On BPS World Volume, RR Couplings and their $\alpha'$ Corrections in type IIB

We compute the asymmetric and symmetric correlation functions of a four point amplitude of a gauge field, a scalar field and a closed string Ramond-Ramond (RR) for different non-vanishing BPS branes. All world volume, Taylor and pull-back couplings and their all order $\alpha'$ corrections have also been explored. Due to various symmetry structures, different restricted BPS Bianchi identities have also been constructed. The prescription of exploring all the corrections of two closed string RR couplings in type IIB is given. We obtain the closed form of the entire S-matrix elements of two closed string RR and a gauge field on the world volume of BPS branes in type IIB. All the correlation functions of $$ are also revealed accordingly. The algebraic forms for the most general case of the integrations $\int d^2z |z-i|^{a} |z+i|^{b} (z - \bar{z})^{c} (z + \bar{z})^{d}$ on upper half plane are derived in terms of Pochhammer and some analytic functions. Lastly, we generate various singularity structures in both effective field theory and IIB string theory, producing different contact interactions as well as their $\alpha'$ higher derivative corrections.


Introduction
The fundamental objects are called D p -branes that are long known to be existed. The late Joe Polchinski has given a life to the D-branes as dynamical objects and they are assumed to be sources for Ramond-Ramond (RR) closed string for all sorts of the stabilised BPS branes [1,2].
These RR couplings have various contributions in many different areas of theoretical high energy physics, ranging from pure String Theory to Mathematics, K-Theory as well as phenomenology. For example, one may point out to the dissolving branes [3] , K-theory [4,5], the well known Dielectric or Myers effect [6] where some of their α ′ corrections are also derived in detail in [7].
To proceed with their dynamics, one needs to know all sorts of effective actions where various potentially interesting references are given in [8]. We would like to take advantage of Conformal Field Theory (CFT) and try to release more information about the structures of the BPS effective actions. Indeed one of our aims is to work out with CFT to get more data and increase our knowledge of deriving various string theory couplings and likewise various techniques to Effective Field Theory (EFT) couplings along the way can also be explored.
One can just mention different applications to some of the known couplings, like the famous N 3 phenomenon for M5 branes, dS solutions as well as entropy growth [9]. It is also known that RR plays the key role for all kinds of BPS and unstable branes [10] where one can study some of its analysis as well as its dynamics in [11,12]. All the standard approaches to get to EFT couplings were explained in detail in [13], where S-matrix computations play the most fundamental role in getting the exact form of string couplings and their precise coefficients are even computed in the presence of higher derivative α ′ corrections. We would like to illustrate just some of the BPS string calculations as given in [14]. For the sake of comprehensiveness and for a review of open strings as well as their properties, we just highlight the original papers that are known and appeared in [15].
In the first part of the paper, we calculate both asymmetric and symmetric S-matrices of a four point amplitude of a gauge, a scalar field and a closed string RR for different non-vanishing traces of all different BPS branes. All world volume, Taylor and pull-back couplings and their α ′ corrections have also been figured out. Due to various symmetry structures, various restricted BPS Bianchi identities can also be derived. We then try to demonstrate a prescription of exploring all the corrections of two closed string RR couplings in type IIB as well. To proceed further, we go head to obtain the closed form of the entire S-matrix elements of two closed string RR's and a gauge field on the world volume of BPS branes just in type IIB string theory.
We first derive all the correlation functions of < V A 0 (x 1 ) V C −1 (z 1 ,z 1 ) V C −1 (z 2 ,z 2 ) > and then reveal the algebraic forms of the integrals for the most general integrations on the upper half plane that are of the sort of d 2 z|z − i| a |z + i| b (z −z) c (z +z) d and the outcome is written down in terms of Pochhammer and some analytic functions. Finally, we try to reconstruct various singularity structures in both Effective Field Theory (EFT) and IIB string theory. We also work out with different contact term analysis and reproduce different contact interactions of the same S-matrix as well as their α ′ higher derivative corrections and lastly point out to different remarks as well.
The lower order supersymmetric generalisation of Wess-Zumino (WZ) action has been found in [16] in both IIB and IIA. It was also highlighted that all structures as well as the coefficients of the α ′ corrections in type IIB are different from their type IIA couplings.
It is worth taking into account the reference [17] that deals with potentially different areas where important remarks on perturbative string amplitude calculations have been given. Indeed to do so, a systematic setup was revealed. It is highlighted that to ignore some spurious singularities one needs to employ the vertical integration formalism with great care. Although this conjecture has potential overlaps with string calculations of IIB analysis, however, the role of the world volume couplings is not clarified in detail, nor the bulk singularity structures are given. In the upcoming paper [18] we address in detail the world volume and bulk singularity analysis of IIA. In this paper, we just would like to show the method of deriving algebraic forms of all the integrals in terms of Pochhammer and consequently argue about the role of the world volume couplings just in IIB as well as their explicit corrections.
2 All order Corrections to < V C −2 V φ 0V A 0 > In this section using direct CFT methods [19] we would like to explore the entire S-matrix elements of an asymmetric RR, a scalar field and a gauge field where its all order α ′ higher derivative corrections can also be examined accordingly. It will be given by exploring all its correlation function as The related vertex operators are read off from [20] and [21] as follows Note that, the total background charge of the world-sheet with topology of a disk must be -2, hence one needs to consider the symmetric and asymmetric picture of RR as illustrated in (2). For our notation we use µ, ν = 0, 1, .., 9 where world volume indices run by a, b, c = 0, 1, ..., p and finally transverse indices are represented by i, j = p + 1, ..., 9.
Other notations for spinors and projector are given by the following formulae where in type IIA (type IIB) field strength of RR takes the value of n = 2, 4,a n = i (n = 1, 3, 5,a n = 1). In order to use the holomorphic parts of the world-sheet fields, we apply the doubling trick which means that, some change of variables is taken into account and the following matrices are needed Now one is able to pick up just the following propagators for the whole world sheet fields of the kind of X µ , ψ µ , φ, as Replacing the related vertex operators inside (1), exploring correlation functions, fixing the SL(2,R) symmetry by gauge fixing as (x 1 , x 2 , z,z) = (x, −x, i, −i) and introducing t = − α ′ 2 (k 1 + k 2 ) 2 one finds out the non zero part of the asymmetric amplitude as follows where the traces can be explored, note that p > 3, H n = * H 10−n , n ≥ 5. and the compact form of the asymmetric amplitude is derived to be We are dealing with massless strings, hence the expansion is low energy expansion 2 , that is, t = −p a p a → 0 and the expansion is found as follows In order to produce the first term (5) in an EFT, one needs to consider the mixed Chern-Simons effective action and Taylor expansion of the scalar field as below Now if we consider the covariant derivative of the scalar field from pull-back of brane and employ the following new effective action then one can show that S 2 precisely produces the second term of (5).
2 Note that we removed the over all factor (2i) −2t−1 However, as can be observed the expansion of the amplitude consists of many contact interaction terms and one reconstructs all the contact terms of the S-matrix in an EFT by imposing an infinite higher derivative corrections to the above S 1 and S 2 effective actions. Therefore all contact terms for the first term of the asymmetric amplitude can be reconstructed by applying all order corrections to S 1 as follows: Likewise all order extensions of S 2 are read off On the other hand, the symmetric result of the amplitude One can also read off Using momentum conservation along the world volume of brane (k 1 + k 2 + p) a = 0, due to symmetry structures and to get consistent result for both above symmetric amplitudes, one gets to derive the following restricted Bianchi identity for RR's field strength as below In the next section we would like to deal with more complicated analysis.
The complete form of S-matrix element of two closed string RR field V has been carried out in [16] which has the following form It is shown that by choosing the gauge fixing as (x 1 , x 2 , x 4 , x 5 ) = (iy, −iy, i, −i) for the moduli space , one maps the moduli space to unit disk and the final amplitude of two closed string RR's in IIB was read off as follows It was also shown that the only massless pole can be reconstructed by using the following sub amplitude in an EFT where k 2 = (p 1 + D.p 1 ) 2 = −s is taken inside the propagator while t = −2p 1 .p 2 . Due to having the entire and closed form of S-matrix of two RR's in (11) one can apply properly higher derivative corrections on C 1 and C 2 so that (ts) n can be produced by the following all order α ′ corrections to two closed string RR of IIB ∞ n,m=0 (t + s) m can also be produced by the following all order α ′ corrections of IIB Now in this section we would like to carry out direct CFT methods to derive the entire S-matrix elements of two closed string RR's and a gauge field on the world volume of BPS branes in IIB. The aim is to explore singularity structures as well as α ′ corrections and also to see if the above prescription holds or not. Hence, the five point function V (z 2 ,z 2 ) in type IIB string theory (z 1 = x 2 + ix 3 , z 2 = x 4 + ix 5 ; x ij = x i − x j ): is given by the following correlation functions where I, and J a take the form with the definition of Mandelstam variables as Substituting the definition of Mandelstam variables into I, we obtain it as The correlation function of four spin operators in IIB is read by Correlation function of four spin operators and one current has been obtained in [22] to be S ca αβγδ (x 1 , x 2 , x 3 , x 4 , x 5 ) := : S α (x 2 ) :: S β (x 3 ) :: S γ (x 4 ) :: S δ (x 5 ) :: ψ c ψ a (x 1 ) : Gauge fixing of SL(2, R) invariance for V (0) The Jacobian for this transformation will be Jac = −2ix 2 1 . After gauge fixing, the expressions for the amplitude in (15) get simplified to The various gauge fixed quantities appearing in the above amplitudes are summarised 3 in the following formulae (z = x + iy; y ≥ 0) Now if we apply on-shell condition for the gauge field k 1 .ξ 1 = 0, then one gets to derive Let us define the following integral where a, b.c are written down in terms of Mandelstam variables and d = 0, 1 for this amplitude, hence the final result for the the amplitude V (0) in IIB can be expressed as where A 1 , A 2 , A 3 , A 4 , A 5 , A 6 are given by One can show that A 5 and A 6 have no contribution to our S-matrix , due to the fact that their integrations are zero on upper half plane.

World-Volume Singularity Structures of IIB
The low energy expansions of all the functions can be found by using the package of HypExp [23] and we just point out to some of the expansions. For instance for n = 0 ( see Appendix) and at first order of the expansion one gets the following values Let us deal with the singularities of the S-matrix. The amplitude makes sense for C p−3 , C p−1 , C p+1 cases. One can summarise the expansions of the functions in terms of Aǫseries[n, ǫorder] accordingly. In particular, for the other cases we find the following expansions with the explicit coefficients as written in the tables 1-4. Table 1: A1ǫseries [1,1] ǫ order coefficient -1 0 0 0 Table 2: A2ǫseries [1,1] One can show that the expansions of A3ǫseries [1,2], A3ǫseries [1,3] and A3ǫseries [1,4] have no poles and made out of just contact interactions of the sort of the following form Given the above structure, for p = n case, now one can apply α ′ higher derivative corrections to explore corrections in type IIB as follows ∞ m,p,q=0 where (−2s + v + w) is an over all factor and p 2 .D.p 2 can be constructed out by applying the sum of momenta as 1 2 (D a D a )C 2 ∧ F . Note that for all the other functions starting from n = 3, ǫ = 0, 1, 2 the only non zero values for the expansion would be at first order expansion and have the following non zero values π 2 12 (2s − v − w), iπ 2 24 v and iπ 2 12 v for A1 [3,1], A2[3,1] and A4 [3,1] accordingly. Now given the low energy expansion and in order to produce the world volume singularity structures, we try to extract the traces and carry out algebraic simplifications.  Table 4: A4ǫseries [1,1] Indeed if λ in the S-matrix takes world volume index λ = d then p 1 .D.p 1 channel pole in IIB can produced by the following EFT sub amplitude (µ 1p , µ 2 p−2 ) are RR charges and it is renormalised by 1 2 6 . Hence if λ picks up world volume index then all the traces have non-zero contributions for C 1 p−1 as well as for C 2 p−3 cases.
Note that this obviously confirms that we do have a gauge field singularity structure and also all various α ′ higher derivative corrections to two closed string RR in type IIB that can be constructed out later on.
The gauge singularity structure for this particular case is regenerated by the following EFT sub amplitude where the Chern-Simons coupling i(2πα ′ )µ 1p Σ p+1 C 1p−1 ∧ F is needed, also note that it is shown that, this coupling does not receive any corrections either.
One can reveal all the simple propagators by employing the kinetic terms appeared in DBI action as (2πα ′ ) 2 F ab F ab and (2πα ′ ) 2 2 Tr (D a φ i D a φ i ) where the kinetic terms of gauge fields and scalars will receive no correction either, because they are fixed in the low energy DBI action as well. One readily gets to derive the EFT vertex operators for the above amplitude as follows The singularity structure for this case is also produced by the following EFT counterpart where V i α (C 1 p+1 , φ) was obtained from i(2πα ′ )µ 1p Σ p+1 ∂ i C 1 p+1 φ i which is indeed the Taylor expansion in EFT. We clarify the EFT vertex operators for the above amplitude as follows where the mixed WZ and CS interaction i(2πα ′ ) 2 µ 2p Σ p+1 ∂ i C 2 p−1 ∧ F φ i for the second coupling is taken. Having set that 5 , one regenerates p 1 .p 2 singularity structure in EFT which is the same pole as appeared in (40) then one would be able to regenerate the singularity in an EFT side as well.

Conclusion
In this paper, first we have computed both asymmetric and symmetric S-matrices of a four point amplitude of a gauge, a scalar field and a closed string RR for different non-vanishing traces of all different BPS branes. We then figured out all world volume, Taylor and pullback couplings as well as their α ′ corrections. Thanks to symmetry structures, various restricted BPS Bianchi identities are also revealed. A prescription for the corrections of two closed string RR couplings in type IIB was found out. We have also gained the closed form of all the correlators of two closed string RR and a guage field in type IIB string theory.
The algebraic forms of the integrals for the most general integrations on the upper half plane that are of the sort of d 2 z|z − i| a |z + i| b (z −z) c (z +z) d are explored where the outcome is written down in terms of Pochhammer and some analytic functions. Lastly, various singularity structures in both EFT and IIB string theory are reconstructed. We have also worked out with different contact term analysis and reproduced different contact interactions of the S-matrices as well as their α ′ higher derivative corrections and eventually some concrete points are clarified in detail. Various world volume couplings just in IIB as well as their explicit corrections are discovered as well.
Now let us just address in detail the technical issues related to solving integrals in the Appendix.

Appendix
Solving the integrals of two RR's and an NS field We did gauge fixing as (x 1 → ∞, z 1 = i,z 1 = −i, z 2 = z,z 2 =z) and eventually one needs to take integrations on the location of the second closed string on upper half plane as follows where d = 0, 1 and a, b, c are written down in terms of three independent Mandelstam variables. We first use the following transformations where z = x + iy and the integration on x is readily done as Using simple algebraic analysis and change of variables one writes down the integration on y as follows Now one can make use of Pochhammer definition as follows If we consider (46) then one gets to derive the whole y-integration. Now one can collect all the results of x and y integration inside I to get to If we use Γ(z) = (z − 1)! then one reads off the final answer to be where the integration 1 0 dss (n+a)/2 (1−s) (n+b)/2 (1−2s) c−n can also be computed in terms of Hypergeometric function as follows