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Physics Letters B

www.elsevier.com/locate/physletb

On RR couplings and bulk singularity structures of non-BPS branes

Ehsan Hatefia,b'*

a Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom b Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria

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A R T I C L E I N F 0

Article history:

Received 27 January 2016

Received in revised form 4 July 2016

Accepted 14 July 2016

Available online 19 July 2016

Editor: N. Lambert

A B S T R A C T

We compute the five point world sheet scattering amplitude of a symmetric closed string Ramond-Ramond, a transverse scalar field, a world volume gauge field and a real tachyon in both world volume and transverse directions of brane in type IIA and IIB superstring theory. We provide the complete analysis of < CA0 T> S-matrix and show that both u' = u + 1 and t channel bulk singularity structures can also be examined by this S-matrix. Various remarks about new restricted Bianchi identities on world volume for the other pictures have also been made.

© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

No matter we talk about stable BPS or unstable (non-BPS) branes, Dp -branes are supposed to be thought of sources for closed string Ramond-Ramond (RR) field [1]. It has been proven that making use of RR couplings, one can investigate or try to address various remarkable issues about string theory, whereas we highlight some of the most crucial ones as follows. The so called brane embeddings [2], through D-brane language the K-theory [3], also the so called Dielectric effect or Myers effect [4] and in particular the way of looking for all order a' higher derivative corrections to BPS or non-BPS branes [5,6] are explored. Note that for the definitions of non-BPS branes, Dp-branes with p odd (even) in IIA (IIB) are taken in which p stands for the spatial dimension of branes. Based on various symmetries, a universal conjecture for all order a' higher derivative corrections to both BPS and non-BPS branes was proposed in [7] and naturally it has been applied at practical levels to various higher point fermionic S-matrices [8] as well.

It was argued in [9] in detail that, to deal with effective theory of unstable branes after integrating out all the massive modes, one is just left with massless and tachyon states, and also to work out the dynamics of branes not only DBI action but also Wess-Zumino terms are indeed needed. To obtain effective actions, either Boundary String Field theory (BSFT) [10] or S-matrix formalism should be employed where the latter has a very strong potential to be taken so that upon applying that, all the coefficients of higher derivative corrections for all orders in a' can be found.

The Wess-Zumino effective action has been given in [11] as

Swz = fip j C A Str ei2naF, (1)

with p being normalisation constant and with the so called the curvature of super connection as follows

iF - ß12T2 ß1DT ß'DT iF - ß12T2

— I ol ßT ,T ol2 T2 )• (2)

* Correspondence to: Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

E-mail addresses: e.hatefi@qmul.ac.uk, ehsan.hatefi@tuwien.ac.at, ehsan.hatefi@cern.ch. http://dx.doi.org/10.1016Zj.physletb.2016.07.038

0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

Now if we expand the exponential, one produces various couplings such as

iii^i (2na')2 \

Swz = 2ji'¡'p(2na )Tr i Cp A DT + (2na')Cp-2 A DT A F + -—Cp-4 A F A F A DT J (3)

Based on the internal Chan-Paton matrix some partial selection rules for superstring amplitudes have been released in [12], and to make sense of S-matrix computations, one needs to keep track of internal CP matrix of tachyons around unstable point of tachyon DBI action. The final form of tachyon DBI up to some orders with its all ingredients is demonstrated in [11] to be as follows

Sdbi dp+1a STr^V(TiTi1 + 1 [Ti, Tj][Tj, Ti])J- det(Vab + 2na'Fab + 2na'DaTi(Q-1 )ijDbTi^ , (4)

V(TiTi) = e-ntT/2, Qij = ISij - i[Ti, Tj], (5)

where i, j = 1, 2, i.e., T1 = Ta1, T2 = Ta2. The entire information about this effective action is given in [12]. This action (4) is also consistent with the entire S-matrix calculations of < VCVTV$V$ > of [11].

Around the stable point of tachyon potential this action gets reduced to tachyon DBI action [13] with T4V(T2) potential and by taking the limit of the tachyon to infinity, the term T4 V (TT) is sent to zero and this can be well described from condensation of an unstable brane.

By making use of the S-matrix of < VCV$VTVT > in [14] and < VCVAVTVT > in [15], we have also explored not only the entire form of D-brane -anti D-brane effective actions but also its all order a' corrections of two scalar (two gauge field)-two tachyon couplings.

Various applications in the literature in favour of those higher derivative string couplings have been pointed out. For instance, employing either some new BPS (non-BPS) couplings, their corrections or Myers terms to M-theory [16] one can actually interpret and get to the phenomenon of N3 entropy growth of M5 branes or discuss various combinations of M2, M5 branes [17]. Evaluating some of (non)-BPS couplings, one derives (AdS)-dS brane world solutions [18] and could further elaborate on the point that despite the fact that we are dealing with non-supersymmetric case, as long as the EFT holds, all the large volume scenario minima will become stable [19]. Applying CFT techniques [20] the tachyonic DBI supersymmetrised action was suggested [21] to be as follows

L = -TpV(T- det(nab + 2na'Fab - 2na'VybdaV + n2a'2 Vy^daVVy^dbV + 2na'daTdbT) (6)

One of the reasons we work with non-BPS branes is due to its direct relationship with realising the properties IIA (IIB) string theory in almost time-dependent backgrounds [22].

The source of instability in flat empty space is tight to the presence of tachyons and by using their effective action one hopes to get diverse results, such as the evaluation of these non-BPS branes and indeed tachyonic action can well describe decay properties of unstable Dp -branes.

The cosmological applications through these effective actions can also be addressed, for example if one considers, the action of D-brane anti D-brane then one is able to explain inflation through string theory [23] whereas several other applications to non-BPS branes have also been released [24]. Tachyons and their corrections can also be employed for models such as holographic like QCD [25] and finally brane anti brane system as a background was used in Sakai-Sugimoto models [26].

The outline of the paper is as follows.

In the next section, we first introduce all the vertex operators with their CP matrix, apply CFT techniques to actually obtain the entire form of a symmetric RR, a scalar, a gauge and a tachyon of < VC-1 V$o VAo VT-1 > S-matrix in both transverse and world volume directions of non-BPS branes of IIA(IIB) superstring theory. We then make use of the expansion for non-BPS branes and reveal that apart from an infinite u' = u + 4 and (t) channel tachyon (scalar) singularities, this S-matrix clearly does involve an infinite u', (t) channel singularity structures in the bulk as well. Indeed we show that A3, A4 of this S-matrix do carry an infinite p.^1 singular terms, whose momenta are located in the bulk directions and from now on due to the presence of pi , we are going to call them bulk singularities and start producing them in effective field theory (EFT) as well.

We also write down the results for symmetric < VC-1 V$-1 VAoVT0 >, < VC-1 V$0 VA-1 VT0 > and asymmetric < VC-2 V$oVAo VT0 > S-matrix and start comparing them which leads to finding out various generalised Bianchi Identities in the presence of non-BPS branes. In section 2 and 3 we produce an infinite number of u', t channel tachyon, scalar singularities. At the end we apply all order a' higher derivative corrections properly to Chern-Simons couplings in such a way that even we are able to produce an infinite number of u', t channel bulk singularity structures. This obviously confirms that those bulk singularities are also needed in the entire S-matrix as they even can be constructed in EFT side as well.

Note that based on various results of this paper, we reveal the important fact as follows. A-priori without knowing any restricted Bianchi identities on world volume of D-branes for RR, there seems to be no chance to even see all the needed bulk singularity structures of A3 and A4 of (18) from the other symmetric analysis (unlike asymmetric analysis) as we go through them in detail in the next sections.

1.1. The entire < C-1$0 A0 T-1 > S-matrix

Here we would like to explore the entire form of the S-matrix elements (both in transverse and world volume direction) of an RR, a real massless scalar field, a gauge field and a real tachyon of type IIA (IIB) superstring theory, which is indeed a five point non-BPS S-matrix from the world sheet point of view. First we are going to explain our notations so that all ¡i, v,... show the entire ten dimensional space-time, on the other hand all a, b, c,.. and i, j, k,.. are taken to be employed for world volume and transverse directions appropriately.

It is discussed in [11] that in the presence of non-BPS branes one needs to consider the Chan-Paton matrices inside the vertices as follows

vf \x) = a'ik-f(x)ea'ikX(x)X ® 01, v(-1)(x) = e-0(x)ea'ikX(x)X ® 02 V(fV)(x) = e-0(x)fi fi(x)ea'iqX(x)X ® 03 V(-1)(x) = e-0(x)fafa(x)ea'iqX(x)X ® 03

V0)(x) = f1i (d Xi (x) + ia'k.ffi (x))eaik'X(x) ® I V0-2)(x) = e-20(x) V(°\x) ® I

v(°)(x) = f1a (d Xa (x) + ia'k.ffa (x))ea'iL X (x) ® I (7)

v(c-2,-2)(z, z) = (P-H(n)Mp)ape-0(z)/2Sa(z)eiapx(z)e-0(i)/2Sp(z)eiap X<z)x ® 0301, v(c-3,-1)(z, z) = (Pn-1)Mp)ape-30(z)/2Sa(z)eitpx(z)e-0(i)/2Sp(z)eitp D• X(z)x ® 01

Note that the CP factors of RR in the presence of dD system for symmetric and asymmetric pictures are a3 and I accordingly. Hence < C-100A0 T-1 > S-matrix shall be taken as follows

A<c-V° A° T-1> „J dx1dx2dx3dzdz(V(°)(x1)V A0)(x2)VT-1)(x3)VRR2,-2 )(z, z», (8)

where we are dealing with the disk level amplitude and mass-shell conditions for k\, k2, p, k3 are

where we set a' = 2. In order to just use the holomorphic correlators, one needs to keep track of the following notations for projection, RR field strength and for spinor as well.

k2 = k2 = p2 = 0, k3 = -a, k-.f- = k2.f1 = k1.f1 = k3.f1 = ° (9)

P- = -(1 - Y11), H(n) = ^HIM..,,nY11 ...y^n, (P-H(n))ap = caS(P-H(n))Sfi. For type IIA (type IIB) n = 2, 4, an = i (n = 1, 3, 5, an = 1).

We also apply the doubling trick to make use of just holomorphic parts of all the world sheet fields as below

XCI(z) ^ DlXv(z), f *(z) ^ Dlfv(z), 0(z) ^ 0(z), and Sa(z) ^ MapSp(z) , with the following definitions for the aforementioned matrices

-19 0 \ [ ^Yi1 Yi2 ...Yip+ €iv..ip+1 forp even

D =(-Tp 1 ° , and Mp =

1p+V I ^Yi1 Yi2 ...Yip+ Y11€i1...ip+i forp odd

A<C 0 A° T-1> „

Having set that, we can now go ahead with the correct form of the correlations for all X1, f1, 0 fields, as follows

{X1 (z)Xv(w» = --nIv log(z - w), a'

{f1 (z)fv(w» = ---niv(z - w)-1 ,

{0(z)0(w)) = log(z - w). (10)

Substituting the above vertex operators into the S-matrix, the amplitude reads off

-1> ~J dx1dx-dx3dx4dx5 (P-H(n)Mp)apfuf2ax451/4(x34x35)-1/2

X(I1 + 12 + I3 + I4) Tr(X1X-X3) Tr(aзala2), (11)

so that one needs to go over the following correlation functions

I1 = <: dXi(x1)ea'ik1.X(x1) : dXa(x-)ea'ik-.X(x-) : ea'ik3.Xx) : eaipXx) : eaip D Xx) :><: SaM : Sp(x5) :>, I- = <: dXi(x1)ea'ik1.X(x1) : ea'ik-.X(x-) : ea': eaip.X(x4) : eaipD^ :>a,ik2b<: Sa(x4) : Sp(x5) : fbfa(x-) :>

13 = <: ea'ik1.X(x1) : dXa(x-)ea'ik-.X(x-) : ea'ik3.X^ : eItv^M : eaip D X:>a'il<1c<: Sa(x4): Sp(x5): fcfi(x1) :>

14 = <: ea'ik1.X(x1) : ea'ik-.X(x-) : ea'ik3.X(x3) : eaip.X(x4) : e^ip.D.X(x5) :>

x (-(a')2k1ck2b)<: Sa(x4) : Spte): fcfi(x1): fbf a(x-) :> (12)

Having taken the Wick-theorem and (10), we were able to compute all the correlators of X. However, to get to fermionic correlations, involving the spin operators, one employs the so called Wick-like rule [27] as follows

4c = <: Sa(X4) : Sp(X5) : fcfi(X1) :>= 2-1X451/4(X14X15)-1(ricC

¡3b = <: Sa(x4) : S¡(x5) : fbfaX) :>= 2-1X-51/4(x24X25)-1(rabC(13)

where xij = xi - Xj, x4 = z, x5 = z. For the calculations of two spin operators and two currents one applies the generalisation of Wick-like rule [28] to indeed find out the fermionic correlations as follows

¡4bic = <: Sa(x4): S¡(X5): fcfi(xt) : fbfaX) :>

-abicC+ a' MX14 X25] (nCb(TaiC- nac (rbiC-%))2-2 X3/54(X14X15X24X25)-1

= (rabicC-i)ap + a - - (riCb(raiC-i)ap - nac (rbiC-l)ap) 2-2 x45* (X14 X15 X24 X25)-1 (14)

[ r X12X45 H r J 45

Considering all the bosonic and fermionic correlators into (11), one reveals the whole closed part of the amplitude as follows

,<C-1$0A0T-'> ,

-V ¿0 T-1> „J dxidX2dx3dxAdX5{ P -H(n) Mp )aß I^u(2i)xJ/4 (X34X35)-1/2

^a1a2x455/4C-ß + a'ik2baii I3b + a' ikic4 I? - (a'fkul^blf^ Tr(M*2*3)

such that

I =|xi2|a'2klk2 |xi3|a' 2kl-k3 |xi4xi5| V kl p |x23|a' 2k2'k3 |x24x25| V k2'P |x34x35| V k3'P |x45 |^ PDP, x54

ai = ip

xi4xi5

a2 = ik°( + -X*L_) + il^i-^ + Y (15)

V X12X24 X12X25J V X23X24 X23 X25J

One is able now to precisely check out the SL(2,R) invariance of the above S-matrix, and we do gauge fixing by fixing all the positions of open strings as x1 = 0, x2 = 1, x3 so that at the end, one has to come over the following sort of integration on the upper half

complex plane

d2z|i - z|a|z|b(z - z)c(z + z)d. (i6)

Note that all a, b, c are some combinatoric parts of the defined Mandelstam variables as follows

s = -(k1 + k3)2, t = -(k1 + k2)2, u = -(k2 + k3)2.

Results of integrations for both d = 0, 1 and for d = 2 have been appropriately explored in [29,11], so that the compact and ultimate results for the entire S-matrix in both transverse and world volume directions of brane are discovered as

A<C-1$0 A° t-1 > = A1 + A2 + A3 + A4 + A5 + A6, (17)

A1 - 2^ - &akxck2bTr(P-H(n)Mprabic) + £1.p^ak2b Tr(P-H(n)MprabL1, A2 - 2i£uTr(P-H(n)Mprai)(t)(u + 1 )L3,

A3 - 4ik1.£2£1.pTr(P-H(n)Mp)(u + 4)L3, A4 - -4ik3.£2^1.pTr(P-H(n)Mp)(t)Ls, A5 - 4ik3.£2£1ik1cTr(P-H(n)Mpric)(t)Ls,

A6 - 4ik1.£2(k1c + k2c)£1i Tr( P -H(n) Mp Tic)(u + 1 )Ls, (18)

L = (2)-2(t+s+u)n r(-u + 4)T(-s + 4)F(-t + 1 )F(-t - s - u + 1) 1 r(-u - t + 4)F(-t - s + 4)T(-s - u + 2) '

L = (2)-2(t+s+u)-1n r(-u - 1 )r(-s + 4)r(-t)r(-t - s - u) 3 r(-u - t + 4)r(-t - s + 4)T(-s - u + 1).

This S-matrix does satisfy the Ward identity associated to the gauge field, so that by replacing ^ k2a the whole S-matrix vanishes. One can also use (k1 + k2)c = (k3 + p)c in A6.

Note that, if we just change the picture of scalar field in the presence of a symmetric RR, one ends up having the final form of the S-matrix of < C-1$-1 A0T0 > [12] as follows

A<C~ $1A0 T0> = A1 + A2 + A3, (19)

A1 - 2£ubak3ck2dTr(P-H(n)MpFcadi)L1,

A2 - j- Tr(P-H(n)Mpy.&y.h)(u + 4) - 2k3.&Tr(P-H(n)Mpy.k2Y^1)jLs(2t)

+ Tr( P -H(n) Mp y.ksy^ 2t(ks.b) + 2(-u - 4 )k1.fej (-2L3). (20)

If one applies the momentum conservation along the world volume of brane, one then realises the fact that A1 of (20) produces the first term A1 of (18). The first term A2 of (20) exactly generates A2 of (18), the sum of the second and third term A2 of (20), reconstructs A5 of (18) and finally the last term A2 of (20) generates A6 of (18). Thus there seems to be no chance to produce all the needed bulk singularities A3 and A4 of (18). It is also important to stress that the second term A1 of (18) was also overlooked in (20).

Note that since the momentum along the brane is just conserved, from now on instead of Bianchi identities, we use Bianchi identities restricted on D-brane directions or restricted Bianchi identities on world volume. Consider the following restricted Bianchi identity on world volume as follows

pi£a°• • apHa0• • •ap - pc^'' ^H^..^ = 0 (21)

Note that H in (21) is (p + 1) form field strength of Cp form and this is obviously true from the traces of gamma matrices which appear in all A2 terms of (20), basically all the traces for A2 terms of (20) are non-zero just for n = p + 1 case.

Upon taking into account the above restricted Bianchi identity and applying momentum conservation along the world volume of brane (k1 + k2 + k3 + p)a = 0 to the sum of the 2nd and 3rd term of A2 (and also to the last term of A2) of (20), one is able to actually produce precisely all infinite u' (t) channel bulk singularities A4 (and A3) of (18) accordingly.

While a priori without knowing any restricted Bianchi identity on world volume for RR, there seems to be no chance to even see all the needed bulk singularities of A3 and A4 of (18) from < C-1$-1 A0T0 > S-matrix.

Let us see what happens in the other picture of the S-matrix. One reads off the S-matrix < C-1$0A-1 T0 > [30] as follows

A<CA-' T0> = A1 + A2, (22)

A1 - baksckuTr(P-H(n)Mprcaid) - £1.p(2k^2a)Tr(P-H(n)Mprca)^JL1, A2 - Jt£1.p(4ka.£2)Tr(P-H(n)Mp) + 4(u + 4)ksc£uTr(P-H(n)MpTci)kx.£2

- 4tk3.£2k1b£1i Tr(P-H(n)Mprib) - 2t(u + 4)£u£2a Tr(P-H(n)Mprai^L3 (23)

By comparisons of the elements of (23) with (18), we are able to produce all the terms inside (18) except its A3. In the other words again in this < C-1$0A-1 T0 > S-matrix, there seems to be no chance to produce A3 bulk singularities of (18).

Taking into account the above restricted Bianchi identity (21) and applying momentum conservation along the world volume of brane to the 2nd term A2 of (23), one is able to indeed construct exactly all infinite t channel bulk singularities A3 of (18). Meanwhile in this particular picture of S-matrix (< C-1$0A-1 T0 >) one could already see that the infinite u' channel bulk singularities A of (18) have been shown up in the entire S-matrix.

While a priori without knowing any restricted Bianchi identity for RR, there seems to be no chance to even observe all the needed t channel bulk singularities A3 of (18) from < C-1$0 A-1 T 0 > S-matrix.

Note that if we would consider the Ward identity associated to the gauge field (£2a ^ k2a), we would reveal that due to presence of the 2nd term of A1 of (23) and 1st term A2 of (23), the S-matrix is not gauge invariant any more. In order to restore gauge invariance, one needs to consider further remarks. Basically if we replace (£2a ^ k2a) in all four terms A2 of (23), apply momentum conservation along the world volume of brane as well as simultaneously consider the restricted Bianchi identity (21), then one observes that all four terms A2 of (23) respect Ward identity.

Finally if one replaces (£2a ^ k2a) in all two terms A1 of (23), apply momentum conservation along the world volume of brane as well as take into account the following restricted Bianchi identity on world volume directions

£1ik3ck2a(-pd ea° • • ap-3cadHa 0 • • • ap-3 + pi ea° • • ap-2acHa0 • • ap-2) = 0 (24)

then one obviously clarifies that all two terms A1 of (23) are also now respecting Ward identity associated to gauge field, therefore based on applying those restricted Bianchi identities on world volume, now the whole S-matrix respects Ward identity.

Note that the form that appears in (24) is H of (p - 1) form field strength of Cp-2 form and this is true from the traces of gamma matrices which appear in all A1 terms of (23), basically all the traces for A1 terms of (23) are non-zero just for n + 1 = p case. Hence, the terms in (21) and (24) do correspond to different RR field couplings.

It is worth mentioning that unlike the symmetric picture, in the asymmetric picture of RR and in the presence of non-BPS branes, without using any further restricted Bianchi identity the ultimate result of amplitude does satisfy Ward identity associated to the gauge

field. Eventually one can compute the same S-matrix but in asymmetric picture of RR so that the result of < C 200A0T0 > is found in [30] to be

A<Ca0 t 0> = Ai + A2 + A3 + A4 (25)

Ai - 23/2^2ak3ck2bL^piTr(Pn-1)Mprcab) - kuTr(P-/(n-i)Mprcabid)^j A2 - 23/2ifi.PL3Tr(P-/(n-i)Mpyc)(2tk3.&[-k3c - ''2c] + 2ki.fcu'k3c - tu'^ A3 - 23/2i^iiL3 Tr(P-/n-i)MpTcid)

- 2ki.%2u'k3c(kid + k2d) + 2tks .&kid(ksc + k2c)

A4 - 23/2%iiL3tu'baTr(P-/(n-i)Mprcai)(k3c + kic + k2c) (26)

As we can see in this asymmetric picture we can precisely produce even all the bulk singularities A3 and A4 of (i8).

Considering all the definitions of H(n), Mp, Fcadi, one knows that the S-matrix is non-zero just for p = n + i and p + i = n cases. Keeping in mind the momentum conservation along the world volume of brane, one obtains

s + t + u = -papa - i, (27)

Standard scattering of RR on BPS branes, with three massless open string states, does not restrict papa momentum invariant of the closed string state to a specific value. Obviously for non-BPS branes due to the presence of the tachyon the kinematic invariants are more restricted. Indeed for an RR and a tachyon momentum conservation leads to papa = k2 = 4. It is also discussed in [ii] that for an RR, two massless open strings and a tachyon, using the on-shell relations k2 = k2 = 0 and l<2 = i, we are able to rewrite the momentum expansion

(ki + k2)2 ^ 0, ki.k3 ^ 0, k2.k3 ^ 0, (28)

just in terms of t ^ 0, s ^ -4, u ^ -4. In fact the constraint (27) clearly confirms that papa must be sent to 4 or papa ^ 4 and this just makes sense only for euclidean brane. This is also consistent with the observation that has been pointed out in [3i], which means that on-shell conditions do impose to us the fact that the amplitude must be carried out just for non-BPS SD-branes [32]. The other point which is worth mentioning is as follows. It is shown in [ii] that the constraint papa ^ i is valid for all three, four and five point non-BPS functions and more importantly it is checked that by using the constraint papa = 4, one is able to precisely produce all infinite (t + s + u + 2) tachyon poles of an RR, two scalar fields and a tachyon of [ii] as the final form of amplitude clearly involves the factor T(-1 - s - u - 2) (for more information see equation (20) and section 4 of [ii]).

Now using papa ^ i for non-BPS branes, also taking into account the non-zero vertex operator of two scalars and a gauge field, one immediately gets to know that t ^ 0, s ^--4, u is the only unique expansion of the S-matrix. Given the facts that

the S-matrix does not include the coefficients of T(-s - 4), r(-1 - s - u - i), and also < VA0 VT0V^-i V^-i >, < VA0 VT0VA-i V^-i > have zero contribution (based on applying CP matrices, as Tr(/aia3a3) = 0), we understand that, this S-matrix does not have any s' = s + i /4, (s' + t + u') poles at all, thus we are left over with an infinite number of u', t channel poles.

Let us analyse all infinite tachyon u' channel poles, then we reconstruct all infinite t-channel scalar poles accordingly. Indeed, by taking the momentum expansion into considerations, we realise that Ai, A2 in (i7) are all contact interactions, while A3, A6 and A4, A5 are related to all infinite t and u' singularities of string amplitude appropriately. Also note that A3, A4 do carry p.^-i singular terms, whose momenta are located in the bulk directions and from now on due to the presence of pi , we are going to call them bulk singularities and start producing them in effective field theory (EFT) as well.

2. An infinite u' channel tachyon singularities

In order to deal with all singularities of S-matrix, we first need to have the entire expansion of Gamma functions. The expansion of tL3 around t ^ 0, s i, u is given by

Cn (s' + t)n+i + J2 fpnm (u')p (ts')n(t + s')mJ ,

n=-i p,n,m=0

:n3/2( -> , Cn(s' + t)n+i + ^J fpnm(u')p(ts')n(t + s')m ), (29)

p,n,m=0

with some of the coefficients as

i 2 i 2

C-i = i, C0 = 0, Ci = -n2, f0,0,i = r-n2, fi,0,i = f0,0,2 = 6f(3). 6 3

The results for the trace that include y 41 can also be true for the following

p > 3, Hn = *Hi0-n, n > 5. We first extract the trace in A5 and write down all tachyon singularities as follows

16 ^ 1 (4ik3.fc)kic£iii—^n3'2)^1/2)Ha0...„„_, e"0"ap-lCJ2 Cn~(s' + t)n+1Tx(XiX2X3) (30)

(p + U- n=—1 u

where we used p'n1/2) as normalisation constant to the S-matrix. To reconstruct all infinite tachyon singularities, one has to consider the following sub amplitude in field theory

A = Va(Cp,01, T)Gafi(T)VP(T, T3, A2). (31)

Note that tachyon kinetic term 2na'DaTDaT has already been fixed in the DBI action and it has no correction, also V, T3, A2) vertex operator comes from tachyon kinetic term by taking (DaT = daT — i[Aa, T]) so it has no correction either. Both Taylor expansion and pull-back of Ca0...ap—1 are needed. A field strength in the string amplitude H'a0 ap—1 being reproduced by pull-back and Taylor expansion of the Ca0...ap—1. Consider the following coupling

2i@(2na')2 J Tr(diCp A DT0) (32)

Above coupling is the mixing Chern-Simons and Taylor expansion of scalar field. We need to take into account 2ip'^'p (2na')2 £a°..apCia0,.,ap—2 Dap—10iDapT coupling as well. If we take integration by parts Dap—1 can just act on C-field. Because

of the e tensor it gives zero result if it acts on DapT. Now if we take into account ea0"ap9[ap—1Ca0...ap—2]i0iDapT where the bracket corresponds to antisymmetrisation of the corresponding world-volume indices and use the definition H'ao ap1 = pdiCa0.ap—1 + 9[ap—1 Ca0...ap—2]i, then we clarify that both terms are needed to reproduce the string amplitude term H'a0 ap1 kap £i.

Note that the contributions from Taylor expansion and the other coupling in momentum space would be ea0..appiCa0...ap—1 kap£i and (—eao..dppap1 Ca o ap—2 kap £i) accordingly, thus one obtains the following counterparts for the vertices in EFT

VP(T, T3, A2) = 2Tp(2na')k3.£2Tr^^A^), ,(2na')2

Va(Cp,01, T) = a'ea°• • apH'a0..ap-1 kap£iiTr(XiAa),

G aa(T ^ =-=--(33)

( ; (2na')Tp(k2 + m2) (2na')Tp(u'), y '

where k in the above is momentum of off-shell tachyon k = k2 + k3 = —(p + k1). Now by replacing (33) inside (31) one obtains

(4iks.£2)£li (-+Un2yva'Ha^• ap—1 (p + k1)ap €a° • • ap - (34)

where if one uses the identity papea°• •ap = 0,1 then one reveals that the first — channel tachyon pole of (30) can be precisely produced. However, in (30) we do have infinite poles and the only way to regenerate them is to induce the higher derivative corrections as follows

p p(2na')2diCp ATr ^ cn(a')n+1 D0l • • • Da„+1 DTD"1 ...Da"+ 0ij (35)

One needs to be reminded about the facts that the tachyon propagator and the vertex of two tachyon and one gauge field, do not receive any correction as they have been derived from the kinetic term of tachyons (as it has been already fixed), that is why we claim that, taking (35) is the only way of reconstructing all the singularities.

Having set (35), we were able to actually obtain the extension of the vertex operator to all orders as follows

2yp a' (2na')2ka„ £1i ~

Va(Cp, T ,01) = yp a( ( J ap ea° • "pH\o ••ap—cm (a% •k)m+1Tr(X1Aa). (36)

(p)' m=—1

Substituting (36) to (31), considering the fixed propagator (the fixed two tachyons, a gauge field vertex operator) and making use of the following identity Ylm=-1 cm(a'k1 • k)m+1 1 cm(t + s')m+1, one exactly reconstructs all infinite — tachyon poles of (30) in the

effective field theory side as well. It is also worth pointing out the fact that by comparisons we get to know that there was no residual contact interactions to be left over in the EFT. Now we turn to infinite t channel scalar singularities.

1 Note that, the derivation of the identity paea0-ap—1a = 0 can be found from various equations of [30], for instance it is clarified in formula (9) of [30], namely, to actually obtain the same result for the amplitude of a three point function of one RR and a scalar field in both symmetric and asymmetric S-matrix, this identity paea0-"p—1a = 0 should be true. The other example to prove the above identity is as follows. We have just shown it in section 5 of [30], basically to get to the same result for the S-matrix of a four point function of < C—1T00—1 > and < C—2T000 >, that identity should be employed (notice to the footnote 19 of [30]). Finally, we have shown that if and only if that identity holds, then certainly the S-matrix of < C—1A—1T0T0 > of [14] does satisfy Ward identity related to the gauge field.

3. An infinite t channel scalar singularities

Having expanded u'L3 around the same t ^ 0, s ^ — 4, u ^ — 1, we get

'1 " "

tj2cn(u' + s')n+1 + J2 fpnmtp (u's')n(u' + s')mj

n=-1 p,n,m=0 ^

P (Us')n(u' + s')m 1 (37)

where the coefficients of cn, fp,n,m are read.2 After extracting the trace in A6 one writes down all scalar t-channel singularities of the S-matrix as follows

16uPpn2 ~ 1 .

4iki.§2k3cfri p cnt(s' + u')n+1 Hla0...ap_lea° -ap-1cTr(M^3). (38)

( P + '' n=—1

Later on we consider the restricted Bianchi identity to actually write down pcH'a0..ap—1 ea°'ap—1C in terms of p'Ha0•••ap''ap and generate all the bulk singularities of the S-matrix in EFT as well. To construct all infinite scalar singularities, one must consider the following sub amplitude in field theory

A = va(Cp, T3,t)Gaf(t)Vf (0,01, A2). (39)

Note that the scalar kinetic term has already been fixed in the DBI action and it has no correction, also VP(0, 01, A2) can be derived just from scalar kinetic term by extracting the covariant derivative of scalar field (Da0' = da0' — i[Aa, 0']) so it has no correction either. One derives the following vertices in EFT

Vj(0,01, A2) = —2(2na')2Tpki.^1jTr(XiX2Ap), (40)

(2na')2

Vf(Cp,0, T3) = 2^'pP'^ea° • • apH'a0 ..ap—1 k3ap Tr(^3Aa),

G *e(0)= —'saps'_ —'SapSij ij (0) (2na')2 Tp (k2) (2na')2 Tp (t), ( )

where k3 in the above is momentum of on-shell tachyon and we used momentum conservation along the world volume direction as well. Now by replacing (41) inside (39) one concludes that the first t-channel scalar pole of (38) can be precisely produced. However, as it is clear in (38) we do have infinite poles and the only way to reproduce them is to propose the higher derivative corrections to the actions. Taking (35) into account one can get the all order extensions of the vertex operator as

Va(Cp,0, T3) = 1-• apH{0^k3ap£ Cm(a'k3 • k)m+1 Tr(X3Aa). (42)

m=—1

Using momentum conservation, one gets, (a'k3 • k) = (u' + s'). If we substitute (42) inside (39) and keeping fixed all the other vertices, one clarifies the fact that all the infinite t-channel poles of (38) are precisely gained in the effective field theory side as well, so the higher derivative corrections of (35) are exact.

To generate all the contact interactions, we just highlight the following references [11,7]. Also notice to the point that for this S-matrix we do have external gauge field as well as an external scalar and a real tachyon therefore in the action of (32), one needs to first consider the presence of commutator inside the covariant derivative of tachyon, so that external gauge field shows up and then try to apply higher derivative corrections properly. Lets turn to the main point of the paper which is indeed dealing with all the bulk singularity structures of the S-matrix.

4. An infinite u' bulk singularity structures

Apart from an infinite number of u' channel tachyon poles, we have an infinite number of bulk singularity structures that can be accommodated in EFT as well.

Consider the expansion of (29) and do replace it into the entire A4 of (18), extract the related trace as well as normalise this fourth part of S-matrix so that the whole bulk singularities are now found out as follows

16uP p'n2 ~ 1

p.^(—4'ks.b) , * • apefl° • • ap V c„-(s' + t)n+1Tr(X1 X2X3) (43)

(p +1)! 0 p u'

C2 = 2Z(3), f2,0,0 = f0,1,0 = 2Z(3), f 1,0,0 = -n2, f 1,0,2 = — n4

f0,0,1 = 1 n2, f0,0,3 = f2,0,1 = 90n4, f1,1,0 = f0, 1, 1 = — n4.

We call them bulk singularities, because they do involve all the infinite momenta of RR in the bulk directions (due to an infinite number of p.£1 terms). In the other words, we can keep track of these terms that carry the scalar product of momentum of RR in the bulk and scalar polarisation and claim that these bulk singularities are needed in the entire S-matrix as we are going to reconstruct them in EFT as well.

To reconstruct all these infinite bulk u' channel singularities, one has to once more, consider the following sub amplitude in field theory side

A = Va(Cp,01, T)GaP(T)VP(T, T3, A2). (44)

It is worth highlighting the remark that both tachyon propagator and VP(T, T3, A2) vertex operator, will not receive any correction.

To actually produce P.£1 terms in EFT one needs to consider the following integrations

2iP'ii'p(2na')2 j ( — Tr(3idflpCfl0...flp_1 T0{) — Tr(3iCfl0...flp_1 Tdap0{

where we suppose all the fields are zero at infinity. We have already taken into account the contribution of the second term of the above action and were able to produce all infinite u' channel tachyon singularities, so here we just need to consider the contribution of the first term of (45) to actually derive the following vertex operator

(2na')2

Va(Cp,01, T) = 2MpP'I-ea0 • • apHa0•••flpP.£1Tr(X1Aa) (46)

Keeping fixed VP(T, T3, A2) and the tachyon propagator and replacing (46) inside (44) one gets 16uPP 'n 2

(—4ik3.£2)p.£1 u ".„.Hae• • apea°• •"p ^1^3) (47)

u'(p + 1)! 0 p

One can observe the fact that (47) is indeed the first bulk singularity of the S-matrix (consider n = —1 inside (43)). To actually even produce all infinite bulk singularities, one needs to apply all higher derivative corrections to the first part of the above action as follows

2iP ^Rin-rr2 I I Tr- \ x im'\r n , , , n rn«1 (48)

(p + 1)! (2na')2 f Cn (a,)n+1didapCa0...ap_1 Da,• • • D^, TDa1 ...D^1 0'J

n=—1

Having set (48), we are able to precisely reconstruct all order extensions of the above vertex operator as follows

2a'p p' (2na')2 p.£x ~

V"(Cp, T,01) = P( ea°• • a"Ha0• • cm(t + s')m+1Tr(MAa). (49)

(P + )! m=—1

Substituting (49) to (44), considering the fixed propagator (and fixed two tachyons, a gauge field vertex operator), one is able to exactly regenerate all infinite u' bulk poles of (43) in the effective field theory side as well. Eventually in the next section, we try to produce an infinite t channel bulk Singularity structures in the effective field theory side too.

5. An infinite t channel bulk singularity structures

Apart from an infinite number of t channel scalar poles, we do have an infinite number of t channel bulk singularity structures that can be found out in EFT as well.

Consider the expansion of (37) and substitute it into the whole A3 of (18), extract the trace as well as normalise this third part of S-matrix, to indeed get to the whole bulk t-channel singularities from the string theory point of view as follows

16u'p p 'n2 ~ 1

p.£1 (4ikx.£2) " Ha0•••ape"0-"" Cn-(s' + u')n+1 ^1^3) (50)

(P + )! n=—1

These are also bulk singularities, because all the infinite singularity terms that involve momentum of RR in the bulk have been embedded into the S-matrix and in below we want to show that they even can be reconstructed in EFT as well. To regenerate all these infinite bulk t channel scalar singularities, one needs to deal with the following sub amplitude in field theory side

A = Va (Cp, T3,0)GaijP(0)Vp (0,01, A2), (51)

Note that both scalar propagator and VP(0, 01, A2) vertex operator, do not get corrected. To actually produce p.£1 terms in EFT one should make use of Taylor expansion and take integration by parts as well. Indeed in (41), we have considered the contribution of the second term of (45) and precisely produced all infinite t channel scalar singularities of A6, in the meantime, here we just need to consider the contribution from the first term of (45) to actually explore

(2na ')2

V"(Cp,0, T3) = —2pifipP'ea0 • • "pH"0• • ap Tr(XsAa) (52)

Keeping fixed VP(0, 01, A2) and the scalar propagator and replacing (52) inside (51) one gains 16u/p p 'n 2

(4'kt.fc)t(pp+ 1)! Ha0• • ap•ap TKX1X2X3) (53)

One can observe the fact that (53) is indeed the first bulk singularity of the S-matrix (consider n = — 1 inside (50)). To be able to produce all infinite bulk singularities, one needs to apply all higher derivative corrections properly. Having set (48), we are able to precisely reconstruct all order extensions of Va(Cp, 0, T3) vertex operator as follows

V*(Cp,0, T3) = —2p'fi'pP'l^e"0• ^„0 •• apJ2 km(u' + s')m+1Tr(X3Aa). (54)

(p + )! m=—1

Substituting (54) to (51), considering the fixed scalar propagator (and the fixed two scalars, a gauge field vertex operator), one is able to precisely regenerate all infinite t channel bulk singularities of (50) in the effective field theory side as well. Therefore, we were able to even produce all the infinite bulk singularities u', t in the EFT and that evidently confirms that the presence of bulk singularities is needed inside the entire S-matrix. Note that, it also has the important consequences for which one of them is the essential appearances of new restricted Bianchi identities on world volume directions of D-branes that we got in this particular S-matrix.

Acknowledgements

The author would like to thank C. Hull, A. Tseytlin and D. Waldram for useful discussions at Imperial College in London, he also thanks N. Lambert for having several valuable discussions at Kings College London. He warmly thanks P. Vanhove, T. Damour, S. Shatashvili, M. Kontsevich and V. Pestun at IHES and his colleagues at QMUL. In particular he thanks C. Vafa, R. Russo for various discussions, C. Papageorgakis, S. Thomas, D. Young, R. Russo, A. Brandhuber, G. Travaglini, D. Berman, S. Ramgoolam, A. Sen, R. Myers, K. Narain, L. Alvarez-Gaume, W. Lerche, J. Polchinski, N. Arkani-Hamed, W. Siegel, E. Witten, H. Steinacker, A. Rebhan, D. Grumiller, T. Wrase, and P. Anastasopoulos for valuable discussions. Some parts of this work have been done at Caltech, Simons Centre, UC Berkeley, IAS at Princeton, Harvard, CERN and IHES. He is very grateful to those institutes for providing such an exciting and challenging environment. This work was supported by the FWF project P26731-N27.

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