Pongwit Srisangyingcharoen

# Relations between closed string amplitudes and mixed string amplitudes at
tree-level

Aphiwat Yuenyong<EMAIL_ADDRESS><EMAIL_ADDRESS>The Institute for
Fundamental Study, Naresuan University, Phitsanulok, 65000, Thailand

###### Abstract

This paper investigates the relationships between closed and mixed string
amplitudes at the tree level in string theory. Through the analytic
continuation of complex variables, we establish a factorization of closed
string amplitudes into those involving ($n-2$) open strings and a single
closed string. Expressions for four-, five-, and six-point amplitudes are
provided, along with systematic formulations for $n$ strings. The paper
addresses possible correction terms arising from integration along infinite
tubes during the Wick rotation process. In the field theory limit, the
correction terms become negligible due to being of subleading order.

###### keywords:

string theory, string scattering amplitudes, closed string amplitudes, mixed
string amplitudes

##  1 Introduction

String theory is often considered a candidate for the theory that describes
all known interactions including gravity. Despite being debatable due to its
peculiar features such as extradimensions and supersymmetries, the study of
string theory is still useful, particularly in the area of scattering
amplitudes. It has been established that in the limit of low energies,
amplitudes in string theory reproduce those in QFT such as Yang-Mills [1] and
Einstein theory [2, 3] plus $\alpha^{\prime}$ corrections [4, 5, 6].
Understanding structures of string amplitudes does not only deepens our
understanding of string theory but also contributes to advancements in field
theories. A great example is a striking discovery made by Kawai, Lewellen and
Tyle (KLT) which provides a relation between closed and open string amplitudes
at tree level [7]. Also, see [8] for a review. The relation relates closed
string amplitudes in terms of products of two open string amplitudes which
provide alternative descriptions of gravity as the square of gauge theory in
the low energy limit [9, 10, 11, 12].

Another interesting structure is known as monodromy relation which relates
partial open string amplitudes all together. This allow us to reduce the
number of independent color-ordered amplitudes from $(n-1)!$ down to $(n-3)!$.
A chosen set of $(n-3)!$ open string amplitudes are called minimal basis where
the remaining amplitudes can be written in terms of this set of basis [13]. In
field theory limit, $\alpha^{\prime}$ $\to$ 0, the monodromy relations reduce
to the BCJ relations of Bern, Carrasco and Johansson [10] and the Kleiss-Kuijf
relation[14]. The monodromy relations among partial open string amplitudes can
be captured by polygons in the complex plane [15].

String amplitudes are described by the correlation functions evaluated on the
Riemann surfaces on which the string states are introduced as local operators
called vertex operators. At tree level, closed string states are inserted on a
sphere which can be conformally mapped to the full complex plane while open
string states are placed along the boundary of a disk or the upper half-plane.
In this paper, we seek to find connections between closed string amplitudes
and the amplitudes that involve scattering between both closed and open
strings known as mixed string amplitudes. The general expression of $n$-point
closed string amplitudes at tree-level are given as the integral [16]

$\displaystyle\mathcal{A}_{n}^{\text{cl}}=C_{S^{2}}(2\pi)^{D}\delta^{D}(\sum_{i}k_{i})\int\frac{\absolutevalue{z_{ab}z_{ac}z_{bc}}^{2}}{dz_{a}dz_{b}dz_{c}}\prod_{i=1}^{n}d^{2}z_{i}\prod_{1\leq
j<l\leq n}\absolutevalue{z_{j}-z_{l}}^{\alpha^{\prime}k_{j}\cdot
k_{l}}F_{n}(z_{i},\bar{z}_{i}),$ (1)

where $C_{S^{2}}$ is a normalization constant and $z_{ij}$ = $z_{i}-z_{j}$.
The function $F_{n}$ is a branch-free function that contains polarization and
kinematic factors of the external closed string states. According to conformal
symmetry, the points $z_{a},z_{b}$ and $z_{c}$ are freely fixed to arbitrary
points in the complex plane.

A mixed string amplitude contains both closed string and open string vertex
operators inserted on a disk worldsheet which can be mapped into the upper
half plane $H_{+}=\\{z\in\mathbb{C}|\text{Im}(z)\geq 0$}. Closed string vertex
operators are placed onto the bulk while those of open string are inserted
along the worldsheet boundary. Unlike the KLT relations, the mixed string
amplitudes involving $N_{c}$ closed strings and $N_{o}$ open strings can be
expressed as linear combinations of $N_{o}+2N_{c}$ color-ordered open string
amplitudes [17]. At low energies, This relates to Einstein-Yang-Mills
amplitudes as a linear combination of pure Yang-Mills amplitudes in the
collinear limit [18, 19].

In this paper, we will only focus on the mixed amplitudes of ($n-2$) open
strings and a single closed string scattering whose expression takes the form
[18]

$\displaystyle\mathcal{M}_{n}(1,\dots,n-2;k)=$ $\displaystyle
C_{D_{2}}(2\pi)^{D}\delta(\sum_{i=1}^{n-2}p_{i}+k)\int_{\mathcal{I}}\prod_{i=1}^{n-2}dx_{i}\prod_{1\leq
r<s\leq n-2}\absolutevalue{x_{r}-x_{s}}^{2\alpha^{\prime}p_{r}p_{s}}$
$\displaystyle\times\int_{H_{+}}d^{2}z|z-\bar{z}|^{\frac{1}{2}\alpha^{\prime}k\cdot
k}\prod_{i=1}^{n-2}\absolutevalue{x_{i}-z}^{2\alpha^{\prime}p_{i}k}K_{n}(x_{i},z_{i},\bar{z}_{i})$
(2)

where $C_{D_{2}}$ is a normalization constant and the function $K_{n}$
contains polarization and kinematic factor of the external states. Notice that
we used letters $p_{i}$ and $k$ to denote the momentum of open strings and a
closed string respectively. The open string coordinates $x_{i}$ obey the
ordering of the integration region
$\mathcal{I}=\\{x_{i}\in\mathbb{R}|x_{1}<x_{2}<\ldots<x_{n-2}\\}$. The partial
amplitudes are associated with a group factor $\Tr{T^{1}T^{2}\dots T^{n-2}}$
with $T^{a}$ being a Chan-Paton factor. This is because the open strings are
subject to boundary conditions. The introduction of Chan-Paton factors refers
to the fact that each endpoint of the open strings is confined to hyperplanes
called D-branes. The factors specify which branes the open strings are
attached to. For the sake of computations, we split the closed string momentum
$k$ into left- and right-moving spacetime momenta $q_{1}$ and $q_{2}$, i.e.
$k=q_{1}+q_{2}$, which are assumed to be unrelated at first. Therefore, we
introduce an integral of the form

$\displaystyle\mathcal{F}_{n}(1,\dots,n-2;q_{1},q_{2})$
$\displaystyle=C_{D_{2}}\int_{\mathcal{I}}\prod_{i=1}^{n-2}dx_{i}\prod_{1\leq
r<s\leq n-2}\absolutevalue{x_{r}-x_{s}}^{2\alpha^{\prime}p_{r}p_{s}+n_{rs}}$
$\displaystyle\times\int_{H_{+}}d^{2}z(z-\bar{z})^{2\alpha^{\prime}q_{1}q_{2}+n}\prod_{i=1}^{n-2}(x_{i}-z)^{2\alpha^{\prime}p_{i}q_{1}+n_{i}}(x_{i}-\bar{z})^{2\alpha^{\prime}p_{i}q_{2}+\bar{n}_{i}}.$
(3)

The integers $n_{rs}$, $n_{i}$, $\bar{n}_{i}$, and $n$ are determined by
external states. Notice that when $q_{1}$ = $q_{2}$ = $k/2$, the integral (3)
reproduces the mixed string amplitude (2) when the contents of external states
are identified.

In this paper, we establish a connection between closed and mixed string
amplitudes at the tree level. The organization of the paper is as follows: In
section two, we establish the relation between closed string and mixed string
amplitudes, ranging from 4-point to 6-point ones. Then, a general expression
for $n$-point amplitudes is presented in section three. Section 4 is dedicated
to a discussion of correction terms pertinent to the results presented in the
previous section. Finally, in Section 5, we conclude from our findings,
summarizing the key aspects of our work.

##  2 Factorization of closed string amplitudes

In this section, we would like to formulate expressions for tree-level closed
string amplitudes as products of mixed string amplitudes involving open
strings and a single closed string. The procedure uses the analytic
continuation of complex variables to factorize the closed string amplitudes
into the product of those of mixed strings. The calculations in this section
will be executed up to the six-point string amplitudes just for the readers to
be able to observe patterns of the relations before we generalize them to
include arbitrary $n$ strings in the next section. For simplicity, we begin by
discussing the relations that arise at four points.

###  2.1 The four-point amplitudes

According to the $PSL(2,\mathbb{C})$ symmetry, we choose $z_{2}=i$, $z_{3}=-i$
and $z_{4}=\infty$. The closed string amplitude takes the form

$\displaystyle\mathcal{A}_{4}^{\text{cl}}=4C_{S^{2}}\int
d^{2}z\absolutevalue{z-i}^{2s_{12}}\absolutevalue{z+i}^{2s_{13}}\absolutevalue{2i}^{2s_{23}}F_{4}(z,\bar{z})$
(4)

where $s_{ij}=\frac{\alpha^{\prime}}{2}k_{i}\cdot k_{j}$. Note that the factor
$(2\pi)^{D}\delta^{D}(\sum_{i}k_{i})$ has been dropped for convenience.
Without loss of generality, we can decompose $F_{4}$ =
$f_{4}(z)\bar{f}_{4}(\bar{z})$. By writing $z$ = $x+iy$ and then rotating the
contour integral of the variable $y$ from the real axis to the imaginary axis

$\displaystyle y\to iy,$ (5)

the integrands transform as follows:

$\displaystyle|z-i|^{2s_{12}}$
$\displaystyle\rightarrow(x-y-i)^{s_{12}}(x-y+i)^{s_{12}},$ (6)
$\displaystyle|z+i|^{2s_{13}}$
$\displaystyle\rightarrow(x-y-i)^{s_{13}}(x-y+i)^{s_{13}}.$ (7)

We then define new variables

$\xi=x+y\hskip 28.45274pt\text{and}\hskip 28.45274pt\eta=x-y.$ (8)

Accordingly, the amplitude becomes

$\displaystyle\mathcal{A}_{4}^{\text{cl}}=4C_{S^{2}}\int
d\xi(\xi+i)^{s_{12}}(\xi-i)^{s_{13}}(-2i)^{s_{23}}\bar{f}_{4}(\xi)\int
d\eta(\eta-i)^{s_{12}}(\eta_{1}+i)^{s_{13}}(2i)^{s_{23}}f_{4}(\eta).$ (9)

The integrals are exactly those of mixed string amplitudes $\mathcal{F}_{4}$.
If we set $C_{S^{2}}=(C_{D_{2}})^{2}$ and identify the functions $f_{4}$ and
$\bar{f}_{4}$ as the external-state dependent function $K_{4}$, we obtain

$\displaystyle\mathcal{A}_{4}^{\text{cl}}=\mathcal{F}_{4}(1,4;2,3)\widetilde{\mathcal{F}}_{4}(1,4;3,2)$
(10)

providing an expression for four-point closed string amplitudes as a product
of mixed string amplitudes in the limit $k_{2}=k_{3}$, i.e.

$\displaystyle\mathcal{A}_{4}^{\text{cl}}\Big{|}_{k_{2}=k_{3}=k}=\mathcal{M}_{4}(1,4;k)\mathcal{M}_{4}(1,4;k).$
(11)

###  2.2 The five-point amplitudes

Let’s add one more external state into consideration. For a scattering of five
strings, when the points $z_{3}$, $z_{4}$ and $z_{5}$ are fixed to the points
$i$, $-i$ and $\infty$ respectively in the complex plane, the five-point
closed string amplitude reads

$\displaystyle\mathcal{A}^{\text{c}l}_{5}=4C_{S^{2}}\int d^{2}z_{1}d^{2}z_{2}$
$\displaystyle\absolutevalue{z_{1}-i}^{2s_{13}}\absolutevalue{z_{1}+i}^{2s_{14}}\absolutevalue{z_{2}-i}^{2s_{23}}\absolutevalue{z_{2}+i}^{2s_{24}}\absolutevalue{2i}^{2s_{34}}$
$\displaystyle\times\absolutevalue{z_{1}-z_{2}}^{2s_{12}}F_{5}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2}).$
(12)

We proceed with the same procedure as in the four-point case by rewriting
$z_{j}=x_{j}+iy_{j}$ for $j=1,2$ and then the variables $y_{j}$ are
analytically continued to complex variables which allows us to transform

$\displaystyle y_{j}\to ie^{-i\epsilon}y_{j}\approx iy_{j}+\epsilon y_{j}.$
(13)

Notice that we rotate the variables $y_{j}$ from the real axis into almost the
imaginary axis to avoid the possible branch points of
$\absolutevalue{z_{1}-z_{2}}^{2s_{12}}$ after the rotations. This changes the
integrand

$\displaystyle\absolutevalue{z_{j}-i}^{2s_{j3}}\rightarrow\quad$
$\displaystyle(x_{j}-y_{j}-i+i\epsilon y_{j})^{s_{j3}}(x_{j}+y_{j}+i-i\epsilon
y_{j})^{s_{j3}}$
$\displaystyle\absolutevalue{z_{j}+i}^{2s_{j4}}\rightarrow\quad$
$\displaystyle(x_{j}-y_{j}+i+i\epsilon y_{j})^{s_{j4}}(x_{j}+y_{j}-i-i\epsilon
y_{j})^{s_{j4}}$
$\displaystyle\absolutevalue{z_{1}-z_{2}}^{2s_{12}}\rightarrow\quad$
$\displaystyle((x_{1}-y_{1})-(x_{2}-y_{2})+i\epsilon(y_{1}-y_{2}))^{s_{12}}$
$\displaystyle\times((x_{1}+y_{1})-(x_{2}+y_{2})-i\epsilon(y_{1}-y_{2}))^{s_{12}}$
(14)

for $j=1,2$. If we introduce new variables

$\xi_{j}=x_{j}+y_{j}\hskip 28.45274pt\text{and}\hskip
28.45274pt\eta_{j}=x_{j}-y_{j},$ (15)

the integral (12) can be written as

$\displaystyle\mathcal{A}^{\text{cl}}_{5}=$ $\displaystyle 4C_{S^{2}}\int
d\xi_{1}d\eta_{1}d\xi_{2}d\eta_{2}(\eta_{1}-i)^{s_{13}}(\eta_{1}+i)^{s_{14}}(\eta_{2}-i)^{s_{23}}(\eta_{2}+i)^{s_{24}}$
$\displaystyle\times(\xi_{1}+i)^{s_{13}}(\xi_{1}-i)^{s_{14}}(\xi_{2}+i)^{s_{23}}(\xi_{2}-i)^{s_{24}}\absolutevalue{2i}^{2s_{34}}$
$\displaystyle\times(\eta_{1}-\eta_{2}+i\epsilon\delta)^{s_{12}}(\xi_{1}-\xi_{2}-i\epsilon\delta)^{s_{12}}f_{5}(\eta_{1},\eta_{2})\bar{f}_{5}(\xi_{1},\xi_{2})$
(16)

where $\delta=y_{1}-y_{2}$ and
$F_{5}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})=f_{5}(z_{1},z_{2})\bar{f}_{5}(\bar{z}_{1}\bar{z}_{2})$.
The above expression resembles the integral of mixed string amplitudes
$\mathcal{F}_{5}$ albeit a possible phase factor correction due to the
intertwining of integrating variables $\xi_{j}$ and $\eta_{j}$ of the
integrand.

The factor $i\epsilon\delta$ determines deformations of the contours around
the branch points, i.e. the points when $\eta_{1}\sim\eta_{2}$ and
$\xi_{1}\sim\xi_{2}$. One needs to be careful when approaching these points.
Let’s investigate the behavior of $i\epsilon\delta$ near the branch points.
When $\eta_{1}$ approaches $\eta_{2}$, $i\epsilon\delta\sim
i\frac{\epsilon}{2}(\xi_{1}-\xi_{2})$. This implies that the position of the
branch point depends on the sign of $\xi_{1}-\xi_{2}$. In case
$\xi_{1}>\xi_{2}$, the branch point is located slightly below the real line.
On the other hand, if $\xi_{1}<\xi_{2}$, the branch point is slightly above
the real line. Consequently, to avoid the branch point, one can slightly shift
the $\eta_{1}$-contour above the real axis when $\xi_{1}>\xi_{2}$ and slightly
shift the contour down for otherwise. The contours of $\eta_{1}$ are depicted
in figure 1 and 1.

A similar pattern also appears for the term
$(\xi_{1}-\xi_{2}-i\epsilon\delta)^{s_{12}}$ when $\xi_{1}$ approaches
$\xi_{2}$. The factor $i\epsilon\delta$ is positive when $\eta_{1}<\eta_{2}$
and is negative when $\eta_{1}>\eta_{2}$. This leads to a slight shift in the
branch point at $\xi_{1}\sim\xi_{2}$ to be above and below the real axis of
$\xi_{1}$ for the case $\eta_{1}>\eta_{2}$ and $\eta_{1}<\eta_{2}$
respectively. As a result, we obtain the $\xi_{1}$-contours for both cases
shown in figure 1 and 1.

$\eta_{1}$$\eta_{2}$$(a)$

$\eta_{1}$$\eta_{2}$$(b)$

$\xi_{1}$$\xi_{2}$$(d)$

$\xi_{1}$$\xi_{2}$$(c)$

Figure 1: Contours of integration for the variable $\eta_{1}$ in the cases
when (a) $\xi_{1}>\xi_{2}$ and (b) $\xi_{1}<\xi_{2}$, and contours of
integration for the variable $\xi_{1}$ for the cases where (c)
$\eta_{1}>\eta_{2}$ and (d) $\eta_{1}<\eta_{2}$.

To proceed, we separate the integral (16) into four terms by inserting the
identity

$1=(\Theta(\xi_{1}-\xi_{2})+\Theta(\xi_{2}-\xi_{1}))(\Theta(\eta_{1}-\eta_{2})+\Theta(\eta_{2}-\eta_{1}))$
(17)

giving four different integral regions among variables. $\Theta(x-y)$ is a
Heaviside step function. It turns out that we can relate the integral in each
term to a product of $\mathcal{F}_{5}$. However, one needs to make sure that
the integrand matches with the expression (3). For this reason, the following
relations are useful to correct the integrand:

$(\eta_{1}-\eta_{2})^{c}=\begin{cases}e^{i\pi c}(\eta_{2}-\eta_{1})^{c},\hskip
19.91684pt\xi_{1}>\xi_{2}\\\ e^{-i\pi c}(\eta_{2}-\eta_{1})^{c},\hskip
14.22636pt\xi_{1}<\xi_{2}\end{cases}$ (18)

when $\text{Re}(\eta_{1})<\text{Re}(\eta_{2})$ and

$(\xi_{1}-\xi_{2})^{c}=\begin{cases}e^{i\pi c}(\xi_{2}-\xi_{1})^{c},\hskip
19.91684pt\eta_{1}>\eta_{2}.\\\ e^{-i\pi c}(\xi_{2}-\xi_{1})^{c},\hskip
14.22636pt\eta_{1}<\eta_{2}\end{cases}$ (19)

when $\text{Re}(\xi_{1})<\text{Re}(\xi_{2})$.

Using the relations (18) and (19), one obtains

$\displaystyle\mathcal{A}^{\text{cl}}_{5}=$ $\displaystyle e^{-i2\pi
s_{12}}\mathcal{F}_{5}(1,2,5;3,4)\widetilde{\mathcal{F}}_{5}(1,2,5;4,3)+e^{i\pi
s_{12}}\mathcal{F}_{5}(1,2,5;3,4)\widetilde{\mathcal{F}}_{5}(2,1,5;4,3)$
$\displaystyle+e^{i\pi
s_{12}}\mathcal{F}_{5}(2,1,5;3,4)\widetilde{\mathcal{F}}_{5}(1,2,5;4,3)+\mathcal{F}_{5}(2,1,5;3,4)\widetilde{\mathcal{F}}_{5}(2,1,5;4,3)$
(20)

if the constant $C_{S^{2}}=(C_{D_{2}})^{2}$ is fixed. When the
$k_{3}=k_{4}=k/2$, the integral $\mathcal{F}_{5}$ reduces to the corresponding
mixed string amplitude $\mathcal{M}_{5}$ with the closed string momentum $k$.

###  2.3 The six-point amplitudes

Before we generalize the relations to an arbitrary number of strings, we would
like to consider another non-trivial case regarding six-point string
amplitudes, allowing us to observe more patterns in the relations. The
expression for an amplitude involving six closed strings reads

$\displaystyle\mathcal{A}^{cl}_{6}=4C_{S^{2}}\int\prod_{j=1}^{3}d^{2}z_{j}\absolutevalue{z_{j}-i}^{2s_{j4}}\absolutevalue{z_{j}+i}^{2s_{j5}}\absolutevalue{2i}^{2s_{45}}F_{6}(z_{j},\bar{z}_{j})\prod_{1\leq
l<m\leq 3}\absolutevalue{z_{l}-z_{m}}^{2s_{lm}}$ (21)

where we choose $z_{4}=i,z_{5}=-i$ and $z_{6}=\infty$ to fix the
$PSL(2,\mathbb{C})$ symmetry. We follow the same steps as the previous
examples by writing $z_{j}=x_{j}+iy_{j}$ and then rotating $y_{j}$ using the
transformations (13). By introducing the new variables defined

$\xi_{j}=x_{j}+y_{j}\hskip 28.45274pt\text{and}\hskip
28.45274pt\eta_{j}=x_{j}-y_{j}$ (22)

while now $j$ runs from 1 to 3, the amplitude takes the form

$\displaystyle\mathcal{A}^{\text{cl}}_{6}=$ $\displaystyle
4C_{S^{2}}\int\prod_{j=1}^{3}d\eta_{j}d\xi_{j}(\eta_{j}-i)^{s_{j4}}(\eta_{j}+i)^{s_{j5}}(\xi_{j}+i)^{s_{j4}}(\xi_{j}-i)^{s_{j5}}|2i|^{2s_{45}}$
$\displaystyle\times f_{6}(\xi_{j})\bar{f}_{6}(\eta_{j})\prod_{1\leq l<m\leq
3}(\eta_{l}-\eta_{m}+i\epsilon\delta_{lm})^{s_{lm}}(\xi_{l}-\xi_{m}-i\epsilon\delta_{lm})^{s_{lm}}$
(23)

where $\delta_{ij}=y_{i}-y_{j}$.

Similar to the four- and five-point cases, the above expression can be
rewritten in terms of products of $\mathcal{F}_{6}$ with appropriate phase
factors. The appearance of $i\epsilon\delta_{lm}$ in the integrand dictates
how the contours of the integrating variables are deformed. According to the
term $(\eta_{l}-\eta_{m}+i\epsilon\delta_{lm})^{s_{lm}}$, the
$\eta_{l}$-contour is shifted slightly above the real axis when
$\xi_{l}>\xi_{m}$ and the contour slightly moves down from the real axis when
$\xi_{l}<\xi_{m}$. Likewise, the contour of $\xi_{l}$ regarding the term
$(\xi_{l}-\xi_{m}+i\epsilon\delta_{lm})^{s_{lm}}$ is translated upward from
the real axis if $\eta_{l}>\eta_{m}$ and is shifted downward from the real
axis if $\eta_{l}<\eta_{m}$. As the contours are deformed differently
depending on how the values of the integrating variables are relative to each
other, we will separate the integration domain into 36 pieces based on
comparative values of the integration variables by inserting

$1=\sum_{i\neq j\neq
k}^{3}\Theta(\xi_{i}-\xi_{j})\Theta\xi_{j}-\xi_{k})\sum_{l\neq m\neq
n}^{3}\Theta(\eta_{l}-\eta_{m})\Theta(\eta_{m}-\eta_{n})$ (24)

to the expression (23).

In each term, one can utilize the relations

$(\eta_{i}-\eta_{j})^{c}=\begin{cases}e^{i\pi c}(\eta_{j}-\eta_{i})^{c},\hskip
19.91684pt\xi_{i}>\xi_{j}\\\ e^{-i\pi c}(\eta_{j}-\eta_{i})^{c},\hskip
14.22636pt\xi_{i}<\xi_{j}\end{cases}$ (25)

when $\text{Re}(\eta_{i})<\text{Re}(\eta_{j})$, and

$(\xi_{i}-\xi_{j})^{c}=\begin{cases}e^{i\pi c}(\xi_{j}-\xi_{i})^{c},\hskip
19.91684pt\eta_{i}>\eta_{j}.\\\ e^{-i\pi c}(\xi_{j}-\xi_{i})^{c},\hskip
14.22636pt\eta_{i}<\eta_{j}\end{cases}$ (26)

when $\text{Re}(\xi_{i})<\text{Re}(\xi_{j})$ to make sure that the integrand
is of the correct form of $\mathcal{F}_{6}.$ After a careful examination, we
obtain the expression

$\mathcal{A}_{6}^{\text{cl}}=\sum_{\sigma,\sigma^{\prime}}e^{i\pi(\Lambda(\sigma,\sigma^{\prime})+\Omega(\sigma,\sigma^{\prime}))}\mathcal{F}_{6}(\sigma(1,2,3),6;4,5)\widetilde{\mathcal{F}}_{6}(\sigma^{\prime}(1,2,3),6;5,4).$
(27)

The expression is summed over the ordering $\sigma$ and $\sigma^{\prime}$
referring to the order of open string vertex operators in which the objects
inside the bracket are to be permuted. The functions
$\Lambda(\sigma,\sigma^{\prime})$ and $\Omega(\sigma,\sigma^{\prime})$ contain
kinematics variables depending on the ordering $\sigma$ and $\sigma^{\prime}$
respectively. The values of both functions are presented in table 1.

$\sigma$ | $\sigma^{\prime}$ | $\Lambda(\sigma,\sigma^{\prime})$ | $\Omega(\sigma,\sigma^{\prime})$
---|---|---|---
1,2,3 | 1,2,3 | $-s_{12}-s_{13}-s_{23}$ | $-s_{12}-s_{13}-s_{23}$
1,2,3 | 2,1,3 | $s_{12}-s_{13}-s_{23}$ | $-s_{13}-s_{23}$
1,2,3 | 2,3,1 | $s_{12}+s_{13}-s_{23}$ | $-s_{23}$
1,2,3 | 3,2,1 | $s_{12}+s_{13}+s_{23}$ | $0$
1,2,3 | 3,1,2 | $-s_{12}+s_{13}+s_{23}$ | $-s_{12}$
1,2,3 | 1,3,2 | $-s_{12}-s_{13}+s_{23}$ | $-s_{12}-s_{13}$
2,1,3 | 1,2,3 | $-s_{13}-s_{23}$ | $s_{12}-s_{13}-s_{23}$
2,1,3 | 2,1,3 | $-s_{13}-s_{23}$ | $-s_{13}-s_{23}$
2,1,3 | 2,3,1 | $s_{13}-s_{23}$ | $-s_{23}$
2,1,3 | 3,2,1 | $s_{13}+s_{23}$ | $0$
2,1,3 | 3,1,2 | $s_{13}+s_{23}$ | $s_{12}$
2,1,3 | 1,3,2 | $-s_{13}+s_{23}$ | $s_{12}-s_{13}$
2,3,1 | 1,2,3 | $-s_{23}$ | $s_{12}+s_{13}-s_{23}$
2,3,1 | 2,1,3 | $-s_{23}$ | $s_{13}-s_{23}$
2,3,1 | 2,3,1 | $-s_{23}$ | $-s_{23}$
2,3,1 | 3,2,1 | $s_{23}$ | $0$
2,3,1 | 3,1,2 | $s_{23}$ | $s_{12}$
2,3,1 | 1,3,2 | $s_{23}$ | $s_{12}+s_{13}$
3,2,1 | 1,2,3 | $0$ | $s_{12}+s_{13}+s_{23}$
3,2,1 | 2,1,3 | $0$ | $s_{13}+s_{23}$
3,2,1 | 2,3,1 | $0$ | $s_{23}$
3,2,1 | 3,2,1 | $0$ | $0$
3,2,1 | 3,1,2 | $0$ | $s_{12}$
3,2,1 | 1,3,2 | $0$ | $s_{12}+s_{13}$
3,1,2 | 1,2,3 | $-s_{12}$ | $-s_{12}+s_{13}+s_{23}$
3,1,2 | 2,1,3 | $s_{12}$ | $s_{13}+s_{23}$
3,1,2 | 2,3,1 | $s_{12}$ | $s_{23}$
3,1,2 | 3,2,1 | $s_{12}$ | $0$
3,1,2 | 3,1,2 | $-s_{12}$ | $-s_{12}$
3,1,2 | 1,3,2 | $-s_{12}$ | $-s_{12}+s_{13}$
1,3,2 | 1,2,3 | $-s_{12}-s_{13}$ | $-s_{12}-s_{13}+s_{23}$
1,3,2 | 2,1,3 | $s_{12}-s_{13}$ | $-s_{13}+s_{23}$
1,3,2 | 2,3,1 | $s_{12}+s_{13}$ | $s_{23}$
1,3,2 | 3,2,1 | $s_{12}+s_{13}$ | $0$
1,3,2 | 3,1,2 | $-s_{12}+s_{13}$ | $-s_{12}$
1,3,2 | 1,3,2 | $-s_{12}-s_{13}$ | $-s_{12}-s_{13}$
Table 1: Values of functions $\Lambda(\sigma,\sigma^{\prime})$ and
$\Omega(\sigma,\sigma^{\prime})$ corresponding to the ordering $\sigma$ and
$\sigma^{\prime}$.

##  3 Relations between closed and mixed string amplitudes

In this section, we would like to generalize the results formulated in the
previous section to include an arbitrary number of strings. To do that, we
repeat the same procedure used in the last section to rewrite the expressions
for $n$-point closed string amplitudes,

$\displaystyle\mathcal{A}^{\text{cl}}_{n}=$ $\displaystyle
4C_{S^{2}}\int\prod_{j=1}^{n-3}d^{2}z_{j}\absolutevalue{z_{j}-i}^{2s_{j,n-2}}\absolutevalue{z_{j}+i}^{2s_{j,n-1}}\absolutevalue{2i}^{2s_{n-2,n-1}}F_{n}(z_{j},\bar{z}_{j})$
$\displaystyle\prod_{1\leq l<m\leq n-3}\absolutevalue{z_{l}-z_{m}}^{2s_{lm}}$
(28)

with $z_{n-2}=i,z_{n-1}=-i$ and $z_{n}=\infty$, into products of mixed string
amplitudes. Again, we rewrite $z_{j}=x_{j}+iy_{j}$ and then transform the
variable $y_{j}$ to $ie^{-i\epsilon}y_{j}$. It yields

$\displaystyle\mathcal{A}^{\text{cl}}_{n}=$ $\displaystyle
4C_{S^{2}}\int\prod_{j=1}^{n-3}d\eta_{j}d\xi_{j}(\eta_{j}-i)^{s_{j,n-2}}(\eta_{j}+i)^{s_{j,n-1}}(\xi_{j}+i)^{s_{j,n-2}}(\xi_{j}-i)^{s_{j,n-1}}$
$\displaystyle f_{n}(\xi_{j})\bar{f}_{n}(\eta_{j})\prod_{1\leq l<m\leq
n-3}(\eta_{l}-\eta_{m}+i\epsilon\delta_{lm})^{s_{lm}}(\xi_{l}-\xi_{m}-i\epsilon\delta_{lm})^{s_{lm}}|2i|^{2s_{n-2,n-1}}$
(29)

where the variables $\xi_{j}$ and $\eta_{j}$ are defined in (22).

By a careful examination of branch cuts and contour deformations, we can write
the general expression for the closed string amplitudes as

$\displaystyle\mathcal{A}_{n}^{\text{cl}}=$
$\displaystyle\sum_{\sigma,\sigma^{\prime}}\mathcal{P}(\sigma(1,2,3,\dots,n-3)|\sigma^{\prime}(1,2,3,\dots,n-3))$
$\displaystyle\times\mathcal{F}_{n}(\sigma(1,2,3,\dots,n-3),n;n-2,n-1)$
$\displaystyle\times\widetilde{\mathcal{F}}_{n}(\sigma^{\prime}(1,2,3,\dots,n-3),n;n-1,n-2)$
(30)

with the help of the relations (25) and (26). This resembles the relation (27)
where the function $\mathcal{P}(\sigma|\sigma^{\prime})$ encapsulates all the
phase factors appearing when correcting the integrand of (29). We define the
function $\mathcal{P}(\sigma|\sigma^{\prime})$ by introducing two new
functions $\theta(i_{j},i_{k})$ and $\beta(i_{j},i_{k})$. The function
$\theta(i_{j},i_{k})$ is defined as

$\theta(i_{p},i_{q})=\begin{cases}1\hskip 14.22636pt;\hskip
14.22636pt(i_{p},i_{q})\hskip 2.84544pt\text{has the same ordering as
set}\hskip 2.84544ptI_{n}\\\ 0\hskip 14.22636pt;\hskip
14.22636pt(i_{p},i_{q})\hskip 2.84544pt\text{has the opposite ordering as
set}\hskip 2.84544ptI_{n}\\\ \end{cases}$ (31)

where the set $I_{n}$ are given by

$I_{n}=\\{1,2,3,\dots,n-3\\}.$ (32)

For example, $\theta(1,2)$ = 1, $\theta(2,1)$ = 0, $\theta(4,7)$ = 0, etc. The
function $\beta(i_{j},i_{k})$ is defined as

$\beta(i_{p},i_{q}|j_{r},j_{s})=\begin{cases}-1\hskip 14.22636pt;\hskip
14.22636pt(i_{p},i_{q})\hskip 2.84544pt\text{and}\hskip
2.84544pt(j_{r},j_{s})\hskip 2.84544pt\text{have the same ordering},\\\
+1\hskip 14.22636pt;\hskip 14.22636pt(i_{p},i_{q})\hskip
2.84544pt\text{and}\hskip 2.84544pt(j_{r},j_{s})\hskip 2.84544pt\text{have the
opposite ordering },\\\ 0\hskip 22.76228pt;\hskip
14.22636pt(i_{p},i_{q})\hskip 2.84544pt\text{and}\hskip
2.84544pt(j_{r},j_{s})\hskip 2.84544pt\text{are different. }\\\ \end{cases}$
(33)

For example, $\beta(1,2|1,2)$ = -1, $\beta(1,2|2,1)$ = +1, $\beta(1,2|1,3)$ =
0, etc. We then construct the function $\mathcal{P}(\sigma|\sigma^{\prime})$
to be

$\displaystyle\mathcal{P}(i_{1},i_{2},\dots,i_{k}|j_{1},j_{2},\dots,j_{k})=$
$\displaystyle\exp\Big{\\{}i\pi\big{(}\sum_{p<q}^{k}\sum_{r<s}^{k}\theta(i_{p},i_{q})\beta(i_{p},i_{q}|j_{r},j_{s})s_{i_{p}i_{q}}$
$\displaystyle+\theta(j_{r},j_{s})\beta(j_{r},j_{s}|i_{p},i_{q})s_{j_{r}j_{s}}\big{)}\Big{\\}}.$
(34)

The expression (30) contains $(n-3)!\times(n-3)!$ terms in the summation
regarding the permutations of $n-3$ objects inside the ordering $\sigma$ and
$\sigma^{\prime}$. In the case where $k_{n-2}=k_{n-1}=k/2$, the relation
reduces to

$\displaystyle\mathcal{A}_{n}^{\text{cl}}\Big{|}_{k_{n-2}=k_{n-1}=k/2}=$
$\displaystyle\sum_{\sigma,\sigma^{\prime}}\mathcal{P}(\sigma(1,2,3,\dots,n-3)|\sigma^{\prime}(1,2,3,\dots,n-3))$
$\displaystyle\times\mathcal{M}_{n}(\sigma(1,2,3,\dots,n-3),n;k)$
$\displaystyle\times\widetilde{\mathcal{M}}_{n}(\sigma^{\prime}(1,2,3,\dots,n-3),n;k).$
(35)

Remind that the polarizations of closed strings $\chi_{i}$ corresponding to
the physical states of $\mathcal{A}_{n}^{\text{cl}}$ are decomposed into open
string polarizations $\zeta_{i}$ and $\tilde{\zeta}_{i}$ regarding the
amplitudes $\mathcal{M}_{n}$ and $\widetilde{\mathcal{M}}_{n}$ respectively
through $\chi_{i}=\zeta_{i}\otimes\tilde{\zeta}_{i}$. However, the
polarizations of the closed string in the mixed amplitudes $\mathcal{M}_{n}$
and $\widetilde{\mathcal{M}}_{n}$ read $\zeta_{n-2}\otimes\zeta_{n-1}$ and
$\tilde{\zeta}_{n-2}\otimes\tilde{\zeta}_{n-1}$ respectively. More detail can
be found in appendix A.

##  4 Correction terms

Although we have formulated the relations between closed string amplitudes and
those of mixed string amplitudes presented in (30) and $(\ref{e30-1})$, the
calculations require Wick rotations of the variables $y_{j}$ via (13). This
step naively assumes no poles or branch points when implementing the contour
integral. In this section, we will take a good care of such possible points
that can exist in the bulk of the complex plane resulting in a possible
correction to the proposed relations. For simplicity, we will content
ourselves to consider only the four-point relation yet provide comments on the
higher-point cases at the end of the section.

$C$$T_{1}$$T_{2}$$T_{3}$$T_{4}$$ix$$-ix$1-1$\Im{y}$$\Re{y}$ Figure 2: Contour
integral in the complex $y$ plane.

It is clear that the expression of the four-point closed string amplitude (4)
contains the terms $|z-i|^{2s_{12}}$ and $|z+i|^{2s_{13}}$ which implies an
existence of branch points at $y=\pm(1\pm ix)$ inside the worldsheet. To
transform $y$ into an imaginary axis, one needs to make sure that the
$y$-contour avoids intersecting any cuts in the complex plane. To address
this, we define the contour of $y$ as shown in figure 2. Following this
contour, additional terms emerge from the integration around the infinite
tubes, namely paths $T_{1},T_{2},T_{3}$, and $T_{4}$. As there are no poles
within the contour, the variable $y$ undergoes transformation using the Cauchy
theorem:

$\displaystyle\Big{(}\int_{-\infty}^{\infty}+\int_{T_{1}+T_{2}}+\int_{T_{3}+T_{4}}+\int_{C}\Big{)}dy\
I(y)=0.$ (36)

Here, $I(y)$ represents an arbitrary analytic function. It is important to
note that we disregard contributions from the integral along infinite arcs,
assuming that the function $I(y)\rightarrow 0$ when $|y|\rightarrow\infty$
which holds true for our integrand. Therefore, the tree-level four-point
closed string amplitude becomes

$\displaystyle\mathcal{A}_{4}^{\text{cl}}=\mathcal{F}_{4}(1,4;2,3)\widetilde{\mathcal{F}}_{4}(1,4;3,2)+8iC_{S^{2}}\int_{-\infty}^{\infty}dx\Big{(}\int_{T_{1}+T_{2}}+\int_{T_{3}+T_{4}}\Big{)}dy\
\mathcal{I}(x,y)$ (37)

where

$\displaystyle\mathcal{I}(x,y)=(x+iy-i)^{s_{12}}(x-iy+i)^{s_{12}}(x+iy+i)^{s_{13}}(x-iy-i)^{s_{13}}|2i|^{2s_{23}}F_{4}(x,y).$
(38)

The first term of (37) emerges from the contour integration along the
imaginary line (path $C$) we already discussed.

Now let’s delve into the additional terms resulting from the integration along
the infinite tubes by investigating the following integrals:

$\displaystyle\int_{T_{1}+T_{2}}dy\
\mathcal{I}(x,y)\qquad\text{and}\qquad\int_{T_{3}+T_{4}}dy\ \mathcal{I}(x,y)$
(39)

where we will denote these integrals $A_{1}$ and $A_{2}$ respectively. For
convenience, we set $F_{4}=1$. To compute, we will examine the additional
integral in two cases: when $x>0$ and $x<0$.

$y^{\prime}$$1$$-1$$\widetilde{T}_{1}$$\widetilde{T}_{2}$

$\tilde{y}$$1$$-1$$\widetilde{T}_{3}$$\widetilde{T}_{4}$

Figure 3: Contour along the infinite tubes of (a) the variable $y^{\prime}$
and (b) the variable $\tilde{y}$.

For the first case $x>0$, by changing the variable $y$ to $y^{\prime}$ where

$y=y^{\prime}+ix,$ (40)

the integral $A_{1}$ becomes

$\displaystyle A_{1}\Big{|}_{x>0}=$ $\displaystyle
i^{(s_{12}+s_{13})}4^{s_{23}}\Big{(}\int_{\widetilde{T}_{1}}+\int_{\widetilde{T}_{2}}\Big{)}dy^{\prime}(y^{\prime}-1)^{s_{12}}(x-iy^{\prime}+x+i)^{s_{12}}$
$\displaystyle\times(y^{\prime}+1)^{s_{13}}(x-iy^{\prime}+x-i)^{s_{13}}.$ (41)

The contour paths $\widetilde{T}_{1}$ and $\widetilde{T}_{2}$ of the contour
$y^{\prime}$ are illustrated in figure 3. Using the relation

$(y^{\prime}-1)^{c}=\begin{cases}\absolutevalue{y^{\prime}-1}^{c}e^{2\pi
ic};\hskip 8.5359pt\text{for path }\widetilde{T}_{1}\\\
\absolutevalue{y^{\prime}-1}^{c};\hskip 28.45274pt\text{for path
}\widetilde{T}_{2},\end{cases}$ (42)

one can write

$\displaystyle A_{1}\Big{|}_{x>0}=$
$\displaystyle-(2i)i^{(s_{12}+s_{13})}4^{s_{23}}\sin{(\pi s_{12})}e^{i\pi
s_{12}}\int^{\infty}_{1}dy^{\prime}\absolutevalue{y^{\prime}-1}^{s_{12}}\absolutevalue{y^{\prime}+1}^{s_{13}}$
$\displaystyle\times(x-iy^{\prime}+x+i)^{s_{12}}(x-iy^{\prime}+x-i)^{s_{13}}.$
(43)

Likewise, one can compute $A_{2}$ for the case $x>0$ to be

$\displaystyle A_{2}\Big{|}_{x>0}=$ $\displaystyle
i^{(s_{12}+s_{13})}4^{s_{23}}\Big{(}\int_{\widetilde{T}_{3}}+\int_{\widetilde{T}_{4}}\Big{)}d\tilde{y}(\tilde{y}-1)^{s_{13}}(x-i\tilde{y}+x+i)^{s_{13}}$
$\displaystyle\times(\tilde{y}+1)^{s_{12}}(x-i\tilde{y}+x-i)^{s_{12}}$ (44)

where we define

$y=-\tilde{y}-ix.$ (45)

The figure 3 illustrates paths $\widetilde{T}_{3}$ and $\widetilde{T}_{4}$ of
the $\tilde{y}$-contour along the infinite tube. Again, we then apply the
relation

$(\tilde{y}-1)^{c}=\begin{cases}\absolutevalue{\tilde{y}-1}^{c}e^{2\pi
ic};\hskip 8.5359pt\text{for path }\widetilde{T}_{3}\\\
\absolutevalue{\tilde{y}-1}^{c};\hskip 28.45274pt\text{for path
}\widetilde{T}_{4}\end{cases}$ (46)

to rewrite $A_{2}$ in the form

$\displaystyle A_{2}\Big{|}_{x>0}=$
$\displaystyle-(2i)i^{(s_{12}+s_{13})}4^{s_{23}}\sin{(\pi s_{13})}e^{i\pi
s_{13}}\int^{\infty}_{1}d\tilde{y}\absolutevalue{\tilde{y}-1}^{s_{13}}\absolutevalue{\tilde{y}+1}^{s_{12}}$
$\displaystyle\times(x-i\tilde{y}+x+i)^{s_{13}}(x-i\tilde{y}+x-i)^{s_{12}}.$
(47)

Without much effort, one can compute expressions for $A_{1}$ and $A_{2}$ for
the latter case where $x<0$ and find out that

$\int_{0}^{\infty}dx\Big{(}A_{1}\Big{|}_{x>0}+A_{2}\Big{|}_{x>0}\Big{)}=\int_{-\infty}^{0}dx\Big{(}A_{1}\Big{|}_{x<0}+A_{2}\Big{|}_{x<0}\Big{)}.$
(48)

At last, the four-point closed string amplitude (37) takes the form

$\displaystyle\mathcal{A}_{4}^{\text{cl}}=\mathcal{F}_{4}(1,4;2,3)\widetilde{\mathcal{F}}_{4}(1,4;3,2)+32C_{S^{2}}i^{(s_{12}+s_{13})}(2^{s_{23}})\Big{(}\mathcal{B}(s_{12},s_{13})+\mathcal{B}(s_{13},s_{12})\Big{)}$
(49)

where

$\displaystyle\mathcal{B}(s_{12},s_{13})=$ $\displaystyle\sin{(\pi
s_{12})}e^{i\pi
s_{12}}\int_{0}^{\infty}dx\int^{\infty}_{1}dy^{\prime}\absolutevalue{y-1}^{s_{12}}\absolutevalue{y+1}^{s_{13}}$
$\displaystyle\times(x-iy+x+i)^{s_{12}}(x-iy+x-i)^{s_{13}}2^{s_{23}}.$ (50)

Remember that the order of the arguments of $\mathcal{B}(x,y)$ matters, i.e.
$\mathcal{B}(x,y)\neq\mathcal{B}(y,x)$.

$\Re{x}$$\Im{x}$$i$$iy$

$x$$1$$y$$-1$$L$$L^{*}$

Figure 4: (a) Contour of integration providing a Wick rotation of variable $x$
and (b) contour path of the variable $x$ after the rotation

Furthermore, we noticed that the integral $\mathcal{B}(x,y)$ can be related to
color-ordered open string amplitudes. This can be done by implementing
binomial expansions to the expression (50), giving

$\displaystyle\mathcal{B}(s_{12},s_{13})=$ $\displaystyle\sin{(\pi
s_{12})}e^{i\pi
s_{12}}\sum_{a,b=0}^{\infty}\binom{s_{12}}{a}\binom{s_{13}}{b}\int_{1}^{\infty}dy\
\absolutevalue{y-1}^{s_{12}}\absolutevalue{y+1}^{s_{13}}$
$\displaystyle\times\int_{0}^{\infty}dx\
(x+i)^{s_{12}-a}(x-iy)^{a+b}(x-i)^{s_{13}-b}2^{s_{23}}.$ (51)

If we sent $x$ $\to$ $ie^{-i\epsilon}x$, $\mathcal{B}(s_{12},s_{13})$ becomes

$\displaystyle\mathcal{B}(s_{12},s_{13})=$ $\displaystyle i\sin{(\pi
s_{12})}e^{i\frac{\pi}{2}(3s_{12}+s_{13})}\sum_{a,b=0}^{\infty}\binom{s_{12}}{a}\binom{s_{13}}{b}\int_{1}^{\infty}dy\absolutevalue{y-1}^{s_{12}}\absolutevalue{y+1}^{s_{13}}$
$\displaystyle\times\int_{L}dx(x+1)^{s_{12}-a}(x-y)^{a+b}(x-1)^{s_{13}-b}2^{s_{23}}.$
(52)

The parameter $\epsilon$ was there to avoid intersecting branch points on the
imaginary axis. Figure 4 shows the contour of integration we followed so that
the variable $x$ is rotated to the imaginary axis. After the rotation, the new
variable $x$ (the rotated one) follows the contour $L$ shown in figure 4. Now,
we would like to refer $\mathcal{B}(s_{12},s_{13})$ as
$\mathcal{B}_{L}(s_{12},s_{13})$ to signify the path of integration. By
assigning the branch cuts to be along the real axis shown in figure 4, one can
write

$\displaystyle\mathcal{B}_{L}(s_{12},s_{13})=$ $\displaystyle i\sin{(\pi
s_{12})}e^{i\frac{\pi}{2}(3s_{12}+s_{13})}\sum_{a,b=0}^{\infty}\binom{s_{12}}{a}\binom{s_{13}}{b}\int_{1}^{\infty}dy\absolutevalue{y-1}^{s_{12}}\absolutevalue{y+1}^{s_{13}}$
$\displaystyle\times e^{\pi i(s_{13}+a)}\bigg{(}\int_{0}^{1}dx+e^{\pi
i(s_{13}-b)}\int_{1}^{y}dx+e^{\pi i(s_{13}+a)}\int_{y}^{\infty}dx\bigg{)}$
$\displaystyle\times\absolutevalue{x+1}^{s_{12}-a}\absolutevalue{x-y}^{a+b}\absolutevalue{x-1}^{s_{13}-b}2^{s_{23}}.$
(53)

It is not hard to see that the integrals that involve the integration region
$x\in(1,y)$ and $x\in(y,\infty)$ (the last two terms in the second line of
(53)) are the integral forms of color-ordered open string amplitudes
considering that the open string vertex positions are chosen to be at $-1,1$
and $\infty$ while gauge fix. According to [16], the integral describing tree-
level five-point partial string amplitude is

$\displaystyle\mathcal{I}^{\text{op}}_{5}(2,3,1,4,5;\\{n_{ij}\\})=$
$\displaystyle\int_{1}^{\infty}dx_{4}\int_{1}^{x_{4}}dx_{1}(x_{1}+1)^{s_{12}+n_{12}}(x_{1}-1)^{s_{13}+n_{13}}$
$\displaystyle\times(x_{4}+1)^{s_{24}+n_{24}}(x_{4}-1)^{s_{34}+n_{34}}(x_{4}-x_{1})^{s_{14}+n_{14}}2^{s_{23}}$
(54)

with a set of integers $n_{ij}$. The three points of open string vertices are
fixed to be $x_{2}=-1$, $x_{3}=1$, and $x_{5}=\infty$. Note that if one would
like to identify the integral (54) to the open string amplitude, the integers
$n_{ij}$ are to be fixed to certain values associated with external string
states.

To get rid of the terms involving the integration of $x$ from 0 to 1 of (53),
we consider the function $\mathcal{B}_{L^{*}}(s_{12},s_{13})$ whose path of
integration $L^{*}$ is defined as shown in figure 4. The function
$\mathcal{B}_{L^{*}}(s_{12},s_{13})$ is in the form

$\displaystyle\mathcal{B}_{L^{*}}(s_{12},s_{13})=$ $\displaystyle i\sin{(\pi
s_{12})}e^{i\frac{\pi}{2}(3s_{12}+s_{13})}\sum_{a,b=0}^{\infty}\binom{s_{12}}{a}\binom{s_{13}}{b}\int_{1}^{\infty}dy\absolutevalue{y-1}^{s_{12}}\absolutevalue{y+1}^{s_{13}}$
$\displaystyle\times e^{\pi i(s_{13}+a)}\bigg{(}\int_{0}^{1}dx+e^{-\pi
i(s_{13}-b)}\int_{1}^{y}dx+e^{-\pi i(s_{13}+a)}\int_{y}^{\infty}dx\bigg{)}$
$\displaystyle\times\absolutevalue{x+1}^{s_{12}-a}\absolutevalue{x-y}^{a+b}\absolutevalue{x-1}^{s_{13}-b}2^{s_{23}}.$
(55)

If we subtract $\mathcal{B}_{L}(s_{12},s_{13})$ by
$\mathcal{B}_{L^{*}}(s_{12},s_{13})$, one yields

$\displaystyle i\sin{(\pi
s_{12})}e^{i\frac{\pi}{2}(3s_{12}+s_{13})}\sum_{a,b=0}^{\infty}\binom{s_{12}}{a}\binom{s_{13}}{b}\int_{1}^{\infty}dy\absolutevalue{y-1}^{s_{12}}\absolutevalue{y+1}^{s_{13}}$
$\displaystyle\times 2ie^{\pi
i(s_{13}+a)}\bigg{(}\sin{(\pi(s_{13}-b))}\int_{1}^{y}dx+\sin{(\pi(s_{13}+a))}\int_{y}^{\infty}dx\bigg{)}$
$\displaystyle\times\absolutevalue{x+1}^{s_{12}-a}\absolutevalue{x-y}^{a+b}\absolutevalue{x-1}^{s_{13}-b}$
(56)

which can be written in terms of $\mathcal{I}^{\text{op}}_{5}$ as

$\displaystyle\mathcal{B}_{L}(s_{12},s_{13})$
$\displaystyle-\mathcal{B}_{L^{*}}(s_{12},s_{13})=-2\sin{(\pi
s_{12})}e^{i\frac{3}{2}\pi(s_{12}+s_{13})}\sum_{a,b=0}^{\infty}\binom{s_{12}}{a}\binom{s_{13}}{b}(-1)^{a}$
$\displaystyle\times\bigg{(}\sin{(\pi(s_{13}-b))}\mathcal{I}^{\text{op}}_{5}(2,3,1,4,5;\\{n_{ij}\\})\bigg{|}_{p_{5}=s_{14}=0}$
$\displaystyle+\sin{(\pi(s_{13}+a))}\mathcal{I}^{\text{op}}_{5}(2,3,4,1,5;\\{n_{ij}\\})\bigg{|}_{p_{5}=s_{14}=0}\bigg{)}.$
(57)

The above expression is evaluated at $s_{14}=0$ and $p_{5}=0$ providing that
$s_{12}=s_{24}$, $s_{13}=s_{34}$ if all the strings are in the same level of
string mass spectrum. The set $\\{n_{ij}\\}$ is
$\\{n_{12}=-a,n_{13}=-b,n_{14}=a+b\\}$.

Besides, we recognize a similarity between the expression of our correction
terms (56) and the infinite tube amplitudes presented in [20] albeit an
infinite sum. For more points of string amplitudes, we would also obtain
correction terms similar to the four-point scenario but with exponentially
more complicated expressions as one needs to deal with multiple contours of
integration together with multiple breach points and cuts in the bulk of the
complex plane. Nevertheless, this issue can be overlooked if we consider the
field theory limit, $\alpha^{\prime}\to 0$, to which it relates Einstein’s
theory with Einstein-Yang-Mills theory. In this limit, the correction terms in
(49) are at the order of $\alpha^{\prime}$ assuming that the open string
amplitudes are at the leading order. Also according to [21], the mixed string
integral $\mathcal{F}_{n}$ is at the same order as the open string amplitudes.
This means that we can ignore the correction terms in this limit. This
argument persists to higher-point string amplitudes as well. This is because
there always appears contribution from contours of integration along infinite
tubes which are there to avoid crossing the branch cuts inside the complex
plane. Such contours would provide a factor of $\sin(\pi s_{ij})$ every time
the contours encircling around the branch cuts. This results in suppression
when taking the field theory limit.

##  5 Conclusions

Using analytic continuation of complex variables, we successfully factorize
closed string amplitudes into those involving ($n-2$) open strings and a
single closed string. Our derivation begins with a detailed examination of
four-point closed string amplitudes and subsequently extends to generalize the
results for arbitrary $n$-point amplitudes. The expressions for four, five,
and six-point amplitudes are presented in (10), (20), and (27), respectively,
while the relations for $n$ strings are expressed in (30).

In addition, our findings reveal the presence of correction terms stemming
from integration along infinite tubes in the complex plane. These corrections
are introduced when considering the existence of branch points in the bulk of
the complex plane during the Wick rotation. The correction terms for the four-
point case are explicitly given in (49). To capture these correction terms, we
define the function $\mathcal{B}(x,y)$, which can be expressed in terms of
integral forms for open string amplitudes, as shown in (57).

Importantly, in the field theory limit, the correction terms become
negligible, being at least of the order of $\alpha^{\prime}$. This justifies
the validity of the established connections in low-energy limits and enhances
the applicability of our results.

In summary, our work achieves the factorization of closed string amplitudes,
provides explicit derivations for specific cases, addresses correction terms
arising from complex variable transformations, and justifies the neglect of
these corrections in the low-energy limit.

##  Appendix A Polarizations and kinematic factors

As mentioned in Section 1, the function $F_{n}(z_{i},\bar{z}_{i})$ contains
the polarizations and kinematic factors of the external closed string states.
During the factorization, the closed string polarizations which are encoded in
the function $F_{n}$ are decomposed to those open and closed strings in the
mixed amplitudes captured by $K_{n}$ and $\widetilde{K}_{n}$.

For tachyons, $F_{n}=K_{n}=\widetilde{K}_{n}=1$. In case of the first excited
states, $F_{n}(z_{i},\bar{z}_{i})$ is given by

$\displaystyle F_{n}(z_{i},\bar{z}_{i})$
$\displaystyle=\text{exp}\Bigg{\\{}\sum_{i>j}\frac{\zeta_{i}\cdot\zeta_{j}}{(z_{i}-z_{j})^{2}}-\sqrt{\alpha^{\prime}}\sum_{i\neq
j}\frac{k_{i}\cdot\zeta_{j}}{(z_{i}-z_{j})}$
$\displaystyle+\sum_{i>j}\frac{\tilde{\zeta}_{i}\cdot\tilde{\zeta}_{j}}{(\bar{z}_{i}-\bar{z}_{j})^{2}}-\sqrt{\alpha^{\prime}}\sum_{i\neq
j}\frac{k_{i}\cdot\tilde{\zeta}_{j}}{(\bar{z}_{i}-\bar{z}_{j})}\Bigg{\\}}\Bigg{|}_{\text{linear
in }\zeta_{i},\tilde{\zeta}_{i}},$ (58)

where $\zeta_{i}\otimes\tilde{\zeta}_{i}$ = $\chi_{i}$ are polarization of
external closed string states. For a scattering including $n-2$ open strings
with polarizations $\zeta_{i}$, for $i\in\\{1,\ldots,n-2\\}$, and one closed
string with a polarization $\chi=\vartheta\otimes\tilde{\vartheta}$, the
function $K_{n}(x_{i})$ reads

$\displaystyle\text{exp}\Bigg{\\{}\sum_{i>j}^{n-2}\frac{\zeta_{i}\cdot\zeta_{j}}{(x_{i}-x_{j})^{2}}-\sqrt{\alpha^{\prime}}\sum_{i\neq
j}^{n-2}\frac{k_{i}\cdot\zeta_{j}}{(x_{i}-x_{j})}+\sum_{j=1}^{n-3}\Bigg{[}\frac{\vartheta\cdot\zeta_{j}}{(x_{j}-z)^{2}}+\frac{\tilde{\vartheta}\cdot\zeta_{j}}{(x_{j}-\bar{z})^{2}}$
$\displaystyle-\sqrt{\alpha^{\prime}}\Bigg{(}\frac{k_{n-1}\cdot\zeta_{j}}{(\bar{z}-\eta_{j})}+\frac{k_{n-2}\cdot\zeta_{j}}{(z-\eta_{j})}+\frac{k_{j}\cdot\vartheta}{(x_{j}-z)}+\frac{k_{j}\cdot\tilde{\vartheta}}{(x_{j}-\bar{z})}\Bigg{)}\Bigg{]}$
$\displaystyle+\frac{\vartheta\cdot\tilde{\vartheta}}{(z-\bar{z})^{2}}-\sqrt{\alpha^{\prime}}\Bigg{(}\frac{k_{n-1}\cdot\vartheta}{(\bar{z}-z)}+\frac{k_{n-2}\cdot\tilde{\vartheta}}{(z-\bar{z})}\Bigg{)}\Bigg{\\}}\Bigg{|}_{\text{linear
in }\zeta_{i},\vartheta,\tilde{\vartheta}}$ (59)

where the closed string momentum is $k_{n-2}+k_{n-1}$.

We can factorize (58) into a product of two functions, i.e.
$F_{n}(z_{i},\bar{z}_{i})$ = $f_{n}(z_{i})\times\tilde{f}_{n}(\bar{z})$. Then,
we follow the similar calculation done in section 2 utilizing the analytic
continuation of complex variables and defining new variables as in (22), we
obtain

$\displaystyle f_{n}(\eta_{i})=$
$\displaystyle\text{exp}\Bigg{\\{}\sum_{i>j}^{n-3}\frac{\zeta_{i}\cdot\zeta_{j}}{(\eta_{i}-\eta_{j})^{2}}-\sqrt{\alpha^{\prime}}\sum_{i\neq
j}^{n-3}\frac{k_{i}\cdot\zeta_{j}}{(\eta_{i}-\eta_{j})}+\sum_{j=1}^{n-3}\Bigg{[}\frac{\zeta_{n-1}\cdot\zeta_{j}}{(\eta_{j}+i)^{2}}+\frac{\zeta_{n-2}\cdot\zeta_{j}}{(\eta_{j}-i)^{2}}$
$\displaystyle-\sqrt{\alpha^{\prime}}\Bigg{(}\frac{k_{n-1}\cdot\zeta_{j}}{(-i-\eta_{j})}+\frac{k_{n-2}\cdot\zeta_{j}}{(i-\eta_{j})}+\frac{k_{j}\cdot\zeta_{n-1}}{(\eta_{j}+i)}+\frac{k_{j}\cdot\zeta_{n-2}}{(\eta_{j}-i)}\Bigg{)}\Bigg{]}$
$\displaystyle+\frac{\zeta_{n-1}\cdot\zeta_{n-2}}{(-2i)^{2}}-\sqrt{\alpha^{\prime}}\Bigg{(}\frac{k_{n-1}\cdot\zeta_{n-2}}{(-2i)}+\frac{k_{n-2}\cdot\zeta_{n-1}}{(2i)}\Bigg{)}\Bigg{\\}}\Bigg{|}_{\text{linear
in }\zeta_{i}}$ (60)

and

$\displaystyle\tilde{f}_{n}(\xi_{i})=$
$\displaystyle\text{exp}\Bigg{\\{}\sum_{i>j}^{n-3}\frac{\tilde{\zeta}_{i}\cdot\tilde{\zeta}_{j}}{(\xi_{i}-\xi_{j})^{2}}-\sqrt{\alpha^{\prime}}\sum_{i\neq
j}^{n-3}\frac{k_{i}\cdot\tilde{\zeta}_{j}}{(\xi_{i}-\xi_{j})}+\sum_{j=1}^{n-3}\Bigg{[}\frac{\tilde{\zeta}_{n-1}\cdot\tilde{\zeta}_{j}}{(\xi_{j}-i)^{2}}+\frac{\tilde{\zeta}_{n-2}\cdot\tilde{\zeta}_{j}}{(\xi_{j}+i)^{2}}$
$\displaystyle-\sqrt{\alpha^{\prime}}\Bigg{(}\frac{k_{n-1}\cdot\tilde{\zeta}_{j}}{(i-\xi_{j})}+\frac{k_{n-2}\cdot\tilde{\zeta}_{j}}{(-i-\xi_{j})}+\frac{k_{j}\cdot\tilde{\zeta}_{n-1}}{(\xi_{j}-i)}+\frac{k_{j}\cdot\tilde{\zeta}_{n-2}}{(\xi_{j}+i)}\Bigg{)}\Bigg{]}$
$\displaystyle+\frac{\tilde{\zeta}_{n-1}\cdot\tilde{\zeta}_{n-2}}{(2i)^{2}}-\sqrt{\alpha^{\prime}}\Bigg{(}\frac{k_{n-1}\cdot\tilde{\zeta}_{n-2}}{(2i)}+\frac{k_{n-2}\cdot\tilde{\zeta}_{n-1}}{(-2i)}\Bigg{)}\Bigg{\\}}\Bigg{|}_{\text{linear
in }\tilde{\zeta}_{i}}.$ (61)

Note that the points $z_{n-2},z_{n-1}$ and $z_{n}$ are fixed to $i,-i$ and
$\infty$ respectively. It is not hard to see that the functions $f_{n}$ and
$\tilde{f}_{n}$ can be identified as $K_{n}$ and $\tilde{K}_{n}$ (59) where
now the closed string polartization regarding $K_{n}$ and $\tilde{K}_{n}$ are
$\zeta_{n-1}\otimes\zeta_{n-2}$ and
$\tilde{\zeta}_{n-1}\otimes\tilde{\zeta}_{n-2}$ respectively in mixed string
amplitudes.

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