On Non-BPS Effective Actions of String Theory

TL;DR

This paper analyzes the detailed structure of non-BPS effective actions in type IIA and IIB superstring theories, focusing on scattering amplitudes, Bianchi identities, and higher derivative corrections to understand their physical implications.

Contribution

It provides a comprehensive calculation of non-BPS scattering amplitudes, including all singularities, contact terms, and higher derivative corrections, revealing universal features and consistency conditions.

Findings

Derived all-order $\alpha'$ corrections to non-BPS amplitudes

Established compatibility of amplitudes in different super-ghost pictures

Identified universal higher derivative expansion in tachyon momenta

Abstract

We discuss some physical prospective of the non-BPS effective actions of type IIA and IIB superstring theories. By dealing with the complete all three and four point functions, including a closed string Ramond-Ramond (in terms of both its field strength and its potential), gauge (scalar) fields as well as a real tachyon and under symmetry structures, we find various restricted world volume and bulk Bianchi identities. The complete forms of the non-BPS scattering amplitudes including their Chan-Paton factors are elaborated. All the singularity structures of the non-BPS amplitudes, their all orders $\alpha'$ higher derivative corrections, their contact terms and various modified Bianchi identities are derived. Finally we show that scattering amplitudes computed in different super-ghost pictures are compatible when suitable Bianchi identities are imposed on the Ramond-Ramond fields. Moreover, we argue that the higher derivative expansion in powers of the momenta of the tachyon is universal.

Equations

$$S=-T_{p}V(T)\sqrt{-\det ({\eta }_{ab}+2\pi {\alpha }^{\prime }{F}_{ab}-2\pi {\alpha }^{\prime }\bar{\Psi }{\gamma }_b{\partial }_a\Psi +{\pi }^2{\alpha }^{\prime 2}\bar{\Psi }{\gamma }^{\mu }{\partial }_a\Psi \bar{\Psi }{\gamma }_{\mu }{\partial }_b\Psi +2\pi {\alpha }^{\prime }{D}_{a}T{D}_{b}T)}$$

$$S_{DBI}$$

$$S_{WZ}$$

$$i{\cal F}=\left(\begin{array}[]{cc}iF-\beta^{\prime 2}T^{2}&\beta^{\prime}DT\\ \beta^{\prime}DT&iF-\beta^{\prime 2}T^{2}\end{array}\right),$$

$$\displaystyle S_{WZ}=2\beta^{\prime}\mu_{p}^{\prime}(2\pi\alpha^{\prime})\mbox{\rm Tr}\,\bigg{(}C_{p}\wedge DT+(2\pi\alpha^{\prime})C_{p-2}\wedge DT\wedge F+\frac{(2\pi\alpha^{\prime})^{2}}{2}C_{p-4}\wedge F\wedge F\wedge DT\bigg{)}$$

$$S_{DBI}$$

$$V_T^{(0)}(x)$$

$$V_T^{(-1)}(x)$$

$$V_A^{(-1)}(x)$$

$$V_A^{(0)}(x)$$

$$V_A^{(-2)}(x)$$

$$V_C^{(-\frac{3}{2},-\frac{1}{2})}(z,\bar{z})$$

$$V_C^{(-\frac{1}{2},-\frac{1}{2})}(z,\bar{z})$$

$$q^2=p^2=0,k_2^2=1/4,q.{\xi }_1=0,$$

$$P_{-}=\mbox{$\frac{1}{2}$}(1-\gamma^{11}),\quad H\!\!\!\!/\,_{(n)}=\frac{a_{n}}{n!}H_{\mu_{1}\ldots\mu_{n}}\gamma^{\mu_{1}}\ldots\gamma^{\mu_{n}},$$

$$(P_{-}H/{}_{(n)}{)}^{\alpha \beta }=C^{\alpha \delta }(P_{-}H/{}_{(n)}{)}_{\delta }{}^{\beta }.$$

$${\tilde{X}}^{\mu }(\bar{z})\to D_{\nu }^{\mu }{X}^{\nu }(\bar{z}),{\tilde{\psi }}^{\mu }(\bar{z})\to D_{\nu }^{\mu }{\psi }^{\nu }(\bar{z}),\tilde{\varphi }(\bar{z})\to \varphi (\bar{z}),\text{mbox}and{\tilde{S}}_{\alpha }(\bar{z})\to M_{\alpha }{}^{\beta }{S}_{\beta }(\bar{z}),$$

$$D=\left(\begin{array}[]{cc}-1_{9-p}&0\\ 0&1_{p+1}\end{array}\right)\ ,\,\,\ \ \ \mbox{and}\ \ \ M_{p}=\left\{\begin{array}[]{cc}\frac{\pm i}{(p+1)!}\gamma^{i_{1}}\gamma^{i_{2}}\ldots\gamma^{i_{p+1}}\epsilon_{i_{1}\ldots i_{p+1}}\,\,\,\,{\rm for\,p\,even}\\ \frac{\pm 1}{(p+1)!}\gamma^{i_{1}}\gamma^{i_{2}}\ldots\gamma^{i_{p+1}}\gamma_{11}\epsilon_{i_{1}\ldots i_{p+1}}\,\,\,\,{\rm for\,p\,odd}\end{array}\right.$$

$$⟨X^{\mu }(z)X^{\nu }(w)⟩$$

$$⟨{\psi }^{\mu }(z){\psi }^{\nu }(w)⟩$$

$$⟨\varphi (z)\varphi (w)⟩$$

$$\int d{x}_{1}d{x}_{2}d{x}_{4}d{x}_5(P_{-}C/{}_{(n-1)}{M}_p{)}^{\alpha \beta }(2{\alpha }^{\prime }i{k}_{2b}{\xi }_{1a})(x_{45}{)}^{-3/4}(I_1+I_2)$$

$$\times |x_{12}{|}^{{\alpha }^{\prime 2}{k}_1.k_2}|x_{14}{x}_{15}{|}^{\frac{{\alpha }^{\prime 2}}{2}{k}_1.p}|x_{24}{x}_{25}{|}^{\frac{{\alpha }^{\prime 2}}{2}{k}_2.p}|x_{45}{|}^{\frac{{\alpha }^{\prime 2}}{4}p.D.p}$$

$$I_1$$

$$I_2=2i{k}_{1c}<:S_{\alpha }(x_4):S_{\beta }(x_5):{\psi }^c{\psi }^a(x_1):{\psi }^b(x_2):>$$

$$I_2$$

$$\displaystyle\int_{-\infty}^{\infty}dx(2x)^{-2t-1/2}(1+x^{2})^{-1/2+2t}\bigg{(}(\frac{1-x^{2}}{2ix})(\eta^{bc}\mbox{\rm Tr}\,(P_{-}C\!\!\!\!/\,_{(n-1)}M_{p}\gamma^{a})$$

$$\displaystyle-\eta^{ab}\mbox{\rm Tr}\,(P_{-}C\!\!\!\!/\,_{(n-1)}M_{p}\gamma^{c}))+\mbox{\rm Tr}\,(P_{-}C\!\!\!\!/\,_{(n-1)}M_{p}\Gamma^{bac})\bigg{)}2^{3/2}k_{1c}k_{2b}\xi_{1a}$$

$${\mathcal{A}}^{T^0,A^0,C^{-2}}$$

$$2i{\beta }^{\prime }{\mu }_p^{\prime }(2\pi {\alpha }^{\prime }{)}^2{\int }_{{\Sigma }_{p+1}}\text{mbox}\mathrm{Tr}(C_{p-2}\wedge F\wedge DT),$$

$${\mathcal{A}}^{A^0,T^{-1},C^{-1}}$$

$${\mathcal{A}}^{A^{-1},T^0,C^{-1}}$$

$$p_b{H}_{a_0...a_{p-2}}{ϵ}^{a_0...a_{p-2}ba}=p_a{H}_{a_0...a_{p-2}}{ϵ}^{a_0...a_{p-2}ba}=0$$

$$\displaystyle 4\xi_{1i}(P_{-}H\!\!\!\!/\,_{(n)}M_{p})^{\alpha\beta}\bigg{(}k_{1a}(\Gamma^{ia}C^{-1})_{\alpha\beta}-p^{i}(C^{-1})_{\alpha\beta}\bigg{)}\int_{-\infty}^{\infty}dx(2x)^{-2t-1/2}(1+x^{2})^{-1/2+2t}$$

$$p^i{ϵ}^{a_0...a_p}{H}_{a_0...a_p}+p^a{ϵ}^{a_0...a_{p-1}a}{H}_{a_0...a_{p-1}}^i=0$$

Full text

On Non-BPS Effective Actions of String Theory

Ehsan Hatefi

Institute for Theoretical Physics, TU Wien

Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria

Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland

Mathematical Institute, Faculty of Mathematics, Charles University, P-18675, CR 111ehsan.hatefi@fuw.edu.pl, ehsanhatefi@gmail.com

Abstract

We discuss some physical prospective of the non-BPS effective actions of type IIA and IIB superstring theories. By dealing with the complete all three and four point functions, including a closed string Ramond-Ramond (in terms of both its field strength and its potential), gauge (scalar) fields as well as a real tachyon and under symmetry structures, we find various restricted world volume and bulk Bianchi identities.

The complete forms of the non-BPS scattering amplitudes including their Chan-Paton factors are elaborated. All the singularity structures of the non-BPS amplitudes, their all orders $\alpha^{\prime}$ higher derivative corrections, their contact terms and various modified Bianchi identities are derived.

Finally we show that scattering amplitudes computed in different super-ghost pictures are compatible when suitable Bianchi identities are imposed on the Ramond-Ramond fields. Moreover, we argue that the higher derivative expansion in powers of the momenta of the tachyon is universal.

1 Introduction

D-branes have been realized to be the sources for Ramond-Ramond (RR) fields [1, 2]. RR couplings actually played important contributions to string theory. For instance to observe some of the application of RR couplings, one may consider the dissolving branes [3], K-theory and Myers effect [4]. The other applications to RR couplings are related to the $N^{3}$ phenomena for M5-branes, dS solutions, entropy growth and geometrical applications to the effective actions [5].

The spectrum of the so called non-BPS (unstable) branes includes massless, tachyon and an infinite number of massive states. There must be an Effective Field theory (EFT) for non-BPS branes where one integrates out all the massive states and hence the spectrum involves just the tachyon and massless states [6]. We will not point out cosmological applications for unstable branes. On general grounds, one might expect that D-branes and SD-branes have similar effective actions. The effective action of these branes has to have two parts. It consists of the extensions of the usual DBI and Wess-Zumino (WZ) actions where the tachyon mode is embedded to these effective actions. By applying the Conformal Field Theory (CFT) methods [7], the leading order effective couplings of the fermions with tachyons were found in [8] as

[TABLE]

In the above action, $F_{ab}$ is the field strength of the gauge field, $\bar{\Psi}\gamma_{\mu}\partial^{\mu}\Psi$ is the kinetic term of fermion fields, $DT$ is the covariant derivative of the tachyon $(D_{a}T=\partial_{a}T-i[A_{a},T])$. On the other hand the Chern-Simons action for BPS branes was constructed in [9]. Using Boundary String Field Theory (BSFT), one talks about tachyon’s kinetic term in DBI part [10] as follows

[TABLE]

The WZ action in BSFT approach is found to be

[TABLE]

where $C_{(p+1)}$ is the RR potential $(p+1)$ form-field and super connection’s curvature would be given by

[TABLE]

$\beta^{\prime}$ is the normalisation constant and $\mu_{p}^{\prime}$ is RR charge of the brane. If we expand the exponential in (2), then we obtain various couplings as follows

[TABLE]

For the sake of the higher derivative corrections, we work with the second approach of exploring effective actions, which is scattering amplitude formalism. In this approach tachyon’s kinetic term is embedded into the DBI action as follows

[TABLE]

where $V({T^{i}T^{i}})=e^{-\pi{T^{i}T^{i}}/2}$, $Q^{ij}=I\delta^{ij}-i[T^{i},T^{j}],T^{1}=T\sigma_{1},T^{2}=T\sigma_{2}$ and $\sigma_{1},\sigma_{2}$ are Pauli matrices. The trace in (1) should be symmetric for all $F_{ab},D_{a}{T^{i}}$, $[T^{i},T^{j}]$ matrices. If all Chan-Paton factors are taken into account, then this action would produce consistent results with all momentum expansions of three and four point functions of a closed string RR and either two, three tachyons or two tachyons two gauge/scalar fields amplitude.

On the stable point, tachyon potential and its effective action get replaced to the known tachyon DBI action [11, 12] with potential $T^{4}V(T^{2})$. The WZ part of the action in this approach has the same formula as appeared in (2). Using S-matrix method the normalisation constants of $\beta^{\prime},\beta$ for non-BPS and brane anti brane system are discovered to be $\beta^{\prime}=\frac{1}{\pi}\sqrt{\frac{6\ln(2)}{\alpha^{\prime}}}$ and $\beta=\frac{1}{\pi}\sqrt{\frac{2\ln(2)}{\alpha^{\prime}}}$ [13]. It is worth mentioning that super conection’s structure for WZ action was found by the S-matrix approach in [14].

The aim of the paper is to show that scattering amplitudes computed in different super-ghost pictures are compatible when suitable Bianchi identities are imposed on the Ramond-Ramond (R-R) fields. Moreover, we argue that the higher-derivative expansion in powers of the momenta of the tachyon is universal.

The outline of this paper is as follows. First we find all three point functions including a gauge field, a tachyon and a closed string RR in all asymmetric and symmetric pictures of the closed string RR. By doing so, not only do we find some restricted Bianchi identities on both world volume and transverse directions of non-BPS branes but also explore all their infinite higher derivative corrections. It is believed that due to supersymmetry transformation BPS S-matrices do not generate Bianchi identity. To get consistent results at the level of singularity structures for four point functions of the two gauge fields, a tachyon and a closed string RR in their all asymmetric and symmetric pictures, we discover various restricted Bianchi identities. Eventually we talk about a universal expansion for tachyon and start constructing all different singularity structures of $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ as well as all order $\alpha^{\prime}$ higher derivative corrections to various couplings of the type IIA, IIB superstring theories.

2 All order $<V_{C^{-2}}V_{A^{0}}V_{T^{0}}>$

In this section we would like to apply CFT methods to derive the complete S-matrix elements of a closed string RR, a gauge field and a tachyon in the world volume of a non-BPS brane. Total super ghost charge for disk level amplitude must be -2. First we choose an asymmetric closed string RR ( that carries total -2 super ghost charge) and hence the gauge field and a tachyon must be put in zero picture. This S-matrix can be achieved if one finds the correlation functions of the following vertex operators

[TABLE]

It is argued in [15] that the vertices of a non-BPS D-brane need to carry internal degrees of freedom or Chan-Paton (CP) matrix. Because if we set the tachyon to zero, then the WZ effective action of non-BPS branes gets reduced to WZ action of BPS branes. Hence, we impose an identity internal CP matrix to all massless fields including gauge (scalar) and RR field in zero picture. It is also discussed in [16] that Picture Changing Operator (PCO) carries CP matrix $\sigma_{3}$. It is also explained in [17] that the tachyon in zero and (-1) picture does carry $\sigma_{1}$ and $\sigma_{2}$ CP factor. We also know that the amplitude of a closed string RR and a tachyon $<V_{C^{-1}}V_{T^{-1}}>$ makes sense in the world volume of non-BPS branes. This fixes the CP factor of RR in $(-1/2,-1/2)$ picture for non-BPS branes to be $\sigma_{3}\sigma_{1}$. By applying PCO to RR in (-1) picture, we derive its CP factor in (-2) picture to be $\sigma_{1}$ and the CP factor for the gauge field in (-1) picture to be $\sigma_{3}$ where $\lambda$ is the external CP matrix for the U(N) gauge group.

We are looking for the disk level amplitude. The closed string will be located in the middle of disk whereas all open strings are located at the boundary of disk. On-shell conditions are

[TABLE]

The definitions of the RR’s field strength and projection operator are

[TABLE]

For type IIA (IIB) $n=2,4$,$a_{n}=i$ ($n=1,3,5$,$a_{n}=1$) and spinor notation is

[TABLE]

We also apply the doubling trick so that all the holomorphic parts of the fields can be used. Thus the following change of variables works

[TABLE]

with

[TABLE]

Now one can use the following two-point functions for $X^{\mu},\psi^{\mu},\phi$, as below

[TABLE]

The amplitude in asymmetric picture is given by

[TABLE]

where $x_{4}=z=x+iy,x_{5}=\bar{z}=x-iy$ and

[TABLE]

One should make use of the Wick-like rule [18] to obtain the correlation function for $I_{2}$

[TABLE]

as follows

[TABLE]

It can be readily shown that the amplitude is $SL(2,R)$ invariant. We use the gauge fixing as $(x_{1},x_{2},z,\bar{z})=(x,-x,i,-i)$ and the Jacobian is $J=-2i(1+x^{2})$. After gauge fixing, one reveals that $I_{1}$ has zero contribution to the S-matrix. Because the integrand is an odd function while the the interval of the integral is symmetric.222 $\alpha^{\prime}=2$ is set. We introduce $t=-\frac{\alpha^{\prime}}{2}(k_{1}+k_{2})^{2}$ and after gauge fixing $I_{2}$ is obtained by

[TABLE]

Note that the last two terms have just non-zero contribution to our amplitude. The final answer for the amplitude is

[TABLE]

To be able to match the leading order of the S-matrix with the following coupling in the EFT

[TABLE]

we use $(\pi\beta^{\prime}\mu_{p}^{\prime}/2)$ as the normalization constant. $\beta^{\prime}$ and $\mu_{p}^{\prime}$ are known to be WZ normalisation constant and RR brane’s charge. On the other hand, the result in symmetric cases (RR is written in (-1)-picture) for ${\cal A}^{A^{0},T^{-1},C^{-1}}$ can be derived as

[TABLE]

Accordingly ${\cal A}^{A^{-1},T^{0},C^{-1}}$ is found to be

[TABLE]

By applying momentum conservation along the world volume of brane $(k_{1}+k_{2}+p)^{a}=0$ and making comparisons between (10) and (11), one clarifies that the following Bianchi identities kept fixed in the presence of RR’s field strength

[TABLE]

Note that all three point functions involving a closed string RR, a tachyon and a scalar field in all symmetric and asymmetric pictures of RR can also be computed. The result in symmetric picture for ${\cal A}^{\phi^{0},T^{-1},C^{-1}}$ is given by

[TABLE]

To get consistent result for the S-matrix, in the presence of all different pictures of closed string RR, the restricted Bianchi identity (12) must get replaced by the following Bianchi identity

[TABLE]

This modified Bianchi identity works in the appearance of a scalar field and holds for all world volume and transverse directions of branes. The below trace is non-zero for $p+1=n+2$ case

[TABLE]

The trace that has $\gamma^{11}$ part, indicates that the following relation holds

[TABLE]

Neither there are massless poles nor tachyon poles for this three point function. It is argued in [19] that the expansion of the non-BPS amplitudes in the presence of a closed string RR makes sense if one applies the following constraint

[TABLE]

Note that for the brane-anti brane configuration the above constraint gets replaced by $p^{a}p_{a}\rightarrow 0$ [20]. Hence, the precise momentum expansion for $CAT$ is $t\rightarrow-1/4$. The expansion for Gamma function is

[TABLE]

with the following coefficients

[TABLE]

An infinite number of the higher derivative corrections to a $C_{p-2}$, a tachyon and a gauge field can be found by producing the contact terms in an EFT and also by applying higher derivative corrections to the WZ effective coupling as follows

[TABLE]

Let us deal with the complete $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$, to see what kinds of restricted Bianchi identities can be explored and also to see whether or not there are bulk singularity structures.

3 The complete $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ amplitude

In order to find out the complete form of the scattering amplitude of a tachyon, a potential RR $(p+1)$ form-field and two massless gauge fields $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ on the world volume of non-BPS branes, one needs to employ all CFT techniques. To achieve all singularities and contact interactions, we make use of the vertex operators that are introduced earlier on. Note that as clarified in [21] the CP factor of RR for brane anti brane system is different from the CP factor of non-BPS branes. RR vertex operators are introduced in [22]. One might hint on some of the BPS and non-BPS scattering amplitudes in [23].

Recently $<V_{C^{-2}}V_{\phi^{0}}V_{\phi^{0}}V_{T^{0}}>$ analysis was done, however, one cannot derive the result for $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ from it, by just exchanging the scalars fields to gauge fields. Because not all world volume couplings nor bulk terms have any effect in our new effective action. More crucially, given the presence of the tachyonic string, we can not compare our effective actions with BPS branes’ s effective action [24, 25, 26]. Let us write down the compact and the closed form of the correlation functions 333 $x_{ij}=x_{i}-x_{j}$, and $\alpha^{\prime}=2$., whereas all the other kinematical relations can be found in [21].

[TABLE]

where

[TABLE]

The last fermionic correlator that has just world volume indices $a_{4}^{cbdae}=<:S_{\alpha}(x_{4}):S_{\beta}(x_{5}):\psi^{e}\psi^{a}(x_{1}):\psi^{d}\psi^{b}(x_{2}):\psi^{c}(x_{3}):>$ can be explored by considering all possible interactions as follows:

[TABLE]

We wrote all the S-matrix elements so that SL(2,R) invariance can be manifestly shown. By fixing three position of vertices, we can get rid of the volume of killing group. In order to get the algebraic answer for the amplitude, we fix the positions of open strings as

[TABLE]

Eventually one needs to take a 2D complex integrals on the location of the closed string RR on the upper half plane as below

[TABLE]

where $d=0,1,2$ and $a,b,c$ are written down in terms of the following Mandelstam variables:

[TABLE]

For $d=0,1$ and $d=2,3$ the algebraic solutions for integrals are gained in [27],[28] accordingly. We just demonstrate the final form of the amplitude as follows

[TABLE]

where

[TABLE]

The functions $L_{1},L_{2},L_{3}$ are

[TABLE]

This amplitude satisfies Ward identities related to both gauge fields. We expand the amplitude in such a way that all tachyon and massless poles can be obtained from the EFT. Finally we start producing all contact interactions to all orders in $\alpha^{\prime}$ as well. One may think that the amplitude in asymmetric case has also non-zero terms for $p+1=n$ case, however, note that these terms that are not gauge invariant, will be disappeared from the final form of the S-matrix. These terms are

[TABLE]

At the end of the day the sum of all coefficients of each parenthesis of (21) is zero. This means that they disappear from the ultimate form of the world volume amplitudes. All $K_{2},M$ functions are written in terms of Gamma functions and for the sake of this paper we will not mention their explicit forms. This also confirms that there is no bulk singularity term for this S-matrix. In the next section we would like to find the symmetric analysis of the same S-matrix.

4 The complete $<V_{C^{-1}}V_{A^{0}}V_{A^{-1}}V_{T^{0}}>$ amplitude

This S-matrix in terms of the field strength of the closed string RR, that is, $<V_{C^{-1}}V_{A^{0}}V_{A^{-1}}V_{T^{0}}>$ has not been calculated yet. Making use of the CFT, we try to explore the complete amplitude of $<V_{C^{-1}}V_{A^{0}}V_{A^{-1}}V_{T^{0}}>$ on the world volume of non-BPS branes as well. It can be found by the following correlations

[TABLE]

[TABLE]

One needs to know the following correlation functions

[TABLE]

One also needs to obtain the correlation function of a current and two fermion fields in two different locations in the presence of two spin operator, that is, $I_{2}^{cbad}=<:S_{\alpha}(x_{4}):S_{\beta}(x_{5}):\psi^{d}\psi^{a}(x_{1}):\psi^{b}(x_{2}):\psi^{c}(x_{3}):>$ as

[TABLE]

We fixed three positions of the open strings as $x_{1}=0,x_{2}=1,x_{3}\rightarrow\infty$,and one essentially takes integration on the position of closed string RR which is a 2D complex integrals on upper half plane. Having set the gauge fixing, one would find the complete form of the integrand for ${\cal A}^{C^{-1}A^{0}A^{-1}T^{0}}$ as follows:

[TABLE]

The final answer is given by

[TABLE]

where

[TABLE]

On the other hand, this amplitude for the following picture $<V_{C^{-1}}V_{A^{0}}V_{A^{0}}V_{T^{-1}}>$ was computed to be

[TABLE]

where

[TABLE]

In the next section we address tachyon’s momentum expansion to be able to expand our complete S-matrix and finally we start generating its non-zero couplings.

5 Tachyon’s momentum expansion

In [19] it is conjectured that the momentum expansion for tachyon is universal. Given the momentum conservation on the world volume of brane for a two point function of a closed string RR and a tachyon, one reveals that $k_{a}+p_{a}=0$, therefore $p^{a}p_{a}$ must be sent to mass of the tachyon ($k^{2}=-m^{2}$). Hence, it is easy to understand the fact that

[TABLE]

and that is just possible for SDbranes or euclidean branes. This means that amplitude makes sense for non-BPS SD-branes [29]. From the kinetic term of the tachyons in DBI action, one reveals that the coupling of the two tachyons and a gauge field is non zero, so to be able to produce all tachyon and massless poles of the EFT, we need to employ a unique expansion for all non-BPS branes. Two Mandelstam variables should be sent to the mass of the tachyon as follows

[TABLE]

$(i\mu^{\prime}_{p}\beta^{\prime}\pi^{1/2})$ is the normalisation constant and the closed form of the expansions are

[TABLE]

some of the above coefficients are found.444

$\displaystyle L_{3}$ $\displaystyle=$ $\displaystyle-{\pi^{5/2}}\sum_{p,n,m=0}^{\infty}\left(c_{n}(s^{\prime}+t+u^{\prime})^{n}+c_{n,m}\frac{s^{\prime n}u^{\prime m}+s^{\prime m}u^{\prime n}}{(t+s^{\prime}+u^{\prime})}+f_{p,n,m}(s^{\prime}+t+u^{\prime})^{p}(s^{\prime}+u^{\prime})^{n}(s^{\prime}u^{\prime})^{m}\right)$

$\displaystyle b_{-1}=1,\,b_{0}=0,\,b_{1}=\frac{1}{6}\pi^{2},e_{1,0,0}=\frac{1}{6}\pi^{2}c_{0}=0,c_{1}=\frac{\pi^{2}}{3},c_{1,0}=c_{0,1}=0,f_{0,0,1}=4\zeta(3)$

Having taken (31), we would understand that $L_{1},L_{2},L_{3}$ have $t$-channel gauge fields, $s^{\prime},u^{\prime}$-channel tachyonic singularities and another kind of $(s^{\prime}+t+u^{\prime})$-channel tachyon poles accordingly. In the next section we first try to make comparisons on singularity structures as well as all contact terms. We then reconstruct all singularities in EFT and get to derive the restricted world volume Bianchi identities for non-BPS branes.

6 Singularities and restricted Bianchi identities

Let us first compare singularity structures between $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ and $<V_{C^{-1}}V_{A^{0}}V_{A^{-1}}V_{T^{0}}>$ with (29). If we use momentum conservation $k_{1a}+k_{2a}+k_{3a}=-p_{a}$ and $pC=H$ to the complete ${\cal A^{\prime}}_{3}$ of (19), then we are able to precisely produce all $(s^{\prime}+t+u^{\prime})$ channel poles ${\cal A}_{3}$ of (29). The first term ${\cal A^{\prime}}_{2}$ exactly produces the 7th term ${\cal A}_{2}$ that has tachyon singularities. Replacing $k_{3c}=-(k_{1c}+k_{2c}+p_{c})$ to the second term of ${\cal A^{\prime}}_{2}$, we gain

[TABLE]

(32) is symmetric under both $k_{1c},k_{2c}$ and also is antisymmetric in terms $\epsilon^{a_{0}...a_{p-3}cde}$ inside the trace, therefore $k_{1c},k_{2c}$ have no contribution to our coupling. Using $pC=H$ we derive the 8th term of ${\cal A}_{2}$ that has $(s^{\prime}+t+u^{\prime})$ tachyon singularities. Above arguments are also held for the third term of ${\cal A^{\prime}}_{2}$ , namely using momentum conservation, the third term of ${\cal A^{\prime}}_{2}$ reconstructs the second term of ${\cal A}_{2}$ that has $s^{\prime}$-channel tachyon poles. If we apply momentum conservation to the 6th term of ${\cal A^{\prime}}_{2}$, we find the following interaction

[TABLE]

$k_{1c}$ has no effect to the above interaction and using $pC=H$, (33) regenerates exactly all $u^{\prime}$ channel tachyon singularities (the 5th term of ${\cal A}_{2}$). Having applied momentum conservation to the 4th term ${\cal A^{\prime}}_{2}$ we obtain

[TABLE]

$k_{2c}$ has no contribution to the above interaction and using $pC=H$, one reproduces exactly the first term of ${\cal A}_{2}$. Now adding the contribution $k_{1c}$ of (34) with the 5th term of ${\cal A^{\prime}}_{2}$ we obtain

[TABLE]

which is precisely the 6th term of ${\cal A}_{2}$. By applying momentum conservation to the 7th term ${\cal A^{\prime}}_{2}$ we get

[TABLE]

once more $k_{2c}$ has no contribution, taking $pC=H$, we generate the third term ${\cal A}_{2}$ of (29). One might suppose that $k_{1c}$ from (36) is an extra singularity, however, we claim that the presence of this term is needed. Indeed if we take into account the contribution $k_{1c}$ from (36) and add it with the last term of ${\cal A^{\prime}}_{2}$ we find

[TABLE]

which is exactly the 3rd term ${\cal A}_{2}$. Therefore, we are able to produce not only all t-channel gauge field singularities (29) but also its all $s^{\prime},u^{\prime},(t+s^{\prime}+u^{\prime})$ channel tachyon singularities of $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ are produced without any ambiguity. Let us deal with singularities that appear in (27). Evidently ${\cal A^{\prime\prime}}_{3}$ is the same as ${\cal A}_{3}$. The 4th and 5th terms of ${\cal A^{\prime\prime}}_{2}$ are equivalent to 5th and 7th terms of ${\cal A}_{2}$. Applying momentum conservation to the 6th term ${\cal A^{\prime\prime}}_{2}$, we would get

[TABLE]

$k_{1c}$ has no contribution to the above interaction. The contribution from $k_{2c}$ produces exactly the 8th term ${\cal A}_{2}$, and to make a consistent result for both symmetric amplitudes, one must impose the following restricted Bianchi identity for the field strength of RR

[TABLE]

Using direct scattering amplitude $<V_{C^{-2}}V_{\phi^{0}}V_{\phi^{0}}V_{T^{0}}>$ the following Bianchi identity holds in terms of the both field strength of RR and RR potential in the complete space-time

[TABLE]

or

[TABLE]

Now if we again apply momentum conservation to the first and third terms ${\cal A^{\prime\prime}}_{2}$ and simultaneously take into account the restricted world volume (39), then we actually reconstruct the sum of first and 6th terms of ${\cal A}_{2}$ as well as the 3rd and 4th terms ${\cal A}_{2}$ accordingly. The same holds for the second term ${\cal A^{\prime\prime}}_{2}$ and we regenerate the 2nd term ${\cal A}_{2}$. Hence, in comparison with (29) and using the restricted world volume Bianchi identities we are able to produce all t-channel gauge field singularities as well as $s^{\prime},u^{\prime},(t+s^{\prime}+u^{\prime})$ channel tachyon singularities of $<V_{C^{-1}}V_{A^{0}}V_{A^{-1}}V_{T^{0}}>$. It is worth emphasizing that unlike $<V_{C^{-2}}V_{\phi^{0}}V_{\phi^{0}}V_{T^{0}}>$ analysis, here we have no bulk singularity structures at all. Hence, on the world volume of non-BPS branes all brane singularities have been matched without producing any extra residual contact interactions.

7 Contact term comparisons

To be able to achieve all the restricted Bianchi identities of non-BPS branes, we try to compare all contact interactions between (19) and (27) with all order contact terms of (29). If we replace $k_{3c}=-(k_{1c}+k_{2c}+p_{c})$ to ${\cal A^{\prime}}_{1}$ of (19) (also to ${\cal A^{\prime\prime}}_{1}$) and use the following Bianchi identity

[TABLE]

then we obviously produce all contact interactions of ${\cal A}_{1}$ for $p=n+3$ case. The leading order couplings can be produced if we would normalize the amplitude by $(\mu^{\prime}_{p}\beta^{\prime}\pi^{1/2})$ and compare it with the following coupling in the EFT

[TABLE]

All order contact interactions can be calculated. Recently the method of getting all order contact interactions has been released in [19, 21], so one can properly apply the higher derivative corrections to EFT couplings to produce all non-leading terms and also derive all order $\alpha^{\prime}$ corrections. For example, if we replace the expansion of $L_{3}$ into the amplitude, then one can derive all order contact interactions of the amplitude for $p=n+3$ case as follows

[TABLE]

Note that both ${\cal A^{\prime\prime}}_{1}$ and ${\cal A^{\prime}}_{1}$ satisfy Ward identity associated to the gauge fields.

Making use of the Bianchi identities we are able to generate all contact interactions $<V_{C^{-2}}V_{A^{0}}V_{A^{0}}V_{T^{0}}>$ from (29) without any ambiguity. For instance, the first contact term of the amplitude for $p=n+1$ case is

[TABLE]

and this contact interaction can be reconstructed by taking into account the following gauge invariant coupling in an EFT

[TABLE]

Notice that (43) is found by expanding the exponential of WZ action and using the multiplication rule of the super matrices. If we consider the expansions of $L_{1},L_{2}$ into the amplitude then one may find the following contact interactions to the next leading order for $p=n+1$ case as below

[TABLE]

All terms in (47) are related to the corrections of the EFT couplings. One can explore the following EFT couplings that regenerate the contact terms in (47) as follows

[TABLE]

where the covariant derivative of tachyon is $D_{a}T=\partial_{a}T-i[A_{a},T]$. Note that by direct scattering amplitude of a closed string RR, a tachyon and a gauge field in section two, we have derived all order $\alpha^{\prime}$ higher derivative corrections to the last coupling of (48). Let us now produce all $(t+s^{\prime}+u^{\prime})$-channel tachyon singularities of the amplitude.

7.1 All $(t+s^{\prime}+u^{\prime})$-channel tachyon singularities

Let us explore all $(t+s^{\prime}+u^{\prime})$- channel tachyon singularities of the amplitude ${\cal A^{\prime}}_{3}$ for $p+1=n$ case. Extracting the trace and normalizing the amplitude we derive them as follows

[TABLE]

which satisfies Ward identity. These poles can be constructed by employing a WZ coupling $2i\mu^{\prime}_{p}\beta^{\prime}(2\pi\alpha^{\prime})\int C_{p}\wedge DT$ and all order higher derivative corrections to two tachyon-two gauge field couplings. In the effective field theory all singularities are derived by the following sub amplitude and vertices

[TABLE]

Replacing the vertex of two tachyon-two gauge field couplings in the above field theory amplitude, we obtain all tachyon singularities in the EFT as follows

[TABLE]

Some of the coefficients are

[TABLE]

These poles (51) are exactly the ones that appeared in S-matrix elements (49).

7.1.1 All $u^{\prime},s^{\prime}$ channel-tachyon singularities

Given the symmetries of the amplitude, we reconstruct all $u^{\prime}$-channel poles in the EFT, and likewise by exchanging momenta and polarizations, all $s^{\prime}$-channel singularities can also be examined.

[TABLE]

All these $u^{\prime}$-channel poles can be constructed by the following rule

[TABLE]

$V^{\beta}(T,T_{3},A_{2})$ should be found from the non-Abelian kinetic term of the tachyons in DBI action which has been fixed. If we employ the corrections that we got from WZ coupling $2i\beta^{\prime}\mu^{\prime}_{p}\int C_{p-2}\wedge F\wedge DT$ in (15), then we obtain the higher order vertex of $V^{\alpha}(C_{p-2},A_{1},T)$ and the other vertices as follows

[TABLE]

$k$ is off-shell ’s tachyon momentum. Replacing (54) inside (53), we obtain all order $u^{\prime}$ channel tachyon poles in an EFT as

[TABLE]

which are precisely those singularities that appeared in (52). Eventually, one can show that all t-channel gauge field singularities are generated by taking into account the following rule and vertices in the EFT

[TABLE]

$k$ is off-shell gauge field’s momentum and $V^{a}(C_{p-2},T_{3},A)$ has been derived from the corrections to the WZ coupling $C_{p-2}\wedge F\wedge DT$ that we derived in the previous section. Notice that the kinetic term of the gauge fields is fixed in DBI action, so one reveals that $V^{b}(A,A_{1},A_{2})$ should not receive any higher derivative corrections. We also come to know that the tachyon expansion that we talked about is consistent with effective field theory, because we are able to produce precisely all tachyon and massless poles of the string amplitude in the EFT as well.

The expansion has also been checked for various other non supersymmetric cases, such as all the other three and four point functions (like $CAT,C\phi\phi T$). That is why we believe that the expansion is universal. This might indicate that tachyon momentum expansion is unique. It would be nice to check it with the higher point functions of non-BPS string amplitudes. The precise form of the solutions for integrals of six point functions is unknown, however, given the exact symmetries of the string theory amplitudes and the universal tachyon expansion in [30] we were able to obtain all the singularity structures of the amplitude of a closed string RR and four tachyons. We hope to overcome some other open questions in near future.

Acknowledgements

This paper was initiated during my 2nd post doc at Queen Mary University of London. Some parts of the paper were carried out at Mathematical institute in Charles University, at KITP in Santa Barbara, UC Berkeley and at Caltech. I am very grateful to L. Alvarez-Gaume, K. Narain, F. Quevedo, D. Francia, A. Sagnotti, B. Jurco, N.Arkani-Hamed, A. Brandhuber, G.Travaglini, P. Horava, G.Veneziano, P. Sulkowski, P. Vasko, L. Mason, H. Steinacker and J.Schwarz for many useful discussions and for sharing their valuable insights with me. This work is supported by ERC Starting Grant no. 335739 ’Quantum fields and knot homologies’, funded by the European Research Council.