# The $SL(K+3,\mathbb{C})$ symmetry of string scatterings from D-branes

Sheng-Hong Lai<EMAIL_ADDRESS>Department of Electrophysics, National
Yang Ming Chiao Tung University, Hsinchu, Taiwan, R.O.C.  Jen-Chi Lee
<EMAIL_ADDRESS>Department of Electrophysics, National Yang Ming Chiao
Tung University, Hsinchu, Taiwan, R.O.C.  Yi Yang<EMAIL_ADDRESS>Department of Electrophysics, National Yang Ming Chiao Tung University,
Hsinchu, Taiwan, R.O.C.

###### Abstract

By using the solvability of Lauricella function
$F_{D}^{(K)}\left(\alpha;\beta_{1},...,\beta_{K};\gamma;x_{1},...,x_{K}\right)$
with nonpositive integer $\beta_{J}$, we show that each scattering or decay
process of string and D-brane states at arbitrary mass levels can be expressed
in terms of a single Lauricella function. This result extends the previous
exact $SL(K+3,\mathbb{C})$ symmetry of tree-level open bosonic string theory
to include the D-brane.

##  I Introduction

Motivated by the previous calculation of high energy symmetry Gross ; Gross1
of string scattering amplitudes (SSA) GM ; GM1 ; GrossManes ,ChanLee ;
ChanLee2 ; CHL ; CHLTY2 ; CHLTY1 , it was shown LSSA  recently that all SSA of
four arbitrary string states of the open bosonic string theory at all
kinematic regimes can be expressed in terms of the $D$-type Lauricella
functions
$F_{D}^{(K)}\left(\alpha;\beta_{1},...,\beta_{K};\gamma;x_{1},...,x_{K}\right)$
with associated exact $SL(K+3,\mathbb{C})$ symmetry (see the definition of $K$
in Eq.(3). On the other hand, a class of polarized fermion SSA (PFSSA) at
arbitrary mass levels of the R-sector of the fermionic string theory can also
be expressed in terms of the $D$-type Lauricella functions pfssa . Indeed, it
can be shown LSSA  that these Lauricella functions form an infinite
dimensional representation of the $SL(K+3,\mathbb{C})$ symmetry group.
Moreover, it was demonstrated that there existed $K+2$ recursion relations
among the $D$-type Lauricella functions. These recursion relations can be used
to reproduce the Cartan subalgebra and simple root system of the
$SL(K+3,\mathbb{C})$ group with rank $K+2$ and vice versa.

For the cases of nonpositive integer $\beta_{J}$ (see Eq.(9)) in the $D$-type
Lauricella functions which correspond to the cases of the SSA or the
Lauricella SSA (LSSA) mentioned above, the $SL(K+3,\mathbb{C})$ group or the
corresponding $K+2$ stringy Ward identities among the LSSA can be used to
solve solve  all the LSSA and express them in terms of one amplitude. These
exact Ward identities among the exact LSSA are generalizations of the linear
relations with constant coefficients among SSA in the hard scattering limit
conjectured by Gross Gross ; Gross1  in 1988 and later corrected and proved in
ChanLee ; ChanLee2 ; CHL ; CHLTY2 ; CHLTY1 .

More recently, by using the string theory extension stringbcfw , bcfw3 ; bcfw4
of the field theory BCFW on-shell recursion relations bcfw1 ; bcfw2  , one can
show that LLY3  the residues of all $n$-point SSA including the Koba-Nielsen
(KN) amplitudes can be expressed in terms of the Lauricella functions with
nonpositive integer $\beta_{J}$. As a result, the above $SL(K+3,\mathbb{C})$
symmetry group of the $4$-point LSSA was extended to the $n$-point LSSA with
arbitrary $n$ exact . It is thus believed that the $SL(K+3,\mathbb{C})$
symmetry is the fundamental symmetry of the whole bosonic string theory, at
least, tree-level of the bosonic string theory.

To justify the conjecture of the fundamental $SL(K+3,\mathbb{C})$ symmetry of
the bosonic string theory, one needs to collect more evidences or more SSA to
support the proposed exact symmetry. In this paper, we will show that the
scattering and decay processes of string and D-brane states at arbitrary mass
levels can again be expressed in terms of the $D$-type Lauricella functions
with nonpositive integer $\beta_{J}$. This result extends the previous exact
$SL(K+3,\mathbb{C})$ symmetry of tree-level open bosonic string theory to
include the D-brane. This is also consistent with the previous results that
the linear relations with constant coefficients among SSA in the hard
scattering limit persist for the processes of D-brane scatterings DD  and
decays decay  as they are all related to the exact $SL(K+3,\mathbb{C})$
symmetry of bosonic string theory. We will see that the calculation will be
greatly simplified by using the solvability solve  of the Lauricella functions
with nonpositive integer $\beta_{J}$.

##  II Review of $SL(K+3,C)$ symmetry

We first review the LSSA of three tachyons and one arbitrary string states in
the $26D$ open bosonic string theory and its associated $SL(K+3,\mathbb{C})$
symmetry. The general states are of the following form LSSA

$\left|r_{n}^{T},r_{m}^{P},r_{l}^{L}\right\rangle=\prod_{n>0}\left(\alpha_{-n}^{T}\right)^{r_{n}^{T}}\prod_{m>0}\left(\alpha_{-m}^{P}\right)^{r_{m}^{P}}\prod_{l>0}\left(\alpha_{-l}^{L}\right)^{r_{l}^{L}}|0,k\rangle$
(1)

where $e^{P}=\frac{1}{M_{2}}(E_{2},\mathrm{k}_{2},0)=\frac{k_{2}}{M_{2}}$ is
the momentum polarization, $e^{L}=\frac{1}{M_{2}}(\mathrm{k}_{2},E_{2},0)$ is
the longitudinal polarization and $e^{T}=(0,0,1)$ is the transverse
polarization on the $\left(2+1\right)$-dimensional scattering plane. In
addition to the mass level $M_{2}^{2}=2(N-1)$ with

$N=\sum_{\begin{subarray}{c}n,m,l>0\\\ \\{\text{ }r_{j}^{X}\neq
0\\}\end{subarray}}\left(nr_{n}^{T}+mr_{m}^{P}+lr_{l}^{L}\right),$ (2)

we define another important index $K$ for the state in Eq.(1)

$K=\sum_{\begin{subarray}{c}n,m,l>0\\\ \\{\text{ }r_{j}^{X}\neq
0\\}\end{subarray}}\left(n+m+l\right)$ (3)

where $X=\left(T,P,L\right)$ and we have put $r_{n}^{T}=r_{m}^{P}=r_{l}^{L}=1$
in Eq.(2) in the definition of $K$. Intuitively, $K$ counts the number of
variaty of the $\alpha_{-j}^{X}$ oscillators in Eq.(1). For later use, we also
define

$k_{j}^{X}\equiv e^{X}\cdot k_{j}\text{ \ for \ }X=\left(T,P,L\right).$ (4)

Note that SSA of three tachyons and one arbitrary string states with
polarizations orthogonal to the scattering plane vanish.

Note that to achieve BRST invariance or physical state conditions in the old
covariant quantization scheme for the state in Eq.(1), one needs to add
polarizations and put on the Virasoro constraints. As an example, let’s
calculate the case of symmetric spin $3$ state of mass level $M_{2}^{2}=4$. We
first note that the three momentum polarizations defined on the scattering
plane above satisfy the completeness relation

$\eta^{\mu\nu}=\underset{\alpha,\beta}{{\textstyle\sum}}e_{\alpha}^{\mu}e_{\beta}^{\nu}\eta^{\alpha\beta}$
(5)

where $\mu,\nu=0,1,2$ and $\alpha,\beta=P,L,T.$ $Diag$
$\eta^{\mu\nu}=(-1,1,1)$. We can use Eq.(5) to transform all $\mu,\nu$
coordinates to coordinates $\alpha,\beta$ on the scattering plane. One gauge
choice of the symmetric spin $3$ state with Virasoro constraints can be
calculated to be

$\epsilon_{\mu\nu\lambda}\alpha_{-1}^{\mu\nu\lambda}\left|0,k\right\rangle;k^{\mu}\epsilon_{\mu\nu\lambda}=0,\eta^{\mu\nu}\epsilon_{\mu\nu\lambda}=0.$
(6)

We can then use the helicity decomposition and writing
$\epsilon_{\mu\nu\lambda}=\Sigma_{\mu,\nu,\lambda}e_{\mu}^{\alpha}e_{\nu}^{\beta}e_{\lambda}^{\delta}u_{\alpha\beta\delta};\alpha,\beta,\delta=P,L,T$
to get

$\epsilon_{\mu\nu\lambda}\alpha_{-1}^{\mu\nu\lambda}\left|0,k\right\rangle=[u_{TTL}(3\alpha_{-1}^{TTL}-\alpha_{-1}^{LLL})+u_{TTT}(\alpha_{-1}^{TTT}-3\alpha_{-1}^{LLT})]\left|0,k\right\rangle.$
(7)

It is now easy to see from Eq.(7) that to achieve BRST invariance the spin $3$
state can be written as a linear combination of states in Eq.(1) with
coefficients $u_{TTL}$ and $u_{TTT}$.

The 4-point LSSA of three tachyons and one string state in Eq.(1) can be
calculated to be LSSA

$A_{4}=B\left(-\frac{t}{2}-1,-\frac{s}{2}-1\right)F_{D}^{(K)}\left(-\frac{t}{2}-1;R_{n}^{X};\frac{u}{2}+2-N;\tilde{Z}_{n}^{X}\right)\prod_{X}\left(\prod_{n=1}\left[-(n-1)!k_{3}^{X}\right]^{r_{n}^{X}}\right)$
(8)

where $B(a,b)$ is the Beta function with $\left(s,t\right)$ being the usual
Mandelstam variables, $k_{i}^{X}$ is the momentum of the $i$th string state
projected on the $X$ polarization. In Eq.(8), we have defined

$R_{l}^{X}\equiv\left\\{-r_{1}^{X}\right\\}^{1},\cdots,\left\\{-r_{l}^{X}\right\\}^{l}\text{
\ with \ }\left\\{a\right\\}^{n}=\underset{n}{\underbrace{a,a,\cdots,a}}.$ (9)

for the $\beta_{J}$ in the Lauricella function and

$Z_{l}^{X}\equiv\left[z_{1}^{X}\right],\cdots,\left[z_{l}^{X}\right]\text{ \ \
with \ \ }\left[z_{l}^{X}\right]=z_{l0}^{X},\cdots,z_{l\left(l-1\right)}^{X}.$
(10)

where in Eq.(10), we have defined

$z_{l}^{X}=\left|\left(-\frac{k_{1}^{X}}{k_{3}^{X}}\right)^{\frac{1}{k}}\right|,\
z_{ll^{\prime}}^{X}=z_{l}^{X}e^{\frac{2\pi il^{\prime}}{l}},\
\tilde{z}_{ll^{\prime}}^{X}\equiv 1-z_{ll^{\prime}}^{X}\text{ \ \ for \ \
}l^{\prime}=0,\cdots,l-1.$ (11)

It is important to note that all $\beta_{j}$ of $F_{D}^{(K)}$ in Eq.(8) are
nonpositive integer. The $D$-type Lauricella function $F_{D}^{(K)}$ in Eq.(8)
is defined to be

$F_{D}^{(K)}\left(\alpha;\beta_{1},...,\beta_{K};\gamma;x_{1},...,x_{K}\right)=\sum_{n_{1},\cdots,n_{K}=0}^{\infty}\frac{\left(\alpha\right)_{n_{1}+\cdots+n_{K}}}{\left(\gamma\right)_{n_{1}+\cdots+n_{K}}}\frac{\left(\beta_{1}\right)_{n_{1}}\cdots\left(\beta_{K}\right)_{n_{K}}}{n_{1}!\cdots
n_{K}!}x_{1}^{n_{1}}\cdots x_{K}^{n_{K}}$ (12)

where
$(\alpha)_{n}=\alpha\cdot\left(\alpha+1\right)\cdots\left(\alpha+n-1\right)$
is the Pochhammer symbol. The result in Eq.(8) can be generalized to LSSA of
four arbitrary string states, and to those of $n$ arbitrary string states (see
Eq.(25)) below LLY3 ; exact .

For illustration, we calculate the Lauricella functions which correspond to
the LSSA for levels $K=1,2$. For $K=1$, there are three type of LSSA
$(\alpha=-\frac{t}{2}-1,\gamma=\frac{u}{2}+2)$

$\displaystyle(\alpha_{-1}^{T})^{p_{1}}\text{,
}F_{D}^{(1)}(\alpha,-p_{1},\gamma-p_{1},1)\text{, }N$ $\displaystyle=p_{1},$
(13) $\displaystyle(\alpha_{-1}^{P})^{q_{1}}\text{,
}F_{D}^{(1)}(\alpha,-q_{1},\gamma-q_{1},\left[\tilde{z}_{1}^{P}\right])\text{,
}N$ $\displaystyle=q_{1},$ (14) $\displaystyle(\alpha_{-1}^{L})^{r_{1}}\text{,
}F_{D}^{(1)}(\alpha,-r_{1},\gamma-r_{1},\left[\tilde{z}_{1}^{L}\right])\text{,
}N$ $\displaystyle=r_{1}.$ (15)

For $K=2$, there are six type of LSSA

$\displaystyle(\alpha_{-1}^{T})^{p_{1}}(\alpha_{-1}^{P})^{q_{1}}\text{,
}F_{D}^{(2)}(\alpha,-p_{1},-q_{1},\gamma-
p_{1}-q_{1},1,\left[\tilde{z}_{1}^{P}\right])\text{,}N$
$\displaystyle=p_{1}+q_{1},$ (16)
$\displaystyle(\alpha_{-1}^{T})^{p_{1}}(\alpha_{-1}^{L})^{r_{1}}\text{,
}F_{D}^{(2)}(\alpha,-p_{1},-r_{1},\gamma-
p_{1}-r_{1},1,\left[\tilde{z}_{1}^{L}\right])\text{,}N$
$\displaystyle=p_{1}+r_{1},$ (17)
$\displaystyle(\alpha_{-1}^{P})^{q_{1}}(\alpha_{-1}^{L})^{r_{1}}\text{,
}F_{D}^{(2)}(\alpha,-q_{1},-r_{1},\gamma-
q_{1}-r_{1},\left[\tilde{z}_{1}^{P}\right],\left[\tilde{z}_{1}^{L}\right])\text{,}N$
$\displaystyle=q_{1}+r_{1},$ (18)
$\displaystyle(\alpha_{-2}^{T})^{p_{2}}\text{,
}F_{D}^{(2)}(\alpha,-p_{2},-p_{2},\gamma-2p_{2},1,1)\text{, }N$
$\displaystyle=2p_{2},$ (19) $\displaystyle(\alpha_{-2}^{P})^{q_{2}}\text{,
}F_{D}^{(2)}(\alpha,-q_{2},-q_{2},\gamma-2q_{2},1-Z_{2}^{P},1-\omega
Z_{2}^{P})\text{, }N$ $\displaystyle=2q_{2},$ (20)
$\displaystyle(\alpha_{-2}^{L})^{r_{2}}\text{,
}F_{D}^{(2)}(\alpha,-r_{2},-r_{2},\gamma-2r_{2},1-Z_{2}^{L},1-\omega
Z_{2}^{L})\text{, }N$ $\displaystyle=2r_{2}.$ (21)

It is important to note that for a given $K$, there are infinite number of
string states with arbitrary higher mass levels. Moreover, each string state
was assigned a particular value of integer $K$, and its associated LSSA is a
basis of the $SL(K+3,\mathbb{C})$ group representation.

To demonstrate the $SL(K+3,\mathbb{C})$ symmetry of the LSSA, one first
defines the basis functions slkc

$\displaystyle f_{ac}^{b_{1}\cdots
b_{K}}\left(\alpha;\beta_{1},\cdots,\beta_{K};\gamma;x_{1},\cdots,x_{K}\right)$
$\displaystyle=B\left(\gamma-\alpha,\alpha\right)F_{D}^{\left(K\right)}\left(\alpha;\beta_{1},\cdots,\beta_{K};\gamma;x_{1},\cdots,x_{K}\right)a^{\alpha}b_{1}^{\beta_{1}}\cdots
b_{K}^{\beta_{K}}c^{\gamma},$ (22)

so that the LSSA in Eq.(8) can be rewritten as Group

$A_{4}=f_{11}^{-(n-1)!k_{3}^{X}}\left(-\frac{t}{2}-1;R_{n}^{X},;\frac{u}{2}+2-N;\tilde{Z}_{n}^{X}\right).$
(23)

One can then introduce the $(K+3)^{2}-1$ generators $\mathcal{E}_{ij}$ of
$SL(K+3,\mathbb{C})$ group slkc ; Group

$\left[\mathcal{E}_{ij},\mathcal{E}_{kl}\right]=\delta_{jk}\mathcal{E}_{il}-\delta_{li}\mathcal{E}_{kj};\text{
\ }1\leqslant i,j\leqslant K+3$ (24)

to operate on the basis functions in Eq.(22). These are $1$ $E^{\alpha}$, $K$
$E^{\beta_{k}}(k=1,2\cdots K)$, $1$ $E^{\gamma}$,$1$ $E^{\alpha\gamma}$, $K$
$E^{\beta_{k}\gamma}$ and $K$ $E^{\alpha\beta_{k}\gamma}$ which sum up to
$3K+3$ raising generators. There are also $3K+3$ corresponding lowering
operators. In addition, there are $K\left(K-1\right)$
$E_{\beta_{p}}^{\beta_{k}}$ and $K+2$
$\left\\{J_{\alpha},J_{\beta_{k}},J_{\gamma}\right\\}$ generators, the Cartan
subalgebra. In sum, the total number of generators are
$2(3K+3)+K(K-1)+K+2=(K+3)^{2}-1$ Group .

For the general $4$-point LSSA, it is straightforward to calculate the
operation of these generators on the basis functions and show the
$SL(K+3,\mathbb{C})$ symmetry Group . For the cases of higher point $(n\geq
5)$ LSSA, one encounters the operation on the sum of products of the
Lauricella functions LLY3

$\text{Residue of }n\text{-point
}LSSA\sim\sum\text{coefficient}\prod(\text{single tensor }4\text{-point
}LSSA).$ (25)

Therefore, one needs to deal with product representations of
$SL(K+3,\mathbb{C})$.

Indeed, we have recently applied the string theory extension of field theory
BCFW on-shell recursion relations bcfw1 ; bcfw2  to show that the $SL(K+3,C)$
symmetry group of the $4$-point LSSA persists for general $n$-point SSA with
arbitrary higher point couplings in string theory exact . We thus have shown
that the $SL(K+3,C)$ symmetry is an exact symmetry of the whole bosonic string
theory, and that all $n$-point SSA of the bosonic string theory form an
infinite dimensional representation of the $SL(K+3,C)$ group. Moreover, all
residues of SSA in the string theory on-shell recursion prescription can be
expressed in terms of the four-point LSSA.

There is an interesting issue of the stringy on-shell Ward identities or
decoupling of zero-norm states associated with the $SL(K+3,C)$ symmetry. For
the $n$-point Ward identities with $n\geq 5$, one can either write down the
linear on-shell Ward identities in terms of $n$-point functions or, through
reduction of stringy BCFW recursion, calculate the non-linear on-shell Ward
identities in terms of $4$-point functions. For the latter case, we conjecture
that the non-linear Ward identities can be reduced to the equivalent linear
$4$-point Ward identities since both forms of Ward identities are associated
with the same $SL(K+3,C)$ group.

##  III Solvability of LSSA

There exist $K+2$ recurrence relations for the $D$-type Lauricella functions
Group . Moreover, these recurrence relations can be used to reproduce the
Cartan subalgebra and simple root system of the $SL(K+3,\mathbb{C})$ group
with rank $K+2$ Group . With the Cartan subalgebra and the simple roots, one
can easily write down the whole Lie algebra of the $SL(K+3,\mathbb{C})$ group.
So one can construct the $SL(K+3,\mathbb{C})$ Lie algebra from the recurrence
relations and vice versa.

On the other hand, one can use the $K+2$ recurrence relations to deduce the
following key recurrence relation solve

$\displaystyle
x_{j}F_{D}^{(K)}\left(\alpha;\beta_{1},.,\beta_{i}-1,..,\beta_{K};\gamma;x_{1},...,x_{K}\right)-x_{i}F_{D}^{(K)}\left(\alpha;\beta_{1},.,\beta_{j}-1,..,\beta_{K};\gamma;x_{1},...,x_{K}\right)$
$\displaystyle+\left(x_{i}-x_{j}\right)F_{D}^{(K)}\left(\alpha;\beta_{1},...,\beta_{K};\gamma;x_{1},...,x_{K}\right)=0,$
(26)

which, for the case of nonpositive $\beta_{j}$, can be repeatedly used to
decrease the value of $K$ and reduce all the Lauricella functions
$F_{D}^{(K)}$ in the LSSA to the Gauss hypergeometric functions $F_{D}^{(1)}=$
${}_{2}F_{1}(\alpha,\beta,\gamma,x)$. Indeed, One can repeatedly apply Eq.(26)
to the Lauricella functions in Eq.(8) and express
$F_{D}^{(K)}\left(\alpha;\beta_{1},...,\beta_{K};\gamma;x_{1},...,x_{K}\right)$
in terms of
$F_{D}^{(K-1)}(\beta_{1},..\beta_{i-1},\beta_{i+1}...\beta_{j}^{\prime},...\beta_{K})$
with
$\beta_{j}^{\prime}=\beta_{j},\beta_{j}-1,...,\beta_{j}-\left|\beta_{i}\right|$
or
$F_{D}^{(K-1)}(\beta_{1},...\beta_{i}^{\prime},...\beta_{j-1},\beta_{j+1},...\beta_{K})$
with
$\beta_{i}^{\prime}=\beta_{i},\beta_{i}-1,...,\beta_{i}-\left|\beta_{j}\right|$
(assuming $i<j$).

For example, for say $K=2$, Eq.(26) reduces to

$\displaystyle
x_{2}F_{D}^{(2)}\left(\alpha;\beta_{1}-1,\beta_{2};\gamma;x_{1},x_{2}\right)-x_{1}F_{D}^{(2)}\left(\alpha;\beta_{1},\beta_{2}-1;\gamma;x_{1},x_{2}\right)$
$\displaystyle+\left(x_{1}-x_{2}\right)F_{D}^{(2)}\left(\alpha;\beta_{1},\beta_{2};\gamma;x_{1},x_{2}\right)=0.$
(27)

For say $\beta_{1}=0$ and $\beta_{2}=-1$, we get

$\displaystyle x_{2}F_{D}^{(2)}\left(\alpha;-1,-1;\gamma;x_{1},x_{2}\right)$
$\displaystyle=x_{1}F_{D}^{(2)}\left(\alpha;0,-2;\gamma;x_{1},x_{2}\right)-\left(x_{1}-x_{2}\right)F_{D}^{(2)}\left(\alpha;0,-1;\gamma;x_{1},x_{2}\right)$
$\displaystyle=x_{1}F_{D}^{(1)}\left(\alpha;-2;\gamma;x_{2}\right)-\left(x_{1}-x_{2}\right)F_{D}^{(1)}\left(\alpha;-1;\gamma;x_{2}\right)$
(28)

which express the Lauricella function with $K=2$ in terms of those of $K=1$.
We can repeat similar process to decrease the value of $K$.

Moreover, one can further reduce the Gauss hypergeometric functions by
deriving a multiplication theorem for them, and then solve solve  all the LSSA
in terms of one single amplitude. This solvability is crucial to show that all
scattering and decay processes of string and D-brane states at arbitrary mass
levels can be expressed in terms of the Lauricella function and thus its
associated $SL(K+3,\mathbb{C})$ symmetry.

##  IV Closed string scattered off D-brane

In this paper, we will consider scattering and decay processes of string and
D-brane states at arbitrary mass levels. These are three classes of processes
Klebanov , D2 ; D3 ; D4 ; D5 ; D6 : (A). Closed string scattered off D-brane,
(B). Closed string decays into two open strings on the brane and (C). Four
open string scattering on the brane. The calculation of process (C) is similar
to that of four open string scattering without D-brane, and thus can be
expressed in terms of the Lauricella function.

In this section, we first consider process (A). In Klebanov , the calculation
was done only for the massless string states. Here we will consider
scatterings of arbitrary massive string states for the bosonic string. The
standard propagators of the left and right moving fields are $\left\langle
X^{\mu}\left(z\right)X^{\nu}\left(w\right)\right\rangle=-\eta^{\mu\nu}\log\left(z-w\right),\left\langle\tilde{X}^{\mu}\left(\bar{z}\right)\tilde{X}^{\nu}\left(\bar{w}\right)\right\rangle=-\eta^{\mu\nu}\log\left(\bar{z}-\bar{w}\right).$
In addition, there are nontrivial correlator as well Klebanov

$\left\langle
X^{\mu}\left(z\right)\tilde{X}^{\nu}\left(\bar{w}\right)\right\rangle=-D^{\mu\nu}\log\left(z-\bar{w}\right)$
(29)

as a result of the Dirichlet boundary condition at the real axis. The diagonal
matrix $D$ in Eq.(29) reverses the sign for fields satisfying Dirichlet
boundary condition. That is, there are $p+1$ Neumann and $25-p$ Dirichlet for
a Dp-brane. We will follow the standard notation and make the following
replacement Klebanov

$\tilde{X}^{\mu}\left(\bar{z}\right)\rightarrow D_{\text{ \
}\nu}^{\mu}X^{\nu}\left(\bar{z}\right)$ (30)

which allows us to use the standard correlators throughout our calculations.
As a warm up exercise, we first consider tachyon to tachyon scattering DD

$\displaystyle A_{tach}$ $\displaystyle=\int d^{2}z_{1}d^{2}z_{2}\left\langle
V_{1}\left(z_{1},\bar{z}_{1}\right)V_{2}\left(z_{2},\bar{z}_{2}\right)\right\rangle$
$\displaystyle=\int
d^{2}z_{1}d^{2}z_{2}\left(z_{1}-\bar{z}_{1}\right)^{k_{1}\cdot D\cdot
k_{1}}\left(z_{2}-\bar{z}_{2}\right)^{k_{2}\cdot D\cdot
k_{2}}\left|z_{1}-z_{2}\right|^{2k_{1}\cdot
k_{2}}\left|z_{1}-\bar{z}_{2}\right|^{2k_{1}\cdot D\cdot k_{2}}.$ (31)

To fix the $SL\left(2,R\right)$ invariance, we set $z_{1}=iy$ and $z_{2}=i$
and, for the contribution of the $(0,1)$ interval, we obtain DD

$\displaystyle A_{tach}^{(0,1)}$
$\displaystyle=4\left(2i\right)^{2a_{0}}\int_{0}^{1}dy\text{
}y^{a_{0}}\left(1-y\right)^{b_{0}}\left(1+y\right)^{c_{0}}$
$\displaystyle=4\left(2i\right)^{2a_{0}}2^{-2a_{0}-1-N}\int_{0}^{1}dx\text{
}x^{b_{0}}\left(1-x\right)^{a_{0}}\left(1+x\right)^{a_{0}+N}.$ (32)

In the above calculations, we have defined

$\displaystyle a_{0}$ $\displaystyle=k_{1}\cdot D\cdot k_{1}=k_{2}\cdot D\cdot
k_{2},$ (33) $\displaystyle b_{0}$ $\displaystyle=2k_{1}\cdot k_{2}+1,$ (34)
$\displaystyle c_{0}$ $\displaystyle=2k_{1}\cdot D\cdot k_{2}+1,$ (35)

so that

$2a_{0}+b_{0}+c_{0}+2=4N_{1}\equiv-N,$ (36)

and
$-k_{1}^{2}=M^{2}\equiv\frac{\alpha_{closed}^{\prime}M_{closed}^{2}}{2}=2(N_{1}-1)$,
$N_{1}=0$ for tachyon. The momentum conservation on the D-brane

$D\cdot k_{1}+k_{1}+D\cdot k_{2}+k_{2}=0$ (37)

is crucial to get the final result Eq.(32). Similarly, For the contribution of
the $(1,\infty)$ interval, we end up with

$\displaystyle A_{tach}^{(1,\infty)}$
$\displaystyle=4\left(2i\right)^{2a_{0}}\int_{1}^{\infty}dy\text{
}y^{a_{0}}\left(y-1\right)^{b_{0}}\left(1+y\right)^{c_{0}}$
$\displaystyle=4\left(2i\right)^{2a_{0}}2^{-2a_{0}-1-N}\int_{0}^{1}dx\text{
}x^{b_{0}}\left(1-x\right)^{a_{0}+N}\left(1+x\right)^{a_{0}}.$ (38)

For the general massive tensor to another massive tensor scattering, the
calculation will be very complicated as there are many new contraction terms.
We will use the solvability of the LSSA discussed above to simplify the
calculation. The strategy is as follows: We can simply calculate a typical
term of a given process. If the result turns out to be a Lauricella function
with nonpositive $\beta_{j}$, we can then use the solvability property to
argue that the final amplitude after summing up all typical terms of the
process is a LSSA. To do the calculation, we first define

$\displaystyle a$ $\displaystyle=k_{1}\cdot D\cdot k_{1}+n_{a},$ (39)
$\displaystyle b$ $\displaystyle=2k_{1}\cdot k_{2}+1+n_{b},$ (40)
$\displaystyle c$ $\displaystyle=2k_{1}\cdot D\cdot k_{2}+1+n_{c},$ (41)

where $n_{a}$, $n_{b}$ and $n_{c}$ are integer and then define
$N^{\prime}=-\left(2n_{a}+n_{b}+n_{c}\right)$, so that

$2a+b+c+2+N^{\prime}=4N_{1}\Longrightarrow 2a+b+c+2=4N_{1}-N^{\prime}\equiv-N$
(42)

where $k_{1}^{2}=2(N_{1}-1)$ and $N_{1}$ is the mass level of $k_{1}$. It is
easy to see that a typical term in the general tensor to tensor scattering can
be calculated to be DD

$\displaystyle I_{\left(0,1\right)}$ $\displaystyle=\int_{0}^{1}dt\text{
}y^{a}\left(1-y\right)^{b}\left(1+y\right)^{c},$
$\displaystyle=2^{-2a-1-N}\int_{0}^{1}dx\text{
}x^{b}\left(1-x\right)^{a}\left(1+x\right)^{a+N}.$ (43)

Similarly, for the $(1,\infty)$ interval, one gets

$\displaystyle I_{\left(1,\infty\right)}$
$\displaystyle=\int_{1}^{\infty}dy\text{
}y^{a}\left(y-1\right)^{b}\left(1+y\right)^{c}$
$\displaystyle=2^{-2a-1-N}\int_{0}^{1}dx\text{
}x^{b}\left(1-x\right)^{a+N}\left(1+x\right)^{a}.$ (44)

The sum of the two channels gives DD

$\displaystyle I$
$\displaystyle=2^{-2a-2-N}\sum_{m=0}^{N}\left[1+\left(-1\right)^{m}\right]\binom{N}{m}\cdot\frac{\Gamma\left(a+1\right)\Gamma\left(\frac{b+1}{2}+\frac{m}{2}\right)}{\Gamma\left(a+\frac{b+3}{2}+\frac{m}{2}\right)}$
$\displaystyle=2^{-2a-1-N}\frac{\Gamma\left(a+1\right)\Gamma\left(\frac{b+1}{2}\right)}{\Gamma\left(a+\frac{b+3}{2}\right)}\sum_{n=0}^{\left[\frac{N}{2}\right]}\binom{N}{2n}\dfrac{\left(\frac{b+1}{2}\right)_{n}}{\left(a+\frac{b+3}{2}\right)_{n}}$
$\displaystyle=2^{-2a-1-N}\cdot B\left(a+1,\frac{b+1}{2}\right)\cdot\text{
}_{3}F_{2}\left(\frac{b+1}{2},-\left[\frac{N}{2}\right],\dfrac{1}{2}-\left[\frac{N}{2}\right];a+\frac{b+3}{2},\dfrac{1}{2};1\right)$
(45)

where ${}_{3}F_{2}$ is a generalized hypergeometric function. For the special
arguments of ${}_{3}F_{2}$ in Eq.(45), the hypergeometric function terminates
to a finite sum and, as a result, the whole scattering amplitudes consistently
reduce to the usual beta function. In calculating Eq.(45), we have used the
identity

$\overset{\left[\frac{N}{2}\right]}{\underset{n=0}{\sum}}\binom{N}{2n}\frac{\left(A\right)_{n}}{\left(C\right)_{n}}=\text{
}_{3}F_{2}\left(A,-\left[\frac{N}{2}\right],\frac{1}{2}-\left[\frac{N}{2}\right];C,\frac{1}{2};1\right),$
(46)

which can be easily proved.

At this point, one might think that the amplitude calculated in Eq.(45) is not
a LSSA, and the $SL(K+3,\mathbb{C})$ group may be just a subgroup of an
unknown larger symmetry group $G\supseteq$ $SL(K+3,\mathbb{C})$ of the bosonic
string theory. However, we will see that this is not the case. To show that
${}_{3}F_{2}$ in the amplitude Eq.(45) is a LSSA, we first do a change of
variable $y=x^{2}$ to get

$\displaystyle I$ $\displaystyle=2^{-2a-1-N}\int_{0}^{1}dx\text{
}x^{b}\left(1-x\right)^{a}\left(1+x\right)^{a}\left[\left(1+x\right)^{N}+\left(1-x\right)^{N}\right]$
$\displaystyle=2^{-2a-1-N}\int_{0}^{1}dy\text{
}y^{\frac{b-1}{2}}\left(1-y\right)^{a}G\left(y\right)$ (47)

where

$G\left(y\right)=G\left(x^{2}\right)=\frac{1}{2}\left[\left(1+x\right)^{N}+\left(1-x\right)^{N}\right].$
(48)

We can solve

$G\left(x^{2}\right)=\frac{1}{2}\left[\left(1+x\right)^{N}+\left(1-x\right)^{N}\right]=0$
(49)

to get

$x=x_{k}=\frac{\left(-1\right)^{\frac{1}{N}}-1}{\left(-1\right)^{\frac{1}{N}}+1}=\frac{w_{N,k}-1}{w_{N,k}+1}=i\tan\frac{\theta_{k}}{2}$
(50)

where

$w_{N,k}=e^{\frac{i\pi}{N}+\frac{2i\pi
k}{N}}=e^{i\theta_{k}},\theta_{k}=\frac{\pi}{N}+\frac{2\pi
k}{N},k=1,2\cdots,N.$ (51)

We can now do the following factorization

$\displaystyle G\left(x^{2}\right)$
$\displaystyle=\underset{k=1}{\overset{N}{{\displaystyle\prod}}}\left(1-\frac{x}{x_{k}}\right)=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1-\frac{x}{x_{k}}\right)\left(1-\frac{x}{\bar{x}_{k}}\right)=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1-\left(\frac{x}{x_{k}}+\frac{x}{\bar{x}_{k}}\right)+\frac{x^{2}}{x_{k}\bar{x}_{k}}\right)$
$\displaystyle=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1+\frac{x^{2}}{x_{k}\bar{x}_{k}}\right)=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1-\frac{x^{2}}{x_{k}^{2}}\right)$
(52)

to obtain

$G\left(y\right)=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1-\frac{y}{x_{k}^{2}}\right)=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1+\frac{y}{\left(\tan\frac{\theta_{k}}{2}\right)^{2}}\right)=\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1+\frac{y}{\left(\tan\left(\frac{\pi}{2N}+\frac{\pi
k}{N}\right)\right)^{2}}\right).$ (53)

Finally, we can use

$\displaystyle\int_{0}^{1}dy\text{
}y^{\frac{b-1}{2}}\left(1-y\right)^{a}G\left(y\right)$
$\displaystyle=\int_{0}^{1}dy\text{
}y^{\frac{b-1}{2}}\left(1-y\right)^{a}\underset{k=1}{\overset{\left[\frac{N}{2}\right]}{{\displaystyle\prod}}}\left(1-\frac{y}{x_{k}^{2}}\right)$
$\displaystyle=B\left(a+1,\frac{b+1}{2}\right)\cdot
F_{D}^{\left(\left[\frac{N}{2}\right]\right)}\left(\frac{b+1}{2};\underset{\left[\frac{N}{2}\right]}{\underbrace{-1,...,-1}};a+\frac{b+3}{2};\frac{1}{x_{1}^{2}},...,\frac{1}{x_{\left[\frac{N}{2}\right]}^{2}}\right)$
(54)

to obtain the identification

$\text{
}_{3}F_{2}\left(\frac{b+1}{2},-\left[\frac{N}{2}\right],\frac{1}{2}-\left[\frac{N}{2}\right];a+\frac{b+3}{2},\frac{1}{2};1\right)=F_{D}^{\left(\left[\frac{N}{2}\right]\right)}\left(\frac{b+1}{2};\underset{\left[\frac{N}{2}\right]}{\underbrace{-1,...,-1}};a+\frac{b+3}{2};\frac{1}{x_{1}^{2}},...,\frac{1}{x_{\left[\frac{N}{2}\right]}^{2}}\right)$
(55)

where $x_{k}$ is defined in Eq.(50). We believe that the identity in Eq.(55)
derived from string theory was not known previously in the literature jack .
In conclusion, we have shown that each amplitude of process (A) can be
expressed in terms of a single Lauricella function with nonpositive integer
$\beta_{j}$ and thus is a LSSA. As a result, all scattering of string at
arbitrary mass levels from D-brane calculated in (A) form a part of an
infinite dimensional representation of the exact $SL(K+3,\mathbb{C})$ symmetry
of the bosonic string theory.

##  V Closed string decays into two open string

In this section, we consider process (B), namely, closed string decays into
two open strings on the brane. We will adapt the same strategy used in the
last section and calculate only a typical term of a given process. We begin
with the kinematics of the decay process. The momentum conservation on the
D-brane reads

$\frac{1}{2}(k_{c}+D\cdot k_{c})+k_{1}+k_{2}=0$ (56)

where $k_{c}$ is the momentum of the closed string state. In the usual three-
point amplitudes, momentum conservation completely constrains the kinematics.
In the presence of D-brane, the non-conservation of momentum in the directions
transverse to the D-brane gives precisely one kinematic variable which can be
defined to be

$t=-\left(k_{1}+k_{2}\right)^{2}.$ (57)

By using Eq.(56) and Eq.(57), one easily gets

$\displaystyle k_{1}\cdot k_{c}$ $\displaystyle=k_{1}\cdot D\cdot
k_{c}=\frac{t+M_{1}^{2}-M_{2}^{2}}{2},$ $\displaystyle k_{2}\cdot k_{c}$
$\displaystyle=k_{2}\cdot D\cdot k_{c}=\frac{t+M_{2}^{2}-M_{1}^{2}}{2},$ (58)

which give

$t=k_{1}\cdot k_{c}+k_{2}\cdot D\cdot k_{c}=k_{2}\cdot k_{c}+k_{1}\cdot D\cdot
k_{c}.$ (59)

We first calculate the amplitude of a closed string tachyon to decay into two
open string tachyons

$\displaystyle A_{tach}$ $\displaystyle=\int dx_{1}dx_{2}d^{2}z\left\langle
e^{ik_{1}\cdot X\left(x_{1}\right)}e^{ik_{2}\cdot
X\left(x_{2}\right)}e^{ik_{c}\cdot
X\left(z\right)}e^{ik_{c}\cdot\bar{X}\left(\bar{z}\right)}\right\rangle$
$\displaystyle=\int
dx_{1}dx_{2}d^{2}z\cdot\left(x_{1}-x_{2}\right)^{k_{1}\cdot
k_{2}}\left(z-\bar{z}\right)^{k_{c}\cdot D\cdot
k_{c}}\left(x_{1}-z\right)^{k_{1}\cdot k_{c}}$
$\displaystyle\cdot\left(x_{1}-\bar{z}\right)^{k_{1}\cdot D\cdot
k_{c}}\left(x_{2}-z\right)^{k_{2}\cdot
k_{c}}\left(x_{2}-\bar{z}\right)^{k_{2}\cdot D\cdot k_{c}}.$ (60)

The next step is to use $\\{z_{1}$, $z_{2}$, $z_{3}$, $\bar{z}_{3}\\}=\\{-x$,
$x$, $i$, $-i\\}$ to fix the $SL\left(2,R\right)$ invariance and obtain

$\displaystyle A_{tach}$ $\displaystyle=\int dx\text{ }x^{k_{1}\cdot
k_{2}}\left(x+i\right)^{k_{1}\cdot k_{c}+k_{2}\cdot D\cdot
k_{c}}\left(x-i\right)^{k_{2}\cdot k_{c}+k_{1}\cdot D\cdot k_{c}}$
$\displaystyle=\int dx\text{ }x^{k_{1}\cdot
k_{2}}\left(x+i\right)^{t}\left(x-i\right)^{t}$ $\displaystyle=\int dx\text{
}x^{b_{0}}\left(1+x^{2}\right)^{a_{0}}$ (61)

where we have used Eq.(59) and defined

$a_{0}=t,b_{0}=k_{1}\cdot k_{2}.$ (62)

We now turn to the general mass level case. We will again use the solvability
of the LSSA discussed above to simplify the calculation as before. A typical
term of an arbitrary massive closed string state decays into two arbitrary
massive open string states can be written as

$A=\int dx\text{ }x^{k_{1}\cdot
k_{2}+n_{b}}\left(x+i\right)^{t+n_{a}}\left(x-i\right)^{t+n_{c}}$ (63)

where $n_{a}$, $n_{b}$ and $n_{c}$ are related to mass levels of $k_{c}$,
$k_{1}$ and $k_{2}$. At this point, we expect after summing up all terms in
the calculation, a real amplitude will be obtained. So we are going to
calculate only the real part of $A$

$\displaystyle A+\bar{A}=\int_{-\infty}^{+\infty}dx\text{ }x^{k_{1}\cdot
k_{2}+n_{b}}\left(x^{2}+1\right)^{t}\left[\left(x+i\right)^{n_{a}}\left(x-i\right)^{n_{c}}+\left(x-i\right)^{n_{a}}\left(x+i\right)^{n_{c}}\right]$
$\displaystyle=\int_{-\infty}^{+\infty}dx\text{ }x^{k_{1}\cdot
k_{2}+n_{b}+N}\left(x^{2}+1\right)^{t+\min\left\\{n_{a},n_{c}\right\\}}\left[\left(1-\frac{i}{x}\right)^{N}+\left(1+\frac{i}{x}\right)^{N}\right]$
(64)

where $N\equiv\left|n_{c}-n_{a}\right|$, and see whether the final answer is a
Lauricella function. Eq.(64) can be further reduced to

$\displaystyle A+\bar{A}$ $\displaystyle=\int_{-\infty}^{+\infty}dx\text{
}x^{b}\left(x^{2}+1\right)^{a}\underset{m=0}{\overset{N}{\sum}}\binom{N}{m}\left[1+\left(-1\right)^{m}\right]\left(\frac{i}{x}\right)^{m}$
$\displaystyle=2\underset{n=0}{\overset{\left[\frac{N}{2}\right]}{\sum}}\binom{N}{2n}\left(-1\right)^{n}\int_{-\infty}^{+\infty}dx\text{
}x^{b-2n}\left(x^{2}+1\right)^{a}$ (65)

where we have defined

$\displaystyle a$ $\displaystyle=t+\min\left\\{n_{a},n_{c}\right\\},$
$\displaystyle b$ $\displaystyle=k_{1}\cdot k_{2}+n_{b}+N,$ (66)

which are higher mass level generalization of Eq.(62). We can use the change
of variable $y=\frac{x^{2}}{x^{2}+1}$ to perform the integral in Eq.(65)

$\displaystyle\int_{-\infty}^{+\infty}dx\text{ }x^{b}\left(1+x^{2}\right)^{a}$
$\displaystyle=\frac{1}{2}\left[1+\left(-1\right)^{b}\right]\int_{0}^{1}dy\text{
}y^{\frac{b}{2}-\frac{1}{2}}\left(1-y\right)^{-a-\frac{b}{2}-\frac{3}{2}}$
$\displaystyle=\frac{1}{2}\left[1+\left(-1\right)^{b}\right]\frac{\Gamma\left(\frac{b}{2}+\frac{1}{2}\right)\Gamma\left(-a-\frac{b}{2}-\frac{1}{2}\right)}{\Gamma\left(-a\right)}.$
(67)

Finally, we obtain

$A+\bar{A}=2\underset{n=0}{\overset{\left[\frac{N}{2}\right]}{\sum}}\binom{N}{2n}\left(-1\right)^{n}\frac{1}{2}\left[1+\left(-1\right)^{b}\right]\frac{\Gamma\left(\frac{b}{2}+\frac{1}{2}+n\right)\Gamma\left(-a-\frac{b}{2}-\frac{1}{2}-n\right)}{\Gamma\left(-a\right)}.$
(68)

To derive a Lauricella function in Eq.(68), we note that

$\frac{\Gamma\left(\frac{b}{2}+\frac{1}{2}+n\right)\Gamma\left(-a-\frac{b}{2}-\frac{1}{2}-n\right)}{\Gamma\left(-a\right)}=\frac{\left(\frac{b}{2}+\frac{1}{2}\right)_{n}\Gamma\left(\frac{b}{2}+\frac{1}{2}\right)\Gamma\left(-a-\frac{b}{2}-\frac{1}{2}\right)}{\left(-1\right)^{n}\left(a+\frac{b}{2}+\frac{3}{2}\right)_{n}\Gamma\left(-a\right)}.$
(69)

So Eq.(68) can be written as

$\displaystyle A+\bar{A}$
$\displaystyle=2\left[1+\left(-1\right)^{b}\right]\frac{\Gamma\left(\frac{b}{2}+\frac{1}{2}\right)\Gamma\left(-a-\frac{b}{2}-\frac{1}{2}\right)}{\Gamma\left(-a\right)}\underset{n=0}{\overset{\left[\frac{N}{2}\right]}{\sum}}\binom{N}{2n}\frac{\left(\frac{b}{2}+\frac{1}{2}\right)_{n}}{\left(a+\frac{b}{2}+\frac{3}{2}\right)_{n}}$
$\displaystyle=2\left[1+\left(-1\right)^{b}\right]B\left(-a-\frac{b+1}{2},\frac{b+1}{2}\right)$
$\displaystyle\times_{3}F_{2}\left(\frac{b+1}{2},-\left[\frac{N}{2}\right],\frac{1}{2}-\left[\frac{N}{2}\right];a+\frac{b+3}{2};\frac{1}{2};1\right)$
$\displaystyle=2\left[1+\left(-1\right)^{b}\right]B\left(-a-\frac{b+1}{2},\frac{b+1}{2}\right)$
$\displaystyle\times
F_{D}^{\left(\left[\frac{N}{2}\right]\right)}\left(\frac{b+1}{2};\underset{\left[\frac{N}{2}\right]}{\underbrace{-1,...,-1}};a+\frac{b+3}{2};\frac{1}{x_{1}^{2}},...,\frac{1}{x_{\left[\frac{N}{2}\right]}^{2}}\right)$
(70)

where we have used the identities in Eq.(46) and Eq.(55). Finally, we can use
the solvability property of $F_{D}^{(\left[\frac{N}{2}\right])}$ with
nonpositive $\beta_{j}$ to argue that the final amplitude after summing up all
typical terms of the decay process is a LSSA.

Incidentally, we note that the factor in the first line of Eq.(70)

$\frac{1}{2}\left[1+\left(-1\right)^{b}\right]\frac{\Gamma\left(\frac{b}{2}+\frac{1}{2}\right)\Gamma\left(-a-\frac{b}{2}-\frac{1}{2}\right)}{\Gamma\left(-a\right)}=\int_{-\infty}^{+\infty}dx\text{
}x^{b}\left(1+x^{2}\right)^{a}$ (71)

with $a=2t-1,b=-2t$ can be calculated to be

$\frac{1}{2}\left[1+\left(1\right)^{-t}\right]\frac{\Gamma\left(-t+\frac{1}{2}\right)\Gamma\left(-2t+1+t-\frac{1}{2}\right)}{\Gamma\left(-2t+1\right)}=\frac{\Gamma^{2}\left(-t+\frac{1}{2}\right)}{\Gamma\left(-2t+1\right)}.$
(72)

By using the duplication formula for the gamma function

$\Gamma\left(-t+\frac{1}{2}\right)=\frac{2^{1+2t}\sqrt{\pi}\Gamma\left(-2t\right)}{\Gamma\left(-t\right)},$
(73)

the result in Eq.(72) can be further reduced to

$\frac{\Gamma^{2}\left(-t+\frac{1}{2}\right)}{\Gamma\left(-2t+1\right)}=\frac{2^{2+4t}\pi\Gamma\left(-2t\right)}{-2t\Gamma^{2}\left(-t\right)}=\frac{4\cdot
16^{t}t\pi\Gamma\left(-2t\right)}{-2\left(-t\Gamma\left(-t\right)\right)^{2}}=-2t\cdot
16^{t}\frac{\pi\Gamma\left(-2t\right)}{\Gamma^{2}\left(-t+1\right)}.$ (74)

The factor $\frac{\Gamma\left(-2t\right)}{\Gamma^{2}\left(-t+1\right)}$ can
also be found in Klebanov ; D3  for the massless string/D-brane decay process.

##  VI Conclusion

In conclusion, in this paper we have shown that each amplitude of processes
(A) and (B) for arbitrary massive string/D-brane states can be expressed in
terms of a single Lauricella function with nonpositive integer $\beta_{j}$ and
thus is a LSSA. To obtain the final results, we have used the solvability of
the LSSA with nonpositive integer $\beta_{j}$ to simplify the calculation.

In addition to the scattering processes calculated in (A), all decay
amplitudes of string/D-brane states at arbitrary mass levels calculated in (B)
also form a part of an infinite dimensional representation of the exact
$SL(K+3,\mathbb{C})$ symmetry of the bosonic string theory. The results in
this paper extends the previous exact $SL(K+3,\mathbb{C})$ symmetry of tree-
level open bosonic string theory to include the D-brane.

###### Acknowledgements.

This work is supported in part by the Ministry of Science and Technology
(MoST) and S.T. Yau center of National Yang Ming Chiao Tung University (NYCU),
Taiwan.

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