On Special Limit of Non-Supersymmetric Effective Actions of type II String theory

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 brane, but also 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 gained.


Introduction
Among several goals of theoretical physicists and in particular string theorists, we may point out to 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 on many properties of various potential different string theories [1,2,3]. Since duality transformation is not promising in this context any more, one needs to be aware of the fact that for non-BPS branes just scattering amplitudes and Conformal Field Theory (CFT) methods [4] would exactly determine all order α ′ corrections of the effective actions of string theory.
Making use of non-supersymmetric branes, the so called Sakai-Sugimoto model [5] as well as the symmetry breaking for holographic QCD models have been known [6]. Tachyons do play crucial role in instability of the aforementioned systems so it would be important to consider tachyons and try to achieve their effective actions in both type IIA and IIB string theories and also explore their new couplings in the Effective Field Theory (EFT).
The leading order non-BPS effective actions including tachyonic modes were proposed in [7,8], where some of their properties such as their decays and tachyon condensation have also been clarified in detail [9]. Following [3], one reveals how to embed the presence of non-BPS branes in the effective actions. We studied D-brane anti-D-brane systems [10]. Recently the generalisation of effective actions of D-brane-anti-D-brane system to all orders in alphaprime for both Chern-Simons and Dirac-Born-Infeld (DBI) effective actions was discovered [11]. Another example would be related to tachyon condensation that has been investigated in [12] 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 non-BPS scattering amplitude formalism as a theoretical framework or laboratory to discover their effective actions, including their all order α ′ corrections 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 and collaborators [13] where various explanations within the context of brane 's effective actions through EFT have been discussed. Not only brane production [14] but also inflation in string theory in the procedure of KKLT [15] can also be mentioned.
In this paper we deal with a non-BPS four point function and explore some Bianchi identities as well as 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 corrections in string theory and reveal new string couplings in both type II string theories. 2 One can try to relate some of the new couplings to AdS/CFT [20]. It is also worth making a remark on D-brane-Anti-D-brane system as they do affect not only in the problem of stability of KKLT model but also string compactifications [21] and in particular in the so called Large Volume Scenario. The relation between D-branes and Ramond-Ramond (RR) charges is well established [22], where one could also take into account some brane's bound states [23]. All the EFT methods of deriving Wess-Zumino (WZ) and DBI effective actions are given in [24,25].
The paper is organised as follows. First we try to study a four point function including a closed string Ramond-Ramond (RR) and a transverse scalar field and a real tachyon on the world volume of non-BPS branes, where an RR and two tachyons has been fully addressed in [26], then we build all order α ′ higher derivative corrections to it and explore a pattern from this calculation to reconstruct all singularity structures of higher point functions of non-BPS branes. we also find a new mixed WZ Taylor expansion as and explore its all order higher derivative corrections too.
Having set all lower point functions of non-BPS branes, we would clarify more hidden symmetries in non-BPS context. Hence we make use of all the CFT techniques to a six point correlation of an RR, a scalar field and three tachyons. We first find out the entire correlators type IIA (IIB) and then we just illustrate the final result in different picture of scalar field, basically we explore < V C −1 V φ 0 V T −1 V T 0 V T 0 > and argue that using this particular case we would be able to precisely obtain all bulk singularity structures that are not present in the other picture. Using selection rules [27] for non-BPS amplitudes, EFT and in a particular decoupling limit, we discover the ultimate answer for the S-matrix.
Having set all symmetries of the S-matrix, we explore the expansion of the S-matrix. Using the soft limit we generate not only all the infinite massless singularities but also an infinite number of u-channel bulk singularity structures can be precisely reconstructed in an EFT and come to a perfect match between string amplitudes and EFT counterparts. Finally we use all the higher derivative corrections of two tachyon two scalar couplings to be able to produce an infinite number of scalar field singularities as well. It is worth to emphasise since there is no coupling between two tachyon and a scalar field, the amplitude (as can be seen from the ultimate result of the S-matrix) has no singularity in t, s, v channels at all. The DBI part of the effective action for non-BPS branes is i, j = 1, 2, i.e., T 1 = T σ 1 , T 2 = T σ 2 . The DBI part of the D-brane-anti-D-brane is given in [11]. If we make kinetic terms symmetrized, find the traces and then use ordinary trace, the action will get replaced to Sen's action [28]. However, in [29,30] by direct CFT computations and scattering amplitudes we have shown that Sen's effective action does not provide consistent result with string amplitudes. The expansion of the S-matrices is consistent with Tachyon's potential V (|T |) = e πα ′ m 2 |T | 2 which comes from BSFT [31,32]. On the other hand WZ action is given by To consider interactions with Tachyons, one can make contact with superconnection of the non-commutative geometry [33,34,35] where curvature is β ′ is WZ normalisation constant. One can find out different types of WZ couplings from the above actions to generate consistent result between string amplitudes and the EFT, such as In order to actually address entire form of a four point function of an RR, a real tachyon and a scalar field in both type IIA, IIB string theories, one must apply conformal field theory methods to the complete S-matrix elements and explore whether or not there are some bulk singularity structures or contact interactions and also to notice how one might be able to find out all order contact interactions. To achieve all the correlation functions, one needs to know vertex operators where their complete forms are shown by This four point function at disk level can be computed if one takes into account the following on-shell conditions Projection operator and closed string RR 's field strength are defined by Spinor notation is given by (P − H / (n) ) αβ = C αδ (P − H / (n) ) δ β and accordingly for IIA (IIB) we pick up n = 2, 4,a n = i (n = 1, 3, 5,a n = 1). If we employ the doubling trick then one is able to just work out with holomorphic parts of the fields. Thus we apply the the following change of variable to our field content Having carried the trick out, we would use the following propagators for all X µ , ψ µ , φ fields as follows Hence, our amplitude in the asymmetric picture of RR is found to be To obtain the other correlation function including two spinors, a current and a fermion field :> we work out the so called Wick-like formula [36] to get to One could precisely show that now the amplitude is SL(2, R) invariant and to remove the volume of conformal killing group we do gauge fixing as ( . Setting the above gauge fixing, we come to know that the second term of I 2 does not have any contribution to the final result of the amplitude due to the fact that integrand is odd while the moduli space is covered on the entire space-time or due to having symmetric interval. u = − α ′ 2 (k 1 + k 2 ) 2 is introduced and the amplitude is resulted by The ultimate result of amplitude is given by µ ′ p is RR charge of brane. All the traces are non zero for p + 1 = n case and can be calculated as The 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 brane k 2 = p a p a = 1 4 [37], one realises that this constraint holds for CT φ amplitude and indeed the proper momentum expansion can be read off as follows where the 1st three coefficients are (2), c 1 = 1 6 (π 2 + 12ln(2) 2 ).
The first term in (8) can be produced by using the following Chern-Simons coupling where the scalar field has been taken from the Taylor expansion Note that the second term of (8) can just be produced if one introduces a new coupling where this time a scalar field comes from pull-back of brane and covariant derivative of tachyon is appeared to cover the entire (p + 1) world volume direction. Hence the second term of (8) can be regenerated by the following new mixed WZ and pull-back coupling As we have seen, the expansion of the amplitude has an infinite contact interaction and all those contact interaction terms related to the first term of (8) can be produced in the EFT by applying all infinite higher derivative corrections to the WZ effective actions of a real tachyon, a scalar field and a C p RR closed string (9) Likewise all the contact interactions related to the second term (8) can be constructed if one applies the same prescription to all higher derivative corrections to S 2 action as follows It is also interesting to revisit the amplitude in the other pictures. The final result of the amplitude can be derived as The trace that includes γ 11 factor, has the special property so that all results are being held for the following relations as well p > 3, H n = * H 10−n , n ≥ 5.
Now if we apply momentum conservation (k 1 + k 2 + p) a = 0 to the above amplitude then we realise that the amplitude (12) can just produce the 1st term of (8), more importantly one finds that a Bianchi identity holds for the world volume of branes in the presence of RR's field strength as Finally the result of the amplitude for Finding the above result and keeping in mind momentum conservation, one understands that to get the consistent result with both string theory and effective field theory parts, the restricted world volume Bianchi identity (13) has to be modified by a new Bianchi identity which will be valid for both world volume and transverse directions of the branes as follows 3 In this section we would like to deal with a non-BPS six point function including an RR, a transverse scalar field and three real tachyons to be able to find not only the proper expansion of the amplitude but also reveal all the bulk singularity structures as well as various restricted Bianchi identities. Given the exact symmetries of string theory amplitudes, tachyon expansion and the particular soft limit, in the following we show that one is able to predict some of the singularity structures of > and determine all the singularities including the bulk singularities that carry momentum of RR in the bulk directions. One needs to provide the correlation function between two spinors and four fermion fields at different locations where just one of them moves along transverse direction of brane so I cbai The amplitude makes sense for p = n + 1, p + 1 = n cases. The algebraic form of the above integrals can be derived in a soft limit 4k 2 .p → 1. Using this limit and appendix B of [37] and [38] one arrives at closed form for the integrals. For the simplicity, we just write down the ultimate result of the amplitude for p = n + 1 case as where M 1 , M 2 are 2 ) The other part of the amplitude holds for C p case and one reveals its final form as follows where M 3 , M 4 are written in terms of ratio of the Gamma functions In this section we would like to produce all the massless bulk singularity structures that carry momentum of RR in the transverse directions. To do so, we deal with the following All the correlation functions can be computed. To shorten the paper we use the same gauge fixing as in the last section. Thus the final form of the amplitude is found where The amplitude makes sense for p = n + 1, p + 1 = n cases. Using soft limit 4k 2 .p → 1, one finds the amplitude for p = n + 1 case in below The second part of the amplitude holds for C p case and one finds it as Now let us deal with all singularities and start to produce them.

Bulk Singularity Structures
To obtain all the singularity structures including the ones that carry momentum of the closed string RR in the bulk direction, one needs to find first the expansion of the amplitude.
The following remarks need to be taken into consideration. Having applied momentum conservation we get s + t + u + v + r + w = −p a p a − 3 2 . Using the constraint p a p a → 1 4 for non-BPS branes, taking the symmetries of our amplitude (given the EFT and the fact that it should be symmetric under exchanging s, t, v), one gains the expansion for the amplitude (20) as follows The expansion of (M 1 M 2 ) for this particular soft limit and around (21) can be read as Given the above expansion, standard EFT methods that propose to us to have an infinite u-channel massless gauge field poles and symmetries [39], one understands k i .k j → 0 for massless strings and p a p a → 1 4 for non-BPS branes. There is no coupling between two tachyon and a scalar field and given the selection rules for non-BPS branes [27], the fact that the kinetic term of tachyon is fixed in DBI action, one clarifies that there is no double pole for p = n+1 case. This point can also be observed from the expansion of the amplitude, hence we have an infinite number of u-channel simple poles. For p = n + 1 case, from EFT and the above expansion, one notices that the S-matrix has an infinite number of u-channel gauge field poles. The first u-channel pole in string theory can be written down as This simple u-channel pole can be reconstructed in an EFT by the following sub amplitude Here α, β are gauge group indices. Now if we use the above vertices in an EFT and make use of momentum conservation (k 1 + k 2 + k 3 + k 4 + p) a = 0, Bianchi identity p a p−2 H a 0 ···a p−3 = 0, the fact that the amplitude is symmetric under k 1a p−1 k 1a p−2 and also due to antisymmetric property of ǫ, we realise that the term k 1a p−1 k 1a p−2 does not have any effect in EFT part of the amplitude. If we multiply (23) by 1 2 µ ′ p β ′ π 1/2 and compare it with above EFT amplitude we then explore that the first simple u-channel gauge field pole is exactly generated.
V a (C p−2 , A, φ 1 , T 4 ) was derived from some part of the mixed WZ coupling and Taylor However, as can be seen from the expansion, the amplitude in (19) has an infinite number of bulk singularity structures concretely, where the first bulk pole in string theory reads as If one wants to extract the couplings from (25), one takes integration by part to arrive at two different contributions (27) where to derive the first simple u-channel gauge pole given in (23), we have already used the contribution from the second term of (27). Now if we use the following relation and plug it into the first term of (27), then we would be able to produce the bulk part of the vertex V b (C p−2 , A, φ 1 , T 4 ). Eventually by replacing it into the same EFT sub amplitude (24) we are able to precisely produce the first bulk singularity u-channel pole (26) which carries p.ξ term as well. As one notices from the expansion of the amplitude, we have an infinite number of u-channel poles and to generate them in an EFT the following remarks are in order. The vertex of V α a (T 2 , T 3 , A) comes from the kinetic term of tachyons in DBI action which is fixed and has no correction. The propagator is also fixed, as it comes from the kinetic term of gauge fields that has been fixed for this case too. Therefore to be able to reconstruct all infinite u-channel poles one must directly apply infinite higher derivative corrections to the mixed WZ coupling (25) as follows Tr Keeping fixed the V α a (T 2 , T 3 , A) and propagator, extracting the modified all order vertex V b (C p−2 , A, φ 1 , T 4 ) from (29) and replacing it in the EFT amplitude (24), then one is able to show that all infinite bulk singularity structures are precisely produced. This clearly confirms that the expansion is consistent with EFT amplitude as well.
The amplitude has an infinite number of massless scalar poles in (t ′ + v ′ + r) 3 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 To generate all scalar poles at first order one needs to employ the following Lagrangian while to produce all the other poles, one needs to know higher derivative corrections to the two tachyon-two scalar field couplings to all orders where The definitions for D nm and D ′ nm higher derivative operators are The all order extended vertex V β j (φ, T 2 , φ 1 , T 4 ) is derived from (33) and in momentum space takes the form (a n,m + b n,m ) (k 2 ·k 1 ) n (k 1 ·k 4 ) m + (k 2 ·k 1 ) m (k 4 ·k 1 ) n +(k 2 ·k 1 ) n (k·k 2 ) m + (k 2 ·k 1 ) m (k·k 2 ) n + (k 1 ·k 4 ) m (k·k 4 ) n + (k 1 ·k 4 ) n (k·k 4 ) m +(k·k 2 ) m (k·k 4 ) n + (k·k 2 ) n (k·k 4 ) m Tr (λ 4 λ 1 λ 2 λ β ) (34) where 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 8iµ ′ p β ′ ǫ a 0 ···ap ξ 1i H i a 0 ···a p−1 k 3ap p!(v ′ + t ′ + r) Tr (λ 1 λ 2 λ 3 λ 4 ) ∞ n,m=0 (a n,m + b n,m )[v ′m t ′n + v ′n t ′m ]v ′ t ′ (35) Eventually, the S-matrix suggests to us that the string amplitude has just a double pole for p + 1 = n case. It emerges from the following Feynman diagram in EFT 4 with the derived vertices taken from the lower order effective actions G(T ) = i (2πα ′ )T p (u + r + w + 1) (38) and k is the off-shell tachyon's momentum. Replacing the above vertices to (36) we would reproduce its double pole as well. Note that By direct calculations, the presence of some new couplings such as F (1) · F (2) or Dφ i(1) · Dφ i (2) has been confirmed in the world volume of D-brane-Anti D-brane systems [29] and [10]. Indeed making string calculation we could produce all massless and tachyon singularities of the amplitudes.
While WZ coupling C p ∧ DT φ will not receive any higher derivative correction, and all the kinetic terms are fixed, hence they do not get any corrections either. Thus all other tachyon singularities give us clues about structures of all order higher derivative corrections to various couplings and in this paper we could consistently fix their coefficients for good.
Note that all these couplings are found in the limit p a p a → 1/4, thus we cannot compare these couplings with BSFT couplings. However, tachyon's potential remains the same as in BSFT (V (T ) = e πα ′ m 2 T 2 [31]) which is where m 2 = −1/(2α ′ ) is tachyon's mass. Tachyon is going to be condensated at T → ∞, therefore its potential will be sent to zero.
We think these corrections play crucial role in determining singularities of the higher point functions of string theories. Veneziano amplitude [40] was generalised in [11], we hope to be able to address supersymmetric generalisation of the D-brane-anti-D-brane system by directly carrying out fermionic amplitudes [41]. We also hope to have progress on the generalisation of the non-supersymmetric DBI and WZ effective actions in near future.