\section{Introduction}
\label{intro}
In nuclear physics,
the mean-field theory has played a central role
in understanding the nuclear ground state properties
as well as the dynamical nuclear excitations not only with
small amplitude but larger one.
Amomg the microscopic methods in the nuclear mean-field theory,
the time-dependent Hartree-Fock (TDHF) is characterized
by the full self-consistency between the mean-field (collective) motion and
the internal nucleonic motion~\cite{RS}.

The first practical numerical calculation of the TDHF was carried out
more than thirty years ago~\cite{BKN}.
Since that time, the TDHF has been applied to the heavy-ion reaction processes
such as fusion, deep-inelastic collision, fission and so on
with the purpose of understanding the microscopic mechanism
of the large amplitude nuclear collective motions in the collision processes~\cite{Negele,Maruhn,Umar}.
In the TDHF calculations, the three-dimensional (3D) Cartesian grid
has been used, combined mainly with the Skyrme effective interactions.
Recent TDHF calculations of the reactions of the heavier nuclei
are carried out by taking advantage of the rapid increase
of the computer power~\cite{Umar2,Iwata,Kedziora}.

The TDHF has been widely used also in the small amplitude domain
of the collective motions such as giant resonances (GR).
It is well known that the TDHF is connected with
the random phase approximation (RPA) in the small amplitude limit,
which is a useful tool to study the collective excitations of the nucleus.
Combined with the linear response theory, the excitation energies
of the collective motions are obtained by calculating the small amplitude responses
of the nucleus to the impulse type perturbation as an initial condition
of the TDHF equations~\cite{LinRes}.
A new method to calculate the RPA energies and amplitudes was proposed
and applied to several nuclei~\cite{FAM1,FAM2}.

The quasiparticle RPA (QRPA) is an extention of the RPA to include
the pairing correlation effects.
In this decade, with the purpose of studying the
excitation mechanism of the unstable neutron rich nuclei,
the QRPA calculations have been carried out based upon the various
formulations
\cite{Matsuo,HagiSaga,Bender02,Khan,Giambrone,Yamagami,Peru05,Terasaki,Bennaceur,Stoitsov,Peru07,Yoshida_C77,Yoshida_C78,Peru08,Tovianen,Losa}.

Just as in the case of the RPA as a small amplitude limit of the TDHF,
the QRPA is the small amplitude limit
of the time-dependent Hartree-Fock Bogoliubov (TDHFB).
The TDHFB is an extension of the TDHF,
in which the pairing correlation in the nucleus is treated
selfconsistently with the particle-hole (ph) correlations
which are taken into account in the TDHF.

A possibility of the practical numerical calculations
of the TDHFB with an effective nuclear force (Gogny force)
was presented in Ref. \cite{YH-KN}.
The type of full TDHFB calculation was carried out in the
study of the pairing vibrations with Skyrme interaction~\cite{Avez}.
Recently, a framework of the TDHFB calculation
in the canonical form with the Skyrme interaction (Cb-TDHFB)
was proposed and applied to the linear responses of
both light and heavy nuclei~\cite{Ebata}.
The cpu time is substantially reduced even in the calculations of
the heavier nuclei by using the Cb-TDHFB, which is a useful method
in studying the systematic property of the
nulei in the wide range of the nuclear chart.

In the HFB calculations with the Skyrme interactions (Skyrme-HFB),
the Skyrme interaction is used for the ph channel,
while another type of interaction is introduced
for the particle-particle (pp) channel.
Since the zero-range interaction is assumed in the Skyrme HFB,
it is necessary to set the appropriate cut-off energy and choose
the optimum parameter set in the pairing part \cite{Borycki}.

In contrast with the Skyrme HFB, in the HFB calculations
with the Gogny interaction (Gogny-HFB), the ph channel and
the pp channel are treated on an equal footing.
In the Gogny-(TD)HFB, a practical cut-off of the energy range of the
physical space is introduced naturally from the terms
with the Gaussian functions of the Gogny interaction.
Therefore, the Gogny interaction is suitable for the formulation
of the self-consistent TDHFB framework combined with a practical numerical
method of integrating the TDHFB equations.
We put emphasis on this advantageous point of the Gogny interaction
and developed a Gogny-TDHFB code.

In this article, we would like to report the first results of the Gogny-TDHFB
calculations, illustrating the feasibility of the formulation and numerical codes.
In sect. 2, the basic equations of the TDHFB are simply given.
A method of solving the TDHFB equations is described.
In sect. 3, taking oxygen isotope ${}^{20}$O as an example,
the feasibility of the numerical code of integrating the TDHFB equations
is illustrated. The conservations of total energy, nucleon numbers, and
center-of-mass are discussed.
In sect. 4, the Gogny-TDHFB is applied to
the linear (small-amplitude) vibrations of spherical nuclei ${}^{18,20,22,24}$O, ${}^{26}$Ne,
and ${}^{44,50,52,54}$Ti as well as deformed nuclei ${}^{24,34}$Mg as typical examples.
The strength functions of the isoscalar quadrupole and isovector dipole excitations
are calculated from the linear responses.
Section 5 is for summary and concluding remarks.
\section{TDHFB equation} \label{sec:1}
The nuclear Hamiltonian is assumed to take the form,
\begin{eqnarray}
  H = \sum_{\alpha \beta} T_{\alpha \beta} C_{\alpha}^{\dagger} C_{\beta}
     + \frac{1}{4} \sum_{\alpha \beta \gamma \delta}
         {\cal V}_{\alpha \beta \gamma \delta} C_{\alpha}^{\dagger} C_{\beta}^{\dagger}
                     C_{\delta} C_{\gamma}, \label{origH}
\end{eqnarray}
where $T_{\alpha \beta}$ is kinetic energy matrix element and
${\cal V}_{\alpha \beta \gamma \delta}$ is antisymmetrized two-body matrix element.
The operator $C_{\alpha}^{\dagger} (C_{\alpha})$ is a nucleon creation (annihilation)
operator of a state labelled with $\alpha$.

The TDHFB equation for the generalized density matrix ${\cal R}$
is written in the form~\cite{RS},
\begin{eqnarray}
   i \hbar \dot{{\cal R}} = \left[ {\cal H}, {\cal R} \right], \label{eq_motion}
\end{eqnarray}
with the HFB Hamiltonian ${\cal H}$,
\begin{eqnarray}
  {\cal H} = \left(
             \begin{array}{cc}
               h & \Delta \cr
              - \Delta^{*} & - h^{*}
             \end{array}
             \right)\, ,  \label{hfbH}
\end{eqnarray}
and the generalized density matrix ${\cal R}$,
\begin{eqnarray}
  {\cal R} = \left(
             \begin{array}{cc}
               \rho & \kappa \cr
              - \kappa^{*} & 1 - \rho^{*}
             \end{array}
             \right)\, , \label{GdenstMat}
\end{eqnarray}
where $\rho$ and $\kappa$ are normal density matrix and pairing tensor, respectively,
\begin{eqnarray}
\rho_{\alpha \beta} = \left( V^{*} V^{T} \right)_{\alpha \beta}, \, \quad
\kappa_{\alpha \beta} = \left( V^{*} U^{T} \right)_{\alpha \beta}. \label{rhoVV_kapVU}
\end{eqnarray}
The matrices $U$ and $V$ are the Bogoliubov transformation matrices,
\begin{eqnarray}
   \beta_{k}^{\dagger} &=&
    \sum_{\alpha} \left( U_{\alpha k} C_{\alpha}^{\dagger}
                       + V_{\alpha k} C_{\alpha} \right), \label{Bog-1} \\
   \beta_{k}           &=&
    \sum_{\alpha} \left( U_{\alpha k}^{*} C_{\alpha}
                       + V_{\alpha k}^{*} C_{\alpha}^{\dagger}   \right),  \label{Bog-2}
\end{eqnarray}
from the particle operators $C_{\alpha}^{\dagger}$  and $C_{\alpha}$
into the quasi-particles $\beta_{k}^{\dagger}$ and $\beta_{k}$.
The notation $U^{T}$ ($V^{T}$) stands for the transposed matrix of $U$ ($V$).
The mean field Hamiltonian $h$ and the pairing mean field $\Delta$ are
introduced through the relations,
\begin{eqnarray}
  h_{\alpha \beta} &=& T_{\alpha \beta} + \Gamma_{\alpha \beta}, \\
 \Gamma_{\alpha \beta} &=& \sum_{\gamma \delta}
      {\cal V}_{\alpha \gamma \beta \delta} \rho_{\delta \gamma}, \quad
 \Delta_{\alpha \beta} = \frac{1}{2}\sum {\cal V}_{\alpha \beta \gamma \delta} \kappa_{\gamma \delta},
     \label{GamDel}
\end{eqnarray}
with the normal density $\rho$ and the pairing tensor $\kappa$.

Instead of the equation (\ref{eq_motion}) of the generalized density matrix ${\cal R}$,
we write the equation of motion in the form~\cite{YH-KN,Bul}
\begin{eqnarray}
  i \hbar \frac{\partial}{\partial t}
    \left(
    \begin{array}{c}
                U(t) \cr
                V(t)
             \end{array}
             \right)
     = {\cal H} \left(
    \begin{array}{c}
                U(t) \cr
                V(t)
             \end{array}
             \right), \label{hfbeq_UV}
\end{eqnarray}
for the matrices $U$ and $V$ in the Bogoliubov transformation (\ref{Bog-1}) and (\ref{Bog-2}).

Noting that the form of the equation (\ref{hfbeq_UV}) is analogous to
the TDHF equation~\cite{FCW},
we follow the numerical method of integrating the TDHF equation
and get the formal solution of the TDHFB equation (\ref{hfbeq_UV}),
\begin{eqnarray}
        \left(
        \begin{array}{c}
                U \cr
                V
             \end{array}
             \right)^{(n+1)} = \exp \left( -i \frac{\Delta t}{\hbar} {\cal H}^{(n+1/2)} \right)
    \left(
    \begin{array}{c}
                U \cr
                V
             \end{array}
             \right)^{(n)}, \label{tdhfb-solution}
\end{eqnarray}
where
$\left( \begin{array}{c} U \cr
                          V
        \end{array} \right)^{(n)}$
is a matrix composed of matrices $U$ and $V$ at a discretized time $t_{n}$ ($n$ = 0, 1, 2, $\cdots$).
The quantity ${\cal H}^{(n+1/2)}$ is an adequate TDHFB Hamiltonian matrix
at every time step from $t_{n}$ to $t_{n+1} = t_{n} + \Delta t$.
The Hamiltonian matrix ${\cal H}^{(n+1/2)}$ is determined so that the expectation value $E$
of the Hamiltonian (\ref{origH}) with respect to the time-dependent HFB state $| \Phi \rangle$,
\begin{eqnarray}
  E &=& \langle \Phi | H | \Phi \rangle \nonumber \\
    &=& \sum_{\alpha \beta} T_{\alpha \beta} \rho_{\beta \alpha}
   + \frac{1}{2} \Gamma_{\alpha \beta} \rho_{\beta \alpha}
   + \frac{1}{2} \kappa_{\alpha \beta}^{*} \Delta_{\alpha \beta}, \label{Ehfb}
\end{eqnarray}
is kept constant in the course of the numerical integrations in (\ref{tdhfb-solution}).

Since in the TDHFB calculation the energy conservation is one
of the most important conditions to be fulfilled, let us see how the
energy in (\ref{Ehfb}) is conserved with respect to a small variation
in the matrices $U$ and $V$.

Let us assume the time step $\Delta t$ is small enough
to approximate the right hand side of Eq. (\ref{tdhfb-solution}) within
the first order of $\Delta t$ in the power series expansion of the
exponential function.
Putting a parameter $\lambda \equiv \frac{\Delta t}{\hbar}$,
the variations of the matrices $U$ and $V$ at a time $t$
into the new matrices $U'$ and $V'$ are written within the
first order of the parameter $\lambda$,
\begin{eqnarray}
 \left( \begin{array}{c}
         U' \\
         V'  \end{array}
 \right)
&=&  \left( \begin{array}{c}
         U \\
         V  \end{array}
 \right)
 - i \lambda \left( \begin{array}{cc}
                    h & \Delta \\
                  - \Delta^{*} & - h^{*}
                   \end{array}
             \right)
      \left( \begin{array}{c}
         U \\
         V  \end{array}
      \right) \nonumber \\
 &=& \left( \begin{array}{c}
        U - i \lambda \left( h U + \Delta V \right) \\
        V + i \lambda \left( h^{*} V + \Delta^{*} U \right)
            \end{array} \right).     \label{UVvariation}
\end{eqnarray}
Here, for the ease of the discussion,
let us assume also that the time increment $\Delta t$ is small enough
so that we can identify the mean field Hamiltonian $h^{(n+1/2)}$
and mean pairing potential $\Delta^{(n+1/2)}$ in the TDHFB
Hamiltonian ${\cal H}^{(n+1/2)}$ with $h$ and $\Delta$ at the time $t$, respectively.

Using the relations in Eq. (\ref{UVvariation}), the variations in the
density matrix $\rho$ and pairing tensor $\kappa$ in Eq. (\ref{rhoVV_kapVU})
are expressed, respectively, as
\begin{eqnarray}
  & & \rho' = V'^{*} V'^{T} \nonumber \\
        &=& \left( V^{*} -i \lambda \left( \Delta U^{*} + h V^{*} \right)  \right)
            \left( V^{T} +i \lambda \left( U^{T} \Delta^{\dagger} + V^{T} h^{\dagger} \right)  \right) \nonumber \\
        &=& \rho -i \lambda [ h, \rho ] - i \lambda \left( - \Delta \kappa^{*} + \kappa \Delta^{*}  \right),
                                                                                \label{varrho}  \\
  & & \kappa' = V'^{*} U'^{T} \nonumber \\
          &=& \left( V^{*} -i \lambda \left( \Delta U^{*} + h V^{*} \right)  \right)
            \left( U^{T} -i \lambda \left( U^{T} h^{T} + V^{T} \Delta^{T} \right)  \right) \nonumber \\
        &=& \kappa -i \lambda \left( \Delta U^{*} U^{T} + h \kappa + \kappa h^{*} - \rho \Delta  \right). \label{varkappa}
\end{eqnarray}

We see that the conservation of the particle number in every time step holds
from Eq. (\ref{varrho}): Taking the trace of both sides in Eq. (\ref{varrho}), we have
\begin{eqnarray}
   & & {\rm Tr} \rho' - {\rm Tr}\rho \nonumber \\
     &=& - i \lambda \left( {\rm Tr}\left( [h, \rho] \right)
                            + {\rm Tr} \left( - \Delta \kappa^{*}
                            + \kappa \Delta^{*} \right)  \right)  \nonumber \\
      &=& - i \frac{\lambda}{2} \sum_{\alpha \beta} \sum_{\mu \nu}
             \left( - {\cal V}_{\alpha \beta \mu \nu} \kappa_{\mu \nu} \kappa^{*}_{\beta \alpha}
                    + \kappa_{\alpha \beta} \left( {\cal V}_{\beta \alpha \mu \nu} \kappa_{\mu \nu} \right)^{*} \right) \nonumber \\
      & =& - i \frac{\lambda}{2} \sum_{\alpha \beta} \sum_{\mu \nu}
             \left(  {\cal V}_{\alpha \beta \mu \nu} \kappa_{\mu \nu} \kappa^{*}_{\alpha \beta}
            - \kappa_{\alpha \beta}  {\cal V}_{\mu \nu \alpha \beta} \kappa_{\mu \nu}^{*}  \right) \nonumber \\
     &=& 0,  \label{num-conserv}
\end{eqnarray}
where the definition of $\Delta$ in Eq. (\ref{GamDel}) and the anti-symmetric property of the matrix elements
${\cal V}_{\alpha \beta \mu \nu}$ and pairing tensor $\kappa_{\alpha \beta}$ are used.
The notation Tr stands for taking the trace of the matrices.
Then, the particle number $N_{\tau} = {\rm Tr} \rho$ is conserved from step to step.
Here, $N_{\tau}$ is the proton (neutron) number Z (N) for proton (neutron) density
matrix $\rho = \rho_{p}$ $(\rho_{n}$), respectively.
\par
Putting the expressions of the density and pairing tensor up to the
first order in the parameter $\lambda$ in Eqs. (\ref{varrho}) and (\ref{varkappa})
into the expression of the energy in Eq. (\ref{Ehfb}),
we have the variation of the energy as follows:
\begin{eqnarray}
  & & \delta E = {\rm Tr} \bigl\{ -i \lambda h [h, \rho]
          - i \lambda h \left( - \Delta \kappa^{*} + \kappa \Delta^{*} \right) \bigr\} \nonumber \\
        & & - \frac{i \lambda}{2}
           {\rm Tr} \bigl\{ - \Delta^{*} \Delta - \Delta^{*} h \kappa - \Delta^{*} \kappa h^{*}
                            + \Delta^{*} \Delta \rho^{*} + \Delta^{*} \rho \Delta \bigr\} \nonumber \\
        & & - \frac{i \lambda}{2}
           {\rm Tr} \bigl\{   \quad \Delta^{*} \Delta + h^{*} \kappa^{*} \Delta + \kappa^{*} h \Delta
                            - \Delta^{*} \rho \Delta - \rho^{*} \Delta^{*} \Delta \bigr\} \nonumber \\
        & & = 0, \label{e-conserv}
\end{eqnarray}
where the relations $h^{*} = h^{T}$, $\kappa^{T} = - \kappa$,
and $\Delta^{T} = - \Delta$ are used.
From Eq. (\ref{e-conserv}), we see that
we can integrate the TDHFB equation (\ref{hfbeq_UV}), conserving the energy
expressed as in Eq. (\ref{Ehfb}), by appropriately setting the time increment
$\Delta t$ and intermediate TDHFB Hamilonian ${\cal H}^{(n+1/2)}$.
In the case of the Gogny interaction \cite{Gogny,Gogny2} with the parameter set D1S,
the contribution of the density-dependent term is
included only in the mean-field Hamiltonian $h$
in the HFB Hamiltonian (\ref{hfbH}), through the variation
of the nucleon density $\rho({\bf x})$ \cite{Girod}.
Then, the energy conservation relation (\ref{e-conserv})
holds when the Gogny interaction with the parameter set D1S
is adopted in the Hamiltonian (\ref{origH}).

\section{Numerical solution and conserved quantities} \label{sec:3}
\subsection{Parameters} \label{sec:3-1}
With the purpose of demonstrating the feasibility of our TDHFB code,
we apply them to the small amplitude (linear) responses of the
oxygen isotope ${}^{20}$O.
We adopt the Gogny interaction \cite{Gogny,Gogny2} for the two-particle interaction part
in the Hamiltonian (\ref{origH}).
The parameter set D1S is used.
The Coulomb part is omitted in all of the calculations in this article.

The initial condition in the present calculation is of the impulse type:
The static HFB solutions $U_0$ and $V_0$ are changed into the initial matrices
$U^{(0)}$ and $V^{(0)}$ by the relations
\begin{eqnarray}
  V^{(0)} &=& \exp \left( i \varepsilon {\bf Q} \right) V_0
           = \sum_{\nu = 1}^{N_{max}}
           \frac{i^\nu \varepsilon^\nu {\bf Q}^\nu}{\nu!} V_0,  \label{ini-1} \\
  U^{(0)} &=& \exp \left( -i \varepsilon {\bf Q}^{*} \right) U_0
           = \sum_{\nu = 1}^{N_{max}}
           \frac{i^\nu (-\varepsilon)^\nu {{\bf Q}^{*}}^\nu}{\nu!} U_0,  \nonumber \\
          & &                                                             \label{ini-2}
\end{eqnarray}
where ${\bf Q}$ stands for the matrix representation
of a multipole operator with respect to the numerical basis states.
$N_{max}$ is the maximum number up to which the exponential function is
expanded into power series.
The parameter $\epsilon$ is chosen so that the linearity is held in the
course of the time integration of the equations (\ref{hfbeq_UV}).

\begin{figure}[htb]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig1.eps}
}
\end{center}
\caption{Time dependence of expectation value of isoscalar quadrupole moment
of ${}^{20}$O.
}
\label{quadosci}
\end{figure}

\begin{figure}[htb]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig2.eps}
}
\end{center}
\caption{Deviation of the excitation energy from the initial (t = 0) value
of the isoscalar quadrupole vibration in fig. \ref{quadosci} with respect to time. }
\label{oscienergy}
\end{figure}

\begin{figure}[htb]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig3.eps}
}
\end{center}
\caption{Deviation of neutron (proton) number expectation value
from the accurate number 12 (8), respectively, in isoscalar quadrupole vibration
in fig. \ref{quadosci}.
}
\label{numb_consv}
\end{figure}

In making the TDHFB hamiltonian ${\cal H}$, we used the same form of energy functional
as that used in the calculation of the HFB ground state, i.e.,
the center-of-mass correction part $E_{CM} = - {\bf P}^2/2Am$
with the nucleon mass $m$ and
the total momentum operator ${\bf P} = \sum_i^{A} {\bf p}_i$ of $A$ nucleons is included.

As the numerical basis, the three-dimensional harmonic oscillator eigenstates were used.
The two-particle matrix elements ${\cal V}_{\alpha \beta \gamma \delta}$
in the Hamiltonian (\ref{origH}) were calculated by following the method
described by Girod and Grammaticos \cite{Girod}.

We used the space of basis states with the relation $0 \leq n_x + n_y + n_z \leq 4$,
where $n_x$ ($n_y$, $n_z$) is the number of quanta
of the harmonic oscillator basis states in the x (y, z) direction, respectively.
The harmonic oscillator parameters were set as follows:
$\hbar \omega_x = \hbar \omega_y = \hbar \omega_z$ = 13.7 MeV.

In the initial conditions (\ref{ini-1}) and (\ref{ini-2}),
we took the multipole operator ${\bf Q}$ to be
an isoscalar quadrupole operator $Q_{\alpha \beta}
= \left( 2 z^2 - x^2 - y^2 \right)_{\alpha \beta}$.
The parameter $\varepsilon$ was put to be $1.0 \times 10^{-3}$,
being small enough so that the linearity of the oscillation
with respect to $\varepsilon$ is satisfied.
The power series expansions of the exponential functions were taken up to
the $N_{max}$-th order with $N_{max}$ put to be ten.

In the course of the time integrations in (\ref{tdhfb-solution}),
a predictor-corrector method was made use of
in making the Hamiltonian matrix ${\cal H}^{(n+1/2)}$ at every time step~\cite{FCW}.

\subsection{Conservation of energy, particle number, and center-of-mass}
\label{sec:3-2}
In fig. \ref{quadosci}, we display the time variation of the
isoscalar quadrupole moment
$Q_{20}(t) = \langle  \sum_{i=1}^{A} r_i^2 Y_{20}(\hat{r}_i)  \rangle$ of ${}^{20}$O.
The time increment $c \Delta t$ is 0.2 fm.
After the initial impulse, regular small-amplitude oscillations take place.

In fig. \ref{oscienergy}, the deviation of the excitation energy
from the initial (t = 0) value in the course of the vibration is shown.
The excitation energy varies from the initial value 0.02554 MeV to
0.02561 MeV after integration time 4500 fm (22500 steps).
The deviation of the excitation energy from the initial value
in the course of the time integration is reduced
when we carry out more numbers of predictor-corrector loops
in each time step.

In fig. \ref{numb_consv}, the deviation of the expectation values
of the nucleon number $N_{\tau}$
from the accurate values (8 protons and 12 neutrons)
is displayed with respect to time.
In ${}^{20}$O, the protons are in the normal state, whereas the
pairing correlation is active on the neutron side.
Therefore, the integration of the equations of motion
of the proton orbitals is equivalent to the
TDHF case. Then, the total number of protons are conserved within
$10^{-11}$.
The neutron number in fig. \ref{numb_consv} is kept
within around $10^{-5}$ in the present integration process.
These results illustrate that the unitarity of the time-displacement operator
in Eq. (\ref{tdhfb-solution}) holds within this accuracy.

\begin{figure}[htb]
\begin{center}
\resizebox{0.45\textwidth}{!}
  \includegraphics{fig4.eps}
}
\end{center}
\caption{Time dependence of expectation values of isovector
dipole moment D(t) (upper panel) and center-of-mass R(t) (lower panel).
}
\label{IVD-oscillation}
\end{figure}

Taking the initial impulse operator $Q_{\alpha \beta}$ to be
the isovector dipole operator
$D_{\alpha \beta} = \frac{N}{A} \left( z \right)_{\alpha \beta}$
 for protons and $ - \frac{Z}{A} \left( z \right)_{\alpha \beta} $ for
neutrons, we have the oscillation pattern in fig. \ref{IVD-oscillation},
where the expectation value
$D(t) = \langle \frac{N}{A} \sum_{i=1}^{Z} z_i -
\frac{Z}{A} \sum_{i=1}^{N} z_i \rangle $ is displayed together with
the center-of-mass expectation value in z direction
$R(t) = \frac{1}{A}\langle \sum_{i = 1}^{Z} z_i + \sum_{i = 1}^{N} z_i \rangle$
in the first 12000 steps of integration.
Here, $N (Z)$ stands for the neutron (proton) number, respectively.
We see that the fluctuation of the center-of-mass motion $R(t)$ is
kept within $2 \times 10^{-5}$ fm.
The effect of the center-of-mass motion on the isovector dipole moment $D(t)$
could be neglected.

When the quadrupole moment operator ($2 z^2 - x^2 - y^2$ or $x^2 - y^2$) is used
as the impulse operator, the expectation values of the center-of-mass and off-diagonal terms
$xy$, $yz$, and $zx$ are kept to be zero,
since the symmetry of the system at the initial time $t = 0$
is kept throughout the integration time.
\section{Applications} \label{sec:4}
\subsection{Spherical nuclei} \label{sec:4-1}
\subsubsection{${}^{18,20,22,24}$O } \label{sec:4-1-1}
Using the impulse type initial conditions (Eqs. (\ref{ini-1}) and (\ref{ini-2}))
with an isoscalar quadrupole operator \\
$Q_{\alpha \beta} = \left( 2 z^2 - x^2 - y^2 \right)_{\alpha \beta}$
and the space of basis functions $0 \leq n_x + n_y + n_z \leq 4$, we applied the method
of integrating the TDHFB equations discussed in the previous section
to the oxygen isotopes ${}^{18,20,22,24}$O.
Taking the Fourier transformation of the quadrupole moment $Q_{20}(t)$,
we calculated the strength functions of the isoscalar quadrupole mode of the oxygen isotopes
in fig. \ref{oxygen_strength}~\cite{LinRes}.

In fig. \ref{oxygen_strength}, we see that the low-energy peaks
are located below 5 MeV.
The energies of the peaks below 5 MeV are shown in fig. \ref{ox-18-24_energies}.
The neutron pairing energies of the oxygen isotopes are shown in table 1.
In the four oxygen isotopes ${}^{18,20,22,24}$O, the magnitude of the neutron pairing energy
is the largest in ${}^{20}$O and zero in ${}^{24}$O.
The neutron number 16 in ${}^{24}$O corresponds to the subshell closure of 2s$\frac{1}{2}$.
The tendency of the energies of the peaks in the TDHFB calculation that the lowest (highest) is
of ${}^{20}$O (${}^{24}$O) is in accordance with the experimental data~\cite{AtomData,Frank,Hoffman}.
In the present TDHFB calculation, the energy of the lowest peak of ${}^{24}$O is 4.1 MeV,
which is similar to the QRPA predictions around 4 MeV~\cite{Khan,Obertelli}.
The experimental results of the energies of the lowest 2${}^{+}$ state in ${}^{24}$O are
around 4.7 MeV \cite{Frank,Hoffman},
which is higher than the QRPA and present TDHFB results by around 0.7 MeV.

Because the Coulomb force is omitted in the present calculations,
we compare our result of the first 2${}^{+}$ energy of the oxygen ${}^{18}$O
with the experimental value of neon ${}^{18}$Ne.
The experimental values of the first 2${}^{+}$ energies of the oxygen ${}^{18}$O and neon ${}^{18}$Ne
are 1.98 MeV and 1.89 MeV~\cite{Tilley}, respectively.
These two nuclei are mirror nuclei to each other and their first 2${}^{+}$ energies
are located within the width of 0.1 MeV.
The present TDHFB result of the oxygen ${}^{18}$O is around 0.5 MeV
higher than these experimental results.

\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig5.eps}
}
\end{center}
\caption{Strength functions of isoscalar quadrupole mode
of oxygen isotopes ${}^{18,20,22,24}$O. Artificial width of 1 MeV is used. }
\label{oxygen_strength}
\end{figure}

Through the Fourier transformation of the isovector dipole moment $D(t)$
of ${}^{20}$O in fig. \ref{IVD-oscillation}, we get the strength function
of the isovector dipole mode in fig. \ref{IVD-strength}.
The peak is located at 25 MeV, which is larger than the observed value
by a few MeV \cite{oxygen-isotopes}.
Since the absolute values of the center-of-mass is kept practically small
in the time integration, the effect of the spurious motion in the strength function
might be neglected.

\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig6.eps}
}
\end{center}
\caption{ Energies of the first peaks of the isoscalar quadrupole strength functions
of oxygen isotopes ${}^{18,20,22,24}$O in fig. \ref{oxygen_strength}.
Closed (open) squares stand for the TDHFB (experimental) results, respectively.
Plus (+) is the first 2$^{+}$ energy of neon ${}^{18}$Ne~\cite{Tilley}.  }
\label{ox-18-24_energies}
\end{figure}

\begin{table}[h]
\caption{
Neutron pairing energy E$_{\rm n}$ in oxygen isotopes ${}^{16,18,20,22,24}$O
in Gogny-HFB.
}
\label{table-1}
\begin{center}
\begin{tabular}{c|ccccc}
                      & ${}^{16}$O    & ${}^{18}$O   & ${}^{20}$O     &   ${}^{22}$O  &  ${}^{24}$O     \\
\hline
E$_{\rm n}$ (MeV)     &     0         &    - 4.56    & - 5.33         &   - 2.39      &      0
\end{tabular}
\end{center}
\end{table}

\begin{figure}[htb]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig7.eps}
}
\end{center}
\caption{ Strength function of isovector dipole mode of $^{20}$O.
Artificial width is 0.5MeV. }
\label{IVD-strength}
\end{figure}

\subsubsection{Pairing energy in ${}^{20}$O} \label{sec:4-1-2}
In fig. \ref{ox-20_epgss_strength}, we show the time dependence of the pairing energy
$E_{p}^{(\rm Gauss)} = \frac{1}{4} \sum_{\alpha \beta \mu \nu} \kappa_{\alpha \beta}^{*}
{\cal V}_{\alpha \beta \mu \nu}^{(\rm Gauss)} \kappa_{\mu \nu}$ \cite{RS}
of the Gaussian part ${\cal V}_{\alpha \beta \mu \nu}^{(\rm Gauss)}$ in the Gogny interaction
in the quadrupole oscillations in fig. \ref{quadosci} of ${}^{20}$O.
Here, $\kappa_{\mu \nu}$ is the pairing tensor of the neutrons.
The magnitude of the pairing energy which comes from the other parts
in the Gogny interaction, i.e.,
spin-orbit, center-of-mass, and Coulomb (if included in the
calculations) terms, is small compared with that from the Gaussian part.
Since the initial impulse is taken to be small enough to ensure the linearity of the
quadrupole oscillation, the fluctuation of the pairing energy
in fig. \ref{ox-20_epgss_strength} is within 0.05 MeV in the integration time up to
4500 fm.

We take the Fourier transformation of the fluctuation of the pairing energy
and show the Fourier (sine) amplitudes in the lower panel
in fig. \ref{ox-20_epgss_strength}.
Corresponding to the giant resonance peak in the strength function in the
energy range from 20 MeV to 25 MeV, we see remarkable fluctuations
in the Fourier amplitudes. In the low energy region under 7 MeV, we find
fluctuations in the Fourier amplitudes corrresponding to the low-energy peak
at 2.3 MeV and a shoulder around 4.4 MeV in the strength function.
On the other hand, there are fluctuations in the Fourier amplitudes to which
there are no corresponding peaks in the strength function curve.

In the present formulation of the TDHFB,
we can not separate out the effects of the pairing correlation,
since they are embedded in the dynamical changes of the matrices
U and V which are defined on the fixed basis of 3D harmonic oscillator
eigenfunctions. We need some theoretical tools to separate out the
effects of the pairing correlation.
We expect that the idea of the Cb-TDHFB will be a good starting point
to improve the present formulation and attack the problem
of making clear the pairing effects in the dynamical processes
of nuclear collective vibrations not only of the quadrupole type
but of the other multipolarity types such as monopole and octupole.

\begin{figure}[h]
\begin{center}
\resizebox{0.45\textwidth}{!}
  \includegraphics{fig8.eps}
}
\end{center}
\caption{(Upper panel) Time dependence of the pairing energy $E_{p}^{\rm (Gauss)}$
of the Gaussian part in the Gogny interaction in the quadrupole oscillation
in fig. \ref{quadosci} of ${}^{20}$O.
(Lower panel) Fourier (sine) amplitude of the fluctuations
of the pairing energy $E_{p}^{\rm (Gauss)}$ in the upper panel (solid curve, left axis).
Artificial width of 0.25 MeV is used.
The isoscalar quadrupole strength function is plotted in dashed curve (right axis).}
\label{ox-20_epgss_strength}
\end{figure}

\subsubsection{${}^{44,50,52,54}$Ti} \label{sec:4-1-3}
The next example of the TDHFB calculations in the spherical nuclei
is the application of the method to the isoscalar quadrupole vibrations
in the titanium isotopes ${}^{44,50,52,54}$Ti.
Here and hereafter, the space of the basis functions $0 \leq n_x + n_y + n_z \leq 5$
was used in the calculations.

The strength functions of the titanium isotopes are shown in fig. \ref{ti-44-54}.
From the strength functions, we got the energies of the first low-energy peaks,
which are shown in fig. \ref{ti-energies}.
In fig. \ref{ti-energies}, we see that the TDHFB results of the lowest-energy 2${}^{+}$ peaks are
along the experimental values~\cite{Simpson,Janssens}:
The lowest 2${}^{+}$ energies in ${}^{50}$Ti and ${}^{54}$Ti
are higher than the other two nuclei ${}^{44}$Ti and ${}^{52}$Ti.

This tendency of the lowest 2${}^{+}$ energies of the four nuclei
is in accordance with the tendency of the neutron pairing energies
and sushell closures. In table 2, we show the pairing energies of protons and neutrons
in the four nuclei ${}^{44,50,52,54}$Ti in their HFB ground states.
The proton pairing energies in ${}^{44,50,52,54}$Ti are almost constant around - 4.8 MeV.
In ${}^{50}$Ti, the neutron number is 28 and 1f$\frac{7}{2}$ orbitals are filled.
The pairing energy in ${}^{50}$Ti is zero.
Likely, in  ${}^{54}$Ti, 2p$\frac{3}{2}$ orbitals are filled
and the pairing energy is comparatively small.

\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig9.eps}
}
\end{center}
\caption{Strength functions of isoscalar quadrupole mode of titanium isotopes ${}^{44,50,52,54}$Ti.
Artificial width of 1 MeV is used. }
\label{ti-44-54}
\end{figure}

\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig10.eps}
}
\end{center}
\caption{Energies of the first peaks of the strength functions
of the isoscalar quadrupole mode of the titanium isotopes ${}^{44,50,52,54}$Ti
in fig. \ref{ti-44-54}.
Closed (open) squares stand for the TDHFB (experimental)
results, respectively~\cite{Simpson,Janssens}. }
\label{ti-energies}
\end{figure}

\begin{table}[h]
\caption{
Proton (neutron) pairing energy E$_{\rm p}$ (E$_{\rm n}$) in titanium isotopes ${}^{44,50,52,54}$Ti
in Gogny-HFB.
}
\label{table-2}
\begin{center}
\begin{tabular}{c|cccc}
                                 & ${}^{44}$Ti    & ${}^{50}$Ti   & ${}^{52}$Ti     &   ${}^{54}$Ti     \\
\hline
E$_{\rm p}$ (MeV)                 & - 4.92         &  - 4.72        &  - 4.79         & - 4.80       \\
E$_{\rm n}$ (MeV)                 & - 4.92         &    0            & - 2.75          & - 1.17
\end{tabular}
\end{center}
\end{table}

\subsubsection{${}^{26}$Ne} \label{sec:4-1-4}
The mean-field of the ground state of ${}^{26}$Ne is spherical in the Gogny-HFB
calclation. The protons are in the paired state
whereas the neutron pairing energy vanishes.
${}^{26}$Ne is one of the neutron-rich nuclei, and a pygmy state is known
as the low-energy components in the isovector dipole resonance~\cite{Gibelin}.

Taking the isovector dipole operator as the impulse operator, we calculated the
expectation values of the isovector dipole moment in ${}^{26}$Ne.
We show the strength function in fig. \ref{ne26-strength}.

The energy of the small 1${}^{-}$ peak near 10 MeV is 11.9 MeV
in the present TDHFB calculation.
In the QRPA calculation by Peru at al.~\cite{Peru07},
they got the value 10.64 MeV for the 1${}^{-}$ peak of the pygmy state.
The experimental value of the pygmy state is around 9 MeV~\cite{Gibelin}.
The corresponding energy by the present TDHFB calculation is larger than
the QRPA (experimental) result by 1.3 (3) MeV, respectively.

\begin{figure}[htb]
\begin{center}
\resizebox{0.38\textwidth}{!}
  \includegraphics{fig11.eps}
}
\end{center}
\caption{Strength function of isovector dipole mode of ${}^{26}$Ne.}
\label{ne26-strength}
\end{figure}

\subsection{Deformed nuclei} \label{sec:4-2}
As the typical examples of the deformed nuclei, we took magnesium isotopes  ${}^{24}$Mg
and ${}^{34}$Mg.
In ${}^{24}$Mg, the pairing energies of both protons and neutrons are zero
in the HFB ground state, in which the mean-field is in a prolate form.
${}^{34}$Mg is also prolately deformed in the HFB ground state,
in which the neutron pairing energy is finite.
The deformation parameter $\beta$ of ${}^{24}$Mg (${}^{34}$Mg) is
$\beta$ = 0.51 (0.43), respectively~\cite{HilGirod}.

Calculating the time dependence of the
expectation values of the $K$ = 0 and 2 components of the isoscalar quadrupole moment,
we get the strength functions in fig. \ref{mg-24-34}.
Here we used the isoscalar quadrupole operator $x^2 - y^2$ as the initial impulse operator
to excite the $K =$ 2 component of the isoscalar quadrupole vibrations.

The giant resonance peaks with $K = $ 0 and 2 are seen in the energy region
from 15 MeV to 30 MeV in both of the nuclei ${}^{24}$Mg and ${}^{34}$Mg.
The peaks with $K =$ 2 are located in the higher energy region than those
with $K =$ 0, reflecting the prolate deformation of the mean-fields in
the ground states of the two nuclei.

The energy of the lowest $K = $ 0 (2) peak in ${}^{24}$Mg is 8.06 (3.3) MeV,
which is comparable with an experimental value 6.43 (4.23) MeV, respectively~\cite{Endt}.
The characteristic point that the $K =$ 2 peak comes lower in energy
than that with $K =$ 0 is closely related with the softness of  ${}^{24}$Mg
with respect to the direction of the gamma deformation.

In ${^{34}}$Mg, the energy of the lowest $K =$ 0 (2) peak is
2.07 (3.0) MeV. In the QRPA calculations with the Skyrme interaction,
the energy of the corresponding peak is 2.65 (around 3.0) MeV~\cite{Yoshida_C78,Losa}.
The energies of the two peaks with $K = $ 0 and 2 by the TDHFB calculations
are comparable with those by the QRPA calculations, though the former tend to
be smaller than the latter.

The strength functions of the isovector dipole modes of
${}^{24}$Mg are shown in the left panel in fig. \ref{mg-24-IVD}.
The energy of the $K = $ 0 (1) peak in fig. \ref{mg-24-IVD}
is 20.2 (28.9) MeV, respectively.
The corresponding value obtained by using the QRPA by Peru et al.
is 19.35 (27.44) MeV~\cite{Peru08}.
The present TDHFB result of the $K =$ 0 (1) peak energy is higher by 0.9 (1.5) MeV
than the QRPA result, respectively.

The strength functions of the isovector dipole modes of
${}^{24}$Mg are shown in the left panel in fig. \ref{mg-24-IVD}.
The energy of the $K = $ 0 (1) peak in fig. \ref{mg-24-IVD}
is 20.2 (28.9) MeV, respectively.
The corresponding value obtained by using the QRPA by Peru et al.
is 19.35 (27.44) MeV~\cite{Peru08}.
The present TDHFB result of the $K =$ 0 (1) peak energy is higher by 0.9 (1.5) MeV
than the QRPA result, respectively.

From the isovector dipole strength functions,
we calculated the photoabsorption cross section $\sigma(E)$
based on the strong coupling scheme~\cite{LinRes,BM}
\begin{widetext}
\begin{eqnarray}
  \sigma(E) &=& \frac{4 \pi^2 e^2 E}{c \hbar} \sum_{K = -1}^{1} \frac{dB(E, f_{1 K})}{d E}\, , \\
  \frac{dB(E, f_{1 K})}{d E} &=& - \frac{1}{\pi \hbar \varepsilon}
   {\rm Im} \int_{0}^{\infty} dt\,  \langle \Psi (t) | f_{1 K} |\Psi (t) \rangle
   e^{ i E t / \hbar - \gamma t }\, ,  \nonumber \\
 & &
\end{eqnarray}
\end{widetext}
where $\frac{dB(E, f_{1 K})}{d E}$ is a transition strength with isovector dipole operators
$f_{1 K} = \frac{N}{A} \sum_{i = 1}^{Z} \left( r Y_{1 K} \right)_i$ for protons and
$f_{1 K} = - \frac{Z}{A} \sum_{i = 1}^{N} \left( r Y_{1 K} \right)_i$ for neutrons
with $K = \pm 1, 0$.
The nuclear state $| \Psi (t) \rangle$ is the TDHFB state.
$\gamma$ is an artificial width used in the strength function and $\varepsilon$ is
the small parameter used in the initial conditions (\ref{ini-1}) and (\ref{ini-2})
of the TDHFB equations.

In the right panel in fig. \ref{mg-24-IVD} we compare the photoabsorption cross section
with the experimental results.
The $K = $ 0 peak comes near the one of the experimental cross section,
whereas the $|K| = 1$ peak is located in the higher energy region than
the distribution of the peaks in the experimental data~\cite{Varlamov}.

The difference of the TDHFB results presented above in this section
from the RPA results and experimental data might come partly
from the restricted space of the basis functions
($n_x + n_y + n_z \leq 5$) and the omission of the Coulomb force in the
present TDHFB calculations.
It is necessary to study the effects of the size of the space of basis functions
and Coulomb force on the TDHFB results.

\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig12.eps}
}
\end{center}
\caption{Strength functions of isoscalar quadrupole mode of magnesium isotopes ${}^{24,34}$Mg.
Solid (dashed) curve stands for $K$ = 0 (2) component, respectively.
Artificial width of 1 MeV is used.}
\label{mg-24-34}
\end{figure}

\begin{figure}[h]
\begin{center}
\resizebox{0.4\textwidth}{!}
  \includegraphics{fig13.eps}
}
\end{center}
\caption{Strength functions of isovector dipole mode (a)) and
photoabsorption cross sections (b)) of ${}^{24}$Mg.
Solid (dashed) curve stands for $|K|$ = 0 (1) component, respectively.
Artificial width of 1 MeV is used in the strength functions.
Experimental data of cross sections are shown with plus (+)~\cite{Varlamov}. }
\label{mg-24-IVD}
\end{figure}

\section{Summary and concluding remarks} \label{sec:5}
In this article, we presented a practical numerical method of integrating the TDHFB equations
with Gogny interaction. The three-dimensional harmonic oscillator eigen functions were used
as the numerical basis functions.
The conservation property of the energy, particle numbers, and center-of-mass of the
method was illustrated.
The method was applied to several typical spherical nuclei as well as axially deformed ones
and the strength functions were obtained in the framework of the linear response theory.
It was shown that the tendency of the energies of the low-energy peaks in the present TDHFB calculations
was similar to the one of the experimental results,
though the energies obtained in the TDHFB calculations tend to be lower than the QRPA and experimental
results. On the other hand, the energies of the typical peaks in the energy region
of the giant resonances tend to be larger than the QRPA and experimental results.
We will study carefully how the numerical results depend on the space of basis functions
and Coulomb force.

The present calculations were carried out under the conditions:
i) The maximum major shell quantum number
of the space of the harmonic oscillator basis functions
was restricted to be four or five.
ii) The Coulomb force was omitted in both HFB and TDHFB calculations.
These restrictions were introduced to save the cpu time and
check the feasibility of the present numerical codes of TDHFB.

It is necessary to use larger space of basis functions and to include
the Coulomb force so that we can deal with the problems of heavier, unstable nuclei
in terms of the present TDHFB method.
When the larger space of the basis functions is taken,
we need efficient numerical methods to save the cpu time of the iterations
in the full Gogny-TDHFB calculations with the Coulomb force.
The acceleration of the calculations of the Coulomb part
might be realized by introducing a set of small number of {\it optimal} integration points
and weights in the calculations of the Coulomb mean-field matrix elements.
The new numerical codes by which we could save the cpu time are now being developed.

The author thanks Professors K.~Yabana and T.~Nakatsukasa for discussions and comments.
He thanks Professor K.~Matsuyanagi for his warm encouragements.
The author is grateful to Professor J.~A.~Maruhn for his suggestions and comments.
The author is also grateful to Dr. S.~Ebata for discussions on the numerical methods.
Part of the numerical calculations was carried out on SR16000 at YITP
in Kyoto University.


\section{Introduction}

D-brane physics \cite{Polchinski:1995mt}\cite{Witten:1995im}\cite{Polchinski:1996na}
 has played a central role in theoretical high energy
physics for more than one and a half decades by now. In the open string
description, a D$_p$-brane with a $(p+1)$-dimensional world volume
is realized as a hypersurface in flat
spacetime with the appropriate boundary condition on the string coordinates:
Dirichlet boundary conditions on the directions transverse
 to the D$_p$-brane and Neumann boundary conditions along the worldvolume of the D$_p$-brane \cite{Polchinski:1994fq}.
The bosonic action for multiple D$_p$-branes  was given by Myers \cite{Myers:1999ps}.
Note that a supersymmetric generalization is still unknown; see however \cite{Howe:2006rv}. The effective action for a single bosonic D$_p$-brane  was found in \cite{Leigh:1989jq}. The supersymmetric action for a single D$_p$-brane  was derived in \cite{Cederwall:1996pv}. See \cite{Hatefi:2010ik} for more details on Born-Infeld, Chern-Simons actions and their higher derivative corrections.
 Section 5 of  \cite{Hatefi:2012wj} has a review of Chern-Simons action.

\vskip 0.1in
 The advent of D$_p$-brane physics
has greatly promoted the significance of an open
 string bringing numerous new results, and is likely to
continue
bringing exciting new physics in the future.
 A conjecture put forward in \cite{Park:2007mc} may be an
example: quantum
effects of open strings moving on
D$_p$-branes should produce the curvature of the host
D$_p$-branes.

\vskip 0.1in

Having concrete tools for various multi-point string amplitudes
is important for many purposes, including the first-principle
derivation of AdS/CFT.
Previous works on scattering that involve D$_p$-branes
and some applications of D$_p$-branes include \cite{Hashimoto:1996bf}.
Given that a close interplay between an
open string and a closed string must be behind AdS/CFT, amplitudes
involving a mixture of open string states and closed string states
should be especially worth studying.
In this work, we continue our previous
endeavours of computing amplitudes of one Ramond-Ramond
$C$-field vertex
and three massless open string vertices \cite{Hatefi:2010ik}.
 The amplitudes that we specifically consider are
$<V_{C}V_{\phi} V_A>$ and $<V_{C}V_{\phi} V_AV_A>$.\footnote{
One may wonder whether it would be possible by applying string
T-duality to deduce
the result of $<V_{C}V_{\phi} V_AV_A>$ from \cite{Hatefi:2010ik} in which
$<V_{C}V_A V_AV_A>$ was analysed. Applying T-duality in present
 types of computations can be subtle. It is definitely so in loop-level
computations \cite{Park:2008sg}. Although the present computations are all at tree-level
the presence of a closed string state can make things subtle, and
indeed does so.
}
(We focus on  $<V_{C}V_{\phi} V_AV_A>$ in the main text presenting
the simpler case $<V_{C}V_{\phi} V_A>$ in Appendix B.)

\vskip 0.1in

Applying the same methodology as that of the previous works \cite{Hatefi:2010ik},\cite{Garousi:2007fk} and \cite{Hatefi:2008ab},
we find below the precise (up to an on-shell ambiguity) forms
of the two scalar two gauge field vertices such that the infinite number of massless poles of the field theory amplitude exactly match the infinite number of massless poles of S-matrix elements of $<V_{C}V_{\phi} V_AV_A>$ to all
orders in $\alpha'$.
 To obtain the infinite number of massless
  poles for this case, one needs to determine the higher derivative couplings of  two scalar two gauge field couplings to all
orders in $\alpha'$. (This is analogous to the situation in \cite{Hatefi:2010ik} where
$<V_{C}V_{A} V_AV_A>$ was analyzed. The precise form of the four gauge field
vertices was determined for the field theory amplitudes to match with the corresponding string theory computations.)
We obtain all the infinite number of massless poles in field theory
 and find consistent results with the string theory counterpart.

\vskip 0.1in

Additionally, remarks on a subtlety are in order. The subtlety may
be tied with the profound relation
between an open and closed string.
The present work is another demonstration that the pure SYM vertices (such as two scalar two gauge field couplings of this work) produce
the same massless poles as the corresponding
correlator of one RR  field vertex $V_C$ and three SYM vertices.

\vskip 0.1in

This phenomenon seems quite universal and
must have deep origins going back to the intertwinedness
of an open string and a closed string. The intertwinedness should
originate from the composite nature of a closed string state in terms of open string
states. The compositeness is (implicitly) exploited in a common
practice in literature, identification of both sets of closed oscillators
with those of an open string's.
  We will briefly comment on this issue in section 2 and contemplate
the issue further in the conclusion.

\vspace{.3in}
The organization of the paper is as follows. In section 2, we calculate a tree-level four-point string scattering of one RR vertex, one
scalar field vertex operator and two gauge field vertex operators, $<V_{C}V_{\phi} V_AV_A>$.
 In section 3, we consider
the low energy effective field theory and determine
the interaction vertices that, with Myers' terms,
produce the same massless poles as those of the string amplitude.
In the conclusion, we summarize the results and ponder on the profound relation between
an open string and a closed string as a reason for the matching. We end with comments on the future
directions.
In Appendix A, a summary of our conventions is presented.
Appendix B contains a parallel analysis for the case of
$<V_{C}V_{\phi} V_A>$.

\section{String amplitude computations }

By applying the conformal field theory technique, we carry out the string scattering amplitude of one closed string Ramond-Ramond  field,
one scalar field  and two gauge fields on the world volume of
BPS D$_p$-branes in type II super string theory within  a flat background.
Some efforts for the tree level scattering amplitudes have been done
\cite{Stieberger:2009hq,Kennedy:1999nn,Chandia:2003sh,Hatefi:2010ik,Garousi:2007fk,Hatefi:2008ab}.
To compute a S-matrix element, one has to know  the picture of the vertex operators so that the sum of the super ghost charges must be -2 for disk level amplitudes.

The vertex operators that we will need are given by\footnote{We keep $\alpha'$
explicitly in this work. One may set $\alpha'=2$ to simplify the expressions.}
\begin{eqnarray}
V_{\phi}^{(0)}(x) &=& \xi_{i}\bigg(\partial
X^i(x)+\alpha' ik{\cdot}\psi\psi^i(x)\bigg)e^{\alpha' ik{\cdot} X(x)},
\nonumber\\
V_{\phi}^{(-1)}(y) &=&\xi.\psi(y) e^{-\phi(y)} e^{\alpha' ik{\cdot} X(y)},
\nonumber\\
V_{A}^{(0)}(x) &=& \xi_{a}\bigg(\partial
X^a(x)+ \alpha'iq{\cdot}\psi\psi^a(x)\bigg)e^{ \alpha' iq{\cdot} X(x)},
\nonumber\\
V_{A}^{(-1)}(y) &=&\xi_a\psi^a(y) e^{-\phi(y)} e^{\alpha'iq{\cdot} X(y)}
\nonumber\\
V_{C}^{(-\frac{1}{2},-\frac{1}{2})}(z,\bar{z})&=&(P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}e^{-\phi(z)/2}
S_{\alpha}(z)e^{i\frac{\alpha'}{2}p{\cdot} X(z)}e^{-\phi(\bar{z})/2} S_{\beta}(\bar{z})
e^{i\frac{\alpha'}{2}p{\cdot} D {\cdot} X(\bar{z})},
\label{d4Vs}
\end{eqnarray}
where $(k,q,p)$ are the momenta
of the scalar field, gauge field and $C$-field respectively; they satisfy the on-shell condition
$k^2=q^2=p^2=0$.
Our notation is such that
 the spinorial
indices are raised by the charge conjugation matrix, $C^{\alpha\beta}$
\begin{eqnarray}
(P_{-}H\!\!\!\!/\,_{(n)})^{\alpha\beta} =
C^{\alpha\delta}(P_{-}H\!\!\!\!/\,_{(n)})_{\delta}{}^{\beta}
\end{eqnarray}
In particular, the trace is defined by
\begin{eqnarray}
\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\gamma^{k})&\equiv & (P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}(\gamma^{k}C^{-1})_{\alpha\beta}
\nonumber\\
\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\Gamma^{jai})&\equiv &
(P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}(\Gamma^{jai}C^{-1})_{\alpha\beta}
\end{eqnarray}
where $P_{-}$ is a projection operator, $P_{-} = \mbox{$\frac{1}{2}$} (1-\gamma^{11})$, and
\begin{displaymath}
H\!\!\!\!/\,_{(n)} = \frac{a
_n}{n!}H_{\mu_{1}\ldots\mu_{n}}\gamma^{\mu_{1}}\ldots
\gamma^{\mu_{n}}
\ ,
\nonumber\end{displaymath}
with $n=2,4$ for type IIA and $n=1,3,5$ for type IIB. $a_n=i$ for IIA and $a_n=1$ for IIB theory.
To employ standard holomorphic worldsheet correlators,
we implement the usual doubling trick. For more details see Appendix A of \cite{Hatefi:2012wj}.

\vskip 0.2in

\subsection{Computation of  $<V_{C}V_{\phi} V_A V_A>$ }

The S-matrix element of one closed string C field, one scalar field and two gauge fields
is given by the following correlation function

\begin{eqnarray}
{\cal A}^{C \phi AA} & \sim & \int dx_{1}dx_{2}dx_{3}dzd\bar{z}\,
  \langle V_{\phi}^{(-1)}{(x_{1})}
V_{A}^{(0)}{(x_{2})}V_A^{(0)}{(x_{3})}
V_{RR}^{(-\frac{1}{2},-\frac{1}{2})}(z,\bar{z})\rangle,\labell{sstring}\end{eqnarray}
The open string vertex operators are inserted
at the boundary of the disk worldsheet while the closed string
vertex operator is inserted inside.
The  amplitude  reduces to the following
correlators for 123 ordering
\begin{eqnarray} {\cal A}^{C \phi AA}&\sim& \int
 dx_{1}dx_{2}dx_{3}dx_{4} dx_{5}\,
(P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}\xi_{1i}\xi_{2a}\xi_{3b}x_{45}^{-1/4}(x_{14}x_{15})^{-1/2}\nonumber\\&&
\times(I_1+I_2+I_3+I_4)\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3),\labell{125}\end{eqnarray} where
$x_{ij}=x_i-x_j$, and
\begin{eqnarray}
I_1&=&{<:e^{\alpha' ik_1.X(x_1)}:\partial X^a(x_2)e^{\alpha' ik_2.X(x_2)}
:\partial X^b(x_3)e^{\alpha' ik_3.X(x_3)}:e^{i\frac{\alpha'}{2}p.X(x_4)}:e^{i\frac{\alpha'}{2}p.D.X(x_5)}:>}
 \  \nonumber \\&&\times{<:S_{\alpha}(x_4):S_{\beta}(x_5):\psi^i(x_1):>},\nonumber\\
I_2&=&{<:e^{\alpha' ik_1.X(x_1)}:e^{\alpha' ik_2.X(x_2)}
:\partial X^b(x_3)e^{\alpha' ik_3.X(x_3)}:e^{i\frac{\alpha'}{2}p.X(x_4)}:e^{i\frac{\alpha'}{2}p.D.X(x_5)}:>}
 \  \nonumber \\&&\times{<:S_{\alpha}(x_4):S_{\beta}(x_5)::\psi^i(x_1):\alpha' ik_2.\psi\psi^{a}(x_2)>},\nonumber\\
 I_3&=&{<: e^{\alpha' ik_1.X(x_1)}:\partial X^a(x_2)e^{\alpha' ik_2.X(x_2)}
:e^{\alpha' ik_3.X(x_3)}:e^{i\frac{\alpha'}{2}p.X(x_4)}:e^{i\frac{\alpha'}{2}p.D.X(x_5)}:>}
 \  \nonumber \\&&\times{<:S_{\alpha}(x_4):S_{\beta}(x_5)::\psi^i(x_1):\alpha' ik_3.\psi\psi^{b}(x_3)>},\nonumber\\
 I_4&=&{<: e^{\alpha' ik_1.X(x_1)}:e^{\alpha' ik_2.X(x_2)}
:e^{\alpha' ik_3.X(x_3)}:e^{i\frac{\alpha'}{2}p.X(x_4)}:e^{i\frac{\alpha'}{2}p.D.X(x_5)}:>}
 \  \nonumber \\&&\times{<:S_{\alpha}(x_4):S_{\beta}(x_5):\psi^i(x_1)
:\alpha' ik_{2}{\cdot}\psi\psi^a(x_2):\alpha' ik_{3}{\cdot}\psi\psi^b(x_3):>}.
\label{i1234}
\end{eqnarray}

\vskip 0.1in

These correlators can be computed in straightforward fashion using Wick's theorem.
To obtain correlation function between two spin operators and several fermion fields and currents see Appendix A of \cite{Hatefi:2012wj}.

 The correlation function between two spin fields, two currents
and one worldsheet fermion -which we call $I_6^{bdaci}$ - is more complicated:

\begin{eqnarray}
I_6^{bdaci}&=&<:S_{\alpha}(x_4):S_{\beta}(x_5)::\psi^i(x_1):\psi^c\psi^a(x_2):\psi^d\psi^b(x_3)>\nonumber\\
&=&\bigg\{(\Gamma^{bdaci}C^{-1})_{{\alpha\beta}}+\alpha' r_1\frac{Re[x_{24}x_{35}]}{x_{23}x_{45}}+\alpha'^2 r_2\bigg(\frac{Re[x_{24}x_{35}]}{x_{23}x_{45}}\bigg)^{2}
\bigg\}\nonumber\\&&
2^{-5/2}x_{45}^{5/4}(x_{24}x_{25}x_{34}x_{35})^{-1}(x_{14}x_{15})^{-1/2},
\label{hh}
\end{eqnarray}
where
\begin{eqnarray}
r_1&=&\bigg(\eta^{cd}(\Gamma^{bai}C^{-1})_{\alpha\beta}
-\eta^{cb}(\Gamma^{dai}C^{-1})_{\alpha\beta}-\eta^{ad}(\Gamma^{bci}C^{-1})_{\alpha\beta}+\eta^{ab}(\Gamma^{dci}C^{-1})_{\alpha\beta}\bigg),\nonumber\\
r_2&=&\bigg((-\eta^{cd}\eta^{ab}+\eta^{ad}\eta^{cb})(\gamma^{i}C^{-1})_{\alpha\beta}\bigg)
\end{eqnarray}
Substituting these results, one finds, after some algebra,
\begin{eqnarray}
{\cal A}^{C\phi AA}&\!\!\!\!\sim\!\!\!\!\!&\int dx_{1}dx_{2} dx_{3}dx_{4}dx_{5}(P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}I\xi_{1i}\xi_{2a}\xi_{3b}x_{45}^{-1/4}(x_{14}x_{15})^{-1/2}\nonumber\\&&\times
\bigg(I_7^i(-\eta^{ab}x_{23}^{-2}+a^a_1a^b_2)+a^a_1a^{bi}_3+a^b_2a^{ai}_4-\alpha'^2 k_{2c}k_{3d}I_6^{bdaci}\bigg)\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3)
\labell{amp3q},\end{eqnarray}
where
\begin{eqnarray}
I&=&|x_{12}|^{\alpha'^2 k_1.k_2}|x_{13}|^{\alpha'^2 k_1.k_3}|x_{14}x_{15}|^{\frac{\alpha'^2}{2} k_1.p}|x_{23}|^{\alpha'^2 k_2.k_3}|
x_{24}x_{25}|^{\frac{\alpha'^2}{2} k_2.p}
|x_{34}x_{35}|^{\frac{\alpha'^2}{2} k_3.p}|x_{45}|^{\frac{\alpha'^2}{4}p.D.p},\nonumber\\
a^a_1&=&ik_1^{a}\bigg(\frac{x_{14}}{x_{12}x_{24}}+\frac{x_{15}}{x_{25}x_{12}}\bigg)
+ik_3^{a}\bigg(\frac{x_{43}}{x_{24}x_{23}}+\frac{x_{53}}{x_{25}x_{23}}\bigg),\nonumber\\
a^b_2&=&ik_1^{b}\bigg(\frac{x_{14}}{x_{13}x_{34}}+\frac{x_{15}}{x_{35}x_{13}}\bigg)
+ik_2^{b}\bigg(\frac{x_{24}}{x_{34}x_{23}}+\frac{x_{25}}{x_{35}x_{23}}\bigg),\nonumber\\
a^{bi}_3&=&\alpha' ik_{3d}2^{-3/2}x_{45}^{1/4}(x_{34}x_{35})^{-1}(x_{14}x_{15})^{-1/2} \bigg\{(\Gamma^{bdi}C^{-1})_{\alpha\beta}\bigg\} ,\nonumber\\
a^{ai}_4&=&\alpha' ik_{2c}2^{-3/2}x_{45}^{1/4}(x_{24}x_{25})^{-1}(x_{14}x_{15})^{-1/2} \bigg\{(\Gamma^{aci}C^{-1})_{\alpha\beta}\bigg\}
,\nonumber\\
I_7^i&=&<:S_{\alpha}(x_4):S_{\beta}(x_5):\psi^i(x_1):>=2^{-1/2}x_{45}^{-3/4}(x_{14}x_{15})^{-1/2}
(\gamma^{i}C^{-1})_{\alpha\beta}.\nonumber
\end{eqnarray}

\vskip 0.2in
Explicitly evaluating the integrals over the closed string location(to deal with the integrals and to provide more details see Appendix B of \cite{Hatefi:2012wj} ),
one can write the amplitude (\ref{amp3q}) as
\begin{eqnarray} {\cal A}^{\phi A A C}&=&{\cal A}_{1}+{\cal A}_{2}+{\cal A}_{3}+{\cal A}_{4}+{\cal A}_{5}\labell{711u}\end{eqnarray}
where
\begin{eqnarray}
{\cal A}_{1}&\!\!\!\sim\!\!\!&-2^{-1/2}\xi_{1i}\xi_{2a}\xi_{3b}
\bigg[k_{3d}k_{2c}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\Gamma^{bdaci})
\bigg]
L_1,
\nonumber\\
{\cal A}_{2}&\sim&2^{-1/2}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bdi})\xi_{1i}\xi_{3b}k_{3d}
\bigg\{2k_1.\xi_2 L_2-2k_3.\xi_2 L_5
\bigg\}\nonumber\\
{\cal A}_{3}&\sim&-2^{-1/2}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{aci})\xi_{1i}\xi_{2a}k_{2c}
\bigg\{-2k_2.\xi_3 L_5+2k_1.\xi_3 L_3
\bigg\}\\
{\cal A}_{4}&\sim&-2^{-1/2}L_5
\bigg\{ \xi_{3b}\xi_{1i}\xi_{2a}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bai})u +2k_2.\xi_3 k_{3d}\xi_{1i}\xi_{2a}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{dai})
\nonumber\\&&+2k_3.\xi_2 k_{2c}\xi_{1i}\xi_{3b}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bci})-\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{dci})k_{3d}k_{2c}\xi_{1i}(2\xi_2.\xi_3)
\bigg\}\nonumber\\
{\cal A}_{5}&\sim&2^{-1/2}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \gamma^{i})\xi_{1i}
\bigg\{
\xi_{3}.\xi_{2}(2ts)+2k_1.\xi_3(2k_3.\xi_2)t-4u k_1.\xi_2(k_1.\xi_3)+4sk_2.\xi_3k_1.\xi_2\bigg\}L_6\nonumber
\labell{483}\end{eqnarray}
where the functions
 $L_1,L_2,L_3,L_5,L_6$ are
\begin{eqnarray}
L_1&=&(2)^{-2(t+s+u)+1}\pi{\frac{\Gamma(-u+\frac{1}{2})
\Gamma(-s+\frac{1}{2})\Gamma(-t+\frac{1}{2})\Gamma(-t-s-u+1)}
{\Gamma(-u-t+1)\Gamma(-t-s+1)\Gamma(-s-u+1)}},\nonumber\\
L_2&=&(2)^{-2(t+s+u)}\pi{\frac{\Gamma(-u+1)
\Gamma(-s+1)\Gamma(-t)\Gamma(-t-s-u+\frac{1}{2})}
{\Gamma(-u-t+1)\Gamma(-t-s+1)\Gamma(-s-u+1)}}
\nonumber\\
L_3&=&(2)^{-2(t+s+u)}\pi{\frac{\Gamma(-u+1)
\Gamma(-s)\Gamma(-t+1)\Gamma(-t-s-u+\frac{1}{2})}{\Gamma(-u-t+1)
\Gamma(-t-s+1)\Gamma(-s-u+1)}}
,\nonumber\\
L_5&=&(2)^{-2(t+s+u)}\pi{\frac{\Gamma(-u)
\Gamma(-s+1)\Gamma(-t+1)\Gamma(-t-s-u+\frac{1}{2})}
{\Gamma(-u-t+1)\Gamma(-t-s+1)\Gamma(-s-u+1)}}
,\nonumber\\
L_6&=&(2)^{-2(t+s+u)-1}\pi{\frac{\Gamma(-u+\frac{1}{2})
\Gamma(-s+\frac{1}{2})\Gamma(-t+\frac{1}{2})\Gamma(-t-s-u)}
{\Gamma(-u-t+1)\Gamma(-t-s+1)\Gamma(-s-u+1)}},
\label{Ls}
\end{eqnarray}
It is possible to further simplify the result above:
\begin{eqnarray} {\cal A}^{\phi AA C}&=&{\hat{\cal A}}_{1}+\hat{{\cal A}}_{2}
+\hat{{\cal A}}_{3}+\hat{{\cal A}}_{4},\labell{141u}\end{eqnarray}
where
\begin{eqnarray}
\hat{{\cal A}}_{1}&\!\!\!\sim\!\!\!& 2^{1/2}\xi_{1i}\xi_{2a}\xi_{3b}
\bigg[k_{3d}k_{2c}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\Gamma^{bdaci})
\bigg]
(t+s+u)\hat{L}_1,
\nonumber\\
\hat{{\cal A}}_{2}&\sim&2^{-1/2}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bdi})\xi_{1i}
\bigg\{2k_1.\xi_2 k_{3d}\xi_{3b}\hat{L}_2+2k_3.\xi_2 \xi_{3b} k_{1d}\hat{L}_5
-[2\leftrightarrow 3]
\bigg\} \\
\hat{{\cal A}}_{3}&\sim&-2^{-1/2}\hat{L}_5
\bigg\{ \xi_{3b}\xi_{1i}\xi_{2a}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bai})u -\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{dci})k_{3d}k_{2c}\xi_{1i}(2\xi_2.\xi_3)
\bigg\}\nonumber\\
\hat{{\cal A}}_{4}&\sim&2^{-3/2}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \gamma^{i})\xi_{1i}
\bigg\{
\xi_{3}.\xi_{2}(2ts)+2k_1.\xi_3(2k_3.\xi_2)t-4u k_1.\xi_2(k_1.\xi_3)+4sk_2.\xi_3k_1.\xi_2\bigg\}\hat{L}_1\nonumber
\labell{4798}
\end{eqnarray}
The functions $\hat{L}_1,\hat{L}_2$ are ($\hat{L_5}=L_5$)
\begin{eqnarray}
\hat{L}_1&=&(2)^{-2(t+s+u)}\pi{\frac{\Gamma(-u+\frac{1}{2})
\Gamma(-s+\frac{1}{2})\Gamma(-t+\frac{1}{2})\Gamma(-t-s-u)}
{\Gamma(-u-t+1)\Gamma(-t-s+1)\Gamma(-s-u+1)}},\nonumber\\
\hat{L}_2&=&(2)^{-2(t+s+u)}\pi{\frac{\Gamma(-u+1)
\Gamma(-s+1)\Gamma(-t)\Gamma(-t-s-u+\frac{1}{2})}
{\Gamma(-u-t+1)\Gamma(-t-s+1)\Gamma(-s-u+1)}}
\nonumber\\
\label{Ls}
\end{eqnarray}

The amplitude satisfies ward identities as it should:  by replacing
$\xi_{2a}\rightarrow k_{2a}$ and $\xi_{3b}\rightarrow k_{3b}$, it vanishes.
As in previous works, the amplitude is non-vanishing for certain values of $p$ and $n$: $n=p-2,  n=p+2$ and $p=n$.
The amplitude has an infinite number of massless scalar and gauge field poles and in addition it has infinite contact interactions.
We must expand it at low energy limit where the momentum expansions have been introduced in detail in \cite{Hatefi:2010ik}.

\vskip 0.2in

Expansion of the functions $\hat{L}_1,\hat{L}_2$
around the $t,s,u \rightarrow 0$ is
\begin{eqnarray}
\hat{L}_1 &=&-{\pi^{5/2}}\left( \sum_{n=0}^{\infty}c_n(s+t+u)^n\right.
\left.+\frac{\sum_{n,m=0}^{\infty}c_{n,m}[s^n t^m +s^m t^n]}{(t+s+u)}\right.\nonumber\\
&&\left.+\sum_{p,n,m=0}^{\infty}f_{p,n,m}(s+t+u)^p[(s+t)^{n}(st)^{m}]\right)\nonumber\\
\hat{L}_2 &=&-\pi^{3/2}\sum_{n=-1}^{\infty}b_n \frac{1}{t}(u+s)^{n+1}+\sum_{p,n,m=0}^{\infty}e_{p,n,m}t^{p}(su)^{n}(s+u)^m
\labell{highcaap}.
\end{eqnarray}

where some of the coefficients $b_n,\,e_{p,n,m},\,c_n,\,c_{n,m}$ and $f_{p,n,m}$ are
\begin{eqnarray}
&&b_{-1}=1,\,b_0=0,\,b_1=\frac{1}{6}\pi^2,\,b_2=2\zeta(3),c_0=0,c_1=-\frac{\pi^2}{6},\nonumber\\
&&e_{2,0,0}=e_{0,1,0}=2\zeta(3),e_{1,0,0}=\frac{1}{6}\pi^2,e_{1,0,2}=\frac{19}{60}\pi^4,e_{1,0,1}=e_{0,0,2}=6\zeta(3),\nonumber\\
&&e_{0,0,1}=\frac{1}{3}\pi^2,e_{3,0,0}=\frac{19}{360}\pi^4,e_{0,0,3}=e_{2,0,1}=\frac{19}{90}\pi^4,e_{1,1,0}=e_{0,1,1}=\frac{1}{30}\pi^4,\labell{577}\\
&&c_2=-2\zeta(3),
\,c_{1,1}=\frac{\pi^2}{6},\,c_{0,0}=\frac{1}{2},c_{3,1}=c_{1,3}=\frac{2}{15}\pi^4,c_{2,2}=\frac{1}{5}\pi^4,\nonumber\\
&&c_{1,0}=c_{0,1}=0
c_{3,0}=c_{0,3}=0\,
,\,c_{2,0}=c_{0,2}=\frac{\pi^2}{6},c_{1,2}=c_{2,1}=-4\zeta(3),\nonumber\\
&&f_{0,1,0}=\frac{\pi^2}{3},\,f_{0,2,0}=-f_{1,1,0}=-6\zeta(3),f_{0,0,1}=-2\zeta(3),c_{4,0}=c_{0,4}=\frac{1}{15}\pi^4.\, \nonumber
\end{eqnarray}

Note that the coefficients $b_n$ are exactly the coefficients that appear in the momentum expansion of the S-matrix element of
$CAAA$
\cite{Hatefi:2010ik}.
 $c_n,\,c_{n,m},f_{p,n,m}$ are different from those coefficients that appeared in \cite {Hatefi:2008ab}.

\section{Two scalar and two  gauge field couplings }

The closed form result of  $<V_{RR}V_{\phi} V_A V_A>$ above can be
momentum-expanded to be compared with the corresponding effective field theory computations.
In the earlier work \cite{Hatefi:2010ik}, it was shown that the string
theory result of $<V_{RR}V_A V_A V_A>$ is reproduced by the following
SYM vertices
\begin{eqnarray}
-T_p (2\pi\alpha')^4 S\mbox{\rm Tr}\,\left(-\frac{1}{8}F_{bd}F^{df}F_{fh}F^{hb}+\frac{1}{32}(F_{ab}F^{ba})^2\right).
\label{55}
\end{eqnarray}
Extension of these interaction vertices to higher derivative
couplings reproduced both the massless pole terms and
the contact terms.

Below, we will show that the same is true for the couplings two scalar fields and two gauge fields.
 First, we show that the correct couplings between two gauge fields and two scalar fields are
\begin{eqnarray}
&&- \frac{T_p(2\pi\alpha')^4}{2}{\rm STr}
\left(D_a\phi^iD^b\phi_iF^{ac}F_{bc}-\frac{1}{4}
(D_a\phi^i D^a\phi_iF^{bc}F_{bc})\right).\labell{a011}
\end{eqnarray}
In order to find the infinite higher derivative corrections to two scalar fields and two
gauge fields, one must
find the amplitude of two scalars and two gauge fields.
Computing this amplitude and working out details
, we could find the higher derivative
corrections of two scalars and two gauge fields to all orders of $\alpha'$ as the following:
\begin{eqnarray}
(2\pi\alpha')^4\frac{1}{ 2 \pi^2}T_p\left(\alpha'\right)^{n+m}\sum_{m,n=0}^{\infty}(\cL_{1}^{nm}+\cL_{2}^{nm}+\cL_{3}^{nm}),\labell{highder}\end{eqnarray}
\begin{eqnarray}
&&\cL_{1}^{nm}=-
\mbox{\rm Tr}\,\left(\frac{}{}a_{n,m}\cD_{nm}[D_a \phi^i D^b \phi_i F^{ac}F_{bc}]+ b_{n,m}\cD'_{nm}[D_a \phi^i F^{ac} D^b \phi_i F_{bc}]+h.c.\frac{}{}\right),\nonumber\\
&&\cL_{2}^{nm}=-\mbox{\rm Tr}\,\left(\frac{}{}a_{n,m}\cD_{nm}[D_a \phi^i D^b \phi_i F_{bc}F^{ac}]+\frac{}{}b_{n,m}\cD'_{nm}[D_a \phi^i F_{bc} D^b \phi_i F^{ac}]+h.c.\frac{}{}\right),\nonumber\\
&&\cL_{3}^{nm}=\frac{1}{2}\mbox{\rm Tr}\,\left(\frac{}{}a_{n,m}\cD_{nm}[D_a \phi^i D^a \phi_i F^{bc}F_{bc}]+\frac{}{}b_{n,m}\cD'_{nm}[D_a \phi^i F_{bc} D^a \phi_i F^{bc}]+h.c\frac{}{}\right),\nonumber\end{eqnarray}
where the higher derivative operators
$D_{nm} $ and $ D'_{nm}$ are defined \cite{Hatefi:2010ik} as
\begin{eqnarray}
\cD_{nm}(EFGH)&\equiv&D_{b_1}\cdots D_{b_m}D_{a_1}\cdots D_{a_n}E  F D^{a_1}\cdots D^{a_n}GD^{b_1}\cdots D^{b_m}H,\nonumber\\
\cD'_{nm}(EFGH)&\equiv&D_{b_1}\cdots D_{b_m}D_{a_1}\cdots D_{a_n}E   D^{a_1}\cdots D^{a_n}F G D^{b_1}\cdots D^{b_m}H.\nonumber
\end{eqnarray}
As usual, the above couplings are valid up to total derivative
terms and terms such as $\prt_a\prt^aFFD\phi D\phi$ that vanish on-shell.
 They have no effect on the massless poles of S-matrix elements,
because by canceling $k^2$ with the massless propagator, they produce
 contact terms.
 We now turn to verification of (\ref{highder}) and the terms
coming from the DBI part.

\subsection{Infinite number of massless scalar poles for $p+2=n$ case }

In this subsection, we check that the two gauge two scalar
interaction vertices \reef{highder} produce an infinite number of massless scalar poles of the string theory S-matrix element that are in the $(s+t+u)$-channel.
Specifically, the goal is to show that the massless poles of
the string computation are reproduced by the following
WZ coupling (that was found in \cite{Garousi:2000ea}),
\begin{eqnarray}
\lambda\mu_p\int d^{p+1}\sigma {1\over (p+1)!}
(\veps^v)^{a_0\cdots a_{p}}\,\mbox{\rm Tr}\,\left(\phi^i\right)\,
H^{(p+2)}_{ia_0\cdots a_{p}}(\sigma) \end{eqnarray}
and by the higher derivative two gauge field and two scalar couplings that have been found in \reef{highder}. To that end, let us consider the amplitude of the decay of one R-R field to one scalar and two gauge fields in the world volume theory of the BPS branes. In the (Feynman gauge) Feynman diagrammatic rules, it is given by
\begin{eqnarray}
{\cal A}&=&V_{\alpha}^{i}(C_{p+1},\phi)G_{\alpha\beta}^{ij}(\phi)V_{\beta}^{j}(\phi,\phi_1,
A_2,A_3),\labell{amp549}
\end{eqnarray}
where
\begin{eqnarray}
G_{\alpha\beta}^{ij}(\phi) &=&\frac{-i\delta_{\alpha\beta}\delta^{ij}}{T_p(2\pi\alpha')^2
k^2}=\frac{-i\delta_{\alpha\beta}\delta^{ij}}{T_p(2\pi\alpha')^2
(t+s+u)},\nonumber\\
V_{\alpha}^{i}(C_{p+1},\phi)&=&i(2\pi\alpha')\mu_p\frac{1}{(p+1)!}(\veps^v)^{a_0\cdots a_{p}}
 H^{i(p+2)}_{a_0\cdots a_{p}}\mbox{\rm Tr}\,(\lambda_{\alpha}).
\labell{Fey}
\end{eqnarray}
We have replaced $k^2$
 by $(t+s+u)$ in the first equation of (\ref{Fey}). In the second equation of  (\ref{Fey}), $\mbox{\rm Tr}\,(\lambda_{\alpha})$ is non-zero for the abelian matrix $\lambda_{\alpha}$.
Noting the fact that the off-shell scalar field must be abelian, one finds the higher derivative vertex  $ V_{\beta}^{j}(\phi,\phi_1,
A_2,A_3)$  from the higher derivative couplings
in \reef{highder}:
\begin{eqnarray}
V_{\beta}^{j}(\phi,\phi_1,
A_2,A_3)&=&\xi_{1}^j\frac{I_8}{2\pi^2}(\alpha')^{n+m}(a_{n,m}+b_{n,m})
\bigg(\frac{}{}(k_3\!\cdot\! k_1)^m(k_1\!\cdot\! k_2)^n+(k_3\!\cdot\! k)^m(k_2\!\cdot\! k)^n
\nonumber\\&&+(k_1\!\cdot\! k_3)^n(k_1\!\cdot\! k_2)^m+(k\!\cdot\! k_3)^n (k\!\cdot\! k_2)^m
\bigg),\labell{verppaa}\end{eqnarray}
where  $k$ is the momentum of the off-shell scalar field, and
\begin{eqnarray}
I_8&=&(2\pi\alpha')^4T_{p}\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3\lambda_{\beta})\bigg[\frac{st}{2}\xi_{2}.\xi_3+t k_1.\xi_3 k_3.\xi_2+s k_1.\xi_2 k_2.\xi_3-u k_1.\xi_2 k_1.\xi_3\bigg]\end{eqnarray}
For the reason explained in \cite{Hatefi:2010ik}, $b_{n,m}$ is symmetric.
 Note that we must consider two permutations,
\begin{eqnarray}
\mbox{\rm Tr}\,(\lambda_1\lambda_{\beta}\lambda_2\lambda_3),
\mbox{\rm Tr}\,(\lambda_{\beta}\lambda_1\lambda_2\lambda_3)
\end{eqnarray}
to obtain the desired 123 ordering
of the amplitude. Let us list some of the coefficients $a_{n,m}$
and $b_{n,m}$ for convenience:
\begin{eqnarray}
&&a_{0,0}=-\frac{\pi^2}{6},\,b_{0,0}=-\frac{\pi^2}{12},a_{1,0}=2\zeta(3),\,a_{0,1}=0,\,b_{0,1}=-\zeta(3),a_{1,1}=a_{0,2}=-7\pi^4/90,\nonumber\\
&&a_{2,2}=(-83\pi^6-7560\zeta(3)^2)/945,b_{2,2}=-(23\pi^6-15120\zeta(3)^2)/1890,a_{1,3}=-62\pi^6/945,\nonumber\\
&&\,a_{2,0}=-4\pi^4/90,\,b_{1,1}=-\pi^4/180,\,b_{0,2}=-\pi^4/45,a_{0,4}=-31\pi^6/945,a_{4,0}=-16\pi^6/945,\nonumber\\
&&a_{1,2}=a_{2,1}=8\zeta(5)+4\pi^2\zeta(3)/3,\,a_{0,3}=0,\,a_{3,0}=8\zeta(5),b_{1,3}=-(12\pi^6-7560\zeta(3)^2)/1890,\nonumber\\
&&a_{3,1}=(-52\pi^6-7560\zeta(3)^2)/945, b_{0,3}=-4\zeta(5),\,b_{1,2}=-8\zeta(5)+2\pi^2\zeta(3)/3,\nonumber\\
&&b_{0,4}=-16\pi^6/1890.\end{eqnarray}
(They were computed in \cite{Hatefi:2010ik}
for the four gauge field amplitude. In retrospect, it must be due to field theory T-duality
that they remain valid for the present two scalars  and two gauge fields amplitude.)
Now one can write  $k_3\!\cdot\! k=k_2.k_1-(k^2)/2$ and $k_2\!\cdot\! k=k_1.k_3-(k^2)/2$.
The terms $k^2$ in the vertex \reef{verppaa} will be cancelled with the $k^2$ in the denominator of the scalar field propagator producing
contact terms. They will not be explicitly considered in this work. Setting them aside, one finds the following result of an infinite number of massless poles,
\begin{eqnarray}
&&16\pi\mu_p\frac{\epsilon^{a_{0}\cdots a_{p}}\xi_1^i
H^{i(p+2)}_{a_0\cdots a_{p}}}{(p+1)!(s+t+u)}\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3)
\sum_{n,m=0}^{\infty}\bigg((a_{n,m}+b_{n,m})[s^{m}t^{n}+s^{n}t^{m}]\nonumber\\&&
\bigg[2st\xi_{2}.\xi_3+4t k_1.\xi_3 k_3.\xi_2+4s k_1.\xi_2 k_2.\xi_3-4u k_1.\xi_2 k_1.\xi_3\bigg]
\label{amphigh8}\end{eqnarray}
\\
As a check of our calculations let us compare the above amplitude with the infinite  number of massless poles in the string theory result. We will take several
 values of $n,m$ for illustrations. Common factors of the
string and field theory amplitudes will be
omitted. For $n=m=0$, the amplitude \reef{amphigh8} has the following numerical factor
\begin{eqnarray}
-2(a_{0,0}+b_{0,0})&=&-2(\frac{-\pi^2}{6}+\frac{-\pi^2}{12})=\frac{\pi^2}{2}\nonumber\end{eqnarray}
There is a corresponding term in the string amplitude, and it has the numerical factor  $(2\frac{\pi^2}{2}c_{0,0})$ which is equal to the number above.  At the order of $\alpha'$, the amplitude \reef{amphigh8} has the following numerical factor
\begin{eqnarray}
-(a_{1,0}+a_{0,1}+b_{1,0}+b_{0,1})(s+t)&=&0\nonumber\end{eqnarray}
The corresponding term in string amplitude is proportional to   $\frac{\pi^2}{2}(c_{1,0}+c_{0,1})(s+t)$ which indeed vanishes.  At the  order of $(\alpha')^2$, the amplitude \reef{amphigh8} has the following factor
\begin{eqnarray}
&&-2(a_{1,1}+b_{1,1})st-(a_{0,2}+a_{2,0}+b_{0,2}+b_{2,0})[s^2+t^2]\nonumber\\
&&=\frac{\pi^4}{6}(st)+\frac{\pi^4}{6}(s^2+t^2)
\nonumber\end{eqnarray}
The string result has the numerical factor  $\frac{\pi^2}{2}[c_{1,1}(2st)+(c_{2,0}+c_{0,2})(s^2+t^2)]$, which is equal to the above factor using the coefficients in \reef{577}.
At the order of $\alpha'^3$, this amplitude has the following factor
\begin{eqnarray}
&&- (a_{3,0}+a_{0,3}+b_{0,3}+b_{3,0})[s^3+t^3]- (a_{1,2}+a_{2,1}+b_{1,2}+b_{2,1})[st(s+t)]\nonumber\\
&&=-4\pi^2\zeta(3)st(s+t)
\nonumber\end{eqnarray}
which is equal to the corresponding term  , {\it i.e.,}\  $\frac{\pi^2}{2}[(c_{0,3}+c_{3,0})[s^3+t^3]+(c_{2,1}+c_{1,2})st(s+t)]$.
At the  order of $(\alpha')^4$, the amplitude \reef{amphigh8} has the following factor
\begin{eqnarray}
&&- (a_{4,0}+a_{0,4}+b_{0,4}+b_{4,0})(s^4+t^4)- (a_{3,1}+a_{1,3}+b_{3,1}+b_{1,3})[st(s^2+t^2)]\nonumber\\
&&-2(a_{2,2}+b_{2,2}) s^2t^2=\frac{\pi^6}{15}(s^4+t^4+2(s^3t+t^3s)+3s^2t^2)
\nonumber\end{eqnarray}
The string result has the numerical factor  $\frac{\pi^2}{2}[(c_{4,0}+c_{0,4})(s^4+t^4)+(c_{1,3}+c_{3,1})(s^3t+t^3s)+2c_{2,2} s^2t^2]$ which is equal to the above factor using the coefficients in \reef{577}. Note that  the string amplitude has been rescaled by  $2^{1/2}\pi^{1/2}\mu_p$.

Similar comparisons can be carried out\footnote{The same checks have been done for finding an infinite number of massless poles of $<V_CV_AV_AV_A>$ in \cite{Hatefi:2010ik}.
} for all higher orders
of $\alpha'$: the field theory amplitude \reef{amphigh8} exactly reproduces the infinite number of massless scalar poles of the string theory amplitude of $<V_CV_\phi V_AV_A>$.
 This shows that the higher derivative couplings of two scalar two gauge fields
are exact up to terms that vanish on-shell. They are also consistent with the momentum expansion of the amplitude of $AA\phi\phi$.

\subsection{Infinite number of massless scalar  poles for $p=n$ case }

Substituting the expansion of $\hat{L}_2$ mentioned in \reef{highcaap} into the amplitude, it is possible to obtain all massless scalar poles in the string theory side:
\begin{eqnarray}
\hat{{\cal A}}_{2}&\sim&2^{-1/2}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bdi})\xi_{1i}
\bigg\{2k_1.\xi_2 \xi_{3b} k_{3d}L_2-[2\leftrightarrow 3]
\bigg\}
\end{eqnarray}
 The trace can be calculated straightforwardly:
\begin{eqnarray}
\mbox{\rm Tr}\,\bigg(H\!\!\!\!/\,_{(n)}M_p\Gamma^{bdi}
\bigg)&=&\pm\frac{32}{p!}\epsilon^{a_{0}\cdots a_{p-2}bd}H^{i}_{a_{0}\cdots a_{p-2}},
\nonumber\end{eqnarray}
Substituting it into the amplitude and keeping
all the scalar poles, one gets
\begin{eqnarray}
A^{C\phi AA}&=&\mp\frac{32}{2 p!}\mu_p \pi^2\xi_{1i}\epsilon^{a_{0}\cdots a_{p-2}bd}H^{i}_{a_{0}\cdots a_{p-2}}\nonumber\\&&\bigg\{\bigg[\sum_{n=-1}^{\infty}\frac{1}{t}{b_n(u+s)^{n+1}}(2k_1.\xi_2)\xi_{3b}k_{3d}\bigg]
-\bigg[2\leftrightarrow 3\bigg]
\bigg\}\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3).\label{UI}
\end{eqnarray}
where the amplitude is rescaled by $2^{1/2}\pi^{1/2}\mu_{p}$. The amplitude is antisymmetric under the interchange of the gauge fields; therefore, the whole amplitude vanishes for the abelian gauge group. The amplitude satisfies the Ward identity under $\xi_{3b} \rightarrow k_{3b}$. Note that we kept only
the massless poles; the other terms are
contact terms infinite in number. Let us analyze all
orders of the massless scalar poles.

Because the amplitudes in s and t-channels are similar, we
analyze the amplitude in the t-channel in detail; by interchanging the labels of momentum and polarization vectors, $(2\leftrightarrow 3)$, the other massless poles in the s channel can easily be obtained. The relevant vertices
for this case in the field theory side is
\begin{eqnarray}
S^{(1)}&=&i\lambda\mu_p\int \mbox{\rm STr}\,\left(F P\left[
C^{(p-1)}(\sigma,\phi)\right]\right)
\labell{interac}\\
&=&{i}\lambda^2\mu_p\int d^{p+1}\sigma {1\over p!}(\veps^v)^{a_0\cdots a_{p}}
\left[{p}\mbox{\rm Tr}\,\left(F_{a_{0}a_{1}}\phi^k\right)
\prt_kC^{(p-1)}_{a_2\cdots a_{p}}(\sigma)\right]\,\,\, .
\nonumber
\end{eqnarray}
Where the scalar field comes from Taylor expansion (see section 5 of \cite{Hatefi:2012wj}). With this vertex, the massless poles in the t-channel should be reproduced as
\begin{eqnarray}
{\cal A}&=&V^i_{\alpha}(C_{p-1},A_3,\phi)G^{ij}_{\alpha\beta}(\phi)V^j_{\beta}(\phi,A_2,\phi_1),\labell{amp642}
\end{eqnarray}
The vertices and propagator above are
\begin{eqnarray}
V^i_{\alpha}(C_{p-1},A_3,\phi)&=&\frac{iN\lambda^2\mu_p}{(p)!}(\veps^v)^{a_0\cdots a_p}(H^{(p)})^i{}_{a_2\cdots a_p}\xi_{3a_{1}}k_{3a_{0}}\mbox{\rm Tr}\,(\lambda_3\lambda_\alpha)\sum_{n=-1}^{\infty}b_n(\alpha'k_3.k)^{n+1},\nonumber\\
\end{eqnarray}
with
\begin{eqnarray}
({V}^{\phi\phi_1 A_2})^j_{\beta}&=&-2i\lambda^2T_p\mbox{\rm Tr}\,
(\xi_2\!\cdot\! k_1[\xi_1^j,T_{\beta}])
\labell{fvertex}\\
({G}^{\phi})^{ij}_{\alpha\beta}&=&-\frac{i}{N\lambda^2T_p}\frac{\delta^{ij}
\delta_{\alpha\beta}}{k^2}\,\,\, ,
\nonumber
\end{eqnarray}
where $k$ is the momentum of the off-shell scalar field and it
satisfies
 $k^2=(k_1+k_2)^2=-t$. We also have written $\zeta^i={\zeta}^i_{\alpha}T_{\alpha}$
where $T_{\alpha}$ are the $U(N)$ generators with normalization
$\mbox{\rm Tr}\,(T_{\alpha}T_{\beta})=N\delta_{\alpha\beta}$.
The propagator is derived from the standard gauge kinetic term resulting in the expansion of the Born-Infeld action. The vertex $V^j_{\beta}(\phi,A_2,\phi_1)$ has found from the standard non-abelian kinetic term of
the scalar field; the vertex $V^i_{\alpha}(C_{p-1},A_3,\phi)$ is found from the higher derivative extension of the WZ coupling $\mbox{\rm Tr}\,(\partial_{k}C_{p-1}\wedge F\phi^k)$. The important point is that the vertex $V^j_{\beta}(\phi,A_2,\phi_1)$ has no higher derivative correction as it comes from the kinetic term of the scalar field.

\vskip 0.1in

Substituting all vertices in the amplitude \reef{amp642}, one finds
\begin{eqnarray}
{\cal A}&=&\mu_p(2\pi\alpha')^{2}\frac{1}{(p)!t}\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3)\epsilon^{a_{0}\cdots a_{p-2}ba}H^{i}_{a_{0}\cdots a_{p-2}}\sum_{n=-1}^{\infty}b_n\bigg(\frac{\alpha'}{2}\bigg)^{n+1}(s+u)^{n+1}\nonumber\\&&\times\bigg[-2k_1.\xi_2\xi_{1i}\xi_{3b} k_{3a}\bigg]\end{eqnarray}
It describes exactly the same infinite number of massless poles of the string theory amplitude \reef{UI} in t-channel; there is precise agreement between the field theory calculation and string result.
A similar comparison in $s$-channel also yields agreement.

\subsection{Infinite number of massless gauge field poles for $p=n$ case }

The expansion of the function $\hat{L}_5$ is given by
\begin{eqnarray}
\hat{L}_5&=&-\pi^{3/2}\bigg(\frac{1}{u}\sum_{n=-1}^{\infty}b_n(t+s)^{n+1}+\sum_{p,n,m=0}^{\infty}e_{p,n,m}u^{p}(st)^{n}(s+t)^m\bigg).\labell{high}\end{eqnarray}
One can read off the infinite massless gauge field poles for the amplitude of $<V_CV_\phi V_A V_A>$:
\begin{eqnarray}
\hat{{\cal A}}_{2}&\sim&2^{-1/2}L_5\bigg(\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bdi})\xi_{1i}\xi_{3b}k_{3d}
(-2k_3.\xi_2 )-\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{aci})\xi_{1i}\xi_{2a}k_{2c}
(-2k_2.\xi_3 )\nonumber\\&&
-2k_2.\xi_3 k_{3d}\xi_{1i}\xi_{2a}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{dai})
-2k_3.\xi_2 k_{2c}\xi_{1i}\xi_{3b}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{bci})\nonumber\\&&+\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p \Gamma^{dci})k_{3d}k_{2c}\xi_{1i}(2\xi_2.\xi_3)\bigg)
\end{eqnarray}

Working out the trace, one finds the infinite
gauge field poles
\begin{eqnarray}
A^{C\phi AA}&=&\mp\frac{32}{2 p!}\mu_p \pi^2\xi_{1i}\epsilon^{a_{0}\cdots a_{p-2}dc}H^{i}_{a_{0}\cdots a_{p-2}}\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3)\sum_{n=-1}^{\infty}\frac{1}{u}{b_n(t+s)^{n+1}}\bigg\{\bigg[2\xi_2.\xi_3 k_{3d}k_{2c}\nonumber\\&&-2k_3.\xi_2 \xi_{3d}(k_{2c}+k_{3c})+2k_2.\xi_3 \xi_{2d}(k_{2c}+k_{3c}) \bigg]
\bigg\}.\label{UIiop}\end{eqnarray}
The amplitude is antisymmetric under the interchange of the gauge
fields; it vanishes for the abelian gauge group.
 We have kept the entire massless poles; the
other terms are infinite contact terms.
Let us  focus on the massless gauge field poles. (A similar analysis can
be done for the contact terms.)
The relevant WZ coupling is
\begin{eqnarray}
S^{(2)}&=&i\lambda\mu_p\int \mbox{\rm STr}\,\left(F P\left[
C^{(p-1)}(\sigma,\phi)\right]\right)
\labell{interac}\\
&=&{i}\lambda^2\mu_p\int d^{p+1}\sigma {1\over p!}(\veps^v)^{a_0\cdots a_{p}}
\left[{p} \mbox{\rm Tr}\,\left(F_{a_{0}a_{1}}\phi^k\right)
\prt_kC^{(p-1)}_{a_2\cdots a_{p}}(\sigma)\right]\,\,\, .
\nonumber
\end{eqnarray}
An infinite number of massless poles in the u-channel should be reproduced in field theory according to the following Feynman rule;
the $u$-channel amplitude can be written as
\begin{eqnarray}
{\cal A}&=&V^a_{\alpha}(C_{p-1},\phi_1,A)G^{ab}_{\alpha\beta}(A)V^b_{\beta}(A,A_2,A_3),\labell{amp60942}\end{eqnarray}
The vertices and propagator can
be read from the effective action
 are
\begin{eqnarray}
V^a_{\alpha}(C_{p-1},\phi_1,A)&=&\frac{i\lambda^2\mu_p}{(p)!}(\veps^v)^{a_0\cdots a_{p-1}a}(H^{(p)})^k{}_{a_1\cdots a_{p-1}}\xi_{1k}k_{a_{0}}\mbox{\rm Tr}\,(\lambda_\alpha\lambda_1)\sum_{n=-1}^{\infty}b_n(\alpha'k_1.k)^{n+1},\nonumber\\
\end{eqnarray}
with
\begin{eqnarray}
V^b_{\beta}(A,A_2,A_3)&=&-iT_p(2\pi\alpha')^{2}\mbox{\rm Tr}\,(\lambda_2\lambda_3\lambda_\beta)\bigg[\xi_2^b(k_2-k).\xi_3+
\xi_3^b(k-k_3).\xi_2+\xi_3.\xi_2(k_3-k_2)^b\bigg],\nonumber\\
G_{\alpha\beta}^{ab}(A)&=&\frac{i\delta_{\alpha\beta}\delta^{ab}}{(2\pi\alpha')^2 T_p
(k)^2},\nonumber\end{eqnarray}
where  $k$ is the momentum of the off-shell gauge field
and satisfies $k^2=(k_2+k_3)^2=-u$.
The vertex $V^a_{\alpha}(C_{p-1},\phi_1,A)$ has found from the higher derivative extension of the WZ coupling $\mbox{\rm Tr}\,(\partial_{k}C_{p-1}\wedge F\phi^k)$.
Substituting these vertices in the amplitude, one gets
\begin{eqnarray}
{\cal A}&=&\mu_p(2\pi\alpha')^{2}\frac{1}{(p)!u}\mbox{\rm Tr}\,(\lambda_1\lambda_2\lambda_3)\epsilon^{a_{0}\cdots a_{p-2}bd}
(H^{(p)})^i{}_{a_0\cdots a_{p-2}}\sum_{n=-1}^{\infty}b_n\bigg(\frac{\alpha'}{2}\bigg)^{n+1}(s+t)^{n+1}\nonumber\\&&\times\bigg[2k_3.\xi_2\xi_{1i}\xi_{3b} k_{1d}-2k_2.\xi_3\xi_{1i}\xi_{2b} k_{1d}-2\xi_3.\xi_2\xi_{1i} k_{2b} k_{3d}\bigg]\end{eqnarray}
It again displays precise agreement with the infinite $u$-channel poles in
the string amplitude, \reef{UIiop}.

\section{Conclusion}

Following \cite{Hatefi:2010ik} in which a similar analysis was done
for $<V_{C}V_A V_AV_A>$, we have computed $<V_{C}V_{\phi} V_AV_A>$
in closed form. By performing the momentum expansion,
the corresponding low energy SYM vertices have
been determined in all orders in $\alpha'$. We believe that the result
of this paper will provide the basis for future research on, e.g.,
next-to-leading order dielectric effect and
other related topics in string theory \cite{Hatefi:2012sy}.

\vskip 0.1in

 For the simple scalar poles, since there is no correction to   $\mbox{\rm Tr}\,\left(\phi^i\right)\,
H^{i(p+2)}_{a_0\cdots a_{p}}$, the non-leading scalar poles should give  information about the higher derivative corrections to the couplings of two  scalars and two gauge fields where we found them up
 to all orders of $\alpha'$ in \reef{highder}.

\vskip 0.1in

Although we did not analyse the contact
terms of the string amplitude, they would give information about the higher derivative corrections
to $\mbox{\rm Tr}\,(\partial_{k}C_{p-3}\wedge F\wedge F\phi^k)$.
There are several other cases in which one could carry out
analogous analyses.

Another line of research
is associated with the subtlety brought up
 in the introduction with regards to any amplitude that contains
both open string vertex operators and closed string vertex
operators. In this work, we have followed a step that
seems to be the commonly implemented in literature, which we are
getting to now. Closed string coordinates have two sets of
oscillators; let us denote them by $\alpha_n$ and
$\tilde{\alpha}_n$
collectively. For an amplitude that involves
both open string states and closed string states, the
computations are typically done in a path integral setup where the
Green function is determined using conformal field
theory techniques.

This is for a good reason. Once one attempts the computations
in the oscillator, what to do with the second
set of a closed string oscillators, $\tilde{\alpha}_n$ is not clear in the framework of the first
quantized string.
The only viable option seems to be to use one set of
oscillators as was commented below (3.4) in
\cite{Billo:2006jm}. In other words, the two sets of
the closed string oscillators are identified with each other,
and in turn identified with the open string oscillators as well.\footnote{The
reason stated in \cite{Billo:2006jm} for the identification
was the presence of fractional branes. However, the same
practice seems to be adopted by other groups in later related works where fractional
branes are not present.}
As far as we can see, a certain "analytic continuation" is
involved in the prescription of identifying the
second set of the closed string oscillators as the first set.
We believe that there is a room for better understanding and
systematic study of this step.
In effect, the identification makes the closed string state
a composite state of the open string fields.
 At the effective field theory level,
this means that the supergravity background fields that
are present in the DBI action should become
composite, namely, they must be functions of the SYM fields.\footnote{
To determine the forms of the functions, that is, the proper
background, it would be necessary to rely on the full open string
setup and to go beyond the tree level: it is expected that quantum effects would play an important role as proposed in \cite{Park:2007mc}.}
Those background fields can then be "Taylor-expanded" as
discussed in the Myers' work \cite{Myers:1999ps}.

\newpage

\renewcommand{\theequation}{A.\arabic{equation}}
 \setcounter{equation}{0}
  \section*{Appendix A: Conventions
  }

 Our index conventions are such that lowercase
Greek indices take values in the whole ten-dimensional spacetime, e.g.,
\begin{eqnarray}
\mu,\nu = 0, 1,..., 9
\end{eqnarray}
 Early
Latin indices run along the world-volume,
\begin{eqnarray}
 a, b, c = 0, 1,..., p
\end{eqnarray}
while middle Latin indices
represent the transverse space
\begin{eqnarray}
i,j = p + 1,...,9.
\end{eqnarray}
 Doubling trick is implemented according to
\begin{displaymath}
\tilde{X}^{\mu}(\bar{z}) \rightarrow D^{\mu}_{\nu}X^{\nu}(\bar{z}) \ ,
\ \ \
\tilde{\psi}^{\mu}(\bar{z}) \rightarrow
D^{\mu}_{\nu}\psi^{\nu}(\bar{z}) \ ,
\ \ \
\tilde{\phi}(\bar{z}) \rightarrow \phi(\bar{z})\,, \spa\mbox{and}\spa
\tilde{S}_{\alpha}(\bar{z}) \rightarrow M_{\alpha}{}^{\beta}{S}_{\beta}(\bar{z})
 ,
\end{displaymath}
where
\begin{displaymath}
D = \left( \begin{array}{cc}
-1_{9-p} & 0 \\
0 & 1_{p+1}
\end{array}
\right) \ ,\,\, \spa\mbox{and}\spa
M_p = \left\{\begin{array}{cc}\frac{\pm i}{(p+1)!}\gamma^{a_{1}}\gamma^{a_{2}}\ldots \gamma^{a_{p+1}}
\epsilon_{a_{1}\ldots a_{p+1}}\,\,\,\,{\rm for\, p \,even}\\ \frac{\pm 1}{(p+1)!}\gamma^{a_{1}}\gamma^{a_{2}}\ldots \gamma^{a_{p+1}}\gamma_{11}
\epsilon_{a_{1}\ldots a_{p+1}} \,\,\,\,{\rm for\, p \,odd}\end{array}\right.
\end{displaymath}
 The basic holomorphic correlators for the world-sheet fields $X^{\mu},\psi^{\mu}, \phi$ are
\begin{eqnarray}
\langle X^{\mu}(z)X^{\nu}(w)\rangle & = & -\frac{\alpha'}{2}\eta^{\mu\nu}\log(z-w) , \nonumber \\
\langle \psi^{\mu}(z)\psi^{\nu}(w) \rangle & = & -\frac{\alpha'}{2}\eta^{\mu\nu}(z-w)^{-1} \ ,\nonumber \\
\langle\phi(z)\phi(w)\rangle & = & -\frac{\alpha'}{2}\log(z-w) \ .
\labell{prop}\end{eqnarray}
For convenience, we introduce
\begin{eqnarray}
 x_{4}\equiv\ z=x+iy\quad,\quad x_{5}\equiv\bz=x-iy
\end{eqnarray}
SL(2,R) symmetry is fixed
by choosing the positions of the open string vertex operators
 \begin{eqnarray}
 x_{1}=0 ,\qquad x_{2}=1,\qquad x_{3}\rightarrow \infty,
 \qquad dx_1dx_2dx_3\rightarrow x_3^{2}.
\label{x123int34}
 \end{eqnarray}
 In section 2, one encounters the following integral
\begin{eqnarray}
 \int d^2 \!z |1-z|^{a} |z|^{b} (z - \bar{z})^{c}
(z + \bar{z})^{d},
 \end{eqnarray}
 where $d=0,1,2$. The region of integration is the upper half
of the complex plane. For $d=0,1$ the integral was evaluated  in \cite{Fotopoulos:2001pt}
and for $d=2$ in \cite{Hatefi:2010ik}.

 The Mandelstam variables are defined by
\begin{eqnarray*}
s&=&-\frac{\alpha'}{2}(k_1+k_3)^2,\qquad t=-\frac{\alpha'}{2}(k_1+k_2)^2,\qquad u=-\frac{\alpha'}{2}(k_2+k_3)^2.
\qquad\end{eqnarray*}
$T_{\alpha}$ denotes the $U(N)$ generators with normalization
\begin{eqnarray}
\mbox{\rm Tr}\,(T_{\alpha}T_{\beta})=N\delta_{\alpha\beta}
\end{eqnarray}

\renewcommand{\theequation}{B.\arabic{equation}}
 \setcounter{equation}{0}
  \section*{Appendix B: Analysis of $<V_{C}V_{\phi} V_A>$
  }

In the main text, we have analyzed $<V_{C}V_{\phi} V_AV_A>$.
Here we carry out a parallel analysis for the simpler case of $<V_{C}V_{\phi} V_A>$ and we find all infinite contact terms of this amplitude.
The S-matrix element
is given by the following correlation function
\begin{eqnarray}
{\cal A}^{C\phi A} & \sim & \int dx_{1}dx_{2}dzd\bar{z}\,
  \langle V_{\phi}^{(0)}{(x_{1})}
V_{A}^{(-1)}{(x_{2})}
V_{C}^{(-\frac{1}{2},-\frac{1}{2})}(z,\bar{z})\rangle,\labell{cpastring}\end{eqnarray}
Substituting the vertex operators, \reef{cpastring} can be written as
\begin{eqnarray} {\cal A}^{C\phi A}&\sim& \int
 dx_{1}dx_{2}dx_{4} dx_{5}\,
(P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}\xi_{1i}\xi_{2a}x_{45}^{-1/4}(x_{24}x_{25})^{-1/2}\nonumber\\&&
\times(I_1+I_2)\,\labell{cpa125}\end{eqnarray}
with
\begin{eqnarray}
I_1&=&{<:\partial X^i(x_1)e^{\alpha'ik_1.X(x_1)}:e^{\alpha'ik_2.X(x_2)}
:e^{i\frac{\alpha'}{2}p.X(x_4)}:e^{i\frac{\alpha'}{2}p.D.X(x_5)}:>}
 \  \nonumber \\&&\times{<:S_{\alpha}(x_4):S_{\beta}(x_5):\psi^a(x_2):>},\nonumber\\
I_2&=&{<:e^{\alpha'ik_1.X(x_1)}:e^{\alpha'ik_2.X(x_2)}
:e^{i\frac{\alpha'}{2}p.X(x_4)}:e^{i\frac{\alpha'}{2}p.D.X(x_5)}:>}
 \  \nonumber \\&&\times{<:S_{\alpha}(x_4):S_{\beta}(x_5):\alpha'ik_{1b}\psi^{b}\psi^{i}(x_1):\psi^a(x_2):>}.
\label{cpp1234}
\end{eqnarray}
Using Wick's theorem, one can show that
\begin{eqnarray}
I_1^a&=&<:S_{\alpha}(x_4):S_{\beta}(x_5):\psi^a(x_2):>=2^{-1/2}x_{45}^{-3/4}(x_{24}x_{25})^{-1/2}
(\gamma^{a}C^{-1})_{\alpha\beta}.\label{1opa2}
\end{eqnarray}
 The Wick-like rule \cite{Liu:2001qa,Kostelecky:1986xg} has been generalized to find the correlation function of two spin operators and some number of currents \cite{Hatefi:2010ik,Hatefi:2012wj}.

 Using these results, one can obtain the correlation function between two spin operators, one current and one world-sheet fermion as follows
\begin{eqnarray}
I_2^{aib}&=&<:S_{\alpha}(x_4):S_{\beta}(x_5):\psi^b\psi^i(x_1)::\psi^a(x_2)>\nonumber\\
&=&\bigg\{(\Gamma^{aib}C^{-1})_{\alpha\beta}
+\frac{\alpha' Re[x_{14}x_{25}]}{x_{12}x_{45}}\bigg(\eta^{ab}(\gamma^{i}C^{-1})_{\alpha\beta}\bigg)\bigg\}
\nonumber\\&&\times2^{-3/2}x_{45}^{1/4}(x_{14}x_{15})^{-1}(x_{24}x_{25})^{-1/2}.
\label{6cpp8}\end{eqnarray}
Replacing the above spin correlators in the amplitude and performing the correlators over $X$, one finds:
\begin{eqnarray}
{\cal A}^{C\phi A}&\!\!\!\!\sim\!\!\!\!\!&\int dx_{1}dx_{2}dx_{4}dx_{5}(P_{-}H\!\!\!\!/\,_{(n)}M_p)^{\alpha\beta}I\xi_{1i}\xi_{2a}x_{45}^{-1/4}
(x_{24}x_{25})^{-1/2}\nonumber\\&&\times\bigg(I_1^a(a^i_1)+i\alpha' k_{1b}I_2^{aib}\bigg)\labell{amp3cpa},\end{eqnarray}
where
\begin{eqnarray}
I&=&|x_{12}|^{ \alpha'^2k_1.k_2}|x_{14}x_{15}|^{\frac{\alpha'^2}{2} k_1.p}|x_{24}x_{25}|^{\frac{\alpha'^2}{2} k_2.p}
|x_{45}|^{\frac{\alpha'^2}{4}p.D.p},\nonumber\\
a^i_1&=&ip^{i}\frac{x_{54}}{x_{14}x_{15}}.
\label{icpa}
\end{eqnarray}
The amplitude has $SL(2,R)$ invariance; let us gauge fix it as
\begin{eqnarray}
(x_1,x_2,x_4,x_5)&=&(x,-x,i,-i) ,dx_{1}dx_{2}dx_{4}dx_{5}=-2i(1+x^2)dx
\end{eqnarray}
With this gauge fixing, the amplitude takes
\begin{eqnarray} {\cal A}^{C\phi A}&=&\int_{-\infty}^{\infty} dx (x^2+1)^{2t-1} (2x)^{-2t} (2\xi_{1i} \xi_{2a} 2^{-1/2})\nonumber\\&&\times
\bigg[-p^i\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\gamma^{a})
+k_{1b}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\Gamma^{bai})\bigg]
\labell{11lcpa}\end{eqnarray}
Note that the term $\frac{\alpha' Re[x_{14}x_{25}]}{x_{12}x_{45}}$ does not have any contribution to the amplitude because the integrand is odd
but we have to take integration over the whole space time.

Having found the integral, the final result for the amplitude is given by
\begin{eqnarray} {\cal A}^{C\phi A}&=& (2\xi_{1i} \xi_{2a} 2^{-1/2}\pi^{1/2})\frac{\Gamma(-t+\frac{1}{2})}{\Gamma(-t+1)}\nonumber\\&&\times
\bigg[-p^i\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\gamma^{a})
+k_{1b}\mbox{\rm Tr}\,(P_{-}H\!\!\!\!/\,_{(n)}M_p\Gamma^{bai})\bigg]
\labell{11lcpa}\end{eqnarray}
where $t=-(k_1+k_2)^2$ and we used the momentum conservation as well. The expansion is low energy expansion which can be
achieved by sending the Mandelstam variable $t$ to zero which means that we took  the $\alpha'\rightarrow 0$ limit of the string amplitude.
 One can then understand that the amplitude is non zero for $n=p$.
Thus it is clear from the  gamma function  that  the scattering amplitude has just infinite contact interactions.
The expansion of the over all factor for the above amplitude is as follows
\begin{eqnarray}
\sqrt{\pi}\frac{\Gamma(-t+\frac{1}{2})}{\Gamma(-t+1)}&=&\pi \sum_{n=-1}^{\infty} c_n(t)^{n+1}\end{eqnarray}
where some coefficients may be written down as
\begin{eqnarray}
c_{-1}=1,\quad c_{0}=2ln(2),\quad c_{1}=\frac{\pi^2}{6}+2ln(2)^2
\end{eqnarray}
To begin we focus on the Chern-Simons action (see eq. (3) of \cite{Hatefi:2012wj}).
The  interactions  will include a bulk RR field ($p$-1)-form potential , one scalar field and one gauge field in the world-volume of brane.
Let us extract the first term in  \reef{11lcpa}
\begin{eqnarray} {\cal A}_{1}^{C\phi A}&=& (2^{-1/2}\pi\mu_{p})(2\xi_{1i} \xi_{2a} 2^{-1/2}\pi) \sum_{n=-1}^{\infty} c_n(t)^{n+1}\nonumber\\&&\times
\bigg(\frac{32}{2p!}p^i\bigg)(\veps^v)^{a_0\cdots a_{p-1}a}H^{(p)}_{a_0\cdots a_{p-1}}
\labell{11lcpa2}\end{eqnarray}
where $(2^{-1/2}\pi\mu_{p})$ is the normalization factor. The leading contact interactions can be produced by the following
coupling
\begin{eqnarray}
S^{(3)}
&=& \frac{\lambda^2\mu_p}{(p)!}\int d^{p+1}\sigma (\veps^v)^{a_0\cdots a_{p}}
 p\mbox{\rm Tr}\,(F_{a_{0}a_{1}}\phi^i)
\prt_{i} C^{(p-1)}_{a_2\cdots a_{p}}(\sigma)
\nonumber\end{eqnarray}
The scalar comes from the Taylor expansion and field strength comes from Chern-Simons action (for more details see introduction and section 5 of \cite{Hatefi:2012wj}). Note with an integration by parts,
one can re express the above coupling as
\begin{eqnarray}
S^{(3)}
&=& \frac{\lambda^2\mu_p}{(p)!}\int d^{p+1}\sigma (\veps^v)^{a_0\cdots a_{p}}
\mbox{\rm Tr}\,(A_{a_{0}}\phi^i)
p^i  H^{(p)}_{a_1\cdots a_{p}}(\sigma)
\nonumber\end{eqnarray}
Having normalized the amplitude,  we are able to produce all infinite contact interactions for the first term of
this amplitude \reef{11lcpa} by the following field theory vertices
\begin{eqnarray}
S^{(3)}
&=& \frac{\lambda^2\mu_p}{(p)!}\int d^{p+1}\sigma (\veps^v)^{a_0\cdots a_{p}}
\sum_{n=0}^{\infty}c_{n}(\alpha')^{n+1}\mbox{\rm Tr}\,(\prt^{a_{1}}...\prt^{a_{n+1}} A_{a_{0}}\prt_{a_{1}}...\prt_{a_{n+1}}\phi^i)
p^i  H^{(p)}_{a_1\cdots a_{p}}(\sigma)
\nonumber\end{eqnarray}
where $H^{(p)}=dC^{(p-1)}$. To confirm all infinite contact interactions of the second term of \reef{11lcpa}
 we have to take into account the following contact interactions  in the low energy effective action. Indeed
the leading low energy terms of the amplitude will be
reproduced by
\begin{eqnarray}
S^{(4)}
&=& \frac{\lambda^2\mu_p} {(p)!}\int d^{p+1}\sigma (\veps^v)^{a_0\cdots a_{p}}
\mbox{\rm Tr}\,(F_{a_{0}a_{1}}D_{a_{2}}\Phi^i)
C^{(p-1)}_{ia_3\cdots a_{p}}(\sigma)p(p-1)
\nonumber\end{eqnarray}
Taking integration by parts and considering the antisymmetric property of $(\veps^v)^{a_0\cdots a_{p}} $
we can show that the all infinite contact terms of the second term of \reef{11lcpa} are reproduced by the following contact interactions
\begin{eqnarray}
S^{(4)}
&=& \frac{\lambda^2\mu_p} {(p)!}\int d^{p+1}\sigma (\veps^v)^{a_0\cdots a_{p}} (p)H^{(p)}_{ia_2\cdots a_{p}}(\sigma)\nonumber\\&&\times
\sum_{n=-1}^{\infty}c_{n}(\alpha')^{n+1}\mbox{\rm Tr}\,(\prt_{a_{1}}...\prt_{a_{n+1}}A_{a_{0}}  \prt^{a_{1}}...\prt^{a_{n+1}} D_{a_{1}}\Phi^i)
\nonumber\end{eqnarray}

\vspace{.3in}
\section*{Acknowledgment}
E.H would like to thank K.S.Narain for very valuable discussions. He would also like to thank F.Quevedo, L.Alvarez-Gaume, N.Arkani-Hamed, G.Veneziano, I.Antoniadis and N.Lambert  for very helpful conversations.