M.M. Musakhanov$^{1}$\footnote{E-mail address: 
musakhanov@nuuz.uzsci.net}   
and Hyun-Chul Kim $^{2}$\footnote{E-mail address: hchkim@pnu.edu}}
\affiliation{(1)Theoretical  Physics Dept, \\
Uzbekistan National University,\\
 Tashkent 700174, Uzbekistan,\\
(2)Department of Physics, \\
Pusan National University, \\
609-735 Pusan, \\
Republic of Korea}
\date{May 2002}

\begin{abstract}
We revisit the QCD$+$QED axial anomaly low-energy 
theorems which give an exact relation between  the matrix elements
of the gluon and photon parts of the axial anomaly operator equation 
within the framework of the nonlocal chiral quark model derived from
the instanton vacuum.  The matrix elements between the vacuum and two 
photon states and between the vacuum  and two gluon states are
investigated for arbitrary $N_f $ in an {\em effective action} approach  
from the instanton vacuum in the chiral limit.
Having gauged the effective action properly, we show that the model 
does exactly satisfy the low-energy theorems.
\vspace{1cm}

\noindent
{\bf Keywords:} Axial anomaly, Low energy theorems, Instanton vacuum, 
nonlocal chiral quark model.
\end{abstract}
\pacs{11.30.Rd, 12.38.Lg}
\maketitle
%\section { Introduction}
{\bf 1.} The quantum fluctuations may destroy the symmetries of the initial 
classical Lagrangian.  One of the most important
examples is the famous axial anomaly in gauge theories.
The axial anomaly leads to many interesting nonperturbative features
intimately related to the topologically nontrivial structure of the vacuum.

The axial anomaly in the divergence of the singlet axial-vector current in
QCD $+$ QED leads to a  low-energy  theorem for  the matrix elements
of this operator equation over the vacuum and two--photon states
(see, for example, a review~\cite{Shifman:1988zk}):
\begin{equation}
\left \langle 0\left|N_f \frac{g^2}{32\pi^2}G\tilde G 
\right| 2\gamma \right\rangle =
N_c \frac{e^2}{8\pi^2} \sum_{f}{Q^{2}_{f}} F^{(1)}\tilde F^{(2)}
\label{theorem}
\end{equation}
at $q^2 \ll m_{\eta'}$~\cite{Shifman:1988zk}.  $m_{\eta'}\sim 1\, {\rm
  GeV}$ is the mass of the $\eta'$-meson, which is 
only the possible one-particle intermediate 
state with the quantum number of the divergence of the singlet 
axial-vector current.  $N_f$ denotes the number of flavors, 
$g$ is the QCD coupling constant
with
$G\tilde G= \frac{1}{2}\epsilon^{\mu\nu\lambda\sigma}
G^{a}_{\mu\nu}  G^{a}_{\lambda\sigma}$ and $N_c$ is the number of 
colors. Here, $G^{a}_{\mu\nu}$ stands for the operator of the gluon 
field strengths.  $e$ and $Q_{f}$ are the QED coupling constant and the 
electric charges of the quarks, respectively.  The photon part of 
Eq.(\ref{theorem}) can be expressed explicitly as 
\begin{equation}
2F^{(1)}\tilde F^{(2)}\; =\; \epsilon^{\mu\nu\lambda\sigma}
 F^{(1)}_{\mu\nu}  F^{(2)}_{\lambda\sigma},\;\;\;
F^{(i)}_{\mu\nu} \;=\; \epsilon^{(i)}_\mu q_{i\nu} -
\epsilon^{(i)}_\nu q_{i\mu},    
\end{equation}
where $\epsilon^{(1,2)}_\mu$ and $q_{1,2}$ are, respectively, 
polarizations and momenta of  photons, and $q=q_{1} + q_{2}$.  This
relation is a consequence of the absence of a massless singlet
pseudoscalar boson.  In the present work, 
the contribution of the quark masses is neglected.

It is evident that gluons can interact with photons only through quark
loops.  In perturbation theory, it leads to at least results of order 
$e^2\sim g^{4}$ for the left hand side of Eq.~(\ref{theorem}). 

Another nontrivial low-energy theorem concerns the matrix element
between the vacuum and two-gluon states:
\begin{equation}
 \langle 0| g^2 G\tilde G | 2 \mbox{gluons} \rangle = 0
 \label{theorem1}
\end{equation}
at the same limit as in Eq.(\ref{theorem}).  Hence, the solution of
these theorems given in Eqs.(\ref{theorem},\ref{theorem1}) is 
only pertinent to the nonperturbative phenomena related to the
topologically nontrivial structure of the QCD vacuum.

Without any doubt instantons represent a very important topologically 
nontrivial component of the QCD vacuum.  Their properties are
described by the average instanton size
$\rho$ and inter-instanton distance $R$. In 1982, Shuryak
\cite{Shuryak:1981ff,Shuryak:dp} estimated these two parameters 
phenomenologically as
\begin{equation}
\rho \,=\, 1/3 \, {\rm fm},\, \, \, \, R\,=\, 1 \, {\rm fm}.
\label{rhoR}
\end{equation}
It was confirmed by theoretical variational calculations
\cite{Diakonov:1983hh,Diakonov:1985eg} and recent lattice simulations 
of the QCD vacuum~\cite{DeGrand:2001tm}.  The presence of 
instantons in the QCD vacuum affects very strongly light quarks 
and leads to the formation of the massive constituent
interacting quarks.  This is the realization of spontaneous breaking
of chiral symmetry (SB$\chi$S) in the instanton vacuum, 
which brings about the collective massless excitations of the QCD
vacuum, {\em i.e.} pions.  These Goldstone bosons are the most 
important degrees of freedom in low-energy QCD.  Hence, instantons 
play a pivotal and more significant role in describing the lightest 
hadrons and their interactions, compared with confinement 
forces~\cite{Diakonov:2002mb,Chu:vi}.

The features of light quarks from the instanton vacuum
are characterized in the fermionic determinant ${\det}_N$ (in the field of 
$N_+$ instantons and  $N_-$ antiinstantons) calculated by Lee and 
Bardeen(LB)~\cite{Lee:sm}:
\begin{equation}
{\det}_N=\det B, \,\, B_{ij}=
im\delta_{ij} + a_{ji}, 
\label{det_N}
\end{equation}
where $a_{ij}$ represents the overlapping matrix element of the 
quark zero-modes $\Phi_{\pm , 0} $ generated by instantons(antiinstantons).
This matrix element is nonzero only between instantons and antiinstantons
(and vice versa) due to specific chiral properties of the zero-modes
and equal to
\begin{equation}
a_{-+}=-<\Phi_{- , 0} | i\rlap{/}{\partial} |\Phi_{+ , 0} > .
\label{a}  
\end{equation}
The overlapping of the quark zero-modes provides the propagation of the quarks
by jumping from one instanton to another one.  Thus, the determinant
of the infinite matrix was reduced to the
determinant of the finite matrix in the space of {\it only zero-modes}.
From Eqs.(\ref{det_N},\ref{a})
it is clear that for  $N_{+}\neq N_{-}$
$
{\det}_N \sim m^{|N_{+}-N_{-}|}
$
which will strongly suppress the fluctuations of  $|N_{+}-N_{-}|.$
Therefore in the final expression we will assume $ N_{+}=N_{-}=N/2 .$

The fermionic determinant ${\det}_N$ averaged over
instanton/anti-instanton positions, orientations and sizes
can be considered as a partition function of light quarks $Z_N$.
Then the properties of the hadrons and their interactions are encoded
in the low-energy QCD effective chiral action.

Some years ago, one of the authors showed that the low-energy QCD 
effective chiral action from the instanton vacuum satisfies the low energy
theorems with an accuracy of about 17 $\%$~\cite{Musakhanov:1996qf}.
This discrepancy is due to the fact that the vector and axial-vector 
currents are not conserved because of the momentum-dependent dynamical 
quark mass in the 
effective action~\cite{Bowler:ir,Plant:1997jr,Broniowski:1999bz}.  

The nonconservation of the N\"other currents breaks various Ward
identities in the model.  In order to remedy these problems, 
one has to gauge the low-energy QCD effective chiral action properly.  
By doing that, the conserved currents can be 
derived~\cite{Bowler:ir,Plant:1997jr,Broniowski:1999bz,LeeKim}.

In the present paper, we shall continue to test the low-energy 
QCD effective chiral action from the instanton vacuum in the chiral
limit aiming at resolving the discrepancy existing in 
Ref.~\cite{Musakhanov:1996qf}.  Having gauged the effective action, 
we shall show that it does exactly satisfy these theorems as it should be. 

%\section{Low-energy QCD effective chiral action in the chiral limit}
{\bf 2.} Following the procedure in~\cite{Tokarev,Salvo:1997nf}, we obtain 
the fermionization of $LB$'s result, i.e.,
\begin{eqnarray}
{\rm det}_N &=& \int D\psi D\psi^{\dagger} \exp(\int d^4 x
\sum_{f}\psi_{f}^{\dagger}i\rlap{/}{\partial} \psi_{f})     \cr  
&\times& \prod_{f}(\prod_{+}^{N_{+}}(im_{f} - V_{+}[\psi_{f}^{\dagger} 
,\psi_{f}]) \prod_{-}^{N_{-}}(im_{f} - V_{-}[\psi_{f}^{\dagger} ,\psi_{f}])),
\label{part-func}
\end{eqnarray}
where 
\begin{equation}
V_{\pm}[\psi_{f}^{\dagger} ,\psi_{f}]= 
\int d^4 x (\psi_{f}^{\dagger} (x) i\rlap{/}{\partial}
\Phi_{\pm , 0} (x; \xi_{\pm}))
\int d^4 y 
(\Phi_{\pm , 0} ^\dagger (y; \xi_{\pm} )  
i\rlap{/}{\partial} \psi_{f} (y)).
\end{equation}
Eq.(\ref{part-func}) coincides with the Ansatz 
for the fixed $N$ partition function
postulated by Diakonov and Petrov~\cite{Diakonov:1983hh,Diakonov:1985eg}.  

Now averaging over collective coordinates $\xi_\pm$ becomes a trivial
problem, since the low density of the instanton media 
allows us to average over positions and orientations 
of the instantons independently.  In the following, we assume the
chiral limit.  Then Eq.(\ref{part-func}) gives us the partition function
\begin{equation}
 Z_N = \int D\psi D\psi^\dagger \exp  (\int d^4 x \, \psi^\dagger 
i \rlap{/}{\partial}  \psi )  \,  W_{+}^{N_+}  \, W_{-}^{N_-}, 
\label{Z_NW}
\end{equation}
where
\begin{eqnarray}
W_\pm &= & \int d^4 \xi_{\pm}\prod_{f} (V_{\pm}[\psi_{f}^{\dagger} 
\,\psi_{f}] ) \cr
&=& (i)^{N_{f}}\left(  \frac{4\pi^2\bar\rho^2}{N_c} \right)^{N_f}
\int \frac{d^4 z}{V} 
\det_{f}i J_\pm (z)
\end{eqnarray}
and
\begin{equation}
J_\pm (z)_{fg} \;=\; \int \frac {d^4 kd^4 l}{(2\pi )^8 } \exp ( -i(k - l)z)
\, F(k^2) F(l^2) \, \psi^\dagger_f (k) \frac12 (1 \pm \gamma_5 ) \psi_g (l) .
\label{J_pm}  
\end{equation}
The form factor $F$ is related to the 
zero--mode wave function in momentum space $\Phi_\pm (k; \xi_{\pm}) $ 
and turns out to be
\begin{equation}
F(k^2) = - t \frac{d}{dt} \left[ I_0 (t) K_0 (t) - I_1 (t) K_1 (t)
\right], \,\, t =\frac{1}{2} \sqrt{k^2} \bar\rho.  
\label{ff}
\end{equation}
The partittion function in Eq.(\ref{Z_NW}) can be written as follows:
\begin{equation}
 Z_N = \int D\psi D\psi^\dagger \exp
(\int  \psi^\dagger i \rlap{/}{\partial}  \psi
+ Y_{+} + Y_{-}),
\label{Z_NY}  
\end{equation}
where
\begin{equation}
Y_{\pm}=  (i )^{N_f} \lambda
\int d^4 z \, \det J_\pm (z) =  \left(\frac{2V}{N}\right)^{N_f - 1}
(i M)^{N_f}
\int d^4 z \, \det J_\pm (z) .
\label{Y_pm}  
\end{equation}
The  self-consistency condition at the saddle point in 
Eqs.(\ref{Z_NY},\ref{Y_pm}) yields
\begin{equation}
4 N_c V \int \frac{d^4 k}{(2\pi )^4} \frac{M^2 F^4 (k)}{M^2 F^4 (k) +
k^2} =  N .
\label{selfconsist}  
\end{equation}
These formulas, being derived from Lee and Bardeen fermionic 
determinant~\cite{Lee:sm}, completely agree with Ref.~\cite{Diakonov:1995qy}.

%\subsection {Gauged effective action for the
%calculation with the gluon operators }
In the quasiclassical(saddle-point) approximation, any gluon operator
receives its main contribution from the instanton background.
As an example, for one  instanton (anti-instanton)$I (\bar I)$,
the following gluonic operators can be expressed in terms of the
instanton background:
\begin{eqnarray}
{g^2}G^2 (x) &=&
\frac{192 \rho^4}{\left[ \rho^2 + (x - z)^2 \right]^4} = f(x-z),
\label{GG}\cr
{g^2}G\tilde G(x) & = &\pm f(x-z) .
\label{GtildeG}  
\end{eqnarray}
The derivation of the correlators with the gluonic
operator $G\tilde G$ can be reduced to the calculation of the 
following partition function:
\begin{equation}
\hat Z_{N}  [\kappa ] =
Z_{N}^{-1} \int  D\psi D\psi^\dagger \exp  (- \hat S_{\rm eff}  ) ,
\label{hatZ_N}  
\end{equation}
where the effective chiral action, $\hat S_{\rm eff},$ in the presence of 
an external field $\kappa (x)$ is given by
\begin{equation}
 -\hat S_{\rm eff}  =  \int  \psi^\dagger i \rlap{/}{\partial}  \psi
+ Y_{+} + Y_{-} + \int dx \left(Y_{G\tilde G +} (x) + Y_{G\tilde G -}
(x)
\right) \kappa (x)  
\label{hatS_ef1}
\end{equation}
with
\begin{equation}
Y_{G\tilde G\pm}(x) = \pm \left(\frac{2V}{N}\right)^{N_f - 1} (i
M)^{N_f}
\int d^4 z\, f(x-z) \, \det J_\pm (z) .  
\label{Y_GtildeGQ}
\end{equation}
Eq.(\ref{hatS_ef1}) can be rewritten as:
\begin{eqnarray}
-\hat S_{\rm eff}  &=&
\int \psi^\dagger i\rlap{/}{\partial} \psi +
\left(\frac{2V}{N}\right)^{N_f - 1}
\int dz \,  \det (i  M_{+} (z) J_{+} (z)) \cr
&& \hspace{47pt} +
\left(\frac{2V}{N}\right)^{N_f - 1}
\int dz \, \det (i  M_{-} (z) J_{-} (z)) ,
\label{hatS_ef3}
\end{eqnarray}
where
\begin{equation}
M_{\pm} (z) =
 \left( 1 \pm \int dx \kappa (x) f(x-z)\right)^{(N_f - 1)^{-1} }M.  
\end{equation}

Using the following relation, which is valid in the saddle-point approximation,
\begin{equation}
\exp (\lambda \det [i A] ) =
\int d{\cal M} \exp\left[ - (N_f - 1) \lambda^{-\frac{1}{N_f - 1}}
(\det{\cal M} )^{\frac{1}{N_f - 1}} + i {\rm tr} ({\cal M} A) \right],
\label{expA}  
\end{equation}
one can easily show that the effective bosonized action
$\hat S_{\rm eff}[ {\cal M}_{\pm}, \kappa ]$ in the presence of the external
field $\kappa$ becomes
\begin{eqnarray}
 -\hat S_{\rm eff}[ {\cal M}_{\pm}, \kappa ] &=& \int dz \left( - (N_f - 1)
\left(\frac{2V}{N}\right)^{ - 1}
(\det{\cal M}_{\pm} )^{\frac{1}{N_f - 1}}\right) \cr
&+& {\rm Tr} \ln \left[i\rlap{/}{p} + i M F (p^2 ){\cal M}_{+}F(p^2) 
\left(1 +
(\kappa f)\right)^{N_{f}^{-1}} \frac{1}{2}(1+\gamma_5)\right.  \cr
&& + \left. i M F(p^2 ) {\cal M}_{-}F(p^2 ) 
\left( 1 -  (\kappa f)\right)^{N_{f}^{-1}} \frac {1}{2}(1- \gamma_5 )
\right].
\label{hat S_ef4}  
\end{eqnarray}
The matrices ${\cal M}_\pm$ describe the pseudoscalar meson fields.  
Here, $p_\mu = i\partial_\mu$ and $F(p^2)$ denotes the form factor 
defined in Eq.(\ref{ff}), and $(\kappa  f) =  \int dx \kappa (x) f(x-z) .$

Let us define the covariant momentum operator:
\begin{eqnarray}
P_\mu & :=& p_\mu  + e Q_{f} V_\mu,
\label{codev}  
\end{eqnarray}
where $ V_\mu$ is an external electromagnetic field.  The usual
procedure of gauging an action under a $\mbox{U(1)}_V$ local gauge 
transformation is performed by replacing the usual derivative by the
covariant derivative defined in Eq.(\ref{codev}).  Since our effective
action contains the nonlocal coupling, {\em i.e.} the
momentum-dependent dynamical quark mass $MF^2 (p^2 )$, we also need to
introduce the covariant derivative in the form factor $F(p^2)$:
$F(p^2 )\rightarrow F(P^2 )$~\cite{Ball:1993ak}.
It is easy to prove that with this prescription we derive the gauge-invariant
effective chiral action:
\begin{eqnarray}
-\hat S_{\rm eff}[{\cal M}_{\pm},V,\kappa] &=& \int dz\left(-(N_f - 1)
\left(\frac{2V}{N}\right)^{ - 1}
(\det{\cal M}_{\pm} )^{\frac{1}{N_f - 1}}\right) \cr
&+& 
{\rm Tr} \ln \left[ \rlap{/}{P} + i M F(P^2) {\cal M}_{+}F(P^2) \left( 1 +
( \kappa  f)\right)^{N_{f}^{-1}} \frac {1}{2}(1+\gamma_5 )\right.  \cr
&& + \left.i M F(P^2) {\cal M}_{-}F(P^2)
\left( 1 - (\kappa f)\right)^{N_{f}^{-1}} \frac {1}{2}(1- \gamma_5 )\right]
\label{hat S_ef5}
\end{eqnarray}
in the presence of the external fields $V_\mu$ and $\kappa$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection { The low-energy theorems for the matrix 
%element between the vacuum and two-photons state}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since the matrix element between the vacuum and two-photon state is 
defined by the following functional derivative:
\begin{equation}
\left. \frac {\delta \hat Z_{N}  [\kappa , V] }
{\delta \kappa (x) \delta V_\mu (x_1) \delta V_\nu (x_2) }
\right|_{ \kappa , V = 0},
\end{equation}
we need to keep the terms of order ${\cal O}(\kappa)$ and ${\cal O}(V^2)$ in 
$\hat S_{\rm eff}[ {\cal M}_{\pm}=1, V, \kappa]$ defined in 
Eq.(\ref{hat S_ef5}).

We will employ for the explicit calculation the Schwinger method~\cite{Schw}, 
which was developed for quantum electrodynamics and later extended to
QCD~\cite{VZNS83}.  Using this method, we can make a derivative 
expansion of the effective action with regard to $(V\cdot\partial)/M$.

We introduce first the coordinate and momentum operators,
$X_\mu$ and $p_\mu$, respectively, which satisfy
$[p_\mu, X_\nu]=i\delta_{\mu\nu}\ ,
\ [p_\mu,p_\nu]=[X_\mu,X_\nu]=0$.
We take then the covariant momentum operator $P_\mu$
satisfying the following commutation relations,
\begin{equation}
[P_\mu, X_\nu]=i\delta_{\mu\nu}\;,\;
[P_\mu,P_\nu]=ieF_{\mu\nu},
\label{eq:op1}
\end{equation}
where  $F_{\mu\nu}$ is the photon field strength tensor.
Moreover, we employ a complete set of eigenstates $|x\rangle$
of the coordinate operator $X_\mu$,
\begin{equation}
X_\mu|x\rangle = x_\mu |x\rangle\ ,
\end{equation}
which satisfies 
\begin{equation}
\langle y | x\rangle = \delta^{(4)}(x-y)\;\;,\;\;
\int d^4 x|x \rangle\langle x |=1\ .
\end{equation}
In this basis, the operator $P_\mu$ acts 
as a covariant derivative $iD_\mu$,
\begin{equation}
\langle y | P_\mu | x \rangle = iD_\mu\langle y | x\rangle
=\left(i\partial_\mu+eQ V_\mu (x)\right)\delta^{(4)}(x-y)\ .
\end{equation}
The algebra given in Eq.(\ref{eq:op1}) is 
the basic tool of the Schwinger formalism.
We expand the  $\hat S_{\rm eff}$ in the derivatives of 
the photon background field, 
and need to use only this algebra in each order of the expansion.
Taking ${\cal M}_{\pm}=1$, considering the first order on  $\kappa$,
and defining $M(P)=MF^2 (P)$, we get
\begin{eqnarray}
 -\hat S_{\rm eff}[ {\cal M}_\pm=1, V, \kappa ] &=&
{\rm Tr} \ln [  \rlap{/}{P} + i M (P) ( 1 + \frac{1}{N_f} 
(\kappa  f)\gamma_5 )]\cr
&=&{\rm Tr} \ln [ \rlap{/}{P} + i M (P)] +
\frac{i}{N_f} {\rm Tr}\left[(\rlap{/}{P} + i M (P))^{-1}M (P)(\kappa
  f)\gamma_5\right], 
\label{hat S_ef6}  
\end{eqnarray}
where ${\rm Tr} = {\rm tr}_{cf\gamma}\int d^4 x\langle x|..... |x\rangle$ and 
${\rm tr}_{cf\gamma}$ denotes the trace over color, flavor, and Dirac 
spaces.  The quark propagator with the covariant derivative can be
treated as follows:
\begin{eqnarray}
( \rlap{/}{P} + i M (P))^{-1} &=& 
[( \rlap{/}{P} - i M (P))(\rlap{/}{P} + i M (P)]^{-1}
( \rlap{/}{P} - i M (P)) \cr
&=& \left( P^2 +  M^2 (P) + \frac{eQ}{2}\sigma \cdot F 
+i[ \rlap{/}{P},M(P)]\right)^{-1} (\rlap{/}{P} - i M (P)),
\label{paropagator}  
\end{eqnarray}
where $\sigma\cdot F=\sigma_{\mu\nu} F_{\mu\nu}$.
We need at least four $\gamma$ matrices on account of $\gamma_5$, so
that we obtain a nonvanishing result with trace over Dirac space
(${\rm tr}_\gamma$).  Thus, the relevant nonvanishing part of the 
effective action is
\begin{eqnarray}
-S_{kf} &=& \frac{i}{N_f} {\rm Tr}\left[( \rlap{/}{P} + i M (P))^{-1}M (P) 
( \kappa  f)\gamma_5\right] \cr
&=&
\frac{1}{N_f} {\rm Tr}\left\{ \left[( P^2 +  M^2 (P))^{-1} 
\left(\frac{eQ}{2}\sigma\cdot F 
+i[ \rlap{/}{P},M(P)]\right)\right]^2\right. \cr
&\times&\left.
( P^2 +  M^2 (P))^{-1}(M^2(P) + i \rlap{/}{P} M(P))
(\kappa f)\gamma_5 \right\}.
\label{S_kf}  
\end{eqnarray}
We observe that $[ \rlap{/}{P},M(P )]$ is of order ${\cal O}(e)$ and 
$M$ is a function of $P^2$.  Then
\begin{eqnarray}
-S_{kf} &=& \frac{1}{N_f} {\rm Tr} \left[( p^2 +  M^2 (p^2))^{-1} 
\left(\frac{eQ}{2}\sigma\cdot F\right)^2
( p^2 +  M^2 (p^2 ))^{-1} M^2(p^2 )(\kappa f)\gamma_5\right. \cr
&-&\left\{ ( p^2 +  M^2 (p^2 ))^{-1}\frac{eQ}{2}\sigma\cdot F,\,\, 
 ( p^2 +  M^2 (p^2 ))^{-1} [ \rlap{/}{P},M(P^2 )]\right\}\cr 
&\times& 
( p^2 +  M^2 (p^2 ))^{-1} \rlap{/}{p} M(p^2 ) (kf)\gamma_5 \Big]
 \,+\, {\cal O}(e^3 ).
\label{S_kf1}  
\end{eqnarray}
Here, we take into account the term to order ${\cal O}(e^2 )$ only.
Then, what we need is to calculate $[\rlap{/}{P},M(P^2)]$: 
\begin{equation}
[ \rlap{/}{P},M(P^2 )] = eQ \left([ \rlap{/}{V}, M(p^2 )] +
\left[\rlap{/}{p},\{p_\mu , V_\mu \}\frac{dM(p^2 )}{dp^2} \right]\right).
\label{commutator}  
\end{equation}
By fixing the gauge condition $[p_\mu , V_\mu ]=0$, we obtain
$\{p_\mu , V_\mu \} = 2 V_\mu p_\mu$.  Hence, the commutator 
(\ref{commutator}) becomes:
\begin{equation}
[ \rlap{/}{P},M(P^2 )]= -2ieQ F_{\mu\nu} \gamma_\nu p_\mu  
\frac{dM(p^2 )}{dp^2}+O(e^2, \partial^2 ) .
\label{commutator1}  
\end{equation}
Putting them together, we finally arrive at the expression:
\begin{equation}
-S_{kf} = \int \frac{d^4 p}{(2\pi )^4}
\frac{M^2 (p^2 ) - p^2 \frac{d M^2 (p^2)}{dp^2}}{(p^2 
+ M^2 (p^2))^3} {\rm tr} \left(\frac{eQ}{2}\sigma\cdot F\right)^2 
(kf)\gamma_5 .
\label{S_kf2}  
\end{equation}
It was shown in the previous paper~\cite{Musakhanov:1996qf} 
that if we neglect the momentum dependence of the $M$, 
then we exactly reproduce the low energy theorem in Eq.(\ref{theorem}).
Now, let us calculate the ratio
\begin{equation}
R = \frac{J_1}{J_2}
\end{equation}
with
\begin{eqnarray}
J_1 &=& \int p^2 dp^2
\frac{M^2 (p^2 ) - p^2 \frac{d M^2 (p^2)}{dp^2}}{(p^2 
+ M^2 (p^2))^3}, \cr
J_2 &=& 
\int dp^2 p^2 \frac{M^2(0)}{(p^2 +M^2(0) )^3}.
\end{eqnarray}
Replacing the variable~\cite{Plant:1997jr} by 
\begin{equation}
s\;=\; \frac{M^2 (p^2)}{p^2},
\end{equation}
we immediately have the ratio $R=1$.  
As a result, the nonlocal chiral quark model from the instanton vacuum
exactly satisfies the low-energy theorem given in Eq.(\ref{theorem}). 

%\subsection{ The low-energy theorem for the matrix element between
%vacuum and two-gluons states }
%
Now, we briefly present some calculations related to Eq.(\ref{theorem1}).
This matrix element can be written in the form:
\begin{eqnarray}
&& \langle 0| g^2 G\tilde G | g(\epsilon^{(1)} , q_1 ), g(\epsilon^{(2)}
, q_2 )\rangle  \cr 
&=& \epsilon^{(1)a_1}_{\mu_1}  \epsilon^{(2)a_2}_{\mu_2}  
\int  \partial^{2}_{2} \,\partial^{2}_{1}\, \langle 0|
T{g^2}G\tilde
G A_{\mu_1}^{a_1}(x_1 ) A_{\mu_2}^{a_2}(x_2)
|0 \rangle  \exp i(q_1 x_1 + q_2 x_2 ) dx_1 dx_2,
\label{me2} 
\end{eqnarray}
where $A_{\mu}^{a}(x )$ is a total gluon field,
$\epsilon^{(i)a_i}_{\mu_i}$ and $ q_i $ are the polarization vectors and
the momentum of gluons, respectively.

As usual, we expand the total field $A_{\mu}^{a}(x )$ around  the
instanton background. The main term in Eq.(\ref{me2}) is the
contribution of the instanton background and is of order ${\cal O}(g^{-2})$.
The next term is due to the perturbative fluctuations 
over the instanton background and corresponds to the contributon of 
order ${\cal O} (g^{2})$.  It is easy to see from previous 
considerations that the term to order $ {\cal O}(g^{-2})$ is given 
by the following expression:
\begin{equation}
Z_{N}^{-1} \int D\psi D\psi^\dagger \exp  (- S_{\rm eff} )
\left( \left( Y_{G\tilde GAA +} (x) + Y_{G\tilde GAA -} (x)\right)
Q\right) ,
\label{GtildeGAAQ2}  
\end{equation}
where
\begin{eqnarray}
Y_{G\tilde GAA \pm} &=& \pm \left(\frac{2V}{N}\right)^{N_f - 1} (i
M)^{N_f}
\int d^4 z\, f(x-z)  \cr
&\times& \int dO
(-\partial^{2}_{1})A_{\mu_1}^{I(\bar I)a_1}(x_1)
(-\partial^{2}_{i})A_{\mu_2}^{I(\bar I)a_2}(x_2)
 \det J_\pm (z) ,  
\label{Y_GtildeGAAQ}
\end{eqnarray}
where the instanton(anti-instanton) is located at the point $z$ with
its orientation $O$.

Repeating the bosonization trick, we obtain the result for the
${\cal O} (g^{-2})$ contribution which is proportional to
\begin{equation}
{\rm Tr} [ (i\rlap{/}{\partial} + iMF^2)^{-1}iM F^2 \gamma_5 ].  
\end{equation}
Hence, one can easily show that the contribution from the term to
order ${\cal O}(g^{-2})$ vanishes.

The contribution from the next order $({\cal O}(g^{2}))$ comes from
two different diagrams.  The first diagram is the direct contribution of
the operator $g^2 G\tilde G$ which is equal to
$- g^2 G^{(1)} \tilde G^{(2)}$, where 
\begin{eqnarray}
2G^{(1)} \tilde G^{(2)} &=& \epsilon^{\mu\nu\lambda\sigma}
 G^{(1)a}_{\mu\nu} G^{(2)a}_{\lambda\sigma},\cr 
G^{(i)a}_{\mu\nu} &=&
\epsilon^{(i)}_{\mu}  q_{i\nu} - \epsilon^{(i)}_{\nu} q_{i\mu}.
\end{eqnarray}

The factors at the vertices of the second diagram are
$g \gamma_{\mu} \lambda_{a} /2$ and $i M f
F^{2}\gamma_{5}N_{f}^{-1}$.  The coefficient $N_{f}$
represents the contribution from all flavors.

A comparison with previous calculations   
ends up with the result that the contribution from the second
loop--diagram is equal in magnitude but opposite in sign to  
that from the first diagram at $q^2 = 0$.  Because of this cancellation, 
the contribution from order ${\cal O}(g^{2})$ vanishes in the  limit
$q^2 \rightarrow 0$.  Hence, we arrive exactly at the same conclusion
as in the previous case.  

From the above calculations, we conclude that the nonlocal chiral quark
model from the instanton vacuum does exactly satisfy the low-energy
theorems, once we have properly gauged the effective chiral action.

%\section{ Conclusion }
{\bf 3.} The solution of the low-energy theorems of the axial anomaly
is nontrivially related to the nonperturbative instanton
structure of the QCD vacuum.  The effective action approach based on the
instanton vacuum exactly satisfies these low-energy theorems.

This approach provides a solid ground for the investigation into
different amplitudes of the nonperturbative conversion of gluons
into hadrons and photons.  Further studies relevant to all of 
these problems are under way.

\section{ Acknowledgments }
One of the authors(MM) acknowledges the warm
hospitality of the Nuclear Theory Group of PNU, 
where this work was initiated.  HCK is grateful to D.K. Hong
for useful discussion.  The present work is supported by   
the KOSEF grant number (R01--2001--00014).
The work of MM is supported in part by INTAS, BMBF and SNF grants. 

\begin{thebibliography}{99}
%\cite{Shifman:1988zk}
\bibitem{Shifman:1988zk}
M.~A.~Shifman,
%``Anomalies And Low-Energy Theorems Of Quantum Chromodynamics,''
Phys.\ Rept.\  {\bf 209} (1991) 341
[Sov.\ Phys.\ Usp.\  {\bf 32} (1989\ UFNAA,157,561-598.1989) 289].
%%CITATION = PRPLC,209,341;%%
%\cite{Shuryak:1981ff}
\bibitem{Shuryak:1981ff}
E.~V.~Shuryak,
%``The Role Of Instantons In Quantum Chromodynamics. 1. Physical Vacuum,''
Nucl.\ Phys.\ B {\bf 203} (1982) 93.
%%CITATION = NUPHA,B203,93;%%
%\cite{Shuryak:dp}
\bibitem{Shuryak:dp}
E.~V.~Shuryak,
%``The Role Of Instantons In Quantum Chromodynamics. 2. Hadronic Structure,''
Nucl.\ Phys.\ B {\bf 203} (1982) 116.
%%CITATION = NUPHA,B203,116;%%
%\cite{Diakonov:1983hh}
\bibitem{Diakonov:1983hh}
D.~Diakonov and V.~Y.~Petrov,
%``Instanton Based Vacuum From Feynman Variational Principle,''
Nucl.\ Phys.\ B {\bf 245} (1984) 259.
%%CITATION = NUPHA,B245,259;%%
%\cite{Diakonov:1985eg}
\bibitem{Diakonov:1985eg}
D.~Diakonov and V.~Y.~Petrov,
%``A Theory Of Light Quarks In The Instanton Vacuum,''
Nucl.\ Phys.\ B {\bf 272} (1986) 457.
%%CITATION = NUPHA,B272,457;%%
\bibitem{DeGrand:2001tm}
T.~DeGrand,
%``Short distance current correlators: Comparing lattice simulations to  the instanton liquid,''
Phys.\ Rev.\ D {\bf 64} (2001) 094508
.
%%CITATION = ;%%
%\cite{Diakonov:2002mb}
\bibitem{Diakonov:2002mb}
D.~Diakonov,
%``Instantons and baryon dynamics,''
arXiv:.
%%CITATION = ;%%
%\cite{DeGrand:2001tm}
%\cite{Chu:vi}
\bibitem{Chu:vi}
M.~C.~Chu, J.~M.~Grandy, S.~Huang and J.~W.~Negele,
%``Evidence For The Role Of Instantons In Hadron Structure From Lattice QCD,''
Phys.\ Rev.\ D {\bf 49} (1994) 6039
.
%%CITATION = ;%%
%\cite{Lee:sm}
\bibitem{Lee:sm}
C.~k.~Lee and W.~A.~Bardeen,
%``Interaction Of Massless Fermions With Instantons,''
Nucl.\ Phys.\ B {\bf 153} (1979) 210.
%%CITATION = NUPHA,B153,210;%%
\bibitem{Musakhanov:1996qf}
M.~M.~Musakhanov and F.~C.~Khanna,
%``A test of the instanton vacuum chiral quark model with axial anomaly  low-energy theorems,''
Phys.\ Lett.\ B {\bf 395} (1997) 298
.
%%CITATION = ;%%
%\cite{Bowler:ir}
\bibitem{Bowler:ir}
R.~D.~Bowler and M.~C.~Birse,
%``A Nonlocal, Covariant Generalization Of The Njl Model,''
Nucl.\ Phys.\ A {\bf 582} (1995) 655
.
%%CITATION = ;%%
%\cite{Plant:1997jr}
\bibitem{Plant:1997jr}
R.~S.~Plant and M.~C.~Birse,
%``Meson properties in an extended non-local NJL model,''
Nucl.\ Phys.\ A {\bf 628} (1998) 607
.
%%CITATION = ;%%
%\cite{Broniowski:1999bz}
\bibitem{Broniowski:1999bz} 
W.~Broniowski,
%``Gauging non-local quark models,''
arXiv:.
%%CITATION = ;%%
\bibitem{LeeKim} J.~H.~Lee and H.-Ch.~Kim, in preparation.  
\bibitem{Tokarev} V. F. Tokarev, Instantons and Colour Screening,
Soviet\ J.\ Theor.\ Math.\ Phys., {\bf  73}, 223 (1987). 

%\cite{Salvo:1997nf}
\bibitem{Salvo:1997nf}
E.~D.~Salvo and M.~M.~Musakhanov,
%``Axial anomaly and low energy tests for instanton vacuum models in QCD. DPBN(9)=Eur. Phys. J. C5 (1998) 501-506,''
Europ.Phys.J. {\bf C5} (1998) 501 .
%%CITATION = ;%%
%\cite{Diakonov:1995qy}
\bibitem{Diakonov:1995qy}
D.~Diakonov, M.~V.~Polyakov and C.~Weiss,
%``Hadronic matrix elements of gluon operators in the instanton vacuum,''
Nucl.\ Phys.\ B {\bf 461} (1996) 539
.
%%CITATION = ;%%
%\cite{Ball:1993ak}
\bibitem{Ball:1993ak}
R.~D.~Ball and G.~Ripka,
%``The Regularization of the fermion determinant in chiral quark models,''
arXiv:.
%%CITATION = ;%%
\bibitem{Schw} J. Schwinger, Phys.\ Rev.\ {\bf 82}, 664 (1951). 

\bibitem{VZNS83}
A. I. Vainshtein, V. I. Zakharov, V. A. Novikov, M. A. Shifman,
                Sov.J.Nucl.Phys. {\bf 39}, 77 (1984).
\end{thebibliography}
\end{document}






