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Nuclear Physics B 903 (2016) 104-117

www. elsevier. com/locate/nuclphysb

Disk scattering of open and closed strings (I)

Stephan Stieberger , lomasz r. xayioi

a Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany b Department of Physics, Northeastern University, Boston, MA 02115, USA

Received 8 October 2015; accepted 6 December 2015

Available online 14 December 2015

Editor: Herman Verlinde

At the tree level, the scattering processes involving open and closed strings are described by a disk world-sheet with vertex operator insertions at the boundary and in the bulk. Such amplitudes can be decomposed as certain linear combinations of pure open string amplitudes. While previous relations have been established on the double cover (complex sphere) in this letter we derive them on the disk (upper complex half plane) allowing for different momenta of the left- and right-movers of the closed string. Formally, the computation of disk amplitudes involving both open and closed strings is reduced to considering the monodromies on the underlying string world-sheet.

© 2015 The Authors. 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.

The relationship between open and closed string amplitudes is important from both mathematical and physical points of view because it helps in understanding what features of the closed string can be implemented by pure open string properties. At tree-level, Kawai, Lewellen and Tye (KLT) [1] derived a formula which expresses any closed string tree amplitude in terms of a sum of the products of appropriate open string tree amplitudes. This formula gives rise to a striking relation between gravity and gauge amplitudes at tree-level. Another description has been developed in [2,3], by constructing tree-level closed superstring amplitudes through the "single-valued" projection of open superstring amplitudes. This projection yields linear relations between the functions encompassing effects of massive closed and open superstring excitations,

* Corresponding author.

E-mail address: stephan.stieberger@mpp.mpg.de (S. Stieberger).

http://dx.doi.org/10.1016/j.nuclphysb.2015.12.002

0550-3213/© 2015 The Authors. 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.

Abstract

to all orders in the inverse string tension a'. They reveal a deeper connection between gauge and gravity string amplitudes than what is implied by the KLT relations. Furthermore, in [4] tree-level string amplitudes involving both open and closed strings have been expressed as linear combinations of pure open string amplitudes. This correspondence gives a relation between Einstein-Yang-Mills (EYM) theory and pure gauge amplitudes at tree-level [5] with interesting consequences for constructing gravity amplitudes from gauge amplitudes [6]. Scattering amplitudes of open and closed strings describe the couplings of brane and bulk fields thus probing the effective D-brane action. Hence, these amplitudes are important for many studies related to D-brane effects.

Tree-level amplitudes involving both open and closed strings are described by a disk world-sheet, which is an oriented manifold with one boundary. The latter can be mapped to the upper half plane:

H+ = {z e C | Im (z) > 0 } . (1)

Open string vertex operator insertions are placed at the boundary of the disk and closed string positions at the bulk. The integration over the latter can be extended from the half-plane covering the disk to the full complex plane if the closed strings are world-sheet symmetric closed string states (such as graviton or dilaton). However, for arbitrary closed string states and generic D-brane and orientifold configurations this world-sheet symmetry is not furnished and one has to perform the computations on the disk. The techniques for evaluating generic disk integrals involving both open and (world-sheet symmetric) closed string states have been developed in [4]. Moreover, in [5] a closed and compact expression for the amplitude involving one closed and any number of open strings has been derived. In this letter we want to extend these results to generic closed string states, i.e. perform the amplitude computation on the disk rather than on its double cover. The amplitudes can be decomposed as certain linear combinations of pure open string amplitudes. Formally, the computation of disk amplitudes involving both open and closed strings is reduced to considering the monodromies on the underlying string world-sheet.

In the following we shall consider disk amplitudes with one bulk and N - 2 boundary op-erators.1 This yields the leading order amplitude for either the absorption of a closed string by a D-brane or the decay of an excited D-brane into a massless closed string state and the unex-cited D-brane [8-10]. Open string vertices with momenta pi, i = 1, ..., N - 2 are inserted on the real axis of (1) at xi e R, while a single closed string vertex operator is inserted at complex z e H+. For the latter we assume different left- and right-moving space-time momenta qi and q2, respectively. This is the most general setup for scattering both open and closed strings in the presence of D-branes and orientifold planes. Due to the boundary at the real axis there are non-trivial correlators between left- and right-movers. In order to compute the amplitudes, it is convenient to use the "doubling trick," [9,11] to convert disk correlators to the standard holo-morphic ones. This method accommodates the boundary conditions by extending the definition of holomorphic fields to the entire complex plane such that their operator product expansions (OPEs) on the complex plane reproduce all the OPEs among holomorphic and anti-holomorphic fields on H+ [12].

By the boundary conditions on the D-brane world-volume the open string momenta pi are restricted to lie within the world-volume directions. On the other hand, the closed string momentum q has generic directions. All strings are massless and their momenta are on-hell, i.e.

1 Disk amplitudes with an arbitrary number of bulk and boundary operators will be considered in [7].

pf = q2 = 0. Since D-branes are infinitely heavy objects they can absorb momentum in the transverse direction, which in turn implies that only along the world-volume directions momentum conservation is furnished. This can be taken into account by choosing

qi = f q , qf = f Dq , (2)

with q the closed string momentum and D a matrix accounting for the specific boundary conditions in d space-time dimensions. Then, the longitudinal closed string momentum is given by

q11 = qi + qf = - (q + Dq) , (3)

while normal to the brane we have the remaining momentum 1

q± = - (q - Dq) , (4)

and total momentum conservation along the D-brane world volume reads:

J2pi+q11=o • (5)

Typically, in flat space-time2 the matrix D^v is a diagonal matrix, equal to Minkowski metric in directions along the D-brane (Neumann boundary conditions) and to in directions orthogonal to the brane (Dirichlet boundary conditions). Then, the left- and right-moving momenta qi define on-shell momenta:

qf = 0 • (6)

In what follows, we shall assume that (6) holds3 for the left- and right-moving closed string momenta (f).

The disk amplitudes involve integrals of the form

N-2 „N-2

l r . o „,' „„

Fn = VC-KG pi + qi + qf) I n dxi n |xr - xs\2a'prps (xr - xs)n

i = 1 i = 1 1<r<s<N-f

d2z (z - z)2"'^2 +n n (Xi - zfa'piq1+ni (Xi - z)

X I d2z (z - z)2a j I (Xi - z)2a piqi+ni (Xi - z)2a piq2+ni , (7)

H+ i = 1

where we included the momentum-conserving (along the D-brane world-volume) (3) delta function and divided by the volume Vckg of the conformal Killing group. The powers nrs, ni, ni, n are some integer numbers. To be specific, we focus on the amplitude associated to one particular Chan-Paton factor (partial amplitude), Tr(T 1T2 ...TN-2), with the integral over ordered x1 < x2 < ••• < xN-2. Note, that in (7), the momenta q1 and q2 are assumed to be unrelated, i.e. in (2) the matrix D is a generic matrix such that the condition (6) is fulfilled.

2 The most general expression for D is given by D = -g 1 + 2 (g + b) 1, with the metric g and the anti-symmetric tensor b [13].

3 Note, that this assumption is obeyed by generic four-dimensional string compactifications with internal metric g and two-form fluxes b without warping for which a CFT description is available.

S. Stieberger, T.R. Taylor /Nuclear Physics B 903 (2016) 104-117 Xra(z2)

+i(Xj - Zi)

+i(Xi - 0i)

-i(xi - Zi)

Ke{z2)

' -i(Xj - «i)

Fig. 1. Branch cut structure and contour deformation in complex Z2-plane.

For the concrete case (7), we write the integral over the complex upper half-plane H+ as an integral over holomorphic and anti-holomorphic coordinates, by following the method proposed in [1]. After writing z = z1 + iz2, the integrand becomes an analytic function of z2 with 2(N — 2) branch points at ±i(xi — z1). We then deform the z2-integral along the real axis Im(z2) = 0 to the pure imaginary axis Re(z2) = 0 with Im(z2) > 0, as depicted in Fig. 1. In this way, the variables

H = zi + i Z2 = z , n = Z1 - i Z2 = z become real, subject to:

n-H>0 •

After changing the integration variables (z1, z2) ^ (H, n) (with the Jacobian det dZ,z,i) = 2), Eq. (7) becomes an integral over N real positions xi, H, n

Fn = VCKGs(!2ki)j n dxi J dh]dn n

i = 1 i = 1 _t 1<r<s<

\xr xs I r S (xr xs)

1<r<s<N-2

x 2 (H - n)n \H - n\2a'kN-1kN V(H,n)

x n U(xi,H,n) x - HI2a'kikN-1 x - n\2a'kikN(xi - H)ni(xi - n)ni ,

with the open string momenta kr = pr, r = 1,..., N — 2 and the closed string momentum split into left- and right-moving parts

kN-1 = q1 , kN = q2

respectively. Eq. (10) resembles a generic open string integral involving N open strings with external momenta ki supplemented by the extra phase factors

U(H,n) = e

_ 2niafkN-1kN 0(i]-H)

U(xf,H,n) = e

_0-2nia/kikN-1 O(H-xi) g2niakikN O(n-xi)

where 0 denotes the Heaviside step function. These monodromy factors (12) account for the correct branch of the integrand, making the integral well defined. Note that the phases, which are independent of the integers nrs, ni, ni, n do not depend on the particular values of integration variables, but only on the ordering of f and n with respect to the original N — 2 vertex positions. In this way, the original integral becomes a weighted (by phase factors) sum of integrals, each of them having the same form as the integrals appearing in N-point (partial) open string amplitudes, with the vertices inserted at xl, l = 1, ..., N, where we identified xN-1 = f and xN = n. Note that the order of the original N — 2 positions remains unchanged. Since the closed string vertex factorizes into two gauge bosons inserted at z = f = xN—1 and z = n = xN, we conclude that the amplitude

Fn = A(1, 2,...,N — 2; qi,q2) (13)

describing closed string decay into N — 2 gauge bosons can be written as a weighted sum of pure open string amplitudes with the closed string replaced by a pair of gauge bosons. The latter carry the left- and right-moving momenta Eq. (11) of the closed string, respectively.

In order to express the partial amplitude A(1, 2, ..., N — 2; qi, q2) in terms of N -point open string amplitudes, we need to analyze the phase factors. For f e R the phase factor (12) in the integrand can be accommodated by considering respective contours in the complex n -plane. After fixing the position of the first open string vertex at x1 = —cc we have the situation depicted in Fig. 2. For the case of interest n > f quite generally, around all open string vertex positions xl > f the contour goes anti-clockwise. The last case f > xN—2 contributes the single term

2 exp {ni(s1,N—1 + sn—1,N — S1,n)} A(1, 2,...,N — 2,N — 1,N), (14)

while the first case f <x2 gives rise to:

— l-A(1,N,N — 1, 2,...,N — 2). (15)

In the latter case we could reduce the contour to a single contribution thanks to string mon-

odromy relations [4]. Eventually, in the second case xl—1 < f < xl with l = 2,..., N — 2 string

monodromy relations can be applied to deform the contour to the left. This is accomplished for

x2 < f < xn^ to obtain a minimal set of integration regions. Each case xl < f < xl+1 with 1 2 1

l = 2,..., \N1 — 1 contributes a residual contour of l arcs starting from x1 = —cc and passing the l points x2, ..., xl and f:

. rN1-1 /+i

- 2 T, EexP

l=2 i=2

(i-1 I

J2sj>n -12 sJ,N-i j =2 j=2

x A(1,...,i - 1,N,i,...,l,N - 1,l + 1,...,N - 2) . (16)

On the other hand, for ^^ <H < xN-2 we leave the contour as depicted in Fig. 2 and obtain

contributions from each region xl < Ç < xl+1 with l = [N1,..., N - 3. Each giving rise to a contour from f to infinity with N - 1 -1 arcs at the points f and xl+1, ..., xN-2:

. N-3 N-2

+2 T, Eexp

l=rfl i=l

N-2 N-2

ni j S1,N-1 + SN-1,N - S1,N + Sj,N-1 - Sj,N

j=l+1 j=i+1

x A(1,...,l,N - 1,l + 1,...,i,N,i + 1,...,N - 2) . (17)

X2 X3 ..... Zff-2

......

gi7rtt2 gl7Ta3

Xl ..... ZjV-2

zj-i < { <

pVK(x piKoti piTrajv-2

£ > rCjV-2

£2 £3 #JV-2

Fig. 2. Complex ^-plane and contour integrations. Here ai = a'piq2 = la'kik^ and a = 2a'qiq2 = 2a'kN—\kN.

In total we obtain [Ni (TN1 — 1) terms:

A(1,...,N — 2; q\,q2) i r N1—11+1

= — 2 E Eexp

i —1 l

ni ( J2SJ,N — J2 Sj,N — 1 j=2 j=2

l = 1 i=2

x A(1,...,i — 1,N,i,...,i,N — 1,i + 1,...,N — 2)

N-2N-2

N-2 N-2

ni\S\,N — l + sN — 1,N — s1,N + Sj,N — 1 — Sj,N

j=i+1 j=i+1

+ 2 T, Eexp

i=r N i i=l

x A(1,...,i,N — 1,i + 1,...,i,N,i + 1,...,N — 2) .

On the r.h.s., according to (11) the N open string momenta are given by ki, i = 1,..., N — 2, kN—1 = qi and kN = q2, respectively. Furthermore, with (2) and (3) we may express the kinematic invariants

sN—1,N = a' (q11 )2 ,

si,N—1 = a' piq11 , si,N = a' piDq11 , i = 1,...,N — 2 ,

in terms of invariants of the D-brane world-volume (i.e. using only momenta parallel to the D-brane world-volume) with the parallel closed string momentum q11 defined in (3). It is easy to see, that for q1 = q2, i.e. s1,N—1 = s1,N and sN—1,N = 0 the real part of (18) reduces to the formula given in [5] describing the result in the double cover.

As an example we display the case N = 5 for which Eq. (18) yields the following four terms: A(1, 2, 3; q1,q2) = { eni(si4-si5+s45) A(1, 2, 3, 4, 5) - eni(-m+s25) A(1, 2, 5, 4, 3)

- e-nis24 A(1, 5, 2,4, 3) - A(1, 5,4, 2, 3) } . (20)

By applying string monodromy relations [4,14] the expression (18) can be expressed in terms of the minimal set of (N - 3)! open string basis amplitudes:

A(1, 2,...,N - 2; q\,q2)

= (-1)N e-ni(s1,N+S2,N-1) J2(-1)' sin(nsUN-1) eni(-l)l sl,N-

LN-3 J J! T2k+1 (P)

x J2 e k=1 S(p)A(1,p,N - 1,N). (21)

pe[OP(a,p'),l}

The second sum involves all permutations p comprising the element l and the ordered set of permutations OP(a, p') of the merged sets:

a = [2,...,l - 1} , p = [l + 1,...,N - 2} . (22)

This ordered set corresponds to all permutations of a U p' which keep the order of elements of a and p', respectively. Besides, p' denotes reversal of the elements in p. Furthermore, in (21) the following string kernel S(p) enters

N-2 N-2

S(p) = S [p(2,...,N - 2) Yl exp {ni eco-1« - p-1(j)) su}, (23)

i=2 j=i + 1

with si,j = sij = 2a'kikj. Some other variants of string KLT kernels occur for pure closed string amplitudes in [1,15]. Finally, we have:

f(p) = \ sign(p-1(i) - P-1(i + 1)) (si,N-1 + Si+1,N-1) , 3 < i < N - 3 , (24) i [sN-2,N-1 , i = N - 2.

In (21) the double sum gives rise to J2 ( N-2 ) = 2N-4 terms. It can be evidenced along the

l=2 l-2

lines of [16,17], that (21) provides the correct soft-limits for kN-2 ^ 0 and collinear limits.

Note, that the leading term in the «'-expansion of (21) starts linearly at a', while the leading term in (18) appears at a'0. As a consequence, the latter must vanish, giving rise to relations similar to (and following from) U(\) decoupling (Kleiss and Kuijf) [18] conditions,

r N1-1 1+1

0 ^ £>YM(1,...,i - 1,N,i,...,l,N - 1,l + 1,...,N - 2) l=1 i=2 N-2 N-2

^^ J2 AYM(1,...,l, N - 1,l + 1,...,i,N,i + 1,...,N - 2), (25)

l=r N1 i=l

involving L N J (r N1 - 1) SYM subamplitudes.

To illustrate the result (21) let us consider some examples with a small number of external particles. The case N = 4 is not new and has already been studied in [10,11]. For completeness we display the latter and (21) yields:

A(1, 2; q1,q2) = e—nis23 sin(ns23) A(1, 2, 3, 4). (26)

In the appendix we explicitly demonstrate how to cast a generic mixed amplitude of two open and one closed string into the form (26). The real part gives the corresponding relation in the double cover (in this case we have u = s23 = s24 and s = s12 = —2s24):

A(1, 2; q,q) = - cos(nu) sin(nu) A(1, 2, 3, 4) = sin(2nu) A(1, 2, 3, 4)

= —sin(ns) A(1, 2, 3,4). (27)

For N = 5 our formula (21) yields A(1, 2, 3; q1,q2)

e—nis51 sin(ns34) A(1, 2, 3,4, 5) — sin(ns24) A(1, 3, 2, 4, 5) 1 , (28)

_ e—nis24

which agrees with (20). Again, in the appendix we explicitly demonstrate how to cast a generic mixed amplitude of three open and one closed string into the form (28). The real part of (28) gives the corresponding relation in the double cover. After using open string relations we obtain (in this case we have s12 = 2s + a, s23 = 2u + a, s34 = s, s45 = a, s51 = u, i.e. a1 = a2 = s and 31 = ¡$2 = —s — u — a = t):

A(1, 2, 3; q,q) = —- sin(na) A(1, 2, 3,4, 5) — - sin(nt) A(1, 5, 2, 4, 3) , (29)

in agreement with Eq. (3.19) of [4]. For N = 6 we find:

A(1, 2, 3, 4; q1,q2) = e—nis25 { e—ni(s61+s35) sin(ns45) A(1, 2, 3, 4, 5, 6)

+ sin(ns25) A(1, 4, 3, 2, 5, 6) — eni(—s61+s34+s45) sin(ns35) x A(1, 2, 4, 3, 5, 6) + enis24 sin(ns35) A(1,4, 2, 3, 5, 6) ] } .

For N = 7 we obtain:

A(1, 2, 3,4, 5; q1,q2) = e—nis26{ e—ni(s17+s36+s46) sin(ns56) A(1, 2, 3,4, 5, 6, 7)

— Sin(ns26) A(1, 5,4, 3, 2, 6, 7) — e—ni(si1+s36—s45) sin(ns46) x [ enis56 A(1, 2, 3, 5, 4, 6, 7) — eni(s56+s35) A(1, 2, 5, 3, 4, 6, 7)

—eni(s56+s25+s35) A(1, 5, 2, 3, 4, 6, 7) 1 + e—nis36 sin(ns36)

eni(s12+s27) A(1, 2, 5, 4, 3, 6, 7) + e—ni(s23+s24+s26) A(1, 5, 2,4, 3, 6, 7)

+e—ni(s23+s26) A(1, 5, 4, 2, 3, 6, 7)]} . (31)

Note that until this point, we did not make any assumption how the total closed string momentum q was distributed among left- and right-movers. In particular, we did not use any specific

form of the D matrix, see Eq. (2). We used the on-shell condition (6) and the total momentum conservation (5) only. This should be contrasted with the computations on the disk double cover which utilize left-right symmetric (half-half) momentum distribution. In order to make contact with the results of [19], we consider the case of a D-brane filling four space-time dimensions and the closed string carrying a purely four-dimensional momentum

P = q = q1 + q2 , (32)

which is on-shell, P2 = 0, and now split as:

q1 = kN—1 = xP , q2 = kN = (1 — x) P . (33)

Hence all results from before can be used for this case. For the invariants (19), we have

sN — 1,N = 0 ,

si,N—1 = xsiP, si,N = (1 — x)siP , i = 1,...,N — 2 , (34)

where siP = a'piP. We are interested in the field theory limit of the amplitudes, i.e. in the Einstein-Yang-Mills (EYM) limit which corresponds to the leading a' order of Eqs. (18) and (21). As mentioned before, at the a'0 order, the r.h.s. of Eq. (18) vanishes as a result of (25). At the leading a' order, Eq. (18) yields:

Aeym(1, 2,...,N — 2; P) rr fl —1 i i

XXX/P AYM(1,...,i — 1,N,i,...,l,N — 1,l + 1,...,N — 2)

i=2 i=2 j=i N—3 N—2 i

N—3 N—2 i 1

£ £ (£ sjP^jAYM(1,...,l,N — 1,l + 1,...,i,N,i + 1,...,N — 2) =rNn i=l+1 j=l+1 >

l=rN1 i=l+1 j=l+1 n

+ y (2x — 1)

if N1—11+1 i—1

J2 Aym(1,...,i — 1,N,i,...,l,N — 1,l + 1,...,N — 2)

l=2 i=2 j=2

N—2 N—2 i ]

j2 j2 (!>p) Aym(1,...,1,N — 1,l + 1,...,i,N,i + 1,...,N — 2U .

l = rN1 i=l j=2 -I

By a repeated use of Bern, Carrasco and Johansson [20] and Kleiss and Kuijf [18] relations, one can show that the terms enclosed by the second curly bracket are equal to (—x) times the terms enclosed by the first bracket. In this way, we obtain:

Aeym(1, 2,...,N — 2; P)

= n x(1 — x)

r^1—1 l l x Es;p)Aym(1,

.,i — 1,N,i,...,l,N — 1,l + 1,...,N — 2)

l=2 i=2 j=i

N—3 N—2 i 1

+ J2 J2 ( J2 sjP^ Aym(1,...,1,N — 1,i + 1,...,i,N,i + 1,...,N — 2) . i=rNi i=i+1 j =i+1 f

The above result reproduces Eq. (8) of Ref. [19], modulo the n factor which together with a' combine into the gravitational coupling constant. Similarly, the leading a' order of Eq. (21) has the same form as Eq. (18) of Ref. [19]. The set of permutations appearing in the sum is specified in Eq. (22).

The advantage of the formalism developed in this work is that it allows for a left-right asymmetric partition of the closed string momentum. By considering the monodromy properties of the amplitude on the underlying world-sheet, we derived Eq. (21) which shows that the full-fledged string disk amplitude involving one closed string and any number of open strings can be expressed as a linear combination of pure open string amplitudes with the original closed string momentum arbitrarily split between two open strings.

Acknowledgements

This material is based in part upon work supported by the National Science Foundation under grant No. PHY-1314774. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Appendix A

In this appendix we demonstrate how performing a direct complex world-sheet integration for the cases N = 4 and N = 5 readily leads to the results (26) and (28), respectively.

Let us consider the world-sheet disk integral involving two open and one closed string

F4 = j dx (2i)a0 (x — i)a1 (x + i)a2 = —n e—nia1

r(-1 - ai - a2)

. e —1

T(1 + ao) T(1 + a1)

r(-aO r(-a2)

= sin(na1) e—nia1 --^ , (37)

r(—a2)

corresponding to the choice of vertex positions:

z1 = —c^ , z2 = x, z3 = i, z3 = —i . (38)

Above we have imposed the constraint:

a0 + a1 + a2 = —2 . (39)

The integral (37) can be computed by considering the contours in the complex x-plane as shown in Fig. 3. After deforming the integration from the real axis to the imaginary axis we obtain

(2i)ao (enia1 - e-nia1) j dy(y - 1)a1 (y + 1)a2 ia1+a2+1 1

= (2i)a1+a2+ao+2 F(1 + e^-1 - a1 - ^

T(-a2)

Fig. 3. Contour integration in the complex x-plane.

which yields4 (37) subject to the branching factor e~nia1. We can relate the mixed amplitude (37) to a pure open string amplitude involving four open strings. The generic open string disk amplitude reads:

A(1, 2n, 3n, 4n) = V—KgJ dzi \zij\sj zj . (41)

D(n) i<j

Note, that due to conformal invariance the integers nij must fulfill the conditions:

J^nij = —2 ,i = 1,...,4 . (42)

With the choice

Z1 = — c&, Z2 = 0, Z3 = 1, Z4 = x —1 , (43)

the canonical subamplitude becomes

A(1 2 3 1)n23+n24+n34 F(— 1 — s24 — s34) F(1 +s34)

( , , , ) ( ) r( —^24)

= (—1)n23+n24+n34 F(1 + a0)T(—1 — a0 — a2) (44)

r(—a2) ,

sij = sij + nij , (45)

and the following identifications

4 Later we will use the integral:

[ dx(x - i)a (x + i)b = -n (2i)2+a+b e-nia F(1ab) . J ( ) ( + ) ( ) r( - a) r( - b)

a0 = s34 + «34 , a1 = s23 + «23 ,

a2 = s24 + «24 , (46)

which fulfill (39), i.e. «23 + «24 + «34 = —2. Comparing (37) with (44) gives the relation (26) subject to (13).

Now, let us consider the world-sheet disk integral involving three open and one closed string5

F5 = (2i)a0 ( — 1)«23 j dx2 J dx3 (x2 — if2 (x2 + if1 (x3 — i)a2 (x3 + i)a1

—c x2

x (x3 — x2f° = — (—1)«23 e—ni(a2+/2) V(—2 — a1 — a2 — / — /2 — /0)

T(1 + /30) V(— 1 — a1 — /30) r(2 + /0 + a1 + /1)

x j sin[n(p0 + a1 + p1)]

r(-a1)V(-p2 - a2)

[-«2, 1 + p0, 2 + p0 + «1 + p1 t x 3 F2 \ ; 1

L 2 + p0 + a1, -a2 - p2

^ -ni(a1+p0) ■ , « , r(1 + p1)T(-1 - a1 - a2 - pp)r(1 + p0 + «1) + e j^m+p0) sin(np1) -

r(-«2) r(-1 - p0 - p2 - «1 - a2)

-a1, -1 - a1 - a2 - p0, 1 + p1 -a1 - p0, -1 - p0 - p2 - «1 - «2

corresponding6 to the choice of vertex positions:

dx (x + p)C x + Y)d xe = pc Y 1+e+d ^ + ^' "J" " ' ^

1(-c - d)

-c, 1 + e , Y

1 ; 1 — -

-c-d p_

z1 = —cc , z2 = x2, z3 = x3, z4 = i, z4 = —i . (48)

Above we have used (40) and the following integral:

In addition, we have imposed the constraint:

a0 + a1 + a2 + /0 + /1 + /2 = — 3 . (49)

We can relate the mixed amplitude (47) to a pure open string amplitude involving five open strings. The generic open string disk amplitude reads:

A(1, 2n, 3n, 4n, 5n) = VC—KG f dzi \zij\sj j . (50)

D(n) i<j

Note, that due to conformal invariance the integers «ij must fulfill the conditions:

5 This specific integral has already been computed in [21] without making reference to the underlying five-point open string amplitude. This link will be established in the sequel.

6 The factor (— 1)«23 has been introduced for later convenience.

J2n'J = -2 = !'•••'5 • (51)

The choice

zi = -œ, z2 = 0, Z3 = 1, Z4 = (xy)-1, Z5 = x-1 (52)

gives rise to the subamplitude A(1, 2, 3, 5, 4) A(1, 2, 3, 5,4)

„23+„24+„25+«34+«35 ^ + fa) T(1 + «1) T(2 + «1 + 01 + ft) T(1 + «0)

( ) r(2 + 00 + «1)T(-«2 - 02)

' -«2, 1 + 00, 2 + «1 + 01 + 00

2 + 00 + «1, -«2 - 02 with the following identifications:

«0 = S45 + «45 , 00 = S23 + «23 , «1 = S35 + «35 , 01 = S25 + n25 ,

«2 = S34 + «34 , 02 = S24 + «24 , (54)

which fulfills (49), iff «23 + «24 + «25 + «34 + «35 + «45 = -3. On the other hand, the choice

Z1 = -œ, Z2 = 0, Z3 = (xy)~l, Z4 = x-1, Z5 = 1 (55)

gives rise to the subamplitude A(1, 2, 5, 4, 3):

T(1 + «0) T(1 + 01) T(-1 - 00 - «1 - «2) T(1 + «2)

A(1, 2, 5,4, 3) = (-1)«23 +n24+n25 x 3F2

T(-1 - 00 - 02 - «1 - «2) r(-«1 - 00) -«1, 1 + 01, -1 - «1 - «2 - 00 -00 - «1, -1 - 00 - 02 - «1 - «2.

Comparing (47) with (53) and (56) gives the relation:

F5 = e—ni(s24+s34) J sin[n(s35 + s25 + s23)] . sin(ns35\ A(1, 2, 3, 5, 4)

I sin[n(s35 + s23)]

— e—ni(s35+s23) sin(ns25)-Sin(ns34)-A(1, 2, 5, 4, 3) 1 . (57)

sin[n(s35 + s23 )] I

Eventually, after applying string monodromy relations involving five open strings [4] the expression (57) can be cast into (28) subject to (13).

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