# On special limit of non-supersymmetric effective actions of type II string theory

## Abstract

In this paper we first address four point functions of string amplitudes in both type IIA and IIB string theories. Making use of non-BPS scattering amplitudes, we explore not only several Bianchi identities that hold in both transverse and world volume directions of the brane, but also we reveal various new couplings. These couplings can just be found by taking into account the mixed pull-back and Taylor couplings where their all order alpha-prime higher derivative corrections have been derived as well. For the first time, we also explore the complete form of a six point non-BPS amplitude, involving three open string tachyons, a scalar field and a Ramond–Ramond closed string in both IIA, IIB. In a special limit of the amplitude and using the proper expansion we obtain an infinite number of bulk singularities that are being constructed in the effective field theory. Finally, using new couplings we construct all the other massless and tachyon singularities in type IIA, IIB string theories. All higher derivative corrections to these new couplings to all orders in \(\alpha '\) and new restricted Bianchi identities have also been obtained.

## 1 Introduction

Among several goals of theoretical physicists and in particular string theorists, we may point out two common interests in uncovering more information: about how the supersymmetry gets broken as well as working out new couplings/ interactions on time dependent backgrounds. If we try to deal with non-supersymmetric (unstable) branes, then one may be able to properly address some of the open questions and also might be able to deepen insight in many properties of various different string theories [1, 2, 3, 4, 5, 6, 7, 8, 9]. Since the duality transformation is not promising in this context anymore, one needs to be aware of the fact that for non-BPS branes just scattering amplitudes and Conformal Field Theory (CFT) methods [10] would exactly determine corrections to all orders in \(\alpha '\) of the effective actions of string theory.

Making use of non-supersymmetric branes, the so-called Sakai–Sugimoto model [11, 12] and the symmetry breaking for holographic QCD models have become known [13, 14]. Tachyons do play a crucial role in the instability of the aforementioned systems so it would be important to consider tachyons and try to obtain their effective actions in both type IIA and IIB string theories and also explore their new couplings in Effective Field Theory (EFT).

The leading order non-BPS effective actions including tachyonic modes were proposed in [15, 16], where some of their properties such as their decays and tachyon condensation have also been clarified in detail [17]. Following [8, 9], one reveals how to embed non-BPS branes in the effective actions. We studied D-brane–anti-D-brane systems [19, 20]. Recently, the generalization of effective actions of D-brane–anti-D-brane system to all orders in alpha-prime for both Chern–Simons and Dirac–Born–Infeld (DBI) effective actions was discovered [21]. Another example would be related to tachyon condensation that has been investigated in [22] in detail. For the D-brane–anti-D-brane system, once the distance between brane and anti-brane becomes smaller than the string length scale, two real tachyonic strings would appear. They are related to strings stretched from D-brane to anti-D-brane and vice versa.

Here we would like to deal with N coincident non-BPS branes and try to embed tachyonic modes and their corrections in EFT. We take the non-BPS scattering amplitude formalism as a theoretical framework or laboratory to discover their effective actions, including their corrections to all order in \(\alpha '\) in string theory in an efficient and consistent way of matching string results with EFT. To deal with the dynamics of unstable branes, we highlight the recent work done by Polchinski et al. [23] where various explanations within the context of the brane ’s effective actions through EFT have been discussed. Not only brane production [24, 25, 26, 27, 28, 29, 30, 31], but also inflation in string theory in the procedure of KKLT [32, 33, 34] can be mentioned. To observe a review of open strings and their features we point out to [35, 36, 37, 38, 39, 40, 41]. For reviews see [42, 43]

In this paper we deal with a non-BPS four point function and explore some Bianchi identities and new EFT couplings that come from the mixed pull-back formalism and Taylor expansion and then try to use the lower point functions to exactly build for the first time a non-BPS six point function. Having used the scattering amplitude methods, we would also fix some of the ambiguities of the corrections in string theory and reveal new string couplings in both type II string theories.^{1}

One can try to relate some of the new couplings to the AdS/CFT correspondence [45, 46]. It is also worth making a remark on the D-brane–anti-D-brane systems, as they do affect not only the problem of stability of KKLT model but also string compactifications [47, 48] and in particular the so-called large volume scenario. The relation between D-branes and Ramond–Ramond (RR) charges is well established [49], where one could also take into account some brane’s bound states [50]. All the EFT methods of deriving the Wess–Zumino (WZ) and DBI effective actions are given in [51, 52].

The paper is organized as follows. First we study a four point function including a closed string RR and a transverse scalar field and a real tachyon on the world volume of non-BPS branes; an RR and two tachyons have been fully addressed in [53]. Then we build all order \(\alpha '\) higher derivative corrections to it and explore a pattern from this calculation to reconstruct all singularity structures of the higher point functions of non-BPS branes.

Our notations for indices are summarized by the following.

\(\mu , \nu = 0, 1, \ldots ,9 \) represent the whole ten dimensional space-time, \(a, b, c = 0, 1, \ldots , p\) show the world volume indices, and finally for transverse directions of the brane \( i, j = p + 1, \ldots , 9\) are taken accordingly.

We establish a new coupling among RR, the tachyon field living on the world volume of a non-BPS brane and one massless scalar field representing a transverse direction of the brane.

Note that the integration should be taken on \((p+1)\) world volume directions and in order to cover the whole world volume indices we extract the coupling and write it as (1). We also explore its corrections to all orders in the higher derivatives too.

*t*,

*s*,

*v*channels at all. The DBI part of the effective action for non-BPS branes is

*i.e.,*\(T^1=T\sigma _1\), \(T^2=T\sigma _2\). The DBI part of the D-brane–anti-D-brane is given in [21]. If we make the kinetic terms symmetrized, find the traces and then use the ordinary trace, the action will get replaced by Sen’s action [55]. However, in [56, 57] by direct CFT computations and scattering amplitudes we have shown that Sen’s effective action does not provide a result consistent with the string amplitudes. The expansion of the S-matrices is consistent with the tachyon’s potential \(V(|T|)={\text {e}}^{\pi \alpha 'm^2|T|^2}\), which comes from BSFT [58, 59, 60]. On the other hand, the WZ action is given by

## 2 All order \(\alpha '\) corrections to \(<V_{C^{-2}} V_{T^{0}} V_{\phi ^{0}}>\)

*SL*(2,

*R*) invariant and to remove the volume of conformal Killing group we do gauge fixing as \((x_1,x_2,z,{{\bar{z}}})=(x,-x,i,-i)\) with the Jacobian \(J=-2i(1+x^2)\). Setting the above gauge fixing, we see that the second term of \(I_2\) does not have any contribution to the final result of the amplitude due to the fact that the integrand is odd, while the moduli space is covered on the entire space-time or due to having a symmetric interval. \(u = -\frac{\alpha '}{2}(k_1+k_2)^2\) is introduced and the amplitude results,The ultimate result of the amplitude is given by\( \mu _p' \) We have the RR charge of the brane. All the traces are nonzero for the \(p+1= n\) case and can be calculated asThe correct expansion of the amplitude can be found by dealing with either massless or tachyon poles of the amplitude. From a three point function including an RR and a real tachyon and using its momentum conservation along the world volume of the brane, \(k^2=p^ap_a=\frac{1}{4} \) [66], one realizes that this constraint holds for the \(CT\phi \) amplitude and indeed the proper momentum expansion can be read off as follows:

## 3 \(<V_{C^{-1}}V_{\phi ^{-1}}V_{T^{0}}V_{T^{0}}V_{T^{0}}>\) amplitude

## 4 \(<V_{C^{-1}(z,{{\bar{z}}})}V_{\phi ^{0} (x_1)}V_{T^{-1}(x_2)}V_{T^{0}(x_3)}V_{T^{0}(x_4)}>\) amplitude

## 5 Bulk Singularity Structures

*s*,

*t*,

*v*), one obtains the expansion for the amplitude (20) as follows:

^{2}channel that correspond to the extensions of higher derivative corrections of two tachyon–two scalar field couplings. These corrections originate from the second part of the amplitude in (20). They are reconstructed by the following EFT prescription:

*k*is the momentum of the off-shell scalar field. Substituting the above vertex in the EFT amplitude (30), we produce all infinite scalar poles as

^{3}

*k*is the off-shell tachyon’s momentum. Replacing the above vertices by (36) we would reproduce its double pole too.

Note that by direct calculations, the presence of some new couplings such as \(F^{(1)}\cdot F^{(2)}\) or \(D\phi ^{i(1)}\cdot D\phi _{i(2)}\) has been confirmed in the world volume of D-brane–anti D-brane systems [19, 20, 56]. Indeed making a string calculation we could produce all massless and tachyon singularities of the amplitudes.

While WZ coupling \(C_{p}\wedge {\text {DT}} \phi \) will not receive any higher derivative correction, and all the kinetic terms are fixed, they do not get any corrections either. Thus all other tachyon singularities give us clues about the structures of all order higher derivative corrections to various couplings and in this paper we could consistently fix their coefficients for good.

We think these corrections play a crucial role in determining singularities of the higher point functions of string theories. The Veneziano amplitude [69] was generalized in [21], and we hope to be able to address a supersymmetric generalization of the D-brane–anti-D-brane system by directly carrying out fermionic amplitudes [70]. We also hope to make progress on the generalization of the non-supersymmetric DBI and WZ effective actions in the near future.

## Footnotes

## Notes

### Acknowledgements

Some parts of the paper were written at Vienna University of Technology and at Queen Mary University of London. EH would like to thank K. Narain, L. Alvarez-Gaume for discussions and supports, also thanks to Mathematical Institute at Charles university for the hospitality. He is also grateful to IHES, CERN, UC Berkeley and Caltech for the warm hospitality. We thank A. Sagnotti, J. Polchinski, B. Jurco, O. Lechtenfeld, N. Arkani-Hamed, P. Horava, P. Sulkowski, G. Veneziano, J. Schwarz and W. Siegel for their insights and fruitful discussions. This work was supported by an ERC Starting Grant no. 335739 “Quantum fields and knot homologies”, funded by the European Research Council under the European Union’s 7th Framework Programme. EH was also supported in part by Scuola Normale Superiore and by INFN.

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