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\begin{document}
%\Tuesday, July 2, 2002 at 23:57
\vspace*{2cm}
\setcounter{footnote}{1}
\def\thefootnote{\fnsymbol{footnote}}
\begin{center}
\hfill ITEP-TH-88/02\\
%\hfill hep-th/0303200\\
\vspace{0.4in}
{\LARGE\bfseries Comments on BPS Bound State Decay}
\end{center}
\vspace{0.2in}
\centerline{{\large Anatoly Dymarsky\footnote{
dymarsky@gate.itep.ru
}
and Dmitry Melnikov\footnote{
%location\\
melnikov@gate.itep.ru}}
}
\vspace{0.3in}
\begin{center}
\large
\it Department of Physics at Moscow State University
\\ Vorobjevy Gory, 119899 Moscow, Russia
\vspace{0.3in}
\\ Institute for Theoretical and Experimental Physics
\\ B.Cheremushkinskaya 25, 117259 Moscow, Russia\\
\vspace{0.3in}
\rm March, 2003
\end{center}



\bigskip

\begin{abstract}
{\footnotesize
In N=2 SYM theory with matter non-conservation of the S-charge of a BPS state under
monodromies leads to so-called BPS
state ``decay". A mechanism for such a behavior on the semiclassical level could be
established through consideration of soliton-fermion bound state. Solutions
to classical equations of motion allow to observe the BPS state ``decay" \it in
vivo.}
\end{abstract}

\begin{center}
\rule{5cm}{1pt}
\end{center}



\section{Introduction}
In the work of Seiberg and Witten \cite{SW2} concerning N=2 SYM with matter in
fundamental representation of SU(2) the following formula for
central charge is established:
\be
\label{Z formula}
Z=n_ma_D+n_ea+\sum\limits_{i=1}^{N_f}\frac{m_iS_i}{\sqrt{2}}
\ee
where besides the magnetic and electric charges an abelian global charge appears.
This charge, called the S-charge or the spectral asymmetry, arises due to global $SU(N_f)$
symmetry which is generally broken down to $U(1)^{N_f}$ by the hypermultiplet mass term in the
action. The S-charge is carried only by the fields from hypermultiplet and
on equal footing with $n_e$ and $n_m$ it parameterizes the BPS spectrum
of the N=2 theory.

The moduli space possesses a certain number of singularities. In this paper
we are going to consider those singularities that correspond to massless
quarks. From the central charge formula (\ref{Z formula}) it follows that
$Z$ vanishes for the states with
$$n_m=0, \qquad n_e=\pm 1, \qquad S=\pm 1 $$
at a special point $a_0=\pm m/\sqrt{2}$ on the moduli space.

The complex coupling $\tau$ has a logarithmic singularity at this point
\be
\label{tau sing}
\tau(a)=-\frac{i}{2\pi}ln(a-a_0),
\\
\\ \ds a_D=c-\frac{i}{2\pi}(a-a_0)ln(a-a_0).
\ee
Considering $a$ and $a_D$ as the functions on the complex $a$ plane and
investigating their monodromies around the point $a_0$ we find
$$a\rightarrow a, \qquad a_D\rightarrow a+a_D-\frac{m}{\sqrt{2}}.$$
Since the central charge should be invariant, the charges of a quantum
state consisted of a monopole of magnetic charge $n_m$ and quark with
charges $n_e$ and $S$ should be shifted by this rotation:
$$ (n_m,n_e,S)\rightarrow (n_m,n_e-n_m,S+n_m)$$
By multiple action of the monodromy in the presence of a magnetic charge
$n_m\neq 0$ in principle we could make the value of the S-charge arbitrarily
large.

On the other hand for the quark-monopole bound state the value of the S-charge is
bounded, since the only fields carrying S-charge are fermion modes, that obey
the Pauli exclusive principle. Therefore starting from one particle state and
encircling the singularity we come to two particle state.

In the limit $m\gg \Lambda$ the singularity becomes semiclassical and hence
the simple dynamical analysis of the theory near the singularity is appropriate.
Below we consider the fermion equations
of motion in the background of monopole fields. We find explicitly a
normalizable fermion mode corresponding to the bound state, which carries
S-charge. It is from the analysis of the properties of the classical solution
that the mechanism of the BPS state decay becomes explicit.

Somewhat similar considerations were made by Heningson in \cite{HEN}, where
the conditions for the existence of normalizable solution were investigated.
The latter result is consistent with simple analysis of Marginal Stability
from BPS Mass Formula performed in \cite{VainsteinR}.

In this paper we rederive the result of \cite{HEN} and present an obvious
picture which describes classically the phenomenon of the BPS state decay.

%-------------------------------------------------------------------------------

\section{Derivation of the fermion mode}
\subsection{Solution to Dirac equation}



Consider $N=2$ SYM theory with $SU(2)$ gauge group and matter hypermultiplets
in fundamental representation.\footnote{Here we restrict ourselves
to the case of $N_f=1$. For more detailed review of $N=2$ actions see for
example \cite{reviews}.}
This theory contains monopoles in its spectrum. There are also solutions
that correspond to matter fermions in the external monopole field. The
question is whether these solutions can be normalizable. In other words,
whether there exist soliton-fermion bound states to which we refer as to one
particle states.
As we shall see below, such normalizable
solutions exist only in some region of the moduli space. The
point of singularity for the massless quark happens to belong to the boundary of this
region, providing a natural explanation for the bound state decay described
above.

The fermionic components of the hypermultiplet satisfy the Dirac equation
\be
\label{Dirac eq}
i\gm D_\mu \Psi - (\hat{m}+\sqrt{2}\hat{A})\Psi = 0,
\\
\\ \hat{m}+\sqrt{2}\hat{A}=\Re e(m+\sqrt{2}A) +i\gamma^5\Im m(m+\sqrt{2}A).
\ee
where the Dirac spinor $\Psi$ is composed of two Weyl spinors as follows:
\be
\label{Psi}
\Psi=\left(
\begin{array}{cc}
 \pq
\\ \bar{\pt}
\end{array}
\right).
\ee
These two Weyl spinors belong to fundamental and antifundamental N=1 chiral multiplets.

Working in the Gauss gauge $A_0=0$, substitute $D_0$ by the energy
eigenvalue $iE$. Now notice that the hamiltonian commutes with the operator
$\Gamma=\gamma^0 \gamma^5$ in the case $\Im m(m)+\sqrt{2}\Im m(A)=0$. Thus general
solution to (\ref{Dirac eq}) could be found as the sum of $\Gamma$
eigenfunctions $\Psi=\Psi^+ +\Psi^ -$.

Substituting $\Psi^+$ instead of $\Psi$ we split the equation (\ref{Dirac eq}):
\be
\label{chiral eq}
\bigl[\sigma^k D_k -\Re e(m+\sqrt{2}A)\bigr] \psi^+=0,
\\
\\ \bigl[ E - \Im m(m+\sqrt{2}A)\bigr] \psi^+ =0.
\ee
Here $\psi^+$ is a chiral Weyl component of $\Psi$.

In the consideration above we treat $m$ as a real constant in contrast with
scalar field $A$ which is generally complex. However it would be more
useful for the future computations to keep $A$ real. Since there is a $U(1)_R$
symmetry acting on the fields, we could choose such a U(1) rotation
\be
A\rightarrow e^{2i\alpha} A,
\\
\\ m\rightarrow e^{2i\alpha} m,
\\
\\ \Psi\rightarrow e^{-i\alpha\gamma^5}\Psi,
\ee
under which complexity of $A$ flows into complexity of $m$. Although this
symmetry is in general anomalous, this is not the case in the semiclassical region.
Then solution to the second equation of (\ref{chiral eq}) implies that
the bound-state mass spectrum satisfies
$$E= \Im m({\cal{M}}),$$
where ${\cal{M}}$ now defines a complex mass parameter.

In the case of real scalar field $A$ we could use the standard radially symmetric
solution for 't~Hooft-Polyakov monopole \cite{'tHooft:1974qc}
in the BPS limit\footnote{We neglect the back
reaction of matter fields to monopole fields assuming that in semiclassical limit the
ratio of corresponding masses is small.}:
\be
\label{BPS mon}
\sqrt{2}A^i=an^i(1-F(r)),
\\
\\ \ds A^a_i=\epsilon^{aij}\frac{n^j}{r}(1-H(r)),
\ee
where $n^i$ is a unit vector, $F$ and $H$ are known functions of radius
$r$. For future convenience we chose the normalization of the scalar field $A$
in (\ref{BPS mon}) slightly different from that in \cite{SW2}. In our
conventions it has asymptotic $A\rightarrow a/\sqrt{2}$.
Furthermore Seiberg and Witten renormalize the charges of the BPS states in
such a way that $a\rightarrow a/2$. So our $a$ will be different from
that of Seiberg and Witten by the factor of $(2\sqrt{2})^{-1}$.


Substituting (\ref{BPS mon}) into (\ref{chiral eq}) we find a solution in
the form
\be
\label{solution}
(\psi^+)^{\alpha}_{~a}=\delta^{\alpha}_{~a}\chi_0\xi + n^i
(\sigma^i)^\alpha_{~\beta} \delta^{\beta}_{~a}\eta_0\zeta,
%\\ (\pbq)^{\alpha}_{~a}= -i(\pq)^{\alpha}_{~a} =-i\epsilon^{\alpha}_{~a}
%\chi_0\xi -i n^i (\sigma^i)^\alpha_{~\beta} \epsilon^{\beta}_{~a}\eta_0\zeta.
\ee
where $\alpha$ and $a$ are spinor and color indices respectively.
Functions $\chi_0$ and $\eta_0$, solving the equations at zero $m$
are defined as follows:\footnote{Here we define $\rho=\frac{r}{r_0},
\quad r_0^{-1}=a$.}
\be
\label{sol homog}
\chi_0 = \frac{1}{\sqrt{\rho {\rm sh}\rho}}{\rm th}\frac{\rho}{2},
\\
\\ \ds
\eta_0 = \frac{1}{\sqrt{\rho {\rm sh}\rho}}{\rm cth}\frac{\rho}{2}.
\ee
Functions $\xi$ and $\zeta$ satisfy the system of the first order differential
equations
\be
\label{xieq}
\xi'=r_0m\zeta {\rm cth}^2({\rho \over 2}),
\\
\\ \ds
\zeta'=r_0m\xi {\rm th}^2({\rho \over 2}),
\ee
which implies the pair of second order equations
\be
\label{der eq2}
\xi^{\prime\prime} + \frac{2\xi^\prime}{{\rm sh}\rho}-r_0^2m^2\xi=0,
\\
\\ \ds \zeta^{\prime\prime} - \frac{2\zeta^\prime}{{\rm sh}\rho}-r_0^2m^2\zeta=0.
\ee
Explicit solutions to this equations could be found in terms of
hypergeometric functions. We shall find normalizable solutions after an
analysis of asymptotic behavior of the general solutions to (\ref{chiral
eq}).

The solution presented above is one of positive chirality. It could be
shown as well that for negative chirality the similar solution could not be made
normalizable.
It is also natural to assume \cite{HEN} that there are no other discrete spectrum
solutions to (\ref{Dirac eq}).

%-----------------------------------------------------------------------------------
\subsection{Analysis of the Solution}
In this subsection we investigate the asymptotic of the solution to
(\ref{Dirac eq}) and rederive the conditions under which this solution exists
\cite{HEN}. Using the asymptotic we also derive the explicit solution in
terms of hypergeometric functions.

Introduce a pair of functions $\chi,\eta$ that will characterize the
asymptotic behavior of solution (\ref{solution}):
\be
\chi=\chi_0\xi, \qquad \eta=\eta_0 \zeta.
\ee
Since we are interested in soliton-fermion bound states,
the fermion mode we are looking for should be a normalizable one, i.e.\ it
should decrease fast enough at infinity and be regular at the origin,
namely
$$\chi(0)={\rm const},\qquad \eta(0)=0$$
In the $\rho \rightarrow 0$ limit the fermion mode behaves as the solution
to the equations
\be
\label{zero asympt1}
\xi^{\prime\prime} + \frac{2\xi^\prime}{\rho}-r_0^2m^2\xi=0,
\\
\\ \zeta^{\prime\prime} - \frac{2\zeta^\prime}{\rho}-r_0^2m^2\zeta=0.
\ee
The regular solutions to (\ref{zero asympt1}) are as follows:
\be
\label{zero asympt2}
\xi=C\frac{{\rm sh}(r_0m\rho)}{\rho},
\\
\\ \zeta = C(r_0m\rho-1)e^{r_0m\rho}+ C(r_0m\rho+1)e^{-r_0m\rho},
\ee
with the coefficients fixed by asymptotical behavior of (\ref{xieq}).

Assume that
we have a solution that is regular at the origin. From the asymptotic
(\ref{zero asympt2}) it follows that the first derivative of this solution
is zero and the second derivative is positive at this point. Furthermore,
the second derivative in exact equations (\ref{der eq2}) is positive wherever
first derivative vanishes. Therefore such solutions could have only minima
and cannot be normalizable.
This means that
solution to (\ref{der eq2}) is either regular at the origin and divergent at
infinity or regular at infinity and divergent at the origin. However if
we look at (\ref{sol homog}), we shall see that under certain conditions
we could construct a normalizable zero-mode. The regular at the origin and
increasing at infinity solution to (\ref{der eq2}) has the asymptotic
\be
\label{inf asympt1}
\xi,\zeta \sim e^{r_0m\rho}.
\ee
It follows from (\ref{sol homog}) that the functions
$\chi_0$,$\eta_0$ have the asymptotic
\be
\label{inf asympt2}
\chi_0, \eta_0 \sim \frac{e^{-\rho}}{\sqrt{\rho }},
\ee
and that for the functions $\chi$,$\eta$
\be
\label{inf asympt3}
\chi,\eta \sim \sqrt{\frac{1}{\rho }}e^{(r_0m-\frac{1}{2})\rho}.
\ee
We see that for the existence of normalizable zero mode the following conditions
should be satisfied:
\be
\label{norm cond}
2r_0 m < 1, \quad {\rm or} \quad \frac{2m}{a}< 1,
\ee
or $a>m/\sqrt{2}$ in the standard normalization of \cite{SW2}. This result was
originally derived in \cite{HEN}, and then was confirmed by CMS considerations
in \cite{VainsteinR} in the weak coupling limit.

Now turn to the equations (\ref{der eq2}). Changing variables from
$\rho$ to $x=({\rm ch}\rho+1)/2$ we obtain a pair of hypergeometric
equations
\be
\label{hypgeom2}
x(x-1)y^{\prime\prime}_1+(x+\frac{1}{2})y_1^\prime-r_0^2m^2y_1=0,
\\
\\ \ds x(x-1)y^{\prime\prime}_2+(x-\frac{3}{2})y_2^\prime-r_0^2m^2y_2=0.
\ee
Solving (\ref{hypgeom2}) and expressing the solution in terms of original
variable $\rho$ we find the unknown functions $\xi$ and $\zeta$:
\be
\label{sol xi,zeta}
\xi(\rho)=CF(\frac{m}{a},-\frac{m}{a},-\frac{1}{2},{\rm ch}^2\frac{\rho}{2}),
\\
\\ \ds \zeta(\rho)= \tilde{C}
F(\frac{m}{a},-\frac{m}{a},\frac{3}{2},{\rm ch}^2\frac{\rho}{2}).
\ee
Here we took into account the asymptotic of solution determined above.
The asymptotic of (\ref{xieq}) also fixes the relation $C=\tilde{C}$.

%------------------------------------------------------------------------------
\subsection{BPS State Decay}
From the considerations above we found that normalizable fermion mode with
the energy defined by complex mass parameter $E=\Im m{\cal{M}}$ exists only
in the region of moduli space defined as\footnote{In the fermion equations $a$
was treated as $vev$ of real field $A$. To define this condition for all complex
moduli plane we substitute the latter by absolute value of complex coordinate
$|a|$.}
\be
\label{norm cond2}
\Re e{\cal{M}}=m<\frac{|a|}{2}.
\ee
Now for complete understanding of the process we need one more thing, namely
to find what region of the moduli space corresponds to the solutions to Dirac
equation (\ref{Dirac eq}) from continuous spectrum, i.e.\ non-bound BPS
states. Considering asymptotic of (\ref{Dirac eq}) at infinity we obtain the
system of linear differential equations, neglecting exponentially decaying
non constant coefficients. The corresponding characteristic equation gives
for fixed energy $E$
\be
\label{contin spec}
E=\pm\left|m-\frac{a}{2}\right|.
\ee

This condition defines a cone in the $E-a$ or equivalently in the $E-{\cal{M}}$
parameter space with the singularity at $a=2m$ (Fig.1). Further we substitute the
investigation on the complex $a$ plane by that on the $\cal{M}$ plane, which
is equivalent due to $U(1)$ symmetry. Continuous spectrum
belongs to the interior of this cone. The region of discrete spectrum is
an inclined plane $E=\Im m{\cal{M}}$, which is bounded due to the condition (\ref{norm
cond2}).

\begin{figure}[]
\epsffile{figure.eps}
\caption{The region of discrete spectrum (inclined semiplane) is continuously
attached to the region of continuous spectrum (cone) by the line $E=\Im m {\cal{M}}$
which goes through the point $a=2m$.}
\end{figure}

Now a closed contour on the complex $\cal{M}$ corresponds to some 3
dimensional curve, which is restricted to belong either to inclined plane of
discrete spectrum or to the interior of the continuous spectrum cone. Since
the contour must be closed and smooth, this curve must have a spiral form.
This means that encircling the singular point we cannot return to the
initial value of energy. Furthermore the final point of a spiral curve
belongs to the state from continuous spectrum, which means a decay of the
initial bound state.

To clarify what happens when we encircle the singularity consider the
dependence of energy $E$ on the angle of rotation $\varphi$ in the
$\cal{M}$ plane (Fig.2). We start at some real mass value $m_0$ and hence at zero
energy.
At $\varphi=\pi/2$ we reach the continuous spectrum. For $\varphi$
from $\pi/2$ to $3\pi/2$ there is no any bound state, i.e. the
initial bound state has decayed. After a full rotation $\varphi=2\pi$ the
mode from continuous spectrum cannot descend to discrete spectrum and the only
allowed values of energy will be $E\geq |m_0-a/2|$. However if we
move in the opposite direction starting from $\varphi=2\pi$ we shall also come
to continuous spectrum at $\varphi=3\pi/2$, but in the part of it which
is below zero level. Since we move adiabatically and all the states for fermion below
zero are occupied we conclude that after encircling the singularity one fermion
mode runs to continuous spectrum and another one comes from the Dirac sea below.

\begin{figure}[h]
\label{Fig2}

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\put(50,50){\makebox(0,0)[cc]{E}}

\emline{40}{45.00}{1}{120}{45}{2}
\emline{40}{5.00}{3}{120}{5}{4}
\emline{70}{3.00}{5}{70}{47}{6}
\emline{85}{3.00}{7}{85}{47}{8}
\emline{100}{3.00}{9}{100}{47}{10}
\emline{115}{3.00}{11}{115}{47}{12}

\put(52,28){\makebox(0,0)[cc]{0}}
\put(67,28){\makebox(0,0)[cc]{$\frac{\pi}{2}$}}
\put(82,28){\makebox(0,0)[cc]{$\pi$}}
\put(97,28){\makebox(0,0)[cc]{$\frac{3\pi}{2}$}}
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\put(55,0){\vector(0,1){50}}

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\bezier{150}(55,25)(53,18)(52,15)
\bezier{150}(118,35)(117,32)(115,25)

\end{picture}
\caption{The initial bound state decays after crossing the point of
stability
at $\varphi = \pi/2$. Another fermion mode replaces it rising from the
continuous spectrum below.
}
\end{figure}
%-----------------------------------------------------------------------------

\section{Conclusions}

Investigations of the properties of the classical solutions to (\ref{Dirac
eq}) tells us that the adiabatic motion around the singularity is severely
restricted in acceptable values of energy $E$. During this motion we
necessarily cross the Curve of Marginal Stability for bound state. For
an initial state this happens only once and that is why this state remains
finally unbounded. On the other hand another fermion line comes from the
continuous spectrum of Dirac sea below and forms a new bound state, thus
confirming the predicted shift in the S-charge
$$S\rightarrow S+n_m.$$

\vspace{0.5cm}
We are grateful to K.Selivanov for initiating this work and various support
and useful discussions at the different stages. We would also like to thank
D.Gal'tsov, A.Gorsky and V.Ch.Zhukovsky for numerous stimulating
discussions. A special thank should be said to S.Dubovsky, A.Vainshtein,
A.Solovyov and K.Zarembo
for clarifying some subtle questions. This work would not have appeared without
support of A.Alexandrov, I.Gordeli, S.Klevtsov, G.Nozadze and V.Poberezhny.
A.D. would like to thank the Carg\`ese 2002 ASI, where the part of work
was done. Analogously D.M. would like to thank for hospitality Department of Theoretical
Physics at Uppsala
University. The work was supported in part by grants  RFBR 01-02-17682, RFBR
00-02-06026, INTAS 00-334, by the Russian President's  grant 00-15-99296 (A.D.),
and by grants RFBR 01-01-00549 , INTAS 00-561, the Russian
President's  grant 00-15-99296 (D.M.).

\begin{thebibliography}{0}
%\bibitem{SW1}
%N.~Seiberg and E.~Witten,
%%%``Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,''
%%Nucl.\ Phys.\ B {\bf 431} (1994) 484
%%[arXiv:hep-th/9408099].

\bibitem{SW2}
N.~Seiberg and E.~Witten,
%``Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,''
Nucl.\ Phys.\ B {\bf 431} (1994) 484
[arXiv:hep-th/9408099].


\bibitem{HEN}
M.~Henningson,
%``Discontinuous BPS spectra in $N = 2$ gauge theory,''
Nucl.\ Phys.\ B {\bf 461}, 101 (1996)
[arXiv:hep-th/9510138].


%\bibitem{Rubakov}
%V.~Rubakov, lectures for students.

\bibitem{VainsteinR}
A.~Ritz, A.~Vainshtein,
%''Long Range Forces and Supersymmetric Bound States,''
Nucl.\ Phys.\ B {\bf 617}, 43 (2001)
[arXiv:hep-th/0102121].


\bibitem{reviews}
L.~Alvarez-Gaume and S.~F.~Hassan Fortsch.Phys. 45 (1997) 159-236
[arXiv:hep-th/9701069];
\\ A.~Bilal hep-th/0101055.

\bibitem{'tHooft:1974qc}
G.~'t Hooft,
%``Magnetic Monopoles In Unified Gauge Theories,''
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\\ A.~M.~Polyakov,
%``Particle Spectrum In Quantum Field Theory,''
JETP Lett.\  {\bf 20} (1974) 194.

%\bibitem{Goldstone}
%J.~Goldstone and F.~Wilczek,
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%Phys.\ Rev.\ Lett.\  {\bf 47} (1981) 986.
%
%\bibitem{Ferrari}
%Ferrari
%
%\bibitem{Semenoff}
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\end{thebibliography}
\end{document}
