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\begin{document}

\title{Exact Results Versus Anomalies}

\tocauthor{P.~I.~Pronin and K.~V.~Stepanyantz}
\author{P.~I.~Pronin and K.~V.~Stepanyantz\\
\small\em Moscow State University, Physics Faculty,\\
\small\em Department of Theoretical Physics.\\
\small\em $117234$, Moscow, Russian Federation}
\date{}

\maketitle

%\begin{center}
%\end{center}


\begin{abstract}
We investigate nonperturbative effects in the N=1 SUSY Yang-Mills
theory with matter using a relation between perturbative and exact
anomalies as a starting point. We show, that the superpotential is
always generated by instantons and should contain gluino condensate.
The exact expression for the superpotential is found.
\end{abstract}

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\section{Introduction}
\hspace{\parindent}

One of the most interesting QFT problem is the investigation of
nonperturbative dynamics. It is well known \cite{thooft,shifman}, that
except for perturbative corrections there is a series of instanton
contributions. The first exact nonperturbative result was found in
\cite{seiberg} for N=2 Yang-Mills theory in the constant field limit.
It was checked, that its assimptotic reproduces one-instanton contribution
correctly \cite{finnell,yung}.

Attempts to construct exact results for N=1 SUSY theories were made in
\cite{affleck,seiberg2}. However, the corresponding results can not be
considered as satisfactory, because for $N_f<N_c-1$ they contradict to
instanton calculations \cite{affleck}. For $N_f\ge N_c$ the prove of
agreement is not beautiful and natural, for $N_f=N_c$ again the
Lagrange multiplier arises unnaturally. Moreover, Affleck-Dine-Seiberg
results do not produce correct expressions for anomalies \cite{hep}.

In principle, at the perturbative level the anomalies is correctly
reproduced by the Veneziano-Yankielowitch effective Lagerangian
\cite{veneziano}, containing gluino condensate. However, this result
is not applicable beyond the frames of perturbation theory.

The purpose of the present paper is to obtain true expression for the
nonperturbative effective Lagrangian of N=1 SUSY Yang-Mills theory with
matter. It should correctly produce anomalies, be in agreement with
instanton calculations and have a structure at least similar to the
one for N=2 case (because N=2 SUSY theories have also N=1 SUSY).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Supersymmetric Yang-Mills theories}
\label{susy}
\hspace{\parindent}

The massless N=1 supersymmetric Yang-Mills theory with $SU(N_c)$ gauge group
and $N_f$ matter multiplets is described by the action

\begin{equation}\label{N1_action}
S=\frac{1}{16\pi} \mbox{tr\ Im}\left(\!\tau\! \int\! d^4x d^2\theta\ W^2\right)
+\frac{1}{4}\! \int\! d^4x d^4\theta \sum\limits_{A=1}^{N_f}
\left(\phi^{+}_A e^{-2V}\phi^A + \tilde\phi^{+A} e^{2V} \tilde \phi_A
\right)
\end{equation}

\noindent
where the matter superfields $\phi$ and $\tilde\phi$ belong to fundamental
and antifundamental representations of the gauge group $SU(N_c)$.

In the massless case the action is invariant under the transformations

\begin{eqnarray}
U(1)_x: &&W(\theta) \to e^{i\alpha} W(e^{-i\alpha\gamma_5}\theta);\nonumber\\
&&\phi(\theta)\to e^{ix\alpha} \phi(e^{-i\alpha\gamma_5}\theta);\qquad
\tilde\phi(\theta)\to e^{ix\alpha} \tilde\phi(e^{-i\alpha\gamma_5}\theta).
\end{eqnarray}

\noindent
where $x$ is an arbitrary constant.

It is easy to see, that this symmetry produces chiral transformations
for fermions. In the perturbation theory the anomaly of correspondent
current is

\begin{eqnarray}\label{pertanomaly}
&&\partial_\mu J^\mu_x = \Big(-N_f + N_c + x N_f\Big) \frac{1}{16\pi^2}
\varepsilon^{\mu\nu\alpha\beta} \mbox{tr} F_{\mu\nu} F_{\alpha\beta}
\nonumber\\
&&\qquad\qquad\qquad\qquad
= \Big(N_f - N_c - x N_f\Big) \frac{1}{16\pi^2}
\mbox{Im\ tr}\int d^2\theta\ W^2 .
\end{eqnarray}

\noindent
In particular, for $x=(N_f-N_c)/N_f$ we obtain anomaly free
$U(1)_R$-symmetry.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Instanton contributions to anomalies and the structure of
the effective potential}\label{instanton}
\hspace{\parindent}

Now let us find the general structure of instanton corrections to
superpotential and anomalies.

The effective Lagrangian can be split into the following parts
\cite{veneziano}

\begin{equation}
L_{eff} = L_{k} +  L_{a} + L_m,
\end{equation}

\noindent
where

\begin{eqnarray}
&& L_k = \int d^4\theta\ K(S, S^{*},\Phi, \Phi^{*}); \nonumber\\
&& L_a = \mbox{Re} \int d^2\theta\ w(S,\Phi)
\end{eqnarray}

\noindent
and $S\equiv\mbox{tr} W^2$

Here $L_k$ denotes kinetic terms, that do not contribute to the anomaly,
$L_{a}$ is a holomorphic part of the superpotential and $L_m$ is a mass
term. Below we will consider only massless case ($L_m=0$). Therefore,
the only nontrivial contribution to anomalies comes from $L_a$ and it is
the only part, that we are able to investigate. (Our method can not give
any information about a possible kinetic term.)

Performing $U(1)_x$ transformation in the effective action we obtain

\begin{equation}\label{N1anomaly1}
\langle\partial_\mu J^\mu_x\rangle =-\frac{\partial\Gamma}{\partial\alpha}
=- \mbox{Im} \int d^2\theta \Big(2w-2\frac{\partial w}{\partial S} S
- x \frac{\partial w}{\partial v} v\Big),
\end{equation}

\noindent
where we substituted $\phi$ and $\tilde\phi$ by their vacuum expectation
values $v$.

On the other hand, the anomaly can be found from the transformation law of
the collective coordinate measure. Comparing the result with \ref{N1anomaly1}
we obtain \cite{hep}

\begin{equation}\label{superpotential0}
w^{(n)}
=  W^2 g_n\left(\frac{v^3}{W^2}\right)
\left(\frac{\Lambda}{v}\right)^{n(3N_c-N_f)}
=  S g_n\left(\frac{v^3}{S}\right)
\left(\frac{\Lambda}{v}\right)^{n(3N_c-N_f)}
\end{equation}

\noindent
where $g_n$ is an arbitrary function. Its explicit form will be found below
>from the relation between perturbative and exact anomalies.

Of course, (\ref{superpotential0}) is in a complete agreement with
dimensional arguments and does not depend on the particular choice of
symmetry (i.e. $x$).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The relation between perturbative and exact anomalies}
\label{rel}
\hspace{\parindent}

So, there are nonperturbative corrections to anomalies. Nevertheless,
there is a way to treat nonperturbative expressions, anomalies, in
particular. It is exact results. For example, starting with the
Seiberg-Witten expressions \cite{seiberg} it is possible to find R-anomaly
\cite{matone,howe}. The result is the following: Exact anomaly is the
vacuum expectation value of the perturbative one. This result is not
unexpected. Really, performing, for example, chiral transformation in the
generating functional we have

\begin{eqnarray}
&&0 = \left.\frac{1}{Z} \frac{\delta Z}{\delta\alpha}\right|_{\alpha=0}
=\left.\frac{1}{Z} \frac{\delta}{\delta\alpha} \int DA D\bar\psi' D\psi'
\mbox{exp} \Big(i S - \partial_\mu\alpha j^\mu_5 \Big)\right|_{\alpha=0}
\nonumber\\
&&=\left.\frac{1}{Z} \frac{\delta}{\delta\alpha} \int DA D\bar\psi D\psi
\mbox{exp} \Big(i S - \partial_\mu\alpha j^\mu_5 - \alpha\mbox{\bf A}\Big)
\right|_{\alpha=0}\!\!\!\!
=\langle  \partial_\mu j^\mu_5 - \mbox{\bf A}  \rangle,
\end{eqnarray}

\noindent
where {\bf A} denotes the perturbative anomaly, produced by the measure
noninvariance \cite{fujikava}. Finally

\begin{equation}\label{relation}
\langle \partial_\mu j^\mu_5\rangle = \langle\mbox{\bf A}  \rangle.
\end{equation}

\noindent
(R-transformation are considered similarly).

Now, let us note, that the derivation of, say, \cite{matone} essentially
use the form of exact results. Therefore, we are tempted to reverse the
arguments and try to obtain exact results from anomalies. Here we will
consider only the most interesting case of N=1 supersymmetric theories.
The detailed description of the method and some simple examples can be found
in \cite{hep}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\section{Effective Lagrangian for N=1 supersymmetric theories.
General structure}\label{general_structure}
\hspace{\parindent}

Let us apply (\ref{relation}) to N=1 supersymmetric $SU(N_c)$ Yang-Mills
theory with $N_f$ matter supermultiplets. The vacuum expectation value
of the perturbative anomaly (\ref{pertanomaly}) is given by

\begin{eqnarray}\label{N1anomaly}
\langle \partial_\mu J^\mu_x\rangle
= \Big(N_f - N_c - x N_f\Big) \frac{1}{16\pi^2}
\mbox{Im}\int d^2\theta\ u.
\end{eqnarray}

\noindent
where $u\equiv \langle \mbox{tr} W^2\rangle$. Therefore, the effective
Lagrangian should depend in particular on $S= \mbox{tr} W^2$. This
result is not new. At the perturbative level the similar investigation
was made in \cite{veneziano}. However, in this paper we do not intend to
restrict ourselves by the frames of perturbation theory. Therefore, we
can not assume, that $u = \mbox{tr} W^2$.

Comparing (\ref{N1anomaly}) with (\ref{N1anomaly1}) and taking into account,
that the equality should be satisfied for all $x$, we obtain

\begin{eqnarray}\label{2anomaly}
&&2w-2\frac{\partial w}{\partial S}S = - \frac{1}{16\pi^2}(N_f-N_c) u;
\nonumber\\
&&\frac{\partial w}{\partial v} v = - \frac{1}{16\pi^2} N_f u.
\end{eqnarray}

\noindent
This equation is very similar to the results of \cite{veneziano}.
Nevertheless, there is a crucial difference: $u\ne S$. Therefore,
one can only conclude that

\begin{equation}\label{ranom}
2w -2 \frac{\partial w}{\partial S} S
- \frac{N_f-N_c}{N_f}\frac{\partial w}{\partial v} = 0.
\end{equation}

This equation corresponds to the exact conservation of R-symmetry
at nonperturbative level. The similar condition was used in
\cite{affleck,seiberg2}, although the dependence $w=w(S)$ was ignored.
Of course, it is quite clear, that integrating out $S$ yields ADS
superpotential \cite{intriligator} and corresponds to imposing the
condition

\begin{equation}
\frac{\partial w}{\partial S} = 0.
\end{equation}

\noindent
Nevertheless, the presence of $S$-field is a strict consequence of
(\ref{2anomaly}) and integrating it out should be propely substantiated
(if it is possible).

The solution of (\ref{ranom}) should agree with instanton calculations.
It is easily verified, that the only solution of (\ref{ranom}), agreeing
with (\ref{superpotential0})\footnote{It corresponds to
${\displaystyle g_n(x) = \frac{1}{32\pi^2} c_n x^{n(N_c-N_f)},
\quad n\ge 1}$
in (\ref{superpotential0}).}, is

\begin{equation}\label{solution}
w = \frac{1}{32\pi^2} S f(z);\qquad
f(z) = f_{pert}(z) + \sum\limits_{n=1}^\infty c_n z^n,
\end{equation}

\noindent
where

\begin{equation}
z\equiv\frac{\Lambda^{3N_c-N_f}}{v^{2N_f} S^{N_c-N_f}}
\end{equation}

\noindent
is a dimensionless parameter.

\noindent

In the final result $z$ should be written in terms of gauge invariant
variables. The derivation, made in the \cite{hep}, gives

\begin{eqnarray}
&&\hspace{-7mm}
z=\frac{\Lambda^{3N_c-N_f}}{\mbox{det} M\ S^{N_c-N_f}}, \qquad N_f<N_c;
\nonumber\\
&&\hspace{-7mm}
z=\frac{\Lambda^{3N_c-N_f}S^{N_f-N_c}}{
\mbox{det} M - (\tilde B^{A_1 A_2\ldots A_{N_f-N_c}} M_{A_1}\!\!{}^{B_1}
M_{A_2}\!\!{}^{B_2}\!\! \ldots M_{A_{N_f-N_c}}\!\!\!\!{}^{B_{N_f-N_c}}
B_{B_1 B_2\ldots B_{N_f-N_c}})},\nonumber\\
&&\hspace{85mm}
N_f\ge N_c.
\end{eqnarray}

In order to define $f_{pert}$ we note, that at the perturbative level $u=S$.
Therefore, in this case (\ref{2anomaly}) gives

\begin{equation}\label{fpert}
\frac{\partial f_{pert}}{\partial z} z = 1,
\end{equation}

\noindent
so that

\begin{equation}\label{expansion}
w = \frac{1}{32\pi^2} S \Big(\ln z +
\sum\limits_{n=1}^\infty c_n z^n\Big).
\end{equation}

\noindent
Substituting it to (\ref{2anomaly}), we obtain

\begin{equation}\label{u_expansion}
u = S\Big(1 + \sum\limits_{n=1}^\infty n c_n z^n \Big),
\end{equation}

\noindent
that defines all anomalies in the theory according to (\ref{N1anomaly}).
At the perturbative level both (\ref{expansion}) and (\ref{u_expansion})
are certainly in agreement with \cite{veneziano}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Effective Lagrangian for N=1 supersymmetric theories.
Exact result}\label{exact_result}
\hspace{\parindent}

Let us try to define $f$ exactly. The general structure of the holomorphic
superpotential, found in section \ref{general_structure}, is similar to the
structure of the nonperturbative prepotential in the N=2 supersymmetric
Yang-Mills theory \cite{seiberg1}. In the latter case the relation between
perturbative and nonperturbative anomalies leads to Picard-Fuchs equations
\cite{hep}, that can be used for derivation of exact results. Is it possible
to extend this approach to the case of N=1 supersymmetry?

First we substitute (\ref{solution}) into (\ref{2anomaly}), that gives

\begin{equation}\label{forf}
S \frac{df}{dz} z = u
\end{equation}

\noindent
(and therefore $u/S$ depends only on $z$).

The way to solve this equation is indicated by the analogy with N=2
supersymmetric $SU(2)$ Yang-Mills theory. In terms of N=1 superfields its
action is written as

\begin{equation}\label{N2versusN1}
\frac{1}{16\pi} \mbox{Im} \int d^4x d^2\theta \left(
\frac{d^2 F}{d\phi^2}W^2
+ \frac{1}{2}\int d^2\bar\theta \frac{dF}{d\phi}\phi^{+}\right).
\end{equation}

Let us compare it with

\begin{equation}\label{Sa}
\Gamma_a = \frac{1}{32\pi^2} \mbox{Re}\int d^4x d^2\theta\ S f(z)
\end{equation}

\noindent
and introduce $a\equiv z^{- 1/4}$ (this choice of the power will be
explained below). The first term in (\ref{N2versusN1}) will coincide
with (\ref{Sa}) if

\begin{equation}\label{new}
- 2\pi i \frac{d^2F}{da^2}\equiv f;
\qquad \frac{d^2U}{da^2}\equiv \frac{u}{S}.
\end{equation}

\noindent
Then (\ref{forf}) takes the form

\begin{equation}
F+F_D=\frac{2i}{\pi} U,
\end{equation}

\noindent
where

\begin{equation}
F_D=F-a a_D;\qquad a_D=\frac{dF}{da}.
\end{equation}

This equation immediately leads to Picard-Fuchs equation \cite{hep}

\begin{equation}\label{pf}
\left(\frac{d^2}{da^2}+ L(U)\right)\left(\begin{array}{c}a\\a_D\end{array}
\right)=0,
\end{equation}

\noindent
where $L(U)$ is a function depending on the structure of singularities
in the $U$-plane. At the perturbative level

\begin{equation}
L(U)=\frac{1}{4 U^2}
\end{equation}

\noindent
but the perturbative solution does not satisfy the requirement
\cite{seiberg,bilal}

\begin{equation}
\mbox{Im}\ \tau > 0,\qquad \mbox{where} \qquad
\tau = \frac{d^2F}{da^2} = \frac{da_D}{da} =\frac{i}{2\pi} f,
\end{equation}

\noindent
that is derived exactly as in the N=2 case. Therefore, two singularities
(at $U=0$ and $U=\infty$) are impossible.

To find the structure of singularities let us note, that the solution
(\ref{expansion}) should contain all positive powers of z and, therefore,
is invariant under $Z_4$ transformations $a \to e^{i\pi k/2} a$. Taking into
account (\ref{new}) and (\ref{u_expansion}) we conclude, that the
corresponding transformations in the $U$-plane are $U \to e^{i\pi k} U$.
Thus, the singularities should come in pairs: for each singularity at
$U=U_0$ there is another one at $U=-U_0$.

Therefore, the considered model is completely equivalent to N=2
supersymmetric SU(2) Yang-Mills theory without matter and the only
possible form of Picard-Fuchs equation (up to the redefinition of
$\Lambda$) is

\begin{equation}\label{N1pf}
\left(\frac{d^2}{dU^2}+\frac{1}{4 (U^2-1)} \right)
\left(\begin{array}{c}a\\a_D\end{array}\right) = 0
\end{equation}

\noindent
with the solution \cite{seiberg,bilal}

\begin{eqnarray}\label{final}
&&a(U)=\frac{\sqrt{2}}{\pi}\int\limits_{-1}^{1} dx
\frac{\sqrt{x-U}}{\sqrt{x^2-1}};
\nonumber\\
&&a_D(U)=\frac{\sqrt{2}}{\pi}\int\limits_{1}^{U} dx
\frac{\sqrt{x-U}}{\sqrt{x^2-1}}.
\end{eqnarray}

\noindent
Its uniqueness and, therefore, the uniqueness of the choice (\ref{N1pf})
was proven in \cite{unique}.

The function $F$ can be found by

\begin{equation}
\frac{dF}{dU} = a_D \frac{da}{dU}.
\end{equation}

\noindent
Its general structure is well known to be

\begin{equation}
F = - \frac{i}{\pi} a^2\Big(\ln a
+ \sum\limits_{n=0}^\infty F_n a^{-4n}\Big),
\end{equation}

\noindent
so that

\begin{equation}
f = -2\pi i \frac{d^2 F}{da^2} =
-4\ln a + \sum\limits_{n=0}^\infty f_n a^{-4n} =  \ln z +
\sum\limits_{n=0}^\infty f_n z^n.
\end{equation}

And now it is quite clear, that the choice $a = z^{-1/4}$ was made to
obtain the true structure of instanton corrections (\ref{solution}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion.}
\hspace{\parindent}

We present a new method for obtaining exact results using the relation
between perturbative and exact anomalies as a starting point. Its
application to N=1 SUSY theories allowed to solve the long standed
problem of agreement between instanton calculations and exact results.
(The disagreement was present for $N_c > N_f-1$). The proposed approach
authomatically leads to the presence of gluino condensate, that was
ignored in all papers devoted both to exact results and instanton
calculations in N=1 SUSY theories. It is this dependence, that gives
the required agreement. It is proved by the investigation of the
collective coordinate measure tranformation law.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}

