All-Order Quartic Couplings in Highly
Symmetric D-brane-Anti-D-brane Systems

Ehsan<EMAIL_ADDRESS><EMAIL_ADDRESS>and Per
Sundellc,222<EMAIL_ADDRESS><EMAIL_ADDRESS>

a Scuola Normale Superiore and I.N.F.N
Piazza dei Cavalieri 7, 56126, Pisa, Italy

b Center for Theoretical Physics and College of Physics
Jilin University, Changchun, 130012, China

c Departamento de Ciencias Físicas, Universidad Andres Bello
Sazié 2212, Santiago, Chile

### Abstract:

We compute six-point string amplitudes for the scattering of one closed string
Ramond-Ramond state, two tachyons and two gauge fields in the worldvolume of
D-brane-anti-D-brane systems in the Type II superstring theories. From the
resulting S-matrix elements, we read off the precise form of the couplings,
including their exact numerical coefficients, of two tachyons and two gauge
fields in the corresponding highly symmetric effective field eheory (EFT)
Lagrangian in the worldvolume of D-brane-Anti-D-brane to all orders in
$\alpha^{\prime}$, which modify and complete previous proposals. We verify
that the EFT reproduces the infinite collection of stringy gauge field
singularities in dual channels. Inspired by interesting similarities between
the all-order highly symmetric EFTs and holographic duals of Vasiliev’s higher
spin gravities à là Nilsson and Vasiliev, we make a proposal for tensionless
limits of D-brane-anti-D-brane systems.

###### Contents

  1. 1 Introduction
  2. 2 Highly Symmetric D-Brane-Anti-D-Brane EFT
  3. 3 From S-matrix to All-Order EFT Couplings
  4. 4 EFT Couplings from Six Point Functions
  5. 5 All-Order Two-Tachyon-Two-Gauge Field Couplings
  6. 6 Conclusion

##  1 Introduction

In this paper, we shall study the effects of stringy corrections to gauge
theory and gravity by extracting effective field theory (EFT) actions
involving such corrections to all orders alias all-order EFTs. As these
theories reproduce the string amplitudes with their characteristic channel
dualities, they are also sometimes referred to as highly symmetric EFTs. More
precisely, we shall focus our attention to all-order EFTs on worldvolumes of
D-brane-anti-D-brane (D$\overline{\rm D}$) systems in which all stringy modes
have been integrated out save massless gauge modes and tachyons. These models
have a number of interesting applications ranging from the study of tachyon
condensation in string theory via open-closed string duality to holography in
non-BPS backgrounds and in high-energy limits.

String theory provides an ultraviolet completion of gravity by systematically
adding higher-derivative corrections to its low-energy approximation given by
Einstein’s theory. Its power relies on the the fact that it intertwines two
quantum theories in a nontrivial fashion such that the stringy corrections to
gravity can be organized into a double perturbative expansion. Thus, viewing
gravity as a quantum field theory with an S-matrix, its stringy quantum
corrections appear weighted by a string coupling constant $g_{s}$, which one
may thus identify as $\hbar$, while at each order in $g_{s}$, the S-matrix
elements are computed using a first-quantized string described by sigma model
on a two-dimensional surface, alias worldsheet, of fixed genus, with its own
coupling constant, often denoted by $\alpha^{\prime}$. Physically speaking,
$\alpha^{\prime}$ sets the unit of length for the spacetime background metric,
which yields an identification of the Planck length as a combination of
positive powers of $g_{s}$ and $\alpha^{\prime}$, which is arguably one of the
most exciting results of string theory.

Another remarkable feature of the double perturbative expansion, is that it
admits a natural extension so as to include non-perturbative effects by means
of geometric objects permeating the spacetime background, known as Dirichlet-
branes, or D-branes. These may be thought of in two dual fashions as either
defects in spacetime, serving as sources for stringy analogs of electric and
magnetic fluxes, or defects on the worldsheet, serving as sources for modified
boundary conditions arising by means of two-dimensional analogs of electric-
magnetic duality rotations.

In this paper, we shall attempt to attain a deeper understanding of the
$\alpha^{\prime}$-corrections to gravity and obtain a more complete control of
their all-order behavior by focusing on D$\overline{\rm D}$ systems. The basic
motivation behind studying these systems is that the aforementioned stringy
dualities have so far been spelled out mainly in perturbative expansions
around supersymmetric string vacua, while supersymmetry is broken throughout
the bulk of the moduli space of string theory. To explore non-supersymmetric
string backgrounds, D$\overline{\rm D}$ systems are important tools as their
underlying first-quantized descriptions remain tractable and amenable to
quantitative studies even in the absence of supersymmetry. The detailed link
between closed-string and open-string constructions originates from [1].

More broadly, in view of work on instabilities, and in particular on the
importance of non-BPS branes in the Swampland [2] and the related gravity
conjecture 333This conjecture confirms the presence of the light elementary
electric and magnetic objects so that their mass/charge ratio is smaller than
the relevant ratio for macroscopic extremal black holes and hence allows the
decay of extremal black holes.  [3], one would like to gather additional
insights into supersymmetry breaking on time–dependent backgrounds [4, 5, 6].
For instance, if we use the Sakai–Sugimoto model of D$\overline{\rm D}$
systems [7] to analyse spontaneous supersymmetry breaking in various
holographic models [8], the model will be granted as a probe if and only if
the number of flavours is much less than the number of colours,
$N_{f}<<N_{c}$. Hence, we shall keep all tachyonic string states in both
worldsheet and EFT descriptions of the D$\overline{\rm D}$ system.

In [9, 10], various authors have found tachyonic effective actions that also
describe the decays of non-BPS branes [11]. Subsequently, in [6], it was
argued how to embed non-BPS branes and massless states into the EFT, whereby
tachyon condensation for brane-anti-brane systems can be discussed in great
detail [12].

In the case of a single D-brane and a single anti-D-brane separated by a
distance below the string length, two real tachyon modes emerge in the EFT. In
[13], the dynamics of this system was analyzed using EFT methods, taking into
account string loop divergences. As applications of tachyonic EFTs, we would
like to highlight brane production [14] and inflation in string theory, which
later on became important ingredients the context of KKLT [15]. Various
thermodynamic aspects of this system at finite temperature have been studied,
whereby the system gets stabilised and can be related to black holes, leading
to applications in AdS/CFT correspondence in string and M-theory [16].
Properties of D$\overline{\rm D}$ systems facilitate stability analyses in the
context of KKLT and the Large Volume Scenario of string compactifications
[17]. Including D-brane sources [18], one may furthermore consider
D$\overline{\rm D}$ bound states within the context of K-theory [19].

In the aforementioned context, where the dualities of the supersymmetric
M-theory web do not hold anymore, worldsheet Conformal Field Theory (CFT)
methods are nonetheless still available, and remarkably they suffice for
constructing string corrections and new couplings [20, 21]. Proceeding with
this program, all standard EFT methods of effective actions for both non-BPS
and BPS branes have been clarified with full details in [22]. More precisely,
the string corrections to DBI, Chern-Simons, Wess-Zumino effective actions are
derived from scattering amplitudes, as, to the best of our knowledge, at least
in the context of non-BPS branes, this approach does not introduce any
ambiguities into $\alpha^{\prime}$-corrections, and one explores all-order
coefficients of string corrections in the Type IIB and IIA superstrings
accordingly.

Finally, in the Conclusions we point to interesting similarities between the
all-order EFT Lagrangian for D$\overline{\rm D}$ systems and the systems of
matter fields coupled to topological conformal higher spin gravities that have
been proposed [23, 24, 25] as duals of bulk higher spin gravity a la Vasiliev
[26] in an attempt to refine the original holography proposal [27, 28, 29,
30]. Motivated by this, within the context of the Frobenius–Chern–Simons (FCS)
extension [31] of Vasiliev’s higher spin gravity, we propose that the
counterparts of D$\overline{\rm D}$ systems are contained as branches of
fractional spin gravities [32].

The paper is structured as follows: We first concentrate on the
D$\overline{\rm D}$ system and explain briefly the details on four- [33] and
five-point functions. We then explore more hidden symmetries in amplitudes of
non-BPS branes. Having taken the selection rules [34], EFT and other
symmetries into account, we explore the expansion of the amplitudes and start
to exhibit all singularities of the type IIA and IIB string theories. This
leads to the construction of the all-order $\alpha^{\prime}$ higher-derivative
corrections to two-tachyon and two-gauge field couplings in the
D$\overline{\rm D}$ system as well.

This result shows that not only the structures of the higher derivative
corrections but also their coefficients are different from the ones of non-BPS
branes, as we shall clarify in detail in section five.

##  2 Highly Symmetric D-Brane-Anti-D-Brane EFT

The D${}_{p}\overline{\rm D}_{p}$-brane system is constructed by
$(-1)^{F_{L}}$-projecting the two branes which yields an unstable
configuration containing tachyonic states. The resulting EFT of tachyons and
massless gauge fields is formulated in terms of a superconnection of a
noncommutative geometry [35, 36, 37], viz.

$i{\cal A}:=\left(\begin{array}[]{cc}iA^{(1)}&\beta T^{*}\\\ \beta
T&iA^{(2)}\end{array}\right)\ ,$

where $A^{(1)}$ and $A^{(2)}$ are gauge fields associated to the Dp\- and
$\overline{\rm D}_{p}$-brane, respectively, and
$T=\frac{1}{\sqrt{2}}(T_{1}+iT_{2})$ is a complex tachyon with normalization
$\beta$. The resulting supercurvature

$\displaystyle{\cal F}$ $\displaystyle:=$ $\displaystyle d{\cal A}-i{\cal
A}\wedge\cal
A=\left(\begin{array}[]{cc}iF^{(1)}-\beta^{2}|T|^{2}&\beta(DT)^{*}\\\ \beta
DT&iF^{(2)}-\beta^{2}|T|^{2}\end{array}\right)\ ,$ (3)

where $F^{(i)}:=dA^{(i)}-iA^{(i)}\wedge
A^{(i)}=\frac{1}{2}F^{(i)}_{ab}dx^{a}\wedge dx^{b}$ and
$DT:=dT-i(A^{(1)}T-TA^{(2)})=D_{a}Tdx^{a}$. Using trivial Chan–Paton factors,
such that $F^{(i)}_{ab}=\partial_{a}A^{(i)}_{b}-\partial_{b}A^{(i)}_{a}$ and
$D_{a}T=\partial_{a}T-i(A^{(1)}_{a}-A^{(2)}_{a})T$, and following [38], the
kinetic terms of the EFT action can be assembled into a DBI-like Lagrangian,
viz.

$\displaystyle S_{DBI}$ $\displaystyle=$ $\displaystyle-\int d^{p+1}\sigma{\rm
TrS}\left(V({\cal
T})\sqrt{-\det(\eta_{ab}+2\pi\alpha^{\prime}F_{ab}+2\pi\alpha^{\prime}D_{a}{\cal
T}D_{b}{\cal T})}\right)\,\,,$ (4)

where $V$ is the Tachyon potential,

$\displaystyle F_{ab}:=\pmatrix{F^{(1)}_{ab}&0\cr 0&F^{(2)}_{ab}}\ ,\qquad
D_{a}{\cal T}:=\pmatrix{0&D_{a}T\cr(D_{a}T)^{*}&0}\ ,\qquad{\cal
T}=\pmatrix{0&T\cr T^{*}&0}\ ,$ (5)

and ${\rm TrS}$ denotes the symmetrized trace over ${\rm mat}_{2}$. The EFT
action also contains couplings to the spacetime RR potentials packaged into
$C:=\sum_{n}(-i)^{\frac{p-n+1}{2}}C_{n}$ which can be assembled into a
Wess–Zumino term

$\displaystyle S_{WZ}$ $\displaystyle=$
$\displaystyle\mu_{p}\int_{\Sigma_{(p+1)}}C\wedge{\rm
STr}\,e^{2\pi\alpha^{\prime}i\cal F}\ ,$ (6)

where ${\rm STr}$ is the supertrace. Indeed, setting the tachyon modes to
zero, the WZ term reduces to the Chern–Simons-like term [39]444The
nomenclature stems from the fact that $S_{\rm CS}$ consists of couplings of
the form $\int C_{n}\wedge dA\wedge\cdots\wedge dA$, though since $C_{n}$
comprises the pull-back $\frac{1}{n!}C_{M_{1}\dots
M_{n}}dX^{M_{1}}\wedge\cdots dX^{M_{n}}$, one may also think of $S_{\rm CS}$
as packaging worldvolume Quillen–Chern classes of a superconnection with
components $(A,X^{M},..)$ whose extension by $T$ thus deforms $S_{\rm CS}$ to
$S_{\rm WZ}$.

$\displaystyle S_{\rm CS}:=S_{\rm
WZ}|_{T=0}=\mu_{p}\int_{\Sigma_{(p+1)}}C\wedge\left(e^{2\pi\alpha^{\prime}iF^{(1)}}-e^{2\pi\alpha^{\prime}iF^{(2)}}\right)\
,$ (7)

as expected by taking the naive sum of a separately supersymmetric $D$\- and
$\overline{\rm D}$-brane.

### Comparison to disc amplitudes.

Expanding the Lagrangian of (6), one derives

$\displaystyle C\wedge\mbox{\rm STr}\,i{\cal F}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!$ $\displaystyle
C_{p-1}\wedge(F^{(1)}-F^{(2)})$ (8) $\displaystyle C\wedge\mbox{\rm
STr}\,i{\cal F}\wedge i{\cal F}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle C_{p-3}\wedge\left\\{F^{(1)}\wedge F^{(1)}-F^{(2)}\wedge
F^{(2)}\right\\}$ (9)
$\displaystyle+C_{p-1}\wedge\left\\{-2\beta^{2}|T|^{2}(F^{(1)}-F^{(2)})+2i\beta^{2}DT\wedge(DT)^{*}\right\\}$

the resulting couplings can be matched to disc amplitudes computed using CFT
methods [40]; for details, see for example [34]. To this end, the relevant
open string vertex operators for real tachyons and gauge photons carrying
Chan–Paton factors $\lambda$ are given by

$\displaystyle V_{T}^{(-1)}(x_{1})$ $\displaystyle=$ $\displaystyle
e^{-\phi(x_{1})}e^{\alpha^{\prime}ik_{1}{\cdot}X(x_{1})}\lambda\otimes\sigma_{2}\
,$ (10) $\displaystyle V_{T}^{(0)}(x_{2})$ $\displaystyle=$
$\displaystyle\alpha^{\prime}ik_{2}{\cdot}\psi(x_{2})e^{\alpha^{\prime}ik_{2}{\cdot}X(x_{2})}\lambda\otimes\sigma_{1}\
,$ (11) $\displaystyle V_{A}^{(-1)}(x_{3})$ $\displaystyle=$
$\displaystyle{\xi}{\cdot}{\psi(x_{3})}e^{-\phi(x_{3})}\,e^{\alpha^{\prime}ik_{3}{\cdot}X(x_{3})}\lambda\otimes\sigma_{3}\
,$ (12) $\displaystyle V_{A}^{(0)}(x_{4})$ $\displaystyle=$
$\displaystyle{\xi_{2a}}{\bigg{(}\partial
X^{a}(x_{4})+\alpha^{\prime}ik{\cdot}\psi(x_{4})\psi^{a}(x_{4})\bigg{)}}e^{\alpha^{\prime}ik_{4}{\cdot}X(x_{4})}\lambda\otimes
I\ ,$ (13)

where $x_{i}$ belong to the boundary of the disc and $k_{i}$ are worldvolume
momenta obeying $k_{3}^{2}=k_{4}^{2}=0$ and $k_{1}^{2}=k_{2}^{2}=-m^{2}$,
where $m^{2}=\frac{-1}{2\alpha^{\prime}}$ is the tachyon mass-squared. The
closed string vertex operator for the RR particle is given by

$\displaystyle V_{RR}^{(-1)}(z,\bar{z})$ $\displaystyle=$
$\displaystyle(P_{-}H\\!\\!\\!\\!/\,_{n}M_{p})^{\alpha\beta}e^{-\phi(z)/2}S_{\alpha}(z)e^{i\frac{\alpha^{\prime}}{2}p{\cdot}X(z)}e^{-\phi(\bar{z})/2}S_{\beta}(\bar{z})e^{i\frac{\alpha^{\prime}}{2}p{\cdot}D{\cdot}X(\bar{z})}\otimes\sigma_{3}\
,\quad$ (14)

where $(z,\bar{z})$ is a point inside the disc and $p$ a massless spacetime
momentum, and555We raise and lower spinor indices using a charge conjugation
matrix $C^{\alpha\beta}$ and the conventions
$S^{\alpha}=C^{\alpha\beta}S_{\beta}$ and
$S_{\alpha}=S^{\beta}(C^{-1})_{\beta\alpha}$.

$\displaystyle P_{-}$ $\displaystyle:=$
$\displaystyle\mbox{$\frac{1}{2}$}(1-\gamma^{11})\ ,$ (15) $\displaystyle
H\\!\\!\\!\\!/\,_{n}$ $\displaystyle:=$
$\displaystyle\frac{m_{n}}{n!}H_{\mu_{1}\ldots\mu_{n}}\gamma^{\mu_{1}}\ldots\gamma^{\mu_{n}}\
,$ (16)

with $n$ even and $m_{n}=i$ in the Type IIA model, and $n$ odd and $m_{n}=1$
in the Type IIB model. Using the doubling trick, the disc amplitude is mapped
by

$\displaystyle\tilde{X}^{\mu}(\bar{z})\rightarrow
D^{\mu}_{\nu}X^{\nu}(\bar{z})\ ,\ \ \ \tilde{\psi}^{\mu}(\bar{z})\rightarrow
D^{\mu}_{\nu}\psi^{\nu}(\bar{z})\ ,\ \ \
\tilde{\phi}(\bar{z})\rightarrow\phi(\bar{z})\ ,\ \ \
\tilde{S}_{\alpha}(\bar{z})\rightarrow
M_{\alpha}{}^{\beta}{S}_{\beta}(\bar{z})\ ,$ (17)

where the matrices

$\displaystyle D:=\left(\begin{array}[]{cc}-1_{9-p}&0\\\
0&1_{p+1}\end{array}\right)\ ,\qquad M_{p}:=\frac{\pm
m_{p}}{(p+1)!}\gamma^{a_{0}}\ldots\gamma^{a_{p}}(\gamma_{11})^{p}\epsilon_{a_{0}\ldots
a_{p}}\ ,$ (20)

to an amplitude computed on the entire complex plane for a single holomorphic
sector with two-point functions

$\displaystyle\langle X^{\mu}(z)X^{\nu}(w)\rangle$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{\prime}}{2}\eta^{\mu\nu}\log(z-w)\ ,$ (21)
$\displaystyle\langle\psi^{\mu}(z)\psi^{\nu}(w)\rangle$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{\prime}}{2}\eta^{\mu\nu}(z-w)^{-1}\ ,$ (22)
$\displaystyle\langle\phi(z)\phi(w)\rangle$ $\displaystyle=$
$\displaystyle-\log(z-w)\ .$ (23)

Letting $u:=-\frac{\alpha^{\prime}}{2}(k_{1}+k_{2})^{2}$ and setting
$\alpha^{\prime}=2$, the complete disc amplitude for the scattering of two
real tachyons and one RR particle on the $D_{p}\bar{D}_{p}$-brane reads

$\displaystyle{\cal A}^{T^{(-1)}T^{(0)}C^{(-1)}_{n-1}}_{\rm disc}$
$\displaystyle=$
$\displaystyle\frac{i\mu_{p}}{4}2\pi\frac{\Gamma(-2u)}{\Gamma(1/2-u)^{2}}\mbox{\rm
Tr}\,(P_{-}H\\!\\!\\!\\!/\,_{n}M_{p}\gamma^{a})k_{2a}\ ,$ (24)

where $\mu_{p}$ is RR charge and $\gamma^{11}$ is chosen such that
$H_{n}=*H_{10-n}$ for $p>3$ and $n\geq 5$.

In the limit $u=-p^{a}p_{a}\rightarrow 0$, the u-channel massless gauge field
pole is reproduced in the EFT as

$\displaystyle{\cal A}^{TTC_{p-1}}_{\rm EFT}|_{\rm u-ch}$ $\displaystyle=$
$\displaystyle V^{a}(C_{p-1},A^{(1)})G_{ab}(A)V^{b}(A^{(1)},T_{1},T_{2})$ (25)
$\displaystyle+V^{a}(C_{p-1},A^{(2)})G_{ab}(A)V^{b}(A^{(2)},T_{1},T_{2})\ ,$

where $G_{ab}(A)$ is the gauge field propagator, and $V_{a}(C_{p-1},A^{(i)})$
and $V_{a}(A^{(i)},T_{1},T_{2})$ are vertices read off, respectively, from the
Chern-Simons coupling

$\displaystyle
i\mu_{p}(2\pi\alpha^{\prime})\int_{\Sigma_{p+1}}\epsilon^{a_{0}...a_{p}}\mbox{\rm
Tr}\,(C_{a_{0}...a_{p-2}}\partial_{a_{p-1}}(A^{(1)}_{a_{p}}-A^{(2)}_{a_{p}}))\
,$ (26)

and the DBI-term. Consequently, by matching the full EFT-amplitude to the disc
amplitude, the required contact terms are contained in

$\displaystyle
i\mu_{p}(2\pi\alpha^{\prime})^{2}\int_{\Sigma_{p+1}}C_{p-1}\wedge\mbox{\rm
Tr}\,\left(\sum_{m=-1}^{\infty}c_{m}(\alpha^{\prime}(D^{a}D_{a}))^{m+1}DT_{1}\wedge
DT_{2}\right)\ ,$ (27)

which thus provide the desired EFT couplings of two real tachyons and one RR
potential to all orders in $\alpha^{\prime}$.

In what follows, we shall include one gauge photon into the amplitude and
exhibit the resulting all-order bosonic four-field corrections to the
$D_{p}\bar{D}_{p}$-brane EFT.

##  3 From S-matrix to All-Order EFT Couplings

In this section, we contextualize our analysis by reviewing the five point
function, which also serves as a warm up for the computations of the new six
point functions in the following section.

At tree level, the string scattering S-matrix element on a
$D_{p}\bar{D}_{p}$-brane of two real tachyons with momenta $k_{2}$ and
$k_{3}$, a gauge photon with momentum $k_{1}$, and an RR particle with
spacetime momentum $p$ is given by the disc amplitude [41]

$\displaystyle{\cal A}^{A^{(-1)}T^{(0)}T^{(0)}C^{(-1)}_{n-1}}_{\rm disc}$
$\displaystyle=$ $\displaystyle\frac{i\mu_{p}}{2\sqrt{\pi}}\int
dzd\bar{z}\,k_{2b}k_{3c}{\xi_{a}}(z-\bar{z})^{-2(t+s+u)-1}|z|^{2t+2s-1}|1-z|^{2u+2t-1/2}$
(28)
$\displaystyle\times\bigg{\\{}(\Gamma^{cba})_{\alpha\beta}-2\eta^{ab}\frac{-x+|z|^{2}}{z-\bar{z}}(\Gamma^{c})_{\alpha\beta}-2\eta^{ac}\frac{x}{z-\bar{z}}(\Gamma^{b})_{\alpha\beta}$
$\displaystyle+2\eta^{bc}\frac{x-1}{z-\bar{z}}(\Gamma^{a})_{\alpha\beta}\bigg{\\}}(P_{-}H\\!\\!\\!\\!/\,_{n}M_{p})^{\alpha\beta}\
,$

where we have gauge fixed
$(x_{1},x_{2},x_{3},z,\bar{z})=(0,1,\infty,z,\bar{z})$, and introduced the
Mandelstam variables

$\displaystyle s$ $\displaystyle:=$
$\displaystyle-\frac{\alpha^{\prime}}{2}(k_{1}+k_{3})^{2},\quad
t:=-\frac{\alpha^{\prime}}{2}(k_{1}+k_{2})^{2},\quad
u:=-\frac{\alpha^{\prime}}{2}(k_{2}+k_{3})^{2}\ .$ (29)

Note that for five point functions the integration over the position of the RR
potential vertex operator involves the prototype integral

$\displaystyle\int\int
d^{2}\\!z|1-z|^{a}|z|^{b}(z-\bar{z})^{c}(z+\bar{z})^{d}\ .$ (30)

These integrals were computed for $d=0,1$ in [42] and for $d=2$ in [43] where
the tricks are explained in [43]. Including a normalization
$i\mu_{p}/2\sqrt{2\pi}$, the final form of the amplitude is

$\displaystyle{\cal A}^{T^{(0)}T^{(0)}A^{(-1)}C^{(-1)}_{n-1}}_{\rm disc}$
$\displaystyle=$ $\displaystyle\frac{i\mu_{p}}{2\sqrt{2\pi}}\bigg{(}\mbox{\rm
Tr}\,((P_{-}H\\!\\!\\!\\!/\,_{n}M_{p})(k_{3}{\cdot}\gamma)(k_{2}{\cdot}\gamma)(\xi{\cdot}\gamma))I+\mbox{\rm
Tr}\,((P_{-}H\\!\\!\\!\\!/\,_{n}M_{p})\gamma^{a})J$
$\displaystyle\times\bigg{[}k_{2a}(t+1/4)(2\xi{\cdot}k_{3})+k_{3a}(s+1/4)(2\xi{\cdot}k_{2})-\xi_{a}(s+1/4)(t+1/4)\bigg{]}\bigg{)}\
,$

where

$\displaystyle I$ $\displaystyle:=$ $\displaystyle
2^{-2(t+s+u)-1/2}\pi{\frac{\Gamma(-u)\Gamma(-s+1/4)\Gamma(-t+1/4)\Gamma(-t-s-u)}{\Gamma(-u-t+1/4)\Gamma(-t-s+1/2)\Gamma(-s-u+1/4)}}\
,$ (31) $\displaystyle J$ $\displaystyle:=$ $\displaystyle
2^{-2(t+s+u)-3/2}\pi{\frac{\Gamma(-u+1/2)\Gamma(-s-1/4)\Gamma(-t-1/4)\Gamma(-t-s-u-1/2}{\Gamma(-u-t+1/4)\Gamma(-t-s+1/2)\Gamma(-s-u+1/4)}}\
.$ (32)

Given the on-shell condition,

$\displaystyle s+t+u$ $\displaystyle=$ $\displaystyle-p_{a}p^{a}-\frac{1}{2}\
.$ (33)

The EFT formalism, which provide the guide of generating massless
singularities, implies that $k_{i}.k_{j}\rightarrow 0$, which yields the
following expansion for $CTTA$ amplitude:

$\displaystyle u\rightarrow 0\ ,\qquad s,t\rightarrow-1/4\ .$ (34)

In [41], some tachyonic and massless singularities of the S-matrix are spelled
out. For $p=n+2$, the amplitude has infinitely many massless poles, and for
$p=n$ there are infinitely many tachyon and massless poles. It does have an
infinite number of gauge field poles in $(s+t+u+1/2)-$ channel poles as well.

As for the infinite collection of massless gauge poles for $p=n+2$, the
corresponding string amplitude is

$\displaystyle{\cal A}^{AT_{1}T_{2}C}$ $\displaystyle=$
$\displaystyle\pm\frac{8i\mu_{p}}{\sqrt{2\pi}(p-2)!}\epsilon^{a_{0}\cdots
a_{p}}H_{a_{0}\cdots a_{p-3}}k_{3a_{p-2}}k_{2a_{p-1}}\xi_{a_{p}}I\ ,$ (35)

which are produced by the following sub amplitude in an EFT

$\displaystyle{\cal A}_{\rm EFT}$ $\displaystyle=$ $\displaystyle
V_{a}(C_{p-3},A^{(1)},A^{(1)})G_{ab}(A^{(1)})V_{b}(A^{(1)},T_{1},T_{2})\ ,$
(36)

where the expansion of $I$ around (34) is

$\displaystyle I$ $\displaystyle=$
$\displaystyle\pi\sqrt{2\pi}\left(-\frac{1}{u}\sum_{n=-1}^{\infty}b_{n}(s^{\prime}+t^{\prime})^{n+1}+\sum_{p,n,m=0}^{\infty}c_{p,n,m}u^{p}\left(s^{\prime}t^{\prime}\right)^{n}(s^{\prime}+t^{\prime})^{m}\right)\
,$ (37)

where $s^{\prime}=s+1/4,t^{\prime}=t+1/4$; the algebraic structure of the
coefficients is represented well by the first few ones, which are

$\displaystyle c_{0,0,2}=\frac{2}{3}\pi^{2}\ln(2)\ ,\qquad
c_{0,1,0}=-14\zeta(3)\ ,\qquad c_{0,0,3}=8\zeta(3)\ln(2)\ .$ (38)

Since the gauge field propagator has no correction, nor the couplings of two
tachyons and a gauge field, it is necessary to add higher-derivative couplings
between the RR potential two gauge fields, which can thus be determined to all
orders, with the result

$\displaystyle\mu_{p}(2\pi\alpha^{\prime})^{2}\sum_{n=-1}^{\infty}b_{n}(\alpha^{\prime})^{n+1}C_{p-3}\wedge
D^{a_{1}}...D^{a_{n}}F\wedge D_{a_{1}}...D_{a_{n}}F\ .$ (39)

The infinite set massless poles can then be derived using the following data:

$\displaystyle G_{ab}(A^{(1)})$ $\displaystyle=$
$\displaystyle\frac{i\delta_{ab}}{(2\pi\alpha^{\prime})^{2}T_{p}\left(u\right)}\
,$ (40) $\displaystyle V_{b}(A^{(1)},T_{1},T_{2})$ $\displaystyle=$
$\displaystyle T_{p}(2\pi\alpha^{\prime})(k_{2}-k_{3})_{b}$ (41)
$\displaystyle V_{a}(C_{p-3},A^{(1)},A^{(1)})$ $\displaystyle=$
$\displaystyle\frac{\mu_{p}(2\pi\alpha^{\prime})^{2}}{(p-2)!}\epsilon_{a_{0}\cdots
a_{p-1}a}H^{a_{0}\cdots
a_{p-3}}k_{1}^{a_{p-2}}\xi^{a_{p-1}}\sum_{n=-1}^{\infty}b_{n}(\alpha^{\prime}k_{1}\cdot
k)^{n+1}\ ,$ (42)

where $k$ is the momentum of the off-shell gauge field. Indeed, it follows
that all string poles are reproduced in the EFT, and the final form of the EFT
amplitude (36) is

$\displaystyle{\cal A}_{\rm
EFT}=\mu_{p}(2\pi\alpha^{\prime})\frac{2i}{(p-2)!u}\epsilon_{a_{0}\cdots
a_{p-1}a}H^{a_{0}\cdots
a_{p-3}}k_{2}^{a_{p-2}}k_{3}^{a_{p-1}}\xi^{a}\sum_{n=-1}^{\infty}b_{n}\left(\frac{\alpha^{\prime}}{2}\right)^{n+1}(s^{\prime}+t^{\prime})^{n+1}\
.$ (43)

On the other hand for $p=n$, one explores the following string amplitude:

$\displaystyle{\cal A}^{AT_{1}T_{1}C}$ $\displaystyle=$
$\displaystyle\pm\frac{8i\mu_{p}}{\sqrt{2\pi}p!}\epsilon^{a_{0}\cdots
a_{p-1}a}H_{a_{0}\cdots
a_{p-1}}J\bigg{\\{}k_{2a}t^{\prime}(2\xi.k_{3})+k_{3a}s^{\prime}(2\xi.k_{2})-\xi_{a}s^{\prime}t^{\prime}\bigg{\\}}\
.$ (44)

Its first simple gauge field pole is reproduced by the following EFT
amplitude:

$\displaystyle{\cal A}_{EFT}$ $\displaystyle=$ $\displaystyle
V_{a}(C_{p-1},A)G_{ab}(A)V_{b}(A,T_{1},T_{1},A^{(1)})\ ,$ (45)

where the vertices and gauge field propagators, which can be read off from the
DBI action and CS term, are given by

$\displaystyle G_{ab}(A)$ $\displaystyle=$
$\displaystyle\frac{i\delta_{ab}}{(2\pi\alpha^{\prime})^{2}T_{p}\left(u+t^{\prime}+s^{\prime}\right)}\
,$ (46) $\displaystyle V_{a}(C_{p-1},A^{(1)})$ $\displaystyle=$ $\displaystyle
i\mu_{p}(2\pi\alpha^{\prime})\frac{1}{p!}\epsilon_{a_{0}\cdots
a_{p-1}a}H^{a_{0}\cdots a_{p-1}}\ ,$ (47) $\displaystyle
V_{a}(C_{p-1},A^{(2)})$ $\displaystyle=$
$\displaystyle-i\mu_{p}(2\pi\alpha^{\prime})\frac{1}{p!}\epsilon_{a_{0}\cdots
a_{p-1}a}H^{a_{0}\cdots a_{p-1}}\ ,$ (48) $\displaystyle
V_{b}(A^{1},T_{1},T_{1},A^{1})$ $\displaystyle=$
$\displaystyle-2iT_{p}(2\pi\alpha^{\prime})\xi_{b}\ ,$ (49) $\displaystyle
V_{b}(A^{(2)},T_{1},T_{1},A^{(1)})$ $\displaystyle=$ $\displaystyle
2iT_{p}(2\pi\alpha^{\prime})\xi_{b}\ .$ (50)

On the other hand, tachyonic singularities are also produced by the following
sub-amplitude:

$\displaystyle{\cal A}$ $\displaystyle=$ $\displaystyle
V(C_{p-1},T_{1},T_{2})G(T_{2})V(T_{2},T_{1},A^{(1)})\ ,$ (51)

where we recall that the vertex of two tachyons and one gauge field has no
correction as the kinetic term of tachyon is fixed and the propagator receives
no correction either; hence

$\displaystyle V(T_{2},T_{1},A^{(1)})$ $\displaystyle=$ $\displaystyle
T_{p}(2\pi\alpha^{\prime})(k_{3}-k)\\!\cdot\\!\xi\ ,$ (52) $\displaystyle
G(T_{2})$ $\displaystyle=$
$\displaystyle\frac{i}{(2\pi\alpha^{\prime})T_{p}(s+1/4)}\ ,$ (53)
$\displaystyle V(C_{p-1},T_{1},T_{2})$ $\displaystyle=$
$\displaystyle(2\pi\alpha^{\prime})^{2}\mu_{p}\sum_{n=0}^{\infty}b_{n}\frac{1}{p!}\epsilon^{a_{0}\cdots
a_{p-1}a}H_{a_{0}\cdots a_{p-1}}(\alpha^{\prime}k_{1}.k)^{n+1}k_{2a}\ ,$ (54)

where $k_{2a}$ is the momentum of the off shell tachyon.

In order to derive all infinite massless poles , we need to obtain all-order
$\alpha^{\prime}$ corrections of two tachyons and two gauge fields in the
worldvolume of brane-anti brane system and to do so we start to properly
address a six point function including an RR, two tachyons and two gauge
fields in the next section.

##  4 EFT Couplings from Six Point Functions

Having warmed up with the five point function, we now proceed by first
computing the S-matrix element for a closed-string RR state and two gauge
fields and two tachyons on the worldvolume of the D$\overline{\rm D}$ system.
We then factorize it in a specific limit over gauge field poles that match to
all orders in $\alpha^{\prime}$ with corresponding higher-derivative couplings
of the EFT of the D$\overline{\rm D}$ system.

We start with the disk correlation functions

$\displaystyle(I_{1}^{cba})_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\langle:S_{\alpha}(z):S_{\beta}(\bar{z}):\psi^{a}(x_{2}):\psi^{b}(x_{3}):\psi^{c}(x_{4}):\rangle\,,$
(55) $\displaystyle(I_{2}^{cbaed})_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\langle:S_{\alpha}(z):S_{\beta}(\bar{z}):\psi^{d}(x_{1})\psi^{e}(x_{1}):\psi^{a}(x_{2}):\psi^{b}(x_{3}):\psi^{c}(x_{4}):\rangle\,,$
(56)

with the result

$\displaystyle(I_{1}^{cba})_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\bigg{\\{}(\Gamma^{cba})_{{\alpha\beta}}-\alpha^{\prime}\eta^{ab}(\Gamma^{c})_{\alpha\beta}\,\frac{\Re(x_{25}x_{36})}{x_{23}x_{56}}+\alpha^{\prime}\eta^{ac}(\Gamma^{b})_{\alpha\beta}\,\frac{\Re(x_{25}x_{46})}{x_{24}x_{56}}$
(57)
$\displaystyle-\alpha^{\prime}\eta^{bc}(\Gamma^{a})_{\alpha\beta}\,\frac{\Re(x_{35}x_{46})}{x_{34}x_{56}}\bigg{)}\bigg{\\}}\,2^{-\frac{3}{2}}\,x_{56}^{\frac{1}{4}}\,(x_{25}x_{26}x_{35}x_{36}x_{45}x_{46})^{-\frac{1}{2}}\,,$

where $x_{5}\equiv z=x+iy\,,x_{6}\equiv\bar{z}$ and $x_{ij}=x_{i}-x_{j}$, and

$\displaystyle(I_{2}^{cbaed})_{\alpha\beta}$
$\displaystyle=2^{-\frac{5}{2}}\,x_{56}^{\frac{5}{4}}\,(x_{25}x_{26}x_{35}x_{36}x_{45}x_{46})^{-\frac{1}{2}}\,(x_{15}x_{16})^{-1}$
(58) $\displaystyle\times\bigg{\\{}(\Gamma^{cbaed})_{{\alpha\beta}}$
$\displaystyle\quad{}-\frac{1}{2}\,\alpha^{\prime}\,l_{1}\,\frac{\Re(x_{15}x_{26})}{x_{12}x_{56}}-\frac{1}{2}\,\alpha^{\prime}\,l_{2}\,\frac{\Re(x_{15}x_{36})}{x_{13}x_{56}}+\frac{1}{2}\,\alpha^{\prime}\,l_{3}\frac{\Re(x_{15}x_{46})}{x_{14}x_{56}}$
$\displaystyle\quad{}-\frac{1}{2}\,\alpha^{\prime}\,l_{4}\,\frac{\Re(x_{25}x_{36})}{x_{23}x_{56}}+\frac{1}{2}\,\alpha^{\prime}\,l_{5}\,\frac{\Re(x_{25}x_{46})}{x_{24}x_{56}}-\frac{1}{2}\,\alpha^{\prime}\,l_{6}\,\frac{\Re(x_{35}x_{46})}{x_{34}x_{56}}$
$\displaystyle-\frac{1}{4}\,(\alpha^{\prime})^{2}\,l_{7}\bigg{(}\frac{\Re(x_{15}x_{26})}{x_{12}x_{56}}\bigg{)}\bigg{(}\frac{\Re(x_{35}x_{46})}{x_{34}x_{56}}\bigg{)}$
$\displaystyle+\frac{1}{4}\,(\alpha^{\prime})^{2}\,l_{8}\bigg{(}\frac{\Re(x_{15}x_{36})}{x_{13}x_{56}}\bigg{)}\bigg{(}\frac{\Re(x_{25}x_{46})}{x_{24}x_{56}}\bigg{)}$
$\displaystyle-\frac{1}{4}\,(\alpha^{\prime})^{2}\,l_{9}\bigg{(}\frac{\Re(x_{15}x_{46})}{x_{14}x_{56}}\bigg{)}\bigg{(}\frac{\Re(x_{25}x_{36})}{x_{23}x_{56}}\bigg{)}$
$\displaystyle+\frac{1}{4}\,(\alpha^{\prime})^{2}\,l_{10}\bigg{(}\frac{\Re(x_{15}x_{26})}{x_{12}x_{56}}\bigg{)}\bigg{(}\frac{\Re(x_{15}x_{36})}{x_{13}x_{56}}\bigg{)}$
$\displaystyle+\frac{1}{4}\,(\alpha^{\prime})^{2}\,l_{11}\bigg{(}\frac{\Re(x_{15}x_{26})}{x_{12}x_{56}}\bigg{)}\bigg{(}\frac{\Re(x_{15}x_{46})}{x_{14}x_{56}}\bigg{)}$
$\displaystyle+\frac{1}{4}\,(\alpha^{\prime})^{2}\,l_{12}\bigg{(}\frac{\Re(x_{15}x_{26})}{x_{12}x_{56}}\bigg{)}\bigg{(}\frac{\Re(x_{35}x_{46})}{x_{34}x_{56}}\bigg{)}\bigg{\\}}\
,$

where indices have been suppressed and we have defined the following tensor-
bispinors:

$\displaystyle l_{1}$ $\displaystyle=$
$\displaystyle-2\eta^{ad}\Gamma^{cbe}+2\eta^{ea}\Gamma^{cbd}\ ,\qquad
l_{2}=2\eta^{db}\Gamma^{cae}-2\eta^{eb}\Gamma^{cad}\ ,$ (59) $\displaystyle
l_{3}$ $\displaystyle=$ $\displaystyle
2\eta^{dc}\Gamma^{bae}-2\eta^{ec}\Gamma^{bad}\ ,\qquad
l_{4}=2\eta^{ab}\Gamma^{ced}\ ,$ (60) $\displaystyle l_{5}$ $\displaystyle=$
$\displaystyle 2\eta^{ac}\Gamma^{bed}\ ,\qquad l_{6}=2\,\eta^{bc}\Gamma^{aed}\
,$ (61) $\displaystyle l_{7}$ $\displaystyle=$ $\displaystyle
4\eta^{ad}\eta^{bc}\Gamma^{e}-4\eta^{ea}\eta^{bc}\Gamma^{d}\ ,$ (62)
$\displaystyle l_{8}$ $\displaystyle=$ $\displaystyle
4\,\eta^{db}\eta^{ac}\Gamma^{e}-4\,\eta^{eb}\eta^{ac}\Gamma^{d}\ ,$ (63)
$\displaystyle l_{9}$ $\displaystyle=$ $\displaystyle
4\eta^{dc}\eta^{ab}\Gamma^{e}-4\eta^{ec}\eta^{ab}\Gamma^{d}\ ,$ (64)
$\displaystyle l_{10}$ $\displaystyle=$ $\displaystyle
4\eta^{da}\eta^{eb}\Gamma^{c}+4\eta^{ea}\eta^{db}\Gamma^{c}\ ,$ (65)
$\displaystyle l_{11}$ $\displaystyle=$ $\displaystyle
4\eta^{da}\eta^{ec}\Gamma^{b}-4\eta^{ea}\eta^{dc}\Gamma^{b}\ ,$ (66)
$\displaystyle l_{12}$ $\displaystyle=$
$\displaystyle-4\eta^{da}\eta^{bc}\Gamma^{d}+4\eta^{ea}\eta^{bc}\Gamma^{d}\ .$
(67)

The desired S-matrix element can then be computed, with the result

$\displaystyle A^{C^{(-1)}A^{(0)}A^{(-1)}T^{(0)}T^{(0)}}$ $\displaystyle=$
$\displaystyle 2i\mbox{\rm
Tr}\,(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})(P_{-}H\\!\\!\\!\\!/\,_{n}M_{p})^{\alpha\beta}\,\xi_{1e}\,\xi_{2a}$
(68) $\displaystyle\quad{}\times\int
dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}dx_{6}\,I\,x_{56}^{-\frac{1}{4}}\,(x_{25}x_{26})^{-\frac{1}{2}}$
$\displaystyle\quad{}\times\bigg{(}i(\alpha^{\prime})^{2}\,k_{3b}k_{4c}\bigg{[}p^{e}(\frac{x_{25}}{x_{15}x_{12}}+\frac{x_{26}}{x_{16}x_{12}})+2k^{e}_{3}\frac{x_{23}}{x_{13}x_{12}}+2k^{e}_{4}\frac{x_{24}}{x_{14}x_{12}}\bigg{]}I_{1}^{cba}$
$\displaystyle-i\,(\alpha^{\prime})^{3}k_{1e}k_{3b}k_{4c}\,I_{2}^{cbaed}\bigg{)}\
,$

where

$\displaystyle I$
$\displaystyle=|x_{12}|^{-2t}|x_{13}|^{-2s-\frac{1}{2}}|x_{14}|^{-2v-\frac{1}{2}}|x_{23}|^{-2u-\frac{1}{2}}|x_{24}|^{-2r-\frac{1}{2}}|x_{34}|^{-2w-1}|x_{15}x_{16}|^{t+s+v+\frac{1}{2}}$
(69)
$\displaystyle\quad{}\times|x_{25}x_{26}|^{t+u+r+\frac{1}{2}}|x_{35}x_{36}|^{s+u+w+\frac{1}{2}}|x_{45}x_{46}|^{v+r+w+\frac{1}{2}}|x_{56}|^{-2(s+t+u+v+r+w)-2}\,,$

in terms of the six independent Mandelstam variables

$\displaystyle s$ $\displaystyle=$ $\displaystyle-(\frac{1}{4}+2k_{1}\cdot
k_{3})\ ,\qquad t=-2k_{1}\cdot k_{2}\ ,\qquad v=-(\frac{1}{4}+2k_{1}\cdot
k_{4})\ ,$ (70) $\displaystyle u$ $\displaystyle=$
$\displaystyle-(\frac{1}{4}+2k_{2}\cdot k_{3})\ ,\qquad
r=-(\frac{1}{4}+2k_{2}\cdot k_{4})\ ,\qquad w=-(\frac{1}{2}+2k_{3}\cdot
k_{4})\ .$ (71)

Since the amplitude has been written so that it respects $SL(2,R)$ invariance
manifestly, we can fix the positions of three open-string vertex operator; we
choose

$x_{1}=0\,,\qquad 0\leq x_{2}\leq 1\,,\qquad x_{3}=1\,,\qquad x_{4}=\infty\,,$
(72)

which brings the final amplitude to the form

$\displaystyle-8k_{3b}k_{4c}\xi_{1e}\xi_{2a}2^{-\frac{1}{2}}(P_{-}H\\!\\!\\!\\!/\,_{n}M_{p})^{\alpha\beta}\int_{0}^{1}dx_{2}x_{2}^{-2t}(1-x_{2})^{-2u-\frac{1}{2}}$
$\displaystyle\times\int
d^{2}z|1-z|^{2s+2u+2w}\,|z|^{2t+2s+2v-1}(z-\bar{z})^{-2(t+s+u+v+r+w)-2}|x_{2}-z|^{2t+2u+2r-1}$
$\displaystyle\times\bigg{\\{}\bigg{(}p^{e}x^{-1}_{2}\frac{x_{2}(z+\bar{z})-2|z|^{2}}{|z|^{2}}+2k^{e}_{3}(x_{2}-1)-2k^{e}_{4}\bigg{)}$
(73)
$\displaystyle\times\bigg{(}(z-\bar{z})(\Gamma^{cba})_{{\alpha\beta}}+2\eta^{ab}\,(\Gamma^{c})_{\alpha\beta}\frac{x_{2}-x\,x_{2}-x+|z|^{2}}{(1-x_{2})}$
$\displaystyle+2\eta^{ac}(\Gamma^{b})_{\alpha\beta}(x-x_{2})+2\,\eta^{bc}\,(\Gamma^{a})_{\alpha\beta}(1-x)\bigg{)}$
$\displaystyle-2k_{1d}\bigg{[}(z-\bar{z})(\Gamma^{cbaed})_{\alpha\beta}+l_{1}\,\frac{-x\,x_{2}+|z|^{2}}{x_{2}}+l_{2}\left(-x+|z|^{2}\right)+l_{3}\,x$
$\displaystyle+l_{4}\,\frac{x_{2}-x\,x_{2}-x+|z|^{2}}{(1-x_{2})}+l_{5}\left(x-x_{2}\right)+l_{6}\left(1-x\right)$
$\displaystyle+(z-\bar{z})^{-1}\bigg{(}l_{7}\,(-1+x)\,\frac{-x\,x_{2}+|z|^{2}}{x_{2}}+l_{8}\left(-x+|z|^{2}\right)(x_{2}-x)$
$\displaystyle+l_{9}x\frac{x_{2}-xx_{2}-x+|z|^{2}}{(1-x_{2})}\bigg{)}+l_{10}x\frac{-x+|z|^{2}}{(-1+x_{2})}$
$\displaystyle+l_{11}x\frac{x-|z|^{2}}{(x_{2})}+l_{12}(-x+|z|^{2}+x^{2}-x|z|^{2})\bigg{]}\bigg{\\}}\mbox{\rm
Tr}\,(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})\,.$ (74)

The total amplitude can be expressed in terms of the integrals

$\displaystyle A_{xz}(a,b,c,d\mid\alpha,\beta\mid\epsilon)$
$\displaystyle\equiv\int
d^{2}z\int_{0}^{1}dx\,|{1-z}|^{a}|{z}|^{b}(z-\bar{z})^{c}(z+\bar{z})^{d}$ (75)
$\displaystyle\times(1-x)^{\alpha}x^{\beta}\,|{x-z}|^{\epsilon},$

whose properties were described in detail in [44]. As explained in [44], one
can work with the soft limit of the six point function as $4k_{2}\cdot
p\rightarrow 1$ which gives an analytic expression for the amplitude, and some
of the integrals can be found in Appendix B of [42]. They are subject to the
on-shell condition

$s+t+u+v+r+w=-\,p^{a}p_{a}-1\,,$ (76)

In what follows, we shall determine quartic couplings in the EFT action by
subtracting EFT exchange graphs with massless poles from the $TTAAC$ disc
amplitude in the limit where $k_{i}{\cdot}k_{j}\rightarrow 0$ for $i\neq j$.
To investigate the structure of poles, we consider the correct limit of the
Mandelstam variables to set up the low-energy expansion where the correct
expansion is

$\displaystyle t\rightarrow 0\,,\quad
s\,,v\,,-\,p^{a}p_{a}\rightarrow-\frac{1}{4}\,,\quad
w\rightarrow-\frac{1}{2}\,,$ $\displaystyle\frac{1}{2}\left[(u\rightarrow
0\,,r\rightarrow-\frac{1}{4})+(u\rightarrow-\frac{1}{4}\,,r\rightarrow
0)\right]\,.$ (77)

where in the last line we took the averaged over the two limits and (4) is
consistent with momentum conservation, and conforms to the symmetries of the
S-matrix. Note that in the limit $-p^{a}p_{a}\rightarrow-\frac{1}{4}$, we
conform to the RR-tachyon two-point function, for which
$p^{a}p_{a}=\frac{1}{4}$ is a constraint.

Next, we derive the precise form of the all-order corrections to the couplings
of two tachyons and two gauge fields with exact coefficients in the
worldvolume of the D$\overline{\rm D}$ system. We will then show that they
reproduce the all-order gauge field poles in an EFT analysis. By algebraic
computations, we will also show that the corresponding existing proposal in
the literature is inconsistent.

The string amplitude is non-vanishing in the case of RR potentials given by
$C_{p-5}$, $C_{p-3}$ and $C_{p-1}$. In the $C_{p-5}$ case, the amplitude has
just contact terms as follows:

$\displaystyle
A_{1}=\frac{64i2^{-\frac{1}{2}}\pi}{(p-4)!}N_{1}N_{2}k_{4c}k_{3b}\xi_{1e}\xi_{2a}k_{1d}\,\epsilon^{a_{0}\ldots
a_{p-5}cbaed}H_{a_{0}\ldots a_{p-5}}\ ,$ (78)

where

$\displaystyle N_{1}$ $\displaystyle=$ $\displaystyle
2^{-2(t+s+u+v+r+w)-1}\frac{\Gamma(-2u+\frac{1}{2})\
\Gamma(-2t+1)}{\Gamma(-2t-2u+\frac{3}{2})}\ ,$ (79) $\displaystyle N_{2}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(-u-r-w)\,\Gamma(-t-v-r+\frac{1}{2})\,\Gamma(r-s)\,\Gamma(-t-s-
u-v-r-w)}{\Gamma(-u-s-w)\,\Gamma(-t-s-v+\frac{1}{2})\,\Gamma(-u-w-
t-v-2r+\frac{1}{2})}\ .$ (80)

It is readily seen that the leading contact terms can be reproduced by the EFT
couplings

$\displaystyle\mu_{p}(2\pi\alpha^{\prime})^{4}\int\sum_{n=-1}^{\infty}b_{n}(\alpha^{\prime})^{n+1}C_{p-5}\wedge
F\wedge F\wedge DT_{1}\wedge DT_{2}\ .$ (81)

Other contact terms can be reproduced by properly adding higher derivative
corrections; for more details see [44]. The amplitude has infinitely many
massless poles in the $C_{p-3}$ case, and there are infinitely many tachyon
poles in the $C_{p-1}$ case, where, for the sake of obtaining correct higher
derivative corrections, it is sufficient to reconstruct the infinite set of
gauge field poles.

For $p=n+2$, the amplitude is given by

$\displaystyle{\cal A}^{AATTC_{p-1}}$ $\displaystyle=$
$\displaystyle\frac{8i\mu_{p}}{\sqrt{2\pi}(p-2)!}\epsilon^{a_{0}\cdots
a_{p-1}a}H_{a_{0}\cdots a_{p-3}}k_{2a_{p-2}}\xi_{a_{p-1}}$ (82)
$\displaystyle\times\bigg{(}k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}\bigg{)}N_{3}N_{4}\
,$

where $N_{3},N_{4}$ are the multiplications of various Gamma functions; for
their explicit forms, see Appendix B of [44].

Having taken the proper expansion, one can show that all massless poles in the
string amplitude are given by

$\displaystyle\frac{8i\mu_{p}}{\sqrt{2\pi}(p-2)!}\frac{1}{(w^{\prime}+r^{\prime}+u^{\prime})}\epsilon^{a_{0}\cdots
a_{p-1}a}H_{a_{0}\cdots a_{p-3}}k_{2a_{p-2}}\xi_{a_{p-1}}$
$\displaystyle\bigg{(}k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}\bigg{)}\sum_{n,m=0}^{\infty}d_{n,m}(u^{\prime}+r^{\prime})^{n}(u^{\prime}r^{\prime})^{m}\
,$ (83)

using momentum conservation in directions parallel to the branes,
$(k_{1}+k_{2}+k_{3}+k_{4}+p)^{a}=0$, and the Bianchi identity

$p_{b}\,H_{a_{0}\cdots a_{p-4}}\epsilon^{a_{0}\cdots a_{p-1}b}=0\,.$ (84)

##  5 All-Order Two-Tachyon-Two-Gauge Field Couplings

In this section, we determine the all-order part of the EFT Lagrangian on the
worldvolume of a D$\overline{\rm D}$ system that is separately quadratic in
gauge fields and tachyons, following the procedure used in [45]666The S-matrix
element for scattering of one RR particle and four tachyons on a
D$\overline{\rm D}$-brane is given in [45], and its relation to the Veneziano
amplitude [46] is spelled out in there as well. Moreover, the all-order
corrections to fermionic couplings [47] as well as a power series
representation of the corresponding string amplitudes were obtained well
beyond the factorized limit in [44] . to compute the highly-symmetric four-
tachyon vertices to all orders in $\alpha^{\prime}$ in the non-BPS case; we
shall then verify the result by comparing the resulting gauge field poles
against those of the corresponding string amplitude. Our results modify and
complete previous simpler proposals [41], whose subtle shortcomings we shall
thus begin by highlighting before turning to the main part of this section.

### Old proposal.

The vertices $V_{a}(A,A^{(1)},T_{1},T_{2})$ stem from the part of the EFT
Lagrangian of the D$\overline{\rm D}$ system that is separately quadratic in
gauge fields and tachyons. In what follows, we will first review the tachyon-
two gauge field EFT couplings of non-BPS branes that have been proposed in the
literature [41], and show that these do not reproduce the desired string disc
amplitude in the case of the D$\overline{\rm D}$ system, after which we shall
provide a remedy in the form of a modified set of EFT higher-derivative
couplings for this system. To spell out the couplings, it is useful to define
the quadri-linear higher-derivative operators777If $D$ is a differential
operator acting in an algebra of functions, then we use the convention
$Dfg\equiv(Df)g$.

$\displaystyle D_{nm}(E,F,G,H)$ $\displaystyle:=$ $\displaystyle
D_{b_{1}}\cdots D_{b_{m}}D_{a_{1}}\cdots D_{a_{n}}EFD^{a_{1}}\cdots
D^{a_{n}}GD^{b_{1}}\cdots D^{b_{m}}H\ ,$ (85) $\displaystyle
D^{\prime}_{nm}(E,F,G,H)$ $\displaystyle:=$ $\displaystyle D_{b_{1}}\cdots
D_{b_{m}}D_{a_{1}}\cdots D_{a_{n}}ED^{a_{1}}\cdots D^{a_{n}}FGD^{b_{1}}\cdots
D^{b_{m}}H\ .$ (86)

The existing proposal for the EFT Lagrangian of D$\overline{\rm D}$ systems
consists of the leading couplings

$\displaystyle T_{p}(\pi\alpha^{\prime})^{3}{\rm TrS}\left(\frac{}{}m^{2}{\cal
T}^{2}F_{\mu\nu}F^{\nu\mu}+\frac{}{}D^{\alpha}{\cal T}D_{\alpha}{\cal
T}F_{\mu\nu}F^{\nu\mu}-4F^{\mu\alpha}F_{\alpha\beta}D^{\beta}{\cal
T}D_{\mu}{\cal T}\right)\ ,$ (87)

at order $(\alpha^{\prime})^{3}$ and where TrS denotes the symmetrised trace,
combined with the higher-order couplings

$\displaystyle{\cal L}^{TTAA}_{\rm old\,EFT}$ $\displaystyle=$ $\displaystyle-
T_{p}(\pi\alpha^{\prime})\sum_{n,m=0}^{\infty}(\alpha^{\prime})^{2+n+m}({\cal
L}_{{\rm old}\,1}^{nm}+{\cal L}_{{\rm old}\,2}^{nm}+{\cal L}_{{\rm
old}\,3}^{nm}+{\cal L}^{nm}_{{\rm old}\,4})\ ,$ (88)

where888The first few $a_{n,m}$ and $b_{n,m}$ coefficients are given by
$\displaystyle a_{0,0}=-\frac{\pi^{2}}{6},\,b_{0,0}=-\frac{\pi^{2}}{12}$ (89)
$\displaystyle a_{1,0}=2\zeta(3),\,a_{0,1}=0,\,b_{0,1}=b_{1,0}=-\zeta(3)$
$\displaystyle
a_{1,1}=a_{0,2}=-7\pi^{4}/90,\,a_{2,0}=-4\pi^{4}/90,\,b_{1,1}=-\pi^{4}/180,\,b_{0,2}=b_{2,0}=-\pi^{4}/45$
$\displaystyle
a_{1,2}=a_{2,1}=8\zeta(5)+4\pi^{2}\zeta(3)/3,\,a_{0,3}=0,\,a_{3,0}=8\zeta(5),$
$\displaystyle\qquad\qquad\qquad\qquad\qquad
b_{0,3}=-4\zeta(5),\,b_{1,2}=-8\zeta(5)+2\pi^{2}\zeta(3)/3$

$\displaystyle{\cal L}_{{\rm old}\,1}^{nm}$ $\displaystyle=$ $\displaystyle
m^{2}{\rm TrS}\left(\frac{}{}a_{n,m}[D_{nm}({\cal T},{\cal
T},F_{\mu\nu},{F}^{\nu\mu})+D_{nm}(F_{\mu\nu},{F}^{\nu\mu},{\cal T},{\cal
T})]\right.$ (90) $\displaystyle\left.+\frac{}{}b_{n,m}[D^{\prime}_{nm}({\cal
T},F_{\mu\nu},{\cal T},F^{\nu\mu})+D^{\prime}_{nm}(F_{\mu\nu},{\cal
T},F^{\nu\mu},{\cal T})]+c.c.\right)\ ,$ $\displaystyle{\cal L}_{{\rm
old}\,2}^{nm}$ $\displaystyle=$ $\displaystyle{\rm
TrS}\left(\frac{}{}a_{n,m}[D_{nm}(D^{\alpha}{\cal T},D_{\alpha}{\cal
T},F_{\mu\nu},{F}^{\nu\mu})+D_{nm}(F_{\mu\nu},{F}^{\nu\mu},D^{\alpha}{\cal
T},D_{\alpha}{\cal T})]\right.$ (91)
$\displaystyle\left.+\frac{}{}b_{n,m}[D^{\prime}_{nm}(D^{\alpha}{\cal
T},F_{\mu\nu},D_{\alpha}{\cal
T},F^{\nu\mu})+D^{\prime}_{nm}(F_{\mu\nu},D_{\alpha}{\cal
T},F^{\nu\mu},D^{\alpha}{\cal T})]+c.c.\right)\ ,$ $\displaystyle{\cal
L}_{{\rm old}\,3}^{nm}$ $\displaystyle=$ $\displaystyle-2{\rm
TrS}\left(\frac{}{}a_{n,m}[D_{nm}(D^{\beta}{\cal T},D_{\mu}{\cal
T},{F}^{\mu\alpha},F_{\alpha\beta})+D_{nm}({F}^{\mu\alpha},F_{\alpha\beta},D^{\beta}{\cal
T},D_{\mu}{\cal T})]\right.$ (92)
$\displaystyle\left.+\frac{}{}b_{n,m}[D^{\prime}_{nm}(D^{\beta}{\cal
T},{F}^{\mu\alpha},D_{\mu}{\cal
T},F_{\alpha\beta})+D^{\prime}_{nm}({F}^{\mu\alpha},D_{\mu}{\cal
T},F_{\alpha\beta},D^{\beta}{\cal T})]+c.c.\right)\ ,\quad$
$\displaystyle{\cal L}_{{\rm old}\,4}^{nm}$ $\displaystyle=$
$\displaystyle-2{\rm TrS}\left(\frac{}{}a_{n,m}[D_{nm}(D^{\beta}{\cal
T},D_{\mu}{\cal
T},F_{\alpha\beta},{F}^{\mu\alpha})+D_{nm}(F_{\alpha\beta},{F}^{\mu\alpha},D^{\beta}{\cal
T},D_{\mu}{\cal T})]\right.$ (93)
$\displaystyle\left.+\frac{}{}b_{n,m}[D^{\prime}_{nm}(D^{\beta}{\cal
T},F_{\alpha\beta},D_{\mu}{\cal
T},{F}^{\mu\alpha})+D^{\prime}_{nm}(F_{\alpha\beta},D_{\mu}{\cal
T},{F}^{\mu\alpha},D^{\beta}{\cal T})]+c.c.\right)\ .$

Using ${\cal L}^{TTAA}_{\rm old}$, the EFT sub-amplitude

$\displaystyle{\cal A}^{CAATT}_{\rm gauge-ch}$ $\displaystyle=$ $\displaystyle
V^{a}(C_{p-3},A,A)G_{ab}(A)V^{b}(A,A,T,T)$ (94)

contains the following infinite set of gauge poles

$\displaystyle 8i\mu_{p}\epsilon^{a_{0}\cdots a_{p-1}a}H_{a_{0}\cdots
a_{p-3}}k_{1a_{p-2}}\xi_{1a_{p-1}}\frac{1}{(p-2)!(u^{\prime}+r^{\prime}+w^{\prime})}\sum_{n,m=0}^{\infty}(a_{n,m}+b_{n,m})$
(95) $\displaystyle(u^{\prime m}r^{\prime n}+u^{\prime n}r^{\prime
m})[k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}]\
.$

At zeroth order in $\alpha^{\prime}$, the residue is given by

$\displaystyle 4u^{\prime}r^{\prime}(a_{0,0}+b_{0,0})$ $\displaystyle=$
$\displaystyle-\pi^{2}u^{\prime}r^{\prime}\ ,$ (96)

while the corresponding quantity in the disc amplitude is given by

$\displaystyle
d_{0,0}u^{\prime}r^{\prime}=\frac{-\pi^{2}}{3}u^{\prime}r^{\prime}=d_{0,0}u^{\prime}r^{\prime}\
.$ (97)

Clearly, the old EFT proposal does not reproduce the string result. Indeed,
this discrepancy prevails at higher orders in the $\alpha^{\prime}$-expansion;
for example, at the first order in $\alpha^{\prime}$, the EFT residue is given
by

$\displaystyle
2u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})(a_{1,0}+a_{0,1}+b_{1,0}+b_{0,1})$
$\displaystyle=$ $\displaystyle 0\ ,$ (98)

in disagreement with the string theory results, which reads

$\displaystyle
d_{1,0}u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})=8\xi(3)u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})\
.$ (99)

In other words, the proposed EFT coulings do not apply to the D$\overline{\rm
D}$ configuration at hand.

### New proposal.

An improved EFT Lagrangian for the D$\overline{\rm D}$ system can be found
using the method of [45]; the sought for all-order expression for the DBI–SYM
Lagrangian is given by

$\displaystyle{\cal L}_{DBI}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-
T_{p}(2\pi\alpha^{\prime})\left(m^{2}|T|^{2}+DT\cdot(DT)^{*}-\frac{\pi\alpha^{\prime}}{2}\left(F^{(1)}\cdot{F^{(1)}}+F^{(2)}\cdot{F^{(2)}}\right)\right)+T_{p}(\pi\alpha^{\prime})^{3}$
(100)
$\displaystyle\times\left(\frac{2}{3}DT\cdot(DT)^{*}\left(F^{(1)}\cdot{F^{(1)}}+F^{(1)}\cdot{F^{(2)}}+F^{(2)}\cdot{F^{(2)}}\right)\right.$
$\displaystyle\left.+\frac{2m^{2}}{3}|T|^{2}\left(F^{(1)}\cdot{F^{(1)}}+F^{(1)}\cdot{F^{(2)}}+F^{(2)}\cdot{F^{(2)}}\right)\right.$
$\displaystyle-\left.\frac{4}{3}\left((D^{\mu}T)^{*}D_{\beta}T+D^{\mu}T(D_{\beta}T)^{*}\right)\left({F^{(1)}}^{\mu\alpha}F^{(1)}_{\alpha\beta}+{F^{(1)}}^{\mu\alpha}F^{(2)}_{\alpha\beta}+{F^{(2)}}^{\mu\alpha}F^{(2)}_{\alpha\beta}\right)\right)\
,$

at the leading order in the $\alpha^{\prime}$-expansion, and

$\displaystyle{\cal L}^{TTAA}_{\rm EFT}$ $\displaystyle=$ $\displaystyle-
T_{p}(\pi\alpha^{\prime})\sum_{n,m=0}^{\infty}(\alpha^{\prime})^{2+n+m}({\cal
L}_{1}^{nm}+{\cal L}_{2}^{nm}+{\cal L}_{3}^{nm}+{\cal L}^{nm}_{4})\ ,$ (101)

to higher orders, where

$\displaystyle{\cal L}_{1}^{nm}$ $\displaystyle=$ $\displaystyle m^{2}{\rm
TrS}\left(\frac{}{}a_{n,m}[D_{nm}(T,T^{*},F^{(1)}_{\mu\nu},{F}^{(1)\nu\mu})+D_{nm}(F^{(1)}_{\mu\nu},{F}^{(1)\nu\mu},T,T^{*})]\right.$
(102)
$\displaystyle\left.-\frac{}{}b_{n,m}[D^{\prime}_{nm}(T,F^{(2)}_{\mu\nu},T^{*},F^{(1)\nu\mu})+D^{\prime}_{nm}(F^{(1)}_{\mu\nu},T,F^{(2)\nu\mu},T^{*})]+c.c.\right)\
,$ $\displaystyle{\cal L}_{2}^{nm}$ $\displaystyle=$ $\displaystyle{\rm
TrS}\left(\frac{}{}a_{n,m}[D_{nm}(D^{\alpha}T,D_{\alpha}T^{*},F^{(1)}_{\mu\nu},{F}^{(1)\nu\mu})+D_{nm}(F^{(1)}_{\mu\nu},{F}^{(1)\nu\mu}D^{\alpha}T,D_{\alpha}T^{*})]\right.$
(103)
$\displaystyle\left.-\frac{}{}b_{n,m}[D^{\prime}_{nm}(D^{\alpha}T,F^{(2)}_{\mu\nu},D_{\alpha}T^{*},F^{(1)\nu\mu})+D^{\prime}_{nm}(F^{(1)}_{\mu\nu},D_{\alpha}T,F^{(2)\nu\mu},D^{\alpha}T^{*})]+c.c.\right)\
,$ $\displaystyle{\cal L}_{3}^{nm}$ $\displaystyle=$ $\displaystyle-2{\rm
TrS}\left(\frac{}{}a_{n,m}[D_{nm}(D^{\beta}T,D_{\mu}T^{*},{F}^{(1)\mu\alpha},F^{(1)}_{\alpha\beta})+D_{nm}({F}^{(1)\mu\alpha},F^{(1)}_{\alpha\beta},D^{\beta}T,D_{\mu}T^{*})]\right.$
(104)
$\displaystyle\left.-\frac{}{}b_{n,m}[D^{\prime}_{nm}(D^{\beta}T,{F}^{(2)\mu\alpha},D_{\mu}T^{*},F^{(1)}_{\alpha\beta})+D^{\prime}_{nm}({F}^{(1)\mu\alpha},D_{\mu}T,F^{(2)}_{\alpha\beta},D^{\beta}T^{*})]+c.c.\right)\
,$ $\displaystyle{\cal L}_{4}^{nm}$ $\displaystyle=$ $\displaystyle-2{\rm
TrS}\left(\frac{}{}a_{n,m}[D_{nm}(D^{\beta}T,D_{\mu}T^{*},F^{(1)}_{\alpha\beta},{F}^{(1)\mu\alpha})+D_{nm}(F^{(1)}_{\alpha\beta},{F}^{(1)\mu\alpha},D^{\beta}T,D_{\mu}T^{*})]\right.$
(105)
$\displaystyle\left.-\frac{}{}b_{n,m}[D^{\prime}_{nm}(D^{\beta}T,F^{(2)}_{\alpha\beta},D_{\mu}T^{*},{F}^{(1)\mu\alpha})+D^{\prime}_{nm}(F^{(1)}_{\alpha\beta},D_{\mu}T,{F}^{(2)\mu\alpha},D^{\beta}T^{*})]+c.c.\right)\
,\qquad\quad$

where $b_{n,m}=b_{m,n}$, and $F^{(1)}$ and $F^{(2)}$, respectively, are the
gauge field strengths on the D-brane and $\overline{D}$-brane. We note that
unless the field strengths on the D- and $\overline{D}$-brane are treated
separately, the EFT residues in the gauge channel cannot be made to match the
string results even to the leading order in $\alpha^{\prime}$.

The modification of the EFT Lagrangian proposed here can also be motivated by
the structure of EFT couplings on the D$\overline{\rm D}$ system including
fermions, viz. $\bar{\psi}^{2}\gamma^{a}T\psi^{1}D_{a}T$ and
$\psi^{1}D_{a}T\bar{\psi}^{2}\gamma^{a}T$, found recently in [47].

### Reproduction of gauge field poles.

Let us consider again the EFT sub-amplitudes with massless gauge fields poles,
viz.

$\displaystyle{\cal A}^{CAATT}_{\rm EFT}$ $\displaystyle=$ $\displaystyle
V_{a}(C_{p-3},A,A^{(1)})G_{ab}(A)V_{b}(A,A^{(1)},T_{1},T_{1})\ ,$ (106)

where, working perturbatively, the vertex $V_{a}(C_{p-1},A)$ and propagator
$G_{ab}(A)$ are given, provided that one uses the following gauge propagator
and vertices:

$\displaystyle G_{ab}(A)$ $\displaystyle=$
$\displaystyle\frac{i\delta_{ab}}{(2\pi\alpha^{\prime})^{2}T_{p}(u^{\prime}+r^{\prime}+w^{\prime})}\
,$ (107) $\displaystyle V_{a}(C_{p-3},A,A^{(1)})$ $\displaystyle=$
$\displaystyle\frac{\mu_{p}(2\pi\alpha^{\prime})^{2}}{(p-2)!}\epsilon_{a_{0}\cdots
a_{p-1}a}H^{a_{0}\cdots
a_{p-3}}k_{1}^{a_{p-2}}\xi^{a_{p-1}}\sum_{n=-1}^{\infty}b_{n}(\alpha^{\prime}k_{1}\cdot
k)^{n+1}\ ,$ (108) $\displaystyle V_{a}(C_{p-3},A,A^{(2)})$ $\displaystyle=$
$\displaystyle-\frac{\mu_{p}(2\pi\alpha^{\prime})^{2}}{(p-2)!}\epsilon_{a_{0}\cdots
a_{p-1}a}H^{a_{0}\cdots
a_{p-3}}k_{2}^{a_{p-2}}\xi^{a_{p-1}}\sum_{n=-1}^{\infty}b_{n}(\alpha^{\prime}k_{2}\cdot
k)^{n+1}\ ,\quad$ (109)

where $k$ is the momentum of the off-shell gauge field. Thus, the all-order
form of the vertex $V_{a}(C_{p-3},A,A^{(1)})$ is determined from

$\displaystyle\mu_{p}(2\pi\alpha^{\prime})^{2}\sum_{n=-1}^{\infty}b_{n}(\alpha^{\prime})^{n+1}C_{p-3}\wedge
D^{a_{0}}...D^{a_{n}}F\wedge D_{a_{0}}...D_{a_{n}}F\ .$ (110)

To the leading order in the $\alpha^{\prime}$-expansion, we take the vertex
$V_{a}(A,A^{(1)},T_{1},T_{2})$ from the second line of (100), with the result

$\displaystyle V_{a}(A^{(1)},A^{(1)},T_{1},T_{1})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
2iT_{p}(\pi\alpha^{\prime})^{3}\left[\frac{2}{3}k_{a}\left[r^{\prime}(2k_{3}\cdot\xi)+u^{\prime}(2k_{4}\cdot\xi)\right]\right.$
$\displaystyle\left.\phantom{\frac{2}{3}}+k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}\right]\
,$ $\displaystyle V_{a}(A^{(2)},A^{(1)},T_{1},T_{2})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
2iT_{p}(\pi\alpha^{\prime})^{3}\left[\frac{1}{3}k_{a}\left[r^{\prime}(2k_{3}\cdot\xi)+u^{\prime}(2k_{4}\cdot\xi)\right]+\right.$
$\displaystyle\left.\phantom{\frac{1}{3}}+k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}\right]\
,$

where $k^{a}$ is the momentum of the off-shell gauge field. Indeed,
substituting the above vertices into (106) reproduces the first massless
singularity of (83).

Note that the correction to the vertex is read off by

$\displaystyle-2\alpha^{\prime}\mu_{p}\sum_{n=0}^{\infty}a_{n}\left(\frac{\alpha^{\prime}}{2}\right)^{n}C_{p-1}\wedge(D^{a}D_{a})^{n}[(F^{(1)}-F^{(2)})|T|^{2}]\
,$ (113)

which is the extension of the couplings that are given in (8).

Proceeding to higher orders, we use the two-tachyon-two-gauge field couplings
from (101), which read

$\displaystyle 4iT_{p}(\alpha^{\prime})^{n+m}(a_{n,m}-b_{n,m})\mbox{\rm
Tr}\,(\lambda_{2}\lambda_{3}\lambda_{4}\Lambda^{\beta})\bigg{[}\bigg{(}(k_{2}\\!\cdot\\!k_{3})^{m}(k_{2}\\!\cdot\\!k_{4})^{n}+(k_{2}\\!\cdot\\!k_{3})^{n}(k_{2}\\!\cdot\\!k_{4})^{m}$
$\displaystyle(k_{1}\\!\cdot\\!k)^{m}(k_{2}\\!\cdot\\!k)^{n}+(k_{3}\\!\cdot\\!k)^{n}(k_{2}\\!\cdot\\!k)^{m}\bigg{]}[k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}]\
,$ (114)

where $k^{a}$ is the off-shell gauge field momentum. The above vertices yield
the following infinite set of massless poles in the EFT:

$\displaystyle 8i\mu_{p}\epsilon^{a_{0}\cdots a_{p-1}a}H_{a_{0}\cdots
a_{p-3}}k_{1a_{p-2}}\xi_{1a_{p-1}}\frac{1}{(p-2)!(u^{\prime}+r^{\prime}+w^{\prime})}\sum_{n,m=0}^{\infty}(a_{n,m}-b_{n,m})$
(115) $\displaystyle(u^{\prime m}r^{\prime n}+u^{\prime n}r^{\prime
m})[k_{3a}u^{\prime}(2\xi.k_{4})+k_{4a}r^{\prime}(2\xi.k_{3})-\xi_{a}u^{\prime}r^{\prime}]\
,$

which indeed agree with the massless singularities of the corresponding string
amplitude (83); for example, in the leading order, that is, for $n=m=0$, the
string amplitude produces $4d_{0,0}u^{\prime}r^{\prime}$, while at zeroth
order in $\alpha^{\prime}$ the EFT yields

$\displaystyle 16u^{\prime}r^{\prime}(a_{0,0}-b_{0,0})$ $\displaystyle=$
$\displaystyle\frac{-4\pi^{2}}{3}u^{\prime}r^{\prime}\ ,$ (116)

which is identical to $4d_{0,0}u^{\prime}r^{\prime}$. At first order of
$\alpha^{\prime}$, the string amplitude yields
$4d_{1,0}u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})$, and in the modified EFT
we obtain

$\displaystyle
8u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})(a_{1,0}+a_{0,1}-b_{1,0}-b_{0,1})$
$\displaystyle=$ $\displaystyle
32\xi(3)u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})=4d_{1,0}u^{\prime}r^{\prime}(u^{\prime}+r^{\prime})\
,$ (117)

in perfect agreement with worldsheet computations. One can check that the
equivalence holds to all orders. Hence, all gauge field poles of the
D$\overline{\rm D}$ system are reconstructed from the proposal (101) for the
all-order two-tachyon-two-gauge field couplings of EFT Lagrangian in an exact
and ambiguitiy-free fashion.

In particular, we have found that it is necessary to embed the mixed coupling

$\displaystyle F^{(1)}\cdot F^{(2)}$ (118)

into the tachyon-extended DBI–SYM action (100) to match EFT and string
amplitudes.

##  6 Conclusion

In summary, using worldsheet methods, we have obtained the two-tachyon-two-
gauge field couplings in (101) of the EFT on the worldvolume of
D$\overline{\rm D}$ systems to all orders in $\alpha^{\prime}$.

Working perturbatively, whereby all kinetic terms remain uncorrected (to
leading order in $g_{s}$), the massless and tachyonic singularities shed light
on the structure of higher-derivative corrections to couplings, and help in
determining the exact coefficients of the aforementioned couplings. In
particular, we have found that already at the leading order, the DBI-SYM
action (100) contains $F^{(1)}F^{(2)}$-couplings in order for the EFT to be
consistent with the worldsheet999In previous works, it has been shown that
$D\phi^{i(1)}\cdot D\phi_{i(2)}$-couplings are required on the worldvolume of
D$\overline{\rm D}$ systems for consistency between EFT and worldsheet
descriptions [48].. We have also found that the $C_{p-1}\wedge F$ coupling of
the WZ term does not receive any higher-derivative correction.

Let us comment on how our results may provide a guide along future directions
towards new ideas in the context of non-BPS branes.

Note that if one uses the ordinary trace effective action [49] instead of (4),
the resulting effective action cannot generate all-order singularities of the
string amplitudes.

We note that the tachyon potential that we used is the same as in boundary
string field theory (BSFT), i.e.,

$\displaystyle V(T^{i}T^{i})$ $\displaystyle=$ $\displaystyle
1+\pi\alpha^{\prime}m^{2}{T^{i}T^{i}}+\frac{1}{2}(\pi\alpha^{\prime}m^{2}{T^{i}T^{i}})^{2}+\cdots\
,$ (119)

where $m_{T}^{2}=-1/(2\alpha^{\prime})$ and $T_{p}$ is the tension of a
$p$-brane. The expansion complies with the first couple of terms of the
potential $V(|T|)=e^{\pi\alpha^{\prime}m^{2}|T|^{2}}$ taken from BSFT [50,
51].

If one just uses the derived effective actions of this paper, one can
precisely generate all infinite singularities and contact terms of string
theory. Indeed we also discussed that the presence of new mixed couplings like
$F^{(1)}\cdot{F^{(2)}}$, as well as $D\phi^{(1)}\cdot{D\phi^{(2)}}$, are
inevitable and necessary to derive the actual and consistent results that are
demanded when matching string computations with EFT.

Combining EFT techniques with stringy symmetries [52], whereby the S-matrix
must be symmetric under exchanging $s$ and $t$, and have massless $u$-channel
poles, one learns that all $k_{i}.k_{j}\rightarrow 0$, and hence we revealed
that the unique expansion $(u\rightarrow 0,\,s,t\rightarrow-1/4)$ for the five
point function.

Carrying out the symmetrized trace in DBI action over $\sigma$ factors, one
finds

$\displaystyle\frac{1}{2}{\rm
TrS}\left(V({T^{i}T^{i}})\sqrt{1+[T^{i},T^{j}][T^{j},T^{i}]}\right)$
$\displaystyle=$
$\displaystyle\left(1-\frac{\pi}{2}T^{2}+\frac{\pi^{2}}{24}T^{4}+\cdots\right)\left(1+T^{4}+\cdots\right)\
.$ (120)

Hence, the tachyon gets condensated at $T\rightarrow\infty$, thereby the
tachyon potential tends to zero precisely.

It is natural to think of the all-order highly symmetric EFTs as counterparts
to the holographic duals of Vasiliev’s higher spin gravitys [23] in their turn
thought of as consistent truncations of FCS models; for a review, see [53].
The latter are formulated in terms of horizontal Quillen superconnections on
noncommutative fibered manifolds with star-product local FCS action
functionals in which all semi-classical nonlocalities have been incorporated
into differential graded algebras. The differential graded algebras serve as
natural configuration spaces for noncommutative sigma models of
Alexandrov–Kontsevich–Schwarz–Zaboronsky (AKSZ) type [54]; Vasiliev’s higher
spin geometries subject to boundary conditions then arise on-shell as strict
operator realizations of these algebras in representations.

The FCS models contain defects of co-dimension one in their turn containing
different sub-defects in various co-dimensions; thus, the FCS models serve as
“grandparents” for cascades of noncommutative gauge theories of AKSZ type. It
is natural to expect these cascades to induce chains of dualities of
topological field theory type between sub-algebras inhabiting the sub-defects,
treated as objects of a geometric category, connected by algebraic morphisms
mediated by the theory inhabitating the parent defect, treated as morphisms of
the geometric category.

In these schemes, Vasiliev’s theory plays the role of a parent theory arising
in the FCS model in co-dimension one in its turn containing different sub-
defects in various co-subdimensions. In particular, in co-dimension there are
sub-defects harbouring fractional spin gravity given by extensions of the
original101010The motivation behind [32] was to construct field theories
describing the coupling of gravity and gauge fields in three dimensions to
fractional spin fields; indeed, the unfolded conformal matter fields sit in
infinite-dimensional representations whose spins, viewed as labels of the
conformal group $SO(2,3)$, can be deformed continuously within the higher spin
extension, while the deformation breaks of $SO(2,3)$ down to $SO(2,2)$.
fractional spin gravity of Chern–Simons type [32] by unfolded conformal matter
fields [23, 24, 25] arising in the noncommutative AKSZ sigma model in zero-
and two-forms in bi-fundamental representations of conformal higher spin and
color groups with topological gauge fields111111As shown in [32], color gauge
fields are crucial for Cartan integrability; under extra assumptions the color
group can be reduced to $U(1)$, suggesting that an extra gauge field is
required in order to complete the Noether procedures set up in [23, 24, 25] to
all orders..

Thinking of the FCS model as a tensionless analog of an open-closed string
field theory, with Vasiliev’s higher spin gravity and fractional spin gravity,
respectively, playing role of closed- and open-string sectors, the counterpart
of open-closed string duality in the case of a single stack of D-branes
becomes the refined holography proposal involving developed largely
independently by Vasiliev [23] and Nilsson [24, 25], thus representing a
morphism from a single sub-defect to the empty set. By this reasoning, a
simple proposal for a tensionless analog of a D$\overline{\rm D}$ system is
thus a higher spin gravity model with two fractional spin gravity duals, thus
representing a morphism from between two sub-defect.

The derivative structure of cubic and quartic vertices of the D$\overline{\rm
D}$ EFT and of effective Fronsdal actions for higher spin gravity are similar.
Indeed, the cubic vertices only have a finite number of derivatives for a
given set of spins, while the quartic vertices are given by all-order
derivative expansions weighted by $\alpha^{\prime}$ and the cosmological
constant in the D$\overline{\rm D}$ EFT and higher spin gravity, respectively.
One may thus propose that there exists a geometrization of the multi-linear
Moyal-like couplings of the EFT at high energies by extending its Quillen
superconnection to that of a suitable fractional spin gravity theory
(including propagating SYM fields in its matter sector), and correspondingly
switching from the metric-like DBI-SYM plus WZ action to a dual noncommutative
AKSZ action. The resulting configuration spaces of strictly quantized
differential graded algebras could provide a framework for resolving classical
singularities in SYM gauge theory; for related results in higher spin gravity,
see [55].

## Acknowledgments

EH would like to thank A. Sagnotti, L. Alvarez-Gaume, R. Antonelli, A. Kuntz,
K. Narain, N. Arkani-Hamed, J. Schwarz, A. Sen and W. Siegel for fruitful
discussions. This work is supported by INFN (ISCSN4-GSS-PI), by Scuola Normale
Superiore, and by MIUR-PRIN contract 2017CC72MK003. PS would like to thank C.
Arias, R. Aros, N. Boulanger, F. Diaz, E. Sezgin, M. Valenzuela and B. Vallilo
for discussions.

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