On RR couplings, singularity structures and all order α′ contact interactions to BPS string amplitudes

We evaluate five point world-sheet string theory amplitudes of one transverse scalar field, two world volume gauge fields (and two transverse scalars, a gauge field) in the presence of a closed string Ramond-Ramond vertex operator in its symmetric picture. We carry out all the entire S-matrix elements of five point mixed RR-scalars/gauge fields , ,  and  in detail and start comparing all order α′ contact interactions and singularities in both transverse and world volume directions. We explore the presence of various new couplings in string theory effective actions and find out their all order α′ higher derivative corrections in both type IIA and IIB. Ultimately we make various remarks for the singularities and contact terms whose RR momenta are embedded in transverse directions. α′ corrections to some of Myers terms are also addressed.


Introduction
The fundamental objects in string theory or the so called D-branes have been playing a key ingredient in various research topics on theoretical high energy physics as well as in super string theory [1][2][3][4].
Indeed either a BPS or non-BPS D p -brane includes (p + 1)-dimensional world volume fields which must be thought of a hypersurface like in a ten dimensional flat space time. We need to take into account some special boundary conditions to them, namely either Neumann or Dirichlet, depending on whether we apply those boundary conditions through transverse or its world volume fields [5,6]. Note that recently some remarks for brane-anti brane have also been mentioned in [7].
To have more complete picture of the effective actions of string theory and what has been carried out up to now, we just point out to various papers that are important to the author. Myers in [8] did explore the form of a bosonic action which holds for multiple D p -brane configurations and the generalization of Myers action with its all order α ′ corrections (using the mixed open-closed scattering amplitudes) has been done in [9].

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Having performed [9], some new couplings were obtained. These new couplings are not inside Effective field theory (EFT) and their importance has played the fundamental role not only in performing the ADM reduction of IIB and exploring dS brane world-volume solutions [10] but also in deriving N 3 entropy of M 5 branes. These couplings could have some specific role in super gravity solutions as well where the particular emphasis is paid on the near-extremal black-branes to actually get to n 3 entropy growth analysis [11].
A remarkable paper [12] on supersymmetrized version of that action was given. A part of the supersymmetric action is known, in fact it involves symmetric traces of the non-abelian fields and what needs exploration is further terms which do not belong to the category that we are looking for in this paper. Whereas the effective action for a bosonic brane given by [13] and naturally its supersymmetric one was written down by [14][15][16][17][18]. One could read off a review of all the DBI, Wess-Zumino and Chern-Simons action just for BPS branes from [19]. On the other hand, to reveal more about three standard ways of effective field theory of the D-brane action (which contain Taylor expansion-Myers Terms and Pull back), and to learn more about all sorts of higher derivative corrections of non-BPS and BPS branes , we advise the section five of [20].
It is also important to have some tools to actually deal with the mixed open-closed higher point functions of string amplitudes, where one can refer to some of the pioneer works on either effective actions or scattering amplitudes that are involved with several D p -brane configurations as well as their string applications [21][22][23][24][25][26][27][28][29][30][31][32][33].
The paper is constructed as follows. In the next section we just introduce vertex operators with all details and notations and then we try to work out Type II super string computations with all order α ′ D-brane S-matrix of a Ramond-Ramond (RR) in symmetric picture, a scalar field in zero picture with two world volume gauge fields on different pictures where we try to address the entire S-matrix and explain the whole techniques that are involved in that particular amplitude. 1 Afterwards we start comparing all the contact interactions and singularity structures of V C V φ V A V A S-matrix in two different pictures in the presence of a symmetric RR vertex operator. Basically we compare both all order α ′ contact interactions and all the singularity structures of C −1 φ 0 A −1 A 0 with C −1 φ −1 A 0 A 0 , where the superscripts refer to the chosen picture of each string operator. Although we regenerate all t, s, u, (t + s + u)channel poles in effective field theory, we also find out some new contact interaction and singularities in the C −1 φ 0 A −1 A 0 S-matrix and for the first time , we explore their all order α ′ couplings in effective field theory as well. 2

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It is also worth reporting some sort of new singularities and new sort of Myers terms that appear in this particular picture of C −1 φ 0 A −1 A 0 S-matrix where those new terms are actually the terms that carry momentum of RR in transverse direction and do involve p.ξ terms inside the S-matrix elements.
Note that these p.ξ terms are derived by direct analysis of C −1 φ 0 A −1 A 0 , due to non zero correlation function of RR field by the first term of scalar field's vertex operator in zero picture, that is, all e ip.x(z) ∂ i x i (x 1 ) terms are indeed non-zero. Therefore since scalar field's polarization is in the bulk , one expects to be concerned about all p.ξ terms and p i , p j terms whose momenta of RR are carried in transverse directions. It is worth pointing out the following fact as follows. Recently, it is shown in [34] that, if one does not know all the Bianchi identities of RR in the bulk, then certainly there will be no chance to explore all the bulk singularities of non-BPS branes.
We perform full comparisons at each order of α ′ for all contact interactions as well, and that leads to finding out new couplings that can be derived by just S-matrix analysis not by any other tools to our knowledge.
The profound relation of open-closed string plays the crucial role in matching out all the singularities of string theory with EFT, as it has been shown that all order α ′ higher derivative corrections to SYM couplings produce all massless poles at (t + s + u)-channel poles through a RR coupling with various BPS open strings. It has also been emphasised that, this phenomenon could have played the major role for finding the universality conjecture on α ′ corrections of string theory [37].
We carry out the same analysis (this time for an RR, two scalars and a gauge field) in type IIA and IIB super string theory for both C −1 A 0 φ −1 φ 0 and C −1 A −1 φ 0 φ 0 Smatrices where we seem to find out the same t, s, u, (t + s + u)-singularity structures in the presence of an RR, even number of scalar fields. However, we claim that various new contact interactions appear in the S-matrix by considering both scalar fields in zero picture. Indeed we derive these new couplings, show that these couplings can just be discovered from C −1 A −1 φ 0 φ 0 S-matrix and explore their all order α ′ corrections in effective field theory side. Finally we conclude by mentioning various remarks about these S-matrices in the conclusion section.
2 Type II Super string computations with all order α ′ D-brane couplings In this section we would like to carry out the Conformal Field Theory (CFT) technique to be able to explore not only all the singularities but also all the infinite contact interactions of the mixture of a closed string RR (in its symmetric picture) and various BPS open string fields. Indeed our calculation makes sense at the level of a world-sheet five point mixed closed-open string amplitude which must be done on the upper half-complex plane. We find the entire S-matrix elements which hold on both world-volume and transverse component of D-branes.
we believe that not all the new couplings are redundant.

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In order to find out the effective action of string theory one needs to deal with or calculate the scattering amplitudes and naturally the first step to do so, is to fix a particular picture of the vertices. Namely, the sum of the superghost charges must have been (-2) for disk amplitudes.
In our notations we use µ, ν = 0, 1, . . . , 9 for the whole spacetime, while a, b, c = 0, 1, . . . , p for world volume space and i, j = p + 1, . . . , 9 for transverse directions. Here we would like to insist on the calculations in the presence of symmetric picture of RR but for the completeness we point out all the different vertex operators in various pictures as follows: To our knowledge the vertex of RR in asymmetric picture has been first shown by an interesting paper on open string theory [50] and then it was argued with some more details in [51,52] where the following kinematic relations are also considered We also apply Doubling trick to make use of holomorphic components of world sheet fields as well, that is, Although all the details of spinor part have been verified in [20], we just clarify the definitions of projector and RR's field strength as follows and H / (n) = a n n! H µ 1 ...µn γ µ 1 . . . γ µn , where for IIA and IIB we use n = 2, 4, a n = i and n = 1, 3, a n = 1 appropriately.
Here we just work out with the holomorphic parts of correlations but the interested reader can easily find out all the tricks in the appendix part of [20] as well.

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The complete form of the S-matrix element of a closed string RR (in its symmetric picture) n-form field strength and a transverse scalar field in zero picture and two world volume gauge fields C −1 φ 0 A −1 A 0 can be found by the following correlation functions We just look for a special ordering. Setting the Wick theorem, the amplitude is written down as follows (2.5) We actually use the standard propagators, as follows We also need to take into account the Wick's theorem to be able to investigate all the bosonic correlators. To see further details , the section 3 of [53] is strongly suggested.
Let us just address the most complicated fermionic correlation function of two spin operators/ two different currents and a fermion field, where all the possible contractions have to be considered.
Once again we use x 4 = z, x 5 =z. Note that unlike the open string correlator where integration is on the real line x 4 , x 5 are integrated on the upper half plane. It is only for the purposes of the Wick contractions that we can forget the complex conjugation of one JHEP12(2015)124 variable to another, in order to simplify things.
Replacing the above correlators and performing some simple algebraic computations, one can further simplify the amplitude and write it down in a closed form as follows Tr (λ 1 λ 2 λ 3 ), fixing as (x 1 = 0, x 2 = 1, x 3 = ∞), one needs to address the following integration on the upper half plane over the position of RR where a, b, c are the combinations of the following Mandelstam variables and the results of the integrations for d = 0, 1 and d = 2 were obtained accordingly in [54] and [20]. Therefore, the final form of the S-matrix in this particular picture to all orders in α ′ is obtained as follows where the functions L 1 , L 2 , are As we have expected by interchanging ξ 2a → k 2a and also ξ 3b → k 3b , the whole S-matrix vanishes, which means that the amplitude does satisfy all the associated Ward identities JHEP12(2015)124 and the amplitude is non zero for various p, n cases. Notice also we are dealing with all massless BPS strings so the expansion is low energy expansion. This S-matrix does have all t, s, u and particularly (t + s + u) channel poles and in particular it has some extra singularities that are precisely carrying momentum of RR in the bulk direction where we will show that these terms cannot be derived in the other picture ( C −1 φ −1 A 0 A 0 ) S-matrix and we argue about them in the next section.
More significantly, in C −1 φ 0 A −1 A 0 S-matrix we discover the new form of contact interactions to all orders in α ′ that cannot be found in the other picture. For the precise definitions of the expansions and more kinematical definitions and identities, one needs to look at [19,35].
Note that by sending t, s, u → 0, one finds the expansion of the functions L 1 , utL 2 as follows with the following coefficients

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Meanwhile the result of the S-matrix in different picture of scalar field, that is, [35] to be as follows Let us first compare the results of the same S-matrix in different pictures and then start producing all the singularity structures as well as new contact interactions.

Comparison on singularity structures of
First of all note that both A 5 's in two different pictures are exactly matched. The first term and apply momentum conservation along the world volume of brane we get Now if we use the identity that has been found in [7], that is, then we get to know the fact that the first term of above equation precisely produces the 2nd term One can see the derivation of (3.1) in various equations of [7]. For example, it is shown in equation (9) of [7] that, in order to get to the same result of three point function of one RR and a scalar field in both C −1 φ −1 and C −2 φ 0 S-matrix, the equation (9) of [7] or (3.1) must hold. Another example to prove that (3.1) holds is as follows. It is shown in section five of [7] that, to get to the same result of four point function of C −1 T 0 φ −1 and C −2 T 0 φ 0 S-matrix, the equation (3.1) must hold ( see the footnote of 19 in page 14 JHEP12(2015)124 of [7]). It has also been discovered that the amplitude of C −1 A −1 T 0 T 0 satisfies Ward identity associated to the gauge field if and only if the above identity (3.1) holds.
Likewise if we add the 1st term A 3 with the 2nd term A 4 of C −1 φ −1 A 0 A 0 and also apply momentum conservation we find the following elements Once more one needs to apply the equation p c ǫ a 0 ...a p−2 ac = 0 so that the first term of above precisely produces the 2nd term Simultaneously if we add the first term A 2 with the 2nd term A 3 of C −1 φ 0 A −1 A 0 with keeping in mind momentum conservation and p c ǫ a 0 ...a p−2 ac = 0, we then precisely produce the 2nd term Therefore the upshot is that we can precisely produce all the singularities of matrix (as we will show later on), however, we have also some extra contact interactions and other singularities of C −1 φ 0 A −1 A 0 S-matrix (in the zero picture of scalar field in the presence of a symmetric RR) that are absent in C −1 φ −1 A 0 A 0 S-matrix and we will argue about them in a moment.
In fact from the direct calculations we observe the facts that at pole levels the whole plane, as we further elaborate on this coupling in the other section. Let us first produce the different singularity structures.
We do have massless scalar poles in t,s and (t+s+u) channels as well as u channel gauge field poles. Here we just produce the s-channel scalar poles and finally by interchanging 2 ↔ 3 and exchanging the respected momenta and polarisations we can produce t-channel poles as well. If we replace the desired expansion of utL 2 , we then obtain all s-channel poles of string amplitude as follows (normalization constant is (2π In order to produce all these massless s-channel scalars, one has to consider a field theory sub amplitude as where by taking into account the kinetic term of scalar fields (2πα ′ ) 2 2 D a φ i D a φ i one obtains the following vertex as well as scalar propagator Now we need to consider the mixed Chern-Simons coupling and the so called Taylor expended of scalar field as follows to actually derive the following vertex operator of an RR, an on-shell gauge field and an off-shell scalar field as is derived from the kinetic term of the scalar field and in particular it has no correction, hence to be able to produce all s-channel poles we need to propose the higher derivative corrections to (3.5) as follows Having taken (3.7), we were able to derive all order vertex operator of (3.6) as Now if we replace (3.8) and (3.4) inside (3.3), then one is able to precisely regenerate all order s-channel singularities of string amplitude (3.2) in the effective field theory as well.
All the u-channel gauge field poles of the string amplitude are given as Note that these u-channel gauge field poles can be reconstructed in the effective field theory by the following field theory sub amplitude where the vertices are

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where all order corrections to V a α (C p−1 , φ 1 , A) have been derived from (3.7). By replacing these vertices into the field theory amplitude (3.10), one exactly produces all u-channel gauge field poles that appeared in (3.9).
For the completeness we just produce all the (s + t + u)-channel singularities of the S-matrix in the field theory as well. To do so, first we replace the part of the expansion of L 1 (which has poles) inside A 5 so that one gets all the poles in string amplitude as In order to produce these poles, one has to consider the following sub amplitude in field theory side where the scalar propagator can be found by taking the kinetic term of scalar fields is obtained by taking the following effective action through Taylor expansion of scalar field (3.14) To be able to produce all scalar massless poles of the string amplitude to all orders in α ′ , one needs to know all order vertex operator of two scalar two gauge field couplings V j β (φ, φ 1 , A 2 , A 3 ). This vertex operator can be found by employing all order α ′ SYM couplings [35] as follows where the following definitions for all higher derivative operators have been considered [19] D nm (EF GH) ≡ D b 1 · · · D bm D a 1 · · · D an EF D a 1 · · · D an GD b 1 · · · D bm H,

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Since the off-shell scalar field is abelin, one needs to consider just two permutations of Tr (λ 1 λ β λ 2 λ 3 ), Tr (λ β λ 1 λ 2 λ 3 ) to be able to derive all order vertex of V j β (φ, φ 1 , A 2 , A 3 ) from the above corrections (3.15) as below with the following definition for I 8 where k is off-shell scalar field's momentum, and some of the coefficients a n,m and b n,m (b n,m is symmetric [19]) are They are computed in [19]. Now if we use momentum conservation, we get k 3 ·k = k 2 .k 1 − (k 2 )/2 and k 2 · k = k 1 .k 3 − (k 2 )/2, whereas k 2 in (3.16) is cancelled with the k 2 in the denominator of the propagator. Since we just want to produce singularities, we are ignoring those contact terms and considering (3.16) and (3.14) inside (3.13), one explores the sub amplitude in field theory as follows (a n,m + b n,m )[s m t n + s n t m ] Now we show that all the poles in field theory of (3.18) can be matched with string amplitude poles that appeared in (3.12). After omitting the common factors of both string

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and field theory we compare string amplitude with sub amplitude in field theory for various cases of n, m. For n = m = 0, the amplitude (3.18) does carry the following factor −4(a 0,0 + b 0,0 ) = −4 −π 2 6 + −π 2 12 = π 2 where the corresponding term for the string amplitude carries (2π 2 c 0,0 ) which is exactly equivalent to the factor of π 2 in field theory sub amplitude. At α ′ order, (3.18) carries the following coefficient where the corresponding term for the string amplitude is now proportional to π 2 (c 1,0 + c 0,1 )(s+t) which is zero as appeared in the field theory sub amplitude. At (α ′ ) 2 order, (3.18) has the following numerical factor where the corresponding term for the string amplitude is now proportional to π 2 [c 1,1 (2st)+ (c 2,0 + c 0,2 )(s 2 + t 2 )], which is exactly equivalent to the factor of field theory sub amplitude. The comparisons at orders of (α ′ ) 3 , (α ′ ) 4 are also done in [35]. Hence, one can keep comparing to all orders and show that indeed all singularities of (t + s + u) channels of string amplitude can be precisely reconstructed by the above field theory sub amplitudes. Before further analysis let us compare all order contact interactions on two different pictures, start finding new coupling in the string theory effective action and also explore its all order α ′ corrections.

Comparison on contact interactions to all α ′ orders
If we look at the whole S-matrix elements in two different pictures and apply momentum conservation to the 1st term A 1 of C −1 φ 0 A −1 A 0 we obtain the following elements ..a p−4 bcad = 0, moreover without any further attempts, one reveals that the first term however, in below we show that there is some other contact interaction that can just be found by C −1 φ 0 A −1 A 0 S-matrix.

Other contact interaction to all orders in α ′
From the direct computations we observed the fact that at the level of contact interactions there is an extra contact term inside the S-matrix of C −1 φ 0 A −1 A 0 that has been overlooked from the direct calculations of C −1 φ −1 A 0 A 0 S-matrix. Indeed the 2nd contact JHEP12(2015)124 interaction A 1 of C −1 φ 0 A −1 A 0 is extra term that is not only needed into the entire amplitude but also cannot be derived from direct computations of C −1 φ −1 A 0 A 0 on upper half plane.
Hence the following coupling is extra contact interaction to all orders in α ′ , which must have been appeared in S-matrix because it stands correctly on the field theory side to all orders as well. Thus let us first write it down where we normalized the S-matrix by a coefficient of (2π) 1/2 µ p .
Having taken the expansion of L 1 inside (5.1), we first produce the leading term of string amplitude by the EFT coupling, in fact it can be produced by mixing Chern-Simons coupling and Taylor expansion as follows Therefore one can explore the next order to the above coupling, which is α ′4 . Indeed all order α ′ corrections to the above coupling can be discovered by applying the proper higher derivative corrections on the above coupling and the coefficients can just be fixed by taking the elements in the expansion of L 1 , so that all order corrections to above couplings are (s + t) n HφAA = (α ′ ) n HD a 1 · · · D an Φ∂ a 1 · · · ∂ an (AA), (s) n t m HφAA = (α ′ ) n+m HD a 1 · · · D an D a 1 · · · D am Φ∂ a 1 · · · ∂ am A∂ a 1 · · · ∂ an A.
Note that in above couplings, inside the covariant derivative terms the connections or commutator terms do not appear and to get them, one needs to compute higher point amplitudes like CφAAA . Therefore, we argue that for higher point function of string theory amplitudes, involving the mixed RR, a scalar and two gauge fields, there is a subtle issue as follows.
Indeed to be able to get to all the corrected and all order contact interactions as well as singularities of the mixed string theory amplitudes, one should consider the scalar field in zero picture as it has just been clarified in detail by the comparisons of It would be nice to generalize this idea to see what happens for the mixed amplitudes of closed string RR, two scalar fields and one gauge field which we carry it out in the next section.
It would be even nicer if we could do it on asymmetric picture of RR ,that is, find out C −2 φ 0 A 0 A 0 to actually generalize the rules and symmetries of string theory, where we leave it for the future works, although partial results for simpler systems, like for brane anti brane have already been announced in [7]. Let us now generate all the other singularity structures of C −1 φ 0 A −1 A 0 in the effective field theory.

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6 Other singularities of C −1 φ 0 A −1 A 0 Having produced some of the singularities and contact interactions, we are now ready to derive some other singularities of C −1 φ 0 A −1 A 0 S-matrix. These singularities do exist for this five point world-sheet S-matrix which includes a symmetric RR, a scalar field in the zero picture and two gauge fields. In fact by direct calculations we have shown that besides having some other contact interactions, even at pole levels the whole A 6 and A 7 of C −1 φ 0 A −1 A 0 are also extra singularities that cannot be derived from the direct computations of C −1 φ −1 A 0 A 0 S-matrix. Let us write them down as follows First we try to produce all these new s-channel poles of A 7 . By considering the desired expansion, one gets all new s-channel poles (note that normalization constant is (2π In order to produce all these new massless s-channel scalars, one has to apply the following field theory sub amplitude where to follow the related vertices, the kinetic term of scalar fields (2πα ′ ) 2 2 D a φ i D a φ i has to be taken into account, so that we obtain Now one needs to re-consider the mixed Chern-Simons coupling and Taylor expended of scalar field, where the extremely important point has to be pointed out as follows. This turn, we take integration by parts and employ the momentum of external gauge field directly to RR (p-1) form potential to be able to produce the necessary field strength of RR whereas the total derivative terms are indeed zero at infinity, hence we find out the following effective action Having set the above action, we obtain the following vertex in the effective field theory

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Note that V j β (φ, A 3 , φ 1 ) was derived from the kinetic term of the scalar field and it has no correction, that is why to produce all the singularities we need to propose all the higher derivative corrections to the new action of (6.5) as follows now we are allowed to actually reveal all order vertex operator of V i α (C p−1 , A 2 , φ) as Replacing (6.8) and (6.4) to (6.3), we are then able to precisely regenerate all order new s-channel singularities (6.2) in the field theory side too. Finally let us reconstruct all new u-channel singularities.
Having replaced the desired expansion, we get all new u-channel poles (normalisation constant is (2π) 1/2 m p ) of string amplitude as follows (6.9) All these u-channel gauge poles are also produced by considering the following sub amplitude in the field theory Here we consider the mixed Chern-Simons coupling and Taylor expended of scalar field, and not only this time we take integration by parts but also we do apply the momentum of external gauge field directly to RR potential to be able to produce the necessary field strength of RR, keeping in mind the above remarks, we obtain the following vertex where V b β (A, A 2 , A 3 ) has no correction, so the only way of obtaining all the poles is to actually impose all infinite higher derivative corrections to the mixed Chern-Simons Taylor expansion of scalar field, so that now we can derive the generalization of above vertex to all orders as Now by taking into account (6.12), the known V b β (A, A 2 , A 3 ) and gauge field propagator G ab αβ (A) = iδ αβ δ ab (2πα ′ ) 2 Tpu inside the sub amplitude (6.10) we are then able to precisely reconstruct all order new u-channel singularities in the effective field theory side as well.
In the next section we further generalize our knowledge by dealing with the mixed RR scalars/ gauge field S-matrices to see what happens to the S-matrix in the presence of two scalar fields (in different pictures), a gauge field and a symmetric RR field strength.

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In this section we would like to see what is going on for the mixed higher point function of a symmetric RR, two transverse scalar fields (in two different pictures) and a gauge field. We do the whole details to get to the entire S-matrix to all orders in α ′ so the C −1 A 0 φ −1 φ 0 S-matrix is shown by Further simplification can be done to get to the closed form of S-matrix as follows (6.14) where x ij = x i − x j , x 4 = z,x 5 =z, and also If we work with all possible contractions, then one finds out the compact form of the following fermionic correlation function as follows Substituting the closed form of the correlators into the amplitude we now claim the final answer for the S-matrix can be written down by Tr (λ 1 λ 2 λ 3 ),

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where It now becomes clear that the S-matrix of (6.18) is SL(2,R) invariant and after gauge fixing over the position of open strings one needs to come over the integrals on upper half complex plane on the location of RR. By evaluating those integrals one eventually writes down the complete form of the S-matrix to all orders as follows where the functions L 1 , L 2 are given in (2.13).
On the other hand if we actually consider both scalar fields in zero picture in the presence of a symmetric RR, then we get the whole S-matrix as

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Having done all integrals, one could find the final answer ( for further details, look at [9]) for the entire S-matrix of a symmetric RR with both transverse scalars in zero picture and a gauge field as follows where the functions L 1 , L 2 are already appeared in (2.13). It is worth highlighting the point that, this S-matrix also satisfies Ward identity, that is, by substituting ξ 1a → k 1a , the entire amplitude vanishes and the amplitude holds for various p, n cases. Let us do the comparisons C −1 A 0 φ −1 φ 0 with C −1 A −1 φ 0 φ 0 Smatrix at both level of singularity structures and contact interactions, find out various new couplings and in particular find out their corrections and eventually get to the conclusion. 7 Comparison on singularity structure of In this section we are going to compare all the singularities of Now if we add the second term A 6 of C −1 A 0 φ −1 φ 0 with the third term A 2 of C −1 A 0 φ −1 φ 0 and make use of momentum conservation along the world volume of branes, we obtain

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Now by applying the following equation p b ǫ a 0 ...a p−1 b = 0, we then realize the fact that the first term in above equation precisely produces the A 9 term of C −1 A −1 φ 0 φ 0 .
Meanwhile A 5 of C −1 A 0 φ −1 φ 0 can be written down as where the first term has no contribution to S-matrix. Because of the antisymmetric property of ǫ and the fact that it is symmetric with respect to k 3 so the result for the first term is zero. More evidently the third term in above equation has no contribution because p b ǫ a 0 ...a p−3 cab = 0 and the second term precisely produces The same so happens to the other terms, namely if we add the 2nd terms of A 2 and A 4 of C −1 A 0 φ −1 φ 0 and apply the momentum conservation, then we are able to precisely produce A 4 of C −1 A −1 φ 0 φ 0 which is related to all s-channel poles.
Indeed without any further details the first term so that all t-channel poles are then reproduced in both pictures. By considering the 2nd term Finally to be able to produce all the second kind of s-channel poles one has to subtract the first term of A 1 of C −1 A 0 φ −1 φ 0 from A 7 term of C −1 A −1 φ 0 φ 0 such that upon considering the following identity Henceforth, we could precisely produce all the singularities of this five point function in two different pictures. However, note that we have some extra contact interactions in C −1 A −1 φ 0 φ 0 amplitude while they are absent in C −1 A 0 φ −1 φ 0 S-matrix. These extra contact interactions are needed by symmetries of string theory amplitudes as we point out/hint them in a moment.
For the completeness we first would like to produce all the singularities. This amplitude has u-channel gauge poles that can be read off from the string amplitude as follows (7.1) where these u-channel poles should be produced by the following sub amplitude in the effective field theory Considering the kinetic terms of scalars iT p (2πα ′ ) 2 2 Tr (D a φ i D a φ i ) and gauge fields we obtain the following vertices 3)

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The kinetic terms have no corrections so we need to apply all higher derivative corrections to Chern-Simons couplings as follows Now if one considers S 6 , then one is able to obtain the following vertex operator to all orders in α ′ as follows Replacing above vertices (7.5) and (7.3) into (7.2), we are then able to exactly produce all u-channel gauge poles in the field theory side.
On the other hand, if we employ all order α ′ SYM couplings as appeared in (3.15), and also apply a following sub amplitude of field theory then we will be able to produce all (t + s + u) gauge field poles. Note that this task has been completely done in section four of [9] and in order to avoid rewriting the old contents of the paper, we refer the interested reader to that section four of [9]. Let us reconstruct all t-channel poles and finally by interchanging 1 ↔ 2 for all the momenta, the polarisations and t to s, we are able to produce all s-channel poles as well. All the t-channel poles of the string amplitude are given by 16ξ 2i ξ 3j k 2 .ξ 1 π 2 µ p t(p + 1)! 2p j ǫ a 0 ···ap H i a 0 ···ap − 2(p + 1)k 3a ǫ a 0 ···a p−1 a H ij a 0 ···a p−1 ∞ n=−1 b n (α ′ k 3 .k) n (7.6) These t-channel poles can be regenerated in the field theory side, and to do so one needs to take into account the following sub amplitude and vertices in the field theory as Consider the Taylor expansion of the two scalar fields as

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and then work out with pull-back and both mixing term involving Taylor and pull-back as follows where one needs to also add the following Myers terms with S 8 and take all the integrations by parts to actually get to the following action Eventually in order to produce the first t-channel pole, one must consider the summation of the Taylor expansion and S 10 as follows From (7.10) we now look for the vertex of V i α (C p+1 , φ 3 , φ) as follows However, to produce all the other t-channel poles, one needs to apply all order higher derivative corrections to (7.10) as below to indeed obtain the following vertex to all orders in α ′ as follows Now if we replace (7.13) inside (7.7) then we are exactly able to regenerate all order tchannel singularities in the field theory side as well.
Note that all of the new couplings that we have discovered, can just be derived with scattering computations not by any duality transformation. Because the coefficients of these couplings can just be fixed without any ambiguity by S-matrix analysis. We now turn to contact interaction terms.

JHEP12(2015)124 8 Comparison on contact interactions
If we look at the precise computations of the S-matrices in two different pictures, we then realize the fact that the first term A 4 of C −1 A 0 φ −1 φ 0 is exactly the term that has been shown up in As we can readily observe, we have just left with two contact terms in A 3 of Now if we apply the momentum conservation to the 2nd term A 3 of C −1 A 0 φ −1 φ 0 and apply the Bianchi equation that we have already got, that is, p b ǫ a 0 ...a p−3 cba = 0 then we are able to precisely produce the first term Eventually we apply momentum conservation to the only remaining term of C −1 A 0 φ −1 φ 0 which is its first A 3 term and do subtract it from the second and third terms A 1 of C −1 A −1 φ 0 φ 0 such that upon holding the following equation, we are able to generate the second and third term Once more p b ǫ a 0 ...a p−2 ab = 0, whereas up to a sign the third term A 1 of C −1 A −1 φ 0 φ 0 is also produced.
However, note to the important point that there is no chance to actually produce even the leading order α ′ of the fourth contact interaction A 1 of C −1 A −1 φ 0 φ 0 . The reason is that, there is no left over term inside C −1 A 0 φ −1 φ 0 S-matrix to be compared with that fourth term A 1 of C −1 A −1 φ 0 φ 0 S-matrix. Therefore, let us further elaborate on the needed contact interactions of this string amplitude. 9 The needed contact interaction for C −1 φ −1 φ 0 A 0 As we have seen above, we were able to produce all the first three contact terms A 1 of C −1 A −1 φ 0 φ 0 to all orders, however, we have evidently observed that indeed there is no chance to produce the fourth term contact interaction In fact we claim that this extra contact interaction must be appeared in the entire S-matrix as it plays the crucial role in all order α ′ contact interaction terms in both type IIA and IIB super string theory. Let us first write it down and then we try to construct its all order α ′ higher derivative couplings.
Hence we figure out the following term inside is indeed needed. We normalized the S-matrix by a coefficient of (2π) 1/2 µ p and considered the expansion of L 1 (with the aforementioned coefficients) given in (2.14). Thus the first leading term of L 1 can be produced by Chern-Simons coupling and Taylor expanded of both scalar fields through closed string RR as follows

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Therefore one explores the next order term which is α ′4 and indeed all order α ′ corrections to the above coupling with exact coefficients can be discovered by applying the proper higher derivative corrections. For example the (st) m HAφφ and (s + t) n HAφφ contact terms of the S-matrix (inside the expansion of L 1 ) can be shown to be matched to all orders by the following couplings (s + t) n HAΦΦ = (α ′ ) n H∂ a 1 · · · ∂ an AD a 1 · · · D an (ΦΦ), (st) m HAΦΦ = (α ′ ) 2m H∂ a 1 · · · ∂ a 2m AD a 1 · · · D am ΦD a m+1 · · · D a 2m Φ Note that the first correction to the above coupling (9.2) and the other new coupling in (5.2) is of α ′4 order.
It is also worth keeping in mind the fact that by expanding the string amplitude of C −1 A −1 φ 0 φ 0 , we could also explore new couplings at leading order as follows.
Let us write down the explicit form of the string amplitude, indeed if we extract the related trace, consider the expansion of stL 2 inside A 5 of C −1 A −1 φ 0 φ 0 S-matrix, we then obtain the following elements of string amplitude −2ξ 3 .ξ 2 k 2b k 3c ξ 1a π 2 µ p 16 (p − 2)! ǫ a 0 ···a p−3 cba H a 0 ···a p−3 e p,n,m u p (st) n (s + t) m (9.3) where we have already produced all the u-channel poles, now to obtain the new couplings, we need to focus on the second term in (9.3) as − 2ξ 3 .ξ 2 k 2b k 3c ξ 1a π 2 µ p 16 (p − 2)! ǫ a 0 ···a p−3 cba H a 0 ···a p−3 ∞ p,n,m=0 e p,n,m u p (st) n (s + t) m (9.4) where (9.4) satisfies the Ward identity associated to the gauge field, which means that by replacing ξ 1a to k 1a , apply the momentum conservation and taking the following identity for RR p a ǫ a 0 ···a p−3 cba = 0 the amplitude vanishes. Thus we understand that (9.4) has to be reconstructed by new coupling and the structure of this new coupling is shown by p+1 d p+1 σ Tr (C p−3 ∧ F ∧ Dφ i ∧ Dφ i ) (9.5) Note that (9.5) is considered by the fact that it has to cover up the whole world volume space and more crucially it has to be antisymmetric with respect to interchanging the momenta of both scalar fields. We now apply e 1,0,0 = π 2 6 and e 0,0,1 = π 2 3 to (9.4) to be able to start constructing new couplings at order of α ′3 .

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interesting proposal for picture changing operator has been appeared in [55], however, we find it complicated to be applied to the real string amplitudes, nevertheless, it would be great to find the deep connections behind those topics as well.

Conclusion
In this paper, we have evaluated the five point world-sheet string theory amplitudes of the mixed RR, scalar and gauge fields, namely we have carried out with entire details the whole We have regenerated all t, s, u, (t + s + u)-channel poles in effective field theory. We also found out new contact interactions as well as some new singularities that appear in C −1 φ 0 A −1 A 0 S-matrix where those new terms were actually the terms that carry momentum of RR in transverse direction and involved p.ξ terms inside the S-matrix elements. These p.ξ terms are needed in the entire form of S-matrix , due to non zero correlation function of RR field by the first term of scalar field vertex operator in zero picture. Indeed all e ip.x(z) ∂ i x i (x 1 ) terms are non-zero so we have reconstructed the S-matrices such that by considering all the scalar fields in zero pictures in the presence of RR, we were able to produce all p.ξ terms as well as p i , p j terms ( inside the S-matrices) whose momenta of RR are carried in transverse directions.
By comparing C −1 φ 0 A −1 A 0 with C −1 φ −1 A 0 A 0 S-matrix we found a coupling inside the C −1 φ 0 A −1 A 0 S-matrix as follows where this coupling can be explained by the effective field theory ways as, the mixed Chern-Simons and Taylor expansion of scalar field was needed. We then generalized its all order higher derivative corrections. We produced all the new singularities of this S-matrix in section six of this paper as well.
We also compared C −1 A 0 φ −1 φ 0 with C −1 A −1 φ 0 φ 0 S-matrix for all order α ′ contact interactions as well as singularities in both transverse and world volume directions of the S-matrices and that leads to finding out various new couplings in string theory effective actions. First we found the following coupling and claimed that this coupling can be verified just by C −1 A −1 φ 0 φ 0 S-matrix where from field theory we employed the Taylor expansion of scalar fields and then we generalized its all order corrections. Basically, we claim that various new contact interactions appear in the S-matrix by considering both scalar fields in zero picture. Indeed we derived the following new couplings a 0 ···a p−4 Tr F a p−3 a p−2 (D a D a ) D a p−1 φ i D ap φ i

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as well as These couplings are needed in order to consider the symmetries of the S-matrix with respect to interchanging of the scalar fields and their all order α ′ corrections generalized in (9.8).
Note that these two above couplings can not be derived by the standard effective field theory ways of Taylor, Myers terms nor by pull-back formalism. Indeed not only the structures of the above new couplings but also their coefficients can just be explored by C −1 A −1 φ 0 φ 0 S-matrix analysis and not by any other tools.
Note that there is no Ward identity for the amplitudes of scalar fields in the presence of RR, thus we argue that for two scalars and a gauge field in the presence of RR, there was a subtle issue. Indeed to be able to get to new couplings as well as the corrected all order contact interactions of higher point functions of string theory amplitudes, one needs to consider both scalar fields in zero picture as we have clarified in detail in this paper. Eventually we have made use of Myers terms and the terms whose RR momenta are embedded in transverse directions, to be able to derive all the singularity structures of an RR, two scalars and a gauge field amplitude.