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%%%%%%%% Gauge Fixing and Scattering Amplitudes in      %%%%%%%%%%
%%%%%%%% String Field Theory around Universal Solutions %%%%%%%%%%
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%%%%%%%% T. Takahashi and S. Zeze                       %%%%%%%%%%
%%%%%%%% April, 2003                                    %%%%%%%%%%
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%\pubinfo{Vol.~10X, No.~X, Mmmmm YYYY}%Editorial Office will fill in this.
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%KUNS-1325\\PTPTeX ver.0.8\\ August, 1997}
hep-th/0304261}
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\title{%        %You can use \\ for explicit line-break
Gauge Fixing and Scattering Amplitudes in String\\
Field Theory around Universal Solutions
}

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\author{%       %Use \scshape  for the family name
Tomohiko \textsc{Takahashi}$^{1,}$\footnote{E-mail:
tomo@asuka.phys.nara-wu.ac.jp} 
and Syoji \textsc{Zeze}$^{2,}$\footnote{E-mail:
zeze@sci.osaka-cu.ac.jp.}
}

\inst{%         %Affiliation, neglected when [addenda] or [errata]
$^1$Department of Physics, Nara Women's University, Japan\\
$^2$Department of Physics, Osaka City University,
Japan
}

%\publishedin{%         %Write this ONLY in cases of addenda and errata
%Prog.~Theor.~Phys.\ \textbf{XX} (19YY), page.}

%\recdate{Mmmmm DD, YYYY}%            %Editorial Office will fill in this.

\abst{%         %this abstract is neglected when [addenda] or [errata]
We study a gauge fixed action of open string field theory
around the universal solution which has been found as an analytic
classical solution with one parameter $a$. 
For $a>-1/2$, we can reproduce open string scattering amplitudes in the
theory fixed in the Siegel gauge.
At $a=-1/2$, all scattering amplitudes vanish and there is no open
string excitation in the gauge fixed theory.
These results support the conjecture that the universal
solution can be regarded as pure gauge or the tachyon vacuum solution.
}

\begin{document}

\maketitle

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\section{Introduction}

It has been proposed that
the tachyon vacuum
solution\cite{rf:KS-tachyon,rf:SZ-tachyon,rf:MT,rf:GR} in open string
field theory has been 
analytically constructed as the universal solution which is found in the
universal subspace of string 
fields.\cite{rf:TT2}
This conjecture is strongly supported by the fact that
the modified BRS charge in the theory expanded around the universal
solution has vanishing cohomology in the Hilbert space of ghost number
one,\cite{rf:KT} and that a non-perturbative vacuum numerically
disappears in the theory.\cite{rf:tomo} This fact
implies that there is no open string excitation perturbatively
around the solution and we have already stayed on the tachyon
vacuum in the theory.

There is another approach to describing the tachyon vacuum
instead of finding the tachyon vacuum solution, that is vacuum
string field theory.\cite{rf:RSZ-VSFT,rf:GRSZ} 
Though much progress has been
made in vacuum string field theory,\cite{rf:VSFT} we need a kind 
of regularization due to a pure ghost midpoint operator in
the kinetic term.\cite{rf:GRSZ}
On the other hand, string field theory around the universal solution has
the kinetic operator which is less singular than the pure ghost
midpoint operator. Moreover, as Drukker pointed out,\cite{rf:Drukker2}
the kinetic term around the universal solution may become a
pure ghost midpoint operator through a certain singular limit of it.
Hence, the theory around the universal solution is much more general and
less singular than vacuum string field theory.

In order to prove the equivalence of the tachyon vacuum and the
universal solutions,
we must show that the energy density of the universal solution cancels
the D-brane tension exactly. In addition, we should understand the
existence of closed strings in the theory around the universal
solution. At present, the exact cancellation seems difficult to be
proved due to 
technical problems\cite{rf:tomo} and then
we should adopt other approaches to understand the relation between
these solutions. To find closed strings, we should firstly
discuss gauge fixing and scattering amplitudes in the theory.

The universal solution $\Psi_0(a)$ given
in ref.~\citen{rf:TT2} is expected to be regarded as pure gauge for $a>-1/2$
and the tachyon vacuum solution at $a=-1/2$.
In ref.~\citen{rf:Drukker2}, Drukker studies the
theory around this solution fixed in the Siegel gauge and suggests that
original open string 
amplitudes can be reproduced around the solution for $a>-1/2$.
In this paper, 
we will investigate this gauge-fixed theory more precisely and show more
explicitly the correspondence of physical
states and amplitudes between the original and expanded theories.
Moreover, solving the cohomology under
the Siegel gauge condition, we will prove that amplitudes become zero in
the theory around the non-trivial solution at $a=-1/2$.

In section 2,
we solve the equation of motion in the Siegel gauge
for $a>-1/2$ and we find
a one-to-one correspondence of spectra between the original and
expanded theories. Next, we derive the
Feynman rule in the expanded theory. We find that this
Feynman rule reproduces the same scattering amplitudes as in the original
theory. In section 3, we analyze the theory
around the solution at $a=-1/2$ and we find that
the resulting scattering amplitudes become zero.
We give some discussion in section 4.

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\section{String Field Theory around Pure Gauge Solutions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pure gauge solutions}

The equation of motion in cubic open string field theory\cite{rf:CSFT}
is given by
\begin{eqnarray}
 \Q\Psi+\Psi*\Psi=0.
\end{eqnarray}
An analytic solution of the equation has been found as\cite{rf:TT2}
\begin{eqnarray}
\label{Eq:solution}
\Psi_0=Q_\rL(e^h-1)I-C_\rL((\partial h)^2 e^h) I,
\end{eqnarray}
where $I$ stands for the identity string field and the operators $Q_\rL$
and $C_\rL$ are defined as
\begin{eqnarray}
 Q_\rL(f)=\int_{C_{\rm left}} \frac{dw}{2\pi i}\,f(w)\,J_{\rm B}(w),
\ \ \ 
 C_\rL(f)=\int_{C_{\rm left}} \frac{dw}{2\pi i}\,f(w)\,c(w).
\end{eqnarray}
Here $J_{\rm B}(w)$ and $c(w)$ are the BRS current and the ghost field,
respectively. The function $h(w)$ in the solution satisfies
$h(-1/w)=h(w)$ and $h(\pm i)=0$. The solution can be expressed by the
matter Virasoro generators and the ghost and anti-ghost oscillators
acting on the $SL(2,R)$ invariant vacuum, because it is constructed by
the BRS current, the ghost field and the identity string
field.\cite{rf:TT2} Then, we call it {\it universal solution}, since its
expression does not depend on any specific background.

If we expand the string field $\Psi$ as
\begin{eqnarray}
 \Psi=\Psi_0+\Phi,
\end{eqnarray}
the action for the fluctuation $\Phi$ becomes
\begin{eqnarray}
 S[\Phi]=-\frac{1}{g^2}\int\left(
\frac{1}{2}\Phi*\Q'\Phi+\frac{1}{3}\Phi*\Phi*\Phi\right),
\end{eqnarray}
where the modified BRS charge is given by\cite{rf:TT2}
\begin{eqnarray}
\label{Eq:modBRS1}
 \Q'=Q(e^h)-C((\partial h)^2 e^h).
\end{eqnarray}
The operators $Q(f)$ and $C(f)$ are defined as
\begin{eqnarray}
 Q(f)=\oint \frac{dw}{2\pi i} f(w) J_{\rm B}(w),\ \ \ 
 C(f)=\oint \frac{dw}{2\pi i} f(w) c(w).
\end{eqnarray}

Let us consider the action around the universal solution
generated by the function
\begin{eqnarray}
\label{Eq:ha}
h_a(w)=\log\left(1+\frac{a}{2}\left(w+\frac{1}{w}\right)^2\right),
\end{eqnarray}
where $a$ is a parameter larger than or equal to $-1/2$.\cite{rf:TT2}
Substituting this function into (\ref{Eq:modBRS1}),
we find that the modified BRS charge is expanded as\cite{rf:TT2}
\begin{eqnarray}
\label{Eq:modBRS2}
\Q'(a)&=&(1+a)\Q+\frac{a}{2}(Q_2+Q_{-2})+4aZ(a)\,c_0
-2aZ(a)^2(c_2+c_{-2})\nn
&&-2a(1-Z(a)^2)
\sum_{n=2}^\infty (-1)^n Z(a)^{n-1}(c_{2n}+c_{-2n}),
\end{eqnarray}
where we expand the BRS current and the ghost field as $J_{\rm
B}(w)=\sum_n Q_n w^{-n-1}$ and $c(w)=\sum_n c_n w^{-n+1}$. The function
$Z(a)$ is defined by $Z(a)=(1+a-\sqrt{1+2a})/a$ and it varies
from $-1$ to $1$ for $a\geq -1/2$.

If the parameter $a$ is not equal to $-1/2$, the action can be
transformed into the action with the ordinary BRS charge through the string
field redefinition\cite{rf:TT2,rf:tomo}
\begin{eqnarray}
 \Phi'=e^{K(h_a)}\Phi,
\end{eqnarray}
where the operator $K(f)$ is defined by using the ghost number current
$J_{\rm gh}=cb$ as
\begin{eqnarray}
 K(f)=\oint\frac{dw}{2\pi i}f(w)\,
\left(J_{\rm gh}(w)-\frac{3}{2}\,w^{-1}\right).
\end{eqnarray}
Substituting the function (\ref{Eq:ha}) into this definition,
we find the mode expansion form of $K(h_a)$ as
\begin{eqnarray}
 K(h_a)=-\tilde{q}_0 \log(1-Z(a))^2
-\sum_{n=1}^\infty \frac{(-1)^n}{n}(q_{2n}+q_{-2n}) Z(a)^n,
\end{eqnarray}
where $\tilde{q}_0$ and $q_n$ are written by the ghost oscillators as
\begin{eqnarray}
 \tilde{q}_0&=&\frac{1}{2}(c_0 b_0-b_0 c_0)
+\sum_{n=1}^\infty (c_{-n} b_n-b_{-n} c_n),\nn
 q_n &=& \sum_{m=-\infty}^\infty c_{n-m}b_m\ \ \ (n\neq 0).
\end{eqnarray}
Under the string field  redefinition, the modified BRS charge transforms
into the original one as follows,\cite{rf:TT2,rf:tomo}
\begin{eqnarray}
\label{Eq:simtrans}
\Q=e^{-K(h_a)}\,\Q'(a)\,e^{K(h_a)}.
\end{eqnarray}
Hence we expect that the universal solution is pure gauge for
$a>-1/2$ as discussed in ref.~\citen{rf:TT2}. Numerical analyses
strongly support this expectation.\cite{rf:tomo}

Let us consider the BRS cohomology in
the theory around the universal solution for $a>-1/2$.
As the cohomology of the Kato-Ogawa BRS charge, we know that any state
$\ket{\psi}$ satisfying $\Q\ket{\psi}=0$ can be written as 
\begin{eqnarray}
\label{Eq:cohomology1}
 \ket{\psi}=\ket{P}\otimes c_1 \ket{0}+
\ket{P'}\otimes c_0\,c_1 \ket{0}+\Q\ket{\phi},
\end{eqnarray}
where $\ket{P}$ and $\ket{P'}$ are positive norm states in the matter
sector,  and if we consider flat backgrounds
they are DDF states.\cite{rf:KatoOgawa,rf:Henneaux,rf:FGZ,rf:AsanoNatsu}
The perturbative equation of motion for the fluctuation is given by
$\Q'(a)\Phi=0$.
Since the modified and original BRS charges are related by the
similarity transformation (\ref{Eq:simtrans}), we can solve the equation
of motion as 
\begin{eqnarray}
\label{Eq:cohomology2}
 \ket{\Phi}=\ket{P}\otimes e^{K(h_a)}\,c_1 \ket{0}+
\ket{P'}\otimes e^{K(h_a)}\,c_0\,c_1 \ket{0}+\Q'(a)\ket{\phi}.
\end{eqnarray}
In this solution, the non-trivial cohomology parts possess the same
ghost number as the physical states in the original theory, and
$\ket{P}$ and $\ket{P'}$ in the matter sector do not change. 
This result suggests that the 
physical spectrum dose not change from the original theory and 
it is natural if the universal solution for $a>-1/2$
corresponds to pure gauge.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gauge fixing and equation of motion}

Though the cohomology is solved in the above subsection,
we must fix the gauge to determine the
physical spectrum precisely. Here, we apply the Siegel gauge
condition to the theory around the universal solution:
\begin{eqnarray}
 b_0\Phi=0.
\end{eqnarray}
Under the Siegel gauge condition, the equation of motion is given by
\begin{eqnarray}
\label{Eq:eqmotion}
L(a)\,\Phi=0,\ \ \ L(a)=\{\Q'(a),\,b_0\}.
\end{eqnarray}
From (\ref{Eq:modBRS2}), we can write the operator $L(a)$ by an oscillator
expression as\cite{rf:tomo}
\begin{eqnarray}
\label{Eq:La1}
 L(a)=(1+a)L_0 +\frac{a}{2}(L_2+L_{-2})+a(q_2-q_{-2})+4aZ(a),
\end{eqnarray}
where $L_n$ denote the total Virasoro generators. We can rewrite it by
the twisted ghost Virasoro generators
$L_n'=L_n+nq_n+\delta_{n,0}$:\cite{rf:GRSZ}
\begin{eqnarray}
\label{Eq:La2}
 L(a)=(1+a)L_0'+\frac{a}{2}(L_2'+L_{-2}')+4aZ(a)-1-a.
\end{eqnarray}
Since the twisted ghost conformal field theory has the central charge
$c'=24$, the three operators $L_0'+3$, $L_{\pm2}'$ satisfy $SL(2,R)$
algebra. Then, it is useful to rewrite $L(a)$ as
\begin{eqnarray}
\label{Eq:La3}
 L(a)=2(1+a)l_0+a(l_2+l_{-2})+4aZ(a)-4(1+a),
\end{eqnarray}
where the operators $l_0,\,l_{\pm 2}$ are defined by
\begin{eqnarray}
 l_0=\frac{1}{2}(L_0'+3),\ \ \ l_{\pm 2}=\frac{1}{2}L_{\pm 2}',
\end{eqnarray}
and these satisfy the algebra
\begin{eqnarray}
\label{Eq:SLcomrel}
\left[l_0,\,l_{\pm 2}\right] = \mp l_{\pm 2},\ \ \ 
\left[l_2,\,l_{-2}\right] = 2l_0.
\end{eqnarray}

In order to solve the gauge fixed equation of motion
(\ref{Eq:eqmotion}), we 
try to diagonalize the operator $L(a)$. Since the operator
$L(a)$ is expressed by the $SL(2,R)$ generators as in
(\ref{Eq:La3}), it may be diagonalized through $SL(2,R)$
transformations. If we restrict the $SL(2,R)$ group to a subgroup
written by `normal ordered operators', arbitrary elements in the
subgroup can be represented as
\begin{eqnarray}
\label{Eq:SLelemnt}
 U(s,t,u)=\exp(s\,l_{-2})\,\exp(t\,l_0)\,\exp(u\,l_2),
\end{eqnarray}
where $s,t,u$ are real parameters. Using the algebra
(\ref{Eq:SLcomrel}), we can calculate the $SL(2,R)$ transformation of
the operator $L(a)$ as
\begin{eqnarray}
 U(s,t,u)L(a)U(s,t,u)^{-1}
&=& \left\{2(1+a+au)-2s(au^2+2(1+a)u+a)\,e^{-t}\right\}\,l_0\nn
&&\hspace{-.5cm}
   +(au^2+2(1+a)u+a)\,e^{-t}\,l_2\nn
&&\hspace{-.5cm}
   +\left\{
    a\,e^t -2a su -2(1+a) s +s^2 (au^2+2(1+a)u+a)\,e^{-t}
    \right\}\,l_{-2}\nn
&&\hspace{-.5cm}
   +4aZ(a)-4(1+a).
\end{eqnarray}
If the coefficients of $l_{\pm 2}$ vanish, the
parameters $s,t,u$ satisfy
\begin{eqnarray}
\label{Eq:stu1}
&& au^2+2(1+a)u+a=0, \\
\label{Eq:stu2}
&& a\,e^t-2a su -2(1+a) s =0.
\end{eqnarray}
From (\ref{Eq:stu1}), $u$ is equal to $-Z(a)$ or $-1/Z(a)$. Substituting
these values of $u$ into (\ref{Eq:stu2}), we obtain
\begin{eqnarray}
\label{Eq:stu3}
 a\,e^t \mp 2\sqrt{1+2a}\,s=0.
\end{eqnarray}
It should be noted that there is no solution of the
equation (\ref{Eq:stu3}) if $a=-1/2$ and $t$ is a finite real
number. This fact implies that the operator $L(a)$ cannot be transformed
into the linear form of $l_0$ through
regular $SL(2,R)$ transformations. This difference between the cases
$a>-1/2$ and $a=-1/2$ is a natural consequence, because the theories
should have different physical 
spectrum in each case if our conjecture for the universal solution is
true.

In the case $a>-1/2$, we can set $s=-u$ in order to
diagonalize  $L(a)$. 
As a result, the operator $L(a)$ can be transformed into
$L_0$ as
\begin{eqnarray}
\label{Eq:Ldiag}
 U'(a)L(a)U'(a)^{-1}=\sqrt{1+2a}\,L_0,
\end{eqnarray}
where $U'(a)$ is given by
\begin{eqnarray}
\label{Eq:Ua}
 U'(a)&=&\exp\left\{\frac{1}{2}\,Z(a)L_{-2}'\right\}
      \exp\left\{\frac{1}{2}(L_0'+3)\log(1-Z(a)^2)\right\}
      \exp\left\{-\frac{1}{2}\,Z(a)L_2'\right\}\nn
     &=&\exp\left\{
     -\frac{1}{4}\left(L_2'-L_{-2}'\right)
    \log\left(\frac{1+Z(a)}{1-Z(a)}\right)\right\}.
\end{eqnarray}
Therefore we can solve the gauge fixed equation of motion
(\ref{Eq:eqmotion}) as 
$\Phi={U'(a)}^{-1} \Phi_0$, where $\Phi_0$ satisfies $L_0 \Phi_0=0$.

For later convenience we provide another procedure for the
diagonalization of $L(a)$ referring to conformal field theory.
In the twisted ghost conformal field theory, the operator $U'(a)$ induces
a conformal mapping represented by the function\cite{rf:LPP1}
\begin{eqnarray}
\label{Eq:fa}
f_a(w)=\left(\frac{w^2+Z(a)}{Z(a)\,w^2+1}\right)^{\frac{1}{2}}.
\end{eqnarray}
Namely, if $\phi(w)$ is a primary field of dimension $h$, the
transformation of $\phi(w)$ by $U'(a)$ is given by
\begin{eqnarray}
 U'(a)\phi(w)U'(a)^{-1} = \left(\frac{df_a(w)}{dw}\right)^h
\phi\left(f_a(w)\right).
\end{eqnarray}
For $0<Z(a)<1\ (a>0)$, this conformal mapping is depicted in
fig.~\ref{fig:map1}. In the $z$ plane, there is a branch cut connecting
$-\sqrt{Z(a)}$ and $\sqrt{Z(a)}$. The branch cut corresponds to a line
segment along the imaginary axis between $-i\sqrt{Z(a)}$ and
$i \sqrt{Z(a)}$ in the $w$ plane. There is another branch cut from $\pm
1/\sqrt{Z(a)}$ to infinity in the $z$ plane. 
For $-1<Z(a)<0\ (-1/2<a<0)$, the mapping is represented by the figure
which is given by rotating fig.~\ref{fig:map1} clockwise $90$
degree. 

\begin{figure}
\centerline{\includegraphics[width=14cm]{map1.eps}}
\caption{Conformal mapping of the $w$ plane to $z=f_a(w)$ in
the case $0<Z(a)<1\ (a>0)$.}
\label{fig:map1}
\end{figure}

From (\ref{Eq:La2}), the operator $L(a)$ can be expressed in terms of the
twisted energy momentum tensor $T'(w)=\sum L_n' w^{-n-2}$:
\begin{eqnarray}
\label{Eq:LT}
 L(a)=\oint \frac{dw}{2\pi i}\,w\,e^{h_a(w)}\,T'(w)
+4Z(a)-1-a,
\end{eqnarray}
where the integration contour is a unit circle.
Generally, the kinetic operator can be written by using the twisted
energy momentum tensor in the theory around universal
solutions in the Siegel gauge.\cite{rf:Drukker2} 
Then the transformation of $L(a)$ by $U'(a)$ is given by
\begin{eqnarray}
\label{Eq:ULTU}
 U'(a)L(a)U'(a)^{-1} 
=
\oint \frac{dw}{2\pi i}\,w\,e^{h_a(w)}\,U'(a)T'(w)U'(a)^{-1}
+4Z(a)-1-a.
\end{eqnarray}
Through the conformal mapping $z=f_a(w)$,
the twisted energy momentum tensor is transformed to
\begin{eqnarray}
\label{Eq:UTU}
 U'(a)T'(w)U'(a)^{-1}=
 \left(\frac{df_a(w)}{dw}\right)^2 T'(f_a(w))+\frac{c'}{12} S(f_a,w),
\end{eqnarray}
where the central charge $c'$ is $24$ and $S(f,w)$ denotes the
Schwartzian derivative 
\begin{eqnarray}
 S(f,w)=\frac{\partial^3 f(w)}{\partial f(w)}-\frac{3}{2}\left(
\frac{\partial^2 f(w)}{\partial f(w)}\right)^2.
\end{eqnarray}
Using (\ref{Eq:ha}) and (\ref{Eq:fa}), we see that
\begin{eqnarray}
&&
 e^{h_a(w)}=\frac{1}{(1-Z(a))^2}\,w^{-2}\,(w^2+Z(a))(Z(a)w^2+1), \\
&&
 S(f_a,w)=-\frac{3Z(a)}{2}\,w^{-2}\,
\frac{(1+2Z(a)w^2+w^4)(Z(a)+2w^2+Z(a)w^4)}{
(w^2+Z(a))^2(Z(a)w^2+a)^2},
\end{eqnarray}
and then
\begin{eqnarray}
\label{Eq:ehf}
&&
 w\,e^{h_a(w)}\,f_a'(w)=\frac{1+Z(a)}{1-Z(a)}\,f_a(w), \\
\label{Eq:ehS}
&&
 w\,e^{h_a(w)}\,S(f_a,w)
=-\frac{3Z(a)}{2(1-Z(a))^2}\nn
&&\hspace{4cm}\times
\frac{(1+2Z(a)w^2+w^4)(Z(a)+2w^2+Z(a)w^4)}{w^3\,
(w^2+Z(a))(Z(a)w^2+a)}.
\end{eqnarray}
Substituting (\ref{Eq:ULTU}) into (\ref{Eq:UTU}) and then using
(\ref{Eq:ehf}) and (\ref{Eq:ehS}), we find
\begin{eqnarray}
 U'(a)L(a)U'(a)^{-1}
&=&
\frac{1+Z(a)}{1-Z(a)}\,
\oint \frac{dw}{2\pi i}\,f_a(w)\,\frac{df_a(w)}{dw}\,T'(f_a(w)) \nn
&&
-\frac{3Z(a)}{(1-Z(a))^2}
\oint \frac{dw}{2\pi i}\,
\frac{(1+2Z(a)w^2+w^4)(Z(a)+2w^2+Z(a)w^4)}{w^3\,
(w^2+Z(a))(Z(a)w^2+a)} 
\nn
&&
+4Z(a)-1-a.
\end{eqnarray}
In the first term in the right-hand side, we can change the variable of
integration to $z=f_a(w)$, since the integration contour
does not across the branch cut in the $z$ plane as seen in
Fig.~\ref{fig:map1}.  Then the first term becomes
\begin{eqnarray}
\label{Eq:term1}
 \frac{1+Z(a)}{1-Z(a)}\,
\oint \frac{dz}{2\pi i}\,z\,T'(z)
=\sqrt{1+2a}\,L_0'. 
\end{eqnarray}
If $1>Z(a)>0$, the integral in the second term can be reduced to the
summation of the residues at $w=0$ and $\pm i \sqrt{Z(a)}$,
each of which can be evaluated as
\begin{eqnarray}
\label{Eq:term2}
&&
\oint_{C_0} = \frac{1+Z(a)^2}{Z(a)},
\ \ \ 
\oint_{C_{i\,\sqrt{Z(a)}}}=
\oint_{C_{-i\,\sqrt{Z(a)}}}
=\frac{-1+Z(a)^2}{2 Z(a)}.
\end{eqnarray}
If $-1<Z(a)<0$, the residues at $\pm i \sqrt{Z(a)}$ are replaced by the
ones at $\pm \sqrt{-Z(a)}$, but their values are unchanged.
Finally, substituting (\ref{Eq:term1}) and (\ref{Eq:term2}) into
(\ref{Eq:ULTU}), we can derive the same result as obtained from operator
expressions:
\begin{eqnarray}
 U'(a)L(a)U'(a)^{-1}
&=&\sqrt{1+2a}\,L_0'-\frac{3 Z(a)}{(1-Z(a))^2}\,2Z(a)
+4Z(a)-1-a \nn
&=& \sqrt{1+2a}\,L_0.
\end{eqnarray}
Note that this result can not be applied to the case of $a=-1/2$,
because the operator $U'(a)$ becomes singular at $a=-1/2$ as seen in
(\ref{Eq:Ua}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Physical subspace}

First, we show that the modified BRS charge is transformed
into the original one through the similarity transformation by $U'(a)$:
\begin{eqnarray}
\label{Eq:UQU1}
 U'(a)\Q'(a)U'(a)^{-1}=\sqrt{1+2a}\,\Q.
\end{eqnarray}

The commutation relations of $L_m$ and $q_m$ with $Q_n$ are given by
\begin{eqnarray}
&& \left[L_m,\,Q_n\right]=-nQ_{m+n},\nn
&&
\left[q_m,\,Q_n\right]=Q_{m+n}-2m n\,c_{m+n}.
\end{eqnarray}
Then we obtain the commutation relation of $L_m'$ with $Q_n$ as
\begin{eqnarray}
 \left[L_m',\,Q_n\right]=
(m-n)Q_{m+n}-2m^2 n\,c_{m+n}.
\end{eqnarray}
The first term in the left-hand side implies that $Q_n$ is transformed as
oscillators of 
a dimension two field in the twisted ghost theory.
Due to the second term, $Q_n$
is not a oscillator of a primary field.
This is a natural result from the fact that
the BRS current is written by $J_{\rm B}(w)=w c'(w) T_X(w)+\cdots$,
where $c'(w)$ and $T_X(w)$ denote the ghost field with dimension zero and
the matter energy momentum tensor with dimension two in the twisted
theory, and the abbreviations stands for the terms containing no matter
oscillators.

As a result, since $\Q'(a)$ and $L(a)$ have the similar oscillator
expressions 
\begin{eqnarray}
 \Q'(a)&=&(1+a)\Q+\frac{a}{2}(Q_2+Q_{-2})+\cdots,\\
 L(a)&=&(1+a)L_0'+\frac{a}{2}(L_2'+L_{-2}')+\cdots,
\end{eqnarray} 
and $L_n'$ are oscillators of dimension 2 field, we can obtain the
transformation of $\Q'(a)$ by referring to the previous
result for $L(a)$ as follows,
\begin{eqnarray}
\label{Eq:UQU2}
 U'(a)\,\Q'(a)U'(a)^{-1}
= \sqrt{1+2a}\,\Q +\cdots,
\end{eqnarray}
where the abbreviations denote pure ghost contributions.
Since the right-hand side of (\ref{Eq:UQU2}) is nilpotent, the pure ghost
contributions are determined to be zero. Because the operator of the
form $Q(f)+C(g)$ must be $Q(e^f)-C((\partial f)^2 e^f)$ or $C(f)$ if it
is nilpotent.\cite{rf:KT} Therefore we can show that the transformation
law of (\ref{Eq:UQU1}) holds.

Here, we should discuss the cohomology without the Siegel gauge
condition. Using the equation (\ref{Eq:UQU1}), we show that the
state $\ket{\Phi}$ with the condition $\Q'(a)\ket{\Phi}=0$ can be
written as
\begin{eqnarray}
\label{Eq:cohomology4}
 \ket{\Phi}=U'(a)^{-1}
(\ket{P}\otimes \,c_1 \ket{0})+
U'(a)^{-1}
(\ket{P'}\otimes \,c_0\,c_1 \ket{0})+\Q'(a)\ket{\phi}.
\end{eqnarray}
However, this expression is different from the state
(\ref{Eq:cohomology2}), though these states are imposed by the same
condition. Naturally, we can show that the apparent difference of these
states reduces merely to a BRS-exact state.

In order to see this fact, let us prove that
\begin{eqnarray}
\label{Eq:UfUfK}
 U'_f={\rm const.}\times U_f\times \exp K\left(
\log \frac{w\,\partial f(w)}{f(w)}\right),
\end{eqnarray}
where $U'_f$ and $U_f$ are the operators that implement the conformal
transformation $z=f(w)$ in the twisted and untwisted ghost conformal
field theory, respectively.

In the untwisted theory, the ghost and anti-ghost fields
are expanded in a unit disc as 
\begin{eqnarray}
 c(w)=\sum_{n=-\infty}^\infty c_n w^{-n+1},\ \ \ 
 b(w)=\sum_{n=-\infty}^\infty b_n w^{-n-2},
\end{eqnarray}
and in the twisted theory they are expanded as
\begin{eqnarray}
  c'(w)=\sum_{n=-\infty}^\infty c_n w^{-n},\ \ \ 
 b'(w)=\sum_{n=-\infty}^\infty b_n w^{-n-1}.
\end{eqnarray}
Then the ghost and anti-ghost fields in the both theories can be
connected by the relations 
\begin{eqnarray}
\label{Eq:bcrel}
 c'(w)=w^{-1}\,c(w),\ \ \ b'(w)=w\,b(w).
\end{eqnarray}
For these ghost fields, the operators $U_f$ and $U'_f$ induce the
following transformations,
\begin{eqnarray}
\label{Eq:UcU1}
 U_f\,c(w)\,U_f^{-1} &=&
   \left(\partial f(w)\right)^{-1} c\left(f(w)\right),\\
\label{Eq:UbU1}
 U_f\,b(w)\,U_f^{-1} &=&
   \left(\partial f(w)\right)^2 b\left(f(w)\right),\\
\label{Eq:UcU2}
 U'_f\,c'(w)\,{U'_f}^{-1} &=&
   \left(\partial f(w)\right)^0 c'\left(f(w)\right),\\
\label{Eq:UbU2}
 U'_f\,b'(w)\,{U'_f}^{-1} &=&
   \left(\partial f(w)\right)^1 b'\left(f(w)\right).
\end{eqnarray}
Combining (\ref{Eq:bcrel}) and (\ref{Eq:UcU1}), we can
find the transformation of $c'(w)$ through $U_f$:
\begin{eqnarray}
\label{Eq:UcU3}
 U_f\,c'(w)\,U_f^{-1} &=& 
 w^{-1}\,U_f\,c(w)\,U_f^{-1}\nn
&=&
 w^{-1}\,(\partial f(w))^{-1}\,c(f(w))\nn
&=&
 \frac{f(w)}{w\partial f(w)}\,c'(f(w)).
\end{eqnarray}
Similarly, from (\ref{Eq:bcrel}) and (\ref{Eq:UcU1}), the transformation
of $b(w)$ is given by 
\begin{eqnarray}
\label{Eq:UbU3}
 U_f\,b'(w)\,{U_f}^{-1} &=& \frac{w(\partial f(w))^2}{f(w)}
\,b'(f(w)).
\end{eqnarray}
We can realize the same transformations of (\ref{Eq:UcU3}) and
(\ref{Eq:UbU3}) by $U_f\,e^{K(g)}$.
For an arbitrary function $g(w)$, the operator $K(g)$ generates
the transformations\cite{rf:TT2,rf:Drukker2,rf:tomo}
\begin{eqnarray}
\label{Eq:KcK}
 e^{K(g)}\,c'(w)\,e^{-K(g)} &=& e^{g(w)}\,c'(w),\\
\label{Eq:KbK}
 e^{K(g)}\,b'(w)\,e^{-K(g)} &=& e^{-g(w)}\,b'(w).
\end{eqnarray}
Combining (\ref{Eq:UcU3}), (\ref{Eq:UbU3}), (\ref{Eq:KcK}) and
(\ref{Eq:KbK}), we find that 
\begin{eqnarray}
\label{Eq:UKcKU}
 U_f e^{K(g)}\,c'(w)\,e^{-K(g)} U_f^{-1} &=&
 e^{g(w)}\,\frac{f(w)}{w(\partial f(w))}\,c'(f(w)), \\
\label{Eq:UKbKU}
 U_f e^{K(g)}\,b'(w)\,e^{-K(g)} U_f^{-1} &=&
 e^{-g(w)}\,\frac{w(\partial f(w))^2}{f(w)} b'(f(w)).
\end{eqnarray}
Comparing (\ref{Eq:UcU2}) and (\ref{Eq:UbU2}) with (\ref{Eq:UKcKU}) and
(\ref{Eq:UKbKU}), we find that these transformation laws are equivalent
to each other if 
$g(w)$ is given by
\begin{eqnarray}
 g(w)=\log\left(\frac{w\,\partial f(w)}{f(w)}\right).
\end{eqnarray}
The ghost part of $U'_f$
is uniquely determined by the transformation laws
(\ref{Eq:UcU2}) and (\ref{Eq:UbU2}) up to a multiplicative constant.
The matter parts of $U_f$ and $U'_f$ have a same form since they are
uncharged by the twist operation. 
Therefore the equation (\ref{Eq:UfUfK}) is proved.
In appendix A, we generalize the equation to the case of general
background charge.

We apply the formula (\ref{Eq:UfUfK}) to the mapping $f(w)=f_a(w)$
given by (\ref{Eq:fa}). We can calculate the function in the operator
$K$ as
\begin{eqnarray}
 \log\left(
\frac{w \partial f_a(w)}{f_a(w)}
\right) &=&
\log\left\{
   (1-Z(a)^2) \frac{w^2}{(w^2+Z(a))(Z(a)w^2+1)}
\right\} \nn
&=& \log\left\{
   (1-Z(a))^2 \frac{w^2}{(w^2+Z(a))(Z(a)w^2+1)}
\right\}+\log\frac{1+Z(a)}{1-Z(a)}
\nn
&=& -h_a(w)+\log\frac{1+Z(a)}{1-Z(a)}.
\end{eqnarray}
Therefore we can represent the operator $U'(a)$ in terms of the
untwisted Virasoro generators as
\begin{eqnarray}
\label{Eq:UaUaK}
 U'(a)= U(a)\,e^{-K(h_a)}\,\exp\left(\tilde{q}_0
\log\frac{1+Z(a)}{1-Z(a)}\right),
\end{eqnarray}
where the operator $U(a)$ is defined as
\begin{eqnarray}
\label{Eq:Ua2}
 U(a) &=&\exp\left\{
     -\frac{1}{4}\left(L_2-L_{-2}\right)
    \log\left(\frac{1+Z(a)}{1-Z(a)}\right)\right\} \nn
&=&\exp\left\{\frac{1}{2}\,Z(a)L_{-2}\right\}
      \exp\left\{\frac{1}{2}\,L_0\log(1-Z(a)^2)\right\}
      \exp\left\{-\frac{1}{2}\,Z(a)L_2\right\}.
\end{eqnarray}
Here, the multiplicative constant is determined to be one by the fact
that $U'(a)$ is a unitary operator.\footnote{The hermiticity property
is given by $(q_n)^\dagger=-q_{-n}\ (n\neq 0)$,
$(\tilde{q}_0)^\dagger=-\tilde{q}_0$ and $(L_n)^\dagger=L_{-n}$.}
Since the untwisted Virasoro generators commute with the BRS charge, it
is easily seen from (\ref{Eq:UaUaK}) that (\ref{Eq:simtrans}) and
(\ref{Eq:UQU1}) are realized simultaneously.

Using (\ref{Eq:UaUaK}), we can express the operator $U'(a)^{-1}$ as
\begin{eqnarray}
 U'(a)^{-1} &=&
 e^{K(h_a)}\,U(a)^{-1}\exp\left(-\tilde{q}_0
\log\frac{1+Z(a)}{1-Z(a)}\right)
\nn
&=&
 e^{K(h_a)}\times \sum_{n=0}^\infty
\frac{1}{n!}(L_2-L_{-2})^n \theta^n \times
e^{-4 \tilde{q}_0\,\theta}
\ \ \ \left(
\theta=
\frac{1}{4}\log\frac{1+Z(a)}{1-Z(a)}
\right)
\nn
&=&
 e^{K(h_a)}\,e^{-4 \tilde{q}_0 \theta} \nn
&& + e^{K(h_a)}\times
\left\{\Q,\,
\sum_{n=1}^\infty
\frac{1}{n!}(b_2-b_{-2})(L_2-L_{-2})^{n-1} \theta^n 
\right\}
\times
e^{-4 \tilde{q}_0 \theta}
\end{eqnarray}
Then we can rewrite the first term of (\ref{Eq:cohomology4}) as
\begin{eqnarray}
\label{Eq:UPKP1}
 U'(a)^{-1}(\ket{P}\otimes c_1 \ket{0}) &=&
e^{2\theta}\ket{P}\otimes e^{K(h_a)} c_1\ket{0}\nn
&&
+e^{K(h_a)}\Q
\sum_{n=1}^\infty
\frac{1}{n!}(b_2-b_{-2})(L_2-L_{-2})^{n-1} \theta^n 
(e^{2\theta}\ket{P}\otimes c_1\ket{0}) \nn
&=&
e^{2\theta}
\ket{P}\otimes e^{K(h_a)} c_1\ket{0}
+\Q'(a)\ket{\phi},
\end{eqnarray}
where $\ket{\phi}$ is given by
\begin{eqnarray}
\ket{\phi}= e^{K(h_a)}\sum_{n=1}^\infty
\frac{1}{n!}(b_2-b_{-2})(L_2-L_{-2})^{n-1} \theta^n 
(e^{2\theta}\ket{P}\otimes c_1\ket{0}).
\end{eqnarray}
Similarly, the second term of (\ref{Eq:cohomology4}) can be
written as
\begin{eqnarray}
\label{Eq:UPKP2}
 U'(a)^{-1}(\ket{P'}\otimes c_0 c_1\ket{0})
&=& 
e^{-2\theta}
\ket{P'}\otimes e^{K(h_a)} c_0 c_1\ket{0}
+\Q'(a)\ket{\phi'}.
\end{eqnarray}
Hence, from (\ref{Eq:UPKP1}) and (\ref{Eq:UPKP2}),
we show that the difference between the cohomologies
(\ref{Eq:cohomology2}) and (\ref{Eq:cohomology4})
can be expressed by a BRS-exact state as stated before.

Now that the cohomology of $\Q'(a)$ has been established, we can
obtain the physical subspace specified by the conditions
\begin{eqnarray}
\label{Eq:Qbphi}
 \Q'(a)\ket{\Phi}=0,\ \ \ b_0\ket{\Phi}=0.
\end{eqnarray}
Since, through the same similarity transformation,  $\Q'(a)$ and $L(a)$
are transformed into $\Q$ and $L_0$, the state $\ket{\Phi}$
satisfying (\ref{Eq:Qbphi}) can be written as
\begin{eqnarray}
\label{Eq:cohomology3}
 \ket{\Phi}=U'(a)^{-1}\left(
\ket{P}\otimes c_1\ket{0}
+ \tQ \ket{\phi} \right),
\end{eqnarray}
where $\tQ$ denotes the terms of $\Q$ which do not contain the ghost and
anti-ghost zero modes, $c_0$ and $b_0$.
Here, we have used the cohomology for the Kato-Ogawa BRS charge
(\ref{Eq:cohomology1}) and the commutation relations $[L_n',\,b_0]=0$.
Hence, there is a one-to-one correspondence between the spectra of the
original theory and of the
theory around the universal solution for $a>-1/2$. They are connected
through the similarity 
transformation by $U'(a)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Scattering amplitudes}

We consider scattering amplitudes in the theory around
the universal solution. 
The general amplitude is calculated by evaluating an expression of
the form
\begin{eqnarray}
 {\cal A} = 
\left(\prod \bra{V}\right)\left(\prod \frac{b_0}{L(a)}\right)
\left(\prod \ket{R}\right)
 \left(\prod\ket{\rm external}' \right),
\end{eqnarray}
where $\bra{V}$, $\ket{R}$ and $\ket{\rm external}'$ are vertices,
reflectors and external states, respectively. $b_0/L(a)$ is
the propagator in the theory around the solution. From (\ref{Eq:Ldiag})
and $[L_n',\,b_0]=0$, we 
can rewrite the propagator as
\begin{eqnarray}
\label{Eq:prop}
 \frac{b_0}{L(a)} 
&=& \frac{1}{\sqrt{1+2a}}\times {U'(a)}^{-1}\,\frac{b_0}{L_0}\,
U'(a).
\end{eqnarray}
From (\ref{Eq:cohomology3}), the external states can be
written as the 
similarity transform of the external states in the original theory:
\begin{eqnarray}
\label{Eq:extstate}
\ket{\rm external}'={U'(a)}^{-1}\ket{\rm external}.
\end{eqnarray}
It can be easily seen that on the reflector the operator
$K'_n=L'_n-(-1)^n L'_{-n}$
satisfies
\begin{eqnarray}
 {}_{12}\bra{R}({K'_n}^{(1)}+{K'_n}^{(2)})=0,
\end{eqnarray}
and then we find
\begin{eqnarray}
\label{Eq:reflector}
{}_{12}\bra{R}\prod_{r=1}^2\,(U'(a)^{(r)})^{-1}
={}_{12}\bra{R}.
\end{eqnarray}
Using (\ref{Eq:prop}), (\ref{Eq:extstate}) and (\ref{Eq:reflector}), 
we can rewrite the amplitude to the form
\begin{eqnarray}
 {\cal A} = 
\left(\prod \bra{V'} \right)\left(\prod \frac{1}{\sqrt{1+2a}} 
\frac{b_0}{L_0}\right)
\left(\prod \ket{R}\right)
 \left(\prod\ket{\rm external} \right).
\end{eqnarray}
The difference from the original amplitude is the normalization factor
of the propagator, and the change of the vertex to
\begin{eqnarray}
{}_{123}\bra{V'} ={}_{123}\bra{V}\prod_{r=1}^3{U'(a)^{(r)}}^{-1}.
\end{eqnarray}

The operator $K_n=L_n-(-1)^n L_{-n}$ is conserved on the
original vertex.\cite{rf:SSFT}
Moreover we can find that the operator $K(h_a)$ is conserved on the
original vertex.\cite{rf:TT2} Using these conservation laws and the
expression  
(\ref{Eq:UaUaK}) of $U'(a)$,
we find that the
modified vertex is rewritten as
\begin{eqnarray}
 {}_{123}\bra{V'}=
{}_{123}\bra{V}\prod_{r=1}^3\,
\exp\left(-\log \sqrt{1+2a}\,{\tilde{q}_0}^{(r)}\right).
\end{eqnarray}
Under the SL(2,R) normal ordering, the zero mode $q_0$ of the ghost
number current has the anomalous contribution on the vertex:\cite{rf:RZ}
\begin{eqnarray}
 {}_{123}\bra{V}\sum_{r=1}^3{q_0}^{(r)}=
 3\times{}_{123}\bra{V}.
\end{eqnarray}
Since $\tilde{q}_0=q_0-3/2$,  we find
\begin{eqnarray}
{}_{123} \bra{V'}=\left(
\sqrt{1+2a}\right)^{\textstyle \frac{3}{2}}\times{}_{123}\bra{V}.
\end{eqnarray}

As a result, the perturbative amplitude in the theory around the solution
is given by the form
\begin{eqnarray}
 {\cal A} = 
\left(\prod \left(
\sqrt{1+2a}\right)^{\textstyle \frac{3}{2}}
\bra{V} \right)\left(\prod \frac{1}{\sqrt{1+2a}} 
\frac{b_0}{L_0}\right)
\left(\prod \ket{R}\right)
 \left(\prod\ket{\rm external} \right). 
\end{eqnarray}
The normalization factors of the propagators and the vertices
are canceled out each other and then the remained factors can be
absorbed into the normalization of external states as
$(\sqrt{1+2a})^{1/2}\times \ket{\rm external}$. 
After all,  the amplitude becomes equal to the corresponding amplitude
in the original theory. 
Hence, we conclude that the theory for $a>-1/2$
describes the same physics of the theory with the original BRS charge.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{String Field Theory around Non-trivial Solutions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Non-trivial solutions}

In this section we consider the theory expanded around the solution
which is obtained by setting the parameter $a$ to $-1/2$.
At $a=-1/2$, the function (\ref{Eq:ha}) is given by
\begin{eqnarray}
 h(w)=\log\left(-\frac{1}{4}\left(w-\frac{1}{w}\right)^2\right),
\end{eqnarray}
and the modified BRS charge (\ref{Eq:modBRS2}) becomes
\begin{eqnarray}
 \bQ &\equiv& \Q'(-1/2) \nn
   &=&
\frac{1}{2}\Q-\frac{1}{4}(Q_2+Q_{-2})
+2c_0 +c_2+c_{-2}.
\end{eqnarray}

Through a similarity transformation, this modified BRS charge can be
transformed into the form which contains
specific level oscillators:\cite{rf:KT}
\begin{eqnarray}
\label{Eq:KbQK}
 e^{K(\rho)}\bQ e^{-K(\rho)}=
-\frac{1}{4}Q_2+c_2,
\end{eqnarray}
where $\rho(w)$ and $K(\rho)$ are given by
\begin{eqnarray}
\label{Eq:rho}
&&
 \rho(w)=-2\log(1-w^{-2})
=2\sum_{n=1}^\infty \frac{1}{n}w^{-2n}, \\
\label{Eq:Krho}
&&
 K(\rho)=2\sum_{n=1}^\infty \frac{1}{n}q_{-2n}.
\end{eqnarray}
Here we introduce the operator $O^{(k)}$ to a certain operator
$O$ which is defined by replacing the ghost oscillator modes $c_n$
and $b_n$ in $O$ by $c_n^{(k)}=c_{n+k}$ and $b_n^{(k)}=b_{n-k}$
without changing their order.\cite{rf:KT}\footnote{
Notice that this notation $O^{(k)}$ does not mean the operator $O$ of
{\it r}-th string.}
Under this definition, we can rewrite the equation (\ref{Eq:KbQK}) as
\begin{eqnarray}
\label{Eq:KbQK2}
\bQ =
-\frac{1}{4}\,e^{-K(\rho)}\,\Q^{(2)}\,e^{K(\rho)}.
\end{eqnarray}

For this $bc$-shift transformation, it is important that
the original algebra of the operator $O$ is realized to
the operator $O^{(k)}$. This follows from
the anti-commutation relations
$\{c_m^{(k)},\,b_n^{(k)}\}=\delta_{m+n,0}$. Consequently, through the
relation (\ref{Eq:KbQK2}), we can determine 
the cohomology of $\bQ$ by referring the original cohomology of $\Q$. We
find that any state $\ket{\psi}$ satisfying $\bQ\ket{\psi}=0$ can be
written as\cite{rf:KT}
\begin{eqnarray}
\label{Eq:cohom1}
 \ket{\psi}=\ket{P}\otimes e^{-K(\rho)}b_{-2}\ket{0}
+\ket{P'}\otimes e^{-K(\rho)}\ket{0}+\bQ\ket{\phi}.
\end{eqnarray}
Here, the states $b_{-2}\ket{0}$ and $\ket{0}$ correspond to 
$c_1\ket{0}$ and $c_0 c_1\ket{0}$ in the original cohomology
(\ref{Eq:cohomology1}), respectively.

In gauge unfixed string field theory, all component fields of the string
field correspond to states with the ghost number one. Therefore the
resulting cohomology (\ref{Eq:cohom1}) implies that all of on-shell
modes are reduced to gauge degrees of freedom in the gauge unfixed
theory.\cite{rf:KT}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{No open string theorem}

First, we show the formula
\begin{eqnarray}
\label{Eq:UfUfKk}
 U_f^{(k)}={\rm const.}\times U'_f\times
\exp K\left(\log\,\frac{f(w)^{k+1}}{w^{k+1} \partial f(w)}\right).
\end{eqnarray}

Since the $bc$-shift transformation preserves the form of commutation
relations, the operators $c^{(k)}(w)$ and $b^{(k)}(w)$ are transformed
as primary fields of dimension $-1$ and $2$ through the similarity
transformation by $U_f^{(k)}$:
\begin{eqnarray}
\label{Eq:UcUk}
&& U_f^{(k)}\,c^{(k)}(w)\,{U_f^{(k)}}^{-1}
=(\partial f(w))^{-1}\,c^{(k)}(w),\\ 
\label{Eq:UbUk}
&& U_f^{(k)}\,b^{(k)}(w)\,{U_f^{(k)}}^{2}
=(\partial f(w))^{2}\,b^{(k)}(w).
\end{eqnarray}
The operators $c^{(k)}(w)$ and $b^{(k)}(w)$ can be written by the
twisted operators as
\begin{eqnarray}
\label{Eq:cck}
&&
 c^{(k)}(w)=\sum_{n=-\infty}^\infty c_{n+k} w^{-n+1}
%           =\sum_{n=-\infty}^\infty c_{n} w^{-n+k+1}
           =w^{k+1}\,c'(w),\\
\label{Eq:bbk}
&&
 b^{(k)}(w)=\sum_{n=-\infty}^\infty b_{n-k} w^{-n-2}
%           =\sum_{n=-\infty}^\infty b_{n} w^{-n-k-2}
           =w^{-k-1}\,b'(w).
\end{eqnarray}
Then, from these relations and the equations (\ref{Eq:UcU1}),
(\ref{Eq:UbU1}), (\ref{Eq:KcK}) and (\ref{Eq:KbK}), it follows that
\begin{eqnarray}
 U'_f\,e^{K(g)}\,c^{(k)}(w)\,e^{-K(g)}\,{U'_f}^{-1}
%&=&  w^{k+1} e^{g(w)} U'_f\,c'(w)\,{U'_f}^{-1} \\
&=&  w^{k+1} e^{g(w)} (f(w))^{-k-1} c^{(k)}(f(w)), \\
 U'_f\,e^{K(g)}\,b^{(k)}(w)\,e^{-K(g)}\,{U'_f}^{-1}
%&=& w^{-k-1} e^{-g(w)} U'_f\,b'(w)\,{U'_f}^{-1} \\
%&=& w^{-k-1} e^{-g(w)} (\partial f(w))^2\,b'(f(w)) \\
&=& w^{-k-1} e^{-g(w)}\,\partial f(w)\, (f(w))^{k+1} \,b^{(k)}(f(w)).
\end{eqnarray}
If the function $g(w)$ is given by
\begin{eqnarray}
\label{Eq:gk}
 g(w)=\log\,\frac{f(w)^{k+1}}{w^{k+1} \partial f(w)},
\end{eqnarray}
these transformation laws coincide with the similarity transformation by
$U_f^{(k)}$. Since the $bc$-shift transformation does not affect on
matter oscillators, the operators $U_f^{(k)}$ and $U_f e^{K(g)}$, in which
$g(w)$ is given by (\ref{Eq:gk}), are equal up to a multiplicative
constant. Hence, the formula (\ref{Eq:UfUfKk}) is proved.

We consider the case of $f(w)=w/\sqrt{1-w^2}$. For this function,
$U_f$ is given by\cite{rf:GRSZ2}
\begin{eqnarray}
 U_f = \exp\left(\,
\frac{1}{2}\,L_2\,\right).
\end{eqnarray}
For $k=-2$, the function $g(w)$ is calculated as
\begin{eqnarray}
 g(w)=2\log(1-w^2).
\end{eqnarray}
Then, applying the formula (\ref{Eq:UfUfKk}) to this case, we obtain the
equation
\begin{eqnarray}
\label{Eq:UfUfq} 
U_f^{(-2)}= U'_f\times \exp\left(
\sum_{n=1}^\infty \frac{2}{n}\,q_{2n}
\right),
\end{eqnarray}
where the multiplicative constant can be determined to be one by
expanding the both 
sides and using the relations $L_n'=L_n+nq_n+\delta_{n,0}$ and
$L_n^{(k)}=L_n+kq_n+(k^2-3k)\delta_{n,0}/2$.\cite{rf:KT} 
Noting that $(L_n^{(k)})^\dagger=L_{-n}^{(-k)}\ \ (n\neq 0)$, we obtain
the following 
relation by the Hermitian conjugation of (\ref{Eq:UfUfq}):
\begin{eqnarray}
\label{Eq:UFKUF}
U'_F = e^{-K(\rho)}\,U_F^{(2)},
\end{eqnarray}
where the operator $K(\rho)$ is given by (\ref{Eq:Krho}) and $U_F$ is
defined as
\begin{eqnarray}
 U_F = (U_f)^\dagger =\exp\left(\,\frac{1}{2}\,L_{-2}\,\right).
\end{eqnarray}
The function corresponding to $U_F$ is given by $F(w)=\sqrt{w^2+1}$.

Since $\Q^{(2)}$ and $L_n^{(2)}$ are commutable,
the operator $\Q^{(2)}$ is transformed to $\bQ$
through the similarity transformation by $U'_F$: 
\begin{eqnarray}
 -\frac{1}{4}\,
U'_F\,\Q^{(2)}\,{U'_F}^{-1} 
=-\frac{1}{4}\,e^{-K(\rho)}\,U_F^{(2)}\,\Q^{(2)}\,
(U_F^{(2)})^{-1}\,e^{K(\rho)}=\bQ,
\end{eqnarray}
where we have used the equation (\ref{Eq:KbQK2}).
Consequently, 
from a similar derivation of (\ref{Eq:cohom1}), it follows that if a
state $\ket{\psi}$ satisfies $\bQ\ket{\psi}=0$, 
the state can be written as
\begin{eqnarray}
\label{Eq:cohom2}
 \ket{\psi}=U'_F\,(\ket{P}\otimes b_{-2}\ket{0})
+ U'_F\,(\ket{P'}\otimes \ket{0}) +\bQ\ket{\phi}.
\end{eqnarray}
As the case of $a>-1/2$, the difference of the cohomologies
(\ref{Eq:cohom1}) and (\ref{Eq:cohom2}) turns out to be a merely
BRS-exact state by using the relation (\ref{Eq:UFKUF}).

At $a=-1/2$, the operator $L(a)$ becomes
\begin{eqnarray}
 \tilde{L}=L(-1/2)=
  \frac{1}{2}L'_0-\frac{1}{4}(L'_2+L'_{-2})+\frac{3}{2}.
\end{eqnarray}
Though it is impossible to transform the operator $\tilde{L}$ to $L_0$
through the similarity transformation by $U(s,t,u)$ as seen in the
previous section, we can connect it to $L'_2$ through the
transformation by $U'_F$:
\begin{eqnarray}
 \tilde{L}=-\frac{1}{4}\,U'_F\,L'_2\,{U'_F}^{-1}.
\end{eqnarray}

Finally, we consider scattering amplitudes in the theory around the
non-trivial solution.
Since $b_0$ commutes with $L'_n$, the cohomology in the Siegel
gauge is given by the form
\begin{eqnarray}
 U'_F\,(\ket{P}\otimes b_{-2}\ket{0})
+ U'_F\,(\ket{P'}\otimes \ket{0}).
\end{eqnarray}
These states have the ghost number $-1$ or $0$.
If we calculate a amplitude with these as external states, the
amplitude becomes zero since its total ghost number does not match with
the necessary ghost number to realize non-zero amplitudes.
Hence, we conclude that the theory around the universal solution at
$a=-1/2$ does not contain open strings perturbatively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}

In this paper, we study open string field theory expanded around the
universal solution under the Siegel gauge condition. In the theory for
$a>-1/2$, we derive physical states and the Feynman rule, and then we
show that open string amplitudes in the original theory can be
reproduced. In the theory at $a=-1/2$, we prove no open string theorem,
which states that open string does not exist around the
non-trivial universal solution since perturbative amplitudes become
zero. These results provide another evidence for the conjecture that the
universal solutions correspond to pure gauge or the tachyon vacuum.
Though the function in the solution is restricted to the form of
(\ref{Eq:ha}), it is possible to extend it to other functions, for
example to the function given in ref.~\citen{rf:KT} in a straightforward
way. 

For $a>-1/2$, through a string field redefinition,
we can understand the coincidence of amplitudes between
the original and expanded theories.
The action of the gauge unfixed theory around the solution can be
transformed into the original action through the similarity
transformation generated by the operator $K(h_a)$.\cite{rf:TT2}
Similarly, as a result of subsection 2.4, 
if we perform a string field redefinition by the operator $U'(a)$ and
we change the normalization of the string field,
we can transform the action in the Siegel gauge to the
original action fixed in the same gauge. Consequently, amplitudes in
both theories agree with each other. Here, this scale transformation of
the string field provides the factor absorbed into external states of
amplitudes. 

It is plausible that the universal solution at $a=-1/2$ is regarded as
the tachyon vacuum solution. If so, we should be able to observe closed
strings, at least as on-shell states, in the theory around the
solution. This is an important open problem in order to prove the
equivalence of these solutions and
furthermore Sen's conjectures.\cite{rf:Sen-1,rf:Sen-2}
As Drukker pointed out,\cite{rf:Drukker2,rf:Drukker} it would be
possible to reproduce closed string scattering amplitudes 
by the Feynman rule around the non-trivial solution.

\section*{Acknowledgements}
We would like to thank Hiroshi Itoyama for his discussions and
encouragement. 

\newpage
\appendix

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Generalization of (\ref{Eq:UfUfK})}

In the twisted ghost conformal field theory with a general background
charge,\cite{rf:GRSZ,rf:FMS} the energy momentum tensor is given by
\begin{eqnarray}
 T'(w)=T(w)+(\lambda-2)\,\partial J_{\rm gh}(w),
\end{eqnarray}
where $\lambda$ denotes a real number. 
The theory of $\lambda=1$ corresponds to the previous case.
The mode expansions of the ghost and anti-ghost fields in the twisted
theory are given by
\begin{eqnarray}
 c'(w)=\sum_{n=-\infty}^\infty c_n w^{-n-1+\lambda},\ \ \ 
 b'(w)=\sum_{n=-\infty}^\infty b_n w^{-n-\lambda}.
\end{eqnarray}
Then, the twisted and untwisted ghost fields have the relations
\begin{eqnarray}
\label{Eq:bclambda}
 c'(w)=w^{\lambda-2}\,c(w),\ \ \ 
 b'(w)=w^{-(\lambda-2)}\,b(w).
\end{eqnarray}
The twisted Virasoro generators can be written by the untwisted
operators:
\begin{eqnarray}
 L_n'=L_n+(2-\lambda)q_n+a_n\,\delta_{n,0},
\end{eqnarray}
where $a_n=-(\lambda-2)(\lambda+1)/2$ if $\lambda=0,\,\pm 1,\,\pm 2,\cdots$.

If the operators $U_f$ and
$U'_f$ implement the conformal mapping $z=f(w)$ in the untwisted and
twisted theories respectively, the primary fields $\phi(w)$ with
dimension $h$ and $\phi'(w)$ with dimension $h'$ in each theory satisfy
\begin{eqnarray}
\label{Eq:UphiU}
U_f \phi(w)U_f^{-1}=(\partial f(w))^h\,\phi(f(w)),\ \ \ 
U_f' \phi'(w){U_f'}^{-1}=(\partial f(w))^{h'}\,\phi'(f(w)).
\end{eqnarray}
The ghost operators $c(w)$, $b(w)$, $c'(w)$ and $b'(w)$
have dimensions $h=-1$, $h=2$,
$h'=1-\lambda$ and $h'=\lambda$, respectively.  
Using (\ref{Eq:bclambda}) and (\ref{Eq:UphiU}), we find that
\begin{eqnarray}
 U_f e^{K(g)}\,c'(w)\,e^{-K(g)} U_f^{-1} &=&
 w^{\lambda-2}\,U_f e^{K(g)}\,c(w)\,e^{-K(g)} U_f^{-1} \nn
 &=&
 w^{\lambda-2}\,e^{g(w)}\,U_f \,c(w)\,U_f^{-1} \nn
&=&
 w^{\lambda-2}\,e^{g(w)}\,(\partial f(w))^{-1}
 \,c(f(w)) \nn
&=&
 w^{\lambda-2}\,e^{g(w)}\,(\partial f(w))^{-1}
 \,(f(w))^{2-\lambda} c'(f(w)).
\end{eqnarray}
Similarly, we obtain the equation for the anti-ghost as
\begin{eqnarray}
 U_f e^{K(g)}\,b'(w)\,e^{-K(g)} U_f^{-1} &=&
% w^{-\lambda+2}\,U_f e^{K(g)}\,b(w)\,e^{-K(g)} U_f^{-1}\nn
% &=&
% w^{-\lambda+2}\,e^{-g(w)}\,U_f \,b(w)\,U_f^{-1} \nn
%&=&
% w^{-\lambda+2}\,e^{-g(w)}\,(\partial f(w))^2
% \,b(f(w)) \nn
%&=&
 w^{-\lambda+2}\,e^{-g(w)}\,(\partial f(w))^2
 \,(f(w))^{\lambda-2} b'(f(w)).
\end{eqnarray}
If these transformation laws coincide with the conformal mapping
$z=f(w)$ in the twisted theory, the function $g(w)$ satisfies
\begin{eqnarray}
&&
 w^{\lambda-2}\,e^{g(w)}\,(\partial f(w))^{-1}
 \,(f(w))^{2-\lambda} = (\partial f(w))^{1-\lambda},\nn
&&
 w^{-\lambda+2}\,e^{-g(w)}\,(\partial f(w))^2
 \,(f(w))^{\lambda-2} 
= (\partial f(w))^\lambda,
\end{eqnarray}
then $g(w)$ is given by
\begin{eqnarray}
 g(w)=\log \left(\frac{w \partial f(w)}{f(w)}\right)^{2-\lambda}.
\end{eqnarray}
Since the operator $U'_f$ which induces the conformal mapping $z=f(w)$
can be uniquely determined up to a multiplicative constant, we obtain
the formula 
\begin{eqnarray}
 U_f'={\rm const.}\times
U_f\times \exp
K\left(\log\left(
\frac{w\,\partial f(w)}{f(w)}
\right)^{2-\lambda}\right).
\end{eqnarray}
If we set $\lambda=1$, this formula reduces to the equation
(\ref{Eq:UfUfK}). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Some macros are available for the bibliography:
%  o for general use
%    \JL : general journals                 \andvol : Vol (Year) Page
%  o for individual journal 
%    \AJ   : Astrophys. J.           \NC         : Nuovo Cim.
%    \ANN  : Ann. of Phys.           \NPA, \NPB  : Nucl. Phys. [A,B]
%    \CMP  : Commun. Math. Phys.     \PLA, \PLB  : Phys. Lett. [A,B]
%    \IJMP : Int. J. Mod. Phys.      \PRA - \PRE : Phys. Rev. [A-E]     
%    \JHEP : J. High Energy Phys.    \PRL        : Phys. Rev. Lett.
%    \JMP  : J. Math. Phys.          \PRP        : Phys. Rep.
%    \JP   : J. of Phys.             \PTP        : Prog. Theor. Phys.     
%    \JPSJ : J. Phys. Soc. Jpn.      \PTPS       : Prog. Theor. Phys. Suppl.
% Usage:
%  \PRD{45,1990,345}          ==> Phys.~Rev.\ \textbf{D45} (1990), 345
%  \JL{Nature,418,2002,123}   ==> Nature \textbf{418} (2002), 123
%  \andvol{B123,1995,1020}    ==> \textbf{B123} (1995), 1020
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}







