# On non-BPS effective actions of string theory

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## Abstract

We discuss some physical prospective of the non-BPS effective actions of type IIA and IIB superstring theories. By dealing with all complete three and four point functions, including a closed Ramond–Ramond string (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 order \(\alpha '\) 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.

## 1 Introduction

D-branes have been realized to be the sources for Ramond–Ramond (RR) fields [1, 2]. RR couplings 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 the Myers effect [4, 5, 6]. 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 [7, 8, 9].

*DT*is the covariant derivative of the tachyon \((D_aT=\partial _a T-i[A_a,T])\). On the other hand the Chern–Simons action for BPS branes was constructed in [14]. Using Boundary String Field Theory (BSFT), one has the tachyon’s kinetic term in the DBI part [15] as follows:

On the stable point, the tachyon potential and its effective action get replaced by the well-known tachyon DBI action [16, 17] with potential \(T^4V(T^2)\). The WZ part of the action in this approach has the same formula as appearing in (2). Using the S-matrix method the normalization constants of \(\beta ',\beta \) for the non-BPS and brane–antibrane system are discovered to be \(\beta '=\frac{1}{\pi }\sqrt{\frac{6\ln (2)}{\alpha '}}\) and \(\beta =\frac{1}{\pi }\sqrt{\frac{2\ln (2)}{\alpha '}}\) [18]. It is worth mentioning that the super-connection’s structure for the WZ action was found by the S-matrix approach in [19].

The aim of the paper is to show that the scattering amplitudes computed in different super-ghost pictures are compatible when suitable Bianchi identities are imposed on the RR 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 we explore all their infinite higher-derivative corrections. It is believed that due to a supersymmetry transformation BPS the S-matrices do not generate a Bianchi identity. To get consistent results for four point functions of the two gauge fields, a tachyon and a closed string RR field in their asymmetric and symmetric pictures, we discover various restricted Bianchi identities. Eventually we have to do with a universal expansion for tachyon and construct all different singularity structures of \(\langle V_{C^{-2}} V_{A^{0}} V_{A^{0}} V_{T^{0}} \rangle \) as well as all order \(\alpha '\) higher-derivative corrections to the various couplings of the type IIA, IIB superstring theories.

## 2 All order \(\langle V_{C^{-2}} V_{A^{0}} V_{T^{0}} \rangle \)

*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 \(J=-2i(1+x^2)\). One reveals that \(I_1\) has zero contribution to the S-matrix. Because the integrand is an odd function while the interval of the integral is symmetric.

^{1}We introduce \(t = -\frac{\alpha '}{2}(k_1+k_2)^2\) and \(I_2\) is obtained byThe last two terms have just non-zero contributions to our amplitude. The final answer for the amplitude isTo be able to match the leading order of the S-matrix with the following coupling in the EFT:

*CAT*is \(t\rightarrow -1/4\). The expansion for the gamma function is

## 3 The complete \(\langle V_{C^{-2}} V_{A^0} V_{A^0} V_{T^0}\rangle \) amplitude

In order to find the complete form of the scattering amplitude of a tachyon, a potential RR \((p+1)\) form-field and two gauge fields \(\langle V_{C^{-2}} V_{A^0} V_{A^0} V_{T^0}\rangle \), one needs to employ all CFT techniques. To achieve all singularities and contact interactions, we use the vertex operators. Note that, as clarified in [26], the CP factor of RR for the brane–antibrane system is different from the CP factor of non-BPS branes. RR vertex operators are introduced in [27]. One might refer for some of the BPS and non-BPS scattering amplitudes to [28, 29, 30, 31, 32, 33].

^{2}and all the other kinematical relations can be found in [26]:where

*a*,

*b*,

*c*are written in terms of the following Mandelstam variables:

## 4 The complete \(\langle V_{C^{-1}} V_{A^{0}} V_{A^{-1}} V_{T^{0}}\rangle \) amplitude

## 5 Tachyon’s momentum expansion

^{3}Having taken (31), we would understand that \(L_1,L_2,L_3\) have

*t*-channel gauge fields and \(s', u',(s'+t+u')\) tachyonic singularities. We make comparisons of the singularity structures as well as all contact terms. We then reconstruct all singularities in EFT and derive the restricted world volume Bianchi identities for non-BPS branes.

## 6 Singularities and restricted Bianchi identities

## 7 Contact term comparisons

### 7.1 All \((t+s'+u')\)-channel tachyon singularities

#### 7.1.1 All \(u',s'\) channel tachyon singularities

*k*is the off-shell tachyon momentum. Replacing (54) inside (53), we obtain all order \(u'\) channel tachyon poles in an EFT:

*k*is the off-shell gauge field’s momentum and \(V^{a}(C_{p-2},T_3,A)\) was derived from the corrections to the WZ coupling \(C_{p-2}\wedge F\wedge DT\). Notice that the kinetic term of the gauge fields is fixed in DBI action, so one finds that \(V^{b}(A,A_1,A_2)\) should not receive any higher-derivative corrections. The tachyon expansion that we talked about is also consistent with effective field theory. This is because we are able to produce 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 the 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. Given the exact symmetries of the amplitudes and the universal tachyon expansion in [42], 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 the near future.

## Footnotes

- 1.
\(\alpha '=2\) is set.

- 2.
\(x_{ij}=x_i-x_j\), and \(\alpha '=2\).

- 3.$$\begin{aligned} L_3= & {} -{\pi ^{5/2}}\sum _{p,n,m=0}^{\infty }\Bigg (c_n(s'+t+u')^n+c_{n,m}\frac{s'^nu'^m +s'^mu'^n}{(t+s'+u')}\\&+\,f_{p,n,m}(s'+t+u')^p (s'+u')^n(s'u')^{m}\Bigg )\nonumber \\&b_{-1}=1,\,b_0=0,\,b_1=\frac{1}{6}\pi ^2 ,e_{1,0,0}=\frac{1}{6}\pi ^2\\&c_0=0,c_1=\frac{\pi ^2}{3},c_{1,0}=c_{0,1}=0, f_{0,0,1} =4\zeta (3). \end{aligned}$$

## Notes

### Acknowledgements

This paper was initiated during my second 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.

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