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\begin{flushright} KIAS-P03004\\hepth/0301075\end{flushright}
%\centerline{\Large \bf DRAFT}

\vskip 1cm
\centerline{\Large \bf 
Thermodynamic Bethe Ansatz }
\centerline{\Large \bf 
for boundary sine-Gordon model}
\vskip 1cm

\centerline{\large Taejun Lee and Chaiho Rim} 
\vskip .5cm
\centerline{\it Department of Physics, Chonbuk National University}
\centerline{\it Chonju 561-756, Korea}
\centerline{\it email: tjun@pine.chonuk.ac.kr, rim@mail.chonbuk.ac.kr}
\vskip 1cm
%\centerline{\small PACS: 11.25.Hf, 11.55.Ds}
\vskip 2cm
\centerline{\bf Abstract}
(R-channel) TBA is elaborated 
to find the effective central charge dependence  
on the boundary parameters
for the massless boundary sine-Gordon model 
with the coupling constant $(8\pi) /\beta^2 = 1+ \lambda $ 
with $\lambda$ a positive integer. 
Numerical analysis 
of the massless boundary TBA demonstrates that 
excited state contributions 
should be included to have the right behavior of 
the effective central charge.

\section{Introduction} 
\noindent

The low dimensional quantum system 
such as a quantum wire with boundaries 
is not easy to study in terms of mean field approach
due to large quantum fluctuations.
The system is also strongly affected by the 
the existence of boundaries. 
For example, one needs a good knowledge of the  
the low dimensional quantum field theory
to study the quantum Hall edge tunnelling \cite{pls} .

In this work, the massless 
Tomonaga-Luttinger liquid with boundaries 
is studied motivated by SNS junction super-conductor analysis\cite{css,iaff}. 
This system is summarized in terms of the boundary sine-Gordon model(bSG).
\beq
{\cal A}= 
\int d^2x \,\frac1{4\pi}(\partial_{a}\varphi)^2
-\mu^{(1)}_{B} \int_{y=0} \!\! dx \, 2 
\cos(b(\varphi- \varphi^{(1)}_0))
-\mu^{(2)}_{B} \int_{y=R} \!\! dx \,
2 \cos( b (\varphi-\varphi^{(2)}_0))\,.
\label{massless-sG-action}
\eeq
Our task is to find the free energy of the finite system 
as the function of the boundary parameter 
$\chi = b ( \varphi^{(2)}_0 - \varphi^{(1)}_0)$.
To do this, we first consider the massive sine-Gordon model with boundaries 
and put the bulk mass vanish \cite{FSW}.  

The massive bSG is written as 
\beaq
{\cal A}&=&\int d^2x\left[\frac1{4\pi}(\partial_{a}\varphi)^2
-2\mu(\cos(2 b \varphi)-1)\right]
\nn\\
&&- \mu^{(1)}_{B} \int_{y=0}\!\!dx \,
2 \cos( b (\varphi-\varphi^{(1)}_0))
-\mu^{(2)}_{B} \int_{y=R} \!\! dx \, 
2 \cos( b (\varphi-\varphi^{(2)}_0))\,,
\label{massive-sG-action}
\eeaq
The coupling constant $b^2$ is restricted to be less than 1.   
(Note that  $b^2 $ is scaled by $8\pi$ from the conventional choice 
$\beta^2 = 8\pi b^2$ ).

The bulk sine-Gordon model (SG) 
belongs to the category of two dimensional 
integrable quantum field theories
and allows an exact treatment of the system
\cite{bulk-sG}.
Integrable quantum systems 
have been studied systematically 
after the pioneering work of Zamolodchikov \cite{sasha}.
The system is regarded as a conformal system  
perturbed by an interaction and infinite number of conserved 
quantities are obtained.
The integrability of the bSG was demonstrated in \cite{GhoZam}. 


The scale dependence of the system can be studied
by the method, thermodynamic Bethe ansatz (TBA)
\cite{alyosha,yy}.  Suppose a system lies along the y-axis 
with a  finite size $R$ 
and appropriate boundary conditions are imposed at
each end as in Fig~(\ref{fig1}).
The $x$-direction is periodic and its size $L$ is put to $\infty$ 
in the thermodynamic limit. 

\begin{figure}[ht]
\begin{picture}(50,100) (-130,-10 )
\put(50,0){\line(0,1) {80}}
\put(50,0){\line(1,0) {20}}
\put(50,0.5){\line(1,0) {20}}
\put(70,0){\line(0,1) {80}}
\put(50,80){\line(1,0) {20}}
\put(50,80.5){\line(1,0) {20}}
\put(30,50){\footnotesize{($\alpha$) }}
\put(10,40){\footnotesize{boundary}}
\put(10,30){\footnotesize{condition}}
\put(70,50){\footnotesize{($\beta$) }}
\put(70,40){\footnotesize{boundary}}
\put(70,30){\footnotesize{condition}}
\put(70,10){L}
\put(60,-10){R}
\end{picture}
\caption{space with two boundaries ($\alpha$) and ($\beta$)} 
\label{fig1}
\end{figure}
\noindent
The partition function with this boundaries is given as 
\bea
Z_{\alpha\beta} = \mbox{Tr} e^{-L H_{\alpha\beta}} = e^{-L E_{\alpha\beta}(R)}
\eea
where $H_{\alpha\beta}$ is the Hamiltonian of the system of size $R$ 
with boundary ($\alpha$) and ($\beta$). 
$E_{\alpha\beta}(R)$ is the ground state energy in the thermodynamic limit,
which depends on the size $R$.

The same system can be viewed as the one with initial state $|B_\beta \rangle$ and  
final state $|B_\alpha \rangle$.
In this picture, the bulk is periodic in $L$ 
as in Fig.~(\ref{fig2}).

\begin{figure}[ht]
\begin{picture}(100,30) (-150,-5)
\put(0,0){\line(1,0) {100}}
\put(0,0.5){\line(1,0) {100}}
\put(0,0){\line(0,1) {20}}
\put(0,20){\line(1,0) {100}}
\put(0,20.5){\line(1,0) {100}}
\put(100,0){\line(0,1) {20}}
\put(60,23){\footnotesize{ final state $|B_\alpha \rangle$ }}
\put(60,-8){\footnotesize{ initial state $| B_\beta \rangle$ }}
\put(-3,10){\tiny$\wedge $}
\put(-3,11){\tiny$\wedge $}
\put(97,10){\tiny$\wedge $}
\put(97,11){\tiny$\wedge $}
\put(-10,5){R}
\put(40,-10){L}
\end{picture}
\caption{space with two states $|B_\alpha \rangle$ 
and $| B_\beta \rangle$ }
\label{fig2}
\end{figure}
\noindent
Then the same partition function is evaluated 
using the bulk Hamiltonian $H$ of the system.
\beq
Z_{\alpha\beta}  \equiv e^{-RL f_{\alpha \beta} (R) }
= <B_\alpha | e^{-RH}|B_\beta > 
= \left(  \sum_{\{A\}} 
\frac{<B_a|A> <A|B_b> e^{-R E_A} }{<A|A> }
\right) \,,
\label{massive-free-energy}
\eeq 
where $f_{\alpha \beta} (R) $ is the free energy density per length
and $\{A\}$ is the complete set of the bulk Hamiltonian eigenstates.
The finite size effect of the SG (with boundary)
was analyzed in \cite{FSW,LMSS} for diagonal case
using thermodynamic Bethe ansatz (R-channel TBA). 

In section 2, we summarize the massive (R-channel) 
TBA for the bulk sine-Gordon model 
with boundary sine-Gordon interaction.
In this analysis, we restrict the coupling constant 
$\lambda = 1 /b^2 -1 \equiv n_b +1$ to a positive integer
($n_b \ge 0$) so that the bulk scattering matrix is diagonal
but non-diagonal boundary scattering is allowed. 
The topological charge violation at the boundary 
is incorporated following the suggestion given in 
\cite{css}. This TBA has the bulk and boundary scale dependence 
as well as the boundary parameter dependence. 

In section 3,  massless (R-channel) TBA 
for the massless boundary sine-Gordon model
is obtained 
as the massless limit of the massive TBA. 
It is demonstrated that outside a certain parameter range
($\chi \ge b^2 \pi$), 
the naive TBA for the ground state energy does not work.
A modified massless TBA is proposed so that 
excited state contribution is included 
according to the suggestion in \cite{css,DT,lyTBA}.
The numerical analysis of this modified TBA
confirms the energy dependence 
on the boundary parameters as expected,
which also supports the relation between 
boundary scattering parameters 
and action parameters.  
Section 4 is the conclusion.

\section{Summary of Massive TBA for boundary sine-Gordon} 
\noindent

The bulk sine-Gordon periodic potential allows 
a soliton with topological charge $+1$ 
and an antisoliton with the charge $-1$.  
Both of them have the same mass $M$ 
as the result of  
the charge conjugation symmetry, $\phi \to -\phi$.
The mass is given in terms of  $\mu$ \cite{mass-mu}, 
\beq
\frac{\pi \mu}{\gamma(b^2)}  =
\left[M \frac{ \sqrt{\pi} }2 \,
\frac{\Gamma(\frac1{2 -2 b^2}) } 
{ \Gamma( \frac{b^2}{2-2b^2}) } \right]^{2-2b^2}
\label{sG-m-mu}
\eeq
In addition, there are  
topologically neutral particles, breathers 
(interpreted as the soliton-antisoliton bound states).
Their masses are given as 
\beq
m_a =2  M  \sin 
\left( \frac{\pi a}{2\lambda } \right)
\,\quad a=1,2,\cdots, n_b\,.
\eeq
$n_b$ is the number of breather species, 
$ n_b = $ positive integer less than $ \lambda $.

The free energy density in Eq.~(\ref{massive-free-energy})
is expanded in terms of the bulk Hamiltonian eigenstates, 
{\it i.e.,} solitons, antisolitons and breathers.
These states are uniquely identified 
in terms of mass and rapidity
due to the Fermi-statistics.

The presence of the boundary 
forces two restrictions on the states. 
First, the pair creation at the boundary 
forces the rapidity paired $(\theta, -\theta)$
and therefore, one can count $\theta$ positive  
and make energy eigenvalue doubled: 
$E_{A} (\theta)  = 2 m_A  \cosh \theta $ 
where $m_A $ is the single particle mass. 

\begin{figure}[ht]
\begin{picture}(200,30) (-120,0)
\put(0,0){\line(1,0) {200}}
\put(0,0.5){\line(1,0) {200}}
\put(40,0){\vector(-1,1) {20}}
\put(40,0){\vector(1,1) {20}}
\put(10,25){\footnotesize{$- \theta_i$ }}
\put(60,25){\footnotesize{$\theta_i$ }}
\put(150,0){\vector(-1,1) {20}}
\put(150,0){\vector(1,1) {20}}
\put(120,25){\footnotesize{$- \theta_j$ }}
\put(170,25){\footnotesize{$\theta_j$ }}
\end{picture}
\caption{pair creation at the boundary} 
\end{figure}

Second, at the boundary an in-coming soliton is allowed  
to be scattered away as an antisoliton and vice versa
since the soliton number is not conserved in general. 
To take care of this, soliton and antisoliton are regarded 
as a constituent of a doublet 
of identical particles \cite{css}.
When the partition function in Eq.~(\ref{massive-free-energy}) is 
written in terms of spectral density $\rho$,
$Z_{\alpha \beta} = \int [d\rho]
\exp(-RL \,f_{\alpha \beta}(L, R))$,
the spectral density should include 
not only one-particle density of 
topological particle (denoted as $\rho_0$) 
(soliton is indistinguishable  from antisoliton)
and its hole density  ($\rho_{h\, 0}$),
one-particle density of a breather ($\rho_a$),  
its hole  density ( $\rho_{h\, a}$ with $1 \le a \le n_b$)
but also should include 
two-particle density of topological particles 
($ \rho_d$) ({\it i.e.} soliton and antisoliton pair).
The densities are  summarized in the table:
\begin{table}[h]
\center{ \begin{tabular}{||c|c||} \hline
\rule[-.4cm]{0cm}{1.cm}
species  &  density \\ \hline \hline
soliton or/and antisoliton   & $\rho_0\,, \rho_d\,, \rho_{h 0}$ \\ \hline
breather & $\rho_a\,, \rho_{h a}\,, \quad  a=1, \cdots, n_b$\\ 
\hline
\end{tabular}}
\caption{ particle species and the corresponding densities.} 
\end{table}

Then the free energy density can be written as 
\beq
R f_{\alpha \beta}(R) =  \int_0^\infty  \!\! 
d\theta \left\{  
\sum_{A=0}^{n_b}  \rho_A   
( 2m_A \cosh \theta - \ln \lambda_{ab}^A (\theta) )
+ \rho_d \,( 2 m_d \cosh \theta 
- \ln \lambda^d_{\alpha \beta} ) -{\cal S_B} 
\right\}\,,
\eeq 
where $m_0 =M$ and ${\cal S}_B$ is the entropy density, 
\beaq
{\cal S}_B &=& \int_0^\infty d\theta \left[ \, 
(\rho_0 + \rho_d + \rho_{h0})  \ln (\rho_0 + \rho_d  + \rho_{h0})  
-\rho_0 \ln \rho_0 -\rho_d \ln \rho_d - \rho_{h0} \ln \rho_{h0} \right.
\nn\\
&& \quad + \sum_{a=1,\cdots, n_b} \{  \left.
(\rho_a + \rho_{ha})  \ln (\rho_a +  \rho_{ha})  
-\rho_a \ln \rho_a - \rho_{ha} \ln \rho_{ha} \} \,\right] \,.
\eeaq
$\lambda_{\alpha \beta}^A  (\theta) 
= \langle B_\alpha| \theta A \rangle \, 
\langle \theta A | B_\beta  \rangle $ 
is the boundary state contribution, 
which is given in terms of the boundary scattering amplitude 
$R(u)$ with $u = -i \theta$:
\beaq
\lambda_{\alpha \beta}^a 
&=& \overline {K^a_{\alpha} (u) } K^b_{\beta} (u)
\qquad 
\textrm{ for } a=1,2, \cdots, n
\nn\\
\lambda_{\alpha \beta}^0 
&=& \overline {K_\alpha ^{++} } K_\beta^{++} 
+ \overline {K_\alpha^{+-}}  K_\beta^{+-} 
+ \overline {K_\alpha^{-+}} K_\beta^{-+} 
+ \overline {K_\alpha^{--}} K_\beta^{--} 
=\mbox{Tr}( \bar K_\alpha K_\beta )
\nn\\
\lambda_{\alpha \beta}^d 
&=& ( \overline { K_\alpha^{++} \, K_\alpha^{--}}
 -  \overline {K_\alpha^{+-} \, K_\alpha^{-+}) }
(K_\beta ^{++} K_\beta ^{--}
 -  K_\beta ^{+-} K_\beta ^{-+}) 
= \textrm{Det} ( \overline {K_\alpha} K_\beta )
\eeaq   
where $K (u) \equiv R (\pi/2 - u)$.


The boundary scattering amplitude 
(modulo CDD-type factors),
can be found in \cite{GhoZam}, which 
satisfies the boundary version of the Yang
Baxter equation, unitarity condition, 
and analyticity-crossing symmetry. 
\beaq
R(\eta ,\vartheta ,u) 
& = & 
\left( \begin{array}{cc}
R^{++}(\eta ,\vartheta ,u) &  R^{+-}(\eta ,\vartheta ,u)\\
R^{-+}Q(\eta , \vartheta , u) &  R^{--}(\eta , \vartheta , u)
\end{array}\right) 
\nn \\
 & = & \left( \begin{array}{cc}
P_{0}^{+}(\eta ,\vartheta ,u) & Q_{0}(u)\\
Q_{0}(u) & P_{0}^{-}(\eta ,\vartheta ,u)
\end{array}\right) R_{0}(u)\frac{\sigma (\eta ,u)}
{\cos (\eta )}\frac{\sigma (i\vartheta ,u)}{\cosh (\vartheta )}\, \, \, ,
\nn \\
P_{0}^{\pm }(\eta ,\vartheta ,u) 
& = & \cos (\lambda u)\cos (\eta )\cosh (\vartheta )\mp 
\sin (\lambda u)\sin (\eta )\sinh (\vartheta )
\nn \\
Q_{0}(u) & = & -\sin (\lambda u)\cos (\lambda u)\,.
\label{Rsas} 
\eeaq
Here $R_{0}$ is the boundary condition independent part,
$$
R_{0}(u)=\prod ^{\infty }_{l=1}
\left[ \frac{\Gamma (4l\lambda -\frac{2\lambda u}{\pi })
\Gamma (4\lambda (l-1)+1-\frac{2\lambda u}{\pi })}
{\Gamma ((4l-3)\lambda -\frac{2\lambda u}{\pi })
\Gamma ((4l-1)\lambda +1-\frac{2\lambda u}{\pi })}
/(u\to -u)\right] 
$$
and $\sigma (x,u)$ is the boundary condition dependence part, 
$$
\sigma (x,u)=\frac{\cos x}{\cos (x+\lambda u)}
\prod ^{\infty }_{l=1}\left[ 
\frac{\Gamma (\frac{1}{2}+\frac{x}{\pi }+(2l-1)
\lambda -\frac{\lambda u}{\pi })
\Gamma (\frac{1}{2}-\frac{x}{\pi }+(2l-1)
\lambda -\frac{\lambda u}{\pi })}
{\Gamma (\frac{1}{2}-\frac{x}{\pi }+(2l-2)
\lambda -\frac{\lambda u}{\pi })\Gamma (\frac{1}{2}+\frac{x}{\pi }+2l
\lambda -\frac{\lambda u}{\pi })}/(u\to -u)\right] \,.
$$
The scattering parameters, $\eta$ and $\vartheta$ are related   
with the action parameters, $\mu_B$ and $\varphi_0$ \cite{parameter,BPT}:
\beaq
\cos ( b^2 \eta)\, 
\cosh (b^2 \vartheta ) \, 
&=&  \mu_B \sqrt{\sin(b^2 \pi)} 
\cos  ( b \varphi_0 )/\sqrt \mu
\nn\\
\sin ( b^2 \eta)\, 
\sinh (b^2 \vartheta ) \, 
&=& \mu_B \sqrt{\sin(b^2 \pi)} 
\sin ( b\varphi_0 )/\sqrt \mu
\label{sG-parameter}
\eeaq

The boundary scattering amplitude of breathers is given as
\beq
R^{(k)}(\eta ,\vartheta ,u)
=R_{0}^{(k)}(u)\, S^{(k)}(\eta ,u)\,
S^{(k)}(i\vartheta ,u)\,,
\qquad 1 \le k \le n_b\,.
\eeq
$R_{0}^{(k)}$ is the boundary independent part
and $S^{(k)}$ the boundary dependent one:
\bea
R_{0}^{(k)}(u)=\frac{\left( \frac{1}{2}\right) 
\left( \frac{k}{2\lambda }+1\right) }
{\left( \frac{k}{2\lambda }+\frac{3}{2}\right) }
\prod ^{k-1}_{l=1}\frac{\left( \frac{l}{2\lambda }\right) 
\left( \frac{l}{2\lambda }+1\right) }
{\left( \frac{l}{2\lambda }+\frac{3}{2}\right) ^{2}} \,,
\quad
S^{(k)}(x,u)=\prod ^{k-1}_{l=0}
\frac{\left( \frac{x}{\lambda \pi }-\frac{1}{2}
+\frac{k-2l-1}{2\lambda }\right) }
{\left( \frac{x}{\lambda \pi }
+\frac{1}{2}+\frac{k-2l-1}{2\lambda }\right) }\,,
\eea
where the notation $(x)$ stands for 
\bea
 (x)=\frac{\sin \left( \frac{u}{2}+\frac{x\pi }{2}\right) }
{\sin \left( \frac{u}{2}-\frac{x\pi }{2}\right)} \,.
\eea

The hole and the particle densities are not independent each other. 
This relation is obtained from the 
bulk scattering amplitude. 
The bulk-scattering amplitude of solitons and antisolitons 
\cite{bulk-sG} are given as 
\beaq
S^{++}_{++}(u )&=& S_{--}^{--}(u)= s( u)
\nonumber \\
S^{+-}_{+-}(u)&=& S_{-+}^{-+}(u)= 
\frac{\sin (\lambda u)}{\sin (\lambda (\pi -u))}s(u)
\nonumber \\
S^{-+}_{+-}(u) &=& S_{-+}^{+-}(u)= 
\frac{\sin (\lambda \pi )}{\sin (\lambda (\pi -u))}s(u) \,
\label{soliton-s} 
\eeaq  
where $s(u)$ is given as  
\bea
s(u) =  
-\prod ^{\infty }_{l=1}
\left[ \frac{\Gamma (2(l-1)\lambda-\frac{\lambda u}{\pi })
\Gamma (2l\lambda +1-\frac{\lambda u}{\pi })}
{\Gamma ((2l-1)\lambda -\frac{\lambda u}{\pi })
\Gamma ((2l-1)\lambda +1-\frac{\lambda u}{\pi })}/(u\to -u)\right] 
\eea

Due to the restriction of  $\lambda$,
the bulk scattering amplitudes are diagonal,
$ S^{-+}_{+-}(u) =0 $. 
This restriction makes our analysis 
not too much complicated \cite{sGTBA}.  
The diagonal scattering amplitude for soliton and antisoliton 
turns out to be equal up to a phase difference:
$ S^{++}_{++}(u)= (-1)^{\lambda -1} S^{+-}_{+-}(u )\,$.

The scattering amplitude of the breathers $B^a$ and $B^b$ 
with $b \le a \le n_b$ takes the form
\beq
S^{a\, b}(u)=\{a+b-1\}\{a+b-3\}\dots
\{a-b+3\}\{a-b+1\}\,,
\eeq
where the notation $\{y\}$ is defined as 
\bea
\{y\}=\frac{\left( \frac{y+1}{2\lambda }\right) \left(
\frac{y-1}{2\lambda }\right) }{\left( \frac{y+1}{2\lambda }-1\right)
\left( \frac{y-1}{2\lambda }+1\right) }
\eea
and satisfies the relations
$\{y\}\{-y\}=1\,$ and $\,\{y+2\lambda \}=\{-y\}\,$.
The scattering amplitude of the soliton (antisoliton) and breather  
$ S^{(a)}(u)  = S_{a\,+}^{a\, +}(u) = S_{a\, -}^{a\, -}(u) $
is given as  
\beq
S^{(a)}(u) =\{a-1+\lambda \}\{a-3+\lambda \}\cdots 
\left\{ \begin{array}{c}
\{1+\lambda \}\quad \textrm{if } a \textrm{ is even}\\
-\sqrt{\{\lambda \}}\quad \textrm{if }a\textrm{ is odd}
\end{array}\right. 
\eeq
where we followed the notation in \cite{BPT}.

Demanding the wave function periodic in $L$
we have the constraints between 
hole densities with particle densities.
For soliton states ($n_0\,, n_d\,, n_{h0}$)
we have 
\beaq
&&\exp(iL m_0 \sinh \theta_i^0 ) 
\prod_{j=1, \ne i}^N \{ 
S_{00}(\theta_i^0 -\theta_j^0) 
S_{00}(\theta_i^0 +\theta_j^0)) 
S_{00}(2\theta_i^0) 
S_{0d}(\theta_i^0 -\theta_j^d) 
S_{0d}(\theta_i^0 +\theta_j^d)) 
\nn\\
&&\quad 
\prod_{a=1}^n ( S_{0a}(\theta_i^0 -\theta_j^a) 
S_{0a}(\theta_i^0 +\theta_j^a)) 
\}  =\pm e^{ 2\pi i (n_0 (\theta_i^0)+ n_d (\theta_i^0)+n_{h0}(\theta_i^0))  }\,.
\eeaq
For breathers ($n_a, n_{ah}$, $a=1,2, \cdots, n_b$):
\beaq
&&\exp(iL m_a \sinh \theta_i^a ) 
\prod_{j=1, \ne i}^N \{ 
S_{aa}(\theta_i^a -\theta_j^a) 
S_{aa}(\theta_i^a +\theta_j^a)) 
S_{aa}(2\theta_i^a) 
S_{a0}(\theta_i^a -\theta_j^0) 
S_{a0}(\theta_i^a +\theta_j^0)) 
\nn\\
&&\qquad\qquad\qquad\qquad
S_{ad}(\theta_i^a -\theta_j^d) 
S_{ad}(\theta_i^a +\theta_j^d)) 
\prod_{b=1}^n ( S_{ab}(\theta_i^a -\theta_j^b) 
S_{ab}(\theta_i^a +\theta_j^b)) 
\}  
\nn\\
&&\qquad\quad 
=\pm e^{ 2\pi i  (n_a (\theta_i^a) +n_{ha} (\theta_i^a) ) }\,.
\eeaq
Differentiating with respect to the rapidity, 
we have the relations of hole and particle spectral densities:
\beaq
&& 
m_A \cosh \theta + 
\sum_{B=0,1, \cdots, n} \int_0^\infty d\theta^B
\, \rho_B(\theta^B)\, 
( \phi_{AB}(\theta^A - \theta^B) + \phi_{AB}(\theta^A + \theta^B) ) 
\nn\\
&& \qquad \qquad \qquad \qquad 
+ \int_0^\infty d\theta^d
\, \rho_d(\theta^d)\, 
( \phi_{Ad}(\theta^A - \theta^d) + \phi_{Ad}(\theta^A + \theta^d) ) 
\nn\\
&&\qquad 
= 2\pi (\rho_A(\theta ) + \rho_{hA} (\theta) + \delta_{A0} \, \rho_d (\theta) ) \quad \textrm{ for } A=0, 1, \cdots, n, d,
\eeaq
where $\phi_{AB}(\theta) = -i \frac{d \ln S_{AB} (\theta)}{d \theta}$ and 
$\phi_{Ad} (\theta) = 2 \phi_{A0}(\theta)$.


Introducing pseudo energies,  $\epsilon$ 
\bea
e^{-\epsilon_a} =\frac{\rho_a}{\rho_{ha}} 
\quad \textrm{ for } a =1, \cdots, n\,, \quad 
e^{-\epsilon_0} = \frac{\rho_0}{\rho_{h0}}  \,,\quad
e^{-\epsilon_d} = \frac{\rho_d}{\rho_{h0}}   \,,
\eea 
and minimizing $f_{\alpha\beta}(R)$ we have 
the massive TBA:
\beaq
\epsilon_A &=& 2 m_A R \cosh \theta - \ln \lambda_{\alpha \beta}^A 
-\frac1{2\pi}\sum_{B=0,1,\cdots, n_b} 
 \int_{-\infty}^\infty d \theta' 
\phi_{AB}(\theta -\theta') L_B(\theta') \,,
\nn\\
\epsilon_d &=& 
2(\epsilon_0  +  \ln \lambda_{\alpha\beta}^0 ) 
- \ln \lambda_{\alpha\beta}^d  \,,
\eeaq
where $L_0 = \ln (1 + e^{- \epsilon_0} +e^{-\epsilon_d})\,,\,\,$
$L_a = \ln (1 + e^{- \epsilon_a})\,$ ( $a =1, \cdots, n_b $).
The free energy has the form, 
\beq
R\,f(R) = - \frac1{4\pi} \int_{-\infty}^\infty d \theta
\,\sum_{A=0,1,\cdots, n_b} m_A \,\cosh \theta \,  L_A(\theta)\,.
\label{sG-energy}
\eeq
(We skip the bulk and boundary energy term, the details of which can be found 
in \cite{BPT}).

This TBA can be written in a more compact form.
To do this, we extend the index 
to include the doublet as $+$, $A' =0,1,\cdots, n_b, +$,  
and shift $\epsilon \to \epsilon - \ln \lambda$,  
\beq
\epsilon_{A'} 
= d_{A'}-\frac1{2\pi}\sum_{ B' } 
 \int_{-\infty}^\infty d \theta' 
\phi_{A'\, B'}(\theta -\theta') 
\widetilde {L_{B'}}(\theta') \,.
\label{bsG-TBA}
\eeq
Here $\epsilon_+ \equiv \epsilon_d /2 $,  
$\, m_+ \equiv m_0$, $\, \widetilde {L_+} \equiv 0\,$,
$\widetilde {L_0} \equiv  \ln (1 + \lambda^0_{\alpha\beta}\,
e^{- \epsilon_0} + \lambda^d_{\alpha\beta} \,e^{-2\epsilon_0})\,$ 
and 
\bea
\widetilde {L_a} \equiv  
\ln (1 + \lambda^a_{\alpha \beta} \, e^{- \epsilon_a}) \quad 
\textrm{ for } a =1, \cdots, n_b\,.
\eea
Then using an identity of $\mathcal N_{A'B'} 
\equiv - \int_{-\infty}^{\infty} \frac{d\theta}{2\pi}  
 \phi_{A'B'} (\theta) $,
\beq
\sum_{C'} (\mathcal N_{A'\,C'} + \delta_{A'\,C'} ) 
\mathcal I_{C'\,B'}  
=  \mathcal N_{A'B'} \,,
\eeq
where $\mathcal I_{A'\,B'}$  is the incidence matrix  
\cite{FSW,KM,ade}
%\vspace{1cm}
\newline 
\noindent 
\centerline
{\hbox{\rlap{\raise28pt\hbox{$\hskip6.5cm\bigcirc\hskip.25cm 0$}}
\rlap{\lower27pt\hbox{$\hskip6.4cm\bigcirc\hskip.3cm +$}}
\rlap{\raise15pt\hbox{$\hskip6.1cm\Big/$}}
\rlap{\lower14pt\hbox{$\hskip6.0cm\Big\backslash$}}
\rlap{\raise15pt\hbox{$1\hskip1.1cm 2
	\hskip1.5cm s\hskip 0.7cm \ n_b-1$}}
$\bigcirc$------$\bigcirc$-- -- --
--$\bigcirc$-- -- --$\bigcirc$------$\bigcirc$\hskip.1cm $\ n_b$ }} 
\bigskip
we can put the TBA  in a reduced form when  $n_b \ge 1$,
\beq
\epsilon_{A'} - d_{A'} = \sum_{B'} \mathcal I_{A'\,B'} \,
K* (L_{B'} +\epsilon_{B'} -d_{B'})\,,
\label{r-bsG-TBA}
\eeq
where $K (\theta)$ is a new kernel,
\bea
K (\theta) = \frac {\lambda}{\cosh(\lambda \theta) }\,.
\eea

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

\section{Massless TBA for boundary sine-Gordon model} 
\noindent 

The massless TBA corresponding to bSG action 
Eq.~(\ref{massless-sG-action}) is obtained 
by taking the limit $\mu \to 0$ 
of the massive TBA in Eq.~(\ref{bsG-TBA},\ref{r-bsG-TBA}). 
Even though the soliton mass $M$ vanishes,
one may introduce a finite renormalized mass scale $M_R$  as
$ M_R = (M/2) e^{\theta_0} $
if a large parameter $\theta_0$ is defined as 
\bea
e^{-\theta_0} =  (C_0\, \mu)^{\frac {\lambda +1}{2 \lambda}}/M_R \,,
\qquad 
C_0= \frac{\pi}{\gamma(b^2)} 
\left(
\frac{\Gamma (\frac{b^2}{2-2 b^2}) }
{\sqrt{\pi} \Gamma (\frac{1}{2-2 b^2})  }
\right)^{2\lambda/(\lambda+1)}\,.
\eea
In this limit, the rapidity is rescaled 
into renormalized one, $\theta_R $
as $\theta = \theta_R + \theta_0$.
The boundary scattering parameter, $\vartheta $ 
is also rescaled as $\vartheta_R $ 
maintaining a relation
$ \vartheta - \lambda \theta  = \vartheta_R - \lambda \theta_R  $.
Then $\vartheta_R $ is written in terms of the action parameters,  
\beq 
(m_R)^{b^2 \lambda} \,
e^{b^2 \vartheta_R  } 
=  2\mu_B \sqrt{C_0 \,\sin (b^2 \pi)}  \,.
\eeq
On the other hand, $\eta$ is not rescaled but is identified as 
\beq
b^2  \eta = \left\{
\begin{array}{ll}
b \phi_0  \qquad  & {\rm for} 
\qquad 0 \leq b \phi_0  \leq \pi/2 \,,
\\
\pi - b \phi_0  \qquad &{\rm  for}
\qquad \pi/2 \leq b \phi_0  \leq \pi \,.
\end{array}
\right.
\label{eta-parameter}
\eeq
This identification is justified 
from the numerical analysis later on.

In terms of this rescaled parameter 
(we omit hereafter the subscript $R$ 
standing for the renormalized one)
the solitonic boundary scattering amplitudes are given as 
\beq
K(\eta ,\vartheta ,u) 
\to 
\left( \begin{array}{cc}
 e^{-i \eta} e^{\frac12( \vartheta +i\lambda \tilde u)}& 
-i e^{-\frac12( \vartheta +i\lambda \tilde u)} \\
-i e^{-\frac12( \vartheta +i\lambda \tilde u)} &  
e^{i \eta} e^{\frac12( \vartheta +i\lambda \tilde u)}
\end{array}\right) 
e^{ -i 3 \pi \lambda/ 4 } \, k(u) 
\eeq
where $\tilde u = \pi/2 -u$ and 
$ k( u)^{-1}  =\prod_{k=0}^{n_b} {2 \cos \left(\frac{\pi}{2 \lambda}
\left(\frac12 + k+ 
\frac{i\vartheta -  \lambda \tilde u }{\pi} \right)\right)}\,$,
and the breather boundary reflection amplitudes are given as 
\beq
K^{(k)}(\vartheta, \eta, u)  \to 
\prod_{l=0}^{k-1}
\frac { \sin(\frac12(\tilde u - 
\frac{i\vartheta}\lambda 
- \frac\pi2 - \pi \frac{k-2l -1}{2\lambda})) }
{\sin(\frac12(\tilde u - \frac{i\vartheta}\lambda
 + \frac\pi2 - \pi \frac{k-2l -1}{2\lambda}))  }
\eeq
The boundary $ \lambda_{\alpha\beta} $ is given as
\beaq
\lambda^0_{\alpha\beta} &=& 2 \left\{ \cos \eta\, 
e^{(\frac{\vartheta_\alpha +\vartheta_\beta}2 - \lambda \theta)} 
+ e^{-(\frac{\vartheta_\alpha 
+\vartheta_\beta}2 - \lambda \theta)} 
\right\} \,\overline{ k_\alpha (u)} \,k_\beta (u) \,,
\nn\\
\lambda^d_{\alpha\beta} &=&
\left\{ e^{(\vartheta_\alpha 
-\lambda \theta - i\pi\lambda/2 ) } 
+e^{-(\vartheta_\alpha -\lambda \theta - i\pi\lambda/2 ) }
\right\}\,
\nn\\
&&\qquad 
\left\{ 
e^{(\vartheta_\beta -\lambda \theta +i\pi\lambda/2 )} 
+ e^{-(\vartheta_\beta -\lambda \theta + i\pi\lambda/2 ) } 
\right\} \, 
\left(\overline{ k_\alpha(u)} \, k_\beta(u) \right)^2
\label{massless-lambda}
\eeaq

From this massless scattering data 
the massless TBA has the form  
\beaq
\epsilon_{A'} &=&
D_{A'}-\frac1{2\pi}\sum_{ B' } 
 \int_{-\infty}^\infty d \theta' 
\phi_{A'\, B'}(\theta -\theta') 
\widetilde {L_{B'}}(\theta') \,,
\nn\\
&=& 
D_{A'} 
+ \sum_{B'}  \mathcal I_{A'\,B'} \,
K* (\widetilde {L_{B'}} +\epsilon_{B'} -D_{A'})\,,
\label{m0sG-TBA}
\eeaq
where $D_{A'} =m_{A'} e^\theta $ 
and the soliton mass $M$ in $m_{A'}$ 
is replaced by the renormalized mass $M_R$.
The free energy is given as 
\beq
R\,f(R) = - \frac1{2\pi} \int_{-\infty}^\infty d \theta
\,\sum_{A=0,1,\cdots, n_b} m_{A } \,e^\theta \, \widetilde {L_A}(\theta)\,.
\label{m0sG-energy}
\eeq

Let us investigate the parametric dependence of 
the ground state energy 
Eq.~(\ref{m0sG-energy}) using the TBA Eq.~(\ref{m0sG-TBA}).
When $\lambda =1 ( n_b=0 )$ the boundary contribution is given 
explicitly as, 
\beaq
\lambda^0_{\alpha\beta} &=& 
\frac{
2\left[\cos \eta\, 
e^{(\frac{\vartheta_\alpha +\vartheta_\beta}2 -  \theta)} 
+ e^{-(\frac{\vartheta_\alpha +\vartheta_\beta}2 - \theta)} 
 \right]}
{4 \cosh (\frac{\vartheta_\alpha - \theta}2 )
\cosh (\frac{\vartheta_\beta - \theta}2 )}  \,,
\nn\\
\lambda^d_{\alpha\beta} &=&
\frac{4 \sinh (\vartheta_\alpha - \theta )
\sinh (\vartheta_\beta - \theta )}
{ \left[ 4 \cosh (\frac{\vartheta_\alpha -\theta}{2} )
\cosh (\frac{\vartheta_\beta -\theta}{2})
\right]^2}  \,.
\nn
\eeaq
and the TBA is trivial since the kernel $\phi(\theta)=0$. 
The energy is obtained numerically and is plotted 
$c_{\rm eff} $ v.s. $\chi$ in Fig.~\ref{fig4}. 
$c_{\rm eff}$ is the effective central charge,
$ c_{\rm eff} = - 24 R\,f(R) /\pi $ and 
$\chi = \beta (\phi_0^{(2)} -\phi_0^{(1)})$.
We put the boundary scale parameters at $y=0$ and $y=R$ 
into the same $\vartheta$ for simplicity. 

\begin{figure}[h]
\begin{minipage}[t]{7.5cm}
{\scalebox{0.45} {\includegraphics{fig4.eps}}}
\caption{$c_{\rm eff}$ vs. $\!\chi\,$  
when $\lambda=1$ and $\vartheta = 10 $ 
before modifing the massless TBA.}  
\label{fig4} 
\end{minipage}
\ $\qquad$ \
\begin{minipage}[t]{7.5cm}
{\scalebox{0.45} {\includegraphics{fig5.eps}}}
\caption{$c_{\rm eff}$ vs. $\!\chi\,$ when $\lambda=1$. 
Modified TBA shows the correct behavior (in stars)
for $\vartheta=10$ (sold) and $\vartheta=0$ (dashed).}  
\label{fig5}
\end{minipage}
\end{figure}
\vspace{.5cm}


We note that the boundary contribution 
$\lambda_{\alpha\beta}^0$ in 
Eq.~(\ref{massless-lambda}) is $2\pi$-periodic 
in $\eta =\eta_1 -\eta_2$ and other $\lambda_{\alpha\beta}$'s are 
$\eta$ independent.  This explains the 
$\pi$-periodic in $\chi$ in Fig.~\ref{fig4}.
(Generally, $c_{\rm eff}$ will be 
$2\pi b^2$- periodic in $\chi$).

However, the periodicity of the energy in $\chi$ 
is not acceptable as pointed out in \cite{css}.
When $\chi >  \pi/2$, the boundary term in the Lagrangian 
effectively changes the relative sign; one can 
equivalently put $\mu^{(1)}_B \to -\mu^{(1)}_B$ 
and $\mu^{(2)}_B \to \mu^{(2)}_B$ while $\chi \to  \pi - \chi $.
This relative sign change of the boundary term
should be reflected in the $c_{\rm eff}$ value. 
The same problem of $c_{\rm eff}$ due to  
the periodicity of a boundary parameter 
was also observed in boundary Lee-Yang model \cite{lyTBA}.

To cure this disease it has been proposed in \cite{css,lyTBA}
that excited state contributions should be properly taken care of.  
According to this proposal, the  $c_{\rm eff}$ is recalculated 
and is presented in Fig.~\ref{fig5} when $\lambda=1$.

In the analysis for $\chi>\pi/2$ two things are considered: 
First, one needs to find the zeroes of $e^{\tilde L_0}$ 
in terms of 
the complex rapidity $\tilde \theta$ following \cite{DT}
with the parameter identification in Eq.~(\ref{eta-parameter}).
\beq 
1 + \lambda^0_{\alpha\beta}\, 
e^{- \epsilon_0 ( \tilde \theta )} 
+ \lambda^d_{\alpha\beta} 
\,e^{-2\epsilon_0 ( \tilde \theta )} =0\,.
\label{zer0}
\eeq
The complex rapidity is of the form  
$ \tilde \theta = i \pi/2 + \theta_p$ 
with $\theta_p$ real.
Second, the free energy should incorporate this branch singularity
into the contour integration and becomes
\beq
R\,f(R) = -im_0 \,e^{\tilde \theta}
- \frac1{2\pi} \int_{-\infty}^\infty d \theta
\, m_0 \,e^\theta \, \tilde L_0(\theta)\,.
\eeq

We generalize this result into the case $\lambda $ is a positive integer.
Excited state contribution is taken for $ \chi < b^2 \pi $.
We give here how to modify  TBA for all parameter range,
$0\le \chi \le \pi $. 

The excited state contribution should be obtained 
from the zeroes of $e^{\widetilde {L_0}}$:
\beq
1 + \lambda^0_{\alpha\beta}\, e^{- \epsilon_0 ( \tilde \theta )} 
+ \lambda^d_{\alpha\beta} \,e^{-2\epsilon_0 ( \tilde \theta )} =0\,,
\label{1branch}
\eeq
with the rapidity $ \tilde \theta = i \pi/2 + \theta_p$ 
and $\theta_p$ real. 
This solution will contribute to the free energy:
\beq
R\,f(R) = -im_0 \,e^{\tilde \theta}
- \frac1{2\pi} \int_{-\infty}^\infty d \theta
\,\sum_{A=0,1,\cdots, n_b} m_{A } \,
e^\theta \, \tilde L_A(\theta)\,.
\eeq

The TBA in Eq.~(\ref{m0sG-TBA}) is accordingly  modified: 
\beaq
\epsilon_{A'} &=& d_{A'}+ \ln S_{A'0}(\theta -\tilde \theta ) 
-\frac1{2\pi} \sum_{B'=0,1,\cdots, n_b,+} 
\int_{-\infty}^\infty d \theta' 
\phi_{A'B\,'}(\theta -\theta') \tilde L_{B'}(\theta')
\nn\\
&=& d_{A'}+ \ln S_{A'0}(\theta -\tilde \theta ) 
+ \sum_{B'}  \mathcal I_{A'\,B'} \,
K* (\tilde L_{B'} +\epsilon_{ B'} -d_{B'})\,.
\label{1e-bsG-TBA}
\eeaq
Here we use the branch-cut information in 
the original TBA, Eq.~(\ref{bsG-TBA});
\bea
\phi_{A0}* \tilde L (\theta) &=& 
-\frac1{2\pi} \int d\theta' \phi_{A0} (\theta -\theta') 
\tilde L(\theta') 
\\
&\to&   \ln S_{A0} (\theta - \tilde \theta) 
-\frac1{2\pi} \int_P d\theta' \phi_{A0} (\theta -\theta') 
\tilde L(\theta') 
\eea
where $\int_P$ represents the principle value of the integration.   

The consistency of the branch-cut solution $\tilde \theta $ 
requires 
\beq
\epsilon_0(\tilde \theta)
=  e^{\tilde \theta} 
+  i \left( \frac {1-(-1)^{n_b}}{2}\right) \left( \frac{\pi}2 \right)
-  \sum_{B'=0,1,\cdots, n_b} 
\phi_{0 B\,'}* \tilde L_{B'}(\tilde \theta) \,.
\label{consistency}
\eeq
A special care has been done for $S_{00}$:
\bea
\ln S_{00} (\theta) 
= \frac12 \ln \{ S_{++} (\theta) \, S_{+-} (\theta) \}
=\frac12 \ln \{(-1)^{n_b} S_{++}^2(\theta) \} \,,
\eea
which accounts for the phase difference of the scattering 
amplitude of soliton and antisoliton. 

We give the numerical result for $\lambda = 2 (n_b=1) $ case.
The boundary $\lambda_{\alpha \beta}$ is given as
\beaq
\lambda^0_{\alpha\beta} &=& 
\frac{ 2 \left[\cos \eta\, 
e^{(\frac{\vartheta_\alpha +\vartheta_\beta}2 -2  \theta)} 
+ e^{-(\frac{\vartheta_\alpha +\vartheta_\beta}2 - 2\theta)} \right]}
{\left[
4 \cosh (\frac{\vartheta_\alpha - 2\theta}4
 + \frac{i\pi}8)
\cosh (\frac{\vartheta_\alpha - 2\theta}4
-\frac{i\pi}8 )
\right] \left[
4 \cosh (\frac{\vartheta_\beta - 2\theta}4
 + \frac{i\pi}8)
\cosh (\frac{\vartheta_\beta - 2\theta}4
-\frac{i\pi}8 ) \right]} \,,
\nn\\
\lambda^d_{\alpha\beta} &=&
\frac{
4 \cosh (\vartheta_\alpha - 2\theta )
\cosh (\vartheta_\beta - 2\theta )}
{\left( 
4 \cosh (\frac{\vartheta_\alpha -2\theta}{4} + \frac{i\pi}8) 
\cosh (\frac{\vartheta_\alpha -2\theta}{4} - \frac{i\pi}8) 
\right)^2
\left(
4 \cosh (\frac{\vartheta_\beta -2\theta}{4} + \frac{i\pi}8) 
\cosh (\frac{\vartheta_\beta -2\theta}{4} - \frac{i\pi}8) \right)^2 } 
\,,
\nn\\
\lambda^1_{\alpha \beta} &=& 
\tanh ( \theta/2  - \vartheta_\alpha /4 )\,
\tanh ( \theta/2  - \vartheta_\beta /4 )\,.
\eeaq
Note that since $|\lambda^{d}| = 1$ and $\rm{Im} 
({\lambda^0}/{\sqrt{\lambda^d}})=0$,
we may put the constraint  Eq.~(\ref{1branch}) as 
\beq
\cosh(\epsilon_0( \tilde \theta )  + i \tau)
 = - \frac{\lambda^0}{2 \sqrt{\lambda^d}}\,.
\label{n=1constraint}
\eeq
where $\lambda^d \equiv \exp(-2i  \tau)$.    
The kernel is given as 
\beq
\phi_{00}(\theta) = \frac12 \phi_{11}(\theta) 
=- \frac 1{2 \pi \cosh \theta} 
\quad 
\phi_{01}(\theta) = 
- \frac {\sqrt2 \cosh(\theta)}{\pi \cosh(2\theta)}\,.
\eeq

$c_{\rm eff}$ is given in Figs.~(\ref{fig6},\ref{fig7}). 
Fig.~\ref{fig6} is the one obtained from 
TBA, Eq.~(\ref{m0sG-TBA}) and Fig.~\ref{fig7} from 
TBA, Eq.~(\ref{1e-bsG-TBA}) modified for $\chi > \pi/3$.

\begin{figure}[h]
\begin{minipage}[t]{7.5cm}
{\scalebox{0.45} {\includegraphics{fig6.eps}}}
\caption{$c_{\rm eff}$ vs. $\chi $ when  $\lambda=2$ 
and $\vartheta= 10 $
obtained from TBA before modification.}
\label{fig6}
\end{minipage}
\ $\qquad$ \
\begin{minipage}[t]{7.5cm}
{\scalebox{0.45} {\includegraphics{fig7.eps}}}
\caption{$c_{\rm eff}$ vs. $\chi $
when $\lambda=2$. 
$\vartheta = 10 $ is drawn as a solid curve 
($\vartheta = 0 $ as dashed)
after corrected.}
\label{fig7} 
\end{minipage}
\end{figure}
\vspace{.5cm}
\noindent
In Fig.~\ref{fig7}, the $\eta$ parameter identification,
Eq.~(\ref{eta-parameter}) is used.  
In addition, we have for $\chi>\pi/3$ 
\beaq
&&S_{00} (\theta- \tilde \theta) =
\tanh \left( \frac{\theta -\theta_p}2 \right) 
\,,\quad
S_{01} (\theta- \tilde \theta) 
= \frac {\sqrt2\,\cosh (\theta-\theta_p)-1}
{\sqrt2\,\cosh (\theta-\theta_p)+1} \,,
\nn\\
&&\epsilon_0(\tilde \theta ) = i \left( 2\, e^{\theta_p} +   
\frac{\pi}2  - \phi_{0 0}* \widetilde {L_0}(\tilde\theta) 
-\phi_{0 1}^e* \widetilde {L_1}(\tilde \theta) \right)  \,.
\nn
\eeaq

One can check the correction of Fig.~\ref{fig7}
by considering the Dirichlet boundary condition,
$\phi(x, 0)= \phi_0^{(1)}$ and $\phi(x, R)= \phi_0^{(2)}$,
whose condition is obtained 
when $\mu_{1B} = \mu_{2B} \to \infty\,$ 
($\vartheta \to \infty$).
The boundary parameters are reduced into
$\lambda^0 = 2\cos \eta $ and $\lambda^d =\lambda^1=1$.
The phase $\chi=0$ corresponds to the c=1 conformal theory.
One can check this using the standard 
Rogers dilogarithmic function \cite{KM,dilog}.
An analysis of a bosonic free theory
also gives the Virasoro conformal dimension 
$\Delta(\chi) = (\frac {\chi}{2b \pi})^2 $
due to the zero mode at the Dirichlet boundary condition.
Therefore, one expects the ground state energy is 
given as $E(\chi)= -\frac{\pi}{24 R}(1- 24 \Delta(\chi))$ 
as seen in Figs.~(\ref{fig4}, \ref{fig7}).

For reference, the boundary scale dependence 
of $c_{\rm eff}$ is also given in 
Figs.~(\ref{fig8}, \ref{fig9}).  
The $c=1$ conformal limit is reached  
for $\chi=0$ 
both when $\mu_B =0$ ($\vartheta \to -\infty$) 
and $\mu_B \to \infty$
($\vartheta \to \infty$).

\begin{figure}[h]
\begin{minipage}[t]{7.5cm}
{\scalebox{0.45} {\includegraphics{fig8.eps}}}
\caption{$c_{\rm eff}$ vs. $\!\vartheta $ when  $\lambda=1$.
$\chi=0$ is drawn as dashed, 
$\chi= \pi/2$ as solid, 
and $\chi=\pi$ as dotted.} 
\label{fig8}
\end{minipage}
\ $\qquad$ \
\begin{minipage}[t]{7.5cm}
{\scalebox{0.45} {\includegraphics{fig9.eps}}}
\caption{$c_{\rm eff}$ vs. $\!\vartheta $ 
when $\lambda=2$.
$\chi=0$ is drawn as dashed, 
$\chi= \pi/3$ as solid, 
and $\chi=\pi$ as dotted.}
\label{fig9} 
\end{minipage}
\end{figure}
\vspace{.5cm}



\section{Conclusion} 

We analyzed the massless (R-channel) TBA  for the boundary
sine-Gordon theory with coupling parameter 
$\lambda =$ positive integer.  
The violation of the topological charge is 
incorporated into the analysis and 
the boundary effect on the effective central charge is investigated. 
Modifying the massless TBA 
using the excited state contribution
(the same is also expected for the massive case),
we obtain the right behavior of the ground state energy.  
In this way, we also confirm the parameter identification
of the boundary action parameters with 
the scattering amplitude parameters 
for the massless case.   

From the parametric dependence of the energy,
one can confirm the exact behavior of Josephson current 
\bea
I(\chi) = 2 e \frac{\partial}{\partial \chi}
E_0 (\mu_{b}^{(1)} ,{\mu_b}^{(2)},\chi ) 
\eea
as expected in \cite{iaff} 
which accounts for the Andreev scattering.

Even though in our analysis the coupling parameter 
$\lambda $ is restricted to 
a positive integer, the $c_{\rm eff}$ should behave
the same way for arbitrary coupling constant.  
However, the TBA of massless/massive
boundary sine-Gordon theory with arbitrary coupling 
is not feasible at this moment 
when the bulk Hamiltonian eigenstates are used
because  this will result in the infinitely 
coupled TBA equations \cite{sGTBA}. 
Instead of this approach,  
DDV type equation is expected to be more suitable,
which does not impose string hypothesis 
for the structure of the roots of Bethe ansatz
\cite{ddv}. 
Further investigation 
will be carried on this arbitrary coupling cases 
and also on the scale dependence as well as 
parametric dependence of the massive TBA 
in a separate paper.

Finally, it is noted that 
the sign change effect of the boundary term 
of boundary Liouville theory 
and boundary sinh-Gordon model 
\cite{bshG} can be explained 
using the analytically continued boundary parameter. 
On the other hand, 
integrable boundary ADE-affine Toda theories \cite{afftoda} 
has discrete boundary conditions,
(+), (--) and Neumann condition.
Among the three, (--) boundary condition
is not yet fully understood.
It remains to be seen that
the relative sign change of the boundary term
$\mu_B \to -\mu_B$ will induce 
the exicted state contribution to $c_{\rm eff}$.

\section*{\bf Acknowledgement}

We thank C. Ahn, P. Dorey, K. Moon and R. Tateo 
for valuable discussions and KIAS for hospitality.
This work is supported in part 
by the Basic Research Program of the Korea Science 
and Engineering Foundation Grant number R01-1999-00018-0(2002)
and by Korea Research Foundation 2002-070-C00025.

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