On New Bulk Singularity Structures, RR Couplings in Asymmetric Picture and Their All Order $\alpha'$ Corrections

We have analyzed in detail four and five point functions of the string theory amplitudes, including a closed string Ramond-Ramond (RR) in an asymmetric picture and either two or three transverse scalar fields in both IIA and IIB. The complete forms of these S-matrices are derived and these asymmetric S-matrices are also compared with their own symmetric results. This leads us to explore two different kinds of bulk singularity structures as well as various new couplings in asymmetric picture of the amplitude in type II string theory. All order $\alpha'$ higher derivative corrections to these new couplings have been discovered as well. Several remarks for these two new bulk singularity structures and for contact interactions of the S-Matrix have also been made.


Introduction
By now it is widely known that D-branes in super string theory play the most important role in this area of research. Indeed there is no doubt they continue to have more contributions even to other topics of high energy physics as well [1,2,3] 2 .
Although we have referred in [5] to various fascinating papers about the subject of string theory's effective actions, for the entire self-completeness, we point out some of the remarks that are of high importance to the author as follows.
To our knowledge Myers in [6] discovered more or less the complete form of a single bosonic action that can be generalized for diverse D p -brane systems. Not really consequently but after while we started finding out the generalization of that action with emphasis on exploring all order α ′ corrections to D-brane effective actions. This involves to deal with both Chern-Simons, DBI effective actions and mixed open-RR S-matrices. Besides those things , some of the new couplings and or Myers terms have been derived in [7]. It is also worth highlighting the fact that applications to some of the new couplings have already been released in the literature. For instance we have applied some of new couplings of type IIA,IIB to get to de Sitter or Anti de Sitter brane world solutions [8]. Besides that, it is noted that some of those new couplings will certainly play the key contribution not only in exploring N 3 entropy for just a single M5-brane [9] or in M-theory [10] but also have portions in various M2-M5 or some other configurations in super gravity solutions for black branes, in order to actually explore n 3 entropy growth production as well.
Various remarks for D-brane anti D-brane system [11] involving their corrections [12] have been given. In addition to the efforts in [13], the entire form of super symmetric Myers action has not been concluded yet. On the other hand, the action for a single brane was understood in [14] where its generalization could be found in [15].
We invite the reader for the complete review of Chern-Simons effective actions and their remarks to have a look at [16,17,18]. One can find out a very brief review of all DBI and new Wess-Zumino terms of BPS branes in [19].
To observe all the three ways of exploring couplings in Effective Field Theory (EFT), (which are either Taylor or pull-back, Myers terms) for both super symmetric and non super symmetric cases we suggest [20].
Behind AdS/CFT there is a close relation between an open and a closed string so one might be interested in gainning the mixed open-closed string amplitudes, which of particular interest to them in string theory is indeed the mixed RR potential (C-field)-open strings. In fact one may hope to address various issues (involving the AdS/CFT) by dealing with these mixed higher point functions of RR-scalar fields of type II. Let us just consider some of the works that are in correspondence either with S-matrix formalism in the presence of D p -branes or are related to D-brane physics applications [21].
The paper is organized as follows. In section 2 , we talk about the conventions and then try to provide the complete calculations for the four and five point functions of type IIA,IIB of string theory including an RR in asymmetric picture and either two or three real transverse scalar fields. Indeed for < V C −2 V φ 0 V φ 0 > S-Matrix, we modify the so called Wicklike rule to have the gauge invariance, particularly we show that in asymmetric picture of RR one finds out new term in the amplitude and hence new couplings in an EFT can be constructed out. In order to be able to produce all string contact interactions in an EFT, one needs to employ mixing couplings where the first scalar comes from Taylor expansion and the second one comes from pull-back method. We then find out all order α ′ higher derivative correction to those couplings as well.
In the next sections we perform the entire analysis of a five point function of a C-field with three transverse scalar fields in zero picture, that is , we deal with We also compare the exact form of the S-matrix in asymmetric picture with its own result in symmetric picture < V C −1 V φ −1 V φ 0 V φ 0 > at both level of contact interaction and singularity structures. We obtain various new contact interactions as well as two kinds of new bulk singularity structures with various new couplings in These two different kinds of bulk singularity structures of string amplitude are t, s, u as well as (t + s + u)-channels bulk singularity structures that can just be explored in an asymmetric picture of the amplitude. These bulk singularity structures carry momentum of RR in transverse directions which are related to winding modes, however, winding modes are not inserted in vertex of RR in symmetric picture in ten dimension. Hence it makes sense to have these new bulk singularities as well as new couplings just in asymmetric picture of the amplitude. Indeed we are also able to produce these two different bulk singularity structures of string amplitude in field theory by taking into account various new couplings in effective field theory side as well.
We also generalize all order α ′ higher derivative corrections to those new couplings that are appeared in an asymmetric picture of the amplitude.
The important point must be noted. These new couplings are discovered by just scattering amplitude formalism in an antisymmetric picture and not any other tools such as T-duality can be employed to get to these couplings. Because these new couplings carry momentum of RR in transverse (or bulk) directions, while winding modes are not embedded in ten dimension of RR vertex operator. We then construct all order bulk singularity structures of t, s, u, (t + s + u) channels in an EFT where the universal conjecture of corrections [22] plays the fundamental ingredient in producing all the infinite singularity structures.
These new bulk singularity structures of string theory amplitude that are just shown up can be generated by taking into account various new EFT couplings.
All those new terms do carry the scalar products of momentum of RR in the bulk with the polarization of scalar fields accordingly. We think that the importance of these results will be provided in future research topics , such as all order Myers effect and various other subjects in type II string theory [23]. We have also observed that at the level of EFT the super gravity background fields in DBI action must be some functions of super Yang-Mills. Some particular Taylor expansion for the background fields should also be taken into account as was notified in Dielectric effect [6].
In this section we take into account some Conformal Field Theory (CFT) tools to get to the Having expanded the elements of scattering amplitude and taken some patters for string corrections, we would start generating α ′ corrections.
Our S-matrix computations are valid at world-sheet level of four and five point functions at the disk level which covers both transverse and world volume directions. Although it is impossible to address all the attempts that have been carried out in this area , we can highlight several efforts that are worth considering for super symmetric and non super symmetric cases [24,25,26,27].
One first needs to apply the general structure of vertices where in this paper we just insist on employing RR potential (which is a C p+1 -form field in asymmetric picture), therefore all the other two or three transverse scalars have to be considered at zero picture. The vertex of RR in asymmetric picture was first proposed by [28]. A new paper about picture changing operators has been recently released [29], however, to our knowledge it is not understood how to deal with all RR closed-open string amplitudes. That is why we try to come up with direct calculations, although the computations in an asymmetric picture is very long and tedious. It would be very nice to work out more to actually understand whether or not the proposal in [29] can be applied to higher point functions of string theory including RR (in an asymmetric picture) and scalar fields. The three point function of a closed string RR and a transverse scalar field (describing oscillation of brane) in both symmetric and asymmetric picture of RR has been accordingly computed in detail [30].
Let us as a warm-up just mention the results also where definitions could be read off from [30]. We have chosen the following notations for entire ten dimensional space-time, world volume and transverse directions appropriately µ, ν = 0, 1, ..., 9 a, b, c = 0, 1, ..., p i, j = p + 1, ..., 9 To be able to produce A φ 0 ,C −2 in an EFT part, we first apply to the second term of (2) momentum conservation on world volume (k a 1 + p a = 0) and in particular consider the following Bianchi identity so that the 2nd term in (2) has zero contribution to the S-Matrix. The 1st term in (2) can be produced in an EFT by considering Taylor expansion of a real scalar field through RR coupling as follows The four point function in an asymmetric picture of a RR and two real scalar fields in zero picture can be done by The scalar field and RR vertex operators in symmetric and asymmetric pictures are given by where k 2 = p 2 = 0, k.ξ 1 = 0. We also consider x 4 ≡ z = x + iy, x 5 ≡z = x − iy and use the doubling trick to make use of standard propagators as follows Having set the Wick theorem, the amplitude would be written down as The exponential factors are In order to be able to explore all the above the fermionic correlators including the correlation functions of various currents with spin operators , one has to consider the Wick-Like rule [26] and modify it as follows where one has to consider all various contractions to make sense of the corrected Smatrix elements. The important point one has to hold, is that in all the above equations, Γ µn...µ 1 must be antisymmetric with respect to all the gamma matrices. The x ′ i s are real and extremely importantly point that has been over looked in the literature is as follows. The Wick-like formula must be modified with a minus sign, in order to respect the gauge invariance of the higher point function of BPS or non-BPS S-matrices. Hence, the corrected Wick-like formula for the definition of two point function is Now we apply the generalisation of Wick-like rule to amlitude so that one is able to derive all the correlators as By applying (7) into this four point amplitude we can easily determine that the S-matrix is SL(2, R) invariant. We do the proper gauge fixing as (x 1 , x 2 , z,z) = (x, −x, i, −i), taking t = − α ′ 2 (k 1 + k 2 ) 2 to get to the S-matrix as where the sixth and seventh terms do not have any contribution to the S-matrix because the integration is taken on the whole space meanwhile the integrand is odd so the result vanishes. More crucially, after using kinematical relations we come to know that the sum of the first term and the last term of (8) is zero. One now explores the final answer for the 3rd and 4th term of (8) as follows Eventually one can find out the result for the 2nd term of asymmetric S-matrix as follows The trace is non-zero for p + 1 = n − 1 case and (10) does not include any poles, because the expansion is low energy expansion (t → 0). It has been emphasized in [20] that for a particular string S-matrix involving scalar fields one needs to take into account three different ways to actually re-generate all the string couplings in an EFT side. The first way was imposed by Myers in [6], while the second and the third ways of EFT have been completely mentioned in [20] to be either pull-back method or Taylor expansion of the scalar fields. This S-matrix in symmetric picture can be readily computed as Carrying out the integrals explicitly and using momentum conservation, we obtain The closed form of the expansion is where all c n are related to higher derivative corrections of the scalar fields through either Taylor expansions or Pull-back of branes.
Now we work out the related super Yang-Mills vertices to reconstruct all string couplings to all orders in α ′ . In order to produce all infinite contact interactions for first term of (12) and also to produce part, one needs to deal with the Taylor expansion of the two real scalar fields through an RR (p+1)-form field as and subsequently all order α ′ corrections can be derived as In the meantime to be able to produce at the leading order the second term of (12) as well as the 5th term of A φ 0 φ 0 C −2 , one must employ the so called Pull-back formalism as follows Essentially one can insist on producing all infinite corrections by imposing the higher derivative corrections to the pull-back and fix all their coefficients without any ambiguities in string theory as below There is no external gauge field in our S-matrix so one could propose the covariant derivatives of scalar fields to above higher derivative couplings to be able to keep track of the gauge invariance of the S-matrix as well.

3.1
The other RR couplings of type II string theory Now in order to be able to produce the term that has been explicitly appeared by asymmetric amplitude in (9), one needs to write down sort of new couplings, in the sense that in this turn, neither both scalar fields come from pull-back nor Taylor expansion.
Indeed for the first time, we just confirm the presence of sort of mixed couplings in string theory so that the first scalar field comes from pull-back and the second scalar comes through Taylor expansion. Thus the presence of mixed coupling in EFT is now being discovered. Note that this fact has become apparent by just dealing with Let us write down the effective coupling at leading order which is α ′2 and then generalize it to all orders in α ′ . To be able to produce (9), at leading order one has to write down the following new coupling As it is clear from (13), we have many contact interactions which are related to higher derivative corrections of the scalar fields. These corrections without any ambiguities can be fixed by just S-matrix method. One can now apply the higher derivative corrections to scalar fields to actually get to all order α ′ corrections of the above couplings as follows Once more one could restore the gauge invariance by replacing the covariant derivatives in (18). Now let us deal with the five point function of an RR and three scalar fields in an asymmetric picture.
The S-matrix elements of three transverse scalar fields and one RR in an asymmetric picture One can extract the whole S-matrix and divide it out to various correlation functions.
In order to be able to explore all fermionic correlations including the correlations of various currents with spin operators, one has to reconsider the modified Wick-like rule as mentioned in the last section. Let us simplify the S-matrix further 3 where x ij = x i − x j , and also where all n ′ i s as well as I kbjaic 8 are given in the Appendix 1. One finds that the amplitude is SL(2,R) invariant so to remove the volume of conformal killing group as well as for practical reasons we fix the position of open strings at zero, one and infinity, that is x 1 = 0, x 2 = 1, x 3 → ∞. Now if we come over the evaluation of all the integrals on the location of closed string RR on upper half plane as explained in Appendix 1, then one finally reads off all the elements of the string amplitude in an asymmetric picture of RR as follows where where the functions Q 1 , Q 2 , Q 3 , Q 4 , Q 5 , Q 6 , Q 7 , L 2 , L 3 , L 5 , L 6 are given in Appendix 2. Let us write down the same amplitude in its symmetric picture, that is, which means that we consider the amplitude in terms of field strength of RR and start comparing these two results. Working out in detail and making use of the integrals that presented in [31] and [20], one constructs the S-matrix in symmetric picture of RR as follows with where L 2 , L 3 , L 5 , L 6 are introduced in (A.3). Let us compare the amplitudes in asymmetric picture with its symmetric result and explore all new bulk singularity structures as well as new couplings (with their all order α ′ corrections) that are just appeared in asymmetric picture of the amplitude.

Singularity Comparisons between asymmetric and symmetric pictures
In this section we try to produce all the singularity structures of symmetric picture by dealing with the S-matrix element in asymmetric picture of RR. We add the 1st term of A 810 of asymmetric amplitude with first term of A 87 and apply momentum conservation along the world volume of brane to actually reach to the following couplings now if we use pC / = H / then we see that above coupling precisely generates A ′ 10 of symmetric picture ( 4 . The first term of above equation will be precisely cancelled off by sum of the first term of A 3 of asymmetric picture and the whole A 89 .
Likewise what we did previously, here we try to add the second terms of A 86 and A 87 and in particular we take into account the momentum conservation to actually arrive at the following singularities The first term of (26) will be removed by the sum of the first term of A 5 and the whole A 85 of asymmetric S-matrix . The same holds as follows. 4 up to a normalisation constant 2 1/2 i Adding the 2nd term of A 810 and the first term of A 86 also paying particular attention to momentum conservation give rise to the following elements −2ituL 6 ξ 3 .ξ 1 ξ 2j Tr (P − C / (n−1) M p Γ bj )(−k 2b − p b ) (27) which is precisely A ′ 8 . Notice to the point that the first term of (27) has been equivalently cancelled off by the sum of first term of A 2 and the entire Having taken the second term of A 72 and having applied the momentum conservation to it we seem to get evidently the first term above has no contribution, because there is an antisymmetric ǫ tensor while the whole element is symmetric with respect to interchanging k 3 so the answer turns out to be zero while the second term in (28) precisely generates A ′ 7 . The the last term in (28) remains to be explored later on.
By applying the same tricks to the second term of A 42 we obtain obviously the first term above has zero contribution to S-Matrix and the second term reconstructs A ′ 6 , meanwhile the last term will be considered in the next section. We also need to take into account the 2nd term of A 84 and apply the momentum conservation to it to get to The first and second term in (30) have zero contribution and the third term produces A ′ 5 . Once more one needs to deal with the 2nd term of A 83 and draw attention to momentum conservation in such a way that the following singularities turn out to be produced clearly the first and second term of (31) have no contribution to amplitude as there is an antisymmetric ǫ tensor while the whole singularity is symmetric with respect to interchanging k 2 , k 3 so the answer turns out to be zero while the third term in above singularity re-builds A ′ 4 .
We need to carry out the same tricks to the second term of A 82 to be able to recreate exactly A ′ 2 of symmetric amplitude.
Eventually we need to add the 1st terms of A 82 , A 83 and A 84 to actually derive the following singularities which is nothing but exactly A ′ 3 part of symmetric S-matrix and L 22 = −uL 5 .From the above comparisons we come to the following points.
By doing careful analysis of asymmetric elements we were able to re-generate all order α ′ singularity structures of symmetric amplitude ( However, the important point that must be emphasized is as follows. Regarding our true comparisons and the remarks that have already been pointed out in this section, we have got some extra contact interactions as well as two extra kinds of bulk singularity structures in asymmetric amplitude that can not be shown up by symmetric analysis and all of them will be highlighted in the next section. We also try to introduce new couplings in an EFT to be able to produce all those new bulk singularities as well.

Bulk singularity structures in Asymmetric Picture
As we have already argued, the S-matrix of an RR and three transverse scalars in its asymmetric picture (in addition to all the singularities of symmetric picture) generates two different kinds of bulk singularity structures. For instance the second term of A 62 of asymmetric picture (< V C −2 V φ 0 V φ 0 V φ 0 >) has got a new kind of infinite u-channel bulk singularities which can not be obtained from the symmetric picture of Indeed we would have expected to have these bulk singularities in asymmetric picture, because of the symmetries with respect to interchanging all three scalars as well as symmetries of string amplitude. Therefore let us point out the first kind of u-channel bulk singularity structure in asymmetric analysis (which is the 2nd term of A 62 ) of ( as well as t, s-channel bulk singularities coming from the 2nd terms of A 42 or (29) and A 72 or (28) appropriately as follows where in the first and second equations one can use momentum conservation to actually write k 3c (k 2c ) in terms of k 1c . Regarding the symmetries of S-matrix we just produce uchannel bulk singularities then by interchanging momenta and polarizations we can easily produce t, s-channel bulk singularities as well.
Let us first generate these new u-channel bulk singularities of the S-matrix. We need to consider the following rule The kinetic terms of scalar field and gauge field have been taken into account to reach at the following vertex and propagator In order to find out V a α (C p−1 , φ 1 , A) , one has to employ Taylor expansion as follows then take the integration by parts. The gauge field here is Abelian, taking (36) to momentum space and consider the following equation Both propagator and V b β (A, φ 2 , φ 3 ) are derived from kinetic terms thus there is no any correction to these terms. Therefore in order to explore all infinite u-channel bulk poles one should impose all infinite higher derivative correction to (36) as to get to all order extension of the above vertex operator as Now if we replace (39) and (35) inside (34) then we are able to precisely produce all infinite u-channel bulk singularities of this amplitude as follows The second kind of new bulk singularity structure is as follows.
We consider the following bulk singularity structures of asymmetric picture as well which can be eventually simplified, by various algebraic calculations as follows. Indeed If we add the 2nd term of A 1 and the entire Likewise if we add up the 3rd term of A 1 and the entire A 73 of asymmetric picture we get to obtain Finally we must add up the 4th term of A 1 and the whole A 63 of asymmetric S-matrix to be able to gain All the above terms must be added up as follows Having extracted the trace and replacing L 6 expansion in string amplitude, one finds out the second kind of new bulk singularity structure of BPS branes as follows This second new bulk singularity structure is related to new structures of an infinite (t + s + u) bulk singularities. Indeed L 6 does have infinite (t + s + u) channel singularities and in order to produce them in an EFT one has to consider the following rule The following coupling in an EFT is needed where in (47) the scalar field has been taken from pull-back. The trace in (44) shows the RR potential has to be p + 1 form field. (47) is a new coupling that plays the crucial rule for matching all the infinite new bulk singularity structures of string amplitude with effective field theory. If we take into account (47) and p a 0 C i a 1 ···ap = p i C a 0 ···ap as well as the kinetic term of the scalar field ((2πα ′ ) 2 /2)D a φ i D a φ i we obtain One needs to also impose the infinite higher derivative corrections to four real scalar field couplings that are derived in [32] as to actually obtain the following vertex Now if we replace (50),(48) inside (46) then we are exactly able to produce all order (t + s + u) channel bulk singularity structures of an RR and three scalar fields of string amplitude (45) in effective field theory side as well. Note that the important point was to derive the new coupling of a scalar field and a p + 1 potential RR field as explored in (47).
Note that there had been another kind of (t + s + u) channel poles which have already been discovered in [22]. For the completeness let us very briefly just produce all infinite u and t-channel poles accordingly through Field strength of RR. Eventually one can exchange the momenta and polarisations to get to all infinite s-channel poles as well.
Now if we impose the mixed couplings of RR's field strength, a gauge field and a scalar field through pull back as follows and if we take integration by parts on scalar field we come to know that D a 2 can just act on C-field, because of the antisymmetric property of (ε v ) it cannot act on F. Thus one can explore the following vertex operator k is the momentum of off-shell gauge field k = k 2 + k 3 and one imposes higher derivative corrections 5 to be able to get to all order extension of vertex as follows Implementing (52) and (35) inside (34) one can exactly generate these infinite u-channel poles. Eventually one reads off all infinite t-channel poles as follows Having considered Taylor expansion for the scalar field and taken integration by parts on the location of gauge field, we got the following action

..ap
We now apply the higher derivative corrections to (54) so that the following vertex can be achieved Taking the rule as A = V a α (C p−1 , φ 3 , A)G ab αβ (A)V b β (A, φ 1 , φ 2 ) and the vertices as and also considering momentum conservation we obtain Now replacing (56) and (55) inside the above rule and making use of the fact that (k 3 + p) 2 = (k 1 + k 2 ) 2 = t we are exactly able to produce all infinite t-channel poles of (53). Exchanging the momenta and polarizations we are also able to construct all infinite s-channel poles as well. Let us discuss contact interactions.

All order α ′ contact interactions of asymmetric S-Matrix
To begin with, we start generating the contact terms of symmetric picture by comparing them with contact terms of asymmetric picture. We address other contact interactions that are appeared just in asymmetric S-Matrix and essentially we write down new EFT couplings and explore their all order α ′ higher derivative corrections accordingly. First let us apply the the momentum conservation to A 81 . It is easy to show that the first term of where due to antisymmetric property of ǫ tensor the first term in (57) has no contribution to the contact interactions and the 3rd term does generate the 2nd contact term interaction of A ′ 1 , meanwhile the 2nd term in (57) will be an extra contact interaction in asymmetric picture for which we consider it in the next section.
Holding the same arguments to A 41 we get to obtain needless to say that the first term in (58) has no contribution to amplitude and the 3rd term above can produce precisely 3rd contact term interaction of A ′ 1 while the 2nd term in above equation should be regarded as an extra contact interaction in asymmetric picture that will be taken into account in the next section.
Finally let us produce the last contact interaction of A ′ 1 of symmetric picture as follows. In order to do so, one has to apply momentum conservation to the 2nd term of A 3 of asymmetric picture to reach to where the last term in (59) produces the the last contact interaction of A ′ 1 of symmetric picture (the 4th term of A ′ 1 ) meanwhile the other terms in (59) remain to be extra contact interactions in asymmetric amplitude.
Henceforth, up to now we have been able to precisely construct or generate the entire contact terms that have been appearing in symmetric picture. However, as we have revealed the other terms of asymmetric S-matrix elements lead us to conclude that those terms are extra contact interactions that can just be explored in asymmetric picture of the amplitude for which we are going to consider them in the next section.

5.3
All order contact terms of Asymmetric S-Matrix In previous section we compared the contact terms of the S-matrix element of an RR and three transverse scalars in both symmetric and asymmetric pictures. This leads us to explore various new contact interactions in an asymmetric picture of the amplitude. Therefore without further explanations we first write down all the other contact interactions that are just appeared in asymmetric picture as follows where A ′′ 11 ∼ iTr (P − C / (n−1) M p ) p.ξ 1 p.ξ 2 p.ξ 3 Q 1 , where these contact terms can be generated in a new standard form of effective field theory couplings that we address it now. Note that all the leading term of the above couplings appear to be at (α ′ ) 3 order. One can obtain their all order α ′ contact interactions as well.
The expansion for Q 1 is given in Appendix 2. If we consider A ′′ 11 , extract the trace and employ the Taylor expansion for all three scalar fields, one can show that the leading contact interaction can be precisely obtained by the following coupling Now all A ′′ 22 ,A ′′ 52 and A ′′ 3 have the same structure so we just produce A ′′ 22 . Therefore to produce this term in an EFT, one needs to extract the trace and show that the second scalar comes from pull-back, while the first and third scalar come from the Taylor expansion.
Hence, the leading contact interaction can be precisely derived by the following coupling All order α ′ corrections to (62) can be derived by considering all the infinite terms appearing in the expansion of Q 1 as shown in Appendix 2.
Note that all A ′′ 5 ,A ′′ 61 and A ′′ 71 have the same structure so we just produce A ′′ 5 . Therefore for this term one needs to first extract the trace and show that the first and the second scalar come from pull-back, while this time the third scalar comes from the Taylor expansion. Thus the leading contact interaction can be explored by the following coupling where its all order α ′ corrections have already been given in Appendix 2. Finally all A ′′ 421 ,A ′′ 621 and A ′′ 721 have the same structure and also tL 2 = sL 3 = uL 5 = L 22 , the expansion of L 22 is also given is Appendix 2. We just produce A ′′ 421 . For this term one needs to first extract the trace and show that the third scalar comes from Taylor expansion, while the first and second scalar neither come from Taylor nor pull-back. The leading contact interaction can be precisely derived by the following coupling where its all order α ′ corrections can also be derived as we did in (A.5).
Therefore in this section not only did we derive new couplings of RR in an EFT approach but also we have been able to find out their all order α ′ higher derivative interactions on both world volume and transverse directions.

Conclusion
In this paper we have analyzed in detail the four and five point functions of the string theory, including an RR in an asymmetric picture and either two or three real transverse scalar fields. We also compared the exact form of the S-matrix in asymmetric picture with its own result in symmetric picture. We have obtained two different kinds of new bulk singularity structures as well as various new couplings in asymmetric picture that are absent in its symmetric picture. We have also generalized their all order α ′ higher derivative interactions as well.
These two different kinds of bulk singularity structures of string amplitude are u, t, s (appeared in (32),(33)) as well as (t + s + u)-channel bulk singularity structures (45) that can just be explored in an asymmetric picture of the amplitude. These bulk singularity structures carry momentum of RR in transverse directions which are related to winding modes, however, winding modes are not inserted in vertex of RR in symmetric picture in ten dimension. Hence it makes sense to have these new bulk singularities as well as new couplings just in asymmetric picture of the amplitude. Indeed we were able to produce these two different bulk singularity structures of string amplitude in field theory by taking into account various new couplings in effective field theory side as well. We think that the importance of these results will be provided in future research topics, such as all order Myers effect and various other subjects in type II super string theory [23].
We have also observed that at the level of EFT the super gravity background fields in DBI action must be some functions of super Yang-Mills. We have also shown that some particular Taylor expansion for the background fields should be taken into account as was noticed in Dielectric effect [6].These results might be important both in constructing higher point functions of string theory amplitudes as well as discovering symmetries or mathematical results behind scattering amplitudes. We hope to address these issues in near future. n 11 = − η ac η ik (Γ bj C −1 ) αβ + η bc η ij (Γ ka C −1 ) αβ , n 12 = η ac η jk (Γ bi C −1 ) αβ − η ab η ij (Γ kc C −1 ) αβ , n 13 = − η bc η ik (Γ ja C −1 ) αβ , n 14 = − η ab η jk (Γ ic C −1 ) αβ , n 15 = − η bc η jk (Γ ai C −1 ) αβ + η ik η ab (Γ jc C −1 ) αβ The integral could be carried over just in terms of Gamma functions and no longer any hypergeometric function appears, where one needs to employ the following sort of integrations on upper half plane [31]: where a, b, c are arbitrary Mandelstam variable where for d = 0, 1 the result is given [31] and for d = 2 one needs to work out [20] with the following Mandelstam definitions One can obtain all order α ′ corrections to (61) by considering all the higher derivative terms appearing in the expansion of Q 1 as explained in the following.