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\begin{document}  
  
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\hbox{hep-th/0303255}}  
  
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\begin{center}
{\Large \bf D--brane --- Anti--D--brane Forces}\\\bigskip{\Large \bf in Plane Wave Backgrounds:}\\
\bigskip
{\Large \bf A Fall From Grace}
  \end{center}
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\centerline{\bf Clifford V. Johnson, Harald G. Svendsen}
  
  
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\bigskip  
  

\centerline{\it Centre  
for Particle Theory}
  \centerline{\it Department of Mathematical Sciences}  
\centerline{\it University of  
Durham}
\centerline{\it Durham, DH1 3LE, U.K.}  

\centerline{$\phantom{and}$}  

\bigskip  
 
  
\centerline{\small \tt  
  c.v.johnson@durham.ac.uk, h.g.svendsen@durham.ac.uk}  
  
\bigskip  
\bigskip  
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%\maketitle  
  
\begin{abstract}  
  \vskip 4pt We study the nature of the force between a D--brane and
  an anti-D--brane in the maximally supersymmetric plane wave
  background of type~IIB superstring theory, which is equipped with a
  mass parameter $\mu$.  An early such study in flat spacetime ($\mu=0$)
  served to sharpen intuition about D--brane interactions, showing in
  particular the key role of the ``stringy halo'' that surrounds a
  D--brane. The halo marks the edge of the region within which tachyon
  condensation occurs, opening a gateway to new non--trivial vacua of
  the theory.  It seems pertinent to study the fate of the halo for
  non--zero $\mu$.  Focusing on the simplest cases of $p=\pm1$, we
  find here that for branes located at the origin, the radius of the
  halo shrinks for increasing~$\mu$. For branes away from the origin,
  this shrinking persists, and is accompanied by a shift of the centre
  of the halo away from the D--brane. In fact, we observe that for
  large enough $\mu$, or beyond a critical distance from the origin,
  D--branes can lose their halo entirely! We suspect that the
  consequences of this physics for key notions about the dynamics,
  stability and classification of D--branes (such as tachyon
  condensation, K--theory, {\it etc.,}) may well be quite significant.

\end{abstract}  
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\section{Introduction} 

A D--brane and its ``anti--particle'', an anti--D--brane, upon
approaching each other, will annihilate. The generic product of this
annihilation process is expected to be a state of closed strings, which
carry no net R--R charge.  This expectation is supported by field
theory intuition and knowledge of which objects are the carriers of
the available conserved charges in perturbative string theory. From
experience with field theory one expects to be able to see the
beginnings of the process of annihilation {\it via} the opening up of
new decay channels at coincidence. These can be seen by studying the
amplitude for exchange of quanta between the two branes, which gives a
potential.  At small separations, the behaviour of the potential
---and the resulting exchange force that can be derived from it--- can
signal new physics. Basically, a divergence in the exchange force as
the objects are brought together can signal the opening up of a new
channel (or new channels) not included in the computation of the
amplitude away from the divergent regime.

In field theory, for  a separation $X$ of the two objects, the
divergence follows simply from the fact that the amplitude for
exchange is controlled by the position space propagator $\Delta(X)$
which (for more than two transverse directions) is divergent at $X=0$.
This is where the new channels can open up, which can include the
processes for complete annihilation into a new sector, if permitted by
the symmetries of the theory.

For D--branes in superstring theory, such a divergence does indeed
show up, but there is an important new feature\cite{Banks:1995ch}. The
divergence occurs when the D--branes are finitely separated, by an
amount set by $X_H^2=2\pi^2 \alpha^\prime$, where $\alpha^\prime$ is
the characteristic length scale set by a fundamental string's tension.
This is interpreted as the fact that in addition to the many special
features of D--branes, they have a ``stringy halo'' originating in the
fact that the bulk of the open strings which (by definition) end on
them can reach out in the transverse directions, forming a region of
potential activity of size set by $X_H$. This halo means that the
D--branes can interact with each other before zero separation, as
there is an enhancement of the physics of interaction by new light
states formed by the entanglement of the halos, and the crossover into
the annihilation channel begins before the branes are coincident.

Recall that the amplitude of exchange can be thought of using two
equivalent pictures: Either as tree level exchange of closed string
quanta between the branes, or (after a modular transformation) as the
one--loop vacuum diagram for open strings stretched between the two
D--branes. In the open string description, at separation $X_H$, the
lightest open string becomes massless, and for any closer separation
it becomes tachyonic, signalling that the entire vacuum configuration
is unstable and wishes to roll to another vacuum. It is this tachyon
which produces the divergence of the exchange force, converting a
decaying exponential into a growing one, spoiling the convergence of
the amplitude in the infra--red (IR) region.

The D--branes annihilate {\it via} conversion to closed strings in the
generic situation, but the tachyon picture can be exploited in a
beautiful way to produce more
structure\cite{Sen:1998rg,Sen:1998ii,Sen:1998sm,Sen:1998tt}. For the
$G=U(N)\times U(N)$ gauge theory on the $(p+1)$--dimensional
world--volume on $N$ D$p$--branes and $N$ anti--D$p$--branes, the tachyon
field, transforming as the $({\mathbf N},{\bar {\mathbf N}})$, can be put into a
configuration endowed with non--trivial topological charge, and the
tachyon potential need not yield a runaway to a sector containing only
closed strings.  Having such topological vacuum solutions in the
tachyon sector allows for the possibility of a stable remnant ---
interpreted as a D--brane of lower dimension--- of the annihilation
process after the debris that is the closed string products has
cleared. It turns out that the spectrum of hypermultiplets in the
$U(N)\times U(N)$ world--volume theory supplies a set of variables
which is isomorphic to those needed to perform a K--theoretic analysis
of the topology of $G$--vector bundles over the world--volume, and so
the classification of all D--branes which can appear on a spacetime is
apparently elegantly and economically by using the results of the
appropriate K--theory of the spacetime which the D$p$--branes and
anti--D$p$--branes
fill\cite{Minasian:1997mm,Witten:1998cd,Horava:1998jy}. The case of
$p=9$ for Minkowski spacetime yields the entire classification of
D--branes in the most familiar symmetric vacuum of type~IIB superstring theory.

This is all well understood for the case of flat ten dimensional
spacetime. So when one encounters another background  which enjoys the
same maximal supersymmetry as flat spacetime --- a  plane wave with
R--R flux\cite{Blau:2001ne}:
\begin{eqnarray}
 ds^2 &=& 2dx^+dx^- - \mu^2 x^2(dx^+)^2 
  + \sum_{i=1}^{4} dx^i dx^i + \sum_{i=5}^{8} dx^i dx^i\ ,\nonumber \\
&&F_{+1234}=F_{+5678}=2\mu\ , \qquad x^2=\sum_{i=1}^8 x^i x^i\ ,
  \qquad x^\pm = \frac{1}{\sqrt{2}}(x^9\pm x^0)\ ,
  \label{eq:planewave}
\end{eqnarray}
which also yields an exactly solvable string
model\cite{Metsaev:2001bj} (in light--cone gauge defined by relating
worldsheet time $\tau$ to $x^+$ {\it via} $x^+ = 2\pi\alpha' p^+\tau$,
where $p^+$ is the $+$ component of spacetime momentum):
\begin{equation}
  \mathcal{L} = \frac{1}{4\pi\alpha'}
  (\partial_+ x^i\partial_- x^i - M^2x^2)
  +\frac{i}{2\pi\alpha'}
  (S^a\partial_ + S^a + \tilde{S}^a\partial_-\tilde{S}^a 
  - 2MS^a\Pi_{ab}\tilde{S}^a)\ ,
\labell{exactlysolvable}
\end{equation}
with a mass parameter $M=2\pi\alpha' p^+\mu$ --- it is inevitable that
questions about the key lessons which were learned about D--branes
will spring to mind\footnote{There has been a number of papers
  studying D--branes in plane wave and pp--wave backgrounds. Some of
  them are
  refs.\cite{Dabholkar:2002zc,Chu:2002in,Skenderis:2002vf,Skenderis:2002wx,Skenderis:2002ps,Billo:2002ff,Bergman:2002hv,Gaberdiel:2002hh,Gaberdiel:2002vz,Takayanagi:2002pi,Alishahiha:2002rw,Bain:2002nq,Bain:2002tq,Takayanagi:2002je,Balasubramanian:2002sa,Bak:2002rq,Kumar:2002ps,Lee:2002cu,Berenstein:2002zw,Kim:2003zw,Bonelli:2003sd,Alday:2003tj,Hyun:2002xe,Hikida:2002qk,Nayak:2002ty,Michishita:2002jp,Biswas:2002yz,Panigrahi:2003rh}.}.
Is the picture of D--branes as Dirichlet open string boundary
conditions as powerful in this context as it has been in flat spacetime?
In particular, do the dynamics hidden within a halo's breadth of the
branes bear any similarity do the flat spacetime case?  Are all D--branes
classified by K--theory, now of the new background?

Quite generally, one can ask, ``Will the things that we learned about
D--branes in flat spacetime teach us  about properties of this new background,
and/or will this new background teach us new facts about
D--branes?''.  A positive answer to either part of this question would
be very welcome, and we think that the observations which we make in
this paper do indeed constitute new information about D--branes which
may be part of a general lesson. In particular, the stringy halo of a
D--brane can be severely distorted, and in fact beyond certain
critical values of natural ratios which arise in the problem, the halo
can disappear altogether!

While we have no deep insights at present concerning the nature of the
K--theory of this particular plane wave background, it seems safe to
suppose that we must be careful in what we say about the K--theory
classification of all D--branes, since if there are regions where
there is no halo (and hence no tachyon) then not all of the key moving
parts of the K--theory machine are in evidence\footnote{This is in the
  context of brane --- anti--brane systems. There is of course the
  possibility of studying variants of K--theory which arise from the
  study of bundles of branes which are intrinsically unstable, such as
  D9--branes in type~IIA in flat spacetime, as done in
  ref.\cite{Horava:1998jy}.}. In particular, the tachyon is crucial in
forming a dynamical realisation of a chain of exact sequences which
extracts the required topology of the endstate of the annihilation
process.  However, the loss of the halo suggests that there is no
available tachyon for a large range of parameters in this background.
Whether this means that K--theory is no longer relevant in those cases
and must be supplemented by something else, or whether the familiar
brane -- anti--brane open string hypermultiplet variables must be replaced by other
variables while the K--theory remains intact and relevant is an
intriguing matter for further investigation.



\section{The Interaction Force}

It is convenient\cite{Skenderis:2002vf,Gaberdiel:2002hh} to label
D--branes in the plane wave background given in
equation~\reef{eq:planewave} as $(r,s)$, if they are Euclidean, where
$r$ denotes the spatial extent in directions $i=1,2,3,4$ and $s$
denotes the spatial extent in directions $i=5,6,7,8$. A D$p$--brane
would then have $r+s=p+1$. If the D--branes are Lorentzian, then their
worldvolume extends in the $x^+$ and $x^-$ direction, and the notation
is $(+,-,r,s)$. In that case, a D$p$--brane has $r+s=p-1$.


The string theory diagram of interest is a cylinder, representing
either the tree level exchange of closed string quanta between two
D--branes, or the one--loop vacuum process involving the circulation
of open strings with ends on either D--brane. % The length of the
% cylinder is controlled by the parameter $t$. 
See figure~\ref{cylinder_exchange}.
\begin{figure}[htbp]
  \centering
  \includegraphics[height=10cm]{cylinder_exchange.eps}
  \caption{\small Cylinder diagram for computing the amplitude of interaction
  between two branes. The parameter $t$ is open string propagation
  time, and is the modulus of the cylinder.}
  \label{cylinder_exchange}
\end{figure}

We will focus on the results for the simplest branes in the Euclidean
and Lorentzian classes. These are the D$(-1)$--branes (or
$(0,0)$--branes), and the D1--branes (or $(+,-,0,0)$--branes), discussed
in ref.\cite{Gaberdiel:2002hh}. The former requires the time
direction, in which the branes are also pointlike, to be Euclidean.

The results are reasonably simple for these cases, compared to other
$(r,s)$ with $r\neq s\neq 0$, and it would be interesting to explore those
other cases in detail.  We expect that the key
observations made in this paper for these $r=0=s$ cases will be quite
generic, although there may be additional features to be deduced from
studying other cases in detail.

\subsection{The Amplitude and Potential}

We consider a D$p$--brane and its antiparticle for $p=\pm1$. If
$p=-1$, it is an instanton, (a $(0,0)$--brane) and we consider it to
be pointlike in Euclidean time.  If $p=+1$ it is a string, (a
$(+,-,0,0)$--brane) and the theory is Lorentzian. 

So we place a D$p$--brane at position $y_1^i$ in the $x^i$ directions
($i=1,\ldots,8$), and a $\overline{{\rm D}p}$--brane (antibrane) at
position $y_2^i$, with a separation $X^\pm$ in the $x^\pm$ directions
if $p=-1$. The cylinder amplitude $A$ is\cite{Gaberdiel:2002hh}:
\begin{equation}
  A = \int_0^\infty \frac{dt}{2t} t^{-\left(\frac{p+1}{2}\right)}e^{- t\frac{X^+ X^-}{2\pi\alpha'}}
 \hat{h}_0({t};y_1,y_2) \frac{\hat{g}_4^{({m})}({t})^4}
  {f_1^{({m})}({t})^8}\ .
\labell{amplitudeform}
\end{equation}
For $p=+1$, the factor $\exp(-X^+X^-/2\pi\alpha^\prime t)$ is not
present. For higher $p$, (which we will not be considering here) there
are no additional powers of $t$ in the integrand. These are normally
due to integration over continuous zero modes in the flat spacetime case.
The plane wave background has no such modes for the direction $x^i$,
(the zero modes are instead themselves harmonic
oscillators\cite{Amati:1989sa,Horowitz:1990bv,Horowitz:1990sr,Jofre:1994hd})
and so such $t^{-1}$ factors beyond those appearing here are present.
See below equation~\reef{marker} for some further discussion of how to
read this expression.

In the above, % for the closed string expression %
we have the functions:
\begin{align}
%  h_0(t;y_1,y_2) = & \exp\Bigl(-\frac{m(m+q^m)(y_1^2+y_2^2)}{\alpha'2(1-q^m)} +
 % \frac{2mq^{\frac{m}{2}} y_1\cdot y_2}{\alpha'(1-q^m)} \Bigr),
 % \nonumber\\
  f_1^{(m)}(t) = & q^{-\Delta_m}(1-q^m)^\half \prod_{n=1}^{\infty}
  ( 1-q^{\omega_n} ),
  \nonumber \\
 % g_2^{(m)}(t) = & 4\pi m q^{-2\Delta_m} q^{\frac{m}{2}}
 % \prod_{n=1}^{\infty} 
 % \Bigl(1+\frac{\omega_n+m}{\omega_n-m} q^{\omega_n} \Bigr)
 % \Bigl(1+\frac{\omega_n-m}{\omega_n+m} q^{\omega_n} \Bigr),
 % \nonumber \\
%\end{align}
%while for the open string expression:
%\begin{align}
  \hat{h}_0({t};y_1,y_2) = & \exp\Bigl(
  -\frac{{m}{t}}{2\alpha'\sinh(\pi{m})}
  [\cosh(\pi{m})(y_1^2+y_2^2) - 2 y_1\cdot y_2] \Bigr),
  \nonumber \\
  \hat{g}_4^{({m})}({t}) =
  &{q}^{-\hat{\Delta}_{({m})}} 
  \prod_{l\in\mathcal{P}_+}\Bigl(1-{q}^{|{\omega}_l|}\Bigr)^\half
  \prod_{l\in\mathcal{P}_-}\Bigl(1-{q}^{|{\omega}_l|}\Bigr)^\half,
  \nonumber \\
 \Delta_m = & - \frac{1}{(2\pi)^2}\sum_{p=1}^\infty \int_0^\infty ds
  \,\, e^{-p^2 s - \frac{\pi^2 m^2}{s}} \ ,\nonumber \\  
\hat{\Delta}_{{m}} = & - \frac{1}{(2\pi)^2}\sum_{p=1}^\infty 
  (-1)^p\sum_{r=0}^\infty c_r^p {m} 
  \frac{\partial^r}{(\partial {m}^2)^r} \frac{1}{{m}}
  \int_0^\infty ds \left(\frac{-s}{\pi^2}\right)^r
  e^{-p^2s-\frac{\pi^2{m}^2}{s}}\ ,
\end{align}
and the parameter $q$ and the deformed harmonic oscillator frequencies
are defined as:
\begin{equation}
  q = e^{-2\pi t}\ ,\qquad \omega_n = \mathrm{sign}(n)\sqrt{n^2+m^2} \ .
%  \qquad \hat{\omega}_n = \mathrm{sign}(n)\sqrt{n^2+\hat{m}^2}\ .
\end{equation}
Note that $\Delta_m$ and $\hat{\Delta}_{{m}}$ are zero--point energies
which arise naturally in the closed and open string sectors,
respectively.  The coefficients $c_r^p$ in $\hat{\Delta}_{{m}}$ are
the coefficients of a specific Taylor expansion:
\begin{equation}
  \Bigl(\frac{x+1}{x-1}\Bigr)^p + \Bigl(\frac{x-1}{x+1}\Bigr)^p =
  \sum_{r=0}^\infty c_r^p x^{2r}\ . 
\end{equation}
The sets $\mathcal{P}_-$ and $\mathcal{P}_+$ are given as solutions of
the equations 
\begin{align}
  l\in \mathcal{P}_- :\quad %\Longleftrightarrow \quad 
  \frac{l-im}{l+im}+e^{2\pi i l}=0\ ,
  \qquad
  l\in \mathcal{P}+ :\quad %\Longleftrightarrow \quad 
  \frac{l+im}{l-im}+e^{2\pi i l}=0\ .
\label{marker}
\end{align}
The details of the derivation of these amplitudes can be found in
ref.\cite{Gaberdiel:2002hh}. We will not need them all here, and refer
the reader there for more information.  Some comments are in order
however.  For $p=-1$, the above expression was computed first with a
boundary state formalism with closed string light cone gauge
$x^+=2\pi\alpha^\prime p^+ \tau$, we have $M=2\pi\alpha^\prime p^+\mu$
and the propagation time in the closed string channel is $1/t$. Then a
modular transformation gives the expression above, with
\begin{equation}
%  \quad \hat{q} = e^{-2\pi\hat{t}},
 % \quad \hat{t}=1/t,
  \quad t = \frac{2\pi\alpha' p^+}{X^+}\ ,
  \quad {m}=\mu X^+   =Mt\ .
\labell{massone}
\end{equation}
For the case $p=1$, the computation was done directly in terms of the open
string channel, with open string light cone gauge
$x^+=2\pi\alpha^\prime p^+ \tau$, so we have
\begin{equation}
%  \quad \hat{q} = e^{-2\pi\hat{t}},
 % \quad \hat{t}=1/t,
  \quad t = \frac{X^+}{2\pi\alpha' p^+}\ ,
  \quad {m}=2\pi\alpha^\prime p^+\mu\ .
\label{masstwo}
\end{equation}



What is important for our discussion is the structure of the full
amplitude for the cylinder diagram, given above in
equation~\reef{amplitudeform} as an integral over the modulus $t$.  
% At
% $t\to 0$, (the IR regime from the closed string perspective, and the
% UV regime for the open string) we have $q\to 1$. In fact, it is easy
% to see in the closed stirng channel (after a modular transformation)
% that the integral is finite there. At ${t}\to\infty$), (UV of closed
% string and IR of open string) we have ${q}\to 0$, and we must
% carefully study possible divergent behaviour of this integral, it is
% therefore more convenient to look at the open string expressions which
% we have here.
% So let us consider our amplitude, which
It can be written as:
\begin{equation}
  A =% \int_0^\infty \frac{d{t}}{2{t}}~ 
  % e^{-\frac{{t}X^+ X^-}{2\pi\alpha'}}
 % \hat{h}_0(y_1,y_2) \frac{\hat{g}_4^{({m})}({t})^4}
 % {f_1^{({m})}({t})^8}
 \int_0^\infty \frac{dt}{2t} t^{-\left(\frac{p+1}{2}\right)}\exp\Bigl\{-2\pi t
Z(m,y_1,y_2)\Bigl\} G(t),
\end{equation} 
where the the exponent $Z(m,y_1,y_2))$ is defined as (delete the
$X^+X^-$ term to get the D1--brane result):
\begin{equation}
\begin{split}
  Z(m,y_1,y_2) = %&
\frac{m\pi}{4\pi^2\alpha'\sinh(m\pi)}
         \left[ \cosh(m\pi)(y_1^2+y_2^2)-2y_1\cdot y_2\right] 
   % \\ &
    - 4(\hat{\Delta}_m-2\Delta_m)
    +\frac{X^+X^-}{4\pi^2\alpha'}.
\end{split}
\end{equation} and the
function $G(t)$ is defined as:
\begin{equation}
\begin{split}
  G(t) %&
= \frac{\prod_{l\in\mathcal{P}_+}(1-q^{|\omega_l|})^2 
    \prod_{l\in\mathcal{P}_-}(1-q^{|\omega_l|})^2}
  {(1-q^m)^4 \prod_{n=1}^{\infty}(1-q^{\omega_n})^8}
 % \\
  %&
= \prod_{n=1}^{\infty}(1-q^{\omega_n})^{-8}
  \prod_{l\in\mathcal{P}_+, l>0} (1-q^{\omega_l})^4
  \prod_{l\in\mathcal{P}_-, l>0} (1-q^{\omega_l})^4  
\end{split}
\end{equation}
For our discussion, the only important fact about the function $G(t)$
is that its behaviour at large and small $t$ is such that generically,
the amplitude is convergent.  That $A$ is finite as $t\to 0$ follows
from the fact that small $t$ is the closed string IR limit, where this
amplitude should reproduce simple low energy field theory results for
massless exchange at tree level. The $t\to\infty$ limit is also well
behaved generically, since this is the open string IR limit,
which is fine --- away from special circumstances which will not show
up in the oscillator contributions since their energies are higher
than the lowest lying states. In fact, it is clear that $G(t)\to 1$ as
$t\to\infty$, and so whether $A$ is finite as $t\to\infty$ depends on
the sign of the exponent $Z$, which (as we shall review shortly)
controls those lowest lying states.

Writing $y_2^i=y_1^i+z^i$, where $z^i$ is the separation between the
branes in the eight directions~$x^i$, the expression for $Z$
becomes
\begin{equation}
  Z(m,y_1,z) = \frac{1}{4\pi^2\alpha'}\frac{m\pi}{\tanh(m\pi)} \Bigl[
      (z+ a)^2 +\frac{\tanh(m\pi)}{m\pi}X^+X^-- b^2 \Bigr],
\end{equation}
where we have defined three crucial parameters:
\begin{align}
  a =  \frac{\cosh(m\pi)-1}{\cosh(m\pi)}y_1\ ,\qquad
  b =  \tanh(m\pi)\sqrt{y_*^2-y_1^2}\ ,\qquad
  y_*^2 =  \frac{16\pi^2\alpha' (\hat{\Delta}_m - 2\Delta_m)}
  {m\pi \tanh(m\pi)}\ .
\labell{crucial}
\end{align}

\subsection{The Force}

Let us now consider the force between the brane and the antibrane. In
the $y^i$ directions it is given as the derivative with respect to the
$z^i$,
\begin{align}
  \nonumber
  F_i & = -\frac{\partial A}{\partial z^i}
  = -\int_0^\infty \frac{dt}{2t} e^{-2\pi t Z}(-2\pi t)\frac{\partial
  Z}{\partial z^i} G(t)
  \\
  &= \int_0^\infty \frac{dt}{2t} e^{-2\pi t Z}(2\pi t)
  \frac{1}{4\pi^2\alpha'}\frac{m\pi}{\tanh(m\pi)} 2(z^i+a^i) G(t)
\end{align}
Defining $u = 2\pi t Z$ (and assuming $Z>0$), this gives
\begin{equation}
  F_i = \frac{1}{4\pi^2\alpha'}\frac{m\pi}{\tanh(m\pi)} (z^i+a^i)
  \frac{1}{Z }
  \int_0^{\infty} du e^{-u} G\left(\frac{u}{2\pi Z}\right)\ .
\end{equation}
When $Z\to 0$, we have $G\left(\frac{u}{2\pi Z}\right)\to 1$, so the
integral approaches unity. In this limit the force clearly diverges,
since it goes like
\begin{equation}
  F_i \sim \frac{m\pi}{\tanh(m\pi)} \frac{z^i+a^i}{Z}\ .
\end{equation}
A similar conclusion can be made for the force in the light--cone
directions (for $p=-1$):
\begin{equation}
  F_\pm = -\frac{\partial A}{\partial X^\pm} 
  = -\frac{1}{4\pi^2\alpha'} \frac{X^\mp}{Z} \int_0^\infty
  du~e^{-u}G\left(\frac{u}{2\pi Z}\right) 
  \sim \frac{X^\mp}{Z}\ .
\end{equation}

\section{Review: The Halo in The Flat Spacetime Limit}

\subsection{Divergence}
If we send $m\to 0$, we recover flat spacetime (see the
solution~\reef{eq:planewave}, and recall that $m$ is proportional to
$\mu$), and in this limit we ought to recover flat spacetime results.
Examining the expressions~\reef{crucial} as $m\to 0$ we see that $a\to
0$, $y_*^2 \to \infty$, $b^2\to 2\pi^2\alpha'$, and
\begin{equation}
  Z\longrightarrow \frac{1}{4\pi^2\alpha'}(z^2+X^+X^--2\pi^2\alpha') \ ,
\end{equation}
(delete the $X^+X^-$ term to give the result for the D1--brane
case).  So, combining this with our observation of the previous
section, we recover the well known\cite{Banks:1995ch} divergence at
separation given by  $X_H^2=2\pi^2\alpha'$.

\subsection{Tachyon}
 
Consider the RNS formulation in the flat spacetime background. Then the
worldsheet Hamiltonian is given as $H=L_0=\alpha' p^2 + N +
a_{R(NS)}$, where the constant $a_{R(NS)}$ is the zero point energy
and~$N$ is the total number operator.  The z.p.e. is $a_R=0$ in the
Ramond sector, and $a_{NS}=-\half$ in NS sector.

For strings stretched between two D--branes, we have $p^m =
{x^m}/{2\pi\alpha'}$ for transverse (to the branes) directions
$x^m$.  So, splitting transverse (labelled $m$) and parallel (labelled
$i$) directions we can write
\begin{equation}
  L_0 = \alpha' p^i p_i + N + \frac{z^2}{4\pi^2\alpha'} + a_{R(NS)}\ .
\end{equation}
This gives a mass spectrum
\begin{equation}
  M^2 = -p^ip_i = \frac{1}{\alpha'}
\left(N +a_{R(NS)} + \frac{z^2}{4\pi^2\alpha'}\right)\ .
\end{equation}
The NS ground state ($N=0$, $a_{NS}=-\half$) has mass squared
\begin{equation}
M_0^2 = \frac{1}{2\alpha'}\left(\frac{z^2}{2\pi^2\alpha'}-1\right)\ .
\end{equation}
This is a tachyon if $z^2 < 2\pi^2\alpha'$.

In the usual case this ground state is eliminated by the GSO
projection $P=\frac{1+(-1)^F}{2}$ in superstring theory.  The
partition function ${\cal Z}$ is roughly given by $\mathrm{Tr}
Pe^{-tL_0}$, where the trace is over everything, in both the R and NS
sectors. There is an overall factor of $\exp\left({-t(a_{R(NS)}
    +\frac{z^2}{4\pi^2\alpha'})}\right)$.  When we consider a
brane--antibrane system,  we can write
\begin{equation}
 {\cal Z}=\mathrm{Tr}( P e^{-tL_0})={\cal Z}_R+{\cal Z}_{NS}=2{\cal Z}_{NS}\ , 
\end{equation}
and so there will be an overall factor in the path integral of
\begin{equation}
  e^{-t\left(a_{NS} +\frac{z^2}{4\pi^2\alpha'}\right)}
  =e^{\frac{t}{2}\left(1-\frac{z^2}{2\pi^2\alpha'}\right)}
  = e^{-t\alpha' M_0^2}\ .
\end{equation}
The other factors will be non--divergent, so whether the integral over
the modulus $t$ is divergent or not just depends of the sign
of the exponent in this factor. That is, the integral is divergent if
$z^2<2\pi^2\alpha'$ ({\it i.e.,} $M_0^2<0$).

When we consider a brane--antibrane system, we are effectively
reversing the GSO projection in the partition function, giving
$P=\frac{1-(-1)^F}{2}$, since antibranes come with a minus sign.  This
means that the NS ground state ($N=0$) will now survive, and the
possible tachyon above is present in the spectrum. So for $z^2<
2\pi^2\alpha'$ there is a tachyon, and so there is a 1--1
correspondence between the tachyon's appearance and divergence of the
integral.

For the case when all of the directions are transverse, as is the case
for the instantons we have been studying here, the tachyon
interpretation follows from continuation and T--duality.




\section{The  Halo in the Plane Wave and the Fall from Grace}

Clearly, we must seek for the places where $Z\to0$ in the more general
case. Wherever this happens, there is a divergence of the force in
exactly the same way as in the flat spacetime case. This divergence is
purely stringy in origin, and signals a place where the lightest open
string is becoming massless and a tachyon is possibly appearing in the
spectrum.

Let us first note that if $y_1^2> y_*^2$, or alternatively, if $m$ is
greater than a certain value we shall call $m_*$, then $b$ is
imaginary, and $Z$ is always positive. In this case the force is never
divergent --- regardless of the separation of the branes. So we
observe our first interesting result: If the branes are far away
enough from the origin of the plane wave background, the halo
disappears! This must not be interpreted as the failure of the
brane--anti--brane pair to annihilate, of course. Field theory
intuition tells us only that a divergence signals the opening up of
new channels, but the converse is not true. The amplitude computed in
position space in field theory will have its divergence controlled by
the position space propagator for massless exchange, which goes
roughly as $x^{2-D}$, where $D$ is the number of flat transverse
directions. So one way to avoid the divergence of the force is if
$D\leq 2$. Notice (see equation~\reef{eq:planewave}) that eight of the
transverse directions $x^i$ $(i=1,\ldots,8)$ of the plane wave are not
like flat space. There is a confining potential $\mu^2x^2$.

So let us consider the case $y_1^2<y_*^2$.  Then we have  
\begin{equation}
  Z=0 \quad \Longleftrightarrow \quad (z+a)^2 + \frac{\tanh(m\pi)}{m\pi}
  X^+X^- = b^2\ .
\end{equation}
This equation describes ellipsoids (see figure \ref{halo_parameters}) in
the space of $z^i$ and $X^\pm$. The radius in the $z^i$ directions is
$b$, and the radius in the $X^\pm$ directions is
$h=({m\pi}/{\tanh(m\pi)})^{1/2}b>b$. Notice that the centre in the
$z^i$ directions of the ellipsoid is shifted by $-a$. For the case of
the Lorentzian D1--brane, we must simply delete the $X^+X^-$ term. We
then have spheres in the eight transverse directions, of radius $b$
and shifted by $-a$. These shapes (spheres or ellipsoids) are loci of
points forming the ``stringy halo'' of the D--branes. They are the
places where there is a sharp sign that there is new physics appearing
as the branes approach each other: The factor $Z$ is vanishing, due to
the appearance of a massless low--lying stretched open string state,
just as in the flat spacetime case.

\begin{figure}
  \centering
\includegraphics[height=7cm]{halo_parameters.eps}
  \caption{\small \label{halo_parameters}
    There is a brane at relative position $z=\sqrt{z^iz^i}=0$. For
    D$(-1)$--branes, it is also at relative position $\sqrt{X^+X^-}=0$.
    Bringing up another brane will produce a divergence on an
    ellipsoid ($b\neq h$) in ten dimensions (for D$(-1)$--branes) or a
    sphere of radius $b$ in eight dimensions (for a D1--brane). This
    sphere or ellipsoid is interpreted as the ``stringy halo''. 
     Shown is the parameterisation of the halo. Its radius is set by
    $b$ in the plane wave's eight transverse directions, and $h>b$ in
    the $X^\pm$ directions if it is a D$(-1)$--brane. It is shifted
    towards the origin by an amount $a$.}
\end{figure}



It is important to note that as long as $0 \le y_1^2 \le y_*^2$, ({\it
  i.e.}, as long as a halo is present) then $a\le
\sqrt{2\pi^2\alpha^\prime}$. Also $b^2 \le 2\pi^2\alpha'$. Another
interesting feature is that the shift $a$ increases while the radius
$b$ decreases with increasing $y_1$ (or increasing~$m$).  Note also
that $a+b \le 2\pi^2\alpha'$, and therefore the size of the halo in
the $x^i$ directions is smaller than that in flat space. The string
halo shrinks with increasing~$m$ or $y_1$.

Because of the shift $a$, there is another interesting scale, marking 
where $a^2 = b^2$. This happens for 
\begin{equation}
  y_1^2 = \frac{\sinh^2(m\pi)}{2\cosh(m\pi)(\cosh(m\pi)-1)}y_*^2
  \equiv y_{**}^2\ .
\labell{starstar}
\end{equation}
It is straightforward to see that $y_{**}^2 \le y_*^2$. (This also
implicitly defines a mass scale $m_{**}$.) The significance of this is
that not only can the centre of the halo shift, but in fact it can
shift so far towards the origin that it can be completely away from
the $y_1$ brane! So while the halo survives, it no longer surrounds a
brane at all.

\subsection{Summary of Possibilites}
\label{summary}
It is useful to list the possible situations. The different cases
are illustrated in figures~\ref{fig:summary2} and~\ref{spheres}.  
\begin{itemize}
\item{$a=0$:} This happens for $m=0$ (case {\it (i)} in
  figures~\ref{fig:summary2} and~\ref{spheres}) in which case we
  simply recover the flat spacetime result that the force diverges when
  the separation between the branes is $\sqrt{2\pi^2\alpha'}$.  The
  case $a=0$ also happens for $m\neq 0$ and $y_1=0$.  In this case
  $a=0$, but $b<h<\sqrt{2\pi^2\alpha'}$.
\item{$b^2>a^2>0$:} This ``small mass'' situation (case {\it (ii)} in
  figure~\ref{fig:summary2} and~\ref{spheres}) happens for
  $0<y_1^2<y_{**}^2$, or equivalently for $0<m<m_{**}$.
\item{$a^2=b^2$} This happens when $y_1^2=y_{**}^2$, or equivalently
  $m=m_{**}$, as defined in equation~\reef{starstar}.
  Since $y_{**}^2$ is a decreasing function of $m$, we can say
  $m>m_{**} \Longleftrightarrow y_{**}^2(m)<y_1^2$, and
  $m<m_{**} \Longleftrightarrow y_{**}^2(m)>y_1^2$.
\item{$a^2>b^2>0$:} This ``medium mass'' situation (case {\it (iii)}
  in figures~\ref{fig:summary2} and~\ref{spheres}) happens for
  $y_{**}^2<y_{1}^2<y_{*}^2$, or equivalently $m_{**}<m<m_{*}$.  The
  brane is no longer surrounded by the string halo. The second brane
  can be brought in to annihilate with the first without encountering
  the halo.
  
\item{$b^2=0$:} This happens when $y_1^2=y_{*}^2$, or equivalently,
  $m=m_{*}$, as defined in equation~\reef{crucial}.
  Since $y_{*}^2$ is a decreasing function of $m$, we can say
  $m>m_{*} \Longleftrightarrow y_{*}^2(m)<y_1^2$, and
  $m<m_{*} \Longleftrightarrow y_{*}^2(m)>y_1^2$.
  
\item{$b^2<0$:} This happens for $y_{*}^2<y_{1}^2$, 
  or equivalently $m_{*}<m$. In this case there is no divergence of
  the force for any separations. This ``large mass'' situation is case {\it (iv)}.
\end{itemize}

\begin{figure}
  \centering
\includegraphics[height=7cm]{halo_possibilities.eps}
 % \input{braneforcediv2.pstex_t}
  \caption{\small \label{fig:summary2}
    There is a brane at relative position $z=\sqrt{z^iz^i}=0$. For
    D$(-1)$--branes, it is also at relative position
    $\sqrt{X^+X^-}=0$.  Bringing up another brane will produce a
    divergence on an ellipsoid ($b\neq h$) in ten dimensions (for
    D$-1$--branes) or a sphere of radius $b$ in eight dimensions (for
    a D1--brane). This sphere or ellipsoid is interpreted as the
    ``stringy halo''. Shown are three characteristic cases that can
    occur for the halo for different values of the parameters. See the
    text for further explanation. See also
    figures~\ref{halo_parameters} and~\ref{spheres}.}
\end{figure}

\section{Discussion}
The maximally supersymmetric plane wave background\cite{Blau:2001ne}
allows for an exactly solvable string
model\cite{Metsaev:2001bj,Metsaev:2002re} in which to examine the
properties of D--branes in some detail.  D--branes in flat spacetime
exhibit an important feature in that they are sensitive to each
other's presence upon approaching to a distance scale set by the
tension of the open strings that can end on them. It is natural
therefore to look for new physics in this regime for D--branes in the
plane wave background, since there is a mass parameter $\mu$ naturally
associated to the background.

It is possible to account for some of the features that we saw in our
amplitudes by examining the worldsheet model, given in
equation~\reef{exactlysolvable}. The mass parameter $\mu$ controls a
mass $m$ in the two dimensional string model (either closed or
open\cite{Bergman:2002hv,Gaberdiel:2002hh}; see
equations~\reef{massone} and~\reef{masstwo}) for the modes in the
$x^i$ ($i=1,\ldots,8$) directions. This means that for non--zero $m$,
it is energetically favourable to keep $x$ as small as possible. This
nicely explains two phenomena about the halo. For increasing $m$, we
saw that $b$, which sets the size of the halo, decreases. The halo is
made of fundamental strings, whose bulk behaviour is controlled by the
model in equation~\reef{exactlysolvable}. So the decrease in the size
of the halo is to be expected, since the open strings wish to stay
closer to the origin $x=0$ of the spacetime.  This also explains why
the halo gets shifted towards the origin when the D--brane is located
at $y_1\neq 0$. The potential $m^2x^2$ in the plane wave model
produces a ``drift force'' on the halo, distorting it sideways
towards~$x=0$. This is shown more graphically in figure~\ref{spheres}.


\begin{figure}
  \centering
\includegraphics[height=7cm]{halo_sphere.eps}
  \caption{\small \label{spheres} This figure depicts a brane (large
    dot) and its accompanying halo in four different situations. (The
    labels {\it (i)} --- {\it (iv)} correspond to the labelling in the
    text and in figure~\ref{fig:summary2}.) This case is for
    fixed~$m$, while the brane is placed at increasing distance from
    the origin. The halo shrinks, shifts off the brane, and eventually
    disappears altogether. Fixing the brane's position and increasing
    $m$ will also shrink the halo and cause it to disappear.}
\end{figure}

The disappearance of the halo past a critical distance, or
equivalently a critical value of $m$, is intriguing.  It originates in
the fact that there is a competition between the classical desire of
the open strings to reduce their length to zero since they have
tension, and the fact that they have a quantum mechanical negative
zero point energy, and so they stretch to soak up the energy, and form
the halo. The background's mass parameter, $m$, tilts the battle in
the favour of the tension, since energy can go into the string's mass
term at the expense of stretching. This results in the shrinking of
the halo, while also shifting its centre as we saw above, since
smaller $x$ is less costly in the presence of the mass term.  At a
critical value of $m$ however, $m_{*}$, the contribution of $m$
overwhelms the quantum mechanical zero point energy to the extent that
there is no longer any energetic reason for the strings to stretch out
at all. In this case, there is no region where there is a massless
state which is about to become tachyonic at all. As mentioned in the
introduction, this is interesting, since while nothing we have done
here shows that the D--branes can't annihilate when there is no
halo\footnote{We've been looking specifically for the halo, which
  marks the opening up of new annihilation channels. In the absence of
  a halo, one expects the annihilation to be akin to that in ordinary
  field theory.  Further analysis is needed to demonstrate anything
  conclusive about the full annihilation process.}, there is no
accompanying open string tachyon for a range of parameters. This leads
one to speculate (as we did in the introduction) about whether or not
the spectrum of branes in this background can be captured by
K--theory, of which the tachyon condensation mechanism for
constructing lower dimensional branes by brane --- anti--brane
annihilation\cite{Sen:1998rg,Sen:1998ii,Sen:1998sm,Sen:1998tt} seems
to supply a physical realisation in flat spacetime for
type~IIB\cite{Minasian:1997mm,Witten:1998cd,Horava:1998jy}. (The
possibility of a K--theory classification arising from studying only
intrinsically unstable branes, as was done for type~IIA in flat
spacetime\cite{Horava:1998jy}, remains open.)

That there is a critical mass $m$ beyond which the branes lose their
halo (or equivalently, a critical radius from the origin outside of
which branes lose their halo) for this plane wave background may have a
number of important consequences beyond the matters of classification
mentioned above.  For example, given that strings in such a background
capture aspects of (a sector of) large~$N$ gauge
theory\cite{Berenstein:2002jq}, there may be consequences for gauge
theory dynamics to be uncovered.



 

\section*{Acknowledgements}
C.V.J. would like to thank the EPSRC and the PPARC for financial
support. H.G.S. was supported by a doctoral student fellowship from
the Research Council of Norway, by an ORS award, and by the University
of Durham. This paper is report number DCPT-03/13.

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