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\bc
{\large\bf EFFECTIVE QUARK LAGRANGIAN
\\[0.2cm] IN THE INSTANTON VACUUM WITH NONZERO MODES
INCLUDED}\\[0.4cm]
{\it B.O.Kerbikov, D.S.Kuzmenko,
Yu.A.Simonov\\ Institute of Theoretical and Experimental Physics,\\
117259 Moscow, Russia}
\vspace{0.8cm}

\parbox{14cm}{~~~~~A new approach to effective theory of quarks in the
instanton vacuum is presented. Exact equations for the quark
propagator and Lagrangian are derived which contain contributions of
all quark modes with known coefficients. The resulting effective
Lagrangian differs from the standard one and  resembles that of the
Nambu--Jona--Lasinio model.  }

\ec
\vspace{0.8cm}

PACS: 02.30.Jr, 11.15.-q\\

1. The recent lattice data [1] provided evidence that instantons may
be responsible for nonperturbative behavior of $q\bar q$ correlators
[2] which makes the study of the quark dynamics in the instantonic
vacuum [3--5] an important and fundamental problem.

To date practically all papers on the subject have relied upon the
use of the so--called zero--mode approximation (ZMA) which amounts to
comprising only zero quark mode in a single--instanton fermion
propagator [3]. Correspondingly an ansatz for the partition function
and effective quark Lagrangian (EQL) containing zero modes only have
been proposed [5] and widely used in literature [6,7].

The purpose of this letter is to give a complete normal mode
expansion of the EQL and of the quark propagator.
Keeping only zero-mode
 coefficients in this expansion one retrieves the ansatz
[5] for EQL.
 Naively one would expect that this choice of the coefficients yields
 dominant contribution to physical quantities, and thus
 justify ZMA. However  the  exact calculation
 of EQL
 presented below does not show this dominance.
   More intricate is
 the analysis of the quark propagator $S$ in the
 instanton--anti--instanton vacuum.  Here the  zero--mode term
survives but  higher  modes enter with coefficients of the same
order.
  The similar feature can be seen in the quark partition function
$<det S^{-1}>$, where the average $<\ldots>$ is defined below. Thus
the new quark dynamics associated with nonzero modes emerges. The
main features of this dynamics are outlined in what follows.

2. To make discussion transparent consider an ideal instanton gas
with the superposition ansatz [8--10] and with
zero net topological charge i.e. equal number of
instantons and anti--instantons, $N_+=N_-=N/2$:

\be
A_{\mu}(x)=\sum^N_{i=1}A^{(i)}_{\mu}(x-R_i),
\ee

\be
gA^{(i)}_{\mu}=\frac{\bar\eta_{a\mu\nu}(x-R_i)_{\nu}\rho^2\Omega^+_i
\tau_a\Omega_i}{(x-R_i)^2[(x-R_i)^2+\rho^2]},
\ee
where $\Omega_i$, $R_i$, and $\rho$ are color orientation, position
and scale--size of the $i$--th instanton.

The EQL is obtained from the Euclidean partition function after
averaging over $\{\Omega_i,R_i\}$:

\be
Z=\int D\psi D\psi^+\e^{-\int dx\psi^+S^{-1}\psi}\prod^N_{i=1}
\frac{dR_i}{V}d\Omega_i=\int D\psi D\psi^+\e^{-L_{eff}},
\ee
where definitions here and in what follows are:

\be
\ba{l}
S^{-1}_0=(-i\hat\partial-im_f),\\[0.3cm]
S^{-1}_i=(-i\hat\partial-g\hat A^{(i)}-im_f),\\[0.3cm]
S^{-1}=(-i\hat\partial-g\hat A-im_f).
\ea
\ee
Next we introduce the standard set of eigenfunctions $\{u^i_n\}$,
$n=0,1,2,\ldots$

\be
(-i\hat\partial-g\hat A^{(i)})|u^i_n>=\lambda_n|u^i_n>.
\ee
Then $S^{-1}$ given by (4) has a formal representation as a sum
over normal modes

\be
S^{-1}=S^{-1}_0+\sum_{i,m,n}S^{-1}_0|u^i_m>\varepsilon^i_{mn}
<u^i_n|S^{-1}_0,
\ee
where $\hat\varepsilon$ can be represented either as

\be
\varepsilon^i_{mn}=
-<u^i_m|(S_i-S_0)[1+S^{-1}_0(S_i-S_0)]^{-1}|u^i_n>,
\ee
or simply as

\be
\varepsilon^i_{mn}=-g<u^i_m|S_0\hat A^{(i)}S_0|u^i_n>.
\ee
Performing in (3) averaging with the help of cumulant or cluster
expansion, one obtains $L_{eff}$ in the form

\be
L_{eff}=\int dx\psi^+S^{-1}_0\psi+\sum^{\infty}_{n=2}
\left(-\frac {2V}{N}\right)^{n-1}\sum_{fmm'}\int d\Gamma_n
det^{(n)}_{k,l}J_{kl},
\ee
where

\be
d\Gamma_n=
(2\pi)^4\delta\left(\sum_j(p_j-p'_j)\right)
\prod^n_{j=1}\left(\frac{d^4p_j}{(2\pi)^4}\frac{d^4p'_j}{(2\pi)^4}\right),
\ee

\be
J_{kl}=\left(\psi^{f_k}(p_k)\right)^+M^{f_kf_l}_{m_km'_l}(p_k,p'_l)
\psi^{f_l}(p'_l),
\ee
and similarly to [5,6] we have introduced the vertices

\be
M^{gr}_{mm'}(p,p')=\frac{N}{2VN_c}(\hat
p-im_g)\varphi_m(p)\varepsilon^i_{mm'}\varphi^+_{m'}(p')(\hat
p'-im_r),
\ee
with $\varphi_m(p)$ being the form factor of $u^i_m$ in momentum
space.

Summation in (9) starts from $n=2$ since the $n=1$ term drops out as
a result of integration over color orientations.

The EQL in (9) is a sum of $n\times n$ determinants. If one confines
oneself to ZMA i.e. puts $\varepsilon^i_{00}$ finite and
$\varepsilon^i_{m>0,n>0}$ equal to zero the sum runs only over $n\le
N_f$. This restriction is due to Grassmann nature of $J_{kl}$. Thus
even in ZMA one obtains e.g. for $N_f=3$ three $2\times2$
determinants and one $3\times3$ determinant. Only the last one is
present in the ansatz [5] with the identification
$\varepsilon^i_{00}\equiv\varepsilon$, $M_{00}\equiv M$. Therefore
our results are in contrast to the common lore according to which for
a given number of flavors $N_f$ the only vertex appearing in the
chiral limit contains $2N_f$ quark operators. We can reproduce this
result for $N_f=2$ if only $\varepsilon^i_{00}$ is kept nonzero,
while for $N_f=3$ this conjecture does not suffice and we get
additional $4q$ terms.

Consider now
 $\varepsilon^i_{00}$ using (8). In the chiral limit the operator
$S_0\hat A^{(i)}S_0$ is chirally odd while instanton zero mode has
definite chirality  and therefore $\varepsilon^i_{00}$ vanishes, for
$m_f\ne0$ one has

\be
\varepsilon^i_{00}=O(m_f),\qquad m_f\to0.
\ee
At the same time nonzero modes $u^i_{mn}$ do not have definite
chirality and hence matrix elements $\varepsilon^i_{mn}$ do not
vanish as $m_f\to0$. Thus
ZMA in the naive sense of dominance of zero-mode  terms in the EQL is
not supported by our calculations. In the next section we discuss
what it means in terms of quark propagator.

3. Now we turn to the quark propagator, expressing it again through
$\varepsilon^i_{mn}$. Inverting (6) one finds

\be
S=S_0-\sum_{ijmn}|u^i_m>\left(\frac{1}{\hat\varepsilon^{-1}+\hat
V}\right)^{ij}_{mn}<u^j_n|,
\ee
where $(\hat\varepsilon)^{ij}_{mn}=\delta_{ij}\varepsilon^i_{mn}$,
and

\be
(\hat V)^{ij}_{mn}=<u^i_m|S^{-1}_0|u^j_n>.
\ee
Note that summation in (14) extends over different instantons and
hence over $u^i_0$ and $u^j_0$ of different chiralities. Equation
(14) has to be compared to the following expression common to most
papers on the subject [3,5,6]

\be
S=S_0-\sum_{i,j}|u^i_0>\left(\frac{1}{2im+V}\right)^{ij}_{00}<u^j_0|,
\ee
which contains only zero modes contributions. To derive (16) one
starts with the following approximation for the quark propagator in a
single instanton field [3,5]:

\be
S_i=(-i\hat\partial)^{-1}+\frac{|u^i_0><u^i_0|}{-im}.
\ee
Introducing this ansatz into expression (7) for $\varepsilon^i_{mn}$
we get

\be
\varepsilon^i_{00}=\frac{1}{2im},\qquad
\varepsilon^i_{m>0,n>0}=0.
\ee
Using this form of $\hat\varepsilon$ in (14) one recovers the
standard ZMA (16). Now, comparing (18) to (13) we conclude that
ansatz (17) is unjustified. Actually when
$\varepsilon^i_{00}$ vanishes in the chiral limit in line with (13),
the propagator (14) still contains terms $|u^i_0><u^j_0|$, but with
coefficients depending upon higher modes contributions $V^{ij}_{mn}$.
This can be seen expanding (14) in series in powers of $\varepsilon$,
i.e.

\be
S=S_0-\sum_{i,j,m,n}|u^i_m>(\hat\varepsilon-
\hat\varepsilon\hat V\hat\varepsilon+
\hat\varepsilon\hat V\hat\varepsilon\hat V\hat\varepsilon-
\ldots)^{ij}_{mn}<u^j_n|.
\ee
If one neglects nonzero modes in $V_{mn}$ in (19), then the
coefficient of $|u^i_0><u^j_0|$ automatically vanishes.
To make contact with popular instantonic technique [3,5,6] where only
zero modes  are kept in the quark wave  functions of instantons $I $
and anti-instantons $\bar{I}$, we rearrange the series for the quark
propagator and partition function, using the
relation $\hat{\varepsilon}\hat{V}=S_0\hat A$ and separate out the
terms containing the overlap of $I\bar I$ zero modes.

In the standard ZMA these terms are  assumed to be dominant while
the overlaps of nonzero modes are neglected.  Our expression (19)
 includes both types of contributions and
 does not show zero--mode  dominance. Therefore we propose to study
 the new EQL derived above and calculate physical quantities like
 chiral quark mass and chiral condensate in order to estimate the
 contribution of nonzero modes.



It is worth noting that the  consistency of the approximation (17) was
questioned in [9] in connection to the calculation of the two--point
correlation function. It was shown in [9] that it is absolutely
necessary to keep the order $\sim m$ terms in $S_i$. However since
for massive fermions the single  instanton propagator $S_i$ is not
explicitly known the effects of higher modes and finite mass have
been investigated only numerically [11].

Finally let us see the effect of nonzero modes in the quark partition
function, which is obtained from (3) integrating first over quark fields.
Using (6) for $S^{-1}$  one easily  obtains

\be
Z/Z_0=\prod^{N_f}_{f=1}det(1+\hat\varepsilon\hat V),
\ee
where $\hat\varepsilon$ and $\hat V$ are the same matrices as in (14), (15).
 We may now repeat arguments presented after (19) to demonstrate the
presence  of nonzero modes and the  absence of zero-modes dominance.



4. One may wonder, why ZMA i.e. keeping only zero--modes in EQL might
 be  invalid even though phenomenologically it
looks like giving reasonable results [5-6, 12]. One of the reasons might be
that $\varepsilon^i_{00}\equiv\varepsilon$ has been treated as a parameter
connected to the properties of the instanton vacuum via the relation
$\varepsilon\sim(\frac{N_cV}{N\rho^2})^{1/2}$ , while the properties
of the vacuum have been in turn adjusted to the correct value of the
gluon condensate.

Our results are at first sight in contradiction to the Banks--Casher relation
[13] which connects the chiral condensate with the density of global (quasi)
zero modes. The standard picture suggests that the later originate from
individual zero modes, and hence would disappear as soon as (13) holds.
However here the standard picture may fail. An insight into its
possible
failure is provided by quantum mechanics of collective levels in $N$
potential wells in $4d$. If each of the well has one loosely bound
level and continuum (equivalent to zero mode and  nonzero modes),
then the approximation of keeping only the bound state poles in the
Green's functions of each well is known to give an  inadequate
description of collective bound states [14].  More than that, the
pole approximation is a poor one even for the Green's function of the
individual well, and instead the so--called unitary pole
approximation (UPA) has to be used [15].

5. To summarize , we have outlined the new approach to the effective
theory of quarks in the instanton vacuum. Our EQL is similar to that
of NJL [16], namely it starts from $4q$ term which might play an
important role in phenomenology. Analogy to NJL model calls for
construction of gap equation yielding chiral quark  mass and quark
condensate. Also, the bosonization procedure has to be
performed yielding the effective chiral Lagrangian for
Nambu--Goldstone modes. Finally, the low  density limit  deserves a
special discussion. This program is in progress now and will be
reported elsewhere.

The authors are thankful to S.V.Bashinsky, Yu.M.Makeenko,
V.A.Novikov, V.A.Rubakov and A.V.Smilga  for useful discussions.
The work was supported by INTAS Grant 94--2851 and by the Russian
Fund for Fundamental Research Grant 96--02--19184a.\\

\cen{\bf\un{\hspace{5cm}}}
\vspace{0.3cm}

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\end{document}

